LMFDB - Embedded newform 50.12.b.f.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,12,Mod(49,50)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	50.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$38.4171590280$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1969})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 985x^{2} + 242064 $ x^4 + 985*x^2 + 242064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-22.6867i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.12.b.f.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.0000i q^{2} +745.734i q^{3} -1024.00 q^{4} +23863.5 q^{6} +71494.9i q^{7} +32768.0i q^{8} -378972. q^{9} -345651. q^{11} -763632. i q^{12} +1.50956e6i q^{13} +2.28784e6 q^{14} +1.04858e6 q^{16} +5.39291e6i q^{17} +1.21271e7i q^{18} +1.11633e7 q^{19} -5.33162e7 q^{21} +1.10608e7i q^{22} -5.27646e6i q^{23} -2.44362e7 q^{24} +4.83060e7 q^{26} -1.50508e8i q^{27} -7.32108e7i q^{28} +1.86291e7 q^{29} +7.10448e7 q^{31} -3.35544e7i q^{32} -2.57763e8i q^{33} +1.72573e8 q^{34} +3.88068e8 q^{36} -3.23164e8i q^{37} -3.57225e8i q^{38} -1.12573e9 q^{39} -9.11277e8 q^{41} +1.70612e9i q^{42} -1.16431e9i q^{43} +3.53946e8 q^{44} -1.68847e8 q^{46} -2.81949e8i q^{47} +7.81959e8i q^{48} -3.13420e9 q^{49} -4.02168e9 q^{51} -1.54579e9i q^{52} +4.05957e9i q^{53} -4.81626e9 q^{54} -2.34275e9 q^{56} +8.32485e9i q^{57} -5.96132e8i q^{58} -4.89828e9 q^{59} +1.07565e10 q^{61} -2.27343e9i q^{62} -2.70946e10i q^{63} -1.07374e9 q^{64} -8.24843e9 q^{66} -3.70812e9i q^{67} -5.52234e9i q^{68} +3.93484e9 q^{69} +3.45274e9 q^{71} -1.24182e10i q^{72} -2.21136e10i q^{73} -1.03413e10 q^{74} -1.14312e10 q^{76} -2.47123e10i q^{77} +3.60235e10i q^{78} -7.02672e9 q^{79} +4.51052e10 q^{81} +2.91609e10i q^{82} +5.55656e10i q^{83} +5.45958e10 q^{84} -3.72580e10 q^{86} +1.38924e10i q^{87} -1.13263e10i q^{88} +9.29706e9 q^{89} -1.07926e11 q^{91} +5.40310e9i q^{92} +5.29805e10i q^{93} -9.02235e9 q^{94} +2.50227e10 q^{96} -4.71888e10i q^{97} +1.00294e11i q^{98} +1.30992e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4096 q^{4} + 38656 q^{6} - 443828 q^{9} + 843168 q^{11} - 901888 q^{14} + 4194304 q^{16} + 57794800 q^{19} - 130893632 q^{21} - 39583744 q^{24} + 110753536 q^{26} - 116452440 q^{29} + 82826768 q^{31}+ \cdots + 502985449824 q^{99}+O(q^{100}) $ 4 * q - 4096 * q^4 + 38656 * q^6 - 443828 * q^9 + 843168 * q^11 - 901888 * q^14 + 4194304 * q^16 + 57794800 * q^19 - 130893632 * q^21 - 39583744 * q^24 + 110753536 * q^26 - 116452440 * q^29 + 82826768 * q^31 + 404712192 * q^34 + 454479872 * q^36 - 2188831696 * q^39 - 1570542072 * q^41 - 863404032 * q^44 - 2895108864 * q^46 - 16963997892 * q^49 - 7779524112 * q^51 - 12664092160 * q^54 + 923533312 * q^56 - 17354204880 * q^59 + 2230997528 * q^61 - 4294967296 * q^64 - 23456437248 * q^66 - 3457605696 * q^69 + 27599665968 * q^71 - 24144348928 * q^74 - 59181875200 * q^76 + 25273860640 * q^79 + 132427867844 * q^81 + 134035079168 * q^84 - 92991183104 * q^86 - 26417480040 * q^89 - 178029806192 * q^91 + 82440175872 * q^94 + 40533753856 * q^96 + 502985449824 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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 $

$2$	− 32.0000i	− 0.707107i
$2$	− 32.0000i	− 0.707107i
$3$	745.734i	1.77181i	0.463867	+	0.885905i	$0.346462\pi$
$3$	745.734i	1.77181i	−0.463867	+	0.885905i	$0.653538\pi$
$4$	−1024.00	−0.500000
$5$	0	0
$5$	0	0
$6$	23863.5	1.25286
$7$	71494.9i	1.60782i	0.594754	+	0.803908i	$0.297249\pi$
$7$	71494.9i	1.60782i	−0.594754	+	0.803908i	$0.702751\pi$
$8$	32768.0i	0.353553i
$9$	−378972.	−2.13931
$10$	0	0
$11$	−345651.	−0.647109	−0.323555	−	0.946209i	$-0.604878\pi$
$11$	−345651.	−0.647109	−0.323555	+	0.946209i	$0.604878\pi$
$12$	− 763632.i	− 0.885905i
$13$	1.50956e6i	1.12762i	0.825904	+	0.563810i	$0.190665\pi$
$13$	1.50956e6i	1.12762i	−0.825904	+	0.563810i	$0.809335\pi$
$14$	2.28784e6	1.13690
$15$	0	0
$16$	1.04858e6	0.250000
$17$	5.39291e6i	0.921200i	0.887608	+	0.460600i	$0.152366\pi$
$17$	5.39291e6i	0.921200i	−0.887608	+	0.460600i	$0.847634\pi$
$18$	1.21271e7i	1.51272i
$19$	1.11633e7	1.03430	0.517151	−	0.855894i	$-0.326993\pi$
$19$	1.11633e7	1.03430	0.517151	+	0.855894i	$0.326993\pi$
$20$	0	0
$21$	−5.33162e7	−2.84874
$22$	1.10608e7i	0.457575i
$23$	− 5.27646e6i	− 0.170938i	−0.996341	−	0.0854692i	$-0.972761\pi$
$23$	− 5.27646e6i	− 0.170938i	0.996341	−	0.0854692i	$-0.0272389\pi$
$24$	−2.44362e7	−0.626429
$25$	0	0
$26$	4.83060e7	0.797348
$27$	− 1.50508e8i	− 2.01864i
$28$	− 7.32108e7i	− 0.803908i
$29$	1.86291e7	0.168657	0.0843284	−	0.996438i	$-0.473126\pi$
$29$	1.86291e7	0.168657	0.0843284	+	0.996438i	$0.473126\pi$
$30$	0	0
$31$	7.10448e7	0.445700	0.222850	−	0.974853i	$-0.428464\pi$
$31$	7.10448e7	0.445700	0.222850	+	0.974853i	$0.428464\pi$
$32$	− 3.35544e7i	− 0.176777i
$33$	− 2.57763e8i	− 1.14655i
$34$	1.72573e8	0.651387
$35$	0	0
$36$	3.88068e8	1.06966
$37$	− 3.23164e8i	− 0.766150i	−0.923717	−	0.383075i	$-0.874865\pi$
$37$	− 3.23164e8i	− 0.766150i	0.923717	−	0.383075i	$-0.125135\pi$
$38$	− 3.57225e8i	− 0.731362i
$39$	−1.12573e9	−1.99793
$40$	0	0
$41$	−9.11277e8	−1.22840	−0.614199	−	0.789151i	$-0.710521\pi$
$41$	−9.11277e8	−1.22840	−0.614199	+	0.789151i	$0.710521\pi$
$42$	1.70612e9i	2.01437i
$43$	− 1.16431e9i	− 1.20779i	−0.797063	−	0.603897i	$-0.793614\pi$
$43$	− 1.16431e9i	− 1.20779i	0.797063	−	0.603897i	$-0.206386\pi$
$44$	3.53946e8	0.323555
$45$	0	0
$46$	−1.68847e8	−0.120872
$47$	− 2.81949e8i	− 0.179321i	−0.995972	−	0.0896606i	$-0.971422\pi$
$47$	− 2.81949e8i	− 0.179321i	0.995972	−	0.0896606i	$-0.0285782\pi$
$48$	7.81959e8i	0.442952i
$49$	−3.13420e9	−1.58507
$50$	0	0
$51$	−4.02168e9	−1.63219
$52$	− 1.54579e9i	− 0.563810i
$53$	4.05957e9i	1.33341i	0.745323	+	0.666704i	$0.232295\pi$
$53$	4.05957e9i	1.33341i	−0.745323	+	0.666704i	$0.767705\pi$
$54$	−4.81626e9	−1.42740
$55$	0	0
$56$	−2.34275e9	−0.568449
$57$	8.32485e9i	1.83259i
$58$	− 5.96132e8i	− 0.119258i
$59$	−4.89828e9	−0.891986	−0.445993	−	0.895036i	$-0.647149\pi$
