LMFDB - Embedded newform 8024.2.a.bb.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67764 q^{3} -0.206110 q^{5} -1.52712 q^{7} -0.185517 q^{9} +O(q^{10})$	$q-1.67764 q^{3} -0.206110 q^{5} -1.52712 q^{7} -0.185517 q^{9} -0.924767 q^{11} +6.25352 q^{13} +0.345779 q^{15} +1.00000 q^{17} +6.31312 q^{19} +2.56195 q^{21} -2.29592 q^{23} -4.95752 q^{25} +5.34416 q^{27} +7.27772 q^{29} -3.54732 q^{31} +1.55143 q^{33} +0.314754 q^{35} +4.46972 q^{37} -10.4912 q^{39} -1.14821 q^{41} -13.0979 q^{43} +0.0382369 q^{45} +3.24961 q^{47} -4.66792 q^{49} -1.67764 q^{51} +12.6772 q^{53} +0.190604 q^{55} -10.5912 q^{57} +1.00000 q^{59} +3.30794 q^{61} +0.283306 q^{63} -1.28891 q^{65} -8.09469 q^{67} +3.85173 q^{69} +7.68152 q^{71} -1.24187 q^{73} +8.31694 q^{75} +1.41223 q^{77} +5.05149 q^{79} -8.40903 q^{81} -1.86511 q^{83} -0.206110 q^{85} -12.2094 q^{87} -9.61501 q^{89} -9.54986 q^{91} +5.95113 q^{93} -1.30120 q^{95} -10.4511 q^{97} +0.171560 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

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	0
$2$	0	0
$3$	−1.67764	−0.968587	−0.484294	−	0.874906i	$-0.660923\pi$
$3$	−1.67764	−0.968587	−0.484294	+	0.874906i	$0.660923\pi$
$4$	0	0
$5$	−0.206110	−0.0921751	−0.0460876	−	0.998937i	$-0.514675\pi$
$5$	−0.206110	−0.0921751	−0.0460876	+	0.998937i	$0.514675\pi$
$6$	0	0
$7$	−1.52712	−0.577196	−0.288598	−	0.957450i	$-0.593189\pi$
$7$	−1.52712	−0.577196	−0.288598	+	0.957450i	$0.593189\pi$
$8$	0	0
$9$	−0.185517	−0.0618390
$10$	0	0
$11$	−0.924767	−0.278828	−0.139414	−	0.990234i	$-0.544522\pi$
$11$	−0.924767	−0.278828	−0.139414	+	0.990234i	$0.544522\pi$
$12$	0	0
$13$	6.25352	1.73441	0.867207	−	0.497947i	$-0.165913\pi$
$13$	6.25352	1.73441	0.867207	+	0.497947i	$0.165913\pi$
$14$	0	0
$15$	0.345779	0.0892797
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	6.31312	1.44833	0.724164	−	0.689628i	$-0.242226\pi$
$19$	6.31312	1.44833	0.724164	+	0.689628i	$0.242226\pi$
$20$	0	0
$21$	2.56195	0.559064
$22$	0	0
$23$	−2.29592	−0.478732	−0.239366	−	0.970929i	$-0.576940\pi$
$23$	−2.29592	−0.478732	−0.239366	+	0.970929i	$0.576940\pi$
$24$	0	0
$25$	−4.95752	−0.991504
$26$	0	0
$27$	5.34416	1.02848
$28$	0	0
$29$	7.27772	1.35144	0.675719	−	0.737159i	$-0.263833\pi$
$29$	7.27772	1.35144	0.675719	+	0.737159i	$0.263833\pi$
$30$	0	0
$31$	−3.54732	−0.637118	−0.318559	−	0.947903i	$-0.603199\pi$
$31$	−3.54732	−0.637118	−0.318559	+	0.947903i	$0.603199\pi$
$32$	0	0
$33$	1.55143	0.270069
$34$	0	0
$35$	0.314754	0.0532031
$36$	0	0
$37$	4.46972	0.734818	0.367409	−	0.930060i	$-0.380245\pi$
$37$	4.46972	0.734818	0.367409	+	0.930060i	$0.380245\pi$
$38$	0	0
$39$	−10.4912	−1.67993
$40$	0	0
$41$	−1.14821	−0.179321	−0.0896603	−	0.995972i	$-0.528578\pi$
$41$	−1.14821	−0.179321	−0.0896603	+	0.995972i	$0.528578\pi$
$42$	0	0
$43$	−13.0979	−1.99742	−0.998708	−	0.0508126i	$-0.983819\pi$
$43$	−13.0979	−1.99742	−0.998708	+	0.0508126i	$0.983819\pi$
$44$	0	0
$45$	0.0382369	0.00570001
$46$	0	0
$47$	3.24961	0.474004	0.237002	−	0.971509i	$-0.423835\pi$
$47$	3.24961	0.474004	0.237002	+	0.971509i	$0.423835\pi$
$48$	0	0
$49$	−4.66792	−0.666845
$50$	0	0
$51$	−1.67764	−0.234917
$52$	0	0
$53$	12.6772	1.74135	0.870675	−	0.491859i	$-0.163683\pi$
$53$	12.6772	1.74135	0.870675	+	0.491859i	$0.163683\pi$
$54$	0	0
$55$	0.190604	0.0257010
$56$	0	0
$57$	−10.5912	−1.40283
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	3.30794	0.423538	0.211769	−	0.977320i	$-0.432078\pi$
$61$	3.30794	0.423538	0.211769	+	0.977320i	$0.432078\pi$
$62$	0	0
$63$	0.283306	0.0356932
$64$	0	0
$65$	−1.28891	−0.159870
$66$	0	0
$67$	−8.09469	−0.988923	−0.494462	−	0.869199i	$-0.664635\pi$
$67$	−8.09469	−0.988923	−0.494462	+	0.869199i	$0.664635\pi$
$68$	0	0
$69$	3.85173	0.463694
$70$	0	0
$71$	7.68152	0.911629	0.455814	−	0.890075i	$-0.349348\pi$
$71$	7.68152	0.911629	0.455814	+	0.890075i	$0.349348\pi$
$72$	0	0
$73$	−1.24187	−0.145349	−0.0726747	−	0.997356i	$-0.523153\pi$
$73$	−1.24187	−0.145349	−0.0726747	+	0.997356i	$0.523153\pi$
$74$	0	0
$75$	8.31694	0.960358
$76$	0	0
$77$	1.41223	0.160938
$78$	0	0
$79$	5.05149	0.568337	0.284168	−	0.958774i	$-0.408283\pi$
$79$	5.05149	0.568337	0.284168	+	0.958774i	$0.408283\pi$
$80$	0	0
$81$	−8.40903	−0.934337
$82$	0	0
$83$	−1.86511	−0.204722	−0.102361	−	0.994747i	$-0.532640\pi$
$83$	−1.86511	−0.204722	−0.102361	+	0.994747i	$0.532640\pi$
$84$	0	0
$85$	−0.206110	−0.0223558
$86$	0	0
$87$	−12.2094	−1.30899
