LMFDB - Embedded newform 117.10.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,10,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	117.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:	$60.2591928312$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580$ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}\cdot 3$
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-24.3176$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.3176 q^{2} -57.5590 q^{4} +1277.14 q^{5} -2277.98 q^{7} -12141.6 q^{8} +27225.6 q^{10} +7177.94 q^{11} +28561.0 q^{13} -48561.1 q^{14} -229361. q^{16} +447890. q^{17} -528333. q^{19} -73510.8 q^{20} +153017. q^{22} -2.24354e6 q^{23} -322041. q^{25} +608853. q^{26} +131118. q^{28} -5.98542e6 q^{29} +169630. q^{31} +1.32709e6 q^{32} +9.54795e6 q^{34} -2.90930e6 q^{35} +1.26336e7 q^{37} -1.12628e7 q^{38} -1.55066e7 q^{40} +2.76549e7 q^{41} -2.27606e7 q^{43} -413155. q^{44} -4.78270e7 q^{46} -5.32103e7 q^{47} -3.51644e7 q^{49} -6.86516e6 q^{50} -1.64394e6 q^{52} -3.18756e7 q^{53} +9.16722e6 q^{55} +2.76584e7 q^{56} -1.27595e8 q^{58} -1.14800e8 q^{59} -7.80352e7 q^{61} +3.61610e6 q^{62} +1.45723e8 q^{64} +3.64764e7 q^{65} +8.40538e7 q^{67} -2.57801e7 q^{68} -6.20193e7 q^{70} -1.25752e8 q^{71} -1.88250e8 q^{73} +2.69318e8 q^{74} +3.04103e7 q^{76} -1.63512e7 q^{77} -4.28673e8 q^{79} -2.92926e8 q^{80} +5.89536e8 q^{82} -2.43067e8 q^{83} +5.72018e8 q^{85} -4.85202e8 q^{86} -8.71520e7 q^{88} -2.92716e8 q^{89} -6.50614e7 q^{91} +1.29136e8 q^{92} -1.13432e9 q^{94} -6.74754e8 q^{95} +1.14275e9 q^{97} -7.49622e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 15 q^{2} + 361 q^{4} - 1803 q^{5} + 10099 q^{7} - 23151 q^{8} + 84505 q^{10} - 121746 q^{11} + 142805 q^{13} - 8475 q^{14} - 322463 q^{16} + 495669 q^{17} - 840738 q^{19} + 1595607 q^{20} - 2023594 q^{22}+ \cdots - 1570614816 q^{98}+O(q^{100})$ 5 * q - 15 * q^2 + 361 * q^4 - 1803 * q^5 + 10099 * q^7 - 23151 * q^8 + 84505 * q^10 - 121746 * q^11 + 142805 * q^13 - 8475 * q^14 - 322463 * q^16 + 495669 * q^17 - 840738 * q^19 + 1595607 * q^20 - 2023594 * q^22 + 592152 * q^23 + 1670362 * q^25 - 428415 * q^26 + 2587955 * q^28 - 10678182 * q^29 + 12885296 * q^31 - 3282927 * q^32 - 9934079 * q^34 - 8380731 * q^35 + 7171823 * q^37 + 25568814 * q^38 - 54359445 * q^40 - 9294012 * q^41 + 12831975 * q^43 + 41479074 * q^44 - 59319696 * q^46 - 43354215 * q^47 + 25249488 * q^49 + 16270770 * q^50 + 10310521 * q^52 - 93231780 * q^53 + 99448846 * q^55 - 199599225 * q^56 + 151020970 * q^58 - 246496182 * q^59 - 132232612 * q^61 - 158135724 * q^62 + 91019105 * q^64 - 51495483 * q^65 - 369388534 * q^67 - 238172073 * q^68 - 144857425 * q^70 - 212150457 * q^71 - 252729806 * q^73 - 192105957 * q^74 - 953775990 * q^76 - 449666118 * q^77 - 1247271728 * q^79 - 900649725 * q^80 + 169559388 * q^82 - 1696894296 * q^83 - 775363765 * q^85 - 3291621459 * q^86 - 220227222 * q^88 + 753854382 * q^89 + 288437539 * q^91 - 13876128 * q^92 + 272071215 * q^94 - 1442632962 * q^95 + 3824606 * q^97 - 1570614816 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	21.3176	0.942115	0.471057	−	0.882103i	$-0.343872\pi$
$2$	21.3176	0.942115	0.471057	+	0.882103i	$0.343872\pi$
$3$	0	0
$3$	0	0
$4$	−57.5590	−0.112420
$5$	1277.14	0.913846	0.456923	−	0.889506i	$-0.348951\pi$
$5$	1277.14	0.913846	0.456923	+	0.889506i	$0.348951\pi$
$6$	0	0
$7$	−2277.98	−0.358599	−0.179299	−	0.983795i	$-0.557383\pi$
$7$	−2277.98	−0.358599	−0.179299	+	0.983795i	$0.557383\pi$
$8$	−12141.6	−1.04803
$9$	0	0
$10$	27225.6	0.860948
$11$	7177.94	0.147820	0.0739099	−	0.997265i	$-0.476452\pi$
$11$	7177.94	0.147820	0.0739099	+	0.997265i	$0.476452\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	−48561.1	−0.337841
$15$	0	0
$16$	−229361.	−0.874942
$17$	447890.	1.30062	0.650311	−	0.759668i	$-0.274639\pi$
$17$	447890.	1.30062	0.650311	+	0.759668i	$0.274639\pi$
$18$	0	0
$19$	−528333.	−0.930071	−0.465036	−	0.885292i	$-0.653959\pi$
$19$	−528333.	−0.930071	−0.465036	+	0.885292i	$0.653959\pi$
$20$	−73510.8	−0.102734
$21$	0	0
$22$	153017.	0.139263
$23$	−2.24354e6	−1.67170	−0.835851	−	0.548957i	$-0.815025\pi$
$23$	−2.24354e6	−1.67170	−0.835851	+	0.548957i	$0.815025\pi$
$24$	0	0
$25$	−322041.	−0.164885
$26$	608853.	0.261296
$27$	0	0
$28$	131118.	0.0403136
$29$	−5.98542e6	−1.57146	−0.785730	−	0.618569i	$-0.787713\pi$
$29$	−5.98542e6	−1.57146	−0.785730	+	0.618569i	$0.787713\pi$
$30$	0	0
$31$	169630.	0.0329894	0.0164947	−	0.999864i	$-0.494749\pi$
$31$	169630.	0.0329894	0.0164947	+	0.999864i	$0.494749\pi$
$32$	1.32709e6	0.223731
$33$	0	0
$34$	9.54795e6	1.22534
$35$	−2.90930e6	−0.327704
$36$	0	0
$37$	1.26336e7	1.10820	0.554100	−	0.832450i	$-0.313062\pi$
$37$	1.26336e7	1.10820	0.554100	+	0.832450i	$0.313062\pi$
$38$	−1.12628e7	−0.876234
$39$	0	0
$40$	−1.55066e7	−0.957736
$41$	2.76549e7	1.52843	0.764213	−	0.644964i	$-0.223128\pi$
$41$	2.76549e7	1.52843	0.764213	+	0.644964i	$0.223128\pi$
$42$	0	0
$43$	−2.27606e7	−1.01526	−0.507628	−	0.861576i	$-0.669478\pi$
$43$	−2.27606e7	−1.01526	−0.507628	+	0.861576i	$0.669478\pi$
$44$	−413155.	−0.0166179
$45$	0	0
$46$	−4.78270e7	−1.57493
$47$	−5.32103e7	−1.59058	−0.795289	−	0.606230i	$-0.792681\pi$
