LMFDB - Embedded newform 1200.3.bg.j.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, not 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:	$32.6976317232$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.j.1057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(-7.34847 + 7.34847i) q^{7} +3.00000i q^{9} +18.0000 q^{11} +(-7.34847 - 7.34847i) q^{13} +(-4.89898 + 4.89898i) q^{17} -10.0000i q^{19} +18.0000 q^{21} +(19.5959 + 19.5959i) q^{23} +(3.67423 - 3.67423i) q^{27} -22.0000 q^{31} +(-22.0454 - 22.0454i) q^{33} +(7.34847 - 7.34847i) q^{37} +18.0000i q^{39} -18.0000 q^{41} +(-29.3939 - 29.3939i) q^{43} +(-44.0908 + 44.0908i) q^{47} -59.0000i q^{49} +12.0000 q^{51} +(4.89898 + 4.89898i) q^{53} +(-12.2474 + 12.2474i) q^{57} -90.0000i q^{59} +2.00000 q^{61} +(-22.0454 - 22.0454i) q^{63} +(-44.0908 + 44.0908i) q^{67} -48.0000i q^{69} -72.0000 q^{71} +(-44.0908 - 44.0908i) q^{73} +(-132.272 + 132.272i) q^{77} +70.0000i q^{79} -9.00000 q^{81} +(-53.8888 - 53.8888i) q^{83} +90.0000i q^{89} +108.000 q^{91} +(26.9444 + 26.9444i) q^{93} +(-102.879 + 102.879i) q^{97} +54.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 72 q^{11} + 72 q^{21} - 88 q^{31} - 72 q^{41} + 48 q^{51} + 8 q^{61} - 288 q^{71} - 36 q^{81} + 432 q^{91}+O(q^{100}) $ 4 * q + 72 * q^11 + 72 * q^21 - 88 * q^31 - 72 * q^41 + 48 * q^51 + 8 * q^61 - 288 * q^71 - 36 * q^81 + 432 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−7.34847	+	7.34847i	−1.04978	+	1.04978i	−0.0510871	+	0.998694i	$0.516269\pi$
$7$	−7.34847	+	7.34847i	−1.04978	+	1.04978i	−0.998694	+	0.0510871i	$0.983731\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	18.0000			1.63636			0.818182	−	0.574960i	$-0.194982\pi$
$11$	18.0000			1.63636			0.818182	+	0.574960i	$0.194982\pi$
$12$		0			0
$13$	−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	$0.880887\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−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	$0.315320\pi$
$18$		0			0
$19$		−	10.0000i		−	0.526316i	−0.964753	−	0.263158i	$-0.915236\pi$
$19$		−	10.0000i		−	0.526316i	0.964753	−	0.263158i	$-0.0847640\pi$
$20$		0			0
$21$	18.0000			0.857143
$22$		0			0
$23$	19.5959	+	19.5959i	0.851996	+	0.851996i	0.990379	−	0.138382i	$-0.0441903\pi$
$23$	19.5959	+	19.5959i	0.851996	+	0.851996i	−0.138382	+	0.990379i	$0.544190\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−22.0000			−0.709677			−0.354839	−	0.934928i	$-0.615464\pi$
$31$	−22.0000			−0.709677			−0.354839	+	0.934928i	$0.615464\pi$
$32$		0			0
$33$	−22.0454	−	22.0454i	−0.668043	−	0.668043i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	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	$0.205149\pi$
$38$		0			0
$39$			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	$0.570447\pi$
$42$		0			0
$43$	−29.3939	−	29.3939i	−0.683579	−	0.683579i	0.277226	−	0.960805i	$-0.410585\pi$
$43$	−29.3939	−	29.3939i	−0.683579	−	0.683579i	−0.960805	+	0.277226i	$0.910585\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−44.0908	+	44.0908i	−0.938102	+	0.938102i	−0.998193	−	0.0600905i	$-0.980861\pi$
$47$	−44.0908	+	44.0908i	−0.938102	+	0.938102i	0.0600905	+	0.998193i	$0.480861\pi$
$48$		0			0
$49$		−	59.0000i		−	1.20408i
$50$		0			0
$51$	12.0000			0.235294
$52$		0			0
$53$	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	$0.729180\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−12.2474	+	12.2474i	−0.214868	+	0.214868i
$58$		0			0
$59$		−	90.0000i		−	1.52542i	−0.646738	−	0.762712i	$-0.723867\pi$
$59$		−	90.0000i		−	1.52542i	0.646738	−	0.762712i	$-0.276133\pi$
$60$		0			0
$61$	2.00000			0.0327869			0.0163934	−	0.999866i	$-0.494782\pi$
$61$	2.00000			0.0327869			0.0163934	+	0.999866i	$0.494782\pi$
$62$		0			0
$63$	−22.0454	−	22.0454i	−0.349927	−	0.349927i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−44.0908	+	44.0908i	−0.658072	+	0.658072i	−0.954924	−	0.296852i	$-0.904063\pi$
$67$	−44.0908	+	44.0908i	−0.658072	+	0.658072i	0.296852	+	0.954924i	$0.404063\pi$
$68$		0			0
$69$		−	48.0000i		−	0.695652i
$70$		0			0
$71$	−72.0000			−1.01408			−0.507042	−	0.861921i	$-0.669261\pi$
$71$	−72.0000			−1.01408			−0.507042	+	0.861921i	$0.669261\pi$
$72$		0			0
$73$	−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	$0.890458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−132.272	+	132.272i	−1.71782	+	1.71782i
$78$		0			0
$79$			70.0000i			0.886076i	0.896503	+	0.443038i	$0.146099\pi$
$79$			70.0000i			0.886076i	−0.896503	+	0.443038i	$0.853901\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−53.8888	−	53.8888i	−0.649262	−	0.649262i	0.303552	−	0.952815i	$-0.401827\pi$
