LMFDB - Embedded newform 8024.2.a.v.1.12

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 9 x^{17} - 2 x^{16} + 212 x^{15} - 289 x^{14} - 2094 x^{13} + 3933 x^{12} + 11326 x^{11} + \cdots + 1136 $ x^18 - 9x^17 - 2x^16 + 212x^15 - 289x^14 - 2094x^13 + 3933x^12 + 11326x^11 - 23166x^10 - 36429x^9 + 72042x^8 + 69272x^7 - 119982x^6 - 69890x^5 + 98783x^4 + 26872x^3 - 30932x^2 + 1600*x + 1136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.947778$ of defining polynomial
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)

$f(q)$	$=$	$q+1.94778 q^{3} -1.72233 q^{5} -3.84469 q^{7} +0.793839 q^{9} +O(q^{10})$	$q+1.94778 q^{3} -1.72233 q^{5} -3.84469 q^{7} +0.793839 q^{9} -4.20176 q^{11} -4.80843 q^{13} -3.35471 q^{15} +1.00000 q^{17} +0.230905 q^{19} -7.48861 q^{21} +8.04201 q^{23} -2.03359 q^{25} -4.29711 q^{27} +0.273143 q^{29} -9.13252 q^{31} -8.18410 q^{33} +6.62182 q^{35} -1.72404 q^{37} -9.36576 q^{39} -1.39955 q^{41} -10.5587 q^{43} -1.36725 q^{45} +13.3891 q^{47} +7.78167 q^{49} +1.94778 q^{51} +8.55027 q^{53} +7.23681 q^{55} +0.449751 q^{57} -1.00000 q^{59} +7.63672 q^{61} -3.05207 q^{63} +8.28169 q^{65} +0.689884 q^{67} +15.6640 q^{69} +9.67564 q^{71} +9.81657 q^{73} -3.96098 q^{75} +16.1545 q^{77} -11.2169 q^{79} -10.7513 q^{81} +9.75456 q^{83} -1.72233 q^{85} +0.532022 q^{87} -5.91323 q^{89} +18.4869 q^{91} -17.7881 q^{93} -0.397693 q^{95} -12.6867 q^{97} -3.33552 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) $ 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9	$ 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) $ 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9 - q^11 + 15 * q^13 + 3 * q^15 + 18 * q^17 + 26 * q^19 - 4 * q^21 + 22 * q^23 + 42 * q^25 + 45 * q^27 + 6 * q^29 + 13 * q^31 - 5 * q^33 + 4 * q^35 + 4 * q^37 + 36 * q^39 - 15 * q^41 + 12 * q^43 + 14 * q^45 + 8 * q^47 - 13 * q^49 + 9 * q^51 - 11 * q^53 + 55 * q^55 - 20 * q^57 - 18 * q^59 + 53 * q^61 + 29 * q^63 - 26 * q^65 + 2 * q^67 + 32 * q^69 + 8 * q^71 - 42 * q^73 + 72 * q^75 + 6 * q^77 - 9 * q^79 + 42 * q^81 - 4 * q^83 + 2 * q^85 + 36 * q^87 + 13 * q^89 + 68 * q^91 + q^93 + 3 * q^95 - 56 * q^97 - 13 * 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
$3$	1.94778	1.12455	0.562275	−	0.826950i	$-0.309926\pi$
$3$	1.94778	1.12455	0.562275	+	0.826950i	$0.309926\pi$
$4$	0	0
$5$	−1.72233	−0.770248	−0.385124	−	0.922865i	$-0.625841\pi$
$5$	−1.72233	−0.770248	−0.385124	+	0.922865i	$0.625841\pi$
$6$	0	0
$7$	−3.84469	−1.45316	−0.726579	−	0.687083i	$-0.758891\pi$
$7$	−3.84469	−1.45316	−0.726579	+	0.687083i	$0.758891\pi$
$8$	0	0
$9$	0.793839	0.264613
$10$	0	0
$11$	−4.20176	−1.26688	−0.633439	−	0.773792i	$-0.718357\pi$
$11$	−4.20176	−1.26688	−0.633439	+	0.773792i	$0.718357\pi$
$12$	0	0
$13$	−4.80843	−1.33362	−0.666810	−	0.745228i	$-0.732341\pi$
$13$	−4.80843	−1.33362	−0.666810	+	0.745228i	$0.732341\pi$
$14$	0	0
$15$	−3.35471	−0.866183
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.230905	0.0529731	0.0264866	−	0.999649i	$-0.491568\pi$
$19$	0.230905	0.0529731	0.0264866	+	0.999649i	$0.491568\pi$
$20$	0	0
$21$	−7.48861	−1.63415
$22$	0	0
$23$	8.04201	1.67687	0.838437	−	0.544999i	$-0.183470\pi$
$23$	8.04201	1.67687	0.838437	+	0.544999i	$0.183470\pi$
$24$	0	0
$25$	−2.03359	−0.406718
$26$	0	0
$27$	−4.29711	−0.826980
$28$	0	0
$29$	0.273143	0.0507214	0.0253607	−	0.999678i	$-0.491927\pi$
$29$	0.273143	0.0507214	0.0253607	+	0.999678i	$0.491927\pi$
$30$	0	0
$31$	−9.13252	−1.64025	−0.820125	−	0.572185i	$-0.806096\pi$
$31$	−9.13252	−1.64025	−0.820125	+	0.572185i	$0.806096\pi$
$32$	0	0
$33$	−8.18410	−1.42467
$34$	0	0
$35$	6.62182	1.11929
$36$	0	0
$37$	−1.72404	−0.283430	−0.141715	−	0.989908i	$-0.545262\pi$
$37$	−1.72404	−0.283430	−0.141715	+	0.989908i	$0.545262\pi$
$38$	0	0
$39$	−9.36576	−1.49972
$40$	0	0
$41$	−1.39955	−0.218573	−0.109286	−	0.994010i	$-0.534857\pi$
$41$	−1.39955	−0.218573	−0.109286	+	0.994010i	$0.534857\pi$
$42$	0	0
$43$	−10.5587	−1.61018	−0.805090	−	0.593153i	$-0.797883\pi$
$43$	−10.5587	−1.61018	−0.805090	+	0.593153i	$0.797883\pi$
$44$	0	0
$45$	−1.36725	−0.203818
$46$	0	0
$47$	13.3891	1.95300	0.976498	−	0.215525i	$-0.0691461\pi$
$47$	13.3891	1.95300	0.976498	+	0.215525i	$0.0691461\pi$
$48$	0	0
$49$	7.78167	1.11167
$50$	0	0
$51$	1.94778	0.272743
$52$	0	0
$53$	8.55027	1.17447	0.587235	−	0.809416i	$-0.300216\pi$
$53$	8.55027	1.17447	0.587235	+	0.809416i	$0.300216\pi$
$54$	0	0
$55$	7.23681	0.975811
$56$	0	0
$57$	0.449751	0.0595710
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	7.63672	0.977782	0.488891	−	0.872345i	$-0.337402\pi$
$61$	7.63672	0.977782	0.488891	+	0.872345i	$0.337402\pi$
$62$	0	0
