LMFDB - Embedded newform 315.10.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("315.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.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:	$162.236288392$
Analytic rank:	$0$
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} - 2x^{4} - 1694x^{3} + 7420x^{2} + 583584x - 2721600 $ x^5 - 2x^4 - 1694x^3 + 7420x^2 + 583584x - 2721600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-36.4619$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+36.4619 q^{2} +817.473 q^{4} +625.000 q^{5} +2401.00 q^{7} +11138.1 q^{8} +22788.7 q^{10} +75602.9 q^{11} -49331.2 q^{13} +87545.1 q^{14} -12428.3 q^{16} +522841. q^{17} +442202. q^{19} +510921. q^{20} +2.75663e6 q^{22} -2.11792e6 q^{23} +390625. q^{25} -1.79871e6 q^{26} +1.96275e6 q^{28} +5.29101e6 q^{29} -5.41211e6 q^{31} -6.15588e6 q^{32} +1.90638e7 q^{34} +1.50062e6 q^{35} +1.68073e7 q^{37} +1.61236e7 q^{38} +6.96133e6 q^{40} -2.98305e6 q^{41} -1.11225e7 q^{43} +6.18033e7 q^{44} -7.72234e7 q^{46} -1.58154e7 q^{47} +5.76480e6 q^{49} +1.42429e7 q^{50} -4.03269e7 q^{52} +2.78375e7 q^{53} +4.72518e7 q^{55} +2.67427e7 q^{56} +1.92920e8 q^{58} -1.71244e7 q^{59} +1.93532e8 q^{61} -1.97336e8 q^{62} -2.18092e8 q^{64} -3.08320e7 q^{65} +1.93697e8 q^{67} +4.27409e8 q^{68} +5.47157e7 q^{70} -1.41665e8 q^{71} -2.36977e7 q^{73} +6.12828e8 q^{74} +3.61488e8 q^{76} +1.81522e8 q^{77} +2.51611e8 q^{79} -7.76766e6 q^{80} -1.08768e8 q^{82} -2.06819e8 q^{83} +3.26776e8 q^{85} -4.05548e8 q^{86} +8.42074e8 q^{88} +6.94584e8 q^{89} -1.18444e8 q^{91} -1.73134e9 q^{92} -5.76662e8 q^{94} +2.76376e8 q^{95} +1.39893e9 q^{97} +2.10196e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 832 * q^4 + 3125 * q^5 + 12005 * q^7 + 14136 * q^8 - 1250 * q^10 - 10312 * q^11 + 158638 * q^13 - 4802 * q^14 - 526696 * q^16 - 31614 * q^17 + 1655376 * q^19 + 520000 * q^20 + 3659464 * q^22 - 796104 * q^23 + 1953125 * q^25 - 10219060 * q^26 + 1997632 * q^28 + 3353726 * q^29 + 2678120 * q^31 + 130400 * q^32 + 21548612 * q^34 + 7503125 * q^35 + 50994846 * q^37 + 34907600 * q^38 + 8835000 * q^40 - 6330194 * q^41 + 6149468 * q^43 + 71225892 * q^44 - 142845472 * q^46 + 6897780 * q^47 + 28824005 * q^49 - 781250 * q^50 - 22199468 * q^52 + 70886738 * q^53 - 6445000 * q^55 + 33940536 * q^56 - 31955924 * q^58 + 152335168 * q^59 + 257015698 * q^61 + 215582768 * q^62 - 193988432 * q^64 + 99148750 * q^65 + 133467828 * q^67 + 268709380 * q^68 - 3001250 * q^70 - 522788960 * q^71 + 159370858 * q^73 + 244178868 * q^74 - 345056616 * q^76 - 24759112 * q^77 + 464174900 * q^79 - 329185000 * q^80 + 308947620 * q^82 - 2207636832 * q^83 - 19758750 * q^85 - 694560616 * q^86 + 930528736 * q^88 + 1106708326 * q^89 + 380889838 * q^91 - 205160216 * q^92 - 3602501744 * q^94 + 1034610000 * q^95 + 1956142254 * q^97 - 11529602 * 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 $

$2$	36.4619	1.61141	0.805703	−	0.592320i	$-0.201788\pi$
$2$	36.4619	1.61141	0.805703	+	0.592320i	$0.201788\pi$
$3$	0	0
$3$	0	0
$4$	817.473	1.59663
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	11138.1	0.961407
$9$	0	0
$10$	22788.7	0.720642
$11$	75602.9	1.55694	0.778469	−	0.627684i	$-0.215997\pi$
$11$	75602.9	1.55694	0.778469	+	0.627684i	$0.215997\pi$
$12$	0	0
$13$	−49331.2	−0.479045	−0.239523	−	0.970891i	$-0.576991\pi$
$13$	−49331.2	−0.479045	−0.239523	+	0.970891i	$0.576991\pi$
$14$	87545.1	0.609054
$15$	0	0
$16$	−12428.3	−0.0474100
$17$	522841.	1.51827	0.759137	−	0.650931i	$-0.225621\pi$
$17$	522841.	1.51827	0.759137	+	0.650931i	$0.225621\pi$
$18$	0	0
$19$	442202.	0.778448	0.389224	−	0.921143i	$-0.372743\pi$
$19$	442202.	0.778448	0.389224	+	0.921143i	$0.372743\pi$
$20$	510921.	0.714033
$21$	0	0
$22$	2.75663e6	2.50886
$23$	−2.11792e6	−1.57810	−0.789049	−	0.614330i	$-0.789426\pi$
$23$	−2.11792e6	−1.57810	−0.789049	+	0.614330i	$0.789426\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−1.79871e6	−0.771936
$27$	0	0
$28$	1.96275e6	0.603468
$29$	5.29101e6	1.38914	0.694572	−	0.719423i	$-0.255594\pi$
$29$	5.29101e6	1.38914	0.694572	+	0.719423i	$0.255594\pi$
$30$	0	0
$31$	−5.41211e6	−1.05254	−0.526270	−	0.850317i	$-0.676410\pi$
$31$	−5.41211e6	−1.05254	−0.526270	+	0.850317i	$0.676410\pi$
$32$	−6.15588e6	−1.03780
$33$	0	0
$34$	1.90638e7	2.44655
$35$	1.50062e6	0.169031
$36$	0	0
$37$	1.68073e7	1.47432	0.737159	−	0.675719i	$-0.236167\pi$
$37$	1.68073e7	1.47432	0.737159	+	0.675719i	$0.236167\pi$
$38$	1.61236e7	1.25440
$39$	0	0
$40$	6.96133e6	0.429954
$41$	−2.98305e6	−0.164867	−0.0824333	−	0.996597i	$-0.526269\pi$
$41$	−2.98305e6	−0.164867	−0.0824333	+	0.996597i	$0.526269\pi$
$42$	0	0
$43$	−1.11225e7	−0.496129	−0.248065	−	0.968744i	$-0.579794\pi$
$43$	−1.11225e7	−0.496129	−0.248065	+	0.968744i	$0.579794\pi$
$44$	6.18033e7	2.48585
$45$	0	0
$46$	−7.72234e7	−2.54296
$47$	−1.58154e7	−0.472760	−0.236380	−	0.971661i	$-0.575961\pi$
$47$	−1.58154e7	−0.472760	−0.236380	+	0.971661i	$0.575961\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	1.42429e7	0.322281
$51$	0	0
$52$	−4.03269e7	−0.764856
$53$	2.78375e7	0.484605	0.242303	−	0.970201i	$-0.422097\pi$
$53$	2.78375e7	0.484605	0.242303	+	0.970201i	$0.422097\pi$
$54$	0	0
$55$	4.72518e7	0.696283
$56$	2.67427e7	0.363378
$57$	0	0
$58$	1.92920e8	2.23847
$59$	−1.71244e7	−0.183985	−0.0919925	−	0.995760i	$-0.529324\pi$
$59$	−1.71244e7	−0.183985	−0.0919925	+	0.995760i	$0.529324\pi$
$60$	0	0
$61$	1.93532e8	1.78965	0.894826	−	0.446416i	$-0.147300\pi$
