LMFDB - Embedded newform 75.18.b.e.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 = [6,0,0,14174] 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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 364793x^{4} + 33276143056x^{2} + 15375182054400 $ x^6 + 364793x^4 + 33276143056x^2 + 15375182054400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$21.5502i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.18.b.e.49.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+105.550i q^{2} +6561.00i q^{3} +119931. q^{4} -692515. q^{6} +2.39385e7i q^{7} +2.64934e7i q^{8} -4.30467e7 q^{9} -5.09412e8 q^{11} +7.86868e8i q^{12} +4.71682e9i q^{13} -2.52671e9 q^{14} +1.29232e10 q^{16} +4.44993e10i q^{17} -4.54359e9i q^{18} +4.25101e10 q^{19} -1.57060e11 q^{21} -5.37686e10i q^{22} +4.70013e11i q^{23} -1.73823e11 q^{24} -4.97862e11 q^{26} -2.82430e11i q^{27} +2.87097e12i q^{28} -3.15724e12 q^{29} +3.46790e12 q^{31} +4.83660e12i q^{32} -3.34226e12i q^{33} -4.69691e12 q^{34} -5.16264e12 q^{36} -3.18740e13i q^{37} +4.48695e12i q^{38} -3.09471e13 q^{39} +8.13537e13 q^{41} -1.65778e13i q^{42} -1.24496e14i q^{43} -6.10944e13 q^{44} -4.96099e13 q^{46} +7.05130e13i q^{47} +8.47893e13i q^{48} -3.40421e14 q^{49} -2.91960e14 q^{51} +5.65694e14i q^{52} +2.51778e14i q^{53} +2.98105e13 q^{54} -6.34213e14 q^{56} +2.78909e14i q^{57} -3.33247e14i q^{58} +6.39615e14 q^{59} -4.96148e13 q^{61} +3.66037e14i q^{62} -1.03047e15i q^{63} +1.18337e15 q^{64} +3.52776e14 q^{66} +2.75976e15i q^{67} +5.33686e15i q^{68} -3.08375e15 q^{69} +2.50612e15 q^{71} -1.14046e15i q^{72} +8.85474e14i q^{73} +3.36430e15 q^{74} +5.09829e15 q^{76} -1.21946e16i q^{77} -3.26647e15i q^{78} +6.85544e13 q^{79} +1.85302e15 q^{81} +8.58690e15i q^{82} +3.38359e16i q^{83} -1.88364e16 q^{84} +1.31405e16 q^{86} -2.07146e16i q^{87} -1.34961e16i q^{88} +3.37871e16 q^{89} -1.12914e17 q^{91} +5.63692e16i q^{92} +2.27529e16i q^{93} -7.44266e15 q^{94} -3.17329e16 q^{96} -6.48330e16i q^{97} -3.59315e16i q^{98} +2.19285e16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14174 q^{4} - 3319866 q^{6} - 258280326 q^{9} + 1887127360 q^{11} - 3694967976 q^{14} - 43887742718 q^{16} + 156244985992 q^{19} - 56850855048 q^{21} + 212425667586 q^{24} + 5331480861772 q^{26} - 11550537176156 q^{29}+ \cdots - 81\!\cdots\!60 q^{99}+O(q^{100}) $ 6 * q + 14174 * q^4 - 3319866 * q^6 - 258280326 * q^9 + 1887127360 * q^11 - 3694967976 * q^14 - 43887742718 * q^16 + 156244985992 * q^19 - 56850855048 * q^21 + 212425667586 * q^24 + 5331480861772 * q^26 - 11550537176156 * q^29 + 11130926298208 * q^31 - 43987108720612 * q^34 - 610144223454 * q^36 - 55860526554300 * q^39 + 334922914576508 * q^41 - 216237355136384 * q^44 - 389063211759552 * q^46 - 137940665027894 * q^49 + 40194910362708 * q^51 + 142909345459386 * q^54 - 2202986365674360 * q^56 - 931203894392512 * q^59 - 4635619353020956 * q^61 + 1326983159085438 * q^64 + 2410726615076736 * q^66 - 3074592254042736 * q^69 + 12931216967981312 * q^71 - 20946687257609036 * q^74 - 336922991545608 * q^76 - 17481017881317760 * q^79 + 11118121133111046 * q^81 - 50706088359713784 * q^84 - 28915112504777240 * q^86 + 34435510717452852 * q^89 - 221670523226066512 * q^91 + 401443493056274240 * q^94 + 56516732556322302 * q^96 - 81234644957386560 * 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 $

$2$	105.550i	0.291544i	0.989318	+	0.145772i	$0.0465666\pi$
$2$	105.550i	0.291544i	−0.989318	+	0.145772i	$0.953433\pi$
$3$	6561.00i	0.577350i
$3$	6561.00i	0.577350i
$4$	119931.	0.915002
$5$	0	0
$5$	0	0
$6$	−692515.	−0.168323
$7$	2.39385e7i	1.56951i	0.619807	+	0.784754i	$0.287211\pi$
$7$	2.39385e7i	1.56951i	−0.619807	+	0.784754i	$0.712789\pi$
$8$	2.64934e7i	0.558307i
$9$	−4.30467e7	−0.333333
$10$	0	0
$11$	−5.09412e8	−0.716526	−0.358263	−	0.933621i	$-0.616631\pi$
$11$	−5.09412e8	−0.716526	−0.358263	+	0.933621i	$0.616631\pi$
$12$	7.86868e8i	0.528277i
$13$	4.71682e9i	1.60373i	0.597506	+	0.801865i	$0.296159\pi$
$13$	4.71682e9i	1.60373i	−0.597506	+	0.801865i	$0.703841\pi$
$14$	−2.52671e9	−0.457580
$15$	0	0
$16$	1.29232e10	0.752231
$17$	4.44993e10i	1.54717i	0.633693	+	0.773584i	$0.281538\pi$
$17$	4.44993e10i	1.54717i	−0.633693	+	0.773584i	$0.718462\pi$
$18$	− 4.54359e9i	− 0.0971813i
$19$	4.25101e10	0.574231	0.287116	−	0.957896i	$-0.407304\pi$
$19$	4.25101e10	0.574231	0.287116	+	0.957896i	$0.407304\pi$
$20$	0	0
$21$	−1.57060e11	−0.906156
$22$	− 5.37686e10i	− 0.208899i
$23$	4.70013e11i	1.25148i	0.780033	+	0.625739i	$0.215202\pi$
$23$	4.70013e11i	1.25148i	−0.780033	+	0.625739i	$0.784798\pi$
$24$	−1.73823e11	−0.322339
$25$	0	0
$26$	−4.97862e11	−0.467558
$27$	− 2.82430e11i	− 0.192450i
$28$	2.87097e12i	1.43610i
$29$	−3.15724e12	−1.17199	−0.585996	−	0.810314i	$-0.699296\pi$
$29$	−3.15724e12	−1.17199	−0.585996	+	0.810314i	$0.699296\pi$
$30$	0	0
$31$	3.46790e12	0.730285	0.365142	−	0.930952i	$-0.381020\pi$
$31$	3.46790e12	0.730285	0.365142	+	0.930952i	$0.381020\pi$
$32$	4.83660e12i	0.777616i
$33$	− 3.34226e12i	− 0.413686i
$34$	−4.69691e12	−0.451068
$35$	0	0
$36$	−5.16264e12	−0.305001
$37$	− 3.18740e13i	− 1.49184i	−0.666037	−	0.745919i	$-0.732011\pi$
$37$	− 3.18740e13i	− 1.49184i	0.666037	−	0.745919i	$-0.267989\pi$
$38$	4.48695e12i	0.167414i
$39$	−3.09471e13	−0.925914
$40$	0	0
$41$	8.13537e13	1.59116	0.795581	−	0.605847i	$-0.207166\pi$
$41$	8.13537e13	1.59116	0.795581	+	0.605847i	$0.207166\pi$
$42$	− 1.65778e13i	− 0.264184i
$43$	− 1.24496e14i	− 1.62432i	−0.583433	−	0.812161i	$-0.698291\pi$
$43$	− 1.24496e14i	− 1.62432i	0.583433	−	0.812161i	$-0.301709\pi$
$44$	−6.10944e13	−0.655623
$45$	0	0
$46$	−4.96099e13	−0.364861
$47$	7.05130e13i	0.431954i	0.976398	+	0.215977i	$0.0692936\pi$
$47$	7.05130e13i	0.431954i	−0.976398	+	0.215977i	$0.930706\pi$
$48$	8.47893e13i	0.434301i
$49$	−3.40421e14	−1.46336
$50$	0	0
$51$	−2.91960e14	−0.893258
$52$	5.65694e14i	1.46742i
$53$	2.51778e14i	0.555485i	0.960656	+	0.277743i	$0.0895862\pi$
$53$	2.51778e14i	0.555485i	−0.960656	+	0.277743i	$0.910414\pi$
$54$	2.98105e13	0.0561076
$55$	0	0
$56$	−6.34213e14	−0.876268
