LMFDB - Embedded newform 75.18.b.b.49.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,18,Mod(49,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.49"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-702752] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$137.416565508$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{849})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 425x^{2} + 44944 $ x^4 + 425*x^2 + 44944
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$15.0688i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.18.b.b.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+702.477i q^{2} +6561.00i q^{3} -362402. q^{4} -4.60895e6 q^{6} +2.67481e6i q^{7} -1.62504e8i q^{8} -4.30467e7 q^{9} -9.86334e8 q^{11} -2.37772e9i q^{12} +2.81488e9i q^{13} -1.87899e9 q^{14} +6.66544e10 q^{16} -1.28501e10i q^{17} -3.02393e10i q^{18} +1.22560e11 q^{19} -1.75494e10 q^{21} -6.92877e11i q^{22} +1.14646e11i q^{23} +1.06619e12 q^{24} -1.97739e12 q^{26} -2.82430e11i q^{27} -9.69355e11i q^{28} +4.90027e12 q^{29} +7.64406e12 q^{31} +2.55235e13i q^{32} -6.47134e12i q^{33} +9.02688e12 q^{34} +1.56002e13 q^{36} -3.18983e12i q^{37} +8.60956e13i q^{38} -1.84684e13 q^{39} -5.53137e12 q^{41} -1.23281e13i q^{42} +2.42435e13i q^{43} +3.57449e14 q^{44} -8.05365e13 q^{46} +5.82161e13i q^{47} +4.37320e14i q^{48} +2.25476e14 q^{49} +8.43094e13 q^{51} -1.02012e15i q^{52} -6.45746e14i q^{53} +1.98400e14 q^{54} +4.34667e14 q^{56} +8.04117e14i q^{57} +3.44233e15i q^{58} +1.04433e12 q^{59} -1.56088e15 q^{61} +5.36978e15i q^{62} -1.15142e14i q^{63} -9.19314e15 q^{64} +4.54597e15 q^{66} -1.31151e15i q^{67} +4.65689e15i q^{68} -7.52195e14 q^{69} -7.09230e15 q^{71} +6.99526e15i q^{72} -1.17619e16i q^{73} +2.24078e15 q^{74} -4.44160e16 q^{76} -2.63826e15i q^{77} -1.29737e16i q^{78} +7.14674e15 q^{79} +1.85302e15 q^{81} -3.88566e15i q^{82} +9.04707e15i q^{83} +6.35994e15 q^{84} -1.70305e16 q^{86} +3.21507e16i q^{87} +1.60283e17i q^{88} +1.16786e16 q^{89} -7.52927e15 q^{91} -4.15481e16i q^{92} +5.01527e16i q^{93} -4.08955e16 q^{94} -1.67460e17 q^{96} +8.32538e16i q^{97} +1.58392e17i q^{98} +4.24585e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 702752 q^{4} - 4671432 q^{6} - 172186884 q^{9} - 2263259824 q^{11} + 8770444032 q^{14} + 102082634240 q^{16} + 388752254432 q^{19} - 272341231344 q^{21} + 2778418267776 q^{24} - 6214885009072 q^{26}+ \cdots + 97\!\cdots\!04 q^{99}+O(q^{100}) $ 4 * q - 702752 * q^4 - 4671432 * q^6 - 172186884 * q^9 - 2263259824 * q^11 + 8770444032 * q^14 + 102082634240 * q^16 + 388752254432 * q^19 - 272341231344 * q^21 + 2778418267776 * q^24 - 6214885009072 * q^26 + 9474125251784 * q^29 + 4512814149744 * q^31 + 12640450603856 * q^34 + 30251169276192 * q^36 + 5861236859928 * q^39 - 26103091718456 * q^41 + 711694545463424 * q^44 - 45232853381376 * q^46 + 262458605334940 * q^49 + 271127040034392 * q^51 + 201089829974472 * q^54 + 2649594618961920 * q^56 - 1926784846232912 * q^59 - 5717586429945320 * q^61 - 23202311714971648 * q^64 + 8431349467781664 * q^66 - 3697976988787248 * q^69 + 4413073310120608 * q^71 + 30034887627397232 * q^74 - 87248335332469888 * q^76 - 9995782195868880 * q^79 + 7412080755407364 * q^81 + 10104100651911168 * q^84 - 73832062872187744 * q^86 + 121714515304433592 * q^89 + 102877589806297824 * q^91 - 180547794550988992 * q^94 - 321225945838331904 * q^96 + 97425914194237104 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	702.477i	1.94034i	0.242432	+	0.970168i	$0.422055\pi$
$2$	702.477i	1.94034i	−0.242432	+	0.970168i	$0.577945\pi$
$3$	6561.00i	0.577350i
$3$	6561.00i	0.577350i
$4$	−362402.	−2.76491
$5$	0	0
$5$	0	0
$6$	−4.60895e6	−1.12025
$7$	2.67481e6i	0.175372i	0.996148	+	0.0876858i	$0.0279472\pi$
$7$	2.67481e6i	0.175372i	−0.996148	+	0.0876858i	$0.972053\pi$
$8$	− 1.62504e8i	− 3.42451i
$9$	−4.30467e7	−0.333333
$10$	0	0
$11$	−9.86334e8	−1.38735	−0.693675	−	0.720288i	$-0.744010\pi$
$11$	−9.86334e8	−1.38735	−0.693675	+	0.720288i	$0.744010\pi$
$12$	− 2.37772e9i	− 1.59632i
$13$	2.81488e9i	0.957065i	0.878070	+	0.478533i	$0.158831\pi$
$13$	2.81488e9i	0.957065i	−0.878070	+	0.478533i	$0.841169\pi$
$14$	−1.87899e9	−0.340280
$15$	0	0
$16$	6.66544e10	3.87980
$17$	− 1.28501e10i	− 0.446776i	−0.974730	−	0.223388i	$-0.928288\pi$
$17$	− 1.28501e10i	− 0.446776i	0.974730	−	0.223388i	$-0.0717117\pi$
$18$	− 3.02393e10i	− 0.646779i
$19$	1.22560e11	1.65555	0.827777	−	0.561057i	$-0.189605\pi$
$19$	1.22560e11	1.65555	0.827777	+	0.561057i	$0.189605\pi$
$20$	0	0
$21$	−1.75494e10	−0.101251
$22$	− 6.92877e11i	− 2.69193i
$23$	1.14646e11i	0.305263i	0.988283	+	0.152631i	$0.0487747\pi$
$23$	1.14646e11i	0.305263i	−0.988283	+	0.152631i	$0.951225\pi$
$24$	1.06619e12	1.97714
$25$	0	0
$26$	−1.97739e12	−1.85703
$27$	− 2.82430e11i	− 0.192450i
$28$	− 9.69355e11i	− 0.484886i
$29$	4.90027e12	1.81902	0.909510	−	0.415682i	$-0.136457\pi$
$29$	4.90027e12	1.81902	0.909510	+	0.415682i	$0.136457\pi$
$30$	0	0
$31$	7.64406e12	1.60972	0.804859	−	0.593466i	$-0.202241\pi$
$31$	7.64406e12	1.60972	0.804859	+	0.593466i	$0.202241\pi$
$32$	2.55235e13i	4.10361i
$33$	− 6.47134e12i	− 0.800987i
$34$	9.02688e12	0.866896
$35$	0	0
$36$	1.56002e13	0.921635
$37$	− 3.18983e12i	− 0.149298i	−0.997210	−	0.0746488i	$-0.976216\pi$
$37$	− 3.18983e12i	− 0.149298i	0.997210	−	0.0746488i	$-0.0237836\pi$
$38$	8.60956e13i	3.21233i
$39$	−1.84684e13	−0.552562
$40$	0	0
$41$	−5.53137e12	−0.108186	−0.0540928	−	0.998536i	$-0.517227\pi$
$41$	−5.53137e12	−0.108186	−0.0540928	+	0.998536i	$0.517227\pi$
$42$	− 1.23281e13i	− 0.196461i
$43$	2.42435e13i	0.316311i	0.987414	+	0.158155i	$0.0505547\pi$
$43$	2.42435e13i	0.316311i	−0.987414	+	0.158155i	$0.949445\pi$
$44$	3.57449e14	3.83589
$45$	0	0
$46$	−8.05365e13	−0.592313
$47$	5.82161e13i	0.356625i	0.983974	+	0.178312i	$0.0570638\pi$
$47$	5.82161e13i	0.356625i	−0.983974	+	0.178312i	$0.942936\pi$
$48$	4.37320e14i	2.24000i
$49$	2.25476e14	0.969245
$50$	0	0
$51$	8.43094e13	0.257946
$52$	− 1.02012e15i	− 2.64620i
$53$	− 6.45746e14i	− 1.42468i	−0.701835	−	0.712339i	$-0.747636\pi$