$59$	−4.89828e9	−0.891986	−0.445993	+	0.895036i	$0.647149\pi$
$60$	0	0
$61$	1.07565e10	1.63064	0.815320	−	0.579011i	$-0.196561\pi$
$61$	1.07565e10	1.63064	0.815320	+	0.579011i	$0.196561\pi$
$62$	− 2.27343e9i	− 0.315158i
$63$	− 2.70946e10i	− 3.43962i
$64$	−1.07374e9	−0.125000
$65$	0	0
$66$	−8.24843e9	−0.810736
$67$	− 3.70812e9i	− 0.335539i	−0.985826	−	0.167769i	$-0.946344\pi$
$67$	− 3.70812e9i	− 0.335539i	0.985826	−	0.167769i	$-0.0536564\pi$
$68$	− 5.52234e9i	− 0.460600i
$69$	3.93484e9	0.302870
$70$	0	0
$71$	3.45274e9	0.227113	0.113557	−	0.993532i	$-0.463776\pi$
$71$	3.45274e9	0.227113	0.113557	+	0.993532i	$0.463776\pi$
$72$	− 1.24182e10i	− 0.756360i
$73$	− 2.21136e10i	− 1.24848i	−0.781231	−	0.624242i	$-0.785408\pi$
$73$	− 2.21136e10i	− 1.24848i	0.781231	−	0.624242i	$-0.214592\pi$
$74$	−1.03413e10	−0.541750
$75$	0	0
$76$	−1.14312e10	−0.517151
$77$	− 2.47123e10i	− 1.04043i
$78$	3.60235e10i	1.41275i
$79$	−7.02672e9	−0.256923	−0.128462	−	0.991714i	$-0.541004\pi$
$79$	−7.02672e9	−0.256923	−0.128462	+	0.991714i	$0.541004\pi$
$80$	0	0
$81$	4.51052e10	1.43734
$82$	2.91609e10i	0.868609i
$83$	5.55656e10i	1.54838i	0.632956	+	0.774188i	$0.281841\pi$
$83$	5.55656e10i	1.54838i	−0.632956	+	0.774188i	$0.718159\pi$
$84$	5.45958e10	1.42437
$85$	0	0
$86$	−3.72580e10	−0.854039
$87$	1.38924e10i	0.298828i
$88$	− 1.13263e10i	− 0.228788i
$89$	9.29706e9	0.176482	0.0882410	−	0.996099i	$-0.471875\pi$
$89$	9.29706e9	0.176482	0.0882410	+	0.996099i	$0.471875\pi$
$90$	0	0
$91$	−1.07926e11	−1.81301
$92$	5.40310e9i	0.0854692i
$93$	5.29805e10i	0.789696i
$94$	−9.02235e9	−0.126799
$95$	0	0
$96$	2.50227e10	0.313215
$97$	− 4.71888e10i	− 0.557949i	−0.960298	−	0.278975i	$-0.910005\pi$
$97$	− 4.71888e10i	− 0.557949i	0.960298	−	0.278975i	$-0.0899945\pi$
$98$	1.00294e11i	1.12081i
$99$	1.30992e11	1.38437
$100$	0	0
$101$	1.61228e11	1.52642	0.763209	−	0.646151i	$-0.223622\pi$
$101$	1.61228e11	1.52642	0.763209	+	0.646151i	$0.223622\pi$
$102$	1.28694e11i	1.15413i
$103$	− 1.85579e11i	− 1.57733i	−0.614822	−	0.788666i	$-0.710772\pi$
$103$	− 1.85579e11i	− 1.57733i	0.614822	−	0.788666i	$-0.289228\pi$
$104$	−4.94654e10	−0.398674
$105$	0	0
$106$	1.29906e11	0.942861
$107$	5.45807e10i	0.376208i	0.982149	+	0.188104i	$0.0602343\pi$
$107$	5.45807e10i	0.376208i	−0.982149	+	0.188104i	$0.939766\pi$
$108$	1.54120e11i	1.00932i
$109$	3.54035e10	0.220394	0.110197	−	0.993910i	$-0.464852\pi$
$109$	3.54035e10	0.220394	0.110197	+	0.993910i	$0.464852\pi$
$110$	0	0
$111$	2.40995e11	1.35747
$112$	7.49679e10i	0.401954i
$113$	1.20314e11i	0.614308i	0.951660	+	0.307154i	$0.0993767\pi$
$113$	1.20314e11i	0.614308i	−0.951660	+	0.307154i	$0.900623\pi$
$114$	2.66395e11	1.29583
$115$	0	0
$116$	−1.90762e10	−0.0843284
$117$	− 5.72083e11i	− 2.41233i
$118$	1.56745e11i	0.630729i
$119$	−3.85566e11	−1.48112
$120$	0	0
$121$	−1.65837e11	−0.581250
$122$	− 3.44209e11i	− 1.15304i
$123$	− 6.79571e11i	− 2.17649i
$124$	−7.27499e10	−0.222850
$125$	0	0
$126$	−8.67028e11	−2.43218
$127$	5.60687e11i	1.50591i	0.658070	+	0.752957i	$0.271373\pi$
$127$	5.60687e11i	1.50591i	−0.658070	+	0.752957i	$0.728627\pi$
$128$	3.43597e10i	0.0883883i
$129$	8.68267e11	2.13998
$130$	0	0
$131$	4.11999e11	0.933048	0.466524	−	0.884508i	$-0.345506\pi$
$131$	4.11999e11	0.933048	0.466524	+	0.884508i	$0.345506\pi$
$132$	2.63950e11i	0.573277i
$133$	7.98119e11i	1.66297i
$134$	−1.18660e11	−0.237262
$135$	0	0
$136$	−1.76715e11	−0.325693
$137$	7.86259e11i	1.39188i	0.718099	+	0.695941i	$0.245013\pi$
$137$	7.86259e11i	1.39188i	−0.718099	+	0.695941i	$0.754987\pi$
$138$	− 1.25915e11i	− 0.214162i
$139$	−8.57152e11	−1.40112	−0.700562	−	0.713592i	$-0.747067\pi$
$139$	−8.57152e11	−1.40112	−0.700562	+	0.713592i	$0.747067\pi$
$140$	0	0
$141$	2.10259e11	0.317723
$142$	− 1.10488e11i	− 0.160593i
$143$	− 5.21782e11i	− 0.729694i
$144$	−3.97381e11	−0.534828
$145$	0	0
$146$	−7.07634e11	−0.882812
$147$	− 2.33728e12i	− 2.80844i
$148$	3.30920e11i	0.383075i
$149$	2.38606e11	0.266168	0.133084	−	0.991105i	$-0.457512\pi$
$149$	2.38606e11	0.266168	0.133084	+	0.991105i	$0.457512\pi$
$150$	0	0
$151$	−9.57309e10	−0.0992382	−0.0496191	−	0.998768i	$-0.515801\pi$
$151$	−9.57309e10	−0.0992382	−0.0496191	+	0.998768i	$0.515801\pi$
$152$	3.65799e11i	0.365681i
$153$	− 2.04376e12i	− 1.97073i
$154$	−7.90793e11	−0.735697
$155$	0	0
$156$	1.15275e12	0.998965
$157$	1.79935e12i	1.50545i	0.658333	+	0.752727i	$0.271262\pi$
$157$	1.79935e12i	1.50545i	−0.658333	+	0.752727i	$0.728738\pi$
$158$	2.24855e11i	0.181672i
$159$	−3.02736e12	−2.36254
$160$	0	0
$161$	3.77240e11	0.274837
$162$	− 1.44337e12i	− 1.01635i
$163$	− 2.50619e12i	− 1.70601i	−0.521903	−	0.853005i	$-0.674778\pi$
$163$	− 2.50619e12i	− 1.70601i	0.521903	−	0.853005i	$-0.325222\pi$
$164$	9.33148e11	0.614199
$165$	0	0
$166$	1.77810e12	1.09487
$167$	2.69855e12i	1.60764i	0.594871	+	0.803822i	$0.297203\pi$
$167$	2.69855e12i	1.60764i	−0.594871	+	0.803822i	$0.702797\pi$
$168$	− 1.74707e12i	− 1.00718i
$169$	−4.86623e11	−0.271529
$170$	0	0
$171$	−4.23058e12	−2.21269
$172$	1.19225e12i	0.603897i
$173$	3.07796e12i	1.51011i	0.655659	+	0.755057i	$0.272391\pi$
$173$	3.07796e12i	1.51011i	−0.655659	+	0.755057i	$0.727609\pi$
$174$	4.44556e11	0.211303
$175$	0	0
$176$	−3.62441e11	−0.161777
$177$	− 3.65282e12i	− 1.58043i
$178$	− 2.97506e11i	− 0.124792i
$179$	−3.51154e11	−0.142826	−0.0714128	−	0.997447i	$-0.522751\pi$
$179$	−3.51154e11	−0.142826	−0.0714128	+	0.997447i	$0.522751\pi$
$180$	0	0
$181$	6.34106e11	0.242622	0.121311	−	0.992615i	$-0.461290\pi$
$181$	6.34106e11	0.242622	0.121311	+	0.992615i	$0.461290\pi$
$182$	3.45364e12i	1.28199i
$183$	8.02151e12i	2.88918i
$184$	1.72899e11	0.0604359
$185$	0	0
$186$	1.69538e12	0.558399
$187$	− 1.86406e12i	− 0.596117i
$188$	2.88715e11i	0.0896606i
$189$	1.07606e13	3.24560
$190$	0	0
$191$	−2.61584e12	−0.744609	−0.372304	−	0.928111i	$-0.621432\pi$
$191$	−2.61584e12	−0.744609	−0.372304	+	0.928111i	$0.621432\pi$
$192$	− 8.00726e11i	− 0.221476i
$193$	1.59819e12i	0.429598i	0.976658	+	0.214799i	$0.0689097\pi$
