LMFDB - Embedded newform 315.10.a.d.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:	$1$
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			1.2
Root			$8.41677$ 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)

$f(q)$	$=$	\(q-9.41677 q^{2} -423.324 q^{4} +625.000 q^{5} -2401.00 q^{7} +8807.73 q^{8} -5885.48 q^{10} -81635.7 q^{11} +142991. q^{13} +22609.7 q^{14} +133802. q^{16} -186341. q^{17} -568546. q^{19} -264578. q^{20} +768745. q^{22} +2.05879e6 q^{23} +390625. q^{25} -1.34652e6 q^{26} +1.01640e6 q^{28} -196707. q^{29} -661063. q^{31} -5.76954e6 q^{32} +1.75473e6 q^{34} -1.50062e6 q^{35} -6.34279e6 q^{37} +5.35387e6 q^{38} +5.50483e6 q^{40} +1.86291e7 q^{41} +4.79937e6 q^{43} +3.45584e7 q^{44} -1.93872e7 q^{46} -1.96227e7 q^{47} +5.76480e6 q^{49} -3.67842e6 q^{50} -6.05318e7 q^{52} -5.89666e7 q^{53} -5.10223e7 q^{55} -2.11474e7 q^{56} +1.85234e6 q^{58} +1.68723e8 q^{59} +9.77330e7 q^{61} +6.22507e6 q^{62} -1.41761e7 q^{64} +8.93696e7 q^{65} -9.57587e7 q^{67} +7.88828e7 q^{68} +1.41310e7 q^{70} +1.52066e8 q^{71} +1.65675e8 q^{73} +5.97286e7 q^{74} +2.40679e8 q^{76} +1.96007e8 q^{77} +2.10472e8 q^{79} +8.36261e7 q^{80} -1.75426e8 q^{82} +5.68738e8 q^{83} -1.16463e8 q^{85} -4.51946e7 q^{86} -7.19026e8 q^{88} +4.82100e7 q^{89} -3.43322e8 q^{91} -8.71537e8 q^{92} +1.84782e8 q^{94} -3.55341e8 q^{95} -9.50202e8 q^{97} -5.42858e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.41677	−0.416166	−0.208083	−	0.978111i	$-0.566722\pi$
$2$	−9.41677	−0.416166	−0.208083	+	0.978111i	$0.566722\pi$
$3$	0	0
$3$	0	0
$4$	−423.324	−0.826806
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	8807.73	0.760255
$9$	0	0
$10$	−5885.48	−0.186115
$11$	−81635.7	−1.68118	−0.840588	−	0.541675i	$-0.817790\pi$
$11$	−81635.7	−1.68118	−0.840588	+	0.541675i	$0.817790\pi$
$12$	0	0
$13$	142991.	1.38856	0.694280	−	0.719705i	$-0.255723\pi$
$13$	142991.	1.38856	0.694280	+	0.719705i	$0.255723\pi$
$14$	22609.7	0.157296
$15$	0	0
$16$	133802.	0.510413
$17$	−186341.	−0.541114	−0.270557	−	0.962704i	$-0.587208\pi$
$17$	−186341.	−0.541114	−0.270557	+	0.962704i	$0.587208\pi$
$18$	0	0
$19$	−568546.	−1.00086	−0.500431	−	0.865776i	$-0.666825\pi$
$19$	−568546.	−1.00086	−0.500431	+	0.865776i	$0.666825\pi$
$20$	−264578.	−0.369759
$21$	0	0
$22$	768745.	0.699649
$23$	2.05879e6	1.53404	0.767021	−	0.641622i	$-0.221738\pi$
$23$	2.05879e6	1.53404	0.767021	+	0.641622i	$0.221738\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−1.34652e6	−0.577872
$27$	0	0
$28$	1.01640e6	0.312503
$29$	−196707.	−0.0516450	−0.0258225	−	0.999667i	$-0.508220\pi$
$29$	−196707.	−0.0516450	−0.0258225	+	0.999667i	$0.508220\pi$
$30$	0	0
$31$	−661063.	−0.128563	−0.0642814	−	0.997932i	$-0.520476\pi$
$31$	−661063.	−0.128563	−0.0642814	+	0.997932i	$0.520476\pi$
$32$	−5.76954e6	−0.972672
$33$	0	0
$34$	1.75473e6	0.225193
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	−6.34279e6	−0.556381	−0.278191	−	0.960526i	$-0.589735\pi$
$37$	−6.34279e6	−0.556381	−0.278191	+	0.960526i	$0.589735\pi$
$38$	5.35387e6	0.416525
$39$	0	0
$40$	5.50483e6	0.339996
$41$	1.86291e7	1.02959	0.514796	−	0.857312i	$-0.327868\pi$
$41$	1.86291e7	1.02959	0.514796	+	0.857312i	$0.327868\pi$
$42$	0	0
$43$	4.79937e6	0.214080	0.107040	−	0.994255i	$-0.465863\pi$
$43$	4.79937e6	0.214080	0.107040	+	0.994255i	$0.465863\pi$
$44$	3.45584e7	1.39001
$45$	0	0
$46$	−1.93872e7	−0.638417
$47$	−1.96227e7	−0.586567	−0.293284	−	0.956026i	$-0.594748\pi$
$47$	−1.96227e7	−0.586567	−0.293284	+	0.956026i	$0.594748\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−3.67842e6	−0.0832333
$51$	0	0
$52$	−6.05318e7	−1.14807
$53$	−5.89666e7	−1.02651	−0.513256	−	0.858235i	$-0.671561\pi$
$53$	−5.89666e7	−1.02651	−0.513256	+	0.858235i	$0.671561\pi$
$54$	0	0
$55$	−5.10223e7	−0.751845
$56$	−2.11474e7	−0.287349
$57$	0	0
$58$	1.85234e6	0.0214929
$59$	1.68723e8	1.81276	0.906378	−	0.422468i	$-0.138836\pi$
$59$	1.68723e8	1.81276	0.906378	+	0.422468i	$0.138836\pi$
$60$	0	0
$61$	9.77330e7	0.903768	0.451884	−	0.892077i	$-0.350752\pi$
$61$	9.77330e7	0.903768	0.451884	+	0.892077i	$0.350752\pi$
$62$	6.22507e6	0.0535035
$63$	0	0
$64$	−1.41761e7	−0.105620
$65$	8.93696e7	0.620983
$66$	0	0
$67$	−9.57587e7	−0.580552	−0.290276	−	0.956943i	$-0.593747\pi$
$67$	−9.57587e7	−0.580552	−0.290276	+	0.956943i	$0.593747\pi$
$68$	7.88828e7	0.447396
$69$	0	0
$70$	1.41310e7	0.0703449
$71$	1.52066e8	0.710180	0.355090	−	0.934832i	$-0.384450\pi$
$71$	1.52066e8	0.710180	0.355090	+	0.934832i	$0.384450\pi$
$72$	0	0
$73$	1.65675e8	0.682815	0.341407	−	0.939915i	$-0.389096\pi$
$73$	1.65675e8	0.682815	0.341407	+	0.939915i	$0.389096\pi$
$74$	5.97286e7	0.231547
$75$	0	0
$76$	2.40679e8	0.827519
$77$	1.96007e8	0.635425
$78$	0	0
$79$	2.10472e8	0.607957	0.303979	−	0.952679i	$-0.401685\pi$
$79$	2.10472e8	0.607957	0.303979	+	0.952679i	$0.401685\pi$
$80$	8.36261e7	0.228264
$81$	0	0
$82$	−1.75426e8	−0.428482
$83$	5.68738e8	1.31541	0.657704	−	0.753276i	$-0.271528\pi$
$83$	5.68738e8	1.31541	0.657704	+	0.753276i	$0.271528\pi$
$84$	0	0
$85$	−1.16463e8	−0.241994
$86$	−4.51946e7	−0.0890929
$87$	0	0
