LMFDB - Embedded newform 81.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,10,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.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:	$41.7179027293$
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} - x^{3} - 1314x^{2} + 10232x + 106624 $ x^4 - x^3 - 1314x^2 + 10232x + 106624
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-38.3405$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-30.3405 q^{2} +408.549 q^{4} +231.735 q^{5} -1699.35 q^{7} +3138.77 q^{8} -7030.96 q^{10} -36802.4 q^{11} -13021.3 q^{13} +51559.1 q^{14} -304409. q^{16} +359721. q^{17} -337522. q^{19} +94675.0 q^{20} +1.11661e6 q^{22} +1.14334e6 q^{23} -1.89942e6 q^{25} +395074. q^{26} -694266. q^{28} +6.10721e6 q^{29} +9.83838e6 q^{31} +7.62888e6 q^{32} -1.09141e7 q^{34} -393798. q^{35} -3.37388e6 q^{37} +1.02406e7 q^{38} +727362. q^{40} +1.22428e7 q^{41} -2.33934e7 q^{43} -1.50356e7 q^{44} -3.46894e7 q^{46} -2.24131e7 q^{47} -3.74658e7 q^{49} +5.76296e7 q^{50} -5.31985e6 q^{52} -6.19733e7 q^{53} -8.52841e6 q^{55} -5.33385e6 q^{56} -1.85296e8 q^{58} -2.47418e7 q^{59} +6.51623e7 q^{61} -2.98502e8 q^{62} -7.56071e7 q^{64} -3.01750e6 q^{65} -1.54834e8 q^{67} +1.46963e8 q^{68} +1.19480e7 q^{70} -3.00576e8 q^{71} +1.59289e7 q^{73} +1.02365e8 q^{74} -1.37894e8 q^{76} +6.25400e7 q^{77} -3.37962e8 q^{79} -7.05422e7 q^{80} -3.71453e8 q^{82} -1.95275e8 q^{83} +8.33599e7 q^{85} +7.09769e8 q^{86} -1.15514e8 q^{88} +3.80741e8 q^{89} +2.21277e7 q^{91} +4.67109e8 q^{92} +6.80027e8 q^{94} -7.82156e7 q^{95} -1.71747e9 q^{97} +1.13673e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) $ 4 * q + 33 * q^2 + 853 * q^4 + 570 * q^5 - 3238 * q^7 + 4791 * q^8 - 9723 * q^10 - 96690 * q^11 - 141118 * q^13 + 3036 * q^14 - 244463 * q^16 + 285156 * q^17 - 465166 * q^19 - 1041711 * q^20 - 2244480 * q^22 - 237318 * q^23 + 3417712 * q^25 - 6433737 * q^26 + 769292 * q^28 + 2806926 * q^29 - 8244448 * q^31 + 3896415 * q^32 + 1386189 * q^34 - 31668402 * q^35 - 8204014 * q^37 - 755556 * q^38 - 54180645 * q^40 + 28889292 * q^41 - 39586246 * q^43 - 89567616 * q^44 - 93870624 * q^46 - 17514588 * q^47 - 67913400 * q^49 + 233832372 * q^50 - 207295237 * q^52 - 155826504 * q^53 - 185126142 * q^55 + 133671300 * q^56 - 164356143 * q^58 + 341256108 * q^59 - 238298254 * q^61 - 729569796 * q^62 - 286454975 * q^64 + 379881528 * q^65 - 72601486 * q^67 + 762353217 * q^68 - 780989460 * q^70 - 667543266 * q^71 - 520588288 * q^73 + 183473031 * q^74 - 413336884 * q^76 + 721530228 * q^77 - 137324662 * q^79 - 1181367123 * q^80 - 275042586 * q^82 - 278546844 * q^83 - 1408511358 * q^85 + 420627720 * q^86 - 30700416 * q^88 + 237712932 * q^89 - 762950942 * q^91 - 1238743680 * q^92 - 507333516 * q^94 - 811663194 * q^95 + 700071356 * q^97 + 735501321 * 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 $

$2$	−30.3405	−1.34088	−0.670438	−	0.741966i	$-0.733894\pi$
$2$	−30.3405	−1.34088	−0.670438	+	0.741966i	$0.733894\pi$
$3$	0	0
$3$	0	0
$4$	408.549	0.797947
$5$	231.735	0.165816	0.0829080	−	0.996557i	$-0.473579\pi$
$5$	231.735	0.165816	0.0829080	+	0.996557i	$0.473579\pi$
$6$	0	0
$7$	−1699.35	−0.267510	−0.133755	−	0.991014i	$-0.542704\pi$
$7$	−1699.35	−0.267510	−0.133755	+	0.991014i	$0.542704\pi$
$8$	3138.77	0.270928
$9$	0	0
$10$	−7030.96	−0.222339
$11$	−36802.4	−0.757895	−0.378948	−	0.925418i	$-0.623714\pi$
$11$	−36802.4	−0.757895	−0.378948	+	0.925418i	$0.623714\pi$
$12$	0	0
$13$	−13021.3	−0.126447	−0.0632237	−	0.997999i	$-0.520138\pi$
$13$	−13021.3	−0.126447	−0.0632237	+	0.997999i	$0.520138\pi$
$14$	51559.1	0.358698
$15$	0	0
$16$	−304409.	−1.16123
$17$	359721.	1.04459	0.522294	−	0.852765i	$-0.325076\pi$
$17$	359721.	1.04459	0.522294	+	0.852765i	$0.325076\pi$
$18$	0	0
$19$	−337522.	−0.594170	−0.297085	−	0.954851i	$-0.596014\pi$
$19$	−337522.	−0.594170	−0.297085	+	0.954851i	$0.596014\pi$
$20$	94675.0	0.132312
$21$	0	0
$22$	1.11661e6	1.01624
$23$	1.14334e6	0.851920	0.425960	−	0.904742i	$-0.359936\pi$
$23$	1.14334e6	0.851920	0.425960	+	0.904742i	$0.359936\pi$
$24$	0	0
$25$	−1.89942e6	−0.972505
$26$	395074.	0.169550
$27$	0	0
$28$	−694266.	−0.213459
$29$	6.10721e6	1.60344	0.801718	−	0.597702i	$-0.203920\pi$
$29$	6.10721e6	1.60344	0.801718	+	0.597702i	$0.203920\pi$
$30$	0	0
$31$	9.83838e6	1.91336	0.956679	−	0.291146i	$-0.0940365\pi$
$31$	9.83838e6	1.91336	0.956679	+	0.291146i	$0.0940365\pi$
$32$	7.62888e6	1.28613
$33$	0	0
$34$	−1.09141e7	−1.40066
$35$	−393798.	−0.0443575
$36$	0	0
$37$	−3.37388e6	−0.295952	−0.147976	−	0.988991i	$-0.547276\pi$
$37$	−3.37388e6	−0.295952	−0.147976	+	0.988991i	$0.547276\pi$
$38$	1.02406e7	0.796708
$39$	0	0
$40$	727362.	0.0449242
$41$	1.22428e7	0.676633	0.338316	−	0.941032i	$-0.390143\pi$
$41$	1.22428e7	0.676633	0.338316	+	0.941032i	$0.390143\pi$
$42$	0	0
$43$	−2.33934e7	−1.04348	−0.521741	−	0.853104i	$-0.674717\pi$
$43$	−2.33934e7	−1.04348	−0.521741	+	0.853104i	$0.674717\pi$
$44$	−1.50356e7	−0.604760
$45$	0	0
$46$	−3.46894e7	−1.14232
$47$	−2.24131e7	−0.669981	−0.334991	−	0.942221i	$-0.608733\pi$
$47$	−2.24131e7	−0.669981	−0.334991	+	0.942221i	$0.608733\pi$
$48$	0	0
$49$	−3.74658e7	−0.928438
$50$	5.76296e7	1.30401
$51$	0	0
$52$	−5.31985e6	−0.100898
$53$	−6.19733e7	−1.07886	−0.539428	−	0.842032i	$-0.681359\pi$
$53$	−6.19733e7	−1.07886	−0.539428	+	0.842032i	$0.681359\pi$
$54$	0	0
$55$	−8.52841e6	−0.125671
$56$	−5.33385e6	−0.0724761
$57$	0	0
$58$	−1.85296e8	−2.15001
$59$	−2.47418e7	−0.265825	−0.132913	−	0.991128i	$-0.542433\pi$
