LMFDB - Embedded newform 2340.2.y.b.53.8

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			53.8
Character	$\chi$	$=$	2340.53
Dual form			2340.2.y.b.1457.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.25978 + 1.84742i) q^{5} +(-1.73394 + 1.73394i) q^{7} +0.384834i q^{11} +(-0.707107 - 0.707107i) q^{13} +(-4.01077 - 4.01077i) q^{17} +1.31687i q^{19} +(-3.09168 + 3.09168i) q^{23} +(-1.82588 + 4.65469i) q^{25} -5.58151 q^{29} -4.45438 q^{31} +(-5.38771 - 1.01892i) q^{35} +(7.95020 - 7.95020i) q^{37} +7.37309i q^{41} +(-6.63893 - 6.63893i) q^{43} +(9.03757 + 9.03757i) q^{47} +0.986880i q^{49} +(4.78645 - 4.78645i) q^{53} +(-0.710948 + 0.484808i) q^{55} -8.10867 q^{59} -15.0948 q^{61} +(0.415517 - 2.19712i) q^{65} +(5.47501 - 5.47501i) q^{67} +1.23609i q^{71} +(-7.10730 - 7.10730i) q^{73} +(-0.667281 - 0.667281i) q^{77} -11.1545i q^{79} +(-5.98851 + 5.98851i) q^{83} +(2.35685 - 12.4623i) q^{85} -10.1275 q^{89} +2.45217 q^{91} +(-2.43281 + 1.65897i) q^{95} +(3.76737 - 3.76737i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.25978	+	1.84742i	0.563393	+	0.826189i
$5$	1.25978	+	1.84742i	0.563393	+	0.826189i
$6$		0			0
$7$	−1.73394	+	1.73394i	−0.655369	+	0.655369i	−0.954281	−	0.298912i	$-0.903376\pi$
$7$	−1.73394	+	1.73394i	−0.655369	+	0.655369i	0.298912	+	0.954281i	$0.403376\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.384834i			0.116032i	0.998316	+	0.0580159i	$0.0184774\pi$
$11$			0.384834i			0.116032i	−0.998316	+	0.0580159i	$0.981523\pi$
$12$		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
$17$	−4.01077	−	4.01077i	−0.972754	−	0.972754i	0.0268843	−	0.999639i	$-0.491441\pi$
$17$	−4.01077	−	4.01077i	−0.972754	−	0.972754i	−0.999639	+	0.0268843i	$0.991441\pi$
$18$		0			0
$19$			1.31687i			0.302111i	0.988525	+	0.151055i	$0.0482672\pi$
$19$			1.31687i			0.302111i	−0.988525	+	0.151055i	$0.951733\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.09168	+	3.09168i	−0.644659	+	0.644659i	−0.951697	−	0.307038i	$-0.900662\pi$
$23$	−3.09168	+	3.09168i	−0.644659	+	0.644659i	0.307038	+	0.951697i	$0.400662\pi$
$24$		0			0
$25$	−1.82588	+	4.65469i	−0.365177	+	0.930938i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.58151			−1.03646			−0.518231	−	0.855241i	$-0.673409\pi$
$29$	−5.58151			−1.03646			−0.518231	+	0.855241i	$0.673409\pi$
$30$		0			0
$31$	−4.45438			−0.800031			−0.400015	−	0.916508i	$-0.630995\pi$
$31$	−4.45438			−0.800031			−0.400015	+	0.916508i	$0.630995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.38771	−	1.01892i	−0.910689	−	0.172229i
$36$		0			0
$37$	7.95020	−	7.95020i	1.30700	−	1.30700i	0.383438	−	0.923567i	$-0.374740\pi$
$37$	7.95020	−	7.95020i	1.30700	−	1.30700i	0.923567	−	0.383438i	$-0.125260\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.37309i			1.15148i	0.817632	+	0.575742i	$0.195287\pi$
$41$			7.37309i			1.15148i	−0.817632	+	0.575742i	$0.804713\pi$
$42$		0			0
$43$	−6.63893	−	6.63893i	−1.01243	−	1.01243i	−0.999922	−	0.0125057i	$-0.996019\pi$
$43$	−6.63893	−	6.63893i	−1.01243	−	1.01243i	−0.0125057	−	0.999922i	$-0.503981\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.03757	+	9.03757i	1.31827	+	1.31827i	0.915149	+	0.403117i	$0.132073\pi$
$47$	9.03757	+	9.03757i	1.31827	+	1.31827i	0.403117	+	0.915149i	$0.367927\pi$
$48$		0			0
$49$			0.986880i			0.140983i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.78645	−	4.78645i	0.657470	−	0.657470i	−0.297311	−	0.954781i	$-0.596090\pi$
$53$	4.78645	−	4.78645i	0.657470	−	0.657470i	0.954781	+	0.297311i	$0.0960898\pi$
$54$		0			0
$55$	−0.710948	+	0.484808i	−0.0958643	+	0.0653715i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.10867			−1.05566			−0.527829	−	0.849351i	$-0.676994\pi$
$59$	−8.10867			−1.05566			−0.527829	+	0.849351i	$0.676994\pi$
$60$		0			0
$61$	−15.0948			−1.93269			−0.966347	−	0.257242i	$-0.917186\pi$
$61$	−15.0948			−1.93269			−0.966347	+	0.257242i	$0.917186\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.415517	−	2.19712i	0.0515386	−	0.272519i
$66$		0			0
$67$	5.47501	−	5.47501i	0.668879	−	0.668879i	−0.288578	−	0.957456i	$-0.593182\pi$
$67$	5.47501	−	5.47501i	0.668879	−	0.668879i	0.957456	+	0.288578i	$0.0931824\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.23609i			0.146697i	0.997306	+	0.0733486i	$0.0233686\pi$
$71$			1.23609i			0.146697i	−0.997306	+	0.0733486i	$0.976631\pi$
$72$		0			0
$73$	−7.10730	−	7.10730i	−0.831846	−	0.831846i	0.155923	−	0.987769i	$-0.450165\pi$
$73$	−7.10730	−	7.10730i	−0.831846	−	0.831846i	−0.987769	+	0.155923i	$0.950165\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.667281	−	0.667281i	−0.0760437	−	0.0760437i
$78$		0			0
$79$		−	11.1545i		−	1.25498i	−0.778626	−	0.627488i	$-0.784083\pi$
$79$		−	11.1545i		−	1.25498i	0.778626	−	0.627488i	$-0.215917\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.98851	+	5.98851i	−0.657324	+	0.657324i	−0.954746	−	0.297422i	$-0.903873\pi$
$83$	−5.98851	+	5.98851i	−0.657324	+	0.657324i	0.297422	+	0.954746i	$0.403873\pi$
$84$		0			0
$85$	2.35685	−	12.4623i	0.255636	−	1.35172i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.1275			−1.07351			−0.536755	−	0.843738i	$-0.680350\pi$
$89$	−10.1275			−1.07351			−0.536755	+	0.843738i	$0.680350\pi$
$90$		0			0
$91$	2.45217			0.257057
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.43281	+	1.65897i	−0.249601	+	0.170207i
$96$		0			0
$97$	3.76737	−	3.76737i	0.382519	−	0.382519i	−0.489490	−	0.872009i	$-0.662817\pi$