$47$	−5.32103e7	−1.59058	−0.795289	+	0.606230i	$0.792681\pi$
$48$	0	0
$49$	−3.51644e7	−0.871407
$50$	−6.86516e6	−0.155341
$51$	0	0
$52$	−1.64394e6	−0.0311797
$53$	−3.18756e7	−0.554904	−0.277452	−	0.960740i	$-0.589490\pi$
$53$	−3.18756e7	−0.554904	−0.277452	+	0.960740i	$0.589490\pi$
$54$	0	0
$55$	9.16722e6	0.135085
$56$	2.76584e7	0.375821
$57$	0	0
$58$	−1.27595e8	−1.48050
$59$	−1.14800e8	−1.23341	−0.616705	−	0.787194i	$-0.711533\pi$
$59$	−1.14800e8	−1.23341	−0.616705	+	0.787194i	$0.711533\pi$
$60$	0	0
$61$	−7.80352e7	−0.721616	−0.360808	−	0.932640i	$-0.617499\pi$
$61$	−7.80352e7	−0.721616	−0.360808	+	0.932640i	$0.617499\pi$
$62$	3.61610e6	0.0310798
$63$	0	0
$64$	1.45723e8	1.08572
$65$	3.64764e7	0.253455
$66$	0	0
$67$	8.40538e7	0.509590	0.254795	−	0.966995i	$-0.417992\pi$
$67$	8.40538e7	0.509590	0.254795	+	0.966995i	$0.417992\pi$
$68$	−2.57801e7	−0.146216
$69$	0	0
$70$	−6.20193e7	−0.308735
$71$	−1.25752e8	−0.587288	−0.293644	−	0.955915i	$-0.594868\pi$
$71$	−1.25752e8	−0.587288	−0.293644	+	0.955915i	$0.594868\pi$
$72$	0	0
$73$	−1.88250e8	−0.775859	−0.387929	−	0.921689i	$-0.626810\pi$
$73$	−1.88250e8	−0.775859	−0.387929	+	0.921689i	$0.626810\pi$
$74$	2.69318e8	1.04405
$75$	0	0
$76$	3.04103e7	0.104558
$77$	−1.63512e7	−0.0530080
$78$	0	0
$79$	−4.28673e8	−1.23824	−0.619120	−	0.785297i	$-0.712510\pi$
$79$	−4.28673e8	−1.23824	−0.619120	+	0.785297i	$0.712510\pi$
$80$	−2.92926e8	−0.799562
$81$	0	0
$82$	5.89536e8	1.43995
$83$	−2.43067e8	−0.562180	−0.281090	−	0.959681i	$-0.590696\pi$
$83$	−2.43067e8	−0.562180	−0.281090	+	0.959681i	$0.590696\pi$
$84$	0	0
$85$	5.72018e8	1.18857
$86$	−4.85202e8	−0.956488
$87$	0	0
$88$	−8.71520e7	−0.154919
$89$	−2.92716e8	−0.494529	−0.247265	−	0.968948i	$-0.579532\pi$
$89$	−2.92716e8	−0.494529	−0.247265	+	0.968948i	$0.579532\pi$
$90$	0	0
$91$	−6.50614e7	−0.0994573
$92$	1.29136e8	0.187932
$93$	0	0
$94$	−1.13432e9	−1.49851
$95$	−6.74754e8	−0.849942
$96$	0	0
$97$	1.14275e9	1.31063	0.655313	−	0.755358i	$-0.272537\pi$
$97$	1.14275e9	1.31063	0.655313	+	0.755358i	$0.272537\pi$
$98$	−7.49622e8	−0.820965
$99$	0	0
$100$	1.85364e7	0.0185364
$101$	8.98629e8	0.859279	0.429640	−	0.903000i	$-0.358641\pi$
$101$	8.98629e8	0.859279	0.429640	+	0.903000i	$0.358641\pi$
$102$	0	0
$103$	6.13518e8	0.537106	0.268553	−	0.963265i	$-0.413455\pi$
$103$	6.13518e8	0.537106	0.268553	+	0.963265i	$0.413455\pi$
$104$	−3.46777e8	−0.290670
$105$	0	0
$106$	−6.79513e8	−0.522783
$107$	−1.46257e9	−1.07868	−0.539338	−	0.842090i	$-0.681325\pi$
$107$	−1.46257e9	−1.07868	−0.539338	+	0.842090i	$0.681325\pi$
$108$	0	0
$109$	7.17715e8	0.487005	0.243502	−	0.969900i	$-0.421704\pi$
$109$	7.17715e8	0.487005	0.243502	+	0.969900i	$0.421704\pi$
$110$	1.95423e8	0.127265
$111$	0	0
$112$	5.22479e8	0.313753
$113$	−9.27444e7	−0.0535099	−0.0267550	−	0.999642i	$-0.508517\pi$
$113$	−9.27444e7	−0.0535099	−0.0267550	+	0.999642i	$0.508517\pi$
$114$	0	0
$115$	−2.86531e9	−1.52768
$116$	3.44514e8	0.176663
$117$	0	0
$118$	−2.44726e9	−1.16201
$119$	−1.02028e9	−0.466401
$120$	0	0
$121$	−2.30642e9	−0.978149
$122$	−1.66352e9	−0.679845
$123$	0	0
$124$	−9.76371e6	−0.00370866
$125$	−2.90570e9	−1.06453
$126$	0	0
$127$	4.40060e9	1.50105	0.750524	−	0.660843i	$-0.229801\pi$
$127$	4.40060e9	1.50105	0.750524	+	0.660843i	$0.229801\pi$
$128$	2.42700e9	0.799144
$129$	0	0
$130$	7.77589e8	0.238784
$131$	1.21951e9	0.361798	0.180899	−	0.983502i	$-0.442099\pi$
$131$	1.21951e9	0.361798	0.180899	+	0.983502i	$0.442099\pi$
$132$	0	0
$133$	1.20353e9	0.333522
$134$	1.79183e9	0.480092
$135$	0	0
$136$	−5.43812e9	−1.36309
$137$	6.73258e9	1.63282	0.816411	−	0.577472i	$-0.195961\pi$
$137$	6.73258e9	1.63282	0.816411	+	0.577472i	$0.195961\pi$
$138$	0	0
$139$	4.12694e8	0.0937695	0.0468847	−	0.998900i	$-0.485071\pi$
$139$	4.12694e8	0.0937695	0.0468847	+	0.998900i	$0.485071\pi$
$140$	1.67456e8	0.0368404
$141$	0	0
$142$	−2.68073e9	−0.553293
$143$	2.05009e8	0.0409978
$144$	0	0
$145$	−7.64421e9	−1.43607
$146$	−4.01305e9	−0.730948
$147$	0	0
$148$	−7.27175e8	−0.124584
$149$	−7.17339e9	−1.19230	−0.596151	−	0.802872i	$-0.703304\pi$
$149$	−7.17339e9	−1.19230	−0.596151	+	0.802872i	$0.703304\pi$
$150$	0	0
$151$	9.30445e9	1.45645	0.728223	−	0.685340i	$-0.240346\pi$
$151$	9.30445e9	1.45645	0.728223	+	0.685340i	$0.240346\pi$
$152$	6.41483e9	0.974740
$153$	0	0
$154$	−3.48569e8	−0.0499396
$155$	2.16641e8	0.0301472
$156$	0	0
$157$	−5.86131e9	−0.769921	−0.384961	−	0.922933i	$-0.625785\pi$
$157$	−5.86131e9	−0.769921	−0.384961	+	0.922933i	$0.625785\pi$
$158$	−9.13830e9	−1.16656
$159$	0	0
$160$	1.69488e9	0.204456
$161$	5.11074e9	0.599470
$162$	0	0
$163$	1.60676e10	1.78282	0.891409	−	0.453199i	$-0.149717\pi$
$163$	1.60676e10	1.78282	0.891409	+	0.453199i	$0.149717\pi$
$164$	−1.59179e9	−0.171825
$165$	0	0
$166$	−5.18162e9	−0.529638
$167$	1.49042e9	0.148281	0.0741404	−	0.997248i	$-0.476379\pi$
$167$	1.49042e9	0.148281	0.0741404	+	0.997248i	$0.476379\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	1.21941e10	1.11977
$171$	0	0
$172$	1.31008e9	0.114135
$173$	6.62250e9	0.562101	0.281050	−	0.959693i	$-0.409317\pi$
$173$	6.62250e9	0.562101	0.281050	+	0.959693i	$0.409317\pi$
$174$	0	0
$175$	7.33604e8	0.0591276
$176$	−1.64634e9	−0.129334
