LMFDB - Embedded newform 315.8.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,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, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
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:	$98.4012830275$
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} - 466x^{2} + 520x + 23440 $ x^4 - x^3 - 466x^2 + 520x + 23440
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			1.3
Root			$8.26635$ 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)

$f(q)$	$=$	\(q+8.26635 q^{2} -59.6674 q^{4} -125.000 q^{5} -343.000 q^{7} -1551.33 q^{8} -1033.29 q^{10} +3547.11 q^{11} +4408.70 q^{13} -2835.36 q^{14} -5186.37 q^{16} +16990.2 q^{17} +25713.7 q^{19} +7458.43 q^{20} +29321.6 q^{22} +24466.1 q^{23} +15625.0 q^{25} +36443.9 q^{26} +20465.9 q^{28} -92513.3 q^{29} -93621.2 q^{31} +155697. q^{32} +140447. q^{34} +42875.0 q^{35} -126496. q^{37} +212559. q^{38} +193916. q^{40} -543623. q^{41} +343395. q^{43} -211647. q^{44} +202246. q^{46} -1.23676e6 q^{47} +117649. q^{49} +129162. q^{50} -263056. q^{52} -1.06010e6 q^{53} -443388. q^{55} +532104. q^{56} -764748. q^{58} -2.05102e6 q^{59} +1.65799e6 q^{61} -773906. q^{62} +1.95090e6 q^{64} -551088. q^{65} +2.34375e6 q^{67} -1.01376e6 q^{68} +354420. q^{70} -3.66791e6 q^{71} +628165. q^{73} -1.04566e6 q^{74} -1.53427e6 q^{76} -1.21666e6 q^{77} -417813. q^{79} +648296. q^{80} -4.49378e6 q^{82} +1.45178e6 q^{83} -2.12378e6 q^{85} +2.83862e6 q^{86} -5.50271e6 q^{88} -692594. q^{89} -1.51219e6 q^{91} -1.45983e6 q^{92} -1.02235e7 q^{94} -3.21421e6 q^{95} -1.77040e7 q^{97} +972528. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 421 q^{4} - 500 q^{5} - 1372 q^{7} - 417 q^{8} - 125 q^{10} - 7852 q^{11} + 18532 q^{13} - 343 q^{14} + 47601 q^{16} - 33976 q^{17} + 22188 q^{19} - 52625 q^{20} + 18076 q^{22} + 22736 q^{23}+ \cdots + 117649 q^{98}+O(q^{100}) $ 4 * q + q^2 + 421 * q^4 - 500 * q^5 - 1372 * q^7 - 417 * q^8 - 125 * q^10 - 7852 * q^11 + 18532 * q^13 - 343 * q^14 + 47601 * q^16 - 33976 * q^17 + 22188 * q^19 - 52625 * q^20 + 18076 * q^22 + 22736 * q^23 + 62500 * q^25 + 235686 * q^26 - 144403 * q^28 - 12808 * q^29 - 64908 * q^31 - 111705 * q^32 + 45198 * q^34 + 171500 * q^35 - 315576 * q^37 - 1505204 * q^38 + 52125 * q^40 + 325240 * q^41 + 736184 * q^43 - 2151396 * q^44 - 1289480 * q^46 - 1661296 * q^47 + 470596 * q^49 + 15625 * q^50 + 2618462 * q^52 - 2569036 * q^53 + 981500 * q^55 + 143031 * q^56 - 553222 * q^58 - 3128416 * q^59 + 1392912 * q^61 + 1590240 * q^62 + 2081297 * q^64 - 2316500 * q^65 + 1654088 * q^67 - 8005658 * q^68 + 42875 * q^70 - 2863060 * q^71 + 4095068 * q^73 - 8950482 * q^74 - 8510596 * q^76 + 2693236 * q^77 + 9277704 * q^79 - 5950125 * q^80 - 2529402 * q^82 + 1362736 * q^83 + 4247000 * q^85 + 10602132 * q^86 - 14814892 * q^88 + 11110608 * q^89 - 6356476 * q^91 + 11222472 * q^92 + 101360 * q^94 - 2773500 * q^95 - 8721516 * q^97 + 117649 * 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 $

$2$	8.26635	0.730649	0.365325	−	0.930880i	$-0.380958\pi$
$2$	8.26635	0.730649	0.365325	+	0.930880i	$0.380958\pi$
$3$	0	0
$3$	0	0
$4$	−59.6674	−0.466152
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−1551.33	−1.07124
$9$	0	0
$10$	−1033.29	−0.326756
$11$	3547.11	0.803526	0.401763	−	0.915744i	$-0.368398\pi$
$11$	3547.11	0.803526	0.401763	+	0.915744i	$0.368398\pi$
$12$	0	0
$13$	4408.70	0.556556	0.278278	−	0.960501i	$-0.410236\pi$
$13$	4408.70	0.556556	0.278278	+	0.960501i	$0.410236\pi$
$14$	−2835.36	−0.276159
$15$	0	0
$16$	−5186.37	−0.316551
$17$	16990.2	0.838740	0.419370	−	0.907815i	$-0.362251\pi$
$17$	16990.2	0.838740	0.419370	+	0.907815i	$0.362251\pi$
$18$	0	0
$19$	25713.7	0.860057	0.430028	−	0.902815i	$-0.358504\pi$
$19$	25713.7	0.860057	0.430028	+	0.902815i	$0.358504\pi$
$20$	7458.43	0.208469
$21$	0	0
$22$	29321.6	0.587095
$23$	24466.1	0.419293	0.209647	−	0.977777i	$-0.432769\pi$
$23$	24466.1	0.419293	0.209647	+	0.977777i	$0.432769\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	36443.9	0.406648
$27$	0	0
$28$	20465.9	0.176189
$29$	−92513.3	−0.704387	−0.352193	−	0.935927i	$-0.614564\pi$
$29$	−92513.3	−0.704387	−0.352193	+	0.935927i	$0.614564\pi$
$30$	0	0
$31$	−93621.2	−0.564428	−0.282214	−	0.959351i	$-0.591069\pi$
$31$	−93621.2	−0.564428	−0.282214	+	0.959351i	$0.591069\pi$
$32$	155697.	0.839955
$33$	0	0
$34$	140447.	0.612825
$35$	42875.0	0.169031
$36$	0	0
$37$	−126496.	−0.410554	−0.205277	−	0.978704i	$-0.565810\pi$
$37$	−126496.	−0.410554	−0.205277	+	0.978704i	$0.565810\pi$
$38$	212559.	0.628400
$39$	0	0
$40$	193916.	0.479074
$41$	−543623.	−1.23184	−0.615920	−	0.787808i	$-0.711216\pi$
$41$	−543623.	−1.23184	−0.615920	+	0.787808i	$0.711216\pi$
$42$	0	0
$43$	343395.	0.658650	0.329325	−	0.944217i	$-0.393179\pi$
$43$	343395.	0.658650	0.329325	+	0.944217i	$0.393179\pi$
$44$	−211647.	−0.374565
$45$	0	0
$46$	202246.	0.306356
$47$	−1.23676e6	−1.73758	−0.868788	−	0.495184i	$-0.835101\pi$
$47$	−1.23676e6	−1.73758	−0.868788	+	0.495184i	$0.835101\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	129162.	0.146130
$51$	0	0
$52$	−263056.	−0.259440
$53$	−1.06010e6	−0.978092	−0.489046	−	0.872258i	$-0.662655\pi$
$53$	−1.06010e6	−0.978092	−0.489046	+	0.872258i	$0.662655\pi$
$54$	0	0
$55$	−443388.	−0.359348
$56$	532104.	0.404892
$57$	0	0
$58$	−764748.	−0.514660
$59$	−2.05102e6	−1.30013	−0.650067	−	0.759877i	$-0.725259\pi$
