LMFDB - Embedded newform 315.8.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,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, 8, names="a")

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
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:	$98.4012830275$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 466x^{2} + 520x + 23440 $ x^4 - x^3 - 466x^2 + 520x + 23440
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.94744$ 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)

$f(q)$	$=$	\(q-6.94744 q^{2} -79.7331 q^{4} -125.000 q^{5} -343.000 q^{7} +1443.21 q^{8} +868.429 q^{10} -3452.43 q^{11} +2793.64 q^{13} +2382.97 q^{14} +179.217 q^{16} -20038.4 q^{17} +18051.5 q^{19} +9966.64 q^{20} +23985.5 q^{22} -35049.2 q^{23} +15625.0 q^{25} -19408.7 q^{26} +27348.5 q^{28} +63613.1 q^{29} +129406. q^{31} -185976. q^{32} +139216. q^{34} +42875.0 q^{35} +571412. q^{37} -125412. q^{38} -180402. q^{40} +49281.3 q^{41} -66688.6 q^{43} +275273. q^{44} +243502. q^{46} +954271. q^{47} +117649. q^{49} -108554. q^{50} -222746. q^{52} -148161. q^{53} +431553. q^{55} -495022. q^{56} -441948. q^{58} -1.07400e6 q^{59} +719682. q^{61} -899039. q^{62} +1.26912e6 q^{64} -349205. q^{65} +3.83495e6 q^{67} +1.59773e6 q^{68} -297871. q^{70} -3.15060e6 q^{71} -4.07755e6 q^{73} -3.96985e6 q^{74} -1.43930e6 q^{76} +1.18418e6 q^{77} -193472. q^{79} -22402.2 q^{80} -342379. q^{82} +2.97097e6 q^{83} +2.50480e6 q^{85} +463315. q^{86} -4.98258e6 q^{88} -1.06557e6 q^{89} -958220. q^{91} +2.79458e6 q^{92} -6.62973e6 q^{94} -2.25644e6 q^{95} +1.20795e6 q^{97} -817359. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 421 q^{4} - 500 q^{5} - 1372 q^{7} - 417 q^{8} - 125 q^{10} - 7852 q^{11} + 18532 q^{13} - 343 q^{14} + 47601 q^{16} - 33976 q^{17} + 22188 q^{19} - 52625 q^{20} + 18076 q^{22} + 22736 q^{23}+ \cdots + 117649 q^{98}+O(q^{100}) $ 4 * q + q^2 + 421 * q^4 - 500 * q^5 - 1372 * q^7 - 417 * q^8 - 125 * q^10 - 7852 * q^11 + 18532 * q^13 - 343 * q^14 + 47601 * q^16 - 33976 * q^17 + 22188 * q^19 - 52625 * q^20 + 18076 * q^22 + 22736 * q^23 + 62500 * q^25 + 235686 * q^26 - 144403 * q^28 - 12808 * q^29 - 64908 * q^31 - 111705 * q^32 + 45198 * q^34 + 171500 * q^35 - 315576 * q^37 - 1505204 * q^38 + 52125 * q^40 + 325240 * q^41 + 736184 * q^43 - 2151396 * q^44 - 1289480 * q^46 - 1661296 * q^47 + 470596 * q^49 + 15625 * q^50 + 2618462 * q^52 - 2569036 * q^53 + 981500 * q^55 + 143031 * q^56 - 553222 * q^58 - 3128416 * q^59 + 1392912 * q^61 + 1590240 * q^62 + 2081297 * q^64 - 2316500 * q^65 + 1654088 * q^67 - 8005658 * q^68 + 42875 * q^70 - 2863060 * q^71 + 4095068 * q^73 - 8950482 * q^74 - 8510596 * q^76 + 2693236 * q^77 + 9277704 * q^79 - 5950125 * q^80 - 2529402 * q^82 + 1362736 * q^83 + 4247000 * q^85 + 10602132 * q^86 - 14814892 * q^88 + 11110608 * q^89 - 6356476 * q^91 + 11222472 * q^92 + 101360 * q^94 - 2773500 * q^95 - 8721516 * q^97 + 117649 * 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 $

$2$	−6.94744	−0.614072	−0.307036	−	0.951698i	$-0.599337\pi$
$2$	−6.94744	−0.614072	−0.307036	+	0.951698i	$0.599337\pi$
$3$	0	0
$3$	0	0
$4$	−79.7331	−0.622915
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	1443.21	0.996587
$9$	0	0
$10$	868.429	0.274621
$11$	−3452.43	−0.782078	−0.391039	−	0.920374i	$-0.627884\pi$
$11$	−3452.43	−0.782078	−0.391039	+	0.920374i	$0.627884\pi$
$12$	0	0
$13$	2793.64	0.352671	0.176335	−	0.984330i	$-0.443576\pi$
$13$	2793.64	0.352671	0.176335	+	0.984330i	$0.443576\pi$
$14$	2382.97	0.232098
$15$	0	0
$16$	179.217	0.0109386
$17$	−20038.4	−0.989219	−0.494610	−	0.869115i	$-0.664689\pi$
$17$	−20038.4	−0.989219	−0.494610	+	0.869115i	$0.664689\pi$
$18$	0	0
$19$	18051.5	0.603777	0.301888	−	0.953343i	$-0.402383\pi$
$19$	18051.5	0.603777	0.301888	+	0.953343i	$0.402383\pi$
$20$	9966.64	0.278576
$21$	0	0
$22$	23985.5	0.480252
$23$	−35049.2	−0.600663	−0.300332	−	0.953835i	$-0.597097\pi$
$23$	−35049.2	−0.600663	−0.300332	+	0.953835i	$0.597097\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−19408.7	−0.216565
$27$	0	0
$28$	27348.5	0.235440
$29$	63613.1	0.484344	0.242172	−	0.970233i	$-0.422140\pi$
$29$	63613.1	0.484344	0.242172	+	0.970233i	$0.422140\pi$
$30$	0	0
$31$	129406.	0.780168	0.390084	−	0.920779i	$-0.372446\pi$
$31$	129406.	0.780168	0.390084	+	0.920779i	$0.372446\pi$
$32$	−185976.	−1.00330
$33$	0	0
$34$	139216.	0.607452
$35$	42875.0	0.169031
$36$	0	0
$37$	571412.	1.85457	0.927286	−	0.374354i	$-0.122136\pi$
$37$	571412.	1.85457	0.927286	+	0.374354i	$0.122136\pi$
$38$	−125412.	−0.370762
$39$	0	0
$40$	−180402.	−0.445687
$41$	49281.3	0.111671	0.0558353	−	0.998440i	$-0.482218\pi$
$41$	49281.3	0.111671	0.0558353	+	0.998440i	$0.482218\pi$
$42$	0	0
$43$	−66688.6	−0.127912	−0.0639561	−	0.997953i	$-0.520372\pi$
$43$	−66688.6	−0.127912	−0.0639561	+	0.997953i	$0.520372\pi$
$44$	275273.	0.487168
$45$	0	0
$46$	243502.	0.368851
$47$	954271.	1.34069	0.670346	−	0.742049i	$-0.266146\pi$
$47$	954271.	1.34069	0.670346	+	0.742049i	$0.266146\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−108554.	−0.122814
$51$	0	0
$52$	−222746.	−0.219684
$53$	−148161.	−0.136700	−0.0683501	−	0.997661i	$-0.521773\pi$
$53$	−148161.	−0.136700	−0.0683501	+	0.997661i	$0.521773\pi$
$54$	0	0
$55$	431553.	0.349756
$56$	−495022.	−0.376675
$57$	0	0
$58$	−441948.	−0.297422
$59$	−1.07400e6	−0.680802	−0.340401	−	0.940280i	$-0.610563\pi$