$97$	3.76737	−	3.76737i	0.382519	−	0.382519i	0.872009	+	0.489490i	$0.162817\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.4622i		−	1.33954i	−0.742570	−	0.669768i	$-0.766393\pi$
$101$		−	13.4622i		−	1.33954i	0.742570	−	0.669768i	$-0.233607\pi$
$102$		0			0
$103$	−10.0329	−	10.0329i	−0.988570	−	0.988570i	0.0113656	−	0.999935i	$-0.496382\pi$
$103$	−10.0329	−	10.0329i	−0.988570	−	0.988570i	−0.999935	+	0.0113656i	$0.996382\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.8795	+	11.8795i	1.14843	+	1.14843i	0.986861	+	0.161573i	$0.0516567\pi$
$107$	11.8795	+	11.8795i	1.14843	+	1.14843i	0.161573	+	0.986861i	$0.448343\pi$
$108$		0			0
$109$		−	4.78381i		−	0.458205i	−0.973402	−	0.229103i	$-0.926421\pi$
$109$		−	4.78381i		−	0.458205i	0.973402	−	0.229103i	$-0.0735792\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.68677	−	5.68677i	0.534966	−	0.534966i	−0.387080	−	0.922046i	$-0.626516\pi$
$113$	5.68677	−	5.68677i	0.534966	−	0.534966i	0.922046	+	0.387080i	$0.126516\pi$
$114$		0			0
$115$	−9.60646	−	1.81676i	−0.895807	−	0.169414i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.9089			1.27503
$120$		0			0
$121$	10.8519			0.986537
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.8994	+	2.49074i	−0.974869	+	0.222779i
$126$		0			0
$127$	−11.7920	+	11.7920i	−1.04637	+	1.04637i	−0.0474990	+	0.998871i	$0.515125\pi$
$127$	−11.7920	+	11.7920i	−1.04637	+	1.04637i	−0.998871	+	0.0474990i	$0.984875\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			22.0526i			1.92674i	0.268171	+	0.963371i	$0.413581\pi$
$131$			22.0526i			1.92674i	−0.268171	+	0.963371i	$0.586419\pi$
$132$		0			0
$133$	−2.28338	−	2.28338i	−0.197994	−	0.197994i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.400904	−	0.400904i	−0.0342516	−	0.0342516i	0.689774	−	0.724025i	$-0.257710\pi$
$137$	−0.400904	−	0.400904i	−0.0342516	−	0.0342516i	−0.724025	+	0.689774i	$0.757710\pi$
$138$		0			0
$139$			21.6285i			1.83451i	0.398303	+	0.917254i	$0.369599\pi$
$139$			21.6285i			1.83451i	−0.398303	+	0.917254i	$0.630401\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.272119	−	0.272119i	0.0227557	−	0.0227557i
$144$		0			0
$145$	−7.03151	−	10.3114i	−0.583935	−	0.856313i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−19.9175			−1.63171			−0.815854	−	0.578258i	$-0.803733\pi$
$149$	−19.9175			−1.63171			−0.815854	+	0.578258i	$0.803733\pi$
$150$		0			0
$151$	0.310475			0.0252661			0.0126331	−	0.999920i	$-0.495979\pi$
$151$	0.310475			0.0252661			0.0126331	+	0.999920i	$0.495979\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.61156	−	8.22909i	−0.450732	−	0.660977i
$156$		0			0
$157$	−5.28851	+	5.28851i	−0.422069	+	0.422069i	−0.885916	−	0.463846i	$-0.846469\pi$
$157$	−5.28851	+	5.28851i	−0.422069	+	0.422069i	0.463846	+	0.885916i	$0.346469\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	10.7216i		−	0.844979i
$162$		0			0
$163$	6.40253	+	6.40253i	0.501485	+	0.501485i	0.911899	−	0.410414i	$-0.134616\pi$
$163$	6.40253	+	6.40253i	0.501485	+	0.501485i	−0.410414	+	0.911899i	$0.634616\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.74597	+	8.74597i	0.676783	+	0.676783i	0.959271	−	0.282488i	$-0.0911596\pi$
$167$	8.74597	+	8.74597i	0.676783	+	0.676783i	−0.282488	+	0.959271i	$0.591160\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.20511	+	7.20511i	−0.547795	+	0.547795i	−0.925802	−	0.378008i	$-0.876609\pi$
$173$	−7.20511	+	7.20511i	−0.547795	+	0.547795i	0.378008	+	0.925802i	$0.376609\pi$
$174$		0			0
$175$	−4.90499	−	11.2370i	−0.370782	−	0.849434i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.2227			−1.06305			−0.531527	−	0.847042i	$-0.678381\pi$
$179$	−14.2227			−1.06305			−0.531527	+	0.847042i	$0.678381\pi$
$180$		0			0
$181$	4.51283			0.335436			0.167718	−	0.985835i	$-0.446360\pi$
$181$	4.51283			0.335436			0.167718	+	0.985835i	$0.446360\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	24.7029	+	4.67178i	1.81619	+	0.343476i
$186$		0			0
$187$	1.54348	−	1.54348i	0.112870	−	0.112870i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			17.0540i			1.23399i	0.786969	+	0.616993i	$0.211649\pi$
$191$			17.0540i			1.23399i	−0.786969	+	0.616993i	$0.788351\pi$
$192$		0			0
$193$	−3.86139	−	3.86139i	−0.277949	−	0.277949i	0.554341	−	0.832290i	$-0.312970\pi$
$193$	−3.86139	−	3.86139i	−0.277949	−	0.277949i	−0.832290	+	0.554341i	$0.812970\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.39560	+	1.39560i	0.0994323	+	0.0994323i	0.755073	−	0.655641i	$-0.227601\pi$
$197$	1.39560	+	1.39560i	0.0994323	+	0.0994323i	−0.655641	+	0.755073i	$0.727601\pi$
$198$		0			0
$199$		−	20.3702i		−	1.44401i	−0.691890	−	0.722003i	$-0.743222\pi$
$199$		−	20.3702i		−	1.44401i	0.691890	−	0.722003i	$-0.256778\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.67803	−	9.67803i	0.679265	−	0.679265i
$204$		0			0
$205$	−13.6212	+	9.28851i	−0.951343	+	0.648738i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.506777			−0.0350545
$210$		0			0
$211$	−1.27375			−0.0876887			−0.0438444	−	0.999038i	$-0.513961\pi$
$211$	−1.27375			−0.0876887			−0.0438444	+	0.999038i	$0.513961\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.90124	−	20.6285i	0.266062	−	1.40685i
$216$		0			0
$217$	7.72365	−	7.72365i	0.524315	−	0.524315i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			5.67208i			0.381546i
$222$		0			0
$223$	7.76181	+	7.76181i	0.519769	+	0.519769i	0.917502	−	0.397732i	$-0.130203\pi$
