LMFDB - Embedded newform 81.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,8,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, 8, names="a")

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

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
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:	$25.3031870642$
Analytic rank:	$0$
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} - 342x^{2} - 352x + 2512 $ x^4 - x^3 - 342x^2 - 352x + 2512
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			$19.3080$ 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)

$f(q)$	$=$	\(q-15.3080 q^{2} +106.336 q^{4} -53.3856 q^{5} +243.109 q^{7} +331.630 q^{8} +817.229 q^{10} -5815.00 q^{11} +12738.2 q^{13} -3721.52 q^{14} -18687.6 q^{16} -26051.9 q^{17} -5144.15 q^{19} -5676.82 q^{20} +89016.2 q^{22} -25734.4 q^{23} -75275.0 q^{25} -194997. q^{26} +25851.3 q^{28} -28600.4 q^{29} +125660. q^{31} +243623. q^{32} +398803. q^{34} -12978.5 q^{35} +413062. q^{37} +78746.9 q^{38} -17704.2 q^{40} +691526. q^{41} +596215. q^{43} -618345. q^{44} +393943. q^{46} +1.01509e6 q^{47} -764441. q^{49} +1.15231e6 q^{50} +1.35454e6 q^{52} -413919. q^{53} +310437. q^{55} +80622.0 q^{56} +437817. q^{58} -27439.1 q^{59} -2.09485e6 q^{61} -1.92361e6 q^{62} -1.33737e6 q^{64} -680037. q^{65} +3.88197e6 q^{67} -2.77026e6 q^{68} +198675. q^{70} +3.26085e6 q^{71} +3.25581e6 q^{73} -6.32317e6 q^{74} -547010. q^{76} -1.41368e6 q^{77} -7.88994e6 q^{79} +997650. q^{80} -1.05859e7 q^{82} +4.21391e6 q^{83} +1.39079e6 q^{85} -9.12688e6 q^{86} -1.92843e6 q^{88} +3.81191e6 q^{89} +3.09677e6 q^{91} -2.73650e6 q^{92} -1.55391e7 q^{94} +274624. q^{95} +4.75971e6 q^{97} +1.17021e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 15 q^{2} + 229 q^{4} + 192 q^{5} + 800 q^{7} + 2505 q^{8} + 5469 q^{10} - 5016 q^{11} - 2200 q^{13} + 19452 q^{14} + 40849 q^{16} - 19620 q^{17} + 11240 q^{19} + 96951 q^{20} - 41280 q^{22} + 154560 q^{23}+ \cdots + 15771015 q^{98}+O(q^{100}) $ 4 * q + 15 * q^2 + 229 * q^4 + 192 * q^5 + 800 * q^7 + 2505 * q^8 + 5469 * q^10 - 5016 * q^11 - 2200 * q^13 + 19452 * q^14 + 40849 * q^16 - 19620 * q^17 + 11240 * q^19 + 96951 * q^20 - 41280 * q^22 + 154560 * q^23 + 164656 * q^25 - 397095 * q^26 + 363860 * q^28 + 118680 * q^29 + 450464 * q^31 + 952065 * q^32 + 664101 * q^34 - 907608 * q^35 + 711800 * q^37 - 202740 * q^38 + 1962723 * q^40 + 1027776 * q^41 + 1259000 * q^43 - 3842544 * q^44 + 1971888 * q^46 + 561840 * q^47 + 1370940 * q^49 + 3258420 * q^50 - 303205 * q^52 - 3858840 * q^53 - 425712 * q^55 + 1932180 * q^56 + 1352145 * q^58 + 341472 * q^59 + 2271896 * q^61 + 1571340 * q^62 + 1961473 * q^64 - 517764 * q^65 - 1649560 * q^67 + 1494135 * q^68 + 413508 * q^70 + 6584688 * q^71 + 358100 * q^73 - 5278983 * q^74 - 13960972 * q^76 - 15209160 * q^77 + 1436336 * q^79 + 23326203 * q^80 - 19970490 * q^82 + 1394160 * q^83 - 4525776 * q^85 - 6164088 * q^86 - 36138720 * q^88 + 14935932 * q^89 - 9244808 * q^91 - 479760 * q^92 - 19330980 * q^94 - 38927136 * q^95 + 4456880 * q^97 + 15771015 * 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 $

$2$	−15.3080	−1.35305	−0.676526	−	0.736418i	$-0.736516\pi$
$2$	−15.3080	−1.35305	−0.676526	+	0.736418i	$0.736516\pi$
$3$	0	0
$3$	0	0
$4$	106.336	0.830752
$5$	−53.3856	−0.190998	−0.0954990	−	0.995430i	$-0.530445\pi$
$5$	−53.3856	−0.190998	−0.0954990	+	0.995430i	$0.530445\pi$
$6$	0	0
$7$	243.109	0.267890	0.133945	−	0.990989i	$-0.457235\pi$
$7$	243.109	0.267890	0.133945	+	0.990989i	$0.457235\pi$
$8$	331.630	0.229002
$9$	0	0
$10$	817.229	0.258430
$11$	−5815.00	−1.31727	−0.658636	−	0.752462i	$-0.728866\pi$
$11$	−5815.00	−1.31727	−0.658636	+	0.752462i	$0.728866\pi$
$12$	0	0
$13$	12738.2	1.60808	0.804039	−	0.594576i	$-0.202680\pi$
$13$	12738.2	1.60808	0.804039	+	0.594576i	$0.202680\pi$
$14$	−3721.52	−0.362470
$15$	0	0
$16$	−18687.6	−1.14060
$17$	−26051.9	−1.28608	−0.643040	−	0.765833i	$-0.722327\pi$
$17$	−26051.9	−1.28608	−0.643040	+	0.765833i	$0.722327\pi$
$18$	0	0
$19$	−5144.15	−0.172059	−0.0860293	−	0.996293i	$-0.527418\pi$
$19$	−5144.15	−0.172059	−0.0860293	+	0.996293i	$0.527418\pi$
$20$	−5676.82	−0.158672
$21$	0	0
$22$	89016.2	1.78234
$23$	−25734.4	−0.441028	−0.220514	−	0.975384i	$-0.570773\pi$
$23$	−25734.4	−0.441028	−0.220514	+	0.975384i	$0.570773\pi$
$24$	0	0
$25$	−75275.0	−0.963520
$26$	−194997.	−2.17582
$27$	0	0
$28$	25851.3	0.222550
$29$	−28600.4	−0.217761	−0.108880	−	0.994055i	$-0.534727\pi$
$29$	−28600.4	−0.217761	−0.108880	+	0.994055i	$0.534727\pi$
$30$	0	0
$31$	125660.	0.757586	0.378793	−	0.925481i	$-0.376339\pi$
$31$	125660.	0.757586	0.378793	+	0.925481i	$0.376339\pi$
$32$	243623.	1.31429
$33$	0	0
$34$	398803.	1.74013
$35$	−12978.5	−0.0511665
$36$	0	0
$37$	413062.	1.34063	0.670315	−	0.742076i	$-0.266159\pi$
$37$	413062.	1.34063	0.670315	+	0.742076i	$0.266159\pi$
$38$	78746.9	0.232804
$39$	0	0
$40$	−17704.2	−0.0437389
$41$	691526.	1.56698	0.783492	−	0.621401i	$-0.213436\pi$
$41$	691526.	1.56698	0.783492	+	0.621401i	$0.213436\pi$
$42$	0	0
$43$	596215.	1.14357	0.571785	−	0.820403i	$-0.306251\pi$
$43$	596215.	1.14357	0.571785	+	0.820403i	$0.306251\pi$
$44$	−618345.	−1.09433
$45$	0	0
$46$	393943.	0.596734
$47$	1.01509e6	1.42614	0.713072	−	0.701091i	$-0.247303\pi$
$47$	1.01509e6	1.42614	0.713072	+	0.701091i	$0.247303\pi$
$48$	0	0
$49$	−764441.	−0.928235
$50$	1.15231e6	1.30369
$51$	0	0
$52$	1.35454e6	1.33591
