LMFDB - Embedded newform 2340.2.y.b.53.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(53,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.53"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.y (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.9
Character	$\chi$	$=$	2340.53
Dual form			2340.2.y.b.1457.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.80799 - 1.31574i) q^{5} +(-0.602482 + 0.602482i) q^{7} -1.84724i q^{11} +(-0.707107 - 0.707107i) q^{13} +(3.70666 + 3.70666i) q^{17} -7.06758i q^{19} +(-0.688512 + 0.688512i) q^{23} +(1.53765 - 4.75769i) q^{25} -5.94680 q^{29} -0.288098 q^{31} +(-0.296571 + 1.88199i) q^{35} +(5.56293 - 5.56293i) q^{37} -5.63302i q^{41} +(5.76366 + 5.76366i) q^{43} +(-4.61312 - 4.61312i) q^{47} +6.27403i q^{49} +(7.46699 - 7.46699i) q^{53} +(-2.43049 - 3.33979i) q^{55} -9.07941 q^{59} +9.04798 q^{61} +(-2.20881 - 0.348072i) q^{65} +(0.873903 - 0.873903i) q^{67} -6.96605i q^{71} +(-4.12147 - 4.12147i) q^{73} +(1.11293 + 1.11293i) q^{77} -6.25959i q^{79} +(3.77666 - 3.77666i) q^{83} +(11.5786 + 1.82460i) q^{85} -9.52370 q^{89} +0.852038 q^{91} +(-9.29910 - 12.7781i) q^{95} +(7.80599 - 7.80599i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.80799	−	1.31574i	0.808557	−	0.588417i
$5$	1.80799	−	1.31574i	0.808557	−	0.588417i
$6$		0			0
$7$	−0.602482	+	0.602482i	−0.227717	+	0.227717i	−0.811738	−	0.584021i	$-0.801478\pi$
$7$	−0.602482	+	0.602482i	−0.227717	+	0.227717i	0.584021	+	0.811738i	$0.301478\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.84724i		−	0.556964i	−0.960441	−	0.278482i	$-0.910169\pi$
$11$		−	1.84724i		−	0.556964i	0.960441	−	0.278482i	$-0.0898313\pi$
$12$		0			0
$13$	−0.707107	−	0.707107i	−0.196116	−	0.196116i
$13$	−0.707107	−	0.707107i	−0.196116	−	0.196116i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.70666	+	3.70666i	0.898997	+	0.898997i	0.995347	−	0.0963503i	$-0.0307169\pi$
$17$	3.70666	+	3.70666i	0.898997	+	0.898997i	−0.0963503	+	0.995347i	$0.530717\pi$
$18$		0			0
$19$		−	7.06758i		−	1.62141i	−0.585452	−	0.810707i	$-0.699083\pi$
$19$		−	7.06758i		−	1.62141i	0.585452	−	0.810707i	$-0.300917\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.688512	+	0.688512i	−0.143565	+	0.143565i	−0.775236	−	0.631672i	$-0.782369\pi$
$23$	−0.688512	+	0.688512i	−0.143565	+	0.143565i	0.631672	+	0.775236i	$0.282369\pi$
$24$		0			0
$25$	1.53765	−	4.75769i	0.307530	−	0.951538i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.94680			−1.10429			−0.552147	−	0.833747i	$-0.686191\pi$
$29$	−5.94680			−1.10429			−0.552147	+	0.833747i	$0.686191\pi$
$30$		0			0
$31$	−0.288098			−0.0517440			−0.0258720	−	0.999665i	$-0.508236\pi$
$31$	−0.288098			−0.0517440			−0.0258720	+	0.999665i	$0.508236\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.296571	+	1.88199i	−0.0501296	+	0.318115i
$36$		0			0
$37$	5.56293	−	5.56293i	0.914540	−	0.914540i	−0.0820856	−	0.996625i	$-0.526158\pi$
$37$	5.56293	−	5.56293i	0.914540	−	0.914540i	0.996625	+	0.0820856i	$0.0261581\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.63302i		−	0.879731i	−0.898064	−	0.439865i	$-0.855026\pi$
$41$		−	5.63302i		−	0.879731i	0.898064	−	0.439865i	$-0.144974\pi$
$42$		0			0
$43$	5.76366	+	5.76366i	0.878950	+	0.878950i	0.993426	−	0.114476i	$-0.0365188\pi$
$43$	5.76366	+	5.76366i	0.878950	+	0.878950i	−0.114476	+	0.993426i	$0.536519\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.61312	−	4.61312i	−0.672893	−	0.672893i	0.285489	−	0.958382i	$-0.407844\pi$
$47$	−4.61312	−	4.61312i	−0.672893	−	0.672893i	−0.958382	+	0.285489i	$0.907844\pi$
$48$		0			0
$49$			6.27403i			0.896290i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.46699	−	7.46699i	1.02567	−	1.02567i	0.0260084	−	0.999662i	$-0.491720\pi$
$53$	7.46699	−	7.46699i	1.02567	−	1.02567i	0.999662	−	0.0260084i	$-0.00827967\pi$
$54$		0			0
$55$	−2.43049	−	3.33979i	−0.327727	−	0.450337i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.07941			−1.18204			−0.591019	−	0.806657i	$-0.701274\pi$
$59$	−9.07941			−1.18204			−0.591019	+	0.806657i	$0.701274\pi$
$60$		0			0
$61$	9.04798			1.15848			0.579238	−	0.815159i	$-0.303350\pi$
$61$	9.04798			1.15848			0.579238	+	0.815159i	$0.303350\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.20881	−	0.348072i	−0.273969	−	0.0431731i
$66$		0			0
$67$	0.873903	−	0.873903i	0.106764	−	0.106764i	−0.651707	−	0.758471i	$-0.725947\pi$
$67$	0.873903	−	0.873903i	0.106764	−	0.106764i	0.758471	+	0.651707i	$0.225947\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.96605i		−	0.826719i	−0.910568	−	0.413359i	$-0.864355\pi$
$71$		−	6.96605i		−	0.826719i	0.910568	−	0.413359i	$-0.135645\pi$
$72$		0			0
$73$	−4.12147	−	4.12147i	−0.482382	−	0.482382i	0.423510	−	0.905892i	$-0.360798\pi$
$73$	−4.12147	−	4.12147i	−0.482382	−	0.482382i	−0.905892	+	0.423510i	$0.860798\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.11293	+	1.11293i	0.126830	+	0.126830i
$78$		0			0
$79$		−	6.25959i		−	0.704259i	−0.935951	−	0.352130i	$-0.885458\pi$
$79$		−	6.25959i		−	0.704259i	0.935951	−	0.352130i	$-0.114542\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.77666	−	3.77666i	0.414542	−	0.414542i	−0.468775	−	0.883317i	$-0.655305\pi$
$83$	3.77666	−	3.77666i	0.414542	−	0.414542i	0.883317	+	0.468775i	$0.155305\pi$
$84$		0			0
$85$	11.5786	+	1.82460i	1.25588	+	0.197905i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.52370			−1.00951			−0.504755	−	0.863262i	$-0.668417\pi$