$63$	−3.05207	−0.384524
$64$	0	0
$65$	8.28169	1.02722
$66$	0	0
$67$	0.689884	0.0842827	0.0421414	−	0.999112i	$-0.486582\pi$
$67$	0.689884	0.0842827	0.0421414	+	0.999112i	$0.486582\pi$
$68$	0	0
$69$	15.6640	1.88573
$70$	0	0
$71$	9.67564	1.14829	0.574144	−	0.818755i	$-0.305335\pi$
$71$	9.67564	1.14829	0.574144	+	0.818755i	$0.305335\pi$
$72$	0	0
$73$	9.81657	1.14894	0.574471	−	0.818525i	$-0.305208\pi$
$73$	9.81657	1.14894	0.574471	+	0.818525i	$0.305208\pi$
$74$	0	0
$75$	−3.96098	−0.457374
$76$	0	0
$77$	16.1545	1.84097
$78$	0	0
$79$	−11.2169	−1.26200	−0.631001	−	0.775782i	$-0.717356\pi$
$79$	−11.2169	−1.26200	−0.631001	+	0.775782i	$0.717356\pi$
$80$	0	0
$81$	−10.7513	−1.19459
$82$	0	0
$83$	9.75456	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$83$	9.75456	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$84$	0	0
$85$	−1.72233	−0.186813
$86$	0	0
$87$	0.532022	0.0570388
$88$	0	0
$89$	−5.91323	−0.626801	−0.313400	−	0.949621i	$-0.601468\pi$
$89$	−5.91323	−0.626801	−0.313400	+	0.949621i	$0.601468\pi$
$90$	0	0
$91$	18.4869	1.93796
$92$	0	0
$93$	−17.7881	−1.84454
$94$	0	0
$95$	−0.397693	−0.0408025
$96$	0	0
$97$	−12.6867	−1.28814	−0.644070	−	0.764966i	$-0.722756\pi$
$97$	−12.6867	−1.28814	−0.644070	+	0.764966i	$0.722756\pi$
$98$	0	0
$99$	−3.33552	−0.335233
$100$	0	0
$101$	−5.58032	−0.555263	−0.277631	−	0.960688i	$-0.589549\pi$
$101$	−5.58032	−0.555263	−0.277631	+	0.960688i	$0.589549\pi$
$102$	0	0
$103$	14.8369	1.46192	0.730959	−	0.682421i	$-0.239073\pi$
$103$	14.8369	1.46192	0.730959	+	0.682421i	$0.239073\pi$
$104$	0	0
$105$	12.8978	1.25870
$106$	0	0
$107$	7.62453	0.737091	0.368545	−	0.929610i	$-0.379856\pi$
$107$	7.62453	0.737091	0.368545	+	0.929610i	$0.379856\pi$
$108$	0	0
$109$	2.31786	0.222011	0.111005	−	0.993820i	$-0.464593\pi$
$109$	2.31786	0.222011	0.111005	+	0.993820i	$0.464593\pi$
$110$	0	0
$111$	−3.35804	−0.318731
$112$	0	0
$113$	−11.7866	−1.10879	−0.554397	−	0.832252i	$-0.687051\pi$
$113$	−11.7866	−1.10879	−0.554397	+	0.832252i	$0.687051\pi$
$114$	0	0
$115$	−13.8510	−1.29161
$116$	0	0
$117$	−3.81712	−0.352893
$118$	0	0
$119$	−3.84469	−0.352442
$120$	0	0
$121$	6.65479	0.604981
$122$	0	0
$123$	−2.72601	−0.245796
$124$	0	0
$125$	12.1141	1.08352
$126$	0	0
$127$	3.57621	0.317338	0.158669	−	0.987332i	$-0.449280\pi$
$127$	3.57621	0.317338	0.158669	+	0.987332i	$0.449280\pi$
$128$	0	0
$129$	−20.5659	−1.81073
$130$	0	0
$131$	4.95080	0.432554	0.216277	−	0.976332i	$-0.430609\pi$
$131$	4.95080	0.432554	0.216277	+	0.976332i	$0.430609\pi$
$132$	0	0
$133$	−0.887757	−0.0769783
$134$	0	0
$135$	7.40103	0.636980
$136$	0	0
$137$	−19.5603	−1.67115	−0.835574	−	0.549378i	$-0.814865\pi$
$137$	−19.5603	−1.67115	−0.835574	+	0.549378i	$0.814865\pi$
$138$	0	0
$139$	−6.04531	−0.512756	−0.256378	−	0.966577i	$-0.582529\pi$
$139$	−6.04531	−0.512756	−0.256378	+	0.966577i	$0.582529\pi$
$140$	0	0
$141$	26.0789	2.19624
$142$	0	0
$143$	20.2039	1.68953
$144$	0	0
$145$	−0.470442	−0.0390681
$146$	0	0
$147$	15.1570	1.25012
$148$	0	0
$149$	22.1651	1.81584	0.907919	−	0.419146i	$-0.137671\pi$
$149$	22.1651	1.81584	0.907919	+	0.419146i	$0.137671\pi$
$150$	0	0
$151$	−6.63853	−0.540236	−0.270118	−	0.962827i	$-0.587063\pi$
$151$	−6.63853	−0.540236	−0.270118	+	0.962827i	$0.587063\pi$
$152$	0	0
$153$	0.793839	0.0641781
$154$	0	0
$155$	15.7292	1.26340
$156$	0	0
$157$	−22.5762	−1.80177	−0.900887	−	0.434054i	$-0.857083\pi$
$157$	−22.5762	−1.80177	−0.900887	+	0.434054i	$0.857083\pi$
$158$	0	0
$159$	16.6540	1.32075
$160$	0	0
$161$	−30.9190	−2.43676
$162$	0	0
$163$	9.71928	0.761272	0.380636	−	0.924725i	$-0.375705\pi$
$163$	9.71928	0.761272	0.380636	+	0.924725i	$0.375705\pi$
$164$	0	0
$165$	14.0957	1.09735
$166$	0	0
$167$	−0.902033	−0.0698014	−0.0349007	−	0.999391i	$-0.511112\pi$
$167$	−0.902033	−0.0698014	−0.0349007	+	0.999391i	$0.511112\pi$
$168$	0	0
$169$	10.1210	0.778540
$170$	0	0
$171$	0.183301	0.0140174
$172$	0	0
$173$	22.1438	1.68356	0.841780	−	0.539821i	$-0.181508\pi$
$173$	22.1438	1.68356	0.841780	+	0.539821i	$0.181508\pi$
$174$	0	0
$175$	7.81852	0.591025
$176$	0	0
$177$	−1.94778	−0.146404
$178$	0	0
$179$	−1.22270	−0.0913885	−0.0456943	−	0.998955i	$-0.514550\pi$
$179$	−1.22270	−0.0913885	−0.0456943	+	0.998955i	$0.514550\pi$
$180$	0	0
$181$	5.04429	0.374939	0.187469	−	0.982270i	$-0.439971\pi$
$181$	5.04429	0.374939	0.187469	+	0.982270i	$0.439971\pi$
$182$	0	0
$183$	14.8746	1.09957
$184$	0	0
$185$	2.96936	0.218311
$186$	0	0
$187$	−4.20176	−0.307263
$188$	0	0
$189$	16.5211	1.20173
$190$	0	0
$191$	−2.25066	−0.162852	−0.0814260	−	0.996679i	$-0.525947\pi$
$191$	−2.25066	−0.162852	−0.0814260	+	0.996679i	$0.525947\pi$
$192$	0	0