$53$	−413919.	−0.381900	−0.190950	−	0.981600i	$-0.561157\pi$
$53$	−413919.	−0.381900	−0.190950	+	0.981600i	$0.561157\pi$
$54$	0	0
$55$	310437.	0.251596
$56$	80622.0	0.0613473
$57$	0	0
$58$	437817.	0.294642
$59$	−27439.1	−0.0173936	−0.00869678	−	0.999962i	$-0.502768\pi$
$59$	−27439.1	−0.0173936	−0.00869678	+	0.999962i	$0.502768\pi$
$60$	0	0
$61$	−2.09485e6	−1.18167	−0.590837	−	0.806791i	$-0.701202\pi$
$61$	−2.09485e6	−1.18167	−0.590837	+	0.806791i	$0.701202\pi$
$62$	−1.92361e6	−1.02505
$63$	0	0
$64$	−1.33737e6	−0.637707
$65$	−680037.	−0.307140
$66$	0	0
$67$	3.88197e6	1.57685	0.788425	−	0.615131i	$-0.210897\pi$
$67$	3.88197e6	1.57685	0.788425	+	0.615131i	$0.210897\pi$
$68$	−2.77026e6	−1.06841
$69$	0	0
$70$	198675.	0.0692310
$71$	3.26085e6	1.08125	0.540625	−	0.841264i	$-0.318188\pi$
$71$	3.26085e6	1.08125	0.540625	+	0.841264i	$0.318188\pi$
$72$	0	0
$73$	3.25581e6	0.979556	0.489778	−	0.871847i	$-0.337078\pi$
$73$	3.25581e6	0.979556	0.489778	+	0.871847i	$0.337078\pi$
$74$	−6.32317e6	−1.81394
$75$	0	0
$76$	−547010.	−0.142938
$77$	−1.41368e6	−0.352884
$78$	0	0
$79$	−7.88994e6	−1.80044	−0.900220	−	0.435435i	$-0.856595\pi$
$79$	−7.88994e6	−1.80044	−0.900220	+	0.435435i	$0.856595\pi$
$80$	997650.	0.217853
$81$	0	0
$82$	−1.05859e7	−2.12021
$83$	4.21391e6	0.808931	0.404466	−	0.914553i	$-0.367458\pi$
$83$	4.21391e6	0.808931	0.404466	+	0.914553i	$0.367458\pi$
$84$	0	0
$85$	1.39079e6	0.245639
$86$	−9.12688e6	−1.54731
$87$	0	0
$88$	−1.92843e6	−0.301657
$89$	3.81191e6	0.573162	0.286581	−	0.958056i	$-0.407481\pi$
$89$	3.81191e6	0.573162	0.286581	+	0.958056i	$0.407481\pi$
$90$	0	0
$91$	3.09677e6	0.430789
$92$	−2.73650e6	−0.366385
$93$	0	0
$94$	−1.55391e7	−1.92965
$95$	274624.	0.0328628
$96$	0	0
$97$	4.75971e6	0.529517	0.264758	−	0.964315i	$-0.414708\pi$
$97$	4.75971e6	0.529517	0.264758	+	0.964315i	$0.414708\pi$
$98$	1.17021e7	1.25595
$99$	0	0
$100$	−8.00446e6	−0.800446
$101$	1.33780e7	1.29201	0.646006	−	0.763332i	$-0.276438\pi$
$101$	1.33780e7	1.29201	0.646006	+	0.763332i	$0.276438\pi$
$102$	0	0
$103$	−1.28643e7	−1.15999	−0.579995	−	0.814620i	$-0.696946\pi$
$103$	−1.28643e7	−1.15999	−0.579995	+	0.814620i	$0.696946\pi$
$104$	4.22438e6	0.368253
$105$	0	0
$106$	6.33629e6	0.516731
$107$	1.25617e7	0.991303	0.495652	−	0.868521i	$-0.334929\pi$
$107$	1.25617e7	0.991303	0.495652	+	0.868521i	$0.334929\pi$
$108$	0	0
$109$	8.06992e6	0.596866	0.298433	−	0.954431i	$-0.403536\pi$
$109$	8.06992e6	0.596866	0.298433	+	0.954431i	$0.403536\pi$
$110$	−4.75218e6	−0.340423
$111$	0	0
$112$	−4.54313e6	−0.305557
$113$	−7.03422e6	−0.458608	−0.229304	−	0.973355i	$-0.573645\pi$
$113$	−7.03422e6	−0.458608	−0.229304	+	0.973355i	$0.573645\pi$
$114$	0	0
$115$	1.37384e6	0.0842354
$116$	−3.04126e6	−0.180905
$117$	0	0
$118$	420039.	0.0235344
$119$	−6.33343e6	−0.344528
$120$	0	0
$121$	1.43270e7	0.735203
$122$	3.20680e7	1.59887
$123$	0	0
$124$	1.33622e7	0.629366
$125$	8.18934e6	0.375028
$126$	0	0
$127$	1.58283e7	0.685680	0.342840	−	0.939394i	$-0.388611\pi$
$127$	1.58283e7	0.685680	0.342840	+	0.939394i	$0.388611\pi$
$128$	−1.07112e7	−0.451444
$129$	0	0
$130$	1.04100e7	0.415576
$131$	−1.44103e7	−0.560047	−0.280024	−	0.959993i	$-0.590342\pi$
$131$	−1.44103e7	−0.560047	−0.280024	+	0.959993i	$0.590342\pi$
$132$	0	0
$133$	−1.25059e6	−0.0460928
$134$	−5.94254e7	−2.13356
$135$	0	0
$136$	−8.63958e6	−0.294514
$137$	4.15192e7	1.37952	0.689759	−	0.724039i	$-0.257716\pi$
$137$	4.15192e7	1.37952	0.689759	+	0.724039i	$0.257716\pi$
$138$	0	0
$139$	3.92075e7	1.23828	0.619138	−	0.785282i	$-0.287482\pi$
$139$	3.92075e7	1.23828	0.619138	+	0.785282i	$0.287482\pi$
$140$	−1.38008e6	−0.0425067
$141$	0	0
$142$	−4.99172e7	−1.46299
$143$	−7.40728e7	−2.11828
$144$	0	0
$145$	1.52685e6	0.0415919
$146$	−4.98401e7	−1.32539
$147$	0	0
$148$	4.39235e7	1.11373
$149$	−1.17542e7	−0.291099	−0.145549	−	0.989351i	$-0.546495\pi$
$149$	−1.17542e7	−0.291099	−0.145549	+	0.989351i	$0.546495\pi$
$150$	0	0
$151$	2.98077e7	0.704546	0.352273	−	0.935897i	$-0.385409\pi$
$151$	2.98077e7	0.704546	0.352273	+	0.935897i	$0.385409\pi$
$152$	−1.70595e6	−0.0394017
$153$	0	0
$154$	2.16406e7	0.477471
$155$	−6.70844e6	−0.144697
$156$	0	0
$157$	4.19779e7	0.865709	0.432854	−	0.901464i	$-0.357506\pi$
$157$	4.19779e7	0.865709	0.432854	+	0.901464i	$0.357506\pi$
$158$	1.20780e8	2.43609
$159$	0	0
$160$	−1.30059e7	−0.251028
$161$	−6.25624e6	−0.118147
$162$	0	0
$163$	4.55192e6	0.0823261	0.0411631	−	0.999152i	$-0.486894\pi$
$163$	4.55192e6	0.0823261	0.0411631	+	0.999152i	$0.486894\pi$
$164$	7.35342e7	1.30178
$165$	0	0
$166$	−6.45067e7	−1.09453
$167$	−1.06013e8	−1.76138	−0.880688	−	0.473697i	$-0.842919\pi$
$167$	−1.06013e8	−1.76138	−0.880688	+	0.473697i	$0.842919\pi$
$168$	0	0
$169$	9.95140e7	1.58592
$170$	−2.12903e7	−0.332362
$171$	0	0
$172$	6.33992e7	0.950024
$173$	−5.95255e7	−0.874061	−0.437030	−	0.899447i	$-0.643970\pi$
$173$	−5.95255e7	−0.874061	−0.437030	+	0.899447i	$0.643970\pi$
$174$	0	0
$175$	−1.83000e7	−0.258118
$176$	1.08669e8	1.50248
$177$	0	0
$178$	−5.83529e7	−0.775519
$179$	1.24002e8	1.61600	0.808002	−	0.589180i	$-0.200549\pi$
$179$	1.24002e8	1.61600	0.808002	+	0.589180i	$0.200549\pi$
$180$	0	0
$181$	−5.13583e7	−0.643778	−0.321889	−	0.946777i	$-0.604318\pi$
$181$	−5.13583e7	−0.643778	−0.321889	+	0.946777i	$0.604318\pi$
$182$	−4.74055e7	−0.582880
$183$	0	0
$184$	−8.53428e6	−0.100996
$185$	−2.20515e7	−0.256058
$186$	0	0
$187$	1.51492e8	1.69411
$188$	1.07941e8	1.18477
$189$	0	0
$190$	−4.20395e6	−0.0444652