$88$	0	0
$89$	−9.61501	−1.01919	−0.509594	−	0.860415i	$-0.670205\pi$
$89$	−9.61501	−1.01919	−0.509594	+	0.860415i	$0.670205\pi$
$90$	0	0
$91$	−9.54986	−1.00110
$92$	0	0
$93$	5.95113	0.617104
$94$	0	0
$95$	−1.30120	−0.133500
$96$	0	0
$97$	−10.4511	−1.06115	−0.530576	−	0.847637i	$-0.678024\pi$
$97$	−10.4511	−1.06115	−0.530576	+	0.847637i	$0.678024\pi$
$98$	0	0
$99$	0.171560	0.0172424
$100$	0	0
$101$	−12.2876	−1.22266	−0.611330	−	0.791376i	$-0.709365\pi$
$101$	−12.2876	−1.22266	−0.611330	+	0.791376i	$0.709365\pi$
$102$	0	0
$103$	2.35296	0.231844	0.115922	−	0.993258i	$-0.463018\pi$
$103$	2.35296	0.231844	0.115922	+	0.993258i	$0.463018\pi$
$104$	0	0
$105$	−0.528044	−0.0515318
$106$	0	0
$107$	−11.2802	−1.09050	−0.545248	−	0.838275i	$-0.683565\pi$
$107$	−11.2802	−1.09050	−0.545248	+	0.838275i	$0.683565\pi$
$108$	0	0
$109$	12.1164	1.16054	0.580272	−	0.814422i	$-0.302946\pi$
$109$	12.1164	1.16054	0.580272	+	0.814422i	$0.302946\pi$
$110$	0	0
$111$	−7.49859	−0.711735
$112$	0	0
$113$	−5.63283	−0.529893	−0.264946	−	0.964263i	$-0.585354\pi$
$113$	−5.63283	−0.529893	−0.264946	+	0.964263i	$0.585354\pi$
$114$	0	0
$115$	0.473212	0.0441272
$116$	0	0
$117$	−1.16013	−0.107254
$118$	0	0
$119$	−1.52712	−0.139991
$120$	0	0
$121$	−10.1448	−0.922255
$122$	0	0
$123$	1.92629	0.173688
$124$	0	0
$125$	2.05234	0.183567
$126$	0	0
$127$	−17.1548	−1.52224	−0.761122	−	0.648609i	$-0.775351\pi$
$127$	−17.1548	−1.52224	−0.761122	+	0.648609i	$0.775351\pi$
$128$	0	0
$129$	21.9736	1.93467
$130$	0	0
$131$	−12.2046	−1.06632	−0.533159	−	0.846015i	$-0.678995\pi$
$131$	−12.2046	−1.06632	−0.533159	+	0.846015i	$0.678995\pi$
$132$	0	0
$133$	−9.64087	−0.835969
$134$	0	0
$135$	−1.10148	−0.0948006
$136$	0	0
$137$	17.1391	1.46429	0.732144	−	0.681150i	$-0.238520\pi$
$137$	17.1391	1.46429	0.732144	+	0.681150i	$0.238520\pi$
$138$	0	0
$139$	16.0518	1.36150	0.680748	−	0.732517i	$-0.261655\pi$
$139$	16.0518	1.36150	0.680748	+	0.732517i	$0.261655\pi$
$140$	0	0
$141$	−5.45168	−0.459114
$142$	0	0
$143$	−5.78305	−0.483603
$144$	0	0
$145$	−1.50001	−0.124569
$146$	0	0
$147$	7.83109	0.645898
$148$	0	0
$149$	6.35763	0.520837	0.260419	−	0.965496i	$-0.416139\pi$
$149$	6.35763	0.520837	0.260419	+	0.965496i	$0.416139\pi$
$150$	0	0
$151$	13.8475	1.12689	0.563446	−	0.826153i	$-0.309475\pi$
$151$	13.8475	1.12689	0.563446	+	0.826153i	$0.309475\pi$
$152$	0	0
$153$	−0.185517	−0.0149981
$154$	0	0
$155$	0.731138	0.0587264
$156$	0	0
$157$	−7.79033	−0.621736	−0.310868	−	0.950453i	$-0.600620\pi$
$157$	−7.79033	−0.621736	−0.310868	+	0.950453i	$0.600620\pi$
$158$	0	0
$159$	−21.2678	−1.68665
$160$	0	0
$161$	3.50614	0.276322
$162$	0	0
$163$	16.4743	1.29037	0.645184	−	0.764028i	$-0.276781\pi$
$163$	16.4743	1.29037	0.645184	+	0.764028i	$0.276781\pi$
$164$	0	0
$165$	−0.319765	−0.0248936
$166$	0	0
$167$	1.57868	0.122162	0.0610812	−	0.998133i	$-0.480545\pi$
$167$	1.57868	0.122162	0.0610812	+	0.998133i	$0.480545\pi$
$168$	0	0
$169$	26.1065	2.00819
$170$	0	0
$171$	−1.17119	−0.0895631
$172$	0	0
$173$	20.2029	1.53600	0.767999	−	0.640451i	$-0.221252\pi$
$173$	20.2029	1.53600	0.767999	+	0.640451i	$0.221252\pi$
$174$	0	0
$175$	7.57071	0.572292
$176$	0	0
$177$	−1.67764	−0.126099
$178$	0	0
$179$	18.6684	1.39534	0.697670	−	0.716419i	$-0.254220\pi$
$179$	18.6684	1.39534	0.697670	+	0.716419i	$0.254220\pi$
$180$	0	0
$181$	24.3081	1.80681	0.903404	−	0.428790i	$-0.141060\pi$
$181$	24.3081	1.80681	0.903404	+	0.428790i	$0.141060\pi$
$182$	0	0
$183$	−5.54953	−0.410233
$184$	0	0
$185$	−0.921254	−0.0677319
$186$	0	0
$187$	−0.924767	−0.0676256
$188$	0	0
$189$	−8.16115	−0.593636
$190$	0	0
$191$	−13.1752	−0.953324	−0.476662	−	0.879087i	$-0.658153\pi$
$191$	−13.1752	−0.953324	−0.476662	+	0.879087i	$0.658153\pi$
$192$	0	0
$193$	14.4503	1.04016	0.520078	−	0.854119i	$-0.325903\pi$
$193$	14.4503	1.04016	0.520078	+	0.854119i	$0.325903\pi$
$194$	0	0
$195$	2.16233	0.154848
$196$	0	0
$197$	6.82149	0.486011	0.243005	−	0.970025i	$-0.421867\pi$
$197$	6.82149	0.486011	0.243005	+	0.970025i	$0.421867\pi$
$198$	0	0
$199$	−25.0885	−1.77848	−0.889239	−	0.457442i	$-0.848766\pi$
$199$	−25.0885	−1.77848	−0.889239	+	0.457442i	$0.848766\pi$
$200$	0	0
$201$	13.5800	0.957858
$202$	0	0
$203$	−11.1139	−0.780044
$204$	0	0
$205$	0.236658	0.0165289
$206$	0	0
$207$	0.425932	0.0296043
$208$	0	0
$209$	−5.83816	−0.403834
$210$	0	0
$211$	2.77550	0.191073	0.0955366	−	0.995426i	$-0.469543\pi$