$88$	−7.19026e8	−1.27812
$89$	4.82100e7	0.0814483	0.0407241	−	0.999170i	$-0.487034\pi$
$89$	4.82100e7	0.0814483	0.0407241	+	0.999170i	$0.487034\pi$
$90$	0	0
$91$	−3.43322e8	−0.524827
$92$	−8.71537e8	−1.26835
$93$	0	0
$94$	1.84782e8	0.244109
$95$	−3.55341e8	−0.447599
$96$	0	0
$97$	−9.50202e8	−1.08979	−0.544895	−	0.838504i	$-0.683431\pi$
$97$	−9.50202e8	−1.08979	−0.544895	+	0.838504i	$0.683431\pi$
$98$	−5.42858e7	−0.0594523
$99$	0	0
$100$	−1.65361e8	−0.165361
$101$	−6.86462e8	−0.656403	−0.328202	−	0.944608i	$-0.606443\pi$
$101$	−6.86462e8	−0.656403	−0.328202	+	0.944608i	$0.606443\pi$
$102$	0	0
$103$	1.44857e8	0.126815	0.0634077	−	0.997988i	$-0.479803\pi$
$103$	1.44857e8	0.126815	0.0634077	+	0.997988i	$0.479803\pi$
$104$	1.25943e9	1.05566
$105$	0	0
$106$	5.55274e8	0.427200
$107$	−8.85421e7	−0.0653014	−0.0326507	−	0.999467i	$-0.510395\pi$
$107$	−8.85421e7	−0.0653014	−0.0326507	+	0.999467i	$0.510395\pi$
$108$	0	0
$109$	−2.78795e9	−1.89176	−0.945881	−	0.324514i	$-0.894799\pi$
$109$	−2.78795e9	−1.89176	−0.945881	+	0.324514i	$0.894799\pi$
$110$	4.80465e8	0.312892
$111$	0	0
$112$	−3.21258e8	−0.192918
$113$	3.34793e9	1.93163	0.965813	−	0.259241i	$-0.0834726\pi$
$113$	3.34793e9	1.93163	0.965813	+	0.259241i	$0.0834726\pi$
$114$	0	0
$115$	1.28675e9	0.686045
$116$	8.32708e7	0.0427004
$117$	0	0
$118$	−1.58882e9	−0.754408
$119$	4.47405e8	0.204522
$120$	0	0
$121$	4.30644e9	1.82635
$122$	−9.20329e8	−0.376118
$123$	0	0
$124$	2.79844e8	0.106296
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−2.49430e9	−0.850810	−0.425405	−	0.905003i	$-0.639868\pi$
$127$	−2.49430e9	−0.850810	−0.425405	+	0.905003i	$0.639868\pi$
$128$	3.08750e9	1.01663
$129$	0	0
$130$	−8.41573e8	−0.258432
$131$	1.80018e9	0.534067	0.267034	−	0.963687i	$-0.413957\pi$
$131$	1.80018e9	0.534067	0.267034	+	0.963687i	$0.413957\pi$
$132$	0	0
$133$	1.36508e9	0.378291
$134$	9.01737e8	0.241606
$135$	0	0
$136$	−1.64124e9	−0.411385
$137$	−4.24238e9	−1.02889	−0.514443	−	0.857525i	$-0.672001\pi$
$137$	−4.24238e9	−1.02889	−0.514443	+	0.857525i	$0.672001\pi$
$138$	0	0
$139$	−3.50196e9	−0.795692	−0.397846	−	0.917452i	$-0.630242\pi$
$139$	−3.50196e9	−0.795692	−0.397846	+	0.917452i	$0.630242\pi$
$140$	6.35251e8	0.139756
$141$	0	0
$142$	−1.43197e9	−0.295553
$143$	−1.16732e10	−2.33441
$144$	0	0
$145$	−1.22942e8	−0.0230963
$146$	−1.56012e9	−0.284164
$147$	0	0
$148$	2.68506e9	0.460019
$149$	4.96552e9	0.825328	0.412664	−	0.910883i	$-0.364598\pi$
$149$	4.96552e9	0.825328	0.412664	+	0.910883i	$0.364598\pi$
$150$	0	0
$151$	1.12718e10	1.76440	0.882202	−	0.470870i	$-0.156060\pi$
$151$	1.12718e10	1.76440	0.882202	+	0.470870i	$0.156060\pi$
$152$	−5.00760e9	−0.760911
$153$	0	0
$154$	−1.84576e9	−0.264442
$155$	−4.13164e8	−0.0574950
$156$	0	0
$157$	−1.00503e10	−1.32018	−0.660088	−	0.751188i	$-0.729481\pi$
$157$	−1.00503e10	−1.32018	−0.660088	+	0.751188i	$0.729481\pi$
$158$	−1.98197e9	−0.253011
$159$	0	0
$160$	−3.60596e9	−0.434992
$161$	−4.94316e9	−0.579814
$162$	0	0
$163$	6.51839e9	0.723263	0.361632	−	0.932321i	$-0.382220\pi$
$163$	6.51839e9	0.723263	0.361632	+	0.932321i	$0.382220\pi$
$164$	−7.88617e9	−0.851273
$165$	0	0
$166$	−5.35567e9	−0.547428
$167$	2.61874e9	0.260536	0.130268	−	0.991479i	$-0.458416\pi$
$167$	2.61874e9	0.260536	0.130268	+	0.991479i	$0.458416\pi$
$168$	0	0
$169$	9.84204e9	0.928101
$170$	1.09671e9	0.100710
$171$	0	0
$172$	−2.03169e9	−0.177003
$173$	−1.64438e10	−1.39571	−0.697856	−	0.716238i	$-0.745863\pi$
$173$	−1.64438e10	−1.39571	−0.697856	+	0.716238i	$0.745863\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−1.09230e10	−0.858094
$177$	0	0
$178$	−4.53982e8	−0.0338960
$179$	−6.28033e9	−0.457240	−0.228620	−	0.973516i	$-0.573421\pi$
$179$	−6.28033e9	−0.457240	−0.228620	+	0.973516i	$0.573421\pi$
$180$	0	0
$181$	−5.12651e9	−0.355033	−0.177516	−	0.984118i	$-0.556806\pi$
$181$	−5.12651e9	−0.355033	−0.177516	+	0.984118i	$0.556806\pi$
$182$	3.23299e9	0.218415
$183$	0	0
$184$	1.81333e10	1.16626
$185$	−3.96424e9	−0.248821
$186$	0	0
$187$	1.52121e10	0.909708
$188$	8.30676e9	0.484977
$189$	0	0
$190$	3.34617e9	0.186276
$191$	−4.75403e9	−0.258471	−0.129236	−	0.991614i	$-0.541252\pi$
$191$	−4.75403e9	−0.258471	−0.129236	+	0.991614i	$0.541252\pi$
$192$	0	0
$193$	−2.27139e10	−1.17838	−0.589189	−	0.807996i	$-0.700552\pi$
$193$	−2.27139e10	−1.17838	−0.589189	+	0.807996i	$0.700552\pi$
$194$	8.94783e9	0.453534
$195$	0	0
$196$	−2.44038e9	−0.118115
$197$	1.93880e9	0.0917140	0.0458570	−	0.998948i	$-0.485398\pi$
$197$	1.93880e9	0.0917140	0.0458570	+	0.998948i	$0.485398\pi$
$198$	0	0
$199$	−2.30672e10	−1.04269	−0.521345	−	0.853346i	$-0.674570\pi$
$199$	−2.30672e10	−1.04269	−0.521345	+	0.853346i	$0.674570\pi$
$200$	3.44052e9	0.152051
$201$	0	0
$202$	6.46426e9	0.273173
$203$	4.72293e8	0.0195200
$204$	0	0
$205$	1.16432e10	0.460448
$206$	−1.36408e9	−0.0527763
$207$	0	0
$208$	1.91325e10	0.708740
$209$	4.64137e10	1.68263
$210$	0	0
$211$	−5.19377e10	−1.80390	−0.901948	−	0.431844i	$-0.857863\pi$
$211$	−5.19377e10	−1.80390	−0.901948	+	0.431844i	$0.857863\pi$
$212$	2.49620e10	0.848727
$213$	0	0
$214$	8.33780e8	0.0271763