$193$	8.00499	0.576212	0.288106	−	0.957599i	$-0.406975\pi$
$193$	8.00499	0.576212	0.288106	+	0.957599i	$0.406975\pi$
$194$	0	0
$195$	16.1309	1.15516
$196$	0	0
$197$	19.8462	1.41398	0.706991	−	0.707222i	$-0.250052\pi$
$197$	19.8462	1.41398	0.706991	+	0.707222i	$0.250052\pi$
$198$	0	0
$199$	−4.80698	−0.340757	−0.170379	−	0.985379i	$-0.554499\pi$
$199$	−4.80698	−0.340757	−0.170379	+	0.985379i	$0.554499\pi$
$200$	0	0
$201$	1.34374	0.0947801
$202$	0	0
$203$	−1.05015	−0.0737062
$204$	0	0
$205$	2.41048	0.168355
$206$	0	0
$207$	6.38406	0.443723
$208$	0	0
$209$	−0.970206	−0.0671105
$210$	0	0
$211$	−2.91570	−0.200725	−0.100363	−	0.994951i	$-0.532000\pi$
$211$	−2.91570	−0.200725	−0.100363	+	0.994951i	$0.532000\pi$
$212$	0	0
$213$	18.8460	1.29131
$214$	0	0
$215$	18.1855	1.24024
$216$	0	0
$217$	35.1118	2.38354
$218$	0	0
$219$	19.1205	1.29204
$220$	0	0
$221$	−4.80843	−0.323450
$222$	0	0
$223$	−17.1761	−1.15020	−0.575100	−	0.818083i	$-0.695037\pi$
$223$	−17.1761	−1.15020	−0.575100	+	0.818083i	$0.695037\pi$
$224$	0	0
$225$	−1.61434	−0.107623
$226$	0	0
$227$	−19.4405	−1.29031	−0.645156	−	0.764051i	$-0.723208\pi$
$227$	−19.4405	−1.29031	−0.645156	+	0.764051i	$0.723208\pi$
$228$	0	0
$229$	−2.45597	−0.162295	−0.0811475	−	0.996702i	$-0.525858\pi$
$229$	−2.45597	−0.162295	−0.0811475	+	0.996702i	$0.525858\pi$
$230$	0	0
$231$	31.4653	2.07027
$232$	0	0
$233$	28.1427	1.84369	0.921845	−	0.387560i	$-0.126682\pi$
$233$	28.1427	1.84369	0.921845	+	0.387560i	$0.126682\pi$
$234$	0	0
$235$	−23.0604	−1.50429
$236$	0	0
$237$	−21.8481	−1.41918
$238$	0	0
$239$	−10.9707	−0.709639	−0.354819	−	0.934935i	$-0.615458\pi$
$239$	−10.9707	−0.709639	−0.354819	+	0.934935i	$0.615458\pi$
$240$	0	0
$241$	1.22186	0.0787068	0.0393534	−	0.999225i	$-0.487470\pi$
$241$	1.22186	0.0787068	0.0393534	+	0.999225i	$0.487470\pi$
$242$	0	0
$243$	−8.04988	−0.516400
$244$	0	0
$245$	−13.4026	−0.856259
$246$	0	0
$247$	−1.11029	−0.0706460
$248$	0	0
$249$	18.9997	1.20406
$250$	0	0
$251$	18.0780	1.14107	0.570537	−	0.821272i	$-0.306735\pi$
$251$	18.0780	1.14107	0.570537	+	0.821272i	$0.306735\pi$
$252$	0	0
$253$	−33.7906	−2.12440
$254$	0	0
$255$	−3.35471	−0.210080
$256$	0	0
$257$	19.8608	1.23888	0.619440	−	0.785044i	$-0.287360\pi$
$257$	19.8608	1.23888	0.619440	+	0.785044i	$0.287360\pi$
$258$	0	0
$259$	6.62839	0.411868
$260$	0	0
$261$	0.216832	0.0134215
$262$	0	0
$263$	10.9353	0.674298	0.337149	−	0.941451i	$-0.390537\pi$
$263$	10.9353	0.674298	0.337149	+	0.941451i	$0.390537\pi$
$264$	0	0
$265$	−14.7264	−0.904634
$266$	0	0
$267$	−11.5177	−0.704869
$268$	0	0
$269$	8.22006	0.501186	0.250593	−	0.968093i	$-0.419374\pi$
$269$	8.22006	0.501186	0.250593	+	0.968093i	$0.419374\pi$
$270$	0	0
$271$	5.40811	0.328519	0.164260	−	0.986417i	$-0.447477\pi$
$271$	5.40811	0.328519	0.164260	+	0.986417i	$0.447477\pi$
$272$	0	0
$273$	36.0085	2.17933
$274$	0	0
$275$	8.54465	0.515262
$276$	0	0
$277$	14.8242	0.890700	0.445350	−	0.895357i	$-0.353079\pi$
$277$	14.8242	0.890700	0.445350	+	0.895357i	$0.353079\pi$
$278$	0	0
$279$	−7.24976	−0.434031
$280$	0	0
$281$	−29.0275	−1.73164	−0.865819	−	0.500358i	$-0.833202\pi$
$281$	−29.0275	−1.73164	−0.865819	+	0.500358i	$0.833202\pi$
$282$	0	0
$283$	13.7771	0.818962	0.409481	−	0.912319i	$-0.365710\pi$
$283$	13.7771	0.818962	0.409481	+	0.912319i	$0.365710\pi$
$284$	0	0
$285$	−0.774618	−0.0458844
$286$	0	0
$287$	5.38083	0.317621
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−24.7109	−1.44858
$292$	0	0
$293$	−25.0540	−1.46367	−0.731835	−	0.681482i	$-0.761336\pi$
$293$	−25.0540	−1.46367	−0.731835	+	0.681482i	$0.761336\pi$
$294$	0	0
$295$	1.72233	0.100278
$296$	0	0
$297$	18.0554	1.04768
$298$	0	0
$299$	−38.6694	−2.23631
$300$	0	0
$301$	40.5948	2.33984
$302$	0	0
$303$	−10.8692	−0.624421
$304$	0	0
$305$	−13.1529	−0.753135
$306$	0	0
$307$	−10.9391	−0.624328	−0.312164	−	0.950028i	$-0.601054\pi$
$307$	−10.9391	−0.624328	−0.312164	+	0.950028i	$0.601054\pi$
$308$	0	0
$309$	28.8989	1.64400
$310$	0	0
$311$	−31.6636	−1.79548	−0.897739	−	0.440528i	$-0.854791\pi$
$311$	−31.6636	−1.79548	−0.897739	+	0.440528i	$0.854791\pi$
$312$	0	0
$313$	23.2836	1.31606	0.658032	−	0.752990i	$-0.271389\pi$
$313$	23.2836	1.31606	0.658032	+	0.752990i	$0.271389\pi$
$314$	0	0
$315$	5.25666	0.296179
$316$	0	0
$317$	−4.15341	−0.233279	−0.116639	−	0.993174i	$-0.537212\pi$
$317$	−4.15341	−0.233279	−0.116639	+	0.993174i	$0.537212\pi$
$318$	0	0
$319$	−1.14768	−0.0642579
$320$	0	0
$321$	14.8509	0.828896
$322$	0	0
$323$	0.230905	0.0128479
$324$	0	0
$325$	9.77837	0.542406
$326$	0	0
$327$	4.51468	0.249662
$328$	0	0
$329$	−51.4769	−2.83801
$330$	0	0
$331$	14.1639	0.778519	0.389259	−	0.921128i	$-0.372731\pi$