$59$	−1.07400e6	−0.680802	−0.340401	+	0.940280i	$0.610563\pi$
$60$	0	0
$61$	719682.	0.405963	0.202981	−	0.979183i	$-0.434937\pi$
$61$	719682.	0.405963	0.202981	+	0.979183i	$0.434937\pi$
$62$	−899039.	−0.479079
$63$	0	0
$64$	1.26912e6	0.605163
$65$	−349205.	−0.157719
$66$	0	0
$67$	3.83495e6	1.55775	0.778874	−	0.627180i	$-0.215791\pi$
$67$	3.83495e6	1.55775	0.778874	+	0.627180i	$0.215791\pi$
$68$	1.59773e6	0.616200
$69$	0	0
$70$	−297871.	−0.103797
$71$	−3.15060e6	−1.04470	−0.522348	−	0.852733i	$-0.674944\pi$
$71$	−3.15060e6	−1.04470	−0.522348	+	0.852733i	$0.674944\pi$
$72$	0	0
$73$	−4.07755e6	−1.22679	−0.613393	−	0.789778i	$-0.710196\pi$
$73$	−4.07755e6	−1.22679	−0.613393	+	0.789778i	$0.710196\pi$
$74$	−3.96985e6	−1.13884
$75$	0	0
$76$	−1.43930e6	−0.376102
$77$	1.18418e6	0.295598
$78$	0	0
$79$	−193472.	−0.0441492	−0.0220746	−	0.999756i	$-0.507027\pi$
$79$	−193472.	−0.0441492	−0.0220746	+	0.999756i	$0.507027\pi$
$80$	−22402.2	−0.00489187
$81$	0	0
$82$	−342379.	−0.0685738
$83$	2.97097e6	0.570329	0.285165	−	0.958479i	$-0.407952\pi$
$83$	2.97097e6	0.570329	0.285165	+	0.958479i	$0.407952\pi$
$84$	0	0
$85$	2.50480e6	0.442392
$86$	463315.	0.0785473
$87$	0	0
$88$	−4.98258e6	−0.779409
$89$	−1.06557e6	−0.160220	−0.0801102	−	0.996786i	$-0.525527\pi$
$89$	−1.06557e6	−0.160220	−0.0801102	+	0.996786i	$0.525527\pi$
$90$	0	0
$91$	−958220.	−0.133297
$92$	2.79458e6	0.374162
$93$	0	0
$94$	−6.62973e6	−0.823282
$95$	−2.25644e6	−0.270017
$96$	0	0
$97$	1.20795e6	0.134384	0.0671921	−	0.997740i	$-0.478596\pi$
$97$	1.20795e6	0.134384	0.0671921	+	0.997740i	$0.478596\pi$
$98$	−817359.	−0.0877246
$99$	0	0
$100$	−1.24583e6	−0.124583
$101$	1.06399e7	1.02757	0.513785	−	0.857919i	$-0.328243\pi$
$101$	1.06399e7	1.02757	0.513785	+	0.857919i	$0.328243\pi$
$102$	0	0
$103$	−3.41075e6	−0.307553	−0.153777	−	0.988106i	$-0.549144\pi$
$103$	−3.41075e6	−0.307553	−0.153777	+	0.988106i	$0.549144\pi$
$104$	4.03182e6	0.351467
$105$	0	0
$106$	1.02934e6	0.0839438
$107$	2.18626e7	1.72528	0.862639	−	0.505821i	$-0.168810\pi$
$107$	2.18626e7	1.72528	0.862639	+	0.505821i	$0.168810\pi$
$108$	0	0
$109$	−1.20019e7	−0.887679	−0.443839	−	0.896106i	$-0.646384\pi$
$109$	−1.20019e7	−0.887679	−0.443839	+	0.896106i	$0.646384\pi$
$110$	−2.99819e6	−0.214775
$111$	0	0
$112$	−61471.6	−0.00413439
$113$	−1.35164e6	−0.0881223	−0.0440612	−	0.999029i	$-0.514030\pi$
$113$	−1.35164e6	−0.0881223	−0.0440612	+	0.999029i	$0.514030\pi$
$114$	0	0
$115$	4.38115e6	0.268625
$116$	−5.07207e6	−0.301705
$117$	0	0
$118$	7.46152e6	0.418062
$119$	6.87318e6	0.373890
$120$	0	0
$121$	−7.56793e6	−0.388354
$122$	−4.99994e6	−0.249290
$123$	0	0
$124$	−1.03179e7	−0.485978
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.77606e7	−1.63578	−0.817891	−	0.575373i	$-0.804857\pi$
$127$	−3.77606e7	−1.63578	−0.817891	+	0.575373i	$0.804857\pi$
$128$	1.49878e7	0.631691
$129$	0	0
$130$	2.42608e6	0.0968509
$131$	−2.35027e7	−0.913415	−0.456708	−	0.889617i	$-0.650971\pi$
$131$	−2.35027e7	−0.913415	−0.456708	+	0.889617i	$0.650971\pi$
$132$	0	0
$133$	−6.19167e6	−0.228206
$134$	−2.66430e7	−0.956570
$135$	0	0
$136$	−2.89197e7	−0.985843
$137$	−3.19349e7	−1.06107	−0.530534	−	0.847664i	$-0.678008\pi$
$137$	−3.19349e7	−1.06107	−0.530534	+	0.847664i	$0.678008\pi$
$138$	0	0
$139$	−4.48717e7	−1.41717	−0.708584	−	0.705627i	$-0.750665\pi$
$139$	−4.48717e7	−1.41717	−0.708584	+	0.705627i	$0.750665\pi$
$140$	−3.41856e6	−0.105292
$141$	0	0
$142$	2.18886e7	0.641518
$143$	−9.64485e6	−0.275816
$144$	0	0
$145$	−7.95164e6	−0.216605
$146$	2.83285e7	0.753336
$147$	0	0
$148$	−4.55605e7	−1.15524
$149$	3.92484e7	0.972009	0.486004	−	0.873956i	$-0.338454\pi$
$149$	3.92484e7	0.972009	0.486004	+	0.873956i	$0.338454\pi$
$150$	0	0
$151$	4.49467e7	1.06238	0.531188	−	0.847254i	$-0.321746\pi$
$151$	4.49467e7	1.06238	0.531188	+	0.847254i	$0.321746\pi$
$152$	2.60522e7	0.601716
$153$	0	0
$154$	−8.22703e6	−0.181518
$155$	−1.61757e7	−0.348902
$156$	0	0
$157$	−2.20027e7	−0.453762	−0.226881	−	0.973922i	$-0.572853\pi$
$157$	−2.20027e7	−0.453762	−0.226881	+	0.973922i	$0.572853\pi$
$158$	1.34413e6	0.0271108
$159$	0	0
$160$	2.32470e7	0.448691
$161$	1.20219e7	0.227029
$162$	0	0
$163$	−2.65265e7	−0.479759	−0.239880	−	0.970803i	$-0.577108\pi$
$163$	−2.65265e7	−0.479759	−0.239880	+	0.970803i	$0.577108\pi$
$164$	−3.92935e6	−0.0695613
$165$	0	0
$166$	−2.06407e7	−0.350223
$167$	−9.48493e7	−1.57589	−0.787946	−	0.615744i	$-0.788855\pi$
$167$	−9.48493e7	−1.57589	−0.787946	+	0.615744i	$0.788855\pi$
$168$	0	0
$169$	−5.49441e7	−0.875623
$170$	−1.74020e7	−0.271661
$171$	0	0
$172$	5.31729e6	0.0796784
$173$	−5.00570e7	−0.735027	−0.367513	−	0.930018i	$-0.619791\pi$
$173$	−5.00570e7	−0.735027	−0.367513	+	0.930018i	$0.619791\pi$
$174$	0	0
$175$	−5.35938e6	−0.0755929
$176$	−618735.	−0.00855481
$177$	0	0
$178$	7.40299e6	0.0983869
$179$	−5.16805e7	−0.673505	−0.336753	−	0.941593i	$-0.609329\pi$
$179$	−5.16805e7	−0.673505	−0.336753	+	0.941593i	$0.609329\pi$
$180$	0	0
$181$	1.21438e8	1.52223	0.761117	−	0.648615i	$-0.224651\pi$
$181$	1.21438e8	1.52223	0.761117	+	0.648615i	$0.224651\pi$
$182$	6.65717e6	0.0818540
$183$	0	0
$184$	−5.05835e7	−0.598613
$185$	−7.14265e7	−0.829390
$186$	0	0
$187$	6.91812e7	0.773646
$188$	−7.60870e7	−0.835137
$189$	0	0
$190$	1.56765e7	0.165810
$191$	−1.25895e8	−1.30735	−0.653674	−	0.756776i	$-0.726773\pi$
$191$	−1.25895e8	−1.30735	−0.653674	+	0.756776i	$0.726773\pi$
