LMFDB - Embedded newform 2880.2.bc.a.593.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(593,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2880.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bc (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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$48$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.14
Character	$\chi$	$=$	2880.593
Dual form			2880.2.bc.a.1457.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.29671 - 1.82168i) q^{5} +(-2.16011 - 2.16011i) q^{7} +O(q^{10})$	$q+(-1.29671 - 1.82168i) q^{5} +(-2.16011 - 2.16011i) q^{7} +(1.59473 + 1.59473i) q^{11} +4.91566i q^{13} +(4.89313 + 4.89313i) q^{17} +(4.17377 - 4.17377i) q^{19} +(-3.03688 - 3.03688i) q^{23} +(-1.63706 + 4.72441i) q^{25} +(-2.23830 - 2.23830i) q^{29} -1.27948 q^{31} +(-1.13399 + 6.73607i) q^{35} -4.85382i q^{37} +7.09526 q^{41} +0.799620 q^{43} +(8.91231 + 8.91231i) q^{47} +2.33212i q^{49} +10.4382i q^{53} +(0.837185 - 4.97301i) q^{55} +(-0.727229 + 0.727229i) q^{59} +(1.14236 + 1.14236i) q^{61} +(8.95479 - 6.37421i) q^{65} -12.3294 q^{67} -0.557059 q^{71} +(6.14521 - 6.14521i) q^{73} -6.88959i q^{77} -13.3789i q^{79} +11.9944 q^{83} +(2.56874 - 15.2587i) q^{85} -12.4431i q^{89} +(10.6184 - 10.6184i) q^{91} +(-13.0155 - 2.19110i) q^{95} +(11.5790 - 11.5790i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q+O(q^{10}) $ 96 * q	$ 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) $ 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.29671	−	1.82168i	−0.579908	−	0.814682i
$5$	−1.29671	−	1.82168i	−0.579908	−	0.814682i
$6$		0			0
$7$	−2.16011	−	2.16011i	−0.816443	−	0.816443i	0.169147	−	0.985591i	$-0.445899\pi$
$7$	−2.16011	−	2.16011i	−0.816443	−	0.816443i	−0.985591	+	0.169147i	$0.945899\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.59473	+	1.59473i	0.480830	+	0.480830i	0.905397	−	0.424567i	$-0.139574\pi$
$11$	1.59473	+	1.59473i	0.480830	+	0.480830i	−0.424567	+	0.905397i	$0.639574\pi$
$12$		0			0
$13$			4.91566i			1.36336i	0.731650	+	0.681680i	$0.238750\pi$
$13$			4.91566i			1.36336i	−0.731650	+	0.681680i	$0.761250\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.89313	+	4.89313i	1.18676	+	1.18676i	0.977958	+	0.208799i	$0.0669556\pi$
$17$	4.89313	+	4.89313i	1.18676	+	1.18676i	0.208799	+	0.977958i	$0.433044\pi$
$18$		0			0
$19$	4.17377	−	4.17377i	0.957529	−	0.957529i	−0.0416055	−	0.999134i	$-0.513247\pi$
$19$	4.17377	−	4.17377i	0.957529	−	0.957529i	0.999134	+	0.0416055i	$0.0132473\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.03688	−	3.03688i	−0.633232	−	0.633232i	0.315645	−	0.948877i	$-0.397779\pi$
$23$	−3.03688	−	3.03688i	−0.633232	−	0.633232i	−0.948877	+	0.315645i	$0.897779\pi$
$24$		0			0
$25$	−1.63706	+	4.72441i	−0.327412	+	0.944882i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.23830	−	2.23830i	−0.415641	−	0.415641i	0.468057	−	0.883698i	$-0.344954\pi$
$29$	−2.23830	−	2.23830i	−0.415641	−	0.415641i	−0.883698	+	0.468057i	$0.844954\pi$
$30$		0			0
$31$	−1.27948			−0.229802			−0.114901	−	0.993377i	$-0.536655\pi$
$31$	−1.27948			−0.229802			−0.114901	+	0.993377i	$0.536655\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.13399	+	6.73607i	−0.191679	+	1.13860i
$36$		0			0
$37$		−	4.85382i		−	0.797963i	−0.916959	−	0.398981i	$-0.869364\pi$
$37$		−	4.85382i		−	0.797963i	0.916959	−	0.398981i	$-0.130636\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.09526			1.10809			0.554046	−	0.832486i	$-0.313083\pi$
$41$	7.09526			1.10809			0.554046	+	0.832486i	$0.313083\pi$
$42$		0			0
$43$	0.799620			0.121941			0.0609704	−	0.998140i	$-0.480580\pi$
$43$	0.799620			0.121941			0.0609704	+	0.998140i	$0.480580\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.91231	+	8.91231i	1.29999	+	1.29999i	0.928392	+	0.371602i	$0.121191\pi$
$47$	8.91231	+	8.91231i	1.29999	+	1.29999i	0.371602	+	0.928392i	$0.378809\pi$
$48$		0			0
$49$			2.33212i			0.333160i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			10.4382i			1.43380i	0.697178	+	0.716899i	$0.254439\pi$
$53$			10.4382i			1.43380i	−0.697178	+	0.716899i	$0.745561\pi$
$54$		0			0
$55$	0.837185	−	4.97301i	0.112886	−	0.670561i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.727229	+	0.727229i	−0.0946771	+	0.0946771i	−0.752859	−	0.658182i	$-0.771326\pi$
$59$	−0.727229	+	0.727229i	−0.0946771	+	0.0946771i	0.658182	+	0.752859i	$0.271326\pi$
$60$		0			0
$61$	1.14236	+	1.14236i	0.146264	+	0.146264i	0.776447	−	0.630183i	$-0.217020\pi$
$61$	1.14236	+	1.14236i	0.146264	+	0.146264i	−0.630183	+	0.776447i	$0.717020\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.95479	−	6.37421i	1.11070	−	0.790624i
$66$		0			0
$67$	−12.3294			−1.50627			−0.753136	−	0.657865i	$-0.771460\pi$
$67$	−12.3294			−1.50627			−0.753136	+	0.657865i	$0.771460\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.557059			−0.0661107			−0.0330554	−	0.999454i	$-0.510524\pi$
$71$	−0.557059			−0.0661107			−0.0330554	+	0.999454i	$0.510524\pi$
$72$		0			0
$73$	6.14521	−	6.14521i	0.719242	−	0.719242i	−0.249208	−	0.968450i	$-0.580170\pi$
$73$	6.14521	−	6.14521i	0.719242	−	0.719242i	0.968450	+	0.249208i	$0.0801702\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	6.88959i		−	0.785141i
$78$		0			0
$79$		−	13.3789i		−	1.50525i	−0.658452	−	0.752623i	$-0.728788\pi$
$79$		−	13.3789i		−	1.50525i	0.658452	−	0.752623i	$-0.271212\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.9944			1.31656			0.658280	−	0.752773i	$-0.271284\pi$
$83$	11.9944			1.31656			0.658280	+	0.752773i	$0.271284\pi$
$84$		0			0
$85$	2.56874	−	15.2587i	0.278619	−	1.65504i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.4431i		−	1.31896i	−0.751722	−	0.659481i	$-0.770776\pi$