$59$	−2.47418e7	−0.265825	−0.132913	+	0.991128i	$0.542433\pi$
$60$	0	0
$61$	6.51623e7	0.602576	0.301288	−	0.953533i	$-0.402583\pi$
$61$	6.51623e7	0.602576	0.301288	+	0.953533i	$0.402583\pi$
$62$	−2.98502e8	−2.56557
$63$	0	0
$64$	−7.56071e7	−0.563317
$65$	−3.01750e6	−0.0209670
$66$	0	0
$67$	−1.54834e8	−0.938704	−0.469352	−	0.883011i	$-0.655512\pi$
$67$	−1.54834e8	−0.938704	−0.469352	+	0.883011i	$0.655512\pi$
$68$	1.46963e8	0.833526
$69$	0	0
$70$	1.19480e7	0.0594779
$71$	−3.00576e8	−1.40376	−0.701879	−	0.712296i	$-0.747655\pi$
$71$	−3.00576e8	−1.40376	−0.701879	+	0.712296i	$0.747655\pi$
$72$	0	0
$73$	1.59289e7	0.0656497	0.0328249	−	0.999461i	$-0.489550\pi$
$73$	1.59289e7	0.0656497	0.0328249	+	0.999461i	$0.489550\pi$
$74$	1.02365e8	0.396835
$75$	0	0
$76$	−1.37894e8	−0.474116
$77$	6.25400e7	0.202745
$78$	0	0
$79$	−3.37962e8	−0.976217	−0.488109	−	0.872783i	$-0.662313\pi$
$79$	−3.37962e8	−0.976217	−0.488109	+	0.872783i	$0.662313\pi$
$80$	−7.05422e7	−0.192550
$81$	0	0
$82$	−3.71453e8	−0.907280
$83$	−1.95275e8	−0.451643	−0.225822	−	0.974169i	$-0.572507\pi$
$83$	−1.95275e8	−0.451643	−0.225822	+	0.974169i	$0.572507\pi$
$84$	0	0
$85$	8.33599e7	0.173210
$86$	7.09769e8	1.39918
$87$	0	0
$88$	−1.15514e8	−0.205335
$89$	3.80741e8	0.643243	0.321622	−	0.946868i	$-0.395772\pi$
$89$	3.80741e8	0.643243	0.321622	+	0.946868i	$0.395772\pi$
$90$	0	0
$91$	2.21277e7	0.0338260
$92$	4.67109e8	0.679787
$93$	0	0
$94$	6.80027e8	0.898361
$95$	−7.82156e7	−0.0985229
$96$	0	0
$97$	−1.71747e9	−1.96977	−0.984887	−	0.173197i	$-0.944590\pi$
$97$	−1.71747e9	−1.96977	−0.984887	+	0.173197i	$0.944590\pi$
$98$	1.13673e9	1.24492
$99$	0	0
$100$	−7.76007e8	−0.776007
$101$	4.02896e8	0.385254	0.192627	−	0.981272i	$-0.438299\pi$
$101$	4.02896e8	0.385254	0.192627	+	0.981272i	$0.438299\pi$
$102$	0	0
$103$	1.67242e9	1.46412	0.732061	−	0.681239i	$-0.238558\pi$
$103$	1.67242e9	1.46412	0.732061	+	0.681239i	$0.238558\pi$
$104$	−4.08709e7	−0.0342582
$105$	0	0
$106$	1.88030e9	1.44661
$107$	−3.79101e8	−0.279594	−0.139797	−	0.990180i	$-0.544645\pi$
$107$	−3.79101e8	−0.279594	−0.139797	+	0.990180i	$0.544645\pi$
$108$	0	0
$109$	2.43370e9	1.65138	0.825692	−	0.564122i	$-0.190785\pi$
$109$	2.43370e9	1.65138	0.825692	+	0.564122i	$0.190785\pi$
$110$	2.58757e8	0.168509
$111$	0	0
$112$	5.17296e8	0.310640
$113$	−1.65738e9	−0.956242	−0.478121	−	0.878294i	$-0.658682\pi$
$113$	−1.65738e9	−0.956242	−0.478121	+	0.878294i	$0.658682\pi$
$114$	0	0
$115$	2.64951e8	0.141262
$116$	2.49509e9	1.27946
$117$	0	0
$118$	7.50678e8	0.356439
$119$	−6.11290e8	−0.279438
$120$	0	0
$121$	−1.00353e9	−0.425594
$122$	−1.97706e9	−0.807979
$123$	0	0
$124$	4.01946e9	1.52676
$125$	−8.92770e8	−0.327073
$126$	0	0
$127$	7.89177e8	0.269189	0.134595	−	0.990901i	$-0.457027\pi$
$127$	7.89177e8	0.269189	0.134595	+	0.990901i	$0.457027\pi$
$128$	−1.61203e9	−0.530796
$129$	0	0
$130$	9.15525e7	0.0281142
$131$	−5.21040e9	−1.54579	−0.772895	−	0.634534i	$-0.781192\pi$
$131$	−5.21040e9	−1.54579	−0.772895	+	0.634534i	$0.781192\pi$
$132$	0	0
$133$	5.73566e8	0.158947
$134$	4.69774e9	1.25868
$135$	0	0
$136$	1.12908e9	0.283008
$137$	−5.79176e9	−1.40465	−0.702325	−	0.711856i	$-0.747855\pi$
$137$	−5.79176e9	−1.40465	−0.702325	+	0.711856i	$0.747855\pi$
$138$	0	0
$139$	−5.50037e9	−1.24976	−0.624879	−	0.780722i	$-0.714852\pi$
$139$	−5.50037e9	−1.24976	−0.624879	+	0.780722i	$0.714852\pi$
$140$	−1.60886e8	−0.0353949
$141$	0	0
$142$	9.11965e9	1.88226
$143$	4.79216e8	0.0958340
$144$	0	0
$145$	1.41525e9	0.265875
$146$	−4.83292e8	−0.0880281
$147$	0	0
$148$	−1.37839e9	−0.236154
$149$	−6.32833e9	−1.05184	−0.525921	−	0.850533i	$-0.676279\pi$
$149$	−6.32833e9	−1.05184	−0.525921	+	0.850533i	$0.676279\pi$
$150$	0	0
$151$	−8.47040e9	−1.32589	−0.662946	−	0.748668i	$-0.730694\pi$
$151$	−8.47040e9	−1.32589	−0.662946	+	0.748668i	$0.730694\pi$
$152$	−1.05940e9	−0.160977
$153$	0	0
$154$	−1.89750e9	−0.271856
$155$	2.27990e9	0.317265
$156$	0	0
$157$	7.68862e9	1.00995	0.504975	−	0.863134i	$-0.331502\pi$
$157$	7.68862e9	1.00995	0.504975	+	0.863134i	$0.331502\pi$
$158$	1.02540e10	1.30899
$159$	0	0
$160$	1.76788e9	0.213262
$161$	−1.94292e9	−0.227897
$162$	0	0
$163$	5.49272e9	0.609458	0.304729	−	0.952439i	$-0.401434\pi$
$163$	5.49272e9	0.609458	0.304729	+	0.952439i	$0.401434\pi$
$164$	5.00178e9	0.539917
$165$	0	0
$166$	5.92475e9	0.605597
$167$	4.50814e9	0.448512	0.224256	−	0.974530i	$-0.428005\pi$
$167$	4.50814e9	0.448512	0.224256	+	0.974530i	$0.428005\pi$
$168$	0	0
$169$	−1.04349e10	−0.984011
$170$	−2.52918e9	−0.232252
$171$	0	0
$172$	−9.55734e9	−0.832644
$173$	1.74009e10	1.47695	0.738473	−	0.674283i	$-0.235547\pi$
$173$	1.74009e10	1.47695	0.738473	+	0.674283i	$0.235547\pi$
$174$	0	0
$175$	3.22778e9	0.260155
$176$	1.12030e10	0.880089
$177$	0	0
$178$	−1.15519e10	−0.862509
$179$	−8.60652e9	−0.626598	−0.313299	−	0.949655i	$-0.601434\pi$
$179$	−8.60652e9	−0.626598	−0.313299	+	0.949655i	$0.601434\pi$
$180$	0	0
$181$	−1.49085e10	−1.03248	−0.516238	−	0.856445i	$-0.672668\pi$
$181$	−1.49085e10	−1.03248	−0.516238	+	0.856445i	$0.672668\pi$
$182$	−6.71368e8	−0.0453565
$183$	0	0
$184$	3.58867e9	0.230809
$185$	−7.81845e8	−0.0490736
$186$	0	0
$187$	−1.32386e10	−0.791689
$188$	−9.15686e9	−0.534609
$189$	0	0
$190$	2.37310e9	0.132107
$191$	−2.93575e9	−0.159613	−0.0798067	−	0.996810i	$-0.525430\pi$
$191$	−2.93575e9	−0.159613	−0.0798067	+	0.996810i	$0.525430\pi$
$192$	0	0
$193$	−9.42902e9	−0.489169	−0.244584	−	0.969628i	$-0.578651\pi$
