LMFDB - Embedded newform 75.14.b.e.49.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,14,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, 14); 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, 14, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-12482] 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:	$80.4231967139$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{3121})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1561x^{2} + 608400 $ x^4 + 1561*x^2 + 608400
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$27.4330i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.14.b.e.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+149.299i q^{2} -729.000i q^{3} -14098.2 q^{4} +108839. q^{6} +384225. i q^{7} -881782. i q^{8} -531441. q^{9} +6.10216e6 q^{11} +1.02776e7i q^{12} -1.13332e6i q^{13} -5.73644e7 q^{14} +1.61570e7 q^{16} -2.69912e7i q^{17} -7.93435e7i q^{18} +3.29943e8 q^{19} +2.80100e8 q^{21} +9.11046e8i q^{22} -4.70204e8i q^{23} -6.42819e8 q^{24} +1.69203e8 q^{26} +3.87420e8i q^{27} -5.41687e9i q^{28} +4.24890e9 q^{29} +7.67054e9 q^{31} -4.81134e9i q^{32} -4.44848e9i q^{33} +4.02976e9 q^{34} +7.49234e9 q^{36} -2.83184e10i q^{37} +4.92601e10i q^{38} -8.26190e8 q^{39} -2.11336e10 q^{41} +4.18187e10i q^{42} +6.93928e10i q^{43} -8.60292e10 q^{44} +7.02010e10 q^{46} +1.01233e11i q^{47} -1.17784e10i q^{48} -5.07401e10 q^{49} -1.96766e10 q^{51} +1.59777e10i q^{52} +1.81846e11i q^{53} -5.78414e10 q^{54} +3.38803e11 q^{56} -2.40528e11i q^{57} +6.34357e11i q^{58} -3.10938e11 q^{59} +1.76596e11 q^{61} +1.14520e12i q^{62} -2.04193e11i q^{63} +8.50686e11 q^{64} +6.64152e11 q^{66} +6.25757e10i q^{67} +3.80526e11i q^{68} -3.42779e11 q^{69} +1.13419e12 q^{71} +4.68615e11i q^{72} +1.17191e11i q^{73} +4.22790e12 q^{74} -4.65158e12 q^{76} +2.34461e12i q^{77} -1.23349e11i q^{78} -3.99281e12 q^{79} +2.82430e11 q^{81} -3.15523e12i q^{82} -8.58275e11i q^{83} -3.94890e12 q^{84} -1.03603e13 q^{86} -3.09745e12i q^{87} -5.38077e12i q^{88} +9.67991e11 q^{89} +4.35450e11 q^{91} +6.62901e12i q^{92} -5.59183e12i q^{93} -1.51139e13 q^{94} -3.50747e12 q^{96} +2.50052e12i q^{97} -7.57544e12i q^{98} -3.24294e12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12482 q^{4} + 190998 q^{6} - 2125764 q^{9} + 12491776 q^{11} - 110628168 q^{14} + 150297410 q^{16} - 76950688 q^{19} + 723564576 q^{21} - 1713825054 q^{24} + 315428908 q^{26} - 2252949416 q^{29} + 12200208624 q^{31}+ \cdots - 6638641929216 q^{99}+O(q^{100}) $ 4 * q - 12482 * q^4 + 190998 * q^6 - 2125764 * q^9 + 12491776 * q^11 - 110628168 * q^14 + 150297410 * q^16 - 76950688 * q^19 + 723564576 * q^21 - 1713825054 * q^24 + 315428908 * q^26 - 2252949416 * q^29 + 12200208624 * q^31 + 10809576476 * q^34 + 6633446562 * q^36 - 2567777112 * q^39 - 59215819576 * q^41 - 169799823776 * q^44 + 130737909984 * q^46 + 67188889180 * q^49 + 70205038632 * q^51 - 101504168118 * q^54 + 743417798760 * q^56 - 1445536360192 * q^59 - 459968804360 * q^61 + 2540337082142 * q^64 + 1324470131424 * q^66 - 1070557810128 * q^69 + 1473091336448 * q^71 + 7778702225812 * q^74 - 15092602309288 * q^76 - 7052697180240 * q^79 + 1129718145924 * q^81 - 6614216830872 * q^84 - 20947533908504 * q^86 - 11826242919528 * q^89 + 1011595853184 * q^91 - 30089779392352 * q^94 - 12096530386434 * q^96 - 6638641929216 * 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 $

$2$	149.299i	1.64954i	0.565472	+	0.824768i	$0.308694\pi$
$2$	149.299i	1.64954i	−0.565472	+	0.824768i	$0.691306\pi$
$3$	− 729.000i	− 0.577350i
$3$	− 729.000i	− 0.577350i
$4$	−14098.2	−1.72097
$5$	0	0
$5$	0	0
$6$	108839.	0.952359
$7$	384225.i	1.23438i	0.786814	+	0.617190i	$0.211729\pi$
$7$	384225.i	1.23438i	−0.786814	+	0.617190i	$0.788271\pi$
$8$	− 881782.i	− 1.18926i
$9$	−531441.	−0.333333
$10$	0	0
$11$	6.10216e6	1.03856	0.519280	−	0.854604i	$-0.326200\pi$
$11$	6.10216e6	1.03856	0.519280	+	0.854604i	$0.326200\pi$
$12$	1.02776e7i	0.993600i
$13$	− 1.13332e6i	− 0.0651209i	−0.999470	−	0.0325605i	$-0.989634\pi$
$13$	− 1.13332e6i	− 0.0651209i	0.999470	−	0.0325605i	$-0.0103661\pi$
$14$	−5.73644e7	−2.03615
$15$	0	0
$16$	1.61570e7	0.240757
$17$	− 2.69912e7i	− 0.271209i	−0.990763	−	0.135605i	$-0.956702\pi$
$17$	− 2.69912e7i	− 0.271209i	0.990763	−	0.135605i	$-0.0432977\pi$
$18$	− 7.93435e7i	− 0.549845i
$19$	3.29943e8	1.60894	0.804471	−	0.593992i	$-0.202449\pi$
$19$	3.29943e8	1.60894	0.804471	+	0.593992i	$0.202449\pi$
$20$	0	0
$21$	2.80100e8	0.712669
$22$	9.11046e8i	1.71314i
$23$	− 4.70204e8i	− 0.662301i	−0.943578	−	0.331151i	$-0.892563\pi$
$23$	− 4.70204e8i	− 0.662301i	0.943578	−	0.331151i	$-0.107437\pi$
$24$	−6.42819e8	−0.686619
$25$	0	0
$26$	1.69203e8	0.107419
$27$	3.87420e8i	0.192450i
$28$	− 5.41687e9i	− 2.12433i
$29$	4.24890e9	1.32645	0.663224	−	0.748421i	$-0.269188\pi$
$29$	4.24890e9	1.32645	0.663224	+	0.748421i	$0.269188\pi$
$30$	0	0
$31$	7.67054e9	1.55230	0.776149	−	0.630550i	$-0.217170\pi$
$31$	7.67054e9	1.55230	0.776149	+	0.630550i	$0.217170\pi$
$32$	− 4.81134e9i	− 0.792121i
$33$	− 4.44848e9i	− 0.599612i
$34$	4.02976e9	0.447369
$35$	0	0
$36$	7.49234e9	0.573655
$37$	− 2.83184e10i	− 1.81450i	−0.420590	−	0.907251i	$-0.638177\pi$
$37$	− 2.83184e10i	− 1.81450i	0.420590	−	0.907251i	$-0.361823\pi$
$38$	4.92601e10i	2.65401i
$39$	−8.26190e8	−0.0375976
$40$	0	0
$41$	−2.11336e10	−0.694831	−0.347415	−	0.937711i	$-0.612941\pi$
$41$	−2.11336e10	−0.694831	−0.347415	+	0.937711i	$0.612941\pi$
$42$	4.18187e10i	1.17557i
$43$	6.93928e10i	1.67405i	0.547162	+	0.837026i	$0.315708\pi$
$43$	6.93928e10i	1.67405i	−0.547162	+	0.837026i	$0.684292\pi$
$44$	−8.60292e10	−1.78732
$45$	0	0
$46$	7.02010e10	1.09249
$47$	1.01233e11i	1.36989i	0.728597	+	0.684943i	$0.240173\pi$
$47$	1.01233e11i	1.36989i	−0.728597	+	0.684943i	$0.759827\pi$
$48$	− 1.17784e10i	− 0.139001i
$49$	−5.07401e10	−0.523693
$50$	0	0
$51$	−1.96766e10	−0.156583
$52$	1.59777e10i	0.112071i
$53$	1.81846e11i	1.12697i	0.826127	+	0.563484i	$0.190539\pi$
$53$	1.81846e11i	1.12697i	−0.826127	+	0.563484i	$0.809461\pi$
$54$	−5.78414e10	−0.317453
