LMFDB - Embedded newform 2340.2.y.b.53.7

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			53.7
Character	$\chi$	$=$	2340.53
Dual form			2340.2.y.b.1457.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.17829 - 1.90043i) q^{5} +(-0.865311 + 0.865311i) q^{7} +3.06508i q^{11} +(0.707107 + 0.707107i) q^{13} +(3.38200 + 3.38200i) q^{17} -1.24957i q^{19} +(3.64321 - 3.64321i) q^{23} +(-2.22328 - 4.47851i) q^{25} -5.13854 q^{29} +8.10633 q^{31} +(0.624880 + 2.66405i) q^{35} +(-2.71262 + 2.71262i) q^{37} +12.0573i q^{41} +(-2.22668 - 2.22668i) q^{43} +(5.86992 + 5.86992i) q^{47} +5.50247i q^{49} +(-7.51155 + 7.51155i) q^{53} +(5.82497 + 3.61154i) q^{55} +8.12784 q^{59} +7.13179 q^{61} +(2.17698 - 0.510633i) q^{65} +(7.19877 - 7.19877i) q^{67} -15.5275i q^{71} +(4.63049 + 4.63049i) q^{73} +(-2.65224 - 2.65224i) q^{77} -11.2727i q^{79} +(-0.0469907 + 0.0469907i) q^{83} +(10.4122 - 2.44229i) q^{85} -9.53727 q^{89} -1.22373 q^{91} +(-2.37473 - 1.47235i) q^{95} +(9.95328 - 9.95328i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.17829	−	1.90043i	0.526946	−	0.849899i
$5$	1.17829	−	1.90043i	0.526946	−	0.849899i
$6$		0			0
$7$	−0.865311	+	0.865311i	−0.327057	+	0.327057i	−0.851466	−	0.524409i	$-0.824286\pi$
$7$	−0.865311	+	0.865311i	−0.327057	+	0.327057i	0.524409	+	0.851466i	$0.324286\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.06508i			0.924155i	0.886839	+	0.462078i	$0.152896\pi$
$11$			3.06508i			0.924155i	−0.886839	+	0.462078i	$0.847104\pi$
$12$		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
$17$	3.38200	+	3.38200i	0.820255	+	0.820255i	0.986144	−	0.165889i	$-0.0530494\pi$
$17$	3.38200	+	3.38200i	0.820255	+	0.820255i	−0.165889	+	0.986144i	$0.553049\pi$
$18$		0			0
$19$		−	1.24957i		−	0.286671i	−0.989674	−	0.143336i	$-0.954217\pi$
$19$		−	1.24957i		−	0.286671i	0.989674	−	0.143336i	$-0.0457829\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.64321	−	3.64321i	0.759661	−	0.759661i	−0.216600	−	0.976261i	$-0.569497\pi$
$23$	3.64321	−	3.64321i	0.759661	−	0.759661i	0.976261	+	0.216600i	$0.0694966\pi$
$24$		0			0
$25$	−2.22328	−	4.47851i	−0.444656	−	0.895701i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.13854			−0.954203			−0.477101	−	0.878848i	$-0.658313\pi$
$29$	−5.13854			−0.954203			−0.477101	+	0.878848i	$0.658313\pi$
$30$		0			0
$31$	8.10633			1.45594			0.727970	−	0.685609i	$-0.240464\pi$
$31$	8.10633			1.45594			0.727970	+	0.685609i	$0.240464\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.624880	+	2.66405i	0.105624	+	0.450306i
$36$		0			0
$37$	−2.71262	+	2.71262i	−0.445952	+	0.445952i	−0.894006	−	0.448054i	$-0.852117\pi$
$37$	−2.71262	+	2.71262i	−0.445952	+	0.445952i	0.448054	+	0.894006i	$0.352117\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			12.0573i			1.88303i	0.336967	+	0.941516i	$0.390599\pi$
$41$			12.0573i			1.88303i	−0.336967	+	0.941516i	$0.609401\pi$
$42$		0			0
$43$	−2.22668	−	2.22668i	−0.339565	−	0.339565i	0.516638	−	0.856204i	$-0.327183\pi$
$43$	−2.22668	−	2.22668i	−0.339565	−	0.339565i	−0.856204	+	0.516638i	$0.827183\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.86992	+	5.86992i	0.856216	+	0.856216i	0.990890	−	0.134674i	$-0.0429986\pi$
$47$	5.86992	+	5.86992i	0.856216	+	0.856216i	−0.134674	+	0.990890i	$0.542999\pi$
$48$		0			0
$49$			5.50247i			0.786068i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.51155	+	7.51155i	−1.03179	+	1.03179i	−0.0323131	+	0.999478i	$0.510287\pi$
$53$	−7.51155	+	7.51155i	−1.03179	+	1.03179i	−0.999478	+	0.0323131i	$0.989713\pi$
$54$		0			0
$55$	5.82497	+	3.61154i	0.785438	+	0.486980i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.12784			1.05815			0.529077	−	0.848574i	$-0.322538\pi$
$59$	8.12784			1.05815			0.529077	+	0.848574i	$0.322538\pi$
$60$		0			0
$61$	7.13179			0.913132			0.456566	−	0.889690i	$-0.349079\pi$
$61$	7.13179			0.913132			0.456566	+	0.889690i	$0.349079\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.17698	−	0.510633i	0.270021	−	0.0633363i
$66$		0			0
$67$	7.19877	−	7.19877i	0.879469	−	0.879469i	−0.114010	−	0.993480i	$-0.536370\pi$
$67$	7.19877	−	7.19877i	0.879469	−	0.879469i	0.993480	+	0.114010i	$0.0363696\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	15.5275i		−	1.84278i	−0.388642	−	0.921389i	$-0.627056\pi$
$71$		−	15.5275i		−	1.84278i	0.388642	−	0.921389i	$-0.372944\pi$
$72$		0			0
$73$	4.63049	+	4.63049i	0.541958	+	0.541958i	0.924103	−	0.382145i	$-0.124814\pi$
$73$	4.63049	+	4.63049i	0.541958	+	0.541958i	−0.382145	+	0.924103i	$0.624814\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.65224	−	2.65224i	−0.302251	−	0.302251i
$78$		0			0
$79$		−	11.2727i		−	1.26828i	−0.773218	−	0.634141i	$-0.781354\pi$
$79$		−	11.2727i		−	1.26828i	0.773218	−	0.634141i	$-0.218646\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.0469907	+	0.0469907i	−0.00515790	+	0.00515790i	−0.709681	−	0.704523i	$-0.751161\pi$
$83$	−0.0469907	+	0.0469907i	−0.00515790	+	0.00515790i	0.704523	+	0.709681i	$0.251161\pi$
$84$		0			0
$85$	10.4122	−	2.44229i	1.12936	−	0.264904i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.53727			−1.01095			−0.505474	−	0.862842i	$-0.668682\pi$
$89$	−9.53727			−1.01095			−0.505474	+	0.862842i	$0.668682\pi$
$90$		0			0
$91$	−1.22373			−0.128282
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.37473	−	1.47235i	−0.243642	−	0.151060i
$96$		0			0
$97$	9.95328	−	9.95328i	1.01060	−	1.01060i	0.0106593	−	0.999943i	$-0.496607\pi$
