LMFDB - Embedded newform 75.14.b.e.49.2

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			49.2
Root			$-28.4330i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.14.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)

$f(q)$	$=$	\(q-18.2989i q^{2} -729.000i q^{3} +7857.15 q^{4} -13339.9 q^{6} +112047. i q^{7} -293681. i q^{8} -531441. q^{9} +143727. q^{11} -5.72786e6i q^{12} -627845. i q^{13} +2.05033e6 q^{14} +5.89918e7 q^{16} +7.51428e7i q^{17} +9.72477e6i q^{18} -3.68418e8 q^{19} +8.16820e7 q^{21} -2.63004e6i q^{22} -2.64060e8i q^{23} -2.14094e8 q^{24} -1.14888e7 q^{26} +3.87420e8i q^{27} +8.80367e8i q^{28} -5.37538e9 q^{29} -1.57044e9 q^{31} -3.48532e9i q^{32} -1.04777e8i q^{33} +1.37503e9 q^{34} -4.17561e9 q^{36} -1.85013e10i q^{37} +6.74163e9i q^{38} -4.57699e8 q^{39} -8.47428e9 q^{41} -1.49469e9i q^{42} -6.20298e9i q^{43} +1.12928e9 q^{44} -4.83201e9 q^{46} +3.77183e9i q^{47} -4.30050e10i q^{48} +8.43346e10 q^{49} +5.47791e10 q^{51} -4.93307e9i q^{52} -2.30272e11i q^{53} +7.08936e9 q^{54} +3.29060e10 q^{56} +2.68577e11i q^{57} +9.83633e10i q^{58} -4.11830e11 q^{59} -4.06580e11 q^{61} +2.87372e10i q^{62} -5.95462e10i q^{63} +4.19483e11 q^{64} -1.91730e9 q^{66} +2.96486e11i q^{67} +5.90409e11i q^{68} -1.92500e11 q^{69} -3.97640e11 q^{71} +1.56074e11i q^{72} +1.37728e12i q^{73} -3.38553e11 q^{74} -2.89472e12 q^{76} +1.61041e10i q^{77} +8.37537e9i q^{78} +4.66466e11 q^{79} +2.82430e11 q^{81} +1.55070e11i q^{82} -4.77278e12i q^{83} +6.41788e11 q^{84} -1.13508e11 q^{86} +3.91865e12i q^{87} -4.22098e10i q^{88} -6.88111e12 q^{89} +7.03479e10 q^{91} -2.07476e12i q^{92} +1.14485e12i q^{93} +6.90203e10 q^{94} -2.54080e12 q^{96} +1.38827e13i q^{97} -1.54323e12i q^{98} -7.63823e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 18.2989i	− 0.202176i	−0.994877	−	0.101088i	$-0.967768\pi$
$2$	− 18.2989i	− 0.202176i	0.994877	−	0.101088i	$-0.0322323\pi$
$3$	− 729.000i	− 0.577350i
$3$	− 729.000i	− 0.577350i
$4$	7857.15	0.959125
$5$	0	0
$5$	0	0
$6$	−13339.9	−0.116726
$7$	112047.i	0.359966i	0.983670	+	0.179983i	$0.0576043\pi$
$7$	112047.i	0.359966i	−0.983670	+	0.179983i	$0.942396\pi$
$8$	− 293681.i	− 0.396088i
$9$	−531441.	−0.333333
$10$	0	0
$11$	143727.	0.0244616	0.0122308	−	0.999925i	$-0.496107\pi$
$11$	143727.	0.0244616	0.0122308	+	0.999925i	$0.496107\pi$
$12$	− 5.72786e6i	− 0.553751i
$13$	− 627845.i	− 0.0360762i	−0.999837	−	0.0180381i	$-0.994258\pi$
$13$	− 627845.i	− 0.0360762i	0.999837	−	0.0180381i	$-0.00574201\pi$
$14$	2.05033e6	0.0727764
$15$	0	0
$16$	5.89918e7	0.879046
$17$	7.51428e7i	0.755039i	0.926002	+	0.377520i	$0.123223\pi$
$17$	7.51428e7i	0.755039i	−0.926002	+	0.377520i	$0.876777\pi$
$18$	9.72477e6i	0.0673919i
$19$	−3.68418e8	−1.79656	−0.898282	−	0.439420i	$-0.855184\pi$
$19$	−3.68418e8	−1.79656	−0.898282	+	0.439420i	$0.855184\pi$
$20$	0	0
$21$	8.16820e7	0.207827
$22$	− 2.63004e6i	− 0.00494555i
$23$	− 2.64060e8i	− 0.371939i	−0.982555	−	0.185970i	$-0.940457\pi$
$23$	− 2.64060e8i	− 0.371939i	0.982555	−	0.185970i	$-0.0595426\pi$
$24$	−2.14094e8	−0.228681
$25$	0	0
$26$	−1.14888e7	−0.00729373
$27$	3.87420e8i	0.192450i
$28$	8.80367e8i	0.345252i
$29$	−5.37538e9	−1.67812	−0.839058	−	0.544042i	$-0.816893\pi$
$29$	−5.37538e9	−1.67812	−0.839058	+	0.544042i	$0.816893\pi$
$30$	0	0
$31$	−1.57044e9	−0.317812	−0.158906	−	0.987294i	$-0.550797\pi$
$31$	−1.57044e9	−0.317812	−0.158906	+	0.987294i	$0.550797\pi$
$32$	− 3.48532e9i	− 0.573809i
$33$	− 1.04777e8i	− 0.0141229i
$34$	1.37503e9	0.152651
$35$	0	0
$36$	−4.17561e9	−0.319708
$37$	− 1.85013e10i	− 1.18547i	−0.805397	−	0.592735i	$-0.798048\pi$
$37$	− 1.85013e10i	− 1.18547i	0.805397	−	0.592735i	$-0.201952\pi$
$38$	6.74163e9i	0.363222i
$39$	−4.57699e8	−0.0208286
$40$	0	0
$41$	−8.47428e9	−0.278617	−0.139309	−	0.990249i	$-0.544488\pi$
$41$	−8.47428e9	−0.278617	−0.139309	+	0.990249i	$0.544488\pi$
$42$	− 1.49469e9i	− 0.0420175i
$43$	− 6.20298e9i	− 0.149643i	−0.997197	−	0.0748214i	$-0.976161\pi$
$43$	− 6.20298e9i	− 0.149643i	0.997197	−	0.0748214i	$-0.0238387\pi$
$44$	1.12928e9	0.0234617
$45$	0	0
$46$	−4.83201e9	−0.0751972
$47$	3.77183e9i	0.0510407i	0.999674	+	0.0255203i	$0.00812426\pi$
$47$	3.77183e9i	0.0510407i	−0.999674	+	0.0255203i	$0.991876\pi$
$48$	− 4.30050e10i	− 0.507517i
$49$	8.43346e10	0.870424
$50$	0	0
$51$	5.47791e10	0.435922
$52$	− 4.93307e9i	− 0.0346016i
$53$	− 2.30272e11i	− 1.42708i	−0.700614	−	0.713540i	$-0.747091\pi$
$53$	− 2.30272e11i	− 1.42708i	0.700614	−	0.713540i	$-0.252909\pi$
$54$	7.08936e9	0.0389088
$55$	0	0
$56$	3.29060e10	0.142578
$57$	2.68577e11i	1.03725i
$58$	9.83633e10i	0.339275i
$59$	−4.11830e11	−1.27110	−0.635551	−	0.772059i	$-0.719227\pi$
$59$	−4.11830e11	−1.27110	−0.635551	+	0.772059i	$0.719227\pi$
$60$	0	0
$61$	−4.06580e11	−1.01042	−0.505210	−	0.862996i	$-0.668585\pi$
$61$	−4.06580e11	−1.01042	−0.505210	+	0.862996i	$0.668585\pi$
$62$	2.87372e10i	0.0642538i
$63$	− 5.95462e10i	− 0.119989i
$64$	4.19483e11	0.763035
$65$	0	0
$66$	−1.91730e9	−0.00285531
$67$	2.96486e11i	0.400423i	0.979753	+	0.200211i	$0.0641629\pi$
$67$	2.96486e11i	0.400423i	−0.979753	+	0.200211i	$0.935837\pi$
$68$	5.90409e11i	0.724177i
$69$	−1.92500e11	−0.214739
$70$	0	0
$71$	−3.97640e11	−0.368393	−0.184196	−	0.982889i	$-0.558968\pi$
$71$	−3.97640e11	−0.368393	−0.184196	+	0.982889i	$0.558968\pi$
$72$	1.56074e11i	0.132029i
$73$	1.37728e12i	1.06518i	0.846373	+	0.532591i	$0.178781\pi$
$73$	1.37728e12i	1.06518i	−0.846373	+	0.532591i	$0.821219\pi$
$74$	−3.38553e11	−0.239673
$75$	0	0
$76$	−2.89472e12	−1.72313
$77$	1.61041e10i	0.00880535i
$78$	8.37537e9i	0.00421104i
$79$	4.66466e11	0.215896	0.107948	−	0.994157i	$-0.465572\pi$
$79$	4.66466e11	0.215896	0.107948	+	0.994157i	$0.465572\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	1.55070e11i	0.0563297i