$57$	2.78909e14i	0.331533i
$58$	− 3.33247e14i	− 0.341687i
$59$	6.39615e14	0.567122	0.283561	−	0.958954i	$-0.408484\pi$
$59$	6.39615e14	0.567122	0.283561	+	0.958954i	$0.408484\pi$
$60$	0	0
$61$	−4.96148e13	−0.0331366	−0.0165683	−	0.999863i	$-0.505274\pi$
$61$	−4.96148e13	−0.0331366	−0.0165683	+	0.999863i	$0.505274\pi$
$62$	3.66037e14i	0.212910i
$63$	− 1.03047e15i	− 0.523169i
$64$	1.18337e15	0.525522
$65$	0	0
$66$	3.52776e14	0.120608
$67$	2.75976e15i	0.830300i	0.909753	+	0.415150i	$0.136271\pi$
$67$	2.75976e15i	0.830300i	−0.909753	+	0.415150i	$0.863729\pi$
$68$	5.33686e15i	1.41566i
$69$	−3.08375e15	−0.722541
$70$	0	0
$71$	2.50612e15	0.460582	0.230291	−	0.973122i	$-0.426032\pi$
$71$	2.50612e15	0.460582	0.230291	+	0.973122i	$0.426032\pi$
$72$	− 1.14046e15i	− 0.186102i
$73$	8.85474e14i	0.128508i	0.997934	+	0.0642542i	$0.0204669\pi$
$73$	8.85474e14i	0.128508i	−0.997934	+	0.0642542i	$0.979533\pi$
$74$	3.36430e15	0.434936
$75$	0	0
$76$	5.09829e15	0.525423
$77$	− 1.21946e16i	− 1.12459i
$78$	− 3.26647e15i	− 0.269944i
$79$	6.85544e13	0.00508400	0.00254200	−	0.999997i	$-0.499191\pi$
$79$	6.85544e13	0.00508400	0.00254200	+	0.999997i	$0.499191\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	8.58690e15i	0.463894i
$83$	3.38359e16i	1.64897i	0.565881	+	0.824487i	$0.308536\pi$
$83$	3.38359e16i	1.64897i	−0.565881	+	0.824487i	$0.691464\pi$
$84$	−1.88364e16	−0.829135
$85$	0	0
$86$	1.31405e16	0.473561
$87$	− 2.07146e16i	− 0.676650i
$88$	− 1.34961e16i	− 0.400041i
$89$	3.37871e16	0.909778	0.454889	−	0.890548i	$-0.349679\pi$
$89$	3.37871e16	0.909778	0.454889	+	0.890548i	$0.349679\pi$
$90$	0	0
$91$	−1.12914e17	−2.51707
$92$	5.63692e16i	1.14510i
$93$	2.27529e16i	0.421630i
$94$	−7.44266e15	−0.125934
$95$	0	0
$96$	−3.17329e16	−0.448957
$97$	− 6.48330e16i	− 0.839917i	−0.907543	−	0.419958i	$-0.862045\pi$
$97$	− 6.48330e16i	− 0.839917i	0.907543	−	0.419958i	$-0.137955\pi$
$98$	− 3.59315e16i	− 0.426632i
$99$	2.19285e16	0.238842
$100$	0	0
$101$	2.50991e16	0.230636	0.115318	−	0.993329i	$-0.463211\pi$
$101$	2.50991e16	0.230636	0.115318	+	0.993329i	$0.463211\pi$
$102$	− 3.08164e16i	− 0.260424i
$103$	− 1.88107e17i	− 1.46315i	−0.681760	−	0.731575i	$-0.738785\pi$
$103$	− 1.88107e17i	− 1.46315i	0.681760	−	0.731575i	$-0.261215\pi$
$104$	−1.24965e17	−0.895374
$105$	0	0
$106$	−2.65752e16	−0.161948
$107$	− 1.04046e17i	− 0.585414i	−0.956202	−	0.292707i	$-0.905444\pi$
$107$	− 1.04046e17i	− 0.585414i	0.956202	−	0.292707i	$-0.0945560\pi$
$108$	− 3.38721e16i	− 0.176092i
$109$	3.25424e17	1.56431	0.782157	−	0.623082i	$-0.214120\pi$
$109$	3.25424e17	1.56431	0.782157	+	0.623082i	$0.214120\pi$
$110$	0	0
$111$	2.09125e17	0.861313
$112$	3.09363e17i	1.18063i
$113$	1.16099e17i	0.410829i	0.978675	+	0.205415i	$0.0658543\pi$
$113$	1.16099e17i	0.410829i	−0.978675	+	0.205415i	$0.934146\pi$
$114$	−2.94389e16	−0.0966563
$115$	0	0
$116$	−3.78651e17	−1.07237
$117$	− 2.03044e17i	− 0.534577i
$118$	6.75115e16i	0.165341i
$119$	−1.06525e18	−2.42829
$120$	0	0
$121$	−2.45946e17	−0.486591
$122$	− 5.23685e15i	− 0.00966077i
$123$	5.33762e17i	0.918658i
$124$	4.15909e17	0.668212
$125$	0	0
$126$	1.08767e17	0.152527
$127$	− 1.11976e18i	− 1.46823i	−0.679024	−	0.734116i	$-0.737597\pi$
$127$	− 1.11976e18i	− 1.46823i	0.679024	−	0.734116i	$-0.262403\pi$
$128$	7.58847e17i	0.930828i
$129$	8.16816e17	0.937803
$130$	0	0
$131$	3.59952e17	0.362609	0.181304	−	0.983427i	$-0.441968\pi$
$131$	3.59952e17	0.362609	0.181304	+	0.983427i	$0.441968\pi$
$132$	− 4.00841e17i	− 0.378524i
$133$	1.01763e18i	0.901260i
$134$	−2.91293e17	−0.242069
$135$	0	0
$136$	−1.17894e18	−0.863795
$137$	9.75314e17i	0.671459i	0.941958	+	0.335730i	$0.108983\pi$
$137$	9.75314e17i	0.671459i	−0.941958	+	0.335730i	$0.891017\pi$
$138$	− 3.25491e17i	− 0.210652i
$139$	−6.02829e17	−0.366917	−0.183459	−	0.983027i	$-0.558729\pi$
$139$	−6.02829e17	−0.366917	−0.183459	+	0.983027i	$0.558729\pi$
$140$	0	0
$141$	−4.62636e17	−0.249389
$142$	2.64522e17i	0.134280i
$143$	− 2.40281e18i	− 1.14911i
$144$	−5.56303e17	−0.250744
$145$	0	0
$146$	−9.34620e16	−0.0374659
$147$	− 2.23350e18i	− 0.844869i
$148$	− 3.82268e18i	− 1.36504i
$149$	−4.51405e18	−1.52224	−0.761120	−	0.648612i	$-0.775350\pi$
$149$	−4.51405e18	−1.52224	−0.761120	+	0.648612i	$0.775350\pi$
$150$	0	0
$151$	−5.57256e17	−0.167784	−0.0838921	−	0.996475i	$-0.526735\pi$
$151$	−5.57256e17	−0.167784	−0.0838921	+	0.996475i	$0.526735\pi$
$152$	1.12624e18i	0.320597i
$153$	− 1.91555e18i	− 0.515723i
$154$	1.28714e18	0.327868
$155$	0	0
$156$	−3.71152e18	−0.847213
$157$	− 5.67610e18i	− 1.22716i	−0.789631	−	0.613582i	$-0.789728\pi$
$157$	− 5.67610e18i	− 1.22716i	0.789631	−	0.613582i	$-0.210272\pi$
$158$	7.23593e15i	0.00148221i
$159$	−1.65191e18	−0.320709
$160$	0	0
$161$	−1.12514e19	−1.96420
$162$	1.95587e17i	0.0323938i
$163$	4.52801e18i	0.711726i	0.934538	+	0.355863i	$0.115813\pi$
$163$	4.52801e18i	0.711726i	−0.934538	+	0.355863i	$0.884187\pi$
$164$	9.75684e18	1.45592
$165$	0	0
$166$	−3.57138e18	−0.480748
$167$	− 9.90189e18i	− 1.26657i	−0.773920	−	0.633283i	$-0.781707\pi$
$167$	− 9.90189e18i	− 1.26657i	0.773920	−	0.633283i	$-0.218293\pi$
$168$	− 4.16107e18i	− 0.505913i
$169$	−1.35980e19	−1.57195
$170$	0	0
$171$	−1.82992e18	−0.191410
$172$	− 1.49309e19i	− 1.48626i
$173$	1.49801e19i	1.41946i	0.704475	+	0.709729i	$0.251183\pi$
$173$	1.49801e19i	1.41946i	−0.704475	+	0.709729i	$0.748817\pi$
$174$	2.18643e18	0.197273
$175$	0	0
$176$	−6.58326e18	−0.538993
$177$	4.19652e18i	0.327428i
$178$	3.56623e18i	0.265240i
$179$	−4.67407e18	−0.331470	−0.165735	−	0.986170i	$-0.553000\pi$
$179$	−4.67407e18	−0.331470	−0.165735	+	0.986170i	$0.553000\pi$
$180$	0	0
$181$	1.00176e17	0.00646394	0.00323197	−	0.999995i	$-0.498971\pi$
$181$	1.00176e17	0.00646394	0.00323197	+	0.999995i	$0.498971\pi$
$182$	− 1.19181e19i	− 0.733835i
$183$	− 3.25523e17i	− 0.0191314i
$184$	−1.24523e19	−0.698709
$185$	0	0
$186$	−2.40157e18	−0.122924
$187$	− 2.26685e19i	− 1.10859i
$188$	8.45671e18i	0.395239i
$189$	6.76094e18	0.302052
$190$	0	0
$191$	−1.79795e19	−0.734503	−0.367252	−	0.930122i	$-0.619701\pi$
