LMFDB - Embedded newform 2340.2.y.b.1457.6

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			1457.6
Character	$\chi$	$=$	2340.1457
Dual form			2340.2.y.b.53.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.686017 - 2.12823i) q^{5} +(0.778541 + 0.778541i) q^{7} -5.13019i q^{11} +(-0.707107 + 0.707107i) q^{13} +(2.51872 - 2.51872i) q^{17} -0.758530i q^{19} +(-0.142183 - 0.142183i) q^{23} +(-4.05876 + 2.92001i) q^{25} +2.35974 q^{29} -2.70016 q^{31} +(1.12283 - 2.19101i) q^{35} +(-3.12130 - 3.12130i) q^{37} +5.42303i q^{41} +(-1.79109 + 1.79109i) q^{43} +(-2.72344 + 2.72344i) q^{47} -5.78775i q^{49} +(-7.53548 - 7.53548i) q^{53} +(-10.9183 + 3.51940i) q^{55} +4.73391 q^{59} +5.36238 q^{61} +(1.98998 + 1.01980i) q^{65} +(-3.20584 - 3.20584i) q^{67} -5.57259i q^{71} +(-10.3790 + 10.3790i) q^{73} +(3.99407 - 3.99407i) q^{77} -15.2434i q^{79} +(8.68060 + 8.68060i) q^{83} +(-7.08831 - 3.63254i) q^{85} -14.8667 q^{89} -1.10102 q^{91} +(-1.61433 + 0.520365i) q^{95} +(-2.00993 - 2.00993i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

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

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
$5$	−0.686017	−	2.12823i	−0.306796	−	0.951775i
$5$	−0.686017	−	2.12823i	−0.306796	−	0.951775i
$6$		0			0
$7$	0.778541	+	0.778541i	0.294261	+	0.294261i	0.838761	−	0.544500i	$-0.183281\pi$
$7$	0.778541	+	0.778541i	0.294261	+	0.294261i	−0.544500	+	0.838761i	$0.683281\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.13019i		−	1.54681i	−0.633911	−	0.773406i	$-0.718551\pi$
$11$		−	5.13019i		−	1.54681i	0.633911	−	0.773406i	$-0.281449\pi$
$12$		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
$17$	2.51872	−	2.51872i	0.610879	−	0.610879i	−0.332296	−	0.943175i	$-0.607823\pi$
$17$	2.51872	−	2.51872i	0.610879	−	0.610879i	0.943175	+	0.332296i	$0.107823\pi$
$18$		0			0
$19$		−	0.758530i		−	0.174019i	−0.996207	−	0.0870094i	$-0.972269\pi$
$19$		−	0.758530i		−	0.174019i	0.996207	−	0.0870094i	$-0.0277310\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.142183	−	0.142183i	−0.0296472	−	0.0296472i	0.692128	−	0.721775i	$-0.256673\pi$
$23$	−0.142183	−	0.142183i	−0.0296472	−	0.0296472i	−0.721775	+	0.692128i	$0.756673\pi$
$24$		0			0
$25$	−4.05876	+	2.92001i	−0.811752	+	0.584002i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.35974			0.438192			0.219096	−	0.975703i	$-0.429689\pi$
$29$	2.35974			0.438192			0.219096	+	0.975703i	$0.429689\pi$
$30$		0			0
$31$	−2.70016			−0.484964			−0.242482	−	0.970156i	$-0.577961\pi$
$31$	−2.70016			−0.484964			−0.242482	+	0.970156i	$0.577961\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.12283	−	2.19101i	0.189792	−	0.370349i
$36$		0			0
$37$	−3.12130	−	3.12130i	−0.513139	−	0.513139i	0.402348	−	0.915487i	$-0.368194\pi$
$37$	−3.12130	−	3.12130i	−0.513139	−	0.513139i	−0.915487	+	0.402348i	$0.868194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.42303i			0.846935i	0.905911	+	0.423468i	$0.139187\pi$
$41$			5.42303i			0.846935i	−0.905911	+	0.423468i	$0.860813\pi$
$42$		0			0
$43$	−1.79109	+	1.79109i	−0.273139	+	0.273139i	−0.830363	−	0.557223i	$-0.811867\pi$
$43$	−1.79109	+	1.79109i	−0.273139	+	0.273139i	0.557223	+	0.830363i	$0.311867\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.72344	+	2.72344i	−0.397255	+	0.397255i	−0.877264	−	0.480009i	$-0.840634\pi$
$47$	−2.72344	+	2.72344i	−0.397255	+	0.397255i	0.480009	+	0.877264i	$0.340634\pi$
$48$		0			0
$49$		−	5.78775i		−	0.826821i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.53548	−	7.53548i	−1.03508	−	1.03508i	−0.999362	−	0.0357156i	$-0.988629\pi$
$53$	−7.53548	−	7.53548i	−1.03508	−	1.03508i	−0.0357156	−	0.999362i	$-0.511371\pi$
$54$		0			0
$55$	−10.9183	+	3.51940i	−1.47222	+	0.474556i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.73391			0.616303			0.308151	−	0.951337i	$-0.400290\pi$
$59$	4.73391			0.616303			0.308151	+	0.951337i	$0.400290\pi$
$60$		0			0
$61$	5.36238			0.686582			0.343291	−	0.939229i	$-0.388458\pi$
$61$	5.36238			0.686582			0.343291	+	0.939229i	$0.388458\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.98998	+	1.01980i	0.246826	+	0.126491i
$66$		0			0
$67$	−3.20584	−	3.20584i	−0.391656	−	0.391656i	0.483622	−	0.875277i	$-0.339321\pi$
$67$	−3.20584	−	3.20584i	−0.391656	−	0.391656i	−0.875277	+	0.483622i	$0.839321\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.57259i		−	0.661344i	−0.943746	−	0.330672i	$-0.892725\pi$
$71$		−	5.57259i		−	0.661344i	0.943746	−	0.330672i	$-0.107275\pi$
$72$		0			0
$73$	−10.3790	+	10.3790i	−1.21477	+	1.21477i	−0.245327	+	0.969440i	$0.578895\pi$
$73$	−10.3790	+	10.3790i	−1.21477	+	1.21477i	−0.969440	+	0.245327i	$0.921105\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.99407	−	3.99407i	0.455166	−	0.455166i
$78$		0			0
$79$		−	15.2434i		−	1.71502i	−0.514466	−	0.857511i	$-0.672010\pi$
$79$		−	15.2434i		−	1.71502i	0.514466	−	0.857511i	$-0.327990\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.68060	+	8.68060i	0.952820	+	0.952820i	0.998936	−	0.0461163i	$-0.0146845\pi$
$83$	8.68060	+	8.68060i	0.952820	+	0.952820i	−0.0461163	+	0.998936i	$0.514684\pi$
$84$		0			0
$85$	−7.08831	−	3.63254i	−0.768835	−	0.394004i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.8667			−1.57586			−0.787932	−	0.615763i	$-0.788848\pi$
$89$	−14.8667			−1.57586			−0.787932	+	0.615763i	$0.788848\pi$
$90$		0			0
$91$	−1.10102			−0.115419
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.61433	+	0.520365i	−0.165627	+	0.0533883i
