LMFDB - Embedded newform 315.10.a.e.1.3

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			1.3
Root			$5.50890$ 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)

$f(q)$	$=$	\(q+8.50890 q^{2} -439.599 q^{4} +625.000 q^{5} +2401.00 q^{7} -8097.06 q^{8} +5318.06 q^{10} +84700.6 q^{11} +189194. q^{13} +20429.9 q^{14} +156177. q^{16} +238796. q^{17} +621510. q^{19} -274749. q^{20} +720709. q^{22} +211156. q^{23} +390625. q^{25} +1.60983e6 q^{26} -1.05548e6 q^{28} -3.28769e6 q^{29} -7.79081e6 q^{31} +5.47459e6 q^{32} +2.03189e6 q^{34} +1.50062e6 q^{35} -8.19742e6 q^{37} +5.28836e6 q^{38} -5.06066e6 q^{40} -1.82676e7 q^{41} +8.54852e6 q^{43} -3.72343e7 q^{44} +1.79671e6 q^{46} +2.44728e7 q^{47} +5.76480e6 q^{49} +3.32379e6 q^{50} -8.31693e7 q^{52} +3.12652e7 q^{53} +5.29379e7 q^{55} -1.94410e7 q^{56} -2.79746e7 q^{58} +1.53039e8 q^{59} +9.34546e7 q^{61} -6.62913e7 q^{62} -3.33800e7 q^{64} +1.18246e8 q^{65} -2.45728e6 q^{67} -1.04974e8 q^{68} +1.27687e7 q^{70} +2.39479e8 q^{71} -3.22135e8 q^{73} -6.97510e7 q^{74} -2.73215e8 q^{76} +2.03366e8 q^{77} -6.41950e8 q^{79} +9.76108e7 q^{80} -1.55437e8 q^{82} +7.07851e8 q^{83} +1.49247e8 q^{85} +7.27385e7 q^{86} -6.85826e8 q^{88} -3.67866e8 q^{89} +4.54254e8 q^{91} -9.28240e7 q^{92} +2.08236e8 q^{94} +3.88443e8 q^{95} -5.71196e8 q^{97} +4.90521e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	8.50890	0.376044	0.188022	−	0.982165i	$-0.439792\pi$
$2$	8.50890	0.376044	0.188022	+	0.982165i	$0.439792\pi$
$3$	0	0
$3$	0	0
$4$	−439.599	−0.858591
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−8097.06	−0.698912
$9$	0	0
$10$	5318.06	0.168172
$11$	84700.6	1.74429	0.872146	−	0.489245i	$-0.162728\pi$
$11$	84700.6	1.74429	0.872146	+	0.489245i	$0.162728\pi$
$12$	0	0
$13$	189194.	1.83722	0.918611	−	0.395163i	$-0.129312\pi$
$13$	189194.	1.83722	0.918611	+	0.395163i	$0.129312\pi$
$14$	20429.9	0.142131
$15$	0	0
$16$	156177.	0.595769
$17$	238796.	0.693435	0.346718	−	0.937970i	$-0.387296\pi$
$17$	238796.	0.693435	0.346718	+	0.937970i	$0.387296\pi$
$18$	0	0
$19$	621510.	1.09410	0.547049	−	0.837100i	$-0.315751\pi$
$19$	621510.	1.09410	0.547049	+	0.837100i	$0.315751\pi$
$20$	−274749.	−0.383974
$21$	0	0
$22$	720709.	0.655931
$23$	211156.	0.157336	0.0786681	−	0.996901i	$-0.474933\pi$
$23$	211156.	0.157336	0.0786681	+	0.996901i	$0.474933\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	1.60983e6	0.690876
$27$	0	0
$28$	−1.05548e6	−0.324517
$29$	−3.28769e6	−0.863176	−0.431588	−	0.902071i	$-0.642047\pi$
$29$	−3.28769e6	−0.863176	−0.431588	+	0.902071i	$0.642047\pi$
$30$	0	0
$31$	−7.79081e6	−1.51515	−0.757574	−	0.652749i	$-0.773616\pi$
$31$	−7.79081e6	−1.51515	−0.757574	+	0.652749i	$0.773616\pi$
$32$	5.47459e6	0.922947
$33$	0	0
$34$	2.03189e6	0.260762
$35$	1.50062e6	0.169031
$36$	0	0
$37$	−8.19742e6	−0.719067	−0.359533	−	0.933132i	$-0.617064\pi$
$37$	−8.19742e6	−0.719067	−0.359533	+	0.933132i	$0.617064\pi$
$38$	5.28836e6	0.411429
$39$	0	0
$40$	−5.06066e6	−0.312563
$41$	−1.82676e7	−1.00961	−0.504805	−	0.863233i	$-0.668436\pi$
$41$	−1.82676e7	−1.00961	−0.504805	+	0.863233i	$0.668436\pi$
$42$	0	0
$43$	8.54852e6	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$43$	8.54852e6	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$44$	−3.72343e7	−1.49763
$45$	0	0
$46$	1.79671e6	0.0591653
$47$	2.44728e7	0.731548	0.365774	−	0.930704i	$-0.380804\pi$
$47$	2.44728e7	0.731548	0.365774	+	0.930704i	$0.380804\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	3.32379e6	0.0752088
$51$	0	0
$52$	−8.31693e7	−1.57742
$53$	3.12652e7	0.544276	0.272138	−	0.962258i	$-0.412269\pi$
$53$	3.12652e7	0.544276	0.272138	+	0.962258i	$0.412269\pi$
$54$	0	0
$55$	5.29379e7	0.780071
$56$	−1.94410e7	−0.264164
$57$	0	0
$58$	−2.79746e7	−0.324592
$59$	1.53039e8	1.64426	0.822128	−	0.569303i	$-0.192787\pi$
$59$	1.53039e8	1.64426	0.822128	+	0.569303i	$0.192787\pi$
$60$	0	0
$61$	9.34546e7	0.864204	0.432102	−	0.901825i	$-0.357772\pi$
$61$	9.34546e7	0.864204	0.432102	+	0.901825i	$0.357772\pi$
$62$	−6.62913e7	−0.569762
$63$	0	0
$64$	−3.33800e7	−0.248701
$65$	1.18246e8	0.821631
$66$	0	0
$67$	−2.45728e6	−0.0148976	−0.00744881	−	0.999972i	$-0.502371\pi$
$67$	−2.45728e6	−0.0148976	−0.00744881	+	0.999972i	$0.502371\pi$
$68$	−1.04974e8	−0.595377
$69$	0	0
$70$	1.27687e7	0.0635630
$71$	2.39479e8	1.11842	0.559211	−	0.829026i	$-0.311104\pi$
$71$	2.39479e8	1.11842	0.559211	+	0.829026i	$0.311104\pi$
$72$	0	0
$73$	−3.22135e8	−1.32765	−0.663827	−	0.747886i	$-0.731069\pi$
$73$	−3.22135e8	−1.32765	−0.663827	+	0.747886i	$0.731069\pi$
$74$	−6.97510e7	−0.270401
$75$	0	0
$76$	−2.73215e8	−0.939383
$77$	2.03366e8	0.659281
$78$	0	0
$79$	−6.41950e8	−1.85430	−0.927149	−	0.374693i	$-0.877748\pi$
$79$	−6.41950e8	−1.85430	−0.927149	+	0.374693i	$0.877748\pi$
$80$	9.76108e7	0.266436
$81$	0	0
$82$	−1.55437e8	−0.379658
$83$	7.07851e8	1.63716	0.818579	−	0.574394i	$-0.194762\pi$
$83$	7.07851e8	1.63716	0.818579	+	0.574394i	$0.194762\pi$
$84$	0	0
$85$	1.49247e8	0.310114
$86$	7.27385e7	0.143391
$87$	0	0
$88$	−6.85826e8	−1.21911
$89$	−3.67866e8	−0.621490	−0.310745	−	0.950493i	$-0.600579\pi$
