LMFDB - Embedded newform 315.10.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,10,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$162.236288392$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1495x^{2} + 193x + 99862 $ x^4 - x^3 - 1495x^2 + 193x + 99862
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$38.2066$ 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)

$f(q)$	$=$	\(q-39.2066 q^{2} +1025.16 q^{4} +625.000 q^{5} -2401.00 q^{7} -20119.2 q^{8} -24504.1 q^{10} -28077.9 q^{11} -138547. q^{13} +94135.1 q^{14} +263924. q^{16} +91618.0 q^{17} +48423.5 q^{19} +640724. q^{20} +1.10084e6 q^{22} +1.37292e6 q^{23} +390625. q^{25} +5.43195e6 q^{26} -2.46140e6 q^{28} +3.84163e6 q^{29} -7.06204e6 q^{31} -46532.4 q^{32} -3.59203e6 q^{34} -1.50062e6 q^{35} +8.92275e6 q^{37} -1.89852e6 q^{38} -1.25745e7 q^{40} +1.03789e7 q^{41} +1.48851e6 q^{43} -2.87842e7 q^{44} -5.38276e7 q^{46} +3.79844e7 q^{47} +5.76480e6 q^{49} -1.53151e7 q^{50} -1.42032e8 q^{52} -3.86334e7 q^{53} -1.75487e7 q^{55} +4.83061e7 q^{56} -1.50617e8 q^{58} -4.07713e7 q^{59} -2.10751e8 q^{61} +2.76879e8 q^{62} -1.33305e8 q^{64} -8.65917e7 q^{65} +4.12391e7 q^{67} +9.39229e7 q^{68} +5.88344e7 q^{70} -1.45423e7 q^{71} +2.20493e8 q^{73} -3.49831e8 q^{74} +4.96417e7 q^{76} +6.74150e7 q^{77} +1.88522e8 q^{79} +1.64952e8 q^{80} -4.06923e8 q^{82} -6.49021e8 q^{83} +5.72612e7 q^{85} -5.83593e7 q^{86} +5.64904e8 q^{88} +6.31366e8 q^{89} +3.32651e8 q^{91} +1.40746e9 q^{92} -1.48924e9 q^{94} +3.02647e7 q^{95} +1.00264e9 q^{97} -2.26018e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 + 949 * q^4 + 2500 * q^5 - 9604 * q^7 - 7767 * q^8 - 3125 * q^10 - 64546 * q^11 - 29390 * q^13 + 12005 * q^14 + 554577 * q^16 - 278788 * q^17 - 929142 * q^19 + 593125 * q^20 + 2767732 * q^22 + 526064 * q^23 + 1562500 * q^25 + 3920706 * q^26 - 2278549 * q^28 - 365860 * q^29 - 4977954 * q^31 - 7227279 * q^32 - 2976030 * q^34 - 6002500 * q^35 + 22846008 * q^37 - 27001832 * q^38 - 4854375 * q^40 + 28257844 * q^41 + 23603420 * q^43 + 9817860 * q^44 - 136600280 * q^46 + 30058700 * q^47 + 23059204 * q^49 - 1953125 * q^50 - 183429274 * q^52 - 113767294 * q^53 - 40341250 * q^55 + 18648567 * q^56 - 323185726 * q^58 + 151786220 * q^59 - 191130108 * q^61 + 292486944 * q^62 - 423209695 * q^64 - 18368750 * q^65 - 147812356 * q^67 + 266317150 * q^68 + 7503125 * q^70 + 37100486 * q^71 + 163444574 * q^73 + 255454086 * q^74 - 720141040 * q^76 + 154974946 * q^77 - 327433200 * q^79 + 346610625 * q^80 - 686338410 * q^82 - 216775352 * q^83 - 174242500 * q^85 + 739989240 * q^86 - 127433212 * q^88 + 792987912 * q^89 + 70565390 * q^91 + 31184232 * q^92 - 1357334188 * q^94 - 580713750 * q^95 - 640730334 * q^97 - 28824005 * 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 $

$2$	−39.2066	−1.73270	−0.866352	−	0.499434i	$-0.833541\pi$
$2$	−39.2066	−1.73270	−0.866352	+	0.499434i	$0.833541\pi$
$3$	0	0
$3$	0	0
$4$	1025.16	2.00226
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−20119.2	−1.73662
$9$	0	0
$10$	−24504.1	−0.774889
$11$	−28077.9	−0.578225	−0.289113	−	0.957295i	$-0.593360\pi$
$11$	−28077.9	−0.578225	−0.289113	+	0.957295i	$0.593360\pi$
$12$	0	0
$13$	−138547.	−1.34540	−0.672700	−	0.739916i	$-0.734865\pi$
$13$	−138547.	−1.34540	−0.672700	+	0.739916i	$0.734865\pi$
$14$	94135.1	0.654900
$15$	0	0
$16$	263924.	1.00679
$17$	91618.0	0.266048	0.133024	−	0.991113i	$-0.457531\pi$
$17$	91618.0	0.266048	0.133024	+	0.991113i	$0.457531\pi$
$18$	0	0
$19$	48423.5	0.0852441	0.0426221	−	0.999091i	$-0.486429\pi$
$19$	48423.5	0.0852441	0.0426221	+	0.999091i	$0.486429\pi$
$20$	640724.	0.895438
$21$	0	0
$22$	1.10084e6	1.00189
$23$	1.37292e6	1.02299	0.511494	−	0.859287i	$-0.329092\pi$
$23$	1.37292e6	1.02299	0.511494	+	0.859287i	$0.329092\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	5.43195e6	2.33118
$27$	0	0
$28$	−2.46140e6	−0.756784
$29$	3.84163e6	1.00861	0.504306	−	0.863525i	$-0.331748\pi$
$29$	3.84163e6	1.00861	0.504306	+	0.863525i	$0.331748\pi$
$30$	0	0
$31$	−7.06204e6	−1.37342	−0.686709	−	0.726933i	$-0.740945\pi$
$31$	−7.06204e6	−1.37342	−0.686709	+	0.726933i	$0.740945\pi$
$32$	−46532.4	−0.00784478
$33$	0	0
$34$	−3.59203e6	−0.460983
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	8.92275e6	0.782692	0.391346	−	0.920244i	$-0.372010\pi$
$37$	8.92275e6	0.782692	0.391346	+	0.920244i	$0.372010\pi$
$38$	−1.89852e6	−0.147703
$39$	0	0
$40$	−1.25745e7	−0.776641
$41$	1.03789e7	0.573621	0.286811	−	0.957987i	$-0.407405\pi$
$41$	1.03789e7	0.573621	0.286811	+	0.957987i	$0.407405\pi$
$42$	0	0
$43$	1.48851e6	0.0663961	0.0331981	−	0.999449i	$-0.489431\pi$
$43$	1.48851e6	0.0663961	0.0331981	+	0.999449i	$0.489431\pi$
$44$	−2.87842e7	−1.15776
$45$	0	0
$46$	−5.38276e7	−1.77253
$47$	3.79844e7	1.13544	0.567721	−	0.823221i	$-0.307825\pi$
$47$	3.79844e7	1.13544	0.567721	+	0.823221i	$0.307825\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−1.53151e7	−0.346541
$51$	0	0
$52$	−1.42032e8	−2.69384
$53$	−3.86334e7	−0.672545	−0.336273	−	0.941765i	$-0.609166\pi$
$53$	−3.86334e7	−0.672545	−0.336273	+	0.941765i	$0.609166\pi$
$54$	0	0
$55$	−1.75487e7	−0.258590
$56$	4.83061e7	0.656381
$57$	0	0
$58$	−1.50617e8	−1.74763
$59$	−4.07713e7	−0.438047	−0.219023	−	0.975720i	$-0.570287\pi$
$59$	−4.07713e7	−0.438047	−0.219023	+	0.975720i	$0.570287\pi$
$60$	0	0
$61$	−2.10751e8	−1.94888	−0.974440	−	0.224647i	$-0.927877\pi$
