LMFDB - Embedded newform 2520.2.k.a.1889.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(1889,2520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2520.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.20
Character	$\chi$	$=$	2520.1889
Dual form			2520.2.k.a.1889.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.52549 + 1.63489i) q^{5} +(0.114732 + 2.64326i) q^{7} +O(q^{10})$	$q+(1.52549 + 1.63489i) q^{5} +(0.114732 + 2.64326i) q^{7} -3.82134i q^{11} -6.36634 q^{13} +0.444278i q^{17} -4.39251i q^{19} -4.65767 q^{23} +(-0.345740 + 4.98803i) q^{25} +9.53809i q^{29} +4.88703i q^{31} +(-4.14642 + 4.21985i) q^{35} -5.61070i q^{37} -8.01459 q^{41} +3.33239i q^{43} +12.8236i q^{47} +(-6.97367 + 0.606534i) q^{49} -1.27706 q^{53} +(6.24747 - 5.82943i) q^{55} -5.81310 q^{59} -5.28930i q^{61} +(-9.71181 - 10.4083i) q^{65} -9.42429i q^{67} +2.35006i q^{71} -10.5206 q^{73} +(10.1008 - 0.438430i) q^{77} +15.4110 q^{79} -1.09521i q^{83} +(-0.726347 + 0.677744i) q^{85} -10.6896 q^{89} +(-0.730423 - 16.8279i) q^{91} +(7.18127 - 6.70074i) q^{95} -4.80920 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) $ 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$1$	$-1$

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
$5$	1.52549	+	1.63489i	0.682221	+	0.731146i
$5$	1.52549	+	1.63489i	0.682221	+	0.731146i
$6$		0			0
$7$	0.114732	+	2.64326i	0.0433646	+	0.999059i
$7$	0.114732	+	2.64326i	0.0433646	+	0.999059i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.82134i		−	1.15218i	−0.817387	−	0.576088i	$-0.804578\pi$
$11$		−	3.82134i		−	1.15218i	0.817387	−	0.576088i	$-0.195422\pi$
$12$		0			0
$13$	−6.36634			−1.76571			−0.882853	−	0.469650i	$-0.844380\pi$
$13$	−6.36634			−1.76571			−0.882853	+	0.469650i	$0.844380\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.444278i			0.107753i	0.998548	+	0.0538767i	$0.0171578\pi$
$17$			0.444278i			0.107753i	−0.998548	+	0.0538767i	$0.982842\pi$
$18$		0			0
$19$		−	4.39251i		−	1.00771i	−0.863788	−	0.503855i	$-0.831915\pi$
$19$		−	4.39251i		−	1.00771i	0.863788	−	0.503855i	$-0.168085\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.65767			−0.971191			−0.485595	−	0.874184i	$-0.661397\pi$
$23$	−4.65767			−0.971191			−0.485595	+	0.874184i	$0.661397\pi$
$24$		0			0
$25$	−0.345740	+	4.98803i	−0.0691480	+	0.997606i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			9.53809i			1.77118i	0.464469	+	0.885589i	$0.346245\pi$
$29$			9.53809i			1.77118i	−0.464469	+	0.885589i	$0.653755\pi$
$30$		0			0
$31$			4.88703i			0.877737i	0.898551	+	0.438869i	$0.144621\pi$
$31$			4.88703i			0.877737i	−0.898551	+	0.438869i	$0.855379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.14642	+	4.21985i	−0.700874	+	0.713285i
$36$		0			0
$37$		−	5.61070i		−	0.922394i	−0.887298	−	0.461197i	$-0.847420\pi$
$37$		−	5.61070i		−	0.922394i	0.887298	−	0.461197i	$-0.152580\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.01459			−1.25167			−0.625834	−	0.779956i	$-0.715241\pi$
$41$	−8.01459			−1.25167			−0.625834	+	0.779956i	$0.715241\pi$
$42$		0			0
$43$			3.33239i			0.508184i	0.967180	+	0.254092i	$0.0817767\pi$
$43$			3.33239i			0.508184i	−0.967180	+	0.254092i	$0.918223\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.8236i			1.87051i	0.353972	+	0.935256i	$0.384831\pi$
$47$			12.8236i			1.87051i	−0.353972	+	0.935256i	$0.615169\pi$
$48$		0			0
$49$	−6.97367	+	0.606534i	−0.996239	+	0.0866477i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.27706			−0.175418			−0.0877089	−	0.996146i	$-0.527955\pi$
$53$	−1.27706			−0.175418			−0.0877089	+	0.996146i	$0.527955\pi$
$54$		0			0
$55$	6.24747	−	5.82943i	0.842409	−	0.786040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.81310			−0.756801			−0.378401	−	0.925642i	$-0.623526\pi$
$59$	−5.81310			−0.756801			−0.378401	+	0.925642i	$0.623526\pi$
$60$		0			0
$61$		−	5.28930i		−	0.677226i	−0.940926	−	0.338613i	$-0.890042\pi$
$61$		−	5.28930i		−	0.677226i	0.940926	−	0.338613i	$-0.109958\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.71181	−	10.4083i	−1.20460	−	1.29099i
$66$		0			0
$67$		−	9.42429i		−	1.15136i	−0.817675	−	0.575680i	$-0.804737\pi$
$67$		−	9.42429i		−	1.15136i	0.817675	−	0.575680i	$-0.195263\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.35006i			0.278901i	0.990229	+	0.139451i	$0.0445336\pi$
$71$			2.35006i			0.278901i	−0.990229	+	0.139451i	$0.955466\pi$
$72$		0			0
$73$	−10.5206			−1.23134			−0.615672	−	0.788003i	$-0.711115\pi$
$73$	−10.5206			−1.23134			−0.615672	+	0.788003i	$0.711115\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.1008	−	0.438430i	1.15109	−	0.0499637i
$78$		0			0
$79$	15.4110			1.73387			0.866937	−	0.498418i	$-0.166086\pi$
$79$	15.4110			1.73387			0.866937	+	0.498418i	$0.166086\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1.09521i		−	0.120215i	−0.998192	−	0.0601077i	$-0.980856\pi$
$83$		−	1.09521i		−	0.120215i	0.998192	−	0.0601077i	$-0.0191444\pi$
$84$		0			0
$85$	−0.726347	+	0.677744i	−0.0787834	+	0.0735116i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.6896			−1.13309			−0.566547	−	0.824029i	$-0.691721\pi$
$89$	−10.6896			−1.13309			−0.566547	+	0.824029i	$0.691721\pi$