$96$		0			0
$97$	−2.00993	−	2.00993i	−0.204077	−	0.204077i	0.597667	−	0.801744i	$-0.296095\pi$
$97$	−2.00993	−	2.00993i	−0.204077	−	0.204077i	−0.801744	+	0.597667i	$0.796095\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	16.9142i		−	1.68303i	−0.540235	−	0.841514i	$-0.681665\pi$
$101$		−	16.9142i		−	1.68303i	0.540235	−	0.841514i	$-0.318335\pi$
$102$		0			0
$103$	1.58678	−	1.58678i	0.156351	−	0.156351i	−0.624597	−	0.780947i	$-0.714737\pi$
$103$	1.58678	−	1.58678i	0.156351	−	0.156351i	0.780947	+	0.624597i	$0.214737\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.299369	+	0.299369i	−0.0289411	+	0.0289411i	−0.721429	−	0.692488i	$-0.756514\pi$
$107$	−0.299369	+	0.299369i	−0.0289411	+	0.0289411i	0.692488	+	0.721429i	$0.256514\pi$
$108$		0			0
$109$			7.22041i			0.691590i	0.938310	+	0.345795i	$0.112391\pi$
$109$			7.22041i			0.691590i	−0.938310	+	0.345795i	$0.887609\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.98107	−	7.98107i	−0.750796	−	0.750796i	0.223832	−	0.974628i	$-0.428143\pi$
$113$	−7.98107	−	7.98107i	−0.750796	−	0.750796i	−0.974628	+	0.223832i	$0.928143\pi$
$114$		0			0
$115$	−0.205058	+	0.400138i	−0.0191218	+	0.0373131i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.92185			0.359516
$120$		0			0
$121$	−15.3189			−1.39263
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.99885	+	6.63481i	0.804881	+	0.593436i
$126$		0			0
$127$	3.91374	+	3.91374i	0.347288	+	0.347288i	0.859099	−	0.511810i	$-0.171025\pi$
$127$	3.91374	+	3.91374i	0.347288	+	0.347288i	−0.511810	+	0.859099i	$0.671025\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	5.78460i		−	0.505403i	−0.967544	−	0.252701i	$-0.918681\pi$
$131$		−	5.78460i		−	0.505403i	0.967544	−	0.252701i	$-0.0813190\pi$
$132$		0			0
$133$	0.590547	−	0.590547i	0.0512069	−	0.0512069i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.867994	−	0.867994i	0.0741577	−	0.0741577i	−0.669055	−	0.743213i	$-0.733301\pi$
$137$	0.867994	−	0.867994i	0.0741577	−	0.0741577i	0.743213	+	0.669055i	$0.233301\pi$
$138$		0			0
$139$			3.44197i			0.291944i	0.989289	+	0.145972i	$0.0466310\pi$
$139$			3.44197i			0.291944i	−0.989289	+	0.145972i	$0.953369\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.62760	+	3.62760i	0.303355	+	0.303355i
$144$		0			0
$145$	−1.61882	−	5.02207i	−0.134436	−	0.417060i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.8941			−1.05633			−0.528163	−	0.849143i	$-0.677119\pi$
$149$	−12.8941			−1.05633			−0.528163	+	0.849143i	$0.677119\pi$
$150$		0			0
$151$	−21.3913			−1.74080			−0.870401	−	0.492344i	$-0.836140\pi$
$151$	−21.3913			−1.74080			−0.870401	+	0.492344i	$0.836140\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.85236	+	5.74658i	0.148785	+	0.461576i
$156$		0			0
$157$	4.05669	+	4.05669i	0.323759	+	0.323759i	0.850207	−	0.526448i	$-0.176477\pi$
$157$	4.05669	+	4.05669i	0.323759	+	0.323759i	−0.526448	+	0.850207i	$0.676477\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.221390i		−	0.0174480i
$162$		0			0
$163$	−4.04609	+	4.04609i	−0.316914	+	0.316914i	−0.847581	−	0.530666i	$-0.821942\pi$
$163$	−4.04609	+	4.04609i	−0.316914	+	0.316914i	0.530666	+	0.847581i	$0.321942\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.8454	−	12.8454i	0.994006	−	0.994006i	−0.00597581	−	0.999982i	$-0.501902\pi$
$167$	12.8454	−	12.8454i	0.994006	−	0.994006i	0.999982	+	0.00597581i	$0.00190217\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.77705	−	3.77705i	−0.287164	−	0.287164i	0.548794	−	0.835958i	$-0.315087\pi$
$173$	−3.77705	−	3.77705i	−0.287164	−	0.287164i	−0.835958	+	0.548794i	$0.815087\pi$
$174$		0			0
$175$	−5.43326	−	0.886563i	−0.410716	−	0.0670179i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.58934			−0.343024			−0.171512	−	0.985182i	$-0.554865\pi$
$179$	−4.58934			−0.343024			−0.171512	+	0.985182i	$0.554865\pi$
$180$		0			0
$181$	−2.61145			−0.194107			−0.0970537	−	0.995279i	$-0.530942\pi$
$181$	−2.61145			−0.194107			−0.0970537	+	0.995279i	$0.530942\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.50159	+	8.78413i	−0.330964	+	0.645822i
$186$		0			0
$187$	−12.9215	−	12.9215i	−0.944915	−	0.944915i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.6654i		−	0.771722i	−0.922557	−	0.385861i	$-0.873904\pi$
$191$		−	10.6654i		−	0.771722i	0.922557	−	0.385861i	$-0.126096\pi$
$192$		0			0
$193$	−8.61898	+	8.61898i	−0.620408	+	0.620408i	−0.945636	−	0.325228i	$-0.894559\pi$
$193$	−8.61898	+	8.61898i	−0.620408	+	0.620408i	0.325228	+	0.945636i	$0.394559\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.30581	−	2.30581i	0.164282	−	0.164282i	−0.620178	−	0.784461i	$-0.712940\pi$
$197$	2.30581	−	2.30581i	0.164282	−	0.164282i	0.784461	+	0.620178i	$0.212940\pi$
$198$		0			0
$199$		−	6.80773i		−	0.482587i	−0.970452	−	0.241293i	$-0.922428\pi$
$199$		−	6.80773i		−	0.482587i	0.970452	−	0.241293i	$-0.0775716\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.83715	+	1.83715i	0.128943	+	0.128943i
$204$		0			0
$205$	11.5415	−	3.72029i	0.806092	−	0.259837i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.89141			−0.269174
$210$		0			0
$211$	14.6035			1.00534			0.502672	−	0.864477i	$-0.332350\pi$
$211$	14.6035			1.00534			0.502672	+	0.864477i	$0.332350\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.04059	+	2.58314i	0.343765	+	0.176169i
$216$		0			0
$217$	−2.10219	−	2.10219i	−0.142706	−	0.142706i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3.56201i			0.239606i
