LMFDB - Embedded newform 75.3.f.a.43.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(7,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.04360198270$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.43
Dual form			75.3.f.a.7.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.22474 - 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} +1.00000i q^{4} +3.00000 q^{6} +(7.34847 - 7.34847i) q^{7} +(6.12372 + 6.12372i) q^{8} +3.00000i q^{9} -18.0000 q^{11} +(-1.22474 + 1.22474i) q^{12} +(-7.34847 - 7.34847i) q^{13} -18.0000i q^{14} +11.0000 q^{16} +(-4.89898 + 4.89898i) q^{17} +(3.67423 + 3.67423i) q^{18} +10.0000i q^{19} +18.0000 q^{21} +(-22.0454 + 22.0454i) q^{22} +(-19.5959 - 19.5959i) q^{23} +15.0000i q^{24} -18.0000 q^{26} +(-3.67423 + 3.67423i) q^{27} +(7.34847 + 7.34847i) q^{28} +22.0000 q^{31} +(-11.0227 + 11.0227i) q^{32} +(-22.0454 - 22.0454i) q^{33} +12.0000i q^{34} -3.00000 q^{36} +(7.34847 - 7.34847i) q^{37} +(12.2474 + 12.2474i) q^{38} -18.0000i q^{39} -18.0000 q^{41} +(22.0454 - 22.0454i) q^{42} +(29.3939 + 29.3939i) q^{43} -18.0000i q^{44} -48.0000 q^{46} +(44.0908 - 44.0908i) q^{47} +(13.4722 + 13.4722i) q^{48} -59.0000i q^{49} -12.0000 q^{51} +(7.34847 - 7.34847i) q^{52} +(4.89898 + 4.89898i) q^{53} +9.00000i q^{54} +90.0000 q^{56} +(-12.2474 + 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(26.9444 - 26.9444i) q^{62} +(22.0454 + 22.0454i) q^{63} +71.0000i q^{64} -54.0000 q^{66} +(44.0908 - 44.0908i) q^{67} +(-4.89898 - 4.89898i) q^{68} -48.0000i q^{69} +72.0000 q^{71} +(-18.3712 + 18.3712i) q^{72} +(-44.0908 - 44.0908i) q^{73} -18.0000i q^{74} -10.0000 q^{76} +(-132.272 + 132.272i) q^{77} +(-22.0454 - 22.0454i) q^{78} -70.0000i q^{79} -9.00000 q^{81} +(-22.0454 + 22.0454i) q^{82} +(53.8888 + 53.8888i) q^{83} +18.0000i q^{84} +72.0000 q^{86} +(-110.227 - 110.227i) q^{88} +90.0000i q^{89} -108.000 q^{91} +(19.5959 - 19.5959i) q^{92} +(26.9444 + 26.9444i) q^{93} -108.000i q^{94} -27.0000 q^{96} +(-102.879 + 102.879i) q^{97} +(-72.2599 - 72.2599i) q^{98} -54.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{6} - 72 q^{11} + 44 q^{16} + 72 q^{21} - 72 q^{26} + 88 q^{31} - 12 q^{36} - 72 q^{41} - 192 q^{46} - 48 q^{51} + 360 q^{56} + 8 q^{61} - 216 q^{66} + 288 q^{71} - 40 q^{76} - 36 q^{81} + 288 q^{86}+ \cdots - 108 q^{96}+O(q^{100}) $ 4 * q + 12 * q^6 - 72 * q^11 + 44 * q^16 + 72 * q^21 - 72 * q^26 + 88 * q^31 - 12 * q^36 - 72 * q^41 - 192 * q^46 - 48 * q^51 + 360 * q^56 + 8 * q^61 - 216 * q^66 + 288 * q^71 - 40 * q^76 - 36 * q^81 + 288 * q^86 - 432 * q^91 - 108 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.22474	−	1.22474i	0.612372	−	0.612372i	−0.331191	−	0.943564i	$-0.607451\pi$
$2$	1.22474	−	1.22474i	0.612372	−	0.612372i	0.943564	+	0.331191i	$0.107451\pi$
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$			1.00000i			0.250000i
$5$		0			0
$5$		0			0
$6$	3.00000			0.500000
$7$	7.34847	−	7.34847i	1.04978	−	1.04978i	0.0510871	−	0.998694i	$-0.483731\pi$
$7$	7.34847	−	7.34847i	1.04978	−	1.04978i	0.998694	−	0.0510871i	$-0.0162686\pi$
$8$	6.12372	+	6.12372i	0.765466	+	0.765466i
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−18.0000			−1.63636			−0.818182	−	0.574960i	$-0.805018\pi$
$11$	−18.0000			−1.63636			−0.818182	+	0.574960i	$0.805018\pi$
$12$	−1.22474	+	1.22474i	−0.102062	+	0.102062i
$13$	−7.34847	−	7.34847i	−0.565267	−	0.565267i	0.365532	−	0.930799i	$-0.380887\pi$
$13$	−7.34847	−	7.34847i	−0.565267	−	0.565267i	−0.930799	+	0.365532i	$0.880887\pi$
$14$		−	18.0000i		−	1.28571i
$15$		0			0
$16$	11.0000			0.687500
$17$	−4.89898	+	4.89898i	−0.288175	+	0.288175i	−0.836358	−	0.548183i	$-0.815320\pi$
$17$	−4.89898	+	4.89898i	−0.288175	+	0.288175i	0.548183	+	0.836358i	$0.315320\pi$
$18$	3.67423	+	3.67423i	0.204124	+	0.204124i
$19$			10.0000i			0.526316i	0.964753	+	0.263158i	$0.0847640\pi$
$19$			10.0000i			0.526316i	−0.964753	+	0.263158i	$0.915236\pi$
$20$		0			0
$21$	18.0000			0.857143
$22$	−22.0454	+	22.0454i	−1.00206	+	1.00206i
$23$	−19.5959	−	19.5959i	−0.851996	−	0.851996i	0.138382	−	0.990379i	$-0.455810\pi$
$23$	−19.5959	−	19.5959i	−0.851996	−	0.851996i	−0.990379	+	0.138382i	$0.955810\pi$
$24$			15.0000i			0.625000i
$25$		0			0
$26$	−18.0000			−0.692308
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$	7.34847	+	7.34847i	0.262445	+	0.262445i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	22.0000			0.709677			0.354839	−	0.934928i	$-0.384536\pi$
$31$	22.0000			0.709677			0.354839	+	0.934928i	$0.384536\pi$
$32$	−11.0227	+	11.0227i	−0.344459	+	0.344459i
$33$	−22.0454	−	22.0454i	−0.668043	−	0.668043i
$34$			12.0000i			0.352941i
$35$		0			0
$36$	−3.00000			−0.0833333
$37$	7.34847	−	7.34847i	0.198607	−	0.198607i	−0.600795	−	0.799403i	$-0.705149\pi$
$37$	7.34847	−	7.34847i	0.198607	−	0.198607i	0.799403	+	0.600795i	$0.205149\pi$
$38$	12.2474	+	12.2474i	0.322301	+	0.322301i
$39$		−	18.0000i		−	0.461538i
$40$		0			0
$41$	−18.0000			−0.439024			−0.219512	−	0.975610i	$-0.570447\pi$
$41$	−18.0000			−0.439024			−0.219512	+	0.975610i	$0.570447\pi$
$42$	22.0454	−	22.0454i	0.524891	−	0.524891i
$43$	29.3939	+	29.3939i	0.683579	+	0.683579i	0.960805	−	0.277226i	$-0.0894151\pi$
$43$	29.3939	+	29.3939i	0.683579	+	0.683579i	−0.277226	+	0.960805i	$0.589415\pi$
$44$		−	18.0000i		−	0.409091i
$45$		0			0
$46$	−48.0000			−1.04348
$47$	44.0908	−	44.0908i	0.938102	−	0.938102i	−0.0600905	−	0.998193i	$-0.519139\pi$
$47$	44.0908	−	44.0908i	0.938102	−	0.938102i	0.998193	+	0.0600905i	$0.0191389\pi$
$48$	13.4722	+	13.4722i	0.280671	+	0.280671i
$49$		−	59.0000i		−	1.20408i
$50$		0			0
$51$	−12.0000			−0.235294
$52$	7.34847	−	7.34847i	0.141317	−	0.141317i
$53$	4.89898	+	4.89898i	0.0924336	+	0.0924336i	0.751812	−	0.659378i	$-0.229180\pi$
$53$	4.89898	+	4.89898i	0.0924336	+	0.0924336i	−0.659378	+	0.751812i	$0.729180\pi$
$54$			9.00000i			0.166667i
$55$		0			0
$56$	90.0000			1.60714
$57$	−12.2474	+	12.2474i	−0.214868	+	0.214868i
$58$		0			0
$59$			90.0000i			1.52542i	0.646738	+	0.762712i	$0.276133\pi$
$59$			90.0000i			1.52542i	−0.646738	+	0.762712i	$0.723867\pi$
$60$		0			0
$61$	2.00000			0.0327869			0.0163934	−	0.999866i	$-0.494782\pi$
$61$	2.00000			0.0327869			0.0163934	+	0.999866i	$0.494782\pi$
$62$	26.9444	−	26.9444i	0.434587	−	0.434587i
$63$	22.0454	+	22.0454i	0.349927	+	0.349927i
$64$			71.0000i			1.10938i
$65$		0			0
$66$	−54.0000			−0.818182
$67$	44.0908	−	44.0908i	0.658072	−	0.658072i	−0.296852	−	0.954924i	$-0.595937\pi$
$67$	44.0908	−	44.0908i	0.658072	−	0.658072i	0.954924	+	0.296852i	$0.0959367\pi$
$68$	−4.89898	−	4.89898i	−0.0720438	−	0.0720438i
$69$		−	48.0000i		−	0.695652i
$70$		0			0
$71$	72.0000			1.01408			0.507042	−	0.861921i	$-0.330739\pi$
$71$	72.0000			1.01408			0.507042	+	0.861921i	$0.330739\pi$
$72$	−18.3712	+	18.3712i	−0.255155	+	0.255155i
$73$	−44.0908	−	44.0908i	−0.603984	−	0.603984i	0.337384	−	0.941367i	$-0.390458\pi$
$73$	−44.0908	−	44.0908i	−0.603984	−	0.603984i	−0.941367	+	0.337384i	$0.890458\pi$
$74$		−	18.0000i		−	0.243243i
$75$		0			0
$76$	−10.0000			−0.131579
