LMFDB - Embedded newform 8001.2.a.y.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.487315 q^{2} -1.76252 q^{4} -0.708639 q^{5} -1.00000 q^{7} -1.83353 q^{8} +O(q^{10})$	\(q+0.487315 q^{2} -1.76252 q^{4} -0.708639 q^{5} -1.00000 q^{7} -1.83353 q^{8} -0.345330 q^{10} -4.44553 q^{11} -4.01971 q^{13} -0.487315 q^{14} +2.63154 q^{16} -1.16643 q^{17} +0.231089 q^{19} +1.24899 q^{20} -2.16638 q^{22} -6.97361 q^{23} -4.49783 q^{25} -1.95886 q^{26} +1.76252 q^{28} -0.276783 q^{29} -6.86349 q^{31} +4.94946 q^{32} -0.568417 q^{34} +0.708639 q^{35} -3.38389 q^{37} +0.112613 q^{38} +1.29931 q^{40} -7.55759 q^{41} +5.27833 q^{43} +7.83536 q^{44} -3.39835 q^{46} +5.93194 q^{47} +1.00000 q^{49} -2.19186 q^{50} +7.08483 q^{52} -10.2985 q^{53} +3.15028 q^{55} +1.83353 q^{56} -0.134880 q^{58} -9.69255 q^{59} +9.65994 q^{61} -3.34468 q^{62} -2.85113 q^{64} +2.84852 q^{65} -5.37876 q^{67} +2.05585 q^{68} +0.345330 q^{70} -2.44937 q^{71} -10.1674 q^{73} -1.64902 q^{74} -0.407300 q^{76} +4.44553 q^{77} +9.67284 q^{79} -1.86481 q^{80} -3.68293 q^{82} +4.65761 q^{83} +0.826575 q^{85} +2.57221 q^{86} +8.15104 q^{88} -2.71373 q^{89} +4.01971 q^{91} +12.2912 q^{92} +2.89073 q^{94} -0.163759 q^{95} +5.35612 q^{97} +0.487315 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0.487315	0.344584	0.172292	−	0.985046i	$-0.444883\pi$
$2$	0.487315	0.344584	0.172292	+	0.985046i	$0.444883\pi$
$3$	0	0
$3$	0	0
$4$	−1.76252	−0.881262
$5$	−0.708639	−0.316913	−0.158456	−	0.987366i	$-0.550652\pi$
$5$	−0.708639	−0.316913	−0.158456	+	0.987366i	$0.550652\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.83353	−0.648252
$9$	0	0
$10$	−0.345330	−0.109203
$11$	−4.44553	−1.34038	−0.670189	−	0.742190i	$-0.733787\pi$
$11$	−4.44553	−1.34038	−0.670189	+	0.742190i	$0.733787\pi$
$12$	0	0
$13$	−4.01971	−1.11487	−0.557433	−	0.830222i	$-0.688214\pi$
$13$	−4.01971	−1.11487	−0.557433	+	0.830222i	$0.688214\pi$
$14$	−0.487315	−0.130240
$15$	0	0
$16$	2.63154	0.657885
$17$	−1.16643	−0.282900	−0.141450	−	0.989945i	$-0.545176\pi$
$17$	−1.16643	−0.282900	−0.141450	+	0.989945i	$0.545176\pi$
$18$	0	0
$19$	0.231089	0.0530155	0.0265078	−	0.999649i	$-0.491561\pi$
$19$	0.231089	0.0530155	0.0265078	+	0.999649i	$0.491561\pi$
$20$	1.24899	0.279283
$21$	0	0
$22$	−2.16638	−0.461873
$23$	−6.97361	−1.45410	−0.727049	−	0.686585i	$-0.759109\pi$
$23$	−6.97361	−1.45410	−0.727049	+	0.686585i	$0.759109\pi$
$24$	0	0
$25$	−4.49783	−0.899566
$26$	−1.95886	−0.384165
$27$	0	0
$28$	1.76252	0.333086
$29$	−0.276783	−0.0513973	−0.0256986	−	0.999670i	$-0.508181\pi$
$29$	−0.276783	−0.0513973	−0.0256986	+	0.999670i	$0.508181\pi$
$30$	0	0
$31$	−6.86349	−1.23272	−0.616360	−	0.787465i	$-0.711393\pi$
$31$	−6.86349	−1.23272	−0.616360	+	0.787465i	$0.711393\pi$
$32$	4.94946	0.874949
$33$	0	0
$34$	−0.568417	−0.0974827
$35$	0.708639	0.119782
$36$	0	0
$37$	−3.38389	−0.556307	−0.278154	−	0.960537i	$-0.589722\pi$
$37$	−3.38389	−0.556307	−0.278154	+	0.960537i	$0.589722\pi$
$38$	0.112613	0.0182683
$39$	0	0
$40$	1.29931	0.205440
$41$	−7.55759	−1.18030	−0.590148	−	0.807295i	$-0.700931\pi$
$41$	−7.55759	−1.18030	−0.590148	+	0.807295i	$0.700931\pi$
$42$	0	0
$43$	5.27833	0.804938	0.402469	−	0.915434i	$-0.368152\pi$
$43$	5.27833	0.804938	0.402469	+	0.915434i	$0.368152\pi$
$44$	7.83536	1.18122
$45$	0	0
$46$	−3.39835	−0.501059
$47$	5.93194	0.865263	0.432632	−	0.901571i	$-0.357585\pi$
$47$	5.93194	0.865263	0.432632	+	0.901571i	$0.357585\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.19186	−0.309976
$51$	0	0
$52$	7.08483	0.982489
$53$	−10.2985	−1.41461	−0.707305	−	0.706908i	$-0.750089\pi$
$53$	−10.2985	−1.41461	−0.707305	+	0.706908i	$0.750089\pi$
$54$	0	0
$55$	3.15028	0.424783
$56$	1.83353	0.245016
$57$	0	0
$58$	−0.134880	−0.0177107
$59$	−9.69255	−1.26186	−0.630932	−	0.775839i	$-0.717327\pi$
$59$	−9.69255	−1.26186	−0.630932	+	0.775839i	$0.717327\pi$
$60$	0	0
$61$	9.65994	1.23683	0.618414	−	0.785852i	$-0.287775\pi$
$61$	9.65994	1.23683	0.618414	+	0.785852i	$0.287775\pi$
$62$	−3.34468	−0.424775
$63$	0	0
$64$	−2.85113	−0.356391
$65$	2.84852	0.353316
$66$	0	0
$67$	−5.37876	−0.657120	−0.328560	−	0.944483i	$-0.606563\pi$
$67$	−5.37876	−0.657120	−0.328560	+	0.944483i	$0.606563\pi$
$68$	2.05585	0.249309
$69$	0	0
$70$	0.345330	0.0412749
$71$	−2.44937	−0.290687	−0.145344	−	0.989381i	$-0.546429\pi$
$71$	−2.44937	−0.290687	−0.145344	+	0.989381i	$0.546429\pi$
$72$	0	0
$73$	−10.1674	−1.19000	−0.595000	−	0.803725i	$-0.702848\pi$
$73$	−10.1674	−1.19000	−0.595000	+	0.803725i	$0.702848\pi$
$74$	−1.64902	−0.191695
$75$	0	0
$76$	−0.407300	−0.0467206
$77$	4.44553	0.506615
$78$	0	0
$79$	9.67284	1.08828	0.544140	−	0.838995i	$-0.316856\pi$
$79$	9.67284	1.08828	0.544140	+	0.838995i	$0.316856\pi$
$80$	−1.86481	−0.208492
$81$	0	0
$82$	−3.68293	−0.406711