$193$	−9.42902e9	−0.489169	−0.244584	+	0.969628i	$0.578651\pi$
$194$	5.21090e10	2.64122
$195$	0	0
$196$	−1.53066e10	−0.740844
$197$	1.63450e10	0.773191	0.386596	−	0.922249i	$-0.373651\pi$
$197$	1.63450e10	0.773191	0.386596	+	0.922249i	$0.373651\pi$
$198$	0	0
$199$	−1.92599e10	−0.870594	−0.435297	−	0.900287i	$-0.643357\pi$
$199$	−1.92599e10	−0.870594	−0.435297	+	0.900287i	$0.643357\pi$
$200$	−5.96185e9	−0.263479
$201$	0	0
$202$	−1.22241e10	−0.516578
$203$	−1.03783e10	−0.428936
$204$	0	0
$205$	2.83708e9	0.112197
$206$	−5.07421e10	−1.96321
$207$	0	0
$208$	3.96381e9	0.146834
$209$	1.24216e10	0.450319
$210$	0	0
$211$	−4.35260e10	−1.51174	−0.755871	−	0.654721i	$-0.772786\pi$
$211$	−4.35260e10	−1.51174	−0.755871	+	0.654721i	$0.772786\pi$
$212$	−2.53191e10	−0.860869
$213$	0	0
$214$	1.15021e10	0.374901
$215$	−5.42107e9	−0.173026
$216$	0	0
$217$	−1.67188e10	−0.511843
$218$	−7.38398e10	−2.21430
$219$	0	0
$220$	−3.48427e9	−0.100279
$221$	−4.68404e9	−0.132086
$222$	0	0
$223$	4.31472e10	1.16837	0.584185	−	0.811620i	$-0.301414\pi$
$223$	4.31472e10	1.16837	0.584185	+	0.811620i	$0.301414\pi$
$224$	−1.29641e10	−0.344054
$225$	0	0
$226$	5.02857e10	1.28220
$227$	6.55392e10	1.63827	0.819134	−	0.573602i	$-0.194454\pi$
$227$	6.55392e10	1.63827	0.819134	+	0.573602i	$0.194454\pi$
$228$	0	0
$229$	−3.00286e10	−0.721564	−0.360782	−	0.932650i	$-0.617490\pi$
$229$	−3.00286e10	−0.721564	−0.360782	+	0.932650i	$0.617490\pi$
$230$	−8.03876e9	−0.189415
$231$	0	0
$232$	1.91691e10	0.434416
$233$	−4.03023e10	−0.895835	−0.447917	−	0.894075i	$-0.647834\pi$
$233$	−4.03023e10	−0.895835	−0.447917	+	0.894075i	$0.647834\pi$
$234$	0	0
$235$	−5.19391e9	−0.111094
$236$	−1.01082e10	−0.212114
$237$	0	0
$238$	1.85469e10	0.374692
$239$	−6.11330e10	−1.21195	−0.605976	−	0.795483i	$-0.707217\pi$
$239$	−6.11330e10	−1.21195	−0.605976	+	0.795483i	$0.707217\pi$
$240$	0	0
$241$	1.79546e10	0.342846	0.171423	−	0.985197i	$-0.445163\pi$
$241$	1.79546e10	0.342846	0.171423	+	0.985197i	$0.445163\pi$
$242$	3.04476e10	0.570669
$243$	0	0
$244$	2.66220e10	0.480824
$245$	−8.68214e9	−0.153950
$246$	0	0
$247$	4.39498e9	0.0751313
$248$	3.08804e10	0.518382
$249$	0	0
$250$	2.70871e10	0.438564
$251$	1.13782e11	1.80942	0.904711	−	0.426026i	$-0.140087\pi$
$251$	1.13782e11	1.80942	0.904711	+	0.426026i	$0.140087\pi$
$252$	0	0
$253$	−4.20775e10	−0.645666
$254$	−2.39441e10	−0.360949
$255$	0	0
$256$	8.76206e10	1.27505
$257$	−5.87971e10	−0.840731	−0.420366	−	0.907355i	$-0.638098\pi$
$257$	−5.87971e10	−0.840731	−0.420366	+	0.907355i	$0.638098\pi$
$258$	0	0
$259$	5.73338e9	0.0791702
$260$	−1.23279e9	−0.0167306
$261$	0	0
$262$	1.58086e11	2.07271
$263$	2.10338e10	0.271092	0.135546	−	0.990771i	$-0.456721\pi$
$263$	2.10338e10	0.271092	0.135546	+	0.990771i	$0.456721\pi$
$264$	0	0
$265$	−1.43614e10	−0.178892
$266$	−1.74023e10	−0.213128
$267$	0	0
$268$	−6.32571e10	−0.749036
$269$	1.46308e11	1.70366	0.851832	−	0.523816i	$-0.175492\pi$
$269$	1.46308e11	1.70366	0.851832	+	0.523816i	$0.175492\pi$
$270$	0	0
$271$	−1.33223e11	−1.50044	−0.750220	−	0.661188i	$-0.770052\pi$
$271$	−1.33223e11	−1.50044	−0.750220	+	0.661188i	$0.770052\pi$
$272$	−1.09502e11	−1.21301
$273$	0	0
$274$	1.75725e11	1.88346
$275$	6.99034e10	0.737057
$276$	0	0
$277$	−7.02879e10	−0.717335	−0.358667	−	0.933465i	$-0.616769\pi$
$277$	−7.02879e10	−0.717335	−0.358667	+	0.933465i	$0.616769\pi$
$278$	1.66884e11	1.67577
$279$	0	0
$280$	−1.23604e9	−0.0120177
$281$	2.00193e11	1.91545	0.957724	−	0.287689i	$-0.0928871\pi$
$281$	2.00193e11	1.91545	0.957724	+	0.287689i	$0.0928871\pi$
$282$	0	0
$283$	8.46940e10	0.784899	0.392450	−	0.919774i	$-0.371628\pi$
$283$	8.46940e10	0.784899	0.392450	+	0.919774i	$0.371628\pi$
$284$	−1.22800e11	−1.12012
$285$	0	0
$286$	−1.45397e10	−0.128501
$287$	−2.08047e10	−0.181006
$288$	0	0
$289$	1.08111e10	0.0911653
$290$	−4.29396e10	−0.356506
$291$	0	0
$292$	6.50773e9	0.0523850
$293$	−3.10322e10	−0.245985	−0.122992	−	0.992408i	$-0.539249\pi$
$293$	−3.10322e10	−0.245985	−0.122992	+	0.992408i	$0.539249\pi$
$294$	0	0
$295$	−5.73353e9	−0.0440781
$296$	−1.05898e10	−0.0801817
$297$	0	0
$298$	1.92005e11	1.41039
$299$	−1.48878e10	−0.107723
$300$	0	0
$301$	3.97535e10	0.279142
$302$	2.56997e11	1.77785
$303$	0	0
$304$	1.02745e11	0.689966
$305$	1.51004e10	0.0999168
$306$	0	0
$307$	−2.77619e10	−0.178372	−0.0891860	−	0.996015i	$-0.528427\pi$
$307$	−2.77619e10	−0.178372	−0.0891860	+	0.996015i	$0.528427\pi$
$308$	2.55507e10	0.161780
$309$	0	0
$310$	−6.91733e10	−0.425413
$311$	−2.56530e11	−1.55495	−0.777474	−	0.628915i	$-0.783499\pi$
$311$	−2.56530e11	−1.55495	−0.777474	+	0.628915i	$0.783499\pi$
$312$	0	0
$313$	1.04794e11	0.617146	0.308573	−	0.951201i	$-0.400149\pi$
$313$	1.04794e11	0.617146	0.308573	+	0.951201i	$0.400149\pi$
$314$	−2.33277e11	−1.35422
$315$	0	0
$316$	−1.38074e11	−0.778970
$317$	9.90769e10	0.551069	0.275534	−	0.961291i	$-0.411145\pi$
$317$	9.90769e10	0.551069	0.275534	+	0.961291i	$0.411145\pi$
$318$	0	0
$319$	−2.24760e11	−1.21524
$320$	−1.75208e10	−0.0934070
$321$	0	0
$322$	5.89494e10	0.305582
$323$	−1.21414e11	−0.620663
$324$	0	0
$325$	2.47330e10	0.122971
$326$	−1.66652e11	−0.817207
$327$	0	0
$328$	3.84273e10	0.183319
$329$	3.80877e10	0.179227
$330$	0	0
$331$	−5.92352e10	−0.271240	−0.135620	−	0.990761i	$-0.543303\pi$
$331$	−5.92352e10	−0.271240	−0.135620	+	0.990761i	$0.543303\pi$
$332$	−7.97794e10	−0.360387
$333$	0	0
$334$	−1.36780e11	−0.601398
$335$	−3.58804e10	−0.155652
$336$	0	0
$337$	−3.32234e11	−1.40317	−0.701583	−	0.712588i	$-0.747523\pi$
