LMFDB - Embedded newform 126.16.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,16,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("126.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	126.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:	$179.793816426$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3744535x^{2} - 1793673244x + 1139587195320 $ x^4 - 2x^3 - 3744535x^2 - 1793673244*x + 1139587195320
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8}\cdot 5\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1295.66$ of defining polynomial
Character	$\chi$	$=$	126.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+128.000 q^{2} +16384.0 q^{4} -189588. q^{5} -823543. q^{7} +2.09715e6 q^{8} +O(q^{10})$	\(q+128.000 q^{2} +16384.0 q^{4} -189588. q^{5} -823543. q^{7} +2.09715e6 q^{8} -2.42673e7 q^{10} -7.67208e7 q^{11} +4.06587e8 q^{13} -1.05414e8 q^{14} +2.68435e8 q^{16} -1.76163e9 q^{17} +5.22515e9 q^{19} -3.10622e9 q^{20} -9.82026e9 q^{22} +1.58764e10 q^{23} +5.42617e9 q^{25} +5.20431e10 q^{26} -1.34929e10 q^{28} -3.08391e10 q^{29} +2.29164e11 q^{31} +3.43597e10 q^{32} -2.25489e11 q^{34} +1.56134e11 q^{35} -3.81992e11 q^{37} +6.68819e11 q^{38} -3.97596e11 q^{40} -2.37380e12 q^{41} +3.28678e12 q^{43} -1.25699e12 q^{44} +2.03217e12 q^{46} -8.64905e11 q^{47} +6.78223e11 q^{49} +6.94550e11 q^{50} +6.66152e12 q^{52} -1.02882e12 q^{53} +1.45454e13 q^{55} -1.72709e12 q^{56} -3.94741e12 q^{58} +1.37466e13 q^{59} -1.48054e13 q^{61} +2.93330e13 q^{62} +4.39805e12 q^{64} -7.70842e13 q^{65} -5.49468e13 q^{67} -2.88626e13 q^{68} +1.99852e13 q^{70} -9.08156e13 q^{71} -1.74174e14 q^{73} -4.88950e13 q^{74} +8.56088e13 q^{76} +6.31829e13 q^{77} -1.63619e13 q^{79} -5.08922e13 q^{80} -3.03847e14 q^{82} +4.63192e14 q^{83} +3.33985e14 q^{85} +4.20708e14 q^{86} -1.60895e14 q^{88} -2.32519e14 q^{89} -3.34842e14 q^{91} +2.60118e14 q^{92} -1.10708e14 q^{94} -9.90627e14 q^{95} -9.47035e14 q^{97} +8.68126e13 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 512 q^{2} + 65536 q^{4} - 55216 q^{5} - 3294172 q^{7} + 8388608 q^{8}+O(q^{10}) $ 4 * q + 512 * q^2 + 65536 * q^4 - 55216 * q^5 - 3294172 * q^7 + 8388608 * q^8	\( 4 q + 512 q^{2} + 65536 q^{4} - 55216 q^{5} - 3294172 q^{7} + 8388608 q^{8} - 7067648 q^{10} - 42410192 q^{11} + 225416856 q^{13} - 421654016 q^{14} + 1073741824 q^{16} - 1067868368 q^{17} + 1490154848 q^{19} - 904658944 q^{20} - 5428504576 q^{22} + 17651202640 q^{23} - 24170778484 q^{25} + 28853357568 q^{26} - 53971714048 q^{28} + 48603833216 q^{29} - 66948471072 q^{31} + 137438953472 q^{32} - 136687151104 q^{34} + 45472750288 q^{35} - 165011511304 q^{37} + 190739820544 q^{38} - 115796344832 q^{40} - 759932074608 q^{41} + 939420947792 q^{43} - 694848585728 q^{44} + 2259353937920 q^{46} - 3606135631392 q^{47} + 2712892291396 q^{49} - 3093859645952 q^{50} + 3693229768704 q^{52} - 15571497448800 q^{53} - 760393960480 q^{55} - 6908379398144 q^{56} + 6221290651648 q^{58} - 36433798220512 q^{59} - 31107553629432 q^{61} - 8569404297216 q^{62} + 17592186044416 q^{64} - 74820511464672 q^{65} - 66339747458960 q^{67} - 17495955341312 q^{68} + 5820512036864 q^{70} - 35524074310928 q^{71} - 214521476058344 q^{73} - 21121473446912 q^{74} + 24414697029632 q^{76} + 34926616750256 q^{77} - 231821535874560 q^{79} - 14821932138496 q^{80} - 97271305549824 q^{82} + 359126111005312 q^{83} - 75573047434624 q^{85} + 120245881317376 q^{86} - 88940618973184 q^{88} + 188429028383568 q^{89} - 185640473840808 q^{91} + 289197304053760 q^{92} - 461585360818176 q^{94} + 346710605250112 q^{95} - 11\!\cdots\!76 q^{97}+ \cdots + 347250213298688 q^{98}+O(q^{100}) \) 4 * q + 512 * q^2 + 65536 * q^4 - 55216 * q^5 - 3294172 * q^7 + 8388608 * q^8 - 7067648 * q^10 - 42410192 * q^11 + 225416856 * q^13 - 421654016 * q^14 + 1073741824 * q^16 - 1067868368 * q^17 + 1490154848 * q^19 - 904658944 * q^20 - 5428504576 * q^22 + 17651202640 * q^23 - 24170778484 * q^25 + 28853357568 * q^26 - 53971714048 * q^28 + 48603833216 * q^29 - 66948471072 * q^31 + 137438953472 * q^32 - 136687151104 * q^34 + 45472750288 * q^35 - 165011511304 * q^37 + 190739820544 * q^38 - 115796344832 * q^40 - 759932074608 * q^41 + 939420947792 * q^43 - 694848585728 * q^44 + 2259353937920 * q^46 - 3606135631392 * q^47 + 2712892291396 * q^49 - 3093859645952 * q^50 + 3693229768704 * q^52 - 15571497448800 * q^53 - 760393960480 * q^55 - 6908379398144 * q^56 + 6221290651648 * q^58 - 36433798220512 * q^59 - 31107553629432 * q^61 - 8569404297216 * q^62 + 17592186044416 * q^64 - 74820511464672 * q^65 - 66339747458960 * q^67 - 17495955341312 * q^68 + 5820512036864 * q^70 - 35524074310928 * q^71 - 214521476058344 * q^73 - 21121473446912 * q^74 + 24414697029632 * q^76 + 34926616750256 * q^77 - 231821535874560 * q^79 - 14821932138496 * q^80 - 97271305549824 * q^82 + 359126111005312 * q^83 - 75573047434624 * q^85 + 120245881317376 * q^86 - 88940618973184 * q^88 + 188429028383568 * q^89 - 185640473840808 * q^91 + 289197304053760 * q^92 - 461585360818176 * q^94 + 346710605250112 * q^95 - 1107546032911976 * q^97 + 347250213298688 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	128.000	0.707107
$2$	128.000	0.707107
$3$	0	0
$3$	0	0
$4$	16384.0	0.500000
$5$	−189588.	−1.08527	−0.542634	−	0.839970i	$-0.682573\pi$
$5$	−189588.	−1.08527	−0.542634	+	0.839970i	$0.682573\pi$
$6$	0	0
$7$	−823543.	−0.377964
$7$	−823543.	−0.377964
$8$	2.09715e6	0.353553
$9$	0	0
$10$	−2.42673e7	−0.767400
$11$	−7.67208e7	−1.18705	−0.593524	−	0.804817i	$-0.702264\pi$
$11$	−7.67208e7	−1.18705	−0.593524	+	0.804817i	$0.702264\pi$
$12$	0	0
$13$	4.06587e8	1.79713	0.898563	−	0.438845i	$-0.144612\pi$
$13$	4.06587e8	1.79713	0.898563	+	0.438845i	$0.144612\pi$
$14$	−1.05414e8	−0.267261
$15$	0	0
$16$	2.68435e8	0.250000
$17$	−1.76163e9	−1.04123	−0.520617	−	0.853790i	$-0.674298\pi$
$17$	−1.76163e9	−1.04123	−0.520617	+	0.853790i	$0.674298\pi$
$18$	0	0
$19$	5.22515e9	1.34106	0.670528	−	0.741885i	$-0.266068\pi$