$59$	−2.05102e6	−1.30013	−0.650067	+	0.759877i	$0.725259\pi$
$60$	0	0
$61$	1.65799e6	0.935250	0.467625	−	0.883927i	$-0.345110\pi$
$61$	1.65799e6	0.935250	0.467625	+	0.883927i	$0.345110\pi$
$62$	−773906.	−0.412399
$63$	0	0
$64$	1.95090e6	0.930263
$65$	−551088.	−0.248900
$66$	0	0
$67$	2.34375e6	0.952025	0.476013	−	0.879438i	$-0.342082\pi$
$67$	2.34375e6	0.952025	0.476013	+	0.879438i	$0.342082\pi$
$68$	−1.01376e6	−0.390980
$69$	0	0
$70$	354420.	0.123502
$71$	−3.66791e6	−1.21623	−0.608113	−	0.793851i	$-0.708073\pi$
$71$	−3.66791e6	−1.21623	−0.608113	+	0.793851i	$0.708073\pi$
$72$	0	0
$73$	628165.	0.188992	0.0944961	−	0.995525i	$-0.469876\pi$
$73$	628165.	0.188992	0.0944961	+	0.995525i	$0.469876\pi$
$74$	−1.04566e6	−0.299971
$75$	0	0
$76$	−1.53427e6	−0.400917
$77$	−1.21666e6	−0.303704
$78$	0	0
$79$	−417813.	−0.0953427	−0.0476714	−	0.998863i	$-0.515180\pi$
$79$	−417813.	−0.0953427	−0.0476714	+	0.998863i	$0.515180\pi$
$80$	648296.	0.141566
$81$	0	0
$82$	−4.49378e6	−0.900043
$83$	1.45178e6	0.278695	0.139347	−	0.990244i	$-0.455500\pi$
$83$	1.45178e6	0.278695	0.139347	+	0.990244i	$0.455500\pi$
$84$	0	0
$85$	−2.12378e6	−0.375096
$86$	2.83862e6	0.481242
$87$	0	0
$88$	−5.50271e6	−0.860771
$89$	−692594.	−0.104139	−0.0520696	−	0.998643i	$-0.516582\pi$
$89$	−692594.	−0.104139	−0.0520696	+	0.998643i	$0.516582\pi$
$90$	0	0
$91$	−1.51219e6	−0.210359
$92$	−1.45983e6	−0.195454
$93$	0	0
$94$	−1.02235e7	−1.26956
$95$	−3.21421e6	−0.384629
$96$	0	0
$97$	−1.77040e7	−1.96956	−0.984781	−	0.173799i	$-0.944396\pi$
$97$	−1.77040e7	−1.96956	−0.984781	+	0.173799i	$0.944396\pi$
$98$	972528.	0.104378
$99$	0	0
$100$	−932303.	−0.0932303
$101$	1.52926e7	1.47692	0.738461	−	0.674296i	$-0.235553\pi$
$101$	1.52926e7	1.47692	0.738461	+	0.674296i	$0.235553\pi$
$102$	0	0
$103$	−5.57731e6	−0.502915	−0.251458	−	0.967868i	$-0.580910\pi$
$103$	−5.57731e6	−0.502915	−0.251458	+	0.967868i	$0.580910\pi$
$104$	−6.83933e6	−0.596207
$105$	0	0
$106$	−8.76313e6	−0.714642
$107$	−2.49679e7	−1.97033	−0.985163	−	0.171621i	$-0.945099\pi$
$107$	−2.49679e7	−1.97033	−0.985163	+	0.171621i	$0.945099\pi$
$108$	0	0
$109$	7.44376e6	0.550554	0.275277	−	0.961365i	$-0.411230\pi$
$109$	7.44376e6	0.550554	0.275277	+	0.961365i	$0.411230\pi$
$110$	−3.66520e6	−0.262557
$111$	0	0
$112$	1.77892e6	0.119645
$113$	6.62703e6	0.432060	0.216030	−	0.976387i	$-0.430689\pi$
$113$	6.62703e6	0.432060	0.216030	+	0.976387i	$0.430689\pi$
$114$	0	0
$115$	−3.05827e6	−0.187514
$116$	5.52003e6	0.328351
$117$	0	0
$118$	−1.69545e7	−0.949942
$119$	−5.82764e6	−0.317014
$120$	0	0
$121$	−6.90521e6	−0.354346
$122$	1.37055e7	0.683340
$123$	0	0
$124$	5.58614e6	0.263109
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	3.00350e7	1.30111	0.650555	−	0.759459i	$-0.274536\pi$
$127$	3.00350e7	1.30111	0.650555	+	0.759459i	$0.274536\pi$
$128$	−3.80239e6	−0.160259
$129$	0	0
$130$	−4.55549e6	−0.181858
$131$	1.94728e7	0.756798	0.378399	−	0.925643i	$-0.376475\pi$
$131$	1.94728e7	0.756798	0.378399	+	0.925643i	$0.376475\pi$
$132$	0	0
$133$	−8.81980e6	−0.325071
$134$	1.93742e7	0.695597
$135$	0	0
$136$	−2.63573e7	−0.898494
$137$	−2.05782e7	−0.683731	−0.341865	−	0.939749i	$-0.611059\pi$
$137$	−2.05782e7	−0.683731	−0.341865	+	0.939749i	$0.611059\pi$
$138$	0	0
$139$	2.02312e6	0.0638954	0.0319477	−	0.999490i	$-0.489829\pi$
$139$	2.02312e6	0.0638954	0.0319477	+	0.999490i	$0.489829\pi$
$140$	−2.55824e6	−0.0787940
$141$	0	0
$142$	−3.03202e7	−0.888635
$143$	1.56381e7	0.447207
$144$	0	0
$145$	1.15642e7	0.315011
$146$	5.19263e6	0.138087
$147$	0	0
$148$	7.54768e6	0.191380
$149$	−7.53498e6	−0.186608	−0.0933039	−	0.995638i	$-0.529743\pi$
$149$	−7.53498e6	−0.186608	−0.0933039	+	0.995638i	$0.529743\pi$
$150$	0	0
$151$	−7.66069e7	−1.81071	−0.905354	−	0.424657i	$-0.860395\pi$
$151$	−7.66069e7	−1.81071	−0.905354	+	0.424657i	$0.860395\pi$
$152$	−3.98903e7	−0.921329
$153$	0	0
$154$	−1.00573e7	−0.221901
$155$	1.17027e7	0.252420
$156$	0	0
$157$	3.68625e7	0.760215	0.380107	−	0.924942i	$-0.375887\pi$
$157$	3.68625e7	0.760215	0.380107	+	0.924942i	$0.375887\pi$
$158$	−3.45379e6	−0.0696621
$159$	0	0
$160$	−1.94622e7	−0.375639
$161$	−8.39188e6	−0.158478
$162$	0	0
$163$	6.02804e7	1.09023	0.545116	−	0.838360i	$-0.316485\pi$
$163$	6.02804e7	1.09023	0.545116	+	0.838360i	$0.316485\pi$
$164$	3.24366e7	0.574225
$165$	0	0
$166$	1.20010e7	0.203628
$167$	−6.08740e6	−0.101140	−0.0505701	−	0.998721i	$-0.516104\pi$
$167$	−6.08740e6	−0.101140	−0.0505701	+	0.998721i	$0.516104\pi$
$168$	0	0
$169$	−4.33118e7	−0.690245
$170$	−1.75559e7	−0.274064
$171$	0	0
$172$	−2.04895e7	−0.307031
$173$	−1.03270e8	−1.51639	−0.758196	−	0.652027i	$-0.773919\pi$
$173$	−1.03270e8	−1.51639	−0.758196	+	0.652027i	$0.773919\pi$
$174$	0	0
$175$	−5.35938e6	−0.0755929
$176$	−1.83966e7	−0.254357
$177$	0	0
$178$	−5.72523e6	−0.0760892
$179$	4.48749e7	0.584815	0.292407	−	0.956294i	$-0.405544\pi$
$179$	4.48749e7	0.584815	0.292407	+	0.956294i	$0.405544\pi$
$180$	0	0
$181$	−4.66260e7	−0.584458	−0.292229	−	0.956348i	$-0.594397\pi$
$181$	−4.66260e7	−0.584458	−0.292229	+	0.956348i	$0.594397\pi$
$182$	−1.25003e7	−0.153698
$183$	0	0
$184$	−3.79549e7	−0.449165
$185$	1.58120e7	0.183605
$186$	0	0
$187$	6.02661e7	0.673949
$188$	7.37945e7	0.809974
$189$	0	0
$190$	−2.65698e7	−0.281029
$191$	−4.20911e7	−0.437092	−0.218546	−	0.975827i	$-0.570131\pi$
$191$	−4.20911e7	−0.437092	−0.218546	+	0.975827i	$0.570131\pi$