$337$	−3.32234e11	−1.40317	−0.701583	+	0.712588i	$0.747523\pi$
$338$	3.16602e11	1.31944
$339$	0	0
$340$	3.40566e10	0.138212
$341$	−3.62076e11	−1.45012
$342$	0	0
$343$	1.32242e11	0.515877
$344$	−7.34264e10	−0.282709
$345$	0	0
$346$	−5.27953e11	−1.98040
$347$	−1.84410e11	−0.682813	−0.341407	−	0.939916i	$-0.610903\pi$
$347$	−1.84410e11	−0.682813	−0.341407	+	0.939916i	$0.610903\pi$
$348$	0	0
$349$	−2.15169e11	−0.776365	−0.388183	−	0.921582i	$-0.626897\pi$
$349$	−2.15169e11	−0.776365	−0.388183	+	0.921582i	$0.626897\pi$
$350$	−9.79325e10	−0.348836
$351$	0	0
$352$	−2.80761e11	−0.974755
$353$	−4.55334e11	−1.56079	−0.780395	−	0.625287i	$-0.784982\pi$
$353$	−4.55334e11	−1.56079	−0.780395	+	0.625287i	$0.784982\pi$
$354$	0	0
$355$	−6.96540e10	−0.232766
$356$	1.55551e11	0.513274
$357$	0	0
$358$	2.61127e11	0.840190
$359$	7.69706e10	0.244568	0.122284	−	0.992495i	$-0.460978\pi$
$359$	7.69706e10	0.244568	0.122284	+	0.992495i	$0.460978\pi$
$360$	0	0
$361$	−2.08767e11	−0.646962
$362$	4.52332e11	1.38442
$363$	0	0
$364$	9.04026e9	0.0269914
$365$	3.69128e9	0.0108858
$366$	0	0
$367$	−5.19222e11	−1.49402	−0.747009	−	0.664813i	$-0.768511\pi$
$367$	−5.19222e11	−1.49402	−0.747009	+	0.664813i	$0.768511\pi$
$368$	−3.48042e11	−0.989273
$369$	0	0
$370$	2.37216e10	0.0658016
$371$	1.05314e11	0.288605
$372$	0	0
$373$	3.36003e11	0.898779	0.449389	−	0.893336i	$-0.351642\pi$
$373$	3.36003e11	0.898779	0.449389	+	0.893336i	$0.351642\pi$
$374$	4.01666e11	1.06156
$375$	0	0
$376$	−7.03496e10	−0.181517
$377$	−7.95239e10	−0.202750
$378$	0	0
$379$	3.52078e11	0.876522	0.438261	−	0.898848i	$-0.355595\pi$
$379$	3.52078e11	0.876522	0.438261	+	0.898848i	$0.355595\pi$
$380$	−3.19549e10	−0.0786160
$381$	0	0
$382$	8.90723e10	0.214022
$383$	5.91054e11	1.40357	0.701783	−	0.712391i	$-0.252388\pi$
$383$	5.91054e11	1.40357	0.701783	+	0.712391i	$0.252388\pi$
$384$	0	0
$385$	1.44927e10	0.0336183
$386$	2.86082e11	0.655914
$387$	0	0
$388$	−7.01670e11	−1.57178
$389$	3.45521e11	0.765070	0.382535	−	0.923941i	$-0.375051\pi$
$389$	3.45521e11	0.765070	0.382535	+	0.923941i	$0.375051\pi$
$390$	0	0
$391$	4.11282e11	0.889906
$392$	−1.17597e11	−0.251540
$393$	0	0
$394$	−4.95916e11	−1.03675
$395$	−7.83177e10	−0.161872
$396$	0	0
$397$	2.14905e11	0.434199	0.217099	−	0.976150i	$-0.430340\pi$
$397$	2.14905e11	0.434199	0.217099	+	0.976150i	$0.430340\pi$
$398$	5.84356e11	1.16736
$399$	0	0
$400$	5.78201e11	1.12930
$401$	−4.24741e11	−0.820304	−0.410152	−	0.912017i	$-0.634524\pi$
$401$	−4.24741e11	−0.820304	−0.410152	+	0.912017i	$0.634524\pi$
$402$	0	0
$403$	−1.28109e11	−0.241939
$404$	1.64603e11	0.307412
$405$	0	0
$406$	3.14882e11	0.575149
$407$	1.24167e11	0.224301
$408$	0	0
$409$	3.70072e11	0.653930	0.326965	−	0.945036i	$-0.393974\pi$
$409$	3.70072e11	0.653930	0.326965	+	0.945036i	$0.393974\pi$
$410$	−8.60786e10	−0.150442
$411$	0	0
$412$	6.83265e11	1.16829
$413$	4.20448e10	0.0711110
$414$	0	0
$415$	−4.52521e10	−0.0748897
$416$	−9.93382e10	−0.162628
$417$	0	0
$418$	−3.76879e11	−0.603821
$419$	2.04429e11	0.324026	0.162013	−	0.986789i	$-0.448201\pi$
$419$	2.04429e11	0.324026	0.162013	+	0.986789i	$0.448201\pi$
$420$	0	0
$421$	−6.16297e11	−0.956139	−0.478069	−	0.878322i	$-0.658663\pi$
$421$	−6.16297e11	−0.956139	−0.478069	+	0.878322i	$0.658663\pi$
$422$	1.32060e12	2.02706
$423$	0	0
$424$	−1.94520e11	−0.292292
$425$	−6.83262e11	−1.01587
$426$	0	0
$427$	−1.10733e11	−0.161195
$428$	−1.54881e11	−0.223101
$429$	0	0
$430$	1.64478e11	0.232007
$431$	2.91473e11	0.406865	0.203433	−	0.979089i	$-0.434790\pi$
$431$	2.91473e11	0.406865	0.203433	+	0.979089i	$0.434790\pi$
$432$	0	0
$433$	1.01533e12	1.38808	0.694038	−	0.719939i	$-0.255830\pi$
$433$	1.01533e12	1.38808	0.694038	+	0.719939i	$0.255830\pi$
$434$	5.07258e11	0.686317
$435$	0	0
$436$	9.94285e11	1.31772
$437$	−3.85901e11	−0.506185
$438$	0	0
$439$	5.92099e11	0.760859	0.380429	−	0.924810i	$-0.375776\pi$
$439$	5.92099e11	0.760859	0.380429	+	0.924810i	$0.375776\pi$
$440$	−2.67687e10	−0.0340479
$441$	0	0
$442$	1.42116e11	0.177110
$443$	−1.41822e12	−1.74955	−0.874775	−	0.484528i	$-0.838991\pi$
$443$	−1.41822e12	−1.74955	−0.874775	+	0.484528i	$0.838991\pi$
$444$	0	0
$445$	8.82311e10	0.106660
$446$	−1.30911e12	−1.56664
$447$	0	0
$448$	1.28483e11	0.150693
$449$	1.36934e11	0.159002	0.0795011	−	0.996835i	$-0.474667\pi$
$449$	1.36934e11	0.159002	0.0795011	+	0.996835i	$0.474667\pi$
$450$	0	0
$451$	−4.50564e11	−0.512817
$452$	−6.77119e11	−0.763031
$453$	0	0
$454$	−1.98850e12	−2.19671
$455$	5.12777e9	0.00560889
$456$	0	0
$457$	1.14631e12	1.22936	0.614680	−	0.788776i	$-0.289285\pi$
$457$	1.14631e12	1.22936	0.614680	+	0.788776i	$0.289285\pi$
$458$	9.11083e11	0.967528
$459$	0	0
$460$	1.08245e11	0.112720
$461$	1.02462e11	0.105659	0.0528297	−	0.998604i	$-0.483176\pi$
$461$	1.02462e11	0.105659	0.0528297	+	0.998604i	$0.483176\pi$
$462$	0	0
$463$	−3.95784e10	−0.0400262	−0.0200131	−	0.999800i	$-0.506371\pi$
$463$	−3.95784e10	−0.0400262	−0.0200131	+	0.999800i	$0.506371\pi$
$464$	−1.85909e12	−1.86195
$465$	0	0
$466$	1.22279e12	1.20120
$467$	−5.16020e11	−0.502042	−0.251021	−	0.967982i	$-0.580766\pi$
$467$	−5.16020e11	−0.502042	−0.251021	+	0.967982i	$0.580766\pi$
$468$	0	0
$469$	2.63116e11	0.251113
$470$	1.57586e11	0.148963
$471$	0	0
$472$	−7.76586e10	−0.0720196
$473$	8.60934e11	0.790851
$474$	0	0
$475$	6.41097e11	0.577833
$476$	−2.49742e11	−0.222977
$477$	0	0
$478$	1.85481e12	1.62508
$479$	−2.82352e11	−0.245065	−0.122533	−	0.992464i	$-0.539102\pi$