$97$	9.95328	−	9.95328i	1.01060	−	1.01060i	0.999943	−	0.0106593i	$-0.00339303\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.85216i			0.781320i	0.920535	+	0.390660i	$0.127753\pi$
$101$			7.85216i			0.781320i	−0.920535	+	0.390660i	$0.872247\pi$
$102$		0			0
$103$	13.2313	+	13.2313i	1.30371	+	1.30371i	0.925868	+	0.377846i	$0.123335\pi$
$103$	13.2313	+	13.2313i	1.30371	+	1.30371i	0.377846	+	0.925868i	$0.376665\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.73518	+	8.73518i	0.844462	+	0.844462i	0.989436	−	0.144974i	$-0.0463097\pi$
$107$	8.73518	+	8.73518i	0.844462	+	0.844462i	−0.144974	+	0.989436i	$0.546310\pi$
$108$		0			0
$109$		−	12.6193i		−	1.20871i	−0.796715	−	0.604355i	$-0.793431\pi$
$109$		−	12.6193i		−	1.20871i	0.796715	−	0.604355i	$-0.206569\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.62243	−	1.62243i	0.152625	−	0.152625i	−0.626664	−	0.779289i	$-0.715580\pi$
$113$	1.62243	−	1.62243i	0.152625	−	0.152625i	0.779289	+	0.626664i	$0.215580\pi$
$114$		0			0
$115$	−2.63092	−	11.2164i	−0.245335	−	1.04594i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.85296			−0.536540
$120$		0			0
$121$	1.60531			0.145937
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1308	−	1.05177i	−0.995565	−	0.0940735i
$126$		0			0
$127$	13.5510	−	13.5510i	1.20246	−	1.20246i	0.229043	−	0.973416i	$-0.426440\pi$
$127$	13.5510	−	13.5510i	1.20246	−	1.20246i	0.973416	−	0.229043i	$-0.0735597\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			11.5447i			1.00867i	0.863508	+	0.504335i	$0.168262\pi$
$131$			11.5447i			1.00867i	−0.863508	+	0.504335i	$0.831738\pi$
$132$		0			0
$133$	1.08127	+	1.08127i	0.0937579	+	0.0937579i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.03574	+	2.03574i	0.173925	+	0.173925i	0.788701	−	0.614776i	$-0.210754\pi$
$137$	2.03574	+	2.03574i	0.173925	+	0.173925i	−0.614776	+	0.788701i	$0.710754\pi$
$138$		0			0
$139$			4.84984i			0.411358i	0.978619	+	0.205679i	$0.0659403\pi$
$139$			4.84984i			0.411358i	−0.978619	+	0.205679i	$0.934060\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.16734	+	2.16734i	−0.181242	+	0.181242i
$144$		0			0
$145$	−6.05467	+	9.76544i	−0.502813	+	0.810976i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.89860			−0.565156			−0.282578	−	0.959244i	$-0.591190\pi$
$149$	−6.89860			−0.565156			−0.282578	+	0.959244i	$0.591190\pi$
$150$		0			0
$151$	−8.82917			−0.718508			−0.359254	−	0.933240i	$-0.616969\pi$
$151$	−8.82917			−0.718508			−0.359254	+	0.933240i	$0.616969\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.55158	−	15.4055i	0.767201	−	1.23740i
$156$		0			0
$157$	−10.5659	+	10.5659i	−0.843247	+	0.843247i	−0.989280	−	0.146033i	$-0.953349\pi$
$157$	−10.5659	+	10.5659i	−0.843247	+	0.843247i	0.146033	+	0.989280i	$0.453349\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			6.30501i			0.496905i
$162$		0			0
$163$	0.975758	+	0.975758i	0.0764273	+	0.0764273i	0.744287	−	0.667860i	$-0.232789\pi$
$163$	0.975758	+	0.975758i	0.0764273	+	0.0764273i	−0.667860	+	0.744287i	$0.732789\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.6837	+	11.6837i	0.904112	+	0.904112i	0.995789	−	0.0916766i	$-0.0292226\pi$
$167$	11.6837	+	11.6837i	0.904112	+	0.904112i	−0.0916766	+	0.995789i	$0.529223\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.0509	−	18.0509i	1.37238	−	1.37238i	0.515481	−	0.856901i	$-0.327613\pi$
$173$	18.0509	−	18.0509i	1.37238	−	1.37238i	0.856901	−	0.515481i	$-0.172387\pi$
$174$		0			0
$175$	5.79913	+	1.95147i	0.438373	+	0.147517i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.9797			−0.820660			−0.410330	−	0.911937i	$-0.634586\pi$
$179$	−10.9797			−0.820660			−0.410330	+	0.911937i	$0.634586\pi$
$180$		0			0
$181$	19.3924			1.44143			0.720713	−	0.693233i	$-0.243814\pi$
$181$	19.3924			1.44143			0.720713	+	0.693233i	$0.243814\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.95890	+	8.35140i	0.144022	+	0.614007i
$186$		0			0
$187$	−10.3661	+	10.3661i	−0.758043	+	0.758043i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			7.27678i			0.526530i	0.964724	+	0.263265i	$0.0847993\pi$
$191$			7.27678i			0.526530i	−0.964724	+	0.263265i	$0.915201\pi$
$192$		0			0
$193$	−9.86068	−	9.86068i	−0.709787	−	0.709787i	0.256703	−	0.966490i	$-0.417364\pi$
$193$	−9.86068	−	9.86068i	−0.709787	−	0.709787i	−0.966490	+	0.256703i	$0.917364\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.64779	+	6.64779i	0.473635	+	0.473635i	0.903089	−	0.429454i	$-0.141294\pi$
$197$	6.64779	+	6.64779i	0.473635	+	0.473635i	−0.429454	+	0.903089i	$0.641294\pi$
$198$		0			0
$199$		−	20.6855i		−	1.46635i	−0.680038	−	0.733177i	$-0.738037\pi$
$199$		−	20.6855i		−	1.46635i	0.680038	−	0.733177i	$-0.261963\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.44643	−	4.44643i	0.312078	−	0.312078i
$204$		0			0
$205$	22.9141	+	14.2069i	1.60039	+	0.992256i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.83003			0.264929
$210$		0			0
$211$	−11.4271			−0.786676			−0.393338	−	0.919394i	$-0.628680\pi$
$211$	−11.4271			−0.786676			−0.393338	+	0.919394i	$0.628680\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.85532	+	1.60798i	−0.467529	+	0.109664i
$216$		0			0
$217$	−7.01449	+	7.01449i	−0.476175	+	0.476175i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.78287i			0.321731i
$222$		0			0
$223$	−6.47685	−	6.47685i	−0.433722	−	0.433722i	0.456170	−	0.889892i	$-0.349221\pi$
