LMFDB - Embedded newform 8024.2.a.bb.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.35639 q^{3} +1.88610 q^{5} -3.68573 q^{7} +2.55257 q^{9} +O(q^{10})$	$q-2.35639 q^{3} +1.88610 q^{5} -3.68573 q^{7} +2.55257 q^{9} -0.273123 q^{11} +6.67712 q^{13} -4.44438 q^{15} +1.00000 q^{17} +2.11712 q^{19} +8.68501 q^{21} +7.76966 q^{23} -1.44264 q^{25} +1.05431 q^{27} +5.02363 q^{29} -1.71833 q^{31} +0.643585 q^{33} -6.95164 q^{35} -0.885476 q^{37} -15.7339 q^{39} +9.14712 q^{41} +10.0637 q^{43} +4.81440 q^{45} -12.3258 q^{47} +6.58459 q^{49} -2.35639 q^{51} -9.87684 q^{53} -0.515137 q^{55} -4.98876 q^{57} +1.00000 q^{59} +4.00598 q^{61} -9.40810 q^{63} +12.5937 q^{65} +4.45927 q^{67} -18.3084 q^{69} -7.29781 q^{71} -6.47408 q^{73} +3.39942 q^{75} +1.00666 q^{77} -6.94163 q^{79} -10.1421 q^{81} -7.46117 q^{83} +1.88610 q^{85} -11.8376 q^{87} +6.36854 q^{89} -24.6100 q^{91} +4.04905 q^{93} +3.99309 q^{95} +13.6966 q^{97} -0.697168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.35639	−1.36046	−0.680231	−	0.732998i	$-0.738121\pi$
$3$	−2.35639	−1.36046	−0.680231	+	0.732998i	$0.738121\pi$
$4$	0	0
$5$	1.88610	0.843488	0.421744	−	0.906715i	$-0.361418\pi$
$5$	1.88610	0.843488	0.421744	+	0.906715i	$0.361418\pi$
$6$	0	0
$7$	−3.68573	−1.39307	−0.696537	−	0.717521i	$-0.745277\pi$
$7$	−3.68573	−1.39307	−0.696537	+	0.717521i	$0.745277\pi$
$8$	0	0
$9$	2.55257	0.850858
$10$	0	0
$11$	−0.273123	−0.0823498	−0.0411749	−	0.999152i	$-0.513110\pi$
$11$	−0.273123	−0.0823498	−0.0411749	+	0.999152i	$0.513110\pi$
$12$	0	0
$13$	6.67712	1.85190	0.925950	−	0.377647i	$-0.123267\pi$
$13$	6.67712	1.85190	0.925950	+	0.377647i	$0.123267\pi$
$14$	0	0
$15$	−4.44438	−1.14753
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.11712	0.485700	0.242850	−	0.970064i	$-0.421918\pi$
$19$	2.11712	0.485700	0.242850	+	0.970064i	$0.421918\pi$
$20$	0	0
$21$	8.68501	1.89523
$22$	0	0
$23$	7.76966	1.62009	0.810043	−	0.586370i	$-0.199444\pi$
$23$	7.76966	1.62009	0.810043	+	0.586370i	$0.199444\pi$
$24$	0	0
$25$	−1.44264	−0.288527
$26$	0	0
$27$	1.05431	0.202902
$28$	0	0
$29$	5.02363	0.932865	0.466432	−	0.884557i	$-0.345539\pi$
$29$	5.02363	0.932865	0.466432	+	0.884557i	$0.345539\pi$
$30$	0	0
$31$	−1.71833	−0.308621	−0.154310	−	0.988022i	$-0.549316\pi$
$31$	−1.71833	−0.308621	−0.154310	+	0.988022i	$0.549316\pi$
$32$	0	0
$33$	0.643585	0.112034
$34$	0	0
$35$	−6.95164	−1.17504
$36$	0	0
$37$	−0.885476	−0.145571	−0.0727857	−	0.997348i	$-0.523189\pi$
$37$	−0.885476	−0.145571	−0.0727857	+	0.997348i	$0.523189\pi$
$38$	0	0
$39$	−15.7339	−2.51944
$40$	0	0
$41$	9.14712	1.42854	0.714270	−	0.699870i	$-0.246759\pi$
$41$	9.14712	1.42854	0.714270	+	0.699870i	$0.246759\pi$
$42$	0	0
$43$	10.0637	1.53470	0.767350	−	0.641229i	$-0.221575\pi$
$43$	10.0637	1.53470	0.767350	+	0.641229i	$0.221575\pi$
$44$	0	0
$45$	4.81440	0.717689
$46$	0	0
$47$	−12.3258	−1.79791	−0.898954	−	0.438044i	$-0.855672\pi$
$47$	−12.3258	−1.79791	−0.898954	+	0.438044i	$0.855672\pi$
$48$	0	0
$49$	6.58459	0.940656
$50$	0	0
$51$	−2.35639	−0.329961
$52$	0	0
$53$	−9.87684	−1.35669	−0.678344	−	0.734745i	$-0.737302\pi$
$53$	−9.87684	−1.35669	−0.678344	+	0.734745i	$0.737302\pi$
$54$	0	0
$55$	−0.515137	−0.0694611
$56$	0	0
$57$	−4.98876	−0.660777
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	4.00598	0.512913	0.256456	−	0.966556i	$-0.417445\pi$
$61$	4.00598	0.512913	0.256456	+	0.966556i	$0.417445\pi$
$62$	0	0
$63$	−9.40810	−1.18531
$64$	0	0
$65$	12.5937	1.56206
$66$	0	0
$67$	4.45927	0.544787	0.272394	−	0.962186i	$-0.412185\pi$
$67$	4.45927	0.544787	0.272394	+	0.962186i	$0.412185\pi$
$68$	0	0
$69$	−18.3084	−2.20407
$70$	0	0
$71$	−7.29781	−0.866091	−0.433046	−	0.901372i	$-0.642561\pi$
$71$	−7.29781	−0.866091	−0.433046	+	0.901372i	$0.642561\pi$
$72$	0	0
$73$	−6.47408	−0.757733	−0.378867	−	0.925451i	$-0.623686\pi$
$73$	−6.47408	−0.757733	−0.378867	+	0.925451i	$0.623686\pi$
$74$	0	0
$75$	3.39942	0.392531
$76$	0	0
$77$	1.00666	0.114719
$78$	0	0
$79$	−6.94163	−0.780995	−0.390497	−	0.920604i	$-0.627697\pi$
$79$	−6.94163	−0.780995	−0.390497	+	0.920604i	$0.627697\pi$
$80$	0	0
$81$	−10.1421	−1.12690
$82$	0	0
$83$	−7.46117	−0.818969	−0.409485	−	0.912317i	$-0.634291\pi$
$83$	−7.46117	−0.818969	−0.409485	+	0.912317i	$0.634291\pi$
$84$	0	0
$85$	1.88610	0.204576
$86$	0	0
$87$	−11.8376	−1.26913
$88$	0	0
$89$	6.36854	0.675064	0.337532	−	0.941314i	$-0.390408\pi$