$211$	2.77550	0.191073	0.0955366	+	0.995426i	$0.469543\pi$
$212$	0	0
$213$	−12.8868	−0.882992
$214$	0	0
$215$	2.69961	0.184112
$216$	0	0
$217$	5.41717	0.367742
$218$	0	0
$219$	2.08341	0.140784
$220$	0	0
$221$	6.25352	0.420657
$222$	0	0
$223$	−4.38436	−0.293598	−0.146799	−	0.989166i	$-0.546897\pi$
$223$	−4.38436	−0.293598	−0.146799	+	0.989166i	$0.546897\pi$
$224$	0	0
$225$	0.919703	0.0613136
$226$	0	0
$227$	13.9934	0.928777	0.464388	−	0.885632i	$-0.346274\pi$
$227$	13.9934	0.928777	0.464388	+	0.885632i	$0.346274\pi$
$228$	0	0
$229$	26.2064	1.73177	0.865885	−	0.500244i	$-0.166756\pi$
$229$	26.2064	1.73177	0.865885	+	0.500244i	$0.166756\pi$
$230$	0	0
$231$	−2.36921	−0.155883
$232$	0	0
$233$	19.5210	1.27886	0.639431	−	0.768848i	$-0.279170\pi$
$233$	19.5210	1.27886	0.639431	+	0.768848i	$0.279170\pi$
$234$	0	0
$235$	−0.669777	−0.0436914
$236$	0	0
$237$	−8.47459	−0.550484
$238$	0	0
$239$	−7.45616	−0.482299	−0.241150	−	0.970488i	$-0.577524\pi$
$239$	−7.45616	−0.482299	−0.241150	+	0.970488i	$0.577524\pi$
$240$	0	0
$241$	24.8126	1.59832	0.799159	−	0.601120i	$-0.205279\pi$
$241$	24.8126	1.59832	0.799159	+	0.601120i	$0.205279\pi$
$242$	0	0
$243$	−1.92512	−0.123497
$244$	0	0
$245$	0.962103	0.0614665
$246$	0	0
$247$	39.4792	2.51200
$248$	0	0
$249$	3.12899	0.198292
$250$	0	0
$251$	−29.4842	−1.86103	−0.930513	−	0.366258i	$-0.880639\pi$
$251$	−29.4842	−1.86103	−0.930513	+	0.366258i	$0.880639\pi$
$252$	0	0
$253$	2.12319	0.133484
$254$	0	0
$255$	0.345779	0.0216535
$256$	0	0
$257$	−9.20906	−0.574445	−0.287223	−	0.957864i	$-0.592732\pi$
$257$	−9.20906	−0.574445	−0.287223	+	0.957864i	$0.592732\pi$
$258$	0	0
$259$	−6.82579	−0.424134
$260$	0	0
$261$	−1.35014	−0.0835715
$262$	0	0
$263$	−18.5331	−1.14280	−0.571399	−	0.820672i	$-0.693599\pi$
$263$	−18.5331	−1.14280	−0.571399	+	0.820672i	$0.693599\pi$
$264$	0	0
$265$	−2.61290	−0.160509
$266$	0	0
$267$	16.1305	0.987173
$268$	0	0
$269$	−25.7016	−1.56705	−0.783527	−	0.621357i	$-0.786582\pi$
$269$	−25.7016	−1.56705	−0.783527	+	0.621357i	$0.786582\pi$
$270$	0	0
$271$	10.2304	0.621453	0.310727	−	0.950499i	$-0.399428\pi$
$271$	10.2304	0.621453	0.310727	+	0.950499i	$0.399428\pi$
$272$	0	0
$273$	16.0212	0.969650
$274$	0	0
$275$	4.58455	0.276459
$276$	0	0
$277$	−18.9461	−1.13836	−0.569179	−	0.822214i	$-0.692739\pi$
$277$	−18.9461	−1.13836	−0.569179	+	0.822214i	$0.692739\pi$
$278$	0	0
$279$	0.658088	0.0393987
$280$	0	0
$281$	−8.61372	−0.513851	−0.256926	−	0.966431i	$-0.582710\pi$
$281$	−8.61372	−0.513851	−0.256926	+	0.966431i	$0.582710\pi$
$282$	0	0
$283$	−1.57466	−0.0936036	−0.0468018	−	0.998904i	$-0.514903\pi$
$283$	−1.57466	−0.0936036	−0.0468018	+	0.998904i	$0.514903\pi$
$284$	0	0
$285$	2.18294	0.129306
$286$	0	0
$287$	1.75345	0.103503
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	17.5333	1.02782
$292$	0	0
$293$	9.56839	0.558991	0.279496	−	0.960147i	$-0.409833\pi$
$293$	9.56839	0.558991	0.279496	+	0.960147i	$0.409833\pi$
$294$	0	0
$295$	−0.206110	−0.0120002
$296$	0	0
$297$	−4.94210	−0.286770
$298$	0	0
$299$	−14.3576	−0.830320
$300$	0	0
$301$	20.0021	1.15290
$302$	0	0
$303$	20.6141	1.18425
$304$	0	0
$305$	−0.681798	−0.0390397
$306$	0	0
$307$	9.90718	0.565432	0.282716	−	0.959204i	$-0.408765\pi$
$307$	9.90718	0.565432	0.282716	+	0.959204i	$0.408765\pi$
$308$	0	0
$309$	−3.94743	−0.224562
$310$	0	0
$311$	13.3566	0.757381	0.378691	−	0.925523i	$-0.376374\pi$
$311$	13.3566	0.757381	0.378691	+	0.925523i	$0.376374\pi$
$312$	0	0
$313$	2.68202	0.151597	0.0757984	−	0.997123i	$-0.475849\pi$
$313$	2.68202	0.151597	0.0757984	+	0.997123i	$0.475849\pi$
$314$	0	0
$315$	−0.0583921	−0.00329002
$316$	0	0
$317$	23.8159	1.33763	0.668816	−	0.743428i	$-0.266801\pi$
$317$	23.8159	1.33763	0.668816	+	0.743428i	$0.266801\pi$
$318$	0	0
$319$	−6.73019	−0.376818
$320$	0	0
$321$	18.9241	1.05624
$322$	0	0
$323$	6.31312	0.351271
$324$	0	0
$325$	−31.0020	−1.71968
$326$	0	0
$327$	−20.3271	−1.12409
$328$	0	0
$329$	−4.96253	−0.273593
$330$	0	0
$331$	−5.26556	−0.289421	−0.144711	−	0.989474i	$-0.546225\pi$
$331$	−5.26556	−0.289421	−0.144711	+	0.989474i	$0.546225\pi$
$332$	0	0
$333$	−0.829209	−0.0454404
$334$	0	0
$335$	1.66839	0.0911542
$336$	0	0
$337$	−5.46246	−0.297559	−0.148780	−	0.988870i	$-0.547535\pi$
$337$	−5.46246	−0.297559	−0.148780	+	0.988870i	$0.547535\pi$
$338$	0	0
$339$	9.44988	0.513247
$340$	0	0
$341$	3.28044	0.177646
$342$	0	0
$343$	17.8183	0.962096
$344$	0	0
$345$	−0.793880	−0.0427410
$346$	0	0