$177$	0	0
$178$	−6.24002e9	−0.465903
$179$	−1.41019e10	−1.02669	−0.513344	−	0.858183i	$-0.671594\pi$
$179$	−1.41019e10	−1.02669	−0.513344	+	0.858183i	$0.671594\pi$
$180$	0	0
$181$	2.38898e10	1.65447	0.827234	−	0.561857i	$-0.189913\pi$
$181$	2.38898e10	1.65447	0.827234	+	0.561857i	$0.189913\pi$
$182$	−1.38695e9	−0.0937002
$183$	0	0
$184$	2.72403e10	1.75199
$185$	1.61348e10	1.01272
$186$	0	0
$187$	3.21493e9	0.192258
$188$	3.06273e9	0.178813
$189$	0	0
$190$	−1.43842e10	−0.800743
$191$	2.32674e10	1.26502	0.632509	−	0.774553i	$-0.282025\pi$
$191$	2.32674e10	1.26502	0.632509	+	0.774553i	$0.282025\pi$
$192$	0	0
$193$	8.87539e9	0.460447	0.230224	−	0.973138i	$-0.426054\pi$
$193$	8.87539e9	0.460447	0.230224	+	0.973138i	$0.426054\pi$
$194$	2.43607e10	1.23476
$195$	0	0
$196$	2.02403e9	0.0979634
$197$	9.96936e9	0.471595	0.235798	−	0.971802i	$-0.424230\pi$
$197$	9.96936e9	0.471595	0.235798	+	0.971802i	$0.424230\pi$
$198$	0	0
$199$	−5.92044e9	−0.267618	−0.133809	−	0.991007i	$-0.542721\pi$
$199$	−5.92044e9	−0.267618	−0.133809	+	0.991007i	$0.542721\pi$
$200$	3.91011e9	0.172804
$201$	0	0
$202$	1.91566e10	0.809540
$203$	1.36347e10	0.563524
$204$	0	0
$205$	3.53191e10	1.39675
$206$	1.30787e10	0.506015
$207$	0	0
$208$	−6.55077e9	−0.242665
$209$	−3.79234e9	−0.137483
$210$	0	0
$211$	−3.18290e10	−1.10548	−0.552741	−	0.833353i	$-0.686418\pi$
$211$	−3.18290e10	−1.10548	−0.552741	+	0.833353i	$0.686418\pi$
$212$	1.83473e9	0.0623822
$213$	0	0
$214$	−3.11786e10	−1.01624
$215$	−2.90685e10	−0.927788
$216$	0	0
$217$	−3.86413e8	−0.0118300
$218$	1.53000e10	0.458814
$219$	0	0
$220$	−5.27656e8	−0.0151862
$221$	1.27922e10	0.360728
$222$	0	0
$223$	−2.33567e10	−0.632470	−0.316235	−	0.948681i	$-0.602419\pi$
$223$	−2.33567e10	−0.632470	−0.316235	+	0.948681i	$0.602419\pi$
$224$	−3.02309e9	−0.0802298
$225$	0	0
$226$	−1.97709e9	−0.0504125
$227$	−2.96928e10	−0.742223	−0.371112	−	0.928588i	$-0.621023\pi$
$227$	−2.96928e10	−0.742223	−0.371112	+	0.928588i	$0.621023\pi$
$228$	0	0
$229$	4.24937e10	1.02109	0.510546	−	0.859850i	$-0.329443\pi$
$229$	4.24937e10	1.02109	0.510546	+	0.859850i	$0.329443\pi$
$230$	−6.10817e10	−1.43925
$231$	0	0
$232$	7.26728e10	1.64693
$233$	3.84693e10	0.855092	0.427546	−	0.903994i	$-0.359378\pi$
$233$	3.84693e10	0.855092	0.427546	+	0.903994i	$0.359378\pi$
$234$	0	0
$235$	−6.79569e10	−1.45354
$236$	6.60776e9	0.138660
$237$	0	0
$238$	−2.17500e10	−0.439403
$239$	1.98946e10	0.394407	0.197203	−	0.980363i	$-0.436814\pi$
$239$	1.98946e10	0.394407	0.197203	+	0.980363i	$0.436814\pi$
$240$	0	0
$241$	−5.31240e10	−1.01441	−0.507206	−	0.861825i	$-0.669322\pi$
$241$	−5.31240e10	−1.01441	−0.507206	+	0.861825i	$0.669322\pi$
$242$	−4.91675e10	−0.921529
$243$	0	0
$244$	4.49162e9	0.0811240
$245$	−4.49098e10	−0.796332
$246$	0	0
$247$	−1.50897e10	−0.257955
$248$	−2.05958e9	−0.0345738
$249$	0	0
$250$	−6.19427e10	−1.00291
$251$	−1.26269e10	−0.200801	−0.100400	−	0.994947i	$-0.532012\pi$
$251$	−1.26269e10	−0.200801	−0.100400	+	0.994947i	$0.532012\pi$
$252$	0	0
$253$	−1.61040e10	−0.247111
$254$	9.38102e10	1.41416
$255$	0	0
$256$	−2.28724e10	−0.332837
$257$	−1.11000e11	−1.58717	−0.793583	−	0.608461i	$-0.791787\pi$
$257$	−1.11000e11	−1.58717	−0.793583	+	0.608461i	$0.791787\pi$
$258$	0	0
$259$	−2.87790e10	−0.397399
$260$	−2.09954e9	−0.0284934
$261$	0	0
$262$	2.59971e10	0.340855
$263$	−7.01020e10	−0.903503	−0.451752	−	0.892144i	$-0.649201\pi$
$263$	−7.01020e10	−0.903503	−0.451752	+	0.892144i	$0.649201\pi$
$264$	0	0
$265$	−4.07096e10	−0.507097
$266$	2.56564e10	0.314216
$267$	0	0
$268$	−4.83805e9	−0.0572880
$269$	−7.41364e10	−0.863269	−0.431635	−	0.902049i	$-0.642063\pi$
$269$	−7.41364e10	−0.863269	−0.431635	+	0.902049i	$0.642063\pi$
$270$	0	0
$271$	8.92477e10	1.00516	0.502580	−	0.864531i	$-0.332384\pi$
$271$	8.92477e10	1.00516	0.502580	+	0.864531i	$0.332384\pi$
$272$	−1.02728e11	−1.13797
$273$	0	0
$274$	1.43523e11	1.53831
$275$	−2.31159e9	−0.0243733
$276$	0	0
$277$	−6.68313e10	−0.682057	−0.341028	−	0.940053i	$-0.610775\pi$
$277$	−6.68313e10	−0.682057	−0.341028	+	0.940053i	$0.610775\pi$
$278$	8.79765e9	0.0883416
$279$	0	0
$280$	3.53236e10	0.343443
$281$	8.13670e10	0.778520	0.389260	−	0.921128i	$-0.372731\pi$
$281$	8.13670e10	0.778520	0.389260	+	0.921128i	$0.372731\pi$
$282$	0	0
$283$	−1.19236e11	−1.10502	−0.552508	−	0.833507i	$-0.686329\pi$
$283$	−1.19236e11	−1.10502	−0.552508	+	0.833507i	$0.686329\pi$
$284$	7.23814e9	0.0660229
$285$	0	0
$286$	4.37031e9	0.0386247
$287$	−6.29972e10	−0.548091
$288$	0	0
$289$	8.20174e10	0.691617
$290$	−1.62956e11	−1.35295
$291$	0	0
$292$	1.08355e10	0.0872219
$293$	6.95594e10	0.551381	0.275690	−	0.961246i	$-0.411094\pi$
$293$	6.95594e10	0.551381	0.275690	+	0.961246i	$0.411094\pi$
$294$	0	0
$295$	−1.46615e11	−1.12715
$296$	−1.53392e11	−1.16142
$297$	0	0
$298$	−1.52920e11	−1.12328
$299$	−6.40778e10	−0.463647
$300$	0	0
$301$	5.18482e10	0.364069
$302$	1.98349e11	1.37214
$303$	0	0
$304$	1.21179e11	0.813759
$305$	−9.96618e10	−0.659446
$306$	0	0
$307$	−4.02660e10	−0.258711	−0.129356	−	0.991598i	$-0.541291\pi$
$307$	−4.02660e10	−0.258711	−0.129356	+	0.991598i	$0.541291\pi$
$308$	9.41157e8	0.00595915
$309$	0	0
$310$	4.61827e9	0.0284022