$55$	0	0
$56$	3.38803e11	1.46800
$57$	− 2.40528e11i	− 0.928923i
$58$	6.34357e11i	2.18802i
$59$	−3.10938e11	−0.959701	−0.479850	−	0.877350i	$-0.659309\pi$
$59$	−3.10938e11	−0.959701	−0.479850	+	0.877350i	$0.659309\pi$
$60$	0	0
$61$	1.76596e11	0.438870	0.219435	−	0.975627i	$-0.429579\pi$
$61$	1.76596e11	0.438870	0.219435	+	0.975627i	$0.429579\pi$
$62$	1.14520e12i	2.56057i
$63$	− 2.04193e11i	− 0.411460i
$64$	8.50686e11	1.54739
$65$	0	0
$66$	6.64152e11	0.989082
$67$	6.25757e10i	0.0845123i	0.999107	+	0.0422561i	$0.0134546\pi$
$67$	6.25757e10i	0.0845123i	−0.999107	+	0.0422561i	$0.986545\pi$
$68$	3.80526e11i	0.466742i
$69$	−3.42779e11	−0.382380
$70$	0	0
$71$	1.13419e12	1.05076	0.525382	−	0.850867i	$-0.323923\pi$
$71$	1.13419e12	1.05076	0.525382	+	0.850867i	$0.323923\pi$
$72$	4.68615e11i	0.396419i
$73$	1.17191e11i	0.0906350i	0.998973	+	0.0453175i	$0.0144299\pi$
$73$	1.17191e11i	0.0906350i	−0.998973	+	0.0453175i	$0.985570\pi$
$74$	4.22790e12	2.99308
$75$	0	0
$76$	−4.65158e12	−2.76893
$77$	2.34461e12i	1.28198i
$78$	− 1.23349e11i	− 0.0620185i
$79$	−3.99281e12	−1.84800	−0.924002	−	0.382387i	$-0.875102\pi$
$79$	−3.99281e12	−1.84800	−0.924002	+	0.382387i	$0.875102\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	− 3.15523e12i	− 1.14615i
$83$	− 8.58275e11i	− 0.288150i	−0.989567	−	0.144075i	$-0.953979\pi$
$83$	− 8.58275e11i	− 0.288150i	0.989567	−	0.144075i	$-0.0460207\pi$
$84$	−3.94890e12	−1.22648
$85$	0	0
$86$	−1.03603e13	−2.76141
$87$	− 3.09745e12i	− 0.765825i
$88$	− 5.38077e12i	− 1.23512i
$89$	9.67991e11	0.206460	0.103230	−	0.994657i	$-0.467082\pi$
$89$	9.67991e11	0.206460	0.103230	+	0.994657i	$0.467082\pi$
$90$	0	0
$91$	4.35450e11	0.0803839
$92$	6.62901e12i	1.13980i
$93$	− 5.59183e12i	− 0.896220i
$94$	−1.51139e13	−2.25967
$95$	0	0
$96$	−3.50747e12	−0.457331
$97$	2.50052e12i	0.304799i	0.988319	+	0.152400i	$0.0487000\pi$
$97$	2.50052e12i	0.304799i	−0.988319	+	0.152400i	$0.951300\pi$
$98$	− 7.57544e12i	− 0.863850i
$99$	−3.24294e12	−0.346186
$100$	0	0
$101$	−3.51016e12	−0.329032	−0.164516	−	0.986374i	$-0.552606\pi$
$101$	−3.51016e12	−0.329032	−0.164516	+	0.986374i	$0.552606\pi$
$102$	− 2.93769e12i	− 0.258289i
$103$	1.08186e13i	0.892747i	0.894847	+	0.446373i	$0.147285\pi$
$103$	1.08186e13i	0.892747i	−0.894847	+	0.446373i	$0.852715\pi$
$104$	−9.99340e11	−0.0774456
$105$	0	0
$106$	−2.71495e13	−1.85897
$107$	1.49010e13i	0.959892i	0.877298	+	0.479946i	$0.159344\pi$
$107$	1.49010e13i	0.959892i	−0.877298	+	0.479946i	$0.840656\pi$
$108$	− 5.46191e12i	− 0.331200i
$109$	2.01251e13	1.14939	0.574694	−	0.818369i	$-0.305121\pi$
$109$	2.01251e13	1.14939	0.574694	+	0.818369i	$0.305121\pi$
$110$	0	0
$111$	−2.06441e13	−1.04760
$112$	6.20791e12i	0.297186i
$113$	− 2.87905e13i	− 1.30088i	−0.759556	−	0.650442i	$-0.774583\pi$
$113$	− 2.87905e13i	− 1.30088i	0.759556	−	0.650442i	$-0.225417\pi$
$114$	3.59106e13	1.53229
$115$	0	0
$116$	−5.99017e13	−2.28277
$117$	6.02292e11i	0.0217070i
$118$	− 4.64227e13i	− 1.58306i
$119$	1.03707e13	0.334775
$120$	0	0
$121$	2.71366e12	0.0786051
$122$	2.63655e13i	0.723932i
$123$	1.54064e13i	0.401161i
$124$	−1.08140e14	−2.67145
$125$	0	0
$126$	3.04858e13	0.678718
$127$	7.01983e13i	1.48458i	0.670081	+	0.742288i	$0.266259\pi$
$127$	7.01983e13i	1.48458i	−0.670081	+	0.742288i	$0.733741\pi$
$128$	8.75919e13i	1.76035i
$129$	5.05873e13	0.966515
$130$	0	0
$131$	9.26965e13	1.60251	0.801254	−	0.598325i	$-0.204167\pi$
$131$	9.26965e13	1.60251	0.801254	+	0.598325i	$0.204167\pi$
$132$	6.27153e13i	1.03191i
$133$	1.26772e14i	1.98604i
$134$	−9.34249e12	−0.139406
$135$	0	0
$136$	−2.38004e13	−0.322538
$137$	− 4.89202e13i	− 0.632127i	−0.948738	−	0.316064i	$-0.897639\pi$
$137$	− 4.89202e13i	− 0.632127i	0.948738	−	0.316064i	$-0.102361\pi$
$138$	− 5.11765e13i	− 0.630749i
$139$	−4.48733e13	−0.527706	−0.263853	−	0.964563i	$-0.584993\pi$
$139$	−4.48733e13	−0.527706	−0.263853	+	0.964563i	$0.584993\pi$
$140$	0	0
$141$	7.37986e13	0.790904
$142$	1.69333e14i	1.73327i
$143$	− 6.91570e12i	− 0.0676319i
$144$	−8.58647e12	−0.0802525
$145$	0	0
$146$	−1.74965e13	−0.149506
$147$	3.69895e13i	0.302354i
$148$	3.99237e14i	3.12270i
$149$	−1.30203e14	−0.974789	−0.487395	−	0.873182i	$-0.662053\pi$
$149$	−1.30203e14	−0.974789	−0.487395	+	0.873182i	$0.662053\pi$
$150$	0	0
$151$	−1.94653e14	−1.33632	−0.668160	−	0.744018i	$-0.732918\pi$
$151$	−1.94653e14	−1.33632	−0.668160	+	0.744018i	$0.732918\pi$
$152$	− 2.90938e14i	− 1.91345i
$153$	1.43442e13i	0.0904031i
$154$	−3.50047e14	−2.11466
$155$	0	0
$156$	1.16477e13	0.0647041
$157$	− 7.93587e13i	− 0.422909i	−0.977388	−	0.211454i	$-0.932180\pi$
$157$	− 7.93587e13i	− 0.422909i	0.977388	−	0.211454i	$-0.0678200\pi$
$158$	− 5.96123e14i	− 3.04835i
$159$	1.32566e14	0.650655
$160$	0	0
$161$	1.80664e14	0.817531
$162$	4.21664e13i	0.183282i
$163$	1.62086e14i	0.676904i	0.940984	+	0.338452i	$0.109903\pi$
$163$	1.62086e14i	0.676904i	−0.940984	+	0.338452i	$0.890097\pi$
$164$	2.97945e14	1.19578
$165$	0	0
$166$	1.28139e14	0.475314
$167$	2.62269e14i	0.935600i	0.883834	+	0.467800i	$0.154953\pi$
$167$	2.62269e14i	0.935600i	−0.883834	+	0.467800i	$0.845047\pi$
$168$	− 2.46987e14i	− 0.847548i
$169$	3.01591e14	0.995759
$170$	0	0
$171$	−1.75345e14	−0.536314
$172$	− 9.78310e14i	− 2.88099i
$173$	− 7.87595e13i	− 0.223359i	−0.993744	−	0.111679i	$-0.964377\pi$
$173$	− 7.87595e13i	− 0.223359i	0.993744	−	0.111679i	$-0.0356230\pi$
$174$	4.62446e14	1.26325
$175$	0	0
$176$	9.85923e13	0.250041
$177$	2.26674e14i	0.554083i
$178$	1.44520e14i	0.340563i
$179$	7.53329e14	1.71175	0.855874	−	0.517184i	$-0.173020\pi$
$179$	7.53329e14	1.71175	0.855874	+	0.517184i	$0.173020\pi$
$180$	0	0
$181$	−4.79674e14	−1.01399	−0.506997	−	0.861948i	$-0.669245\pi$
$181$	−4.79674e14	−1.01399	−0.506997	+	0.861948i	$0.669245\pi$
$182$	6.50122e13i	0.132596i
$183$	− 1.28738e14i	− 0.253382i
$184$	−4.14618e14	−0.787647
$185$	0	0
$186$	8.34853e14	1.47835
$187$	− 1.64705e14i	− 0.281667i
$188$	− 1.42719e15i	− 2.35753i
$189$	−1.48857e14	−0.237556
$190$	0	0
$191$	4.70960e14	0.701888	0.350944	−	0.936397i	$-0.385861\pi$