$89$	−3.67866e8	−0.621490	−0.310745	+	0.950493i	$0.600579\pi$
$90$	0	0
$91$	4.54254e8	0.694405
$92$	−9.28240e7	−0.135087
$93$	0	0
$94$	2.08236e8	0.275094
$95$	3.88443e8	0.489296
$96$	0	0
$97$	−5.71196e8	−0.655107	−0.327553	−	0.944833i	$-0.606224\pi$
$97$	−5.71196e8	−0.655107	−0.327553	+	0.944833i	$0.606224\pi$
$98$	4.90521e7	0.0537206
$99$	0	0
$100$	−1.71718e8	−0.171718
$101$	−1.64426e8	−0.157226	−0.0786131	−	0.996905i	$-0.525049\pi$
$101$	−1.64426e8	−0.157226	−0.0786131	+	0.996905i	$0.525049\pi$
$102$	0	0
$103$	−4.33218e8	−0.379262	−0.189631	−	0.981855i	$-0.560729\pi$
$103$	−4.33218e8	−0.379262	−0.189631	+	0.981855i	$0.560729\pi$
$104$	−1.53191e9	−1.28406
$105$	0	0
$106$	2.66032e8	0.204672
$107$	1.12984e9	0.833280	0.416640	−	0.909072i	$-0.363208\pi$
$107$	1.12984e9	0.833280	0.416640	+	0.909072i	$0.363208\pi$
$108$	0	0
$109$	1.28549e9	0.872268	0.436134	−	0.899882i	$-0.356347\pi$
$109$	1.28549e9	0.872268	0.436134	+	0.899882i	$0.356347\pi$
$110$	4.50443e8	0.293341
$111$	0	0
$112$	3.74982e8	0.225180
$113$	1.13778e9	0.656456	0.328228	−	0.944598i	$-0.393549\pi$
$113$	1.13778e9	0.656456	0.328228	+	0.944598i	$0.393549\pi$
$114$	0	0
$115$	1.31973e8	0.0703629
$116$	1.44526e9	0.741115
$117$	0	0
$118$	1.30220e9	0.618313
$119$	5.73348e8	0.262094
$120$	0	0
$121$	4.81624e9	2.04256
$122$	7.95196e8	0.324979
$123$	0	0
$124$	3.42483e9	1.30089
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	2.72140e8	0.0928273	0.0464136	−	0.998922i	$-0.485221\pi$
$127$	2.72140e8	0.0928273	0.0464136	+	0.998922i	$0.485221\pi$
$128$	−3.08702e9	−1.01647
$129$	0	0
$130$	1.00614e9	0.308969
$131$	3.49321e9	1.03634	0.518172	−	0.855276i	$-0.326613\pi$
$131$	3.49321e9	1.03634	0.518172	+	0.855276i	$0.326613\pi$
$132$	0	0
$133$	1.49224e9	0.413531
$134$	−2.09087e7	−0.00560216
$135$	0	0
$136$	−1.93354e9	−0.484650
$137$	−1.58837e9	−0.385219	−0.192610	−	0.981275i	$-0.561695\pi$
$137$	−1.58837e9	−0.385219	−0.192610	+	0.981275i	$0.561695\pi$
$138$	0	0
$139$	−6.80759e9	−1.54677	−0.773387	−	0.633934i	$-0.781439\pi$
$139$	−6.80759e9	−1.54677	−0.773387	+	0.633934i	$0.781439\pi$
$140$	−6.59673e8	−0.145128
$141$	0	0
$142$	2.03771e9	0.420576
$143$	1.60248e10	3.20465
$144$	0	0
$145$	−2.05480e9	−0.386024
$146$	−2.74101e9	−0.499256
$147$	0	0
$148$	3.60357e9	0.617384
$149$	−5.22131e9	−0.867843	−0.433922	−	0.900951i	$-0.642871\pi$
$149$	−5.22131e9	−0.867843	−0.433922	+	0.900951i	$0.642871\pi$
$150$	0	0
$151$	7.29225e9	1.14147	0.570736	−	0.821134i	$-0.306658\pi$
$151$	7.29225e9	1.14147	0.570736	+	0.821134i	$0.306658\pi$
$152$	−5.03240e9	−0.764679
$153$	0	0
$154$	1.73042e9	0.247919
$155$	−4.86926e9	−0.677595
$156$	0	0
$157$	−1.17911e9	−0.154884	−0.0774420	−	0.996997i	$-0.524675\pi$
$157$	−1.17911e9	−0.154884	−0.0774420	+	0.996997i	$0.524675\pi$
$158$	−5.46229e9	−0.697298
$159$	0	0
$160$	3.42162e9	0.412755
$161$	5.06986e8	0.0594675
$162$	0	0
$163$	−3.90232e9	−0.432991	−0.216495	−	0.976284i	$-0.569463\pi$
$163$	−3.90232e9	−0.432991	−0.216495	+	0.976284i	$0.569463\pi$
$164$	8.03040e9	0.866842
$165$	0	0
$166$	6.02304e9	0.615643
$167$	−1.75237e10	−1.74342	−0.871708	−	0.490027i	$-0.836987\pi$
$167$	−1.75237e10	−1.74342	−0.871708	+	0.490027i	$0.836987\pi$
$168$	0	0
$169$	2.51898e10	2.37538
$170$	1.26993e9	0.116616
$171$	0	0
$172$	−3.75792e9	−0.327393
$173$	1.31696e10	1.11780	0.558902	−	0.829234i	$-0.311223\pi$
$173$	1.31696e10	1.11780	0.558902	+	0.829234i	$0.311223\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	1.32283e10	1.03920
$177$	0	0
$178$	−3.13013e9	−0.233708
$179$	−8.29958e9	−0.604251	−0.302126	−	0.953268i	$-0.597696\pi$
$179$	−8.29958e9	−0.604251	−0.302126	+	0.953268i	$0.597696\pi$
$180$	0	0
$181$	−2.35265e10	−1.62931	−0.814654	−	0.579948i	$-0.803073\pi$
$181$	−2.35265e10	−1.62931	−0.814654	+	0.579948i	$0.803073\pi$
$182$	3.86520e9	0.261127
$183$	0	0
$184$	−1.70975e9	−0.109964
$185$	−5.12338e9	−0.321576
$186$	0	0
$187$	2.02261e10	1.20955
$188$	−1.07582e10	−0.628100
$189$	0	0
$190$	3.30523e9	0.183997
$191$	−1.46919e10	−0.798779	−0.399389	−	0.916781i	$-0.630778\pi$
$191$	−1.46919e10	−0.798779	−0.399389	+	0.916781i	$0.630778\pi$
$192$	0	0
$193$	1.81312e10	0.940631	0.470316	−	0.882498i	$-0.344140\pi$
$193$	1.81312e10	0.940631	0.470316	+	0.882498i	$0.344140\pi$
$194$	−4.86025e9	−0.246349
$195$	0	0
$196$	−2.53420e9	−0.122656
$197$	7.70706e9	0.364578	0.182289	−	0.983245i	$-0.441649\pi$
$197$	7.70706e9	0.364578	0.182289	+	0.983245i	$0.441649\pi$
$198$	0	0
$199$	3.03335e10	1.37115	0.685573	−	0.728004i	$-0.259552\pi$
$199$	3.03335e10	1.37115	0.685573	+	0.728004i	$0.259552\pi$
$200$	−3.16291e9	−0.139782
$201$	0	0
$202$	−1.39909e9	−0.0591240
$203$	−7.89373e9	−0.326250
$204$	0	0
$205$	−1.14172e10	−0.451511
$206$	−3.68621e9	−0.142619
$207$	0	0
$208$	2.95478e10	1.09456
$209$	5.26422e10	1.90843
$210$	0	0
$211$	−3.92883e10	−1.36456	−0.682279	−	0.731092i	$-0.739011\pi$
$211$	−3.92883e10	−1.36456	−0.682279	+	0.731092i	$0.739011\pi$
$212$	−1.37441e10	−0.467311
$213$	0	0
$214$	9.61372e9	0.313350
$215$	5.34282e9	0.170529
$216$	0	0
$217$	−1.87057e10	−0.572672
$218$	1.09381e10	0.328011
