LMFDB - Embedded newform 27.10.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,10,Mod(1,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.9059675764$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.203942560.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 83x^{2} + 1440 $ x^4 - 83*x^2 + 1440
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.96988$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-12.7978 q^{2} -348.216 q^{4} -2019.77 q^{5} -665.864 q^{7} +11008.9 q^{8} +25848.6 q^{10} -85435.1 q^{11} +84838.5 q^{13} +8521.60 q^{14} +37397.1 q^{16} +139510. q^{17} +993145. q^{19} +703317. q^{20} +1.09338e6 q^{22} -758529. q^{23} +2.12635e6 q^{25} -1.08575e6 q^{26} +231865. q^{28} +4.04867e6 q^{29} -3.01466e6 q^{31} -6.11515e6 q^{32} -1.78543e6 q^{34} +1.34489e6 q^{35} +2.04252e6 q^{37} -1.27101e7 q^{38} -2.22354e7 q^{40} +7.85245e6 q^{41} -2.72306e7 q^{43} +2.97499e7 q^{44} +9.70751e6 q^{46} +3.29865e7 q^{47} -3.99102e7 q^{49} -2.72126e7 q^{50} -2.95421e7 q^{52} -1.02265e8 q^{53} +1.72559e8 q^{55} -7.33042e6 q^{56} -5.18141e7 q^{58} +1.15657e8 q^{59} +1.29214e8 q^{61} +3.85811e7 q^{62} +5.91132e7 q^{64} -1.71354e8 q^{65} -1.95575e7 q^{67} -4.85797e7 q^{68} -1.72117e7 q^{70} +2.18178e8 q^{71} +4.41427e8 q^{73} -2.61398e7 q^{74} -3.45829e8 q^{76} +5.68882e7 q^{77} -1.09562e7 q^{79} -7.55335e7 q^{80} -1.00494e8 q^{82} +2.86843e8 q^{83} -2.81779e8 q^{85} +3.48492e8 q^{86} -9.40544e8 q^{88} -7.64398e8 q^{89} -5.64909e7 q^{91} +2.64132e8 q^{92} -4.22154e8 q^{94} -2.00593e9 q^{95} +815669. q^{97} +5.10764e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2236 q^{4} + 11852 q^{7} - 41760 q^{10} + 179684 q^{13} + 2348680 q^{16} + 1011428 q^{19} + 3473568 q^{22} + 2554060 q^{25} + 19793924 q^{28} + 889136 q^{31} - 43111008 q^{34} - 3805156 q^{37} - 133649280 q^{40}+ \cdots + 1935734516 q^{97}+O(q^{100}) $ 4 * q + 2236 * q^4 + 11852 * q^7 - 41760 * q^10 + 179684 * q^13 + 2348680 * q^16 + 1011428 * q^19 + 3473568 * q^22 + 2554060 * q^25 + 19793924 * q^28 + 889136 * q^31 - 43111008 * q^34 - 3805156 * q^37 - 133649280 * q^40 + 11585024 * q^43 + 13428000 * q^46 - 73622328 * q^49 - 44411860 * q^52 + 314722080 * q^55 - 407685888 * q^58 + 219882572 * q^61 + 1519285360 * q^64 + 390561236 * q^67 - 650481120 * q^70 + 1197949508 * q^73 - 2121017572 * q^76 + 49814324 * q^79 - 1767126528 * q^82 + 370444320 * q^85 - 653198976 * q^88 - 47017172 * q^91 - 4146926688 * q^94 + 1935734516 * q^97

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 $

$2$	−12.7978	−0.565589	−0.282794	−	0.959181i	$-0.591261\pi$
$2$	−12.7978	−0.565589	−0.282794	+	0.959181i	$0.591261\pi$
$3$	0	0
$3$	0	0
$4$	−348.216	−0.680110
$5$	−2019.77	−1.44523	−0.722615	−	0.691251i	$-0.757060\pi$
$5$	−2019.77	−1.44523	−0.722615	+	0.691251i	$0.757060\pi$
$6$	0	0
$7$	−665.864	−0.104820	−0.0524100	−	0.998626i	$-0.516690\pi$
$7$	−665.864	−0.104820	−0.0524100	+	0.998626i	$0.516690\pi$
$8$	11008.9	0.950251
$9$	0	0
$10$	25848.6	0.817406
$11$	−85435.1	−1.75942	−0.879709	−	0.475512i	$-0.842263\pi$
$11$	−85435.1	−1.75942	−0.879709	+	0.475512i	$0.842263\pi$
$12$	0	0
$13$	84838.5	0.823850	0.411925	−	0.911218i	$-0.364857\pi$
$13$	84838.5	0.823850	0.411925	+	0.911218i	$0.364857\pi$
$14$	8521.60	0.0592851
$15$	0	0
$16$	37397.1	0.142658
$17$	139510.	0.405122	0.202561	−	0.979270i	$-0.435074\pi$
$17$	139510.	0.405122	0.202561	+	0.979270i	$0.435074\pi$
$18$	0	0
$19$	993145.	1.74832	0.874161	−	0.485636i	$-0.161412\pi$
$19$	993145.	1.74832	0.874161	+	0.485636i	$0.161412\pi$
$20$	703317.	0.982915
$21$	0	0
$22$	1.09338e6	0.995107
$23$	−758529.	−0.565193	−0.282597	−	0.959239i	$-0.591196\pi$
$23$	−758529.	−0.565193	−0.282597	+	0.959239i	$0.591196\pi$
$24$	0	0
$25$	2.12635e6	1.08869
$26$	−1.08575e6	−0.465960
$27$	0	0
$28$	231865.	0.0712891
$29$	4.04867e6	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$29$	4.04867e6	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$30$	0	0
$31$	−3.01466e6	−0.586288	−0.293144	−	0.956068i	$-0.594702\pi$
$31$	−3.01466e6	−0.586288	−0.293144	+	0.956068i	$0.594702\pi$
$32$	−6.11515e6	−1.03094
$33$	0	0
$34$	−1.78543e6	−0.229133
$35$	1.34489e6	0.151489
$36$	0	0
$37$	2.04252e6	0.179168	0.0895838	−	0.995979i	$-0.471446\pi$
$37$	2.04252e6	0.179168	0.0895838	+	0.995979i	$0.471446\pi$
$38$	−1.27101e7	−0.988831
$39$	0	0
$40$	−2.22354e7	−1.37333
$41$	7.85245e6	0.433988	0.216994	−	0.976173i	$-0.430375\pi$
$41$	7.85245e6	0.433988	0.216994	+	0.976173i	$0.430375\pi$
$42$	0	0
$43$	−2.72306e7	−1.21464	−0.607322	−	0.794456i	$-0.707756\pi$
$43$	−2.72306e7	−1.21464	−0.607322	+	0.794456i	$0.707756\pi$
$44$	2.97499e7	1.19660
$45$	0	0
$46$	9.70751e6	0.319667
$47$	3.29865e7	0.986042	0.493021	−	0.870017i	$-0.335893\pi$
$47$	3.29865e7	0.986042	0.493021	+	0.870017i	$0.335893\pi$
$48$	0	0
$49$	−3.99102e7	−0.989013
$50$	−2.72126e7	−0.615751
$51$	0	0
$52$	−2.95421e7	−0.560308
$53$	−1.02265e8	−1.78026	−0.890131	−	0.455704i	$-0.849388\pi$
$53$	−1.02265e8	−1.78026	−0.890131	+	0.455704i	$0.849388\pi$
$54$	0	0
$55$	1.72559e8	2.54276
$56$	−7.33042e6	−0.0996054
$57$	0	0
$58$	−5.18141e7	−0.601204
$59$	1.15657e8	1.24262	0.621308	−	0.783566i	$-0.286601\pi$
$59$	1.15657e8	1.24262	0.621308	+	0.783566i	$0.286601\pi$
$60$	0	0
$61$	1.29214e8	1.19488	0.597439	−	0.801914i	$-0.296185\pi$
$61$	1.29214e8	1.19488	0.597439	+	0.801914i	$0.296185\pi$
$62$	3.85811e7	0.331598
$63$	0	0