$19$	5.22515e9	1.34106	0.670528	+	0.741885i	$0.266068\pi$
$20$	−3.10622e9	−0.542634
$21$	0	0
$22$	−9.82026e9	−0.839369
$23$	1.58764e10	0.972282	0.486141	−	0.873880i	$-0.338404\pi$
$23$	1.58764e10	0.972282	0.486141	+	0.873880i	$0.338404\pi$
$24$	0	0
$25$	5.42617e9	0.177805
$26$	5.20431e10	1.27076
$27$	0	0
$28$	−1.34929e10	−0.188982
$29$	−3.08391e10	−0.331984	−0.165992	−	0.986127i	$-0.553083\pi$
$29$	−3.08391e10	−0.331984	−0.165992	+	0.986127i	$0.553083\pi$
$30$	0	0
$31$	2.29164e11	1.49601	0.748004	−	0.663694i	$-0.231012\pi$
$31$	2.29164e11	1.49601	0.748004	+	0.663694i	$0.231012\pi$
$32$	3.43597e10	0.176777
$33$	0	0
$34$	−2.25489e11	−0.736264
$35$	1.56134e11	0.410192
$36$	0	0
$37$	−3.81992e11	−0.661518	−0.330759	−	0.943715i	$-0.607305\pi$
$37$	−3.81992e11	−0.661518	−0.330759	+	0.943715i	$0.607305\pi$
$38$	6.68819e11	0.948269
$39$	0	0
$40$	−3.97596e11	−0.383700
$41$	−2.37380e12	−1.90356	−0.951778	−	0.306788i	$-0.900746\pi$
$41$	−2.37380e12	−1.90356	−0.951778	+	0.306788i	$0.900746\pi$
$42$	0	0
$43$	3.28678e12	1.84398	0.921992	−	0.387209i	$-0.126561\pi$
$43$	3.28678e12	1.84398	0.921992	+	0.387209i	$0.126561\pi$
$44$	−1.25699e12	−0.593524
$45$	0	0
$46$	2.03217e12	0.687507
$47$	−8.64905e11	−0.249020	−0.124510	−	0.992218i	$-0.539736\pi$
$47$	−8.64905e11	−0.249020	−0.124510	+	0.992218i	$0.539736\pi$
$48$	0	0
$49$	6.78223e11	0.142857
$50$	6.94550e11	0.125727
$51$	0	0
$52$	6.66152e12	0.898563
$53$	−1.02882e12	−0.120302	−0.0601508	−	0.998189i	$-0.519158\pi$
$53$	−1.02882e12	−0.120302	−0.0601508	+	0.998189i	$0.519158\pi$
$54$	0	0
$55$	1.45454e13	1.28826
$56$	−1.72709e12	−0.133631
$57$	0	0
$58$	−3.94741e12	−0.234748
$59$	1.37466e13	0.719128	0.359564	−	0.933120i	$-0.382925\pi$
$59$	1.37466e13	0.719128	0.359564	+	0.933120i	$0.382925\pi$
$60$	0	0
$61$	−1.48054e13	−0.603180	−0.301590	−	0.953438i	$-0.597517\pi$
$61$	−1.48054e13	−0.603180	−0.301590	+	0.953438i	$0.597517\pi$
$62$	2.93330e13	1.05784
$63$	0	0
$64$	4.39805e12	0.125000
$65$	−7.70842e13	−1.95036
$66$	0	0
$67$	−5.49468e13	−1.10760	−0.553798	−	0.832651i	$-0.686822\pi$
$67$	−5.49468e13	−1.10760	−0.553798	+	0.832651i	$0.686822\pi$
$68$	−2.88626e13	−0.520617
$69$	0	0
$70$	1.99852e13	0.290050
$71$	−9.08156e13	−1.18501	−0.592506	−	0.805566i	$-0.701861\pi$
$71$	−9.08156e13	−1.18501	−0.592506	+	0.805566i	$0.701861\pi$
$72$	0	0
$73$	−1.74174e14	−1.84528	−0.922641	−	0.385661i	$-0.873973\pi$
$73$	−1.74174e14	−1.84528	−0.922641	+	0.385661i	$0.873973\pi$
$74$	−4.88950e13	−0.467764
$75$	0	0
$76$	8.56088e13	0.670528
$77$	6.31829e13	0.448662
$78$	0	0
$79$	−1.63619e13	−0.0958585	−0.0479292	−	0.998851i	$-0.515262\pi$
$79$	−1.63619e13	−0.0958585	−0.0479292	+	0.998851i	$0.515262\pi$
$80$	−5.08922e13	−0.271317
$81$	0	0
$82$	−3.03847e14	−1.34602
$83$	4.63192e14	1.87359	0.936797	−	0.349873i	$-0.113775\pi$
$83$	4.63192e14	1.87359	0.936797	+	0.349873i	$0.113775\pi$
$84$	0	0
$85$	3.33985e14	1.13002
$86$	4.20708e14	1.30389
$87$	0	0
$88$	−1.60895e14	−0.419685
$89$	−2.32519e14	−0.557229	−0.278615	−	0.960403i	$-0.589875\pi$
$89$	−2.32519e14	−0.557229	−0.278615	+	0.960403i	$0.589875\pi$
$90$	0	0
$91$	−3.34842e14	−0.679250
$92$	2.60118e14	0.486141
$93$	0	0
$94$	−1.10708e14	−0.176084
$95$	−9.90627e14	−1.45540
$96$	0	0
$97$	−9.47035e14	−1.19008	−0.595042	−	0.803694i	$-0.702865\pi$
$97$	−9.47035e14	−1.19008	−0.595042	+	0.803694i	$0.702865\pi$
$98$	8.68126e13	0.101015
$99$	0	0
$100$	8.89024e13	0.0889024
$101$	−5.81656e14	−0.539828	−0.269914	−	0.962884i	$-0.586995\pi$
$101$	−5.81656e14	−0.539828	−0.269914	+	0.962884i	$0.586995\pi$
$102$	0	0
$103$	4.79839e14	0.384429	0.192215	−	0.981353i	$-0.438433\pi$
$103$	4.79839e14	0.384429	0.192215	+	0.981353i	$0.438433\pi$
$104$	8.52675e14	0.635380
$105$	0	0
$106$	−1.31689e14	−0.0850661
$107$	4.12439e14	0.248303	0.124151	−	0.992263i	$-0.460379\pi$
$107$	4.12439e14	0.248303	0.124151	+	0.992263i	$0.460379\pi$
$108$	0	0
$109$	−1.20350e14	−0.0630593	−0.0315296	−	0.999503i	$-0.510038\pi$
$109$	−1.20350e14	−0.0630593	−0.0315296	+	0.999503i	$0.510038\pi$
$110$	1.86181e15	0.910940
$111$	0	0
$112$	−2.21068e14	−0.0944911
$113$	−4.04207e15	−1.61627	−0.808137	−	0.588994i	$-0.799524\pi$
$113$	−4.04207e15	−1.61627	−0.808137	+	0.588994i	$0.799524\pi$
$114$	0	0
$115$	−3.00997e15	−1.05519
$116$	−5.05268e14	−0.165992
$117$	0	0
$118$	1.75957e15	0.508501
$119$	1.45078e15	0.393550
$120$	0	0
$121$	1.70884e15	0.409082
$122$	−1.89509e15	−0.426513
$123$	0	0
$124$	3.75463e15	0.748004
$125$	4.75704e15	0.892301
$126$	0	0
$127$	−3.41431e15	−0.568558	−0.284279	−	0.958742i	$-0.591754\pi$
$127$	−3.41431e15	−0.568558	−0.284279	+	0.958742i	$0.591754\pi$
$128$	5.62950e14	0.0883883
$129$	0	0
$130$	−9.86677e15	−1.37911
$131$	4.13554e15	0.545755	0.272877	−	0.962049i	$-0.412025\pi$
$131$	4.13554e15	0.545755	0.272877	+	0.962049i	$0.412025\pi$
$132$	0	0
$133$	−4.30313e15	−0.506871
$134$	−7.03319e15	−0.783189
$135$	0	0
$136$	−3.69441e15	−0.368132
$137$	−2.93950e15	−0.277248	−0.138624	−	0.990345i	$-0.544268\pi$
$137$	−2.93950e15	−0.277248	−0.138624	+	0.990345i	$0.544268\pi$
$138$	0	0
$139$	−3.10360e15	−0.262576	−0.131288	−	0.991344i	$-0.541911\pi$
$139$	−3.10360e15	−0.262576	−0.131288	+	0.991344i	$0.541911\pi$
$140$	2.55810e15	0.205096
$141$	0	0
$142$	−1.16244e16	−0.837930
$143$	−3.11937e16	−2.13327
$144$	0	0
$145$	5.84674e15	0.360291
$146$	−2.22943e16	−1.30481
$147$	0	0
$148$	−6.25856e15	−0.330759
$149$	3.40543e16	1.71110	0.855549	−	0.517722i	$-0.173220\pi$
$149$	3.40543e16	1.71110	0.855549	+	0.517722i	$0.173220\pi$
$150$	0	0
$151$	−3.03452e16	−1.37963	−0.689815	−	0.723986i	$-0.742308\pi$
$151$	−3.03452e16	−1.37963	−0.689815	+	0.723986i	$0.742308\pi$
$152$	1.09579e16	0.474135
$153$	0	0
$154$	8.08741e15	0.317252
$155$	−4.34469e16	−1.62357
$156$	0	0
$157$	−2.68592e14	−0.00911687	−0.00455844	−	0.999990i	$-0.501451\pi$
$157$	−2.68592e14	−0.00911687	−0.00455844	+	0.999990i	$0.501451\pi$
$158$	−2.09432e15	−0.0677822
$159$	0	0
$160$	−6.51421e15	−0.191850
$161$	−1.30749e16	−0.367488
$162$	0	0