$61$	−2.10751e8	−1.94888	−0.974440	+	0.224647i	$0.927877\pi$
$62$	2.76879e8	2.37973
$63$	0	0
$64$	−1.33305e8	−0.993196
$65$	−8.65917e7	−0.601681
$66$	0	0
$67$	4.12391e7	0.250018	0.125009	−	0.992156i	$-0.460104\pi$
$67$	4.12391e7	0.250018	0.125009	+	0.992156i	$0.460104\pi$
$68$	9.39229e7	0.532698
$69$	0	0
$70$	5.88344e7	0.292880
$71$	−1.45423e7	−0.0679159	−0.0339580	−	0.999423i	$-0.510811\pi$
$71$	−1.45423e7	−0.0679159	−0.0339580	+	0.999423i	$0.510811\pi$
$72$	0	0
$73$	2.20493e8	0.908746	0.454373	−	0.890812i	$-0.349863\pi$
$73$	2.20493e8	0.908746	0.454373	+	0.890812i	$0.349863\pi$
$74$	−3.49831e8	−1.35617
$75$	0	0
$76$	4.96417e7	0.170681
$77$	6.74150e7	0.218549
$78$	0	0
$79$	1.88522e8	0.544554	0.272277	−	0.962219i	$-0.412223\pi$
$79$	1.88522e8	0.544554	0.272277	+	0.962219i	$0.412223\pi$
$80$	1.64952e8	0.450250
$81$	0	0
$82$	−4.06923e8	−0.993915
$83$	−6.49021e8	−1.50109	−0.750546	−	0.660818i	$-0.770209\pi$
$83$	−6.49021e8	−1.50109	−0.750546	+	0.660818i	$0.770209\pi$
$84$	0	0
$85$	5.72612e7	0.118980
$86$	−5.83593e7	−0.115045
$87$	0	0
$88$	5.64904e8	1.00416
$89$	6.31366e8	1.06666	0.533330	−	0.845907i	$-0.320940\pi$
$89$	6.31366e8	1.06666	0.533330	+	0.845907i	$0.320940\pi$
$90$	0	0
$91$	3.32651e8	0.508513
$92$	1.40746e9	2.04829
$93$	0	0
$94$	−1.48924e9	−1.96739
$95$	3.02647e7	0.0381223
$96$	0	0
$97$	1.00264e9	1.14993	0.574967	−	0.818177i	$-0.305015\pi$
$97$	1.00264e9	1.14993	0.574967	+	0.818177i	$0.305015\pi$
$98$	−2.26018e8	−0.247529
$99$	0	0
$100$	4.00452e8	0.400452
$101$	1.26854e9	1.21299	0.606494	−	0.795088i	$-0.292575\pi$
$101$	1.26854e9	1.21299	0.606494	+	0.795088i	$0.292575\pi$
$102$	0	0
$103$	1.63442e9	1.43086	0.715429	−	0.698686i	$-0.246231\pi$
$103$	1.63442e9	1.43086	0.715429	+	0.698686i	$0.246231\pi$
$104$	2.78745e9	2.33645
$105$	0	0
$106$	1.51468e9	1.16532
$107$	1.11463e9	0.822060	0.411030	−	0.911622i	$-0.365169\pi$
$107$	1.11463e9	0.822060	0.411030	+	0.911622i	$0.365169\pi$
$108$	0	0
$109$	−1.79976e9	−1.22123	−0.610613	−	0.791929i	$-0.709077\pi$
$109$	−1.79976e9	−1.22123	−0.610613	+	0.791929i	$0.709077\pi$
$110$	6.88024e8	0.448060
$111$	0	0
$112$	−6.33681e8	−0.380530
$113$	−1.21360e9	−0.700200	−0.350100	−	0.936712i	$-0.613852\pi$
$113$	−1.21360e9	−0.700200	−0.350100	+	0.936712i	$0.613852\pi$
$114$	0	0
$115$	8.58076e8	0.457494
$116$	3.93827e9	2.01950
$117$	0	0
$118$	1.59850e9	0.759005
$119$	−2.19975e8	−0.100557
$120$	0	0
$121$	−1.56958e9	−0.665655
$122$	8.26283e9	3.37683
$123$	0	0
$124$	−7.23971e9	−2.74994
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	3.50005e9	1.19387	0.596936	−	0.802289i	$-0.296384\pi$
$127$	3.50005e9	1.19387	0.596936	+	0.802289i	$0.296384\pi$
$128$	5.25024e9	1.72876
$129$	0	0
$130$	3.39497e9	1.04253
$131$	−7.41508e8	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$131$	−7.41508e8	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$132$	0	0
$133$	−1.16265e8	−0.0322193
$134$	−1.61684e9	−0.433208
$135$	0	0
$136$	−1.84328e9	−0.462025
$137$	−4.91157e9	−1.19118	−0.595591	−	0.803288i	$-0.703082\pi$
$137$	−4.91157e9	−1.19118	−0.595591	+	0.803288i	$0.703082\pi$
$138$	0	0
$139$	3.47398e9	0.789334	0.394667	−	0.918824i	$-0.370860\pi$
$139$	3.47398e9	0.789334	0.394667	+	0.918824i	$0.370860\pi$
$140$	−1.53838e9	−0.338444
$141$	0	0
$142$	5.70156e8	0.117678
$143$	3.89010e9	0.777944
$144$	0	0
$145$	2.40102e9	0.451065
$146$	−8.64479e9	−1.57459
$147$	0	0
$148$	9.14723e9	1.56715
$149$	−4.29101e9	−0.713216	−0.356608	−	0.934254i	$-0.616067\pi$
$149$	−4.29101e9	−0.713216	−0.356608	+	0.934254i	$0.616067\pi$
$150$	0	0
$151$	6.16724e9	0.965372	0.482686	−	0.875794i	$-0.339661\pi$
$151$	6.16724e9	0.965372	0.482686	+	0.875794i	$0.339661\pi$
$152$	−9.74240e8	−0.148037
$153$	0	0
$154$	−2.64311e9	−0.378680
$155$	−4.41378e9	−0.614211
$156$	0	0
$157$	6.09092e8	0.0800082	0.0400041	−	0.999200i	$-0.487263\pi$
$157$	6.09092e8	0.0800082	0.0400041	+	0.999200i	$0.487263\pi$
$158$	−7.39133e9	−0.943551
$159$	0	0
$160$	−2.90827e7	−0.00350829
$161$	−3.29638e9	−0.386653
$162$	0	0
$163$	−1.55925e10	−1.73010	−0.865048	−	0.501689i	$-0.832712\pi$
$163$	−1.55925e10	−1.73010	−0.865048	+	0.501689i	$0.832712\pi$
$164$	1.06400e10	1.14854
$165$	0	0
$166$	2.54459e10	2.60095
$167$	7.32353e9	0.728612	0.364306	−	0.931279i	$-0.381306\pi$
$167$	7.32353e9	0.728612	0.364306	+	0.931279i	$0.381306\pi$
$168$	0	0
$169$	8.59070e9	0.810099
$170$	−2.24502e9	−0.206158
$171$	0	0
$172$	1.52595e9	0.132942
$173$	−1.07081e10	−0.908875	−0.454437	−	0.890779i	$-0.650160\pi$
$173$	−1.07081e10	−0.908875	−0.454437	+	0.890779i	$0.650160\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−7.41041e9	−0.582151
$177$	0	0
$178$	−2.47537e10	−1.84820
$179$	1.04408e9	0.0760139	0.0380070	−	0.999277i	$-0.487899\pi$
$179$	1.04408e9	0.0760139	0.0380070	+	0.999277i	$0.487899\pi$
$180$	0	0
$181$	−1.16175e10	−0.804559	−0.402280	−	0.915517i	$-0.631782\pi$
$181$	−1.16175e10	−0.804559	−0.402280	+	0.915517i	$0.631782\pi$
$182$	−1.30421e10	−0.881102
$183$	0	0
$184$	−2.76220e10	−1.77654
$185$	5.57672e9	0.350031
$186$	0	0
$187$	−2.57244e9	−0.153836
$188$	3.89400e10	2.27345
$189$	0	0
$190$	−1.18657e9	−0.0660547
$191$	−2.41935e10	−1.31537	−0.657687	−	0.753291i	$-0.728465\pi$
$191$	−2.41935e10	−1.31537	−0.657687	+	0.753291i	$0.728465\pi$
$192$	0	0
$193$	2.57532e10	1.33605	0.668025	−	0.744139i	$-0.267140\pi$
$193$	2.57532e10	1.33605	0.668025	+	0.744139i	$0.267140\pi$
