LMFDB - Embedded newform 8024.2.a.x.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:	$22$
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)

$f(q)$	$=$	$q+2.29019 q^{3} +0.921405 q^{5} -4.02395 q^{7} +2.24495 q^{9} +O(q^{10})$	$q+2.29019 q^{3} +0.921405 q^{5} -4.02395 q^{7} +2.24495 q^{9} -1.08425 q^{11} -2.96543 q^{13} +2.11019 q^{15} -1.00000 q^{17} +6.49971 q^{19} -9.21559 q^{21} +7.97535 q^{23} -4.15101 q^{25} -1.72921 q^{27} +4.72654 q^{29} -5.00678 q^{31} -2.48314 q^{33} -3.70769 q^{35} +10.2510 q^{37} -6.79139 q^{39} -9.32311 q^{41} -10.9408 q^{43} +2.06851 q^{45} -1.14710 q^{47} +9.19218 q^{49} -2.29019 q^{51} -3.37966 q^{53} -0.999036 q^{55} +14.8855 q^{57} +1.00000 q^{59} -11.7199 q^{61} -9.03356 q^{63} -2.73237 q^{65} -2.70640 q^{67} +18.2650 q^{69} -10.1919 q^{71} +0.737506 q^{73} -9.50659 q^{75} +4.36298 q^{77} -5.24791 q^{79} -10.6951 q^{81} -10.9501 q^{83} -0.921405 q^{85} +10.8247 q^{87} -5.18284 q^{89} +11.9328 q^{91} -11.4665 q^{93} +5.98887 q^{95} +17.3578 q^{97} -2.43409 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	2.29019	1.32224	0.661120	−	0.750281i	$-0.270082\pi$
$3$	2.29019	1.32224	0.661120	+	0.750281i	$0.270082\pi$
$4$	0	0
$5$	0.921405	0.412065	0.206033	−	0.978545i	$-0.433945\pi$
$5$	0.921405	0.412065	0.206033	+	0.978545i	$0.433945\pi$
$6$	0	0
$7$	−4.02395	−1.52091	−0.760455	−	0.649390i	$-0.775024\pi$
$7$	−4.02395	−1.52091	−0.760455	+	0.649390i	$0.775024\pi$
$8$	0	0
$9$	2.24495	0.748316
$10$	0	0
$11$	−1.08425	−0.326914	−0.163457	−	0.986550i	$-0.552265\pi$
$11$	−1.08425	−0.326914	−0.163457	+	0.986550i	$0.552265\pi$
$12$	0	0
$13$	−2.96543	−0.822464	−0.411232	−	0.911531i	$-0.634901\pi$
$13$	−2.96543	−0.822464	−0.411232	+	0.911531i	$0.634901\pi$
$14$	0	0
$15$	2.11019	0.544849
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.49971	1.49114	0.745568	−	0.666430i	$-0.232178\pi$
$19$	6.49971	1.49114	0.745568	+	0.666430i	$0.232178\pi$
$20$	0	0
$21$	−9.21559	−2.01101
$22$	0	0
$23$	7.97535	1.66298	0.831488	−	0.555543i	$-0.187490\pi$
$23$	7.97535	1.66298	0.831488	+	0.555543i	$0.187490\pi$
$24$	0	0
$25$	−4.15101	−0.830202
$26$	0	0
$27$	−1.72921	−0.332786
$28$	0	0
$29$	4.72654	0.877697	0.438848	−	0.898561i	$-0.355387\pi$
$29$	4.72654	0.877697	0.438848	+	0.898561i	$0.355387\pi$
$30$	0	0
$31$	−5.00678	−0.899245	−0.449622	−	0.893219i	$-0.648441\pi$
$31$	−5.00678	−0.899245	−0.449622	+	0.893219i	$0.648441\pi$
$32$	0	0
$33$	−2.48314	−0.432259
$34$	0	0
$35$	−3.70769	−0.626714
$36$	0	0