$223$	−6.47685	−	6.47685i	−0.433722	−	0.433722i	−0.889892	+	0.456170i	$0.849221\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.03155	+	6.03155i	0.400328	+	0.400328i	0.878349	−	0.478020i	$-0.158646\pi$
$227$	6.03155	+	6.03155i	0.400328	+	0.400328i	−0.478020	+	0.878349i	$0.658646\pi$
$228$		0			0
$229$		−	12.0174i		−	0.794131i	−0.917790	−	0.397065i	$-0.870029\pi$
$229$		−	12.0174i		−	0.794131i	0.917790	−	0.397065i	$-0.129971\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.45100	+	1.45100i	−0.0950581	+	0.0950581i	−0.753037	−	0.657979i	$-0.771412\pi$
$233$	−1.45100	+	1.45100i	−0.0950581	+	0.0950581i	0.657979	+	0.753037i	$0.271412\pi$
$234$		0			0
$235$	18.0718	−	4.23893i	1.17888	−	0.276518i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.36939			0.541371			0.270686	−	0.962668i	$-0.412750\pi$
$239$	8.36939			0.541371			0.270686	+	0.962668i	$0.412750\pi$
$240$		0			0
$241$	−16.8303			−1.08413			−0.542067	−	0.840335i	$-0.682358\pi$
$241$	−16.8303			−1.08413			−0.542067	+	0.840335i	$0.682358\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.4571	+	6.48349i	0.668078	+	0.414215i
$246$		0			0
$247$	0.883581	−	0.883581i	0.0562209	−	0.0562209i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			9.70452i			0.612544i	0.951944	+	0.306272i	$0.0990816\pi$
$251$			9.70452i			0.612544i	−0.951944	+	0.306272i	$0.900918\pi$
$252$		0			0
$253$	11.1667	+	11.1667i	0.702045	+	0.702045i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.61723	−	5.61723i	−0.350393	−	0.350393i	0.509863	−	0.860256i	$-0.329696\pi$
$257$	−5.61723	−	5.61723i	−0.350393	−	0.350393i	−0.860256	+	0.509863i	$0.829696\pi$
$258$		0			0
$259$		−	4.69452i		−	0.291703i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.97154	+	9.97154i	−0.614871	+	0.614871i	−0.944211	−	0.329340i	$-0.893174\pi$
$263$	−9.97154	+	9.97154i	−0.614871	+	0.614871i	0.329340	+	0.944211i	$0.393174\pi$
$264$		0			0
$265$	5.42443	+	23.1260i	0.333220	+	1.42062i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−16.5322			−1.00798			−0.503992	−	0.863708i	$-0.668136\pi$
$269$	−16.5322			−1.00798			−0.503992	+	0.863708i	$0.668136\pi$
$270$		0			0
$271$	−26.9455			−1.63682			−0.818412	−	0.574632i	$-0.805145\pi$
$271$	−26.9455			−1.63682			−0.818412	+	0.574632i	$0.805145\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.7270	−	6.81452i	0.827767	−	0.410931i
$276$		0			0
$277$	−17.2129	+	17.2129i	−1.03422	+	1.03422i	−0.0348267	+	0.999393i	$0.511088\pi$
$277$	−17.2129	+	17.2129i	−1.03422	+	1.03422i	−0.999393	+	0.0348267i	$0.988912\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6.53372i		−	0.389769i	−0.980826	−	0.194884i	$-0.937567\pi$
$281$		−	6.53372i		−	0.389769i	0.980826	−	0.194884i	$-0.0624332\pi$
$282$		0			0
$283$	0.859435	+	0.859435i	0.0510881	+	0.0510881i	0.732189	−	0.681101i	$-0.238499\pi$
$283$	0.859435	+	0.859435i	0.0510881	+	0.0510881i	−0.681101	+	0.732189i	$0.738499\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.4333	−	10.4333i	−0.615859	−	0.615859i
$288$		0			0
$289$			5.87583i			0.345637i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.51928	−	1.51928i	0.0887574	−	0.0887574i	−0.661334	−	0.750091i	$-0.730009\pi$
$293$	1.51928	−	1.51928i	0.0887574	−	0.0887574i	0.750091	+	0.661334i	$0.230009\pi$
$294$		0			0
$295$	9.57692	−	15.4464i	0.557590	−	0.899324i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.15227			0.297964
$300$		0			0
$301$	3.85354			0.222114
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.40329	−	13.5535i	0.481171	−	0.776070i
$306$		0			0
$307$	−13.0908	+	13.0908i	−0.747133	+	0.747133i	−0.973940	−	0.226807i	$-0.927171\pi$
$307$	−13.0908	+	13.0908i	−0.747133	+	0.747133i	0.226807	+	0.973940i	$0.427171\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.08663i			0.118322i	0.998248	+	0.0591611i	$0.0188426\pi$
$311$			2.08663i			0.118322i	−0.998248	+	0.0591611i	$0.981157\pi$
$312$		0			0
$313$	12.1702	+	12.1702i	0.687899	+	0.687899i	0.961767	−	0.273868i	$-0.0883032\pi$
$313$	12.1702	+	12.1702i	0.687899	+	0.687899i	−0.273868	+	0.961767i	$0.588303\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.54760	+	5.54760i	0.311584	+	0.311584i	0.845523	−	0.533939i	$-0.179289\pi$
$317$	5.54760	+	5.54760i	0.311584	+	0.311584i	−0.533939	+	0.845523i	$0.679289\pi$
$318$		0			0
$319$		−	15.7500i		−	0.881831i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.22605	−	4.22605i	0.235144	−	0.235144i
$324$		0			0
$325$	1.59469	−	4.73888i	0.0884573	−	0.262866i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.1586			−0.560063
$330$		0			0
$331$	−15.0141			−0.825250			−0.412625	−	0.910901i	$-0.635388\pi$
$331$	−15.0141			−0.825250			−0.412625	+	0.910901i	$0.635388\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.19855	−	22.1630i	−0.284027	−	1.21089i
$336$		0			0
$337$	−0.165005	+	0.165005i	−0.00898839	+	0.00898839i	−0.711587	−	0.702598i	$-0.752023\pi$
$337$	−0.165005	+	0.165005i	−0.00898839	+	0.00898839i	0.702598	+	0.711587i	$0.252023\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			24.8465i			1.34551i
$342$		0			0
$343$	−10.8185	−	10.8185i	−0.584146	−	0.584146i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.7898	−	10.7898i	−0.579227	−	0.579227i	0.355463	−	0.934690i	$-0.384323\pi$
$347$	−10.7898	−	10.7898i	−0.579227	−	0.579227i	−0.934690	+	0.355463i	$0.884323\pi$
$348$		0			0
$349$			3.76246i			0.201400i	0.994917	+	0.100700i	$0.0321082\pi$