$347$	25.9583	1.39352	0.696758	−	0.717307i	$-0.254625\pi$
$347$	25.9583	1.39352	0.696758	+	0.717307i	$0.254625\pi$
$348$	0	0
$349$	19.0858	1.02164	0.510819	−	0.859688i	$-0.329342\pi$
$349$	19.0858	1.02164	0.510819	+	0.859688i	$0.329342\pi$
$350$	0	0
$351$	33.4198	1.78382
$352$	0	0
$353$	−34.6010	−1.84163	−0.920813	−	0.390005i	$-0.872473\pi$
$353$	−34.6010	−1.84163	−0.920813	+	0.390005i	$0.872473\pi$
$354$	0	0
$355$	−1.58324	−0.0840295
$356$	0	0
$357$	2.56195	0.135593
$358$	0	0
$359$	−6.58154	−0.347361	−0.173680	−	0.984802i	$-0.555566\pi$
$359$	−6.58154	−0.347361	−0.173680	+	0.984802i	$0.555566\pi$
$360$	0	0
$361$	20.8555	1.09766
$362$	0	0
$363$	17.0194	0.893284
$364$	0	0
$365$	0.255961	0.0133976
$366$	0	0
$367$	30.8416	1.60992	0.804958	−	0.593331i	$-0.202188\pi$
$367$	30.8416	1.60992	0.804958	+	0.593331i	$0.202188\pi$
$368$	0	0
$369$	0.213013	0.0110890
$370$	0	0
$371$	−19.3596	−1.00510
$372$	0	0
$373$	5.36781	0.277934	0.138967	−	0.990297i	$-0.455622\pi$
$373$	5.36781	0.277934	0.138967	+	0.990297i	$0.455622\pi$
$374$	0	0
$375$	−3.44310	−0.177801
$376$	0	0
$377$	45.5114	2.34395
$378$	0	0
$379$	−6.72331	−0.345353	−0.172677	−	0.984979i	$-0.555242\pi$
$379$	−6.72331	−0.345353	−0.172677	+	0.984979i	$0.555242\pi$
$380$	0	0
$381$	28.7796	1.47443
$382$	0	0
$383$	8.41188	0.429827	0.214913	−	0.976633i	$-0.431053\pi$
$383$	8.41188	0.429827	0.214913	+	0.976633i	$0.431053\pi$
$384$	0	0
$385$	−0.291074	−0.0148345
$386$	0	0
$387$	2.42989	0.123518
$388$	0	0
$389$	−35.8968	−1.82004	−0.910020	−	0.414565i	$-0.863934\pi$
$389$	−35.8968	−1.82004	−0.910020	+	0.414565i	$0.863934\pi$
$390$	0	0
$391$	−2.29592	−0.116110
$392$	0	0
$393$	20.4749	1.03282
$394$	0	0
$395$	−1.04116	−0.0523865
$396$	0	0
$397$	9.90952	0.497345	0.248672	−	0.968588i	$-0.420006\pi$
$397$	9.90952	0.497345	0.248672	+	0.968588i	$0.420006\pi$
$398$	0	0
$399$	16.1739	0.809709
$400$	0	0
$401$	−30.1133	−1.50379	−0.751894	−	0.659284i	$-0.770860\pi$
$401$	−30.1133	−1.50379	−0.751894	+	0.659284i	$0.770860\pi$
$402$	0	0
$403$	−22.1832	−1.10503
$404$	0	0
$405$	1.73318	0.0861226
$406$	0	0
$407$	−4.13345	−0.204888
$408$	0	0
$409$	23.4023	1.15717	0.578585	−	0.815622i	$-0.303605\pi$
$409$	23.4023	1.15717	0.578585	+	0.815622i	$0.303605\pi$
$410$	0	0
$411$	−28.7532	−1.41829
$412$	0	0
$413$	−1.52712	−0.0751445
$414$	0	0
$415$	0.384418	0.0188703
$416$	0	0
$417$	−26.9292	−1.31873
$418$	0	0
$419$	−35.2594	−1.72253	−0.861267	−	0.508153i	$-0.830328\pi$
$419$	−35.2594	−1.72253	−0.861267	+	0.508153i	$0.830328\pi$
$420$	0	0
$421$	8.43406	0.411051	0.205525	−	0.978652i	$-0.434110\pi$
$421$	8.43406	0.411051	0.205525	+	0.978652i	$0.434110\pi$
$422$	0	0
$423$	−0.602857	−0.0293119
$424$	0	0
$425$	−4.95752	−0.240475
$426$	0	0
$427$	−5.05160	−0.244464
$428$	0	0
$429$	9.70189	0.468411
$430$	0	0
$431$	15.8211	0.762076	0.381038	−	0.924559i	$-0.375567\pi$
$431$	15.8211	0.762076	0.381038	+	0.924559i	$0.375567\pi$
$432$	0	0
$433$	−28.9220	−1.38990	−0.694952	−	0.719056i	$-0.744574\pi$
$433$	−28.9220	−1.38990	−0.694952	+	0.719056i	$0.744574\pi$
$434$	0	0
$435$	2.51648	0.120656
$436$	0	0
$437$	−14.4944	−0.693361
$438$	0	0
$439$	22.6194	1.07956	0.539782	−	0.841805i	$-0.318507\pi$
$439$	22.6194	1.07956	0.539782	+	0.841805i	$0.318507\pi$
$440$	0	0
$441$	0.865977	0.0412370
$442$	0	0
$443$	4.79654	0.227891	0.113945	−	0.993487i	$-0.463651\pi$
$443$	4.79654	0.227891	0.113945	+	0.993487i	$0.463651\pi$
$444$	0	0
$445$	1.98175	0.0939439
$446$	0	0
$447$	−10.6658	−0.504476
$448$	0	0
$449$	36.8620	1.73963	0.869813	−	0.493382i	$-0.164240\pi$
$449$	36.8620	1.73963	0.869813	+	0.493382i	$0.164240\pi$
$450$	0	0
$451$	1.06183	0.0499995
$452$	0	0
$453$	−23.2311	−1.09149
$454$	0	0
$455$	1.96832	0.0922762
$456$	0	0
$457$	32.1078	1.50194	0.750969	−	0.660337i	$-0.229587\pi$
$457$	32.1078	1.50194	0.750969	+	0.660337i	$0.229587\pi$
$458$	0	0
$459$	5.34416	0.249444
$460$	0	0
$461$	−2.20188	−0.102552	−0.0512758	−	0.998685i	$-0.516329\pi$
$461$	−2.20188	−0.102552	−0.0512758	+	0.998685i	$0.516329\pi$
$462$	0	0
$463$	32.9458	1.53112	0.765561	−	0.643364i	$-0.222462\pi$
$463$	32.9458	1.53112	0.765561	+	0.643364i	$0.222462\pi$
$464$	0	0
$465$	−1.22659	−0.0568816
$466$	0	0
$467$	22.0985	1.02260	0.511299	−	0.859403i	$-0.329165\pi$
$467$	22.0985	1.02260	0.511299	+	0.859403i	$0.329165\pi$
$468$	0	0
$469$	12.3615	0.570802
$470$	0	0
$471$	13.0694	0.602206
$472$	0	0
$473$	12.1125	0.556935
$474$	0	0
$475$	−31.2974	−1.43602