$193$	1.59819e12i	0.429598i	−0.976658	+	0.214799i	$0.931090\pi$
$194$	−1.51004e12	−0.394530
$195$	0	0
$196$	3.20942e12	0.792535
$197$	6.16281e10i	0.0147984i	0.999973	+	0.00739920i	$0.00235526\pi$
$197$	6.16281e10i	0.0147984i	−0.999973	+	0.00739920i	$0.997645\pi$
$198$	− 4.19175e12i	− 0.978896i
$199$	−2.36544e12	−0.537303	−0.268652	−	0.963237i	$-0.586578\pi$
$199$	−2.36544e12	−0.537303	−0.268652	+	0.963237i	$0.586578\pi$
$200$	0	0
$201$	2.76527e12	0.594510
$202$	− 5.15931e12i	− 1.07934i
$203$	1.33189e12i	0.271169i
$204$	4.11820e12	0.816096
$205$	0	0
$206$	−5.93851e12	−1.11534
$207$	1.99963e12i	0.365690i
$208$	1.58289e12i	0.281905i
$209$	−3.85860e12	−0.669307
$210$	0	0
$211$	5.39734e12	0.888436	0.444218	−	0.895919i	$-0.353482\pi$
$211$	5.39734e12	0.888436	0.444218	+	0.895919i	$0.353482\pi$
$212$	− 4.15700e12i	− 0.666704i
$213$	2.57482e12i	0.402402i
$214$	1.74658e12	0.266020
$215$	0	0
$216$	4.93185e12	0.713698
$217$	5.07934e12i	0.716604i
$218$	− 1.13291e12i	− 0.155842i
$219$	1.64908e13	2.21208
$220$	0	0
$221$	−8.14094e12	−1.03876
$222$	− 7.71183e12i	− 0.959878i
$223$	− 9.37124e12i	− 1.13794i	−0.822357	−	0.568971i	$-0.807342\pi$
$223$	− 9.37124e12i	− 1.13794i	0.822357	−	0.568971i	$-0.192658\pi$
$224$	2.39897e12	0.284224
$225$	0	0
$226$	3.85006e12	0.434382
$227$	6.51561e12i	0.717485i	0.933437	+	0.358742i	$0.116794\pi$
$227$	6.51561e12i	0.717485i	−0.933437	+	0.358742i	$0.883206\pi$
$228$	− 8.52464e12i	− 0.916294i
$229$	1.74808e12	0.183429	0.0917144	−	0.995785i	$-0.470765\pi$
$229$	1.74808e12	0.183429	0.0917144	+	0.995785i	$0.470765\pi$
$230$	0	0
$231$	1.84288e13	1.84345
$232$	6.10440e11i	0.0596292i
$233$	8.64493e12i	0.824715i	0.911022	+	0.412358i	$0.135295\pi$
$233$	8.64493e12i	0.824715i	−0.911022	+	0.412358i	$0.864705\pi$
$234$	−1.83067e13	−1.70578
$235$	0	0
$236$	5.01584e12	0.445993
$237$	− 5.24006e12i	− 0.455219i
$238$	1.23381e13i	1.04731i
$239$	−6.71025e12	−0.556609	−0.278304	−	0.960493i	$-0.589772\pi$
$239$	−6.71025e12	−0.556609	−0.278304	+	0.960493i	$0.589772\pi$
$240$	0	0
$241$	−1.20329e13	−0.953404	−0.476702	−	0.879065i	$-0.658168\pi$
$241$	−1.20329e13	−0.953404	−0.476702	+	0.879065i	$0.658168\pi$
$242$	5.30679e12i	0.411006i
$243$	6.97444e12i	0.528050i
$244$	−1.10147e13	−0.815320
$245$	0	0
$246$	−2.17463e13	−1.53901
$247$	1.68517e13i	1.16630i
$248$	2.32800e12i	0.157579i
$249$	−4.14371e13	−2.74343
$250$	0	0
$251$	−2.23212e13	−1.41420	−0.707101	−	0.707113i	$-0.749997\pi$
$251$	−2.23212e13	−1.41420	−0.707101	+	0.707113i	$0.749997\pi$
$252$	2.77449e13i	1.71981i
$253$	1.82381e12i	0.110616i
$254$	1.79420e13	1.06484
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	− 1.80055e13i	− 1.00178i	−0.865511	−	0.500890i	$-0.833006\pi$
$257$	− 1.80055e13i	− 1.00178i	0.865511	−	0.500890i	$-0.166994\pi$
$258$	− 2.77845e13i	− 1.51319i
$259$	2.31046e13	1.23183
$260$	0	0
$261$	−7.05993e12	−0.360809
$262$	− 1.31840e13i	− 0.659765i
$263$	1.38894e13i	0.680653i	0.940307	+	0.340326i	$0.110538\pi$
$263$	1.38894e13i	0.680653i	−0.940307	+	0.340326i	$0.889462\pi$
$264$	8.44639e12	0.405368
$265$	0	0
$266$	2.55398e13	1.17590
$267$	6.93313e12i	0.312693i
$268$	3.79711e12i	0.167769i
$269$	−2.71611e13	−1.17574	−0.587868	−	0.808957i	$-0.700033\pi$
$269$	−2.71611e13	−1.17574	−0.587868	+	0.808957i	$0.700033\pi$
$270$	0	0
$271$	−7.24542e12	−0.301115	−0.150558	−	0.988601i	$-0.548107\pi$
$271$	−7.24542e12	−0.301115	−0.150558	+	0.988601i	$0.548107\pi$
$272$	5.65488e12i	0.230300i
$273$	− 8.04842e13i	− 3.21230i
$274$	2.51603e13	0.984209
$275$	0	0
$276$	−4.02928e12	−0.151435
$277$	− 3.97300e13i	− 1.46380i	−0.681415	−	0.731898i	$-0.738635\pi$
$277$	− 3.97300e13i	− 1.46380i	0.681415	−	0.731898i	$-0.261365\pi$
$278$	2.74289e13i	0.990744i
$279$	−2.69240e13	−0.953491
$280$	0	0
$281$	5.95255e12	0.202683	0.101342	−	0.994852i	$-0.467686\pi$
$281$	5.95255e12	0.202683	0.101342	+	0.994852i	$0.467686\pi$
$282$	− 6.72828e12i	− 0.224664i
$283$	− 3.85782e13i	− 1.26333i	−0.775241	−	0.631665i	$-0.782372\pi$
$283$	− 3.85782e13i	− 1.26333i	0.775241	−	0.631665i	$-0.217628\pi$
$284$	−3.53560e12	−0.113557
$285$	0	0
$286$	−1.66970e13	−0.515971
$287$	− 6.51517e13i	− 1.97504i
$288$	1.27162e13i	0.378180i
$289$	5.18843e12	0.151390
$290$	0	0
$291$	3.51903e13	0.988580
$292$	2.26443e13i	0.624242i
$293$	− 5.32529e13i	− 1.44069i	−0.693615	−	0.720346i	$-0.743983\pi$
$293$	− 5.32529e13i	− 1.44069i	0.693615	−	0.720346i	$-0.256017\pi$
$294$	−7.47930e13	−1.98587
$295$	0	0
$296$	1.05895e13	0.270875
$297$	5.20232e13i	1.30628i
$298$	− 7.63539e12i	− 0.188210i
$299$	7.96516e12	0.192754
$300$	0	0
$301$	8.32424e13	1.94191
$302$	3.06339e12i	0.0701720i
$303$	1.20233e14i	2.70452i
$304$	1.17056e13	0.258576
$305$	0	0
$306$	−6.54004e13	−1.39352
$307$	5.74336e13i	1.20200i	0.799249	+	0.601001i	$0.205231\pi$
$307$	5.74336e13i	1.20200i	−0.799249	+	0.601001i	$0.794769\pi$
$308$	2.53054e13i	0.520216i
$309$	1.38392e14	2.79473
$310$	0	0
$311$	−2.58616e13	−0.504050	−0.252025	−	0.967721i	$-0.581097\pi$
$311$	−2.58616e13	−0.504050	−0.252025	+	0.967721i	$0.581097\pi$
$312$	− 3.68880e13i	− 0.706375i
$313$	− 3.57307e13i	− 0.672277i	−0.941813	−	0.336138i	$-0.890879\pi$
$313$	− 3.57307e13i	− 0.672277i	0.941813	−	0.336138i	$-0.109121\pi$
$314$	5.75792e13	1.06452
$315$	0	0
$316$	7.19536e12	0.128462
$317$	1.78631e13i	0.313422i	0.987644	+	0.156711i	$0.0500892\pi$
$317$	1.78631e13i	0.313422i	−0.987644	+	0.156711i	$0.949911\pi$
$318$	9.68755e13i	1.67057i
$319$	−6.43917e12	−0.109139
$320$	0	0
$321$	−4.07027e13	−0.666570
$322$	− 1.20717e13i	− 0.194339i
$323$	6.02026e13i	0.952800i
$324$	−4.61877e13	−0.718669
$325$	0	0
$326$	−8.01980e13	−1.20633
$327$	2.64016e13i	0.390496i
$328$	− 2.98607e13i	− 0.434305i
$329$	2.01579e13	0.288315
$330$	0	0
$331$	7.78278e13	1.07667	0.538333	−	0.842732i	$-0.319054\pi$
$331$	7.78278e13	1.07667	0.538333	+	0.842732i	$0.319054\pi$
$332$	− 5.68992e13i	− 0.774188i
$333$	1.22470e14i	1.63903i
$334$	8.63535e13	1.13678
$335$	0	0
$336$	−5.59061e13	−0.712186