$349$			3.76246i			0.201400i	−0.994917	+	0.100700i	$0.967892\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.2688	+	18.2688i	−0.972352	+	0.972352i	−0.999628	−	0.0272758i	$-0.991317\pi$
$353$	−18.2688	+	18.2688i	−0.972352	+	0.972352i	0.0272758	+	0.999628i	$0.491317\pi$
$354$		0			0
$355$	−29.5090	−	18.2959i	−1.56617	−	0.971044i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.49074			0.131456			0.0657281	−	0.997838i	$-0.479063\pi$
$359$	2.49074			0.131456			0.0657281	+	0.997838i	$0.479063\pi$
$360$		0			0
$361$	17.4386			0.917819
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.2560	−	3.34388i	0.746192	−	0.175027i
$366$		0			0
$367$	5.73353	−	5.73353i	0.299288	−	0.299288i	−0.541447	−	0.840735i	$-0.682123\pi$
$367$	5.73353	−	5.73353i	0.299288	−	0.299288i	0.840735	+	0.541447i	$0.182123\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	12.9997i		−	0.674908i
$372$		0			0
$373$	15.1506	+	15.1506i	0.784468	+	0.784468i	0.980581	−	0.196114i	$-0.0628321\pi$
$373$	15.1506	+	15.1506i	0.784468	+	0.784468i	−0.196114	+	0.980581i	$0.562832\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.63350	−	3.63350i	−0.187135	−	0.187135i
$378$		0			0
$379$			24.9721i			1.28273i	0.767236	+	0.641364i	$0.221631\pi$
$379$			24.9721i			1.28273i	−0.767236	+	0.641364i	$0.778369\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.3288	−	21.3288i	1.08985	−	1.08985i	0.0943091	−	0.995543i	$-0.469936\pi$
$383$	21.3288	−	21.3288i	1.08985	−	1.08985i	0.995543	−	0.0943091i	$-0.0300642\pi$
$384$		0			0
$385$	−8.16551	+	1.91530i	−0.416153	+	0.0976129i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.5799			−0.942039			−0.471019	−	0.882123i	$-0.656114\pi$
$389$	−18.5799			−0.942039			−0.471019	+	0.882123i	$0.656114\pi$
$390$		0			0
$391$	24.6426			1.24623
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−21.4231	−	13.2825i	−1.07791	−	0.668316i
$396$		0			0
$397$	2.38571	−	2.38571i	0.119735	−	0.119735i	−0.644700	−	0.764435i	$-0.723018\pi$
$397$	2.38571	−	2.38571i	0.119735	−	0.119735i	0.764435	+	0.644700i	$0.223018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	28.9113i		−	1.44376i	−0.692018	−	0.721881i	$-0.743278\pi$
$401$		−	28.9113i		−	1.44376i	0.692018	−	0.721881i	$-0.256722\pi$
$402$		0			0
$403$	5.73204	+	5.73204i	0.285533	+	0.285533i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.31439	−	8.31439i	−0.412129	−	0.412129i
$408$		0			0
$409$		−	20.1947i		−	0.998564i	−0.866440	−	0.499282i	$-0.833597\pi$
$409$		−	20.1947i		−	0.998564i	0.866440	−	0.499282i	$-0.166403\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.03311	+	7.03311i	−0.346076	+	0.346076i
$414$		0			0
$415$	0.0339341	+	0.144671i	0.00166576	+	0.00710163i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.4859			−0.805387			−0.402694	−	0.915335i	$-0.631926\pi$
$419$	−16.4859			−0.805387			−0.402694	+	0.915335i	$0.631926\pi$
$420$		0			0
$421$	−26.9618			−1.31404			−0.657018	−	0.753875i	$-0.728183\pi$
$421$	−26.9618			−1.31404			−0.657018	+	0.753875i	$0.728183\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.62717	−	22.6654i	0.369972	−	1.09944i
$426$		0			0
$427$	−6.17121	+	6.17121i	−0.298646	+	0.298646i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			28.8674i			1.39049i	0.718771	+	0.695247i	$0.244705\pi$
$431$			28.8674i			1.39049i	−0.718771	+	0.695247i	$0.755295\pi$
$432$		0			0
$433$	−14.6619	−	14.6619i	−0.704607	−	0.704607i	0.260789	−	0.965396i	$-0.416017\pi$
$433$	−14.6619	−	14.6619i	−0.704607	−	0.704607i	−0.965396	+	0.260789i	$0.916017\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.55245	−	4.55245i	−0.217773	−	0.217773i
$438$		0			0
$439$		−	6.42120i		−	0.306467i	−0.988190	−	0.153234i	$-0.951031\pi$
$439$		−	6.42120i		−	0.306467i	0.988190	−	0.153234i	$-0.0489687\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.1994	−	16.1994i	0.769656	−	0.769656i	−0.208390	−	0.978046i	$-0.566822\pi$
$443$	16.1994	−	16.1994i	0.769656	−	0.769656i	0.978046	+	0.208390i	$0.0668223\pi$
$444$		0			0
$445$	−11.2376	+	18.1249i	−0.532715	+	0.859204i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−39.1796			−1.84900			−0.924500	−	0.381181i	$-0.875518\pi$
$449$	−39.1796			−1.84900			−0.924500	+	0.381181i	$0.875518\pi$
$450$		0			0
$451$	−36.9565			−1.74021
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.44191	+	2.32562i	−0.0675978	+	0.109027i
$456$		0			0
$457$	−7.85447	+	7.85447i	−0.367416	+	0.367416i	−0.866534	−	0.499118i	$-0.833658\pi$
$457$	−7.85447	+	7.85447i	−0.367416	+	0.367416i	0.499118	+	0.866534i	$0.333658\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	30.4284i		−	1.41719i	−0.705614	−	0.708596i	$-0.749329\pi$
$461$		−	30.4284i		−	1.41719i	0.705614	−	0.708596i	$-0.250671\pi$
$462$		0			0
$463$	−11.0289	−	11.0289i	−0.512555	−	0.512555i	0.402753	−	0.915309i	$-0.368053\pi$
$463$	−11.0289	−	11.0289i	−0.512555	−	0.512555i	−0.915309	+	0.402753i	$0.868053\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.50756	+	4.50756i	0.208585	+	0.208585i	0.803666	−	0.595081i	$-0.202880\pi$
$467$	4.50756	+	4.50756i	0.208585	+	0.208585i	−0.595081	+	0.803666i	$0.702880\pi$
$468$		0			0
$469$			12.4583i			0.575273i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.82494	−	6.82494i	0.313811	−	0.313811i
$474$		0			0
$475$	−5.59622	+	2.77815i	−0.256772	+	0.127470i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.78468			0.264309			0.132154	−	0.991229i	$-0.457811\pi$
$479$	5.78468			0.264309			0.132154	+	0.991229i	$0.457811\pi$
$480$		0			0
$481$	−3.83623			−0.174917