$191$	−1.79795e19	−0.734503	−0.367252	+	0.930122i	$0.619701\pi$
$192$	7.76409e18i	0.303410i
$193$	1.20664e19i	0.451171i	0.974223	+	0.225585i	$0.0724295\pi$
$193$	1.20664e19i	0.451171i	−0.974223	+	0.225585i	$0.927571\pi$
$194$	6.84313e18	0.244873
$195$	0	0
$196$	−4.08271e19	−1.33897
$197$	− 1.55077e19i	− 0.487062i	−0.969893	−	0.243531i	$-0.921694\pi$
$197$	− 1.55077e19i	− 0.487062i	0.969893	−	0.243531i	$-0.0783058\pi$
$198$	2.31456e18i	0.0696329i
$199$	1.56769e19	0.451865	0.225932	−	0.974143i	$-0.427457\pi$
$199$	1.56769e19	0.451865	0.225932	+	0.974143i	$0.427457\pi$
$200$	0	0
$201$	−1.81068e19	−0.479374
$202$	2.64922e18i	0.0672405i
$203$	− 7.55796e19i	− 1.83945i
$204$	−3.50151e19	−0.817333
$205$	0	0
$206$	1.98548e19	0.426573
$207$	− 2.02325e19i	− 0.417159i
$208$	6.09566e19i	1.20638i
$209$	−2.16552e19	−0.411451
$210$	0	0
$211$	−4.67993e19	−0.820046	−0.410023	−	0.912075i	$-0.634479\pi$
$211$	−4.67993e19	−0.820046	−0.410023	+	0.912075i	$0.634479\pi$
$212$	3.01960e19i	0.508270i
$213$	1.64427e19i	0.265917i
$214$	1.09821e19	0.170674
$215$	0	0
$216$	7.48253e18	0.107446
$217$	8.30163e19i	1.14619i
$218$	3.43485e19i	0.456066i
$219$	−5.80960e18	−0.0741944
$220$	0	0
$221$	−2.09895e20	−2.48124
$222$	2.20732e19i	0.251111i
$223$	− 1.70487e19i	− 0.186681i	−0.995634	−	0.0933407i	$-0.970245\pi$
$223$	− 1.70487e19i	− 0.186681i	0.995634	−	0.0933407i	$-0.0297546\pi$
$224$	−1.15781e20	−1.22047
$225$	0	0
$226$	−1.22543e19	−0.119775
$227$	− 2.03225e20i	− 1.91318i	−0.291429	−	0.956592i	$-0.594131\pi$
$227$	− 2.03225e20i	− 1.91318i	0.291429	−	0.956592i	$-0.405869\pi$
$228$	3.34499e19i	0.303353i
$229$	−1.73884e19	−0.151935	−0.0759677	−	0.997110i	$-0.524205\pi$
$229$	−1.73884e19	−0.151935	−0.0759677	+	0.997110i	$0.524205\pi$
$230$	0	0
$231$	8.00086e19	0.649284
$232$	− 8.36461e19i	− 0.654331i
$233$	− 7.22288e19i	− 0.544735i	−0.962193	−	0.272368i	$-0.912193\pi$
$233$	− 7.22288e19i	− 0.544735i	0.962193	−	0.272368i	$-0.0878067\pi$
$234$	2.14313e19	0.155853
$235$	0	0
$236$	7.67098e19	0.518918
$237$	4.49786e17i	0.00293525i
$238$	− 1.12437e20i	− 0.707954i
$239$	4.28634e19	0.260438	0.130219	−	0.991485i	$-0.458432\pi$
$239$	4.28634e19	0.260438	0.130219	+	0.991485i	$0.458432\pi$
$240$	0	0
$241$	2.14062e20	1.21170	0.605848	−	0.795580i	$-0.292834\pi$
$241$	2.14062e20	1.21170	0.605848	+	0.795580i	$0.292834\pi$
$242$	− 2.59596e19i	− 0.141863i
$243$	1.21577e19i	0.0641500i
$244$	−5.95036e18	−0.0303201
$245$	0	0
$246$	−5.63386e19	−0.267829
$247$	2.00513e20i	0.920912i
$248$	9.18766e19i	0.407723i
$249$	−2.21997e20	−0.952035
$250$	0	0
$251$	4.02872e20	1.61414	0.807068	−	0.590458i	$-0.201053\pi$
$251$	4.02872e20	1.61414	0.807068	+	0.590458i	$0.201053\pi$
$252$	− 1.23586e20i	− 0.478701i
$253$	− 2.39430e20i	− 0.896716i
$254$	1.18191e20	0.428054
$255$	0	0
$256$	7.50103e19	0.254145
$257$	− 7.39189e19i	− 0.242283i	−0.992635	−	0.121142i	$-0.961344\pi$
$257$	− 7.39189e19i	− 0.242283i	0.992635	−	0.121142i	$-0.0386555\pi$
$258$	8.62150e19i	0.273411i
$259$	7.63015e20	2.34145
$260$	0	0
$261$	1.35909e20	0.390664
$262$	3.79930e19i	0.105716i
$263$	1.87464e20i	0.505004i	0.967596	+	0.252502i	$0.0812535\pi$
$263$	1.87464e20i	0.505004i	−0.967596	+	0.252502i	$0.918747\pi$
$264$	8.85478e19	0.230964
$265$	0	0
$266$	−1.07411e20	−0.262757
$267$	2.21677e20i	0.525261i
$268$	3.30981e20i	0.759726i
$269$	4.94362e20	1.09939	0.549694	−	0.835366i	$-0.314744\pi$
$269$	4.94362e20	1.09939	0.549694	+	0.835366i	$0.314744\pi$
$270$	0	0
$271$	−2.66243e20	−0.555954	−0.277977	−	0.960588i	$-0.589664\pi$
$271$	−2.66243e20	−0.555954	−0.277977	+	0.960588i	$0.589664\pi$
$272$	5.75075e20i	1.16383i
$273$	− 7.40827e20i	− 1.45323i
$274$	−1.02945e20	−0.195760
$275$	0	0
$276$	−3.69838e20	−0.661127
$277$	− 7.36150e19i	− 0.127611i	−0.997962	−	0.0638055i	$-0.979676\pi$
$277$	− 7.36150e19i	− 0.127611i	0.997962	−	0.0638055i	$-0.0203237\pi$
$278$	− 6.36287e19i	− 0.106973i
$279$	−1.49282e20	−0.243428
$280$	0	0
$281$	7.23193e20	1.10982	0.554908	−	0.831912i	$-0.312754\pi$
$281$	7.23193e20	1.10982	0.554908	+	0.831912i	$0.312754\pi$
$282$	− 4.88313e19i	− 0.0727078i
$283$	2.78845e20i	0.402883i	0.979501	+	0.201441i	$0.0645626\pi$
$283$	2.78845e20i	0.402883i	−0.979501	+	0.201441i	$0.935437\pi$
$284$	3.00562e20	0.421433
$285$	0	0
$286$	2.53617e20	0.335017
$287$	1.94749e21i	2.49734i
$288$	− 2.08200e20i	− 0.259205i
$289$	−1.15295e21	−1.39373
$290$	0	0
$291$	4.25369e20	0.484926
$292$	1.06196e20i	0.117586i
$293$	3.59066e20i	0.386188i	0.981180	+	0.193094i	$0.0618523\pi$
$293$	3.59066e20i	0.386188i	−0.981180	+	0.193094i	$0.938148\pi$
$294$	2.35747e20	0.246316
$295$	0	0
$296$	8.44451e20	0.832904
$297$	1.43873e20i	0.137895i
$298$	− 4.76459e20i	− 0.443800i
$299$	−2.21697e21	−2.00703
$300$	0	0
$301$	2.98024e21	2.54939
$302$	− 5.88185e19i	− 0.0489165i
$303$	1.64675e20i	0.133158i
$304$	5.49368e20	0.431955
$305$	0	0
$306$	2.02187e20	0.150356
$307$	3.57691e20i	0.258721i	0.991598	+	0.129361i	$0.0412925\pi$
$307$	3.57691e20i	0.258721i	−0.991598	+	0.129361i	$0.958708\pi$
$308$	− 1.46251e21i	− 1.02900i
$309$	1.23417e21	0.844751
$310$	0	0
$311$	−2.19139e21	−1.41989	−0.709947	−	0.704255i	$-0.751281\pi$
$311$	−2.19139e21	−1.41989	−0.709947	+	0.704255i	$0.751281\pi$
$312$	− 8.19894e20i	− 0.516944i
$313$	1.02616e21i	0.629637i	0.949152	+	0.314818i	$0.101944\pi$
$313$	1.02616e21i	0.629637i	−0.949152	+	0.314818i	$0.898056\pi$
$314$	5.99113e20	0.357772
$315$	0	0
$316$	8.22181e18	0.00465187
$317$	− 8.06555e19i	− 0.0444253i	−0.999753	−	0.0222126i	$-0.992929\pi$
$317$	− 8.06555e19i	− 0.0444253i	0.999753	−	0.0222126i	$-0.00707109\pi$
$318$	− 1.74360e20i	− 0.0935009i
$319$	1.60834e21	0.839762
$320$	0	0
$321$	6.82645e20	0.337989
$322$	− 1.18759e21i	− 0.572652i
$323$	1.89167e21i	0.888432i
$324$	2.22235e20	0.101667
$325$	0	0
$326$	−4.77933e20	−0.207499
$327$	2.13510e21i	0.903157i
$328$	2.15534e21i	0.888357i
$329$	−1.68798e21	−0.677955
$330$	0	0
$331$	1.60438e21	0.612025	0.306012	−	0.952028i	$-0.401005\pi$
$331$	1.60438e21	0.612025	0.306012	+	0.952028i	$0.401005\pi$
$332$	4.05798e21i	1.50881i
$333$	1.37207e21i	0.497279i
