LMFDB - Embedded newform 8024.2.a.bc.1.5

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:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.77346 q^{3} +1.76876 q^{5} -4.75466 q^{7} +4.69208 q^{9} +O(q^{10})$	$q-2.77346 q^{3} +1.76876 q^{5} -4.75466 q^{7} +4.69208 q^{9} +1.21748 q^{11} +1.19934 q^{13} -4.90559 q^{15} -1.00000 q^{17} -2.51517 q^{19} +13.1869 q^{21} -7.49522 q^{23} -1.87149 q^{25} -4.69293 q^{27} +5.63543 q^{29} -7.04157 q^{31} -3.37663 q^{33} -8.40986 q^{35} +8.00212 q^{37} -3.32633 q^{39} -8.01527 q^{41} +3.67675 q^{43} +8.29917 q^{45} -0.175842 q^{47} +15.6068 q^{49} +2.77346 q^{51} -5.21044 q^{53} +2.15343 q^{55} +6.97572 q^{57} -1.00000 q^{59} -3.24950 q^{61} -22.3093 q^{63} +2.12135 q^{65} +4.05440 q^{67} +20.7877 q^{69} +4.37143 q^{71} -10.5703 q^{73} +5.19050 q^{75} -5.78870 q^{77} -4.84649 q^{79} -1.06061 q^{81} +9.63284 q^{83} -1.76876 q^{85} -15.6297 q^{87} +1.09193 q^{89} -5.70247 q^{91} +19.5295 q^{93} -4.44873 q^{95} -13.9845 q^{97} +5.71251 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−2.77346	−1.60126	−0.800629	−	0.599160i	$-0.795501\pi$
$3$	−2.77346	−1.60126	−0.800629	+	0.599160i	$0.795501\pi$
$4$	0	0
$5$	1.76876	0.791013	0.395507	−	0.918463i	$-0.370569\pi$
$5$	1.76876	0.791013	0.395507	+	0.918463i	$0.370569\pi$
$6$	0	0
$7$	−4.75466	−1.79709	−0.898547	−	0.438878i	$-0.855376\pi$
$7$	−4.75466	−1.79709	−0.898547	+	0.438878i	$0.855376\pi$
$8$	0	0
$9$	4.69208	1.56403
$10$	0	0
$11$	1.21748	0.367084	0.183542	−	0.983012i	$-0.441244\pi$
$11$	1.21748	0.367084	0.183542	+	0.983012i	$0.441244\pi$
$12$	0	0
$13$	1.19934	0.332638	0.166319	−	0.986072i	$-0.446812\pi$
$13$	1.19934	0.332638	0.166319	+	0.986072i	$0.446812\pi$
$14$	0	0
$15$	−4.90559	−1.26662
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.51517	−0.577019	−0.288509	−	0.957477i	$-0.593160\pi$
$19$	−2.51517	−0.577019	−0.288509	+	0.957477i	$0.593160\pi$
$20$	0	0
$21$	13.1869	2.87761
$22$	0	0
$23$	−7.49522	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$23$	−7.49522	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$24$	0	0
$25$	−1.87149	−0.374298
$26$	0	0
$27$	−4.69293	−0.903154
$28$	0	0
$29$	5.63543	1.04647	0.523237	−	0.852187i	$-0.324724\pi$
$29$	5.63543	1.04647	0.523237	+	0.852187i	$0.324724\pi$
$30$	0	0
$31$	−7.04157	−1.26470	−0.632352	−	0.774681i	$-0.717910\pi$
$31$	−7.04157	−1.26470	−0.632352	+	0.774681i	$0.717910\pi$
$32$	0	0
$33$	−3.37663	−0.587796
$34$	0	0
$35$	−8.40986	−1.42153
$36$	0	0
$37$	8.00212	1.31554	0.657771	−	0.753218i	$-0.271500\pi$
$37$	8.00212	1.31554	0.657771	+	0.753218i	$0.271500\pi$
$38$	0	0
$39$	−3.32633	−0.532639
$40$	0	0
$41$	−8.01527	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$41$	−8.01527	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$42$	0	0
$43$	3.67675	0.560699	0.280349	−	0.959898i	$-0.409550\pi$
$43$	3.67675	0.560699	0.280349	+	0.959898i	$0.409550\pi$
$44$	0	0
$45$	8.29917	1.23717
$46$	0	0
$47$	−0.175842	−0.0256491	−0.0128246	−	0.999918i	$-0.504082\pi$
$47$	−0.175842	−0.0256491	−0.0128246	+	0.999918i	$0.504082\pi$
$48$	0	0
$49$	15.6068	2.22955
$50$	0	0
$51$	2.77346	0.388362
$52$	0	0
$53$	−5.21044	−0.715710	−0.357855	−	0.933777i	$-0.616492\pi$
$53$	−5.21044	−0.715710	−0.357855	+	0.933777i	$0.616492\pi$
$54$	0	0
$55$	2.15343	0.290368
$56$	0	0
$57$	6.97572	0.923956
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−3.24950	−0.416056	−0.208028	−	0.978123i	$-0.566705\pi$
$61$	−3.24950	−0.416056	−0.208028	+	0.978123i	$0.566705\pi$
$62$	0	0
$63$	−22.3093	−2.81070
$64$	0	0
$65$	2.12135	0.263121
$66$	0	0
$67$	4.05440	0.495324	0.247662	−	0.968847i	$-0.420338\pi$
$67$	4.05440	0.495324	0.247662	+	0.968847i	$0.420338\pi$
$68$	0	0
$69$	20.7877	2.50255
$70$	0	0
$71$	4.37143	0.518793	0.259397	−	0.965771i	$-0.416476\pi$
$71$	4.37143	0.518793	0.259397	+	0.965771i	$0.416476\pi$
$72$	0	0
$73$	−10.5703	−1.23716	−0.618578	−	0.785723i	$-0.712291\pi$
$73$	−10.5703	−1.23716	−0.618578	+	0.785723i	$0.712291\pi$
$74$	0	0
$75$	5.19050	0.599347
$76$	0	0
$77$	−5.78870	−0.659684
$78$	0	0
$79$	−4.84649	−0.545272	−0.272636	−	0.962117i	$-0.587896\pi$
$79$	−4.84649	−0.545272	−0.272636	+	0.962117i	$0.587896\pi$
$80$	0	0
$81$	−1.06061	−0.117845
$82$	0	0
$83$	9.63284	1.05734	0.528671	−	0.848827i	$-0.322691\pi$
$83$	9.63284	1.05734	0.528671	+	0.848827i	$0.322691\pi$
$84$	0	0
$85$	−1.76876	−0.191849
$86$	0	0
$87$	−15.6297	−1.67567
$88$	0	0