$191$	1.80930e8	1.87886	0.939430	−	0.342740i	$-0.111355\pi$
$191$	1.80930e8	1.87886	0.939430	+	0.342740i	$0.111355\pi$
$192$	0	0
$193$	−1.76110e8	−1.76333	−0.881665	−	0.471876i	$-0.843577\pi$
$193$	−1.76110e8	−1.76333	−0.881665	+	0.471876i	$0.843577\pi$
$194$	−7.28619e7	−0.716464
$195$	0	0
$196$	−8.12878e7	−0.771133
$197$	−8.41058e7	−0.783780	−0.391890	−	0.920012i	$-0.628179\pi$
$197$	−8.41058e7	−0.783780	−0.391890	+	0.920012i	$0.628179\pi$
$198$	0	0
$199$	−4.10771e6	−0.0369500	−0.0184750	−	0.999829i	$-0.505881\pi$
$199$	−4.10771e6	−0.0369500	−0.0184750	+	0.999829i	$0.505881\pi$
$200$	−2.49634e7	−0.220648
$201$	0	0
$202$	−2.04791e8	−1.74816
$203$	−6.95301e6	−0.0583360
$204$	0	0
$205$	−3.69175e7	−0.299291
$206$	1.96927e8	1.56953
$207$	0	0
$208$	−2.38047e8	−1.83418
$209$	2.99132e7	0.226648
$210$	0	0
$211$	1.82652e8	1.33856	0.669278	−	0.743012i	$-0.266604\pi$
$211$	1.82652e8	1.33856	0.669278	+	0.743012i	$0.266604\pi$
$212$	−4.40146e7	−0.317264
$213$	0	0
$214$	−1.92296e8	−1.34129
$215$	−3.18293e7	−0.218420
$216$	0	0
$217$	3.05491e7	0.202950
$218$	−1.23535e8	−0.807591
$219$	0	0
$220$	3.30107e7	0.209014
$221$	−3.31855e8	−2.06812
$222$	0	0
$223$	−1.48526e8	−0.896882	−0.448441	−	0.893812i	$-0.648021\pi$
$223$	−1.48526e8	−0.896882	−0.448441	+	0.893812i	$0.648021\pi$
$224$	5.92268e7	0.352087
$225$	0	0
$226$	1.07680e8	0.620520
$227$	−2.13713e8	−1.21267	−0.606333	−	0.795211i	$-0.707360\pi$
$227$	−2.13713e8	−1.21267	−0.606333	+	0.795211i	$0.707360\pi$
$228$	0	0
$229$	1.57057e8	0.864235	0.432118	−	0.901817i	$-0.357766\pi$
$229$	1.57057e8	0.864235	0.432118	+	0.901817i	$0.357766\pi$
$230$	−2.10309e7	−0.113975
$231$	0	0
$232$	−9.48476e6	−0.0498676
$233$	1.33939e8	0.693686	0.346843	−	0.937923i	$-0.387254\pi$
$233$	1.33939e8	0.693686	0.346843	+	0.937923i	$0.387254\pi$
$234$	0	0
$235$	−5.41913e7	−0.272391
$236$	−2.91777e6	−0.0144497
$237$	0	0
$238$	9.69525e7	0.466165
$239$	2.41326e6	0.0114344	0.00571718	−	0.999984i	$-0.498180\pi$
$239$	2.41326e6	0.0114344	0.00571718	+	0.999984i	$0.498180\pi$
$240$	0	0
$241$	7.41613e6	0.0341285	0.0170643	−	0.999854i	$-0.494568\pi$
$241$	7.41613e6	0.0341285	0.0170643	+	0.999854i	$0.494568\pi$
$242$	−2.19319e8	−0.994769
$243$	0	0
$244$	−2.22758e8	−0.981678
$245$	4.08101e7	0.177291
$246$	0	0
$247$	−6.55274e7	−0.276684
$248$	4.16727e7	0.173489
$249$	0	0
$250$	−1.25363e8	−0.507433
$251$	−2.17451e7	−0.0867966	−0.0433983	−	0.999058i	$-0.513818\pi$
$251$	−2.17451e7	−0.0867966	−0.0433983	+	0.999058i	$0.513818\pi$
$252$	0	0
$253$	1.49645e8	0.580953
$254$	−2.42300e8	−0.927761
$255$	0	0
$256$	3.35151e8	1.24853
$257$	8.85681e7	0.325470	0.162735	−	0.986670i	$-0.447968\pi$
$257$	8.85681e7	0.325470	0.162735	+	0.986670i	$0.447968\pi$
$258$	0	0
$259$	1.00419e8	0.359142
$260$	−7.23126e7	−0.255157
$261$	0	0
$262$	2.20594e8	0.757774
$263$	5.15791e7	0.174835	0.0874175	−	0.996172i	$-0.472139\pi$
$263$	5.15791e7	0.174835	0.0874175	+	0.996172i	$0.472139\pi$
$264$	0	0
$265$	2.20973e7	0.0729422
$266$	1.91441e7	0.0623661
$267$	0	0
$268$	4.12794e8	1.30997
$269$	−1.09104e8	−0.341749	−0.170875	−	0.985293i	$-0.554659\pi$
$269$	−1.09104e8	−0.341749	−0.170875	+	0.985293i	$0.554659\pi$
$270$	0	0
$271$	2.22968e7	0.0680534	0.0340267	−	0.999421i	$-0.489167\pi$
$271$	2.22968e7	0.0680534	0.0340267	+	0.999421i	$0.489167\pi$
$272$	4.86848e8	1.46691
$273$	0	0
$274$	−6.35579e8	−1.86656
$275$	4.37724e8	1.26922
$276$	0	0
$277$	−3.30652e8	−0.934743	−0.467371	−	0.884061i	$-0.654799\pi$
$277$	−3.30652e8	−0.934743	−0.467371	+	0.884061i	$0.654799\pi$
$278$	−6.00190e8	−1.67545
$279$	0	0
$280$	−4.30405e6	−0.0117172
$281$	6.38855e8	1.71763	0.858816	−	0.512285i	$-0.171201\pi$
$281$	6.38855e8	1.71763	0.858816	+	0.512285i	$0.171201\pi$
$282$	0	0
$283$	−3.02101e8	−0.792319	−0.396159	−	0.918182i	$-0.629657\pi$
$283$	−3.02101e8	−0.792319	−0.396159	+	0.918182i	$0.629657\pi$
$284$	3.46746e8	0.898251
$285$	0	0
$286$	1.13391e9	2.86614
$287$	1.68116e8	0.419780
$288$	0	0
$289$	2.68361e8	0.654000
$290$	−2.33731e7	−0.0562760
$291$	0	0
$292$	3.46211e8	0.813768
$293$	7.25966e8	1.68609	0.843043	−	0.537846i	$-0.180762\pi$
$293$	7.25966e8	1.68609	0.843043	+	0.537846i	$0.180762\pi$
$294$	0	0
$295$	1.46485e6	0.00332213
$296$	1.36984e8	0.307007
$297$	0	0
$298$	1.79933e8	0.393872
$299$	−3.27810e8	−0.709208
$300$	0	0
$301$	1.44945e8	0.306352
$302$	−4.56298e8	−0.953288
$303$	0	0
$304$	9.61321e7	0.196251
$305$	1.11835e8	0.225697
$306$	0	0
$307$	8.47111e8	1.67092	0.835461	−	0.549550i	$-0.185201\pi$
$307$	8.47111e8	1.67092	0.835461	+	0.549550i	$0.185201\pi$
$308$	−1.50325e8	−0.293159
$309$	0	0
$310$	1.02693e8	0.195783
$311$	2.16037e8	0.407256	0.203628	−	0.979048i	$-0.434727\pi$
$311$	2.16037e8	0.407256	0.203628	+	0.979048i	$0.434727\pi$
$312$	0	0
$313$	−1.07774e7	−0.0198659	−0.00993295	−	0.999951i	$-0.503162\pi$
$313$	−1.07774e7	−0.0198659	−0.00993295	+	0.999951i	$0.503162\pi$
$314$	−6.42599e8	−1.17135
$315$	0	0
$316$	−8.38986e8	−1.49572
$317$	−1.88986e8	−0.333214	−0.166607	−	0.986023i	$-0.553281\pi$
$317$	−1.88986e8	−0.333214	−0.166607	+	0.986023i	$0.553281\pi$
$318$	0	0
$319$	1.66311e8	0.286850
$320$	7.13962e7	0.121801
$321$	0	0
$322$	9.57709e7	0.159859
$323$	1.34015e8	0.221281
$324$	0	0
$325$	−9.58870e8	−1.54942
$326$	−6.96809e7	−0.111392
$327$	0	0
$328$	2.29331e8	0.358842
$329$	2.46778e8	0.382050
$330$	0	0
$331$	−8.64349e8	−1.31006	−0.655030	−	0.755603i	$-0.727344\pi$
$331$	−8.64349e8	−1.31006	−0.655030	+	0.755603i	$0.727344\pi$