$334$	1.04515e21	0.369260
$335$	0	0
$336$	−2.02973e21	−0.681639
$337$	1.51531e21i	0.496190i	0.968736	+	0.248095i	$0.0798046\pi$
$337$	1.51531e21i	0.496190i	−0.968736	+	0.248095i	$0.920195\pi$
$338$	− 1.43527e21i	− 0.458292i
$339$	−7.61725e20	−0.237192
$340$	0	0
$341$	−1.76659e21	−0.523268
$342$	− 1.93149e20i	− 0.0558045i
$343$	− 2.58034e21i	− 0.727240i
$344$	3.29832e21	0.906871
$345$	0	0
$346$	−1.58115e21	−0.413834
$347$	− 4.99855e21i	− 1.27657i	−0.769801	−	0.638283i	$-0.779645\pi$
$347$	− 4.99855e21i	− 1.27657i	0.769801	−	0.638283i	$-0.220355\pi$
$348$	− 2.48433e21i	− 0.619136i
$349$	7.23627e21	1.75994	0.879971	−	0.475027i	$-0.157562\pi$
$349$	7.23627e21	1.75994	0.879971	+	0.475027i	$0.157562\pi$
$350$	0	0
$351$	1.33217e21	0.308638
$352$	− 2.46382e21i	− 0.557182i
$353$	3.25282e20i	0.0718083i	0.999355	+	0.0359041i	$0.0114311\pi$
$353$	3.25282e20i	0.0718083i	−0.999355	+	0.0359041i	$0.988569\pi$
$354$	−4.42943e20	−0.0954597
$355$	0	0
$356$	4.05212e21	0.832449
$357$	− 6.98909e21i	− 1.40198i
$358$	− 4.93348e20i	− 0.0966380i
$359$	1.95689e21	0.374338	0.187169	−	0.982328i	$-0.440069\pi$
$359$	1.95689e21	0.374338	0.187169	+	0.982328i	$0.440069\pi$
$360$	0	0
$361$	−3.67328e21	−0.670259
$362$	1.05736e19i	0.00188452i
$363$	− 1.61365e21i	− 0.280933i
$364$	−1.35419e22	−2.30312
$365$	0	0
$366$	3.43590e19	0.00557765
$367$	2.21711e21i	0.351662i	0.984420	+	0.175831i	$0.0562612\pi$
$367$	2.21711e21i	0.351662i	−0.984420	+	0.175831i	$0.943739\pi$
$368$	6.07409e21i	0.941401i
$369$	−3.50201e21	−0.530388
$370$	0	0
$371$	−6.02718e21	−0.871838
$372$	2.72878e21i	0.385792i
$373$	− 1.07577e22i	− 1.48660i	−0.668961	−	0.743298i	$-0.733261\pi$
$373$	− 1.07577e22i	− 1.48660i	0.668961	−	0.743298i	$-0.266739\pi$
$374$	2.39267e21	0.323201
$375$	0	0
$376$	−1.86813e21	−0.241163
$377$	− 1.48921e22i	− 1.87956i
$378$	7.13618e20i	0.0880614i
$379$	−7.81839e21	−0.943374	−0.471687	−	0.881766i	$-0.656355\pi$
$379$	−7.81839e21	−0.943374	−0.471687	+	0.881766i	$0.656355\pi$
$380$	0	0
$381$	7.34677e21	0.847684
$382$	− 1.89774e21i	− 0.214140i
$383$	9.49308e21i	1.04765i	0.851825	+	0.523827i	$0.175496\pi$
$383$	9.49308e21i	1.04765i	−0.851825	+	0.523827i	$0.824504\pi$
$384$	−4.97880e21	−0.537414
$385$	0	0
$386$	−1.27361e21	−0.131536
$387$	5.35913e21i	0.541441i
$388$	− 7.77549e21i	− 0.768526i
$389$	−7.30323e21	−0.706225	−0.353113	−	0.935581i	$-0.614877\pi$
$389$	−7.30323e21	−0.706225	−0.353113	+	0.935581i	$0.614877\pi$
$390$	0	0
$391$	−2.09153e22	−1.93625
$392$	− 9.01892e21i	− 0.817002i
$393$	2.36164e21i	0.209352i
$394$	1.63684e21	0.142000
$395$	0	0
$396$	2.62991e21	0.218541
$397$	6.92001e21i	0.562843i	0.959584	+	0.281422i	$0.0908059\pi$
$397$	6.92001e21i	0.562843i	−0.959584	+	0.281422i	$0.909194\pi$
$398$	1.65470e21i	0.131738i
$399$	−6.67666e21	−0.520343
$400$	0	0
$401$	−1.52720e22	−1.14070	−0.570348	−	0.821403i	$-0.693192\pi$
$401$	−1.52720e22	−1.14070	−0.570348	+	0.821403i	$0.693192\pi$
$402$	− 1.91117e21i	− 0.139759i
$403$	1.63575e22i	1.17118i
$404$	3.01017e21	0.211032
$405$	0	0
$406$	7.97743e21	0.536280
$407$	1.62370e22i	1.06894i
$408$	− 7.73502e21i	− 0.498712i
$409$	−2.22167e21	−0.140292	−0.0701458	−	0.997537i	$-0.522346\pi$
$409$	−2.22167e21	−0.140292	−0.0701458	+	0.997537i	$0.522346\pi$
$410$	0	0
$411$	−6.39903e21	−0.387667
$412$	− 2.25599e22i	− 1.33879i
$413$	1.53114e22i	0.890103i
$414$	2.13555e21	0.121620
$415$	0	0
$416$	−2.28134e22	−1.24709
$417$	− 3.95516e21i	− 0.211840i
$418$	− 2.28571e21i	− 0.119956i
$419$	1.64936e22	0.848198	0.424099	−	0.905616i	$-0.360591\pi$
$419$	1.64936e22	0.848198	0.424099	+	0.905616i	$0.360591\pi$
$420$	0	0
$421$	6.48638e21	0.320335	0.160168	−	0.987090i	$-0.448797\pi$
$421$	6.48638e21	0.320335	0.160168	+	0.987090i	$0.448797\pi$
$422$	− 4.93967e21i	− 0.239079i
$423$	− 3.03535e21i	− 0.143985i
$424$	−6.67045e21	−0.310131
$425$	0	0
$426$	−1.73553e21	−0.0775265
$427$	− 1.18770e21i	− 0.0520082i
$428$	− 1.24784e22i	− 0.535655i
$429$	1.57648e22	0.663441
$430$	0	0
$431$	−2.94810e22	−1.19257	−0.596286	−	0.802772i	$-0.703358\pi$
$431$	−2.94810e22	−1.19257	−0.596286	+	0.802772i	$0.703358\pi$
$432$	− 3.64990e21i	− 0.144767i
$433$	− 4.61864e22i	− 1.79625i	−0.439741	−	0.898125i	$-0.644930\pi$
$433$	− 4.61864e22i	− 1.79625i	0.439741	−	0.898125i	$-0.355070\pi$
$434$	−8.76239e21	−0.334164
$435$	0	0
$436$	3.90284e22	1.43135
$437$	1.99803e22i	0.718638i
$438$	− 6.13204e20i	− 0.0216309i
$439$	5.08029e21	0.175768	0.0878840	−	0.996131i	$-0.471990\pi$
$439$	5.08029e21	0.175768	0.0878840	+	0.996131i	$0.471990\pi$
$440$	0	0
$441$	1.46540e22	0.487785
$442$	− 2.21545e22i	− 0.723390i
$443$	4.56719e22i	1.46291i	0.681890	+	0.731454i	$0.261158\pi$
$443$	4.56719e22i	1.46291i	−0.681890	+	0.731454i	$0.738842\pi$
$444$	2.50806e22	0.788103
$445$	0	0
$446$	1.79950e21	0.0544258
$447$	− 2.96167e22i	− 0.878865i
$448$	2.83281e22i	0.824811i
$449$	−9.26468e21	−0.264689	−0.132345	−	0.991204i	$-0.542251\pi$
$449$	−9.26468e21	−0.264689	−0.132345	+	0.991204i	$0.542251\pi$
$450$	0	0
$451$	−4.14426e22	−1.14011
$452$	1.39239e22i	0.375910i
$453$	− 3.65616e21i	− 0.0968703i
$454$	2.14504e22	0.557777
$455$	0	0
$456$	−7.38925e21	−0.185097
$457$	− 2.45492e22i	− 0.603601i	−0.953371	−	0.301801i	$-0.902412\pi$
$457$	− 2.45492e22i	− 0.603601i	0.953371	−	0.301801i	$-0.0975877\pi$
$458$	− 1.83535e21i	− 0.0442958i
$459$	1.25679e22	0.297753
$460$	0	0
$461$	4.22070e22	0.963667	0.481834	−	0.876263i	$-0.339971\pi$
$461$	4.22070e22	0.963667	0.481834	+	0.876263i	$0.339971\pi$
$462$	8.44492e21i	0.189295i
$463$	2.72110e22i	0.598832i	0.954123	+	0.299416i	$0.0967919\pi$
$463$	2.72110e22i	0.598832i	−0.954123	+	0.299416i	$0.903208\pi$
$464$	−4.08017e22	−0.881608
$465$	0	0
$466$	7.62377e21	0.158814
$467$	4.73325e22i	0.968202i	0.875012	+	0.484101i	$0.160853\pi$
$467$	4.73325e22i	0.968202i	−0.875012	+	0.484101i	$0.839147\pi$
$468$	− 2.43513e22i	− 0.489139i
$469$	−6.60644e22	−1.30316
$470$	0	0
$471$	3.72409e22	0.708504
$472$	1.69456e22i	0.316628i
$473$	6.34196e22i	1.16387i
$474$	−4.74749e19	−0.000855753 0
$475$	0	0
$476$	−1.27756e23	−2.22189
$477$	− 1.08382e22i	− 0.185162i