$53$	− 6.45746e14i	− 1.42468i	0.701835	−	0.712339i	$-0.252364\pi$
$54$	1.98400e14	0.373418
$55$	0	0
$56$	4.34667e14	0.600562
$57$	8.04117e14i	0.955835i
$58$	3.44233e15i	3.52951i
$59$	1.04433e12	0.000925970 0	0.000462985	−	1.00000i	$-0.499853\pi$
$59$	1.04433e12	0.000925970 0	0.000462985		1.00000i	$0.499853\pi$
$60$	0	0
$61$	−1.56088e15	−1.04247	−0.521237	−	0.853412i	$-0.674529\pi$
$61$	−1.56088e15	−1.04247	−0.521237	+	0.853412i	$0.674529\pi$
$62$	5.36978e15i	3.12340i
$63$	− 1.15142e14i	− 0.0584572i
$64$	−9.19314e15	−4.08258
$65$	0	0
$66$	4.54597e15	1.55419
$67$	− 1.31151e15i	− 0.394581i	−0.980345	−	0.197290i	$-0.936786\pi$
$67$	− 1.31151e15i	− 0.394581i	0.980345	−	0.197290i	$-0.0632142\pi$
$68$	4.65689e15i	1.23529i
$69$	−7.52195e14	−0.176244
$70$	0	0
$71$	−7.09230e15	−1.30344	−0.651720	−	0.758459i	$-0.725952\pi$
$71$	−7.09230e15	−1.30344	−0.651720	+	0.758459i	$0.725952\pi$
$72$	6.99526e15i	1.14150i
$73$	− 1.17619e16i	− 1.70700i	−0.521091	−	0.853501i	$-0.674475\pi$
$73$	− 1.17619e16i	− 1.70700i	0.521091	−	0.853501i	$-0.325525\pi$
$74$	2.24078e15	0.289688
$75$	0	0
$76$	−4.44160e16	−4.57745
$77$	− 2.63826e15i	− 0.243302i
$78$	− 1.29737e16i	− 1.07216i
$79$	7.14674e15	0.530003	0.265001	−	0.964248i	$-0.414628\pi$
$79$	7.14674e15	0.530003	0.265001	+	0.964248i	$0.414628\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	− 3.88566e15i	− 0.209917i
$83$	9.04707e15i	0.440904i	0.975398	+	0.220452i	$0.0707532\pi$
$83$	9.04707e15i	0.440904i	−0.975398	+	0.220452i	$0.929247\pi$
$84$	6.35994e15	0.279949
$85$	0	0
$86$	−1.70305e16	−0.613749
$87$	3.21507e16i	1.05021i
$88$	1.60283e17i	4.75100i
$89$	1.16786e16	0.314467	0.157233	−	0.987561i	$-0.449743\pi$
$89$	1.16786e16	0.314467	0.157233	+	0.987561i	$0.449743\pi$
$90$	0	0
$91$	−7.52927e15	−0.167842
$92$	− 4.15481e16i	− 0.844023i
$93$	5.01527e16i	0.929371i
$94$	−4.08955e16	−0.691972
$95$	0	0
$96$	−1.67460e17	−2.36922
$97$	8.32538e16i	1.07856i	0.842126	+	0.539280i	$0.181304\pi$
$97$	8.32538e16i	1.07856i	−0.842126	+	0.539280i	$0.818696\pi$
$98$	1.58392e17i	1.88066i
$99$	4.24585e16	0.462450
$100$	0	0
$101$	−1.36210e16	−0.125164	−0.0625818	−	0.998040i	$-0.519933\pi$
$101$	−1.36210e16	−0.125164	−0.0625818	+	0.998040i	$0.519933\pi$
$102$	5.92254e16i	0.500503i
$103$	1.19030e17i	0.925847i	0.886398	+	0.462923i	$0.153199\pi$
$103$	1.19030e17i	0.925847i	−0.886398	+	0.462923i	$0.846801\pi$
$104$	4.57429e17	3.27748
$105$	0	0
$106$	4.53622e17	2.76436
$107$	− 3.05375e17i	− 1.71819i	−0.511814	−	0.859096i	$-0.671026\pi$
$107$	− 3.05375e17i	− 1.71819i	0.511814	−	0.859096i	$-0.328974\pi$
$108$	1.02353e17i	0.532106i
$109$	2.94297e17	1.41469	0.707344	−	0.706869i	$-0.249893\pi$
$109$	2.94297e17	1.41469	0.707344	+	0.706869i	$0.249893\pi$
$110$	0	0
$111$	2.09285e16	0.0861970
$112$	1.78288e17i	0.680407i
$113$	− 7.25034e16i	− 0.256562i	−0.991738	−	0.128281i	$-0.959054\pi$
$113$	− 7.25034e16i	− 0.256562i	0.991738	−	0.128281i	$-0.0409459\pi$
$114$	−5.64873e17	−1.85464
$115$	0	0
$116$	−1.77587e18	−5.02942
$117$	− 1.21171e17i	− 0.319022i
$118$	7.33620e14i	0.00179669i
$119$	3.43715e16	0.0783519
$120$	0	0
$121$	4.67408e17	0.924742
$122$	− 1.09648e18i	− 2.02275i
$123$	− 3.62913e16i	− 0.0624610i
$124$	−2.77022e18	−4.45072
$125$	0	0
$126$	8.08844e16	0.113427
$127$	1.00107e18i	1.31261i	0.754497	+	0.656304i	$0.227881\pi$
$127$	1.00107e18i	1.31261i	−0.754497	+	0.656304i	$0.772119\pi$
$128$	− 3.11255e18i	− 3.81797i
$129$	−1.59062e17	−0.182622
$130$	0	0
$131$	6.36150e17	0.640846	0.320423	−	0.947275i	$-0.396175\pi$
$131$	6.36150e17	0.640846	0.320423	+	0.947275i	$0.396175\pi$
$132$	2.34522e18i	2.21465i
$133$	3.27825e17i	0.290337i
$134$	9.21306e17	0.765620
$135$	0	0
$136$	−2.08819e18	−1.52999
$137$	− 2.75383e17i	− 0.189588i	−0.995497	−	0.0947942i	$-0.969781\pi$
$137$	− 2.75383e17i	− 0.189588i	0.995497	−	0.0947942i	$-0.0302193\pi$
$138$	− 5.28400e17i	− 0.341972i
$139$	−3.61048e17	−0.219755	−0.109878	−	0.993945i	$-0.535046\pi$
$139$	−3.61048e17	−0.219755	−0.109878	+	0.993945i	$0.535046\pi$
$140$	0	0
$141$	−3.81956e17	−0.205897
$142$	− 4.98218e18i	− 2.52911i
$143$	− 2.77641e18i	− 1.32779i
$144$	−2.86926e18	−1.29327
$145$	0	0
$146$	8.26248e18	3.31216
$147$	1.47935e18i	0.559594i
$148$	1.15600e18i	0.412794i
$149$	4.02824e18	1.35841	0.679207	−	0.733947i	$-0.262324\pi$
$149$	4.02824e18	1.35841	0.679207	+	0.733947i	$0.262324\pi$
$150$	0	0
$151$	−3.02464e18	−0.910689	−0.455345	−	0.890315i	$-0.650484\pi$
$151$	−3.02464e18	−0.910689	−0.455345	+	0.890315i	$0.650484\pi$
$152$	− 1.99165e19i	− 5.66947i
$153$	5.53154e17i	0.148925i
$154$	1.85331e18	0.472088
$155$	0	0
$156$	6.69300e18	1.52778
$157$	3.80909e18i	0.823520i	0.911292	+	0.411760i	$0.135086\pi$
$157$	3.80909e18i	0.823520i	−0.911292	+	0.411760i	$0.864914\pi$
$158$	5.02042e18i	1.02838i
$159$	4.23674e18	0.822539
$160$	0	0
$161$	−3.06657e17	−0.0535345
$162$	1.30170e18i	0.215593i
$163$	− 7.01276e18i	− 1.10229i	−0.834411	−	0.551143i	$-0.814192\pi$
$163$	− 7.01276e18i	− 1.10229i	0.834411	−	0.551143i	$-0.185808\pi$
$164$	2.00458e18	0.299123
$165$	0	0
$166$	−6.35535e18	−0.855502
$167$	5.20936e18i	0.666337i	0.942867	+	0.333169i	$0.108118\pi$
$167$	5.20936e18i	0.666337i	−0.942867	+	0.333169i	$0.891882\pi$
$168$	2.85185e18i	0.346735i
$169$	7.26857e17	0.0840256
$170$	0	0
$171$	−5.27581e18	−0.551851
$172$	− 8.78590e18i	− 0.874570i
$173$	4.55714e18i	0.431818i	0.976413	+	0.215909i	$0.0692715\pi$
$173$	4.55714e18i	0.431818i	−0.976413	+	0.215909i	$0.930729\pi$
$174$	−2.25851e19	−2.03776
$175$	0	0
$176$	−6.57436e19	−5.38264
$177$	6.85187e15i	0 0.000534609i
$178$	8.20393e18i	0.610171i
$179$	−1.27111e19	−0.901428	−0.450714	−	0.892668i	$-0.648831\pi$
$179$	−1.27111e19	−0.901428	−0.450714	+	0.892668i	$0.648831\pi$
$180$	0	0
$181$	1.67839e19	1.08299	0.541494	−	0.840704i	$-0.317859\pi$
$181$	1.67839e19	1.08299	0.541494	+	0.840704i	$0.317859\pi$
$182$	− 5.28914e18i	− 0.325670i
$183$	− 1.02409e19i	− 0.601872i
$184$	1.86305e19	1.04538
$185$	0	0
$186$	−3.52311e19	−1.80329
$187$	1.26745e19i	0.619835i
$188$	− 2.10976e19i	− 0.986034i
$189$	7.55445e17	0.0337503