$83$	4.65761	0.511239	0.255619	−	0.966777i	$-0.417721\pi$
$83$	4.65761	0.511239	0.255619	+	0.966777i	$0.417721\pi$
$84$	0	0
$85$	0.826575	0.0896546
$86$	2.57221	0.277369
$87$	0	0
$88$	8.15104	0.868904
$89$	−2.71373	−0.287655	−0.143827	−	0.989603i	$-0.545941\pi$
$89$	−2.71373	−0.287655	−0.143827	+	0.989603i	$0.545941\pi$
$90$	0	0
$91$	4.01971	0.421380
$92$	12.2912	1.28144
$93$	0	0
$94$	2.89073	0.298156
$95$	−0.163759	−0.0168013
$96$	0	0
$97$	5.35612	0.543831	0.271916	−	0.962321i	$-0.412343\pi$
$97$	5.35612	0.543831	0.271916	+	0.962321i	$0.412343\pi$
$98$	0.487315	0.0492263
$99$	0	0
$100$	7.92753	0.792753
$101$	−8.03179	−0.799193	−0.399596	−	0.916691i	$-0.630850\pi$
$101$	−8.03179	−0.799193	−0.399596	+	0.916691i	$0.630850\pi$
$102$	0	0
$103$	−12.1212	−1.19433	−0.597167	−	0.802117i	$-0.703707\pi$
$103$	−12.1212	−1.19433	−0.597167	+	0.802117i	$0.703707\pi$
$104$	7.37028	0.722715
$105$	0	0
$106$	−5.01862	−0.487452
$107$	11.1977	1.08252	0.541262	−	0.840854i	$-0.317947\pi$
$107$	11.1977	1.08252	0.541262	+	0.840854i	$0.317947\pi$
$108$	0	0
$109$	−10.4591	−1.00180	−0.500900	−	0.865505i	$-0.666997\pi$
$109$	−10.4591	−1.00180	−0.500900	+	0.865505i	$0.666997\pi$
$110$	1.53518	0.146373
$111$	0	0
$112$	−2.63154	−0.248657
$113$	7.87867	0.741163	0.370582	−	0.928800i	$-0.379158\pi$
$113$	7.87867	0.741163	0.370582	+	0.928800i	$0.379158\pi$
$114$	0	0
$115$	4.94177	0.460823
$116$	0.487836	0.0452945
$117$	0	0
$118$	−4.72333	−0.434818
$119$	1.16643	0.106926
$120$	0	0
$121$	8.76276	0.796614
$122$	4.70743	0.426191
$123$	0	0
$124$	12.0971	1.08635
$125$	6.73053	0.601997
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.2883	−0.997756
$129$	0	0
$130$	1.38813	0.121747
$131$	−6.58055	−0.574946	−0.287473	−	0.957789i	$-0.592815\pi$
$131$	−6.58055	−0.574946	−0.287473	+	0.957789i	$0.592815\pi$
$132$	0	0
$133$	−0.231089	−0.0200380
$134$	−2.62115	−0.226433
$135$	0	0
$136$	2.13868	0.183391
$137$	19.1658	1.63744	0.818721	−	0.574191i	$-0.194683\pi$
$137$	19.1658	1.63744	0.818721	+	0.574191i	$0.194683\pi$
$138$	0	0
$139$	−5.15283	−0.437057	−0.218529	−	0.975831i	$-0.570126\pi$
$139$	−5.15283	−0.437057	−0.218529	+	0.975831i	$0.570126\pi$
$140$	−1.24899	−0.105559
$141$	0	0
$142$	−1.19362	−0.100166
$143$	17.8697	1.49434
$144$	0	0
$145$	0.196139	0.0162885
$146$	−4.95471	−0.410055
$147$	0	0
$148$	5.96418	0.490252
$149$	−4.97869	−0.407870	−0.203935	−	0.978984i	$-0.565373\pi$
$149$	−4.97869	−0.407870	−0.203935	+	0.978984i	$0.565373\pi$
$150$	0	0
$151$	−22.9375	−1.86663	−0.933314	−	0.359061i	$-0.883097\pi$
$151$	−22.9375	−1.86663	−0.933314	+	0.359061i	$0.883097\pi$
$152$	−0.423710	−0.0343674
$153$	0	0
$154$	2.16638	0.174571
$155$	4.86374	0.390665
$156$	0	0
$157$	16.3926	1.30827	0.654135	−	0.756377i	$-0.273033\pi$
$157$	16.3926	1.30827	0.654135	+	0.756377i	$0.273033\pi$
$158$	4.71372	0.375003
$159$	0	0
$160$	−3.50738	−0.277283
$161$	6.97361	0.549598
$162$	0	0
$163$	−20.5481	−1.60945	−0.804724	−	0.593649i	$-0.797687\pi$
$163$	−20.5481	−1.60945	−0.804724	+	0.593649i	$0.797687\pi$
$164$	13.3204	1.04015
$165$	0	0
$166$	2.26972	0.176165
$167$	−9.17092	−0.709667	−0.354834	−	0.934929i	$-0.615462\pi$
$167$	−9.17092	−0.709667	−0.354834	+	0.934929i	$0.615462\pi$
$168$	0	0
$169$	3.15805	0.242927
$170$	0.402802	0.0308935
$171$	0	0
$172$	−9.30319	−0.709361
$173$	−12.6056	−0.958386	−0.479193	−	0.877710i	$-0.659071\pi$
$173$	−12.6056	−0.958386	−0.479193	+	0.877710i	$0.659071\pi$
$174$	0	0
$175$	4.49783	0.340004
$176$	−11.6986	−0.881814
$177$	0	0
$178$	−1.32244	−0.0991212
$179$	3.02510	0.226107	0.113053	−	0.993589i	$-0.463937\pi$
$179$	3.02510	0.226107	0.113053	+	0.993589i	$0.463937\pi$
$180$	0	0
$181$	11.3615	0.844494	0.422247	−	0.906481i	$-0.361241\pi$
$181$	11.3615	0.844494	0.422247	+	0.906481i	$0.361241\pi$
$182$	1.95886	0.145201
$183$	0	0
$184$	12.7864	0.942623
$185$	2.39795	0.176301
$186$	0	0
$187$	5.18538	0.379193
$188$	−10.4552	−0.762523
$189$	0	0
$190$	−0.0798022	−0.00578946
$191$	−12.3592	−0.894278	−0.447139	−	0.894464i	$-0.647557\pi$
$191$	−12.3592	−0.894278	−0.447139	+	0.894464i	$0.647557\pi$
$192$	0	0
$193$	21.9860	1.58259	0.791294	−	0.611436i	$-0.209408\pi$
$193$	21.9860	1.58259	0.791294	+	0.611436i	$0.209408\pi$
$194$	2.61012	0.187395
$195$	0	0
$196$	−1.76252	−0.125895
$197$	10.2498	0.730270	0.365135	−	0.930955i	$-0.381023\pi$
$197$	10.2498	0.730270	0.365135	+	0.930955i	$0.381023\pi$
$198$	0	0
$199$	11.1527	0.790595	0.395298	−	0.918553i	$-0.370641\pi$
$199$	11.1527	0.790595	0.395298	+	0.918553i	$0.370641\pi$
$200$	8.24693	0.583146
$201$	0	0
$202$	−3.91401	−0.275389
$203$	0.276783	0.0194263
$204$	0	0
$205$	5.35560	0.374051
$206$	−5.90683	−0.411549
$207$	0	0
$208$	−10.5780	−0.733454
$209$	−1.02732	−0.0710609
$210$	0	0
$211$	16.8010	1.15663	0.578315	−	0.815814i	$-0.303710\pi$
$211$	16.8010	1.15663	0.578315	+	0.815814i	$0.303710\pi$