$89$		−	12.4431i		−	1.31896i	0.751722	−	0.659481i	$-0.229224\pi$
$90$		0			0
$91$	10.6184	−	10.6184i	1.11311	−	1.11311i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.0155	−	2.19110i	−1.33536	−	0.224802i
$96$		0			0
$97$	11.5790	−	11.5790i	1.17567	−	1.17567i	0.194837	−	0.980836i	$-0.437582\pi$
$97$	11.5790	−	11.5790i	1.17567	−	1.17567i	0.980836	−	0.194837i	$-0.0624178\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.21009	+	1.21009i	0.120409	+	0.120409i	0.764743	−	0.644335i	$-0.222866\pi$
$101$	1.21009	+	1.21009i	0.120409	+	0.120409i	−0.644335	+	0.764743i	$0.722866\pi$
$102$		0			0
$103$	−0.387230	+	0.387230i	−0.0381549	+	0.0381549i	−0.725927	−	0.687772i	$-0.758589\pi$
$103$	−0.387230	+	0.387230i	−0.0381549	+	0.0381549i	0.687772	+	0.725927i	$0.258589\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.69761			0.260788			0.130394	−	0.991462i	$-0.458376\pi$
$107$	2.69761			0.260788			0.130394	+	0.991462i	$0.458376\pi$
$108$		0			0
$109$	4.05978	−	4.05978i	0.388857	−	0.388857i	−0.485423	−	0.874280i	$-0.661334\pi$
$109$	4.05978	−	4.05978i	0.388857	−	0.388857i	0.874280	+	0.485423i	$0.161334\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.30077	−	5.30077i	0.498655	−	0.498655i	−0.412364	−	0.911019i	$-0.635297\pi$
$113$	5.30077	−	5.30077i	0.498655	−	0.498655i	0.911019	+	0.412364i	$0.135297\pi$
$114$		0			0
$115$	−1.59427	+	9.47019i	−0.148666	+	0.883100i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	21.1394i		−	1.93784i
$120$		0			0
$121$		−	5.91365i		−	0.537605i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.7292	−	3.14400i	0.959647	−	0.281208i
$126$		0			0
$127$	6.22633	−	6.22633i	0.552497	−	0.552497i	−0.374663	−	0.927161i	$-0.622242\pi$
$127$	6.22633	−	6.22633i	0.552497	−	0.552497i	0.927161	+	0.374663i	$0.122242\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.26708	+	9.26708i	−0.809669	+	0.809669i	−0.984584	−	0.174915i	$-0.944035\pi$
$131$	−9.26708	+	9.26708i	−0.809669	+	0.809669i	0.174915	+	0.984584i	$0.444035\pi$
$132$		0			0
$133$	−18.0316			−1.56354
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.11255	−	6.11255i	0.522231	−	0.522231i	−0.396014	−	0.918245i	$-0.629607\pi$
$137$	6.11255	−	6.11255i	0.522231	−	0.522231i	0.918245	+	0.396014i	$0.129607\pi$
$138$		0			0
$139$	−15.7875	−	15.7875i	−1.33908	−	1.33908i	−0.896950	−	0.442132i	$-0.854222\pi$
$139$	−15.7875	−	15.7875i	−1.33908	−	1.33908i	−0.442132	−	0.896950i	$-0.645778\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.83917	+	7.83917i	−0.655545	+	0.655545i
$144$		0			0
$145$	−1.17504	+	6.97990i	−0.0975814	+	0.579649i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.28174	+	8.28174i	−0.678467	+	0.678467i	−0.959653	−	0.281186i	$-0.909272\pi$
$149$	−8.28174	+	8.28174i	−0.678467	+	0.678467i	0.281186	+	0.959653i	$0.409272\pi$
$150$		0			0
$151$		−	4.27955i		−	0.348265i	−0.984722	−	0.174132i	$-0.944288\pi$
$151$		−	4.27955i		−	0.348265i	0.984722	−	0.174132i	$-0.0557120\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.65912	+	2.33081i	0.133264	+	0.187215i
$156$		0			0
$157$	13.0703			1.04312			0.521560	−	0.853214i	$-0.325350\pi$
$157$	13.0703			1.04312			0.521560	+	0.853214i	$0.325350\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			13.1200i			1.03400i
$162$		0			0
$163$			7.02503i			0.550242i	0.961410	+	0.275121i	$0.0887180\pi$
$163$			7.02503i			0.550242i	−0.961410	+	0.275121i	$0.911282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.7629	−	13.7629i	1.06500	−	1.06500i	0.0672663	−	0.997735i	$-0.478572\pi$
$167$	13.7629	−	13.7629i	1.06500	−	1.06500i	0.997735	−	0.0672663i	$-0.0214277\pi$
$168$		0			0
$169$	−11.1638			−0.858751
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.77568			0.287060			0.143530	−	0.989646i	$-0.454155\pi$
$173$	3.77568			0.287060			0.143530	+	0.989646i	$0.454155\pi$
$174$		0			0
$175$	13.7414	−	6.66899i	1.03876	−	0.504129i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.50611	+	8.50611i	0.635776	+	0.635776i	0.949511	−	0.313735i	$-0.101580\pi$
$179$	8.50611	+	8.50611i	0.635776	+	0.635776i	−0.313735	+	0.949511i	$0.601580\pi$
$180$		0			0
$181$	4.01676	−	4.01676i	0.298563	−	0.298563i	−0.541888	−	0.840451i	$-0.682290\pi$
$181$	4.01676	−	4.01676i	0.298563	−	0.298563i	0.840451	+	0.541888i	$0.182290\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.84212	+	6.29402i	−0.650086	+	0.462745i
$186$		0			0
$187$			15.6065i			1.14126i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.1097i		−	0.803873i	−0.915668	−	0.401937i	$-0.868337\pi$
$191$		−	11.1097i		−	0.803873i	0.915668	−	0.401937i	$-0.131663\pi$
$192$		0			0
$193$	2.14007	+	2.14007i	0.154046	+	0.154046i	0.779922	−	0.625876i	$-0.215259\pi$
$193$	2.14007	+	2.14007i	0.154046	+	0.154046i	−0.625876	+	0.779922i	$0.715259\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.29714			0.306158			0.153079	−	0.988214i	$-0.451081\pi$
$197$	4.29714			0.306158			0.153079	+	0.988214i	$0.451081\pi$
$198$		0			0
$199$	15.6025			1.10603			0.553015	−	0.833171i	$-0.313477\pi$
$199$	15.6025			1.10603			0.553015	+	0.833171i	$0.313477\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			9.66992i			0.678695i
$204$		0			0
$205$	−9.20052	−	12.9253i	−0.642592	−	0.902743i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.3121			0.920817
$210$		0			0
$211$	6.32913	+	6.32913i	0.435715	+	0.435715i	0.890567	−	0.454852i	$-0.150308\pi$
$211$	6.32913	+	6.32913i	0.435715	+	0.435715i	−0.454852	+	0.890567i	$0.650308\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.03688	−	1.45665i	−0.0707145	−	0.0993430i
$216$		0			0