$311$	1.78556e11	1.08231	0.541157	−	0.840921i	$-0.317986\pi$
$311$	1.78556e11	1.08231	0.541157	+	0.840921i	$0.317986\pi$
$312$	0	0
$313$	9.44550e9	0.0556257	0.0278128	−	0.999613i	$-0.491146\pi$
$313$	9.44550e9	0.0556257	0.0278128	+	0.999613i	$0.491146\pi$
$314$	−1.24949e11	−0.725354
$315$	0	0
$316$	2.46740e10	0.139203
$317$	−5.42593e10	−0.301792	−0.150896	−	0.988550i	$-0.548216\pi$
$317$	−5.42593e10	−0.301792	−0.150896	+	0.988550i	$0.548216\pi$
$318$	0	0
$319$	−4.29630e10	−0.232293
$320$	1.86109e11	0.992183
$321$	0	0
$322$	1.08949e11	0.564769
$323$	−2.36635e11	−1.20967
$324$	0	0
$325$	−9.19783e9	−0.0457309
$326$	3.42523e11	1.67962
$327$	0	0
$328$	−3.35776e11	−1.60183
$329$	1.21212e11	0.570379
$330$	0	0
$331$	6.66790e10	0.305325	0.152663	−	0.988278i	$-0.451215\pi$
$331$	6.66790e10	0.305325	0.152663	+	0.988278i	$0.451215\pi$
$332$	1.39907e10	0.0632001
$333$	0	0
$334$	3.17722e10	0.139697
$335$	1.07348e11	0.465687
$336$	0	0
$337$	2.13457e11	0.901522	0.450761	−	0.892645i	$-0.351153\pi$
$337$	2.13457e11	0.901522	0.450761	+	0.892645i	$0.351153\pi$
$338$	1.73894e10	0.0724704
$339$	0	0
$340$	−3.29247e10	−0.133619
$341$	1.21759e9	0.00487649
$342$	0	0
$343$	1.72028e11	0.671084
$344$	2.76351e11	1.06402
$345$	0	0
$346$	1.41176e11	0.529564
$347$	9.49978e10	0.351748	0.175874	−	0.984413i	$-0.443725\pi$
$347$	9.49978e10	0.351748	0.175874	+	0.984413i	$0.443725\pi$
$348$	0	0
$349$	2.52647e11	0.911590	0.455795	−	0.890085i	$-0.349355\pi$
$349$	2.52647e11	0.911590	0.455795	+	0.890085i	$0.349355\pi$
$350$	1.56387e10	0.0557050
$351$	0	0
$352$	9.52580e9	0.0330719
$353$	−4.50376e11	−1.54379	−0.771896	−	0.635749i	$-0.780691\pi$
$353$	−4.50376e11	−1.54379	−0.771896	+	0.635749i	$0.780691\pi$
$354$	0	0
$355$	−1.60602e11	−0.536691
$356$	1.68485e10	0.0555949
$357$	0	0
$358$	−3.00619e11	−0.967258
$359$	−3.93740e11	−1.25108	−0.625539	−	0.780193i	$-0.715121\pi$
$359$	−3.93740e11	−1.25108	−0.625539	+	0.780193i	$0.715121\pi$
$360$	0	0
$361$	−4.35522e10	−0.134967
$362$	5.09273e11	1.55870
$363$	0	0
$364$	3.74486e9	0.0111810
$365$	−2.40422e11	−0.709015
$366$	0	0
$367$	−2.26305e11	−0.651173	−0.325586	−	0.945512i	$-0.605562\pi$
$367$	−2.26305e11	−0.651173	−0.325586	+	0.945512i	$0.605562\pi$
$368$	5.14580e11	1.46264
$369$	0	0
$370$	3.43956e11	0.954103
$371$	7.26120e10	0.198988
$372$	0	0
$373$	4.75023e11	1.27065	0.635323	−	0.772246i	$-0.280867\pi$
$373$	4.75023e11	1.27065	0.635323	+	0.772246i	$0.280867\pi$
$374$	6.85346e10	0.181129
$375$	0	0
$376$	6.46060e11	1.66697
$377$	−1.70950e11	−0.435845
$378$	0	0
$379$	−1.67293e11	−0.416486	−0.208243	−	0.978077i	$-0.566775\pi$
$379$	−1.67293e11	−0.416486	−0.208243	+	0.978077i	$0.566775\pi$
$380$	3.88381e10	0.0955504
$381$	0	0
$382$	4.96005e11	1.19179
$383$	5.36589e11	1.27423	0.637114	−	0.770769i	$-0.280128\pi$
$383$	5.36589e11	1.27423	0.637114	+	0.770769i	$0.280128\pi$
$384$	0	0
$385$	−2.08827e10	−0.0484411
$386$	1.89202e11	0.433794
$387$	0	0
$388$	−6.57755e10	−0.147340
$389$	−5.95613e11	−1.31884	−0.659418	−	0.751776i	$-0.729197\pi$
$389$	−5.95613e11	−1.31884	−0.659418	+	0.751776i	$0.729197\pi$
$390$	0	0
$391$	−1.00486e12	−2.17425
$392$	4.26954e11	0.913258
$393$	0	0
$394$	2.12523e11	0.444297
$395$	−5.47475e11	−1.13156
$396$	0	0
$397$	−6.68797e11	−1.35125	−0.675627	−	0.737243i	$-0.736127\pi$
$397$	−6.68797e11	−1.35125	−0.675627	+	0.737243i	$0.736127\pi$
$398$	−1.26210e11	−0.252127
$399$	0	0
$400$	7.38637e10	0.144265
$401$	−4.27579e11	−0.825785	−0.412892	−	0.910780i	$-0.635481\pi$
$401$	−4.27579e11	−0.825785	−0.412892	+	0.910780i	$0.635481\pi$
$402$	0	0
$403$	4.84480e9	0.00914962
$404$	−5.17241e10	−0.0966000
$405$	0	0
$406$	2.90658e11	0.530904
$407$	9.06830e10	0.163814
$408$	0	0
$409$	−4.37893e10	−0.0773772	−0.0386886	−	0.999251i	$-0.512318\pi$
$409$	−4.37893e10	−0.0773772	−0.0386886	+	0.999251i	$0.512318\pi$
$410$	7.52919e11	1.31589
$411$	0	0
$412$	−3.53135e10	−0.0603813
$413$	2.61512e11	0.442299
$414$	0	0
$415$	−3.10431e11	−0.513746
$416$	3.79031e10	0.0620519
$417$	0	0
$418$	−8.08437e10	−0.129525
$419$	1.14243e11	0.181079	0.0905395	−	0.995893i	$-0.471141\pi$
$419$	1.14243e11	0.181079	0.0905395	+	0.995893i	$0.471141\pi$
$420$	0	0
$421$	−5.06520e11	−0.785828	−0.392914	−	0.919575i	$-0.628533\pi$
$421$	−5.06520e11	−0.785828	−0.392914	+	0.919575i	$0.628533\pi$
$422$	−6.78518e11	−1.04149
$423$	0	0
$424$	3.87023e11	0.581554
$425$	−1.44239e11	−0.214453
$426$	0	0
$427$	1.77763e11	0.258770
$428$	8.41842e10	0.121264
$429$	0	0
$430$	−6.19670e11	−0.874083
$431$	1.32605e11	0.185103	0.0925513	−	0.995708i	$-0.470498\pi$
$431$	1.32605e11	0.185103	0.0925513	+	0.995708i	$0.470498\pi$
$432$	0	0
$433$	1.21346e12	1.65894	0.829469	−	0.558553i	$-0.188643\pi$
$433$	1.21346e12	1.65894	0.829469	+	0.558553i	$0.188643\pi$
$434$	−8.23741e9	−0.0111452
$435$	0	0
$436$	−4.13110e10	−0.0547490
$437$	1.18534e12	1.55480
$438$	0	0
$439$	−7.26961e11	−0.934159	−0.467079	−	0.884215i	$-0.654694\pi$
$439$	−7.26961e11	−0.934159	−0.467079	+	0.884215i	$0.654694\pi$
$440$	−1.11305e11	−0.141572
$441$	0	0
$442$	2.72699e11	0.339847
$443$	1.02161e12	1.26028	0.630142	−	0.776480i	$-0.282997\pi$
$443$	1.02161e12	1.26028	0.630142	+	0.776480i	$0.282997\pi$
$444$	0	0
$445$	−3.73840e11	−0.451924