$83$	− 4.77278e12i	− 1.60237i	−0.598415	−	0.801186i	$-0.704203\pi$
$83$	− 4.77278e12i	− 1.60237i	0.598415	−	0.801186i	$-0.295797\pi$
$84$	6.41788e11	0.199332
$85$	0	0
$86$	−1.13508e11	−0.0302541
$87$	3.91865e12i	0.968861i
$88$	− 4.22098e10i	− 0.00968894i
$89$	−6.88111e12	−1.46765	−0.733827	−	0.679336i	$-0.762268\pi$
$89$	−6.88111e12	−1.46765	−0.733827	+	0.679336i	$0.762268\pi$
$90$	0	0
$91$	7.03479e10	0.0129862
$92$	− 2.07476e12i	− 0.356736i
$93$	1.14485e12i	0.183489i
$94$	6.90203e10	0.0103192
$95$	0	0
$96$	−2.54080e12	−0.331289
$97$	1.38827e13i	1.69223i	0.533004	+	0.846113i	$0.321063\pi$
$97$	1.38827e13i	1.69223i	−0.533004	+	0.846113i	$0.678937\pi$
$98$	− 1.54323e12i	− 0.175979i
$99$	−7.63823e10	−0.00815387
$100$	0	0
$101$	−1.02752e13	−0.963162	−0.481581	−	0.876402i	$-0.659937\pi$
$101$	−1.02752e13	−0.963162	−0.481581	+	0.876402i	$0.659937\pi$
$102$	− 1.00240e12i	− 0.0881329i
$103$	− 1.05918e13i	− 0.874036i	−0.899453	−	0.437018i	$-0.856035\pi$
$103$	− 1.05918e13i	− 0.874036i	0.899453	−	0.437018i	$-0.143965\pi$
$104$	−1.84386e11	−0.0142893
$105$	0	0
$106$	−4.21372e12	−0.288521
$107$	1.26172e13i	0.812774i	0.913701	+	0.406387i	$0.133211\pi$
$107$	1.26172e13i	0.812774i	−0.913701	+	0.406387i	$0.866789\pi$
$108$	3.04402e12i	0.184584i
$109$	−1.49988e13	−0.856612	−0.428306	−	0.903634i	$-0.640890\pi$
$109$	−1.49988e13	−0.856612	−0.428306	+	0.903634i	$0.640890\pi$
$110$	0	0
$111$	−1.34874e13	−0.684432
$112$	6.60983e12i	0.316427i
$113$	3.11837e13i	1.40902i	0.709693	+	0.704512i	$0.248834\pi$
$113$	3.11837e13i	1.40902i	−0.709693	+	0.704512i	$0.751166\pi$
$114$	4.91465e12	0.209706
$115$	0	0
$116$	−4.22352e13	−1.60952
$117$	3.33662e11i	0.0120254i
$118$	7.53603e12i	0.256986i
$119$	−8.41950e12	−0.271789
$120$	0	0
$121$	−3.45021e13	−0.999402
$122$	7.43996e12i	0.204283i
$123$	6.17775e12i	0.160860i
$124$	−1.23392e13	−0.304821
$125$	0	0
$126$	−1.08963e12	−0.0242588
$127$	6.92694e13i	1.46493i	0.680804	+	0.732466i	$0.261631\pi$
$127$	6.92694e13i	1.46493i	−0.680804	+	0.732466i	$0.738369\pi$
$128$	− 3.62278e13i	− 0.728077i
$129$	−4.52198e12	−0.0863963
$130$	0	0
$131$	−3.60740e13	−0.623636	−0.311818	−	0.950142i	$-0.600938\pi$
$131$	−3.60740e13	−0.623636	−0.311818	+	0.950142i	$0.600938\pi$
$132$	− 8.23247e11i	− 0.0135456i
$133$	− 4.12800e13i	− 0.646702i
$134$	5.42537e12	0.0809558
$135$	0	0
$136$	2.20680e13	0.299062
$137$	8.46431e13i	1.09372i	0.837223	+	0.546862i	$0.184178\pi$
$137$	8.46431e13i	1.09372i	−0.837223	+	0.546862i	$0.815822\pi$
$138$	3.52253e12i	0.0434151i
$139$	−1.23811e13	−0.145600	−0.0728002	−	0.997347i	$-0.523194\pi$
$139$	−1.23811e13	−0.145600	−0.0728002	+	0.997347i	$0.523194\pi$
$140$	0	0
$141$	2.74967e12	0.0294683
$142$	7.27637e12i	0.0744801i
$143$	− 9.02380e10i	0 0.000882481i
$144$	−3.13506e13	−0.293015
$145$	0	0
$146$	2.52027e13	0.215354
$147$	− 6.14799e13i	− 0.502540i
$148$	− 1.45367e14i	− 1.13701i
$149$	1.01554e14	0.760303	0.380152	−	0.924924i	$-0.375872\pi$
$149$	1.01554e14	0.760303	0.380152	+	0.924924i	$0.375872\pi$
$150$	0	0
$151$	−6.26055e13	−0.429796	−0.214898	−	0.976636i	$-0.568942\pi$
$151$	−6.26055e13	−0.429796	−0.214898	+	0.976636i	$0.568942\pi$
$152$	1.08198e14i	0.711597i
$153$	− 3.99340e13i	− 0.251680i
$154$	2.94687e11	0.00178023
$155$	0	0
$156$	−3.59621e12	−0.0199772
$157$	− 3.46935e14i	− 1.84885i	−0.381368	−	0.924423i	$-0.624547\pi$
$157$	− 3.46935e14i	− 1.84885i	0.381368	−	0.924423i	$-0.375453\pi$
$158$	− 8.53580e12i	− 0.0436489i
$159$	−1.67868e14	−0.823925
$160$	0	0
$161$	2.95871e13	0.133886
$162$	− 5.16814e12i	− 0.0224640i
$163$	4.29452e14i	1.79347i	0.442565	+	0.896736i	$0.354069\pi$
$163$	4.29452e14i	1.79347i	−0.442565	+	0.896736i	$0.645931\pi$
$164$	−6.65837e13	−0.267229
$165$	0	0
$166$	−8.73364e13	−0.323961
$167$	− 3.09663e14i	− 1.10467i	−0.833623	−	0.552334i	$-0.813737\pi$
$167$	− 3.09663e14i	− 1.10467i	0.833623	−	0.552334i	$-0.186263\pi$
$168$	− 2.39885e13i	− 0.0823175i
$169$	3.02481e14	0.998699
$170$	0	0
$171$	1.95793e14	0.598855
$172$	− 4.87378e13i	− 0.143526i
$173$	− 4.50613e13i	− 0.127792i	−0.997957	−	0.0638960i	$-0.979647\pi$
$173$	− 4.50613e13i	− 0.127792i	0.997957	−	0.0638960i	$-0.0203526\pi$
$174$	7.17069e13	0.195880
$175$	0	0
$176$	8.47869e12	0.0215029
$177$	3.00224e14i	0.733871i
$178$	1.25917e14i	0.296724i
$179$	−1.74245e14	−0.395926	−0.197963	−	0.980209i	$-0.563433\pi$
$179$	−1.74245e14	−0.395926	−0.197963	+	0.980209i	$0.563433\pi$
$180$	0	0
$181$	6.89956e14	1.45851	0.729257	−	0.684240i	$-0.239866\pi$
$181$	6.89956e14	1.45851	0.729257	+	0.684240i	$0.239866\pi$
$182$	− 1.28729e12i	− 0.00262549i
$183$	2.96397e14i	0.583367i
$184$	−7.75496e13	−0.147321
$185$	0	0
$186$	2.09494e13	0.0370970
$187$	1.08000e13i	0.0184695i
$188$	2.96359e13i	0.0489544i
$189$	−4.34092e13	−0.0692755
$190$	0	0
$191$	7.52391e14	1.12131	0.560656	−	0.828049i	$-0.310549\pi$
$191$	7.52391e14	1.12131	0.560656	+	0.828049i	$0.310549\pi$
$192$	− 3.05803e14i	− 0.440539i
$193$	− 4.18449e14i	− 0.582801i	−0.956601	−	0.291400i	$-0.905879\pi$
$193$	− 4.18449e14i	− 0.582801i	0.956601	−	0.291400i	$-0.0941212\pi$
$194$	2.54038e14	0.342127
$195$	0	0
$196$	6.62629e14	0.834846
$197$	− 5.04365e14i	− 0.614772i	−0.951585	−	0.307386i	$-0.900546\pi$
$197$	− 5.04365e14i	− 0.614772i	0.951585	−	0.307386i	$-0.0994543\pi$
$198$	1.39771e12i	0.00164852i
$199$	1.01359e15	1.15696	0.578479	−	0.815697i	$-0.303646\pi$
$199$	1.01359e15	1.15696	0.578479	+	0.815697i	$0.303646\pi$
$200$	0	0
$201$	2.16139e14	0.231184
$202$	1.88024e14i	0.194728i
$203$	− 6.02293e14i	− 0.604065i
$204$	4.30408e14	0.418104
$205$	0	0
$206$	−1.93819e14	−0.176709
$207$	1.40333e14i	0.123980i
$208$	− 3.70377e13i	− 0.0317126i
$209$	−5.29515e13	−0.0439468
$210$	0	0
$211$	6.68511e14	0.521522	0.260761	−	0.965403i	$-0.416027\pi$
$211$	6.68511e14	0.521522	0.260761	+	0.965403i	$0.416027\pi$