$37$	10.2510	1.68525	0.842626	−	0.538499i	$-0.181009\pi$
$37$	10.2510	1.68525	0.842626	+	0.538499i	$0.181009\pi$
$38$	0	0
$39$	−6.79139	−1.08749
$40$	0	0
$41$	−9.32311	−1.45603	−0.728013	−	0.685564i	$-0.759556\pi$
$41$	−9.32311	−1.45603	−0.728013	+	0.685564i	$0.759556\pi$
$42$	0	0
$43$	−10.9408	−1.66846	−0.834232	−	0.551414i	$-0.814089\pi$
$43$	−10.9408	−1.66846	−0.834232	+	0.551414i	$0.814089\pi$
$44$	0	0
$45$	2.06851	0.308355
$46$	0	0
$47$	−1.14710	−0.167322	−0.0836610	−	0.996494i	$-0.526661\pi$
$47$	−1.14710	−0.167322	−0.0836610	+	0.996494i	$0.526661\pi$
$48$	0	0
$49$	9.19218	1.31317
$50$	0	0
$51$	−2.29019	−0.320690
$52$	0	0
$53$	−3.37966	−0.464232	−0.232116	−	0.972688i	$-0.574565\pi$
$53$	−3.37966	−0.464232	−0.232116	+	0.972688i	$0.574565\pi$
$54$	0	0
$55$	−0.999036	−0.134710
$56$	0	0
$57$	14.8855	1.97164
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−11.7199	−1.50057	−0.750287	−	0.661113i	$-0.770085\pi$
$61$	−11.7199	−1.50057	−0.750287	+	0.661113i	$0.770085\pi$
$62$	0	0
$63$	−9.03356	−1.13812
$64$	0	0
$65$	−2.73237	−0.338908
$66$	0	0
$67$	−2.70640	−0.330639	−0.165319	−	0.986240i	$-0.552866\pi$
$67$	−2.70640	−0.330639	−0.165319	+	0.986240i	$0.552866\pi$
$68$	0	0
$69$	18.2650	2.19885
$70$	0	0
$71$	−10.1919	−1.20955	−0.604776	−	0.796395i	$-0.706737\pi$
$71$	−10.1919	−1.20955	−0.604776	+	0.796395i	$0.706737\pi$
$72$	0	0
$73$	0.737506	0.0863186	0.0431593	−	0.999068i	$-0.486258\pi$
$73$	0.737506	0.0863186	0.0431593	+	0.999068i	$0.486258\pi$
$74$	0	0
$75$	−9.50659	−1.09773
$76$	0	0
$77$	4.36298	0.497207
$78$	0	0
$79$	−5.24791	−0.590436	−0.295218	−	0.955430i	$-0.595392\pi$
$79$	−5.24791	−0.590436	−0.295218	+	0.955430i	$0.595392\pi$
$80$	0	0
$81$	−10.6951	−1.18834
$82$	0	0
$83$	−10.9501	−1.20193	−0.600964	−	0.799276i	$-0.705217\pi$
$83$	−10.9501	−1.20193	−0.600964	+	0.799276i	$0.705217\pi$
$84$	0	0
$85$	−0.921405	−0.0999405
$86$	0	0
$87$	10.8247	1.16052
$88$	0	0
$89$	−5.18284	−0.549379	−0.274690	−	0.961533i	$-0.588575\pi$
$89$	−5.18284	−0.549379	−0.274690	+	0.961533i	$0.588575\pi$
$90$	0	0
$91$	11.9328	1.25089
$92$	0	0
$93$	−11.4665	−1.18902
$94$	0	0
$95$	5.98887	0.614445
$96$	0	0
$97$	17.3578	1.76241	0.881207	−	0.472731i	$-0.156732\pi$
$97$	17.3578	1.76241	0.881207	+	0.472731i	$0.156732\pi$
$98$	0	0
$99$	−2.43409	−0.244635
$100$	0	0
$101$	7.49521	0.745801	0.372901	−	0.927871i	$-0.378363\pi$
$101$	7.49521	0.745801	0.372901	+	0.927871i	$0.378363\pi$
$102$	0	0
$103$	5.21597	0.513944	0.256972	−	0.966419i	$-0.417275\pi$
$103$	5.21597	0.513944	0.256972	+	0.966419i	$0.417275\pi$