$446$	−4.97910e11	−0.595859
$447$	0	0
$448$	−3.31954e11	−0.389339
$449$	−5.91722e11	−0.687083	−0.343542	−	0.939137i	$-0.611627\pi$
$449$	−5.91722e11	−0.687083	−0.343542	+	0.939137i	$0.611627\pi$
$450$	0	0
$451$	1.98505e11	0.225932
$452$	5.33827e9	0.00601558
$453$	0	0
$454$	−6.32979e11	−0.699259
$455$	−8.30924e10	−0.0908887
$456$	0	0
$457$	−8.30664e11	−0.890845	−0.445423	−	0.895320i	$-0.646947\pi$
$457$	−8.30664e11	−0.890845	−0.445423	+	0.895320i	$0.646947\pi$
$458$	9.05865e11	0.961986
$459$	0	0
$460$	1.64924e11	0.171741
$461$	−8.73246e11	−0.900497	−0.450249	−	0.892903i	$-0.648665\pi$
$461$	−8.73246e11	−0.900497	−0.450249	+	0.892903i	$0.648665\pi$
$462$	0	0
$463$	7.70415e11	0.779130	0.389565	−	0.920999i	$-0.372625\pi$
$463$	7.70415e11	0.779130	0.389565	+	0.920999i	$0.372625\pi$
$464$	1.37282e12	1.37494
$465$	0	0
$466$	8.20074e11	0.805595
$467$	−4.50995e11	−0.438779	−0.219390	−	0.975637i	$-0.570407\pi$
$467$	−4.50995e11	−0.438779	−0.219390	+	0.975637i	$0.570407\pi$
$468$	0	0
$469$	−1.91473e11	−0.182738
$470$	−1.44868e12	−1.36940
$471$	0	0
$472$	1.39386e12	1.29265
$473$	−1.63374e11	−0.150075
$474$	0	0
$475$	1.70145e11	0.153355
$476$	5.87265e10	0.0524327
$477$	0	0
$478$	4.24105e11	0.371576
$479$	5.26966e11	0.457376	0.228688	−	0.973500i	$-0.426556\pi$
$479$	5.26966e11	0.457376	0.228688	+	0.973500i	$0.426556\pi$
$480$	0	0
$481$	3.60827e11	0.307360
$482$	−1.13248e12	−0.955692
$483$	0	0
$484$	1.32755e11	0.109963
$485$	1.45945e12	1.19771
$486$	0	0
$487$	1.88076e12	1.51514	0.757570	−	0.652754i	$-0.226386\pi$
$487$	1.88076e12	1.51514	0.757570	+	0.652754i	$0.226386\pi$
$488$	9.47475e11	0.756273
$489$	0	0
$490$	−9.57371e11	−0.750236
$491$	−1.68889e12	−1.31140	−0.655699	−	0.755022i	$-0.727626\pi$
$491$	−1.68889e12	−1.31140	−0.655699	+	0.755022i	$0.727626\pi$
$492$	0	0
$493$	−2.68081e12	−2.04388
$494$	−3.21677e11	−0.243024
$495$	0	0
$496$	−3.89064e10	−0.0288638
$497$	2.86460e11	0.210601
$498$	0	0
$499$	1.60699e12	1.16027	0.580137	−	0.814519i	$-0.302999\pi$
$499$	1.60699e12	1.16027	0.580137	+	0.814519i	$0.302999\pi$
$500$	1.67249e11	0.119674
$501$	0	0
$502$	−2.69176e11	−0.189177
$503$	−2.73565e12	−1.90548	−0.952740	−	0.303786i	$-0.901749\pi$
$503$	−2.73565e12	−1.90548	−0.952740	+	0.303786i	$0.901749\pi$
$504$	0	0
$505$	1.14767e12	0.785249
$506$	−3.43299e11	−0.232807
$507$	0	0
$508$	−2.53294e11	−0.168748
$509$	−7.49256e11	−0.494766	−0.247383	−	0.968918i	$-0.579571\pi$
$509$	−7.49256e11	−0.494766	−0.247383	+	0.968918i	$0.579571\pi$
$510$	0	0
$511$	4.28830e11	0.278222
$512$	−1.73021e12	−1.11271
$513$	0	0
$514$	−2.36625e12	−1.49529
$515$	7.83548e11	0.490832
$516$	0	0
$517$	−3.81940e11	−0.235119
$518$	−6.13500e11	−0.374396
$519$	0	0
$520$	−4.42883e11	−0.265628
$521$	−1.07862e12	−0.641355	−0.320678	−	0.947188i	$-0.603911\pi$
$521$	−1.07862e12	−0.641355	−0.320678	+	0.947188i	$0.603911\pi$
$522$	0	0
$523$	−2.69236e12	−1.57353	−0.786765	−	0.617253i	$-0.788246\pi$
$523$	−2.69236e12	−1.57353	−0.786765	+	0.617253i	$0.788246\pi$
$524$	−7.01939e10	−0.0406732
$525$	0	0
$526$	−1.49441e12	−0.851204
$527$	7.59755e10	0.0429067
$528$	0	0
$529$	3.23232e12	1.79459
$530$	−8.67832e11	−0.477743
$531$	0	0
$532$	−6.92740e10	−0.0374945
$533$	7.89851e11	0.423909
$534$	0	0
$535$	−1.86791e12	−0.985743
$536$	−1.02055e12	−0.534064
$537$	0	0
$538$	−1.58041e12	−0.813299
$539$	−2.52408e11	−0.128811
$540$	0	0
$541$	1.94584e12	0.976604	0.488302	−	0.872675i	$-0.337616\pi$
$541$	1.94584e12	0.976604	0.488302	+	0.872675i	$0.337616\pi$
$542$	1.90255e12	0.946976
$543$	0	0
$544$	5.94392e11	0.290990
$545$	9.16622e11	0.445047
$546$	0	0
$547$	−7.49274e11	−0.357847	−0.178924	−	0.983863i	$-0.557261\pi$
$547$	−7.49274e11	−0.357847	−0.178924	+	0.983863i	$0.557261\pi$
$548$	−3.87520e11	−0.183562
$549$	0	0
$550$	−4.92777e10	−0.0229625
$551$	3.16229e12	1.46157
$552$	0	0
$553$	9.76509e11	0.444031
$554$	−1.42468e12	−0.642576
$555$	0	0
$556$	−2.37542e10	−0.0105415
$557$	−2.43252e12	−1.07080	−0.535399	−	0.844599i	$-0.679839\pi$
$557$	−2.43252e12	−1.07080	−0.535399	+	0.844599i	$0.679839\pi$
$558$	0	0
$559$	−6.50066e11	−0.281581
$560$	6.67278e11	0.286722
$561$	0	0
$562$	1.73455e12	0.733455
$563$	2.26566e12	0.950401	0.475200	−	0.879878i	$-0.342376\pi$
$563$	2.26566e12	0.950401	0.475200	+	0.879878i	$0.342376\pi$
$564$	0	0
$565$	−1.18447e11	−0.0488999
$566$	−2.54183e12	−1.04105
$567$	0	0
$568$	1.52683e12	0.615494
$569$	−4.39567e12	−1.75800	−0.879002	−	0.476817i	$-0.841790\pi$
$569$	−4.39567e12	−1.75800	−0.879002	+	0.476817i	$0.841790\pi$
$570$	0	0
$571$	2.81010e12	1.10626	0.553132	−	0.833094i	$-0.313433\pi$
$571$	2.81010e12	1.10626	0.553132	+	0.833094i	$0.313433\pi$
$572$	−1.18001e10	−0.00460897
$573$	0	0
$574$	−1.34295e12	−0.516365
$575$	7.22513e11	0.275639
$576$	0	0
$577$	−3.06576e12	−1.15146	−0.575728	−	0.817642i	$-0.695281\pi$
$577$	−3.06576e12	−1.15146	−0.575728	+	0.817642i	$0.695281\pi$
$578$	1.74842e12	0.651583
$579$	0	0
$580$	4.39993e11	0.161443
$581$	5.53702e11	0.201597
$582$	0	0
$583$	−2.28801e11	−0.0820257
$584$	2.28567e12	0.813121
$585$	0	0
$586$	1.48284e12	0.519464
$587$	−1.53677e12	−0.534240	−0.267120	−	0.963663i	$-0.586072\pi$
$587$	−1.53677e12	−0.534240	−0.267120	+	0.963663i	$0.586072\pi$
