LMFDB - Embedded newform 315.10.a.e.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:	$4$
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} - 1253x^{2} - 1039x + 42996 $ x^4 - x^3 - 1253x^2 - 1039x + 42996
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-33.9278$ 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)

$f(q)$	$=$	\(q-30.9278 q^{2} +444.529 q^{4} +625.000 q^{5} +2401.00 q^{7} +2086.72 q^{8} -19329.9 q^{10} +51657.7 q^{11} -129921. q^{13} -74257.7 q^{14} -292137. q^{16} +259513. q^{17} +90316.4 q^{19} +277831. q^{20} -1.59766e6 q^{22} -751692. q^{23} +390625. q^{25} +4.01818e6 q^{26} +1.06731e6 q^{28} +2.20510e6 q^{29} +3.10032e6 q^{31} +7.96675e6 q^{32} -8.02616e6 q^{34} +1.50062e6 q^{35} -1.42524e7 q^{37} -2.79329e6 q^{38} +1.30420e6 q^{40} +1.45691e7 q^{41} +1.89091e7 q^{43} +2.29633e7 q^{44} +2.32482e7 q^{46} +1.43175e7 q^{47} +5.76480e6 q^{49} -1.20812e7 q^{50} -5.77538e7 q^{52} +3.44220e7 q^{53} +3.22860e7 q^{55} +5.01022e6 q^{56} -6.81989e7 q^{58} +3.61330e7 q^{59} -9.33007e7 q^{61} -9.58860e7 q^{62} -9.68200e7 q^{64} -8.12008e7 q^{65} +2.05762e8 q^{67} +1.15361e8 q^{68} -4.64110e7 q^{70} +1.92397e8 q^{71} +8.11669e7 q^{73} +4.40795e8 q^{74} +4.01483e7 q^{76} +1.24030e8 q^{77} +1.07122e8 q^{79} -1.82585e8 q^{80} -4.50592e8 q^{82} -3.54922e8 q^{83} +1.62195e8 q^{85} -5.84818e8 q^{86} +1.07795e8 q^{88} -4.42433e8 q^{89} -3.11941e8 q^{91} -3.34149e8 q^{92} -4.42810e8 q^{94} +5.64478e7 q^{95} -6.40463e8 q^{97} -1.78293e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) $ 4 * q + 13 * q^2 + 501 * q^4 + 2500 * q^5 + 9604 * q^7 + 16263 * q^8 + 8125 * q^10 + 87062 * q^11 + 39494 * q^13 + 31213 * q^14 + 328849 * q^16 + 291756 * q^17 + 50482 * q^19 + 313125 * q^20 - 1003016 * q^22 + 1048656 * q^23 + 1562500 * q^25 + 7622594 * q^26 + 1202901 * q^28 + 8018996 * q^29 - 1148130 * q^31 + 9765119 * q^32 + 1255358 * q^34 + 6002500 * q^35 - 23416328 * q^37 - 11334892 * q^38 + 10164375 * q^40 + 8000860 * q^41 + 27520044 * q^43 - 42192 * q^44 + 90166880 * q^46 + 78277092 * q^47 + 23059204 * q^49 + 5078125 * q^50 - 62776870 * q^52 - 36168514 * q^53 + 54413750 * q^55 + 39047463 * q^56 - 32819954 * q^58 + 132924868 * q^59 - 119014876 * q^61 + 76291172 * q^62 - 400777983 * q^64 + 24683750 * q^65 - 170018492 * q^67 + 345488158 * q^68 + 19508125 * q^70 + 743169022 * q^71 - 360015502 * q^73 + 547147386 * q^74 - 472117316 * q^76 + 209035862 * q^77 - 613400256 * q^79 + 205530625 * q^80 - 714522510 * q^82 + 1222551024 * q^83 + 182347500 * q^85 + 593721752 * q^86 - 855674336 * q^88 - 835315808 * q^89 + 94825094 * q^91 + 1276931776 * q^92 + 950602540 * q^94 + 31551250 * q^95 - 704297762 * q^97 + 74942413 * 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 $

$2$	−30.9278	−1.36683	−0.683414	−	0.730031i	$-0.739506\pi$
$2$	−30.9278	−1.36683	−0.683414	+	0.730031i	$0.739506\pi$
$3$	0	0
$3$	0	0
$4$	444.529	0.868221
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	2086.72	0.180119
$9$	0	0
$10$	−19329.9	−0.611264
$11$	51657.7	1.06382	0.531909	−	0.846801i	$-0.321475\pi$
$11$	51657.7	1.06382	0.531909	+	0.846801i	$0.321475\pi$
$12$	0	0
$13$	−129921.	−1.26164	−0.630820	−	0.775930i	$-0.717281\pi$
$13$	−129921.	−1.26164	−0.630820	+	0.775930i	$0.717281\pi$
$14$	−74257.7	−0.516613
$15$	0	0
$16$	−292137.	−1.11441
$17$	259513.	0.753595	0.376798	−	0.926296i	$-0.377025\pi$
$17$	259513.	0.753595	0.376798	+	0.926296i	$0.377025\pi$
$18$	0	0
$19$	90316.4	0.158992	0.0794961	−	0.996835i	$-0.474669\pi$
$19$	90316.4	0.158992	0.0794961	+	0.996835i	$0.474669\pi$
$20$	277831.	0.388280
$21$	0	0
$22$	−1.59766e6	−1.45406
$23$	−751692.	−0.560099	−0.280049	−	0.959986i	$-0.590351\pi$
$23$	−751692.	−0.560099	−0.280049	+	0.959986i	$0.590351\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	4.01818e6	1.72444
$27$	0	0
$28$	1.06731e6	0.328157
$29$	2.20510e6	0.578945	0.289472	−	0.957186i	$-0.406520\pi$
$29$	2.20510e6	0.578945	0.289472	+	0.957186i	$0.406520\pi$
$30$	0	0
$31$	3.10032e6	0.602946	0.301473	−	0.953475i	$-0.402522\pi$
$31$	3.10032e6	0.602946	0.301473	+	0.953475i	$0.402522\pi$
$32$	7.96675e6	1.34309
$33$	0	0
$34$	−8.02616e6	−1.03004
$35$	1.50062e6	0.169031
$36$	0	0
$37$	−1.42524e7	−1.25020	−0.625101	−	0.780544i	$-0.714942\pi$
$37$	−1.42524e7	−1.25020	−0.625101	+	0.780544i	$0.714942\pi$
$38$	−2.79329e6	−0.217315
$39$	0	0
$40$	1.30420e6	0.0805517
$41$	1.45691e7	0.805205	0.402603	−	0.915375i	$-0.368106\pi$
$41$	1.45691e7	0.805205	0.402603	+	0.915375i	$0.368106\pi$
$42$	0	0
$43$	1.89091e7	0.843458	0.421729	−	0.906722i	$-0.361423\pi$
$43$	1.89091e7	0.843458	0.421729	+	0.906722i	$0.361423\pi$
$44$	2.29633e7	0.923630
$45$	0	0
$46$	2.32482e7	0.765559
$47$	1.43175e7	0.427985	0.213992	−	0.976835i	$-0.431353\pi$
$47$	1.43175e7	0.427985	0.213992	+	0.976835i	$0.431353\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−1.20812e7	−0.273366
$51$	0	0
$52$	−5.77538e7	−1.09538
$53$	3.44220e7	0.599232	0.299616	−	0.954060i	$-0.403142\pi$
$53$	3.44220e7	0.599232	0.299616	+	0.954060i	$0.403142\pi$
$54$	0	0
$55$	3.22860e7	0.475754
$56$	5.01022e6	0.0680786
$57$	0	0
$58$	−6.81989e7	−0.791319
$59$	3.61330e7	0.388213	0.194107	−	0.980980i	$-0.437819\pi$
$59$	3.61330e7	0.388213	0.194107	+	0.980980i	$0.437819\pi$
$60$	0	0
$61$	−9.33007e7	−0.862781	−0.431390	−	0.902165i	$-0.641977\pi$
$61$	−9.33007e7	−0.862781	−0.431390	+	0.902165i	$0.641977\pi$
$62$	−9.58860e7	−0.824124