$332$	4.48091e8	0.672021
$333$	0	0
$334$	1.62285e9	2.38323
$335$	−2.07241e8	−0.301175
$336$	0	0
$337$	−1.03292e8	−0.147015	−0.0735075	−	0.997295i	$-0.523419\pi$
$337$	−1.03292e8	−0.147015	−0.0735075	+	0.997295i	$0.523419\pi$
$338$	−1.52336e9	−2.14583
$339$	0	0
$340$	1.47892e8	0.204065
$341$	−7.30714e8	−0.997947
$342$	0	0
$343$	−3.86053e8	−0.516556
$344$	1.97723e8	0.261880
$345$	0	0
$346$	9.11219e8	1.18265
$347$	−9.26206e8	−1.19002	−0.595010	−	0.803718i	$-0.702852\pi$
$347$	−9.26206e8	−1.19002	−0.595010	+	0.803718i	$0.702852\pi$
$348$	0	0
$349$	−1.00869e9	−1.27019	−0.635097	−	0.772432i	$-0.719040\pi$
$349$	−1.00869e9	−1.27019	−0.635097	+	0.772432i	$0.719040\pi$
$350$	2.80137e8	0.349247
$351$	0	0
$352$	−1.41667e9	−1.73128
$353$	−1.00115e9	−1.21140	−0.605700	−	0.795693i	$-0.707107\pi$
$353$	−1.00115e9	−1.21140	−0.605700	+	0.795693i	$0.707107\pi$
$354$	0	0
$355$	−1.74082e8	−0.206517
$356$	4.05344e8	0.476156
$357$	0	0
$358$	−1.89822e9	−2.18654
$359$	4.86556e8	0.555011	0.277506	−	0.960724i	$-0.410492\pi$
$359$	4.86556e8	0.555011	0.277506	+	0.960724i	$0.410492\pi$
$360$	0	0
$361$	−8.67409e8	−0.970396
$362$	7.86196e8	0.871065
$363$	0	0
$364$	3.29299e8	0.357879
$365$	−1.73813e8	−0.187093
$366$	0	0
$367$	3.88196e8	0.409939	0.204970	−	0.978768i	$-0.434290\pi$
$367$	3.88196e8	0.409939	0.204970	+	0.978768i	$0.434290\pi$
$368$	4.80915e8	0.503038
$369$	0	0
$370$	3.37566e8	0.346460
$371$	−1.00627e8	−0.102307
$372$	0	0
$373$	−9.50838e8	−0.948693	−0.474346	−	0.880338i	$-0.657316\pi$
$373$	−9.50838e8	−0.948693	−0.474346	+	0.880338i	$0.657316\pi$
$374$	−2.31904e9	−2.29223
$375$	0	0
$376$	3.36635e8	0.326589
$377$	−3.64319e8	−0.350177
$378$	0	0
$379$	−1.95384e8	−0.184354	−0.0921769	−	0.995743i	$-0.529383\pi$
$379$	−1.95384e8	−0.184354	−0.0921769	+	0.995743i	$0.529383\pi$
$380$	2.92024e7	0.0273009
$381$	0	0
$382$	−2.76969e9	−2.54220
$383$	1.84603e9	1.67897	0.839484	−	0.543384i	$-0.182857\pi$
$383$	1.84603e9	1.67897	0.839484	+	0.543384i	$0.182857\pi$
$384$	0	0
$385$	7.54699e7	0.0674002
$386$	2.69590e9	2.38588
$387$	0	0
$388$	5.06130e8	0.439897
$389$	−8.00684e7	−0.0689664	−0.0344832	−	0.999405i	$-0.510979\pi$
$389$	−8.00684e7	−0.0689664	−0.0344832	+	0.999405i	$0.510979\pi$
$390$	0	0
$391$	6.70428e8	0.567197
$392$	−2.53511e8	−0.212567
$393$	0	0
$394$	1.28750e9	1.06050
$395$	4.21209e8	0.343880
$396$	0	0
$397$	1.75856e8	0.141055	0.0705277	−	0.997510i	$-0.477532\pi$
$397$	1.75856e8	0.141055	0.0705277	+	0.997510i	$0.477532\pi$
$398$	6.28811e7	0.0499953
$399$	0	0
$400$	1.40671e9	1.09899
$401$	2.00769e8	0.155486	0.0777431	−	0.996973i	$-0.475229\pi$
$401$	2.00769e8	0.155486	0.0777431	+	0.996973i	$0.475229\pi$
$402$	0	0
$403$	1.60069e9	1.21826
$404$	1.42257e9	1.07334
$405$	0	0
$406$	1.06437e8	0.0789317
$407$	−2.40195e9	−1.76597
$408$	0	0
$409$	−2.46797e9	−1.78364	−0.891821	−	0.452389i	$-0.850572\pi$
$409$	−2.46797e9	−1.78364	−0.891821	+	0.452389i	$0.850572\pi$
$410$	5.65135e8	0.404956
$411$	0	0
$412$	−1.36794e9	−0.963664
$413$	−6.67069e6	−0.00465956
$414$	0	0
$415$	−2.24962e8	−0.154504
$416$	3.10332e9	2.11349
$417$	0	0
$418$	−4.57913e8	−0.306666
$419$	−5.30383e8	−0.352242	−0.176121	−	0.984369i	$-0.556355\pi$
$419$	−5.30383e8	−0.352242	−0.176121	+	0.984369i	$0.556355\pi$
$420$	0	0
$421$	9.52227e8	0.621946	0.310973	−	0.950419i	$-0.399345\pi$
$421$	9.52227e8	0.621946	0.310973	+	0.950419i	$0.399345\pi$
$422$	−2.79605e9	−1.81114
$423$	0	0
$424$	−1.37268e8	−0.0874558
$425$	1.96105e9	1.23916
$426$	0	0
$427$	−5.09275e8	−0.316559
$428$	1.33577e9	0.823527
$429$	0	0
$430$	4.87244e8	0.295533
$431$	1.32253e9	0.795673	0.397836	−	0.917456i	$-0.369761\pi$
$431$	1.32253e9	0.795673	0.397836	+	0.917456i	$0.369761\pi$
$432$	0	0
$433$	−6.31440e8	−0.373787	−0.186893	−	0.982380i	$-0.559842\pi$
$433$	−6.31440e8	−0.373787	−0.186893	+	0.982380i	$0.559842\pi$
$434$	−4.67647e8	−0.274602
$435$	0	0
$436$	8.58125e8	0.495847
$437$	1.32382e8	0.0758826
$438$	0	0
$439$	7.08702e8	0.399795	0.199898	−	0.979817i	$-0.435939\pi$
$439$	7.08702e8	0.399795	0.199898	+	0.979817i	$0.435939\pi$
$440$	1.02950e8	0.0576159
$441$	0	0
$442$	5.08005e9	2.79827
$443$	−7.35403e8	−0.401895	−0.200947	−	0.979602i	$-0.564402\pi$
$443$	−7.35403e8	−0.401895	−0.200947	+	0.979602i	$0.564402\pi$
$444$	0	0
$445$	−2.03501e8	−0.109473
$446$	2.27364e9	1.21353
$447$	0	0
$448$	−3.25126e8	−0.170836
$449$	−3.34839e9	−1.74571	−0.872857	−	0.487975i	$-0.837736\pi$
$449$	−3.34839e9	−1.74571	−0.872857	+	0.487975i	$0.837736\pi$
$450$	0	0
$451$	−4.02122e9	−2.06414
$452$	−7.47992e8	−0.380989
$453$	0	0
$454$	3.27153e9	1.64080
$455$	−1.65323e8	−0.0822798
$456$	0	0
$457$	−1.20876e9	−0.592424	−0.296212	−	0.955122i	$-0.595723\pi$
$457$	−1.20876e9	−0.592424	−0.296212	+	0.955122i	$0.595723\pi$
$458$	−2.40423e9	−1.16936
$459$	0	0
$460$	1.46089e8	0.0699787
$461$	5.70954e8	0.271424	0.135712	−	0.990748i	$-0.456668\pi$
$461$	5.70954e8	0.271424	0.135712	+	0.990748i	$0.456668\pi$
$462$	0	0
$463$	2.09945e9	0.983044	0.491522	−	0.870865i	$-0.336441\pi$
$463$	2.09945e9	0.983044	0.491522	+	0.870865i	$0.336441\pi$
$464$	5.34475e8	0.248379
$465$	0	0
$466$	−2.05035e9	−0.938593
$467$	−3.74055e9	−1.69952	−0.849760	−	0.527170i	$-0.823253\pi$
$467$	−3.74055e9	−1.69952	−0.849760	+	0.527170i	$0.823253\pi$
$468$	0	0
$469$	9.43740e8	0.422423
$470$	8.29563e8	0.368559
$471$	0	0
$472$	−9.09963e6	−0.00398315
$473$	−3.46699e9	−1.50639
$474$	0	0
$475$	3.87226e8	0.165782
$476$	−6.73473e8	−0.286217