$212$	18.1514	1.24664
$213$	0	0
$214$	5.45682	0.373021
$215$	−3.74043	−0.255095
$216$	0	0
$217$	6.86349	0.465924
$218$	−5.09687	−0.345204
$219$	0	0
$220$	−5.55244	−0.374345
$221$	4.68869	0.315396
$222$	0	0
$223$	14.3679	0.962148	0.481074	−	0.876680i	$-0.340247\pi$
$223$	14.3679	0.962148	0.481074	+	0.876680i	$0.340247\pi$
$224$	−4.94946	−0.330700
$225$	0	0
$226$	3.83940	0.255393
$227$	−25.0501	−1.66263	−0.831316	−	0.555801i	$-0.812412\pi$
$227$	−25.0501	−1.66263	−0.831316	+	0.555801i	$0.812412\pi$
$228$	0	0
$229$	12.3121	0.813609	0.406804	−	0.913515i	$-0.366643\pi$
$229$	12.3121	0.813609	0.406804	+	0.913515i	$0.366643\pi$
$230$	2.40820	0.158792
$231$	0	0
$232$	0.507491	0.0333184
$233$	−12.9128	−0.845944	−0.422972	−	0.906143i	$-0.639013\pi$
$233$	−12.9128	−0.845944	−0.422972	+	0.906143i	$0.639013\pi$
$234$	0	0
$235$	−4.20361	−0.274213
$236$	17.0834	1.11203
$237$	0	0
$238$	0.568417	0.0368450
$239$	22.9286	1.48313	0.741564	−	0.670883i	$-0.234085\pi$
$239$	22.9286	1.48313	0.741564	+	0.670883i	$0.234085\pi$
$240$	0	0
$241$	3.93591	0.253534	0.126767	−	0.991933i	$-0.459540\pi$
$241$	3.93591	0.253534	0.126767	+	0.991933i	$0.459540\pi$
$242$	4.27022	0.274500
$243$	0	0
$244$	−17.0259	−1.08997
$245$	−0.708639	−0.0452733
$246$	0	0
$247$	−0.928912	−0.0591052
$248$	12.5844	0.799113
$249$	0	0
$250$	3.27989	0.207438
$251$	−24.9566	−1.57524	−0.787622	−	0.616159i	$-0.788688\pi$
$251$	−24.9566	−1.57524	−0.787622	+	0.616159i	$0.788688\pi$
$252$	0	0
$253$	31.0014	1.94904
$254$	0.487315	0.0305769
$255$	0	0
$256$	0.201294	0.0125809
$257$	1.87577	0.117007	0.0585037	−	0.998287i	$-0.481367\pi$
$257$	1.87577	0.117007	0.0585037	+	0.998287i	$0.481367\pi$
$258$	0	0
$259$	3.38389	0.210264
$260$	−5.02059	−0.311364
$261$	0	0
$262$	−3.20680	−0.198117
$263$	22.3178	1.37618	0.688088	−	0.725628i	$-0.258450\pi$
$263$	22.3178	1.37618	0.688088	+	0.725628i	$0.258450\pi$
$264$	0	0
$265$	7.29793	0.448308
$266$	−0.112613	−0.00690477
$267$	0	0
$268$	9.48019	0.579095
$269$	3.14560	0.191790	0.0958952	−	0.995391i	$-0.469429\pi$
$269$	3.14560	0.191790	0.0958952	+	0.995391i	$0.469429\pi$
$270$	0	0
$271$	23.4088	1.42198	0.710991	−	0.703201i	$-0.248247\pi$
$271$	23.4088	1.42198	0.710991	+	0.703201i	$0.248247\pi$
$272$	−3.06949	−0.186115
$273$	0	0
$274$	9.33977	0.564236
$275$	19.9953	1.20576
$276$	0	0
$277$	−12.3914	−0.744525	−0.372263	−	0.928127i	$-0.621418\pi$
$277$	−12.3914	−0.744525	−0.372263	+	0.928127i	$0.621418\pi$
$278$	−2.51105	−0.150603
$279$	0	0
$280$	−1.29931	−0.0776489
$281$	0.509059	0.0303679	0.0151840	−	0.999885i	$-0.495167\pi$
$281$	0.509059	0.0303679	0.0151840	+	0.999885i	$0.495167\pi$
$282$	0	0
$283$	14.4598	0.859547	0.429773	−	0.902937i	$-0.358593\pi$
$283$	14.4598	0.859547	0.429773	+	0.902937i	$0.358593\pi$
$284$	4.31708	0.256172
$285$	0	0
$286$	8.70820	0.514926
$287$	7.55759	0.446110
$288$	0	0
$289$	−15.6395	−0.919968
$290$	0.0955815	0.00561274
$291$	0	0
$292$	17.9202	1.04870
$293$	−10.0307	−0.586002	−0.293001	−	0.956112i	$-0.594654\pi$
$293$	−10.0307	−0.586002	−0.293001	+	0.956112i	$0.594654\pi$
$294$	0	0
$295$	6.86852	0.399901
$296$	6.20447	0.360628
$297$	0	0
$298$	−2.42619	−0.140545
$299$	28.0319	1.62113
$300$	0	0
$301$	−5.27833	−0.304238
$302$	−11.1778	−0.643210
$303$	0	0
$304$	0.608120	0.0348781
$305$	−6.84541	−0.391967
$306$	0	0
$307$	1.70954	0.0975688	0.0487844	−	0.998809i	$-0.484465\pi$
$307$	1.70954	0.0975688	0.0487844	+	0.998809i	$0.484465\pi$
$308$	−7.83536	−0.446461
$309$	0	0
$310$	2.37017	0.134617
$311$	10.1740	0.576912	0.288456	−	0.957493i	$-0.406858\pi$
$311$	10.1740	0.576912	0.288456	+	0.957493i	$0.406858\pi$
$312$	0	0
$313$	10.2723	0.580627	0.290314	−	0.956932i	$-0.406240\pi$
$313$	10.2723	0.580627	0.290314	+	0.956932i	$0.406240\pi$
$314$	7.98836	0.450809
$315$	0	0
$316$	−17.0486	−0.959059
$317$	−9.81675	−0.551364	−0.275682	−	0.961249i	$-0.588904\pi$
$317$	−9.81675	−0.551364	−0.275682	+	0.961249i	$0.588904\pi$
$318$	0	0
$319$	1.23045	0.0688918
$320$	2.02042	0.112945
$321$	0	0
$322$	3.39835	0.189382
$323$	−0.269549	−0.0149981
$324$	0	0
$325$	18.0800	1.00290
$326$	−10.0134	−0.554590
$327$	0	0
$328$	13.8571	0.765130
$329$	−5.93194	−0.327039
$330$	0	0
$331$	14.1848	0.779669	0.389834	−	0.920885i	$-0.372532\pi$
$331$	14.1848	0.779669	0.389834	+	0.920885i	$0.372532\pi$
$332$	−8.20914	−0.450535
$333$	0	0
$334$	−4.46913	−0.244540
$335$	3.81160	0.208250
$336$	0	0
$337$	−9.58219	−0.521975	−0.260988	−	0.965342i	$-0.584048\pi$
$337$	−9.58219	−0.521975	−0.260988	+	0.965342i	$0.584048\pi$
$338$	1.53897	0.0837088
$339$	0	0
$340$	−1.45686	−0.0790092
$341$	30.5119	1.65231
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−9.67801	−0.521803
$345$	0	0
$346$	−6.14290	−0.330244
$347$	−6.57274	−0.352843	−0.176421	−	0.984315i	$-0.556452\pi$
$347$	−6.57274	−0.352843	−0.176421	+	0.984315i	$0.556452\pi$