$63$	0	0
$64$	−9.68200e7	−0.721365
$65$	−8.12008e7	−0.564222
$66$	0	0
$67$	2.05762e8	1.24746	0.623732	−	0.781638i	$-0.285616\pi$
$67$	2.05762e8	1.24746	0.623732	+	0.781638i	$0.285616\pi$
$68$	1.15361e8	0.654288
$69$	0	0
$70$	−4.64110e7	−0.231036
$71$	1.92397e8	0.898535	0.449268	−	0.893397i	$-0.351685\pi$
$71$	1.92397e8	0.898535	0.449268	+	0.893397i	$0.351685\pi$
$72$	0	0
$73$	8.11669e7	0.334523	0.167262	−	0.985913i	$-0.446508\pi$
$73$	8.11669e7	0.334523	0.167262	+	0.985913i	$0.446508\pi$
$74$	4.40795e8	1.70881
$75$	0	0
$76$	4.01483e7	0.138040
$77$	1.24030e8	0.402086
$78$	0	0
$79$	1.07122e8	0.309427	0.154713	−	0.987959i	$-0.450555\pi$
$79$	1.07122e8	0.309427	0.154713	+	0.987959i	$0.450555\pi$
$80$	−1.82585e8	−0.498381
$81$	0	0
$82$	−4.50592e8	−1.10058
$83$	−3.54922e8	−0.820883	−0.410441	−	0.911887i	$-0.634625\pi$
$83$	−3.54922e8	−0.820883	−0.410441	+	0.911887i	$0.634625\pi$
$84$	0	0
$85$	1.62195e8	0.337018
$86$	−5.84818e8	−1.15286
$87$	0	0
$88$	1.07795e8	0.191614
$89$	−4.42433e8	−0.747468	−0.373734	−	0.927536i	$-0.621923\pi$
$89$	−4.42433e8	−0.747468	−0.373734	+	0.927536i	$0.621923\pi$
$90$	0	0
$91$	−3.11941e8	−0.476855
$92$	−3.34149e8	−0.486290
$93$	0	0
$94$	−4.42810e8	−0.584982
$95$	5.64478e7	0.0711034
$96$	0	0
$97$	−6.40463e8	−0.734549	−0.367275	−	0.930113i	$-0.619709\pi$
$97$	−6.40463e8	−0.734549	−0.367275	+	0.930113i	$0.619709\pi$
$98$	−1.78293e8	−0.195261
$99$	0	0
$100$	1.73644e8	0.173644
$101$	−5.75536e8	−0.550334	−0.275167	−	0.961396i	$-0.588733\pi$
$101$	−5.75536e8	−0.550334	−0.275167	+	0.961396i	$0.588733\pi$
$102$	0	0
$103$	5.97246e8	0.522860	0.261430	−	0.965222i	$-0.415806\pi$
$103$	5.97246e8	0.522860	0.261430	+	0.965222i	$0.415806\pi$
$104$	−2.71110e8	−0.227245
$105$	0	0
$106$	−1.06460e9	−0.819047
$107$	−1.37917e9	−1.01716	−0.508580	−	0.861015i	$-0.669829\pi$
$107$	−1.37917e9	−1.01716	−0.508580	+	0.861015i	$0.669829\pi$
$108$	0	0
$109$	−1.97995e9	−1.34349	−0.671744	−	0.740783i	$-0.734455\pi$
$109$	−1.97995e9	−1.34349	−0.671744	+	0.740783i	$0.734455\pi$
$110$	−9.98536e8	−0.650275
$111$	0	0
$112$	−7.01420e8	−0.421209
$113$	2.56021e9	1.47714	0.738572	−	0.674175i	$-0.235501\pi$
$113$	2.56021e9	1.47714	0.738572	+	0.674175i	$0.235501\pi$
$114$	0	0
$115$	−4.69807e8	−0.250484
$116$	9.80231e8	0.502652
$117$	0	0
$118$	−1.11752e9	−0.530621
$119$	6.23090e8	0.284832
$120$	0	0
$121$	3.10567e8	0.131711
$122$	2.88559e9	1.17927
$123$	0	0
$124$	1.37818e9	0.523491
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−5.69352e9	−1.94207	−0.971034	−	0.238943i	$-0.923199\pi$
$127$	−5.69352e9	−1.94207	−0.971034	+	0.238943i	$0.923199\pi$
$128$	−1.08454e9	−0.357110
$129$	0	0
$130$	2.51136e9	0.771195
$131$	−2.61624e7	−0.00776170	−0.00388085	−	0.999992i	$-0.501235\pi$
$131$	−2.61624e7	−0.00776170	−0.00388085	+	0.999992i	$0.501235\pi$
$132$	0	0
$133$	2.16850e8	0.0600934
$134$	−6.36376e9	−1.70507
$135$	0	0
$136$	5.41531e8	0.135737
$137$	−4.42139e9	−1.07230	−0.536150	−	0.844123i	$-0.680122\pi$
$137$	−4.42139e9	−1.07230	−0.536150	+	0.844123i	$0.680122\pi$
$138$	0	0
$139$	3.73682e9	0.849055	0.424527	−	0.905415i	$-0.360440\pi$
$139$	3.73682e9	0.849055	0.424527	+	0.905415i	$0.360440\pi$
$140$	6.67072e8	0.146756
$141$	0	0
$142$	−5.95041e9	−1.22814
$143$	−6.71143e9	−1.34216
$144$	0	0
$145$	1.37819e9	0.258912
$146$	−2.51032e9	−0.457236
$147$	0	0
$148$	−6.33561e9	−1.08545
$149$	−4.21470e9	−0.700532	−0.350266	−	0.936650i	$-0.613909\pi$
$149$	−4.21470e9	−0.700532	−0.350266	+	0.936650i	$0.613909\pi$
$150$	0	0
$151$	1.01846e10	1.59422	0.797110	−	0.603834i	$-0.206361\pi$
$151$	1.01846e10	1.59422	0.797110	+	0.603834i	$0.206361\pi$
$152$	1.88465e8	0.0286375
$153$	0	0
$154$	−3.83598e9	−0.549582
$155$	1.93770e9	0.269646
$156$	0	0
$157$	6.91306e9	0.908075	0.454038	−	0.890983i	$-0.349983\pi$
$157$	6.91306e9	0.908075	0.454038	+	0.890983i	$0.349983\pi$
$158$	−3.31305e9	−0.422933
$159$	0	0
$160$	4.97922e9	0.600649
$161$	−1.80481e9	−0.211697
$162$	0	0
$163$	−1.38846e9	−0.154060	−0.0770298	−	0.997029i	$-0.524544\pi$
$163$	−1.38846e9	−0.154060	−0.0770298	+	0.997029i	$0.524544\pi$
$164$	6.47641e9	0.699096
$165$	0	0
$166$	1.09769e10	1.12201
$167$	−2.76615e9	−0.275202	−0.137601	−	0.990488i	$-0.543939\pi$
$167$	−2.76615e9	−0.275202	−0.137601	+	0.990488i	$0.543939\pi$
$168$	0	0
$169$	6.27504e9	0.591733
$170$	−5.01635e9	−0.460646
$171$	0	0
$172$	8.40567e9	0.732309
$173$	−1.57511e9	−0.133691	−0.0668457	−	0.997763i	$-0.521294\pi$
$173$	−1.57511e9	−0.133691	−0.0668457	+	0.997763i	$0.521294\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−1.50911e10	−1.18553
$177$	0	0
$178$	1.36835e10	1.02166
$179$	6.62999e9	0.482697	0.241348	−	0.970439i	$-0.422410\pi$
$179$	6.62999e9	0.482697	0.241348	+	0.970439i	$0.422410\pi$
$180$	0	0
$181$	2.73231e10	1.89224	0.946119	−	0.323819i	$-0.104967\pi$
$181$	2.73231e10	1.89224	0.946119	+	0.323819i	$0.104967\pi$
$182$	9.64765e9	0.651779
$183$	0	0
$184$	−1.56857e9	−0.100884
$185$	−8.90775e9	−0.559107
$186$	0	0
$187$	1.34058e10	0.801689
$188$	6.36457e9	0.371585
$189$	0	0
$190$	−1.74581e9	−0.0971862
$191$	−2.04415e10	−1.11138	−0.555689	−	0.831390i	$-0.687545\pi$
$191$	−2.04415e10	−1.11138	−0.555689	+	0.831390i	$0.687545\pi$
$192$	0	0
$193$	−2.46642e10	−1.27955	−0.639777	−	0.768561i	$-0.720973\pi$
$193$	−2.46642e10	−1.27955	−0.639777	+	0.768561i	$0.720973\pi$
$194$	1.98081e10	1.00400
$195$	0	0
$196$	2.56262e9	0.124032
