LMFDB - Embedded newform 315.10.a.l.1.2

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			1.2
Root			$-36.8299$ 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)

$f(q)$	$=$	\(q-39.8299 q^{2} +1074.42 q^{4} -625.000 q^{5} -2401.00 q^{7} -22401.1 q^{8} +24893.7 q^{10} +50463.3 q^{11} +3922.21 q^{13} +95631.6 q^{14} +342132. q^{16} +589429. q^{17} +796460. q^{19} -671513. q^{20} -2.00995e6 q^{22} +436793. q^{23} +390625. q^{25} -156221. q^{26} -2.57968e6 q^{28} -1.20833e6 q^{29} +3.38466e6 q^{31} -2.15768e6 q^{32} -2.34769e7 q^{34} +1.50062e6 q^{35} +1.70047e7 q^{37} -3.17229e7 q^{38} +1.40007e7 q^{40} -2.58643e7 q^{41} +1.44925e7 q^{43} +5.42188e7 q^{44} -1.73974e7 q^{46} +5.55921e7 q^{47} +5.76480e6 q^{49} -1.55586e7 q^{50} +4.21410e6 q^{52} +7.67285e7 q^{53} -3.15395e7 q^{55} +5.37851e7 q^{56} +4.81276e7 q^{58} +1.15070e8 q^{59} -1.54502e8 q^{61} -1.34811e8 q^{62} -8.92311e7 q^{64} -2.45138e6 q^{65} +2.72000e8 q^{67} +6.33295e8 q^{68} -5.97697e7 q^{70} +2.24602e7 q^{71} -8.21761e6 q^{73} -6.77297e8 q^{74} +8.55733e8 q^{76} -1.21162e8 q^{77} +5.20375e8 q^{79} -2.13832e8 q^{80} +1.03017e9 q^{82} -9.36372e7 q^{83} -3.68393e8 q^{85} -5.77233e8 q^{86} -1.13043e9 q^{88} -1.94318e8 q^{89} -9.41722e6 q^{91} +4.69299e8 q^{92} -2.21423e9 q^{94} -4.97788e8 q^{95} +1.48287e8 q^{97} -2.29611e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−39.8299	−1.76025	−0.880125	−	0.474743i	$-0.842541\pi$
$2$	−39.8299	−1.76025	−0.880125	+	0.474743i	$0.842541\pi$
$3$	0	0
$3$	0	0
$4$	1074.42	2.09848
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−22401.1	−1.93359
$9$	0	0
$10$	24893.7	0.787207
$11$	50463.3	1.03922	0.519611	−	0.854403i	$-0.326077\pi$
$11$	50463.3	1.03922	0.519611	+	0.854403i	$0.326077\pi$
$12$	0	0
$13$	3922.21	0.0380878	0.0190439	−	0.999819i	$-0.493938\pi$
$13$	3922.21	0.0380878	0.0190439	+	0.999819i	$0.493938\pi$
$14$	95631.6	0.665312
$15$	0	0
$16$	342132.	1.30513
$17$	589429.	1.71164	0.855819	−	0.517276i	$-0.173054\pi$
$17$	589429.	1.71164	0.855819	+	0.517276i	$0.173054\pi$
$18$	0	0
$19$	796460.	1.40208	0.701040	−	0.713122i	$-0.252719\pi$
$19$	796460.	1.40208	0.701040	+	0.713122i	$0.252719\pi$
$20$	−671513.	−0.938467
$21$	0	0
$22$	−2.00995e6	−1.82929
$23$	436793.	0.325462	0.162731	−	0.986670i	$-0.447970\pi$
$23$	436793.	0.325462	0.162731	+	0.986670i	$0.447970\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−156221.	−0.0670439
$27$	0	0
$28$	−2.57968e6	−0.793150
$29$	−1.20833e6	−0.317244	−0.158622	−	0.987339i	$-0.550705\pi$
$29$	−1.20833e6	−0.317244	−0.158622	+	0.987339i	$0.550705\pi$
$30$	0	0
$31$	3.38466e6	0.658244	0.329122	−	0.944287i	$-0.393247\pi$
$31$	3.38466e6	0.658244	0.329122	+	0.944287i	$0.393247\pi$
$32$	−2.15768e6	−0.363758
$33$	0	0
$34$	−2.34769e7	−3.01291
$35$	1.50062e6	0.169031
$36$	0	0
$37$	1.70047e7	1.49163	0.745817	−	0.666151i	$-0.232059\pi$
$37$	1.70047e7	1.49163	0.745817	+	0.666151i	$0.232059\pi$
$38$	−3.17229e7	−2.46801
$39$	0	0
$40$	1.40007e7	0.864729
$41$	−2.58643e7	−1.42946	−0.714732	−	0.699398i	$-0.753451\pi$
$41$	−2.58643e7	−1.42946	−0.714732	+	0.699398i	$0.753451\pi$
$42$	0	0
$43$	1.44925e7	0.646449	0.323224	−	0.946322i	$-0.395233\pi$
$43$	1.44925e7	0.646449	0.323224	+	0.946322i	$0.395233\pi$
$44$	5.42188e7	2.18078
$45$	0	0
$46$	−1.73974e7	−0.572894
$47$	5.55921e7	1.66178	0.830889	−	0.556438i	$-0.187832\pi$
$47$	5.55921e7	1.66178	0.830889	+	0.556438i	$0.187832\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−1.55586e7	−0.352050
$51$	0	0
$52$	4.21410e6	0.0799263
$53$	7.67285e7	1.33572	0.667860	−	0.744287i	$-0.267210\pi$
$53$	7.67285e7	1.33572	0.667860	+	0.744287i	$0.267210\pi$
$54$	0	0
$55$	−3.15395e7	−0.464754
$56$	5.37851e7	0.730829
$57$	0	0
$58$	4.81276e7	0.558429
$59$	1.15070e8	1.23631	0.618156	−	0.786056i	$-0.287880\pi$
$59$	1.15070e8	1.23631	0.618156	+	0.786056i	$0.287880\pi$
$60$	0	0
$61$	−1.54502e8	−1.42873	−0.714366	−	0.699772i	$-0.753285\pi$
$61$	−1.54502e8	−1.42873	−0.714366	+	0.699772i	$0.753285\pi$
$62$	−1.34811e8	−1.15867
$63$	0	0
$64$	−8.92311e7	−0.664823
$65$	−2.45138e6	−0.0170334
$66$	0	0
$67$	2.72000e8	1.64904	0.824522	−	0.565829i	$-0.191444\pi$
$67$	2.72000e8	1.64904	0.824522	+	0.565829i	$0.191444\pi$
$68$	6.33295e8	3.59183
$69$	0	0
$70$	−5.97697e7	−0.297536
$71$	2.24602e7	0.104894	0.0524470	−	0.998624i	$-0.483298\pi$
$71$	2.24602e7	0.104894	0.0524470	+	0.998624i	$0.483298\pi$
$72$	0	0
$73$	−8.21761e6	−0.0338682	−0.0169341	−	0.999857i	$-0.505391\pi$
$73$	−8.21761e6	−0.0338682	−0.0169341	+	0.999857i	$0.505391\pi$
$74$	−6.77297e8	−2.62565
$75$	0	0
$76$	8.55733e8	2.94223
$77$	−1.21162e8	−0.392789
$78$	0	0
$79$	5.20375e8	1.50312	0.751561	−	0.659663i	$-0.229301\pi$
$79$	5.20375e8	1.50312	0.751561	+	0.659663i	$0.229301\pi$
$80$	−2.13832e8	−0.583671
$81$	0	0
$82$	1.03017e9	2.51621
$83$	−9.36372e7	−0.216569	−0.108285	−	0.994120i	$-0.534536\pi$
$83$	−9.36372e7	−0.216569	−0.108285	+	0.994120i	$0.534536\pi$
$84$	0	0
$85$	−3.68393e8	−0.765467
$86$	−5.77233e8	−1.13791
$87$	0	0
$88$	−1.13043e9	−2.00943
$89$	−1.94318e8	−0.328289	−0.164145	−	0.986436i	$-0.552486\pi$
$89$	−1.94318e8	−0.328289	−0.164145	+	0.986436i	$0.552486\pi$
