LMFDB - Embedded newform 90.12.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,12,Mod(1,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("90.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

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:	yes
Analytic conductor:	$69.1508862504$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1969}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 492 $ x^2 - x - 492
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-21.6867$ of defining polynomial
Character	$\chi$	$=$	90.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.0000 q^{2} +1024.00 q^{4} -3125.00 q^{5} -71494.9 q^{7} -32768.0 q^{8} +100000. q^{10} +345651. q^{11} +1.50956e6 q^{13} +2.28784e6 q^{14} +1.04858e6 q^{16} +5.39291e6 q^{17} -1.11633e7 q^{19} -3.20000e6 q^{20} -1.10608e7 q^{22} +5.27646e6 q^{23} +9.76562e6 q^{25} -4.83060e7 q^{26} -7.32108e7 q^{28} +1.86291e7 q^{29} +7.10448e7 q^{31} -3.35544e7 q^{32} -1.72573e8 q^{34} +2.23422e8 q^{35} +3.23164e8 q^{37} +3.57225e8 q^{38} +1.02400e8 q^{40} +9.11277e8 q^{41} -1.16431e9 q^{43} +3.53946e8 q^{44} -1.68847e8 q^{46} -2.81949e8 q^{47} +3.13420e9 q^{49} -3.12500e8 q^{50} +1.54579e9 q^{52} -4.05957e9 q^{53} -1.08016e9 q^{55} +2.34275e9 q^{56} -5.96132e8 q^{58} -4.89828e9 q^{59} +1.07565e10 q^{61} -2.27343e9 q^{62} +1.07374e9 q^{64} -4.71739e9 q^{65} +3.70812e9 q^{67} +5.52234e9 q^{68} -7.14949e9 q^{70} -3.45274e9 q^{71} -2.21136e10 q^{73} -1.03413e10 q^{74} -1.14312e10 q^{76} -2.47123e10 q^{77} +7.02672e9 q^{79} -3.27680e9 q^{80} -2.91609e10 q^{82} -5.55656e10 q^{83} -1.68528e10 q^{85} +3.72580e10 q^{86} -1.13263e10 q^{88} +9.29706e9 q^{89} -1.07926e11 q^{91} +5.40310e9 q^{92} +9.02235e9 q^{94} +3.48853e10 q^{95} +4.71888e10 q^{97} -1.00294e11 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 64 q^{2} + 2048 q^{4} - 6250 q^{5} + 14092 q^{7} - 65536 q^{8} + 200000 q^{10} - 421584 q^{11} + 1730524 q^{13} - 450944 q^{14} + 2097152 q^{16} + 6323628 q^{17} - 28897400 q^{19} - 6400000 q^{20}+ \cdots - 271423966272 q^{98}+O(q^{100}) $ 2 * q - 64 * q^2 + 2048 * q^4 - 6250 * q^5 + 14092 * q^7 - 65536 * q^8 + 200000 * q^10 - 421584 * q^11 + 1730524 * q^13 - 450944 * q^14 + 2097152 * q^16 + 6323628 * q^17 - 28897400 * q^19 - 6400000 * q^20 + 13490688 * q^22 + 45236076 * q^23 + 19531250 * q^25 - 55376768 * q^26 + 14430208 * q^28 - 58226220 * q^29 + 41413384 * q^31 - 67108864 * q^32 - 202356096 * q^34 - 44037500 * q^35 + 377255452 * q^37 + 924716800 * q^38 + 204800000 * q^40 + 785271036 * q^41 - 1452987236 * q^43 - 431702016 * q^44 - 1447554432 * q^46 + 1288127748 * q^47 + 8481998946 * q^49 - 625000000 * q^50 + 1772056576 * q^52 + 30490836 * q^53 + 1317450000 * q^55 - 461766656 * q^56 + 1863239040 * q^58 - 8677102440 * q^59 + 1115498764 * q^61 - 1325228288 * q^62 + 2147483648 * q^64 - 5407887500 * q^65 - 12673769708 * q^67 + 6475395072 * q^68 + 1409200000 * q^70 - 13799832984 * q^71 - 17842079516 * q^73 - 12072174464 * q^74 - 29590937600 * q^76 - 90377541264 * q^77 - 12636930320 * q^79 - 6553600000 * q^80 - 25128673152 * q^82 - 41986488924 * q^83 - 19761337500 * q^85 + 46495591552 * q^86 + 13814464512 * q^88 - 13208740020 * q^89 - 89014903096 * q^91 + 46321741824 * q^92 - 41220087936 * q^94 + 90304375000 * q^95 - 61787462828 * q^97 - 271423966272 * q^98

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 $

$2$	−32.0000	−0.707107
$2$	−32.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	−3125.00	−0.447214
$5$	−3125.00	−0.447214
$6$	0	0
$7$	−71494.9	−1.60782	−0.803908	−	0.594754i	$-0.797249\pi$
$7$	−71494.9	−1.60782	−0.803908	+	0.594754i	$0.797249\pi$
$8$	−32768.0	−0.353553
$9$	0	0
$10$	100000.	0.316228
$11$	345651.	0.647109	0.323555	−	0.946209i	$-0.395122\pi$
$11$	345651.	0.647109	0.323555	+	0.946209i	$0.395122\pi$
$12$	0	0
$13$	1.50956e6	1.12762	0.563810	−	0.825904i	$-0.309335\pi$
$13$	1.50956e6	1.12762	0.563810	+	0.825904i	$0.309335\pi$
$14$	2.28784e6	1.13690
$15$	0	0
$16$	1.04858e6	0.250000
$17$	5.39291e6	0.921200	0.460600	−	0.887608i	$-0.347634\pi$
$17$	5.39291e6	0.921200	0.460600	+	0.887608i	$0.347634\pi$
$18$	0	0
$19$	−1.11633e7	−1.03430	−0.517151	−	0.855894i	$-0.673007\pi$
$19$	−1.11633e7	−1.03430	−0.517151	+	0.855894i	$0.673007\pi$
$20$	−3.20000e6	−0.223607
$21$	0	0
$22$	−1.10608e7	−0.457575
$23$	5.27646e6	0.170938	0.0854692	−	0.996341i	$-0.472761\pi$
$23$	5.27646e6	0.170938	0.0854692	+	0.996341i	$0.472761\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	−4.83060e7	−0.797348
$27$	0	0
$28$	−7.32108e7	−0.803908
$29$	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	$0.428464\pi$
$32$	−3.35544e7	−0.176777
$33$	0	0
$34$	−1.72573e8	−0.651387
$35$	2.23422e8	0.719037
$36$	0	0
$37$	3.23164e8	0.766150	0.383075	−	0.923717i	$-0.374865\pi$
$37$	3.23164e8	0.766150	0.383075	+	0.923717i	$0.374865\pi$
$38$	3.57225e8	0.731362
$39$	0	0
$40$	1.02400e8	0.158114
$41$	9.11277e8	1.22840	0.614199	−	0.789151i	$-0.289479\pi$
$41$	9.11277e8	1.22840	0.614199	+	0.789151i	$0.289479\pi$
$42$	0	0
$43$	−1.16431e9	−1.20779	−0.603897	−	0.797063i	$-0.706386\pi$
$43$	−1.16431e9	−1.20779	−0.603897	+	0.797063i	$0.706386\pi$