$192$	0	0
$193$	−1.01607e8	−1.01736	−0.508679	−	0.860956i	$-0.669866\pi$
$193$	−1.01607e8	−1.01736	−0.508679	+	0.860956i	$0.669866\pi$
$194$	−8.39216e6	−0.0825216
$195$	0	0
$196$	−9.38052e6	−0.0889879
$197$	−1.64940e8	−1.53707	−0.768537	−	0.639805i	$-0.779015\pi$
$197$	−1.64940e8	−1.53707	−0.768537	+	0.639805i	$0.779015\pi$
$198$	0	0
$199$	426671.	0.00383802	0.00191901	−	0.999998i	$-0.499389\pi$
$199$	426671.	0.00383802	0.00191901	+	0.999998i	$0.499389\pi$
$200$	2.25502e7	0.199317
$201$	0	0
$202$	−7.39198e7	−0.631003
$203$	−2.18193e7	−0.183065
$204$	0	0
$205$	−6.16016e6	−0.0499406
$206$	2.36960e7	0.188860
$207$	0	0
$208$	500669.	0.00385771
$209$	−6.23215e7	−0.472200
$210$	0	0
$211$	−6.72814e7	−0.493067	−0.246534	−	0.969134i	$-0.579292\pi$
$211$	−6.72814e7	−0.493067	−0.246534	+	0.969134i	$0.579292\pi$
$212$	1.18134e7	0.0851526
$213$	0	0
$214$	−1.51889e8	−1.05944
$215$	8.33607e6	0.0572041
$216$	0	0
$217$	−4.43862e7	−0.294876
$218$	8.33821e7	0.545099
$219$	0	0
$220$	−3.44091e7	−0.217868
$221$	−5.59802e7	−0.348869
$222$	0	0
$223$	1.52978e8	0.923764	0.461882	−	0.886941i	$-0.347174\pi$
$223$	1.52978e8	0.923764	0.461882	+	0.886941i	$0.347174\pi$
$224$	6.37899e7	0.379213
$225$	0	0
$226$	9.39042e6	0.0541135
$227$	−1.04776e8	−0.594527	−0.297264	−	0.954795i	$-0.596074\pi$
$227$	−1.04776e8	−0.594527	−0.297264	+	0.954795i	$0.596074\pi$
$228$	0	0
$229$	−1.79834e7	−0.0989570	−0.0494785	−	0.998775i	$-0.515756\pi$
$229$	−1.79834e7	−0.0989570	−0.0494785	+	0.998775i	$0.515756\pi$
$230$	−3.04378e7	−0.164955
$231$	0	0
$232$	9.18072e7	0.482691
$233$	1.69751e8	0.879158	0.439579	−	0.898204i	$-0.355128\pi$
$233$	1.69751e8	0.879158	0.439579	+	0.898204i	$0.355128\pi$
$234$	0	0
$235$	−1.19284e8	−0.599576
$236$	8.56332e7	0.424082
$237$	0	0
$238$	−4.77510e7	−0.229595
$239$	8.37721e7	0.396924	0.198462	−	0.980109i	$-0.436405\pi$
$239$	8.37721e7	0.396924	0.198462	+	0.980109i	$0.436405\pi$
$240$	0	0
$241$	−1.31188e8	−0.603718	−0.301859	−	0.953353i	$-0.597607\pi$
$241$	−1.31188e8	−0.603718	−0.301859	+	0.953353i	$0.597607\pi$
$242$	5.25777e7	0.238478
$243$	0	0
$244$	−5.73825e7	−0.252880
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	5.04295e7	0.212934
$248$	1.86760e8	0.777505
$249$	0	0
$250$	1.35692e7	0.0549243
$251$	7.04199e7	0.281085	0.140542	−	0.990075i	$-0.455115\pi$
$251$	7.04199e7	0.281085	0.140542	+	0.990075i	$0.455115\pi$
$252$	0	0
$253$	1.21005e8	0.469765
$254$	2.62339e8	1.00449
$255$	0	0
$256$	−2.66574e8	−0.993067
$257$	−4.12149e8	−1.51457	−0.757284	−	0.653086i	$-0.773474\pi$
$257$	−4.12149e8	−1.51457	−0.757284	+	0.653086i	$0.773474\pi$
$258$	0	0
$259$	−1.95994e8	−0.700962
$260$	2.78432e7	0.0982456
$261$	0	0
$262$	1.63283e8	0.560903
$263$	4.74805e8	1.60942	0.804712	−	0.593665i	$-0.202320\pi$
$263$	4.74805e8	1.60942	0.804712	+	0.593665i	$0.202320\pi$
$264$	0	0
$265$	1.85201e7	0.0611342
$266$	4.30162e7	0.140135
$267$	0	0
$268$	−3.05772e8	−0.970345
$269$	7.22505e7	0.226312	0.113156	−	0.993577i	$-0.463904\pi$
$269$	7.22505e7	0.226312	0.113156	+	0.993577i	$0.463904\pi$
$270$	0	0
$271$	2.53457e8	0.773590	0.386795	−	0.922166i	$-0.373582\pi$
$271$	2.53457e8	0.773590	0.386795	+	0.922166i	$0.373582\pi$
$272$	−3.59124e6	−0.0108206
$273$	0	0
$274$	2.21865e8	0.651572
$275$	−5.39442e7	−0.156416
$276$	0	0
$277$	5.05351e8	1.42861	0.714306	−	0.699834i	$-0.246743\pi$
$277$	5.05351e8	1.42861	0.714306	+	0.699834i	$0.246743\pi$
$278$	3.11743e8	0.870243
$279$	0	0
$280$	6.18777e7	0.168454
$281$	4.71397e7	0.126740	0.0633701	−	0.997990i	$-0.479815\pi$
$281$	4.71397e7	0.126740	0.0633701	+	0.997990i	$0.479815\pi$
$282$	0	0
$283$	−1.29781e8	−0.340377	−0.170188	−	0.985412i	$-0.554438\pi$
$283$	−1.29781e8	−0.340377	−0.170188	+	0.985412i	$0.554438\pi$
$284$	2.51208e8	0.650757
$285$	0	0
$286$	6.70070e7	0.169371
$287$	−1.69035e7	−0.0422075
$288$	0	0
$289$	−8.79979e6	−0.0214452
$290$	5.52435e7	0.133011
$291$	0	0
$292$	3.25116e8	0.764184
$293$	−4.59414e8	−1.06701	−0.533504	−	0.845797i	$-0.679125\pi$
$293$	−4.59414e8	−1.06701	−0.533504	+	0.845797i	$0.679125\pi$
$294$	0	0
$295$	1.34250e8	0.304464
$296$	8.24669e8	1.84824
$297$	0	0
$298$	−2.72676e8	−0.596884
$299$	−9.79150e7	−0.211836
$300$	0	0
$301$	2.28742e7	0.0483463
$302$	−3.12264e8	−0.652376
$303$	0	0
$304$	3.23515e6	0.00660445
$305$	−8.99602e7	−0.181552
$306$	0	0
$307$	5.63910e8	1.11231	0.556155	−	0.831079i	$-0.312276\pi$
$307$	5.63910e8	1.11231	0.556155	+	0.831079i	$0.312276\pi$
$308$	−9.44186e7	−0.184132
$309$	0	0
$310$	1.12380e8	0.214251
$311$	8.71276e8	1.64246	0.821229	−	0.570599i	$-0.193289\pi$
$311$	8.71276e8	1.64246	0.821229	+	0.570599i	$0.193289\pi$
$312$	0	0
$313$	−5.05856e8	−0.932442	−0.466221	−	0.884668i	$-0.654385\pi$
$313$	−5.05856e8	−0.932442	−0.466221	+	0.884668i	$0.654385\pi$
$314$	1.52863e8	0.278643
$315$	0	0
$316$	1.54261e7	0.0275012
$317$	−9.91394e7	−0.174799	−0.0873995	−	0.996173i	$-0.527856\pi$
$317$	−9.91394e7	−0.174799	−0.0873995	+	0.996173i	$0.527856\pi$
$318$	0	0
$319$	−2.19620e8	−0.378794
$320$	−1.58640e8	−0.270637
$321$	0	0
$322$	−8.35212e7	−0.139412
$323$	−3.61724e8	−0.597267
$324$	0	0
$325$	4.36507e7	0.0705341
$326$	1.84291e8	0.294607
$327$	0	0
$328$	7.11234e7	0.111289
$329$	−3.27315e8	−0.506734
$330$	0	0
$331$	5.09577e8	0.772346	0.386173	−	0.922426i	$-0.373797\pi$
$331$	5.09577e8	0.772346	0.386173	+	0.922426i	$0.373797\pi$
$332$	−2.36885e8	−0.355267
$333$	0	0
$334$	6.58959e8	0.967712
