LMFDB - Embedded newform 2880.2.bl.b.1871.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1871.8
Character	$\chi$	$=$	2880.1871
Dual form			2880.2.bl.b.431.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} +4.13654 q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{5} +4.13654 q^{7} +(-1.99565 - 1.99565i) q^{11} +(0.516505 - 0.516505i) q^{13} +3.26086i q^{17} +(3.73322 + 3.73322i) q^{19} -7.20935i q^{23} -1.00000i q^{25} +(-1.21770 - 1.21770i) q^{29} -9.55357i q^{31} +(2.92498 - 2.92498i) q^{35} +(6.37513 + 6.37513i) q^{37} +7.26417 q^{41} +(-0.127818 + 0.127818i) q^{43} +8.87784 q^{47} +10.1110 q^{49} +(-9.72788 + 9.72788i) q^{53} -2.82228 q^{55} +(-2.86405 - 2.86405i) q^{59} +(-4.64789 + 4.64789i) q^{61} -0.730448i q^{65} +(-0.264363 - 0.264363i) q^{67} -1.88076i q^{71} -2.96795i q^{73} +(-8.25510 - 8.25510i) q^{77} -6.08712i q^{79} +(4.98155 - 4.98155i) q^{83} +(2.30577 + 2.30577i) q^{85} -6.90265 q^{89} +(2.13654 - 2.13654i) q^{91} +5.27957 q^{95} -10.2393 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{7}+O(q^{10}) $ 24 * q + 8 * q^7	$ 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$6$		0			0
$7$	4.13654			1.56347			0.781733	−	0.623613i	$-0.214336\pi$
$7$	4.13654			1.56347			0.781733	+	0.623613i	$0.214336\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.99565	−	1.99565i	−0.601712	−	0.601712i	0.339055	−	0.940767i	$-0.389893\pi$
$11$	−1.99565	−	1.99565i	−0.601712	−	0.601712i	−0.940767	+	0.339055i	$0.889893\pi$
$12$		0			0
$13$	0.516505	−	0.516505i	0.143253	−	0.143253i	−0.631843	−	0.775096i	$-0.717701\pi$
$13$	0.516505	−	0.516505i	0.143253	−	0.143253i	0.775096	+	0.631843i	$0.217701\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.26086i			0.790874i	0.918493	+	0.395437i	$0.129407\pi$
$17$			3.26086i			0.790874i	−0.918493	+	0.395437i	$0.870593\pi$
$18$		0			0
$19$	3.73322	+	3.73322i	0.856460	+	0.856460i	0.990919	−	0.134459i	$-0.0429297\pi$
$19$	3.73322	+	3.73322i	0.856460	+	0.856460i	−0.134459	+	0.990919i	$0.542930\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.20935i		−	1.50325i	−0.659589	−	0.751627i	$-0.729269\pi$
$23$		−	7.20935i		−	1.50325i	0.659589	−	0.751627i	$-0.270731\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.21770	−	1.21770i	−0.226120	−	0.226120i	0.584949	−	0.811070i	$-0.301114\pi$
$29$	−1.21770	−	1.21770i	−0.226120	−	0.226120i	−0.811070	+	0.584949i	$0.801114\pi$
$30$		0			0
$31$		−	9.55357i		−	1.71587i	−0.513757	−	0.857936i	$-0.671747\pi$
$31$		−	9.55357i		−	1.71587i	0.513757	−	0.857936i	$-0.328253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.92498	−	2.92498i	0.494412	−	0.494412i
$36$		0			0
$37$	6.37513	+	6.37513i	1.04807	+	1.04807i	0.998785	+	0.0492801i	$0.0156927\pi$
$37$	6.37513	+	6.37513i	1.04807	+	1.04807i	0.0492801	+	0.998785i	$0.484307\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.26417			1.13447			0.567236	−	0.823555i	$-0.308013\pi$
$41$	7.26417			1.13447			0.567236	+	0.823555i	$0.308013\pi$
$42$		0			0
$43$	−0.127818	+	0.127818i	−0.0194921	+	0.0194921i	−0.716786	−	0.697294i	$-0.754387\pi$
$43$	−0.127818	+	0.127818i	−0.0194921	+	0.0194921i	0.697294	+	0.716786i	$0.254387\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.87784			1.29497			0.647483	−	0.762080i	$-0.275822\pi$
$47$	8.87784			1.29497			0.647483	+	0.762080i	$0.275822\pi$
$48$		0			0
$49$	10.1110			1.44443
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.72788	+	9.72788i	−1.33623	+	1.33623i	−0.436544	+	0.899683i	$0.643798\pi$
$53$	−9.72788	+	9.72788i	−1.33623	+	1.33623i	−0.899683	+	0.436544i	$0.856202\pi$
$54$		0			0
$55$	−2.82228			−0.380556
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.86405	−	2.86405i	−0.372868	−	0.372868i	0.495653	−	0.868521i	$-0.334929\pi$
$59$	−2.86405	−	2.86405i	−0.372868	−	0.372868i	−0.868521	+	0.495653i	$0.834929\pi$
$60$		0			0
$61$	−4.64789	+	4.64789i	−0.595102	+	0.595102i	−0.939005	−	0.343903i	$-0.888251\pi$
$61$	−4.64789	+	4.64789i	−0.595102	+	0.595102i	0.343903	+	0.939005i	$0.388251\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.730448i		−	0.0906009i
$66$		0			0
$67$	−0.264363	−	0.264363i	−0.0322970	−	0.0322970i	0.690774	−	0.723071i	$-0.257270\pi$
$67$	−0.264363	−	0.264363i	−0.0322970	−	0.0322970i	−0.723071	+	0.690774i	$0.757270\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.88076i		−	0.223206i	−0.993753	−	0.111603i	$-0.964402\pi$
$71$		−	1.88076i		−	0.223206i	0.993753	−	0.111603i	$-0.0355984\pi$
$72$		0			0
$73$		−	2.96795i		−	0.347372i	−0.984801	−	0.173686i	$-0.944432\pi$
$73$		−	2.96795i		−	0.347372i	0.984801	−	0.173686i	$-0.0555678\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.25510	−	8.25510i	−0.940756	−	0.940756i
$78$		0			0
$79$		−	6.08712i		−	0.684855i	−0.939544	−	0.342427i	$-0.888751\pi$
$79$		−	6.08712i		−	0.684855i	0.939544	−	0.342427i	$-0.111249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.98155	−	4.98155i	0.546797	−	0.546797i	−0.378716	−	0.925513i	$-0.623634\pi$
$83$	4.98155	−	4.98155i	0.546797	−	0.546797i	0.925513	+	0.378716i	$0.123634\pi$
$84$		0			0
$85$	2.30577	+	2.30577i	0.250096	+	0.250096i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.90265			−0.731679			−0.365840	−	0.930678i	$-0.619218\pi$
$89$	−6.90265			−0.731679			−0.365840	+	0.930678i	$0.619218\pi$
$90$		0			0
$91$	2.13654	−	2.13654i	0.223971	−	0.223971i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.27957			0.541673
$96$		0			0