$191$	4.70960e14	0.701888	0.350944	+	0.936397i	$0.385861\pi$
$192$	− 6.20150e14i	− 0.893385i
$193$	1.24350e14i	0.173191i	0.996244	+	0.0865954i	$0.0275987\pi$
$193$	1.24350e14i	0.173191i	−0.996244	+	0.0865954i	$0.972401\pi$
$194$	−3.73324e14	−0.502777
$195$	0	0
$196$	7.15342e14	0.901258
$197$	− 4.20205e12i	− 0.00512190i	−0.999997	−	0.00256095i	$-0.999185\pi$
$197$	− 4.20205e12i	− 0.00512190i	0.999997	−	0.00256095i	$-0.000815177\pi$
$198$	− 4.84167e14i	− 0.571047i
$199$	−1.09925e15	−1.25474	−0.627370	−	0.778722i	$-0.715868\pi$
$199$	−1.09925e15	−1.25474	−0.627370	+	0.778722i	$0.715868\pi$
$200$	0	0
$201$	4.56177e13	0.0487932
$202$	− 5.24063e14i	− 0.542749i
$203$	1.63254e15i	1.63734i
$204$	2.77404e14	0.269474
$205$	0	0
$206$	−1.61520e15	−1.47262
$207$	2.49886e14i	0.220767i
$208$	− 1.83110e13i	− 0.0156783i
$209$	2.01336e15	1.67098
$210$	0	0
$211$	−4.43736e14	−0.346170	−0.173085	−	0.984907i	$-0.555373\pi$
$211$	−4.43736e14	−0.346170	−0.173085	+	0.984907i	$0.555373\pi$
$212$	− 2.56370e15i	− 1.93947i
$213$	− 8.26822e14i	− 0.606659i
$214$	−2.22471e15	−1.58337
$215$	0	0
$216$	3.41620e14	0.228873
$217$	2.94722e15i	1.91613i
$218$	3.00466e15i	1.89595i
$219$	8.54323e13	0.0523281
$220$	0	0
$221$	−3.05897e13	−0.0176614
$222$	− 3.08214e15i	− 1.72806i
$223$	2.15493e15i	1.17342i	0.809799	+	0.586708i	$0.199576\pi$
$223$	2.15493e15i	1.17342i	−0.809799	+	0.586708i	$0.800424\pi$
$224$	1.84864e15	0.977777
$225$	0	0
$226$	4.29838e15	2.14586
$227$	3.86902e15i	1.87686i	0.345466	+	0.938431i	$0.387721\pi$
$227$	3.86902e15i	1.87686i	−0.345466	+	0.938431i	$0.612279\pi$
$228$	3.39100e15i	1.59864i
$229$	1.09844e15	0.503324	0.251662	−	0.967815i	$-0.419023\pi$
$229$	1.09844e15	0.503324	0.251662	+	0.967815i	$0.419023\pi$
$230$	0	0
$231$	1.70922e15	0.740149
$232$	− 3.74661e15i	− 1.57749i
$233$	− 4.34127e15i	− 1.77747i	−0.458417	−	0.888737i	$-0.651583\pi$
$233$	− 4.34127e15i	− 1.77747i	0.458417	−	0.888737i	$-0.348417\pi$
$234$	−8.99216e13	−0.0358064
$235$	0	0
$236$	4.38365e15	1.65161
$237$	2.91076e15i	1.06695i
$238$	1.54834e15i	0.552224i
$239$	1.70803e15	0.592801	0.296400	−	0.955064i	$-0.404214\pi$
$239$	1.70803e15	0.592801	0.296400	+	0.955064i	$0.404214\pi$
$240$	0	0
$241$	2.87167e15	0.944114	0.472057	−	0.881568i	$-0.343512\pi$
$241$	2.87167e15	0.944114	0.472057	+	0.881568i	$0.343512\pi$
$242$	4.05146e14i	0.129662i
$243$	− 2.05891e14i	− 0.0641500i
$244$	−2.48967e15	−0.755281
$245$	0	0
$246$	−2.30016e15	−0.661729
$247$	− 3.73931e14i	− 0.104776i
$248$	− 6.76374e15i	− 1.84608i
$249$	−6.25683e14	−0.166364
$250$	0	0
$251$	−9.41402e14	−0.237627	−0.118813	−	0.992917i	$-0.537909\pi$
$251$	−9.41402e14	−0.237627	−0.118813	+	0.992917i	$0.537909\pi$
$252$	2.87875e15i	0.708108i
$253$	− 2.86926e15i	− 0.687839i
$254$	−1.04805e16	−2.44886
$255$	0	0
$256$	−6.10855e15	−1.35637
$257$	− 3.31316e15i	− 0.717260i	−0.933480	−	0.358630i	$-0.883244\pi$
$257$	− 3.31316e15i	− 0.717260i	0.933480	−	0.358630i	$-0.116756\pi$
$258$	7.55263e15i	1.59430i
$259$	1.08806e16	2.23978
$260$	0	0
$261$	−2.25804e15	−0.442149
$262$	1.38395e16i	2.64339i
$263$	3.12580e15i	0.582437i	0.956657	+	0.291219i	$0.0940607\pi$
$263$	3.12580e15i	0.582437i	−0.956657	+	0.291219i	$0.905939\pi$
$264$	−3.92258e15	−0.713094
$265$	0	0
$266$	−1.89270e16	−3.27605
$267$	− 7.05666e14i	− 0.119200i
$268$	− 8.82202e14i	− 0.145443i
$269$	−2.51897e15	−0.405352	−0.202676	−	0.979246i	$-0.564964\pi$
$269$	−2.51897e15	−0.405352	−0.202676	+	0.979246i	$0.564964\pi$
$270$	0	0
$271$	−9.26522e14	−0.142087	−0.0710436	−	0.997473i	$-0.522633\pi$
$271$	−9.26522e14	−0.142087	−0.0710436	+	0.997473i	$0.522633\pi$
$272$	− 4.36096e14i	− 0.0652957i
$273$	− 3.17443e14i	− 0.0464097i
$274$	7.30373e15	1.04272
$275$	0	0
$276$	4.83255e15	0.658063
$277$	− 4.03418e14i	− 0.0536583i	−0.999640	−	0.0268292i	$-0.991459\pi$
$277$	− 4.03418e14i	− 0.0536583i	0.999640	−	0.0268292i	$-0.00854101\pi$
$278$	− 6.69953e15i	− 0.870469i
$279$	−4.07644e15	−0.517433
$280$	0	0
$281$	−8.56272e15	−1.03758	−0.518789	−	0.854903i	$-0.673617\pi$
$281$	−8.56272e15	−1.03758	−0.518789	+	0.854903i	$0.673617\pi$
$282$	1.10180e16i	1.30462i
$283$	3.74874e15i	0.433784i	0.976196	+	0.216892i	$0.0695919\pi$
$283$	3.74874e15i	0.433784i	−0.976196	+	0.216892i	$0.930408\pi$
$284$	−1.59899e16	−1.80833
$285$	0	0
$286$	1.03251e15	0.111561
$287$	− 8.12008e15i	− 0.857685i
$288$	2.55694e15i	0.264040i
$289$	9.17605e15	0.926445
$290$	0	0
$291$	1.82288e15	0.175976
$292$	− 1.65218e15i	− 0.155980i
$293$	− 2.47408e15i	− 0.228441i	−0.993455	−	0.114221i	$-0.963563\pi$
$293$	− 2.47408e15i	− 0.228441i	0.993455	−	0.114221i	$-0.0364371\pi$
$294$	−5.52250e15	−0.498744
$295$	0	0
$296$	−2.49706e16	−2.15791
$297$	2.36410e15i	0.199871i
$298$	− 1.94392e16i	− 1.60795i
$299$	−5.32892e14	−0.0431297
$300$	0	0
$301$	−2.66625e16	−2.06642
$302$	− 2.90614e16i	− 2.20431i
$303$	2.55891e15i	0.189967i
$304$	5.33087e15	0.387365
$305$	0	0
$306$	−2.14158e15	−0.149123
$307$	3.12228e15i	0.212849i	0.994321	+	0.106425i	$0.0339403\pi$
$307$	3.12228e15i	0.212849i	−0.994321	+	0.106425i	$0.966060\pi$
$308$	− 3.30546e16i	− 2.20624i
$309$	7.88675e15	0.515428
$310$	0	0
$311$	8.71900e15	0.546417	0.273209	−	0.961955i	$-0.411915\pi$
$311$	8.71900e15	0.546417	0.273209	+	0.961955i	$0.411915\pi$
$312$	7.28519e14i	0.0447132i
$313$	− 1.94308e16i	− 1.16803i	−0.811744	−	0.584013i	$-0.801482\pi$
$313$	− 1.94308e16i	− 1.16803i	0.811744	−	0.584013i	$-0.198518\pi$
$314$	1.18482e16	0.697603
$315$	0	0
$316$	5.62913e16	3.18035
$317$	2.73728e16i	1.51508i	0.652791	+	0.757538i	$0.273598\pi$
$317$	2.73728e16i	1.51508i	−0.652791	+	0.757538i	$0.726402\pi$
$318$	1.97920e16i	1.07328i
$319$	2.59275e16	1.37759
$320$	0	0
$321$	1.08629e16	0.554194
$322$	2.69730e16i	1.34855i
$323$	− 8.90556e15i	− 0.436360i
$324$	−3.98173e15	−0.191218
$325$	0	0
$326$	−2.41993e16	−1.11658
$327$	− 1.46712e16i	− 0.663599i
$328$	1.86352e16i	0.826333i
$329$	−3.88961e16	−1.69096
$330$	0	0
$331$	1.48805e16	0.621923	0.310961	−	0.950423i	$-0.399349\pi$
$331$	1.48805e16	0.621923	0.310961	+	0.950423i	$0.399349\pi$