$90$		0			0
$91$	−0.730423	−	16.8279i	−0.0765692	−	1.76404i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.18127	−	6.70074i	0.736783	−	0.687481i
$96$		0			0
$97$	−4.80920			−0.488301			−0.244150	−	0.969737i	$-0.578509\pi$
$97$	−4.80920			−0.488301			−0.244150	+	0.969737i	$0.578509\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−15.8936			−1.58148			−0.790738	−	0.612155i	$-0.790303\pi$
$101$	−15.8936			−1.58148			−0.790738	+	0.612155i	$0.790303\pi$
$102$		0			0
$103$	−6.50081			−0.640544			−0.320272	−	0.947326i	$-0.603774\pi$
$103$	−6.50081			−0.640544			−0.320272	+	0.947326i	$0.603774\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.8312			1.04709			0.523547	−	0.851997i	$-0.324608\pi$
$107$	10.8312			1.04709			0.523547	+	0.851997i	$0.324608\pi$
$108$		0			0
$109$	0.772179			0.0739613			0.0369807	−	0.999316i	$-0.488226\pi$
$109$	0.772179			0.0739613			0.0369807	+	0.999316i	$0.488226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.6270			1.47006			0.735032	−	0.678033i	$-0.237167\pi$
$113$	15.6270			1.47006			0.735032	+	0.678033i	$0.237167\pi$
$114$		0			0
$115$	−7.10524	−	7.61478i	−0.662567	−	0.710082i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.17434	+	0.0509730i	−0.107652	+	0.00467268i
$120$		0			0
$121$	−3.60263			−0.327512
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.68231	+	7.04396i	−0.776570	+	0.630031i
$126$		0			0
$127$		−	9.42595i		−	0.836418i	−0.908351	−	0.418209i	$-0.862658\pi$
$127$		−	9.42595i		−	0.836418i	0.908351	−	0.418209i	$-0.137342\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	17.0174			1.48682			0.743409	−	0.668837i	$-0.233207\pi$
$131$	17.0174			1.48682			0.743409	+	0.668837i	$0.233207\pi$
$132$		0			0
$133$	11.6105	−	0.503961i	1.00676	−	0.0436990i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.4339			−1.14773			−0.573866	−	0.818949i	$-0.694557\pi$
$137$	−13.4339			−1.14773			−0.573866	+	0.818949i	$0.694557\pi$
$138$		0			0
$139$		−	7.18137i		−	0.609116i	−0.952494	−	0.304558i	$-0.901491\pi$
$139$		−	7.18137i		−	0.609116i	0.952494	−	0.304558i	$-0.0985087\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			24.3279i			2.03440i
$144$		0			0
$145$	−15.5937	+	14.5503i	−1.29499	+	1.20834i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			10.5160i			0.861503i	0.902471	+	0.430752i	$0.141751\pi$
$149$			10.5160i			0.861503i	−0.902471	+	0.430752i	$0.858249\pi$
$150$		0			0
$151$	−20.8931			−1.70025			−0.850127	−	0.526577i	$-0.823475\pi$
$151$	−20.8931			−1.70025			−0.850127	+	0.526577i	$0.823475\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.98977	+	7.45514i	−0.641754	+	0.598811i
$156$		0			0
$157$	−10.9905			−0.877138			−0.438569	−	0.898697i	$-0.644515\pi$
$157$	−10.9905			−0.877138			−0.438569	+	0.898697i	$0.644515\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.534384	−	12.3114i	−0.0421154	−	0.970277i
$162$		0			0
$163$			2.40231i			0.188163i	0.995564	+	0.0940815i	$0.0299914\pi$
$163$			2.40231i			0.188163i	−0.995564	+	0.0940815i	$0.970009\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.41244i			0.341445i	0.985319	+	0.170723i	$0.0546102\pi$
$167$			4.41244i			0.341445i	−0.985319	+	0.170723i	$0.945390\pi$
$168$		0			0
$169$	27.5303			2.11772
$170$		0			0
$171$		0			0
$172$		0			0
$173$			13.1123i			0.996913i	0.866915	+	0.498457i	$0.166100\pi$
$173$			13.1123i			0.996913i	−0.866915	+	0.498457i	$0.833900\pi$
$174$		0			0
$175$	−13.2243	−	0.341594i	−0.999667	−	0.0258221i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			3.92328i			0.293240i	0.989193	+	0.146620i	$0.0468395\pi$
$179$			3.92328i			0.293240i	−0.989193	+	0.146620i	$0.953161\pi$
$180$		0			0
$181$		−	5.48099i		−	0.407399i	−0.979033	−	0.203699i	$-0.934703\pi$
$181$		−	5.48099i		−	0.407399i	0.979033	−	0.203699i	$-0.0652965\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.17289	−	8.55909i	0.674404	−	0.629277i
$186$		0			0
$187$	1.69774			0.124151
$188$		0			0
$189$		0			0
$190$		0			0
$191$			6.90336i			0.499510i	0.968309	+	0.249755i	$0.0803500\pi$
$191$			6.90336i			0.499510i	−0.968309	+	0.249755i	$0.919650\pi$
$192$		0			0
$193$		−	9.80846i		−	0.706028i	−0.935618	−	0.353014i	$-0.885157\pi$
$193$		−	9.80846i		−	0.706028i	0.935618	−	0.353014i	$-0.114843\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.37038			−0.525118			−0.262559	−	0.964916i	$-0.584566\pi$
$197$	−7.37038			−0.525118			−0.262559	+	0.964916i	$0.584566\pi$
$198$		0			0
$199$		−	16.0524i		−	1.13792i	−0.822364	−	0.568962i	$-0.807345\pi$
$199$		−	16.0524i		−	1.13792i	0.822364	−	0.568962i	$-0.192655\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−25.2117	+	1.09432i	−1.76951	+	0.0768065i
$204$		0			0
$205$	−12.2262	−	13.1030i	−0.853915	−	0.915152i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.7853			−1.16106
$210$		0			0
$211$	21.3515			1.46990			0.734950	−	0.678121i	$-0.237206\pi$
$211$	21.3515			1.46990			0.734950	+	0.678121i	$0.237206\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.44809	+	5.08354i	−0.371557	+	0.346694i
$216$		0			0
$217$	−12.9177	+	0.560700i	−0.876912	+	0.0380628i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.82843i		−	0.190261i
$222$		0			0
