LMFDB - Embedded newform 7500.2.d.g.1249.12

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			1249.12
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.62675i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +4.62675i q^{7} -1.00000 q^{9} -4.94880 q^{11} +3.76277i q^{13} -2.69040i q^{17} -5.87106 q^{19} +4.62675 q^{21} -6.67829i q^{23} +1.00000i q^{27} -1.20049 q^{29} +3.30235 q^{31} +4.94880i q^{33} +1.87725i q^{37} +3.76277 q^{39} -3.03290 q^{41} +10.6626i q^{43} +0.259214i q^{47} -14.4068 q^{49} -2.69040 q^{51} -9.79659i q^{53} +5.87106i q^{57} +9.62455 q^{59} +6.27369 q^{61} -4.62675i q^{63} +2.56053i q^{67} -6.67829 q^{69} +8.67729 q^{71} -4.87481i q^{73} -22.8968i q^{77} -12.4732 q^{79} +1.00000 q^{81} +8.89025i q^{83} +1.20049i q^{87} -14.5774 q^{89} -17.4094 q^{91} -3.30235i q^{93} -3.98689i q^{97} +4.94880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.62675i	1.74875i	0.485254	+	0.874373i	$0.338727\pi$
$7$	4.62675i	1.74875i	−0.485254	+	0.874373i	$0.661273\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.94880	−1.49212	−0.746059	−	0.665880i	$-0.768056\pi$
$11$	−4.94880	−1.49212	−0.746059	+	0.665880i	$0.768056\pi$
$12$	0	0
$13$	3.76277i	1.04360i	0.853066	+	0.521802i	$0.174740\pi$
$13$	3.76277i	1.04360i	−0.853066	+	0.521802i	$0.825260\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.69040i	− 0.652517i	−0.945281	−	0.326258i	$-0.894212\pi$
$17$	− 2.69040i	− 0.652517i	0.945281	−	0.326258i	$-0.105788\pi$
$18$	0	0
$19$	−5.87106	−1.34691	−0.673457	−	0.739227i	$-0.735191\pi$
$19$	−5.87106	−1.34691	−0.673457	+	0.739227i	$0.735191\pi$
$20$	0	0
$21$	4.62675	1.00964
$22$	0	0
$23$	− 6.67829i	− 1.39252i	−0.717790	−	0.696260i	$-0.754846\pi$
$23$	− 6.67829i	− 1.39252i	0.717790	−	0.696260i	$-0.245154\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−1.20049	−0.222926	−0.111463	−	0.993769i	$-0.535554\pi$
$29$	−1.20049	−0.222926	−0.111463	+	0.993769i	$0.535554\pi$
$30$	0	0
$31$	3.30235	0.593119	0.296560	−	0.955014i	$-0.404161\pi$
$31$	3.30235	0.593119	0.296560	+	0.955014i	$0.404161\pi$
$32$	0	0
$33$	4.94880i	0.861475i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.87725i	0.308618i	0.988023	+	0.154309i	$0.0493151\pi$
$37$	1.87725i	0.308618i	−0.988023	+	0.154309i	$0.950685\pi$
$38$	0	0
$39$	3.76277	0.602525
$40$	0	0
$41$	−3.03290	−0.473659	−0.236829	−	0.971551i	$-0.576108\pi$
$41$	−3.03290	−0.473659	−0.236829	+	0.971551i	$0.576108\pi$
$42$	0	0
$43$	10.6626i	1.62603i	0.582244	+	0.813014i	$0.302175\pi$
$43$	10.6626i	1.62603i	−0.582244	+	0.813014i	$0.697825\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.259214i	0.0378103i	0.999821	+	0.0189051i	$0.00601805\pi$
$47$	0.259214i	0.0378103i	−0.999821	+	0.0189051i	$0.993982\pi$
$48$	0	0
$49$	−14.4068	−2.05811
$50$	0	0
$51$	−2.69040	−0.376731
$52$	0	0
$53$	− 9.79659i	− 1.34566i	−0.739795	−	0.672832i	$-0.765078\pi$
$53$	− 9.79659i	− 1.34566i	0.739795	−	0.672832i	$-0.234922\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.87106i	0.777641i
$58$	0	0
$59$	9.62455	1.25301	0.626505	−	0.779417i	$-0.284485\pi$
$59$	9.62455	1.25301	0.626505	+	0.779417i	$0.284485\pi$
$60$	0	0
$61$	6.27369	0.803264	0.401632	−	0.915801i	$-0.368443\pi$
$61$	6.27369	0.803264	0.401632	+	0.915801i	$0.368443\pi$
$62$	0	0
$63$	− 4.62675i	− 0.582915i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.56053i	0.312818i	0.987692	+	0.156409i	$0.0499918\pi$
$67$	2.56053i	0.312818i	−0.987692	+	0.156409i	$0.950008\pi$
$68$	0	0
$69$	−6.67829	−0.803971
$70$	0	0
$71$	8.67729	1.02980	0.514902	−	0.857249i	$-0.327828\pi$
$71$	8.67729	1.02980	0.514902	+	0.857249i	$0.327828\pi$
$72$	0	0
$73$	− 4.87481i	− 0.570554i	−0.958445	−	0.285277i	$-0.907914\pi$
$73$	− 4.87481i	− 0.570554i	0.958445	−	0.285277i	$-0.0920856\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 22.8968i	− 2.60934i
$78$	0	0
$79$	−12.4732	−1.40334	−0.701672	−	0.712500i	$-0.747563\pi$
$79$	−12.4732	−1.40334	−0.701672	+	0.712500i	$0.747563\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.89025i	0.975832i	0.872891	+	0.487916i	$0.162243\pi$
$83$	8.89025i	0.975832i	−0.872891	+	0.487916i	$0.837757\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.20049i	0.128706i
$88$	0	0
$89$	−14.5774	−1.54521	−0.772603	−	0.634889i	$-0.781046\pi$
$89$	−14.5774	−1.54521	−0.772603	+	0.634889i	$0.781046\pi$
$90$	0	0
$91$	−17.4094	−1.82500
$92$	0	0
$93$	− 3.30235i	− 0.342437i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.98689i	− 0.404808i	−0.979302	−	0.202404i	$-0.935125\pi$
$97$	− 3.98689i	− 0.404808i	0.979302	−	0.202404i	$-0.0648754\pi$
$98$	0	0
$99$	4.94880	0.497373
