LMFDB - Embedded newform 1200.3.bg.j.1057.2

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			1057.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.1057
Dual form			1200.3.bg.j.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(7.34847 + 7.34847i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.22474 - 1.22474i) q^{3} +(7.34847 + 7.34847i) q^{7} -3.00000i q^{9} +18.0000 q^{11} +(7.34847 - 7.34847i) q^{13} +(4.89898 + 4.89898i) q^{17} +10.0000i q^{19} +18.0000 q^{21} +(-19.5959 + 19.5959i) q^{23} +(-3.67423 - 3.67423i) q^{27} -22.0000 q^{31} +(22.0454 - 22.0454i) q^{33} +(-7.34847 - 7.34847i) q^{37} -18.0000i q^{39} -18.0000 q^{41} +(29.3939 - 29.3939i) q^{43} +(44.0908 + 44.0908i) q^{47} +59.0000i q^{49} +12.0000 q^{51} +(-4.89898 + 4.89898i) q^{53} +(12.2474 + 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(22.0454 - 22.0454i) q^{63} +(44.0908 + 44.0908i) q^{67} +48.0000i q^{69} -72.0000 q^{71} +(44.0908 - 44.0908i) q^{73} +(132.272 + 132.272i) q^{77} -70.0000i q^{79} -9.00000 q^{81} +(53.8888 - 53.8888i) q^{83} -90.0000i q^{89} +108.000 q^{91} +(-26.9444 + 26.9444i) q^{93} +(102.879 + 102.879i) q^{97} -54.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * 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

Display 100 coefficients 10 coefficients

Character values

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

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

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	0.998694	+	0.0510871i	$0.0162686\pi$
$7$	7.34847	+	7.34847i	1.04978	+	1.04978i	0.0510871	+	0.998694i	$0.483731\pi$
$8$		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	$-0.619113\pi$
$13$	7.34847	−	7.34847i	0.565267	−	0.565267i	0.930799	+	0.365532i	$0.119113\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.89898	+	4.89898i	0.288175	+	0.288175i	0.836358	−	0.548183i	$-0.184680\pi$
$17$	4.89898	+	4.89898i	0.288175	+	0.288175i	−0.548183	+	0.836358i	$0.684680\pi$
$18$		0			0
$19$			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
$22$		0			0
$23$	−19.5959	+	19.5959i	−0.851996	+	0.851996i	−0.990379	−	0.138382i	$-0.955810\pi$
$23$	−19.5959	+	19.5959i	−0.851996	+	0.851996i	0.138382	+	0.990379i	$0.455810\pi$
$24$		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	$-0.294851\pi$
$37$	−7.34847	−	7.34847i	−0.198607	−	0.198607i	−0.799403	+	0.600795i	$0.794851\pi$
$38$		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	$-0.589415\pi$
$43$	29.3939	−	29.3939i	0.683579	−	0.683579i	0.960805	+	0.277226i	$0.0894151\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	44.0908	+	44.0908i	0.938102	+	0.938102i	0.998193	−	0.0600905i	$-0.0191389\pi$
$47$	44.0908	+	44.0908i	0.938102	+	0.938102i	−0.0600905	+	0.998193i	$0.519139\pi$
$48$		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	$-0.770820\pi$
$53$	−4.89898	+	4.89898i	−0.0924336	+	0.0924336i	0.659378	+	0.751812i	$0.270820\pi$
$54$		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	$0.276133\pi$
$59$			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	$0.494782\pi$
$62$		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	$-0.0959367\pi$
$67$	44.0908	+	44.0908i	0.658072	+	0.658072i	−0.296852	+	0.954924i	$0.595937\pi$
$68$		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	$-0.609542\pi$
$73$	44.0908	−	44.0908i	0.603984	−	0.603984i	0.941367	+	0.337384i	$0.109542\pi$