$197$	3.76002e10	1.77866	0.889329	−	0.457267i	$-0.151172\pi$
$197$	3.76002e10	1.77866	0.889329	+	0.457267i	$0.151172\pi$
$198$	0	0
$199$	1.12694e10	0.509403	0.254702	−	0.967020i	$-0.418023\pi$
$199$	1.12694e10	0.509403	0.254702	+	0.967020i	$0.418023\pi$
$200$	8.15126e8	0.0360238
$201$	0	0
$202$	1.78001e10	0.752212
$203$	5.29444e9	0.218821
$204$	0	0
$205$	9.10571e9	0.360099
$206$	−1.84715e10	−0.714661
$207$	0	0
$208$	3.79548e10	1.40599
$209$	4.66554e9	0.169139
$210$	0	0
$211$	2.96721e10	1.03057	0.515285	−	0.857019i	$-0.327686\pi$
$211$	2.96721e10	1.03057	0.515285	+	0.857019i	$0.327686\pi$
$212$	1.53016e10	0.520266
$213$	0	0
$214$	4.26546e10	1.39028
$215$	1.18182e10	0.377206
$216$	0	0
$217$	7.44386e9	0.227892
$218$	6.12354e10	1.83632
$219$	0	0
$220$	1.43521e10	0.413060
$221$	−3.37162e10	−0.950766
$222$	0	0
$223$	−1.53280e10	−0.415061	−0.207531	−	0.978229i	$-0.566543\pi$
$223$	−1.53280e10	−0.415061	−0.207531	+	0.978229i	$0.566543\pi$
$224$	1.91282e10	0.507641
$225$	0	0
$226$	−7.91817e10	−2.01900
$227$	1.51208e10	0.377972	0.188986	−	0.981980i	$-0.439480\pi$
$227$	1.51208e10	0.377972	0.188986	+	0.981980i	$0.439480\pi$
$228$	0	0
$229$	−5.99252e10	−1.43996	−0.719979	−	0.693996i	$-0.755848\pi$
$229$	−5.99252e10	−1.43996	−0.719979	+	0.693996i	$0.755848\pi$
$230$	1.45301e10	0.342368
$231$	0	0
$232$	4.60143e9	0.104279
$233$	6.20541e10	1.37933	0.689666	−	0.724128i	$-0.257757\pi$
$233$	6.20541e10	1.37933	0.689666	+	0.724128i	$0.257757\pi$
$234$	0	0
$235$	8.94847e9	0.191401
$236$	1.60622e10	0.337055
$237$	0	0
$238$	−1.92708e10	−0.389317
$239$	5.37734e10	1.06605	0.533024	−	0.846100i	$-0.321055\pi$
$239$	5.37734e10	1.06605	0.533024	+	0.846100i	$0.321055\pi$
$240$	0	0
$241$	−2.31709e10	−0.442452	−0.221226	−	0.975223i	$-0.571006\pi$
$241$	−2.31709e10	−0.442452	−0.221226	+	0.975223i	$0.571006\pi$
$242$	−9.60515e9	−0.180026
$243$	0	0
$244$	−4.14749e10	−0.749085
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−1.17340e10	−0.200591
$248$	6.46950e9	0.108602
$249$	0	0
$250$	−7.55073e9	−0.122253
$251$	−1.41545e10	−0.225093	−0.112547	−	0.993646i	$-0.535901\pi$
$251$	−1.41545e10	−0.225093	−0.112547	+	0.993646i	$0.535901\pi$
$252$	0	0
$253$	−3.88306e10	−0.595844
$254$	1.76088e11	2.65447
$255$	0	0
$256$	8.31144e10	1.20947
$257$	1.28261e11	1.83399	0.916995	−	0.398899i	$-0.130608\pi$
$257$	1.28261e11	1.83399	0.916995	+	0.398899i	$0.130608\pi$
$258$	0	0
$259$	−3.42200e10	−0.472532
$260$	−3.60961e10	−0.489870
$261$	0	0
$262$	8.09146e8	0.0106089
$263$	1.53155e10	0.197392	0.0986960	−	0.995118i	$-0.468533\pi$
$263$	1.53155e10	0.197392	0.0986960	+	0.995118i	$0.468533\pi$
$264$	0	0
$265$	2.15138e10	0.267985
$266$	−6.70669e9	−0.0821373
$267$	0	0
$268$	9.14671e10	1.08307
$269$	1.42724e11	1.66193	0.830964	−	0.556326i	$-0.187789\pi$
$269$	1.42724e11	1.66193	0.830964	+	0.556326i	$0.187789\pi$
$270$	0	0
$271$	1.19856e11	1.34989	0.674944	−	0.737869i	$-0.264168\pi$
$271$	1.19856e11	1.34989	0.674944	+	0.737869i	$0.264168\pi$
$272$	−7.58132e10	−0.839817
$273$	0	0
$274$	1.36744e11	1.46565
$275$	2.01788e10	0.212764
$276$	0	0
$277$	3.81727e9	0.0389578	0.0194789	−	0.999810i	$-0.493799\pi$
$277$	3.81727e9	0.0389578	0.0194789	+	0.999810i	$0.493799\pi$
$278$	−1.15572e11	−1.16051
$279$	0	0
$280$	3.13139e9	0.0304457
$281$	1.43663e11	1.37457	0.687283	−	0.726390i	$-0.258803\pi$
$281$	1.43663e11	1.37457	0.687283	+	0.726390i	$0.258803\pi$
$282$	0	0
$283$	6.91570e9	0.0640910	0.0320455	−	0.999486i	$-0.489798\pi$
$283$	6.91570e9	0.0640910	0.0320455	+	0.999486i	$0.489798\pi$
$284$	8.55260e10	0.780127
$285$	0	0
$286$	2.07570e11	1.83450
$287$	3.49805e10	0.304339
$288$	0	0
$289$	−5.12411e10	−0.432094
$290$	−4.26243e10	−0.353888
$291$	0	0
$292$	3.60811e10	0.290440
$293$	−2.23244e11	−1.76960	−0.884802	−	0.465967i	$-0.845707\pi$
$293$	−2.23244e11	−1.76960	−0.884802	+	0.465967i	$0.845707\pi$
$294$	0	0
$295$	2.25832e10	0.173614
$296$	−2.97408e10	−0.225185
$297$	0	0
$298$	1.30351e11	0.957508
$299$	9.76607e10	0.706642
$300$	0	0
$301$	4.54008e10	0.318797
$302$	−3.14988e11	−2.17903
$303$	0	0
$304$	−2.63847e10	−0.177183
$305$	−5.83129e10	−0.385847
$306$	0	0
$307$	2.42296e11	1.55677	0.778383	−	0.627790i	$-0.216040\pi$
$307$	2.42296e11	1.55677	0.778383	+	0.627790i	$0.216040\pi$
$308$	5.51350e10	0.349099
$309$	0	0
$310$	−5.99287e10	−0.368559
$311$	1.48593e11	0.900694	0.450347	−	0.892854i	$-0.351300\pi$
$311$	1.48593e11	0.900694	0.450347	+	0.892854i	$0.351300\pi$
$312$	0	0
$313$	−1.97735e11	−1.16449	−0.582243	−	0.813015i	$-0.697825\pi$
$313$	−1.97735e11	−1.16449	−0.582243	+	0.813015i	$0.697825\pi$
$314$	−2.13806e11	−1.24118
$315$	0	0
$316$	4.76190e10	0.268651
$317$	9.37774e10	0.521593	0.260796	−	0.965394i	$-0.416015\pi$
$317$	9.37774e10	0.521593	0.260796	+	0.965394i	$0.416015\pi$
$318$	0	0
$319$	1.13910e11	0.615892
$320$	−6.05125e10	−0.322604
$321$	0	0
$322$	5.58189e10	0.289354
$323$	2.34383e10	0.119816
$324$	0	0
$325$	−5.07505e10	−0.252328
$326$	4.29420e10	0.210573
$327$	0	0
$328$	3.04017e10	0.145033
$329$	3.43764e10	0.161763
$330$	0	0
$331$	−1.37421e11	−0.629257	−0.314628	−	0.949215i	$-0.601880\pi$
$331$	−1.37421e11	−0.629257	−0.314628	+	0.949215i	$0.601880\pi$
$332$	−1.57773e11	−0.712708
$333$	0	0
$334$	8.55509e10	0.376154
$335$	1.28601e11	0.557883
$336$	0	0
$337$	−4.35827e11	−1.84069	−0.920344	−	0.391111i	$-0.872091\pi$
$337$	−4.35827e11	−1.84069	−0.920344	+	0.391111i	$0.872091\pi$
$338$	−1.94073e11	−0.808798
