LMFDB - Embedded newform 7500.2.d.g.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (of order $2$, degree $1$, not 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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -0.957526i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -0.957526i q^{7} -1.00000 q^{9} -5.41590 q^{11} +2.02271i q^{13} -0.642866i q^{17} +5.04105 q^{19} -0.957526 q^{21} -3.51960i q^{23} +1.00000i q^{27} -10.1408 q^{29} +3.69178 q^{31} +5.41590i q^{33} -11.3153i q^{37} +2.02271 q^{39} +3.52277 q^{41} -0.766348i q^{43} +4.93421i q^{47} +6.08314 q^{49} -0.642866 q^{51} -5.94634i q^{53} -5.04105i q^{57} -4.71878 q^{59} +4.34485 q^{61} +0.957526i q^{63} +9.51778i q^{67} -3.51960 q^{69} -11.9450 q^{71} +5.43304i q^{73} +5.18586i q^{77} -11.8494 q^{79} +1.00000 q^{81} +1.39408i q^{83} +10.1408i q^{87} +1.70812 q^{89} +1.93680 q^{91} -3.69178i q^{93} +14.5878i q^{97} +5.41590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^9	$ 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) $ 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.957526i	− 0.361911i	−0.983491	−	0.180955i	$-0.942081\pi$
$7$	− 0.957526i	− 0.361911i	0.983491	−	0.180955i	$-0.0579190\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.41590	−1.63295	−0.816477	−	0.577378i	$-0.804076\pi$
$11$	−5.41590	−1.63295	−0.816477	+	0.577378i	$0.804076\pi$
$12$	0	0
$13$	2.02271i	0.560999i	0.959854	+	0.280499i	$0.0905001\pi$
$13$	2.02271i	0.560999i	−0.959854	+	0.280499i	$0.909500\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.642866i	− 0.155918i	−0.996957	−	0.0779590i	$-0.975160\pi$
$17$	− 0.642866i	− 0.155918i	0.996957	−	0.0779590i	$-0.0248403\pi$
$18$	0	0
$19$	5.04105	1.15650	0.578248	−	0.815861i	$-0.303737\pi$
$19$	5.04105	1.15650	0.578248	+	0.815861i	$0.303737\pi$
$20$	0	0
$21$	−0.957526	−0.208949
$22$	0	0
$23$	− 3.51960i	− 0.733888i	−0.930243	−	0.366944i	$-0.880404\pi$
$23$	− 3.51960i	− 0.733888i	0.930243	−	0.366944i	$-0.119596\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−10.1408	−1.88309	−0.941546	−	0.336884i	$-0.890627\pi$
$29$	−10.1408	−1.88309	−0.941546	+	0.336884i	$0.890627\pi$
$30$	0	0
$31$	3.69178	0.663063	0.331531	−	0.943444i	$-0.392435\pi$
$31$	3.69178	0.663063	0.331531	+	0.943444i	$0.392435\pi$
$32$	0	0
$33$	5.41590i	0.942787i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11.3153i	− 1.86023i	−0.367272	−	0.930114i	$-0.619708\pi$
$37$	− 11.3153i	− 1.86023i	0.367272	−	0.930114i	$-0.380292\pi$
$38$	0	0
$39$	2.02271	0.323893
$40$	0	0
$41$	3.52277	0.550164	0.275082	−	0.961421i	$-0.411295\pi$
$41$	3.52277	0.550164	0.275082	+	0.961421i	$0.411295\pi$
$42$	0	0
$43$	− 0.766348i	− 0.116867i	−0.998291	−	0.0584335i	$-0.981389\pi$
$43$	− 0.766348i	− 0.116867i	0.998291	−	0.0584335i	$-0.0186106\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.93421i	0.719729i	0.933005	+	0.359865i	$0.117177\pi$
$47$	4.93421i	0.719729i	−0.933005	+	0.359865i	$0.882823\pi$
$48$	0	0
$49$	6.08314	0.869021
$50$	0	0
$51$	−0.642866	−0.0900193
$52$	0	0
$53$	− 5.94634i	− 0.816793i	−0.912805	−	0.408396i	$-0.866088\pi$
$53$	− 5.94634i	− 0.816793i	0.912805	−	0.408396i	$-0.133912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 5.04105i	− 0.667703i
$58$	0	0
$59$	−4.71878	−0.614333	−0.307166	−	0.951656i	$-0.599381\pi$
$59$	−4.71878	−0.614333	−0.307166	+	0.951656i	$0.599381\pi$
$60$	0	0
$61$	4.34485	0.556301	0.278150	−	0.960538i	$-0.410279\pi$
$61$	4.34485	0.556301	0.278150	+	0.960538i	$0.410279\pi$
$62$	0	0
$63$	0.957526i	0.120637i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.51778i	1.16278i	0.813624	+	0.581391i	$0.197491\pi$
$67$	9.51778i	1.16278i	−0.813624	+	0.581391i	$0.802509\pi$
$68$	0	0
$69$	−3.51960	−0.423710
$70$	0	0
$71$	−11.9450	−1.41761	−0.708803	−	0.705406i	$-0.750765\pi$
$71$	−11.9450	−1.41761	−0.708803	+	0.705406i	$0.750765\pi$
$72$	0	0
$73$	5.43304i	0.635889i	0.948109	+	0.317944i	$0.102993\pi$
$73$	5.43304i	0.635889i	−0.948109	+	0.317944i	$0.897007\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.18586i	0.590984i
$78$	0	0
$79$	−11.8494	−1.33316	−0.666581	−	0.745432i	$-0.732243\pi$
$79$	−11.8494	−1.33316	−0.666581	+	0.745432i	$0.732243\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.39408i	0.153020i	0.997069	+	0.0765101i	$0.0243778\pi$
$83$	1.39408i	0.153020i	−0.997069	+	0.0765101i	$0.975622\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.1408i	1.08720i
$88$	0	0
$89$	1.70812	0.181061	0.0905303	−	0.995894i	$-0.471144\pi$
$89$	1.70812	0.181061	0.0905303	+	0.995894i	$0.471144\pi$
$90$	0	0
$91$	1.93680	0.203032
$92$	0	0