$190$	0	0
$191$	1.93607e19	0.790930	0.395465	−	0.918481i	$-0.370583\pi$
$191$	1.93607e19	0.790930	0.395465	+	0.918481i	$0.370583\pi$
$192$	− 6.03162e19i	− 2.35708i
$193$	5.41103e18i	0.202322i	0.994870	+	0.101161i	$0.0322557\pi$
$193$	5.41103e18i	0.202322i	−0.994870	+	0.101161i	$0.967744\pi$
$194$	−5.84839e19	−2.09277
$195$	0	0
$196$	−8.17129e19	−2.67987
$197$	− 1.69857e19i	− 0.533483i	−0.963768	−	0.266741i	$-0.914053\pi$
$197$	− 1.69857e19i	− 0.533483i	0.963768	−	0.266741i	$-0.0859469\pi$
$198$	2.98261e19i	0.897309i
$199$	8.38703e18	0.241745	0.120872	−	0.992668i	$-0.461431\pi$
$199$	8.38703e18	0.241745	0.120872	+	0.992668i	$0.461431\pi$
$200$	0	0
$201$	8.60482e18	0.227811
$202$	− 9.56845e18i	− 0.242860i
$203$	1.31073e19i	0.319004i
$204$	−3.05539e19	−0.713197
$205$	0	0
$206$	−8.36156e19	−1.79645
$207$	− 4.93515e18i	− 0.101754i
$208$	1.87624e20i	3.71322i
$209$	−1.20885e20	−2.29683
$210$	0	0
$211$	2.60946e19	0.457245	0.228623	−	0.973515i	$-0.426578\pi$
$211$	2.60946e19	0.457245	0.228623	+	0.973515i	$0.426578\pi$
$212$	2.34020e20i	3.93910i
$213$	− 4.65326e19i	− 0.752542i
$214$	2.14519e20	3.33387
$215$	0	0
$216$	−4.58959e19	−0.659048
$217$	2.04464e19i	0.282299i
$218$	2.06737e20i	2.74497i
$219$	7.71700e19	0.985539
$220$	0	0
$221$	3.61715e19	0.427594
$222$	1.47018e19i	0.167251i
$223$	1.19753e19i	0.131128i	0.997848	+	0.0655641i	$0.0208847\pi$
$223$	1.19753e19i	0.131128i	−0.997848	+	0.0655641i	$0.979115\pi$
$224$	−6.82705e19	−0.719656
$225$	0	0
$226$	5.09320e19	0.497816
$227$	− 1.24200e20i	− 1.16924i	−0.811308	−	0.584619i	$-0.801244\pi$
$227$	− 1.24200e20i	− 1.16924i	0.811308	−	0.584619i	$-0.198756\pi$
$228$	− 2.91413e20i	− 2.64279i
$229$	−1.03099e20	−0.900853	−0.450426	−	0.892814i	$-0.648728\pi$
$229$	−1.03099e20	−0.900853	−0.450426	+	0.892814i	$0.648728\pi$
$230$	0	0
$231$	1.73096e19	0.140470
$232$	− 7.96313e20i	− 6.22925i
$233$	1.13357e20i	0.854919i	0.904035	+	0.427459i	$0.140591\pi$
$233$	1.13357e20i	0.854919i	−0.904035	+	0.427459i	$0.859409\pi$
$234$	8.51201e19	0.619010
$235$	0	0
$236$	−3.78468e17	−0.00256022
$237$	4.68898e19i	0.305997i
$238$	2.41452e19i	0.152029i
$239$	8.29065e19	0.503740	0.251870	−	0.967761i	$-0.418954\pi$
$239$	8.29065e19	0.503740	0.251870	+	0.967761i	$0.418954\pi$
$240$	0	0
$241$	−1.62688e20	−0.920896	−0.460448	−	0.887687i	$-0.652311\pi$
$241$	−1.62688e20	−0.920896	−0.460448	+	0.887687i	$0.652311\pi$
$242$	3.28343e20i	1.79431i
$243$	1.21577e19i	0.0641500i
$244$	5.65664e20	2.88234
$245$	0	0
$246$	2.54938e19	0.121195
$247$	3.44992e20i	1.58447i
$248$	− 1.24219e21i	− 5.51250i
$249$	−5.93578e19	−0.254556
$250$	0	0
$251$	4.10409e20	1.64433	0.822167	−	0.569247i	$-0.192765\pi$
$251$	4.10409e20	1.64433	0.822167	+	0.569247i	$0.192765\pi$
$252$	4.17276e19i	0.161629i
$253$	− 1.13080e20i	− 0.423507i
$254$	−7.03232e20	−2.54690
$255$	0	0
$256$	9.81533e20	3.32556
$257$	1.69582e20i	0.555839i	0.960604	+	0.277920i	$0.0896449\pi$
$257$	1.69582e20i	0.555839i	−0.960604	+	0.277920i	$0.910355\pi$
$258$	− 1.11737e20i	− 0.354348i
$259$	8.53218e18	0.0261826
$260$	0	0
$261$	−2.10941e20	−0.606340
$262$	4.46881e20i	1.24346i
$263$	− 3.50624e20i	− 0.944534i	−0.881456	−	0.472267i	$-0.843436\pi$
$263$	− 3.50624e20i	− 0.944534i	0.881456	−	0.472267i	$-0.156564\pi$
$264$	−1.05162e21	−2.74299
$265$	0	0
$266$	−2.30289e20	−0.563352
$267$	7.66231e19i	0.181557i
$268$	4.75294e20i	1.09098i
$269$	−2.92843e20	−0.651240	−0.325620	−	0.945501i	$-0.605573\pi$
$269$	−2.92843e20	−0.651240	−0.325620	+	0.945501i	$0.605573\pi$
$270$	0	0
$271$	5.64043e20	1.17780	0.588902	−	0.808204i	$-0.299560\pi$
$271$	5.64043e20	1.17780	0.588902	+	0.808204i	$0.299560\pi$
$272$	− 8.56515e20i	− 1.73340i
$273$	− 4.93995e19i	− 0.0969037i
$274$	1.93450e20	0.367865
$275$	0	0
$276$	2.72597e20	0.487297
$277$	− 7.92843e20i	− 1.37439i	−0.726474	−	0.687194i	$-0.758842\pi$
$277$	− 7.92843e20i	− 1.37439i	0.726474	−	0.687194i	$-0.241158\pi$
$278$	− 2.53628e20i	− 0.426399i
$279$	−3.29052e20	−0.536573
$280$	0	0
$281$	−4.07531e20	−0.625399	−0.312700	−	0.949852i	$-0.601233\pi$
$281$	−4.07531e20	−0.625399	−0.312700	+	0.949852i	$0.601233\pi$
$282$	− 2.68315e20i	− 0.399510i
$283$	9.05800e20i	1.30872i	0.756182	+	0.654362i	$0.227063\pi$
$283$	9.05800e20i	1.30872i	−0.756182	+	0.654362i	$0.772937\pi$
$284$	2.57026e21	3.60389
$285$	0	0
$286$	1.95037e21	2.57635
$287$	− 1.47953e19i	− 0.0189727i
$288$	− 1.09870e21i	− 1.36787i
$289$	6.62116e20	0.800391
$290$	0	0
$291$	−5.46228e20	−0.622707
$292$	4.26254e21i	4.71970i
$293$	1.22920e21i	1.32205i	0.750364	+	0.661025i	$0.229878\pi$
$293$	1.22920e21i	1.32205i	−0.750364	+	0.661025i	$0.770122\pi$
$294$	−1.03921e21	−1.08580
$295$	0	0
$296$	−5.18359e20	−0.511271
$297$	2.78570e20i	0.266996i
$298$	2.82974e21i	2.63578i
$299$	−3.22716e20	−0.292157
$300$	0	0
$301$	−6.48468e19	−0.0554719
$302$	− 2.12474e21i	− 1.76704i
$303$	− 8.93675e19i	− 0.0722632i
$304$	8.16918e21	6.42322
$305$	0	0
$306$	−3.88578e20	−0.288965
$307$	1.33769e21i	0.967565i	0.875188	+	0.483783i	$0.160738\pi$
$307$	1.33769e21i	0.967565i	−0.875188	+	0.483783i	$0.839262\pi$
$308$	9.56108e20i	0.672707i
$309$	−7.80954e20	−0.534538
$310$	0	0
$311$	−8.26556e19	−0.0535561	−0.0267781	−	0.999641i	$-0.508525\pi$
$311$	−8.26556e19	−0.0535561	−0.0267781	+	0.999641i	$0.508525\pi$
$312$	3.00119e21i	1.89226i
$313$	− 2.88314e20i	− 0.176905i	−0.996080	−	0.0884523i	$-0.971808\pi$
$313$	− 2.88314e20i	− 0.176905i	0.996080	−	0.0884523i	$-0.0281921\pi$
$314$	−2.67580e21	−1.59791
$315$	0	0
$316$	−2.58999e21	−1.46541
$317$	3.17162e21i	1.74694i	0.486882	+	0.873468i	$0.338134\pi$
$317$	3.17162e21i	1.74694i	−0.486882	+	0.873468i	$0.661866\pi$
$318$	2.97621e21i	1.59600i
$319$	−4.83331e21	−2.52362
$320$	0	0
$321$	2.00357e21	0.991999
$322$	− 2.15420e20i	− 0.103875i
$323$	− 1.57491e21i	− 0.739662i
$324$	−6.71538e20	−0.307212
$325$	0	0
$326$	4.92630e21	2.13881
$327$	1.93088e21i	0.816771i
$328$	8.98868e20i	0.370483i
$329$	−1.55717e20	−0.0625419
$330$	0	0
$331$	−1.58984e21	−0.606478	−0.303239	−	0.952915i	$-0.598068\pi$
$331$	−1.58984e21	−0.606478	−0.303239	+	0.952915i	$0.598068\pi$
