LMFDB - Embedded newform 450.8.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,8,Mod(1,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	450.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:	$140.573261468$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} +713.000 q^{7} +512.000 q^{8} -3810.00 q^{11} -391.000 q^{13} +5704.00 q^{14} +4096.00 q^{16} +4182.00 q^{17} -1561.00 q^{19} -30480.0 q^{22} -114150. q^{23} -3128.00 q^{26} +45632.0 q^{28} +83214.0 q^{29} -83167.0 q^{31} +32768.0 q^{32} +33456.0 q^{34} -231334. q^{37} -12488.0 q^{38} +124656. q^{41} +193757. q^{43} -243840. q^{44} -913200. q^{46} -319290. q^{47} -315174. q^{49} -25024.0 q^{52} -1.64543e6 q^{53} +365056. q^{56} +665712. q^{58} +38610.0 q^{59} -1.97390e6 q^{61} -665336. q^{62} +262144. q^{64} +4.40975e6 q^{67} +267648. q^{68} -124080. q^{71} +3.96763e6 q^{73} -1.85067e6 q^{74} -99904.0 q^{76} -2.71653e6 q^{77} +7.10799e6 q^{79} +997248. q^{82} -8.11769e6 q^{83} +1.55006e6 q^{86} -1.95072e6 q^{88} -6.72787e6 q^{89} -278783. q^{91} -7.30560e6 q^{92} -2.55432e6 q^{94} -1.42687e7 q^{97} -2.52139e6 q^{98} +O(q^{100})$

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	713.000	0.785681	0.392841	−	0.919607i	$-0.371492\pi$
$7$	713.000	0.785681	0.392841	+	0.919607i	$0.371492\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	0	0
$11$	−3810.00	−0.863079	−0.431540	−	0.902094i	$-0.642029\pi$
$11$	−3810.00	−0.863079	−0.431540	+	0.902094i	$0.642029\pi$
$12$	0	0
$13$	−391.000	−0.0493600	−0.0246800	−	0.999695i	$-0.507857\pi$
$13$	−391.000	−0.0493600	−0.0246800	+	0.999695i	$0.507857\pi$
$14$	5704.00	0.555561
$15$	0	0
$16$	4096.00	0.250000
$17$	4182.00	0.206449	0.103225	−	0.994658i	$-0.467084\pi$
$17$	4182.00	0.206449	0.103225	+	0.994658i	$0.467084\pi$
$18$	0	0
$19$	−1561.00	−0.0522114	−0.0261057	−	0.999659i	$-0.508311\pi$
$19$	−1561.00	−0.0522114	−0.0261057	+	0.999659i	$0.508311\pi$
$20$	0	0
$21$	0	0
$22$	−30480.0	−0.610289
$23$	−114150.	−1.95627	−0.978134	−	0.207974i	$-0.933313\pi$
$23$	−114150.	−1.95627	−0.978134	+	0.207974i	$0.933313\pi$
$24$	0	0
$25$	0	0
$26$	−3128.00	−0.0349028
$27$	0	0
$28$	45632.0	0.392841
$29$	83214.0	0.633583	0.316791	−	0.948495i	$-0.397394\pi$
$29$	83214.0	0.633583	0.316791	+	0.948495i	$0.397394\pi$
$30$	0	0
$31$	−83167.0	−0.501401	−0.250700	−	0.968065i	$-0.580661\pi$
$31$	−83167.0	−0.501401	−0.250700	+	0.968065i	$0.580661\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	33456.0	0.145981
$35$	0	0
$36$	0	0
$37$	−231334.	−0.750816	−0.375408	−	0.926860i	$-0.622497\pi$
$37$	−231334.	−0.750816	−0.375408	+	0.926860i	$0.622497\pi$
$38$	−12488.0	−0.0369190
$39$	0	0
$40$	0	0
$41$	124656.	0.282468	0.141234	−	0.989976i	$-0.454893\pi$
$41$	124656.	0.282468	0.141234	+	0.989976i	$0.454893\pi$
$42$	0	0
$43$	193757.	0.371636	0.185818	−	0.982584i	$-0.440507\pi$
$43$	193757.	0.371636	0.185818	+	0.982584i	$0.440507\pi$
$44$	−243840.	−0.431540
$45$	0	0
$46$	−913200.	−1.38329
$47$	−319290.	−0.448583	−0.224292	−	0.974522i	$-0.572007\pi$
$47$	−319290.	−0.448583	−0.224292	+	0.974522i	$0.572007\pi$
$48$	0	0
$49$	−315174.	−0.382705
$50$	0	0
$51$	0	0
$52$	−25024.0	−0.0246800
$53$	−1.64543e6	−1.51815	−0.759073	−	0.651006i	$-0.774347\pi$
$53$	−1.64543e6	−1.51815	−0.759073	+	0.651006i	$0.774347\pi$
$54$	0	0
$55$	0	0
$56$	365056.	0.277780
$57$	0	0
$58$	665712.	0.448011
$59$	38610.0	0.0244747	0.0122374	−	0.999925i	$-0.496105\pi$
$59$	38610.0	0.0244747	0.0122374	+	0.999925i	$0.496105\pi$
$60$	0	0
$61$	−1.97390e6	−1.11345	−0.556726	−	0.830696i	$-0.687943\pi$
$61$	−1.97390e6	−1.11345	−0.556726	+	0.830696i	$0.687943\pi$
$62$	−665336.	−0.354544
$63$	0	0
$64$	262144.	0.125000
$65$	0	0
$66$	0	0
$67$	4.40975e6	1.79123	0.895617	−	0.444825i	$-0.146734\pi$
$67$	4.40975e6	1.79123	0.895617	+	0.444825i	$0.146734\pi$
$68$	267648.	0.103225
$69$	0	0
$70$	0	0
$71$	−124080.	−0.0411432	−0.0205716	−	0.999788i	$-0.506549\pi$
$71$	−124080.	−0.0411432	−0.0205716	+	0.999788i	$0.506549\pi$
$72$	0	0
$73$	3.96763e6	1.19372	0.596859	−	0.802346i	$-0.296415\pi$
$73$	3.96763e6	1.19372	0.596859	+	0.802346i	$0.296415\pi$
$74$	−1.85067e6	−0.530907
$75$	0	0
$76$	−99904.0	−0.0261057
$77$	−2.71653e6	−0.678105
$78$	0	0
$79$	7.10799e6	1.62200	0.811002	−	0.585043i	$-0.198922\pi$
$79$	7.10799e6	1.62200	0.811002	+	0.585043i	$0.198922\pi$
$80$	0	0
$81$	0	0
$82$	997248.	0.199735
$83$	−8.11769e6	−1.55833	−0.779165	−	0.626819i	$-0.784357\pi$
$83$	−8.11769e6	−1.55833	−0.779165	+	0.626819i	$0.784357\pi$
$84$	0	0
$85$	0	0
$86$	1.55006e6	0.262786
$87$	0	0
$88$	−1.95072e6	−0.305145
$89$	−6.72787e6	−1.01161	−0.505805	−	0.862648i	$-0.668804\pi$