$90$	0	0
$91$	−9.41722e6	−0.0143958
$92$	4.69299e8	0.682974
$93$	0	0
$94$	−2.21423e9	−2.92514
$95$	−4.97788e8	−0.627030
$96$	0	0
$97$	1.48287e8	0.170071	0.0850355	−	0.996378i	$-0.472900\pi$
$97$	1.48287e8	0.170071	0.0850355	+	0.996378i	$0.472900\pi$
$98$	−2.29611e8	−0.251464
$99$	0	0
$100$	4.19695e8	0.419695
$101$	−1.15408e9	−1.10355	−0.551774	−	0.833994i	$-0.686049\pi$
$101$	−1.15408e9	−1.10355	−0.551774	+	0.833994i	$0.686049\pi$
$102$	0	0
$103$	−1.93967e9	−1.69809	−0.849043	−	0.528324i	$-0.822821\pi$
$103$	−1.93967e9	−1.69809	−0.849043	+	0.528324i	$0.822821\pi$
$104$	−8.78619e7	−0.0736462
$105$	0	0
$106$	−3.05609e9	−2.35120
$107$	−2.63127e9	−1.94061	−0.970306	−	0.241881i	$-0.922236\pi$
$107$	−2.63127e9	−1.94061	−0.970306	+	0.241881i	$0.922236\pi$
$108$	0	0
$109$	5.52674e8	0.375016	0.187508	−	0.982263i	$-0.439959\pi$
$109$	5.52674e8	0.375016	0.187508	+	0.982263i	$0.439959\pi$
$110$	1.25622e9	0.818083
$111$	0	0
$112$	−8.21458e8	−0.493292
$113$	−1.76808e9	−1.02011	−0.510056	−	0.860141i	$-0.670375\pi$
$113$	−1.76808e9	−1.02011	−0.510056	+	0.860141i	$0.670375\pi$
$114$	0	0
$115$	−2.72995e8	−0.145551
$116$	−1.29825e9	−0.665730
$117$	0	0
$118$	−4.58322e9	−2.17622
$119$	−1.41522e9	−0.646938
$120$	0	0
$121$	1.88594e8	0.0799824
$122$	6.15381e9	2.51493
$123$	0	0
$124$	3.63654e9	1.38131
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	2.81244e9	0.959327	0.479664	−	0.877452i	$-0.340759\pi$
$127$	2.81244e9	0.959327	0.479664	+	0.877452i	$0.340759\pi$
$128$	4.65880e9	1.53401
$129$	0	0
$130$	9.76382e7	0.0299830
$131$	1.16151e9	0.344591	0.172295	−	0.985045i	$-0.444882\pi$
$131$	1.16151e9	0.344591	0.172295	+	0.985045i	$0.444882\pi$
$132$	0	0
$133$	−1.91230e9	−0.529937
$134$	−1.08337e10	−2.90273
$135$	0	0
$136$	−1.32039e10	−3.30961
$137$	3.47682e8	0.0843217	0.0421609	−	0.999111i	$-0.486576\pi$
$137$	3.47682e8	0.0843217	0.0421609	+	0.999111i	$0.486576\pi$
$138$	0	0
$139$	−2.03085e9	−0.461436	−0.230718	−	0.973021i	$-0.574108\pi$
$139$	−2.03085e9	−0.461436	−0.230718	+	0.973021i	$0.574108\pi$
$140$	1.61230e9	0.354707
$141$	0	0
$142$	−8.94586e8	−0.184640
$143$	1.97927e8	0.0395816
$144$	0	0
$145$	7.55205e8	0.141876
$146$	3.27307e8	0.0596166
$147$	0	0
$148$	1.82702e10	3.13016
$149$	−1.97067e9	−0.327549	−0.163774	−	0.986498i	$-0.552367\pi$
$149$	−1.97067e9	−0.327549	−0.163774	+	0.986498i	$0.552367\pi$
$150$	0	0
$151$	2.47989e9	0.388183	0.194091	−	0.980983i	$-0.437824\pi$
$151$	2.47989e9	0.388183	0.194091	+	0.980983i	$0.437824\pi$
$152$	−1.78416e10	−2.71105
$153$	0	0
$154$	4.82588e9	0.691406
$155$	−2.11541e9	−0.294376
$156$	0	0
$157$	−1.54180e8	−0.0202525	−0.0101262	−	0.999949i	$-0.503223\pi$
$157$	−1.54180e8	−0.0202525	−0.0101262	+	0.999949i	$0.503223\pi$
$158$	−2.07265e10	−2.64587
$159$	0	0
$160$	1.34855e9	0.162678
$161$	−1.04874e9	−0.123013
$162$	0	0
$163$	5.00929e9	0.555817	0.277909	−	0.960607i	$-0.410359\pi$
$163$	5.00929e9	0.555817	0.277909	+	0.960607i	$0.410359\pi$
$164$	−2.77891e10	−2.99970
$165$	0	0
$166$	3.72956e9	0.381216
$167$	4.54470e9	0.452148	0.226074	−	0.974110i	$-0.427411\pi$
$167$	4.54470e9	0.452148	0.226074	+	0.974110i	$0.427411\pi$
$168$	0	0
$169$	−1.05891e10	−0.998549
$170$	1.46731e10	1.34741
$171$	0	0
$172$	1.55710e10	1.35656
$173$	1.12736e10	0.956877	0.478438	−	0.878121i	$-0.341203\pi$
$173$	1.12736e10	0.956877	0.478438	+	0.878121i	$0.341203\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	1.72651e10	1.35632
$177$	0	0
$178$	7.73964e9	0.577871
$179$	−9.15200e8	−0.0666312	−0.0333156	−	0.999445i	$-0.510607\pi$
$179$	−9.15200e8	−0.0666312	−0.0333156	+	0.999445i	$0.510607\pi$
$180$	0	0
$181$	1.27927e10	0.885947	0.442974	−	0.896535i	$-0.353924\pi$
$181$	1.27927e10	0.885947	0.442974	+	0.896535i	$0.353924\pi$
$182$	3.75087e8	0.0253402
$183$	0	0
$184$	−9.78465e9	−0.629311
$185$	−1.06280e10	−0.667079
$186$	0	0
$187$	2.97445e10	1.77877
$188$	5.97293e10	3.48720
$189$	0	0
$190$	1.98268e10	1.10373
$191$	−2.16679e10	−1.17806	−0.589028	−	0.808112i	$-0.700489\pi$
$191$	−2.16679e10	−1.17806	−0.589028	+	0.808112i	$0.700489\pi$
$192$	0	0
$193$	−2.49369e10	−1.29370	−0.646850	−	0.762617i	$-0.723914\pi$
$193$	−2.49369e10	−1.29370	−0.646850	+	0.762617i	$0.723914\pi$
$194$	−5.90625e9	−0.299367
$195$	0	0
$196$	6.19382e9	0.299782
$197$	−1.60189e10	−0.757767	−0.378883	−	0.925444i	$-0.623692\pi$
$197$	−1.60189e10	−0.757767	−0.378883	+	0.925444i	$0.623692\pi$
$198$	0	0
$199$	−5.80085e9	−0.262212	−0.131106	−	0.991368i	$-0.541853\pi$
$199$	−5.80085e9	−0.262212	−0.131106	+	0.991368i	$0.541853\pi$
$200$	−8.75044e9	−0.386719
$201$	0	0
$202$	4.59670e10	1.94252
$203$	2.90119e9	0.119907
$204$	0	0
$205$	1.61652e10	0.639276
$206$	7.72567e10	2.98905
$207$	0	0
$208$	1.34191e9	0.0497094
$209$	4.01920e10	1.45707
$210$	0	0
$211$	−5.24698e10	−1.82238	−0.911188	−	0.411991i	$-0.864834\pi$
$211$	−5.24698e10	−1.82238	−0.911188	+	0.411991i	$0.864834\pi$
$212$	8.24387e10	2.80298
$213$	0	0
$214$	1.04803e11	3.41596
$215$	−9.05779e9	−0.289101
$216$	0	0
$217$	−8.12656e9	−0.248793
$218$	−2.20129e10	−0.660122
$219$	0	0
$220$	−3.38867e10	−0.975276
$221$	2.31186e9	0.0651924
$222$	0	0