$222$		0			0
$223$	−0.914357	+	0.914357i	−0.0612299	+	0.0612299i	−0.737059	−	0.675829i	$-0.763786\pi$
$223$	−0.914357	+	0.914357i	−0.0612299	+	0.0612299i	0.675829	+	0.737059i	$0.263786\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.9786	−	16.9786i	1.12691	−	1.12691i	0.136230	−	0.990677i	$-0.456501\pi$
$227$	16.9786	−	16.9786i	1.12691	−	1.12691i	0.990677	−	0.136230i	$-0.0434985\pi$
$228$		0			0
$229$			6.92261i			0.457459i	0.973490	+	0.228730i	$0.0734572\pi$
$229$			6.92261i			0.457459i	−0.973490	+	0.228730i	$0.926543\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.53548	−	4.53548i	−0.297129	−	0.297129i	0.542759	−	0.839888i	$-0.317380\pi$
$233$	−4.53548	−	4.53548i	−0.297129	−	0.297129i	−0.839888	+	0.542759i	$0.817380\pi$
$234$		0			0
$235$	7.66446	+	3.92780i	0.499974	+	0.256221i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.77740			−0.438393			−0.219197	−	0.975681i	$-0.570344\pi$
$239$	−6.77740			−0.438393			−0.219197	+	0.975681i	$0.570344\pi$
$240$		0			0
$241$	12.1896			0.785201			0.392600	−	0.919709i	$-0.371576\pi$
$241$	12.1896			0.785201			0.392600	+	0.919709i	$0.371576\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.3177	+	3.97050i	−0.786948	+	0.253666i
$246$		0			0
$247$	0.536362	+	0.536362i	0.0341279	+	0.0341279i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	19.0872i		−	1.20477i	−0.798204	−	0.602387i	$-0.794216\pi$
$251$		−	19.0872i		−	1.20477i	0.798204	−	0.602387i	$-0.205784\pi$
$252$		0			0
$253$	−0.729425	+	0.729425i	−0.0458586	+	0.0458586i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.57385	−	2.57385i	0.160552	−	0.160552i	−0.622259	−	0.782811i	$-0.713785\pi$
$257$	2.57385	−	2.57385i	0.160552	−	0.160552i	0.782811	+	0.622259i	$0.213785\pi$
$258$		0			0
$259$		−	4.86013i		−	0.301993i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.1667	+	20.1667i	1.24353	+	1.24353i	0.958526	+	0.285007i	$0.0919958\pi$
$263$	20.1667	+	20.1667i	1.24353	+	1.24353i	0.285007	+	0.958526i	$0.408004\pi$
$264$		0			0
$265$	−10.8678	+	21.2067i	−0.667603	+	1.30272i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.3802			−1.36455			−0.682273	−	0.731098i	$-0.739008\pi$
$269$	−22.3802			−1.36455			−0.682273	+	0.731098i	$0.739008\pi$
$270$		0			0
$271$	−4.01955			−0.244170			−0.122085	−	0.992520i	$-0.538958\pi$
$271$	−4.01955			−0.244170			−0.122085	+	0.992520i	$0.538958\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.9802	+	20.8222i	0.903342	+	1.25563i
$276$		0			0
$277$	13.4025	+	13.4025i	0.805279	+	0.805279i	0.983915	−	0.178636i	$-0.0571685\pi$
$277$	13.4025	+	13.4025i	0.805279	+	0.805279i	−0.178636	+	0.983915i	$0.557168\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.49504i		−	0.208496i	−0.994551	−	0.104248i	$-0.966756\pi$
$281$		−	3.49504i		−	0.208496i	0.994551	−	0.104248i	$-0.0332436\pi$
$282$		0			0
$283$	18.9358	−	18.9358i	1.12562	−	1.12562i	0.134736	−	0.990882i	$-0.456982\pi$
$283$	18.9358	−	18.9358i	1.12562	−	1.12562i	0.990882	−	0.134736i	$-0.0430185\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.22205	+	4.22205i	−0.249220	+	0.249220i
$288$		0			0
$289$			4.31212i			0.253654i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.65762	+	3.65762i	0.213680	+	0.213680i	0.805829	−	0.592148i	$-0.201720\pi$
$293$	3.65762	+	3.65762i	0.213680	+	0.213680i	−0.592148	+	0.805829i	$0.701720\pi$
$294$		0			0
$295$	−3.24755	−	10.0749i	−0.189079	−	0.586582i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.201077			0.0116286
$300$		0			0
$301$	−2.78888			−0.160748
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.67868	−	11.4124i	−0.210641	−	0.653472i
$306$		0			0
$307$	21.3299	+	21.3299i	1.21736	+	1.21736i	0.968553	+	0.248809i	$0.0800392\pi$
$307$	21.3299	+	21.3299i	1.21736	+	1.21736i	0.248809	+	0.968553i	$0.419961\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.2396i		−	0.750749i	−0.926873	−	0.375374i	$-0.877514\pi$
$311$		−	13.2396i		−	0.750749i	0.926873	−	0.375374i	$-0.122486\pi$
$312$		0			0
$313$	7.30455	−	7.30455i	0.412877	−	0.412877i	−0.469862	−	0.882740i	$-0.655696\pi$
$313$	7.30455	−	7.30455i	0.412877	−	0.412877i	0.882740	+	0.469862i	$0.155696\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.3214	−	15.3214i	0.860538	−	0.860538i	−0.130863	−	0.991401i	$-0.541775\pi$
$317$	15.3214	−	15.3214i	0.860538	−	0.860538i	0.991401	+	0.130863i	$0.0417746\pi$
$318$		0			0
$319$		−	12.1059i		−	0.677800i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.91052	−	1.91052i	−0.106304	−	0.106304i
$324$		0			0
$325$	0.805217	−	4.93474i	0.0446654	−	0.273730i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.24063			−0.233793
$330$		0			0
$331$	31.9053			1.75367			0.876837	−	0.480788i	$-0.159649\pi$
$331$	31.9053			1.75367			0.876837	+	0.480788i	$0.159649\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.62351	+	9.02204i	−0.252610	+	0.492927i
$336$		0			0
$337$	19.3880	+	19.3880i	1.05613	+	1.05613i	0.998328	+	0.0578015i	$0.0184091\pi$
$337$	19.3880	+	19.3880i	1.05613	+	1.05613i	0.0578015	+	0.998328i	$0.481591\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			13.8524i			0.750147i
$342$		0			0
$343$	9.95579	−	9.95579i	0.537562	−	0.537562i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.3148	−	11.3148i	0.607411	−	0.607411i	−0.334857	−	0.942269i	$-0.608688\pi$
$347$	11.3148	−	11.3148i	0.607411	−	0.607411i	0.942269	+	0.334857i	$0.108688\pi$
$348$		0			0
$349$			7.73939i			0.414280i	0.978311	+	0.207140i	$0.0664156\pi$
