LMFDB - Embedded newform 315.10.a.l.1.1

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3018x^{4} + 3368x^{3} + 2066979x^{2} - 6329061x - 14714266 $ x^6 - 3x^5 - 3018x^4 + 3368x^3 + 2066979x^2 - 6329061*x - 14714266
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-39.7818$ 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)

$f(q)$	$=$	\(q-42.7818 q^{2} +1318.28 q^{4} -625.000 q^{5} -2401.00 q^{7} -34494.1 q^{8} +26738.6 q^{10} -41568.7 q^{11} +103859. q^{13} +102719. q^{14} +800760. q^{16} -355256. q^{17} -349126. q^{19} -823925. q^{20} +1.77838e6 q^{22} -228204. q^{23} +390625. q^{25} -4.44327e6 q^{26} -3.16519e6 q^{28} -4.02216e6 q^{29} -3.29476e6 q^{31} -1.65969e7 q^{32} +1.51985e7 q^{34} +1.50062e6 q^{35} -2.13083e7 q^{37} +1.49362e7 q^{38} +2.15588e7 q^{40} -1.05289e7 q^{41} +3.89712e6 q^{43} -5.47991e7 q^{44} +9.76296e6 q^{46} +3.31846e7 q^{47} +5.76480e6 q^{49} -1.67116e7 q^{50} +1.36915e8 q^{52} +6.31880e7 q^{53} +2.59804e7 q^{55} +8.28203e7 q^{56} +1.72075e8 q^{58} -5.86733e7 q^{59} +1.37845e8 q^{61} +1.40956e8 q^{62} +3.00057e8 q^{64} -6.49118e7 q^{65} +8.36796e7 q^{67} -4.68327e8 q^{68} -6.41994e7 q^{70} -1.50169e8 q^{71} -2.74817e8 q^{73} +9.11609e8 q^{74} -4.60246e8 q^{76} +9.98063e7 q^{77} -2.75964e8 q^{79} -5.00475e8 q^{80} +4.50444e8 q^{82} -2.32312e8 q^{83} +2.22035e8 q^{85} -1.66726e8 q^{86} +1.43387e9 q^{88} -2.71482e8 q^{89} -2.49365e8 q^{91} -3.00837e8 q^{92} -1.41969e9 q^{94} +2.18204e8 q^{95} -9.76607e8 q^{97} -2.46628e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8} + 9375 q^{10} + 47796 q^{11} + 102168 q^{13} + 36015 q^{14} + 2371065 q^{16} + 38472 q^{17} + 361056 q^{19} - 1880625 q^{20} + 2068680 q^{22}+ \cdots - 86472015 q^{98}+O(q^{100}) $ 6 * q - 15 * q^2 + 3009 * q^4 - 3750 * q^5 - 14406 * q^7 - 22041 * q^8 + 9375 * q^10 + 47796 * q^11 + 102168 * q^13 + 36015 * q^14 + 2371065 * q^16 + 38472 * q^17 + 361056 * q^19 - 1880625 * q^20 + 2068680 * q^22 - 697032 * q^23 + 2343750 * q^25 - 10847622 * q^26 - 7224609 * q^28 - 16028184 * q^29 + 1362912 * q^31 + 1148823 * q^32 - 9216294 * q^34 + 9003750 * q^35 - 3912924 * q^37 - 39877140 * q^38 + 13775625 * q^40 - 22452756 * q^41 - 29998992 * q^43 + 94378728 * q^44 + 49589520 * q^46 + 121271508 * q^47 + 34588806 * q^49 - 5859375 * q^50 - 44528550 * q^52 + 20308596 * q^53 - 29872500 * q^55 + 52920441 * q^56 + 256759494 * q^58 - 120280392 * q^59 - 87693540 * q^61 + 344802528 * q^62 + 640647105 * q^64 - 63855000 * q^65 + 495050664 * q^67 + 141443706 * q^68 - 22509375 * q^70 - 253762512 * q^71 + 187195308 * q^73 + 418806054 * q^74 - 584719476 * q^76 - 114758196 * q^77 + 831079500 * q^79 - 1481915625 * q^80 + 2286848814 * q^82 + 767650536 * q^83 - 24045000 * q^85 - 566199948 * q^86 + 2952495444 * q^88 - 582579684 * q^89 - 245305368 * q^91 + 4259382384 * q^92 - 1115946660 * q^94 - 225660000 * q^95 - 1184506872 * q^97 - 86472015 * 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 $

$2$	−42.7818	−1.89071	−0.945353	−	0.326050i	$-0.894282\pi$
$2$	−42.7818	−1.89071	−0.945353	+	0.326050i	$0.894282\pi$
$3$	0	0
$3$	0	0
$4$	1318.28	2.57477
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−34494.1	−2.97742
$9$	0	0
$10$	26738.6	0.845549
$11$	−41568.7	−0.856050	−0.428025	−	0.903767i	$-0.640790\pi$
$11$	−41568.7	−0.856050	−0.428025	+	0.903767i	$0.640790\pi$
$12$	0	0
$13$	103859.	1.00855	0.504277	−	0.863542i	$-0.331759\pi$
$13$	103859.	1.00855	0.504277	+	0.863542i	$0.331759\pi$
$14$	102719.	0.714619
$15$	0	0
$16$	800760.	3.05466
$17$	−355256.	−1.03162	−0.515812	−	0.856702i	$-0.672510\pi$
$17$	−355256.	−1.03162	−0.515812	+	0.856702i	$0.672510\pi$
$18$	0	0
$19$	−349126.	−0.614597	−0.307299	−	0.951613i	$-0.599425\pi$
$19$	−349126.	−0.614597	−0.307299	+	0.951613i	$0.599425\pi$
$20$	−823925.	−1.15147
$21$	0	0
$22$	1.77838e6	1.61854
$23$	−228204.	−0.170039	−0.0850193	−	0.996379i	$-0.527095\pi$
$23$	−228204.	−0.170039	−0.0850193	+	0.996379i	$0.527095\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−4.44327e6	−1.90688
$27$	0	0
$28$	−3.16519e6	−0.973170
$29$	−4.02216e6	−1.05601	−0.528005	−	0.849241i	$-0.677060\pi$
$29$	−4.02216e6	−1.05601	−0.528005	+	0.849241i	$0.677060\pi$
$30$	0	0
$31$	−3.29476e6	−0.640761	−0.320381	−	0.947289i	$-0.603811\pi$
$31$	−3.29476e6	−0.640761	−0.320381	+	0.947289i	$0.603811\pi$
$32$	−1.65969e7	−2.79803
$33$	0	0
$34$	1.51985e7	1.95050
$35$	1.50062e6	0.169031
$36$	0	0
$37$	−2.13083e7	−1.86914	−0.934570	−	0.355778i	$-0.884216\pi$
$37$	−2.13083e7	−1.86914	−0.934570	+	0.355778i	$0.884216\pi$
$38$	1.49362e7	1.16202
$39$	0	0
$40$	2.15588e7	1.33154
$41$	−1.05289e7	−0.581909	−0.290954	−	0.956737i	$-0.593973\pi$
$41$	−1.05289e7	−0.581909	−0.290954	+	0.956737i	$0.593973\pi$
$42$	0	0
$43$	3.89712e6	0.173835	0.0869173	−	0.996216i	$-0.472298\pi$
$43$	3.89712e6	0.173835	0.0869173	+	0.996216i	$0.472298\pi$
$44$	−5.47991e7	−2.20413
$45$	0	0
$46$	9.76296e6	0.321493
$47$	3.31846e7	0.991964	0.495982	−	0.868333i	$-0.334808\pi$
$47$	3.31846e7	0.991964	0.495982	+	0.868333i	$0.334808\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−1.67116e7	−0.378141
$51$	0	0
$52$	1.36915e8	2.59679
$53$	6.31880e7	1.10000	0.550001	−	0.835164i	$-0.314627\pi$
$53$	6.31880e7	1.10000	0.550001	+	0.835164i	$0.314627\pi$
$54$	0	0
$55$	2.59804e7	0.382837
$56$	8.28203e7	1.12536
$57$	0	0
$58$	1.72075e8	1.99660
$59$	−5.86733e7	−0.630386	−0.315193	−	0.949028i	$-0.602069\pi$
$59$	−5.86733e7	−0.630386	−0.315193	+	0.949028i	$0.602069\pi$