$479$	−2.82352e11	−0.245065	−0.122533	+	0.992464i	$0.539102\pi$
$480$	0	0
$481$	4.39323e10	0.0374224
$482$	−5.44753e11	−0.459714
$483$	0	0
$484$	−4.09991e11	−0.339602
$485$	−3.97998e11	−0.326620
$486$	0	0
$487$	9.63445e11	0.776151	0.388076	−	0.921627i	$-0.373140\pi$
$487$	9.63445e11	0.776151	0.388076	+	0.921627i	$0.373140\pi$
$488$	2.04529e11	0.163255
$489$	0	0
$490$	2.63421e11	0.206428
$491$	−7.78305e11	−0.604342	−0.302171	−	0.953254i	$-0.597711\pi$
$491$	−7.78305e11	−0.604342	−0.302171	+	0.953254i	$0.597711\pi$
$492$	0	0
$493$	2.19689e12	1.67493
$494$	−1.33346e11	−0.100742
$495$	0	0
$496$	−2.99489e12	−2.22184
$497$	5.10783e11	0.375520
$498$	0	0
$499$	4.30792e10	0.0311039	0.0155520	−	0.999879i	$-0.495049\pi$
$499$	4.30792e10	0.0311039	0.0155520	+	0.999879i	$0.495049\pi$
$500$	−3.64740e11	−0.260987
$501$	0	0
$502$	−3.45219e12	−2.42621
$503$	9.51004e11	0.662409	0.331205	−	0.943559i	$-0.392545\pi$
$503$	9.51004e11	0.662409	0.331205	+	0.943559i	$0.392545\pi$
$504$	0	0
$505$	9.33651e10	0.0638813
$506$	1.27666e12	0.865758
$507$	0	0
$508$	3.22417e11	0.214799
$509$	2.49605e12	1.64825	0.824127	−	0.566405i	$-0.191666\pi$
$509$	2.49605e12	1.64825	0.824127	+	0.566405i	$0.191666\pi$
$510$	0	0
$511$	−2.70687e10	−0.0175620
$512$	−1.83310e12	−1.17888
$513$	0	0
$514$	1.78394e12	1.12732
$515$	3.87558e11	0.242775
$516$	0	0
$517$	8.24858e11	0.507776
$518$	−1.73954e11	−0.106157
$519$	0	0
$520$	−9.47122e9	−0.00568055
$521$	−4.54898e11	−0.270486	−0.135243	−	0.990812i	$-0.543181\pi$
$521$	−4.54898e11	−0.270486	−0.135243	+	0.990812i	$0.543181\pi$
$522$	0	0
$523$	−2.02470e12	−1.18332	−0.591660	−	0.806188i	$-0.701527\pi$
$523$	−2.02470e12	−1.18332	−0.591660	+	0.806188i	$0.701527\pi$
$524$	−2.12870e12	−1.23346
$525$	0	0
$526$	−6.38178e11	−0.363501
$527$	3.53907e12	1.99867
$528$	0	0
$529$	−4.93935e11	−0.274233
$530$	4.35732e11	0.239871
$531$	0	0
$532$	2.34330e11	0.126831
$533$	−1.59417e11	−0.0855585
$534$	0	0
$535$	−8.78510e10	−0.0463612
$536$	−4.85987e11	−0.254321
$537$	0	0
$538$	−4.43907e12	−2.28440
$539$	1.37883e12	0.703659
$540$	0	0
$541$	−2.30600e12	−1.15737	−0.578685	−	0.815551i	$-0.696434\pi$
$541$	−2.30600e12	−1.15737	−0.578685	+	0.815551i	$0.696434\pi$
$542$	4.04207e12	2.01190
$543$	0	0
$544$	2.74427e12	1.34348
$545$	5.63973e11	0.273826
$546$	0	0
$547$	−8.29913e11	−0.396360	−0.198180	−	0.980166i	$-0.563503\pi$
$547$	−8.29913e11	−0.396360	−0.198180	+	0.980166i	$0.563503\pi$
$548$	−2.36622e12	−1.12084
$549$	0	0
$550$	−2.12091e12	−0.988302
$551$	−2.06132e12	−0.952713
$552$	0	0
$553$	5.74315e11	0.261148
$554$	2.13257e12	0.961856
$555$	0	0
$556$	−2.24717e12	−0.997240
$557$	2.17270e12	0.956424	0.478212	−	0.878244i	$-0.341285\pi$
$557$	2.17270e12	0.956424	0.478212	+	0.878244i	$0.341285\pi$
$558$	0	0
$559$	3.04613e11	0.131946
$560$	1.19876e11	0.0515092
$561$	0	0
$562$	−6.07396e12	−2.56838
$563$	2.40236e12	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$563$	2.40236e12	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$564$	0	0
$565$	−3.84072e11	−0.158560
$566$	−2.56966e12	−1.05245
$567$	0	0
$568$	−9.43439e11	−0.380317
$569$	3.14119e12	1.25629	0.628143	−	0.778098i	$-0.283815\pi$
$569$	3.14119e12	1.25629	0.628143	+	0.778098i	$0.283815\pi$
$570$	0	0
$571$	3.18984e12	1.25576	0.627880	−	0.778310i	$-0.283923\pi$
$571$	3.18984e12	1.25576	0.627880	+	0.778310i	$0.283923\pi$
$572$	1.95783e11	0.0764704
$573$	0	0
$574$	6.31227e11	0.242707
$575$	−2.17168e12	−0.828496
$576$	0	0
$577$	1.32352e12	0.497097	0.248548	−	0.968620i	$-0.420047\pi$
$577$	1.32352e12	0.497097	0.248548	+	0.968620i	$0.420047\pi$
$578$	−3.28015e11	−0.122241
$579$	0	0
$580$	5.78200e11	0.212154
$581$	3.31840e11	0.120819
$582$	0	0
$583$	2.28077e12	0.817660
$584$	4.99971e10	0.0177864
$585$	0	0
$586$	9.41533e11	0.329835
$587$	−3.94222e11	−0.137047	−0.0685234	−	0.997650i	$-0.521829\pi$
$587$	−3.94222e11	−0.137047	−0.0685234	+	0.997650i	$0.521829\pi$
$588$	0	0
$589$	−3.32067e12	−1.13686
$590$	1.73958e11	0.0591032
$591$	0	0
$592$	1.02704e12	0.343668
$593$	−4.59838e12	−1.52707	−0.763535	−	0.645767i	$-0.776538\pi$
$593$	−4.59838e12	−1.52707	−0.763535	+	0.645767i	$0.776538\pi$
$594$	0	0
$595$	−1.41657e11	−0.0463353
$596$	−2.58543e12	−0.839314
$597$	0	0
$598$	4.51703e11	0.144443
$599$	−3.67774e12	−1.16724	−0.583620	−	0.812027i	$-0.698364\pi$
$599$	−3.67774e12	−1.16724	−0.583620	+	0.812027i	$0.698364\pi$
$600$	0	0
$601$	3.42104e12	1.06960	0.534802	−	0.844977i	$-0.320386\pi$
$601$	3.42104e12	1.06960	0.534802	+	0.844977i	$0.320386\pi$
$602$	−1.20614e12	−0.374295
$603$	0	0
$604$	−3.46057e12	−1.05799
$605$	−2.32553e11	−0.0705704
$606$	0	0
$607$	−4.81992e12	−1.44109	−0.720544	−	0.693409i	$-0.756108\pi$
$607$	−4.81992e12	−1.44109	−0.720544	+	0.693409i	$0.756108\pi$
$608$	−2.57491e12	−0.764182
$609$	0	0
$610$	−4.58154e11	−0.133976
$611$	2.91849e11	0.0847174
$612$	0	0
$613$	3.32772e12	0.951862	0.475931	−	0.879483i	$-0.342111\pi$
$613$	3.32772e12	0.951862	0.475931	+	0.879483i	$0.342111\pi$
$614$	8.42312e11	0.239175
$615$	0	0
$616$	1.96299e11	0.0549293
$617$	9.32994e11	0.259176	0.129588	−	0.991568i	$-0.458634\pi$
$617$	9.32994e11	0.259176	0.129588	+	0.991568i	$0.458634\pi$
$618$	0	0
$619$	6.92588e11	0.189613	0.0948063	−	0.995496i	$-0.469777\pi$
$619$	6.92588e11	0.189613	0.0948063	+	0.995496i	$0.469777\pi$
$620$	9.31449e11	0.253161
$621$	0	0
$622$	7.78325e12	2.08499
$623$	−6.47011e11	−0.172074
$624$	0	0
$625$	3.50293e12	0.918271
$626$	−3.17951e12	−0.827516
$627$	0	0