$337$	− 4.95598e12i	− 0.0621105i	−0.999518	−	0.0310552i	$-0.990113\pi$
$337$	− 4.95598e12i	− 0.0621105i	0.999518	−	0.0310552i	$-0.00988678\pi$
$338$	1.55719e13i	0.192000i
$339$	−8.97226e13	−1.08844
$340$	0	0
$341$	−2.45567e13	−0.288417
$342$	1.35379e14i	1.56461i
$343$	− 8.27106e13i	− 0.940684i
$344$	3.81522e13	0.427019
$345$	0	0
$346$	9.84948e13	1.06781
$347$	6.55706e13i	0.699676i	0.936810	+	0.349838i	$0.113763\pi$
$347$	6.55706e13i	0.699676i	−0.936810	+	0.349838i	$0.886237\pi$
$348$	− 1.42258e13i	− 0.149414i
$349$	−1.14750e14	−1.18635	−0.593177	−	0.805072i	$-0.702127\pi$
$349$	−1.14750e14	−1.18635	−0.593177	+	0.805072i	$0.702127\pi$
$350$	0	0
$351$	2.27202e14	2.27626
$352$	1.15981e13i	0.114394i
$353$	− 5.10272e13i	− 0.495497i	−0.968824	−	0.247749i	$-0.920309\pi$
$353$	− 5.10272e13i	− 0.495497i	0.968824	−	0.247749i	$-0.0796907\pi$
$354$	−1.16890e14	−1.11753
$355$	0	0
$356$	−9.52019e12	−0.0882410
$357$	− 2.87530e14i	− 2.62426i
$358$	1.12369e13i	0.100993i
$359$	1.16657e14	1.03251	0.516253	−	0.856436i	$-0.327326\pi$
$359$	1.16657e14	1.03251	0.516253	+	0.856436i	$0.327326\pi$
$360$	0	0
$361$	8.12884e12	0.0697813
$362$	− 2.02914e13i	− 0.171559i
$363$	− 1.23671e14i	− 1.02986i
$364$	1.10516e14	0.906503
$365$	0	0
$366$	2.56688e14	2.04296
$367$	− 1.44058e14i	− 1.12947i	−0.825274	−	0.564733i	$-0.808979\pi$
$367$	− 1.44058e14i	− 1.12947i	0.825274	−	0.564733i	$-0.191021\pi$
$368$	− 5.53277e12i	− 0.0427346i
$369$	3.45349e14	2.62793
$370$	0	0
$371$	−2.90239e14	−2.14387
$372$	− 5.42521e13i	− 0.394848i
$373$	6.90594e13i	0.495250i	0.968856	+	0.247625i	$0.0796501\pi$
$373$	6.90594e13i	0.495250i	−0.968856	+	0.247625i	$0.920350\pi$
$374$	−5.96500e13	−0.421518
$375$	0	0
$376$	9.23889e12	0.0633996
$377$	2.81219e13i	0.190181i
$378$	− 3.44338e14i	− 2.29499i
$379$	−2.28579e14	−1.50149	−0.750743	−	0.660595i	$-0.770304\pi$
$379$	−2.28579e14	−1.50149	−0.750743	+	0.660595i	$0.770304\pi$
$380$	0	0
$381$	−4.18123e14	−2.66819
$382$	8.37070e13i	0.526518i
$383$	− 7.16725e13i	− 0.444385i	−0.975003	−	0.222192i	$-0.928679\pi$
$383$	− 7.16725e13i	− 0.444385i	0.975003	−	0.222192i	$-0.0713213\pi$
$384$	−2.56232e13	−0.156607
$385$	0	0
$386$	5.11420e13	0.303772
$387$	4.41242e14i	2.58385i
$388$	4.83214e13i	0.278975i
$389$	−5.31179e13	−0.302356	−0.151178	−	0.988507i	$-0.548307\pi$
$389$	−5.31179e13	−0.302356	−0.151178	+	0.988507i	$0.548307\pi$
$390$	0	0
$391$	2.84555e13	0.157469
$392$	− 1.02701e14i	− 0.560407i
$393$	3.07242e14i	1.65318i
$394$	1.97210e12	0.0104641
$395$	0	0
$396$	−1.34136e14	−0.692184
$397$	− 9.39721e12i	− 0.0478246i	−0.999714	−	0.0239123i	$-0.992388\pi$
$397$	− 9.39721e12i	− 0.0478246i	0.999714	−	0.0239123i	$-0.00761225\pi$
$398$	7.56940e13i	0.379931i
$399$	−5.95185e14	−2.94646
$400$	0	0
$401$	2.20216e14	1.06061	0.530303	−	0.847808i	$-0.322078\pi$
$401$	2.20216e14	1.06061	0.530303	+	0.847808i	$0.322078\pi$
$402$	− 8.84887e13i	− 0.420382i
$403$	1.07247e14i	0.502581i
$404$	−1.65098e14	−0.763209
$405$	0	0
$406$	4.26204e13	0.191745
$407$	1.11702e14i	0.495783i
$408$	− 1.31782e14i	− 0.577067i
$409$	3.05103e14	1.31816	0.659080	−	0.752072i	$-0.270946\pi$
$409$	3.05103e14	1.31816	0.659080	+	0.752072i	$0.270946\pi$
$410$	0	0
$411$	−5.86340e14	−2.46615
$412$	1.90032e14i	0.788666i
$413$	− 3.50203e14i	− 1.43415i
$414$	6.39883e13	0.258582
$415$	0	0
$416$	5.06526e13	0.199337
$417$	− 6.39208e14i	− 2.48253i
$418$	1.23475e14i	0.473271i
$419$	−4.21440e14	−1.59426	−0.797130	−	0.603808i	$-0.793649\pi$
$419$	−4.21440e14	−1.59426	−0.797130	+	0.603808i	$0.793649\pi$
$420$	0	0
$421$	1.11120e12	0.00409487	0.00204744	−	0.999998i	$-0.499348\pi$
$421$	1.11120e12	0.00409487	0.00204744	+	0.999998i	$0.499348\pi$
$422$	− 1.72715e14i	− 0.628219i
$423$	1.06851e14i	0.383624i
$424$	−1.33024e14	−0.471431
$425$	0	0
$426$	8.23944e13	0.284541
$427$	7.69037e14i	2.62177i
$428$	− 5.58907e13i	− 0.188104i
$429$	3.89110e14	1.29288
$430$	0	0
$431$	2.29204e14	0.742332	0.371166	−	0.928567i	$-0.378958\pi$
$431$	2.29204e14	0.742332	0.371166	+	0.928567i	$0.378958\pi$
$432$	− 1.57819e14i	− 0.504660i
$433$	1.54032e14i	0.486325i	0.969986	+	0.243162i	$0.0781848\pi$
$433$	1.54032e14i	0.486325i	−0.969986	+	0.243162i	$0.921815\pi$
$434$	1.62539e14	0.506715
$435$	0	0
$436$	−3.62531e13	−0.110197
$437$	− 5.89027e13i	− 0.176802i
$438$	− 5.27707e14i	− 1.56417i
$439$	1.25797e14	0.368227	0.184114	−	0.982905i	$-0.441059\pi$
$439$	1.25797e14	0.368227	0.184114	+	0.982905i	$0.441059\pi$
$440$	0	0
$441$	1.18778e15	3.39096
$442$	2.60510e14i	0.734517i
$443$	− 1.34623e14i	− 0.374885i	−0.982276	−	0.187443i	$-0.939980\pi$
$443$	− 1.34623e14i	− 0.374885i	0.982276	−	0.187443i	$-0.0600198\pi$
$444$	−2.46779e14	−0.678736
$445$	0	0
$446$	−2.99880e14	−0.804647
$447$	1.77937e14i	0.471600i
$448$	− 7.67671e13i	− 0.200977i
$449$	−4.28972e14	−1.10937	−0.554683	−	0.832062i	$-0.687160\pi$
$449$	−4.28972e14	−1.10937	−0.554683	+	0.832062i	$0.687160\pi$
$450$	0	0
$451$	3.14984e14	0.794908
$452$	− 1.23202e14i	− 0.307154i
$453$	− 7.13898e13i	− 0.175831i
$454$	2.08499e14	0.507338
$455$	0	0
$456$	−2.72789e14	−0.647917
$457$	7.89196e14i	1.85202i	0.377497	+	0.926011i	$0.376785\pi$
$457$	7.89196e14i	1.85202i	−0.377497	+	0.926011i	$0.623215\pi$
$458$	− 5.59387e13i	− 0.129704i
$459$	8.11677e14	1.85957
$460$	0	0
$461$	−6.01739e14	−1.34602	−0.673012	−	0.739632i	$-0.735000\pi$
$461$	−6.01739e14	−1.34602	−0.673012	+	0.739632i	$0.735000\pi$
$462$	− 5.89721e14i	− 1.30351i
$463$	2.36873e14i	0.517393i	0.965959	+	0.258696i	$0.0832929\pi$
$463$	2.36873e14i	0.517393i	−0.965959	+	0.258696i	$0.916707\pi$
$464$	1.95341e13	0.0421642
$465$	0	0
$466$	2.76638e14	0.583162
$467$	− 8.17060e14i	− 1.70220i	−0.525002	−	0.851101i	$-0.675936\pi$
$467$	− 8.17060e14i	− 1.70220i	0.525002	−	0.851101i	$-0.324064\pi$
$468$	5.85813e14i	1.20617i
$469$	2.65112e14	0.539484
$470$	0	0
$471$	−1.34184e15	−2.66738
$472$	− 1.60507e14i	− 0.315365i
$473$	4.02445e14i	0.781574i
$474$	−1.67682e14	−0.321889
$475$	0	0
$476$	3.94819e14	0.740560
$477$	− 1.53847e15i	− 2.85257i