$478$	4.52424e21i	0.0759292i
$479$	3.54183e22	0.583950	0.291975	−	0.956426i	$-0.405688\pi$
$479$	3.54183e22	0.583950	0.291975	+	0.956426i	$0.405688\pi$
$480$	0	0
$481$	1.50344e23	2.39251
$482$	2.25942e22i	0.353263i
$483$	− 7.38204e22i	− 1.13403i
$484$	−2.94966e22	−0.445232
$485$	0	0
$486$	−1.28324e21	−0.0187025
$487$	2.32584e22i	0.333107i	0.986032	+	0.166553i	$0.0532638\pi$
$487$	2.32584e22i	0.333107i	−0.986032	+	0.166553i	$0.946736\pi$
$488$	− 1.31447e21i	− 0.0185004i
$489$	−2.97083e22	−0.410915
$490$	0	0
$491$	2.08629e22	0.278728	0.139364	−	0.990241i	$-0.455494\pi$
$491$	2.08629e22	0.278728	0.139364	+	0.990241i	$0.455494\pi$
$492$	6.40147e22i	0.840574i
$493$	− 1.40495e23i	− 1.81327i
$494$	−2.11642e22	−0.268486
$495$	0	0
$496$	4.48165e22	0.549343
$497$	5.99929e22i	0.722887i
$498$	− 2.34318e22i	− 0.277560i
$499$	1.49399e23	1.73978	0.869890	−	0.493246i	$-0.164190\pi$
$499$	1.49399e23	1.73978	0.869890	+	0.493246i	$0.164190\pi$
$500$	0	0
$501$	6.49663e22	0.731252
$502$	4.25232e22i	0.470592i
$503$	1.02620e23i	1.11662i	0.829633	+	0.558309i	$0.188550\pi$
$503$	1.02620e23i	1.11662i	−0.829633	+	0.558309i	$0.811450\pi$
$504$	2.73008e22	0.292089
$505$	0	0
$506$	2.52719e22	0.261432
$507$	− 8.92165e22i	− 0.907565i
$508$	− 1.34295e23i	− 1.34344i
$509$	1.41389e23	1.39096	0.695481	−	0.718545i	$-0.255191\pi$
$509$	1.41389e23	1.39096	0.695481	+	0.718545i	$0.255191\pi$
$510$	0	0
$511$	−2.11969e22	−0.201695
$512$	1.07381e23i	1.00492i
$513$	− 1.20061e22i	− 0.110511i
$514$	7.80215e21	0.0706362
$515$	0	0
$516$	9.79617e22	0.858092
$517$	− 3.59202e22i	− 0.309506i
$518$	8.05364e22i	0.682636i
$519$	−9.82844e22	−0.819524
$520$	0	0
$521$	1.43662e23	1.15937	0.579685	−	0.814841i	$-0.303176\pi$
$521$	1.43662e23	1.15937	0.579685	+	0.814841i	$0.303176\pi$
$522$	1.43452e22i	0.113896i
$523$	− 1.08226e23i	− 0.845412i	−0.906267	−	0.422706i	$-0.861080\pi$
$523$	− 1.08226e23i	− 0.845412i	0.906267	−	0.422706i	$-0.138920\pi$
$524$	4.31694e22	0.331788
$525$	0	0
$526$	−1.97869e22	−0.147231
$527$	1.54319e23i	1.12987i
$528$	− 4.31927e22i	− 0.311188i
$529$	−7.98621e22	−0.566197
$530$	0	0
$531$	−2.75333e22	−0.189041
$532$	1.22045e23i	0.824655i
$533$	3.83731e23i	2.55179i
$534$	−2.33981e22	−0.153137
$535$	0	0
$536$	−7.31154e22	−0.463562
$537$	− 3.06665e22i	− 0.191374i
$538$	5.21800e22i	0.320520i
$539$	1.73415e23	1.04853
$540$	0	0
$541$	1.33856e23	0.784265	0.392132	−	0.919909i	$-0.371738\pi$
$541$	1.33856e23	0.784265	0.392132	+	0.919909i	$0.371738\pi$
$542$	− 2.81019e22i	− 0.162085i
$543$	6.57257e20i	0.00373196i
$544$	−2.15225e23	−1.20310
$545$	0	0
$546$	7.81944e22	0.423680
$547$	6.84817e22i	0.365327i	0.983175	+	0.182664i	$0.0584720\pi$
$547$	6.84817e22i	0.365327i	−0.983175	+	0.182664i	$0.941528\pi$
$548$	1.16971e23i	0.614387i
$549$	2.13576e21	0.0110455
$550$	0	0
$551$	−1.34215e23	−0.672994
$552$	− 8.16992e22i	− 0.403400i
$553$	1.64109e21i	0.00797937i
$554$	7.77007e21	0.0372042
$555$	0	0
$556$	−7.22980e22	−0.335730
$557$	− 2.95445e23i	− 1.35116i	−0.737286	−	0.675580i	$-0.763893\pi$
$557$	− 2.95445e23i	− 1.35116i	0.737286	−	0.675580i	$-0.236107\pi$
$558$	− 1.57567e22i	− 0.0709700i
$559$	5.87224e23	2.60497
$560$	0	0
$561$	1.48728e23	0.640042
$562$	7.63332e22i	0.323560i
$563$	− 4.12155e23i	− 1.72084i	−0.509590	−	0.860418i	$-0.670203\pi$
$563$	− 4.12155e23i	− 1.72084i	0.509590	−	0.860418i	$-0.329797\pi$
$564$	−5.54845e22	−0.228191
$565$	0	0
$566$	−2.94322e22	−0.117458
$567$	4.43585e22i	0.174390i
$568$	6.63958e22i	0.257146i
$569$	−5.03050e23	−1.91936	−0.959681	−	0.281092i	$-0.909303\pi$
$569$	−5.03050e23	−1.91936	−0.959681	+	0.281092i	$0.909303\pi$
$570$	0	0
$571$	5.27359e23	1.95299	0.976493	−	0.215547i	$-0.0691536\pi$
$571$	5.27359e23	1.95299	0.976493	+	0.215547i	$0.0691536\pi$
$572$	− 2.88172e23i	− 1.05144i
$573$	− 1.17963e23i	− 0.424066i
$574$	−2.05557e23	−0.728085
$575$	0	0
$576$	−5.09402e22	−0.175174
$577$	− 2.31429e23i	− 0.784192i	−0.919924	−	0.392096i	$-0.871750\pi$
$577$	− 2.31429e23i	− 0.784192i	0.919924	−	0.392096i	$-0.128250\pi$
$578$	− 1.21694e23i	− 0.406334i
$579$	−7.91677e22	−0.260484
$580$	0	0
$581$	−8.09980e23	−2.58808
$582$	4.48978e22i	0.141377i
$583$	− 1.28259e23i	− 0.398019i
$584$	−2.34593e22	−0.0717472
$585$	0	0
$586$	−3.78994e22	−0.112591
$587$	1.67583e23i	0.490688i	0.969436	+	0.245344i	$0.0789010\pi$
$587$	1.67583e23i	0.490688i	−0.969436	+	0.245344i	$0.921099\pi$
$588$	− 2.67867e23i	− 0.773057i
$589$	1.47421e23	0.419352
$590$	0	0
$591$	1.01746e23	0.281206
$592$	− 4.11915e23i	− 1.12221i
$593$	4.62003e23i	1.24074i	0.784310	+	0.620369i	$0.213017\pi$
$593$	4.62003e23i	1.24074i	−0.784310	+	0.620369i	$0.786983\pi$
$594$	−1.51858e22	−0.0402026
$595$	0	0
$596$	−5.41375e23	−1.39285
$597$	1.02856e23i	0.260884i
$598$	− 2.34001e23i	− 0.585138i
$599$	−6.81228e21	−0.0167944	−0.00839720	−	0.999965i	$-0.502673\pi$
$599$	−6.81228e21	−0.0167944	−0.00839720	+	0.999965i	$0.502673\pi$
$600$	0	0
$601$	−7.15068e23	−1.71362	−0.856809	−	0.515633i	$-0.827557\pi$
$601$	−7.15068e23	−1.71362	−0.856809	+	0.515633i	$0.827557\pi$
$602$	3.14565e23i	0.743258i
$603$	− 1.18798e23i	− 0.276767i
$604$	−6.68324e22	−0.153523
$605$	0	0
$606$	−1.73815e22	−0.0388213
$607$	2.99308e23i	0.659195i	0.944122	+	0.329597i	$0.106913\pi$
$607$	2.99308e23i	0.659195i	−0.944122	+	0.329597i	$0.893087\pi$
$608$	2.05604e23i	0.446531i
$609$	4.95877e23	1.06201
$610$	0	0
$611$	−3.32597e23	−0.692737
$612$	− 2.29734e23i	− 0.471888i
$613$	7.66386e23i	1.55251i	0.630421	+	0.776253i	$0.282882\pi$
$613$	7.66386e23i	1.55251i	−0.630421	+	0.776253i	$0.717118\pi$
$614$	−3.77544e22	−0.0754286
$615$	0	0
$616$	3.23076e23	0.627868
$617$	1.04769e23i	0.200820i	0.994946	+	0.100410i	$0.0320155\pi$
$617$	1.04769e23i	0.200820i	−0.994946	+	0.100410i	$0.967984\pi$
$618$	1.30267e23i	0.246282i
$619$	−7.62993e23	−1.42282	−0.711410	−	0.702777i	$-0.751943\pi$
$619$	−7.62993e23	−1.42282	−0.711410	+	0.702777i	$0.751943\pi$
$620$	0	0
$621$	1.32746e23	0.240847
$622$	− 2.31301e23i	− 0.413961i
$623$	8.08812e23i	1.42790i
$624$	−3.99936e23	−0.696501
$625$	0	0
$626$	−1.08312e23	−0.183567