$89$	−6.72787e6	−1.01161	−0.505805	+	0.862648i	$0.668804\pi$
$90$	0	0
$91$	−278783.	−0.0387812
$92$	−7.30560e6	−0.978134
$93$	0	0
$94$	−2.55432e6	−0.317196
$95$	0	0
$96$	0	0
$97$	−1.42687e7	−1.58739	−0.793693	−	0.608318i	$-0.791844\pi$
$97$	−1.42687e7	−1.58739	−0.793693	+	0.608318i	$0.791844\pi$
$98$	−2.52139e6	−0.270613
$99$	0	0
$100$	0	0
$101$	−6.93119e6	−0.669396	−0.334698	−	0.942326i	$-0.608634\pi$
$101$	−6.93119e6	−0.669396	−0.334698	+	0.942326i	$0.608634\pi$
$102$	0	0
$103$	1.34707e7	1.21467	0.607337	−	0.794444i	$-0.292238\pi$
$103$	1.34707e7	1.21467	0.607337	+	0.794444i	$0.292238\pi$
$104$	−200192.	−0.0174514
$105$	0	0
$106$	−1.31634e7	−1.07349
$107$	−1.05625e7	−0.833533	−0.416767	−	0.909014i	$-0.636837\pi$
$107$	−1.05625e7	−0.833533	−0.416767	+	0.909014i	$0.636837\pi$
$108$	0	0
$109$	6.74796e6	0.499091	0.249546	−	0.968363i	$-0.419719\pi$
$109$	6.74796e6	0.499091	0.249546	+	0.968363i	$0.419719\pi$
$110$	0	0
$111$	0	0
$112$	2.92045e6	0.196420
$113$	−730764.	−0.0476434	−0.0238217	−	0.999716i	$-0.507583\pi$
$113$	−730764.	−0.0476434	−0.0238217	+	0.999716i	$0.507583\pi$
$114$	0	0
$115$	0	0
$116$	5.32570e6	0.316791
$117$	0	0
$118$	308880.	0.0173062
$119$	2.98177e6	0.162203
$120$	0	0
$121$	−4.97107e6	−0.255095
$122$	−1.57912e7	−0.787330
$123$	0	0
$124$	−5.32269e6	−0.250700
$125$	0	0
$126$	0	0
$127$	−3.53961e7	−1.53335	−0.766676	−	0.642034i	$-0.778091\pi$
$127$	−3.53961e7	−1.53335	−0.766676	+	0.642034i	$0.778091\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	0	0
$131$	−2.58059e7	−1.00293	−0.501464	−	0.865178i	$-0.667205\pi$
$131$	−2.58059e7	−1.00293	−0.501464	+	0.865178i	$0.667205\pi$
$132$	0	0
$133$	−1.11299e6	−0.0410215
$134$	3.52780e7	1.26659
$135$	0	0
$136$	2.14118e6	0.0729907
$137$	−1.09054e7	−0.362344	−0.181172	−	0.983451i	$-0.557989\pi$
$137$	−1.09054e7	−0.362344	−0.181172	+	0.983451i	$0.557989\pi$
$138$	0	0
$139$	−5.32412e7	−1.68150	−0.840748	−	0.541426i	$-0.817885\pi$
$139$	−5.32412e7	−1.68150	−0.840748	+	0.541426i	$0.817885\pi$
$140$	0	0
$141$	0	0
$142$	−992640.	−0.0290926
$143$	1.48971e6	0.0426016
$144$	0	0
$145$	0	0
$146$	3.17411e7	0.844086
$147$	0	0
$148$	−1.48054e7	−0.375408
$149$	3.39941e7	0.841884	0.420942	−	0.907088i	$-0.361700\pi$
$149$	3.39941e7	0.841884	0.420942	+	0.907088i	$0.361700\pi$
$150$	0	0
$151$	3.13410e7	0.740786	0.370393	−	0.928875i	$-0.379223\pi$
$151$	3.13410e7	0.740786	0.370393	+	0.928875i	$0.379223\pi$
$152$	−799232.	−0.0184595
$153$	0	0
$154$	−2.17322e7	−0.479493
$155$	0	0
$156$	0	0
$157$	−6.83485e7	−1.40955	−0.704775	−	0.709431i	$-0.748952\pi$
$157$	−6.83485e7	−1.40955	−0.704775	+	0.709431i	$0.748952\pi$
$158$	5.68639e7	1.14693
$159$	0	0
$160$	0	0
$161$	−8.13890e7	−1.53700
$162$	0	0
$163$	4.62777e7	0.836980	0.418490	−	0.908221i	$-0.362560\pi$
$163$	4.62777e7	0.836980	0.418490	+	0.908221i	$0.362560\pi$
$164$	7.97798e6	0.141234
$165$	0	0
$166$	−6.49416e7	−1.10191
$167$	3.99639e7	0.663988	0.331994	−	0.943282i	$-0.392279\pi$
$167$	3.99639e7	0.663988	0.331994	+	0.943282i	$0.392279\pi$
$168$	0	0
$169$	−6.25956e7	−0.997564
$170$	0	0
$171$	0	0
$172$	1.24004e7	0.185818
$173$	9.07312e7	1.33228	0.666139	−	0.745827i	$-0.267946\pi$
$173$	9.07312e7	1.33228	0.666139	+	0.745827i	$0.267946\pi$
$174$	0	0
$175$	0	0
$176$	−1.56058e7	−0.215770
$177$	0	0
$178$	−5.38230e7	−0.715316
$179$	−1.30570e8	−1.70160	−0.850802	−	0.525486i	$-0.823883\pi$
$179$	−1.30570e8	−1.70160	−0.850802	+	0.525486i	$0.823883\pi$
$180$	0	0
$181$	1.11281e8	1.39490	0.697452	−	0.716632i	$-0.254317\pi$
$181$	1.11281e8	1.39490	0.697452	+	0.716632i	$0.254317\pi$
$182$	−2.23026e6	−0.0274225
$183$	0	0
$184$	−5.84448e7	−0.691645
$185$	0	0
$186$	0	0
$187$	−1.59334e7	−0.178182
$188$	−2.04346e7	−0.224292
$189$	0	0
$190$	0	0
$191$	−1.39416e8	−1.44776	−0.723880	−	0.689926i	$-0.757643\pi$
$191$	−1.39416e8	−1.44776	−0.723880	+	0.689926i	$0.757643\pi$
$192$	0	0
$193$	−2.37893e7	−0.238194	−0.119097	−	0.992883i	$-0.538000\pi$
$193$	−2.37893e7	−0.238194	−0.119097	+	0.992883i	$0.538000\pi$
$194$	−1.14149e8	−1.12245
$195$	0	0
$196$	−2.01711e7	−0.191352
$197$	−4.67913e7	−0.436047	−0.218024	−	0.975943i	$-0.569961\pi$
$197$	−4.67913e7	−0.436047	−0.218024	+	0.975943i	$0.569961\pi$
$198$	0	0
$199$	−1.16854e8	−1.05113	−0.525566	−	0.850753i	$-0.676146\pi$
$199$	−1.16854e8	−1.05113	−0.525566	+	0.850753i	$0.676146\pi$
$200$	0	0
$201$	0	0
$202$	−5.54495e7	−0.473334
$203$	5.93316e7	0.497794
$204$	0	0
$205$	0	0
$206$	1.07766e8	0.858904
$207$	0	0
$208$	−1.60154e6	−0.0123400
$209$	5.94741e6	0.0450626
$210$	0	0
$211$	−1.83300e8	−1.34330	−0.671651	−	0.740867i	$-0.734415\pi$
$211$	−1.83300e8	−1.34330	−0.671651	+	0.740867i	$0.734415\pi$