$331$	14.1639	0.778519	0.389259	+	0.921128i	$0.372731\pi$
$332$	0	0
$333$	−1.36861	−0.0749993
$334$	0	0
$335$	−1.18821	−0.0649186
$336$	0	0
$337$	−26.7664	−1.45806	−0.729030	−	0.684482i	$-0.760028\pi$
$337$	−26.7664	−1.45806	−0.729030	+	0.684482i	$0.760028\pi$
$338$	0	0
$339$	−22.9578	−1.24689
$340$	0	0
$341$	38.3727	2.07800
$342$	0	0
$343$	−3.00527	−0.162269
$344$	0	0
$345$	−26.9786	−1.45248
$346$	0	0
$347$	7.86732	0.422340	0.211170	−	0.977449i	$-0.432273\pi$
$347$	7.86732	0.422340	0.211170	+	0.977449i	$0.432273\pi$
$348$	0	0
$349$	11.4119	0.610865	0.305433	−	0.952214i	$-0.401199\pi$
$349$	11.4119	0.610865	0.305433	+	0.952214i	$0.401199\pi$
$350$	0	0
$351$	20.6624	1.10288
$352$	0	0
$353$	−33.9454	−1.80673	−0.903367	−	0.428869i	$-0.858912\pi$
$353$	−33.9454	−1.80673	−0.903367	+	0.428869i	$0.858912\pi$
$354$	0	0
$355$	−16.6646	−0.884466
$356$	0	0
$357$	−7.48861	−0.396339
$358$	0	0
$359$	−12.6788	−0.669161	−0.334581	−	0.942367i	$-0.608595\pi$
$359$	−12.6788	−0.669161	−0.334581	+	0.942367i	$0.608595\pi$
$360$	0	0
$361$	−18.9467	−0.997194
$362$	0	0
$363$	12.9620	0.680331
$364$	0	0
$365$	−16.9074	−0.884971
$366$	0	0
$367$	20.1408	1.05134	0.525670	−	0.850689i	$-0.323815\pi$
$367$	20.1408	1.05134	0.525670	+	0.850689i	$0.323815\pi$
$368$	0	0
$369$	−1.11102	−0.0578372
$370$	0	0
$371$	−32.8732	−1.70669
$372$	0	0
$373$	26.7139	1.38319	0.691596	−	0.722284i	$-0.256908\pi$
$373$	26.7139	1.38319	0.691596	+	0.722284i	$0.256908\pi$
$374$	0	0
$375$	23.5957	1.21847
$376$	0	0
$377$	−1.31339	−0.0676430
$378$	0	0
$379$	18.0012	0.924659	0.462329	−	0.886708i	$-0.347014\pi$
$379$	18.0012	0.924659	0.462329	+	0.886708i	$0.347014\pi$
$380$	0	0
$381$	6.96567	0.356862
$382$	0	0
$383$	33.6008	1.71692	0.858461	−	0.512878i	$-0.171421\pi$
$383$	33.6008	1.71692	0.858461	+	0.512878i	$0.171421\pi$
$384$	0	0
$385$	−27.8233	−1.41801
$386$	0	0
$387$	−8.38187	−0.426075
$388$	0	0
$389$	19.8089	1.00435	0.502175	−	0.864766i	$-0.332533\pi$
$389$	19.8089	1.00435	0.502175	+	0.864766i	$0.332533\pi$
$390$	0	0
$391$	8.04201	0.406702
$392$	0	0
$393$	9.64306	0.486428
$394$	0	0
$395$	19.3192	0.972054
$396$	0	0
$397$	−0.863145	−0.0433200	−0.0216600	−	0.999765i	$-0.506895\pi$
$397$	−0.863145	−0.0433200	−0.0216600	+	0.999765i	$0.506895\pi$
$398$	0	0
$399$	−1.72915	−0.0865660
$400$	0	0
$401$	−37.6463	−1.87996	−0.939982	−	0.341224i	$-0.889159\pi$
$401$	−37.6463	−1.87996	−0.939982	+	0.341224i	$0.889159\pi$
$402$	0	0
$403$	43.9131	2.18747
$404$	0	0
$405$	18.5173	0.920133
$406$	0	0
$407$	7.24399	0.359071
$408$	0	0
$409$	−0.767399	−0.0379454	−0.0189727	−	0.999820i	$-0.506040\pi$
$409$	−0.767399	−0.0379454	−0.0189727	+	0.999820i	$0.506040\pi$
$410$	0	0
$411$	−38.0991	−1.87929
$412$	0	0
$413$	3.84469	0.189185
$414$	0	0
$415$	−16.8005	−0.824706
$416$	0	0
$417$	−11.7749	−0.576620
$418$	0	0
$419$	−2.59677	−0.126861	−0.0634304	−	0.997986i	$-0.520204\pi$
$419$	−2.59677	−0.126861	−0.0634304	+	0.997986i	$0.520204\pi$
$420$	0	0
$421$	5.80494	0.282916	0.141458	−	0.989944i	$-0.454821\pi$
$421$	5.80494	0.282916	0.141458	+	0.989944i	$0.454821\pi$
$422$	0	0
$423$	10.6288	0.516788
$424$	0	0
$425$	−2.03359	−0.0986435
$426$	0	0
$427$	−29.3609	−1.42087
$428$	0	0
$429$	39.3527	1.89996
$430$	0	0
$431$	12.5763	0.605779	0.302890	−	0.953026i	$-0.402049\pi$
$431$	12.5763	0.605779	0.302890	+	0.953026i	$0.402049\pi$
$432$	0	0
$433$	−35.7923	−1.72007	−0.860034	−	0.510237i	$-0.829558\pi$
$433$	−35.7923	−1.72007	−0.860034	+	0.510237i	$0.829558\pi$
$434$	0	0
$435$	−0.916317	−0.0439340
$436$	0	0
$437$	1.85694	0.0888293
$438$	0	0
$439$	−18.4783	−0.881923	−0.440962	−	0.897526i	$-0.645362\pi$
$439$	−18.4783	−0.881923	−0.440962	+	0.897526i	$0.645362\pi$
$440$	0	0
$441$	6.17739	0.294161
$442$	0	0
$443$	−21.1522	−1.00497	−0.502485	−	0.864586i	$-0.667581\pi$
$443$	−21.1522	−1.00497	−0.502485	+	0.864586i	$0.667581\pi$
$444$	0	0
$445$	10.1845	0.482792
$446$	0	0
$447$	43.1727	2.04200
$448$	0	0
$449$	22.2391	1.04953	0.524764	−	0.851248i	$-0.324154\pi$
$449$	22.2391	1.04953	0.524764	+	0.851248i	$0.324154\pi$
$450$	0	0
$451$	5.88056	0.276905
$452$	0	0
$453$	−12.9304	−0.607522
$454$	0	0
$455$	−31.8406	−1.49271
$456$	0	0
$457$	15.0158	0.702410	0.351205	−	0.936299i	$-0.385772\pi$
$457$	15.0158	0.702410	0.351205	+	0.936299i	$0.385772\pi$
$458$	0	0
$459$	−4.29711	−0.200572
$460$	0	0
$461$	−20.0940	−0.935871	−0.467936	−	0.883762i	$-0.655002\pi$
$461$	−20.0940	−0.935871	−0.467936	+	0.883762i	$0.655002\pi$
$462$	0	0
$463$	20.7201	0.962945	0.481473	−	0.876461i	$-0.340102\pi$
$463$	20.7201	0.962945	0.481473	+	0.876461i	$0.340102\pi$
$464$	0	0
$465$	30.6370	1.42076
$466$	0	0
$467$	−7.92087	−0.366534	−0.183267	−	0.983063i	$-0.558667\pi$