$349$			7.73939i			0.414280i	−0.978311	+	0.207140i	$0.933584\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.0426	+	20.0426i	1.06676	+	1.06676i	0.997606	+	0.0691538i	$0.0220299\pi$
$353$	20.0426	+	20.0426i	1.06676	+	1.06676i	0.0691538	+	0.997606i	$0.477970\pi$
$354$		0			0
$355$	−11.8598	+	3.82289i	−0.629451	+	0.202898i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.1393			−1.11569			−0.557846	−	0.829945i	$-0.688372\pi$
$359$	−21.1393			−1.11569			−0.557846	+	0.829945i	$0.688372\pi$
$360$		0			0
$361$	18.4246			0.969717
$362$		0			0
$363$		0			0
$364$		0			0
$365$	29.2091	+	14.9687i	1.52887	+	0.783499i
$366$		0			0
$367$	−24.7535	−	24.7535i	−1.29212	−	1.29212i	−0.933471	−	0.358654i	$-0.883236\pi$
$367$	−24.7535	−	24.7535i	−1.29212	−	1.29212i	−0.358654	−	0.933471i	$-0.616764\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	11.7334i		−	0.609166i
$372$		0			0
$373$	−12.5014	+	12.5014i	−0.647298	+	0.647298i	−0.952339	−	0.305041i	$-0.901330\pi$
$373$	−12.5014	+	12.5014i	−0.647298	+	0.647298i	0.305041	+	0.952339i	$0.401330\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.66858	+	1.66858i	−0.0859365	+	0.0859365i
$378$		0			0
$379$		−	20.2385i		−	1.03958i	−0.854293	−	0.519792i	$-0.826009\pi$
$379$		−	20.2385i		−	1.03958i	0.854293	−	0.519792i	$-0.173991\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.41348	−	8.41348i	−0.429909	−	0.429909i	0.458689	−	0.888597i	$-0.348319\pi$
$383$	−8.41348	−	8.41348i	−0.429909	−	0.429909i	−0.888597	+	0.458689i	$0.848319\pi$
$384$		0			0
$385$	−11.2403	−	5.76031i	−0.572859	−	0.293573i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.5963			−0.638656			−0.319328	−	0.947644i	$-0.603457\pi$
$389$	−12.5963			−0.638656			−0.319328	+	0.947644i	$0.603457\pi$
$390$		0			0
$391$	−0.716237			−0.0362217
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−32.4416	+	10.4573i	−1.63232	+	0.526162i
$396$		0			0
$397$	−6.90354	−	6.90354i	−0.346479	−	0.346479i	0.512317	−	0.858796i	$-0.328787\pi$
$397$	−6.90354	−	6.90354i	−0.346479	−	0.346479i	−0.858796	+	0.512317i	$0.828787\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			26.2841i			1.31256i	0.754516	+	0.656282i	$0.227872\pi$
$401$			26.2841i			1.31256i	−0.754516	+	0.656282i	$0.772128\pi$
$402$		0			0
$403$	1.90930	−	1.90930i	0.0951092	−	0.0951092i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.0129	+	16.0129i	−0.793729	+	0.793729i
$408$		0			0
$409$			5.80896i			0.287234i	0.989633	+	0.143617i	$0.0458734\pi$
$409$			5.80896i			0.287234i	−0.989633	+	0.143617i	$0.954127\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.68555	+	3.68555i	0.181354	+	0.181354i
$414$		0			0
$415$	12.5193	−	24.4294i	0.614549	−	1.19919i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.46952			−0.267203			−0.133602	−	0.991035i	$-0.542654\pi$
$419$	−5.46952			−0.267203			−0.133602	+	0.991035i	$0.542654\pi$
$420$		0			0
$421$	18.8828			0.920289			0.460144	−	0.887844i	$-0.347798\pi$
$421$	18.8828			0.920289			0.460144	+	0.887844i	$0.347798\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.86819	+	17.5776i	−0.139127	+	0.852637i
$426$		0			0
$427$	4.17483	+	4.17483i	0.202034	+	0.202034i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			17.0150i			0.819586i	0.912179	+	0.409793i	$0.134399\pi$
$431$			17.0150i			0.819586i	−0.912179	+	0.409793i	$0.865601\pi$
$432$		0			0
$433$	−26.0681	+	26.0681i	−1.25275	+	1.25275i	−0.298269	+	0.954482i	$0.596409\pi$
$433$	−26.0681	+	26.0681i	−1.25275	+	1.25275i	−0.954482	+	0.298269i	$0.903591\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.107850	+	0.107850i	−0.00515916	+	0.00515916i
$438$		0			0
$439$		−	25.3848i		−	1.21155i	−0.795635	−	0.605776i	$-0.792863\pi$
$439$		−	25.3848i		−	1.21155i	0.795635	−	0.605776i	$-0.207137\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.3418	+	21.3418i	1.01398	+	1.01398i	0.999901	+	0.0140806i	$0.00448213\pi$
$443$	21.3418	+	21.3418i	1.01398	+	1.01398i	0.0140806	+	0.999901i	$0.495518\pi$
$444$		0			0
$445$	10.1988	+	31.6397i	0.483469	+	1.49987i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−28.0453			−1.32354			−0.661769	−	0.749708i	$-0.730194\pi$
$449$	−28.0453			−1.32354			−0.661769	+	0.749708i	$0.730194\pi$
$450$		0			0
$451$	27.8212			1.31005
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.755322	+	2.34324i	0.0354100	+	0.109853i
$456$		0			0
$457$	23.5875	+	23.5875i	1.10338	+	1.10338i	0.994000	+	0.109376i	$0.0348852\pi$
$457$	23.5875	+	23.5875i	1.10338	+	1.10338i	0.109376	+	0.994000i	$0.465115\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			5.61612i			0.261569i	0.991411	+	0.130784i	$0.0417496\pi$
$461$			5.61612i			0.261569i	−0.991411	+	0.130784i	$0.958250\pi$
$462$		0			0
$463$	−15.4155	+	15.4155i	−0.716417	+	0.716417i	−0.967870	−	0.251452i	$-0.919092\pi$
$463$	−15.4155	+	15.4155i	−0.716417	+	0.716417i	0.251452	+	0.967870i	$0.419092\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.9097	+	11.9097i	−0.551113	+	0.551113i	−0.926762	−	0.375649i	$-0.877420\pi$
$467$	−11.9097	+	11.9097i	−0.551113	+	0.551113i	0.375649	+	0.926762i	$0.377420\pi$
$468$		0			0
$469$		−	4.99176i		−	0.230498i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.18865	+	9.18865i	0.422495	+	0.422495i
$474$		0			0
$475$	2.21492	+	3.07869i	0.101627	+	0.141260i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.6094			0.758902			0.379451	−	0.925212i	$-0.376113\pi$
$479$	16.6094			0.758902			0.379451	+	0.925212i	$0.376113\pi$
$480$		0			0