$212$	−1.05307e8	−0.759073
$213$	0	0
$214$	−8.44999e7	−0.589397
$215$	0	0
$216$	0	0
$217$	−5.92981e7	−0.393941
$218$	5.39837e7	0.352911
$219$	0	0
$220$	0	0
$221$	−1.63516e6	−0.0101903
$222$	0	0
$223$	1.91650e8	1.15729	0.578644	−	0.815580i	$-0.303582\pi$
$223$	1.91650e8	1.15729	0.578644	+	0.815580i	$0.303582\pi$
$224$	2.33636e7	0.138890
$225$	0	0
$226$	−5.84611e6	−0.0336890
$227$	1.11119e8	0.630521	0.315260	−	0.949005i	$-0.397908\pi$
$227$	1.11119e8	0.630521	0.315260	+	0.949005i	$0.397908\pi$
$228$	0	0
$229$	−8.92220e7	−0.490962	−0.245481	−	0.969401i	$-0.578946\pi$
$229$	−8.92220e7	−0.490962	−0.245481	+	0.969401i	$0.578946\pi$
$230$	0	0
$231$	0	0
$232$	4.26056e7	0.224005
$233$	5.04671e7	0.261374	0.130687	−	0.991424i	$-0.458282\pi$
$233$	5.04671e7	0.261374	0.130687	+	0.991424i	$0.458282\pi$
$234$	0	0
$235$	0	0
$236$	2.47104e6	0.0122374
$237$	0	0
$238$	2.38541e7	0.114695
$239$	−8.16563e7	−0.386899	−0.193449	−	0.981110i	$-0.561967\pi$
$239$	−8.16563e7	−0.386899	−0.193449	+	0.981110i	$0.561967\pi$
$240$	0	0
$241$	−2.40323e8	−1.10595	−0.552976	−	0.833197i	$-0.686508\pi$
$241$	−2.40323e8	−1.10595	−0.552976	+	0.833197i	$0.686508\pi$
$242$	−3.97686e7	−0.180379
$243$	0	0
$244$	−1.26330e8	−0.556726
$245$	0	0
$246$	0	0
$247$	610351.	0.00257715
$248$	−4.25815e7	−0.177272
$249$	0	0
$250$	0	0
$251$	3.23741e8	1.29223	0.646115	−	0.763240i	$-0.276393\pi$
$251$	3.23741e8	1.29223	0.646115	+	0.763240i	$0.276393\pi$
$252$	0	0
$253$	4.34912e8	1.68841
$254$	−2.83169e8	−1.08424
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	8.10385e7	0.297800	0.148900	−	0.988852i	$-0.452427\pi$
$257$	8.10385e7	0.297800	0.148900	+	0.988852i	$0.452427\pi$
$258$	0	0
$259$	−1.64941e8	−0.589902
$260$	0	0
$261$	0	0
$262$	−2.06448e8	−0.709178
$263$	−2.11263e8	−0.716106	−0.358053	−	0.933701i	$-0.616559\pi$
$263$	−2.11263e8	−0.716106	−0.358053	+	0.933701i	$0.616559\pi$
$264$	0	0
$265$	0	0
$266$	−8.90394e6	−0.0290066
$267$	0	0
$268$	2.82224e8	0.895617
$269$	−4.20832e8	−1.31818	−0.659092	−	0.752062i	$-0.729059\pi$
$269$	−4.20832e8	−1.31818	−0.659092	+	0.752062i	$0.729059\pi$
$270$	0	0
$271$	−2.13859e8	−0.652731	−0.326366	−	0.945244i	$-0.605824\pi$
$271$	−2.13859e8	−0.652731	−0.326366	+	0.945244i	$0.605824\pi$
$272$	1.71295e7	0.0516123
$273$	0	0
$274$	−8.72435e7	−0.256216
$275$	0	0
$276$	0	0
$277$	−2.94664e8	−0.833007	−0.416503	−	0.909134i	$-0.636745\pi$
$277$	−2.94664e8	−0.833007	−0.416503	+	0.909134i	$0.636745\pi$
$278$	−4.25930e8	−1.18900
$279$	0	0
$280$	0	0
$281$	2.81001e8	0.755503	0.377752	−	0.925907i	$-0.376697\pi$
$281$	2.81001e8	0.755503	0.377752	+	0.925907i	$0.376697\pi$
$282$	0	0
$283$	5.81465e8	1.52500	0.762502	−	0.646986i	$-0.223971\pi$
$283$	5.81465e8	1.52500	0.762502	+	0.646986i	$0.223971\pi$
$284$	−7.94112e6	−0.0205716
$285$	0	0
$286$	1.19177e7	0.0301239
$287$	8.88797e7	0.221930
$288$	0	0
$289$	−3.92850e8	−0.957379
$290$	0	0
$291$	0	0
$292$	2.53929e8	0.596859
$293$	7.92049e8	1.83957	0.919783	−	0.392427i	$-0.128365\pi$
$293$	7.92049e8	1.83957	0.919783	+	0.392427i	$0.128365\pi$
$294$	0	0
$295$	0	0
$296$	−1.18443e8	−0.265453
$297$	0	0
$298$	2.71953e8	0.595302
$299$	4.46326e7	0.0965614
$300$	0	0
$301$	1.38149e8	0.291987
$302$	2.50728e8	0.523815
$303$	0	0
$304$	−6.39386e6	−0.0130528
$305$	0	0
$306$	0	0
$307$	2.03330e8	0.401067	0.200533	−	0.979687i	$-0.435732\pi$
$307$	2.03330e8	0.401067	0.200533	+	0.979687i	$0.435732\pi$
$308$	−1.73858e8	−0.339053
$309$	0	0
$310$	0	0
$311$	−3.29428e8	−0.621011	−0.310505	−	0.950572i	$-0.600498\pi$
$311$	−3.29428e8	−0.621011	−0.310505	+	0.950572i	$0.600498\pi$
$312$	0	0
$313$	1.61495e8	0.297684	0.148842	−	0.988861i	$-0.452445\pi$
$313$	1.61495e8	0.297684	0.148842	+	0.988861i	$0.452445\pi$
$314$	−5.46788e8	−0.996702
$315$	0	0
$316$	4.54911e8	0.811002
$317$	−6.39147e8	−1.12692	−0.563460	−	0.826143i	$-0.690530\pi$
$317$	−6.39147e8	−1.12692	−0.563460	+	0.826143i	$0.690530\pi$
$318$	0	0
$319$	−3.17045e8	−0.546832
$320$	0	0
$321$	0	0
$322$	−6.51112e8	−1.08683
$323$	−6.52810e6	−0.0107790
$324$	0	0
$325$	0	0
$326$	3.70221e8	0.591834
$327$	0	0
$328$	6.38239e7	0.0998676
$329$	−2.27654e8	−0.352443
$330$	0	0
$331$	−5.39401e8	−0.817548	−0.408774	−	0.912636i	$-0.634044\pi$
$331$	−5.39401e8	−0.817548	−0.408774	+	0.912636i	$0.634044\pi$
$332$	−5.19532e8	−0.779165
$333$	0	0
$334$	3.19711e8	0.469510
$335$	0	0
$336$	0	0
$337$	7.72336e8	1.09926	0.549632	−	0.835407i	$-0.314768\pi$
$337$	7.72336e8	1.09926	0.549632	+	0.835407i	$0.314768\pi$
$338$	−5.00765e8	−0.705384
$339$	0	0
$340$	0	0
$341$	3.16866e8	0.432749
$342$	0	0
$343$	−8.11905e8	−1.08637
$344$	9.92036e7	0.131393