$467$	−7.92087	−0.366534	−0.183267	+	0.983063i	$0.558667\pi$
$468$	0	0
$469$	−2.65239	−0.122476
$470$	0	0
$471$	−43.9733	−2.02618
$472$	0	0
$473$	44.3649	2.03990
$474$	0	0
$475$	−0.469565	−0.0215451
$476$	0	0
$477$	6.78754	0.310780
$478$	0	0
$479$	−10.7533	−0.491331	−0.245666	−	0.969355i	$-0.579007\pi$
$479$	−10.7533	−0.491331	−0.245666	+	0.969355i	$0.579007\pi$
$480$	0	0
$481$	8.28991	0.377988
$482$	0	0
$483$	−60.2234	−2.74026
$484$	0	0
$485$	21.8507	0.992188
$486$	0	0
$487$	−25.8517	−1.17145	−0.585725	−	0.810510i	$-0.699190\pi$
$487$	−25.8517	−1.17145	−0.585725	+	0.810510i	$0.699190\pi$
$488$	0	0
$489$	18.9310	0.856089
$490$	0	0
$491$	37.8186	1.70673	0.853365	−	0.521314i	$-0.174558\pi$
$491$	37.8186	1.70673	0.853365	+	0.521314i	$0.174558\pi$
$492$	0	0
$493$	0.273143	0.0123018
$494$	0	0
$495$	5.74486	0.258212
$496$	0	0
$497$	−37.1999	−1.66864
$498$	0	0
$499$	33.7933	1.51279	0.756397	−	0.654113i	$-0.226958\pi$
$499$	33.7933	1.51279	0.756397	+	0.654113i	$0.226958\pi$
$500$	0	0
$501$	−1.75696	−0.0784952
$502$	0	0
$503$	−4.30287	−0.191855	−0.0959277	−	0.995388i	$-0.530582\pi$
$503$	−4.30287	−0.191855	−0.0959277	+	0.995388i	$0.530582\pi$
$504$	0	0
$505$	9.61114	0.427690
$506$	0	0
$507$	19.7135	0.875507
$508$	0	0
$509$	−30.0018	−1.32981	−0.664903	−	0.746930i	$-0.731527\pi$
$509$	−30.0018	−1.32981	−0.664903	+	0.746930i	$0.731527\pi$
$510$	0	0
$511$	−37.7417	−1.66959
$512$	0	0
$513$	−0.992223	−0.0438077
$514$	0	0
$515$	−25.5539	−1.12604
$516$	0	0
$517$	−56.2577	−2.47421
$518$	0	0
$519$	43.1312	1.89325
$520$	0	0
$521$	26.5619	1.16370	0.581849	−	0.813297i	$-0.302330\pi$
$521$	26.5619	1.16370	0.581849	+	0.813297i	$0.302330\pi$
$522$	0	0
$523$	30.6954	1.34222	0.671108	−	0.741359i	$-0.265819\pi$
$523$	30.6954	1.34222	0.671108	+	0.741359i	$0.265819\pi$
$524$	0	0
$525$	15.2287	0.664637
$526$	0	0
$527$	−9.13252	−0.397819
$528$	0	0
$529$	41.6738	1.81191
$530$	0	0
$531$	−0.793839	−0.0344497
$532$	0	0
$533$	6.72963	0.291493
$534$	0	0
$535$	−13.1319	−0.567743
$536$	0	0
$537$	−2.38154	−0.102771
$538$	0	0
$539$	−32.6967	−1.40835
$540$	0	0
$541$	−0.354048	−0.0152217	−0.00761085	−	0.999971i	$-0.502423\pi$
$541$	−0.354048	−0.0152217	−0.00761085	+	0.999971i	$0.502423\pi$
$542$	0	0
$543$	9.82515	0.421638
$544$	0	0
$545$	−3.99212	−0.171004
$546$	0	0
$547$	5.58694	0.238880	0.119440	−	0.992841i	$-0.461890\pi$
$547$	5.58694	0.238880	0.119440	+	0.992841i	$0.461890\pi$
$548$	0	0
$549$	6.06233	0.258734
$550$	0	0
$551$	0.0630700	0.00268687
$552$	0	0
$553$	43.1256	1.83389
$554$	0	0
$555$	5.78365	0.245502
$556$	0	0
$557$	−4.77708	−0.202411	−0.101206	−	0.994866i	$-0.532270\pi$
$557$	−4.77708	−0.202411	−0.101206	+	0.994866i	$0.532270\pi$
$558$	0	0
$559$	50.7706	2.14737
$560$	0	0
$561$	−8.18410	−0.345533
$562$	0	0
$563$	26.5112	1.11731	0.558656	−	0.829399i	$-0.311317\pi$
$563$	26.5112	1.11731	0.558656	+	0.829399i	$0.311317\pi$
$564$	0	0
$565$	20.3005	0.854047
$566$	0	0
$567$	41.3356	1.73593
$568$	0	0
$569$	32.6793	1.36999	0.684994	−	0.728548i	$-0.259805\pi$
$569$	32.6793	1.36999	0.684994	+	0.728548i	$0.259805\pi$
$570$	0	0
$571$	−28.6278	−1.19804	−0.599018	−	0.800736i	$-0.704442\pi$
$571$	−28.6278	−1.19804	−0.599018	+	0.800736i	$0.704442\pi$
$572$	0	0
$573$	−4.38378	−0.183135
$574$	0	0
$575$	−16.3541	−0.682014
$576$	0	0
$577$	−46.6849	−1.94352	−0.971759	−	0.235975i	$-0.924172\pi$
$577$	−46.6849	−1.94352	−0.971759	+	0.235975i	$0.924172\pi$
$578$	0	0
$579$	15.5919	0.647979
$580$	0	0
$581$	−37.5033	−1.55590
$582$	0	0
$583$	−35.9262	−1.48791
$584$	0	0
$585$	6.57433	0.271815
$586$	0	0
$587$	19.3834	0.800039	0.400019	−	0.916507i	$-0.369003\pi$
$587$	19.3834	0.800039	0.400019	+	0.916507i	$0.369003\pi$
$588$	0	0
$589$	−2.10874	−0.0868892
$590$	0	0
$591$	38.6560	1.59009
$592$	0	0
$593$	−0.857596	−0.0352172	−0.0176086	−	0.999845i	$-0.505605\pi$
$593$	−0.857596	−0.0352172	−0.0176086	+	0.999845i	$0.505605\pi$
$594$	0	0
$595$	6.62182	0.271468
$596$	0	0
$597$	−9.36292	−0.383199
$598$	0	0
$599$	9.57231	0.391114	0.195557	−	0.980692i	$-0.437349\pi$
$599$	9.57231	0.391114	0.195557	+	0.980692i	$0.437349\pi$
$600$	0	0
$601$	−20.9791	−0.855755	−0.427877	−	0.903837i	$-0.640738\pi$
$601$	−20.9791	−0.855755	−0.427877	+	0.903837i	$0.640738\pi$
$602$	0	0
$603$	0.547657	0.0223023
$604$	0	0
$605$	−11.4617	−0.465985
$606$	0	0
$607$	0.158485	0.00643272	0.00321636	−	0.999995i	$-0.498976\pi$
$607$	0.158485	0.00643272	0.00321636	+	0.999995i	$0.498976\pi$
$608$	0	0
$609$	−2.04546	−0.0828863
$610$	0	0
$611$	−64.3804	−2.60455
$612$	0	0
$613$	−17.4496	−0.704781	−0.352391	−	0.935853i	$-0.614631\pi$
$613$	−17.4496	−0.704781	−0.352391	+	0.935853i	$0.614631\pi$
$614$	0	0