$223$	7.76181	+	7.76181i	0.519769	+	0.519769i	−0.397732	+	0.917502i	$0.630203\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.73039	−	9.73039i	−0.645829	−	0.645829i	0.306154	−	0.951982i	$-0.400958\pi$
$227$	−9.73039	−	9.73039i	−0.645829	−	0.645829i	−0.951982	+	0.306154i	$0.900958\pi$
$228$		0			0
$229$		−	4.69055i		−	0.309960i	−0.987918	−	0.154980i	$-0.950469\pi$
$229$		−	4.69055i		−	0.309960i	0.987918	−	0.154980i	$-0.0495314\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.55451	+	7.55451i	−0.494912	+	0.494912i	−0.909850	−	0.414938i	$-0.863803\pi$
$233$	−7.55451	+	7.55451i	−0.494912	+	0.494912i	0.414938	+	0.909850i	$0.363803\pi$
$234$		0			0
$235$	−5.31075	+	28.0815i	−0.346435	+	1.83184i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.67798			−0.367278			−0.183639	−	0.982994i	$-0.558788\pi$
$239$	−5.67798			−0.367278			−0.183639	+	0.982994i	$0.558788\pi$
$240$		0			0
$241$	−5.00783			−0.322582			−0.161291	−	0.986907i	$-0.551566\pi$
$241$	−5.00783			−0.322582			−0.161291	+	0.986907i	$0.551566\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.82318	+	1.24326i	−0.116478	+	0.0794287i
$246$		0			0
$247$	0.931168	−	0.931168i	0.0592488	−	0.0592488i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	12.9735i		−	0.818881i	−0.912337	−	0.409440i	$-0.865724\pi$
$251$		−	12.9735i		−	0.818881i	0.912337	−	0.409440i	$-0.134276\pi$
$252$		0			0
$253$	−1.18978	−	1.18978i	−0.0748010	−	0.0748010i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.08065	+	3.08065i	0.192165	+	0.192165i	0.796631	−	0.604466i	$-0.206613\pi$
$257$	3.08065	+	3.08065i	0.192165	+	0.192165i	−0.604466	+	0.796631i	$0.706613\pi$
$258$		0			0
$259$			27.5704i			1.71314i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.87145	+	6.87145i	−0.423712	+	0.423712i	−0.886479	−	0.462768i	$-0.846856\pi$
$263$	−6.87145	+	6.87145i	−0.423712	+	0.423712i	0.462768	+	0.886479i	$0.346856\pi$
$264$		0			0
$265$	14.8725	+	2.81266i	0.913608	+	0.172781i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.6283			−0.769960			−0.384980	−	0.922925i	$-0.625792\pi$
$269$	−12.6283			−0.769960			−0.384980	+	0.922925i	$0.625792\pi$
$270$		0			0
$271$	−0.726979			−0.0441608			−0.0220804	−	0.999756i	$-0.507029\pi$
$271$	−0.726979			−0.0441608			−0.0220804	+	0.999756i	$0.507029\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.79128	−	0.702663i	−0.108018	−	0.0423722i
$276$		0			0
$277$	17.4237	−	17.4237i	1.04689	−	1.04689i	0.0480410	−	0.998845i	$-0.484702\pi$
$277$	17.4237	−	17.4237i	1.04689	−	1.04689i	0.998845	−	0.0480410i	$-0.0152978\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			26.2012i			1.56303i	0.623886	+	0.781515i	$0.285553\pi$
$281$			26.2012i			1.56303i	−0.623886	+	0.781515i	$0.714447\pi$
$282$		0			0
$283$	4.23917	+	4.23917i	0.251992	+	0.251992i	0.821787	−	0.569795i	$-0.192977\pi$
$283$	4.23917	+	4.23917i	0.251992	+	0.251992i	−0.569795	+	0.821787i	$0.692977\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.7845	−	12.7845i	−0.754647	−	0.754647i
$288$		0			0
$289$			15.1725i			0.892502i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.57405	+	2.57405i	−0.150377	+	0.150377i	−0.778287	−	0.627909i	$-0.783911\pi$
$293$	−2.57405	+	2.57405i	−0.150377	+	0.150377i	0.627909	+	0.778287i	$0.283911\pi$
$294$		0			0
$295$	−10.2152	−	14.9801i	−0.594750	−	0.872174i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.37229			0.252856
$300$		0			0
$301$	23.0231			1.32703
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.0162	−	27.8864i	−1.08887	−	1.59677i
$306$		0			0
$307$	−1.90730	+	1.90730i	−0.108855	+	0.108855i	−0.759437	−	0.650581i	$-0.774525\pi$
$307$	−1.90730	+	1.90730i	−0.108855	+	0.108855i	0.650581	+	0.759437i	$0.274525\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4.43080i			0.251248i	0.992078	+	0.125624i	$0.0400932\pi$
$311$			4.43080i			0.251248i	−0.992078	+	0.125624i	$0.959907\pi$
$312$		0			0
$313$	1.12790	+	1.12790i	0.0637524	+	0.0637524i	0.738264	−	0.674512i	$-0.235646\pi$
$313$	1.12790	+	1.12790i	0.0637524	+	0.0637524i	−0.674512	+	0.738264i	$0.735646\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.3499	+	14.3499i	0.805968	+	0.805968i	0.984021	−	0.178053i	$-0.0569797\pi$
$317$	14.3499	+	14.3499i	0.805968	+	0.805968i	−0.178053	+	0.984021i	$0.556980\pi$
$318$		0			0
$319$		−	2.14796i		−	0.120263i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.28166	−	5.28166i	0.293879	−	0.293879i
$324$		0			0
$325$	4.58246	−	2.00027i	0.254189	−	0.110955i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−31.3413			−1.72790
$330$		0			0
$331$	−14.0552			−0.772546			−0.386273	−	0.922385i	$-0.626238\pi$
$331$	−14.0552			−0.772546			−0.386273	+	0.922385i	$0.626238\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.0119	+	3.21728i	0.929462	+	0.175779i
$336$		0			0
$337$	−5.63536	+	5.63536i	−0.306977	+	0.306977i	−0.843736	−	0.536758i	$-0.819649\pi$
$337$	−5.63536	+	5.63536i	−0.306977	+	0.306977i	0.536758	+	0.843736i	$0.319649\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1.71420i		−	0.0928291i
$342$		0			0
$343$	−13.8488	−	13.8488i	−0.747765	−	0.747765i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.39937	+	3.39937i	0.182488	+	0.182488i	0.792439	−	0.609951i	$-0.208811\pi$
$347$	3.39937	+	3.39937i	0.182488	+	0.182488i	−0.609951	+	0.792439i	$0.708811\pi$
$348$		0			0
$349$		−	24.0905i		−	1.28953i	−0.764380	−	0.644766i	$-0.776955\pi$