$44$	3.53946e8	0.323555
$45$	0	0
$46$	−1.68847e8	−0.120872
$47$	−2.81949e8	−0.179321	−0.0896606	−	0.995972i	$-0.528578\pi$
$47$	−2.81949e8	−0.179321	−0.0896606	+	0.995972i	$0.528578\pi$
$48$	0	0
$49$	3.13420e9	1.58507
$50$	−3.12500e8	−0.141421
$51$	0	0
$52$	1.54579e9	0.563810
$53$	−4.05957e9	−1.33341	−0.666704	−	0.745323i	$-0.732295\pi$
$53$	−4.05957e9	−1.33341	−0.666704	+	0.745323i	$0.732295\pi$
$54$	0	0
$55$	−1.08016e9	−0.289396
$56$	2.34275e9	0.568449
$57$	0	0
$58$	−5.96132e8	−0.119258
$59$	−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	$0.196561\pi$
$62$	−2.27343e9	−0.315158
$63$	0	0
$64$	1.07374e9	0.125000
$65$	−4.71739e9	−0.504287
$66$	0	0
$67$	3.70812e9	0.335539	0.167769	−	0.985826i	$-0.446344\pi$
$67$	3.70812e9	0.335539	0.167769	+	0.985826i	$0.446344\pi$
$68$	5.52234e9	0.460600
$69$	0	0
$70$	−7.14949e9	−0.508436
$71$	−3.45274e9	−0.227113	−0.113557	−	0.993532i	$-0.536224\pi$
$71$	−3.45274e9	−0.227113	−0.113557	+	0.993532i	$0.536224\pi$
$72$	0	0
$73$	−2.21136e10	−1.24848	−0.624242	−	0.781231i	$-0.714592\pi$
$73$	−2.21136e10	−1.24848	−0.624242	+	0.781231i	$0.714592\pi$
$74$	−1.03413e10	−0.541750
$75$	0	0
$76$	−1.14312e10	−0.517151
$77$	−2.47123e10	−1.04043
$78$	0	0
$79$	7.02672e9	0.256923	0.128462	−	0.991714i	$-0.458996\pi$
$79$	7.02672e9	0.256923	0.128462	+	0.991714i	$0.458996\pi$
$80$	−3.27680e9	−0.111803
$81$	0	0
$82$	−2.91609e10	−0.868609
$83$	−5.55656e10	−1.54838	−0.774188	−	0.632956i	$-0.781841\pi$
$83$	−5.55656e10	−1.54838	−0.774188	+	0.632956i	$0.781841\pi$
$84$	0	0
$85$	−1.68528e10	−0.411973
$86$	3.72580e10	0.854039
$87$	0	0
$88$	−1.13263e10	−0.228788
$89$	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
$92$	5.40310e9	0.0854692
$93$	0	0
$94$	9.02235e9	0.126799
$95$	3.48853e10	0.462554
$96$	0	0
$97$	4.71888e10	0.557949	0.278975	−	0.960298i	$-0.410005\pi$
$97$	4.71888e10	0.557949	0.278975	+	0.960298i	$0.410005\pi$
$98$	−1.00294e11	−1.12081
$99$	0	0
$100$	1.00000e10	0.100000
$101$	−1.61228e11	−1.52642	−0.763209	−	0.646151i	$-0.776378\pi$
$101$	−1.61228e11	−1.52642	−0.763209	+	0.646151i	$0.776378\pi$
$102$	0	0
$103$	−1.85579e11	−1.57733	−0.788666	−	0.614822i	$-0.789228\pi$
$103$	−1.85579e11	−1.57733	−0.788666	+	0.614822i	$0.789228\pi$
$104$	−4.94654e10	−0.398674
$105$	0	0
$106$	1.29906e11	0.942861
$107$	5.45807e10	0.376208	0.188104	−	0.982149i	$-0.439766\pi$
$107$	5.45807e10	0.376208	0.188104	+	0.982149i	$0.439766\pi$
$108$	0	0
$109$	−3.54035e10	−0.220394	−0.110197	−	0.993910i	$-0.535148\pi$
$109$	−3.54035e10	−0.220394	−0.110197	+	0.993910i	$0.535148\pi$
$110$	3.45651e10	0.204634
$111$	0	0
$112$	−7.49679e10	−0.401954
$113$	−1.20314e11	−0.614308	−0.307154	−	0.951660i	$-0.599377\pi$
$113$	−1.20314e11	−0.614308	−0.307154	+	0.951660i	$0.599377\pi$
$114$	0	0
$115$	−1.64889e10	−0.0764460
$116$	1.90762e10	0.0843284
$117$	0	0
$118$	1.56745e11	0.630729
$119$	−3.85566e11	−1.48112
$120$	0	0
$121$	−1.65837e11	−0.581250
$122$	−3.44209e11	−1.15304
$123$	0	0
$124$	7.27499e10	0.222850
$125$	−3.05176e10	−0.0894427
$126$	0	0
$127$	−5.60687e11	−1.50591	−0.752957	−	0.658070i	$-0.771373\pi$
$127$	−5.60687e11	−1.50591	−0.752957	+	0.658070i	$0.771373\pi$
$128$	−3.43597e10	−0.0883883
$129$	0	0
$130$	1.50956e11	0.356585
$131$	−4.11999e11	−0.933048	−0.466524	−	0.884508i	$-0.654494\pi$
$131$	−4.11999e11	−0.933048	−0.466524	+	0.884508i	$0.654494\pi$
$132$	0	0
$133$	7.98119e11	1.66297
$134$	−1.18660e11	−0.237262
$135$	0	0
$136$	−1.76715e11	−0.325693
$137$	7.86259e11	1.39188	0.695941	−	0.718099i	$-0.254987\pi$
$137$	7.86259e11	1.39188	0.695941	+	0.718099i	$0.254987\pi$
$138$	0	0
$139$	8.57152e11	1.40112	0.700562	−	0.713592i	$-0.252933\pi$
$139$	8.57152e11	1.40112	0.700562	+	0.713592i	$0.252933\pi$
$140$	2.28784e11	0.359518
$141$	0	0
$142$	1.10488e11	0.160593
$143$	5.21782e11	0.729694
$144$	0	0
$145$	−5.82160e10	−0.0754256
$146$	7.07634e11	0.882812
$147$	0	0
$148$	3.30920e11	0.383075
$149$	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	$0.515801\pi$
$152$	3.65799e11	0.365681
$153$	0	0
$154$	7.90793e11	0.735697
$155$	−2.22015e11	−0.199323
$156$	0	0
$157$	−1.79935e12	−1.50545	−0.752727	−	0.658333i	$-0.771262\pi$
$157$	−1.79935e12	−1.50545	−0.752727	+	0.658333i	$0.771262\pi$
$158$	−2.24855e11	−0.181672
$159$	0	0
$160$	1.04858e11	0.0790569
$161$	−3.77240e11	−0.274837
$162$	0	0
$163$	−2.50619e12	−1.70601	−0.853005	−	0.521903i	$-0.825222\pi$
$163$	−2.50619e12	−1.70601	−0.853005	+	0.521903i	$0.825222\pi$
$164$	9.33148e11	0.614199
$165$	0	0
$166$	1.77810e12	1.09487
$167$	2.69855e12	1.60764	0.803822	−	0.594871i	$-0.202797\pi$
$167$	2.69855e12	1.60764	0.803822	+	0.594871i	$0.202797\pi$
$168$	0	0
$169$	4.86623e11	0.271529
$170$	5.39291e11	0.291309
$171$	0	0
$172$	−1.19225e12	−0.603897
$173$	−3.07796e12	−1.51011	−0.755057	−	0.655659i	$-0.772391\pi$
$173$	−3.07796e12	−1.51011	−0.755057	+	0.655659i	$0.772391\pi$
$174$	0	0
$175$	−6.98193e11	−0.321563
$176$	3.62441e11	0.161777
$177$	0	0
$178$	−2.97506e11	−0.124792
$179$	−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	$0.461290\pi$
$182$	3.45364e12	1.28199
$183$	0	0
$184$	−1.72899e11	−0.0604359