$223$	5.96053e10	1.61403	0.807017	−	0.590528i	$-0.201080\pi$
$223$	5.96053e10	1.61403	0.807017	+	0.590528i	$0.201080\pi$
$224$	5.18060e9	0.137488
$225$	0	0
$226$	7.04223e10	1.79565
$227$	−2.06726e9	−0.0516749	−0.0258375	−	0.999666i	$-0.508225\pi$
$227$	−2.06726e9	−0.0516749	−0.0258375	+	0.999666i	$0.508225\pi$
$228$	0	0
$229$	−8.49346e9	−0.204092	−0.102046	−	0.994780i	$-0.532539\pi$
$229$	−8.49346e9	−0.204092	−0.102046	+	0.994780i	$0.532539\pi$
$230$	1.08734e10	0.256206
$231$	0	0
$232$	2.70679e10	0.613421
$233$	8.34761e10	1.85550	0.927749	−	0.373205i	$-0.121741\pi$
$233$	8.34761e10	1.85550	0.927749	+	0.373205i	$0.121741\pi$
$234$	0	0
$235$	−3.47451e10	−0.743170
$236$	1.23633e11	2.59437
$237$	0	0
$238$	5.63681e10	1.13877
$239$	−5.10768e9	−0.101259	−0.0506294	−	0.998718i	$-0.516123\pi$
$239$	−5.10768e9	−0.101259	−0.0506294	+	0.998718i	$0.516123\pi$
$240$	0	0
$241$	4.21633e10	0.805115	0.402557	−	0.915395i	$-0.368121\pi$
$241$	4.21633e10	0.805115	0.402557	+	0.915395i	$0.368121\pi$
$242$	−7.51169e9	−0.140789
$243$	0	0
$244$	−1.66000e11	−2.99816
$245$	−3.60300e9	−0.0638877
$246$	0	0
$247$	3.12388e9	0.0534021
$248$	−7.58202e10	−1.27278
$249$	0	0
$250$	9.72409e9	0.157441
$251$	8.09322e10	1.28703	0.643516	−	0.765432i	$-0.277475\pi$
$251$	8.09322e10	1.28703	0.643516	+	0.765432i	$0.277475\pi$
$252$	0	0
$253$	2.20420e10	0.338227
$254$	−1.12019e11	−1.68866
$255$	0	0
$256$	−1.39873e11	−2.03542
$257$	−8.44717e10	−1.20785	−0.603924	−	0.797042i	$-0.706397\pi$
$257$	−8.44717e10	−1.20785	−0.603924	+	0.797042i	$0.706397\pi$
$258$	0	0
$259$	−4.08284e10	−0.563785
$260$	−2.63381e9	−0.0357441
$261$	0	0
$262$	−4.62629e10	−0.606565
$263$	9.05095e10	1.16652	0.583261	−	0.812284i	$-0.301776\pi$
$263$	9.05095e10	1.16652	0.583261	+	0.812284i	$0.301776\pi$
$264$	0	0
$265$	−4.79553e10	−0.597352
$266$	7.61668e10	0.932821
$267$	0	0
$268$	2.92242e11	3.46048
$269$	−7.22101e10	−0.840838	−0.420419	−	0.907330i	$-0.638117\pi$
$269$	−7.22101e10	−0.840838	−0.420419	+	0.907330i	$0.638117\pi$
$270$	0	0
$271$	7.70659e10	0.867961	0.433981	−	0.900922i	$-0.357109\pi$
$271$	7.70659e10	0.867961	0.433981	+	0.900922i	$0.357109\pi$
$272$	2.01662e11	2.23391
$273$	0	0
$274$	−1.38481e10	−0.148427
$275$	1.97122e10	0.207844
$276$	0	0
$277$	−1.06723e11	−1.08917	−0.544587	−	0.838704i	$-0.683314\pi$
$277$	−1.06723e11	−1.08917	−0.544587	+	0.838704i	$0.683314\pi$
$278$	8.08886e10	0.812243
$279$	0	0
$280$	−3.36157e10	−0.326837
$281$	−6.23369e10	−0.596440	−0.298220	−	0.954497i	$-0.596393\pi$
$281$	−6.23369e10	−0.596440	−0.298220	+	0.954497i	$0.596393\pi$
$282$	0	0
$283$	−1.05619e11	−0.978822	−0.489411	−	0.872053i	$-0.662788\pi$
$283$	−1.05619e11	−0.978822	−0.489411	+	0.872053i	$0.662788\pi$
$284$	2.41317e10	0.220118
$285$	0	0
$286$	−7.88343e9	−0.0696735
$287$	6.21002e10	0.540287
$288$	0	0
$289$	2.28839e11	1.92970
$290$	−3.00797e10	−0.249737
$291$	0	0
$292$	−8.82917e9	−0.0710717
$293$	9.77547e10	0.774878	0.387439	−	0.921895i	$-0.373360\pi$
$293$	9.77547e10	0.774878	0.387439	+	0.921895i	$0.373360\pi$
$294$	0	0
$295$	−7.19187e10	−0.552895
$296$	−3.80926e11	−2.88421
$297$	0	0
$298$	7.84917e10	0.576568
$299$	1.71319e9	0.0123961
$300$	0	0
$301$	−3.47964e10	−0.244335
$302$	−9.87738e10	−0.683299
$303$	0	0
$304$	2.72494e11	1.82990
$305$	9.65640e10	0.638949
$306$	0	0
$307$	−2.48236e11	−1.59493	−0.797467	−	0.603363i	$-0.793827\pi$
$307$	−2.48236e11	−1.59493	−0.797467	+	0.603363i	$0.793827\pi$
$308$	−1.30179e11	−0.824259
$309$	0	0
$310$	8.42566e10	0.518175
$311$	1.63257e11	0.989578	0.494789	−	0.869013i	$-0.335245\pi$
$311$	1.63257e11	0.989578	0.494789	+	0.869013i	$0.335245\pi$
$312$	0	0
$313$	1.54423e11	0.909415	0.454707	−	0.890641i	$-0.349744\pi$
$313$	1.54423e11	0.909415	0.454707	+	0.890641i	$0.349744\pi$
$314$	6.14096e9	0.0356494
$315$	0	0
$316$	5.59101e11	3.15427
$317$	−1.41436e11	−0.786669	−0.393335	−	0.919395i	$-0.628679\pi$
$317$	−1.41436e11	−0.786669	−0.393335	+	0.919395i	$0.628679\pi$
$318$	0	0
$319$	−6.09762e10	−0.329687
$320$	5.57694e10	0.297318
$321$	0	0
$322$	4.17712e10	0.216533
$323$	4.69457e11	2.39985
$324$	0	0
$325$	1.53211e9	0.00761755
$326$	−1.99520e11	−0.978377
$327$	0	0
$328$	5.79390e11	2.76400
$329$	−1.33477e11	−0.628093
$330$	0	0
$331$	1.93663e11	0.886791	0.443395	−	0.896326i	$-0.353774\pi$
$331$	1.93663e11	0.886791	0.443395	+	0.896326i	$0.353774\pi$
$332$	−1.00606e11	−0.454466
$333$	0	0
$334$	−1.81015e11	−0.795893
$335$	−1.70000e11	−0.737475
$336$	0	0
$337$	2.00520e10	0.0846881	0.0423440	−	0.999103i	$-0.486517\pi$
$337$	2.00520e10	0.0846881	0.0423440	+	0.999103i	$0.486517\pi$
$338$	4.21763e11	1.75770
$339$	0	0
$340$	−3.95809e11	−1.60632
$341$	1.70801e11	0.684062
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−3.24648e11	−1.24997
$345$	0	0
$346$	−4.49027e11	−1.68434
$347$	4.57822e11	1.69517	0.847587	−	0.530656i	$-0.178055\pi$
$347$	4.57822e11	1.69517	0.847587	+	0.530656i	$0.178055\pi$
$348$	0	0
$349$	1.79006e11	0.645880	0.322940	−	0.946419i	$-0.395329\pi$
$349$	1.79006e11	0.645880	0.322940	+	0.946419i	$0.395329\pi$
$350$	3.73561e10	0.133062
$351$	0	0
$352$	−1.08884e11	−0.378026
$353$	2.73777e11	0.938448	0.469224	−	0.883079i	$-0.344534\pi$