$339$	0	0
$340$	7.21006e10	0.292606
$341$	1.60155e11	0.641425
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	3.94581e10	0.151923
$345$	0	0
$346$	4.87147e10	0.182733
$347$	5.95838e10	0.220620	0.110310	−	0.993897i	$-0.464816\pi$
$347$	5.95838e10	0.220620	0.110310	+	0.993897i	$0.464816\pi$
$348$	0	0
$349$	1.58456e11	0.571733	0.285866	−	0.958269i	$-0.407719\pi$
$349$	1.58456e11	0.571733	0.285866	+	0.958269i	$0.407719\pi$
$350$	−2.90069e10	−0.103323
$351$	0	0
$352$	4.11544e11	1.42881
$353$	−4.78124e11	−1.63891	−0.819453	−	0.573146i	$-0.805723\pi$
$353$	−4.78124e11	−1.63891	−0.819453	+	0.573146i	$0.805723\pi$
$354$	0	0
$355$	1.20248e11	0.401837
$356$	−1.96674e11	−0.648967
$357$	0	0
$358$	−2.05051e11	−0.659764
$359$	5.64725e11	1.79437	0.897185	−	0.441655i	$-0.145609\pi$
$359$	5.64725e11	1.79437	0.897185	+	0.441655i	$0.145609\pi$
$360$	0	0
$361$	−3.14531e11	−0.974722
$362$	−8.45043e11	−2.58637
$363$	0	0
$364$	−1.38667e11	−0.414015
$365$	5.07293e10	0.149603
$366$	0	0
$367$	2.93967e11	0.845865	0.422933	−	0.906161i	$-0.361001\pi$
$367$	2.93967e11	0.845865	0.422933	+	0.906161i	$0.361001\pi$
$368$	2.19597e11	0.624181
$369$	0	0
$370$	2.75497e11	0.764204
$371$	8.26472e10	0.226488
$372$	0	0
$373$	−7.23302e11	−1.93477	−0.967386	−	0.253307i	$-0.918482\pi$
$373$	−7.23302e11	−1.93477	−0.967386	+	0.253307i	$0.918482\pi$
$374$	−4.14612e11	−1.09577
$375$	0	0
$376$	2.98767e10	0.0770882
$377$	−2.86489e11	−0.730419
$378$	0	0
$379$	−7.65506e10	−0.190578	−0.0952888	−	0.995450i	$-0.530377\pi$
$379$	−7.65506e10	−0.190578	−0.0952888	+	0.995450i	$0.530377\pi$
$380$	2.50927e10	0.0617335
$381$	0	0
$382$	6.32210e11	1.51906
$383$	3.85038e10	0.0914343	0.0457171	−	0.998954i	$-0.485443\pi$
$383$	3.85038e10	0.0914343	0.0457171	+	0.998954i	$0.485443\pi$
$384$	0	0
$385$	7.75188e10	0.179818
$386$	7.62809e11	1.74893
$387$	0	0
$388$	−2.84704e11	−0.637751
$389$	3.16636e11	0.701112	0.350556	−	0.936542i	$-0.385993\pi$
$389$	3.16636e11	0.701112	0.350556	+	0.936542i	$0.385993\pi$
$390$	0	0
$391$	−1.95073e11	−0.422088
$392$	1.20295e10	0.0257313
$393$	0	0
$394$	−1.16289e12	−2.43112
$395$	6.69514e10	0.138380
$396$	0	0
$397$	8.72503e11	1.76283	0.881414	−	0.472345i	$-0.156592\pi$
$397$	8.72503e11	1.76283	0.881414	+	0.472345i	$0.156592\pi$
$398$	−3.48538e11	−0.696267
$399$	0	0
$400$	−1.14116e11	−0.222883
$401$	−2.01947e11	−0.390020	−0.195010	−	0.980801i	$-0.562474\pi$
$401$	−2.01947e11	−0.390020	−0.195010	+	0.980801i	$0.562474\pi$
$402$	0	0
$403$	−4.02797e11	−0.760700
$404$	−2.55843e11	−0.477812
$405$	0	0
$406$	−1.63746e11	−0.299090
$407$	−7.36246e11	−1.32999
$408$	0	0
$409$	6.00556e11	1.06120	0.530602	−	0.847621i	$-0.321966\pi$
$409$	6.00556e11	1.06120	0.530602	+	0.847621i	$0.321966\pi$
$410$	−2.81620e11	−0.492193
$411$	0	0
$412$	2.65493e11	0.453959
$413$	8.67554e10	0.146731
$414$	0	0
$415$	−2.21826e11	−0.367110
$416$	−1.03505e12	−1.69450
$417$	0	0
$418$	−1.44295e11	−0.231184
$419$	−7.70658e10	−0.122151	−0.0610757	−	0.998133i	$-0.519453\pi$
$419$	−7.70658e10	−0.122151	−0.0610757	+	0.998133i	$0.519453\pi$
$420$	0	0
$421$	−2.92063e10	−0.0453113	−0.0226557	−	0.999743i	$-0.507212\pi$
$421$	−2.92063e10	−0.0453113	−0.0226557	+	0.999743i	$0.507212\pi$
$422$	−9.17694e11	−1.40861
$423$	0	0
$424$	7.18291e10	0.107933
$425$	1.01372e11	0.150719
$426$	0	0
$427$	−2.24015e11	−0.326101
$428$	−6.13079e11	−0.883120
$429$	0	0
$430$	−3.65511e11	−0.515576
$431$	8.55400e11	1.19405	0.597024	−	0.802224i	$-0.296350\pi$
$431$	8.55400e11	1.19405	0.597024	+	0.802224i	$0.296350\pi$
$432$	0	0
$433$	−1.27919e12	−1.74879	−0.874395	−	0.485214i	$-0.838742\pi$
$433$	−1.27919e12	−1.74879	−0.874395	+	0.485214i	$0.838742\pi$
$434$	−2.30222e11	−0.311490
$435$	0	0
$436$	−8.80144e11	−1.16645
$437$	−6.78901e10	−0.0890513
$438$	0	0
$439$	−6.24487e11	−0.802478	−0.401239	−	0.915973i	$-0.631420\pi$
$439$	−6.24487e11	−0.802478	−0.401239	+	0.915973i	$0.631420\pi$
$440$	6.73720e10	0.0856924
$441$	0	0
$442$	1.04277e12	1.29953
$443$	−3.08157e11	−0.380151	−0.190075	−	0.981769i	$-0.560873\pi$
$443$	−3.08157e11	−0.380151	−0.190075	+	0.981769i	$0.560873\pi$
$444$	0	0
$445$	−2.76521e11	−0.334278
$446$	4.74060e11	0.567318
$447$	0	0
$448$	−2.32465e11	−0.272650
$449$	−2.51528e11	−0.292064	−0.146032	−	0.989280i	$-0.546650\pi$
$449$	−2.51528e11	−0.292064	−0.146032	+	0.989280i	$0.546650\pi$
$450$	0	0
$451$	7.52608e11	0.856592
$452$	1.13809e12	1.28249
$453$	0	0
$454$	−4.67654e11	−0.516623
$455$	−1.94963e11	−0.213256
$456$	0	0
$457$	−7.28476e11	−0.781254	−0.390627	−	0.920549i	$-0.627742\pi$
$457$	−7.28476e11	−0.781254	−0.390627	+	0.920549i	$0.627742\pi$
$458$	1.85335e12	1.96818
$459$	0	0
$460$	−2.08843e11	−0.217475
$461$	−7.37088e10	−0.0760090	−0.0380045	−	0.999278i	$-0.512100\pi$
$461$	−7.37088e10	−0.0760090	−0.0380045	+	0.999278i	$0.512100\pi$
$462$	0	0
$463$	−4.60918e11	−0.466132	−0.233066	−	0.972461i	$-0.574876\pi$
$463$	−4.60918e11	−0.466132	−0.233066	+	0.972461i	$0.574876\pi$
$464$	−6.44190e11	−0.645184
$465$	0	0
$466$	−1.91920e12	−1.88531
$467$	−6.43236e11	−0.625813	−0.312907	−	0.949784i	$-0.601303\pi$
$467$	−6.43236e11	−0.625813	−0.312907	+	0.949784i	$0.601303\pi$
$468$	0	0
$469$	4.94034e11	0.471497
$470$	−2.76756e11	−0.261612
$471$	0	0
$472$	7.53996e10	0.0699246
$473$	9.76802e11	0.897287
$474$	0	0
$475$	3.52799e10	0.0317984
$476$	2.76982e11	0.247297
$477$	0	0
$478$	−1.66309e12	−1.45711
$479$	2.06775e12	1.79468	0.897342	−	0.441336i	$-0.145495\pi$