$348$	0	0
$349$	−12.4702	−0.667513	−0.333757	−	0.942659i	$-0.608316\pi$
$349$	−12.4702	−0.667513	−0.333757	+	0.942659i	$0.608316\pi$
$350$	2.19186	0.117160
$351$	0	0
$352$	−22.0030	−1.17276
$353$	−30.6990	−1.63394	−0.816971	−	0.576679i	$-0.804348\pi$
$353$	−30.6990	−1.63394	−0.816971	+	0.576679i	$0.804348\pi$
$354$	0	0
$355$	1.73572	0.0921225
$356$	4.78301	0.253499
$357$	0	0
$358$	1.47418	0.0779127
$359$	10.9313	0.576930	0.288465	−	0.957490i	$-0.406855\pi$
$359$	10.9313	0.576930	0.288465	+	0.957490i	$0.406855\pi$
$360$	0	0
$361$	−18.9466	−0.997189
$362$	5.53663	0.290999
$363$	0	0
$364$	−7.08483	−0.371346
$365$	7.20499	0.377127
$366$	0	0
$367$	14.3619	0.749687	0.374843	−	0.927088i	$-0.377697\pi$
$367$	14.3619	0.749687	0.374843	+	0.927088i	$0.377697\pi$
$368$	−18.3513	−0.956629
$369$	0	0
$370$	1.16856	0.0607505
$371$	10.2985	0.534672
$372$	0	0
$373$	6.52175	0.337683	0.168842	−	0.985643i	$-0.445997\pi$
$373$	6.52175	0.337683	0.168842	+	0.985643i	$0.445997\pi$
$374$	2.52692	0.130664
$375$	0	0
$376$	−10.8764	−0.560909
$377$	1.11259	0.0573011
$378$	0	0
$379$	−6.56819	−0.337385	−0.168693	−	0.985669i	$-0.553954\pi$
$379$	−6.56819	−0.337385	−0.168693	+	0.985669i	$0.553954\pi$
$380$	0.288629	0.0148063
$381$	0	0
$382$	−6.02281	−0.308154
$383$	−17.8013	−0.909604	−0.454802	−	0.890592i	$-0.650290\pi$
$383$	−17.8013	−0.909604	−0.454802	+	0.890592i	$0.650290\pi$
$384$	0	0
$385$	−3.15028	−0.160553
$386$	10.7141	0.545334
$387$	0	0
$388$	−9.44028	−0.479258
$389$	−26.8008	−1.35885	−0.679427	−	0.733743i	$-0.737772\pi$
$389$	−26.8008	−1.35885	−0.679427	+	0.733743i	$0.737772\pi$
$390$	0	0
$391$	8.13420	0.411364
$392$	−1.83353	−0.0926075
$393$	0	0
$394$	4.99489	0.251639
$395$	−6.85455	−0.344890
$396$	0	0
$397$	14.3047	0.717932	0.358966	−	0.933351i	$-0.383129\pi$
$397$	14.3047	0.717932	0.358966	+	0.933351i	$0.383129\pi$
$398$	5.43489	0.272426
$399$	0	0
$400$	−11.8362	−0.591811
$401$	−7.72247	−0.385642	−0.192821	−	0.981234i	$-0.561764\pi$
$401$	−7.72247	−0.385642	−0.192821	+	0.981234i	$0.561764\pi$
$402$	0	0
$403$	27.5892	1.37432
$404$	14.1562	0.704298
$405$	0	0
$406$	0.134880	0.00669400
$407$	15.0432	0.745662
$408$	0	0
$409$	−2.41265	−0.119298	−0.0596488	−	0.998219i	$-0.518998\pi$
$409$	−2.41265	−0.119298	−0.0596488	+	0.998219i	$0.518998\pi$
$410$	2.60987	0.128892
$411$	0	0
$412$	21.3639	1.05252
$413$	9.69255	0.476939
$414$	0	0
$415$	−3.30056	−0.162018
$416$	−19.8954	−0.975451
$417$	0	0
$418$	−0.500626	−0.0244864
$419$	−18.1299	−0.885705	−0.442852	−	0.896594i	$-0.646033\pi$
$419$	−18.1299	−0.885705	−0.442852	+	0.896594i	$0.646033\pi$
$420$	0	0
$421$	22.2093	1.08242	0.541208	−	0.840889i	$-0.317967\pi$
$421$	22.2093	1.08242	0.541208	+	0.840889i	$0.317967\pi$
$422$	8.18739	0.398556
$423$	0	0
$424$	18.8827	0.917024
$425$	5.24639	0.254487
$426$	0	0
$427$	−9.65994	−0.467477
$428$	−19.7363	−0.953988
$429$	0	0
$430$	−1.82277	−0.0879017
$431$	4.43053	0.213411	0.106706	−	0.994291i	$-0.465970\pi$
$431$	4.43053	0.213411	0.106706	+	0.994291i	$0.465970\pi$
$432$	0	0
$433$	−11.7205	−0.563251	−0.281625	−	0.959524i	$-0.590874\pi$
$433$	−11.7205	−0.563251	−0.281625	+	0.959524i	$0.590874\pi$
$434$	3.34468	0.160550
$435$	0	0
$436$	18.4344	0.882848
$437$	−1.61153	−0.0770898
$438$	0	0
$439$	29.6460	1.41492	0.707462	−	0.706751i	$-0.249840\pi$
$439$	29.6460	1.41492	0.707462	+	0.706751i	$0.249840\pi$
$440$	−5.77614	−0.275367
$441$	0	0
$442$	2.28487	0.108680
$443$	−25.1045	−1.19275	−0.596376	−	0.802706i	$-0.703393\pi$
$443$	−25.1045	−1.19275	−0.596376	+	0.802706i	$0.703393\pi$
$444$	0	0
$445$	1.92305	0.0911615
$446$	7.00171	0.331541
$447$	0	0
$448$	2.85113	0.134703
$449$	−0.786336	−0.0371095	−0.0185548	−	0.999828i	$-0.505907\pi$
$449$	−0.786336	−0.0371095	−0.0185548	+	0.999828i	$0.505907\pi$
$450$	0	0
$451$	33.5975	1.58204
$452$	−13.8864	−0.653159
$453$	0	0
$454$	−12.2073	−0.572916
$455$	−2.84852	−0.133541
$456$	0	0
$457$	10.5485	0.493440	0.246720	−	0.969087i	$-0.420647\pi$
$457$	10.5485	0.493440	0.246720	+	0.969087i	$0.420647\pi$
$458$	5.99989	0.280356
$459$	0	0
$460$	−8.70999	−0.406105
$461$	−29.7725	−1.38664	−0.693322	−	0.720628i	$-0.743853\pi$
$461$	−29.7725	−1.38664	−0.693322	+	0.720628i	$0.743853\pi$
$462$	0	0
$463$	16.0991	0.748187	0.374093	−	0.927391i	$-0.377954\pi$
$463$	16.0991	0.748187	0.374093	+	0.927391i	$0.377954\pi$
$464$	−0.728364	−0.0338135
$465$	0	0
$466$	−6.29259	−0.291499
$467$	23.4876	1.08688	0.543438	−	0.839450i	$-0.317122\pi$
$467$	23.4876	1.08688	0.543438	+	0.839450i	$0.317122\pi$
$468$	0	0
$469$	5.37876	0.248368
$470$	−2.04848	−0.0944894
$471$	0	0
$472$	17.7716	0.818006
$473$	−23.4650	−1.07892
$474$	0	0
$475$	−1.03940	−0.0476910
$476$	−2.05585	−0.0942299
$477$	0	0
$478$	11.1735	0.511062
$479$	−29.7558	−1.35958	−0.679788	−	0.733409i	$-0.737928\pi$