$93$	− 3.69178i	− 0.382819i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.5878i	1.48117i	0.671963	+	0.740585i	$0.265451\pi$
$97$	14.5878i	1.48117i	−0.671963	+	0.740585i	$0.734549\pi$
$98$	0	0
$99$	5.41590	0.544318
$100$	0	0
$101$	−10.2832	−1.02322	−0.511610	−	0.859218i	$-0.670951\pi$
$101$	−10.2832	−1.02322	−0.511610	+	0.859218i	$0.670951\pi$
$102$	0	0
$103$	− 13.9848i	− 1.37797i	−0.724777	−	0.688983i	$-0.758057\pi$
$103$	− 13.9848i	− 1.37797i	0.724777	−	0.688983i	$-0.241943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.06727i	0.876566i	0.898837	+	0.438283i	$0.144413\pi$
$107$	9.06727i	0.876566i	−0.898837	+	0.438283i	$0.855587\pi$
$108$	0	0
$109$	−2.37535	−0.227518	−0.113759	−	0.993508i	$-0.536289\pi$
$109$	−2.37535	−0.227518	−0.113759	+	0.993508i	$0.536289\pi$
$110$	0	0
$111$	−11.3153	−1.07400
$112$	0	0
$113$	13.4374i	1.26408i	0.774935	+	0.632041i	$0.217782\pi$
$113$	13.4374i	1.26408i	−0.774935	+	0.632041i	$0.782218\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.02271i	− 0.187000i
$118$	0	0
$119$	−0.615561	−0.0564284
$120$	0	0
$121$	18.3319	1.66654
$122$	0	0
$123$	− 3.52277i	− 0.317637i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.2629i	1.26563i	0.774303	+	0.632815i	$0.218101\pi$
$127$	14.2629i	1.26563i	−0.774303	+	0.632815i	$0.781899\pi$
$128$	0	0
$129$	−0.766348	−0.0674732
$130$	0	0
$131$	−0.128666	−0.0112416	−0.00562082	−	0.999984i	$-0.501789\pi$
$131$	−0.128666	−0.0112416	−0.00562082	+	0.999984i	$0.501789\pi$
$132$	0	0
$133$	− 4.82694i	− 0.418548i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.7022i	1.17066i	0.810795	+	0.585331i	$0.199035\pi$
$137$	13.7022i	1.17066i	−0.810795	+	0.585331i	$0.800965\pi$
$138$	0	0
$139$	−23.4718	−1.99085	−0.995425	−	0.0955488i	$-0.969539\pi$
$139$	−23.4718	−1.99085	−0.995425	+	0.0955488i	$0.969539\pi$
$140$	0	0
$141$	4.93421	0.415536
$142$	0	0
$143$	− 10.9548i	− 0.916086i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 6.08314i	− 0.501729i
$148$	0	0
$149$	10.6938	0.876071	0.438035	−	0.898958i	$-0.355674\pi$
$149$	10.6938	0.876071	0.438035	+	0.898958i	$0.355674\pi$
$150$	0	0
$151$	7.37520	0.600185	0.300092	−	0.953910i	$-0.402982\pi$
$151$	7.37520	0.600185	0.300092	+	0.953910i	$0.402982\pi$
$152$	0	0
$153$	0.642866i	0.0519727i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.0329i	1.04014i	0.854124	+	0.520070i	$0.174094\pi$
$157$	13.0329i	1.04014i	−0.854124	+	0.520070i	$0.825906\pi$
$158$	0	0
$159$	−5.94634	−0.471575
$160$	0	0
$161$	−3.37011	−0.265602
$162$	0	0
$163$	− 7.39602i	− 0.579301i	−0.957133	−	0.289650i	$-0.906461\pi$
$163$	− 7.39602i	− 0.579301i	0.957133	−	0.289650i	$-0.0935390\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.1265i	1.55744i	0.627374	+	0.778718i	$0.284130\pi$
$167$	20.1265i	1.55744i	−0.627374	+	0.778718i	$0.715870\pi$
$168$	0	0
$169$	8.90864	0.685280
$170$	0	0
$171$	−5.04105	−0.385499
$172$	0	0
$173$	2.22112i	0.168869i	0.996429	+	0.0844344i	$0.0269084\pi$
$173$	2.22112i	0.168869i	−0.996429	+	0.0844344i	$0.973092\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.71878i	0.354685i
$178$	0	0
$179$	0.0385919	0.00288449	0.00144225	−	0.999999i	$-0.499541\pi$
$179$	0.0385919	0.00288449	0.00144225	+	0.999999i	$0.499541\pi$
$180$	0	0
$181$	0.146945	0.0109223	0.00546116	−	0.999985i	$-0.498262\pi$
$181$	0.146945	0.0109223	0.00546116	+	0.999985i	$0.498262\pi$
$182$	0	0
$183$	− 4.34485i	− 0.321180i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.48170i	0.254607i
$188$	0	0
$189$	0.957526	0.0696498
$190$	0	0
$191$	−0.459682	−0.0332614	−0.0166307	−	0.999862i	$-0.505294\pi$
$191$	−0.459682	−0.0332614	−0.0166307	+	0.999862i	$0.505294\pi$
$192$	0	0
$193$	19.0231i	1.36932i	0.728864	+	0.684658i	$0.240048\pi$
$193$	19.0231i	1.36932i	−0.728864	+	0.684658i	$0.759952\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.1151i	0.863164i	0.902074	+	0.431582i	$0.142045\pi$
$197$	12.1151i	0.863164i	−0.902074	+	0.431582i	$0.857955\pi$
$198$	0	0
$199$	−16.4872	−1.16875	−0.584375	−	0.811484i	$-0.698660\pi$
$199$	−16.4872	−1.16875	−0.584375	+	0.811484i	$0.698660\pi$
$200$	0	0
$201$	9.51778	0.671333
$202$	0	0
$203$	9.71005i	0.681512i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.51960i	0.244629i
$208$	0	0
$209$	−27.3018	−1.88850
$210$	0	0
$211$	18.1862	1.25199	0.625996	−	0.779827i	$-0.284693\pi$
$211$	18.1862	1.25199	0.625996	+	0.779827i	$0.284693\pi$
$212$	0	0
$213$	11.9450i	0.818455i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.53497i	− 0.239970i
$218$	0	0
$219$	5.43304	0.367131
$220$	0	0
$221$	1.30033	0.0874698
$222$	0	0
$223$	23.5103i	1.57436i	0.616721	+	0.787182i	$0.288461\pi$