$89$	6.36854	0.675064	0.337532	+	0.941314i	$0.390408\pi$
$90$	0	0
$91$	−24.6100	−2.57983
$92$	0	0
$93$	4.04905	0.419867
$94$	0	0
$95$	3.99309	0.409683
$96$	0	0
$97$	13.6966	1.39068	0.695340	−	0.718681i	$-0.255254\pi$
$97$	13.6966	1.39068	0.695340	+	0.718681i	$0.255254\pi$
$98$	0	0
$99$	−0.697168	−0.0700680
$100$	0	0
$101$	10.9583	1.09040	0.545198	−	0.838307i	$-0.316454\pi$
$101$	10.9583	1.09040	0.545198	+	0.838307i	$0.316454\pi$
$102$	0	0
$103$	−11.2738	−1.11084	−0.555420	−	0.831570i	$-0.687443\pi$
$103$	−11.2738	−1.11084	−0.555420	+	0.831570i	$0.687443\pi$
$104$	0	0
$105$	16.3808	1.59860
$106$	0	0
$107$	17.7536	1.71630	0.858152	−	0.513397i	$-0.171613\pi$
$107$	17.7536	1.71630	0.858152	+	0.513397i	$0.171613\pi$
$108$	0	0
$109$	−3.70076	−0.354468	−0.177234	−	0.984169i	$-0.556715\pi$
$109$	−3.70076	−0.354468	−0.177234	+	0.984169i	$0.556715\pi$
$110$	0	0
$111$	2.08653	0.198044
$112$	0	0
$113$	−5.04259	−0.474367	−0.237184	−	0.971465i	$-0.576224\pi$
$113$	−5.04259	−0.474367	−0.237184	+	0.971465i	$0.576224\pi$
$114$	0	0
$115$	14.6543	1.36652
$116$	0	0
$117$	17.0438	1.57570
$118$	0	0
$119$	−3.68573	−0.337870
$120$	0	0
$121$	−10.9254	−0.993219
$122$	0	0
$123$	−21.5542	−1.94348
$124$	0	0
$125$	−12.1514	−1.08686
$126$	0	0
$127$	8.31427	0.737772	0.368886	−	0.929475i	$-0.379739\pi$
$127$	8.31427	0.737772	0.368886	+	0.929475i	$0.379739\pi$
$128$	0	0
$129$	−23.7140	−2.08790
$130$	0	0
$131$	16.9961	1.48496	0.742478	−	0.669871i	$-0.233650\pi$
$131$	16.9961	1.48496	0.742478	+	0.669871i	$0.233650\pi$
$132$	0	0
$133$	−7.80312	−0.676617
$134$	0	0
$135$	1.98853	0.171145
$136$	0	0
$137$	−0.316848	−0.0270701	−0.0135351	−	0.999908i	$-0.504308\pi$
$137$	−0.316848	−0.0270701	−0.0135351	+	0.999908i	$0.504308\pi$
$138$	0	0
$139$	−21.3369	−1.80977	−0.904884	−	0.425658i	$-0.860043\pi$
$139$	−21.3369	−1.80977	−0.904884	+	0.425658i	$0.860043\pi$
$140$	0	0
$141$	29.0445	2.44599
$142$	0	0
$143$	−1.82368	−0.152504
$144$	0	0
$145$	9.47506	0.786861
$146$	0	0
$147$	−15.5159	−1.27973
$148$	0	0
$149$	7.26752	0.595379	0.297689	−	0.954663i	$-0.403784\pi$
$149$	7.26752	0.595379	0.297689	+	0.954663i	$0.403784\pi$
$150$	0	0
$151$	−16.1770	−1.31647	−0.658235	−	0.752813i	$-0.728697\pi$
$151$	−16.1770	−1.31647	−0.658235	+	0.752813i	$0.728697\pi$
$152$	0	0
$153$	2.55257	0.206363
$154$	0	0
$155$	−3.24093	−0.260318
$156$	0	0
$157$	11.9482	0.953571	0.476785	−	0.879020i	$-0.341802\pi$
$157$	11.9482	0.953571	0.476785	+	0.879020i	$0.341802\pi$
$158$	0	0
$159$	23.2737	1.84572
$160$	0	0
$161$	−28.6369	−2.25690
$162$	0	0
$163$	7.48569	0.586324	0.293162	−	0.956063i	$-0.405292\pi$
$163$	7.48569	0.586324	0.293162	+	0.956063i	$0.405292\pi$
$164$	0	0
$165$	1.21386	0.0944992
$166$	0	0
$167$	−23.4821	−1.81710	−0.908551	−	0.417774i	$-0.862810\pi$
$167$	−23.4821	−1.81710	−0.908551	+	0.417774i	$0.862810\pi$
$168$	0	0
$169$	31.5839	2.42953
$170$	0	0
$171$	5.40410	0.413262
$172$	0	0
$173$	2.13249	0.162130	0.0810652	−	0.996709i	$-0.474168\pi$
$173$	2.13249	0.162130	0.0810652	+	0.996709i	$0.474168\pi$
$174$	0	0
$175$	5.31717	0.401940
$176$	0	0
$177$	−2.35639	−0.177117
$178$	0	0
$179$	−6.79857	−0.508149	−0.254075	−	0.967185i	$-0.581771\pi$
$179$	−6.79857	−0.508149	−0.254075	+	0.967185i	$0.581771\pi$
$180$	0	0
$181$	18.9942	1.41183	0.705915	−	0.708297i	$-0.250536\pi$
$181$	18.9942	1.41183	0.705915	+	0.708297i	$0.250536\pi$
$182$	0	0
$183$	−9.43964	−0.697798
$184$	0	0
$185$	−1.67009	−0.122788
$186$	0	0
$187$	−0.273123	−0.0199728
$188$	0	0
$189$	−3.88589	−0.282657
$190$	0	0
$191$	16.8859	1.22182	0.610910	−	0.791700i	$-0.290804\pi$
$191$	16.8859	1.22182	0.610910	+	0.791700i	$0.290804\pi$
$192$	0	0
$193$	−13.3801	−0.963122	−0.481561	−	0.876412i	$-0.659930\pi$
$193$	−13.3801	−0.963122	−0.481561	+	0.876412i	$0.659930\pi$
$194$	0	0
$195$	−29.6757	−2.12512
$196$	0	0
$197$	25.6741	1.82920	0.914602	−	0.404356i	$-0.132504\pi$
$197$	25.6741	1.82920	0.914602	+	0.404356i	$0.132504\pi$
$198$	0	0
$199$	10.0605	0.713171	0.356586	−	0.934263i	$-0.383941\pi$
$199$	10.0605	0.713171	0.356586	+	0.934263i	$0.383941\pi$
$200$	0	0
$201$	−10.5078	−0.741162
$202$	0	0
$203$	−18.5157	−1.29955
$204$	0	0
$205$	17.2524	1.20496
$206$	0	0
$207$	19.8326	1.37846
$208$	0	0
$209$	−0.578234	−0.0399973
$210$	0	0
$211$	26.0001	1.78992	0.894961	−	0.446144i	$-0.147203\pi$
$211$	26.0001	1.78992	0.894961	+	0.446144i	$0.147203\pi$
$212$	0	0
$213$	17.1965	1.17828
$214$	0	0
$215$	18.9811	1.29450