$74$		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	$-0.853901\pi$
$79$		−	70.0000i		−	0.886076i	0.896503	−	0.443038i	$-0.146099\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	53.8888	−	53.8888i	0.649262	−	0.649262i	−0.303552	−	0.952815i	$-0.598173\pi$
$83$	53.8888	−	53.8888i	0.649262	−	0.649262i	0.952815	+	0.303552i	$0.0981727\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	90.0000i		−	1.01124i	−0.862757	−	0.505618i	$-0.831265\pi$
$89$		−	90.0000i		−	1.01124i	0.862757	−	0.505618i	$-0.168735\pi$
$90$		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	0.998041	+	0.0625628i	$0.0199274\pi$
$97$	102.879	+	102.879i	1.06060	+	1.06060i	0.0625628	+	0.998041i	$0.480073\pi$
$98$		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	$-0.599985\pi$
$103$	66.1362	−	66.1362i	0.642099	−	0.642099i	0.951071	+	0.308972i	$0.0999849\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	44.0908	+	44.0908i	0.412064	+	0.412064i	0.882457	−	0.470393i	$-0.155888\pi$
$107$	44.0908	+	44.0908i	0.412064	+	0.412064i	−0.470393	+	0.882457i	$0.655888\pi$
$108$		0			0
$109$		−	170.000i		−	1.55963i	−0.626008	−	0.779817i	$-0.715312\pi$
$109$		−	170.000i		−	1.55963i	0.626008	−	0.779817i	$-0.284688\pi$
$110$		0			0
$111$	−18.0000			−0.162162
$112$		0			0
$113$	44.0908	−	44.0908i	0.390184	−	0.390184i	−0.484569	−	0.874753i	$-0.661024\pi$
$113$	44.0908	−	44.0908i	0.390184	−	0.390184i	0.874753	+	0.484569i	$0.161024\pi$
$114$		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	$-0.236973\pi$
$127$	7.34847	+	7.34847i	0.0578620	+	0.0578620i	−0.677584	+	0.735446i	$0.736973\pi$
$128$		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	−0.999288	−	0.0377220i	$-0.987990\pi$
$137$	−142.070	−	142.070i	−1.03701	−	1.03701i	−0.0377220	−	0.999288i	$-0.512010\pi$
$138$		0			0
$139$		−	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
$142$		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	$0.206438\pi$
$149$			180.000i			1.20805i	−0.796964	+	0.604027i	$0.793562\pi$
$150$		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	$-0.0335326\pi$
$157$	139.621	+	139.621i	0.889305	+	0.889305i	−0.105151	+	0.994456i	$0.533533\pi$
$158$		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	$-0.920374\pi$
$163$	−117.576	+	117.576i	−0.721322	+	0.721322i	0.247552	+	0.968875i	$0.420374\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.5959	+	19.5959i	0.117341	+	0.117341i	0.763339	−	0.645998i	$-0.223559\pi$
$167$	19.5959	+	19.5959i	0.117341	+	0.117341i	−0.645998	+	0.763339i	$0.723559\pi$
$168$		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	$-0.552784\pi$
$173$	142.070	−	142.070i	0.821216	−	0.821216i	0.986282	+	0.165066i	$0.0527838\pi$
$174$		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	$-0.919110\pi$
$179$		−	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	$0.587261\pi$
$182$		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	0.511292	−	0.859407i	$-0.329167\pi$
$193$	264.545	−	264.545i	1.37070	−	1.37070i	0.859407	−	0.511292i	$-0.170833\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	127.373	+	127.373i	0.646566	+	0.646566i	0.952161	−	0.305596i	$-0.0988556\pi$
$197$	127.373	+	127.373i	0.646566	+	0.646566i	−0.305596	+	0.952161i	$0.598856\pi$
$198$		0			0
$199$			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