$64$	5.91132e7	0.440428
$65$	−1.71354e8	−1.19065
$66$	0	0
$67$	−1.95575e7	−0.118570	−0.0592852	−	0.998241i	$-0.518882\pi$
$67$	−1.95575e7	−0.118570	−0.0592852	+	0.998241i	$0.518882\pi$
$68$	−4.85797e7	−0.275528
$69$	0	0
$70$	−1.72117e7	−0.0856806
$71$	2.18178e8	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$71$	2.18178e8	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$72$	0	0
$73$	4.41427e8	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$73$	4.41427e8	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$74$	−2.61398e7	−0.101335
$75$	0	0
$76$	−3.45829e8	−1.18905
$77$	5.68882e7	0.184422
$78$	0	0
$79$	−1.09562e7	−0.0316475	−0.0158237	−	0.999875i	$-0.505037\pi$
$79$	−1.09562e7	−0.0316475	−0.0158237	+	0.999875i	$0.505037\pi$
$80$	−7.55335e7	−0.206174
$81$	0	0
$82$	−1.00494e8	−0.245459
$83$	2.86843e8	0.663427	0.331714	−	0.943380i	$-0.392373\pi$
$83$	2.86843e8	0.663427	0.331714	+	0.943380i	$0.392373\pi$
$84$	0	0
$85$	−2.81779e8	−0.585495
$86$	3.48492e8	0.686988
$87$	0	0
$88$	−9.40544e8	−1.67189
$89$	−7.64398e8	−1.29141	−0.645706	−	0.763586i	$-0.723437\pi$
$89$	−7.64398e8	−1.29141	−0.645706	+	0.763586i	$0.723437\pi$
$90$	0	0
$91$	−5.64909e7	−0.0863560
$92$	2.64132e8	0.384393
$93$	0	0
$94$	−4.22154e8	−0.557694
$95$	−2.00593e9	−2.52673
$96$	0	0
$97$	815669.	0.000935494 0	0.000467747	−	1.00000i	$-0.499851\pi$
$97$	815669.	0.000935494 0	0.000467747		1.00000i	$0.499851\pi$
$98$	5.10764e8	0.559374
$99$	0	0
$100$	−7.40429e8	−0.740429
$101$	8.80693e8	0.842129	0.421064	−	0.907031i	$-0.361657\pi$
$101$	8.80693e8	0.842129	0.421064	+	0.907031i	$0.361657\pi$
$102$	0	0
$103$	1.47976e9	1.29546	0.647730	−	0.761870i	$-0.275718\pi$
$103$	1.47976e9	1.29546	0.647730	+	0.761870i	$0.275718\pi$
$104$	9.33977e8	0.782864
$105$	0	0
$106$	1.30876e9	1.00690
$107$	1.64812e9	1.21552	0.607759	−	0.794122i	$-0.292069\pi$
$107$	1.64812e9	1.21552	0.607759	+	0.794122i	$0.292069\pi$
$108$	0	0
$109$	−3.21670e8	−0.218268	−0.109134	−	0.994027i	$-0.534808\pi$
$109$	−3.21670e8	−0.218268	−0.109134	+	0.994027i	$0.534808\pi$
$110$	−2.20838e9	−1.43816
$111$	0	0
$112$	−2.49014e7	−0.0149535
$113$	1.91951e9	1.10748	0.553741	−	0.832689i	$-0.313200\pi$
$113$	1.91951e9	1.10748	0.553741	+	0.832689i	$0.313200\pi$
$114$	0	0
$115$	1.53206e9	0.816835
$116$	−1.40981e9	−0.722937
$117$	0	0
$118$	−1.48015e9	−0.702810
$119$	−9.28949e7	−0.0424650
$120$	0	0
$121$	4.94120e9	2.09555
$122$	−1.65365e9	−0.675810
$123$	0	0
$124$	1.04975e9	0.398740
$125$	−3.49874e8	−0.128179
$126$	0	0
$127$	−2.81305e9	−0.959535	−0.479767	−	0.877396i	$-0.659279\pi$
$127$	−2.81305e9	−0.959535	−0.479767	+	0.877396i	$0.659279\pi$
$128$	2.37444e9	0.781836
$129$	0	0
$130$	2.19296e9	0.673419
$131$	7.34372e8	0.217869	0.108934	−	0.994049i	$-0.465256\pi$
$131$	7.34372e8	0.217869	0.108934	+	0.994049i	$0.465256\pi$
$132$	0	0
$133$	−6.61300e8	−0.183259
$134$	2.50293e8	0.0670621
$135$	0	0
$136$	1.53585e9	0.384968
$137$	−1.00181e9	−0.242964	−0.121482	−	0.992594i	$-0.538765\pi$
$137$	−1.00181e9	−0.242964	−0.121482	+	0.992594i	$0.538765\pi$
$138$	0	0
$139$	−2.12304e9	−0.482382	−0.241191	−	0.970478i	$-0.577538\pi$
$139$	−2.12304e9	−0.482382	−0.241191	+	0.970478i	$0.577538\pi$
$140$	−4.68313e8	−0.103029
$141$	0	0
$142$	−2.79221e9	−0.576302
$143$	−7.24818e9	−1.44950
$144$	0	0
$145$	−8.17739e9	−1.53624
$146$	−5.64930e9	−1.02898
$147$	0	0
$148$	−7.11240e8	−0.121854
$149$	6.02059e9	1.00069	0.500346	−	0.865825i	$-0.333206\pi$
$149$	6.02059e9	1.00069	0.500346	+	0.865825i	$0.333206\pi$
$150$	0	0
$151$	−8.86951e9	−1.38836	−0.694182	−	0.719800i	$-0.744234\pi$
$151$	−8.86951e9	−1.38836	−0.694182	+	0.719800i	$0.744234\pi$
$152$	1.09334e10	1.66135
$153$	0	0
$154$	−7.28044e8	−0.104307
$155$	6.08893e9	0.847321
$156$	0	0
$157$	−2.51721e9	−0.330652	−0.165326	−	0.986239i	$-0.552868\pi$
$157$	−2.51721e9	−0.330652	−0.165326	+	0.986239i	$0.552868\pi$
$158$	1.40216e8	0.0178994
$159$	0	0
$160$	1.23512e10	1.48994
$161$	5.05078e8	0.0592436
$162$	0	0
$163$	−6.17153e8	−0.0684776	−0.0342388	−	0.999414i	$-0.510901\pi$
$163$	−6.17153e8	−0.0684776	−0.0342388	+	0.999414i	$0.510901\pi$
$164$	−2.73435e9	−0.295160
$165$	0	0
$166$	−3.67096e9	−0.375227
$167$	−1.06520e10	−1.05976	−0.529882	−	0.848072i	$-0.677764\pi$
$167$	−1.06520e10	−1.05976	−0.529882	+	0.848072i	$0.677764\pi$
$168$	0	0
$169$	−3.40693e9	−0.321272
$170$	3.60615e9	0.331149
$171$	0	0
$172$	9.48212e9	0.826090
$173$	−4.55410e8	−0.0386540	−0.0193270	−	0.999813i	$-0.506152\pi$
$173$	−4.55410e8	−0.0386540	−0.0193270	+	0.999813i	$0.506152\pi$
$174$	0	0
$175$	−1.41586e9	−0.114117
$176$	−3.19502e9	−0.250996
$177$	0	0
$178$	9.78262e9	0.730408
$179$	−9.54934e9	−0.695240	−0.347620	−	0.937636i	$-0.613010\pi$
$179$	−9.54934e9	−0.695240	−0.347620	+	0.937636i	$0.613010\pi$
$180$	0	0
$181$	2.10546e10	1.45812	0.729061	−	0.684449i	$-0.239957\pi$
$181$	2.10546e10	1.45812	0.729061	+	0.684449i	$0.239957\pi$
$182$	7.22960e8	0.0488420
$183$	0	0
$184$	−8.35056e9	−0.537075
$185$	−4.12543e9	−0.258938
$186$	0	0
$187$	−1.19191e10	−0.712779
$188$	−1.14864e10	−0.670617
$189$	0	0
$190$	2.56715e10	1.42909
$191$	2.71245e10	1.47473	0.737364	−	0.675496i	$-0.236070\pi$
$191$	2.71245e10	1.47473	0.737364	+	0.675496i	$0.236070\pi$
$192$	0	0
$193$	5.06145e9	0.262583	0.131292	−	0.991344i	$-0.458088\pi$
$193$	5.06145e9	0.262583	0.131292	+	0.991344i	$0.458088\pi$
$194$	−1.04388e7	−0.000529105 0
$195$	0	0
$196$	1.38974e10	0.672637