$332$	1.21001e16i	0.495897i
$333$	1.50496e16i	0.604834i
$334$	−3.91565e16	−1.54330
$335$	0	0
$336$	4.52557e15	0.171580
$337$	− 9.51124e15i	− 0.353706i	−0.984237	−	0.176853i	$-0.943408\pi$
$337$	− 9.51124e15i	− 0.353706i	0.984237	−	0.176853i	$-0.0565917\pi$
$338$	4.50271e16i	1.64254i
$339$	−2.09883e16	−0.751066
$340$	0	0
$341$	4.68069e16	1.61215
$342$	− 2.61788e16i	− 0.884668i
$343$	1.77316e16i	0.587943i
$344$	6.11893e16	1.99088
$345$	0	0
$346$	1.17587e16	0.368438
$347$	2.02295e16i	0.622076i	0.950398	+	0.311038i	$0.100677\pi$
$347$	2.02295e16i	0.622076i	−0.950398	+	0.311038i	$0.899323\pi$
$348$	4.36683e16i	1.31796i
$349$	−8.80222e15	−0.260751	−0.130376	−	0.991465i	$-0.541618\pi$
$349$	−8.80222e15	−0.260751	−0.130376	+	0.991465i	$0.541618\pi$
$350$	0	0
$351$	4.39071e14	0.0125325
$352$	− 2.93596e16i	− 0.822664i
$353$	− 1.67276e16i	− 0.460148i	−0.973173	−	0.230074i	$-0.926103\pi$
$353$	− 1.67276e16i	− 0.460148i	0.973173	−	0.230074i	$-0.0738968\pi$
$354$	−3.38421e16	−0.913980
$355$	0	0
$356$	−1.36469e16	−0.355311
$357$	− 7.56025e15i	− 0.193283i
$358$	1.12471e17i	2.82359i
$359$	7.94820e16	1.95954	0.979772	−	0.200119i	$-0.0641328\pi$
$359$	7.94820e16	1.95954	0.979772	+	0.200119i	$0.0641328\pi$
$360$	0	0
$361$	6.68093e16	1.58869
$362$	− 7.16148e16i	− 1.67262i
$363$	− 1.97826e15i	− 0.0453827i
$364$	−6.13904e15	−0.138338
$365$	0	0
$366$	1.92205e16	0.417962
$367$	− 6.97003e16i	− 1.48904i	−0.667602	−	0.744518i	$-0.732679\pi$
$367$	− 6.97003e16i	− 1.48904i	0.667602	−	0.744518i	$-0.267321\pi$
$368$	− 7.59707e15i	− 0.159454i
$369$	1.12313e16	0.231610
$370$	0	0
$371$	−6.98700e16	−1.39111
$372$	7.88344e16i	1.54236i
$373$	− 2.07893e16i	− 0.399698i	−0.979827	−	0.199849i	$-0.935955\pi$
$373$	− 2.07893e16i	− 0.399698i	0.979827	−	0.199849i	$-0.0640452\pi$
$374$	2.45902e16	0.464619
$375$	0	0
$376$	8.92650e16	1.62915
$377$	− 4.81537e15i	− 0.0863795i
$378$	− 2.22241e16i	− 0.391858i
$379$	−7.77269e16	−1.34715	−0.673576	−	0.739118i	$-0.735243\pi$
$379$	−7.77269e16	−1.34715	−0.673576	+	0.739118i	$0.735243\pi$
$380$	0	0
$381$	5.11746e16	0.857120
$382$	7.03139e16i	1.15779i
$383$	− 6.62740e16i	− 1.07288i	−0.843938	−	0.536440i	$-0.819769\pi$
$383$	− 6.62740e16i	− 1.07288i	0.843938	−	0.536440i	$-0.180231\pi$
$384$	6.38545e16	1.01634
$385$	0	0
$386$	−1.85654e16	−0.285684
$387$	− 3.68782e16i	− 0.558018i
$388$	− 3.52527e16i	− 0.524549i
$389$	3.03230e15	0.0443711	0.0221855	−	0.999754i	$-0.492938\pi$
$389$	3.03230e15	0.0443711	0.0221855	+	0.999754i	$0.492938\pi$
$390$	0	0
$391$	−1.26914e16	−0.179622
$392$	4.47417e16i	0.622806i
$393$	− 6.75758e16i	− 0.925208i
$394$	6.27362e14	0.00844876
$395$	0	0
$396$	4.57194e16	0.595775
$397$	3.47691e16i	0.445712i	0.974851	+	0.222856i	$0.0715380\pi$
$397$	3.47691e16i	0.445712i	−0.974851	+	0.222856i	$0.928462\pi$
$398$	− 1.64118e17i	− 2.06974i
$399$	9.24171e16	1.14664
$400$	0	0
$401$	4.47291e16	0.537220	0.268610	−	0.963249i	$-0.413436\pi$
$401$	4.47291e16	0.537220	0.268610	+	0.963249i	$0.413436\pi$
$402$	6.81067e15i	0.0804861i
$403$	− 8.69317e15i	− 0.101087i
$404$	4.94867e16	0.566252
$405$	0	0
$406$	−2.43736e17	−2.70085
$407$	− 1.72803e17i	− 1.88447i
$408$	1.73505e16i	0.186217i
$409$	−4.58740e16	−0.484579	−0.242290	−	0.970204i	$-0.577898\pi$
$409$	−4.58740e16	−0.484579	−0.242290	+	0.970204i	$0.577898\pi$
$410$	0	0
$411$	−3.56628e16	−0.364959
$412$	− 1.52522e17i	− 1.53639i
$413$	− 1.19470e17i	− 1.18463i
$414$	−3.73077e16	−0.364163
$415$	0	0
$416$	−5.45279e15	−0.0515836
$417$	3.27126e16i	0.304671i
$418$	3.00593e17i	2.75634i
$419$	−7.04588e16	−0.636128	−0.318064	−	0.948069i	$-0.603033\pi$
$419$	−7.04588e16	−0.636128	−0.318064	+	0.948069i	$0.603033\pi$
$420$	0	0
$421$	−2.11625e16	−0.185239	−0.0926196	−	0.995702i	$-0.529524\pi$
$421$	−2.11625e16	−0.185239	−0.0926196	+	0.995702i	$0.529524\pi$
$422$	− 6.62493e16i	− 0.571019i
$423$	− 5.37991e16i	− 0.456628i
$424$	1.60349e17	1.34026
$425$	0	0
$426$	1.23444e17	1.00070
$427$	6.78526e16i	0.541733i
$428$	− 2.10077e17i	− 1.65194i
$429$	−5.04154e15	−0.0390473
$430$	0	0
$431$	4.09241e16	0.307523	0.153761	−	0.988108i	$-0.450861\pi$
$431$	4.09241e16	0.307523	0.153761	+	0.988108i	$0.450861\pi$
$432$	6.25954e15i	0.0463338i
$433$	− 8.57212e16i	− 0.625053i	−0.949909	−	0.312527i	$-0.898825\pi$
$433$	− 8.57212e16i	− 0.625053i	0.949909	−	0.312527i	$-0.101175\pi$
$434$	−4.40016e17	−3.16072
$435$	0	0
$436$	−2.83727e17	−1.97806
$437$	− 1.55141e17i	− 1.06560i
$438$	1.27549e16i	0.0863171i
$439$	−3.70635e16	−0.247131	−0.123565	−	0.992336i	$-0.539433\pi$
$439$	−3.70635e16	−0.247131	−0.123565	+	0.992336i	$0.539433\pi$
$440$	0	0
$441$	2.69654e16	0.174564
$442$	− 4.56700e15i	− 0.0291331i
$443$	1.55015e16i	0.0974431i	0.998812	+	0.0487215i	$0.0155147\pi$
$443$	1.55015e16i	0.0974431i	−0.998812	+	0.0487215i	$0.984485\pi$
$444$	2.91044e17	1.80289
$445$	0	0
$446$	−3.21729e17	−1.93559
$447$	9.49181e16i	0.562795i
$448$	3.26855e17i	1.91006i
$449$	2.75973e17	1.58952	0.794759	−	0.606925i	$-0.207597\pi$
$449$	2.75973e17	1.58952	0.794759	+	0.606925i	$0.207597\pi$
$450$	0	0
$451$	−1.28961e17	−0.721623
$452$	4.05892e17i	2.23878i
$453$	1.41902e17i	0.771524i
$454$	−5.77640e17	−3.09595
$455$	0	0
$456$	−2.12093e17	−1.10473
$457$	2.05141e17i	1.05341i	0.850049	+	0.526703i	$0.176572\pi$
$457$	2.05141e17i	1.05341i	−0.850049	+	0.526703i	$0.823428\pi$
$458$	1.63997e17i	0.830250i
$459$	1.04570e16	0.0521943
$460$	0	0
$461$	2.29523e17	1.11370	0.556852	−	0.830612i	$-0.312009\pi$
$461$	2.29523e17	1.11370	0.556852	+	0.830612i	$0.312009\pi$
$462$	2.55184e17i	1.22090i
$463$	− 1.90465e17i	− 0.898544i	−0.893395	−	0.449272i	$-0.851683\pi$
$463$	− 1.90465e17i	− 0.898544i	0.893395	−	0.449272i	$-0.148317\pi$
$464$	6.86493e16	0.319352
$465$	0	0
$466$	6.48147e17	2.93201
$467$	− 1.67739e17i	− 0.748297i	−0.927369	−	0.374148i	$-0.877935\pi$
$467$	− 1.67739e17i	− 0.748297i	0.927369	−	0.374148i	$-0.122065\pi$
$468$	− 8.49121e15i	− 0.0373570i
$469$	−2.40432e16	−0.104320
$470$	0	0
$471$	−5.78525e16	−0.244166
$472$	2.74179e17i	1.14133i
$473$	4.23446e17i	1.73860i
$474$	−4.34573e17	−1.75996