$223$	23.5103i	1.57436i	−0.616721	+	0.787182i	$0.711539\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19.6803i	− 1.30623i	−0.757260	−	0.653114i	$-0.773462\pi$
$227$	− 19.6803i	− 1.30623i	0.757260	−	0.653114i	$-0.226538\pi$
$228$	0	0
$229$	−5.30698	−0.350695	−0.175347	−	0.984507i	$-0.556105\pi$
$229$	−5.30698	−0.350695	−0.175347	+	0.984507i	$0.556105\pi$
$230$	0	0
$231$	5.18586	0.341205
$232$	0	0
$233$	23.2693i	1.52442i	0.647327	+	0.762212i	$0.275887\pi$
$233$	23.2693i	1.52442i	−0.647327	+	0.762212i	$0.724113\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.8494i	0.769702i
$238$	0	0
$239$	15.0750	0.975118	0.487559	−	0.873090i	$-0.337887\pi$
$239$	15.0750	0.975118	0.487559	+	0.873090i	$0.337887\pi$
$240$	0	0
$241$	−18.9133	−1.21831	−0.609157	−	0.793049i	$-0.708492\pi$
$241$	−18.9133	−1.21831	−0.609157	+	0.793049i	$0.708492\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.1966i	0.648793i
$248$	0	0
$249$	1.39408	0.0883463
$250$	0	0
$251$	−4.56761	−0.288305	−0.144153	−	0.989555i	$-0.546046\pi$
$251$	−4.56761	−0.288305	−0.144153	+	0.989555i	$0.546046\pi$
$252$	0	0
$253$	19.0618i	1.19841i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.2556i	1.26351i	0.775169	+	0.631754i	$0.217665\pi$
$257$	20.2556i	1.26351i	−0.775169	+	0.631754i	$0.782335\pi$
$258$	0	0
$259$	−10.8347	−0.673237
$260$	0	0
$261$	10.1408	0.627698
$262$	0	0
$263$	− 30.7789i	− 1.89791i	−0.315418	−	0.948953i	$-0.602145\pi$
$263$	− 30.7789i	− 1.89791i	0.315418	−	0.948953i	$-0.397855\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.70812i	− 0.104535i
$268$	0	0
$269$	26.6739	1.62634	0.813168	−	0.582030i	$-0.197741\pi$
$269$	26.6739	1.62634	0.813168	+	0.582030i	$0.197741\pi$
$270$	0	0
$271$	−5.52945	−0.335890	−0.167945	−	0.985796i	$-0.553713\pi$
$271$	−5.52945	−0.335890	−0.167945	+	0.985796i	$0.553713\pi$
$272$	0	0
$273$	− 1.93680i	− 0.117220i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.9233i	1.85800i	0.370077	+	0.929001i	$0.379331\pi$
$277$	30.9233i	1.85800i	−0.370077	+	0.929001i	$0.620669\pi$
$278$	0	0
$279$	−3.69178	−0.221021
$280$	0	0
$281$	−16.8489	−1.00512	−0.502560	−	0.864542i	$-0.667608\pi$
$281$	−16.8489	−1.00512	−0.502560	+	0.864542i	$0.667608\pi$
$282$	0	0
$283$	− 11.2627i	− 0.669501i	−0.942307	−	0.334750i	$-0.891348\pi$
$283$	− 11.2627i	− 0.669501i	0.942307	−	0.334750i	$-0.108652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.37314i	− 0.199110i
$288$	0	0
$289$	16.5867	0.975690
$290$	0	0
$291$	14.5878	0.855154
$292$	0	0
$293$	− 11.1995i	− 0.654284i	−0.944975	−	0.327142i	$-0.893914\pi$
$293$	− 11.1995i	− 0.654284i	0.944975	−	0.327142i	$-0.106086\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.41590i	− 0.314262i
$298$	0	0
$299$	7.11914	0.411710
$300$	0	0
$301$	−0.733798	−0.0422954
$302$	0	0
$303$	10.2832i	0.590756i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 24.1289i	− 1.37711i	−0.725185	−	0.688554i	$-0.758246\pi$
$307$	− 24.1289i	− 1.37711i	0.725185	−	0.688554i	$-0.241754\pi$
$308$	0	0
$309$	−13.9848	−0.795569
$310$	0	0
$311$	2.07312	0.117556	0.0587778	−	0.998271i	$-0.481280\pi$
$311$	2.07312	0.117556	0.0587778	+	0.998271i	$0.481280\pi$
$312$	0	0
$313$	− 8.51966i	− 0.481560i	−0.970580	−	0.240780i	$-0.922597\pi$
$313$	− 8.51966i	− 0.481560i	0.970580	−	0.240780i	$-0.0774032\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.6070i	− 0.764243i	−0.924112	−	0.382122i	$-0.875194\pi$
$317$	− 13.6070i	− 0.764243i	0.924112	−	0.382122i	$-0.124806\pi$
$318$	0	0
$319$	54.9213	3.07500
$320$	0	0
$321$	9.06727	0.506086
$322$	0	0
$323$	− 3.24072i	− 0.180318i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.37535i	0.131357i
$328$	0	0
$329$	4.72464	0.260478
$330$	0	0
$331$	−22.6320	−1.24397	−0.621985	−	0.783029i	$-0.713673\pi$
$331$	−22.6320	−1.24397	−0.621985	+	0.783029i	$0.713673\pi$
$332$	0	0
$333$	11.3153i	0.620076i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	35.9900i	1.96050i	0.197762	+	0.980250i	$0.436633\pi$
$337$	35.9900i	1.96050i	−0.197762	+	0.980250i	$0.563367\pi$
$338$	0	0
$339$	13.4374	0.729818
$340$	0	0
$341$	−19.9943	−1.08275
$342$	0	0
$343$	− 12.5275i	− 0.676419i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.96344i	− 0.481183i	−0.970627	−	0.240591i	$-0.922659\pi$
$347$	− 8.96344i	− 0.481183i	0.970627	−	0.240591i	$-0.0773414\pi$
$348$	0	0
$349$	11.8276	0.633114	0.316557	−	0.948573i	$-0.397473\pi$
$349$	11.8276	0.633114	0.316557	+	0.948573i	$0.397473\pi$
$350$	0	0
$351$	−2.02271	−0.107964
$352$	0	0
$353$	− 23.0886i	− 1.22888i	−0.788962	−	0.614442i	$-0.789381\pi$