$60$	0	0
$61$	1.37845e8	1.27470	0.637348	−	0.770576i	$-0.280032\pi$
$61$	1.37845e8	1.27470	0.637348	+	0.770576i	$0.280032\pi$
$62$	1.40956e8	1.21149
$63$	0	0
$64$	3.00057e8	2.23560
$65$	−6.49118e7	−0.451039
$66$	0	0
$67$	8.36796e7	0.507321	0.253660	−	0.967293i	$-0.418365\pi$
$67$	8.36796e7	0.507321	0.253660	+	0.967293i	$0.418365\pi$
$68$	−4.68327e8	−2.65619
$69$	0	0
$70$	−6.41994e7	−0.319588
$71$	−1.50169e8	−0.701322	−0.350661	−	0.936503i	$-0.614043\pi$
$71$	−1.50169e8	−0.701322	−0.350661	+	0.936503i	$0.614043\pi$
$72$	0	0
$73$	−2.74817e8	−1.13264	−0.566318	−	0.824187i	$-0.691633\pi$
$73$	−2.74817e8	−1.13264	−0.566318	+	0.824187i	$0.691633\pi$
$74$	9.11609e8	3.53399
$75$	0	0
$76$	−4.60246e8	−1.58244
$77$	9.98063e7	0.323556
$78$	0	0
$79$	−2.75964e8	−0.797133	−0.398566	−	0.917139i	$-0.630492\pi$
$79$	−2.75964e8	−0.797133	−0.398566	+	0.917139i	$0.630492\pi$
$80$	−5.00475e8	−1.36608
$81$	0	0
$82$	4.50444e8	1.10022
$83$	−2.32312e8	−0.537304	−0.268652	−	0.963237i	$-0.586578\pi$
$83$	−2.32312e8	−0.537304	−0.268652	+	0.963237i	$0.586578\pi$
$84$	0	0
$85$	2.22035e8	0.461356
$86$	−1.66726e8	−0.328670
$87$	0	0
$88$	1.43387e9	2.54882
$89$	−2.71482e8	−0.458654	−0.229327	−	0.973349i	$-0.573653\pi$
$89$	−2.71482e8	−0.458654	−0.229327	+	0.973349i	$0.573653\pi$
$90$	0	0
$91$	−2.49365e8	−0.381197
$92$	−3.00837e8	−0.437810
$93$	0	0
$94$	−1.41969e9	−1.87551
$95$	2.18204e8	0.274856
$96$	0	0
$97$	−9.76607e8	−1.12007	−0.560037	−	0.828467i	$-0.689213\pi$
$97$	−9.76607e8	−1.12007	−0.560037	+	0.828467i	$0.689213\pi$
$98$	−2.46628e8	−0.270101
$99$	0	0
$100$	5.14953e8	0.514953
$101$	−1.11904e9	−1.07004	−0.535019	−	0.844840i	$-0.679696\pi$
$101$	−1.11904e9	−1.07004	−0.535019	+	0.844840i	$0.679696\pi$
$102$	0	0
$103$	1.01154e9	0.885559	0.442779	−	0.896631i	$-0.353993\pi$
$103$	1.01154e9	0.885559	0.442779	+	0.896631i	$0.353993\pi$
$104$	−3.58252e9	−3.00289
$105$	0	0
$106$	−2.70329e9	−2.07978
$107$	1.47411e9	1.08719	0.543593	−	0.839349i	$-0.317064\pi$
$107$	1.47411e9	1.08719	0.543593	+	0.839349i	$0.317064\pi$
$108$	0	0
$109$	−1.79852e9	−1.22038	−0.610190	−	0.792255i	$-0.708907\pi$
$109$	−1.79852e9	−1.22038	−0.610190	+	0.792255i	$0.708907\pi$
$110$	−1.11149e9	−0.723832
$111$	0	0
$112$	−1.92262e9	−1.15455
$113$	−1.09680e9	−0.632812	−0.316406	−	0.948624i	$-0.602476\pi$
$113$	−1.09680e9	−0.632812	−0.316406	+	0.948624i	$0.602476\pi$
$114$	0	0
$115$	1.42627e8	0.0760436
$116$	−5.30233e9	−2.71898
$117$	0	0
$118$	2.51015e9	1.19187
$119$	8.52970e8	0.389917
$120$	0	0
$121$	−6.29995e8	−0.267179
$122$	−5.89725e9	−2.41007
$123$	0	0
$124$	−4.34342e9	−1.64981
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	−3.23678e8	−0.110407	−0.0552035	−	0.998475i	$-0.517581\pi$
$127$	−3.23678e8	−0.110407	−0.0552035	+	0.998475i	$0.517581\pi$
$128$	−4.33936e9	−1.42883
$129$	0	0
$130$	2.77704e9	0.852781
$131$	4.60246e9	1.36543	0.682714	−	0.730686i	$-0.260800\pi$
$131$	4.60246e9	1.36543	0.682714	+	0.730686i	$0.260800\pi$
$132$	0	0
$133$	8.38251e8	0.232296
$134$	−3.57996e9	−0.959194
$135$	0	0
$136$	1.22542e10	3.07158
$137$	3.07931e9	0.746812	0.373406	−	0.927668i	$-0.378190\pi$
$137$	3.07931e9	0.746812	0.373406	+	0.927668i	$0.378190\pi$
$138$	0	0
$139$	−6.80655e9	−1.54654	−0.773268	−	0.634079i	$-0.781379\pi$
$139$	−6.80655e9	−1.54654	−0.773268	+	0.634079i	$0.781379\pi$
$140$	1.97824e9	0.435215
$141$	0	0
$142$	6.42449e9	1.32599
$143$	−4.31728e9	−0.863371
$144$	0	0
$145$	2.51385e9	0.472262
$146$	1.17572e10	2.14148
$147$	0	0
$148$	−2.80904e10	−4.81260
$149$	−1.11783e10	−1.85796	−0.928981	−	0.370127i	$-0.879314\pi$
$149$	−1.11783e10	−1.85796	−0.928981	+	0.370127i	$0.879314\pi$
$150$	0	0
$151$	−4.87685e9	−0.763384	−0.381692	−	0.924290i	$-0.624658\pi$
$151$	−4.87685e9	−0.763384	−0.381692	+	0.924290i	$0.624658\pi$
$152$	1.20428e10	1.82991
$153$	0	0
$154$	−4.26989e9	−0.611750
$155$	2.05923e9	0.286557
$156$	0	0
$157$	−5.01169e9	−0.658318	−0.329159	−	0.944274i	$-0.606765\pi$
$157$	−5.01169e9	−0.658318	−0.329159	+	0.944274i	$0.606765\pi$
$158$	1.18062e10	1.50714
$159$	0	0
$160$	1.03731e10	1.25132
$161$	5.47917e8	0.0642686
$162$	0	0
$163$	−1.45416e10	−1.61349	−0.806746	−	0.590899i	$-0.798773\pi$
$163$	−1.45416e10	−1.61349	−0.806746	+	0.590899i	$0.798773\pi$
$164$	−1.38800e10	−1.49828
$165$	0	0
$166$	9.93872e9	1.01588
$167$	4.62247e8	0.0459886	0.0229943	−	0.999736i	$-0.492680\pi$
$167$	4.62247e8	0.0459886	0.0229943	+	0.999736i	$0.492680\pi$
$168$	0	0
$169$	1.82180e8	0.0171795
$170$	−9.49906e9	−0.872289
$171$	0	0
$172$	5.13750e9	0.447583
$173$	−1.02318e9	−0.0868449	−0.0434224	−	0.999057i	$-0.513826\pi$
$173$	−1.02318e9	−0.0868449	−0.0434224	+	0.999057i	$0.513826\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−3.32865e10	−2.61494
$177$	0	0
$178$	1.16145e10	0.867180
$179$	−1.42460e10	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$179$	−1.42460e10	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$180$	0	0
$181$	1.83877e10	1.27343	0.636714	−	0.771100i	$-0.280293\pi$
$181$	1.83877e10	1.27343	0.636714	+	0.771100i	$0.280293\pi$
$182$	1.06683e10	0.720732
$183$	0	0
$184$	7.87168e9	0.506276
$185$	1.33177e10	0.835905
$186$	0	0
$187$	1.47675e10	0.883121
$188$	4.37466e10	2.55408
$189$	0	0
$190$	−9.33514e9	−0.519672
$191$	2.65507e10	1.44353	0.721766	−	0.692137i	$-0.243331\pi$
$191$	2.65507e10	1.44353	0.721766	+	0.692137i	$0.243331\pi$
$192$	0	0
$193$	5.84127e9	0.303040	0.151520	−	0.988454i	$-0.451583\pi$
$193$	5.84127e9	0.303040	0.151520	+	0.988454i	$0.451583\pi$