$475$	0	0
$476$	−1.46208e17	−0.576137
$477$	− 9.66406e16i	− 0.375656i
$478$	2.55007e17i	0.977845i
$479$	3.71727e17	1.40619	0.703095	−	0.711096i	$-0.251801\pi$
$479$	3.71727e17	1.40619	0.703095	+	0.711096i	$0.251801\pi$
$480$	0	0
$481$	−3.20938e16	−0.118162
$482$	4.28738e17i	1.55735i
$483$	− 1.31704e17i	− 0.472002i
$484$	−3.82576e16	−0.135277
$485$	0	0
$486$	3.07393e16	0.105818
$487$	4.87760e16i	0.165679i	0.996563	+	0.0828397i	$0.0263989\pi$
$487$	4.87760e16i	0.165679i	−0.996563	+	0.0828397i	$0.973601\pi$
$488$	− 1.55719e17i	− 0.521930i
$489$	1.18161e17	0.390810
$490$	0	0
$491$	−6.19444e16	−0.199513	−0.0997567	−	0.995012i	$-0.531806\pi$
$491$	−6.19444e16	−0.199513	−0.0997567	+	0.995012i	$0.531806\pi$
$492$	− 2.17202e17i	− 0.690384i
$493$	− 1.14683e17i	− 0.359745i
$494$	5.58274e16	0.172831
$495$	0	0
$496$	1.23933e17	0.373727
$497$	4.35783e17i	1.29704i
$498$	− 9.34137e16i	− 0.274423i
$499$	2.79726e17	0.811108	0.405554	−	0.914071i	$-0.367079\pi$
$499$	2.79726e17	0.811108	0.405554	+	0.914071i	$0.367079\pi$
$500$	0	0
$501$	1.91194e17	0.540169
$502$	− 1.40550e17i	− 0.391974i
$503$	− 3.38759e17i	− 0.932609i	−0.884624	−	0.466304i	$-0.845585\pi$
$503$	− 3.38759e17i	− 0.932609i	0.884624	−	0.466304i	$-0.154415\pi$
$504$	−1.80054e17	−0.489332
$505$	0	0
$506$	4.28378e17	1.13461
$507$	− 2.19860e17i	− 0.574902i
$508$	− 9.89666e17i	− 2.55490i
$509$	3.81496e17	0.972354	0.486177	−	0.873860i	$-0.338391\pi$
$509$	3.81496e17	0.972354	0.486177	+	0.873860i	$0.338391\pi$
$510$	0	0
$511$	−4.50278e16	−0.111878
$512$	− 1.94447e17i	− 0.477032i
$513$	1.27827e17i	0.309641i
$514$	4.94651e17	1.18315
$515$	0	0
$516$	−7.13188e17	−1.66334
$517$	6.17738e17i	1.42271i
$518$	1.62447e18i	3.69460i
$519$	−5.74157e16	−0.128956
$520$	0	0
$521$	−3.97604e17	−0.870975	−0.435487	−	0.900195i	$-0.643424\pi$
$521$	−3.97604e17	−0.870975	−0.435487	+	0.900195i	$0.643424\pi$
$522$	− 3.37123e17i	− 0.729340i
$523$	− 6.78177e16i	− 0.144905i	−0.997372	−	0.0724523i	$-0.976918\pi$
$523$	− 6.78177e16i	− 0.144905i	0.997372	−	0.0724523i	$-0.0230825\pi$
$524$	−1.30685e18	−2.75786
$525$	0	0
$526$	−4.66679e17	−0.960750
$527$	− 2.07037e17i	− 0.420998i
$528$	− 7.18738e16i	− 0.144361i
$529$	2.82944e17	0.561357
$530$	0	0
$531$	1.65245e17	0.319900
$532$	− 1.78726e18i	− 3.41792i
$533$	2.39511e16i	0.0452480i
$534$	1.05355e17	0.196624
$535$	0	0
$536$	5.51781e16	0.100507
$537$	− 5.49177e17i	− 0.988278i
$538$	− 3.76079e17i	− 0.668643i
$539$	−3.09624e17	−0.543886
$540$	0	0
$541$	−9.53886e17	−1.63574	−0.817870	−	0.575403i	$-0.804845\pi$
$541$	−9.53886e17	−1.63574	−0.817870	+	0.575403i	$0.804845\pi$
$542$	− 1.38329e17i	− 0.234378i
$543$	3.49683e17i	0.585430i
$544$	−1.29864e17	−0.214830
$545$	0	0
$546$	4.73939e16	0.0765544
$547$	5.54825e17i	0.885601i	0.896620	+	0.442800i	$0.146015\pi$
$547$	5.54825e17i	0.885601i	−0.896620	+	0.442800i	$0.853985\pi$
$548$	6.89684e17i	1.08787i
$549$	−9.38502e16	−0.146290
$550$	0	0
$551$	1.40190e18	2.13418
$552$	3.02256e17i	0.454748i
$553$	− 1.53414e18i	− 2.28114i
$554$	6.02299e16	0.0885113
$555$	0	0
$556$	6.32630e17	0.908163
$557$	8.83747e17i	1.25392i	0.779052	+	0.626959i	$0.215701\pi$
$557$	8.83747e17i	1.25392i	−0.779052	+	0.626959i	$0.784299\pi$
$558$	− 6.08608e17i	− 0.853523i
$559$	7.86442e16	0.109016
$560$	0	0
$561$	−1.20070e17	−0.162620
$562$	− 1.27840e18i	− 1.71152i
$563$	− 1.34034e18i	− 1.77383i	−0.461937	−	0.886913i	$-0.652846\pi$
$563$	− 1.34034e18i	− 1.77383i	0.461937	−	0.886913i	$-0.347154\pi$
$564$	−1.04042e18	−1.36112
$565$	0	0
$566$	−5.59682e17	−0.715541
$567$	1.08517e17i	0.137153i
$568$	− 1.00010e18i	− 1.24963i
$569$	2.62408e16	0.0324151	0.0162075	−	0.999869i	$-0.494841\pi$
$569$	2.62408e16	0.0324151	0.0162075	+	0.999869i	$0.494841\pi$
$570$	0	0
$571$	6.67128e17	0.805517	0.402758	−	0.915306i	$-0.368051\pi$
$571$	6.67128e17	0.805517	0.402758	+	0.915306i	$0.368051\pi$
$572$	9.74986e16i	0.116392i
$573$	− 3.43330e17i	− 0.405235i
$574$	1.21232e18	1.41478
$575$	0	0
$576$	−4.52089e17	−0.515796
$577$	− 1.37034e17i	− 0.154591i	−0.997008	−	0.0772955i	$-0.975372\pi$
$577$	− 1.37034e17i	− 0.154591i	0.997008	−	0.0772955i	$-0.0246285\pi$
$578$	1.36997e18i	1.52820i
$579$	9.06515e16	0.0999918
$580$	0	0
$581$	3.29771e17	0.355687
$582$	2.72153e17i	0.290278i
$583$	1.10966e18i	1.17042i
$584$	1.03337e17	0.107788
$585$	0	0
$586$	3.69378e17	0.376822
$587$	1.19496e17i	0.120561i	0.998181	+	0.0602805i	$0.0191995\pi$
$587$	1.19496e17i	0.120561i	−0.998181	+	0.0602805i	$0.980800\pi$
$588$	− 5.21484e17i	− 0.520342i
$589$	2.53084e18	2.49756
$590$	0	0
$591$	−3.06330e15	−0.00295713
$592$	− 4.57539e17i	− 0.436855i
$593$	1.87526e18i	1.77095i	0.464687	+	0.885475i	$0.346167\pi$
$593$	1.87526e18i	1.77095i	−0.464687	+	0.885475i	$0.653833\pi$
$594$	−3.52958e17	−0.329694
$595$	0	0
$596$	1.83562e18	1.67758
$597$	8.01357e17i	0.724424i
$598$	− 7.95601e16i	− 0.0711439i
$599$	5.19643e17	0.459654	0.229827	−	0.973231i	$-0.426184\pi$
$599$	5.19643e17	0.459654	0.229827	+	0.973231i	$0.426184\pi$
$600$	0	0
$601$	5.26704e17	0.455913	0.227957	−	0.973671i	$-0.426796\pi$
$601$	5.26704e17	0.455913	0.227957	+	0.973671i	$0.426796\pi$
$602$	− 3.98067e18i	− 3.40863i
$603$	− 3.32553e16i	− 0.0281708i
$604$	2.74424e18	2.29976
$605$	0	0
$606$	−3.82042e17	−0.313356
$607$	1.87571e18i	1.52209i	0.648701	+	0.761043i	$0.275312\pi$
$607$	1.87571e18i	1.52209i	−0.648701	+	0.761043i	$0.724688\pi$
$608$	− 1.58747e18i	− 1.27448i
$609$	1.19012e18	0.945318
$610$	0	0
$611$	1.14729e17	0.0892082
$612$	− 2.02227e17i	− 0.155581i
$613$	4.47233e17i	0.340440i	0.985406	+	0.170220i	$0.0544479\pi$
$613$	4.47233e17i	0.340440i	−0.985406	+	0.170220i	$0.945552\pi$
$614$	−4.66152e17	−0.351102
$615$	0	0
$616$	2.06743e18	1.52460
$617$	− 8.44420e17i	− 0.616176i	−0.951358	−	0.308088i	$-0.900311\pi$
$617$	− 8.44420e17i	− 0.616176i	0.951358	−	0.308088i	$-0.0996892\pi$
$618$	1.17748e18i	0.850216i
$619$	−2.36925e18	−1.69286	−0.846430	−	0.532500i	$-0.821253\pi$
$619$	−2.36925e18	−1.69286	−0.846430	+	0.532500i	$0.821253\pi$
$620$	0	0
$621$	1.82167e17	0.127460
$622$	1.30174e18i	0.901334i