$349$		−	24.0905i		−	1.28953i	0.764380	−	0.644766i	$-0.223045\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.3257	−	18.3257i	0.975380	−	0.975380i	−0.0243238	−	0.999704i	$-0.507743\pi$
$353$	18.3257	−	18.3257i	0.975380	−	0.975380i	0.999704	+	0.0243238i	$0.00774327\pi$
$354$		0			0
$355$	−2.28358	+	1.55721i	−0.121200	+	0.0826482i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.254736			−0.0134445			−0.00672224	−	0.999977i	$-0.502140\pi$
$359$	−0.254736			−0.0134445			−0.00672224	+	0.999977i	$0.502140\pi$
$360$		0			0
$361$	17.2659			0.908729
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.17646	−	22.0838i	0.218606	−	1.15592i
$366$		0			0
$367$	−8.93710	+	8.93710i	−0.466513	+	0.466513i	−0.900783	−	0.434270i	$-0.857006\pi$
$367$	−8.93710	+	8.93710i	−0.466513	+	0.466513i	0.434270	+	0.900783i	$0.357006\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			16.5989i			0.861771i
$372$		0			0
$373$	15.1876	+	15.1876i	0.786385	+	0.786385i	0.980900	−	0.194514i	$-0.0623131\pi$
$373$	15.1876	+	15.1876i	0.786385	+	0.786385i	−0.194514	+	0.980900i	$0.562313\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.94673	+	3.94673i	0.203267	+	0.203267i
$378$		0			0
$379$			29.8373i			1.53264i	0.642461	+	0.766318i	$0.277913\pi$
$379$			29.8373i			1.53264i	−0.642461	+	0.766318i	$0.722087\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.3978	−	12.3978i	0.633495	−	0.633495i	−0.315448	−	0.948943i	$-0.602155\pi$
$383$	12.3978	−	12.3978i	0.633495	−	0.633495i	0.948943	+	0.315448i	$0.102155\pi$
$384$		0			0
$385$	0.392114	−	2.07337i	0.0199840	−	0.105669i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.10223			0.207991			0.103996	−	0.994578i	$-0.466837\pi$
$389$	4.10223			0.207991			0.103996	+	0.994578i	$0.466837\pi$
$390$		0			0
$391$	24.8000			1.25419
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.6069	−	14.0522i	1.03685	−	0.707045i
$396$		0			0
$397$	18.2078	−	18.2078i	0.913823	−	0.913823i	−0.0827474	−	0.996571i	$-0.526369\pi$
$397$	18.2078	−	18.2078i	0.913823	−	0.913823i	0.996571	+	0.0827474i	$0.0263695\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7.08373i			0.353745i	0.984234	+	0.176872i	$0.0565980\pi$
$401$			7.08373i			0.353745i	−0.984234	+	0.176872i	$0.943402\pi$
$402$		0			0
$403$	3.14972	+	3.14972i	0.156899	+	0.156899i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.05951	+	3.05951i	0.151654	+	0.151654i
$408$		0			0
$409$			34.4019i			1.70106i	0.525924	+	0.850531i	$0.323719\pi$
$409$			34.4019i			1.70106i	−0.525924	+	0.850531i	$0.676281\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14.0600	−	14.0600i	0.691846	−	0.691846i
$414$		0			0
$415$	−18.6075	−	3.51903i	−0.913406	−	0.172742i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	39.0049			1.90552			0.952758	−	0.303730i	$-0.0982320\pi$
$419$	39.0049			1.90552			0.952758	+	0.303730i	$0.0982320\pi$
$420$		0			0
$421$	21.6018			1.05281			0.526405	−	0.850234i	$-0.323540\pi$
$421$	21.6018			1.05281			0.526405	+	0.850234i	$0.323540\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	25.9921	−	11.3457i	1.26080	−	0.550347i
$426$		0			0
$427$	26.1736	−	26.1736i	1.26663	−	1.26663i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9.91513i			0.477595i	0.971069	+	0.238798i	$0.0767532\pi$
$431$			9.91513i			0.477595i	−0.971069	+	0.238798i	$0.923247\pi$
$432$		0			0
$433$	−16.0952	−	16.0952i	−0.773488	−	0.773488i	0.205226	−	0.978715i	$-0.434207\pi$
$433$	−16.0952	−	16.0952i	−0.773488	−	0.773488i	−0.978715	+	0.205226i	$0.934207\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.07134	−	4.07134i	−0.194758	−	0.194758i
$438$		0			0
$439$			34.4090i			1.64225i	0.570746	+	0.821127i	$0.306654\pi$
$439$			34.4090i			1.64225i	−0.570746	+	0.821127i	$0.693346\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.3175	−	11.3175i	0.537711	−	0.537711i	−0.385145	−	0.922856i	$-0.625849\pi$
$443$	11.3175	−	11.3175i	0.537711	−	0.537711i	0.922856	+	0.385145i	$0.125849\pi$
$444$		0			0
$445$	−12.7584	−	18.7097i	−0.604808	−	0.886923i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.48334			0.305968			0.152984	−	0.988229i	$-0.451112\pi$
$449$	6.48334			0.305968			0.152984	+	0.988229i	$0.451112\pi$
$450$		0			0
$451$	−2.83742			−0.133609
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.08920	+	4.53017i	0.144824	+	0.212378i
$456$		0			0
$457$	−4.20816	+	4.20816i	−0.196850	+	0.196850i	−0.798648	−	0.601798i	$-0.794451\pi$
$457$	−4.20816	+	4.20816i	−0.196850	+	0.196850i	0.601798	+	0.798648i	$0.294451\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			9.12753i			0.425112i	0.977149	+	0.212556i	$0.0681787\pi$
$461$			9.12753i			0.425112i	−0.977149	+	0.212556i	$0.931821\pi$
$462$		0			0
$463$	−6.99801	−	6.99801i	−0.325225	−	0.325225i	0.525542	−	0.850768i	$-0.323862\pi$
$463$	−6.99801	−	6.99801i	−0.325225	−	0.325225i	−0.850768	+	0.525542i	$0.823862\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.81099	−	5.81099i	−0.268901	−	0.268901i	0.559756	−	0.828657i	$-0.310895\pi$
$467$	−5.81099	−	5.81099i	−0.268901	−	0.268901i	−0.828657	+	0.559756i	$0.810895\pi$
$468$		0			0
$469$			18.9867i			0.876725i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.55489	−	2.55489i	0.117474	−	0.117474i
$474$		0			0
$475$	−6.12962	−	2.40445i	−0.281246	−	0.110324i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.1406			−0.874558			−0.437279	−	0.899326i	$-0.644058\pi$
$479$	−19.1406			−0.874558			−0.437279	+	0.899326i	$0.644058\pi$
$480$		0			0
$481$	−11.2433			−0.512649
