LMFDB - Embedded newform 8024.2.a.z.1.19

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:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+1.93554 q^{3} +0.236703 q^{5} +1.77739 q^{7} +0.746320 q^{9} +O(q^{10})$	$q+1.93554 q^{3} +0.236703 q^{5} +1.77739 q^{7} +0.746320 q^{9} -1.42951 q^{11} -1.08346 q^{13} +0.458149 q^{15} +1.00000 q^{17} -8.54684 q^{19} +3.44022 q^{21} +4.01824 q^{23} -4.94397 q^{25} -4.36209 q^{27} -3.82086 q^{29} +4.61799 q^{31} -2.76687 q^{33} +0.420714 q^{35} +4.58788 q^{37} -2.09709 q^{39} -10.4605 q^{41} -6.57363 q^{43} +0.176656 q^{45} +0.108052 q^{47} -3.84087 q^{49} +1.93554 q^{51} -2.98723 q^{53} -0.338369 q^{55} -16.5428 q^{57} +1.00000 q^{59} +9.13557 q^{61} +1.32650 q^{63} -0.256459 q^{65} +1.87199 q^{67} +7.77746 q^{69} -11.4708 q^{71} -0.474887 q^{73} -9.56926 q^{75} -2.54079 q^{77} -10.0185 q^{79} -10.6820 q^{81} +0.303718 q^{83} +0.236703 q^{85} -7.39543 q^{87} +10.2766 q^{89} -1.92574 q^{91} +8.93832 q^{93} -2.02306 q^{95} -3.83061 q^{97} -1.06687 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	1.93554	1.11749	0.558743	−	0.829341i	$-0.311284\pi$
$3$	1.93554	1.11749	0.558743	+	0.829341i	$0.311284\pi$
$4$	0	0
$5$	0.236703	0.105857	0.0529284	−	0.998598i	$-0.483144\pi$
$5$	0.236703	0.105857	0.0529284	+	0.998598i	$0.483144\pi$
$6$	0	0
$7$	1.77739	0.671792	0.335896	−	0.941899i	$-0.390961\pi$
$7$	1.77739	0.671792	0.335896	+	0.941899i	$0.390961\pi$
$8$	0	0
$9$	0.746320	0.248773
$10$	0	0
$11$	−1.42951	−0.431012	−0.215506	−	0.976502i	$-0.569140\pi$
$11$	−1.42951	−0.431012	−0.215506	+	0.976502i	$0.569140\pi$
$12$	0	0
$13$	−1.08346	−0.300499	−0.150249	−	0.988648i	$-0.548008\pi$
$13$	−1.08346	−0.300499	−0.150249	+	0.988648i	$0.548008\pi$
$14$	0	0
$15$	0.458149	0.118293
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−8.54684	−1.96078	−0.980390	−	0.197066i	$-0.936859\pi$
$19$	−8.54684	−1.96078	−0.980390	+	0.197066i	$0.936859\pi$
$20$	0	0
$21$	3.44022	0.750717
$22$	0	0
$23$	4.01824	0.837860	0.418930	−	0.908018i	$-0.362405\pi$
$23$	4.01824	0.837860	0.418930	+	0.908018i	$0.362405\pi$
$24$	0	0
$25$	−4.94397	−0.988794
$26$	0	0
$27$	−4.36209	−0.839485
$28$	0	0
$29$	−3.82086	−0.709516	−0.354758	−	0.934958i	$-0.615437\pi$
$29$	−3.82086	−0.709516	−0.354758	+	0.934958i	$0.615437\pi$
$30$	0	0
$31$	4.61799	0.829416	0.414708	−	0.909955i	$-0.363884\pi$
$31$	4.61799	0.829416	0.414708	+	0.909955i	$0.363884\pi$
$32$	0	0
$33$	−2.76687	−0.481650
$34$	0	0
$35$	0.420714	0.0711137
$36$	0	0
$37$	4.58788	0.754242	0.377121	−	0.926164i	$-0.376914\pi$
$37$	4.58788	0.754242	0.377121	+	0.926164i	$0.376914\pi$
$38$	0	0
$39$	−2.09709	−0.335803
$40$	0	0
$41$	−10.4605	−1.63366	−0.816830	−	0.576879i	$-0.804270\pi$
$41$	−10.4605	−1.63366	−0.816830	+	0.576879i	$0.804270\pi$
$42$	0	0
$43$	−6.57363	−1.00247	−0.501235	−	0.865311i	$-0.667121\pi$
$43$	−6.57363	−1.00247	−0.501235	+	0.865311i	$0.667121\pi$
$44$	0	0
$45$	0.176656	0.0263344
$46$	0	0
$47$	0.108052	0.0157610	0.00788048	−	0.999969i	$-0.497492\pi$
$47$	0.108052	0.0157610	0.00788048	+	0.999969i	$0.497492\pi$
$48$	0	0
$49$	−3.84087	−0.548696
$50$	0	0
$51$	1.93554	0.271030
$52$	0	0
$53$	−2.98723	−0.410327	−0.205164	−	0.978728i	$-0.565773\pi$
$53$	−2.98723	−0.410327	−0.205164	+	0.978728i	$0.565773\pi$
$54$	0	0
$55$	−0.338369	−0.0456256
$56$	0	0
$57$	−16.5428	−2.19114
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	9.13557	1.16969	0.584845	−	0.811145i	$-0.301155\pi$
$61$	9.13557	1.16969	0.584845	+	0.811145i	$0.301155\pi$
$62$	0	0
$63$	1.32650	0.167124
$64$	0	0
$65$	−0.256459	−0.0318099
$66$	0	0
$67$	1.87199	0.228700	0.114350	−	0.993441i	$-0.463521\pi$
$67$	1.87199	0.228700	0.114350	+	0.993441i	$0.463521\pi$
$68$	0	0
$69$	7.77746	0.936297
$70$	0	0
$71$	−11.4708	−1.36134	−0.680669	−	0.732591i	$-0.738311\pi$
$71$	−11.4708	−1.36134	−0.680669	+	0.732591i	$0.738311\pi$
$72$	0	0
$73$	−0.474887	−0.0555814	−0.0277907	−	0.999614i	$-0.508847\pi$
$73$	−0.474887	−0.0555814	−0.0277907	+	0.999614i	$0.508847\pi$
$74$	0	0
$75$	−9.56926	−1.10496
$76$	0	0
$77$	−2.54079	−0.289550
$78$	0	0
$79$	−10.0185	−1.12717	−0.563583	−	0.826059i	$-0.690578\pi$
$79$	−10.0185	−1.12717	−0.563583	+	0.826059i	$0.690578\pi$
$80$	0	0
$81$	−10.6820	−1.18689
$82$	0	0
$83$	0.303718	0.0333374	0.0166687	−	0.999861i	$-0.494694\pi$
$83$	0.303718	0.0333374	0.0166687	+	0.999861i	$0.494694\pi$
$84$	0	0
$85$	0.236703	0.0256741
$86$	0	0
$87$	−7.39543	−0.792874
$88$	0	0