$202$		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	$-0.832508\pi$
$223$	−80.8332	+	80.8332i	−0.362481	+	0.362481i	0.502245	+	0.864725i	$0.332508\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−53.8888	−	53.8888i	−0.237395	−	0.237395i	0.578375	−	0.815771i	$-0.303687\pi$
$227$	−53.8888	−	53.8888i	−0.237395	−	0.237395i	−0.815771	+	0.578375i	$0.803687\pi$
$228$		0			0
$229$			50.0000i			0.218341i	0.994023	+	0.109170i	$0.0348194\pi$
$229$			50.0000i			0.218341i	−0.994023	+	0.109170i	$0.965181\pi$
$230$		0			0
$231$	324.000			1.40260
$232$		0			0
$233$	44.0908	−	44.0908i	0.189231	−	0.189231i	−0.606133	−	0.795364i	$-0.707280\pi$
$233$	44.0908	−	44.0908i	0.189231	−	0.189231i	0.795364	+	0.606133i	$0.207280\pi$
$234$		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	$0.122896\pi$
$239$			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	$0.620401\pi$
$242$		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	−0.991030	−	0.133638i	$-0.957334\pi$
$257$	−289.040	−	289.040i	−1.12467	−	1.12467i	−0.133638	−	0.991030i	$-0.542666\pi$
$258$		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	−0.00589147	+	0.999983i	$0.501875\pi$
$263$	−264.545	+	264.545i	−1.00587	+	1.00587i	−0.999983	+	0.00589147i	$0.998125\pi$
$264$		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	$-0.316157\pi$
$277$	−80.8332	−	80.8332i	−0.291816	−	0.291816i	−0.837798	+	0.545981i	$0.816157\pi$
$278$		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	$-0.535395\pi$
$283$	249.848	−	249.848i	0.882855	−	0.882855i	0.993824	+	0.110969i	$0.0353954\pi$
$284$		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	−0.325017	+	0.945708i	$0.605370\pi$
$293$	−372.322	+	372.322i	−1.27073	+	1.27073i	−0.945708	+	0.325017i	$0.894630\pi$
$294$		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	−0.933195	−	0.359369i	$-0.882992\pi$
$307$	−396.817	−	396.817i	−1.29256	−	1.29256i	−0.359369	−	0.933195i	$-0.617008\pi$
$308$		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	$-0.664424\pi$
$313$	117.576	−	117.576i	0.375641	−	0.375641i	0.869527	+	0.493886i	$0.164424\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	274.343	+	274.343i	0.865435	+	0.865435i	0.991963	−	0.126528i	$-0.0403834\pi$
$317$	274.343	+	274.343i	0.865435	+	0.865435i	−0.126528	+	0.991963i	$0.540383\pi$
$318$		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	$-0.381299\pi$
$337$	−191.060	−	191.060i	−0.566944	−	0.566944i	−0.931271	+	0.364327i	$0.881299\pi$
$338$		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	$-0.221362\pi$
$347$	44.0908	+	44.0908i	0.127063	+	0.127063i	−0.640716	+	0.767778i	$0.721362\pi$
$348$		0			0
$349$		−	70.0000i		−	0.200573i	−0.994959	−	0.100287i	$-0.968024\pi$
$349$		−	70.0000i		−	0.200573i	0.994959	−	0.100287i	$-0.0319759\pi$
$350$		0			0
$351$	−54.0000			−0.153846
$352$		0			0
$353$	44.0908	−	44.0908i	0.124903	−	0.124903i	−0.641892	−	0.766795i	$-0.721850\pi$
$353$	44.0908	−	44.0908i	0.124903	−	0.124903i	0.766795	+	0.641892i	$0.221850\pi$
$354$		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	$-0.729046\pi$
$359$		−	540.000i		−	1.50418i	0.659061	−	0.752089i	$-0.270954\pi$
$360$		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	$-0.290672\pi$
$367$	−66.1362	−	66.1362i	−0.180208	−	0.180208i	−0.791446	+	0.611239i	$0.790672\pi$
$368$		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	$-0.882377\pi$