$335$	−4.79368e8	−0.696646
$336$	0	0
$337$	1.18599e9	1.68801	0.844005	−	0.536336i	$-0.180192\pi$
$337$	1.18599e9	1.68801	0.844005	+	0.536336i	$0.180192\pi$
$338$	3.81720e8	0.537696
$339$	0	0
$340$	−1.99716e8	−0.275573
$341$	−4.46764e8	−0.610152
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	−9.62458e7	−0.127476
$345$	0	0
$346$	3.47767e8	0.451360
$347$	7.02435e8	0.902512	0.451256	−	0.892395i	$-0.350976\pi$
$347$	7.02435e8	0.902512	0.451256	+	0.892395i	$0.350976\pi$
$348$	0	0
$349$	−5.46813e8	−0.688572	−0.344286	−	0.938865i	$-0.611879\pi$
$349$	−5.46813e8	−0.688572	−0.344286	+	0.938865i	$0.611879\pi$
$350$	3.72339e7	0.0464195
$351$	0	0
$352$	6.42069e8	0.784662
$353$	1.66629e8	0.201622	0.100811	−	0.994906i	$-0.467856\pi$
$353$	1.66629e8	0.201622	0.100811	+	0.994906i	$0.467856\pi$
$354$	0	0
$355$	3.93826e8	0.467202
$356$	8.49614e7	0.0998037
$357$	0	0
$358$	3.59047e8	0.413581
$359$	6.10709e8	0.696633	0.348316	−	0.937377i	$-0.386754\pi$
$359$	6.10709e8	0.696633	0.348316	+	0.937377i	$0.386754\pi$
$360$	0	0
$361$	−5.68014e8	−0.635454
$362$	−8.43686e8	−0.934762
$363$	0	0
$364$	7.64019e7	0.0830327
$365$	5.09693e8	0.548636
$366$	0	0
$367$	−1.43810e9	−1.51865	−0.759325	−	0.650712i	$-0.774471\pi$
$367$	−1.43810e9	−1.51865	−0.759325	+	0.650712i	$0.774471\pi$
$368$	−6.28143e6	−0.00657039
$369$	0	0
$370$	4.96231e8	0.509305
$371$	5.08193e7	0.0516678
$372$	0	0
$373$	−3.01792e8	−0.301111	−0.150555	−	0.988602i	$-0.548106\pi$
$373$	−3.01792e8	−0.301111	−0.150555	+	0.988602i	$0.548106\pi$
$374$	−4.80632e8	−0.475075
$375$	0	0
$376$	1.37722e9	1.33612
$377$	1.77712e8	0.170814
$378$	0	0
$379$	−7.89548e8	−0.744975	−0.372487	−	0.928037i	$-0.621495\pi$
$379$	−7.89548e8	−0.744975	−0.372487	+	0.928037i	$0.621495\pi$
$380$	1.79913e8	0.168198
$381$	0	0
$382$	8.74647e8	0.802807
$383$	−2.00883e9	−1.82703	−0.913517	−	0.406800i	$-0.866645\pi$
$383$	−2.00883e9	−1.82703	−0.913517	+	0.406800i	$0.866645\pi$
$384$	0	0
$385$	−1.48023e8	−0.132195
$386$	7.05909e8	0.624732
$387$	0	0
$388$	−9.63138e7	−0.0837100
$389$	1.44433e9	1.24407	0.622033	−	0.782991i	$-0.286307\pi$
$389$	1.44433e9	1.24407	0.622033	+	0.782991i	$0.286307\pi$
$390$	0	0
$391$	7.02331e8	0.594187
$392$	1.69793e8	0.142370
$393$	0	0
$394$	1.14591e9	0.943875
$395$	2.41840e7	0.0197441
$396$	0	0
$397$	7.42825e8	0.595826	0.297913	−	0.954593i	$-0.403709\pi$
$397$	7.42825e8	0.595826	0.297913	+	0.954593i	$0.403709\pi$
$398$	−2.96427e6	−0.00235682
$399$	0	0
$400$	2.80027e6	0.00218771
$401$	1.60084e8	0.123978	0.0619889	−	0.998077i	$-0.480256\pi$
$401$	1.60084e8	0.123978	0.0619889	+	0.998077i	$0.480256\pi$
$402$	0	0
$403$	3.61514e8	0.275142
$404$	−8.48351e8	−0.640089
$405$	0	0
$406$	1.51588e8	0.112415
$407$	−1.97276e9	−1.45042
$408$	0	0
$409$	−1.66930e9	−1.20643	−0.603216	−	0.797578i	$-0.706114\pi$
$409$	−1.66930e9	−1.20643	−0.603216	+	0.797578i	$0.706114\pi$
$410$	4.27973e7	0.0306671
$411$	0	0
$412$	2.71950e8	0.191579
$413$	3.68381e8	0.257319
$414$	0	0
$415$	−3.71372e8	−0.255059
$416$	−5.19552e8	−0.353836
$417$	0	0
$418$	4.32975e8	0.289965
$419$	5.13361e8	0.340937	0.170468	−	0.985363i	$-0.445472\pi$
$419$	5.13361e8	0.340937	0.170468	+	0.985363i	$0.445472\pi$
$420$	0	0
$421$	1.95356e9	1.27597	0.637984	−	0.770050i	$-0.279769\pi$
$421$	1.95356e9	1.27597	0.637984	+	0.770050i	$0.279769\pi$
$422$	4.67433e8	0.302779
$423$	0	0
$424$	−2.13828e8	−0.136234
$425$	−3.13101e8	−0.197844
$426$	0	0
$427$	−2.46851e8	−0.153439
$428$	−1.74318e9	−1.07470
$429$	0	0
$430$	−5.79143e7	−0.0351274
$431$	1.53975e9	0.926361	0.463180	−	0.886264i	$-0.346708\pi$
$431$	1.53975e9	0.926361	0.463180	+	0.886264i	$0.346708\pi$
$432$	0	0
$433$	−1.72864e9	−1.02328	−0.511642	−	0.859199i	$-0.670963\pi$
$433$	−1.72864e9	−1.02328	−0.511642	+	0.859199i	$0.670963\pi$
$434$	3.08370e8	0.181075
$435$	0	0
$436$	9.56946e8	0.552948
$437$	−6.32692e8	−0.362666
$438$	0	0
$439$	2.80622e9	1.58306	0.791528	−	0.611132i	$-0.209286\pi$
$439$	2.80622e9	1.58306	0.791528	+	0.611132i	$0.209286\pi$
$440$	6.22823e8	0.348562
$441$	0	0
$442$	3.88919e8	0.214231
$443$	−6.73039e8	−0.367813	−0.183907	−	0.982944i	$-0.558874\pi$
$443$	−6.73039e8	−0.367813	−0.183907	+	0.982944i	$0.558874\pi$
$444$	0	0
$445$	1.33196e8	0.0716527
$446$	−1.06280e9	−0.567258
$447$	0	0
$448$	−4.35308e8	−0.228730
$449$	−8.59815e8	−0.448273	−0.224137	−	0.974558i	$-0.571956\pi$
$449$	−8.59815e8	−0.448273	−0.224137	+	0.974558i	$0.571956\pi$
$450$	0	0
$451$	−1.70140e8	−0.0873351
$452$	1.07770e8	0.0548927
$453$	0	0
$454$	7.27925e8	0.365083
$455$	1.19777e8	0.0596122
$456$	0	0
$457$	−1.85328e9	−0.908312	−0.454156	−	0.890922i	$-0.650059\pi$
$457$	−1.85328e9	−0.908312	−0.454156	+	0.890922i	$0.650059\pi$
$458$	1.24938e8	0.0607668
$459$	0	0
$460$	−3.49323e8	−0.167330
$461$	5.84090e8	0.277668	0.138834	−	0.990316i	$-0.455665\pi$
$461$	5.84090e8	0.277668	0.138834	+	0.990316i	$0.455665\pi$
$462$	0	0
$463$	−3.10502e9	−1.45389	−0.726945	−	0.686696i	$-0.759060\pi$
$463$	−3.10502e9	−1.45389	−0.726945	+	0.686696i	$0.759060\pi$
$464$	1.14006e7	0.00529802
$465$	0	0
$466$	−1.17933e9	−0.539866
$467$	2.39617e9	1.08870	0.544350	−	0.838858i	$-0.316776\pi$
$467$	2.39617e9	1.08870	0.544350	+	0.838858i	$0.316776\pi$
$468$	0	0
$469$	−1.31539e9	−0.588774
$470$	8.28717e8	0.368183
$471$	0	0
$472$	−1.55001e9	−0.678479
$473$	2.30237e8	0.100037
$474$	0	0
$475$	2.82055e8	0.120755
$476$	−5.48021e8	−0.232902
$477$	0	0