$100$	0	0
$101$	9.36896	0.932246	0.466123	−	0.884720i	$-0.345650\pi$
$101$	9.36896	0.932246	0.466123	+	0.884720i	$0.345650\pi$
$102$	0	0
$103$	− 10.4241i	− 1.02712i	−0.858054	−	0.513559i	$-0.828327\pi$
$103$	− 10.4241i	− 1.02712i	0.858054	−	0.513559i	$-0.171673\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.220683i	− 0.0213342i	−0.999943	−	0.0106671i	$-0.996604\pi$
$107$	− 0.220683i	− 0.0213342i	0.999943	−	0.0106671i	$-0.00339551\pi$
$108$	0	0
$109$	−6.68640	−0.640441	−0.320220	−	0.947343i	$-0.603757\pi$
$109$	−6.68640	−0.640441	−0.320220	+	0.947343i	$0.603757\pi$
$110$	0	0
$111$	1.87725	0.178181
$112$	0	0
$113$	− 9.60864i	− 0.903905i	−0.892042	−	0.451952i	$-0.850728\pi$
$113$	− 9.60864i	− 0.903905i	0.892042	−	0.451952i	$-0.149272\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.76277i	− 0.347868i
$118$	0	0
$119$	12.4478	1.14109
$120$	0	0
$121$	13.4906	1.22642
$122$	0	0
$123$	3.03290i	0.273467i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.9923i	1.41909i	0.704662	+	0.709543i	$0.251098\pi$
$127$	15.9923i	1.41909i	−0.704662	+	0.709543i	$0.748902\pi$
$128$	0	0
$129$	10.6626	0.938788
$130$	0	0
$131$	−12.3228	−1.07665	−0.538324	−	0.842738i	$-0.680942\pi$
$131$	−12.3228	−1.07665	−0.538324	+	0.842738i	$0.680942\pi$
$132$	0	0
$133$	− 27.1639i	− 2.35541i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.69454i	− 0.486518i	−0.969961	−	0.243259i	$-0.921784\pi$
$137$	− 5.69454i	− 0.486518i	0.969961	−	0.243259i	$-0.0782164\pi$
$138$	0	0
$139$	17.9969	1.52648	0.763240	−	0.646115i	$-0.223607\pi$
$139$	17.9969	1.52648	0.763240	+	0.646115i	$0.223607\pi$
$140$	0	0
$141$	0.259214	0.0218298
$142$	0	0
$143$	− 18.6212i	− 1.55718i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.4068i	1.18825i
$148$	0	0
$149$	1.09001	0.0892972	0.0446486	−	0.999003i	$-0.485783\pi$
$149$	1.09001	0.0892972	0.0446486	+	0.999003i	$0.485783\pi$
$150$	0	0
$151$	11.3789	0.926004	0.463002	−	0.886357i	$-0.346772\pi$
$151$	11.3789	0.926004	0.463002	+	0.886357i	$0.346772\pi$
$152$	0	0
$153$	2.69040i	0.217506i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.98415i	− 0.317970i	−0.987281	−	0.158985i	$-0.949178\pi$
$157$	− 3.98415i	− 0.317970i	0.987281	−	0.158985i	$-0.0508221\pi$
$158$	0	0
$159$	−9.79659	−0.776920
$160$	0	0
$161$	30.8988	2.43516
$162$	0	0
$163$	− 21.8314i	− 1.70997i	−0.518657	−	0.854983i	$-0.673568\pi$
$163$	− 21.8314i	− 1.70997i	0.518657	−	0.854983i	$-0.326432\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 15.4689i	− 1.19702i	−0.801116	−	0.598509i	$-0.795760\pi$
$167$	− 15.4689i	− 1.19702i	0.801116	−	0.598509i	$-0.204240\pi$
$168$	0	0
$169$	−1.15844	−0.0891104
$170$	0	0
$171$	5.87106	0.448971
$172$	0	0
$173$	− 17.1704i	− 1.30544i	−0.757600	−	0.652719i	$-0.773628\pi$
$173$	− 17.1704i	− 1.30544i	0.757600	−	0.652719i	$-0.226372\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.62455i	− 0.723426i
$178$	0	0
$179$	14.4456	1.07971	0.539856	−	0.841757i	$-0.318479\pi$
$179$	14.4456	1.07971	0.539856	+	0.841757i	$0.318479\pi$
$180$	0	0
$181$	12.3964	0.921420	0.460710	−	0.887551i	$-0.347595\pi$
$181$	12.3964	0.921420	0.460710	+	0.887551i	$0.347595\pi$
$182$	0	0
$183$	− 6.27369i	− 0.463765i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.3142i	0.973632i
$188$	0	0
$189$	−4.62675	−0.336546
$190$	0	0
$191$	−1.65546	−0.119785	−0.0598925	−	0.998205i	$-0.519076\pi$
$191$	−1.65546	−0.119785	−0.0598925	+	0.998205i	$0.519076\pi$
$192$	0	0
$193$	− 16.3253i	− 1.17512i	−0.809181	−	0.587560i	$-0.800089\pi$
$193$	− 16.3253i	− 1.17512i	0.809181	−	0.587560i	$-0.199911\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.6324i	− 0.971267i	−0.874162	−	0.485634i	$-0.838589\pi$
$197$	− 13.6324i	− 0.971267i	0.874162	−	0.485634i	$-0.161411\pi$
$198$	0	0
$199$	6.07817	0.430870	0.215435	−	0.976518i	$-0.430883\pi$
$199$	6.07817	0.430870	0.215435	+	0.976518i	$0.430883\pi$
$200$	0	0
$201$	2.56053	0.180606
$202$	0	0
$203$	− 5.55437i	− 0.389840i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.67829i	0.464173i
$208$	0	0
$209$	29.0547	2.00975
$210$	0	0
$211$	17.0825	1.17601	0.588003	−	0.808859i	$-0.299914\pi$
$211$	17.0825	1.17601	0.588003	+	0.808859i	$0.299914\pi$
$212$	0	0
$213$	− 8.67729i	− 0.594558i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.2791i	1.03721i
$218$	0	0
$219$	−4.87481	−0.329409
$220$	0	0
$221$	10.1233	0.680969
$222$	0	0
$223$	− 0.783790i	− 0.0524865i	−0.999656	−	0.0262432i	$-0.991646\pi$
$223$	− 0.783790i	− 0.0524865i	0.999656	−	0.0262432i	$-0.00835444\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.9494i	1.58958i	0.606886	+	0.794789i	$0.292419\pi$