$478$	2.14728e14i	0.393582i
$479$	−2.80917e14	−0.509018	−0.254509	−	0.967070i	$-0.581914\pi$
$479$	−2.80917e14	−0.509018	−0.254509	+	0.967070i	$0.581914\pi$
$480$	0	0
$481$	4.87837e14	0.863927
$482$	3.85053e14i	0.674159i
$483$	2.81321e14i	0.486960i
$484$	1.69817e14	0.290625
$485$	0	0
$486$	2.23182e14	0.373387
$487$	4.06414e14i	0.672294i	0.941810	+	0.336147i	$0.109124\pi$
$487$	4.06414e14i	0.672294i	−0.941810	+	0.336147i	$0.890876\pi$
$488$	3.52470e14i	0.576518i
$489$	1.86895e15	3.02273
$490$	0	0
$491$	1.02479e15	1.62065	0.810324	−	0.585982i	$-0.199291\pi$
$491$	1.02479e15	1.62065	0.810324	+	0.585982i	$0.199291\pi$
$492$	6.95880e14i	1.08824i
$493$	1.00465e14i	0.155367i
$494$	5.39255e14	0.824699
$495$	0	0
$496$	7.44959e13	0.111425
$497$	2.46853e14i	0.365156i
$498$	1.32599e15i	1.93990i
$499$	−1.01593e14	−0.146997	−0.0734987	−	0.997295i	$-0.523416\pi$
$499$	−1.01593e14	−0.146997	−0.0734987	+	0.997295i	$0.523416\pi$
$500$	0	0
$501$	−2.01240e15	−2.84844
$502$	7.14277e14i	0.999991i
$503$	9.67924e14i	1.34035i	0.742204	+	0.670174i	$0.233780\pi$
$503$	9.67924e14i	1.34035i	−0.742204	+	0.670174i	$0.766220\pi$
$504$	8.87836e14	1.21609
$505$	0	0
$506$	5.83620e13	0.0782172
$507$	− 3.62891e14i	− 0.481097i
$508$	− 5.74143e14i	− 0.752957i
$509$	−9.40088e14	−1.21961	−0.609805	−	0.792552i	$-0.708752\pi$
$509$	−9.40088e14	−1.21961	−0.609805	+	0.792552i	$0.708752\pi$
$510$	0	0
$511$	1.58101e15	2.00733
$512$	− 3.51844e13i	− 0.0441942i
$513$	− 1.68017e15i	− 2.08789i
$514$	−5.76175e14	−0.708365
$515$	0	0
$516$	−8.89105e14	−1.06999
$517$	9.74557e13i	0.116040i
$518$	− 7.39348e14i	− 0.871034i
$519$	−2.29534e15	−2.67564
$520$	0	0
$521$	−1.00574e15	−1.14783	−0.573916	−	0.818914i	$-0.694576\pi$
$521$	−1.00574e15	−1.14783	−0.573916	+	0.818914i	$0.694576\pi$
$522$	2.25918e14i	0.255131i
$523$	− 1.24911e15i	− 1.39586i	−0.716165	−	0.697931i	$-0.754104\pi$
$523$	− 1.24911e15i	− 1.39586i	0.716165	−	0.697931i	$-0.245896\pi$
$524$	−4.21887e14	−0.466524
$525$	0	0
$526$	4.44460e14	0.481294
$527$	3.83138e14i	0.410579i
$528$	− 2.70285e14i	− 0.286639i
$529$	9.24969e14	0.970780
$530$	0	0
$531$	1.85631e15	1.90823
$532$	− 8.17274e14i	− 0.831483i
$533$	− 1.37563e15i	− 1.38517i
$534$	2.21860e14	0.221107
$535$	0	0
$536$	1.21508e14	0.118631
$537$	− 2.61868e14i	− 0.253060i
$538$	8.69155e14i	0.831371i
$539$	1.08334e15	1.02571
$540$	0	0
$541$	1.47684e15	1.37009	0.685046	−	0.728500i	$-0.259782\pi$
$541$	1.47684e15	1.37009	0.685046	+	0.728500i	$0.259782\pi$
$542$	2.31853e14i	0.212921i
$543$	4.72874e14i	0.429880i
$544$	1.80956e14	0.162847
$545$	0	0
$546$	−2.57550e15	−2.27144
$547$	2.32000e14i	0.202562i	0.994858	+	0.101281i	$0.0322941\pi$
$547$	2.32000e14i	0.202562i	−0.994858	+	0.101281i	$0.967706\pi$
$548$	− 8.05129e14i	− 0.695941i
$549$	−4.07643e15	−3.48844
$550$	0	0
$551$	2.07962e14	0.174442
$552$	1.28937e14i	0.107081i
$553$	− 5.02375e14i	− 0.413085i
$554$	−1.27136e15	−1.03506
$555$	0	0
$556$	8.77724e14	0.700562
$557$	− 2.34238e15i	− 1.85120i	−0.378498	−	0.925602i	$-0.623559\pi$
$557$	− 2.34238e15i	− 1.85120i	0.378498	−	0.925602i	$-0.376441\pi$
$558$	8.61568e14i	0.674220i
$559$	1.75760e15	1.36193
$560$	0	0
$561$	1.39010e15	1.05621
$562$	− 1.90482e14i	− 0.143319i
$563$	2.32309e15i	1.73089i	0.501002	+	0.865446i	$0.332965\pi$
$563$	2.32309e15i	1.73089i	−0.501002	+	0.865446i	$0.667035\pi$
$564$	−2.15305e14	−0.158862
$565$	0	0
$566$	−1.23450e15	−0.893309
$567$	3.22479e15i	2.31098i
$568$	1.13139e14i	0.0802967i
$569$	2.48598e14	0.174735	0.0873676	−	0.996176i	$-0.472155\pi$
$569$	2.48598e14	0.174735	0.0873676	+	0.996176i	$0.472155\pi$
$570$	0	0
$571$	−2.44700e15	−1.68708	−0.843540	−	0.537066i	$-0.819533\pi$
$571$	−2.44700e15	−1.68708	−0.843540	+	0.537066i	$0.819533\pi$
$572$	5.34305e14i	0.364847i
$573$	− 1.95072e15i	− 1.31931i
$574$	−2.08486e15	−1.39656
$575$	0	0
$576$	4.06919e14	0.267414
$577$	2.05338e15i	1.33660i	0.743890	+	0.668302i	$0.232979\pi$
$577$	2.05338e15i	1.33660i	−0.743890	+	0.668302i	$0.767021\pi$
$578$	− 1.66030e14i	− 0.107049i
$579$	−1.19182e15	−0.761166
$580$	0	0
$581$	−3.97266e15	−2.48950
$582$	− 1.12609e15i	− 0.699032i
$583$	− 1.40319e15i	− 0.862860i
$584$	7.24617e14	0.441406
$585$	0	0
$586$	−1.70409e15	−1.01872
$587$	− 3.00291e14i	− 0.177841i	−0.996039	−	0.0889207i	$-0.971658\pi$
$587$	− 3.00291e14i	− 0.177841i	0.996039	−	0.0889207i	$-0.0283418\pi$
$588$	2.39338e15i	1.40422i
$589$	7.93094e14	0.460989
$590$	0	0
$591$	−4.59582e13	−0.0262200
$592$	− 3.38862e14i	− 0.191538i
$593$	− 2.71460e15i	− 1.52022i	−0.649796	−	0.760108i	$-0.725146\pi$
$593$	− 2.71460e15i	− 1.52022i	0.649796	−	0.760108i	$-0.274854\pi$
$594$	1.66474e15	0.923681
$595$	0	0
$596$	−2.44332e14	−0.133084
$597$	− 1.76399e15i	− 0.951999i
$598$	− 2.54885e14i	− 0.136298i
$599$	2.10006e15	1.11272	0.556358	−	0.830943i	$-0.312198\pi$
$599$	2.10006e15	1.11272	0.556358	+	0.830943i	$0.312198\pi$
$600$	0	0
$601$	2.84187e15	1.47841	0.739204	−	0.673481i	$-0.235202\pi$
$601$	2.84187e15	1.47841	0.739204	+	0.673481i	$0.235202\pi$
$602$	− 2.66376e15i	− 1.37314i
$603$	1.40528e15i	0.717821i
$604$	9.80284e13	0.0496191
$605$	0	0
$606$	3.84747e15	1.91239
$607$	3.78069e14i	0.186223i	0.995656	+	0.0931114i	$0.0296813\pi$
$607$	3.78069e14i	0.186223i	−0.995656	+	0.0931114i	$0.970319\pi$
$608$	− 3.74578e14i	− 0.182841i
$609$	−9.93235e14	−0.480460
$610$	0	0
$611$	4.25619e14	0.202206
$612$	2.09281e15i	0.985367i
$613$	2.97141e15i	1.38653i	0.720681	+	0.693267i	$0.243829\pi$
$613$	2.97141e15i	1.38653i	−0.720681	+	0.693267i	$0.756171\pi$
$614$	1.83788e15	0.849943
$615$	0	0
$616$	8.09772e14	0.367848
$617$	1.19545e15i	0.538222i	0.963109	+	0.269111i	$0.0867299\pi$
$617$	1.19545e15i	0.538222i	−0.963109	+	0.269111i	$0.913270\pi$
$618$	− 4.42855e15i	− 1.97617i
$619$	−3.17735e14	−0.140529	−0.0702645	−	0.997528i	$-0.522384\pi$
$619$	−3.17735e14	−0.140529	−0.0702645	+	0.997528i	$0.522384\pi$
$620$	0	0
$621$	−7.94151e14	−0.345064
$622$	8.27572e14i	0.356417i
$623$	6.64693e14i	0.283750i
$624$	−1.18042e15	−0.499482
$625$	0	0