$215$	2.99961e9	0.0957395
$216$	0	0
$217$	1.58721e9	0.0485921
$218$	2.62535e10	0.787287
$219$	0	0
$220$	2.15990e10	0.621629
$221$	−2.66452e10	−0.751370
$222$	0	0
$223$	−6.75023e9	−0.182788	−0.0913938	−	0.995815i	$-0.529132\pi$
$223$	−6.75023e9	−0.182788	−0.0913938	+	0.995815i	$0.529132\pi$
$224$	1.38527e10	0.367635
$225$	0	0
$226$	−3.15266e10	−0.803877
$227$	1.55162e9	0.0387854	0.0193927	−	0.999812i	$-0.493827\pi$
$227$	1.55162e9	0.0387854	0.0193927	+	0.999812i	$0.493827\pi$
$228$	0	0
$229$	−7.09078e10	−1.70386	−0.851930	−	0.523655i	$-0.824568\pi$
$229$	−7.09078e10	−1.70386	−0.851930	+	0.523655i	$0.824568\pi$
$230$	−1.21170e10	−0.285509
$231$	0	0
$232$	−1.73254e9	−0.0392634
$233$	−7.24517e10	−1.61045	−0.805224	−	0.592971i	$-0.797955\pi$
$233$	−7.24517e10	−1.61045	−0.805224	+	0.592971i	$0.797955\pi$
$234$	0	0
$235$	−1.22642e10	−0.262321
$236$	−7.14244e10	−1.49880
$237$	0	0
$238$	−4.21311e9	−0.0851151
$239$	−4.54381e10	−0.900802	−0.450401	−	0.892826i	$-0.648719\pi$
$239$	−4.54381e10	−0.900802	−0.450401	+	0.892826i	$0.648719\pi$
$240$	0	0
$241$	−9.03673e10	−1.72558	−0.862789	−	0.505564i	$-0.831284\pi$
$241$	−9.03673e10	−1.72558	−0.862789	+	0.505564i	$0.831284\pi$
$242$	−4.05528e10	−0.760066
$243$	0	0
$244$	−4.13728e10	−0.747241
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−8.12972e10	−1.38976
$248$	−5.82246e9	−0.0977404
$249$	0	0
$250$	−2.29902e9	−0.0372230
$251$	5.21785e10	0.829775	0.414887	−	0.909873i	$-0.363821\pi$
$251$	5.21785e10	0.829775	0.414887	+	0.909873i	$0.363821\pi$
$252$	0	0
$253$	−1.68071e11	−2.57900
$254$	2.34883e10	0.354079
$255$	0	0
$256$	−2.18161e10	−0.317466
$257$	−4.84258e9	−0.0692433	−0.0346216	−	0.999400i	$-0.511023\pi$
$257$	−4.84258e9	−0.0692433	−0.0346216	+	0.999400i	$0.511023\pi$
$258$	0	0
$259$	1.52290e10	0.210292
$260$	−3.78324e10	−0.513432
$261$	0	0
$262$	−1.69519e10	−0.222261
$263$	7.87714e10	1.01524	0.507619	−	0.861582i	$-0.330526\pi$
$263$	7.87714e10	1.01524	0.507619	+	0.861582i	$0.330526\pi$
$264$	0	0
$265$	−3.68541e10	−0.459071
$266$	−1.28546e10	−0.157432
$267$	0	0
$268$	4.05370e10	0.480004
$269$	−5.14223e9	−0.0598778	−0.0299389	−	0.999552i	$-0.509531\pi$
$269$	−5.14223e9	−0.0598778	−0.0299389	+	0.999552i	$0.509531\pi$
$270$	0	0
$271$	−1.41745e10	−0.159642	−0.0798209	−	0.996809i	$-0.525435\pi$
$271$	−1.41745e10	−0.159642	−0.0798209	+	0.996809i	$0.525435\pi$
$272$	−2.49328e10	−0.276192
$273$	0	0
$274$	3.99495e10	0.428188
$275$	−3.18890e10	−0.336235
$276$	0	0
$277$	6.40169e10	0.653335	0.326668	−	0.945139i	$-0.394074\pi$
$277$	6.40169e10	0.653335	0.326668	+	0.945139i	$0.394074\pi$
$278$	3.29772e10	0.331140
$279$	0	0
$280$	−1.32171e10	−0.128507
$281$	−9.28031e10	−0.887941	−0.443970	−	0.896041i	$-0.646430\pi$
$281$	−9.28031e10	−0.887941	−0.443970	+	0.896041i	$0.646430\pi$
$282$	0	0
$283$	−1.21652e11	−1.12741	−0.563704	−	0.825977i	$-0.690624\pi$
$283$	−1.21652e11	−1.12741	−0.563704	+	0.825977i	$0.690624\pi$
$284$	−6.43731e10	−0.587181
$285$	0	0
$286$	1.09924e11	0.971505
$287$	−4.47286e10	−0.389149
$288$	0	0
$289$	−8.38648e10	−0.707196
$290$	1.15771e9	0.00961192
$291$	0	0
$292$	−7.01341e10	−0.564555
$293$	−5.64775e10	−0.447684	−0.223842	−	0.974625i	$-0.571860\pi$
$293$	−5.64775e10	−0.447684	−0.223842	+	0.974625i	$0.571860\pi$
$294$	0	0
$295$	1.05452e11	0.810689
$296$	−5.58656e10	−0.422992
$297$	0	0
$298$	−4.67592e10	−0.343474
$299$	2.94390e11	2.13011
$300$	0	0
$301$	−1.15233e10	−0.0809147
$302$	−1.06144e11	−0.734286
$303$	0	0
$304$	−7.60725e10	−0.510853
$305$	6.10831e10	0.404177
$306$	0	0
$307$	−3.31660e10	−0.213093	−0.106547	−	0.994308i	$-0.533979\pi$
$307$	−3.31660e10	−0.213093	−0.106547	+	0.994308i	$0.533979\pi$
$308$	−8.29747e10	−0.525373
$309$	0	0
$310$	3.89067e9	0.0239275
$311$	1.96048e11	1.18834	0.594170	−	0.804340i	$-0.297481\pi$
$311$	1.96048e11	1.18834	0.594170	+	0.804340i	$0.297481\pi$
$312$	0	0
$313$	−2.73808e11	−1.61249	−0.806244	−	0.591583i	$-0.798503\pi$
$313$	−2.73808e11	−1.61249	−0.806244	+	0.591583i	$0.798503\pi$
$314$	9.46416e10	0.549413
$315$	0	0
$316$	−8.90981e10	−0.502662
$317$	−3.33062e11	−1.85250	−0.926250	−	0.376909i	$-0.876987\pi$
$317$	−3.33062e11	−1.85250	−0.926250	+	0.376909i	$0.876987\pi$
$318$	0	0
$319$	1.60583e10	0.0868243
$320$	−8.86005e9	−0.0472347
$321$	0	0
$322$	4.65486e10	0.241299
$323$	1.05944e11	0.541581
$324$	0	0
$325$	5.58560e10	0.277712
$326$	−6.13822e10	−0.300998
$327$	0	0
$328$	1.64081e11	0.782753
$329$	4.71140e10	0.221702
$330$	0	0
$331$	2.82425e11	1.29324	0.646618	−	0.762814i	$-0.276183\pi$
$331$	2.82425e11	1.29324	0.646618	+	0.762814i	$0.276183\pi$
$332$	−2.40761e11	−1.08759
$333$	0	0
$334$	−2.46600e10	−0.108426
$335$	−5.98492e10	−0.259631
$336$	0	0
$337$	3.36843e11	1.42263	0.711317	−	0.702871i	$-0.248099\pi$
$337$	3.36843e11	1.42263	0.711317	+	0.702871i	$0.248099\pi$
$338$	−9.26802e10	−0.386244
$339$	0	0
$340$	4.93017e10	0.200082
$341$	5.39663e10	0.216137
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	4.22716e10	0.162755
$345$	0	0
$346$	1.54848e11	0.580849
$347$	−9.03801e10	−0.334649	−0.167325	−	0.985902i	$-0.553513\pi$