$77$	−132.272	+	132.272i	−1.71782	+	1.71782i
$78$	−22.0454	−	22.0454i	−0.282633	−	0.282633i
$79$		−	70.0000i		−	0.886076i	−0.896503	−	0.443038i	$-0.853901\pi$
$79$		−	70.0000i		−	0.886076i	0.896503	−	0.443038i	$-0.146099\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$	−22.0454	+	22.0454i	−0.268846	+	0.268846i
$83$	53.8888	+	53.8888i	0.649262	+	0.649262i	0.952815	−	0.303552i	$-0.0981727\pi$
$83$	53.8888	+	53.8888i	0.649262	+	0.649262i	−0.303552	+	0.952815i	$0.598173\pi$
$84$			18.0000i			0.214286i
$85$		0			0
$86$	72.0000			0.837209
$87$		0			0
$88$	−110.227	−	110.227i	−1.25258	−	1.25258i
$89$			90.0000i			1.01124i	0.862757	+	0.505618i	$0.168735\pi$
$89$			90.0000i			1.01124i	−0.862757	+	0.505618i	$0.831265\pi$
$90$		0			0
$91$	−108.000			−1.18681
$92$	19.5959	−	19.5959i	0.212999	−	0.212999i
$93$	26.9444	+	26.9444i	0.289725	+	0.289725i
$94$		−	108.000i		−	1.14894i
$95$		0			0
$96$	−27.0000			−0.281250
$97$	−102.879	+	102.879i	−1.06060	+	1.06060i	−0.0625628	+	0.998041i	$0.519927\pi$
$97$	−102.879	+	102.879i	−1.06060	+	1.06060i	−0.998041	+	0.0625628i	$0.980073\pi$
$98$	−72.2599	−	72.2599i	−0.737346	−	0.737346i
$99$		−	54.0000i		−	0.545455i
$100$		0			0
$101$	−108.000			−1.06931			−0.534653	−	0.845071i	$-0.679558\pi$
$101$	−108.000			−1.06931			−0.534653	+	0.845071i	$0.679558\pi$
$102$	−14.6969	+	14.6969i	−0.144088	+	0.144088i
$103$	66.1362	+	66.1362i	0.642099	+	0.642099i	0.951071	−	0.308972i	$-0.0999849\pi$
$103$	66.1362	+	66.1362i	0.642099	+	0.642099i	−0.308972	+	0.951071i	$0.599985\pi$
$104$		−	90.0000i		−	0.865385i
$105$		0			0
$106$	12.0000			0.113208
$107$	44.0908	−	44.0908i	0.412064	−	0.412064i	−0.470393	−	0.882457i	$-0.655888\pi$
$107$	44.0908	−	44.0908i	0.412064	−	0.412064i	0.882457	+	0.470393i	$0.155888\pi$
$108$	−3.67423	−	3.67423i	−0.0340207	−	0.0340207i
$109$			170.000i			1.55963i	0.626008	+	0.779817i	$0.284688\pi$
$109$			170.000i			1.55963i	−0.626008	+	0.779817i	$0.715312\pi$
$110$		0			0
$111$	18.0000			0.162162
$112$	80.8332	−	80.8332i	0.721725	−	0.721725i
$113$	−44.0908	−	44.0908i	−0.390184	−	0.390184i	0.484569	−	0.874753i	$-0.338976\pi$
$113$	−44.0908	−	44.0908i	−0.390184	−	0.390184i	−0.874753	+	0.484569i	$0.838976\pi$
$114$			30.0000i			0.263158i
$115$		0			0
$116$		0			0
$117$	22.0454	−	22.0454i	0.188422	−	0.188422i
$118$	110.227	+	110.227i	0.934127	+	0.934127i
$119$			72.0000i			0.605042i
$120$		0			0
$121$	203.000			1.67769
$122$	2.44949	−	2.44949i	0.0200778	−	0.0200778i
$123$	−22.0454	−	22.0454i	−0.179231	−	0.179231i
$124$			22.0000i			0.177419i
$125$		0			0
$126$	54.0000			0.428571
$127$	7.34847	−	7.34847i	0.0578620	−	0.0578620i	−0.677584	−	0.735446i	$-0.736973\pi$
$127$	7.34847	−	7.34847i	0.0578620	−	0.0578620i	0.735446	+	0.677584i	$0.236973\pi$
$128$	42.8661	+	42.8661i	0.334891	+	0.334891i
$129$			72.0000i			0.558140i
$130$		0			0
$131$	−18.0000			−0.137405			−0.0687023	−	0.997637i	$-0.521886\pi$
$131$	−18.0000			−0.137405			−0.0687023	+	0.997637i	$0.521886\pi$
$132$	22.0454	−	22.0454i	0.167011	−	0.167011i
$133$	73.4847	+	73.4847i	0.552516	+	0.552516i
$134$		−	108.000i		−	0.805970i
$135$		0			0
$136$	−60.0000			−0.441176
$137$	142.070	−	142.070i	1.03701	−	1.03701i	0.0377220	−	0.999288i	$-0.487990\pi$
$137$	142.070	−	142.070i	1.03701	−	1.03701i	0.999288	−	0.0377220i	$-0.0120101\pi$
$138$	−58.7878	−	58.7878i	−0.425998	−	0.425998i
$139$		−	170.000i		−	1.22302i	−0.791236	−	0.611511i	$-0.790562\pi$
$139$		−	170.000i		−	1.22302i	0.791236	−	0.611511i	$-0.209438\pi$
$140$		0			0
$141$	108.000			0.765957
$142$	88.1816	−	88.1816i	0.620997	−	0.620997i
$143$	132.272	+	132.272i	0.924982	+	0.924982i
$144$			33.0000i			0.229167i
$145$		0			0
$146$	−108.000			−0.739726
$147$	72.2599	−	72.2599i	0.491564	−	0.491564i
$148$	7.34847	+	7.34847i	0.0496518	+	0.0496518i
$149$		−	180.000i		−	1.20805i	−0.796964	−	0.604027i	$-0.793562\pi$
$149$		−	180.000i		−	1.20805i	0.796964	−	0.604027i	$-0.206438\pi$
$150$		0			0
$151$	22.0000			0.145695			0.0728477	−	0.997343i	$-0.476791\pi$
$151$	22.0000			0.145695			0.0728477	+	0.997343i	$0.476791\pi$
$152$	−61.2372	+	61.2372i	−0.402877	+	0.402877i
$153$	−14.6969	−	14.6969i	−0.0960584	−	0.0960584i
$154$			324.000i			2.10390i
$155$		0			0
$156$	18.0000			0.115385
$157$	−139.621	+	139.621i	−0.889305	+	0.889305i	−0.994456	−	0.105151i	$-0.966467\pi$
$157$	−139.621	+	139.621i	−0.889305	+	0.889305i	0.105151	+	0.994456i	$0.466467\pi$
$158$	−85.7321	−	85.7321i	−0.542608	−	0.542608i
$159$			12.0000i			0.0754717i
$160$		0			0
$161$	−288.000			−1.78882
$162$	−11.0227	+	11.0227i	−0.0680414	+	0.0680414i
$163$	−117.576	−	117.576i	−0.721322	−	0.721322i	0.247552	−	0.968875i	$-0.420374\pi$
$163$	−117.576	−	117.576i	−0.721322	−	0.721322i	−0.968875	+	0.247552i	$0.920374\pi$
$164$		−	18.0000i		−	0.109756i
$165$		0			0
$166$	132.000			0.795181
$167$	19.5959	−	19.5959i	0.117341	−	0.117341i	−0.645998	−	0.763339i	$-0.723559\pi$
$167$	19.5959	−	19.5959i	0.117341	−	0.117341i	0.763339	+	0.645998i	$0.223559\pi$
$168$	110.227	+	110.227i	0.656113	+	0.656113i
$169$		−	61.0000i		−	0.360947i
$170$		0			0
$171$	−30.0000			−0.175439
$172$	−29.3939	+	29.3939i	−0.170895	+	0.170895i
$173$	−142.070	−	142.070i	−0.821216	−	0.821216i	0.165066	−	0.986282i	$-0.447216\pi$
$173$	−142.070	−	142.070i	−0.821216	−	0.821216i	−0.986282	+	0.165066i	$0.947216\pi$
$174$		0			0
$175$		0			0
$176$	−198.000			−1.12500
$177$	−110.227	+	110.227i	−0.622752	+	0.622752i
$178$	110.227	+	110.227i	0.619253	+	0.619253i
$179$		−	90.0000i		−	0.502793i	−0.967884	−	0.251397i	$-0.919110\pi$
$179$		−	90.0000i		−	0.502793i	0.967884	−	0.251397i	$-0.0808899\pi$
$180$		0			0
$181$	−98.0000			−0.541436			−0.270718	−	0.962659i	$-0.587261\pi$
$181$	−98.0000			−0.541436			−0.270718	+	0.962659i	$0.587261\pi$
$182$	−132.272	+	132.272i	−0.726772	+	0.726772i
$183$	2.44949	+	2.44949i	0.0133852	+	0.0133852i
$184$		−	240.000i		−	1.30435i
$185$		0			0
$186$	66.0000			0.354839
$187$	88.1816	−	88.1816i	0.471560	−	0.471560i
$188$	44.0908	+	44.0908i	0.234526	+	0.234526i
$189$			54.0000i			0.285714i
$190$		0			0
$191$	252.000			1.31937			0.659686	−	0.751541i	$-0.270689\pi$
$191$	252.000			1.31937			0.659686	+	0.751541i	$0.270689\pi$
$192$	−86.9569	+	86.9569i	−0.452900	+	0.452900i
$193$	−264.545	−	264.545i	−1.37070	−	1.37070i	−0.859407	−	0.511292i	$-0.829167\pi$
$193$	−264.545	−	264.545i	−1.37070	−	1.37070i	−0.511292	−	0.859407i	$-0.670833\pi$
$194$			252.000i			1.29897i
$195$		0			0
$196$	59.0000			0.301020
$197$	−127.373	+	127.373i	−0.646566	+	0.646566i	−0.952161	−	0.305596i	$-0.901144\pi$
$197$	−127.373	+	127.373i	−0.646566	+	0.646566i	0.305596	+	0.952161i	$0.401144\pi$
$198$	−66.1362	−	66.1362i	−0.334021	−	0.334021i
$199$			290.000i			1.45729i	0.684893	+	0.728643i	$0.259849\pi$
$199$			290.000i			1.45729i	−0.684893	+	0.728643i	$0.740151\pi$
$200$		0			0
$201$	108.000			0.537313
$202$	−132.272	+	132.272i	−0.654814	+	0.654814i
$203$		0			0
$204$		−	12.0000i		−	0.0588235i
$205$		0			0
$206$	162.000			0.786408
$207$	58.7878	−	58.7878i	0.283999	−	0.283999i
$208$	−80.8332	−	80.8332i	−0.388621	−	0.388621i