$479$	−29.7558	−1.35958	−0.679788	+	0.733409i	$0.737928\pi$
$480$	0	0
$481$	13.6022	0.620208
$482$	1.91803	0.0873637
$483$	0	0
$484$	−15.4446	−0.702026
$485$	−3.79555	−0.172347
$486$	0	0
$487$	−4.06984	−0.184422	−0.0922110	−	0.995739i	$-0.529393\pi$
$487$	−4.06984	−0.184422	−0.0922110	+	0.995739i	$0.529393\pi$
$488$	−17.7118	−0.801777
$489$	0	0
$490$	−0.345330	−0.0156004
$491$	21.5351	0.971867	0.485934	−	0.873996i	$-0.338480\pi$
$491$	21.5351	0.971867	0.485934	+	0.873996i	$0.338480\pi$
$492$	0	0
$493$	0.322847	0.0145403
$494$	−0.452673	−0.0203667
$495$	0	0
$496$	−18.0615	−0.810987
$497$	2.44937	0.109869
$498$	0	0
$499$	15.9904	0.715830	0.357915	−	0.933754i	$-0.383488\pi$
$499$	15.9904	0.715830	0.357915	+	0.933754i	$0.383488\pi$
$500$	−11.8627	−0.530517
$501$	0	0
$502$	−12.1617	−0.542804
$503$	2.47442	0.110329	0.0551645	−	0.998477i	$-0.482432\pi$
$503$	2.47442	0.110329	0.0551645	+	0.998477i	$0.482432\pi$
$504$	0	0
$505$	5.69164	0.253274
$506$	15.1075	0.671609
$507$	0	0
$508$	−1.76252	−0.0781994
$509$	−12.3113	−0.545689	−0.272845	−	0.962058i	$-0.587965\pi$
$509$	−12.3113	−0.545689	−0.272845	+	0.962058i	$0.587965\pi$
$510$	0	0
$511$	10.1674	0.449778
$512$	22.6747	1.00209
$513$	0	0
$514$	0.914092	0.0403189
$515$	8.58954	0.378500
$516$	0	0
$517$	−26.3707	−1.15978
$518$	1.64902	0.0724537
$519$	0	0
$520$	−5.22286	−0.229038
$521$	−10.6667	−0.467315	−0.233657	−	0.972319i	$-0.575069\pi$
$521$	−10.6667	−0.467315	−0.233657	+	0.972319i	$0.575069\pi$
$522$	0	0
$523$	−33.8109	−1.47845	−0.739223	−	0.673461i	$-0.764807\pi$
$523$	−33.8109	−1.47845	−0.739223	+	0.673461i	$0.764807\pi$
$524$	11.5984	0.506678
$525$	0	0
$526$	10.8758	0.474208
$527$	8.00575	0.348736
$528$	0	0
$529$	25.6313	1.11440
$530$	3.55639	0.154480
$531$	0	0
$532$	0.407300	0.0176587
$533$	30.3793	1.31587
$534$	0	0
$535$	−7.93514	−0.343066
$536$	9.86214	0.425980
$537$	0	0
$538$	1.53290	0.0660879
$539$	−4.44553	−0.191483
$540$	0	0
$541$	29.9095	1.28591	0.642955	−	0.765904i	$-0.277708\pi$
$541$	29.9095	1.28591	0.642955	+	0.765904i	$0.277708\pi$
$542$	11.4074	0.489992
$543$	0	0
$544$	−5.77318	−0.247523
$545$	7.41172	0.317483
$546$	0	0
$547$	4.75633	0.203366	0.101683	−	0.994817i	$-0.467577\pi$
$547$	4.75633	0.203366	0.101683	+	0.994817i	$0.467577\pi$
$548$	−33.7801	−1.44302
$549$	0	0
$550$	9.74399	0.415485
$551$	−0.0639615	−0.00272485
$552$	0	0
$553$	−9.67284	−0.411331
$554$	−6.03850	−0.256551
$555$	0	0
$556$	9.08199	0.385162
$557$	−21.0394	−0.891467	−0.445733	−	0.895166i	$-0.647057\pi$
$557$	−21.0394	−0.891467	−0.445733	+	0.895166i	$0.647057\pi$
$558$	0	0
$559$	−21.2174	−0.897399
$560$	1.86481	0.0788026
$561$	0	0
$562$	0.248072	0.0104643
$563$	0.979301	0.0412726	0.0206363	−	0.999787i	$-0.493431\pi$
$563$	0.979301	0.0412726	0.0206363	+	0.999787i	$0.493431\pi$
$564$	0	0
$565$	−5.58313	−0.234884
$566$	7.04649	0.296186
$567$	0	0
$568$	4.49101	0.188439
$569$	−30.7127	−1.28754	−0.643772	−	0.765217i	$-0.722632\pi$
$569$	−30.7127	−1.28754	−0.643772	+	0.765217i	$0.722632\pi$
$570$	0	0
$571$	−38.9175	−1.62865	−0.814324	−	0.580410i	$-0.802892\pi$
$571$	−38.9175	−1.62865	−0.814324	+	0.580410i	$0.802892\pi$
$572$	−31.4959	−1.31691
$573$	0	0
$574$	3.68293	0.153722
$575$	31.3661	1.30806
$576$	0	0
$577$	−2.38437	−0.0992628	−0.0496314	−	0.998768i	$-0.515805\pi$
$577$	−2.38437	−0.0992628	−0.0496314	+	0.998768i	$0.515805\pi$
$578$	−7.62134	−0.317006
$579$	0	0
$580$	−0.345700	−0.0143544
$581$	−4.65761	−0.193230
$582$	0	0
$583$	45.7824	1.89611
$584$	18.6422	0.771421
$585$	0	0
$586$	−4.88813	−0.201927
$587$	40.0906	1.65472	0.827358	−	0.561675i	$-0.189843\pi$
$587$	40.0906	1.65472	0.827358	+	0.561675i	$0.189843\pi$
$588$	0	0
$589$	−1.58608	−0.0653533
$590$	3.34713	0.137799
$591$	0	0
$592$	−8.90482	−0.365986
$593$	14.8133	0.608309	0.304155	−	0.952623i	$-0.401626\pi$
$593$	14.8133	0.608309	0.304155	+	0.952623i	$0.401626\pi$
$594$	0	0
$595$	−0.826575	−0.0338863
$596$	8.77506	0.359440
$597$	0	0
$598$	13.6604	0.558614
$599$	−8.30624	−0.339384	−0.169692	−	0.985497i	$-0.554277\pi$
$599$	−8.30624	−0.339384	−0.169692	+	0.985497i	$0.554277\pi$
$600$	0	0
$601$	−28.6167	−1.16730	−0.583649	−	0.812006i	$-0.698376\pi$
$601$	−28.6167	−1.16730	−0.583649	+	0.812006i	$0.698376\pi$
$602$	−2.57221	−0.104836
$603$	0	0
$604$	40.4279	1.64499
$605$	−6.20963	−0.252457
$606$	0	0
$607$	−9.28776	−0.376979	−0.188489	−	0.982075i	$-0.560359\pi$
$607$	−9.28776	−0.376979	−0.188489	+	0.982075i	$0.560359\pi$
$608$	1.14377	0.0463859
$609$	0	0
$610$	−3.33587	−0.135065
$611$	−23.8447	−0.964653
$612$	0	0
$613$	−3.36950	−0.136093	−0.0680465	−	0.997682i	$-0.521677\pi$
$613$	−3.36950	−0.136093	−0.0680465	+	0.997682i	$0.521677\pi$
$614$	0.833087	0.0336206
$615$	0	0
$616$	−8.15104	−0.328415
$617$	−5.44174	−0.219076	−0.109538	−	0.993983i	$-0.534937\pi$
$617$	−5.44174	−0.219076	−0.109538	+	0.993983i	$0.534937\pi$