$588$	0	0
$589$	−8.96210e10	−0.0306825
$590$	−3.12549e12	−1.06190
$591$	0	0
$592$	−2.89765e12	−0.969611
$593$	3.34336e12	1.11029	0.555146	−	0.831753i	$-0.312662\pi$
$593$	3.34336e12	1.11029	0.555146	+	0.831753i	$0.312662\pi$
$594$	0	0
$595$	−1.30304e12	−0.426219
$596$	4.12893e11	0.134038
$597$	0	0
$598$	−1.36599e12	−0.436808
$599$	9.97101e11	0.316460	0.158230	−	0.987402i	$-0.449421\pi$
$599$	9.97101e11	0.316460	0.158230	+	0.987402i	$0.449421\pi$
$600$	0	0
$601$	−2.24332e12	−0.701386	−0.350693	−	0.936491i	$-0.614054\pi$
$601$	−2.24332e12	−0.701386	−0.350693	+	0.936491i	$0.614054\pi$
$602$	1.10528e12	0.342995
$603$	0	0
$604$	−5.35555e11	−0.163733
$605$	−2.94562e12	−0.893878
$606$	0	0
$607$	−5.75690e12	−1.72123	−0.860617	−	0.509253i	$-0.829922\pi$
$607$	−5.75690e12	−1.72123	−0.860617	+	0.509253i	$0.829922\pi$
$608$	−7.01147e11	−0.208086
$609$	0	0
$610$	−2.12455e12	−0.621274
$611$	−1.51974e12	−0.441147
$612$	0	0
$613$	3.44873e12	0.986479	0.493239	−	0.869894i	$-0.335813\pi$
$613$	3.44873e12	0.986479	0.493239	+	0.869894i	$0.335813\pi$
$614$	−8.58375e11	−0.243736
$615$	0	0
$616$	1.98530e11	0.0555538
$617$	5.13402e12	1.42618	0.713089	−	0.701073i	$-0.247295\pi$
$617$	5.13402e12	1.42618	0.713089	+	0.701073i	$0.247295\pi$
$618$	0	0
$619$	1.83851e12	0.503336	0.251668	−	0.967814i	$-0.419021\pi$
$619$	1.83851e12	0.503336	0.251668	+	0.967814i	$0.419021\pi$
$620$	−1.24696e10	−0.00338915
$621$	0	0
$622$	3.80640e12	1.01966
$623$	6.66802e11	0.177338
$624$	0	0
$625$	−3.08200e12	−0.807928
$626$	2.01356e11	0.0524058
$627$	0	0
$628$	3.37371e11	0.0865544
$629$	5.65845e12	1.44135
$630$	0	0
$631$	3.01780e12	0.757805	0.378903	−	0.925437i	$-0.376301\pi$
$631$	3.01780e12	0.757805	0.378903	+	0.925437i	$0.376301\pi$
$632$	5.20480e12	1.29771
$633$	0	0
$634$	−1.15668e12	−0.284322
$635$	5.62017e12	1.37173
$636$	0	0
$637$	−1.00433e12	−0.241685
$638$	−9.15868e11	−0.218847
$639$	0	0
$640$	3.09962e12	0.730295
$641$	−2.47728e12	−0.579580	−0.289790	−	0.957090i	$-0.593585\pi$
$641$	−2.47728e12	−0.579580	−0.289790	+	0.957090i	$0.593585\pi$
$642$	0	0
$643$	5.96725e12	1.37665	0.688327	−	0.725401i	$-0.258346\pi$
$643$	5.96725e12	1.37665	0.688327	+	0.725401i	$0.258346\pi$
$644$	−2.94169e11	−0.0673923
$645$	0	0
$646$	−5.04449e12	−1.13965
$647$	4.85153e12	1.08845	0.544227	−	0.838938i	$-0.316823\pi$
$647$	4.85153e12	1.08845	0.544227	+	0.838938i	$0.316823\pi$
$648$	0	0
$649$	−8.24027e11	−0.182322
$650$	−1.96076e11	−0.0430838
$651$	0	0
$652$	−9.24835e11	−0.200424
$653$	1.50523e12	0.323961	0.161980	−	0.986794i	$-0.448212\pi$
$653$	1.50523e12	0.323961	0.161980	+	0.986794i	$0.448212\pi$
$654$	0	0
$655$	1.55749e12	0.330628
$656$	−6.34294e12	−1.33728
$657$	0	0
$658$	2.58395e12	0.537362
$659$	3.29009e12	0.679552	0.339776	−	0.940506i	$-0.389649\pi$
$659$	3.29009e12	0.679552	0.339776	+	0.940506i	$0.389649\pi$
$660$	0	0
$661$	7.79224e12	1.58765	0.793827	−	0.608144i	$-0.208086\pi$
$661$	7.79224e12	1.58765	0.793827	+	0.608144i	$0.208086\pi$
$662$	1.42144e12	0.287652
$663$	0	0
$664$	2.95124e12	0.589179
$665$	1.53708e12	0.304788
$666$	0	0
$667$	1.34285e13	2.62701
$668$	−8.57871e10	−0.0166697
$669$	0	0
$670$	2.28841e12	0.438730
$671$	−5.60132e11	−0.106669
$672$	0	0
$673$	−6.76161e12	−1.27052	−0.635261	−	0.772297i	$-0.719108\pi$
$673$	−6.76161e12	−1.27052	−0.635261	+	0.772297i	$0.719108\pi$
$674$	4.55040e12	0.849337
$675$	0	0
$676$	−4.69526e10	−0.00864768
$677$	−6.04459e12	−1.10590	−0.552952	−	0.833213i	$-0.686499\pi$
$677$	−6.04459e12	−1.10590	−0.552952	+	0.833213i	$0.686499\pi$
$678$	0	0
$679$	−2.60316e12	−0.469988
$680$	−6.94523e12	−1.24565
$681$	0	0
$682$	2.59562e10	0.00459421
$683$	−3.24587e12	−0.570740	−0.285370	−	0.958417i	$-0.592116\pi$
$683$	−3.24587e12	−0.570740	−0.285370	+	0.958417i	$0.592116\pi$
$684$	0	0
$685$	8.59843e12	1.49215
$686$	3.66724e12	0.632238
$687$	0	0
$688$	5.22039e12	0.888290
$689$	−9.10400e11	−0.153903
$690$	0	0
$691$	−4.14584e12	−0.691769	−0.345885	−	0.938277i	$-0.612421\pi$
$691$	−4.14584e12	−0.691769	−0.345885	+	0.938277i	$0.612421\pi$
$692$	−3.81184e11	−0.0631913
$693$	0	0
$694$	2.02513e12	0.331387
$695$	5.27067e11	0.0856908
$696$	0	0
$697$	1.23863e13	1.98790
$698$	5.38583e12	0.858822
$699$	0	0
$700$	−4.22255e10	−0.00664711
$701$	5.47240e12	0.855947	0.427974	−	0.903791i	$-0.359228\pi$
$701$	5.47240e12	0.855947	0.427974	+	0.903791i	$0.359228\pi$
$702$	0	0
$703$	−6.67473e12	−1.03071
$704$	1.04599e12	0.160491
$705$	0	0
$706$	−9.60094e12	−1.45443
$707$	−2.04706e12	−0.308136
$708$	0	0
$709$	1.68810e12	0.250895	0.125447	−	0.992100i	$-0.459963\pi$
$709$	1.68810e12	0.250895	0.125447	+	0.992100i	$0.459963\pi$
$710$	−3.42366e12	−0.505625
$711$	0	0
$712$	3.55406e12	0.518280
$713$	−3.80571e11	−0.0551484
$714$	0	0
$715$	2.61825e11	0.0374657
$716$	8.11690e11	0.115420
$717$	0	0
$718$	−8.39360e12	−1.17866
$719$	3.99409e12	0.557363	0.278681	−	0.960384i	$-0.410103\pi$
$719$	3.99409e12	0.557363	0.278681	+	0.960384i	$0.410103\pi$
$720$	0	0
$721$	−1.39758e12	−0.192605
$722$	−9.28430e11	−0.127154
$723$	0	0
$724$	−1.37507e12	−0.185995
$725$	1.92755e12	0.259111
$726$	0	0
$727$	−1.96928e12	−0.261459	−0.130730	−	0.991418i	$-0.541732\pi$
$727$	−1.96928e12	−0.261459	−0.130730	+	0.991418i	$0.541732\pi$