$163$	−7.16783e16	−1.83646	−0.918228	−	0.396053i	$-0.870380\pi$
$163$	−7.16783e16	−1.83646	−0.918228	+	0.396053i	$0.870380\pi$
$164$	−3.88924e16	−0.951778
$165$	0	0
$166$	5.92886e16	1.32483
$167$	7.29315e16	1.55791	0.778953	−	0.627082i	$-0.215751\pi$
$167$	7.29315e16	1.55791	0.778953	+	0.627082i	$0.215751\pi$
$168$	0	0
$169$	1.14127e17	2.22966
$170$	4.27501e16	0.799043
$171$	0	0
$172$	5.38506e16	0.921992
$173$	−7.34373e16	−1.20385	−0.601923	−	0.798554i	$-0.705598\pi$
$173$	−7.34373e16	−1.20385	−0.601923	+	0.798554i	$0.705598\pi$
$174$	0	0
$175$	−4.46868e15	−0.0672039
$176$	−2.05946e16	−0.296762
$177$	0	0
$178$	−2.97625e16	−0.394021
$179$	−6.39993e16	−0.812414	−0.406207	−	0.913781i	$-0.633149\pi$
$179$	−6.39993e16	−0.812414	−0.406207	+	0.913781i	$0.633149\pi$
$180$	0	0
$181$	−1.98249e15	−0.0231538	−0.0115769	−	0.999933i	$-0.503685\pi$
$181$	−1.98249e15	−0.0231538	−0.0115769	+	0.999933i	$0.503685\pi$
$182$	−4.28598e16	−0.480302
$183$	0	0
$184$	3.32951e16	0.343753
$185$	7.24213e16	0.717924
$186$	0	0
$187$	1.35154e17	1.23599
$188$	−1.41706e16	−0.124510
$189$	0	0
$190$	−1.26800e17	−1.02913
$191$	−2.11130e17	−1.64740	−0.823702	−	0.567024i	$-0.808095\pi$
$191$	−2.11130e17	−1.64740	−0.823702	+	0.567024i	$0.808095\pi$
$192$	0	0
$193$	6.75387e16	0.487386	0.243693	−	0.969852i	$-0.421641\pi$
$193$	6.75387e16	0.487386	0.243693	+	0.969852i	$0.421641\pi$
$194$	−1.21220e17	−0.841517
$195$	0	0
$196$	1.11120e16	0.0714286
$197$	2.39124e17	1.47954	0.739769	−	0.672860i	$-0.234935\pi$
$197$	2.39124e17	1.47954	0.739769	+	0.672860i	$0.234935\pi$
$198$	0	0
$199$	−4.64033e16	−0.266165	−0.133082	−	0.991105i	$-0.542487\pi$
$199$	−4.64033e16	−0.266165	−0.133082	+	0.991105i	$0.542487\pi$
$200$	1.13795e16	0.0628635
$201$	0	0
$202$	−7.44519e16	−0.381716
$203$	2.53973e16	0.125478
$204$	0	0
$205$	4.50045e17	2.06587
$206$	6.14194e16	0.271832
$207$	0	0
$208$	1.09142e17	0.449281
$209$	−4.00878e17	−1.59190
$210$	0	0
$211$	2.28118e17	0.843416	0.421708	−	0.906732i	$-0.361431\pi$
$211$	2.28118e17	0.843416	0.421708	+	0.906732i	$0.361431\pi$
$212$	−1.68562e16	−0.0601508
$213$	0	0
$214$	5.27922e16	0.175577
$215$	−6.23135e17	−2.00122
$216$	0	0
$217$	−1.88727e17	−0.565438
$218$	−1.54049e16	−0.0445896
$219$	0	0
$220$	2.38311e17	0.644132
$221$	−7.16257e17	−1.87123
$222$	0	0
$223$	−2.66294e17	−0.650241	−0.325121	−	0.945673i	$-0.605405\pi$
$223$	−2.66294e17	−0.650241	−0.325121	+	0.945673i	$0.605405\pi$
$224$	−2.82967e16	−0.0668153
$225$	0	0
$226$	−5.17384e17	−1.14288
$227$	−6.29794e17	−1.34587	−0.672937	−	0.739699i	$-0.734968\pi$
$227$	−6.29794e17	−1.34587	−0.672937	+	0.739699i	$0.734968\pi$
$228$	0	0
$229$	−9.23084e17	−1.84704	−0.923518	−	0.383556i	$-0.874699\pi$
$229$	−9.23084e17	−1.84704	−0.923518	+	0.383556i	$0.874699\pi$
$230$	−3.85277e17	−0.746129
$231$	0	0
$232$	−6.46743e16	−0.117374
$233$	−5.46848e17	−0.960942	−0.480471	−	0.877011i	$-0.659534\pi$
$233$	−5.46848e17	−0.960942	−0.480471	+	0.877011i	$0.659534\pi$
$234$	0	0
$235$	1.63976e17	0.270253
$236$	2.25225e17	0.359564
$237$	0	0
$238$	1.85700e17	0.278282
$239$	1.03752e17	0.150666	0.0753328	−	0.997158i	$-0.475998\pi$
$239$	1.03752e17	0.150666	0.0753328	+	0.997158i	$0.475998\pi$
$240$	0	0
$241$	−5.61702e17	−0.766263	−0.383132	−	0.923694i	$-0.625154\pi$
$241$	−5.61702e17	−0.766263	−0.383132	+	0.923694i	$0.625154\pi$
$242$	2.18731e17	0.289264
$243$	0	0
$244$	−2.42572e17	−0.301590
$245$	−1.28583e17	−0.155038
$246$	0	0
$247$	2.12448e18	2.41004
$248$	4.80592e17	0.528919
$249$	0	0
$250$	6.08901e17	0.630952
$251$	−8.60824e17	−0.865688	−0.432844	−	0.901469i	$-0.642490\pi$
$251$	−8.60824e17	−0.865688	−0.432844	+	0.901469i	$0.642490\pi$
$252$	0	0
$253$	−1.21805e18	−1.15414
$254$	−4.37031e17	−0.402031
$255$	0	0
$256$	7.20576e16	0.0625000
$257$	−1.42228e18	−1.19809	−0.599043	−	0.800717i	$-0.704452\pi$
$257$	−1.42228e18	−1.19809	−0.599043	+	0.800717i	$0.704452\pi$
$258$	0	0
$259$	3.14587e17	0.250030
$260$	−1.26295e18	−0.975181
$261$	0	0
$262$	5.29349e17	0.385907
$263$	1.32328e18	0.937526	0.468763	−	0.883324i	$-0.344700\pi$
$263$	1.32328e18	0.937526	0.468763	+	0.883324i	$0.344700\pi$
$264$	0	0
$265$	1.95053e17	0.130559
$266$	−5.50801e17	−0.358412
$267$	0	0
$268$	−9.00249e17	−0.553798
$269$	−1.95302e18	−1.16833	−0.584164	−	0.811635i	$-0.698578\pi$
$269$	−1.95302e18	−1.16833	−0.584164	+	0.811635i	$0.698578\pi$
$270$	0	0
$271$	2.33894e17	0.132358	0.0661788	−	0.997808i	$-0.478919\pi$
$271$	2.33894e17	0.132358	0.0661788	+	0.997808i	$0.478919\pi$
$272$	−4.72885e17	−0.260309
$273$	0	0
$274$	−3.76256e17	−0.196044
$275$	−4.16300e17	−0.211063
$276$	0	0
$277$	3.69031e18	1.77200	0.886001	−	0.463683i	$-0.153472\pi$
$277$	3.69031e18	1.77200	0.886001	+	0.463683i	$0.153472\pi$
$278$	−3.97261e17	−0.185669
$279$	0	0
$280$	3.27437e17	0.145025
$281$	−3.16403e18	−1.36441	−0.682203	−	0.731163i	$-0.738978\pi$
$281$	−3.16403e18	−1.36441	−0.682203	+	0.731163i	$0.738978\pi$
$282$	0	0
$283$	−4.85640e18	−1.98571	−0.992856	−	0.119318i	$-0.961929\pi$
$283$	−4.85640e18	−1.98571	−0.992856	+	0.119318i	$0.961929\pi$
$284$	−1.48792e18	−0.592506
$285$	0	0
$286$	−3.99279e18	−1.50845
$287$	1.95493e18	0.719477
$288$	0	0
$289$	2.40928e17	0.0841693
$290$	7.48382e17	0.254764
$291$	0	0
$292$	−2.85367e18	−0.922641
$293$	−4.81728e17	−0.151808	−0.0759040	−	0.997115i	$-0.524184\pi$
$293$	−4.81728e17	−0.151808	−0.0759040	+	0.997115i	$0.524184\pi$
$294$	0	0
$295$	−2.60620e18	−0.780446
$296$	−8.01096e17	−0.233882
$297$	0	0
$298$	4.35895e18	1.20993
$299$	6.45512e18	1.74731
$300$	0	0
$301$	−2.70680e18	−0.696961
$302$	−3.88418e18	−0.975546
$303$	0	0
$304$	1.40262e18	0.335264
$305$	2.80694e18	0.654612
$306$	0	0
$307$	−4.46511e18	−0.991504	−0.495752	−	0.868464i	$-0.665107\pi$
$307$	−4.46511e18	−0.991504	−0.495752	+	0.868464i	$0.665107\pi$
$308$	1.03519e18	0.224331
$309$	0	0
$310$	−5.56120e18	−1.14804
$311$	−2.23728e18	−0.450834	−0.225417	−	0.974262i	$-0.572375\pi$
$311$	−2.23728e18	−0.450834	−0.225417	+	0.974262i	$0.572375\pi$
$312$	0	0
$313$	5.12022e18	0.983345	0.491673	−	0.870780i	$-0.336386\pi$