$219$	0	0
$220$	−2.32714e10	−0.669762
$221$	4.51786e10	1.27399
$222$	0	0
$223$	6.57832e10	1.78133	0.890663	−	0.454664i	$-0.150241\pi$
$223$	6.57832e10	1.78133	0.890663	+	0.454664i	$0.150241\pi$
$224$	1.31445e10	0.348841
$225$	0	0
$226$	9.68127e9	0.246856
$227$	−1.71346e10	−0.428310	−0.214155	−	0.976800i	$-0.568700\pi$
$227$	−1.71346e10	−0.428310	−0.214155	+	0.976800i	$0.568700\pi$
$228$	0	0
$229$	−6.28548e10	−1.51035	−0.755177	−	0.655521i	$-0.772449\pi$
$229$	−6.28548e10	−1.51035	−0.755177	+	0.655521i	$0.772449\pi$
$230$	1.12294e9	0.0264595
$231$	0	0
$232$	2.66206e10	0.603284
$233$	7.49874e9	0.166681	0.0833406	−	0.996521i	$-0.473441\pi$
$233$	7.49874e9	0.166681	0.0833406	+	0.996521i	$0.473441\pi$
$234$	0	0
$235$	1.52955e10	0.327158
$236$	−6.72759e10	−1.41174
$237$	0	0
$238$	4.87856e9	0.0985589
$239$	−1.17440e10	−0.232823	−0.116412	−	0.993201i	$-0.537139\pi$
$239$	−1.17440e10	−0.232823	−0.116412	+	0.993201i	$0.537139\pi$
$240$	0	0
$241$	7.32711e9	0.139912	0.0699562	−	0.997550i	$-0.477714\pi$
$241$	7.32711e9	0.139912	0.0699562	+	0.997550i	$0.477714\pi$
$242$	4.09809e10	0.768091
$243$	0	0
$244$	−4.10825e10	−0.741998
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	1.17586e11	2.01010
$248$	6.30827e10	1.05896
$249$	0	0
$250$	2.07737e9	0.0336344
$251$	1.01275e11	1.61053	0.805265	−	0.592915i	$-0.202023\pi$
$251$	1.01275e11	1.61053	0.805265	+	0.592915i	$0.202023\pi$
$252$	0	0
$253$	1.78851e10	0.274440
$254$	2.31561e9	0.0349071
$255$	0	0
$256$	−9.17658e9	−0.133537
$257$	−1.90121e10	−0.271851	−0.135925	−	0.990719i	$-0.543401\pi$
$257$	−1.90121e10	−0.271851	−0.135925	+	0.990719i	$0.543401\pi$
$258$	0	0
$259$	−1.96820e10	−0.271782
$260$	−5.19808e10	−0.705445
$261$	0	0
$262$	2.97234e10	0.389711
$263$	8.69752e10	1.12097	0.560486	−	0.828164i	$-0.310615\pi$
$263$	8.69752e10	1.12097	0.560486	+	0.828164i	$0.310615\pi$
$264$	0	0
$265$	1.95407e10	0.243408
$266$	1.26974e10	0.155506
$267$	0	0
$268$	1.08021e9	0.0127910
$269$	−4.30691e10	−0.501511	−0.250756	−	0.968050i	$-0.580679\pi$
$269$	−4.30691e10	−0.501511	−0.250756	+	0.968050i	$0.580679\pi$
$270$	0	0
$271$	−2.87351e10	−0.323632	−0.161816	−	0.986821i	$-0.551735\pi$
$271$	−2.87351e10	−0.323632	−0.161816	+	0.986821i	$0.551735\pi$
$272$	3.72945e10	0.413128
$273$	0	0
$274$	−1.35153e10	−0.144859
$275$	3.30862e10	0.348859
$276$	0	0
$277$	1.04061e11	1.06201	0.531005	−	0.847369i	$-0.321815\pi$
$277$	1.04061e11	1.06201	0.531005	+	0.847369i	$0.321815\pi$
$278$	−5.79252e10	−0.581655
$279$	0	0
$280$	−1.21507e10	−0.118138
$281$	1.41171e11	1.35072	0.675361	−	0.737487i	$-0.263988\pi$
$281$	1.41171e11	1.35072	0.675361	+	0.737487i	$0.263988\pi$
$282$	0	0
$283$	−1.59424e11	−1.47746	−0.738728	−	0.674003i	$-0.764573\pi$
$283$	−1.59424e11	−1.47746	−0.738728	+	0.674003i	$0.764573\pi$
$284$	−1.05275e11	−0.960266
$285$	0	0
$286$	1.36354e11	1.20509
$287$	−4.38605e10	−0.381597
$288$	0	0
$289$	−6.15646e10	−0.519147
$290$	−1.74841e10	−0.145162
$291$	0	0
$292$	1.41610e11	1.13991
$293$	1.36625e11	1.08299	0.541496	−	0.840703i	$-0.317858\pi$
$293$	1.36625e11	1.08299	0.541496	+	0.840703i	$0.317858\pi$
$294$	0	0
$295$	9.56497e10	0.735334
$296$	6.63750e10	0.502564
$297$	0	0
$298$	−4.44276e10	−0.326347
$299$	3.99494e10	0.289062
$300$	0	0
$301$	2.05250e10	0.144123
$302$	6.20491e10	0.429244
$303$	0	0
$304$	9.70657e10	0.651831
$305$	5.84091e10	0.386484
$306$	0	0
$307$	9.46778e10	0.608311	0.304155	−	0.952622i	$-0.401626\pi$
$307$	9.46778e10	0.608311	0.304155	+	0.952622i	$0.401626\pi$
$308$	−8.93995e10	−0.566052
$309$	0	0
$310$	−4.14321e10	−0.254806
$311$	−3.38383e10	−0.205110	−0.102555	−	0.994727i	$-0.532702\pi$
$311$	−3.38383e10	−0.205110	−0.102555	+	0.994727i	$0.532702\pi$
$312$	0	0
$313$	3.57622e10	0.210608	0.105304	−	0.994440i	$-0.466418\pi$
$313$	3.57622e10	0.210608	0.105304	+	0.994440i	$0.466418\pi$
$314$	−1.00330e10	−0.0582432
$315$	0	0
$316$	2.82200e11	1.59208
$317$	9.29341e10	0.516902	0.258451	−	0.966024i	$-0.416788\pi$
$317$	9.29341e10	0.516902	0.258451	+	0.966024i	$0.416788\pi$
$318$	0	0
$319$	−2.78469e11	−1.50563
$320$	−2.08625e10	−0.111222
$321$	0	0
$322$	4.31390e9	0.0223624
$323$	1.48414e11	0.758687
$324$	0	0
$325$	7.39038e10	0.367444
$326$	−3.32045e10	−0.162824
$327$	0	0
$328$	1.47914e11	0.705629
$329$	5.87591e10	0.276499
$330$	0	0
$331$	7.12840e10	0.326412	0.163206	−	0.986592i	$-0.447816\pi$
$331$	7.12840e10	0.326412	0.163206	+	0.986592i	$0.447816\pi$
$332$	−3.11170e11	−1.40565
$333$	0	0
$334$	−1.49107e11	−0.655601
$335$	−1.53580e9	−0.00666242
$336$	0	0
$337$	−4.12351e10	−0.174153	−0.0870767	−	0.996202i	$-0.527753\pi$
$337$	−4.12351e10	−0.174153	−0.0870767	+	0.996202i	$0.527753\pi$
$338$	2.14337e11	0.893249
$339$	0	0
$340$	−6.56089e10	−0.266261
$341$	−6.59887e11	−2.64286
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	−6.92179e10	−0.266505
$345$	0	0
$346$	1.12059e11	0.420343
$347$	1.15561e11	0.427887	0.213944	−	0.976846i	$-0.431369\pi$
$347$	1.15561e11	0.427887	0.213944	+	0.976846i	$0.431369\pi$
$348$	0	0
$349$	−1.48436e11	−0.535581	−0.267790	−	0.963477i	$-0.586294\pi$