$194$	−3.93101e10	−1.99249
$195$	0	0
$196$	5.90983e9	0.286037
$197$	1.98421e10	0.938621	0.469310	−	0.883033i	$-0.344503\pi$
$197$	1.98421e10	0.938621	0.469310	+	0.883033i	$0.344503\pi$
$198$	0	0
$199$	−3.20087e10	−1.44687	−0.723435	−	0.690392i	$-0.757438\pi$
$199$	−3.20087e10	−1.44687	−0.723435	+	0.690392i	$0.757438\pi$
$200$	−7.85905e9	−0.347324
$201$	0	0
$202$	−4.97350e10	−2.10175
$203$	−9.22374e9	−0.381220
$204$	0	0
$205$	6.48683e9	0.256531
$206$	−6.40801e10	−2.47925
$207$	0	0
$208$	−3.65658e10	−1.35453
$209$	−1.35963e9	−0.0492903
$210$	0	0
$211$	2.97389e9	0.103289	0.0516446	−	0.998666i	$-0.483554\pi$
$211$	2.97389e9	0.103289	0.0516446	+	0.998666i	$0.483554\pi$
$212$	−3.96053e10	−1.34661
$213$	0	0
$214$	−4.37008e10	−1.42439
$215$	9.30317e8	0.0296933
$216$	0	0
$217$	1.69560e10	0.519103
$218$	7.05626e10	2.11602
$219$	0	0
$220$	−1.79902e10	−0.517765
$221$	−1.26934e10	−0.357941
$222$	0	0
$223$	−4.72551e10	−1.27961	−0.639803	−	0.768539i	$-0.720984\pi$
$223$	−4.72551e10	−1.27961	−0.639803	+	0.768539i	$0.720984\pi$
$224$	1.11724e8	0.00296505
$225$	0	0
$226$	4.75811e10	1.21324
$227$	−1.83337e10	−0.458284	−0.229142	−	0.973393i	$-0.573592\pi$
$227$	−1.83337e10	−0.458284	−0.229142	+	0.973393i	$0.573592\pi$
$228$	0	0
$229$	1.03989e10	0.249878	0.124939	−	0.992164i	$-0.460127\pi$
$229$	1.03989e10	0.249878	0.124939	+	0.992164i	$0.460127\pi$
$230$	−3.36422e10	−0.792701
$231$	0	0
$232$	−7.72903e10	−1.75158
$233$	−4.69902e10	−1.04449	−0.522247	−	0.852794i	$-0.674906\pi$
$233$	−4.69902e10	−1.04449	−0.522247	+	0.852794i	$0.674906\pi$
$234$	0	0
$235$	2.37403e10	0.507785
$236$	−4.17970e10	−0.877084
$237$	0	0
$238$	8.62446e9	0.174235
$239$	2.32518e10	0.460962	0.230481	−	0.973077i	$-0.425970\pi$
$239$	2.32518e10	0.460962	0.230481	+	0.973077i	$0.425970\pi$
$240$	0	0
$241$	5.51119e10	1.05237	0.526186	−	0.850370i	$-0.323622\pi$
$241$	5.51119e10	1.05237	0.526186	+	0.850370i	$0.323622\pi$
$242$	6.15379e10	1.15338
$243$	0	0
$244$	−2.16053e11	−3.90217
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−6.70891e9	−0.114687
$248$	1.42082e11	2.38511
$249$	0	0
$250$	−9.57192e9	−0.154978
$251$	−4.85810e10	−0.772564	−0.386282	−	0.922381i	$-0.626241\pi$
$251$	−4.85810e10	−0.772564	−0.386282	+	0.922381i	$0.626241\pi$
$252$	0	0
$253$	−3.85487e10	−0.591517
$254$	−1.37225e11	−2.06863
$255$	0	0
$256$	−1.37592e11	−2.00223
$257$	1.10113e11	1.57449	0.787244	−	0.616642i	$-0.211507\pi$
$257$	1.10113e11	1.57449	0.787244	+	0.616642i	$0.211507\pi$
$258$	0	0
$259$	−2.14235e10	−0.295830
$260$	−8.87702e10	−1.20472
$261$	0	0
$262$	2.90720e10	0.381171
$263$	7.87453e10	1.01490	0.507451	−	0.861681i	$-0.330588\pi$
$263$	7.87453e10	1.01490	0.507451	+	0.861681i	$0.330588\pi$
$264$	0	0
$265$	−2.41459e10	−0.300771
$266$	4.55834e9	0.0558264
$267$	0	0
$268$	4.22765e10	0.500602
$269$	−4.18782e10	−0.487643	−0.243822	−	0.969820i	$-0.578401\pi$
$269$	−4.18782e10	−0.487643	−0.243822	+	0.969820i	$0.578401\pi$
$270$	0	0
$271$	−5.02529e10	−0.565978	−0.282989	−	0.959123i	$-0.591326\pi$
$271$	−5.02529e10	−0.565978	−0.282989	+	0.959123i	$0.591326\pi$
$272$	2.41801e10	0.267854
$273$	0	0
$274$	1.92566e11	2.06397
$275$	−1.09679e10	−0.115645
$276$	0	0
$277$	−1.61787e11	−1.65114	−0.825572	−	0.564298i	$-0.809147\pi$
$277$	−1.61787e11	−1.65114	−0.825572	+	0.564298i	$0.809147\pi$
$278$	−1.36203e11	−1.36768
$279$	0	0
$280$	3.01913e10	0.293543
$281$	9.51948e10	0.910825	0.455412	−	0.890281i	$-0.349492\pi$
$281$	9.51948e10	0.910825	0.455412	+	0.890281i	$0.349492\pi$
$282$	0	0
$283$	6.79205e10	0.629451	0.314725	−	0.949183i	$-0.398088\pi$
$283$	6.79205e10	0.629451	0.314725	+	0.949183i	$0.398088\pi$
$284$	−1.49082e10	−0.135985
$285$	0	0
$286$	−1.52518e11	−1.34795
$287$	−2.49198e10	−0.216808
$288$	0	0
$289$	−1.10194e11	−0.929218
$290$	−9.41357e10	−0.781562
$291$	0	0
$292$	2.26040e11	1.81955
$293$	1.60446e11	1.27181	0.635907	−	0.771766i	$-0.280626\pi$
$293$	1.60446e11	1.27181	0.635907	+	0.771766i	$0.280626\pi$
$294$	0	0
$295$	−2.54821e10	−0.195901
$296$	−1.79518e11	−1.35924
$297$	0	0
$298$	1.68236e11	1.23579
$299$	−1.90214e11	−1.37633
$300$	0	0
$301$	−3.57390e9	−0.0250954
$302$	−2.41797e11	−1.67270
$303$	0	0
$304$	1.27801e10	0.0858228
$305$	−1.31719e11	−0.871566
$306$	0	0
$307$	−1.68354e11	−1.08169	−0.540843	−	0.841124i	$-0.681895\pi$
$307$	−1.68354e11	−1.08169	−0.540843	+	0.841124i	$0.681895\pi$
$308$	6.91110e10	0.437591
$309$	0	0
$310$	1.73049e11	1.06425
$311$	1.84159e11	1.11627	0.558136	−	0.829749i	$-0.311517\pi$
$311$	1.84159e11	1.11627	0.558136	+	0.829749i	$0.311517\pi$
$312$	0	0
$313$	2.62929e11	1.54842	0.774210	−	0.632928i	$-0.218147\pi$
$313$	2.62929e11	1.54842	0.774210	+	0.632928i	$0.218147\pi$
$314$	−2.38804e10	−0.138630
$315$	0	0
$316$	1.93265e11	1.09034
$317$	−3.00462e11	−1.67118	−0.835589	−	0.549354i	$-0.814874\pi$
$317$	−3.00462e11	−1.67118	−0.835589	+	0.549354i	$0.814874\pi$
$318$	0	0
$319$	−1.07865e11	−0.583205
$320$	−8.33153e10	−0.444171
$321$	0	0
$322$	1.29240e11	0.669955
$323$	4.43646e9	0.0226791
$324$	0	0
$325$	−5.41198e10	−0.269080
$326$	6.11327e11	2.99774
$327$	0	0
$328$	−2.08815e11	−0.996163
$329$	−9.12006e10	−0.429157
$330$	0	0
$331$	−3.14048e11	−1.43804	−0.719018	−	0.694992i	$-0.755408\pi$
$331$	−3.14048e11	−1.43804	−0.719018	+	0.694992i	$0.755408\pi$
$332$	−6.65349e11	−3.00558
$333$	0	0
$334$	−2.87131e11	−1.26247
$335$	2.57744e10	0.111812
$336$	0	0
$337$	−4.50544e11	−1.90284	−0.951421	−	0.307894i	$-0.900376\pi$