$194$	4.17810e10	2.11773
$195$	0	0
$196$	7.59962e9	0.367824
$197$	2.06491e10	0.976793	0.488397	−	0.872622i	$-0.337582\pi$
$197$	2.06491e10	0.976793	0.488397	+	0.872622i	$0.337582\pi$
$198$	0	0
$199$	−2.66301e10	−1.20374	−0.601871	−	0.798593i	$-0.705578\pi$
$199$	−2.66301e10	−1.20374	−0.601871	+	0.798593i	$0.705578\pi$
$200$	−1.34743e10	−0.595484
$201$	0	0
$202$	4.78745e10	2.02313
$203$	9.65720e9	0.399134
$204$	0	0
$205$	6.58055e9	0.260238
$206$	−4.32756e10	−1.67433
$207$	0	0
$208$	8.31660e10	3.08078
$209$	1.45127e10	0.526126
$210$	0	0
$211$	1.35165e10	0.469454	0.234727	−	0.972061i	$-0.424580\pi$
$211$	1.35165e10	0.469454	0.234727	+	0.972061i	$0.424580\pi$
$212$	8.32995e10	2.83225
$213$	0	0
$214$	−6.30652e10	−2.05555
$215$	−2.43570e9	−0.0777412
$216$	0	0
$217$	7.91072e9	0.242185
$218$	7.69437e10	2.30738
$219$	0	0
$220$	3.42495e10	0.985716
$221$	−3.68965e10	−1.04045
$222$	0	0
$223$	4.47296e10	1.21122	0.605610	−	0.795762i	$-0.292929\pi$
$223$	4.47296e10	1.21122	0.605610	+	0.795762i	$0.292929\pi$
$224$	3.98492e10	1.05756
$225$	0	0
$226$	4.69231e10	1.19646
$227$	−2.46775e10	−0.616858	−0.308429	−	0.951247i	$-0.599803\pi$
$227$	−2.46775e10	−0.616858	−0.308429	+	0.951247i	$0.599803\pi$
$228$	0	0
$229$	−5.44437e10	−1.30824	−0.654121	−	0.756390i	$-0.726961\pi$
$229$	−5.44437e10	−1.30824	−0.654121	+	0.756390i	$0.726961\pi$
$230$	−6.10185e9	−0.143776
$231$	0	0
$232$	1.38741e11	3.14418
$233$	5.82731e10	1.29529	0.647644	−	0.761943i	$-0.275754\pi$
$233$	5.82731e10	1.29529	0.647644	+	0.761943i	$0.275754\pi$
$234$	0	0
$235$	−2.07404e10	−0.443620
$236$	−7.73478e10	−1.62310
$237$	0	0
$238$	−3.64916e10	−0.737219
$239$	−5.53754e10	−1.09781	−0.548904	−	0.835885i	$-0.684955\pi$
$239$	−5.53754e10	−1.09781	−0.548904	+	0.835885i	$0.684955\pi$
$240$	0	0
$241$	6.01280e10	1.14815	0.574077	−	0.818802i	$-0.305361\pi$
$241$	6.01280e10	1.14815	0.574077	+	0.818802i	$0.305361\pi$
$242$	2.69523e10	0.505157
$243$	0	0
$244$	1.81718e11	3.28204
$245$	−3.60300e9	−0.0638877
$246$	0	0
$247$	−3.62598e10	−0.619854
$248$	1.13650e11	1.90781
$249$	0	0
$250$	1.04448e10	0.169110
$251$	−1.01611e9	−0.0161588	−0.00807938	−	0.999967i	$-0.502572\pi$
$251$	−1.01611e9	−0.0161588	−0.00807938	+	0.999967i	$0.502572\pi$
$252$	0	0
$253$	9.48612e9	0.145561
$254$	1.38475e10	0.208747
$255$	0	0
$256$	3.20161e10	0.465895
$257$	1.10965e11	1.58667	0.793334	−	0.608787i	$-0.208343\pi$
$257$	1.10965e11	1.58667	0.793334	+	0.608787i	$0.208343\pi$
$258$	0	0
$259$	5.11613e10	0.706469
$260$	−8.55720e10	−1.16132
$261$	0	0
$262$	−1.96901e11	−2.58162
$263$	−1.21172e11	−1.56171	−0.780856	−	0.624711i	$-0.785217\pi$
$263$	−1.21172e11	−1.56171	−0.780856	+	0.624711i	$0.785217\pi$
$264$	0	0
$265$	−3.94925e10	−0.491936
$266$	−3.58619e10	−0.439203
$267$	0	0
$268$	1.10313e11	1.30623
$269$	−5.16654e10	−0.601609	−0.300805	−	0.953686i	$-0.597255\pi$
$269$	−5.16654e10	−0.601609	−0.300805	+	0.953686i	$0.597255\pi$
$270$	0	0
$271$	−2.49166e10	−0.280626	−0.140313	−	0.990107i	$-0.544811\pi$
$271$	−2.49166e10	−0.280626	−0.140313	+	0.990107i	$0.544811\pi$
$272$	−2.84475e11	−3.15126
$273$	0	0
$274$	−1.31738e11	−1.41200
$275$	−1.62378e10	−0.171210
$276$	0	0
$277$	−2.22882e10	−0.227466	−0.113733	−	0.993511i	$-0.536281\pi$
$277$	−2.22882e10	−0.227466	−0.113733	+	0.993511i	$0.536281\pi$
$278$	2.91196e11	2.92404
$279$	0	0
$280$	−5.17627e10	−0.503276
$281$	−9.03151e10	−0.864135	−0.432068	−	0.901841i	$-0.642216\pi$
$281$	−9.03151e10	−0.864135	−0.432068	+	0.901841i	$0.642216\pi$
$282$	0	0
$283$	−9.63268e10	−0.892706	−0.446353	−	0.894857i	$-0.647277\pi$
$283$	−9.63268e10	−0.892706	−0.446353	+	0.894857i	$0.647277\pi$
$284$	−1.97965e11	−1.80574
$285$	0	0
$286$	1.84701e11	1.63238
$287$	2.52798e10	0.219941
$288$	0	0
$289$	7.61912e9	0.0642487
$290$	−1.07547e11	−0.892908
$291$	0	0
$292$	−3.62286e11	−2.91627
$293$	4.21047e10	0.333754	0.166877	−	0.985978i	$-0.446632\pi$
$293$	4.21047e10	0.333754	0.166877	+	0.985978i	$0.446632\pi$
$294$	0	0
$295$	3.66708e10	0.281917
$296$	7.35012e11	5.56521
$297$	0	0
$298$	4.78227e11	3.51286
$299$	−2.37010e10	−0.171493
$300$	0	0
$301$	−9.35700e9	−0.0657033
$302$	2.08640e11	1.44333
$303$	0	0
$304$	−2.79566e11	−1.87738
$305$	−8.61531e10	−0.570061
$306$	0	0
$307$	9.28980e10	0.596875	0.298438	−	0.954429i	$-0.403534\pi$
$307$	9.28980e10	0.596875	0.298438	+	0.954429i	$0.403534\pi$
$308$	1.31573e11	0.833082
$309$	0	0
$310$	−8.80973e10	−0.541795
$311$	1.56495e11	0.948588	0.474294	−	0.880366i	$-0.342703\pi$
$311$	1.56495e11	0.948588	0.474294	+	0.880366i	$0.342703\pi$
$312$	0	0
$313$	8.88488e10	0.523242	0.261621	−	0.965171i	$-0.415743\pi$
$313$	8.88488e10	0.523242	0.261621	+	0.965171i	$0.415743\pi$
$314$	2.14409e11	1.24469
$315$	0	0
$316$	−3.63798e11	−2.05243
$317$	−6.19880e10	−0.344779	−0.172389	−	0.985029i	$-0.555149\pi$
$317$	−6.19880e10	−0.344779	−0.172389	+	0.985029i	$0.555149\pi$
$318$	0	0
$319$	1.67196e11	0.903997
$320$	−1.87536e11	−0.999791
$321$	0	0
$322$	−2.34409e10	−0.121513
$323$	1.24029e11	0.634034
$324$	0	0
$325$	4.05699e10	0.201711
$326$	6.22114e11	3.05064
$327$	0	0
$328$	3.63184e11	1.73259
$329$	−7.96762e10	−0.374927
$330$	0	0
$331$	3.65322e11	1.67282	0.836410	−	0.548104i	$-0.184650\pi$
$331$	3.65322e11	1.67282	0.836410	+	0.548104i	$0.184650\pi$
$332$	−3.06252e11	−1.38343
$333$	0	0
$334$	−1.97757e10	−0.0869508
$335$	−5.22997e10	−0.226881
$336$	0	0
$337$	1.45738e11	0.615514	0.307757	−	0.951465i	$-0.400422\pi$
$337$	1.45738e11	0.615514	0.307757	+	0.951465i	$0.400422\pi$