$97$	−10.2393			−1.03964			−0.519822	−	0.854275i	$-0.674002\pi$
$97$	−10.2393			−1.03964			−0.519822	+	0.854275i	$0.674002\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.82574	−	7.82574i	0.778690	−	0.778690i	−0.200918	−	0.979608i	$-0.564393\pi$
$101$	7.82574	−	7.82574i	0.778690	−	0.778690i	0.979608	+	0.200918i	$0.0643925\pi$
$102$		0			0
$103$	10.1803			1.00309			0.501546	−	0.865131i	$-0.332765\pi$
$103$	10.1803			1.00309			0.501546	+	0.865131i	$0.332765\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.52289	−	9.52289i	−0.920613	−	0.920613i	0.0764598	−	0.997073i	$-0.475638\pi$
$107$	−9.52289	−	9.52289i	−0.920613	−	0.920613i	−0.997073	+	0.0764598i	$0.975638\pi$
$108$		0			0
$109$	8.83148	−	8.83148i	0.845903	−	0.845903i	−0.143716	−	0.989619i	$-0.545905\pi$
$109$	8.83148	−	8.83148i	0.845903	−	0.845903i	0.989619	+	0.143716i	$0.0459052\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.46386i		−	0.325853i	−0.986638	−	0.162926i	$-0.947907\pi$
$113$		−	3.46386i		−	0.325853i	0.986638	−	0.162926i	$-0.0520933\pi$
$114$		0			0
$115$	−5.09778	−	5.09778i	−0.475371	−	0.475371i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			13.4887i			1.23650i
$120$		0			0
$121$		−	3.03475i		−	0.275886i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$			21.3548i			1.89493i	0.319856	+	0.947466i	$0.396366\pi$
$127$			21.3548i			1.89493i	−0.319856	+	0.947466i	$0.603634\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.26817	−	3.26817i	0.285541	−	0.285541i	−0.549773	−	0.835314i	$-0.685286\pi$
$131$	3.26817	−	3.26817i	0.285541	−	0.285541i	0.835314	+	0.549773i	$0.185286\pi$
$132$		0			0
$133$	15.4426	+	15.4426i	1.33905	+	1.33905i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.20493			0.615559			0.307779	−	0.951458i	$-0.400414\pi$
$137$	7.20493			0.615559			0.307779	+	0.951458i	$0.400414\pi$
$138$		0			0
$139$	−13.1704	+	13.1704i	−1.11710	+	1.11710i	−0.124932	+	0.992165i	$0.539871\pi$
$139$	−13.1704	+	13.1704i	−1.11710	+	1.11710i	−0.992165	+	0.124932i	$0.960129\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.06153			−0.172394
$144$		0			0
$145$	−1.72208			−0.143011
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.9496	+	15.9496i	−1.30664	+	1.30664i	−0.382817	+	0.923824i	$0.625046\pi$
$149$	−15.9496	+	15.9496i	−1.30664	+	1.30664i	−0.923824	+	0.382817i	$0.874954\pi$
$150$		0			0
$151$	11.6263			0.946132			0.473066	−	0.881027i	$-0.343147\pi$
$151$	11.6263			0.946132			0.473066	+	0.881027i	$0.343147\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.75539	−	6.75539i	−0.542606	−	0.542606i
$156$		0			0
$157$	13.4861	−	13.4861i	1.07631	−	1.07631i	0.0794735	−	0.996837i	$-0.474676\pi$
$157$	13.4861	−	13.4861i	1.07631	−	1.07631i	0.996837	−	0.0794735i	$-0.0253239\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	29.8218i		−	2.35029i
$162$		0			0
$163$	−1.19882	−	1.19882i	−0.0938991	−	0.0938991i	0.658597	−	0.752496i	$-0.271150\pi$
$163$	−1.19882	−	1.19882i	−0.0938991	−	0.0938991i	−0.752496	+	0.658597i	$0.771150\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			6.51068i			0.503811i	0.967752	+	0.251906i	$0.0810573\pi$
$167$			6.51068i			0.503811i	−0.967752	+	0.251906i	$0.918943\pi$
$168$		0			0
$169$			12.4664i			0.958957i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.24028	+	7.24028i	0.550468	+	0.550468i	0.926576	−	0.376108i	$-0.122738\pi$
$173$	7.24028	+	7.24028i	0.550468	+	0.550468i	−0.376108	+	0.926576i	$0.622738\pi$
$174$		0			0
$175$		−	4.13654i		−	0.312693i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.11091	−	4.11091i	0.307264	−	0.307264i	−0.536583	−	0.843847i	$-0.680285\pi$
$179$	4.11091	−	4.11091i	0.307264	−	0.307264i	0.843847	+	0.536583i	$0.180285\pi$
$180$		0			0
$181$	11.7469	+	11.7469i	0.873142	+	0.873142i	0.992814	−	0.119672i	$-0.0381842\pi$
$181$	11.7469	+	11.7469i	0.873142	+	0.873142i	−0.119672	+	0.992814i	$0.538184\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.01580			0.662855
$186$		0			0
$187$	6.50753	−	6.50753i	0.475878	−	0.475878i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.86217			−0.424171			−0.212086	−	0.977251i	$-0.568026\pi$
$191$	−5.86217			−0.424171			−0.212086	+	0.977251i	$0.568026\pi$
$192$		0			0
$193$	23.4308			1.68659			0.843293	−	0.537453i	$-0.180614\pi$
$193$	23.4308			1.68659			0.843293	+	0.537453i	$0.180614\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.18092	−	3.18092i	0.226631	−	0.226631i	−0.584653	−	0.811284i	$-0.698769\pi$
$197$	3.18092	−	3.18092i	0.226631	−	0.226631i	0.811284	+	0.584653i	$0.198769\pi$
$198$		0			0
$199$	21.5758			1.52947			0.764736	−	0.644344i	$-0.222870\pi$
$199$	21.5758			1.52947			0.764736	+	0.644344i	$0.222870\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.03705	−	5.03705i	−0.353532	−	0.353532i
$204$		0			0
$205$	5.13654	−	5.13654i	0.358752	−	0.358752i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	14.9004i		−	1.03068i
$210$		0			0
$211$	−4.30351	−	4.30351i	−0.296266	−	0.296266i	0.543283	−	0.839549i	$-0.317181\pi$
$211$	−4.30351	−	4.30351i	−0.296266	−	0.296266i	−0.839549	+	0.543283i	$0.817181\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.180762i			0.0123279i
$216$		0			0
$217$		−	39.5188i		−	2.68271i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.68425	+	1.68425i	0.113295	+	0.113295i
$222$		0			0
$223$		−	10.9977i		−	0.736460i	−0.929735	−	0.368230i	$-0.879964\pi$