$217$	2.76382	+	2.76382i	0.187620	+	0.187620i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.0530	+	24.0530i	−1.61798	+	1.61798i
$222$		0			0
$223$	15.9816	+	15.9816i	1.07021	+	1.07021i	0.997342	+	0.0728635i	$0.0232138\pi$
$223$	15.9816	+	15.9816i	1.07021	+	1.07021i	0.0728635	+	0.997342i	$0.476786\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.3722i			0.821175i	0.911821	+	0.410587i	$0.134676\pi$
$227$			12.3722i			0.821175i	−0.911821	+	0.410587i	$0.865324\pi$
$228$		0			0
$229$	3.96094	+	3.96094i	0.261746	+	0.261746i	0.825763	−	0.564017i	$-0.190745\pi$
$229$	3.96094	+	3.96094i	0.261746	+	0.261746i	−0.564017	+	0.825763i	$0.690745\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.78605	+	7.78605i	0.510081	+	0.510081i	0.914551	−	0.404470i	$-0.132544\pi$
$233$	7.78605	+	7.78605i	0.510081	+	0.510081i	−0.404470	+	0.914551i	$0.632544\pi$
$234$		0			0
$235$	4.67869	−	27.7921i	0.305204	−	1.81296i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.26350			0.405152			0.202576	−	0.979267i	$-0.435069\pi$
$239$	6.26350			0.405152			0.202576	+	0.979267i	$0.435069\pi$
$240$		0			0
$241$	5.99300			0.386043			0.193022	−	0.981194i	$-0.438171\pi$
$241$	5.99300			0.386043			0.193022	+	0.981194i	$0.438171\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.24838	−	3.02409i	0.271419	−	0.193202i
$246$		0			0
$247$	20.5169	+	20.5169i	1.30546	+	1.30546i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.36118	−	6.36118i	−0.401514	−	0.401514i	0.477252	−	0.878766i	$-0.341633\pi$
$251$	−6.36118	−	6.36118i	−0.401514	−	0.401514i	−0.878766	+	0.477252i	$0.841633\pi$
$252$		0			0
$253$		−	9.68601i		−	0.608954i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.29740	+	8.29740i	0.517577	+	0.517577i	0.916838	−	0.399260i	$-0.130733\pi$
$257$	8.29740	+	8.29740i	0.517577	+	0.517577i	−0.399260	+	0.916838i	$0.630733\pi$
$258$		0			0
$259$	−10.4848	+	10.4848i	−0.651492	+	0.651492i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.90559	+	4.90559i	0.302492	+	0.302492i	0.841988	−	0.539496i	$-0.181385\pi$
$263$	4.90559	+	4.90559i	0.302492	+	0.302492i	−0.539496	+	0.841988i	$0.681385\pi$
$264$		0			0
$265$	19.0151	−	13.5354i	1.16809	−	0.831471i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.03484	−	1.03484i	−0.0630953	−	0.0630953i	0.674855	−	0.737950i	$-0.264206\pi$
$269$	−1.03484	−	1.03484i	−0.0630953	−	0.0630953i	−0.737950	+	0.674855i	$0.764206\pi$
$270$		0			0
$271$	−20.4126			−1.23998			−0.619989	−	0.784611i	$-0.712863\pi$
$271$	−20.4126			−1.23998			−0.619989	+	0.784611i	$0.712863\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.1448	+	4.92349i	−0.611757	+	0.296898i
$276$		0			0
$277$		−	28.0012i		−	1.68243i	−0.540702	−	0.841214i	$-0.681841\pi$
$277$		−	28.0012i		−	1.68243i	0.540702	−	0.841214i	$-0.318159\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−17.8349			−1.06394			−0.531970	−	0.846763i	$-0.678548\pi$
$281$	−17.8349			−1.06394			−0.531970	+	0.846763i	$0.678548\pi$
$282$		0			0
$283$	−26.3812			−1.56820			−0.784100	−	0.620634i	$-0.786875\pi$
$283$	−26.3812			−1.56820			−0.784100	+	0.620634i	$0.786875\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−15.3265	−	15.3265i	−0.904695	−	0.904695i
$288$		0			0
$289$			30.8854i			1.81679i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			4.67822i			0.273305i	0.990619	+	0.136652i	$0.0436343\pi$
$293$			4.67822i			0.273305i	−0.990619	+	0.136652i	$0.956366\pi$
$294$		0			0
$295$	2.26779	+	0.381773i	0.132036	+	0.0222277i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.9283	−	14.9283i	0.863324	−	0.863324i
$300$		0			0
$301$	−1.72726	−	1.72726i	−0.0995578	−	0.0995578i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.599701	−	3.56232i	0.0343388	−	0.203978i
$306$		0			0
$307$	−20.6419			−1.17809			−0.589047	−	0.808099i	$-0.700497\pi$
$307$	−20.6419			−1.17809			−0.589047	+	0.808099i	$0.700497\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.68911			0.549419			0.274710	−	0.961527i	$-0.411418\pi$
$311$	9.68911			0.549419			0.274710	+	0.961527i	$0.411418\pi$
$312$		0			0
$313$	−16.9050	+	16.9050i	−0.955526	+	0.955526i	−0.999052	−	0.0435263i	$-0.986141\pi$
$313$	−16.9050	+	16.9050i	−0.955526	+	0.955526i	0.0435263	+	0.999052i	$0.486141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.11296i			0.0625101i	0.999511	+	0.0312550i	$0.00995041\pi$
$317$			1.11296i			0.0625101i	−0.999511	+	0.0312550i	$0.990050\pi$
$318$		0			0
$319$		−	7.13897i		−	0.399706i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	40.8456			2.27271
$324$		0			0
$325$	−23.2236	−	8.04725i	−1.28821	−	0.446381i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	38.5031i		−	2.12274i
$330$		0			0
$331$	−1.59706	+	1.59706i	−0.0877826	+	0.0877826i	−0.749635	−	0.661852i	$-0.769771\pi$
$331$	−1.59706	+	1.59706i	−0.0877826	+	0.0877826i	0.661852	+	0.749635i	$0.269771\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.9877	+	22.4602i	0.873499	+	1.22713i
$336$		0			0
$337$	−21.8501	+	21.8501i	−1.19025	+	1.19025i	−0.213253	+	0.976997i	$0.568406\pi$
$337$	−21.8501	+	21.8501i	−1.19025	+	1.19025i	−0.976997	+	0.213253i	$0.931594\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.04043	−	2.04043i	−0.110496	−	0.110496i
$342$		0			0
$343$	−10.0831	+	10.0831i	−0.544437	+	0.544437i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.2699			0.551318			0.275659	−	0.961255i	$-0.411104\pi$
$347$	10.2699			0.551318			0.275659	+	0.961255i	$0.411104\pi$
$348$		0			0
$349$	−3.49875	+	3.49875i	−0.187284	+	0.187284i	−0.794521	−	0.607237i	$-0.792278\pi$
$349$	−3.49875	+	3.49875i	−0.187284	+	0.187284i	0.607237	+	0.794521i	$0.292278\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.24233	+	9.24233i	−0.491920	+	0.491920i	−0.908911	−	0.416991i	$-0.863085\pi$