$223$	18.3648			1.22979			0.614897	−	0.788607i	$-0.289197\pi$
$223$	18.3648			1.22979			0.614897	+	0.788607i	$0.289197\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.64572i			0.441092i	0.975377	+	0.220546i	$0.0707839\pi$
$227$			6.64572i			0.441092i	−0.975377	+	0.220546i	$0.929216\pi$
$228$		0			0
$229$			23.3322i			1.54184i	0.636935	+	0.770918i	$0.280202\pi$
$229$			23.3322i			1.54184i	−0.636935	+	0.770918i	$0.719798\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.83404			0.447713			0.223856	−	0.974622i	$-0.428135\pi$
$233$	6.83404			0.447713			0.223856	+	0.974622i	$0.428135\pi$
$234$		0			0
$235$	−20.9652	+	19.5623i	−1.36762	+	1.27610i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	0.394262i		−	0.0255027i	−0.999919	−	0.0127513i	$-0.995941\pi$
$239$		−	0.394262i		−	0.0255027i	0.999919	−	0.0127513i	$-0.00405899\pi$
$240$		0			0
$241$			28.1397i			1.81264i	0.422595	+	0.906319i	$0.361119\pi$
$241$			28.1397i			1.81264i	−0.422595	+	0.906319i	$0.638881\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−11.6299	−	10.4759i	−0.743008	−	0.669283i
$246$		0			0
$247$			27.9642i			1.77932i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.3019			−0.965846			−0.482923	−	0.875663i	$-0.660425\pi$
$251$	−15.3019			−0.965846			−0.482923	+	0.875663i	$0.660425\pi$
$252$		0			0
$253$			17.7985i			1.11898i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.1657i			1.00839i	0.863590	+	0.504194i	$0.168210\pi$
$257$			16.1657i			1.00839i	−0.863590	+	0.504194i	$0.831790\pi$
$258$		0			0
$259$	14.8306	−	0.643727i	0.921526	−	0.0399993i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.99497			−0.184678			−0.0923390	−	0.995728i	$-0.529434\pi$
$263$	−2.99497			−0.184678			−0.0923390	+	0.995728i	$0.529434\pi$
$264$		0			0
$265$	−1.94815	−	2.08786i	−0.119674	−	0.128256i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	27.1797			1.65718			0.828589	−	0.559858i	$-0.189144\pi$
$269$	27.1797			1.65718			0.828589	+	0.559858i	$0.189144\pi$
$270$		0			0
$271$		−	16.9027i		−	1.02677i	−0.858159	−	0.513384i	$-0.828392\pi$
$271$		−	16.9027i		−	1.02677i	0.858159	−	0.513384i	$-0.171608\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	19.0610	+	1.32119i	1.14942	+	0.0796707i
$276$		0			0
$277$			15.0261i			0.902829i	0.892314	+	0.451414i	$0.149080\pi$
$277$			15.0261i			0.902829i	−0.892314	+	0.451414i	$0.850920\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	14.7234i		−	0.878323i	−0.898408	−	0.439162i	$-0.855276\pi$
$281$		−	14.7234i		−	0.878323i	0.898408	−	0.439162i	$-0.144724\pi$
$282$		0			0
$283$	21.1123			1.25499			0.627497	−	0.778619i	$-0.284079\pi$
$283$	21.1123			1.25499			0.627497	+	0.778619i	$0.284079\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.919530	−	21.1847i	−0.0542782	−	1.25049i
$288$		0			0
$289$	16.8026			0.988389
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	3.45996i		−	0.202133i	−0.994880	−	0.101066i	$-0.967775\pi$
$293$		−	3.45996i		−	0.202133i	0.994880	−	0.101066i	$-0.0322255\pi$
$294$		0			0
$295$	−8.86785	−	9.50379i	−0.516306	−	0.553332i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29.6523			1.71484
$300$		0			0
$301$	−8.80838	+	0.382332i	−0.507706	+	0.0220372i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.64744	−	8.06880i	0.495151	−	0.462018i
$306$		0			0
$307$	−3.26315			−0.186238			−0.0931188	−	0.995655i	$-0.529684\pi$
$307$	−3.26315			−0.186238			−0.0931188	+	0.995655i	$0.529684\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.62685			−0.489184			−0.244592	−	0.969626i	$-0.578654\pi$
$311$	−8.62685			−0.489184			−0.244592	+	0.969626i	$0.578654\pi$
$312$		0			0
$313$	−19.0715			−1.07799			−0.538993	−	0.842311i	$-0.681195\pi$
$313$	−19.0715			−1.07799			−0.538993	+	0.842311i	$0.681195\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.45062			−0.418468			−0.209234	−	0.977866i	$-0.567097\pi$
$317$	−7.45062			−0.418468			−0.209234	+	0.977866i	$0.567097\pi$
$318$		0			0
$319$	36.4483			2.04071
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.95150			0.108584
$324$		0			0
$325$	2.20110	−	31.7555i	0.122095	−	1.76148i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−33.8961	+	1.47128i	−1.86875	+	0.0811141i
$330$		0			0
$331$	−29.7822			−1.63698			−0.818488	−	0.574524i	$-0.805187\pi$
$331$	−29.7822			−1.63698			−0.818488	+	0.574524i	$0.805187\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.4077	−	14.3767i	0.841812	−	0.785482i
$336$		0			0
$337$		−	2.14777i		−	0.116996i	−0.998288	−	0.0584981i	$-0.981369\pi$
$337$		−	2.14777i		−	0.116996i	0.998288	−	0.0584981i	$-0.0186312\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.6750			1.01131
$342$		0			0
$343$	−2.40333	−	18.3637i	−0.129768	−	0.991544i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.13394			0.436653			0.218326	−	0.975876i	$-0.429940\pi$
$347$	8.13394			0.436653			0.218326	+	0.975876i	$0.429940\pi$
$348$		0			0
$349$		−	11.2639i		−	0.602941i	−0.953476	−	0.301470i	$-0.902523\pi$
$349$		−	11.2639i		−	0.602941i	0.953476	−	0.301470i	$-0.0974774\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	30.2542i		−	1.61027i	−0.593091	−	0.805135i	$-0.702093\pi$