$478$	−5.82001e8	−0.243740
$479$	−3.11093e6	−0.00129335	−0.000646675	−	1.00000i	$-0.500206\pi$
$479$	−3.11093e6	−0.00129335	−0.000646675		1.00000i	$0.500206\pi$
$480$	0	0
$481$	1.59632e9	0.654053
$482$	9.11420e8	0.370727
$483$	0	0
$484$	6.03415e8	0.241912
$485$	−1.50994e8	−0.0600985
$486$	0	0
$487$	−3.45835e9	−1.35681	−0.678403	−	0.734690i	$-0.737328\pi$
$487$	−3.45835e9	−1.35681	−0.678403	+	0.734690i	$0.737328\pi$
$488$	1.03865e9	0.404577
$489$	0	0
$490$	1.02170e8	0.0392316
$491$	−1.87982e9	−0.716690	−0.358345	−	0.933589i	$-0.616659\pi$
$491$	−1.87982e9	−0.716690	−0.358345	+	0.933589i	$0.616659\pi$
$492$	0	0
$493$	−1.27471e9	−0.479122
$494$	−3.50356e8	−0.130757
$495$	0	0
$496$	2.31918e7	0.00853391
$497$	1.08066e9	0.394858
$498$	0	0
$499$	−2.98642e9	−1.07597	−0.537984	−	0.842955i	$-0.680814\pi$
$499$	−2.98642e9	−1.07597	−0.537984	+	0.842955i	$0.680814\pi$
$500$	1.55729e8	0.0557152
$501$	0	0
$502$	−4.89238e8	−0.172606
$503$	−2.95022e9	−1.03363	−0.516817	−	0.856096i	$-0.672883\pi$
$503$	−2.95022e9	−1.03363	−0.516817	+	0.856096i	$0.672883\pi$
$504$	0	0
$505$	−1.32998e9	−0.459544
$506$	−8.40673e8	−0.288470
$507$	0	0
$508$	3.01077e9	1.01895
$509$	2.57436e9	0.865280	0.432640	−	0.901567i	$-0.357582\pi$
$509$	2.57436e9	0.865280	0.432640	+	0.901567i	$0.357582\pi$
$510$	0	0
$511$	1.39860e9	0.463682
$512$	−6.64372e7	−0.0218759
$513$	0	0
$514$	2.86338e9	0.930054
$515$	4.26344e8	0.137542
$516$	0	0
$517$	−3.29455e9	−1.04853
$518$	1.36166e9	0.430441
$519$	0	0
$520$	−5.03978e8	−0.157181
$521$	2.20184e9	0.682109	0.341054	−	0.940044i	$-0.389216\pi$
$521$	2.20184e9	0.682109	0.341054	+	0.940044i	$0.389216\pi$
$522$	0	0
$523$	6.06489e8	0.185382	0.0926909	−	0.995695i	$-0.470453\pi$
$523$	6.06489e8	0.185382	0.0926909	+	0.995695i	$0.470453\pi$
$524$	1.87394e9	0.568980
$525$	0	0
$526$	−3.29868e9	−0.988303
$527$	−2.59309e9	−0.771757
$528$	0	0
$529$	−2.17638e9	−0.639204
$530$	−1.28668e8	−0.0375408
$531$	0	0
$532$	4.93682e8	0.142153
$533$	1.37674e8	0.0393829
$534$	0	0
$535$	−2.73283e9	−0.771567
$536$	5.53464e9	1.55243
$537$	0	0
$538$	−5.01955e8	−0.138972
$539$	−4.06174e8	−0.111725
$540$	0	0
$541$	2.42481e7	0.00658395	0.00329197	−	0.999995i	$-0.498952\pi$
$541$	2.42481e7	0.00658395	0.00329197	+	0.999995i	$0.498952\pi$
$542$	−1.76087e9	−0.475040
$543$	0	0
$544$	3.72667e9	0.992488
$545$	1.50023e9	0.396982
$546$	0	0
$547$	−5.54093e8	−0.144753	−0.0723764	−	0.997377i	$-0.523058\pi$
$547$	−5.54093e8	−0.144753	−0.0723764	+	0.997377i	$0.523058\pi$
$548$	2.54627e9	0.660955
$549$	0	0
$550$	3.74774e8	0.0960505
$551$	1.14831e9	0.292435
$552$	0	0
$553$	6.63609e7	0.0166868
$554$	−3.51089e9	−0.877271
$555$	0	0
$556$	3.57776e9	0.882775
$557$	5.51883e9	1.35317	0.676587	−	0.736363i	$-0.263458\pi$
$557$	5.51883e9	1.35317	0.676587	+	0.736363i	$0.263458\pi$
$558$	0	0
$559$	−1.86304e8	−0.0451109
$560$	7.68395e6	0.00184895
$561$	0	0
$562$	−3.27500e8	−0.0778276
$563$	−7.95448e9	−1.87859	−0.939296	−	0.343109i	$-0.888520\pi$
$563$	−7.95448e9	−1.87859	−0.939296	+	0.343109i	$0.888520\pi$
$564$	0	0
$565$	1.68955e8	0.0394095
$566$	9.01648e8	0.209016
$567$	0	0
$568$	−4.54699e9	−1.04113
$569$	3.19008e9	0.725953	0.362976	−	0.931798i	$-0.381761\pi$
$569$	3.19008e9	0.725953	0.362976	+	0.931798i	$0.381761\pi$
$570$	0	0
$571$	2.73377e9	0.614520	0.307260	−	0.951626i	$-0.400588\pi$
$571$	2.73377e9	0.614520	0.307260	+	0.951626i	$0.400588\pi$
$572$	7.69014e8	0.171810
$573$	0	0
$574$	1.17436e8	0.0259185
$575$	−5.47644e8	−0.120133
$576$	0	0
$577$	−6.38537e9	−1.38379	−0.691895	−	0.721998i	$-0.743224\pi$
$577$	−6.38537e9	−1.38379	−0.691895	+	0.721998i	$0.743224\pi$
$578$	6.11360e7	0.0131689
$579$	0	0
$580$	6.34009e8	0.134927
$581$	−1.01904e9	−0.215564
$582$	0	0
$583$	5.11516e8	0.106910
$584$	−5.88477e9	−1.22260
$585$	0	0
$586$	3.19175e9	0.655220
$587$	−6.65567e9	−1.35818	−0.679091	−	0.734054i	$-0.737626\pi$
$587$	−6.65567e9	−1.35818	−0.679091	+	0.734054i	$0.737626\pi$
$588$	0	0
$589$	2.33597e9	0.471047
$590$	−9.32691e8	−0.186963
$591$	0	0
$592$	1.02407e8	0.0202863
$593$	−1.46920e9	−0.289327	−0.144663	−	0.989481i	$-0.546210\pi$
$593$	−1.46920e9	−0.289327	−0.144663	+	0.989481i	$0.546210\pi$
$594$	0	0
$595$	−8.59148e8	−0.167209
$596$	−3.12940e9	−0.605479
$597$	0	0
$598$	6.80258e8	0.130083
$599$	−9.43186e9	−1.79310	−0.896548	−	0.442947i	$-0.853933\pi$
$599$	−9.43186e9	−1.79310	−0.896548	+	0.442947i	$0.853933\pi$
$600$	0	0
$601$	8.07554e8	0.151744	0.0758719	−	0.997118i	$-0.475826\pi$
$601$	8.07554e8	0.151744	0.0758719	+	0.997118i	$0.475826\pi$
$602$	−1.58917e8	−0.0296881
$603$	0	0
$604$	−3.58374e9	−0.661771
$605$	9.45991e8	0.173677
$606$	0	0
$607$	−7.93485e9	−1.44005	−0.720027	−	0.693947i	$-0.755870\pi$
$607$	−7.93485e9	−1.44005	−0.720027	+	0.693947i	$0.755870\pi$
$608$	−3.35716e9	−0.605772
$609$	0	0
$610$	6.24993e8	0.111486
$611$	2.66589e9	0.472823
$612$	0	0
$613$	−1.79108e8	−0.0314053	−0.0157027	−	0.999877i	$-0.504999\pi$
$613$	−1.79108e8	−0.0314053	−0.0157027	+	0.999877i	$0.504999\pi$
$614$	−3.91773e9	−0.683039
$615$	0	0
$616$	1.70903e9	0.294589
$617$	−4.58991e9	−0.786695	−0.393347	−	0.919390i	$-0.628683\pi$
$617$	−4.58991e9	−0.786695	−0.393347	+	0.919390i	$0.628683\pi$
$618$	0	0
$619$	4.19664e9	0.711189	0.355594	−	0.934640i	$-0.384278\pi$
$619$	4.19664e9	0.711189	0.355594	+	0.934640i	$0.384278\pi$
$620$	1.28974e9	0.217336
$621$	0	0
$622$	−6.05313e9	−1.00859
$623$	3.65491e8	0.0605576