$353$	− 23.0886i	− 1.22888i	0.788962	−	0.614442i	$-0.210619\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.615561i	0.0325790i
$358$	0	0
$359$	−21.1188	−1.11461	−0.557304	−	0.830309i	$-0.688164\pi$
$359$	−21.1188	−1.11461	−0.557304	+	0.830309i	$0.688164\pi$
$360$	0	0
$361$	6.41217	0.337482
$362$	0	0
$363$	− 18.3319i	− 0.962177i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.39044i	0.437977i	0.975727	+	0.218989i	$0.0702758\pi$
$367$	8.39044i	0.437977i	−0.975727	+	0.218989i	$0.929724\pi$
$368$	0	0
$369$	−3.52277	−0.183388
$370$	0	0
$371$	−5.69378	−0.295606
$372$	0	0
$373$	− 6.32863i	− 0.327684i	−0.986487	−	0.163842i	$-0.947611\pi$
$373$	− 6.32863i	− 0.327684i	0.986487	−	0.163842i	$-0.0523887\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 20.5118i	− 1.05641i
$378$	0	0
$379$	−15.0788	−0.774546	−0.387273	−	0.921965i	$-0.626583\pi$
$379$	−15.0788	−0.774546	−0.387273	+	0.921965i	$0.626583\pi$
$380$	0	0
$381$	14.2629	0.730712
$382$	0	0
$383$	− 32.5782i	− 1.66467i	−0.554276	−	0.832333i	$-0.687005\pi$
$383$	− 32.5782i	− 1.66467i	0.554276	−	0.832333i	$-0.312995\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.766348i	0.0389557i
$388$	0	0
$389$	5.47168	0.277425	0.138713	−	0.990333i	$-0.455704\pi$
$389$	5.47168	0.277425	0.138713	+	0.990333i	$0.455704\pi$
$390$	0	0
$391$	−2.26263	−0.114426
$392$	0	0
$393$	0.128666i	0.00649036i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.2686i	− 0.716120i	−0.933699	−	0.358060i	$-0.883438\pi$
$397$	− 14.2686i	− 0.716120i	0.933699	−	0.358060i	$-0.116562\pi$
$398$	0	0
$399$	−4.82694	−0.241649
$400$	0	0
$401$	17.8291	0.890342	0.445171	−	0.895446i	$-0.353143\pi$
$401$	17.8291	0.890342	0.445171	+	0.895446i	$0.353143\pi$
$402$	0	0
$403$	7.46740i	0.371977i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	61.2826i	3.03767i
$408$	0	0
$409$	−24.0428	−1.18884	−0.594420	−	0.804154i	$-0.702618\pi$
$409$	−24.0428	−1.18884	−0.594420	+	0.804154i	$0.702618\pi$
$410$	0	0
$411$	13.7022	0.675882
$412$	0	0
$413$	4.51835i	0.222334i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.4718i	1.14942i
$418$	0	0
$419$	28.0809	1.37184	0.685922	−	0.727675i	$-0.259399\pi$
$419$	28.0809	1.37184	0.685922	+	0.727675i	$0.259399\pi$
$420$	0	0
$421$	20.1952	0.984254	0.492127	−	0.870523i	$-0.336220\pi$
$421$	20.1952	0.984254	0.492127	+	0.870523i	$0.336220\pi$
$422$	0	0
$423$	− 4.93421i	− 0.239910i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.16030i	− 0.201331i
$428$	0	0
$429$	−10.9548	−0.528902
$430$	0	0
$431$	−4.15210	−0.200000	−0.0999999	−	0.994987i	$-0.531884\pi$
$431$	−4.15210	−0.200000	−0.0999999	+	0.994987i	$0.531884\pi$
$432$	0	0
$433$	− 30.3201i	− 1.45709i	−0.684998	−	0.728545i	$-0.740197\pi$
$433$	− 30.3201i	− 1.45709i	0.684998	−	0.728545i	$-0.259803\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 17.7425i	− 0.848738i
$438$	0	0
$439$	3.18451	0.151988	0.0759941	−	0.997108i	$-0.475787\pi$
$439$	3.18451	0.151988	0.0759941	+	0.997108i	$0.475787\pi$
$440$	0	0
$441$	−6.08314	−0.289674
$442$	0	0
$443$	30.4607i	1.44723i	0.690204	+	0.723615i	$0.257521\pi$
$443$	30.4607i	1.44723i	−0.690204	+	0.723615i	$0.742479\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 10.6938i	− 0.505800i
$448$	0	0
$449$	−1.35787	−0.0640820	−0.0320410	−	0.999487i	$-0.510201\pi$
$449$	−1.35787	−0.0640820	−0.0320410	+	0.999487i	$0.510201\pi$
$450$	0	0
$451$	−19.0789	−0.898392
$452$	0	0
$453$	− 7.37520i	− 0.346517i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.89208i	− 0.462732i	−0.972867	−	0.231366i	$-0.925680\pi$
$457$	− 9.89208i	− 0.462732i	0.972867	−	0.231366i	$-0.0743195\pi$
$458$	0	0
$459$	0.642866	0.0300064
$460$	0	0
$461$	−21.0835	−0.981954	−0.490977	−	0.871172i	$-0.663360\pi$
$461$	−21.0835	−0.981954	−0.490977	+	0.871172i	$0.663360\pi$
$462$	0	0
$463$	− 0.514142i	− 0.0238942i	−0.999929	−	0.0119471i	$-0.996197\pi$
$463$	− 0.514142i	− 0.0238942i	0.999929	−	0.0119471i	$-0.00380297\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.5740i	0.859502i	0.902947	+	0.429751i	$0.141399\pi$
$467$	18.5740i	0.859502i	−0.902947	+	0.429751i	$0.858601\pi$
$468$	0	0
$469$	9.11353	0.420824
$470$	0	0
$471$	13.0329	0.600525
$472$	0	0
$473$	4.15046i	0.190838i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.94634i	0.272264i
$478$	0	0
$479$	−22.7247	−1.03832	−0.519159	−	0.854677i	$-0.673755\pi$
$479$	−22.7247	−1.03832	−0.519159	+	0.854677i	$0.673755\pi$
$480$	0	0
$481$	22.8876	1.04359
$482$	0	0
$483$	3.37011i	0.153345i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.5852i	1.20469i	0.798235	+	0.602346i	$0.205767\pi$
$487$	26.5852i	1.20469i	−0.798235	+	0.602346i	$0.794233\pi$
$488$	0	0
$489$	−7.39602	−0.334459