$627$	− 1.42080e23i	− 0.237552i
$628$	− 6.80741e23i	− 1.12286i
$629$	1.41837e24	2.30813
$630$	0	0
$631$	2.67118e22	0.0423110	0.0211555	−	0.999776i	$-0.493265\pi$
$631$	2.67118e22	0.0423110	0.0211555	+	0.999776i	$0.493265\pi$
$632$	1.81624e21i	0.00283843i
$633$	− 3.07050e23i	− 0.473454i
$634$	8.51320e21	0.0129519
$635$	0	0
$636$	−1.98116e23	−0.293450
$637$	− 1.60571e24i	− 2.34683i
$638$	1.69760e23i	0.244827i
$639$	−1.07880e23	−0.153527
$640$	0	0
$641$	−7.71926e23	−1.06975	−0.534875	−	0.844931i	$-0.679641\pi$
$641$	−7.71926e23	−1.06975	−0.534875	+	0.844931i	$0.679641\pi$
$642$	7.20533e22i	0.0985386i
$643$	5.74619e23i	0.775509i	0.921763	+	0.387754i	$0.126749\pi$
$643$	5.74619e23i	0.775509i	−0.921763	+	0.387754i	$0.873251\pi$
$644$	−1.34939e24	−1.79725
$645$	0	0
$646$	−1.99666e23	−0.259017
$647$	6.99931e23i	0.896125i	0.894002	+	0.448063i	$0.147886\pi$
$647$	6.99931e23i	0.896125i	−0.894002	+	0.448063i	$0.852114\pi$
$648$	4.90929e22i	0.0620341i
$649$	−3.25828e23	−0.406358
$650$	0	0
$651$	−5.44670e23	−0.661752
$652$	5.43050e23i	0.651231i
$653$	1.29872e24i	1.53728i	0.639682	+	0.768640i	$0.279066\pi$
$653$	1.29872e24i	1.53728i	−0.639682	+	0.768640i	$0.720934\pi$
$654$	−2.25361e23	−0.263310
$655$	0	0
$656$	1.05135e24	1.19692
$657$	− 3.81168e22i	− 0.0428362i
$658$	− 1.78166e23i	− 0.197654i
$659$	4.10190e23	0.449220	0.224610	−	0.974449i	$-0.427889\pi$
$659$	4.10190e23	0.449220	0.224610	+	0.974449i	$0.427889\pi$
$660$	0	0
$661$	1.61344e24	1.72202	0.861012	−	0.508584i	$-0.169831\pi$
$661$	1.61344e24	1.72202	0.861012	+	0.508584i	$0.169831\pi$
$662$	1.69342e23i	0.178432i
$663$	− 1.37712e24i	− 1.43254i
$664$	−8.96428e23	−0.920634
$665$	0	0
$666$	−1.44822e23	−0.144979
$667$	− 1.48394e24i	− 1.46672i
$668$	− 1.18754e24i	− 1.15891i
$669$	1.11857e23	0.107781
$670$	0	0
$671$	2.52744e22	0.0237432
$672$	− 7.59638e23i	− 0.704641i
$673$	− 3.46816e23i	− 0.317666i	−0.987305	−	0.158833i	$-0.949227\pi$
$673$	− 3.46816e23i	− 0.317666i	0.987305	−	0.158833i	$-0.0507731\pi$
$674$	−1.59942e23	−0.144661
$675$	0	0
$676$	−1.63083e24	−1.43834
$677$	− 2.06445e24i	− 1.79805i	−0.437899	−	0.899024i	$-0.644277\pi$
$677$	− 2.06445e24i	− 1.79805i	0.437899	−	0.899024i	$-0.355723\pi$
$678$	− 8.04002e22i	− 0.0691520i
$679$	1.55200e24	1.31826
$680$	0	0
$681$	1.33336e24	1.10458
$682$	− 1.86464e23i	− 0.152556i
$683$	1.79202e22i	0.0144799i	0.999974	+	0.00723997i	$0.00230457\pi$
$683$	1.79202e22i	0.0144799i	−0.999974	+	0.00723997i	$0.997695\pi$
$684$	−2.19465e23	−0.175141
$685$	0	0
$686$	2.72356e23	0.212022
$687$	− 1.14086e23i	− 0.0877199i
$688$	− 1.60889e24i	− 1.22187i
$689$	−1.18759e24	−0.890848
$690$	0	0
$691$	−1.82328e24	−1.33441	−0.667207	−	0.744873i	$-0.732510\pi$
$691$	−1.82328e24	−1.33441	−0.667207	+	0.744873i	$0.732510\pi$
$692$	1.79658e24i	1.29881i
$693$	5.24936e23i	0.374864i
$694$	5.27598e23	0.372175
$695$	0	0
$696$	5.48802e23	0.377778
$697$	3.62019e24i	2.46180i
$698$	7.63789e23i	0.513101i
$699$	4.73893e23	0.314503
$700$	0	0
$701$	−2.61207e24	−1.69193	−0.845965	−	0.533239i	$-0.820975\pi$
$701$	−2.61207e24	−1.69193	−0.845965	+	0.533239i	$0.820975\pi$
$702$	1.40611e23i	0.0899815i
$703$	− 1.35497e24i	− 0.856660i
$704$	−6.02824e23	−0.376550
$705$	0	0
$706$	−3.43335e22	−0.0209353
$707$	6.00835e23i	0.361985i
$708$	5.03293e23i	0.299598i
$709$	1.84759e24	1.08671	0.543353	−	0.839504i	$-0.317155\pi$
$709$	1.84759e24	1.08671	0.543353	+	0.839504i	$0.317155\pi$
$710$	0	0
$711$	−2.95104e21	−0.00169467
$712$	8.95136e23i	0.507936i
$713$	1.62996e24i	0.913935i
$714$	7.37699e23	0.408738
$715$	0	0
$716$	−5.60566e23	−0.303296
$717$	2.81227e23i	0.150364i
$718$	2.06550e23i	0.109136i
$719$	9.54585e23	0.498447	0.249224	−	0.968446i	$-0.419825\pi$
$719$	9.54585e23	0.498447	0.249224	+	0.968446i	$0.419825\pi$
$720$	0	0
$721$	4.50300e24	2.29643
$722$	− 3.87715e23i	− 0.195410i
$723$	1.40446e24i	0.699573i
$724$	1.20143e22	0.00591452
$725$	0	0
$726$	1.70321e23	0.0819044
$727$	− 3.28829e23i	− 0.156289i	−0.996942	−	0.0781443i	$-0.975101\pi$
$727$	− 3.28829e23i	− 0.156289i	0.996942	−	0.0781443i	$-0.0248995\pi$
$728$	− 2.99147e24i	− 1.40530i
$729$	−7.97664e22	−0.0370370
$730$	0	0
$731$	5.53997e24	2.51310
$732$	− 3.90403e22i	− 0.0175053i
$733$	1.68096e24i	0.745031i	0.928026	+	0.372515i	$0.121505\pi$
$733$	1.68096e24i	0.745031i	−0.928026	+	0.372515i	$0.878495\pi$
$734$	−2.34016e23	−0.102525
$735$	0	0
$736$	−2.27326e24	−0.973169
$737$	− 1.40585e24i	− 0.594931i
$738$	− 3.69638e23i	− 0.154631i
$739$	−1.71501e24	−0.709231	−0.354616	−	0.935012i	$-0.615388\pi$
$739$	−1.71501e24	−0.709231	−0.354616	+	0.935012i	$0.615388\pi$
$740$	0	0
$741$	−1.31556e24	−0.531689
$742$	− 6.36170e23i	− 0.254179i
$743$	− 1.92121e24i	− 0.758873i	−0.925218	−	0.379436i	$-0.876118\pi$
$743$	− 1.92121e24i	− 0.758873i	0.925218	−	0.379436i	$-0.123882\pi$
$744$	−6.02802e23	−0.235399
$745$	0	0
$746$	1.13547e24	0.433408
$747$	− 1.45652e24i	− 0.549658i
$748$	− 2.71866e24i	− 1.01436i
$749$	2.49070e24	0.918812
$750$	0	0
$751$	3.66527e24	1.32180	0.660902	−	0.750472i	$-0.270174\pi$
$751$	3.66527e24	1.32180	0.660902	+	0.750472i	$0.270174\pi$
$752$	9.11256e23i	0.324929i
$753$	2.64324e24i	0.931922i
$754$	1.57187e24	0.547974
$755$	0	0
$756$	8.10847e23	0.276378
$757$	2.09505e24i	0.706121i	0.935600	+	0.353061i	$0.114859\pi$
$757$	2.09505e24i	0.706121i	−0.935600	+	0.353061i	$0.885141\pi$
$758$	− 8.25233e23i	− 0.275035i
$759$	1.57090e24	0.517719
$760$	0	0
$761$	3.96049e24	1.27638	0.638189	−	0.769880i	$-0.279684\pi$
$761$	3.96049e24	1.27638	0.638189	+	0.769880i	$0.279684\pi$
$762$	7.75453e23i	0.247137i
$763$	7.79015e24i	2.45520i
$764$	−2.15630e24	−0.672072
$765$	0	0
$766$	−1.00200e24	−0.305437
$767$	3.01695e24i	0.909511i
$768$	4.92142e23i	0.146731i
$769$	1.03312e24	0.304632	0.152316	−	0.988332i	$-0.451327\pi$
$769$	1.03312e24	0.304632	0.152316	+	0.988332i	$0.451327\pi$
$770$	0	0
$771$	4.84982e23	0.139882
$772$	1.44714e24i	0.412822i
$773$	5.97296e23i	0.168525i	0.996444	+	0.0842624i	$0.0268534\pi$
$773$	5.97296e23i	0.168525i	−0.996444	+	0.0842624i	$0.973147\pi$
$774$	−5.65657e23	−0.157854