$338$	−7.79400e9	−0.0324815
$339$	0	0
$340$	2.92705e11	1.18788
$341$	1.36959e11	0.548523
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−1.34428e11	−0.517578
$345$	0	0
$346$	4.37734e10	0.164198
$347$	−2.96112e11	−1.09641	−0.548206	−	0.836343i	$-0.684689\pi$
$347$	−2.96112e11	−1.09641	−0.548206	+	0.836343i	$0.684689\pi$
$348$	0	0
$349$	3.01854e11	1.08914	0.544569	−	0.838716i	$-0.316693\pi$
$349$	3.01854e11	1.08914	0.544569	+	0.838716i	$0.316693\pi$
$350$	4.01246e10	0.142924
$351$	0	0
$352$	6.89912e11	2.39526
$353$	1.99229e11	0.682916	0.341458	−	0.939897i	$-0.389079\pi$
$353$	1.99229e11	0.682916	0.341458	+	0.939897i	$0.389079\pi$
$354$	0	0
$355$	9.38555e10	0.313641
$356$	−3.57889e11	−1.18093
$357$	0	0
$358$	6.09469e11	1.96100
$359$	6.48945e10	0.206197	0.103099	−	0.994671i	$-0.467124\pi$
$359$	6.48945e10	0.206197	0.103099	+	0.994671i	$0.467124\pi$
$360$	0	0
$361$	−2.00799e11	−0.622270
$362$	−7.86660e11	−2.40768
$363$	0	0
$364$	−3.28733e11	−0.981494
$365$	1.71761e11	0.506530
$366$	0	0
$367$	−4.27450e10	−0.122995	−0.0614976	−	0.998107i	$-0.519588\pi$
$367$	−4.27450e10	−0.122995	−0.0614976	+	0.998107i	$0.519588\pi$
$368$	−1.82736e11	−0.519409
$369$	0	0
$370$	−5.69756e11	−1.58045
$371$	−1.51714e11	−0.415761
$372$	0	0
$373$	5.72742e11	1.53204	0.766018	−	0.642819i	$-0.222235\pi$
$373$	5.72742e11	1.53204	0.766018	+	0.642819i	$0.222235\pi$
$374$	−6.31781e11	−1.66972
$375$	0	0
$376$	−1.14467e12	−2.95349
$377$	−4.17737e11	−1.06504
$378$	0	0
$379$	−2.28088e11	−0.567840	−0.283920	−	0.958848i	$-0.591635\pi$
$379$	−2.28088e11	−0.567840	−0.283920	+	0.958848i	$0.591635\pi$
$380$	2.87654e11	0.707691
$381$	0	0
$382$	−1.13589e12	−2.72929
$383$	−8.33542e11	−1.97940	−0.989699	−	0.143162i	$-0.954273\pi$
$383$	−8.33542e11	−1.97940	−0.989699	+	0.143162i	$0.954273\pi$
$384$	0	0
$385$	−6.23790e10	−0.144699
$386$	−2.49900e11	−0.572959
$387$	0	0
$388$	−1.28744e12	−2.88393
$389$	3.47070e11	0.768501	0.384251	−	0.923229i	$-0.374460\pi$
$389$	3.47070e11	0.768501	0.384251	+	0.923229i	$0.374460\pi$
$390$	0	0
$391$	8.10708e10	0.175416
$392$	−1.98852e11	−0.425346
$393$	0	0
$394$	−8.83404e11	−1.84683
$395$	1.72478e11	0.356489
$396$	0	0
$397$	5.31999e11	1.07486	0.537432	−	0.843307i	$-0.319394\pi$
$397$	5.31999e11	1.07486	0.537432	+	0.843307i	$0.319394\pi$
$398$	1.13928e12	2.27592
$399$	0	0
$400$	3.12797e11	0.610931
$401$	1.10007e11	0.212456	0.106228	−	0.994342i	$-0.466123\pi$
$401$	1.10007e11	0.212456	0.106228	+	0.994342i	$0.466123\pi$
$402$	0	0
$403$	−3.42190e11	−0.646242
$404$	−1.47521e12	−2.75510
$405$	0	0
$406$	−4.13152e11	−0.754645
$407$	8.85759e11	1.60008
$408$	0	0
$409$	5.36319e11	0.947694	0.473847	−	0.880607i	$-0.342865\pi$
$409$	5.36319e11	0.947694	0.473847	+	0.880607i	$0.342865\pi$
$410$	−2.81528e11	−0.492032
$411$	0	0
$412$	1.33350e12	2.28011
$413$	1.40875e11	0.238263
$414$	0	0
$415$	1.45195e11	0.240290
$416$	−1.72374e12	−2.82197
$417$	0	0
$418$	−6.20879e11	−0.994749
$419$	−2.77847e11	−0.440395	−0.220198	−	0.975455i	$-0.570670\pi$
$419$	−2.77847e11	−0.440395	−0.220198	+	0.975455i	$0.570670\pi$
$420$	0	0
$421$	5.42301e11	0.841339	0.420670	−	0.907214i	$-0.361795\pi$
$421$	5.42301e11	0.841339	0.420670	+	0.907214i	$0.361795\pi$
$422$	−5.78260e11	−0.887599
$423$	0	0
$424$	−2.17961e12	−3.27516
$425$	−1.38772e11	−0.206325
$426$	0	0
$427$	−3.30966e11	−0.481790
$428$	1.94329e12	2.79925
$429$	0	0
$430$	1.04204e11	0.146986
$431$	5.57666e11	0.778442	0.389221	−	0.921144i	$-0.372744\pi$
$431$	5.57666e11	0.778442	0.389221	+	0.921144i	$0.372744\pi$
$432$	0	0
$433$	2.13337e11	0.291656	0.145828	−	0.989310i	$-0.453415\pi$
$433$	2.13337e11	0.291656	0.145828	+	0.989310i	$0.453415\pi$
$434$	−3.38435e11	−0.457900
$435$	0	0
$436$	−2.37095e12	−3.14219
$437$	7.96718e10	0.104505
$438$	0	0
$439$	−1.05316e12	−1.35334	−0.676668	−	0.736289i	$-0.736577\pi$
$439$	−1.05316e12	−1.35334	−0.676668	+	0.736289i	$0.736577\pi$
$440$	−8.96171e11	−1.13987
$441$	0	0
$442$	1.57850e12	1.96718
$443$	7.98551e11	0.985113	0.492556	−	0.870281i	$-0.336063\pi$
$443$	7.98551e11	0.985113	0.492556	+	0.870281i	$0.336063\pi$
$444$	0	0
$445$	1.69676e11	0.205117
$446$	−1.91361e12	−2.29006
$447$	0	0
$448$	−7.20438e11	−0.844978
$449$	9.80369e11	1.13836	0.569182	−	0.822211i	$-0.307260\pi$
$449$	9.80369e11	1.13836	0.569182	+	0.822211i	$0.307260\pi$
$450$	0	0
$451$	4.37671e11	0.498143
$452$	−1.44589e12	−1.62934
$453$	0	0
$454$	1.05575e12	1.16630
$455$	1.55853e11	0.170477
$456$	0	0
$457$	7.38737e11	0.792259	0.396130	−	0.918195i	$-0.370353\pi$
$457$	7.38737e11	0.792259	0.396130	+	0.918195i	$0.370353\pi$
$458$	2.32920e12	2.47350
$459$	0	0
$460$	1.88023e11	0.195794
$461$	8.57026e11	0.883771	0.441885	−	0.897072i	$-0.354310\pi$
$461$	8.57026e11	0.883771	0.441885	+	0.897072i	$0.354310\pi$
$462$	0	0
$463$	−8.66003e11	−0.875800	−0.437900	−	0.899024i	$-0.644278\pi$
$463$	−8.66003e11	−0.875800	−0.437900	+	0.899024i	$0.644278\pi$
$464$	−3.22078e12	−3.22575
$465$	0	0
$466$	−2.49303e12	−2.44901
$467$	1.74446e12	1.69721	0.848606	−	0.529025i	$-0.177442\pi$
$467$	1.74446e12	1.69721	0.848606	+	0.529025i	$0.177442\pi$
$468$	0	0
$469$	−2.00915e11	−0.191749
$470$	8.87309e11	0.838754
$471$	0	0
$472$	2.02388e12	1.87692
$473$	−1.61998e11	−0.148811
$474$	0	0
$475$	−1.36377e11	−0.122919
$476$	1.12445e12	1.00395
$477$	0	0
$478$	2.36906e12	2.07563
$479$	6.51076e11	0.565096	0.282548	−	0.959253i	$-0.408820\pi$
$479$	6.51076e11	0.565096	0.282548	+	0.959253i	$0.408820\pi$
$480$	0	0