$185$	−1.00989e12	−0.342633
$186$	0	0
$187$	1.86406e12	0.596117
$188$	−2.88715e11	−0.0896606
$189$	0	0
$190$	−1.11633e12	−0.327075
$191$	2.61584e12	0.744609	0.372304	−	0.928111i	$-0.378568\pi$
$191$	2.61584e12	0.744609	0.372304	+	0.928111i	$0.378568\pi$
$192$	0	0
$193$	1.59819e12	0.429598	0.214799	−	0.976658i	$-0.431090\pi$
$193$	1.59819e12	0.429598	0.214799	+	0.976658i	$0.431090\pi$
$194$	−1.51004e12	−0.394530
$195$	0	0
$196$	3.20942e12	0.792535
$197$	6.16281e10	0.0147984	0.00739920	−	0.999973i	$-0.497645\pi$
$197$	6.16281e10	0.0147984	0.00739920	+	0.999973i	$0.497645\pi$
$198$	0	0
$199$	2.36544e12	0.537303	0.268652	−	0.963237i	$-0.413422\pi$
$199$	2.36544e12	0.537303	0.268652	+	0.963237i	$0.413422\pi$
$200$	−3.20000e11	−0.0707107
$201$	0	0
$202$	5.15931e12	1.07934
$203$	−1.33189e12	−0.271169
$204$	0	0
$205$	−2.84774e12	−0.549357
$206$	5.93851e12	1.11534
$207$	0	0
$208$	1.58289e12	0.281905
$209$	−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	$0.353482\pi$
$212$	−4.15700e12	−0.666704
$213$	0	0
$214$	−1.74658e12	−0.266020
$215$	3.63847e12	0.540142
$216$	0	0
$217$	−5.07934e12	−0.716604
$218$	1.13291e12	0.155842
$219$	0	0
$220$	−1.10608e12	−0.144698
$221$	8.14094e12	1.03876
$222$	0	0
$223$	−9.37124e12	−1.13794	−0.568971	−	0.822357i	$-0.692658\pi$
$223$	−9.37124e12	−1.13794	−0.568971	+	0.822357i	$0.692658\pi$
$224$	2.39897e12	0.284224
$225$	0	0
$226$	3.85006e12	0.434382
$227$	6.51561e12	0.717485	0.358742	−	0.933437i	$-0.383206\pi$
$227$	6.51561e12	0.717485	0.358742	+	0.933437i	$0.383206\pi$
$228$	0	0
$229$	−1.74808e12	−0.183429	−0.0917144	−	0.995785i	$-0.529235\pi$
$229$	−1.74808e12	−0.183429	−0.0917144	+	0.995785i	$0.529235\pi$
$230$	5.27646e11	0.0540555
$231$	0	0
$232$	−6.10440e11	−0.0596292
$233$	−8.64493e12	−0.824715	−0.412358	−	0.911022i	$-0.635295\pi$
$233$	−8.64493e12	−0.824715	−0.412358	+	0.911022i	$0.635295\pi$
$234$	0	0
$235$	8.81089e11	0.0801949
$236$	−5.01584e12	−0.445993
$237$	0	0
$238$	1.23381e13	1.04731
$239$	−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	$0.658168\pi$
$242$	5.30679e12	0.411006
$243$	0	0
$244$	1.10147e13	0.815320
$245$	−9.79438e12	−0.708865
$246$	0	0
$247$	−1.68517e13	−1.16630
$248$	−2.32800e12	−0.157579
$249$	0	0
$250$	9.76562e11	0.0632456
$251$	2.23212e13	1.41420	0.707101	−	0.707113i	$-0.250003\pi$
$251$	2.23212e13	1.41420	0.707101	+	0.707113i	$0.250003\pi$
$252$	0	0
$253$	1.82381e12	0.110616
$254$	1.79420e13	1.06484
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	−1.80055e13	−1.00178	−0.500890	−	0.865511i	$-0.666994\pi$
$257$	−1.80055e13	−1.00178	−0.500890	+	0.865511i	$0.666994\pi$
$258$	0	0
$259$	−2.31046e13	−1.23183
$260$	−4.83060e12	−0.252144
$261$	0	0
$262$	1.31840e13	0.659765
$263$	−1.38894e13	−0.680653	−0.340326	−	0.940307i	$-0.610538\pi$
$263$	−1.38894e13	−0.680653	−0.340326	+	0.940307i	$0.610538\pi$
$264$	0	0
$265$	1.26862e13	0.596318
$266$	−2.55398e13	−1.17590
$267$	0	0
$268$	3.79711e12	0.167769
$269$	−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	$0.548107\pi$
$272$	5.65488e12	0.230300
$273$	0	0
$274$	−2.51603e13	−0.984209
$275$	3.37549e12	0.129422
$276$	0	0
$277$	3.97300e13	1.46380	0.731898	−	0.681415i	$-0.238635\pi$
$277$	3.97300e13	1.46380	0.731898	+	0.681415i	$0.238635\pi$
$278$	−2.74289e13	−0.990744
$279$	0	0
$280$	−7.32108e12	−0.254218
$281$	−5.95255e12	−0.202683	−0.101342	−	0.994852i	$-0.532314\pi$
$281$	−5.95255e12	−0.202683	−0.101342	+	0.994852i	$0.532314\pi$
$282$	0	0
$283$	−3.85782e13	−1.26333	−0.631665	−	0.775241i	$-0.717628\pi$
$283$	−3.85782e13	−1.26333	−0.631665	+	0.775241i	$0.717628\pi$
$284$	−3.53560e12	−0.113557
$285$	0	0
$286$	−1.66970e13	−0.515971
$287$	−6.51517e13	−1.97504
$288$	0	0
$289$	−5.18843e12	−0.151390
$290$	1.86291e12	0.0533339
$291$	0	0
$292$	−2.26443e13	−0.624242
$293$	5.32529e13	1.44069	0.720346	−	0.693615i	$-0.243983\pi$
$293$	5.32529e13	1.44069	0.720346	+	0.693615i	$0.243983\pi$
$294$	0	0
$295$	1.53071e13	0.398908
$296$	−1.05895e13	−0.270875
$297$	0	0
$298$	−7.63539e12	−0.188210
$299$	7.96516e12	0.192754
$300$	0	0
$301$	8.32424e13	1.94191
$302$	3.06339e12	0.0701720
$303$	0	0
$304$	−1.17056e13	−0.258576
$305$	−3.36142e13	−0.729244
$306$	0	0
$307$	−5.74336e13	−1.20200	−0.601001	−	0.799249i	$-0.705231\pi$
$307$	−5.74336e13	−1.20200	−0.601001	+	0.799249i	$0.705231\pi$
$308$	−2.53054e13	−0.520216
$309$	0	0
$310$	7.10448e12	0.140943
$311$	2.58616e13	0.504050	0.252025	−	0.967721i	$-0.418903\pi$
$311$	2.58616e13	0.504050	0.252025	+	0.967721i	$0.418903\pi$
$312$	0	0
$313$	−3.57307e13	−0.672277	−0.336138	−	0.941813i	$-0.609121\pi$
$313$	−3.57307e13	−0.672277	−0.336138	+	0.941813i	$0.609121\pi$
$314$	5.75792e13	1.06452
$315$	0	0
$316$	7.19536e12	0.128462
$317$	1.78631e13	0.313422	0.156711	−	0.987644i	$-0.449911\pi$
$317$	1.78631e13	0.313422	0.156711	+	0.987644i	$0.449911\pi$
$318$	0	0
$319$	6.43917e12	0.109139
$320$	−3.35544e12	−0.0559017
$321$	0	0
$322$	1.20717e13	0.194339
$323$	−6.02026e13	−0.952800
$324$	0	0
$325$	1.47418e13	0.225524
$326$	8.01980e13	1.20633
$327$	0	0
$328$	−2.98607e13	−0.434305
$329$	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	$0.319054\pi$