$476$	0	0
$477$	−2.35184	−0.107683
$478$	0	0
$479$	−19.1404	−0.874546	−0.437273	−	0.899329i	$-0.644056\pi$
$479$	−19.1404	−0.874546	−0.437273	+	0.899329i	$0.644056\pi$
$480$	0	0
$481$	27.9515	1.27448
$482$	0	0
$483$	−5.88204	−0.267642
$484$	0	0
$485$	2.15408	0.0978118
$486$	0	0
$487$	37.0602	1.67936	0.839680	−	0.543082i	$-0.182743\pi$
$487$	37.0602	1.67936	0.839680	+	0.543082i	$0.182743\pi$
$488$	0	0
$489$	−27.6380	−1.24983
$490$	0	0
$491$	−17.6906	−0.798365	−0.399183	−	0.916871i	$-0.630706\pi$
$491$	−17.6906	−0.798365	−0.399183	+	0.916871i	$0.630706\pi$
$492$	0	0
$493$	7.27772	0.327772
$494$	0	0
$495$	−0.0353602	−0.00158932
$496$	0	0
$497$	−11.7306	−0.526188
$498$	0	0
$499$	35.6483	1.59583	0.797917	−	0.602767i	$-0.205935\pi$
$499$	35.6483	1.59583	0.797917	+	0.602767i	$0.205935\pi$
$500$	0	0
$501$	−2.64847	−0.118325
$502$	0	0
$503$	2.85185	0.127158	0.0635789	−	0.997977i	$-0.479749\pi$
$503$	2.85185	0.127158	0.0635789	+	0.997977i	$0.479749\pi$
$504$	0	0
$505$	2.53259	0.112699
$506$	0	0
$507$	−43.7974	−1.94511
$508$	0	0
$509$	39.3060	1.74221	0.871104	−	0.491098i	$-0.163404\pi$
$509$	39.3060	1.74221	0.871104	+	0.491098i	$0.163404\pi$
$510$	0	0
$511$	1.89647	0.0838951
$512$	0	0
$513$	33.7383	1.48958
$514$	0	0
$515$	−0.484969	−0.0213703
$516$	0	0
$517$	−3.00513	−0.132165
$518$	0	0
$519$	−33.8933	−1.48775
$520$	0	0
$521$	−16.0848	−0.704688	−0.352344	−	0.935871i	$-0.614615\pi$
$521$	−16.0848	−0.704688	−0.352344	+	0.935871i	$0.614615\pi$
$522$	0	0
$523$	13.6258	0.595817	0.297908	−	0.954594i	$-0.403711\pi$
$523$	13.6258	0.595817	0.297908	+	0.954594i	$0.403711\pi$
$524$	0	0
$525$	−12.7009	−0.554314
$526$	0	0
$527$	−3.54732	−0.154524
$528$	0	0
$529$	−17.7288	−0.770815
$530$	0	0
$531$	−0.185517	−0.00805075
$532$	0	0
$533$	−7.18037	−0.311016
$534$	0	0
$535$	2.32496	0.100517
$536$	0	0
$537$	−31.3189	−1.35151
$538$	0	0
$539$	4.31673	0.185935
$540$	0	0
$541$	12.2639	0.527265	0.263632	−	0.964623i	$-0.415079\pi$
$541$	12.2639	0.527265	0.263632	+	0.964623i	$0.415079\pi$
$542$	0	0
$543$	−40.7803	−1.75005
$544$	0	0
$545$	−2.49732	−0.106973
$546$	0	0
$547$	−8.55878	−0.365947	−0.182974	−	0.983118i	$-0.558572\pi$
$547$	−8.55878	−0.365947	−0.182974	+	0.983118i	$0.558572\pi$
$548$	0	0
$549$	−0.613678	−0.0261911
$550$	0	0
$551$	45.9451	1.95733
$552$	0	0
$553$	−7.71421	−0.328042
$554$	0	0
$555$	1.54553	0.0656043
$556$	0	0
$557$	−11.6034	−0.491652	−0.245826	−	0.969314i	$-0.579059\pi$
$557$	−11.6034	−0.491652	−0.245826	+	0.969314i	$0.579059\pi$
$558$	0	0
$559$	−81.9082	−3.46435
$560$	0	0
$561$	1.55143	0.0655013
$562$	0	0
$563$	−21.8958	−0.922800	−0.461400	−	0.887192i	$-0.652653\pi$
$563$	−21.8958	−0.922800	−0.461400	+	0.887192i	$0.652653\pi$
$564$	0	0
$565$	1.16098	0.0488429
$566$	0	0
$567$	12.8416	0.539295
$568$	0	0
$569$	5.98551	0.250926	0.125463	−	0.992098i	$-0.459958\pi$
$569$	5.98551	0.250926	0.125463	+	0.992098i	$0.459958\pi$
$570$	0	0
$571$	14.0904	0.589663	0.294831	−	0.955549i	$-0.404736\pi$
$571$	14.0904	0.589663	0.294831	+	0.955549i	$0.404736\pi$
$572$	0	0
$573$	22.1033	0.923377
$574$	0	0
$575$	11.3821	0.474665
$576$	0	0
$577$	24.6759	1.02727	0.513636	−	0.858008i	$-0.328298\pi$
$577$	24.6759	1.02727	0.513636	+	0.858008i	$0.328298\pi$
$578$	0	0
$579$	−24.2425	−1.00748
$580$	0	0
$581$	2.84824	0.118165
$582$	0	0
$583$	−11.7235	−0.485536
$584$	0	0
$585$	0.239115	0.00988619
$586$	0	0
$587$	−10.1882	−0.420510	−0.210255	−	0.977647i	$-0.567430\pi$
$587$	−10.1882	−0.420510	−0.210255	+	0.977647i	$0.567430\pi$
$588$	0	0
$589$	−22.3947	−0.922756
$590$	0	0
$591$	−11.4440	−0.470744
$592$	0	0
$593$	42.3608	1.73955	0.869775	−	0.493449i	$-0.164264\pi$
$593$	42.3608	1.73955	0.869775	+	0.493449i	$0.164264\pi$
$594$	0	0
$595$	0.314754	0.0129036
$596$	0	0
$597$	42.0896	1.72261
$598$	0	0
$599$	36.1022	1.47510	0.737548	−	0.675295i	$-0.235984\pi$
$599$	36.1022	1.47510	0.737548	+	0.675295i	$0.235984\pi$
$600$	0	0
$601$	23.2825	0.949712	0.474856	−	0.880064i	$-0.342500\pi$
$601$	23.2825	0.949712	0.474856	+	0.880064i	$0.342500\pi$
$602$	0	0
$603$	1.50170	0.0611540
$604$	0	0
$605$	2.09094	0.0850090
$606$	0	0
$607$	−15.7426	−0.638973	−0.319487	−	0.947591i	$-0.603510\pi$
$607$	−15.7426	−0.638973	−0.319487	+	0.947591i	$0.603510\pi$
$608$	0	0
$609$	18.6452	0.755541
$610$	0	0
$611$	20.3215	0.822120
$612$	0	0
$613$	9.86043	0.398259	0.199129	−	0.979973i	$-0.436189\pi$
$613$	9.86043	0.398259	0.199129	+	0.979973i	$0.436189\pi$