$353$	2.73777e11	0.938448	0.469224	+	0.883079i	$0.344534\pi$
$354$	0	0
$355$	−1.40376e10	−0.0469100
$356$	−2.08779e11	−0.688908
$357$	0	0
$358$	3.64523e10	0.117287
$359$	−7.70608e9	−0.0244855	−0.0122427	−	0.999925i	$-0.503897\pi$
$359$	−7.70608e9	−0.0244855	−0.0122427	+	0.999925i	$0.503897\pi$
$360$	0	0
$361$	3.11661e11	0.965830
$362$	−5.09531e11	−1.55949
$363$	0	0
$364$	−1.01180e10	−0.0302093
$365$	5.13601e9	0.0151463
$366$	0	0
$367$	−1.03107e11	−0.296682	−0.148341	−	0.988936i	$-0.547393\pi$
$367$	−1.03107e11	−0.296682	−0.148341	+	0.988936i	$0.547393\pi$
$368$	1.49441e11	0.424769
$369$	0	0
$370$	4.23311e11	1.17423
$371$	−1.84225e11	−0.504855
$372$	0	0
$373$	−1.33782e11	−0.357856	−0.178928	−	0.983862i	$-0.557263\pi$
$373$	−1.33782e11	−0.357856	−0.178928	+	0.983862i	$0.557263\pi$
$374$	−1.18472e12	−3.13108
$375$	0	0
$376$	−1.24533e12	−3.21320
$377$	−4.73931e9	−0.0120831
$378$	0	0
$379$	−3.30128e11	−0.821875	−0.410937	−	0.911664i	$-0.634798\pi$
$379$	−3.30128e11	−0.821875	−0.410937	+	0.911664i	$0.634798\pi$
$380$	−5.34833e11	−1.31581
$381$	0	0
$382$	8.63029e11	2.07367
$383$	−3.48502e11	−0.827583	−0.413791	−	0.910372i	$-0.635796\pi$
$383$	−3.48502e11	−0.827583	−0.413791	+	0.910372i	$0.635796\pi$
$384$	0	0
$385$	7.57264e10	0.175661
$386$	9.93232e11	2.27724
$387$	0	0
$388$	1.59323e11	0.356890
$389$	−1.72150e11	−0.381183	−0.190591	−	0.981669i	$-0.561041\pi$
$389$	−1.72150e11	−0.381183	−0.190591	+	0.981669i	$0.561041\pi$
$390$	0	0
$391$	2.57458e11	0.557072
$392$	−1.29138e11	−0.276228
$393$	0	0
$394$	6.38032e11	1.33386
$395$	−3.25234e11	−0.672217
$396$	0	0
$397$	−6.89676e11	−1.39344	−0.696719	−	0.717344i	$-0.745357\pi$
$397$	−6.89676e11	−1.39344	−0.696719	+	0.717344i	$0.745357\pi$
$398$	2.31047e11	0.461559
$399$	0	0
$400$	1.33645e11	0.261026
$401$	−3.89787e11	−0.752797	−0.376399	−	0.926458i	$-0.622838\pi$
$401$	−3.89787e11	−0.752797	−0.376399	+	0.926458i	$0.622838\pi$
$402$	0	0
$403$	1.32753e10	0.0250710
$404$	−1.23997e12	−2.31577
$405$	0	0
$406$	−1.15554e11	−0.211066
$407$	8.58115e11	1.55014
$408$	0	0
$409$	−5.70543e11	−1.00817	−0.504084	−	0.863654i	$-0.668170\pi$
$409$	−5.70543e11	−1.00817	−0.504084	+	0.863654i	$0.668170\pi$
$410$	−6.43858e11	−1.12528
$411$	0	0
$412$	−2.08402e12	−3.56339
$413$	−2.76283e11	−0.467282
$414$	0	0
$415$	5.85233e10	0.0968528
$416$	−8.46288e9	−0.0138547
$417$	0	0
$418$	−1.60084e12	−2.56481
$419$	3.80518e11	0.603132	0.301566	−	0.953445i	$-0.402491\pi$
$419$	3.80518e11	0.603132	0.301566	+	0.953445i	$0.402491\pi$
$420$	0	0
$421$	3.64852e11	0.566041	0.283020	−	0.959114i	$-0.408664\pi$
$421$	3.64852e11	0.566041	0.283020	+	0.959114i	$0.408664\pi$
$422$	2.08986e12	3.20784
$423$	0	0
$424$	−1.71881e12	−2.58274
$425$	2.30246e11	0.342327
$426$	0	0
$427$	3.70960e11	0.540010
$428$	−2.82709e12	−4.07233
$429$	0	0
$430$	3.60771e11	0.508889
$431$	−7.18123e11	−1.00242	−0.501211	−	0.865325i	$-0.667112\pi$
$431$	−7.18123e11	−1.00242	−0.501211	+	0.865325i	$0.667112\pi$
$432$	0	0
$433$	8.28850e11	1.13313	0.566566	−	0.824017i	$-0.308272\pi$
$433$	8.28850e11	1.13313	0.566566	+	0.824017i	$0.308272\pi$
$434$	3.23680e11	0.437938
$435$	0	0
$436$	5.93804e11	0.786962
$437$	3.47888e11	0.456324
$438$	0	0
$439$	2.28406e10	0.0293506	0.0146753	−	0.999892i	$-0.495329\pi$
$439$	2.28406e10	0.0293506	0.0146753	+	0.999892i	$0.495329\pi$
$440$	7.06522e11	0.898645
$441$	0	0
$442$	−9.20813e10	−0.114755
$443$	6.10715e11	0.753393	0.376697	−	0.926337i	$-0.377060\pi$
$443$	6.10715e11	0.753393	0.376697	+	0.926337i	$0.377060\pi$
$444$	0	0
$445$	1.21448e11	0.146815
$446$	−2.37407e12	−2.84110
$447$	0	0
$448$	2.14244e11	0.251280
$449$	7.13843e11	0.828885	0.414442	−	0.910076i	$-0.363977\pi$
$449$	7.13843e11	0.828885	0.414442	+	0.910076i	$0.363977\pi$
$450$	0	0
$451$	−1.30520e12	−1.48553
$452$	−1.89966e12	−2.14068
$453$	0	0
$454$	8.23389e10	0.0909607
$455$	5.88576e9	0.00643801
$456$	0	0
$457$	8.16711e11	0.875882	0.437941	−	0.899004i	$-0.355708\pi$
$457$	8.16711e11	0.875882	0.437941	+	0.899004i	$0.355708\pi$
$458$	3.38294e11	0.359252
$459$	0	0
$460$	−2.93312e11	−0.305435
$461$	1.15418e11	0.119020	0.0595100	−	0.998228i	$-0.481046\pi$
$461$	1.15418e11	0.119020	0.0595100	+	0.998228i	$0.481046\pi$
$462$	0	0
$463$	1.89097e12	1.91236	0.956181	−	0.292776i	$-0.0945790\pi$
$463$	1.89097e12	1.91236	0.956181	+	0.292776i	$0.0945790\pi$
$464$	−4.13407e11	−0.414044
$465$	0	0
$466$	−3.32484e12	−3.26614
$467$	1.41788e12	1.37948	0.689739	−	0.724058i	$-0.257725\pi$
$467$	1.41788e12	1.37948	0.689739	+	0.724058i	$0.257725\pi$
$468$	0	0
$469$	−6.53072e11	−0.623280
$470$	1.38389e12	1.30816
$471$	0	0
$472$	−2.57770e12	−2.39052
$473$	7.31337e11	0.671804
$474$	0	0
$475$	3.11117e11	0.280416
$476$	−1.52054e12	−1.35758
$477$	0	0
$478$	2.03438e11	0.178241
$479$	5.70763e11	0.495388	0.247694	−	0.968838i	$-0.420327\pi$
$479$	5.70763e11	0.495388	0.247694	+	0.968838i	$0.420327\pi$
$480$	0	0
$481$	6.66961e10	0.0568130
$482$	−1.67936e12	−1.41720
$483$	0	0
$484$	2.02629e11	0.167841
$485$	−9.26794e10	−0.0760580
$486$	0	0
$487$	−4.98222e11	−0.401367	−0.200684	−	0.979656i	$-0.564316\pi$
$487$	−4.98222e11	−0.401367	−0.200684	+	0.979656i	$0.564316\pi$