$353$	−9.24233	+	9.24233i	−0.491920	+	0.491920i	0.416991	+	0.908911i	$0.363085\pi$
$354$		0			0
$355$	0.722347	+	1.01479i	0.0383382	+	0.0538592i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.29265i			0.226557i	0.993563	+	0.113279i	$0.0361353\pi$
$359$			4.29265i			0.226557i	−0.993563	+	0.113279i	$0.963865\pi$
$360$		0			0
$361$		−	15.8407i		−	0.833722i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.1632	−	3.22604i	−1.00305	−	0.168859i
$366$		0			0
$367$	−2.17487	+	2.17487i	−0.113527	+	0.113527i	−0.761588	−	0.648061i	$-0.775580\pi$
$367$	−2.17487	+	2.17487i	−0.113527	+	0.113527i	0.648061	+	0.761588i	$0.275580\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	22.5476	−	22.5476i	1.17061	−	1.17061i
$372$		0			0
$373$	−15.0439			−0.778944			−0.389472	−	0.921038i	$-0.627343\pi$
$373$	−15.0439			−0.778944			−0.389472	+	0.921038i	$0.627343\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	11.0027	−	11.0027i	0.566669	−	0.566669i
$378$		0			0
$379$	19.3492	+	19.3492i	0.993901	+	0.993901i	0.999982	−	0.00608010i	$-0.00193537\pi$
$379$	19.3492	+	19.3492i	0.993901	+	0.993901i	−0.00608010	+	0.999982i	$0.501935\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.71102	+	8.71102i	−0.445112	+	0.445112i	−0.893726	−	0.448613i	$-0.851918\pi$
$383$	−8.71102	+	8.71102i	−0.445112	+	0.445112i	0.448613	+	0.893726i	$0.351918\pi$
$384$		0			0
$385$	−12.5506	+	8.93383i	−0.639640	+	0.455310i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.4942	+	23.4942i	−1.19120	+	1.19120i	−0.214475	+	0.976730i	$0.568804\pi$
$389$	−23.4942	+	23.4942i	−1.19120	+	1.19120i	−0.976730	+	0.214475i	$0.931196\pi$
$390$		0			0
$391$		−	29.7196i		−	1.50299i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−24.3722	+	17.3486i	−1.22630	+	0.872905i
$396$		0			0
$397$	12.2248			0.613544			0.306772	−	0.951783i	$-0.400751\pi$
$397$	12.2248			0.613544			0.306772	+	0.951783i	$0.400751\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			8.35952i			0.417454i	0.977974	+	0.208727i	$0.0669321\pi$
$401$			8.35952i			0.417454i	−0.977974	+	0.208727i	$0.933068\pi$
$402$		0			0
$403$		−	6.28951i		−	0.313303i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.74054	−	7.74054i	0.383685	−	0.383685i
$408$		0			0
$409$	24.6191			1.21733			0.608667	−	0.793426i	$-0.291705\pi$
$409$	24.6191			1.21733			0.608667	+	0.793426i	$0.291705\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.14178			0.154597
$414$		0			0
$415$	−15.5533	−	21.8501i	−0.763484	−	1.07258i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.4381	+	16.4381i	0.803052	+	0.803052i	0.983571	−	0.180519i	$-0.0577778\pi$
$419$	16.4381	+	16.4381i	0.803052	+	0.803052i	−0.180519	+	0.983571i	$0.557778\pi$
$420$		0			0
$421$	2.95236	−	2.95236i	0.143889	−	0.143889i	−0.631493	−	0.775382i	$-0.717557\pi$
$421$	2.95236	−	2.95236i	0.143889	−	0.143889i	0.775382	+	0.631493i	$0.217557\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−31.1275	+	15.1068i	−1.50990	+	0.732786i
$426$		0			0
$427$		−	4.93522i		−	0.238832i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	37.4103i		−	1.80199i	−0.433829	−	0.900995i	$-0.642838\pi$
$431$		−	37.4103i		−	1.80199i	0.433829	−	0.900995i	$-0.357162\pi$
$432$		0			0
$433$	−15.6331	−	15.6331i	−0.751280	−	0.751280i	0.223438	−	0.974718i	$-0.428272\pi$
$433$	−15.6331	−	15.6331i	−0.751280	−	0.751280i	−0.974718	+	0.223438i	$0.928272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−25.3505			−1.21268
$438$		0			0
$439$	32.5235			1.55226			0.776132	−	0.630571i	$-0.217179\pi$
$439$	32.5235			1.55226			0.776132	+	0.630571i	$0.217179\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			11.1463i			0.529575i	0.964307	+	0.264788i	$0.0853018\pi$
$443$			11.1463i			0.529575i	−0.964307	+	0.264788i	$0.914698\pi$
$444$		0			0
$445$	−22.6673	+	16.1351i	−1.07453	+	0.764877i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.4968			1.43923			0.719617	−	0.694371i	$-0.244317\pi$
$449$	30.4968			1.43923			0.719617	+	0.694371i	$0.244317\pi$
$450$		0			0
$451$	11.3150	+	11.3150i	0.532804	+	0.532804i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−33.1123	−	5.57431i	−1.55233	−	0.261328i
$456$		0			0
$457$	14.3454	+	14.3454i	0.671049	+	0.671049i	0.957958	−	0.286909i	$-0.0926277\pi$
$457$	14.3454	+	14.3454i	0.671049	+	0.671049i	−0.286909	+	0.957958i	$0.592628\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	24.4883	−	24.4883i	1.14053	−	1.14053i	0.152180	−	0.988353i	$-0.451371\pi$
$461$	24.4883	−	24.4883i	1.14053	−	1.14053i	0.988353	−	0.152180i	$-0.0486292\pi$
$462$		0			0
$463$	13.7438	+	13.7438i	0.638728	+	0.638728i	0.950242	−	0.311514i	$-0.100836\pi$
$463$	13.7438	+	13.7438i	0.638728	+	0.638728i	−0.311514	+	0.950242i	$0.600836\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	34.2210i		−	1.58356i	−0.610806	−	0.791780i	$-0.709154\pi$
$467$		−	34.2210i		−	1.58356i	0.610806	−	0.791780i	$-0.290846\pi$
$468$		0			0
$469$	26.6327	+	26.6327i	1.22979	+	1.22979i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.27518	+	1.27518i	0.0586328	+	0.0586328i
$474$		0			0
$475$	12.8859	+	26.5513i	0.591244	+	1.21826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.3091			−0.973636			−0.486818	−	0.873503i	$-0.661843\pi$
$479$	−21.3091			−0.973636			−0.486818	+	0.873503i	$0.661843\pi$
$480$		0			0
$481$	23.8597			1.08791
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−36.1080	−	6.07863i	−1.63958	−	0.276016i
$486$		0			0
$487$	−8.31120	−	8.31120i	−0.376616	−	0.376616i	0.493264	−	0.869880i	$-0.335804\pi$