$216$	0	0
$217$	6.33329	0.429931
$218$	0	0
$219$	15.2555	1.03087
$220$	0	0
$221$	6.67712	0.449152
$222$	0	0
$223$	11.7925	0.789685	0.394842	−	0.918749i	$-0.370799\pi$
$223$	11.7925	0.789685	0.394842	+	0.918749i	$0.370799\pi$
$224$	0	0
$225$	−3.68244	−0.245496
$226$	0	0
$227$	−6.40125	−0.424866	−0.212433	−	0.977176i	$-0.568139\pi$
$227$	−6.40125	−0.424866	−0.212433	+	0.977176i	$0.568139\pi$
$228$	0	0
$229$	−3.26525	−0.215774	−0.107887	−	0.994163i	$-0.534408\pi$
$229$	−3.26525	−0.215774	−0.107887	+	0.994163i	$0.534408\pi$
$230$	0	0
$231$	−2.37208	−0.156071
$232$	0	0
$233$	−6.96459	−0.456266	−0.228133	−	0.973630i	$-0.573262\pi$
$233$	−6.96459	−0.456266	−0.228133	+	0.973630i	$0.573262\pi$
$234$	0	0
$235$	−23.2477	−1.51651
$236$	0	0
$237$	16.3572	1.06251
$238$	0	0
$239$	−8.26654	−0.534718	−0.267359	−	0.963597i	$-0.586151\pi$
$239$	−8.26654	−0.534718	−0.267359	+	0.963597i	$0.586151\pi$
$240$	0	0
$241$	−4.46669	−0.287725	−0.143862	−	0.989598i	$-0.545952\pi$
$241$	−4.46669	−0.287725	−0.143862	+	0.989598i	$0.545952\pi$
$242$	0	0
$243$	20.7358	1.33020
$244$	0	0
$245$	12.4192	0.793432
$246$	0	0
$247$	14.1363	0.899468
$248$	0	0
$249$	17.5814	1.11418
$250$	0	0
$251$	1.85167	0.116876	0.0584381	−	0.998291i	$-0.481388\pi$
$251$	1.85167	0.116876	0.0584381	+	0.998291i	$0.481388\pi$
$252$	0	0
$253$	−2.12208	−0.133414
$254$	0	0
$255$	−4.44438	−0.278318
$256$	0	0
$257$	5.44455	0.339622	0.169811	−	0.985477i	$-0.445684\pi$
$257$	5.44455	0.339622	0.169811	+	0.985477i	$0.445684\pi$
$258$	0	0
$259$	3.26362	0.202792
$260$	0	0
$261$	12.8232	0.793736
$262$	0	0
$263$	12.1973	0.752118	0.376059	−	0.926596i	$-0.377279\pi$
$263$	12.1973	0.752118	0.376059	+	0.926596i	$0.377279\pi$
$264$	0	0
$265$	−18.6287	−1.14435
$266$	0	0
$267$	−15.0068	−0.918399
$268$	0	0
$269$	7.90886	0.482212	0.241106	−	0.970499i	$-0.422490\pi$
$269$	7.90886	0.482212	0.241106	+	0.970499i	$0.422490\pi$
$270$	0	0
$271$	−26.5554	−1.61312	−0.806562	−	0.591149i	$-0.798674\pi$
$271$	−26.5554	−1.61312	−0.806562	+	0.591149i	$0.798674\pi$
$272$	0	0
$273$	57.9909	3.50977
$274$	0	0
$275$	0.394018	0.0237602
$276$	0	0
$277$	2.10265	0.126336	0.0631681	−	0.998003i	$-0.479880\pi$
$277$	2.10265	0.126336	0.0631681	+	0.998003i	$0.479880\pi$
$278$	0	0
$279$	−4.38616	−0.262592
$280$	0	0
$281$	−14.9881	−0.894116	−0.447058	−	0.894505i	$-0.647528\pi$
$281$	−14.9881	−0.894116	−0.447058	+	0.894505i	$0.647528\pi$
$282$	0	0
$283$	−26.9460	−1.60177	−0.800887	−	0.598816i	$-0.795638\pi$
$283$	−26.9460	−1.60177	−0.800887	+	0.598816i	$0.795638\pi$
$284$	0	0
$285$	−9.40928	−0.557358
$286$	0	0
$287$	−33.7138	−1.99006
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−32.2746	−1.89197
$292$	0	0
$293$	−1.16499	−0.0680593	−0.0340297	−	0.999421i	$-0.510834\pi$
$293$	−1.16499	−0.0680593	−0.0340297	+	0.999421i	$0.510834\pi$
$294$	0	0
$295$	1.88610	0.109813
$296$	0	0
$297$	−0.287956	−0.0167089
$298$	0	0
$299$	51.8789	3.00024
$300$	0	0
$301$	−37.0920	−2.13795
$302$	0	0
$303$	−25.8221	−1.48344
$304$	0	0
$305$	7.55566	0.432636
$306$	0	0
$307$	18.1428	1.03546	0.517732	−	0.855543i	$-0.326777\pi$
$307$	18.1428	1.03546	0.517732	+	0.855543i	$0.326777\pi$
$308$	0	0
$309$	26.5655	1.51126
$310$	0	0
$311$	26.3066	1.49171	0.745854	−	0.666109i	$-0.232042\pi$
$311$	26.3066	1.49171	0.745854	+	0.666109i	$0.232042\pi$
$312$	0	0
$313$	−13.6090	−0.769225	−0.384612	−	0.923078i	$-0.625665\pi$
$313$	−13.6090	−0.769225	−0.384612	+	0.923078i	$0.625665\pi$
$314$	0	0
$315$	−17.7446	−0.999794
$316$	0	0
$317$	−26.4797	−1.48725	−0.743623	−	0.668599i	$-0.766894\pi$
$317$	−26.4797	−1.48725	−0.743623	+	0.668599i	$0.766894\pi$
$318$	0	0
$319$	−1.37207	−0.0768212
$320$	0	0
$321$	−41.8344	−2.33497
$322$	0	0
$323$	2.11712	0.117800
$324$	0	0
$325$	−9.63266	−0.534324
$326$	0	0
$327$	8.72042	0.482241
$328$	0	0
$329$	45.4297	2.50462
$330$	0	0
$331$	8.76108	0.481553	0.240776	−	0.970581i	$-0.422598\pi$
$331$	8.76108	0.481553	0.240776	+	0.970581i	$0.422598\pi$
$332$	0	0
$333$	−2.26024	−0.123861
$334$	0	0
$335$	8.41063	0.459522
$336$	0	0
$337$	10.2681	0.559340	0.279670	−	0.960096i	$-0.409775\pi$
$337$	10.2681	0.559340	0.279670	+	0.960096i	$0.409775\pi$
$338$	0	0
$339$	11.8823	0.645359
$340$	0	0
$341$	0.469315	0.0254148
$342$	0	0
$343$	1.53109	0.0826713
$344$	0	0
$345$	−34.5313	−1.85910
$346$	0	0
$347$	−11.7384	−0.630148	−0.315074	−	0.949067i	$-0.602029\pi$
$347$	−11.7384	−0.630148	−0.315074	+	0.949067i	$0.602029\pi$