$481$	4.41419			0.201270
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.89875	+	5.65645i	−0.131626	+	0.256846i
$486$		0			0
$487$	−6.46072	−	6.46072i	−0.292763	−	0.292763i	0.545408	−	0.838171i	$-0.316375\pi$
$487$	−6.46072	−	6.46072i	−0.292763	−	0.292763i	−0.838171	+	0.545408i	$0.816375\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			21.8307i			0.985208i	0.870254	+	0.492604i	$0.163955\pi$
$491$			21.8307i			0.985208i	−0.870254	+	0.492604i	$0.836045\pi$
$492$		0			0
$493$	5.94351	−	5.94351i	0.267682	−	0.267682i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.33849	−	4.33849i	0.194608	−	0.194608i
$498$		0			0
$499$		−	24.9183i		−	1.11550i	−0.830010	−	0.557749i	$-0.811665\pi$
$499$		−	24.9183i		−	1.11550i	0.830010	−	0.557749i	$-0.188335\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20.7963	+	20.7963i	0.927262	+	0.927262i	0.997528	−	0.0702659i	$-0.0223848\pi$
$503$	20.7963	+	20.7963i	0.927262	+	0.927262i	−0.0702659	+	0.997528i	$0.522385\pi$
$504$		0			0
$505$	−35.9974	+	11.6035i	−1.60186	+	0.516347i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.00020			−0.398927			−0.199463	−	0.979905i	$-0.563920\pi$
$509$	−9.00020			−0.398927			−0.199463	+	0.979905i	$0.563920\pi$
$510$		0			0
$511$	−16.1609			−0.714917
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.46561	−	2.28849i	−0.196778	−	0.100843i
$516$		0			0
$517$	13.9718	+	13.9718i	0.614479	+	0.614479i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3.83534i			0.168029i	0.996465	+	0.0840147i	$0.0267743\pi$
$521$			3.83534i			0.168029i	−0.996465	+	0.0840147i	$0.973226\pi$
$522$		0			0
$523$	21.4876	−	21.4876i	0.939589	−	0.939589i	−0.0586875	−	0.998276i	$-0.518692\pi$
$523$	21.4876	−	21.4876i	0.939589	−	0.939589i	0.998276	+	0.0586875i	$0.0186915\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.80095	+	6.80095i	−0.296254	+	0.296254i
$528$		0			0
$529$		−	22.9596i		−	0.998242i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.83466	−	3.83466i	−0.166098	−	0.166098i
$534$		0			0
$535$	0.842500	+	0.431755i	0.0364245	+	0.0186664i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−29.6923			−1.27894
$540$		0			0
$541$	−11.9171			−0.512358			−0.256179	−	0.966629i	$-0.582464\pi$
$541$	−11.9171			−0.512358			−0.256179	+	0.966629i	$0.582464\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15.3667	−	4.95333i	0.658238	−	0.212177i
$546$		0			0
$547$	25.0963	+	25.0963i	1.07304	+	1.07304i	0.997113	+	0.0759274i	$0.0241917\pi$
$547$	25.0963	+	25.0963i	1.07304	+	1.07304i	0.0759274	+	0.997113i	$0.475808\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.78993i		−	0.0762536i
$552$		0			0
$553$	11.8677	−	11.8677i	0.504664	−	0.504664i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.99097	+	2.99097i	−0.126731	+	0.126731i	−0.767628	−	0.640896i	$-0.778563\pi$
$557$	−2.99097	+	2.99097i	−0.126731	+	0.126731i	0.640896	+	0.767628i	$0.278563\pi$
$558$		0			0
$559$		−	2.53299i		−	0.107134i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.91667	−	3.91667i	−0.165068	−	0.165068i	0.619740	−	0.784808i	$-0.287238\pi$
$563$	−3.91667	−	3.91667i	−0.165068	−	0.165068i	−0.784808	+	0.619740i	$0.787238\pi$
$564$		0			0
$565$	−11.5104	+	22.4607i	−0.484247	+	0.944930i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.3200			0.642249			0.321124	−	0.947037i	$-0.395939\pi$
$569$	15.3200			0.642249			0.321124	+	0.947037i	$0.395939\pi$
$570$		0			0
$571$	24.8305			1.03912			0.519561	−	0.854433i	$-0.326095\pi$
$571$	24.8305			1.03912			0.519561	+	0.854433i	$0.326095\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.992261	+	0.161910i	0.0413802	+	0.00675213i
$576$		0			0
$577$	30.5771	+	30.5771i	1.27294	+	1.27294i	0.944538	+	0.328403i	$0.106510\pi$
$577$	30.5771	+	30.5771i	1.27294	+	1.27294i	0.328403	+	0.944538i	$0.393490\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			13.5164i			0.560755i
$582$		0			0
$583$	−38.6585	+	38.6585i	−1.60107	+	1.60107i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.60293	−	5.60293i	0.231258	−	0.231258i	−0.581960	−	0.813217i	$-0.697714\pi$
$587$	5.60293	−	5.60293i	0.231258	−	0.231258i	0.813217	+	0.581960i	$0.197714\pi$
$588$		0			0
$589$			2.04816i			0.0843928i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−23.8695	−	23.8695i	−0.980204	−	0.980204i	0.0196036	−	0.999808i	$-0.493760\pi$
$593$	−23.8695	−	23.8695i	−0.980204	−	0.980204i	−0.999808	+	0.0196036i	$0.993760\pi$
$594$		0			0
$595$	−2.69046	−	8.34662i	−0.110298	−	0.342178i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.4255			0.875420			0.437710	−	0.899116i	$-0.355790\pi$
$599$	21.4255			0.875420			0.437710	+	0.899116i	$0.355790\pi$
$600$		0			0
$601$	38.8554			1.58494			0.792472	−	0.609908i	$-0.208794\pi$
$601$	38.8554			1.58494			0.792472	+	0.609908i	$0.208794\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.5090	+	32.6022i	0.427253	+	1.32547i
$606$		0			0
$607$	21.4068	+	21.4068i	0.868874	+	0.868874i	0.992348	−	0.123474i	$-0.0394035\pi$
$607$	21.4068	+	21.4068i	0.868874	+	0.868874i	−0.123474	+	0.992348i	$0.539404\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	3.85153i		−	0.155816i
$612$		0			0
$613$	24.7085	−	24.7085i	0.997967	−	0.997967i	−0.00203135	−	0.999998i	$-0.500647\pi$
$613$	24.7085	−	24.7085i	0.997967	−	0.997967i	0.999998	+	0.00203135i	$0.000646600\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.81012	−	8.81012i	0.354682	−	0.354682i	−0.507166	−	0.861848i	$-0.669307\pi$