$349$	−1.48436e11	−0.535581	−0.267790	+	0.963477i	$0.586294\pi$
$350$	7.98042e9	0.0284263
$351$	0	0
$352$	4.63701e11	1.60989
$353$	−2.97608e11	−1.02014	−0.510068	−	0.860134i	$-0.670380\pi$
$353$	−2.97608e11	−1.02014	−0.510068	+	0.860134i	$0.670380\pi$
$354$	0	0
$355$	1.49675e11	0.500173
$356$	1.61713e11	0.533606
$357$	0	0
$358$	−7.06203e10	−0.227225
$359$	−1.05804e11	−0.336183	−0.168091	−	0.985771i	$-0.553760\pi$
$359$	−1.05804e11	−0.336183	−0.168091	+	0.985771i	$0.553760\pi$
$360$	0	0
$361$	6.35864e10	0.197052
$362$	−2.00184e11	−0.612691
$363$	0	0
$364$	−1.99689e11	−0.596210
$365$	−2.01334e11	−0.593745
$366$	0	0
$367$	−5.06458e11	−1.45729	−0.728645	−	0.684892i	$-0.759849\pi$
$367$	−5.06458e11	−1.45729	−0.728645	+	0.684892i	$0.759849\pi$
$368$	3.29778e10	0.0937361
$369$	0	0
$370$	−4.35944e10	−0.120927
$371$	7.50677e10	0.205717
$372$	0	0
$373$	3.83248e11	1.02516	0.512578	−	0.858640i	$-0.328690\pi$
$373$	3.83248e11	1.02516	0.512578	+	0.858640i	$0.328690\pi$
$374$	1.72102e11	0.454846
$375$	0	0
$376$	−1.98157e11	−0.511287
$377$	−6.22010e11	−1.58585
$378$	0	0
$379$	2.92901e11	0.729196	0.364598	−	0.931165i	$-0.381206\pi$
$379$	2.92901e11	0.729196	0.364598	+	0.931165i	$0.381206\pi$
$380$	−1.70759e11	−0.420105
$381$	0	0
$382$	−1.25012e11	−0.300376
$383$	2.49849e11	0.593312	0.296656	−	0.954984i	$-0.404129\pi$
$383$	2.49849e11	0.593312	0.296656	+	0.954984i	$0.404129\pi$
$384$	0	0
$385$	1.27104e11	0.294839
$386$	1.54277e11	0.353719
$387$	0	0
$388$	2.51097e11	0.562469
$389$	5.76843e11	1.27728	0.638638	−	0.769508i	$-0.279498\pi$
$389$	5.76843e11	1.27728	0.638638	+	0.769508i	$0.279498\pi$
$390$	0	0
$391$	5.04232e10	0.109103
$392$	−4.66779e10	−0.0998446
$393$	0	0
$394$	6.55787e10	0.137097
$395$	−4.01219e11	−0.829267
$396$	0	0
$397$	3.17997e11	0.642489	0.321244	−	0.946996i	$-0.395899\pi$
$397$	3.17997e11	0.642489	0.321244	+	0.946996i	$0.395899\pi$
$398$	2.58105e11	0.515611
$399$	0	0
$400$	6.10068e10	0.119154
$401$	7.92446e11	1.53045	0.765227	−	0.643761i	$-0.222627\pi$
$401$	7.92446e11	1.53045	0.765227	+	0.643761i	$0.222627\pi$
$402$	0	0
$403$	−1.47397e12	−2.78366
$404$	7.22815e10	0.134993
$405$	0	0
$406$	−6.71670e10	−0.122684
$407$	−6.94326e11	−1.25426
$408$	0	0
$409$	−9.24061e11	−1.63285	−0.816424	−	0.577453i	$-0.804047\pi$
$409$	−9.24061e11	−1.63285	−0.816424	+	0.577453i	$0.804047\pi$
$410$	−9.71482e10	−0.169788
$411$	0	0
$412$	1.90442e11	0.325631
$413$	3.67448e11	0.621470
$414$	0	0
$415$	4.42407e11	0.732159
$416$	1.03576e12	1.69566
$417$	0	0
$418$	4.47928e11	0.717653
$419$	1.25893e11	0.199543	0.0997717	−	0.995010i	$-0.468189\pi$
$419$	1.25893e11	0.199543	0.0997717	+	0.995010i	$0.468189\pi$
$420$	0	0
$421$	2.45108e11	0.380267	0.190133	−	0.981758i	$-0.439108\pi$
$421$	2.45108e11	0.380267	0.190133	+	0.981758i	$0.439108\pi$
$422$	−3.34300e11	−0.513134
$423$	0	0
$424$	−2.53156e11	−0.380401
$425$	9.32795e10	0.138687
$426$	0	0
$427$	2.24384e11	0.326638
$428$	−4.96677e11	−0.715446
$429$	0	0
$430$	4.54616e10	0.0641263
$431$	−4.05916e11	−0.566615	−0.283308	−	0.959029i	$-0.591432\pi$
$431$	−4.05916e11	−0.566615	−0.283308	+	0.959029i	$0.591432\pi$
$432$	0	0
$433$	1.01105e10	0.0138221	0.00691107	−	0.999976i	$-0.497800\pi$
$433$	1.01105e10	0.0138221	0.00691107	+	0.999976i	$0.497800\pi$
$434$	−1.59165e11	−0.215350
$435$	0	0
$436$	−5.65100e11	−0.748922
$437$	1.31236e11	0.172141
$438$	0	0
$439$	9.62543e11	1.23689	0.618443	−	0.785830i	$-0.287764\pi$
$439$	9.62543e11	1.23689	0.618443	+	0.785830i	$0.287764\pi$
$440$	−4.28641e11	−0.545201
$441$	0	0
$442$	3.84420e11	0.479078
$443$	1.60851e12	1.98430	0.992151	−	0.125048i	$-0.0399085\pi$
$443$	1.60851e12	1.98430	0.992151	+	0.125048i	$0.0399085\pi$
$444$	0	0
$445$	−2.29916e11	−0.277939
$446$	5.59743e11	0.669857
$447$	0	0
$448$	−8.01454e10	−0.0940000
$449$	3.14682e11	0.365396	0.182698	−	0.983169i	$-0.441517\pi$
$449$	3.14682e11	0.365396	0.182698	+	0.983169i	$0.441517\pi$
$450$	0	0
$451$	−1.54728e12	−1.76106
$452$	−5.00167e11	−0.563627
$453$	0	0
$454$	−1.45797e11	−0.161063
$455$	2.83909e11	0.310547
$456$	0	0
$457$	−1.81581e12	−1.94736	−0.973682	−	0.227909i	$-0.926811\pi$
$457$	−1.81581e12	−1.94736	−0.973682	+	0.227909i	$0.926811\pi$
$458$	−5.34826e11	−0.567960
$459$	0	0
$460$	−5.80150e10	−0.0604130
$461$	−5.49214e11	−0.566353	−0.283177	−	0.959068i	$-0.591388\pi$
$461$	−5.49214e11	−0.566353	−0.283177	+	0.959068i	$0.591388\pi$
$462$	0	0
$463$	8.97351e11	0.907502	0.453751	−	0.891128i	$-0.350086\pi$
$463$	8.97351e11	0.907502	0.453751	+	0.891128i	$0.350086\pi$
$464$	−5.13462e11	−0.514254
$465$	0	0
$466$	6.38061e10	0.0626795
$467$	−4.77151e10	−0.0464227	−0.0232113	−	0.999731i	$-0.507389\pi$
$467$	−4.77151e10	−0.0464227	−0.0232113	+	0.999731i	$0.507389\pi$
$468$	0	0
$469$	−5.89992e9	−0.00563077
$470$	1.30148e11	0.123026
$471$	0	0
$472$	−1.23917e12	−1.14919
$473$	7.24065e11	0.665123
$474$	0	0
$475$	2.42777e11	0.218820
$476$	−2.52043e11	−0.225032
$477$	0	0
$478$	−9.99288e10	−0.0875518
$479$	1.70191e12	1.47716	0.738579	−	0.674167i	$-0.235497\pi$
$479$	1.70191e12	1.47716	0.738579	+	0.674167i	$0.235497\pi$
$480$	0	0
$481$	−1.55090e12	−1.32109