$479$	2.06775e12	1.79468	0.897342	+	0.441336i	$0.145495\pi$
$480$	0	0
$481$	1.85169e12	1.57730
$482$	7.16626e11	0.604757
$483$	0	0
$484$	1.38056e11	0.114354
$485$	−4.00289e11	−0.328500
$486$	0	0
$487$	8.68183e11	0.699408	0.349704	−	0.936860i	$-0.386282\pi$
$487$	8.68183e11	0.699408	0.349704	+	0.936860i	$0.386282\pi$
$488$	−1.94693e11	−0.155403
$489$	0	0
$490$	−1.11433e11	−0.0873235
$491$	5.62296e11	0.436615	0.218307	−	0.975880i	$-0.429946\pi$
$491$	5.62296e11	0.436615	0.218307	+	0.975880i	$0.429946\pi$
$492$	0	0
$493$	5.72251e11	0.436290
$494$	3.62908e11	0.274173
$495$	0	0
$496$	−9.05716e11	−0.671931
$497$	4.61945e11	0.339614
$498$	0	0
$499$	1.13252e12	0.817702	0.408851	−	0.912601i	$-0.365930\pi$
$499$	1.13252e12	0.817702	0.408851	+	0.912601i	$0.365930\pi$
$500$	1.08528e11	0.0776561
$501$	0	0
$502$	4.37768e11	0.307664
$503$	−5.02176e11	−0.349784	−0.174892	−	0.984588i	$-0.555958\pi$
$503$	−5.02176e11	−0.349784	−0.174892	+	0.984588i	$0.555958\pi$
$504$	0	0
$505$	−3.59710e11	−0.246117
$506$	1.20095e12	0.814416
$507$	0	0
$508$	−2.53094e12	−1.68614
$509$	6.17949e11	0.408059	0.204029	−	0.978965i	$-0.434596\pi$
$509$	6.17949e11	0.408059	0.204029	+	0.978965i	$0.434596\pi$
$510$	0	0
$511$	1.94882e11	0.126438
$512$	−2.01526e12	−1.29603
$513$	0	0
$514$	−3.96684e12	−2.50675
$515$	3.73279e11	0.233830
$516$	0	0
$517$	7.39611e11	0.455298
$518$	1.05835e12	0.645870
$519$	0	0
$520$	−1.69443e11	−0.101627
$521$	2.25896e12	1.34319	0.671597	−	0.740916i	$-0.265609\pi$
$521$	2.25896e12	1.34319	0.671597	+	0.740916i	$0.265609\pi$
$522$	0	0
$523$	2.96359e12	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$523$	2.96359e12	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$524$	−1.16300e10	−0.00673887
$525$	0	0
$526$	−4.73674e11	−0.269801
$527$	8.04571e11	0.454377
$528$	0	0
$529$	−1.23611e12	−0.686289
$530$	−6.65373e11	−0.366289
$531$	0	0
$532$	9.63961e10	0.0521743
$533$	−1.89284e12	−1.01588
$534$	0	0
$535$	−8.61979e11	−0.454888
$536$	4.29367e11	0.224692
$537$	0	0
$538$	−4.41415e12	−2.27157
$539$	2.97796e11	0.151974
$540$	0	0
$541$	3.40796e12	1.71043	0.855217	−	0.518271i	$-0.173424\pi$
$541$	3.40796e12	1.71043	0.855217	+	0.518271i	$0.173424\pi$
$542$	−3.70688e12	−1.84507
$543$	0	0
$544$	2.06747e12	1.01215
$545$	−1.23747e12	−0.600826
$546$	0	0
$547$	3.30764e12	1.57970	0.789852	−	0.613298i	$-0.210157\pi$
$547$	3.30764e12	1.57970	0.789852	+	0.613298i	$0.210157\pi$
$548$	−1.96544e12	−0.930994
$549$	0	0
$550$	−6.24085e11	−0.290812
$551$	1.99157e11	0.0920477
$552$	0	0
$553$	2.57200e11	0.116952
$554$	−1.18060e11	−0.0532486
$555$	0	0
$556$	1.66113e12	0.737167
$557$	4.15545e12	1.82923	0.914617	−	0.404322i	$-0.132492\pi$
$557$	4.15545e12	1.82923	0.914617	+	0.404322i	$0.132492\pi$
$558$	0	0
$559$	−2.45670e12	−1.06414
$560$	−4.38388e11	−0.188370
$561$	0	0
$562$	−4.44317e12	−1.87880
$563$	−2.31187e12	−0.969787	−0.484894	−	0.874573i	$-0.661142\pi$
$563$	−2.31187e12	−0.969787	−0.484894	+	0.874573i	$0.661142\pi$
$564$	0	0
$565$	1.60013e12	0.660599
$566$	−2.13887e11	−0.0876014
$567$	0	0
$568$	4.01478e11	0.161843
$569$	4.00934e12	1.60349	0.801747	−	0.597663i	$-0.203904\pi$
$569$	4.00934e12	1.60349	0.801747	+	0.597663i	$0.203904\pi$
$570$	0	0
$571$	2.40366e12	0.946260	0.473130	−	0.880993i	$-0.343124\pi$
$571$	2.40366e12	0.946260	0.473130	+	0.880993i	$0.343124\pi$
$572$	−2.98343e12	−1.16529
$573$	0	0
$574$	−1.08187e12	−0.415979
$575$	−2.93630e11	−0.112020
$576$	0	0
$577$	−2.40214e12	−0.902208	−0.451104	−	0.892471i	$-0.648970\pi$
$577$	−2.40214e12	−0.902208	−0.451104	+	0.892471i	$0.648970\pi$
$578$	1.58477e12	0.590598
$579$	0	0
$580$	6.12645e11	0.224793
$581$	−8.52167e11	−0.310264
$582$	0	0
$583$	1.77816e12	0.637474
$584$	1.69373e11	0.0602540
$585$	0	0
$586$	6.90446e12	2.41875
$587$	2.71458e12	0.943694	0.471847	−	0.881681i	$-0.343588\pi$
$587$	2.71458e12	0.943694	0.471847	+	0.881681i	$0.343588\pi$
$588$	0	0
$589$	2.80010e11	0.0958637
$590$	−6.98447e11	−0.237301
$591$	0	0
$592$	4.16365e12	1.39324
$593$	2.44735e12	0.812736	0.406368	−	0.913710i	$-0.366795\pi$
$593$	2.44735e12	0.812736	0.406368	+	0.913710i	$0.366795\pi$
$594$	0	0
$595$	3.89431e11	0.127381
$596$	−1.87356e12	−0.608217
$597$	0	0
$598$	−3.02043e12	−0.965859
$599$	1.24301e11	0.0394506	0.0197253	−	0.999805i	$-0.493721\pi$
$599$	1.24301e11	0.0394506	0.0197253	+	0.999805i	$0.493721\pi$
$600$	0	0
$601$	3.71305e12	1.16090	0.580451	−	0.814295i	$-0.302876\pi$
$601$	3.71305e12	1.16090	0.580451	+	0.814295i	$0.302876\pi$
$602$	−1.40415e12	−0.435741
$603$	0	0
$604$	4.52736e12	1.38414
$605$	1.94104e11	0.0589028
$606$	0	0
$607$	8.89382e11	0.265913	0.132956	−	0.991122i	$-0.457553\pi$
$607$	8.89382e11	0.265913	0.132956	+	0.991122i	$0.457553\pi$
$608$	7.19528e11	0.213541
$609$	0	0
$610$	1.80349e12	0.527387
$611$	−1.86015e12	−0.539962
$612$	0	0
$613$	1.53860e12	0.440103	0.220052	−	0.975488i	$-0.429377\pi$
$613$	1.53860e12	0.440103	0.220052	+	0.975488i	$0.429377\pi$
$614$	−7.49368e12	−2.12783
$615$	0	0
$616$	2.58816e11	0.0724233
$617$	−3.70682e12	−1.02972	−0.514859	−	0.857275i	$-0.672156\pi$
$617$	−3.70682e12	−1.02972	−0.514859	+	0.857275i	$0.672156\pi$
$618$	0	0
$619$	−5.27045e12	−1.44291	−0.721456	−	0.692460i	$-0.756527\pi$
$619$	−5.27045e12	−1.44291	−0.721456	+	0.692460i	$0.756527\pi$
$620$	8.61363e11	0.234112
$621$	0	0
$622$	−4.59566e12	−1.23109
$623$	−1.06228e12	−0.282516