$332$	−5.68992e13	−0.774188
$333$	0	0
$334$	−8.63535e13	−1.13678
$335$	−1.15879e13	−0.150057
$336$	0	0
$337$	4.95598e12	0.0621105	0.0310552	−	0.999518i	$-0.490113\pi$
$337$	4.95598e12	0.0621105	0.0310552	+	0.999518i	$0.490113\pi$
$338$	−1.55719e13	−0.192000
$339$	0	0
$340$	−1.72573e13	−0.205987
$341$	2.45567e13	0.288417
$342$	0	0
$343$	−8.27106e13	−0.940684
$344$	3.81522e13	0.427019
$345$	0	0
$346$	9.84948e13	1.06781
$347$	6.55706e13	0.699676	0.349838	−	0.936810i	$-0.386237\pi$
$347$	6.55706e13	0.699676	0.349838	+	0.936810i	$0.386237\pi$
$348$	0	0
$349$	1.14750e14	1.18635	0.593177	−	0.805072i	$-0.297873\pi$
$349$	1.14750e14	1.18635	0.593177	+	0.805072i	$0.297873\pi$
$350$	2.23422e13	0.227379
$351$	0	0
$352$	−1.15981e13	−0.114394
$353$	5.10272e13	0.495497	0.247749	−	0.968824i	$-0.420309\pi$
$353$	5.10272e13	0.495497	0.247749	+	0.968824i	$0.420309\pi$
$354$	0	0
$355$	1.07898e13	0.101568
$356$	9.52019e12	0.0882410
$357$	0	0
$358$	1.12369e13	0.100993
$359$	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
$362$	−2.02914e13	−0.171559
$363$	0	0
$364$	−1.10516e14	−0.906503
$365$	6.91049e13	0.558339
$366$	0	0
$367$	1.44058e14	1.12947	0.564733	−	0.825274i	$-0.308979\pi$
$367$	1.44058e14	1.12947	0.564733	+	0.825274i	$0.308979\pi$
$368$	5.53277e12	0.0427346
$369$	0	0
$370$	3.23164e13	0.242278
$371$	2.90239e14	2.14387
$372$	0	0
$373$	6.90594e13	0.495250	0.247625	−	0.968856i	$-0.420350\pi$
$373$	6.90594e13	0.495250	0.247625	+	0.968856i	$0.420350\pi$
$374$	−5.96500e13	−0.421518
$375$	0	0
$376$	9.23889e12	0.0633996
$377$	2.81219e13	0.190181
$378$	0	0
$379$	2.28579e14	1.50149	0.750743	−	0.660595i	$-0.229696\pi$
$379$	2.28579e14	1.50149	0.750743	+	0.660595i	$0.229696\pi$
$380$	3.57225e13	0.231277
$381$	0	0
$382$	−8.37070e13	−0.526518
$383$	7.16725e13	0.444385	0.222192	−	0.975003i	$-0.428679\pi$
$383$	7.16725e13	0.444385	0.222192	+	0.975003i	$0.428679\pi$
$384$	0	0
$385$	7.72259e13	0.465295
$386$	−5.11420e13	−0.303772
$387$	0	0
$388$	4.83214e13	0.278975
$389$	−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
$392$	−1.02701e14	−0.560407
$393$	0	0
$394$	−1.97210e12	−0.0104641
$395$	−2.19585e13	−0.114900
$396$	0	0
$397$	9.39721e12	0.0478246	0.0239123	−	0.999714i	$-0.492388\pi$
$397$	9.39721e12	0.0478246	0.0239123	+	0.999714i	$0.492388\pi$
$398$	−7.56940e13	−0.379931
$399$	0	0
$400$	1.02400e13	0.0500000
$401$	−2.20216e14	−1.06061	−0.530303	−	0.847808i	$-0.677922\pi$
$401$	−2.20216e14	−1.06061	−0.530303	+	0.847808i	$0.677922\pi$
$402$	0	0
$403$	1.07247e14	0.502581
$404$	−1.65098e14	−0.763209
$405$	0	0
$406$	4.26204e13	0.191745
$407$	1.11702e14	0.495783
$408$	0	0
$409$	−3.05103e14	−1.31816	−0.659080	−	0.752072i	$-0.729054\pi$
$409$	−3.05103e14	−1.31816	−0.659080	+	0.752072i	$0.729054\pi$
$410$	9.11277e13	0.388454
$411$	0	0
$412$	−1.90032e14	−0.788666
$413$	3.50203e14	1.43415
$414$	0	0
$415$	1.73642e14	0.692455
$416$	−5.06526e13	−0.199337
$417$	0	0
$418$	1.23475e14	0.473271
$419$	−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	$0.499348\pi$
$422$	−1.72715e14	−0.628219
$423$	0	0
$424$	1.33024e14	0.471431
$425$	5.26651e13	0.184240
$426$	0	0
$427$	−7.69037e14	−2.62177
$428$	5.58907e13	0.188104
$429$	0	0
$430$	−1.16431e14	−0.381938
$431$	−2.29204e14	−0.742332	−0.371166	−	0.928567i	$-0.621042\pi$
$431$	−2.29204e14	−0.742332	−0.371166	+	0.928567i	$0.621042\pi$
$432$	0	0
$433$	1.54032e14	0.486325	0.243162	−	0.969986i	$-0.421815\pi$
$433$	1.54032e14	0.486325	0.243162	+	0.969986i	$0.421815\pi$
$434$	1.62539e14	0.506715
$435$	0	0
$436$	−3.62531e13	−0.110197
$437$	−5.89027e13	−0.176802
$438$	0	0
$439$	−1.25797e14	−0.368227	−0.184114	−	0.982905i	$-0.558941\pi$
$439$	−1.25797e14	−0.368227	−0.184114	+	0.982905i	$0.558941\pi$
$440$	3.53946e13	0.102317
$441$	0	0
$442$	−2.60510e14	−0.734517
$443$	1.34623e14	0.374885	0.187443	−	0.982276i	$-0.439980\pi$
$443$	1.34623e14	0.374885	0.187443	+	0.982276i	$0.439980\pi$
$444$	0	0
$445$	−2.90533e13	−0.0789252
$446$	2.99880e14	0.804647
$447$	0	0
$448$	−7.67671e13	−0.200977
$449$	−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
$452$	−1.23202e14	−0.307154
$453$	0	0
$454$	−2.08499e14	−0.507338
$455$	3.37269e14	0.810801
$456$	0	0
$457$	−7.89196e14	−1.85202	−0.926011	−	0.377497i	$-0.876785\pi$
$457$	−7.89196e14	−1.85202	−0.926011	+	0.377497i	$0.876785\pi$
$458$	5.59387e13	0.129704
$459$	0	0
$460$	−1.68847e13	−0.0382230
$461$	6.01739e14	1.34602	0.673012	−	0.739632i	$-0.265000\pi$
$461$	6.01739e14	1.34602	0.673012	+	0.739632i	$0.265000\pi$
$462$	0	0
$463$	2.36873e14	0.517393	0.258696	−	0.965959i	$-0.416707\pi$
$463$	2.36873e14	0.517393	0.258696	+	0.965959i	$0.416707\pi$
$464$	1.95341e13	0.0421642
$465$	0	0
$466$	2.76638e14	0.583162
$467$	−8.17060e14	−1.70220	−0.851101	−	0.525002i	$-0.824064\pi$
$467$	−8.17060e14	−1.70220	−0.851101	+	0.525002i	$0.824064\pi$
$468$	0	0
$469$	−2.65112e14	−0.539484
$470$	−2.81949e13	−0.0567063
$471$	0	0
$472$	1.60507e14	0.315365
$473$	−4.02445e14	−0.781574
$474$	0	0
$475$	−1.09017e14	−0.206860
$476$	−3.94819e14	−0.740560
$477$	0	0
$478$	2.14728e14	0.393582
$479$	−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