$227$	23.9494i	1.58958i	−0.606886	+	0.794789i	$0.707581\pi$
$228$	0	0
$229$	5.53383	0.365686	0.182843	−	0.983142i	$-0.441470\pi$
$229$	5.53383	0.365686	0.182843	+	0.983142i	$0.441470\pi$
$230$	0	0
$231$	−22.8968	−1.50650
$232$	0	0
$233$	− 0.907742i	− 0.0594682i	−0.999558	−	0.0297341i	$-0.990534\pi$
$233$	− 0.907742i	− 0.0594682i	0.999558	−	0.0297341i	$-0.00946605\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.4732i	0.810221i
$238$	0	0
$239$	11.2387	0.726969	0.363484	−	0.931600i	$-0.381587\pi$
$239$	11.2387	0.726969	0.363484	+	0.931600i	$0.381587\pi$
$240$	0	0
$241$	−20.2517	−1.30453	−0.652264	−	0.757992i	$-0.726181\pi$
$241$	−20.2517	−1.30453	−0.652264	+	0.757992i	$0.726181\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 22.0914i	− 1.40564i
$248$	0	0
$249$	8.89025	0.563397
$250$	0	0
$251$	−30.6919	−1.93725	−0.968627	−	0.248520i	$-0.920056\pi$
$251$	−30.6919	−1.93725	−0.968627	+	0.248520i	$0.920056\pi$
$252$	0	0
$253$	33.0495i	2.07780i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.77543i	0.297883i	0.988846	+	0.148941i	$0.0475866\pi$
$257$	4.77543i	0.297883i	−0.988846	+	0.148941i	$0.952413\pi$
$258$	0	0
$259$	−8.68556	−0.539694
$260$	0	0
$261$	1.20049	0.0743085
$262$	0	0
$263$	− 1.20966i	− 0.0745910i	−0.999304	−	0.0372955i	$-0.988126\pi$
$263$	− 1.20966i	− 0.0745910i	0.999304	−	0.0372955i	$-0.0118743\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.5774i	0.892125i
$268$	0	0
$269$	25.5249	1.55628	0.778140	−	0.628090i	$-0.216163\pi$
$269$	25.5249	1.55628	0.778140	+	0.628090i	$0.216163\pi$
$270$	0	0
$271$	−5.87342	−0.356785	−0.178393	−	0.983959i	$-0.557090\pi$
$271$	−5.87342	−0.356785	−0.178393	+	0.983959i	$0.557090\pi$
$272$	0	0
$273$	17.4094i	1.05366i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.90539i	− 0.294737i	−0.989082	−	0.147368i	$-0.952920\pi$
$277$	− 4.90539i	− 0.294737i	0.989082	−	0.147368i	$-0.0470803\pi$
$278$	0	0
$279$	−3.30235	−0.197706
$280$	0	0
$281$	−1.29526	−0.0772686	−0.0386343	−	0.999253i	$-0.512301\pi$
$281$	−1.29526	−0.0772686	−0.0386343	+	0.999253i	$0.512301\pi$
$282$	0	0
$283$	− 30.2297i	− 1.79697i	−0.439006	−	0.898484i	$-0.644669\pi$
$283$	− 30.2297i	− 1.79697i	0.439006	−	0.898484i	$-0.355331\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 14.0324i	− 0.828309i
$288$	0	0
$289$	9.76177	0.574222
$290$	0	0
$291$	−3.98689	−0.233716
$292$	0	0
$293$	− 8.06831i	− 0.471356i	−0.971831	−	0.235678i	$-0.924269\pi$
$293$	− 8.06831i	− 0.471356i	0.971831	−	0.235678i	$-0.0757310\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.94880i	− 0.287158i
$298$	0	0
$299$	25.1289	1.45324
$300$	0	0
$301$	−49.3331	−2.84351
$302$	0	0
$303$	− 9.36896i	− 0.538233i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 14.7750i	− 0.843255i	−0.906769	−	0.421628i	$-0.861459\pi$
$307$	− 14.7750i	− 0.843255i	0.906769	−	0.421628i	$-0.138541\pi$
$308$	0	0
$309$	−10.4241	−0.593007
$310$	0	0
$311$	−9.15373	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$311$	−9.15373	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$312$	0	0
$313$	13.2553i	0.749235i	0.927179	+	0.374618i	$0.122226\pi$
$313$	13.2553i	0.749235i	−0.927179	+	0.374618i	$0.877774\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.790550i	− 0.0444017i	−0.999754	−	0.0222009i	$-0.992933\pi$
$317$	− 0.790550i	− 0.0444017i	0.999754	−	0.0222009i	$-0.00706734\pi$
$318$	0	0
$319$	5.94099	0.332631
$320$	0	0
$321$	−0.220683	−0.0123173
$322$	0	0
$323$	15.7955i	0.878883i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.68640i	0.369759i
$328$	0	0
$329$	−1.19932	−0.0661205
$330$	0	0
$331$	−3.37370	−0.185435	−0.0927176	−	0.995692i	$-0.529555\pi$
$331$	−3.37370	−0.185435	−0.0927176	+	0.995692i	$0.529555\pi$
$332$	0	0
$333$	− 1.87725i	− 0.102873i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 18.0660i	− 0.984116i	−0.870563	−	0.492058i	$-0.836245\pi$
$337$	− 18.0660i	− 0.984116i	0.870563	−	0.492058i	$-0.163755\pi$
$338$	0	0
$339$	−9.60864	−0.521870
$340$	0	0
$341$	−16.3426	−0.885004
$342$	0	0
$343$	− 34.2694i	− 1.85037i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.05223i	− 0.163852i	−0.996638	−	0.0819262i	$-0.973893\pi$
$347$	− 3.05223i	− 0.163852i	0.996638	−	0.0819262i	$-0.0261072\pi$
$348$	0	0
$349$	0.628744	0.0336559	0.0168280	−	0.999858i	$-0.494643\pi$
$349$	0.628744	0.0336559	0.0168280	+	0.999858i	$0.494643\pi$
$350$	0	0
$351$	−3.76277	−0.200842
$352$	0	0
$353$	− 18.8896i	− 1.00539i	−0.864463	−	0.502696i	$-0.832342\pi$
$353$	− 18.8896i	− 1.00539i	0.864463	−	0.502696i	$-0.167658\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.4478i	− 0.658807i