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	6.11552e12	1.59165
$627$	0	0
$628$	3.07306e12	0.788410
$629$	−3.69868e12	−0.942146
$630$	0	0
$631$	1.99691e12	0.501448	0.250724	−	0.968059i	$-0.419331\pi$
$631$	1.99691e12	0.501448	0.250724	+	0.968059i	$0.419331\pi$
$632$	2.23534e11	0.0557336
$633$	0	0
$634$	−2.90033e12	−0.712928
$635$	−3.55845e12	−0.868519
$636$	0	0
$637$	−7.48970e11	−0.180234
$638$	−3.52300e12	−0.841820
$639$	0	0
$640$	−6.77840e11	−0.159705
$641$	2.45489e12	0.574342	0.287171	−	0.957879i	$-0.407285\pi$
$641$	2.45489e12	0.574342	0.287171	+	0.957879i	$0.407285\pi$
$642$	0	0
$643$	−4.85617e12	−1.12033	−0.560163	−	0.828383i	$-0.689261\pi$
$643$	−4.85617e12	−1.12033	−0.560163	+	0.828383i	$0.689261\pi$
$644$	−8.02292e11	−0.183800
$645$	0	0
$646$	−7.24894e11	−0.163768
$647$	4.02189e12	0.902320	0.451160	−	0.892443i	$-0.351010\pi$
$647$	4.02189e12	0.902320	0.451160	+	0.892443i	$0.351010\pi$
$648$	0	0
$649$	1.86655e12	0.412989
$650$	1.56960e12	0.344889
$651$	0	0
$652$	−6.17210e11	−0.133758
$653$	−2.63594e12	−0.567317	−0.283659	−	0.958925i	$-0.591548\pi$
$653$	−2.63594e12	−0.567317	−0.283659	+	0.958925i	$0.591548\pi$
$654$	0	0
$655$	−1.63515e10	−0.00347114
$656$	−4.25618e12	−0.897331
$657$	0	0
$658$	−1.06319e12	−0.221102
$659$	−9.36892e12	−1.93511	−0.967554	−	0.252663i	$-0.918694\pi$
$659$	−9.36892e12	−1.93511	−0.967554	+	0.252663i	$0.918694\pi$
$660$	0	0
$661$	8.25438e11	0.168181	0.0840907	−	0.996458i	$-0.473201\pi$
$661$	8.25438e11	0.168181	0.0840907	+	0.996458i	$0.473201\pi$
$662$	4.25014e12	0.860086
$663$	0	0
$664$	−7.40623e11	−0.147857
$665$	1.35531e11	0.0268746
$666$	0	0
$667$	−1.65756e12	−0.324266
$668$	−1.22963e12	−0.238936
$669$	0	0
$670$	−3.97735e12	−0.762530
$671$	−4.81970e12	−0.917843
$672$	0	0
$673$	−3.23621e12	−0.608092	−0.304046	−	0.952657i	$-0.598338\pi$
$673$	−3.23621e12	−0.608092	−0.304046	+	0.952657i	$0.598338\pi$
$674$	1.34792e13	2.51590
$675$	0	0
$676$	2.78944e12	0.513756
$677$	7.40697e12	1.35516	0.677581	−	0.735448i	$-0.263028\pi$
$677$	7.40697e12	1.35516	0.677581	+	0.735448i	$0.263028\pi$
$678$	0	0
$679$	−1.53775e12	−0.277634
$680$	3.38457e11	0.0607034
$681$	0	0
$682$	−4.95325e12	−0.876719
$683$	3.59414e12	0.631978	0.315989	−	0.948763i	$-0.397664\pi$
$683$	3.59414e12	0.631978	0.315989	+	0.948763i	$0.397664\pi$
$684$	0	0
$685$	−2.76337e12	−0.479547
$686$	−4.28081e11	−0.0738018
$687$	0	0
$688$	−5.52405e12	−0.939961
$689$	−4.47215e12	−0.756014
$690$	0	0
$691$	2.54498e11	0.0424652	0.0212326	−	0.999775i	$-0.493241\pi$
$691$	2.54498e11	0.0424652	0.0212326	+	0.999775i	$0.493241\pi$
$692$	−7.00183e11	−0.116074
$693$	0	0
$694$	−1.84280e12	−0.301550
$695$	2.33551e12	0.379709
$696$	0	0
$697$	3.78087e12	0.606799
$698$	−4.90068e12	−0.781461
$699$	0	0
$700$	4.16920e11	0.0656314
$701$	2.85836e11	0.0447081	0.0223540	−	0.999750i	$-0.492884\pi$
$701$	2.85836e11	0.0447081	0.0223540	+	0.999750i	$0.492884\pi$
$702$	0	0
$703$	−1.28723e12	−0.198772
$704$	−5.00150e12	−0.767402
$705$	0	0
$706$	1.47873e13	2.24010
$707$	−1.38186e12	−0.208007
$708$	0	0
$709$	1.36881e12	0.203440	0.101720	−	0.994813i	$-0.467565\pi$
$709$	1.36881e12	0.203440	0.101720	+	0.994813i	$0.467565\pi$
$710$	−3.71901e12	−0.549243
$711$	0	0
$712$	−9.23234e11	−0.134633
$713$	−2.33048e12	−0.337709
$714$	0	0
$715$	−4.19464e12	−0.600230
$716$	2.94722e12	0.419087
$717$	0	0
$718$	−1.74657e13	−2.45260
$719$	1.27505e13	1.77929	0.889645	−	0.456653i	$-0.150952\pi$
$719$	1.27505e13	1.77929	0.889645	+	0.456653i	$0.150952\pi$
$720$	0	0
$721$	1.43399e12	0.197623
$722$	9.72774e12	1.33228
$723$	0	0
$724$	1.21459e13	1.64288
$725$	8.61367e11	0.115789
$726$	0	0
$727$	−7.20369e12	−0.956423	−0.478212	−	0.878245i	$-0.658715\pi$
$727$	−7.20369e12	−0.956423	−0.478212	+	0.878245i	$0.658715\pi$
$728$	−6.50934e11	−0.0858906
$729$	0	0
$730$	−1.56895e12	−0.204482
$731$	4.90716e12	0.635626
$732$	0	0
$733$	−1.27852e11	−0.0163583	−0.00817917	−	0.999967i	$-0.502604\pi$
$733$	−1.27852e11	−0.0163583	−0.00817917	+	0.999967i	$0.502604\pi$
$734$	−9.09175e12	−1.15615
$735$	0	0
$736$	−5.98854e12	−0.752265
$737$	1.06292e13	1.32708
$738$	0	0
$739$	−3.13217e12	−0.386318	−0.193159	−	0.981167i	$-0.561873\pi$
$739$	−3.13217e12	−0.386318	−0.193159	+	0.981167i	$0.561873\pi$
$740$	−3.95975e12	−0.485429
$741$	0	0
$742$	−2.55610e12	−0.309571
$743$	−4.86428e12	−0.585556	−0.292778	−	0.956180i	$-0.594580\pi$
$743$	−4.86428e12	−0.585556	−0.292778	+	0.956180i	$0.594580\pi$
$744$	0	0
$745$	−2.63419e12	−0.313287
$746$	2.23701e13	2.64450
$747$	0	0
$748$	5.95928e12	0.696044
$749$	−3.31138e12	−0.384450
$750$	0	0
$751$	1.18619e13	1.36074	0.680370	−	0.732869i	$-0.261819\pi$
$751$	1.18619e13	1.36074	0.680370	+	0.732869i	$0.261819\pi$
$752$	−4.18268e12	−0.476952
$753$	0	0
$754$	8.86049e12	0.998358
$755$	6.36538e12	0.712957
$756$	0	0
$757$	1.71751e13	1.90094	0.950471	−	0.310813i	$-0.100601\pi$
$757$	1.71751e13	1.90094	0.950471	+	0.310813i	$0.100601\pi$
$758$	2.36754e12	0.260487
$759$	0	0
$760$	1.17791e11	0.0128071
$761$	−6.68211e12	−0.722241	−0.361121	−	0.932519i	$-0.617606\pi$
$761$	−6.68211e12	−0.722241	−0.361121	+	0.932519i	$0.617606\pi$
$762$	0	0
$763$	−4.75385e12	−0.507791
$764$	−9.08683e12	−0.964922
$765$	0	0
$766$	−1.19084e12	−0.124975
$767$	−4.69445e12	−0.489785
$768$	0	0
$769$	1.07137e13	1.10477	0.552385	−	0.833589i	$-0.313718\pi$
$769$	1.07137e13	1.10477	0.552385	+	0.833589i	$0.313718\pi$
$770$	−2.39749e12	−0.245781
$771$	0	0