$488$	3.46103e12	2.76259
$489$	0	0
$490$	1.43507e11	0.112458
$491$	2.24762e12	1.74525	0.872624	−	0.488393i	$-0.162417\pi$
$491$	2.24762e12	1.74525	0.872624	+	0.488393i	$0.162417\pi$
$492$	0	0
$493$	−7.12224e11	−0.543007
$494$	−1.24424e11	−0.0940010
$495$	0	0
$496$	1.15800e12	0.859093
$497$	−5.39269e10	−0.0396462
$498$	0	0
$499$	1.47577e11	0.106553	0.0532767	−	0.998580i	$-0.483033\pi$
$499$	1.47577e11	0.106553	0.0532767	+	0.998580i	$0.483033\pi$
$500$	−2.62310e11	−0.187693
$501$	0	0
$502$	−3.22352e12	−2.26550
$503$	3.17462e11	0.221124	0.110562	−	0.993869i	$-0.464735\pi$
$503$	3.17462e11	0.221124	0.110562	+	0.993869i	$0.464735\pi$
$504$	0	0
$505$	7.21302e11	0.493521
$506$	−8.77930e11	−0.595364
$507$	0	0
$508$	3.02174e12	2.01313
$509$	−2.74584e12	−1.81320	−0.906598	−	0.421995i	$-0.861330\pi$
$509$	−2.74584e12	−1.81320	−0.906598	+	0.421995i	$0.861330\pi$
$510$	0	0
$511$	1.97305e10	0.0128010
$512$	3.18583e12	2.04884
$513$	0	0
$514$	3.36450e12	2.12611
$515$	1.21229e12	0.759407
$516$	0	0
$517$	2.80536e12	1.72696
$518$	1.62619e12	0.992402
$519$	0	0
$520$	5.49137e10	0.0329356
$521$	2.48978e12	1.48044	0.740220	−	0.672364i	$-0.234721\pi$
$521$	2.48978e12	1.48044	0.740220	+	0.672364i	$0.234721\pi$
$522$	0	0
$523$	9.38103e11	0.548268	0.274134	−	0.961692i	$-0.411609\pi$
$523$	9.38103e11	0.548268	0.274134	+	0.961692i	$0.411609\pi$
$524$	1.24795e12	0.723115
$525$	0	0
$526$	−3.60498e12	−2.05337
$527$	1.99502e12	1.12668
$528$	0	0
$529$	−1.61036e12	−0.894075
$530$	1.91006e12	1.05149
$531$	0	0
$532$	−2.05462e12	−1.11206
$533$	−1.01445e11	−0.0544451
$534$	0	0
$535$	1.64454e12	0.867868
$536$	−6.09311e12	−3.18858
$537$	0	0
$538$	2.87612e12	1.48008
$539$	2.90911e11	0.148460
$540$	0	0
$541$	1.53540e12	0.770609	0.385304	−	0.922790i	$-0.374096\pi$
$541$	1.53540e12	0.770609	0.385304	+	0.922790i	$0.374096\pi$
$542$	−3.06952e12	−1.52783
$543$	0	0
$544$	−1.27180e12	−0.622622
$545$	−3.45421e11	−0.167712
$546$	0	0
$547$	7.20289e11	0.344004	0.172002	−	0.985097i	$-0.444976\pi$
$547$	7.20289e11	0.344004	0.172002	+	0.985097i	$0.444976\pi$
$548$	3.73556e11	0.176947
$549$	0	0
$550$	−7.85135e11	−0.365858
$551$	−9.62385e11	−0.444802
$552$	0	0
$553$	−1.24942e12	−0.568127
$554$	4.25075e12	1.91722
$555$	0	0
$556$	−2.18199e12	−0.968313
$557$	1.48402e11	0.0653269	0.0326634	−	0.999466i	$-0.489601\pi$
$557$	1.48402e11	0.0653269	0.0326634	+	0.999466i	$0.489601\pi$
$558$	0	0
$559$	5.68424e10	0.0246218
$560$	5.13411e11	0.220607
$561$	0	0
$562$	2.48287e12	1.04988
$563$	2.14987e12	0.901828	0.450914	−	0.892567i	$-0.351098\pi$
$563$	2.14987e12	0.901828	0.450914	+	0.892567i	$0.351098\pi$
$564$	0	0
$565$	1.10505e12	0.456208
$566$	4.20680e12	1.72297
$567$	0	0
$568$	−5.03133e11	−0.202822
$569$	−1.79224e12	−0.716788	−0.358394	−	0.933570i	$-0.616676\pi$
$569$	−1.79224e12	−0.716788	−0.358394	+	0.933570i	$0.616676\pi$
$570$	0	0
$571$	−1.07595e12	−0.423575	−0.211787	−	0.977316i	$-0.567928\pi$
$571$	−1.07595e12	−0.423575	−0.211787	+	0.977316i	$0.567928\pi$
$572$	2.12657e11	0.0830611
$573$	0	0
$574$	−2.47344e12	−0.951039
$575$	1.70622e11	0.0650923
$576$	0	0
$577$	1.49614e12	0.561928	0.280964	−	0.959718i	$-0.409346\pi$
$577$	1.49614e12	0.561928	0.280964	+	0.959718i	$0.409346\pi$
$578$	−9.11464e12	−3.39676
$579$	0	0
$580$	8.11407e11	0.297723
$581$	2.24823e11	0.0818555
$582$	0	0
$583$	3.87197e12	1.38811
$584$	1.84084e11	0.0654874
$585$	0	0
$586$	−3.89356e12	−1.36398
$587$	−1.36578e12	−0.474797	−0.237399	−	0.971412i	$-0.576295\pi$
$587$	−1.36578e12	−0.474797	−0.237399	+	0.971412i	$0.576295\pi$
$588$	0	0
$589$	2.69575e12	0.922912
$590$	2.86451e12	0.973233
$591$	0	0
$592$	5.81786e12	1.94677
$593$	−6.11529e11	−0.203082	−0.101541	−	0.994831i	$-0.532377\pi$
$593$	−6.11529e11	−0.203082	−0.101541	+	0.994831i	$0.532377\pi$
$594$	0	0
$595$	8.84513e11	0.289319
$596$	−2.11733e12	−0.687354
$597$	0	0
$598$	−6.82362e10	−0.0218202
$599$	1.09965e12	0.349007	0.174504	−	0.984657i	$-0.444168\pi$
$599$	1.09965e12	0.349007	0.174504	+	0.984657i	$0.444168\pi$
$600$	0	0
$601$	2.88372e12	0.901607	0.450804	−	0.892623i	$-0.351137\pi$
$601$	2.88372e12	0.901607	0.450804	+	0.892623i	$0.351137\pi$
$602$	1.38594e12	0.430090
$603$	0	0
$604$	2.66445e12	0.814593
$605$	−1.17871e11	−0.0357692
$606$	0	0
$607$	−4.15300e12	−1.24169	−0.620844	−	0.783934i	$-0.713210\pi$
$607$	−4.15300e12	−1.24169	−0.620844	+	0.783934i	$0.713210\pi$
$608$	−1.71851e12	−0.510018
$609$	0	0
$610$	−3.84613e12	−1.12471
$611$	2.18044e11	0.0632934
$612$	0	0
$613$	3.47765e12	0.994751	0.497375	−	0.867535i	$-0.334297\pi$
$613$	3.47765e12	0.994751	0.497375	+	0.867535i	$0.334297\pi$
$614$	9.88722e12	2.80748
$615$	0	0
$616$	2.71417e12	0.759494
$617$	−6.57646e12	−1.82687	−0.913437	−	0.406979i	$-0.866582\pi$
$617$	−6.57646e12	−1.82687	−0.913437	+	0.406979i	$0.866582\pi$
$618$	0	0
$619$	9.01386e11	0.246776	0.123388	−	0.992359i	$-0.460624\pi$
$619$	9.01386e11	0.246776	0.123388	+	0.992359i	$0.460624\pi$
$620$	−2.27284e12	−0.617741
$621$	0	0
$622$	−6.50251e12	−1.74190
$623$	4.66556e11	0.124082
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−6.15064e12	−1.60080
$627$	0	0
$628$	−1.65654e11	−0.0424994