$83$	−53.8888	−	53.8888i	−0.649262	−	0.649262i	−0.952815	+	0.303552i	$0.901827\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			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
$92$		0			0
$93$	26.9444	+	26.9444i	0.289725	+	0.289725i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−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	$0.980073\pi$
$98$		0			0
$99$			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	$0.679558\pi$
$102$		0			0
$103$	−66.1362	−	66.1362i	−0.642099	−	0.642099i	0.308972	−	0.951071i	$-0.400015\pi$
$103$	−66.1362	−	66.1362i	−0.642099	−	0.642099i	−0.951071	+	0.308972i	$0.900015\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−44.0908	+	44.0908i	−0.412064	+	0.412064i	−0.882457	−	0.470393i	$-0.844112\pi$
$107$	−44.0908	+	44.0908i	−0.412064	+	0.412064i	0.470393	+	0.882457i	$0.344112\pi$
$108$		0			0
$109$			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
$112$		0			0
$113$	−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	$0.838976\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	22.0454	−	22.0454i	0.188422	−	0.188422i
$118$		0			0
$119$		−	72.0000i		−	0.605042i
$120$		0			0
$121$	203.000			1.67769
$122$		0			0
$123$	22.0454	+	22.0454i	0.179231	+	0.179231i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.34847	+	7.34847i	−0.0578620	+	0.0578620i	−0.735446	−	0.677584i	$-0.763027\pi$
$127$	−7.34847	+	7.34847i	−0.0578620	+	0.0578620i	0.677584	+	0.735446i	$0.263027\pi$
$128$		0			0
$129$			72.0000i			0.558140i
$130$		0			0
$131$	18.0000			0.137405			0.0687023	−	0.997637i	$-0.478114\pi$
$131$	18.0000			0.137405			0.0687023	+	0.997637i	$0.478114\pi$
$132$		0			0
$133$	73.4847	+	73.4847i	0.552516	+	0.552516i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	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	$-0.0120101\pi$
$138$		0			0
$139$			170.000i			1.22302i	0.791236	+	0.611511i	$0.209438\pi$
$139$			170.000i			1.22302i	−0.791236	+	0.611511i	$0.790562\pi$
$140$		0			0
$141$	108.000			0.765957
$142$		0			0
$143$	−132.272	−	132.272i	−0.924982	−	0.924982i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−72.2599	+	72.2599i	−0.491564	+	0.491564i
$148$		0			0
$149$		−	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	$-0.523209\pi$
$151$	−22.0000			−0.145695			−0.0728477	+	0.997343i	$0.523209\pi$
$152$		0			0
$153$	−14.6969	−	14.6969i	−0.0960584	−	0.0960584i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−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	$0.466467\pi$
$158$		0			0
$159$		−	12.0000i		−	0.0754717i
$160$		0			0
$161$	−288.000			−1.78882
$162$		0			0
$163$	117.576	+	117.576i	0.721322	+	0.721322i	0.968875	−	0.247552i	$-0.0796262\pi$
$163$	117.576	+	117.576i	0.721322	+	0.721322i	−0.247552	+	0.968875i	$0.579626\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−19.5959	+	19.5959i	−0.117341	+	0.117341i	−0.763339	−	0.645998i	$-0.776441\pi$
$167$	−19.5959	+	19.5959i	−0.117341	+	0.117341i	0.645998	+	0.763339i	$0.276441\pi$
$168$		0			0
$169$		−	61.0000i		−	0.360947i
$170$		0			0
$171$	30.0000			0.175439
$172$		0			0
$173$	−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
$176$		0			0
$177$	−110.227	+	110.227i	−0.622752	+	0.622752i
$178$		0			0
$179$			90.0000i			0.502793i	0.967884	+	0.251397i	$0.0808899\pi$
$179$			90.0000i			0.502793i	−0.967884	+	0.251397i	$0.919110\pi$
$180$		0			0
$181$	−98.0000			−0.541436			−0.270718	−	0.962659i	$-0.587261\pi$
$181$	−98.0000			−0.541436			−0.270718	+	0.962659i	$0.587261\pi$
$182$		0			0
$183$	−2.44949	−	2.44949i	−0.0133852	−	0.0133852i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−88.1816	+	88.1816i	−0.471560	+	0.471560i
$188$		0			0
$189$			54.0000i			0.285714i
$190$		0			0
$191$	−252.000			−1.31937			−0.659686	−	0.751541i	$-0.729311\pi$
$191$	−252.000			−1.31937			−0.659686	+	0.751541i	$0.729311\pi$
$192$		0			0
$193$	−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	$-0.670833\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−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	$0.401144\pi$
$198$		0			0
$199$		−	290.000i		−	1.45729i	−0.684893	−	0.728643i	$-0.740151\pi$
$199$		−	290.000i		−	1.45729i	0.684893	−	0.728643i	$-0.259849\pi$
$200$		0			0
$201$	108.000			0.537313
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−58.7878	+	58.7878i	−0.283999	+	0.283999i
$208$		0			0
$209$		−	180.000i		−	0.861244i
$210$		0			0
$211$	−122.000			−0.578199			−0.289100	−	0.957299i	$-0.593356\pi$
$211$	−122.000			−0.578199			−0.289100	+	0.957299i	$0.593356\pi$
$212$		0			0