$617$	8.81012	−	8.81012i	0.354682	−	0.354682i	0.861848	+	0.507166i	$0.169307\pi$
$618$		0			0
$619$		−	29.1781i		−	1.17276i	−0.810034	−	0.586382i	$-0.800552\pi$
$619$		−	29.1781i		−	1.17276i	0.810034	−	0.586382i	$-0.199448\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.5743	−	11.5743i	−0.463715	−	0.463715i
$624$		0			0
$625$	7.94707	−	23.7033i	0.317883	−	0.948130i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.7234			−0.626931
$630$		0			0
$631$	30.3157			1.20685			0.603425	−	0.797420i	$-0.293802\pi$
$631$	30.3157			1.20685			0.603425	+	0.797420i	$0.293802\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.64446	−	11.0142i	0.223994	−	0.437087i
$636$		0			0
$637$	4.09255	+	4.09255i	0.162153	+	0.162153i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	45.4250i		−	1.79418i	−0.441850	−	0.897089i	$-0.645678\pi$
$641$		−	45.4250i		−	1.79418i	0.441850	−	0.897089i	$-0.354322\pi$
$642$		0			0
$643$	15.9826	−	15.9826i	0.630291	−	0.630291i	−0.317850	−	0.948141i	$-0.602961\pi$
$643$	15.9826	−	15.9826i	0.630291	−	0.630291i	0.948141	+	0.317850i	$0.102961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.4945	−	11.4945i	0.451896	−	0.451896i	−0.444087	−	0.895983i	$-0.646472\pi$
$647$	11.4945	−	11.4945i	0.451896	−	0.451896i	0.895983	+	0.444087i	$0.146472\pi$
$648$		0			0
$649$		−	24.2859i		−	0.953305i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.0649	+	12.0649i	0.472138	+	0.472138i	0.902606	−	0.430468i	$-0.141652\pi$
$653$	12.0649	+	12.0649i	0.472138	+	0.472138i	−0.430468	+	0.902606i	$0.641652\pi$
$654$		0			0
$655$	−12.3110	+	3.96834i	−0.481030	+	0.155056i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	23.0216			0.896795			0.448398	−	0.893834i	$-0.351995\pi$
$659$	23.0216			0.896795			0.448398	+	0.893834i	$0.351995\pi$
$660$		0			0
$661$	−19.2309			−0.747994			−0.373997	−	0.927430i	$-0.622013\pi$
$661$	−19.2309			−0.747994			−0.373997	+	0.927430i	$0.622013\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.66195	−	0.851697i	−0.0644476	−	0.0330274i
$666$		0			0
$667$	−0.335514	−	0.335514i	−0.0129911	−	0.0129911i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	27.5100i		−	1.06201i
$672$		0			0
$673$	0.471549	−	0.471549i	0.0181769	−	0.0181769i	−0.697960	−	0.716137i	$-0.745909\pi$
$673$	0.471549	−	0.471549i	0.0181769	−	0.0181769i	0.716137	+	0.697960i	$0.245909\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.6135	+	14.6135i	−0.561644	+	0.561644i	−0.929774	−	0.368131i	$-0.879998\pi$
$677$	−14.6135	+	14.6135i	−0.561644	+	0.561644i	0.368131	+	0.929774i	$0.379998\pi$
$678$		0			0
$679$		−	3.12963i		−	0.120104i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.55682	−	6.55682i	−0.250890	−	0.250890i	0.570446	−	0.821335i	$-0.306771\pi$
$683$	−6.55682	−	6.55682i	−0.250890	−	0.250890i	−0.821335	+	0.570446i	$0.806771\pi$
$684$		0			0
$685$	−2.44275	−	1.25184i	−0.0933328	−	0.0478302i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.6568			0.405991
$690$		0			0
$691$	44.9588			1.71032			0.855158	−	0.518368i	$-0.173460\pi$
$691$	44.9588			1.71032			0.855158	+	0.518368i	$0.173460\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.32532	−	2.36125i	0.277865	−	0.0895674i
$696$		0			0
$697$	13.6591	+	13.6591i	0.517375	+	0.517375i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	28.2722i		−	1.06783i	−0.845539	−	0.533914i	$-0.820721\pi$
$701$		−	28.2722i		−	1.06783i	0.845539	−	0.533914i	$-0.179279\pi$
$702$		0			0
$703$	−2.36760	+	2.36760i	−0.0892958	+	0.0892958i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.1684	−	13.1684i	0.495250	−	0.495250i
$708$		0			0
$709$			41.1125i			1.54401i	0.635614	+	0.772007i	$0.280747\pi$
$709$			41.1125i			1.54401i	−0.635614	+	0.772007i	$0.719253\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.383917	+	0.383917i	0.0143778	+	0.0143778i
$714$		0			0
$715$	5.23178	−	10.2090i	0.195657	−	0.381794i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.7951			−1.29764			−0.648819	−	0.760943i	$-0.724737\pi$
$719$	−34.7951			−1.29764			−0.648819	+	0.760943i	$0.724737\pi$
$720$		0			0
$721$	2.47076			0.0920157
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.57760	+	6.89045i	−0.355703	+	0.255905i
$726$		0			0
$727$	−0.799528	−	0.799528i	−0.0296529	−	0.0296529i	0.692125	−	0.721778i	$-0.256675\pi$
$727$	−0.799528	−	0.799528i	−0.0296529	−	0.0296529i	−0.721778	+	0.692125i	$0.756675\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			9.02252i			0.333710i
$732$		0			0
$733$	9.87358	−	9.87358i	0.364689	−	0.364689i	−0.500847	−	0.865536i	$-0.666978\pi$
$733$	9.87358	−	9.87358i	0.364689	−	0.364689i	0.865536	+	0.500847i	$0.166978\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.4466	+	16.4466i	−0.605818	+	0.605818i
$738$		0			0
$739$			36.3597i			1.33751i	0.743481	+	0.668757i	$0.233173\pi$
$739$			36.3597i			1.33751i	−0.743481	+	0.668757i	$0.766827\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.8529	+	14.8529i	0.544901	+	0.544901i	0.924962	−	0.380060i	$-0.124097\pi$
$743$	14.8529	+	14.8529i	0.544901	+	0.544901i	−0.380060	+	0.924962i	$0.624097\pi$
$744$		0			0
$745$	8.84557	+	27.4416i	0.324077	+	1.00538i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.466143			−0.0170325
$750$		0			0
$751$	27.0024			0.985330			0.492665	−	0.870219i	$-0.336023\pi$
$751$	27.0024			0.985330			0.492665	+	0.870219i	$0.336023\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.6748	+	45.5257i	0.534071	+	1.65685i
$756$		0			0
