LMFDB - Embedded newform 27.10.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,10,Mod(1,27)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.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:	$13.9059675764$
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			1.3
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	$-0.408739\pi$
$2$	12.7978	0.565589	0.282794	+	0.959181i	$0.408739\pi$
$3$	0	0
$3$	0	0
$4$	−348.216	−0.680110
$5$	2019.77	1.44523	0.722615	−	0.691251i	$-0.242940\pi$
$5$	2019.77	1.44523	0.722615	+	0.691251i	$0.242940\pi$
$6$	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	$-0.157737\pi$
$11$	85435.1	1.75942	0.879709	+	0.475512i	$0.157737\pi$
$12$	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	$-0.564926\pi$
$17$	−139510.	−0.405122	−0.202561	+	0.979270i	$0.564926\pi$
$18$	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	$-0.408804\pi$
$23$	758529.	0.565193	0.282597	+	0.959239i	$0.408804\pi$
$24$	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	$-0.678366\pi$
$29$	−4.04867e6	−1.06297	−0.531486	+	0.847067i	$0.678366\pi$
$30$	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	$-0.569625\pi$
$41$	−7.85245e6	−0.433988	−0.216994	+	0.976173i	$0.569625\pi$
$42$	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	$-0.664107\pi$
$47$	−3.29865e7	−0.986042	−0.493021	+	0.870017i	$0.664107\pi$
$48$	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	$-0.150612\pi$
$53$	1.02265e8	1.78026	0.890131	+	0.455704i	$0.150612\pi$
$54$	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	$-0.713399\pi$
$59$	−1.15657e8	−1.24262	−0.621308	+	0.783566i	$0.713399\pi$
$60$	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	$-0.670159\pi$
$71$	−2.18178e8	−1.01894	−0.509471	+	0.860488i	$0.670159\pi$
$72$	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	$-0.607627\pi$
$83$	−2.86843e8	−0.663427	−0.331714	+	0.943380i	$0.607627\pi$
$84$	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	$-0.276563\pi$
$89$	7.64398e8	1.29141	0.645706	+	0.763586i	$0.276563\pi$
$90$	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	$-0.638343\pi$
$101$	−8.80693e8	−0.842129	−0.421064	+	0.907031i	$0.638343\pi$
$102$	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	$-0.707931\pi$
$107$	−1.64812e9	−1.21552	−0.607759	+	0.794122i	$0.707931\pi$
$108$	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	$-0.686800\pi$
$113$	−1.91951e9	−1.10748	−0.553741	+	0.832689i	$0.686800\pi$
$114$	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	$-0.534744\pi$
$131$	−7.34372e8	−0.217869	−0.108934	+	0.994049i	$0.534744\pi$
$132$	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	$-0.461235\pi$
$137$	1.00181e9	0.242964	0.121482	+	0.992594i	$0.461235\pi$
$138$	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	$-0.666794\pi$
$149$	−6.02059e9	−1.00069	−0.500346	+	0.865825i	$0.666794\pi$
$150$	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	$-0.322236\pi$
$167$	1.06520e10	1.05976	0.529882	+	0.848072i	$0.322236\pi$
$168$	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	$-0.493848\pi$
$173$	4.55410e8	0.0386540	0.0193270	+	0.999813i	$0.493848\pi$
$174$	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	$-0.386990\pi$
$179$	9.54934e9	0.695240	0.347620	+	0.937636i	$0.386990\pi$
$180$	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	$-0.763930\pi$
$191$	−2.71245e10	−1.47473	−0.737364	+	0.675496i	$0.763930\pi$
$192$	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	$-0.594322\pi$
$197$	−1.23458e10	−0.584010	−0.292005	+	0.956417i	$0.594322\pi$
$198$	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	$-0.539351\pi$
$227$	−9.86598e9	−0.246617	−0.123309	+	0.992368i	$0.539351\pi$
$228$	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	$-0.366977\pi$
$233$	3.65168e10	0.811693	0.405846	+	0.913941i	$0.366977\pi$
$234$	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	$-0.481723\pi$
$239$	5.78935e9	0.114773	0.0573865	+	0.998352i	$0.481723\pi$
$240$	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	$-0.463376\pi$
$251$	1.44384e10	0.229608	0.114804	+	0.993388i	$0.463376\pi$
$252$	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	$-0.746303\pi$
$257$	−9.77486e10	−1.39769	−0.698846	+	0.715272i	$0.746303\pi$
$258$	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	$-0.439215\pi$