$332$	− 3.27867e21i	− 1.21906i
$333$	1.37312e20i	0.0497659i
$334$	−3.65945e21	−1.29292
$335$	0	0
$336$	−1.16975e21	−0.392833
$337$	3.01551e20i	0.0987432i	0.998780	+	0.0493716i	$0.0157219\pi$
$337$	3.01551e20i	0.0987432i	−0.998780	+	0.0493716i	$0.984278\pi$
$338$	5.10600e20i	0.163038i
$339$	4.75695e20	0.148126
$340$	0	0
$341$	−7.53960e21	−2.23324
$342$	− 3.70613e21i	− 1.07078i
$343$	1.22535e21i	0.345350i
$344$	3.93967e21	1.08321
$345$	0	0
$346$	−3.20129e21	−0.837873
$347$	3.81902e21i	0.975331i	0.873031	+	0.487665i	$0.162151\pi$
$347$	3.81902e21i	0.975331i	−0.873031	+	0.487665i	$0.837849\pi$
$348$	− 1.16515e22i	− 2.90374i
$349$	3.13624e20	0.0762769	0.0381384	−	0.999272i	$-0.487857\pi$
$349$	3.13624e20	0.0762769	0.0381384	+	0.999272i	$0.487857\pi$
$350$	0	0
$351$	7.95006e20	0.184187
$352$	− 2.51747e22i	− 5.69314i
$353$	− 4.65586e21i	− 1.02781i	−0.857846	−	0.513907i	$-0.828198\pi$
$353$	− 4.65586e21i	− 1.02781i	0.857846	−	0.513907i	$-0.171802\pi$
$354$	−4.81328e18	−0.00103732
$355$	0	0
$356$	−4.23234e21	−0.869471
$357$	2.25511e20i	0.0452365i
$358$	− 8.92923e21i	− 1.74907i
$359$	−3.57780e21	−0.684405	−0.342203	−	0.939626i	$-0.611173\pi$
$359$	−3.57780e21	−0.684405	−0.342203	+	0.939626i	$0.611173\pi$
$360$	0	0
$361$	9.54059e21	1.74086
$362$	1.17903e22i	2.10136i
$363$	3.06667e21i	0.533900i
$364$	2.72862e21	0.464068
$365$	0	0
$366$	7.19400e21	1.16783
$367$	5.72348e21i	0.907818i	0.891048	+	0.453909i	$0.149971\pi$
$367$	5.72348e21i	0.907818i	−0.891048	+	0.453909i	$0.850029\pi$
$368$	7.64170e21i	1.18436i
$369$	2.38107e20	0.0360619
$370$	0	0
$371$	1.72725e21	0.249848
$372$	− 1.81754e22i	− 2.56962i
$373$	7.79326e21i	1.07695i	0.842643	+	0.538473i	$0.180999\pi$
$373$	7.79326e21i	1.07695i	−0.842643	+	0.538473i	$0.819001\pi$
$374$	−8.90352e21	−1.20269
$375$	0	0
$376$	9.46034e21	1.22127
$377$	1.37937e22i	1.74092i
$378$	5.30683e20i	0.0654869i
$379$	−4.41273e21	−0.532444	−0.266222	−	0.963912i	$-0.585775\pi$
$379$	−4.41273e21	−0.532444	−0.266222	+	0.963912i	$0.585775\pi$
$380$	0	0
$381$	−6.56805e21	−0.757834
$382$	1.36005e22i	1.53467i
$383$	7.42094e21i	0.818973i	0.912316	+	0.409486i	$0.134292\pi$
$383$	7.42094e21i	0.818973i	−0.912316	+	0.409486i	$0.865708\pi$
$384$	2.04215e22	2.20430
$385$	0	0
$386$	−3.80112e21	−0.392572
$387$	− 1.04360e21i	− 0.105437i
$388$	− 3.01713e22i	− 2.98212i
$389$	−4.44977e21	−0.430295	−0.215147	−	0.976582i	$-0.569023\pi$
$389$	−4.44977e21	−0.430295	−0.215147	+	0.976582i	$0.569023\pi$
$390$	0	0
$391$	1.47322e21	0.136384
$392$	− 3.66407e22i	− 3.31919i
$393$	4.17378e21i	0.369993i
$394$	1.19321e22	1.03514
$395$	0	0
$396$	−1.53870e22	−1.27863
$397$	4.94222e20i	0.0401979i	0.999798	+	0.0200989i	$0.00639812\pi$
$397$	4.94222e20i	0.0401979i	−0.999798	+	0.0200989i	$0.993602\pi$
$398$	5.89170e21i	0.469066i
$399$	−2.15086e21	−0.167626
$400$	0	0
$401$	2.49341e22	1.86237	0.931187	−	0.364542i	$-0.118775\pi$
$401$	2.49341e22	1.86237	0.931187	+	0.364542i	$0.118775\pi$
$402$	6.04469e21i	0.442031i
$403$	2.15171e22i	1.54061i
$404$	4.93628e21	0.346066
$405$	0	0
$406$	−9.20757e21	−0.618976
$407$	3.14624e21i	0.207128i
$408$	− 1.37006e22i	− 0.883340i
$409$	−7.82942e21	−0.494404	−0.247202	−	0.968964i	$-0.579511\pi$
$409$	−7.82942e21	−0.494404	−0.247202	+	0.968964i	$0.579511\pi$
$410$	0	0
$411$	1.80679e21	0.109459
$412$	− 4.31366e22i	− 2.55988i
$413$	2.79339e18i	0 0.000162389i
$414$	3.46683e21	0.197438
$415$	0	0
$416$	−7.18457e22	−3.92742
$417$	− 2.36884e21i	− 0.126876i
$418$	− 8.49191e22i	− 4.45663i
$419$	1.37314e20	0.00706150	0.00353075	−	0.999994i	$-0.498876\pi$
$419$	1.37314e20	0.00706150	0.00353075	+	0.999994i	$0.498876\pi$
$420$	0	0
$421$	−1.84158e21	−0.0909481	−0.0454740	−	0.998966i	$-0.514480\pi$
$421$	−1.84158e21	−0.0909481	−0.0454740	+	0.998966i	$0.514480\pi$
$422$	1.83308e22i	0.887209i
$423$	− 2.50601e21i	− 0.118875i
$424$	−1.04936e23	−4.87883
$425$	0	0
$426$	3.26881e22	1.46018
$427$	− 4.17505e21i	− 0.182820i
$428$	1.10669e23i	4.75064i
$429$	1.82161e22	0.766597
$430$	0	0
$431$	−3.30639e21	−0.133751	−0.0668756	−	0.997761i	$-0.521303\pi$
$431$	−3.30639e21	−0.133751	−0.0668756	+	0.997761i	$0.521303\pi$
$432$	− 1.88252e22i	− 0.746668i
$433$	1.49902e22i	0.582990i	0.956572	+	0.291495i	$0.0941526\pi$
$433$	1.49902e22i	0.582990i	−0.956572	+	0.291495i	$0.905847\pi$
$434$	−1.43631e22	−0.547755
$435$	0	0
$436$	−1.06654e23	−3.91148
$437$	1.40511e22i	0.505379i
$438$	5.42102e22i	1.91228i
$439$	−5.27352e22	−1.82453	−0.912267	−	0.409597i	$-0.865669\pi$
$439$	−5.27352e22	−1.82453	−0.912267	+	0.409597i	$0.865669\pi$
$440$	0	0
$441$	−9.70600e21	−0.323082
$442$	2.54096e22i	0.829676i
$443$	2.21332e22i	0.708944i	0.935067	+	0.354472i	$0.115339\pi$
$443$	2.21332e22i	0.708944i	−0.935067	+	0.354472i	$0.884661\pi$
$444$	−7.58452e21	−0.238327
$445$	0	0
$446$	−8.41239e21	−0.254433
$447$	2.64293e22i	0.784280i
$448$	− 2.45899e22i	− 0.715968i
$449$	−1.01171e22	−0.289041	−0.144521	−	0.989502i	$-0.546164\pi$
$449$	−1.01171e22	−0.289041	−0.144521	+	0.989502i	$0.546164\pi$
$450$	0	0
$451$	5.45578e21	0.150091
$452$	2.62754e22i	0.709369i
$453$	− 1.98447e22i	− 0.525787i
$454$	8.72479e22	2.26872
$455$	0	0
$456$	1.30672e23	3.27327
$457$	7.09706e22i	1.74498i	0.488632	+	0.872490i	$0.337496\pi$
$457$	7.09706e22i	1.74498i	−0.488632	+	0.872490i	$0.662504\pi$
$458$	− 7.24248e22i	− 1.74796i
$459$	−3.62924e21	−0.0859821
$460$	0	0
$461$	1.09889e22	0.250899	0.125449	−	0.992100i	$-0.459963\pi$
$461$	1.09889e22	0.250899	0.125449	+	0.992100i	$0.459963\pi$
$462$	1.21596e22i	0.272560i
$463$	1.78028e22i	0.391787i	0.980625	+	0.195893i	$0.0627606\pi$
$463$	1.78028e22i	0.391787i	−0.980625	+	0.195893i	$0.937239\pi$
$464$	3.26625e23	7.05743
$465$	0	0
$466$	−7.96310e22	−1.65883
$467$	− 7.82704e22i	− 1.60105i	−0.599301	−	0.800524i	$-0.704555\pi$
$467$	− 7.82704e22i	− 1.60105i	0.599301	−	0.800524i	$-0.295445\pi$
$468$	4.39127e22i	0.882065i
$469$	3.50804e21	0.0691983
$470$	0	0
$471$	−2.49914e22	−0.475459
$472$	− 1.69708e20i	− 0.00317100i
$473$	− 2.39122e22i	− 0.438834i
$474$	−3.29390e22	−0.593737
$475$	0	0
$476$	−1.24563e22	−0.216636