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11.7060	+	2.21382i	0.531541	+	0.100524i
$486$		0			0
$487$	−12.3303	+	12.3303i	−0.558738	+	0.558738i	−0.928948	−	0.370210i	$-0.879286\pi$
$487$	−12.3303	+	12.3303i	−0.558738	+	0.558738i	0.370210	+	0.928948i	$0.379286\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			1.78011i			0.0803350i	0.999193	+	0.0401675i	$0.0127892\pi$
$491$			1.78011i			0.0803350i	−0.999193	+	0.0401675i	$0.987211\pi$
$492$		0			0
$493$	22.3862	+	22.3862i	1.00822	+	1.00822i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.14331	−	2.14331i	−0.0961408	−	0.0961408i
$498$		0			0
$499$			4.33708i			0.194154i	0.995277	+	0.0970771i	$0.0309493\pi$
$499$			4.33708i			0.194154i	−0.995277	+	0.0970771i	$0.969051\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.82220	+	1.82220i	−0.0812477	+	0.0812477i	−0.746563	−	0.665315i	$-0.768297\pi$
$503$	−1.82220	+	1.82220i	−0.0812477	+	0.0812477i	0.665315	+	0.746563i	$0.268297\pi$
$504$		0			0
$505$	24.8702	−	16.9594i	1.10671	−	0.754685i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.75196			0.343600			0.171800	−	0.985132i	$-0.445042\pi$
$509$	7.75196			0.343600			0.171800	+	0.985132i	$0.445042\pi$
$510$		0			0
$511$	24.6473			1.09033
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.89563	−	31.1742i	0.259792	−	1.37370i
$516$		0			0
$517$	−3.47797	+	3.47797i	−0.152961	+	0.152961i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.91813i		−	0.259278i	−0.991561	−	0.129639i	$-0.958618\pi$
$521$		−	5.91813i		−	0.259278i	0.991561	−	0.129639i	$-0.0413818\pi$
$522$		0			0
$523$	24.4439	+	24.4439i	1.06886	+	1.06886i	0.997447	+	0.0714116i	$0.0227504\pi$
$523$	24.4439	+	24.4439i	1.06886	+	1.06886i	0.0714116	+	0.997447i	$0.477250\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.8655	+	17.8655i	0.778233	+	0.778233i
$528$		0			0
$529$			3.88307i			0.168829i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	5.21356	−	5.21356i	0.225825	−	0.225825i
$534$		0			0
$535$	−6.98075	+	36.9119i	−0.301804	+	1.59584i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.379785			−0.0163585
$540$		0			0
$541$	20.6242			0.886701			0.443351	−	0.896348i	$-0.353790\pi$
$541$	20.6242			0.886701			0.443351	+	0.896348i	$0.353790\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.83767	−	6.02656i	0.378564	−	0.258150i
$546$		0			0
$547$	−19.0133	+	19.0133i	−0.812950	+	0.812950i	−0.985075	−	0.172125i	$-0.944937\pi$
$547$	−19.0133	+	19.0133i	−0.812950	+	0.812950i	0.172125	+	0.985075i	$0.444937\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	7.35013i		−	0.313126i
$552$		0			0
$553$	19.3412	+	19.3412i	0.822473	+	0.822473i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.8754	−	13.8754i	−0.587920	−	0.587920i	0.349148	−	0.937068i	$-0.386471\pi$
$557$	−13.8754	−	13.8754i	−0.587920	−	0.587920i	−0.937068	+	0.349148i	$0.886471\pi$
$558$		0			0
$559$			9.38887i			0.397107i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.64918	+	1.64918i	−0.0695049	+	0.0695049i	−0.741005	−	0.671500i	$-0.765650\pi$
$563$	−1.64918	+	1.64918i	−0.0695049	+	0.0695049i	0.671500	+	0.741005i	$0.265650\pi$
$564$		0			0
$565$	17.6699	+	3.34172i	0.743380	+	0.140587i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.77209			0.158134			0.0790672	−	0.996869i	$-0.474806\pi$
$569$	3.77209			0.158134			0.0790672	+	0.996869i	$0.474806\pi$
$570$		0			0
$571$	−1.09477			−0.0458148			−0.0229074	−	0.999738i	$-0.507292\pi$
$571$	−1.09477			−0.0458148			−0.0229074	+	0.999738i	$0.507292\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.74575	−	20.0358i	−0.364723	−	0.835552i
$576$		0			0
$577$	−7.39782	+	7.39782i	−0.307975	+	0.307975i	−0.844124	−	0.536149i	$-0.819879\pi$
$577$	−7.39782	+	7.39782i	−0.307975	+	0.307975i	0.536149	+	0.844124i	$0.319879\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	20.7675i		−	0.861580i
$582$		0			0
$583$	1.84199	+	1.84199i	0.0762874	+	0.0762874i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25.9668	+	25.9668i	1.07177	+	1.07177i	0.997217	+	0.0745491i	$0.0237517\pi$
$587$	25.9668	+	25.9668i	1.07177	+	1.07177i	0.0745491	+	0.997217i	$0.476248\pi$
$588$		0			0
$589$		−	5.86584i		−	0.241698i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.3048	+	24.3048i	−0.998078	+	0.998078i	−0.999998	−	0.00192013i	$-0.999389\pi$
$593$	−24.3048	+	24.3048i	−0.998078	+	0.998078i	0.00192013	+	0.999998i	$0.499389\pi$
$594$		0			0
$595$	17.5222	+	25.6955i	0.718341	+	1.05341i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.3406			0.667659			0.333829	−	0.942633i	$-0.391659\pi$
$599$	16.3406			0.667659			0.333829	+	0.942633i	$0.391659\pi$
$600$		0			0
$601$	46.2265			1.88562			0.942808	−	0.333336i	$-0.108174\pi$
$601$	46.2265			1.88562			0.942808	+	0.333336i	$0.108174\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.6711	+	20.0480i	0.555808	+	0.815066i
$606$		0			0
$607$	−3.69609	+	3.69609i	−0.150020	+	0.150020i	−0.778127	−	0.628107i	$-0.783830\pi$
$607$	−3.69609	+	3.69609i	−0.150020	+	0.150020i	0.628107	+	0.778127i	$0.283830\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	12.7811i		−	0.517066i
$612$		0			0
$613$	−26.2759	−	26.2759i	−1.06127	−	1.06127i	−0.997996	−	0.0632783i	$-0.979844\pi$
$613$	−26.2759	−	26.2759i	−1.06127	−	1.06127i	−0.0632783	−	0.997996i	$-0.520156\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.86473	−	7.86473i	−0.316622	−	0.316622i	0.530846	−	0.847468i	$-0.321874\pi$