$347$	−9.03801e10	−0.334649	−0.167325	+	0.985902i	$0.553513\pi$
$348$	0	0
$349$	−4.05067e11	−1.46155	−0.730773	−	0.682620i	$-0.760840\pi$
$349$	−4.05067e11	−1.46155	−0.730773	+	0.682620i	$0.760840\pi$
$350$	8.83190e9	0.0314592
$351$	0	0
$352$	4.71001e11	1.63523
$353$	−2.70274e11	−0.926440	−0.463220	−	0.886243i	$-0.653306\pi$
$353$	−2.70274e11	−0.926440	−0.463220	+	0.886243i	$0.653306\pi$
$354$	0	0
$355$	9.50410e10	0.317602
$356$	−2.04085e10	−0.0673419
$357$	0	0
$358$	5.91404e10	0.190288
$359$	1.05345e11	0.334726	0.167363	−	0.985895i	$-0.446475\pi$
$359$	1.05345e11	0.334726	0.167363	+	0.985895i	$0.446475\pi$
$360$	0	0
$361$	5.56967e8	0.00172602
$362$	4.82752e10	0.147753
$363$	0	0
$364$	1.45337e11	0.433930
$365$	1.03547e11	0.305364
$366$	0	0
$367$	1.59156e11	0.457958	0.228979	−	0.973431i	$-0.426461\pi$
$367$	1.59156e11	0.457958	0.228979	+	0.973431i	$0.426461\pi$
$368$	2.75470e11	0.782995
$369$	0	0
$370$	3.73304e10	0.103551
$371$	1.41579e11	0.387985
$372$	0	0
$373$	−3.67998e11	−0.984364	−0.492182	−	0.870492i	$-0.663800\pi$
$373$	−3.67998e11	−0.984364	−0.492182	+	0.870492i	$0.663800\pi$
$374$	−1.43249e11	−0.378590
$375$	0	0
$376$	−1.72831e11	−0.445940
$377$	−2.81274e10	−0.0717122
$378$	0	0
$379$	4.62007e11	1.15020	0.575098	−	0.818084i	$-0.304964\pi$
$379$	4.62007e11	1.15020	0.575098	+	0.818084i	$0.304964\pi$
$380$	1.50425e11	0.370078
$381$	0	0
$382$	4.47676e10	0.107567
$383$	−3.55193e11	−0.843471	−0.421735	−	0.906719i	$-0.638579\pi$
$383$	−3.55193e11	−0.843471	−0.421735	+	0.906719i	$0.638579\pi$
$384$	0	0
$385$	1.22505e11	0.284171
$386$	2.13892e11	0.490401
$387$	0	0
$388$	4.02244e11	0.901045
$389$	3.70667e11	0.820749	0.410375	−	0.911917i	$-0.365398\pi$
$389$	3.70667e11	0.820749	0.410375	+	0.911917i	$0.365398\pi$
$390$	0	0
$391$	−3.83638e11	−0.830092
$392$	5.07748e10	0.108608
$393$	0	0
$394$	−1.82573e10	−0.0381683
$395$	1.31545e11	0.271887
$396$	0	0
$397$	4.80651e10	0.0971119	0.0485560	−	0.998820i	$-0.484538\pi$
$397$	4.80651e10	0.0971119	0.0485560	+	0.998820i	$0.484538\pi$
$398$	2.17218e11	0.433932
$399$	0	0
$400$	5.22663e10	0.102083
$401$	−7.92241e10	−0.153006	−0.0765028	−	0.997069i	$-0.524375\pi$
$401$	−7.92241e10	−0.153006	−0.0765028	+	0.997069i	$0.524375\pi$
$402$	0	0
$403$	−9.45263e10	−0.178517
$404$	2.90596e11	0.542718
$405$	0	0
$406$	−4.44747e9	−0.00812356
$407$	5.17798e11	0.935375
$408$	0	0
$409$	−4.79197e11	−0.846757	−0.423379	−	0.905953i	$-0.639156\pi$
$409$	−4.79197e11	−0.846757	−0.423379	+	0.905953i	$0.639156\pi$
$410$	−1.09641e11	−0.191623
$411$	0	0
$412$	−6.13215e10	−0.104852
$413$	−4.05103e11	−0.685157
$414$	0	0
$415$	3.55461e11	0.588268
$416$	−8.24995e11	−1.35061
$417$	0	0
$418$	−4.37067e11	−0.700252
$419$	2.43897e11	0.386584	0.193292	−	0.981141i	$-0.438084\pi$
$419$	2.43897e11	0.386584	0.193292	+	0.981141i	$0.438084\pi$
$420$	0	0
$421$	−1.06824e12	−1.65730	−0.828648	−	0.559769i	$-0.810890\pi$
$421$	−1.06824e12	−1.65730	−0.828648	+	0.559769i	$0.810890\pi$
$422$	4.89085e11	0.750721
$423$	0	0
$424$	−5.19362e11	−0.780411
$425$	−7.27895e10	−0.108223
$426$	0	0
$427$	−2.34657e11	−0.341592
$428$	3.74820e10	0.0539916
$429$	0	0
$430$	−2.82466e10	−0.0398436
$431$	4.43324e11	0.618833	0.309416	−	0.950927i	$-0.399866\pi$
$431$	4.43324e11	0.618833	0.309416	+	0.950927i	$0.399866\pi$
$432$	0	0
$433$	1.09500e12	1.49699	0.748494	−	0.663141i	$-0.230777\pi$
$433$	1.09500e12	1.49699	0.748494	+	0.663141i	$0.230777\pi$
$434$	−1.49464e10	−0.0202224
$435$	0	0
$436$	1.18021e12	1.56412
$437$	−1.17052e12	−1.53537
$438$	0	0
$439$	−4.14241e11	−0.532308	−0.266154	−	0.963931i	$-0.585753\pi$
$439$	−4.14241e11	−0.532308	−0.266154	+	0.963931i	$0.585753\pi$
$440$	−4.49391e11	−0.571594
$441$	0	0
$442$	2.50912e11	0.312695
$443$	4.12784e11	0.509220	0.254610	−	0.967044i	$-0.418053\pi$
$443$	4.12784e11	0.509220	0.254610	+	0.967044i	$0.418053\pi$
$444$	0	0
$445$	3.01312e10	0.0364248
$446$	6.35654e10	0.0760701
$447$	0	0
$448$	3.40368e10	0.0399206
$449$	1.58278e12	1.83786	0.918929	−	0.394423i	$-0.129055\pi$
$449$	1.58278e12	1.83786	0.918929	+	0.394423i	$0.129055\pi$
$450$	0	0
$451$	−1.52080e12	−1.73093
$452$	−1.41726e12	−1.59708
$453$	0	0
$454$	−1.46112e10	−0.0161412
$455$	−2.14576e11	−0.234710
$456$	0	0
$457$	1.51321e12	1.62284	0.811421	−	0.584462i	$-0.198694\pi$
$457$	1.51321e12	1.62284	0.811421	+	0.584462i	$0.198694\pi$
$458$	6.67722e11	0.709089
$459$	0	0
$460$	−5.44711e11	−0.567226
$461$	2.69025e11	0.277420	0.138710	−	0.990333i	$-0.455704\pi$
$461$	2.69025e11	0.277420	0.138710	+	0.990333i	$0.455704\pi$
$462$	0	0
$463$	1.58258e12	1.60048	0.800242	−	0.599678i	$-0.204704\pi$
$463$	1.58258e12	1.60048	0.800242	+	0.599678i	$0.204704\pi$
$464$	−2.63197e10	−0.0263603
$465$	0	0
$466$	6.82260e11	0.670214
$467$	5.45148e10	0.0530382	0.0265191	−	0.999648i	$-0.491558\pi$
$467$	5.45148e10	0.0530382	0.0265191	+	0.999648i	$0.491558\pi$
$468$	0	0
$469$	2.29917e11	0.219428
$470$	1.15489e11	0.109169
$471$	0	0
$472$	1.48606e12	1.37816
$473$	−3.91800e11	−0.359906
$474$	0	0
$475$	−2.22088e11	−0.200173
$476$	−1.89398e11	−0.169100
$477$	0	0
$478$	4.27880e11	0.374883
$479$	1.04188e12	0.904291	0.452146	−	0.891944i	$-0.350659\pi$