$628$	3.14118e12	0.805886
$629$	−1.21365e12	−0.309148
$630$	0	0
$631$	−4.77434e12	−1.19890	−0.599448	−	0.800414i	$-0.704613\pi$
$631$	−4.77434e12	−1.19890	−0.599448	+	0.800414i	$0.704613\pi$
$632$	−1.06079e12	−0.264485
$633$	0	0
$634$	−3.00605e12	−0.738915
$635$	1.82880e11	0.0446359
$636$	0	0
$637$	4.87855e11	0.117399
$638$	6.81934e12	1.62948
$639$	0	0
$640$	−3.73563e11	−0.0880144
$641$	−5.00673e12	−1.17137	−0.585684	−	0.810540i	$-0.699174\pi$
$641$	−5.00673e12	−1.17137	−0.585684	+	0.810540i	$0.699174\pi$
$642$	0	0
$643$	−5.87210e12	−1.35470	−0.677352	−	0.735659i	$-0.736872\pi$
$643$	−5.87210e12	−1.35470	−0.677352	+	0.735659i	$0.736872\pi$
$644$	−7.93779e11	−0.181850
$645$	0	0
$646$	3.68375e12	0.832232
$647$	5.86641e12	1.31614	0.658071	−	0.752956i	$-0.271372\pi$
$647$	5.86641e12	1.31614	0.658071	+	0.752956i	$0.271372\pi$
$648$	0	0
$649$	9.10556e11	0.201468
$650$	−7.50413e11	−0.164889
$651$	0	0
$652$	2.24405e12	0.486315
$653$	−7.14975e12	−1.53880	−0.769399	−	0.638769i	$-0.779444\pi$
$653$	−7.14975e12	−1.53880	−0.769399	+	0.638769i	$0.779444\pi$
$654$	0	0
$655$	−1.20743e12	−0.256317
$656$	−3.72681e12	−0.785725
$657$	0	0
$658$	−1.15560e12	−0.240321
$659$	−2.66321e12	−0.550074	−0.275037	−	0.961434i	$-0.588690\pi$
$659$	−2.66321e12	−0.550074	−0.275037	+	0.961434i	$0.588690\pi$
$660$	0	0
$661$	7.12379e12	1.45146	0.725729	−	0.687981i	$-0.241503\pi$
$661$	7.12379e12	1.45146	0.725729	+	0.687981i	$0.241503\pi$
$662$	1.79723e12	0.363699
$663$	0	0
$664$	−6.12923e11	−0.122363
$665$	1.32915e11	0.0263559
$666$	0	0
$667$	6.98259e12	1.36600
$668$	1.84180e12	0.357888
$669$	0	0
$670$	1.08863e12	0.208710
$671$	−2.39813e12	−0.456690
$672$	0	0
$673$	−6.42882e12	−1.20799	−0.603995	−	0.796988i	$-0.706425\pi$
$673$	−6.42882e12	−1.20799	−0.603995	+	0.796988i	$0.706425\pi$
$674$	1.00801e13	1.88147
$675$	0	0
$676$	−4.26318e12	−0.785188
$677$	7.94372e12	1.45337	0.726683	−	0.686973i	$-0.241061\pi$
$677$	7.94372e12	1.45337	0.726683	+	0.686973i	$0.241061\pi$
$678$	0	0
$679$	2.91858e12	0.526935
$680$	2.61647e11	0.0469273
$681$	0	0
$682$	1.09856e13	1.94444
$683$	7.51902e12	1.32211	0.661056	−	0.750336i	$-0.270108\pi$
$683$	7.51902e12	1.32211	0.661056	+	0.750336i	$0.270108\pi$
$684$	0	0
$685$	−1.34215e12	−0.232914
$686$	−4.01230e12	−0.691727
$687$	0	0
$688$	7.12116e12	1.21172
$689$	8.06975e11	0.136419
$690$	0	0
$691$	−8.50242e12	−1.41870	−0.709351	−	0.704855i	$-0.751012\pi$
$691$	−8.50242e12	−1.41870	−0.709351	+	0.704855i	$0.751012\pi$
$692$	7.10912e12	1.17852
$693$	0	0
$694$	5.59510e12	0.915568
$695$	−1.27463e12	−0.207230
$696$	0	0
$697$	4.40398e12	0.706803
$698$	6.52836e12	1.04101
$699$	0	0
$700$	1.31870e12	0.207590
$701$	−8.88313e12	−1.38942	−0.694712	−	0.719288i	$-0.744468\pi$
$701$	−8.88313e12	−1.38942	−0.694712	+	0.719288i	$0.744468\pi$
$702$	0	0
$703$	1.13876e12	0.175846
$704$	2.78253e12	0.426935
$705$	0	0
$706$	1.38151e13	2.09282
$707$	−6.84660e11	−0.103059
$708$	0	0
$709$	−1.22025e12	−0.181359	−0.0906797	−	0.995880i	$-0.528904\pi$
$709$	−1.22025e12	−0.181359	−0.0906797	+	0.995880i	$0.528904\pi$
$710$	2.11334e12	0.312110
$711$	0	0
$712$	1.19506e12	0.174273
$713$	1.12486e13	1.63003
$714$	0	0
$715$	1.11051e11	0.0158908
$716$	−3.51618e12	−0.499992
$717$	0	0
$718$	−2.33533e12	−0.327935
$719$	−2.93448e12	−0.409497	−0.204749	−	0.978815i	$-0.565638\pi$
$719$	−2.93448e12	−0.409497	−0.204749	+	0.978815i	$0.565638\pi$
$720$	0	0
$721$	−2.84202e12	−0.391668
$722$	6.33410e12	0.867496
$723$	0	0
$724$	−6.09085e12	−0.823861
$725$	−1.16002e13	−1.55935
$726$	0	0
$727$	−8.52049e12	−1.13125	−0.565627	−	0.824661i	$-0.691366\pi$
$727$	−8.52049e12	−1.13125	−0.565627	+	0.824661i	$0.691366\pi$
$728$	6.94538e10	0.00916442
$729$	0	0
$730$	−1.11996e11	−0.0145965
$731$	−8.41509e12	−1.09001
$732$	0	0
$733$	−7.03067e12	−0.899557	−0.449778	−	0.893140i	$-0.648497\pi$
$733$	−7.03067e12	−0.899557	−0.449778	+	0.893140i	$0.648497\pi$
$734$	1.57535e13	2.00329
$735$	0	0
$736$	8.72238e12	1.09568
$737$	5.69825e12	0.711439
$738$	0	0
$739$	1.29746e12	0.160027	0.0800136	−	0.996794i	$-0.474504\pi$
$739$	1.29746e12	0.160027	0.0800136	+	0.996794i	$0.474504\pi$
$740$	−3.19422e11	−0.0391581
$741$	0	0
$742$	−3.19529e12	−0.386983
$743$	−3.91322e12	−0.471069	−0.235535	−	0.971866i	$-0.575684\pi$
$743$	−3.91322e12	−0.471069	−0.235535	+	0.971866i	$0.575684\pi$
$744$	0	0
$745$	−1.46649e12	−0.174412
$746$	−1.01945e13	−1.20515
$747$	0	0
$748$	−5.40861e12	−0.631726
$749$	6.44224e11	0.0747943
$750$	0	0
$751$	3.30065e12	0.378634	0.189317	−	0.981916i	$-0.439373\pi$
$751$	3.30065e12	0.378634	0.189317	+	0.981916i	$0.439373\pi$
$752$	6.82276e12	0.778001
$753$	0	0
$754$	2.41280e12	0.271863
$755$	−1.96289e12	−0.219854
$756$	0	0
$757$	1.36242e13	1.50792	0.753960	−	0.656920i	$-0.228141\pi$
$757$	1.36242e13	1.50792	0.753960	+	0.656920i	$0.228141\pi$
$758$	−1.06822e13	−1.17531
$759$	0	0
$760$	−2.45500e11	−0.0266926
$761$	2.23239e12	0.241290	0.120645	−	0.992696i	$-0.461504\pi$
$761$	2.23239e12	0.241290	0.120645	+	0.992696i	$0.461504\pi$
$762$	0	0
$763$	−4.13570e12	−0.441762
$764$	−1.19940e12	−0.127363
$765$	0	0
$766$	−1.79329e13	−1.88201
$767$	3.22170e11	0.0336129
$768$	0	0
$769$	−1.16863e13	−1.20505	−0.602527	−	0.798098i	$-0.705840\pi$
$769$	−1.16863e13	−1.20505	−0.602527	+	0.798098i	$0.705840\pi$
$770$	−4.39717e11	−0.0450780
$771$	0	0
$772$	−3.85221e12	−0.390330
$773$	−9.53346e12	−0.960379	−0.480190	−	0.877165i	$-0.659432\pi$
$773$	−9.53346e12	−0.960379	−0.480190	+	0.877165i	$0.659432\pi$
$774$	0	0