$345$	0	0
$346$	7.25849e8	0.942063
$347$	−5.21028e8	−0.669435	−0.334717	−	0.942319i	$-0.608641\pi$
$347$	−5.21028e8	−0.669435	−0.334717	+	0.942319i	$0.608641\pi$
$348$	0	0
$349$	5.22374e8	0.657798	0.328899	−	0.944365i	$-0.393322\pi$
$349$	5.22374e8	0.657798	0.328899	+	0.944365i	$0.393322\pi$
$350$	0	0
$351$	0	0
$352$	−1.24846e8	−0.152572
$353$	1.02256e9	1.23731	0.618656	−	0.785662i	$-0.287677\pi$
$353$	1.02256e9	1.23731	0.618656	+	0.785662i	$0.287677\pi$
$354$	0	0
$355$	0	0
$356$	−4.30584e8	−0.505805
$357$	0	0
$358$	−1.04456e9	−1.20322
$359$	1.17896e9	1.34484	0.672418	−	0.740171i	$-0.265256\pi$
$359$	1.17896e9	1.34484	0.672418	+	0.740171i	$0.265256\pi$
$360$	0	0
$361$	−8.91435e8	−0.997274
$362$	8.90244e8	0.986346
$363$	0	0
$364$	−1.78421e7	−0.0193906
$365$	0	0
$366$	0	0
$367$	−7.85348e8	−0.829337	−0.414668	−	0.909973i	$-0.636102\pi$
$367$	−7.85348e8	−0.829337	−0.414668	+	0.909973i	$0.636102\pi$
$368$	−4.67558e8	−0.489067
$369$	0	0
$370$	0	0
$371$	−1.17319e9	−1.19278
$372$	0	0
$373$	−7.41463e8	−0.739791	−0.369895	−	0.929073i	$-0.620606\pi$
$373$	−7.41463e8	−0.739791	−0.369895	+	0.929073i	$0.620606\pi$
$374$	−1.27467e8	−0.125994
$375$	0	0
$376$	−1.63476e8	−0.158598
$377$	−3.25367e7	−0.0312736
$378$	0	0
$379$	−4.13198e8	−0.389871	−0.194936	−	0.980816i	$-0.562450\pi$
$379$	−4.13198e8	−0.389871	−0.194936	+	0.980816i	$0.562450\pi$
$380$	0	0
$381$	0	0
$382$	−1.11533e9	−1.02372
$383$	2.07784e9	1.88980	0.944899	−	0.327361i	$-0.106159\pi$
$383$	2.07784e9	1.88980	0.944899	+	0.327361i	$0.106159\pi$
$384$	0	0
$385$	0	0
$386$	−1.90314e8	−0.168428
$387$	0	0
$388$	−9.13195e8	−0.793693
$389$	1.62801e9	1.40227	0.701137	−	0.713027i	$-0.252676\pi$
$389$	1.62801e9	1.40227	0.701137	+	0.713027i	$0.252676\pi$
$390$	0	0
$391$	−4.77375e8	−0.403870
$392$	−1.61369e8	−0.135307
$393$	0	0
$394$	−3.74331e8	−0.308332
$395$	0	0
$396$	0	0
$397$	6.54372e8	0.524877	0.262439	−	0.964949i	$-0.415473\pi$
$397$	6.54372e8	0.524877	0.262439	+	0.964949i	$0.415473\pi$
$398$	−9.34830e8	−0.743262
$399$	0	0
$400$	0	0
$401$	−1.91730e9	−1.48486	−0.742430	−	0.669924i	$-0.766327\pi$
$401$	−1.91730e9	−1.48486	−0.742430	+	0.669924i	$0.766327\pi$
$402$	0	0
$403$	3.25183e7	0.0247491
$404$	−4.43596e8	−0.334698
$405$	0	0
$406$	4.74653e8	0.351994
$407$	8.81383e8	0.648013
$408$	0	0
$409$	1.46590e9	1.05943	0.529717	−	0.848175i	$-0.322298\pi$
$409$	1.46590e9	1.05943	0.529717	+	0.848175i	$0.322298\pi$
$410$	0	0
$411$	0	0
$412$	8.62124e8	0.607337
$413$	2.75289e7	0.0192293
$414$	0	0
$415$	0	0
$416$	−1.28123e7	−0.00872570
$417$	0	0
$418$	4.75793e7	0.0318640
$419$	1.87001e9	1.24192	0.620961	−	0.783841i	$-0.286742\pi$
$419$	1.87001e9	1.24192	0.620961	+	0.783841i	$0.286742\pi$
$420$	0	0
$421$	1.79884e9	1.17491	0.587456	−	0.809256i	$-0.300129\pi$
$421$	1.79884e9	1.17491	0.587456	+	0.809256i	$0.300129\pi$
$422$	−1.46640e9	−0.949859
$423$	0	0
$424$	−8.42459e8	−0.536745
$425$	0	0
$426$	0	0
$427$	−1.40739e9	−0.874819
$428$	−6.75999e8	−0.416767
$429$	0	0
$430$	0	0
$431$	−3.61421e7	−0.0217442	−0.0108721	−	0.999941i	$-0.503461\pi$
$431$	−3.61421e7	−0.0217442	−0.0108721	+	0.999941i	$0.503461\pi$
$432$	0	0
$433$	−1.54727e9	−0.915924	−0.457962	−	0.888972i	$-0.651420\pi$
$433$	−1.54727e9	−0.915924	−0.457962	+	0.888972i	$0.651420\pi$
$434$	−4.74385e8	−0.278559
$435$	0	0
$436$	4.31870e8	0.249546
$437$	1.78188e8	0.102140
$438$	0	0
$439$	−5.66134e8	−0.319370	−0.159685	−	0.987168i	$-0.551048\pi$
$439$	−5.66134e8	−0.319370	−0.159685	+	0.987168i	$0.551048\pi$
$440$	0	0
$441$	0	0
$442$	−1.30813e7	−0.00720564
$443$	−9.51833e8	−0.520173	−0.260087	−	0.965585i	$-0.583751\pi$
$443$	−9.51833e8	−0.520173	−0.260087	+	0.965585i	$0.583751\pi$
$444$	0	0
$445$	0	0
$446$	1.53320e9	0.818326
$447$	0	0
$448$	1.86909e8	0.0982102
$449$	1.12013e9	0.583989	0.291995	−	0.956420i	$-0.405681\pi$
$449$	1.12013e9	0.583989	0.291995	+	0.956420i	$0.405681\pi$
$450$	0	0
$451$	−4.74939e8	−0.243792
$452$	−4.67689e7	−0.0238217
$453$	0	0
$454$	8.88955e8	0.445845
$455$	0	0
$456$	0	0
$457$	2.91044e9	1.42644	0.713218	−	0.700943i	$-0.247237\pi$
$457$	2.91044e9	1.42644	0.713218	+	0.700943i	$0.247237\pi$
$458$	−7.13776e8	−0.347162
$459$	0	0
$460$	0	0
$461$	−4.59640e8	−0.218507	−0.109253	−	0.994014i	$-0.534846\pi$
$461$	−4.59640e8	−0.218507	−0.109253	+	0.994014i	$0.534846\pi$
$462$	0	0
$463$	5.60051e8	0.262237	0.131118	−	0.991367i	$-0.458143\pi$
$463$	5.60051e8	0.262237	0.131118	+	0.991367i	$0.458143\pi$
$464$	3.40845e8	0.158396
$465$	0	0
$466$	4.03736e8	0.184819
$467$	3.70796e9	1.68471	0.842356	−	0.538921i	$-0.181168\pi$
$467$	3.70796e9	1.68471	0.842356	+	0.538921i	$0.181168\pi$
$468$	0	0