$337$	−4.50544e11	−1.90284	−0.951421	+	0.307894i	$0.900376\pi$
$338$	−3.36812e11	−1.40366
$339$	0	0
$340$	5.87018e10	0.238230
$341$	1.98287e11	0.794145
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−2.99475e10	−0.115305
$345$	0	0
$346$	4.19827e11	1.57481
$347$	3.20391e11	1.18631	0.593154	−	0.805089i	$-0.297882\pi$
$347$	3.20391e11	1.18631	0.593154	+	0.805089i	$0.297882\pi$
$348$	0	0
$349$	−4.15333e11	−1.49859	−0.749294	−	0.662237i	$-0.769607\pi$
$349$	−4.15333e11	−1.49859	−0.749294	+	0.662237i	$0.769607\pi$
$350$	3.67715e10	0.130980
$351$	0	0
$352$	1.30653e9	0.00453605
$353$	3.30609e11	1.13326	0.566629	−	0.823973i	$-0.308247\pi$
$353$	3.30609e11	1.13326	0.566629	+	0.823973i	$0.308247\pi$
$354$	0	0
$355$	−9.08896e9	−0.0303729
$356$	6.47249e11	2.13573
$357$	0	0
$358$	−4.09346e10	−0.131710
$359$	−3.49111e9	−0.0110927	−0.00554637	−	0.999985i	$-0.501765\pi$
$359$	−3.49111e9	−0.0110927	−0.00554637	+	0.999985i	$0.501765\pi$
$360$	0	0
$361$	−3.20343e11	−0.992733
$362$	4.55482e11	1.39406
$363$	0	0
$364$	3.41019e11	1.01818
$365$	1.37808e11	0.406403
$366$	0	0
$367$	−2.27821e11	−0.655536	−0.327768	−	0.944758i	$-0.606296\pi$
$367$	−2.27821e11	−0.655536	−0.327768	+	0.944758i	$0.606296\pi$
$368$	3.62346e11	1.02993
$369$	0	0
$370$	−2.18644e11	−0.606499
$371$	9.27588e10	0.254198
$372$	0	0
$373$	3.04601e11	0.814783	0.407391	−	0.913254i	$-0.366439\pi$
$373$	3.04601e11	0.814783	0.407391	+	0.913254i	$0.366439\pi$
$374$	1.00857e11	0.266552
$375$	0	0
$376$	−7.64215e11	−1.97183
$377$	−5.32245e11	−1.35699
$378$	0	0
$379$	−2.24607e10	−0.0559174	−0.0279587	−	0.999609i	$-0.508901\pi$
$379$	−2.24607e10	−0.0559174	−0.0279587	+	0.999609i	$0.508901\pi$
$380$	3.10260e10	0.0763309
$381$	0	0
$382$	9.48547e11	2.27915
$383$	2.85686e11	0.678412	0.339206	−	0.940712i	$-0.389842\pi$
$383$	2.85686e11	0.678412	0.339206	+	0.940712i	$0.389842\pi$
$384$	0	0
$385$	4.21344e10	0.0977379
$386$	−1.00969e12	−2.31498
$387$	0	0
$388$	1.02787e12	2.30247
$389$	−2.65038e11	−0.586861	−0.293430	−	0.955980i	$-0.594797\pi$
$389$	−2.65038e11	−0.586861	−0.293430	+	0.955980i	$0.594797\pi$
$390$	0	0
$391$	1.25784e11	0.272164
$392$	−1.15983e11	−0.248089
$393$	0	0
$394$	−7.77942e11	−1.62635
$395$	1.17827e11	0.243532
$396$	0	0
$397$	7.82739e11	1.58146	0.790732	−	0.612162i	$-0.209700\pi$
$397$	7.82739e11	1.58146	0.790732	+	0.612162i	$0.209700\pi$
$398$	1.25495e12	2.50700
$399$	0	0
$400$	1.03095e11	0.201358
$401$	−1.12972e11	−0.218183	−0.109092	−	0.994032i	$-0.534794\pi$
$401$	−1.12972e11	−0.218183	−0.109092	+	0.994032i	$0.534794\pi$
$402$	0	0
$403$	9.78423e11	1.84779
$404$	1.30045e12	2.42872
$405$	0	0
$406$	3.61632e11	0.660540
$407$	−2.50532e11	−0.452572
$408$	0	0
$409$	2.84614e10	0.0502923	0.0251461	−	0.999684i	$-0.491995\pi$
$409$	2.84614e10	0.0502923	0.0251461	+	0.999684i	$0.491995\pi$
$410$	−2.54327e11	−0.444492
$411$	0	0
$412$	1.67554e12	2.86495
$413$	9.78919e10	0.165566
$414$	0	0
$415$	−4.05638e11	−0.671309
$416$	6.44691e9	0.0105544
$417$	0	0
$418$	5.33064e10	0.0854055
$419$	2.66541e11	0.422475	0.211237	−	0.977435i	$-0.432251\pi$
$419$	2.66541e11	0.422475	0.211237	+	0.977435i	$0.432251\pi$
$420$	0	0
$421$	−2.12857e11	−0.330232	−0.165116	−	0.986274i	$-0.552800\pi$
$421$	−2.12857e11	−0.330232	−0.165116	+	0.986274i	$0.552800\pi$
$422$	−1.16596e11	−0.178969
$423$	0	0
$424$	7.77272e11	1.16796
$425$	3.57883e10	0.0532097
$426$	0	0
$427$	5.06013e11	0.736608
$428$	1.14267e12	1.64598
$429$	0	0
$430$	−3.64746e10	−0.0514496
$431$	3.90617e11	0.545259	0.272630	−	0.962119i	$-0.412107\pi$
$431$	3.90617e11	0.545259	0.272630	+	0.962119i	$0.412107\pi$
$432$	0	0
$433$	−8.60576e11	−1.17651	−0.588253	−	0.808677i	$-0.700184\pi$
$433$	−8.60576e11	−1.17651	−0.588253	+	0.808677i	$0.700184\pi$
$434$	−6.64786e11	−0.899452
$435$	0	0
$436$	−1.84504e12	−2.44521
$437$	6.64816e10	0.0872037
$438$	0	0
$439$	2.62511e11	0.337332	0.168666	−	0.985673i	$-0.446054\pi$
$439$	2.62511e11	0.337332	0.168666	+	0.985673i	$0.446054\pi$
$440$	3.53065e11	0.449073
$441$	0	0
$442$	4.97664e11	0.620206
$443$	−1.06295e11	−0.131128	−0.0655641	−	0.997848i	$-0.520885\pi$
$443$	−1.06295e11	−0.131128	−0.0655641	+	0.997848i	$0.520885\pi$
$444$	0	0
$445$	3.94603e11	0.477025
$446$	1.85271e12	2.21718
$447$	0	0
$448$	3.20064e11	0.375393
$449$	−9.74168e11	−1.13116	−0.565582	−	0.824692i	$-0.691348\pi$
$449$	−9.74168e11	−1.13116	−0.565582	+	0.824692i	$0.691348\pi$
$450$	0	0
$451$	−2.91418e11	−0.331682
$452$	−1.24413e12	−1.40198
$453$	0	0
$454$	7.18803e11	0.794070
$455$	2.07907e11	0.227414
$456$	0	0
$457$	1.56535e12	1.67876	0.839381	−	0.543544i	$-0.182918\pi$
$457$	1.56535e12	1.67876	0.839381	+	0.543544i	$0.182918\pi$
$458$	−4.07706e11	−0.432964
$459$	0	0
$460$	8.79663e11	0.916022
$461$	−1.60192e12	−1.65191	−0.825955	−	0.563737i	$-0.809363\pi$
$461$	−1.60192e12	−1.65191	−0.825955	+	0.563737i	$0.809363\pi$
$462$	0	0
$463$	−1.48926e12	−1.50610	−0.753052	−	0.657961i	$-0.771419\pi$
$463$	−1.48926e12	−1.50610	−0.753052	+	0.657961i	$0.771419\pi$
$464$	1.01390e12	1.01546
$465$	0	0
$466$	1.84233e12	1.80980
$467$	−9.17563e11	−0.892709	−0.446355	−	0.894856i	$-0.647278\pi$
$467$	−9.17563e11	−0.892709	−0.446355	+	0.894856i	$0.647278\pi$
$468$	0	0
$469$	−9.90150e10	−0.0944981
$470$	−9.30775e11	−0.879842
$471$	0	0
$472$	8.20285e11	0.760722
$473$	−4.17941e10	−0.0383919
$474$	0	0
$475$	1.89154e10	0.0170488
$476$	−2.25509e11	−0.201341
