LMFDB - Embedded newform 8.20.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,20,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	8.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:	$18.3053357245$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2519x + 43659 $ x^3 - x^2 - 2519*x + 43659
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$20.6663$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+54241.2 q^{3} +7.60339e6 q^{5} +7.17920e7 q^{7} +1.77984e9 q^{9} +O(q^{10})$	$q+54241.2 q^{3} +7.60339e6 q^{5} +7.17920e7 q^{7} +1.77984e9 q^{9} -4.34624e9 q^{11} -7.43111e10 q^{13} +4.12417e11 q^{15} +1.24277e11 q^{17} +3.27915e11 q^{19} +3.89408e12 q^{21} +4.60752e12 q^{23} +3.87380e13 q^{25} +3.34984e13 q^{27} +5.90175e12 q^{29} -2.14979e14 q^{31} -2.35745e14 q^{33} +5.45862e14 q^{35} -1.90271e13 q^{37} -4.03072e15 q^{39} +1.20083e15 q^{41} -3.64834e15 q^{43} +1.35328e16 q^{45} +7.52366e15 q^{47} -6.24480e15 q^{49} +6.74095e15 q^{51} +2.09104e16 q^{53} -3.30462e16 q^{55} +1.77865e16 q^{57} -1.11875e17 q^{59} +8.86226e16 q^{61} +1.27779e17 q^{63} -5.65016e17 q^{65} -1.47802e17 q^{67} +2.49917e17 q^{69} +3.98293e17 q^{71} -7.71257e15 q^{73} +2.10119e18 q^{75} -3.12026e17 q^{77} -1.54385e18 q^{79} -2.51651e17 q^{81} -5.86011e17 q^{83} +9.44929e17 q^{85} +3.20118e17 q^{87} +2.54352e18 q^{89} -5.33495e18 q^{91} -1.16607e19 q^{93} +2.49327e18 q^{95} +7.18112e18 q^{97} -7.73564e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9}+O(q^{10}) $ 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9	$ 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} - 297392964 q^{11} - 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} + 3212269666884 q^{19} + 11192319829728 q^{21} + 24948509305560 q^{23} + 72340360289109 q^{25} + 64092343553864 q^{27} + 77667139511058 q^{29} - 248431735193568 q^{31} - 252071696774128 q^{33} - 13\!\cdots\!08 q^{35}+ \cdots - 11\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9 - 297392964 * q^11 - 14862401022 * q^13 + 292635653528 * q^15 + 803332464534 * q^17 + 3212269666884 * q^19 + 11192319829728 * q^21 + 24948509305560 * q^23 + 72340360289109 * q^25 + 64092343553864 * q^27 + 77667139511058 * q^29 - 248431735193568 * q^31 - 252071696774128 * q^33 - 1384378114232208 * q^35 - 414866302559142 * q^37 - 5522045976027016 * q^39 + 2818737880869678 * q^41 - 6663230715469860 * q^43 + 22824972038174722 * q^45 + 1500497644728624 * q^47 + 39641123057321787 * q^49 - 1085042177122520 * q^51 + 56067344774978154 * q^53 - 56817855560205432 * q^55 - 6408983732955152 * q^57 - 154317270851496852 * q^59 - 134994376571654862 * q^61 - 78435010097874072 * q^63 - 563892790723886724 * q^65 + 151032181904450292 * q^67 - 153590366326625632 * q^69 + 1210541848845584136 * q^71 - 81876123599662770 * q^73 + 2698278154129173484 * q^75 + 97235023193490336 * q^77 + 1439028483035907408 * q^79 - 318667537468381733 * q^81 - 983438102798849916 * q^83 - 1618537843962618540 * q^85 - 4280411824494387144 * q^87 - 1312857528832070946 * q^89 - 11488143382330124496 * q^91 - 4308257105281185920 * q^93 - 10926153348157720968 * q^95 + 14033245412567998566 * q^97 - 11415749644087184660 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	54241.2	1.59103	0.795513	−	0.605937i	$-0.207202\pi$
$3$	54241.2	1.59103	0.795513	+	0.605937i	$0.207202\pi$
$4$	0	0
$5$	7.60339e6	1.74097	0.870486	−	0.492193i	$-0.163804\pi$
$5$	7.60339e6	1.74097	0.870486	+	0.492193i	$0.163804\pi$
$6$	0	0
$7$	7.17920e7	0.672426	0.336213	−	0.941786i	$-0.390854\pi$
$7$	7.17920e7	0.672426	0.336213	+	0.941786i	$0.390854\pi$
$8$	0	0
$9$	1.77984e9	1.53136
$10$	0	0
$11$	−4.34624e9	−0.555755	−0.277878	−	0.960616i	$-0.589631\pi$
$11$	−4.34624e9	−0.555755	−0.277878	+	0.960616i	$0.589631\pi$
$12$	0	0
$13$	−7.43111e10	−1.94353	−0.971767	−	0.235944i	$-0.924182\pi$
$13$	−7.43111e10	−1.94353	−0.971767	+	0.235944i	$0.924182\pi$
$14$	0	0
$15$	4.12417e11	2.76993
$16$	0	0
$17$	1.24277e11	0.254172	0.127086	−	0.991892i	$-0.459438\pi$
$17$	1.24277e11	0.254172	0.127086	+	0.991892i	$0.459438\pi$
$18$	0	0
$19$	3.27915e11	0.233132	0.116566	−	0.993183i	$-0.462811\pi$
$19$	3.27915e11	0.233132	0.116566	+	0.993183i	$0.462811\pi$
$20$	0	0
$21$	3.89408e12	1.06985
$22$	0	0
$23$	4.60752e12	0.533399	0.266700	−	0.963780i	$-0.414067\pi$
$23$	4.60752e12	0.533399	0.266700	+	0.963780i	$0.414067\pi$
$24$	0	0
$25$	3.87380e13	2.03099
$26$	0	0
$27$	3.34984e13	0.845412
$28$	0	0
$29$	5.90175e12	0.0755439	0.0377720	−	0.999286i	$-0.487974\pi$
$29$	5.90175e12	0.0755439	0.0377720	+	0.999286i	$0.487974\pi$
$30$	0	0
$31$	−2.14979e14	−1.46036	−0.730181	−	0.683254i	$-0.760564\pi$
$31$	−2.14979e14	−1.46036	−0.730181	+	0.683254i	$0.760564\pi$
$32$	0	0
$33$	−2.35745e14	−0.884221
$34$	0	0
$35$	5.45862e14	1.17068
$36$	0	0
$37$	−1.90271e13	−0.0240689	−0.0120344	−	0.999928i	$-0.503831\pi$
$37$	−1.90271e13	−0.0240689	−0.0120344	+	0.999928i	$0.503831\pi$
$38$	0	0
$39$	−4.03072e15	−3.09221
$40$	0	0
$41$	1.20083e15	0.572841	0.286420	−	0.958104i	$-0.407535\pi$
$41$	1.20083e15	0.572841	0.286420	+	0.958104i	$0.407535\pi$
$42$	0	0
$43$	−3.64834e15	−1.10699	−0.553496	−	0.832852i	$-0.686707\pi$
$43$	−3.64834e15	−1.10699	−0.553496	+	0.832852i	$0.686707\pi$