$615$	4.69508	0.189324
$616$	0	0
$617$	−40.6203	−1.63531	−0.817656	−	0.575707i	$-0.804727\pi$
$617$	−40.6203	−1.63531	−0.817656	+	0.575707i	$0.804727\pi$
$618$	0	0
$619$	−6.79244	−0.273011	−0.136506	−	0.990639i	$-0.543587\pi$
$619$	−6.79244	−0.273011	−0.136506	+	0.990639i	$0.543587\pi$
$620$	0	0
$621$	−34.5574	−1.38674
$622$	0	0
$623$	22.7345	0.910840
$624$	0	0
$625$	−10.6966	−0.427863
$626$	0	0
$627$	−1.88975	−0.0754692
$628$	0	0
$629$	−1.72404	−0.0687419
$630$	0	0
$631$	10.6522	0.424057	0.212028	−	0.977263i	$-0.431993\pi$
$631$	10.6522	0.424057	0.212028	+	0.977263i	$0.431993\pi$
$632$	0	0
$633$	−5.67914	−0.225725
$634$	0	0
$635$	−6.15941	−0.244429
$636$	0	0
$637$	−37.4176	−1.48254
$638$	0	0
$639$	7.68090	0.303852
$640$	0	0
$641$	2.71658	0.107298	0.0536492	−	0.998560i	$-0.482915\pi$
$641$	2.71658	0.107298	0.0536492	+	0.998560i	$0.482915\pi$
$642$	0	0
$643$	3.36385	0.132657	0.0663286	−	0.997798i	$-0.478871\pi$
$643$	3.36385	0.132657	0.0663286	+	0.997798i	$0.478871\pi$
$644$	0	0
$645$	35.4212	1.39471
$646$	0	0
$647$	23.9619	0.942040	0.471020	−	0.882123i	$-0.343886\pi$
$647$	23.9619	0.942040	0.471020	+	0.882123i	$0.343886\pi$
$648$	0	0
$649$	4.20176	0.164934
$650$	0	0
$651$	68.3899	2.68041
$652$	0	0
$653$	−29.3850	−1.14992	−0.574962	−	0.818180i	$-0.694983\pi$
$653$	−29.3850	−1.14992	−0.574962	+	0.818180i	$0.694983\pi$
$654$	0	0
$655$	−8.52690	−0.333174
$656$	0	0
$657$	7.79278	0.304025
$658$	0	0
$659$	−24.1096	−0.939178	−0.469589	−	0.882885i	$-0.655598\pi$
$659$	−24.1096	−0.939178	−0.469589	+	0.882885i	$0.655598\pi$
$660$	0	0
$661$	−43.9476	−1.70936	−0.854681	−	0.519154i	$-0.826247\pi$
$661$	−43.9476	−1.70936	−0.854681	+	0.519154i	$0.826247\pi$
$662$	0	0
$663$	−9.36576	−0.363736
$664$	0	0
$665$	1.52901	0.0592924
$666$	0	0
$667$	2.19662	0.0850534
$668$	0	0
$669$	−33.4553	−1.29346
$670$	0	0
$671$	−32.0877	−1.23873
$672$	0	0
$673$	25.8499	0.996439	0.498219	−	0.867051i	$-0.333987\pi$
$673$	25.8499	0.996439	0.498219	+	0.867051i	$0.333987\pi$
$674$	0	0
$675$	8.73856	0.336347
$676$	0	0
$677$	−21.6681	−0.832771	−0.416386	−	0.909188i	$-0.636703\pi$
$677$	−21.6681	−0.832771	−0.416386	+	0.909188i	$0.636703\pi$
$678$	0	0
$679$	48.7765	1.87187
$680$	0	0
$681$	−37.8658	−1.45102
$682$	0	0
$683$	−39.0708	−1.49500	−0.747501	−	0.664260i	$-0.768747\pi$
$683$	−39.0708	−1.49500	−0.747501	+	0.664260i	$0.768747\pi$
$684$	0	0
$685$	33.6892	1.28720
$686$	0	0
$687$	−4.78368	−0.182509
$688$	0	0
$689$	−41.1134	−1.56630
$690$	0	0
$691$	21.9548	0.835201	0.417600	−	0.908631i	$-0.362871\pi$
$691$	21.9548	0.835201	0.417600	+	0.908631i	$0.362871\pi$
$692$	0	0
$693$	12.8241	0.487146
$694$	0	0
$695$	10.4120	0.394950
$696$	0	0
$697$	−1.39955	−0.0530117
$698$	0	0
$699$	54.8157	2.07332
$700$	0	0
$701$	−20.4671	−0.773033	−0.386517	−	0.922282i	$-0.626322\pi$
$701$	−20.4671	−0.773033	−0.386517	+	0.922282i	$0.626322\pi$
$702$	0	0
$703$	−0.398088	−0.0150142
$704$	0	0
$705$	−44.9165	−1.69165
$706$	0	0
$707$	21.4546	0.806884
$708$	0	0
$709$	11.4439	0.429783	0.214892	−	0.976638i	$-0.431060\pi$
$709$	11.4439	0.429783	0.214892	+	0.976638i	$0.431060\pi$
$710$	0	0
$711$	−8.90443	−0.333942
$712$	0	0
$713$	−73.4438	−2.75049
$714$	0	0
$715$	−34.7977	−1.30136
$716$	0	0
$717$	−21.3686	−0.798024
$718$	0	0
$719$	37.1036	1.38373	0.691865	−	0.722027i	$-0.256789\pi$
$719$	37.1036	1.38373	0.691865	+	0.722027i	$0.256789\pi$
$720$	0	0
$721$	−57.0431	−2.12440
$722$	0	0
$723$	2.37991	0.0885097
$724$	0	0
$725$	−0.555461	−0.0206293
$726$	0	0
$727$	16.6857	0.618838	0.309419	−	0.950926i	$-0.399865\pi$
$727$	16.6857	0.618838	0.309419	+	0.950926i	$0.399865\pi$
$728$	0	0
$729$	16.5746	0.613875
$730$	0	0
$731$	−10.5587	−0.390526
$732$	0	0
$733$	46.6860	1.72439	0.862194	−	0.506579i	$-0.169090\pi$
$733$	46.6860	1.72439	0.862194	+	0.506579i	$0.169090\pi$
$734$	0	0
$735$	−26.1052	−0.962906
$736$	0	0
$737$	−2.89873	−0.106776
$738$	0	0
$739$	6.02167	0.221511	0.110755	−	0.993848i	$-0.464673\pi$
$739$	6.02167	0.221511	0.110755	+	0.993848i	$0.464673\pi$
$740$	0	0
$741$	−2.16260	−0.0794450
$742$	0	0
$743$	34.5967	1.26923	0.634615	−	0.772829i	$-0.281159\pi$
$743$	34.5967	1.26923	0.634615	+	0.772829i	$0.281159\pi$
$744$	0	0
$745$	−38.1756	−1.39865
$746$	0	0
$747$	7.74355	0.283322
$748$	0	0
$749$	−29.3140	−1.07111
$750$	0	0
$751$	−23.7994	−0.868452	−0.434226	−	0.900804i	$-0.642978\pi$
$751$	−23.7994	−0.868452	−0.434226	+	0.900804i	$0.642978\pi$
$752$	0	0
$753$	35.2120	1.28320
$754$	0	0
$755$	11.4337	0.416116
$756$	0	0
$757$	17.2778	0.627971	0.313986	−	0.949428i	$-0.398336\pi$
$757$	17.2778	0.627971	0.313986	+	0.949428i	$0.398336\pi$
$758$	0	0
$759$	−65.8165	−2.38899
$760$	0	0