$477$	0	0
$478$	−9.11623e11	−0.798711
$479$	1.09370e12	0.949268	0.474634	−	0.880183i	$-0.342581\pi$
$479$	1.09370e12	0.949268	0.474634	+	0.880183i	$0.342581\pi$
$480$	0	0
$481$	−1.23622e12	−1.05303
$482$	−2.16075e12	−1.82345
$483$	0	0
$484$	−1.60907e12	−1.33282
$485$	6.26651e11	0.514266
$486$	0	0
$487$	−1.67694e12	−1.35094	−0.675470	−	0.737387i	$-0.736059\pi$
$487$	−1.67694e12	−1.35094	−0.675470	+	0.737387i	$0.736059\pi$
$488$	4.24013e12	3.38447
$489$	0	0
$490$	−1.41261e11	−0.110698
$491$	−1.42105e12	−1.10342	−0.551711	−	0.834035i	$-0.686025\pi$
$491$	−1.42105e12	−1.10342	−0.551711	+	0.834035i	$0.686025\pi$
$492$	0	0
$493$	3.51962e11	0.268339
$494$	2.63034e11	0.198719
$495$	0	0
$496$	−1.86384e12	−1.38274
$497$	3.49162e10	0.0256698
$498$	0	0
$499$	−1.38700e12	−1.00144	−0.500718	−	0.865610i	$-0.666931\pi$
$499$	−1.38700e12	−1.00144	−0.500718	+	0.865610i	$0.666931\pi$
$500$	2.50283e11	0.179088
$501$	0	0
$502$	1.90469e12	1.33862
$503$	−1.41176e12	−0.983341	−0.491670	−	0.870781i	$-0.663614\pi$
$503$	−1.41176e12	−0.983341	−0.491670	+	0.870781i	$0.663614\pi$
$504$	0	0
$505$	7.92835e11	0.542465
$506$	1.51136e12	1.02492
$507$	0	0
$508$	3.58811e12	2.39045
$509$	3.93550e9	0.00259878	0.00129939	−	0.999999i	$-0.499586\pi$
$509$	3.93550e9	0.00259878	0.00129939	+	0.999999i	$0.499586\pi$
$510$	0	0
$511$	−5.29404e11	−0.343474
$512$	2.70640e12	1.74051
$513$	0	0
$514$	−4.31715e12	−2.72812
$515$	1.02151e12	0.639899
$516$	0	0
$517$	−1.06652e12	−0.656542
$518$	8.39944e11	0.512585
$519$	0	0
$520$	1.74215e12	1.04489
$521$	−3.05974e12	−1.81934	−0.909670	−	0.415331i	$-0.863666\pi$
$521$	−3.05974e12	−1.81934	−0.909670	+	0.415331i	$0.863666\pi$
$522$	0	0
$523$	1.86991e12	1.09285	0.546427	−	0.837507i	$-0.315988\pi$
$523$	1.86991e12	1.09285	0.546427	+	0.837507i	$0.315988\pi$
$524$	−7.60163e11	−0.440470
$525$	0	0
$526$	−3.08734e12	−1.75852
$527$	−6.47010e11	−0.365395
$528$	0	0
$529$	8.37600e10	0.0465036
$530$	9.46678e11	0.521148
$531$	0	0
$532$	−1.19190e11	−0.0645114
$533$	−1.43797e12	−0.771749
$534$	0	0
$535$	6.96644e11	0.367636
$536$	−8.29696e11	−0.434187
$537$	0	0
$538$	1.64190e12	0.844941
$539$	−1.61863e11	−0.0826036
$540$	0	0
$541$	−1.49190e12	−0.748775	−0.374388	−	0.927272i	$-0.622147\pi$
$541$	−1.49190e12	−0.748775	−0.374388	+	0.927272i	$0.622147\pi$
$542$	1.97025e12	0.980672
$543$	0	0
$544$	−4.26320e9	−0.00208709
$545$	−1.12485e12	−0.546149
$546$	0	0
$547$	−2.26281e12	−1.08070	−0.540349	−	0.841441i	$-0.681708\pi$
$547$	−2.26281e12	−1.08070	−0.540349	+	0.841441i	$0.681708\pi$
$548$	−5.03514e12	−2.38506
$549$	0	0
$550$	4.30015e11	0.200379
$551$	1.86025e11	0.0859783
$552$	0	0
$553$	−4.52642e11	−0.205822
$554$	6.34312e12	2.86094
$555$	0	0
$556$	3.56138e12	1.58045
$557$	−3.04930e12	−1.34231	−0.671153	−	0.741319i	$-0.734201\pi$
$557$	−3.04930e12	−1.34231	−0.671153	+	0.741319i	$0.734201\pi$
$558$	0	0
$559$	−2.06228e11	−0.0893293
$560$	−3.96050e11	−0.170178
$561$	0	0
$562$	−3.73227e12	−1.57819
$563$	−3.84056e12	−1.61104	−0.805521	−	0.592568i	$-0.798114\pi$
$563$	−3.84056e12	−1.61104	−0.805521	+	0.592568i	$0.798114\pi$
$564$	0	0
$565$	−7.58499e11	−0.313139
$566$	−2.66293e12	−1.09065
$567$	0	0
$568$	2.92580e11	0.117944
$569$	−2.65494e12	−1.06182	−0.530908	−	0.847429i	$-0.678149\pi$
$569$	−2.65494e12	−1.06182	−0.530908	+	0.847429i	$0.678149\pi$
$570$	0	0
$571$	4.90270e10	0.0193007	0.00965035	−	0.999953i	$-0.496928\pi$
$571$	4.90270e10	0.0193007	0.00965035	+	0.999953i	$0.496928\pi$
$572$	3.98796e12	1.55765
$573$	0	0
$574$	9.77021e11	0.375665
$575$	5.36297e11	0.204598
$576$	0	0
$577$	2.83897e12	1.06628	0.533138	−	0.846028i	$-0.321013\pi$
$577$	2.83897e12	1.06628	0.533138	+	0.846028i	$0.321013\pi$
$578$	4.32033e12	1.61006
$579$	0	0
$580$	2.46142e12	0.903150
$581$	1.55830e12	0.567360
$582$	0	0
$583$	1.08474e12	0.388883
$584$	−4.43614e12	−1.57815
$585$	0	0
$586$	−6.29052e12	−2.20367
$587$	−5.44405e12	−1.89257	−0.946283	−	0.323340i	$-0.895194\pi$
$587$	−5.44405e12	−1.89257	−0.946283	+	0.323340i	$0.895194\pi$
$588$	0	0
$589$	−3.41968e11	−0.117076
$590$	9.99065e11	0.339437
$591$	0	0
$592$	2.35492e12	0.788006
$593$	−5.23689e12	−1.73911	−0.869555	−	0.493836i	$-0.835594\pi$
$593$	−5.23689e12	−1.73911	−0.869555	+	0.493836i	$0.835594\pi$
$594$	0	0
$595$	−1.37484e11	−0.0449704
$596$	−4.39896e12	−1.42805
$597$	0	0
$598$	7.45763e12	2.38477
$599$	2.07026e12	0.657060	0.328530	−	0.944494i	$-0.393447\pi$
$599$	2.07026e12	0.657060	0.328530	+	0.944494i	$0.393447\pi$
$600$	0	0
$601$	−4.05959e12	−1.26925	−0.634624	−	0.772821i	$-0.718845\pi$
$601$	−4.05959e12	−1.26925	−0.634624	+	0.772821i	$0.718845\pi$
$602$	1.40121e11	0.0434828
$603$	0	0
$604$	6.32239e12	1.93293
$605$	−9.80988e11	−0.297690
$606$	0	0
$607$	−4.29680e12	−1.28468	−0.642342	−	0.766418i	$-0.722037\pi$
$607$	−4.29680e12	−1.28468	−0.642342	+	0.766418i	$0.722037\pi$
$608$	−2.25326e9	−0.000668721 0
$609$	0	0
$610$	5.16427e12	1.51017
$611$	−5.26262e12	−1.52762
$612$	0	0
$613$	5.36589e11	0.153486	0.0767432	−	0.997051i	$-0.475548\pi$
$613$	5.36589e11	0.153486	0.0767432	+	0.997051i	$0.475548\pi$
$614$	6.60059e12	1.87424
$615$	0	0
$616$	−1.35633e12	−0.379536
$617$	5.55886e12	1.54420	0.772099	−	0.635503i	$-0.219207\pi$
$617$	5.55886e12	1.54420	0.772099	+	0.635503i	$0.219207\pi$
$618$	0	0
$619$	3.75981e12	1.02934	0.514669	−	0.857389i	$-0.327915\pi$
$619$	3.75981e12	1.02934	0.514669	+	0.857389i	$0.327915\pi$