$89$	10.2766	1.08932	0.544660	−	0.838657i	$-0.316659\pi$
$89$	10.2766	1.08932	0.544660	+	0.838657i	$0.316659\pi$
$90$	0	0
$91$	−1.92574	−0.201873
$92$	0	0
$93$	8.93832	0.926860
$94$	0	0
$95$	−2.02306	−0.207562
$96$	0	0
$97$	−3.83061	−0.388939	−0.194470	−	0.980909i	$-0.562299\pi$
$97$	−3.83061	−0.388939	−0.194470	+	0.980909i	$0.562299\pi$
$98$	0	0
$99$	−1.06687	−0.107224
$100$	0	0
$101$	3.80422	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$101$	3.80422	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$102$	0	0
$103$	2.47184	0.243558	0.121779	−	0.992557i	$-0.461140\pi$
$103$	2.47184	0.243558	0.121779	+	0.992557i	$0.461140\pi$
$104$	0	0
$105$	0.814310	0.0794685
$106$	0	0
$107$	−11.5082	−1.11254	−0.556270	−	0.831002i	$-0.687768\pi$
$107$	−11.5082	−1.11254	−0.556270	+	0.831002i	$0.687768\pi$
$108$	0	0
$109$	−5.12285	−0.490680	−0.245340	−	0.969437i	$-0.578900\pi$
$109$	−5.12285	−0.490680	−0.245340	+	0.969437i	$0.578900\pi$
$110$	0	0
$111$	8.88003	0.842855
$112$	0	0
$113$	7.02569	0.660922	0.330461	−	0.943820i	$-0.392796\pi$
$113$	7.02569	0.660922	0.330461	+	0.943820i	$0.392796\pi$
$114$	0	0
$115$	0.951129	0.0886933
$116$	0	0
$117$	−0.808611	−0.0747561
$118$	0	0
$119$	1.77739	0.162933
$120$	0	0
$121$	−8.95651	−0.814228
$122$	0	0
$123$	−20.2468	−1.82559
$124$	0	0
$125$	−2.35377	−0.210527
$126$	0	0
$127$	−1.12530	−0.0998544	−0.0499272	−	0.998753i	$-0.515899\pi$
$127$	−1.12530	−0.0998544	−0.0499272	+	0.998753i	$0.515899\pi$
$128$	0	0
$129$	−12.7235	−1.12025
$130$	0	0
$131$	−8.66705	−0.757243	−0.378622	−	0.925552i	$-0.623602\pi$
$131$	−8.66705	−0.757243	−0.378622	+	0.925552i	$0.623602\pi$
$132$	0	0
$133$	−15.1911	−1.31724
$134$	0	0
$135$	−1.03252	−0.0888652
$136$	0	0
$137$	−0.762625	−0.0651555	−0.0325777	−	0.999469i	$-0.510372\pi$
$137$	−0.762625	−0.0651555	−0.0325777	+	0.999469i	$0.510372\pi$
$138$	0	0
$139$	12.0138	1.01899	0.509497	−	0.860473i	$-0.329832\pi$
$139$	12.0138	1.01899	0.509497	+	0.860473i	$0.329832\pi$
$140$	0	0
$141$	0.209139	0.0176126
$142$	0	0
$143$	1.54882	0.129519
$144$	0	0
$145$	−0.904409	−0.0751071
$146$	0	0
$147$	−7.43417	−0.613160
$148$	0	0
$149$	−18.6514	−1.52798	−0.763992	−	0.645226i	$-0.776763\pi$
$149$	−18.6514	−1.52798	−0.763992	+	0.645226i	$0.776763\pi$
$150$	0	0
$151$	21.3235	1.73528	0.867642	−	0.497190i	$-0.165635\pi$
$151$	21.3235	1.73528	0.867642	+	0.497190i	$0.165635\pi$
$152$	0	0
$153$	0.746320	0.0603364
$154$	0	0
$155$	1.09309	0.0877994
$156$	0	0
$157$	−11.7604	−0.938585	−0.469293	−	0.883043i	$-0.655491\pi$
$157$	−11.7604	−0.938585	−0.469293	+	0.883043i	$0.655491\pi$
$158$	0	0
$159$	−5.78190	−0.458535
$160$	0	0
$161$	7.14199	0.562868
$162$	0	0
$163$	8.40299	0.658173	0.329086	−	0.944300i	$-0.393259\pi$
$163$	8.40299	0.658173	0.329086	+	0.944300i	$0.393259\pi$
$164$	0	0
$165$	−0.654926	−0.0509859
$166$	0	0
$167$	17.7985	1.37729	0.688645	−	0.725099i	$-0.258206\pi$
$167$	17.7985	1.37729	0.688645	+	0.725099i	$0.258206\pi$
$168$	0	0
$169$	−11.8261	−0.909700
$170$	0	0
$171$	−6.37868	−0.487790
$172$	0	0
$173$	10.5613	0.802961	0.401480	−	0.915868i	$-0.368496\pi$
$173$	10.5613	0.802961	0.401480	+	0.915868i	$0.368496\pi$
$174$	0	0
$175$	−8.78738	−0.664264
$176$	0	0
$177$	1.93554	0.145484
$178$	0	0
$179$	−2.68745	−0.200869	−0.100435	−	0.994944i	$-0.532023\pi$
$179$	−2.68745	−0.200869	−0.100435	+	0.994944i	$0.532023\pi$
$180$	0	0
$181$	−17.3766	−1.29159	−0.645796	−	0.763510i	$-0.723474\pi$
$181$	−17.3766	−1.29159	−0.645796	+	0.763510i	$0.723474\pi$
$182$	0	0
$183$	17.6823	1.30711
$184$	0	0
$185$	1.08596	0.0798417
$186$	0	0
$187$	−1.42951	−0.104536
$188$	0	0
$189$	−7.75315	−0.563959
$190$	0	0
$191$	−1.63907	−0.118599	−0.0592995	−	0.998240i	$-0.518887\pi$
$191$	−1.63907	−0.118599	−0.0592995	+	0.998240i	$0.518887\pi$
$192$	0	0
$193$	16.5667	1.19249	0.596247	−	0.802801i	$-0.296658\pi$
$193$	16.5667	1.19249	0.596247	+	0.802801i	$0.296658\pi$
$194$	0	0
$195$	−0.496387	−0.0355470
$196$	0	0
$197$	−25.0871	−1.78738	−0.893692	−	0.448681i	$-0.851894\pi$
$197$	−25.0871	−1.78738	−0.893692	+	0.448681i	$0.851894\pi$
$198$	0	0
$199$	25.7302	1.82397	0.911983	−	0.410228i	$-0.134551\pi$
$199$	25.7302	1.82397	0.911983	+	0.410228i	$0.134551\pi$
$200$	0	0
$201$	3.62332	0.255569
$202$	0	0
$203$	−6.79117	−0.476647
$204$	0	0
$205$	−2.47604	−0.172934
$206$	0	0
$207$	2.99889	0.208437
$208$	0	0
$209$	12.2178	0.845121
$210$	0	0
$211$	−17.7277	−1.22043	−0.610213	−	0.792237i	$-0.708916\pi$
$211$	−17.7277	−1.22043	−0.610213	+	0.792237i	$0.708916\pi$