$617$	−7.86473	−	7.86473i	−0.316622	−	0.316622i	−0.847468	+	0.530846i	$0.821874\pi$
$618$		0			0
$619$		−	27.8932i		−	1.12112i	−0.828113	−	0.560561i	$-0.810586\pi$
$619$		−	27.8932i		−	1.12112i	0.828113	−	0.560561i	$-0.189414\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	17.5605	−	17.5605i	0.703546	−	0.703546i
$624$		0			0
$625$	−18.3323	−	16.9979i	−0.733292	−	0.679914i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−63.7728			−2.54279
$630$		0			0
$631$	2.88186			0.114725			0.0573626	−	0.998353i	$-0.481731\pi$
$631$	2.88186			0.114725			0.0573626	+	0.998353i	$0.481731\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−36.6401	−	6.92933i	−1.45402	−	0.274982i
$636$		0			0
$637$	0.697829	−	0.697829i	0.0276490	−	0.0276490i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	9.99809i		−	0.394901i	−0.980313	−	0.197450i	$-0.936734\pi$
$641$		−	9.99809i		−	0.394901i	0.980313	−	0.197450i	$-0.0632662\pi$
$642$		0			0
$643$	−17.1161	−	17.1161i	−0.674995	−	0.674995i	0.283868	−	0.958863i	$-0.408382\pi$
$643$	−17.1161	−	17.1161i	−0.674995	−	0.674995i	−0.958863	+	0.283868i	$0.908382\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.5831	+	20.5831i	0.809205	+	0.809205i	0.984514	−	0.175308i	$-0.0560922\pi$
$647$	20.5831	+	20.5831i	0.809205	+	0.809205i	−0.175308	+	0.984514i	$0.556092\pi$
$648$		0			0
$649$		−	3.12049i		−	0.122490i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.2022	−	15.2022i	0.594909	−	0.594909i	−0.344044	−	0.938953i	$-0.611797\pi$
$653$	15.2022	−	15.2022i	0.594909	−	0.594909i	0.938953	+	0.344044i	$0.111797\pi$
$654$		0			0
$655$	−40.7403	+	27.7815i	−1.59185	+	1.08551i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.34993			−0.208403			−0.104202	−	0.994556i	$-0.533229\pi$
$659$	−5.34993			−0.208403			−0.104202	+	0.994556i	$0.533229\pi$
$660$		0			0
$661$	−36.2838			−1.41127			−0.705637	−	0.708573i	$-0.749339\pi$
$661$	−36.2838			−1.41127			−0.705637	+	0.708573i	$0.749339\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.34178	−	7.09491i	0.0520321	−	0.275129i
$666$		0			0
$667$	17.2562	−	17.2562i	0.668164	−	0.668164i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	5.80900i		−	0.224254i
$672$		0			0
$673$	16.8311	+	16.8311i	0.648792	+	0.648792i	0.952701	−	0.303909i	$-0.0982919\pi$
$673$	16.8311	+	16.8311i	0.648792	+	0.648792i	−0.303909	+	0.952701i	$0.598292\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.6084	+	31.6084i	1.21481	+	1.21481i	0.969426	+	0.245384i	$0.0789141\pi$
$677$	31.6084	+	31.6084i	1.21481	+	1.21481i	0.245384	+	0.969426i	$0.421086\pi$
$678$		0			0
$679$			13.0648i			0.501382i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.08577	−	1.08577i	0.0415459	−	0.0415459i	−0.686029	−	0.727575i	$-0.740648\pi$
$683$	1.08577	−	1.08577i	0.0415459	−	0.0415459i	0.727575	+	0.686029i	$0.240648\pi$
$684$		0			0
$685$	0.235584	−	1.24569i	0.00900118	−	0.0475954i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.76906			−0.257881
$690$		0			0
$691$	6.62996			0.252216			0.126108	−	0.992017i	$-0.459751\pi$
$691$	6.62996			0.252216			0.126108	+	0.992017i	$0.459751\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−39.9569	+	27.2473i	−1.51565	+	1.03355i
$696$		0			0
$697$	29.5718	−	29.5718i	1.12011	−	1.12011i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	20.7488i		−	0.783671i	−0.920035	−	0.391836i	$-0.871840\pi$
$701$		−	20.7488i		−	0.783671i	0.920035	−	0.391836i	$-0.128160\pi$
$702$		0			0
$703$	10.4694	+	10.4694i	0.394860	+	0.394860i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	23.3426	+	23.3426i	0.877891	+	0.877891i
$708$		0			0
$709$			44.5642i			1.67364i	0.547475	+	0.836822i	$0.315589\pi$
$709$			44.5642i			1.67364i	−0.547475	+	0.836822i	$0.684411\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.7715	−	13.7715i	0.515747	−	0.515747i
$714$		0			0
$715$	0.845528	+	0.159905i	0.0316209	+	0.00598012i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−43.5519			−1.62421			−0.812106	−	0.583511i	$-0.801679\pi$
$719$	−43.5519			−1.62421			−0.812106	+	0.583511i	$0.801679\pi$
$720$		0			0
$721$	34.7929			1.29576
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.1912	−	25.9802i	0.378492	−	0.964881i
$726$		0			0
$727$	17.3632	−	17.3632i	0.643967	−	0.643967i	−0.307562	−	0.951528i	$-0.599513\pi$
$727$	17.3632	−	17.3632i	0.643967	−	0.643967i	0.951528	+	0.307562i	$0.0995130\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			53.2544i			1.96969i
$732$		0			0
$733$	6.72801	+	6.72801i	0.248505	+	0.248505i	0.820357	−	0.571852i	$-0.193775\pi$
$733$	6.72801	+	6.72801i	0.248505	+	0.248505i	−0.571852	+	0.820357i	$0.693775\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.10697	+	2.10697i	0.0776112	+	0.0776112i
$738$		0			0
$739$			14.5894i			0.536678i	0.963325	+	0.268339i	$0.0864748\pi$
$739$			14.5894i			0.536678i	−0.963325	+	0.268339i	$0.913525\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.1474	−	15.1474i	0.555703	−	0.555703i	−0.372378	−	0.928081i	$-0.621457\pi$
$743$	15.1474	−	15.1474i	0.555703	−	0.555703i	0.928081	+	0.372378i	$0.121457\pi$
$744$		0			0
$745$	−25.0918	−	36.7959i	−0.919293	−	1.34810i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−41.1967			−1.50530
$750$		0			0
$751$	−45.2657			−1.65177			−0.825884	−	0.563840i	$-0.809323\pi$
$751$	−45.2657			−1.65177			−0.825884	+	0.563840i	$0.809323\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.391132	+	0.573577i	0.0142347	+	0.0208746i
$756$		0			0