$223$		−	10.9977i		−	0.736460i	0.929735	−	0.368230i	$-0.120036\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.9364	−	10.9364i	0.725876	−	0.725876i	−0.243919	−	0.969796i	$-0.578433\pi$
$227$	10.9364	−	10.9364i	0.725876	−	0.725876i	0.969796	+	0.243919i	$0.0784332\pi$
$228$		0			0
$229$	−5.16994	−	5.16994i	−0.341639	−	0.341639i	0.515344	−	0.856983i	$-0.327664\pi$
$229$	−5.16994	−	5.16994i	−0.341639	−	0.341639i	−0.856983	+	0.515344i	$0.827664\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.98003			−0.326252			−0.163126	−	0.986605i	$-0.552158\pi$
$233$	−4.98003			−0.326252			−0.163126	+	0.986605i	$0.552158\pi$
$234$		0			0
$235$	6.27758	−	6.27758i	0.409504	−	0.409504i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.5353			−0.746156			−0.373078	−	0.927800i	$-0.621698\pi$
$239$	−11.5353			−0.746156			−0.373078	+	0.927800i	$0.621698\pi$
$240$		0			0
$241$	−6.58992			−0.424494			−0.212247	−	0.977216i	$-0.568078\pi$
$241$	−6.58992			−0.424494			−0.212247	+	0.977216i	$0.568078\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.14956	−	7.14956i	0.456768	−	0.456768i
$246$		0			0
$247$	3.85645			0.245380
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.76507	−	6.76507i	−0.427007	−	0.427007i	0.460600	−	0.887608i	$-0.347634\pi$
$251$	−6.76507	−	6.76507i	−0.427007	−	0.427007i	−0.887608	+	0.460600i	$0.847634\pi$
$252$		0			0
$253$	−14.3874	+	14.3874i	−0.904525	+	0.904525i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	25.6146i		−	1.59780i	−0.601466	−	0.798899i	$-0.705416\pi$
$257$		−	25.6146i		−	1.59780i	0.601466	−	0.798899i	$-0.294584\pi$
$258$		0			0
$259$	26.3710	+	26.3710i	1.63862	+	1.63862i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.56169i			0.281286i	0.990060	+	0.140643i	$0.0449169\pi$
$263$			4.56169i			0.281286i	−0.990060	+	0.140643i	$0.955083\pi$
$264$		0			0
$265$			13.7573i			0.845104i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.80272	−	9.80272i	−0.597682	−	0.597682i	0.342013	−	0.939695i	$-0.388891\pi$
$269$	−9.80272	−	9.80272i	−0.597682	−	0.597682i	−0.939695	+	0.342013i	$0.888891\pi$
$270$		0			0
$271$			2.97033i			0.180435i	0.995922	+	0.0902174i	$0.0287562\pi$
$271$			2.97033i			0.180435i	−0.995922	+	0.0902174i	$0.971244\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.99565	+	1.99565i	−0.120342	+	0.120342i
$276$		0			0
$277$	−12.6705	−	12.6705i	−0.761297	−	0.761297i	0.215260	−	0.976557i	$-0.430940\pi$
$277$	−12.6705	−	12.6705i	−0.761297	−	0.761297i	−0.976557	+	0.215260i	$0.930940\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.7093			1.35472			0.677362	−	0.735650i	$-0.263123\pi$
$281$	22.7093			1.35472			0.677362	+	0.735650i	$0.263123\pi$
$282$		0			0
$283$	−15.6922	+	15.6922i	−0.932804	+	0.932804i	−0.997880	−	0.0650764i	$-0.979271\pi$
$283$	−15.6922	+	15.6922i	−0.932804	+	0.932804i	0.0650764	+	0.997880i	$0.479271\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	30.0486			1.77371
$288$		0			0
$289$	6.36682			0.374519
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.60395	+	4.60395i	−0.268965	+	0.268965i	−0.828683	−	0.559718i	$-0.810910\pi$
$293$	−4.60395	+	4.60395i	−0.268965	+	0.268965i	0.559718	+	0.828683i	$0.310910\pi$
$294$		0			0
$295$	−4.05038			−0.235822
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.72366	−	3.72366i	−0.215345	−	0.215345i
$300$		0			0
$301$	−0.528725	+	0.528725i	−0.0304752	+	0.0304752i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			6.57312i			0.376376i
$306$		0			0
$307$	−6.49416	−	6.49416i	−0.370641	−	0.370641i	0.497069	−	0.867711i	$-0.334409\pi$
$307$	−6.49416	−	6.49416i	−0.370641	−	0.370641i	−0.867711	+	0.497069i	$0.834409\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			26.5023i			1.50280i	0.659844	+	0.751402i	$0.270622\pi$
$311$			26.5023i			1.50280i	−0.659844	+	0.751402i	$0.729378\pi$
$312$		0			0
$313$		−	16.2292i		−	0.917327i	−0.888610	−	0.458664i	$-0.848328\pi$
$313$		−	16.2292i		−	0.917327i	0.888610	−	0.458664i	$-0.151672\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.45683	+	3.45683i	0.194155	+	0.194155i	0.797489	−	0.603334i	$-0.206161\pi$
$317$	3.45683	+	3.45683i	0.194155	+	0.194155i	−0.603334	+	0.797489i	$0.706161\pi$
$318$		0			0
$319$			4.86019i			0.272119i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.1735	+	12.1735i	−0.677352	+	0.677352i
$324$		0			0
$325$	−0.516505	−	0.516505i	−0.0286505	−	0.0286505i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	36.7236			2.02464
$330$		0			0
$331$	0.270581	−	0.270581i	0.0148725	−	0.0148725i	−0.699631	−	0.714504i	$-0.746652\pi$
$331$	0.270581	−	0.270581i	0.0148725	−	0.0148725i	0.714504	+	0.699631i	$0.246652\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.373865			−0.0204264
$336$		0			0
$337$	−34.5747			−1.88340			−0.941702	−	0.336447i	$-0.890775\pi$
$337$	−34.5747			−1.88340			−0.941702	+	0.336447i	$0.890775\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−19.0656	+	19.0656i	−1.03246	+	1.03246i
$342$		0			0
$343$	12.8688			0.694850
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.32453	+	2.32453i	0.124787	+	0.124787i	0.766742	−	0.641955i	$-0.221876\pi$
$347$	2.32453	+	2.32453i	0.124787	+	0.124787i	−0.641955	+	0.766742i	$0.721876\pi$
$348$		0			0
$349$	−17.1089	+	17.1089i	−0.915819	+	0.915819i	−0.996722	−	0.0809032i	$-0.974220\pi$
$349$	−17.1089	+	17.1089i	−0.915819	+	0.915819i	0.0809032	+	0.996722i	$0.474220\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			0.666944i			0.0354978i	0.999842	+	0.0177489i	$0.00564995\pi$