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.18771	−	30.6433i	−0.326377	−	1.39144i
$486$		0			0
$487$	−14.4622	+	14.4622i	−0.655346	+	0.655346i	−0.954275	−	0.298929i	$-0.903371\pi$
$487$	−14.4622	+	14.4622i	−0.655346	+	0.655346i	0.298929	+	0.954275i	$0.403371\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			19.1212i			0.862928i	0.902130	+	0.431464i	$0.142003\pi$
$491$			19.1212i			0.862928i	−0.902130	+	0.431464i	$0.857997\pi$
$492$		0			0
$493$	−17.3785	−	17.3785i	−0.782690	−	0.782690i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.4361	+	13.4361i	0.602693	+	0.602693i
$498$		0			0
$499$		−	14.0808i		−	0.630343i	−0.949035	−	0.315172i	$-0.897938\pi$
$499$		−	14.0808i		−	0.630343i	0.949035	−	0.315172i	$-0.102062\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.78932	−	2.78932i	0.124370	−	0.124370i	−0.642182	−	0.766552i	$-0.721971\pi$
$503$	2.78932	−	2.78932i	0.124370	−	0.124370i	0.766552	+	0.642182i	$0.221971\pi$
$504$		0			0
$505$	14.9225	+	9.25210i	0.664043	+	0.411713i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−17.6597			−0.782755			−0.391377	−	0.920230i	$-0.628001\pi$
$509$	−17.6597			−0.782755			−0.391377	+	0.920230i	$0.628001\pi$
$510$		0			0
$511$	−8.01363			−0.354502
$512$		0			0
$513$		0			0
$514$		0			0
$515$	40.7353	−	9.55488i	1.79501	−	0.421038i
$516$		0			0
$517$	−17.9918	+	17.9918i	−0.791277	+	0.791277i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	3.27382i		−	0.143429i	−0.997425	−	0.0717144i	$-0.977153\pi$
$521$		−	3.27382i		−	0.143429i	0.997425	−	0.0717144i	$-0.0228470\pi$
$522$		0			0
$523$	4.65202	+	4.65202i	0.203419	+	0.203419i	0.801463	−	0.598044i	$-0.204055\pi$
$523$	4.65202	+	4.65202i	0.203419	+	0.203419i	−0.598044	+	0.801463i	$0.704055\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	27.4156	+	27.4156i	1.19424	+	1.19424i
$528$		0			0
$529$		−	3.54590i		−	0.154170i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.52579	+	8.52579i	−0.369293	+	0.369293i
$534$		0			0
$535$	26.8932	−	6.30807i	1.16269	−	0.272721i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.8655			−0.726449
$540$		0			0
$541$	−26.4084			−1.13539			−0.567693	−	0.823241i	$-0.692164\pi$
$541$	−26.4084			−1.13539			−0.567693	+	0.823241i	$0.692164\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−23.9821	−	14.8692i	−1.02728	−	0.636925i
$546$		0			0
$547$	−17.1741	+	17.1741i	−0.734311	+	0.734311i	−0.971471	−	0.237159i	$-0.923784\pi$
$547$	−17.1741	+	17.1741i	−0.734311	+	0.734311i	0.237159	+	0.971471i	$0.423784\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			6.42097i			0.273543i
$552$		0			0
$553$	9.75442	+	9.75442i	0.414800	+	0.414800i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.83852	−	9.83852i	−0.416871	−	0.416871i	0.467253	−	0.884124i	$-0.345244\pi$
$557$	−9.83852	−	9.83852i	−0.416871	−	0.416871i	−0.884124	+	0.467253i	$0.845244\pi$
$558$		0			0
$559$		−	3.14900i		−	0.133188i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.5333	−	21.5333i	0.907519	−	0.907519i	−0.0885528	−	0.996071i	$-0.528224\pi$
$563$	21.5333	−	21.5333i	0.907519	−	0.907519i	0.996071	+	0.0885528i	$0.0282242\pi$
$564$		0			0
$565$	−1.17163	−	4.99499i	−0.0492907	−	0.210141i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.8521			0.664554			0.332277	−	0.943182i	$-0.392183\pi$
$569$	15.8521			0.664554			0.332277	+	0.943182i	$0.392183\pi$
$570$		0			0
$571$	−32.8273			−1.37378			−0.686890	−	0.726761i	$-0.741025\pi$
$571$	−32.8273			−1.37378			−0.686890	+	0.726761i	$0.741025\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.4160	−	8.21626i	−1.01822	−	0.342642i
$576$		0			0
$577$	19.9730	−	19.9730i	0.831487	−	0.831487i	−0.156233	−	0.987720i	$-0.549935\pi$
$577$	19.9730	−	19.9730i	0.831487	−	0.831487i	0.987720	+	0.156233i	$0.0499352\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.0813232i		−	0.00337385i
$582$		0			0
$583$	−23.0235	−	23.0235i	−0.953535	−	0.953535i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.5045	+	16.5045i	0.681212	+	0.681212i	0.960273	−	0.279061i	$-0.0900232\pi$
$587$	16.5045	+	16.5045i	0.681212	+	0.681212i	−0.279061	+	0.960273i	$0.590023\pi$
$588$		0			0
$589$		−	10.1294i		−	0.417376i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.2909	+	15.2909i	−0.627923	+	0.627923i	−0.947545	−	0.319622i	$-0.896444\pi$
$593$	−15.2909	+	15.2909i	−0.627923	+	0.627923i	0.319622	+	0.947545i	$0.396444\pi$
$594$		0			0
$595$	−6.89647	+	11.1232i	−0.282728	+	0.456005i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−38.0039			−1.55280			−0.776399	−	0.630242i	$-0.782956\pi$
$599$	−38.0039			−1.55280			−0.776399	+	0.630242i	$0.782956\pi$
$600$		0			0
$601$	17.5200			0.714657			0.357328	−	0.933979i	$-0.383688\pi$
$601$	17.5200			0.714657			0.357328	+	0.933979i	$0.383688\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.89151	−	3.05078i	0.0769010	−	0.124032i
$606$		0			0
$607$	23.1121	−	23.1121i	0.938091	−	0.938091i	−0.0601011	−	0.998192i	$-0.519142\pi$
$607$	23.1121	−	23.1121i	0.938091	−	0.938091i	0.998192	+	0.0601011i	$0.0191423\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			8.30133i			0.335836i
$612$		0			0
$613$	28.0956	+	28.0956i	1.13477	+	1.13477i	0.989373	+	0.145399i	$0.0464467\pi$
$613$	28.0956	+	28.0956i	1.13477	+	1.13477i	0.145399	+	0.989373i	$0.453553\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.98532	−	1.98532i	−0.0799258	−	0.0799258i	0.666014	−	0.745939i	$-0.267999\pi$