$775$	0	0
$776$	1.71765e24	0.468932
$777$	5.00614e24i	1.35184i
$778$	− 7.70857e23i	− 0.205896i
$779$	3.45836e24	0.913695
$780$	0	0
$781$	−1.27665e24	−0.330019
$782$	− 2.20761e24i	− 0.564501i
$783$	8.91698e23i	0.225550i
$784$	−4.39934e24	−1.10078
$785$	0	0
$786$	−2.49272e23	−0.0610354
$787$	4.68780e24i	1.13549i	0.823204	+	0.567746i	$0.192184\pi$
$787$	4.68780e24i	1.13549i	−0.823204	+	0.567746i	$0.807816\pi$
$788$	− 1.85986e24i	− 0.445663i
$789$	−1.22995e24	−0.291564
$790$	0	0
$791$	−2.77923e24	−0.644800
$792$	5.80962e23i	0.133347i
$793$	− 2.34024e23i	− 0.0531421i
$794$	−7.30408e23	−0.164093
$795$	0	0
$796$	1.88015e24	0.413457
$797$	4.23199e24i	0.920764i	0.887721	+	0.460382i	$0.152288\pi$
$797$	4.23199e24i	0.920764i	−0.887721	+	0.460382i	$0.847712\pi$
$798$	− 7.04723e23i	− 0.151703i
$799$	−3.13778e24	−0.668306
$800$	0	0
$801$	−1.45442e24	−0.303259
$802$	− 1.61197e24i	− 0.332563i
$803$	− 4.51072e23i	− 0.0920796i
$804$	−2.17157e24	−0.438628
$805$	0	0
$806$	−1.72653e24	−0.341450
$807$	3.24351e24i	0.634732i
$808$	6.64962e23i	0.128766i
$809$	6.89799e24	1.32178	0.660891	−	0.750482i	$-0.270179\pi$
$809$	6.89799e24	1.32178	0.660891	+	0.750482i	$0.270179\pi$
$810$	0	0
$811$	−5.76842e24	−1.08238	−0.541190	−	0.840900i	$-0.682026\pi$
$811$	−5.76842e24	−1.08238	−0.541190	+	0.840900i	$0.682026\pi$
$812$	− 9.06434e24i	− 1.68310i
$813$	− 1.74682e24i	− 0.320980i
$814$	−1.71382e24	−0.311643
$815$	0	0
$816$	−3.77307e24	−0.671937
$817$	− 5.29232e24i	− 0.932737i
$818$	− 2.34498e23i	− 0.0409012i
$819$	4.86056e24	0.839022
$820$	0	0
$821$	4.82230e24	0.815338	0.407669	−	0.913130i	$-0.366342\pi$
$821$	4.82230e24	0.815338	0.407669	+	0.913130i	$0.366342\pi$
$822$	− 6.75419e23i	− 0.113022i
$823$	− 4.94083e24i	− 0.818278i	−0.912472	−	0.409139i	$-0.865829\pi$
$823$	− 4.94083e24i	− 0.818278i	0.912472	−	0.409139i	$-0.134171\pi$
$824$	4.98361e24	0.816888
$825$	0	0
$826$	−1.61612e24	−0.259504
$827$	− 3.01636e24i	− 0.479386i	−0.970849	−	0.239693i	$-0.922953\pi$
$827$	− 3.01636e24i	− 0.479386i	0.970849	−	0.239693i	$-0.0770468\pi$
$828$	− 2.42651e24i	− 0.381702i
$829$	−9.18768e23	−0.143051	−0.0715257	−	0.997439i	$-0.522787\pi$
$829$	−9.18768e23	−0.143051	−0.0715257	+	0.997439i	$0.522787\pi$
$830$	0	0
$831$	4.82988e23	0.0736763
$832$	5.58175e24i	0.842795i
$833$	− 1.51485e25i	− 2.26406i
$834$	4.17468e23	0.0617606
$835$	0	0
$836$	−2.59713e24	−0.376479
$837$	− 9.79437e23i	− 0.140543i
$838$	1.74091e24i	0.247287i
$839$	−7.98403e24	−1.12265	−0.561326	−	0.827595i	$-0.689709\pi$
$839$	−7.98403e24	−1.12265	−0.561326	+	0.827595i	$0.689709\pi$
$840$	0	0
$841$	2.71101e24	0.373564
$842$	6.84639e23i	0.0933918i
$843$	4.74487e24i	0.640752i
$844$	−5.61269e24	−0.750344
$845$	0	0
$846$	3.20382e23	0.0419779
$847$	− 5.88758e24i	− 0.763708i
$848$	3.25378e24i	0.417853i
$849$	−1.82950e24	−0.232604
$850$	0	0
$851$	1.49812e25	1.86700
$852$	1.97199e24i	0.243315i
$853$	5.88324e24i	0.718704i	0.933202	+	0.359352i	$0.117002\pi$
$853$	5.88324e24i	0.718704i	−0.933202	+	0.359352i	$0.882998\pi$
$854$	1.25362e23	0.0151627
$855$	0	0
$856$	2.75653e24	0.326841
$857$	− 1.15530e25i	− 1.35630i	−0.734922	−	0.678152i	$-0.762781\pi$
$857$	− 1.15530e25i	− 1.35630i	0.734922	−	0.678152i	$-0.237219\pi$
$858$	1.66398e24i	0.193422i
$859$	8.55025e24	0.984095	0.492047	−	0.870568i	$-0.336249\pi$
$859$	8.55025e24	0.984095	0.492047	+	0.870568i	$0.336249\pi$
$860$	0	0
$861$	−1.27775e25	−1.44184
$862$	− 3.11172e24i	− 0.347687i
$863$	− 7.97738e24i	− 0.882610i	−0.897357	−	0.441305i	$-0.854516\pi$
$863$	− 7.97738e24i	− 0.882610i	0.897357	−	0.441305i	$-0.145484\pi$
$864$	1.36600e24	0.149652
$865$	0	0
$866$	4.87498e24	0.523685
$867$	− 7.56451e24i	− 0.804671i
$868$	9.95624e24i	1.04876i
$869$	−3.49225e22	−0.00364281
$870$	0	0
$871$	−1.30173e25	−1.33158
$872$	8.62159e24i	0.873367i
$873$	2.79085e24i	0.279972i
$874$	−2.10892e24	−0.209514
$875$	0	0
$876$	−6.96752e23	−0.0678880
$877$	− 1.63184e25i	− 1.57464i	−0.616543	−	0.787321i	$-0.711467\pi$
$877$	− 1.63184e25i	− 1.57464i	0.616543	−	0.787321i	$-0.288533\pi$
$878$	5.36225e23i	0.0512441i
$879$	−2.35583e24	−0.222966
$880$	0	0
$881$	−3.38146e24	−0.313913	−0.156956	−	0.987606i	$-0.550168\pi$
$881$	−3.38146e24	−0.313913	−0.156956	+	0.987606i	$0.550168\pi$
$882$	1.54673e24i	0.142211i
$883$	1.13867e25i	1.03689i	0.855112	+	0.518443i	$0.173488\pi$
$883$	1.13867e25i	1.03689i	−0.855112	+	0.518443i	$0.826512\pi$
$884$	−2.51730e25	−2.27034
$885$	0	0
$886$	−4.82068e24	−0.426502
$887$	− 4.89146e24i	− 0.428635i	−0.976764	−	0.214317i	$-0.931247\pi$
$887$	− 4.89146e24i	− 0.428635i	0.976764	−	0.214317i	$-0.0687527\pi$
$888$	5.54044e24i	0.480877i
$889$	2.68055e25	2.30440
$890$	0	0
$891$	−9.43952e23	−0.0796140
$892$	− 2.04467e24i	− 0.170814i
$893$	2.99752e24i	0.248041i
$894$	3.12604e24	0.256228
$895$	0	0
$896$	−1.81657e25	−1.46094
$897$	− 1.45455e25i	− 1.15876i
$898$	− 9.77889e23i	− 0.0771686i
$899$	−1.09490e25	−0.855887
$900$	0	0
$901$	−1.12039e25	−0.859429
$902$	− 4.37427e24i	− 0.332392i
$903$	1.95533e25i	1.47189i
$904$	−3.07586e24	−0.229369
$905$	0	0
$906$	3.85908e23	0.0282419
$907$	2.47170e25i	1.79198i	0.444074	+	0.895990i	$0.353533\pi$
$907$	2.47170e25i	1.79198i	−0.444074	+	0.895990i	$0.646467\pi$
$908$	− 2.43730e25i	− 1.75057i
$909$	−1.08044e24	−0.0768786
$910$	0	0
$911$	7.05356e24	0.492609	0.246304	−	0.969193i	$-0.420784\pi$
$911$	7.05356e24	0.492609	0.246304	+	0.969193i	$0.420784\pi$
$912$	3.60440e24i	0.249389i
$913$	− 1.72364e25i	− 1.18153i
$914$	2.59118e24	0.175976
$915$	0	0
$916$	−2.08542e24	−0.139021
$917$	8.61670e24i	0.569117i
$918$	1.32655e24i	0.0868080i
$919$	−5.70217e24	−0.369707	−0.184854	−	0.982766i	$-0.559181\pi$
$919$	−5.70217e24	−0.369707	−0.184854	+	0.982766i	$0.559181\pi$
$920$	0	0
$921$	−2.34681e24	−0.149373
$922$	4.45496e24i	0.280951i
$923$	1.18209e25i	0.738649i
$924$	9.59552e24	0.594096
$925$	0	0
$926$	−2.87212e24	−0.174586
$927$	8.09740e24i	0.487717i
$928$	− 1.52703e25i	− 0.911359i
$929$	−1.10885e25	−0.655752	−0.327876	−	0.944721i	$-0.606333\pi$
$929$	−1.10885e25	−0.655752	−0.327876	+	0.944721i	$0.606333\pi$
$930$	0	0