$620$	−4.52482e12	−1.22981
$621$	0	0
$622$	−7.22023e12	−1.93417
$623$	−1.51591e12	−0.403159
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−1.03086e13	−2.68295
$627$	0	0
$628$	6.24415e11	0.160197
$629$	8.17484e11	0.208234
$630$	0	0
$631$	3.36353e12	0.844624	0.422312	−	0.906451i	$-0.361219\pi$
$631$	3.36353e12	0.844624	0.422312	+	0.906451i	$0.361219\pi$
$632$	−3.79292e12	−0.945685
$633$	0	0
$634$	1.17801e13	2.89566
$635$	2.18753e12	0.533916
$636$	0	0
$637$	−7.98694e11	−0.192200
$638$	4.22901e12	1.01052
$639$	0	0
$640$	3.28140e12	0.773124
$641$	−7.22975e12	−1.69146	−0.845730	−	0.533610i	$-0.820835\pi$
$641$	−7.22975e12	−1.69146	−0.845730	+	0.533610i	$0.820835\pi$
$642$	0	0
$643$	2.87463e12	0.663181	0.331591	−	0.943423i	$-0.392415\pi$
$643$	2.87463e12	0.663181	0.331591	+	0.943423i	$0.392415\pi$
$644$	−3.37931e12	−0.774180
$645$	0	0
$646$	−1.73938e11	−0.0392961
$647$	1.80534e12	0.405031	0.202516	−	0.979279i	$-0.435088\pi$
$647$	1.80534e12	0.405031	0.202516	+	0.979279i	$0.435088\pi$
$648$	0	0
$649$	1.14477e12	0.253290
$650$	2.12185e12	0.466236
$651$	0	0
$652$	−1.59847e13	−3.46410
$653$	4.26401e12	0.917717	0.458858	−	0.888509i	$-0.348258\pi$
$653$	4.26401e12	0.917717	0.458858	+	0.888509i	$0.348258\pi$
$654$	0	0
$655$	−4.63443e11	−0.0983808
$656$	2.73924e12	0.577515
$657$	0	0
$658$	3.57567e12	0.743602
$659$	8.24905e12	1.70380	0.851901	−	0.523702i	$-0.175450\pi$
$659$	8.24905e12	1.70380	0.851901	+	0.523702i	$0.175450\pi$
$660$	0	0
$661$	3.52424e12	0.718058	0.359029	−	0.933326i	$-0.383108\pi$
$661$	3.52424e12	0.718058	0.359029	+	0.933326i	$0.383108\pi$
$662$	1.23127e13	2.49169
$663$	0	0
$664$	1.30578e13	2.60683
$665$	−7.26654e10	−0.0144089
$666$	0	0
$667$	5.27425e12	1.03180
$668$	7.50777e12	1.45887
$669$	0	0
$670$	−1.01053e12	−0.193736
$671$	5.91744e12	1.12689
$672$	0	0
$673$	−3.66959e12	−0.689524	−0.344762	−	0.938690i	$-0.612040\pi$
$673$	−3.66959e12	−0.689524	−0.344762	+	0.938690i	$0.612040\pi$
$674$	1.76643e13	3.29706
$675$	0	0
$676$	8.80682e12	1.62203
$677$	5.92398e11	0.108384	0.0541919	−	0.998531i	$-0.482742\pi$
$677$	5.92398e11	0.108384	0.0541919	+	0.998531i	$0.482742\pi$
$678$	0	0
$679$	−2.40734e12	−0.434634
$680$	−1.15205e12	−0.206624
$681$	0	0
$682$	−7.77416e12	−1.37602
$683$	9.65921e12	1.69843	0.849217	−	0.528044i	$-0.177074\pi$
$683$	9.65921e12	1.69843	0.849217	+	0.528044i	$0.177074\pi$
$684$	0	0
$685$	−3.06973e12	−0.532713
$686$	5.42670e11	0.0935572
$687$	0	0
$688$	3.92852e11	0.0668469
$689$	5.35253e12	0.904842
$690$	0	0
$691$	5.57234e11	0.0929794	0.0464897	−	0.998919i	$-0.485197\pi$
$691$	5.57234e11	0.0929794	0.0464897	+	0.998919i	$0.485197\pi$
$692$	−1.09775e13	−1.81980
$693$	0	0
$694$	−1.25614e13	−2.05552
$695$	2.17124e12	0.353001
$696$	0	0
$697$	9.50896e11	0.152611
$698$	1.62838e13	2.59661
$699$	0	0
$700$	−9.61486e11	−0.151357
$701$	−2.08541e12	−0.326183	−0.163091	−	0.986611i	$-0.552147\pi$
$701$	−2.08541e12	−0.326183	−0.163091	+	0.986611i	$0.552147\pi$
$702$	0	0
$703$	4.32070e11	0.0667199
$704$	3.74291e12	0.574291
$705$	0	0
$706$	−1.29621e13	−1.96360
$707$	−3.04575e12	−0.458466
$708$	0	0
$709$	−9.44610e12	−1.40393	−0.701963	−	0.712213i	$-0.747693\pi$
$709$	−9.44610e12	−1.40393	−0.701963	+	0.712213i	$0.747693\pi$
$710$	3.56347e11	0.0526273
$711$	0	0
$712$	−1.27026e13	−1.85238
$713$	−9.69563e12	−1.40499
$714$	0	0
$715$	2.43131e12	0.347907
$716$	1.07034e12	0.152200
$717$	0	0
$718$	1.36875e11	0.0192204
$719$	−3.04469e12	−0.424876	−0.212438	−	0.977175i	$-0.568140\pi$
$719$	−3.04469e12	−0.424876	−0.212438	+	0.977175i	$0.568140\pi$
$720$	0	0
$721$	−3.92425e12	−0.540813
$722$	1.25596e13	1.72011
$723$	0	0
$724$	−1.19097e13	−1.61094
$725$	1.50064e12	0.201722
$726$	0	0
$727$	−2.99307e11	−0.0397385	−0.0198693	−	0.999803i	$-0.506325\pi$
$727$	−2.99307e11	−0.0397385	−0.0198693	+	0.999803i	$0.506325\pi$
$728$	−6.69266e12	−0.883095
$729$	0	0
$730$	−5.40299e12	−0.704177
$731$	1.36374e11	0.0176646
$732$	0	0
$733$	−2.93313e12	−0.375287	−0.187643	−	0.982237i	$-0.560085\pi$
$733$	−2.93313e12	−0.375287	−0.187643	+	0.982237i	$0.560085\pi$
$734$	8.93209e12	1.13585
$735$	0	0
$736$	−6.38853e10	−0.00802511
$737$	−1.15790e12	−0.144567
$738$	0	0
$739$	−1.55228e13	−1.91457	−0.957284	−	0.289149i	$-0.906628\pi$
$739$	−1.55228e13	−1.91457	−0.957284	+	0.289149i	$0.906628\pi$
$740$	5.71702e12	0.700853
$741$	0	0
$742$	−3.63676e12	−0.440450
$743$	−1.44068e13	−1.73427	−0.867136	−	0.498072i	$-0.834042\pi$
$743$	−1.44068e13	−1.73427	−0.867136	+	0.498072i	$0.834042\pi$
$744$	0	0
$745$	−2.68188e12	−0.318960
$746$	−1.19424e13	−1.41178
$747$	0	0
$748$	−2.63715e12	−0.308020
$749$	−2.67623e12	−0.310710
$750$	0	0
$751$	−1.55303e12	−0.178156	−0.0890778	−	0.996025i	$-0.528392\pi$
$751$	−1.55303e12	−0.178156	−0.0890778	+	0.996025i	$0.528392\pi$
$752$	1.00250e13	1.14315
$753$	0	0
$754$	2.08675e13	2.35125
$755$	3.85453e12	0.431727
$756$	0	0
$757$	−1.90963e12	−0.211357	−0.105679	−	0.994400i	$-0.533701\pi$
$757$	−1.90963e12	−0.211357	−0.105679	+	0.994400i	$0.533701\pi$
$758$	8.80608e11	0.0968882
$759$	0	0
$760$	−6.08900e11	−0.0662041
$761$	5.14749e12	0.556371	0.278186	−	0.960527i	$-0.410267\pi$
$761$	5.14749e12	0.556371	0.278186	+	0.960527i	$0.410267\pi$
$762$	0	0
$763$	4.32123e12	0.461580
$764$	−2.48022e13	−2.63372
$765$	0	0
$766$	−1.12008e13	−1.17549
$767$	5.64873e12	0.589348
$768$	0	0
$769$	1.11707e12	0.115189	0.0575946	−	0.998340i	$-0.481657\pi$