$761$	7.64784	0.277234	0.138617	−	0.990346i	$-0.455734\pi$
$761$	7.64784	0.277234	0.138617	+	0.990346i	$0.455734\pi$
$762$	0	0
$763$	−8.91147	−0.322617
$764$	0	0
$765$	−1.36725	−0.0494331
$766$	0	0
$767$	4.80843	0.173622
$768$	0	0
$769$	−11.2928	−0.407230	−0.203615	−	0.979051i	$-0.565269\pi$
$769$	−11.2928	−0.407230	−0.203615	+	0.979051i	$0.565269\pi$
$770$	0	0
$771$	38.6843	1.39318
$772$	0	0
$773$	−5.61449	−0.201939	−0.100970	−	0.994890i	$-0.532194\pi$
$773$	−5.61449	−0.201939	−0.100970	+	0.994890i	$0.532194\pi$
$774$	0	0
$775$	18.5718	0.667119
$776$	0	0
$777$	12.9106	0.463167
$778$	0	0
$779$	−0.323162	−0.0115785
$780$	0	0
$781$	−40.6547	−1.45474
$782$	0	0
$783$	−1.17373	−0.0419456
$784$	0	0
$785$	38.8835	1.38781
$786$	0	0
$787$	38.1407	1.35957	0.679784	−	0.733412i	$-0.262074\pi$
$787$	38.1407	1.35957	0.679784	+	0.733412i	$0.262074\pi$
$788$	0	0
$789$	21.2995	0.758282
$790$	0	0
$791$	45.3160	1.61125
$792$	0	0
$793$	−36.7207	−1.30399
$794$	0	0
$795$	−28.6837	−1.01731
$796$	0	0
$797$	4.48244	0.158776	0.0793882	−	0.996844i	$-0.474703\pi$
$797$	4.48244	0.158776	0.0793882	+	0.996844i	$0.474703\pi$
$798$	0	0
$799$	13.3891	0.473671
$800$	0	0
$801$	−4.69415	−0.165860
$802$	0	0
$803$	−41.2469	−1.45557
$804$	0	0
$805$	53.2527	1.87691
$806$	0	0
$807$	16.0109	0.563609
$808$	0	0
$809$	4.34580	0.152790	0.0763951	−	0.997078i	$-0.475659\pi$
$809$	4.34580	0.152790	0.0763951	+	0.997078i	$0.475659\pi$
$810$	0	0
$811$	44.6564	1.56810	0.784050	−	0.620698i	$-0.213151\pi$
$811$	44.6564	1.56810	0.784050	+	0.620698i	$0.213151\pi$
$812$	0	0
$813$	10.5338	0.369436
$814$	0	0
$815$	−16.7398	−0.586369
$816$	0	0
$817$	−2.43804	−0.0852963
$818$	0	0
$819$	14.6757	0.512809
$820$	0	0
$821$	3.32421	0.116016	0.0580078	−	0.998316i	$-0.481525\pi$
$821$	3.32421	0.116016	0.0580078	+	0.998316i	$0.481525\pi$
$822$	0	0
$823$	10.5842	0.368942	0.184471	−	0.982838i	$-0.440943\pi$
$823$	10.5842	0.368942	0.184471	+	0.982838i	$0.440943\pi$
$824$	0	0
$825$	16.6431	0.579438
$826$	0	0
$827$	18.8814	0.656571	0.328285	−	0.944579i	$-0.393529\pi$
$827$	18.8814	0.656571	0.328285	+	0.944579i	$0.393529\pi$
$828$	0	0
$829$	−17.6637	−0.613487	−0.306744	−	0.951792i	$-0.599239\pi$
$829$	−17.6637	−0.613487	−0.306744	+	0.951792i	$0.599239\pi$
$830$	0	0
$831$	28.8743	1.00164
$832$	0	0
$833$	7.78167	0.269619
$834$	0	0
$835$	1.55360	0.0537644
$836$	0	0
$837$	39.2435	1.35645
$838$	0	0
$839$	−19.3585	−0.668331	−0.334165	−	0.942514i	$-0.608454\pi$
$839$	−19.3585	−0.668331	−0.334165	+	0.942514i	$0.608454\pi$
$840$	0	0
$841$	−28.9254	−0.997427
$842$	0	0
$843$	−56.5392	−1.94731
$844$	0	0
$845$	−17.4317	−0.599669
$846$	0	0
$847$	−25.5856	−0.879132
$848$	0	0
$849$	26.8347	0.920964
$850$	0	0
$851$	−13.8647	−0.475276
$852$	0	0
$853$	−26.7832	−0.917038	−0.458519	−	0.888685i	$-0.651620\pi$
$853$	−26.7832	−0.917038	−0.458519	+	0.888685i	$0.651620\pi$
$854$	0	0
$855$	−0.315704	−0.0107969
$856$	0	0
$857$	56.8554	1.94214	0.971071	−	0.238790i	$-0.0767507\pi$
$857$	56.8554	1.94214	0.971071	+	0.238790i	$0.0767507\pi$
$858$	0	0
$859$	12.6948	0.433141	0.216571	−	0.976267i	$-0.430513\pi$
$859$	12.6948	0.433141	0.216571	+	0.976267i	$0.430513\pi$
$860$	0	0
$861$	10.4807	0.357180
$862$	0	0
$863$	−17.2055	−0.585683	−0.292841	−	0.956161i	$-0.594601\pi$
$863$	−17.2055	−0.585683	−0.292841	+	0.956161i	$0.594601\pi$
$864$	0	0
$865$	−38.1388	−1.29676
$866$	0	0
$867$	1.94778	0.0661500
$868$	0	0
$869$	47.1308	1.59880
$870$	0	0
$871$	−3.31726	−0.112401
$872$	0	0
$873$	−10.0712	−0.340859
$874$	0	0
$875$	−46.5752	−1.57453
$876$	0	0
$877$	−40.1550	−1.35594	−0.677969	−	0.735090i	$-0.737140\pi$
$877$	−40.1550	−1.35594	−0.677969	+	0.735090i	$0.737140\pi$
$878$	0	0
$879$	−48.7996	−1.64597
$880$	0	0
$881$	1.82117	0.0613567	0.0306784	−	0.999529i	$-0.490233\pi$
$881$	1.82117	0.0613567	0.0306784	+	0.999529i	$0.490233\pi$
$882$	0	0
$883$	−43.6459	−1.46880	−0.734401	−	0.678716i	$-0.762537\pi$
$883$	−43.6459	−1.46880	−0.734401	+	0.678716i	$0.762537\pi$
$884$	0	0
$885$	3.35471	0.112767
$886$	0	0
$887$	2.81697	0.0945847	0.0472923	−	0.998881i	$-0.484941\pi$
$887$	2.81697	0.0945847	0.0472923	+	0.998881i	$0.484941\pi$
$888$	0	0
$889$	−13.7494	−0.461142
$890$	0	0
$891$	45.1745	1.51340
$892$	0	0
$893$	3.09160	0.103456
$894$	0	0
$895$	2.10588	0.0703919
$896$	0	0
$897$	−75.3195	−2.51484
$898$	0	0
$899$	−2.49449	−0.0831958
$900$	0	0
$901$	8.55027	0.284851
$902$	0	0
$903$	79.0696	2.63127
$904$	0	0
$905$	−8.68791	−0.288796
$906$	0	0
$907$	−32.9572	−1.09433	−0.547163	−	0.837026i	$-0.684292\pi$
$907$	−32.9572	−1.09433	−0.547163	+	0.837026i	$0.684292\pi$
$908$	0	0
$909$	−4.42988	−0.146930
$910$	0	0
$911$	−38.3691	−1.27122	−0.635612	−	0.772009i	$-0.719252\pi$