$213$	88.1816	+	88.1816i	0.413998	+	0.413998i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	161.666	−	161.666i	0.745006	−	0.745006i
$218$		0			0
$219$			108.000i			0.493151i
$220$		0			0
$221$	72.0000			0.325792
$222$		0			0
$223$	80.8332	+	80.8332i	0.362481	+	0.362481i	0.864725	−	0.502245i	$-0.167492\pi$
$223$	80.8332	+	80.8332i	0.362481	+	0.362481i	−0.502245	+	0.864725i	$0.667492\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	53.8888	−	53.8888i	0.237395	−	0.237395i	−0.578375	−	0.815771i	$-0.696313\pi$
$227$	53.8888	−	53.8888i	0.237395	−	0.237395i	0.815771	+	0.578375i	$0.196313\pi$
$228$		0			0
$229$		−	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	$0.792720\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	85.7321	−	85.7321i	0.361739	−	0.361739i
$238$		0			0
$239$		−	180.000i		−	0.753138i	−0.926389	−	0.376569i	$-0.877104\pi$
$239$		−	180.000i		−	0.753138i	0.926389	−	0.376569i	$-0.122896\pi$
$240$		0			0
$241$	−178.000			−0.738589			−0.369295	−	0.929312i	$-0.620401\pi$
$241$	−178.000			−0.738589			−0.369295	+	0.929312i	$0.620401\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−73.4847	+	73.4847i	−0.297509	+	0.297509i
$248$		0			0
$249$			132.000i			0.530120i
$250$		0			0
$251$	−342.000			−1.36255			−0.681275	−	0.732028i	$-0.738574\pi$
$251$	−342.000			−1.36255			−0.681275	+	0.732028i	$0.738574\pi$
$252$		0			0
$253$	352.727	+	352.727i	1.39418	+	1.39418i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	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	$-0.0426660\pi$
$258$		0			0
$259$			108.000i			0.416988i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	264.545	+	264.545i	1.00587	+	1.00587i	0.999983	+	0.00589147i	$0.00187532\pi$
$263$	264.545	+	264.545i	1.00587	+	1.00587i	0.00589147	+	0.999983i	$0.498125\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	110.227	−	110.227i	0.412835	−	0.412835i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	478.000			1.76384			0.881919	−	0.471401i	$-0.156252\pi$
$271$	478.000			1.76384			0.881919	+	0.471401i	$0.156252\pi$
$272$		0			0
$273$	−132.272	−	132.272i	−0.484514	−	0.484514i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	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	$0.183843\pi$
$278$		0			0
$279$		−	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	$0.406925\pi$
$282$		0			0
$283$	−249.848	−	249.848i	−0.882855	−	0.882855i	0.110969	−	0.993824i	$-0.464605\pi$
$283$	−249.848	−	249.848i	−0.882855	−	0.882855i	−0.993824	+	0.110969i	$0.964605\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	132.272	−	132.272i	0.460880	−	0.460880i
$288$		0			0
$289$			241.000i			0.833910i
$290$		0			0
$291$	252.000			0.865979
$292$		0			0
$293$	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	$0.394630\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	66.1362	−	66.1362i	0.222681	−	0.222681i
$298$		0			0
$299$		−	288.000i		−	0.963211i
$300$		0			0
$301$	432.000			1.43522
$302$		0			0
$303$	132.272	+	132.272i	0.436543	+	0.436543i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	396.817	−	396.817i	1.29256	−	1.29256i	0.359369	−	0.933195i	$-0.382992\pi$
$307$	396.817	−	396.817i	1.29256	−	1.29256i	0.933195	−	0.359369i	$-0.117008\pi$
$308$		0			0
$309$			162.000i			0.524272i
$310$		0			0
$311$	−252.000			−0.810289			−0.405145	−	0.914253i	$-0.632779\pi$
$311$	−252.000			−0.810289			−0.405145	+	0.914253i	$0.632779\pi$
$312$		0			0
$313$	−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	$0.835576\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−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	$0.459617\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	108.000			0.336449
$322$		0			0
$323$	48.9898	+	48.9898i	0.151671	+	0.151671i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	208.207	−	208.207i	0.636718	−	0.636718i
$328$		0			0
$329$		−	648.000i		−	1.96960i
$330$		0			0
$331$	418.000			1.26284			0.631420	−	0.775441i	$-0.282472\pi$
$331$	418.000			1.26284			0.631420	+	0.775441i	$0.282472\pi$
$332$		0			0
$333$	22.0454	+	22.0454i	0.0662024	+	0.0662024i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	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	$0.118701\pi$
$338$		0			0
$339$			108.000i			0.318584i
$340$		0			0
$341$	−396.000			−1.16129
$342$		0			0
$343$	73.4847	+	73.4847i	0.214241	+	0.214241i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−44.0908	+	44.0908i	−0.127063	+	0.127063i	−0.767778	−	0.640716i	$-0.778638\pi$
$347$	−44.0908	+	44.0908i	−0.127063	+	0.127063i	0.640716	+	0.767778i	$0.278638\pi$
$348$		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