$358$	0	0
$359$	−28.9347	−1.52712	−0.763558	−	0.645739i	$-0.776550\pi$
$359$	−28.9347	−1.52712	−0.763558	+	0.645739i	$0.776550\pi$
$360$	0	0
$361$	15.4693	0.814175
$362$	0	0
$363$	− 13.4906i	− 0.708073i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.38295i	− 0.280988i	−0.990082	−	0.140494i	$-0.955131\pi$
$367$	− 5.38295i	− 0.280988i	0.990082	−	0.140494i	$-0.0448690\pi$
$368$	0	0
$369$	3.03290	0.157886
$370$	0	0
$371$	45.3263	2.35323
$372$	0	0
$373$	− 10.9975i	− 0.569428i	−0.958613	−	0.284714i	$-0.908101\pi$
$373$	− 10.9975i	− 0.569428i	0.958613	−	0.284714i	$-0.0918986\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.51717i	− 0.232646i
$378$	0	0
$379$	−7.06775	−0.363046	−0.181523	−	0.983387i	$-0.558103\pi$
$379$	−7.06775	−0.363046	−0.181523	+	0.983387i	$0.558103\pi$
$380$	0	0
$381$	15.9923	0.819309
$382$	0	0
$383$	8.10135i	0.413960i	0.978345	+	0.206980i	$0.0663635\pi$
$383$	8.10135i	0.413960i	−0.978345	+	0.206980i	$0.933637\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.6626i	− 0.542009i
$388$	0	0
$389$	−33.0432	−1.67536	−0.837678	−	0.546164i	$-0.816088\pi$
$389$	−33.0432	−1.67536	−0.837678	+	0.546164i	$0.816088\pi$
$390$	0	0
$391$	−17.9672	−0.908642
$392$	0	0
$393$	12.3228i	0.621603i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.3668i	0.620671i	0.950627	+	0.310336i	$0.100441\pi$
$397$	12.3668i	0.620671i	−0.950627	+	0.310336i	$0.899559\pi$
$398$	0	0
$399$	−27.1639	−1.35990
$400$	0	0
$401$	−14.7983	−0.738993	−0.369496	−	0.929232i	$-0.620470\pi$
$401$	−14.7983	−0.738993	−0.369496	+	0.929232i	$0.620470\pi$
$402$	0	0
$403$	12.4260i	0.618982i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9.29012i	− 0.460494i
$408$	0	0
$409$	23.6853	1.17116	0.585581	−	0.810614i	$-0.300866\pi$
$409$	23.6853	1.17116	0.585581	+	0.810614i	$0.300866\pi$
$410$	0	0
$411$	−5.69454	−0.280891
$412$	0	0
$413$	44.5304i	2.19120i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 17.9969i	− 0.881314i
$418$	0	0
$419$	−1.54379	−0.0754193	−0.0377096	−	0.999289i	$-0.512006\pi$
$419$	−1.54379	−0.0754193	−0.0377096	+	0.999289i	$0.512006\pi$
$420$	0	0
$421$	−17.5965	−0.857599	−0.428799	−	0.903400i	$-0.641063\pi$
$421$	−17.5965	−0.857599	−0.428799	+	0.903400i	$0.641063\pi$
$422$	0	0
$423$	− 0.259214i	− 0.0126034i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	29.0268i	1.40471i
$428$	0	0
$429$	−18.6212	−0.899039
$430$	0	0
$431$	15.6473	0.753702	0.376851	−	0.926274i	$-0.377007\pi$
$431$	15.6473	0.753702	0.376851	+	0.926274i	$0.377007\pi$
$432$	0	0
$433$	− 0.228980i	− 0.0110041i	−0.999985	−	0.00550203i	$-0.998249\pi$
$433$	− 0.228980i	− 0.0110041i	0.999985	−	0.00550203i	$-0.00175136\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	39.2086i	1.87560i
$438$	0	0
$439$	−19.4199	−0.926863	−0.463431	−	0.886133i	$-0.653382\pi$
$439$	−19.4199	−0.926863	−0.463431	+	0.886133i	$0.653382\pi$
$440$	0	0
$441$	14.4068	0.686038
$442$	0	0
$443$	12.7980i	0.608051i	0.952664	+	0.304026i	$0.0983309\pi$
$443$	12.7980i	0.608051i	−0.952664	+	0.304026i	$0.901669\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.09001i	− 0.0515558i
$448$	0	0
$449$	21.1499	0.998124	0.499062	−	0.866566i	$-0.333678\pi$
$449$	21.1499	0.998124	0.499062	+	0.866566i	$0.333678\pi$
$450$	0	0
$451$	15.0092	0.706755
$452$	0	0
$453$	− 11.3789i	− 0.534629i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.97118i	− 0.419654i	−0.977739	−	0.209827i	$-0.932710\pi$
$457$	− 8.97118i	− 0.419654i	0.977739	−	0.209827i	$-0.0672901\pi$
$458$	0	0
$459$	2.69040	0.125577
$460$	0	0
$461$	27.5351	1.28244	0.641218	−	0.767358i	$-0.278429\pi$
$461$	27.5351	1.28244	0.641218	+	0.767358i	$0.278429\pi$
$462$	0	0
$463$	− 29.4011i	− 1.36638i	−0.730239	−	0.683191i	$-0.760591\pi$
$463$	− 29.4011i	− 1.36638i	0.730239	−	0.683191i	$-0.239409\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.2609i	0.613643i	0.951767	+	0.306821i	$0.0992654\pi$
$467$	13.2609i	0.613643i	−0.951767	+	0.306821i	$0.900735\pi$
$468$	0	0
$469$	−11.8469	−0.547040
$470$	0	0
$471$	−3.98415	−0.183580
$472$	0	0
$473$	− 52.7669i	− 2.42623i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.79659i	0.448555i
$478$	0	0
$479$	18.0591	0.825142	0.412571	−	0.910925i	$-0.364631\pi$
$479$	18.0591	0.825142	0.412571	+	0.910925i	$0.364631\pi$
$480$	0	0
$481$	−7.06365	−0.322075
$482$	0	0
$483$	− 30.8988i	− 1.40594i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.0571i	0.636990i	0.947925	+	0.318495i	$0.103177\pi$
$487$	14.0571i	0.636990i	−0.947925	+	0.318495i	$0.896823\pi$
$488$	0	0
$489$	−21.8314	−0.987249
$490$	0	0
$491$	−27.2319	−1.22896	−0.614478	−	0.788934i	$-0.710634\pi$