$487$	−8.31120	−	8.31120i	−0.376616	−	0.376616i	−0.869880	+	0.493264i	$0.835804\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.27142	−	9.27142i	−0.418413	−	0.418413i	0.466243	−	0.884657i	$-0.345607\pi$
$491$	−9.27142	−	9.27142i	−0.418413	−	0.418413i	−0.884657	+	0.466243i	$0.845607\pi$
$492$		0			0
$493$		−	21.9045i		−	0.986531i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.20331	+	1.20331i	0.0539757	+	0.0539757i
$498$		0			0
$499$	14.8544	−	14.8544i	0.664976	−	0.664976i	−0.291573	−	0.956549i	$-0.594179\pi$
$499$	14.8544	−	14.8544i	0.664976	−	0.664976i	0.956549	+	0.291573i	$0.0941786\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.0738	+	10.0738i	0.449167	+	0.449167i	0.895078	−	0.445910i	$-0.147120\pi$
$503$	10.0738	+	10.0738i	0.449167	+	0.449167i	−0.445910	+	0.895078i	$0.647120\pi$
$504$		0			0
$505$	0.635260	−	3.77355i	0.0282687	−	0.167921i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.3449	−	15.3449i	−0.680153	−	0.680153i	0.279882	−	0.960034i	$-0.409705\pi$
$509$	−15.3449	−	15.3449i	−0.680153	−	0.680153i	−0.960034	+	0.279882i	$0.909705\pi$
$510$		0			0
$511$	−26.5486			−1.17444
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.20754	+	0.203284i	0.0532104	+	0.00895774i
$516$		0			0
$517$			28.4255i			1.25015i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.9098			0.740831			0.370416	−	0.928866i	$-0.379215\pi$
$521$	16.9098			0.740831			0.370416	+	0.928866i	$0.379215\pi$
$522$		0			0
$523$	0.746648			0.0326486			0.0163243	−	0.999867i	$-0.494804\pi$
$523$	0.746648			0.0326486			0.0163243	+	0.999867i	$0.494804\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.26067	−	6.26067i	−0.272719	−	0.272719i
$528$		0			0
$529$		−	4.55476i		−	0.198033i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			34.8779i			1.51073i
$534$		0			0
$535$	−3.49803	−	4.91419i	−0.151233	−	0.212459i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.71910	+	3.71910i	−0.160193	+	0.160193i
$540$		0			0
$541$	26.9288	+	26.9288i	1.15776	+	1.15776i	0.984956	+	0.172804i	$0.0552826\pi$
$541$	26.9288	+	26.9288i	1.15776	+	1.15776i	0.172804	+	0.984956i	$0.444717\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.6600	−	2.13126i	−0.542296	−	0.0912931i
$546$		0			0
$547$	27.9592			1.19545			0.597726	−	0.801701i	$-0.296071\pi$
$547$	27.9592			1.19545			0.597726	+	0.801701i	$0.296071\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.6843			−0.795977
$552$		0			0
$553$	−28.8999	+	28.8999i	−1.22895	+	1.22895i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.27808i			0.308382i	0.988041	+	0.154191i	$0.0492772\pi$
$557$			7.27808i			0.308382i	−0.988041	+	0.154191i	$0.950723\pi$
$558$		0			0
$559$			3.93066i			0.166249i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−39.6135			−1.66951			−0.834754	−	0.550622i	$-0.814390\pi$
$563$	−39.6135			−1.66951			−0.834754	+	0.550622i	$0.814390\pi$
$564$		0			0
$565$	−16.5299	−	2.78274i	−0.695419	−	0.117071i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			5.68667i			0.238398i	0.992870	+	0.119199i	$0.0380326\pi$
$569$			5.68667i			0.238398i	−0.992870	+	0.119199i	$0.961967\pi$
$570$		0			0
$571$	−21.9804	+	21.9804i	−0.919850	+	0.919850i	−0.997018	−	0.0771677i	$-0.975412\pi$
$571$	−21.9804	+	21.9804i	−0.919850	+	0.919850i	0.0771677	+	0.997018i	$0.475412\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	19.3190	−	9.37589i	0.805658	−	0.391002i
$576$		0			0
$577$	0.214556	−	0.214556i	0.00893208	−	0.00893208i	−0.702627	−	0.711559i	$-0.747990\pi$
$577$	0.214556	−	0.214556i	0.00893208	−	0.00893208i	0.711559	+	0.702627i	$0.247990\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−25.9092	−	25.9092i	−1.07490	−	1.07490i
$582$		0			0
$583$	−16.6461	+	16.6461i	−0.689413	+	0.689413i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−34.8325			−1.43769			−0.718846	−	0.695169i	$-0.755329\pi$
$587$	−34.8325			−1.43769			−0.718846	+	0.695169i	$0.755329\pi$
$588$		0			0
$589$	−5.34027	+	5.34027i	−0.220042	+	0.220042i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	30.0143	−	30.0143i	1.23254	−	1.23254i	0.269553	−	0.962986i	$-0.413124\pi$
$593$	30.0143	−	30.0143i	1.23254	−	1.23254i	0.962986	−	0.269553i	$-0.0868760\pi$
$594$		0			0
$595$	−38.5092	+	27.4117i	−1.57872	+	1.12377i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	45.5010i		−	1.85912i	−0.368673	−	0.929559i	$-0.620188\pi$
$599$		−	45.5010i		−	1.85912i	0.368673	−	0.929559i	$-0.379812\pi$
$600$		0			0
$601$		−	18.9893i		−	0.774590i	−0.921956	−	0.387295i	$-0.873410\pi$
$601$		−	18.9893i		−	0.774590i	0.921956	−	0.387295i	$-0.126590\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.7728	+	7.66832i	−0.437977	+	0.311762i
$606$		0			0
$607$	−12.0763	+	12.0763i	−0.490162	+	0.490162i	−0.908357	−	0.418195i	$-0.862663\pi$
$607$	−12.0763	+	12.0763i	−0.490162	+	0.490162i	0.418195	+	0.908357i	$0.362663\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−43.8099	+	43.8099i	−1.77236	+	1.77236i
$612$		0			0
$613$	39.3086			1.58766			0.793830	−	0.608140i	$-0.208084\pi$
$613$	39.3086			1.58766			0.793830	+	0.608140i	$0.208084\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.483007	+	0.483007i	−0.0194451	+	0.0194451i	−0.716763	−	0.697317i	$-0.754377\pi$
$617$	−0.483007	+	0.483007i	−0.0194451	+	0.0194451i	0.697317	+	0.716763i	$0.254377\pi$
$618$		0			0
$619$	7.19546	+	7.19546i	0.289210	+	0.289210i	0.836768	−	0.547558i	$-0.184442\pi$
$619$	7.19546	+	7.19546i	0.289210	+	0.289210i	−0.547558	+	0.836768i	$0.684442\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−26.8783	+	26.8783i	−1.07686	+	1.07686i
$624$		0			0
$625$	−19.6401	−	15.4683i	−0.785602	−	0.618732i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	23.7504	−	23.7504i	0.946989	−	0.946989i