$482$	6.23457e10	0.0526132
$483$	0	0
$484$	−2.11721e12	−1.75372
$485$	−3.56997e11	−0.292973
$486$	0	0
$487$	1.19913e12	0.966023	0.483012	−	0.875614i	$-0.339543\pi$
$487$	1.19913e12	0.966023	0.483012	+	0.875614i	$0.339543\pi$
$488$	−7.56707e11	−0.604003
$489$	0	0
$490$	3.06576e10	0.0240246
$491$	−2.54813e12	−1.97859	−0.989294	−	0.145938i	$-0.953380\pi$
$491$	−2.54813e12	−1.97859	−0.989294	+	0.145938i	$0.953380\pi$
$492$	0	0
$493$	−7.85085e11	−0.598557
$494$	1.00053e12	0.755887
$495$	0	0
$496$	−1.21675e12	−0.902679
$497$	5.74990e11	0.422723
$498$	0	0
$499$	−1.05425e12	−0.761189	−0.380595	−	0.924742i	$-0.624281\pi$
$499$	−1.05425e12	−0.761189	−0.380595	+	0.924742i	$0.624281\pi$
$500$	−1.07324e11	−0.0767947
$501$	0	0
$502$	8.61736e11	0.605630
$503$	1.45930e12	1.01646	0.508230	−	0.861222i	$-0.330300\pi$
$503$	1.45930e12	1.01646	0.508230	+	0.861222i	$0.330300\pi$
$504$	0	0
$505$	−1.02766e11	−0.0703137
$506$	1.52182e11	0.103202
$507$	0	0
$508$	−1.19632e11	−0.0797007
$509$	−1.40901e12	−0.930430	−0.465215	−	0.885198i	$-0.654023\pi$
$509$	−1.40901e12	−0.930430	−0.465215	+	0.885198i	$0.654023\pi$
$510$	0	0
$511$	−7.73446e11	−0.501806
$512$	1.50247e12	0.966254
$513$	0	0
$514$	−1.61772e11	−0.102228
$515$	−2.70761e11	−0.169611
$516$	0	0
$517$	2.07286e12	1.27603
$518$	−1.67472e11	−0.102202
$519$	0	0
$520$	−9.57446e11	−0.574247
$521$	−1.65259e12	−0.982643	−0.491321	−	0.870978i	$-0.663486\pi$
$521$	−1.65259e12	−0.982643	−0.491321	+	0.870978i	$0.663486\pi$
$522$	0	0
$523$	−1.76435e12	−1.03116	−0.515581	−	0.856841i	$-0.672424\pi$
$523$	−1.76435e12	−1.03116	−0.515581	+	0.856841i	$0.672424\pi$
$524$	−1.53561e12	−0.889795
$525$	0	0
$526$	7.40063e11	0.421534
$527$	−1.86041e12	−1.05066
$528$	0	0
$529$	−1.75657e12	−0.975245
$530$	1.66270e11	0.0915320
$531$	0	0
$532$	−6.55988e11	−0.355054
$533$	−3.45611e12	−1.85488
$534$	0	0
$535$	7.06151e11	0.372654
$536$	1.98967e10	0.0104121
$537$	0	0
$538$	−3.66471e11	−0.188590
$539$	4.88282e11	0.249185
$540$	0	0
$541$	−8.50067e11	−0.426644	−0.213322	−	0.976982i	$-0.568428\pi$
$541$	−8.50067e11	−0.426644	−0.213322	+	0.976982i	$0.568428\pi$
$542$	−2.44504e11	−0.121700
$543$	0	0
$544$	1.30731e12	0.640004
$545$	8.03433e11	0.390090
$546$	0	0
$547$	2.05072e12	0.979407	0.489704	−	0.871889i	$-0.337105\pi$
$547$	2.05072e12	0.979407	0.489704	+	0.871889i	$0.337105\pi$
$548$	6.98244e11	0.330746
$549$	0	0
$550$	2.81527e11	0.131186
$551$	−2.04333e12	−0.944400
$552$	0	0
$553$	−1.54132e12	−0.700859
$554$	8.85444e11	0.399362
$555$	0	0
$556$	2.99261e12	1.32805
$557$	−2.68111e12	−1.18023	−0.590115	−	0.807319i	$-0.700918\pi$
$557$	−2.68111e12	−1.18023	−0.590115	+	0.807319i	$0.700918\pi$
$558$	0	0
$559$	1.61733e12	0.700559
$560$	2.34364e11	0.100703
$561$	0	0
$562$	1.20121e12	0.507931
$563$	−1.46222e12	−0.613374	−0.306687	−	0.951810i	$-0.599220\pi$
$563$	−1.46222e12	−0.613374	−0.306687	+	0.951810i	$0.599220\pi$
$564$	0	0
$565$	7.11113e11	0.293576
$566$	−1.35652e12	−0.555589
$567$	0	0
$568$	−1.93908e12	−0.781678
$569$	−1.87869e12	−0.751364	−0.375682	−	0.926749i	$-0.622592\pi$
$569$	−1.87869e12	−0.751364	−0.375682	+	0.926749i	$0.622592\pi$
$570$	0	0
$571$	−3.56191e12	−1.40223	−0.701117	−	0.713046i	$-0.747315\pi$
$571$	−3.56191e12	−1.40223	−0.701117	+	0.713046i	$0.747315\pi$
$572$	−7.04449e12	−2.75149
$573$	0	0
$574$	−3.73205e11	−0.143497
$575$	8.24829e10	0.0314673
$576$	0	0
$577$	−3.51615e12	−1.32062	−0.660308	−	0.750995i	$-0.729574\pi$
$577$	−3.51615e12	−1.32062	−0.660308	+	0.750995i	$0.729574\pi$
$578$	−5.23847e11	−0.195222
$579$	0	0
$580$	9.03289e11	0.331437
$581$	1.69955e12	0.618787
$582$	0	0
$583$	2.64818e12	0.949377
$584$	2.60835e12	0.927913
$585$	0	0
$586$	1.16253e12	0.407253
$587$	−2.39384e12	−0.832194	−0.416097	−	0.909320i	$-0.636602\pi$
$587$	−2.39384e12	−0.832194	−0.416097	+	0.909320i	$0.636602\pi$
$588$	0	0
$589$	−4.84207e12	−1.65772
$590$	8.13874e11	0.276518
$591$	0	0
$592$	−1.28025e12	−0.428398
$593$	−4.92157e12	−1.63440	−0.817198	−	0.576357i	$-0.804474\pi$
$593$	−4.92157e12	−1.63440	−0.817198	+	0.576357i	$0.804474\pi$
$594$	0	0
$595$	3.58343e11	0.117212
$596$	2.29528e12	0.745122
$597$	0	0
$598$	3.39926e11	0.108700
$599$	4.51212e12	1.43206	0.716028	−	0.698071i	$-0.245958\pi$
$599$	4.51212e12	1.43206	0.716028	+	0.698071i	$0.245958\pi$
$600$	0	0
$601$	4.33139e12	1.35423	0.677115	−	0.735877i	$-0.263230\pi$
$601$	4.33139e12	1.35423	0.677115	+	0.735877i	$0.263230\pi$
$602$	1.74645e11	0.0541967
$603$	0	0
$604$	−3.20566e12	−0.980058
$605$	3.01015e12	0.913459
$606$	0	0
$607$	−1.88763e12	−0.564376	−0.282188	−	0.959359i	$-0.591060\pi$
$607$	−1.88763e12	−0.564376	−0.282188	+	0.959359i	$0.591060\pi$
$608$	3.40251e12	1.00980
$609$	0	0
$610$	4.96998e11	0.145335
$611$	4.63009e12	1.34402
$612$	0	0
$613$	3.50333e12	1.00210	0.501048	−	0.865419i	$-0.332948\pi$
$613$	3.50333e12	1.00210	0.501048	+	0.865419i	$0.332948\pi$
$614$	8.05604e11	0.228752
$615$	0	0
$616$	−1.64667e12	−0.460779
$617$	6.96071e12	1.93362	0.966808	−	0.255503i	$-0.0822410\pi$
$617$	6.96071e12	1.93362	0.966808	+	0.255503i	$0.0822410\pi$
$618$	0	0
$619$	4.80705e12	1.31604	0.658022	−	0.752998i	$-0.271393\pi$