$490$	0	0
$491$	5.39133	0.243307	0.121654	−	0.992573i	$-0.461180\pi$
$491$	5.39133	0.243307	0.121654	+	0.992573i	$0.461180\pi$
$492$	0	0
$493$	6.51915i	0.293608i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.4376i	0.513047i
$498$	0	0
$499$	14.5574	0.651677	0.325839	−	0.945425i	$-0.394353\pi$
$499$	14.5574	0.651677	0.325839	+	0.945425i	$0.394353\pi$
$500$	0	0
$501$	20.1265	0.899187
$502$	0	0
$503$	33.9160i	1.51224i	0.654432	+	0.756121i	$0.272908\pi$
$503$	33.9160i	1.51224i	−0.654432	+	0.756121i	$0.727092\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8.90864i	− 0.395647i
$508$	0	0
$509$	16.7914	0.744265	0.372132	−	0.928180i	$-0.378627\pi$
$509$	16.7914	0.744265	0.372132	+	0.928180i	$0.378627\pi$
$510$	0	0
$511$	5.20228	0.230135
$512$	0	0
$513$	5.04105i	0.222568i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 26.7232i	− 1.17528i
$518$	0	0
$519$	2.22112	0.0974965
$520$	0	0
$521$	−17.0913	−0.748782	−0.374391	−	0.927271i	$-0.622148\pi$
$521$	−17.0913	−0.748782	−0.374391	+	0.927271i	$0.622148\pi$
$522$	0	0
$523$	− 20.6505i	− 0.902983i	−0.892275	−	0.451491i	$-0.850892\pi$
$523$	− 20.6505i	− 0.902983i	0.892275	−	0.451491i	$-0.149108\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.37332i	− 0.103383i
$528$	0	0
$529$	10.6124	0.461409
$530$	0	0
$531$	4.71878	0.204778
$532$	0	0
$533$	7.12554i	0.308641i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 0.0385919i	− 0.00166536i
$538$	0	0
$539$	−32.9457	−1.41907
$540$	0	0
$541$	−12.1666	−0.523082	−0.261541	−	0.965192i	$-0.584231\pi$
$541$	−12.1666	−0.523082	−0.261541	+	0.965192i	$0.584231\pi$
$542$	0	0
$543$	− 0.146945i	− 0.00630600i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.1670i	0.477465i	0.971085	+	0.238733i	$0.0767320\pi$
$547$	11.1670i	0.477465i	−0.971085	+	0.238733i	$0.923268\pi$
$548$	0	0
$549$	−4.34485	−0.185434
$550$	0	0
$551$	−51.1201	−2.17779
$552$	0	0
$553$	11.3461i	0.482486i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.9282i	0.547787i	0.961760	+	0.273894i	$0.0883116\pi$
$557$	12.9282i	0.547787i	−0.961760	+	0.273894i	$0.911688\pi$
$558$	0	0
$559$	1.55010	0.0655622
$560$	0	0
$561$	3.48170	0.146997
$562$	0	0
$563$	31.7249i	1.33705i	0.743692	+	0.668523i	$0.233073\pi$
$563$	31.7249i	1.33705i	−0.743692	+	0.668523i	$0.766927\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.957526i	− 0.0402123i
$568$	0	0
$569$	−28.8066	−1.20764	−0.603818	−	0.797122i	$-0.706354\pi$
$569$	−28.8066	−1.20764	−0.603818	+	0.797122i	$0.706354\pi$
$570$	0	0
$571$	−18.1701	−0.760396	−0.380198	−	0.924905i	$-0.624144\pi$
$571$	−18.1701	−0.760396	−0.380198	+	0.924905i	$0.624144\pi$
$572$	0	0
$573$	0.459682i	0.0192035i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	40.7956i	1.69834i	0.528116	+	0.849172i	$0.322899\pi$
$577$	40.7956i	1.69834i	−0.528116	+	0.849172i	$0.677101\pi$
$578$	0	0
$579$	19.0231	0.790575
$580$	0	0
$581$	1.33487	0.0553797
$582$	0	0
$583$	32.2048i	1.33379i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.20488i	− 0.379926i	−0.981791	−	0.189963i	$-0.939163\pi$
$587$	− 9.20488i	− 0.379926i	0.981791	−	0.189963i	$-0.0608368\pi$
$588$	0	0
$589$	18.6104	0.766829
$590$	0	0
$591$	12.1151	0.498348
$592$	0	0
$593$	12.3856i	0.508614i	0.967124	+	0.254307i	$0.0818474\pi$
$593$	12.3856i	0.508614i	−0.967124	+	0.254307i	$0.918153\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.4872i	0.674778i
$598$	0	0
$599$	18.1732	0.742536	0.371268	−	0.928526i	$-0.378923\pi$
$599$	18.1732	0.742536	0.371268	+	0.928526i	$0.378923\pi$
$600$	0	0
$601$	29.8155	1.21620	0.608099	−	0.793861i	$-0.291932\pi$
$601$	29.8155	1.21620	0.608099	+	0.793861i	$0.291932\pi$
$602$	0	0
$603$	− 9.51778i	− 0.387594i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9.16747i	− 0.372096i	−0.982541	−	0.186048i	$-0.940432\pi$
$607$	− 9.16747i	− 0.372096i	0.982541	−	0.186048i	$-0.0595680\pi$
$608$	0	0
$609$	9.71005	0.393471
$610$	0	0
$611$	−9.98049	−0.403767
$612$	0	0
$613$	14.0540i	0.567638i	0.958878	+	0.283819i	$0.0916014\pi$
$613$	14.0540i	0.567638i	−0.958878	+	0.283819i	$0.908399\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.2343i	1.45874i	0.684121	+	0.729368i	$0.260186\pi$
$617$	36.2343i	1.45874i	−0.684121	+	0.729368i	$0.739814\pi$
$618$	0	0
$619$	−25.6751	−1.03197	−0.515986	−	0.856597i	$-0.672574\pi$
$619$	−25.6751	−1.03197	−0.515986	+	0.856597i	$0.672574\pi$
$620$	0	0
$621$	3.51960	0.141237
$622$	0	0
$623$	− 1.63557i	− 0.0655278i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	27.3018i	1.09033i
$628$	0	0
$629$	−7.27424	−0.290043
$630$	0	0
$631$	−24.6723	−0.982190	−0.491095	−	0.871106i	$-0.663403\pi$