$89$	−9.52370			−1.00951			−0.504755	+	0.863262i	$0.668417\pi$
$90$		0			0
$91$	0.852038			0.0893179
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.29910	−	12.7781i	−0.954068	−	1.31101i
$96$		0			0
$97$	7.80599	−	7.80599i	0.792579	−	0.792579i	−0.189334	−	0.981913i	$-0.560633\pi$
$97$	7.80599	−	7.80599i	0.792579	−	0.792579i	0.981913	+	0.189334i	$0.0606329\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12.2033i			1.21427i	0.794599	+	0.607135i	$0.207681\pi$
$101$			12.2033i			1.21427i	−0.794599	+	0.607135i	$0.792319\pi$
$102$		0			0
$103$	−5.92683	−	5.92683i	−0.583988	−	0.583988i	0.352009	−	0.935997i	$-0.385499\pi$
$103$	−5.92683	−	5.92683i	−0.583988	−	0.583988i	−0.935997	+	0.352009i	$0.885499\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.63355	−	1.63355i	−0.157922	−	0.157922i	0.623723	−	0.781645i	$-0.285619\pi$
$107$	−1.63355	−	1.63355i	−0.157922	−	0.157922i	−0.781645	+	0.623723i	$0.785619\pi$
$108$		0			0
$109$			15.6407i			1.49810i	0.662511	+	0.749052i	$0.269491\pi$
$109$			15.6407i			1.49810i	−0.662511	+	0.749052i	$0.730509\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.40592	+	9.40592i	−0.884834	+	0.884834i	−0.994021	−	0.109187i	$-0.965175\pi$
$113$	−9.40592	+	9.40592i	−0.884834	+	0.884834i	0.109187	+	0.994021i	$0.465175\pi$
$114$		0			0
$115$	−0.338919	+	2.15072i	−0.0316043	+	0.200556i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.46639			−0.409434
$120$		0			0
$121$	7.58770			0.689791
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.47983	−	10.6250i	−0.311246	−	0.950330i
$126$		0			0
$127$	12.5316	−	12.5316i	1.11200	−	1.11200i	0.119121	−	0.992880i	$-0.461992\pi$
$127$	12.5316	−	12.5316i	1.11200	−	1.11200i	0.992880	−	0.119121i	$-0.0380076\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			17.7964i			1.55488i	0.628959	+	0.777438i	$0.283481\pi$
$131$			17.7964i			1.55488i	−0.628959	+	0.777438i	$0.716519\pi$
$132$		0			0
$133$	4.25809	+	4.25809i	0.369223	+	0.369223i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.37742	−	3.37742i	−0.288553	−	0.288553i	0.547955	−	0.836508i	$-0.315406\pi$
$137$	−3.37742	−	3.37742i	−0.288553	−	0.288553i	−0.836508	+	0.547955i	$0.815406\pi$
$138$		0			0
$139$		−	17.4464i		−	1.47978i	−0.672727	−	0.739891i	$-0.734877\pi$
$139$		−	17.4464i		−	1.47978i	0.672727	−	0.739891i	$-0.265123\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.30620	+	1.30620i	−0.109230	+	0.109230i
$144$		0			0
$145$	−10.7518	+	7.82445i	−0.892885	+	0.649785i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.07747			0.743655			0.371828	−	0.928302i	$-0.378731\pi$
$149$	9.07747			0.743655			0.371828	+	0.928302i	$0.378731\pi$
$150$		0			0
$151$	9.41290			0.766011			0.383005	−	0.923746i	$-0.374889\pi$
$151$	9.41290			0.766011			0.383005	+	0.923746i	$0.374889\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.520879	+	0.379063i	−0.0418380	+	0.0304471i
$156$		0			0
$157$	−0.252047	+	0.252047i	−0.0201156	+	0.0201156i	−0.717093	−	0.696977i	$-0.754528\pi$
$157$	−0.252047	+	0.252047i	−0.0201156	+	0.0201156i	0.696977	+	0.717093i	$0.254528\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.829632i		−	0.0653841i
$162$		0			0
$163$	−9.55132	−	9.55132i	−0.748117	−	0.748117i	0.226009	−	0.974125i	$-0.427432\pi$
$163$	−9.55132	−	9.55132i	−0.748117	−	0.748117i	−0.974125	+	0.226009i	$0.927432\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.83233	−	9.83233i	−0.760849	−	0.760849i	0.215627	−	0.976476i	$-0.430820\pi$
$167$	−9.83233	−	9.83233i	−0.760849	−	0.760849i	−0.976476	+	0.215627i	$0.930820\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.76048	−	5.76048i	0.437961	−	0.437961i	−0.453364	−	0.891325i	$-0.649776\pi$
$173$	5.76048	−	5.76048i	0.437961	−	0.437961i	0.891325	+	0.453364i	$0.149776\pi$
$174$		0			0
$175$	1.94002	+	3.79283i	0.146651	+	0.286711i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.45699			0.407875			0.203937	−	0.978984i	$-0.434626\pi$
$179$	5.45699			0.407875			0.203937	+	0.978984i	$0.434626\pi$
$180$		0			0
$181$	14.7259			1.09456			0.547282	−	0.836948i	$-0.315662\pi$
$181$	14.7259			1.09456			0.547282	+	0.836948i	$0.315662\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.73834	−	17.3771i	0.201327	−	1.27759i
$186$		0			0
$187$	6.84709	−	6.84709i	0.500709	−	0.500709i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	6.91868i		−	0.500618i	−0.968166	−	0.250309i	$-0.919468\pi$
$191$		−	6.91868i		−	0.500618i	0.968166	−	0.250309i	$-0.0805322\pi$
$192$		0			0
$193$	5.42143	+	5.42143i	0.390243	+	0.390243i	0.874774	−	0.484531i	$-0.161010\pi$
$193$	5.42143	+	5.42143i	0.390243	+	0.390243i	−0.484531	+	0.874774i	$0.661010\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.27122	−	3.27122i	−0.233065	−	0.233065i	0.580906	−	0.813971i	$-0.302698\pi$
$197$	−3.27122	−	3.27122i	−0.233065	−	0.233065i	−0.813971	+	0.580906i	$0.802698\pi$
$198$		0			0
$199$			7.36799i			0.522303i	0.965298	+	0.261151i	$0.0841022\pi$
$199$			7.36799i			0.522303i	−0.965298	+	0.261151i	$0.915898\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.58284	−	3.58284i	0.251466	−	0.251466i
$204$		0			0
$205$	−7.41160	−	10.1844i	−0.517649	−	0.711313i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.0555			−0.903069
$210$		0			0
$211$	−24.1786			−1.66452			−0.832262	−	0.554383i	$-0.812954\pi$
$211$	−24.1786			−1.66452			−0.832262	+	0.554383i	$0.812954\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	18.0041	+	2.83715i	1.22787	+	0.193492i
$216$		0			0