$209$		−	180.000i		−	0.861244i
$210$		0			0
$211$	122.000			0.578199			0.289100	−	0.957299i	$-0.406644\pi$
$211$	122.000			0.578199			0.289100	+	0.957299i	$0.406644\pi$
$212$	−4.89898	+	4.89898i	−0.0231084	+	0.0231084i
$213$	88.1816	+	88.1816i	0.413998	+	0.413998i
$214$		−	108.000i		−	0.504673i
$215$		0			0
$216$	−45.0000			−0.208333
$217$	161.666	−	161.666i	0.745006	−	0.745006i
$218$	208.207	+	208.207i	0.955076	+	0.955076i
$219$		−	108.000i		−	0.493151i
$220$		0			0
$221$	72.0000			0.325792
$222$	22.0454	−	22.0454i	0.0993036	−	0.0993036i
$223$	−80.8332	−	80.8332i	−0.362481	−	0.362481i	0.502245	−	0.864725i	$-0.332508\pi$
$223$	−80.8332	−	80.8332i	−0.362481	−	0.362481i	−0.864725	+	0.502245i	$0.832508\pi$
$224$			162.000i			0.723214i
$225$		0			0
$226$	−108.000			−0.477876
$227$	−53.8888	+	53.8888i	−0.237395	+	0.237395i	−0.815771	−	0.578375i	$-0.803687\pi$
$227$	−53.8888	+	53.8888i	−0.237395	+	0.237395i	0.578375	+	0.815771i	$0.303687\pi$
$228$	−12.2474	−	12.2474i	−0.0537169	−	0.0537169i
$229$		−	50.0000i		−	0.218341i	−0.994023	−	0.109170i	$-0.965181\pi$
$229$		−	50.0000i		−	0.218341i	0.994023	−	0.109170i	$-0.0348194\pi$
$230$		0			0
$231$	−324.000			−1.40260
$232$		0			0
$233$	−44.0908	−	44.0908i	−0.189231	−	0.189231i	0.606133	−	0.795364i	$-0.292720\pi$
$233$	−44.0908	−	44.0908i	−0.189231	−	0.189231i	−0.795364	+	0.606133i	$0.792720\pi$
$234$		−	54.0000i		−	0.230769i
$235$		0			0
$236$	−90.0000			−0.381356
$237$	85.7321	−	85.7321i	0.361739	−	0.361739i
$238$	88.1816	+	88.1816i	0.370511	+	0.370511i
$239$			180.000i			0.753138i	0.926389	+	0.376569i	$0.122896\pi$
$239$			180.000i			0.753138i	−0.926389	+	0.376569i	$0.877104\pi$
$240$		0			0
$241$	−178.000			−0.738589			−0.369295	−	0.929312i	$-0.620401\pi$
$241$	−178.000			−0.738589			−0.369295	+	0.929312i	$0.620401\pi$
$242$	248.623	−	248.623i	1.02737	−	1.02737i
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$			2.00000i			0.00819672i
$245$		0			0
$246$	−54.0000			−0.219512
$247$	73.4847	−	73.4847i	0.297509	−	0.297509i
$248$	134.722	+	134.722i	0.543234	+	0.543234i
$249$			132.000i			0.530120i
$250$		0			0
$251$	342.000			1.36255			0.681275	−	0.732028i	$-0.261426\pi$
$251$	342.000			1.36255			0.681275	+	0.732028i	$0.261426\pi$
$252$	−22.0454	+	22.0454i	−0.0874818	+	0.0874818i
$253$	352.727	+	352.727i	1.39418	+	1.39418i
$254$		−	18.0000i		−	0.0708661i
$255$		0			0
$256$	−179.000			−0.699219
$257$	289.040	−	289.040i	1.12467	−	1.12467i	0.133638	−	0.991030i	$-0.457334\pi$
$257$	289.040	−	289.040i	1.12467	−	1.12467i	0.991030	−	0.133638i	$-0.0426660\pi$
$258$	88.1816	+	88.1816i	0.341789	+	0.341789i
$259$		−	108.000i		−	0.416988i
$260$		0			0
$261$		0			0
$262$	−22.0454	+	22.0454i	−0.0841428	+	0.0841428i
$263$	−264.545	−	264.545i	−1.00587	−	1.00587i	−0.999983	−	0.00589147i	$-0.998125\pi$
$263$	−264.545	−	264.545i	−1.00587	−	1.00587i	−0.00589147	−	0.999983i	$-0.501875\pi$
$264$		−	270.000i		−	1.02273i
$265$		0			0
$266$	180.000			0.676692
$267$	−110.227	+	110.227i	−0.412835	+	0.412835i
$268$	44.0908	+	44.0908i	0.164518	+	0.164518i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−478.000			−1.76384			−0.881919	−	0.471401i	$-0.843748\pi$
$271$	−478.000			−1.76384			−0.881919	+	0.471401i	$0.843748\pi$
$272$	−53.8888	+	53.8888i	−0.198120	+	0.198120i
$273$	−132.272	−	132.272i	−0.484514	−	0.484514i
$274$		−	348.000i		−	1.27007i
$275$		0			0
$276$	48.0000			0.173913
$277$	80.8332	−	80.8332i	0.291816	−	0.291816i	−0.545981	−	0.837798i	$-0.683843\pi$
$277$	80.8332	−	80.8332i	0.291816	−	0.291816i	0.837798	+	0.545981i	$0.183843\pi$
$278$	−208.207	−	208.207i	−0.748945	−	0.748945i
$279$			66.0000i			0.236559i
$280$		0			0
$281$	162.000			0.576512			0.288256	−	0.957553i	$-0.406925\pi$
$281$	162.000			0.576512			0.288256	+	0.957553i	$0.406925\pi$
$282$	132.272	−	132.272i	0.469051	−	0.469051i
$283$	249.848	+	249.848i	0.882855	+	0.882855i	0.993824	−	0.110969i	$-0.0353954\pi$
$283$	249.848	+	249.848i	0.882855	+	0.882855i	−0.110969	+	0.993824i	$0.535395\pi$
$284$			72.0000i			0.253521i
$285$		0			0
$286$	324.000			1.13287
$287$	−132.272	+	132.272i	−0.460880	+	0.460880i
$288$	−33.0681	−	33.0681i	−0.114820	−	0.114820i
$289$			241.000i			0.833910i
$290$		0			0
$291$	−252.000			−0.865979
$292$	44.0908	−	44.0908i	0.150996	−	0.150996i
$293$	372.322	+	372.322i	1.27073	+	1.27073i	0.945708	+	0.325017i	$0.105370\pi$
$293$	372.322	+	372.322i	1.27073	+	1.27073i	0.325017	+	0.945708i	$0.394630\pi$
$294$		−	177.000i		−	0.602041i
$295$		0			0
$296$	90.0000			0.304054
$297$	66.1362	−	66.1362i	0.222681	−	0.222681i
$298$	−220.454	−	220.454i	−0.739779	−	0.739779i
$299$			288.000i			0.963211i
$300$		0			0
$301$	432.000			1.43522
$302$	26.9444	−	26.9444i	0.0892198	−	0.0892198i
$303$	−132.272	−	132.272i	−0.436543	−	0.436543i
$304$			110.000i			0.361842i
$305$		0			0
$306$	−36.0000			−0.117647
$307$	−396.817	+	396.817i	−1.29256	+	1.29256i	−0.359369	+	0.933195i	$0.617008\pi$
$307$	−396.817	+	396.817i	−1.29256	+	1.29256i	−0.933195	+	0.359369i	$0.882992\pi$
$308$	−132.272	−	132.272i	−0.429456	−	0.429456i
$309$			162.000i			0.524272i
$310$		0			0
$311$	252.000			0.810289			0.405145	−	0.914253i	$-0.367221\pi$
$311$	252.000			0.810289			0.405145	+	0.914253i	$0.367221\pi$
$312$	110.227	−	110.227i	0.353292	−	0.353292i
$313$	−117.576	−	117.576i	−0.375641	−	0.375641i	0.493886	−	0.869527i	$-0.335576\pi$
$313$	−117.576	−	117.576i	−0.375641	−	0.375641i	−0.869527	+	0.493886i	$0.835576\pi$
$314$			342.000i			1.08917i
$315$		0			0
$316$	70.0000			0.221519
$317$	−274.343	+	274.343i	−0.865435	+	0.865435i	−0.991963	−	0.126528i	$-0.959617\pi$
$317$	−274.343	+	274.343i	−0.865435	+	0.865435i	0.126528	+	0.991963i	$0.459617\pi$
$318$	14.6969	+	14.6969i	0.0462168	+	0.0462168i
$319$		0			0
$320$		0			0
$321$	108.000			0.336449
$322$	−352.727	+	352.727i	−1.09542	+	1.09542i
$323$	−48.9898	−	48.9898i	−0.151671	−	0.151671i
$324$		−	9.00000i		−	0.0277778i
$325$		0			0
$326$	−288.000			−0.883436
$327$	−208.207	+	208.207i	−0.636718	+	0.636718i
$328$	−110.227	−	110.227i	−0.336058	−	0.336058i
$329$		−	648.000i		−	1.96960i
$330$		0			0
$331$	−418.000			−1.26284			−0.631420	−	0.775441i	$-0.717528\pi$
$331$	−418.000			−1.26284			−0.631420	+	0.775441i	$0.717528\pi$
$332$	−53.8888	+	53.8888i	−0.162316	+	0.162316i
$333$	22.0454	+	22.0454i	0.0662024	+	0.0662024i
$334$		−	48.0000i		−	0.143713i
$335$		0			0
$336$	198.000			0.589286
$337$	191.060	−	191.060i	0.566944	−	0.566944i	−0.364327	−	0.931271i	$-0.618701\pi$
$337$	191.060	−	191.060i	0.566944	−	0.566944i	0.931271	+	0.364327i	$0.118701\pi$
$338$	−74.7094	−	74.7094i	−0.221034	−	0.221034i
$339$		−	108.000i		−	0.318584i
$340$		0			0
$341$	−396.000			−1.16129
$342$	−36.7423	+	36.7423i	−0.107434	+	0.107434i
$343$	−73.4847	−	73.4847i	−0.214241	−	0.214241i
$344$			360.000i			1.04651i
$345$		0			0
$346$	−348.000			−1.00578
$347$	44.0908	−	44.0908i	0.127063	−	0.127063i	−0.640716	−	0.767778i	$-0.721362\pi$
$347$	44.0908	−	44.0908i	0.127063	−	0.127063i	0.767778	+	0.640716i	$0.221362\pi$
$348$		0			0
$349$			70.0000i			0.200573i	0.994959	+	0.100287i	$0.0319759\pi$