$911$	−38.3691	−1.27122	−0.635612	+	0.772009i	$0.719252\pi$
$912$	0	0
$913$	−40.9863	−1.35645
$914$	0	0
$915$	−25.6190	−0.846938
$916$	0	0
$917$	−19.0343	−0.628568
$918$	0	0
$919$	−2.22802	−0.0734957	−0.0367479	−	0.999325i	$-0.511700\pi$
$919$	−2.22802	−0.0734957	−0.0367479	+	0.999325i	$0.511700\pi$
$920$	0	0
$921$	−21.3070	−0.702088
$922$	0	0
$923$	−46.5247	−1.53138
$924$	0	0
$925$	3.50598	0.115276
$926$	0	0
$927$	11.7781	0.386843
$928$	0	0
$929$	−22.0731	−0.724196	−0.362098	−	0.932140i	$-0.617939\pi$
$929$	−22.0731	−0.724196	−0.362098	+	0.932140i	$0.617939\pi$
$930$	0	0
$931$	1.79682	0.0588885
$932$	0	0
$933$	−61.6737	−2.01910
$934$	0	0
$935$	7.23681	0.236669
$936$	0	0
$937$	33.4820	1.09381	0.546905	−	0.837194i	$-0.315806\pi$
$937$	33.4820	1.09381	0.546905	+	0.837194i	$0.315806\pi$
$938$	0	0
$939$	45.3512	1.47998
$940$	0	0
$941$	2.64760	0.0863091	0.0431546	−	0.999068i	$-0.486259\pi$
$941$	2.64760	0.0863091	0.0431546	+	0.999068i	$0.486259\pi$
$942$	0	0
$943$	−11.2552	−0.366519
$944$	0	0
$945$	−28.4547	−0.925632
$946$	0	0
$947$	37.9913	1.23455	0.617276	−	0.786747i	$-0.288236\pi$
$947$	37.9913	1.23455	0.617276	+	0.786747i	$0.288236\pi$
$948$	0	0
$949$	−47.2023	−1.53225
$950$	0	0
$951$	−8.08991	−0.262333
$952$	0	0
$953$	34.3376	1.11231	0.556153	−	0.831080i	$-0.312277\pi$
$953$	34.3376	1.11231	0.556153	+	0.831080i	$0.312277\pi$
$954$	0	0
$955$	3.87637	0.125436
$956$	0	0
$957$	−2.23543	−0.0722612
$958$	0	0
$959$	75.2033	2.42844
$960$	0	0
$961$	52.4030	1.69042
$962$	0	0
$963$	6.05265	0.195044
$964$	0	0
$965$	−13.7872	−0.443826
$966$	0	0
$967$	34.0962	1.09646	0.548230	−	0.836328i	$-0.315302\pi$
$967$	34.0962	1.09646	0.548230	+	0.836328i	$0.315302\pi$
$968$	0	0
$969$	0.449751	0.0144481
$970$	0	0
$971$	−8.48874	−0.272417	−0.136208	−	0.990680i	$-0.543492\pi$
$971$	−8.48874	−0.272417	−0.136208	+	0.990680i	$0.543492\pi$
$972$	0	0
$973$	23.2423	0.745116
$974$	0	0
$975$	19.0461	0.609963
$976$	0	0
$977$	1.54378	0.0493899	0.0246950	−	0.999695i	$-0.492139\pi$
$977$	1.54378	0.0493899	0.0246950	+	0.999695i	$0.492139\pi$
$978$	0	0
$979$	24.8460	0.794080
$980$	0	0
$981$	1.84001	0.0587470
$982$	0	0
$983$	50.3909	1.60722	0.803611	−	0.595155i	$-0.202909\pi$
$983$	50.3909	1.60722	0.803611	+	0.595155i	$0.202909\pi$
$984$	0	0
$985$	−34.1816	−1.08912
$986$	0	0
$987$	−100.266	−3.19149
$988$	0	0
$989$	−84.9128	−2.70007
$990$	0	0
$991$	−45.5789	−1.44786	−0.723931	−	0.689872i	$-0.757667\pi$
$991$	−45.5789	−1.44786	−0.723931	+	0.689872i	$0.757667\pi$
$992$	0	0
$993$	27.5881	0.875483
$994$	0	0
$995$	8.27919	0.262468
$996$	0	0
$997$	55.1687	1.74721	0.873605	−	0.486636i	$-0.161776\pi$
$997$	55.1687	1.74721	0.873605	+	0.486636i	$0.161776\pi$
$998$	0	0
$999$	7.40838	0.234391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.v.1.12	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.v.1.12	✓	18	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)

\(f(q)\)	\(=\)	\(q+1.94778 q^{3} -1.72233 q^{5} -3.84469 q^{7} +0.793839 q^{9} +O(q^{10})\)	\(q+1.94778 q^{3} -1.72233 q^{5} -3.84469 q^{7} +0.793839 q^{9} -4.20176 q^{11} -4.80843 q^{13} -3.35471 q^{15} +1.00000 q^{17} +0.230905 q^{19} -7.48861 q^{21} +8.04201 q^{23} -2.03359 q^{25} -4.29711 q^{27} +0.273143 q^{29} -9.13252 q^{31} -8.18410 q^{33} +6.62182 q^{35} -1.72404 q^{37} -9.36576 q^{39} -1.39955 q^{41} -10.5587 q^{43} -1.36725 q^{45} +13.3891 q^{47} +7.78167 q^{49} +1.94778 q^{51} +8.55027 q^{53} +7.23681 q^{55} +0.449751 q^{57} -1.00000 q^{59} +7.63672 q^{61} -3.05207 q^{63} +8.28169 q^{65} +0.689884 q^{67} +15.6640 q^{69} +9.67564 q^{71} +9.81657 q^{73} -3.96098 q^{75} +16.1545 q^{77} -11.2169 q^{79} -10.7513 q^{81} +9.75456 q^{83} -1.72233 q^{85} +0.532022 q^{87} -5.91323 q^{89} +18.4869 q^{91} -17.7881 q^{93} -0.397693 q^{95} -12.6867 q^{97} -3.33552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) \) 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9	\( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) \) 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9 - q^11 + 15 * q^13 + 3 * q^15 + 18 * q^17 + 26 * q^19 - 4 * q^21 + 22 * q^23 + 42 * q^25 + 45 * q^27 + 6 * q^29 + 13 * q^31 - 5 * q^33 + 4 * q^35 + 4 * q^37 + 36 * q^39 - 15 * q^41 + 12 * q^43 + 14 * q^45 + 8 * q^47 - 13 * q^49 + 9 * q^51 - 11 * q^53 + 55 * q^55 - 20 * q^57 - 18 * q^59 + 53 * q^61 + 29 * q^63 - 26 * q^65 + 2 * q^67 + 32 * q^69 + 8 * q^71 - 42 * q^73 + 72 * q^75 + 6 * q^77 - 9 * q^79 + 42 * q^81 - 4 * q^83 + 2 * q^85 + 36 * q^87 + 13 * q^89 + 68 * q^91 + q^93 + 3 * q^95 - 56 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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