$217$	0.173574	−	0.173574i	0.0117830	−	0.0117830i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	5.24201i		−	0.352616i
$222$		0			0
$223$	−9.59310	−	9.59310i	−0.642401	−	0.642401i	0.308744	−	0.951145i	$-0.400091\pi$
$223$	−9.59310	−	9.59310i	−0.642401	−	0.642401i	−0.951145	+	0.308744i	$0.900091\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.5278	+	16.5278i	1.09699	+	1.09699i	0.994761	+	0.102230i	$0.0325978\pi$
$227$	16.5278	+	16.5278i	1.09699	+	1.09699i	0.102230	+	0.994761i	$0.467402\pi$
$228$		0			0
$229$			2.99487i			0.197907i	0.995092	+	0.0989533i	$0.0315495\pi$
$229$			2.99487i			0.197907i	−0.995092	+	0.0989533i	$0.968451\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.6481	−	12.6481i	0.828604	−	0.828604i	−0.158720	−	0.987324i	$-0.550737\pi$
$233$	12.6481	−	12.6481i	0.828604	−	0.828604i	0.987324	+	0.158720i	$0.0507367\pi$
$234$		0			0
$235$	−14.4101	−	2.27080i	−0.940014	−	0.148131i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.7098			−1.53366			−0.766830	−	0.641850i	$-0.778167\pi$
$239$	−23.7098			−1.53366			−0.766830	+	0.641850i	$0.778167\pi$
$240$		0			0
$241$	9.45090			0.608786			0.304393	−	0.952547i	$-0.401546\pi$
$241$	9.45090			0.608786			0.304393	+	0.952547i	$0.401546\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.25500	+	11.3434i	0.527393	+	0.724702i
$246$		0			0
$247$	−4.99753	+	4.99753i	−0.317985	+	0.317985i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	3.37759i		−	0.213192i	−0.994302	−	0.106596i	$-0.966005\pi$
$251$		−	3.37759i		−	0.213192i	0.994302	−	0.106596i	$-0.0339951\pi$
$252$		0			0
$253$	1.27185	+	1.27185i	0.0799603	+	0.0799603i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.6865	+	18.6865i	1.16563	+	1.16563i	0.983223	+	0.182407i	$0.0583889\pi$
$257$	18.6865	+	18.6865i	1.16563	+	1.16563i	0.182407	+	0.983223i	$0.441611\pi$
$258$		0			0
$259$			6.70313i			0.416512i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20.3943	+	20.3943i	−1.25757	+	1.25757i	−0.305314	+	0.952252i	$0.598761\pi$
$263$	−20.3943	+	20.3943i	−1.25757	+	1.25757i	−0.952252	+	0.305314i	$0.901239\pi$
$264$		0			0
$265$	3.67562	−	23.3249i	0.225791	−	1.43284i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.53030			−0.215246			−0.107623	−	0.994192i	$-0.534324\pi$
$269$	−3.53030			−0.215246			−0.107623	+	0.994192i	$0.534324\pi$
$270$		0			0
$271$	18.3887			1.11703			0.558517	−	0.829493i	$-0.311370\pi$
$271$	18.3887			1.11703			0.558517	+	0.829493i	$0.311370\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.78860	−	2.84041i	−0.529973	−	0.171283i
$276$		0			0
$277$	21.6436	−	21.6436i	1.30044	−	1.30044i	0.372338	−	0.928097i	$-0.378556\pi$
$277$	21.6436	−	21.6436i	1.30044	−	1.30044i	0.928097	−	0.372338i	$-0.121444\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5.20362i		−	0.310422i	−0.987881	−	0.155211i	$-0.950394\pi$
$281$		−	5.20362i		−	0.310422i	0.987881	−	0.155211i	$-0.0496057\pi$
$282$		0			0
$283$	−6.10331	−	6.10331i	−0.362804	−	0.362804i	0.502040	−	0.864844i	$-0.332583\pi$
$283$	−6.10331	−	6.10331i	−0.362804	−	0.362804i	−0.864844	+	0.502040i	$0.832583\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.39380	+	3.39380i	0.200329	+	0.200329i
$288$		0			0
$289$			10.4787i			0.616392i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.10324	−	9.10324i	0.531817	−	0.531817i	−0.389296	−	0.921113i	$-0.627282\pi$
$293$	9.10324	−	9.10324i	0.531817	−	0.531817i	0.921113	+	0.389296i	$0.127282\pi$
$294$		0			0
$295$	−16.4155	+	11.9462i	−0.955746	+	0.695532i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.973702			0.0563107
$300$		0			0
$301$	−6.94501			−0.400303
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.3587	−	11.9048i	0.936694	−	0.681667i
$306$		0			0
$307$	−4.99152	+	4.99152i	−0.284881	+	0.284881i	−0.835052	−	0.550171i	$-0.814563\pi$
$307$	−4.99152	+	4.99152i	−0.284881	+	0.284881i	0.550171	+	0.835052i	$0.314563\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			28.5978i			1.62163i	0.585303	+	0.810815i	$0.300976\pi$
$311$			28.5978i			1.62163i	−0.585303	+	0.810815i	$0.699024\pi$
$312$		0			0
$313$	10.9280	+	10.9280i	0.617686	+	0.617686i	0.944937	−	0.327251i	$-0.106122\pi$
$313$	10.9280	+	10.9280i	0.617686	+	0.617686i	−0.327251	+	0.944937i	$0.606122\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.52481	+	2.52481i	0.141807	+	0.141807i	0.774447	−	0.632639i	$-0.218028\pi$
$317$	2.52481	+	2.52481i	0.141807	+	0.141807i	−0.632639	+	0.774447i	$0.718028\pi$
$318$		0			0
$319$			10.9852i			0.615052i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	26.1971	−	26.1971i	1.45765	−	1.45765i
$324$		0			0
$325$	−4.45148	+	2.27691i	−0.246924	+	0.126300i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.55865			0.306458
$330$		0			0
$331$	30.2207			1.66108			0.830540	−	0.556959i	$-0.188032\pi$
$331$	30.2207			1.66108			0.830540	+	0.556959i	$0.188032\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.430177	−	2.72984i	0.0235031	−	0.149147i
$336$		0			0
$337$	−9.34379	+	9.34379i	−0.508989	+	0.508989i	−0.914216	−	0.405227i	$-0.867192\pi$
$337$	−9.34379	+	9.34379i	−0.508989	+	0.508989i	0.405227	+	0.914216i	$0.367192\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			0.532187i			0.0288195i
$342$		0			0
$343$	−7.99737	−	7.99737i	−0.431817	−	0.431817i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.9364	+	21.9364i	1.17761	+	1.17761i	0.980352	+	0.197254i	$0.0632025\pi$
$347$	21.9364	+	21.9364i	1.17761	+	1.17761i	0.197254	+	0.980352i	$0.436798\pi$