$728$	7.89952e11	0.104234
$729$	0	0
$730$	−5.12522e12	−0.667974
$731$	−1.01942e13	−1.32046
$732$	0	0
$733$	2.48265e12	0.317649	0.158825	−	0.987307i	$-0.449230\pi$
$733$	2.48265e12	0.317649	0.158825	+	0.987307i	$0.449230\pi$
$734$	−4.82428e12	−0.613479
$735$	0	0
$736$	−2.97739e12	−0.374012
$737$	6.03333e11	0.0753275
$738$	0	0
$739$	−8.83198e12	−1.08933	−0.544664	−	0.838655i	$-0.683343\pi$
$739$	−8.83198e12	−1.08933	−0.544664	+	0.838655i	$0.683343\pi$
$740$	−9.28704e11	−0.113850
$741$	0	0
$742$	1.54792e12	0.187469
$743$	−5.88065e12	−0.707906	−0.353953	−	0.935263i	$-0.615163\pi$
$743$	−5.88065e12	−0.707906	−0.353953	+	0.935263i	$0.615163\pi$
$744$	0	0
$745$	−9.16141e12	−1.08958
$746$	1.01264e13	1.19710
$747$	0	0
$748$	−1.85048e11	−0.0216136
$749$	3.33171e12	0.386811
$750$	0	0
$751$	−9.39265e12	−1.07748	−0.538739	−	0.842473i	$-0.681099\pi$
$751$	−9.39265e12	−1.07748	−0.538739	+	0.842473i	$0.681099\pi$
$752$	1.22043e13	1.39166
$753$	0	0
$754$	−3.64424e12	−0.410616
$755$	1.18831e13	1.33097
$756$	0	0
$757$	1.20690e13	1.33579	0.667897	−	0.744254i	$-0.267195\pi$
$757$	1.20690e13	1.33579	0.667897	+	0.744254i	$0.267195\pi$
$758$	−3.56628e12	−0.392378
$759$	0	0
$760$	8.19263e12	0.890762
$761$	−1.73414e12	−0.187436	−0.0937182	−	0.995599i	$-0.529875\pi$
$761$	−1.73414e12	−0.187436	−0.0937182	+	0.995599i	$0.529875\pi$
$762$	0	0
$763$	−1.63494e12	−0.174639
$764$	−1.33924e12	−0.142213
$765$	0	0
$766$	1.14388e13	1.20047
$767$	−3.27880e12	−0.342086
$768$	0	0
$769$	−1.87490e13	−1.93335	−0.966675	−	0.256006i	$-0.917593\pi$
$769$	−1.87490e13	−1.93335	−0.966675	+	0.256006i	$0.917593\pi$
$770$	−4.45170e11	−0.0456371
$771$	0	0
$772$	−5.10858e11	−0.0517634
$773$	2.23308e12	0.224956	0.112478	−	0.993654i	$-0.464121\pi$
$773$	2.23308e12	0.224956	0.112478	+	0.993654i	$0.464121\pi$
$774$	0	0
$775$	−5.46278e10	−0.00543947
$776$	−1.38749e13	−1.37357
$777$	0	0
$778$	−1.26971e13	−1.24250
$779$	−1.46110e13	−1.42154
$780$	0	0
$781$	−9.02638e11	−0.0868129
$782$	−2.14212e13	−2.04839
$783$	0	0
$784$	8.06534e12	0.762431
$785$	−7.48571e12	−0.703590
$786$	0	0
$787$	7.05497e12	0.655555	0.327777	−	0.944755i	$-0.393700\pi$
$787$	7.05497e12	0.655555	0.327777	+	0.944755i	$0.393700\pi$
$788$	−5.73826e11	−0.0530166
$789$	0	0
$790$	−1.16709e13	−1.06606
$791$	2.11270e11	0.0191886
$792$	0	0
$793$	−2.22876e12	−0.200140
$794$	−1.42572e13	−1.27304
$795$	0	0
$796$	3.40774e11	0.0300855
$797$	−1.16654e13	−1.02409	−0.512045	−	0.858959i	$-0.671112\pi$
$797$	−1.16654e13	−1.02409	−0.512045	+	0.858959i	$0.671112\pi$
$798$	0	0
$799$	−2.38323e13	−2.06874
$800$	−4.27379e11	−0.0368900
$801$	0	0
$802$	−9.11497e12	−0.777984
$803$	−1.35125e12	−0.114687
$804$	0	0
$805$	6.52712e12	0.547823
$806$	1.03280e11	0.00861999
$807$	0	0
$808$	−1.09108e13	−0.900548
$809$	1.37043e13	1.12483	0.562415	−	0.826855i	$-0.309872\pi$
$809$	1.37043e13	1.12483	0.562415	+	0.826855i	$0.309872\pi$
$810$	0	0
$811$	2.07440e12	0.168383	0.0841917	−	0.996450i	$-0.473169\pi$
$811$	2.07440e12	0.168383	0.0841917	+	0.996450i	$0.473169\pi$
$812$	−7.84797e11	−0.0633512
$813$	0	0
$814$	1.93315e12	0.154332
$815$	2.05206e13	1.62922
$816$	0	0
$817$	1.20252e13	0.944261
$818$	−9.33483e11	−0.0728982
$819$	0	0
$820$	−2.03293e12	−0.157022
$821$	1.51952e13	1.16724	0.583622	−	0.812026i	$-0.301635\pi$
$821$	1.51952e13	1.16724	0.583622	+	0.812026i	$0.301635\pi$
$822$	0	0
$823$	8.32881e12	0.632825	0.316412	−	0.948622i	$-0.397522\pi$
$823$	8.32881e12	0.632825	0.316412	+	0.948622i	$0.397522\pi$
$824$	−7.44912e12	−0.562901
$825$	0	0
$826$	5.57481e12	0.416697
$827$	1.50997e13	1.12252	0.561258	−	0.827641i	$-0.310317\pi$
$827$	1.50997e13	1.12252	0.561258	+	0.827641i	$0.310317\pi$
$828$	0	0
$829$	−9.65383e12	−0.709912	−0.354956	−	0.934883i	$-0.615504\pi$
$829$	−9.65383e12	−0.709912	−0.354956	+	0.934883i	$0.615504\pi$
$830$	−6.61764e12	−0.484007
$831$	0	0
$832$	4.16200e12	0.301125
$833$	−1.57498e13	−1.13337
$834$	0	0
$835$	1.90347e12	0.135506
$836$	2.18283e11	0.0154558
$837$	0	0
$838$	2.43540e12	0.170597
$839$	−1.96235e13	−1.36725	−0.683623	−	0.729835i	$-0.739597\pi$
$839$	−1.96235e13	−1.36725	−0.683623	+	0.729835i	$0.739597\pi$
$840$	0	0
$841$	2.13181e13	1.46949
$842$	−1.07978e13	−0.740340
$843$	0	0
$844$	1.83204e12	0.124278
$845$	1.04180e12	0.0702959
$846$	0	0
$847$	5.25399e12	0.350763
$848$	7.31102e12	0.485508
$849$	0	0
$850$	−3.07483e12	−0.202040
$851$	−2.83439e13	−1.85258
$852$	0	0
$853$	−1.66671e13	−1.07793	−0.538963	−	0.842329i	$-0.681184\pi$
$853$	−1.66671e13	−1.07793	−0.538963	+	0.842329i	$0.681184\pi$
$854$	3.78947e12	0.243791
$855$	0	0
$856$	1.77580e13	1.13048
$857$	8.56434e12	0.542351	0.271175	−	0.962530i	$-0.412588\pi$
$857$	8.56434e12	0.542351	0.271175	+	0.962530i	$0.412588\pi$
$858$	0	0
$859$	1.87649e12	0.117592	0.0587958	−	0.998270i	$-0.481274\pi$
$859$	1.87649e12	0.117592	0.0587958	+	0.998270i	$0.481274\pi$
$860$	1.67315e12	0.104302
$861$	0	0
$862$	2.82683e12	0.174388
$863$	−2.54184e11	−0.0155991	−0.00779956	−	0.999970i	$-0.502483\pi$
$863$	−2.54184e11	−0.0155991	−0.00779956	+	0.999970i	$0.502483\pi$
$864$	0	0
$865$	8.45785e12	0.513674
$866$	2.58681e13	1.56291
$867$	0	0
$868$	2.22415e10	0.00132992