$192$	0	0
$193$	5.03754e7	0.504391	0.252196	−	0.967676i	$-0.418847\pi$
$193$	5.03754e7	0.504391	0.252196	+	0.967676i	$0.418847\pi$
$194$	−1.46347e8	−1.43906
$195$	0	0
$196$	−7.01981e6	−0.0665931
$197$	−9.44022e7	−0.879732	−0.439866	−	0.898063i	$-0.644974\pi$
$197$	−9.44022e7	−0.879732	−0.439866	+	0.898063i	$0.644974\pi$
$198$	0	0
$199$	−1.69674e8	−1.52626	−0.763130	−	0.646245i	$-0.776338\pi$
$199$	−1.69674e8	−1.52626	−0.763130	+	0.646245i	$0.776338\pi$
$200$	−2.42395e7	−0.214249
$201$	0	0
$202$	1.26414e8	1.07911
$203$	3.17321e7	0.266233
$204$	0	0
$205$	6.79529e7	0.550896
$206$	−4.61040e7	−0.367455
$207$	0	0
$208$	−2.28652e7	−0.176178
$209$	9.12093e7	0.691078
$210$	0	0
$211$	4.76245e7	0.349013	0.174506	−	0.984656i	$-0.444167\pi$
$211$	4.76245e7	0.349013	0.174506	+	0.984656i	$0.444167\pi$
$212$	6.32532e7	0.455939
$213$	0	0
$214$	−2.06393e8	−1.43962
$215$	−4.29244e7	−0.294557
$216$	0	0
$217$	3.21121e7	0.213334
$218$	6.15328e7	0.402262
$219$	0	0
$220$	2.64558e7	0.167511
$221$	7.49048e7	0.466806
$222$	0	0
$223$	−2.01841e8	−1.21883	−0.609415	−	0.792851i	$-0.708596\pi$
$223$	−2.01841e8	−1.21883	−0.609415	+	0.792851i	$0.708596\pi$
$224$	−5.34042e7	−0.317473
$225$	0	0
$226$	5.47813e7	0.315684
$227$	−1.28556e8	−0.729461	−0.364730	−	0.931113i	$-0.618839\pi$
$227$	−1.28556e8	−0.729461	−0.364730	+	0.931113i	$0.618839\pi$
$228$	0	0
$229$	−1.37958e8	−0.759141	−0.379571	−	0.925163i	$-0.623928\pi$
$229$	−1.37958e8	−0.759141	−0.379571	+	0.925163i	$0.623928\pi$
$230$	−2.52807e7	−0.137007
$231$	0	0
$232$	1.43518e8	0.754569
$233$	1.56981e8	0.813019	0.406510	−	0.913646i	$-0.366746\pi$
$233$	1.56981e8	0.813019	0.406510	+	0.913646i	$0.366746\pi$
$234$	0	0
$235$	1.54595e8	0.777068
$236$	1.22379e8	0.606059
$237$	0	0
$238$	−4.81733e7	−0.231626
$239$	−2.34755e8	−1.11230	−0.556149	−	0.831082i	$-0.687722\pi$
$239$	−2.34755e8	−1.11230	−0.556149	+	0.831082i	$0.687722\pi$
$240$	0	0
$241$	3.50057e7	0.161094	0.0805469	−	0.996751i	$-0.474333\pi$
$241$	3.50057e7	0.161094	0.0805469	+	0.996751i	$0.474333\pi$
$242$	−5.70809e7	−0.258903
$243$	0	0
$244$	−9.89280e7	−0.435968
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	1.13364e8	0.478670
$248$	1.45237e8	0.604639
$249$	0	0
$250$	−1.61452e7	−0.0653513
$251$	−6.95867e7	−0.277759	−0.138880	−	0.990309i	$-0.544350\pi$
$251$	−6.95867e7	−0.277759	−0.138880	+	0.990309i	$0.544350\pi$
$252$	0	0
$253$	8.67840e7	0.336913
$254$	2.48280e8	0.950656
$255$	0	0
$256$	−2.81148e8	−1.04736
$257$	−3.25923e8	−1.19770	−0.598852	−	0.800860i	$-0.704376\pi$
$257$	−3.25923e8	−1.19770	−0.598852	+	0.800860i	$0.704376\pi$
$258$	0	0
$259$	4.33881e7	0.155175
$260$	3.28820e7	0.116025
$261$	0	0
$262$	1.60969e8	0.552954
$263$	9.75945e7	0.330811	0.165406	−	0.986226i	$-0.447107\pi$
$263$	9.75945e7	0.330811	0.165406	+	0.986226i	$0.447107\pi$
$264$	0	0
$265$	1.32512e8	0.437416
$266$	−7.29076e7	−0.237513
$267$	0	0
$268$	−1.39845e8	−0.443788
$269$	1.97478e8	0.618566	0.309283	−	0.950970i	$-0.399911\pi$
$269$	1.97478e8	0.618566	0.309283	+	0.950970i	$0.399911\pi$
$270$	0	0
$271$	2.08346e8	0.635904	0.317952	−	0.948107i	$-0.397005\pi$
$271$	2.08346e8	0.635904	0.317952	+	0.948107i	$0.397005\pi$
$272$	−8.81175e7	−0.265504
$273$	0	0
$274$	−1.70107e8	−0.499567
$275$	5.54235e7	0.160705
$276$	0	0
$277$	−5.69876e8	−1.61102	−0.805511	−	0.592581i	$-0.798109\pi$
$277$	−5.69876e8	−1.61102	−0.805511	+	0.592581i	$0.798109\pi$
$278$	1.67238e7	0.0466851
$279$	0	0
$280$	−6.65131e7	−0.181073
$281$	−3.82839e8	−1.02930	−0.514652	−	0.857399i	$-0.672079\pi$
$281$	−3.82839e8	−1.02930	−0.514652	+	0.857399i	$0.672079\pi$
$282$	0	0
$283$	−2.15239e8	−0.564505	−0.282252	−	0.959340i	$-0.591082\pi$
$283$	−2.15239e8	−0.564505	−0.282252	+	0.959340i	$0.591082\pi$
$284$	2.18855e8	0.566946
$285$	0	0
$286$	1.29270e8	0.326752
$287$	1.86463e8	0.465592
$288$	0	0
$289$	−1.21671e8	−0.296515
$290$	9.55935e7	0.230163
$291$	0	0
$292$	−3.74810e7	−0.0880990
$293$	2.04076e8	0.473974	0.236987	−	0.971513i	$-0.423840\pi$
$293$	2.04076e8	0.473974	0.236987	+	0.971513i	$0.423840\pi$
$294$	0	0
$295$	2.56378e8	0.581437
$296$	1.96236e8	0.439803
$297$	0	0
$298$	−6.22868e7	−0.136345
$299$	1.07864e8	0.233360
$300$	0	0
$301$	−1.17785e8	−0.248946
$302$	−6.33260e8	−1.32299
$303$	0	0
$304$	−1.33361e8	−0.272252
$305$	−2.07249e8	−0.418257
$306$	0	0
$307$	−3.52627e6	−0.00695555	−0.00347778	−	0.999994i	$-0.501107\pi$
$307$	−3.52627e6	−0.00695555	−0.00347778	+	0.999994i	$0.501107\pi$
$308$	7.25948e7	0.141572
$309$	0	0
$310$	9.67383e7	0.184430
$311$	−1.66245e8	−0.313391	−0.156696	−	0.987647i	$-0.550084\pi$
$311$	−1.66245e8	−0.313391	−0.156696	+	0.987647i	$0.550084\pi$
$312$	0	0
$313$	8.83677e8	1.62888	0.814439	−	0.580249i	$-0.197045\pi$
$313$	8.83677e8	1.62888	0.814439	+	0.580249i	$0.197045\pi$
$314$	3.04719e8	0.555450
$315$	0	0
$316$	2.49299e7	0.0444442
$317$	4.23678e8	0.747014	0.373507	−	0.927627i	$-0.378155\pi$
$317$	4.23678e8	0.747014	0.373507	+	0.927627i	$0.378155\pi$
$318$	0	0
$319$	−3.28155e8	−0.565993
$320$	−2.43863e8	−0.416026
$321$	0	0
$322$	−6.93703e7	−0.115792
$323$	4.36881e8	0.721364
$324$	0	0
$325$	6.88860e7	0.111311
$326$	4.98299e8	0.796578
$327$	0	0
$328$	8.43336e8	1.31960
$329$	4.24210e8	0.656742
$330$	0	0
$331$	1.28843e7	0.0195282	0.00976410	−	0.999952i	$-0.496892\pi$
$331$	1.28843e7	0.0195282	0.00976410	+	0.999952i	$0.496892\pi$
$332$	−8.66242e7	−0.129914
$333$	0	0
$334$	−5.03206e7	−0.0738981