$353$		−	30.2542i		−	1.61027i	0.593091	−	0.805135i	$-0.297907\pi$
$354$		0			0
$355$	−3.84210	+	3.58500i	−0.203917	+	0.190272i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			13.0372i			0.688079i	0.938955	+	0.344040i	$0.111795\pi$
$359$			13.0372i			0.688079i	−0.938955	+	0.344040i	$0.888205\pi$
$360$		0			0
$361$	−0.294107			−0.0154793
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.0491	−	17.2001i	−0.840049	−	0.900292i
$366$		0			0
$367$	31.6788			1.65362			0.826809	−	0.562483i	$-0.190154\pi$
$367$	31.6788			1.65362			0.826809	+	0.562483i	$0.190154\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.146520	−	3.37561i	−0.00760693	−	0.175253i
$372$		0			0
$373$			21.2799i			1.10183i	0.834561	+	0.550916i	$0.185721\pi$
$373$			21.2799i			1.10183i	−0.834561	+	0.550916i	$0.814279\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	60.7227i		−	3.12738i
$378$		0			0
$379$	9.47047			0.486466			0.243233	−	0.969968i	$-0.421792\pi$
$379$	9.47047			0.486466			0.243233	+	0.969968i	$0.421792\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			3.00398i			0.153496i	0.997051	+	0.0767481i	$0.0244537\pi$
$383$			3.00398i			0.153496i	−0.997051	+	0.0767481i	$0.975546\pi$
$384$		0			0
$385$	16.1255	+	15.8449i	0.821831	+	0.807530i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			14.4364i			0.731957i	0.930623	+	0.365978i	$0.119266\pi$
$389$			14.4364i			0.731957i	−0.930623	+	0.365978i	$0.880734\pi$
$390$		0			0
$391$		−	2.06930i		−	0.104649i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	23.5094	+	25.1953i	1.18289	+	1.26771i
$396$		0			0
$397$	−2.92897			−0.147001			−0.0735004	−	0.997295i	$-0.523417\pi$
$397$	−2.92897			−0.147001			−0.0735004	+	0.997295i	$0.523417\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			20.5059i			1.02402i	0.858981	+	0.512008i	$0.171098\pi$
$401$			20.5059i			1.02402i	−0.858981	+	0.512008i	$0.828902\pi$
$402$		0			0
$403$		−	31.1125i		−	1.54983i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.4404			−1.06276
$408$		0			0
$409$			3.57788i			0.176915i	0.996080	+	0.0884574i	$0.0281937\pi$
$409$			3.57788i			0.176915i	−0.996080	+	0.0884574i	$0.971806\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.666949	−	15.3656i	−0.0328184	−	0.756089i
$414$		0			0
$415$	1.79056	−	1.67074i	0.0878949	−	0.0820135i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.18681			−0.448805			−0.224402	−	0.974497i	$-0.572043\pi$
$419$	−9.18681			−0.448805			−0.224402	+	0.974497i	$0.572043\pi$
$420$		0			0
$421$	−33.0694			−1.61171			−0.805853	−	0.592115i	$-0.798293\pi$
$421$	−33.0694			−1.61171			−0.805853	+	0.592115i	$0.798293\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.21607	−	0.153605i	−0.107495	−	0.00745092i
$426$		0			0
$427$	13.9810	−	0.606853i	0.676589	−	0.0293677i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.80366i		−	0.0868791i	−0.999056	−	0.0434395i	$-0.986168\pi$
$431$		−	1.80366i		−	0.0868791i	0.999056	−	0.0434395i	$-0.0138316\pi$
$432$		0			0
$433$	0.596107			0.0286471			0.0143235	−	0.999897i	$-0.495441\pi$
$433$	0.596107			0.0286471			0.0143235	+	0.999897i	$0.495441\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			20.4588i			0.978679i
$438$		0			0
$439$			39.8636i			1.90259i	0.308289	+	0.951293i	$0.400244\pi$
$439$			39.8636i			1.90259i	−0.308289	+	0.951293i	$0.599756\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.76075			0.368724			0.184362	−	0.982858i	$-0.440978\pi$
$443$	7.76075			0.368724			0.184362	+	0.982858i	$0.440978\pi$
$444$		0			0
$445$	−16.3069	−	17.4763i	−0.773021	−	0.828457i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			34.0393i			1.60642i	0.595699	+	0.803208i	$0.296875\pi$
$449$			34.0393i			1.60642i	−0.595699	+	0.803208i	$0.703125\pi$
$450$		0			0
$451$			30.6265i			1.44214i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	26.3975	−	26.8650i	1.23754	−	1.25945i
$456$		0			0
$457$		−	29.7092i		−	1.38974i	−0.719136	−	0.694869i	$-0.755462\pi$
$457$		−	29.7092i		−	1.38974i	0.719136	−	0.694869i	$-0.244538\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.1657			0.659761			0.329880	−	0.944023i	$-0.392992\pi$
$461$	14.1657			0.659761			0.329880	+	0.944023i	$0.392992\pi$
$462$		0			0
$463$		−	27.7185i		−	1.28819i	−0.764947	−	0.644093i	$-0.777235\pi$
$463$		−	27.7185i		−	1.28819i	0.764947	−	0.644093i	$-0.222765\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			5.27507i			0.244101i	0.992524	+	0.122051i	$0.0389470\pi$
$467$			5.27507i			0.244101i	−0.992524	+	0.122051i	$0.961053\pi$
$468$		0			0
$469$	24.9109	−	1.08127i	1.15028	−	0.0499283i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.7342			0.585518
$474$		0			0
$475$	21.9100	+	1.51866i	1.00530	+	0.0696811i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.1395			0.691742			0.345871	−	0.938282i	$-0.387583\pi$
$479$	15.1395			0.691742			0.345871	+	0.938282i	$0.387583\pi$
$480$		0			0
$481$			35.7196i			1.62868i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.33641	−	7.86253i	−0.333129	−	0.357019i
$486$		0			0
$487$			37.0652i			1.67959i	0.542906	+	0.839793i	$0.317324\pi$
$487$			37.0652i			1.67959i	−0.542906	+	0.839793i	$0.682676\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	4.54982i		−	0.205330i	−0.994716	−	0.102665i	$-0.967263\pi$