$348$		0			0
$349$			22.4475i			1.20159i	0.799404	+	0.600794i	$0.205149\pi$
$349$			22.4475i			1.20159i	−0.799404	+	0.600794i	$0.794851\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.83825	+	7.83825i	−0.417188	+	0.417188i	−0.884233	−	0.467045i	$-0.845318\pi$
$353$	−7.83825	+	7.83825i	−0.417188	+	0.417188i	0.467045	+	0.884233i	$0.345318\pi$
$354$		0			0
$355$	−9.16552	−	12.5946i	−0.486455	−	0.668449i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.8249			−0.887986			−0.443993	−	0.896030i	$-0.646439\pi$
$359$	−16.8249			−0.887986			−0.443993	+	0.896030i	$0.646439\pi$
$360$		0			0
$361$	−30.9507			−1.62898
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.8744	−	2.02879i	−0.673876	−	0.106192i
$366$		0			0
$367$	0.897642	−	0.897642i	0.0468565	−	0.0468565i	−0.683290	−	0.730147i	$-0.739452\pi$
$367$	0.897642	−	0.897642i	0.0468565	−	0.0468565i	0.730147	+	0.683290i	$0.239452\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			8.99746i			0.467125i
$372$		0			0
$373$	1.22114	+	1.22114i	0.0632282	+	0.0632282i	0.738014	−	0.674786i	$-0.235764\pi$
$373$	1.22114	+	1.22114i	0.0632282	+	0.0632282i	−0.674786	+	0.738014i	$0.735764\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.20503	+	4.20503i	0.216570	+	0.216570i
$378$		0			0
$379$			19.8416i			1.01919i	0.860413	+	0.509597i	$0.170205\pi$
$379$			19.8416i			1.01919i	−0.860413	+	0.509597i	$0.829795\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−23.4176	+	23.4176i	−1.19658	+	1.19658i	−0.221398	+	0.975183i	$0.571062\pi$
$383$	−23.4176	+	23.4176i	−1.19658	+	1.19658i	−0.975183	+	0.221398i	$0.928938\pi$
$384$		0			0
$385$	3.47649	+	0.547838i	0.177178	+	0.0279204i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.66508			−0.490039			−0.245020	−	0.969518i	$-0.578794\pi$
$389$	−9.66508			−0.490039			−0.245020	+	0.969518i	$0.578794\pi$
$390$		0			0
$391$	−5.10416			−0.258128
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.23600	−	11.3173i	−0.414398	−	0.569434i
$396$		0			0
$397$	−17.3437	+	17.3437i	−0.870454	+	0.870454i	−0.992522	−	0.122068i	$-0.961047\pi$
$397$	−17.3437	+	17.3437i	−0.870454	+	0.870454i	0.122068	+	0.992522i	$0.461047\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	18.4796i		−	0.922829i	−0.887185	−	0.461414i	$-0.847342\pi$
$401$		−	18.4796i		−	0.922829i	0.887185	−	0.461414i	$-0.152658\pi$
$402$		0			0
$403$	0.203716	+	0.203716i	0.0101478	+	0.0101478i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.2761	−	10.2761i	−0.509366	−	0.509366i
$408$		0			0
$409$			11.6714i			0.577111i	0.957463	+	0.288556i	$0.0931750\pi$
$409$			11.6714i			0.577111i	−0.957463	+	0.288556i	$0.906825\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.47018	−	5.47018i	0.269170	−	0.269170i
$414$		0			0
$415$	1.85905	−	11.7973i	0.0912574	−	0.579105i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.09067			0.199842			0.0999211	−	0.994995i	$-0.468141\pi$
$419$	4.09067			0.199842			0.0999211	+	0.994995i	$0.468141\pi$
$420$		0			0
$421$	−19.0414			−0.928020			−0.464010	−	0.885830i	$-0.653590\pi$
$421$	−19.0414			−0.928020			−0.464010	+	0.885830i	$0.653590\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	23.3347	−	11.9356i	1.13190	−	0.578961i
$426$		0			0
$427$	−5.45125	+	5.45125i	−0.263804	+	0.263804i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			29.8642i			1.43851i	0.694748	+	0.719253i	$0.255516\pi$
$431$			29.8642i			1.43851i	−0.694748	+	0.719253i	$0.744484\pi$
$432$		0			0
$433$	28.6328	+	28.6328i	1.37601	+	1.37601i	0.851259	+	0.524746i	$0.175840\pi$
$433$	28.6328	+	28.6328i	1.37601	+	1.37601i	0.524746	+	0.851259i	$0.324160\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.86611	+	4.86611i	0.232778	+	0.232778i
$438$		0			0
$439$		−	10.9991i		−	0.524958i	−0.964938	−	0.262479i	$-0.915460\pi$
$439$		−	10.9991i		−	0.524958i	0.964938	−	0.262479i	$-0.0845401\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.63354	+	5.63354i	−0.267658	+	0.267658i	−0.828156	−	0.560498i	$-0.810610\pi$
$443$	−5.63354	+	5.63354i	−0.267658	+	0.267658i	0.560498	+	0.828156i	$0.310610\pi$
$444$		0			0
$445$	−17.2188	+	12.5307i	−0.816247	+	0.594013i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.26470			−0.248456			−0.124228	−	0.992254i	$-0.539645\pi$
$449$	−5.26470			−0.248456			−0.124228	+	0.992254i	$0.539645\pi$
$450$		0			0
$451$	−10.4056			−0.489978
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.54048	−	1.12106i	0.0722186	−	0.0525562i
$456$		0			0
$457$	−22.5847	+	22.5847i	−1.05647	+	1.05647i	−0.0581605	+	0.998307i	$0.518524\pi$
$457$	−22.5847	+	22.5847i	−1.05647	+	1.05647i	−0.998307	+	0.0581605i	$0.981476\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			11.4945i			0.535351i	0.963509	+	0.267676i	$0.0862555\pi$
$461$			11.4945i			0.535351i	−0.963509	+	0.267676i	$0.913744\pi$
$462$		0			0
$463$	9.39919	+	9.39919i	0.436817	+	0.436817i	0.890939	−	0.454122i	$-0.150047\pi$
$463$	9.39919	+	9.39919i	0.436817	+	0.436817i	−0.454122	+	0.890939i	$0.650047\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.1247	+	12.1247i	0.561065	+	0.561065i	0.929610	−	0.368545i	$-0.120144\pi$
$467$	12.1247	+	12.1247i	0.561065	+	0.561065i	−0.368545	+	0.929610i	$0.620144\pi$
$468$		0			0
$469$			1.05302i			0.0486240i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.6469	−	10.6469i	0.489544	−	0.489544i
$474$		0			0
$475$	−33.6254	−	10.8675i	−1.54284	−	0.498634i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−29.7812			−1.36074			−0.680368	−	0.732871i	$-0.738180\pi$