$61$	1.93532e8	1.78965	0.894826	+	0.446416i	$0.147300\pi$
$62$	−1.97336e8	−1.69607
$63$	0	0
$64$	−2.18092e8	−1.62491
$65$	−3.08320e7	−0.214236
$66$	0	0
$67$	1.93697e8	1.17432	0.587160	−	0.809471i	$-0.300246\pi$
$67$	1.93697e8	1.17432	0.587160	+	0.809471i	$0.300246\pi$
$68$	4.27409e8	2.42412
$69$	0	0
$70$	5.47157e7	0.272377
$71$	−1.41665e8	−0.661605	−0.330802	−	0.943700i	$-0.607319\pi$
$71$	−1.41665e8	−0.661605	−0.330802	+	0.943700i	$0.607319\pi$
$72$	0	0
$73$	−2.36977e7	−0.0976683	−0.0488342	−	0.998807i	$-0.515551\pi$
$73$	−2.36977e7	−0.0976683	−0.0488342	+	0.998807i	$0.515551\pi$
$74$	6.12828e8	2.37572
$75$	0	0
$76$	3.61488e8	1.24289
$77$	1.81522e8	0.588467
$78$	0	0
$79$	2.51611e8	0.726788	0.363394	−	0.931636i	$-0.381618\pi$
$79$	2.51611e8	0.726788	0.363394	+	0.931636i	$0.381618\pi$
$80$	−7.76766e6	−0.0212024
$81$	0	0
$82$	−1.08768e8	−0.265667
$83$	−2.06819e8	−0.478343	−0.239171	−	0.970977i	$-0.576876\pi$
$83$	−2.06819e8	−0.478343	−0.239171	+	0.970977i	$0.576876\pi$
$84$	0	0
$85$	3.26776e8	0.678992
$86$	−4.05548e8	−0.799465
$87$	0	0
$88$	8.42074e8	1.49685
$89$	6.94584e8	1.17346	0.586732	−	0.809781i	$-0.300414\pi$
$89$	6.94584e8	1.17346	0.586732	+	0.809781i	$0.300414\pi$
$90$	0	0
$91$	−1.18444e8	−0.181062
$92$	−1.73134e9	−2.51963
$93$	0	0
$94$	−5.76662e8	−0.761808
$95$	2.76376e8	0.348133
$96$	0	0
$97$	1.39893e9	1.60443	0.802217	−	0.597033i	$-0.203654\pi$
$97$	1.39893e9	1.60443	0.802217	+	0.597033i	$0.203654\pi$
$98$	2.10196e8	0.230201
$99$	0	0
$100$	3.19325e8	0.319325
$101$	9.82882e8	0.939843	0.469922	−	0.882708i	$-0.344282\pi$
$101$	9.82882e8	0.939843	0.469922	+	0.882708i	$0.344282\pi$
$102$	0	0
$103$	−1.53146e9	−1.34072	−0.670359	−	0.742037i	$-0.733860\pi$
$103$	−1.53146e9	−1.34072	−0.670359	+	0.742037i	$0.733860\pi$
$104$	−5.49457e8	−0.460558
$105$	0	0
$106$	1.01501e9	0.780895
$107$	5.19394e8	0.383063	0.191531	−	0.981487i	$-0.438655\pi$
$107$	5.19394e8	0.383063	0.191531	+	0.981487i	$0.438655\pi$
$108$	0	0
$109$	−1.24526e9	−0.844967	−0.422483	−	0.906371i	$-0.638842\pi$
$109$	−1.24526e9	−0.844967	−0.422483	+	0.906371i	$0.638842\pi$
$110$	1.72289e9	1.12199
$111$	0	0
$112$	−2.98402e7	−0.0179193
$113$	−1.12841e9	−0.651048	−0.325524	−	0.945534i	$-0.605541\pi$
$113$	−1.12841e9	−0.651048	−0.325524	+	0.945534i	$0.605541\pi$
$114$	0	0
$115$	−1.32370e9	−0.705747
$116$	4.32526e9	2.21795
$117$	0	0
$118$	−6.24391e8	−0.296474
$119$	1.25534e9	0.573853
$120$	0	0
$121$	3.35784e9	1.42405
$122$	7.05655e9	2.88385
$123$	0	0
$124$	−4.42425e9	−1.68051
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−3.31282e9	−1.13001	−0.565004	−	0.825088i	$-0.691125\pi$
$127$	−3.31282e9	−1.13001	−0.565004	+	0.825088i	$0.691125\pi$
$128$	−4.80025e9	−1.58059
$129$	0	0
$130$	−1.12419e9	−0.345220
$131$	−4.83380e9	−1.43406	−0.717031	−	0.697042i	$-0.754499\pi$
$131$	−4.83380e9	−1.43406	−0.717031	+	0.697042i	$0.754499\pi$
$132$	0	0
$133$	1.06173e9	0.294226
$134$	7.06257e9	1.89231
$135$	0	0
$136$	5.82348e9	1.45968
$137$	3.43281e9	0.832543	0.416272	−	0.909240i	$-0.363337\pi$
$137$	3.43281e9	0.832543	0.416272	+	0.909240i	$0.363337\pi$
$138$	0	0
$139$	3.96923e9	0.901860	0.450930	−	0.892559i	$-0.351092\pi$
$139$	3.96923e9	0.901860	0.450930	+	0.892559i	$0.351092\pi$
$140$	1.22672e9	0.269879
$141$	0	0
$142$	−5.16537e9	−1.06611
$143$	−3.72958e9	−0.745843
$144$	0	0
$145$	3.30688e9	0.621244
$146$	−8.64065e8	−0.157383
$147$	0	0
$148$	1.37395e10	2.35394
$149$	−2.72837e8	−0.0453487	−0.0226744	−	0.999743i	$-0.507218\pi$
$149$	−2.72837e8	−0.0453487	−0.0226744	+	0.999743i	$0.507218\pi$
$150$	0	0
$151$	3.77089e9	0.590265	0.295133	−	0.955456i	$-0.404636\pi$
$151$	3.77089e9	0.590265	0.295133	+	0.955456i	$0.404636\pi$
$152$	4.92531e9	0.748406
$153$	0	0
$154$	6.61866e9	0.948259
$155$	−3.38257e9	−0.470710
$156$	0	0
$157$	9.54797e9	1.25419	0.627094	−	0.778944i	$-0.284244\pi$
$157$	9.54797e9	1.25419	0.627094	+	0.778944i	$0.284244\pi$
$158$	9.17422e9	1.17115
$159$	0	0
$160$	−3.84743e9	−0.464120
$161$	−5.08512e9	−0.596465
$162$	0	0
$163$	−6.35428e9	−0.705054	−0.352527	−	0.935802i	$-0.614678\pi$
$163$	−6.35428e9	−0.705054	−0.352527	+	0.935802i	$0.614678\pi$
$164$	−2.43856e9	−0.263230
$165$	0	0
$166$	−7.54102e9	−0.770804
$167$	2.04016e9	0.202974	0.101487	−	0.994837i	$-0.467640\pi$
$167$	2.04016e9	0.202974	0.101487	+	0.994837i	$0.467640\pi$
$168$	0	0
$169$	−8.17093e9	−0.770516
$170$	1.19149e10	1.09413
$171$	0	0
$172$	−9.09235e9	−0.792133
$173$	1.72853e10	1.46713	0.733564	−	0.679620i	$-0.237855\pi$
$173$	1.72853e10	1.46713	0.733564	+	0.679620i	$0.237855\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−9.39611e8	−0.0738144
$177$	0	0
$178$	2.53259e10	1.89092
$179$	−6.95395e8	−0.0506282	−0.0253141	−	0.999680i	$-0.508059\pi$
$179$	−6.95395e8	−0.0506282	−0.0253141	+	0.999680i	$0.508059\pi$
$180$	0	0
$181$	1.80182e10	1.24784	0.623919	−	0.781489i	$-0.285539\pi$
$181$	1.80182e10	1.24784	0.623919	+	0.781489i	$0.285539\pi$
$182$	−4.31870e9	−0.291764
$183$	0	0
$184$	−2.35897e10	−1.51720
$185$	1.05046e10	0.659335
$186$	0	0
$187$	3.95283e10	2.36386
$188$	−1.29287e10	−0.754822
$189$	0	0
$190$	1.00772e10	0.560983
$191$	2.60774e10	1.41780	0.708899	−	0.705310i	$-0.249192\pi$
$191$	2.60774e10	1.41780	0.708899	+	0.705310i	$0.249192\pi$
$192$	0	0
$193$	9.94176e9	0.515769	0.257884	−	0.966176i	$-0.416975\pi$
$193$	9.94176e9	0.515769	0.257884	+	0.966176i	$0.416975\pi$
$194$	5.10075e10	2.58539
$195$	0	0
$196$	4.71257e9	0.228090