$373$	−213.106	+	213.106i	−0.571329	+	0.571329i	0.361171	+	0.932500i	$0.382377\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	170.000i		−	0.448549i	−0.974526	−	0.224274i	$-0.927999\pi$
$379$		−	170.000i		−	0.448549i	0.974526	−	0.224274i	$-0.0720012\pi$
$380$		0			0
$381$	18.0000			0.0472441
$382$		0			0
$383$	151.868	−	151.868i	0.396523	−	0.396523i	−0.480482	−	0.877005i	$-0.659538\pi$
$383$	151.868	−	151.868i	0.396523	−	0.396523i	0.877005	+	0.480482i	$0.159538\pi$
$384$		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	$0.153128\pi$
$389$			360.000i			0.925450i	−0.886502	+	0.462725i	$0.846872\pi$
$390$		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	$-0.430303\pi$
$397$	−301.287	−	301.287i	−0.758910	−	0.758910i	−0.976124	+	0.217214i	$0.930303\pi$
$398$		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	$-0.694489\pi$
$409$		−	670.000i		−	1.63814i	0.573692	−	0.819071i	$-0.305511\pi$
$410$		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	$0.270809\pi$
$419$			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	$0.446061\pi$
$422$		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	$-0.803732\pi$
$433$	−102.879	+	102.879i	−0.237595	+	0.237595i	0.578259	+	0.815853i	$0.303732\pi$
$434$		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	$-0.837088\pi$
$439$		−	430.000i		−	0.979499i	0.871863	−	0.489749i	$-0.162912\pi$
$440$		0			0
$441$	177.000			0.401361
$442$		0			0
$443$	−582.979	+	582.979i	−1.31598	+	1.31598i	−0.399049	+	0.916930i	$0.630660\pi$
$443$	−582.979	+	582.979i	−1.31598	+	1.31598i	−0.916930	+	0.399049i	$0.869340\pi$
$444$		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	$-0.968044\pi$
$449$		−	90.0000i		−	0.200445i	0.994965	−	0.100223i	$-0.0319555\pi$
$450$		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	$-0.345527\pi$
$457$	−191.060	−	191.060i	−0.418075	−	0.418075i	−0.884540	+	0.466465i	$0.845527\pi$
$458$		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	$-0.753572\pi$
$463$	−7.34847	+	7.34847i	−0.0158714	+	0.0158714i	0.699127	+	0.714998i	$0.253572\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−347.828	−	347.828i	−0.744813	−	0.744813i	0.228687	−	0.973500i	$-0.426557\pi$
$467$	−347.828	−	347.828i	−0.744813	−	0.744813i	−0.973500	+	0.228687i	$0.926557\pi$
$468$		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	$-0.877374\pi$
$479$		−	360.000i		−	0.751566i	0.926708	−	0.375783i	$-0.122626\pi$
$480$		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	−0.976490	−	0.215561i	$-0.930842\pi$
$487$	−580.529	−	580.529i	−1.19205	−	1.19205i	−0.215561	−	0.976490i	$-0.569158\pi$
$488$		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	$0.201339\pi$
$499$			590.000i			1.18236i	−0.806538	+	0.591182i	$0.798661\pi$
$500$		0			0
$501$	48.0000			0.0958084
$502$		0			0
$503$	−93.0806	+	93.0806i	−0.185051	+	0.185051i	−0.793553	−	0.608502i	$-0.791771\pi$
$503$	−93.0806	+	93.0806i	−0.185051	+	0.185051i	0.608502	+	0.793553i	$0.291771\pi$
$504$		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	$-0.822022\pi$
$509$		−	540.000i		−	1.06090i	0.847715	−	0.530452i	$-0.177978\pi$
$510$		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	$-0.866429\pi$
$523$	−264.545	+	264.545i	−0.505822	+	0.505822i	0.407419	+	0.913241i	$0.366429\pi$
$524$		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	$-0.138903\pi$