$212$	0	0
$213$	−22.2023	−1.52127
$214$	0	0
$215$	−1.55600	−0.106118
$216$	0	0
$217$	8.20799	0.557195
$218$	0	0
$219$	−0.919164	−0.0621113
$220$	0	0
$221$	−1.08346	−0.0728817
$222$	0	0
$223$	−4.91131	−0.328885	−0.164443	−	0.986387i	$-0.552583\pi$
$223$	−4.91131	−0.328885	−0.164443	+	0.986387i	$0.552583\pi$
$224$	0	0
$225$	−3.68979	−0.245986
$226$	0	0
$227$	−21.9080	−1.45408	−0.727041	−	0.686594i	$-0.759105\pi$
$227$	−21.9080	−1.45408	−0.727041	+	0.686594i	$0.759105\pi$
$228$	0	0
$229$	−9.22114	−0.609350	−0.304675	−	0.952456i	$-0.598548\pi$
$229$	−9.22114	−0.609350	−0.304675	+	0.952456i	$0.598548\pi$
$230$	0	0
$231$	−4.91781	−0.323568
$232$	0	0
$233$	6.05214	0.396489	0.198244	−	0.980153i	$-0.436476\pi$
$233$	6.05214	0.396489	0.198244	+	0.980153i	$0.436476\pi$
$234$	0	0
$235$	0.0255762	0.00166841
$236$	0	0
$237$	−19.3912	−1.25959
$238$	0	0
$239$	2.26497	0.146508	0.0732542	−	0.997313i	$-0.476662\pi$
$239$	2.26497	0.146508	0.0732542	+	0.997313i	$0.476662\pi$
$240$	0	0
$241$	28.1608	1.81399	0.906997	−	0.421137i	$-0.138369\pi$
$241$	28.1608	1.81399	0.906997	+	0.421137i	$0.138369\pi$
$242$	0	0
$243$	−7.58912	−0.486842
$244$	0	0
$245$	−0.909147	−0.0580832
$246$	0	0
$247$	9.26020	0.589212
$248$	0	0
$249$	0.587859	0.0372541
$250$	0	0
$251$	0.508509	0.0320968	0.0160484	−	0.999871i	$-0.494891\pi$
$251$	0.508509	0.0320968	0.0160484	+	0.999871i	$0.494891\pi$
$252$	0	0
$253$	−5.74410	−0.361128
$254$	0	0
$255$	0.458149	0.0286904
$256$	0	0
$257$	−4.36044	−0.271997	−0.135998	−	0.990709i	$-0.543424\pi$
$257$	−4.36044	−0.271997	−0.135998	+	0.990709i	$0.543424\pi$
$258$	0	0
$259$	8.15446	0.506694
$260$	0	0
$261$	−2.85159	−0.176509
$262$	0	0
$263$	−20.1592	−1.24307	−0.621536	−	0.783386i	$-0.713491\pi$
$263$	−20.1592	−1.24307	−0.621536	+	0.783386i	$0.713491\pi$
$264$	0	0
$265$	−0.707086	−0.0434360
$266$	0	0
$267$	19.8908	1.21730
$268$	0	0
$269$	31.1604	1.89988	0.949940	−	0.312432i	$-0.101144\pi$
$269$	31.1604	1.89988	0.949940	+	0.312432i	$0.101144\pi$
$270$	0	0
$271$	−10.3388	−0.628037	−0.314019	−	0.949417i	$-0.601675\pi$
$271$	−10.3388	−0.628037	−0.314019	+	0.949417i	$0.601675\pi$
$272$	0	0
$273$	−3.72735	−0.225590
$274$	0	0
$275$	7.06744	0.426183
$276$	0	0
$277$	6.29750	0.378380	0.189190	−	0.981940i	$-0.439414\pi$
$277$	6.29750	0.378380	0.189190	+	0.981940i	$0.439414\pi$
$278$	0	0
$279$	3.44650	0.206337
$280$	0	0
$281$	−25.5030	−1.52138	−0.760689	−	0.649116i	$-0.775139\pi$
$281$	−25.5030	−1.52138	−0.760689	+	0.649116i	$0.775139\pi$
$282$	0	0
$283$	30.8857	1.83596	0.917982	−	0.396622i	$-0.129818\pi$
$283$	30.8857	1.83596	0.917982	+	0.396622i	$0.129818\pi$
$284$	0	0
$285$	−3.91572	−0.231948
$286$	0	0
$287$	−18.5925	−1.09748
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.41430	−0.434634
$292$	0	0
$293$	22.3293	1.30449	0.652245	−	0.758008i	$-0.273827\pi$
$293$	22.3293	1.30449	0.652245	+	0.758008i	$0.273827\pi$
$294$	0	0
$295$	0.236703	0.0137814
$296$	0	0
$297$	6.23564	0.361828
$298$	0	0
$299$	−4.35361	−0.251776
$300$	0	0
$301$	−11.6839	−0.673451
$302$	0	0
$303$	7.36322	0.423006
$304$	0	0
$305$	2.16242	0.123820
$306$	0	0
$307$	−11.8140	−0.674258	−0.337129	−	0.941458i	$-0.609456\pi$
$307$	−11.8140	−0.674258	−0.337129	+	0.941458i	$0.609456\pi$
$308$	0	0
$309$	4.78435	0.272172
$310$	0	0
$311$	20.7840	1.17855	0.589275	−	0.807933i	$-0.299413\pi$
$311$	20.7840	1.17855	0.589275	+	0.807933i	$0.299413\pi$
$312$	0	0
$313$	6.35044	0.358948	0.179474	−	0.983763i	$-0.442560\pi$
$313$	6.35044	0.358948	0.179474	+	0.983763i	$0.442560\pi$
$314$	0	0
$315$	0.313988	0.0176912
$316$	0	0
$317$	30.9787	1.73994	0.869969	−	0.493107i	$-0.164139\pi$
$317$	30.9787	1.73994	0.869969	+	0.493107i	$0.164139\pi$
$318$	0	0
$319$	5.46194	0.305810
$320$	0	0
$321$	−22.2746	−1.24325
$322$	0	0
$323$	−8.54684	−0.475559
$324$	0	0
$325$	5.35661	0.297132
$326$	0	0
$327$	−9.91549	−0.548328
$328$	0	0
$329$	0.192050	0.0105881
$330$	0	0
$331$	−5.25720	−0.288962	−0.144481	−	0.989508i	$-0.546151\pi$
$331$	−5.25720	−0.288962	−0.144481	+	0.989508i	$0.546151\pi$
$332$	0	0
$333$	3.42403	0.187635
$334$	0	0
$335$	0.443106	0.0242095
$336$	0	0
$337$	−3.04694	−0.165978	−0.0829888	−	0.996550i	$-0.526447\pi$
$337$	−3.04694	−0.165978	−0.0829888	+	0.996550i	$0.526447\pi$
$338$	0	0
$339$	13.5985	0.738570
$340$	0	0
$341$	−6.60145	−0.357489
$342$	0	0
$343$	−19.2685	−1.04040
$344$	0	0
$345$	1.84095	0.0991134
$346$	0	0
$347$	−6.38419	−0.342721	−0.171361	−	0.985208i	$-0.554816\pi$