$313$	5.12022e18	0.983345	0.491673	+	0.870780i	$0.336386\pi$
$314$	−3.43798e16	−0.00644660
$315$	0	0
$316$	−2.68073e17	−0.0479292
$317$	−7.69668e18	−1.34387	−0.671937	−	0.740608i	$-0.734538\pi$
$317$	−7.69668e18	−1.34387	−0.671937	+	0.740608i	$0.734538\pi$
$318$	0	0
$319$	2.36600e18	0.394081
$320$	−8.33818e17	−0.135658
$321$	0	0
$322$	−1.67358e18	−0.259853
$323$	−9.20479e18	−1.39635
$324$	0	0
$325$	2.20621e18	0.319537
$326$	−9.17482e18	−1.29857
$327$	0	0
$328$	−4.97822e18	−0.673009
$329$	7.12286e17	0.0941207
$330$	0	0
$331$	3.11653e18	0.393515	0.196758	−	0.980452i	$-0.436959\pi$
$331$	3.11653e18	0.393515	0.196758	+	0.980452i	$0.436959\pi$
$332$	7.58894e18	0.936797
$333$	0	0
$334$	9.33523e18	1.10161
$335$	1.04173e19	1.20204
$336$	0	0
$337$	5.02717e17	0.0554752	0.0277376	−	0.999615i	$-0.491170\pi$
$337$	5.02717e17	0.0554752	0.0277376	+	0.999615i	$0.491170\pi$
$338$	1.46083e19	1.57661
$339$	0	0
$340$	5.47201e18	0.565009
$341$	−1.75817e19	−1.77583
$342$	0	0
$343$	−5.58546e17	−0.0539949
$344$	6.89288e18	0.651947
$345$	0	0
$346$	−9.39997e18	−0.851247
$347$	1.97823e18	0.175310	0.0876549	−	0.996151i	$-0.472063\pi$
$347$	1.97823e18	0.175310	0.0876549	+	0.996151i	$0.472063\pi$
$348$	0	0
$349$	1.69806e19	1.44132	0.720662	−	0.693286i	$-0.243838\pi$
$349$	1.69806e19	1.44132	0.720662	+	0.693286i	$0.243838\pi$
$350$	−5.71992e17	−0.0475203
$351$	0	0
$352$	−2.63611e18	−0.209842
$353$	−2.15841e18	−0.168199	−0.0840996	−	0.996457i	$-0.526801\pi$
$353$	−2.15841e18	−0.168199	−0.0840996	+	0.996457i	$0.526801\pi$
$354$	0	0
$355$	1.72176e19	1.28605
$356$	−3.80960e18	−0.278615
$357$	0	0
$358$	−8.19190e18	−0.574463
$359$	2.77278e19	1.90418	0.952089	−	0.305822i	$-0.0989311\pi$
$359$	2.77278e19	1.90418	0.952089	+	0.305822i	$0.0989311\pi$
$360$	0	0
$361$	1.21211e19	0.798429
$362$	−2.53759e17	−0.0163722
$363$	0	0
$364$	−5.48605e18	−0.339625
$365$	3.30214e19	2.00262
$366$	0	0
$367$	−3.00540e18	−0.174947	−0.0874737	−	0.996167i	$-0.527879\pi$
$367$	−3.00540e18	−0.174947	−0.0874737	+	0.996167i	$0.527879\pi$
$368$	4.26178e18	0.243070
$369$	0	0
$370$	9.26992e18	0.507649
$371$	8.47280e17	0.0454697
$372$	0	0
$373$	2.27402e19	1.17214	0.586069	−	0.810261i	$-0.300675\pi$
$373$	2.27402e19	1.17214	0.586069	+	0.810261i	$0.300675\pi$
$374$	1.72997e19	0.873980
$375$	0	0
$376$	−1.81384e18	−0.0880419
$377$	−1.25388e19	−0.596617
$378$	0	0
$379$	2.13865e19	0.978016	0.489008	−	0.872279i	$-0.337359\pi$
$379$	2.13865e19	0.978016	0.489008	+	0.872279i	$0.337359\pi$
$380$	−1.62304e19	−0.727702
$381$	0	0
$382$	−2.70247e19	−1.16489
$383$	−4.96908e18	−0.210032	−0.105016	−	0.994471i	$-0.533489\pi$
$383$	−4.96908e18	−0.210032	−0.105016	+	0.994471i	$0.533489\pi$
$384$	0	0
$385$	−1.19787e19	−0.486918
$386$	8.64496e18	0.344634
$387$	0	0
$388$	−1.55162e19	−0.595042
$389$	3.93308e19	1.47948	0.739742	−	0.672890i	$-0.234947\pi$
$389$	3.93308e19	1.47948	0.739742	+	0.672890i	$0.234947\pi$
$390$	0	0
$391$	−2.79683e19	−1.01237
$392$	1.42234e18	0.0505076
$393$	0	0
$394$	3.06079e19	1.04619
$395$	3.10203e18	0.104032
$396$	0	0
$397$	2.19663e19	0.709299	0.354649	−	0.934999i	$-0.384600\pi$
$397$	2.19663e19	0.709299	0.354649	+	0.934999i	$0.384600\pi$
$398$	−5.93962e18	−0.188207
$399$	0	0
$400$	1.45658e18	0.0444512
$401$	1.90871e19	0.571686	0.285843	−	0.958276i	$-0.407726\pi$
$401$	1.90871e19	0.571686	0.285843	+	0.958276i	$0.407726\pi$
$402$	0	0
$403$	9.31752e19	2.68851
$404$	−9.52985e18	−0.269914
$405$	0	0
$406$	3.25086e18	0.0887264
$407$	2.93068e19	0.785253
$408$	0	0
$409$	−3.69984e19	−0.955562	−0.477781	−	0.878479i	$-0.658559\pi$
$409$	−3.69984e19	−0.955562	−0.477781	+	0.878479i	$0.658559\pi$
$410$	5.76058e19	1.46079
$411$	0	0
$412$	7.86168e18	0.192215
$413$	−1.13209e19	−0.271805
$414$	0	0
$415$	−8.78159e19	−2.03335
$416$	1.39702e19	0.317690
$417$	0	0
$418$	−5.13123e19	−1.12564
$419$	3.51913e19	0.758281	0.379141	−	0.925339i	$-0.376220\pi$
$419$	3.51913e19	0.758281	0.379141	+	0.925339i	$0.376220\pi$
$420$	0	0
$421$	−3.21949e19	−0.669378	−0.334689	−	0.942329i	$-0.608631\pi$
$421$	−3.21949e19	−0.669378	−0.334689	+	0.942329i	$0.608631\pi$
$422$	2.91992e19	0.596385
$423$	0	0
$424$	−2.15760e18	−0.0425331
$425$	−9.55892e18	−0.185136
$426$	0	0
$427$	1.21929e19	0.227981
$428$	6.75741e18	0.124151
$429$	0	0
$430$	−7.97613e19	−1.41507
$431$	7.69773e19	1.34210	0.671048	−	0.741414i	$-0.265845\pi$
$431$	7.69773e19	1.34210	0.671048	+	0.741414i	$0.265845\pi$
$432$	0	0
$433$	−3.99126e19	−0.672126	−0.336063	−	0.941839i	$-0.609096\pi$
$433$	−3.99126e19	−0.672126	−0.336063	+	0.941839i	$0.609096\pi$
$434$	−2.41570e19	−0.399825
$435$	0	0
$436$	−1.97182e18	−0.0315296
$437$	8.29563e19	1.30388
$438$	0	0
$439$	5.04532e19	0.766311	0.383156	−	0.923684i	$-0.374837\pi$
$439$	5.04532e19	0.766311	0.383156	+	0.923684i	$0.374837\pi$
$440$	3.05039e19	0.455470
$441$	0	0
$442$	−9.16809e19	−1.32316
$443$	4.18794e19	0.594254	0.297127	−	0.954838i	$-0.403972\pi$
$443$	4.18794e19	0.594254	0.297127	+	0.954838i	$0.403972\pi$
$444$	0	0
$445$	4.40830e19	0.604743
$446$	−3.40856e19	−0.459790
$447$	0	0
$448$	−3.62198e18	−0.0472456
$449$	−1.20910e20	−1.55101	−0.775506	−	0.631341i	$-0.782505\pi$
$449$	−1.20910e20	−1.55101	−0.775506	+	0.631341i	$0.782505\pi$
$450$	0	0
$451$	1.82120e20	2.25961
$452$	−6.62252e19	−0.808137
$453$	0	0
$454$	−8.06136e19	−0.951677
$455$	6.34821e19	0.737167
$456$	0	0
$457$	7.86043e19	0.883232	0.441616	−	0.897204i	$-0.354405\pi$
$457$	7.86043e19	0.883232	0.441616	+	0.897204i	$0.354405\pi$
$458$	−1.18155e20	−1.30605
$459$	0	0
$460$	−4.93154e19	−0.527593
$461$	2.34130e19	0.246434	0.123217	−	0.992380i	$-0.460679\pi$
$461$	2.34130e19	0.246434	0.123217	+	0.992380i	$0.460679\pi$
$462$	0	0
$463$	6.84962e19	0.697925	0.348963	−	0.937137i	$-0.386534\pi$
$463$	6.84962e19	0.697925	0.348963	+	0.937137i	$0.386534\pi$
$464$	−8.27831e18	−0.0829960
$465$	0	0
$466$	−6.99965e19	−0.679488
$467$	1.91317e19	0.182758	0.0913791	−	0.995816i	$-0.470872\pi$
$467$	1.91317e19	0.182758	0.0913791	+	0.995816i	$0.470872\pi$