$197$	1.23458e10	0.584010	0.292005	−	0.956417i	$-0.405678\pi$
$197$	1.23458e10	0.584010	0.292005	+	0.956417i	$0.405678\pi$
$198$	0	0
$199$	1.20900e10	0.546497	0.273249	−	0.961943i	$-0.411902\pi$
$199$	1.20900e10	0.546497	0.273249	+	0.961943i	$0.411902\pi$
$200$	2.34087e10	1.03453
$201$	0	0
$202$	−1.12709e10	−0.476299
$203$	−2.69587e9	−0.111421
$204$	0	0
$205$	−1.58602e10	−0.627213
$206$	−1.89377e10	−0.732698
$207$	0	0
$208$	3.17271e9	0.117529
$209$	−8.48494e10	−3.07603
$210$	0	0
$211$	−1.72325e10	−0.598516	−0.299258	−	0.954172i	$-0.596739\pi$
$211$	−1.72325e10	−0.598516	−0.299258	+	0.954172i	$0.596739\pi$
$212$	3.56102e10	1.21077
$213$	0	0
$214$	−2.10923e10	−0.687483
$215$	5.49995e10	1.75544
$216$	0	0
$217$	2.00736e9	0.0614548
$218$	4.11667e9	0.123450
$219$	0	0
$220$	−6.00879e10	−1.72936
$221$	1.18358e10	0.333760
$222$	0	0
$223$	−3.12781e10	−0.846972	−0.423486	−	0.905903i	$-0.639194\pi$
$223$	−3.12781e10	−0.846972	−0.423486	+	0.905903i	$0.639194\pi$
$224$	4.07186e9	0.108063
$225$	0	0
$226$	−2.45655e10	−0.626379
$227$	9.86598e9	0.246617	0.123309	−	0.992368i	$-0.460649\pi$
$227$	9.86598e9	0.246617	0.123309	+	0.992368i	$0.460649\pi$
$228$	0	0
$229$	4.08107e10	0.980651	0.490325	−	0.871539i	$-0.336878\pi$
$229$	4.08107e10	0.980651	0.490325	+	0.871539i	$0.336878\pi$
$230$	−1.96070e10	−0.461992
$231$	0	0
$232$	4.45713e10	1.01009
$233$	−3.65168e10	−0.811693	−0.405846	−	0.913941i	$-0.633023\pi$
$233$	−3.65168e10	−0.811693	−0.405846	+	0.913941i	$0.633023\pi$
$234$	0	0
$235$	−6.66251e10	−1.42506
$236$	−4.02736e10	−0.845115
$237$	0	0
$238$	1.18885e9	0.0240177
$239$	−5.78935e9	−0.114773	−0.0573865	−	0.998352i	$-0.518277\pi$
$239$	−5.78935e9	−0.114773	−0.0573865	+	0.998352i	$0.518277\pi$
$240$	0	0
$241$	2.33566e10	0.445998	0.222999	−	0.974819i	$-0.428415\pi$
$241$	2.33566e10	0.445998	0.222999	+	0.974819i	$0.428415\pi$
$242$	−6.32366e10	−1.18522
$243$	0	0
$244$	−4.49942e10	−0.812649
$245$	8.06095e10	1.42935
$246$	0	0
$247$	8.42570e10	1.44035
$248$	−3.31881e10	−0.557121
$249$	0	0
$250$	4.47761e9	0.0724964
$251$	−1.44384e10	−0.229608	−0.114804	−	0.993388i	$-0.536624\pi$
$251$	−1.44384e10	−0.229608	−0.114804	+	0.993388i	$0.536624\pi$
$252$	0	0
$253$	6.48050e10	0.994411
$254$	3.60009e10	0.542702
$255$	0	0
$256$	−6.06535e10	−0.882625
$257$	9.77486e10	1.39769	0.698846	−	0.715272i	$-0.253697\pi$
$257$	9.77486e10	1.39769	0.698846	+	0.715272i	$0.253697\pi$
$258$	0	0
$259$	−1.36004e9	−0.0187804
$260$	5.96683e10	0.809774
$261$	0	0
$262$	−9.39835e9	−0.123224
$263$	−2.94535e10	−0.379609	−0.189804	−	0.981822i	$-0.560785\pi$
$263$	−2.94535e10	−0.379609	−0.189804	+	0.981822i	$0.560785\pi$
$264$	0	0
$265$	2.06551e11	2.57289
$266$	8.46319e9	0.103649
$267$	0	0
$268$	6.81023e9	0.0806409
$269$	6.86011e10	0.798814	0.399407	−	0.916774i	$-0.369216\pi$
$269$	6.86011e10	0.798814	0.399407	+	0.916774i	$0.369216\pi$
$270$	0	0
$271$	−6.74978e10	−0.760200	−0.380100	−	0.924945i	$-0.624110\pi$
$271$	−6.74978e10	−0.760200	−0.380100	+	0.924945i	$0.624110\pi$
$272$	5.21728e9	0.0577941
$273$	0	0
$274$	1.28209e10	0.137417
$275$	−1.81665e11	−1.91546
$276$	0	0
$277$	6.24779e10	0.637628	0.318814	−	0.947817i	$-0.396715\pi$
$277$	6.24779e10	0.637628	0.318814	+	0.947817i	$0.396715\pi$
$278$	2.71702e10	0.272830
$279$	0	0
$280$	1.48058e10	0.143953
$281$	−8.68104e10	−0.830603	−0.415301	−	0.909684i	$-0.636324\pi$
$281$	−8.68104e10	−0.830603	−0.415301	+	0.909684i	$0.636324\pi$
$282$	0	0
$283$	1.37271e11	1.27216	0.636078	−	0.771625i	$-0.280556\pi$
$283$	1.37271e11	1.27216	0.636078	+	0.771625i	$0.280556\pi$
$284$	−7.59733e10	−0.692992
$285$	0	0
$286$	9.27609e10	0.819818
$287$	−5.22867e9	−0.0454907
$288$	0	0
$289$	−9.91248e10	−0.835876
$290$	1.04653e11	0.868879
$291$	0	0
$292$	−1.53712e11	−1.23733
$293$	1.06733e11	0.846050	0.423025	−	0.906118i	$-0.360968\pi$
$293$	1.06733e11	0.846050	0.423025	+	0.906118i	$0.360968\pi$
$294$	0	0
$295$	−2.33600e11	−1.79587
$296$	2.24859e10	0.170254
$297$	0	0
$298$	−7.70503e10	−0.565980
$299$	−6.43525e10	−0.465634
$300$	0	0
$301$	1.81319e10	0.127319
$302$	1.13510e11	0.785243
$303$	0	0
$304$	3.71407e10	0.249413
$305$	−2.60982e11	−1.72688
$306$	0	0
$307$	−1.13729e11	−0.730716	−0.365358	−	0.930867i	$-0.619054\pi$
$307$	−1.13729e11	−0.730716	−0.365358	+	0.930867i	$0.619054\pi$
$308$	−1.98094e10	−0.125427
$309$	0	0
$310$	−7.79249e10	−0.479235
$311$	1.66986e9	0.0101218	0.00506091	−	0.999987i	$-0.498389\pi$
$311$	1.66986e9	0.0101218	0.00506091	+	0.999987i	$0.498389\pi$
$312$	0	0
$313$	5.22551e10	0.307736	0.153868	−	0.988091i	$-0.450827\pi$
$313$	5.22551e10	0.307736	0.153868	+	0.988091i	$0.450827\pi$
$314$	3.22148e10	0.187013
$315$	0	0
$316$	3.81513e9	0.0215237
$317$	1.76926e11	0.984070	0.492035	−	0.870575i	$-0.336253\pi$
$317$	1.76926e11	0.984070	0.492035	+	0.870575i	$0.336253\pi$
$318$	0	0
$319$	−3.45898e11	−1.87021
$320$	−1.19395e11	−0.636519
$321$	0	0
$322$	−6.46389e9	−0.0335075
$323$	1.38554e11	0.708284
$324$	0	0
$325$	1.80396e11	0.896918
$326$	7.89820e9	0.0387301
$327$	0	0
$328$	8.64467e10	0.412398
$329$	−2.19645e10	−0.103357
$330$	0	0
$331$	−1.59523e11	−0.730461	−0.365231	−	0.930917i	$-0.619010\pi$
$331$	−1.59523e11	−0.730461	−0.365231	+	0.930917i	$0.619010\pi$
$332$	−9.98834e10	−0.451203
$333$	0	0
$334$	1.36323e11	0.599390
$335$	3.95017e10	0.171362
$336$	0	0
$337$	1.07204e11	0.452769	0.226384	−	0.974038i	$-0.427309\pi$