$353$			0.666944i			0.0354978i	−0.999842	+	0.0177489i	$0.994350\pi$
$354$		0			0
$355$	−1.32990	−	1.32990i	−0.0705838	−	0.0705838i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			29.8378i			1.57478i	0.616456	+	0.787389i	$0.288568\pi$
$359$			29.8378i			1.57478i	−0.616456	+	0.787389i	$0.711432\pi$
$360$		0			0
$361$			8.87391i			0.467048i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.09866	−	2.09866i	−0.109849	−	0.109849i
$366$		0			0
$367$		−	21.9584i		−	1.14622i	−0.819478	−	0.573111i	$-0.805737\pi$
$367$		−	21.9584i		−	1.14622i	0.819478	−	0.573111i	$-0.194263\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−40.2398	+	40.2398i	−2.08915	+	2.08915i
$372$		0			0
$373$	−8.34820	−	8.34820i	−0.432253	−	0.432253i	0.457141	−	0.889394i	$-0.348873\pi$
$373$	−8.34820	−	8.34820i	−0.432253	−	0.432253i	−0.889394	+	0.457141i	$0.848873\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.25789			−0.0647847
$378$		0			0
$379$	14.5127	−	14.5127i	0.745469	−	0.745469i	−0.228156	−	0.973625i	$-0.573270\pi$
$379$	14.5127	−	14.5127i	0.745469	−	0.745469i	0.973625	+	0.228156i	$0.0732696\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.6064			1.10404			0.552019	−	0.833832i	$-0.313858\pi$
$383$	21.6064			1.10404			0.552019	+	0.833832i	$0.313858\pi$
$384$		0			0
$385$	−11.6745			−0.594986
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.3468	+	23.3468i	−1.18373	+	1.18373i	−0.204958	+	0.978771i	$0.565706\pi$
$389$	−23.3468	+	23.3468i	−1.18373	+	1.18373i	−0.978771	+	0.204958i	$0.934294\pi$
$390$		0			0
$391$	23.5086			1.18888
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.30424	−	4.30424i	−0.216570	−	0.216570i
$396$		0			0
$397$	−12.1620	+	12.1620i	−0.610394	+	0.610394i	−0.943049	−	0.332655i	$-0.892056\pi$
$397$	−12.1620	+	12.1620i	−0.610394	+	0.610394i	0.332655	+	0.943049i	$0.392056\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			35.4000i			1.76779i	0.467683	+	0.883896i	$0.345089\pi$
$401$			35.4000i			1.76779i	−0.467683	+	0.883896i	$0.654911\pi$
$402$		0			0
$403$	−4.93446	−	4.93446i	−0.245803	−	0.245803i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	25.4451i		−	1.26127i
$408$		0			0
$409$			5.10272i			0.252313i	0.992010	+	0.126157i	$0.0402642\pi$
$409$			5.10272i			0.252313i	−0.992010	+	0.126157i	$0.959736\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11.8473	−	11.8473i	−0.582967	−	0.582967i
$414$		0			0
$415$		−	7.04498i		−	0.345825i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.52922	−	5.52922i	0.270120	−	0.270120i	−0.559028	−	0.829148i	$-0.688826\pi$
$419$	5.52922	−	5.52922i	0.270120	−	0.270120i	0.829148	+	0.559028i	$0.188826\pi$
$420$		0			0
$421$	14.8492	+	14.8492i	0.723708	+	0.723708i	0.969358	−	0.245651i	$-0.0790016\pi$
$421$	14.8492	+	14.8492i	0.723708	+	0.723708i	−0.245651	+	0.969358i	$0.579002\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.26086			0.158175
$426$		0			0
$427$	−19.2262	+	19.2262i	−0.930422	+	0.930422i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.0747			−0.870626			−0.435313	−	0.900279i	$-0.643362\pi$
$431$	−18.0747			−0.870626			−0.435313	+	0.900279i	$0.643362\pi$
$432$		0			0
$433$	−20.6052			−0.990223			−0.495112	−	0.868829i	$-0.664873\pi$
$433$	−20.6052			−0.990223			−0.495112	+	0.868829i	$0.664873\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.9141	−	26.9141i	1.28748	−	1.28748i
$438$		0			0
$439$	−34.9204			−1.66666			−0.833330	−	0.552776i	$-0.813568\pi$
$439$	−34.9204			−1.66666			−0.833330	+	0.552776i	$0.813568\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.35145	−	3.35145i	−0.159232	−	0.159232i	0.622994	−	0.782226i	$-0.285916\pi$
$443$	−3.35145	−	3.35145i	−0.159232	−	0.159232i	−0.782226	+	0.622994i	$0.785916\pi$
$444$		0			0
$445$	−4.88091	+	4.88091i	−0.231377	+	0.231377i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			35.4234i			1.67173i	0.548931	+	0.835867i	$0.315035\pi$
$449$			35.4234i			1.67173i	−0.548931	+	0.835867i	$0.684965\pi$
$450$		0			0
$451$	−14.4968	−	14.4968i	−0.682625	−	0.682625i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.02153i		−	0.141652i
$456$		0			0
$457$		−	18.3448i		−	0.858135i	−0.903272	−	0.429067i	$-0.858842\pi$
$457$		−	18.3448i		−	0.858135i	0.903272	−	0.429067i	$-0.141158\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.16968	+	6.16968i	0.287351	+	0.287351i	0.836032	−	0.548681i	$-0.184870\pi$
$461$	6.16968	+	6.16968i	0.287351	+	0.287351i	−0.548681	+	0.836032i	$0.684870\pi$
$462$		0			0
$463$			16.5392i			0.768641i	0.923200	+	0.384320i	$0.125564\pi$
$463$			16.5392i			0.768641i	−0.923200	+	0.384320i	$0.874436\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.7412	−	17.7412i	0.820964	−	0.820964i	−0.165282	−	0.986246i	$-0.552853\pi$
$467$	17.7412	−	17.7412i	0.820964	−	0.820964i	0.986246	+	0.165282i	$0.0528534\pi$
$468$		0			0
$469$	−1.09355	−	1.09355i	−0.0504953	−	0.0504953i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.510161			0.0234572
$474$		0			0
$475$	3.73322	−	3.73322i	0.171292	−	0.171292i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.04478			−0.276193			−0.138096	−	0.990419i	$-0.544098\pi$
$479$	−6.04478			−0.276193			−0.138096	+	0.990419i	$0.544098\pi$
$480$		0			0
$481$	6.58557			0.300276
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.24028	+	7.24028i	−0.328764	+	0.328764i
$486$		0			0
$487$	−27.1826			−1.23176			−0.615882	−	0.787839i	$-0.711200\pi$