$769$	1.11707e12	0.115189	0.0575946	+	0.998340i	$0.481657\pi$
$770$	−1.65195e12	−0.169351
$771$	0	0
$772$	2.64011e13	2.67512
$773$	−9.04418e12	−0.911090	−0.455545	−	0.890213i	$-0.650556\pi$
$773$	−9.04418e12	−0.911090	−0.455545	+	0.890213i	$0.650556\pi$
$774$	0	0
$775$	−2.75861e12	−0.274684
$776$	−2.01723e13	−1.99700
$777$	0	0
$778$	1.03912e13	1.01686
$779$	5.02584e11	0.0488978
$780$	0	0
$781$	4.08318e11	0.0392707
$782$	−4.93157e12	−0.471580
$783$	0	0
$784$	1.52147e12	0.143827
$785$	3.80682e11	0.0357807
$786$	0	0
$787$	−1.07182e13	−0.995948	−0.497974	−	0.867192i	$-0.665922\pi$
$787$	−1.07182e13	−0.995948	−0.497974	+	0.867192i	$0.665922\pi$
$788$	2.03413e13	1.87936
$789$	0	0
$790$	−4.61958e12	−0.421969
$791$	2.91385e12	0.264651
$792$	0	0
$793$	2.91988e13	2.62202
$794$	−3.06885e13	−2.74021
$795$	0	0
$796$	−3.28140e13	−2.89701
$797$	−5.64269e12	−0.495364	−0.247682	−	0.968841i	$-0.579669\pi$
$797$	−5.64269e12	−0.495364	−0.247682	+	0.968841i	$0.579669\pi$
$798$	0	0
$799$	3.48006e12	0.302083
$800$	−1.81767e10	−0.00156896
$801$	0	0
$802$	4.42925e12	0.378047
$803$	−6.19098e12	−0.525460
$804$	0	0
$805$	−2.06024e12	−0.172916
$806$	−3.83606e13	−3.20168
$807$	0	0
$808$	−2.55219e13	−2.10650
$809$	8.37646e12	0.687531	0.343765	−	0.939056i	$-0.388298\pi$
$809$	8.37646e12	0.687531	0.343765	+	0.939056i	$0.388298\pi$
$810$	0	0
$811$	−1.54387e13	−1.25319	−0.626595	−	0.779345i	$-0.715552\pi$
$811$	−1.54387e13	−1.25319	−0.626595	+	0.779345i	$0.715552\pi$
$812$	−9.45579e12	−0.763301
$813$	0	0
$814$	9.82250e12	0.784174
$815$	−9.74528e12	−0.773722
$816$	0	0
$817$	7.20786e10	0.00565988
$818$	−1.11587e12	−0.0871416
$819$	0	0
$820$	6.65002e12	0.513642
$821$	−1.66183e12	−0.127657	−0.0638283	−	0.997961i	$-0.520331\pi$
$821$	−1.66183e12	−0.127657	−0.0638283	+	0.997961i	$0.520331\pi$
$822$	0	0
$823$	1.41746e13	1.07699	0.538496	−	0.842628i	$-0.318993\pi$
$823$	1.41746e13	1.07699	0.538496	+	0.842628i	$0.318993\pi$
$824$	−3.28832e13	−2.48486
$825$	0	0
$826$	−3.83801e12	−0.286877
$827$	3.24750e12	0.241421	0.120710	−	0.992688i	$-0.461483\pi$
$827$	3.24750e12	0.241421	0.120710	+	0.992688i	$0.461483\pi$
$828$	0	0
$829$	−2.56064e13	−1.88301	−0.941505	−	0.337000i	$-0.890588\pi$
$829$	−2.56064e13	−1.88301	−0.941505	+	0.337000i	$0.890588\pi$
$830$	1.59037e13	1.16318
$831$	0	0
$832$	1.84689e13	1.33625
$833$	5.28159e11	0.0380069
$834$	0	0
$835$	4.57720e12	0.325845
$836$	−1.39383e12	−0.0986921
$837$	0	0
$838$	−1.04502e13	−0.732024
$839$	2.55511e12	0.178025	0.0890126	−	0.996031i	$-0.471629\pi$
$839$	2.55511e12	0.178025	0.0890126	+	0.996031i	$0.471629\pi$
$840$	0	0
$841$	2.50947e11	0.0172982
$842$	8.34541e12	0.572194
$843$	0	0
$844$	3.04871e12	0.206812
$845$	5.36919e12	0.362287
$846$	0	0
$847$	3.76856e12	0.251594
$848$	−1.01963e13	−0.677111
$849$	0	0
$850$	−1.40314e12	−0.0921965
$851$	1.22502e13	0.800684
$852$	0	0
$853$	−2.07928e12	−0.134475	−0.0672375	−	0.997737i	$-0.521419\pi$
$853$	−2.07928e12	−0.134475	−0.0672375	+	0.997737i	$0.521419\pi$
$854$	−1.98390e13	−1.27632
$855$	0	0
$856$	−2.24254e13	−1.42761
$857$	−5.42353e12	−0.343454	−0.171727	−	0.985145i	$-0.554935\pi$
$857$	−5.42353e12	−0.343454	−0.171727	+	0.985145i	$0.554935\pi$
$858$	0	0
$859$	2.52127e11	0.0157997	0.00789986	−	0.999969i	$-0.497485\pi$
$859$	2.52127e11	0.0157997	0.00789986	+	0.999969i	$0.497485\pi$
$860$	9.53721e11	0.0594536
$861$	0	0
$862$	−1.53148e13	−0.944773
$863$	3.07023e12	0.188418	0.0942089	−	0.995552i	$-0.469968\pi$
$863$	3.07023e12	0.188418	0.0942089	+	0.995552i	$0.469968\pi$
$864$	0	0
$865$	−6.69255e12	−0.406461
$866$	3.37403e13	2.03853
$867$	0	0
$868$	1.73825e13	1.03938
$869$	−5.29331e12	−0.314875
$870$	0	0
$871$	−5.71354e12	−0.336375
$872$	3.62097e13	2.12081
$873$	0	0
$874$	−2.60652e12	−0.151098
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	2.69391e13	1.53775	0.768875	−	0.639399i	$-0.220817\pi$
$877$	2.69391e13	1.53775	0.768875	+	0.639399i	$0.220817\pi$
$878$	−1.02922e13	−0.584497
$879$	0	0
$880$	−4.63151e12	−0.260346
$881$	2.00884e13	1.12345	0.561724	−	0.827325i	$-0.310138\pi$
$881$	2.00884e13	1.12345	0.561724	+	0.827325i	$0.310138\pi$
$882$	0	0
$883$	7.72699e12	0.427747	0.213874	−	0.976861i	$-0.431392\pi$
$883$	7.72699e12	0.427747	0.213874	+	0.976861i	$0.431392\pi$
$884$	−1.30127e13	−0.716692
$885$	0	0
$886$	4.16747e12	0.227206
$887$	2.70164e13	1.46545	0.732726	−	0.680524i	$-0.238248\pi$
$887$	2.70164e13	1.46545	0.732726	+	0.680524i	$0.238248\pi$
$888$	0	0
$889$	−8.40363e12	−0.451241
$890$	−1.54711e13	−0.826542
$891$	0	0
$892$	−4.84439e13	−2.56211
$893$	1.83934e12	0.0967899
$894$	0	0
$895$	6.52547e11	0.0339945
$896$	−1.26058e13	−0.653409
$897$	0	0
$898$	3.81938e13	1.95997
$899$	−2.71297e13	−1.38525
$900$	0	0
$901$	−3.53951e12	−0.178929
$902$	1.14255e13	0.574707
$903$	0	0
$904$	2.44166e13	1.21598
$905$	−7.26092e12	−0.359810
$906$	0	0
$907$	−2.26412e13	−1.11088	−0.555439	−	0.831558i	$-0.687450\pi$
$907$	−2.26412e13	−1.11088	−0.555439	+	0.831558i	$0.687450\pi$
$908$	−1.87950e13	−0.917604
$909$	0	0
$910$	−8.15131e12	−0.394041
$911$	−8.27670e12	−0.398130	−0.199065	−	0.979986i	$-0.563790\pi$
$911$	−8.27670e12	−0.398130	−0.199065	+	0.979986i	$0.563790\pi$
$912$	0	0
$913$	1.82231e13	0.867970
$914$	−6.13721e13	−2.90880
$915$	0	0
$916$	1.06605e13	0.500321
$917$	1.78036e12	0.0831469
$918$	0	0
$919$	6.89361e12	0.318806	0.159403	−	0.987214i	$-0.449043\pi$
$919$	6.89361e12	0.318806	0.159403	+	0.987214i	$0.449043\pi$