$630$		0			0
$631$			47.3200i			1.88378i	0.335921	+	0.941890i	$0.390953\pi$
$631$			47.3200i			1.88378i	−0.335921	+	0.941890i	$0.609047\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.4162	−	3.26863i	−0.770507	−	0.129712i
$636$		0			0
$637$	−11.4639			−0.454216
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	25.4001i		−	1.00324i	−0.865087	−	0.501621i	$-0.832737\pi$
$641$		−	25.4001i		−	1.00324i	0.865087	−	0.501621i	$-0.167263\pi$
$642$		0			0
$643$		−	2.65375i		−	0.104654i	−0.998630	−	0.0523269i	$-0.983336\pi$
$643$		−	2.65375i		−	0.104654i	0.998630	−	0.0523269i	$-0.0166638\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.19649	−	4.19649i	0.164981	−	0.164981i	−0.619788	−	0.784769i	$-0.712781\pi$
$647$	4.19649	−	4.19649i	0.164981	−	0.164981i	0.784769	+	0.619788i	$0.212781\pi$
$648$		0			0
$649$	−2.31947			−0.0910472
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16.0192			0.626879			0.313439	−	0.949608i	$-0.398519\pi$
$653$	16.0192			0.626879			0.313439	+	0.949608i	$0.398519\pi$
$654$		0			0
$655$	28.8985	+	4.86493i	1.12916	+	0.190089i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.400776	+	0.400776i	0.0156120	+	0.0156120i	0.714870	−	0.699258i	$-0.246486\pi$
$659$	0.400776	+	0.400776i	0.0156120	+	0.0156120i	−0.699258	+	0.714870i	$0.746486\pi$
$660$		0			0
$661$	−27.6021	+	27.6021i	−1.07360	+	1.07360i	−0.0765303	+	0.997067i	$0.524384\pi$
$661$	−27.6021	+	27.6021i	−1.07360	+	1.07360i	−0.997067	+	0.0765303i	$0.975616\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.3818	+	32.8478i	0.906708	+	1.27378i
$666$		0			0
$667$			13.5949i			0.526395i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			3.64351i			0.140656i
$672$		0			0
$673$	−1.32866	−	1.32866i	−0.0512160	−	0.0512160i	0.681035	−	0.732251i	$-0.261530\pi$
$673$	−1.32866	−	1.32866i	−0.0512160	−	0.0512160i	−0.732251	+	0.681035i	$0.761530\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	45.7582			1.75863			0.879315	−	0.476241i	$-0.158001\pi$
$677$	45.7582			1.75863			0.879315	+	0.476241i	$0.158001\pi$
$678$		0			0
$679$	−50.0239			−1.91974
$680$		0			0
$681$		0			0
$682$		0			0
$683$			15.6090i			0.597261i	0.954369	+	0.298631i	$0.0965299\pi$
$683$			15.6090i			0.597261i	−0.954369	+	0.298631i	$0.903470\pi$
$684$		0			0
$685$	−19.0614	−	3.20890i	−0.728298	−	0.122606i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−51.3107			−1.95478
$690$		0			0
$691$	−20.5405	−	20.5405i	−0.781399	−	0.781399i	0.198668	−	0.980067i	$-0.436339\pi$
$691$	−20.5405	−	20.5405i	−0.781399	−	0.781399i	−0.980067	+	0.198668i	$0.936339\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.28797	+	49.2319i	−0.314381	+	1.86747i
$696$		0			0
$697$	34.7180	+	34.7180i	1.31504	+	1.31504i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.09850	−	6.09850i	0.230337	−	0.230337i	−0.582496	−	0.812833i	$-0.697924\pi$
$701$	6.09850	−	6.09850i	0.230337	−	0.230337i	0.812833	+	0.582496i	$0.197924\pi$
$702$		0			0
$703$	−20.2587	−	20.2587i	−0.764072	−	0.764072i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	5.22785i		−	0.196614i
$708$		0			0
$709$	−3.11088	−	3.11088i	−0.116832	−	0.116832i	0.646274	−	0.763106i	$-0.276326\pi$
$709$	−3.11088	−	3.11088i	−0.116832	−	0.116832i	−0.763106	+	0.646274i	$0.776326\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.88563	+	3.88563i	0.145518	+	0.145518i
$714$		0			0
$715$	24.4457	+	4.11532i	0.914216	+	0.153904i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.8532			0.516637			0.258318	−	0.966060i	$-0.416832\pi$
$719$	13.8532			0.516637			0.258318	+	0.966060i	$0.416832\pi$
$720$		0			0
$721$	1.67291			0.0623026
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.2389	−	6.91040i	0.528818	−	0.256646i
$726$		0			0
$727$	31.8937	+	31.8937i	1.18287	+	1.18287i	0.978997	+	0.203873i	$0.0653529\pi$
$727$	31.8937	+	31.8937i	1.18287	+	1.18287i	0.203873	+	0.978997i	$0.434647\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.91264	+	3.91264i	0.144714	+	0.144714i
$732$		0			0
$733$			2.98113i			0.110111i	0.998483	+	0.0550553i	$0.0175335\pi$
$733$			2.98113i			0.110111i	−0.998483	+	0.0550553i	$0.982466\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.6620	−	19.6620i	−0.724261	−	0.724261i
$738$		0			0
$739$	33.5628	−	33.5628i	1.23463	−	1.23463i	0.272460	−	0.962167i	$-0.412163\pi$
$739$	33.5628	−	33.5628i	1.23463	−	1.23463i	0.962167	−	0.272460i	$-0.0878372\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.57830	−	1.57830i	−0.0579021	−	0.0579021i	0.677563	−	0.735465i	$-0.263036\pi$
$743$	−1.57830	−	1.57830i	−0.0579021	−	0.0579021i	−0.735465	+	0.677563i	$0.763036\pi$
$744$		0			0
$745$	25.8258	+	4.34766i	0.946183	+	0.159286i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.82712	−	5.82712i	−0.212918	−	0.212918i
$750$		0			0
$751$	7.14641			0.260776			0.130388	−	0.991463i	$-0.458378\pi$
$751$	7.14641			0.260776			0.130388	+	0.991463i	$0.458378\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.79598	+	5.54935i	−0.283725	+	0.201962i
$756$		0			0
$757$			22.6897i			0.824672i	0.911032	+	0.412336i	$0.135287\pi$
$757$			22.6897i			0.824672i	−0.911032	+	0.412336i	$0.864713\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.10507			−0.257559			−0.128779	−	0.991673i	$-0.541106\pi$
$761$	−7.10507			−0.257559			−0.128779	+	0.991673i	$0.541106\pi$
$762$		0			0
$763$	−17.5391			−0.634959
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.57481	−	3.57481i	−0.129079	−	0.129079i
$768$		0			0
$769$		−	6.34941i		−	0.228966i	−0.993425	−	0.114483i	$-0.963479\pi$
$769$		−	6.34941i		−	0.228966i	0.993425	−	0.114483i	$-0.0365211\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			34.7116i			1.24849i	0.781229	+	0.624244i	$0.214593\pi$