$481$	−2.21306e12	−1.88513
$482$	−2.57238e12	−2.17082
$483$	0	0
$484$	−8.30510e11	−0.687924
$485$	6.10379e11	0.500913
$486$	0	0
$487$	9.91101e8	0.000798432 0	0.000399216	−	1.00000i	$-0.499873\pi$
$487$	9.91101e8	0.000798432 0	0.000399216		1.00000i	$0.499873\pi$
$488$	−4.75484e12	−3.79530
$489$	0	0
$490$	1.54143e11	0.120793
$491$	−8.38408e11	−0.651012	−0.325506	−	0.945540i	$-0.605535\pi$
$491$	−8.38408e11	−0.651012	−0.325506	+	0.945540i	$0.605535\pi$
$492$	0	0
$493$	1.42890e12	1.08941
$494$	1.55126e12	1.17196
$495$	0	0
$496$	−2.63831e12	−1.95730
$497$	3.60555e11	0.265075
$498$	0	0
$499$	2.48735e12	1.79591	0.897955	−	0.440088i	$-0.145053\pi$
$499$	2.48735e12	1.79591	0.897955	+	0.440088i	$0.145053\pi$
$500$	−3.21846e11	−0.230294
$501$	0	0
$502$	4.34709e10	0.0305515
$503$	1.40863e12	0.981163	0.490582	−	0.871395i	$-0.336784\pi$
$503$	1.40863e12	0.981163	0.490582	+	0.871395i	$0.336784\pi$
$504$	0	0
$505$	6.99400e11	0.478536
$506$	−4.05833e11	−0.275214
$507$	0	0
$508$	−4.26699e11	−0.284272
$509$	8.28720e11	0.547240	0.273620	−	0.961838i	$-0.411779\pi$
$509$	8.28720e11	0.547240	0.273620	+	0.961838i	$0.411779\pi$
$510$	0	0
$511$	6.59835e11	0.428096
$512$	8.52047e11	0.547960
$513$	0	0
$514$	−4.74727e12	−2.99992
$515$	−6.32215e11	−0.396034
$516$	0	0
$517$	−1.37944e12	−0.849170
$518$	−2.18877e12	−1.33572
$519$	0	0
$520$	2.23908e12	1.34293
$521$	8.58369e11	0.510393	0.255196	−	0.966889i	$-0.417860\pi$
$521$	8.58369e11	0.510393	0.255196	+	0.966889i	$0.417860\pi$
$522$	0	0
$523$	1.43567e12	0.839069	0.419534	−	0.907739i	$-0.362193\pi$
$523$	1.43567e12	0.839069	0.419534	+	0.907739i	$0.362193\pi$
$524$	6.06733e12	3.51566
$525$	0	0
$526$	5.18395e12	2.95274
$527$	1.17048e12	0.661025
$528$	0	0
$529$	−1.74908e12	−0.971087
$530$	1.68956e12	0.930105
$531$	0	0
$532$	1.10505e12	0.598108
$533$	−1.09352e12	−0.586886
$534$	0	0
$535$	−9.21320e11	−0.486204
$536$	−2.88645e12	−1.51051
$537$	0	0
$538$	2.21034e12	1.13747
$539$	−2.39635e11	−0.122293
$540$	0	0
$541$	3.25599e12	1.63416	0.817081	−	0.576523i	$-0.195591\pi$
$541$	3.25599e12	1.63416	0.817081	+	0.576523i	$0.195591\pi$
$542$	1.06598e12	0.530581
$543$	0	0
$544$	5.89616e12	2.88652
$545$	1.12407e12	0.545770
$546$	0	0
$547$	−1.31051e12	−0.625887	−0.312944	−	0.949772i	$-0.601315\pi$
$547$	−1.31051e12	−0.625887	−0.312944	+	0.949772i	$0.601315\pi$
$548$	4.05940e12	1.92287
$549$	0	0
$550$	6.94680e11	0.323707
$551$	1.40424e12	0.649021
$552$	0	0
$553$	6.62590e11	0.301288
$554$	9.53528e11	0.430070
$555$	0	0
$556$	−8.97294e12	−3.98197
$557$	−6.12448e11	−0.269600	−0.134800	−	0.990873i	$-0.543039\pi$
$557$	−6.12448e11	−0.269600	−0.134800	+	0.990873i	$0.543039\pi$
$558$	0	0
$559$	4.04751e11	0.175321
$560$	1.20164e12	0.516331
$561$	0	0
$562$	3.86384e12	1.63383
$563$	−2.74733e12	−1.15245	−0.576227	−	0.817290i	$-0.695476\pi$
$563$	−2.74733e12	−1.15245	−0.576227	+	0.817290i	$0.695476\pi$
$564$	0	0
$565$	6.85500e11	0.283002
$566$	4.12103e12	1.68784
$567$	0	0
$568$	5.17994e12	2.08813
$569$	7.10003e11	0.283958	0.141979	−	0.989870i	$-0.454653\pi$
$569$	7.10003e11	0.283958	0.141979	+	0.989870i	$0.454653\pi$
$570$	0	0
$571$	2.87190e12	1.13060	0.565298	−	0.824887i	$-0.308761\pi$
$571$	2.87190e12	1.13060	0.565298	+	0.824887i	$0.308761\pi$
$572$	−5.69138e12	−2.22298
$573$	0	0
$574$	−1.08152e12	−0.415843
$575$	−8.91421e10	−0.0340077
$576$	0	0
$577$	−5.13479e12	−1.92855	−0.964277	−	0.264897i	$-0.914662\pi$
$577$	−5.13479e12	−1.92855	−0.964277	+	0.264897i	$0.914662\pi$
$578$	−3.25959e11	−0.121475
$579$	0	0
$580$	3.31396e12	1.21596
$581$	5.57781e11	0.203082
$582$	0	0
$583$	−2.62664e12	−0.941656
$584$	9.47956e12	3.37233
$585$	0	0
$586$	−1.80131e12	−0.631030
$587$	−2.46529e12	−0.857031	−0.428515	−	0.903535i	$-0.640963\pi$
$587$	−2.46529e12	−0.857031	−0.428515	+	0.903535i	$0.640963\pi$
$588$	0	0
$589$	1.15029e12	0.393810
$590$	−1.56884e12	−0.533022
$591$	0	0
$592$	−1.70629e13	−5.70958
$593$	1.51279e12	0.502379	0.251189	−	0.967938i	$-0.419178\pi$
$593$	1.51279e12	0.502379	0.251189	+	0.967938i	$0.419178\pi$
$594$	0	0
$595$	−5.33106e11	−0.174376
$596$	−1.47361e13	−4.78382
$597$	0	0
$598$	1.01397e12	0.324243
$599$	5.89069e12	1.86959	0.934794	−	0.355191i	$-0.115584\pi$
$599$	5.89069e12	1.86959	0.934794	+	0.355191i	$0.115584\pi$
$600$	0	0
$601$	−3.78551e12	−1.18356	−0.591779	−	0.806101i	$-0.701574\pi$
$601$	−3.78551e12	−1.18356	−0.591779	+	0.806101i	$0.701574\pi$
$602$	4.00309e11	0.124226
$603$	0	0
$604$	−6.42905e12	−1.96553
$605$	3.93747e11	0.119486
$606$	0	0
$607$	4.65336e12	1.39129	0.695645	−	0.718386i	$-0.255119\pi$
$607$	4.65336e12	1.39129	0.695645	+	0.718386i	$0.255119\pi$
$608$	5.79442e12	1.71966
$609$	0	0
$610$	3.68578e12	1.07782
$611$	3.44651e12	1.00045
$612$	0	0
$613$	5.65330e12	1.61707	0.808537	−	0.588445i	$-0.200260\pi$
$613$	5.65330e12	1.61707	0.808537	+	0.588445i	$0.200260\pi$
$614$	−3.97434e12	−1.12852
$615$	0	0
$616$	−3.44273e12	−0.963363
$617$	4.91335e12	1.36488	0.682441	−	0.730941i	$-0.260919\pi$
$617$	4.91335e12	1.36488	0.682441	+	0.730941i	$0.260919\pi$
$618$	0	0
$619$	1.31898e12	0.361101	0.180551	−	0.983566i	$-0.442212\pi$
$619$	1.31898e12	0.361101	0.180551	+	0.983566i	$0.442212\pi$
$620$	2.71464e12	0.737818
$621$	0	0
$622$	−6.69512e12	−1.79350
$623$	6.51828e11	0.173355
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−3.80111e12	−0.989296
$627$	0	0
$628$	−6.60681e12	−1.69502
$629$	7.56992e12	1.92825
$630$	0	0
$631$	−1.26560e12	−0.317808	−0.158904	−	0.987294i	$-0.550796\pi$