$89$	1.09193	0.115745	0.0578723	−	0.998324i	$-0.481568\pi$
$89$	1.09193	0.115745	0.0578723	+	0.998324i	$0.481568\pi$
$90$	0	0
$91$	−5.70247	−0.597781
$92$	0	0
$93$	19.5295	2.02512
$94$	0	0
$95$	−4.44873	−0.456430
$96$	0	0
$97$	−13.9845	−1.41991	−0.709953	−	0.704249i	$-0.751284\pi$
$97$	−13.9845	−1.41991	−0.709953	+	0.704249i	$0.751284\pi$
$98$	0	0
$99$	5.71251	0.574129
$100$	0	0
$101$	4.87057	0.484640	0.242320	−	0.970196i	$-0.422092\pi$
$101$	4.87057	0.484640	0.242320	+	0.970196i	$0.422092\pi$
$102$	0	0
$103$	19.9719	1.96789	0.983946	−	0.178465i	$-0.0571130\pi$
$103$	19.9719	1.96789	0.983946	+	0.178465i	$0.0571130\pi$
$104$	0	0
$105$	23.3244	2.27623
$106$	0	0
$107$	−9.19235	−0.888658	−0.444329	−	0.895864i	$-0.646558\pi$
$107$	−9.19235	−0.888658	−0.444329	+	0.895864i	$0.646558\pi$
$108$	0	0
$109$	−10.8951	−1.04356	−0.521779	−	0.853081i	$-0.674731\pi$
$109$	−10.8951	−1.04356	−0.521779	+	0.853081i	$0.674731\pi$
$110$	0	0
$111$	−22.1936	−2.10652
$112$	0	0
$113$	−10.3077	−0.969668	−0.484834	−	0.874606i	$-0.661120\pi$
$113$	−10.3077	−0.969668	−0.484834	+	0.874606i	$0.661120\pi$
$114$	0	0
$115$	−13.2572	−1.23624
$116$	0	0
$117$	5.62741	0.520255
$118$	0	0
$119$	4.75466	0.435859
$120$	0	0
$121$	−9.51775	−0.865250
$122$	0	0
$123$	22.2300	2.00441
$124$	0	0
$125$	−12.1540	−1.08709
$126$	0	0
$127$	−12.9240	−1.14682	−0.573412	−	0.819267i	$-0.694380\pi$
$127$	−12.9240	−1.14682	−0.573412	+	0.819267i	$0.694380\pi$
$128$	0	0
$129$	−10.1973	−0.897823
$130$	0	0
$131$	−0.372333	−0.0325309	−0.0162654	−	0.999868i	$-0.505178\pi$
$131$	−0.372333	−0.0325309	−0.0162654	+	0.999868i	$0.505178\pi$
$132$	0	0
$133$	11.9588	1.03696
$134$	0	0
$135$	−8.30066	−0.714407
$136$	0	0
$137$	−10.4262	−0.890768	−0.445384	−	0.895340i	$-0.646933\pi$
$137$	−10.4262	−0.890768	−0.445384	+	0.895340i	$0.646933\pi$
$138$	0	0
$139$	22.9294	1.94485	0.972423	−	0.233223i	$-0.0749273\pi$
$139$	22.9294	1.94485	0.972423	+	0.233223i	$0.0749273\pi$
$140$	0	0
$141$	0.487690	0.0410709
$142$	0	0
$143$	1.46017	0.122106
$144$	0	0
$145$	9.96773	0.827775
$146$	0	0
$147$	−43.2849	−3.57008
$148$	0	0
$149$	14.8086	1.21317	0.606584	−	0.795019i	$-0.292539\pi$
$149$	14.8086	1.21317	0.606584	+	0.795019i	$0.292539\pi$
$150$	0	0
$151$	5.62522	0.457774	0.228887	−	0.973453i	$-0.426491\pi$
$151$	5.62522	0.457774	0.228887	+	0.973453i	$0.426491\pi$
$152$	0	0
$153$	−4.69208	−0.379332
$154$	0	0
$155$	−12.4549	−1.00040
$156$	0	0
$157$	−17.5193	−1.39819	−0.699097	−	0.715026i	$-0.746415\pi$
$157$	−17.5193	−1.39819	−0.699097	+	0.715026i	$0.746415\pi$
$158$	0	0
$159$	14.4510	1.14604
$160$	0	0
$161$	35.6373	2.80861
$162$	0	0
$163$	−10.6375	−0.833194	−0.416597	−	0.909091i	$-0.636777\pi$
$163$	−10.6375	−0.833194	−0.416597	+	0.909091i	$0.636777\pi$
$164$	0	0
$165$	−5.97245	−0.464954
$166$	0	0
$167$	−5.02299	−0.388691	−0.194345	−	0.980933i	$-0.562258\pi$
$167$	−5.02299	−0.388691	−0.194345	+	0.980933i	$0.562258\pi$
$168$	0	0
$169$	−11.5616	−0.889352
$170$	0	0
$171$	−11.8014	−0.902473
$172$	0	0
$173$	−9.95393	−0.756784	−0.378392	−	0.925646i	$-0.623523\pi$
$173$	−9.95393	−0.756784	−0.378392	+	0.925646i	$0.623523\pi$
$174$	0	0
$175$	8.89830	0.672648
$176$	0	0
$177$	2.77346	0.208466
$178$	0	0
$179$	0.798839	0.0597080	0.0298540	−	0.999554i	$-0.490496\pi$
$179$	0.798839	0.0597080	0.0298540	+	0.999554i	$0.490496\pi$
$180$	0	0
$181$	19.6802	1.46282	0.731409	−	0.681940i	$-0.238863\pi$
$181$	19.6802	1.46282	0.731409	+	0.681940i	$0.238863\pi$
$182$	0	0
$183$	9.01237	0.666213
$184$	0	0
$185$	14.1538	1.04061
$186$	0	0
$187$	−1.21748	−0.0890309
$188$	0	0
$189$	22.3133	1.62305
$190$	0	0
$191$	−0.462974	−0.0334996	−0.0167498	−	0.999860i	$-0.505332\pi$
$191$	−0.462974	−0.0334996	−0.0167498	+	0.999860i	$0.505332\pi$
$192$	0	0
$193$	−22.3192	−1.60657	−0.803284	−	0.595596i	$-0.796916\pi$
$193$	−22.3192	−1.60657	−0.803284	+	0.595596i	$0.796916\pi$
$194$	0	0
$195$	−5.88348	−0.421324
$196$	0	0
$197$	−4.58026	−0.326330	−0.163165	−	0.986599i	$-0.552170\pi$
$197$	−4.58026	−0.326330	−0.163165	+	0.986599i	$0.552170\pi$
$198$	0	0
$199$	14.8664	1.05385	0.526927	−	0.849910i	$-0.323344\pi$
$199$	14.8664	1.05385	0.526927	+	0.849910i	$0.323344\pi$
$200$	0	0
$201$	−11.2447	−0.793141
$202$	0	0
$203$	−26.7946	−1.88061
$204$	0	0
$205$	−14.1771	−0.990170
$206$	0	0
$207$	−35.1682	−2.44436
$208$	0	0
$209$	−3.06216	−0.211814
$210$	0	0
$211$	10.9823	0.756050	0.378025	−	0.925795i	$-0.376603\pi$