$44$	0	0
$45$	1.35328e16	2.66606
$46$	0	0
$47$	7.52366e15	0.980617	0.490309	−	0.871549i	$-0.336884\pi$
$47$	7.52366e15	0.980617	0.490309	+	0.871549i	$0.336884\pi$
$48$	0	0
$49$	−6.24480e15	−0.547843
$50$	0	0
$51$	6.74095e15	0.404394
$52$	0	0
$53$	2.09104e16	0.870444	0.435222	−	0.900323i	$-0.356670\pi$
$53$	2.09104e16	0.870444	0.435222	+	0.900323i	$0.356670\pi$
$54$	0	0
$55$	−3.30462e16	−0.967555
$56$	0	0
$57$	1.77865e16	0.370919
$58$	0	0
$59$	−1.11875e17	−1.68128	−0.840640	−	0.541595i	$-0.817821\pi$
$59$	−1.11875e17	−1.68128	−0.840640	+	0.541595i	$0.817821\pi$
$60$	0	0
$61$	8.86226e16	0.970311	0.485155	−	0.874428i	$-0.338763\pi$
$61$	8.86226e16	0.970311	0.485155	+	0.874428i	$0.338763\pi$
$62$	0	0
$63$	1.27779e17	1.02973
$64$	0	0
$65$	−5.65016e17	−3.38364
$66$	0	0
$67$	−1.47802e17	−0.663696	−0.331848	−	0.943333i	$-0.607672\pi$
$67$	−1.47802e17	−0.663696	−0.331848	+	0.943333i	$0.607672\pi$
$68$	0	0
$69$	2.49917e17	0.848652
$70$	0	0
$71$	3.98293e17	1.03097	0.515487	−	0.856897i	$-0.327611\pi$
$71$	3.98293e17	1.03097	0.515487	+	0.856897i	$0.327611\pi$
$72$	0	0
$73$	−7.71257e15	−0.0153332	−0.00766658	−	0.999971i	$-0.502440\pi$
$73$	−7.71257e15	−0.0153332	−0.00766658	+	0.999971i	$0.502440\pi$
$74$	0	0
$75$	2.10119e18	3.23135
$76$	0	0
$77$	−3.12026e17	−0.373705
$78$	0	0
$79$	−1.54385e18	−1.44927	−0.724633	−	0.689135i	$-0.757990\pi$
$79$	−1.54385e18	−1.44927	−0.724633	+	0.689135i	$0.757990\pi$
$80$	0	0
$81$	−2.51651e17	−0.186291
$82$	0	0
$83$	−5.86011e17	−0.344084	−0.172042	−	0.985090i	$-0.555036\pi$
$83$	−5.86011e17	−0.344084	−0.172042	+	0.985090i	$0.555036\pi$
$84$	0	0
$85$	9.44929e17	0.442506
$86$	0	0
$87$	3.20118e17	0.120192
$88$	0	0
$89$	2.54352e18	0.769538	0.384769	−	0.923013i	$-0.374281\pi$
$89$	2.54352e18	0.769538	0.384769	+	0.923013i	$0.374281\pi$
$90$	0	0
$91$	−5.33495e18	−1.30688
$92$	0	0
$93$	−1.16607e19	−2.32347
$94$	0	0
$95$	2.49327e18	0.405877
$96$	0	0
$97$	7.18112e18	0.959094	0.479547	−	0.877516i	$-0.340801\pi$
$97$	7.18112e18	0.959094	0.479547	+	0.877516i	$0.340801\pi$
$98$	0	0
$99$	−7.73564e18	−0.851063
$100$	0	0
$101$	7.25107e18	0.659704	0.329852	−	0.944033i	$-0.393001\pi$
$101$	7.25107e18	0.659704	0.329852	+	0.944033i	$0.393001\pi$
$102$	0	0
$103$	3.67861e18	0.277799	0.138899	−	0.990307i	$-0.455644\pi$
$103$	3.67861e18	0.277799	0.138899	+	0.990307i	$0.455644\pi$
$104$	0	0
$105$	2.96082e19	1.86258
$106$	0	0
$107$	−7.99646e18	−0.420487	−0.210243	−	0.977649i	$-0.567426\pi$
$107$	−7.99646e18	−0.420487	−0.210243	+	0.977649i	$0.567426\pi$
$108$	0	0
$109$	3.36141e19	1.48242	0.741208	−	0.671275i	$-0.234253\pi$
$109$	3.36141e19	1.48242	0.741208	+	0.671275i	$0.234253\pi$
$110$	0	0
$111$	−1.03205e18	−0.0382942
$112$	0	0
$113$	−3.36352e19	−1.05329	−0.526646	−	0.850085i	$-0.676550\pi$
$113$	−3.36352e19	−1.05329	−0.526646	+	0.850085i	$0.676550\pi$
$114$	0	0
$115$	3.50327e19	0.928634
$116$	0	0
$117$	−1.32262e20	−2.97625
$118$	0	0
$119$	8.92213e18	0.170912
$120$	0	0
$121$	−4.22692e19	−0.691136
$122$	0	0
$123$	6.51343e19	0.911404
$124$	0	0
$125$	1.49517e20	1.79492
$126$	0	0
$127$	−3.95683e19	−0.408519	−0.204259	−	0.978917i	$-0.565479\pi$
$127$	−3.95683e19	−0.408519	−0.204259	+	0.978917i	$0.565479\pi$
$128$	0	0
$129$	−1.97890e20	−1.76125
$130$	0	0
$131$	−2.27948e20	−1.75291	−0.876454	−	0.481486i	$-0.840097\pi$
$131$	−2.27948e20	−1.75291	−0.876454	+	0.481486i	$0.840097\pi$
$132$	0	0
$133$	2.35417e19	0.156764
$134$	0	0
$135$	2.54701e20	1.47184
$136$	0	0
$137$	8.06460e19	0.405264	0.202632	−	0.979255i	$-0.435051\pi$
$137$	8.06460e19	0.405264	0.202632	+	0.979255i	$0.435051\pi$
$138$	0	0
$139$	1.20721e20	0.528619	0.264309	−	0.964438i	$-0.414856\pi$
$139$	1.20721e20	0.528619	0.264309	+	0.964438i	$0.414856\pi$
$140$	0	0
$141$	4.08092e20	1.56019
$142$	0	0
$143$	3.22974e20	1.08013
$144$	0	0
$145$	4.48733e19	0.131520
$146$	0	0
$147$	−3.38725e20	−0.871632
$148$	0	0
$149$	−2.79576e20	−0.632747	−0.316374	−	0.948635i	$-0.602465\pi$
$149$	−2.79576e20	−0.632747	−0.316374	+	0.948635i	$0.602465\pi$
$150$	0	0
$151$	−2.72950e20	−0.544255	−0.272127	−	0.962261i	$-0.587727\pi$
$151$	−2.72950e20	−0.544255	−0.272127	+	0.962261i	$0.587727\pi$
$152$	0	0
$153$	2.21194e20	0.389229
$154$	0	0
$155$	−1.63457e21	−2.54245
$156$	0	0
$157$	−2.54300e19	−0.0350186	−0.0175093	−	0.999847i	$-0.505574\pi$
$157$	−2.54300e19	−0.0350186	−0.0175093	+	0.999847i	$0.505574\pi$
$158$	0	0
$159$	1.13420e21	1.38490
$160$	0	0
$161$	3.30783e20	0.358672
$162$	0	0
$163$	−4.89681e20	−0.472205	−0.236103	−	0.971728i	$-0.575870\pi$
$163$	−4.89681e20	−0.472205	−0.236103	+	0.971728i	$0.575870\pi$
$164$	0	0
$165$	−1.79246e21	−1.53940
$166$	0	0
$167$	−1.28556e21	−0.984661	−0.492331	−	0.870408i	$-0.663855\pi$
$167$	−1.28556e21	−0.984661	−0.492331	+	0.870408i	$0.663855\pi$
$168$	0	0
$169$	4.06022e21	2.77732
$170$	0	0
$171$	5.83638e20	0.357010
$172$	0	0
$173$	−1.13445e21	−0.621367	−0.310684	−	0.950513i	$-0.600558\pi$
$173$	−1.13445e21	−0.621367	−0.310684	+	0.950513i	$0.600558\pi$
$174$	0	0
$175$	2.78108e21	1.36569
$176$	0	0
$177$	−6.06825e21	−2.67496
$178$	0	0
$179$	3.16357e21	1.25335	0.626676	−	0.779280i	$-0.284415\pi$
$179$	3.16357e21	1.25335	0.626676	+	0.779280i	$0.284415\pi$
$180$	0	0
$181$	1.73143e21	0.617246	0.308623	−	0.951184i	$-0.400132\pi$