$631$	−1.26560e12	−0.317808	−0.158904	+	0.987294i	$0.550796\pi$
$632$	9.51913e12	2.37340
$633$	0	0
$634$	2.65196e12	0.651875
$635$	2.02299e11	0.0493755
$636$	0	0
$637$	5.98726e11	0.144079
$638$	−7.15293e12	−1.70919
$639$	0	0
$640$	2.71210e12	0.638992
$641$	−7.10592e12	−1.66249	−0.831246	−	0.555905i	$-0.812372\pi$
$641$	−7.10592e12	−1.66249	−0.831246	+	0.555905i	$0.812372\pi$
$642$	0	0
$643$	4.17010e12	0.962049	0.481024	−	0.876707i	$-0.340265\pi$
$643$	4.17010e12	0.962049	0.481024	+	0.876707i	$0.340265\pi$
$644$	7.22309e11	0.165477
$645$	0	0
$646$	−5.30619e12	−1.19877
$647$	5.36916e12	1.20458	0.602292	−	0.798276i	$-0.294254\pi$
$647$	5.36916e12	1.20458	0.602292	+	0.798276i	$0.294254\pi$
$648$	0	0
$649$	2.43897e12	0.539641
$650$	−1.73565e12	−0.381375
$651$	0	0
$652$	−1.91699e13	−4.15436
$653$	−2.71691e12	−0.584744	−0.292372	−	0.956305i	$-0.594445\pi$
$653$	−2.71691e12	−0.584744	−0.292372	+	0.956305i	$0.594445\pi$
$654$	0	0
$655$	−2.87653e12	−0.610638
$656$	−8.43110e12	−1.77753
$657$	0	0
$658$	3.40869e12	0.708877
$659$	−6.36955e12	−1.31560	−0.657801	−	0.753192i	$-0.728513\pi$
$659$	−6.36955e12	−1.31560	−0.657801	+	0.753192i	$0.728513\pi$
$660$	0	0
$661$	−5.58167e12	−1.13725	−0.568627	−	0.822596i	$-0.692525\pi$
$661$	−5.58167e12	−1.13725	−0.568627	+	0.822596i	$0.692525\pi$
$662$	−1.56291e13	−3.16281
$663$	0	0
$664$	8.01340e12	1.59978
$665$	−5.23907e11	−0.103886
$666$	0	0
$667$	9.17871e11	0.179562
$668$	6.09371e11	0.118410
$669$	0	0
$670$	2.23748e12	0.428965
$671$	−5.73003e12	−1.09120
$672$	0	0
$673$	4.49826e12	0.845234	0.422617	−	0.906308i	$-0.361112\pi$
$673$	4.49826e12	0.845234	0.422617	+	0.906308i	$0.361112\pi$
$674$	−6.23492e12	−1.16376
$675$	0	0
$676$	2.40165e11	0.0442333
$677$	−5.35625e12	−0.979969	−0.489984	−	0.871731i	$-0.662998\pi$
$677$	−5.35625e12	−0.979969	−0.489984	+	0.871731i	$0.662998\pi$
$678$	0	0
$679$	2.34483e12	0.423349
$680$	−7.65890e12	−1.37365
$681$	0	0
$682$	−5.85934e12	−1.03710
$683$	2.20240e12	0.387261	0.193630	−	0.981075i	$-0.437974\pi$
$683$	2.20240e12	0.387261	0.193630	+	0.981075i	$0.437974\pi$
$684$	0	0
$685$	−1.92457e12	−0.333984
$686$	5.92155e11	0.102088
$687$	0	0
$688$	3.12066e12	0.531005
$689$	6.56264e12	1.10941
$690$	0	0
$691$	4.30201e12	0.717827	0.358913	−	0.933371i	$-0.383147\pi$
$691$	4.30201e12	0.717827	0.358913	+	0.933371i	$0.383147\pi$
$692$	−1.34884e12	−0.223605
$693$	0	0
$694$	1.26682e13	2.07299
$695$	4.25409e12	0.691632
$696$	0	0
$697$	3.74045e12	0.600311
$698$	−1.29139e13	−2.05924
$699$	0	0
$700$	−1.23640e12	−0.194634
$701$	−9.24701e12	−1.44634	−0.723170	−	0.690670i	$-0.757316\pi$
$701$	−9.24701e12	−1.44634	−0.723170	+	0.690670i	$0.757316\pi$
$702$	0	0
$703$	7.43929e12	1.14877
$704$	−1.24730e13	−1.91379
$705$	0	0
$706$	−8.52339e12	−1.29119
$707$	2.68681e12	0.404436
$708$	0	0
$709$	−1.08956e13	−1.61936	−0.809680	−	0.586872i	$-0.800359\pi$
$709$	−1.08956e13	−1.61936	−0.809680	+	0.586872i	$0.800359\pi$
$710$	−4.01531e12	−0.593002
$711$	0	0
$712$	9.36452e12	1.36561
$713$	7.51877e11	0.108954
$714$	0	0
$715$	2.69830e12	0.386111
$716$	−1.87802e13	−2.67050
$717$	0	0
$718$	−2.77630e12	−0.389858
$719$	6.34998e12	0.886120	0.443060	−	0.896492i	$-0.353893\pi$
$719$	6.34998e12	0.886120	0.443060	+	0.896492i	$0.353893\pi$
$720$	0	0
$721$	−2.42872e12	−0.334710
$722$	8.59053e12	1.17653
$723$	0	0
$724$	2.42402e13	3.27878
$725$	−1.57115e12	−0.211202
$726$	0	0
$727$	4.81017e12	0.638639	0.319319	−	0.947647i	$-0.396546\pi$
$727$	4.81017e12	0.638639	0.319319	+	0.947647i	$0.396546\pi$
$728$	8.60163e12	1.13498
$729$	0	0
$730$	−7.34822e12	−0.957700
$731$	−1.38448e12	−0.179332
$732$	0	0
$733$	−1.10839e13	−1.41816	−0.709082	−	0.705126i	$-0.750890\pi$
$733$	−1.10839e13	−1.41816	−0.709082	+	0.705126i	$0.750890\pi$
$734$	1.82871e12	0.232548
$735$	0	0
$736$	3.78748e12	0.475774
$737$	−3.47845e12	−0.434292
$738$	0	0
$739$	−1.33603e13	−1.64784	−0.823922	−	0.566704i	$-0.808218\pi$
$739$	−1.33603e13	−1.64784	−0.823922	+	0.566704i	$0.808218\pi$
$740$	1.75565e13	2.15226
$741$	0	0
$742$	6.49061e12	0.786082
$743$	−2.12151e12	−0.255384	−0.127692	−	0.991814i	$-0.540757\pi$
$743$	−2.12151e12	−0.255384	−0.127692	+	0.991814i	$0.540757\pi$
$744$	0	0
$745$	6.98643e12	0.830906
$746$	−2.45029e13	−2.89663
$747$	0	0
$748$	1.94677e13	2.27383
$749$	−3.53934e12	−0.410918
$750$	0	0
$751$	2.36099e12	0.270841	0.135420	−	0.990788i	$-0.456762\pi$
$751$	2.36099e12	0.270841	0.135420	+	0.990788i	$0.456762\pi$
$752$	2.65729e13	3.03011
$753$	0	0
$754$	1.78715e13	2.01368
$755$	3.04803e12	0.341396
$756$	0	0
$757$	6.53431e12	0.723217	0.361608	−	0.932330i	$-0.382228\pi$
$757$	6.53431e12	0.723217	0.361608	+	0.932330i	$0.382228\pi$
$758$	9.75801e12	1.07362
$759$	0	0
$760$	−7.52674e12	−0.818363
$761$	7.47260e12	0.807682	0.403841	−	0.914829i	$-0.367675\pi$
$761$	7.47260e12	0.807682	0.403841	+	0.914829i	$0.367675\pi$
$762$	0	0
$763$	4.31824e12	0.461260
$764$	3.50013e13	3.71676
$765$	0	0
$766$	3.56604e13	3.74246
$767$	−6.09374e12	−0.635777
$768$	0	0
$769$	3.53111e12	0.364119	0.182059	−	0.983288i	$-0.441724\pi$
$769$	3.53111e12	0.364119	0.182059	+	0.983288i	$0.441724\pi$
$770$	2.66868e12	0.273583
$771$	0	0
$772$	7.70043e12	0.780256
$773$	−4.38854e12	−0.442092	−0.221046	−	0.975263i	$-0.570947\pi$
$773$	−4.38854e12	−0.442092	−0.221046	+	0.975263i	$0.570947\pi$
$774$	0	0
$775$	−1.28702e12	−0.128152
$776$	3.36872e13	3.33493
$777$	0	0
$778$	−1.48483e13	−1.45301
$779$	3.67591e12	0.357640
$780$	0	0
$781$	6.24232e12	0.600366
$782$	−3.46835e12	−0.331660