$773$			34.7116i			1.24849i	−0.781229	+	0.624244i	$0.785407\pi$
$774$		0			0
$775$	2.09459	−	6.04480i	0.0752400	−	0.217136i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29.6140	−	29.6140i	1.06103	−	1.06103i
$780$		0			0
$781$	−0.888360	−	0.888360i	−0.0317880	−	0.0317880i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.9484	−	23.8099i	−0.604914	−	0.849811i
$786$		0			0
$787$	−36.4455			−1.29914			−0.649570	−	0.760302i	$-0.725051\pi$
$787$	−36.4455			−1.29914			−0.649570	+	0.760302i	$0.725051\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−22.9005			−0.814247
$792$		0			0
$793$	−5.61544	+	5.61544i	−0.199410	+	0.199410i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	33.8743i		−	1.19989i	−0.800042	−	0.599944i	$-0.795189\pi$
$797$		−	33.8743i		−	1.19989i	0.800042	−	0.599944i	$-0.204811\pi$
$798$		0			0
$799$			87.2182i			3.08556i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.5999			0.691667
$804$		0			0
$805$	23.9004	−	17.0128i	0.842378	−	0.599624i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			30.9188i			1.08705i	0.839394	+	0.543523i	$0.182910\pi$
$809$			30.9188i			1.08705i	−0.839394	+	0.543523i	$0.817090\pi$
$810$		0			0
$811$	22.3532	−	22.3532i	0.784927	−	0.784927i	−0.195730	−	0.980658i	$-0.562708\pi$
$811$	22.3532	−	22.3532i	0.784927	−	0.784927i	0.980658	+	0.195730i	$0.0627078\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.7974	−	9.10945i	0.448272	−	0.319090i
$816$		0			0
$817$	3.33743	−	3.33743i	0.116762	−	0.116762i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.2769	−	28.2769i	−0.986871	−	0.986871i	0.0130442	−	0.999915i	$-0.495848\pi$
$821$	−28.2769	−	28.2769i	−0.986871	−	0.986871i	−0.999915	+	0.0130442i	$0.995848\pi$
$822$		0			0
$823$	19.5029	−	19.5029i	0.679830	−	0.679830i	−0.280132	−	0.959962i	$-0.590378\pi$
$823$	19.5029	−	19.5029i	0.679830	−	0.679830i	0.959962	+	0.280132i	$0.0903782\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.91722			0.205762			0.102881	−	0.994694i	$-0.467194\pi$
$827$	5.91722			0.205762			0.102881	+	0.994694i	$0.467194\pi$
$828$		0			0
$829$	21.1678	−	21.1678i	0.735188	−	0.735188i	−0.236455	−	0.971643i	$-0.575985\pi$
$829$	21.1678	−	21.1678i	0.735188	−	0.735188i	0.971643	+	0.236455i	$0.0759855\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.4113	+	11.4113i	−0.395380	+	0.395380i
$834$		0			0
$835$	−42.9181	−	7.22507i	−1.48524	−	0.250034i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1.25753i			0.0434148i	0.999764	+	0.0217074i	$0.00691023\pi$
$839$			1.25753i			0.0434148i	−0.999764	+	0.0217074i	$0.993090\pi$
$840$		0			0
$841$		−	18.9801i		−	0.654485i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.4762	+	20.3368i	0.497997	+	0.699608i
$846$		0			0
$847$	−12.7741	+	12.7741i	−0.438924	+	0.438924i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.7404	+	14.7404i	−0.505296	+	0.505296i
$852$		0			0
$853$	13.8859			0.475445			0.237723	−	0.971333i	$-0.423599\pi$
$853$	13.8859			0.475445			0.237723	+	0.971333i	$0.423599\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.98575	+	9.98575i	−0.341107	+	0.341107i	−0.856783	−	0.515677i	$-0.827541\pi$
$857$	−9.98575	+	9.98575i	−0.341107	+	0.341107i	0.515677	+	0.856783i	$0.327541\pi$
$858$		0			0
$859$	−4.92165	−	4.92165i	−0.167924	−	0.167924i	0.618142	−	0.786066i	$-0.287886\pi$
$859$	−4.92165	−	4.92165i	−0.167924	−	0.167924i	−0.786066	+	0.618142i	$0.787886\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.01064	−	4.01064i	0.136524	−	0.136524i	−0.635542	−	0.772066i	$-0.719223\pi$
$863$	4.01064	−	4.01064i	0.136524	−	0.136524i	0.772066	+	0.635542i	$0.219223\pi$
$864$		0			0
$865$	−4.89598	−	6.87809i	−0.166468	−	0.233862i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	21.3358	−	21.3358i	0.723767	−	0.723767i
$870$		0			0
$871$		−	60.6070i		−	2.05359i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−29.9675	−	16.3848i	−1.01309	−	0.553907i
$876$		0			0
$877$	−15.5968			−0.526666			−0.263333	−	0.964705i	$-0.584822\pi$
$877$	−15.5968			−0.526666			−0.263333	+	0.964705i	$0.584822\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	10.1457i		−	0.341819i	−0.985287	−	0.170909i	$-0.945329\pi$
$881$		−	10.1457i		−	0.341819i	0.985287	−	0.170909i	$-0.0546706\pi$
$882$		0			0
$883$		−	0.488312i		−	0.0164330i	−0.999966	−	0.00821651i	$-0.997385\pi$
$883$		−	0.488312i		−	0.0164330i	0.999966	−	0.00821651i	$-0.00261543\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−30.7573	+	30.7573i	−1.03273	+	1.03273i	−0.0332830	+	0.999446i	$0.510596\pi$
$887$	−30.7573	+	30.7573i	−1.03273	+	1.03273i	−0.999446	+	0.0332830i	$0.989404\pi$
$888$		0			0
$889$	−26.8991			−0.902166
$890$		0			0
$891$		0			0
$892$		0			0
$893$	74.3959			2.48956
$894$		0			0
$895$	4.46544	−	26.5254i	0.149263	−	0.886647i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.86386	+	2.86386i	0.0955152	+	0.0955152i
$900$		0			0
$901$	−51.0755	+	51.0755i	−1.70157	+	1.70157i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.5259	−	2.10867i	−0.416373	−	0.0700947i
$906$		0			0
$907$			30.1189i			1.00008i	0.866002	+	0.500041i	$0.166682\pi$
$907$			30.1189i			1.00008i	−0.866002	+	0.500041i	$0.833318\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.51837i			0.0834373i	0.999129	+	0.0417187i	$0.0132833\pi$
$911$			2.51837i			0.0834373i	−0.999129	+	0.0417187i	$0.986717\pi$
$912$		0			0
$913$	19.1279	+	19.1279i	0.633041	+	0.633041i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.0358			1.32210
$918$		0			0
$919$	−1.27745			−0.0421391			−0.0210696	−	0.999778i	$-0.506707\pi$