$212$	− 1.80928e15i	− 1.36875i
$213$	2.89880e14i	0.212692i
$214$	2.30881e14	0.164323
$215$	0	0
$216$	1.13778e14	0.0762271
$217$	− 1.75962e14i	− 0.114401i
$218$	2.74461e14i	0.173186i
$219$	1.00404e15	0.614983
$220$	0	0
$221$	4.71780e13	0.0272389
$222$	2.46805e14i	0.138376i
$223$	6.64343e14i	0.361752i	0.983506	+	0.180876i	$0.0578933\pi$
$223$	6.64343e14i	0.361752i	−0.983506	+	0.180876i	$0.942107\pi$
$224$	3.90518e14	0.206552
$225$	0	0
$226$	5.70627e14	0.284870
$227$	− 2.74872e15i	− 1.33341i	−0.745324	−	0.666703i	$-0.767705\pi$
$227$	− 2.74872e15i	− 1.33341i	0.745324	−	0.666703i	$-0.232295\pi$
$228$	2.11025e15i	0.994849i
$229$	−1.93108e15	−0.884851	−0.442426	−	0.896805i	$-0.645882\pi$
$229$	−1.93108e15	−0.884851	−0.442426	+	0.896805i	$0.645882\pi$
$230$	0	0
$231$	1.17399e13	0.00508377
$232$	1.57865e15i	0.664681i
$233$	− 3.04503e15i	− 1.24674i	−0.781925	−	0.623372i	$-0.785762\pi$
$233$	− 3.04503e15i	− 1.24674i	0.781925	−	0.623372i	$-0.214238\pi$
$234$	6.10564e12	0.00243124
$235$	0	0
$236$	−3.23581e15	−1.21914
$237$	− 3.40054e14i	− 0.124647i
$238$	1.54067e14i	0.0549491i
$239$	−1.63875e15	−0.568756	−0.284378	−	0.958712i	$-0.591787\pi$
$239$	−1.63875e15	−0.568756	−0.284378	+	0.958712i	$0.591787\pi$
$240$	0	0
$241$	−2.33624e14	−0.0768081	−0.0384041	−	0.999262i	$-0.512227\pi$
$241$	−2.33624e14	−0.0768081	−0.0384041	+	0.999262i	$0.512227\pi$
$242$	6.31348e14i	0.202055i
$243$	− 2.05891e14i	− 0.0641500i
$244$	−3.19456e15	−0.969120
$245$	0	0
$246$	1.13046e14	0.0325219
$247$	2.31309e14i	0.0648131i
$248$	4.61208e14i	0.125881i
$249$	−3.47935e15	−0.925130
$250$	0	0
$251$	−3.58149e15	−0.904033	−0.452016	−	0.892010i	$-0.649295\pi$
$251$	−3.58149e15	−0.904033	−0.452016	+	0.892010i	$0.649295\pi$
$252$	− 4.67863e14i	− 0.115084i
$253$	− 3.79525e13i	− 0.00909824i
$254$	1.26755e15	0.296174
$255$	0	0
$256$	2.77348e15	0.615836
$257$	7.60591e15i	1.64659i	0.567612	+	0.823296i	$0.307867\pi$
$257$	7.60591e15i	1.64659i	−0.567612	+	0.823296i	$0.692133\pi$
$258$	8.27470e13i	0.0174672i
$259$	2.07301e15	0.426729
$260$	0	0
$261$	2.85670e15	0.559372
$262$	6.60113e14i	0.126084i
$263$	− 7.86154e15i	− 1.46486i	−0.680844	−	0.732428i	$-0.738387\pi$
$263$	− 7.86154e15i	− 1.46486i	0.680844	−	0.732428i	$-0.261613\pi$
$264$	−3.07710e13	−0.00559391
$265$	0	0
$266$	−7.55378e14	−0.130748
$267$	5.01633e15i	0.847350i
$268$	2.32954e15i	0.384055i
$269$	6.04452e15	0.972685	0.486343	−	0.873768i	$-0.338331\pi$
$269$	6.04452e15	0.972685	0.486343	+	0.873768i	$0.338331\pi$
$270$	0	0
$271$	−1.14219e16	−1.75162	−0.875809	−	0.482658i	$-0.839672\pi$
$271$	−1.14219e16	−1.75162	−0.875809	+	0.482658i	$0.839672\pi$
$272$	4.43281e15i	0.663714i
$273$	− 5.12836e13i	− 0.00749758i
$274$	1.54887e15	0.221125
$275$	0	0
$276$	−1.51250e15	−0.205962
$277$	− 1.65973e15i	− 0.220760i	−0.993889	−	0.110380i	$-0.964793\pi$
$277$	− 1.65973e15i	− 0.220760i	0.993889	−	0.110380i	$-0.0352068\pi$
$278$	2.26560e14i	0.0294369i
$279$	8.34595e14	0.105937
$280$	0	0
$281$	−1.28234e16	−1.55386	−0.776930	−	0.629587i	$-0.783224\pi$
$281$	−1.28234e16	−1.55386	−0.776930	+	0.629587i	$0.783224\pi$
$282$	− 5.03158e13i	− 0.00595779i
$283$	1.55354e16i	1.79767i	0.438283	+	0.898837i	$0.355587\pi$
$283$	1.55354e16i	1.79767i	−0.438283	+	0.898837i	$0.644413\pi$
$284$	−3.12432e15	−0.353335
$285$	0	0
$286$	−1.65125e12	−0.000178416 0
$287$	− 9.49515e14i	− 0.100293i
$288$	1.85224e15i	0.191270i
$289$	4.25813e15	0.429916
$290$	0	0
$291$	1.01205e16	0.977007
$292$	1.08215e16i	1.02164i
$293$	− 1.56366e16i	− 1.44378i	−0.692007	−	0.721891i	$-0.743273\pi$
$293$	− 1.56366e16i	− 1.44378i	0.692007	−	0.721891i	$-0.256727\pi$
$294$	−1.12501e15	−0.101601
$295$	0	0
$296$	−5.43348e15	−0.469550
$297$	5.56827e13i	0.00470764i
$298$	− 1.85833e15i	− 0.153715i
$299$	−1.65789e14	−0.0134182
$300$	0	0
$301$	6.95024e14	0.0538663
$302$	1.14561e15i	0.0868944i
$303$	7.49059e15i	0.556082i
$304$	−2.17336e16	−1.57926
$305$	0	0
$306$	−7.30747e14	−0.0508836
$307$	− 4.73925e15i	− 0.323080i	−0.986866	−	0.161540i	$-0.948354\pi$
$307$	− 4.73925e15i	− 0.323080i	0.986866	−	0.161540i	$-0.0516461\pi$
$308$	1.26532e14i	0.00844543i
$309$	−7.72145e15	−0.504625
$310$	0	0
$311$	−2.24003e16	−1.40382	−0.701911	−	0.712265i	$-0.747670\pi$
$311$	−2.24003e16	−1.40382	−0.701911	+	0.712265i	$0.747670\pi$
$312$	1.34418e14i	0.00824995i
$313$	2.48799e15i	0.149558i	0.997200	+	0.0747791i	$0.0238252\pi$
$313$	2.48799e15i	0.149558i	−0.997200	+	0.0747791i	$0.976175\pi$
$314$	−6.34852e15	−0.373792
$315$	0	0
$316$	3.66510e15	0.207071
$317$	− 7.02646e14i	− 0.0388912i	−0.999811	−	0.0194456i	$-0.993810\pi$
$317$	− 7.02646e14i	− 0.0388912i	0.999811	−	0.0194456i	$-0.00619012\pi$
$318$	3.07180e15i	0.166578i
$319$	−7.72585e14	−0.0410494
$320$	0	0
$321$	9.19797e15	0.469255
$322$	− 5.41410e14i	− 0.0270684i
$323$	− 2.76840e16i	− 1.35648i
$324$	2.21909e15	0.106569
$325$	0	0
$326$	7.85848e15	0.362597
$327$	1.09341e16i	0.494565i
$328$	2.48874e15i	0.110357i
$329$	−4.22621e14	−0.0183729
$330$	0	0
$331$	2.99883e16	1.25334	0.626672	−	0.779283i	$-0.284417\pi$
$331$	2.99883e16	1.25334	0.626672	+	0.779283i	$0.284417\pi$
$332$	− 3.75004e16i	− 1.53688i
$333$	9.83234e15i	0.395157i
$334$	−5.66648e15	−0.223337
$335$	0	0
$336$	4.81856e15	0.182689
$337$	2.60291e16i	0.967975i	0.875075	+	0.483987i	$0.160812\pi$
$337$	2.60291e16i	0.967975i	−0.875075	+	0.483987i	$0.839188\pi$
$338$	− 5.53506e15i	− 0.201913i
$339$	2.27329e16	0.813500
$340$	0	0
$341$	−2.25714e14	−0.00777419
$342$	− 3.58278e15i	− 0.121074i
$343$	2.03055e16i	0.673289i
$344$	−1.82170e15	−0.0592717
$345$	0	0
$346$	−8.24570e14	−0.0258365
$347$	− 5.96465e16i	− 1.83419i	−0.398674	−	0.917093i	$-0.630530\pi$
$347$	− 5.96465e16i	− 1.83419i	0.398674	−	0.917093i	$-0.369470\pi$
$348$	3.07894e16i	0.929259i
$349$	2.72283e16	0.806595	0.403298	−	0.915069i	$-0.367864\pi$