$479$	−29.7812			−1.36074			−0.680368	+	0.732871i	$0.738180\pi$
$480$		0			0
$481$	−7.86717			−0.358712
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.84249	−	24.3838i	0.174478	−	1.10721i
$486$		0			0
$487$	5.76865	−	5.76865i	0.261403	−	0.261403i	−0.564221	−	0.825624i	$-0.690823\pi$
$487$	5.76865	−	5.76865i	0.261403	−	0.261403i	0.825624	+	0.564221i	$0.190823\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			32.7290i			1.47704i	0.674231	+	0.738520i	$0.264475\pi$
$491$			32.7290i			1.47704i	−0.674231	+	0.738520i	$0.735525\pi$
$492$		0			0
$493$	−22.0428	−	22.0428i	−0.992757	−	0.992757i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.19692	+	4.19692i	0.188258	+	0.188258i
$498$		0			0
$499$			13.7824i			0.616984i	0.951227	+	0.308492i	$0.0998243\pi$
$499$			13.7824i			0.616984i	−0.951227	+	0.308492i	$0.900176\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.0930	−	19.0930i	0.851316	−	0.851316i	−0.138980	−	0.990295i	$-0.544382\pi$
$503$	19.0930	−	19.0930i	0.851316	−	0.851316i	0.990295	+	0.138980i	$0.0443823\pi$
$504$		0			0
$505$	16.0563	+	22.0634i	0.714498	+	0.981807i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−26.6927			−1.18313			−0.591566	−	0.806257i	$-0.701490\pi$
$509$	−26.6927			−1.18313			−0.591566	+	0.806257i	$0.701490\pi$
$510$		0			0
$511$	4.96623			0.219693
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−18.5138	−	2.91747i	−0.815816	−	0.128559i
$516$		0			0
$517$	−8.52155	+	8.52155i	−0.374777	+	0.374777i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			25.5958i			1.12137i	0.828029	+	0.560686i	$0.189462\pi$
$521$			25.5958i			1.12137i	−0.828029	+	0.560686i	$0.810538\pi$
$522$		0			0
$523$	−20.1731	−	20.1731i	−0.882109	−	0.882109i	0.111640	−	0.993749i	$-0.464390\pi$
$523$	−20.1731	−	20.1731i	−0.882109	−	0.882109i	−0.993749	+	0.111640i	$0.964390\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.06788	−	1.06788i	−0.0465177	−	0.0465177i
$528$		0			0
$529$			22.0519i			0.958778i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.98315	+	3.98315i	−0.172529	+	0.172529i
$534$		0			0
$535$	−5.10278	−	0.804114i	−0.220612	−	0.0347649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.5896			0.499201
$540$		0			0
$541$	22.9905			0.988439			0.494219	−	0.869337i	$-0.335454\pi$
$541$	22.9905			0.988439			0.494219	+	0.869337i	$0.335454\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20.5791	+	28.2782i	0.881510	+	1.21130i
$546$		0			0
$547$	−22.1104	+	22.1104i	−0.945372	+	0.945372i	−0.998583	−	0.0532109i	$-0.983054\pi$
$547$	−22.1104	+	22.1104i	−0.945372	+	0.945372i	0.0532109	+	0.998583i	$0.483054\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			42.0295i			1.79052i
$552$		0			0
$553$	3.77129	+	3.77129i	0.160372	+	0.160372i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.02215	+	3.02215i	0.128053	+	0.128053i	0.768228	−	0.640176i	$-0.221138\pi$
$557$	3.02215	+	3.02215i	0.128053	+	0.128053i	−0.640176	+	0.768228i	$0.721138\pi$
$558$		0			0
$559$		−	8.15105i		−	0.344753i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.11590	−	1.11590i	0.0470295	−	0.0470295i	−0.683201	−	0.730230i	$-0.739413\pi$
$563$	1.11590	−	1.11590i	0.0470295	−	0.0470295i	0.730230	+	0.683201i	$0.239413\pi$
$564$		0			0
$565$	−4.63005	+	29.3816i	−0.194788	+	1.23609i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.7682			−0.954494			−0.477247	−	0.878769i	$-0.658365\pi$
$569$	−22.7682			−0.954494			−0.477247	+	0.878769i	$0.658365\pi$
$570$		0			0
$571$	9.81292			0.410658			0.205329	−	0.978693i	$-0.434174\pi$
$571$	9.81292			0.410658			0.205329	+	0.978693i	$0.434174\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.21703	+	4.33442i	0.0924567	+	0.180758i
$576$		0			0
$577$	6.19318	−	6.19318i	0.257825	−	0.257825i	−0.566344	−	0.824169i	$-0.691643\pi$
$577$	6.19318	−	6.19318i	0.257825	−	0.257825i	0.824169	+	0.566344i	$0.191643\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			4.55074i			0.188796i
$582$		0			0
$583$	−13.7933	−	13.7933i	−0.571261	−	0.571261i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.8904	+	20.8904i	0.862237	+	0.862237i	0.991598	−	0.129360i	$-0.0412924\pi$
$587$	20.8904	+	20.8904i	0.862237	+	0.862237i	−0.129360	+	0.991598i	$0.541292\pi$
$588$		0			0
$589$			2.03616i			0.0838984i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.5358	−	12.5358i	0.514784	−	0.514784i	−0.401204	−	0.915989i	$-0.631408\pi$
$593$	12.5358	−	12.5358i	0.514784	−	0.514784i	0.915989	+	0.401204i	$0.131408\pi$
$594$		0			0
$595$	−8.07519	+	5.87662i	−0.331051	+	0.240918i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27.7828			1.13517			0.567586	−	0.823314i	$-0.307877\pi$
$599$	27.7828			1.13517			0.567586	+	0.823314i	$0.307877\pi$
$600$		0			0
$601$	−5.32627			−0.217263			−0.108631	−	0.994082i	$-0.534647\pi$
$601$	−5.32627			−0.217263			−0.108631	+	0.994082i	$0.534647\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.7185	−	9.98345i	0.557736	−	0.405885i
$606$		0			0
$607$	17.0913	−	17.0913i	0.693713	−	0.693713i	−0.269334	−	0.963047i	$-0.586804\pi$
$607$	17.0913	−	17.0913i	0.693713	−	0.693713i	0.963047	+	0.269334i	$0.0868036\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			6.52394i			0.263930i
$612$		0			0
$613$	−14.9039	−	14.9039i	−0.601964	−	0.601964i	0.338869	−	0.940833i	$-0.389956\pi$