$626$	−1.14338e15	−0.475372
$627$	− 2.87749e15i	− 1.18588i
$628$	− 1.84253e15i	− 0.752727i
$629$	1.74280e15	0.705778
$630$	0	0
$631$	−7.14141e14	−0.284199	−0.142099	−	0.989852i	$-0.545385\pi$
$631$	−7.14141e14	−0.284199	−0.142099	+	0.989852i	$0.545385\pi$
$632$	− 2.30251e14i	− 0.0908361i
$633$	4.02498e15i	1.57414i
$634$	5.71618e14	0.221623
$635$	0	0
$636$	3.10002e15	1.18127
$637$	− 4.73128e15i	− 1.78736i
$638$	2.06054e14i	0.0771732i
$639$	−1.30849e15	−0.485866
$640$	0	0
$641$	3.74739e15	1.36776	0.683880	−	0.729594i	$-0.260291\pi$
$641$	3.74739e15	1.36776	0.683880	+	0.729594i	$0.260291\pi$
$642$	1.30249e15i	0.471336i
$643$	3.72691e14i	0.133718i	0.997762	+	0.0668588i	$0.0212977\pi$
$643$	3.72691e14i	0.133718i	−0.997762	+	0.0668588i	$0.978702\pi$
$644$	−3.86294e14	−0.137419
$645$	0	0
$646$	1.92648e15	0.673731
$647$	− 3.55565e15i	− 1.23295i	−0.787374	−	0.616475i	$-0.788560\pi$
$647$	− 3.55565e15i	− 1.23295i	0.787374	−	0.616475i	$-0.211440\pi$
$648$	1.47801e15i	0.508176i
$649$	1.69310e15	0.577212
$650$	0	0
$651$	−3.78784e15	−1.26969
$652$	2.56633e15i	0.853005i
$653$	5.07282e15i	1.67196i	0.548757	+	0.835982i	$0.315101\pi$
$653$	5.07282e15i	1.67196i	−0.548757	+	0.835982i	$0.684899\pi$
$654$	8.44850e14	0.276123
$655$	0	0
$656$	−9.55544e14	−0.307100
$657$	8.38043e15i	2.67090i
$658$	− 6.45053e14i	− 0.203870i
$659$	−1.66566e15	−0.522054	−0.261027	−	0.965331i	$-0.584061\pi$
$659$	−1.66566e15	−0.522054	−0.261027	+	0.965331i	$0.584061\pi$
$660$	0	0
$661$	3.06978e14	0.0946235	0.0473118	−	0.998880i	$-0.484935\pi$
$661$	3.06978e14	0.0946235	0.0473118	+	0.998880i	$0.484935\pi$
$662$	− 2.49049e15i	− 0.761318i
$663$	− 6.07098e15i	− 1.84049i
$664$	−1.82077e15	−0.547433
$665$	0	0
$666$	3.91905e15	1.15897
$667$	− 9.82960e13i	− 0.0288299i
$668$	− 2.76331e15i	− 0.803822i
$669$	6.98845e15	2.01622
$670$	0	0
$671$	−3.71800e15	−1.05520
$672$	1.78900e15i	0.503591i
$673$	3.20107e15i	0.893742i	0.894599	+	0.446871i	$0.147462\pi$
$673$	3.20107e15i	0.893742i	−0.894599	+	0.446871i	$0.852538\pi$
$674$	−1.58591e14	−0.0439187
$675$	0	0
$676$	4.98302e14	0.135764
$677$	4.88443e15i	1.32001i	0.751262	+	0.660004i	$0.229445\pi$
$677$	4.88443e15i	1.32001i	−0.751262	+	0.660004i	$0.770555\pi$
$678$	2.87112e15i	0.769642i
$679$	3.37376e15	0.897079
$680$	0	0
$681$	−4.85891e15	−1.27125
$682$	7.85814e14i	0.203941i
$683$	3.43048e15i	0.883163i	0.897221	+	0.441582i	$0.145582\pi$
$683$	3.43048e15i	0.883163i	−0.897221	+	0.441582i	$0.854418\pi$
$684$	4.33211e15	1.10635
$685$	0	0
$686$	−2.64674e15	−0.665164
$687$	1.30361e15i	0.325001i
$688$	− 1.22087e15i	− 0.301948i
$689$	−6.12818e15	−1.50358
$690$	0	0
$691$	1.26067e15	0.304420	0.152210	−	0.988348i	$-0.451361\pi$
$691$	1.26067e15	0.304420	0.152210	+	0.988348i	$0.451361\pi$
$692$	− 3.15183e15i	− 0.755057i
$693$	9.36527e15i	2.22581i
$694$	2.09826e15	0.494746
$695$	0	0
$696$	−4.55226e14	−0.105652
$697$	− 4.91444e15i	− 1.13160i
$698$	3.67201e15i	0.838879i
$699$	−6.44682e15	−1.46124
$700$	0	0
$701$	−1.46416e15	−0.326693	−0.163347	−	0.986569i	$-0.552229\pi$
$701$	−1.46416e15	−0.326693	−0.163347	+	0.986569i	$0.552229\pi$
$702$	− 7.27045e15i	− 1.60956i
$703$	− 3.60758e15i	− 0.792431i
$704$	3.71140e14	0.0808887
$705$	0	0
$706$	−1.63287e15	−0.350369
$707$	1.15270e16i	2.45420i
$708$	3.74049e15i	0.790215i
$709$	2.00407e15	0.420105	0.210053	−	0.977690i	$-0.432636\pi$
$709$	2.00407e15	0.420105	0.210053	+	0.977690i	$0.432636\pi$
$710$	0	0
$711$	2.66293e15	0.549639
$712$	3.04646e14i	0.0623958i
$713$	− 3.74865e14i	− 0.0761873i
$714$	−9.20095e15	−1.85563
$715$	0	0
$716$	3.59582e14	0.0714128
$717$	− 5.00406e15i	− 0.986205i
$718$	− 3.73303e15i	− 0.730092i
$719$	6.94101e15	1.34714	0.673571	−	0.739122i	$-0.264759\pi$
$719$	6.94101e15	1.34714	0.673571	+	0.739122i	$0.264759\pi$
$720$	0	0
$721$	1.32679e16	2.53606
$722$	− 2.60123e14i	− 0.0493428i
$723$	− 8.97336e15i	− 1.68925i
$724$	−6.49324e14	−0.121311
$725$	0	0
$726$	−3.95746e15	−0.728224
$727$	8.96895e14i	0.163796i	0.996641	+	0.0818978i	$0.0260981\pi$
$727$	8.96895e14i	0.163796i	−0.996641	+	0.0818978i	$0.973902\pi$
$728$	− 3.53653e15i	− 0.640994i
$729$	2.78918e15	0.501735
$730$	0	0
$731$	6.27903e15	1.11262
$732$	− 8.21403e15i	− 1.44459i
$733$	− 1.11078e16i	− 1.93890i	−0.245286	−	0.969451i	$-0.578882\pi$
$733$	− 1.11078e16i	− 1.93890i	0.245286	−	0.969451i	$-0.421118\pi$
$734$	−4.60985e15	−0.798653
$735$	0	0
$736$	−1.77049e14	−0.0302179
$737$	1.28171e15i	0.217130i
$738$	− 1.10512e16i	− 1.85822i
$739$	−5.03625e15	−0.840548	−0.420274	−	0.907397i	$-0.638066\pi$
$739$	−5.03625e15	−0.840548	−0.420274	+	0.907397i	$0.638066\pi$
$740$	0	0
$741$	−1.25669e16	−2.06646
$742$	9.28764e15i	1.51595i
$743$	1.36645e15i	0.221389i	0.993854	+	0.110695i	$0.0353075\pi$
$743$	1.36645e15i	0.221389i	−0.993854	+	0.110695i	$0.964692\pi$
$744$	−1.73607e15	−0.279200
$745$	0	0
$746$	2.20990e15	0.350194
$747$	− 2.10578e16i	− 3.31246i
$748$	1.90880e15i	0.298059i
$749$	−3.90225e15	−0.604874
$750$	0	0
$751$	−9.78301e15	−1.49435	−0.747176	−	0.664627i	$-0.768591\pi$
$751$	−9.78301e15	−1.49435	−0.747176	+	0.664627i	$0.768591\pi$
$752$	− 2.95644e14i	− 0.0448303i
$753$	− 1.66456e16i	− 2.50570i
$754$	8.99900e14	0.134478
$755$	0	0
$756$	−1.10188e16	−1.62280
$757$	1.39502e15i	0.203964i	0.994786	+	0.101982i	$0.0325184\pi$
$757$	1.39502e15i	0.203964i	−0.994786	+	0.101982i	$0.967482\pi$
$758$	7.31454e15i	1.06171i
$759$	−1.36008e15	−0.195990
$760$	0	0
$761$	6.09995e15	0.866385	0.433192	−	0.901301i	$-0.357387\pi$
$761$	6.09995e15	0.866385	0.433192	+	0.901301i	$0.357387\pi$
$762$	1.33799e16i	1.88670i
$763$	2.53117e15i	0.354353i
$764$	2.67862e15	0.372304
$765$	0	0
$766$	−2.29352e15	−0.314228
$767$	− 7.39427e15i	− 1.00582i
$768$	8.19943e14i	0.110738i
$769$	−6.60626e15	−0.885851	−0.442925	−	0.896558i	$-0.646059\pi$
$769$	−6.60626e15	−0.885851	−0.442925	+	0.896558i	$0.646059\pi$
$770$	0	0
$771$	1.34273e16	1.77496
$772$	− 1.63654e15i	− 0.214799i
$773$	3.74520e15i	0.488076i	0.969766	+	0.244038i	$0.0784722\pi$