$757$	4.49276	−	4.49276i	0.163292	−	0.163292i	−0.620731	−	0.784023i	$-0.713164\pi$
$757$	4.49276	−	4.49276i	0.163292	−	0.163292i	0.784023	+	0.620731i	$0.213164\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41.1807i		−	1.49280i	−0.665499	−	0.746399i	$-0.731781\pi$
$761$		−	41.1807i		−	1.49280i	0.665499	−	0.746399i	$-0.268219\pi$
$762$		0			0
$763$	8.29485	+	8.29485i	0.300294	+	0.300294i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.73369	+	5.73369i	0.207032	+	0.207032i
$768$		0			0
$769$			37.0674i			1.33668i	0.743854	+	0.668342i	$0.232996\pi$
$769$			37.0674i			1.33668i	−0.743854	+	0.668342i	$0.767004\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.0552	−	10.0552i	0.361662	−	0.361662i	−0.502763	−	0.864424i	$-0.667683\pi$
$773$	10.0552	−	10.0552i	0.361662	−	0.361662i	0.864424	+	0.502763i	$0.167683\pi$
$774$		0			0
$775$	8.13319	−	20.7338i	0.292153	−	0.744779i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.70940			−0.347876
$780$		0			0
$781$	−0.475691			−0.0170216
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.4325	−	3.10769i	−0.586500	−	0.110918i
$786$		0			0
$787$	−27.3799	+	27.3799i	−0.975988	+	0.975988i	−0.999718	−	0.0237300i	$-0.992446\pi$
$787$	−27.3799	+	27.3799i	−0.975988	+	0.975988i	0.0237300	+	0.999718i	$0.492446\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			19.7211i			0.701201i
$792$		0			0
$793$	10.6737	+	10.6737i	0.379033	+	0.379033i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.535126	+	0.535126i	0.0189551	+	0.0189551i	0.716521	−	0.697566i	$-0.245733\pi$
$797$	0.535126	+	0.535126i	0.0189551	+	0.0189551i	−0.697566	+	0.716521i	$0.745733\pi$
$798$		0			0
$799$		−	72.4952i		−	2.56470i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.73513	−	2.73513i	0.0965207	−	0.0965207i
$804$		0			0
$805$	19.8072	−	13.5069i	0.698113	−	0.476055i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−29.9749			−1.05386			−0.526930	−	0.849909i	$-0.676657\pi$
$809$	−29.9749			−1.05386			−0.526930	+	0.849909i	$0.676657\pi$
$810$		0			0
$811$	40.9976			1.43962			0.719811	−	0.694170i	$-0.244228\pi$
$811$	40.9976			1.43962			0.719811	+	0.694170i	$0.244228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.76232	+	19.8940i	−0.131788	+	0.696855i
$816$		0			0
$817$	8.74261	−	8.74261i	0.305865	−	0.305865i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			9.53271i			0.332694i	0.986067	+	0.166347i	$0.0531971\pi$
$821$			9.53271i			0.332694i	−0.986067	+	0.166347i	$0.946803\pi$
$822$		0			0
$823$	−21.6615	−	21.6615i	−0.755072	−	0.755072i	0.220349	−	0.975421i	$-0.429280\pi$
$823$	−21.6615	−	21.6615i	−0.755072	−	0.755072i	−0.975421	+	0.220349i	$0.929280\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−29.4969	−	29.4969i	−1.02571	−	1.02571i	−0.999661	−	0.0260467i	$-0.991708\pi$
$827$	−29.4969	−	29.4969i	−1.02571	−	1.02571i	−0.0260467	−	0.999661i	$-0.508292\pi$
$828$		0			0
$829$			43.8951i			1.52454i	0.647258	+	0.762271i	$0.275915\pi$
$829$			43.8951i			1.52454i	−0.647258	+	0.762271i	$0.724085\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.95815	−	3.95815i	0.137142	−	0.137142i
$834$		0			0
$835$	−5.13940	+	27.1755i	−0.177856	+	0.940446i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.9864			−0.517389			−0.258694	−	0.965959i	$-0.583292\pi$
$839$	−14.9864			−0.517389			−0.258694	+	0.965959i	$0.583292\pi$
$840$		0			0
$841$	2.15331			0.0742520
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.84742	+	1.25978i	−0.0635530	+	0.0433379i
$846$		0			0
$847$	−18.8166	+	18.8166i	−0.646546	+	0.646546i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			49.1589i			1.68515i
$852$		0			0
$853$	−20.1898	−	20.1898i	−0.691287	−	0.691287i	0.271228	−	0.962515i	$-0.412570\pi$
$853$	−20.1898	−	20.1898i	−0.691287	−	0.691287i	−0.962515	+	0.271228i	$0.912570\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13.2518	+	13.2518i	0.452674	+	0.452674i	0.896241	−	0.443567i	$-0.146287\pi$
$857$	13.2518	+	13.2518i	0.452674	+	0.452674i	−0.443567	+	0.896241i	$0.646287\pi$
$858$		0			0
$859$			37.0940i			1.26563i	0.774302	+	0.632816i	$0.218101\pi$
$859$			37.0940i			1.26563i	−0.774302	+	0.632816i	$0.781899\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.7665	−	13.7665i	0.468617	−	0.468617i	−0.432850	−	0.901466i	$-0.642492\pi$
$863$	13.7665	−	13.7665i	0.468617	−	0.468617i	0.901466	+	0.432850i	$0.142492\pi$
$864$		0			0
$865$	−22.3877	−	4.23394i	−0.761206	−	0.143958i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.29262			0.145617
$870$		0			0
$871$	−7.74283			−0.262356
$872$		0			0
$873$		0			0
$874$		0			0
$875$	14.5801	−	23.2177i	0.492897	−	0.784901i
$876$		0			0
$877$	31.3254	−	31.3254i	1.05778	−	1.05778i	0.0595574	−	0.998225i	$-0.481031\pi$
$877$	31.3254	−	31.3254i	1.05778	−	1.05778i	0.998225	−	0.0595574i	$-0.0189689\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	9.43955i		−	0.318027i	−0.987276	−	0.159013i	$-0.949169\pi$
$881$		−	9.43955i		−	0.318027i	0.987276	−	0.159013i	$-0.0508313\pi$
$882$		0			0
$883$	−5.12917	−	5.12917i	−0.172610	−	0.172610i	0.615515	−	0.788125i	$-0.288948\pi$
$883$	−5.12917	−	5.12917i	−0.172610	−	0.172610i	−0.788125	+	0.615515i	$0.788948\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.4582	−	24.4582i	−0.821225	−	0.821225i	0.165059	−	0.986284i	$-0.447219\pi$
$887$	−24.4582	−	24.4582i	−0.821225	−	0.821225i	−0.986284	+	0.165059i	$0.947219\pi$
$888$		0			0