$211$	10.9823	0.756050	0.378025	+	0.925795i	$0.376603\pi$
$212$	0	0
$213$	−12.1240	−0.830722
$214$	0	0
$215$	6.50328	0.443520
$216$	0	0
$217$	33.4803	2.27279
$218$	0	0
$219$	29.3162	1.98101
$220$	0	0
$221$	−1.19934	−0.0806765
$222$	0	0
$223$	19.3391	1.29504	0.647520	−	0.762048i	$-0.275806\pi$
$223$	19.3391	1.29504	0.647520	+	0.762048i	$0.275806\pi$
$224$	0	0
$225$	−8.78118	−0.585412
$226$	0	0
$227$	−1.31265	−0.0871235	−0.0435618	−	0.999051i	$-0.513871\pi$
$227$	−1.31265	−0.0871235	−0.0435618	+	0.999051i	$0.513871\pi$
$228$	0	0
$229$	−10.7782	−0.712242	−0.356121	−	0.934440i	$-0.615901\pi$
$229$	−10.7782	−0.712242	−0.356121	+	0.934440i	$0.615901\pi$
$230$	0	0
$231$	16.0547	1.05632
$232$	0	0
$233$	22.8626	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$233$	22.8626	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$234$	0	0
$235$	−0.311022	−0.0202888
$236$	0	0
$237$	13.4415	0.873122
$238$	0	0
$239$	10.5966	0.685439	0.342720	−	0.939438i	$-0.388652\pi$
$239$	10.5966	0.685439	0.342720	+	0.939438i	$0.388652\pi$
$240$	0	0
$241$	24.7142	1.59198	0.795992	−	0.605308i	$-0.206950\pi$
$241$	24.7142	1.59198	0.795992	+	0.605308i	$0.206950\pi$
$242$	0	0
$243$	17.0203	1.09185
$244$	0	0
$245$	27.6047	1.76360
$246$	0	0
$247$	−3.01655	−0.191938
$248$	0	0
$249$	−26.7163	−1.69308
$250$	0	0
$251$	25.7440	1.62495	0.812473	−	0.582998i	$-0.198121\pi$
$251$	25.7440	1.62495	0.812473	+	0.582998i	$0.198121\pi$
$252$	0	0
$253$	−9.12527	−0.573701
$254$	0	0
$255$	4.90559	0.307200
$256$	0	0
$257$	16.7052	1.04204	0.521020	−	0.853545i	$-0.325552\pi$
$257$	16.7052	1.04204	0.521020	+	0.853545i	$0.325552\pi$
$258$	0	0
$259$	−38.0474	−2.36415
$260$	0	0
$261$	26.4419	1.63671
$262$	0	0
$263$	−31.2728	−1.92837	−0.964183	−	0.265237i	$-0.914550\pi$
$263$	−31.2728	−1.92837	−0.964183	+	0.265237i	$0.914550\pi$
$264$	0	0
$265$	−9.21602	−0.566136
$266$	0	0
$267$	−3.02843	−0.185337
$268$	0	0
$269$	23.5228	1.43421	0.717106	−	0.696964i	$-0.245466\pi$
$269$	23.5228	1.43421	0.717106	+	0.696964i	$0.245466\pi$
$270$	0	0
$271$	−17.3115	−1.05160	−0.525799	−	0.850609i	$-0.676234\pi$
$271$	−17.3115	−1.05160	−0.525799	+	0.850609i	$0.676234\pi$
$272$	0	0
$273$	15.8156	0.957202
$274$	0	0
$275$	−2.27850	−0.137399
$276$	0	0
$277$	25.2096	1.51470	0.757348	−	0.653011i	$-0.226495\pi$
$277$	25.2096	1.51470	0.757348	+	0.653011i	$0.226495\pi$
$278$	0	0
$279$	−33.0396	−1.97803
$280$	0	0
$281$	−4.61038	−0.275032	−0.137516	−	0.990500i	$-0.543912\pi$
$281$	−4.61038	−0.275032	−0.137516	+	0.990500i	$0.543912\pi$
$282$	0	0
$283$	−0.225816	−0.0134234	−0.00671168	−	0.999977i	$-0.502136\pi$
$283$	−0.225816	−0.0134234	−0.00671168	+	0.999977i	$0.502136\pi$
$284$	0	0
$285$	12.3384	0.730862
$286$	0	0
$287$	38.1099	2.24956
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	38.7853	2.27364
$292$	0	0
$293$	1.77506	0.103700	0.0518500	−	0.998655i	$-0.483488\pi$
$293$	1.77506	0.103700	0.0518500	+	0.998655i	$0.483488\pi$
$294$	0	0
$295$	−1.76876	−0.102981
$296$	0	0
$297$	−5.71354	−0.331533
$298$	0	0
$299$	−8.98934	−0.519867
$300$	0	0
$301$	−17.4817	−1.00763
$302$	0	0
$303$	−13.5083	−0.776033
$304$	0	0
$305$	−5.74759	−0.329106
$306$	0	0
$307$	−5.48525	−0.313060	−0.156530	−	0.987673i	$-0.550031\pi$
$307$	−5.48525	−0.313060	−0.156530	+	0.987673i	$0.550031\pi$
$308$	0	0
$309$	−55.3914	−3.15110
$310$	0	0
$311$	−20.5804	−1.16701	−0.583505	−	0.812110i	$-0.698319\pi$
$311$	−20.5804	−1.16701	−0.583505	+	0.812110i	$0.698319\pi$
$312$	0	0
$313$	19.0852	1.07876	0.539379	−	0.842063i	$-0.318659\pi$
$313$	19.0852	1.07876	0.539379	+	0.842063i	$0.318659\pi$
$314$	0	0
$315$	−39.4597	−2.22330
$316$	0	0
$317$	12.7199	0.714419	0.357210	−	0.934024i	$-0.383728\pi$
$317$	12.7199	0.714419	0.357210	+	0.934024i	$0.383728\pi$
$318$	0	0
$319$	6.86102	0.384143
$320$	0	0
$321$	25.4946	1.42297
$322$	0	0
$323$	2.51517	0.139948
$324$	0	0
$325$	−2.24456	−0.124506
$326$	0	0
$327$	30.2170	1.67101
$328$	0	0
$329$	0.836068	0.0460939
$330$	0	0
$331$	−11.0342	−0.606492	−0.303246	−	0.952912i	$-0.598070\pi$
$331$	−11.0342	−0.606492	−0.303246	+	0.952912i	$0.598070\pi$
$332$	0	0
$333$	37.5466	2.05754
$334$	0	0
$335$	7.17126	0.391808
$336$	0	0
$337$	4.05315	0.220789	0.110395	−	0.993888i	$-0.464789\pi$
$337$	4.05315	0.220789	0.110395	+	0.993888i	$0.464789\pi$
$338$	0	0
$339$	28.5880	1.55269
$340$	0	0
$341$	−8.57297	−0.464252
$342$	0	0
$343$	−40.9225	−2.20961
$344$	0	0
$345$	36.7684	1.97955
$346$	0	0