$617$	−1.98532	−	1.98532i	−0.0799258	−	0.0799258i	−0.745939	+	0.666014i	$0.767999\pi$
$618$		0			0
$619$		−	26.8920i		−	1.08088i	−0.841382	−	0.540441i	$-0.818258\pi$
$619$		−	26.8920i		−	1.08088i	0.841382	−	0.540441i	$-0.181742\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.25270	−	8.25270i	0.330638	−	0.330638i
$624$		0			0
$625$	−15.1141	+	19.9140i	−0.604562	+	0.796558i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.3482			−0.731589
$630$		0			0
$631$	−5.75241			−0.229000			−0.114500	−	0.993423i	$-0.536527\pi$
$631$	−5.75241			−0.229000			−0.114500	+	0.993423i	$0.536527\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.78580	−	41.7198i	−0.388338	−	1.65560i
$636$		0			0
$637$	−3.89084	+	3.89084i	−0.154161	+	0.154161i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	14.4478i		−	0.570653i	−0.958430	−	0.285326i	$-0.907898\pi$
$641$		−	14.4478i		−	0.570653i	0.958430	−	0.285326i	$-0.0921020\pi$
$642$		0			0
$643$	−27.3776	−	27.3776i	−1.07967	−	1.07967i	−0.996539	−	0.0831304i	$-0.973508\pi$
$643$	−27.3776	−	27.3776i	−1.07967	−	1.07967i	−0.0831304	−	0.996539i	$-0.526492\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.6926	−	20.6926i	−0.813511	−	0.813511i	0.171647	−	0.985158i	$-0.445091\pi$
$647$	−20.6926	−	20.6926i	−0.813511	−	0.813511i	−0.985158	+	0.171647i	$0.945091\pi$
$648$		0			0
$649$			24.9124i			0.977899i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	30.8427	−	30.8427i	1.20697	−	1.20697i	0.234962	−	0.972005i	$-0.424504\pi$
$653$	30.8427	−	30.8427i	1.20697	−	1.20697i	0.972005	−	0.234962i	$-0.0754965\pi$
$654$		0			0
$655$	21.9400	+	13.6030i	0.857267	+	0.531514i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.84765			0.149883			0.0749416	−	0.997188i	$-0.476123\pi$
$659$	3.84765			0.149883			0.0749416	+	0.997188i	$0.476123\pi$
$660$		0			0
$661$	−42.5150			−1.65364			−0.826822	−	0.562464i	$-0.809853\pi$
$661$	−42.5150			−1.65364			−0.826822	+	0.562464i	$0.809853\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.32892	−	0.780832i	0.129090	−	0.0302794i
$666$		0			0
$667$	−18.7208	+	18.7208i	−0.724871	+	0.724871i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			21.8595i			0.843875i
$672$		0			0
$673$	28.1298	+	28.1298i	1.08432	+	1.08432i	0.996101	+	0.0882236i	$0.0281190\pi$
$673$	28.1298	+	28.1298i	1.08432	+	1.08432i	0.0882236	+	0.996101i	$0.471881\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.8645	+	22.8645i	0.878754	+	0.878754i	0.993406	−	0.114652i	$-0.0365753\pi$
$677$	22.8645	+	22.8645i	0.878754	+	0.878754i	−0.114652	+	0.993406i	$0.536575\pi$
$678$		0			0
$679$			17.2254i			0.661049i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.8197	−	10.8197i	0.414004	−	0.414004i	−0.469127	−	0.883131i	$-0.655431\pi$
$683$	10.8197	−	10.8197i	0.414004	−	0.414004i	0.883131	+	0.469127i	$0.155431\pi$
$684$		0			0
$685$	6.26747	−	1.47010i	0.239468	−	0.0561696i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.6229			−0.404702
$690$		0			0
$691$	16.6128			0.631982			0.315991	−	0.948762i	$-0.397663\pi$
$691$	16.6128			0.631982			0.315991	+	0.948762i	$0.397663\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.21679	+	5.71451i	0.349613	+	0.216764i
$696$		0			0
$697$	−40.7778	+	40.7778i	−1.54457	+	1.54457i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			15.5585i			0.587636i	0.955861	+	0.293818i	$0.0949260\pi$
$701$			15.5585i			0.587636i	−0.955861	+	0.293818i	$0.905074\pi$
$702$		0			0
$703$	3.38962	+	3.38962i	0.127842	+	0.127842i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.79456	−	6.79456i	−0.255536	−	0.255536i
$708$		0			0
$709$			28.6522i			1.07606i	0.842927	+	0.538029i	$0.180831\pi$
$709$			28.6522i			1.07606i	−0.842927	+	0.538029i	$0.819169\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	29.5330	−	29.5330i	1.10602	−	1.10602i
$714$		0			0
$715$	1.56513	+	6.67262i	0.0585326	+	0.249542i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	33.1174			1.23507			0.617536	−	0.786543i	$-0.288131\pi$
$719$	33.1174			1.23507			0.617536	+	0.786543i	$0.288131\pi$
$720$		0			0
$721$	−22.8983			−0.852777
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.4244	+	23.0130i	0.424292	+	0.854681i
$726$		0			0
$727$	6.79544	−	6.79544i	0.252029	−	0.252029i	−0.569773	−	0.821802i	$-0.692969\pi$
$727$	6.79544	−	6.79544i	0.252029	−	0.252029i	0.821802	+	0.569773i	$0.192969\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	15.0612i		−	0.557060i
$732$		0			0
$733$	−9.37355	−	9.37355i	−0.346220	−	0.346220i	0.512480	−	0.858699i	$-0.328727\pi$
$733$	−9.37355	−	9.37355i	−0.346220	−	0.346220i	−0.858699	+	0.512480i	$0.828727\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	22.0648	+	22.0648i	0.812766	+	0.812766i
$738$		0			0
$739$		−	25.2311i		−	0.928140i	−0.885799	−	0.464070i	$-0.846389\pi$
$739$		−	25.2311i		−	0.928140i	0.885799	−	0.464070i	$-0.153611\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.13825	+	4.13825i	−0.151818	+	0.151818i	−0.778929	−	0.627112i	$-0.784237\pi$
$743$	−4.13825	+	4.13825i	−0.151818	+	0.151818i	0.627112	+	0.778929i	$0.284237\pi$
$744$		0			0
$745$	−8.12853	+	13.1103i	−0.297806	+	0.480325i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−15.1173			−0.552374
$750$		0			0
$751$	41.4041			1.51086			0.755429	−	0.655230i	$-0.227428\pi$
$751$	41.4041			1.51086			0.755429	+	0.655230i	$0.227428\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.4033	+	16.7792i	−0.378615	+	0.610659i
$756$		0			0