$775$	−1.86873e13	−1.86075
$776$	−5.39074e12	−0.533667
$777$	0	0
$778$	−1.04833e13	−1.02586
$779$	−4.13221e12	−0.402035
$780$	0	0
$781$	1.10619e13	1.06390
$782$	−1.24785e13	−1.19325
$783$	0	0
$784$	1.14049e13	1.07813
$785$	1.78172e12	0.167466
$786$	0	0
$787$	−1.86618e13	−1.73407	−0.867034	−	0.498248i	$-0.833977\pi$
$787$	−1.86618e13	−1.73407	−0.867034	+	0.498248i	$0.833977\pi$
$788$	6.67773e12	0.616965
$789$	0	0
$790$	2.37620e12	0.217051
$791$	2.81645e12	0.255805
$792$	0	0
$793$	−8.48499e11	−0.0761942
$794$	−6.52032e12	−0.582206
$795$	0	0
$796$	−7.86861e12	−0.694687
$797$	1.78187e12	0.156428	0.0782139	−	0.996937i	$-0.475078\pi$
$797$	1.78187e12	0.156428	0.0782139	+	0.996937i	$0.475078\pi$
$798$	0	0
$799$	−8.06247e12	−0.699855
$800$	−1.44905e13	−1.25077
$801$	0	0
$802$	1.28869e13	1.09993
$803$	−5.86222e11	−0.0497556
$804$	0	0
$805$	−4.50243e11	−0.0377890
$806$	3.88689e12	0.324410
$807$	0	0
$808$	1.26460e12	0.104376
$809$	−1.10802e12	−0.0909451	−0.0454726	−	0.998966i	$-0.514479\pi$
$809$	−1.10802e12	−0.0909451	−0.0454726	+	0.998966i	$0.514479\pi$
$810$	0	0
$811$	−1.53607e13	−1.24685	−0.623427	−	0.781881i	$-0.714260\pi$
$811$	−1.53607e13	−1.24685	−0.623427	+	0.781881i	$0.714260\pi$
$812$	−4.24002e12	−0.342268
$813$	0	0
$814$	−3.76729e12	−0.300759
$815$	1.27286e12	0.101058
$816$	0	0
$817$	7.89578e12	0.620006
$818$	−1.12282e13	−0.876839
$819$	0	0
$820$	1.15909e12	0.0895269
$821$	1.39731e13	1.07337	0.536685	−	0.843783i	$-0.319676\pi$
$821$	1.39731e13	1.07337	0.536685	+	0.843783i	$0.319676\pi$
$822$	0	0
$823$	−8.54103e12	−0.648950	−0.324475	−	0.945894i	$-0.605188\pi$
$823$	−8.54103e12	−0.648950	−0.324475	+	0.945894i	$0.605188\pi$
$824$	5.24933e12	0.396672
$825$	0	0
$826$	−1.27566e12	−0.0953510
$827$	−1.41887e13	−1.05480	−0.527398	−	0.849618i	$-0.676832\pi$
$827$	−1.41887e13	−1.05480	−0.527398	+	0.849618i	$0.676832\pi$
$828$	0	0
$829$	2.47597e12	0.182075	0.0910373	−	0.995847i	$-0.470982\pi$
$829$	2.47597e12	0.182075	0.0910373	+	0.995847i	$0.470982\pi$
$830$	1.37297e12	0.100418
$831$	0	0
$832$	9.84505e11	0.0712300
$833$	−1.34772e13	−0.969836
$834$	0	0
$835$	1.04469e12	0.0743704
$836$	5.07484e12	0.359330
$837$	0	0
$838$	−6.20250e12	−0.434479
$839$	−1.76017e12	−0.122638	−0.0613192	−	0.998118i	$-0.519531\pi$
$839$	−1.76017e12	−0.122638	−0.0613192	+	0.998118i	$0.519531\pi$
$840$	0	0
$841$	2.27908e13	1.57101
$842$	1.86988e13	1.28206
$843$	0	0
$844$	−1.77825e13	−1.20629
$845$	−2.41814e12	−0.163165
$846$	0	0
$847$	1.70534e12	0.113851
$848$	1.88652e13	1.25280
$849$	0	0
$850$	2.07305e13	1.36215
$851$	−3.85747e12	−0.252127
$852$	0	0
$853$	−3.65983e12	−0.236696	−0.118348	−	0.992972i	$-0.537760\pi$
$853$	−3.65983e12	−0.236696	−0.118348	+	0.992972i	$0.537760\pi$
$854$	3.35971e12	0.216143
$855$	0	0
$856$	−1.18991e12	−0.0757499
$857$	2.33292e13	1.47736	0.738681	−	0.674055i	$-0.235449\pi$
$857$	2.33292e13	1.47736	0.738681	+	0.674055i	$0.235449\pi$
$858$	0	0
$859$	−1.05637e12	−0.0661982	−0.0330991	−	0.999452i	$-0.510538\pi$
$859$	−1.05637e12	−0.0661982	−0.0330991	+	0.999452i	$0.510538\pi$
$860$	−2.21477e12	−0.138066
$861$	0	0
$862$	−8.84345e12	−0.545556
$863$	3.03140e13	1.86035	0.930176	−	0.367113i	$-0.119654\pi$
$863$	3.03140e13	1.86035	0.930176	+	0.367113i	$0.119654\pi$
$864$	0	0
$865$	4.03240e12	0.244901
$866$	−3.08058e13	−1.86124
$867$	0	0
$868$	−6.83045e12	−0.408423
$869$	1.24378e13	0.739871
$870$	0	0
$871$	2.01614e12	0.118697
$872$	7.63882e12	0.447406
$873$	0	0
$874$	1.17084e13	0.678731
$875$	1.51713e12	0.0874954
$876$	0	0
$877$	3.79827e12	0.216814	0.108407	−	0.994107i	$-0.465425\pi$
$877$	3.79827e12	0.216814	0.108407	+	0.994107i	$0.465425\pi$
$878$	−1.79646e13	−1.02022
$879$	0	0
$880$	2.59612e12	0.145933
$881$	−1.93222e13	−1.08060	−0.540299	−	0.841473i	$-0.681689\pi$
$881$	−1.93222e13	−1.08060	−0.540299	+	0.841473i	$0.681689\pi$
$882$	0	0
$883$	−1.55128e13	−0.858751	−0.429376	−	0.903126i	$-0.641266\pi$
$883$	−1.55128e13	−0.858751	−0.429376	+	0.903126i	$0.641266\pi$
$884$	−1.91366e12	−0.105397
$885$	0	0
$886$	4.30295e13	2.34593
$887$	2.28162e13	1.23762	0.618808	−	0.785542i	$-0.287616\pi$
$887$	2.28162e13	1.23762	0.618808	+	0.785542i	$0.287616\pi$
$888$	0	0
$889$	−1.34108e12	−0.0720109
$890$	−2.67698e12	−0.143018
$891$	0	0
$892$	1.76277e13	0.932298
$893$	7.56493e12	0.398082
$894$	0	0
$895$	−1.99443e12	−0.103900
$896$	2.73939e12	0.141993
$897$	0	0
$898$	−4.15466e12	−0.213202
$899$	6.00850e13	3.06795
$900$	0	0
$901$	−2.22931e13	−1.12696
$902$	1.36704e13	0.687624
$903$	0	0
$904$	−5.20212e12	−0.259073
$905$	−3.45482e12	−0.171201
$906$	0	0
$907$	−7.83380e11	−0.0384361	−0.0192181	−	0.999815i	$-0.506118\pi$
$907$	−7.83380e11	−0.0384361	−0.0192181	+	0.999815i	$0.506118\pi$
$908$	2.67760e13	1.30725
$909$	0	0
$910$	−1.55579e11	−0.00752083
$911$	−3.53016e13	−1.69809	−0.849047	−	0.528317i	$-0.822823\pi$
$911$	−3.53016e13	−1.69809	−0.849047	+	0.528317i	$0.822823\pi$
$912$	0	0
$913$	7.18660e12	0.342298
$914$	−3.47797e13	−1.64842
$915$	0	0
$916$	−1.22681e13	−0.575770
$917$	8.85427e12	0.413515
$918$	0	0
$919$	−1.06193e13	−0.491106	−0.245553	−	0.969383i	$-0.578970\pi$
$919$	−1.06193e13	−0.491106	−0.245553	+	0.969383i	$0.578970\pi$
$920$	8.31619e11	0.0382718
$921$	0	0
$922$	−3.10875e12	−0.141676
$923$	3.91390e12	0.177502
$924$	0	0
$925$	6.40842e12	0.287815
$926$	1.20083e12	0.0536701
$927$	0	0
$928$	4.65912e13	2.06223
$929$	2.30491e13	1.01527	0.507637	−	0.861571i	$-0.330519\pi$
$929$	2.30491e13	1.01527	0.507637	+	0.861571i	$0.330519\pi$