$773$	3.74520e15i	0.488076i	−0.969766	+	0.244038i	$0.921528\pi$
$774$	1.41197e16	1.82705
$775$	0	0
$776$	1.54628e15	0.197265
$777$	1.72299e16i	2.18257i
$778$	1.69977e15i	0.213798i
$779$	−1.01729e16	−1.27054
$780$	0	0
$781$	−1.19344e15	−0.146967
$782$	− 9.10576e14i	− 0.111347i
$783$	− 2.80384e15i	− 0.340458i
$784$	−3.28645e15	−0.396267
$785$	0	0
$786$	9.83174e15	1.16898
$787$	1.01107e16i	1.19377i	0.802326	+	0.596885i	$0.203595\pi$
$787$	1.01107e16i	1.19377i	−0.802326	+	0.596885i	$0.796405\pi$
$788$	− 6.31072e13i	− 0.00739920i
$789$	−1.03578e16	−1.20599
$790$	0	0
$791$	−8.60188e15	−0.987694
$792$	4.29235e15i	0.489448i
$793$	1.62377e16i	1.83874i
$794$	−3.00711e14	−0.0338171
$795$	0	0
$796$	2.42221e15	0.268652
$797$	1.40032e16i	1.54244i	0.636570	+	0.771219i	$0.280353\pi$
$797$	1.40032e16i	1.54244i	−0.636570	+	0.771219i	$0.719647\pi$
$798$	1.90459e16i	2.08346i
$799$	1.52052e15	0.165191
$800$	0	0
$801$	−3.52333e15	−0.377550
$802$	− 7.04691e15i	− 0.749962i
$803$	7.64357e15i	0.807906i
$804$	−2.83164e15	−0.297255
$805$	0	0
$806$	3.43189e15	0.355378
$807$	− 2.02550e16i	− 2.08318i
$808$	5.28313e15i	0.539671i
$809$	−2.05016e15	−0.208004	−0.104002	−	0.994577i	$-0.533165\pi$
$809$	−2.05016e15	−0.208004	−0.104002	+	0.994577i	$0.533165\pi$
$810$	0	0
$811$	−1.45631e16	−1.45760	−0.728801	−	0.684725i	$-0.759922\pi$
$811$	−1.45631e16	−1.45760	−0.728801	+	0.684725i	$0.759922\pi$
$812$	− 1.36385e15i	− 0.135584i
$813$	− 5.40316e15i	− 0.533519i
$814$	3.57446e15	0.350571
$815$	0	0
$816$	−4.21703e15	−0.408048
$817$	− 1.29975e16i	− 1.24922i
$818$	− 9.76330e15i	− 0.932080i
$819$	4.09011e16	3.87858
$820$	0	0
$821$	−5.13429e15	−0.480389	−0.240195	−	0.970725i	$-0.577211\pi$
$821$	−5.13429e15	−0.480389	−0.240195	+	0.970725i	$0.577211\pi$
$822$	1.87629e16i	1.74383i
$823$	− 6.16638e15i	− 0.569287i	−0.958633	−	0.284643i	$-0.908125\pi$
$823$	− 6.16638e15i	− 0.569287i	0.958633	−	0.284643i	$-0.0918752\pi$
$824$	6.08104e15	0.557671
$825$	0	0
$826$	−1.12065e16	−1.01410
$827$	2.77673e15i	0.249605i	0.992182	+	0.124803i	$0.0398298\pi$
$827$	2.77673e15i	0.249605i	−0.992182	+	0.124803i	$0.960170\pi$
$828$	− 2.04763e15i	− 0.182845i
$829$	−1.33865e16	−1.18745	−0.593726	−	0.804667i	$-0.702344\pi$
$829$	−1.33865e16	−1.18745	−0.593726	+	0.804667i	$0.702344\pi$
$830$	0	0
$831$	2.96281e16	2.59357
$832$	− 1.62088e15i	− 0.140953i
$833$	− 1.69025e16i	− 1.46017i
$834$	−2.04546e16	−1.75541
$835$	0	0
$836$	3.95121e15	0.334653
$837$	− 1.06928e16i	− 0.899709i
$838$	1.34861e16i	1.12731i
$839$	−1.47887e16	−1.22812	−0.614059	−	0.789260i	$-0.710464\pi$
$839$	−1.47887e16	−1.22812	−0.614059	+	0.789260i	$0.710464\pi$
$840$	0	0
$841$	−1.18535e16	−0.971555
$842$	− 3.55584e13i	− 0.00289551i
$843$	4.43902e15i	0.359116i
$844$	−5.52687e15	−0.444218
$845$	0	0
$846$	3.41922e15	0.271263
$847$	− 1.18565e16i	− 0.934542i
$848$	4.25677e15i	0.333352i
$849$	2.87691e16	2.23838
$850$	0	0
$851$	−1.70517e15	−0.130965
$852$	− 2.63662e15i	− 0.201201i
$853$	1.97554e16i	1.49784i	0.662658	+	0.748922i	$0.269428\pi$
$853$	1.97554e16i	1.49784i	−0.662658	+	0.748922i	$0.730572\pi$
$854$	2.46092e16	1.85387
$855$	0	0
$856$	−1.78850e15	−0.133010
$857$	2.32233e16i	1.71605i	0.513612	+	0.858023i	$0.328307\pi$
$857$	2.32233e16i	1.71605i	−0.513612	+	0.858023i	$0.671693\pi$
$858$	− 1.24515e16i	− 0.914203i
$859$	1.67344e16	1.22081	0.610405	−	0.792089i	$-0.291007\pi$
$859$	1.67344e16	1.22081	0.610405	+	0.792089i	$0.291007\pi$
$860$	0	0
$861$	4.85859e16	3.49939
$862$	− 7.33454e15i	− 0.524908i
$863$	− 2.46024e16i	− 1.74951i	−0.484563	−	0.874757i	$-0.661021\pi$
$863$	− 2.46024e16i	− 1.74951i	0.484563	−	0.874757i	$-0.338979\pi$
$864$	−5.05021e15	−0.356849
$865$	0	0
$866$	4.92901e15	0.343883
$867$	3.86919e15i	0.268234i
$868$	− 5.20125e15i	− 0.358302i
$869$	2.42879e15	0.166257
$870$	0	0
$871$	5.59764e15	0.378360
$872$	1.16010e15i	0.0779211i
$873$	1.78833e16i	1.19363i
$874$	−1.88489e15	−0.125018
$875$	0	0
$876$	−1.68866e16	−1.10604
$877$	4.55865e15i	0.296714i	0.988934	+	0.148357i	$0.0473985\pi$
$877$	4.55865e15i	0.296714i	−0.988934	+	0.148357i	$0.952601\pi$
$878$	− 4.02551e15i	− 0.260376i
$879$	3.97125e16	2.55263
$880$	0	0
$881$	1.67834e16	1.06540	0.532701	−	0.846304i	$-0.321177\pi$
$881$	1.67834e16	1.06540	0.532701	+	0.846304i	$0.321177\pi$
$882$	− 3.80088e16i	− 2.39777i
$883$	− 1.13628e16i	− 0.712362i	−0.934417	−	0.356181i	$-0.884079\pi$
$883$	− 1.13628e16i	− 0.712362i	0.934417	−	0.356181i	$-0.115921\pi$
$884$	8.33632e15	0.519382
$885$	0	0
$886$	−4.30793e15	−0.265084
$887$	− 1.55603e16i	− 0.951563i	−0.879563	−	0.475782i	$-0.842165\pi$
$887$	− 1.55603e16i	− 0.951563i	0.879563	−	0.475782i	$-0.157835\pi$
$888$	7.89691e15i	0.479939i
$889$	−4.00863e16	−2.42123
$890$	0	0
$891$	−1.55906e16	−0.930115
$892$	9.59615e15i	0.568971i
$893$	− 3.14747e15i	− 0.185472i
$894$	5.69397e15	0.333471
$895$	0	0
$896$	−2.45655e15	−0.142112
$897$	5.93989e15i	0.341523i
$898$	1.37271e16i	0.784440i
$899$	1.32350e15	0.0751704
$900$	0	0
$901$	−2.18929e16	−1.22834
$902$	− 1.00795e16i	− 0.562085i
$903$	6.20767e16i	3.44069i
$904$	−3.94246e15	−0.217191
$905$	0	0
$906$	−2.28447e15	−0.124331
$907$	− 3.40764e16i	− 1.84338i	−0.387932	−	0.921688i	$-0.626811\pi$
$907$	− 3.40764e16i	− 1.84338i	0.387932	−	0.921688i	$-0.373189\pi$
$908$	− 6.67198e15i	− 0.358742i
$909$	−6.11011e16	−3.26548
$910$	0	0
$911$	1.60732e16	0.848697	0.424348	−	0.905499i	$-0.360503\pi$
$911$	1.60732e16	0.848697	0.424348	+	0.905499i	$0.360503\pi$
$912$	8.72924e15i	0.458147i
$913$	− 1.92063e16i	− 1.00197i
$914$	2.52543e16	1.30958
$915$	0	0
$916$	−1.79004e15	−0.0917144
$917$	2.94559e16i	1.50017i
$918$	− 2.59737e16i	− 1.31492i
$919$	7.32080e15	0.368403	0.184201	−	0.982889i	$-0.441030\pi$
$919$	7.32080e15	0.368403	0.184201	+	0.982889i	$0.441030\pi$
$920$	0	0
$921$	−4.28302e16	−2.12972
$922$	1.92556e16i	0.951783i
$923$	5.21213e15i	0.256098i
$924$	−1.88711e16	−0.921724
$925$	0	0
$926$	7.57994e15	0.365852
$927$	7.03292e16i	3.37440i
$928$	− 6.25090e14i	− 0.0298146i
$929$	1.41658e16	0.671666	0.335833	−	0.941922i	$-0.390982\pi$