$349$			70.0000i			0.200573i	−0.994959	+	0.100287i	$0.968024\pi$
$350$		0			0
$351$	54.0000			0.153846
$352$	198.409	−	198.409i	0.563661	−	0.563661i
$353$	−44.0908	−	44.0908i	−0.124903	−	0.124903i	0.641892	−	0.766795i	$-0.278150\pi$
$353$	−44.0908	−	44.0908i	−0.124903	−	0.124903i	−0.766795	+	0.641892i	$0.778150\pi$
$354$			270.000i			0.762712i
$355$		0			0
$356$	−90.0000			−0.252809
$357$	−88.1816	+	88.1816i	−0.247007	+	0.247007i
$358$	−110.227	−	110.227i	−0.307897	−	0.307897i
$359$		−	540.000i		−	1.50418i	−0.659061	−	0.752089i	$-0.729046\pi$
$359$		−	540.000i		−	1.50418i	0.659061	−	0.752089i	$-0.270954\pi$
$360$		0			0
$361$	261.000			0.722992
$362$	−120.025	+	120.025i	−0.331561	+	0.331561i
$363$	248.623	+	248.623i	0.684912	+	0.684912i
$364$		−	108.000i		−	0.296703i
$365$		0			0
$366$	6.00000			0.0163934
$367$	−66.1362	+	66.1362i	−0.180208	+	0.180208i	−0.791446	−	0.611239i	$-0.790672\pi$
$367$	−66.1362	+	66.1362i	−0.180208	+	0.180208i	0.611239	+	0.791446i	$0.290672\pi$
$368$	−215.555	−	215.555i	−0.585748	−	0.585748i
$369$		−	54.0000i		−	0.146341i
$370$		0			0
$371$	72.0000			0.194070
$372$	−26.9444	+	26.9444i	−0.0724311	+	0.0724311i
$373$	213.106	+	213.106i	0.571329	+	0.571329i	0.932500	−	0.361171i	$-0.117623\pi$
$373$	213.106	+	213.106i	0.571329	+	0.571329i	−0.361171	+	0.932500i	$0.617623\pi$
$374$		−	216.000i		−	0.577540i
$375$		0			0
$376$	540.000			1.43617
$377$		0			0
$378$	66.1362	+	66.1362i	0.174964	+	0.174964i
$379$		−	170.000i		−	0.448549i	−0.974526	−	0.224274i	$-0.927999\pi$
$379$		−	170.000i		−	0.448549i	0.974526	−	0.224274i	$-0.0720012\pi$
$380$		0			0
$381$	18.0000			0.0472441
$382$	308.636	−	308.636i	0.807947	−	0.807947i
$383$	151.868	+	151.868i	0.396523	+	0.396523i	0.877005	−	0.480482i	$-0.159538\pi$
$383$	151.868	+	151.868i	0.396523	+	0.396523i	−0.480482	+	0.877005i	$0.659538\pi$
$384$			105.000i			0.273438i
$385$		0			0
$386$	−648.000			−1.67876
$387$	−88.1816	+	88.1816i	−0.227860	+	0.227860i
$388$	−102.879	−	102.879i	−0.265151	−	0.265151i
$389$		−	360.000i		−	0.925450i	−0.886502	−	0.462725i	$-0.846872\pi$
$389$		−	360.000i		−	0.925450i	0.886502	−	0.462725i	$-0.153128\pi$
$390$		0			0
$391$	192.000			0.491049
$392$	361.300	−	361.300i	0.921683	−	0.921683i
$393$	−22.0454	−	22.0454i	−0.0560952	−	0.0560952i
$394$			312.000i			0.791878i
$395$		0			0
$396$	54.0000			0.136364
$397$	301.287	−	301.287i	0.758910	−	0.758910i	−0.217214	−	0.976124i	$-0.569697\pi$
$397$	301.287	−	301.287i	0.758910	−	0.758910i	0.976124	+	0.217214i	$0.0696970\pi$
$398$	355.176	+	355.176i	0.892402	+	0.892402i
$399$			180.000i			0.451128i
$400$		0			0
$401$	−558.000			−1.39152			−0.695761	−	0.718274i	$-0.744933\pi$
$401$	−558.000			−1.39152			−0.695761	+	0.718274i	$0.744933\pi$
$402$	132.272	−	132.272i	0.329036	−	0.329036i
$403$	−161.666	−	161.666i	−0.401157	−	0.401157i
$404$		−	108.000i		−	0.267327i
$405$		0			0
$406$		0			0
$407$	−132.272	+	132.272i	−0.324994	+	0.324994i
$408$	−73.4847	−	73.4847i	−0.180110	−	0.180110i
$409$			670.000i			1.63814i	0.573692	+	0.819071i	$0.305511\pi$
$409$			670.000i			1.63814i	−0.573692	+	0.819071i	$0.694489\pi$
$410$		0			0
$411$	348.000			0.846715
$412$	−66.1362	+	66.1362i	−0.160525	+	0.160525i
$413$	661.362	+	661.362i	1.60136	+	1.60136i
$414$		−	144.000i		−	0.347826i
$415$		0			0
$416$	162.000			0.389423
$417$	208.207	−	208.207i	0.499296	−	0.499296i
$418$	−220.454	−	220.454i	−0.527402	−	0.527402i
$419$			630.000i			1.50358i	0.659403	+	0.751790i	$0.270809\pi$
$419$			630.000i			1.50358i	−0.659403	+	0.751790i	$0.729191\pi$
$420$		0			0
$421$	142.000			0.337292			0.168646	−	0.985677i	$-0.446061\pi$
$421$	142.000			0.337292			0.168646	+	0.985677i	$0.446061\pi$
$422$	149.419	−	149.419i	0.354073	−	0.354073i
$423$	132.272	+	132.272i	0.312701	+	0.312701i
$424$			60.0000i			0.141509i
$425$		0			0
$426$	216.000			0.507042
$427$	14.6969	−	14.6969i	0.0344191	−	0.0344191i
$428$	44.0908	+	44.0908i	0.103016	+	0.103016i
$429$			324.000i			0.755245i
$430$		0			0
$431$	612.000			1.41995			0.709977	−	0.704225i	$-0.248705\pi$
$431$	612.000			1.41995			0.709977	+	0.704225i	$0.248705\pi$
$432$	−40.4166	+	40.4166i	−0.0935569	+	0.0935569i
$433$	102.879	+	102.879i	0.237595	+	0.237595i	0.815853	−	0.578259i	$-0.196268\pi$
$433$	102.879	+	102.879i	0.237595	+	0.237595i	−0.578259	+	0.815853i	$0.696268\pi$
$434$		−	396.000i		−	0.912442i
$435$		0			0
$436$	−170.000			−0.389908
$437$	195.959	−	195.959i	0.448419	−	0.448419i
$438$	−132.272	−	132.272i	−0.301992	−	0.301992i
$439$		−	430.000i		−	0.979499i	−0.871863	−	0.489749i	$-0.837088\pi$
$439$		−	430.000i		−	0.979499i	0.871863	−	0.489749i	$-0.162912\pi$
$440$		0			0
$441$	177.000			0.401361
$442$	88.1816	−	88.1816i	0.199506	−	0.199506i
$443$	−582.979	−	582.979i	−1.31598	−	1.31598i	−0.916930	−	0.399049i	$-0.869340\pi$
$443$	−582.979	−	582.979i	−1.31598	−	1.31598i	−0.399049	−	0.916930i	$-0.630660\pi$
$444$			18.0000i			0.0405405i
$445$		0			0
$446$	−198.000			−0.443946
$447$	220.454	−	220.454i	0.493186	−	0.493186i
$448$	521.741	+	521.741i	1.16460	+	1.16460i
$449$			90.0000i			0.200445i	0.994965	+	0.100223i	$0.0319555\pi$
$449$			90.0000i			0.200445i	−0.994965	+	0.100223i	$0.968044\pi$
$450$		0			0
$451$	324.000			0.718404
$452$	44.0908	−	44.0908i	0.0975461	−	0.0975461i
$453$	26.9444	+	26.9444i	0.0594799	+	0.0594799i
$454$			132.000i			0.290749i
$455$		0			0
$456$	−150.000			−0.328947
$457$	191.060	−	191.060i	0.418075	−	0.418075i	−0.466465	−	0.884540i	$-0.654473\pi$
$457$	191.060	−	191.060i	0.418075	−	0.418075i	0.884540	+	0.466465i	$0.154473\pi$
$458$	−61.2372	−	61.2372i	−0.133706	−	0.133706i
$459$		−	36.0000i		−	0.0784314i
$460$		0			0
$461$	−828.000			−1.79610			−0.898048	−	0.439898i	$-0.855015\pi$
$461$	−828.000			−1.79610			−0.898048	+	0.439898i	$0.855015\pi$
$462$	−396.817	+	396.817i	−0.858912	+	0.858912i
$463$	−7.34847	−	7.34847i	−0.0158714	−	0.0158714i	0.699127	−	0.714998i	$-0.253572\pi$
$463$	−7.34847	−	7.34847i	−0.0158714	−	0.0158714i	−0.714998	+	0.699127i	$0.753572\pi$
$464$		0			0
$465$		0			0
$466$	−108.000			−0.231760
$467$	−347.828	+	347.828i	−0.744813	+	0.744813i	−0.973500	−	0.228687i	$-0.926557\pi$
$467$	−347.828	+	347.828i	−0.744813	+	0.744813i	0.228687	+	0.973500i	$0.426557\pi$
$468$	22.0454	+	22.0454i	0.0471056	+	0.0471056i
$469$		−	648.000i		−	1.38166i
$470$		0			0
$471$	−342.000			−0.726115
$472$	−551.135	+	551.135i	−1.16766	+	1.16766i
$473$	−529.090	−	529.090i	−1.11858	−	1.11858i
$474$		−	210.000i		−	0.443038i
$475$		0			0
$476$	−72.0000			−0.151261
$477$	−14.6969	+	14.6969i	−0.0308112	+	0.0308112i
$478$	220.454	+	220.454i	0.461201	+	0.461201i
$479$		−	360.000i		−	0.751566i	−0.926708	−	0.375783i	$-0.877374\pi$
$479$		−	360.000i		−	0.751566i	0.926708	−	0.375783i	$-0.122626\pi$
$480$		0			0
$481$	−108.000			−0.224532
$482$	−218.005	+	218.005i	−0.452292	+	0.452292i
$483$	−352.727	−	352.727i	−0.730283	−	0.730283i
$484$			203.000i			0.419421i
$485$		0			0
$486$	−27.0000			−0.0555556
$487$	−580.529	+	580.529i	−1.19205	+	1.19205i	−0.215561	+	0.976490i	$0.569158\pi$