$491$		−	4.54982i		−	0.205330i	0.994716	−	0.102665i	$-0.0327370\pi$
$492$		0			0
$493$	−4.23757			−0.190850
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.21183	+	0.269627i	−0.278639	+	0.0120944i
$498$		0			0
$499$	1.87024			0.0837234			0.0418617	−	0.999123i	$-0.486671\pi$
$499$	1.87024			0.0837234			0.0418617	+	0.999123i	$0.486671\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	28.1955i		−	1.25717i	−0.777739	−	0.628587i	$-0.783634\pi$
$503$		−	28.1955i		−	1.25717i	0.777739	−	0.628587i	$-0.216366\pi$
$504$		0			0
$505$	−24.2456	−	25.9844i	−1.07892	−	1.15629i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	36.8373			1.63279			0.816393	−	0.577497i	$-0.195970\pi$
$509$	36.8373			1.63279			0.816393	+	0.577497i	$0.195970\pi$
$510$		0			0
$511$	−1.20705	−	27.8087i	−0.0533968	−	1.23019i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.91695	−	10.6281i	−0.436993	−	0.468331i
$516$		0			0
$517$	49.0033			2.15516
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19.4209			0.850843			0.425422	−	0.904995i	$-0.360126\pi$
$521$	19.4209			0.850843			0.425422	+	0.904995i	$0.360126\pi$
$522$		0			0
$523$	−31.4983			−1.37733			−0.688663	−	0.725082i	$-0.741802\pi$
$523$	−31.4983			−1.37733			−0.688663	+	0.725082i	$0.741802\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.17120			−0.0945791
$528$		0			0
$529$	−1.30613			−0.0567881
$530$		0			0
$531$		0			0
$532$		0			0
$533$	51.0236			2.21008
$534$		0			0
$535$	16.5230	+	17.7079i	0.714350	+	0.765578i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.31777	+	26.6488i	0.0998335	+	1.14784i
$540$		0			0
$541$	21.9864			0.945267			0.472634	−	0.881259i	$-0.343303\pi$
$541$	21.9864			0.945267			0.472634	+	0.881259i	$0.343303\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.17795	+	1.26243i	0.0504580	+	0.0540765i
$546$		0			0
$547$			22.6015i			0.966370i	0.875518	+	0.483185i	$0.160520\pi$
$547$			22.6015i			0.966370i	−0.875518	+	0.483185i	$0.839480\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	41.8961			1.78483
$552$		0			0
$553$	1.76814	+	40.7353i	0.0751888	+	1.73224i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.5421			0.658541			0.329271	−	0.944236i	$-0.393197\pi$
$557$	15.5421			0.658541			0.329271	+	0.944236i	$0.393197\pi$
$558$		0			0
$559$		−	21.2151i		−	0.897304i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	29.1339i		−	1.22785i	−0.789366	−	0.613923i	$-0.789591\pi$
$563$		−	29.1339i		−	1.22785i	0.789366	−	0.613923i	$-0.210409\pi$
$564$		0			0
$565$	23.8389	+	25.5484i	1.00291	+	1.07483i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1.85789i			0.0778867i	0.999241	+	0.0389433i	$0.0123992\pi$
$569$			1.85789i			0.0778867i	−0.999241	+	0.0389433i	$0.987601\pi$
$570$		0			0
$571$	19.9726			0.835826			0.417913	−	0.908487i	$-0.362762\pi$
$571$	19.9726			0.835826			0.417913	+	0.908487i	$0.362762\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.61034	−	23.2326i	0.0671559	−	0.968866i
$576$		0			0
$577$	−36.6669			−1.52646			−0.763231	−	0.646126i	$-0.776388\pi$
$577$	−36.6669			−1.52646			−0.763231	+	0.646126i	$0.776388\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.89494	−	0.125656i	0.120102	−	0.00521310i
$582$		0			0
$583$			4.88008i			0.202112i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			24.7183i			1.02023i	0.860105	+	0.510117i	$0.170398\pi$
$587$			24.7183i			1.02023i	−0.860105	+	0.510117i	$0.829602\pi$
$588$		0			0
$589$	21.4663			0.884505
$590$		0			0
$591$		0			0
$592$		0			0
$593$			17.2021i			0.706404i	0.935547	+	0.353202i	$0.114907\pi$
$593$			17.2021i			0.706404i	−0.935547	+	0.353202i	$0.885093\pi$
$594$		0			0
$595$	−1.87479	−	1.84217i	−0.0768589	−	0.0755215i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			34.8399i			1.42352i	0.702423	+	0.711760i	$0.252102\pi$
$599$			34.8399i			1.42352i	−0.702423	+	0.711760i	$0.747898\pi$
$600$		0			0
$601$			16.6091i			0.677501i	0.940876	+	0.338751i	$0.110004\pi$
$601$			16.6091i			0.677501i	−0.940876	+	0.338751i	$0.889996\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.49578	−	5.88990i	−0.223435	−	0.239459i
$606$		0			0
$607$	−30.4020			−1.23398			−0.616990	−	0.786971i	$-0.711648\pi$
$607$	−30.4020			−1.23398			−0.616990	+	0.786971i	$0.711648\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	81.6393i		−	3.30277i
$612$		0			0
$613$		−	27.8087i		−	1.12318i	−0.827414	−	0.561592i	$-0.810189\pi$
$613$		−	27.8087i		−	1.12318i	0.827414	−	0.561592i	$-0.189811\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.73050			−0.0696671			−0.0348336	−	0.999393i	$-0.511090\pi$
$617$	−1.73050			−0.0696671			−0.0348336	+	0.999393i	$0.511090\pi$
$618$		0			0
$619$			37.8450i			1.52112i	0.649269	+	0.760559i	$0.275075\pi$
$619$			37.8450i			1.52112i	−0.649269	+	0.760559i	$0.724925\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.22644	−	28.2554i	−0.0491362	−	1.13203i
$624$		0			0
$625$	−24.7609	−	3.44912i	−0.990437	−	0.137965i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.49271			0.0993910
$630$		0			0
$631$	−25.4218			−1.01203			−0.506013	−	0.862526i	$-0.668881\pi$