$919$	−1.27745			−0.0421391			−0.0210696	+	0.999778i	$0.506707\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2.73832i		−	0.0901327i
$924$		0			0
$925$	22.9314	+	7.94600i	0.753980	+	0.261263i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.91478			−0.259675			−0.129838	−	0.991535i	$-0.541446\pi$
$929$	−7.91478			−0.259675			−0.129838	+	0.991535i	$0.541446\pi$
$930$		0			0
$931$	9.73372	+	9.73372i	0.319010	+	0.319010i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	28.4300	−	20.2371i	0.929762	−	0.661825i
$936$		0			0
$937$	−11.8863	−	11.8863i	−0.388309	−	0.388309i	0.485775	−	0.874084i	$-0.338538\pi$
$937$	−11.8863	−	11.8863i	−0.388309	−	0.388309i	−0.874084	+	0.485775i	$0.838538\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−37.7787	+	37.7787i	−1.23155	+	1.23155i	−0.268184	+	0.963368i	$0.586424\pi$
$941$	−37.7787	+	37.7787i	−1.23155	+	1.23155i	−0.963368	+	0.268184i	$0.913576\pi$
$942$		0			0
$943$	−21.5474	−	21.5474i	−0.701680	−	0.701680i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	21.8687i		−	0.710638i	−0.934745	−	0.355319i	$-0.884372\pi$
$947$		−	21.8687i		−	0.710638i	0.934745	−	0.355319i	$-0.115628\pi$
$948$		0			0
$949$	30.2078	+	30.2078i	0.980586	+	0.980586i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13.3744	−	13.3744i	−0.433239	−	0.433239i	0.456490	−	0.889729i	$-0.349106\pi$
$953$	−13.3744	−	13.3744i	−0.433239	−	0.433239i	−0.889729	+	0.456490i	$0.849106\pi$
$954$		0			0
$955$	−20.2384	+	14.4062i	−0.654901	+	0.466173i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−26.4075			−0.852744
$960$		0			0
$961$	−29.3629			−0.947191
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.12347	−	6.67360i	0.0361659	−	0.214831i
$966$		0			0
$967$	−19.1923	−	19.1923i	−0.617183	−	0.617183i	0.327625	−	0.944808i	$-0.393752\pi$
$967$	−19.1923	−	19.1923i	−0.617183	−	0.617183i	−0.944808	+	0.327625i	$0.893752\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.0246	+	30.0246i	0.963536	+	0.963536i	0.999358	−	0.0358222i	$-0.0114050\pi$
$971$	30.0246	+	30.0246i	0.963536	+	0.963536i	−0.0358222	+	0.999358i	$0.511405\pi$
$972$		0			0
$973$			68.2055i			2.18657i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.3015	+	24.3015i	0.777474	+	0.777474i	0.979401	−	0.201927i	$-0.0647204\pi$
$977$	24.3015	+	24.3015i	0.777474	+	0.777474i	−0.201927	+	0.979401i	$0.564720\pi$
$978$		0			0
$979$	19.8433	−	19.8433i	0.634196	−	0.634196i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	31.4111	+	31.4111i	1.00186	+	1.00186i	0.999998	+	0.00186130i	$0.000592470\pi$
$983$	31.4111	+	31.4111i	1.00186	+	1.00186i	0.00186130	+	0.999998i	$0.499408\pi$
$984$		0			0
$985$	−5.57216	−	7.82803i	−0.177544	−	0.249422i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.42835	−	2.42835i	−0.0772169	−	0.0772169i
$990$		0			0
$991$	−60.1276			−1.91002			−0.955008	−	0.296580i	$-0.904154\pi$
$991$	−60.1276			−1.91002			−0.955008	+	0.296580i	$0.904154\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.2320	−	28.4228i	−0.641396	−	0.901062i
$996$		0			0
$997$		−	7.08895i		−	0.224509i	−0.993679	−	0.112255i	$-0.964193\pi$
$997$		−	7.08895i		−	0.224509i	0.993679	−	0.112255i	$-0.0358072\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bc.a.593.14		96
3.2	odd	2	inner	2880.2.bc.a.593.35		96
4.3	odd	2		720.2.bc.a.413.33	yes	96
5.2	odd	4		2880.2.bg.a.17.11		96
12.11	even	2		720.2.bc.a.413.16	yes	96
15.2	even	4		2880.2.bg.a.17.38		96
16.5	even	4		2880.2.bg.a.2033.38		96
16.11	odd	4		720.2.bg.a.53.40	yes	96
20.7	even	4		720.2.bg.a.557.9	yes	96
48.5	odd	4		2880.2.bg.a.2033.11		96
48.11	even	4		720.2.bg.a.53.9	yes	96
60.47	odd	4		720.2.bg.a.557.40	yes	96
80.27	even	4		720.2.bc.a.197.16	✓	96
80.37	odd	4	inner	2880.2.bc.a.1457.35		96
240.107	odd	4		720.2.bc.a.197.33	yes	96
240.197	even	4	inner	2880.2.bc.a.1457.14		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bc.a.197.16	✓	96	80.27	even	4
720.2.bc.a.197.33	yes	96	240.107	odd	4
720.2.bc.a.413.16	yes	96	12.11	even	2
720.2.bc.a.413.33	yes	96	4.3	odd	2
720.2.bg.a.53.9	yes	96	48.11	even	4
720.2.bg.a.53.40	yes	96	16.11	odd	4
720.2.bg.a.557.9	yes	96	20.7	even	4
720.2.bg.a.557.40	yes	96	60.47	odd	4
2880.2.bc.a.593.14		96	1.1	even	1	trivial
2880.2.bc.a.593.35		96	3.2	odd	2	inner
2880.2.bc.a.1457.14		96	240.197	even	4	inner
2880.2.bc.a.1457.35		96	80.37	odd	4	inner
2880.2.bg.a.17.11		96	5.2	odd	4
2880.2.bg.a.17.38		96	15.2	even	4
2880.2.bg.a.2033.11		96	48.5	odd	4
2880.2.bg.a.2033.38		96	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bc (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.29671 - 1.82168i) q^{5} +(-2.16011 - 2.16011i) q^{7} +O(q^{10})\)	\(q+(-1.29671 - 1.82168i) q^{5} +(-2.16011 - 2.16011i) q^{7} +(1.59473 + 1.59473i) q^{11} +4.91566i q^{13} +(4.89313 + 4.89313i) q^{17} +(4.17377 - 4.17377i) q^{19} +(-3.03688 - 3.03688i) q^{23} +(-1.63706 + 4.72441i) q^{25} +(-2.23830 - 2.23830i) q^{29} -1.27948 q^{31} +(-1.13399 + 6.73607i) q^{35} -4.85382i q^{37} +7.09526 q^{41} +0.799620 q^{43} +(8.91231 + 8.91231i) q^{47} +2.33212i q^{49} +10.4382i q^{53} +(0.837185 - 4.97301i) q^{55} +(-0.727229 + 0.727229i) q^{59} +(1.14236 + 1.14236i) q^{61} +(8.95479 - 6.37421i) q^{65} -12.3294 q^{67} -0.557059 q^{71} +(6.14521 - 6.14521i) q^{73} -6.88959i q^{77} -13.3789i q^{79} +11.9944 q^{83} +(2.56874 - 15.2587i) q^{85} -12.4431i q^{89} +(10.6184 - 10.6184i) q^{91} +(-13.0155 - 2.19110i) q^{95} +(11.5790 - 11.5790i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q+O(q^{10}) \) 96 * q	\( 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) \) 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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