$491$	−27.2319	−1.22896	−0.614478	+	0.788934i	$0.710634\pi$
$492$	0	0
$493$	3.22980i	0.145463i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.1476i	1.80087i
$498$	0	0
$499$	8.17654	0.366032	0.183016	−	0.983110i	$-0.441414\pi$
$499$	8.17654	0.366032	0.183016	+	0.983110i	$0.441414\pi$
$500$	0	0
$501$	−15.4689	−0.691099
$502$	0	0
$503$	− 9.94479i	− 0.443416i	−0.975113	−	0.221708i	$-0.928837\pi$
$503$	− 9.94479i	− 0.443416i	0.975113	−	0.221708i	$-0.0711632\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.15844i	0.0514479i
$508$	0	0
$509$	−17.2750	−0.765700	−0.382850	−	0.923811i	$-0.625057\pi$
$509$	−17.2750	−0.765700	−0.382850	+	0.923811i	$0.625057\pi$
$510$	0	0
$511$	22.5545	0.997754
$512$	0	0
$513$	− 5.87106i	− 0.259214i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.28280i	− 0.0564174i
$518$	0	0
$519$	−17.1704	−0.753695
$520$	0	0
$521$	−11.1252	−0.487402	−0.243701	−	0.969850i	$-0.578362\pi$
$521$	−11.1252	−0.487402	−0.243701	+	0.969850i	$0.578362\pi$
$522$	0	0
$523$	3.75282i	0.164099i	0.996628	+	0.0820496i	$0.0261466\pi$
$523$	3.75282i	0.164099i	−0.996628	+	0.0820496i	$0.973853\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.88462i	− 0.387020i
$528$	0	0
$529$	−21.5995	−0.939109
$530$	0	0
$531$	−9.62455	−0.417670
$532$	0	0
$533$	− 11.4121i	− 0.494312i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 14.4456i	− 0.623372i
$538$	0	0
$539$	71.2963	3.07095
$540$	0	0
$541$	18.8812	0.811764	0.405882	−	0.913925i	$-0.366964\pi$
$541$	18.8812	0.811764	0.405882	+	0.913925i	$0.366964\pi$
$542$	0	0
$543$	− 12.3964i	− 0.531982i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.155307i	0.00664046i	0.999994	+	0.00332023i	$0.00105686\pi$
$547$	0.155307i	0.00664046i	−0.999994	+	0.00332023i	$0.998943\pi$
$548$	0	0
$549$	−6.27369	−0.267755
$550$	0	0
$551$	7.04815	0.300261
$552$	0	0
$553$	− 57.7103i	− 2.45409i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19.0371i	− 0.806626i	−0.915062	−	0.403313i	$-0.867859\pi$
$557$	− 19.0371i	− 0.806626i	0.915062	−	0.403313i	$-0.132141\pi$
$558$	0	0
$559$	−40.1208	−1.69693
$560$	0	0
$561$	13.3142	0.562127
$562$	0	0
$563$	31.0709i	1.30948i	0.755854	+	0.654741i	$0.227222\pi$
$563$	31.0709i	1.30948i	−0.755854	+	0.654741i	$0.772778\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.62675i	0.194305i
$568$	0	0
$569$	−39.3417	−1.64929	−0.824646	−	0.565649i	$-0.808626\pi$
$569$	−39.3417	−1.64929	−0.824646	+	0.565649i	$0.808626\pi$
$570$	0	0
$571$	28.4331	1.18989	0.594944	−	0.803767i	$-0.297174\pi$
$571$	28.4331	1.18989	0.594944	+	0.803767i	$0.297174\pi$
$572$	0	0
$573$	1.65546i	0.0691579i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.4595i	− 0.935002i	−0.883992	−	0.467501i	$-0.845154\pi$
$577$	− 22.4595i	− 0.935002i	0.883992	−	0.467501i	$-0.154846\pi$
$578$	0	0
$579$	−16.3253	−0.678456
$580$	0	0
$581$	−41.1330	−1.70648
$582$	0	0
$583$	48.4813i	2.00789i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.0830i	− 0.663816i	−0.943312	−	0.331908i	$-0.892308\pi$
$587$	− 16.0830i	− 0.663816i	0.943312	−	0.331908i	$-0.107692\pi$
$588$	0	0
$589$	−19.3883	−0.798880
$590$	0	0
$591$	−13.6324	−0.560761
$592$	0	0
$593$	− 20.4648i	− 0.840389i	−0.907434	−	0.420194i	$-0.861962\pi$
$593$	− 20.4648i	− 0.840389i	0.907434	−	0.420194i	$-0.138038\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.07817i	− 0.248763i
$598$	0	0
$599$	−18.5688	−0.758699	−0.379349	−	0.925253i	$-0.623852\pi$
$599$	−18.5688	−0.758699	−0.379349	+	0.925253i	$0.623852\pi$
$600$	0	0
$601$	47.2047	1.92552	0.962761	−	0.270355i	$-0.0871409\pi$
$601$	47.2047	1.92552	0.962761	+	0.270355i	$0.0871409\pi$
$602$	0	0
$603$	− 2.56053i	− 0.104273i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.0786576i	0.00319261i	0.999999	+	0.00159631i	$0.000508121\pi$
$607$	0.0786576i	0.00319261i	−0.999999	+	0.00159631i	$0.999492\pi$
$608$	0	0
$609$	−5.55437	−0.225074
$610$	0	0
$611$	−0.975363	−0.0394590
$612$	0	0
$613$	− 5.31081i	− 0.214502i	−0.994232	−	0.107251i	$-0.965795\pi$
$613$	− 5.31081i	− 0.214502i	0.994232	−	0.107251i	$-0.0342048\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.4323i	1.22516i	0.790409	+	0.612580i	$0.209868\pi$
$617$	30.4323i	1.22516i	−0.790409	+	0.612580i	$0.790132\pi$
$618$	0	0
$619$	31.6461	1.27196	0.635982	−	0.771704i	$-0.280595\pi$
$619$	31.6461	1.27196	0.635982	+	0.771704i	$0.280595\pi$
$620$	0	0
$621$	6.67829	0.267990
$622$	0	0
$623$	− 67.4462i	− 2.70217i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 29.0547i	− 1.16033i
$628$	0	0
$629$	5.05054	0.201378
$630$	0	0