$469$	3.14415e9	1.40734
$470$	0	0
$471$	0	0
$472$	1.97683e7	0.00865312
$473$	−7.38214e8	−0.320751
$474$	0	0
$475$	0	0
$476$	1.90833e8	0.0811016
$477$	0	0
$478$	−6.53251e8	−0.273579
$479$	3.96322e9	1.64768	0.823842	−	0.566820i	$-0.191826\pi$
$479$	3.96322e9	1.64768	0.823842	+	0.566820i	$0.191826\pi$
$480$	0	0
$481$	9.04516e7	0.0370603
$482$	−1.92259e9	−0.782026
$483$	0	0
$484$	−3.18149e8	−0.127547
$485$	0	0
$486$	0	0
$487$	1.33008e9	0.521825	0.260913	−	0.965362i	$-0.415977\pi$
$487$	1.33008e9	0.521825	0.260913	+	0.965362i	$0.415977\pi$
$488$	−1.01064e9	−0.393665
$489$	0	0
$490$	0	0
$491$	2.55785e9	0.975191	0.487596	−	0.873070i	$-0.337874\pi$
$491$	2.55785e9	0.975191	0.487596	+	0.873070i	$0.337874\pi$
$492$	0	0
$493$	3.48001e8	0.130803
$494$	4.88281e6	0.00182232
$495$	0	0
$496$	−3.40652e8	−0.125350
$497$	−8.84690e7	−0.0323254
$498$	0	0
$499$	4.31855e9	1.55592	0.777959	−	0.628316i	$-0.216255\pi$
$499$	4.31855e9	1.55592	0.777959	+	0.628316i	$0.216255\pi$
$500$	0	0
$501$	0	0
$502$	2.58993e9	0.913745
$503$	−6.29536e8	−0.220563	−0.110281	−	0.993900i	$-0.535175\pi$
$503$	−6.29536e8	−0.220563	−0.110281	+	0.993900i	$0.535175\pi$
$504$	0	0
$505$	0	0
$506$	3.47929e9	1.19389
$507$	0	0
$508$	−2.26535e9	−0.766676
$509$	−5.79723e9	−1.94853	−0.974267	−	0.225397i	$-0.927632\pi$
$509$	−5.79723e9	−1.94853	−0.974267	+	0.225397i	$0.927632\pi$
$510$	0	0
$511$	2.82892e9	0.937882
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	6.48308e8	0.210577
$515$	0	0
$516$	0	0
$517$	1.21649e9	0.387163
$518$	−1.31953e9	−0.417124
$519$	0	0
$520$	0	0
$521$	−1.39059e9	−0.430790	−0.215395	−	0.976527i	$-0.569104\pi$
$521$	−1.39059e9	−0.430790	−0.215395	+	0.976527i	$0.569104\pi$
$522$	0	0
$523$	1.35994e9	0.415684	0.207842	−	0.978162i	$-0.433356\pi$
$523$	1.35994e9	0.415684	0.207842	+	0.978162i	$0.433356\pi$
$524$	−1.65158e9	−0.501464
$525$	0	0
$526$	−1.69010e9	−0.506363
$527$	−3.47804e8	−0.103514
$528$	0	0
$529$	9.62540e9	2.82699
$530$	0	0
$531$	0	0
$532$	−7.12316e7	−0.0205108
$533$	−4.87405e7	−0.0139426
$534$	0	0
$535$	0	0
$536$	2.25779e9	0.633297
$537$	0	0
$538$	−3.36666e9	−0.932097
$539$	1.20081e9	0.330305
$540$	0	0
$541$	6.15055e8	0.167003	0.0835013	−	0.996508i	$-0.473390\pi$
$541$	6.15055e8	0.167003	0.0835013	+	0.996508i	$0.473390\pi$
$542$	−1.71087e9	−0.461551
$543$	0	0
$544$	1.37036e8	0.0364954
$545$	0	0
$546$	0	0
$547$	3.38341e8	0.0883890	0.0441945	−	0.999023i	$-0.485928\pi$
$547$	3.38341e8	0.0883890	0.0441945	+	0.999023i	$0.485928\pi$
$548$	−6.97948e8	−0.181172
$549$	0	0
$550$	0	0
$551$	−1.29897e8	−0.0330802
$552$	0	0
$553$	5.06800e9	1.27438
$554$	−2.35732e9	−0.589025
$555$	0	0
$556$	−3.40744e9	−0.840748
$557$	−1.09577e9	−0.268674	−0.134337	−	0.990936i	$-0.542890\pi$
$557$	−1.09577e9	−0.268674	−0.134337	+	0.990936i	$0.542890\pi$
$558$	0	0
$559$	−7.57590e7	−0.0183439
$560$	0	0
$561$	0	0
$562$	2.24801e9	0.534222
$563$	−7.36019e8	−0.173824	−0.0869119	−	0.996216i	$-0.527700\pi$
$563$	−7.36019e8	−0.173824	−0.0869119	+	0.996216i	$0.527700\pi$
$564$	0	0
$565$	0	0
$566$	4.65172e9	1.07834
$567$	0	0
$568$	−6.35290e7	−0.0145463
$569$	−9.08142e8	−0.206662	−0.103331	−	0.994647i	$-0.532950\pi$
$569$	−9.08142e8	−0.206662	−0.103331	+	0.994647i	$0.532950\pi$
$570$	0	0
$571$	1.02179e9	0.229685	0.114843	−	0.993384i	$-0.463364\pi$
$571$	1.02179e9	0.229685	0.114843	+	0.993384i	$0.463364\pi$
$572$	9.53414e7	0.0213008
$573$	0	0
$574$	7.11038e8	0.156928
$575$	0	0
$576$	0	0
$577$	9.10202e8	0.197253	0.0986263	−	0.995125i	$-0.468555\pi$
$577$	9.10202e8	0.197253	0.0986263	+	0.995125i	$0.468555\pi$
$578$	−3.14280e9	−0.676969
$579$	0	0
$580$	0	0
$581$	−5.78792e9	−1.22435
$582$	0	0
$583$	6.26908e9	1.31028
$584$	2.03143e9	0.422043
$585$	0	0
$586$	6.33639e9	1.30077
$587$	7.80594e9	1.59291	0.796456	−	0.604697i	$-0.206706\pi$
$587$	7.80594e9	1.59291	0.796456	+	0.604697i	$0.206706\pi$
$588$	0	0
$589$	1.29824e8	0.0261788
$590$	0	0
$591$	0	0
$592$	−9.47544e8	−0.187704
$593$	5.90086e9	1.16205	0.581024	−	0.813886i	$-0.302652\pi$
$593$	5.90086e9	1.16205	0.581024	+	0.813886i	$0.302652\pi$
$594$	0	0
$595$	0	0
$596$	2.17562e9	0.420942
$597$	0	0
$598$	3.57061e8	0.0682792
$599$	3.05005e9	0.579847	0.289924	−	0.957050i	$-0.406370\pi$
$599$	3.05005e9	0.579847	0.289924	+	0.957050i	$0.406370\pi$
$600$	0	0
$601$	6.80081e8	0.127791	0.0638954	−	0.997957i	$-0.479648\pi$
$601$	6.80081e8	0.127791	0.0638954	+	0.997957i	$0.479648\pi$
$602$	1.10519e9	0.206466
$603$	0	0
$604$	2.00582e9	0.370393
$605$	0	0
$606$	0	0
$607$	7.79549e9	1.41476	0.707381	−	0.706833i	$-0.249877\pi$
$607$	7.79549e9	1.41476	0.707381	+	0.706833i	$0.249877\pi$
$608$	−5.11508e7	−0.00922976