$482$	3.85053e14	0.674159
$483$	0	0
$484$	−1.69817e14	−0.290625
$485$	−1.47465e14	−0.249522
$486$	0	0
$487$	−4.06414e14	−0.672294	−0.336147	−	0.941810i	$-0.609124\pi$
$487$	−4.06414e14	−0.672294	−0.336147	+	0.941810i	$0.609124\pi$
$488$	−3.52470e14	−0.576518
$489$	0	0
$490$	3.13420e14	0.501243
$491$	−1.02479e15	−1.62065	−0.810324	−	0.585982i	$-0.800709\pi$
$491$	−1.02479e15	−1.62065	−0.810324	+	0.585982i	$0.800709\pi$
$492$	0	0
$493$	1.00465e14	0.155367
$494$	5.39255e14	0.824699
$495$	0	0
$496$	7.44959e13	0.111425
$497$	2.46853e14	0.365156
$498$	0	0
$499$	1.01593e14	0.146997	0.0734987	−	0.997295i	$-0.476584\pi$
$499$	1.01593e14	0.146997	0.0734987	+	0.997295i	$0.476584\pi$
$500$	−3.12500e13	−0.0447214
$501$	0	0
$502$	−7.14277e14	−0.999991
$503$	−9.67924e14	−1.34035	−0.670174	−	0.742204i	$-0.733780\pi$
$503$	−9.67924e14	−1.34035	−0.670174	+	0.742204i	$0.733780\pi$
$504$	0	0
$505$	5.03838e14	0.682635
$506$	−5.83620e13	−0.0782172
$507$	0	0
$508$	−5.74143e14	−0.752957
$509$	−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
$512$	−3.51844e13	−0.0441942
$513$	0	0
$514$	5.76175e14	0.708365
$515$	5.79933e14	0.705404
$516$	0	0
$517$	−9.74557e13	−0.116040
$518$	7.39348e14	0.871034
$519$	0	0
$520$	1.54579e14	0.178293
$521$	1.00574e15	1.14783	0.573916	−	0.818914i	$-0.305424\pi$
$521$	1.00574e15	1.14783	0.573916	+	0.818914i	$0.305424\pi$
$522$	0	0
$523$	−1.24911e15	−1.39586	−0.697931	−	0.716165i	$-0.745896\pi$
$523$	−1.24911e15	−1.39586	−0.697931	+	0.716165i	$0.745896\pi$
$524$	−4.21887e14	−0.466524
$525$	0	0
$526$	4.44460e14	0.481294
$527$	3.83138e14	0.410579
$528$	0	0
$529$	−9.24969e14	−0.970780
$530$	−4.05957e14	−0.421660
$531$	0	0
$532$	8.17274e14	0.831483
$533$	1.37563e15	1.38517
$534$	0	0
$535$	−1.70565e14	−0.168246
$536$	−1.21508e14	−0.118631
$537$	0	0
$538$	8.69155e14	0.831371
$539$	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	$0.259782\pi$
$542$	2.31853e14	0.212921
$543$	0	0
$544$	−1.80956e14	−0.162847
$545$	1.10636e14	0.0985632
$546$	0	0
$547$	−2.32000e14	−0.202562	−0.101281	−	0.994858i	$-0.532294\pi$
$547$	−2.32000e14	−0.202562	−0.101281	+	0.994858i	$0.532294\pi$
$548$	8.05129e14	0.695941
$549$	0	0
$550$	−1.08016e14	−0.0915151
$551$	−2.07962e14	−0.174442
$552$	0	0
$553$	−5.02375e14	−0.413085
$554$	−1.27136e15	−1.03506
$555$	0	0
$556$	8.77724e14	0.700562
$557$	−2.34238e15	−1.85120	−0.925602	−	0.378498i	$-0.876441\pi$
$557$	−2.34238e15	−1.85120	−0.925602	+	0.378498i	$0.876441\pi$
$558$	0	0
$559$	−1.75760e15	−1.36193
$560$	2.34275e14	0.179759
$561$	0	0
$562$	1.90482e14	0.143319
$563$	−2.32309e15	−1.73089	−0.865446	−	0.501002i	$-0.832965\pi$
$563$	−2.32309e15	−1.73089	−0.865446	+	0.501002i	$0.832965\pi$
$564$	0	0
$565$	3.75983e14	0.274727
$566$	1.23450e15	0.893309
$567$	0	0
$568$	1.13139e14	0.0802967
$569$	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	$0.819533\pi$
$572$	5.34305e14	0.364847
$573$	0	0
$574$	2.08486e15	1.39656
$575$	5.15280e13	0.0341877
$576$	0	0
$577$	−2.05338e15	−1.33660	−0.668302	−	0.743890i	$-0.732979\pi$
$577$	−2.05338e15	−1.33660	−0.668302	+	0.743890i	$0.732979\pi$
$578$	1.66030e14	0.107049
$579$	0	0
$580$	−5.96132e13	−0.0377128
$581$	3.97266e15	2.48950
$582$	0	0
$583$	−1.40319e15	−0.862860
$584$	7.24617e14	0.441406
$585$	0	0
$586$	−1.70409e15	−1.01872
$587$	−3.00291e14	−0.177841	−0.0889207	−	0.996039i	$-0.528342\pi$
$587$	−3.00291e14	−0.177841	−0.0889207	+	0.996039i	$0.528342\pi$
$588$	0	0
$589$	−7.93094e14	−0.460989
$590$	−4.89828e14	−0.282071
$591$	0	0
$592$	3.38862e14	0.191538
$593$	2.71460e15	1.52022	0.760108	−	0.649796i	$-0.225146\pi$
$593$	2.71460e15	1.52022	0.760108	+	0.649796i	$0.225146\pi$
$594$	0	0
$595$	1.20489e15	0.662377
$596$	2.44332e14	0.133084
$597$	0	0
$598$	−2.54885e14	−0.136298
$599$	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	$0.235202\pi$
$602$	−2.66376e15	−1.37314
$603$	0	0
$604$	−9.80284e13	−0.0496191
$605$	5.18242e14	0.259943
$606$	0	0
$607$	−3.78069e14	−0.186223	−0.0931114	−	0.995656i	$-0.529681\pi$
$607$	−3.78069e14	−0.186223	−0.0931114	+	0.995656i	$0.529681\pi$
$608$	3.74578e14	0.182841
$609$	0	0
$610$	1.07565e15	0.515654
$611$	−4.25619e14	−0.202206
$612$	0	0
$613$	2.97141e15	1.38653	0.693267	−	0.720681i	$-0.256171\pi$
$613$	2.97141e15	1.38653	0.693267	+	0.720681i	$0.256171\pi$
$614$	1.83788e15	0.849943
$615$	0	0
$616$	8.09772e14	0.367848
$617$	1.19545e15	0.538222	0.269111	−	0.963109i	$-0.413270\pi$
$617$	1.19545e15	0.538222	0.269111	+	0.963109i	$0.413270\pi$
$618$	0	0
$619$	3.17735e14	0.140529	0.0702645	−	0.997528i	$-0.477616\pi$
$619$	3.17735e14	0.140529	0.0702645	+	0.997528i	$0.477616\pi$
$620$	−2.27343e14	−0.0996616
$621$	0	0
$622$	−8.27572e14	−0.356417
$623$	−6.64693e14	−0.283750
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	1.14338e15	0.475372
$627$	0	0
$628$	−1.84253e15	−0.752727
$629$	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	$0.545385\pi$
$632$	−2.30251e14	−0.0908361
$633$	0	0
$634$	−5.71618e14	−0.221623
$635$	1.75215e15	0.673465
$636$	0	0
$637$	4.73128e15	1.78736
$638$	−2.06054e14	−0.0771732
$639$	0	0
$640$	1.07374e14	0.0395285