$347$	−6.38419	−0.342721	−0.171361	+	0.985208i	$0.554816\pi$
$348$	0	0
$349$	−12.7605	−0.683051	−0.341526	−	0.939872i	$-0.610944\pi$
$349$	−12.7605	−0.683051	−0.341526	+	0.939872i	$0.610944\pi$
$350$	0	0
$351$	4.72617	0.252264
$352$	0	0
$353$	−26.0481	−1.38640	−0.693200	−	0.720745i	$-0.743800\pi$
$353$	−26.0481	−1.38640	−0.693200	+	0.720745i	$0.743800\pi$
$354$	0	0
$355$	−2.71518	−0.144107
$356$	0	0
$357$	3.44022	0.182076
$358$	0	0
$359$	−12.7875	−0.674899	−0.337449	−	0.941344i	$-0.609564\pi$
$359$	−12.7875	−0.674899	−0.337449	+	0.941344i	$0.609564\pi$
$360$	0	0
$361$	54.0485	2.84466
$362$	0	0
$363$	−17.3357	−0.909888
$364$	0	0
$365$	−0.112407	−0.00588367
$366$	0	0
$367$	−17.8860	−0.933640	−0.466820	−	0.884352i	$-0.654600\pi$
$367$	−17.8860	−0.933640	−0.466820	+	0.884352i	$0.654600\pi$
$368$	0	0
$369$	−7.80690	−0.406411
$370$	0	0
$371$	−5.30948	−0.275654
$372$	0	0
$373$	−15.8226	−0.819264	−0.409632	−	0.912251i	$-0.634343\pi$
$373$	−15.8226	−0.819264	−0.409632	+	0.912251i	$0.634343\pi$
$374$	0	0
$375$	−4.55582	−0.235261
$376$	0	0
$377$	4.13976	0.213209
$378$	0	0
$379$	9.71577	0.499066	0.249533	−	0.968366i	$-0.419723\pi$
$379$	9.71577	0.499066	0.249533	+	0.968366i	$0.419723\pi$
$380$	0	0
$381$	−2.17807	−0.111586
$382$	0	0
$383$	5.79352	0.296035	0.148018	−	0.988985i	$-0.452711\pi$
$383$	5.79352	0.296035	0.148018	+	0.988985i	$0.452711\pi$
$384$	0	0
$385$	−0.601414	−0.0306509
$386$	0	0
$387$	−4.90604	−0.249388
$388$	0	0
$389$	−15.1902	−0.770173	−0.385087	−	0.922880i	$-0.625828\pi$
$389$	−15.1902	−0.770173	−0.385087	+	0.922880i	$0.625828\pi$
$390$	0	0
$391$	4.01824	0.203211
$392$	0	0
$393$	−16.7754	−0.846208
$394$	0	0
$395$	−2.37140	−0.119318
$396$	0	0
$397$	−13.2937	−0.667194	−0.333597	−	0.942716i	$-0.608262\pi$
$397$	−13.2937	−0.667194	−0.333597	+	0.942716i	$0.608262\pi$
$398$	0	0
$399$	−29.4030	−1.47199
$400$	0	0
$401$	9.59095	0.478949	0.239474	−	0.970903i	$-0.423025\pi$
$401$	9.59095	0.478949	0.239474	+	0.970903i	$0.423025\pi$
$402$	0	0
$403$	−5.00343	−0.249239
$404$	0	0
$405$	−2.52845	−0.125640
$406$	0	0
$407$	−6.55840	−0.325088
$408$	0	0
$409$	9.98907	0.493928	0.246964	−	0.969025i	$-0.420567\pi$
$409$	9.98907	0.493928	0.246964	+	0.969025i	$0.420567\pi$
$410$	0	0
$411$	−1.47609	−0.0728103
$412$	0	0
$413$	1.77739	0.0874598
$414$	0	0
$415$	0.0718911	0.00352899
$416$	0	0
$417$	23.2531	1.13871
$418$	0	0
$419$	34.4746	1.68419	0.842097	−	0.539325i	$-0.181321\pi$
$419$	34.4746	1.68419	0.842097	+	0.539325i	$0.181321\pi$
$420$	0	0
$421$	−36.5442	−1.78105	−0.890527	−	0.454931i	$-0.849664\pi$
$421$	−36.5442	−1.78105	−0.890527	+	0.454931i	$0.849664\pi$
$422$	0	0
$423$	0.0806412	0.00392091
$424$	0	0
$425$	−4.94397	−0.239818
$426$	0	0
$427$	16.2375	0.785788
$428$	0	0
$429$	2.99780	0.144735
$430$	0	0
$431$	−25.3526	−1.22119	−0.610596	−	0.791942i	$-0.709070\pi$
$431$	−25.3526	−1.22119	−0.610596	+	0.791942i	$0.709070\pi$
$432$	0	0
$433$	4.60461	0.221284	0.110642	−	0.993860i	$-0.464709\pi$
$433$	4.60461	0.221284	0.110642	+	0.993860i	$0.464709\pi$
$434$	0	0
$435$	−1.75052	−0.0839311
$436$	0	0
$437$	−34.3432	−1.64286
$438$	0	0
$439$	−11.3370	−0.541085	−0.270543	−	0.962708i	$-0.587203\pi$
$439$	−11.3370	−0.541085	−0.270543	+	0.962708i	$0.587203\pi$
$440$	0	0
$441$	−2.86652	−0.136501
$442$	0	0
$443$	5.99787	0.284968	0.142484	−	0.989797i	$-0.454491\pi$
$443$	5.99787	0.284968	0.142484	+	0.989797i	$0.454491\pi$
$444$	0	0
$445$	2.43251	0.115312
$446$	0	0
$447$	−36.1006	−1.70750
$448$	0	0
$449$	17.3777	0.820105	0.410052	−	0.912062i	$-0.365510\pi$
$449$	17.3777	0.820105	0.410052	+	0.912062i	$0.365510\pi$
$450$	0	0
$451$	14.9534	0.704127
$452$	0	0
$453$	41.2726	1.93915
$454$	0	0
$455$	−0.455829	−0.0213696
$456$	0	0
$457$	19.9500	0.933219	0.466610	−	0.884463i	$-0.345475\pi$
$457$	19.9500	0.933219	0.466610	+	0.884463i	$0.345475\pi$
$458$	0	0
$459$	−4.36209	−0.203605
$460$	0	0
$461$	12.1640	0.566534	0.283267	−	0.959041i	$-0.408582\pi$
$461$	12.1640	0.566534	0.283267	+	0.959041i	$0.408582\pi$
$462$	0	0
$463$	−31.1980	−1.44990	−0.724948	−	0.688804i	$-0.758136\pi$
$463$	−31.1980	−1.44990	−0.724948	+	0.688804i	$0.758136\pi$
$464$	0	0
$465$	2.11573	0.0981145
$466$	0	0
$467$	37.1277	1.71806	0.859032	−	0.511921i	$-0.171066\pi$
$467$	37.1277	1.71806	0.859032	+	0.511921i	$0.171066\pi$
$468$	0	0
$469$	3.32727	0.153639
$470$	0	0
$471$	−22.7628	−1.04886
$472$	0	0
$473$	9.39705	0.432077
$474$	0	0
$475$	42.2554	1.93881
$476$	0	0
$477$	−2.22943	−0.102079
$478$	0	0