$618$	0	0
$619$	23.7609	0.955032	0.477516	−	0.878623i	$-0.341537\pi$
$619$	23.7609	0.955032	0.477516	+	0.878623i	$0.341537\pi$
$620$	−8.57245	−0.344278
$621$	0	0
$622$	4.95792	0.198795
$623$	2.71373	0.108723
$624$	0	0
$625$	17.7196	0.708786
$626$	5.00587	0.200075
$627$	0	0
$628$	−28.8923	−1.15293
$629$	3.94705	0.157379
$630$	0	0
$631$	8.82905	0.351479	0.175740	−	0.984437i	$-0.443768\pi$
$631$	8.82905	0.351479	0.175740	+	0.984437i	$0.443768\pi$
$632$	−17.7355	−0.705480
$633$	0	0
$634$	−4.78385	−0.189991
$635$	−0.708639	−0.0281215
$636$	0	0
$637$	−4.01971	−0.159267
$638$	0.599615	0.0237390
$639$	0	0
$640$	7.99934	0.316202
$641$	−10.2756	−0.405864	−0.202932	−	0.979193i	$-0.565047\pi$
$641$	−10.2756	−0.405864	−0.202932	+	0.979193i	$0.565047\pi$
$642$	0	0
$643$	21.8800	0.862865	0.431432	−	0.902145i	$-0.358008\pi$
$643$	21.8800	0.862865	0.431432	+	0.902145i	$0.358008\pi$
$644$	−12.2912	−0.484339
$645$	0	0
$646$	−0.131355	−0.00516810
$647$	−39.7972	−1.56459	−0.782295	−	0.622908i	$-0.785951\pi$
$647$	−39.7972	−1.56459	−0.782295	+	0.622908i	$0.785951\pi$
$648$	0	0
$649$	43.0886	1.69137
$650$	8.81064	0.345582
$651$	0	0
$652$	36.2164	1.41835
$653$	19.3461	0.757073	0.378537	−	0.925586i	$-0.376427\pi$
$653$	19.3461	0.757073	0.378537	+	0.925586i	$0.376427\pi$
$654$	0	0
$655$	4.66324	0.182208
$656$	−19.8881	−0.776499
$657$	0	0
$658$	−2.89073	−0.112692
$659$	23.2178	0.904438	0.452219	−	0.891907i	$-0.350632\pi$
$659$	23.2178	0.904438	0.452219	+	0.891907i	$0.350632\pi$
$660$	0	0
$661$	43.0273	1.67357	0.836784	−	0.547533i	$-0.184433\pi$
$661$	43.0273	1.67357	0.836784	+	0.547533i	$0.184433\pi$
$662$	6.91248	0.268661
$663$	0	0
$664$	−8.53989	−0.331412
$665$	0.163759	0.00635030
$666$	0	0
$667$	1.93018	0.0747367
$668$	16.1640	0.625403
$669$	0	0
$670$	1.85745	0.0717595
$671$	−42.9436	−1.65782
$672$	0	0
$673$	−28.5855	−1.10189	−0.550945	−	0.834542i	$-0.685733\pi$
$673$	−28.5855	−1.10189	−0.550945	+	0.834542i	$0.685733\pi$
$674$	−4.66955	−0.179864
$675$	0	0
$676$	−5.56615	−0.214083
$677$	30.3041	1.16468	0.582340	−	0.812945i	$-0.302137\pi$
$677$	30.3041	1.16468	0.582340	+	0.812945i	$0.302137\pi$
$678$	0	0
$679$	−5.35612	−0.205549
$680$	−1.51555	−0.0581188
$681$	0	0
$682$	14.8689	0.569359
$683$	−21.5333	−0.823949	−0.411974	−	0.911195i	$-0.635161\pi$
$683$	−21.5333	−0.823949	−0.411974	+	0.911195i	$0.635161\pi$
$684$	0	0
$685$	−13.5816	−0.518927
$686$	−0.487315	−0.0186058
$687$	0	0
$688$	13.8901	0.529557
$689$	41.3970	1.57710
$690$	0	0
$691$	36.2050	1.37730	0.688651	−	0.725093i	$-0.258203\pi$
$691$	36.2050	1.37730	0.688651	+	0.725093i	$0.258203\pi$
$692$	22.2177	0.844589
$693$	0	0
$694$	−3.20299	−0.121584
$695$	3.65150	0.138509
$696$	0	0
$697$	8.81537	0.333906
$698$	−6.07690	−0.230014
$699$	0	0
$700$	−7.92753	−0.299633
$701$	−28.5639	−1.07884	−0.539422	−	0.842035i	$-0.681357\pi$
$701$	−28.5639	−1.07884	−0.539422	+	0.842035i	$0.681357\pi$
$702$	0	0
$703$	−0.781980	−0.0294929
$704$	12.6748	0.477699
$705$	0	0
$706$	−14.9601	−0.563030
$707$	8.03179	0.302066
$708$	0	0
$709$	38.3923	1.44185	0.720927	−	0.693011i	$-0.243716\pi$
$709$	38.3923	1.44185	0.720927	+	0.693011i	$0.243716\pi$
$710$	0.845843	0.0317439
$711$	0	0
$712$	4.97572	0.186473
$713$	47.8633	1.79250
$714$	0	0
$715$	−12.6632	−0.473577
$716$	−5.33181	−0.199259
$717$	0	0
$718$	5.32697	0.198801
$719$	19.6881	0.734241	0.367121	−	0.930173i	$-0.380344\pi$
$719$	19.6881	0.734241	0.367121	+	0.930173i	$0.380344\pi$
$720$	0	0
$721$	12.1212	0.451416
$722$	−9.23296	−0.343615
$723$	0	0
$724$	−20.0249	−0.744220
$725$	1.24492	0.0462352
$726$	0	0
$727$	−44.1351	−1.63688	−0.818440	−	0.574592i	$-0.805161\pi$
$727$	−44.1351	−1.63688	−0.818440	+	0.574592i	$0.805161\pi$
$728$	−7.37028	−0.273161
$729$	0	0
$730$	3.51110	0.129952
$731$	−6.15679	−0.227717
$732$	0	0
$733$	−1.12685	−0.0416213	−0.0208106	−	0.999783i	$-0.506625\pi$
$733$	−1.12685	−0.0416213	−0.0208106	+	0.999783i	$0.506625\pi$
$734$	6.99879	0.258330
$735$	0	0
$736$	−34.5156	−1.27226
$737$	23.9114	0.880789
$738$	0	0
$739$	0.372615	0.0137069	0.00685344	−	0.999977i	$-0.497818\pi$
$739$	0.372615	0.0137069	0.00685344	+	0.999977i	$0.497818\pi$
$740$	−4.22645	−0.155367
$741$	0	0
$742$	5.01862	0.184239
$743$	−43.0807	−1.58048	−0.790239	−	0.612798i	$-0.790044\pi$
$743$	−43.0807	−1.58048	−0.790239	+	0.612798i	$0.790044\pi$
$744$	0	0
$745$	3.52809	0.129259
$746$	3.17815	0.116360
$747$	0	0
$748$	−9.13936	−0.334168
$749$	−11.1977	−0.409156
$750$	0	0
$751$	−36.9812	−1.34946	−0.674732	−	0.738063i	$-0.735741\pi$
$751$	−36.9812	−1.34946	−0.674732	+	0.738063i	$0.735741\pi$
$752$	15.6101	0.569243
$753$	0	0
$754$	0.542180	0.0197450
$755$	16.2544	0.591559
$756$	0	0
$757$	−12.1174	−0.440416	−0.220208	−	0.975453i	$-0.570674\pi$
$757$	−12.1174	−0.440416	−0.220208	+	0.975453i	$0.570674\pi$