$349$	2.72283e16	0.806595	0.403298	+	0.915069i	$0.367864\pi$
$350$	0	0
$351$	2.43240e14	0.00694286
$352$	− 5.00933e14i	− 0.0140363i
$353$	− 5.67802e16i	− 1.56193i	−0.624576	−	0.780964i	$-0.714728\pi$
$353$	− 5.67802e16i	− 1.56193i	0.624576	−	0.780964i	$-0.285272\pi$
$354$	5.49376e15	0.148371
$355$	0	0
$356$	−5.40659e16	−1.40766
$357$	6.13782e15i	0.156917i
$358$	3.18848e15i	0.0800467i
$359$	7.02312e16	1.73147	0.865737	−	0.500499i	$-0.166850\pi$
$359$	7.02312e16	1.73147	0.865737	+	0.500499i	$0.166850\pi$
$360$	0	0
$361$	9.36790e16	2.22764
$362$	− 1.26254e16i	− 0.294876i
$363$	2.51520e16i	0.577005i
$364$	5.52734e14	0.0124554
$365$	0	0
$366$	5.42373e15	0.117943
$367$	− 2.11880e16i	− 0.452649i	−0.974052	−	0.226324i	$-0.927329\pi$
$367$	− 2.11880e16i	− 0.452649i	0.974052	−	0.226324i	$-0.0726710\pi$
$368$	− 1.55774e16i	− 0.326952i
$369$	4.50358e15	0.0928724
$370$	0	0
$371$	2.58012e16	0.513700
$372$	8.99526e15i	0.175989i
$373$	− 2.26994e16i	− 0.436421i	−0.975902	−	0.218211i	$-0.929978\pi$
$373$	− 2.26994e16i	− 0.436421i	0.975902	−	0.218211i	$-0.0700220\pi$
$374$	1.97628e14	0.00373408
$375$	0	0
$376$	1.10772e15	0.0202166
$377$	3.37490e15i	0.0605400i
$378$	7.94338e14i	0.0140058i
$379$	3.61052e16	0.625770	0.312885	−	0.949791i	$-0.398704\pi$
$379$	3.61052e16	0.625770	0.312885	+	0.949791i	$0.398704\pi$
$380$	0	0
$381$	5.04974e16	0.845779
$382$	− 1.37679e16i	− 0.226702i
$383$	− 2.14417e16i	− 0.347110i	−0.984824	−	0.173555i	$-0.944475\pi$
$383$	− 2.14417e16i	− 0.347110i	0.984824	−	0.173555i	$-0.0555254\pi$
$384$	−2.64101e16	−0.420355
$385$	0	0
$386$	−7.65714e15	−0.117828
$387$	3.29652e15i	0.0498809i
$388$	1.09079e17i	1.62306i
$389$	−6.52962e16	−0.955466	−0.477733	−	0.878505i	$-0.658541\pi$
$389$	−6.52962e16	−0.955466	−0.477733	+	0.878505i	$0.658541\pi$
$390$	0	0
$391$	1.98422e16	0.280829
$392$	− 2.47675e16i	− 0.344764i
$393$	2.62980e16i	0.360056i
$394$	−9.22930e15	−0.124292
$395$	0	0
$396$	−6.00147e14	−0.00782058
$397$	3.27384e16i	0.419680i	0.977736	+	0.209840i	$0.0672944\pi$
$397$	3.27384e16i	0.419680i	−0.977736	+	0.209840i	$0.932706\pi$
$398$	− 1.85476e16i	− 0.233909i
$399$	−3.00931e16	−0.373374
$400$	0	0
$401$	3.67179e16	0.441001	0.220501	−	0.975387i	$-0.429231\pi$
$401$	3.67179e16	0.441001	0.220501	+	0.975387i	$0.429231\pi$
$402$	− 3.95509e15i	− 0.0467398i
$403$	9.85991e14i	0.0114654i
$404$	−8.07334e16	−0.923793
$405$	0	0
$406$	−1.10213e16	−0.122127
$407$	− 2.65913e15i	− 0.0289985i
$408$	− 1.60876e16i	− 0.172663i
$409$	−2.37728e14	−0.00251119	−0.00125559	−	0.999999i	$-0.500400\pi$
$409$	−2.37728e14	−0.00251119	−0.00125559	+	0.999999i	$0.500400\pi$
$410$	0	0
$411$	6.17048e16	0.631462
$412$	− 8.32217e16i	− 0.838310i
$413$	− 4.61442e16i	− 0.457553i
$414$	2.56793e15	0.0250657
$415$	0	0
$416$	−2.18824e15	−0.0207008
$417$	9.02581e15i	0.0840624i
$418$	9.68953e14i	0.00888499i
$419$	1.07748e17	0.972785	0.486392	−	0.873741i	$-0.338313\pi$
$419$	1.07748e17	0.972785	0.486392	+	0.873741i	$0.338313\pi$
$420$	0	0
$421$	−4.46846e15	−0.0391133	−0.0195566	−	0.999809i	$-0.506225\pi$
$421$	−4.46846e15	−0.0391133	−0.0195566	+	0.999809i	$0.506225\pi$
$422$	− 1.22330e16i	− 0.105439i
$423$	− 2.00451e15i	− 0.0170136i
$424$	−6.76266e16	−0.565249
$425$	0	0
$426$	5.30447e15	0.0430011
$427$	− 4.55559e16i	− 0.363717i
$428$	9.91356e16i	0.779552i
$429$	−6.57835e13	−0.000509501 0
$430$	0	0
$431$	−2.38034e17	−1.78869	−0.894347	−	0.447373i	$-0.852360\pi$
$431$	−2.38034e17	−1.78869	−0.894347	+	0.447373i	$0.852360\pi$
$432$	2.28546e16i	0.169172i
$433$	− 1.52333e16i	− 0.111077i	−0.998457	−	0.0555384i	$-0.982312\pi$
$433$	− 1.52333e16i	− 0.111077i	0.998457	−	0.0555384i	$-0.0176875\pi$
$434$	−3.21991e15	−0.0231292
$435$	0	0
$436$	−1.17848e17	−0.821598
$437$	9.72846e16i	0.668213i
$438$	− 1.83727e16i	− 0.124335i
$439$	1.46731e17	0.978370	0.489185	−	0.872180i	$-0.337294\pi$
$439$	1.46731e17	0.978370	0.489185	+	0.872180i	$0.337294\pi$
$440$	0	0
$441$	−4.48188e16	−0.290141
$442$	− 8.63304e14i	− 0.00550705i
$443$	− 1.80890e17i	− 1.13708i	−0.822656	−	0.568540i	$-0.807508\pi$
$443$	− 1.80890e17i	− 1.13708i	0.822656	−	0.568540i	$-0.192492\pi$
$444$	−1.05973e17	−0.656455
$445$	0	0
$446$	1.21567e16	0.0731375
$447$	− 7.40330e16i	− 0.438961i
$448$	4.70017e16i	0.274667i
$449$	2.86599e17	1.65072	0.825360	−	0.564607i	$-0.190972\pi$
$449$	2.86599e17	1.65072	0.825360	+	0.564607i	$0.190972\pi$
$450$	0	0
$451$	−1.21798e15	−0.00681542
$452$	2.45015e17i	1.35143i
$453$	4.56394e16i	0.248143i
$454$	−5.02984e16	−0.269582
$455$	0	0
$456$	7.88760e16	0.410841
$457$	2.00560e17i	1.02989i	0.857224	+	0.514943i	$0.172187\pi$
$457$	2.00560e17i	1.02989i	−0.857224	+	0.514943i	$0.827813\pi$
$458$	3.53366e16i	0.178895i
$459$	−2.91119e16	−0.145307
$460$	0	0
$461$	−7.03451e16	−0.341332	−0.170666	−	0.985329i	$-0.554592\pi$
$461$	−7.03451e16	−0.341332	−0.170666	+	0.985329i	$0.554592\pi$
$462$	− 2.14827e14i	− 0.00102782i
$463$	− 1.71885e17i	− 0.810891i	−0.914119	−	0.405446i	$-0.867116\pi$
$463$	− 1.71885e17i	− 0.810891i	0.914119	−	0.405446i	$-0.132884\pi$
$464$	−3.17103e17	−1.47514
$465$	0	0
$466$	−5.57205e16	−0.252062
$467$	5.37684e16i	0.239865i	0.992782	+	0.119933i	$0.0382678\pi$
$467$	5.37684e16i	0.239865i	−0.992782	+	0.119933i	$0.961732\pi$
$468$	2.62164e15i	0.0115339i
$469$	−3.32203e16	−0.144139
$470$	0	0
$471$	−2.52916e17	−1.06743
$472$	1.20947e17i	0.503467i
$473$	− 8.91535e14i	− 0.00366050i
$474$	−6.22260e15	−0.0252007
$475$	0	0
$476$	−6.61533e16	−0.260679
$477$	1.22376e17i	0.475693i
$478$	2.99872e16i	0.114989i
$479$	−1.45707e17	−0.551188	−0.275594	−	0.961274i	$-0.588875\pi$
$479$	−1.45707e17	−0.551188	−0.275594	+	0.961274i	$0.588875\pi$
$480$	0	0
$481$	−1.16159e16	−0.0427672
$482$	4.27506e15i	0.0155287i
$483$	− 2.15690e16i	− 0.0772989i
$484$	−2.71088e17	−0.958551