$352$		0			0
$353$	−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	$0.778150\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−88.1816	+	88.1816i	−0.247007	+	0.247007i
$358$		0			0
$359$			540.000i			1.50418i	0.659061	+	0.752089i	$0.270954\pi$
$359$			540.000i			1.50418i	−0.659061	+	0.752089i	$0.729046\pi$
$360$		0			0
$361$	261.000			0.722992
$362$		0			0
$363$	−248.623	−	248.623i	−0.684912	−	0.684912i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	66.1362	−	66.1362i	0.180208	−	0.180208i	−0.611239	−	0.791446i	$-0.709328\pi$
$367$	66.1362	−	66.1362i	0.180208	−	0.180208i	0.791446	+	0.611239i	$0.209328\pi$
$368$		0			0
$369$		−	54.0000i		−	0.146341i
$370$		0			0
$371$	−72.0000			−0.194070
$372$		0			0
$373$	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	$0.617623\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			170.000i			0.448549i	0.974526	+	0.224274i	$0.0720012\pi$
$379$			170.000i			0.448549i	−0.974526	+	0.224274i	$0.927999\pi$
$380$		0			0
$381$	18.0000			0.0472441
$382$		0			0
$383$	−151.868	−	151.868i	−0.396523	−	0.396523i	0.480482	−	0.877005i	$-0.340462\pi$
$383$	−151.868	−	151.868i	−0.396523	−	0.396523i	−0.877005	+	0.480482i	$0.840462\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	88.1816	−	88.1816i	0.227860	−	0.227860i
$388$		0			0
$389$		−	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
$392$		0			0
$393$	−22.0454	−	22.0454i	−0.0560952	−	0.0560952i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	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	$0.0696970\pi$
$398$		0			0
$399$		−	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	$0.744933\pi$
$402$		0			0
$403$	161.666	+	161.666i	0.401157	+	0.401157i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	132.272	−	132.272i	0.324994	−	0.324994i
$408$		0			0
$409$			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
$412$		0			0
$413$	661.362	+	661.362i	1.60136	+	1.60136i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	208.207	−	208.207i	0.499296	−	0.499296i
$418$		0			0
$419$		−	630.000i		−	1.50358i	−0.659403	−	0.751790i	$-0.729191\pi$
$419$		−	630.000i		−	1.50358i	0.659403	−	0.751790i	$-0.270809\pi$
$420$		0			0
$421$	142.000			0.337292			0.168646	−	0.985677i	$-0.446061\pi$
$421$	142.000			0.337292			0.168646	+	0.985677i	$0.446061\pi$
$422$		0			0
$423$	−132.272	−	132.272i	−0.312701	−	0.312701i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−14.6969	+	14.6969i	−0.0344191	+	0.0344191i
$428$		0			0
$429$			324.000i			0.755245i
$430$		0			0
$431$	−612.000			−1.41995			−0.709977	−	0.704225i	$-0.751295\pi$
$431$	−612.000			−1.41995			−0.709977	+	0.704225i	$0.751295\pi$
$432$		0			0
$433$	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	$0.696268\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	195.959	−	195.959i	0.448419	−	0.448419i
$438$		0			0
$439$			430.000i			0.979499i	0.871863	+	0.489749i	$0.162912\pi$
$439$			430.000i			0.979499i	−0.871863	+	0.489749i	$0.837088\pi$
$440$		0			0
$441$	177.000			0.401361
$442$		0			0
$443$	582.979	+	582.979i	1.31598	+	1.31598i	0.916930	+	0.399049i	$0.130660\pi$
$443$	582.979	+	582.979i	1.31598	+	1.31598i	0.399049	+	0.916930i	$0.369340\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−220.454	+	220.454i	−0.493186	+	0.493186i
$448$		0			0
$449$			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
$452$		0			0
$453$	26.9444	+	26.9444i	0.0594799	+	0.0594799i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	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	$0.154473\pi$
$458$		0			0
$459$			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	$0.855015\pi$
$462$		0			0
$463$	7.34847	+	7.34847i	0.0158714	+	0.0158714i	0.714998	−	0.699127i	$-0.246428\pi$
$463$	7.34847	+	7.34847i	0.0158714	+	0.0158714i	−0.699127	+	0.714998i	$0.746428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	347.828	−	347.828i	0.744813	−	0.744813i	−0.228687	−	0.973500i	$-0.573443\pi$
$467$	347.828	−	347.828i	0.744813	−	0.744813i	0.973500	+	0.228687i	$0.0734433\pi$
$468$		0			0
$469$		−	648.000i		−	1.38166i
$470$		0			0
$471$	342.000			0.726115
$472$		0			0
$473$	−529.090	−	529.090i	−1.11858	−	1.11858i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−14.6969	+	14.6969i	−0.0308112	+	0.0308112i
$478$		0			0
$479$			360.000i			0.751566i	0.926708	+	0.375783i	$0.122626\pi$
$479$			360.000i			0.751566i	−0.926708	+	0.375783i	$0.877374\pi$
$480$		0			0
$481$	−108.000			−0.224532
$482$		0			0
$483$	352.727	+	352.727i	0.730283	+	0.730283i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	580.529	−	580.529i	1.19205	−	1.19205i	0.215561	−	0.976490i	$-0.430842\pi$