$479$	1.04188e12	0.904291	0.452146	+	0.891944i	$0.350659\pi$
$480$	0	0
$481$	−9.06964e11	−0.772569
$482$	8.50968e11	0.718127
$483$	0	0
$484$	−1.82302e12	−1.51004
$485$	−5.93876e11	−0.487369
$486$	0	0
$487$	−4.56101e9	−0.00367435	−0.00183718	−	0.999998i	$-0.500585\pi$
$487$	−4.56101e9	−0.00367435	−0.00183718	+	0.999998i	$0.500585\pi$
$488$	8.60807e11	0.687094
$489$	0	0
$490$	−3.39286e10	−0.0265879
$491$	1.32266e12	1.02702	0.513512	−	0.858082i	$-0.328344\pi$
$491$	1.32266e12	1.02702	0.513512	+	0.858082i	$0.328344\pi$
$492$	0	0
$493$	3.66546e10	0.0279458
$494$	7.65557e11	0.578371
$495$	0	0
$496$	−8.84513e10	−0.0656201
$497$	−3.65109e11	−0.268423
$498$	0	0
$499$	1.65126e12	1.19224	0.596118	−	0.802897i	$-0.296709\pi$
$499$	1.65126e12	1.19224	0.596118	+	0.802897i	$0.296709\pi$
$500$	−1.03351e11	−0.0739517
$501$	0	0
$502$	−4.91353e11	−0.345324
$503$	2.24891e12	1.56645	0.783225	−	0.621739i	$-0.213573\pi$
$503$	2.24891e12	1.56645	0.783225	+	0.621739i	$0.213573\pi$
$504$	0	0
$505$	−4.29039e11	−0.293552
$506$	1.58269e12	1.07329
$507$	0	0
$508$	1.05590e12	0.703455
$509$	1.12596e12	0.743518	0.371759	−	0.928329i	$-0.378755\pi$
$509$	1.12596e12	0.743518	0.371759	+	0.928329i	$0.378755\pi$
$510$	0	0
$511$	−3.97784e11	−0.258080
$512$	−1.37536e12	−0.884509
$513$	0	0
$514$	4.56014e10	0.0288167
$515$	9.05356e10	0.0567135
$516$	0	0
$517$	1.60191e12	0.986122
$518$	−1.43408e11	−0.0875166
$519$	0	0
$520$	7.87144e11	0.472106
$521$	−3.16980e12	−1.88479	−0.942393	−	0.334508i	$-0.891430\pi$
$521$	−3.16980e12	−1.88479	−0.942393	+	0.334508i	$0.891430\pi$
$522$	0	0
$523$	−3.20261e12	−1.87174	−0.935872	−	0.352339i	$-0.885386\pi$
$523$	−3.20261e12	−1.87174	−0.935872	+	0.352339i	$0.885386\pi$
$524$	−7.62062e11	−0.441570
$525$	0	0
$526$	−7.41772e11	−0.422508
$527$	1.23183e11	0.0695671
$528$	0	0
$529$	2.43747e12	1.35329
$530$	3.47047e11	0.191050
$531$	0	0
$532$	−5.77871e11	−0.312773
$533$	2.66381e12	1.42965
$534$	0	0
$535$	−5.53388e10	−0.0292037
$536$	−8.43417e11	−0.441368
$537$	0	0
$538$	4.84231e10	0.0249191
$539$	−4.70614e11	−0.240168
$540$	0	0
$541$	−1.26299e12	−0.633886	−0.316943	−	0.948445i	$-0.602656\pi$
$541$	−1.26299e12	−0.633886	−0.316943	+	0.948445i	$0.602656\pi$
$542$	1.33478e11	0.0664375
$543$	0	0
$544$	1.07510e12	0.526326
$545$	−1.74247e12	−0.846021
$546$	0	0
$547$	−1.89310e12	−0.904130	−0.452065	−	0.891985i	$-0.649312\pi$
$547$	−1.89310e12	−0.904130	−0.452065	+	0.891985i	$0.649312\pi$
$548$	1.79590e12	0.850689
$549$	0	0
$550$	3.00291e11	0.139930
$551$	1.11837e11	0.0516895
$552$	0	0
$553$	−5.05344e11	−0.229786
$554$	−6.02833e11	−0.271896
$555$	0	0
$556$	1.48247e12	0.657883
$557$	−3.08955e12	−1.36003	−0.680013	−	0.733200i	$-0.738026\pi$
$557$	−3.08955e12	−1.36003	−0.680013	+	0.733200i	$0.738026\pi$
$558$	0	0
$559$	6.86269e11	0.297263
$560$	−2.00786e11	−0.0862756
$561$	0	0
$562$	8.73905e11	0.369531
$563$	−3.48321e12	−1.46114	−0.730570	−	0.682838i	$-0.760745\pi$
$563$	−3.48321e12	−1.46114	−0.730570	+	0.682838i	$0.760745\pi$
$564$	0	0
$565$	2.09245e12	0.863849
$566$	1.14557e12	0.469189
$567$	0	0
$568$	1.33935e12	0.539918
$569$	−4.84964e12	−1.93956	−0.969782	−	0.243975i	$-0.921549\pi$
$569$	−4.84964e12	−1.93956	−0.969782	+	0.243975i	$0.921549\pi$
$570$	0	0
$571$	3.64312e12	1.43420	0.717102	−	0.696969i	$-0.245468\pi$
$571$	3.64312e12	1.43420	0.717102	+	0.696969i	$0.245468\pi$
$572$	4.94155e12	1.93011
$573$	0	0
$574$	4.21199e11	0.161951
$575$	8.04216e11	0.306808
$576$	0	0
$577$	4.98140e12	1.87094	0.935471	−	0.353404i	$-0.114976\pi$
$577$	4.98140e12	1.87094	0.935471	+	0.353404i	$0.114976\pi$
$578$	7.89736e11	0.294311
$579$	0	0
$580$	5.20442e10	0.0190962
$581$	−1.36554e12	−0.497178
$582$	0	0
$583$	4.81378e12	1.72575
$584$	1.45922e12	0.519113
$585$	0	0
$586$	5.31836e11	0.186311
$587$	−3.70534e12	−1.28812	−0.644061	−	0.764974i	$-0.722752\pi$
$587$	−3.70534e12	−1.28812	−0.644061	+	0.764974i	$0.722752\pi$
$588$	0	0
$589$	3.75845e11	0.128674
$590$	−9.93014e11	−0.337381
$591$	0	0
$592$	−8.48676e11	−0.283984
$593$	1.49825e12	0.497551	0.248776	−	0.968561i	$-0.419972\pi$
$593$	1.49825e12	0.497551	0.248776	+	0.968561i	$0.419972\pi$
$594$	0	0
$595$	2.79628e11	0.0914650
$596$	−2.10203e12	−0.682386
$597$	0	0
$598$	−2.77220e12	−0.886480
$599$	−1.91927e12	−0.609138	−0.304569	−	0.952490i	$-0.598512\pi$
$599$	−1.91927e12	−0.609138	−0.304569	+	0.952490i	$0.598512\pi$
$600$	0	0
$601$	5.04272e12	1.57663	0.788315	−	0.615271i	$-0.210954\pi$
$601$	5.04272e12	1.57663	0.788315	+	0.615271i	$0.210954\pi$
$602$	1.08512e11	0.0336740
$603$	0	0
$604$	−4.77164e12	−1.45882
$605$	2.69153e12	0.816770
$606$	0	0
$607$	−3.22833e12	−0.965226	−0.482613	−	0.875834i	$-0.660312\pi$
$607$	−3.22833e12	−0.965226	−0.482613	+	0.875834i	$0.660312\pi$
$608$	3.28025e12	0.973511
$609$	0	0
$610$	−5.75206e11	−0.168205
$611$	−2.80587e12	−0.814484
$612$	0	0
$613$	2.89802e12	0.828952	0.414476	−	0.910060i	$-0.363965\pi$
$613$	2.89802e12	0.828952	0.414476	+	0.910060i	$0.363965\pi$
$614$	3.12316e11	0.0886822
$615$	0	0
$616$	1.72638e12	0.483085
$617$	−2.66605e12	−0.740602	−0.370301	−	0.928912i	$-0.620745\pi$