$629$	1.00231e13	2.55314
$630$	0	0
$631$	−4.67767e12	−1.17462	−0.587310	−	0.809362i	$-0.699813\pi$
$631$	−4.67767e12	−1.17462	−0.587310	+	0.809362i	$0.699813\pi$
$632$	−1.16570e13	−2.90643
$633$	0	0
$634$	5.63337e12	1.38473
$635$	−1.75778e12	−0.429024
$636$	0	0
$637$	2.26107e10	0.00544111
$638$	2.42867e12	0.580332
$639$	0	0
$640$	−2.91175e12	−0.686031
$641$	−3.76634e12	−0.881167	−0.440583	−	0.897712i	$-0.645228\pi$
$641$	−3.76634e12	−0.881167	−0.440583	+	0.897712i	$0.645228\pi$
$642$	0	0
$643$	−3.77493e12	−0.870882	−0.435441	−	0.900217i	$-0.643408\pi$
$643$	−3.77493e12	−0.870882	−0.435441	+	0.900217i	$0.643408\pi$
$644$	−1.12679e12	−0.258140
$645$	0	0
$646$	−1.86984e13	−4.22434
$647$	−5.06343e12	−1.13599	−0.567996	−	0.823031i	$-0.692281\pi$
$647$	−5.06343e12	−1.13599	−0.567996	+	0.823031i	$0.692281\pi$
$648$	0	0
$649$	5.80681e12	1.28480
$650$	−6.10239e10	−0.0134088
$651$	0	0
$652$	5.38208e12	1.16637
$653$	−3.01881e12	−0.649721	−0.324860	−	0.945762i	$-0.605317\pi$
$653$	−3.01881e12	−0.649721	−0.324860	+	0.945762i	$0.605317\pi$
$654$	0	0
$655$	−7.25946e11	−0.154106
$656$	−8.84899e12	−1.86563
$657$	0	0
$658$	5.31636e12	1.10560
$659$	−1.25826e12	−0.259889	−0.129944	−	0.991521i	$-0.541480\pi$
$659$	−1.25826e12	−0.259889	−0.129944	+	0.991521i	$0.541480\pi$
$660$	0	0
$661$	4.12094e12	0.839634	0.419817	−	0.907609i	$-0.362094\pi$
$661$	4.12094e12	0.839634	0.419817	+	0.907609i	$0.362094\pi$
$662$	−7.71358e12	−1.56097
$663$	0	0
$664$	2.09758e12	0.418757
$665$	1.19519e12	0.236995
$666$	0	0
$667$	−5.27789e11	−0.103251
$668$	4.88291e12	0.948823
$669$	0	0
$670$	6.77108e12	1.29814
$671$	−7.79670e12	−1.48477
$672$	0	0
$673$	−6.41884e12	−1.20612	−0.603058	−	0.797697i	$-0.706051\pi$
$673$	−6.41884e12	−1.20612	−0.603058	+	0.797697i	$0.706051\pi$
$674$	−7.98667e11	−0.149072
$675$	0	0
$676$	−1.13772e13	−2.09543
$677$	7.67465e12	1.40414	0.702068	−	0.712110i	$-0.252260\pi$
$677$	7.67465e12	1.40414	0.702068	+	0.712110i	$0.252260\pi$
$678$	0	0
$679$	−3.56037e11	−0.0642808
$680$	8.25243e12	1.48010
$681$	0	0
$682$	−6.80298e12	−1.20412
$683$	−2.73396e12	−0.480727	−0.240363	−	0.970683i	$-0.577267\pi$
$683$	−2.73396e12	−0.480727	−0.240363	+	0.970683i	$0.577267\pi$
$684$	0	0
$685$	−2.17301e11	−0.0377098
$686$	5.51297e11	0.0950445
$687$	0	0
$688$	4.95833e12	0.843698
$689$	3.00945e11	0.0508746
$690$	0	0
$691$	−7.94059e12	−1.32496	−0.662478	−	0.749081i	$-0.730495\pi$
$691$	−7.94059e12	−1.32496	−0.662478	+	0.749081i	$0.730495\pi$
$692$	1.21126e13	2.00798
$693$	0	0
$694$	−1.82350e13	−2.98393
$695$	1.26928e12	0.206361
$696$	0	0
$697$	−1.52452e13	−2.44672
$698$	−7.12977e12	−1.13691
$699$	0	0
$700$	−1.00769e12	−0.158630
$701$	8.03968e11	0.125750	0.0628750	−	0.998021i	$-0.479973\pi$
$701$	8.03968e11	0.125750	0.0628750	+	0.998021i	$0.479973\pi$
$702$	0	0
$703$	1.35436e13	2.09139
$704$	−4.50289e12	−0.690899
$705$	0	0
$706$	−1.09045e13	−1.65190
$707$	2.77095e12	0.417102
$708$	0	0
$709$	6.49723e12	0.965652	0.482826	−	0.875716i	$-0.339610\pi$
$709$	6.49723e12	0.965652	0.482826	+	0.875716i	$0.339610\pi$
$710$	5.59116e11	0.0825733
$711$	0	0
$712$	4.35293e12	0.634778
$713$	1.47839e12	0.214233
$714$	0	0
$715$	−1.23705e11	−0.0177014
$716$	−9.83310e11	−0.139824
$717$	0	0
$718$	3.06932e11	0.0431005
$719$	6.26229e12	0.873882	0.436941	−	0.899490i	$-0.356062\pi$
$719$	6.26229e12	0.873882	0.436941	+	0.899490i	$0.356062\pi$
$720$	0	0
$721$	4.65714e12	0.641816
$722$	−1.24134e13	−1.70010
$723$	0	0
$724$	1.37447e13	1.85914
$725$	−4.72003e11	−0.0634488
$726$	0	0
$727$	4.99771e11	0.0663538	0.0331769	−	0.999449i	$-0.489438\pi$
$727$	4.99771e11	0.0663538	0.0331769	+	0.999449i	$0.489438\pi$
$728$	2.10956e11	0.0278357
$729$	0	0
$730$	−2.04567e11	−0.0266613
$731$	8.54228e12	1.10649
$732$	0	0
$733$	−1.22506e13	−1.56744	−0.783719	−	0.621115i	$-0.786680\pi$
$733$	−1.22506e13	−1.56744	−0.783719	+	0.621115i	$0.786680\pi$
$734$	4.10674e12	0.522233
$735$	0	0
$736$	−9.42460e11	−0.118389
$737$	1.37260e13	1.71372
$738$	0	0
$739$	1.11519e12	0.137546	0.0687732	−	0.997632i	$-0.478092\pi$
$739$	1.11519e12	0.137546	0.0687732	+	0.997632i	$0.478092\pi$
$740$	−1.14189e13	−1.39985
$741$	0	0
$742$	7.33767e12	0.888670
$743$	1.40083e13	1.68630	0.843149	−	0.537679i	$-0.180699\pi$
$743$	1.40083e13	1.68630	0.843149	+	0.537679i	$0.180699\pi$
$744$	0	0
$745$	1.23167e12	0.146484
$746$	5.32853e12	0.629915
$747$	0	0
$748$	3.19581e13	3.73271
$749$	6.31768e12	0.733482
$750$	0	0
$751$	5.25286e12	0.602582	0.301291	−	0.953532i	$-0.402582\pi$
$751$	5.25286e12	0.602582	0.301291	+	0.953532i	$0.402582\pi$
$752$	1.90198e13	2.16883
$753$	0	0
$754$	1.88766e11	0.0212693
$755$	−1.54993e12	−0.173601
$756$	0	0
$757$	1.56939e13	1.73700	0.868501	−	0.495688i	$-0.165084\pi$
$757$	1.56939e13	1.73700	0.868501	+	0.495688i	$0.165084\pi$
$758$	1.31490e13	1.44670
$759$	0	0
$760$	1.11510e13	1.21242
$761$	−5.73267e12	−0.619621	−0.309811	−	0.950798i	$-0.600266\pi$
$761$	−5.73267e12	−0.619621	−0.309811	+	0.950798i	$0.600266\pi$
$762$	0	0
$763$	−1.32697e12	−0.141743
$764$	−2.32804e13	−2.47212
$765$	0	0
$766$	1.38808e13	1.45675
$767$	4.51328e11	0.0470883
$768$	0	0