$477$	2.77973e22i	0.474893i
$478$	5.82399e22i	0.977425i
$479$	1.17366e23	1.93504	0.967522	−	0.252785i	$-0.0813467\pi$
$479$	1.17366e23	1.93504	0.967522	+	0.252785i	$0.0813467\pi$
$480$	0	0
$481$	8.97899e21	0.142888
$482$	− 1.14285e23i	− 1.78685i
$483$	− 2.01198e21i	− 0.0309081i
$484$	−1.69390e23	−2.55683
$485$	0	0
$486$	−8.54048e21	−0.124473
$487$	− 4.40441e22i	− 0.630800i	−0.948959	−	0.315400i	$-0.897861\pi$
$487$	− 4.40441e22i	− 0.630800i	0.948959	−	0.315400i	$-0.102139\pi$
$488$	2.53648e23i	3.56996i
$489$	4.60107e22	0.636405
$490$	0	0
$491$	−9.19512e21	−0.122847	−0.0614235	−	0.998112i	$-0.519564\pi$
$491$	−9.19512e21	−0.122847	−0.0614235	+	0.998112i	$0.519564\pi$
$492$	1.31520e22i	0.172699i
$493$	− 6.29689e22i	− 0.812695i
$494$	−2.42349e23	−3.07441
$495$	0	0
$496$	5.09511e23	6.24539
$497$	− 1.89705e22i	− 0.228586i
$498$	− 4.16975e22i	− 0.493924i
$499$	9.12259e22	1.06234	0.531170	−	0.847265i	$-0.321753\pi$
$499$	9.12259e22	1.06234	0.531170	+	0.847265i	$0.321753\pi$
$500$	0	0
$501$	−3.41786e22	−0.384710
$502$	2.88303e23i	3.19056i
$503$	− 4.20493e22i	− 0.457542i	−0.973480	−	0.228771i	$-0.926529\pi$
$503$	− 4.20493e22i	− 0.457542i	0.973480	−	0.228771i	$-0.0734708\pi$
$504$	−1.87110e22	−0.200187
$505$	0	0
$506$	7.94359e22	0.821746
$507$	4.76891e21i	0.0485122i
$508$	− 3.62791e23i	− 3.62924i
$509$	−3.12071e22	−0.307010	−0.153505	−	0.988148i	$-0.549056\pi$
$509$	−3.12071e22	−0.307010	−0.153505	+	0.988148i	$0.549056\pi$
$510$	0	0
$511$	3.14609e22	0.299360
$512$	2.81536e23i	2.63475i
$513$	− 3.46146e22i	− 0.318612i
$514$	−1.19128e23	−1.07852
$515$	0	0
$516$	5.76443e22	0.504933
$517$	− 5.74206e22i	− 0.494764i
$518$	5.99366e21i	0.0508030i
$519$	−2.98994e22	−0.249310
$520$	0	0
$521$	−1.71736e23	−1.38593	−0.692965	−	0.720971i	$-0.743696\pi$
$521$	−1.71736e23	−1.38593	−0.692965	+	0.720971i	$0.743696\pi$
$522$	− 1.48181e23i	− 1.17650i
$523$	− 3.54202e22i	− 0.276686i	−0.990384	−	0.138343i	$-0.955822\pi$
$523$	− 3.54202e22i	− 0.276686i	0.990384	−	0.138343i	$-0.0441776\pi$
$524$	−2.30542e23	−1.77188
$525$	0	0
$526$	2.46305e23	1.83271
$527$	− 9.82268e22i	− 0.719184i
$528$	− 4.31344e23i	− 3.10767i
$529$	1.27906e23	0.906815
$530$	0	0
$531$	−4.49551e19	−0.000308657 0
$532$	− 1.18804e23i	− 0.802755i
$533$	− 1.55701e22i	− 0.103541i
$534$	−5.38260e22	−0.352283
$535$	0	0
$536$	−2.13126e23	−1.35125
$537$	− 8.33973e22i	− 0.520440i
$538$	− 2.05716e23i	− 1.26363i
$539$	−2.22395e23	−1.34468
$540$	0	0
$541$	−1.03474e23	−0.606257	−0.303128	−	0.952950i	$-0.598031\pi$
$541$	−1.03474e23	−0.606257	−0.303128	+	0.952950i	$0.598031\pi$
$542$	3.96227e23i	2.28534i
$543$	1.10119e23i	0.625264i
$544$	3.27979e23	1.83339
$545$	0	0
$546$	3.47020e22	0.188026
$547$	− 2.96616e23i	− 1.58235i	−0.611589	−	0.791176i	$-0.709469\pi$
$547$	− 2.96616e23i	− 1.58235i	0.611589	−	0.791176i	$-0.290531\pi$
$548$	9.97992e22i	0.524194i
$549$	6.71906e22	0.347491
$550$	0	0
$551$	6.00578e23	3.01149
$552$	1.22235e23i	0.603548i
$553$	1.91162e22i	0.0929474i
$554$	5.56954e23	2.66678
$555$	0	0
$556$	1.30845e23	0.607603
$557$	5.46640e22i	0.249996i	0.992157	+	0.124998i	$0.0398924\pi$
$557$	5.46640e22i	0.249996i	−0.992157	+	0.124998i	$0.960108\pi$
$558$	− 2.31151e23i	− 1.04113i
$559$	−6.82427e22	−0.302730
$560$	0	0
$561$	−8.31572e22	−0.357862
$562$	− 2.86281e23i	− 1.21349i
$563$	− 1.18516e23i	− 0.494829i	−0.968910	−	0.247414i	$-0.920419\pi$
$563$	− 1.18516e23i	− 0.494829i	0.968910	−	0.247414i	$-0.0795809\pi$
$564$	1.38422e23	0.569287
$565$	0	0
$566$	−6.36304e23	−2.53936
$567$	4.95647e21i	0.0194857i
$568$	1.15253e24i	4.46365i
$569$	4.39785e23	1.67798	0.838988	−	0.544149i	$-0.183147\pi$
$569$	4.39785e23	1.67798	0.838988	+	0.544149i	$0.183147\pi$
$570$	0	0
$571$	−4.89721e22	−0.181360	−0.0906800	−	0.995880i	$-0.528904\pi$
$571$	−4.89721e22	−0.181360	−0.0906800	+	0.995880i	$0.528904\pi$
$572$	1.00618e24i	3.67120i
$573$	1.27026e23i	0.456644i
$574$	1.03934e22	0.0368134
$575$	0	0
$576$	3.95735e23	1.36086
$577$	− 5.15333e23i	− 1.74620i	−0.487543	−	0.873099i	$-0.662107\pi$
$577$	− 5.15333e23i	− 1.74620i	0.487543	−	0.873099i	$-0.337893\pi$
$578$	4.65121e23i	1.55303i
$579$	−3.55018e22	−0.116811
$580$	0	0
$581$	−2.41992e22	−0.0773220
$582$	− 3.83713e23i	− 1.20826i
$583$	6.36922e23i	1.97653i
$584$	−1.91136e24	−5.84565
$585$	0	0
$586$	−8.63485e23	−2.56522
$587$	3.59992e22i	0.105407i	0.998610	+	0.0527035i	$0.0167838\pi$
$587$	3.59992e22i	0.105407i	−0.998610	+	0.0527035i	$0.983216\pi$
$588$	− 5.36118e23i	− 1.54722i
$589$	9.36857e23	2.66498
$590$	0	0
$591$	1.11443e23	0.308006
$592$	− 2.12616e23i	− 0.579245i
$593$	− 3.67652e23i	− 0.987351i	−0.869646	−	0.493676i	$-0.835653\pi$
$593$	− 3.67652e23i	− 0.987351i	0.869646	−	0.493676i	$-0.164347\pi$
$594$	−1.95689e23	−0.518062
$595$	0	0
$596$	−1.45984e24	−3.75588
$597$	5.50273e22i	0.139571i
$598$	− 2.26701e23i	− 0.566882i
$599$	6.97132e23	1.71865	0.859324	−	0.511432i	$-0.170885\pi$
$599$	6.97132e23	1.71865	0.859324	+	0.511432i	$0.170885\pi$
$600$	0	0
$601$	3.44330e23	0.825166	0.412583	−	0.910920i	$-0.364627\pi$
$601$	3.44330e23	0.825166	0.412583	+	0.910920i	$0.364627\pi$
$602$	− 4.55534e22i	− 0.107634i
$603$	5.64562e22i	0.131527i
$604$	1.09614e24	2.51797
$605$	0	0
$606$	6.27786e22	0.140215
$607$	3.08234e23i	0.678855i	0.940632	+	0.339427i	$0.110233\pi$
$607$	3.08234e23i	0.678855i	−0.940632	+	0.339427i	$0.889767\pi$
$608$	3.12816e24i	6.79374i
$609$	−8.59970e22	−0.184177
$610$	0	0
$611$	−1.63872e23	−0.341313
$612$	− 2.00464e23i	− 0.411765i
$613$	− 4.35176e23i	− 0.881558i	−0.897616	−	0.440779i	$-0.854702\pi$
$613$	− 4.35176e23i	− 0.881558i	0.897616	−	0.440779i	$-0.145298\pi$
$614$	−9.39699e23	−1.87740
$615$	0	0
$616$	−4.28727e23	−0.833190
$617$	− 6.04231e23i	− 1.15819i	−0.815261	−	0.579094i	$-0.803406\pi$
$617$	− 6.04231e23i	− 1.15819i	0.815261	−	0.579094i	$-0.196594\pi$
$618$	− 5.48602e23i	− 1.03718i
$619$	3.33503e23	0.621913	0.310956	−	0.950424i	$-0.399351\pi$
$619$	3.33503e23	0.621913	0.310956	+	0.950424i	$0.399351\pi$
$620$	0	0
$621$	3.23795e22	0.0587479
$622$	− 5.80637e22i	− 0.103917i
$623$	3.12379e22i	0.0551485i