$181$	1.73143e21	0.617246	0.308623	+	0.951184i	$0.400132\pi$
$182$	0	0
$183$	4.80699e21	1.54379
$184$	0	0
$185$	−1.44670e20	−0.0419032
$186$	0	0
$187$	−5.40140e20	−0.141257
$188$	0	0
$189$	2.40492e21	0.568477
$190$	0	0
$191$	4.91737e21	1.05176	0.525879	−	0.850559i	$-0.323737\pi$
$191$	4.91737e21	1.05176	0.525879	+	0.850559i	$0.323737\pi$
$192$	0	0
$193$	2.01335e21	0.390055	0.195027	−	0.980798i	$-0.437520\pi$
$193$	2.01335e21	0.390055	0.195027	+	0.980798i	$0.437520\pi$
$194$	0	0
$195$	−3.06471e22	−5.38346
$196$	0	0
$197$	−5.05730e21	−0.806287	−0.403143	−	0.915137i	$-0.632082\pi$
$197$	−5.05730e21	−0.806287	−0.403143	+	0.915137i	$0.632082\pi$
$198$	0	0
$199$	7.67114e21	1.11111	0.555553	−	0.831481i	$-0.312507\pi$
$199$	7.67114e21	1.11111	0.555553	+	0.831481i	$0.312507\pi$
$200$	0	0
$201$	−8.01694e21	−1.05596
$202$	0	0
$203$	4.23698e20	0.0507977
$204$	0	0
$205$	9.13035e21	0.997300
$206$	0	0
$207$	8.20066e21	0.816828
$208$	0	0
$209$	−1.42520e21	−0.129565
$210$	0	0
$211$	1.45561e22	1.20882	0.604409	−	0.796674i	$-0.293409\pi$
$211$	1.45561e22	1.20882	0.604409	+	0.796674i	$0.293409\pi$
$212$	0	0
$213$	2.16039e22	1.64031
$214$	0	0
$215$	−2.77397e22	−1.92724
$216$	0	0
$217$	−1.54338e22	−0.981986
$218$	0	0
$219$	−4.18339e20	−0.0243955
$220$	0	0
$221$	−9.23519e21	−0.493991
$222$	0	0
$223$	2.86567e22	1.40712	0.703559	−	0.710637i	$-0.251593\pi$
$223$	2.86567e22	1.40712	0.703559	+	0.710637i	$0.251593\pi$
$224$	0	0
$225$	6.89476e22	3.11018
$226$	0	0
$227$	−4.29551e22	−1.78143	−0.890716	−	0.454560i	$-0.849796\pi$
$227$	−4.29551e22	−1.78143	−0.890716	+	0.454560i	$0.849796\pi$
$228$	0	0
$229$	1.58539e22	0.604922	0.302461	−	0.953162i	$-0.402192\pi$
$229$	1.58539e22	0.604922	0.302461	+	0.953162i	$0.402192\pi$
$230$	0	0
$231$	−1.69246e22	−0.594574
$232$	0	0
$233$	−2.39783e22	−0.776135	−0.388068	−	0.921631i	$-0.626857\pi$
$233$	−2.39783e22	−0.776135	−0.388068	+	0.921631i	$0.626857\pi$
$234$	0	0
$235$	5.72053e22	1.70723
$236$	0	0
$237$	−8.37403e22	−2.30582
$238$	0	0
$239$	4.53940e22	1.15403	0.577017	−	0.816732i	$-0.304217\pi$
$239$	4.53940e22	1.15403	0.577017	+	0.816732i	$0.304217\pi$
$240$	0	0
$241$	−2.71233e22	−0.637060	−0.318530	−	0.947913i	$-0.603189\pi$
$241$	−2.71233e22	−0.637060	−0.318530	+	0.947913i	$0.603189\pi$
$242$	0	0
$243$	−5.25838e22	−1.14181
$244$	0	0
$245$	−4.74816e22	−0.953779
$246$	0	0
$247$	−2.43677e22	−0.453100
$248$	0	0
$249$	−3.17859e22	−0.547446
$250$	0	0
$251$	3.70408e22	0.591263	0.295631	−	0.955302i	$-0.404470\pi$
$251$	3.70408e22	0.591263	0.295631	+	0.955302i	$0.404470\pi$
$252$	0	0
$253$	−2.00254e22	−0.296440
$254$	0	0
$255$	5.12541e22	0.704039
$256$	0	0
$257$	9.33291e22	1.19029	0.595145	−	0.803618i	$-0.297095\pi$
$257$	9.33291e22	1.19029	0.595145	+	0.803618i	$0.297095\pi$
$258$	0	0
$259$	−1.36599e21	−0.0161845
$260$	0	0
$261$	1.05042e22	0.115685
$262$	0	0
$263$	1.29563e22	0.132709	0.0663544	−	0.997796i	$-0.478863\pi$
$263$	1.29563e22	0.132709	0.0663544	+	0.997796i	$0.478863\pi$
$264$	0	0
$265$	1.58990e23	1.51542
$266$	0	0
$267$	1.37963e23	1.22435
$268$	0	0
$269$	1.06950e23	0.884165	0.442083	−	0.896974i	$-0.354240\pi$
$269$	1.06950e23	0.884165	0.442083	+	0.896974i	$0.354240\pi$
$270$	0	0
$271$	−1.22584e23	−0.944554	−0.472277	−	0.881450i	$-0.656568\pi$
$271$	−1.22584e23	−0.944554	−0.472277	+	0.881450i	$0.656568\pi$
$272$	0	0
$273$	−2.89374e23	−2.07928
$274$	0	0
$275$	−1.68365e23	−1.12873
$276$	0	0
$277$	3.10802e23	1.94503	0.972513	−	0.232848i	$-0.0748046\pi$
$277$	3.10802e23	1.94503	0.972513	+	0.232848i	$0.0748046\pi$
$278$	0	0
$279$	−3.82630e23	−2.23634
$280$	0	0
$281$	−2.37391e22	−0.129645	−0.0648223	−	0.997897i	$-0.520648\pi$
$281$	−2.37391e22	−0.129645	−0.0648223	+	0.997897i	$0.520648\pi$
$282$	0	0
$283$	3.25849e23	1.66359	0.831793	−	0.555085i	$-0.187314\pi$
$283$	3.25849e23	1.66359	0.831793	+	0.555085i	$0.187314\pi$
$284$	0	0
$285$	1.35238e23	0.645761
$286$	0	0
$287$	8.62098e22	0.385193
$288$	0	0
$289$	−2.23628e23	−0.935397
$290$	0	0
$291$	3.89513e23	1.52594
$292$	0	0
$293$	−2.70509e23	−0.992978	−0.496489	−	0.868043i	$-0.665378\pi$
$293$	−2.70509e23	−0.992978	−0.496489	+	0.868043i	$0.665378\pi$
$294$	0	0
$295$	−8.50631e23	−2.92706
$296$	0	0
$297$	−1.45592e23	−0.469842
$298$	0	0
$299$	−3.42390e23	−1.03668
$300$	0	0
$301$	−2.61922e23	−0.744371
$302$	0	0
$303$	3.93306e23	1.04961
$304$	0	0
$305$	6.73832e23	1.68928
$306$	0	0
$307$	1.53436e23	0.361503	0.180751	−	0.983529i	$-0.442147\pi$
$307$	1.53436e23	0.361503	0.180751	+	0.983529i	$0.442147\pi$
$308$	0	0
$309$	1.99532e23	0.441985
$310$	0	0
$311$	5.66445e22	0.118014	0.0590071	−	0.998258i	$-0.481207\pi$
$311$	5.66445e22	0.118014	0.0590071	+	0.998258i	$0.481207\pi$
$312$	0	0
$313$	7.19437e22	0.141033	0.0705166	−	0.997511i	$-0.477535\pi$
$313$	7.19437e22	0.141033	0.0705166	+	0.997511i	$0.477535\pi$
$314$	0	0
$315$	9.71550e23	1.79273
$316$	0	0
$317$	8.52672e23	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$317$	8.52672e23	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$318$	0	0
$319$	−2.56504e22	−0.0419840
$320$	0	0
$321$	−4.33738e23	−0.669005
$322$	0	0
$323$	4.07525e22	0.0592557
$324$	0	0
$325$	−2.87866e24	−3.94729
$326$	0	0
$327$	1.82327e24	2.35856
$328$	0	0