$758$	−3.20078	−0.116257
$759$	0	0
$760$	0.300258	0.0108915
$761$	20.3956	0.739341	0.369670	−	0.929163i	$-0.379471\pi$
$761$	20.3956	0.739341	0.369670	+	0.929163i	$0.379471\pi$
$762$	0	0
$763$	10.4591	0.378645
$764$	21.7833	0.788093
$765$	0	0
$766$	−8.67485	−0.313435
$767$	38.9612	1.40681
$768$	0	0
$769$	51.8100	1.86832	0.934159	−	0.356857i	$-0.116152\pi$
$769$	51.8100	1.86832	0.934159	+	0.356857i	$0.116152\pi$
$770$	−1.53518	−0.0553240
$771$	0	0
$772$	−38.7509	−1.39467
$773$	−7.84456	−0.282149	−0.141075	−	0.989999i	$-0.545056\pi$
$773$	−7.84456	−0.282149	−0.141075	+	0.989999i	$0.545056\pi$
$774$	0	0
$775$	30.8708	1.10891
$776$	−9.82063	−0.352540
$777$	0	0
$778$	−13.0604	−0.468239
$779$	−1.74648	−0.0625741
$780$	0	0
$781$	10.8888	0.389631
$782$	3.96392	0.141749
$783$	0	0
$784$	2.63154	0.0939835
$785$	−11.6164	−0.414608
$786$	0	0
$787$	25.0806	0.894028	0.447014	−	0.894527i	$-0.352487\pi$
$787$	25.0806	0.894028	0.447014	+	0.894527i	$0.352487\pi$
$788$	−18.0656	−0.643559
$789$	0	0
$790$	−3.34033	−0.118843
$791$	−7.87867	−0.280133
$792$	0	0
$793$	−38.8301	−1.37890
$794$	6.97089	0.247388
$795$	0	0
$796$	−19.6569	−0.696722
$797$	−31.9270	−1.13091	−0.565456	−	0.824778i	$-0.691300\pi$
$797$	−31.9270	−1.13091	−0.565456	+	0.824778i	$0.691300\pi$
$798$	0	0
$799$	−6.91917	−0.244783
$800$	−22.2618	−0.787074
$801$	0	0
$802$	−3.76328	−0.132886
$803$	45.1994	1.59505
$804$	0	0
$805$	−4.94177	−0.174175
$806$	13.4446	0.473568
$807$	0	0
$808$	14.7266	0.518079
$809$	−36.2961	−1.27610	−0.638052	−	0.769993i	$-0.720260\pi$
$809$	−36.2961	−1.27610	−0.638052	+	0.769993i	$0.720260\pi$
$810$	0	0
$811$	−31.8823	−1.11954	−0.559770	−	0.828648i	$-0.689111\pi$
$811$	−31.8823	−1.11954	−0.559770	+	0.828648i	$0.689111\pi$
$812$	−0.487836	−0.0171197
$813$	0	0
$814$	7.33077	0.256943
$815$	14.5612	0.510055
$816$	0	0
$817$	1.21977	0.0426742
$818$	−1.17572	−0.0411081
$819$	0	0
$820$	−9.43937	−0.329637
$821$	11.7624	0.410510	0.205255	−	0.978709i	$-0.434198\pi$
$821$	11.7624	0.410510	0.205255	+	0.978709i	$0.434198\pi$
$822$	0	0
$823$	18.3668	0.640226	0.320113	−	0.947379i	$-0.396279\pi$
$823$	18.3668	0.640226	0.320113	+	0.947379i	$0.396279\pi$
$824$	22.2246	0.774231
$825$	0	0
$826$	4.72333	0.164346
$827$	13.4858	0.468946	0.234473	−	0.972123i	$-0.424663\pi$
$827$	13.4858	0.468946	0.234473	+	0.972123i	$0.424663\pi$
$828$	0	0
$829$	−25.1638	−0.873974	−0.436987	−	0.899468i	$-0.643954\pi$
$829$	−25.1638	−0.873974	−0.436987	+	0.899468i	$0.643954\pi$
$830$	−1.60841	−0.0558289
$831$	0	0
$832$	11.4607	0.397329
$833$	−1.16643	−0.0404143
$834$	0	0
$835$	6.49887	0.224903
$836$	1.81067	0.0626232
$837$	0	0
$838$	−8.83499	−0.305200
$839$	5.43112	0.187503	0.0937515	−	0.995596i	$-0.470114\pi$
$839$	5.43112	0.187503	0.0937515	+	0.995596i	$0.470114\pi$
$840$	0	0
$841$	−28.9234	−0.997358
$842$	10.8229	0.372983
$843$	0	0
$844$	−29.6122	−1.01929
$845$	−2.23792	−0.0769868
$846$	0	0
$847$	−8.76276	−0.301092
$848$	−27.1009	−0.930650
$849$	0	0
$850$	2.55664	0.0876922
$851$	23.5979	0.808926
$852$	0	0
$853$	−29.7990	−1.02030	−0.510149	−	0.860086i	$-0.670410\pi$
$853$	−29.7990	−1.02030	−0.510149	+	0.860086i	$0.670410\pi$
$854$	−4.70743	−0.161085
$855$	0	0
$856$	−20.5314	−0.701749
$857$	−28.3224	−0.967474	−0.483737	−	0.875213i	$-0.660721\pi$
$857$	−28.3224	−0.967474	−0.483737	+	0.875213i	$0.660721\pi$
$858$	0	0
$859$	6.73816	0.229903	0.114951	−	0.993371i	$-0.463329\pi$
$859$	6.73816	0.229903	0.114951	+	0.993371i	$0.463329\pi$
$860$	6.59260	0.224806
$861$	0	0
$862$	2.15907	0.0735381
$863$	−41.1313	−1.40013	−0.700063	−	0.714081i	$-0.746845\pi$
$863$	−41.1313	−1.40013	−0.700063	+	0.714081i	$0.746845\pi$
$864$	0	0
$865$	8.93282	0.303725
$866$	−5.71157	−0.194087
$867$	0	0
$868$	−12.0971	−0.410601
$869$	−43.0009	−1.45871
$870$	0	0
$871$	21.6210	0.732601
$872$	19.1771	0.649419
$873$	0	0
$874$	−0.785322	−0.0265639
$875$	−6.73053	−0.227533
$876$	0	0
$877$	35.5769	1.20135	0.600673	−	0.799495i	$-0.294899\pi$
$877$	35.5769	1.20135	0.600673	+	0.799495i	$0.294899\pi$
$878$	14.4469	0.487560
$879$	0	0
$880$	8.29007	0.279458
$881$	55.0792	1.85566	0.927832	−	0.372999i	$-0.121670\pi$
$881$	55.0792	1.85566	0.927832	+	0.372999i	$0.121670\pi$
$882$	0	0
$883$	3.28813	0.110655	0.0553273	−	0.998468i	$-0.482380\pi$
$883$	3.28813	0.110655	0.0553273	+	0.998468i	$0.482380\pi$
$884$	−8.26393	−0.277946
$885$	0	0
$886$	−12.2338	−0.411003
$887$	−50.0083	−1.67912	−0.839558	−	0.543270i	$-0.817186\pi$
$887$	−50.0083	−1.67912	−0.839558	+	0.543270i	$0.817186\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0.937133	0.0314128
$891$	0	0
$892$	−25.3238	−0.847904
$893$	1.37081	0.0458724
$894$	0	0
$895$	−2.14370	−0.0716561
$896$	11.2883	0.377116
$897$	0	0
$898$	−0.383194	−0.0127873
$899$	1.89970	0.0633584
$900$	0	0
$901$	12.0125	0.400193
$902$	16.3726	0.545147