$624$	−1.23100e24	−2.14383
$625$	0	0
$626$	2.02534e23	0.343255
$627$	− 7.93128e23i	− 1.32608i
$628$	− 1.38042e24i	− 2.27695i
$629$	−4.09896e22	−0.0667026
$630$	0	0
$631$	7.67831e23	1.21623	0.608116	−	0.793848i	$-0.291926\pi$
$631$	7.67831e23	1.21623	0.608116	+	0.793848i	$0.291926\pi$
$632$	− 1.16137e24i	− 1.81500i
$633$	1.71206e23i	0.263991i
$634$	−2.22799e24	−3.38964
$635$	0	0
$636$	−1.53540e24	−2.27424
$637$	6.34688e23i	0.927631i
$638$	− 3.39529e24i	− 4.89667i
$639$	3.05300e23	0.434480
$640$	0	0
$641$	−1.34739e24	−1.86724	−0.933620	−	0.358264i	$-0.883369\pi$
$641$	−1.34739e24	−1.86724	−0.933620	+	0.358264i	$0.883369\pi$
$642$	1.40746e24i	1.92481i
$643$	1.07291e24i	1.44800i	0.689801	+	0.723999i	$0.257698\pi$
$643$	1.07291e24i	1.44800i	−0.689801	+	0.723999i	$0.742302\pi$
$644$	1.11133e23	0.148018
$645$	0	0
$646$	1.10634e24	1.43519
$647$	− 2.56767e23i	− 0.328740i	−0.986399	−	0.164370i	$-0.947441\pi$
$647$	− 2.56767e23i	− 0.328740i	0.986399	−	0.164370i	$-0.0525591\pi$
$648$	− 3.01123e23i	− 0.380501i
$649$	−1.03006e21	−0.00128465
$650$	0	0
$651$	−1.34149e23	−0.162985
$652$	2.54144e24i	3.04772i
$653$	9.55573e23i	1.13110i	0.824713	+	0.565551i	$0.191336\pi$
$653$	9.55573e23i	1.13110i	−0.824713	+	0.565551i	$0.808664\pi$
$654$	−1.35640e24	−1.58481
$655$	0	0
$656$	−3.68690e23	−0.419739
$657$	5.06312e23i	0.569001i
$658$	− 1.09388e23i	− 0.121352i
$659$	4.48249e23	0.490901	0.245450	−	0.969409i	$-0.421064\pi$
$659$	4.48249e23	0.490901	0.245450	+	0.969409i	$0.421064\pi$
$660$	0	0
$661$	3.31071e23	0.353353	0.176677	−	0.984269i	$-0.443465\pi$
$661$	3.31071e23	0.353353	0.176677	+	0.984269i	$0.443465\pi$
$662$	− 1.11682e24i	− 1.17677i
$663$	2.37321e23i	0.246872i
$664$	1.47018e24	1.50988
$665$	0	0
$666$	−9.64583e22	−0.0965625
$667$	5.61799e23i	0.555279i
$668$	− 1.88788e24i	− 1.84236i
$669$	−7.85701e22	−0.0757069
$670$	0	0
$671$	1.53955e24	1.44628
$672$	− 4.47923e23i	− 0.415494i
$673$	− 9.61434e23i	− 0.880626i	−0.897844	−	0.440313i	$-0.854868\pi$
$673$	− 9.61434e23i	− 0.880626i	0.897844	−	0.440313i	$-0.145132\pi$
$674$	−2.11833e23	−0.191595
$675$	0	0
$676$	−2.63414e23	−0.232323
$677$	− 1.08092e24i	− 0.941432i	−0.882285	−	0.470716i	$-0.843996\pi$
$677$	− 1.08092e24i	− 0.941432i	0.882285	−	0.470716i	$-0.156004\pi$
$678$	3.34165e23i	0.287414i
$679$	−2.22688e23	−0.189149
$680$	0	0
$681$	8.14879e23	0.675060
$682$	− 5.29640e24i	− 4.33325i
$683$	4.56949e23i	0.369226i	0.982811	+	0.184613i	$0.0591032\pi$
$683$	4.56949e23i	0.369226i	−0.982811	+	0.184613i	$0.940897\pi$
$684$	1.91196e24	1.52582
$685$	0	0
$686$	−8.60778e23	−0.670095
$687$	− 6.76434e23i	− 0.520107i
$688$	1.61594e24i	1.22722i
$689$	1.81770e24	1.36351
$690$	0	0
$691$	−1.38418e24	−1.01305	−0.506523	−	0.862226i	$-0.669070\pi$
$691$	−1.38418e24	−1.01305	−0.506523	+	0.862226i	$0.669070\pi$
$692$	− 1.65152e24i	− 1.19394i
$693$	1.13568e23i	0.0811007i
$694$	−2.68278e24	−1.89247
$695$	0	0
$696$	5.22461e24	3.59646
$697$	7.10785e22i	0.0483348i
$698$	2.20314e23i	0.148003i
$699$	−7.43739e23	−0.493588
$700$	0	0
$701$	9.18978e22	0.0595254	0.0297627	−	0.999557i	$-0.490525\pi$
$701$	9.18978e22	0.0595254	0.0297627	+	0.999557i	$0.490525\pi$
$702$	5.58473e23i	0.357385i
$703$	− 3.90946e23i	− 0.247170i
$704$	9.06751e24	5.66396
$705$	0	0
$706$	3.27063e24	1.99431
$707$	− 3.64336e22i	− 0.0219501i
$708$	− 2.48313e21i	− 0.00147814i
$709$	1.54021e24	0.905913	0.452956	−	0.891533i	$-0.350369\pi$
$709$	1.54021e24	0.905913	0.452956	+	0.891533i	$0.350369\pi$
$710$	0	0
$711$	−3.07644e23	−0.176668
$712$	− 1.89781e24i	− 1.07690i
$713$	8.76365e23i	0.491387i
$714$	−1.58417e23	−0.0877740
$715$	0	0
$716$	4.60651e24	2.49236
$717$	5.43949e23i	0.290834i
$718$	− 2.51332e24i	− 1.32798i
$719$	−3.23522e24	−1.68931	−0.844654	−	0.535313i	$-0.820194\pi$
$719$	−3.23522e24	−1.68931	−0.844654	+	0.535313i	$0.820194\pi$
$720$	0	0
$721$	−3.18382e23	−0.162367
$722$	6.70204e24i	3.37786i
$723$	− 1.06740e24i	− 0.531679i
$724$	−6.08250e24	−2.99436
$725$	0	0
$726$	−2.15426e24	−1.03595
$727$	2.33512e24i	1.10986i	0.831898	+	0.554929i	$0.187255\pi$
$727$	2.33512e24i	1.10986i	−0.831898	+	0.554929i	$0.812745\pi$
$728$	1.22354e24i	0.574777i
$729$	−7.97664e22	−0.0370370
$730$	0	0
$731$	3.11531e23	0.141320
$732$	3.71132e24i	1.66412i
$733$	1.81465e24i	0.804282i	0.915578	+	0.402141i	$0.131734\pi$
$733$	1.81465e24i	0.804282i	−0.915578	+	0.402141i	$0.868266\pi$
$734$	−4.02061e24	−1.76147
$735$	0	0
$736$	−2.92618e24	−1.25268
$737$	1.29359e24i	0.547422i
$738$	1.67265e23i	0.0699722i
$739$	1.21901e24	0.504114	0.252057	−	0.967712i	$-0.418893\pi$
$739$	1.21901e24	0.504114	0.252057	+	0.967712i	$0.418893\pi$
$740$	0	0
$741$	−2.26349e24	−0.914797
$742$	1.21335e24i	0.484790i
$743$	4.70556e23i	0.185869i	0.995672	+	0.0929344i	$0.0296247\pi$
$743$	4.70556e23i	0.185869i	−0.995672	+	0.0929344i	$0.970375\pi$
$744$	8.15001e24	3.18264
$745$	0	0
$746$	−5.47458e24	−2.08964
$747$	− 3.89447e23i	− 0.146968i
$748$	− 4.59325e24i	− 1.71379i
$749$	8.16820e23	0.301322
$750$	0	0
$751$	−3.20027e24	−1.15411	−0.577056	−	0.816705i	$-0.695798\pi$
$751$	−3.20027e24	−1.15411	−0.577056	+	0.816705i	$0.695798\pi$
$752$	3.88036e24i	1.38363i
$753$	2.69269e24i	0.949356i
$754$	−9.68975e24	−3.37797
$755$	0	0
$756$	−2.73775e23	−0.0933164
$757$	5.33142e24i	1.79692i	0.439060	+	0.898458i	$0.355312\pi$
$757$	5.33142e24i	1.79692i	−0.439060	+	0.898458i	$0.644688\pi$
$758$	− 3.09984e24i	− 1.03312i
$759$	7.41916e23	0.244512
$760$	0	0
$761$	3.19775e24	1.03057	0.515283	−	0.857020i	$-0.327687\pi$
$761$	3.19775e24	1.03057	0.515283	+	0.857020i	$0.327687\pi$
$762$	− 4.61390e24i	− 1.47045i
$763$	7.87188e23i	0.248096i
$764$	−7.01636e24	−2.18685
$765$	0	0
$766$	−5.21304e24	−1.58908
$767$	2.93967e21i	0 0.000886214i
$768$	6.43984e24i	1.92001i
$769$	3.24975e24	0.958245	0.479122	−	0.877748i	$-0.340955\pi$
$769$	3.24975e24	0.958245	0.479122	+	0.877748i	$0.340955\pi$
$770$	0	0
$771$	−1.11263e24	−0.320914
$772$	− 1.96097e24i	− 0.559401i
$773$	4.35391e24i	1.22844i	0.789135	+	0.614220i	$0.210529\pi$
$773$	4.35391e24i	1.22844i	−0.789135	+	0.614220i	$0.789471\pi$
$774$	7.33108e23	0.204583