$631$	−25.4218			−1.01203			−0.506013	+	0.862526i	$0.668881\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.4104	−	14.3792i	0.611543	−	0.570622i
$636$		0			0
$637$	44.3968	−	3.86140i	1.75906	−	0.152994i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	10.7517i		−	0.424668i	−0.977197	−	0.212334i	$-0.931893\pi$
$641$		−	10.7517i		−	0.424668i	0.977197	−	0.212334i	$-0.0681065\pi$
$642$		0			0
$643$	37.2282			1.46814			0.734068	−	0.679076i	$-0.237619\pi$
$643$	37.2282			1.46814			0.734068	+	0.679076i	$0.237619\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.5856i		−	1.04519i	−0.852582	−	0.522593i	$-0.824965\pi$
$647$		−	26.5856i		−	1.04519i	0.852582	−	0.522593i	$-0.175035\pi$
$648$		0			0
$649$			22.2138i			0.871969i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.4459			−0.878375			−0.439187	−	0.898395i	$-0.644734\pi$
$653$	−22.4459			−0.878375			−0.439187	+	0.898395i	$0.644734\pi$
$654$		0			0
$655$	25.9600	+	27.8216i	1.01434	+	1.08708i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			29.4188i			1.14599i	0.819558	+	0.572997i	$0.194219\pi$
$659$			29.4188i			1.14599i	−0.819558	+	0.572997i	$0.805781\pi$
$660$		0			0
$661$		−	37.0645i		−	1.44164i	−0.693121	−	0.720821i	$-0.743765\pi$
$661$		−	37.0645i		−	1.44164i	0.693121	−	0.720821i	$-0.256235\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.5357	+	18.2132i	0.718785	+	0.706277i
$666$		0			0
$667$		−	44.4252i		−	1.72015i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−20.2122			−0.780284
$672$		0			0
$673$			33.7804i			1.30214i	0.759017	+	0.651070i	$0.225680\pi$
$673$			33.7804i			1.30214i	−0.759017	+	0.651070i	$0.774320\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.05815i		−	0.0406682i	−0.999793	−	0.0203341i	$-0.993527\pi$
$677$		−	1.05815i		−	0.0406682i	0.999793	−	0.0203341i	$-0.00647299\pi$
$678$		0			0
$679$	−0.551770	−	12.7120i	−0.0211750	−	0.487841i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−40.0158			−1.53116			−0.765581	−	0.643340i	$-0.777548\pi$
$683$	−40.0158			−1.53116			−0.765581	+	0.643340i	$0.777548\pi$
$684$		0			0
$685$	−20.4933	−	21.9629i	−0.783007	−	0.839159i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.13021			0.309736
$690$		0			0
$691$			5.01912i			0.190936i	0.995432	+	0.0954682i	$0.0304348\pi$
$691$			5.01912i			0.190936i	−0.995432	+	0.0954682i	$0.969565\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.7408	−	10.9551i	0.445352	−	0.415552i
$696$		0			0
$697$		−	3.56071i		−	0.134871i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5.12674i			0.193634i	0.995302	+	0.0968172i	$0.0308662\pi$
$701$			5.12674i			0.193634i	−0.995302	+	0.0968172i	$0.969134\pi$
$702$		0			0
$703$	−24.6450			−0.929505
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.82351	−	42.0111i	−0.0685801	−	1.57999i
$708$		0			0
$709$	16.6941			0.626962			0.313481	−	0.949594i	$-0.398505\pi$
$709$	16.6941			0.626962			0.313481	+	0.949594i	$0.398505\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	22.7622i		−	0.852450i
$714$		0			0
$715$	−39.7735	+	37.1121i	−1.48745	+	1.38791i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.4901			0.913328			0.456664	−	0.889639i	$-0.349044\pi$
$719$	24.4901			0.913328			0.456664	+	0.889639i	$0.349044\pi$
$720$		0			0
$721$	−0.745852	−	17.1834i	−0.0277770	−	0.639942i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−47.5763	−	3.29770i	−1.76694	−	0.122473i
$726$		0			0
$727$	−16.4513			−0.610146			−0.305073	−	0.952329i	$-0.598681\pi$
$727$	−16.4513			−0.610146			−0.305073	+	0.952329i	$0.598681\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.48051			−0.0547586
$732$		0			0
$733$	38.0356			1.40488			0.702438	−	0.711745i	$-0.252095\pi$
$733$	38.0356			1.40488			0.702438	+	0.711745i	$0.252095\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.0134			−1.32657
$738$		0			0
$739$	21.8136			0.802426			0.401213	−	0.915985i	$-0.368589\pi$
$739$	21.8136			0.802426			0.401213	+	0.915985i	$0.368589\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−28.5623			−1.04785			−0.523925	−	0.851765i	$-0.675533\pi$
$743$	−28.5623			−1.04785			−0.523925	+	0.851765i	$0.675533\pi$
$744$		0			0
$745$	−17.1925	+	16.0421i	−0.629884	+	0.587736i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.24269	+	28.6298i	0.0454069	+	1.04611i
$750$		0			0
$751$	2.49599			0.0910801			0.0455401	−	0.998963i	$-0.485499\pi$
$751$	2.49599			0.0910801			0.0455401	+	0.998963i	$0.485499\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.8723	−	34.1579i	−1.15995	−	1.24313i
$756$		0			0
$757$			10.2538i			0.372681i	0.982485	+	0.186340i	$0.0596627\pi$
$757$			10.2538i			0.372681i	−0.982485	+	0.186340i	$0.940337\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	26.5529			0.962541			0.481271	−	0.876572i	$-0.340175\pi$
$761$	26.5529			0.962541			0.481271	+	0.876572i	$0.340175\pi$
$762$		0			0
$763$	0.0885937	+	2.04107i	0.00320731	+	0.0738917i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	37.0082			1.33629
$768$		0			0
$769$			15.5747i			0.561638i	0.959761	+	0.280819i	$0.0906061\pi$
$769$			15.5747i			0.561638i	−0.959761	+	0.280819i	$0.909394\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			25.7402i			0.925810i	0.886408	+	0.462905i	$0.153193\pi$