$468$	0	0
$469$	4.52511e19	0.418632
$470$	2.09889e19	0.191098
$471$	0	0
$472$	2.88288e19	0.254250
$473$	−2.52164e20	−2.18890
$474$	0	0
$475$	2.83525e19	0.238446
$476$	2.37696e19	0.196775
$477$	0	0
$478$	1.32803e19	0.106537
$479$	5.13196e19	0.405291	0.202646	−	0.979252i	$-0.435046\pi$
$479$	5.13196e19	0.405291	0.202646	+	0.979252i	$0.435046\pi$
$480$	0	0
$481$	−1.55313e20	−1.18883
$482$	−7.18978e19	−0.541830
$483$	0	0
$484$	2.79976e19	0.204541
$485$	1.79547e20	1.29156
$486$	0	0
$487$	−1.14297e20	−0.797199	−0.398599	−	0.917125i	$-0.630504\pi$
$487$	−1.14297e20	−0.797199	−0.398599	+	0.917125i	$0.630504\pi$
$488$	−3.10492e19	−0.213256
$489$	0	0
$490$	−1.64586e19	−0.109629
$491$	−2.48229e19	−0.162832	−0.0814162	−	0.996680i	$-0.525944\pi$
$491$	−2.48229e19	−0.162832	−0.0814162	+	0.996680i	$0.525944\pi$
$492$	0	0
$493$	5.43272e19	0.345673
$494$	2.71933e20	1.70416
$495$	0	0
$496$	6.15158e19	0.374002
$497$	7.47905e19	0.447892
$498$	0	0
$499$	−8.25214e19	−0.479526	−0.239763	−	0.970831i	$-0.577070\pi$
$499$	−8.25214e19	−0.479526	−0.239763	+	0.970831i	$0.577070\pi$
$500$	7.79393e19	0.446151
$501$	0	0
$502$	−1.10185e20	−0.612134
$503$	−5.16370e19	−0.282619	−0.141309	−	0.989965i	$-0.545131\pi$
$503$	−5.16370e19	−0.282619	−0.141309	+	0.989965i	$0.545131\pi$
$504$	0	0
$505$	1.10275e20	0.585858
$506$	−1.55910e20	−0.816103
$507$	0	0
$508$	−5.59400e19	−0.284279
$509$	1.04953e20	0.525547	0.262774	−	0.964858i	$-0.415363\pi$
$509$	1.04953e20	0.525547	0.262774	+	0.964858i	$0.415363\pi$
$510$	0	0
$511$	1.43440e20	0.697451
$512$	9.22337e18	0.0441942
$513$	0	0
$514$	−1.82052e20	−0.847174
$515$	−9.09718e19	−0.417208
$516$	0	0
$517$	6.63562e19	0.295599
$518$	4.02671e19	0.176798
$519$	0	0
$520$	−1.61657e20	−0.689557
$521$	−3.93750e19	−0.165553	−0.0827765	−	0.996568i	$-0.526379\pi$
$521$	−3.93750e19	−0.165553	−0.0827765	+	0.996568i	$0.526379\pi$
$522$	0	0
$523$	−2.90388e18	−0.0118636	−0.00593180	−	0.999982i	$-0.501888\pi$
$523$	−2.90388e18	−0.0118636	−0.00593180	+	0.999982i	$0.501888\pi$
$524$	6.77567e19	0.272877
$525$	0	0
$526$	1.69380e20	0.662931
$527$	−4.03703e20	−1.55770
$528$	0	0
$529$	−1.45765e19	−0.0546683
$530$	2.49668e19	0.0923194
$531$	0	0
$532$	−7.05026e19	−0.253436
$533$	−9.65157e20	−3.42093
$534$	0	0
$535$	−7.81937e19	−0.269475
$536$	−1.15232e20	−0.391594
$537$	0	0
$538$	−2.49987e20	−0.826133
$539$	−5.20338e19	−0.169578
$540$	0	0
$541$	4.42034e20	1.40112	0.700561	−	0.713592i	$-0.252933\pi$
$541$	4.42034e20	1.40112	0.700561	+	0.713592i	$0.252933\pi$
$542$	2.99384e19	0.0935910
$543$	0	0
$544$	−6.05293e19	−0.184066
$545$	2.28170e19	0.0684362
$546$	0	0
$547$	7.21144e19	0.210435	0.105217	−	0.994449i	$-0.466446\pi$
$547$	7.21144e19	0.210435	0.105217	+	0.994449i	$0.466446\pi$
$548$	−4.81608e19	−0.138624
$549$	0	0
$550$	−5.32864e19	−0.149244
$551$	−1.61139e20	−0.445209
$552$	0	0
$553$	1.34747e19	0.0362311
$554$	4.72359e20	1.25300
$555$	0	0
$556$	−5.08494e19	−0.131288
$557$	1.37561e20	0.350413	0.175206	−	0.984532i	$-0.443941\pi$
$557$	1.37561e20	0.350413	0.175206	+	0.984532i	$0.443941\pi$
$558$	0	0
$559$	1.33636e21	3.31387
$560$	4.19119e19	0.102548
$561$	0	0
$562$	−4.04996e20	−0.964780
$563$	−3.06487e20	−0.720443	−0.360221	−	0.932867i	$-0.617299\pi$
$563$	−3.06487e20	−0.720443	−0.360221	+	0.932867i	$0.617299\pi$
$564$	0	0
$565$	7.66329e20	1.75409
$566$	−6.21619e20	−1.40411
$567$	0	0
$568$	−1.90454e20	−0.418965
$569$	1.55371e20	0.337309	0.168654	−	0.985675i	$-0.446058\pi$
$569$	1.55371e20	0.337309	0.168654	+	0.985675i	$0.446058\pi$
$570$	0	0
$571$	−5.68205e20	−1.20153	−0.600765	−	0.799426i	$-0.705137\pi$
$571$	−5.68205e20	−1.20153	−0.600765	+	0.799426i	$0.705137\pi$
$572$	−5.11077e20	−1.06664
$573$	0	0
$574$	2.50231e20	0.508747
$575$	8.61478e19	0.172876
$576$	0	0
$577$	−5.93919e20	−1.16120	−0.580602	−	0.814187i	$-0.697183\pi$
$577$	−5.93919e20	−1.16120	−0.580602	+	0.814187i	$0.697183\pi$
$578$	3.08388e19	0.0595167
$579$	0	0
$580$	9.57930e19	0.180146
$581$	−3.81459e20	−0.708152
$582$	0	0
$583$	7.89321e19	0.142804
$584$	−3.65270e20	−0.652405
$585$	0	0
$586$	−6.16612e19	−0.107344
$587$	−7.32987e20	−1.25983	−0.629913	−	0.776666i	$-0.716909\pi$
$587$	−7.32987e20	−1.25983	−0.629913	+	0.776666i	$0.716909\pi$
$588$	0	0
$589$	1.19742e21	2.00623
$590$	−3.33594e20	−0.551859
$591$	0	0
$592$	−1.02540e20	−0.165379
$593$	−2.71230e20	−0.431944	−0.215972	−	0.976400i	$-0.569292\pi$
$593$	−2.71230e20	−0.431944	−0.215972	+	0.976400i	$0.569292\pi$
$594$	0	0
$595$	−2.75051e20	−0.427106
$596$	5.57945e20	0.855549
$597$	0	0
$598$	8.26256e20	1.23554
$599$	1.75649e20	0.259385	0.129692	−	0.991554i	$-0.458601\pi$
$599$	1.75649e20	0.259385	0.129692	+	0.991554i	$0.458601\pi$
$600$	0	0
$601$	3.36994e20	0.485360	0.242680	−	0.970106i	$-0.421974\pi$
$601$	3.36994e20	0.485360	0.242680	+	0.970106i	$0.421974\pi$
$602$	−3.46471e20	−0.492826
$603$	0	0
$604$	−4.97176e20	−0.689815
$605$	−3.23975e20	−0.443963
$606$	0	0
$607$	−9.25100e20	−1.23673	−0.618363	−	0.785892i	$-0.712204\pi$
$607$	−9.25100e20	−1.23673	−0.618363	+	0.785892i	$0.712204\pi$
$608$	1.79535e20	0.237067
$609$	0	0
$610$	3.59288e20	0.462880
$611$	−3.51659e20	−0.447520
$612$	0	0
$613$	1.59431e20	0.197979	0.0989894	−	0.995088i	$-0.468439\pi$
$613$	1.59431e20	0.197979	0.0989894	+	0.995088i	$0.468439\pi$
$614$	−5.71534e20	−0.701099
$615$	0	0
$616$	1.32504e20	0.158626
$617$	1.07816e21	1.27511	0.637553	−	0.770407i	$-0.279947\pi$
$617$	1.07816e21	1.27511	0.637553	+	0.770407i	$0.279947\pi$
$618$	0	0
$619$	5.01637e20	0.579041	0.289520	−	0.957172i	$-0.406504\pi$
$619$	5.01637e20	0.579041	0.289520	+	0.957172i	$0.406504\pi$
$620$	−7.11833e20	−0.811784
$621$	0	0
$622$	−2.86372e20	−0.318788
$623$	1.91490e20	0.210613
$624$	0	0
$625$	−1.06747e21	−1.14619
$626$	6.55388e20	0.695330
$627$	0	0
$628$	−4.40061e18	−0.00455844
$629$	6.72930e20	0.688795
$630$	0	0
$631$	7.91618e20	0.791216	0.395608	−	0.918419i	$-0.370534\pi$
$631$	7.91618e20	0.791216	0.395608	+	0.918419i	$0.370534\pi$
$632$	−3.43134e19	−0.0338911
$633$	0	0