$335$	−2.92968e8	−0.425759
$336$	0	0
$337$	5.36738e7	0.0763937	0.0381969	−	0.999270i	$-0.487839\pi$
$337$	5.36738e7	0.0763937	0.0381969	+	0.999270i	$0.487839\pi$
$338$	−3.58031e8	−0.504327
$339$	0	0
$340$	1.26720e8	0.174852
$341$	−3.32084e8	−0.453532
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	−5.32717e8	−0.705574
$345$	0	0
$346$	−8.53663e8	−1.10795
$347$	−6.78354e8	−0.871572	−0.435786	−	0.900050i	$-0.643530\pi$
$347$	−6.78354e8	−0.871572	−0.435786	+	0.900050i	$0.643530\pi$
$348$	0	0
$349$	1.38576e9	1.74501	0.872506	−	0.488603i	$-0.162494\pi$
$349$	1.38576e9	1.74501	0.872506	+	0.488603i	$0.162494\pi$
$350$	−4.43025e7	−0.0552319
$351$	0	0
$352$	5.52275e8	0.674925
$353$	1.07056e9	1.29538	0.647692	−	0.761902i	$-0.275734\pi$
$353$	1.07056e9	1.29538	0.647692	+	0.761902i	$0.275734\pi$
$354$	0	0
$355$	4.58489e8	0.543913
$356$	4.13253e7	0.0485446
$357$	0	0
$358$	3.70952e8	0.427294
$359$	8.78736e8	1.00237	0.501185	−	0.865340i	$-0.332898\pi$
$359$	8.78736e8	1.00237	0.501185	+	0.865340i	$0.332898\pi$
$360$	0	0
$361$	−2.32677e8	−0.260302
$362$	−3.85427e8	−0.427034
$363$	0	0
$364$	9.02282e7	0.0980590
$365$	−7.85206e7	−0.0845199
$366$	0	0
$367$	2.19145e8	0.231420	0.115710	−	0.993283i	$-0.463086\pi$
$367$	2.19145e8	0.231420	0.115710	+	0.993283i	$0.463086\pi$
$368$	−1.26890e8	−0.132728
$369$	0	0
$370$	1.30707e8	0.134151
$371$	3.63613e8	0.369684
$372$	0	0
$373$	−3.22635e8	−0.321907	−0.160953	−	0.986962i	$-0.551457\pi$
$373$	−3.22635e8	−0.321907	−0.160953	+	0.986962i	$0.551457\pi$
$374$	4.98181e8	0.492421
$375$	0	0
$376$	1.91862e9	1.86137
$377$	−4.07864e8	−0.392031
$378$	0	0
$379$	1.60097e9	1.51059	0.755295	−	0.655385i	$-0.227494\pi$
$379$	1.60097e9	1.51059	0.755295	+	0.655385i	$0.227494\pi$
$380$	1.91784e8	0.179295
$381$	0	0
$382$	−3.47940e8	−0.319361
$383$	1.99401e8	0.181356	0.0906781	−	0.995880i	$-0.471097\pi$
$383$	1.99401e8	0.181356	0.0906781	+	0.995880i	$0.471097\pi$
$384$	0	0
$385$	1.52082e8	0.135821
$386$	4.16420e8	0.368533
$387$	0	0
$388$	1.05635e9	0.918115
$389$	−1.06573e9	−0.917956	−0.458978	−	0.888448i	$-0.651784\pi$
$389$	−1.06573e9	−0.917956	−0.458978	+	0.888448i	$0.651784\pi$
$390$	0	0
$391$	4.15685e8	0.351678
$392$	−1.82512e8	−0.153035
$393$	0	0
$394$	−7.80362e8	−0.642776
$395$	5.22267e7	0.0426386
$396$	0	0
$397$	1.50813e9	1.20969	0.604844	−	0.796344i	$-0.293236\pi$
$397$	1.50813e9	1.20969	0.604844	+	0.796344i	$0.293236\pi$
$398$	−1.40258e9	−1.11516
$399$	0	0
$400$	−8.10370e7	−0.0633102
$401$	−1.18481e8	−0.0917579	−0.0458790	−	0.998947i	$-0.514609\pi$
$401$	−1.18481e8	−0.0917579	−0.0458790	+	0.998947i	$0.514609\pi$
$402$	0	0
$403$	−4.12748e8	−0.314136
$404$	−9.12472e8	−0.688470
$405$	0	0
$406$	2.62308e8	0.194523
$407$	−4.48694e8	−0.329891
$408$	0	0
$409$	−4.10749e8	−0.296856	−0.148428	−	0.988923i	$-0.547421\pi$
$409$	−4.10749e8	−0.296856	−0.148428	+	0.988923i	$0.547421\pi$
$410$	5.61723e8	0.402512
$411$	0	0
$412$	3.32784e8	0.234435
$413$	7.03500e8	0.491404
$414$	0	0
$415$	−1.81473e8	−0.124636
$416$	6.86423e8	0.467482
$417$	0	0
$418$	7.53968e8	0.504935
$419$	9.38739e8	0.623442	0.311721	−	0.950174i	$-0.399095\pi$
$419$	9.38739e8	0.623442	0.311721	+	0.950174i	$0.399095\pi$
$420$	0	0
$421$	−2.00748e9	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$421$	−2.00748e9	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$422$	3.93681e8	0.255006
$423$	0	0
$424$	1.64455e9	1.04777
$425$	2.65472e8	0.167748
$426$	0	0
$427$	−5.68691e8	−0.353491
$428$	1.48977e9	0.918471
$429$	0	0
$430$	−3.54828e8	−0.215218
$431$	5.32959e8	0.320644	0.160322	−	0.987065i	$-0.448747\pi$
$431$	5.32959e8	0.320644	0.160322	+	0.987065i	$0.448747\pi$
$432$	0	0
$433$	2.69449e9	1.59503	0.797516	−	0.603297i	$-0.206147\pi$
$433$	2.69449e9	1.59503	0.797516	+	0.603297i	$0.206147\pi$
$434$	2.65450e8	0.155872
$435$	0	0
$436$	−4.44150e8	−0.256642
$437$	6.29115e8	0.360616
$438$	0	0
$439$	−2.57558e9	−1.45295	−0.726473	−	0.687195i	$-0.758842\pi$
$439$	−2.57558e9	−1.45295	−0.726473	+	0.687195i	$0.758842\pi$
$440$	6.87839e8	0.384949
$441$	0	0
$442$	6.19190e8	0.341072
$443$	−5.38925e8	−0.294521	−0.147260	−	0.989098i	$-0.547045\pi$
$443$	−5.38925e8	−0.294521	−0.147260	+	0.989098i	$0.547045\pi$
$444$	0	0
$445$	8.65743e7	0.0465724
$446$	−1.66849e9	−0.890538
$447$	0	0
$448$	−6.69160e8	−0.351606
$449$	2.08562e9	1.08736	0.543681	−	0.839292i	$-0.317030\pi$
$449$	2.08562e9	1.08736	0.543681	+	0.839292i	$0.317030\pi$
$450$	0	0
$451$	−1.92829e9	−0.989816
$452$	−3.95418e8	−0.201406
$453$	0	0
$454$	−1.06269e9	−0.532980
$455$	1.89023e8	0.0940752
$456$	0	0
$457$	−2.76612e9	−1.35570	−0.677852	−	0.735199i	$-0.737089\pi$
$457$	−2.76612e9	−1.35570	−0.677852	+	0.735199i	$0.737089\pi$
$458$	−1.14041e9	−0.554666
$459$	0	0
$460$	1.82479e8	0.0874098
$461$	2.74411e9	1.30451	0.652256	−	0.757998i	$-0.273823\pi$
$461$	2.74411e9	1.30451	0.652256	+	0.757998i	$0.273823\pi$
$462$	0	0
$463$	5.74107e8	0.268819	0.134409	−	0.990926i	$-0.457086\pi$
$463$	5.74107e8	0.268819	0.134409	+	0.990926i	$0.457086\pi$
$464$	4.79808e8	0.222974
$465$	0	0
$466$	1.29766e9	0.594032
$467$	−1.43212e9	−0.650683	−0.325341	−	0.945597i	$-0.605479\pi$
$467$	−1.43212e9	−0.650683	−0.325341	+	0.945597i	$0.605479\pi$
$468$	0	0
$469$	−8.03905e8	−0.359832
$470$	1.27794e9	0.567764
$471$	0	0
$472$	3.18180e9	1.39276
$473$	1.21806e9	0.529242
$474$	0	0
$475$	4.01777e8	0.172011
$476$	3.47720e8	0.147777
$477$	0	0