$337$	1.07204e11	0.452769	0.226384	+	0.974038i	$0.427309\pi$
$338$	4.36012e10	0.181708
$339$	0	0
$340$	9.81199e10	0.398201
$341$	2.57558e11	1.03153
$342$	0	0
$343$	5.34448e10	0.208489
$344$	−2.99778e11	−1.15422
$345$	0	0
$346$	5.82824e9	0.0218623
$347$	4.80474e11	1.77904	0.889522	−	0.456892i	$-0.151037\pi$
$347$	4.80474e11	1.77904	0.889522	+	0.456892i	$0.151037\pi$
$348$	0	0
$349$	−2.05930e11	−0.743027	−0.371514	−	0.928428i	$-0.621161\pi$
$349$	−2.05930e11	−0.743027	−0.371514	+	0.928428i	$0.621161\pi$
$350$	1.81199e10	0.0645431
$351$	0	0
$352$	5.22448e11	1.81385
$353$	−2.84250e11	−0.974348	−0.487174	−	0.873305i	$-0.661972\pi$
$353$	−2.84250e11	−0.974348	−0.487174	+	0.873305i	$0.661972\pi$
$354$	0	0
$355$	−4.40671e11	−1.47261
$356$	2.66176e11	0.878301
$357$	0	0
$358$	1.22211e11	0.393220
$359$	−3.45110e11	−1.09656	−0.548280	−	0.836295i	$-0.684717\pi$
$359$	−3.45110e11	−1.09656	−0.548280	+	0.836295i	$0.684717\pi$
$360$	0	0
$361$	6.63650e11	2.05663
$362$	−2.69453e11	−0.824697
$363$	0	0
$364$	1.96711e10	0.0587315
$365$	−8.91581e11	−2.62932
$366$	0	0
$367$	7.68807e10	0.221218	0.110609	−	0.993864i	$-0.464720\pi$
$367$	7.68807e10	0.221218	0.110609	+	0.993864i	$0.464720\pi$
$368$	−2.83668e10	−0.0806296
$369$	0	0
$370$	5.27965e10	0.146453
$371$	6.80944e10	0.186607
$372$	0	0
$373$	3.76354e11	1.00671	0.503357	−	0.864078i	$-0.332098\pi$
$373$	3.76354e11	1.00671	0.503357	+	0.864078i	$0.332098\pi$
$374$	1.52538e11	0.403140
$375$	0	0
$376$	3.63144e11	0.936987
$377$	3.43483e11	0.875728
$378$	0	0
$379$	−3.18048e11	−0.791801	−0.395900	−	0.918294i	$-0.629567\pi$
$379$	−3.18048e11	−0.791801	−0.395900	+	0.918294i	$0.629567\pi$
$380$	6.98496e11	1.71845
$381$	0	0
$382$	−3.47135e11	−0.834089
$383$	1.21289e11	0.288023	0.144011	−	0.989576i	$-0.454000\pi$
$383$	1.21289e11	0.288023	0.144011	+	0.989576i	$0.454000\pi$
$384$	0	0
$385$	−1.14901e11	−0.266533
$386$	−6.47755e10	−0.148514
$387$	0	0
$388$	−2.84029e8	−0.000636239 0
$389$	7.63532e11	1.69065	0.845325	−	0.534252i	$-0.179407\pi$
$389$	7.63532e11	1.69065	0.845325	+	0.534252i	$0.179407\pi$
$390$	0	0
$391$	−1.05823e11	−0.228972
$392$	−4.39367e11	−0.939810
$393$	0	0
$394$	−1.57999e11	−0.330309
$395$	2.21291e10	0.0457379
$396$	0	0
$397$	−5.37941e11	−1.08687	−0.543435	−	0.839451i	$-0.682876\pi$
$397$	−5.37941e11	−1.08687	−0.543435	+	0.839451i	$0.682876\pi$
$398$	−1.54726e11	−0.309093
$399$	0	0
$400$	7.95192e10	0.155311
$401$	−2.23876e11	−0.432373	−0.216187	−	0.976352i	$-0.569362\pi$
$401$	−2.23876e11	−0.432373	−0.216187	+	0.976352i	$0.569362\pi$
$402$	0	0
$403$	−2.55760e11	−0.483013
$404$	−3.06672e11	−0.572740
$405$	0	0
$406$	3.45012e10	0.0630183
$407$	−1.74503e11	−0.315231
$408$	0	0
$409$	−9.59201e11	−1.69494	−0.847471	−	0.530842i	$-0.821876\pi$
$409$	−9.59201e11	−1.69494	−0.847471	+	0.530842i	$0.821876\pi$
$410$	2.02975e11	0.354745
$411$	0	0
$412$	−5.15277e11	−0.881055
$413$	−7.70117e10	−0.130251
$414$	0	0
$415$	−5.79358e11	−0.958805
$416$	−5.18800e11	−0.849337
$417$	0	0
$418$	1.08589e12	1.73977
$419$	8.85619e11	1.40373	0.701865	−	0.712310i	$-0.252351\pi$
$419$	8.85619e11	1.40373	0.701865	+	0.712310i	$0.252351\pi$
$420$	0	0
$421$	1.06418e12	1.65100	0.825498	−	0.564405i	$-0.190894\pi$
$421$	1.06418e12	1.65100	0.825498	+	0.564405i	$0.190894\pi$
$422$	2.20538e11	0.338514
$423$	0	0
$424$	−1.12582e12	−1.69170
$425$	2.96648e11	0.441053
$426$	0	0
$427$	−8.60387e10	−0.125247
$428$	−5.73901e11	−0.826685
$429$	0	0
$430$	−7.03873e11	−0.992857
$431$	−4.41717e11	−0.616589	−0.308295	−	0.951291i	$-0.599758\pi$
$431$	−4.41717e11	−0.616589	−0.308295	+	0.951291i	$0.599758\pi$
$432$	0	0
$433$	8.91127e11	1.21827	0.609136	−	0.793066i	$-0.291516\pi$
$433$	8.91127e11	1.21827	0.609136	+	0.793066i	$0.291516\pi$
$434$	−2.56898e10	−0.0347581
$435$	0	0
$436$	1.12011e11	0.148446
$437$	−7.53330e11	−0.988140
$438$	0	0
$439$	−3.66122e11	−0.470474	−0.235237	−	0.971938i	$-0.575587\pi$
$439$	−3.66122e11	−0.470474	−0.235237	+	0.971938i	$0.575587\pi$
$440$	1.89968e12	2.41626
$441$	0	0
$442$	−1.51473e11	−0.188771
$443$	−6.97003e11	−0.859840	−0.429920	−	0.902867i	$-0.641458\pi$
$443$	−6.97003e11	−0.859840	−0.429920	+	0.902867i	$0.641458\pi$
$444$	0	0
$445$	1.54391e12	1.86639
$446$	4.00292e11	0.479038
$447$	0	0
$448$	−3.93614e10	−0.0461657
$449$	−8.70295e11	−1.01055	−0.505275	−	0.862958i	$-0.668609\pi$
$449$	−8.70295e11	−1.01055	−0.505275	+	0.862958i	$0.668609\pi$
$450$	0	0
$451$	−6.70875e11	−0.763567
$452$	−6.68403e11	−0.753208
$453$	0	0
$454$	−1.26263e11	−0.139484
$455$	1.14099e11	0.124804
$456$	0	0
$457$	8.32855e11	0.893195	0.446598	−	0.894735i	$-0.352636\pi$
$457$	8.32855e11	0.893195	0.446598	+	0.894735i	$0.352636\pi$
$458$	−5.22287e11	−0.554645
$459$	0	0
$460$	−5.33486e11	−0.555537
$461$	−1.63782e11	−0.168893	−0.0844467	−	0.996428i	$-0.526912\pi$
$461$	−1.63782e11	−0.168893	−0.0844467	+	0.996428i	$0.526912\pi$
$462$	0	0
$463$	7.51302e11	0.759801	0.379901	−	0.925027i	$-0.375958\pi$
$463$	7.51302e11	0.759801	0.379901	+	0.925027i	$0.375958\pi$
$464$	1.51408e11	0.151642
$465$	0	0
$466$	4.67336e11	0.459084
$467$	−3.65892e11	−0.355981	−0.177991	−	0.984032i	$-0.556960\pi$
$467$	−3.65892e11	−0.355981	−0.177991	+	0.984032i	$0.556960\pi$
$468$	0	0
$469$	1.30226e10	0.0124286
$470$	8.52655e11	0.805996
$471$	0	0
$472$	1.27325e12	1.18080
$473$	2.32645e12	2.13707
$474$	0	0
$475$	2.11177e12	1.90338
$476$	3.23475e10	0.0288808
$477$	0	0
$478$	7.40910e10	0.0649142