$487$	−580.529	+	580.529i	−1.19205	+	1.19205i	−0.976490	+	0.215561i	$0.930842\pi$
$488$	12.2474	+	12.2474i	0.0250972	+	0.0250972i
$489$		−	288.000i		−	0.588957i
$490$		0			0
$491$	−18.0000			−0.0366599			−0.0183299	−	0.999832i	$-0.505835\pi$
$491$	−18.0000			−0.0366599			−0.0183299	+	0.999832i	$0.505835\pi$
$492$	22.0454	−	22.0454i	0.0448077	−	0.0448077i
$493$		0			0
$494$		−	180.000i		−	0.364372i
$495$		0			0
$496$	242.000			0.487903
$497$	529.090	−	529.090i	1.06457	−	1.06457i
$498$	161.666	+	161.666i	0.324631	+	0.324631i
$499$			590.000i			1.18236i	0.806538	+	0.591182i	$0.201339\pi$
$499$			590.000i			1.18236i	−0.806538	+	0.591182i	$0.798661\pi$
$500$		0			0
$501$	48.0000			0.0958084
$502$	418.863	−	418.863i	0.834388	−	0.834388i
$503$	−93.0806	−	93.0806i	−0.185051	−	0.185051i	0.608502	−	0.793553i	$-0.291771\pi$
$503$	−93.0806	−	93.0806i	−0.185051	−	0.185051i	−0.793553	+	0.608502i	$0.791771\pi$
$504$			270.000i			0.535714i
$505$		0			0
$506$	864.000			1.70751
$507$	74.7094	−	74.7094i	0.147356	−	0.147356i
$508$	7.34847	+	7.34847i	0.0144655	+	0.0144655i
$509$			540.000i			1.06090i	0.847715	+	0.530452i	$0.177978\pi$
$509$			540.000i			1.06090i	−0.847715	+	0.530452i	$0.822022\pi$
$510$		0			0
$511$	−648.000			−1.26810
$512$	−390.694	+	390.694i	−0.763073	+	0.763073i
$513$	−36.7423	−	36.7423i	−0.0716225	−	0.0716225i
$514$		−	708.000i		−	1.37743i
$515$		0			0
$516$	−72.0000			−0.139535
$517$	−793.635	+	793.635i	−1.53508	+	1.53508i
$518$	−132.272	−	132.272i	−0.255352	−	0.255352i
$519$		−	348.000i		−	0.670520i
$520$		0			0
$521$	342.000			0.656430			0.328215	−	0.944603i	$-0.393553\pi$
$521$	342.000			0.656430			0.328215	+	0.944603i	$0.393553\pi$
$522$		0			0
$523$	−264.545	−	264.545i	−0.505822	−	0.505822i	0.407419	−	0.913241i	$-0.366429\pi$
$523$	−264.545	−	264.545i	−0.505822	−	0.505822i	−0.913241	+	0.407419i	$0.866429\pi$
$524$		−	18.0000i		−	0.0343511i
$525$		0			0
$526$	−648.000			−1.23194
$527$	−107.778	+	107.778i	−0.204511	+	0.204511i
$528$	−242.499	−	242.499i	−0.459279	−	0.459279i
$529$			239.000i			0.451796i
$530$		0			0
$531$	−270.000			−0.508475
$532$	−73.4847	+	73.4847i	−0.138129	+	0.138129i
$533$	132.272	+	132.272i	0.248166	+	0.248166i
$534$			270.000i			0.505618i
$535$		0			0
$536$	540.000			1.00746
$537$	110.227	−	110.227i	0.205265	−	0.205265i
$538$		0			0
$539$			1062.00i			1.97032i
$540$		0			0
$541$	2.00000			0.00369686			0.00184843	−	0.999998i	$-0.499412\pi$
$541$	2.00000			0.00369686			0.00184843	+	0.999998i	$0.499412\pi$
$542$	−585.428	+	585.428i	−1.08013	+	1.08013i
$543$	−120.025	−	120.025i	−0.221041	−	0.221041i
$544$		−	108.000i		−	0.198529i
$545$		0			0
$546$	−324.000			−0.593407
$547$	264.545	−	264.545i	0.483629	−	0.483629i	−0.422660	−	0.906288i	$-0.638903\pi$
$547$	264.545	−	264.545i	0.483629	−	0.483629i	0.906288	+	0.422660i	$0.138903\pi$
$548$	142.070	+	142.070i	0.259253	+	0.259253i
$549$			6.00000i			0.0109290i
$550$		0			0
$551$		0			0
$552$	293.939	−	293.939i	0.532498	−	0.532498i
$553$	−514.393	−	514.393i	−0.930186	−	0.930186i
$554$		−	198.000i		−	0.357401i
$555$		0			0
$556$	170.000			0.305755
$557$	484.999	−	484.999i	0.870734	−	0.870734i	−0.121818	−	0.992552i	$-0.538872\pi$
$557$	484.999	−	484.999i	0.870734	−	0.870734i	0.992552	+	0.121818i	$0.0388725\pi$
$558$	80.8332	+	80.8332i	0.144862	+	0.144862i
$559$		−	432.000i		−	0.772809i
$560$		0			0
$561$	216.000			0.385027
$562$	198.409	−	198.409i	0.353040	−	0.353040i
$563$	396.817	+	396.817i	0.704827	+	0.704827i	0.965443	−	0.260616i	$-0.0839256\pi$
$563$	396.817	+	396.817i	0.704827	+	0.704827i	−0.260616	+	0.965443i	$0.583926\pi$
$564$			108.000i			0.191489i
$565$		0			0
$566$	612.000			1.08127
$567$	−66.1362	+	66.1362i	−0.116642	+	0.116642i
$568$	440.908	+	440.908i	0.776247	+	0.776247i
$569$			630.000i			1.10721i	0.832781	+	0.553603i	$0.186747\pi$
$569$			630.000i			1.10721i	−0.832781	+	0.553603i	$0.813253\pi$
$570$		0			0
$571$	302.000			0.528897			0.264448	−	0.964400i	$-0.414810\pi$
$571$	302.000			0.528897			0.264448	+	0.964400i	$0.414810\pi$
$572$	−132.272	+	132.272i	−0.231246	+	0.231246i
$573$	308.636	+	308.636i	0.538631	+	0.538631i
$574$			324.000i			0.564460i
$575$		0			0
$576$	−213.000			−0.369792
$577$	484.999	−	484.999i	0.840553	−	0.840553i	−0.148378	−	0.988931i	$-0.547405\pi$
$577$	484.999	−	484.999i	0.840553	−	0.840553i	0.988931	+	0.148378i	$0.0474052\pi$
$578$	295.164	+	295.164i	0.510664	+	0.510664i
$579$		−	648.000i		−	1.11917i
$580$		0			0
$581$	792.000			1.36317
$582$	−308.636	+	308.636i	−0.530302	+	0.530302i
$583$	−88.1816	−	88.1816i	−0.151255	−	0.151255i
$584$		−	540.000i		−	0.924658i
$585$		0			0
$586$	912.000			1.55631
$587$	582.979	−	582.979i	0.993149	−	0.993149i	−0.00682753	−	0.999977i	$-0.502173\pi$
$587$	582.979	−	582.979i	0.993149	−	0.993149i	0.999977	+	0.00682753i	$0.00217329\pi$
$588$	72.2599	+	72.2599i	0.122891	+	0.122891i
$589$			220.000i			0.373514i
$590$		0			0
$591$	−312.000			−0.527919
$592$	80.8332	−	80.8332i	0.136543	−	0.136543i
$593$	−533.989	−	533.989i	−0.900487	−	0.900487i	0.0949912	−	0.995478i	$-0.469718\pi$
$593$	−533.989	−	533.989i	−0.900487	−	0.900487i	−0.995478	+	0.0949912i	$0.969718\pi$
$594$		−	162.000i		−	0.272727i
$595$		0			0
$596$	180.000			0.302013
$597$	−355.176	+	355.176i	−0.594935	+	0.594935i
$598$	352.727	+	352.727i	0.589844	+	0.589844i
$599$		−	540.000i		−	0.901503i	−0.892650	−	0.450751i	$-0.851156\pi$
$599$		−	540.000i		−	0.901503i	0.892650	−	0.450751i	$-0.148844\pi$
$600$		0			0
$601$	−758.000			−1.26123			−0.630616	−	0.776095i	$-0.717198\pi$
$601$	−758.000			−1.26123			−0.630616	+	0.776095i	$0.717198\pi$
$602$	529.090	−	529.090i	0.878887	−	0.878887i
$603$	132.272	+	132.272i	0.219357	+	0.219357i
$604$			22.0000i			0.0364238i
$605$		0			0
$606$	−324.000			−0.534653
$607$	668.711	−	668.711i	1.10167	−	1.10167i	0.107455	−	0.994210i	$-0.465730\pi$
$607$	668.711	−	668.711i	1.10167	−	1.10167i	0.994210	−	0.107455i	$-0.0342702\pi$
$608$	−110.227	−	110.227i	−0.181294	−	0.181294i
$609$		0			0
$610$		0			0
$611$	−648.000			−1.06056
$612$	14.6969	−	14.6969i	0.0240146	−	0.0240146i
$613$	66.1362	+	66.1362i	0.107889	+	0.107889i	0.758991	−	0.651101i	$-0.225693\pi$
$613$	66.1362	+	66.1362i	0.107889	+	0.107889i	−0.651101	+	0.758991i	$0.725693\pi$
$614$			972.000i			1.58306i
$615$		0			0
$616$	−1620.00			−2.62987
$617$	−592.777	+	592.777i	−0.960740	+	0.960740i	−0.999258	−	0.0385180i	$-0.987736\pi$
$617$	−592.777	+	592.777i	−0.960740	+	0.960740i	0.0385180	+	0.999258i	$0.487736\pi$
$618$	198.409	+	198.409i	0.321050	+	0.321050i
$619$		−	1030.00i		−	1.66397i	−0.554795	−	0.831987i	$-0.687203\pi$
$619$		−	1030.00i		−	1.66397i	0.554795	−	0.831987i	$-0.312797\pi$
$620$		0			0
$621$	144.000			0.231884
$622$	308.636	−	308.636i	0.496199	−	0.496199i
$623$	661.362	+	661.362i	1.06158	+	1.06158i
$624$		−	198.000i		−	0.317308i
$625$		0			0
$626$	−288.000			−0.460064
$627$	220.454	−	220.454i	0.351601	−	0.351601i
$628$	−139.621	−	139.621i	−0.222326	−	0.222326i
$629$			72.0000i			0.114467i
$630$		0			0
$631$	242.000			0.383518			0.191759	−	0.981442i	$-0.438581\pi$