$634$	−9.85175e20	−0.950263
$635$	6.47313e20	0.617037
$636$	0	0
$637$	2.75757e20	0.256732
$638$	3.02848e20	0.278657
$639$	0	0
$640$	−1.06729e20	−0.0959250
$641$	−6.44000e20	−0.572072	−0.286036	−	0.958219i	$-0.592338\pi$
$641$	−6.44000e20	−0.572072	−0.286036	+	0.958219i	$0.592338\pi$
$642$	0	0
$643$	−7.80183e20	−0.677040	−0.338520	−	0.940959i	$-0.609926\pi$
$643$	−7.80183e20	−0.677040	−0.338520	+	0.940959i	$0.609926\pi$
$644$	−2.14219e20	−0.183744
$645$	0	0
$646$	−1.17821e21	−0.987371
$647$	−6.34530e20	−0.525617	−0.262809	−	0.964848i	$-0.584649\pi$
$647$	−6.34530e20	−0.525617	−0.262809	+	0.964848i	$0.584649\pi$
$648$	0	0
$649$	−1.05465e21	−0.853640
$650$	2.82395e20	0.225947
$651$	0	0
$652$	−1.17438e21	−0.918228
$653$	1.37832e21	1.06537	0.532685	−	0.846314i	$-0.321183\pi$
$653$	1.37832e21	1.06537	0.532685	+	0.846314i	$0.321183\pi$
$654$	0	0
$655$	−7.84050e20	−0.592289
$656$	−6.37213e20	−0.475889
$657$	0	0
$658$	9.11726e19	0.0665534
$659$	−1.53420e21	−1.10724	−0.553620	−	0.832769i	$-0.686754\pi$
$659$	−1.53420e21	−1.10724	−0.553620	+	0.832769i	$0.686754\pi$
$660$	0	0
$661$	−9.17615e20	−0.647365	−0.323683	−	0.946166i	$-0.604921\pi$
$661$	−9.17615e20	−0.647365	−0.323683	+	0.946166i	$0.604921\pi$
$662$	3.98916e20	0.278257
$663$	0	0
$664$	9.71385e20	0.662416
$665$	8.15824e20	0.550091
$666$	0	0
$667$	−4.89613e20	−0.322782
$668$	1.19491e21	0.778953
$669$	0	0
$670$	1.33341e21	0.849969
$671$	1.13588e21	0.716004
$672$	0	0
$673$	−3.02720e21	−1.86607	−0.933035	−	0.359785i	$-0.882850\pi$
$673$	−3.02720e21	−1.86607	−0.933035	+	0.359785i	$0.882850\pi$
$674$	6.43477e19	0.0392269
$675$	0	0
$676$	1.86986e21	1.11483
$677$	2.58425e21	1.52377	0.761885	−	0.647712i	$-0.224274\pi$
$677$	2.58425e21	1.52377	0.761885	+	0.647712i	$0.224274\pi$
$678$	0	0
$679$	7.79924e20	0.449810
$680$	7.00418e20	0.399522
$681$	0	0
$682$	−2.25045e21	−1.25570
$683$	−1.87016e21	−1.03210	−0.516051	−	0.856558i	$-0.672598\pi$
$683$	−1.87016e21	−1.03210	−0.516051	+	0.856558i	$0.672598\pi$
$684$	0	0
$685$	5.57295e20	0.300889
$686$	−7.14939e19	−0.0381802
$687$	0	0
$688$	8.82288e20	0.460996
$689$	−4.18306e20	−0.216197
$690$	0	0
$691$	−2.78363e21	−1.40775	−0.703877	−	0.710322i	$-0.748549\pi$
$691$	−2.78363e21	−1.40775	−0.703877	+	0.710322i	$0.748549\pi$
$692$	−1.20320e21	−0.601923
$693$	0	0
$694$	2.53214e20	0.123963
$695$	5.88407e20	0.284965
$696$	0	0
$697$	4.18177e21	1.98205
$698$	2.17351e21	1.01917
$699$	0	0
$700$	−7.32149e19	−0.0336019
$701$	−4.65618e19	−0.0211419	−0.0105710	−	0.999944i	$-0.503365\pi$
$701$	−4.65618e19	−0.0211419	−0.0105710	+	0.999944i	$0.503365\pi$
$702$	0	0
$703$	−1.99597e21	−0.887132
$704$	−3.37422e20	−0.148381
$705$	0	0
$706$	−2.76277e20	−0.118935
$707$	4.79019e20	0.204036
$708$	0	0
$709$	−5.91716e19	−0.0246755	−0.0123378	−	0.999924i	$-0.503927\pi$
$709$	−5.91716e19	−0.0246755	−0.0123378	+	0.999924i	$0.503927\pi$
$710$	2.20385e21	0.909378
$711$	0	0
$712$	−4.87628e20	−0.197010
$713$	3.63829e21	1.45454
$714$	0	0
$715$	5.91396e21	2.31517
$716$	−1.04856e21	−0.406207
$717$	0	0
$718$	3.54916e21	1.34646
$719$	3.07383e21	1.15402	0.577009	−	0.816738i	$-0.304220\pi$
$719$	3.07383e21	1.15402	0.577009	+	0.816738i	$0.304220\pi$
$720$	0	0
$721$	−3.95168e20	−0.145301
$722$	1.55149e21	0.564574
$723$	0	0
$724$	−3.24811e19	−0.0115769
$725$	−1.67338e20	−0.0590283
$726$	0	0
$727$	−1.47873e21	−0.510952	−0.255476	−	0.966815i	$-0.582232\pi$
$727$	−1.47873e21	−0.510952	−0.255476	+	0.966815i	$0.582232\pi$
$728$	−7.02214e20	−0.240151
$729$	0	0
$730$	4.22674e21	1.41607
$731$	−5.79010e21	−1.92002
$732$	0	0
$733$	5.75915e20	0.187102	0.0935510	−	0.995614i	$-0.470178\pi$
$733$	5.75915e20	0.187102	0.0935510	+	0.995614i	$0.470178\pi$
$734$	−3.84692e20	−0.123706
$735$	0	0
$736$	5.45508e20	0.171877
$737$	4.21556e21	1.31477
$738$	0	0
$739$	1.61727e21	0.494252	0.247126	−	0.968983i	$-0.420514\pi$
$739$	1.61727e21	0.494252	0.247126	+	0.968983i	$0.420514\pi$
$740$	1.18655e21	0.358962
$741$	0	0
$742$	1.08452e20	0.0321520
$743$	−1.30174e21	−0.382040	−0.191020	−	0.981586i	$-0.561180\pi$
$743$	−1.30174e21	−0.382040	−0.191020	+	0.981586i	$0.561180\pi$
$744$	0	0
$745$	−6.45630e21	−1.85700
$746$	2.91075e21	0.828826
$747$	0	0
$748$	2.21436e21	0.617997
$749$	−3.39662e20	−0.0938497
$750$	0	0
$751$	−2.03977e19	−0.00552435	−0.00276218	−	0.999996i	$-0.500879\pi$
$751$	−2.03977e19	−0.00552435	−0.00276218	+	0.999996i	$0.500879\pi$
$752$	−2.32171e20	−0.0622550
$753$	0	0
$754$	−1.60496e21	−0.421872
$755$	5.75309e21	1.49727
$756$	0	0
$757$	−5.17147e21	−1.31946	−0.659728	−	0.751504i	$-0.729329\pi$
$757$	−5.17147e21	−1.31946	−0.659728	+	0.751504i	$0.729329\pi$
$758$	2.73747e21	0.691562
$759$	0	0
$760$	−2.07750e21	−0.514563
$761$	1.24527e21	0.305407	0.152703	−	0.988272i	$-0.451202\pi$
$761$	1.24527e21	0.305407	0.152703	+	0.988272i	$0.451202\pi$
$762$	0	0
$763$	9.91138e19	0.0238342
$764$	−3.45916e21	−0.823702
$765$	0	0
$766$	−6.36042e20	−0.148515
$767$	5.58921e21	1.29236
$768$	0	0
$769$	−1.17019e21	−0.265344	−0.132672	−	0.991160i	$-0.542356\pi$
$769$	−1.17019e21	−0.265344	−0.132672	+	0.991160i	$0.542356\pi$
$770$	−1.53328e21	−0.344303
$771$	0	0
$772$	1.10655e21	0.243693
$773$	2.30876e21	0.503538	0.251769	−	0.967787i	$-0.418988\pi$
$773$	2.30876e21	0.503538	0.251769	+	0.967787i	$0.418988\pi$
$774$	0	0
$775$	1.24348e21	0.265997
$776$	−1.98608e21	−0.420758
$777$	0	0
$778$	5.03434e21	1.04615
$779$	−1.24035e22	−2.55277
$780$	0	0
$781$	6.96744e21	1.40667
$782$	−3.57994e21	−0.715856
$783$	0	0
$784$	1.82059e20	0.0357143
$785$	5.09219e19	0.00989424
$786$	0	0
$787$	−2.94258e21	−0.560941	−0.280470	−	0.959863i	$-0.590490\pi$
$787$	−2.94258e21	−0.560941	−0.280470	+	0.959863i	$0.590490\pi$
$788$	3.91781e21	0.739769
$789$	0	0
$790$	3.97059e20	0.0735618
$791$	3.32882e21	0.610894
$792$	0	0
$793$	−6.01970e21	−1.08399
$794$	2.81169e21	0.501550
$795$	0	0
$796$	−7.60271e20	−0.133082
$797$	−9.10081e21	−1.57813	−0.789065	−	0.614310i	$-0.789435\pi$
$797$	−9.10081e21	−1.57813	−0.789065	+	0.614310i	$0.789435\pi$
$798$	0	0