$347$	11.1143	0.596649	0.298325	−	0.954464i	$-0.403572\pi$
$347$	11.1143	0.596649	0.298325	+	0.954464i	$0.403572\pi$
$348$	0	0
$349$	−32.2030	−1.72379	−0.861894	−	0.507088i	$-0.830722\pi$
$349$	−32.2030	−1.72379	−0.861894	+	0.507088i	$0.830722\pi$
$350$	0	0
$351$	−5.62842	−0.300423
$352$	0	0
$353$	37.0019	1.96941	0.984707	−	0.174221i	$-0.0557406\pi$
$353$	37.0019	1.96941	0.984707	+	0.174221i	$0.0557406\pi$
$354$	0	0
$355$	7.73201	0.410372
$356$	0	0
$357$	−13.1869	−0.697923
$358$	0	0
$359$	−10.6546	−0.562328	−0.281164	−	0.959660i	$-0.590721\pi$
$359$	−10.6546	−0.562328	−0.281164	+	0.959660i	$0.590721\pi$
$360$	0	0
$361$	−12.6739	−0.667049
$362$	0	0
$363$	26.3971	1.38549
$364$	0	0
$365$	−18.6963	−0.978607
$366$	0	0
$367$	−28.9896	−1.51325	−0.756623	−	0.653851i	$-0.773152\pi$
$367$	−28.9896	−1.51325	−0.756623	+	0.653851i	$0.773152\pi$
$368$	0	0
$369$	−37.6083	−1.95781
$370$	0	0
$371$	24.7739	1.28620
$372$	0	0
$373$	−0.741705	−0.0384040	−0.0192020	−	0.999816i	$-0.506113\pi$
$373$	−0.741705	−0.0384040	−0.0192020	+	0.999816i	$0.506113\pi$
$374$	0	0
$375$	33.7087	1.74071
$376$	0	0
$377$	6.75882	0.348097
$378$	0	0
$379$	−29.0958	−1.49455	−0.747275	−	0.664515i	$-0.768638\pi$
$379$	−29.0958	−1.49455	−0.747275	+	0.664515i	$0.768638\pi$
$380$	0	0
$381$	35.8443	1.83636
$382$	0	0
$383$	−10.2239	−0.522419	−0.261209	−	0.965282i	$-0.584121\pi$
$383$	−10.2239	−0.522419	−0.261209	+	0.965282i	$0.584121\pi$
$384$	0	0
$385$	−10.2388	−0.521819
$386$	0	0
$387$	17.2516	0.876948
$388$	0	0
$389$	6.41133	0.325067	0.162534	−	0.986703i	$-0.448033\pi$
$389$	6.41133	0.325067	0.162534	+	0.986703i	$0.448033\pi$
$390$	0	0
$391$	7.49522	0.379050
$392$	0	0
$393$	1.03265	0.0520903
$394$	0	0
$395$	−8.57227	−0.431318
$396$	0	0
$397$	6.65196	0.333852	0.166926	−	0.985969i	$-0.446616\pi$
$397$	6.65196	0.333852	0.166926	+	0.985969i	$0.446616\pi$
$398$	0	0
$399$	−33.1672	−1.66044
$400$	0	0
$401$	2.90023	0.144830	0.0724152	−	0.997375i	$-0.476929\pi$
$401$	2.90023	0.144830	0.0724152	+	0.997375i	$0.476929\pi$
$402$	0	0
$403$	−8.44526	−0.420688
$404$	0	0
$405$	−1.87596	−0.0932171
$406$	0	0
$407$	9.74242	0.482914
$408$	0	0
$409$	19.3064	0.954641	0.477321	−	0.878729i	$-0.341608\pi$
$409$	19.3064	0.954641	0.477321	+	0.878729i	$0.341608\pi$
$410$	0	0
$411$	28.9166	1.42635
$412$	0	0
$413$	4.75466	0.233962
$414$	0	0
$415$	17.0382	0.836372
$416$	0	0
$417$	−63.5938	−3.11420
$418$	0	0
$419$	20.9670	1.02431	0.512153	−	0.858895i	$-0.328848\pi$
$419$	20.9670	1.02431	0.512153	+	0.858895i	$0.328848\pi$
$420$	0	0
$421$	8.02579	0.391153	0.195577	−	0.980688i	$-0.437342\pi$
$421$	8.02579	0.391153	0.195577	+	0.980688i	$0.437342\pi$
$422$	0	0
$423$	−0.825063	−0.0401160
$424$	0	0
$425$	1.87149	0.0907806
$426$	0	0
$427$	15.4503	0.747692
$428$	0	0
$429$	−4.04973	−0.195523
$430$	0	0
$431$	−11.4166	−0.549917	−0.274958	−	0.961456i	$-0.588664\pi$
$431$	−11.4166	−0.549917	−0.274958	+	0.961456i	$0.588664\pi$
$432$	0	0
$433$	3.14065	0.150930	0.0754650	−	0.997148i	$-0.475956\pi$
$433$	3.14065	0.150930	0.0754650	+	0.997148i	$0.475956\pi$
$434$	0	0
$435$	−27.6451	−1.32548
$436$	0	0
$437$	18.8517	0.901801
$438$	0	0
$439$	16.2607	0.776081	0.388040	−	0.921642i	$-0.373152\pi$
$439$	16.2607	0.776081	0.388040	+	0.921642i	$0.373152\pi$
$440$	0	0
$441$	73.2285	3.48707
$442$	0	0
$443$	6.32591	0.300553	0.150277	−	0.988644i	$-0.451984\pi$
$443$	6.32591	0.300553	0.150277	+	0.988644i	$0.451984\pi$
$444$	0	0
$445$	1.93137	0.0915556
$446$	0	0
$447$	−41.0711	−1.94260
$448$	0	0
$449$	36.1713	1.70703	0.853514	−	0.521069i	$-0.174467\pi$
$449$	36.1713	1.70703	0.853514	+	0.521069i	$0.174467\pi$
$450$	0	0
$451$	−9.75842	−0.459506
$452$	0	0
$453$	−15.6013	−0.733014
$454$	0	0
$455$	−10.0863	−0.472853
$456$	0	0
$457$	0.552847	0.0258611	0.0129305	−	0.999916i	$-0.495884\pi$
$457$	0.552847	0.0258611	0.0129305	+	0.999916i	$0.495884\pi$
$458$	0	0
$459$	4.69293	0.219047
$460$	0	0
$461$	40.0253	1.86416	0.932082	−	0.362248i	$-0.117991\pi$
$461$	40.0253	1.86416	0.932082	+	0.362248i	$0.117991\pi$
$462$	0	0
$463$	37.3083	1.73386	0.866932	−	0.498426i	$-0.166088\pi$
$463$	37.3083	1.73386	0.866932	+	0.498426i	$0.166088\pi$
$464$	0	0
$465$	34.5430	1.60189
$466$	0	0
$467$	15.7584	0.729210	0.364605	−	0.931162i	$-0.381204\pi$
$467$	15.7584	0.729210	0.364605	+	0.931162i	$0.381204\pi$
$468$	0	0
$469$	−19.2773	−0.890143
$470$	0	0
$471$	48.5892	2.23887
$472$	0	0
$473$	4.47636	0.205823