$773$			25.7402i			0.925810i	−0.886408	+	0.462905i	$0.846807\pi$
$774$		0			0
$775$	−24.3767	−	1.68964i	−0.875636	−	0.0606938i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			35.2041i			1.26132i
$780$		0			0
$781$	8.98038			0.321343
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.7660	−	17.9683i	−0.598403	−	0.641316i
$786$		0			0
$787$	22.0998			0.787774			0.393887	−	0.919159i	$-0.371130\pi$
$787$	22.0998			0.787774			0.393887	+	0.919159i	$0.371130\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.79292	+	41.3062i	0.0637488	+	1.46868i
$792$		0			0
$793$			33.6735i			1.19578i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	20.9810i		−	0.743185i	−0.928396	−	0.371593i	$-0.878812\pi$
$797$		−	20.9810i		−	0.743185i	0.928396	−	0.371593i	$-0.121188\pi$
$798$		0			0
$799$	−5.69724			−0.201554
$800$		0			0
$801$		0			0
$802$		0			0
$803$			40.2028i			1.41873i
$804$		0			0
$805$	19.3127	−	19.6547i	0.680682	−	0.692736i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			4.57219i			0.160750i	0.996765	+	0.0803748i	$0.0256117\pi$
$809$			4.57219i			0.160750i	−0.996765	+	0.0803748i	$0.974388\pi$
$810$		0			0
$811$		−	11.3075i		−	0.397059i	−0.980095	−	0.198529i	$-0.936384\pi$
$811$		−	11.3075i		−	0.397059i	0.980095	−	0.198529i	$-0.0636165\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.92751	+	3.66470i	−0.137575	+	0.128369i
$816$		0			0
$817$	14.6375			0.512103
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	15.3988i		−	0.537423i	−0.963221	−	0.268711i	$-0.913402\pi$
$821$		−	15.3988i		−	0.537423i	0.963221	−	0.268711i	$-0.0865978\pi$
$822$		0			0
$823$		−	37.9418i		−	1.32257i	−0.750136	−	0.661284i	$-0.770012\pi$
$823$		−	37.9418i		−	1.32257i	0.750136	−	0.661284i	$-0.229988\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	53.6659			1.86614			0.933072	−	0.359689i	$-0.117117\pi$
$827$	53.6659			1.86614			0.933072	+	0.359689i	$0.117117\pi$
$828$		0			0
$829$		−	31.1117i		−	1.08055i	−0.841487	−	0.540277i	$-0.818320\pi$
$829$		−	31.1117i		−	1.08055i	0.841487	−	0.540277i	$-0.181680\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.269470	−	3.09825i	−0.00933658	−	0.107348i
$834$		0			0
$835$	−7.21387	+	6.73115i	−0.249646	+	0.232941i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−37.5213			−1.29538			−0.647690	−	0.761904i	$-0.724265\pi$
$839$	−37.5213			−1.29538			−0.647690	+	0.761904i	$0.724265\pi$
$840$		0			0
$841$	−61.9751			−2.13707
$842$		0			0
$843$		0			0
$844$		0			0
$845$	41.9973	+	45.0090i	1.44475	+	1.54836i
$846$		0			0
$847$	−0.413337	−	9.52269i	−0.0142024	−	0.327203i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			26.1328i			0.895820i
$852$		0			0
$853$	34.5817			1.18405			0.592027	−	0.805918i	$-0.298328\pi$
$853$	34.5817			1.18405			0.592027	+	0.805918i	$0.298328\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	19.8456i		−	0.677912i	−0.940802	−	0.338956i	$-0.889926\pi$
$857$		−	19.8456i		−	0.677912i	0.940802	−	0.338956i	$-0.110074\pi$
$858$		0			0
$859$		−	28.3445i		−	0.967102i	−0.875316	−	0.483551i	$-0.839347\pi$
$859$		−	28.3445i		−	0.967102i	0.875316	−	0.483551i	$-0.160653\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−47.1015			−1.60335			−0.801676	−	0.597758i	$-0.796058\pi$
$863$	−47.1015			−1.60335			−0.801676	+	0.597758i	$0.796058\pi$
$864$		0			0
$865$	−21.4373	+	20.0028i	−0.728889	+	0.680115i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	58.8907i		−	1.99773i
$870$		0			0
$871$			59.9982i			2.03296i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.6152	−	22.1415i	−0.663114	−	0.748518i
$876$		0			0
$877$			5.45843i			0.184318i	0.995744	+	0.0921590i	$0.0293768\pi$
$877$			5.45843i			0.184318i	−0.995744	+	0.0921590i	$0.970623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−54.5149			−1.83665			−0.918327	−	0.395822i	$-0.870460\pi$
$881$	−54.5149			−1.83665			−0.918327	+	0.395822i	$0.870460\pi$
$882$		0			0
$883$			10.5780i			0.355978i	0.984032	+	0.177989i	$0.0569591\pi$
$883$			10.5780i			0.355978i	−0.984032	+	0.177989i	$0.943041\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			14.7147i			0.494070i	0.969007	+	0.247035i	$0.0794563\pi$
$887$			14.7147i			0.494070i	−0.969007	+	0.247035i	$0.920544\pi$
$888$		0			0
$889$	24.9153	−	1.08146i	0.835631	−	0.0362710i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	56.3277			1.88493
$894$		0			0
$895$	−6.41414	+	5.98494i	−0.214401	+	0.200055i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−46.6130			−1.55463
$900$		0			0
$901$		−	0.567371i		−	0.0189019i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.96083	−	8.36122i	0.297868	−	0.277936i
$906$		0			0
$907$			6.72445i			0.223282i	0.993749	+	0.111641i	$0.0356106\pi$
$907$			6.72445i			0.223282i	−0.993749	+	0.111641i	$0.964389\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	22.4753i		−	0.744639i	−0.928105	−	0.372320i	$-0.878563\pi$
$911$		−	22.4753i		−	0.744639i	0.928105	−	0.372320i	$-0.121437\pi$
$912$		0			0
$913$	−4.18518			−0.138509
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.95244	+	44.9815i	0.0644754	+	1.48542i
$918$		0			0