$631$	242.000			0.383518			0.191759	+	0.981442i	$0.438581\pi$
$632$	428.661	−	428.661i	0.678261	−	0.678261i
$633$	149.419	+	149.419i	0.236049	+	0.236049i
$634$			672.000i			1.05994i
$635$		0			0
$636$	−12.0000			−0.0188679
$637$	−433.560	+	433.560i	−0.680627	+	0.680627i
$638$		0			0
$639$			216.000i			0.338028i
$640$		0			0
$641$	162.000			0.252730			0.126365	−	0.991984i	$-0.459669\pi$
$641$	162.000			0.252730			0.126365	+	0.991984i	$0.459669\pi$
$642$	132.272	−	132.272i	0.206032	−	0.206032i
$643$	−264.545	−	264.545i	−0.411423	−	0.411423i	0.470811	−	0.882234i	$-0.343961\pi$
$643$	−264.545	−	264.545i	−0.411423	−	0.411423i	−0.882234	+	0.470811i	$0.843961\pi$
$644$		−	288.000i		−	0.447205i
$645$		0			0
$646$	−120.000			−0.185759
$647$	93.0806	−	93.0806i	0.143865	−	0.143865i	−0.631506	−	0.775371i	$-0.717563\pi$
$647$	93.0806	−	93.0806i	0.143865	−	0.143865i	0.775371	+	0.631506i	$0.217563\pi$
$648$	−55.1135	−	55.1135i	−0.0850517	−	0.0850517i
$649$		−	1620.00i		−	2.49615i
$650$		0			0
$651$	396.000			0.608295
$652$	117.576	−	117.576i	0.180331	−	0.180331i
$653$	−68.5857	−	68.5857i	−0.105032	−	0.105032i	0.652638	−	0.757670i	$-0.273662\pi$
$653$	−68.5857	−	68.5857i	−0.105032	−	0.105032i	−0.757670	+	0.652638i	$0.773662\pi$
$654$			510.000i			0.779817i
$655$		0			0
$656$	−198.000			−0.301829
$657$	132.272	−	132.272i	0.201328	−	0.201328i
$658$	−793.635	−	793.635i	−1.20613	−	1.20613i
$659$			630.000i			0.955994i	0.878361	+	0.477997i	$0.158637\pi$
$659$			630.000i			0.955994i	−0.878361	+	0.477997i	$0.841363\pi$
$660$		0			0
$661$	622.000			0.940998			0.470499	−	0.882400i	$-0.344074\pi$
$661$	622.000			0.940998			0.470499	+	0.882400i	$0.344074\pi$
$662$	−511.943	+	511.943i	−0.773328	+	0.773328i
$663$	88.1816	+	88.1816i	0.133004	+	0.133004i
$664$			660.000i			0.993976i
$665$		0			0
$666$	54.0000			0.0810811
$667$		0			0
$668$	19.5959	+	19.5959i	0.0293352	+	0.0293352i
$669$		−	198.000i		−	0.295964i
$670$		0			0
$671$	−36.0000			−0.0536513
$672$	−198.409	+	198.409i	−0.295251	+	0.295251i
$673$	102.879	+	102.879i	0.152866	+	0.152866i	0.779397	−	0.626531i	$-0.215526\pi$
$673$	102.879	+	102.879i	0.152866	+	0.152866i	−0.626531	+	0.779397i	$0.715526\pi$
$674$		−	468.000i		−	0.694362i
$675$		0			0
$676$	61.0000			0.0902367
$677$	−176.363	+	176.363i	−0.260507	+	0.260507i	−0.825260	−	0.564753i	$-0.808971\pi$
$677$	−176.363	+	176.363i	−0.260507	+	0.260507i	0.564753	+	0.825260i	$0.308971\pi$
$678$	−132.272	−	132.272i	−0.195092	−	0.195092i
$679$			1512.00i			2.22680i
$680$		0			0
$681$	−132.000			−0.193833
$682$	−484.999	+	484.999i	−0.711142	+	0.711142i
$683$	445.807	+	445.807i	0.652719	+	0.652719i	0.953647	−	0.300928i	$-0.0972963\pi$
$683$	445.807	+	445.807i	0.652719	+	0.652719i	−0.300928	+	0.953647i	$0.597296\pi$
$684$		−	30.0000i		−	0.0438596i
$685$		0			0
$686$	−180.000			−0.262391
$687$	61.2372	−	61.2372i	0.0891372	−	0.0891372i
$688$	323.333	+	323.333i	0.469960	+	0.469960i
$689$		−	72.0000i		−	0.104499i
$690$		0			0
$691$	682.000			0.986975			0.493488	−	0.869753i	$-0.335722\pi$
$691$	682.000			0.986975			0.493488	+	0.869753i	$0.335722\pi$
$692$	142.070	−	142.070i	0.205304	−	0.205304i
$693$	−396.817	−	396.817i	−0.572608	−	0.572608i
$694$		−	108.000i		−	0.155620i
$695$		0			0
$696$		0			0
$697$	88.1816	−	88.1816i	0.126516	−	0.126516i
$698$	85.7321	+	85.7321i	0.122825	+	0.122825i
$699$		−	108.000i		−	0.154506i
$700$		0			0
$701$	−468.000			−0.667618			−0.333809	−	0.942641i	$-0.608334\pi$
$701$	−468.000			−0.667618			−0.333809	+	0.942641i	$0.608334\pi$
$702$	66.1362	−	66.1362i	0.0942111	−	0.0942111i
$703$	73.4847	+	73.4847i	0.104530	+	0.104530i
$704$		−	1278.00i		−	1.81534i
$705$		0			0
$706$	−108.000			−0.152975
$707$	−793.635	+	793.635i	−1.12254	+	1.12254i
$708$	−110.227	−	110.227i	−0.155688	−	0.155688i
$709$			310.000i			0.437236i	0.975811	+	0.218618i	$0.0701548\pi$
$709$			310.000i			0.437236i	−0.975811	+	0.218618i	$0.929845\pi$
$710$		0			0
$711$	210.000			0.295359
$712$	−551.135	+	551.135i	−0.774066	+	0.774066i
$713$	−431.110	−	431.110i	−0.604643	−	0.604643i
$714$			216.000i			0.302521i
$715$		0			0
$716$	90.0000			0.125698
$717$	−220.454	+	220.454i	−0.307467	+	0.307467i
$718$	−661.362	−	661.362i	−0.921117	−	0.921117i
$719$			180.000i			0.250348i	0.992135	+	0.125174i	$0.0399489\pi$
$719$			180.000i			0.250348i	−0.992135	+	0.125174i	$0.960051\pi$
$720$		0			0
$721$	972.000			1.34813
$722$	319.658	−	319.658i	0.442740	−	0.442740i
$723$	−218.005	−	218.005i	−0.301528	−	0.301528i
$724$		−	98.0000i		−	0.135359i
$725$		0			0
$726$	609.000			0.838843
$727$	−507.044	+	507.044i	−0.697448	+	0.697448i	−0.963859	−	0.266412i	$-0.914162\pi$
$727$	−507.044	+	507.044i	−0.697448	+	0.697448i	0.266412	+	0.963859i	$0.414162\pi$
$728$	−661.362	−	661.362i	−0.908465	−	0.908465i
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−288.000			−0.393981
$732$	−2.44949	+	2.44949i	−0.00334630	+	0.00334630i
$733$	800.983	+	800.983i	1.09275	+	1.09275i	0.995234	+	0.0975121i	$0.0310885\pi$
$733$	800.983	+	800.983i	1.09275	+	1.09275i	0.0975121	+	0.995234i	$0.468912\pi$
$734$			162.000i			0.220708i
$735$		0			0
$736$	432.000			0.586957
$737$	−793.635	+	793.635i	−1.07684	+	1.07684i
$738$	−66.1362	−	66.1362i	−0.0896155	−	0.0896155i
$739$		−	350.000i		−	0.473613i	−0.971557	−	0.236806i	$-0.923899\pi$
$739$		−	350.000i		−	0.473613i	0.971557	−	0.236806i	$-0.0761007\pi$
$740$		0			0
$741$	180.000			0.242915
$742$	88.1816	−	88.1816i	0.118843	−	0.118843i
$743$	−925.907	−	925.907i	−1.24617	−	1.24617i	−0.957396	−	0.288778i	$-0.906751\pi$
$743$	−925.907	−	925.907i	−1.24617	−	1.24617i	−0.288778	−	0.957396i	$-0.593249\pi$
$744$			330.000i			0.443548i
$745$		0			0
$746$	522.000			0.699732
$747$	−161.666	+	161.666i	−0.216421	+	0.216421i
$748$	88.1816	+	88.1816i	0.117890	+	0.117890i
$749$		−	648.000i		−	0.865154i
$750$		0			0
$751$	−338.000			−0.450067			−0.225033	−	0.974351i	$-0.572249\pi$
$751$	−338.000			−0.450067			−0.225033	+	0.974351i	$0.572249\pi$
$752$	484.999	−	484.999i	0.644945	−	0.644945i
$753$	418.863	+	418.863i	0.556259	+	0.556259i
$754$		0			0
$755$		0			0
$756$	−54.0000			−0.0714286
$757$	−286.590	+	286.590i	−0.378587	+	0.378587i	−0.870592	−	0.492005i	$-0.836264\pi$
$757$	−286.590	+	286.590i	−0.378587	+	0.378587i	0.492005	+	0.870592i	$0.336264\pi$
$758$	−208.207	−	208.207i	−0.274679	−	0.274679i
$759$			864.000i			1.13834i
$760$		0			0
$761$	−1278.00			−1.67937			−0.839685	−	0.543074i	$-0.817260\pi$
$761$	−1278.00			−1.67937			−0.839685	+	0.543074i	$0.817260\pi$
$762$	22.0454	−	22.0454i	0.0289310	−	0.0289310i
$763$	1249.24	+	1249.24i	1.63727	+	1.63727i
$764$			252.000i			0.329843i
$765$		0			0
$766$	372.000			0.485640
$767$	661.362	−	661.362i	0.862271	−	0.862271i
$768$	−219.229	−	219.229i	−0.285455	−	0.285455i
$769$			590.000i			0.767230i	0.923493	+	0.383615i	$0.125321\pi$
$769$			590.000i			0.767230i	−0.923493	+	0.383615i	$0.874679\pi$
$770$		0			0
$771$	708.000			0.918288
$772$	264.545	−	264.545i	0.342675	−	0.342675i
$773$	127.373	+	127.373i	0.164778	+	0.164778i	0.784680	−	0.619902i	$-0.212827\pi$