$474$	0	0
$475$	4.70711	0.215977
$476$	0	0
$477$	−24.4478	−1.11939
$478$	0	0
$479$	38.6354	1.76530	0.882649	−	0.470033i	$-0.155758\pi$
$479$	38.6354	1.76530	0.882649	+	0.470033i	$0.155758\pi$
$480$	0	0
$481$	9.59729	0.437599
$482$	0	0
$483$	−98.8385	−4.49731
$484$	0	0
$485$	−24.7351	−1.12316
$486$	0	0
$487$	−40.3213	−1.82713	−0.913567	−	0.406688i	$-0.866684\pi$
$487$	−40.3213	−1.82713	−0.913567	+	0.406688i	$0.866684\pi$
$488$	0	0
$489$	29.5027	1.33416
$490$	0	0
$491$	27.9844	1.26292	0.631459	−	0.775409i	$-0.282456\pi$
$491$	27.9844	1.26292	0.631459	+	0.775409i	$0.282456\pi$
$492$	0	0
$493$	−5.63543	−0.253807
$494$	0	0
$495$	10.1041	0.454144
$496$	0	0
$497$	−20.7847	−0.932320
$498$	0	0
$499$	−19.2639	−0.862372	−0.431186	−	0.902263i	$-0.641905\pi$
$499$	−19.2639	−0.862372	−0.431186	+	0.902263i	$0.641905\pi$
$500$	0	0
$501$	13.9311	0.622394
$502$	0	0
$503$	20.8812	0.931045	0.465522	−	0.885036i	$-0.345867\pi$
$503$	20.8812	0.931045	0.465522	+	0.885036i	$0.345867\pi$
$504$	0	0
$505$	8.61486	0.383356
$506$	0	0
$507$	32.0656	1.42408
$508$	0	0
$509$	31.3505	1.38958	0.694792	−	0.719210i	$-0.255496\pi$
$509$	31.3505	1.38958	0.694792	+	0.719210i	$0.255496\pi$
$510$	0	0
$511$	50.2580	2.22329
$512$	0	0
$513$	11.8035	0.521137
$514$	0	0
$515$	35.3255	1.55663
$516$	0	0
$517$	−0.214083	−0.00941538
$518$	0	0
$519$	27.6068	1.21181
$520$	0	0
$521$	13.3415	0.584500	0.292250	−	0.956342i	$-0.405596\pi$
$521$	13.3415	0.584500	0.292250	+	0.956342i	$0.405596\pi$
$522$	0	0
$523$	−6.94119	−0.303517	−0.151759	−	0.988418i	$-0.548494\pi$
$523$	−6.94119	−0.303517	−0.151759	+	0.988418i	$0.548494\pi$
$524$	0	0
$525$	−24.6791	−1.07708
$526$	0	0
$527$	7.04157	0.306736
$528$	0	0
$529$	33.1783	1.44254
$530$	0	0
$531$	−4.69208	−0.203619
$532$	0	0
$533$	−9.61305	−0.416387
$534$	0	0
$535$	−16.2591	−0.702940
$536$	0	0
$537$	−2.21555	−0.0956080
$538$	0	0
$539$	19.0010	0.818430
$540$	0	0
$541$	−27.9269	−1.20067	−0.600336	−	0.799748i	$-0.704966\pi$
$541$	−27.9269	−1.20067	−0.600336	+	0.799748i	$0.704966\pi$
$542$	0	0
$543$	−54.5822	−2.34235
$544$	0	0
$545$	−19.2708	−0.825468
$546$	0	0
$547$	45.5435	1.94730	0.973649	−	0.228052i	$-0.0732357\pi$
$547$	45.5435	1.94730	0.973649	+	0.228052i	$0.0732357\pi$
$548$	0	0
$549$	−15.2469	−0.650723
$550$	0	0
$551$	−14.1741	−0.603835
$552$	0	0
$553$	23.0434	0.979906
$554$	0	0
$555$	−39.2551	−1.66629
$556$	0	0
$557$	−3.46072	−0.146635	−0.0733177	−	0.997309i	$-0.523359\pi$
$557$	−3.46072	−0.146635	−0.0733177	+	0.997309i	$0.523359\pi$
$558$	0	0
$559$	4.40968	0.186510
$560$	0	0
$561$	3.37663	0.142561
$562$	0	0
$563$	−16.1008	−0.678569	−0.339285	−	0.940684i	$-0.610185\pi$
$563$	−16.1008	−0.678569	−0.339285	+	0.940684i	$0.610185\pi$
$564$	0	0
$565$	−18.2319	−0.767020
$566$	0	0
$567$	5.04282	0.211779
$568$	0	0
$569$	8.72550	0.365792	0.182896	−	0.983132i	$-0.441453\pi$
$569$	8.72550	0.365792	0.182896	+	0.983132i	$0.441453\pi$
$570$	0	0
$571$	35.2798	1.47641	0.738206	−	0.674576i	$-0.235673\pi$
$571$	35.2798	1.47641	0.738206	+	0.674576i	$0.235673\pi$
$572$	0	0
$573$	1.28404	0.0536415
$574$	0	0
$575$	14.0272	0.584976
$576$	0	0
$577$	−18.1549	−0.755800	−0.377900	−	0.925846i	$-0.623354\pi$
$577$	−18.1549	−0.755800	−0.377900	+	0.925846i	$0.623354\pi$
$578$	0	0
$579$	61.9013	2.57253
$580$	0	0
$581$	−45.8009	−1.90014
$582$	0	0
$583$	−6.34360	−0.262725
$584$	0	0
$585$	9.95354	0.411528
$586$	0	0
$587$	7.77587	0.320945	0.160472	−	0.987040i	$-0.448698\pi$
$587$	7.77587	0.320945	0.160472	+	0.987040i	$0.448698\pi$
$588$	0	0
$589$	17.7107	0.729758
$590$	0	0
$591$	12.7032	0.522538
$592$	0	0
$593$	9.51942	0.390916	0.195458	−	0.980712i	$-0.437381\pi$
$593$	9.51942	0.390916	0.195458	+	0.980712i	$0.437381\pi$
$594$	0	0
$595$	8.40986	0.344771
$596$	0	0
$597$	−41.2315	−1.68749
$598$	0	0
$599$	41.9700	1.71485	0.857424	−	0.514610i	$-0.172063\pi$
$599$	41.9700	1.71485	0.857424	+	0.514610i	$0.172063\pi$
$600$	0	0
$601$	−9.61349	−0.392142	−0.196071	−	0.980590i	$-0.562818\pi$
$601$	−9.61349	−0.392142	−0.196071	+	0.980590i	$0.562818\pi$
$602$	0	0
$603$	19.0236	0.774700
$604$	0	0
$605$	−16.8346	−0.684424
$606$	0	0
$607$	14.0847	0.571679	0.285839	−	0.958278i	$-0.407728\pi$
$607$	14.0847	0.571679	0.285839	+	0.958278i	$0.407728\pi$
$608$	0	0
$609$	74.3137	3.01134
$610$	0	0
$611$	−0.210894	−0.00853187
$612$	0	0
$613$	22.1892	0.896212	0.448106	−	0.893980i	$-0.352099\pi$
$613$	22.1892	0.896212	0.448106	+	0.893980i	$0.352099\pi$
