LMFDB - Embedded newform 2340.2.y.a.53.5

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)

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);

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")

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			53.5
Character	$\chi$	$=$	2340.53
Dual form			2340.2.y.a.1457.5

$q$-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	$-0.981523\pi$
$11$		−	0.384834i		−	0.116032i	0.998316	−	0.0580159i	$-0.0184774\pi$
$12$		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	0.999639	−	0.0268843i	$-0.00855858\pi$
$17$	4.01077	+	4.01077i	0.972754	+	0.972754i	−0.0268843	+	0.999639i	$0.508559\pi$
$18$		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	−0.307038	−	0.951697i	$-0.599338\pi$
$23$	3.09168	−	3.09168i	0.644659	−	0.644659i	0.951697	+	0.307038i	$0.0993379\pi$
$24$		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	$-0.326591\pi$
$29$	5.58151			1.03646			0.518231	+	0.855241i	$0.326591\pi$
$30$		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	$-0.804713\pi$
$41$		−	7.37309i		−	1.15148i	0.817632	−	0.575742i	$-0.195287\pi$
$42$		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	$-0.867927\pi$
$47$	−9.03757	−	9.03757i	−1.31827	−	1.31827i	−0.403117	−	0.915149i	$-0.632073\pi$
$48$		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	−0.954781	−	0.297311i	$-0.903910\pi$
$53$	−4.78645	+	4.78645i	−0.657470	+	0.657470i	0.297311	+	0.954781i	$0.403910\pi$
$54$		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	$-0.323006\pi$
$59$	8.10867			1.05566			0.527829	+	0.849351i	$0.323006\pi$
$60$		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	$-0.976631\pi$
$71$		−	1.23609i		−	0.146697i	0.997306	−	0.0733486i	$-0.0233686\pi$
$72$		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	−0.297422	−	0.954746i	$-0.596127\pi$
$83$	5.98851	−	5.98851i	0.657324	−	0.657324i	0.954746	+	0.297422i	$0.0961268\pi$
$84$		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	$-0.319650\pi$
$89$	10.1275			1.07351			0.536755	+	0.843738i	$0.319650\pi$
$90$		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	$0.233607\pi$
$101$			13.4622i			1.33954i	−0.742570	+	0.669768i	$0.766393\pi$
$102$		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	$-0.948343\pi$
$107$	−11.8795	−	11.8795i	−1.14843	−	1.14843i	−0.161573	−	0.986861i	$-0.551657\pi$
$108$		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	−0.922046	−	0.387080i	$-0.873484\pi$
$113$	−5.68677	+	5.68677i	−0.534966	+	0.534966i	0.387080	+	0.922046i	$0.373484\pi$
$114$		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	$-0.586419\pi$
$131$		−	22.0526i		−	1.92674i	0.268171	−	0.963371i	$-0.413581\pi$
$132$		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	0.724025	−	0.689774i	$-0.242290\pi$
$137$	0.400904	+	0.400904i	0.0342516	+	0.0342516i	−0.689774	+	0.724025i	$0.742290\pi$
$138$		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	$-0.196267\pi$
$149$	19.9175			1.63171			0.815854	+	0.578258i	$0.196267\pi$
$150$		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	0.282488	−	0.959271i	$-0.408840\pi$
$167$	−8.74597	−	8.74597i	−0.676783	−	0.676783i	−0.959271	+	0.282488i	$0.908840\pi$
$168$		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	−0.378008	−	0.925802i	$-0.623391\pi$
$173$	7.20511	−	7.20511i	0.547795	−	0.547795i	0.925802	+	0.378008i	$0.123391\pi$
$174$		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	$-0.321619\pi$
$179$	14.2227			1.06305			0.531527	+	0.847042i	$0.321619\pi$
$180$		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	$-0.788351\pi$
$191$		−	17.0540i		−	1.23399i	0.786969	−	0.616993i	$-0.211649\pi$