$919$	21.8348			0.720264			0.360132	−	0.932901i	$-0.382732\pi$
$919$	21.8348			0.720264			0.360132	+	0.932901i	$0.382732\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	14.9613i		−	0.492457i
$924$		0			0
$925$	27.9864	+	1.93984i	0.920186	+	0.0637816i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.4260			−0.670156			−0.335078	−	0.942190i	$-0.608763\pi$
$929$	−20.4260			−0.670156			−0.335078	+	0.942190i	$0.608763\pi$
$930$		0			0
$931$	2.66420	+	30.6319i	0.0873158	+	1.00392i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.58989	+	2.77562i	0.0846984	+	0.0907724i
$936$		0			0
$937$	4.99830			0.163287			0.0816437	−	0.996662i	$-0.473983\pi$
$937$	4.99830			0.163287			0.0816437	+	0.996662i	$0.473983\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	32.3388			1.05421			0.527107	−	0.849799i	$-0.323277\pi$
$941$	32.3388			1.05421			0.527107	+	0.849799i	$0.323277\pi$
$942$		0			0
$943$	37.3293			1.21561
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.97719			0.161737			0.0808685	−	0.996725i	$-0.474231\pi$
$947$	4.97719			0.161737			0.0808685	+	0.996725i	$0.474231\pi$
$948$		0			0
$949$	66.9778			2.17419
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.49171			0.145501			0.0727504	−	0.997350i	$-0.476822\pi$
$953$	4.49171			0.145501			0.0727504	+	0.997350i	$0.476822\pi$
$954$		0			0
$955$	−11.2862	+	10.5310i	−0.365214	+	0.340776i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.54129	−	35.5092i	−0.0497710	−	1.14665i
$960$		0			0
$961$	7.11690			0.229577
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.0358	−	14.9627i	0.516210	−	0.481668i
$966$		0			0
$967$		−	23.8989i		−	0.768537i	−0.923221	−	0.384268i	$-0.874454\pi$
$967$		−	23.8989i		−	0.768537i	0.923221	−	0.384268i	$-0.125546\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.1129			0.420812			0.210406	−	0.977614i	$-0.432521\pi$
$971$	13.1129			0.420812			0.210406	+	0.977614i	$0.432521\pi$
$972$		0			0
$973$	18.9822	−	0.823933i	0.608543	−	0.0264141i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−29.7866			−0.952957			−0.476479	−	0.879186i	$-0.658087\pi$
$977$	−29.7866			−0.952957			−0.476479	+	0.879186i	$0.658087\pi$
$978$		0			0
$979$			40.8486i			1.30553i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			50.4762i			1.60994i	0.593314	+	0.804971i	$0.297819\pi$
$983$			50.4762i			1.60994i	−0.593314	+	0.804971i	$0.702181\pi$
$984$		0			0
$985$	−11.2435	−	12.0498i	−0.358246	−	0.383937i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	15.5212i		−	0.493544i
$990$		0			0
$991$	−36.1380			−1.14796			−0.573980	−	0.818869i	$-0.694601\pi$
$991$	−36.1380			−1.14796			−0.573980	+	0.818869i	$0.694601\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	26.2439	−	24.4878i	0.831988	−	0.776316i
$996$		0			0
$997$	−32.8539			−1.04049			−0.520246	−	0.854016i	$-0.674160\pi$
$997$	−32.8539			−1.04049			−0.520246	+	0.854016i	$0.674160\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.k.a.1889.20	yes	24
3.2	odd	2		2520.2.k.b.1889.5	yes	24
4.3	odd	2		5040.2.k.i.1889.20		24
5.4	even	2		2520.2.k.b.1889.19	yes	24
7.6	odd	2	inner	2520.2.k.a.1889.5	✓	24
12.11	even	2		5040.2.k.h.1889.5		24
15.14	odd	2	inner	2520.2.k.a.1889.6	yes	24
20.19	odd	2		5040.2.k.h.1889.19		24
21.20	even	2		2520.2.k.b.1889.20	yes	24
28.27	even	2		5040.2.k.i.1889.5		24
35.34	odd	2		2520.2.k.b.1889.6	yes	24
60.59	even	2		5040.2.k.i.1889.6		24
84.83	odd	2		5040.2.k.h.1889.20		24
105.104	even	2	inner	2520.2.k.a.1889.19	yes	24
140.139	even	2		5040.2.k.h.1889.6		24
420.419	odd	2		5040.2.k.i.1889.19		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.2.k.a.1889.5	✓	24	7.6	odd	2	inner
2520.2.k.a.1889.6	yes	24	15.14	odd	2	inner
2520.2.k.a.1889.19	yes	24	105.104	even	2	inner
2520.2.k.a.1889.20	yes	24	1.1	even	1	trivial
2520.2.k.b.1889.5	yes	24	3.2	odd	2
2520.2.k.b.1889.6	yes	24	35.34	odd	2
2520.2.k.b.1889.19	yes	24	5.4	even	2
2520.2.k.b.1889.20	yes	24	21.20	even	2
5040.2.k.h.1889.5		24	12.11	even	2
5040.2.k.h.1889.6		24	140.139	even	2
5040.2.k.h.1889.19		24	20.19	odd	2
5040.2.k.h.1889.20		24	84.83	odd	2
5040.2.k.i.1889.5		24	28.27	even	2
5040.2.k.i.1889.6		24	60.59	even	2
5040.2.k.i.1889.19		24	420.419	odd	2
5040.2.k.i.1889.20		24	4.3	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.52549 + 1.63489i) q^{5} +(0.114732 + 2.64326i) q^{7} +O(q^{10})\)	\(q+(1.52549 + 1.63489i) q^{5} +(0.114732 + 2.64326i) q^{7} -3.82134i q^{11} -6.36634 q^{13} +0.444278i q^{17} -4.39251i q^{19} -4.65767 q^{23} +(-0.345740 + 4.98803i) q^{25} +9.53809i q^{29} +4.88703i q^{31} +(-4.14642 + 4.21985i) q^{35} -5.61070i q^{37} -8.01459 q^{41} +3.33239i q^{43} +12.8236i q^{47} +(-6.97367 + 0.606534i) q^{49} -1.27706 q^{53} +(6.24747 - 5.82943i) q^{55} -5.81310 q^{59} -5.28930i q^{61} +(-9.71181 - 10.4083i) q^{65} -9.42429i q^{67} +2.35006i q^{71} -10.5206 q^{73} +(10.1008 - 0.438430i) q^{77} +15.4110 q^{79} -1.09521i q^{83} +(-0.726347 + 0.677744i) q^{85} -10.6896 q^{89} +(-0.730423 - 16.8279i) q^{91} +(7.18127 - 6.70074i) q^{95} -4.80920 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) \) 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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