$631$	−33.8887	−1.34909	−0.674544	−	0.738234i	$-0.735660\pi$
$631$	−33.8887	−1.34909	−0.674544	+	0.738234i	$0.735660\pi$
$632$	0	0
$633$	− 17.0825i	− 0.678967i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 54.2095i	− 2.14786i
$638$	0	0
$639$	−8.67729	−0.343268
$640$	0	0
$641$	−49.0347	−1.93676	−0.968378	−	0.249489i	$-0.919737\pi$
$641$	−49.0347	−1.93676	−0.968378	+	0.249489i	$0.919737\pi$
$642$	0	0
$643$	14.2509i	0.562000i	0.959708	+	0.281000i	$0.0906660\pi$
$643$	14.2509i	0.562000i	−0.959708	+	0.281000i	$0.909334\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.3343i	1.66433i	0.554526	+	0.832166i	$0.312899\pi$
$647$	42.3343i	1.66433i	−0.554526	+	0.832166i	$0.687101\pi$
$648$	0	0
$649$	−47.6300	−1.86964
$650$	0	0
$651$	15.2791	0.598836
$652$	0	0
$653$	− 47.0292i	− 1.84039i	−0.391456	−	0.920197i	$-0.628029\pi$
$653$	− 47.0292i	− 1.84039i	0.391456	−	0.920197i	$-0.371971\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.87481i	0.190185i
$658$	0	0
$659$	11.1866	0.435767	0.217884	−	0.975975i	$-0.430085\pi$
$659$	11.1866	0.435767	0.217884	+	0.975975i	$0.430085\pi$
$660$	0	0
$661$	−46.6124	−1.81301	−0.906505	−	0.422195i	$-0.861260\pi$
$661$	−46.6124	−1.81301	−0.906505	+	0.422195i	$0.861260\pi$
$662$	0	0
$663$	− 10.1233i	− 0.393158i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.01722i	0.310428i
$668$	0	0
$669$	−0.783790	−0.0303031
$670$	0	0
$671$	−31.0472	−1.19857
$672$	0	0
$673$	13.7340i	0.529407i	0.964330	+	0.264703i	$0.0852741\pi$
$673$	13.7340i	0.529407i	−0.964330	+	0.264703i	$0.914726\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.8210i	1.41514i	0.706641	+	0.707572i	$0.250210\pi$
$677$	36.8210i	1.41514i	−0.706641	+	0.707572i	$0.749790\pi$
$678$	0	0
$679$	18.4464	0.707906
$680$	0	0
$681$	23.9494	0.917744
$682$	0	0
$683$	30.0906i	1.15138i	0.817666	+	0.575692i	$0.195267\pi$
$683$	30.0906i	1.15138i	−0.817666	+	0.575692i	$0.804733\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.53383i	− 0.211129i
$688$	0	0
$689$	36.8623	1.40434
$690$	0	0
$691$	−8.36619	−0.318265	−0.159132	−	0.987257i	$-0.550870\pi$
$691$	−8.36619	−0.318265	−0.159132	+	0.987257i	$0.550870\pi$
$692$	0	0
$693$	22.8968i	0.869779i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.15969i	0.309070i
$698$	0	0
$699$	−0.907742	−0.0343340
$700$	0	0
$701$	−3.42495	−0.129359	−0.0646794	−	0.997906i	$-0.520602\pi$
$701$	−3.42495	−0.129359	−0.0646794	+	0.997906i	$0.520602\pi$
$702$	0	0
$703$	− 11.0214i	− 0.415681i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43.3478i	1.63026i
$708$	0	0
$709$	49.4835	1.85839	0.929197	−	0.369585i	$-0.120500\pi$
$709$	49.4835	1.85839	0.929197	+	0.369585i	$0.120500\pi$
$710$	0	0
$711$	12.4732	0.467781
$712$	0	0
$713$	− 22.0540i	− 0.825930i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 11.2387i	− 0.419716i
$718$	0	0
$719$	−26.1522	−0.975313	−0.487656	−	0.873036i	$-0.662148\pi$
$719$	−26.1522	−0.975313	−0.487656	+	0.873036i	$0.662148\pi$
$720$	0	0
$721$	48.2297	1.79617
$722$	0	0
$723$	20.2517i	0.753170i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 34.6172i	− 1.28388i	−0.766754	−	0.641941i	$-0.778129\pi$
$727$	− 34.6172i	− 1.28388i	0.766754	−	0.641941i	$-0.221871\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	28.6866	1.06101
$732$	0	0
$733$	− 17.4652i	− 0.645091i	−0.946554	−	0.322545i	$-0.895461\pi$
$733$	− 17.4652i	− 0.645091i	0.946554	−	0.322545i	$-0.104539\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.6715i	− 0.466762i
$738$	0	0
$739$	−20.0685	−0.738233	−0.369116	−	0.929383i	$-0.620340\pi$
$739$	−20.0685	−0.738233	−0.369116	+	0.929383i	$0.620340\pi$
$740$	0	0
$741$	−22.0914	−0.811549
$742$	0	0
$743$	− 5.84644i	− 0.214485i	−0.994233	−	0.107243i	$-0.965798\pi$
$743$	− 5.84644i	− 0.214485i	0.994233	−	0.107243i	$-0.0342022\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.89025i	− 0.325277i
$748$	0	0
$749$	1.02104	0.0373081
$750$	0	0
$751$	−32.7925	−1.19662	−0.598308	−	0.801266i	$-0.704160\pi$
$751$	−32.7925	−1.19662	−0.598308	+	0.801266i	$0.704160\pi$
$752$	0	0
$753$	30.6919i	1.11847i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.6371i	0.822759i	0.911464	+	0.411379i	$0.134953\pi$
$757$	22.6371i	0.822759i	−0.911464	+	0.411379i	$0.865047\pi$
$758$	0	0
$759$	33.0495	1.19962
$760$	0	0
$761$	−36.3308	−1.31699	−0.658496	−	0.752584i	$-0.728807\pi$
$761$	−36.3308	−1.31699	−0.658496	+	0.752584i	$0.728807\pi$
$762$	0	0
$763$	− 30.9363i	− 1.11997i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.2150i	1.30765i
$768$	0	0
$769$	−14.6238	−0.527349	−0.263675	−	0.964612i	$-0.584935\pi$
$769$	−14.6238	−0.527349	−0.263675	+	0.964612i	$0.584935\pi$