$485$	0	0
$486$	−3.76757e15	−0.0129696
$487$	2.20545e17i	0.749132i	0.927200	+	0.374566i	$0.122208\pi$
$487$	2.20545e17i	0.749132i	−0.927200	+	0.374566i	$0.877792\pi$
$488$	1.19405e17i	0.400215i
$489$	3.13070e17	1.03546
$490$	0	0
$491$	−2.43820e16	−0.0785309	−0.0392654	−	0.999229i	$-0.512502\pi$
$491$	−2.43820e16	−0.0785309	−0.0392654	+	0.999229i	$0.512502\pi$
$492$	4.85395e16i	0.154285i
$493$	− 4.03921e17i	− 1.26704i
$494$	4.23270e15	0.0131036
$495$	0	0
$496$	−9.26429e16	−0.279371
$497$	− 4.45543e16i	− 0.132609i
$498$	6.36682e16i	0.187039i
$499$	−2.54719e17	−0.738599	−0.369299	−	0.929311i	$-0.620402\pi$
$499$	−2.54719e17	−0.738599	−0.369299	+	0.929311i	$0.620402\pi$
$500$	0	0
$501$	−2.25744e17	−0.637781
$502$	6.55371e16i	0.182774i
$503$	3.72489e17i	1.02547i	0.858548	+	0.512733i	$0.171367\pi$
$503$	3.72489e17i	1.02547i	−0.858548	+	0.512733i	$0.828633\pi$
$504$	−1.74876e16	−0.0475260
$505$	0	0
$506$	−6.94488e14	−0.00183944
$507$	− 2.20509e17i	− 0.576599i
$508$	5.44260e17i	1.40505i
$509$	1.62073e17	0.413091	0.206546	−	0.978437i	$-0.433778\pi$
$509$	1.62073e17	0.413091	0.206546	+	0.978437i	$0.433778\pi$
$510$	0	0
$511$	−1.54320e17	−0.383429
$512$	− 3.47530e17i	− 0.852584i
$513$	− 1.42733e17i	− 0.345749i
$514$	1.39180e17	0.332901
$515$	0	0
$516$	−3.55298e16	−0.0828648
$517$	5.42113e14i	0.00124854i
$518$	− 3.79337e16i	− 0.0862743i
$519$	−3.28497e16	−0.0737808
$520$	0	0
$521$	1.71142e17	0.374896	0.187448	−	0.982275i	$-0.439978\pi$
$521$	1.71142e17	0.374896	0.187448	+	0.982275i	$0.439978\pi$
$522$	− 5.22743e16i	− 0.113092i
$523$	− 4.86625e17i	− 1.03976i	−0.854239	−	0.519880i	$-0.825977\pi$
$523$	− 4.86625e17i	− 1.03976i	0.854239	−	0.519880i	$-0.174023\pi$
$524$	−2.83439e17	−0.598145
$525$	0	0
$526$	−1.43857e17	−0.296159
$527$	− 1.18007e17i	− 0.239960i
$528$	− 6.18096e15i	− 0.0124147i
$529$	4.34308e17	0.861661
$530$	0	0
$531$	2.18863e17	0.423700
$532$	− 3.24343e17i	− 0.620268i
$533$	5.32053e15i	0.0100514i
$534$	9.17932e16	0.171314
$535$	0	0
$536$	8.70725e16	0.158602
$537$	1.27024e17i	0.228588i
$538$	− 1.10608e17i	− 0.196653i
$539$	1.21211e16	0.0212920
$540$	0	0
$541$	2.80840e17	0.481589	0.240795	−	0.970576i	$-0.422592\pi$
$541$	2.80840e17	0.481589	0.240795	+	0.970576i	$0.422592\pi$
$542$	2.09009e17i	0.354135i
$543$	− 5.02978e17i	− 0.842073i
$544$	2.61897e17	0.433249
$545$	0	0
$546$	−9.38432e14	−0.00151583
$547$	− 6.77006e17i	− 1.08062i	−0.841464	−	0.540312i	$-0.818306\pi$
$547$	− 6.77006e17i	− 1.08062i	0.841464	−	0.540312i	$-0.181694\pi$
$548$	6.65054e17i	1.04902i
$549$	2.16073e17	0.336807
$550$	0	0
$551$	1.98039e18	3.01484
$552$	5.65336e16i	0.0850556i
$553$	5.22660e16i	0.0777152i
$554$	−3.03713e16	−0.0446323
$555$	0	0
$556$	−9.72800e16	−0.139649
$557$	− 4.76254e17i	− 0.675740i	−0.941193	−	0.337870i	$-0.890294\pi$
$557$	− 4.76254e17i	− 0.675740i	0.941193	−	0.337870i	$-0.109706\pi$
$558$	− 1.52721e16i	− 0.0214179i
$559$	−3.89451e15	−0.00539854
$560$	0	0
$561$	7.87322e15	0.0106634
$562$	2.34654e17i	0.314153i
$563$	1.40153e17i	0.185480i	0.995690	+	0.0927400i	$0.0295625\pi$
$563$	1.40153e17i	0.185480i	−0.995690	+	0.0927400i	$0.970437\pi$
$564$	2.16045e16	0.0282638
$565$	0	0
$566$	2.84280e17	0.363446
$567$	3.16453e16i	0.0399962i
$568$	1.16780e17i	0.145916i
$569$	−4.19192e17	−0.517825	−0.258912	−	0.965901i	$-0.583364\pi$
$569$	−4.19192e17	−0.517825	−0.258912	+	0.965901i	$0.583364\pi$
$570$	0	0
$571$	4.21896e17	0.509414	0.254707	−	0.967018i	$-0.418021\pi$
$571$	4.21896e17	0.509414	0.254707	+	0.967018i	$0.418021\pi$
$572$	− 7.09014e14i	0 0.000846410i
$573$	− 5.48493e17i	− 0.647390i
$574$	−1.73750e16	−0.0202768
$575$	0	0
$576$	−2.22930e17	−0.254345
$577$	− 2.42814e17i	− 0.273924i	−0.990576	−	0.136962i	$-0.956266\pi$
$577$	− 2.42814e17i	− 0.273924i	0.990576	−	0.136962i	$-0.0437338\pi$
$578$	− 7.79190e16i	− 0.0869185i
$579$	−3.05049e17	−0.336480
$580$	0	0
$581$	5.34774e17	0.576800
$582$	− 1.85194e17i	− 0.197527i
$583$	− 3.30963e16i	− 0.0349087i
$584$	4.04481e17	0.421905
$585$	0	0
$586$	−2.86131e17	−0.291898
$587$	1.71527e18i	1.73055i	0.501299	+	0.865274i	$0.332856\pi$
$587$	1.71527e18i	1.73055i	−0.501299	+	0.865274i	$0.667144\pi$
$588$	− 4.83057e17i	− 0.481998i
$589$	5.78578e17	0.570969
$590$	0	0
$591$	−3.67682e17	−0.354939
$592$	− 1.09142e18i	− 1.04208i
$593$	1.76024e18i	1.66232i	0.556031	+	0.831161i	$0.312324\pi$
$593$	1.76024e18i	1.66232i	−0.556031	+	0.831161i	$0.687676\pi$
$594$	1.01893e15	0.000951771 0
$595$	0	0
$596$	7.97926e17	0.729226
$597$	− 7.38908e17i	− 0.667970i
$598$	3.03375e15i	0.00271283i
$599$	1.01218e18	0.895328	0.447664	−	0.894202i	$-0.352256\pi$
$599$	1.01218e18	0.895328	0.447664	+	0.894202i	$0.352256\pi$
$600$	0	0
$601$	−1.00124e18	−0.866669	−0.433335	−	0.901233i	$-0.642663\pi$
$601$	−1.00124e18	−0.866669	−0.433335	+	0.901233i	$0.642663\pi$
$602$	− 1.27181e16i	− 0.0108905i
$603$	− 1.57565e17i	− 0.133474i
$604$	−4.91901e17	−0.412228
$605$	0	0
$606$	1.37069e17	0.112426
$607$	− 1.81791e18i	− 1.47518i	−0.675249	−	0.737590i	$-0.735964\pi$
$607$	− 1.81791e18i	− 1.47518i	0.675249	−	0.737590i	$-0.264036\pi$
$608$	1.28405e18i	1.03089i
$609$	−4.39072e17	−0.348757
$610$	0	0
$611$	2.36812e15	0.00184135
$612$	− 3.13767e17i	− 0.241392i
$613$	1.45199e18i	1.10527i	0.833422	+	0.552637i	$0.186378\pi$
$613$	1.45199e18i	1.10527i	−0.833422	+	0.552637i	$0.813622\pi$
$614$	−8.67229e16	−0.0653190
$615$	0	0
$616$	4.72947e15	0.00348769
$617$	7.36777e17i	0.537629i	0.963192	+	0.268814i	$0.0866318\pi$
$617$	7.36777e17i	0.537629i	−0.963192	+	0.268814i	$0.913368\pi$
$618$	1.41294e17i	0.102023i
$619$	−3.78289e16	−0.0270293	−0.0135147	−	0.999909i	$-0.504302\pi$
$619$	−3.78289e16	−0.0270293	−0.0135147	+	0.999909i	$0.504302\pi$
$620$	0	0
$621$	1.02302e17	0.0715798
$622$	4.09901e17i	0.283819i
$623$	− 7.71006e17i	− 0.528306i