$479$	28.0753	1.28279	0.641397	−	0.767209i	$-0.278355\pi$
$479$	28.0753	1.28279	0.641397	+	0.767209i	$0.278355\pi$
$480$	0	0
$481$	−4.97080	−0.226649
$482$	0	0
$483$	13.8236	0.628996
$484$	0	0
$485$	−0.906717	−0.0411719
$486$	0	0
$487$	−17.4159	−0.789188	−0.394594	−	0.918856i	$-0.629115\pi$
$487$	−17.4159	−0.789188	−0.394594	+	0.918856i	$0.629115\pi$
$488$	0	0
$489$	16.2643	0.735499
$490$	0	0
$491$	−13.2961	−0.600043	−0.300021	−	0.953932i	$-0.596994\pi$
$491$	−13.2961	−0.600043	−0.300021	+	0.953932i	$0.596994\pi$
$492$	0	0
$493$	−3.82086	−0.172083
$494$	0	0
$495$	−0.252531	−0.0113504
$496$	0	0
$497$	−20.3882	−0.914535
$498$	0	0
$499$	−29.8832	−1.33776	−0.668879	−	0.743372i	$-0.733225\pi$
$499$	−29.8832	−1.33776	−0.668879	+	0.743372i	$0.733225\pi$
$500$	0	0
$501$	34.4497	1.53910
$502$	0	0
$503$	18.7671	0.836786	0.418393	−	0.908266i	$-0.362594\pi$
$503$	18.7671	0.836786	0.418393	+	0.908266i	$0.362594\pi$
$504$	0	0
$505$	0.900470	0.0400704
$506$	0	0
$507$	−22.8899	−1.01658
$508$	0	0
$509$	−4.33885	−0.192316	−0.0961580	−	0.995366i	$-0.530655\pi$
$509$	−4.33885	−0.192316	−0.0961580	+	0.995366i	$0.530655\pi$
$510$	0	0
$511$	−0.844061	−0.0373391
$512$	0	0
$513$	37.2821	1.64605
$514$	0	0
$515$	0.585092	0.0257823
$516$	0	0
$517$	−0.154461	−0.00679317
$518$	0	0
$519$	20.4418	0.897297
$520$	0	0
$521$	35.1422	1.53961	0.769803	−	0.638282i	$-0.220354\pi$
$521$	35.1422	1.53961	0.769803	+	0.638282i	$0.220354\pi$
$522$	0	0
$523$	−5.70489	−0.249458	−0.124729	−	0.992191i	$-0.539806\pi$
$523$	−5.70489	−0.249458	−0.124729	+	0.992191i	$0.539806\pi$
$524$	0	0
$525$	−17.0083	−0.742305
$526$	0	0
$527$	4.61799	0.201163
$528$	0	0
$529$	−6.85377	−0.297990
$530$	0	0
$531$	0.746320	0.0323875
$532$	0	0
$533$	11.3336	0.490913
$534$	0	0
$535$	−2.72403	−0.117770
$536$	0	0
$537$	−5.20166	−0.224468
$538$	0	0
$539$	5.49055	0.236495
$540$	0	0
$541$	−2.53751	−0.109096	−0.0545480	−	0.998511i	$-0.517372\pi$
$541$	−2.53751	−0.109096	−0.0545480	+	0.998511i	$0.517372\pi$
$542$	0	0
$543$	−33.6331	−1.44333
$544$	0	0
$545$	−1.21259	−0.0519418
$546$	0	0
$547$	−25.0384	−1.07057	−0.535283	−	0.844673i	$-0.679795\pi$
$547$	−25.0384	−1.07057	−0.535283	+	0.844673i	$0.679795\pi$
$548$	0	0
$549$	6.81806	0.290988
$550$	0	0
$551$	32.6563	1.39121
$552$	0	0
$553$	−17.8068	−0.757221
$554$	0	0
$555$	2.10193	0.0892219
$556$	0	0
$557$	0.0435093	0.00184355	0.000921774	−	1.00000i	$-0.499707\pi$
$557$	0.0435093	0.00184355	0.000921774		1.00000i	$0.499707\pi$
$558$	0	0
$559$	7.12230	0.301241
$560$	0	0
$561$	−2.76687	−0.116817
$562$	0	0
$563$	−29.4718	−1.24209	−0.621044	−	0.783775i	$-0.713291\pi$
$563$	−29.4718	−1.24209	−0.621044	+	0.783775i	$0.713291\pi$
$564$	0	0
$565$	1.66300	0.0699631
$566$	0	0
$567$	−18.9861	−0.797339
$568$	0	0
$569$	−2.30533	−0.0966444	−0.0483222	−	0.998832i	$-0.515387\pi$
$569$	−2.30533	−0.0966444	−0.0483222	+	0.998832i	$0.515387\pi$
$570$	0	0
$571$	−15.2438	−0.637934	−0.318967	−	0.947766i	$-0.603336\pi$
$571$	−15.2438	−0.637934	−0.318967	+	0.947766i	$0.603336\pi$
$572$	0	0
$573$	−3.17249	−0.132533
$574$	0	0
$575$	−19.8661	−0.828472
$576$	0	0
$577$	−7.39713	−0.307946	−0.153973	−	0.988075i	$-0.549207\pi$
$577$	−7.39713	−0.307946	−0.153973	+	0.988075i	$0.549207\pi$
$578$	0	0
$579$	32.0654	1.33259
$580$	0	0
$581$	0.539827	0.0223958
$582$	0	0
$583$	4.27026	0.176856
$584$	0	0
$585$	−0.191401	−0.00791345
$586$	0	0
$587$	−24.5259	−1.01229	−0.506146	−	0.862448i	$-0.668930\pi$
$587$	−24.5259	−1.01229	−0.506146	+	0.862448i	$0.668930\pi$
$588$	0	0
$589$	−39.4693	−1.62630
$590$	0	0
$591$	−48.5572	−1.99738
$592$	0	0
$593$	−7.72128	−0.317075	−0.158537	−	0.987353i	$-0.550678\pi$
$593$	−7.72128	−0.317075	−0.158537	+	0.987353i	$0.550678\pi$
$594$	0	0
$595$	0.420714	0.0172476
$596$	0	0
$597$	49.8019	2.03826
$598$	0	0
$599$	32.9967	1.34821	0.674103	−	0.738637i	$-0.264530\pi$
$599$	32.9967	1.34821	0.674103	+	0.738637i	$0.264530\pi$
$600$	0	0
$601$	25.9485	1.05846	0.529231	−	0.848478i	$-0.322481\pi$
$601$	25.9485	1.05846	0.529231	+	0.848478i	$0.322481\pi$
$602$	0	0
$603$	1.39711	0.0568946
$604$	0	0
$605$	−2.12003	−0.0861916
$606$	0	0
$607$	−40.4226	−1.64070	−0.820351	−	0.571861i	$-0.806222\pi$
$607$	−40.4226	−1.64070	−0.820351	+	0.571861i	$0.806222\pi$
$608$	0	0
$609$	−13.1446	−0.532646
$610$	0	0
$611$	−0.117070	−0.00473615
$612$	0	0
$613$	41.7320	1.68554	0.842770	−	0.538274i	$-0.180924\pi$
$613$	41.7320	1.68554	0.842770	+	0.538274i	$0.180924\pi$
$614$	0	0
$615$	−4.79247	−0.193251