$487$	−27.1826			−1.23176			−0.615882	+	0.787839i	$0.711200\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.92201	−	3.92201i	−0.176998	−	0.176998i	0.613048	−	0.790046i	$-0.289943\pi$
$491$	−3.92201	−	3.92201i	−0.176998	−	0.176998i	−0.790046	+	0.613048i	$0.789943\pi$
$492$		0			0
$493$	3.97073	−	3.97073i	0.178833	−	0.178833i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.77987i		−	0.348975i
$498$		0			0
$499$	16.8663	+	16.8663i	0.755039	+	0.755039i	0.975415	−	0.220376i	$-0.0707285\pi$
$499$	16.8663	+	16.8663i	0.755039	+	0.755039i	−0.220376	+	0.975415i	$0.570729\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			19.7695i			0.881480i	0.897635	+	0.440740i	$0.145284\pi$
$503$			19.7695i			0.881480i	−0.897635	+	0.440740i	$0.854716\pi$
$504$		0			0
$505$		−	11.0673i		−	0.492487i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.29061	+	5.29061i	0.234502	+	0.234502i	0.814569	−	0.580067i	$-0.196974\pi$
$509$	5.29061	+	5.29061i	0.234502	+	0.234502i	−0.580067	+	0.814569i	$0.696974\pi$
$510$		0			0
$511$		−	12.2771i		−	0.543105i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.19855	−	7.19855i	0.317206	−	0.317206i
$516$		0			0
$517$	−17.7171	−	17.7171i	−0.779196	−	0.779196i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.0355			−1.22826			−0.614129	−	0.789206i	$-0.710493\pi$
$521$	−28.0355			−1.22826			−0.614129	+	0.789206i	$0.710493\pi$
$522$		0			0
$523$	−30.8808	+	30.8808i	−1.35032	+	1.35032i	−0.465024	+	0.885298i	$0.653954\pi$
$523$	−30.8808	+	30.8808i	−1.35032	+	1.35032i	−0.885298	+	0.465024i	$0.846046\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	31.1528			1.35704
$528$		0			0
$529$	−28.9747			−1.25977
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.75198	−	3.75198i	0.162516	−	0.162516i
$534$		0			0
$535$	−13.4674			−0.582247
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.1780	−	20.1780i	−0.869129	−	0.869129i
$540$		0			0
$541$	−26.3056	+	26.3056i	−1.13096	+	1.13096i	−0.140948	+	0.990017i	$0.545015\pi$
$541$	−26.3056	+	26.3056i	−1.13096	+	1.13096i	−0.990017	+	0.140948i	$0.954985\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	12.4896i		−	0.534996i
$546$		0			0
$547$	−10.3884	−	10.3884i	−0.444178	−	0.444178i	0.449236	−	0.893413i	$-0.351696\pi$
$547$	−10.3884	−	10.3884i	−0.444178	−	0.444178i	−0.893413	+	0.449236i	$0.851696\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.09186i		−	0.387326i
$552$		0			0
$553$		−	25.1796i		−	1.07075i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.76163	−	6.76163i	−0.286499	−	0.286499i	0.549195	−	0.835694i	$-0.314934\pi$
$557$	−6.76163	−	6.76163i	−0.286499	−	0.286499i	−0.835694	+	0.549195i	$0.814934\pi$
$558$		0			0
$559$			0.132037i			0.00558458i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.6530	+	18.6530i	−0.786132	+	0.786132i	−0.980858	−	0.194726i	$-0.937618\pi$
$563$	−18.6530	+	18.6530i	−0.786132	+	0.786132i	0.194726	+	0.980858i	$0.437618\pi$
$564$		0			0
$565$	−2.44932	−	2.44932i	−0.103044	−	0.103044i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.7176			−0.616995			−0.308497	−	0.951225i	$-0.599826\pi$
$569$	−14.7176			−0.616995			−0.308497	+	0.951225i	$0.599826\pi$
$570$		0			0
$571$	−15.1005	+	15.1005i	−0.631935	+	0.631935i	−0.948553	−	0.316618i	$-0.897453\pi$
$571$	−15.1005	+	15.1005i	−0.631935	+	0.631935i	0.316618	+	0.948553i	$0.397453\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.20935			−0.300651
$576$		0			0
$577$	17.5374			0.730092			0.365046	−	0.930989i	$-0.381053\pi$
$577$	17.5374			0.730092			0.365046	+	0.930989i	$0.381053\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	20.6064	−	20.6064i	0.854898	−	0.854898i
$582$		0			0
$583$	38.8269			1.60805
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.2367	+	14.2367i	0.587611	+	0.587611i	0.936984	−	0.349373i	$-0.113605\pi$
$587$	14.2367	+	14.2367i	0.587611	+	0.587611i	−0.349373	+	0.936984i	$0.613605\pi$
$588$		0			0
$589$	35.6656	−	35.6656i	1.46958	−	1.46958i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	3.18442i		−	0.130769i	−0.997860	−	0.0653843i	$-0.979173\pi$
$593$		−	3.18442i		−	0.130769i	0.997860	−	0.0653843i	$-0.0208273\pi$
$594$		0			0
$595$	9.53793	+	9.53793i	0.391017	+	0.391017i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			15.3081i			0.625471i	0.949840	+	0.312736i	$0.101245\pi$
$599$			15.3081i			0.625471i	−0.949840	+	0.312736i	$0.898755\pi$
$600$		0			0
$601$			18.5232i			0.755575i	0.925892	+	0.377788i	$0.123315\pi$
$601$			18.5232i			0.755575i	−0.925892	+	0.377788i	$0.876685\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.14589	−	2.14589i	−0.0872429	−	0.0872429i
$606$		0			0
$607$		−	16.7857i		−	0.681312i	−0.940188	−	0.340656i	$-0.889351\pi$
$607$		−	16.7857i		−	0.681312i	0.940188	−	0.340656i	$-0.110649\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.58545	−	4.58545i	0.185507	−	0.185507i
$612$		0			0
$613$	16.4664	+	16.4664i	0.665071	+	0.665071i	0.956571	−	0.291500i	$-0.0941544\pi$
$613$	16.4664	+	16.4664i	0.665071	+	0.665071i	−0.291500	+	0.956571i	$0.594154\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	43.4236			1.74817			0.874084	−	0.485775i	$-0.161463\pi$
$617$	43.4236			1.74817			0.874084	+	0.485775i	$0.161463\pi$
$618$		0			0
$619$	10.4967	−	10.4967i	0.421900	−	0.421900i	−0.463958	−	0.885857i	$-0.653571\pi$
$619$	10.4967	−	10.4967i	0.421900	−	0.421900i	0.885857	+	0.463958i	$0.153571\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.5531			−1.14396