$757$	−14.3842	−	14.3842i	−0.522803	−	0.522803i	0.395614	−	0.918417i	$-0.370532\pi$
$757$	−14.3842	−	14.3842i	−0.522803	−	0.522803i	−0.918417	+	0.395614i	$0.870532\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	16.7167i		−	0.605980i	−0.952994	−	0.302990i	$-0.902015\pi$
$761$		−	16.7167i		−	0.605980i	0.952994	−	0.302990i	$-0.0979848\pi$
$762$		0			0
$763$	−5.62139	+	5.62139i	−0.203508	+	0.203508i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.34738	+	3.34738i	−0.120867	+	0.120867i
$768$		0			0
$769$		−	38.3934i		−	1.38450i	−0.721657	−	0.692251i	$-0.756619\pi$
$769$		−	38.3934i		−	1.38450i	0.721657	−	0.692251i	$-0.243381\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.0119	−	16.0119i	−0.575907	−	0.575907i	0.357866	−	0.933773i	$-0.383504\pi$
$773$	−16.0119	−	16.0119i	−0.575907	−	0.575907i	−0.933773	+	0.357866i	$0.883504\pi$
$774$		0			0
$775$	10.9593	−	7.88451i	0.393670	−	0.283220i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.11353			0.147383
$780$		0			0
$781$	−28.5885			−1.02298
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.85063	−	11.4165i	0.208818	−	0.407474i
$786$		0			0
$787$	−14.2816	−	14.2816i	−0.509084	−	0.509084i	0.405161	−	0.914245i	$-0.367215\pi$
$787$	−14.2816	−	14.2816i	−0.509084	−	0.509084i	−0.914245	+	0.405161i	$0.867215\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	12.4272i		−	0.441860i
$792$		0			0
$793$	−3.79177	+	3.79177i	−0.134650	+	0.134650i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.77509	−	8.77509i	0.310830	−	0.310830i	−0.534401	−	0.845231i	$-0.679463\pi$
$797$	8.77509	−	8.77509i	0.310830	−	0.310830i	0.845231	+	0.534401i	$0.179463\pi$
$798$		0			0
$799$			13.7192i			0.485350i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	53.2462	+	53.2462i	1.87902	+	1.87902i
$804$		0			0
$805$	−0.471171	+	0.151878i	−0.0166066	+	0.00535298i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.8365			−0.802889			−0.401444	−	0.915883i	$-0.631492\pi$
$809$	−22.8365			−0.802889			−0.401444	+	0.915883i	$0.631492\pi$
$810$		0			0
$811$	−15.2844			−0.536707			−0.268353	−	0.963321i	$-0.586480\pi$
$811$	−15.2844			−0.536707			−0.268353	+	0.963321i	$0.586480\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.3867	+	5.83534i	0.398859	+	0.204403i
$816$		0			0
$817$	1.35860	+	1.35860i	0.0475313	+	0.0475313i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			20.5233i			0.716268i	0.933670	+	0.358134i	$0.116587\pi$
$821$			20.5233i			0.716268i	−0.933670	+	0.358134i	$0.883413\pi$
$822$		0			0
$823$	−28.1214	+	28.1214i	−0.980250	+	0.980250i	−0.999809	−	0.0195591i	$-0.993774\pi$
$823$	−28.1214	+	28.1214i	−0.980250	+	0.980250i	0.0195591	+	0.999809i	$0.493774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.24626	+	4.24626i	−0.147657	+	0.147657i	−0.777070	−	0.629414i	$-0.783295\pi$
$827$	−4.24626	+	4.24626i	−0.147657	+	0.147657i	0.629414	+	0.777070i	$0.283295\pi$
$828$		0			0
$829$		−	33.0365i		−	1.14740i	−0.819064	−	0.573702i	$-0.805507\pi$
$829$		−	33.0365i		−	1.14740i	0.819064	−	0.573702i	$-0.194493\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.5777	−	14.5777i	−0.505087	−	0.505087i
$834$		0			0
$835$	−36.1502	−	18.5258i	−1.25103	−	0.641113i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	43.4221			1.49910			0.749549	−	0.661949i	$-0.230271\pi$
$839$	43.4221			1.49910			0.749549	+	0.661949i	$0.230271\pi$
$840$		0			0
$841$	−23.4316			−0.807988
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.12823	+	0.686017i	−0.0732135	+	0.0235997i
$846$		0			0
$847$	−11.9264	−	11.9264i	−0.409796	−	0.409796i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			0.887591i			0.0304262i
$852$		0			0
$853$	−9.51845	+	9.51845i	−0.325906	+	0.325906i	−0.851027	−	0.525122i	$-0.824020\pi$
$853$	−9.51845	+	9.51845i	−0.325906	+	0.325906i	0.525122	+	0.851027i	$0.324020\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.44191	−	6.44191i	0.220051	−	0.220051i	−0.588469	−	0.808520i	$-0.700269\pi$
$857$	6.44191	−	6.44191i	0.220051	−	0.220051i	0.808520	+	0.588469i	$0.200269\pi$
$858$		0			0
$859$		−	20.4149i		−	0.696549i	−0.937393	−	0.348275i	$-0.886768\pi$
$859$		−	20.4149i		−	0.696549i	0.937393	−	0.348275i	$-0.113232\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.49306	−	2.49306i	−0.0848646	−	0.0848646i	0.663400	−	0.748265i	$-0.269113\pi$
$863$	−2.49306	−	2.49306i	−0.0848646	−	0.0848646i	−0.748265	+	0.663400i	$0.769113\pi$
$864$		0			0
$865$	−5.44733	+	10.6296i	−0.185215	+	0.361417i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−78.2018			−2.65282
$870$		0			0
$871$	4.53374			0.153620
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.84050	+	12.1715i	0.0622203	+	0.411470i
$876$		0			0
$877$	−9.95800	−	9.95800i	−0.336258	−	0.336258i	0.518699	−	0.854957i	$-0.326417\pi$
$877$	−9.95800	−	9.95800i	−0.336258	−	0.336258i	−0.854957	+	0.518699i	$0.826417\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			34.6008i			1.16573i	0.812569	+	0.582865i	$0.198068\pi$
$881$			34.6008i			1.16573i	−0.812569	+	0.582865i	$0.801932\pi$
$882$		0			0
$883$	−24.0027	+	24.0027i	−0.807756	+	0.807756i	−0.984294	−	0.176538i	$-0.943510\pi$
$883$	−24.0027	+	24.0027i	−0.807756	+	0.807756i	0.176538	+	0.984294i	$0.443510\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	30.5714	−	30.5714i	1.02649	−	1.02649i	0.0268490	−	0.999640i	$-0.491453\pi$
$887$	30.5714	−	30.5714i	1.02649	−	1.02649i	0.999640	−	0.0268490i	$-0.00854732\pi$
$888$		0			0