$616$	0	0
$617$	−21.3277	−0.858620	−0.429310	−	0.903157i	$-0.641243\pi$
$617$	−21.3277	−0.858620	−0.429310	+	0.903157i	$0.641243\pi$
$618$	0	0
$619$	−3.05299	−0.122710	−0.0613550	−	0.998116i	$-0.519542\pi$
$619$	−3.05299	−0.122710	−0.0613550	+	0.998116i	$0.519542\pi$
$620$	0	0
$621$	−17.5279	−0.703371
$622$	0	0
$623$	18.2656	0.731796
$624$	0	0
$625$	24.1627	0.966509
$626$	0	0
$627$	23.6480	0.944410
$628$	0	0
$629$	4.58788	0.182931
$630$	0	0
$631$	−22.0780	−0.878912	−0.439456	−	0.898264i	$-0.644829\pi$
$631$	−22.0780	−0.878912	−0.439456	+	0.898264i	$0.644829\pi$
$632$	0	0
$633$	−34.3127	−1.36381
$634$	0	0
$635$	−0.266362	−0.0105703
$636$	0	0
$637$	4.16145	0.164883
$638$	0	0
$639$	−8.56092	−0.338665
$640$	0	0
$641$	32.4393	1.28127	0.640637	−	0.767844i	$-0.278670\pi$
$641$	32.4393	1.28127	0.640637	+	0.767844i	$0.278670\pi$
$642$	0	0
$643$	−7.34913	−0.289822	−0.144911	−	0.989445i	$-0.546290\pi$
$643$	−7.34913	−0.289822	−0.144911	+	0.989445i	$0.546290\pi$
$644$	0	0
$645$	−3.01170	−0.118586
$646$	0	0
$647$	−22.8684	−0.899049	−0.449524	−	0.893268i	$-0.648406\pi$
$647$	−22.8684	−0.899049	−0.449524	+	0.893268i	$0.648406\pi$
$648$	0	0
$649$	−1.42951	−0.0561130
$650$	0	0
$651$	15.8869	0.622657
$652$	0	0
$653$	42.0844	1.64689	0.823445	−	0.567396i	$-0.192049\pi$
$653$	42.0844	1.64689	0.823445	+	0.567396i	$0.192049\pi$
$654$	0	0
$655$	−2.05152	−0.0801594
$656$	0	0
$657$	−0.354418	−0.0138272
$658$	0	0
$659$	−11.5845	−0.451269	−0.225634	−	0.974212i	$-0.572445\pi$
$659$	−11.5845	−0.451269	−0.225634	+	0.974212i	$0.572445\pi$
$660$	0	0
$661$	17.1084	0.665441	0.332721	−	0.943025i	$-0.392033\pi$
$661$	17.1084	0.665441	0.332721	+	0.943025i	$0.392033\pi$
$662$	0	0
$663$	−2.09709	−0.0814442
$664$	0	0
$665$	−3.59578	−0.139438
$666$	0	0
$667$	−15.3531	−0.594475
$668$	0	0
$669$	−9.50604	−0.367525
$670$	0	0
$671$	−13.0594	−0.504151
$672$	0	0
$673$	−20.1130	−0.775300	−0.387650	−	0.921807i	$-0.626713\pi$
$673$	−20.1130	−0.775300	−0.387650	+	0.921807i	$0.626713\pi$
$674$	0	0
$675$	21.5661	0.830078
$676$	0	0
$677$	2.39838	0.0921771	0.0460886	−	0.998937i	$-0.485324\pi$
$677$	2.39838	0.0921771	0.0460886	+	0.998937i	$0.485324\pi$
$678$	0	0
$679$	−6.80850	−0.261286
$680$	0	0
$681$	−42.4037	−1.62492
$682$	0	0
$683$	31.0870	1.18951	0.594755	−	0.803907i	$-0.297249\pi$
$683$	31.0870	1.18951	0.594755	+	0.803907i	$0.297249\pi$
$684$	0	0
$685$	−0.180516	−0.00689715
$686$	0	0
$687$	−17.8479	−0.680940
$688$	0	0
$689$	3.23655	0.123303
$690$	0	0
$691$	−5.00389	−0.190357	−0.0951785	−	0.995460i	$-0.530342\pi$
$691$	−5.00389	−0.190357	−0.0951785	+	0.995460i	$0.530342\pi$
$692$	0	0
$693$	−1.89625	−0.0720325
$694$	0	0
$695$	2.84369	0.107867
$696$	0	0
$697$	−10.4605	−0.396221
$698$	0	0
$699$	11.7142	0.443070
$700$	0	0
$701$	−13.0503	−0.492904	−0.246452	−	0.969155i	$-0.579265\pi$
$701$	−13.0503	−0.492904	−0.246452	+	0.969155i	$0.579265\pi$
$702$	0	0
$703$	−39.2119	−1.47890
$704$	0	0
$705$	0.0495038	0.00186442
$706$	0	0
$707$	6.76159	0.254296
$708$	0	0
$709$	33.1253	1.24405	0.622024	−	0.782998i	$-0.286311\pi$
$709$	33.1253	1.24405	0.622024	+	0.782998i	$0.286311\pi$
$710$	0	0
$711$	−7.47699	−0.280409
$712$	0	0
$713$	18.5562	0.694935
$714$	0	0
$715$	0.366610	0.0137104
$716$	0	0
$717$	4.38394	0.163721
$718$	0	0
$719$	−13.4032	−0.499856	−0.249928	−	0.968264i	$-0.580407\pi$
$719$	−13.4032	−0.499856	−0.249928	+	0.968264i	$0.580407\pi$
$720$	0	0
$721$	4.39343	0.163620
$722$	0	0
$723$	54.5063	2.02711
$724$	0	0
$725$	18.8902	0.701565
$726$	0	0
$727$	−43.8300	−1.62557	−0.812783	−	0.582566i	$-0.802049\pi$
$727$	−43.8300	−1.62557	−0.812783	+	0.582566i	$0.802049\pi$
$728$	0	0
$729$	17.3568	0.642846
$730$	0	0
$731$	−6.57363	−0.243135
$732$	0	0
$733$	7.66538	0.283127	0.141564	−	0.989929i	$-0.454787\pi$
$733$	7.66538	0.283127	0.141564	+	0.989929i	$0.454787\pi$
$734$	0	0
$735$	−1.75969	−0.0649072
$736$	0	0
$737$	−2.67603	−0.0985727
$738$	0	0
$739$	−8.84751	−0.325461	−0.162730	−	0.986671i	$-0.552030\pi$
$739$	−8.84751	−0.325461	−0.162730	+	0.986671i	$0.552030\pi$
$740$	0	0
$741$	17.9235	0.658436
$742$	0	0
$743$	−10.8486	−0.397998	−0.198999	−	0.980000i	$-0.563769\pi$
$743$	−10.8486	−0.397998	−0.198999	+	0.980000i	$0.563769\pi$
$744$	0	0
$745$	−4.41485	−0.161748
$746$	0	0
$747$	0.226671	0.00829347
$748$	0	0
$749$	−20.4546	−0.747395
$750$	0	0
$751$	32.8105	1.19727	0.598636	−	0.801022i	$-0.295710\pi$
$751$	32.8105	1.19727	0.598636	+	0.801022i	$0.295710\pi$
$752$	0	0