$631$	−24.6723	−0.982190	−0.491095	+	0.871106i	$0.663403\pi$
$632$	0	0
$633$	− 18.1862i	− 0.722838i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.3044i	0.487520i
$638$	0	0
$639$	11.9450	0.472535
$640$	0	0
$641$	−29.9378	−1.18247	−0.591236	−	0.806499i	$-0.701360\pi$
$641$	−29.9378	−1.18247	−0.591236	+	0.806499i	$0.701360\pi$
$642$	0	0
$643$	− 10.1343i	− 0.399658i	−0.979831	−	0.199829i	$-0.935961\pi$
$643$	− 10.1343i	− 0.399658i	0.979831	−	0.199829i	$-0.0640387\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.2593i	1.77932i	0.456619	+	0.889662i	$0.349060\pi$
$647$	45.2593i	1.77932i	−0.456619	+	0.889662i	$0.650940\pi$
$648$	0	0
$649$	25.5564	1.00318
$650$	0	0
$651$	−3.53497	−0.138547
$652$	0	0
$653$	32.9848i	1.29080i	0.763846	+	0.645398i	$0.223309\pi$
$653$	32.9848i	1.29080i	−0.763846	+	0.645398i	$0.776691\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.43304i	− 0.211963i
$658$	0	0
$659$	−2.55453	−0.0995102	−0.0497551	−	0.998761i	$-0.515844\pi$
$659$	−2.55453	−0.0995102	−0.0497551	+	0.998761i	$0.515844\pi$
$660$	0	0
$661$	−45.4124	−1.76634	−0.883170	−	0.469054i	$-0.844595\pi$
$661$	−45.4124	−1.76634	−0.883170	+	0.469054i	$0.844595\pi$
$662$	0	0
$663$	− 1.30033i	− 0.0505007i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.6915i	1.38198i
$668$	0	0
$669$	23.5103	0.908959
$670$	0	0
$671$	−23.5312	−0.908413
$672$	0	0
$673$	− 11.3427i	− 0.437228i	−0.975811	−	0.218614i	$-0.929847\pi$
$673$	− 11.3427i	− 0.437228i	0.975811	−	0.218614i	$-0.0701535\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.4684i	− 0.594499i	−0.954800	−	0.297249i	$-0.903931\pi$
$677$	− 15.4684i	− 0.594499i	0.954800	−	0.297249i	$-0.0960693\pi$
$678$	0	0
$679$	13.9682	0.536051
$680$	0	0
$681$	−19.6803	−0.754151
$682$	0	0
$683$	− 6.79302i	− 0.259928i	−0.991519	−	0.129964i	$-0.958514\pi$
$683$	− 6.79302i	− 0.259928i	0.991519	−	0.129964i	$-0.0414861\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.30698i	0.202474i
$688$	0	0
$689$	12.0277	0.458220
$690$	0	0
$691$	42.3195	1.60991	0.804954	−	0.593337i	$-0.202190\pi$
$691$	42.3195	1.60991	0.804954	+	0.593337i	$0.202190\pi$
$692$	0	0
$693$	− 5.18586i	− 0.196995i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.26467i	− 0.0857804i
$698$	0	0
$699$	23.2693	0.880127
$700$	0	0
$701$	32.2924	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$701$	32.2924	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$702$	0	0
$703$	− 57.0411i	− 2.15135i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.84647i	0.370314i
$708$	0	0
$709$	2.15883	0.0810767	0.0405383	−	0.999178i	$-0.487093\pi$
$709$	2.15883	0.0810767	0.0405383	+	0.999178i	$0.487093\pi$
$710$	0	0
$711$	11.8494	0.444388
$712$	0	0
$713$	− 12.9936i	− 0.486614i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 15.0750i	− 0.562985i
$718$	0	0
$719$	−1.59152	−0.0593537	−0.0296768	−	0.999560i	$-0.509448\pi$
$719$	−1.59152	−0.0593537	−0.0296768	+	0.999560i	$0.509448\pi$
$720$	0	0
$721$	−13.3908	−0.498701
$722$	0	0
$723$	18.9133i	0.703394i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.33982i	− 0.0496910i	−0.999691	−	0.0248455i	$-0.992091\pi$
$727$	− 1.33982i	− 0.0496910i	0.999691	−	0.0248455i	$-0.00790938\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−0.492659	−0.0182217
$732$	0	0
$733$	43.8507i	1.61966i	0.586664	+	0.809831i	$0.300441\pi$
$733$	43.8507i	1.61966i	−0.586664	+	0.809831i	$0.699559\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 51.5473i	− 1.89877i
$738$	0	0
$739$	−18.3674	−0.675656	−0.337828	−	0.941208i	$-0.609692\pi$
$739$	−18.3674	−0.675656	−0.337828	+	0.941208i	$0.609692\pi$
$740$	0	0
$741$	10.1966	0.374581
$742$	0	0
$743$	− 21.5051i	− 0.788947i	−0.918907	−	0.394474i	$-0.870927\pi$
$743$	− 21.5051i	− 0.788947i	0.918907	−	0.394474i	$-0.129073\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.39408i	− 0.0510068i
$748$	0	0
$749$	8.68215	0.317239
$750$	0	0
$751$	−7.02810	−0.256459	−0.128230	−	0.991745i	$-0.540929\pi$
$751$	−7.02810	−0.256459	−0.128230	+	0.991745i	$0.540929\pi$
$752$	0	0
$753$	4.56761i	0.166453i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 5.39361i	− 0.196034i	−0.995185	−	0.0980171i	$-0.968750\pi$
$757$	− 5.39361i	− 0.196034i	0.995185	−	0.0980171i	$-0.0312500\pi$
$758$	0	0
$759$	19.0618	0.691900
$760$	0	0
$761$	−20.1500	−0.730438	−0.365219	−	0.930922i	$-0.619006\pi$
$761$	−20.1500	−0.730438	−0.365219	+	0.930922i	$0.619006\pi$
$762$	0	0
$763$	2.27446i	0.0823411i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.54472i	− 0.344640i
$768$	0	0
$769$	−30.9167	−1.11488	−0.557442	−	0.830216i	$-0.688217\pi$