$757$	10.4525	−	10.4525i	0.379902	−	0.379902i	−0.491165	−	0.871067i	$-0.663429\pi$
$757$	10.4525	−	10.4525i	0.379902	−	0.379902i	0.871067	+	0.491165i	$0.163429\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			14.3313i			0.519509i	0.965675	+	0.259755i	$0.0836417\pi$
$761$			14.3313i			0.519509i	−0.965675	+	0.259755i	$0.916358\pi$
$762$		0			0
$763$	10.9196	+	10.9196i	0.395317	+	0.395317i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.74725	+	5.74725i	0.207521	+	0.207521i
$768$		0			0
$769$		−	17.8845i		−	0.644932i	−0.946581	−	0.322466i	$-0.895488\pi$
$769$		−	17.8845i		−	0.644932i	0.946581	−	0.322466i	$-0.104512\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.8838	+	26.8838i	−0.966943	+	0.966943i	−0.999471	−	0.0325275i	$-0.989644\pi$
$773$	−26.8838	+	26.8838i	−0.966943	+	0.966943i	0.0325275	+	0.999471i	$0.489644\pi$
$774$		0			0
$775$	−18.0226	−	36.3042i	−0.647392	−	1.30409i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15.0665			0.539812
$780$		0			0
$781$	47.5930			1.70301
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.63007	+	32.5293i	0.272329	+	1.16102i
$786$		0			0
$787$	−19.0346	+	19.0346i	−0.678509	+	0.678509i	−0.959663	−	0.281154i	$-0.909283\pi$
$787$	−19.0346	+	19.0346i	−0.678509	+	0.678509i	0.281154	+	0.959663i	$0.409283\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.80781i			0.0998341i
$792$		0			0
$793$	5.04293	+	5.04293i	0.179080	+	0.179080i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.12038	+	9.12038i	0.323060	+	0.323060i	0.849940	−	0.526880i	$-0.176638\pi$
$797$	9.12038	+	9.12038i	0.323060	+	0.323060i	−0.526880	+	0.849940i	$0.676638\pi$
$798$		0			0
$799$			39.7042i			1.40463i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.1928	+	14.1928i	−0.500853	+	0.500853i
$804$		0			0
$805$	11.9822	+	7.42911i	0.422319	+	0.261842i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.3868			−1.38477			−0.692383	−	0.721530i	$-0.743439\pi$
$809$	−39.3868			−1.38477			−0.692383	+	0.721530i	$0.743439\pi$
$810$		0			0
$811$	−32.6054			−1.14493			−0.572465	−	0.819929i	$-0.694013\pi$
$811$	−32.6054			−1.14493			−0.572465	+	0.819929i	$0.694013\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.00409	−	0.704639i	0.105229	−	0.0246824i
$816$		0			0
$817$	−2.78240	+	2.78240i	−0.0973437	+	0.0973437i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26.9109i		−	0.939197i	−0.882880	−	0.469599i	$-0.844399\pi$
$821$		−	26.9109i		−	0.939197i	0.882880	−	0.469599i	$-0.155601\pi$
$822$		0			0
$823$	−22.7575	−	22.7575i	−0.793278	−	0.793278i	0.188747	−	0.982026i	$-0.439557\pi$
$823$	−22.7575	−	22.7575i	−0.793278	−	0.793278i	−0.982026	+	0.188747i	$0.939557\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.6647	−	18.6647i	−0.649036	−	0.649036i	0.303724	−	0.952760i	$-0.401770\pi$
$827$	−18.6647	−	18.6647i	−0.649036	−	0.649036i	−0.952760	+	0.303724i	$0.901770\pi$
$828$		0			0
$829$		−	15.3470i		−	0.533022i	−0.963832	−	0.266511i	$-0.914129\pi$
$829$		−	15.3470i		−	0.533022i	0.963832	−	0.266511i	$-0.0858709\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.6094	+	18.6094i	−0.644776	+	0.644776i
$834$		0			0
$835$	35.9708	−	8.43733i	1.24482	−	0.291986i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.71723			0.231905			0.115952	−	0.993255i	$-0.463008\pi$
$839$	6.71723			0.231905			0.115952	+	0.993255i	$0.463008\pi$
$840$		0			0
$841$	−2.59542			−0.0894974
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.90043	+	1.17829i	0.0653768	+	0.0405343i
$846$		0			0
$847$	−1.38909	+	1.38909i	−0.0477297	+	0.0477297i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			19.7653i			0.677545i
$852$		0			0
$853$	−25.8954	−	25.8954i	−0.886641	−	0.886641i	0.107558	−	0.994199i	$-0.465697\pi$
$853$	−25.8954	−	25.8954i	−0.886641	−	0.886641i	−0.994199	+	0.107558i	$0.965697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.7282	+	15.7282i	0.537265	+	0.537265i	0.922725	−	0.385460i	$-0.125957\pi$
$857$	15.7282	+	15.7282i	0.537265	+	0.537265i	−0.385460	+	0.922725i	$0.625957\pi$
$858$		0			0
$859$		−	50.0491i		−	1.70765i	−0.520557	−	0.853827i	$-0.674276\pi$
$859$		−	50.0491i		−	1.70765i	0.520557	−	0.853827i	$-0.325724\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.6038	−	15.6038i	0.531158	−	0.531158i	−0.389759	−	0.920917i	$-0.627442\pi$
$863$	15.6038	−	15.6038i	0.531158	−	0.531158i	0.920917	+	0.389759i	$0.127442\pi$
$864$		0			0
$865$	−13.0353	−	55.5735i	−0.443215	−	1.88956i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	34.5518			1.17209
$870$		0			0
$871$	10.1806			0.344956
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.5417	−	8.72146i	0.356374	−	0.294839i
$876$		0			0
$877$	14.4770	−	14.4770i	0.488854	−	0.488854i	−0.419090	−	0.907944i	$-0.637651\pi$
$877$	14.4770	−	14.4770i	0.488854	−	0.488854i	0.907944	+	0.419090i	$0.137651\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			40.8517i			1.37633i	0.725554	+	0.688165i	$0.241583\pi$
$881$			40.8517i			1.37633i	−0.725554	+	0.688165i	$0.758417\pi$
$882$		0			0
$883$	14.6319	+	14.6319i	0.492401	+	0.492401i	0.909062	−	0.416661i	$-0.136800\pi$
$883$	14.6319	+	14.6319i	0.492401	+	0.492401i	−0.416661	+	0.909062i	$0.636800\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.7234	+	24.7234i	0.830131	+	0.830131i	0.987534	−	0.157403i	$-0.0503123\pi$
$887$	24.7234	+	24.7234i	0.830131	+	0.830131i	−0.157403	+	0.987534i	$0.550312\pi$
$888$		0			0