$617$	−2.66605e12	−0.740602	−0.370301	+	0.928912i	$0.620745\pi$
$618$	0	0
$619$	−3.68074e12	−1.00769	−0.503846	−	0.863794i	$-0.668082\pi$
$619$	−3.68074e12	−1.00769	−0.503846	+	0.863794i	$0.668082\pi$
$620$	1.74903e11	0.0475372
$621$	0	0
$622$	−1.84614e12	−0.494547
$623$	−1.15752e11	−0.0307846
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	2.57839e12	0.671063
$627$	0	0
$628$	4.25455e12	1.09153
$629$	1.18192e12	0.301066
$630$	0	0
$631$	−1.51762e12	−0.381093	−0.190547	−	0.981678i	$-0.561026\pi$
$631$	−1.51762e12	−0.381093	−0.190547	+	0.981678i	$0.561026\pi$
$632$	1.85378e12	0.462202
$633$	0	0
$634$	3.13637e12	0.770948
$635$	−1.55894e12	−0.380494
$636$	0	0
$637$	8.24317e11	0.198366
$638$	−1.51217e11	−0.0361334
$639$	0	0
$640$	1.92969e12	0.454649
$641$	−4.22255e12	−0.987901	−0.493951	−	0.869490i	$-0.664448\pi$
$641$	−4.22255e12	−0.987901	−0.493951	+	0.869490i	$0.664448\pi$
$642$	0	0
$643$	−6.88640e12	−1.58870	−0.794352	−	0.607458i	$-0.792189\pi$
$643$	−6.88640e12	−1.58870	−0.794352	+	0.607458i	$0.792189\pi$
$644$	2.09256e12	0.479393
$645$	0	0
$646$	−9.97646e11	−0.225388
$647$	6.62005e12	1.48522	0.742612	−	0.669722i	$-0.233587\pi$
$647$	6.62005e12	1.48522	0.742612	+	0.669722i	$0.233587\pi$
$648$	0	0
$649$	−1.37738e13	−3.04756
$650$	−5.25983e11	−0.115574
$651$	0	0
$652$	−2.75940e12	−0.597998
$653$	1.60282e12	0.344965	0.172482	−	0.985013i	$-0.444821\pi$
$653$	1.60282e12	0.344965	0.172482	+	0.985013i	$0.444821\pi$
$654$	0	0
$655$	1.12511e12	0.238842
$656$	2.49261e12	0.525518
$657$	0	0
$658$	−4.43662e11	−0.0922647
$659$	−7.17733e12	−1.48244	−0.741222	−	0.671260i	$-0.765753\pi$
$659$	−7.17733e12	−1.48244	−0.741222	+	0.671260i	$0.765753\pi$
$660$	0	0
$661$	−8.14522e12	−1.65957	−0.829787	−	0.558081i	$-0.811538\pi$
$661$	−8.14522e12	−1.65957	−0.829787	+	0.558081i	$0.811538\pi$
$662$	−2.65953e12	−0.538201
$663$	0	0
$664$	5.00929e12	1.00005
$665$	8.53174e11	0.169177
$666$	0	0
$667$	−4.04978e11	−0.0792256
$668$	−1.10858e12	−0.215413
$669$	0	0
$670$	5.63586e11	0.108050
$671$	−7.97851e12	−1.51939
$672$	0	0
$673$	−1.03946e12	−0.195316	−0.0976580	−	0.995220i	$-0.531135\pi$
$673$	−1.03946e12	−0.195316	−0.0976580	+	0.995220i	$0.531135\pi$
$674$	−3.17197e12	−0.592052
$675$	0	0
$676$	−4.16638e12	−0.767359
$677$	9.42905e12	1.72512	0.862559	−	0.505957i	$-0.168861\pi$
$677$	9.42905e12	1.72512	0.862559	+	0.505957i	$0.168861\pi$
$678$	0	0
$679$	2.28143e12	0.411902
$680$	−1.02578e12	−0.183977
$681$	0	0
$682$	−5.08188e11	−0.0899487
$683$	−4.76113e12	−0.837176	−0.418588	−	0.908176i	$-0.637475\pi$
$683$	−4.76113e12	−0.837176	−0.418588	+	0.908176i	$0.637475\pi$
$684$	0	0
$685$	−2.65149e12	−0.460132
$686$	1.30340e11	0.0224709
$687$	0	0
$688$	6.42164e11	0.109269
$689$	−8.43171e12	−1.42538
$690$	0	0
$691$	−4.49132e12	−0.749415	−0.374708	−	0.927143i	$-0.622257\pi$
$691$	−4.49132e12	−0.749415	−0.374708	+	0.927143i	$0.622257\pi$
$692$	6.96108e12	1.15398
$693$	0	0
$694$	8.51088e11	0.139270
$695$	−2.18873e12	−0.355844
$696$	0	0
$697$	−3.47138e12	−0.557127
$698$	3.81442e12	0.608246
$699$	0	0
$700$	3.97032e11	0.0625006
$701$	6.11474e12	0.956415	0.478208	−	0.878247i	$-0.341287\pi$
$701$	6.11474e12	0.956415	0.478208	+	0.878247i	$0.341287\pi$
$702$	0	0
$703$	3.60617e12	0.556861
$704$	1.15727e12	0.177566
$705$	0	0
$706$	2.54510e12	0.385553
$707$	1.64820e12	0.248097
$708$	0	0
$709$	−7.22690e12	−1.07410	−0.537049	−	0.843551i	$-0.680461\pi$
$709$	−7.22690e12	−1.07410	−0.537049	+	0.843551i	$0.680461\pi$
$710$	−8.94979e11	−0.132175
$711$	0	0
$712$	4.24621e11	0.0619214
$713$	−1.36099e12	−0.197221
$714$	0	0
$715$	−7.29575e12	−1.04398
$716$	2.65862e12	0.378049
$717$	0	0
$718$	−9.92011e11	−0.139302
$719$	9.43538e12	1.31668	0.658339	−	0.752722i	$-0.271259\pi$
$719$	9.43538e12	1.31668	0.658339	+	0.752722i	$0.271259\pi$
$720$	0	0
$721$	−3.47801e11	−0.0479317
$722$	−5.24483e9	−0.000718313 0
$723$	0	0
$724$	2.17018e12	0.293543
$725$	−7.68386e10	−0.0103290
$726$	0	0
$727$	−1.40060e13	−1.85955	−0.929776	−	0.368127i	$-0.879999\pi$
$727$	−1.40060e13	−1.85955	−0.929776	+	0.368127i	$0.879999\pi$
$728$	−3.02389e12	−0.399002
$729$	0	0
$730$	−9.75074e11	−0.127082
$731$	−8.94321e11	−0.115842
$732$	0	0
$733$	6.77926e12	0.867390	0.433695	−	0.901060i	$-0.357209\pi$
$733$	6.77926e12	0.867390	0.433695	+	0.901060i	$0.357209\pi$
$734$	−1.49873e12	−0.190587
$735$	0	0
$736$	−1.18783e13	−1.49212
$737$	7.81733e12	0.976011
$738$	0	0
$739$	1.26577e13	1.56118	0.780592	−	0.625040i	$-0.214918\pi$
$739$	1.26577e13	1.56118	0.780592	+	0.625040i	$0.214918\pi$
$740$	1.67816e12	0.205727
$741$	0	0
$742$	−1.33321e12	−0.161466
$743$	5.10500e12	0.614534	0.307267	−	0.951623i	$-0.400585\pi$
$743$	5.10500e12	0.614534	0.307267	+	0.951623i	$0.400585\pi$
$744$	0	0
$745$	3.10345e12	0.369098
$746$	3.46535e12	0.409659
$747$	0	0
$748$	−6.43965e12	−0.752152
$749$	2.12589e11	0.0246816
$750$	0	0
$751$	3.15781e12	0.362248	0.181124	−	0.983460i	$-0.442026\pi$
$751$	3.15781e12	0.362248	0.181124	+	0.983460i	$0.442026\pi$
$752$	−2.62555e12	−0.299392
$753$	0	0
$754$	2.64869e11	0.0298442
$755$	7.04489e12	0.789066