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	3.51440e9	0.572587
$627$	0	0
$628$	1.75435e9	0.282655
$629$	−1.14502e10	−1.83458
$630$	0	0
$631$	−8.08881e9	−1.28169	−0.640843	−	0.767672i	$-0.721415\pi$
$631$	−8.08881e9	−1.28169	−0.640843	+	0.767672i	$0.721415\pi$
$632$	−2.79221e8	−0.0439986
$633$	0	0
$634$	6.88765e8	0.107339
$635$	4.72007e9	0.731544
$636$	0	0
$637$	3.28669e8	0.0503815
$638$	1.52579e9	0.232607
$639$	0	0
$640$	−1.87348e9	−0.282501
$641$	1.05667e10	1.58466	0.792329	−	0.610094i	$-0.208868\pi$
$641$	1.05667e10	1.58466	0.792329	+	0.610094i	$0.208868\pi$
$642$	0	0
$643$	9.32183e9	1.38281	0.691405	−	0.722467i	$-0.256992\pi$
$643$	9.32183e9	1.38281	0.691405	+	0.722467i	$0.256992\pi$
$644$	−9.58542e8	−0.141420
$645$	0	0
$646$	2.51306e9	0.366765
$647$	−2.14135e9	−0.310830	−0.155415	−	0.987849i	$-0.549671\pi$
$647$	−2.14135e9	−0.310830	−0.155415	+	0.987849i	$0.549671\pi$
$648$	0	0
$649$	3.70790e9	0.532440
$650$	−3.03260e8	−0.0433130
$651$	0	0
$652$	2.11504e9	0.298849
$653$	3.10122e9	0.435850	0.217925	−	0.975966i	$-0.430071\pi$
$653$	3.10122e9	0.435850	0.217925	+	0.975966i	$0.430071\pi$
$654$	0	0
$655$	2.93784e9	0.408492
$656$	8.83207e6	0.00122152
$657$	0	0
$658$	2.27400e9	0.311171
$659$	−7.00087e9	−0.952912	−0.476456	−	0.879198i	$-0.658079\pi$
$659$	−7.00087e9	−0.952912	−0.476456	+	0.879198i	$0.658079\pi$
$660$	0	0
$661$	−1.17338e10	−1.58028	−0.790139	−	0.612928i	$-0.789991\pi$
$661$	−1.17338e10	−1.58028	−0.790139	+	0.612928i	$0.789991\pi$
$662$	−3.54026e9	−0.474277
$663$	0	0
$664$	4.28775e9	0.568383
$665$	7.73959e8	0.102057
$666$	0	0
$667$	−2.22959e9	−0.290927
$668$	7.56263e9	0.981647
$669$	0	0
$670$	3.33038e9	0.427791
$671$	−2.48465e9	−0.317494
$672$	0	0
$673$	−4.96419e9	−0.627763	−0.313882	−	0.949462i	$-0.601630\pi$
$673$	−4.96419e9	−0.627763	−0.313882	+	0.949462i	$0.601630\pi$
$674$	−8.23956e9	−1.03656
$675$	0	0
$676$	4.38086e9	0.545439
$677$	5.29734e9	0.656142	0.328071	−	0.944653i	$-0.393602\pi$
$677$	5.29734e9	0.656142	0.328071	+	0.944653i	$0.393602\pi$
$678$	0	0
$679$	−4.14327e8	−0.0507925
$680$	3.61497e9	0.440883
$681$	0	0
$682$	3.10386e9	0.374677
$683$	−4.78049e9	−0.574116	−0.287058	−	0.957913i	$-0.592677\pi$
$683$	−4.78049e9	−0.574116	−0.287058	+	0.957913i	$0.592677\pi$
$684$	0	0
$685$	3.99186e9	0.474524
$686$	2.80354e8	0.0331568
$687$	0	0
$688$	−1.19518e7	−0.00139918
$689$	−4.13910e8	−0.0482101
$690$	0	0
$691$	−2.35648e9	−0.271700	−0.135850	−	0.990729i	$-0.543377\pi$
$691$	−2.35648e9	−0.271700	−0.135850	+	0.990729i	$0.543377\pi$
$692$	3.99120e9	0.457859
$693$	0	0
$694$	−4.88012e9	−0.554207
$695$	5.60897e9	0.633776
$696$	0	0
$697$	−9.87521e8	−0.110467
$698$	3.79895e9	0.422833
$699$	0	0
$700$	4.27320e8	0.0470880
$701$	1.65547e10	1.81513	0.907567	−	0.419906i	$-0.137937\pi$
$701$	1.65547e10	1.81513	0.907567	+	0.419906i	$0.137937\pi$
$702$	0	0
$703$	1.03149e10	1.11975
$704$	−4.38154e9	−0.473284
$705$	0	0
$706$	−1.15764e9	−0.123810
$707$	−3.64948e9	−0.388385
$708$	0	0
$709$	−6.42948e9	−0.677507	−0.338754	−	0.940875i	$-0.610005\pi$
$709$	−6.42948e9	−0.677507	−0.338754	+	0.940875i	$0.610005\pi$
$710$	−2.73608e9	−0.286896
$711$	0	0
$712$	−1.53785e9	−0.159674
$713$	−4.53557e9	−0.468618
$714$	0	0
$715$	1.20561e9	0.123349
$716$	4.12065e9	0.419537
$717$	0	0
$718$	−4.24286e9	−0.427783
$719$	9.15944e9	0.919005	0.459502	−	0.888177i	$-0.348028\pi$
$719$	9.15944e9	0.919005	0.459502	+	0.888177i	$0.348028\pi$
$720$	0	0
$721$	1.16989e9	0.116244
$722$	3.94624e9	0.390215
$723$	0	0
$724$	−9.68267e9	−0.948223
$725$	9.93955e8	0.0968687
$726$	0	0
$727$	1.73629e10	1.67591	0.837955	−	0.545739i	$-0.183751\pi$
$727$	1.73629e10	1.67591	0.837955	+	0.545739i	$0.183751\pi$
$728$	−1.38291e9	−0.132842
$729$	0	0
$730$	−3.54106e9	−0.336902
$731$	1.33633e9	0.126533
$732$	0	0
$733$	−1.62448e10	−1.52353	−0.761765	−	0.647853i	$-0.775667\pi$
$733$	−1.62448e10	−1.52353	−0.761765	+	0.647853i	$0.775667\pi$
$734$	9.99110e9	0.932561
$735$	0	0
$736$	6.51832e9	0.602648
$737$	−1.32399e10	−1.21828
$738$	0	0
$739$	1.32542e9	0.120808	0.0604041	−	0.998174i	$-0.480761\pi$
$739$	1.32542e9	0.120808	0.0604041	+	0.998174i	$0.480761\pi$
$740$	5.69506e9	0.516639
$741$	0	0
$742$	−3.53064e8	−0.0317278
$743$	−2.74796e9	−0.245782	−0.122891	−	0.992420i	$-0.539217\pi$
$743$	−2.74796e9	−0.245782	−0.122891	+	0.992420i	$0.539217\pi$
$744$	0	0
$745$	−4.90605e9	−0.434696
$746$	2.09668e9	0.184904
$747$	0	0
$748$	−5.51604e9	−0.481916
$749$	−7.49888e9	−0.652094
$750$	0	0
$751$	1.33321e10	1.14857	0.574287	−	0.818654i	$-0.305279\pi$
$751$	1.33321e10	1.14857	0.574287	+	0.818654i	$0.305279\pi$
$752$	1.71022e8	0.0146652
$753$	0	0
$754$	−1.23464e9	−0.104892
$755$	−5.61834e9	−0.475109
$756$	0	0
$757$	−1.92796e10	−1.61534	−0.807668	−	0.589637i	$-0.799271\pi$
$757$	−1.92796e10	−1.61534	−0.807668	+	0.589637i	$0.799271\pi$
$758$	5.48534e9	0.457468
$759$	0	0
$760$	−3.25652e9	−0.269096
$761$	1.74311e10	1.43377	0.716883	−	0.697193i	$-0.245568\pi$
$761$	1.74311e10	1.43377	0.716883	+	0.697193i	$0.245568\pi$
$762$	0	0
$763$	4.11664e9	0.335511
$764$	1.00380e10	0.814367
$765$	0	0
$766$	1.39562e10	1.12193
$767$	−3.00036e9	−0.240099
$768$	0	0
$769$	−1.68123e10	−1.33317	−0.666585	−	0.745429i	$-0.732245\pi$
$769$	−1.68123e10	−1.33317	−0.666585	+	0.745429i	$0.732245\pi$
$770$	1.02838e9	0.0811775
$771$	0	0
$772$	8.10146e9	0.633728
$773$	−1.92866e10	−1.50185	−0.750927	−	0.660385i	$-0.770393\pi$