$263$	2.94535e10	0.379609	0.189804	+	0.981822i	$0.439215\pi$
$264$	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	$-0.630784\pi$
$269$	−6.86011e10	−0.798814	−0.399407	+	0.916774i	$0.630784\pi$
$270$	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	$-0.363676\pi$
$281$	8.68104e10	0.830603	0.415301	+	0.909684i	$0.363676\pi$
$282$	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	$-0.639032\pi$
$293$	−1.06733e11	−0.846050	−0.423025	+	0.906118i	$0.639032\pi$
$294$	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	$-0.501611\pi$
$311$	−1.66986e9	−0.0101218	−0.00506091	+	0.999987i	$0.501611\pi$
$312$	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	$-0.663747\pi$
$317$	−1.76926e11	−0.984070	−0.492035	+	0.870575i	$0.663747\pi$
$318$	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	$-0.848963\pi$
$347$	−4.80474e11	−1.77904	−0.889522	+	0.456892i	$0.848963\pi$
$348$	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	$-0.338028\pi$
$353$	2.84250e11	0.974348	0.487174	+	0.873305i	$0.338028\pi$
$354$	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	$-0.315283\pi$
$359$	3.45110e11	1.09656	0.548280	+	0.836295i	$0.315283\pi$
$360$	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	$-0.546000\pi$
$383$	−1.21289e11	−0.288023	−0.144011	+	0.989576i	$0.546000\pi$
$384$	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	$-0.820593\pi$
$389$	−7.63532e11	−1.69065	−0.845325	+	0.534252i	$0.820593\pi$
$390$	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	$-0.430638\pi$
$401$	2.23876e11	0.432373	0.216187	+	0.976352i	$0.430638\pi$
$402$	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	$-0.747649\pi$
$419$	−8.85619e11	−1.40373	−0.701865	+	0.712310i	$0.747649\pi$
$420$	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	$-0.400242\pi$
$431$	4.41717e11	0.616589	0.308295	+	0.951291i	$0.400242\pi$
$432$	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	$-0.358542\pi$
$443$	6.97003e11	0.859840	0.429920	+	0.902867i	$0.358542\pi$
$444$	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	$-0.331391\pi$
$449$	8.70295e11	1.01055	0.505275	+	0.862958i	$0.331391\pi$
$450$	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	$-0.473088\pi$
$461$	1.63782e11	0.168893	0.0844467	+	0.996428i	$0.473088\pi$
$462$	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	$-0.443040\pi$
$467$	3.65892e11	0.355981	0.177991	+	0.984032i	$0.443040\pi$
$468$	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	$-0.573994\pi$
$479$	−5.30846e11	−0.460743	−0.230371	+	0.973103i	$0.573994\pi$
$480$	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	$-0.282484\pi$
$491$	1.62628e12	1.26278	0.631391	+	0.775465i	$0.282484\pi$
$492$	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	$-0.574857\pi$
$503$	−6.69047e11	−0.466016	−0.233008	+	0.972475i	$0.574857\pi$
$504$	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	$-0.451659\pi$
$509$	4.58197e11	0.302568	0.151284	+	0.988490i	$0.451659\pi$
$510$	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	$-0.967005\pi$
$521$	−3.34551e12	−1.98926	−0.994632	+	0.103472i	$0.967005\pi$
$522$	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	$-0.177570\pi$
$557$	3.85457e12	1.69679	0.848394	+	0.529366i	$0.177570\pi$
$558$	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	$-0.280066\pi$
$563$	3.03834e12	1.27453	0.637264	+	0.770646i	$0.280066\pi$
$564$	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	$-0.462792\pi$
$569$	5.83227e11	0.233256	0.116628	+	0.993176i	$0.462792\pi$
$570$	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	$-0.602339\pi$
$587$	−1.81796e12	−0.631995	−0.315997	+	0.948760i	$0.602339\pi$
$588$	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	$-0.418901\pi$
$593$	1.51787e12	0.504067	0.252034	+	0.967719i	$0.418901\pi$
$594$	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	$-0.623376\pi$
$599$	−2.38179e12	−0.755932	−0.377966	+	0.925819i	$0.623376\pi$
$600$	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	$-0.544382\pi$
$617$	−1.00060e12	−0.277958	−0.138979	+	0.990295i	$0.544382\pi$
$618$	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	$-0.635389\pi$
$641$	−3.52736e12	−0.825256	−0.412628	+	0.910900i	$0.635389\pi$
$642$	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	$-0.287078\pi$
$647$	5.52821e12	1.24027	0.620134	+	0.784496i	$0.287078\pi$