$329$	5.40139e23	0.659393
$330$	0	0
$331$	1.11532e23	0.128539	0.0642693	−	0.997933i	$-0.479528\pi$
$331$	1.11532e23	0.128539	0.0642693	+	0.997933i	$0.479528\pi$
$332$	0	0
$333$	−3.38652e22	−0.0368581
$334$	0	0
$335$	−1.12379e24	−1.15548
$336$	0	0
$337$	−4.62744e23	−0.449632	−0.224816	−	0.974401i	$-0.572178\pi$
$337$	−4.62744e23	−0.449632	−0.224816	+	0.974401i	$0.572178\pi$
$338$	0	0
$339$	−1.82441e24	−1.67581
$340$	0	0
$341$	9.34353e23	0.811604
$342$	0	0
$343$	−1.26668e24	−1.04081
$344$	0	0
$345$	1.90022e24	1.47748
$346$	0	0
$347$	−3.04145e23	−0.223847	−0.111923	−	0.993717i	$-0.535701\pi$
$347$	−3.04145e23	−0.223847	−0.111923	+	0.993717i	$0.535701\pi$
$348$	0	0
$349$	9.54713e23	0.665321	0.332661	−	0.943047i	$-0.392054\pi$
$349$	9.54713e23	0.665321	0.332661	+	0.943047i	$0.392054\pi$
$350$	0	0
$351$	−2.48930e24	−1.64309
$352$	0	0
$353$	−6.79039e23	−0.424654	−0.212327	−	0.977199i	$-0.568104\pi$
$353$	−6.79039e23	−0.424654	−0.212327	+	0.977199i	$0.568104\pi$
$354$	0	0
$355$	3.02837e24	1.79490
$356$	0	0
$357$	4.83947e23	0.271925
$358$	0	0
$359$	3.83232e23	0.204204	0.102102	−	0.994774i	$-0.467443\pi$
$359$	3.83232e23	0.204204	0.102102	+	0.994774i	$0.467443\pi$
$360$	0	0
$361$	−1.87089e24	−0.945649
$362$	0	0
$363$	−2.29273e24	−1.09962
$364$	0	0
$365$	−5.86416e22	−0.0266946
$366$	0	0
$367$	−1.01361e23	−0.0438069	−0.0219035	−	0.999760i	$-0.506973\pi$
$367$	−1.01361e23	−0.0438069	−0.0219035	+	0.999760i	$0.506973\pi$
$368$	0	0
$369$	2.13729e24	0.877227
$370$	0	0
$371$	1.50120e24	0.585309
$372$	0	0
$373$	4.09193e24	1.51598	0.757992	−	0.652264i	$-0.226181\pi$
$373$	4.09193e24	1.51598	0.757992	+	0.652264i	$0.226181\pi$
$374$	0	0
$375$	8.10996e24	2.85576
$376$	0	0
$377$	−4.38565e23	−0.146822
$378$	0	0
$379$	−4.60553e24	−1.46625	−0.733124	−	0.680095i	$-0.761939\pi$
$379$	−4.60553e24	−1.46625	−0.733124	+	0.680095i	$0.761939\pi$
$380$	0	0
$381$	−2.14623e24	−0.649964
$382$	0	0
$383$	5.13713e24	1.48024	0.740120	−	0.672475i	$-0.234769\pi$
$383$	5.13713e24	1.48024	0.740120	+	0.672475i	$0.234769\pi$
$384$	0	0
$385$	−2.37245e24	−0.650610
$386$	0	0
$387$	−6.49347e24	−1.69521
$388$	0	0
$389$	6.60845e24	1.64278	0.821388	−	0.570370i	$-0.193200\pi$
$389$	6.60845e24	1.64278	0.821388	+	0.570370i	$0.193200\pi$
$390$	0	0
$391$	5.72610e23	0.135575
$392$	0	0
$393$	−1.23642e25	−2.78892
$394$	0	0
$395$	−1.17385e25	−2.52313
$396$	0	0
$397$	−7.93312e24	−1.62530	−0.812651	−	0.582751i	$-0.801976\pi$
$397$	−7.93312e24	−1.62530	−0.812651	+	0.582751i	$0.801976\pi$
$398$	0	0
$399$	1.27693e24	0.249416
$400$	0	0
$401$	4.48702e24	0.835770	0.417885	−	0.908500i	$-0.362771\pi$
$401$	4.48702e24	0.835770	0.417885	+	0.908500i	$0.362771\pi$
$402$	0	0
$403$	1.59754e25	2.83826
$404$	0	0
$405$	−1.91340e24	−0.324327
$406$	0	0
$407$	8.26963e22	0.0133764
$408$	0	0
$409$	7.90722e24	1.22082	0.610411	−	0.792085i	$-0.291004\pi$
$409$	7.90722e24	1.22082	0.610411	+	0.792085i	$0.291004\pi$
$410$	0	0
$411$	4.37434e24	0.644785
$412$	0	0
$413$	−8.03175e24	−1.13054
$414$	0	0
$415$	−4.45567e24	−0.599040
$416$	0	0
$417$	6.54806e24	0.841046
$418$	0	0
$419$	−1.13527e25	−1.39336	−0.696682	−	0.717380i	$-0.745341\pi$
$419$	−1.13527e25	−1.39336	−0.696682	+	0.717380i	$0.745341\pi$
$420$	0	0
$421$	−1.32453e25	−1.55375	−0.776876	−	0.629653i	$-0.783197\pi$
$421$	−1.32453e25	−1.55375	−0.776876	+	0.629653i	$0.783197\pi$
$422$	0	0
$423$	1.33909e25	1.50168
$424$	0	0
$425$	4.81426e24	0.516219
$426$	0	0
$427$	6.36239e24	0.652463
$428$	0	0
$429$	1.75185e25	1.71851
$430$	0	0
$431$	−1.09929e25	−1.03176	−0.515879	−	0.856662i	$-0.672534\pi$
$431$	−1.09929e25	−1.03176	−0.515879	+	0.856662i	$0.672534\pi$
$432$	0	0
$433$	9.77489e24	0.877964	0.438982	−	0.898496i	$-0.355339\pi$
$433$	9.77489e24	0.877964	0.438982	+	0.898496i	$0.355339\pi$
$434$	0	0
$435$	2.43398e24	0.209252
$436$	0	0
$437$	1.51088e24	0.124353
$438$	0	0
$439$	1.10857e25	0.873672	0.436836	−	0.899541i	$-0.356099\pi$
$439$	1.10857e25	0.873672	0.436836	+	0.899541i	$0.356099\pi$
$440$	0	0
$441$	−1.11148e25	−0.838946
$442$	0	0
$443$	−2.09528e25	−1.51498	−0.757491	−	0.652845i	$-0.773575\pi$
$443$	−2.09528e25	−1.51498	−0.757491	+	0.652845i	$0.773575\pi$
$444$	0	0
$445$	1.93393e25	1.33974
$446$	0	0
$447$	−1.51645e25	−1.00672
$448$	0	0
$449$	2.72879e25	1.73632	0.868159	−	0.496285i	$-0.165303\pi$
$449$	2.72879e25	1.73632	0.868159	+	0.496285i	$0.165303\pi$
$450$	0	0
$451$	−5.21909e24	−0.318359
$452$	0	0
$453$	−1.48051e25	−0.865923
$454$	0	0
$455$	−4.05636e25	−2.27525
$456$	0	0
$457$	1.01860e23	0.00548024	0.00274012	−	0.999996i	$-0.499128\pi$
$457$	1.01860e23	0.00548024	0.00274012	+	0.999996i	$0.499128\pi$
$458$	0	0
$459$	4.16309e24	0.214880
$460$	0	0
$461$	−1.12624e25	−0.557790	−0.278895	−	0.960322i	$-0.589968\pi$
$461$	−1.12624e25	−0.557790	−0.278895	+	0.960322i	$0.589968\pi$
$462$	0	0
$463$	3.43863e25	1.63443	0.817216	−	0.576332i	$-0.195517\pi$
$463$	3.43863e25	1.63443	0.817216	+	0.576332i	$0.195517\pi$
$464$	0	0
$465$	−8.86610e25	−4.04510
$466$	0	0
$467$	2.38273e25	1.04367	0.521837	−	0.853045i	$-0.325247\pi$
$467$	2.38273e25	1.04367	0.521837	+	0.853045i	$0.325247\pi$
$468$	0	0
$469$	−1.06110e25	−0.446287
$470$	0	0
$471$	−1.37935e24	−0.0557156
$472$	0	0
$473$	1.58566e25	0.615217
$474$	0	0