$623$	3.71927e17i	0.254850i
$624$	−1.33487e16	−0.00905190
$625$	0	0
$626$	2.90100e18	1.92670
$627$	− 1.46774e18i	− 0.964741i
$628$	1.11881e18i	0.727811i
$629$	−7.64348e17	−0.492110
$630$	0	0
$631$	−2.91193e18	−1.83649	−0.918247	−	0.396009i	$-0.870395\pi$
$631$	−2.91193e18	−1.83649	−0.918247	+	0.396009i	$0.870395\pi$
$632$	3.52079e18i	2.19775i
$633$	3.23484e17i	0.199861i
$634$	−4.08673e18	−2.49917
$635$	0	0
$636$	−1.86894e18	−1.11976
$637$	5.75048e16i	0.0341034i
$638$	3.87095e18i	2.27239i
$639$	−6.02753e17	−0.350255
$640$	0	0
$641$	−3.49512e17	−0.199014	−0.0995072	−	0.995037i	$-0.531727\pi$
$641$	−3.49512e17	−0.199014	−0.0995072	+	0.995037i	$0.531727\pi$
$642$	1.62181e18i	0.914162i
$643$	3.11686e18i	1.73919i	0.493769	+	0.869593i	$0.335619\pi$
$643$	3.11686e18i	1.73919i	−0.493769	+	0.869593i	$0.664381\pi$
$644$	−2.54703e18	−1.40694
$645$	0	0
$646$	1.32959e18	0.719791
$647$	1.99766e18i	1.07064i	0.844649	+	0.535321i	$0.179809\pi$
$647$	1.99766e18i	1.07064i	−0.844649	+	0.535321i	$0.820191\pi$
$648$	− 2.49041e17i	− 0.132140i
$649$	−1.89739e18	−0.996706
$650$	0	0
$651$	2.14852e18	1.10628
$652$	− 2.28512e18i	− 1.16493i
$653$	1.08626e18i	0.548276i	0.961690	+	0.274138i	$0.0883925\pi$
$653$	1.08626e18i	0.548276i	−0.961690	+	0.274138i	$0.911607\pi$
$654$	2.19040e18	1.09463
$655$	0	0
$656$	−3.41455e17	−0.167286
$657$	− 6.22801e16i	− 0.0302117i
$658$	− 5.80715e18i	− 2.78930i
$659$	1.64776e17	0.0783678	0.0391839	−	0.999232i	$-0.487524\pi$
$659$	1.64776e17	0.0783678	0.0391839	+	0.999232i	$0.487524\pi$
$660$	0	0
$661$	−4.20993e18	−1.96320	−0.981601	−	0.190942i	$-0.938846\pi$
$661$	−4.20993e18	−1.96320	−0.981601	+	0.190942i	$0.938846\pi$
$662$	2.22165e18i	1.02588i
$663$	2.22999e16i	0.0101968i
$664$	−7.56811e17	−0.342685
$665$	0	0
$666$	−2.24688e18	−0.997695
$667$	− 1.99785e18i	− 0.878508i
$668$	− 3.69751e18i	− 1.61013i
$669$	1.57094e18	0.677472
$670$	0	0
$671$	1.07762e18	0.455793
$672$	− 1.34766e18i	− 0.564520i
$673$	2.30935e18i	0.958057i	0.877799	+	0.479029i	$0.159011\pi$
$673$	2.30935e18i	0.958057i	−0.877799	+	0.479029i	$0.840989\pi$
$674$	1.42002e18	0.583451
$675$	0	0
$676$	−4.25187e18	−1.71367
$677$	− 4.16216e18i	− 1.66147i	−0.556667	−	0.830736i	$-0.687920\pi$
$677$	− 4.16216e18i	− 1.66147i	0.556667	−	0.830736i	$-0.312080\pi$
$678$	− 3.13352e18i	− 1.23891i
$679$	−9.60762e17	−0.376238
$680$	0	0
$681$	2.82051e18	1.08361
$682$	6.98822e18i	2.65930i
$683$	5.67684e17i	0.213979i	0.994260	+	0.106990i	$0.0341212\pi$
$683$	5.67684e17i	0.213979i	−0.994260	+	0.106990i	$0.965879\pi$
$684$	2.47204e18	0.922978
$685$	0	0
$686$	−2.64730e18	−0.969833
$687$	− 8.00766e17i	− 0.290594i
$688$	1.12118e18i	0.403041i
$689$	2.06090e17	0.0733892
$690$	0	0
$691$	1.60571e18	0.561126	0.280563	−	0.959836i	$-0.409479\pi$
$691$	1.60571e18	0.561126	0.280563	+	0.959836i	$0.409479\pi$
$692$	1.11036e18i	0.384393i
$693$	− 1.24602e18i	− 0.427325i
$694$	−3.02024e18	−1.02614
$695$	0	0
$696$	−2.73128e18	−0.910763
$697$	5.70423e17i	0.188445i
$698$	− 1.31416e18i	− 0.430119i
$699$	−3.16479e18	−1.02623
$700$	0	0
$701$	−2.88624e18	−0.918681	−0.459341	−	0.888260i	$-0.651914\pi$
$701$	−2.88624e18	−0.918681	−0.459341	+	0.888260i	$0.651914\pi$
$702$	6.55528e16i	0.0206728i
$703$	− 9.34345e18i	− 2.91943i
$704$	5.19102e18	1.60705
$705$	0	0
$706$	2.49741e18	0.759030
$707$	− 1.34869e18i	− 0.406150i
$708$	− 3.19568e18i	− 0.953558i
$709$	−3.41704e18	−1.01030	−0.505150	−	0.863032i	$-0.668563\pi$
$709$	−3.41704e18	−1.01030	−0.505150	+	0.863032i	$0.668563\pi$
$710$	0	0
$711$	2.12195e18	0.616002
$712$	− 8.53557e17i	− 0.245535i
$713$	− 3.60672e18i	− 1.02809i
$714$	1.12874e18	0.318826
$715$	0	0
$716$	−1.06205e19	−2.94586
$717$	− 1.24515e18i	− 0.342254i
$718$	1.18666e19i	3.23234i
$719$	4.01107e18	1.08274	0.541368	−	0.840786i	$-0.317907\pi$
$719$	4.01107e18	1.08274	0.541368	+	0.840786i	$0.317907\pi$
$720$	0	0
$721$	−4.15677e18	−1.10199
$722$	9.97455e18i	2.62060i
$723$	− 2.09345e18i	− 0.545084i
$724$	6.76252e18	1.74505
$725$	0	0
$726$	2.95352e17	0.0748603
$727$	− 2.32433e18i	− 0.583881i	−0.956436	−	0.291941i	$-0.905699\pi$
$727$	− 2.32433e18i	− 0.583881i	0.956436	−	0.291941i	$-0.0943010\pi$
$728$	− 3.83972e17i	− 0.0955973i
$729$	−1.50095e17	−0.0370370
$730$	0	0
$731$	1.87300e18	0.454019
$732$	1.81497e18i	0.436062i
$733$	2.96621e17i	0.0706359i	0.999376	+	0.0353180i	$0.0112444\pi$
$733$	2.96621e17i	0.0706359i	−0.999376	+	0.0353180i	$0.988756\pi$
$734$	1.04062e19	2.45622
$735$	0	0
$736$	−2.26231e18	−0.524622
$737$	3.81847e17i	0.0877710i
$738$	1.67682e18i	0.382049i
$739$	−8.38574e18	−1.89388	−0.946940	−	0.321410i	$-0.895843\pi$
$739$	−8.38574e18	−1.89388	−0.946940	+	0.321410i	$0.895843\pi$
$740$	0	0
$741$	−2.72595e17	−0.0604923
$742$	− 1.04315e19i	− 2.29468i
$743$	− 1.06547e18i	− 0.232335i	−0.993230	−	0.116168i	$-0.962939\pi$
$743$	− 1.06547e18i	− 0.232335i	0.993230	−	0.116168i	$-0.0370610\pi$
$744$	−4.93077e18	−1.06584
$745$	0	0
$746$	3.10382e18	0.659316
$747$	4.56123e17i	0.0960501i
$748$	2.32203e18i	0.484739i
$749$	−5.72536e18	−1.18487
$750$	0	0
$751$	−9.57457e18	−1.94742	−0.973710	−	0.227792i	$-0.926849\pi$
$751$	−9.57457e18	−1.94742	−0.973710	+	0.227792i	$0.926849\pi$
$752$	1.63561e18i	0.329810i
$753$	6.86282e17i	0.137194i
$754$	7.18929e17	0.142486
$755$	0	0
$756$	2.09861e18	0.408827
$757$	4.59607e18i	0.887695i	0.896102	+	0.443847i	$0.146387\pi$
$757$	4.59607e18i	0.887695i	−0.896102	+	0.443847i	$0.853613\pi$
$758$	− 1.16045e19i	− 2.22217i
$759$	−2.09169e18	−0.397124
$760$	0	0
$761$	3.11769e18	0.581879	0.290940	−	0.956741i	$-0.406032\pi$
$761$	3.11769e18	0.581879	0.290940	+	0.956741i	$0.406032\pi$
$762$	7.64030e18i	1.41385i
$763$	7.73258e18i	1.41878i
$764$	−6.63967e18	−1.20792
$765$	0	0
$766$	9.89464e18	1.76975
$767$	3.52392e17i	0.0624966i
$768$	4.45313e18i	0.783101i
$769$	−4.30427e18	−0.750547	−0.375274	−	0.926914i	$-0.622451\pi$
$769$	−4.30427e18	−0.750547	−0.375274	+	0.926914i	$0.622451\pi$
$770$	0	0
$771$	−2.41529e18	−0.414110
$772$	− 1.75311e18i	− 0.298056i
$773$	6.51519e18i	1.09840i	0.835691	+	0.549200i	$0.185067\pi$