$487$	580.529	−	580.529i	1.19205	−	1.19205i	0.976490	−	0.215561i	$-0.0691581\pi$
$488$		0			0
$489$		−	288.000i		−	0.588957i
$490$		0			0
$491$	18.0000			0.0366599			0.0183299	−	0.999832i	$-0.494165\pi$
$491$	18.0000			0.0366599			0.0183299	+	0.999832i	$0.494165\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	529.090	−	529.090i	1.06457	−	1.06457i
$498$		0			0
$499$		−	590.000i		−	1.18236i	−0.806538	−	0.591182i	$-0.798661\pi$
$499$		−	590.000i		−	1.18236i	0.806538	−	0.591182i	$-0.201339\pi$
$500$		0			0
$501$	48.0000			0.0958084
$502$		0			0
$503$	93.0806	+	93.0806i	0.185051	+	0.185051i	0.793553	−	0.608502i	$-0.208229\pi$
$503$	93.0806	+	93.0806i	0.185051	+	0.185051i	−0.608502	+	0.793553i	$0.708229\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−74.7094	+	74.7094i	−0.147356	+	0.147356i
$508$		0			0
$509$			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
$512$		0			0
$513$	−36.7423	−	36.7423i	−0.0716225	−	0.0716225i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−793.635	+	793.635i	−1.53508	+	1.53508i
$518$		0			0
$519$			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	0.913241	−	0.407419i	$-0.133571\pi$
$523$	264.545	+	264.545i	0.505822	+	0.505822i	−0.407419	+	0.913241i	$0.633571\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	107.778	−	107.778i	0.204511	−	0.204511i
$528$		0			0
$529$			239.000i			0.451796i
$530$		0			0
$531$	270.000			0.508475
$532$		0			0
$533$	132.272	+	132.272i	0.248166	+	0.248166i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	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	$0.499412\pi$
$542$		0			0
$543$	120.025	+	120.025i	0.221041	+	0.221041i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−264.545	+	264.545i	−0.483629	+	0.483629i	−0.906288	−	0.422660i	$-0.861097\pi$
$547$	−264.545	+	264.545i	−0.483629	+	0.483629i	0.422660	+	0.906288i	$0.361097\pi$
$548$		0			0
$549$			6.00000i			0.0109290i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−514.393	−	514.393i	−0.930186	−	0.930186i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	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	$0.0388725\pi$
$558$		0			0
$559$			432.000i			0.772809i
$560$		0			0
$561$	216.000			0.385027
$562$		0			0
$563$	−396.817	−	396.817i	−0.704827	−	0.704827i	0.260616	−	0.965443i	$-0.416074\pi$
$563$	−396.817	−	396.817i	−0.704827	−	0.704827i	−0.965443	+	0.260616i	$0.916074\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	66.1362	−	66.1362i	0.116642	−	0.116642i
$568$		0			0
$569$			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	$-0.585190\pi$
$571$	−302.000			−0.528897			−0.264448	+	0.964400i	$0.585190\pi$
$572$		0			0
$573$	308.636	+	308.636i	0.538631	+	0.538631i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	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	$0.0474052\pi$
$578$		0			0
$579$			648.000i			1.11917i
$580$		0			0
$581$	792.000			1.36317
$582$		0			0
$583$	88.1816	+	88.1816i	0.151255	+	0.151255i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−582.979	+	582.979i	−0.993149	+	0.993149i	−0.999977	−	0.00682753i	$-0.997827\pi$
$587$	−582.979	+	582.979i	−0.993149	+	0.993149i	0.00682753	+	0.999977i	$0.497827\pi$
$588$		0			0
$589$			220.000i			0.373514i
$590$		0			0
$591$	312.000			0.527919
$592$		0			0
$593$	−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	$0.969718\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−355.176	+	355.176i	−0.594935	+	0.594935i
$598$		0			0
$599$			540.000i			0.901503i	0.892650	+	0.450751i	$0.148844\pi$
$599$			540.000i			0.901503i	−0.892650	+	0.450751i	$0.851156\pi$
$600$		0			0
$601$	−758.000			−1.26123			−0.630616	−	0.776095i	$-0.717198\pi$
$601$	−758.000			−1.26123			−0.630616	+	0.776095i	$0.717198\pi$
$602$		0			0
$603$	−132.272	−	132.272i	−0.219357	−	0.219357i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−668.711	+	668.711i	−1.10167	+	1.10167i	−0.107455	+	0.994210i	$0.534270\pi$
$607$	−668.711	+	668.711i	−1.10167	+	1.10167i	−0.994210	+	0.107455i	$0.965730\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	648.000			1.06056
$612$		0			0
$613$	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	$0.725693\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−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	$0.487736\pi$
$618$		0			0
$619$			1030.00i			1.66397i	0.554795	+	0.831987i	$0.312797\pi$
$619$			1030.00i			1.66397i	−0.554795	+	0.831987i	$0.687203\pi$
$620$		0			0
$621$	144.000			0.231884
$622$		0			0
$623$	−661.362	−	661.362i	−1.06158	−	1.06158i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−220.454	+	220.454i	−0.351601	+	0.351601i