$479$	5.30846e11	0.460743	0.230371	−	0.973103i	$-0.426006\pi$
$479$	5.30846e11	0.460743	0.230371	+	0.973103i	$0.426006\pi$
$480$	0	0
$481$	1.73285e11	0.147607
$482$	−2.98913e11	−0.252251
$483$	0	0
$484$	−1.72061e12	−1.42520
$485$	−1.64746e9	−0.00135200
$486$	0	0
$487$	9.23142e11	0.743684	0.371842	−	0.928296i	$-0.378726\pi$
$487$	9.23142e11	0.743684	0.371842	+	0.928296i	$0.378726\pi$
$488$	1.42250e12	1.13543
$489$	0	0
$490$	−1.03163e12	−0.808425
$491$	−1.62628e12	−1.26278	−0.631391	−	0.775465i	$-0.717516\pi$
$491$	−1.62628e12	−1.26278	−0.631391	+	0.775465i	$0.717516\pi$
$492$	0	0
$493$	5.64831e11	0.430633
$494$	−1.07830e12	−0.814648
$495$	0	0
$496$	−1.12740e11	−0.0836390
$497$	−1.45277e11	−0.106806
$498$	0	0
$499$	−6.59222e11	−0.475970	−0.237985	−	0.971269i	$-0.576487\pi$
$499$	−6.59222e11	−0.475970	−0.237985	+	0.971269i	$0.576487\pi$
$500$	1.21832e11	0.0871756
$501$	0	0
$502$	1.84780e11	0.129864
$503$	6.69047e11	0.466016	0.233008	−	0.972475i	$-0.425143\pi$
$503$	6.69047e11	0.466016	0.233008	+	0.972475i	$0.425143\pi$
$504$	0	0
$505$	−1.77880e12	−1.21707
$506$	−8.29362e11	−0.562428
$507$	0	0
$508$	9.79549e11	0.652589
$509$	−4.58197e11	−0.302568	−0.151284	−	0.988490i	$-0.548341\pi$
$509$	−4.58197e11	−0.302568	−0.151284	+	0.988490i	$0.548341\pi$
$510$	0	0
$511$	−2.93930e11	−0.190700
$512$	−4.39479e11	−0.282633
$513$	0	0
$514$	−1.25097e12	−0.790518
$515$	−2.98878e12	−1.87224
$516$	0	0
$517$	−2.81820e12	−1.73486
$518$	1.74056e10	0.0106220
$519$	0	0
$520$	−1.88642e12	−1.13142
$521$	3.34551e12	1.98926	0.994632	−	0.103472i	$-0.0329952\pi$
$521$	3.34551e12	1.98926	0.994632	+	0.103472i	$0.0329952\pi$
$522$	0	0
$523$	2.03945e12	1.19194	0.595970	−	0.803007i	$-0.296768\pi$
$523$	2.03945e12	1.19194	0.595970	+	0.803007i	$0.296768\pi$
$524$	−2.55720e11	−0.148175
$525$	0	0
$526$	3.76940e11	0.214702
$527$	−4.20577e11	−0.237518
$528$	0	0
$529$	−1.22579e12	−0.680556
$530$	−2.64340e12	−1.45520
$531$	0	0
$532$	2.30275e11	0.124636
$533$	6.66190e11	0.357541
$534$	0	0
$535$	−3.32882e12	−1.75670
$536$	−2.15306e11	−0.112672
$537$	0	0
$538$	−8.77944e11	−0.451800
$539$	3.40973e12	1.74009
$540$	0	0
$541$	3.69312e12	1.85355	0.926777	−	0.375612i	$-0.122567\pi$
$541$	3.69312e12	1.85355	0.926777	+	0.375612i	$0.122567\pi$
$542$	8.63825e11	0.429961
$543$	0	0
$544$	−8.53126e11	−0.417655
$545$	6.49699e11	0.315448
$546$	0	0
$547$	5.88944e11	0.281275	0.140637	−	0.990061i	$-0.455085\pi$
$547$	5.88944e11	0.281275	0.140637	+	0.990061i	$0.455085\pi$
$548$	3.48845e11	0.165242
$549$	0	0
$550$	2.32491e12	1.08336
$551$	4.02092e12	1.85842
$552$	0	0
$553$	7.29536e9	0.00331729
$554$	−7.99580e11	−0.360635
$555$	0	0
$556$	7.39275e11	0.328072
$557$	−3.85457e12	−1.69679	−0.848394	−	0.529366i	$-0.822430\pi$
$557$	−3.85457e12	−1.69679	−0.848394	+	0.529366i	$0.822430\pi$
$558$	0	0
$559$	−2.31020e12	−1.00068
$560$	5.02951e10	0.0216112
$561$	0	0
$562$	1.11098e12	0.469779
$563$	−3.03834e12	−1.27453	−0.637264	−	0.770646i	$-0.719934\pi$
$563$	−3.03834e12	−1.27453	−0.637264	+	0.770646i	$0.719934\pi$
$564$	0	0
$565$	−3.87696e12	−1.60057
$566$	−1.75677e12	−0.719517
$567$	0	0
$568$	2.40190e12	0.968250
$569$	−5.83227e11	−0.233256	−0.116628	−	0.993176i	$-0.537208\pi$
$569$	−5.83227e11	−0.233256	−0.116628	+	0.993176i	$0.537208\pi$
$570$	0	0
$571$	−2.09092e12	−0.823142	−0.411571	−	0.911378i	$-0.635020\pi$
$571$	−2.09092e12	−0.823142	−0.411571	+	0.911378i	$0.635020\pi$
$572$	2.52393e12	0.985816
$573$	0	0
$574$	6.69155e10	0.0257290
$575$	−1.61290e12	−0.615321
$576$	0	0
$577$	3.05766e12	1.14841	0.574206	−	0.818711i	$-0.305311\pi$
$577$	3.05766e12	1.14841	0.574206	+	0.818711i	$0.305311\pi$
$578$	1.26858e12	0.472762
$579$	0	0
$580$	2.84750e12	1.04481
$581$	−1.90999e11	−0.0695405
$582$	0	0
$583$	8.73699e12	3.13223
$584$	4.85962e12	1.72880
$585$	0	0
$586$	−1.36595e12	−0.478516
$587$	1.81796e12	0.631995	0.315997	−	0.948760i	$-0.397661\pi$
$587$	1.81796e12	0.631995	0.315997	+	0.948760i	$0.397661\pi$
$588$	0	0
$589$	−2.99400e12	−1.02502
$590$	2.98957e12	1.01572
$591$	0	0
$592$	7.63844e10	0.0255598
$593$	−1.51787e12	−0.504067	−0.252034	−	0.967719i	$-0.581099\pi$
$593$	−1.51787e12	−0.504067	−0.252034	+	0.967719i	$0.581099\pi$
$594$	0	0
$595$	1.87626e11	0.0613716
$596$	−2.09647e12	−0.680581
$597$	0	0
$598$	8.23571e11	0.263357
$599$	2.38179e12	0.755932	0.377966	−	0.925819i	$-0.376624\pi$
$599$	2.38179e12	0.755932	0.377966	+	0.925819i	$0.376624\pi$
$600$	0	0
$601$	−1.56564e12	−0.489506	−0.244753	−	0.969586i	$-0.578707\pi$
$601$	−1.56564e12	−0.489506	−0.244753	+	0.969586i	$0.578707\pi$
$602$	−2.32048e11	−0.0720102
$603$	0	0
$604$	3.08851e12	0.944239
$605$	−9.98009e12	−3.02855
$606$	0	0
$607$	4.20030e12	1.25583	0.627915	−	0.778282i	$-0.283908\pi$
$607$	4.20030e12	1.25583	0.627915	+	0.778282i	$0.283908\pi$
$608$	−6.07323e12	−1.80241
$609$	0	0
$610$	3.34000e12	0.976701
$611$	2.79852e12	0.812350
$612$	0	0
$613$	−3.01770e12	−0.863186	−0.431593	−	0.902068i	$-0.642048\pi$
$613$	−3.01770e12	−0.863186	−0.431593	+	0.902068i	$0.642048\pi$
$614$	1.45548e12	0.413285
$615$	0	0
$616$	6.26275e11	0.175247
$617$	1.00060e12	0.277958	0.138979	−	0.990295i	$-0.455618\pi$
$617$	1.00060e12	0.277958	0.138979	+	0.990295i	$0.455618\pi$
$618$	0	0
$619$	−4.40437e12	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−4.40437e12	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$620$	−2.12026e12	−0.576271
$621$	0	0
$622$	−2.13706e10	−0.00572479