$769$	2.10268e12	0.216823	0.108412	−	0.994106i	$-0.465424\pi$
$769$	2.10268e12	0.216823	0.108412	+	0.994106i	$0.465424\pi$
$770$	−3.01618e12	−0.309206
$771$	0	0
$772$	−2.67927e13	−2.71480
$773$	6.49634e12	0.654427	0.327213	−	0.944950i	$-0.393890\pi$
$773$	6.49634e12	0.654427	0.327213	+	0.944950i	$0.393890\pi$
$774$	0	0
$775$	1.32213e12	0.131649
$776$	−3.32180e12	−0.328848
$777$	0	0
$778$	6.85670e12	0.670976
$779$	−2.05999e13	−2.00422
$780$	0	0
$781$	1.13341e12	0.109008
$782$	−1.02545e13	−0.980586
$783$	0	0
$784$	1.97232e12	0.186447
$785$	9.63622e10	0.00905719
$786$	0	0
$787$	1.81068e12	0.168250	0.0841252	−	0.996455i	$-0.473190\pi$
$787$	1.81068e12	0.168250	0.0841252	+	0.996455i	$0.473190\pi$
$788$	−1.72111e13	−1.59016
$789$	0	0
$790$	1.29540e13	1.18327
$791$	4.24515e12	0.385566
$792$	0	0
$793$	−6.05990e11	−0.0544172
$794$	2.74697e13	2.45280
$795$	0	0
$796$	−6.23255e12	−0.550246
$797$	1.69941e12	0.149189	0.0745945	−	0.997214i	$-0.476234\pi$
$797$	1.69941e12	0.149189	0.0745945	+	0.997214i	$0.476234\pi$
$798$	0	0
$799$	3.27677e13	2.84436
$800$	−8.42845e11	−0.0727516
$801$	0	0
$802$	1.55252e13	1.32511
$803$	−4.14688e11	−0.0351966
$804$	0	0
$805$	6.55462e11	0.0550131
$806$	−5.28755e11	−0.0441313
$807$	0	0
$808$	2.58528e13	2.13381
$809$	−1.90929e13	−1.56713	−0.783564	−	0.621311i	$-0.786600\pi$
$809$	−1.90929e13	−1.56713	−0.783564	+	0.621311i	$0.786600\pi$
$810$	0	0
$811$	−2.15585e13	−1.74995	−0.874974	−	0.484169i	$-0.839122\pi$
$811$	−2.15585e13	−1.74995	−0.874974	+	0.484169i	$0.839122\pi$
$812$	3.11710e12	0.251622
$813$	0	0
$814$	−3.41786e13	−2.72863
$815$	−3.13081e12	−0.248569
$816$	0	0
$817$	1.15427e13	0.906373
$818$	2.27246e13	1.77463
$819$	0	0
$820$	1.73682e13	1.34151
$821$	−1.81138e13	−1.39144	−0.695720	−	0.718313i	$-0.744915\pi$
$821$	−1.81138e13	−1.39144	−0.695720	+	0.718313i	$0.744915\pi$
$822$	0	0
$823$	2.19003e13	1.66399	0.831994	−	0.554785i	$-0.187200\pi$
$823$	2.19003e13	1.66399	0.831994	+	0.554785i	$0.187200\pi$
$824$	4.34507e13	3.28341
$825$	0	0
$826$	1.10043e13	0.822532
$827$	5.56249e12	0.413518	0.206759	−	0.978392i	$-0.433708\pi$
$827$	5.56249e12	0.413518	0.206759	+	0.978392i	$0.433708\pi$
$828$	0	0
$829$	−3.07036e12	−0.225785	−0.112892	−	0.993607i	$-0.536011\pi$
$829$	−3.07036e12	−0.225785	−0.112892	+	0.993607i	$0.536011\pi$
$830$	−2.33097e12	−0.170485
$831$	0	0
$832$	−3.49983e11	−0.0253216
$833$	3.39794e12	0.244520
$834$	0	0
$835$	−2.84044e12	−0.202207
$836$	4.31831e13	3.05763
$837$	0	0
$838$	−1.51560e13	−1.06166
$839$	−1.76918e13	−1.23266	−0.616331	−	0.787487i	$-0.711382\pi$
$839$	−1.76918e13	−1.23266	−0.616331	+	0.787487i	$0.711382\pi$
$840$	0	0
$841$	−1.30471e13	−0.899356
$842$	−1.45320e13	−0.996373
$843$	0	0
$844$	−5.63746e13	−3.82421
$845$	6.61820e12	0.446565
$846$	0	0
$847$	−4.52815e11	−0.0302305
$848$	2.62513e13	1.74329
$849$	0	0
$850$	−9.17067e12	−0.602582
$851$	7.42754e12	0.485470
$852$	0	0
$853$	−2.59374e13	−1.67748	−0.838738	−	0.544535i	$-0.816706\pi$
$853$	−2.59374e13	−1.67748	−0.838738	+	0.544535i	$0.816706\pi$
$854$	−1.47753e13	−0.950552
$855$	0	0
$856$	5.89435e13	3.75235
$857$	−1.15558e13	−0.731791	−0.365896	−	0.930656i	$-0.619237\pi$
$857$	−1.15558e13	−0.731791	−0.365896	+	0.930656i	$0.619237\pi$
$858$	0	0
$859$	−9.77860e12	−0.612784	−0.306392	−	0.951905i	$-0.599122\pi$
$859$	−9.77860e12	−0.612784	−0.306392	+	0.951905i	$0.599122\pi$
$860$	−9.73187e12	−0.606671
$861$	0	0
$862$	2.86027e13	1.76451
$863$	−4.93535e12	−0.302879	−0.151439	−	0.988467i	$-0.548391\pi$
$863$	−4.93535e12	−0.302879	−0.151439	+	0.988467i	$0.548391\pi$
$864$	0	0
$865$	−7.04601e12	−0.427928
$866$	−3.30130e13	−1.99459
$867$	0	0
$868$	−8.73134e12	−0.522086
$869$	2.62598e13	1.56208
$870$	0	0
$871$	1.06684e12	0.0628084
$872$	−1.23805e13	−0.725128
$873$	0	0
$874$	−1.38563e13	−0.803243
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	1.33258e12	0.0760665	0.0380333	−	0.999276i	$-0.487891\pi$
$877$	1.33258e12	0.0760665	0.0380333	+	0.999276i	$0.487891\pi$
$878$	−9.09737e11	−0.0516643
$879$	0	0
$880$	−1.07907e13	−0.606564
$881$	−7.09860e12	−0.396991	−0.198496	−	0.980102i	$-0.563606\pi$
$881$	−7.09860e12	−0.396991	−0.198496	+	0.980102i	$0.563606\pi$
$882$	0	0
$883$	−3.77620e12	−0.209041	−0.104521	−	0.994523i	$-0.533331\pi$
$883$	−3.77620e12	−0.209041	−0.104521	+	0.994523i	$0.533331\pi$
$884$	2.48391e12	0.136805
$885$	0	0
$886$	−2.43247e13	−1.32616
$887$	2.40016e12	0.130192	0.0650960	−	0.997879i	$-0.479265\pi$
$887$	2.40016e12	0.130192	0.0650960	+	0.997879i	$0.479265\pi$
$888$	0	0
$889$	−6.75267e12	−0.362592
$890$	−4.83728e12	−0.258432
$891$	0	0
$892$	6.40411e13	3.38701
$893$	4.42769e13	2.32995
$894$	0	0
$895$	5.72000e11	0.0297984
$896$	−1.11858e13	−0.579802
$897$	0	0
$898$	−2.84323e13	−1.45904
$899$	−4.08978e12	−0.208824
$900$	0	0
$901$	4.52261e13	2.28627
$902$	5.19859e13	2.61490
$903$	0	0
$904$	3.96069e13	1.97248
$905$	−7.99542e12	−0.396208
$906$	0	0
$907$	−1.05763e13	−0.518920	−0.259460	−	0.965754i	$-0.583545\pi$
$907$	−1.05763e13	−0.518920	−0.259460	+	0.965754i	$0.583545\pi$
$908$	−2.22111e12	−0.108439
$909$	0	0
$910$	−2.34429e11	−0.0113325
$911$	−9.67151e12	−0.465223	−0.232612	−	0.972570i	$-0.574727\pi$