$628$		0			0
$629$			72.0000i			0.114467i
$630$		0			0
$631$	−242.000			−0.383518			−0.191759	−	0.981442i	$-0.561419\pi$
$631$	−242.000			−0.383518			−0.191759	+	0.981442i	$0.561419\pi$
$632$		0			0
$633$	149.419	+	149.419i	0.236049	+	0.236049i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−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	$0.459669\pi$
$642$		0			0
$643$	264.545	+	264.545i	0.411423	+	0.411423i	0.882234	−	0.470811i	$-0.156039\pi$
$643$	264.545	+	264.545i	0.411423	+	0.411423i	−0.470811	+	0.882234i	$0.656039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−93.0806	+	93.0806i	−0.143865	+	0.143865i	−0.775371	−	0.631506i	$-0.782437\pi$
$647$	−93.0806	+	93.0806i	−0.143865	+	0.143865i	0.631506	+	0.775371i	$0.282437\pi$
$648$		0			0
$649$		−	1620.00i		−	2.49615i
$650$		0			0
$651$	−396.000			−0.608295
$652$		0			0
$653$	−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	$0.773662\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	132.272	−	132.272i	0.201328	−	0.201328i
$658$		0			0
$659$		−	630.000i		−	0.955994i	−0.878361	−	0.477997i	$-0.841363\pi$
$659$		−	630.000i		−	0.955994i	0.878361	−	0.477997i	$-0.158637\pi$
$660$		0			0
$661$	622.000			0.940998			0.470499	−	0.882400i	$-0.344074\pi$
$661$	622.000			0.940998			0.470499	+	0.882400i	$0.344074\pi$
$662$		0			0
$663$	−88.1816	−	88.1816i	−0.133004	−	0.133004i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	198.000i		−	0.295964i
$670$		0			0
$671$	36.0000			0.0536513
$672$		0			0
$673$	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	$0.715526\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−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	$0.308971\pi$
$678$		0			0
$679$		−	1512.00i		−	2.22680i
$680$		0			0
$681$	−132.000			−0.193833
$682$		0			0
$683$	−445.807	−	445.807i	−0.652719	−	0.652719i	0.300928	−	0.953647i	$-0.402704\pi$
$683$	−445.807	−	445.807i	−0.652719	−	0.652719i	−0.953647	+	0.300928i	$0.902704\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−61.2372	+	61.2372i	−0.0891372	+	0.0891372i
$688$		0			0
$689$		−	72.0000i		−	0.104499i
$690$		0			0
$691$	−682.000			−0.986975			−0.493488	−	0.869753i	$-0.664278\pi$
$691$	−682.000			−0.986975			−0.493488	+	0.869753i	$0.664278\pi$
$692$		0			0
$693$	−396.817	−	396.817i	−0.572608	−	0.572608i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	88.1816	−	88.1816i	0.126516	−	0.126516i
$698$		0			0
$699$			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	$0.608334\pi$
$702$		0			0
$703$	−73.4847	−	73.4847i	−0.104530	−	0.104530i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	793.635	−	793.635i	1.12254	−	1.12254i
$708$		0			0
$709$			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
$712$		0			0
$713$	−431.110	−	431.110i	−0.604643	−	0.604643i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−220.454	+	220.454i	−0.307467	+	0.307467i
$718$		0			0
$719$		−	180.000i		−	0.250348i	−0.992135	−	0.125174i	$-0.960051\pi$
$719$		−	180.000i		−	0.250348i	0.992135	−	0.125174i	$-0.0399489\pi$
$720$		0			0
$721$	972.000			1.34813
$722$		0			0
$723$	218.005	+	218.005i	0.301528	+	0.301528i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	507.044	−	507.044i	0.697448	−	0.697448i	−0.266412	−	0.963859i	$-0.585838\pi$
$727$	507.044	−	507.044i	0.697448	−	0.697448i	0.963859	+	0.266412i	$0.0858381\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	288.000			0.393981
$732$		0			0
$733$	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	$0.468912\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−793.635	+	793.635i	−1.07684	+	1.07684i
$738$		0			0
$739$			350.000i			0.473613i	0.971557	+	0.236806i	$0.0761007\pi$
$739$			350.000i			0.473613i	−0.971557	+	0.236806i	$0.923899\pi$
$740$		0			0
$741$	180.000			0.242915
$742$		0			0
$743$	925.907	+	925.907i	1.24617	+	1.24617i	0.957396	+	0.288778i	$0.0932488\pi$
$743$	925.907	+	925.907i	1.24617	+	1.24617i	0.288778	+	0.957396i	$0.406751\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	161.666	−	161.666i	0.216421	−	0.216421i
$748$		0			0
$749$		−	648.000i		−	0.865154i
$750$		0			0
$751$	338.000			0.450067			0.225033	−	0.974351i	$-0.427751\pi$
$751$	338.000			0.450067			0.225033	+	0.974351i	$0.427751\pi$
$752$		0			0
$753$	418.863	+	418.863i	0.556259	+	0.556259i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−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	$0.336264\pi$
$758$		0			0
$759$		−	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	$0.817260\pi$
$762$		0			0