$475$	1.27028e25	0.473488
$476$	0	0
$477$	3.72172e25	1.33297
$478$	0	0
$479$	1.91851e25	0.660355	0.330177	−	0.943919i	$-0.392891\pi$
$479$	1.91851e25	0.660355	0.330177	+	0.943919i	$0.392891\pi$
$480$	0	0
$481$	1.41392e24	0.0467786
$482$	0	0
$483$	1.79421e25	0.570656
$484$	0	0
$485$	5.46009e25	1.66976
$486$	0	0
$487$	1.71477e25	0.504292	0.252146	−	0.967689i	$-0.418864\pi$
$487$	1.71477e25	0.504292	0.252146	+	0.967689i	$0.418864\pi$
$488$	0	0
$489$	−2.65609e25	−0.751290
$490$	0	0
$491$	2.04456e25	0.556321	0.278161	−	0.960535i	$-0.410275\pi$
$491$	2.04456e25	0.556321	0.278161	+	0.960535i	$0.410275\pi$
$492$	0	0
$493$	7.33454e23	0.0192011
$494$	0	0
$495$	−5.88170e25	−1.48168
$496$	0	0
$497$	2.85942e25	0.693255
$498$	0	0
$499$	−4.83482e25	−1.12830	−0.564151	−	0.825672i	$-0.690796\pi$
$499$	−4.83482e25	−1.12830	−0.564151	+	0.825672i	$0.690796\pi$
$500$	0	0
$501$	−6.97305e25	−1.56662
$502$	0	0
$503$	−7.87529e25	−1.70361	−0.851806	−	0.523857i	$-0.824493\pi$
$503$	−7.87529e25	−1.70361	−0.851806	+	0.523857i	$0.824493\pi$
$504$	0	0
$505$	5.51327e25	1.14853
$506$	0	0
$507$	2.20231e26	4.41879
$508$	0	0
$509$	−6.17524e25	−1.19353	−0.596767	−	0.802414i	$-0.703548\pi$
$509$	−6.17524e25	−1.19353	−0.596767	+	0.802414i	$0.703548\pi$
$510$	0	0
$511$	−5.53701e23	−0.0103104
$512$	0	0
$513$	1.09846e25	0.197093
$514$	0	0
$515$	2.79699e25	0.483640
$516$	0	0
$517$	−3.26997e25	−0.544983
$518$	0	0
$519$	−6.15341e25	−0.988612
$520$	0	0
$521$	−7.46066e25	−1.15563	−0.577815	−	0.816168i	$-0.696094\pi$
$521$	−7.46066e25	−1.15563	−0.577815	+	0.816168i	$0.696094\pi$
$522$	0	0
$523$	−7.40672e25	−1.10627	−0.553133	−	0.833093i	$-0.686568\pi$
$523$	−7.40672e25	−1.10627	−0.553133	+	0.833093i	$0.686568\pi$
$524$	0	0
$525$	1.50849e26	2.17285
$526$	0	0
$527$	−2.67171e25	−0.371183
$528$	0	0
$529$	−5.33863e25	−0.715485
$530$	0	0
$531$	−1.99120e26	−2.57465
$532$	0	0
$533$	−8.92348e25	−1.11334
$534$	0	0
$535$	−6.08002e25	−0.732056
$536$	0	0
$537$	1.71596e26	1.99411
$538$	0	0
$539$	2.71414e25	0.304467
$540$	0	0
$541$	−9.71062e25	−1.05165	−0.525827	−	0.850592i	$-0.676244\pi$
$541$	−9.71062e25	−1.05165	−0.525827	+	0.850592i	$0.676244\pi$
$542$	0	0
$543$	9.39147e25	0.982054
$544$	0	0
$545$	2.55581e26	2.58085
$546$	0	0
$547$	3.89150e25	0.379522	0.189761	−	0.981830i	$-0.439229\pi$
$547$	3.89150e25	0.379522	0.189761	+	0.981830i	$0.439229\pi$
$548$	0	0
$549$	1.57734e26	1.48590
$550$	0	0
$551$	1.93527e24	0.0176117
$552$	0	0
$553$	−1.10836e26	−0.974524
$554$	0	0
$555$	−7.84708e24	−0.0666691
$556$	0	0
$557$	1.06329e25	0.0873023	0.0436511	−	0.999047i	$-0.486101\pi$
$557$	1.06329e25	0.0873023	0.0436511	+	0.999047i	$0.486101\pi$
$558$	0	0
$559$	2.71112e26	2.15148
$560$	0	0
$561$	−2.92978e25	−0.224744
$562$	0	0
$563$	6.36234e25	0.471831	0.235916	−	0.971774i	$-0.424191\pi$
$563$	6.36234e25	0.471831	0.235916	+	0.971774i	$0.424191\pi$
$564$	0	0
$565$	−2.55741e26	−1.83375
$566$	0	0
$567$	−1.80666e25	−0.125267
$568$	0	0
$569$	−1.05414e26	−0.706855	−0.353427	−	0.935462i	$-0.614984\pi$
$569$	−1.05414e26	−0.706855	−0.353427	+	0.935462i	$0.614984\pi$
$570$	0	0
$571$	−2.86148e26	−1.85587	−0.927935	−	0.372742i	$-0.878418\pi$
$571$	−2.86148e26	−1.85587	−0.927935	+	0.372742i	$0.878418\pi$
$572$	0	0
$573$	2.66724e26	1.67337
$574$	0	0
$575$	1.78486e26	1.08333
$576$	0	0
$577$	6.45786e25	0.379244	0.189622	−	0.981857i	$-0.439274\pi$
$577$	6.45786e25	0.379244	0.189622	+	0.981857i	$0.439274\pi$
$578$	0	0
$579$	1.09207e26	0.620587
$580$	0	0
$581$	−4.20709e25	−0.231371
$582$	0	0
$583$	−9.08815e25	−0.483754
$584$	0	0
$585$	−1.00564e27	−5.18158
$586$	0	0
$587$	2.66775e26	1.33071	0.665355	−	0.746527i	$-0.268280\pi$
$587$	2.66775e26	1.33071	0.665355	+	0.746527i	$0.268280\pi$
$588$	0	0
$589$	−7.04950e25	−0.340458
$590$	0	0
$591$	−2.74314e26	−1.28282
$592$	0	0
$593$	3.33072e26	1.50841	0.754203	−	0.656642i	$-0.228024\pi$
$593$	3.33072e26	1.50841	0.754203	+	0.656642i	$0.228024\pi$
$594$	0	0
$595$	6.78384e25	0.297553
$596$	0	0
$597$	4.16092e26	1.76780
$598$	0	0
$599$	2.21905e26	0.913299	0.456649	−	0.889647i	$-0.349049\pi$
$599$	2.21905e26	0.913299	0.456649	+	0.889647i	$0.349049\pi$
$600$	0	0
$601$	2.52202e26	1.00564	0.502819	−	0.864392i	$-0.332296\pi$
$601$	2.52202e26	1.00564	0.502819	+	0.864392i	$0.332296\pi$
$602$	0	0
$603$	−2.63064e26	−1.01636
$604$	0	0
$605$	−3.21389e26	−1.20325
$606$	0	0
$607$	−2.84239e26	−1.03132	−0.515658	−	0.856794i	$-0.672452\pi$
$607$	−2.84239e26	−1.03132	−0.515658	+	0.856794i	$0.672452\pi$
$608$	0	0
$609$	2.29819e25	0.0808205
$610$	0	0
$611$	−5.59092e26	−1.90586
$612$	0	0
$613$	1.40389e26	0.463938	0.231969	−	0.972723i	$-0.425483\pi$
$613$	1.40389e26	0.463938	0.231969	+	0.972723i	$0.425483\pi$
$614$	0	0
$615$	4.95241e26	1.58673
$616$	0	0
$617$	−2.10067e26	−0.652603	−0.326301	−	0.945266i	$-0.605802\pi$
$617$	−2.10067e26	−0.652603	−0.326301	+	0.945266i	$0.605802\pi$
$618$	0	0
$619$	3.33919e25	0.100596	0.0502979	−	0.998734i	$-0.483983\pi$
$619$	3.33919e25	0.100596	0.0502979	+	0.998734i	$0.483983\pi$
$620$	0	0
$621$	1.54344e26	0.450942
$622$	0	0
$623$	1.82604e26	0.517457
$624$	0	0
$625$	3.97965e26	1.09392
$626$	0	0
$627$	−7.73045e25	−0.206140
$628$	0	0
$629$	−2.36463e24	−0.00611762
$630$	0	0