$756$	0	0
$757$	−4.05450e12	−0.448751	−0.224375	−	0.974503i	$-0.572034\pi$
$757$	−4.05450e12	−0.448751	−0.224375	+	0.974503i	$0.572034\pi$
$758$	−4.35061e12	−0.478673
$759$	0	0
$760$	−3.12975e12	−0.340290
$761$	8.92263e11	0.0964411	0.0482205	−	0.998837i	$-0.484645\pi$
$761$	8.92263e11	0.0964411	0.0482205	+	0.998837i	$0.484645\pi$
$762$	0	0
$763$	6.69388e12	0.715019
$764$	2.01250e12	0.213705
$765$	0	0
$766$	3.34477e12	0.351024
$767$	2.41259e13	2.51712
$768$	0	0
$769$	−1.04962e13	−1.08234	−0.541171	−	0.840912i	$-0.682019\pi$
$769$	−1.04962e13	−1.08234	−0.541171	+	0.840912i	$0.682019\pi$
$770$	−1.15360e12	−0.118262
$771$	0	0
$772$	9.61536e12	0.974289
$773$	−8.06872e12	−0.812825	−0.406413	−	0.913690i	$-0.633220\pi$
$773$	−8.06872e12	−0.812825	−0.406413	+	0.913690i	$0.633220\pi$
$774$	0	0
$775$	−2.58228e11	−0.0257125
$776$	−8.36912e12	−0.828518
$777$	0	0
$778$	−3.49048e12	−0.341568
$779$	−1.05915e13	−1.03048
$780$	0	0
$781$	−1.24140e13	−1.19394
$782$	3.61263e12	0.345456
$783$	0	0
$784$	7.71340e11	0.0729162
$785$	−6.28146e12	−0.590401
$786$	0	0
$787$	8.88787e12	0.825869	0.412935	−	0.910761i	$-0.364504\pi$
$787$	8.88787e12	0.825869	0.412935	+	0.910761i	$0.364504\pi$
$788$	−8.20743e11	−0.0758297
$789$	0	0
$790$	−1.23873e12	−0.113150
$791$	−8.03837e12	−0.730086
$792$	0	0
$793$	1.39750e13	1.25494
$794$	−4.52618e11	−0.0404147
$795$	0	0
$796$	9.76489e12	0.862102
$797$	−1.52691e13	−1.34045	−0.670227	−	0.742156i	$-0.733803\pi$
$797$	−1.52691e13	−1.34045	−0.670227	+	0.742156i	$0.733803\pi$
$798$	0	0
$799$	3.65651e12	0.317400
$800$	−2.25373e12	−0.194534
$801$	0	0
$802$	7.46034e11	0.0636758
$803$	−1.35250e13	−1.14793
$804$	0	0
$805$	−3.08948e12	−0.259300
$806$	8.90132e11	0.0742928
$807$	0	0
$808$	−6.04618e12	−0.499034
$809$	2.10480e13	1.72760	0.863798	−	0.503838i	$-0.168079\pi$
$809$	2.10480e13	1.72760	0.863798	+	0.503838i	$0.168079\pi$
$810$	0	0
$811$	−2.08302e13	−1.69083	−0.845415	−	0.534110i	$-0.820647\pi$
$811$	−2.08302e13	−1.69083	−0.845415	+	0.534110i	$0.820647\pi$
$812$	−1.99933e11	−0.0161392
$813$	0	0
$814$	−4.87598e12	−0.389271
$815$	4.07400e12	0.323453
$816$	0	0
$817$	−2.72866e12	−0.214265
$818$	4.51248e12	0.352392
$819$	0	0
$820$	−4.92886e12	−0.380701
$821$	−8.68337e12	−0.667028	−0.333514	−	0.942745i	$-0.608234\pi$
$821$	−8.68337e12	−0.667028	−0.333514	+	0.942745i	$0.608234\pi$
$822$	0	0
$823$	−6.42176e12	−0.487927	−0.243963	−	0.969784i	$-0.578448\pi$
$823$	−6.42176e12	−0.487927	−0.243963	+	0.969784i	$0.578448\pi$
$824$	1.27586e12	0.0964120
$825$	0	0
$826$	3.81476e12	0.285139
$827$	−1.29801e13	−0.964948	−0.482474	−	0.875910i	$-0.660262\pi$
$827$	−1.29801e13	−0.964948	−0.482474	+	0.875910i	$0.660262\pi$
$828$	0	0
$829$	1.81614e13	1.33553	0.667767	−	0.744370i	$-0.267250\pi$
$829$	1.81614e13	1.33553	0.667767	+	0.744370i	$0.267250\pi$
$830$	−3.34729e12	−0.244817
$831$	0	0
$832$	−2.02706e12	−0.146660
$833$	−1.07422e12	−0.0773020
$834$	0	0
$835$	1.63671e12	0.116515
$836$	−1.96480e13	−1.39120
$837$	0	0
$838$	−2.29672e12	−0.160883
$839$	−1.74914e12	−0.121869	−0.0609347	−	0.998142i	$-0.519408\pi$
$839$	−1.74914e12	−0.121869	−0.0609347	+	0.998142i	$0.519408\pi$
$840$	0	0
$841$	−1.44685e13	−0.997333
$842$	1.00594e13	0.689711
$843$	0	0
$844$	2.19865e13	1.49147
$845$	6.15128e12	0.415059
$846$	0	0
$847$	−1.03398e13	−0.690296
$848$	−7.88983e12	−0.523946
$849$	0	0
$850$	6.85442e11	0.0450387
$851$	−1.30585e13	−0.853512
$852$	0	0
$853$	2.64586e13	1.71118	0.855591	−	0.517652i	$-0.173194\pi$
$853$	2.64586e13	1.71118	0.855591	+	0.517652i	$0.173194\pi$
$854$	2.20971e12	0.142159
$855$	0	0
$856$	−7.79855e11	−0.0496457
$857$	2.59348e12	0.164237	0.0821183	−	0.996623i	$-0.473831\pi$
$857$	2.59348e12	0.164237	0.0821183	+	0.996623i	$0.473831\pi$
$858$	0	0
$859$	−3.03333e12	−0.190086	−0.0950430	−	0.995473i	$-0.530299\pi$
$859$	−3.03333e12	−0.190086	−0.0950430	+	0.995473i	$0.530299\pi$
$860$	−1.26981e12	−0.0791580
$861$	0	0
$862$	−4.17468e12	−0.257537
$863$	−1.11634e13	−0.685092	−0.342546	−	0.939501i	$-0.611289\pi$
$863$	−1.11634e13	−0.685092	−0.342546	+	0.939501i	$0.611289\pi$
$864$	0	0
$865$	−1.02774e13	−0.624182
$866$	−1.03114e13	−0.622996
$867$	0	0
$868$	−6.71905e11	−0.0401763
$869$	−1.71821e13	−1.02208
$870$	0	0
$871$	−1.36927e13	−0.806132
$872$	−2.45556e13	−1.43822
$873$	0	0
$874$	1.10225e13	0.638967
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	−6.53951e12	−0.373291	−0.186645	−	0.982427i	$-0.559762\pi$
$877$	−6.53951e12	−0.373291	−0.186645	+	0.982427i	$0.559762\pi$
$878$	3.90081e12	0.221529
$879$	0	0
$880$	−6.82688e12	−0.383751
$881$	−2.60830e13	−1.45870	−0.729349	−	0.684142i	$-0.760177\pi$
$881$	−2.60830e13	−1.45870	−0.729349	+	0.684142i	$0.760177\pi$
$882$	0	0
$883$	1.37956e13	0.763688	0.381844	−	0.924227i	$-0.375289\pi$
$883$	1.37956e13	0.763688	0.381844	+	0.924227i	$0.375289\pi$
$884$	1.12796e13	0.621237
$885$	0	0
$886$	−3.88709e12	−0.211920
$887$	−2.26012e12	−0.122596	−0.0612978	−	0.998120i	$-0.519524\pi$
$887$	−2.26012e12	−0.122596	−0.0612978	+	0.998120i	$0.519524\pi$
$888$	0	0
$889$	5.98883e12	0.321576
$890$	−2.83739e11	−0.0151588