$770$	0	0
$771$	4.77543	0.171983
$772$	0	0
$773$	− 30.5251i	− 1.09791i	−0.835851	−	0.548956i	$-0.815025\pi$
$773$	− 30.5251i	− 1.09791i	0.835851	−	0.548956i	$-0.184975\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.68556i	0.311593i
$778$	0	0
$779$	17.8063	0.637977
$780$	0	0
$781$	−42.9421	−1.53659
$782$	0	0
$783$	− 1.20049i	− 0.0429020i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.1515i	1.75206i	0.482254	+	0.876031i	$0.339818\pi$
$787$	49.1515i	1.75206i	−0.482254	+	0.876031i	$0.660182\pi$
$788$	0	0
$789$	−1.20966	−0.0430651
$790$	0	0
$791$	44.4567	1.58070
$792$	0	0
$793$	23.6065i	0.838290i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44.5325i	− 1.57742i	−0.614764	−	0.788711i	$-0.710749\pi$
$797$	− 44.5325i	− 1.57742i	0.614764	−	0.788711i	$-0.289251\pi$
$798$	0	0
$799$	0.697388	0.0246718
$800$	0	0
$801$	14.5774	0.515069
$802$	0	0
$803$	24.1245i	0.851334i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 25.5249i	− 0.898519i
$808$	0	0
$809$	30.2911	1.06498	0.532488	−	0.846437i	$-0.321257\pi$
$809$	30.2911	1.06498	0.532488	+	0.846437i	$0.321257\pi$
$810$	0	0
$811$	6.24687	0.219357	0.109679	−	0.993967i	$-0.465018\pi$
$811$	6.24687	0.219357	0.109679	+	0.993967i	$0.465018\pi$
$812$	0	0
$813$	5.87342i	0.205990i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 62.6006i	− 2.19012i
$818$	0	0
$819$	17.4094	0.608333
$820$	0	0
$821$	−19.2716	−0.672582	−0.336291	−	0.941758i	$-0.609173\pi$
$821$	−19.2716	−0.672582	−0.336291	+	0.941758i	$0.609173\pi$
$822$	0	0
$823$	7.55750i	0.263438i	0.991287	+	0.131719i	$0.0420496\pi$
$823$	7.55750i	0.263438i	−0.991287	+	0.131719i	$0.957950\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 31.3740i	− 1.09098i	−0.838117	−	0.545490i	$-0.816344\pi$
$827$	− 31.3740i	− 1.09098i	0.838117	−	0.545490i	$-0.183656\pi$
$828$	0	0
$829$	−27.6886	−0.961665	−0.480832	−	0.876813i	$-0.659665\pi$
$829$	−27.6886	−0.961665	−0.480832	+	0.876813i	$0.659665\pi$
$830$	0	0
$831$	−4.90539	−0.170166
$832$	0	0
$833$	38.7600i	1.34295i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.30235i	0.114146i
$838$	0	0
$839$	37.7794	1.30429	0.652145	−	0.758094i	$-0.273869\pi$
$839$	37.7794	1.30429	0.652145	+	0.758094i	$0.273869\pi$
$840$	0	0
$841$	−27.5588	−0.950304
$842$	0	0
$843$	1.29526i	0.0446110i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	62.4176i	2.14469i
$848$	0	0
$849$	−30.2297	−1.03748
$850$	0	0
$851$	12.5368	0.429756
$852$	0	0
$853$	49.3670i	1.69029i	0.534535	+	0.845147i	$0.320487\pi$
$853$	49.3670i	1.69029i	−0.534535	+	0.845147i	$0.679513\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.7692i	− 0.538667i	−0.963047	−	0.269333i	$-0.913197\pi$
$857$	− 15.7692i	− 0.538667i	0.963047	−	0.269333i	$-0.0868034\pi$
$858$	0	0
$859$	−9.39688	−0.320617	−0.160309	−	0.987067i	$-0.551249\pi$
$859$	−9.39688	−0.320617	−0.160309	+	0.987067i	$0.551249\pi$
$860$	0	0
$861$	−14.0324	−0.478225
$862$	0	0
$863$	17.8705i	0.608318i	0.952621	+	0.304159i	$0.0983754\pi$
$863$	17.8705i	0.608318i	−0.952621	+	0.304159i	$0.901625\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 9.76177i	− 0.331527i
$868$	0	0
$869$	61.7273	2.09395
$870$	0	0
$871$	−9.63468	−0.326459
$872$	0	0
$873$	3.98689i	0.134936i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17.7052i	− 0.597862i	−0.954275	−	0.298931i	$-0.903370\pi$
$877$	− 17.7052i	− 0.597862i	0.954275	−	0.298931i	$-0.0966301\pi$
$878$	0	0
$879$	−8.06831	−0.272137
$880$	0	0
$881$	−31.7784	−1.07064	−0.535321	−	0.844649i	$-0.679809\pi$
$881$	−31.7784	−1.07064	−0.535321	+	0.844649i	$0.679809\pi$
$882$	0	0
$883$	− 34.6098i	− 1.16471i	−0.812934	−	0.582356i	$-0.802131\pi$
$883$	− 34.6098i	− 1.16471i	0.812934	−	0.582356i	$-0.197869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.5204i	0.957622i	0.877918	+	0.478811i	$0.158932\pi$
$887$	28.5204i	0.957622i	−0.877918	+	0.478811i	$0.841068\pi$
$888$	0	0
$889$	−73.9923	−2.48162
$890$	0	0
$891$	−4.94880	−0.165791
$892$	0	0
$893$	− 1.52186i	− 0.0509271i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 25.1289i	− 0.839028i
$898$	0	0
$899$	−3.96444	−0.132221
$900$	0	0
$901$	−26.3567	−0.878069
$902$	0	0
$903$	49.3331i	1.64170i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7.01754i	0.233013i	0.993190	+	0.116507i	$0.0371697\pi$
$907$	7.01754i	0.233013i	−0.993190	+	0.116507i	$0.962830\pi$
$908$	0	0
$909$	−9.36896	−0.310749
$910$	0	0
$911$	−20.9075	−0.692696	−0.346348	−	0.938106i	$-0.612578\pi$
$911$	−20.9075	−0.692696	−0.346348	+	0.938106i	$0.612578\pi$
$912$	0	0
$913$	− 43.9961i	− 1.45606i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 57.0145i	− 1.88278i