$631$	5.88917e26	1.47834	0.739171	−	0.673518i	$-0.235217\pi$
$631$	5.88917e26	1.47834	0.739171	+	0.673518i	$0.235217\pi$
$632$	0	0
$633$	7.89538e26	1.92326
$634$	0	0
$635$	−3.00853e26	−0.711220
$636$	0	0
$637$	4.64058e26	1.06475
$638$	0	0
$639$	7.08899e26	1.57880
$640$	0	0
$641$	−3.23194e26	−0.698734	−0.349367	−	0.936986i	$-0.613603\pi$
$641$	−3.23194e26	−0.698734	−0.349367	+	0.936986i	$0.613603\pi$
$642$	0	0
$643$	−4.39234e26	−0.921916	−0.460958	−	0.887422i	$-0.652494\pi$
$643$	−4.39234e26	−0.921916	−0.460958	+	0.887422i	$0.652494\pi$
$644$	0	0
$645$	−1.50463e27	−3.06629
$646$	0	0
$647$	7.67429e25	0.151861	0.0759307	−	0.997113i	$-0.475807\pi$
$647$	7.67429e25	0.151861	0.0759307	+	0.997113i	$0.475807\pi$
$648$	0	0
$649$	4.86237e26	0.934380
$650$	0	0
$651$	−8.37147e26	−1.56237
$652$	0	0
$653$	−9.29598e25	−0.168508	−0.0842539	−	0.996444i	$-0.526851\pi$
$653$	−9.29598e25	−0.168508	−0.0842539	+	0.996444i	$0.526851\pi$
$654$	0	0
$655$	−1.73318e27	−3.05176
$656$	0	0
$657$	−1.37272e25	−0.0234806
$658$	0	0
$659$	8.74750e26	1.45369	0.726847	−	0.686800i	$-0.240985\pi$
$659$	8.74750e26	1.45369	0.726847	+	0.686800i	$0.240985\pi$
$660$	0	0
$661$	−4.51459e26	−0.728961	−0.364481	−	0.931211i	$-0.618753\pi$
$661$	−4.51459e26	−0.728961	−0.364481	+	0.931211i	$0.618753\pi$
$662$	0	0
$663$	−5.00928e26	−0.785953
$664$	0	0
$665$	1.78997e26	0.272922
$666$	0	0
$667$	2.71924e25	0.0402951
$668$	0	0
$669$	1.55437e27	2.23876
$670$	0	0
$671$	−3.85175e26	−0.539256
$672$	0	0
$673$	5.36661e26	0.730393	0.365197	−	0.930930i	$-0.381002\pi$
$673$	5.36661e26	0.730393	0.365197	+	0.930930i	$0.381002\pi$
$674$	0	0
$675$	1.29766e27	1.71702
$676$	0	0
$677$	−1.23461e27	−1.58832	−0.794160	−	0.607708i	$-0.792089\pi$
$677$	−1.23461e27	−1.58832	−0.794160	+	0.607708i	$0.792089\pi$
$678$	0	0
$679$	5.15547e26	0.644920
$680$	0	0
$681$	−2.32994e27	−2.83430
$682$	0	0
$683$	4.41928e26	0.522824	0.261412	−	0.965227i	$-0.415812\pi$
$683$	4.41928e26	0.522824	0.261412	+	0.965227i	$0.415812\pi$
$684$	0	0
$685$	6.13183e26	0.705553
$686$	0	0
$687$	8.59934e26	0.962446
$688$	0	0
$689$	−1.55387e27	−1.69174
$690$	0	0
$691$	1.64360e27	1.74082	0.870412	−	0.492324i	$-0.163853\pi$
$691$	1.64360e27	1.74082	0.870412	+	0.492324i	$0.163853\pi$
$692$	0	0
$693$	−5.55357e26	−0.572277
$694$	0	0
$695$	9.17889e26	0.920311
$696$	0	0
$697$	1.49236e26	0.145600
$698$	0	0
$699$	−1.30061e27	−1.23485
$700$	0	0
$701$	5.20256e26	0.480725	0.240362	−	0.970683i	$-0.422734\pi$
$701$	5.20256e26	0.480725	0.240362	+	0.970683i	$0.422734\pi$
$702$	0	0
$703$	−6.23926e24	−0.00561123
$704$	0	0
$705$	3.10288e27	2.71624
$706$	0	0
$707$	5.20569e26	0.443602
$708$	0	0
$709$	1.87693e26	0.155708	0.0778538	−	0.996965i	$-0.475193\pi$
$709$	1.87693e26	0.155708	0.0778538	+	0.996965i	$0.475193\pi$
$710$	0	0
$711$	−2.74781e27	−2.21935
$712$	0	0
$713$	−9.90521e26	−0.778956
$714$	0	0
$715$	2.45570e27	1.88048
$716$	0	0
$717$	2.46222e27	1.83610
$718$	0	0
$719$	−1.97885e27	−1.43710	−0.718552	−	0.695474i	$-0.755195\pi$
$719$	−1.97885e27	−1.43710	−0.718552	+	0.695474i	$0.755195\pi$
$720$	0	0
$721$	2.64095e26	0.186799
$722$	0	0
$723$	−1.47120e27	−1.01358
$724$	0	0
$725$	2.28622e26	0.153429
$726$	0	0
$727$	1.00954e27	0.660006	0.330003	−	0.943980i	$-0.392950\pi$
$727$	1.00954e27	0.660006	0.330003	+	0.943980i	$0.392950\pi$
$728$	0	0
$729$	−2.55972e27	−1.63035
$730$	0	0
$731$	−4.53406e26	−0.281366
$732$	0	0
$733$	2.87911e26	0.174089	0.0870443	−	0.996204i	$-0.472258\pi$
$733$	2.87911e26	0.174089	0.0870443	+	0.996204i	$0.472258\pi$
$734$	0	0
$735$	−2.57546e27	−1.51749
$736$	0	0
$737$	6.42383e26	0.368853
$738$	0	0
$739$	−4.82106e26	−0.269786	−0.134893	−	0.990860i	$-0.543069\pi$
$739$	−4.82106e26	−0.269786	−0.134893	+	0.990860i	$0.543069\pi$
$740$	0	0
$741$	−1.32174e27	−0.720894
$742$	0	0
$743$	1.28340e27	0.682289	0.341144	−	0.940011i	$-0.389185\pi$
$743$	1.28340e27	0.682289	0.341144	+	0.940011i	$0.389185\pi$
$744$	0	0
$745$	−2.12572e27	−1.10160
$746$	0	0
$747$	−1.04301e27	−0.526917
$748$	0	0
$749$	−5.74082e26	−0.282746
$750$	0	0
$751$	−9.50810e26	−0.456577	−0.228289	−	0.973593i	$-0.573313\pi$
$751$	−9.50810e26	−0.456577	−0.228289	+	0.973593i	$0.573313\pi$
$752$	0	0
$753$	2.00914e27	0.940714
$754$	0	0
$755$	−2.07534e27	−0.947533
$756$	0	0
$757$	−1.27663e27	−0.568402	−0.284201	−	0.958765i	$-0.591728\pi$
$757$	−1.27663e27	−0.568402	−0.284201	+	0.958765i	$0.591728\pi$
$758$	0	0
$759$	−1.08620e27	−0.471643
$760$	0	0
$761$	3.70731e27	1.57002	0.785009	−	0.619485i	$-0.212658\pi$
$761$	3.70731e27	1.57002	0.785009	+	0.619485i	$0.212658\pi$
$762$	0	0
$763$	2.41323e27	0.996816
$764$	0	0
$765$	1.68183e27	0.677638
$766$	0	0
$767$	8.31357e27	3.26762
$768$	0	0
$769$	3.72681e27	1.42902	0.714508	−	0.699627i	$-0.246650\pi$
$769$	3.72681e27	1.42902	0.714508	+	0.699627i	$0.246650\pi$
$770$	0	0
$771$	5.06228e27	1.89378
$772$	0	0
$773$	−2.80552e27	−1.02402	−0.512011	−	0.858979i	$-0.671099\pi$
$773$	−2.80552e27	−1.02402	−0.512011	+	0.858979i	$0.671099\pi$
$774$	0	0
$775$	−8.32787e27	−2.96598
$776$	0	0
$777$	−7.40930e25	−0.0257500
$778$	0	0
$779$	3.93770e26	0.133548
$780$	0	0
$781$	−1.73108e27	−0.572970
$782$	0	0
$783$	1.97699e26	0.0638657
$784$	0	0
$785$	−1.93354e26	−0.0609665
$786$	0	0
$787$	−2.59134e27	−0.797562	−0.398781	−	0.917046i	$-0.630567\pi$