$192$		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	0.655641	−	0.755073i	$-0.272399\pi$
$197$	−1.39560	−	1.39560i	−0.0994323	−	0.0994323i	−0.755073	+	0.655641i	$0.772399\pi$
$198$		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	0.951982	−	0.306154i	$-0.0990421\pi$
$227$	9.73039	+	9.73039i	0.645829	+	0.645829i	−0.306154	+	0.951982i	$0.599042\pi$
$228$		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	−0.414938	−	0.909850i	$-0.636197\pi$
$233$	7.55451	−	7.55451i	0.494912	−	0.494912i	0.909850	+	0.414938i	$0.136197\pi$
$234$		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	$-0.441212\pi$
$239$	5.67798			0.367278			0.183639	+	0.982994i	$0.441212\pi$
$240$		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	$0.134276\pi$
$251$			12.9735i			0.818881i	−0.912337	+	0.409440i	$0.865724\pi$
$252$		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	0.604466	−	0.796631i	$-0.293387\pi$
$257$	−3.08065	−	3.08065i	−0.192165	−	0.192165i	−0.796631	+	0.604466i	$0.793387\pi$
$258$		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	−0.462768	−	0.886479i	$-0.653144\pi$
$263$	6.87145	−	6.87145i	0.423712	−	0.423712i	0.886479	+	0.462768i	$0.153144\pi$
$264$		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	$-0.374208\pi$
$269$	12.6283			0.769960			0.384980	+	0.922925i	$0.374208\pi$
$270$		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	$-0.714447\pi$
$281$		−	26.2012i		−	1.56303i	0.623886	−	0.781515i	$-0.285553\pi$
$282$		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	−0.627909	−	0.778287i	$-0.716089\pi$
$293$	2.57405	−	2.57405i	0.150377	−	0.150377i	0.778287	+	0.627909i	$0.216089\pi$
$294$		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	$-0.959907\pi$
$311$		−	4.43080i		−	0.251248i	0.992078	−	0.125624i	$-0.0400932\pi$
$312$		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	0.178053	−	0.984021i	$-0.443020\pi$
$317$	−14.3499	−	14.3499i	−0.805968	−	0.805968i	−0.984021	+	0.178053i	$0.943020\pi$
$318$		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	0.609951	−	0.792439i	$-0.291189\pi$
$347$	−3.39937	−	3.39937i	−0.182488	−	0.182488i	−0.792439	+	0.609951i	$0.791189\pi$
$348$		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	−0.999704	−	0.0243238i	$-0.992257\pi$
$353$	−18.3257	+	18.3257i	−0.975380	+	0.975380i	0.0243238	+	0.999704i	$0.492257\pi$
$354$		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	$-0.497860\pi$
$359$	0.254736			0.0134445			0.00672224	+	0.999977i	$0.497860\pi$
$360$		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	−0.948943	−	0.315448i	$-0.897845\pi$
$383$	−12.3978	+	12.3978i	−0.633495	+	0.633495i	0.315448	+	0.948943i	$0.397845\pi$
$384$		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	$-0.533163\pi$
$389$	−4.10223			−0.207991			−0.103996	+	0.994578i	$0.533163\pi$
$390$		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	$-0.943402\pi$
$401$		−	7.08373i		−	0.353745i	0.984234	−	0.176872i	$-0.0565980\pi$
$402$		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	$-0.901768\pi$
$419$	−39.0049			−1.90552			−0.952758	+	0.303730i	$0.901768\pi$
$420$		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	$-0.923247\pi$
$431$		−	9.91513i		−	0.477595i	0.971069	−	0.238798i	$-0.0767532\pi$
$432$		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	−0.922856	−	0.385145i	$-0.874151\pi$
$443$	−11.3175	+	11.3175i	−0.537711	+	0.537711i	0.385145	+	0.922856i	$0.374151\pi$
$444$		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	$-0.548888\pi$
$449$	−6.48334			−0.305968			−0.152984	+	0.988229i	$0.548888\pi$