$769$	−30.9167	−1.11488	−0.557442	+	0.830216i	$0.688217\pi$
$770$	0	0
$771$	20.2556	0.729486
$772$	0	0
$773$	− 31.9913i	− 1.15065i	−0.817927	−	0.575323i	$-0.804876\pi$
$773$	− 31.9913i	− 1.15065i	0.817927	−	0.575323i	$-0.195124\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.8347i	0.388693i
$778$	0	0
$779$	17.7584	0.636262
$780$	0	0
$781$	64.6927	2.31489
$782$	0	0
$783$	− 10.1408i	− 0.362401i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 24.4220i	− 0.870550i	−0.900298	−	0.435275i	$-0.856651\pi$
$787$	− 24.4220i	− 0.870550i	0.900298	−	0.435275i	$-0.143349\pi$
$788$	0	0
$789$	−30.7789	−1.09576
$790$	0	0
$791$	12.8666	0.457485
$792$	0	0
$793$	8.78837i	0.312084i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.6380i	1.75827i	0.476576	+	0.879133i	$0.341878\pi$
$797$	49.6380i	1.75827i	−0.476576	+	0.879133i	$0.658122\pi$
$798$	0	0
$799$	3.17204	0.112219
$800$	0	0
$801$	−1.70812	−0.0603535
$802$	0	0
$803$	− 29.4248i	− 1.03838i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 26.6739i	− 0.938965i
$808$	0	0
$809$	18.8392	0.662350	0.331175	−	0.943569i	$-0.392555\pi$
$809$	18.8392	0.662350	0.331175	+	0.943569i	$0.392555\pi$
$810$	0	0
$811$	18.7021	0.656721	0.328360	−	0.944553i	$-0.393504\pi$
$811$	18.7021	0.656721	0.328360	+	0.944553i	$0.393504\pi$
$812$	0	0
$813$	5.52945i	0.193926i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.86320i	− 0.135156i
$818$	0	0
$819$	−1.93680	−0.0676772
$820$	0	0
$821$	36.7240	1.28168	0.640838	−	0.767676i	$-0.278587\pi$
$821$	36.7240	1.28168	0.640838	+	0.767676i	$0.278587\pi$
$822$	0	0
$823$	− 40.5157i	− 1.41229i	−0.708068	−	0.706145i	$-0.750433\pi$
$823$	− 40.5157i	− 1.41229i	0.708068	−	0.706145i	$-0.249567\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.1733i	0.388534i	0.980949	+	0.194267i	$0.0622327\pi$
$827$	11.1733i	0.388534i	−0.980949	+	0.194267i	$0.937767\pi$
$828$	0	0
$829$	−36.8737	−1.28068	−0.640338	−	0.768093i	$-0.721206\pi$
$829$	−36.8737	−1.28068	−0.640338	+	0.768093i	$0.721206\pi$
$830$	0	0
$831$	30.9233	1.07272
$832$	0	0
$833$	− 3.91065i	− 0.135496i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.69178i	0.127606i
$838$	0	0
$839$	11.4389	0.394913	0.197457	−	0.980312i	$-0.436732\pi$
$839$	11.4389	0.394913	0.197457	+	0.980312i	$0.436732\pi$
$840$	0	0
$841$	73.8351	2.54604
$842$	0	0
$843$	16.8489i	0.580306i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 17.5533i	− 0.603139i
$848$	0	0
$849$	−11.2627	−0.386537
$850$	0	0
$851$	−39.8254	−1.36520
$852$	0	0
$853$	24.6432i	0.843768i	0.906650	+	0.421884i	$0.138631\pi$
$853$	24.6432i	0.843768i	−0.906650	+	0.421884i	$0.861369\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.2342i	0.383752i	0.981419	+	0.191876i	$0.0614571\pi$
$857$	11.2342i	0.383752i	−0.981419	+	0.191876i	$0.938543\pi$
$858$	0	0
$859$	15.5193	0.529511	0.264756	−	0.964316i	$-0.414709\pi$
$859$	15.5193	0.529511	0.264756	+	0.964316i	$0.414709\pi$
$860$	0	0
$861$	−3.37314	−0.114956
$862$	0	0
$863$	19.5189i	0.664432i	0.943203	+	0.332216i	$0.107796\pi$
$863$	19.5189i	0.664432i	−0.943203	+	0.332216i	$0.892204\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.5867i	− 0.563315i
$868$	0	0
$869$	64.1752	2.17699
$870$	0	0
$871$	−19.2517	−0.652320
$872$	0	0
$873$	− 14.5878i	− 0.493723i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 13.7361i	− 0.463836i	−0.972735	−	0.231918i	$-0.925500\pi$
$877$	− 13.7361i	− 0.463836i	0.972735	−	0.231918i	$-0.0745001\pi$
$878$	0	0
$879$	−11.1995	−0.377751
$880$	0	0
$881$	−52.4110	−1.76577	−0.882886	−	0.469588i	$-0.844402\pi$
$881$	−52.4110	−1.76577	−0.882886	+	0.469588i	$0.844402\pi$
$882$	0	0
$883$	16.7017i	0.562056i	0.959700	+	0.281028i	$0.0906754\pi$
$883$	16.7017i	0.562056i	−0.959700	+	0.281028i	$0.909325\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 25.9008i	− 0.869664i	−0.900512	−	0.434832i	$-0.856808\pi$
$887$	− 25.9008i	− 0.869664i	0.900512	−	0.434832i	$-0.143192\pi$
$888$	0	0
$889$	13.6571	0.458045
$890$	0	0
$891$	−5.41590	−0.181439
$892$	0	0
$893$	24.8736i	0.832364i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.11914i	− 0.237701i
$898$	0	0
$899$	−37.4374	−1.24861
$900$	0	0
$901$	−3.82270	−0.127353
$902$	0	0
$903$	0.733798i	0.0244193i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.74821i	0.157662i	0.996888	+	0.0788308i	$0.0251187\pi$
$907$	4.74821i	0.157662i	−0.996888	+	0.0788308i	$0.974881\pi$
$908$	0	0
$909$	10.2832	0.341073
$910$	0	0
$911$	−12.2828	−0.406947	−0.203473	−	0.979080i	$-0.565223\pi$
$911$	−12.2828	−0.406947	−0.203473	+	0.979080i	$0.565223\pi$
$912$	0	0
$913$	− 7.55020i	− 0.249875i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.123201i	0.00406847i