$772$	−1.09639e13	−1.11094
$773$	−4.59483e12	−0.462873	−0.231437	−	0.972850i	$-0.574343\pi$
$773$	−4.59483e12	−0.462873	−0.231437	+	0.972850i	$0.574343\pi$
$774$	0	0
$775$	1.21106e12	0.120589
$776$	−1.33647e12	−0.132306
$777$	0	0
$778$	−9.79286e12	−0.958300
$779$	1.31583e12	0.128021
$780$	0	0
$781$	9.93877e12	0.955879
$782$	6.03319e12	0.576922
$783$	0	0
$784$	−1.68411e12	−0.159202
$785$	4.32066e12	0.406104
$786$	0	0
$787$	1.10128e13	1.02332	0.511662	−	0.859187i	$-0.329030\pi$
$787$	1.10128e13	1.02332	0.511662	+	0.859187i	$0.329030\pi$
$788$	1.67144e13	1.54427
$789$	0	0
$790$	−2.07066e12	−0.189141
$791$	6.14707e12	0.558308
$792$	0	0
$793$	1.21217e13	1.08852
$794$	−2.69846e13	−2.40948
$795$	0	0
$796$	5.00958e12	0.442275
$797$	1.04621e12	0.0918449	0.0459224	−	0.998945i	$-0.485377\pi$
$797$	1.04621e12	0.0918449	0.0459224	+	0.998945i	$0.485377\pi$
$798$	0	0
$799$	3.71558e12	0.322527
$800$	3.11201e12	0.268619
$801$	0	0
$802$	6.24577e12	0.533091
$803$	4.19290e12	0.355872
$804$	0	0
$805$	−1.12801e12	−0.0946740
$806$	1.24576e13	1.03975
$807$	0	0
$808$	−1.20098e12	−0.0991256
$809$	7.20013e12	0.590979	0.295489	−	0.955346i	$-0.404517\pi$
$809$	7.20013e12	0.590979	0.295489	+	0.955346i	$0.404517\pi$
$810$	0	0
$811$	1.58290e13	1.28487	0.642437	−	0.766338i	$-0.277923\pi$
$811$	1.58290e13	1.28487	0.642437	+	0.766338i	$0.277923\pi$
$812$	2.35354e12	0.189985
$813$	0	0
$814$	2.27705e13	1.81787
$815$	−8.67786e11	−0.0688975
$816$	0	0
$817$	1.70781e12	0.134103
$818$	−1.85739e13	−1.45048
$819$	0	0
$820$	4.04776e12	0.312645
$821$	1.62250e13	1.24635	0.623177	−	0.782081i	$-0.285842\pi$
$821$	1.62250e13	1.24635	0.623177	+	0.782081i	$0.285842\pi$
$822$	0	0
$823$	1.27028e13	0.965160	0.482580	−	0.875852i	$-0.339700\pi$
$823$	1.27028e13	0.965160	0.482580	+	0.875852i	$0.339700\pi$
$824$	1.24629e12	0.0941771
$825$	0	0
$826$	−2.68316e12	−0.200556
$827$	−2.21026e13	−1.64312	−0.821558	−	0.570125i	$-0.806895\pi$
$827$	−2.21026e13	−1.64312	−0.821558	+	0.570125i	$0.806895\pi$
$828$	0	0
$829$	7.98616e12	0.587277	0.293638	−	0.955917i	$-0.405134\pi$
$829$	7.98616e12	0.587277	0.293638	+	0.955917i	$0.405134\pi$
$830$	6.86059e12	0.501776
$831$	0	0
$832$	1.25790e13	0.910103
$833$	1.49604e12	0.107656
$834$	0	0
$835$	−1.72884e12	−0.123074
$836$	2.07397e12	0.146850
$837$	0	0
$838$	2.38348e12	0.166960
$839$	2.51460e12	0.175202	0.0876011	−	0.996156i	$-0.472080\pi$
$839$	2.51460e12	0.175202	0.0876011	+	0.996156i	$0.472080\pi$
$840$	0	0
$841$	−9.64468e12	−0.664823
$842$	9.03286e11	0.0619328
$843$	0	0
$844$	1.31901e13	0.894763
$845$	3.92190e12	0.264631
$846$	0	0
$847$	7.45671e11	0.0497820
$848$	−1.00559e13	−0.667792
$849$	0	0
$850$	−3.13522e12	−0.206007
$851$	1.07134e13	0.700236
$852$	0	0
$853$	2.20939e13	1.42890	0.714449	−	0.699687i	$-0.246677\pi$
$853$	2.20939e13	1.42890	0.714449	+	0.699687i	$0.246677\pi$
$854$	6.92829e12	0.445724
$855$	0	0
$856$	−2.87793e12	−0.183210
$857$	−2.65431e13	−1.68088	−0.840441	−	0.541903i	$-0.817704\pi$
$857$	−2.65431e13	−1.68088	−0.840441	+	0.541903i	$0.817704\pi$
$858$	0	0
$859$	−8.96349e12	−0.561704	−0.280852	−	0.959751i	$-0.590617\pi$
$859$	−8.96349e12	−0.561704	−0.280852	+	0.959751i	$0.590617\pi$
$860$	5.25354e12	0.327498
$861$	0	0
$862$	−2.64556e13	−1.63206
$863$	−1.90231e13	−1.16743	−0.583717	−	0.811957i	$-0.698402\pi$
$863$	−1.90231e13	−1.16743	−0.583717	+	0.811957i	$0.698402\pi$
$864$	0	0
$865$	−9.84444e11	−0.0597886
$866$	3.95624e13	2.39030
$867$	0	0
$868$	3.30901e12	0.197861
$869$	5.53368e12	0.329174
$870$	0	0
$871$	−2.67328e13	−1.57385
$872$	−4.13159e12	−0.241988
$873$	0	0
$874$	2.09969e12	0.121718
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−8.89591e12	−0.507799	−0.253900	−	0.967231i	$-0.581713\pi$
$877$	−8.89591e12	−0.507799	−0.253900	+	0.967231i	$0.581713\pi$
$878$	1.93140e13	1.09685
$879$	0	0
$880$	−9.43194e12	−0.530187
$881$	−3.32011e13	−1.85678	−0.928391	−	0.371605i	$-0.878808\pi$
$881$	−3.32011e13	−1.85678	−0.928391	+	0.371605i	$0.878808\pi$
$882$	0	0
$883$	6.33990e12	0.350961	0.175481	−	0.984483i	$-0.443852\pi$
$883$	6.33990e12	0.350961	0.175481	+	0.984483i	$0.443852\pi$
$884$	−1.49878e13	−0.825475
$885$	0	0
$886$	9.53064e12	0.519601
$887$	4.67382e12	0.253522	0.126761	−	0.991933i	$-0.459542\pi$
$887$	4.67382e12	0.253522	0.126761	+	0.991933i	$0.459542\pi$
$888$	0	0
$889$	−1.36701e13	−0.734032
$890$	8.55217e12	0.456900
$891$	0	0
$892$	−6.81372e12	−0.360365
$893$	1.29311e12	0.0680462
$894$	0	0
$895$	4.14374e12	0.215868
$896$	−2.60399e12	−0.134975
$897$	0	0
$898$	7.77920e12	0.399201
$899$	6.83651e12	0.349072
$900$	0	0
$901$	8.93294e12	0.451578
$902$	−2.32765e13	−1.17082
$903$	0	0
$904$	5.34245e12	0.266062
$905$	1.70769e13	0.846235
$906$	0	0
$907$	−1.02073e13	−0.500816	−0.250408	−	0.968140i	$-0.580565\pi$
$907$	−1.02073e13	−0.500816	−0.250408	+	0.968140i	$0.580565\pi$
$908$	6.72165e12	0.328163
$909$	0	0
$910$	6.02978e12	0.291484
$911$	2.05078e13	0.986474	0.493237	−	0.869895i	$-0.335813\pi$
$911$	2.05078e13	0.986474	0.493237	+	0.869895i	$0.335813\pi$
$912$	0	0
$913$	−1.83344e13	−0.873270
$914$	2.25302e13	1.06784
$915$	0	0
$916$	−2.66385e13	−1.25020
$917$	−6.28159e10	−0.00293365
$918$	0	0
$919$	1.80163e13	0.833192	0.416596	−	0.909092i	$-0.363223\pi$
$919$	1.80163e13	0.833192	0.416596	+	0.909092i	$0.363223\pi$
$920$	−9.80357e11	−0.0451169
$921$	0	0
$922$	2.27965e12	0.103891
$923$	−2.49964e13	−1.13363
$924$	0	0
$925$	−5.56734e12	−0.250040