$799$	1.52364e21	0.259288
$800$	1.86442e20	0.0314317
$801$	0	0
$802$	2.44315e21	0.404243
$803$	1.33628e22	2.19044
$804$	0	0
$805$	2.47884e21	0.398823
$806$	1.19264e22	1.90107
$807$	0	0
$808$	−1.21982e21	−0.190858
$809$	−2.14667e21	−0.332776	−0.166388	−	0.986060i	$-0.553210\pi$
$809$	−2.14667e21	−0.332776	−0.166388	+	0.986060i	$0.553210\pi$
$810$	0	0
$811$	−4.84877e21	−0.737862	−0.368931	−	0.929457i	$-0.620276\pi$
$811$	−4.84877e21	−0.737862	−0.368931	+	0.929457i	$0.620276\pi$
$812$	4.16110e20	0.0627391
$813$	0	0
$814$	3.75126e21	0.555258
$815$	1.35894e22	1.99304
$816$	0	0
$817$	1.71739e22	2.47288
$818$	−4.73580e21	−0.675684
$819$	0	0
$820$	7.37354e21	1.03293
$821$	1.50552e21	0.208984	0.104492	−	0.994526i	$-0.466678\pi$
$821$	1.50552e21	0.208984	0.104492	+	0.994526i	$0.466678\pi$
$822$	0	0
$823$	−2.46914e21	−0.336548	−0.168274	−	0.985740i	$-0.553819\pi$
$823$	−2.46914e21	−0.336548	−0.168274	+	0.985740i	$0.553819\pi$
$824$	1.00629e21	0.135916
$825$	0	0
$826$	−1.44908e21	−0.192195
$827$	7.33171e21	0.963639	0.481819	−	0.876271i	$-0.339976\pi$
$827$	7.33171e21	0.963639	0.481819	+	0.876271i	$0.339976\pi$
$828$	0	0
$829$	−5.79915e21	−0.748523	−0.374262	−	0.927323i	$-0.622104\pi$
$829$	−5.79915e21	−0.748523	−0.374262	+	0.927323i	$0.622104\pi$
$830$	−1.12404e22	−1.43780
$831$	0	0
$832$	1.78819e21	0.224641
$833$	−1.19478e21	−0.148748
$834$	0	0
$835$	−1.38270e22	−1.69074
$836$	−6.56798e21	−0.795948
$837$	0	0
$838$	4.50449e21	0.536186
$839$	−1.24501e21	−0.146878	−0.0734391	−	0.997300i	$-0.523397\pi$
$839$	−1.24501e21	−0.146878	−0.0734391	+	0.997300i	$0.523397\pi$
$840$	0	0
$841$	−7.67814e21	−0.889787
$842$	−4.12095e21	−0.473322
$843$	0	0
$844$	3.73749e21	0.421708
$845$	−2.16372e22	−2.41978
$846$	0	0
$847$	−1.40730e21	−0.154618
$848$	−2.76172e20	−0.0300754
$849$	0	0
$850$	−1.22354e21	−0.130911
$851$	−6.06464e21	−0.643182
$852$	0	0
$853$	−9.50800e21	−0.990766	−0.495383	−	0.868675i	$-0.664972\pi$
$853$	−9.50800e21	−0.990766	−0.495383	+	0.868675i	$0.664972\pi$
$854$	1.56069e21	0.161207
$855$	0	0
$856$	8.64948e20	0.0877883
$857$	−1.01384e22	−1.02003	−0.510016	−	0.860165i	$-0.670360\pi$
$857$	−1.01384e22	−1.02003	−0.510016	+	0.860165i	$0.670360\pi$
$858$	0	0
$859$	−1.38736e22	−1.37164	−0.685822	−	0.727769i	$-0.740557\pi$
$859$	−1.38736e22	−1.37164	−0.685822	+	0.727769i	$0.740557\pi$
$860$	−1.02094e22	−1.00061
$861$	0	0
$862$	9.85310e21	0.949005
$863$	1.32935e22	1.26928	0.634640	−	0.772808i	$-0.281148\pi$
$863$	1.32935e22	1.26928	0.634640	+	0.772808i	$0.281148\pi$
$864$	0	0
$865$	1.39228e22	1.30649
$866$	−5.10881e21	−0.475265
$867$	0	0
$868$	−3.09210e21	−0.282719
$869$	1.25530e21	0.113789
$870$	0	0
$871$	−2.23407e22	−1.99049
$872$	−2.52393e20	−0.0222948
$873$	0	0
$874$	1.06184e22	0.921985
$875$	−3.91763e21	−0.337258
$876$	0	0
$877$	−2.70003e21	−0.228493	−0.114246	−	0.993452i	$-0.536445\pi$
$877$	−2.70003e21	−0.228493	−0.114246	+	0.993452i	$0.536445\pi$
$878$	6.45801e21	0.541864
$879$	0	0
$880$	3.90449e21	0.322066
$881$	4.24641e21	0.347298	0.173649	−	0.984808i	$-0.444444\pi$
$881$	4.24641e21	0.347298	0.173649	+	0.984808i	$0.444444\pi$
$882$	0	0
$883$	1.97824e22	1.59065	0.795324	−	0.606185i	$-0.207301\pi$
$883$	1.97824e22	1.59065	0.795324	+	0.606185i	$0.207301\pi$
$884$	−1.17352e22	−0.935614
$885$	0	0
$886$	5.36056e21	0.420201
$887$	−9.03039e21	−0.701906	−0.350953	−	0.936393i	$-0.614142\pi$
$887$	−9.03039e21	−0.701906	−0.350953	+	0.936393i	$0.614142\pi$
$888$	0	0
$889$	2.81183e21	0.214895
$890$	5.64262e21	0.427618
$891$	0	0
$892$	−4.36296e21	−0.325121
$893$	−4.51926e21	−0.333950
$894$	0	0
$895$	1.21335e22	0.881686
$896$	−4.63613e20	−0.0334077
$897$	0	0
$898$	−1.54765e22	−1.09673
$899$	−7.06722e21	−0.496651
$900$	0	0
$901$	1.81241e21	0.125262
$902$	2.33114e22	1.59779
$903$	0	0
$904$	−8.47683e21	−0.571439
$905$	3.75857e20	0.0251280
$906$	0	0
$907$	1.10375e22	0.725797	0.362898	−	0.931829i	$-0.381787\pi$
$907$	1.10375e22	0.725797	0.362898	+	0.931829i	$0.381787\pi$
$908$	−1.03185e22	−0.672937
$909$	0	0
$910$	8.12571e21	0.521256
$911$	−2.59345e22	−1.65002	−0.825011	−	0.565116i	$-0.808831\pi$
$911$	−2.59345e22	−1.65002	−0.825011	+	0.565116i	$0.808831\pi$
$912$	0	0
$913$	−3.55365e22	−2.22405
$914$	1.00614e22	0.624540
$915$	0	0
$916$	−1.51238e22	−0.923518
$917$	−3.40579e21	−0.206276
$918$	0	0
$919$	−9.30007e21	−0.554141	−0.277070	−	0.960850i	$-0.589364\pi$
$919$	−9.30007e21	−0.554141	−0.277070	+	0.960850i	$0.589364\pi$
$920$	−6.31237e21	−0.373064
$921$	0	0
$922$	2.99686e21	0.174255
$923$	−3.69244e22	−2.12961
$924$	0	0
$925$	−2.07275e21	−0.117621
$926$	8.76751e21	0.493508
$927$	0	0
$928$	−1.05962e21	−0.0586870
$929$	−5.64001e21	−0.309857	−0.154929	−	0.987926i	$-0.549515\pi$
$929$	−5.64001e21	−0.309857	−0.154929	+	0.987926i	$0.549515\pi$
$930$	0	0
$931$	3.54382e21	0.191579
$932$	−8.95955e21	−0.480471
$933$	0	0
$934$	2.44886e21	0.129230
$935$	−2.56236e22	−1.34138
$936$	0	0
$937$	2.23878e22	1.15336	0.576679	−	0.816971i	$-0.304349\pi$
$937$	2.23878e22	1.15336	0.576679	+	0.816971i	$0.304349\pi$
$938$	5.79214e21	0.296018
$939$	0	0
$940$	2.68658e21	0.135127
$941$	−2.29462e22	−1.14496	−0.572478	−	0.819920i	$-0.694018\pi$
$941$	−2.29462e22	−1.14496	−0.572478	+	0.819920i	$0.694018\pi$
$942$	0	0
$943$	−3.76873e22	−1.85079
$944$	3.69009e21	0.179782
$945$	0	0
$946$	−3.22770e22	−1.54778
$947$	6.87757e21	0.327198	0.163599	−	0.986527i	$-0.447690\pi$
$947$	6.87757e21	0.327198	0.163599	+	0.986527i	$0.447690\pi$
$948$	0	0
$949$	−7.08170e22	−3.31620
$950$	3.62913e21	0.168607
$951$	0	0
$952$	3.04251e21	0.139141
$953$	−2.97204e22	−1.34852	−0.674260	−	0.738494i	$-0.735537\pi$
$953$	−2.97204e22	−1.34852	−0.674260	+	0.738494i	$0.735537\pi$
$954$	0	0
$955$	4.00278e22	1.78787
$956$	1.69988e21	0.0753328
$957$	0	0
$958$	6.56891e21	0.286584
$959$	2.42080e21	0.104790
$960$	0	0
$961$	2.90509e22	1.23804
$962$	−1.98801e22	−0.840630
$963$	0	0
$964$	−9.20292e21	−0.383132
$965$	−1.28046e22	−0.528944
$966$	0	0
$967$	1.48431e22	0.603708	0.301854	−	0.953354i	$-0.402394\pi$