$753$	0.984240	0.0358677
$754$	0	0
$755$	5.04734	0.183692
$756$	0	0
$757$	−19.4893	−0.708351	−0.354176	−	0.935179i	$-0.615239\pi$
$757$	−19.4893	−0.708351	−0.354176	+	0.935179i	$0.615239\pi$
$758$	0	0
$759$	−11.1179	−0.403555
$760$	0	0
$761$	−17.6618	−0.640240	−0.320120	−	0.947377i	$-0.603723\pi$
$761$	−17.6618	−0.640240	−0.320120	+	0.947377i	$0.603723\pi$
$762$	0	0
$763$	−9.10532	−0.329635
$764$	0	0
$765$	0.176656	0.00638702
$766$	0	0
$767$	−1.08346	−0.0391216
$768$	0	0
$769$	19.9456	0.719255	0.359627	−	0.933096i	$-0.382904\pi$
$769$	19.9456	0.719255	0.359627	+	0.933096i	$0.382904\pi$
$770$	0	0
$771$	−8.43981	−0.303952
$772$	0	0
$773$	−2.24878	−0.0808830	−0.0404415	−	0.999182i	$-0.512876\pi$
$773$	−2.24878	−0.0808830	−0.0404415	+	0.999182i	$0.512876\pi$
$774$	0	0
$775$	−22.8312	−0.820122
$776$	0	0
$777$	15.7833	0.566223
$778$	0	0
$779$	89.4045	3.20325
$780$	0	0
$781$	16.3976	0.586753
$782$	0	0
$783$	16.6669	0.595628
$784$	0	0
$785$	−2.78373	−0.0993556
$786$	0	0
$787$	−1.19661	−0.0426544	−0.0213272	−	0.999773i	$-0.506789\pi$
$787$	−1.19661	−0.0426544	−0.0213272	+	0.999773i	$0.506789\pi$
$788$	0	0
$789$	−39.0190	−1.38911
$790$	0	0
$791$	12.4874	0.444002
$792$	0	0
$793$	−9.89806	−0.351490
$794$	0	0
$795$	−1.36859	−0.0485391
$796$	0	0
$797$	−13.6274	−0.482707	−0.241354	−	0.970437i	$-0.577591\pi$
$797$	−13.6274	−0.482707	−0.241354	+	0.970437i	$0.577591\pi$
$798$	0	0
$799$	0.108052	0.00382260
$800$	0	0
$801$	7.66965	0.270994
$802$	0	0
$803$	0.678854	0.0239562
$804$	0	0
$805$	1.69053	0.0595834
$806$	0	0
$807$	60.3121	2.12309
$808$	0	0
$809$	15.7644	0.554246	0.277123	−	0.960834i	$-0.410619\pi$
$809$	15.7644	0.554246	0.277123	+	0.960834i	$0.410619\pi$
$810$	0	0
$811$	−52.8715	−1.85657	−0.928284	−	0.371872i	$-0.878716\pi$
$811$	−52.8715	−1.85657	−0.928284	+	0.371872i	$0.878716\pi$
$812$	0	0
$813$	−20.0112	−0.701822
$814$	0	0
$815$	1.98901	0.0696721
$816$	0	0
$817$	56.1838	1.96562
$818$	0	0
$819$	−1.43722	−0.0502205
$820$	0	0
$821$	21.7226	0.758123	0.379062	−	0.925371i	$-0.376247\pi$
$821$	21.7226	0.758123	0.379062	+	0.925371i	$0.376247\pi$
$822$	0	0
$823$	23.5660	0.821458	0.410729	−	0.911758i	$-0.365274\pi$
$823$	23.5660	0.821458	0.410729	+	0.911758i	$0.365274\pi$
$824$	0	0
$825$	13.6793	0.476253
$826$	0	0
$827$	−11.0947	−0.385800	−0.192900	−	0.981218i	$-0.561789\pi$
$827$	−11.0947	−0.385800	−0.192900	+	0.981218i	$0.561789\pi$
$828$	0	0
$829$	−38.1262	−1.32418	−0.662089	−	0.749426i	$-0.730330\pi$
$829$	−38.1262	−1.32418	−0.662089	+	0.749426i	$0.730330\pi$
$830$	0	0
$831$	12.1891	0.422834
$832$	0	0
$833$	−3.84087	−0.133078
$834$	0	0
$835$	4.21296	0.145795
$836$	0	0
$837$	−20.1441	−0.696282
$838$	0	0
$839$	43.5981	1.50517	0.752586	−	0.658494i	$-0.228806\pi$
$839$	43.5981	1.50517	0.752586	+	0.658494i	$0.228806\pi$
$840$	0	0
$841$	−14.4010	−0.496587
$842$	0	0
$843$	−49.3620	−1.70012
$844$	0	0
$845$	−2.79928	−0.0962980
$846$	0	0
$847$	−15.9192	−0.546992
$848$	0	0
$849$	59.7805	2.05166
$850$	0	0
$851$	18.4352	0.631950
$852$	0	0
$853$	−36.0323	−1.23372	−0.616862	−	0.787072i	$-0.711596\pi$
$853$	−36.0323	−1.23372	−0.616862	+	0.787072i	$0.711596\pi$
$854$	0	0
$855$	−1.50985	−0.0516359
$856$	0	0
$857$	23.2654	0.794732	0.397366	−	0.917660i	$-0.369924\pi$
$857$	23.2654	0.794732	0.397366	+	0.917660i	$0.369924\pi$
$858$	0	0
$859$	−24.2240	−0.826511	−0.413255	−	0.910615i	$-0.635608\pi$
$859$	−24.2240	−0.826511	−0.413255	+	0.910615i	$0.635608\pi$
$860$	0	0
$861$	−35.9865	−1.22642
$862$	0	0
$863$	37.4064	1.27333	0.636665	−	0.771141i	$-0.280314\pi$
$863$	37.4064	1.27333	0.636665	+	0.771141i	$0.280314\pi$
$864$	0	0
$865$	2.49989	0.0849989
$866$	0	0
$867$	1.93554	0.0657344
$868$	0	0
$869$	14.3215	0.485823
$870$	0	0
$871$	−2.02824	−0.0687242
$872$	0	0
$873$	−2.85886	−0.0967578
$874$	0	0
$875$	−4.18357	−0.141431
$876$	0	0
$877$	−5.73337	−0.193602	−0.0968010	−	0.995304i	$-0.530861\pi$
$877$	−5.73337	−0.193602	−0.0968010	+	0.995304i	$0.530861\pi$
$878$	0	0
$879$	43.2193	1.45775
$880$	0	0
$881$	−21.0233	−0.708292	−0.354146	−	0.935190i	$-0.615228\pi$
$881$	−21.0233	−0.708292	−0.354146	+	0.935190i	$0.615228\pi$
$882$	0	0
$883$	21.9117	0.737387	0.368694	−	0.929551i	$-0.379805\pi$
$883$	21.9117	0.737387	0.368694	+	0.929551i	$0.379805\pi$
$884$	0	0
$885$	0.458149	0.0154005
$886$	0	0
$887$	−35.2715	−1.18430	−0.592151	−	0.805827i	$-0.701721\pi$
$887$	−35.2715	−1.18430	−0.592151	+	0.805827i	$0.701721\pi$
$888$	0	0
$889$	−2.00010	−0.0670813
$890$	0	0