$773$	6.51519e18i	1.09840i	−0.835691	+	0.549200i	$0.814933\pi$
$774$	5.50587e18	0.920470
$775$	0	0
$776$	2.20491e18	0.362485
$777$	− 7.93199e18i	− 1.29314i
$778$	4.52719e17i	0.0731916i
$779$	−6.97289e18	−1.11794
$780$	0	0
$781$	6.92099e18	1.09128
$782$	− 1.89481e18i	− 0.296293i
$783$	1.64611e18i	0.255275i
$784$	−8.19806e17	−0.126083
$785$	0	0
$786$	1.00890e19	1.52616
$787$	− 7.49401e18i	− 1.12429i	−0.827039	−	0.562145i	$-0.809976\pi$
$787$	− 7.49401e18i	− 1.12429i	0.827039	−	0.562145i	$-0.190024\pi$
$788$	5.92412e16i	0.00881462i
$789$	2.27871e18	0.336270
$790$	0	0
$791$	1.10620e19	1.60579
$792$	2.85956e18i	0.411705i
$793$	− 2.00139e17i	− 0.0285796i
$794$	−5.19098e18	−0.735218
$795$	0	0
$796$	1.54975e19	2.15936
$797$	− 8.43590e18i	− 1.16588i	−0.812516	−	0.582938i	$-0.801903\pi$
$797$	− 8.43590e18i	− 1.16588i	0.812516	−	0.582938i	$-0.198097\pi$
$798$	1.37978e19i	1.89143i
$799$	2.73239e18	0.371526
$800$	0	0
$801$	−5.14430e17	−0.0688201
$802$	6.67801e18i	0.886163i
$803$	7.15119e17i	0.0941298i
$804$	−6.43125e17	−0.0839714
$805$	0	0
$806$	1.29788e18	0.166747
$807$	1.83633e18i	0.234030i
$808$	3.09519e18i	0.391304i
$809$	2.98965e18	0.374934	0.187467	−	0.982271i	$-0.439972\pi$
$809$	2.98965e18	0.374934	0.187467	+	0.982271i	$0.439972\pi$
$810$	0	0
$811$	−1.10858e19	−1.36814	−0.684072	−	0.729415i	$-0.739792\pi$
$811$	−1.10858e19	−1.36814	−0.684072	+	0.729415i	$0.739792\pi$
$812$	− 2.30157e19i	− 2.81780i
$813$	6.75434e17i	0.0820341i
$814$	2.57993e19	3.10849
$815$	0	0
$816$	−3.17914e17	−0.0376985
$817$	2.28956e19i	2.69345i
$818$	− 6.84893e18i	− 0.799330i
$819$	−2.31416e17	−0.0267946
$820$	0	0
$821$	1.97742e18	0.225356	0.112678	−	0.993632i	$-0.464057\pi$
$821$	1.97742e18	0.225356	0.112678	+	0.993632i	$0.464057\pi$
$822$	− 5.32442e18i	− 0.602012i
$823$	1.12620e19i	1.26333i	0.775242	+	0.631665i	$0.217628\pi$
$823$	1.12620e19i	1.26333i	−0.775242	+	0.631665i	$0.782372\pi$
$824$	9.53963e18	1.06171
$825$	0	0
$826$	1.78368e19	1.95410
$827$	− 1.16025e19i	− 1.26115i	−0.776129	−	0.630574i	$-0.782819\pi$
$827$	− 1.16025e19i	− 1.26115i	0.776129	−	0.630574i	$-0.217181\pi$
$828$	− 3.52293e18i	− 0.379933i
$829$	4.67038e18	0.499745	0.249872	−	0.968279i	$-0.419611\pi$
$829$	4.67038e18	0.499745	0.249872	+	0.968279i	$0.419611\pi$
$830$	0	0
$831$	−2.94092e17	−0.0309797
$832$	− 9.64098e17i	− 0.100767i
$833$	1.36954e18i	0.142030i
$834$	−4.88396e18	−0.502565
$835$	0	0
$836$	−2.83847e19	−2.87570
$837$	2.97173e18i	0.298740i
$838$	− 1.05194e19i	− 1.04931i
$839$	8.73017e17	0.0864112	0.0432056	−	0.999066i	$-0.486243\pi$
$839$	8.73017e17	0.0864112	0.0432056	+	0.999066i	$0.486243\pi$
$840$	0	0
$841$	7.79256e18	0.759462
$842$	− 3.15953e18i	− 0.305559i
$843$	6.24222e18i	0.599046i
$844$	6.25586e18	0.595746
$845$	0	0
$846$	8.03215e18	0.753225
$847$	1.04266e18i	0.0970285i
$848$	2.93808e18i	0.271326i
$849$	2.73283e18	0.250445
$850$	0	0
$851$	−1.33154e19	−1.20175
$852$	1.16567e19i	1.04404i
$853$	− 1.24079e19i	− 1.10288i	−0.834213	−	0.551442i	$-0.814078\pi$
$853$	− 1.24079e19i	− 1.10288i	0.834213	−	0.551442i	$-0.185922\pi$
$854$	−1.01303e19	−0.893607
$855$	0	0
$856$	1.31395e19	1.14156
$857$	7.04934e18i	0.607817i	0.952701	+	0.303909i	$0.0982918\pi$
$857$	7.04934e18i	0.607817i	−0.952701	+	0.303909i	$0.901708\pi$
$858$	− 7.52697e17i	− 0.0644099i
$859$	2.37855e17	0.0202003	0.0101001	−	0.999949i	$-0.496785\pi$
$859$	2.37855e17	0.0202003	0.0101001	+	0.999949i	$0.496785\pi$
$860$	0	0
$861$	−5.91953e18	−0.495185
$862$	6.10992e18i	0.507269i
$863$	− 1.21210e19i	− 0.998774i	−0.866379	−	0.499387i	$-0.833559\pi$
$863$	− 1.21210e19i	− 0.998774i	0.866379	−	0.499387i	$-0.166441\pi$
$864$	1.86401e18	0.152444
$865$	0	0
$866$	1.27981e19	1.03105
$867$	− 6.68934e18i	− 0.534884i
$868$	− 4.15503e19i	− 3.29759i
$869$	−2.43648e19	−1.91926
$870$	0	0
$871$	7.09183e16	0.00550352
$872$	− 1.77460e19i	− 1.36692i
$873$	− 1.32888e18i	− 0.101600i
$874$	2.31623e19	1.75775
$875$	0	0
$876$	−1.20444e18	−0.0900549
$877$	− 3.74100e18i	− 0.277645i	−0.990317	−	0.138823i	$-0.955668\pi$
$877$	− 3.74100e18i	− 0.277645i	0.990317	−	0.138823i	$-0.0443318\pi$
$878$	− 5.53354e18i	− 0.407651i
$879$	−1.80361e18	−0.131891
$880$	0	0
$881$	−1.36085e19	−0.980544	−0.490272	−	0.871570i	$-0.663102\pi$
$881$	−1.36085e19	−0.980544	−0.490272	+	0.871570i	$0.663102\pi$
$882$	4.02590e18i	0.287950i
$883$	− 9.87005e18i	− 0.700769i	−0.936606	−	0.350384i	$-0.886051\pi$
$883$	− 9.87005e18i	− 0.700769i	0.936606	−	0.350384i	$-0.113949\pi$
$884$	4.31258e17	0.0303947
$885$	0	0
$886$	−2.31436e18	−0.160736
$887$	1.35100e19i	0.931432i	0.884934	+	0.465716i	$0.154203\pi$
$887$	1.35100e19i	0.931432i	−0.884934	+	0.465716i	$0.845797\pi$
$888$	1.82036e19i	1.24587i
$889$	−2.69720e19	−1.83253
$890$	0	0
$891$	1.72343e18	0.115395
$892$	− 3.03805e19i	− 2.01941i
$893$	3.34010e19i	2.20407i
$894$	−1.41712e19	−0.928350
$895$	0	0
$896$	−3.36550e19	−2.17294
$897$	3.88478e17i	0.0249009i
$898$	4.12025e19i	2.62197i
$899$	3.25914e19	2.05904
$900$	0	0
$901$	4.90826e18	0.305644
$902$	− 1.92537e19i	− 1.19034i
$903$	1.94369e19i	1.19305i
$904$	−2.53869e19	−1.54709
$905$	0	0
$906$	−2.11858e19	−1.27266
$907$	− 7.78812e18i	− 0.464500i	−0.972656	−	0.232250i	$-0.925391\pi$
$907$	− 7.78812e18i	− 0.464500i	0.972656	−	0.232250i	$-0.0746087\pi$
$908$	− 5.45460e19i	− 3.23002i
$909$	1.86544e18	0.109677
$910$	0	0
$911$	6.24185e17	0.0361779	0.0180890	−	0.999836i	$-0.494242\pi$
$911$	6.24185e17	0.0361779	0.0180890	+	0.999836i	$0.494242\pi$
$912$	− 3.88621e18i	− 0.223645i
$913$	− 5.23733e18i	− 0.299261i
$914$	−3.06272e19	−1.73763
$915$	0	0
$916$	−1.54860e19	−0.866203
$917$	3.56164e19i	1.97810i
$918$	1.56121e18i	0.0860963i
$919$	3.08457e19	1.68906	0.844528	−	0.535511i	$-0.179881\pi$
$919$	3.08457e19	1.68906	0.844528	+	0.535511i	$0.179881\pi$
$920$	0	0
$921$	2.27614e18	0.122888
$922$	3.42675e19i	1.83709i
$923$	− 1.28539e18i	− 0.0684267i
$924$	−2.40968e19	−1.27377
$925$	0	0
$926$	2.84363e19	1.48218
$927$	− 5.74944e18i	− 0.297582i
$928$	− 2.04429e19i	− 1.05071i
$929$	2.60942e19	1.33181	0.665905	−	0.746037i	$-0.268046\pi$