$609$	0	0
$610$	0	0
$611$	1.24842e8	0.0221421
$612$	0	0
$613$	−9.21168e9	−1.61520	−0.807602	−	0.589728i	$-0.799235\pi$
$613$	−9.21168e9	−1.61520	−0.807602	+	0.589728i	$0.799235\pi$
$614$	1.62664e9	0.283597
$615$	0	0
$616$	−1.39086e9	−0.239746
$617$	5.26863e9	0.903024	0.451512	−	0.892265i	$-0.350885\pi$
$617$	5.26863e9	0.903024	0.451512	+	0.892265i	$0.350885\pi$
$618$	0	0
$619$	−5.84901e9	−0.991208	−0.495604	−	0.868548i	$-0.665053\pi$
$619$	−5.84901e9	−0.991208	−0.495604	+	0.868548i	$0.665053\pi$
$620$	0	0
$621$	0	0
$622$	−2.63542e9	−0.439121
$623$	−4.79697e9	−0.794802
$624$	0	0
$625$	0	0
$626$	1.29196e9	0.210494
$627$	0	0
$628$	−4.37430e9	−0.704775
$629$	−9.67439e8	−0.155005
$630$	0	0
$631$	2.65927e9	0.421366	0.210683	−	0.977554i	$-0.432431\pi$
$631$	2.65927e9	0.421366	0.210683	+	0.977554i	$0.432431\pi$
$632$	3.63929e9	0.573465
$633$	0	0
$634$	−5.11318e9	−0.796853
$635$	0	0
$636$	0	0
$637$	1.23233e8	0.0188903
$638$	−2.53636e9	−0.386669
$639$	0	0
$640$	0	0
$641$	−4.97659e8	−0.0746327	−0.0373163	−	0.999304i	$-0.511881\pi$
$641$	−4.97659e8	−0.0746327	−0.0373163	+	0.999304i	$0.511881\pi$
$642$	0	0
$643$	2.59642e9	0.385156	0.192578	−	0.981282i	$-0.438315\pi$
$643$	2.59642e9	0.385156	0.192578	+	0.981282i	$0.438315\pi$
$644$	−5.20889e9	−0.768502
$645$	0	0
$646$	−5.22248e7	−0.00762190
$647$	4.44711e9	0.645525	0.322762	−	0.946480i	$-0.395389\pi$
$647$	4.44711e9	0.645525	0.322762	+	0.946480i	$0.395389\pi$
$648$	0	0
$649$	−1.47104e8	−0.0211236
$650$	0	0
$651$	0	0
$652$	2.96177e9	0.418490
$653$	1.83971e9	0.258556	0.129278	−	0.991608i	$-0.458734\pi$
$653$	1.83971e9	0.258556	0.129278	+	0.991608i	$0.458734\pi$
$654$	0	0
$655$	0	0
$656$	5.10591e8	0.0706171
$657$	0	0
$658$	−1.82123e9	−0.249215
$659$	3.41580e9	0.464936	0.232468	−	0.972604i	$-0.425320\pi$
$659$	3.41580e9	0.464936	0.232468	+	0.972604i	$0.425320\pi$
$660$	0	0
$661$	−4.71690e9	−0.635260	−0.317630	−	0.948215i	$-0.602887\pi$
$661$	−4.71690e9	−0.635260	−0.317630	+	0.948215i	$0.602887\pi$
$662$	−4.31521e9	−0.578094
$663$	0	0
$664$	−4.15626e9	−0.550953
$665$	0	0
$666$	0	0
$667$	−9.49888e9	−1.23946
$668$	2.55769e9	0.331994
$669$	0	0
$670$	0	0
$671$	7.52058e9	0.960998
$672$	0	0
$673$	−5.83430e9	−0.737796	−0.368898	−	0.929470i	$-0.620265\pi$
$673$	−5.83430e9	−0.737796	−0.368898	+	0.929470i	$0.620265\pi$
$674$	6.17869e9	0.777297
$675$	0	0
$676$	−4.00612e9	−0.498782
$677$	−1.34655e8	−0.0166787	−0.00833935	−	0.999965i	$-0.502655\pi$
$677$	−1.34655e8	−0.0166787	−0.00833935	+	0.999965i	$0.502655\pi$
$678$	0	0
$679$	−1.01736e10	−1.24718
$680$	0	0
$681$	0	0
$682$	2.53493e9	0.305999
$683$	1.33536e10	1.60371	0.801853	−	0.597521i	$-0.203848\pi$
$683$	1.33536e10	1.60371	0.801853	+	0.597521i	$0.203848\pi$
$684$	0	0
$685$	0	0
$686$	−6.49524e9	−0.768176
$687$	0	0
$688$	7.93629e8	0.0929090
$689$	6.43362e8	0.0749356
$690$	0	0
$691$	−3.26642e9	−0.376616	−0.188308	−	0.982110i	$-0.560300\pi$
$691$	−3.26642e9	−0.376616	−0.188308	+	0.982110i	$0.560300\pi$
$692$	5.80679e9	0.666139
$693$	0	0
$694$	−4.16823e9	−0.473362
$695$	0	0
$696$	0	0
$697$	5.21311e8	0.0583153
$698$	4.17899e9	0.465134
$699$	0	0
$700$	0	0
$701$	−7.82925e9	−0.858434	−0.429217	−	0.903201i	$-0.641211\pi$
$701$	−7.82925e9	−0.858434	−0.429217	+	0.903201i	$0.641211\pi$
$702$	0	0
$703$	3.61112e8	0.0392011
$704$	−9.98769e8	−0.107885
$705$	0	0
$706$	8.18052e9	0.874912
$707$	−4.94194e9	−0.525932
$708$	0	0
$709$	−3.26880e9	−0.344450	−0.172225	−	0.985058i	$-0.555096\pi$
$709$	−3.26880e9	−0.344450	−0.172225	+	0.985058i	$0.555096\pi$
$710$	0	0
$711$	0	0
$712$	−3.44467e9	−0.357658
$713$	9.49351e9	0.980875
$714$	0	0
$715$	0	0
$716$	−8.35649e9	−0.850802
$717$	0	0
$718$	9.43170e9	0.950943
$719$	4.69765e9	0.471335	0.235668	−	0.971834i	$-0.424272\pi$
$719$	4.69765e9	0.471335	0.235668	+	0.971834i	$0.424272\pi$
$720$	0	0
$721$	9.60460e9	0.954346
$722$	−7.13148e9	−0.705179
$723$	0	0
$724$	7.12195e9	0.697452
$725$	0	0
$726$	0	0
$727$	−1.10503e10	−1.06661	−0.533304	−	0.845924i	$-0.679050\pi$
$727$	−1.10503e10	−1.06661	−0.533304	+	0.845924i	$0.679050\pi$
$728$	−1.42737e8	−0.0137112
$729$	0	0
$730$	0	0
$731$	8.10292e8	0.0767239
$732$	0	0
$733$	−1.52446e10	−1.42972	−0.714860	−	0.699268i	$-0.753509\pi$
$733$	−1.52446e10	−1.42972	−0.714860	+	0.699268i	$0.753509\pi$
$734$	−6.28279e9	−0.586430
$735$	0	0
$736$	−3.74047e9	−0.345823
$737$	−1.68012e10	−1.54598
$738$	0	0
$739$	−1.54738e10	−1.41039	−0.705197	−	0.709012i	$-0.749141\pi$
$739$	−1.54738e10	−1.41039	−0.705197	+	0.709012i	$0.749141\pi$
$740$	0	0
$741$	0	0
$742$	−9.38552e9	−0.843422
$743$	−1.25550e9	−0.112294	−0.0561471	−	0.998423i	$-0.517882\pi$