$783$	0	0
$784$	4.61622e12	0.436379
$785$	3.13231e12	0.294409
$786$	0	0
$787$	−8.78357e12	−0.816178	−0.408089	−	0.912942i	$-0.633805\pi$
$787$	−8.78357e12	−0.816178	−0.408089	+	0.912942i	$0.633805\pi$
$788$	2.72213e13	2.51501
$789$	0	0
$790$	−7.37890e12	−0.674015
$791$	2.63342e12	0.239180
$792$	0	0
$793$	1.43164e13	1.28560
$794$	−2.27599e13	−2.03225
$795$	0	0
$796$	−3.51059e13	−3.09935
$797$	3.11376e12	0.273352	0.136676	−	0.990616i	$-0.456358\pi$
$797$	3.11376e12	0.273352	0.136676	+	0.990616i	$0.456358\pi$
$798$	0	0
$799$	−1.17890e13	−1.02333
$800$	−6.48318e12	−0.559607
$801$	0	0
$802$	−4.70628e12	−0.401692
$803$	1.14238e13	0.969593
$804$	0	0
$805$	−3.42448e11	−0.0287418
$806$	1.46395e13	1.22185
$807$	0	0
$808$	3.86003e13	3.18595
$809$	−2.27913e13	−1.87068	−0.935341	−	0.353747i	$-0.884907\pi$
$809$	−2.27913e13	−1.87068	−0.935341	+	0.353747i	$0.884907\pi$
$810$	0	0
$811$	6.23133e12	0.505810	0.252905	−	0.967491i	$-0.418614\pi$
$811$	6.23133e12	0.505810	0.252905	+	0.967491i	$0.418614\pi$
$812$	1.27309e13	1.02768
$813$	0	0
$814$	−3.78944e13	−3.02527
$815$	9.08847e12	0.721575
$816$	0	0
$817$	−1.36059e12	−0.106838
$818$	−2.29447e13	−1.79181
$819$	0	0
$820$	8.67501e12	0.670051
$821$	6.34294e12	0.487244	0.243622	−	0.969870i	$-0.421664\pi$
$821$	6.34294e12	0.487244	0.243622	+	0.969870i	$0.421664\pi$
$822$	0	0
$823$	−1.01100e13	−0.768158	−0.384079	−	0.923300i	$-0.625481\pi$
$823$	−1.01100e13	−0.768158	−0.384079	+	0.923300i	$0.625481\pi$
$824$	−3.48923e13	−2.63668
$825$	0	0
$826$	−6.02686e12	−0.450486
$827$	2.09841e11	0.0155997	0.00779984	−	0.999970i	$-0.497517\pi$
$827$	2.09841e11	0.0155997	0.00779984	+	0.999970i	$0.497517\pi$
$828$	0	0
$829$	−1.42099e13	−1.04495	−0.522474	−	0.852655i	$-0.674991\pi$
$829$	−1.42099e13	−1.04495	−0.522474	+	0.852655i	$0.674991\pi$
$830$	−6.21170e12	−0.454317
$831$	0	0
$832$	3.11636e13	2.25472
$833$	−2.04798e12	−0.147375
$834$	0	0
$835$	−2.88904e11	−0.0205667
$836$	1.91318e13	1.35465
$837$	0	0
$838$	1.18868e13	0.832658
$839$	−6.20295e12	−0.432185	−0.216092	−	0.976373i	$-0.569331\pi$
$839$	−6.20295e12	−0.432185	−0.216092	+	0.976373i	$0.569331\pi$
$840$	0	0
$841$	1.67059e12	0.115157
$842$	−2.32006e13	−1.59072
$843$	0	0
$844$	1.78185e13	1.20873
$845$	−1.13863e11	−0.00768292
$846$	0	0
$847$	1.51262e12	0.100984
$848$	5.05984e13	3.36012
$849$	0	0
$850$	5.93691e12	0.390099
$851$	4.86265e12	0.317826
$852$	0	0
$853$	−3.60370e12	−0.233066	−0.116533	−	0.993187i	$-0.537178\pi$
$853$	−3.60370e12	−0.233066	−0.116533	+	0.993187i	$0.537178\pi$
$854$	1.41593e13	0.910922
$855$	0	0
$856$	−5.08482e13	−3.23701
$857$	4.01270e12	0.254110	0.127055	−	0.991896i	$-0.459447\pi$
$857$	4.01270e12	0.254110	0.127055	+	0.991896i	$0.459447\pi$
$858$	0	0
$859$	2.02534e13	1.26919	0.634597	−	0.772843i	$-0.281166\pi$
$859$	2.02534e13	1.26919	0.634597	+	0.772843i	$0.281166\pi$
$860$	−3.21094e12	−0.200165
$861$	0	0
$862$	−2.38579e13	−1.47180
$863$	−1.57831e13	−0.968598	−0.484299	−	0.874903i	$-0.660925\pi$
$863$	−1.57831e13	−0.968598	−0.484299	+	0.874903i	$0.660925\pi$
$864$	0	0
$865$	6.39487e11	0.0388382
$866$	−9.12694e12	−0.551435
$867$	0	0
$868$	1.04285e13	0.623570
$869$	1.14715e13	0.682385
$870$	0	0
$871$	8.69087e12	0.511660
$872$	6.20382e13	3.63358
$873$	0	0
$874$	−3.40850e12	−0.197589
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−8.86665e12	−0.506129	−0.253065	−	0.967449i	$-0.581439\pi$
$877$	−8.86665e12	−0.506129	−0.253065	+	0.967449i	$0.581439\pi$
$878$	4.50562e13	2.55876
$879$	0	0
$880$	2.08041e13	1.16943
$881$	1.22614e13	0.685720	0.342860	−	0.939387i	$-0.388604\pi$
$881$	1.22614e13	0.685720	0.342860	+	0.939387i	$0.388604\pi$
$882$	0	0
$883$	1.90484e13	1.05447	0.527236	−	0.849719i	$-0.323228\pi$
$883$	1.90484e13	1.05447	0.527236	+	0.849719i	$0.323228\pi$
$884$	−4.86400e13	−2.67891
$885$	0	0
$886$	−3.41634e13	−1.86256
$887$	−2.67439e13	−1.45067	−0.725335	−	0.688396i	$-0.758315\pi$
$887$	−2.67439e13	−1.45067	−0.725335	+	0.688396i	$0.758315\pi$
$888$	0	0
$889$	7.77152e11	0.0417300
$890$	−7.25904e12	−0.387815
$891$	0	0
$892$	5.89661e13	3.11861
$893$	−1.15856e13	−0.609659
$894$	0	0
$895$	8.90375e12	0.463841
$896$	1.04188e13	0.540047
$897$	0	0
$898$	−4.19419e13	−2.15231
$899$	1.32520e13	0.676650
$900$	0	0
$901$	−2.24479e13	−1.13479
$902$	−1.87244e13	−0.941841
$903$	0	0
$904$	3.78332e13	1.88415
$905$	−1.14923e13	−0.569495
$906$	0	0
$907$	3.03949e13	1.49131	0.745655	−	0.666332i	$-0.232137\pi$
$907$	3.03949e13	1.49131	0.745655	+	0.666332i	$0.232137\pi$
$908$	−3.25319e13	−1.58827
$909$	0	0
$910$	−6.66768e12	−0.322321
$911$	−1.77272e13	−0.852721	−0.426361	−	0.904553i	$-0.640205\pi$
$911$	−1.77272e13	−0.852721	−0.426361	+	0.904553i	$0.640205\pi$
$912$	0	0
$913$	9.65690e12	0.459959
$914$	−3.16045e13	−1.49793
$915$	0	0
$916$	−7.17721e13	−3.36842
$917$	−1.10505e13	−0.516083
$918$	0	0
$919$	−3.30500e13	−1.52845	−0.764226	−	0.644948i	$-0.776879\pi$
$919$	−3.30500e13	−1.52845	−0.764226	+	0.644948i	$0.776879\pi$
$920$	−4.91980e12	−0.226414
$921$	0	0
$922$	−3.66651e13	−1.67095
$923$	−1.55964e13	−0.707320
$924$	0	0
$925$	−8.32357e12	−0.373828
$926$	3.70491e13	1.65588
$927$	0	0
$928$	6.67555e13	2.95475
$929$	3.62767e13	1.59793	0.798963	−	0.601380i	$-0.205382\pi$
$929$	3.62767e13	1.59793	0.798963	+	0.601380i	$0.205382\pi$
$930$	0	0
$931$	−2.01264e12	−0.0877996
$932$	7.68203e13	3.33506
$933$	0	0
$934$	−7.46313e13	−3.20893
$935$	−9.22970e12	−0.394944
$936$	0	0
$937$	8.07643e11	0.0342288	0.0171144	−	0.999854i	$-0.494552\pi$