$547$	264.545	+	264.545i	0.483629	+	0.483629i	−0.422660	+	0.906288i	$0.638903\pi$
$548$		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	$-0.461128\pi$
$557$	−484.999	−	484.999i	−0.870734	−	0.870734i	−0.992552	+	0.121818i	$0.961128\pi$
$558$		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	$-0.583926\pi$
$563$	396.817	−	396.817i	0.704827	−	0.704827i	0.965443	+	0.260616i	$0.0839256\pi$
$564$		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	$-0.813253\pi$
$569$		−	630.000i		−	1.10721i	0.832781	−	0.553603i	$-0.186747\pi$
$570$		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	$-0.452595\pi$
$577$	−484.999	−	484.999i	−0.840553	−	0.840553i	−0.988931	+	0.148378i	$0.952595\pi$
$578$		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	$-0.00217329\pi$
$587$	582.979	+	582.979i	0.993149	+	0.993149i	−0.00682753	+	0.999977i	$0.502173\pi$
$588$		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	$-0.530282\pi$
$593$	533.989	−	533.989i	0.900487	−	0.900487i	0.995478	+	0.0949912i	$0.0302823\pi$
$594$		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	$-0.851156\pi$
$599$		−	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	$0.717198\pi$
$602$		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	0.994210	+	0.107455i	$0.0342702\pi$
$607$	668.711	+	668.711i	1.10167	+	1.10167i	0.107455	+	0.994210i	$0.465730\pi$
$608$		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	$-0.774307\pi$
$613$	−66.1362	+	66.1362i	−0.107889	+	0.107889i	0.651101	+	0.758991i	$0.274307\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	592.777	+	592.777i	0.960740	+	0.960740i	0.999258	−	0.0385180i	$-0.0122637\pi$
$617$	592.777	+	592.777i	0.960740	+	0.960740i	−0.0385180	+	0.999258i	$0.512264\pi$
$618$		0			0
$619$		−	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
$622$		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	$-0.843961\pi$
$643$	−264.545	+	264.545i	−0.411423	+	0.411423i	0.470811	+	0.882234i	$0.343961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	93.0806	+	93.0806i	0.143865	+	0.143865i	0.775371	−	0.631506i	$-0.217563\pi$
$647$	93.0806	+	93.0806i	0.143865	+	0.143865i	−0.631506	+	0.775371i	$0.717563\pi$
$648$		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	$-0.726338\pi$
$653$	68.5857	−	68.5857i	0.105032	−	0.105032i	0.757670	+	0.652638i	$0.226338\pi$
$654$		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	$0.158637\pi$
$659$			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	$0.344074\pi$
$662$		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	$-0.784474\pi$
$673$	−102.879	+	102.879i	−0.152866	+	0.152866i	0.626531	+	0.779397i	$0.284474\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	176.363	+	176.363i	0.260507	+	0.260507i	0.825260	−	0.564753i	$-0.191029\pi$
$677$	176.363	+	176.363i	0.260507	+	0.260507i	−0.564753	+	0.825260i	$0.691029\pi$
$678$		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	$-0.597296\pi$
$683$	445.807	−	445.807i	0.652719	−	0.652719i	0.953647	+	0.300928i	$0.0972963\pi$
$684$		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	$-0.929845\pi$
$709$		−	310.000i		−	0.437236i	0.975811	−	0.218618i	$-0.0701548\pi$
$710$		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	$0.0399489\pi$
$719$			180.000i			0.250348i	−0.992135	+	0.125174i	$0.960051\pi$