$624$	−2.70004e16	−0.0183093
$625$	0	0
$626$	4.55273e16	0.0302370
$627$	3.86017e16i	0.0253727i
$628$	− 2.72592e18i	− 1.77327i
$629$	1.39024e18	0.895077
$630$	0	0
$631$	1.18325e18	0.746253	0.373127	−	0.927780i	$-0.378286\pi$
$631$	1.18325e18	0.746253	0.373127	+	0.927780i	$0.378286\pi$
$632$	− 1.36992e17i	− 0.0855137i
$633$	− 4.87344e17i	− 0.301101i
$634$	−1.28576e16	−0.00786286
$635$	0	0
$636$	−1.31897e18	−0.790247
$637$	− 5.29490e16i	− 0.0314016i
$638$	1.41374e16i	0.00829920i
$639$	2.11322e17	0.122798
$640$	0	0
$641$	2.00244e17	0.114020	0.0570102	−	0.998374i	$-0.481843\pi$
$641$	2.00244e17	0.114020	0.0570102	+	0.998374i	$0.481843\pi$
$642$	− 1.68312e17i	− 0.0948721i
$643$	− 2.12199e18i	− 1.18406i	−0.805917	−	0.592028i	$-0.798327\pi$
$643$	− 2.12199e18i	− 1.18406i	0.805917	−	0.592028i	$-0.201673\pi$
$644$	2.32470e17	0.128413
$645$	0	0
$646$	−5.06586e17	−0.274247
$647$	− 1.12704e18i	− 0.604032i	−0.953303	−	0.302016i	$-0.902340\pi$
$647$	− 1.12704e18i	− 0.604032i	0.953303	−	0.302016i	$-0.0976595\pi$
$648$	− 8.29443e16i	− 0.0440097i
$649$	−5.91910e16	−0.0310932
$650$	0	0
$651$	−1.28277e17	−0.0660497
$652$	3.37427e18i	1.72016i
$653$	2.48373e18i	1.25363i	0.779169	+	0.626813i	$0.215641\pi$
$653$	2.48373e18i	1.25363i	−0.779169	+	0.626813i	$0.784359\pi$
$654$	2.00082e17	0.0999891
$655$	0	0
$656$	−4.99913e17	−0.244917
$657$	− 7.31943e17i	− 0.355060i
$658$	7.73349e15i	0.00371456i
$659$	−3.56057e18	−1.69342	−0.846709	−	0.532056i	$-0.821420\pi$
$659$	−3.56057e18	−1.69342	−0.846709	+	0.532056i	$0.821420\pi$
$660$	0	0
$661$	−1.25090e18	−0.583329	−0.291664	−	0.956521i	$-0.594209\pi$
$661$	−1.25090e18	−0.583329	−0.291664	+	0.956521i	$0.594209\pi$
$662$	− 5.48752e17i	− 0.253396i
$663$	− 3.43928e16i	− 0.0157264i
$664$	−1.40167e18	−0.634680
$665$	0	0
$666$	1.79921e17	0.0798911
$667$	1.41942e18i	0.624158i
$668$	− 2.43307e18i	− 1.05952i
$669$	4.84306e17	0.208858
$670$	0	0
$671$	−5.84364e16	−0.0247165
$672$	− 2.84688e17i	− 0.119253i
$673$	− 2.74283e18i	− 1.13789i	−0.822375	−	0.568946i	$-0.807352\pi$
$673$	− 2.74283e18i	− 1.13789i	0.822375	−	0.568946i	$-0.192648\pi$
$674$	4.76302e17	0.195701
$675$	0	0
$676$	2.37664e18	0.957877
$677$	4.19537e18i	1.67473i	0.546646	+	0.837364i	$0.315905\pi$
$677$	4.19537e18i	1.67473i	−0.546646	+	0.837364i	$0.684095\pi$
$678$	− 4.15987e17i	− 0.164470i
$679$	−1.55551e18	−0.609144
$680$	0	0
$681$	−2.00382e18	−0.769842
$682$	4.13031e15i	0.00157175i
$683$	2.74346e18i	1.03410i	0.855954	+	0.517052i	$0.172971\pi$
$683$	2.74346e18i	1.03410i	−0.855954	+	0.517052i	$0.827029\pi$
$684$	1.53837e18	0.574376
$685$	0	0
$686$	3.71567e17	0.136123
$687$	1.40776e18i	0.510869i
$688$	− 3.65925e17i	− 0.131543i
$689$	−1.44575e17	−0.0514836
$690$	0	0
$691$	−5.46424e18	−1.90951	−0.954756	−	0.297390i	$-0.903884\pi$
$691$	−5.46424e18	−1.90951	−0.954756	+	0.297390i	$0.903884\pi$
$692$	− 3.54053e17i	− 0.122569i
$693$	− 8.55838e15i	− 0.00293512i
$694$	−1.09146e18	−0.370828
$695$	0	0
$696$	1.15083e18	0.383754
$697$	− 6.36782e17i	− 0.210367i
$698$	− 4.98248e17i	− 0.163074i
$699$	−2.21982e18	−0.719808
$700$	0	0
$701$	−5.09836e18	−1.62279	−0.811397	−	0.584495i	$-0.801293\pi$
$701$	−5.09836e18	−1.62279	−0.811397	+	0.584495i	$0.801293\pi$
$702$	− 4.45101e15i	− 0.00140368i
$703$	6.81621e18i	2.12977i
$704$	6.02909e16	0.0186651
$705$	0	0
$706$	−1.03901e18	−0.315784
$707$	− 1.15130e18i	− 0.346706i
$708$	2.35891e18i	0.703874i
$709$	2.21731e18	0.655580	0.327790	−	0.944751i	$-0.393696\pi$
$709$	2.21731e18	0.655580	0.327790	+	0.944751i	$0.393696\pi$
$710$	0	0
$711$	−2.47899e17	−0.0719653
$712$	2.02085e18i	0.581320i
$713$	4.14691e17i	0.118207i
$714$	1.12315e17	0.0317249
$715$	0	0
$716$	−1.36907e18	−0.379743
$717$	1.19465e18i	0.328371i
$718$	− 1.28515e18i	− 0.350062i
$719$	−5.05135e18	−1.36355	−0.681773	−	0.731563i	$-0.738791\pi$
$719$	−5.05135e18	−1.36355	−0.681773	+	0.731563i	$0.738791\pi$
$720$	0	0
$721$	1.18678e18	0.314623
$722$	− 1.71422e18i	− 0.450375i
$723$	1.70312e17i	0.0443452i
$724$	5.42109e18	1.39890
$725$	0	0
$726$	4.60253e17	0.116656
$727$	2.06443e18i	0.518594i	0.965798	+	0.259297i	$0.0834908\pi$
$727$	2.06443e18i	0.518594i	−0.965798	+	0.259297i	$0.916509\pi$
$728$	− 2.06599e16i	− 0.00514367i
$729$	−1.50095e17	−0.0370370
$730$	0	0
$731$	4.66110e17	0.112986
$732$	2.32884e18i	0.559522i
$733$	2.95043e18i	0.702603i	0.936262	+	0.351301i	$0.114261\pi$
$733$	2.95043e18i	0.702603i	−0.936262	+	0.351301i	$0.885739\pi$
$734$	−3.87717e17	−0.0915147
$735$	0	0
$736$	−9.20335e17	−0.213422
$737$	4.26130e16i	0.00979498i
$738$	− 8.24104e16i	− 0.0187766i
$739$	1.77031e18	0.399816	0.199908	−	0.979815i	$-0.435936\pi$
$739$	1.77031e18	0.399816	0.199908	+	0.979815i	$0.435936\pi$
$740$	0	0
$741$	1.68625e17	0.0374199
$742$	− 4.72133e17i	− 0.103858i
$743$	− 8.75773e18i	− 1.90970i	−0.297094	−	0.954848i	$-0.596017\pi$
$743$	− 8.75773e18i	− 1.90970i	0.297094	−	0.954848i	$-0.403983\pi$
$744$	3.36221e17	0.0726776
$745$	0	0
$746$	−4.15372e17	−0.0882339
$747$	2.53645e18i	0.534124i
$748$	8.48575e16i	0.0177145i
$749$	−1.41372e18	−0.292571
$750$	0	0
$751$	3.32951e18	0.677205	0.338602	−	0.940930i	$-0.390046\pi$
$751$	3.32951e18	0.677205	0.338602	+	0.940930i	$0.390046\pi$
$752$	2.22507e17i	0.0448671i
$753$	2.61090e18i	0.521944i
$754$	6.17569e16	0.0122397
$755$	0	0
$756$	−3.41072e17	−0.0664439
$757$	1.11835e17i	0.0216001i	0.999942	+	0.0108000i	$0.00343783\pi$
$757$	1.11835e17i	0.0216001i	−0.999942	+	0.0108000i	$0.996562\pi$
$758$	− 6.60685e17i	− 0.126516i
$759$	−2.76674e16	−0.00525287
$760$	0	0
$761$	1.10635e18	0.206486	0.103243	−	0.994656i	$-0.467078\pi$
$761$	1.10635e18	0.206486	0.103243	+	0.994656i	$0.467078\pi$
$762$	− 9.24045e17i	− 0.170996i
$763$	− 1.68057e18i	− 0.308351i
$764$	5.91165e18	1.07548
$765$	0	0
$766$	−3.92359e17	−0.0701773