$903$	0	0
$904$	−14.4458	−0.480461
$905$	−8.05120	−0.267631
$906$	0	0
$907$	5.57167	0.185004	0.0925022	−	0.995712i	$-0.470513\pi$
$907$	5.57167	0.185004	0.0925022	+	0.995712i	$0.470513\pi$
$908$	44.1513	1.46521
$909$	0	0
$910$	−1.38813	−0.0460160
$911$	41.4104	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$911$	41.4104	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$912$	0	0
$913$	−20.7055	−0.685254
$914$	5.14046	0.170031
$915$	0	0
$916$	−21.7004	−0.717002
$917$	6.58055	0.217309
$918$	0	0
$919$	−20.9610	−0.691440	−0.345720	−	0.938338i	$-0.612365\pi$
$919$	−20.9610	−0.691440	−0.345720	+	0.938338i	$0.612365\pi$
$920$	−9.06091	−0.298729
$921$	0	0
$922$	−14.5086	−0.477815
$923$	9.84577	0.324077
$924$	0	0
$925$	15.2201	0.500435
$926$	7.84532	0.257813
$927$	0	0
$928$	−1.36992	−0.0449700
$929$	−43.7964	−1.43691	−0.718457	−	0.695571i	$-0.755151\pi$
$929$	−43.7964	−1.43691	−0.718457	+	0.695571i	$0.755151\pi$
$930$	0	0
$931$	0.231089	0.00757365
$932$	22.7591	0.745499
$933$	0	0
$934$	11.4458	0.374520
$935$	−3.67456	−0.120171
$936$	0	0
$937$	−30.5238	−0.997168	−0.498584	−	0.866841i	$-0.666147\pi$
$937$	−30.5238	−0.997168	−0.498584	+	0.866841i	$0.666147\pi$
$938$	2.62115	0.0855836
$939$	0	0
$940$	7.40896	0.241654
$941$	−14.2779	−0.465446	−0.232723	−	0.972543i	$-0.574764\pi$
$941$	−14.2779	−0.465446	−0.232723	+	0.972543i	$0.574764\pi$
$942$	0	0
$943$	52.7037	1.71627
$944$	−25.5063	−0.830160
$945$	0	0
$946$	−11.4349	−0.371779
$947$	−12.8467	−0.417461	−0.208730	−	0.977973i	$-0.566933\pi$
$947$	−12.8467	−0.417461	−0.208730	+	0.977973i	$0.566933\pi$
$948$	0	0
$949$	40.8699	1.32669
$950$	−0.506516	−0.0164335
$951$	0	0
$952$	−2.13868	−0.0693151
$953$	−54.6449	−1.77012	−0.885061	−	0.465475i	$-0.845883\pi$
$953$	−54.6449	−1.77012	−0.885061	+	0.465475i	$0.845883\pi$
$954$	0	0
$955$	8.75819	0.283408
$956$	−40.4122	−1.30702
$957$	0	0
$958$	−14.5004	−0.468488
$959$	−19.1658	−0.618895
$960$	0	0
$961$	16.1075	0.519597
$962$	6.62857	0.213714
$963$	0	0
$964$	−6.93713	−0.223430
$965$	−15.5801	−0.501543
$966$	0	0
$967$	−47.4172	−1.52483	−0.762417	−	0.647086i	$-0.775988\pi$
$967$	−47.4172	−1.52483	−0.762417	+	0.647086i	$0.775988\pi$
$968$	−16.0668	−0.516407
$969$	0	0
$970$	−1.84963	−0.0593880
$971$	−32.8849	−1.05533	−0.527663	−	0.849454i	$-0.676932\pi$
$971$	−32.8849	−1.05533	−0.527663	+	0.849454i	$0.676932\pi$
$972$	0	0
$973$	5.15283	0.165192
$974$	−1.98329	−0.0635488
$975$	0	0
$976$	25.4205	0.813690
$977$	16.2482	0.519827	0.259914	−	0.965632i	$-0.416306\pi$
$977$	16.2482	0.519827	0.259914	+	0.965632i	$0.416306\pi$
$978$	0	0
$979$	12.0640	0.385566
$980$	1.24899	0.0398976
$981$	0	0
$982$	10.4944	0.334890
$983$	48.0077	1.53121	0.765603	−	0.643313i	$-0.222441\pi$
$983$	48.0077	1.53121	0.765603	+	0.643313i	$0.222441\pi$
$984$	0	0
$985$	−7.26342	−0.231432
$986$	0.157328	0.00501034
$987$	0	0
$988$	1.63723	0.0520872
$989$	−36.8090	−1.17046
$990$	0	0
$991$	5.59145	0.177618	0.0888091	−	0.996049i	$-0.471694\pi$
$991$	5.59145	0.177618	0.0888091	+	0.996049i	$0.471694\pi$
$992$	−33.9706	−1.07857
$993$	0	0
$994$	1.19362	0.0378592
$995$	−7.90325	−0.250550
$996$	0	0
$997$	−43.4008	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$997$	−43.4008	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$998$	7.79238	0.246663
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.16	yes	28
3.2	odd	2	inner	8001.2.a.y.1.13	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.13	✓	28	3.2	odd	2	inner
8001.2.a.y.1.16	yes	28	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.487315 q^{2} -1.76252 q^{4} -0.708639 q^{5} -1.00000 q^{7} -1.83353 q^{8} +O(q^{10})\)	\(q+0.487315 q^{2} -1.76252 q^{4} -0.708639 q^{5} -1.00000 q^{7} -1.83353 q^{8} -0.345330 q^{10} -4.44553 q^{11} -4.01971 q^{13} -0.487315 q^{14} +2.63154 q^{16} -1.16643 q^{17} +0.231089 q^{19} +1.24899 q^{20} -2.16638 q^{22} -6.97361 q^{23} -4.49783 q^{25} -1.95886 q^{26} +1.76252 q^{28} -0.276783 q^{29} -6.86349 q^{31} +4.94946 q^{32} -0.568417 q^{34} +0.708639 q^{35} -3.38389 q^{37} +0.112613 q^{38} +1.29931 q^{40} -7.55759 q^{41} +5.27833 q^{43} +7.83536 q^{44} -3.39835 q^{46} +5.93194 q^{47} +1.00000 q^{49} -2.19186 q^{50} +7.08483 q^{52} -10.2985 q^{53} +3.15028 q^{55} +1.83353 q^{56} -0.134880 q^{58} -9.69255 q^{59} +9.65994 q^{61} -3.34468 q^{62} -2.85113 q^{64} +2.84852 q^{65} -5.37876 q^{67} +2.05585 q^{68} +0.345330 q^{70} -2.44937 q^{71} -10.1674 q^{73} -1.64902 q^{74} -0.407300 q^{76} +4.44553 q^{77} +9.67284 q^{79} -1.86481 q^{80} -3.68293 q^{82} +4.65761 q^{83} +0.826575 q^{85} +2.57221 q^{86} +8.15104 q^{88} -2.71373 q^{89} +4.01971 q^{91} +12.2912 q^{92} +2.89073 q^{94} -0.163759 q^{95} +5.35612 q^{97} +0.487315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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