$623$	5.08985e11	0.135366
$624$	0	0
$625$	−3.44636e12	−0.903443
$626$	−6.68750e11	−0.174052
$627$	0	0
$628$	8.76534e11	0.224880
$629$	2.84953e11	0.0725848
$630$	0	0
$631$	−4.71463e11	−0.118390	−0.0591950	−	0.998246i	$-0.518853\pi$
$631$	−4.71463e11	−0.118390	−0.0591950	+	0.998246i	$0.518853\pi$
$632$	−1.20616e11	−0.0300730
$633$	0	0
$634$	−2.26427e12	−0.556579
$635$	5.68172e12	1.38675
$636$	0	0
$637$	−3.38592e12	−0.814798
$638$	4.42674e12	1.05777
$639$	0	0
$640$	−4.79582e12	−1.12993
$641$	3.52736e12	0.825256	0.412628	−	0.910900i	$-0.364611\pi$
$641$	3.52736e12	0.825256	0.412628	+	0.910900i	$0.364611\pi$
$642$	0	0
$643$	4.60256e12	1.06182	0.530909	−	0.847429i	$-0.321851\pi$
$643$	4.60256e12	1.06182	0.530909	+	0.847429i	$0.321851\pi$
$644$	−1.75876e11	−0.0402922
$645$	0	0
$646$	−1.77319e12	−0.400598
$647$	−5.52821e12	−1.24027	−0.620134	−	0.784496i	$-0.712922\pi$
$647$	−5.52821e12	−1.24027	−0.620134	+	0.784496i	$0.712922\pi$
$648$	0	0
$649$	−9.88115e12	−2.18628
$650$	−2.30868e12	−0.507286
$651$	0	0
$652$	2.14902e11	0.0465723
$653$	6.02691e12	1.29714	0.648568	−	0.761157i	$-0.275368\pi$
$653$	6.02691e12	1.29714	0.648568	+	0.761157i	$0.275368\pi$
$654$	0	0
$655$	−1.48326e12	−0.314871
$656$	2.93659e11	0.0619121
$657$	0	0
$658$	2.81098e11	0.0584575
$659$	2.97106e11	0.0613658	0.0306829	−	0.999529i	$-0.490232\pi$
$659$	2.97106e11	0.0613658	0.0306829	+	0.999529i	$0.490232\pi$
$660$	0	0
$661$	5.47716e12	1.11596	0.557981	−	0.829854i	$-0.311576\pi$
$661$	5.47716e12	1.11596	0.557981	+	0.829854i	$0.311576\pi$
$662$	2.04154e12	0.413141
$663$	0	0
$664$	3.15782e12	0.630422
$665$	1.33567e12	0.264852
$666$	0	0
$667$	−3.07103e12	−0.600784
$668$	3.70921e12	0.720755
$669$	0	0
$670$	−5.05535e11	−0.0969202
$671$	−1.10394e13	−2.10229
$672$	0	0
$673$	−4.06690e12	−0.764180	−0.382090	−	0.924125i	$-0.624795\pi$
$673$	−4.06690e12	−0.764180	−0.382090	+	0.924125i	$0.624795\pi$
$674$	−1.37198e12	−0.256081
$675$	0	0
$676$	1.18635e12	0.218500
$677$	4.84588e12	0.886591	0.443295	−	0.896376i	$-0.353809\pi$
$677$	4.84588e12	0.886591	0.443295	+	0.896376i	$0.353809\pi$
$678$	0	0
$679$	−5.43125e8	−9.80586e−5 0
$680$	−3.10207e12	−0.556367
$681$	0	0
$682$	−3.29618e12	−0.583419
$683$	−5.80468e12	−1.02067	−0.510334	−	0.859976i	$-0.670478\pi$
$683$	−5.80468e12	−1.02067	−0.510334	+	0.859976i	$0.670478\pi$
$684$	0	0
$685$	2.02342e12	0.351138
$686$	−6.83977e11	−0.117919
$687$	0	0
$688$	−1.01834e12	−0.173279
$689$	−8.67598e12	−1.46667
$690$	0	0
$691$	9.46733e12	1.57971	0.789853	−	0.613296i	$-0.210157\pi$
$691$	9.46733e12	1.57971	0.789853	+	0.613296i	$0.210157\pi$
$692$	1.58581e11	0.0262890
$693$	0	0
$694$	−6.14901e12	−1.00621
$695$	4.28805e12	0.697153
$696$	0	0
$697$	1.09550e12	0.175818
$698$	2.63545e12	0.420248
$699$	0	0
$700$	4.93025e11	0.0776118
$701$	7.16819e12	1.12119	0.560594	−	0.828091i	$-0.310573\pi$
$701$	7.16819e12	1.12119	0.560594	+	0.828091i	$0.310573\pi$
$702$	0	0
$703$	2.02852e12	0.313243
$704$	−5.05034e12	−0.774896
$705$	0	0
$706$	3.63778e12	0.551080
$707$	−5.86422e11	−0.0882720
$708$	0	0
$709$	3.05585e12	0.454175	0.227088	−	0.973874i	$-0.427080\pi$
$709$	3.05585e12	0.454175	0.227088	+	0.973874i	$0.427080\pi$
$710$	5.63962e12	0.832889
$711$	0	0
$712$	−8.41517e12	−1.22716
$713$	2.28671e12	0.331366
$714$	0	0
$715$	1.46397e13	2.09486
$716$	3.32523e12	0.472839
$717$	0	0
$718$	4.41665e12	0.620202
$719$	−5.43142e12	−0.757938	−0.378969	−	0.925409i	$-0.623721\pi$
$719$	−5.43142e12	−0.757938	−0.378969	+	0.925409i	$0.623721\pi$
$720$	0	0
$721$	−9.85320e11	−0.135790
$722$	−8.49327e12	−1.16321
$723$	0	0
$724$	−7.33155e12	−0.991682
$725$	8.60889e12	1.15725
$726$	0	0
$727$	7.25700e12	0.963502	0.481751	−	0.876308i	$-0.340001\pi$
$727$	7.25700e12	0.963502	0.481751	+	0.876308i	$0.340001\pi$
$728$	−6.21902e11	−0.0820599
$729$	0	0
$730$	1.14103e13	1.48711
$731$	−3.79895e12	−0.492079
$732$	0	0
$733$	8.75487e11	0.112016	0.0560082	−	0.998430i	$-0.482163\pi$
$733$	8.75487e11	0.112016	0.0560082	+	0.998430i	$0.482163\pi$
$734$	−9.83905e11	−0.125118
$735$	0	0
$736$	4.63852e12	0.582679
$737$	1.67090e12	0.208615
$738$	0	0
$739$	3.05185e12	0.376411	0.188206	−	0.982130i	$-0.439733\pi$
$739$	3.05185e12	0.376411	0.188206	+	0.982130i	$0.439733\pi$
$740$	1.43654e12	0.176106
$741$	0	0
$742$	−8.71459e11	−0.105543
$743$	−8.89001e11	−0.107017	−0.0535085	−	0.998567i	$-0.517040\pi$
$743$	−8.89001e11	−0.107017	−0.0535085	+	0.998567i	$0.517040\pi$
$744$	0	0
$745$	−1.21602e13	−1.44623
$746$	−4.81650e12	−0.569386
$747$	0	0
$748$	4.15041e12	0.484768
$749$	−1.09742e12	−0.127411
$750$	0	0
$751$	6.97996e12	0.800707	0.400353	−	0.916361i	$-0.368887\pi$
$751$	6.97996e12	0.800707	0.400353	+	0.916361i	$0.368887\pi$
$752$	1.23360e12	0.140667
$753$	0	0
$754$	−4.39583e12	−0.495302
$755$	1.79144e13	2.00651
$756$	0	0
$757$	1.54121e13	1.70581	0.852905	−	0.522066i	$-0.174838\pi$
$757$	1.54121e13	1.70581	0.852905	+	0.522066i	$0.174838\pi$
$758$	4.07031e12	0.447833
$759$	0	0
$760$	−2.20830e13	−2.40103
$761$	−8.97482e12	−0.970052	−0.485026	−	0.874500i	$-0.661190\pi$
$761$	−8.97482e12	−0.970052	−0.485026	+	0.874500i	$0.661190\pi$
$762$	0	0
$763$	2.14188e11	0.0228789
$764$	−9.44520e12	−1.00298
$765$	0	0
$766$	−1.55223e12	−0.162902
$767$	9.81215e12	1.02373
$768$	0	0
$769$	2.29561e12	0.236718	0.118359	−	0.992971i	$-0.462237\pi$
$769$	2.29561e12	0.236718	0.118359	+	0.992971i	$0.462237\pi$
$770$	1.47048e12	0.150748