$477$	0	0
$478$	−3.69423e7	−0.0154713
$479$	−3.08337e9	−1.28189	−0.640946	−	0.767586i	$-0.721458\pi$
$479$	−3.08337e9	−1.28189	−0.640946	+	0.767586i	$0.721458\pi$
$480$	0	0
$481$	5.26168e9	2.15584
$482$	−1.13526e8	−0.0461777
$483$	0	0
$484$	1.52348e9	0.610771
$485$	−2.54100e8	−0.101137
$486$	0	0
$487$	4.31352e9	1.69231	0.846156	−	0.532935i	$-0.178911\pi$
$487$	4.31352e9	1.69231	0.846156	+	0.532935i	$0.178911\pi$
$488$	−6.94714e8	−0.270605
$489$	0	0
$490$	−6.24723e8	−0.239884
$491$	4.00872e9	1.52834	0.764170	−	0.645015i	$-0.223149\pi$
$491$	4.00872e9	1.52834	0.764170	+	0.645015i	$0.223149\pi$
$492$	0	0
$493$	7.45095e8	0.280058
$494$	1.00310e9	0.374368
$495$	0	0
$496$	−2.34829e9	−0.864105
$497$	7.92740e8	0.289657
$498$	0	0
$499$	1.02319e9	0.368640	0.184320	−	0.982866i	$-0.440992\pi$
$499$	1.02319e9	0.368640	0.184320	+	0.982866i	$0.440992\pi$
$500$	8.70824e8	0.311555
$501$	0	0
$502$	3.32875e8	0.117440
$503$	3.51279e9	1.23074	0.615368	−	0.788240i	$-0.289007\pi$
$503$	3.51279e9	1.23074	0.615368	+	0.788240i	$0.289007\pi$
$504$	0	0
$505$	−7.14193e8	−0.246772
$506$	−2.29078e9	−0.786060
$507$	0	0
$508$	1.68312e9	0.569630
$509$	9.85971e8	0.331400	0.165700	−	0.986176i	$-0.447012\pi$
$509$	9.85971e8	0.331400	0.165700	+	0.986176i	$0.447012\pi$
$510$	0	0
$511$	7.91516e8	0.262414
$512$	−3.75947e9	−1.23789
$513$	0	0
$514$	−1.35581e9	−0.440379
$515$	6.86765e8	0.221556
$516$	0	0
$517$	−5.90276e9	−1.87862
$518$	−1.53722e9	−0.485938
$519$	0	0
$520$	−2.25521e8	−0.0703355
$521$	7.11203e8	0.220324	0.110162	−	0.993914i	$-0.464863\pi$
$521$	7.11203e8	0.220324	0.110162	+	0.993914i	$0.464863\pi$
$522$	0	0
$523$	5.39505e9	1.64907	0.824535	−	0.565810i	$-0.191437\pi$
$523$	5.39505e9	1.64907	0.824535	+	0.565810i	$0.191437\pi$
$524$	−1.53234e9	−0.465260
$525$	0	0
$526$	−7.89575e8	−0.236561
$527$	−3.27369e9	−0.974316
$528$	0	0
$529$	−2.74257e9	−0.805494
$530$	−3.38267e8	−0.0986947
$531$	0	0
$532$	−1.32983e8	−0.0382917
$533$	8.80881e9	2.51984
$534$	0	0
$535$	−6.70615e8	−0.189337
$536$	1.28738e9	0.361101
$537$	0	0
$538$	1.67017e9	0.462405
$539$	4.44522e9	1.22274
$540$	0	0
$541$	−6.33559e9	−1.72027	−0.860134	−	0.510068i	$-0.829620\pi$
$541$	−6.33559e9	−1.72027	−0.860134	+	0.510068i	$0.829620\pi$
$542$	−3.41320e8	−0.0920798
$543$	0	0
$544$	−6.34683e9	−1.69029
$545$	−4.30817e8	−0.114000
$546$	0	0
$547$	−4.89390e8	−0.127850	−0.0639248	−	0.997955i	$-0.520362\pi$
$547$	−4.89390e8	−0.127850	−0.0639248	+	0.997955i	$0.520362\pi$
$548$	4.41500e9	1.14604
$549$	0	0
$550$	−6.70070e9	−1.71732
$551$	1.47125e8	0.0374676
$552$	0	0
$553$	−1.91811e9	−0.482321
$554$	5.06164e9	1.26476
$555$	0	0
$556$	4.16918e9	1.02870
$557$	2.06892e9	0.507283	0.253642	−	0.967298i	$-0.418372\pi$
$557$	2.06892e9	0.507283	0.253642	+	0.967298i	$0.418372\pi$
$558$	0	0
$559$	7.59472e9	1.83895
$560$	2.42537e8	0.0583607
$561$	0	0
$562$	−9.77962e9	−2.32405
$563$	−3.90559e9	−0.922375	−0.461188	−	0.887303i	$-0.652577\pi$
$563$	−3.90559e9	−0.922375	−0.461188	+	0.887303i	$0.652577\pi$
$564$	0	0
$565$	3.75526e8	0.0875931
$566$	4.62458e9	1.07205
$567$	0	0
$568$	1.08139e9	0.247608
$569$	1.76153e9	0.400864	0.200432	−	0.979708i	$-0.435765\pi$
$569$	1.76153e9	0.400864	0.200432	+	0.979708i	$0.435765\pi$
$570$	0	0
$571$	4.96964e9	1.11712	0.558558	−	0.829465i	$-0.311355\pi$
$571$	4.96964e9	1.11712	0.558558	+	0.829465i	$0.311355\pi$
$572$	−7.87662e9	−1.75976
$573$	0	0
$574$	−2.57352e9	−0.567985
$575$	1.93715e9	0.424939
$576$	0	0
$577$	2.19022e8	0.0474649	0.0237324	−	0.999718i	$-0.492445\pi$
$577$	2.19022e8	0.0474649	0.0237324	+	0.999718i	$0.492445\pi$
$578$	−4.10809e9	−0.884896
$579$	0	0
$580$	1.62360e8	0.0345525
$581$	1.02444e9	0.216705
$582$	0	0
$583$	2.40694e9	0.503066
$584$	1.07972e9	0.224320
$585$	0	0
$586$	−1.11131e10	−2.28136
$587$	−6.34220e8	−0.129422	−0.0647108	−	0.997904i	$-0.520612\pi$
$587$	−6.34220e8	−0.129422	−0.0647108	+	0.997904i	$0.520612\pi$
$588$	0	0
$589$	−6.46416e8	−0.130349
$590$	−2.24240e7	−0.00449502
$591$	0	0
$592$	−7.71916e9	−1.52913
$593$	−3.47748e9	−0.684815	−0.342408	−	0.939552i	$-0.611242\pi$
$593$	−3.47748e9	−0.684815	−0.342408	+	0.939552i	$0.611242\pi$
$594$	0	0
$595$	3.38114e8	0.0658042
$596$	−1.24990e9	−0.241831
$597$	0	0
$598$	5.01813e9	0.959595
$599$	−3.85315e9	−0.732525	−0.366263	−	0.930512i	$-0.619363\pi$
$599$	−3.85315e9	−0.732525	−0.366263	+	0.930512i	$0.619363\pi$
$600$	0	0
$601$	−4.68707e9	−0.880725	−0.440363	−	0.897820i	$-0.645150\pi$
$601$	−4.68707e9	−0.880725	−0.440363	+	0.897820i	$0.645150\pi$
$602$	−2.21882e9	−0.414510
$603$	0	0
$604$	3.16964e9	0.585303
$605$	−7.64856e8	−0.140422
$606$	0	0
$607$	2.91530e9	0.529082	0.264541	−	0.964374i	$-0.414780\pi$
$607$	2.91530e9	0.529082	0.264541	+	0.964374i	$0.414780\pi$
$608$	−1.25323e9	−0.226136
$609$	0	0
$610$	−1.71197e9	−0.305381
$611$	1.29305e10	2.29335
$612$	0	0
$613$	5.94387e9	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$613$	5.94387e9	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$614$	−1.29676e10	−2.26084
$615$	0	0
$616$	−4.68817e8	−0.0808111
$617$	−3.68882e7	−0.00632251	−0.00316126	−	0.999995i	$-0.501006\pi$
$617$	−3.68882e7	−0.00632251	−0.00316126	+	0.999995i	$0.501006\pi$
$618$	0	0
$619$	1.92584e9	0.326365	0.163183	−	0.986596i	$-0.447824\pi$
$619$	1.92584e9	0.326365	0.163183	+	0.986596i	$0.447824\pi$
$620$	−7.13351e8	−0.120208
$621$	0	0
$622$	−3.30711e9	−0.551039
$623$	9.26708e8	0.153545
$624$	0	0
$625$	5.44367e9	0.891890