$614$	0	0
$615$	39.3196	1.58552
$616$	0	0
$617$	11.5295	0.464160	0.232080	−	0.972697i	$-0.425447\pi$
$617$	11.5295	0.464160	0.232080	+	0.972697i	$0.425447\pi$
$618$	0	0
$619$	21.6164	0.868835	0.434417	−	0.900712i	$-0.356954\pi$
$619$	21.6164	0.868835	0.434417	+	0.900712i	$0.356954\pi$
$620$	0	0
$621$	35.1745	1.41150
$622$	0	0
$623$	−5.19177	−0.208004
$624$	0	0
$625$	−12.1401	−0.485603
$626$	0	0
$627$	8.49279	0.339169
$628$	0	0
$629$	−8.00212	−0.319066
$630$	0	0
$631$	−4.97929	−0.198222	−0.0991112	−	0.995076i	$-0.531600\pi$
$631$	−4.97929	−0.198222	−0.0991112	+	0.995076i	$0.531600\pi$
$632$	0	0
$633$	−30.4589	−1.21063
$634$	0	0
$635$	−22.8595	−0.907153
$636$	0	0
$637$	18.7179	0.741631
$638$	0	0
$639$	20.5111	0.811407
$640$	0	0
$641$	29.8675	1.17970	0.589848	−	0.807514i	$-0.299188\pi$
$641$	29.8675	1.17970	0.589848	+	0.807514i	$0.299188\pi$
$642$	0	0
$643$	−25.7765	−1.01652	−0.508262	−	0.861202i	$-0.669712\pi$
$643$	−25.7765	−1.01652	−0.508262	+	0.861202i	$0.669712\pi$
$644$	0	0
$645$	−18.0366	−0.710190
$646$	0	0
$647$	−32.8066	−1.28976	−0.644880	−	0.764284i	$-0.723093\pi$
$647$	−32.8066	−1.28976	−0.644880	+	0.764284i	$0.723093\pi$
$648$	0	0
$649$	−1.21748	−0.0477902
$650$	0	0
$651$	−92.8563	−3.63933
$652$	0	0
$653$	−43.4335	−1.69968	−0.849842	−	0.527038i	$-0.823303\pi$
$653$	−43.4335	−1.69968	−0.849842	+	0.527038i	$0.823303\pi$
$654$	0	0
$655$	−0.658567	−0.0257323
$656$	0	0
$657$	−49.5966	−1.93495
$658$	0	0
$659$	5.28798	0.205990	0.102995	−	0.994682i	$-0.467157\pi$
$659$	5.28798	0.205990	0.102995	+	0.994682i	$0.467157\pi$
$660$	0	0
$661$	46.8103	1.82071	0.910354	−	0.413830i	$-0.135809\pi$
$661$	46.8103	1.82071	0.910354	+	0.413830i	$0.135809\pi$
$662$	0	0
$663$	3.32633	0.129184
$664$	0	0
$665$	21.1522	0.820247
$666$	0	0
$667$	−42.2388	−1.63549
$668$	0	0
$669$	−53.6362	−2.07369
$670$	0	0
$671$	−3.95620	−0.152727
$672$	0	0
$673$	−5.31029	−0.204697	−0.102348	−	0.994749i	$-0.532636\pi$
$673$	−5.31029	−0.204697	−0.102348	+	0.994749i	$0.532636\pi$
$674$	0	0
$675$	8.78276	0.338049
$676$	0	0
$677$	−11.2867	−0.433782	−0.216891	−	0.976196i	$-0.569592\pi$
$677$	−11.2867	−0.433782	−0.216891	+	0.976196i	$0.569592\pi$
$678$	0	0
$679$	66.4914	2.55170
$680$	0	0
$681$	3.64058	0.139507
$682$	0	0
$683$	−26.6519	−1.01981	−0.509904	−	0.860231i	$-0.670319\pi$
$683$	−26.6519	−1.01981	−0.509904	+	0.860231i	$0.670319\pi$
$684$	0	0
$685$	−18.4414	−0.704610
$686$	0	0
$687$	29.8929	1.14048
$688$	0	0
$689$	−6.24911	−0.238072
$690$	0	0
$691$	13.2620	0.504509	0.252254	−	0.967661i	$-0.418828\pi$
$691$	13.2620	0.504509	0.252254	+	0.967661i	$0.418828\pi$
$692$	0	0
$693$	−27.1611	−1.03176
$694$	0	0
$695$	40.5566	1.53840
$696$	0	0
$697$	8.01527	0.303600
$698$	0	0
$699$	−63.4085	−2.39833
$700$	0	0
$701$	13.4306	0.507267	0.253633	−	0.967300i	$-0.418374\pi$
$701$	13.4306	0.507267	0.253633	+	0.967300i	$0.418374\pi$
$702$	0	0
$703$	−20.1267	−0.759092
$704$	0	0
$705$	0.862606	0.0324876
$706$	0	0
$707$	−23.1579	−0.870943
$708$	0	0
$709$	−32.2926	−1.21278	−0.606388	−	0.795169i	$-0.707382\pi$
$709$	−32.2926	−1.21278	−0.606388	+	0.795169i	$0.707382\pi$
$710$	0	0
$711$	−22.7401	−0.852821
$712$	0	0
$713$	52.7782	1.97656
$714$	0	0
$715$	2.58270	0.0965874
$716$	0	0
$717$	−29.3893	−1.09757
$718$	0	0
$719$	11.2336	0.418942	0.209471	−	0.977815i	$-0.432826\pi$
$719$	11.2336	0.418942	0.209471	+	0.977815i	$0.432826\pi$
$720$	0	0
$721$	−94.9598	−3.53649
$722$	0	0
$723$	−68.5439	−2.54918
$724$	0	0
$725$	−10.5467	−0.391693
$726$	0	0
$727$	13.8057	0.512026	0.256013	−	0.966673i	$-0.417591\pi$
$727$	13.8057	0.512026	0.256013	+	0.966673i	$0.417591\pi$
$728$	0	0
$729$	−44.0234	−1.63050
$730$	0	0
$731$	−3.67675	−0.135989
$732$	0	0
$733$	−34.5554	−1.27633	−0.638166	−	0.769899i	$-0.720307\pi$
$733$	−34.5554	−1.27633	−0.638166	+	0.769899i	$0.720307\pi$
$734$	0	0
$735$	−76.5606	−2.82398
$736$	0	0
$737$	4.93614	0.181825
$738$	0	0
$739$	43.9025	1.61498	0.807489	−	0.589882i	$-0.200826\pi$
$739$	43.9025	1.61498	0.807489	+	0.589882i	$0.200826\pi$
$740$	0	0
$741$	8.36627	0.307343
$742$	0	0
$743$	−16.6221	−0.609805	−0.304903	−	0.952384i	$-0.598624\pi$
$743$	−16.6221	−0.609805	−0.304903	+	0.952384i	$0.598624\pi$
$744$	0	0
$745$	26.1929	0.959633
$746$	0	0
$747$	45.1981	1.65371
$748$	0	0
$749$	43.7065	1.59700
$750$	0	0
$751$	23.3059	0.850444	0.425222	−	0.905089i	$-0.360196\pi$
$751$	23.3059	0.850444	0.425222	+	0.905089i	$0.360196\pi$