$197$	−3.35442e10	−1.58679	−0.793395	−	0.608707i	$-0.791688\pi$
$197$	−3.35442e10	−1.58679	−0.793395	+	0.608707i	$0.791688\pi$
$198$	0	0
$199$	1.19117e10	0.538435	0.269218	−	0.963079i	$-0.413235\pi$
$199$	1.19117e10	0.538435	0.269218	+	0.963079i	$0.413235\pi$
$200$	4.35083e9	0.192281
$201$	0	0
$202$	3.58378e10	1.51447
$203$	1.27037e10	0.525047
$204$	0	0
$205$	−1.86440e9	−0.0737306
$206$	−5.58399e10	−2.16044
$207$	0	0
$208$	6.13101e8	0.0227115
$209$	3.34318e10	1.21200
$210$	0	0
$211$	−4.45526e10	−1.54740	−0.773699	−	0.633554i	$-0.781596\pi$
$211$	−4.45526e10	−1.54740	−0.773699	+	0.633554i	$0.781596\pi$
$212$	2.27564e10	0.773734
$213$	0	0
$214$	1.89381e10	0.617269
$215$	−6.95157e9	−0.221876
$216$	0	0
$217$	−1.29945e10	−0.397823
$218$	−4.54045e10	−1.36158
$219$	0	0
$220$	3.86270e10	1.11170
$221$	−2.57924e10	−0.727322
$222$	0	0
$223$	−7.31712e10	−1.98138	−0.990691	−	0.136129i	$-0.956534\pi$
$223$	−7.31712e10	−1.98138	−0.990691	+	0.136129i	$0.956534\pi$
$224$	−1.47803e10	−0.392253
$225$	0	0
$226$	−4.11440e10	−1.04910
$227$	−2.44357e10	−0.610813	−0.305406	−	0.952222i	$-0.598792\pi$
$227$	−2.44357e10	−0.610813	−0.305406	+	0.952222i	$0.598792\pi$
$228$	0	0
$229$	−4.16163e10	−1.00001	−0.500004	−	0.866023i	$-0.666668\pi$
$229$	−4.16163e10	−1.00001	−0.500004	+	0.866023i	$0.666668\pi$
$230$	−4.82646e10	−1.13724
$231$	0	0
$232$	5.89319e10	1.33553
$233$	−3.77622e10	−0.839375	−0.419687	−	0.907669i	$-0.637860\pi$
$233$	−3.77622e10	−0.839375	−0.419687	+	0.907669i	$0.637860\pi$
$234$	0	0
$235$	−9.88465e9	−0.211425
$236$	−1.39988e10	−0.293755
$237$	0	0
$238$	4.57722e10	0.924710
$239$	3.14171e10	0.622839	0.311420	−	0.950273i	$-0.399196\pi$
$239$	3.14171e10	0.622839	0.311420	+	0.950273i	$0.399196\pi$
$240$	0	0
$241$	3.12996e10	0.597671	0.298835	−	0.954305i	$-0.403402\pi$
$241$	3.12996e10	0.597671	0.298835	+	0.954305i	$0.403402\pi$
$242$	1.22433e11	2.29473
$243$	0	0
$244$	1.58207e11	2.85740
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−2.18144e10	−0.372912
$248$	−6.02808e10	−1.01192
$249$	0	0
$250$	8.90184e9	0.144128
$251$	−7.69585e10	−1.22384	−0.611920	−	0.790919i	$-0.709603\pi$
$251$	−7.69585e10	−1.22384	−0.611920	+	0.790919i	$0.709603\pi$
$252$	0	0
$253$	−1.60121e11	−2.45700
$254$	−1.20792e11	−1.82090
$255$	0	0
$256$	−6.33632e10	−0.922056
$257$	−7.48449e10	−1.07020	−0.535098	−	0.844790i	$-0.679725\pi$
$257$	−7.48449e10	−1.07020	−0.535098	+	0.844790i	$0.679725\pi$
$258$	0	0
$259$	4.03544e10	0.557240
$260$	−2.52043e10	−0.342054
$261$	0	0
$262$	−1.76250e11	−2.31085
$263$	8.16203e9	0.105196	0.0525978	−	0.998616i	$-0.483250\pi$
$263$	8.16203e9	0.105196	0.0525978	+	0.998616i	$0.483250\pi$
$264$	0	0
$265$	1.73984e10	0.216722
$266$	3.87127e10	0.474117
$267$	0	0
$268$	1.58342e11	1.87495
$269$	6.23895e9	0.0726484	0.0363242	−	0.999340i	$-0.488435\pi$
$269$	6.23895e9	0.0726484	0.0363242	+	0.999340i	$0.488435\pi$
$270$	0	0
$271$	8.25889e10	0.930165	0.465083	−	0.885267i	$-0.346025\pi$
$271$	8.25889e10	0.930165	0.465083	+	0.885267i	$0.346025\pi$
$272$	−6.49801e9	−0.0719814
$273$	0	0
$274$	1.25167e11	1.34156
$275$	2.95324e10	0.311387
$276$	0	0
$277$	−8.98709e10	−0.917192	−0.458596	−	0.888645i	$-0.651647\pi$
$277$	−8.98709e10	−0.917192	−0.458596	+	0.888645i	$0.651647\pi$
$278$	1.44726e11	1.45326
$279$	0	0
$280$	1.67142e10	0.162507
$281$	1.64133e10	0.157042	0.0785211	−	0.996912i	$-0.474980\pi$
$281$	1.64133e10	0.157042	0.0785211	+	0.996912i	$0.474980\pi$
$282$	0	0
$283$	2.99673e10	0.277721	0.138860	−	0.990312i	$-0.455656\pi$
$283$	2.99673e10	0.277721	0.138860	+	0.990312i	$0.455656\pi$
$284$	−1.15807e11	−1.05634
$285$	0	0
$286$	−1.35988e11	−1.20186
$287$	−7.16230e9	−0.0623137
$288$	0	0
$289$	1.54775e11	1.30515
$290$	1.20575e11	1.00108
$291$	0	0
$292$	−1.93722e10	−0.155940
$293$	−1.61513e11	−1.28028	−0.640138	−	0.768260i	$-0.721123\pi$
$293$	−1.61513e11	−1.28028	−0.640138	+	0.768260i	$0.721123\pi$
$294$	0	0
$295$	−1.07028e10	−0.0822806
$296$	1.87202e11	1.41742
$297$	0	0
$298$	−9.94817e9	−0.0730752
$299$	1.04479e11	0.755980
$300$	0	0
$301$	−2.67051e10	−0.187519
$302$	1.37494e11	0.951157
$303$	0	0
$304$	−5.49580e9	−0.0369063
$305$	1.20957e11	0.800356
$306$	0	0
$307$	−1.65779e10	−0.106514	−0.0532570	−	0.998581i	$-0.516960\pi$
$307$	−1.65779e10	−0.106514	−0.0532570	+	0.998581i	$0.516960\pi$
$308$	1.48390e11	0.939562
$309$	0	0
$310$	−1.23335e11	−0.758505
$311$	−3.53749e10	−0.214424	−0.107212	−	0.994236i	$-0.534192\pi$
$311$	−3.53749e10	−0.214424	−0.107212	+	0.994236i	$0.534192\pi$
$312$	0	0
$313$	8.44655e9	0.0497428	0.0248714	−	0.999691i	$-0.492082\pi$
$313$	8.44655e9	0.0497428	0.0248714	+	0.999691i	$0.492082\pi$
$314$	3.48137e11	2.02100
$315$	0	0
$316$	2.05685e11	1.16041
$317$	−2.19980e11	−1.22354	−0.611768	−	0.791037i	$-0.709541\pi$
$317$	−2.19980e11	−1.22354	−0.611768	+	0.791037i	$0.709541\pi$
$318$	0	0
$319$	4.00015e11	2.16281
$320$	−1.36308e11	−0.726683
$321$	0	0
$322$	−1.85413e11	−0.961147
$323$	2.31202e11	1.18190
$324$	0	0
$325$	−1.92700e10	−0.0958091
$326$	−2.31690e11	−1.13613
$327$	0	0
$328$	−3.32256e10	−0.158504
$329$	−3.79729e10	−0.178687
$330$	0	0
$331$	6.96186e10	0.318786	0.159393	−	0.987215i	$-0.449046\pi$
$331$	6.96186e10	0.318786	0.159393	+	0.987215i	$0.449046\pi$
$332$	−1.69069e11	−0.763734
$333$	0	0
$334$	7.43882e10	0.327073
$335$	1.21061e11	0.525172
$336$	0	0
$337$	2.90202e11	1.22565	0.612825	−	0.790219i	$-0.290033\pi$
$337$	2.90202e11	1.22565	0.612825	+	0.790219i	$0.290033\pi$
$338$	−2.97928e11	−1.24161
$339$	0	0