$889$		−	40.8933i		−	1.37152i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11.9013	+	11.9013i	−0.398262	+	0.398262i
$894$		0			0
$895$	−17.9175	−	26.2752i	−0.598917	−	0.878283i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.8622			0.829201
$900$		0			0
$901$	−38.3947			−1.27911
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.68519	+	8.33707i	0.188982	+	0.277133i
$906$		0			0
$907$	24.9524	−	24.9524i	0.828530	−	0.828530i	−0.158784	−	0.987313i	$-0.550757\pi$
$907$	24.9524	−	24.9524i	0.828530	−	0.828530i	0.987313	+	0.158784i	$0.0507572\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			40.0573i			1.32716i	0.748106	+	0.663579i	$0.230963\pi$
$911$			40.0573i			1.32716i	−0.748106	+	0.663579i	$0.769037\pi$
$912$		0			0
$913$	−2.30458	−	2.30458i	−0.0762705	−	0.0762705i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−38.2379	−	38.2379i	−1.26273	−	1.26273i
$918$		0			0
$919$		−	18.9481i		−	0.625041i	−0.949911	−	0.312520i	$-0.898827\pi$
$919$		−	18.9481i		−	0.625041i	0.949911	−	0.312520i	$-0.101173\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.874050	−	0.874050i	0.0287697	−	0.0287697i
$924$		0			0
$925$	22.4896	+	51.5219i	0.739453	+	1.69403i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	59.7002			1.95870			0.979350	−	0.202173i	$-0.0648004\pi$
$929$	59.7002			1.95870			0.979350	+	0.202173i	$0.0648004\pi$
$930$		0			0
$931$	−1.29959			−0.0425924
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.79590	+	0.906996i	0.156843	+	0.0296619i
$936$		0			0
$937$	−4.04485	+	4.04485i	−0.132139	+	0.132139i	−0.770083	−	0.637944i	$-0.779785\pi$
$937$	−4.04485	+	4.04485i	−0.132139	+	0.132139i	0.637944	+	0.770083i	$0.279785\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			16.2029i			0.528201i	0.964495	+	0.264101i	$0.0850751\pi$
$941$			16.2029i			0.528201i	−0.964495	+	0.264101i	$0.914925\pi$
$942$		0			0
$943$	−22.7952	−	22.7952i	−0.742315	−	0.742315i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.7238	+	11.7238i	0.380972	+	0.380972i	0.871452	−	0.490480i	$-0.163179\pi$
$947$	11.7238	+	11.7238i	0.380972	+	0.380972i	−0.490480	+	0.871452i	$0.663179\pi$
$948$		0			0
$949$			10.0512i			0.326277i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−40.9382	+	40.9382i	−1.32612	+	1.32612i	−0.417392	+	0.908726i	$0.637056\pi$
$953$	−40.9382	+	40.9382i	−1.32612	+	1.32612i	−0.908726	+	0.417392i	$0.862944\pi$
$954$		0			0
$955$	−31.5059	+	21.4844i	−1.01951	+	0.695219i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.39029			0.0448948
$960$		0			0
$961$	−11.1585			−0.359951
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.26907	−	11.9981i	0.0730440	−	0.386233i
$966$		0			0
$967$	7.60162	−	7.60162i	0.244452	−	0.244452i	−0.574237	−	0.818689i	$-0.694701\pi$
$967$	7.60162	−	7.60162i	0.244452	−	0.244452i	0.818689	+	0.574237i	$0.194701\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	50.0100i		−	1.60490i	−0.596721	−	0.802449i	$-0.703530\pi$
$971$		−	50.0100i		−	1.60490i	0.596721	−	0.802449i	$-0.296470\pi$
$972$		0			0
$973$	−37.5026	−	37.5026i	−1.20228	−	1.20228i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.3481	+	28.3481i	0.906935	+	0.906935i	0.996024	−	0.0890885i	$-0.0283954\pi$
$977$	28.3481	+	28.3481i	0.906935	+	0.906935i	−0.0890885	+	0.996024i	$0.528395\pi$
$978$		0			0
$979$		−	3.89740i		−	0.124561i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.1533	−	16.1533i	0.515210	−	0.515210i	−0.400908	−	0.916118i	$-0.631305\pi$
$983$	16.1533	−	16.1533i	0.515210	−	0.515210i	0.916118	+	0.400908i	$0.131305\pi$
$984$		0			0
$985$	−0.820096	+	4.33640i	−0.0261304	+	0.138169i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	41.0509			1.30534
$990$		0			0
$991$	−41.3274			−1.31281			−0.656404	−	0.754409i	$-0.727923\pi$
$991$	−41.3274			−1.31281			−0.656404	+	0.754409i	$0.727923\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	37.6322	−	25.6621i	1.19302	−	0.813543i
$996$		0			0
$997$	9.69815	−	9.69815i	0.307144	−	0.307144i	−0.536657	−	0.843801i	$-0.680313\pi$
$997$	9.69815	−	9.69815i	0.307144	−	0.307144i	0.843801	+	0.536657i	$0.180313\pi$
$998$		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.53.8	yes	24
3.2	odd	2		2340.2.y.a.53.5	✓	24
5.2	odd	4		2340.2.y.a.1457.5	yes	24
15.2	even	4	inner	2340.2.y.b.1457.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.5	✓	24	3.2	odd	2
2340.2.y.a.1457.5	yes	24	5.2	odd	4
2340.2.y.b.53.8	yes	24	1.1	even	1	trivial
2340.2.y.b.1457.8	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)

\(f(q)\)	\(=\)	\(q+(1.25978 + 1.84742i) q^{5} +(-1.73394 + 1.73394i) q^{7} +0.384834i q^{11} +(-0.707107 - 0.707107i) q^{13} +(-4.01077 - 4.01077i) q^{17} +1.31687i q^{19} +(-3.09168 + 3.09168i) q^{23} +(-1.82588 + 4.65469i) q^{25} -5.58151 q^{29} -4.45438 q^{31} +(-5.38771 - 1.01892i) q^{35} +(7.95020 - 7.95020i) q^{37} +7.37309i q^{41} +(-6.63893 - 6.63893i) q^{43} +(9.03757 + 9.03757i) q^{47} +0.986880i q^{49} +(4.78645 - 4.78645i) q^{53} +(-0.710948 + 0.484808i) q^{55} -8.10867 q^{59} -15.0948 q^{61} +(0.415517 - 2.19712i) q^{65} +(5.47501 - 5.47501i) q^{67} +1.23609i q^{71} +(-7.10730 - 7.10730i) q^{73} +(-0.667281 - 0.667281i) q^{77} -11.1545i q^{79} +(-5.98851 + 5.98851i) q^{83} +(2.35685 - 12.4623i) q^{85} -10.1275 q^{89} +2.45217 q^{91} +(-2.43281 + 1.65897i) q^{95} +(3.76737 - 3.76737i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 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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