$771$	0	0
$772$	−1.76248e12	−0.178585
$773$	−4.85355e12	−0.488936	−0.244468	−	0.969657i	$-0.578613\pi$
$773$	−4.85355e12	−0.488936	−0.244468	+	0.969657i	$0.578613\pi$
$774$	0	0
$775$	−6.41023e12	−0.638287
$776$	8.97960e9	0.000888954 0
$777$	0	0
$778$	−9.77154e12	−0.956213
$779$	7.79863e12	0.758752
$780$	0	0
$781$	−1.86401e13	−1.79274
$782$	1.35430e12	0.129504
$783$	0	0
$784$	−1.49253e12	−0.141091
$785$	5.08419e12	0.477869
$786$	0	0
$787$	−1.25927e13	−1.17013	−0.585063	−	0.810987i	$-0.698930\pi$
$787$	−1.25927e13	−1.17013	−0.585063	+	0.810987i	$0.698930\pi$
$788$	−4.29899e12	−0.397191
$789$	0	0
$790$	−2.83203e11	−0.0258688
$791$	−1.27813e12	−0.116086
$792$	0	0
$793$	1.09623e13	0.984400
$794$	6.88447e12	0.614721
$795$	0	0
$796$	−4.20994e12	−0.371678
$797$	5.33654e12	0.468487	0.234243	−	0.972178i	$-0.424739\pi$
$797$	5.33654e12	0.468487	0.234243	+	0.972178i	$0.424739\pi$
$798$	0	0
$799$	4.60195e12	0.399468
$800$	−1.30029e13	−1.12237
$801$	0	0
$802$	2.86513e12	0.244545
$803$	−3.77133e13	−3.20092
$804$	0	0
$805$	−1.02014e12	−0.0856207
$806$	3.27316e12	0.273187
$807$	0	0
$808$	9.69545e12	0.800234
$809$	−2.28889e13	−1.87870	−0.939348	−	0.342964i	$-0.888569\pi$
$809$	−2.28889e13	−1.87870	−0.939348	+	0.342964i	$0.888569\pi$
$810$	0	0
$811$	−9.01947e11	−0.0732128	−0.0366064	−	0.999330i	$-0.511655\pi$
$811$	−9.01947e11	−0.0732128	−0.0366064	+	0.999330i	$0.511655\pi$
$812$	9.38744e11	0.0757783
$813$	0	0
$814$	2.23326e12	0.178291
$815$	1.24651e12	0.0989659
$816$	0	0
$817$	−2.70439e13	−2.12359
$818$	1.22757e13	0.958640
$819$	0	0
$820$	5.52276e12	0.426574
$821$	−7.51091e12	−0.576963	−0.288482	−	0.957485i	$-0.593150\pi$
$821$	−7.51091e12	−0.576963	−0.288482	+	0.957485i	$0.593150\pi$
$822$	0	0
$823$	−1.53129e13	−1.16348	−0.581738	−	0.813377i	$-0.697627\pi$
$823$	−1.53129e13	−1.16348	−0.581738	+	0.813377i	$0.697627\pi$
$824$	1.62905e13	1.23101
$825$	0	0
$826$	9.85582e11	0.0736686
$827$	1.24239e13	0.923602	0.461801	−	0.886984i	$-0.347203\pi$
$827$	1.24239e13	0.923602	0.461801	+	0.886984i	$0.347203\pi$
$828$	0	0
$829$	7.14034e12	0.525078	0.262539	−	0.964921i	$-0.415440\pi$
$829$	7.14034e12	0.525078	0.262539	+	0.964921i	$0.415440\pi$
$830$	7.41451e12	0.542289
$831$	0	0
$832$	5.01508e12	0.362846
$833$	−5.56789e12	−0.400671
$834$	0	0
$835$	2.15147e13	1.53160
$836$	2.95459e13	2.09204
$837$	0	0
$838$	−1.13340e13	−0.793934
$839$	1.84306e13	1.28414	0.642068	−	0.766648i	$-0.278077\pi$
$839$	1.84306e13	1.28414	0.642068	+	0.766648i	$0.278077\pi$
$840$	0	0
$841$	1.88459e12	0.129907
$842$	−1.36192e13	−0.933785
$843$	0	0
$844$	6.00062e12	0.407057
$845$	6.88121e12	0.464312
$846$	0	0
$847$	−3.29017e12	−0.219656
$848$	−3.82440e12	−0.253970
$849$	0	0
$850$	−3.79644e12	−0.249455
$851$	−1.54931e12	−0.101264
$852$	0	0
$853$	5.76438e12	0.372805	0.186403	−	0.982473i	$-0.440317\pi$
$853$	5.76438e12	0.372805	0.186403	+	0.982473i	$0.440317\pi$
$854$	1.10111e12	0.0708385
$855$	0	0
$856$	1.81439e13	1.15505
$857$	4.75199e12	0.300927	0.150464	−	0.988616i	$-0.451923\pi$
$857$	4.75199e12	0.300927	0.150464	+	0.988616i	$0.451923\pi$
$858$	0	0
$859$	4.45715e12	0.279311	0.139656	−	0.990200i	$-0.455400\pi$
$859$	4.45715e12	0.279311	0.139656	+	0.990200i	$0.455400\pi$
$860$	−1.91517e13	−1.19389
$861$	0	0
$862$	5.65301e12	0.348736
$863$	−1.65466e13	−1.01545	−0.507727	−	0.861518i	$-0.669514\pi$
$863$	−1.65466e13	−1.01545	−0.507727	+	0.861518i	$0.669514\pi$
$864$	0	0
$865$	9.19823e11	0.0558640
$866$	−1.14045e13	−0.689041
$867$	0	0
$868$	−6.98994e11	−0.0417960
$869$	9.36045e11	0.0556811
$870$	0	0
$871$	−1.65923e12	−0.0976842
$872$	−3.54122e12	−0.207410
$873$	0	0
$874$	9.64097e12	0.558881
$875$	2.32968e11	0.0134357
$876$	0	0
$877$	3.05493e13	1.74382	0.871912	−	0.489663i	$-0.162880\pi$
$877$	3.05493e13	1.74382	0.871912	+	0.489663i	$0.162880\pi$
$878$	4.68556e12	0.266095
$879$	0	0
$880$	6.45321e12	0.362747
$881$	6.84641e12	0.382887	0.191444	−	0.981504i	$-0.438683\pi$
$881$	6.84641e12	0.382887	0.191444	+	0.981504i	$0.438683\pi$
$882$	0	0
$883$	−2.29262e13	−1.26914	−0.634570	−	0.772865i	$-0.718823\pi$
$883$	−2.29262e13	−1.26914	−0.634570	+	0.772865i	$0.718823\pi$
$884$	−4.12143e12	−0.226993
$885$	0	0
$886$	8.92011e12	0.486316
$887$	2.46814e13	1.33879	0.669397	−	0.742905i	$-0.266553\pi$
$887$	2.46814e13	1.33879	0.669397	+	0.742905i	$0.266553\pi$
$888$	0	0
$889$	1.87311e12	0.100578
$890$	−1.97587e13	−1.05561
$891$	0	0
$892$	1.08916e13	0.576034
$893$	3.27604e13	1.72392
$894$	0	0
$895$	1.92875e13	1.00478
$896$	−1.58105e12	−0.0819521
$897$	0	0
$898$	1.11379e13	0.571556
$899$	−1.22054e13	−0.623207
$900$	0	0
$901$	−1.42670e13	−0.721224
$902$	8.58573e12	0.431865
$903$	0	0
$904$	2.11316e13	1.05238
$905$	−4.25255e13	−2.10732
$906$	0	0
$907$	−2.45774e13	−1.20588	−0.602938	−	0.797788i	$-0.706003\pi$
$907$	−2.45774e13	−1.20588	−0.602938	+	0.797788i	$0.706003\pi$
$908$	−3.43549e12	−0.167727
$909$	0	0
$910$	−1.46021e12	−0.0705879
$911$	5.71188e12	0.274756	0.137378	−	0.990519i	$-0.456133\pi$
$911$	5.71188e12	0.274756	0.137378	+	0.990519i	$0.456133\pi$
$912$	0	0
$913$	−2.45065e13	−1.16725
$914$	−1.06587e13	−0.505181
$915$	0	0
$916$	−1.42109e13	−0.666950
$917$	−4.88992e11	−0.0228370
$918$	0	0
$919$	−1.90767e13	−0.882231	−0.441116	−	0.897450i	$-0.645417\pi$
$919$	−1.90767e13	−0.882231	−0.441116	+	0.897450i	$0.645417\pi$
$920$	1.68662e13	0.776198
$921$	0	0
$922$	2.09605e12	0.0955242
$923$	1.85099e13	0.839455