$641$	−3.74739e15	−1.36776	−0.683880	−	0.729594i	$-0.739709\pi$
$641$	−3.74739e15	−1.36776	−0.683880	+	0.729594i	$0.739709\pi$
$642$	0	0
$643$	3.72691e14	0.133718	0.0668588	−	0.997762i	$-0.478702\pi$
$643$	3.72691e14	0.133718	0.0668588	+	0.997762i	$0.478702\pi$
$644$	−3.86294e14	−0.137419
$645$	0	0
$646$	1.92648e15	0.673731
$647$	−3.55565e15	−1.23295	−0.616475	−	0.787374i	$-0.711440\pi$
$647$	−3.55565e15	−1.23295	−0.616475	+	0.787374i	$0.711440\pi$
$648$	0	0
$649$	−1.69310e15	−0.577212
$650$	−4.71739e14	−0.159470
$651$	0	0
$652$	−2.56633e15	−0.853005
$653$	−5.07282e15	−1.67196	−0.835982	−	0.548757i	$-0.815101\pi$
$653$	−5.07282e15	−1.67196	−0.835982	+	0.548757i	$0.815101\pi$
$654$	0	0
$655$	1.28750e15	0.417272
$656$	9.55544e14	0.307100
$657$	0	0
$658$	−6.45053e14	−0.203870
$659$	−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	$0.484935\pi$
$662$	−2.49049e15	−0.761318
$663$	0	0
$664$	1.82077e15	0.547433
$665$	−2.49412e15	−0.743701
$666$	0	0
$667$	9.82960e13	0.0288299
$668$	2.76331e15	0.803822
$669$	0	0
$670$	3.70812e14	0.106107
$671$	3.71800e15	1.05520
$672$	0	0
$673$	3.20107e15	0.893742	0.446871	−	0.894599i	$-0.352538\pi$
$673$	3.20107e15	0.893742	0.446871	+	0.894599i	$0.352538\pi$
$674$	−1.58591e14	−0.0439187
$675$	0	0
$676$	4.98302e14	0.135764
$677$	4.88443e15	1.32001	0.660004	−	0.751262i	$-0.270555\pi$
$677$	4.88443e15	1.32001	0.660004	+	0.751262i	$0.270555\pi$
$678$	0	0
$679$	−3.37376e15	−0.897079
$680$	5.52234e14	0.145655
$681$	0	0
$682$	−7.85814e14	−0.203941
$683$	−3.43048e15	−0.883163	−0.441582	−	0.897221i	$-0.645582\pi$
$683$	−3.43048e15	−0.883163	−0.441582	+	0.897221i	$0.645582\pi$
$684$	0	0
$685$	−2.45706e15	−0.622469
$686$	2.64674e15	0.665164
$687$	0	0
$688$	−1.22087e15	−0.301948
$689$	−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	$0.451361\pi$
$692$	−3.15183e15	−0.755057
$693$	0	0
$694$	−2.09826e15	−0.494746
$695$	−2.67860e15	−0.626602
$696$	0	0
$697$	4.91444e15	1.13160
$698$	−3.67201e15	−0.838879
$699$	0	0
$700$	−7.14949e14	−0.160782
$701$	1.46416e15	0.326693	0.163347	−	0.986569i	$-0.447771\pi$
$701$	1.46416e15	0.326693	0.163347	+	0.986569i	$0.447771\pi$
$702$	0	0
$703$	−3.60758e15	−0.792431
$704$	3.71140e14	0.0808887
$705$	0	0
$706$	−1.63287e15	−0.350369
$707$	1.15270e16	2.45420
$708$	0	0
$709$	−2.00407e15	−0.420105	−0.210053	−	0.977690i	$-0.567364\pi$
$709$	−2.00407e15	−0.420105	−0.210053	+	0.977690i	$0.567364\pi$
$710$	−3.45274e14	−0.0718196
$711$	0	0
$712$	−3.04646e14	−0.0623958
$713$	3.74865e14	0.0761873
$714$	0	0
$715$	−1.63057e15	−0.326329
$716$	−3.59582e14	−0.0714128
$717$	0	0
$718$	−3.73303e15	−0.730092
$719$	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
$722$	−2.60123e14	−0.0493428
$723$	0	0
$724$	6.49324e14	0.121311
$725$	1.81925e14	0.0337313
$726$	0	0
$727$	−8.96895e14	−0.163796	−0.0818978	−	0.996641i	$-0.526098\pi$
$727$	−8.96895e14	−0.163796	−0.0818978	+	0.996641i	$0.526098\pi$
$728$	3.53653e15	0.640994
$729$	0	0
$730$	−2.21136e15	−0.394805
$731$	−6.27903e15	−1.11262
$732$	0	0
$733$	−1.11078e16	−1.93890	−0.969451	−	0.245286i	$-0.921118\pi$
$733$	−1.11078e16	−1.93890	−0.969451	+	0.245286i	$0.921118\pi$
$734$	−4.60985e15	−0.798653
$735$	0	0
$736$	−1.77049e14	−0.0302179
$737$	1.28171e15	0.217130
$738$	0	0
$739$	5.03625e15	0.840548	0.420274	−	0.907397i	$-0.361934\pi$
$739$	5.03625e15	0.840548	0.420274	+	0.907397i	$0.361934\pi$
$740$	−1.03413e15	−0.171316
$741$	0	0
$742$	−9.28764e15	−1.51595
$743$	−1.36645e15	−0.221389	−0.110695	−	0.993854i	$-0.535308\pi$
$743$	−1.36645e15	−0.221389	−0.110695	+	0.993854i	$0.535308\pi$
$744$	0	0
$745$	−7.45643e14	−0.119034
$746$	−2.20990e15	−0.350194
$747$	0	0
$748$	1.90880e15	0.298059
$749$	−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	$0.768591\pi$
$752$	−2.95644e14	−0.0448303
$753$	0	0
$754$	−8.99900e14	−0.134478
$755$	2.99159e14	0.0443807
$756$	0	0
$757$	−1.39502e15	−0.203964	−0.101982	−	0.994786i	$-0.532518\pi$
$757$	−1.39502e15	−0.203964	−0.101982	+	0.994786i	$0.532518\pi$
$758$	−7.31454e15	−1.06171
$759$	0	0
$760$	−1.14312e15	−0.163538
$761$	−6.09995e15	−0.866385	−0.433192	−	0.901301i	$-0.642613\pi$
$761$	−6.09995e15	−0.866385	−0.433192	+	0.901301i	$0.642613\pi$
$762$	0	0
$763$	2.53117e15	0.354353
$764$	2.67862e15	0.372304
$765$	0	0
$766$	−2.29352e15	−0.314228
$767$	−7.39427e15	−1.00582
$768$	0	0
$769$	6.60626e15	0.885851	0.442925	−	0.896558i	$-0.353941\pi$
$769$	6.60626e15	0.885851	0.442925	+	0.896558i	$0.353941\pi$
$770$	−2.47123e15	−0.329013
$771$	0	0
$772$	1.63654e15	0.214799
$773$	−3.74520e15	−0.488076	−0.244038	−	0.969766i	$-0.578472\pi$
$773$	−3.74520e15	−0.488076	−0.244038	+	0.969766i	$0.578472\pi$
$774$	0	0
$775$	6.93797e14	0.0891400
$776$	−1.54628e15	−0.197265
$777$	0	0
$778$	1.69977e15	0.213798
$779$	−1.01729e16	−1.27054
$780$	0	0
$781$	−1.19344e15	−0.146967
$782$	−9.10576e14	−0.111347
$783$	0	0
$784$	3.28645e15	0.396267
$785$	5.62297e15	0.673259
$786$	0	0
$787$	−1.01107e16	−1.19377	−0.596885	−	0.802326i	$-0.703595\pi$
$787$	−1.01107e16	−1.19377	−0.596885	+	0.802326i	$0.703595\pi$
$788$	6.31072e13	0.00739920
$789$	0	0
$790$	7.02672e14	0.0812463
$791$	8.60188e15	0.987694
$792$	0	0