$348$	0	0
$349$	−9.02304	−0.482992	−0.241496	−	0.970402i	$-0.577638\pi$
$349$	−9.02304	−0.482992	−0.241496	+	0.970402i	$0.577638\pi$
$350$	0	0
$351$	7.03974	0.375754
$352$	0	0
$353$	−28.0910	−1.49513	−0.747566	−	0.664188i	$-0.768778\pi$
$353$	−28.0910	−1.49513	−0.747566	+	0.664188i	$0.768778\pi$
$354$	0	0
$355$	−13.7644	−0.730538
$356$	0	0
$357$	8.68501	0.459660
$358$	0	0
$359$	14.5356	0.767160	0.383580	−	0.923508i	$-0.374691\pi$
$359$	14.5356	0.767160	0.383580	+	0.923508i	$0.374691\pi$
$360$	0	0
$361$	−14.5178	−0.764095
$362$	0	0
$363$	25.7445	1.35124
$364$	0	0
$365$	−12.2107	−0.639139
$366$	0	0
$367$	18.8415	0.983516	0.491758	−	0.870732i	$-0.336354\pi$
$367$	18.8415	0.983516	0.491758	+	0.870732i	$0.336354\pi$
$368$	0	0
$369$	23.3487	1.21549
$370$	0	0
$371$	36.4033	1.88997
$372$	0	0
$373$	6.60160	0.341818	0.170909	−	0.985287i	$-0.445330\pi$
$373$	6.60160	0.341818	0.170909	+	0.985287i	$0.445330\pi$
$374$	0	0
$375$	28.6335	1.47863
$376$	0	0
$377$	33.5434	1.72757
$378$	0	0
$379$	−1.08262	−0.0556104	−0.0278052	−	0.999613i	$-0.508852\pi$
$379$	−1.08262	−0.0556104	−0.0278052	+	0.999613i	$0.508852\pi$
$380$	0	0
$381$	−19.5917	−1.00371
$382$	0	0
$383$	21.6461	1.10606	0.553031	−	0.833161i	$-0.313471\pi$
$383$	21.6461	1.10606	0.553031	+	0.833161i	$0.313471\pi$
$384$	0	0
$385$	1.89866	0.0967644
$386$	0	0
$387$	25.6883	1.30581
$388$	0	0
$389$	9.28119	0.470575	0.235288	−	0.971926i	$-0.424397\pi$
$389$	9.28119	0.470575	0.235288	+	0.971926i	$0.424397\pi$
$390$	0	0
$391$	7.76966	0.392929
$392$	0	0
$393$	−40.0494	−2.02023
$394$	0	0
$395$	−13.0926	−0.658760
$396$	0	0
$397$	−1.88103	−0.0944060	−0.0472030	−	0.998885i	$-0.515031\pi$
$397$	−1.88103	−0.0944060	−0.0472030	+	0.998885i	$0.515031\pi$
$398$	0	0
$399$	18.3872	0.920511
$400$	0	0
$401$	10.3783	0.518268	0.259134	−	0.965841i	$-0.416563\pi$
$401$	10.3783	0.518268	0.259134	+	0.965841i	$0.416563\pi$
$402$	0	0
$403$	−11.4735	−0.571534
$404$	0	0
$405$	−19.1290	−0.950526
$406$	0	0
$407$	0.241844	0.0119878
$408$	0	0
$409$	7.76886	0.384145	0.192073	−	0.981381i	$-0.438479\pi$
$409$	7.76886	0.384145	0.192073	+	0.981381i	$0.438479\pi$
$410$	0	0
$411$	0.746617	0.0368279
$412$	0	0
$413$	−3.68573	−0.181363
$414$	0	0
$415$	−14.0725	−0.690791
$416$	0	0
$417$	50.2780	2.46212
$418$	0	0
$419$	29.0642	1.41988	0.709939	−	0.704263i	$-0.248722\pi$
$419$	29.0642	1.41988	0.709939	+	0.704263i	$0.248722\pi$
$420$	0	0
$421$	−23.9458	−1.16704	−0.583522	−	0.812097i	$-0.698326\pi$
$421$	−23.9458	−1.16704	−0.583522	+	0.812097i	$0.698326\pi$
$422$	0	0
$423$	−31.4626	−1.52976
$424$	0	0
$425$	−1.44264	−0.0699782
$426$	0	0
$427$	−14.7649	−0.714525
$428$	0	0
$429$	4.29729	0.207475
$430$	0	0
$431$	27.6434	1.33154	0.665768	−	0.746158i	$-0.268104\pi$
$431$	27.6434	1.33154	0.665768	+	0.746158i	$0.268104\pi$
$432$	0	0
$433$	−8.29706	−0.398731	−0.199366	−	0.979925i	$-0.563888\pi$
$433$	−8.29706	−0.398731	−0.199366	+	0.979925i	$0.563888\pi$
$434$	0	0
$435$	−22.3269	−1.07049
$436$	0	0
$437$	16.4493	0.786876
$438$	0	0
$439$	21.6504	1.03332	0.516659	−	0.856191i	$-0.327175\pi$
$439$	21.6504	1.03332	0.516659	+	0.856191i	$0.327175\pi$
$440$	0	0
$441$	16.8077	0.800365
$442$	0	0
$443$	15.8383	0.752501	0.376251	−	0.926518i	$-0.377213\pi$
$443$	15.8383	0.752501	0.376251	+	0.926518i	$0.377213\pi$
$444$	0	0
$445$	12.0117	0.569408
$446$	0	0
$447$	−17.1251	−0.809990
$448$	0	0
$449$	−6.75120	−0.318609	−0.159304	−	0.987229i	$-0.550925\pi$
$449$	−6.75120	−0.318609	−0.159304	+	0.987229i	$0.550925\pi$
$450$	0	0
$451$	−2.49829	−0.117640
$452$	0	0
$453$	38.1194	1.79101
$454$	0	0
$455$	−46.4169	−2.17606
$456$	0	0
$457$	1.90340	0.0890374	0.0445187	−	0.999009i	$-0.485825\pi$
$457$	1.90340	0.0890374	0.0445187	+	0.999009i	$0.485825\pi$
$458$	0	0
$459$	1.05431	0.0492109
$460$	0	0
$461$	−10.9088	−0.508073	−0.254036	−	0.967195i	$-0.581758\pi$
$461$	−10.9088	−0.508073	−0.254036	+	0.967195i	$0.581758\pi$
$462$	0	0
$463$	1.95571	0.0908897	0.0454449	−	0.998967i	$-0.485529\pi$
$463$	1.95571	0.0908897	0.0454449	+	0.998967i	$0.485529\pi$
$464$	0	0
$465$	7.63690	0.354153
$466$	0	0
$467$	−14.3813	−0.665488	−0.332744	−	0.943017i	$-0.607975\pi$
$467$	−14.3813	−0.665488	−0.332744	+	0.943017i	$0.607975\pi$
$468$	0	0
$469$	−16.4357	−0.758929
$470$	0	0
$471$	−28.1546	−1.29730
$472$	0	0
$473$	−2.74863	−0.126382
$474$	0	0
$475$	−3.05423	−0.140138
$476$	0	0
$477$	−25.2114	−1.15435
$478$	0	0