$767$	2.58565e17i	0.0458565i
$768$	− 2.02186e18i	− 0.355553i
$769$	−7.10729e18	−1.23932	−0.619659	−	0.784871i	$-0.712729\pi$
$769$	−7.10729e18	−1.23932	−0.619659	+	0.784871i	$0.712729\pi$
$770$	0	0
$771$	5.54471e18	0.950661
$772$	− 3.28782e18i	− 0.558979i
$773$	− 4.57751e18i	− 0.771724i	−0.922556	−	0.385862i	$-0.873904\pi$
$773$	− 4.57751e18i	− 0.771724i	0.922556	−	0.385862i	$-0.126096\pi$
$774$	6.03226e16	0.0100847
$775$	0	0
$776$	4.07709e18	0.670270
$777$	− 1.51122e18i	− 0.246372i
$778$	1.19485e18i	0.193172i
$779$	3.12208e18	0.500554
$780$	0	0
$781$	−5.71515e16	−0.00901148
$782$	− 3.63091e17i	− 0.0567768i
$783$	− 2.08253e18i	− 0.322954i
$784$	4.97504e18	0.765143
$785$	0	0
$786$	4.81223e17	0.0727947
$787$	− 8.42386e18i	− 1.26379i	−0.775053	−	0.631896i	$-0.782277\pi$
$787$	− 8.42386e18i	− 1.26379i	0.775053	−	0.631896i	$-0.217723\pi$
$788$	− 3.96287e18i	− 0.589643i
$789$	−5.73106e18	−0.845736
$790$	0	0
$791$	−3.49403e18	−0.507201
$792$	2.24320e16i	0.00322965i
$793$	2.55269e17i	0.0364521i
$794$	5.99075e17	0.0848492
$795$	0	0
$796$	7.96393e18	1.10967
$797$	− 4.95824e18i	− 0.685249i	−0.939472	−	0.342625i	$-0.888684\pi$
$797$	− 4.95824e18i	− 0.685249i	0.939472	−	0.342625i	$-0.111316\pi$
$798$	5.50670e17i	0.0754871i
$799$	−2.83426e17	−0.0385377
$800$	0	0
$801$	3.65691e18	0.489218
$802$	− 6.71897e17i	− 0.0891598i
$803$	1.97952e17i	0.0260560i
$804$	1.69823e18	0.221734
$805$	0	0
$806$	1.80425e16	0.00231803
$807$	− 4.40646e18i	− 0.561580i
$808$	3.01762e18i	0.381497i
$809$	2.91720e18	0.365848	0.182924	−	0.983127i	$-0.441444\pi$
$809$	2.91720e18	0.365848	0.182924	+	0.983127i	$0.441444\pi$
$810$	0	0
$811$	2.27898e18	0.281258	0.140629	−	0.990062i	$-0.455088\pi$
$811$	2.27898e18	0.281258	0.140629	+	0.990062i	$0.455088\pi$
$812$	− 4.73231e18i	− 0.579374i
$813$	8.32659e18i	1.01130i
$814$	−4.86590e16	−0.00586280
$815$	0	0
$816$	3.23152e18	0.383195
$817$	2.28529e18i	0.268843i
$818$	4.35016e15i	0 0.000507702i
$819$	−3.73857e16	−0.00432873
$820$	0	0
$821$	−9.35907e18	−1.06660	−0.533301	−	0.845926i	$-0.679049\pi$
$821$	−9.35907e18	−1.06660	−0.533301	+	0.845926i	$0.679049\pi$
$822$	− 1.12913e18i	− 0.127666i
$823$	1.20581e19i	1.35264i	0.736609	+	0.676319i	$0.236426\pi$
$823$	1.20581e19i	1.35264i	−0.736609	+	0.676319i	$0.763574\pi$
$824$	−3.11063e18	−0.346195
$825$	0	0
$826$	−8.44386e17	−0.0925062
$827$	− 7.53213e18i	− 0.818714i	−0.912374	−	0.409357i	$-0.865753\pi$
$827$	− 7.53213e18i	− 0.818714i	0.912374	−	0.409357i	$-0.134247\pi$
$828$	1.10261e18i	0.118912i
$829$	−1.09464e19	−1.17130	−0.585649	−	0.810565i	$-0.699160\pi$
$829$	−1.09464e19	−1.17130	−0.585649	+	0.810565i	$0.699160\pi$
$830$	0	0
$831$	−1.20995e18	−0.127456
$832$	− 2.63370e17i	− 0.0275274i
$833$	6.33714e18i	0.657205i
$834$	1.65162e17	0.0169954
$835$	0	0
$836$	−4.16048e17	−0.0421505
$837$	− 6.08420e17i	− 0.0611629i
$838$	− 1.97166e18i	− 0.196674i
$839$	1.38440e19	1.37028	0.685139	−	0.728413i	$-0.259742\pi$
$839$	1.38440e19	1.37028	0.685139	+	0.728413i	$0.259742\pi$
$840$	0	0
$841$	1.86341e19	1.81607
$842$	8.17677e16i	0.00790775i
$843$	9.34826e18i	0.897122i
$844$	5.25259e18	0.500204
$845$	0	0
$846$	−3.66802e16	−0.00343973
$847$	− 3.86584e18i	− 0.359751i
$848$	− 1.35842e19i	− 1.25447i
$849$	1.13253e19	1.03789
$850$	0	0
$851$	−4.88546e18	−0.440923
$852$	2.27763e18i	0.203998i
$853$	− 1.10709e19i	− 0.984039i	−0.870584	−	0.492020i	$-0.836259\pi$
$853$	− 1.10709e19i	− 0.984039i	0.870584	−	0.492020i	$-0.163741\pi$
$854$	−8.33622e17	−0.0735348
$855$	0	0
$856$	3.70545e18	0.321930
$857$	− 3.38629e18i	− 0.291977i	−0.989286	−	0.145989i	$-0.953364\pi$
$857$	− 3.38629e18i	− 0.291977i	0.989286	−	0.145989i	$-0.0466363\pi$
$858$	1.20376e15i	0 0.000103009i
$859$	−5.43237e18	−0.461354	−0.230677	−	0.973030i	$-0.574094\pi$
$859$	−5.43237e18	−0.461354	−0.230677	+	0.973030i	$0.574094\pi$
$860$	0	0
$861$	−6.92196e17	−0.0579040
$862$	4.35575e18i	0.361631i
$863$	− 1.86531e18i	− 0.153703i	−0.997043	−	0.0768513i	$-0.975513\pi$
$863$	− 1.86531e18i	− 0.153703i	0.997043	−	0.0768513i	$-0.0244867\pi$
$864$	1.35028e18	0.110430
$865$	0	0
$866$	−2.78752e17	−0.0224570
$867$	− 3.10418e18i	− 0.248212i
$868$	− 1.38256e18i	− 0.109725i
$869$	6.70437e16	0.00528116
$870$	0	0
$871$	1.86147e17	0.0144457
$872$	4.40487e18i	0.339294i
$873$	− 7.37784e18i	− 0.564075i
$874$	1.78020e18	0.135097
$875$	0	0
$876$	7.88887e18	0.589845
$877$	1.78738e19i	1.32654i	0.748381	+	0.663269i	$0.230831\pi$
$877$	1.78738e19i	1.32654i	−0.748381	+	0.663269i	$0.769169\pi$
$878$	− 2.68501e18i	− 0.197803i
$879$	−1.13991e19	−0.833568
$880$	0	0
$881$	3.72462e18	0.268372	0.134186	−	0.990956i	$-0.457158\pi$
$881$	3.72462e18	0.268372	0.134186	+	0.990956i	$0.457158\pi$
$882$	8.20134e17i	0.0586596i
$883$	1.78621e19i	1.26820i	0.773250	+	0.634102i	$0.218630\pi$
$883$	1.78621e19i	1.26820i	−0.773250	+	0.634102i	$0.781370\pi$
$884$	3.70685e17	0.0261255
$885$	0	0
$886$	−3.31009e18	−0.229890
$887$	1.20818e19i	0.832967i	0.909143	+	0.416483i	$0.136738\pi$
$887$	1.20818e19i	0.832967i	−0.909143	+	0.416483i	$0.863262\pi$
$888$	3.96101e18i	0.271095i
$889$	−7.76141e18	−0.527326
$890$	0	0
$891$	4.05927e16	0.00271796
$892$	5.21985e18i	0.346965i
$893$	− 1.38961e18i	− 0.0916978i
$894$	−1.35472e18	−0.0887474
$895$	0	0
$896$	4.05920e18	0.262083
$897$	1.20860e17i	0.00774697i
$898$	− 5.24444e18i	− 0.333736i
$899$	8.44170e18	0.533325
$900$	0	0
$901$	1.73033e19	1.07750
$902$	2.22877e16i	0.00137791i
$903$	− 5.06672e17i	− 0.0310997i
$904$	9.15808e18	0.558097
$905$	0	0
$906$	8.35150e17	0.0501685
$907$	− 1.97495e19i	− 1.17790i	−0.808168	−	0.588951i	$-0.799541\pi$
$907$	− 1.97495e19i	− 1.17790i	0.808168	−	0.588951i	$-0.200459\pi$
$908$	− 2.15971e19i	− 1.27890i
$909$	5.46064e18	0.321054
$910$	0	0
$911$	−2.11213e19	−1.22420	−0.612100	−	0.790781i	$-0.709675\pi$