$340$	2.67130e11	1.08410
$341$	−4.09171e11	−1.63874
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	−1.23884e11	−0.476982
$345$	0	0
$346$	6.30254e11	2.36414
$347$	−1.42619e11	−0.528075	−0.264038	−	0.964512i	$-0.585054\pi$
$347$	−1.42619e11	−0.528075	−0.264038	+	0.964512i	$0.585054\pi$
$348$	0	0
$349$	4.06617e11	1.46714	0.733569	−	0.679615i	$-0.237853\pi$
$349$	4.06617e11	1.46714	0.733569	+	0.679615i	$0.237853\pi$
$350$	3.41973e10	0.121811
$351$	0	0
$352$	−4.65402e11	−1.61580
$353$	−1.80418e11	−0.618435	−0.309217	−	0.950991i	$-0.600067\pi$
$353$	−1.80418e11	−0.618435	−0.309217	+	0.950991i	$0.600067\pi$
$354$	0	0
$355$	−8.85404e10	−0.295879
$356$	5.67803e11	1.87358
$357$	0	0
$358$	−2.53554e10	−0.0815826
$359$	5.04299e11	1.60237	0.801186	−	0.598416i	$-0.204203\pi$
$359$	5.04299e11	1.60237	0.801186	+	0.598416i	$0.204203\pi$
$360$	0	0
$361$	−1.27145e11	−0.394018
$362$	6.56979e11	2.01077
$363$	0	0
$364$	−9.68249e10	−0.289089
$365$	−1.48111e10	−0.0436786
$366$	0	0
$367$	−1.80774e11	−0.520162	−0.260081	−	0.965587i	$-0.583749\pi$
$367$	−1.80774e11	−0.520162	−0.260081	+	0.965587i	$0.583749\pi$
$368$	2.63220e10	0.0748177
$369$	0	0
$370$	3.83018e11	1.06246
$371$	6.68377e10	0.183164
$372$	0	0
$373$	−3.96719e11	−1.06119	−0.530596	−	0.847625i	$-0.678032\pi$
$373$	−3.96719e11	−1.06119	−0.530596	+	0.847625i	$0.678032\pi$
$374$	1.44128e12	3.80913
$375$	0	0
$376$	−1.76154e11	−0.454515
$377$	−2.61012e11	−0.665463
$378$	0	0
$379$	−7.29691e11	−1.81661	−0.908306	−	0.418306i	$-0.862624\pi$
$379$	−7.29691e11	−1.81661	−0.908306	+	0.418306i	$0.862624\pi$
$380$	2.25930e11	0.555838
$381$	0	0
$382$	9.50833e11	2.28465
$383$	−6.45922e11	−1.53386	−0.766930	−	0.641731i	$-0.778217\pi$
$383$	−6.45922e11	−1.53386	−0.766930	+	0.641731i	$0.778217\pi$
$384$	0	0
$385$	1.13452e11	0.263170
$386$	3.62496e11	0.831113
$387$	0	0
$388$	1.14358e12	2.56168
$389$	−4.16712e11	−0.922706	−0.461353	−	0.887217i	$-0.652636\pi$
$389$	−4.16712e11	−0.922706	−0.461353	+	0.887217i	$0.652636\pi$
$390$	0	0
$391$	−1.10734e12	−2.39598
$392$	6.42091e10	0.137344
$393$	0	0
$394$	−1.22309e12	−2.55696
$395$	1.57257e11	0.325029
$396$	0	0
$397$	9.61738e11	1.94312	0.971560	−	0.236793i	$-0.0760963\pi$
$397$	9.61738e11	1.94312	0.971560	+	0.236793i	$0.0760963\pi$
$398$	4.34322e11	0.867637
$399$	0	0
$400$	−4.85479e9	−0.00948200
$401$	−2.17932e10	−0.0420892	−0.0210446	−	0.999779i	$-0.506699\pi$
$401$	−2.17932e10	−0.0420892	−0.0210446	+	0.999779i	$0.506699\pi$
$402$	0	0
$403$	2.66986e11	0.504215
$404$	8.03480e11	1.50058
$405$	0	0
$406$	4.63202e11	0.846064
$407$	1.27068e12	2.29542
$408$	0	0
$409$	−8.91299e11	−1.57496	−0.787478	−	0.616342i	$-0.788614\pi$
$409$	−8.91299e11	−1.57496	−0.787478	+	0.616342i	$0.788614\pi$
$410$	−6.79798e10	−0.118810
$411$	0	0
$412$	−1.25192e12	−2.14063
$413$	−4.11158e10	−0.0695398
$414$	0	0
$415$	−1.29262e11	−0.213921
$416$	3.03677e11	0.497155
$417$	0	0
$418$	1.21899e12	1.95302
$419$	−1.00476e12	−1.59258	−0.796290	−	0.604915i	$-0.793207\pi$
$419$	−1.00476e12	−1.59258	−0.796290	+	0.604915i	$0.793207\pi$
$420$	0	0
$421$	−9.93627e11	−1.54154	−0.770768	−	0.637115i	$-0.780127\pi$
$421$	−9.93627e11	−1.54154	−0.770768	+	0.637115i	$0.780127\pi$
$422$	−1.62447e12	−2.49348
$423$	0	0
$424$	3.10057e11	0.465903
$425$	2.04235e11	0.303655
$426$	0	0
$427$	4.64670e11	0.676425
$428$	4.24590e11	0.611608
$429$	0	0
$430$	−2.53468e11	−0.357532
$431$	5.70989e11	0.797040	0.398520	−	0.917160i	$-0.369524\pi$
$431$	5.70989e11	0.797040	0.398520	+	0.917160i	$0.369524\pi$
$432$	0	0
$433$	1.38637e11	0.189532	0.0947659	−	0.995500i	$-0.469790\pi$
$433$	1.38637e11	0.189532	0.0947659	+	0.995500i	$0.469790\pi$
$434$	−4.73804e11	−0.641054
$435$	0	0
$436$	−1.01796e12	−1.34910
$437$	−9.36549e11	−1.22847
$438$	0	0
$439$	1.11468e12	1.43238	0.716189	−	0.697906i	$-0.245885\pi$
$439$	1.11468e12	1.43238	0.716189	+	0.697906i	$0.245885\pi$
$440$	5.26297e11	0.669412
$441$	0	0
$442$	−9.40441e11	−1.17201
$443$	−8.81863e11	−1.08789	−0.543944	−	0.839121i	$-0.683070\pi$
$443$	−8.81863e11	−1.08789	−0.543944	+	0.839121i	$0.683070\pi$
$444$	0	0
$445$	4.34115e11	0.524789
$446$	−2.66796e12	−3.19281
$447$	0	0
$448$	−5.23639e11	−0.614159
$449$	−9.45977e11	−1.09843	−0.549214	−	0.835681i	$-0.685073\pi$
$449$	−9.45977e11	−1.09843	−0.549214	+	0.835681i	$0.685073\pi$
$450$	0	0
$451$	−2.25527e11	−0.256687
$452$	−9.22443e11	−1.03948
$453$	0	0
$454$	−8.90972e11	−0.984266
$455$	−7.40276e10	−0.0809734
$456$	0	0
$457$	−2.09756e11	−0.224953	−0.112476	−	0.993654i	$-0.535878\pi$
$457$	−2.09756e11	−0.224953	−0.112476	+	0.993654i	$0.535878\pi$
$458$	−1.51741e12	−1.61142
$459$	0	0
$460$	−1.08209e12	−1.12681
$461$	−1.13248e12	−1.16782	−0.583912	−	0.811817i	$-0.698479\pi$
$461$	−1.13248e12	−1.16782	−0.583912	+	0.811817i	$0.698479\pi$
$462$	0	0
$463$	−6.47872e11	−0.655202	−0.327601	−	0.944816i	$-0.606240\pi$
$463$	−6.47872e11	−0.655202	−0.327601	+	0.944816i	$0.606240\pi$
$464$	−6.57580e10	−0.0658594
$465$	0	0
$466$	−1.37688e12	−1.35257
$467$	1.63471e12	1.59043	0.795216	−	0.606327i	$-0.207358\pi$
$467$	1.63471e12	1.59043	0.795216	+	0.606327i	$0.207358\pi$
$468$	0	0
$469$	4.65067e11	0.443851
$470$	−3.60413e11	−0.340691
$471$	0	0
$472$	−1.90734e11	−0.176885
$473$	−8.40893e11	−0.772442
$474$	0	0
$475$	1.72735e11	0.155690
$476$	1.02621e12	0.916229
$477$	0	0
$478$	1.14553e12	1.00365
$479$	1.44714e12	1.25603	0.628016	−	0.778200i	$-0.283867\pi$
$479$	1.44714e12	1.25603	0.628016	+	0.778200i	$0.283867\pi$