$104$	0	0
$105$	−8.49130	−0.828666
$106$	0	0
$107$	−1.36363	−0.131827	−0.0659136	−	0.997825i	$-0.520996\pi$
$107$	−1.36363	−0.131827	−0.0659136	+	0.997825i	$0.520996\pi$
$108$	0	0
$109$	−12.2879	−1.17697	−0.588484	−	0.808509i	$-0.700275\pi$
$109$	−12.2879	−1.17697	−0.588484	+	0.808509i	$0.700275\pi$
$110$	0	0
$111$	23.4767	2.22831
$112$	0	0
$113$	−8.97543	−0.844337	−0.422169	−	0.906517i	$-0.638731\pi$
$113$	−8.97543	−0.844337	−0.422169	+	0.906517i	$0.638731\pi$
$114$	0	0
$115$	7.34853	0.685254
$116$	0	0
$117$	−6.65725	−0.615463
$118$	0	0
$119$	4.02395	0.368875
$120$	0	0
$121$	−9.82440	−0.893127
$122$	0	0
$123$	−21.3517	−1.92521
$124$	0	0
$125$	−8.43179	−0.754162
$126$	0	0
$127$	−12.9627	−1.15025	−0.575127	−	0.818064i	$-0.695047\pi$
$127$	−12.9627	−1.15025	−0.575127	+	0.818064i	$0.695047\pi$
$128$	0	0
$129$	−25.0566	−2.20611
$130$	0	0
$131$	2.55135	0.222913	0.111456	−	0.993769i	$-0.464448\pi$
$131$	2.55135	0.222913	0.111456	+	0.993769i	$0.464448\pi$
$132$	0	0
$133$	−26.1545	−2.26788
$134$	0	0
$135$	−1.59330	−0.137130
$136$	0	0
$137$	−3.89017	−0.332360	−0.166180	−	0.986095i	$-0.553143\pi$
$137$	−3.89017	−0.332360	−0.166180	+	0.986095i	$0.553143\pi$
$138$	0	0
$139$	17.8361	1.51283	0.756417	−	0.654089i	$-0.226948\pi$
$139$	17.8361	1.51283	0.756417	+	0.654089i	$0.226948\pi$
$140$	0	0
$141$	−2.62708	−0.221240
$142$	0	0
$143$	3.21528	0.268875
$144$	0	0
$145$	4.35506	0.361668
$146$	0	0
$147$	21.0518	1.73632
$148$	0	0
$149$	−6.91282	−0.566320	−0.283160	−	0.959073i	$-0.591383\pi$
$149$	−6.91282	−0.566320	−0.283160	+	0.959073i	$0.591383\pi$
$150$	0	0
$151$	7.71235	0.627622	0.313811	−	0.949485i	$-0.398394\pi$
$151$	7.71235	0.627622	0.313811	+	0.949485i	$0.398394\pi$
$152$	0	0
$153$	−2.24495	−0.181493
$154$	0	0
$155$	−4.61328	−0.370547
$156$	0	0
$157$	11.3161	0.903126	0.451563	−	0.892239i	$-0.350867\pi$
$157$	11.3161	0.903126	0.451563	+	0.892239i	$0.350867\pi$
$158$	0	0
$159$	−7.74005	−0.613826
$160$	0	0
$161$	−32.0924	−2.52924
$162$	0	0
$163$	−23.0238	−1.80337	−0.901683	−	0.432397i	$-0.857668\pi$
$163$	−23.0238	−1.80337	−0.901683	+	0.432397i	$0.857668\pi$
$164$	0	0
$165$	−2.28798	−0.178119
$166$	0	0
$167$	20.3434	1.57422	0.787108	−	0.616815i	$-0.211577\pi$
$167$	20.3434	1.57422	0.787108	+	0.616815i	$0.211577\pi$
$168$	0	0
$169$	−4.20620	−0.323554
$170$	0	0
$171$	14.5915	1.11584
$172$	0	0
$173$	3.02617	0.230075	0.115038	−	0.993361i	$-0.463301\pi$
$173$	3.02617	0.230075	0.115038	+	0.993361i	$0.463301\pi$
$174$	0	0
$175$	16.7035	1.26266