$478$	−1.94056e9	−0.812700
$479$	−7.93038e8	−0.329701	−0.164850	−	0.986319i	$-0.552714\pi$
$479$	−7.93038e8	−0.329701	−0.164850	+	0.986319i	$0.552714\pi$
$480$	0	0
$481$	−5.57683e8	−0.228496
$482$	2.89369e8	0.117703
$483$	0	0
$484$	4.12016e8	0.165179
$485$	2.21300e9	0.880815
$486$	0	0
$487$	−3.21542e9	−1.26150	−0.630749	−	0.775987i	$-0.717252\pi$
$487$	−3.21542e9	−1.26150	−0.630749	+	0.775987i	$0.717252\pi$
$488$	−2.57208e9	−1.00188
$489$	0	0
$490$	−1.21566e8	−0.0466795
$491$	2.15616e9	0.822045	0.411023	−	0.911625i	$-0.365172\pi$
$491$	2.15616e9	0.822045	0.411023	+	0.911625i	$0.365172\pi$
$492$	0	0
$493$	−1.57182e9	−0.590798
$494$	9.37108e8	0.349740
$495$	0	0
$496$	4.85554e8	0.178670
$497$	1.25809e9	0.459690
$498$	0	0
$499$	2.40680e9	0.867137	0.433569	−	0.901121i	$-0.357254\pi$
$499$	2.40680e9	0.867137	0.433569	+	0.901121i	$0.357254\pi$
$500$	1.16538e8	0.0416939
$501$	0	0
$502$	−5.75228e8	−0.202944
$503$	−1.32577e9	−0.464495	−0.232248	−	0.972657i	$-0.574608\pi$
$503$	−1.32577e9	−0.464495	−0.232248	+	0.972657i	$0.574608\pi$
$504$	0	0
$505$	−1.91158e9	−0.660500
$506$	7.17387e8	0.246165
$507$	0	0
$508$	−1.79211e9	−0.606515
$509$	−5.72914e9	−1.92565	−0.962824	−	0.270130i	$-0.912933\pi$
$509$	−5.72914e9	−1.92565	−0.962824	+	0.270130i	$0.912933\pi$
$510$	0	0
$511$	−2.15461e8	−0.0714323
$512$	−1.83736e9	−0.604991
$513$	0	0
$514$	−2.69420e9	−0.875102
$515$	6.97164e8	0.224911
$516$	0	0
$517$	−4.38693e9	−1.39619
$518$	3.58661e8	0.113378
$519$	0	0
$520$	8.54917e8	0.266632
$521$	3.58877e9	1.11177	0.555884	−	0.831260i	$-0.312380\pi$
$521$	3.58877e9	1.11177	0.555884	+	0.831260i	$0.312380\pi$
$522$	0	0
$523$	1.61650e9	0.494106	0.247053	−	0.969002i	$-0.420538\pi$
$523$	1.61650e9	0.494106	0.247053	+	0.969002i	$0.420538\pi$
$524$	−1.16189e9	−0.352783
$525$	0	0
$526$	8.06751e8	0.241707
$527$	−1.59064e9	−0.473408
$528$	0	0
$529$	−2.80623e9	−0.824193
$530$	1.09539e9	0.319598
$531$	0	0
$532$	5.26255e8	0.151532
$533$	−2.39667e9	−0.685589
$534$	0	0
$535$	3.12098e9	0.881157
$536$	−3.63591e9	−1.01985
$537$	0	0
$538$	1.63242e9	0.451955
$539$	4.17314e8	0.114789
$540$	0	0
$541$	−9.99263e8	−0.271325	−0.135662	−	0.990755i	$-0.543316\pi$
$541$	−9.99263e8	−0.271325	−0.135662	+	0.990755i	$0.543316\pi$
$542$	1.72226e9	0.464623
$543$	0	0
$544$	2.64533e9	0.704504
$545$	−9.30471e8	−0.246215
$546$	0	0
$547$	−5.11148e9	−1.33534	−0.667668	−	0.744459i	$-0.732708\pi$
$547$	−5.11148e9	−1.33534	−0.667668	+	0.744459i	$0.732708\pi$
$548$	1.22785e9	0.318722
$549$	0	0
$550$	4.58150e8	0.117419
$551$	−2.37886e9	−0.605813
$552$	0	0
$553$	1.43310e8	0.0360362
$554$	−4.71080e9	−1.17709
$555$	0	0
$556$	−1.20714e8	−0.0297849
$557$	−6.05188e9	−1.48387	−0.741937	−	0.670470i	$-0.766093\pi$
$557$	−6.05188e9	−1.48387	−0.741937	+	0.670470i	$0.766093\pi$
$558$	0	0
$559$	1.51393e9	0.366576
$560$	−2.22366e8	−0.0535069
$561$	0	0
$562$	−3.16468e9	−0.752060
$563$	−5.25872e9	−1.24194	−0.620970	−	0.783834i	$-0.713261\pi$
$563$	−5.25872e9	−1.24194	−0.620970	+	0.783834i	$0.713261\pi$
$564$	0	0
$565$	−8.28378e8	−0.193223
$566$	−1.77924e9	−0.412455
$567$	0	0
$568$	5.69012e9	1.30287
$569$	−7.61342e9	−1.73255	−0.866277	−	0.499564i	$-0.833493\pi$
$569$	−7.61342e9	−1.73255	−0.866277	+	0.499564i	$0.833493\pi$
$570$	0	0
$571$	−6.80860e9	−1.53049	−0.765247	−	0.643737i	$-0.777383\pi$
$571$	−6.80860e9	−1.53049	−0.765247	+	0.643737i	$0.777383\pi$
$572$	−9.33088e8	−0.208467
$573$	0	0
$574$	1.54137e9	0.340184
$575$	3.82283e8	0.0838586
$576$	0	0
$577$	−5.69226e9	−1.23359	−0.616793	−	0.787125i	$-0.711569\pi$
$577$	−5.69226e9	−1.23359	−0.616793	+	0.787125i	$0.711569\pi$
$578$	−1.00578e9	−0.216648
$579$	0	0
$580$	−6.90004e8	−0.146843
$581$	−4.97962e8	−0.105337
$582$	0	0
$583$	−3.76027e9	−0.785922
$584$	−9.74488e8	−0.202456
$585$	0	0
$586$	1.68696e9	0.346309
$587$	4.89079e9	0.998034	0.499017	−	0.866592i	$-0.333694\pi$
$587$	4.89079e9	0.998034	0.499017	+	0.866592i	$0.333694\pi$
$588$	0	0
$589$	−2.40735e9	−0.485440
$590$	2.11931e9	0.424827
$591$	0	0
$592$	6.56054e8	0.129961
$593$	7.77823e9	1.53175	0.765877	−	0.642987i	$-0.222305\pi$
$593$	7.77823e9	1.53175	0.765877	+	0.642987i	$0.222305\pi$
$594$	0	0
$595$	7.28455e8	0.141773
$596$	4.49593e8	0.0869876
$597$	0	0
$598$	8.91641e8	0.170505
$599$	−5.27135e9	−1.00214	−0.501070	−	0.865407i	$-0.667060\pi$
$599$	−5.27135e9	−1.00214	−0.501070	+	0.865407i	$0.667060\pi$
$600$	0	0
$601$	−6.15092e9	−1.15579	−0.577896	−	0.816111i	$-0.696126\pi$
$601$	−6.15092e9	−1.15579	−0.577896	+	0.816111i	$0.696126\pi$
$602$	−9.73648e8	−0.181892
$603$	0	0
$604$	4.57094e9	0.844065
$605$	8.63151e8	0.158469
$606$	0	0
$607$	−1.45455e8	−0.0263978	−0.0131989	−	0.999913i	$-0.504201\pi$
$607$	−1.45455e8	−0.0263978	−0.0131989	+	0.999913i	$0.504201\pi$
$608$	4.00355e9	0.722409
$609$	0	0
$610$	−1.71319e9	−0.305599
$611$	−5.45252e9	−0.967060
$612$	0	0
$613$	−5.31052e9	−0.931163	−0.465582	−	0.885005i	$-0.654155\pi$
$613$	−5.31052e9	−0.931163	−0.465582	+	0.885005i	$0.654155\pi$
$614$	−2.91494e7	−0.00508207
$615$	0	0
$616$	1.88743e9	0.325341
$617$	9.43304e8	0.161679	0.0808395	−	0.996727i	$-0.474240\pi$
$617$	9.43304e8	0.161679	0.0808395	+	0.996727i	$0.474240\pi$
$618$	0	0
$619$	−1.99924e9	−0.338804	−0.169402	−	0.985547i	$-0.554184\pi$
$619$	−1.99924e9	−0.338804	−0.169402	+	0.985547i	$0.554184\pi$
$620$	−6.98267e8	−0.117666
$621$	0	0
$622$	−1.37424e9	−0.228979
$623$	2.37560e8	0.0393609