$480$	0	0
$481$	−8.29126e11	−0.706265
$482$	1.14124e12	0.963089
$483$	0	0
$484$	2.74495e12	2.27368
$485$	8.74329e11	0.717525
$486$	0	0
$487$	−7.93797e11	−0.639483	−0.319742	−	0.947505i	$-0.603596\pi$
$487$	−7.93797e11	−0.639483	−0.319742	+	0.947505i	$0.603596\pi$
$488$	2.15558e12	1.72058
$489$	0	0
$490$	1.31372e11	0.102949
$491$	−8.93419e11	−0.693726	−0.346863	−	0.937916i	$-0.612753\pi$
$491$	−8.93419e11	−0.693726	−0.346863	+	0.937916i	$0.612753\pi$
$492$	0	0
$493$	2.76636e12	2.10910
$494$	−7.95394e11	−0.600912
$495$	0	0
$496$	6.72630e10	0.0499010
$497$	−3.40137e11	−0.250063
$498$	0	0
$499$	−2.29839e12	−1.65948	−0.829740	−	0.558150i	$-0.811511\pi$
$499$	−2.29839e12	−1.65948	−0.829740	+	0.558150i	$0.811511\pi$
$500$	1.99578e11	0.142807
$501$	0	0
$502$	−2.80606e12	−1.97210
$503$	1.28863e12	0.897580	0.448790	−	0.893637i	$-0.351855\pi$
$503$	1.28863e12	0.897580	0.448790	+	0.893637i	$0.351855\pi$
$504$	0	0
$505$	6.14301e11	0.420311
$506$	−5.83831e12	−3.95922
$507$	0	0
$508$	−2.70814e12	−1.80420
$509$	−2.18603e12	−1.44353	−0.721765	−	0.692139i	$-0.756669\pi$
$509$	−2.18603e12	−1.44353	−0.721765	+	0.692139i	$0.756669\pi$
$510$	0	0
$511$	−5.68982e10	−0.0369151
$512$	1.47382e11	0.0947826
$513$	0	0
$514$	−2.72899e12	−1.72452
$515$	−9.57161e11	−0.599587
$516$	0	0
$517$	−1.19569e12	−0.736058
$518$	1.47140e12	0.897939
$519$	0	0
$520$	−3.43411e11	−0.205968
$521$	1.84809e12	1.09889	0.549443	−	0.835531i	$-0.314840\pi$
$521$	1.84809e12	1.09889	0.549443	+	0.835531i	$0.314840\pi$
$522$	0	0
$523$	9.46663e11	0.553271	0.276635	−	0.960975i	$-0.410781\pi$
$523$	9.46663e11	0.553271	0.276635	+	0.960975i	$0.410781\pi$
$524$	−3.95150e12	−2.28966
$525$	0	0
$526$	2.97603e11	0.169513
$527$	−2.82967e12	−1.59804
$528$	0	0
$529$	2.68443e12	1.49039
$530$	6.34380e11	0.349227
$531$	0	0
$532$	8.67934e11	0.469769
$533$	1.47157e11	0.0789786
$534$	0	0
$535$	3.24621e11	0.171311
$536$	2.15742e12	1.12900
$537$	0	0
$538$	2.27484e11	0.117066
$539$	4.35835e11	0.222420
$540$	0	0
$541$	−3.43412e12	−1.72357	−0.861783	−	0.507277i	$-0.830652\pi$
$541$	−3.43412e12	−1.72357	−0.861783	+	0.507277i	$0.830652\pi$
$542$	3.01135e12	1.49887
$543$	0	0
$544$	−3.21855e12	−1.57567
$545$	−7.78286e11	−0.377881
$546$	0	0
$547$	−1.17006e12	−0.558809	−0.279405	−	0.960173i	$-0.590137\pi$
$547$	−1.17006e12	−0.558809	−0.279405	+	0.960173i	$0.590137\pi$
$548$	2.80623e12	1.32926
$549$	0	0
$550$	1.07681e12	0.501771
$551$	2.33970e12	1.08138
$552$	0	0
$553$	6.04118e11	0.274700
$554$	−3.27687e12	−1.47797
$555$	0	0
$556$	3.24473e12	1.43993
$557$	−1.14677e12	−0.504812	−0.252406	−	0.967621i	$-0.581222\pi$
$557$	−1.14677e12	−0.504812	−0.252406	+	0.967621i	$0.581222\pi$
$558$	0	0
$559$	5.48687e11	0.237668
$560$	−1.86501e10	−0.00801376
$561$	0	0
$562$	5.98459e11	0.253059
$563$	−1.28982e12	−0.541056	−0.270528	−	0.962712i	$-0.587198\pi$
$563$	−1.28982e12	−0.541056	−0.270528	+	0.962712i	$0.587198\pi$
$564$	0	0
$565$	−7.05255e11	−0.291158
$566$	1.09266e12	0.447520
$567$	0	0
$568$	−1.57788e12	−0.636072
$569$	−2.06212e12	−0.824724	−0.412362	−	0.911020i	$-0.635296\pi$
$569$	−2.06212e12	−0.824724	−0.412362	+	0.911020i	$0.635296\pi$
$570$	0	0
$571$	−4.00715e12	−1.57751	−0.788756	−	0.614706i	$-0.789275\pi$
$571$	−4.00715e12	−1.57751	−0.788756	+	0.614706i	$0.789275\pi$
$572$	−3.04883e12	−1.19083
$573$	0	0
$574$	−2.61151e11	−0.100413
$575$	−8.27312e11	−0.315620
$576$	0	0
$577$	1.71845e12	0.645424	0.322712	−	0.946497i	$-0.395405\pi$
$577$	1.71845e12	0.645424	0.322712	+	0.946497i	$0.395405\pi$
$578$	5.64341e12	2.10313
$579$	0	0
$580$	2.70328e12	0.991895
$581$	−4.96572e11	−0.180796
$582$	0	0
$583$	2.10459e12	0.754500
$584$	−2.63948e11	−0.0938990
$585$	0	0
$586$	−5.88908e12	−2.06304
$587$	−2.80000e12	−0.973389	−0.486695	−	0.873572i	$-0.661798\pi$
$587$	−2.80000e12	−0.973389	−0.486695	+	0.873572i	$0.661798\pi$
$588$	0	0
$589$	−2.39325e12	−0.819348
$590$	−3.90244e11	−0.132587
$591$	0	0
$592$	−2.08886e11	−0.0698975
$593$	−1.10112e12	−0.365669	−0.182834	−	0.983144i	$-0.558527\pi$
$593$	−1.10112e12	−0.365669	−0.182834	+	0.983144i	$0.558527\pi$
$594$	0	0
$595$	7.84589e11	0.256635
$596$	−2.23037e11	−0.0724050
$597$	0	0
$598$	3.80952e12	1.21819
$599$	−2.24275e12	−0.711805	−0.355902	−	0.934523i	$-0.615826\pi$
$599$	−2.24275e12	−0.711805	−0.355902	+	0.934523i	$0.615826\pi$
$600$	0	0
$601$	−9.54770e11	−0.298513	−0.149257	−	0.988798i	$-0.547688\pi$
$601$	−9.54770e11	−0.298513	−0.149257	+	0.988798i	$0.547688\pi$
$602$	−9.73721e11	−0.302169
$603$	0	0
$604$	3.08260e12	0.942434
$605$	2.09865e12	0.636856
$606$	0	0
$607$	5.12222e11	0.153147	0.0765736	−	0.997064i	$-0.475602\pi$
$607$	5.12222e11	0.153147	0.0765736	+	0.997064i	$0.475602\pi$
$608$	−2.72215e12	−0.807877
$609$	0	0
$610$	4.41034e12	1.28970
$611$	7.80195e11	0.226474
$612$	0	0
$613$	−1.54411e12	−0.441680	−0.220840	−	0.975310i	$-0.570880\pi$
$613$	−1.54411e12	−0.441680	−0.220840	+	0.975310i	$0.570880\pi$
$614$	−6.04463e11	−0.171637
$615$	0	0
$616$	2.02182e12	0.565756
$617$	−2.69330e12	−0.748171	−0.374085	−	0.927394i	$-0.622043\pi$
$617$	−2.69330e12	−0.748171	−0.374085	+	0.927394i	$0.622043\pi$
$618$	0	0
$619$	2.16325e12	0.592242	0.296121	−	0.955150i	$-0.404307\pi$
$619$	2.16325e12	0.592242	0.296121	+	0.955150i	$0.404307\pi$
$620$	−2.76516e12	−0.751549
$621$	0	0
$622$	−1.28984e12	−0.345524
$623$	1.66770e12	0.443527
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	3.07978e11	0.0801557
$627$	0	0
$628$	7.80520e12	2.00247
$629$	8.78758e12	2.23842