$479$	−9.97224	−0.455643	−0.227822	−	0.973703i	$-0.573160\pi$
$479$	−9.97224	−0.455643	−0.227822	+	0.973703i	$0.573160\pi$
$480$	0	0
$481$	−5.91243	−0.269584
$482$	0	0
$483$	67.4796	3.07043
$484$	0	0
$485$	25.8331	1.17302
$486$	0	0
$487$	35.5199	1.60956	0.804781	−	0.593572i	$-0.202283\pi$
$487$	35.5199	1.60956	0.804781	+	0.593572i	$0.202283\pi$
$488$	0	0
$489$	−17.6392	−0.797672
$490$	0	0
$491$	−2.80752	−0.126702	−0.0633508	−	0.997991i	$-0.520179\pi$
$491$	−2.80752	−0.126702	−0.0633508	+	0.997991i	$0.520179\pi$
$492$	0	0
$493$	5.02363	0.226253
$494$	0	0
$495$	−1.31493	−0.0591015
$496$	0	0
$497$	26.8978	1.20653
$498$	0	0
$499$	5.93648	0.265753	0.132877	−	0.991133i	$-0.457579\pi$
$499$	5.93648	0.265753	0.132877	+	0.991133i	$0.457579\pi$
$500$	0	0
$501$	55.3331	2.47210
$502$	0	0
$503$	−21.1211	−0.941744	−0.470872	−	0.882201i	$-0.656061\pi$
$503$	−21.1211	−0.941744	−0.470872	+	0.882201i	$0.656061\pi$
$504$	0	0
$505$	20.6685	0.919736
$506$	0	0
$507$	−74.4240	−3.30529
$508$	0	0
$509$	−5.27836	−0.233959	−0.116980	−	0.993134i	$-0.537321\pi$
$509$	−5.27836	−0.233959	−0.116980	+	0.993134i	$0.537321\pi$
$510$	0	0
$511$	23.8617	1.05558
$512$	0	0
$513$	2.23209	0.0985494
$514$	0	0
$515$	−21.2635	−0.936981
$516$	0	0
$517$	3.36647	0.148057
$518$	0	0
$519$	−5.02499	−0.220572
$520$	0	0
$521$	35.6473	1.56173	0.780867	−	0.624697i	$-0.214777\pi$
$521$	35.6473	1.56173	0.780867	+	0.624697i	$0.214777\pi$
$522$	0	0
$523$	−11.4090	−0.498880	−0.249440	−	0.968390i	$-0.580246\pi$
$523$	−11.4090	−0.498880	−0.249440	+	0.968390i	$0.580246\pi$
$524$	0	0
$525$	−12.5293	−0.546824
$526$	0	0
$527$	−1.71833	−0.0748515
$528$	0	0
$529$	37.3676	1.62468
$530$	0	0
$531$	2.55257	0.110772
$532$	0	0
$533$	61.0764	2.64551
$534$	0	0
$535$	33.4850	1.44768
$536$	0	0
$537$	16.0201	0.691318
$538$	0	0
$539$	−1.79840	−0.0774628
$540$	0	0
$541$	−23.1127	−0.993694	−0.496847	−	0.867838i	$-0.665509\pi$
$541$	−23.1127	−0.993694	−0.496847	+	0.867838i	$0.665509\pi$
$542$	0	0
$543$	−44.7578	−1.92074
$544$	0	0
$545$	−6.97998	−0.298990
$546$	0	0
$547$	19.6235	0.839038	0.419519	−	0.907747i	$-0.362199\pi$
$547$	19.6235	0.839038	0.419519	+	0.907747i	$0.362199\pi$
$548$	0	0
$549$	10.2256	0.436416
$550$	0	0
$551$	10.6356	0.453093
$552$	0	0
$553$	25.5850	1.08798
$554$	0	0
$555$	3.93539	0.167048
$556$	0	0
$557$	−1.03378	−0.0438027	−0.0219013	−	0.999760i	$-0.506972\pi$
$557$	−1.03378	−0.0438027	−0.0219013	+	0.999760i	$0.506972\pi$
$558$	0	0
$559$	67.1965	2.84211
$560$	0	0
$561$	0.643585	0.0271722
$562$	0	0
$563$	−27.9390	−1.17749	−0.588743	−	0.808320i	$-0.700377\pi$
$563$	−27.9390	−1.17749	−0.588743	+	0.808320i	$0.700377\pi$
$564$	0	0
$565$	−9.51082	−0.400123
$566$	0	0
$567$	37.3810	1.56985
$568$	0	0
$569$	40.2667	1.68807	0.844035	−	0.536288i	$-0.180174\pi$
$569$	40.2667	1.68807	0.844035	+	0.536288i	$0.180174\pi$
$570$	0	0
$571$	41.9789	1.75676	0.878380	−	0.477963i	$-0.158625\pi$
$571$	41.9789	1.75676	0.878380	+	0.477963i	$0.158625\pi$
$572$	0	0
$573$	−39.7898	−1.66224
$574$	0	0
$575$	−11.2088	−0.467439
$576$	0	0
$577$	−5.22236	−0.217410	−0.108705	−	0.994074i	$-0.534670\pi$
$577$	−5.22236	−0.217410	−0.108705	+	0.994074i	$0.534670\pi$
$578$	0	0
$579$	31.5288	1.31029
$580$	0	0
$581$	27.4998	1.14089
$582$	0	0
$583$	2.69759	0.111723
$584$	0	0
$585$	32.1464	1.32909
$586$	0	0
$587$	9.58387	0.395569	0.197784	−	0.980246i	$-0.436625\pi$
$587$	9.58387	0.395569	0.197784	+	0.980246i	$0.436625\pi$
$588$	0	0
$589$	−3.63790	−0.149897
$590$	0	0
$591$	−60.4982	−2.48856
$592$	0	0
$593$	4.97405	0.204260	0.102130	−	0.994771i	$-0.467434\pi$
$593$	4.97405	0.204260	0.102130	+	0.994771i	$0.467434\pi$
$594$	0	0
$595$	−6.95164	−0.284989
$596$	0	0
$597$	−23.7065	−0.970243
$598$	0	0
$599$	−11.9578	−0.488582	−0.244291	−	0.969702i	$-0.578555\pi$
$599$	−11.9578	−0.488582	−0.244291	+	0.969702i	$0.578555\pi$
$600$	0	0
$601$	40.6335	1.65748	0.828738	−	0.559638i	$-0.189060\pi$
$601$	40.6335	1.65748	0.828738	+	0.559638i	$0.189060\pi$
$602$	0	0
$603$	11.3826	0.463537
$604$	0	0
$605$	−20.6064	−0.837768
$606$	0	0
$607$	−17.4617	−0.708749	−0.354374	−	0.935104i	$-0.615306\pi$
$607$	−17.4617	−0.708749	−0.354374	+	0.935104i	$0.615306\pi$
$608$	0	0
$609$	43.6303	1.76799
$610$	0	0
$611$	−82.3011	−3.32954
$612$	0	0
$613$	−15.3408	−0.619609	−0.309804	−	0.950800i	$-0.600264\pi$
$613$	−15.3408	−0.619609	−0.309804	+	0.950800i	$0.600264\pi$
$614$	0	0
$615$	−40.6533	−1.63930
$616$	0	0