$176$	0	0
$177$	2.29019	0.172141
$178$	0	0
$179$	1.97206	0.147399	0.0736993	−	0.997281i	$-0.476519\pi$
$179$	1.97206	0.147399	0.0736993	+	0.997281i	$0.476519\pi$
$180$	0	0
$181$	1.78378	0.132587	0.0662937	−	0.997800i	$-0.478883\pi$
$181$	1.78378	0.132587	0.0662937	+	0.997800i	$0.478883\pi$
$182$	0	0
$183$	−26.8406	−1.98412
$184$	0	0
$185$	9.44532	0.694433
$186$	0	0
$187$	1.08425	0.0792884
$188$	0	0
$189$	6.95825	0.506138
$190$	0	0
$191$	−16.7049	−1.20872	−0.604361	−	0.796710i	$-0.706572\pi$
$191$	−16.7049	−1.20872	−0.604361	+	0.796710i	$0.706572\pi$
$192$	0	0
$193$	−12.8496	−0.924935	−0.462467	−	0.886636i	$-0.653036\pi$
$193$	−12.8496	−0.924935	−0.462467	+	0.886636i	$0.653036\pi$
$194$	0	0
$195$	−6.25763	−0.448118
$196$	0	0
$197$	−22.6366	−1.61279	−0.806396	−	0.591376i	$-0.798585\pi$
$197$	−22.6366	−1.61279	−0.806396	+	0.591376i	$0.798585\pi$
$198$	0	0
$199$	−6.01590	−0.426456	−0.213228	−	0.977003i	$-0.568398\pi$
$199$	−6.01590	−0.426456	−0.213228	+	0.977003i	$0.568398\pi$
$200$	0	0
$201$	−6.19815	−0.437184
$202$	0	0
$203$	−19.0194	−1.33490
$204$	0	0
$205$	−8.59037	−0.599977
$206$	0	0
$207$	17.9042	1.24443
$208$	0	0
$209$	−7.04732	−0.487474
$210$	0	0
$211$	−2.32211	−0.159861	−0.0799303	−	0.996800i	$-0.525470\pi$
$211$	−2.32211	−0.159861	−0.0799303	+	0.996800i	$0.525470\pi$
$212$	0	0
$213$	−23.3413	−1.59932
$214$	0	0
$215$	−10.0810	−0.687516
$216$	0	0
$217$	20.1471	1.36767
$218$	0	0
$219$	1.68903	0.114134
$220$	0	0
$221$	2.96543	0.199477
$222$	0	0
$223$	22.7910	1.52620	0.763101	−	0.646280i	$-0.223676\pi$
$223$	22.7910	1.52620	0.763101	+	0.646280i	$0.223676\pi$
$224$	0	0
$225$	−9.31881	−0.621254
$226$	0	0
$227$	−5.40342	−0.358638	−0.179319	−	0.983791i	$-0.557389\pi$
$227$	−5.40342	−0.358638	−0.179319	+	0.983791i	$0.557389\pi$
$228$	0	0
$229$	20.2863	1.34056	0.670278	−	0.742111i	$-0.266175\pi$
$229$	20.2863	1.34056	0.670278	+	0.742111i	$0.266175\pi$
$230$	0	0
$231$	9.99203	0.657427
$232$	0	0
$233$	−9.22412	−0.604292	−0.302146	−	0.953262i	$-0.597703\pi$
$233$	−9.22412	−0.604292	−0.302146	+	0.953262i	$0.597703\pi$
$234$	0	0
$235$	−1.05695	−0.0689476
$236$	0	0
$237$	−12.0187	−0.780697
$238$	0	0
$239$	4.47553	0.289498	0.144749	−	0.989468i	$-0.453763\pi$
$239$	4.47553	0.289498	0.144749	+	0.989468i	$0.453763\pi$
$240$	0	0
$241$	−18.5756	−1.19656	−0.598280	−	0.801287i	$-0.704149\pi$
$241$	−18.5756	−1.19656	−0.598280	+	0.801287i	$0.704149\pi$
$242$	0	0
$243$	−19.3060	−1.23848
$244$	0	0
$245$	8.46973	0.541111
$246$	0	0
$247$	−19.2745	−1.22640