$929$	1.41658e16	0.671666	0.335833	+	0.941922i	$0.390982\pi$
$930$	0	0
$931$	−3.49880e16	−1.63944
$932$	− 8.85241e15i	− 0.412358i
$933$	− 1.92859e16i	− 0.893081i
$934$	−2.61459e16	−1.20364
$935$	0	0
$936$	1.87460e16	0.852888
$937$	− 1.51385e16i	− 0.684721i	−0.939569	−	0.342361i	$-0.888774\pi$
$937$	− 1.51385e16i	− 0.684721i	0.939569	−	0.342361i	$-0.111226\pi$
$938$	− 8.48358e15i	− 0.381473i
$939$	2.66456e16	1.19115
$940$	0	0
$941$	8.81297e15	0.389385	0.194693	−	0.980864i	$-0.437629\pi$
$941$	8.81297e15	0.389385	0.194693	+	0.980864i	$0.437629\pi$
$942$	4.29388e16i	1.88612i
$943$	4.80832e15i	0.209981i
$944$	−5.13622e15	−0.222996
$945$	0	0
$946$	1.28782e16	0.552656
$947$	− 6.85225e15i	− 0.292353i	−0.989258	−	0.146177i	$-0.953303\pi$
$947$	− 6.85225e15i	− 0.292353i	0.989258	−	0.146177i	$-0.0466968\pi$
$948$	5.36582e15i	0.227610i
$949$	3.33818e16	1.40782
$950$	0	0
$951$	−1.33211e16	−0.555325
$952$	− 1.26342e16i	− 0.523655i
$953$	3.21185e16i	1.32356i	0.749697	+	0.661781i	$0.230199\pi$
$953$	3.21185e16i	1.32356i	−0.749697	+	0.661781i	$0.769801\pi$
$954$	−4.92309e16	−2.01707
$955$	0	0
$956$	6.87129e15	0.278304
$957$	− 4.80191e15i	− 0.193374i
$958$	8.98935e15i	0.359930i
$959$	−5.62135e16	−2.23789
$960$	0	0
$961$	−2.03611e16	−0.801351
$962$	− 1.56108e16i	− 0.610889i
$963$	− 2.06846e16i	− 0.804827i
$964$	1.23217e16	0.476702
$965$	0	0
$966$	9.00228e15	0.344333
$967$	− 2.01801e16i	− 0.767500i	−0.923437	−	0.383750i	$-0.874633\pi$
$967$	− 2.01801e16i	− 0.767500i	0.923437	−	0.383750i	$-0.125367\pi$
$968$	− 5.43416e15i	− 0.205503i
$969$	−4.48952e16	−1.68818
$970$	0	0
$971$	7.19924e15	0.267658	0.133829	−	0.991004i	$-0.457273\pi$
$971$	7.19924e15	0.267658	0.133829	+	0.991004i	$0.457273\pi$
$972$	− 7.14182e15i	− 0.264025i
$973$	− 6.12821e16i	− 2.25275i
$974$	1.30052e16	0.475383
$975$	0	0
$976$	1.12790e16	0.407660
$977$	− 3.14860e16i	− 1.13161i	−0.824538	−	0.565806i	$-0.808565\pi$
$977$	− 3.14860e16i	− 1.13161i	0.824538	−	0.565806i	$-0.191435\pi$
$978$	− 5.98064e16i	− 2.13739i
$979$	−3.21353e15	−0.114203
$980$	0	0
$981$	−1.34169e16	−0.471491
$982$	− 3.27934e16i	− 1.14597i
$983$	− 1.68059e15i	− 0.0584006i	−0.999574	−	0.0292003i	$-0.990704\pi$
$983$	− 1.68059e15i	− 0.0584006i	0.999574	−	0.0292003i	$-0.00929607\pi$
$984$	2.22682e16	0.769505
$985$	0	0
$986$	3.21489e15	0.109861
$987$	1.50324e16i	0.510840i
$988$	− 1.72561e16i	− 0.583150i
$989$	−6.14345e15	−0.206458
$990$	0	0
$991$	−2.21678e15	−0.0736746	−0.0368373	−	0.999321i	$-0.511728\pi$
$991$	−2.21678e15	−0.0736746	−0.0368373	+	0.999321i	$0.511728\pi$
$992$	− 2.38387e15i	− 0.0787894i
$993$	5.80389e16i	1.90765i
$994$	7.89931e15	0.258205
$995$	0	0
$996$	4.24316e16	1.37171
$997$	− 2.59411e14i	− 0.00833999i	−0.999991	−	0.00416999i	$-0.998673\pi$
$997$	− 2.59411e14i	− 0.00833999i	0.999991	−	0.00416999i	$-0.00132735\pi$
$998$	3.25097e15i	0.103943i
$999$	−4.86389e16	−1.54658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.12.b.f.49.2		4
5.2	odd	4		10.12.a.d.1.2	✓	2
5.3	odd	4		50.12.a.f.1.1		2
5.4	even	2	inner	50.12.b.f.49.3		4
15.2	even	4		90.12.a.l.1.1		2
20.7	even	4		80.12.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.a.d.1.2	✓	2	5.2	odd	4
50.12.a.f.1.1		2	5.3	odd	4
50.12.b.f.49.2		4	1.1	even	1	trivial
50.12.b.f.49.3		4	5.4	even	2	inner
80.12.a.g.1.1		2	20.7	even	4
90.12.a.l.1.1		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-32.0000i q^{2} +745.734i q^{3} -1024.00 q^{4} +23863.5 q^{6} +71494.9i q^{7} +32768.0i q^{8} -378972. q^{9} -345651. q^{11} -763632. i q^{12} +1.50956e6i q^{13} +2.28784e6 q^{14} +1.04858e6 q^{16} +5.39291e6i q^{17} +1.21271e7i q^{18} +1.11633e7 q^{19} -5.33162e7 q^{21} +1.10608e7i q^{22} -5.27646e6i q^{23} -2.44362e7 q^{24} +4.83060e7 q^{26} -1.50508e8i q^{27} -7.32108e7i q^{28} +1.86291e7 q^{29} +7.10448e7 q^{31} -3.35544e7i q^{32} -2.57763e8i q^{33} +1.72573e8 q^{34} +3.88068e8 q^{36} -3.23164e8i q^{37} -3.57225e8i q^{38} -1.12573e9 q^{39} -9.11277e8 q^{41} +1.70612e9i q^{42} -1.16431e9i q^{43} +3.53946e8 q^{44} -1.68847e8 q^{46} -2.81949e8i q^{47} +7.81959e8i q^{48} -3.13420e9 q^{49} -4.02168e9 q^{51} -1.54579e9i q^{52} +4.05957e9i q^{53} -4.81626e9 q^{54} -2.34275e9 q^{56} +8.32485e9i q^{57} -5.96132e8i q^{58} -4.89828e9 q^{59} +1.07565e10 q^{61} -2.27343e9i q^{62} -2.70946e10i q^{63} -1.07374e9 q^{64} -8.24843e9 q^{66} -3.70812e9i q^{67} -5.52234e9i q^{68} +3.93484e9 q^{69} +3.45274e9 q^{71} -1.24182e10i q^{72} -2.21136e10i q^{73} -1.03413e10 q^{74} -1.14312e10 q^{76} -2.47123e10i q^{77} +3.60235e10i q^{78} -7.02672e9 q^{79} +4.51052e10 q^{81} +2.91609e10i q^{82} +5.55656e10i q^{83} +5.45958e10 q^{84} -3.72580e10 q^{86} +1.38924e10i q^{87} -1.13263e10i q^{88} +9.29706e9 q^{89} -1.07926e11 q^{91} +5.40310e9i q^{92} +5.29805e10i q^{93} -9.02235e9 q^{94} +2.50227e10 q^{96} -4.71888e10i q^{97} +1.00294e11i q^{98} +1.30992e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4096 q^{4} + 38656 q^{6} - 443828 q^{9} + 843168 q^{11} - 901888 q^{14} + 4194304 q^{16} + 57794800 q^{19} - 130893632 q^{21} - 39583744 q^{24} + 110753536 q^{26} - 116452440 q^{29} + 82826768 q^{31}+ \cdots + 502985449824 q^{99}+O(q^{100}) \) 4 * q - 4096 * q^4 + 38656 * q^6 - 443828 * q^9 + 843168 * q^11 - 901888 * q^14 + 4194304 * q^16 + 57794800 * q^19 - 130893632 * q^21 - 39583744 * q^24 + 110753536 * q^26 - 116452440 * q^29 + 82826768 * q^31 + 404712192 * q^34 + 454479872 * q^36 - 2188831696 * q^39 - 1570542072 * q^41 - 863404032 * q^44 - 2895108864 * q^46 - 16963997892 * q^49 - 7779524112 * q^51 - 12664092160 * q^54 + 923533312 * q^56 - 17354204880 * q^59 + 2230997528 * q^61 - 4294967296 * q^64 - 23456437248 * q^66 - 3457605696 * q^69 + 27599665968 * q^71 - 24144348928 * q^74 - 59181875200 * q^76 + 25273860640 * q^79 + 132427867844 * q^81 + 134035079168 * q^84 - 92991183104 * q^86 - 26417480040 * q^89 - 178029806192 * q^91 + 82440175872 * q^94 + 40533753856 * q^96 + 502985449824 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more