$619$	4.80705e12	1.31604	0.658022	+	0.752998i	$0.271393\pi$
$620$	2.14052e12	0.581777
$621$	0	0
$622$	−2.87927e11	−0.0771304
$623$	−8.83246e11	−0.234901
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	3.04297e11	0.0791978
$627$	0	0
$628$	5.18336e11	0.132982
$629$	−1.95751e12	−0.498626
$630$	0	0
$631$	4.54293e12	1.14078	0.570392	−	0.821372i	$-0.306791\pi$
$631$	4.54293e12	1.14078	0.570392	+	0.821372i	$0.306791\pi$
$632$	5.19791e12	1.29599
$633$	0	0
$634$	7.90768e11	0.194378
$635$	1.70088e11	0.0415136
$636$	0	0
$637$	1.09066e12	0.262460
$638$	−2.36947e12	−0.566184
$639$	0	0
$640$	−1.92939e12	−0.454579
$641$	−2.57828e12	−0.603210	−0.301605	−	0.953433i	$-0.597522\pi$
$641$	−2.57828e12	−0.603210	−0.301605	+	0.953433i	$0.597522\pi$
$642$	0	0
$643$	5.54902e12	1.28017	0.640083	−	0.768305i	$-0.278900\pi$
$643$	5.54902e12	1.28017	0.640083	+	0.768305i	$0.278900\pi$
$644$	−2.22870e11	−0.0510583
$645$	0	0
$646$	1.26284e12	0.285300
$647$	−5.61451e12	−1.25963	−0.629814	−	0.776746i	$-0.716869\pi$
$647$	−5.61451e12	−1.25963	−0.629814	+	0.776746i	$0.716869\pi$
$648$	0	0
$649$	1.29625e13	2.86806
$650$	6.28840e11	0.138175
$651$	0	0
$652$	1.71545e12	0.371762
$653$	8.36692e12	1.80076	0.900380	−	0.435103i	$-0.143288\pi$
$653$	8.36692e12	1.80076	0.900380	+	0.435103i	$0.143288\pi$
$654$	0	0
$655$	2.18326e12	0.463467
$656$	−2.85298e12	−0.601495
$657$	0	0
$658$	4.99976e11	0.103976
$659$	−6.92847e12	−1.43104	−0.715522	−	0.698591i	$-0.753811\pi$
$659$	−6.92847e12	−1.43104	−0.715522	+	0.698591i	$0.753811\pi$
$660$	0	0
$661$	2.81659e12	0.573875	0.286937	−	0.957949i	$-0.407363\pi$
$661$	2.81659e12	0.573875	0.286937	+	0.957949i	$0.407363\pi$
$662$	6.06549e11	0.122745
$663$	0	0
$664$	−5.73151e12	−1.14423
$665$	9.32653e11	0.184936
$666$	0	0
$667$	−6.94216e11	−0.135809
$668$	7.70338e12	1.49688
$669$	0	0
$670$	−1.30679e10	−0.00250536
$671$	7.91566e12	1.50743
$672$	0	0
$673$	−1.22982e12	−0.231085	−0.115543	−	0.993303i	$-0.536861\pi$
$673$	−1.22982e12	−0.231085	−0.115543	+	0.993303i	$0.536861\pi$
$674$	−3.50865e11	−0.0654893
$675$	0	0
$676$	−1.10734e13	−2.03948
$677$	−2.17691e12	−0.398283	−0.199142	−	0.979971i	$-0.563815\pi$
$677$	−2.17691e12	−0.398283	−0.199142	+	0.979971i	$0.563815\pi$
$678$	0	0
$679$	−1.37144e12	−0.247607
$680$	−1.20846e12	−0.216742
$681$	0	0
$682$	−5.61491e12	−0.993833
$683$	−7.73735e12	−1.36050	−0.680250	−	0.732980i	$-0.738129\pi$
$683$	−7.73735e12	−1.36050	−0.680250	+	0.732980i	$0.738129\pi$
$684$	0	0
$685$	−9.92729e11	−0.172275
$686$	1.17774e11	0.0203045
$687$	0	0
$688$	1.33509e12	0.227175
$689$	5.91517e12	0.999956
$690$	0	0
$691$	3.42836e12	0.572051	0.286025	−	0.958222i	$-0.407666\pi$
$691$	3.42836e12	0.572051	0.286025	+	0.958222i	$0.407666\pi$
$692$	−5.78934e12	−0.959736
$693$	0	0
$694$	9.83299e11	0.160904
$695$	−4.25475e12	−0.691739
$696$	0	0
$697$	−4.36222e12	−0.700100
$698$	−1.26303e12	−0.201402
$699$	0	0
$700$	−4.12295e11	−0.0649034
$701$	1.96070e12	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$701$	1.96070e12	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$702$	0	0
$703$	−5.09477e12	−0.786730
$704$	−2.82731e12	−0.433807
$705$	0	0
$706$	−2.53232e12	−0.383616
$707$	−3.94787e11	−0.0594259
$708$	0	0
$709$	4.18254e12	0.621630	0.310815	−	0.950471i	$-0.399398\pi$
$709$	4.18254e12	0.621630	0.310815	+	0.950471i	$0.399398\pi$
$710$	1.27357e12	0.188087
$711$	0	0
$712$	2.97863e12	0.434367
$713$	−1.64508e12	−0.238388
$714$	0	0
$715$	1.00155e13	1.43316
$716$	3.64848e12	0.518805
$717$	0	0
$718$	−9.00272e11	−0.126419
$719$	5.71065e12	0.796904	0.398452	−	0.917189i	$-0.369548\pi$
$719$	5.71065e12	0.796904	0.398452	+	0.917189i	$0.369548\pi$
$720$	0	0
$721$	−1.04016e12	−0.143348
$722$	5.41051e11	0.0741004
$723$	0	0
$724$	1.03422e13	1.39891
$725$	−1.28425e12	−0.172635
$726$	0	0
$727$	4.83775e12	0.642301	0.321150	−	0.947028i	$-0.395930\pi$
$727$	4.83775e12	0.642301	0.321150	+	0.947028i	$0.395930\pi$
$728$	−3.67812e12	−0.485328
$729$	0	0
$730$	−1.71313e12	−0.223274
$731$	2.04135e12	0.264417
$732$	0	0
$733$	1.47990e13	1.89349	0.946747	−	0.321979i	$-0.104348\pi$
$733$	1.47990e13	1.89349	0.946747	+	0.321979i	$0.104348\pi$
$734$	−4.30940e12	−0.548005
$735$	0	0
$736$	1.15599e12	0.145213
$737$	−2.08133e11	−0.0259858
$738$	0	0
$739$	−1.16451e13	−1.43629	−0.718145	−	0.695894i	$-0.755008\pi$
$739$	−1.16451e13	−1.43629	−0.718145	+	0.695894i	$0.755008\pi$
$740$	2.25223e12	0.276103
$741$	0	0
$742$	6.38744e11	0.0773587
$743$	7.98143e12	0.960796	0.480398	−	0.877051i	$-0.340492\pi$
$743$	7.98143e12	0.960796	0.480398	+	0.877051i	$0.340492\pi$
$744$	0	0
$745$	−3.26332e12	−0.388111
$746$	3.26102e12	0.385504
$747$	0	0
$748$	−8.89137e12	−1.03851
$749$	2.71275e12	0.314950
$750$	0	0
$751$	−5.06543e12	−0.581081	−0.290540	−	0.956863i	$-0.593835\pi$
$751$	−5.06543e12	−0.581081	−0.290540	+	0.956863i	$0.593835\pi$
$752$	3.82209e12	0.435834
$753$	0	0
$754$	−5.29262e12	−0.596348
$755$	4.55766e12	0.510482
$756$	0	0
$757$	−5.91294e12	−0.654443	−0.327222	−	0.944948i	$-0.606112\pi$
$757$	−5.91294e12	−0.654443	−0.327222	+	0.944948i	$0.606112\pi$