$891$	0	0
$892$	2.85754e12	0.151130
$893$	1.11564e13	0.587073
$894$	0	0
$895$	−3.92521e12	−0.204484
$896$	−7.41308e12	−0.384249
$897$	0	0
$898$	−1.49047e13	−0.764855
$899$	1.30035e11	0.00663962
$900$	0	0
$901$	1.09879e13	0.555460
$902$	1.43211e13	0.720353
$903$	0	0
$904$	2.94876e13	1.46853
$905$	−3.20407e12	−0.158775
$906$	0	0
$907$	−3.76970e13	−1.84958	−0.924791	−	0.380475i	$-0.875760\pi$
$907$	−3.76970e13	−1.84958	−0.924791	+	0.380475i	$0.875760\pi$
$908$	−6.56838e11	−0.0320680
$909$	0	0
$910$	2.02062e12	0.0976782
$911$	3.86657e13	1.85991	0.929957	−	0.367669i	$-0.119844\pi$
$911$	3.86657e13	1.85991	0.929957	+	0.367669i	$0.119844\pi$
$912$	0	0
$913$	−4.64293e13	−2.21143
$914$	−1.42495e13	−0.675372
$915$	0	0
$916$	3.00170e13	1.40876
$917$	−4.32224e12	−0.201858
$918$	0	0
$919$	−9.70951e12	−0.449032	−0.224516	−	0.974470i	$-0.572080\pi$
$919$	−9.70951e12	−0.449032	−0.224516	+	0.974470i	$0.572080\pi$
$920$	1.13333e13	0.521569
$921$	0	0
$922$	−2.53334e12	−0.115453
$923$	2.17441e13	0.986127
$924$	0	0
$925$	−2.47765e12	−0.111276
$926$	−1.49028e13	−0.666067
$927$	0	0
$928$	1.13491e12	0.0502336
$929$	−2.04831e13	−0.902248	−0.451124	−	0.892461i	$-0.648977\pi$
$929$	−2.04831e13	−0.902248	−0.451124	+	0.892461i	$0.648977\pi$
$930$	0	0
$931$	−3.27756e12	−0.142980
$932$	3.06706e13	1.33153
$933$	0	0
$934$	−5.13354e11	−0.0220727
$935$	9.50756e12	0.406834
$936$	0	0
$937$	5.81444e12	0.246422	0.123211	−	0.992380i	$-0.460681\pi$
$937$	5.81444e12	0.246422	0.123211	+	0.992380i	$0.460681\pi$
$938$	−2.16507e12	−0.0913186
$939$	0	0
$940$	5.19172e12	0.216888
$941$	−1.99514e13	−0.829508	−0.414754	−	0.909934i	$-0.636132\pi$
$941$	−1.99514e13	−0.829508	−0.414754	+	0.909934i	$0.636132\pi$
$942$	0	0
$943$	3.83535e13	1.57944
$944$	2.25754e13	0.925255
$945$	0	0
$946$	3.68949e12	0.149781
$947$	1.84349e13	0.744843	0.372422	−	0.928064i	$-0.378528\pi$
$947$	1.84349e13	0.744843	0.372422	+	0.928064i	$0.378528\pi$
$948$	0	0
$949$	2.36900e13	0.948130
$950$	2.09135e12	0.0833051
$951$	0	0
$952$	3.94063e12	0.155489
$953$	−4.21018e13	−1.65342	−0.826709	−	0.562629i	$-0.809790\pi$
$953$	−4.21018e13	−1.65342	−0.826709	+	0.562629i	$0.809790\pi$
$954$	0	0
$955$	−2.97127e12	−0.115592
$956$	1.92350e13	0.744788
$957$	0	0
$958$	−9.81115e12	−0.376335
$959$	1.01860e13	0.388882
$960$	0	0
$961$	−2.60026e13	−0.983472
$962$	8.54067e12	0.321517
$963$	0	0
$964$	3.82547e13	1.42672
$965$	−1.41962e13	−0.526986
$966$	0	0
$967$	−2.95695e12	−0.108749	−0.0543744	−	0.998521i	$-0.517316\pi$
$967$	−2.95695e12	−0.108749	−0.0543744	+	0.998521i	$0.517316\pi$
$968$	3.79300e13	1.38849
$969$	0	0
$970$	5.59239e12	0.202827
$971$	−2.93845e13	−1.06080	−0.530398	−	0.847749i	$-0.677957\pi$
$971$	−2.93845e13	−1.06080	−0.530398	+	0.847749i	$0.677957\pi$
$972$	0	0
$973$	8.40822e12	0.300743
$974$	4.29500e10	0.00152914
$975$	0	0
$976$	1.30769e13	0.461295
$977$	4.45820e13	1.56543	0.782716	−	0.622379i	$-0.213834\pi$
$977$	4.45820e13	1.56543	0.782716	+	0.622379i	$0.213834\pi$
$978$	0	0
$979$	−3.93566e12	−0.136929
$980$	−1.52524e12	−0.0528227
$981$	0	0
$982$	−1.24552e13	−0.427413
$983$	−3.77173e12	−0.128840	−0.0644198	−	0.997923i	$-0.520520\pi$
$983$	−3.77173e12	−0.128840	−0.0644198	+	0.997923i	$0.520520\pi$
$984$	0	0
$985$	1.21175e12	0.0410158
$986$	−3.45168e11	−0.0116301
$987$	0	0
$988$	3.44151e13	1.14906
$989$	9.88091e12	0.328408
$990$	0	0
$991$	1.84525e13	0.607749	0.303874	−	0.952712i	$-0.401720\pi$
$991$	1.84525e13	0.607749	0.303874	+	0.952712i	$0.401720\pi$
$992$	3.81403e12	0.125049
$993$	0	0
$994$	3.43815e12	0.111708
$995$	−1.44170e13	−0.466305
$996$	0	0
$997$	3.89046e13	1.24702	0.623509	−	0.781816i	$-0.285706\pi$
$997$	3.89046e13	1.24702	0.623509	+	0.781816i	$0.285706\pi$
$998$	−1.55495e13	−0.496169
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-9.41677 q^{2} -423.324 q^{4} +625.000 q^{5} -2401.00 q^{7} +8807.73 q^{8} -5885.48 q^{10} -81635.7 q^{11} +142991. q^{13} +22609.7 q^{14} +133802. q^{16} -186341. q^{17} -568546. q^{19} -264578. q^{20} +768745. q^{22} +2.05879e6 q^{23} +390625. q^{25} -1.34652e6 q^{26} +1.01640e6 q^{28} -196707. q^{29} -661063. q^{31} -5.76954e6 q^{32} +1.75473e6 q^{34} -1.50062e6 q^{35} -6.34279e6 q^{37} +5.35387e6 q^{38} +5.50483e6 q^{40} +1.86291e7 q^{41} +4.79937e6 q^{43} +3.45584e7 q^{44} -1.93872e7 q^{46} -1.96227e7 q^{47} +5.76480e6 q^{49} -3.67842e6 q^{50} -6.05318e7 q^{52} -5.89666e7 q^{53} -5.10223e7 q^{55} -2.11474e7 q^{56} +1.85234e6 q^{58} +1.68723e8 q^{59} +9.77330e7 q^{61} +6.22507e6 q^{62} -1.41761e7 q^{64} +8.93696e7 q^{65} -9.57587e7 q^{67} +7.88828e7 q^{68} +1.41310e7 q^{70} +1.52066e8 q^{71} +1.65675e8 q^{73} +5.97286e7 q^{74} +2.40679e8 q^{76} +1.96007e8 q^{77} +2.10472e8 q^{79} +8.36261e7 q^{80} -1.75426e8 q^{82} +5.68738e8 q^{83} -1.16463e8 q^{85} -4.51946e7 q^{86} -7.19026e8 q^{88} +4.82100e7 q^{89} -3.43322e8 q^{91} -8.71537e8 q^{92} +1.84782e8 q^{94} -3.55341e8 q^{95} -9.50202e8 q^{97} -5.42858e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) \) 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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