$752$	0	0
$753$	−71.4000	−2.60196
$754$	0	0
$755$	9.94966	0.362105
$756$	0	0
$757$	31.0414	1.12822	0.564110	−	0.825700i	$-0.309219\pi$
$757$	31.0414	1.12822	0.564110	+	0.825700i	$0.309219\pi$
$758$	0	0
$759$	25.3086	0.918643
$760$	0	0
$761$	28.7633	1.04267	0.521335	−	0.853352i	$-0.325434\pi$
$761$	28.7633	1.04267	0.521335	+	0.853352i	$0.325434\pi$
$762$	0	0
$763$	51.8024	1.87537
$764$	0	0
$765$	−8.29917	−0.300057
$766$	0	0
$767$	−1.19934	−0.0433057
$768$	0	0
$769$	12.2045	0.440107	0.220054	−	0.975488i	$-0.429377\pi$
$769$	12.2045	0.440107	0.220054	+	0.975488i	$0.429377\pi$
$770$	0	0
$771$	−46.3311	−1.66857
$772$	0	0
$773$	−41.9662	−1.50942	−0.754710	−	0.656059i	$-0.772222\pi$
$773$	−41.9662	−1.50942	−0.754710	+	0.656059i	$0.772222\pi$
$774$	0	0
$775$	13.1782	0.473376
$776$	0	0
$777$	105.523	3.78562
$778$	0	0
$779$	20.1597	0.722297
$780$	0	0
$781$	5.32212	0.190441
$782$	0	0
$783$	−26.4467	−0.945127
$784$	0	0
$785$	−30.9875	−1.10599
$786$	0	0
$787$	3.13364	0.111702	0.0558511	−	0.998439i	$-0.482213\pi$
$787$	3.13364	0.111702	0.0558511	+	0.998439i	$0.482213\pi$
$788$	0	0
$789$	86.7340	3.08781
$790$	0	0
$791$	49.0097	1.74258
$792$	0	0
$793$	−3.89727	−0.138396
$794$	0	0
$795$	25.5603	0.906530
$796$	0	0
$797$	12.6514	0.448135	0.224067	−	0.974574i	$-0.428066\pi$
$797$	12.6514	0.448135	0.224067	+	0.974574i	$0.428066\pi$
$798$	0	0
$799$	0.175842	0.00622083
$800$	0	0
$801$	5.12344	0.181028
$802$	0	0
$803$	−12.8691	−0.454140
$804$	0	0
$805$	63.0337	2.22165
$806$	0	0
$807$	−65.2397	−2.29654
$808$	0	0
$809$	51.7669	1.82003	0.910014	−	0.414577i	$-0.136071\pi$
$809$	51.7669	1.82003	0.910014	+	0.414577i	$0.136071\pi$
$810$	0	0
$811$	43.8247	1.53889	0.769446	−	0.638712i	$-0.220532\pi$
$811$	43.8247	1.53889	0.769446	+	0.638712i	$0.220532\pi$
$812$	0	0
$813$	48.0127	1.68388
$814$	0	0
$815$	−18.8152	−0.659068
$816$	0	0
$817$	−9.24763	−0.323534
$818$	0	0
$819$	−26.7565	−0.934946
$820$	0	0
$821$	22.6334	0.789911	0.394955	−	0.918700i	$-0.370760\pi$
$821$	22.6334	0.789911	0.394955	+	0.918700i	$0.370760\pi$
$822$	0	0
$823$	−14.3176	−0.499081	−0.249540	−	0.968364i	$-0.580280\pi$
$823$	−14.3176	−0.499081	−0.249540	+	0.968364i	$0.580280\pi$
$824$	0	0
$825$	6.31932	0.220011
$826$	0	0
$827$	9.79248	0.340518	0.170259	−	0.985399i	$-0.445540\pi$
$827$	9.79248	0.340518	0.170259	+	0.985399i	$0.445540\pi$
$828$	0	0
$829$	−0.568503	−0.0197449	−0.00987245	−	0.999951i	$-0.503143\pi$
$829$	−0.568503	−0.0197449	−0.00987245	+	0.999951i	$0.503143\pi$
$830$	0	0
$831$	−69.9177	−2.42542
$832$	0	0
$833$	−15.6068	−0.540744
$834$	0	0
$835$	−8.88446	−0.307460
$836$	0	0
$837$	33.0456	1.14222
$838$	0	0
$839$	−34.0237	−1.17463	−0.587314	−	0.809359i	$-0.699815\pi$
$839$	−34.0237	−1.17463	−0.587314	+	0.809359i	$0.699815\pi$
$840$	0	0
$841$	2.75812	0.0951076
$842$	0	0
$843$	12.7867	0.440397
$844$	0	0
$845$	−20.4497	−0.703489
$846$	0	0
$847$	45.2537	1.55493
$848$	0	0
$849$	0.626292	0.0214943
$850$	0	0
$851$	−59.9777	−2.05601
$852$	0	0
$853$	−18.2927	−0.626328	−0.313164	−	0.949699i	$-0.601389\pi$
$853$	−18.2927	−0.626328	−0.313164	+	0.949699i	$0.601389\pi$
$854$	0	0
$855$	−20.8738	−0.713869
$856$	0	0
$857$	−29.4791	−1.00699	−0.503494	−	0.863999i	$-0.667952\pi$
$857$	−29.4791	−1.00699	−0.503494	+	0.863999i	$0.667952\pi$
$858$	0	0
$859$	−5.21729	−0.178012	−0.0890059	−	0.996031i	$-0.528369\pi$
$859$	−5.21729	−0.178012	−0.0890059	+	0.996031i	$0.528369\pi$
$860$	0	0
$861$	−105.696	−3.60212
$862$	0	0
$863$	37.6873	1.28289	0.641446	−	0.767168i	$-0.278335\pi$
$863$	37.6873	1.28289	0.641446	+	0.767168i	$0.278335\pi$
$864$	0	0
$865$	−17.6061	−0.598626
$866$	0	0
$867$	−2.77346	−0.0941917
$868$	0	0
$869$	−5.90050	−0.200161
$870$	0	0
$871$	4.86261	0.164763
$872$	0	0
$873$	−65.6162	−2.22077
$874$	0	0
$875$	57.7882	1.95360
$876$	0	0
$877$	29.4832	0.995576	0.497788	−	0.867299i	$-0.334146\pi$
$877$	29.4832	0.995576	0.497788	+	0.867299i	$0.334146\pi$
$878$	0	0
$879$	−4.92306	−0.166051
$880$	0	0
$881$	−25.7876	−0.868807	−0.434403	−	0.900718i	$-0.643041\pi$
$881$	−25.7876	−0.868807	−0.434403	+	0.900718i	$0.643041\pi$
$882$	0	0
$883$	−9.08521	−0.305742	−0.152871	−	0.988246i	$-0.548852\pi$
$883$	−9.08521	−0.305742	−0.152871	+	0.988246i	$0.548852\pi$
$884$	0	0
$885$	4.90559	0.164899
$886$	0	0
$887$	−41.0686	−1.37895	−0.689474	−	0.724310i	$-0.742158\pi$
$887$	−41.0686	−1.37895	−0.689474	+	0.724310i	$0.742158\pi$
$888$	0	0
$889$	61.4495	2.06095