$630$	0	0
$631$	−6.47015e11	−0.162473	−0.0812367	−	0.996695i	$-0.525887\pi$
$631$	−6.47015e11	−0.162473	−0.0812367	+	0.996695i	$0.525887\pi$
$632$	2.80248e12	0.698739
$633$	0	0
$634$	−8.02090e12	−1.97161
$635$	−2.07051e12	−0.505355
$636$	0	0
$637$	−2.84385e11	−0.0684350
$638$	1.45853e13	3.48516
$639$	0	0
$640$	−3.00016e12	−0.706861
$641$	−4.33378e12	−1.01392	−0.506962	−	0.861968i	$-0.669232\pi$
$641$	−4.33378e12	−1.01392	−0.506962	+	0.861968i	$0.669232\pi$
$642$	0	0
$643$	3.66638e12	0.845840	0.422920	−	0.906167i	$-0.361005\pi$
$643$	3.66638e12	0.845840	0.422920	+	0.906167i	$0.361005\pi$
$644$	−4.15695e12	−0.952332
$645$	0	0
$646$	8.43006e12	1.90452
$647$	−5.62143e12	−1.26118	−0.630591	−	0.776116i	$-0.717187\pi$
$647$	−5.62143e12	−1.26118	−0.630591	+	0.776116i	$0.717187\pi$
$648$	0	0
$649$	−1.29466e12	−0.286453
$650$	−7.02621e11	−0.154387
$651$	0	0
$652$	−5.19446e12	−1.12571
$653$	7.11463e12	1.53124	0.765620	−	0.643294i	$-0.222433\pi$
$653$	7.11463e12	1.53124	0.765620	+	0.643294i	$0.222433\pi$
$654$	0	0
$655$	−3.02112e12	−0.641332
$656$	3.70741e10	0.00781633
$657$	0	0
$658$	−1.38456e12	−0.287936
$659$	−7.21268e12	−1.48975	−0.744873	−	0.667206i	$-0.767490\pi$
$659$	−7.21268e12	−1.48975	−0.744873	+	0.667206i	$0.767490\pi$
$660$	0	0
$661$	6.13573e12	1.25014	0.625072	−	0.780567i	$-0.285070\pi$
$661$	6.13573e12	1.25014	0.625072	+	0.780567i	$0.285070\pi$
$662$	2.53843e12	0.513693
$663$	0	0
$664$	−2.30358e12	−0.459882
$665$	6.63580e11	0.131582
$666$	0	0
$667$	−1.12059e13	−2.19221
$668$	1.66778e12	0.324074
$669$	0	0
$670$	4.41411e12	0.846265
$671$	1.46316e13	2.78637
$672$	0	0
$673$	5.72507e12	1.07575	0.537877	−	0.843023i	$-0.319226\pi$
$673$	5.72507e12	1.07575	0.537877	+	0.843023i	$0.319226\pi$
$674$	1.05813e13	1.97502
$675$	0	0
$676$	−6.67952e12	−1.23023
$677$	−5.22145e12	−0.955306	−0.477653	−	0.878549i	$-0.658512\pi$
$677$	−5.22145e12	−0.955306	−0.477653	+	0.878549i	$0.658512\pi$
$678$	0	0
$679$	3.35882e12	0.606419
$680$	3.63967e12	0.652788
$681$	0	0
$682$	−1.49192e13	−2.64067
$683$	−8.77130e12	−1.54231	−0.771154	−	0.636649i	$-0.780320\pi$
$683$	−8.77130e12	−1.54231	−0.771154	+	0.636649i	$0.780320\pi$
$684$	0	0
$685$	2.14550e12	0.372325
$686$	5.04680e11	0.0870077
$687$	0	0
$688$	1.38233e11	0.0235215
$689$	−1.37326e12	−0.232148
$690$	0	0
$691$	3.10716e12	0.518457	0.259229	−	0.965816i	$-0.416532\pi$
$691$	3.10716e12	0.518457	0.259229	+	0.965816i	$0.416532\pi$
$692$	1.41302e13	2.34246
$693$	0	0
$694$	−5.20018e12	−0.850943
$695$	2.48077e12	0.403324
$696$	0	0
$697$	−1.55966e12	−0.250313
$698$	1.48260e13	2.36415
$699$	0	0
$700$	7.66700e11	0.120694
$701$	−2.85667e12	−0.446816	−0.223408	−	0.974725i	$-0.571718\pi$
$701$	−2.85667e12	−0.446816	−0.223408	+	0.974725i	$0.571718\pi$
$702$	0	0
$703$	7.43225e12	1.14768
$704$	−1.64884e13	−2.52989
$705$	0	0
$706$	−6.57839e12	−0.996549
$707$	2.35990e12	0.355227
$708$	0	0
$709$	3.61602e12	0.537431	0.268716	−	0.963220i	$-0.413401\pi$
$709$	3.61602e12	0.537431	0.268716	+	0.963220i	$0.413401\pi$
$710$	−3.22835e12	−0.476780
$711$	0	0
$712$	7.73636e12	1.12818
$713$	1.14624e13	1.66101
$714$	0	0
$715$	−2.33099e12	−0.333551
$716$	−5.68466e11	−0.0808344
$717$	0	0
$718$	1.83877e13	2.58207
$719$	1.61890e12	0.225913	0.112956	−	0.993600i	$-0.463968\pi$
$719$	1.61890e12	0.225913	0.112956	+	0.993600i	$0.463968\pi$
$720$	0	0
$721$	−3.67703e12	−0.506744
$722$	−4.63595e12	−0.634923
$723$	0	0
$724$	1.47294e13	1.99233
$725$	2.06680e12	0.277829
$726$	0	0
$727$	9.74494e12	1.29382	0.646911	−	0.762566i	$-0.276061\pi$
$727$	9.74494e12	1.29382	0.646911	+	0.762566i	$0.276061\pi$
$728$	−1.31925e12	−0.174074
$729$	0	0
$730$	−5.40040e11	−0.0703839
$731$	−5.81531e12	−0.753259
$732$	0	0
$733$	1.72463e12	0.220662	0.110331	−	0.993895i	$-0.464809\pi$
$733$	1.72463e12	0.220662	0.110331	+	0.993895i	$0.464809\pi$
$734$	−6.59137e12	−0.838191
$735$	0	0
$736$	1.30377e13	1.63776
$737$	1.46441e13	1.82834
$738$	0	0
$739$	3.28397e12	0.405041	0.202520	−	0.979278i	$-0.435087\pi$
$739$	3.28397e12	0.405041	0.202520	+	0.979278i	$0.435087\pi$
$740$	8.58722e12	1.05271
$741$	0	0
$742$	2.43703e12	0.295151
$743$	−4.22418e12	−0.508502	−0.254251	−	0.967138i	$-0.581829\pi$
$743$	−4.22418e12	−0.508502	−0.254251	+	0.967138i	$0.581829\pi$
$744$	0	0
$745$	−1.70523e11	−0.0202806
$746$	−1.44652e13	−1.71001
$747$	0	0
$748$	3.23133e13	3.77420
$749$	1.24706e12	0.144784
$750$	0	0
$751$	4.20239e12	0.482077	0.241038	−	0.970516i	$-0.422512\pi$
$751$	4.20239e12	0.482077	0.241038	+	0.970516i	$0.422512\pi$
$752$	1.96558e11	0.0224136
$753$	0	0
$754$	−9.51699e12	−1.07233
$755$	2.35680e12	0.263975
$756$	0	0
$757$	1.42679e13	1.57917	0.789587	−	0.613639i	$-0.210295\pi$
$757$	1.42679e13	1.57917	0.789587	+	0.613639i	$0.210295\pi$
$758$	−2.66059e13	−2.92730
$759$	0	0
$760$	3.07832e12	0.334697
$761$	1.04976e13	1.13464	0.567321	−	0.823497i	$-0.307980\pi$
$761$	1.04976e13	1.13464	0.567321	+	0.823497i	$0.307980\pi$
$762$	0	0
$763$	−2.98986e12	−0.319367
$764$	2.13176e13	2.26369
$765$	0	0
$766$	−2.35516e13	−2.47167
$767$	8.44770e11	0.0881372
$768$	0	0
$769$	−4.68130e12	−0.482723	−0.241361	−	0.970435i	$-0.577594\pi$
$769$	−4.68130e12	−0.482723	−0.241361	+	0.970435i	$0.577594\pi$
$770$	4.13666e12	0.424074
$771$	0	0
$772$	8.12712e12	0.823490
$773$	1.76519e13	1.77821	0.889106	−	0.457701i	$-0.151327\pi$
$773$	1.76519e13	1.77821	0.889106	+	0.457701i	$0.151327\pi$
$774$	0	0
$775$	−2.11410e12	−0.210508
$776$	1.55814e13	1.54251
$777$	0	0