$773$	127.373	+	127.373i	0.164778	+	0.164778i	−0.619902	+	0.784680i	$0.712827\pi$
$774$			216.000i			0.279070i
$775$		0			0
$776$	−1260.00			−1.62371
$777$	132.272	−	132.272i	0.170235	−	0.170235i
$778$	−440.908	−	440.908i	−0.566720	−	0.566720i
$779$		−	180.000i		−	0.231065i
$780$		0			0
$781$	−1296.00			−1.65941
$782$	235.151	−	235.151i	0.300705	−	0.300705i
$783$		0			0
$784$		−	649.000i		−	0.827806i
$785$		0			0
$786$	−54.0000			−0.0687023
$787$	264.545	−	264.545i	0.336143	−	0.336143i	−0.518770	−	0.854914i	$-0.673610\pi$
$787$	264.545	−	264.545i	0.336143	−	0.336143i	0.854914	+	0.518770i	$0.173610\pi$
$788$	−127.373	−	127.373i	−0.161641	−	0.161641i
$789$		−	648.000i		−	0.821293i
$790$		0			0
$791$	−648.000			−0.819216
$792$	330.681	−	330.681i	0.417527	−	0.417527i
$793$	−14.6969	−	14.6969i	−0.0185333	−	0.0185333i
$794$		−	738.000i		−	0.929471i
$795$		0			0
$796$	−290.000			−0.364322
$797$	−200.858	+	200.858i	−0.252018	+	0.252018i	−0.821797	−	0.569780i	$-0.807029\pi$
$797$	−200.858	+	200.858i	−0.252018	+	0.252018i	0.569780	+	0.821797i	$0.307029\pi$
$798$	220.454	+	220.454i	0.276258	+	0.276258i
$799$			432.000i			0.540676i
$800$		0			0
$801$	−270.000			−0.337079
$802$	−683.408	+	683.408i	−0.852129	+	0.852129i
$803$	793.635	+	793.635i	0.988337	+	0.988337i
$804$			108.000i			0.134328i
$805$		0			0
$806$	−396.000			−0.491315
$807$		0			0
$808$	−661.362	−	661.362i	−0.818518	−	0.818518i
$809$		−	630.000i		−	0.778739i	−0.921082	−	0.389370i	$-0.872693\pi$
$809$		−	630.000i		−	0.778739i	0.921082	−	0.389370i	$-0.127307\pi$
$810$		0			0
$811$	−218.000			−0.268804			−0.134402	−	0.990927i	$-0.542911\pi$
$811$	−218.000			−0.268804			−0.134402	+	0.990927i	$0.542911\pi$
$812$		0			0
$813$	−585.428	−	585.428i	−0.720084	−	0.720084i
$814$			324.000i			0.398034i
$815$		0			0
$816$	−132.000			−0.161765
$817$	−293.939	+	293.939i	−0.359778	+	0.359778i
$818$	820.579	+	820.579i	1.00315	+	1.00315i
$819$		−	324.000i		−	0.395604i
$820$		0			0
$821$	432.000			0.526188			0.263094	−	0.964770i	$-0.415257\pi$
$821$	432.000			0.526188			0.263094	+	0.964770i	$0.415257\pi$
$822$	426.211	−	426.211i	0.518505	−	0.518505i
$823$	360.075	+	360.075i	0.437515	+	0.437515i	0.891175	−	0.453660i	$-0.149882\pi$
$823$	360.075	+	360.075i	0.437515	+	0.437515i	−0.453660	+	0.891175i	$0.649882\pi$
$824$			810.000i			0.983010i
$825$		0			0
$826$	1620.00			1.96126
$827$	876.917	−	876.917i	1.06036	−	1.06036i	0.0623022	−	0.998057i	$-0.480156\pi$
$827$	876.917	−	876.917i	1.06036	−	1.06036i	0.998057	−	0.0623022i	$-0.0198443\pi$
$828$	58.7878	+	58.7878i	0.0709997	+	0.0709997i
$829$			70.0000i			0.0844391i	0.999108	+	0.0422195i	$0.0134429\pi$
$829$			70.0000i			0.0844391i	−0.999108	+	0.0422195i	$0.986557\pi$
$830$		0			0
$831$	198.000			0.238267
$832$	521.741	−	521.741i	0.627093	−	0.627093i
$833$	289.040	+	289.040i	0.346987	+	0.346987i
$834$		−	510.000i		−	0.611511i
$835$		0			0
$836$	180.000			0.215311
$837$	−80.8332	+	80.8332i	−0.0965749	+	0.0965749i
$838$	771.589	+	771.589i	0.920751	+	0.920751i
$839$		−	540.000i		−	0.643623i	−0.946804	−	0.321812i	$-0.895708\pi$
$839$		−	540.000i		−	0.643623i	0.946804	−	0.321812i	$-0.104292\pi$
$840$		0			0
$841$	841.000			1.00000
$842$	173.914	−	173.914i	0.206548	−	0.206548i
$843$	198.409	+	198.409i	0.235360	+	0.235360i
$844$			122.000i			0.144550i
$845$		0			0
$846$	324.000			0.382979
$847$	1491.74	−	1491.74i	1.76120	−	1.76120i
$848$	53.8888	+	53.8888i	0.0635481	+	0.0635481i
$849$			612.000i			0.720848i
$850$		0			0
$851$	−288.000			−0.338425
$852$	−88.1816	+	88.1816i	−0.103500	+	0.103500i
$853$	−815.680	−	815.680i	−0.956249	−	0.956249i	0.0428336	−	0.999082i	$-0.486361\pi$
$853$	−815.680	−	815.680i	−0.956249	−	0.956249i	−0.999082	+	0.0428336i	$0.986361\pi$
$854$		−	36.0000i		−	0.0421546i
$855$		0			0
$856$	540.000			0.630841
$857$	−298.838	+	298.838i	−0.348702	+	0.348702i	−0.859626	−	0.510924i	$-0.829303\pi$
$857$	−298.838	+	298.838i	−0.348702	+	0.348702i	0.510924	+	0.859626i	$0.329303\pi$
$858$	396.817	+	396.817i	0.462491	+	0.462491i
$859$		−	310.000i		−	0.360885i	−0.983586	−	0.180442i	$-0.942247\pi$
$859$		−	310.000i		−	0.360885i	0.983586	−	0.180442i	$-0.0577529\pi$
$860$		0			0
$861$	−324.000			−0.376307
$862$	749.544	−	749.544i	0.869540	−	0.869540i
$863$	225.353	+	225.353i	0.261128	+	0.261128i	0.825512	−	0.564385i	$-0.190886\pi$
$863$	225.353	+	225.353i	0.261128	+	0.261128i	−0.564385	+	0.825512i	$0.690886\pi$
$864$		−	81.0000i		−	0.0937500i
$865$		0			0
$866$	252.000			0.290993
$867$	−295.164	+	295.164i	−0.340442	+	0.340442i
$868$	161.666	+	161.666i	0.186252	+	0.186252i
$869$			1260.00i			1.44994i
$870$		0			0
$871$	−648.000			−0.743972
$872$	−1041.03	+	1041.03i	−1.19385	+	1.19385i
$873$	−308.636	−	308.636i	−0.353535	−	0.353535i
$874$		−	480.000i		−	0.549199i
$875$		0			0
$876$	108.000			0.123288
$877$	−213.106	+	213.106i	−0.242994	+	0.242994i	−0.818088	−	0.575094i	$-0.804966\pi$
$877$	−213.106	+	213.106i	−0.242994	+	0.242994i	0.575094	+	0.818088i	$0.304966\pi$
$878$	−526.640	−	526.640i	−0.599818	−	0.599818i
$879$			912.000i			1.03754i
$880$		0			0
$881$	1062.00			1.20545			0.602724	−	0.797950i	$-0.294082\pi$
$881$	1062.00			1.20545			0.602724	+	0.797950i	$0.294082\pi$
$882$	216.780	−	216.780i	0.245782	−	0.245782i
$883$	−117.576	−	117.576i	−0.133155	−	0.133155i	0.637388	−	0.770543i	$-0.280015\pi$
$883$	−117.576	−	117.576i	−0.133155	−	0.133155i	−0.770543	+	0.637388i	$0.780015\pi$
$884$			72.0000i			0.0814480i
$885$		0			0
$886$	−1428.00			−1.61174
$887$	509.494	−	509.494i	0.574401	−	0.574401i	−0.358954	−	0.933355i	$-0.616867\pi$
$887$	509.494	−	509.494i	0.574401	−	0.574401i	0.933355	+	0.358954i	$0.116867\pi$
$888$	110.227	+	110.227i	0.124130	+	0.124130i
$889$		−	108.000i		−	0.121485i
$890$		0			0
$891$	162.000			0.181818
$892$	80.8332	−	80.8332i	0.0906201	−	0.0906201i
$893$	440.908	+	440.908i	0.493738	+	0.493738i
$894$		−	540.000i		−	0.604027i
$895$		0			0
$896$	630.000			0.703125
$897$	−352.727	+	352.727i	−0.393229	+	0.393229i
$898$	110.227	+	110.227i	0.122747	+	0.122747i
$899$		0			0
$900$		0			0
$901$	−48.0000			−0.0532741
$902$	396.817	−	396.817i	0.439931	−	0.439931i
$903$	529.090	+	529.090i	0.585924	+	0.585924i
$904$		−	540.000i		−	0.597345i
$905$		0			0
$906$	66.0000			0.0728477
$907$	411.514	−	411.514i	0.453709	−	0.453709i	−0.442874	−	0.896584i	$-0.646041\pi$
$907$	411.514	−	411.514i	0.453709	−	0.453709i	0.896584	+	0.442874i	$0.146041\pi$
$908$	−53.8888	−	53.8888i	−0.0593489	−	0.0593489i
$909$		−	324.000i		−	0.356436i
$910$		0			0
$911$	792.000			0.869374			0.434687	−	0.900582i	$-0.356859\pi$
$911$	792.000			0.869374			0.434687	+	0.900582i	$0.356859\pi$
$912$	−134.722	+	134.722i	−0.147721	+	0.147721i
$913$	−969.998	−	969.998i	−1.06243	−	1.06243i
$914$		−	468.000i		−	0.512035i
$915$		0			0
$916$	50.0000			0.0545852
$917$	−132.272	+	132.272i	−0.144245	+	0.144245i
$918$	−44.0908	−	44.0908i	−0.0480292	−	0.0480292i
$919$			430.000i			0.467900i	0.972249	+	0.233950i	$0.0751652\pi$
$919$			430.000i			0.467900i	−0.972249	+	0.233950i	$0.924835\pi$
$920$		0			0
$921$	−972.000			−1.05537