$793$	1.62377e16	1.83874
$794$	−3.00711e14	−0.0338171
$795$	0	0
$796$	2.42221e15	0.268652
$797$	1.40032e16	1.54244	0.771219	−	0.636570i	$-0.219647\pi$
$797$	1.40032e16	1.54244	0.771219	+	0.636570i	$0.219647\pi$
$798$	0	0
$799$	−1.52052e15	−0.165191
$800$	−3.27680e14	−0.0353553
$801$	0	0
$802$	7.04691e15	0.749962
$803$	−7.64357e15	−0.807906
$804$	0	0
$805$	1.17888e15	0.122911
$806$	−3.43189e15	−0.355378
$807$	0	0
$808$	5.28313e15	0.539671
$809$	−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	$0.759922\pi$
$812$	−1.36385e15	−0.135584
$813$	0	0
$814$	−3.57446e15	−0.350571
$815$	7.83183e15	0.762951
$816$	0	0
$817$	1.29975e16	1.24922
$818$	9.76330e15	0.932080
$819$	0	0
$820$	−2.91609e15	−0.274678
$821$	5.13429e15	0.480389	0.240195	−	0.970725i	$-0.422789\pi$
$821$	5.13429e15	0.480389	0.240195	+	0.970725i	$0.422789\pi$
$822$	0	0
$823$	−6.16638e15	−0.569287	−0.284643	−	0.958633i	$-0.591875\pi$
$823$	−6.16638e15	−0.569287	−0.284643	+	0.958633i	$0.591875\pi$
$824$	6.08104e15	0.557671
$825$	0	0
$826$	−1.12065e16	−1.01410
$827$	2.77673e15	0.249605	0.124803	−	0.992182i	$-0.460170\pi$
$827$	2.77673e15	0.249605	0.124803	+	0.992182i	$0.460170\pi$
$828$	0	0
$829$	1.33865e16	1.18745	0.593726	−	0.804667i	$-0.297656\pi$
$829$	1.33865e16	1.18745	0.593726	+	0.804667i	$0.297656\pi$
$830$	−5.55656e15	−0.489639
$831$	0	0
$832$	1.62088e15	0.140953
$833$	1.69025e16	1.46017
$834$	0	0
$835$	−8.43296e15	−0.718960
$836$	−3.95121e15	−0.334653
$837$	0	0
$838$	1.34861e16	1.12731
$839$	−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
$842$	−3.55584e13	−0.00289551
$843$	0	0
$844$	5.52687e15	0.444218
$845$	−1.52070e15	−0.121431
$846$	0	0
$847$	1.18565e16	0.934542
$848$	−4.25677e15	−0.333352
$849$	0	0
$850$	−1.68528e15	−0.130277
$851$	1.70517e15	0.130965
$852$	0	0
$853$	1.97554e16	1.49784	0.748922	−	0.662658i	$-0.230572\pi$
$853$	1.97554e16	1.49784	0.748922	+	0.662658i	$0.230572\pi$
$854$	2.46092e16	1.85387
$855$	0	0
$856$	−1.78850e15	−0.133010
$857$	2.32233e16	1.71605	0.858023	−	0.513612i	$-0.171693\pi$
$857$	2.32233e16	1.71605	0.858023	+	0.513612i	$0.171693\pi$
$858$	0	0
$859$	−1.67344e16	−1.22081	−0.610405	−	0.792089i	$-0.708993\pi$
$859$	−1.67344e16	−1.22081	−0.610405	+	0.792089i	$0.708993\pi$
$860$	3.72580e15	0.270071
$861$	0	0
$862$	7.33454e15	0.524908
$863$	2.46024e16	1.74951	0.874757	−	0.484563i	$-0.161021\pi$
$863$	2.46024e16	1.74951	0.874757	+	0.484563i	$0.161021\pi$
$864$	0	0
$865$	9.61864e15	0.675344
$866$	−4.92901e15	−0.343883
$867$	0	0
$868$	−5.20125e15	−0.358302
$869$	2.42879e15	0.166257
$870$	0	0
$871$	5.59764e15	0.378360
$872$	1.16010e15	0.0779211
$873$	0	0
$874$	1.88489e15	0.125018
$875$	2.18185e15	0.143807
$876$	0	0
$877$	−4.55865e15	−0.296714	−0.148357	−	0.988934i	$-0.547399\pi$
$877$	−4.55865e15	−0.296714	−0.148357	+	0.988934i	$0.547399\pi$
$878$	4.02551e15	0.260376
$879$	0	0
$880$	−1.13263e15	−0.0723490
$881$	−1.67834e16	−1.06540	−0.532701	−	0.846304i	$-0.678823\pi$
$881$	−1.67834e16	−1.06540	−0.532701	+	0.846304i	$0.678823\pi$
$882$	0	0
$883$	−1.13628e16	−0.712362	−0.356181	−	0.934417i	$-0.615921\pi$
$883$	−1.13628e16	−0.712362	−0.356181	+	0.934417i	$0.615921\pi$
$884$	8.33632e15	0.519382
$885$	0	0
$886$	−4.30793e15	−0.265084
$887$	−1.55603e16	−0.951563	−0.475782	−	0.879563i	$-0.657835\pi$
$887$	−1.55603e16	−0.951563	−0.475782	+	0.879563i	$0.657835\pi$
$888$	0	0
$889$	4.00863e16	2.42123
$890$	9.29706e14	0.0558085
$891$	0	0
$892$	−9.59615e15	−0.568971
$893$	3.14747e15	0.185472
$894$	0	0
$895$	1.09736e15	0.0638736
$896$	2.45655e15	0.142112
$897$	0	0
$898$	1.37271e16	0.784440
$899$	1.32350e15	0.0751704
$900$	0	0
$901$	−2.18929e16	−1.22834
$902$	−1.00795e16	−0.562085
$903$	0	0
$904$	3.94246e15	0.217191
$905$	−1.98158e15	−0.108504
$906$	0	0
$907$	3.40764e16	1.84338	0.921688	−	0.387932i	$-0.126811\pi$
$907$	3.40764e16	1.84338	0.921688	+	0.387932i	$0.126811\pi$
$908$	6.67198e15	0.358742
$909$	0	0
$910$	−1.07926e16	−0.573323
$911$	−1.60732e16	−0.848697	−0.424348	−	0.905499i	$-0.639497\pi$
$911$	−1.60732e16	−0.848697	−0.424348	+	0.905499i	$0.639497\pi$
$912$	0	0
$913$	−1.92063e16	−1.00197
$914$	2.52543e16	1.30958
$915$	0	0
$916$	−1.79004e15	−0.0917144
$917$	2.94559e16	1.50017
$918$	0	0
$919$	−7.32080e15	−0.368403	−0.184201	−	0.982889i	$-0.558970\pi$
$919$	−7.32080e15	−0.368403	−0.184201	+	0.982889i	$0.558970\pi$
$920$	5.40310e14	0.0270277
$921$	0	0
$922$	−1.92556e16	−0.951783
$923$	−5.21213e15	−0.256098
$924$	0	0
$925$	3.15590e15	0.153230
$926$	−7.57994e15	−0.365852
$927$	0	0
$928$	−6.25090e14	−0.0298146
$929$	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
$932$	−8.85241e15	−0.412358
$933$	0	0
$934$	2.61459e16	1.20364
$935$	−5.82520e15	−0.266592
$936$	0	0
$937$	1.51385e16	0.684721	0.342361	−	0.939569i	$-0.388774\pi$
$937$	1.51385e16	0.684721	0.342361	+	0.939569i	$0.388774\pi$
$938$	8.48358e15	0.381473
$939$	0	0
$940$	9.02235e14	0.0400974
$941$	−8.81297e15	−0.389385	−0.194693	−	0.980864i	$-0.562371\pi$
$941$	−8.81297e15	−0.389385	−0.194693	+	0.980864i	$0.562371\pi$
$942$	0	0
$943$	4.80832e15	0.209981
$944$	−5.13622e15	−0.222996
$945$	0	0
$946$	1.28782e16	0.552656