$775$	0	0
$776$	1.35291e25	3.69354
$777$	5.59797e22i	0.0151165i
$778$	− 3.12586e24i	− 0.834917i
$779$	−6.77925e23	−0.179107
$780$	0	0
$781$	6.99538e24	1.80833
$782$	1.03490e24i	0.264631i
$783$	− 1.38398e24i	− 0.350070i
$784$	1.50290e25	3.76048
$785$	0	0
$786$	−2.93199e24	−0.717910
$787$	− 1.67282e24i	− 0.405194i	−0.979262	−	0.202597i	$-0.935062\pi$
$787$	− 1.67282e24i	− 0.405194i	0.979262	−	0.202597i	$-0.0649381\pi$
$788$	6.15565e24i	1.47503i
$789$	2.30044e24	0.545327
$790$	0	0
$791$	1.93933e23	0.0449936
$792$	− 6.89966e24i	− 1.58367i
$793$	− 4.39368e24i	− 0.997715i
$794$	−3.47180e23	−0.0779974
$795$	0	0
$796$	−3.03948e24	−0.668402
$797$	− 9.07285e23i	− 0.197400i	−0.995117	−	0.0987002i	$-0.968532\pi$
$797$	− 9.07285e23i	− 0.197400i	0.995117	−	0.0987002i	$-0.0314685\pi$
$798$	− 1.51093e24i	− 0.325251i
$799$	7.48082e23	0.159331
$800$	0	0
$801$	−5.02724e23	−0.104822
$802$	1.75156e25i	3.61363i
$803$	1.16012e25i	2.36821i
$804$	−3.11840e24	−0.629877
$805$	0	0
$806$	−1.51153e25	−2.98929
$807$	− 1.92135e24i	− 0.375994i
$808$	2.21347e24i	0.428624i
$809$	9.92795e23	0.190238	0.0951189	−	0.995466i	$-0.469677\pi$
$809$	9.92795e23	0.190238	0.0951189	+	0.995466i	$0.469677\pi$
$810$	0	0
$811$	5.63576e24	1.05749	0.528744	−	0.848781i	$-0.322663\pi$
$811$	5.63576e24	1.05749	0.528744	+	0.848781i	$0.322663\pi$
$812$	− 4.75011e24i	− 0.882017i
$813$	3.70068e24i	0.680006i
$814$	−2.21016e24	−0.401898
$815$	0	0
$816$	5.61959e24	1.00078
$817$	2.97129e24i	0.523670i
$818$	− 5.49999e24i	− 0.959310i
$819$	3.24110e23	0.0559474
$820$	0	0
$821$	−9.18781e24	−1.55344	−0.776721	−	0.629845i	$-0.783118\pi$
$821$	−9.18781e24	−1.55344	−0.776721	+	0.629845i	$0.783118\pi$
$822$	1.26923e24i	0.212387i
$823$	6.83078e24i	1.13128i	0.824651	+	0.565642i	$0.191372\pi$
$823$	6.83078e24i	1.13128i	−0.824651	+	0.565642i	$0.808628\pi$
$824$	1.93428e25	3.17057
$825$	0	0
$826$	−1.96229e21	−0.000315089 0
$827$	9.93808e24i	1.57945i	0.613462	+	0.789724i	$0.289776\pi$
$827$	9.93808e24i	1.57945i	−0.613462	+	0.789724i	$0.710224\pi$
$828$	1.78851e24i	0.281341i
$829$	−1.00372e25	−1.56278	−0.781391	−	0.624042i	$-0.785489\pi$
$829$	−1.00372e25	−1.56278	−0.781391	+	0.624042i	$0.785489\pi$
$830$	0	0
$831$	5.20185e24	0.793503
$832$	− 2.58776e25i	− 3.90729i
$833$	− 2.89738e24i	− 0.433035i
$834$	1.66405e24	0.246182
$835$	0	0
$836$	4.38090e25	6.35053
$837$	− 2.15891e24i	− 0.309790i
$838$	9.64602e22i	0.0137017i
$839$	−1.16269e25	−1.63489	−0.817446	−	0.576005i	$-0.804611\pi$
$839$	−1.16269e25	−1.63489	−0.817446	+	0.576005i	$0.804611\pi$
$840$	0	0
$841$	1.67555e25	2.30883
$842$	− 1.29367e24i	− 0.176470i
$843$	− 2.67381e24i	− 0.361074i
$844$	−9.45671e24	−1.26424
$845$	0	0
$846$	1.76042e24	0.230657
$847$	1.25023e24i	0.162174i
$848$	− 4.30419e25i	− 5.52747i
$849$	−5.94296e24	−0.755592
$850$	0	0
$851$	3.65703e23	0.0455750
$852$	1.68635e25i	2.08071i
$853$	1.09678e23i	0.0133984i	0.999978	+	0.00669922i	$0.00213244\pi$
$853$	1.09678e23i	0.0133984i	−0.999978	+	0.00669922i	$0.997868\pi$
$854$	2.93287e24	0.354733
$855$	0	0
$856$	−4.96246e25	−5.88397
$857$	− 1.52818e25i	− 1.79406i	−0.441970	−	0.897030i	$-0.645720\pi$
$857$	− 1.52818e25i	− 1.79406i	0.441970	−	0.897030i	$-0.354280\pi$
$858$	1.27964e25i	1.48746i
$859$	1.62639e25	1.87190	0.935951	−	0.352130i	$-0.114543\pi$
$859$	1.62639e25	1.87190	0.935951	+	0.352130i	$0.114543\pi$
$860$	0	0
$861$	9.70723e22	0.0109539
$862$	− 2.32266e24i	− 0.259522i
$863$	− 8.64833e24i	− 0.956842i	−0.878130	−	0.478421i	$-0.841209\pi$
$863$	− 8.64833e24i	− 0.956842i	0.878130	−	0.478421i	$-0.158791\pi$
$864$	7.20859e24	0.789739
$865$	0	0
$866$	−1.05303e25	−1.13120
$867$	4.34414e24i	0.462106i
$868$	− 7.40981e24i	− 0.780530i
$869$	−7.04908e24	−0.735299
$870$	0	0
$871$	3.69175e24	0.377640
$872$	− 4.78244e25i	− 4.84462i
$873$	− 3.58381e24i	− 0.359520i
$874$	−9.87056e24	−0.980606
$875$	0	0
$876$	−2.79665e25	−2.72492
$877$	− 1.02051e25i	− 0.984740i	−0.870386	−	0.492370i	$-0.836131\pi$
$877$	− 1.02051e25i	− 0.984740i	0.870386	−	0.492370i	$-0.163869\pi$
$878$	− 3.70452e25i	− 3.54021i
$879$	−8.06479e24	−0.763286
$880$	0	0
$881$	−9.18419e23	−0.0852601	−0.0426301	−	0.999091i	$-0.513574\pi$
$881$	−9.18419e23	−0.0852601	−0.0426301	+	0.999091i	$0.513574\pi$
$882$	− 6.81824e24i	− 0.626887i
$883$	8.44090e24i	0.768640i	0.923200	+	0.384320i	$0.125564\pi$
$883$	8.44090e24i	0.768640i	−0.923200	+	0.384320i	$0.874436\pi$
$884$	−1.31086e25	−1.18226
$885$	0	0
$886$	−1.55480e25	−1.37559
$887$	1.20407e25i	1.05512i	0.849519	+	0.527558i	$0.176892\pi$
$887$	1.20407e25i	1.05512i	−0.849519	+	0.527558i	$0.823108\pi$
$888$	− 3.40096e24i	− 0.295183i
$889$	−2.67768e24	−0.230194
$890$	0	0
$891$	−1.82770e24	−0.154150
$892$	− 4.33988e24i	− 0.362557i
$893$	7.13498e24i	0.590412i
$894$	−1.85660e25	−1.52177
$895$	0	0
$896$	8.32548e24	0.669563
$897$	− 2.11734e24i	− 0.168677i
$898$	− 7.10700e24i	− 0.560838i
$899$	3.74580e25	2.92811
$900$	0	0
$901$	−8.29789e24	−0.636512
$902$	3.83256e24i	0.291228i
$903$	− 4.25460e23i	− 0.0320267i
$904$	−1.17821e25	−0.878599
$905$	0	0
$906$	1.39404e25	1.02020
$907$	2.51467e25i	1.82314i	0.411150	+	0.911568i	$0.365127\pi$
$907$	2.51467e25i	1.82314i	−0.411150	+	0.911568i	$0.634873\pi$
$908$	4.50104e25i	3.23284i
$909$	5.86340e23	0.0417212
$910$	0	0
$911$	−8.57548e24	−0.598897	−0.299448	−	0.954112i	$-0.596803\pi$
$911$	−8.57548e24	−0.598897	−0.299448	+	0.954112i	$0.596803\pi$
$912$	5.35980e25i	3.70845i
$913$	− 8.92343e24i	− 0.611688i
$914$	−4.98552e25	−3.38585
$915$	0	0
$916$	3.73633e25	2.49077
$917$	1.70158e24i	0.112386i
$918$	− 2.54946e24i	− 0.166834i
$919$	−1.26569e25	−0.820626	−0.410313	−	0.911945i	$-0.634581\pi$
$919$	−1.26569e25	−0.820626	−0.410313	+	0.911945i	$0.634581\pi$
$920$	0	0
$921$	−8.77661e24	−0.558624
$922$	7.71948e24i	0.486828i
$923$	− 1.99640e25i	− 1.24748i
$924$	−6.27303e24	−0.388388
$925$	0	0
$926$	−1.25060e25	−0.760198
$927$	− 5.12384e24i	− 0.308616i
$928$	1.25072e26i	7.46454i
$929$	1.61298e25	0.953881	0.476941	−	0.878936i	$-0.341746\pi$
$929$	1.61298e25	0.953881	0.476941	+	0.878936i	$0.341746\pi$
$930$	0	0