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.7884	+	20.7884i	−0.828887	+	0.828887i
$630$		0			0
$631$	−27.8165			−1.10736			−0.553679	−	0.832730i	$-0.686777\pi$
$631$	−27.8165			−1.10736			−0.553679	+	0.832730i	$0.686777\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.1001	+	15.1001i	0.599230	+	0.599230i
$636$		0			0
$637$	5.22238	−	5.22238i	0.206918	−	0.206918i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			8.37454i			0.330774i	0.986229	+	0.165387i	$0.0528874\pi$
$641$			8.37454i			0.330774i	−0.986229	+	0.165387i	$0.947113\pi$
$642$		0			0
$643$	−8.42530	−	8.42530i	−0.332261	−	0.332261i	0.521183	−	0.853445i	$-0.325491\pi$
$643$	−8.42530	−	8.42530i	−0.332261	−	0.332261i	−0.853445	+	0.521183i	$0.825491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.1494i		−	1.02804i	−0.857778	−	0.514020i	$-0.828156\pi$
$647$		−	26.1494i		−	1.02804i	0.857778	−	0.514020i	$-0.171844\pi$
$648$		0			0
$649$			11.4313i			0.448718i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	21.2934	+	21.2934i	0.833275	+	0.833275i	0.987963	−	0.154689i	$-0.0494375\pi$
$653$	21.2934	+	21.2934i	0.833275	+	0.833275i	−0.154689	+	0.987963i	$0.549437\pi$
$654$		0			0
$655$		−	4.62189i		−	0.180592i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.8901	+	15.8901i	−0.618991	+	0.618991i	−0.945273	−	0.326282i	$-0.894204\pi$
$659$	−15.8901	+	15.8901i	−0.618991	+	0.618991i	0.326282	+	0.945273i	$0.394204\pi$
$660$		0			0
$661$	15.1989	+	15.1989i	0.591169	+	0.591169i	0.937947	−	0.346778i	$-0.112724\pi$
$661$	15.1989	+	15.1989i	0.591169	+	0.591169i	−0.346778	+	0.937947i	$0.612724\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	21.8392			0.846888
$666$		0			0
$667$	−8.77880	+	8.77880i	−0.339916	+	0.339916i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	18.5512			0.716159
$672$		0			0
$673$	−2.85084			−0.109892			−0.0549460	−	0.998489i	$-0.517499\pi$
$673$	−2.85084			−0.109892			−0.0549460	+	0.998489i	$0.517499\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.5131	+	21.5131i	−0.826814	+	0.826814i	−0.987075	−	0.160261i	$-0.948766\pi$
$677$	−21.5131	+	21.5131i	−0.826814	+	0.826814i	0.160261	+	0.987075i	$0.448766\pi$
$678$		0			0
$679$	−42.3553			−1.62545
$680$		0			0
$681$		0			0
$682$		0			0
$683$	33.3993	+	33.3993i	1.27799	+	1.27799i	0.941791	+	0.336198i	$0.109141\pi$
$683$	33.3993	+	33.3993i	1.27799	+	1.27799i	0.336198	+	0.941791i	$0.390859\pi$
$684$		0			0
$685$	5.09465	−	5.09465i	0.194657	−	0.194657i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			10.0490i			0.382836i
$690$		0			0
$691$	−26.3790	−	26.3790i	−1.00351	−	1.00351i	−0.999994	−	0.00351149i	$-0.998882\pi$
$691$	−26.3790	−	26.3790i	−1.00351	−	1.00351i	−0.00351149	−	0.999994i	$-0.501118\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			18.6257i			0.706515i
$696$		0			0
$697$			23.6874i			0.897225i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5.58696	+	5.58696i	0.211016	+	0.211016i	0.804699	−	0.593683i	$-0.202327\pi$
$701$	5.58696	+	5.58696i	0.211016	+	0.211016i	−0.593683	+	0.804699i	$0.702327\pi$
$702$		0			0
$703$			47.5996i			1.79525i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	32.3715	−	32.3715i	1.21746	−	1.21746i
$708$		0			0
$709$	14.9208	+	14.9208i	0.560362	+	0.560362i	0.929410	−	0.369048i	$-0.120316\pi$
$709$	14.9208	+	14.9208i	0.560362	+	0.560362i	−0.369048	+	0.929410i	$0.620316\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−68.8750			−2.57939
$714$		0			0
$715$	−1.45772	+	1.45772i	−0.0545156	+	0.0545156i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0373			−0.747265			−0.373633	−	0.927577i	$-0.621888\pi$
$719$	−20.0373			−0.747265			−0.373633	+	0.927577i	$0.621888\pi$
$720$		0			0
$721$	42.1112			1.56830
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.21770	+	1.21770i	−0.0452241	+	0.0452241i
$726$		0			0
$727$	−19.3409			−0.717313			−0.358656	−	0.933470i	$-0.616765\pi$
$727$	−19.3409			−0.717313			−0.358656	+	0.933470i	$0.616765\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.416796	−	0.416796i	−0.0154158	−	0.0154158i
$732$		0			0
$733$	−8.24076	+	8.24076i	−0.304379	+	0.304379i	−0.842724	−	0.538345i	$-0.819050\pi$
$733$	−8.24076	+	8.24076i	−0.304379	+	0.304379i	0.538345	+	0.842724i	$0.319050\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.05515i			0.0388670i
$738$		0			0
$739$	−12.2855	−	12.2855i	−0.451930	−	0.451930i	0.444065	−	0.895995i	$-0.353536\pi$
$739$	−12.2855	−	12.2855i	−0.451930	−	0.451930i	−0.895995	+	0.444065i	$0.853536\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	16.2633i		−	0.596644i	−0.954465	−	0.298322i	$-0.903573\pi$
$743$		−	16.2633i		−	0.596644i	0.954465	−	0.298322i	$-0.0964269\pi$
$744$		0			0
$745$			22.5561i			0.826392i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−39.3919	−	39.3919i	−1.43935	−	1.43935i
$750$		0			0
$751$		−	24.6548i		−	0.899667i	−0.893113	−	0.449833i	$-0.851483\pi$
$751$		−	24.6548i		−	0.899667i	0.893113	−	0.449833i	$-0.148517\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.22101	−	8.22101i	0.299193	−	0.299193i
$756$		0			0
$757$	−25.1769	−	25.1769i	−0.915072	−	0.915072i	0.0815940	−	0.996666i	$-0.473999\pi$
$757$	−25.1769	−	25.1769i	−0.915072	−	0.915072i	−0.996666	+	0.0815940i	$0.973999\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−34.1018			−1.23619			−0.618095	−	0.786103i	$-0.712095\pi$
$761$	−34.1018			−1.23619			−0.618095	+	0.786103i	$0.712095\pi$