$758$	2.49226e12	0.274210
$759$	0	0
$760$	−3.14525e12	−0.341975
$761$	1.25566e13	1.35719	0.678597	−	0.734511i	$-0.262588\pi$
$761$	1.25566e13	1.35719	0.678597	+	0.734511i	$0.262588\pi$
$762$	0	0
$763$	3.08647e12	0.329686
$764$	6.45852e12	0.685824
$765$	0	0
$766$	2.12594e12	0.223111
$767$	2.89541e13	3.02086
$768$	0	0
$769$	9.85398e12	1.01612	0.508058	−	0.861323i	$-0.330364\pi$
$769$	9.85398e12	1.01612	0.508058	+	0.861323i	$0.330364\pi$
$770$	1.08151e12	0.110873
$771$	0	0
$772$	−7.97046e12	−0.807618
$773$	4.10104e12	0.413130	0.206565	−	0.978433i	$-0.433772\pi$
$773$	4.10104e12	0.413130	0.206565	+	0.978433i	$0.433772\pi$
$774$	0	0
$775$	−3.04329e12	−0.303030
$776$	4.62501e12	0.457862
$777$	0	0
$778$	4.90830e12	0.480312
$779$	−1.13535e13	−1.10461
$780$	0	0
$781$	2.02840e13	1.95085
$782$	4.29046e11	0.0410273
$783$	0	0
$784$	9.00331e11	0.0851099
$785$	−7.36945e11	−0.0692663
$786$	0	0
$787$	−1.27865e13	−1.18813	−0.594067	−	0.804416i	$-0.702479\pi$
$787$	−1.27865e13	−1.18813	−0.594067	+	0.804416i	$0.702479\pi$
$788$	−3.38801e12	−0.313024
$789$	0	0
$790$	−3.41393e12	−0.311841
$791$	2.73181e12	0.248117
$792$	0	0
$793$	1.76810e13	1.58773
$794$	2.70580e12	0.241604
$795$	0	0
$796$	−1.33346e13	−1.17725
$797$	6.41049e12	0.562767	0.281384	−	0.959595i	$-0.409207\pi$
$797$	6.41049e12	0.562767	0.281384	+	0.959595i	$0.409207\pi$
$798$	0	0
$799$	5.84399e12	0.507281
$800$	2.13851e12	0.184589
$801$	0	0
$802$	6.74285e12	0.575518
$803$	−2.72850e13	−2.31582
$804$	0	0
$805$	3.16866e11	0.0265947
$806$	−1.25419e13	−1.04678
$807$	0	0
$808$	1.33137e12	0.109887
$809$	−1.89503e13	−1.55542	−0.777709	−	0.628624i	$-0.783618\pi$
$809$	−1.89503e13	−1.55542	−0.777709	+	0.628624i	$0.783618\pi$
$810$	0	0
$811$	8.10314e12	0.657748	0.328874	−	0.944374i	$-0.393331\pi$
$811$	8.10314e12	0.657748	0.328874	+	0.944374i	$0.393331\pi$
$812$	3.47007e12	0.280115
$813$	0	0
$814$	−5.90795e12	−0.471658
$815$	−2.43895e12	−0.193639
$816$	0	0
$817$	5.31299e12	0.417195
$818$	−7.86275e12	−0.614023
$819$	0	0
$820$	5.01900e12	0.387664
$821$	9.43162e12	0.724506	0.362253	−	0.932080i	$-0.382008\pi$
$821$	9.43162e12	0.724506	0.362253	+	0.932080i	$0.382008\pi$
$822$	0	0
$823$	−1.45042e13	−1.10204	−0.551018	−	0.834493i	$-0.685761\pi$
$823$	−1.45042e13	−1.10204	−0.551018	+	0.834493i	$0.685761\pi$
$824$	3.50779e12	0.265071
$825$	0	0
$826$	3.12658e12	0.233700
$827$	−6.72790e12	−0.500155	−0.250077	−	0.968226i	$-0.580456\pi$
$827$	−6.72790e12	−0.500155	−0.250077	+	0.968226i	$0.580456\pi$
$828$	0	0
$829$	2.21097e13	1.62588	0.812940	−	0.582348i	$-0.197866\pi$
$829$	2.21097e13	1.62588	0.812940	+	0.582348i	$0.197866\pi$
$830$	3.76440e12	0.275324
$831$	0	0
$832$	−6.31529e12	−0.456918
$833$	1.37661e12	0.0990622
$834$	0	0
$835$	−1.09523e13	−0.779679
$836$	−2.31414e13	−1.63856
$837$	0	0
$838$	1.07121e12	0.0750371
$839$	−1.30193e13	−0.907106	−0.453553	−	0.891229i	$-0.649844\pi$
$839$	−1.30193e13	−0.907106	−0.453553	+	0.891229i	$0.649844\pi$
$840$	0	0
$841$	−3.69827e12	−0.254927
$842$	2.08560e12	0.142997
$843$	0	0
$844$	1.72711e13	1.17160
$845$	1.57436e13	1.06230
$846$	0	0
$847$	1.15638e13	0.772014
$848$	4.88291e12	0.324263
$849$	0	0
$850$	7.93706e11	0.0521524
$851$	−1.73094e12	−0.113135
$852$	0	0
$853$	−1.56823e13	−1.01424	−0.507118	−	0.861877i	$-0.669289\pi$
$853$	−1.56823e13	−1.01424	−0.507118	+	0.861877i	$0.669289\pi$
$854$	1.90927e12	0.122830
$855$	0	0
$856$	−9.14840e12	−0.582389
$857$	8.10782e12	0.513441	0.256720	−	0.966486i	$-0.417358\pi$
$857$	8.10782e12	0.513441	0.256720	+	0.966486i	$0.417358\pi$
$858$	0	0
$859$	2.46876e13	1.54707	0.773534	−	0.633754i	$-0.218487\pi$
$859$	2.46876e13	1.54707	0.773534	+	0.633754i	$0.218487\pi$
$860$	−2.34870e12	−0.146415
$861$	0	0
$862$	−3.45390e12	−0.213072
$863$	−4.86143e12	−0.298343	−0.149171	−	0.988811i	$-0.547661\pi$
$863$	−4.86143e12	−0.298343	−0.149171	+	0.988811i	$0.547661\pi$
$864$	0	0
$865$	8.23101e12	0.499897
$866$	8.60289e10	0.00519773
$867$	0	0
$868$	8.22302e12	0.491691
$869$	−5.43736e13	−3.23444
$870$	0	0
$871$	−4.64901e11	−0.0273703
$872$	−1.04087e13	−0.609639
$873$	0	0
$874$	1.11667e12	0.0647327
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	−1.97011e13	−1.12459	−0.562293	−	0.826938i	$-0.690081\pi$
$877$	−1.97011e13	−1.12459	−0.562293	+	0.826938i	$0.690081\pi$
$878$	8.19018e12	0.465123
$879$	0	0
$880$	8.26770e12	0.464743
$881$	−1.78983e12	−0.100097	−0.0500485	−	0.998747i	$-0.515938\pi$
$881$	−1.78983e12	−0.100097	−0.0500485	+	0.998747i	$0.515938\pi$
$882$	0	0
$883$	2.95803e12	0.163749	0.0818747	−	0.996643i	$-0.473909\pi$
$883$	2.95803e12	0.163749	0.0818747	+	0.996643i	$0.473909\pi$
$884$	−1.98605e13	−1.09384
$885$	0	0
$886$	1.36867e13	0.746185
$887$	1.15469e13	0.626341	0.313170	−	0.949697i	$-0.398609\pi$
$887$	1.15469e13	0.626341	0.313170	+	0.949697i	$0.398609\pi$
$888$	0	0
$889$	6.53408e11	0.0350854
$890$	−1.95633e12	−0.104517
$891$	0	0
$892$	−2.89182e13	−1.52943
$893$	1.52101e13	0.800386
$894$	0	0
$895$	−5.18724e12	−0.270229
$896$	−7.41193e12	−0.384189
$897$	0	0
$898$	2.67760e12	0.137405
$899$	2.56138e13	1.30784
$900$	0	0