$773$	−1.92866e10	−1.50185	−0.750927	+	0.660385i	$0.770393\pi$
$774$	0	0
$775$	2.02197e9	0.156034
$776$	1.74333e9	0.133926
$777$	0	0
$778$	−1.00344e10	−0.763946
$779$	8.89603e8	0.0674241
$780$	0	0
$781$	1.08772e10	0.817033
$782$	−4.87940e9	−0.364874
$783$	0	0
$784$	2.10847e7	0.00156265
$785$	2.75034e9	0.202929
$786$	0	0
$787$	1.04192e8	0.00761946	0.00380973	−	0.999993i	$-0.498787\pi$
$787$	1.04192e8	0.00761946	0.00380973	+	0.999993i	$0.498787\pi$
$788$	1.31512e10	0.957467
$789$	0	0
$790$	−1.68017e8	−0.0121243
$791$	4.63612e8	0.0333071
$792$	0	0
$793$	2.01053e9	0.143171
$794$	−5.16073e9	−0.365880
$795$	0	0
$796$	−3.40198e7	−0.00239076
$797$	−6.82275e9	−0.477370	−0.238685	−	0.971097i	$-0.576716\pi$
$797$	−6.82275e9	−0.477370	−0.238685	+	0.971097i	$0.576716\pi$
$798$	0	0
$799$	−1.91221e10	−1.32624
$800$	−2.90588e9	−0.200661
$801$	0	0
$802$	−1.11218e9	−0.0761313
$803$	1.40774e10	0.959443
$804$	0	0
$805$	−1.50274e9	−0.101531
$806$	−2.51159e9	−0.168957
$807$	0	0
$808$	1.53556e10	1.02406
$809$	−1.93003e10	−1.28158	−0.640788	−	0.767718i	$-0.721392\pi$
$809$	−1.93003e10	−1.28158	−0.640788	+	0.767718i	$0.721392\pi$
$810$	0	0
$811$	−2.17571e10	−1.43228	−0.716139	−	0.697957i	$-0.754092\pi$
$811$	−2.17571e10	−1.43228	−0.716139	+	0.697957i	$0.754092\pi$
$812$	1.73972e9	0.114034
$813$	0	0
$814$	1.37056e10	0.890662
$815$	3.31581e9	0.214555
$816$	0	0
$817$	−1.20383e9	−0.0772304
$818$	1.15973e10	0.740836
$819$	0	0
$820$	4.91169e8	0.0311088
$821$	6.17840e9	0.389650	0.194825	−	0.980838i	$-0.437586\pi$
$821$	6.17840e9	0.389650	0.194825	+	0.980838i	$0.437586\pi$
$822$	0	0
$823$	3.12248e10	1.95254	0.976271	−	0.216553i	$-0.0694814\pi$
$823$	3.12248e10	1.95254	0.976271	+	0.216553i	$0.0694814\pi$
$824$	−4.92244e9	−0.306503
$825$	0	0
$826$	−2.55930e9	−0.158013
$827$	−2.77162e10	−1.70398	−0.851991	−	0.523556i	$-0.824605\pi$
$827$	−2.77162e10	−1.70398	−0.851991	+	0.523556i	$0.824605\pi$
$828$	0	0
$829$	−1.18348e8	−0.00721475	−0.00360737	−	0.999993i	$-0.501148\pi$
$829$	−1.18348e8	−0.00721475	−0.00360737	+	0.999993i	$0.501148\pi$
$830$	2.58008e9	0.156625
$831$	0	0
$832$	3.54546e9	0.213423
$833$	−2.35750e9	−0.141317
$834$	0	0
$835$	1.18562e10	0.704760
$836$	4.96909e9	0.294141
$837$	0	0
$838$	−3.56654e9	−0.209360
$839$	2.11926e10	1.23885	0.619423	−	0.785057i	$-0.287367\pi$
$839$	2.11926e10	1.23885	0.619423	+	0.785057i	$0.287367\pi$
$840$	0	0
$841$	−1.32033e10	−0.765411
$842$	−1.35722e10	−0.783536
$843$	0	0
$844$	5.36456e9	0.307139
$845$	6.86801e9	0.391591
$846$	0	0
$847$	2.59580e9	0.146784
$848$	−2.65531e7	−0.00149530
$849$	0	0
$850$	2.17525e9	0.121490
$851$	−2.00276e10	−1.11397
$852$	0	0
$853$	3.35615e10	1.85148	0.925742	−	0.378156i	$-0.123442\pi$
$853$	3.35615e10	1.85148	0.925742	+	0.378156i	$0.123442\pi$
$854$	1.71498e9	0.0942229
$855$	0	0
$856$	3.15524e10	1.71939
$857$	5.34969e9	0.290333	0.145166	−	0.989407i	$-0.453628\pi$
$857$	5.34969e9	0.290333	0.145166	+	0.989407i	$0.453628\pi$
$858$	0	0
$859$	3.39623e10	1.82819	0.914093	−	0.405504i	$-0.132904\pi$
$859$	3.39623e10	1.82819	0.914093	+	0.405504i	$0.132904\pi$
$860$	−6.64661e8	−0.0356333
$861$	0	0
$862$	−1.06973e10	−0.568853
$863$	9.44338e9	0.500138	0.250069	−	0.968228i	$-0.419547\pi$
$863$	9.44338e9	0.500138	0.250069	+	0.968228i	$0.419547\pi$
$864$	0	0
$865$	6.25712e9	0.328714
$866$	1.20096e10	0.628370
$867$	0	0
$868$	3.53905e9	0.183683
$869$	6.67948e8	0.0345281
$870$	0	0
$871$	1.07135e10	0.549372
$872$	−1.73212e10	−0.884649
$873$	0	0
$874$	4.39558e9	0.222703
$875$	6.69922e8	0.0338062
$876$	0	0
$877$	3.45584e10	1.73004	0.865018	−	0.501740i	$-0.167307\pi$
$877$	3.45584e10	1.73004	0.865018	+	0.501740i	$0.167307\pi$
$878$	−1.94961e10	−0.972111
$879$	0	0
$880$	7.73418e7	0.00382583
$881$	3.78291e10	1.86385	0.931923	−	0.362656i	$-0.118130\pi$
$881$	3.78291e10	1.86385	0.931923	+	0.362656i	$0.118130\pi$
$882$	0	0
$883$	−1.14776e9	−0.0561033	−0.0280517	−	0.999606i	$-0.508930\pi$
$883$	−1.14776e9	−0.0561033	−0.0280517	+	0.999606i	$0.508930\pi$
$884$	4.46348e9	0.217316
$885$	0	0
$886$	4.67589e9	0.225864
$887$	−1.07492e9	−0.0517181	−0.0258590	−	0.999666i	$-0.508232\pi$
$887$	−1.07492e9	−0.0517181	−0.0258590	+	0.999666i	$0.508232\pi$
$888$	0	0
$889$	1.29519e10	0.618268
$890$	−9.25374e8	−0.0439999
$891$	0	0
$892$	−1.21974e10	−0.575427
$893$	1.72260e10	0.809478
$894$	0	0
$895$	6.46006e9	0.301201
$896$	−5.14083e9	−0.238757
$897$	0	0
$898$	5.97351e9	0.275272
$899$	8.23191e9	0.377869
$900$	0	0
$901$	2.96892e9	0.135226
$902$	1.18204e9	0.0536301
$903$	0	0
$904$	−1.95070e9	−0.0878216
$905$	−1.51798e10	−0.680764
$906$	0	0
$907$	−1.23252e10	−0.548489	−0.274245	−	0.961660i	$-0.588428\pi$
$907$	−1.23252e10	−0.548489	−0.274245	+	0.961660i	$0.588428\pi$
$908$	8.35413e9	0.370340
$909$	0	0
$910$	−8.32146e8	−0.0366062
$911$	−1.38378e10	−0.606389	−0.303194	−	0.952929i	$-0.598053\pi$
$911$	−1.38378e10	−0.606389	−0.303194	+	0.952929i	$0.598053\pi$
$912$	0	0
$913$	−1.02571e10	−0.446042
$914$	1.28756e10	0.557769
$915$	0	0
$916$	1.43387e9	0.0616418
$917$	8.06143e9	0.345238
$918$	0	0
$919$	−3.14447e10	−1.33642	−0.668210	−	0.743973i	$-0.732939\pi$
$919$	−3.14447e10	−1.33642	−0.668210	+	0.743973i	$0.732939\pi$
$920$	6.32293e9	0.267708
$921$	0	0
$922$	−4.05792e9	−0.170508
$923$	−8.80167e9	−0.368433
$924$	0	0
$925$	8.92832e9	0.370914
$926$	2.15720e10	0.892794
$927$	0	0
$928$	−1.18305e10	−0.485944