$720$		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	$-0.414162\pi$
$727$	−507.044	−	507.044i	−0.697448	−	0.697448i	−0.963859	+	0.266412i	$0.914162\pi$
$728$		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	−0.0975121	+	0.995234i	$0.531088\pi$
$733$	−800.983	+	800.983i	−1.09275	+	1.09275i	−0.995234	+	0.0975121i	$0.968912\pi$
$734$		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	$-0.923899\pi$
$739$		−	350.000i		−	0.473613i	0.971557	−	0.236806i	$-0.0761007\pi$
$740$		0			0
$741$	180.000			0.242915
$742$		0			0
$743$	−925.907	+	925.907i	−1.24617	+	1.24617i	−0.288778	+	0.957396i	$0.593249\pi$
$743$	−925.907	+	925.907i	−1.24617	+	1.24617i	−0.957396	+	0.288778i	$0.906751\pi$
$744$		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	$-0.163736\pi$
$757$	286.590	+	286.590i	0.378587	+	0.378587i	−0.492005	+	0.870592i	$0.663736\pi$
$758$		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	$-0.874679\pi$
$769$		−	590.000i		−	0.767230i	0.923493	−	0.383615i	$-0.125321\pi$
$770$		0			0
$771$	−708.000			−0.918288
$772$		0			0
$773$	−127.373	+	127.373i	−0.164778	+	0.164778i	−0.784680	−	0.619902i	$-0.787173\pi$
$773$	−127.373	+	127.373i	−0.164778	+	0.164778i	0.619902	+	0.784680i	$0.287173\pi$
$774$		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	$-0.173610\pi$
$787$	264.545	+	264.545i	0.336143	+	0.336143i	−0.518770	+	0.854914i	$0.673610\pi$
$788$		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	$-0.192971\pi$
$797$	200.858	+	200.858i	0.252018	+	0.252018i	−0.569780	+	0.821797i	$0.692971\pi$
$798$		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	$0.127307\pi$
$809$			630.000i			0.778739i	−0.921082	+	0.389370i	$0.872693\pi$
$810$		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	$-0.649882\pi$
$823$	360.075	−	360.075i	0.437515	−	0.437515i	0.891175	+	0.453660i	$0.149882\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	876.917	+	876.917i	1.06036	+	1.06036i	0.998057	+	0.0623022i	$0.0198443\pi$
$827$	876.917	+	876.917i	1.06036	+	1.06036i	0.0623022	+	0.998057i	$0.480156\pi$
$828$		0			0
$829$		−	70.0000i		−	0.0844391i	−0.999108	−	0.0422195i	$-0.986557\pi$
$829$		−	70.0000i		−	0.0844391i	0.999108	−	0.0422195i	$-0.0134429\pi$
$830$		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	$-0.895708\pi$
$839$		−	540.000i		−	0.643623i	0.946804	−	0.321812i	$-0.104292\pi$
$840$		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	$-0.513639\pi$
$853$	815.680	−	815.680i	0.956249	−	0.956249i	0.999082	+	0.0428336i	$0.0136385\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	298.838	+	298.838i	0.348702	+	0.348702i	0.859626	−	0.510924i	$-0.170697\pi$
$857$	298.838	+	298.838i	0.348702	+	0.348702i	−0.510924	+	0.859626i	$0.670697\pi$
$858$		0			0
$859$		−	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
$862$		0			0
$863$	225.353	−	225.353i	0.261128	−	0.261128i	−0.564385	−	0.825512i	$-0.690886\pi$
$863$	225.353	−	225.353i	0.261128	−	0.261128i	0.825512	+	0.564385i	$0.190886\pi$
$864$		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	$-0.195034\pi$
$877$	213.106	+	213.106i	0.242994	+	0.242994i	−0.575094	+	0.818088i	$0.695034\pi$
$878$		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	$-0.780015\pi$
$883$	−117.576	+	117.576i	−0.133155	+	0.133155i	0.637388	+	0.770543i	$0.280015\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	509.494	+	509.494i	0.574401	+	0.574401i	0.933355	−	0.358954i	$-0.116867\pi$