$617$	30.8136	1.24051	0.620254	−	0.784401i	$-0.287030\pi$
$617$	30.8136	1.24051	0.620254	+	0.784401i	$0.287030\pi$
$618$	0	0
$619$	38.8287	1.56066	0.780328	−	0.625371i	$-0.215052\pi$
$619$	38.8287	1.56066	0.780328	+	0.625371i	$0.215052\pi$
$620$	0	0
$621$	8.19161	0.328718
$622$	0	0
$623$	−23.4727	−0.940414
$624$	0	0
$625$	−15.7056	−0.628224
$626$	0	0
$627$	1.36255	0.0544148
$628$	0	0
$629$	−0.885476	−0.0353062
$630$	0	0
$631$	38.1866	1.52018	0.760091	−	0.649816i	$-0.225154\pi$
$631$	38.1866	1.52018	0.760091	+	0.649816i	$0.225154\pi$
$632$	0	0
$633$	−61.2664	−2.43512
$634$	0	0
$635$	15.6815	0.622302
$636$	0	0
$637$	43.9661	1.74200
$638$	0	0
$639$	−18.6282	−0.736921
$640$	0	0
$641$	−2.60973	−0.103078	−0.0515391	−	0.998671i	$-0.516413\pi$
$641$	−2.60973	−0.103078	−0.0515391	+	0.998671i	$0.516413\pi$
$642$	0	0
$643$	−30.0077	−1.18339	−0.591694	−	0.806163i	$-0.701541\pi$
$643$	−30.0077	−1.18339	−0.591694	+	0.806163i	$0.701541\pi$
$644$	0	0
$645$	−44.7269	−1.76112
$646$	0	0
$647$	22.2482	0.874667	0.437333	−	0.899299i	$-0.355923\pi$
$647$	22.2482	0.874667	0.437333	+	0.899299i	$0.355923\pi$
$648$	0	0
$649$	−0.273123	−0.0107210
$650$	0	0
$651$	−14.9237	−0.584906
$652$	0	0
$653$	22.4044	0.876753	0.438376	−	0.898791i	$-0.355554\pi$
$653$	22.4044	0.876753	0.438376	+	0.898791i	$0.355554\pi$
$654$	0	0
$655$	32.0563	1.25254
$656$	0	0
$657$	−16.5256	−0.644724
$658$	0	0
$659$	37.8185	1.47320	0.736600	−	0.676328i	$-0.236430\pi$
$659$	37.8185	1.47320	0.736600	+	0.676328i	$0.236430\pi$
$660$	0	0
$661$	−39.3439	−1.53030	−0.765150	−	0.643852i	$-0.777335\pi$
$661$	−39.3439	−1.53030	−0.765150	+	0.643852i	$0.777335\pi$
$662$	0	0
$663$	−15.7339	−0.611054
$664$	0	0
$665$	−14.7174	−0.570718
$666$	0	0
$667$	39.0319	1.51132
$668$	0	0
$669$	−27.7877	−1.07434
$670$	0	0
$671$	−1.09413	−0.0422382
$672$	0	0
$673$	−25.2670	−0.973972	−0.486986	−	0.873410i	$-0.661904\pi$
$673$	−25.2670	−0.973972	−0.486986	+	0.873410i	$0.661904\pi$
$674$	0	0
$675$	−1.52098	−0.0585427
$676$	0	0
$677$	2.57027	0.0987835	0.0493917	−	0.998779i	$-0.484272\pi$
$677$	2.57027	0.0987835	0.0493917	+	0.998779i	$0.484272\pi$
$678$	0	0
$679$	−50.4820	−1.93732
$680$	0	0
$681$	15.0838	0.578014
$682$	0	0
$683$	3.16286	0.121023	0.0605117	−	0.998167i	$-0.480727\pi$
$683$	3.16286	0.121023	0.0605117	+	0.998167i	$0.480727\pi$
$684$	0	0
$685$	−0.597605	−0.0228333
$686$	0	0
$687$	7.69421	0.293553
$688$	0	0
$689$	−65.9488	−2.51245
$690$	0	0
$691$	34.5375	1.31387	0.656935	−	0.753947i	$-0.271853\pi$
$691$	34.5375	1.31387	0.656935	+	0.753947i	$0.271853\pi$
$692$	0	0
$693$	2.56957	0.0976099
$694$	0	0
$695$	−40.2434	−1.52652
$696$	0	0
$697$	9.14712	0.346472
$698$	0	0
$699$	16.4113	0.620732
$700$	0	0
$701$	23.7909	0.898570	0.449285	−	0.893388i	$-0.351679\pi$
$701$	23.7909	0.898570	0.449285	+	0.893388i	$0.351679\pi$
$702$	0	0
$703$	−1.87466	−0.0707041
$704$	0	0
$705$	54.7807	2.06316
$706$	0	0
$707$	−40.3895	−1.51900
$708$	0	0
$709$	8.57993	0.322226	0.161113	−	0.986936i	$-0.448492\pi$
$709$	8.57993	0.322226	0.161113	+	0.986936i	$0.448492\pi$
$710$	0	0
$711$	−17.7190	−0.664516
$712$	0	0
$713$	−13.3508	−0.499992
$714$	0	0
$715$	−3.43963	−0.128635
$716$	0	0
$717$	19.4792	0.727463
$718$	0	0
$719$	40.3230	1.50379	0.751897	−	0.659280i	$-0.229139\pi$
$719$	40.3230	1.50379	0.751897	+	0.659280i	$0.229139\pi$
$720$	0	0
$721$	41.5522	1.54748
$722$	0	0
$723$	10.5253	0.391438
$724$	0	0
$725$	−7.24728	−0.269157
$726$	0	0
$727$	−0.557292	−0.0206688	−0.0103344	−	0.999947i	$-0.503290\pi$
$727$	−0.557292	−0.0206688	−0.0103344	+	0.999947i	$0.503290\pi$
$728$	0	0
$729$	−18.4354	−0.682791
$730$	0	0
$731$	10.0637	0.372219
$732$	0	0
$733$	−13.6905	−0.505669	−0.252834	−	0.967510i	$-0.581363\pi$
$733$	−13.6905	−0.505669	−0.252834	+	0.967510i	$0.581363\pi$
$734$	0	0
$735$	−29.2644	−1.07943
$736$	0	0
$737$	−1.21793	−0.0448631
$738$	0	0
$739$	−21.7238	−0.799122	−0.399561	−	0.916707i	$-0.630837\pi$
$739$	−21.7238	−0.799122	−0.399561	+	0.916707i	$0.630837\pi$
$740$	0	0
$741$	−33.3105	−1.22369
$742$	0	0
$743$	−51.7948	−1.90017	−0.950083	−	0.311996i	$-0.899002\pi$
$743$	−51.7948	−1.90017	−0.950083	+	0.311996i	$0.899002\pi$
$744$	0	0
$745$	13.7073	0.502195
$746$	0	0
$747$	−19.0452	−0.696827
$748$	0	0
$749$	−65.4348	−2.39094
$750$	0	0
$751$	20.5354	0.749346	0.374673	−	0.927157i	$-0.377755\pi$
$751$	20.5354	0.749346	0.374673	+	0.927157i	$0.377755\pi$
$752$	0	0
$753$	−4.36325	−0.159006