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	7.30479e9	1.19014
$627$	0	0
$628$	−2.19949e9	−0.354375
$629$	−2.14919e9	−0.344348
$630$	0	0
$631$	5.14161e9	0.814697	0.407349	−	0.913273i	$-0.366453\pi$
$631$	5.14161e9	0.814697	0.407349	+	0.913273i	$0.366453\pi$
$632$	6.48165e8	0.102135
$633$	0	0
$634$	3.50227e9	0.545805
$635$	−3.75437e9	−0.581874
$636$	0	0
$637$	5.18680e8	0.0795081
$638$	−2.71264e9	−0.413542
$639$	0	0
$640$	4.75299e8	0.0716699
$641$	8.05402e9	1.20784	0.603920	−	0.797045i	$-0.293604\pi$
$641$	8.05402e9	1.20784	0.603920	+	0.797045i	$0.293604\pi$
$642$	0	0
$643$	1.21205e10	1.79797	0.898983	−	0.437984i	$-0.144307\pi$
$643$	1.21205e10	1.79797	0.898983	+	0.437984i	$0.144307\pi$
$644$	5.00722e8	0.0738748
$645$	0	0
$646$	3.61142e9	0.527064
$647$	−7.75077e9	−1.12507	−0.562536	−	0.826773i	$-0.690174\pi$
$647$	−7.75077e9	−1.12507	−0.562536	+	0.826773i	$0.690174\pi$
$648$	0	0
$649$	−7.27519e9	−1.04469
$650$	5.69436e8	0.0813295
$651$	0	0
$652$	−3.59677e9	−0.508214
$653$	−7.09569e9	−0.997237	−0.498619	−	0.866822i	$-0.666159\pi$
$653$	−7.09569e9	−0.997237	−0.498619	+	0.866822i	$0.666159\pi$
$654$	0	0
$655$	−2.43411e9	−0.338450
$656$	2.81943e9	0.389940
$657$	0	0
$658$	3.50667e9	0.479848
$659$	−4.73697e9	−0.644766	−0.322383	−	0.946609i	$-0.604484\pi$
$659$	−4.73697e9	−0.644766	−0.322383	+	0.946609i	$0.604484\pi$
$660$	0	0
$661$	7.36157e9	0.991438	0.495719	−	0.868483i	$-0.334905\pi$
$661$	7.36157e9	0.991438	0.495719	+	0.868483i	$0.334905\pi$
$662$	1.06506e8	0.0142683
$663$	0	0
$664$	−2.25219e9	−0.298550
$665$	1.10248e9	0.145376
$666$	0	0
$667$	−2.26344e9	−0.295345
$668$	3.63219e8	0.0471467
$669$	0	0
$670$	−2.42178e9	−0.311080
$671$	5.88107e9	0.751497
$672$	0	0
$673$	−9.84049e9	−1.24441	−0.622205	−	0.782854i	$-0.713763\pi$
$673$	−9.84049e9	−1.24441	−0.622205	+	0.782854i	$0.713763\pi$
$674$	4.43687e8	0.0558170
$675$	0	0
$676$	2.58431e9	0.321759
$677$	2.42035e9	0.299790	0.149895	−	0.988702i	$-0.452106\pi$
$677$	2.42035e9	0.299790	0.149895	+	0.988702i	$0.452106\pi$
$678$	0	0
$679$	6.07246e9	0.744425
$680$	3.29467e9	0.401819
$681$	0	0
$682$	−2.74513e9	−0.331373
$683$	−1.33955e10	−1.60874	−0.804368	−	0.594131i	$-0.797496\pi$
$683$	−1.33955e10	−1.60874	−0.804368	+	0.594131i	$0.797496\pi$
$684$	0	0
$685$	2.57227e9	0.305774
$686$	−3.33577e8	−0.0394514
$687$	0	0
$688$	−1.78097e9	−0.208496
$689$	−4.67365e9	−0.544363
$690$	0	0
$691$	9.33422e9	1.07623	0.538115	−	0.842872i	$-0.319137\pi$
$691$	9.33422e9	1.07623	0.538115	+	0.842872i	$0.319137\pi$
$692$	6.16183e9	0.706869
$693$	0	0
$694$	−5.60751e9	−0.636814
$695$	−2.52890e8	−0.0285749
$696$	0	0
$697$	−9.23627e9	−1.03319
$698$	1.14552e10	1.27499
$699$	0	0
$700$	3.19780e8	0.0352378
$701$	4.41877e8	0.0484493	0.0242247	−	0.999707i	$-0.492288\pi$
$701$	4.41877e8	0.0484493	0.0242247	+	0.999707i	$0.492288\pi$
$702$	0	0
$703$	−3.25268e9	−0.353100
$704$	6.92006e9	0.747491
$705$	0	0
$706$	8.84960e9	0.946471
$707$	−5.24537e9	−0.558224
$708$	0	0
$709$	1.18108e10	1.24456	0.622280	−	0.782795i	$-0.286206\pi$
$709$	1.18108e10	1.24456	0.622280	+	0.782795i	$0.286206\pi$
$710$	3.79003e9	0.397409
$711$	0	0
$712$	1.07444e9	0.111558
$713$	−2.29055e9	−0.236661
$714$	0	0
$715$	−1.95477e9	−0.199997
$716$	−2.67757e9	−0.272612
$717$	0	0
$718$	7.26394e9	0.732381
$719$	−2.21789e8	−0.0222530	−0.0111265	−	0.999938i	$-0.503542\pi$
$719$	−2.21789e8	−0.0222530	−0.0111265	+	0.999938i	$0.503542\pi$
$720$	0	0
$721$	1.91302e9	0.190084
$722$	−1.92339e9	−0.190190
$723$	0	0
$724$	2.78205e9	0.272446
$725$	−1.44552e9	−0.140877
$726$	0	0
$727$	−1.48455e10	−1.43293	−0.716463	−	0.697625i	$-0.754240\pi$
$727$	−1.48455e10	−1.43293	−0.716463	+	0.697625i	$0.754240\pi$
$728$	2.34589e9	0.225345
$729$	0	0
$730$	−6.49079e8	−0.0617544
$731$	5.83435e9	0.552436
$732$	0	0
$733$	5.09111e9	0.477473	0.238736	−	0.971084i	$-0.423267\pi$
$733$	5.09111e9	0.477473	0.238736	+	0.971084i	$0.423267\pi$
$734$	1.81153e9	0.169087
$735$	0	0
$736$	3.80931e9	0.352187
$737$	8.31351e9	0.764977
$738$	0	0
$739$	1.21561e10	1.10800	0.553998	−	0.832518i	$-0.313102\pi$
$739$	1.21561e10	1.10800	0.553998	+	0.832518i	$0.313102\pi$
$740$	−9.43460e8	−0.0855879
$741$	0	0
$742$	3.00575e9	0.270109
$743$	−4.05104e9	−0.362331	−0.181166	−	0.983453i	$-0.557987\pi$
$743$	−4.05104e9	−0.362331	−0.181166	+	0.983453i	$0.557987\pi$
$744$	0	0
$745$	9.41872e8	0.0834536
$746$	−2.66701e9	−0.235201
$747$	0	0
$748$	−3.59592e9	−0.314163
$749$	8.56398e9	0.744713
$750$	0	0
$751$	4.19624e9	0.361510	0.180755	−	0.983528i	$-0.442146\pi$
$751$	4.19624e9	0.361510	0.180755	+	0.983528i	$0.442146\pi$
$752$	6.41431e9	0.550032
$753$	0	0
$754$	−3.37155e9	−0.286437
$755$	9.57586e9	0.809774
$756$	0	0
$757$	−7.95828e9	−0.666782	−0.333391	−	0.942789i	$-0.608193\pi$
$757$	−7.95828e9	−0.666782	−0.333391	+	0.942789i	$0.608193\pi$
$758$	1.32342e10	1.10371
$759$	0	0
$760$	4.98629e9	0.412031
$761$	2.19417e10	1.80478	0.902388	−	0.430923i	$-0.141812\pi$
$761$	2.19417e10	1.80478	0.902388	+	0.430923i	$0.141812\pi$
$762$	0	0
$763$	−2.55321e9	−0.208090
$764$	2.51146e9	0.203751
$765$	0	0
$766$	1.64832e9	0.132508
$767$	−9.04234e9	−0.723598
$768$	0	0
$769$	−3.93748e9	−0.312231	−0.156116	−	0.987739i	$-0.549897\pi$
$769$	−3.93748e9	−0.312231	−0.156116	+	0.987739i	$0.549897\pi$
$770$	1.25716e9	0.0992372
$771$	0	0
$772$	−3.00577e9	−0.235123
$773$	9.54261e9	0.743086	0.371543	−	0.928416i	$-0.378829\pi$
$773$	9.54261e9	0.743086	0.371543	+	0.928416i	$0.378829\pi$