$926$	1.42552e13	0.637123
$927$	0	0
$928$	1.75675e13	0.777577
$929$	2.38578e13	1.05090	0.525448	−	0.850826i	$-0.323898\pi$
$929$	2.38578e13	1.05090	0.525448	+	0.850826i	$0.323898\pi$
$930$	0	0
$931$	5.20656e11	0.0227132
$932$	2.75849e13	1.19756
$933$	0	0
$934$	1.98939e13	0.855379
$935$	8.37863e12	0.358526
$936$	0	0
$937$	3.41070e13	1.44549	0.722746	−	0.691114i	$-0.242880\pi$
$937$	3.41070e13	1.44549	0.722746	+	0.691114i	$0.242880\pi$
$938$	−1.52794e13	−0.644456
$939$	0	0
$940$	3.97786e12	0.166178
$941$	8.20908e12	0.341304	0.170652	−	0.985331i	$-0.445413\pi$
$941$	8.20908e12	0.341304	0.170652	+	0.985331i	$0.445413\pi$
$942$	0	0
$943$	−1.09515e13	−0.450994
$944$	−1.05558e13	−0.432630
$945$	0	0
$946$	−3.02103e13	−1.22644
$947$	3.59289e13	1.45167	0.725837	−	0.687867i	$-0.241453\pi$
$947$	3.59289e13	1.45167	0.725837	+	0.687867i	$0.241453\pi$
$948$	0	0
$949$	−1.05453e13	−0.422048
$950$	−1.09113e12	−0.0434630
$951$	0	0
$952$	1.30021e12	0.0513037
$953$	−3.90153e13	−1.53221	−0.766103	−	0.642717i	$-0.777807\pi$
$953$	−3.90153e13	−1.53221	−0.766103	+	0.642717i	$0.777807\pi$
$954$	0	0
$955$	−1.27759e13	−0.497023
$956$	2.39039e13	0.925566
$957$	0	0
$958$	−6.39509e13	−2.45303
$959$	−1.06158e13	−0.405291
$960$	0	0
$961$	−1.68277e13	−0.636456
$962$	−5.72687e13	−2.15590
$963$	0	0
$964$	−1.03001e13	−0.384147
$965$	−1.54151e13	−0.572234
$966$	0	0
$967$	−3.06022e12	−0.112547	−0.0562735	−	0.998415i	$-0.517922\pi$
$967$	−3.06022e12	−0.112547	−0.0562735	+	0.998415i	$0.517922\pi$
$968$	6.48067e11	0.0237236
$969$	0	0
$970$	1.23801e13	0.449004
$971$	2.82554e12	0.102004	0.0510018	−	0.998699i	$-0.483759\pi$
$971$	2.82554e12	0.102004	0.0510018	+	0.998699i	$0.483759\pi$
$972$	0	0
$973$	8.97211e12	0.320912
$974$	−2.68510e13	−0.955972
$975$	0	0
$976$	2.72566e13	0.961494
$977$	4.75492e12	0.166962	0.0834810	−	0.996509i	$-0.473396\pi$
$977$	4.75492e12	0.166962	0.0834810	+	0.996509i	$0.473396\pi$
$978$	0	0
$979$	−2.28550e13	−0.795170
$980$	1.60164e12	0.0554686
$981$	0	0
$982$	−1.73906e13	−0.596778
$983$	−6.98713e12	−0.238676	−0.119338	−	0.992854i	$-0.538077\pi$
$983$	−6.98713e12	−0.238676	−0.119338	+	0.992854i	$0.538077\pi$
$984$	0	0
$985$	2.35001e13	0.795440
$986$	−1.76985e13	−0.596334
$987$	0	0
$988$	−5.21612e12	−0.174157
$989$	−1.42138e13	−0.472420
$990$	0	0
$991$	2.88898e13	0.951508	0.475754	−	0.879578i	$-0.342175\pi$
$991$	2.88898e13	0.951508	0.475754	+	0.879578i	$0.342175\pi$
$992$	2.46994e13	0.809813
$993$	0	0
$994$	−1.42869e13	−0.464195
$995$	7.04337e12	0.227812
$996$	0	0
$997$	−6.04428e13	−1.93739	−0.968694	−	0.248257i	$-0.920142\pi$
$997$	−6.04428e13	−1.93739	−0.968694	+	0.248257i	$0.920142\pi$
$998$	−3.50265e13	−1.11766
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.e.1.1		4
3.2	odd	2		105.10.a.d.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.d.1.4	✓	4	3.2	odd	2
315.10.a.e.1.1		4	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)

\(f(q)\)	\(=\)	\(q-30.9278 q^{2} +444.529 q^{4} +625.000 q^{5} +2401.00 q^{7} +2086.72 q^{8} -19329.9 q^{10} +51657.7 q^{11} -129921. q^{13} -74257.7 q^{14} -292137. q^{16} +259513. q^{17} +90316.4 q^{19} +277831. q^{20} -1.59766e6 q^{22} -751692. q^{23} +390625. q^{25} +4.01818e6 q^{26} +1.06731e6 q^{28} +2.20510e6 q^{29} +3.10032e6 q^{31} +7.96675e6 q^{32} -8.02616e6 q^{34} +1.50062e6 q^{35} -1.42524e7 q^{37} -2.79329e6 q^{38} +1.30420e6 q^{40} +1.45691e7 q^{41} +1.89091e7 q^{43} +2.29633e7 q^{44} +2.32482e7 q^{46} +1.43175e7 q^{47} +5.76480e6 q^{49} -1.20812e7 q^{50} -5.77538e7 q^{52} +3.44220e7 q^{53} +3.22860e7 q^{55} +5.01022e6 q^{56} -6.81989e7 q^{58} +3.61330e7 q^{59} -9.33007e7 q^{61} -9.58860e7 q^{62} -9.68200e7 q^{64} -8.12008e7 q^{65} +2.05762e8 q^{67} +1.15361e8 q^{68} -4.64110e7 q^{70} +1.92397e8 q^{71} +8.11669e7 q^{73} +4.40795e8 q^{74} +4.01483e7 q^{76} +1.24030e8 q^{77} +1.07122e8 q^{79} -1.82585e8 q^{80} -4.50592e8 q^{82} -3.54922e8 q^{83} +1.62195e8 q^{85} -5.84818e8 q^{86} +1.07795e8 q^{88} -4.42433e8 q^{89} -3.11941e8 q^{91} -3.34149e8 q^{92} -4.42810e8 q^{94} +5.64478e7 q^{95} -6.40463e8 q^{97} -1.78293e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) 4 * q + 13 * q^2 + 501 * q^4 + 2500 * q^5 + 9604 * q^7 + 16263 * q^8 + 8125 * q^10 + 87062 * q^11 + 39494 * q^13 + 31213 * q^14 + 328849 * q^16 + 291756 * q^17 + 50482 * q^19 + 313125 * q^20 - 1003016 * q^22 + 1048656 * q^23 + 1562500 * q^25 + 7622594 * q^26 + 1202901 * q^28 + 8018996 * q^29 - 1148130 * q^31 + 9765119 * q^32 + 1255358 * q^34 + 6002500 * q^35 - 23416328 * q^37 - 11334892 * q^38 + 10164375 * q^40 + 8000860 * q^41 + 27520044 * q^43 - 42192 * q^44 + 90166880 * q^46 + 78277092 * q^47 + 23059204 * q^49 + 5078125 * q^50 - 62776870 * q^52 - 36168514 * q^53 + 54413750 * q^55 + 39047463 * q^56 - 32819954 * q^58 + 132924868 * q^59 - 119014876 * q^61 + 76291172 * q^62 - 400777983 * q^64 + 24683750 * q^65 - 170018492 * q^67 + 345488158 * q^68 + 19508125 * q^70 + 743169022 * q^71 - 360015502 * q^73 + 547147386 * q^74 - 472117316 * q^76 + 209035862 * q^77 - 613400256 * q^79 + 205530625 * q^80 - 714522510 * q^82 + 1222551024 * q^83 + 182347500 * q^85 + 593721752 * q^86 - 855674336 * q^88 - 835315808 * q^89 + 94825094 * q^91 + 1276931776 * q^92 + 950602540 * q^94 + 31551250 * q^95 - 704297762 * q^97 + 74942413 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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