$614$	0	0
$615$	−0.397027	−0.0160097
$616$	0	0
$617$	−7.81929	−0.314793	−0.157396	−	0.987536i	$-0.550310\pi$
$617$	−7.81929	−0.314793	−0.157396	+	0.987536i	$0.550310\pi$
$618$	0	0
$619$	−20.3502	−0.817942	−0.408971	−	0.912547i	$-0.634112\pi$
$619$	−20.3502	−0.817942	−0.408971	+	0.912547i	$0.634112\pi$
$620$	0	0
$621$	−12.2698	−0.492368
$622$	0	0
$623$	14.6832	0.588271
$624$	0	0
$625$	24.3646	0.974583
$626$	0	0
$627$	9.79434	0.391148
$628$	0	0
$629$	4.46972	0.178219
$630$	0	0
$631$	23.2999	0.927556	0.463778	−	0.885951i	$-0.346493\pi$
$631$	23.2999	0.927556	0.463778	+	0.885951i	$0.346493\pi$
$632$	0	0
$633$	−4.65629	−0.185071
$634$	0	0
$635$	3.53578	0.140313
$636$	0	0
$637$	−29.1909	−1.15659
$638$	0	0
$639$	−1.42505	−0.0563742
$640$	0	0
$641$	−24.6198	−0.972422	−0.486211	−	0.873841i	$-0.661621\pi$
$641$	−24.6198	−0.972422	−0.486211	+	0.873841i	$0.661621\pi$
$642$	0	0
$643$	−47.7644	−1.88364	−0.941822	−	0.336111i	$-0.890888\pi$
$643$	−47.7644	−1.88364	−0.941822	+	0.336111i	$0.890888\pi$
$644$	0	0
$645$	−4.52899	−0.178329
$646$	0	0
$647$	−19.1004	−0.750913	−0.375456	−	0.926840i	$-0.622514\pi$
$647$	−19.1004	−0.750913	−0.375456	+	0.926840i	$0.622514\pi$
$648$	0	0
$649$	−0.924767	−0.0363003
$650$	0	0
$651$	−9.08807	−0.356190
$652$	0	0
$653$	15.8962	0.622064	0.311032	−	0.950399i	$-0.399325\pi$
$653$	15.8962	0.622064	0.311032	+	0.950399i	$0.399325\pi$
$654$	0	0
$655$	2.51548	0.0982881
$656$	0	0
$657$	0.230387	0.00898826
$658$	0	0
$659$	−8.03610	−0.313042	−0.156521	−	0.987675i	$-0.550028\pi$
$659$	−8.03610	−0.313042	−0.156521	+	0.987675i	$0.550028\pi$
$660$	0	0
$661$	−39.0086	−1.51726	−0.758629	−	0.651523i	$-0.774130\pi$
$661$	−39.0086	−1.51726	−0.758629	+	0.651523i	$0.774130\pi$
$662$	0	0
$663$	−10.4912	−0.407443
$664$	0	0
$665$	1.98708	0.0770556
$666$	0	0
$667$	−16.7090	−0.646977
$668$	0	0
$669$	7.35539	0.284376
$670$	0	0
$671$	−3.05907	−0.118094
$672$	0	0
$673$	39.8095	1.53454	0.767272	−	0.641322i	$-0.221614\pi$
$673$	39.8095	1.53454	0.767272	+	0.641322i	$0.221614\pi$
$674$	0	0
$675$	−26.4938	−1.01975
$676$	0	0
$677$	21.8926	0.841399	0.420700	−	0.907200i	$-0.361785\pi$
$677$	21.8926	0.841399	0.420700	+	0.907200i	$0.361785\pi$
$678$	0	0
$679$	15.9601	0.612492
$680$	0	0
$681$	−23.4760	−0.899601
$682$	0	0
$683$	9.56893	0.366145	0.183072	−	0.983099i	$-0.441396\pi$
$683$	9.56893	0.366145	0.183072	+	0.983099i	$0.441396\pi$
$684$	0	0
$685$	−3.53253	−0.134971
$686$	0	0
$687$	−43.9650	−1.67737
$688$	0	0
$689$	79.2772	3.02022
$690$	0	0
$691$	16.8742	0.641926	0.320963	−	0.947092i	$-0.395994\pi$
$691$	16.8742	0.641926	0.320963	+	0.947092i	$0.395994\pi$
$692$	0	0
$693$	−0.261992	−0.00995224
$694$	0	0
$695$	−3.30844	−0.125496
$696$	0	0
$697$	−1.14821	−0.0434916
$698$	0	0
$699$	−32.7492	−1.23869
$700$	0	0
$701$	8.65555	0.326916	0.163458	−	0.986550i	$-0.447735\pi$
$701$	8.65555	0.326916	0.163458	+	0.986550i	$0.447735\pi$
$702$	0	0
$703$	28.2179	1.06426
$704$	0	0
$705$	1.12365	0.0423189
$706$	0	0
$707$	18.7646	0.705714
$708$	0	0
$709$	37.3133	1.40133	0.700665	−	0.713490i	$-0.252887\pi$
$709$	37.3133	1.40133	0.700665	+	0.713490i	$0.252887\pi$
$710$	0	0
$711$	−0.937136	−0.0351454
$712$	0	0
$713$	8.14436	0.305009
$714$	0	0
$715$	1.19194	0.0445762
$716$	0	0
$717$	12.5088	0.467149
$718$	0	0
$719$	40.6394	1.51559	0.757796	−	0.652491i	$-0.226276\pi$
$719$	40.6394	1.51559	0.757796	+	0.652491i	$0.226276\pi$
$720$	0	0
$721$	−3.59325	−0.133820
$722$	0	0
$723$	−41.6266	−1.54811
$724$	0	0
$725$	−36.0794	−1.33996
$726$	0	0
$727$	−15.1160	−0.560623	−0.280311	−	0.959909i	$-0.590438\pi$
$727$	−15.1160	−0.560623	−0.280311	+	0.959909i	$0.590438\pi$
$728$	0	0
$729$	28.4568	1.05395
$730$	0	0
$731$	−13.0979	−0.484445
$732$	0	0
$733$	−24.7404	−0.913807	−0.456903	−	0.889516i	$-0.651041\pi$
$733$	−24.7404	−0.913807	−0.456903	+	0.889516i	$0.651041\pi$
$734$	0	0
$735$	−1.61407	−0.0595357
$736$	0	0
$737$	7.48570	0.275739
$738$	0	0
$739$	46.7907	1.72122	0.860612	−	0.509261i	$-0.170081\pi$
$739$	46.7907	1.72122	0.860612	+	0.509261i	$0.170081\pi$
$740$	0	0
$741$	−66.2320	−2.43309
$742$	0	0
$743$	−8.43456	−0.309434	−0.154717	−	0.987959i	$-0.549447\pi$
$743$	−8.43456	−0.309434	−0.154717	+	0.987959i	$0.549447\pi$
$744$	0	0
$745$	−1.31037	−0.0480082
$746$	0	0
$747$	0.346009	0.0126598
$748$	0	0
$749$	17.2262	0.629430
$750$	0	0
$751$	37.0366	1.35149	0.675743	−	0.737137i	$-0.263823\pi$
$751$	37.0366	1.35149	0.675743	+	0.737137i	$0.263823\pi$