$743$	−1.25550e9	−0.112294	−0.0561471	+	0.998423i	$0.517882\pi$
$744$	0	0
$745$	0	0
$746$	−5.93171e9	−0.523111
$747$	0	0
$748$	−1.01974e9	−0.0890909
$749$	−7.53105e9	−0.654891
$750$	0	0
$751$	−1.75101e10	−1.50851	−0.754257	−	0.656580i	$-0.772003\pi$
$751$	−1.75101e10	−1.50851	−0.754257	+	0.656580i	$0.772003\pi$
$752$	−1.30781e9	−0.112146
$753$	0	0
$754$	−2.60293e8	−0.0221138
$755$	0	0
$756$	0	0
$757$	1.55877e9	0.130601	0.0653005	−	0.997866i	$-0.479199\pi$
$757$	1.55877e9	0.130601	0.0653005	+	0.997866i	$0.479199\pi$
$758$	−3.30559e9	−0.275681
$759$	0	0
$760$	0	0
$761$	−8.14586e9	−0.670025	−0.335012	−	0.942214i	$-0.608740\pi$
$761$	−8.14586e9	−0.670025	−0.335012	+	0.942214i	$0.608740\pi$
$762$	0	0
$763$	4.81130e9	0.392127
$764$	−8.92264e9	−0.723880
$765$	0	0
$766$	1.66227e10	1.33629
$767$	−1.50965e7	−0.00120807
$768$	0	0
$769$	1.61934e10	1.28409	0.642046	−	0.766666i	$-0.278086\pi$
$769$	1.61934e10	1.28409	0.642046	+	0.766666i	$0.278086\pi$
$770$	0	0
$771$	0	0
$772$	−1.52251e9	−0.119097
$773$	3.85870e9	0.300478	0.150239	−	0.988650i	$-0.451996\pi$
$773$	3.85870e9	0.300478	0.150239	+	0.988650i	$0.451996\pi$
$774$	0	0
$775$	0	0
$776$	−7.30556e9	−0.561226
$777$	0	0
$778$	1.30241e10	0.991557
$779$	−1.94588e8	−0.0147481
$780$	0	0
$781$	4.72745e8	0.0355098
$782$	−3.81900e9	−0.285579
$783$	0	0
$784$	−1.29095e9	−0.0956762
$785$	0	0
$786$	0	0
$787$	−9.76950e9	−0.714431	−0.357216	−	0.934022i	$-0.616274\pi$
$787$	−9.76950e9	−0.714431	−0.357216	+	0.934022i	$0.616274\pi$
$788$	−2.99465e9	−0.218024
$789$	0	0
$790$	0	0
$791$	−5.21035e8	−0.0374325
$792$	0	0
$793$	7.71797e8	0.0549600
$794$	5.23498e9	0.371144
$795$	0	0
$796$	−7.47864e9	−0.525566
$797$	−1.86704e10	−1.30632	−0.653159	−	0.757221i	$-0.726557\pi$
$797$	−1.86704e10	−1.30632	−0.653159	+	0.757221i	$0.726557\pi$
$798$	0	0
$799$	−1.33527e9	−0.0926095
$800$	0	0
$801$	0	0
$802$	−1.53384e10	−1.04995
$803$	−1.51167e10	−1.03027
$804$	0	0
$805$	0	0
$806$	2.60146e8	0.0175003
$807$	0	0
$808$	−3.54877e9	−0.236667
$809$	7.97985e9	0.529877	0.264938	−	0.964265i	$-0.414648\pi$
$809$	7.97985e9	0.529877	0.264938	+	0.964265i	$0.414648\pi$
$810$	0	0
$811$	−1.70481e10	−1.12228	−0.561141	−	0.827720i	$-0.689638\pi$
$811$	−1.70481e10	−1.12228	−0.561141	+	0.827720i	$0.689638\pi$
$812$	3.79722e9	0.248897
$813$	0	0
$814$	7.05106e9	0.458215
$815$	0	0
$816$	0	0
$817$	−3.02455e8	−0.0194036
$818$	1.17272e10	0.749132
$819$	0	0
$820$	0	0
$821$	6.64028e9	0.418779	0.209390	−	0.977832i	$-0.432852\pi$
$821$	6.64028e9	0.418779	0.209390	+	0.977832i	$0.432852\pi$
$822$	0	0
$823$	2.99586e10	1.87336	0.936682	−	0.350180i	$-0.113880\pi$
$823$	2.99586e10	1.87336	0.936682	+	0.350180i	$0.113880\pi$
$824$	6.89699e9	0.429452
$825$	0	0
$826$	2.20231e8	0.0135972
$827$	−3.07056e9	−0.188777	−0.0943883	−	0.995535i	$-0.530090\pi$
$827$	−3.07056e9	−0.188777	−0.0943883	+	0.995535i	$0.530090\pi$
$828$	0	0
$829$	−2.10442e10	−1.28289	−0.641447	−	0.767167i	$-0.721666\pi$
$829$	−2.10442e10	−1.28289	−0.641447	+	0.767167i	$0.721666\pi$
$830$	0	0
$831$	0	0
$832$	−1.02498e8	−0.00617000
$833$	−1.31806e9	−0.0790091
$834$	0	0
$835$	0	0
$836$	3.80634e8	0.0225313
$837$	0	0
$838$	1.49601e10	0.878172
$839$	−1.16048e10	−0.678377	−0.339189	−	0.940718i	$-0.610152\pi$
$839$	−1.16048e10	−0.678377	−0.339189	+	0.940718i	$0.610152\pi$
$840$	0	0
$841$	−1.03253e10	−0.598573
$842$	1.43907e10	0.830789
$843$	0	0
$844$	−1.17312e10	−0.671651
$845$	0	0
$846$	0	0
$847$	−3.54437e9	−0.200423
$848$	−6.73967e9	−0.379536
$849$	0	0
$850$	0	0
$851$	2.64068e10	1.46880
$852$	0	0
$853$	3.08015e10	1.69922	0.849610	−	0.527411i	$-0.176837\pi$
$853$	3.08015e10	1.69922	0.849610	+	0.527411i	$0.176837\pi$
$854$	−1.12592e10	−0.618590
$855$	0	0
$856$	−5.40799e9	−0.294698
$857$	−2.03442e10	−1.10410	−0.552050	−	0.833811i	$-0.686154\pi$
$857$	−2.03442e10	−1.10410	−0.552050	+	0.833811i	$0.686154\pi$
$858$	0	0
$859$	−5.51078e9	−0.296645	−0.148323	−	0.988939i	$-0.547387\pi$
$859$	−5.51078e9	−0.296645	−0.148323	+	0.988939i	$0.547387\pi$
$860$	0	0
$861$	0	0
$862$	−2.89137e8	−0.0153755
$863$	−2.54712e10	−1.34900	−0.674501	−	0.738274i	$-0.735641\pi$
$863$	−2.54712e10	−1.34900	−0.674501	+	0.738274i	$0.735641\pi$
$864$	0	0
$865$	0	0
$866$	−1.23782e10	−0.647656
$867$	0	0
$868$	−3.79508e9	−0.196971
$869$	−2.70814e10	−1.39992
$870$	0	0
$871$	−1.72421e9	−0.0884153
$872$	3.45496e9	0.176455
$873$	0	0
$874$	1.42551e9	0.0722235
$875$	0	0
$876$	0	0
$877$	−1.83885e9	−0.0920549	−0.0460275	−	0.998940i	$-0.514656\pi$
$877$	−1.83885e9	−0.0920549	−0.0460275	+	0.998940i	$0.514656\pi$
$878$	−4.52907e9	−0.225828
$879$	0	0
$880$	0	0
$881$	3.40718e10	1.67873	0.839363	−	0.543571i	$-0.182928\pi$