$947$	−6.85225e15	−0.292353	−0.146177	−	0.989258i	$-0.546697\pi$
$947$	−6.85225e15	−0.292353	−0.146177	+	0.989258i	$0.546697\pi$
$948$	0	0
$949$	−3.33818e16	−1.40782
$950$	3.48853e15	0.146272
$951$	0	0
$952$	1.26342e16	0.523655
$953$	−3.21185e16	−1.32356	−0.661781	−	0.749697i	$-0.730199\pi$
$953$	−3.21185e16	−1.32356	−0.661781	+	0.749697i	$0.730199\pi$
$954$	0	0
$955$	−8.17451e15	−0.332999
$956$	−6.87129e15	−0.278304
$957$	0	0
$958$	8.98935e15	0.359930
$959$	−5.62135e16	−2.23789
$960$	0	0
$961$	−2.03611e16	−0.801351
$962$	−1.56108e16	−0.610889
$963$	0	0
$964$	−1.23217e16	−0.476702
$965$	−4.99433e15	−0.192122
$966$	0	0
$967$	2.01801e16	0.767500	0.383750	−	0.923437i	$-0.374633\pi$
$967$	2.01801e16	0.767500	0.383750	+	0.923437i	$0.374633\pi$
$968$	5.43416e15	0.205503
$969$	0	0
$970$	4.71888e15	0.176439
$971$	−7.19924e15	−0.267658	−0.133829	−	0.991004i	$-0.542727\pi$
$971$	−7.19924e15	−0.267658	−0.133829	+	0.991004i	$0.542727\pi$
$972$	0	0
$973$	−6.12821e16	−2.25275
$974$	1.30052e16	0.475383
$975$	0	0
$976$	1.12790e16	0.407660
$977$	−3.14860e16	−1.13161	−0.565806	−	0.824538i	$-0.691435\pi$
$977$	−3.14860e16	−1.13161	−0.565806	+	0.824538i	$0.691435\pi$
$978$	0	0
$979$	3.21353e15	0.114203
$980$	−1.00294e16	−0.354432
$981$	0	0
$982$	3.27934e16	1.14597
$983$	1.68059e15	0.0584006	0.0292003	−	0.999574i	$-0.490704\pi$
$983$	1.68059e15	0.0584006	0.0292003	+	0.999574i	$0.490704\pi$
$984$	0	0
$985$	−1.92588e14	−0.00661805
$986$	−3.21489e15	−0.109861
$987$	0	0
$988$	−1.72561e16	−0.583150
$989$	−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	$0.511728\pi$
$992$	−2.38387e15	−0.0787894
$993$	0	0
$994$	−7.89931e15	−0.258205
$995$	−7.39199e15	−0.240289
$996$	0	0
$997$	2.59411e14	0.00833999	0.00416999	−	0.999991i	$-0.498673\pi$
$997$	2.59411e14	0.00833999	0.00416999	+	0.999991i	$0.498673\pi$
$998$	−3.25097e15	−0.103943
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.12.a.l.1.1		2
3.2	odd	2		10.12.a.d.1.2	✓	2
12.11	even	2		80.12.a.g.1.1		2
15.2	even	4		50.12.b.f.49.3		4
15.8	even	4		50.12.b.f.49.2		4
15.14	odd	2		50.12.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.a.d.1.2	✓	2	3.2	odd	2
50.12.a.f.1.1		2	15.14	odd	2
50.12.b.f.49.2		4	15.8	even	4
50.12.b.f.49.3		4	15.2	even	4
80.12.a.g.1.1		2	12.11	even	2
90.12.a.l.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

\(f(q)\)	\(=\)	\(q-32.0000 q^{2} +1024.00 q^{4} -3125.00 q^{5} -71494.9 q^{7} -32768.0 q^{8} +100000. q^{10} +345651. q^{11} +1.50956e6 q^{13} +2.28784e6 q^{14} +1.04858e6 q^{16} +5.39291e6 q^{17} -1.11633e7 q^{19} -3.20000e6 q^{20} -1.10608e7 q^{22} +5.27646e6 q^{23} +9.76562e6 q^{25} -4.83060e7 q^{26} -7.32108e7 q^{28} +1.86291e7 q^{29} +7.10448e7 q^{31} -3.35544e7 q^{32} -1.72573e8 q^{34} +2.23422e8 q^{35} +3.23164e8 q^{37} +3.57225e8 q^{38} +1.02400e8 q^{40} +9.11277e8 q^{41} -1.16431e9 q^{43} +3.53946e8 q^{44} -1.68847e8 q^{46} -2.81949e8 q^{47} +3.13420e9 q^{49} -3.12500e8 q^{50} +1.54579e9 q^{52} -4.05957e9 q^{53} -1.08016e9 q^{55} +2.34275e9 q^{56} -5.96132e8 q^{58} -4.89828e9 q^{59} +1.07565e10 q^{61} -2.27343e9 q^{62} +1.07374e9 q^{64} -4.71739e9 q^{65} +3.70812e9 q^{67} +5.52234e9 q^{68} -7.14949e9 q^{70} -3.45274e9 q^{71} -2.21136e10 q^{73} -1.03413e10 q^{74} -1.14312e10 q^{76} -2.47123e10 q^{77} +7.02672e9 q^{79} -3.27680e9 q^{80} -2.91609e10 q^{82} -5.55656e10 q^{83} -1.68528e10 q^{85} +3.72580e10 q^{86} -1.13263e10 q^{88} +9.29706e9 q^{89} -1.07926e11 q^{91} +5.40310e9 q^{92} +9.02235e9 q^{94} +3.48853e10 q^{95} +4.71888e10 q^{97} -1.00294e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64 q^{2} + 2048 q^{4} - 6250 q^{5} + 14092 q^{7} - 65536 q^{8} + 200000 q^{10} - 421584 q^{11} + 1730524 q^{13} - 450944 q^{14} + 2097152 q^{16} + 6323628 q^{17} - 28897400 q^{19} - 6400000 q^{20}+ \cdots - 271423966272 q^{98}+O(q^{100}) \) 2 * q - 64 * q^2 + 2048 * q^4 - 6250 * q^5 + 14092 * q^7 - 65536 * q^8 + 200000 * q^10 - 421584 * q^11 + 1730524 * q^13 - 450944 * q^14 + 2097152 * q^16 + 6323628 * q^17 - 28897400 * q^19 - 6400000 * q^20 + 13490688 * q^22 + 45236076 * q^23 + 19531250 * q^25 - 55376768 * q^26 + 14430208 * q^28 - 58226220 * q^29 + 41413384 * q^31 - 67108864 * q^32 - 202356096 * q^34 - 44037500 * q^35 + 377255452 * q^37 + 924716800 * q^38 + 204800000 * q^40 + 785271036 * q^41 - 1452987236 * q^43 - 431702016 * q^44 - 1447554432 * q^46 + 1288127748 * q^47 + 8481998946 * q^49 - 625000000 * q^50 + 1772056576 * q^52 + 30490836 * q^53 + 1317450000 * q^55 - 461766656 * q^56 + 1863239040 * q^58 - 8677102440 * q^59 + 1115498764 * q^61 - 1325228288 * q^62 + 2147483648 * q^64 - 5407887500 * q^65 - 12673769708 * q^67 + 6475395072 * q^68 + 1409200000 * q^70 - 13799832984 * q^71 - 17842079516 * q^73 - 12072174464 * q^74 - 29590937600 * q^76 - 90377541264 * q^77 - 12636930320 * q^79 - 6553600000 * q^80 - 25128673152 * q^82 - 41986488924 * q^83 - 19761337500 * q^85 + 46495591552 * q^86 + 13814464512 * q^88 - 13208740020 * q^89 - 89014903096 * q^91 + 46321741824 * q^92 - 41220087936 * q^94 + 90304375000 * q^95 - 61787462828 * q^97 - 271423966272 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more