$248$	0	0
$249$	−25.0777	−1.58924
$250$	0	0
$251$	6.86926	0.433584	0.216792	−	0.976218i	$-0.430441\pi$
$251$	6.86926	0.433584	0.216792	+	0.976218i	$0.430441\pi$
$252$	0	0
$253$	−8.64729	−0.543650
$254$	0	0
$255$	−2.11019	−0.132145
$256$	0	0
$257$	−25.8748	−1.61403	−0.807014	−	0.590532i	$-0.798918\pi$
$257$	−25.8748	−1.61403	−0.807014	+	0.590532i	$0.798918\pi$
$258$	0	0
$259$	−41.2495	−2.56312
$260$	0	0
$261$	10.6108	0.656795
$262$	0	0
$263$	−4.52501	−0.279024	−0.139512	−	0.990220i	$-0.544553\pi$
$263$	−4.52501	−0.279024	−0.139512	+	0.990220i	$0.544553\pi$
$264$	0	0
$265$	−3.11404	−0.191294
$266$	0	0
$267$	−11.8697	−0.726411
$268$	0	0
$269$	−9.35067	−0.570121	−0.285060	−	0.958510i	$-0.592014\pi$
$269$	−9.35067	−0.570121	−0.285060	+	0.958510i	$0.592014\pi$
$270$	0	0
$271$	−0.170734	−0.0103714	−0.00518569	−	0.999987i	$-0.501651\pi$
$271$	−0.170734	−0.0103714	−0.00518569	+	0.999987i	$0.501651\pi$
$272$	0	0
$273$	27.3282	1.65398
$274$	0	0
$275$	4.50074	0.271405
$276$	0	0
$277$	−21.4267	−1.28741	−0.643703	−	0.765276i	$-0.722603\pi$
$277$	−21.4267	−1.28741	−0.643703	+	0.765276i	$0.722603\pi$
$278$	0	0
$279$	−11.2400	−0.672920
$280$	0	0
$281$	6.89011	0.411030	0.205515	−	0.978654i	$-0.434113\pi$
$281$	6.89011	0.411030	0.205515	+	0.978654i	$0.434113\pi$
$282$	0	0
$283$	−17.6060	−1.04657	−0.523283	−	0.852159i	$-0.675293\pi$
$283$	−17.6060	−1.04657	−0.523283	+	0.852159i	$0.675293\pi$
$284$	0	0
$285$	13.7156	0.812443
$286$	0	0
$287$	37.5157	2.21448
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	39.7525	2.33033
$292$	0	0
$293$	4.60051	0.268765	0.134382	−	0.990930i	$-0.457095\pi$
$293$	4.60051	0.268765	0.134382	+	0.990930i	$0.457095\pi$
$294$	0	0
$295$	0.921405	0.0536463
$296$	0	0
$297$	1.87490	0.108793
$298$	0	0
$299$	−23.6504	−1.36774
$300$	0	0
$301$	44.0254	2.53758
$302$	0	0
$303$	17.1654	0.986128
$304$	0	0
$305$	−10.7987	−0.618334
$306$	0	0
$307$	26.9304	1.53700	0.768501	−	0.639849i	$-0.221003\pi$
$307$	26.9304	1.53700	0.768501	+	0.639849i	$0.221003\pi$
$308$	0	0
$309$	11.9455	0.679557
$310$	0	0
$311$	25.0219	1.41886	0.709430	−	0.704776i	$-0.248952\pi$
$311$	25.0219	1.41886	0.709430	+	0.704776i	$0.248952\pi$
$312$	0	0
$313$	16.7571	0.947167	0.473584	−	0.880749i	$-0.342960\pi$
$313$	16.7571	0.947167	0.473584	+	0.880749i	$0.342960\pi$
$314$	0	0
$315$	−8.32358	−0.468980
$316$	0	0
$317$	−13.3958	−0.752384	−0.376192	−	0.926542i	$-0.622767\pi$
$317$	−13.3958	−0.752384	−0.376192	+	0.926542i	$0.622767\pi$
$318$	0	0
$319$	−5.12476	−0.286932