$763$	−1249.24	−	1249.24i	−1.63727	−	1.63727i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−661.362	+	661.362i	−0.862271	+	0.862271i
$768$		0			0
$769$			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
$772$		0			0
$773$	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	$0.712827\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	132.272	−	132.272i	0.170235	−	0.170235i
$778$		0			0
$779$			180.000i			0.231065i
$780$		0			0
$781$	−1296.00			−1.65941
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−264.545	+	264.545i	−0.336143	+	0.336143i	−0.854914	−	0.518770i	$-0.826390\pi$
$787$	−264.545	+	264.545i	−0.336143	+	0.336143i	0.518770	+	0.854914i	$0.326390\pi$
$788$		0			0
$789$		−	648.000i		−	0.821293i
$790$		0			0
$791$	648.000			0.819216
$792$		0			0
$793$	−14.6969	−	14.6969i	−0.0185333	−	0.0185333i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−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	$0.307029\pi$
$798$		0			0
$799$		−	432.000i		−	0.540676i
$800$		0			0
$801$	−270.000			−0.337079
$802$		0			0
$803$	−793.635	−	793.635i	−0.988337	−	0.988337i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	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	$-0.457089\pi$
$811$	218.000			0.268804			0.134402	+	0.990927i	$0.457089\pi$
$812$		0			0
$813$	−585.428	−	585.428i	−0.720084	−	0.720084i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−293.939	+	293.939i	−0.359778	+	0.359778i
$818$		0			0
$819$			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	$0.415257\pi$
$822$		0			0
$823$	−360.075	−	360.075i	−0.437515	−	0.437515i	0.453660	−	0.891175i	$-0.350118\pi$
$823$	−360.075	−	360.075i	−0.437515	−	0.437515i	−0.891175	+	0.453660i	$0.850118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−876.917	+	876.917i	−1.06036	+	1.06036i	−0.0623022	+	0.998057i	$0.519844\pi$
$827$	−876.917	+	876.917i	−1.06036	+	1.06036i	−0.998057	+	0.0623022i	$0.980156\pi$
$828$		0			0
$829$			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
$832$		0			0
$833$	289.040	+	289.040i	0.346987	+	0.346987i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−80.8332	+	80.8332i	−0.0965749	+	0.0965749i
$838$		0			0
$839$			540.000i			0.643623i	0.946804	+	0.321812i	$0.104292\pi$
$839$			540.000i			0.643623i	−0.946804	+	0.321812i	$0.895708\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$	−198.409	−	198.409i	−0.235360	−	0.235360i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1491.74	+	1491.74i	−1.76120	+	1.76120i
$848$		0			0
$849$			612.000i			0.720848i
$850$		0			0
$851$	288.000			0.338425
$852$		0			0
$853$	−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	$0.986361\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−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	$0.329303\pi$
$858$		0			0
$859$			310.000i			0.360885i	0.983586	+	0.180442i	$0.0577529\pi$
$859$			310.000i			0.360885i	−0.983586	+	0.180442i	$0.942247\pi$
$860$		0			0
$861$	−324.000			−0.376307
$862$		0			0
$863$	−225.353	−	225.353i	−0.261128	−	0.261128i	0.564385	−	0.825512i	$-0.309114\pi$
$863$	−225.353	−	225.353i	−0.261128	−	0.261128i	−0.825512	+	0.564385i	$0.809114\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	295.164	−	295.164i	0.340442	−	0.340442i
$868$		0			0
$869$			1260.00i			1.44994i
$870$		0			0
$871$	648.000			0.743972
$872$		0			0
$873$	−308.636	−	308.636i	−0.353535	−	0.353535i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−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	$0.304966\pi$
$878$		0			0
$879$		−	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	$0.294082\pi$
$882$		0			0
$883$	117.576	+	117.576i	0.133155	+	0.133155i	0.770543	−	0.637388i	$-0.219985\pi$
$883$	117.576	+	117.576i	0.133155	+	0.133155i	−0.637388	+	0.770543i	$0.719985\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−509.494	+	509.494i	−0.574401	+	0.574401i	−0.933355	−	0.358954i	$-0.883133\pi$
$887$	−509.494	+	509.494i	−0.574401	+	0.574401i	0.358954	+	0.933355i	$0.383133\pi$
$888$		0			0
$889$		−	108.000i		−	0.121485i
$890$		0			0
$891$	−162.000			−0.181818
$892$		0			0
$893$	440.908	+	440.908i	0.493738	+	0.493738i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−352.727	+	352.727i	−0.393229	+	0.393229i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−48.0000			−0.0532741
$902$		0			0
$903$	−529.090	−	529.090i	−0.585924	−	0.585924i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−411.514	+	411.514i	−0.453709	+	0.453709i	−0.896584	−	0.442874i	$-0.853959\pi$
$907$	−411.514	+	411.514i	−0.453709	+	0.453709i	0.442874	+	0.896584i	$0.353959\pi$
$908$		0			0
$909$		−	324.000i		−	0.356436i