$774$	0	0
$775$	−1.46283e9	−0.112886
$776$	2.74646e10	2.10988
$777$	0	0
$778$	−8.80967e9	−0.670704
$779$	−1.39786e10	−1.05945
$780$	0	0
$781$	−1.30105e10	−0.977269
$782$	3.43620e9	0.256953
$783$	0	0
$784$	−6.10171e8	−0.0452216
$785$	−4.60782e9	−0.339978
$786$	0	0
$787$	2.02204e10	1.47870	0.739349	−	0.673323i	$-0.235134\pi$
$787$	2.02204e10	1.47870	0.739349	+	0.673323i	$0.235134\pi$
$788$	5.63274e9	0.410089
$789$	0	0
$790$	4.31724e8	0.0311538
$791$	−2.27307e9	−0.163303
$792$	0	0
$793$	7.30959e9	0.520519
$794$	1.24668e10	0.883857
$795$	0	0
$796$	1.01240e10	0.711468
$797$	1.55713e9	0.108948	0.0544741	−	0.998515i	$-0.482652\pi$
$797$	1.55713e9	0.108948	0.0544741	+	0.998515i	$0.482652\pi$
$798$	0	0
$799$	−2.10129e10	−1.45738
$800$	2.43277e9	0.167991
$801$	0	0
$802$	−9.79406e8	−0.0670428
$803$	2.22817e9	0.151860
$804$	0	0
$805$	1.04899e9	0.0708735
$806$	−3.41192e9	−0.229523
$807$	0	0
$808$	−2.37238e10	−1.58214
$809$	−1.38275e10	−0.918170	−0.459085	−	0.888392i	$-0.651823\pi$
$809$	−1.38275e10	−0.918170	−0.459085	+	0.888392i	$0.651823\pi$
$810$	0	0
$811$	2.98876e10	1.96752	0.983758	−	0.179501i	$-0.0574482\pi$
$811$	2.98876e10	1.96752	0.983758	+	0.179501i	$0.0574482\pi$
$812$	−1.89337e9	−0.124105
$813$	0	0
$814$	−3.70906e9	−0.241034
$815$	−7.53505e9	−0.487567
$816$	0	0
$817$	8.82996e9	0.566476
$818$	−3.39540e9	−0.216897
$819$	0	0
$820$	−4.05457e9	−0.256801
$821$	1.84955e10	1.16645	0.583224	−	0.812311i	$-0.301791\pi$
$821$	1.84955e10	1.16645	0.583224	+	0.812311i	$0.301791\pi$
$822$	0	0
$823$	−2.85886e7	−0.00178769	−0.000893847	−	1.00000i	$-0.500285\pi$
$823$	−2.85886e7	−0.00178769	−0.000893847		1.00000i	$0.500285\pi$
$824$	8.65223e9	0.538744
$825$	0	0
$826$	5.81538e9	0.359044
$827$	−1.63130e10	−1.00291	−0.501457	−	0.865182i	$-0.667203\pi$
$827$	−1.63130e10	−1.00291	−0.501457	+	0.865182i	$0.667203\pi$
$828$	0	0
$829$	1.94409e9	0.118516	0.0592579	−	0.998243i	$-0.481127\pi$
$829$	1.94409e9	0.118516	0.0592579	+	0.998243i	$0.481127\pi$
$830$	−1.50012e9	−0.0910652
$831$	0	0
$832$	8.60096e9	0.517744
$833$	1.99888e9	0.119820
$834$	0	0
$835$	7.60925e8	0.0452313
$836$	−5.44222e9	−0.322147
$837$	0	0
$838$	7.75995e9	0.455517
$839$	7.14687e8	0.0417781	0.0208891	−	0.999782i	$-0.493350\pi$
$839$	7.14687e8	0.0417781	0.0208891	+	0.999782i	$0.493350\pi$
$840$	0	0
$841$	−8.69116e9	−0.503839
$842$	−1.65946e10	−0.958018
$843$	0	0
$844$	−2.84163e9	−0.162693
$845$	5.41398e9	0.308687
$846$	0	0
$847$	2.36849e9	0.133930
$848$	5.49805e9	0.309616
$849$	0	0
$850$	2.19449e9	0.122565
$851$	−3.09486e9	−0.172142
$852$	0	0
$853$	7.55022e9	0.416522	0.208261	−	0.978073i	$-0.433220\pi$
$853$	7.55022e9	0.416522	0.208261	+	0.978073i	$0.433220\pi$
$854$	−4.70100e9	−0.258278
$855$	0	0
$856$	3.87333e10	2.11070
$857$	−3.37192e10	−1.82997	−0.914987	−	0.403483i	$-0.867799\pi$
$857$	−3.37192e10	−1.82997	−0.914987	+	0.403483i	$0.867799\pi$
$858$	0	0
$859$	−2.78638e10	−1.49991	−0.749954	−	0.661490i	$-0.769925\pi$
$859$	−2.78638e10	−1.49991	−0.749954	+	0.661490i	$0.769925\pi$
$860$	2.56119e9	0.137308
$861$	0	0
$862$	4.40562e9	0.234278
$863$	−3.07420e10	−1.62815	−0.814075	−	0.580760i	$-0.802755\pi$
$863$	−3.07420e10	−1.62815	−0.814075	+	0.580760i	$0.802755\pi$
$864$	0	0
$865$	1.29087e10	0.678151
$866$	2.22736e10	1.16541
$867$	0	0
$868$	−1.91605e9	−0.0994459
$869$	−1.48203e9	−0.0766103
$870$	0	0
$871$	1.03329e10	0.529856
$872$	−1.15477e10	−0.589777
$873$	0	0
$874$	5.20049e9	0.263484
$875$	6.69922e8	0.0338062
$876$	0	0
$877$	8.59730e9	0.430391	0.215196	−	0.976571i	$-0.430961\pi$
$877$	8.59730e9	0.430391	0.215196	+	0.976571i	$0.430961\pi$
$878$	−2.12907e10	−1.06159
$879$	0	0
$880$	2.29958e9	0.113752
$881$	6.86087e9	0.338036	0.169018	−	0.985613i	$-0.445940\pi$
$881$	6.86087e9	0.338036	0.169018	+	0.985613i	$0.445940\pi$
$882$	0	0
$883$	−1.34116e10	−0.655569	−0.327784	−	0.944753i	$-0.606302\pi$
$883$	−1.34116e10	−0.655569	−0.327784	+	0.944753i	$0.606302\pi$
$884$	−4.46938e9	−0.217603
$885$	0	0
$886$	−4.45495e9	−0.215191
$887$	−3.23054e10	−1.55433	−0.777163	−	0.629300i	$-0.783342\pi$
$887$	−3.23054e10	−1.55433	−0.777163	+	0.629300i	$0.783342\pi$
$888$	0	0
$889$	−1.03020e10	−0.491774
$890$	7.15654e8	0.0340281
$891$	0	0
$892$	1.20434e10	0.568160
$893$	−3.18018e10	−1.49441
$894$	0	0
$895$	−5.60937e9	−0.261537
$896$	1.30422e9	0.0605721
$897$	0	0
$898$	1.72405e10	0.794480
$899$	8.66121e9	0.397576
$900$	0	0
$901$	−1.80113e10	−0.820365
$902$	−1.59399e10	−0.723208
$903$	0	0
$904$	−1.02807e10	−0.462841
$905$	5.82825e9	0.261378
$906$	0	0
$907$	4.43045e10	1.97161	0.985807	−	0.167883i	$-0.0536931\pi$
$907$	4.43045e10	1.97161	0.985807	+	0.167883i	$0.0536931\pi$
$908$	7.67061e9	0.340039
$909$	0	0
$910$	1.56253e9	0.0687360
$911$	3.77401e10	1.65382	0.826910	−	0.562334i	$-0.190096\pi$
$911$	3.77401e10	1.65382	0.826910	+	0.562334i	$0.190096\pi$
$912$	0	0
$913$	5.14963e9	0.223938
$914$	−2.28657e10	−0.990544
$915$	0	0
$916$	8.23160e9	0.353875
$917$	−6.67918e9	−0.286043
$918$	0	0
$919$	4.52168e10	1.92174	0.960871	−	0.276996i	$-0.0893387\pi$
$919$	4.52168e10	1.92174	0.960871	+	0.276996i	$0.0893387\pi$
$920$	4.74436e9	0.200873
$921$	0	0
$922$	2.26838e10	0.953141
$923$	−1.61707e10	−0.676898
$924$	0	0
$925$	−1.97650e9	−0.0821108
$926$	4.74577e9	0.196412
$927$	0	0
$928$	−1.44041e10	−0.591653
$929$	2.46341e10	1.00805	0.504025	−	0.863689i	$-0.331852\pi$
$929$	2.46341e10	1.00805	0.504025	+	0.863689i	$0.331852\pi$