$918$	0	0
$919$	−20.6802	−0.682177	−0.341089	−	0.940031i	$-0.610796\pi$
$919$	−20.6802	−0.682177	−0.341089	+	0.940031i	$0.610796\pi$
$920$	0	0
$921$	−24.1289	−0.795073
$922$	0	0
$923$	− 24.1612i	− 0.795276i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13.9848i	0.459322i
$928$	0	0
$929$	−24.0728	−0.789803	−0.394902	−	0.918723i	$-0.629221\pi$
$929$	−24.0728	−0.789803	−0.394902	+	0.918723i	$0.629221\pi$
$930$	0	0
$931$	30.6654	1.00502
$932$	0	0
$933$	− 2.07312i	− 0.0678708i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.7993i	1.46353i	0.681557	+	0.731765i	$0.261303\pi$
$937$	44.7993i	1.46353i	−0.681557	+	0.731765i	$0.738697\pi$
$938$	0	0
$939$	−8.51966	−0.278029
$940$	0	0
$941$	−47.0851	−1.53493	−0.767465	−	0.641091i	$-0.778482\pi$
$941$	−47.0851	−1.53493	−0.767465	+	0.641091i	$0.778482\pi$
$942$	0	0
$943$	− 12.3987i	− 0.403759i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17.9753i	− 0.584120i	−0.956400	−	0.292060i	$-0.905659\pi$
$947$	− 17.9753i	− 0.584120i	0.956400	−	0.292060i	$-0.0943407\pi$
$948$	0	0
$949$	−10.9895	−0.356733
$950$	0	0
$951$	−13.6070	−0.441236
$952$	0	0
$953$	− 35.9516i	− 1.16459i	−0.812979	−	0.582293i	$-0.802156\pi$
$953$	− 35.9516i	− 1.16459i	0.812979	−	0.582293i	$-0.197844\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 54.9213i	− 1.77535i
$958$	0	0
$959$	13.1203	0.423675
$960$	0	0
$961$	−17.3708	−0.560348
$962$	0	0
$963$	− 9.06727i	− 0.292189i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 9.59917i	− 0.308689i	−0.988017	−	0.154344i	$-0.950674\pi$
$967$	− 9.59917i	− 0.308689i	0.988017	−	0.154344i	$-0.0493265\pi$
$968$	0	0
$969$	−3.24072	−0.104107
$970$	0	0
$971$	−27.8888	−0.894995	−0.447497	−	0.894285i	$-0.647685\pi$
$971$	−27.8888	−0.894995	−0.447497	+	0.894285i	$0.647685\pi$
$972$	0	0
$973$	22.4748i	0.720510i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 56.7436i	− 1.81539i	−0.419632	−	0.907694i	$-0.637841\pi$
$977$	− 56.7436i	− 1.81539i	0.419632	−	0.907694i	$-0.362159\pi$
$978$	0	0
$979$	−9.25101	−0.295664
$980$	0	0
$981$	2.37535	0.0758392
$982$	0	0
$983$	− 11.2511i	− 0.358855i	−0.983771	−	0.179427i	$-0.942575\pi$
$983$	− 11.2511i	− 0.358855i	0.983771	−	0.179427i	$-0.0574245\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.72464i	− 0.150387i
$988$	0	0
$989$	−2.69724	−0.0857672
$990$	0	0
$991$	34.4601	1.09466	0.547331	−	0.836916i	$-0.315644\pi$
$991$	34.4601	1.09466	0.547331	+	0.836916i	$0.315644\pi$
$992$	0	0
$993$	22.6320i	0.718206i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 51.1902i	− 1.62121i	−0.585592	−	0.810606i	$-0.699138\pi$
$997$	− 51.1902i	− 1.62121i	0.585592	−	0.810606i	$-0.300862\pi$
$998$	0	0
$999$	11.3153	0.358001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.5		24
5.2	odd	4		7500.2.a.m.1.8		12
5.3	odd	4		7500.2.a.n.1.5		12
5.4	even	2	inner	7500.2.d.g.1249.20		24
25.2	odd	20		1500.2.m.d.601.4		24
25.9	even	10		300.2.o.a.169.2	✓	24
25.11	even	5		300.2.o.a.229.2	yes	24
25.12	odd	20		1500.2.m.d.901.4		24
25.13	odd	20		1500.2.m.c.901.3		24
25.14	even	10		1500.2.o.c.649.5		24
25.16	even	5		1500.2.o.c.349.5		24
25.23	odd	20		1500.2.m.c.601.3		24
75.11	odd	10		900.2.w.c.829.4		24
75.59	odd	10		900.2.w.c.469.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.2	✓	24	25.9	even	10
300.2.o.a.229.2	yes	24	25.11	even	5
900.2.w.c.469.4		24	75.59	odd	10
900.2.w.c.829.4		24	75.11	odd	10
1500.2.m.c.601.3		24	25.23	odd	20
1500.2.m.c.901.3		24	25.13	odd	20
1500.2.m.d.601.4		24	25.2	odd	20
1500.2.m.d.901.4		24	25.12	odd	20
1500.2.o.c.349.5		24	25.16	even	5
1500.2.o.c.649.5		24	25.14	even	10
7500.2.a.m.1.8		12	5.2	odd	4
7500.2.a.n.1.5		12	5.3	odd	4
7500.2.d.g.1249.5		24	1.1	even	1	trivial
7500.2.d.g.1249.20		24	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -0.957526i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -0.957526i q^{7} -1.00000 q^{9} -5.41590 q^{11} +2.02271i q^{13} -0.642866i q^{17} +5.04105 q^{19} -0.957526 q^{21} -3.51960i q^{23} +1.00000i q^{27} -10.1408 q^{29} +3.69178 q^{31} +5.41590i q^{33} -11.3153i q^{37} +2.02271 q^{39} +3.52277 q^{41} -0.766348i q^{43} +4.93421i q^{47} +6.08314 q^{49} -0.642866 q^{51} -5.94634i q^{53} -5.04105i q^{57} -4.71878 q^{59} +4.34485 q^{61} +0.957526i q^{63} +9.51778i q^{67} -3.51960 q^{69} -11.9450 q^{71} +5.43304i q^{73} +5.18586i q^{77} -11.8494 q^{79} +1.00000 q^{81} +1.39408i q^{83} +10.1408i q^{87} +1.70812 q^{89} +1.93680 q^{91} -3.69178i q^{93} +14.5878i q^{97} +5.41590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^9	\( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more