$922$	−1014.09	+	1014.09i	−1.09988	+	1.09988i
$923$	−529.090	−	529.090i	−0.573228	−	0.573228i
$924$		−	324.000i		−	0.350649i
$925$		0			0
$926$	−18.0000			−0.0194384
$927$	−198.409	+	198.409i	−0.214033	+	0.214033i
$928$		0			0
$929$			1530.00i			1.64693i	0.567365	+	0.823466i	$0.307963\pi$
$929$			1530.00i			1.64693i	−0.567365	+	0.823466i	$0.692037\pi$
$930$		0			0
$931$	590.000			0.633727
$932$	44.0908	−	44.0908i	0.0473077	−	0.0473077i
$933$	308.636	+	308.636i	0.330799	+	0.330799i
$934$			852.000i			0.912206i
$935$		0			0
$936$	270.000			0.288462
$937$	−323.333	+	323.333i	−0.345072	+	0.345072i	−0.858270	−	0.513198i	$-0.828461\pi$
$937$	−323.333	+	323.333i	−0.345072	+	0.345072i	0.513198	+	0.858270i	$0.328461\pi$
$938$	−793.635	−	793.635i	−0.846092	−	0.846092i
$939$		−	288.000i		−	0.306709i
$940$		0			0
$941$	1152.00			1.22423			0.612115	−	0.790769i	$-0.290319\pi$
$941$	1152.00			1.22423			0.612115	+	0.790769i	$0.290319\pi$
$942$	−418.863	+	418.863i	−0.444653	+	0.444653i
$943$	352.727	+	352.727i	0.374047	+	0.374047i
$944$			990.000i			1.04873i
$945$		0			0
$946$	−1296.00			−1.36998
$947$	−739.746	+	739.746i	−0.781147	+	0.781147i	−0.980024	−	0.198878i	$-0.936270\pi$
$947$	−739.746	+	739.746i	−0.781147	+	0.781147i	0.198878	+	0.980024i	$0.436270\pi$
$948$	85.7321	+	85.7321i	0.0904347	+	0.0904347i
$949$			648.000i			0.682824i
$950$		0			0
$951$	−672.000			−0.706625
$952$	−440.908	+	440.908i	−0.463139	+	0.463139i
$953$	−44.0908	−	44.0908i	−0.0462653	−	0.0462653i	0.683596	−	0.729861i	$-0.260415\pi$
$953$	−44.0908	−	44.0908i	−0.0462653	−	0.0462653i	−0.729861	+	0.683596i	$0.760415\pi$
$954$			36.0000i			0.0377358i
$955$		0			0
$956$	−180.000			−0.188285
$957$		0			0
$958$	−440.908	−	440.908i	−0.460238	−	0.460238i
$959$		−	2088.00i		−	2.17727i
$960$		0			0
$961$	−477.000			−0.496358
$962$	−132.272	+	132.272i	−0.137497	+	0.137497i
$963$	132.272	+	132.272i	0.137355	+	0.137355i
$964$		−	178.000i		−	0.184647i
$965$		0			0
$966$	−864.000			−0.894410
$967$	−727.498	+	727.498i	−0.752325	+	0.752325i	−0.974913	−	0.222588i	$-0.928550\pi$
$967$	−727.498	+	727.498i	−0.752325	+	0.752325i	0.222588	+	0.974913i	$0.428550\pi$
$968$	1243.12	+	1243.12i	1.28421	+	1.28421i
$969$		−	120.000i		−	0.123839i
$970$		0			0
$971$	−1278.00			−1.31617			−0.658084	−	0.752944i	$-0.728633\pi$
$971$	−1278.00			−1.31617			−0.658084	+	0.752944i	$0.728633\pi$
$972$	11.0227	−	11.0227i	0.0113402	−	0.0113402i
$973$	−1249.24	−	1249.24i	−1.28391	−	1.28391i
$974$			1422.00i			1.45996i
$975$		0			0
$976$	22.0000			0.0225410
$977$	−396.817	+	396.817i	−0.406159	+	0.406159i	−0.880397	−	0.474238i	$-0.842724\pi$
$977$	−396.817	+	396.817i	−0.406159	+	0.406159i	0.474238	+	0.880397i	$0.342724\pi$
$978$	−352.727	−	352.727i	−0.360661	−	0.360661i
$979$		−	1620.00i		−	1.65475i
$980$		0			0
$981$	−510.000			−0.519878
$982$	−22.0454	+	22.0454i	−0.0224495	+	0.0224495i
$983$	−827.928	−	827.928i	−0.842246	−	0.842246i	0.146905	−	0.989151i	$-0.453069\pi$
$983$	−827.928	−	827.928i	−0.842246	−	0.842246i	−0.989151	+	0.146905i	$0.953069\pi$
$984$		−	270.000i		−	0.274390i
$985$		0			0
$986$		0			0
$987$	793.635	−	793.635i	0.804088	−	0.804088i
$988$	73.4847	+	73.4847i	0.0743772	+	0.0743772i
$989$		−	1152.00i		−	1.16481i
$990$		0			0
$991$	−118.000			−0.119072			−0.0595358	−	0.998226i	$-0.518962\pi$
$991$	−118.000			−0.119072			−0.0595358	+	0.998226i	$0.518962\pi$
$992$	−242.499	+	242.499i	−0.244455	+	0.244455i
$993$	−511.943	−	511.943i	−0.515552	−	0.515552i
$994$		−	1296.00i		−	1.30382i
$995$		0			0
$996$	−132.000			−0.132530
$997$	815.680	−	815.680i	0.818134	−	0.818134i	−0.167703	−	0.985838i	$-0.553635\pi$
$997$	815.680	−	815.680i	0.818134	−	0.818134i	0.985838	+	0.167703i	$0.0536350\pi$
$998$	722.599	+	722.599i	0.724048	+	0.724048i
$999$			54.0000i			0.0540541i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.3.f.a.43.2	yes	4
3.2	odd	2		225.3.g.f.118.1		4
4.3	odd	2		1200.3.bg.j.193.1		4
5.2	odd	4	inner	75.3.f.a.7.2	yes	4
5.3	odd	4	inner	75.3.f.a.7.1	✓	4
5.4	even	2	inner	75.3.f.a.43.1	yes	4
15.2	even	4		225.3.g.f.82.1		4
15.8	even	4		225.3.g.f.82.2		4
15.14	odd	2		225.3.g.f.118.2		4
20.3	even	4		1200.3.bg.j.1057.2		4
20.7	even	4		1200.3.bg.j.1057.1		4
20.19	odd	2		1200.3.bg.j.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.a.7.1	✓	4	5.3	odd	4	inner
75.3.f.a.7.2	yes	4	5.2	odd	4	inner
75.3.f.a.43.1	yes	4	5.4	even	2	inner
75.3.f.a.43.2	yes	4	1.1	even	1	trivial
225.3.g.f.82.1		4	15.2	even	4
225.3.g.f.82.2		4	15.8	even	4
225.3.g.f.118.1		4	3.2	odd	2
225.3.g.f.118.2		4	15.14	odd	2
1200.3.bg.j.193.1		4	4.3	odd	2
1200.3.bg.j.193.2		4	20.19	odd	2
1200.3.bg.j.1057.1		4	20.7	even	4
1200.3.bg.j.1057.2		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} +1.00000i q^{4} +3.00000 q^{6} +(7.34847 - 7.34847i) q^{7} +(6.12372 + 6.12372i) q^{8} +3.00000i q^{9} -18.0000 q^{11} +(-1.22474 + 1.22474i) q^{12} +(-7.34847 - 7.34847i) q^{13} -18.0000i q^{14} +11.0000 q^{16} +(-4.89898 + 4.89898i) q^{17} +(3.67423 + 3.67423i) q^{18} +10.0000i q^{19} +18.0000 q^{21} +(-22.0454 + 22.0454i) q^{22} +(-19.5959 - 19.5959i) q^{23} +15.0000i q^{24} -18.0000 q^{26} +(-3.67423 + 3.67423i) q^{27} +(7.34847 + 7.34847i) q^{28} +22.0000 q^{31} +(-11.0227 + 11.0227i) q^{32} +(-22.0454 - 22.0454i) q^{33} +12.0000i q^{34} -3.00000 q^{36} +(7.34847 - 7.34847i) q^{37} +(12.2474 + 12.2474i) q^{38} -18.0000i q^{39} -18.0000 q^{41} +(22.0454 - 22.0454i) q^{42} +(29.3939 + 29.3939i) q^{43} -18.0000i q^{44} -48.0000 q^{46} +(44.0908 - 44.0908i) q^{47} +(13.4722 + 13.4722i) q^{48} -59.0000i q^{49} -12.0000 q^{51} +(7.34847 - 7.34847i) q^{52} +(4.89898 + 4.89898i) q^{53} +9.00000i q^{54} +90.0000 q^{56} +(-12.2474 + 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(26.9444 - 26.9444i) q^{62} +(22.0454 + 22.0454i) q^{63} +71.0000i q^{64} -54.0000 q^{66} +(44.0908 - 44.0908i) q^{67} +(-4.89898 - 4.89898i) q^{68} -48.0000i q^{69} +72.0000 q^{71} +(-18.3712 + 18.3712i) q^{72} +(-44.0908 - 44.0908i) q^{73} -18.0000i q^{74} -10.0000 q^{76} +(-132.272 + 132.272i) q^{77} +(-22.0454 - 22.0454i) q^{78} -70.0000i q^{79} -9.00000 q^{81} +(-22.0454 + 22.0454i) q^{82} +(53.8888 + 53.8888i) q^{83} +18.0000i q^{84} +72.0000 q^{86} +(-110.227 - 110.227i) q^{88} +90.0000i q^{89} -108.000 q^{91} +(19.5959 - 19.5959i) q^{92} +(26.9444 + 26.9444i) q^{93} -108.000i q^{94} -27.0000 q^{96} +(-102.879 + 102.879i) q^{97} +(-72.2599 - 72.2599i) q^{98} -54.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{6} - 72 q^{11} + 44 q^{16} + 72 q^{21} - 72 q^{26} + 88 q^{31} - 12 q^{36} - 72 q^{41} - 192 q^{46} - 48 q^{51} + 360 q^{56} + 8 q^{61} - 216 q^{66} + 288 q^{71} - 40 q^{76} - 36 q^{81} + 288 q^{86}+ \cdots - 108 q^{96}+O(q^{100}) \) 4 * q + 12 * q^6 - 72 * q^11 + 44 * q^16 + 72 * q^21 - 72 * q^26 + 88 * q^31 - 12 * q^36 - 72 * q^41 - 192 * q^46 - 48 * q^51 + 360 * q^56 + 8 * q^61 - 216 * q^66 + 288 * q^71 - 40 * q^76 - 36 * q^81 + 288 * q^86 - 432 * q^91 - 108 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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