$931$	−1.44713e25	−0.840304
$932$	− 8.66249e24i	− 0.498434i
$933$	− 1.43777e25i	− 0.819776i
$934$	−4.99595e24	−0.282273
$935$	0	0
$936$	5.37933e24	0.298458
$937$	6.37448e24i	0.350476i	0.984526	+	0.175238i	$0.0560695\pi$
$937$	6.37448e24i	0.350476i	−0.984526	+	0.175238i	$0.943931\pi$
$938$	− 6.97311e24i	− 0.379929i
$939$	−6.73266e24	−0.363521
$940$	0	0
$941$	6.53698e24	0.346630	0.173315	−	0.984866i	$-0.444552\pi$
$941$	6.53698e24	0.346630	0.173315	+	0.984866i	$0.444552\pi$
$942$	3.93078e24i	0.206560i
$943$	3.82373e25i	1.99130i
$944$	8.26590e24	0.426607
$945$	0	0
$946$	−6.69395e24	−0.339319
$947$	− 1.40620e25i	− 0.706434i	−0.935541	−	0.353217i	$-0.885088\pi$
$947$	− 1.40620e25i	− 0.706434i	0.935541	−	0.353217i	$-0.114912\pi$
$948$	5.39433e22i	0.00268576i
$949$	−4.17663e24	−0.206093
$950$	0	0
$951$	5.29181e23	0.0256490
$952$	− 2.82220e25i	− 1.35573i
$953$	− 3.29795e25i	− 1.57020i	−0.619371	−	0.785098i	$-0.712612\pi$
$953$	− 3.29795e25i	− 1.57020i	0.619371	−	0.785098i	$-0.287388\pi$
$954$	1.14397e24	0.0539828
$955$	0	0
$956$	5.14066e24	0.238302
$957$	1.05523e25i	0.484837i
$958$	3.73840e24i	0.170247i
$959$	−2.33475e25	−1.05386
$960$	0	0
$961$	−1.05238e25	−0.466684
$962$	1.58688e25i	0.697520i
$963$	4.47884e24i	0.195138i
$964$	2.56727e25	1.10871
$965$	0	0
$966$	7.79176e24	0.330621
$967$	7.72859e24i	0.325069i	0.986703	+	0.162534i	$0.0519668\pi$
$967$	7.72859e24i	0.325069i	−0.986703	+	0.162534i	$0.948033\pi$
$968$	− 6.51595e24i	− 0.271667i
$969$	−1.24113e25	−0.512937
$970$	0	0
$971$	6.48625e24	0.263409	0.131704	−	0.991289i	$-0.457955\pi$
$971$	6.48625e24	0.263409	0.131704	+	0.991289i	$0.457955\pi$
$972$	1.45808e24i	0.0586974i
$973$	− 1.44308e25i	− 0.575880i
$974$	−2.45493e24	−0.0971153
$975$	0	0
$976$	−6.41184e23	−0.0249264
$977$	− 3.48651e25i	− 1.34365i	−0.740709	−	0.671826i	$-0.765510\pi$
$977$	− 3.48651e25i	− 1.34365i	0.740709	−	0.671826i	$-0.234490\pi$
$978$	− 3.13572e24i	− 0.119800i
$979$	−1.72116e25	−0.651880
$980$	0	0
$981$	−1.40084e25	−0.521438
$982$	2.20208e24i	0.0812616i
$983$	3.69470e25i	1.35168i	0.737048	+	0.675840i	$0.236219\pi$
$983$	3.69470e25i	1.35168i	−0.737048	+	0.675840i	$0.763781\pi$
$984$	−1.41412e25	−0.512893
$985$	0	0
$986$	1.48293e25	0.528647
$987$	− 1.10748e25i	− 0.391418i
$988$	2.40477e25i	0.842636i
$989$	5.85146e25	2.03280
$990$	0	0
$991$	4.08576e25	1.39523	0.697616	−	0.716472i	$-0.254244\pi$
$991$	4.08576e25	1.39523	0.697616	+	0.716472i	$0.254244\pi$
$992$	1.67728e25i	0.567881i
$993$	1.05263e25i	0.353353i
$994$	−6.33226e24	−0.210753
$995$	0	0
$996$	−2.66244e25	−0.871114
$997$	4.40879e25i	1.43024i	0.699000	+	0.715122i	$0.253629\pi$
$997$	4.40879e25i	1.43024i	−0.699000	+	0.715122i	$0.746371\pi$
$998$	1.57691e25i	0.507222i
$999$	−9.00216e24	−0.287104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.18.b.e.49.4		6
5.2	odd	4		15.18.a.c.1.2	✓	3
5.3	odd	4		75.18.a.d.1.2		3
5.4	even	2	inner	75.18.b.e.49.3		6
15.2	even	4		45.18.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.18.a.c.1.2	✓	3	5.2	odd	4
45.18.a.d.1.2		3	15.2	even	4
75.18.a.d.1.2		3	5.3	odd	4
75.18.b.e.49.3		6	5.4	even	2	inner
75.18.b.e.49.4		6	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)

\(f(q)\)	\(=\)	\(q+105.550i q^{2} +6561.00i q^{3} +119931. q^{4} -692515. q^{6} +2.39385e7i q^{7} +2.64934e7i q^{8} -4.30467e7 q^{9} -5.09412e8 q^{11} +7.86868e8i q^{12} +4.71682e9i q^{13} -2.52671e9 q^{14} +1.29232e10 q^{16} +4.44993e10i q^{17} -4.54359e9i q^{18} +4.25101e10 q^{19} -1.57060e11 q^{21} -5.37686e10i q^{22} +4.70013e11i q^{23} -1.73823e11 q^{24} -4.97862e11 q^{26} -2.82430e11i q^{27} +2.87097e12i q^{28} -3.15724e12 q^{29} +3.46790e12 q^{31} +4.83660e12i q^{32} -3.34226e12i q^{33} -4.69691e12 q^{34} -5.16264e12 q^{36} -3.18740e13i q^{37} +4.48695e12i q^{38} -3.09471e13 q^{39} +8.13537e13 q^{41} -1.65778e13i q^{42} -1.24496e14i q^{43} -6.10944e13 q^{44} -4.96099e13 q^{46} +7.05130e13i q^{47} +8.47893e13i q^{48} -3.40421e14 q^{49} -2.91960e14 q^{51} +5.65694e14i q^{52} +2.51778e14i q^{53} +2.98105e13 q^{54} -6.34213e14 q^{56} +2.78909e14i q^{57} -3.33247e14i q^{58} +6.39615e14 q^{59} -4.96148e13 q^{61} +3.66037e14i q^{62} -1.03047e15i q^{63} +1.18337e15 q^{64} +3.52776e14 q^{66} +2.75976e15i q^{67} +5.33686e15i q^{68} -3.08375e15 q^{69} +2.50612e15 q^{71} -1.14046e15i q^{72} +8.85474e14i q^{73} +3.36430e15 q^{74} +5.09829e15 q^{76} -1.21946e16i q^{77} -3.26647e15i q^{78} +6.85544e13 q^{79} +1.85302e15 q^{81} +8.58690e15i q^{82} +3.38359e16i q^{83} -1.88364e16 q^{84} +1.31405e16 q^{86} -2.07146e16i q^{87} -1.34961e16i q^{88} +3.37871e16 q^{89} -1.12914e17 q^{91} +5.63692e16i q^{92} +2.27529e16i q^{93} -7.44266e15 q^{94} -3.17329e16 q^{96} -6.48330e16i q^{97} -3.59315e16i q^{98} +2.19285e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14174 q^{4} - 3319866 q^{6} - 258280326 q^{9} + 1887127360 q^{11} - 3694967976 q^{14} - 43887742718 q^{16} + 156244985992 q^{19} - 56850855048 q^{21} + 212425667586 q^{24} + 5331480861772 q^{26} - 11550537176156 q^{29}+ \cdots - 81\!\cdots\!60 q^{99}+O(q^{100}) \) 6 * q + 14174 * q^4 - 3319866 * q^6 - 258280326 * q^9 + 1887127360 * q^11 - 3694967976 * q^14 - 43887742718 * q^16 + 156244985992 * q^19 - 56850855048 * q^21 + 212425667586 * q^24 + 5331480861772 * q^26 - 11550537176156 * q^29 + 11130926298208 * q^31 - 43987108720612 * q^34 - 610144223454 * q^36 - 55860526554300 * q^39 + 334922914576508 * q^41 - 216237355136384 * q^44 - 389063211759552 * q^46 - 137940665027894 * q^49 + 40194910362708 * q^51 + 142909345459386 * q^54 - 2202986365674360 * q^56 - 931203894392512 * q^59 - 4635619353020956 * q^61 + 1326983159085438 * q^64 + 2410726615076736 * q^66 - 3074592254042736 * q^69 + 12931216967981312 * q^71 - 20946687257609036 * q^74 - 336922991545608 * q^76 - 17481017881317760 * q^79 + 11118121133111046 * q^81 - 50706088359713784 * q^84 - 28915112504777240 * q^86 + 34435510717452852 * q^89 - 221670523226066512 * q^91 + 401443493056274240 * q^94 + 56516732556322302 * q^96 - 81234644957386560 * q^99

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