$889$			23.4517i			0.786545i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.33489	−	7.33489i	0.245453	−	0.245453i
$894$		0			0
$895$	−12.9372	+	20.8661i	−0.432443	+	0.697478i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−41.6547			−1.38926
$900$		0			0
$901$	−50.8081			−1.69266
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.8498	−	36.8540i	0.759554	−	1.22507i
$906$		0			0
$907$	2.46686	−	2.46686i	0.0819109	−	0.0819109i	−0.664964	−	0.746875i	$-0.731553\pi$
$907$	2.46686	−	2.46686i	0.0819109	−	0.0819109i	0.746875	+	0.664964i	$0.231553\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			6.35356i			0.210503i	0.994446	+	0.105251i	$0.0335647\pi$
$911$			6.35356i			0.210503i	−0.994446	+	0.105251i	$0.966435\pi$
$912$		0			0
$913$	−0.144030	−	0.144030i	−0.00476670	−	0.00476670i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.98980	−	9.98980i	−0.329892	−	0.329892i
$918$		0			0
$919$			12.4829i			0.411772i	0.978576	+	0.205886i	$0.0660076\pi$
$919$			12.4829i			0.411772i	−0.978576	+	0.205886i	$0.933992\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.9796	−	10.9796i	0.361398	−	0.361398i
$924$		0			0
$925$	18.1794	+	6.11758i	0.597735	+	0.201145i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−36.9331			−1.21174			−0.605868	−	0.795565i	$-0.707174\pi$
$929$	−36.9331			−1.21174			−0.605868	+	0.795565i	$0.707174\pi$
$930$		0			0
$931$	6.87574			0.225343
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.48581	+	31.9143i	0.244812	+	1.04371i
$936$		0			0
$937$	10.0896	−	10.0896i	0.329613	−	0.329613i	−0.522826	−	0.852439i	$-0.675122\pi$
$937$	10.0896	−	10.0896i	0.329613	−	0.329613i	0.852439	+	0.522826i	$0.175122\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	11.6037i		−	0.378269i	−0.981951	−	0.189134i	$-0.939432\pi$
$941$		−	11.6037i		−	0.378269i	0.981951	−	0.189134i	$-0.0605682\pi$
$942$		0			0
$943$	43.9272	+	43.9272i	1.43047	+	1.43047i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.1508	−	34.1508i	−1.10975	−	1.10975i	−0.993182	−	0.116571i	$-0.962810\pi$
$947$	−34.1508	−	34.1508i	−1.10975	−	1.10975i	−0.116571	−	0.993182i	$-0.537190\pi$
$948$		0			0
$949$			6.54850i			0.212573i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13.2582	+	13.2582i	−0.429475	+	0.429475i	−0.888449	−	0.458974i	$-0.848217\pi$
$953$	−13.2582	+	13.2582i	−0.429475	+	0.429475i	0.458974	+	0.888449i	$0.348217\pi$
$954$		0			0
$955$	13.8290	+	8.57414i	0.447497	+	0.277453i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.52309			−0.113767
$960$		0			0
$961$	34.7125			1.11976
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−30.3583	+	7.12084i	−0.977267	+	0.229228i
$966$		0			0
$967$	11.4930	−	11.4930i	0.369589	−	0.369589i	−0.497738	−	0.867327i	$-0.665836\pi$
$967$	11.4930	−	11.4930i	0.369589	−	0.369589i	0.867327	+	0.497738i	$0.165836\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	19.7690i		−	0.634419i	−0.948356	−	0.317209i	$-0.897254\pi$
$971$		−	19.7690i		−	0.634419i	0.948356	−	0.317209i	$-0.102746\pi$
$972$		0			0
$973$	−4.19662	−	4.19662i	−0.134538	−	0.134538i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.7581	−	13.7581i	−0.440159	−	0.440159i	0.451906	−	0.892065i	$-0.350744\pi$
$977$	−13.7581	−	13.7581i	−0.440159	−	0.440159i	−0.892065	+	0.451906i	$0.850744\pi$
$978$		0			0
$979$		−	29.2325i		−	0.934273i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.4513	−	28.4513i	0.907457	−	0.907457i	−0.0886096	−	0.996066i	$-0.528242\pi$
$983$	28.4513	−	28.4513i	0.907457	−	0.907457i	0.996066	+	0.0886096i	$0.0282423\pi$
$984$		0			0
$985$	20.4667	−	4.80066i	0.652122	−	0.152962i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.2245			−0.515909
$990$		0			0
$991$	19.0657			0.605641			0.302820	−	0.953048i	$-0.402072\pi$
$991$	19.0657			0.605641			0.302820	+	0.953048i	$0.402072\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−39.3113	−	24.3734i	−1.24625	−	0.772689i
$996$		0			0
$997$	−11.5009	+	11.5009i	−0.364238	+	0.364238i	−0.865371	−	0.501133i	$-0.832917\pi$
$997$	−11.5009	+	11.5009i	−0.364238	+	0.364238i	0.501133	+	0.865371i	$0.332917\pi$
$998$		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.53.7	yes	24
3.2	odd	2		2340.2.y.a.53.6	✓	24
5.2	odd	4		2340.2.y.a.1457.6	yes	24
15.2	even	4	inner	2340.2.y.b.1457.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.6	✓	24	3.2	odd	2
2340.2.y.a.1457.6	yes	24	5.2	odd	4
2340.2.y.b.53.7	yes	24	1.1	even	1	trivial
2340.2.y.b.1457.7	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)

\(f(q)\)	\(=\)	\(q+(1.17829 - 1.90043i) q^{5} +(-0.865311 + 0.865311i) q^{7} +3.06508i q^{11} +(0.707107 + 0.707107i) q^{13} +(3.38200 + 3.38200i) q^{17} -1.24957i q^{19} +(3.64321 - 3.64321i) q^{23} +(-2.22328 - 4.47851i) q^{25} -5.13854 q^{29} +8.10633 q^{31} +(0.624880 + 2.66405i) q^{35} +(-2.71262 + 2.71262i) q^{37} +12.0573i q^{41} +(-2.22668 - 2.22668i) q^{43} +(5.86992 + 5.86992i) q^{47} +5.50247i q^{49} +(-7.51155 + 7.51155i) q^{53} +(5.82497 + 3.61154i) q^{55} +8.12784 q^{59} +7.13179 q^{61} +(2.17698 - 0.510633i) q^{65} +(7.19877 - 7.19877i) q^{67} -15.5275i q^{71} +(4.63049 + 4.63049i) q^{73} +(-2.65224 - 2.65224i) q^{77} -11.2727i q^{79} +(-0.0469907 + 0.0469907i) q^{83} +(10.4122 - 2.44229i) q^{85} -9.53727 q^{89} -1.22373 q^{91} +(-2.37473 - 1.47235i) q^{95} +(9.95328 - 9.95328i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 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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