$613$	−14.9039	−	14.9039i	−0.601964	−	0.601964i	−0.940833	+	0.338869i	$0.889956\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.84901	+	9.84901i	0.396506	+	0.396506i	0.876999	−	0.480493i	$-0.159542\pi$
$617$	9.84901	+	9.84901i	0.396506	+	0.396506i	−0.480493	+	0.876999i	$0.659542\pi$
$618$		0			0
$619$			27.5240i			1.10628i	0.833087	+	0.553142i	$0.186571\pi$
$619$			27.5240i			1.10628i	−0.833087	+	0.553142i	$0.813429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.73786	−	5.73786i	0.229883	−	0.229883i
$624$		0			0
$625$	−20.2713	−	14.6313i	−0.810850	−	0.585254i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	41.2398			1.64434
$630$		0			0
$631$	24.6061			0.979552			0.489776	−	0.871848i	$-0.337078\pi$
$631$	24.6061			0.979552			0.489776	+	0.871848i	$0.337078\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.16867	−	39.1454i	0.244796	−	1.55344i
$636$		0			0
$637$	4.43641	−	4.43641i	0.175777	−	0.175777i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	22.4148i		−	0.885333i	−0.896686	−	0.442666i	$-0.854033\pi$
$641$		−	22.4148i		−	0.885333i	0.896686	−	0.442666i	$-0.145967\pi$
$642$		0			0
$643$	−10.9106	−	10.9106i	−0.430272	−	0.430272i	0.458449	−	0.888721i	$-0.348405\pi$
$643$	−10.9106	−	10.9106i	−0.430272	−	0.430272i	−0.888721	+	0.458449i	$0.848405\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.2179	+	11.2179i	0.441021	+	0.441021i	0.892355	−	0.451334i	$-0.149052\pi$
$647$	11.2179	+	11.2179i	0.441021	+	0.441021i	−0.451334	+	0.892355i	$0.649052\pi$
$648$		0			0
$649$			16.7719i			0.658353i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.60922	−	2.60922i	0.102107	−	0.102107i	−0.654208	−	0.756315i	$-0.726998\pi$
$653$	2.60922	−	2.60922i	0.102107	−	0.102107i	0.756315	+	0.654208i	$0.226998\pi$
$654$		0			0
$655$	23.4154	+	32.1757i	0.914916	+	1.25721i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−19.2611			−0.750307			−0.375153	−	0.926963i	$-0.622410\pi$
$659$	−19.2611			−0.750307			−0.375153	+	0.926963i	$0.622410\pi$
$660$		0			0
$661$	23.7674			0.924443			0.462222	−	0.886764i	$-0.347052\pi$
$661$	23.7674			0.924443			0.462222	+	0.886764i	$0.347052\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	13.3011	+	2.09604i	0.515795	+	0.0812809i
$666$		0			0
$667$	4.09444	−	4.09444i	0.158537	−	0.158537i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	16.7138i		−	0.645229i
$672$		0			0
$673$	−5.05692	−	5.05692i	−0.194930	−	0.194930i	0.602892	−	0.797822i	$-0.294015\pi$
$673$	−5.05692	−	5.05692i	−0.194930	−	0.194930i	−0.797822	+	0.602892i	$0.794015\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.2619	+	11.2619i	0.432828	+	0.432828i	0.889589	−	0.456761i	$-0.150991\pi$
$677$	11.2619	+	11.2619i	0.432828	+	0.432828i	−0.456761	+	0.889589i	$0.650991\pi$
$678$		0			0
$679$			9.40594i			0.360967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.0195	+	10.0195i	−0.383387	+	0.383387i	−0.872321	−	0.488934i	$-0.837386\pi$
$683$	−10.0195	+	10.0195i	−0.383387	+	0.383387i	0.488934	+	0.872321i	$0.337386\pi$
$684$		0			0
$685$	−10.5502	−	1.66253i	−0.403101	−	0.0635221i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.5599			−0.402301
$690$		0			0
$691$	−38.7442			−1.47390			−0.736950	−	0.675947i	$-0.763735\pi$
$691$	−38.7442			−1.47390			−0.736950	+	0.675947i	$0.763735\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−22.9549	−	31.5428i	−0.870729	−	1.19649i
$696$		0			0
$697$	20.8797	−	20.8797i	0.790875	−	0.790875i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	25.1981i		−	0.951718i	−0.879522	−	0.475859i	$-0.842137\pi$
$701$		−	25.1981i		−	0.951718i	0.879522	−	0.475859i	$-0.157863\pi$
$702$		0			0
$703$	−39.3164	−	39.3164i	−1.48285	−	1.48285i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.35225	−	7.35225i	−0.276510	−	0.276510i
$708$		0			0
$709$			0.867835i			0.0325922i	0.999867	+	0.0162961i	$0.00518744\pi$
$709$			0.867835i			0.0325922i	−0.999867	+	0.0162961i	$0.994813\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.198359	−	0.198359i	0.00742860	−	0.00742860i
$714$		0			0
$715$	−0.642973	+	4.08021i	−0.0240458	+	0.152591i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.6503			0.844712			0.422356	−	0.906430i	$-0.361203\pi$
$719$	22.6503			0.844712			0.422356	+	0.906430i	$0.361203\pi$
$720$		0			0
$721$	7.14162			0.265968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.14411	+	28.2931i	−0.339604	+	1.05078i
$726$		0			0
$727$	1.04875	−	1.04875i	0.0388959	−	0.0388959i	−0.687391	−	0.726287i	$-0.741244\pi$
$727$	1.04875	−	1.04875i	0.0388959	−	0.0388959i	0.726287	+	0.687391i	$0.241244\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			42.7279i			1.58035i
$732$		0			0
$733$	24.0927	+	24.0927i	0.889883	+	0.889883i	0.994511	−	0.104628i	$-0.0333653\pi$
$733$	24.0927	+	24.0927i	0.889883	+	0.889883i	−0.104628	+	0.994511i	$0.533365\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.61431	−	1.61431i	−0.0594638	−	0.0594638i
$738$		0			0
$739$		−	31.6768i		−	1.16525i	−0.812741	−	0.582625i	$-0.802026\pi$
$739$		−	31.6768i		−	1.16525i	0.812741	−	0.582625i	$-0.197974\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20.5692	−	20.5692i	0.754610	−	0.754610i	−0.220726	−	0.975336i	$-0.570843\pi$
$743$	20.5692	−	20.5692i	0.754610	−	0.754610i	0.975336	+	0.220726i	$0.0708427\pi$
$744$		0			0
$745$	16.4120	−	11.9436i	0.601288	−	0.437579i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.96837			0.0719228
$750$		0			0
$751$	−46.4299			−1.69425			−0.847125	−	0.531394i	$-0.821668\pi$