$901$	7.46598e12	0.377420
$902$	−1.31656e13	−0.662234
$903$	0	0
$904$	−9.21268e12	−0.458805
$905$	−1.47040e13	−0.728648
$906$	0	0
$907$	−7.52299e11	−0.0369111	−0.0184556	−	0.999830i	$-0.505875\pi$
$907$	−7.52299e11	−0.0369111	−0.0184556	+	0.999830i	$0.505875\pi$
$908$	7.53236e12	0.367743
$909$	0	0
$910$	2.41575e12	0.116779
$911$	−4.17434e12	−0.200796	−0.100398	−	0.994947i	$-0.532012\pi$
$911$	−4.17434e12	−0.200796	−0.100398	+	0.994947i	$0.532012\pi$
$912$	0	0
$913$	5.99554e13	2.85568
$914$	−1.54505e13	−0.732295
$915$	0	0
$916$	2.76309e13	1.29678
$917$	8.38720e12	0.391701
$918$	0	0
$919$	−2.40098e13	−1.11037	−0.555187	−	0.831726i	$-0.687353\pi$
$919$	−2.40098e13	−1.11037	−0.555187	+	0.831726i	$0.687353\pi$
$920$	−1.06859e12	−0.0491775
$921$	0	0
$922$	−4.67321e12	−0.212974
$923$	4.53080e13	2.05479
$924$	0	0
$925$	−3.20212e12	−0.143813
$926$	7.63547e12	0.341261
$927$	0	0
$928$	−1.79987e13	−0.796666
$929$	−1.56890e13	−0.691072	−0.345536	−	0.938405i	$-0.612303\pi$
$929$	−1.56890e13	−0.691072	−0.345536	+	0.938405i	$0.612303\pi$
$930$	0	0
$931$	3.58288e12	0.156300
$932$	−3.29644e12	−0.143111
$933$	0	0
$934$	−4.06003e11	−0.0174570
$935$	1.26413e13	0.540929
$936$	0	0
$937$	4.52860e13	1.91927	0.959635	−	0.281250i	$-0.0907490\pi$
$937$	4.52860e13	1.91927	0.959635	+	0.281250i	$0.0907490\pi$
$938$	−5.02018e10	−0.00211742
$939$	0	0
$940$	−6.72387e12	−0.280895
$941$	3.54407e13	1.47350	0.736748	−	0.676167i	$-0.236360\pi$
$941$	3.54407e13	1.47350	0.736748	+	0.676167i	$0.236360\pi$
$942$	0	0
$943$	−3.85732e12	−0.158848
$944$	2.39013e13	0.979597
$945$	0	0
$946$	6.16100e12	0.250116
$947$	−4.57200e13	−1.84727	−0.923636	−	0.383270i	$-0.874798\pi$
$947$	−4.57200e13	−1.84727	−0.923636	+	0.383270i	$0.874798\pi$
$948$	0	0
$949$	−6.09459e13	−2.43919
$950$	2.06577e12	0.0822859
$951$	0	0
$952$	−4.64243e12	−0.183181
$953$	5.29571e12	0.207973	0.103986	−	0.994579i	$-0.466840\pi$
$953$	5.29571e12	0.207973	0.103986	+	0.994579i	$0.466840\pi$
$954$	0	0
$955$	−9.18241e12	−0.357225
$956$	5.16266e12	0.199900
$957$	0	0
$958$	1.44814e13	0.555477
$959$	−3.81367e12	−0.145599
$960$	0	0
$961$	3.42572e13	1.29568
$962$	−1.31965e13	−0.496786
$963$	0	0
$964$	−3.22099e12	−0.120127
$965$	1.13320e13	0.420663
$966$	0	0
$967$	−5.20855e13	−1.91557	−0.957785	−	0.287485i	$-0.907181\pi$
$967$	−5.20855e13	−1.91557	−0.957785	+	0.287485i	$0.907181\pi$
$968$	−3.89974e13	−1.42757
$969$	0	0
$970$	−3.03766e12	−0.110171
$971$	4.97025e13	1.79429	0.897143	−	0.441741i	$-0.145639\pi$
$971$	4.97025e13	1.79429	0.897143	+	0.441741i	$0.145639\pi$
$972$	0	0
$973$	−1.63450e13	−0.584626
$974$	1.02033e13	0.363267
$975$	0	0
$976$	1.45955e13	0.514866
$977$	−4.16458e13	−1.46233	−0.731166	−	0.682199i	$-0.761024\pi$
$977$	−4.16458e13	−1.46233	−0.731166	+	0.682199i	$0.761024\pi$
$978$	0	0
$979$	−3.11584e13	−1.08406
$980$	−1.58387e12	−0.0548534
$981$	0	0
$982$	−2.16818e13	−0.744036
$983$	−1.49379e13	−0.510267	−0.255134	−	0.966906i	$-0.582119\pi$
$983$	−1.49379e13	−0.510267	−0.255134	+	0.966906i	$0.582119\pi$
$984$	0	0
$985$	4.81691e12	0.163044
$986$	−6.68021e12	−0.225084
$987$	0	0
$988$	−5.16905e13	−1.72586
$989$	1.80507e12	0.0599945
$990$	0	0
$991$	3.29026e13	1.08368	0.541838	−	0.840483i	$-0.317729\pi$
$991$	3.29026e13	1.08368	0.541838	+	0.840483i	$0.317729\pi$
$992$	−4.26515e13	−1.39840
$993$	0	0
$994$	4.89253e12	0.158963
$995$	1.89584e13	0.613195
$996$	0	0
$997$	−1.81623e13	−0.582161	−0.291081	−	0.956699i	$-0.594015\pi$
$997$	−1.81623e13	−0.582161	−0.291081	+	0.956699i	$0.594015\pi$
$998$	−8.97054e12	−0.286241
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.e.1.3		4
3.2	odd	2		105.10.a.d.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.d.1.2	✓	4	3.2	odd	2
315.10.a.e.1.3		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)

\(f(q)\)	\(=\)	\(q+8.50890 q^{2} -439.599 q^{4} +625.000 q^{5} +2401.00 q^{7} -8097.06 q^{8} +5318.06 q^{10} +84700.6 q^{11} +189194. q^{13} +20429.9 q^{14} +156177. q^{16} +238796. q^{17} +621510. q^{19} -274749. q^{20} +720709. q^{22} +211156. q^{23} +390625. q^{25} +1.60983e6 q^{26} -1.05548e6 q^{28} -3.28769e6 q^{29} -7.79081e6 q^{31} +5.47459e6 q^{32} +2.03189e6 q^{34} +1.50062e6 q^{35} -8.19742e6 q^{37} +5.28836e6 q^{38} -5.06066e6 q^{40} -1.82676e7 q^{41} +8.54852e6 q^{43} -3.72343e7 q^{44} +1.79671e6 q^{46} +2.44728e7 q^{47} +5.76480e6 q^{49} +3.32379e6 q^{50} -8.31693e7 q^{52} +3.12652e7 q^{53} +5.29379e7 q^{55} -1.94410e7 q^{56} -2.79746e7 q^{58} +1.53039e8 q^{59} +9.34546e7 q^{61} -6.62913e7 q^{62} -3.33800e7 q^{64} +1.18246e8 q^{65} -2.45728e6 q^{67} -1.04974e8 q^{68} +1.27687e7 q^{70} +2.39479e8 q^{71} -3.22135e8 q^{73} -6.97510e7 q^{74} -2.73215e8 q^{76} +2.03366e8 q^{77} -6.41950e8 q^{79} +9.76108e7 q^{80} -1.55437e8 q^{82} +7.07851e8 q^{83} +1.49247e8 q^{85} +7.27385e7 q^{86} -6.85826e8 q^{88} -3.67866e8 q^{89} +4.54254e8 q^{91} -9.28240e7 q^{92} +2.08236e8 q^{94} +3.88443e8 q^{95} -5.71196e8 q^{97} +4.90521e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) 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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