$320$	0	0
$321$	−3.12297	−0.174307
$322$	0	0
$323$	−6.49971	−0.361653
$324$	0	0
$325$	12.3096	0.682811
$326$	0	0
$327$	−28.1416	−1.55623
$328$	0	0
$329$	4.61588	0.254482
$330$	0	0
$331$	26.0670	1.43277	0.716385	−	0.697706i	$-0.245796\pi$
$331$	26.0670	1.43277	0.716385	+	0.697706i	$0.245796\pi$
$332$	0	0
$333$	23.0129	1.26110
$334$	0	0
$335$	−2.49369	−0.136245
$336$	0	0
$337$	−1.29716	−0.0706606	−0.0353303	−	0.999376i	$-0.511248\pi$
$337$	−1.29716	−0.0706606	−0.0353303	+	0.999376i	$0.511248\pi$
$338$	0	0
$339$	−20.5554	−1.11642
$340$	0	0
$341$	5.42862	0.293976
$342$	0	0
$343$	−8.82123	−0.476302
$344$	0	0
$345$	16.8295	0.906070
$346$	0	0
$347$	−2.86939	−0.154037	−0.0770186	−	0.997030i	$-0.524540\pi$
$347$	−2.86939	−0.154037	−0.0770186	+	0.997030i	$0.524540\pi$
$348$	0	0
$349$	−3.08705	−0.165246	−0.0826230	−	0.996581i	$-0.526330\pi$
$349$	−3.08705	−0.165246	−0.0826230	+	0.996581i	$0.526330\pi$
$350$	0	0
$351$	5.12785	0.273704
$352$	0	0
$353$	−13.3836	−0.712339	−0.356169	−	0.934421i	$-0.615917\pi$
$353$	−13.3836	−0.712339	−0.356169	+	0.934421i	$0.615917\pi$
$354$	0	0
$355$	−9.39084	−0.498414
$356$	0	0
$357$	9.21559	0.487741
$358$	0	0
$359$	−15.1530	−0.799744	−0.399872	−	0.916571i	$-0.630945\pi$
$359$	−15.1530	−0.799744	−0.399872	+	0.916571i	$0.630945\pi$
$360$	0	0
$361$	23.2462	1.22348
$362$	0	0
$363$	−22.4997	−1.18093
$364$	0	0
$365$	0.679542	0.0355689
$366$	0	0
$367$	0.901576	0.0470619	0.0235309	−	0.999723i	$-0.492509\pi$
$367$	0.901576	0.0470619	0.0235309	+	0.999723i	$0.492509\pi$
$368$	0	0
$369$	−20.9299	−1.08957
$370$	0	0
$371$	13.5996	0.706055
$372$	0	0
$373$	32.0853	1.66131	0.830656	−	0.556786i	$-0.187966\pi$
$373$	32.0853	1.66131	0.830656	+	0.556786i	$0.187966\pi$
$374$	0	0
$375$	−19.3104	−0.997183
$376$	0	0
$377$	−14.0162	−0.721874
$378$	0	0
$379$	−19.6382	−1.00874	−0.504372	−	0.863486i	$-0.668276\pi$
$379$	−19.6382	−1.00874	−0.504372	+	0.863486i	$0.668276\pi$
$380$	0	0
$381$	−29.6870	−1.52091
$382$	0	0
$383$	−1.58120	−0.0807957	−0.0403979	−	0.999184i	$-0.512863\pi$
$383$	−1.58120	−0.0807957	−0.0403979	+	0.999184i	$0.512863\pi$
$384$	0	0
$385$	4.02007	0.204882
$386$	0	0
$387$	−24.5616	−1.24854
$388$	0	0
$389$	31.8591	1.61532	0.807661	−	0.589647i	$-0.200733\pi$
$389$	31.8591	1.61532	0.807661	+	0.589647i	$0.200733\pi$
$390$	0	0
$391$	−7.97535	−0.403331
$392$	0	0
$393$	5.84307	0.294744
$394$	0	0
$395$	−4.83545	−0.243298
$396$	0	0
$397$	−19.6083	−0.984112	−0.492056	−	0.870563i	$-0.663755\pi$
$397$	−19.6083	−0.984112	−0.492056	+	0.870563i	$0.663755\pi$