$920$	−1.72638e13	−0.794494
$921$	0	0
$922$	6.28058e13	2.86227
$923$	2.01479e12	0.0913741
$924$	0	0
$925$	3.48545e12	0.156538
$926$	5.83887e13	2.60963
$927$	0	0
$928$	−1.78760e11	−0.00791233
$929$	1.18966e13	0.524024	0.262012	−	0.965065i	$-0.415614\pi$
$929$	1.18966e13	0.524024	0.262012	+	0.965065i	$0.415614\pi$
$930$	0	0
$931$	2.79152e11	0.0121777
$932$	−4.81724e13	−2.09135
$933$	0	0
$934$	3.59745e13	1.54680
$935$	−1.60777e12	−0.0687975
$936$	0	0
$937$	5.18752e12	0.219852	0.109926	−	0.993940i	$-0.464939\pi$
$937$	5.18752e12	0.219852	0.109926	+	0.993940i	$0.464939\pi$
$938$	3.88204e12	0.163737
$939$	0	0
$940$	2.43375e13	1.01672
$941$	−2.90694e13	−1.20860	−0.604301	−	0.796756i	$-0.706547\pi$
$941$	−2.90694e13	−1.20860	−0.604301	+	0.796756i	$0.706547\pi$
$942$	0	0
$943$	1.42495e13	0.586807
$944$	−1.07605e13	−0.441021
$945$	0	0
$946$	1.63860e12	0.0665218
$947$	−2.03573e13	−0.822516	−0.411258	−	0.911519i	$-0.634910\pi$
$947$	−2.03573e13	−0.822516	−0.411258	+	0.911519i	$0.634910\pi$
$948$	0	0
$949$	−3.05486e13	−1.22263
$950$	−7.41609e11	−0.0295406
$951$	0	0
$952$	4.42571e12	0.174629
$953$	2.11856e12	0.0831997	0.0415998	−	0.999134i	$-0.486755\pi$
$953$	2.11856e12	0.0831997	0.0415998	+	0.999134i	$0.486755\pi$
$954$	0	0
$955$	−1.51210e13	−0.588253
$956$	2.38367e13	0.922967
$957$	0	0
$958$	−4.28803e13	−1.64480
$959$	1.17927e13	0.450225
$960$	0	0
$961$	2.34328e13	0.886276
$962$	4.84679e13	1.82459
$963$	0	0
$964$	5.64984e13	2.10712
$965$	1.60957e13	0.597500
$966$	0	0
$967$	−2.78323e13	−1.02360	−0.511800	−	0.859105i	$-0.671021\pi$
$967$	−2.78323e13	−1.02360	−0.511800	+	0.859105i	$0.671021\pi$
$968$	3.15787e13	1.15599
$969$	0	0
$970$	−2.45688e13	−0.891070
$971$	8.90071e12	0.321320	0.160660	−	0.987010i	$-0.448638\pi$
$971$	8.90071e12	0.321320	0.160660	+	0.987010i	$0.448638\pi$
$972$	0	0
$973$	−8.34102e12	−0.298340
$974$	6.57470e13	2.34078
$975$	0	0
$976$	−5.56221e13	−1.96211
$977$	2.65380e13	0.931844	0.465922	−	0.884826i	$-0.345723\pi$
$977$	2.65380e13	0.931844	0.465922	+	0.884826i	$0.345723\pi$
$978$	0	0
$979$	−1.77274e13	−0.616770
$980$	3.69364e12	0.127920
$981$	0	0
$982$	5.57144e13	1.91190
$983$	4.64406e12	0.158638	0.0793189	−	0.996849i	$-0.474725\pi$
$983$	4.64406e12	0.158638	0.0793189	+	0.996849i	$0.474725\pi$
$984$	0	0
$985$	1.24013e13	0.419764
$986$	−1.37992e13	−0.464953
$987$	0	0
$988$	−6.87769e12	−0.229634
$989$	2.04360e12	0.0679224
$990$	0	0
$991$	−2.21791e13	−0.730487	−0.365243	−	0.930912i	$-0.619014\pi$
$991$	−2.21791e13	−0.730487	−0.365243	+	0.930912i	$0.619014\pi$
$992$	3.28614e11	0.0107742
$993$	0	0
$994$	−1.36894e12	−0.0444782
$995$	−2.00055e13	−0.647060
$996$	0	0
$997$	2.46898e13	0.791389	0.395695	−	0.918382i	$-0.370504\pi$
$997$	2.46898e13	0.791389	0.395695	+	0.918382i	$0.370504\pi$
$998$	5.43795e13	1.73519
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.e.1.4	✓	4	3.2	odd	2
315.10.a.d.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-39.2066 q^{2} +1025.16 q^{4} +625.000 q^{5} -2401.00 q^{7} -20119.2 q^{8} -24504.1 q^{10} -28077.9 q^{11} -138547. q^{13} +94135.1 q^{14} +263924. q^{16} +91618.0 q^{17} +48423.5 q^{19} +640724. q^{20} +1.10084e6 q^{22} +1.37292e6 q^{23} +390625. q^{25} +5.43195e6 q^{26} -2.46140e6 q^{28} +3.84163e6 q^{29} -7.06204e6 q^{31} -46532.4 q^{32} -3.59203e6 q^{34} -1.50062e6 q^{35} +8.92275e6 q^{37} -1.89852e6 q^{38} -1.25745e7 q^{40} +1.03789e7 q^{41} +1.48851e6 q^{43} -2.87842e7 q^{44} -5.38276e7 q^{46} +3.79844e7 q^{47} +5.76480e6 q^{49} -1.53151e7 q^{50} -1.42032e8 q^{52} -3.86334e7 q^{53} -1.75487e7 q^{55} +4.83061e7 q^{56} -1.50617e8 q^{58} -4.07713e7 q^{59} -2.10751e8 q^{61} +2.76879e8 q^{62} -1.33305e8 q^{64} -8.65917e7 q^{65} +4.12391e7 q^{67} +9.39229e7 q^{68} +5.88344e7 q^{70} -1.45423e7 q^{71} +2.20493e8 q^{73} -3.49831e8 q^{74} +4.96417e7 q^{76} +6.74150e7 q^{77} +1.88522e8 q^{79} +1.64952e8 q^{80} -4.06923e8 q^{82} -6.49021e8 q^{83} +5.72612e7 q^{85} -5.83593e7 q^{86} +5.64904e8 q^{88} +6.31366e8 q^{89} +3.32651e8 q^{91} +1.40746e9 q^{92} -1.48924e9 q^{94} +3.02647e7 q^{95} +1.00264e9 q^{97} -2.26018e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 + 949 * q^4 + 2500 * q^5 - 9604 * q^7 - 7767 * q^8 - 3125 * q^10 - 64546 * q^11 - 29390 * q^13 + 12005 * q^14 + 554577 * q^16 - 278788 * q^17 - 929142 * q^19 + 593125 * q^20 + 2767732 * q^22 + 526064 * q^23 + 1562500 * q^25 + 3920706 * q^26 - 2278549 * q^28 - 365860 * q^29 - 4977954 * q^31 - 7227279 * q^32 - 2976030 * q^34 - 6002500 * q^35 + 22846008 * q^37 - 27001832 * q^38 - 4854375 * q^40 + 28257844 * q^41 + 23603420 * q^43 + 9817860 * q^44 - 136600280 * q^46 + 30058700 * q^47 + 23059204 * q^49 - 1953125 * q^50 - 183429274 * q^52 - 113767294 * q^53 - 40341250 * q^55 + 18648567 * q^56 - 323185726 * q^58 + 151786220 * q^59 - 191130108 * q^61 + 292486944 * q^62 - 423209695 * q^64 - 18368750 * q^65 - 147812356 * q^67 + 266317150 * q^68 + 7503125 * q^70 + 37100486 * q^71 + 163444574 * q^73 + 255454086 * q^74 - 720141040 * q^76 + 154974946 * q^77 - 327433200 * q^79 + 346610625 * q^80 - 686338410 * q^82 - 216775352 * q^83 - 174242500 * q^85 + 739989240 * q^86 - 127433212 * q^88 + 792987912 * q^89 + 70565390 * q^91 + 31184232 * q^92 - 1357334188 * q^94 - 580713750 * q^95 - 640730334 * q^97 - 28824005 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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