$751$	−46.4299			−1.69425			−0.847125	+	0.531394i	$0.821668\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.0184	−	12.3849i	0.619364	−	0.450734i
$756$		0			0
$757$	−6.96008	+	6.96008i	−0.252968	+	0.252968i	−0.822187	−	0.569218i	$-0.807246\pi$
$757$	−6.96008	+	6.96008i	−0.252968	+	0.252968i	0.569218	+	0.822187i	$0.307246\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22.8968i			0.830007i	0.909820	+	0.415003i	$0.136220\pi$
$761$			22.8968i			0.830007i	−0.909820	+	0.415003i	$0.863780\pi$
$762$		0			0
$763$	−9.42322	−	9.42322i	−0.341143	−	0.341143i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.42011	+	6.42011i	0.231817	+	0.231817i
$768$		0			0
$769$			3.62371i			0.130674i	0.997863	+	0.0653372i	$0.0208123\pi$
$769$			3.62371i			0.130674i	−0.997863	+	0.0653372i	$0.979188\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.1093	−	28.1093i	1.01102	−	1.01102i	0.0110835	−	0.999939i	$-0.496472\pi$
$773$	28.1093	−	28.1093i	1.01102	−	1.01102i	0.999939	−	0.0110835i	$-0.00352805\pi$
$774$		0			0
$775$	−0.442995	+	1.37068i	−0.0159128	+	0.0492364i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−39.8118			−1.42641
$780$		0			0
$781$	−12.8680			−0.460452
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.124070	+	0.787327i	−0.00442824	+	0.0281009i
$786$		0			0
$787$	15.6625	−	15.6625i	0.558308	−	0.558308i	−0.370518	−	0.928825i	$-0.620820\pi$
$787$	15.6625	−	15.6625i	0.558308	−	0.558308i	0.928825	+	0.370518i	$0.120820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	11.3338i		−	0.402983i
$792$		0			0
$793$	−6.39789	−	6.39789i	−0.227196	−	0.227196i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.74981	−	2.74981i	−0.0974034	−	0.0974034i	0.656726	−	0.754129i	$-0.271941\pi$
$797$	−2.74981	−	2.74981i	−0.0974034	−	0.0974034i	−0.754129	+	0.656726i	$0.771941\pi$
$798$		0			0
$799$		−	34.1985i		−	1.20986i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.61336	+	7.61336i	−0.268669	+	0.268669i
$804$		0			0
$805$	−1.09158	−	1.49997i	−0.0384732	−	0.0528668i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.24173			0.254606			0.127303	−	0.991864i	$-0.459368\pi$
$809$	7.24173			0.254606			0.127303	+	0.991864i	$0.459368\pi$
$810$		0			0
$811$	−30.3509			−1.06576			−0.532882	−	0.846190i	$-0.678891\pi$
$811$	−30.3509			−1.06576			−0.532882	+	0.846190i	$0.678891\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−29.8357	−	4.70162i	−1.04510	−	0.164691i
$816$		0			0
$817$	40.7351	−	40.7351i	1.42514	−	1.42514i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			23.0636i			0.804927i	0.915436	+	0.402463i	$0.131846\pi$
$821$			23.0636i			0.804927i	−0.915436	+	0.402463i	$0.868154\pi$
$822$		0			0
$823$	35.4963	+	35.4963i	1.23732	+	1.23732i	0.961091	+	0.276231i	$0.0890853\pi$
$823$	35.4963	+	35.4963i	1.23732	+	1.23732i	0.276231	+	0.961091i	$0.410915\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.8095	+	29.8095i	1.03658	+	1.03658i	0.999305	+	0.0372711i	$0.0118665\pi$
$827$	29.8095	+	29.8095i	1.03658	+	1.03658i	0.0372711	+	0.999305i	$0.488133\pi$
$828$		0			0
$829$			22.4727i			0.780509i	0.920707	+	0.390255i	$0.127613\pi$
$829$			22.4727i			0.780509i	−0.920707	+	0.390255i	$0.872387\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23.2557	+	23.2557i	−0.805762	+	0.805762i
$834$		0			0
$835$	−30.7135	−	4.83995i	−1.06289	−	0.167493i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.4975			−0.914797			−0.457398	−	0.889262i	$-0.651219\pi$
$839$	−26.4975			−0.914797			−0.457398	+	0.889262i	$0.651219\pi$
$840$		0			0
$841$	6.36448			0.219465
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.31574	+	1.80799i	0.0452629	+	0.0621967i
$846$		0			0
$847$	−4.57145	+	4.57145i	−0.157077	+	0.157077i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			7.66028i			0.262591i
$852$		0			0
$853$	34.6170	+	34.6170i	1.18526	+	1.18526i	0.978362	+	0.206901i	$0.0663378\pi$
$853$	34.6170	+	34.6170i	1.18526	+	1.18526i	0.206901	+	0.978362i	$0.433662\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.42393	−	3.42393i	−0.116959	−	0.116959i	0.646205	−	0.763164i	$-0.276355\pi$
$857$	−3.42393	−	3.42393i	−0.116959	−	0.116959i	−0.763164	+	0.646205i	$0.776355\pi$
$858$		0			0
$859$			34.0583i			1.16205i	0.813885	+	0.581026i	$0.197349\pi$
$859$			34.0583i			1.16205i	−0.813885	+	0.581026i	$0.802651\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.89592	−	6.89592i	0.234740	−	0.234740i	−0.579928	−	0.814668i	$-0.696919\pi$
$863$	6.89592	−	6.89592i	0.234740	−	0.234740i	0.814668	+	0.579928i	$0.196919\pi$
$864$		0			0
$865$	2.83559	−	17.9942i	0.0964129	−	0.611821i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.5630			−0.392247
$870$		0			0
$871$	−1.23589			−0.0418764
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.49791	+	4.30484i	0.287282	+	0.145530i
$876$		0			0
$877$	17.4440	−	17.4440i	0.589042	−	0.589042i	−0.348330	−	0.937372i	$-0.613251\pi$
$877$	17.4440	−	17.4440i	0.589042	−	0.589042i	0.937372	+	0.348330i	$0.113251\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	37.4924i		−	1.26315i	−0.775314	−	0.631576i	$-0.782408\pi$
$881$		−	37.4924i		−	1.26315i	0.775314	−	0.631576i	$-0.217592\pi$
$882$		0			0
$883$	28.4948	+	28.4948i	0.958925	+	0.958925i	0.999189	−	0.0402637i	$-0.0128198\pi$
$883$	28.4948	+	28.4948i	0.958925	+	0.958925i	−0.0402637	+	0.999189i	$0.512820\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.7368	−	24.7368i	−0.830579	−	0.830579i	0.157017	−	0.987596i	$-0.449812\pi$