$887$	509.494	+	509.494i	0.574401	+	0.574401i	−0.358954	+	0.933355i	$0.616867\pi$
$888$		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	$-0.146041\pi$
$907$	411.514	+	411.514i	0.453709	+	0.453709i	−0.442874	+	0.896584i	$0.646041\pi$
$908$		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	$0.0751652\pi$
$919$			430.000i			0.467900i	−0.972249	+	0.233950i	$0.924835\pi$
$920$		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	$-0.692037\pi$
$929$		−	1530.00i		−	1.64693i	0.567365	−	0.823466i	$-0.307963\pi$
$930$		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	$-0.171539\pi$
$937$	323.333	+	323.333i	0.345072	+	0.345072i	−0.513198	+	0.858270i	$0.671539\pi$
$938$		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	$-0.436270\pi$
$947$	−739.746	−	739.746i	−0.781147	−	0.781147i	−0.980024	+	0.198878i	$0.936270\pi$
$948$		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	$-0.739585\pi$
$953$	44.0908	−	44.0908i	0.0462653	−	0.0462653i	0.729861	+	0.683596i	$0.239585\pi$
$954$		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	$-0.428550\pi$
$967$	−727.498	−	727.498i	−0.752325	−	0.752325i	−0.974913	+	0.222588i	$0.928550\pi$
$968$		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	$-0.157276\pi$
$977$	396.817	+	396.817i	0.406159	+	0.406159i	−0.474238	+	0.880397i	$0.657276\pi$
$978$		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	$-0.953069\pi$
$983$	−827.928	+	827.928i	−0.842246	+	0.842246i	0.146905	+	0.989151i	$0.453069\pi$
$984$		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	$-0.446365\pi$
$997$	−815.680	−	815.680i	−0.818134	−	0.818134i	−0.985838	+	0.167703i	$0.946365\pi$
$998$		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.1057.2		4
4.3	odd	2		75.3.f.a.7.1	✓	4
5.2	odd	4	inner	1200.3.bg.j.193.1		4
5.3	odd	4	inner	1200.3.bg.j.193.2		4
5.4	even	2	inner	1200.3.bg.j.1057.1		4
12.11	even	2		225.3.g.f.82.2		4
20.3	even	4		75.3.f.a.43.1	yes	4
20.7	even	4		75.3.f.a.43.2	yes	4
20.19	odd	2		75.3.f.a.7.2	yes	4
60.23	odd	4		225.3.g.f.118.2		4
60.47	odd	4		225.3.g.f.118.1		4
60.59	even	2		225.3.g.f.82.1		4

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

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)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(7.34847 + 7.34847i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} +(7.34847 + 7.34847i) q^{7} -3.00000i q^{9} +18.0000 q^{11} +(7.34847 - 7.34847i) q^{13} +(4.89898 + 4.89898i) q^{17} +10.0000i q^{19} +18.0000 q^{21} +(-19.5959 + 19.5959i) q^{23} +(-3.67423 - 3.67423i) q^{27} -22.0000 q^{31} +(22.0454 - 22.0454i) q^{33} +(-7.34847 - 7.34847i) q^{37} -18.0000i q^{39} -18.0000 q^{41} +(29.3939 - 29.3939i) q^{43} +(44.0908 + 44.0908i) q^{47} +59.0000i q^{49} +12.0000 q^{51} +(-4.89898 + 4.89898i) q^{53} +(12.2474 + 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(22.0454 - 22.0454i) q^{63} +(44.0908 + 44.0908i) q^{67} +48.0000i q^{69} -72.0000 q^{71} +(44.0908 - 44.0908i) q^{73} +(132.272 + 132.272i) q^{77} -70.0000i q^{79} -9.00000 q^{81} +(53.8888 - 53.8888i) q^{83} -90.0000i q^{89} +108.000 q^{91} +(-26.9444 + 26.9444i) q^{93} +(102.879 + 102.879i) q^{97} -54.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * 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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