$648$	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	$-0.724632\pi$
$653$	−6.02691e12	−1.29714	−0.648568	+	0.761157i	$0.724632\pi$
$654$	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	$-0.509768\pi$
$659$	−2.97106e11	−0.0613658	−0.0306829	+	0.999529i	$0.509768\pi$
$660$	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	$-0.646191\pi$
$677$	−4.84588e12	−0.886591	−0.443295	+	0.896376i	$0.646191\pi$
$678$	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	$-0.329522\pi$
$683$	5.80468e12	1.02067	0.510334	+	0.859976i	$0.329522\pi$
$684$	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	$-0.689427\pi$
$701$	−7.16819e12	−1.12119	−0.560594	+	0.828091i	$0.689427\pi$
$702$	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	$-0.376279\pi$
$719$	5.43142e12	0.757938	0.378969	+	0.925409i	$0.376279\pi$
$720$	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	$-0.482960\pi$
$743$	8.89001e11	0.107017	0.0535085	+	0.998567i	$0.482960\pi$
$744$	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	$-0.338810\pi$
$761$	8.97482e12	0.970052	0.485026	+	0.874500i	$0.338810\pi$
$762$	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	$-0.421387\pi$
$773$	4.85355e12	0.488936	0.244468	+	0.969657i	$0.421387\pi$
$774$	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	$-0.575261\pi$
$797$	−5.33654e12	−0.468487	−0.234243	+	0.972178i	$0.575261\pi$
$798$	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	$-0.111431\pi$
$809$	2.28889e13	1.87870	0.939348	+	0.342964i	$0.111431\pi$
$810$	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	$-0.406850\pi$
$821$	7.51091e12	0.576963	0.288482	+	0.957485i	$0.406850\pi$
$822$	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	$-0.652797\pi$
$827$	−1.24239e13	−0.923602	−0.461801	+	0.886984i	$0.652797\pi$
$828$	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	$-0.721923\pi$
$839$	−1.84306e13	−1.28414	−0.642068	+	0.766648i	$0.721923\pi$
$840$	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	$-0.548077\pi$
$857$	−4.75199e12	−0.300927	−0.150464	+	0.988616i	$0.548077\pi$
$858$	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	$-0.330486\pi$
$863$	1.65466e13	1.01545	0.507727	+	0.861518i	$0.330486\pi$
$864$	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	$-0.561317\pi$
$881$	−6.84641e12	−0.382887	−0.191444	+	0.981504i	$0.561317\pi$
$882$	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	$-0.733447\pi$
$887$	−2.46814e13	−1.33879	−0.669397	+	0.742905i	$0.733447\pi$
$888$	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	$-0.543867\pi$
$911$	−5.71188e12	−0.274756	−0.137378	+	0.990519i	$0.543867\pi$
$912$	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	$-0.294835\pi$
$929$	2.72807e13	1.20167	0.600835	+	0.799373i	$0.294835\pi$
$930$	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	$-0.508860\pi$
$941$	−1.33886e12	−0.0556648	−0.0278324	+	0.999613i	$0.508860\pi$
$942$	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	$-0.169438\pi$
$947$	4.26511e13	1.72328	0.861639	+	0.507521i	$0.169438\pi$
$948$	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	$-0.672195\pi$
$953$	−2.62256e13	−1.02993	−0.514964	+	0.857212i	$0.672195\pi$
$954$	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	$-0.678940\pi$
$971$	−2.95293e13	−1.06602	−0.533012	+	0.846108i	$0.678940\pi$
$972$	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	$-0.833076\pi$
$977$	−4.93042e13	−1.73124	−0.865622	+	0.500699i	$0.833076\pi$
$978$	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	$-0.451469\pi$
$983$	8.89206e12	0.303747	0.151873	+	0.988400i	$0.451469\pi$
$984$	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.1.3	yes	4
3.2	odd	2	inner	27.10.a.d.1.2	✓	4
9.2	odd	6		81.10.c.j.28.3		8
9.4	even	3		81.10.c.j.55.2		8
9.5	odd	6		81.10.c.j.55.3		8
9.7	even	3		81.10.c.j.28.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.d.1.2	✓	4	3.2	odd	2	inner
27.10.a.d.1.3	yes	4	1.1	even	1	trivial
81.10.c.j.28.2		8	9.7	even	3
81.10.c.j.28.3		8	9.2	odd	6
81.10.c.j.55.2		8	9.4	even	3
81.10.c.j.55.3		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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
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

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Varieties

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