$752$	0	0
$753$	49.4640	1.80257
$754$	0	0
$755$	−2.85410	−0.103871
$756$	0	0
$757$	−21.6964	−0.788570	−0.394285	−	0.918988i	$-0.629008\pi$
$757$	−21.6964	−0.788570	−0.394285	+	0.918988i	$0.629008\pi$
$758$	0	0
$759$	−3.56195	−0.129291
$760$	0	0
$761$	41.8396	1.51669	0.758343	−	0.651856i	$-0.226009\pi$
$761$	41.8396	1.51669	0.758343	+	0.651856i	$0.226009\pi$
$762$	0	0
$763$	−18.5032	−0.669861
$764$	0	0
$765$	0.0382369	0.00138246
$766$	0	0
$767$	6.25352	0.225802
$768$	0	0
$769$	31.7728	1.14576	0.572879	−	0.819640i	$-0.305827\pi$
$769$	31.7728	1.14576	0.572879	+	0.819640i	$0.305827\pi$
$770$	0	0
$771$	15.4495	0.556400
$772$	0	0
$773$	52.8519	1.90095	0.950475	−	0.310800i	$-0.100597\pi$
$773$	52.8519	1.90095	0.950475	+	0.310800i	$0.100597\pi$
$774$	0	0
$775$	17.5859	0.631704
$776$	0	0
$777$	11.4512	0.410810
$778$	0	0
$779$	−7.24879	−0.259715
$780$	0	0
$781$	−7.10361	−0.254187
$782$	0	0
$783$	38.8933	1.38993
$784$	0	0
$785$	1.60566	0.0573086
$786$	0	0
$787$	19.5241	0.695957	0.347979	−	0.937502i	$-0.386868\pi$
$787$	19.5241	0.695957	0.347979	+	0.937502i	$0.386868\pi$
$788$	0	0
$789$	31.0919	1.10690
$790$	0	0
$791$	8.60199	0.305852
$792$	0	0
$793$	20.6863	0.734590
$794$	0	0
$795$	4.38351	0.155467
$796$	0	0
$797$	−9.01386	−0.319287	−0.159644	−	0.987175i	$-0.551035\pi$
$797$	−9.01386	−0.319287	−0.159644	+	0.987175i	$0.551035\pi$
$798$	0	0
$799$	3.24961	0.114963
$800$	0	0
$801$	1.78375	0.0630256
$802$	0	0
$803$	1.14844	0.0405274
$804$	0	0
$805$	−0.722649	−0.0254700
$806$	0	0
$807$	43.1181	1.51783
$808$	0	0
$809$	−14.2817	−0.502118	−0.251059	−	0.967972i	$-0.580779\pi$
$809$	−14.2817	−0.502118	−0.251059	+	0.967972i	$0.580779\pi$
$810$	0	0
$811$	−6.92733	−0.243251	−0.121626	−	0.992576i	$-0.538811\pi$
$811$	−6.92733	−0.243251	−0.121626	+	0.992576i	$0.538811\pi$
$812$	0	0
$813$	−17.1630	−0.601932
$814$	0	0
$815$	−3.39552	−0.118940
$816$	0	0
$817$	−82.6888	−2.89292
$818$	0	0
$819$	1.77166	0.0619068
$820$	0	0
$821$	25.0226	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$821$	25.0226	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$822$	0	0
$823$	23.0509	0.803505	0.401752	−	0.915748i	$-0.368401\pi$
$823$	23.0509	0.803505	0.401752	+	0.915748i	$0.368401\pi$
$824$	0	0
$825$	−7.69123	−0.267774
$826$	0	0
$827$	−0.551772	−0.0191870	−0.00959350	−	0.999954i	$-0.503054\pi$
$827$	−0.551772	−0.0191870	−0.00959350	+	0.999954i	$0.503054\pi$
$828$	0	0
$829$	17.6442	0.612807	0.306404	−	0.951902i	$-0.400874\pi$
$829$	17.6442	0.612807	0.306404	+	0.951902i	$0.400874\pi$
$830$	0	0
$831$	31.7847	1.10260
$832$	0	0
$833$	−4.66792	−0.161734
$834$	0	0
$835$	−0.325383	−0.0112603
$836$	0	0
$837$	−18.9574	−0.655265
$838$	0	0
$839$	31.5880	1.09054	0.545269	−	0.838261i	$-0.316427\pi$
$839$	31.5880	1.09054	0.545269	+	0.838261i	$0.316427\pi$
$840$	0	0
$841$	23.9652	0.826385
$842$	0	0
$843$	14.4507	0.497710
$844$	0	0
$845$	−5.38081	−0.185106
$846$	0	0
$847$	15.4923	0.532322
$848$	0	0
$849$	2.64171	0.0906633
$850$	0	0
$851$	−10.2621	−0.351781
$852$	0	0
$853$	3.51115	0.120220	0.0601098	−	0.998192i	$-0.480855\pi$
$853$	3.51115	0.120220	0.0601098	+	0.998192i	$0.480855\pi$
$854$	0	0
$855$	0.241394	0.00825549
$856$	0	0
$857$	−32.2692	−1.10229	−0.551147	−	0.834408i	$-0.685810\pi$
$857$	−32.2692	−1.10229	−0.551147	+	0.834408i	$0.685810\pi$
$858$	0	0
$859$	−15.2467	−0.520209	−0.260105	−	0.965580i	$-0.583757\pi$
$859$	−15.2467	−0.520209	−0.260105	+	0.965580i	$0.583757\pi$
$860$	0	0
$861$	−2.94167	−0.100252
$862$	0	0
$863$	−23.2445	−0.791252	−0.395626	−	0.918412i	$-0.629472\pi$
$863$	−23.2445	−0.791252	−0.395626	+	0.918412i	$0.629472\pi$
$864$	0	0
$865$	−4.16402	−0.141581
$866$	0	0
$867$	−1.67764	−0.0569757
$868$	0	0
$869$	−4.67145	−0.158468
$870$	0	0
$871$	−50.6203	−1.71520
$872$	0	0
$873$	1.93886	0.0656205
$874$	0	0
$875$	−3.13417	−0.105954
$876$	0	0
$877$	−38.7216	−1.30754	−0.653768	−	0.756695i	$-0.726813\pi$
$877$	−38.7216	−1.30754	−0.653768	+	0.756695i	$0.726813\pi$
$878$	0	0
$879$	−16.0523	−0.541432
$880$	0	0
$881$	22.5486	0.759681	0.379841	−	0.925052i	$-0.375979\pi$
$881$	22.5486	0.759681	0.379841	+	0.925052i	$0.375979\pi$
$882$	0	0
$883$	−6.80256	−0.228924	−0.114462	−	0.993428i	$-0.536514\pi$
$883$	−6.80256	−0.228924	−0.114462	+	0.993428i	$0.536514\pi$
$884$	0	0
$885$	0.345779	0.0116232
$886$	0	0
$887$	−51.1038	−1.71590	−0.857949	−	0.513736i	$-0.828261\pi$
$887$	−51.1038	−1.71590	−0.857949	+	0.513736i	$0.828261\pi$