$626$	1.64981e8	0.0268796
$627$	0	0
$628$	4.46377e9	0.719189
$629$	−1.07610e10	−1.72416
$630$	0	0
$631$	−1.97642e9	−0.313168	−0.156584	−	0.987665i	$-0.550048\pi$
$631$	−1.97642e9	−0.313168	−0.156584	+	0.987665i	$0.550048\pi$
$632$	−2.61654e9	−0.412304
$633$	0	0
$634$	2.89301e9	0.450856
$635$	−8.45003e8	−0.130963
$636$	0	0
$637$	−9.73763e9	−1.49267
$638$	−2.54590e9	−0.388123
$639$	0	0
$640$	5.71824e8	0.0862248
$641$	1.15898e10	1.73810	0.869049	−	0.494726i	$-0.164731\pi$
$641$	1.15898e10	1.73810	0.869049	+	0.494726i	$0.164731\pi$
$642$	0	0
$643$	3.80920e9	0.565061	0.282530	−	0.959258i	$-0.408826\pi$
$643$	3.80920e9	0.565061	0.282530	+	0.959258i	$0.408826\pi$
$644$	−6.65266e8	−0.0981509
$645$	0	0
$646$	−2.05151e9	−0.299405
$647$	−6.74654e9	−0.979300	−0.489650	−	0.871919i	$-0.662876\pi$
$647$	−6.74654e9	−0.979300	−0.489650	+	0.871919i	$0.662876\pi$
$648$	0	0
$649$	1.59558e8	0.0229120
$650$	1.46784e10	2.09644
$651$	0	0
$652$	4.84034e8	0.0683926
$653$	−8.55274e9	−1.20201	−0.601007	−	0.799244i	$-0.705234\pi$
$653$	−8.55274e9	−1.20201	−0.601007	+	0.799244i	$0.705234\pi$
$654$	0	0
$655$	7.69304e8	0.106968
$656$	−1.29230e10	−1.78731
$657$	0	0
$658$	−3.77769e9	−0.516934
$659$	−1.15327e10	−1.56976	−0.784881	−	0.619646i	$-0.787276\pi$
$659$	−1.15327e10	−1.56976	−0.784881	+	0.619646i	$0.787276\pi$
$660$	0	0
$661$	1.36320e9	0.183592	0.0917959	−	0.995778i	$-0.470739\pi$
$661$	1.36320e9	0.183592	0.0917959	+	0.995778i	$0.470739\pi$
$662$	1.32315e10	1.77258
$663$	0	0
$664$	1.39746e9	0.185247
$665$	6.67633e7	0.00880364
$666$	0	0
$667$	7.36014e8	0.0960386
$668$	−1.12730e10	−1.46327
$669$	0	0
$670$	3.17246e9	0.407506
$671$	1.21815e10	1.55659
$672$	0	0
$673$	6.31920e8	0.0799115	0.0399557	−	0.999201i	$-0.487278\pi$
$673$	6.31920e8	0.0799115	0.0399557	+	0.999201i	$0.487278\pi$
$674$	1.58120e9	0.198919
$675$	0	0
$676$	1.05819e10	1.31750
$677$	1.04239e10	1.29113	0.645566	−	0.763705i	$-0.276622\pi$
$677$	1.04239e10	1.29113	0.645566	+	0.763705i	$0.276622\pi$
$678$	0	0
$679$	1.15713e9	0.141852
$680$	4.61229e8	0.0562516
$681$	0	0
$682$	1.11858e10	1.35027
$683$	2.80991e9	0.337458	0.168729	−	0.985662i	$-0.446034\pi$
$683$	2.80991e9	0.337458	0.168729	+	0.985662i	$0.446034\pi$
$684$	0	0
$685$	−2.21653e9	−0.263485
$686$	5.90971e9	0.698927
$687$	0	0
$688$	−1.11418e10	−1.30436
$689$	−5.27260e9	−0.614126
$690$	0	0
$691$	1.65847e10	1.91220	0.956102	−	0.293033i	$-0.0946646\pi$
$691$	1.65847e10	1.91220	0.956102	+	0.293033i	$0.0946646\pi$
$692$	−6.32972e9	−0.726127
$693$	0	0
$694$	1.41784e10	1.61016
$695$	−2.09311e9	−0.236508
$696$	0	0
$697$	−1.80155e10	−2.01527
$698$	1.54411e10	1.71864
$699$	0	0
$700$	−1.94595e9	−0.214432
$701$	−1.16139e9	−0.127340	−0.0636701	−	0.997971i	$-0.520281\pi$
$701$	−1.16139e9	−0.127340	−0.0636701	+	0.997971i	$0.520281\pi$
$702$	0	0
$703$	−2.12485e9	−0.230667
$704$	7.77679e9	0.840033
$705$	0	0
$706$	1.53256e10	1.63909
$707$	3.25231e9	0.346118
$708$	0	0
$709$	1.66683e10	1.75642	0.878211	−	0.478274i	$-0.158737\pi$
$709$	1.66683e10	1.75642	0.878211	+	0.478274i	$0.158737\pi$
$710$	2.66486e9	0.279428
$711$	0	0
$712$	1.26414e9	0.131255
$713$	−3.23379e9	−0.334117
$714$	0	0
$715$	3.95442e9	0.404586
$716$	1.31859e10	1.34250
$717$	0	0
$718$	−7.44822e9	−0.750960
$719$	1.21223e10	1.21628	0.608141	−	0.793829i	$-0.291915\pi$
$719$	1.21223e10	1.21628	0.608141	+	0.793829i	$0.291915\pi$
$720$	0	0
$721$	−3.12741e9	−0.310750
$722$	1.32783e10	1.31300
$723$	0	0
$724$	−5.46125e9	−0.534819
$725$	2.15290e9	0.209817
$726$	0	0
$727$	5.08203e9	0.490531	0.245266	−	0.969456i	$-0.421125\pi$
$727$	5.08203e9	0.490531	0.245266	+	0.969456i	$0.421125\pi$
$728$	1.02698e9	0.0986514
$729$	0	0
$730$	2.66074e9	0.253147
$731$	−1.55325e10	−1.47072
$732$	0	0
$733$	−2.43056e8	−0.0227951	−0.0113976	−	0.999935i	$-0.503628\pi$
$733$	−2.43056e8	−0.0227951	−0.0113976	+	0.999935i	$0.503628\pi$
$734$	−5.94252e9	−0.554669
$735$	0	0
$736$	−6.26947e9	−0.579641
$737$	−2.25736e10	−2.07714
$738$	0	0
$739$	3.54378e9	0.323007	0.161503	−	0.986872i	$-0.448366\pi$
$739$	3.54378e9	0.323007	0.161503	+	0.986872i	$0.448366\pi$
$740$	−2.34488e9	−0.212720
$741$	0	0
$742$	1.54041e9	0.138427
$743$	3.06235e9	0.273901	0.136951	−	0.990578i	$-0.456270\pi$
$743$	3.06235e9	0.273901	0.136951	+	0.990578i	$0.456270\pi$
$744$	0	0
$745$	6.27503e8	0.0555993
$746$	1.45555e10	1.28363
$747$	0	0
$748$	1.61090e10	1.40739
$749$	3.05387e9	0.265561
$750$	0	0
$751$	−1.84078e9	−0.158585	−0.0792925	−	0.996851i	$-0.525266\pi$
$751$	−1.84078e9	−0.158585	−0.0792925	+	0.996851i	$0.525266\pi$
$752$	−1.89697e10	−1.62666
$753$	0	0
$754$	5.57701e9	0.473807
$755$	−1.59130e9	−0.134567
$756$	0	0
$757$	−5.79351e9	−0.485407	−0.242703	−	0.970101i	$-0.578034\pi$
$757$	−5.79351e9	−0.485407	−0.242703	+	0.970101i	$0.578034\pi$
$758$	2.99095e9	0.249440
$759$	0	0
$760$	9.10734e7	0.00752565
$761$	1.68000e9	0.138186	0.0690930	−	0.997610i	$-0.477989\pi$
$761$	1.68000e9	0.138186	0.0690930	+	0.997610i	$0.477989\pi$
$762$	0	0
$763$	1.96187e9	0.159895
$764$	1.92394e10	1.56087
$765$	0	0
$766$	−2.82591e10	−2.27173
$767$	−3.49526e8	−0.0279702
$768$	0	0
$769$	−5.46808e8	−0.0433603	−0.0216802	−	0.999765i	$-0.506902\pi$
$769$	−5.46808e8	−0.0433603	−0.0216802	+	0.999765i	$0.506902\pi$
$770$	−1.15530e9	−0.0911960
$771$	0	0
$772$	−1.87269e10	−1.46489
$773$	−9.15860e9	−0.713183	−0.356592	−	0.934260i	$-0.616061\pi$
$773$	−9.15860e9	−0.713183	−0.356592	+	0.934260i	$0.616061\pi$
$774$	0	0
$775$	−9.45908e9	−0.729949