$967$	1.48431e22	0.603708	0.301854	+	0.953354i	$0.402394\pi$
$968$	3.58369e21	0.144632
$969$	0	0
$970$	2.29820e22	0.913271
$971$	3.40633e22	1.34320	0.671602	−	0.740912i	$-0.265606\pi$
$971$	3.40633e22	1.34320	0.671602	+	0.740912i	$0.265606\pi$
$972$	0	0
$973$	2.55595e21	0.0992445
$974$	−1.46300e22	−0.563705
$975$	0	0
$976$	−3.97430e21	−0.150795
$977$	2.13977e22	0.805673	0.402836	−	0.915272i	$-0.368024\pi$
$977$	2.13977e22	0.805673	0.402836	+	0.915272i	$0.368024\pi$
$978$	0	0
$979$	1.78391e22	0.661458
$980$	−2.10671e21	−0.0775191
$981$	0	0
$982$	−3.17733e21	−0.115140
$983$	2.91325e22	1.04767	0.523836	−	0.851819i	$-0.324500\pi$
$983$	2.91325e22	1.04767	0.523836	+	0.851819i	$0.324500\pi$
$984$	0	0
$985$	−4.53351e22	−1.60569
$986$	6.95388e21	0.244428
$987$	0	0
$988$	3.48074e22	1.20502
$989$	5.21821e22	1.79287
$990$	0	0
$991$	2.82631e22	0.956463	0.478231	−	0.878234i	$-0.341278\pi$
$991$	2.82631e22	0.956463	0.478231	+	0.878234i	$0.341278\pi$
$992$	7.87402e21	0.264459
$993$	0	0
$994$	9.57319e21	0.316708
$995$	8.79752e21	0.288860
$996$	0	0
$997$	−1.14133e22	−0.369145	−0.184572	−	0.982819i	$-0.559090\pi$
$997$	−1.14133e22	−0.369145	−0.184572	+	0.982819i	$0.559090\pi$
$998$	−1.05627e22	−0.339076
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.16.a.p.1.1	yes	4
3.2	odd	2		126.16.a.o.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.16.a.o.1.4	✓	4	3.2	odd	2
126.16.a.p.1.1	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

\(f(q)\)	\(=\)	\(q+128.000 q^{2} +16384.0 q^{4} -189588. q^{5} -823543. q^{7} +2.09715e6 q^{8} +O(q^{10})\)	\(q+128.000 q^{2} +16384.0 q^{4} -189588. q^{5} -823543. q^{7} +2.09715e6 q^{8} -2.42673e7 q^{10} -7.67208e7 q^{11} +4.06587e8 q^{13} -1.05414e8 q^{14} +2.68435e8 q^{16} -1.76163e9 q^{17} +5.22515e9 q^{19} -3.10622e9 q^{20} -9.82026e9 q^{22} +1.58764e10 q^{23} +5.42617e9 q^{25} +5.20431e10 q^{26} -1.34929e10 q^{28} -3.08391e10 q^{29} +2.29164e11 q^{31} +3.43597e10 q^{32} -2.25489e11 q^{34} +1.56134e11 q^{35} -3.81992e11 q^{37} +6.68819e11 q^{38} -3.97596e11 q^{40} -2.37380e12 q^{41} +3.28678e12 q^{43} -1.25699e12 q^{44} +2.03217e12 q^{46} -8.64905e11 q^{47} +6.78223e11 q^{49} +6.94550e11 q^{50} +6.66152e12 q^{52} -1.02882e12 q^{53} +1.45454e13 q^{55} -1.72709e12 q^{56} -3.94741e12 q^{58} +1.37466e13 q^{59} -1.48054e13 q^{61} +2.93330e13 q^{62} +4.39805e12 q^{64} -7.70842e13 q^{65} -5.49468e13 q^{67} -2.88626e13 q^{68} +1.99852e13 q^{70} -9.08156e13 q^{71} -1.74174e14 q^{73} -4.88950e13 q^{74} +8.56088e13 q^{76} +6.31829e13 q^{77} -1.63619e13 q^{79} -5.08922e13 q^{80} -3.03847e14 q^{82} +4.63192e14 q^{83} +3.33985e14 q^{85} +4.20708e14 q^{86} -1.60895e14 q^{88} -2.32519e14 q^{89} -3.34842e14 q^{91} +2.60118e14 q^{92} -1.10708e14 q^{94} -9.90627e14 q^{95} -9.47035e14 q^{97} +8.68126e13 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 512 q^{2} + 65536 q^{4} - 55216 q^{5} - 3294172 q^{7} + 8388608 q^{8}+O(q^{10}) \) 4 * q + 512 * q^2 + 65536 * q^4 - 55216 * q^5 - 3294172 * q^7 + 8388608 * q^8	\( 4 q + 512 q^{2} + 65536 q^{4} - 55216 q^{5} - 3294172 q^{7} + 8388608 q^{8} - 7067648 q^{10} - 42410192 q^{11} + 225416856 q^{13} - 421654016 q^{14} + 1073741824 q^{16} - 1067868368 q^{17} + 1490154848 q^{19} - 904658944 q^{20} - 5428504576 q^{22} + 17651202640 q^{23} - 24170778484 q^{25} + 28853357568 q^{26} - 53971714048 q^{28} + 48603833216 q^{29} - 66948471072 q^{31} + 137438953472 q^{32} - 136687151104 q^{34} + 45472750288 q^{35} - 165011511304 q^{37} + 190739820544 q^{38} - 115796344832 q^{40} - 759932074608 q^{41} + 939420947792 q^{43} - 694848585728 q^{44} + 2259353937920 q^{46} - 3606135631392 q^{47} + 2712892291396 q^{49} - 3093859645952 q^{50} + 3693229768704 q^{52} - 15571497448800 q^{53} - 760393960480 q^{55} - 6908379398144 q^{56} + 6221290651648 q^{58} - 36433798220512 q^{59} - 31107553629432 q^{61} - 8569404297216 q^{62} + 17592186044416 q^{64} - 74820511464672 q^{65} - 66339747458960 q^{67} - 17495955341312 q^{68} + 5820512036864 q^{70} - 35524074310928 q^{71} - 214521476058344 q^{73} - 21121473446912 q^{74} + 24414697029632 q^{76} + 34926616750256 q^{77} - 231821535874560 q^{79} - 14821932138496 q^{80} - 97271305549824 q^{82} + 359126111005312 q^{83} - 75573047434624 q^{85} + 120245881317376 q^{86} - 88940618973184 q^{88} + 188429028383568 q^{89} - 185640473840808 q^{91} + 289197304053760 q^{92} - 461585360818176 q^{94} + 346710605250112 q^{95} - 11\!\cdots\!76 q^{97}+ \cdots + 347250213298688 q^{98}+O(q^{100}) \) 4 * q + 512 * q^2 + 65536 * q^4 - 55216 * q^5 - 3294172 * q^7 + 8388608 * q^8 - 7067648 * q^10 - 42410192 * q^11 + 225416856 * q^13 - 421654016 * q^14 + 1073741824 * q^16 - 1067868368 * q^17 + 1490154848 * q^19 - 904658944 * q^20 - 5428504576 * q^22 + 17651202640 * q^23 - 24170778484 * q^25 + 28853357568 * q^26 - 53971714048 * q^28 + 48603833216 * q^29 - 66948471072 * q^31 + 137438953472 * q^32 - 136687151104 * q^34 + 45472750288 * q^35 - 165011511304 * q^37 + 190739820544 * q^38 - 115796344832 * q^40 - 759932074608 * q^41 + 939420947792 * q^43 - 694848585728 * q^44 + 2259353937920 * q^46 - 3606135631392 * q^47 + 2712892291396 * q^49 - 3093859645952 * q^50 + 3693229768704 * q^52 - 15571497448800 * q^53 - 760393960480 * q^55 - 6908379398144 * q^56 + 6221290651648 * q^58 - 36433798220512 * q^59 - 31107553629432 * q^61 - 8569404297216 * q^62 + 17592186044416 * q^64 - 74820511464672 * q^65 - 66339747458960 * q^67 - 17495955341312 * q^68 + 5820512036864 * q^70 - 35524074310928 * q^71 - 214521476058344 * q^73 - 21121473446912 * q^74 + 24414697029632 * q^76 + 34926616750256 * q^77 - 231821535874560 * q^79 - 14821932138496 * q^80 - 97271305549824 * q^82 + 359126111005312 * q^83 - 75573047434624 * q^85 + 120245881317376 * q^86 - 88940618973184 * q^88 + 188429028383568 * q^89 - 185640473840808 * q^91 + 289197304053760 * q^92 - 461585360818176 * q^94 + 346710605250112 * q^95 - 1107546032911976 * q^97 + 347250213298688 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more