$762$		0			0
$763$	36.5318	−	36.5318i	1.32254	−	1.32254i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.95859			−0.106829
$768$		0			0
$769$	−33.1599			−1.19578			−0.597888	−	0.801580i	$-0.703993\pi$
$769$	−33.1599			−1.19578			−0.597888	+	0.801580i	$0.703993\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.69057	−	5.69057i	0.204676	−	0.204676i	−0.597324	−	0.802000i	$-0.703769\pi$
$773$	5.69057	−	5.69057i	0.204676	−	0.204676i	0.802000	+	0.597324i	$0.203769\pi$
$774$		0			0
$775$	−9.55357			−0.343174
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27.1188	+	27.1188i	0.971631	+	0.971631i
$780$		0			0
$781$	−3.75335	+	3.75335i	−0.134305	+	0.134305i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	19.0723i		−	0.680719i
$786$		0			0
$787$	−24.6712	−	24.6712i	−0.879434	−	0.879434i	0.114042	−	0.993476i	$-0.463620\pi$
$787$	−24.6712	−	24.6712i	−0.879434	−	0.879434i	−0.993476	+	0.114042i	$0.963620\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	14.3284i		−	0.509460i
$792$		0			0
$793$			4.80132i			0.170500i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	37.3362	+	37.3362i	1.32251	+	1.32251i	0.911734	+	0.410780i	$0.134744\pi$
$797$	37.3362	+	37.3362i	1.32251	+	1.32251i	0.410780	+	0.911734i	$0.365256\pi$
$798$		0			0
$799$			28.9494i			1.02415i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.92299	+	5.92299i	−0.209018	+	0.209018i
$804$		0			0
$805$	−21.0872	−	21.0872i	−0.743226	−	0.743226i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17.9038			−0.629465			−0.314732	−	0.949180i	$-0.601915\pi$
$809$	−17.9038			−0.629465			−0.314732	+	0.949180i	$0.601915\pi$
$810$		0			0
$811$	−11.6377	+	11.6377i	−0.408656	+	0.408656i	−0.881270	−	0.472614i	$-0.843311\pi$
$811$	−11.6377	+	11.6377i	−0.408656	+	0.408656i	0.472614	+	0.881270i	$0.343311\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.69539			−0.0593870
$816$		0			0
$817$	−0.954347			−0.0333884
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−29.3168	+	29.3168i	−1.02316	+	1.02316i	−0.0234386	+	0.999725i	$0.507461\pi$
$821$	−29.3168	+	29.3168i	−1.02316	+	1.02316i	−0.999725	+	0.0234386i	$0.992539\pi$
$822$		0			0
$823$	−9.89629			−0.344963			−0.172481	−	0.985013i	$-0.555178\pi$
$823$	−9.89629			−0.344963			−0.172481	+	0.985013i	$0.555178\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.195254	+	0.195254i	0.00678966	+	0.00678966i	0.710493	−	0.703704i	$-0.248472\pi$
$827$	0.195254	+	0.195254i	0.00678966	+	0.00678966i	−0.703704	+	0.710493i	$0.748472\pi$
$828$		0			0
$829$	38.9291	−	38.9291i	1.35206	−	1.35206i	0.468712	−	0.883351i	$-0.344718\pi$
$829$	38.9291	−	38.9291i	1.35206	−	1.35206i	0.883351	−	0.468712i	$-0.155282\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			32.9705i			1.14236i
$834$		0			0
$835$	4.60374	+	4.60374i	0.159319	+	0.159319i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			26.1042i			0.901217i	0.892722	+	0.450609i	$0.148793\pi$
$839$			26.1042i			0.901217i	−0.892722	+	0.450609i	$0.851207\pi$
$840$		0			0
$841$		−	26.0344i		−	0.897739i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.81511	+	8.81511i	0.303249	+	0.303249i
$846$		0			0
$847$		−	12.5534i		−	0.431339i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	45.9606	−	45.9606i	1.57551	−	1.57551i
$852$		0			0
$853$	3.88268	+	3.88268i	0.132940	+	0.132940i	0.770446	−	0.637505i	$-0.220034\pi$
$853$	3.88268	+	3.88268i	0.132940	+	0.132940i	−0.637505	+	0.770446i	$0.720034\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.0296			−0.957472			−0.478736	−	0.877959i	$-0.658905\pi$
$857$	−28.0296			−0.957472			−0.478736	+	0.877959i	$0.658905\pi$
$858$		0			0
$859$	9.91482	−	9.91482i	0.338289	−	0.338289i	−0.517434	−	0.855723i	$-0.673113\pi$
$859$	9.91482	−	9.91482i	0.338289	−	0.338289i	0.855723	+	0.517434i	$0.173113\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.8640			−0.437895			−0.218948	−	0.975737i	$-0.570262\pi$
$863$	−12.8640			−0.437895			−0.218948	+	0.975737i	$0.570262\pi$
$864$		0			0
$865$	10.2393			0.348147
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.1478	+	12.1478i	−0.412085	+	0.412085i
$870$		0			0
$871$	−0.273089			−0.00925327
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.92498	−	2.92498i	−0.0988823	−	0.0988823i
$876$		0			0
$877$	−28.2843	+	28.2843i	−0.955091	+	0.955091i	−0.999034	−	0.0439427i	$-0.986008\pi$
$877$	−28.2843	+	28.2843i	−0.955091	+	0.955091i	0.0439427	+	0.999034i	$0.486008\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			15.3589i			0.517456i	0.965950	+	0.258728i	$0.0833033\pi$
$881$			15.3589i			0.517456i	−0.965950	+	0.258728i	$0.916697\pi$
$882$		0			0
$883$	−33.6553	−	33.6553i	−1.13259	−	1.13259i	−0.989745	−	0.142847i	$-0.954374\pi$
$883$	−33.6553	−	33.6553i	−1.13259	−	1.13259i	−0.142847	−	0.989745i	$-0.545626\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22.1187i			0.742675i	0.928498	+	0.371337i	$0.121101\pi$
$887$			22.1187i			0.742675i	−0.928498	+	0.371337i	$0.878899\pi$
$888$		0			0
$889$			88.3351i			2.96266i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	33.1430	+	33.1430i	1.10909	+	1.10909i
$894$		0			0
$895$		−	5.81371i		−	0.194331i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.6333	+	11.6333i	−0.387994	+	0.387994i
$900$		0			0
$901$	−31.7212	−	31.7212i	−1.05679	−	1.05679i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.6127			0.552223
$906$		0			0