$888$	0	0
$889$	26.1974	0.878633
$890$	0	0
$891$	7.77639	0.260519
$892$	0	0
$893$	20.5152	0.686514
$894$	0	0
$895$	−3.84774	−0.128616
$896$	0	0
$897$	24.0869	0.804237
$898$	0	0
$899$	−25.8164	−0.861025
$900$	0	0
$901$	12.6772	0.422339
$902$	0	0
$903$	−33.5563	−1.11668
$904$	0	0
$905$	−5.01014	−0.166543
$906$	0	0
$907$	2.82721	0.0938760	0.0469380	−	0.998898i	$-0.485054\pi$
$907$	2.82721	0.0938760	0.0469380	+	0.998898i	$0.485054\pi$
$908$	0	0
$909$	2.27955	0.0756080
$910$	0	0
$911$	−11.8343	−0.392089	−0.196044	−	0.980595i	$-0.562810\pi$
$911$	−11.8343	−0.392089	−0.196044	+	0.980595i	$0.562810\pi$
$912$	0	0
$913$	1.72479	0.0570823
$914$	0	0
$915$	1.14381	0.0378133
$916$	0	0
$917$	18.6378	0.615475
$918$	0	0
$919$	−22.4135	−0.739354	−0.369677	−	0.929160i	$-0.620532\pi$
$919$	−22.4135	−0.739354	−0.369677	+	0.929160i	$0.620532\pi$
$920$	0	0
$921$	−16.6207	−0.547671
$922$	0	0
$923$	48.0366	1.58114
$924$	0	0
$925$	−22.1587	−0.728575
$926$	0	0
$927$	−0.436515	−0.0143370
$928$	0	0
$929$	−32.4048	−1.06317	−0.531584	−	0.847006i	$-0.678403\pi$
$929$	−32.4048	−1.06317	−0.531584	+	0.847006i	$0.678403\pi$
$930$	0	0
$931$	−29.4691	−0.965811
$932$	0	0
$933$	−22.4075	−0.733590
$934$	0	0
$935$	0.190604	0.00623340
$936$	0	0
$937$	5.29668	0.173035	0.0865174	−	0.996250i	$-0.472426\pi$
$937$	5.29668	0.173035	0.0865174	+	0.996250i	$0.472426\pi$
$938$	0	0
$939$	−4.49947	−0.146835
$940$	0	0
$941$	−53.9794	−1.75968	−0.879839	−	0.475272i	$-0.842350\pi$
$941$	−53.9794	−1.75968	−0.879839	+	0.475272i	$0.842350\pi$
$942$	0	0
$943$	2.63620	0.0858465
$944$	0	0
$945$	1.68209	0.0547185
$946$	0	0
$947$	−17.5103	−0.569010	−0.284505	−	0.958675i	$-0.591829\pi$
$947$	−17.5103	−0.569010	−0.284505	+	0.958675i	$0.591829\pi$
$948$	0	0
$949$	−7.76604	−0.252096
$950$	0	0
$951$	−39.9545	−1.29561
$952$	0	0
$953$	19.7586	0.640043	0.320022	−	0.947410i	$-0.396310\pi$
$953$	19.7586	0.640043	0.320022	+	0.947410i	$0.396310\pi$
$954$	0	0
$955$	2.71554	0.0878727
$956$	0	0
$957$	11.2908	0.364981
$958$	0	0
$959$	−26.1733	−0.845181
$960$	0	0
$961$	−18.4165	−0.594081
$962$	0	0
$963$	2.09266	0.0674352
$964$	0	0
$965$	−2.97835	−0.0958766
$966$	0	0
$967$	−26.9595	−0.866958	−0.433479	−	0.901164i	$-0.642714\pi$
$967$	−26.9595	−0.866958	−0.433479	+	0.901164i	$0.642714\pi$
$968$	0	0
$969$	−10.5912	−0.340237
$970$	0	0
$971$	−10.3828	−0.333201	−0.166601	−	0.986024i	$-0.553279\pi$
$971$	−10.3828	−0.333201	−0.166601	+	0.986024i	$0.553279\pi$
$972$	0	0
$973$	−24.5130	−0.785850
$974$	0	0
$975$	52.0102	1.66566
$976$	0	0
$977$	−33.3218	−1.06606	−0.533029	−	0.846097i	$-0.678946\pi$
$977$	−33.3218	−1.06606	−0.533029	+	0.846097i	$0.678946\pi$
$978$	0	0
$979$	8.89164	0.284178
$980$	0	0
$981$	−2.24780	−0.0717669
$982$	0	0
$983$	−44.1741	−1.40893	−0.704467	−	0.709737i	$-0.748814\pi$
$983$	−44.1741	−1.40893	−0.704467	+	0.709737i	$0.748814\pi$
$984$	0	0
$985$	−1.40598	−0.0447981
$986$	0	0
$987$	8.32535	0.264999
$988$	0	0
$989$	30.0718	0.956228
$990$	0	0
$991$	24.4075	0.775331	0.387665	−	0.921800i	$-0.373282\pi$
$991$	24.4075	0.775331	0.387665	+	0.921800i	$0.373282\pi$
$992$	0	0
$993$	8.83373	0.280330
$994$	0	0
$995$	5.17099	0.163931
$996$	0	0
$997$	−36.6159	−1.15964	−0.579819	−	0.814745i	$-0.696877\pi$
$997$	−36.6159	−1.15964	−0.579819	+	0.814745i	$0.696877\pi$
$998$	0	0
$999$	23.8869	0.755748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.9	✓	32	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67764 q^{3} -0.206110 q^{5} -1.52712 q^{7} -0.185517 q^{9} +O(q^{10})\)	\(q-1.67764 q^{3} -0.206110 q^{5} -1.52712 q^{7} -0.185517 q^{9} -0.924767 q^{11} +6.25352 q^{13} +0.345779 q^{15} +1.00000 q^{17} +6.31312 q^{19} +2.56195 q^{21} -2.29592 q^{23} -4.95752 q^{25} +5.34416 q^{27} +7.27772 q^{29} -3.54732 q^{31} +1.55143 q^{33} +0.314754 q^{35} +4.46972 q^{37} -10.4912 q^{39} -1.14821 q^{41} -13.0979 q^{43} +0.0382369 q^{45} +3.24961 q^{47} -4.66792 q^{49} -1.67764 q^{51} +12.6772 q^{53} +0.190604 q^{55} -10.5912 q^{57} +1.00000 q^{59} +3.30794 q^{61} +0.283306 q^{63} -1.28891 q^{65} -8.09469 q^{67} +3.85173 q^{69} +7.68152 q^{71} -1.24187 q^{73} +8.31694 q^{75} +1.41223 q^{77} +5.05149 q^{79} -8.40903 q^{81} -1.86511 q^{83} -0.206110 q^{85} -12.2094 q^{87} -9.61501 q^{89} -9.54986 q^{91} +5.95113 q^{93} -1.30120 q^{95} -10.4511 q^{97} +0.171560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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