$776$	1.57846e9	0.121260
$777$	0	0
$778$	1.22569e9	0.0933151
$779$	−3.55731e9	−0.269613
$780$	0	0
$781$	−1.89618e10	−1.42430
$782$	−1.02629e10	−0.767447
$783$	0	0
$784$	1.42856e10	1.05875
$785$	−2.24101e9	−0.165349
$786$	0	0
$787$	−4.86176e9	−0.355535	−0.177767	−	0.984073i	$-0.556887\pi$
$787$	−4.86176e9	−0.355535	−0.177767	+	0.984073i	$0.556887\pi$
$788$	−8.94349e9	−0.651127
$789$	0	0
$790$	−6.44788e9	−0.465288
$791$	−1.71008e9	−0.122857
$792$	0	0
$793$	−2.66846e10	−1.90023
$794$	−2.69201e9	−0.190855
$795$	0	0
$796$	−4.36799e8	−0.0306963
$797$	−1.82890e10	−1.27963	−0.639817	−	0.768527i	$-0.720990\pi$
$797$	−1.82890e10	−1.27963	−0.639817	+	0.768527i	$0.720990\pi$
$798$	0	0
$799$	−2.64451e10	−1.83413
$800$	−1.83387e10	−1.26635
$801$	0	0
$802$	−3.07338e9	−0.210381
$803$	−1.89325e10	−1.29034
$804$	0	0
$805$	3.33993e8	0.0225659
$806$	−2.45034e10	−1.64837
$807$	0	0
$808$	4.43655e9	0.295873
$809$	1.42882e10	0.948764	0.474382	−	0.880319i	$-0.342672\pi$
$809$	1.42882e10	0.948764	0.474382	+	0.880319i	$0.342672\pi$
$810$	0	0
$811$	1.91584e10	1.26121	0.630604	−	0.776105i	$-0.282807\pi$
$811$	1.91584e10	1.26121	0.630604	+	0.776105i	$0.282807\pi$
$812$	−7.39357e8	−0.0484628
$813$	0	0
$814$	3.67692e10	2.38946
$815$	−2.43007e8	−0.0157241
$816$	0	0
$817$	−3.06702e9	−0.196761
$818$	3.77797e10	2.41336
$819$	0	0
$820$	−3.92567e9	−0.248637
$821$	−6.44351e9	−0.406370	−0.203185	−	0.979140i	$-0.565129\pi$
$821$	−6.44351e9	−0.406370	−0.203185	+	0.979140i	$0.565129\pi$
$822$	0	0
$823$	1.09792e10	0.686547	0.343273	−	0.939236i	$-0.388464\pi$
$823$	1.09792e10	0.686547	0.343273	+	0.939236i	$0.388464\pi$
$824$	−4.26617e9	−0.265640
$825$	0	0
$826$	1.02115e8	0.00630464
$827$	1.30644e10	0.803196	0.401598	−	0.915816i	$-0.368455\pi$
$827$	1.30644e10	0.803196	0.401598	+	0.915816i	$0.368455\pi$
$828$	0	0
$829$	−2.18728e10	−1.33341	−0.666705	−	0.745322i	$-0.732296\pi$
$829$	−2.18728e10	−1.33341	−0.666705	+	0.745322i	$0.732296\pi$
$830$	3.44373e9	0.209052
$831$	0	0
$832$	−1.70357e10	−1.02548
$833$	1.99151e10	1.19378
$834$	0	0
$835$	5.65957e9	0.336419
$836$	3.18086e9	0.188288
$837$	0	0
$838$	8.11913e9	0.476601
$839$	−2.36638e10	−1.38330	−0.691652	−	0.722231i	$-0.743117\pi$
$839$	−2.36638e10	−1.38330	−0.691652	+	0.722231i	$0.743117\pi$
$840$	0	0
$841$	−1.64319e10	−0.952580
$842$	−1.45767e10	−0.841526
$843$	0	0
$844$	1.94226e10	1.11201
$845$	−5.31261e9	−0.302907
$846$	0	0
$847$	3.48302e9	0.196954
$848$	7.73518e9	0.435597
$849$	0	0
$850$	−3.00199e10	−1.67665
$851$	−1.06299e10	−0.591256
$852$	0	0
$853$	−2.11867e10	−1.16880	−0.584402	−	0.811465i	$-0.698671\pi$
$853$	−2.11867e10	−1.16880	−0.584402	+	0.811465i	$0.698671\pi$
$854$	7.79601e9	0.428321
$855$	0	0
$856$	4.16585e9	0.227010
$857$	2.61839e10	1.42103	0.710513	−	0.703684i	$-0.248463\pi$
$857$	2.61839e10	1.42103	0.710513	+	0.703684i	$0.248463\pi$
$858$	0	0
$859$	−3.47241e9	−0.186920	−0.0934598	−	0.995623i	$-0.529793\pi$
$859$	−3.47241e9	−0.186920	−0.0934598	+	0.995623i	$0.529793\pi$
$860$	−3.38460e9	−0.181453
$861$	0	0
$862$	−2.02453e10	−1.07659
$863$	6.49335e9	0.343899	0.171950	−	0.985106i	$-0.444993\pi$
$863$	6.49335e9	0.343899	0.171950	+	0.985106i	$0.444993\pi$
$864$	0	0
$865$	3.17780e9	0.166944
$866$	9.66611e9	0.505754
$867$	0	0
$868$	3.24848e9	0.168601
$869$	4.58800e10	2.37167
$870$	0	0
$871$	4.94494e10	2.53570
$872$	2.67623e9	0.136683
$873$	0	0
$874$	−2.02650e9	−0.102673
$875$	1.99090e9	0.100466
$876$	0	0
$877$	−1.04649e10	−0.523887	−0.261943	−	0.965083i	$-0.584363\pi$
$877$	−1.04649e10	−0.523887	−0.261943	+	0.965083i	$0.584363\pi$
$878$	−1.08488e10	−0.540944
$879$	0	0
$880$	−5.80133e9	−0.286971
$881$	−2.01241e9	−0.0991521	−0.0495760	−	0.998770i	$-0.515787\pi$
$881$	−2.01241e9	−0.0991521	−0.0495760	+	0.998770i	$0.515787\pi$
$882$	0	0
$883$	1.46717e10	0.717165	0.358582	−	0.933498i	$-0.383260\pi$
$883$	1.46717e10	0.717165	0.358582	+	0.933498i	$0.383260\pi$
$884$	−3.52882e10	−1.71809
$885$	0	0
$886$	1.12576e10	0.543785
$887$	8.20707e9	0.394871	0.197436	−	0.980316i	$-0.436739\pi$
$887$	8.20707e9	0.394871	0.197436	+	0.980316i	$0.436739\pi$
$888$	0	0
$889$	3.84799e9	0.183687
$890$	3.11520e9	0.148123
$891$	0	0
$892$	−1.57937e10	−0.745086
$893$	−5.22179e9	−0.245380
$894$	0	0
$895$	−6.61990e9	−0.308653
$896$	−2.60399e9	−0.120937
$897$	0	0
$898$	5.12572e10	2.36204
$899$	−3.59394e9	−0.164973
$900$	0	0
$901$	1.07834e10	0.491154
$902$	6.15570e10	2.79290
$903$	0	0
$904$	−2.33276e9	−0.105022
$905$	2.74179e9	0.122960
$906$	0	0
$907$	−2.59506e10	−1.15484	−0.577419	−	0.816448i	$-0.695940\pi$
$907$	−2.59506e10	−1.15484	−0.577419	+	0.816448i	$0.695940\pi$
$908$	−2.27255e10	−1.00742
$909$	0	0
$910$	2.53077e9	0.111329
$911$	2.04912e10	0.897951	0.448975	−	0.893544i	$-0.351789\pi$
$911$	2.04912e10	0.897951	0.448975	+	0.893544i	$0.351789\pi$
$912$	0	0
$913$	−2.45039e10	−1.06558
$914$	1.85037e10	0.801581
$915$	0	0
$916$	1.67008e10	0.717965
$917$	−3.50328e9	−0.150031
$918$	0	0
$919$	−7.45645e9	−0.316904	−0.158452	−	0.987367i	$-0.550650\pi$
$919$	−7.45645e9	−0.316904	−0.158452	+	0.987367i	$0.550650\pi$
$920$	4.55607e8	0.0192901
$921$	0	0
$922$	−8.74019e9	−0.367251
$923$	4.15374e10	1.73874
$924$	0	0
$925$	−3.10932e10	−1.29172
$926$	−3.21385e10	−1.33011
$927$	0	0
$928$	−6.96771e9	−0.286202
$929$	8.01831e9	0.328116	0.164058	−	0.986451i	$-0.447542\pi$
$929$	8.01831e9	0.328116	0.164058	+	0.986451i	$0.447542\pi$
$930$	0	0
$931$	3.93240e9	0.159711
$932$	1.42426e10	0.576281