$929$	2.60942e19	1.33181	0.665905	+	0.746037i	$0.268046\pi$
$930$	0	0
$931$	−1.67413e19	−0.842592
$932$	6.12039e19i	3.05897i
$933$	− 6.35615e18i	− 0.315474i
$934$	2.50432e19	1.23434
$935$	0	0
$936$	5.31090e17	0.0258152
$937$	− 1.43219e19i	− 0.691340i	−0.938356	−	0.345670i	$-0.887652\pi$
$937$	− 1.43219e19i	− 0.691340i	0.938356	−	0.345670i	$-0.112348\pi$
$938$	− 3.58962e18i	− 0.172080i
$939$	−1.41651e19	−0.674361
$940$	0	0
$941$	−6.74040e18	−0.316485	−0.158242	−	0.987400i	$-0.550583\pi$
$941$	−6.74040e18	−0.316485	−0.158242	+	0.987400i	$0.550583\pi$
$942$	− 8.63731e18i	− 0.402761i
$943$	9.93712e18i	0.460187i
$944$	−5.02381e18	−0.231055
$945$	0	0
$946$	−6.32200e19	−2.86789
$947$	− 4.26475e18i	− 0.192140i	−0.995375	−	0.0960701i	$-0.969373\pi$
$947$	− 4.26475e18i	− 0.192140i	0.995375	−	0.0960701i	$-0.0306273\pi$
$948$	− 4.10364e19i	− 1.83618i
$949$	1.32815e17	0.00590223
$950$	0	0
$951$	1.99548e19	0.874729
$952$	− 9.14471e18i	− 0.398134i
$953$	− 2.39618e17i	− 0.0103613i	−0.999987	−	0.00518067i	$-0.998351\pi$
$953$	− 2.39618e17i	− 0.0103613i	0.999987	−	0.00518067i	$-0.00164906\pi$
$954$	1.44283e19	0.619658
$955$	0	0
$956$	−2.40800e19	−1.02019
$957$	− 1.89011e19i	− 0.795354i
$958$	5.54985e19i	2.31956i
$959$	1.87964e19	0.780285
$960$	0	0
$961$	3.44197e19	1.40963
$962$	− 4.79157e18i	− 0.194912i
$963$	− 7.91903e18i	− 0.319964i
$964$	−4.04853e19	−1.62479
$965$	0	0
$966$	1.96633e19	0.778584
$967$	1.50588e19i	0.592268i	0.955146	+	0.296134i	$0.0956975\pi$
$967$	1.50588e19i	0.592268i	−0.955146	+	0.296134i	$0.904302\pi$
$968$	− 2.39286e18i	− 0.0934817i
$969$	−6.49215e18	−0.251933
$970$	0	0
$971$	1.51669e19	0.580728	0.290364	−	0.956916i	$-0.406224\pi$
$971$	1.51669e19	0.580728	0.290364	+	0.956916i	$0.406224\pi$
$972$	2.90268e18i	0.110400i
$973$	− 1.72415e19i	− 0.651389i
$974$	−7.28221e18	−0.273294
$975$	0	0
$976$	2.85325e18	0.105661
$977$	5.43050e18i	0.199768i	0.994999	+	0.0998838i	$0.0318471\pi$
$977$	5.43050e18i	0.199768i	−0.994999	+	0.0998838i	$0.968153\pi$
$978$	1.76413e19i	0.644656i
$979$	5.90684e18	0.214421
$980$	0	0
$981$	−1.06953e19	−0.383129
$982$	− 9.24822e18i	− 0.329104i
$983$	4.45646e19i	1.57540i	0.616057	+	0.787702i	$0.288729\pi$
$983$	4.45646e19i	1.57540i	−0.616057	+	0.787702i	$0.711271\pi$
$984$	1.35851e19	0.477084
$985$	0	0
$986$	1.71221e19	0.593412
$987$	2.83553e19i	0.976275i
$988$	5.27173e18i	0.180315i
$989$	3.26288e19	1.10873
$990$	0	0
$991$	8.11044e18	0.271998	0.135999	−	0.990709i	$-0.456576\pi$
$991$	8.11044e18	0.271998	0.135999	+	0.990709i	$0.456576\pi$
$992$	− 3.69056e19i	− 1.22961i
$993$	− 1.08479e19i	− 0.359067i
$994$	−6.50619e19	−2.13951
$995$	0	0
$996$	8.82097e18	0.286306
$997$	3.75569e19i	1.21107i	0.795817	+	0.605537i	$0.207042\pi$
$997$	3.75569e19i	1.21107i	−0.795817	+	0.605537i	$0.792958\pi$
$998$	4.17627e19i	1.33795i
$999$	1.09711e19	0.349201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.b.e.49.4		4
5.2	odd	4		75.14.a.c.1.1		2
5.3	odd	4		15.14.a.c.1.2	✓	2
5.4	even	2	inner	75.14.b.e.49.1		4
15.8	even	4		45.14.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.14.a.c.1.2	✓	2	5.3	odd	4
45.14.a.b.1.1		2	15.8	even	4
75.14.a.c.1.1		2	5.2	odd	4
75.14.b.e.49.1		4	5.4	even	2	inner
75.14.b.e.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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+149.299i q^{2} -729.000i q^{3} -14098.2 q^{4} +108839. q^{6} +384225. i q^{7} -881782. i q^{8} -531441. q^{9} +6.10216e6 q^{11} +1.02776e7i q^{12} -1.13332e6i q^{13} -5.73644e7 q^{14} +1.61570e7 q^{16} -2.69912e7i q^{17} -7.93435e7i q^{18} +3.29943e8 q^{19} +2.80100e8 q^{21} +9.11046e8i q^{22} -4.70204e8i q^{23} -6.42819e8 q^{24} +1.69203e8 q^{26} +3.87420e8i q^{27} -5.41687e9i q^{28} +4.24890e9 q^{29} +7.67054e9 q^{31} -4.81134e9i q^{32} -4.44848e9i q^{33} +4.02976e9 q^{34} +7.49234e9 q^{36} -2.83184e10i q^{37} +4.92601e10i q^{38} -8.26190e8 q^{39} -2.11336e10 q^{41} +4.18187e10i q^{42} +6.93928e10i q^{43} -8.60292e10 q^{44} +7.02010e10 q^{46} +1.01233e11i q^{47} -1.17784e10i q^{48} -5.07401e10 q^{49} -1.96766e10 q^{51} +1.59777e10i q^{52} +1.81846e11i q^{53} -5.78414e10 q^{54} +3.38803e11 q^{56} -2.40528e11i q^{57} +6.34357e11i q^{58} -3.10938e11 q^{59} +1.76596e11 q^{61} +1.14520e12i q^{62} -2.04193e11i q^{63} +8.50686e11 q^{64} +6.64152e11 q^{66} +6.25757e10i q^{67} +3.80526e11i q^{68} -3.42779e11 q^{69} +1.13419e12 q^{71} +4.68615e11i q^{72} +1.17191e11i q^{73} +4.22790e12 q^{74} -4.65158e12 q^{76} +2.34461e12i q^{77} -1.23349e11i q^{78} -3.99281e12 q^{79} +2.82430e11 q^{81} -3.15523e12i q^{82} -8.58275e11i q^{83} -3.94890e12 q^{84} -1.03603e13 q^{86} -3.09745e12i q^{87} -5.38077e12i q^{88} +9.67991e11 q^{89} +4.35450e11 q^{91} +6.62901e12i q^{92} -5.59183e12i q^{93} -1.51139e13 q^{94} -3.50747e12 q^{96} +2.50052e12i q^{97} -7.57544e12i q^{98} -3.24294e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12482 q^{4} + 190998 q^{6} - 2125764 q^{9} + 12491776 q^{11} - 110628168 q^{14} + 150297410 q^{16} - 76950688 q^{19} + 723564576 q^{21} - 1713825054 q^{24} + 315428908 q^{26} - 2252949416 q^{29} + 12200208624 q^{31}+ \cdots - 6638641929216 q^{99}+O(q^{100}) \) 4 * q - 12482 * q^4 + 190998 * q^6 - 2125764 * q^9 + 12491776 * q^11 - 110628168 * q^14 + 150297410 * q^16 - 76950688 * q^19 + 723564576 * q^21 - 1713825054 * q^24 + 315428908 * q^26 - 2252949416 * q^29 + 12200208624 * q^31 + 10809576476 * q^34 + 6633446562 * q^36 - 2567777112 * q^39 - 59215819576 * q^41 - 169799823776 * q^44 + 130737909984 * q^46 + 67188889180 * q^49 + 70205038632 * q^51 - 101504168118 * q^54 + 743417798760 * q^56 - 1445536360192 * q^59 - 459968804360 * q^61 + 2540337082142 * q^64 + 1324470131424 * q^66 - 1070557810128 * q^69 + 1473091336448 * q^71 + 7778702225812 * q^74 - 15092602309288 * q^76 - 7052697180240 * q^79 + 1129718145924 * q^81 - 6614216830872 * q^84 - 20947533908504 * q^86 - 11826242919528 * q^89 + 1011595853184 * q^91 - 30089779392352 * q^94 - 12096530386434 * q^96 - 6638641929216 * q^99

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