$450$		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	$-0.931821\pi$
$461$		−	9.12753i		−	0.425112i	0.977149	−	0.212556i	$-0.0681787\pi$
$462$		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	0.828657	−	0.559756i	$-0.189105\pi$
$467$	5.81099	+	5.81099i	0.268901	+	0.268901i	−0.559756	+	0.828657i	$0.689105\pi$
$468$		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	$-0.355942\pi$
$479$	19.1406			0.874558			0.437279	+	0.899326i	$0.355942\pi$
$480$		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	$-0.987211\pi$
$491$		−	1.78011i		−	0.0803350i	0.999193	−	0.0401675i	$-0.0127892\pi$
$492$		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	−0.665315	−	0.746563i	$-0.731703\pi$
$503$	1.82220	−	1.82220i	0.0812477	−	0.0812477i	0.746563	+	0.665315i	$0.231703\pi$
$504$		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	$-0.554958\pi$
$509$	−7.75196			−0.343600			−0.171800	+	0.985132i	$0.554958\pi$
$510$		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	$0.0413818\pi$
$521$			5.91813i			0.259278i	−0.991561	+	0.129639i	$0.958618\pi$
$522$		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	0.937068	−	0.349148i	$-0.113529\pi$
$557$	13.8754	+	13.8754i	0.587920	+	0.587920i	−0.349148	+	0.937068i	$0.613529\pi$
$558$		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	−0.671500	−	0.741005i	$-0.734350\pi$
$563$	1.64918	−	1.64918i	0.0695049	−	0.0695049i	0.741005	+	0.671500i	$0.234350\pi$
$564$		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	$-0.525194\pi$
$569$	−3.77209			−0.158134			−0.0790672	+	0.996869i	$0.525194\pi$
$570$		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	$-0.976248\pi$
$587$	−25.9668	−	25.9668i	−1.07177	−	1.07177i	−0.0745491	−	0.997217i	$-0.523752\pi$
$588$		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	−0.00192013	−	0.999998i	$-0.500611\pi$
$593$	24.3048	−	24.3048i	0.998078	−	0.998078i	0.999998	+	0.00192013i	$0.000611197\pi$
$594$		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	$-0.608341\pi$
$599$	−16.3406			−0.667659			−0.333829	+	0.942633i	$0.608341\pi$
$600$		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	0.847468	−	0.530846i	$-0.178126\pi$
$617$	7.86473	+	7.86473i	0.316622	+	0.316622i	−0.530846	+	0.847468i	$0.678126\pi$
$618$		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	$0.0632662\pi$
$641$			9.99809i			0.394901i	−0.980313	+	0.197450i	$0.936734\pi$
$642$		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	0.175308	−	0.984514i	$-0.443908\pi$
$647$	−20.5831	−	20.5831i	−0.809205	−	0.809205i	−0.984514	+	0.175308i	$0.943908\pi$
$648$		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	−0.938953	−	0.344044i	$-0.888203\pi$
$653$	−15.2022	+	15.2022i	−0.594909	+	0.594909i	0.344044	+	0.938953i	$0.388203\pi$
$654$		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	$-0.466771\pi$
$659$	5.34993			0.208403			0.104202	+	0.994556i	$0.466771\pi$
$660$		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	$-0.921086\pi$
$677$	−31.6084	−	31.6084i	−1.21481	−	1.21481i	−0.245384	−	0.969426i	$-0.578914\pi$
$678$		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	−0.727575	−	0.686029i	$-0.759352\pi$
$683$	−1.08577	+	1.08577i	−0.0415459	+	0.0415459i	0.686029	+	0.727575i	$0.259352\pi$
$684$		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	$0.128160\pi$
$701$			20.7488i			0.783671i	−0.920035	+	0.391836i	$0.871840\pi$
$702$		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	$-0.198321\pi$
$719$	43.5519			1.62421			0.812106	+	0.583511i	$0.198321\pi$
$720$		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	−0.928081	−	0.372378i	$-0.878543\pi$