$778$	−1.51941e13	−1.48685
$779$	−1.31911e12	−0.128340
$780$	0	0
$781$	−1.07102e13	−1.03008
$782$	−4.03756e13	−3.86090
$783$	0	0
$784$	−7.16464e10	−0.00677286
$785$	5.96748e12	0.560890
$786$	0	0
$787$	−1.53402e12	−0.142542	−0.0712712	−	0.997457i	$-0.522706\pi$
$787$	−1.53402e12	−0.142542	−0.0712712	+	0.997457i	$0.522706\pi$
$788$	−2.74215e13	−2.53351
$789$	0	0
$790$	5.73389e12	0.523754
$791$	−2.70931e12	−0.246073
$792$	0	0
$793$	−9.54716e12	−0.857324
$794$	3.50668e13	3.13115
$795$	0	0
$796$	9.73746e12	0.859680
$797$	6.94555e12	0.609740	0.304870	−	0.952394i	$-0.401387\pi$
$797$	6.94555e12	0.609740	0.304870	+	0.952394i	$0.401387\pi$
$798$	0	0
$799$	−8.26897e12	−0.717779
$800$	−2.40464e12	−0.207561
$801$	0	0
$802$	−7.94621e11	−0.0678228
$803$	−1.79162e12	−0.152063
$804$	0	0
$805$	−3.17820e12	−0.266747
$806$	9.73482e12	0.812494
$807$	0	0
$808$	1.09475e13	0.903572
$809$	−2.20581e12	−0.181050	−0.0905251	−	0.995894i	$-0.528855\pi$
$809$	−2.20581e12	−0.181050	−0.0905251	+	0.995894i	$0.528855\pi$
$810$	0	0
$811$	9.32362e12	0.756816	0.378408	−	0.925639i	$-0.376472\pi$
$811$	9.32362e12	0.756816	0.378408	+	0.925639i	$0.376472\pi$
$812$	1.03849e13	0.838304
$813$	0	0
$814$	4.63316e13	3.69885
$815$	−3.97143e12	−0.315310
$816$	0	0
$817$	−4.91840e12	−0.386211
$818$	−3.24985e13	−2.53789
$819$	0	0
$820$	−1.52410e12	−0.117720
$821$	2.27617e12	0.174848	0.0874238	−	0.996171i	$-0.472137\pi$
$821$	2.27617e12	0.174848	0.0874238	+	0.996171i	$0.472137\pi$
$822$	0	0
$823$	1.21606e13	0.923969	0.461984	−	0.886888i	$-0.347138\pi$
$823$	1.21606e13	0.923969	0.461984	+	0.886888i	$0.347138\pi$
$824$	−1.70576e13	−1.28898
$825$	0	0
$826$	−1.49916e12	−0.112057
$827$	1.00142e13	0.744458	0.372229	−	0.928141i	$-0.378594\pi$
$827$	1.00142e13	0.744458	0.372229	+	0.928141i	$0.378594\pi$
$828$	0	0
$829$	−5.39146e12	−0.396470	−0.198235	−	0.980154i	$-0.563521\pi$
$829$	−5.39146e12	−0.396470	−0.198235	+	0.980154i	$0.563521\pi$
$830$	−4.71314e12	−0.344714
$831$	0	0
$832$	1.07587e13	0.778407
$833$	3.01408e12	0.216896
$834$	0	0
$835$	1.27510e12	0.0907727
$836$	2.73296e13	1.93510
$837$	0	0
$838$	−3.66357e13	−2.56629
$839$	1.69446e13	1.18060	0.590300	−	0.807184i	$-0.299010\pi$
$839$	1.69446e13	1.18060	0.590300	+	0.807184i	$0.299010\pi$
$840$	0	0
$841$	1.34876e13	0.929722
$842$	−3.62296e13	−2.48404
$843$	0	0
$844$	−3.64205e13	−2.47062
$845$	−5.10683e12	−0.344585
$846$	0	0
$847$	8.06218e12	0.538242
$848$	−3.45971e11	−0.0229751
$849$	0	0
$850$	7.44680e12	0.489311
$851$	−3.55966e13	−2.32662
$852$	0	0
$853$	1.32813e13	0.858956	0.429478	−	0.903077i	$-0.358698\pi$
$853$	1.32813e13	0.858956	0.429478	+	0.903077i	$0.358698\pi$
$854$	1.69428e13	1.08999
$855$	0	0
$856$	5.78508e12	0.368279
$857$	8.21516e10	0.00520238	0.00260119	−	0.999997i	$-0.499172\pi$
$857$	8.21516e10	0.00520238	0.00260119	+	0.999997i	$0.499172\pi$
$858$	0	0
$859$	−1.80741e13	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$859$	−1.80741e13	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$860$	−5.68272e12	−0.354253
$861$	0	0
$862$	2.08194e13	1.28436
$863$	1.87322e13	1.14958	0.574791	−	0.818300i	$-0.305083\pi$
$863$	1.87322e13	1.14958	0.574791	+	0.818300i	$0.305083\pi$
$864$	0	0
$865$	1.08033e13	0.656120
$866$	5.05496e12	0.305413
$867$	0	0
$868$	−1.06226e13	−0.635175
$869$	1.90225e13	1.13156
$870$	0	0
$871$	−9.55531e12	−0.562553
$872$	−1.38698e13	−0.812357
$873$	0	0
$874$	−3.41484e13	−1.97956
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	1.95650e13	1.11682	0.558409	−	0.829566i	$-0.311412\pi$
$877$	1.95650e13	1.11682	0.558409	+	0.829566i	$0.311412\pi$
$878$	4.06432e13	2.30814
$879$	0	0
$880$	−5.87257e11	−0.0330108
$881$	−2.30530e12	−0.128925	−0.0644623	−	0.997920i	$-0.520533\pi$
$881$	−2.30530e12	−0.128925	−0.0644623	+	0.997920i	$0.520533\pi$
$882$	0	0
$883$	6.82366e12	0.377741	0.188871	−	0.982002i	$-0.439517\pi$
$883$	6.82366e12	0.377741	0.188871	+	0.982002i	$0.439517\pi$
$884$	−2.10846e13	−1.16126
$885$	0	0
$886$	−3.21544e13	−1.75303
$887$	−6.96936e12	−0.378039	−0.189020	−	0.981973i	$-0.560531\pi$
$887$	−6.96936e12	−0.378039	−0.189020	+	0.981973i	$0.560531\pi$
$888$	0	0
$889$	−7.95409e12	−0.427103
$890$	1.58287e13	0.845647
$891$	0	0
$892$	−5.98155e13	−3.16353
$893$	−6.99362e12	−0.368019
$894$	0	0
$895$	−4.34622e11	−0.0226416
$896$	−1.15254e13	−0.597406
$897$	0	0
$898$	−3.44921e13	−1.77001
$899$	−2.86355e13	−1.46213
$900$	0	0
$901$	1.45546e13	0.735763
$902$	−8.22315e12	−0.413627
$903$	0	0
$904$	−1.25684e13	−0.625923
$905$	1.12614e13	0.558050
$906$	0	0
$907$	2.37471e13	1.16514	0.582571	−	0.812780i	$-0.302047\pi$
$907$	2.37471e13	1.16514	0.582571	+	0.812780i	$0.302047\pi$
$908$	−1.99755e13	−0.975240
$909$	0	0
$910$	−2.69919e12	−0.130481
$911$	−1.81347e13	−0.872326	−0.436163	−	0.899868i	$-0.643663\pi$
$911$	−1.81347e13	−0.872326	−0.436163	+	0.899868i	$0.643663\pi$
$912$	0	0
$913$	−1.56361e13	−0.744749
$914$	−7.64811e12	−0.362490
$915$	0	0
$916$	−3.40202e13	−1.59664
$917$	−1.16060e13	−0.542024
$918$	0	0
$919$	6.30502e12	0.291586	0.145793	−	0.989315i	$-0.453427\pi$
$919$	6.30502e12	0.291586	0.145793	+	0.989315i	$0.453427\pi$
$920$	−1.47435e13	−0.678510
$921$	0	0
$922$	−4.12925e13	−1.88184
$923$	6.98848e12	0.316939
$924$	0	0
$925$	6.56537e12	0.294864
$926$	−2.36227e13	−1.05580
$927$	0	0
$928$	−3.25708e13	−1.44166
$929$	−4.17837e13	−1.84050	−0.920252	−	0.391327i	$-0.872016\pi$
$929$	−4.17837e13	−1.84050	−0.920252	+	0.391327i	$0.872016\pi$
$930$	0	0
$931$	2.54921e12	0.111207