$907$	−36.5189	+	36.5189i	−1.21259	+	1.21259i	−0.242420	+	0.970171i	$0.577941\pi$
$907$	−36.5189	+	36.5189i	−1.21259	+	1.21259i	−0.970171	+	0.242420i	$0.922059\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.60406			0.218802			0.109401	−	0.993998i	$-0.465107\pi$
$911$	6.60406			0.218802			0.109401	+	0.993998i	$0.465107\pi$
$912$		0			0
$913$	−19.8829			−0.658028
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.5189	−	13.5189i	0.446434	−	0.446434i
$918$		0			0
$919$	−15.3917			−0.507727			−0.253863	−	0.967240i	$-0.581701\pi$
$919$	−15.3917			−0.507727			−0.253863	+	0.967240i	$0.581701\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.971424	−	0.971424i	−0.0319748	−	0.0319748i
$924$		0			0
$925$	6.37513	−	6.37513i	0.209613	−	0.209613i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			55.9587i			1.83594i	0.396645	+	0.917972i	$0.370174\pi$
$929$			55.9587i			1.83594i	−0.396645	+	0.917972i	$0.629826\pi$
$930$		0			0
$931$	37.7466	+	37.7466i	1.23710	+	1.23710i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	9.20304i		−	0.300972i
$936$		0			0
$937$		−	15.1509i		−	0.494959i	−0.968893	−	0.247480i	$-0.920398\pi$
$937$		−	15.1509i		−	0.494959i	0.968893	−	0.247480i	$-0.0796023\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	41.6844	+	41.6844i	1.35887	+	1.35887i	0.875309	+	0.483564i	$0.160658\pi$
$941$	41.6844	+	41.6844i	1.35887	+	1.35887i	0.483564	+	0.875309i	$0.339342\pi$
$942$		0			0
$943$		−	52.3700i		−	1.70540i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.30064	+	6.30064i	−0.204743	+	0.204743i	−0.802029	−	0.597285i	$-0.796246\pi$
$947$	−6.30064	+	6.30064i	−0.204743	+	0.204743i	0.597285	+	0.802029i	$0.296246\pi$
$948$		0			0
$949$	−1.53296	−	1.53296i	−0.0497620	−	0.0497620i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.87663			−0.319935			−0.159968	−	0.987122i	$-0.551139\pi$
$953$	−9.87663			−0.319935			−0.159968	+	0.987122i	$0.551139\pi$
$954$		0			0
$955$	−4.14518	+	4.14518i	−0.134135	+	0.134135i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	29.8035			0.962406
$960$		0			0
$961$	−60.2706			−1.94421
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.5681	−	16.5681i	0.533346	−	0.533346i
$966$		0			0
$967$	32.1357			1.03342			0.516708	−	0.856162i	$-0.327157\pi$
$967$	32.1357			1.03342			0.516708	+	0.856162i	$0.327157\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	17.4159	+	17.4159i	0.558902	+	0.558902i	0.928995	−	0.370093i	$-0.120674\pi$
$971$	17.4159	+	17.4159i	0.558902	+	0.558902i	−0.370093	+	0.928995i	$0.620674\pi$
$972$		0			0
$973$	−54.4799	+	54.4799i	−1.74654	+	1.74654i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	19.9741i		−	0.639027i	−0.947582	−	0.319513i	$-0.896481\pi$
$977$		−	19.9741i		−	0.639027i	0.947582	−	0.319513i	$-0.103519\pi$
$978$		0			0
$979$	13.7753	+	13.7753i	0.440260	+	0.440260i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	2.23440i		−	0.0712662i	−0.999365	−	0.0356331i	$-0.988655\pi$
$983$		−	2.23440i		−	0.0712662i	0.999365	−	0.0356331i	$-0.0113448\pi$
$984$		0			0
$985$		−	4.49850i		−	0.143334i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.921486	+	0.921486i	0.0293015	+	0.0293015i
$990$		0			0
$991$			45.3220i			1.43970i	0.694129	+	0.719851i	$0.255790\pi$
$991$			45.3220i			1.43970i	−0.694129	+	0.719851i	$0.744210\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	15.2564	−	15.2564i	0.483661	−	0.483661i
$996$		0			0
$997$	−10.4556	−	10.4556i	−0.331132	−	0.331132i	0.521884	−	0.853016i	$-0.325229\pi$
$997$	−10.4556	−	10.4556i	−0.331132	−	0.331132i	−0.853016	+	0.521884i	$0.825229\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.b.1871.8		24
3.2	odd	2	inner	2880.2.bl.b.1871.6		24
4.3	odd	2		720.2.bl.b.251.2	✓	24
12.11	even	2		720.2.bl.b.251.11	yes	24
16.3	odd	4	inner	2880.2.bl.b.431.6		24
16.13	even	4		720.2.bl.b.611.11	yes	24
48.29	odd	4		720.2.bl.b.611.2	yes	24
48.35	even	4	inner	2880.2.bl.b.431.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.b.251.2	✓	24	4.3	odd	2
720.2.bl.b.251.11	yes	24	12.11	even	2
720.2.bl.b.611.2	yes	24	48.29	odd	4
720.2.bl.b.611.11	yes	24	16.13	even	4
2880.2.bl.b.431.6		24	16.3	odd	4	inner
2880.2.bl.b.431.8		24	48.35	even	4	inner
2880.2.bl.b.1871.6		24	3.2	odd	2	inner
2880.2.bl.b.1871.8		24	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} +4.13654 q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{5} +4.13654 q^{7} +(-1.99565 - 1.99565i) q^{11} +(0.516505 - 0.516505i) q^{13} +3.26086i q^{17} +(3.73322 + 3.73322i) q^{19} -7.20935i q^{23} -1.00000i q^{25} +(-1.21770 - 1.21770i) q^{29} -9.55357i q^{31} +(2.92498 - 2.92498i) q^{35} +(6.37513 + 6.37513i) q^{37} +7.26417 q^{41} +(-0.127818 + 0.127818i) q^{43} +8.87784 q^{47} +10.1110 q^{49} +(-9.72788 + 9.72788i) q^{53} -2.82228 q^{55} +(-2.86405 - 2.86405i) q^{59} +(-4.64789 + 4.64789i) q^{61} -0.730448i q^{65} +(-0.264363 - 0.264363i) q^{67} -1.88076i q^{71} -2.96795i q^{73} +(-8.25510 - 8.25510i) q^{77} -6.08712i q^{79} +(4.98155 - 4.98155i) q^{83} +(2.30577 + 2.30577i) q^{85} -6.90265 q^{89} +(2.13654 - 2.13654i) q^{91} +5.27957 q^{95} -10.2393 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{7}+O(q^{10}) \) 24 * q + 8 * q^7	\( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^7 + 24 * q^19 + 8 * q^37 - 48 * q^43 + 24 * q^49 - 24 * q^55 + 40 * q^61 + 40 * q^67 + 24 * q^85 - 40 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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