$929$	9.95416e9	0.407333	0.203667	−	0.979040i	$-0.434714\pi$
$929$	9.95416e9	0.407333	0.203667	+	0.979040i	$0.434714\pi$
$930$	0	0
$931$	2.12374e9	0.0862538
$932$	−1.35348e10	−0.547641
$933$	0	0
$934$	−1.66472e10	−0.668541
$935$	−8.64765e9	−0.345985
$936$	0	0
$937$	5.58944e9	0.221963	0.110981	−	0.993822i	$-0.464601\pi$
$937$	5.58944e9	0.221963	0.110981	+	0.993822i	$0.464601\pi$
$938$	9.13856e9	0.361550
$939$	0	0
$940$	9.51087e9	0.373485
$941$	6.34397e9	0.248198	0.124099	−	0.992270i	$-0.460396\pi$
$941$	6.34397e9	0.248198	0.124099	+	0.992270i	$0.460396\pi$
$942$	0	0
$943$	−1.72727e9	−0.0670764
$944$	−1.92479e8	−0.00744700
$945$	0	0
$946$	−1.59956e9	−0.0614301
$947$	−1.62315e10	−0.621060	−0.310530	−	0.950564i	$-0.600507\pi$
$947$	−1.62315e10	−0.621060	−0.310530	+	0.950564i	$0.600507\pi$
$948$	0	0
$949$	−1.13912e10	−0.432652
$950$	−1.95956e9	−0.0741525
$951$	0	0
$952$	9.91946e9	0.372614
$953$	−7.68925e9	−0.287779	−0.143889	−	0.989594i	$-0.545961\pi$
$953$	−7.68925e9	−0.287779	−0.143889	+	0.989594i	$0.545961\pi$
$954$	0	0
$955$	1.57369e10	0.584664
$956$	−6.67941e9	−0.247250
$957$	0	0
$958$	2.16130e7	0.000794211 0
$959$	1.09537e10	0.401046
$960$	0	0
$961$	−1.07667e10	−0.391338
$962$	−1.10903e10	−0.401636
$963$	0	0
$964$	1.04600e10	0.376065
$965$	1.27009e10	0.454976
$966$	0	0
$967$	−1.92306e10	−0.683912	−0.341956	−	0.939716i	$-0.611089\pi$
$967$	−1.92306e10	−0.683912	−0.341956	+	0.939716i	$0.611089\pi$
$968$	−1.09221e10	−0.387029
$969$	0	0
$970$	1.04902e9	0.0369048
$971$	3.66365e10	1.28424	0.642120	−	0.766604i	$-0.278055\pi$
$971$	3.66365e10	1.28424	0.642120	+	0.766604i	$0.278055\pi$
$972$	0	0
$973$	1.53910e10	0.535639
$974$	2.40267e10	0.833177
$975$	0	0
$976$	1.28980e8	0.00444065
$977$	−2.73002e10	−0.936558	−0.468279	−	0.883581i	$-0.655126\pi$
$977$	−2.73002e10	−0.936558	−0.468279	+	0.883581i	$0.655126\pi$
$978$	0	0
$979$	3.67881e9	0.125305
$980$	1.17257e9	0.0397966
$981$	0	0
$982$	1.30599e10	0.440099
$983$	6.21169e7	0.00208580	0.00104290	−	0.999999i	$-0.499668\pi$
$983$	6.21169e7	0.00208580	0.00104290	+	0.999999i	$0.499668\pi$
$984$	0	0
$985$	2.06175e10	0.687401
$986$	8.85594e9	0.294216
$987$	0	0
$988$	−4.02090e9	−0.132640
$989$	2.33738e9	0.0768321
$990$	0	0
$991$	−1.82777e10	−0.596573	−0.298287	−	0.954476i	$-0.596415\pi$
$991$	−1.82777e10	−0.596573	−0.298287	+	0.954476i	$0.596415\pi$
$992$	−2.40664e10	−0.782746
$993$	0	0
$994$	−7.50780e9	−0.242471
$995$	−5.33338e7	−0.00171641
$996$	0	0
$997$	−3.93538e10	−1.25763	−0.628817	−	0.777553i	$-0.716460\pi$
$997$	−3.93538e10	−1.25763	−0.628817	+	0.777553i	$0.716460\pi$
$998$	2.07480e10	0.660722
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.i.1.2		4
3.2	odd	2		105.8.a.f.1.3	✓	4
15.14	odd	2		525.8.a.k.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.f.1.3	✓	4	3.2	odd	2
315.8.a.i.1.2		4	1.1	even	1	trivial
525.8.a.k.1.2		4	15.14	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-6.94744 q^{2} -79.7331 q^{4} -125.000 q^{5} -343.000 q^{7} +1443.21 q^{8} +868.429 q^{10} -3452.43 q^{11} +2793.64 q^{13} +2382.97 q^{14} +179.217 q^{16} -20038.4 q^{17} +18051.5 q^{19} +9966.64 q^{20} +23985.5 q^{22} -35049.2 q^{23} +15625.0 q^{25} -19408.7 q^{26} +27348.5 q^{28} +63613.1 q^{29} +129406. q^{31} -185976. q^{32} +139216. q^{34} +42875.0 q^{35} +571412. q^{37} -125412. q^{38} -180402. q^{40} +49281.3 q^{41} -66688.6 q^{43} +275273. q^{44} +243502. q^{46} +954271. q^{47} +117649. q^{49} -108554. q^{50} -222746. q^{52} -148161. q^{53} +431553. q^{55} -495022. q^{56} -441948. q^{58} -1.07400e6 q^{59} +719682. q^{61} -899039. q^{62} +1.26912e6 q^{64} -349205. q^{65} +3.83495e6 q^{67} +1.59773e6 q^{68} -297871. q^{70} -3.15060e6 q^{71} -4.07755e6 q^{73} -3.96985e6 q^{74} -1.43930e6 q^{76} +1.18418e6 q^{77} -193472. q^{79} -22402.2 q^{80} -342379. q^{82} +2.97097e6 q^{83} +2.50480e6 q^{85} +463315. q^{86} -4.98258e6 q^{88} -1.06557e6 q^{89} -958220. q^{91} +2.79458e6 q^{92} -6.62973e6 q^{94} -2.25644e6 q^{95} +1.20795e6 q^{97} -817359. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 421 q^{4} - 500 q^{5} - 1372 q^{7} - 417 q^{8} - 125 q^{10} - 7852 q^{11} + 18532 q^{13} - 343 q^{14} + 47601 q^{16} - 33976 q^{17} + 22188 q^{19} - 52625 q^{20} + 18076 q^{22} + 22736 q^{23}+ \cdots + 117649 q^{98}+O(q^{100}) \) 4 * q + q^2 + 421 * q^4 - 500 * q^5 - 1372 * q^7 - 417 * q^8 - 125 * q^10 - 7852 * q^11 + 18532 * q^13 - 343 * q^14 + 47601 * q^16 - 33976 * q^17 + 22188 * q^19 - 52625 * q^20 + 18076 * q^22 + 22736 * q^23 + 62500 * q^25 + 235686 * q^26 - 144403 * q^28 - 12808 * q^29 - 64908 * q^31 - 111705 * q^32 + 45198 * q^34 + 171500 * q^35 - 315576 * q^37 - 1505204 * q^38 + 52125 * q^40 + 325240 * q^41 + 736184 * q^43 - 2151396 * q^44 - 1289480 * q^46 - 1661296 * q^47 + 470596 * q^49 + 15625 * q^50 + 2618462 * q^52 - 2569036 * q^53 + 981500 * q^55 + 143031 * q^56 - 553222 * q^58 - 3128416 * q^59 + 1392912 * q^61 + 1590240 * q^62 + 2081297 * q^64 - 2316500 * q^65 + 1654088 * q^67 - 8005658 * q^68 + 42875 * q^70 - 2863060 * q^71 + 4095068 * q^73 - 8950482 * q^74 - 8510596 * q^76 + 2693236 * q^77 + 9277704 * q^79 - 5950125 * q^80 - 2529402 * q^82 + 1362736 * q^83 + 4247000 * q^85 + 10602132 * q^86 - 14814892 * q^88 + 11110608 * q^89 - 6356476 * q^91 + 11222472 * q^92 + 101360 * q^94 - 2773500 * q^95 - 8721516 * q^97 + 117649 * q^98

Introduction

L-functions

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