$889$			6.09402i			0.204387i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.06581	+	2.06581i	0.0691298	+	0.0691298i
$894$		0			0
$895$	3.14837	+	9.76720i	0.105238	+	0.326481i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.37167			−0.212507
$900$		0			0
$901$	−37.9595			−1.26461
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.79150	+	5.55777i	0.0595514	+	0.184747i
$906$		0			0
$907$	5.01428	+	5.01428i	0.166496	+	0.166496i	0.785437	−	0.618941i	$-0.212438\pi$
$907$	5.01428	+	5.01428i	0.166496	+	0.166496i	−0.618941	+	0.785437i	$0.712438\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	10.0534i		−	0.333085i	−0.986034	−	0.166543i	$-0.946740\pi$
$911$		−	10.0534i		−	0.333085i	0.986034	−	0.166543i	$-0.0532604\pi$
$912$		0			0
$913$	44.5332	−	44.5332i	1.47383	−	1.47383i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.50355	−	4.50355i	0.148720	−	0.148720i
$918$		0			0
$919$			29.5002i			0.973121i	0.873647	+	0.486560i	$0.161749\pi$
$919$			29.5002i			0.973121i	−0.873647	+	0.486560i	$0.838251\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.94041	+	3.94041i	0.129700	+	0.129700i
$924$		0			0
$925$	21.7829	+	3.55438i	0.716216	+	0.116867i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−37.3370			−1.22499			−0.612494	−	0.790476i	$-0.709833\pi$
$929$	−37.3370			−1.22499			−0.612494	+	0.790476i	$0.709833\pi$
$930$		0			0
$931$	−4.39018			−0.143882
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−18.6356	+	36.3644i	−0.609450	+	1.18924i
$936$		0			0
$937$	2.47400	+	2.47400i	0.0808220	+	0.0808220i	0.746362	−	0.665540i	$-0.231799\pi$
$937$	2.47400	+	2.47400i	0.0808220	+	0.0808220i	−0.665540	+	0.746362i	$0.731799\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			4.85440i			0.158249i	0.996865	+	0.0791245i	$0.0252125\pi$
$941$			4.85440i			0.158249i	−0.996865	+	0.0791245i	$0.974788\pi$
$942$		0			0
$943$	0.771062	−	0.771062i	0.0251092	−	0.0251092i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.9352	−	33.9352i	1.10275	−	1.10275i	0.108669	−	0.994078i	$-0.465341\pi$
$947$	33.9352	−	33.9352i	1.10275	−	1.10275i	0.994078	−	0.108669i	$-0.0346588\pi$
$948$		0			0
$949$		−	14.6781i		−	0.476471i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−30.9144	−	30.9144i	−1.00142	−	1.00142i	−0.999999	−	0.00141682i	$-0.999549\pi$
$953$	−30.9144	−	30.9144i	−1.00142	−	1.00142i	−0.00141682	−	0.999999i	$-0.500451\pi$
$954$		0			0
$955$	−22.6985	+	7.31666i	−0.734506	+	0.236761i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.35154			0.0436435
$960$		0			0
$961$	−23.7091			−0.764810
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.2560	+	12.4304i	0.780828	+	0.400150i
$966$		0			0
$967$	−36.9560	−	36.9560i	−1.18842	−	1.18842i	−0.977503	−	0.210921i	$-0.932354\pi$
$967$	−36.9560	−	36.9560i	−1.18842	−	1.18842i	−0.210921	−	0.977503i	$-0.567646\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	29.8842i		−	0.959028i	−0.877534	−	0.479514i	$-0.840813\pi$
$971$		−	29.8842i		−	0.959028i	0.877534	−	0.479514i	$-0.159187\pi$
$972$		0			0
$973$	−2.67972	+	2.67972i	−0.0859078	+	0.0859078i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.932667	+	0.932667i	−0.0298386	+	0.0298386i	−0.721869	−	0.692030i	$-0.756717\pi$
$977$	−0.932667	+	0.932667i	−0.0298386	+	0.0298386i	0.692030	+	0.721869i	$0.256717\pi$
$978$		0			0
$979$			76.2689i			2.43756i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.3425	−	21.3425i	−0.680720	−	0.680720i	0.279442	−	0.960163i	$-0.409850\pi$
$983$	−21.3425	−	21.3425i	−0.680720	−	0.680720i	−0.960163	+	0.279442i	$0.909850\pi$
$984$		0			0
$985$	−6.48914	−	3.32548i	−0.206761	−	0.105959i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.509325			0.0161956
$990$		0			0
$991$	31.3268			0.995128			0.497564	−	0.867427i	$-0.334228\pi$
$991$	31.3268			0.995128			0.497564	+	0.867427i	$0.334228\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.4884	+	4.67022i	−0.459314	+	0.148056i
$996$		0			0
$997$	−28.0676	−	28.0676i	−0.888909	−	0.888909i	0.105509	−	0.994418i	$-0.466353\pi$
$997$	−28.0676	−	28.0676i	−0.888909	−	0.888909i	−0.994418	+	0.105509i	$0.966353\pi$
$998$		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.1457.6	yes	24
3.2	odd	2		2340.2.y.a.1457.7	yes	24
5.3	odd	4		2340.2.y.a.53.7	✓	24
15.8	even	4	inner	2340.2.y.b.53.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.7	✓	24	5.3	odd	4
2340.2.y.a.1457.7	yes	24	3.2	odd	2
2340.2.y.b.53.6	yes	24	15.8	even	4	inner
2340.2.y.b.1457.6	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(f(q)\)	\(=\)	\(q+(-0.686017 - 2.12823i) q^{5} +(0.778541 + 0.778541i) q^{7} -5.13019i q^{11} +(-0.707107 + 0.707107i) q^{13} +(2.51872 - 2.51872i) q^{17} -0.758530i q^{19} +(-0.142183 - 0.142183i) q^{23} +(-4.05876 + 2.92001i) q^{25} +2.35974 q^{29} -2.70016 q^{31} +(1.12283 - 2.19101i) q^{35} +(-3.12130 - 3.12130i) q^{37} +5.42303i q^{41} +(-1.79109 + 1.79109i) q^{43} +(-2.72344 + 2.72344i) q^{47} -5.78775i q^{49} +(-7.53548 - 7.53548i) q^{53} +(-10.9183 + 3.51940i) q^{55} +4.73391 q^{59} +5.36238 q^{61} +(1.98998 + 1.01980i) q^{65} +(-3.20584 - 3.20584i) q^{67} -5.57259i q^{71} +(-10.3790 + 10.3790i) q^{73} +(3.99407 - 3.99407i) q^{77} -15.2434i q^{79} +(8.68060 + 8.68060i) q^{83} +(-7.08831 - 3.63254i) q^{85} -14.8667 q^{89} -1.10102 q^{91} +(-1.61433 + 0.520365i) q^{95} +(-2.00993 - 2.00993i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 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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