$754$	0	0
$755$	−30.5115	−1.11043
$756$	0	0
$757$	−22.2517	−0.808752	−0.404376	−	0.914593i	$-0.632511\pi$
$757$	−22.2517	−0.808752	−0.404376	+	0.914593i	$0.632511\pi$
$758$	0	0
$759$	5.00044	0.181504
$760$	0	0
$761$	−46.2105	−1.67513	−0.837564	−	0.546339i	$-0.816021\pi$
$761$	−46.2105	−1.67513	−0.837564	+	0.546339i	$0.816021\pi$
$762$	0	0
$763$	13.6400	0.493800
$764$	0	0
$765$	4.81440	0.174065
$766$	0	0
$767$	6.67712	0.241097
$768$	0	0
$769$	5.37767	0.193924	0.0969619	−	0.995288i	$-0.469087\pi$
$769$	5.37767	0.193924	0.0969619	+	0.995288i	$0.469087\pi$
$770$	0	0
$771$	−12.8295	−0.462042
$772$	0	0
$773$	17.6567	0.635067	0.317533	−	0.948247i	$-0.397145\pi$
$773$	17.6567	0.635067	0.317533	+	0.948247i	$0.397145\pi$
$774$	0	0
$775$	2.47892	0.0890455
$776$	0	0
$777$	−7.69037	−0.275891
$778$	0	0
$779$	19.3655	0.693842
$780$	0	0
$781$	1.99320	0.0713224
$782$	0	0
$783$	5.29645	0.189280
$784$	0	0
$785$	22.5355	0.804326
$786$	0	0
$787$	19.1412	0.682309	0.341154	−	0.940007i	$-0.389182\pi$
$787$	19.1412	0.682309	0.341154	+	0.940007i	$0.389182\pi$
$788$	0	0
$789$	−28.7416	−1.02323
$790$	0	0
$791$	18.5856	0.660828
$792$	0	0
$793$	26.7484	0.949863
$794$	0	0
$795$	43.8964	1.55685
$796$	0	0
$797$	−19.8819	−0.704252	−0.352126	−	0.935953i	$-0.614541\pi$
$797$	−19.8819	−0.704252	−0.352126	+	0.935953i	$0.614541\pi$
$798$	0	0
$799$	−12.3258	−0.436057
$800$	0	0
$801$	16.2562	0.574384
$802$	0	0
$803$	1.76822	0.0623992
$804$	0	0
$805$	−54.0119	−1.90367
$806$	0	0
$807$	−18.6364	−0.656031
$808$	0	0
$809$	54.4925	1.91586	0.957928	−	0.287010i	$-0.0926612\pi$
$809$	54.4925	1.91586	0.957928	+	0.287010i	$0.0926612\pi$
$810$	0	0
$811$	−18.9753	−0.666314	−0.333157	−	0.942871i	$-0.608114\pi$
$811$	−18.9753	−0.666314	−0.333157	+	0.942871i	$0.608114\pi$
$812$	0	0
$813$	62.5748	2.19460
$814$	0	0
$815$	14.1187	0.494558
$816$	0	0
$817$	21.3060	0.745404
$818$	0	0
$819$	−62.8190	−2.19507
$820$	0	0
$821$	28.2818	0.987041	0.493521	−	0.869734i	$-0.335710\pi$
$821$	28.2818	0.987041	0.493521	+	0.869734i	$0.335710\pi$
$822$	0	0
$823$	9.12712	0.318151	0.159076	−	0.987266i	$-0.449149\pi$
$823$	9.12712	0.318151	0.159076	+	0.987266i	$0.449149\pi$
$824$	0	0
$825$	−0.928460	−0.0323248
$826$	0	0
$827$	−22.4987	−0.782356	−0.391178	−	0.920315i	$-0.627932\pi$
$827$	−22.4987	−0.782356	−0.391178	+	0.920315i	$0.627932\pi$
$828$	0	0
$829$	−5.68560	−0.197469	−0.0987346	−	0.995114i	$-0.531479\pi$
$829$	−5.68560	−0.197469	−0.0987346	+	0.995114i	$0.531479\pi$
$830$	0	0
$831$	−4.95467	−0.171876
$832$	0	0
$833$	6.58459	0.228142
$834$	0	0
$835$	−44.2896	−1.53270
$836$	0	0
$837$	−1.81165	−0.0626196
$838$	0	0
$839$	29.2209	1.00882	0.504409	−	0.863465i	$-0.331710\pi$
$839$	29.2209	1.00882	0.504409	+	0.863465i	$0.331710\pi$
$840$	0	0
$841$	−3.76313	−0.129763
$842$	0	0
$843$	35.3178	1.21641
$844$	0	0
$845$	59.5703	2.04928
$846$	0	0
$847$	40.2681	1.38363
$848$	0	0
$849$	63.4953	2.17915
$850$	0	0
$851$	−6.87985	−0.235838
$852$	0	0
$853$	25.8758	0.885970	0.442985	−	0.896529i	$-0.353920\pi$
$853$	25.8758	0.885970	0.442985	+	0.896529i	$0.353920\pi$
$854$	0	0
$855$	10.1927	0.348582
$856$	0	0
$857$	30.9161	1.05607	0.528037	−	0.849221i	$-0.322928\pi$
$857$	30.9161	1.05607	0.528037	+	0.849221i	$0.322928\pi$
$858$	0	0
$859$	−33.8090	−1.15355	−0.576774	−	0.816904i	$-0.695689\pi$
$859$	−33.8090	−1.15355	−0.576774	+	0.816904i	$0.695689\pi$
$860$	0	0
$861$	79.4428	2.70740
$862$	0	0
$863$	−17.6669	−0.601388	−0.300694	−	0.953721i	$-0.597218\pi$
$863$	−17.6669	−0.601388	−0.300694	+	0.953721i	$0.597218\pi$
$864$	0	0
$865$	4.02209	0.136755
$866$	0	0
$867$	−2.35639	−0.0800272
$868$	0	0
$869$	1.89592	0.0643148
$870$	0	0
$871$	29.7751	1.00889
$872$	0	0
$873$	34.9616	1.18327
$874$	0	0
$875$	44.7869	1.51407
$876$	0	0
$877$	−47.3115	−1.59760	−0.798798	−	0.601599i	$-0.794531\pi$
$877$	−47.3115	−1.59760	−0.798798	+	0.601599i	$0.794531\pi$
$878$	0	0
$879$	2.74517	0.0925921
$880$	0	0
$881$	−33.6430	−1.13346	−0.566730	−	0.823904i	$-0.691792\pi$
$881$	−33.6430	−1.13346	−0.566730	+	0.823904i	$0.691792\pi$
$882$	0	0
$883$	−25.4756	−0.857323	−0.428661	−	0.903465i	$-0.641015\pi$
$883$	−25.4756	−0.857323	−0.428661	+	0.903465i	$0.641015\pi$
$884$	0	0
$885$	−4.44438	−0.149396
$886$	0	0
$887$	−40.9252	−1.37413	−0.687067	−	0.726594i	$-0.741102\pi$
$887$	−40.9252	−1.37413	−0.687067	+	0.726594i	$0.741102\pi$
$888$	0	0
$889$	−30.6441	−1.02777
$890$	0	0
$891$	2.77004	0.0927998