$887$	−24.7368	−	24.7368i	−0.830579	−	0.830579i	−0.987596	+	0.157017i	$0.949812\pi$
$888$		0			0
$889$			15.1001i			0.506442i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−32.6036	+	32.6036i	−1.09104	+	1.09104i
$894$		0			0
$895$	9.86618	−	7.17999i	0.329790	−	0.240001i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.71326			0.0571406
$900$		0			0
$901$	55.3552			1.84415
$902$		0			0
$903$		0			0
$904$		0			0
$905$	26.6242	−	19.3754i	0.885018	−	0.644060i
$906$		0			0
$907$	−36.5118	+	36.5118i	−1.21235	+	1.21235i	−0.242103	+	0.970251i	$0.577837\pi$
$907$	−36.5118	+	36.5118i	−1.21235	+	1.21235i	−0.970251	+	0.242103i	$0.922163\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	25.1238i		−	0.832389i	−0.909276	−	0.416195i	$-0.863364\pi$
$911$		−	25.1238i		−	0.832389i	0.909276	−	0.416195i	$-0.136636\pi$
$912$		0			0
$913$	−6.97640	−	6.97640i	−0.230885	−	0.230885i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.7220	−	10.7220i	−0.354072	−	0.354072i
$918$		0			0
$919$			13.0767i			0.431360i	0.976464	+	0.215680i	$0.0691968\pi$
$919$			13.0767i			0.431360i	−0.976464	+	0.215680i	$0.930803\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.92574	+	4.92574i	−0.162133	+	0.162133i
$924$		0			0
$925$	−17.9128	−	35.0205i	−0.588971	−	1.15147i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−41.7052			−1.36830			−0.684152	−	0.729340i	$-0.739827\pi$
$929$	−41.7052			−1.36830			−0.684152	+	0.729340i	$0.739827\pi$
$930$		0			0
$931$	44.3422			1.45326
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.37047	−	21.3885i	0.110226	−	0.699478i
$936$		0			0
$937$	−6.66046	+	6.66046i	−0.217588	+	0.217588i	−0.807481	−	0.589893i	$-0.799170\pi$
$937$	−6.66046	+	6.66046i	−0.217588	+	0.217588i	0.589893	+	0.807481i	$0.299170\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	20.7498i		−	0.676423i	−0.941070	−	0.338212i	$-0.890178\pi$
$941$		−	20.7498i		−	0.676423i	0.941070	−	0.338212i	$-0.109822\pi$
$942$		0			0
$943$	3.87840	+	3.87840i	0.126298	+	0.126298i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.2802	+	18.2802i	0.594025	+	0.594025i	0.938716	−	0.344691i	$-0.112016\pi$
$947$	18.2802	+	18.2802i	0.594025	+	0.594025i	−0.344691	+	0.938716i	$0.612016\pi$
$948$		0			0
$949$			5.82865i			0.189206i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.5264	−	14.5264i	0.470556	−	0.470556i	−0.431538	−	0.902095i	$-0.642029\pi$
$953$	14.5264	−	14.5264i	0.470556	−	0.470556i	0.902095	+	0.431538i	$0.142029\pi$
$954$		0			0
$955$	−9.10319	−	12.5089i	−0.294572	−	0.404778i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.06967			0.131417
$960$		0			0
$961$	−30.9170			−0.997323
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.9351	+	2.66869i	0.545160	+	0.0859082i
$966$		0			0
$967$	27.8492	−	27.8492i	0.895571	−	0.895571i	−0.0994693	−	0.995041i	$-0.531714\pi$
$967$	27.8492	−	27.8492i	0.895571	−	0.895571i	0.995041	+	0.0994693i	$0.0317145\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	19.5289i		−	0.626713i	−0.949636	−	0.313356i	$-0.898547\pi$
$971$		−	19.5289i		−	0.626713i	0.949636	−	0.313356i	$-0.101453\pi$
$972$		0			0
$973$	10.5111	+	10.5111i	0.336971	+	0.336971i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.7523	+	24.7523i	0.791897	+	0.791897i	0.981802	−	0.189905i	$-0.0608181\pi$
$977$	24.7523	+	24.7523i	0.791897	+	0.791897i	−0.189905	+	0.981802i	$0.560818\pi$
$978$		0			0
$979$			17.5926i			0.562261i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.1796	−	17.1796i	0.547943	−	0.547943i	−0.377902	−	0.925845i	$-0.623355\pi$
$983$	17.1796	−	17.1796i	0.547943	−	0.547943i	0.925845	+	0.377902i	$0.123355\pi$
$984$		0			0
$985$	−10.2184	−	1.61025i	−0.325586	−	0.0513070i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.93670			−0.252372
$990$		0			0
$991$	45.8404			1.45617			0.728085	−	0.685487i	$-0.240411\pi$
$991$	45.8404			1.45617			0.728085	+	0.685487i	$0.240411\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.69437	+	13.3212i	0.307332	+	0.422312i
$996$		0			0
$997$	−7.26377	+	7.26377i	−0.230046	+	0.230046i	−0.812712	−	0.582666i	$-0.802010\pi$
$997$	−7.26377	+	7.26377i	−0.230046	+	0.230046i	0.582666	+	0.812712i	$0.302010\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.y.b.53.9	yes	24
3.2	odd	2		2340.2.y.a.53.4	✓	24
5.2	odd	4		2340.2.y.a.1457.4	yes	24
15.2	even	4	inner	2340.2.y.b.1457.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.4	✓	24	3.2	odd	2
2340.2.y.a.1457.4	yes	24	5.2	odd	4
2340.2.y.b.53.9	yes	24	1.1	even	1	trivial
2340.2.y.b.1457.9	yes	24	15.2	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.80799 - 1.31574i) q^{5} +(-0.602482 + 0.602482i) q^{7} -1.84724i q^{11} +(-0.707107 - 0.707107i) q^{13} +(3.70666 + 3.70666i) q^{17} -7.06758i q^{19} +(-0.688512 + 0.688512i) q^{23} +(1.53765 - 4.75769i) q^{25} -5.94680 q^{29} -0.288098 q^{31} +(-0.296571 + 1.88199i) q^{35} +(5.56293 - 5.56293i) q^{37} -5.63302i q^{41} +(5.76366 + 5.76366i) q^{43} +(-4.61312 - 4.61312i) q^{47} +6.27403i q^{49} +(7.46699 - 7.46699i) q^{53} +(-2.43049 - 3.33979i) q^{55} -9.07941 q^{59} +9.04798 q^{61} +(-2.20881 - 0.348072i) q^{65} +(0.873903 - 0.873903i) q^{67} -6.96605i q^{71} +(-4.12147 - 4.12147i) q^{73} +(1.11293 + 1.11293i) q^{77} -6.25959i q^{79} +(3.77666 - 3.77666i) q^{83} +(11.5786 + 1.82460i) q^{85} -9.52370 q^{89} +0.852038 q^{91} +(-9.29910 - 12.7781i) q^{95} +(7.80599 - 7.80599i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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