$910$		0			0
$911$	−792.000			−0.869374			−0.434687	−	0.900582i	$-0.643141\pi$
$911$	−792.000			−0.869374			−0.434687	+	0.900582i	$0.643141\pi$
$912$		0			0
$913$	−969.998	−	969.998i	−1.06243	−	1.06243i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−132.272	+	132.272i	−0.144245	+	0.144245i
$918$		0			0
$919$		−	430.000i		−	0.467900i	−0.972249	−	0.233950i	$-0.924835\pi$
$919$		−	430.000i		−	0.467900i	0.972249	−	0.233950i	$-0.0751652\pi$
$920$		0			0
$921$	−972.000			−1.05537
$922$		0			0
$923$	529.090	+	529.090i	0.573228	+	0.573228i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	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
$932$		0			0
$933$	308.636	+	308.636i	0.330799	+	0.330799i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−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	$0.328461\pi$
$938$		0			0
$939$			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	$0.290319\pi$
$942$		0			0
$943$	−352.727	−	352.727i	−0.374047	−	0.374047i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	739.746	−	739.746i	0.781147	−	0.781147i	−0.198878	−	0.980024i	$-0.563730\pi$
$947$	739.746	−	739.746i	0.781147	−	0.781147i	0.980024	+	0.198878i	$0.0637296\pi$
$948$		0			0
$949$			648.000i			0.682824i
$950$		0			0
$951$	672.000			0.706625
$952$		0			0
$953$	−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	$0.760415\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			2088.00i			2.17727i
$960$		0			0
$961$	−477.000			−0.496358
$962$		0			0
$963$	−132.272	−	132.272i	−0.137355	−	0.137355i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	727.498	−	727.498i	0.752325	−	0.752325i	−0.222588	−	0.974913i	$-0.571450\pi$
$967$	727.498	−	727.498i	0.752325	−	0.752325i	0.974913	+	0.222588i	$0.0714503\pi$
$968$		0			0
$969$		−	120.000i		−	0.123839i
$970$		0			0
$971$	1278.00			1.31617			0.658084	−	0.752944i	$-0.271367\pi$
$971$	1278.00			1.31617			0.658084	+	0.752944i	$0.271367\pi$
$972$		0			0
$973$	−1249.24	−	1249.24i	−1.28391	−	1.28391i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−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	$0.342724\pi$
$978$		0			0
$979$			1620.00i			1.65475i
$980$		0			0
$981$	−510.000			−0.519878
$982$		0			0
$983$	827.928	+	827.928i	0.842246	+	0.842246i	0.989151	−	0.146905i	$-0.0469311\pi$
$983$	827.928	+	827.928i	0.842246	+	0.842246i	−0.146905	+	0.989151i	$0.546931\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−793.635	+	793.635i	−0.804088	+	0.804088i
$988$		0			0
$989$		−	1152.00i		−	1.16481i
$990$		0			0
$991$	118.000			0.119072			0.0595358	−	0.998226i	$-0.481038\pi$
$991$	118.000			0.119072			0.0595358	+	0.998226i	$0.481038\pi$
$992$		0			0
$993$	−511.943	−	511.943i	−0.515552	−	0.515552i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	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	$0.0536350\pi$
$998$		0			0
$999$		−	54.0000i		−	0.0540541i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.a.7.1	✓	4	20.3	even	4
75.3.f.a.7.2	yes	4	20.7	even	4
75.3.f.a.43.1	yes	4	20.19	odd	2
75.3.f.a.43.2	yes	4	4.3	odd	2
225.3.g.f.82.1		4	60.47	odd	4
225.3.g.f.82.2		4	60.23	odd	4
225.3.g.f.118.1		4	12.11	even	2
225.3.g.f.118.2		4	60.59	even	2
1200.3.bg.j.193.1		4	1.1	even	1	trivial
1200.3.bg.j.193.2		4	5.4	even	2	inner
1200.3.bg.j.1057.1		4	5.2	odd	4	inner
1200.3.bg.j.1057.2		4	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-7.34847 + 7.34847i) q^{7} +3.00000i q^{9} +18.0000 q^{11} +(-7.34847 - 7.34847i) q^{13} +(-4.89898 + 4.89898i) q^{17} -10.0000i q^{19} +18.0000 q^{21} +(19.5959 + 19.5959i) q^{23} +(3.67423 - 3.67423i) q^{27} -22.0000 q^{31} +(-22.0454 - 22.0454i) q^{33} +(7.34847 - 7.34847i) q^{37} +18.0000i q^{39} -18.0000 q^{41} +(-29.3939 - 29.3939i) q^{43} +(-44.0908 + 44.0908i) q^{47} -59.0000i q^{49} +12.0000 q^{51} +(4.89898 + 4.89898i) q^{53} +(-12.2474 + 12.2474i) q^{57} -90.0000i q^{59} +2.00000 q^{61} +(-22.0454 - 22.0454i) q^{63} +(-44.0908 + 44.0908i) q^{67} -48.0000i q^{69} -72.0000 q^{71} +(-44.0908 - 44.0908i) q^{73} +(-132.272 + 132.272i) q^{77} +70.0000i q^{79} -9.00000 q^{81} +(-53.8888 - 53.8888i) q^{83} +90.0000i q^{89} +108.000 q^{91} +(26.9444 + 26.9444i) q^{93} +(-102.879 + 102.879i) q^{97} +54.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 72 q^{11} + 72 q^{21} - 88 q^{31} - 72 q^{41} + 48 q^{51} + 8 q^{61} - 288 q^{71} - 36 q^{81} + 432 q^{91}+O(q^{100}) \) 4 * q + 72 * q^11 + 72 * q^21 - 88 * q^31 - 72 * q^41 + 48 * q^51 + 8 * q^61 - 288 * q^71 - 36 * q^81 + 432 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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