$911$	−9.67151e12	−0.465223	−0.232612	+	0.972570i	$0.574727\pi$
$912$	0	0
$913$	−4.72524e12	−0.225064
$914$	−3.25295e13	−1.54177
$915$	0	0
$916$	−9.12555e12	−0.428282
$917$	−2.78879e12	−0.130243
$918$	0	0
$919$	−2.02154e13	−0.934896	−0.467448	−	0.884020i	$-0.654827\pi$
$919$	−2.02154e13	−0.934896	−0.467448	+	0.884020i	$0.654827\pi$
$920$	6.11541e12	0.281436
$921$	0	0
$922$	−4.59709e12	−0.209505
$923$	8.80934e10	0.00399518
$924$	0	0
$925$	6.64248e12	0.298327
$926$	−7.53171e13	−3.36623
$927$	0	0
$928$	2.60719e12	0.115400
$929$	−3.64006e13	−1.60338	−0.801692	−	0.597737i	$-0.796067\pi$
$929$	−3.64006e13	−1.60338	−0.801692	+	0.597737i	$0.796067\pi$
$930$	0	0
$931$	4.59144e12	0.200297
$932$	8.96884e13	3.89372
$933$	0	0
$934$	−5.64742e13	−2.42822
$935$	−1.85903e13	−0.795491
$936$	0	0
$937$	1.31569e13	0.557605	0.278803	−	0.960348i	$-0.410063\pi$
$937$	1.31569e13	0.557605	0.278803	+	0.960348i	$0.410063\pi$
$938$	2.60118e13	1.09713
$939$	0	0
$940$	−3.73308e13	−1.55952
$941$	−5.82267e12	−0.242086	−0.121043	−	0.992647i	$-0.538624\pi$
$941$	−5.82267e12	−0.242086	−0.121043	+	0.992647i	$0.538624\pi$
$942$	0	0
$943$	−1.12973e13	−0.465236
$944$	3.93691e13	1.61354
$945$	0	0
$946$	−2.91291e13	−1.18254
$947$	−1.97020e13	−0.796041	−0.398020	−	0.917377i	$-0.630303\pi$
$947$	−1.97020e13	−0.796041	−0.398020	+	0.917377i	$0.630303\pi$
$948$	0	0
$949$	−3.22312e10	−0.00128997
$950$	−1.23918e13	−0.493602
$951$	0	0
$952$	3.17025e13	1.25091
$953$	8.60597e12	0.337973	0.168986	−	0.985618i	$-0.445951\pi$
$953$	8.60597e12	0.337973	0.168986	+	0.985618i	$0.445951\pi$
$954$	0	0
$955$	1.35424e13	0.526843
$956$	−5.48779e12	−0.212489
$957$	0	0
$958$	−2.27334e13	−0.872007
$959$	−8.34784e11	−0.0318706
$960$	0	0
$961$	−1.49837e13	−0.566714
$962$	−2.65650e12	−0.100005
$963$	0	0
$964$	4.53011e13	1.68951
$965$	1.55855e13	0.578561
$966$	0	0
$967$	−3.11450e13	−1.14543	−0.572716	−	0.819754i	$-0.694110\pi$
$967$	−3.11450e13	−1.14543	−0.572716	+	0.819754i	$0.694110\pi$
$968$	−4.22473e12	−0.154653
$969$	0	0
$970$	3.69141e12	0.133881
$971$	2.54818e12	0.0919906	0.0459953	−	0.998942i	$-0.485354\pi$
$971$	2.54818e12	0.0919906	0.0459953	+	0.998942i	$0.485354\pi$
$972$	0	0
$973$	4.87608e12	0.174406
$974$	1.98441e13	0.706507
$975$	0	0
$976$	−5.28601e13	−1.86468
$977$	−2.02788e12	−0.0712061	−0.0356030	−	0.999366i	$-0.511335\pi$
$977$	−2.02788e12	−0.0712061	−0.0356030	+	0.999366i	$0.511335\pi$
$978$	0	0
$979$	−9.80590e12	−0.341166
$980$	−3.87114e12	−0.134067
$981$	0	0
$982$	−8.95226e13	−3.07207
$983$	2.37073e13	0.809825	0.404912	−	0.914355i	$-0.367302\pi$
$983$	2.37073e13	0.809825	0.404912	+	0.914355i	$0.367302\pi$
$984$	0	0
$985$	1.00118e13	0.338884
$986$	2.83678e13	0.955828
$987$	0	0
$988$	3.35636e12	0.112063
$989$	6.33020e12	0.210394
$990$	0	0
$991$	4.70050e13	1.54815	0.774074	−	0.633095i	$-0.218216\pi$
$991$	4.70050e13	1.54815	0.774074	+	0.633095i	$0.218216\pi$
$992$	−7.30302e12	−0.239442
$993$	0	0
$994$	2.14790e12	0.0697872
$995$	3.62553e12	0.117265
$996$	0	0
$997$	3.19557e12	0.102428	0.0512141	−	0.998688i	$-0.483691\pi$
$997$	3.19557e12	0.102428	0.0512141	+	0.998688i	$0.483691\pi$
$998$	−5.87799e12	−0.187560
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.l.1.2		6
3.2	odd	2		35.10.a.e.1.5	✓	6
15.2	even	4		175.10.b.g.99.10		12
15.8	even	4		175.10.b.g.99.3		12
15.14	odd	2		175.10.a.g.1.2		6
21.20	even	2		245.10.a.g.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.e.1.5	✓	6	3.2	odd	2
175.10.a.g.1.2		6	15.14	odd	2
175.10.b.g.99.3		12	15.8	even	4
175.10.b.g.99.10		12	15.2	even	4
245.10.a.g.1.5		6	21.20	even	2
315.10.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q-39.8299 q^{2} +1074.42 q^{4} -625.000 q^{5} -2401.00 q^{7} -22401.1 q^{8} +24893.7 q^{10} +50463.3 q^{11} +3922.21 q^{13} +95631.6 q^{14} +342132. q^{16} +589429. q^{17} +796460. q^{19} -671513. q^{20} -2.00995e6 q^{22} +436793. q^{23} +390625. q^{25} -156221. q^{26} -2.57968e6 q^{28} -1.20833e6 q^{29} +3.38466e6 q^{31} -2.15768e6 q^{32} -2.34769e7 q^{34} +1.50062e6 q^{35} +1.70047e7 q^{37} -3.17229e7 q^{38} +1.40007e7 q^{40} -2.58643e7 q^{41} +1.44925e7 q^{43} +5.42188e7 q^{44} -1.73974e7 q^{46} +5.55921e7 q^{47} +5.76480e6 q^{49} -1.55586e7 q^{50} +4.21410e6 q^{52} +7.67285e7 q^{53} -3.15395e7 q^{55} +5.37851e7 q^{56} +4.81276e7 q^{58} +1.15070e8 q^{59} -1.54502e8 q^{61} -1.34811e8 q^{62} -8.92311e7 q^{64} -2.45138e6 q^{65} +2.72000e8 q^{67} +6.33295e8 q^{68} -5.97697e7 q^{70} +2.24602e7 q^{71} -8.21761e6 q^{73} -6.77297e8 q^{74} +8.55733e8 q^{76} -1.21162e8 q^{77} +5.20375e8 q^{79} -2.13832e8 q^{80} +1.03017e9 q^{82} -9.36372e7 q^{83} -3.68393e8 q^{85} -5.77233e8 q^{86} -1.13043e9 q^{88} -1.94318e8 q^{89} -9.41722e6 q^{91} +4.69299e8 q^{92} -2.21423e9 q^{94} -4.97788e8 q^{95} +1.48287e8 q^{97} -2.29611e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8} + 9375 q^{10} + 47796 q^{11} + 102168 q^{13} + 36015 q^{14} + 2371065 q^{16} + 38472 q^{17} + 361056 q^{19} - 1880625 q^{20} + 2068680 q^{22}+ \cdots - 86472015 q^{98}+O(q^{100}) \) 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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