$931$	2.76343e25	1.60464
$932$	− 4.10810e25i	− 2.36377i
$933$	− 5.42304e23i	− 0.0309206i
$934$	5.49832e25	3.10657
$935$	0	0
$936$	−1.96908e25	−1.09249
$937$	− 2.27726e24i	− 0.125206i	−0.998038	−	0.0626032i	$-0.980060\pi$
$937$	− 2.27726e24i	− 0.125206i	0.998038	−	0.0626032i	$-0.0199402\pi$
$938$	2.46432e24i	0.134268i
$939$	1.89163e24	0.102136
$940$	0	0
$941$	−1.42919e25	−0.757839	−0.378919	−	0.925430i	$-0.623704\pi$
$941$	−1.42919e25	−0.757839	−0.378919	+	0.925430i	$0.623704\pi$
$942$	− 1.75559e25i	− 0.922551i
$943$	− 6.34151e23i	− 0.0330251i
$944$	6.96095e22	0.00359258
$945$	0	0
$946$	1.67978e25	0.851486
$947$	1.82863e24i	0.0918653i	0.998945	+	0.0459326i	$0.0146260\pi$
$947$	1.82863e24i	0.0918653i	−0.998945	+	0.0459326i	$0.985374\pi$
$948$	− 1.69929e25i	− 0.846053i
$949$	3.31084e25	1.63371
$950$	0	0
$951$	−2.08090e25	−1.00859
$952$	− 5.58550e24i	− 0.268317i
$953$	1.66599e25i	0.793201i	0.917991	+	0.396601i	$0.129810\pi$
$953$	1.66599e25i	0.793201i	−0.917991	+	0.396601i	$0.870190\pi$
$954$	−1.95269e25	−0.921452
$955$	0	0
$956$	−3.00455e25	−1.39279
$957$	− 3.17113e25i	− 1.45701i
$958$	8.24470e25i	3.75464i
$959$	7.36596e23	0.0332484
$960$	0	0
$961$	3.58816e25	1.59119
$962$	6.30753e24i	0.277250i
$963$	1.31454e25i	0.572731i
$964$	5.89584e25	2.54619
$965$	0	0
$966$	1.41337e24	0.0599722
$967$	3.59794e25i	1.51331i	0.653812	+	0.756657i	$0.273169\pi$
$967$	3.59794e25i	1.51331i	−0.653812	+	0.756657i	$0.726831\pi$
$968$	− 7.59556e25i	− 3.16679i
$969$	1.03330e25	0.427044
$970$	0	0
$971$	−2.68946e25	−1.09220	−0.546099	−	0.837721i	$-0.683888\pi$
$971$	−2.68946e25	−1.09220	−0.546099	+	0.837721i	$0.683888\pi$
$972$	− 4.40596e24i	− 0.177369i
$973$	− 9.65735e23i	− 0.0385389i
$974$	3.09400e25	1.22396
$975$	0	0
$976$	−1.04039e26	−4.04459
$977$	− 1.26185e25i	− 0.486301i	−0.969989	−	0.243151i	$-0.921819\pi$
$977$	− 1.26185e25i	− 0.486301i	0.969989	−	0.243151i	$-0.0781809\pi$
$978$	3.23215e25i	1.23484i
$979$	−1.15190e25	−0.436276
$980$	0	0
$981$	−1.26685e25	−0.471563
$982$	− 6.45936e24i	− 0.238365i
$983$	− 2.72481e25i	− 0.996852i	−0.866932	−	0.498426i	$-0.833912\pi$
$983$	− 2.72481e25i	− 0.996852i	0.866932	−	0.498426i	$-0.166088\pi$
$984$	−5.89747e24	−0.213898
$985$	0	0
$986$	4.42342e25	1.57690
$987$	− 1.02166e24i	− 0.0361086i
$988$	− 1.25026e26i	− 4.38092i
$989$	−2.77944e24	−0.0965580
$990$	0	0
$991$	−2.34443e25	−0.800593	−0.400296	−	0.916386i	$-0.631093\pi$
$991$	−2.34443e25	−0.800593	−0.400296	+	0.916386i	$0.631093\pi$
$992$	1.95103e26i	6.60565i
$993$	− 1.04309e25i	− 0.350150i
$994$	1.33264e25	0.443535
$995$	0	0
$996$	2.15114e25	0.703823
$997$	− 5.52750e25i	− 1.79316i	−0.442877	−	0.896582i	$-0.646042\pi$
$997$	− 5.52750e25i	− 1.79316i	0.442877	−	0.896582i	$-0.353958\pi$
$998$	6.40841e25i	2.06130i
$999$	−9.00902e23	−0.0287323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.18.b.b.49.4		4
5.2	odd	4		15.18.a.a.1.1	✓	2
5.3	odd	4		75.18.a.c.1.2		2
5.4	even	2	inner	75.18.b.b.49.1		4
15.2	even	4		45.18.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.a.1.1	✓	2	5.2	odd	4
45.18.a.b.1.2		2	15.2	even	4
75.18.a.c.1.2		2	5.3	odd	4
75.18.b.b.49.1		4	5.4	even	2	inner
75.18.b.b.49.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+702.477i q^{2} +6561.00i q^{3} -362402. q^{4} -4.60895e6 q^{6} +2.67481e6i q^{7} -1.62504e8i q^{8} -4.30467e7 q^{9} -9.86334e8 q^{11} -2.37772e9i q^{12} +2.81488e9i q^{13} -1.87899e9 q^{14} +6.66544e10 q^{16} -1.28501e10i q^{17} -3.02393e10i q^{18} +1.22560e11 q^{19} -1.75494e10 q^{21} -6.92877e11i q^{22} +1.14646e11i q^{23} +1.06619e12 q^{24} -1.97739e12 q^{26} -2.82430e11i q^{27} -9.69355e11i q^{28} +4.90027e12 q^{29} +7.64406e12 q^{31} +2.55235e13i q^{32} -6.47134e12i q^{33} +9.02688e12 q^{34} +1.56002e13 q^{36} -3.18983e12i q^{37} +8.60956e13i q^{38} -1.84684e13 q^{39} -5.53137e12 q^{41} -1.23281e13i q^{42} +2.42435e13i q^{43} +3.57449e14 q^{44} -8.05365e13 q^{46} +5.82161e13i q^{47} +4.37320e14i q^{48} +2.25476e14 q^{49} +8.43094e13 q^{51} -1.02012e15i q^{52} -6.45746e14i q^{53} +1.98400e14 q^{54} +4.34667e14 q^{56} +8.04117e14i q^{57} +3.44233e15i q^{58} +1.04433e12 q^{59} -1.56088e15 q^{61} +5.36978e15i q^{62} -1.15142e14i q^{63} -9.19314e15 q^{64} +4.54597e15 q^{66} -1.31151e15i q^{67} +4.65689e15i q^{68} -7.52195e14 q^{69} -7.09230e15 q^{71} +6.99526e15i q^{72} -1.17619e16i q^{73} +2.24078e15 q^{74} -4.44160e16 q^{76} -2.63826e15i q^{77} -1.29737e16i q^{78} +7.14674e15 q^{79} +1.85302e15 q^{81} -3.88566e15i q^{82} +9.04707e15i q^{83} +6.35994e15 q^{84} -1.70305e16 q^{86} +3.21507e16i q^{87} +1.60283e17i q^{88} +1.16786e16 q^{89} -7.52927e15 q^{91} -4.15481e16i q^{92} +5.01527e16i q^{93} -4.08955e16 q^{94} -1.67460e17 q^{96} +8.32538e16i q^{97} +1.58392e17i q^{98} +4.24585e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 702752 q^{4} - 4671432 q^{6} - 172186884 q^{9} - 2263259824 q^{11} + 8770444032 q^{14} + 102082634240 q^{16} + 388752254432 q^{19} - 272341231344 q^{21} + 2778418267776 q^{24} - 6214885009072 q^{26}+ \cdots + 97\!\cdots\!04 q^{99}+O(q^{100}) \) 4 * q - 702752 * q^4 - 4671432 * q^6 - 172186884 * q^9 - 2263259824 * q^11 + 8770444032 * q^14 + 102082634240 * q^16 + 388752254432 * q^19 - 272341231344 * q^21 + 2778418267776 * q^24 - 6214885009072 * q^26 + 9474125251784 * q^29 + 4512814149744 * q^31 + 12640450603856 * q^34 + 30251169276192 * q^36 + 5861236859928 * q^39 - 26103091718456 * q^41 + 711694545463424 * q^44 - 45232853381376 * q^46 + 262458605334940 * q^49 + 271127040034392 * q^51 + 201089829974472 * q^54 + 2649594618961920 * q^56 - 1926784846232912 * q^59 - 5717586429945320 * q^61 - 23202311714971648 * q^64 + 8431349467781664 * q^66 - 3697976988787248 * q^69 + 4413073310120608 * q^71 + 30034887627397232 * q^74 - 87248335332469888 * q^76 - 9995782195868880 * q^79 + 7412080755407364 * q^81 + 10104100651911168 * q^84 - 73832062872187744 * q^86 + 121714515304433592 * q^89 + 102877589806297824 * q^91 - 180547794550988992 * q^94 - 321225945838331904 * q^96 + 97425914194237104 * q^99

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