$890$	0	0
$891$	−1.29127	−0.0432590
$892$	0	0
$893$	0.442271	0.0148000
$894$	0	0
$895$	1.41295	0.0472299
$896$	0	0
$897$	24.9316	0.832441
$898$	0	0
$899$	−39.6823	−1.32348
$900$	0	0
$901$	5.21044	0.173585
$902$	0	0
$903$	48.4848	1.61347
$904$	0	0
$905$	34.8095	1.15711
$906$	0	0
$907$	−2.50348	−0.0831268	−0.0415634	−	0.999136i	$-0.513234\pi$
$907$	−2.50348	−0.0831268	−0.0415634	+	0.999136i	$0.513234\pi$
$908$	0	0
$909$	22.8531	0.757990
$910$	0	0
$911$	−26.7405	−0.885953	−0.442976	−	0.896533i	$-0.646077\pi$
$911$	−26.7405	−0.885953	−0.442976	+	0.896533i	$0.646077\pi$
$912$	0	0
$913$	11.7278	0.388133
$914$	0	0
$915$	15.9407	0.526984
$916$	0	0
$917$	1.77032	0.0584610
$918$	0	0
$919$	23.0117	0.759085	0.379543	−	0.925174i	$-0.376081\pi$
$919$	23.0117	0.759085	0.379543	+	0.925174i	$0.376081\pi$
$920$	0	0
$921$	15.2131	0.501289
$922$	0	0
$923$	5.24284	0.172570
$924$	0	0
$925$	−14.9759	−0.492404
$926$	0	0
$927$	93.7100	3.07784
$928$	0	0
$929$	−40.4233	−1.32625	−0.663123	−	0.748510i	$-0.730770\pi$
$929$	−40.4233	−1.32625	−0.663123	+	0.748510i	$0.730770\pi$
$930$	0	0
$931$	−39.2538	−1.28649
$932$	0	0
$933$	57.0790	1.86868
$934$	0	0
$935$	−2.15343	−0.0704246
$936$	0	0
$937$	0.786420	0.0256912	0.0128456	−	0.999917i	$-0.495911\pi$
$937$	0.786420	0.0256912	0.0128456	+	0.999917i	$0.495911\pi$
$938$	0	0
$939$	−52.9320	−1.72737
$940$	0	0
$941$	−36.7394	−1.19767	−0.598835	−	0.800872i	$-0.704370\pi$
$941$	−36.7394	−1.19767	−0.598835	+	0.800872i	$0.704370\pi$
$942$	0	0
$943$	60.0762	1.95635
$944$	0	0
$945$	39.4668	1.28386
$946$	0	0
$947$	−31.9815	−1.03926	−0.519630	−	0.854391i	$-0.673930\pi$
$947$	−31.9815	−1.03926	−0.519630	+	0.854391i	$0.673930\pi$
$948$	0	0
$949$	−12.6774	−0.411525
$950$	0	0
$951$	−35.2781	−1.14397
$952$	0	0
$953$	−0.977957	−0.0316791	−0.0158396	−	0.999875i	$-0.505042\pi$
$953$	−0.977957	−0.0316791	−0.0158396	+	0.999875i	$0.505042\pi$
$954$	0	0
$955$	−0.818889	−0.0264986
$956$	0	0
$957$	−19.0288	−0.615113
$958$	0	0
$959$	49.5729	1.60079
$960$	0	0
$961$	18.5837	0.599476
$962$	0	0
$963$	−43.1313	−1.38989
$964$	0	0
$965$	−39.4772	−1.27082
$966$	0	0
$967$	55.0501	1.77029	0.885145	−	0.465315i	$-0.154059\pi$
$967$	55.0501	1.77029	0.885145	+	0.465315i	$0.154059\pi$
$968$	0	0
$969$	−6.97572	−0.224092
$970$	0	0
$971$	31.2794	1.00380	0.501902	−	0.864924i	$-0.332634\pi$
$971$	31.2794	1.00380	0.501902	+	0.864924i	$0.332634\pi$
$972$	0	0
$973$	−109.022	−3.49507
$974$	0	0
$975$	6.22519	0.199366
$976$	0	0
$977$	−42.1592	−1.34879	−0.674396	−	0.738370i	$-0.735596\pi$
$977$	−42.1592	−1.34879	−0.674396	+	0.738370i	$0.735596\pi$
$978$	0	0
$979$	1.32941	0.0424880
$980$	0	0
$981$	−51.1206	−1.63215
$982$	0	0
$983$	−11.2607	−0.359160	−0.179580	−	0.983743i	$-0.557474\pi$
$983$	−11.2607	−0.359160	−0.179580	+	0.983743i	$0.557474\pi$
$984$	0	0
$985$	−8.10137	−0.258131
$986$	0	0
$987$	−2.31880	−0.0738082
$988$	0	0
$989$	−27.5580	−0.876295
$990$	0	0
$991$	−16.7613	−0.532439	−0.266219	−	0.963912i	$-0.585775\pi$
$991$	−16.7613	−0.532439	−0.266219	+	0.963912i	$0.585775\pi$
$992$	0	0
$993$	30.6028	0.971150
$994$	0	0
$995$	26.2952	0.833613
$996$	0	0
$997$	35.8846	1.13648	0.568238	−	0.822864i	$-0.307625\pi$
$997$	35.8846	1.13648	0.568238	+	0.822864i	$0.307625\pi$
$998$	0	0
$999$	−37.5534	−1.18814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.5	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.5	✓	33	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.77346 q^{3} +1.76876 q^{5} -4.75466 q^{7} +4.69208 q^{9} +O(q^{10})\)	\(q-2.77346 q^{3} +1.76876 q^{5} -4.75466 q^{7} +4.69208 q^{9} +1.21748 q^{11} +1.19934 q^{13} -4.90559 q^{15} -1.00000 q^{17} -2.51517 q^{19} +13.1869 q^{21} -7.49522 q^{23} -1.87149 q^{25} -4.69293 q^{27} +5.63543 q^{29} -7.04157 q^{31} -3.37663 q^{33} -8.40986 q^{35} +8.00212 q^{37} -3.32633 q^{39} -8.01527 q^{41} +3.67675 q^{43} +8.29917 q^{45} -0.175842 q^{47} +15.6068 q^{49} +2.77346 q^{51} -5.21044 q^{53} +2.15343 q^{55} +6.97572 q^{57} -1.00000 q^{59} -3.24950 q^{61} -22.3093 q^{63} +2.12135 q^{65} +4.05440 q^{67} +20.7877 q^{69} +4.37143 q^{71} -10.5703 q^{73} +5.19050 q^{75} -5.78870 q^{77} -4.84649 q^{79} -1.06061 q^{81} +9.63284 q^{83} -1.76876 q^{85} -15.6297 q^{87} +1.09193 q^{89} -5.70247 q^{91} +19.5295 q^{93} -4.44873 q^{95} -13.9845 q^{97} +5.71251 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

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