$937$	8.07643e11	0.0342288	0.0171144	+	0.999854i	$0.494552\pi$
$938$	8.59548e12	0.362541
$939$	0	0
$940$	−2.73416e13	−1.14222
$941$	2.62828e13	1.09274	0.546371	−	0.837543i	$-0.316009\pi$
$941$	2.62828e13	1.09274	0.546371	+	0.837543i	$0.316009\pi$
$942$	0	0
$943$	2.40273e12	0.0989470
$944$	−4.69832e13	−1.92561
$945$	0	0
$946$	6.93057e12	0.281358
$947$	−1.64435e13	−0.664384	−0.332192	−	0.943212i	$-0.607788\pi$
$947$	−1.64435e13	−0.664384	−0.332192	+	0.943212i	$0.607788\pi$
$948$	0	0
$949$	−2.85422e13	−1.14232
$950$	5.83446e12	0.232405
$951$	0	0
$952$	−2.94224e13	−1.16095
$953$	1.18588e13	0.465717	0.232859	−	0.972511i	$-0.425192\pi$
$953$	1.18588e13	0.465717	0.232859	+	0.972511i	$0.425192\pi$
$954$	0	0
$955$	−1.65942e13	−0.645567
$956$	−7.30003e13	−2.82660
$957$	0	0
$958$	−2.78542e13	−1.06843
$959$	−7.39343e12	−0.282268
$960$	0	0
$961$	−1.55842e13	−0.589425
$962$	9.46787e13	3.56422
$963$	0	0
$964$	7.92655e13	2.95623
$965$	−3.65079e12	−0.135523
$966$	0	0
$967$	1.14207e13	0.420023	0.210011	−	0.977699i	$-0.432650\pi$
$967$	1.14207e13	0.420023	0.210011	+	0.977699i	$0.432650\pi$
$968$	2.17311e13	0.795505
$969$	0	0
$970$	−2.61131e13	−0.947078
$971$	−8.26707e12	−0.298446	−0.149223	−	0.988804i	$-0.547677\pi$
$971$	−8.26707e12	−0.298446	−0.149223	+	0.988804i	$0.547677\pi$
$972$	0	0
$973$	1.63425e13	0.584536
$974$	−4.24011e10	−0.00150960
$975$	0	0
$976$	1.10381e14	3.89375
$977$	5.06889e13	1.77987	0.889934	−	0.456090i	$-0.150750\pi$
$977$	5.06889e13	1.77987	0.889934	+	0.456090i	$0.150750\pi$
$978$	0	0
$979$	1.12851e13	0.392631
$980$	−4.74976e12	−0.164496
$981$	0	0
$982$	3.58686e13	1.23087
$983$	−1.12786e13	−0.385268	−0.192634	−	0.981271i	$-0.561703\pi$
$983$	−1.12786e13	−0.385268	−0.192634	+	0.981271i	$0.561703\pi$
$984$	0	0
$985$	−1.29057e13	−0.436835
$986$	−6.11307e13	−2.05974
$987$	0	0
$988$	−4.78006e13	−1.59598
$989$	−8.89338e11	−0.0295586
$990$	0	0
$991$	5.17189e13	1.70340	0.851702	−	0.524027i	$-0.175571\pi$
$991$	5.17189e13	1.70340	0.851702	+	0.524027i	$0.175571\pi$
$992$	5.46829e13	1.79287
$993$	0	0
$994$	−1.54252e13	−0.501178
$995$	1.66438e13	0.538330
$996$	0	0
$997$	1.92951e13	0.618471	0.309235	−	0.950986i	$-0.399927\pi$
$997$	1.92951e13	0.618471	0.309235	+	0.950986i	$0.399927\pi$
$998$	−1.06413e14	−3.39554
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.l.1.1		6
3.2	odd	2		35.10.a.e.1.6	✓	6
15.2	even	4		175.10.b.g.99.11		12
15.8	even	4		175.10.b.g.99.2		12
15.14	odd	2		175.10.a.g.1.1		6
21.20	even	2		245.10.a.g.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.e.1.6	✓	6	3.2	odd	2
175.10.a.g.1.1		6	15.14	odd	2
175.10.b.g.99.2		12	15.8	even	4
175.10.b.g.99.11		12	15.2	even	4
245.10.a.g.1.6		6	21.20	even	2
315.10.a.l.1.1		6	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)

\(f(q)\)	\(=\)	\(q-42.7818 q^{2} +1318.28 q^{4} -625.000 q^{5} -2401.00 q^{7} -34494.1 q^{8} +26738.6 q^{10} -41568.7 q^{11} +103859. q^{13} +102719. q^{14} +800760. q^{16} -355256. q^{17} -349126. q^{19} -823925. q^{20} +1.77838e6 q^{22} -228204. q^{23} +390625. q^{25} -4.44327e6 q^{26} -3.16519e6 q^{28} -4.02216e6 q^{29} -3.29476e6 q^{31} -1.65969e7 q^{32} +1.51985e7 q^{34} +1.50062e6 q^{35} -2.13083e7 q^{37} +1.49362e7 q^{38} +2.15588e7 q^{40} -1.05289e7 q^{41} +3.89712e6 q^{43} -5.47991e7 q^{44} +9.76296e6 q^{46} +3.31846e7 q^{47} +5.76480e6 q^{49} -1.67116e7 q^{50} +1.36915e8 q^{52} +6.31880e7 q^{53} +2.59804e7 q^{55} +8.28203e7 q^{56} +1.72075e8 q^{58} -5.86733e7 q^{59} +1.37845e8 q^{61} +1.40956e8 q^{62} +3.00057e8 q^{64} -6.49118e7 q^{65} +8.36796e7 q^{67} -4.68327e8 q^{68} -6.41994e7 q^{70} -1.50169e8 q^{71} -2.74817e8 q^{73} +9.11609e8 q^{74} -4.60246e8 q^{76} +9.98063e7 q^{77} -2.75964e8 q^{79} -5.00475e8 q^{80} +4.50444e8 q^{82} -2.32312e8 q^{83} +2.22035e8 q^{85} -1.66726e8 q^{86} +1.43387e9 q^{88} -2.71482e8 q^{89} -2.49365e8 q^{91} -3.00837e8 q^{92} -1.41969e9 q^{94} +2.18204e8 q^{95} -9.76607e8 q^{97} -2.46628e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8} + 9375 q^{10} + 47796 q^{11} + 102168 q^{13} + 36015 q^{14} + 2371065 q^{16} + 38472 q^{17} + 361056 q^{19} - 1880625 q^{20} + 2068680 q^{22}+ \cdots - 86472015 q^{98}+O(q^{100}) \) 6 * q - 15 * q^2 + 3009 * q^4 - 3750 * q^5 - 14406 * q^7 - 22041 * q^8 + 9375 * q^10 + 47796 * q^11 + 102168 * q^13 + 36015 * q^14 + 2371065 * q^16 + 38472 * q^17 + 361056 * q^19 - 1880625 * q^20 + 2068680 * q^22 - 697032 * q^23 + 2343750 * q^25 - 10847622 * q^26 - 7224609 * q^28 - 16028184 * q^29 + 1362912 * q^31 + 1148823 * q^32 - 9216294 * q^34 + 9003750 * q^35 - 3912924 * q^37 - 39877140 * q^38 + 13775625 * q^40 - 22452756 * q^41 - 29998992 * q^43 + 94378728 * q^44 + 49589520 * q^46 + 121271508 * q^47 + 34588806 * q^49 - 5859375 * q^50 - 44528550 * q^52 + 20308596 * q^53 - 29872500 * q^55 + 52920441 * q^56 + 256759494 * q^58 - 120280392 * q^59 - 87693540 * q^61 + 344802528 * q^62 + 640647105 * q^64 - 63855000 * q^65 + 495050664 * q^67 + 141443706 * q^68 - 22509375 * q^70 - 253762512 * q^71 + 187195308 * q^73 + 418806054 * q^74 - 584719476 * q^76 - 114758196 * q^77 + 831079500 * q^79 - 1481915625 * q^80 + 2286848814 * q^82 + 767650536 * q^83 - 24045000 * q^85 - 566199948 * q^86 + 2952495444 * q^88 - 582579684 * q^89 - 245305368 * q^91 + 4259382384 * q^92 - 1115946660 * q^94 - 225660000 * q^95 - 1184506872 * q^97 - 86472015 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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