$881$	3.40718e10	1.67873	0.839363	+	0.543571i	$0.182928\pi$
$882$	0	0
$883$	−2.54935e10	−1.24614	−0.623071	−	0.782165i	$-0.714115\pi$
$883$	−2.54935e10	−1.24614	−0.623071	+	0.782165i	$0.714115\pi$
$884$	−1.04650e8	−0.00509516
$885$	0	0
$886$	−7.61467e9	−0.367818
$887$	−2.29613e10	−1.10475	−0.552374	−	0.833596i	$-0.686278\pi$
$887$	−2.29613e10	−1.10475	−0.552374	+	0.833596i	$0.686278\pi$
$888$	0	0
$889$	−2.52374e10	−1.20473
$890$	0	0
$891$	0	0
$892$	1.22656e10	0.578644
$893$	4.98412e8	0.0234211
$894$	0	0
$895$	0	0
$896$	1.49527e9	0.0694451
$897$	0	0
$898$	8.96101e9	0.412943
$899$	−6.92066e9	−0.317679
$900$	0	0
$901$	−6.88118e9	−0.313420
$902$	−3.79951e9	−0.172387
$903$	0	0
$904$	−3.74151e8	−0.0168445
$905$	0	0
$906$	0	0
$907$	5.32481e9	0.236962	0.118481	−	0.992956i	$-0.462198\pi$
$907$	5.32481e9	0.236962	0.118481	+	0.992956i	$0.462198\pi$
$908$	7.11164e9	0.315260
$909$	0	0
$910$	0	0
$911$	−2.29502e10	−1.00571	−0.502853	−	0.864372i	$-0.667716\pi$
$911$	−2.29502e10	−1.00571	−0.502853	+	0.864372i	$0.667716\pi$
$912$	0	0
$913$	3.09284e10	1.34496
$914$	2.32835e10	1.00864
$915$	0	0
$916$	−5.71021e9	−0.245481
$917$	−1.83996e10	−0.787982
$918$	0	0
$919$	−2.49309e10	−1.05958	−0.529790	−	0.848129i	$-0.677729\pi$
$919$	−2.49309e10	−1.05958	−0.529790	+	0.848129i	$0.677729\pi$
$920$	0	0
$921$	0	0
$922$	−3.67712e9	−0.154507
$923$	4.85153e7	0.00203083
$924$	0	0
$925$	0	0
$926$	4.48040e9	0.185429
$927$	0	0
$928$	2.72676e9	0.112003
$929$	−1.04115e10	−0.426049	−0.213024	−	0.977047i	$-0.568331\pi$
$929$	−1.04115e10	−0.426049	−0.213024	+	0.977047i	$0.568331\pi$
$930$	0	0
$931$	4.91987e8	0.0199816
$932$	3.22989e9	0.130687
$933$	0	0
$934$	2.96637e10	1.19127
$935$	0	0
$936$	0	0
$937$	−7.28154e9	−0.289158	−0.144579	−	0.989493i	$-0.546183\pi$
$937$	−7.28154e9	−0.289158	−0.144579	+	0.989493i	$0.546183\pi$
$938$	2.51532e10	0.995139
$939$	0	0
$940$	0	0
$941$	−2.30278e10	−0.900927	−0.450464	−	0.892795i	$-0.648741\pi$
$941$	−2.30278e10	−0.900927	−0.450464	+	0.892795i	$0.648741\pi$
$942$	0	0
$943$	−1.42295e10	−0.552584
$944$	1.58147e8	0.00611868
$945$	0	0
$946$	−5.90571e9	−0.226805
$947$	4.69600e10	1.79681	0.898407	−	0.439164i	$-0.144725\pi$
$947$	4.69600e10	1.79681	0.898407	+	0.439164i	$0.144725\pi$
$948$	0	0
$949$	−1.55134e9	−0.0589219
$950$	0	0
$951$	0	0
$952$	1.52666e9	0.0573475
$953$	−3.43809e10	−1.28674	−0.643372	−	0.765553i	$-0.722465\pi$
$953$	−3.43809e10	−1.28674	−0.643372	+	0.765553i	$0.722465\pi$
$954$	0	0
$955$	0	0
$956$	−5.22600e9	−0.193449
$957$	0	0
$958$	3.17058e10	1.16509
$959$	−7.77558e9	−0.284687
$960$	0	0
$961$	−2.05959e10	−0.748597
$962$	7.23613e8	0.0262056
$963$	0	0
$964$	−1.53807e10	−0.552976
$965$	0	0
$966$	0	0
$967$	1.14483e10	0.407146	0.203573	−	0.979060i	$-0.434745\pi$
$967$	1.14483e10	0.407146	0.203573	+	0.979060i	$0.434745\pi$
$968$	−2.54519e9	−0.0901895
$969$	0	0
$970$	0	0
$971$	−1.85255e8	−0.00649385	−0.00324693	−	0.999995i	$-0.501034\pi$
$971$	−1.85255e8	−0.00649385	−0.00324693	+	0.999995i	$0.501034\pi$
$972$	0	0
$973$	−3.79610e10	−1.32112
$974$	1.06406e10	0.368986
$975$	0	0
$976$	−8.08511e9	−0.278363
$977$	3.06849e10	1.05267	0.526337	−	0.850276i	$-0.323565\pi$
$977$	3.06849e10	1.05267	0.526337	+	0.850276i	$0.323565\pi$
$978$	0	0
$979$	2.56332e10	0.873099
$980$	0	0
$981$	0	0
$982$	2.04628e10	0.689564
$983$	1.00287e9	0.0336751	0.0168376	−	0.999858i	$-0.494640\pi$
$983$	1.00287e9	0.0336751	0.0168376	+	0.999858i	$0.494640\pi$
$984$	0	0
$985$	0	0
$986$	2.78401e9	0.0924914
$987$	0	0
$988$	3.90625e7	0.00128858
$989$	−2.21174e10	−0.727020
$990$	0	0
$991$	4.82282e10	1.57414	0.787070	−	0.616864i	$-0.211597\pi$
$991$	4.82282e10	1.57414	0.787070	+	0.616864i	$0.211597\pi$
$992$	−2.72522e9	−0.0886360
$993$	0	0
$994$	−7.07752e8	−0.0228575
$995$	0	0
$996$	0	0
$997$	−4.68836e10	−1.49826	−0.749132	−	0.662421i	$-0.769529\pi$
$997$	−4.68836e10	−1.49826	−0.749132	+	0.662421i	$0.769529\pi$
$998$	3.45484e10	1.10020
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.8.a.w.1.1		1
3.2	odd	2		150.8.a.h.1.1	✓	1
5.2	odd	4		450.8.c.e.199.2		2
5.3	odd	4		450.8.c.e.199.1		2
5.4	even	2		450.8.a.d.1.1		1
15.2	even	4		150.8.c.c.49.1		2
15.8	even	4		150.8.c.c.49.2		2
15.14	odd	2		150.8.a.j.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.8.a.h.1.1	✓	1	3.2	odd	2
150.8.a.j.1.1	yes	1	15.14	odd	2
150.8.c.c.49.1		2	15.2	even	4
150.8.c.c.49.2		2	15.8	even	4
450.8.a.d.1.1		1	5.4	even	2
450.8.a.w.1.1		1	1.1	even	1	trivial
450.8.c.e.199.1		2	5.3	odd	4
450.8.c.e.199.2		2	5.2	odd	4

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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