$787$	−2.59134e27	−0.797562	−0.398781	+	0.917046i	$0.630567\pi$
$788$	0	0
$789$	7.02763e26	0.211143
$790$	0	0
$791$	−2.41474e27	−0.708261
$792$	0	0
$793$	−6.58564e27	−1.88583
$794$	0	0
$795$	8.62378e27	2.41107
$796$	0	0
$797$	−8.71868e26	−0.238011	−0.119005	−	0.992894i	$-0.537971\pi$
$797$	−8.71868e26	−0.238011	−0.119005	+	0.992894i	$0.537971\pi$
$798$	0	0
$799$	9.35021e26	0.249245
$800$	0	0
$801$	4.52707e27	1.17844
$802$	0	0
$803$	3.35207e25	0.00852149
$804$	0	0
$805$	2.51507e27	0.624438
$806$	0	0
$807$	5.80108e27	1.40673
$808$	0	0
$809$	4.27931e26	0.101359	0.0506796	−	0.998715i	$-0.483861\pi$
$809$	4.27931e26	0.101359	0.0506796	+	0.998715i	$0.483861\pi$
$810$	0	0
$811$	−3.28332e27	−0.759652	−0.379826	−	0.925058i	$-0.624016\pi$
$811$	−3.28332e27	−0.759652	−0.379826	+	0.925058i	$0.624016\pi$
$812$	0	0
$813$	−6.64912e27	−1.50281
$814$	0	0
$815$	−3.72323e27	−0.822096
$816$	0	0
$817$	−1.19635e27	−0.258076
$818$	0	0
$819$	−9.49537e27	−2.00131
$820$	0	0
$821$	−2.64077e27	−0.543840	−0.271920	−	0.962320i	$-0.587659\pi$
$821$	−2.64077e27	−0.543840	−0.271920	+	0.962320i	$0.587659\pi$
$822$	0	0
$823$	6.55913e27	1.31992	0.659960	−	0.751301i	$-0.270573\pi$
$823$	6.55913e27	1.31992	0.659960	+	0.751301i	$0.270573\pi$
$824$	0	0
$825$	−9.13230e27	−1.79584
$826$	0	0
$827$	−4.39186e27	−0.844008	−0.422004	−	0.906594i	$-0.638673\pi$
$827$	−4.39186e27	−0.844008	−0.422004	+	0.906594i	$0.638673\pi$
$828$	0	0
$829$	1.85945e27	0.349234	0.174617	−	0.984636i	$-0.444131\pi$
$829$	1.85945e27	0.349234	0.174617	+	0.984636i	$0.444131\pi$
$830$	0	0
$831$	1.68582e28	3.09459
$832$	0	0
$833$	−7.76088e26	−0.139246
$834$	0	0
$835$	−9.77463e27	−1.71427
$836$	0	0
$837$	−7.20146e27	−1.23461
$838$	0	0
$839$	−6.62387e27	−1.11013	−0.555064	−	0.831808i	$-0.687306\pi$
$839$	−6.62387e27	−1.11013	−0.555064	+	0.831808i	$0.687306\pi$
$840$	0	0
$841$	−6.06843e27	−0.994293
$842$	0	0
$843$	−1.28764e27	−0.206268
$844$	0	0
$845$	3.08714e28	4.83524
$846$	0	0
$847$	−3.03459e27	−0.464738
$848$	0	0
$849$	1.76744e28	2.64681
$850$	0	0
$851$	−8.76675e25	−0.0128383
$852$	0	0
$853$	8.30838e27	1.18987	0.594937	−	0.803773i	$-0.297177\pi$
$853$	8.30838e27	1.18987	0.594937	+	0.803773i	$0.297177\pi$
$854$	0	0
$855$	4.43762e27	0.621545
$856$	0	0
$857$	−9.02489e27	−1.23630	−0.618151	−	0.786060i	$-0.712118\pi$
$857$	−9.02489e27	−1.23630	−0.618151	+	0.786060i	$0.712118\pi$
$858$	0	0
$859$	−1.64411e27	−0.220290	−0.110145	−	0.993916i	$-0.535132\pi$
$859$	−1.64411e27	−0.220290	−0.110145	+	0.993916i	$0.535132\pi$
$860$	0	0
$861$	4.67612e27	0.612852
$862$	0	0
$863$	−2.01704e27	−0.258590	−0.129295	−	0.991606i	$-0.541271\pi$
$863$	−2.01704e27	−0.258590	−0.129295	+	0.991606i	$0.541271\pi$
$864$	0	0
$865$	−8.62568e27	−1.08178
$866$	0	0
$867$	−1.21298e28	−1.48824
$868$	0	0
$869$	6.70995e27	0.805437
$870$	0	0
$871$	1.09833e28	1.28991
$872$	0	0
$873$	1.27813e28	1.46872
$874$	0	0
$875$	1.07341e28	1.20695
$876$	0	0
$877$	−1.05981e28	−1.16609	−0.583043	−	0.812441i	$-0.698138\pi$
$877$	−1.05981e28	−1.16609	−0.583043	+	0.812441i	$0.698138\pi$
$878$	0	0
$879$	−1.46727e28	−1.57985
$880$	0	0
$881$	1.17187e28	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$881$	1.17187e28	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$882$	0	0
$883$	1.23917e28	1.27792	0.638962	−	0.769239i	$-0.279364\pi$
$883$	1.23917e28	1.27792	0.638962	+	0.769239i	$0.279364\pi$
$884$	0	0
$885$	−4.61392e28	−4.65703
$886$	0	0
$887$	−8.48429e27	−0.838187	−0.419093	−	0.907943i	$-0.637652\pi$
$887$	−8.48429e27	−0.838187	−0.419093	+	0.907943i	$0.637652\pi$
$888$	0	0
$889$	−2.84069e27	−0.274699
$890$	0	0
$891$	1.09374e27	0.103532
$892$	0	0
$893$	2.46712e27	0.228613
$894$	0	0
$895$	2.40538e28	2.18205
$896$	0	0
$897$	−1.85716e28	−1.64938
$898$	0	0
$899$	−1.26875e27	−0.110322
$900$	0	0
$901$	2.59869e27	0.221242
$902$	0	0
$903$	−1.42069e28	−1.18431
$904$	0	0
$905$	1.31647e28	1.07461
$906$	0	0
$907$	−1.89858e28	−1.51761	−0.758806	−	0.651317i	$-0.774217\pi$
$907$	−1.89858e28	−1.51761	−0.758806	+	0.651317i	$0.774217\pi$
$908$	0	0
$909$	1.29058e28	1.01025
$910$	0	0
$911$	−6.32123e27	−0.484593	−0.242296	−	0.970202i	$-0.577901\pi$
$911$	−6.32123e27	−0.484593	−0.242296	+	0.970202i	$0.577901\pi$
$912$	0	0
$913$	2.54695e27	0.191226
$914$	0	0
$915$	3.65494e28	2.68770
$916$	0	0
$917$	−1.63649e28	−1.17870
$918$	0	0
$919$	3.89136e27	0.274539	0.137269	−	0.990534i	$-0.456167\pi$
$919$	3.89136e27	0.274539	0.137269	+	0.990534i	$0.456167\pi$
$920$	0	0
$921$	8.32253e27	0.575161
$922$	0	0
$923$	−2.95976e28	−2.00373
$924$	0	0
$925$	−7.37070e26	−0.0488835
$926$	0	0
$927$	6.54735e27	0.425411
$928$	0	0
$929$	−2.13124e28	−1.35670	−0.678349	−	0.734740i	$-0.737304\pi$
$929$	−2.13124e28	−1.35670	−0.678349	+	0.734740i	$0.737304\pi$
$930$	0	0
$931$	−2.04777e27	−0.127720
$932$	0	0
$933$	3.07247e27	0.187764
$934$	0	0
$935$	−4.10689e27	−0.245925
$936$	0	0
$937$	1.27423e28	0.747692	0.373846	−	0.927491i	$-0.378039\pi$
$937$	1.27423e28	0.747692	0.373846	+	0.927491i	$0.378039\pi$
$938$	0	0
$939$	3.90231e27	0.224388
$940$	0	0
$941$	−4.44618e27	−0.250545	−0.125272	−	0.992122i	$-0.539981\pi$
$941$	−4.44618e27	−0.250545	−0.125272	+	0.992122i	$0.539981\pi$
$942$	0	0
$943$	5.53283e27	0.305553