$924$	0	0
$925$	4.34312e12	0.195058
$926$	−9.61502e12	−0.429735
$927$	0	0
$928$	−2.47582e13	−1.09586
$929$	−2.72807e13	−1.20167	−0.600835	−	0.799373i	$-0.705165\pi$
$929$	−2.72807e13	−1.20167	−0.600835	+	0.799373i	$0.705165\pi$
$930$	0	0
$931$	−3.96367e13	−1.72911
$932$	1.27158e13	0.552040
$933$	0	0
$934$	4.68261e12	0.201339
$935$	2.40738e13	1.03013
$936$	0	0
$937$	1.38541e13	0.587152	0.293576	−	0.955936i	$-0.405155\pi$
$937$	1.38541e13	0.587152	0.293576	+	0.955936i	$0.405155\pi$
$938$	−1.66661e11	−0.00702946
$939$	0	0
$940$	2.31999e13	0.969195
$941$	1.33886e12	0.0556648	0.0278324	−	0.999613i	$-0.491140\pi$
$941$	1.33886e12	0.0556648	0.0278324	+	0.999613i	$0.491140\pi$
$942$	0	0
$943$	−5.95632e12	−0.245287
$944$	4.32522e12	0.177270
$945$	0	0
$946$	−2.97734e13	−1.20870
$947$	−4.26511e13	−1.72328	−0.861639	−	0.507521i	$-0.830562\pi$
$947$	−4.26511e13	−1.72328	−0.861639	+	0.507521i	$0.830562\pi$
$948$	0	0
$949$	3.74500e13	1.49883
$950$	−2.70261e13	−1.07653
$951$	0	0
$952$	−1.02267e12	−0.0403524
$953$	2.62256e13	1.02993	0.514964	−	0.857212i	$-0.327805\pi$
$953$	2.62256e13	1.02993	0.514964	+	0.857212i	$0.327805\pi$
$954$	0	0
$955$	−5.47853e13	−2.13132
$956$	2.01595e12	0.0780581
$957$	0	0
$958$	−6.79366e12	−0.260591
$959$	6.67067e11	0.0254675
$960$	0	0
$961$	−1.73514e13	−0.656266
$962$	−2.21766e12	−0.0834849
$963$	0	0
$964$	−8.13314e12	−0.303328
$965$	−1.02230e13	−0.379493
$966$	0	0
$967$	−1.79584e11	−0.00660463	−0.00330232	−	0.999995i	$-0.501051\pi$
$967$	−1.79584e11	−0.00660463	−0.00330232	+	0.999995i	$0.501051\pi$
$968$	5.43971e13	1.99130
$969$	0	0
$970$	2.10839e10	0.000764679 0
$971$	2.95293e13	1.06602	0.533012	−	0.846108i	$-0.321060\pi$
$971$	2.95293e13	1.06602	0.533012	+	0.846108i	$0.321060\pi$
$972$	0	0
$973$	1.41365e12	0.0505633
$974$	−1.18142e13	−0.420619
$975$	0	0
$976$	4.83221e12	0.170460
$977$	4.93042e13	1.73124	0.865622	−	0.500699i	$-0.166924\pi$
$977$	4.93042e13	1.73124	0.865622	+	0.500699i	$0.166924\pi$
$978$	0	0
$979$	6.53064e13	2.27213
$980$	−2.80695e13	−0.972115
$981$	0	0
$982$	2.08128e13	0.714215
$983$	−8.89206e12	−0.303747	−0.151873	−	0.988400i	$-0.548531\pi$
$983$	−8.89206e12	−0.303747	−0.151873	+	0.988400i	$0.548531\pi$
$984$	0	0
$985$	−2.49356e13	−0.844029
$986$	−7.22860e12	−0.243561
$987$	0	0
$988$	−2.93396e13	−0.979599
$989$	2.06552e13	0.686508
$990$	0	0
$991$	−1.67054e13	−0.550207	−0.275103	−	0.961415i	$-0.588712\pi$
$991$	−1.67054e13	−0.550207	−0.275103	+	0.961415i	$0.588712\pi$
$992$	1.84351e13	0.604426
$993$	0	0
$994$	1.85923e12	0.0604080
$995$	−2.44191e13	−0.789814
$996$	0	0
$997$	2.62911e12	0.0842714	0.0421357	−	0.999112i	$-0.486584\pi$
$997$	2.62911e12	0.0842714	0.0421357	+	0.999112i	$0.486584\pi$
$998$	8.43660e12	0.269203
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.a.d.1.2	✓	4
3.2	odd	2	inner	27.10.a.d.1.3	yes	4
9.2	odd	6		81.10.c.j.28.2		8
9.4	even	3		81.10.c.j.55.3		8
9.5	odd	6		81.10.c.j.55.2		8
9.7	even	3		81.10.c.j.28.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.d.1.2	✓	4	1.1	even	1	trivial
27.10.a.d.1.3	yes	4	3.2	odd	2	inner
81.10.c.j.28.2		8	9.2	odd	6
81.10.c.j.28.3		8	9.7	even	3
81.10.c.j.55.2		8	9.5	odd	6
81.10.c.j.55.3		8	9.4	even	3

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-12.7978 q^{2} -348.216 q^{4} -2019.77 q^{5} -665.864 q^{7} +11008.9 q^{8} +25848.6 q^{10} -85435.1 q^{11} +84838.5 q^{13} +8521.60 q^{14} +37397.1 q^{16} +139510. q^{17} +993145. q^{19} +703317. q^{20} +1.09338e6 q^{22} -758529. q^{23} +2.12635e6 q^{25} -1.08575e6 q^{26} +231865. q^{28} +4.04867e6 q^{29} -3.01466e6 q^{31} -6.11515e6 q^{32} -1.78543e6 q^{34} +1.34489e6 q^{35} +2.04252e6 q^{37} -1.27101e7 q^{38} -2.22354e7 q^{40} +7.85245e6 q^{41} -2.72306e7 q^{43} +2.97499e7 q^{44} +9.70751e6 q^{46} +3.29865e7 q^{47} -3.99102e7 q^{49} -2.72126e7 q^{50} -2.95421e7 q^{52} -1.02265e8 q^{53} +1.72559e8 q^{55} -7.33042e6 q^{56} -5.18141e7 q^{58} +1.15657e8 q^{59} +1.29214e8 q^{61} +3.85811e7 q^{62} +5.91132e7 q^{64} -1.71354e8 q^{65} -1.95575e7 q^{67} -4.85797e7 q^{68} -1.72117e7 q^{70} +2.18178e8 q^{71} +4.41427e8 q^{73} -2.61398e7 q^{74} -3.45829e8 q^{76} +5.68882e7 q^{77} -1.09562e7 q^{79} -7.55335e7 q^{80} -1.00494e8 q^{82} +2.86843e8 q^{83} -2.81779e8 q^{85} +3.48492e8 q^{86} -9.40544e8 q^{88} -7.64398e8 q^{89} -5.64909e7 q^{91} +2.64132e8 q^{92} -4.22154e8 q^{94} -2.00593e9 q^{95} +815669. q^{97} +5.10764e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2236 q^{4} + 11852 q^{7} - 41760 q^{10} + 179684 q^{13} + 2348680 q^{16} + 1011428 q^{19} + 3473568 q^{22} + 2554060 q^{25} + 19793924 q^{28} + 889136 q^{31} - 43111008 q^{34} - 3805156 q^{37} - 133649280 q^{40}+ \cdots + 1935734516 q^{97}+O(q^{100}) \) 4 * q + 2236 * q^4 + 11852 * q^7 - 41760 * q^10 + 179684 * q^13 + 2348680 * q^16 + 1011428 * q^19 + 3473568 * q^22 + 2554060 * q^25 + 19793924 * q^28 + 889136 * q^31 - 43111008 * q^34 - 3805156 * q^37 - 133649280 * q^40 + 11585024 * q^43 + 13428000 * q^46 - 73622328 * q^49 - 44411860 * q^52 + 314722080 * q^55 - 407685888 * q^58 + 219882572 * q^61 + 1519285360 * q^64 + 390561236 * q^67 - 650481120 * q^70 + 1197949508 * q^73 - 2121017572 * q^76 + 49814324 * q^79 - 1767126528 * q^82 + 370444320 * q^85 - 653198976 * q^88 - 47017172 * q^91 - 4146926688 * q^94 + 1935734516 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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