$930$	0	0
$931$	1.26455e13	0.551650
$932$	−1.64654e13	−0.714828
$933$	0	0
$934$	1.56563e13	0.673176
$935$	−3.06784e12	−0.131275
$936$	0	0
$937$	2.83663e13	1.20220	0.601098	−	0.799176i	$-0.294730\pi$
$937$	2.83663e13	1.20220	0.601098	+	0.799176i	$0.294730\pi$
$938$	−7.98308e12	−0.336711
$939$	0	0
$940$	−2.12197e12	−0.0886468
$941$	−1.81784e13	−0.755790	−0.377895	−	0.925848i	$-0.623352\pi$
$941$	−1.81784e13	−0.755790	−0.377895	+	0.925848i	$0.623352\pi$
$942$	0	0
$943$	1.39976e13	0.576437
$944$	7.53161e12	0.308684
$945$	0	0
$946$	−2.61212e13	−1.06043
$947$	−2.75665e12	−0.111380	−0.0556899	−	0.998448i	$-0.517736\pi$
$947$	−2.75665e12	−0.111380	−0.0556899	+	0.998448i	$0.517736\pi$
$948$	0	0
$949$	−2.07415e11	−0.00830124
$950$	−1.94512e13	−0.774802
$951$	0	0
$952$	−1.91870e12	−0.0757077
$953$	−1.37564e13	−0.540240	−0.270120	−	0.962827i	$-0.587063\pi$
$953$	−1.37564e13	−0.540240	−0.270120	+	0.962827i	$0.587063\pi$
$954$	0	0
$955$	−6.80316e11	−0.0264665
$956$	−2.49758e13	−0.967073
$957$	0	0
$958$	8.56672e12	0.328602
$959$	9.84221e12	0.375758
$960$	0	0
$961$	7.03542e13	2.66094
$962$	−1.33293e12	−0.0501788
$963$	0	0
$964$	7.33534e12	0.273573
$965$	−2.18503e12	−0.0811120
$966$	0	0
$967$	−3.31021e13	−1.21741	−0.608704	−	0.793397i	$-0.708310\pi$
$967$	−3.31021e13	−1.21741	−0.608704	+	0.793397i	$0.708310\pi$
$968$	−3.14984e12	−0.115306
$969$	0	0
$970$	1.20755e13	0.437957
$971$	−1.64424e13	−0.593579	−0.296790	−	0.954943i	$-0.595916\pi$
$971$	−1.64424e13	−0.593579	−0.296790	+	0.954943i	$0.595916\pi$
$972$	0	0
$973$	9.34704e12	0.334323
$974$	−2.92314e13	−1.04072
$975$	0	0
$976$	−1.98360e13	−0.699728
$977$	−1.10850e13	−0.389235	−0.194617	−	0.980879i	$-0.562347\pi$
$977$	−1.10850e13	−0.389235	−0.194617	+	0.980879i	$0.562347\pi$
$978$	0	0
$979$	−1.40122e13	−0.487511
$980$	−3.54708e12	−0.122844
$981$	0	0
$982$	2.36142e13	0.810348
$983$	−1.04408e13	−0.356652	−0.178326	−	0.983971i	$-0.557068\pi$
$983$	−1.04408e13	−0.356652	−0.178326	+	0.983971i	$0.557068\pi$
$984$	0	0
$985$	3.78771e12	0.128207
$986$	−6.66548e13	−2.24587
$987$	0	0
$988$	1.79556e12	0.0599508
$989$	−2.67465e13	−0.888964
$990$	0	0
$991$	8.23938e12	0.271371	0.135685	−	0.990752i	$-0.456676\pi$
$991$	8.23938e12	0.271371	0.135685	+	0.990752i	$0.456676\pi$
$992$	7.50559e13	2.46083
$993$	0	0
$994$	−1.54974e13	−0.503525
$995$	−4.46320e12	−0.144358
$996$	0	0
$997$	4.09580e13	1.31284	0.656419	−	0.754397i	$-0.272070\pi$
$997$	4.09580e13	1.31284	0.656419	+	0.754397i	$0.272070\pi$
$998$	−1.30705e12	−0.0417065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.b.1.1	yes	4
3.2	odd	2		81.10.a.a.1.4	✓	4
9.2	odd	6		81.10.c.k.28.1		8
9.4	even	3		81.10.c.i.55.4		8
9.5	odd	6		81.10.c.k.55.1		8
9.7	even	3		81.10.c.i.28.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.10.a.a.1.4	✓	4	3.2	odd	2
81.10.a.b.1.1	yes	4	1.1	even	1	trivial
81.10.c.i.28.4		8	9.7	even	3
81.10.c.i.55.4		8	9.4	even	3
81.10.c.k.28.1		8	9.2	odd	6
81.10.c.k.55.1		8	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-30.3405 q^{2} +408.549 q^{4} +231.735 q^{5} -1699.35 q^{7} +3138.77 q^{8} -7030.96 q^{10} -36802.4 q^{11} -13021.3 q^{13} +51559.1 q^{14} -304409. q^{16} +359721. q^{17} -337522. q^{19} +94675.0 q^{20} +1.11661e6 q^{22} +1.14334e6 q^{23} -1.89942e6 q^{25} +395074. q^{26} -694266. q^{28} +6.10721e6 q^{29} +9.83838e6 q^{31} +7.62888e6 q^{32} -1.09141e7 q^{34} -393798. q^{35} -3.37388e6 q^{37} +1.02406e7 q^{38} +727362. q^{40} +1.22428e7 q^{41} -2.33934e7 q^{43} -1.50356e7 q^{44} -3.46894e7 q^{46} -2.24131e7 q^{47} -3.74658e7 q^{49} +5.76296e7 q^{50} -5.31985e6 q^{52} -6.19733e7 q^{53} -8.52841e6 q^{55} -5.33385e6 q^{56} -1.85296e8 q^{58} -2.47418e7 q^{59} +6.51623e7 q^{61} -2.98502e8 q^{62} -7.56071e7 q^{64} -3.01750e6 q^{65} -1.54834e8 q^{67} +1.46963e8 q^{68} +1.19480e7 q^{70} -3.00576e8 q^{71} +1.59289e7 q^{73} +1.02365e8 q^{74} -1.37894e8 q^{76} +6.25400e7 q^{77} -3.37962e8 q^{79} -7.05422e7 q^{80} -3.71453e8 q^{82} -1.95275e8 q^{83} +8.33599e7 q^{85} +7.09769e8 q^{86} -1.15514e8 q^{88} +3.80741e8 q^{89} +2.21277e7 q^{91} +4.67109e8 q^{92} +6.80027e8 q^{94} -7.82156e7 q^{95} -1.71747e9 q^{97} +1.13673e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) \) 4 * q + 33 * q^2 + 853 * q^4 + 570 * q^5 - 3238 * q^7 + 4791 * q^8 - 9723 * q^10 - 96690 * q^11 - 141118 * q^13 + 3036 * q^14 - 244463 * q^16 + 285156 * q^17 - 465166 * q^19 - 1041711 * q^20 - 2244480 * q^22 - 237318 * q^23 + 3417712 * q^25 - 6433737 * q^26 + 769292 * q^28 + 2806926 * q^29 - 8244448 * q^31 + 3896415 * q^32 + 1386189 * q^34 - 31668402 * q^35 - 8204014 * q^37 - 755556 * q^38 - 54180645 * q^40 + 28889292 * q^41 - 39586246 * q^43 - 89567616 * q^44 - 93870624 * q^46 - 17514588 * q^47 - 67913400 * q^49 + 233832372 * q^50 - 207295237 * q^52 - 155826504 * q^53 - 185126142 * q^55 + 133671300 * q^56 - 164356143 * q^58 + 341256108 * q^59 - 238298254 * q^61 - 729569796 * q^62 - 286454975 * q^64 + 379881528 * q^65 - 72601486 * q^67 + 762353217 * q^68 - 780989460 * q^70 - 667543266 * q^71 - 520588288 * q^73 + 183473031 * q^74 - 413336884 * q^76 + 721530228 * q^77 - 137324662 * q^79 - 1181367123 * q^80 - 275042586 * q^82 - 278546844 * q^83 - 1408511358 * q^85 + 420627720 * q^86 - 30700416 * q^88 + 237712932 * q^89 - 762950942 * q^91 - 1238743680 * q^92 - 507333516 * q^94 - 811663194 * q^95 + 700071356 * q^97 + 735501321 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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