$930$	0	0
$931$	3.02519e9	0.122865
$932$	−9.36664e9	−0.378990
$933$	0	0
$934$	−1.18384e10	−0.475421
$935$	−7.53326e9	−0.301399
$936$	0	0
$937$	1.28125e10	0.508798	0.254399	−	0.967099i	$-0.418122\pi$
$937$	1.28125e10	0.508798	0.254399	+	0.967099i	$0.418122\pi$
$938$	−6.64536e9	−0.262911
$939$	0	0
$940$	−9.22431e9	−0.362232
$941$	3.61748e9	0.141528	0.0707640	−	0.997493i	$-0.477456\pi$
$941$	3.61748e9	0.141528	0.0707640	+	0.997493i	$0.477456\pi$
$942$	0	0
$943$	−1.33004e10	−0.516502
$944$	1.06374e10	0.411558
$945$	0	0
$946$	1.00689e10	0.386690
$947$	3.37196e10	1.29020	0.645101	−	0.764097i	$-0.276815\pi$
$947$	3.37196e10	1.29020	0.645101	+	0.764097i	$0.276815\pi$
$948$	0	0
$949$	2.76939e9	0.105185
$950$	3.32123e9	0.125680
$951$	0	0
$952$	9.04057e9	0.339599
$953$	4.78636e10	1.79135	0.895675	−	0.444709i	$-0.146693\pi$
$953$	4.78636e10	1.79135	0.895675	+	0.444709i	$0.146693\pi$
$954$	0	0
$955$	5.26138e9	0.195474
$956$	1.40072e10	0.518500
$957$	0	0
$958$	−6.55553e9	−0.240896
$959$	7.05832e9	0.258426
$960$	0	0
$961$	−1.87477e10	−0.681421
$962$	−4.61000e9	−0.166951
$963$	0	0
$964$	−2.08870e9	−0.0750941
$965$	−6.29692e9	−0.225571
$966$	0	0
$967$	−4.12808e10	−1.46810	−0.734049	−	0.679096i	$-0.762372\pi$
$967$	−4.12808e10	−1.46810	−0.734049	+	0.679096i	$0.762372\pi$
$968$	1.07122e10	0.379591
$969$	0	0
$970$	1.82934e10	0.643567
$971$	4.22551e10	1.48119	0.740597	−	0.671950i	$-0.234543\pi$
$971$	4.22551e10	1.48119	0.740597	+	0.671950i	$0.234543\pi$
$972$	0	0
$973$	−6.93930e8	−0.0241502
$974$	−2.65798e10	−0.921712
$975$	0	0
$976$	−8.59895e9	−0.296054
$977$	−2.31359e10	−0.793697	−0.396848	−	0.917884i	$-0.629896\pi$
$977$	−2.31359e10	−0.793697	−0.396848	+	0.917884i	$0.629896\pi$
$978$	0	0
$979$	−2.45671e9	−0.0836785
$980$	8.77476e8	0.0297813
$981$	0	0
$982$	1.78236e10	0.600627
$983$	−4.54149e10	−1.52497	−0.762485	−	0.647006i	$-0.776021\pi$
$983$	−4.54149e10	−1.52497	−0.762485	+	0.647006i	$0.776021\pi$
$984$	0	0
$985$	1.18003e10	0.393428
$986$	−1.29932e10	−0.431666
$987$	0	0
$988$	−6.76415e9	−0.223133
$989$	8.40155e9	0.276167
$990$	0	0
$991$	1.57362e10	0.513621	0.256811	−	0.966462i	$-0.417328\pi$
$991$	1.57362e10	0.513621	0.256811	+	0.966462i	$0.417328\pi$
$992$	−1.45766e10	−0.474094
$993$	0	0
$994$	1.03998e10	0.335872
$995$	2.12092e10	0.682564
$996$	0	0
$997$	−2.87843e10	−0.919863	−0.459931	−	0.887954i	$-0.652126\pi$
$997$	−2.87843e10	−0.919863	−0.459931	+	0.887954i	$0.652126\pi$
$998$	1.98955e10	0.633573
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.i.1.3		4
3.2	odd	2		105.8.a.f.1.2	✓	4
15.14	odd	2		525.8.a.k.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.f.1.2	✓	4	3.2	odd	2
315.8.a.i.1.3		4	1.1	even	1	trivial
525.8.a.k.1.3		4	15.14	odd	2

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+8.26635 q^{2} -59.6674 q^{4} -125.000 q^{5} -343.000 q^{7} -1551.33 q^{8} -1033.29 q^{10} +3547.11 q^{11} +4408.70 q^{13} -2835.36 q^{14} -5186.37 q^{16} +16990.2 q^{17} +25713.7 q^{19} +7458.43 q^{20} +29321.6 q^{22} +24466.1 q^{23} +15625.0 q^{25} +36443.9 q^{26} +20465.9 q^{28} -92513.3 q^{29} -93621.2 q^{31} +155697. q^{32} +140447. q^{34} +42875.0 q^{35} -126496. q^{37} +212559. q^{38} +193916. q^{40} -543623. q^{41} +343395. q^{43} -211647. q^{44} +202246. q^{46} -1.23676e6 q^{47} +117649. q^{49} +129162. q^{50} -263056. q^{52} -1.06010e6 q^{53} -443388. q^{55} +532104. q^{56} -764748. q^{58} -2.05102e6 q^{59} +1.65799e6 q^{61} -773906. q^{62} +1.95090e6 q^{64} -551088. q^{65} +2.34375e6 q^{67} -1.01376e6 q^{68} +354420. q^{70} -3.66791e6 q^{71} +628165. q^{73} -1.04566e6 q^{74} -1.53427e6 q^{76} -1.21666e6 q^{77} -417813. q^{79} +648296. q^{80} -4.49378e6 q^{82} +1.45178e6 q^{83} -2.12378e6 q^{85} +2.83862e6 q^{86} -5.50271e6 q^{88} -692594. q^{89} -1.51219e6 q^{91} -1.45983e6 q^{92} -1.02235e7 q^{94} -3.21421e6 q^{95} -1.77040e7 q^{97} +972528. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 421 q^{4} - 500 q^{5} - 1372 q^{7} - 417 q^{8} - 125 q^{10} - 7852 q^{11} + 18532 q^{13} - 343 q^{14} + 47601 q^{16} - 33976 q^{17} + 22188 q^{19} - 52625 q^{20} + 18076 q^{22} + 22736 q^{23}+ \cdots + 117649 q^{98}+O(q^{100}) \) 4 * q + q^2 + 421 * q^4 - 500 * q^5 - 1372 * q^7 - 417 * q^8 - 125 * q^10 - 7852 * q^11 + 18532 * q^13 - 343 * q^14 + 47601 * q^16 - 33976 * q^17 + 22188 * q^19 - 52625 * q^20 + 18076 * q^22 + 22736 * q^23 + 62500 * q^25 + 235686 * q^26 - 144403 * q^28 - 12808 * q^29 - 64908 * q^31 - 111705 * q^32 + 45198 * q^34 + 171500 * q^35 - 315576 * q^37 - 1505204 * q^38 + 52125 * q^40 + 325240 * q^41 + 736184 * q^43 - 2151396 * q^44 - 1289480 * q^46 - 1661296 * q^47 + 470596 * q^49 + 15625 * q^50 + 2618462 * q^52 - 2569036 * q^53 + 981500 * q^55 + 143031 * q^56 - 553222 * q^58 - 3128416 * q^59 + 1392912 * q^61 + 1590240 * q^62 + 2081297 * q^64 - 2316500 * q^65 + 1654088 * q^67 - 8005658 * q^68 + 42875 * q^70 - 2863060 * q^71 + 4095068 * q^73 - 8950482 * q^74 - 8510596 * q^76 + 2693236 * q^77 + 9277704 * q^79 - 5950125 * q^80 - 2529402 * q^82 + 1362736 * q^83 + 4247000 * q^85 + 10602132 * q^86 - 14814892 * q^88 + 11110608 * q^89 - 6356476 * q^91 + 11222472 * q^92 + 101360 * q^94 - 2773500 * q^95 - 8721516 * q^97 + 117649 * q^98

Introduction

L-functions

Modular forms

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Database

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