$892$	0	0
$893$	−26.0952	−0.873244
$894$	0	0
$895$	−12.8228	−0.428618
$896$	0	0
$897$	−122.247	−4.08171
$898$	0	0
$899$	−8.63224	−0.287901
$900$	0	0
$901$	−9.87684	−0.329045
$902$	0	0
$903$	87.4033	2.90860
$904$	0	0
$905$	35.8249	1.19086
$906$	0	0
$907$	−23.2026	−0.770430	−0.385215	−	0.922827i	$-0.625873\pi$
$907$	−23.2026	−0.770430	−0.385215	+	0.922827i	$0.625873\pi$
$908$	0	0
$909$	27.9720	0.927773
$910$	0	0
$911$	42.5222	1.40882	0.704411	−	0.709792i	$-0.251211\pi$
$911$	42.5222	1.40882	0.704411	+	0.709792i	$0.251211\pi$
$912$	0	0
$913$	2.03782	0.0674419
$914$	0	0
$915$	−17.8041	−0.588585
$916$	0	0
$917$	−62.6430	−2.06865
$918$	0	0
$919$	−46.8576	−1.54569	−0.772844	−	0.634596i	$-0.781167\pi$
$919$	−46.8576	−1.54569	−0.772844	+	0.634596i	$0.781167\pi$
$920$	0	0
$921$	−42.7515	−1.40871
$922$	0	0
$923$	−48.7284	−1.60391
$924$	0	0
$925$	1.27742	0.0420013
$926$	0	0
$927$	−28.7772	−0.945168
$928$	0	0
$929$	−18.7852	−0.616322	−0.308161	−	0.951334i	$-0.599714\pi$
$929$	−18.7852	−0.616322	−0.308161	+	0.951334i	$0.599714\pi$
$930$	0	0
$931$	13.9404	0.456877
$932$	0	0
$933$	−61.9885	−2.02941
$934$	0	0
$935$	−0.515137	−0.0168468
$936$	0	0
$937$	22.0088	0.718996	0.359498	−	0.933146i	$-0.382948\pi$
$937$	22.0088	0.718996	0.359498	+	0.933146i	$0.382948\pi$
$938$	0	0
$939$	32.0681	1.04650
$940$	0	0
$941$	9.48480	0.309196	0.154598	−	0.987977i	$-0.450592\pi$
$941$	9.48480	0.309196	0.154598	+	0.987977i	$0.450592\pi$
$942$	0	0
$943$	71.0700	2.31436
$944$	0	0
$945$	−7.32917	−0.238418
$946$	0	0
$947$	58.1651	1.89011	0.945056	−	0.326907i	$-0.106006\pi$
$947$	58.1651	1.89011	0.945056	+	0.326907i	$0.106006\pi$
$948$	0	0
$949$	−43.2282	−1.40325
$950$	0	0
$951$	62.3964	2.02334
$952$	0	0
$953$	20.8238	0.674548	0.337274	−	0.941407i	$-0.390495\pi$
$953$	20.8238	0.674548	0.337274	+	0.941407i	$0.390495\pi$
$954$	0	0
$955$	31.8484	1.03059
$956$	0	0
$957$	3.23313	0.104512
$958$	0	0
$959$	1.16781	0.0377107
$960$	0	0
$961$	−28.0474	−0.904753
$962$	0	0
$963$	45.3173	1.46033
$964$	0	0
$965$	−25.2362	−0.812383
$966$	0	0
$967$	23.5625	0.757720	0.378860	−	0.925454i	$-0.376316\pi$
$967$	23.5625	0.757720	0.378860	+	0.925454i	$0.376316\pi$
$968$	0	0
$969$	−4.98876	−0.160262
$970$	0	0
$971$	55.0019	1.76509	0.882547	−	0.470224i	$-0.155827\pi$
$971$	55.0019	1.76509	0.882547	+	0.470224i	$0.155827\pi$
$972$	0	0
$973$	78.6418	2.52114
$974$	0	0
$975$	22.6983	0.726928
$976$	0	0
$977$	−17.6661	−0.565189	−0.282594	−	0.959239i	$-0.591195\pi$
$977$	−17.6661	−0.565189	−0.282594	+	0.959239i	$0.591195\pi$
$978$	0	0
$979$	−1.73940	−0.0555914
$980$	0	0
$981$	−9.44646	−0.301602
$982$	0	0
$983$	19.4301	0.619722	0.309861	−	0.950782i	$-0.399717\pi$
$983$	19.4301	0.619722	0.309861	+	0.950782i	$0.399717\pi$
$984$	0	0
$985$	48.4238	1.54291
$986$	0	0
$987$	−107.050	−3.40744
$988$	0	0
$989$	78.1915	2.48635
$990$	0	0
$991$	32.4169	1.02976	0.514878	−	0.857264i	$-0.327837\pi$
$991$	32.4169	1.02976	0.514878	+	0.857264i	$0.327837\pi$
$992$	0	0
$993$	−20.6445	−0.655134
$994$	0	0
$995$	18.9751	0.601552
$996$	0	0
$997$	14.5567	0.461015	0.230507	−	0.973071i	$-0.425961\pi$
$997$	14.5567	0.461015	0.230507	+	0.973071i	$0.425961\pi$
$998$	0	0
$999$	−0.933564	−0.0295367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.6	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.6	✓	32	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35639 q^{3} +1.88610 q^{5} -3.68573 q^{7} +2.55257 q^{9} +O(q^{10})\)	\(q-2.35639 q^{3} +1.88610 q^{5} -3.68573 q^{7} +2.55257 q^{9} -0.273123 q^{11} +6.67712 q^{13} -4.44438 q^{15} +1.00000 q^{17} +2.11712 q^{19} +8.68501 q^{21} +7.76966 q^{23} -1.44264 q^{25} +1.05431 q^{27} +5.02363 q^{29} -1.71833 q^{31} +0.643585 q^{33} -6.95164 q^{35} -0.885476 q^{37} -15.7339 q^{39} +9.14712 q^{41} +10.0637 q^{43} +4.81440 q^{45} -12.3258 q^{47} +6.58459 q^{49} -2.35639 q^{51} -9.87684 q^{53} -0.515137 q^{55} -4.98876 q^{57} +1.00000 q^{59} +4.00598 q^{61} -9.40810 q^{63} +12.5937 q^{65} +4.45927 q^{67} -18.3084 q^{69} -7.29781 q^{71} -6.47408 q^{73} +3.39942 q^{75} +1.00666 q^{77} -6.94163 q^{79} -10.1421 q^{81} -7.46117 q^{83} +1.88610 q^{85} -11.8376 q^{87} +6.36854 q^{89} -24.6100 q^{91} +4.04905 q^{93} +3.99309 q^{95} +13.6966 q^{97} -0.697168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 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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