$944$	0	0
$945$	1.82855e28	0.989703
$946$	0	0
$947$	1.55107e28	0.822825	0.411412	−	0.911449i	$-0.365036\pi$
$947$	1.55107e28	0.822825	0.411412	+	0.911449i	$0.365036\pi$
$948$	0	0
$949$	5.73130e26	0.0298005
$950$	0	0
$951$	4.62499e28	2.35720
$952$	0	0
$953$	−2.06322e28	−1.03077	−0.515386	−	0.856958i	$-0.672352\pi$
$953$	−2.06322e28	−1.03077	−0.515386	+	0.856958i	$0.672352\pi$
$954$	0	0
$955$	3.73886e28	1.83108
$956$	0	0
$957$	−1.39131e27	−0.0667975
$958$	0	0
$959$	5.78974e27	0.272510
$960$	0	0
$961$	2.45454e28	1.13266
$962$	0	0
$963$	−1.42325e28	−0.643917
$964$	0	0
$965$	1.53083e28	0.679075
$966$	0	0
$967$	−1.23039e28	−0.535168	−0.267584	−	0.963535i	$-0.586225\pi$
$967$	−1.23039e28	−0.535168	−0.267584	+	0.963535i	$0.586225\pi$
$968$	0	0
$969$	2.21046e27	0.0942773
$970$	0	0
$971$	−1.81059e28	−0.757247	−0.378623	−	0.925551i	$-0.623602\pi$
$971$	−1.81059e28	−0.757247	−0.378623	+	0.925551i	$0.623602\pi$
$972$	0	0
$973$	8.66681e27	0.355457
$974$	0	0
$975$	−1.56142e29	−6.28024
$976$	0	0
$977$	−3.59860e28	−1.41950	−0.709750	−	0.704454i	$-0.751192\pi$
$977$	−3.59860e28	−1.41950	−0.709750	+	0.704454i	$0.751192\pi$
$978$	0	0
$979$	−1.10548e28	−0.427675
$980$	0	0
$981$	5.98279e28	2.27012
$982$	0	0
$983$	−1.14994e27	−0.0427972	−0.0213986	−	0.999771i	$-0.506812\pi$
$983$	−1.14994e27	−0.0427972	−0.0213986	+	0.999771i	$0.506812\pi$
$984$	0	0
$985$	−3.84526e28	−1.40372
$986$	0	0
$987$	2.92978e28	1.04911
$988$	0	0
$989$	−1.68098e28	−0.590469
$990$	0	0
$991$	−7.71249e27	−0.265763	−0.132882	−	0.991132i	$-0.542423\pi$
$991$	−7.71249e27	−0.265763	−0.132882	+	0.991132i	$0.542423\pi$
$992$	0	0
$993$	6.04963e27	0.204508
$994$	0	0
$995$	5.83266e28	1.93441
$996$	0	0
$997$	1.79107e28	0.582784	0.291392	−	0.956604i	$-0.405882\pi$
$997$	1.79107e28	0.582784	0.291392	+	0.956604i	$0.405882\pi$
$998$	0	0
$999$	−6.37376e26	−0.0203481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.20.a.b.1.3	✓	3
3.2	odd	2		72.20.a.f.1.1		3
4.3	odd	2		16.20.a.f.1.1		3
8.3	odd	2		64.20.a.m.1.3		3
8.5	even	2		64.20.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.a.b.1.3	✓	3	1.1	even	1	trivial
16.20.a.f.1.1		3	4.3	odd	2
64.20.a.l.1.1		3	8.5	even	2
64.20.a.m.1.3		3	8.3	odd	2
72.20.a.f.1.1		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q+54241.2 q^{3} +7.60339e6 q^{5} +7.17920e7 q^{7} +1.77984e9 q^{9} +O(q^{10})\)	\(q+54241.2 q^{3} +7.60339e6 q^{5} +7.17920e7 q^{7} +1.77984e9 q^{9} -4.34624e9 q^{11} -7.43111e10 q^{13} +4.12417e11 q^{15} +1.24277e11 q^{17} +3.27915e11 q^{19} +3.89408e12 q^{21} +4.60752e12 q^{23} +3.87380e13 q^{25} +3.34984e13 q^{27} +5.90175e12 q^{29} -2.14979e14 q^{31} -2.35745e14 q^{33} +5.45862e14 q^{35} -1.90271e13 q^{37} -4.03072e15 q^{39} +1.20083e15 q^{41} -3.64834e15 q^{43} +1.35328e16 q^{45} +7.52366e15 q^{47} -6.24480e15 q^{49} +6.74095e15 q^{51} +2.09104e16 q^{53} -3.30462e16 q^{55} +1.77865e16 q^{57} -1.11875e17 q^{59} +8.86226e16 q^{61} +1.27779e17 q^{63} -5.65016e17 q^{65} -1.47802e17 q^{67} +2.49917e17 q^{69} +3.98293e17 q^{71} -7.71257e15 q^{73} +2.10119e18 q^{75} -3.12026e17 q^{77} -1.54385e18 q^{79} -2.51651e17 q^{81} -5.86011e17 q^{83} +9.44929e17 q^{85} +3.20118e17 q^{87} +2.54352e18 q^{89} -5.33495e18 q^{91} -1.16607e19 q^{93} +2.49327e18 q^{95} +7.18112e18 q^{97} -7.73564e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9}+O(q^{10}) \) 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9	\( 3 q + 23732 q^{3} + 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} - 297392964 q^{11} - 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} + 3212269666884 q^{19} + 11192319829728 q^{21} + 24948509305560 q^{23} + 72340360289109 q^{25} + 64092343553864 q^{27} + 77667139511058 q^{29} - 248431735193568 q^{31} - 252071696774128 q^{33} - 13\!\cdots\!08 q^{35}+ \cdots - 11\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 23732 * q^3 + 2140218 * q^5 + 55851720 * q^7 + 646753951 * q^9 - 297392964 * q^11 - 14862401022 * q^13 + 292635653528 * q^15 + 803332464534 * q^17 + 3212269666884 * q^19 + 11192319829728 * q^21 + 24948509305560 * q^23 + 72340360289109 * q^25 + 64092343553864 * q^27 + 77667139511058 * q^29 - 248431735193568 * q^31 - 252071696774128 * q^33 - 1384378114232208 * q^35 - 414866302559142 * q^37 - 5522045976027016 * q^39 + 2818737880869678 * q^41 - 6663230715469860 * q^43 + 22824972038174722 * q^45 + 1500497644728624 * q^47 + 39641123057321787 * q^49 - 1085042177122520 * q^51 + 56067344774978154 * q^53 - 56817855560205432 * q^55 - 6408983732955152 * q^57 - 154317270851496852 * q^59 - 134994376571654862 * q^61 - 78435010097874072 * q^63 - 563892790723886724 * q^65 + 151032181904450292 * q^67 - 153590366326625632 * q^69 + 1210541848845584136 * q^71 - 81876123599662770 * q^73 + 2698278154129173484 * q^75 + 97235023193490336 * q^77 + 1439028483035907408 * q^79 - 318667537468381733 * q^81 - 983438102798849916 * q^83 - 1618537843962618540 * q^85 - 4280411824494387144 * q^87 - 1312857528832070946 * q^89 - 11488143382330124496 * q^91 - 4308257105281185920 * q^93 - 10926153348157720968 * q^95 + 14033245412567998566 * q^97 - 11415749644087184660 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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