$743$	−15.1474	+	15.1474i	−0.555703	+	0.555703i	0.372378	+	0.928081i	$0.378543\pi$
$744$		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	$0.268219\pi$
$761$			41.1807i			1.49280i	−0.665499	+	0.746399i	$0.731781\pi$
$762$		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	−0.864424	−	0.502763i	$-0.832317\pi$
$773$	−10.0552	+	10.0552i	−0.361662	+	0.361662i	0.502763	+	0.864424i	$0.332317\pi$
$774$		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	0.697566	−	0.716521i	$-0.254267\pi$
$797$	−0.535126	−	0.535126i	−0.0189551	−	0.0189551i	−0.716521	+	0.697566i	$0.754267\pi$
$798$		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	$-0.323343\pi$
$809$	29.9749			1.05386			0.526930	+	0.849909i	$0.323343\pi$
$810$		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	$-0.946803\pi$
$821$		−	9.53271i		−	0.332694i	0.986067	−	0.166347i	$-0.0531971\pi$
$822$		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	$0.00829187\pi$
$827$	29.4969	+	29.4969i	1.02571	+	1.02571i	0.0260467	+	0.999661i	$0.491708\pi$
$828$		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	$-0.416708\pi$
$839$	14.9864			0.517389			0.258694	+	0.965959i	$0.416708\pi$
$840$		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	0.443567	−	0.896241i	$-0.353713\pi$
$857$	−13.2518	−	13.2518i	−0.452674	−	0.452674i	−0.896241	+	0.443567i	$0.853713\pi$
$858$		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	−0.901466	−	0.432850i	$-0.857508\pi$
$863$	−13.7665	+	13.7665i	−0.468617	+	0.468617i	0.432850	+	0.901466i	$0.357508\pi$
$864$		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	$0.0508313\pi$
$881$			9.43955i			0.318027i	−0.987276	+	0.159013i	$0.949169\pi$
$882$		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	0.986284	−	0.165059i	$-0.0527813\pi$
$887$	24.4582	+	24.4582i	0.821225	+	0.821225i	−0.165059	+	0.986284i	$0.552781\pi$
$888$		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	$-0.769037\pi$
$911$		−	40.0573i		−	1.32716i	0.748106	−	0.663579i	$-0.230963\pi$
$912$		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	$-0.935200\pi$
$929$	−59.7002			−1.95870			−0.979350	+	0.202173i	$0.935200\pi$
$930$		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	$-0.914925\pi$
$941$		−	16.2029i		−	0.528201i	0.964495	−	0.264101i	$-0.0850751\pi$
$942$		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	0.490480	−	0.871452i	$-0.336821\pi$
$947$	−11.7238	−	11.7238i	−0.380972	−	0.380972i	−0.871452	+	0.490480i	$0.836821\pi$
$948$		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	$-0.362944\pi$
$953$	40.9382	−	40.9382i	1.32612	−	1.32612i	0.908726	−	0.417392i	$-0.137056\pi$
$954$		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	$0.296470\pi$
$971$			50.0100i			1.60490i	−0.596721	+	0.802449i	$0.703530\pi$
$972$		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	0.0890885	−	0.996024i	$-0.471605\pi$
$977$	−28.3481	−	28.3481i	−0.906935	−	0.906935i	−0.996024	+	0.0890885i	$0.971605\pi$
$978$		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	−0.916118	−	0.400908i	$-0.868695\pi$
$983$	−16.1533	+	16.1533i	−0.515210	+	0.515210i	0.400908	+	0.916118i	$0.368695\pi$
$984$		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.a.53.5	✓	24
3.2	odd	2		2340.2.y.b.53.8	yes	24
5.2	odd	4		2340.2.y.b.1457.8	yes	24
15.2	even	4	inner	2340.2.y.a.1457.5	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.5	✓	24	1.1	even	1	trivial
2340.2.y.a.1457.5	yes	24	15.2	even	4	inner
2340.2.y.b.53.8	yes	24	3.2	odd	2
2340.2.y.b.1457.8	yes	24	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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