$933$	0	0
$934$	5.72605e10	2.29954
$935$	−8.08746e9	−0.323573
$936$	0	0
$937$	1.49057e10	0.591920	0.295960	−	0.955200i	$-0.404360\pi$
$937$	1.49057e10	0.591920	0.295960	+	0.955200i	$0.404360\pi$
$938$	−1.44468e10	−0.571560
$939$	0	0
$940$	−5.76250e9	−0.226289
$941$	−3.60500e10	−1.41040	−0.705199	−	0.709010i	$-0.749142\pi$
$941$	−3.60500e10	−1.41040	−0.705199	+	0.709010i	$0.749142\pi$
$942$	0	0
$943$	−1.77960e10	−0.691084
$944$	5.12773e8	0.0198391
$945$	0	0
$946$	5.30728e10	2.03823
$947$	−6.99407e9	−0.267612	−0.133806	−	0.991008i	$-0.542720\pi$
$947$	−6.99407e9	−0.267612	−0.133806	+	0.991008i	$0.542720\pi$
$948$	0	0
$949$	4.14733e10	1.57520
$950$	−5.92767e9	−0.224312
$951$	0	0
$952$	−2.10036e9	−0.0788975
$953$	1.91968e10	0.718464	0.359232	−	0.933248i	$-0.383039\pi$
$953$	1.91968e10	0.718464	0.359232	+	0.933248i	$0.383039\pi$
$954$	0	0
$955$	−9.65907e9	−0.358859
$956$	2.56617e8	0.00949912
$957$	0	0
$958$	4.72004e10	1.73447
$959$	1.00937e10	0.369560
$960$	0	0
$961$	−1.17221e10	−0.426063
$962$	−8.05460e10	−2.91697
$963$	0	0
$964$	7.88603e8	0.0283523
$965$	9.40173e9	0.336792
$966$	0	0
$967$	3.12180e10	1.11023	0.555114	−	0.831774i	$-0.312675\pi$
$967$	3.12180e10	1.11023	0.555114	+	0.831774i	$0.312675\pi$
$968$	4.75127e9	0.168363
$969$	0	0
$970$	3.88977e9	0.136843
$971$	1.62444e10	0.569425	0.284713	−	0.958613i	$-0.408102\pi$
$971$	1.62444e10	0.569425	0.284713	+	0.958613i	$0.408102\pi$
$972$	0	0
$973$	9.53168e9	0.331722
$974$	−6.60316e10	−2.28979
$975$	0	0
$976$	3.91478e10	1.34782
$977$	3.13872e10	1.07677	0.538383	−	0.842701i	$-0.319036\pi$
$977$	3.13872e10	1.07677	0.538383	+	0.842701i	$0.319036\pi$
$978$	0	0
$979$	−2.21662e10	−0.755010
$980$	4.33960e9	0.147285
$981$	0	0
$982$	−6.13656e10	−2.06793
$983$	−4.16663e10	−1.39910	−0.699548	−	0.714586i	$-0.746615\pi$
$983$	−4.16663e10	−1.39910	−0.699548	+	0.714586i	$0.746615\pi$
$984$	0	0
$985$	4.49003e9	0.149700
$986$	−1.14059e10	−0.378933
$987$	0	0
$988$	−6.96794e9	−0.229856
$989$	−1.53432e10	−0.504347
$990$	0	0
$991$	5.54465e10	1.80974	0.904870	−	0.425688i	$-0.139968\pi$
$991$	5.54465e10	1.80974	0.904870	+	0.425688i	$0.139968\pi$
$992$	3.06137e10	0.995692
$993$	0	0
$994$	−1.21353e10	−0.391921
$995$	2.19293e8	0.00705738
$996$	0	0
$997$	3.10920e10	0.993608	0.496804	−	0.867863i	$-0.334507\pi$
$997$	3.10920e10	0.993608	0.496804	+	0.867863i	$0.334507\pi$
$998$	−1.56630e10	−0.498789
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.a.b.1.1	yes	4
3.2	odd	2		81.8.a.a.1.4	✓	4
9.2	odd	6		81.8.c.j.28.1		8
9.4	even	3		81.8.c.i.55.4		8
9.5	odd	6		81.8.c.j.55.1		8
9.7	even	3		81.8.c.i.28.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.8.a.a.1.4	✓	4	3.2	odd	2
81.8.a.b.1.1	yes	4	1.1	even	1	trivial
81.8.c.i.28.4		8	9.7	even	3
81.8.c.i.55.4		8	9.4	even	3
81.8.c.j.28.1		8	9.2	odd	6
81.8.c.j.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-15.3080 q^{2} +106.336 q^{4} -53.3856 q^{5} +243.109 q^{7} +331.630 q^{8} +817.229 q^{10} -5815.00 q^{11} +12738.2 q^{13} -3721.52 q^{14} -18687.6 q^{16} -26051.9 q^{17} -5144.15 q^{19} -5676.82 q^{20} +89016.2 q^{22} -25734.4 q^{23} -75275.0 q^{25} -194997. q^{26} +25851.3 q^{28} -28600.4 q^{29} +125660. q^{31} +243623. q^{32} +398803. q^{34} -12978.5 q^{35} +413062. q^{37} +78746.9 q^{38} -17704.2 q^{40} +691526. q^{41} +596215. q^{43} -618345. q^{44} +393943. q^{46} +1.01509e6 q^{47} -764441. q^{49} +1.15231e6 q^{50} +1.35454e6 q^{52} -413919. q^{53} +310437. q^{55} +80622.0 q^{56} +437817. q^{58} -27439.1 q^{59} -2.09485e6 q^{61} -1.92361e6 q^{62} -1.33737e6 q^{64} -680037. q^{65} +3.88197e6 q^{67} -2.77026e6 q^{68} +198675. q^{70} +3.26085e6 q^{71} +3.25581e6 q^{73} -6.32317e6 q^{74} -547010. q^{76} -1.41368e6 q^{77} -7.88994e6 q^{79} +997650. q^{80} -1.05859e7 q^{82} +4.21391e6 q^{83} +1.39079e6 q^{85} -9.12688e6 q^{86} -1.92843e6 q^{88} +3.81191e6 q^{89} +3.09677e6 q^{91} -2.73650e6 q^{92} -1.55391e7 q^{94} +274624. q^{95} +4.75971e6 q^{97} +1.17021e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 15 q^{2} + 229 q^{4} + 192 q^{5} + 800 q^{7} + 2505 q^{8} + 5469 q^{10} - 5016 q^{11} - 2200 q^{13} + 19452 q^{14} + 40849 q^{16} - 19620 q^{17} + 11240 q^{19} + 96951 q^{20} - 41280 q^{22} + 154560 q^{23}+ \cdots + 15771015 q^{98}+O(q^{100}) \) 4 * q + 15 * q^2 + 229 * q^4 + 192 * q^5 + 800 * q^7 + 2505 * q^8 + 5469 * q^10 - 5016 * q^11 - 2200 * q^13 + 19452 * q^14 + 40849 * q^16 - 19620 * q^17 + 11240 * q^19 + 96951 * q^20 - 41280 * q^22 + 154560 * q^23 + 164656 * q^25 - 397095 * q^26 + 363860 * q^28 + 118680 * q^29 + 450464 * q^31 + 952065 * q^32 + 664101 * q^34 - 907608 * q^35 + 711800 * q^37 - 202740 * q^38 + 1962723 * q^40 + 1027776 * q^41 + 1259000 * q^43 - 3842544 * q^44 + 1971888 * q^46 + 561840 * q^47 + 1370940 * q^49 + 3258420 * q^50 - 303205 * q^52 - 3858840 * q^53 - 425712 * q^55 + 1932180 * q^56 + 1352145 * q^58 + 341472 * q^59 + 2271896 * q^61 + 1571340 * q^62 + 1961473 * q^64 - 517764 * q^65 - 1649560 * q^67 + 1494135 * q^68 + 413508 * q^70 + 6584688 * q^71 + 358100 * q^73 - 5278983 * q^74 - 13960972 * q^76 - 15209160 * q^77 + 1436336 * q^79 + 23326203 * q^80 - 19970490 * q^82 + 1394160 * q^83 - 4525776 * q^85 - 6164088 * q^86 - 36138720 * q^88 + 14935932 * q^89 - 9244808 * q^91 - 479760 * q^92 - 19330980 * q^94 - 38927136 * q^95 + 4456880 * q^97 + 15771015 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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