$932$	−3.08696e13	−1.34017
$933$	0	0
$934$	5.96047e13	2.56283
$935$	2.47052e13	1.05715
$936$	0	0
$937$	−2.80323e13	−1.18804	−0.594019	−	0.804451i	$-0.702460\pi$
$937$	−2.80323e13	−1.18804	−0.594019	+	0.804451i	$0.702460\pi$
$938$	1.69572e13	0.715225
$939$	0	0
$940$	−8.08043e12	−0.337566
$941$	1.31067e13	0.544931	0.272465	−	0.962166i	$-0.412161\pi$
$941$	1.31067e13	0.544931	0.272465	+	0.962166i	$0.412161\pi$
$942$	0	0
$943$	6.31785e12	0.260176
$944$	2.12827e11	0.00872274
$945$	0	0
$946$	−3.06606e13	−1.24472
$947$	−2.61465e13	−1.05643	−0.528213	−	0.849112i	$-0.677138\pi$
$947$	−2.61465e13	−1.05643	−0.528213	+	0.849112i	$0.677138\pi$
$948$	0	0
$949$	1.16904e12	0.0467875
$950$	6.29826e12	0.250879
$951$	0	0
$952$	1.39822e13	0.551707
$953$	−1.54925e13	−0.608420	−0.304210	−	0.952605i	$-0.598392\pi$
$953$	−1.54925e13	−0.608420	−0.304210	+	0.952605i	$0.598392\pi$
$954$	0	0
$955$	1.62984e13	0.634059
$956$	2.56826e13	0.994442
$957$	0	0
$958$	5.27655e13	2.02398
$959$	8.24217e12	0.314672
$960$	0	0
$961$	2.85129e12	0.107842
$962$	−3.02316e13	−1.13808
$963$	0	0
$964$	2.55866e13	0.954257
$965$	6.21360e12	0.230659
$966$	0	0
$967$	−9.68529e12	−0.356200	−0.178100	−	0.984012i	$-0.556995\pi$
$967$	−9.68529e12	−0.356200	−0.178100	+	0.984012i	$0.556995\pi$
$968$	3.74001e13	1.36910
$969$	0	0
$970$	3.18797e13	1.15622
$971$	−2.06318e12	−0.0744820	−0.0372410	−	0.999306i	$-0.511857\pi$
$971$	−2.06318e12	−0.0744820	−0.0372410	+	0.999306i	$0.511857\pi$
$972$	0	0
$973$	9.53011e12	0.340871
$974$	−2.89434e13	−1.03047
$975$	0	0
$976$	−2.40526e12	−0.0848474
$977$	−3.09066e13	−1.08524	−0.542619	−	0.839979i	$-0.682567\pi$
$977$	−3.09066e13	−1.08524	−0.542619	+	0.839979i	$0.682567\pi$
$978$	0	0
$979$	5.25125e13	1.82701
$980$	2.94536e12	0.102005
$981$	0	0
$982$	−3.25758e13	−1.11787
$983$	−3.07721e13	−1.05115	−0.525577	−	0.850746i	$-0.676151\pi$
$983$	−3.07721e13	−1.05115	−0.525577	+	0.850746i	$0.676151\pi$
$984$	0	0
$985$	−2.09651e13	−0.709634
$986$	1.00867e14	3.39862
$987$	0	0
$988$	−1.78327e13	−0.595401
$989$	2.35566e13	0.782940
$990$	0	0
$991$	3.91433e13	1.28922	0.644608	−	0.764513i	$-0.277021\pi$
$991$	3.91433e13	1.28922	0.644608	+	0.764513i	$0.277021\pi$
$992$	3.33163e13	1.09233
$993$	0	0
$994$	−1.24020e13	−0.402953
$995$	7.44479e12	0.240796
$996$	0	0
$997$	3.07488e13	0.985599	0.492800	−	0.870143i	$-0.335974\pi$
$997$	3.07488e13	0.985599	0.492800	+	0.870143i	$0.335974\pi$
$998$	−8.38039e13	−2.67409
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.j.1.5		5
3.2	odd	2		35.10.a.d.1.1	✓	5
15.2	even	4		175.10.b.f.99.1		10
15.8	even	4		175.10.b.f.99.10		10
15.14	odd	2		175.10.a.f.1.5		5
21.20	even	2		245.10.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.d.1.1	✓	5	3.2	odd	2
175.10.a.f.1.5		5	15.14	odd	2
175.10.b.f.99.1		10	15.2	even	4
175.10.b.f.99.10		10	15.8	even	4
245.10.a.f.1.1		5	21.20	even	2
315.10.a.j.1.5		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+36.4619 q^{2} +817.473 q^{4} +625.000 q^{5} +2401.00 q^{7} +11138.1 q^{8} +22788.7 q^{10} +75602.9 q^{11} -49331.2 q^{13} +87545.1 q^{14} -12428.3 q^{16} +522841. q^{17} +442202. q^{19} +510921. q^{20} +2.75663e6 q^{22} -2.11792e6 q^{23} +390625. q^{25} -1.79871e6 q^{26} +1.96275e6 q^{28} +5.29101e6 q^{29} -5.41211e6 q^{31} -6.15588e6 q^{32} +1.90638e7 q^{34} +1.50062e6 q^{35} +1.68073e7 q^{37} +1.61236e7 q^{38} +6.96133e6 q^{40} -2.98305e6 q^{41} -1.11225e7 q^{43} +6.18033e7 q^{44} -7.72234e7 q^{46} -1.58154e7 q^{47} +5.76480e6 q^{49} +1.42429e7 q^{50} -4.03269e7 q^{52} +2.78375e7 q^{53} +4.72518e7 q^{55} +2.67427e7 q^{56} +1.92920e8 q^{58} -1.71244e7 q^{59} +1.93532e8 q^{61} -1.97336e8 q^{62} -2.18092e8 q^{64} -3.08320e7 q^{65} +1.93697e8 q^{67} +4.27409e8 q^{68} +5.47157e7 q^{70} -1.41665e8 q^{71} -2.36977e7 q^{73} +6.12828e8 q^{74} +3.61488e8 q^{76} +1.81522e8 q^{77} +2.51611e8 q^{79} -7.76766e6 q^{80} -1.08768e8 q^{82} -2.06819e8 q^{83} +3.26776e8 q^{85} -4.05548e8 q^{86} +8.42074e8 q^{88} +6.94584e8 q^{89} -1.18444e8 q^{91} -1.73134e9 q^{92} -5.76662e8 q^{94} +2.76376e8 q^{95} +1.39893e9 q^{97} +2.10196e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22}+ \cdots - 11529602 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 832 * q^4 + 3125 * q^5 + 12005 * q^7 + 14136 * q^8 - 1250 * q^10 - 10312 * q^11 + 158638 * q^13 - 4802 * q^14 - 526696 * q^16 - 31614 * q^17 + 1655376 * q^19 + 520000 * q^20 + 3659464 * q^22 - 796104 * q^23 + 1953125 * q^25 - 10219060 * q^26 + 1997632 * q^28 + 3353726 * q^29 + 2678120 * q^31 + 130400 * q^32 + 21548612 * q^34 + 7503125 * q^35 + 50994846 * q^37 + 34907600 * q^38 + 8835000 * q^40 - 6330194 * q^41 + 6149468 * q^43 + 71225892 * q^44 - 142845472 * q^46 + 6897780 * q^47 + 28824005 * q^49 - 781250 * q^50 - 22199468 * q^52 + 70886738 * q^53 - 6445000 * q^55 + 33940536 * q^56 - 31955924 * q^58 + 152335168 * q^59 + 257015698 * q^61 + 215582768 * q^62 - 193988432 * q^64 + 99148750 * q^65 + 133467828 * q^67 + 268709380 * q^68 - 3001250 * q^70 - 522788960 * q^71 + 159370858 * q^73 + 244178868 * q^74 - 345056616 * q^76 - 24759112 * q^77 + 464174900 * q^79 - 329185000 * q^80 + 308947620 * q^82 - 2207636832 * q^83 - 19758750 * q^85 - 694560616 * q^86 + 930528736 * q^88 + 1106708326 * q^89 + 380889838 * q^91 - 205160216 * q^92 - 3602501744 * q^94 + 1034610000 * q^95 + 1956142254 * q^97 - 11529602 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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