$869$	−3.07699e12	−0.183036
$870$	0	0
$871$	2.40066e12	0.141335
$872$	−8.71425e12	−0.510394
$873$	0	0
$874$	2.52685e13	1.46480
$875$	6.61913e12	0.381737
$876$	0	0
$877$	−2.88730e13	−1.64814	−0.824069	−	0.566490i	$-0.808301\pi$
$877$	−2.88730e13	−1.64814	−0.824069	+	0.566490i	$0.808301\pi$
$878$	−1.54971e13	−0.880085
$879$	0	0
$880$	−2.10260e12	−0.118191
$881$	1.43284e13	0.801317	0.400659	−	0.916227i	$-0.368781\pi$
$881$	1.43284e13	0.801317	0.400659	+	0.916227i	$0.368781\pi$
$882$	0	0
$883$	3.54065e12	0.196001	0.0980007	−	0.995186i	$-0.468755\pi$
$883$	3.54065e12	0.196001	0.0980007	+	0.995186i	$0.468755\pi$
$884$	−7.36305e11	−0.0405529
$885$	0	0
$886$	2.17783e13	1.18733
$887$	−1.07018e13	−0.580497	−0.290249	−	0.956951i	$-0.593738\pi$
$887$	−1.07018e13	−0.580497	−0.290249	+	0.956951i	$0.593738\pi$
$888$	0	0
$889$	−1.00245e13	−0.538274
$890$	−7.96937e12	−0.425764
$891$	0	0
$892$	1.34439e12	0.0711022
$893$	2.81127e13	1.47935
$894$	0	0
$895$	−1.80101e13	−0.938235
$896$	−5.52866e12	−0.286572
$897$	0	0
$898$	−1.26141e13	−0.647311
$899$	−1.01531e12	−0.0518416
$900$	0	0
$901$	−1.42768e13	−0.721720
$902$	4.23165e12	0.212853
$903$	0	0
$904$	1.12607e12	0.0560799
$905$	3.05106e13	1.51193
$906$	0	0
$907$	2.74516e13	1.34690	0.673448	−	0.739234i	$-0.264812\pi$
$907$	2.74516e13	1.34690	0.673448	+	0.739234i	$0.264812\pi$
$908$	1.70909e12	0.0834406
$909$	0	0
$910$	−1.77133e12	−0.0856276
$911$	−1.57903e13	−0.759550	−0.379775	−	0.925079i	$-0.623999\pi$
$911$	−1.57903e13	−0.759550	−0.379775	+	0.925079i	$0.623999\pi$
$912$	0	0
$913$	−1.74472e12	−0.0831013
$914$	−1.77078e13	−0.839279
$915$	0	0
$916$	−2.44589e12	−0.114791
$917$	−2.77803e12	−0.129740
$918$	0	0
$919$	3.33575e13	1.54267	0.771335	−	0.636429i	$-0.219589\pi$
$919$	3.33575e13	1.54267	0.771335	+	0.636429i	$0.219589\pi$
$920$	3.47896e13	1.60105
$921$	0	0
$922$	−1.86155e13	−0.848372
$923$	−3.59160e12	−0.162885
$924$	0	0
$925$	−4.06853e12	−0.182726
$926$	1.64234e13	0.734030
$927$	0	0
$928$	−7.94322e12	−0.351585
$929$	5.90953e12	0.260305	0.130152	−	0.991494i	$-0.458453\pi$
$929$	5.90953e12	0.260305	0.130152	+	0.991494i	$0.458453\pi$
$930$	0	0
$931$	1.85785e13	0.810471
$932$	−2.21425e12	−0.0961293
$933$	0	0
$934$	−9.61415e12	−0.413380
$935$	4.10591e12	0.175694
$936$	0	0
$937$	2.94135e13	1.24657	0.623287	−	0.781993i	$-0.285797\pi$
$937$	2.94135e13	1.24657	0.623287	+	0.781993i	$0.285797\pi$
$938$	−4.08175e12	−0.172160
$939$	0	0
$940$	3.91153e12	0.163407
$941$	1.78277e13	0.741212	0.370606	−	0.928790i	$-0.379150\pi$
$941$	1.78277e13	0.741212	0.370606	+	0.928790i	$0.379150\pi$
$942$	0	0
$943$	−6.20448e13	−2.55507
$944$	2.63306e13	1.07916
$945$	0	0
$946$	−3.48275e12	−0.141388
$947$	2.20380e13	0.890427	0.445213	−	0.895425i	$-0.353128\pi$
$947$	2.20380e13	0.890427	0.445213	+	0.895425i	$0.353128\pi$
$948$	0	0
$949$	−5.37661e12	−0.215184
$950$	3.62709e12	0.144478
$951$	0	0
$952$	1.23879e13	0.488801
$953$	−3.10318e13	−1.21868	−0.609340	−	0.792909i	$-0.708565\pi$
$953$	−3.10318e13	−1.21868	−0.609340	+	0.792909i	$0.708565\pi$
$954$	0	0
$955$	2.97156e13	1.15603
$956$	−1.14511e12	−0.0443391
$957$	0	0
$958$	1.12337e13	0.430900
$959$	−1.53367e13	−0.585527
$960$	0	0
$961$	−2.64108e13	−0.998912
$962$	7.69198e12	0.289568
$963$	0	0
$964$	3.05776e12	0.114040
$965$	1.13351e13	0.420778
$966$	0	0
$967$	3.86641e13	1.42196	0.710982	−	0.703210i	$-0.248251\pi$
$967$	3.86641e13	1.42196	0.710982	+	0.703210i	$0.248251\pi$
$968$	2.80038e13	1.02513
$969$	0	0
$970$	3.11120e13	1.12838
$971$	−2.91204e13	−1.05126	−0.525631	−	0.850713i	$-0.676171\pi$
$971$	−2.91204e13	−1.05126	−0.525631	+	0.850713i	$0.676171\pi$
$972$	0	0
$973$	−9.40108e11	−0.0336256
$974$	4.00933e13	1.42744
$975$	0	0
$976$	1.78982e13	0.631372
$977$	2.43957e13	0.856618	0.428309	−	0.903632i	$-0.359109\pi$
$977$	2.43957e13	0.856618	0.428309	+	0.903632i	$0.359109\pi$
$978$	0	0
$979$	−2.10110e12	−0.0731012
$980$	2.58496e12	0.0895235
$981$	0	0
$982$	−3.60031e13	−1.23549
$983$	−1.65480e13	−0.565268	−0.282634	−	0.959228i	$-0.591208\pi$
$983$	−1.65480e13	−0.565268	−0.282634	+	0.959228i	$0.591208\pi$
$984$	0	0
$985$	1.27323e13	0.430965
$986$	−5.71485e13	−1.92557
$987$	0	0
$988$	8.68548e11	0.0289993
$989$	5.10643e13	1.69721
$990$	0	0
$991$	−6.37548e12	−0.209982	−0.104991	−	0.994473i	$-0.533481\pi$
$991$	−6.37548e12	−0.209982	−0.104991	+	0.994473i	$0.533481\pi$
$992$	2.25115e11	0.00738077
$993$	0	0
$994$	6.10664e12	0.198410
$995$	−7.56122e12	−0.244561
$996$	0	0
$997$	1.52049e13	0.487366	0.243683	−	0.969855i	$-0.421644\pi$
$997$	1.52049e13	0.487366	0.243683	+	0.969855i	$0.421644\pi$
$998$	3.42572e13	1.09311
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.10.a.e.1.4		5
3.2	odd	2		13.10.a.b.1.2	✓	5
12.11	even	2		208.10.a.h.1.5		5
15.14	odd	2		325.10.a.b.1.4		5
39.38	odd	2		169.10.a.b.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.2	✓	5	3.2	odd	2
117.10.a.e.1.4		5	1.1	even	1	trivial
169.10.a.b.1.4		5	39.38	odd	2
208.10.a.h.1.5		5	12.11	even	2
325.10.a.b.1.4		5	15.14	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	117.10.a.e.1.4

Level	$117$
Weight	$10$
Character	117.1
Self dual	yes
Analytic conductor	$60.259$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$