$918$	0	0
$919$	−59.7567	−1.97119	−0.985596	−	0.169117i	$-0.945908\pi$
$919$	−59.7567	−1.97119	−0.985596	+	0.169117i	$0.945908\pi$
$920$	0	0
$921$	−14.7750	−0.486854
$922$	0	0
$923$	32.6506i	1.07471i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.4241i	0.342372i
$928$	0	0
$929$	40.2308	1.31993	0.659965	−	0.751296i	$-0.270571\pi$
$929$	40.2308	1.31993	0.659965	+	0.751296i	$0.270571\pi$
$930$	0	0
$931$	84.5832	2.77210
$932$	0	0
$933$	9.15373i	0.299680i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 49.4155i	− 1.61433i	−0.590324	−	0.807167i	$-0.701000\pi$
$937$	− 49.4155i	− 1.61433i	0.590324	−	0.807167i	$-0.299000\pi$
$938$	0	0
$939$	13.2553	0.432571
$940$	0	0
$941$	22.2653	0.725828	0.362914	−	0.931823i	$-0.381782\pi$
$941$	22.2653	0.725828	0.362914	+	0.931823i	$0.381782\pi$
$942$	0	0
$943$	20.2546i	0.659579i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 35.6778i	− 1.15937i	−0.814839	−	0.579687i	$-0.803175\pi$
$947$	− 35.6778i	− 1.15937i	0.814839	−	0.579687i	$-0.196825\pi$
$948$	0	0
$949$	18.3428	0.595433
$950$	0	0
$951$	−0.790550	−0.0256354
$952$	0	0
$953$	− 2.80677i	− 0.0909203i	−0.998966	−	0.0454602i	$-0.985525\pi$
$953$	− 2.80677i	− 0.0909203i	0.998966	−	0.0454602i	$-0.0144754\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.94099i	− 0.192045i
$958$	0	0
$959$	26.3472	0.850796
$960$	0	0
$961$	−20.0945	−0.648210
$962$	0	0
$963$	0.220683i	0.00711140i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11.5283i	− 0.370724i	−0.982670	−	0.185362i	$-0.940654\pi$
$967$	− 11.5283i	− 0.370724i	0.982670	−	0.185362i	$-0.0593458\pi$
$968$	0	0
$969$	15.7955	0.507424
$970$	0	0
$971$	23.1566	0.743130	0.371565	−	0.928407i	$-0.378821\pi$
$971$	23.1566	0.743130	0.371565	+	0.928407i	$0.378821\pi$
$972$	0	0
$973$	83.2673i	2.66943i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.26069i	0.264283i	0.991231	+	0.132141i	$0.0421853\pi$
$977$	8.26069i	0.264283i	−0.991231	+	0.132141i	$0.957815\pi$
$978$	0	0
$979$	72.1408	2.30563
$980$	0	0
$981$	6.68640	0.213480
$982$	0	0
$983$	19.7887i	0.631163i	0.948899	+	0.315581i	$0.102200\pi$
$983$	19.7887i	0.631163i	−0.948899	+	0.315581i	$0.897800\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.19932i	0.0381747i
$988$	0	0
$989$	71.2078	2.26427
$990$	0	0
$991$	21.3884	0.679424	0.339712	−	0.940529i	$-0.389670\pi$
$991$	21.3884	0.679424	0.339712	+	0.940529i	$0.389670\pi$
$992$	0	0
$993$	3.37370i	0.107061i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 17.5832i	− 0.556865i	−0.960456	−	0.278432i	$-0.910185\pi$
$997$	− 17.5832i	− 0.556865i	0.960456	−	0.278432i	$-0.0898148\pi$
$998$	0	0
$999$	−1.87725	−0.0593935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.12		24
5.2	odd	4		7500.2.a.m.1.1		12
5.3	odd	4		7500.2.a.n.1.12		12
5.4	even	2	inner	7500.2.d.g.1249.13		24
25.3	odd	20		1500.2.m.c.1201.6		24
25.4	even	10		300.2.o.a.109.2	✓	24
25.6	even	5		300.2.o.a.289.2	yes	24
25.8	odd	20		1500.2.m.c.301.6		24
25.17	odd	20		1500.2.m.d.301.1		24
25.19	even	10		1500.2.o.c.949.6		24
25.21	even	5		1500.2.o.c.49.6		24
25.22	odd	20		1500.2.m.d.1201.1		24
75.29	odd	10		900.2.w.c.109.3		24
75.56	odd	10		900.2.w.c.289.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.2	✓	24	25.4	even	10
300.2.o.a.289.2	yes	24	25.6	even	5
900.2.w.c.109.3		24	75.29	odd	10
900.2.w.c.289.3		24	75.56	odd	10
1500.2.m.c.301.6		24	25.8	odd	20
1500.2.m.c.1201.6		24	25.3	odd	20
1500.2.m.d.301.1		24	25.17	odd	20
1500.2.m.d.1201.1		24	25.22	odd	20
1500.2.o.c.49.6		24	25.21	even	5
1500.2.o.c.949.6		24	25.19	even	10
7500.2.a.m.1.1		12	5.2	odd	4
7500.2.a.n.1.12		12	5.3	odd	4
7500.2.d.g.1249.12		24	1.1	even	1	trivial
7500.2.d.g.1249.13		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.62675i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +4.62675i q^{7} -1.00000 q^{9} -4.94880 q^{11} +3.76277i q^{13} -2.69040i q^{17} -5.87106 q^{19} +4.62675 q^{21} -6.67829i q^{23} +1.00000i q^{27} -1.20049 q^{29} +3.30235 q^{31} +4.94880i q^{33} +1.87725i q^{37} +3.76277 q^{39} -3.03290 q^{41} +10.6626i q^{43} +0.259214i q^{47} -14.4068 q^{49} -2.69040 q^{51} -9.79659i q^{53} +5.87106i q^{57} +9.62455 q^{59} +6.27369 q^{61} -4.62675i q^{63} +2.56053i q^{67} -6.67829 q^{69} +8.67729 q^{71} -4.87481i q^{73} -22.8968i q^{77} -12.4732 q^{79} +1.00000 q^{81} +8.89025i q^{83} +1.20049i q^{87} -14.5774 q^{89} -17.4094 q^{91} -3.30235i q^{93} -3.98689i q^{97} +4.94880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 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

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