$911$	−2.11213e19	−1.22420	−0.612100	+	0.790781i	$0.709675\pi$
$912$	1.58438e19i	0.911787i
$913$	− 6.85975e17i	− 0.0391966i
$914$	3.67002e18	0.208218
$915$	0	0
$916$	−1.51728e19	−0.848683
$917$	− 4.04197e18i	− 0.224488i
$918$	5.32714e17i	0.0293776i
$919$	2.23735e18	0.122513	0.0612566	−	0.998122i	$-0.480489\pi$
$919$	2.23735e18	0.122513	0.0612566	+	0.998122i	$0.480489\pi$
$920$	0	0
$921$	−3.45491e18	−0.186530
$922$	1.28723e18i	0.0690092i
$923$	2.49656e17i	0.0132902i
$924$	9.22421e16	0.00487597
$925$	0	0
$926$	−3.14531e18	−0.163943
$927$	5.62894e18i	0.291345i
$928$	1.87349e19i	0.962919i
$929$	−2.69207e19	−1.37399	−0.686996	−	0.726662i	$-0.741071\pi$
$929$	−2.69207e19	−1.37399	−0.686996	+	0.726662i	$0.741071\pi$
$930$	0	0
$931$	−3.10704e19	−1.56377
$932$	− 2.39252e19i	− 1.19578i
$933$	1.63298e19i	0.810497i
$934$	9.83900e17	0.0484949
$935$	0	0
$936$	9.79904e16	0.00476311
$937$	1.74551e19i	0.842588i	0.906924	+	0.421294i	$0.138424\pi$
$937$	1.74551e19i	0.842588i	−0.906924	+	0.421294i	$0.861576\pi$
$938$	6.07894e17i	0.0291413i
$939$	1.81374e18	0.0863474
$940$	0	0
$941$	−2.30287e19	−1.08128	−0.540638	−	0.841255i	$-0.681817\pi$
$941$	−2.30287e19	−1.08128	−0.540638	+	0.841255i	$0.681817\pi$
$942$	4.62807e18i	0.215809i
$943$	2.23772e18i	0.103629i
$944$	−2.42946e19	−1.11736
$945$	0	0
$946$	−1.63141e16	−0.000740065 0
$947$	− 2.31518e18i	− 0.104306i	−0.998639	−	0.0521532i	$-0.983392\pi$
$947$	− 2.31518e18i	− 0.104306i	0.998639	−	0.0521532i	$-0.0166084\pi$
$948$	− 2.67186e18i	− 0.119553i
$949$	8.64718e17	0.0384277
$950$	0	0
$951$	−5.12229e17	−0.0224538
$952$	2.47265e18i	0.107652i
$953$	− 1.88776e19i	− 0.816288i	−0.912917	−	0.408144i	$-0.866176\pi$
$953$	− 1.88776e19i	− 0.816288i	0.912917	−	0.408144i	$-0.133824\pi$
$954$	2.23934e18	0.0961737
$955$	0	0
$956$	−1.28759e19	−0.545508
$957$	5.63215e17i	0.0236999i
$958$	2.66627e18i	0.111437i
$959$	−9.48398e18	−0.393704
$960$	0	0
$961$	−2.19513e19	−0.898996
$962$	2.12558e17i	0.00864650i
$963$	− 6.70532e18i	− 0.270925i
$964$	−1.83562e18	−0.0736686
$965$	0	0
$966$	−3.94688e17	−0.0156280
$967$	3.58183e18i	0.140875i	0.997516	+	0.0704374i	$0.0224395\pi$
$967$	3.58183e18i	0.140875i	−0.997516	+	0.0704374i	$0.977561\pi$
$968$	1.01326e19i	0.395851i
$969$	−2.01816e19	−0.783162
$970$	0	0
$971$	3.44871e19	1.32048	0.660240	−	0.751055i	$-0.270455\pi$
$971$	3.44871e19	1.32048	0.660240	+	0.751055i	$0.270455\pi$
$972$	− 1.61772e18i	− 0.0615279i
$973$	− 1.38726e18i	− 0.0524112i
$974$	4.03572e18	0.151456
$975$	0	0
$976$	−2.39849e19	−0.888206
$977$	1.61246e19i	0.593162i	0.955008	+	0.296581i	$0.0958465\pi$
$977$	1.61246e19i	0.593162i	−0.955008	+	0.296581i	$0.904154\pi$
$978$	− 5.72883e18i	− 0.209345i
$979$	−9.89000e17	−0.0359012
$980$	0	0
$981$	7.97098e18	0.285537
$982$	4.46164e17i	0.0158770i
$983$	1.25333e19i	0.443065i	0.975153	+	0.221532i	$0.0711059\pi$
$983$	1.25333e19i	0.443065i	−0.975153	+	0.221532i	$0.928894\pi$
$984$	1.81429e18	0.0637146
$985$	0	0
$986$	−7.39130e18	−0.256166
$987$	3.08091e17i	0.0106076i
$988$	1.81743e18i	0.0621639i
$989$	−1.63796e18	−0.0556581
$990$	0	0
$991$	−5.10124e18	−0.171079	−0.0855396	−	0.996335i	$-0.527261\pi$
$991$	−5.10124e18	−0.171079	−0.0855396	+	0.996335i	$0.527261\pi$
$992$	5.47348e18i	0.182363i
$993$	− 2.18615e19i	− 0.723618i
$994$	−8.15292e17	−0.0268103
$995$	0	0
$996$	−2.73378e19	−0.887315
$997$	2.50551e18i	0.0807939i	0.999184	+	0.0403969i	$0.0128622\pi$
$997$	2.50551e18i	0.0807939i	−0.999184	+	0.0403969i	$0.987138\pi$
$998$	4.66108e18i	0.149327i
$999$	7.16778e18	0.228144

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.b.e.49.2		4
5.2	odd	4		75.14.a.c.1.2		2
5.3	odd	4		15.14.a.c.1.1	✓	2
5.4	even	2	inner	75.14.b.e.49.3		4
15.8	even	4		45.14.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.14.a.c.1.1	✓	2	5.3	odd	4
45.14.a.b.1.2		2	15.8	even	4
75.14.a.c.1.2		2	5.2	odd	4
75.14.b.e.49.2		4	1.1	even	1	trivial
75.14.b.e.49.3		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-18.2989i q^{2} -729.000i q^{3} +7857.15 q^{4} -13339.9 q^{6} +112047. i q^{7} -293681. i q^{8} -531441. q^{9} +143727. q^{11} -5.72786e6i q^{12} -627845. i q^{13} +2.05033e6 q^{14} +5.89918e7 q^{16} +7.51428e7i q^{17} +9.72477e6i q^{18} -3.68418e8 q^{19} +8.16820e7 q^{21} -2.63004e6i q^{22} -2.64060e8i q^{23} -2.14094e8 q^{24} -1.14888e7 q^{26} +3.87420e8i q^{27} +8.80367e8i q^{28} -5.37538e9 q^{29} -1.57044e9 q^{31} -3.48532e9i q^{32} -1.04777e8i q^{33} +1.37503e9 q^{34} -4.17561e9 q^{36} -1.85013e10i q^{37} +6.74163e9i q^{38} -4.57699e8 q^{39} -8.47428e9 q^{41} -1.49469e9i q^{42} -6.20298e9i q^{43} +1.12928e9 q^{44} -4.83201e9 q^{46} +3.77183e9i q^{47} -4.30050e10i q^{48} +8.43346e10 q^{49} +5.47791e10 q^{51} -4.93307e9i q^{52} -2.30272e11i q^{53} +7.08936e9 q^{54} +3.29060e10 q^{56} +2.68577e11i q^{57} +9.83633e10i q^{58} -4.11830e11 q^{59} -4.06580e11 q^{61} +2.87372e10i q^{62} -5.95462e10i q^{63} +4.19483e11 q^{64} -1.91730e9 q^{66} +2.96486e11i q^{67} +5.90409e11i q^{68} -1.92500e11 q^{69} -3.97640e11 q^{71} +1.56074e11i q^{72} +1.37728e12i q^{73} -3.38553e11 q^{74} -2.89472e12 q^{76} +1.61041e10i q^{77} +8.37537e9i q^{78} +4.66466e11 q^{79} +2.82430e11 q^{81} +1.55070e11i q^{82} -4.77278e12i q^{83} +6.41788e11 q^{84} -1.13508e11 q^{86} +3.91865e12i q^{87} -4.22098e10i q^{88} -6.88111e12 q^{89} +7.03479e10 q^{91} -2.07476e12i q^{92} +1.14485e12i q^{93} +6.90203e10 q^{94} -2.54080e12 q^{96} +1.38827e13i q^{97} -1.54323e12i q^{98} -7.63823e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12482 q^{4} + 190998 q^{6} - 2125764 q^{9} + 12491776 q^{11} - 110628168 q^{14} + 150297410 q^{16} - 76950688 q^{19} + 723564576 q^{21} - 1713825054 q^{24} + 315428908 q^{26} - 2252949416 q^{29} + 12200208624 q^{31}+ \cdots - 6638641929216 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more