$398$	0	0
$399$	−59.8987	−2.99868
$400$	0	0
$401$	−13.5960	−0.678953	−0.339477	−	0.940614i	$-0.610250\pi$
$401$	−13.5960	−0.678953	−0.339477	+	0.940614i	$0.610250\pi$
$402$	0	0
$403$	14.8473	0.739596
$404$	0	0
$405$	−9.85448	−0.489673
$406$	0	0
$407$	−11.1147	−0.550933
$408$	0	0
$409$	−14.2356	−0.703905	−0.351952	−	0.936018i	$-0.614482\pi$
$409$	−14.2356	−0.703905	−0.351952	+	0.936018i	$0.614482\pi$
$410$	0	0
$411$	−8.90922	−0.439459
$412$	0	0
$413$	−4.02395	−0.198006
$414$	0	0
$415$	−10.0895	−0.495273
$416$	0	0
$417$	40.8479	2.00033
$418$	0	0
$419$	−14.7131	−0.718781	−0.359391	−	0.933187i	$-0.617015\pi$
$419$	−14.7131	−0.718781	−0.359391	+	0.933187i	$0.617015\pi$
$420$	0	0
$421$	−13.1321	−0.640019	−0.320009	−	0.947414i	$-0.603686\pi$
$421$	−13.1321	−0.640019	−0.320009	+	0.947414i	$0.603686\pi$
$422$	0	0
$423$	−2.57519	−0.125210
$424$	0	0
$425$	4.15101	0.201354
$426$	0	0
$427$	47.1601	2.28224
$428$	0	0
$429$	7.36358	0.355517
$430$	0	0
$431$	9.81834	0.472933	0.236466	−	0.971640i	$-0.424011\pi$
$431$	9.81834	0.472933	0.236466	+	0.971640i	$0.424011\pi$
$432$	0	0
$433$	21.9007	1.05248	0.526240	−	0.850336i	$-0.323601\pi$
$433$	21.9007	1.05248	0.526240	+	0.850336i	$0.323601\pi$
$434$	0	0
$435$	9.97390	0.478212
$436$	0	0
$437$	51.8374	2.47972
$438$	0	0
$439$	14.8188	0.707265	0.353632	−	0.935384i	$-0.384946\pi$
$439$	14.8188	0.707265	0.353632	+	0.935384i	$0.384946\pi$
$440$	0	0
$441$	20.6360	0.982666
$442$	0	0
$443$	−13.7131	−0.651531	−0.325765	−	0.945451i	$-0.605622\pi$
$443$	−13.7131	−0.651531	−0.325765	+	0.945451i	$0.605622\pi$
$444$	0	0
$445$	−4.77549	−0.226380
$446$	0	0
$447$	−15.8316	−0.748811
$448$	0	0
$449$	1.53823	0.0725935	0.0362967	−	0.999341i	$-0.488444\pi$
$449$	1.53823	0.0725935	0.0362967	+	0.999341i	$0.488444\pi$
$450$	0	0
$451$	10.1086	0.475996
$452$	0	0
$453$	17.6627	0.829866
$454$	0	0
$455$	10.9949	0.515449
$456$	0	0
$457$	−12.6369	−0.591130	−0.295565	−	0.955323i	$-0.595508\pi$
$457$	−12.6369	−0.591130	−0.295565	+	0.955323i	$0.595508\pi$
$458$	0	0
$459$	1.72921	0.0807125
$460$	0	0
$461$	11.8362	0.551268	0.275634	−	0.961263i	$-0.411112\pi$
$461$	11.8362	0.551268	0.275634	+	0.961263i	$0.411112\pi$
$462$	0	0
$463$	8.41315	0.390992	0.195496	−	0.980704i	$-0.437368\pi$
$463$	8.41315	0.390992	0.195496	+	0.980704i	$0.437368\pi$
$464$	0	0
$465$	−10.5653	−0.489952
$466$	0	0
$467$	−31.1017	−1.43921	−0.719607	−	0.694381i	$-0.755678\pi$
$467$	−31.1017	−1.43921	−0.719607	+	0.694381i	$0.755678\pi$
$468$	0	0