$891$	15.2699	0.511562
$892$	0	0
$893$	−0.923501	−0.0309038
$894$	0	0
$895$	−0.636127	−0.0212634
$896$	0	0
$897$	−8.42660	−0.281356
$898$	0	0
$899$	−17.6447	−0.588484
$900$	0	0
$901$	−2.98723	−0.0995190
$902$	0	0
$903$	−22.6147	−0.752571
$904$	0	0
$905$	−4.11309	−0.136724
$906$	0	0
$907$	−7.52224	−0.249772	−0.124886	−	0.992171i	$-0.539857\pi$
$907$	−7.52224	−0.249772	−0.124886	+	0.992171i	$0.539857\pi$
$908$	0	0
$909$	2.83916	0.0941691
$910$	0	0
$911$	15.1792	0.502909	0.251455	−	0.967869i	$-0.419091\pi$
$911$	15.1792	0.502909	0.251455	+	0.967869i	$0.419091\pi$
$912$	0	0
$913$	−0.434167	−0.0143688
$914$	0	0
$915$	4.18545	0.138367
$916$	0	0
$917$	−15.4047	−0.508710
$918$	0	0
$919$	−13.4273	−0.442926	−0.221463	−	0.975169i	$-0.571083\pi$
$919$	−13.4273	−0.442926	−0.221463	+	0.975169i	$0.571083\pi$
$920$	0	0
$921$	−22.8664	−0.753474
$922$	0	0
$923$	12.4282	0.409080
$924$	0	0
$925$	−22.6823	−0.745791
$926$	0	0
$927$	1.84479	0.0605907
$928$	0	0
$929$	27.1152	0.889622	0.444811	−	0.895624i	$-0.353271\pi$
$929$	27.1152	0.889622	0.444811	+	0.895624i	$0.353271\pi$
$930$	0	0
$931$	32.8273	1.07587
$932$	0	0
$933$	40.2282	1.31701
$934$	0	0
$935$	−0.338369	−0.0110658
$936$	0	0
$937$	−44.4352	−1.45164	−0.725818	−	0.687887i	$-0.758538\pi$
$937$	−44.4352	−1.45164	−0.725818	+	0.687887i	$0.758538\pi$
$938$	0	0
$939$	12.2915	0.401119
$940$	0	0
$941$	59.5638	1.94172	0.970862	−	0.239639i	$-0.0770290\pi$
$941$	59.5638	1.94172	0.970862	+	0.239639i	$0.0770290\pi$
$942$	0	0
$943$	−42.0329	−1.36878
$944$	0	0
$945$	−1.83519	−0.0596989
$946$	0	0
$947$	−55.4215	−1.80096	−0.900478	−	0.434901i	$-0.856783\pi$
$947$	−55.4215	−1.80096	−0.900478	+	0.434901i	$0.856783\pi$
$948$	0	0
$949$	0.514523	0.0167021
$950$	0	0
$951$	59.9606	1.94435
$952$	0	0
$953$	49.2038	1.59387	0.796933	−	0.604067i	$-0.206454\pi$
$953$	49.2038	1.59387	0.796933	+	0.604067i	$0.206454\pi$
$954$	0	0
$955$	−0.387973	−0.0125545
$956$	0	0
$957$	10.5718	0.341738
$958$	0	0
$959$	−1.35548	−0.0437709
$960$	0	0
$961$	−9.67414	−0.312069
$962$	0	0
$963$	−8.58880	−0.276770
$964$	0	0
$965$	3.92138	0.126234
$966$	0	0
$967$	−30.2649	−0.973254	−0.486627	−	0.873610i	$-0.661773\pi$
$967$	−30.2649	−0.973254	−0.486627	+	0.873610i	$0.661773\pi$
$968$	0	0
$969$	−16.5428	−0.531430
$970$	0	0
$971$	−29.7029	−0.953211	−0.476606	−	0.879117i	$-0.658133\pi$
$971$	−29.7029	−0.953211	−0.476606	+	0.879117i	$0.658133\pi$
$972$	0	0
$973$	21.3532	0.684551
$974$	0	0
$975$	10.3679	0.332040
$976$	0	0
$977$	26.3575	0.843250	0.421625	−	0.906770i	$-0.361460\pi$
$977$	26.3575	0.843250	0.421625	+	0.906770i	$0.361460\pi$
$978$	0	0
$979$	−14.6905	−0.469510
$980$	0	0
$981$	−3.82329	−0.122068
$982$	0	0
$983$	−9.91663	−0.316292	−0.158146	−	0.987416i	$-0.550552\pi$
$983$	−9.91663	−0.316292	−0.158146	+	0.987416i	$0.550552\pi$
$984$	0	0
$985$	−5.93820	−0.189207
$986$	0	0
$987$	0.371722	0.0118320
$988$	0	0
$989$	−26.4144	−0.839930
$990$	0	0
$991$	−0.580644	−0.0184448	−0.00922239	−	0.999957i	$-0.502936\pi$
$991$	−0.580644	−0.0184448	−0.00922239	+	0.999957i	$0.502936\pi$
$992$	0	0
$993$	−10.1755	−0.322910
$994$	0	0
$995$	6.09042	0.193079
$996$	0	0
$997$	33.0420	1.04645	0.523225	−	0.852195i	$-0.324729\pi$
$997$	33.0420	1.04645	0.523225	+	0.852195i	$0.324729\pi$
$998$	0	0
$999$	−20.0127	−0.633175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.19	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.19	✓	24	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)

\(f(q)\)	\(=\)	\(q+1.93554 q^{3} +0.236703 q^{5} +1.77739 q^{7} +0.746320 q^{9} +O(q^{10})\)	\(q+1.93554 q^{3} +0.236703 q^{5} +1.77739 q^{7} +0.746320 q^{9} -1.42951 q^{11} -1.08346 q^{13} +0.458149 q^{15} +1.00000 q^{17} -8.54684 q^{19} +3.44022 q^{21} +4.01824 q^{23} -4.94397 q^{25} -4.36209 q^{27} -3.82086 q^{29} +4.61799 q^{31} -2.76687 q^{33} +0.420714 q^{35} +4.58788 q^{37} -2.09709 q^{39} -10.4605 q^{41} -6.57363 q^{43} +0.176656 q^{45} +0.108052 q^{47} -3.84087 q^{49} +1.93554 q^{51} -2.98723 q^{53} -0.338369 q^{55} -16.5428 q^{57} +1.00000 q^{59} +9.13557 q^{61} +1.32650 q^{63} -0.256459 q^{65} +1.87199 q^{67} +7.77746 q^{69} -11.4708 q^{71} -0.474887 q^{73} -9.56926 q^{75} -2.54079 q^{77} -10.0185 q^{79} -10.6820 q^{81} +0.303718 q^{83} +0.236703 q^{85} -7.39543 q^{87} +10.2766 q^{89} -1.92574 q^{91} +8.93832 q^{93} -2.02306 q^{95} -3.83061 q^{97} -1.06687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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