$469$	10.8904	0.502872
$470$	0	0
$471$	25.9161	1.19415
$472$	0	0
$473$	11.8626	0.545445
$474$	0	0
$475$	−26.9804	−1.23794
$476$	0	0
$477$	−7.58716	−0.347392
$478$	0	0
$479$	28.3918	1.29726	0.648628	−	0.761106i	$-0.275343\pi$
$479$	28.3918	1.29726	0.648628	+	0.761106i	$0.275343\pi$
$480$	0	0
$481$	−30.3986	−1.38606
$482$	0	0
$483$	−73.4976	−3.34426
$484$	0	0
$485$	15.9935	0.726229
$486$	0	0
$487$	37.9391	1.71919	0.859593	−	0.510979i	$-0.170717\pi$
$487$	37.9391	1.71919	0.859593	+	0.510979i	$0.170717\pi$
$488$	0	0
$489$	−52.7289	−2.38448
$490$	0	0
$491$	−32.7091	−1.47614	−0.738070	−	0.674724i	$-0.764263\pi$
$491$	−32.7091	−1.47614	−0.738070	+	0.674724i	$0.764263\pi$
$492$	0	0
$493$	−4.72654	−0.212873
$494$	0	0
$495$	−2.24278	−0.100806
$496$	0	0
$497$	41.0116	1.83962
$498$	0	0
$499$	−7.77337	−0.347984	−0.173992	−	0.984747i	$-0.555667\pi$
$499$	−7.77337	−0.347984	−0.173992	+	0.984747i	$0.555667\pi$
$500$	0	0
$501$	46.5901	2.08149
$502$	0	0
$503$	19.1283	0.852890	0.426445	−	0.904514i	$-0.359766\pi$
$503$	19.1283	0.852890	0.426445	+	0.904514i	$0.359766\pi$
$504$	0	0
$505$	6.90613	0.307319
$506$	0	0
$507$	−9.63297	−0.427815
$508$	0	0
$509$	1.05608	0.0468101	0.0234051	−	0.999726i	$-0.492549\pi$
$509$	1.05608	0.0468101	0.0234051	+	0.999726i	$0.492549\pi$
$510$	0	0
$511$	−2.96769	−0.131283
$512$	0	0
$513$	−11.2393	−0.496229
$514$	0	0
$515$	4.80602	0.211779
$516$	0	0
$517$	1.24375	0.0547000
$518$	0	0
$519$	6.93049	0.304215
$520$	0	0
$521$	−32.1389	−1.40803	−0.704016	−	0.710184i	$-0.748612\pi$
$521$	−32.1389	−1.40803	−0.704016	+	0.710184i	$0.748612\pi$
$522$	0	0
$523$	−10.7694	−0.470913	−0.235457	−	0.971885i	$-0.575659\pi$
$523$	−10.7694	−0.470913	−0.235457	+	0.971885i	$0.575659\pi$
$524$	0	0
$525$	38.2540	1.66954
$526$	0	0
$527$	5.00678	0.218099
$528$	0	0
$529$	40.6062	1.76549
$530$	0	0
$531$	2.24495	0.0974225
$532$	0	0
$533$	27.6471	1.19753
$534$	0	0
$535$	−1.25646	−0.0543214
$536$	0	0
$537$	4.51638	0.194896
$538$	0	0
$539$	−9.96664	−0.429294
$540$	0	0
$541$	12.1830	0.523789	0.261895	−	0.965096i	$-0.415653\pi$
$541$	12.1830	0.523789	0.261895	+	0.965096i	$0.415653\pi$
$542$	0	0
$543$	4.08519	0.175312
$544$	0	0
$545$	−11.3221	−0.484987
$546$	0	0
$547$	−11.4654	−0.490227	−0.245114	−	0.969494i	$-0.578825\pi$
$547$	−11.4654	−0.490227	−0.245114	+	0.969494i	$0.578825\pi$
$548$	0	0
$549$	−26.3105	−1.12290
$550$	0	0
$551$	30.7211	1.30876
$552$	0	0
$553$	21.1173	0.898000
$554$	0	0
$555$	21.6315	0.918207
$556$	0	0
$557$	−5.48710	−0.232496	−0.116248	−	0.993220i	$-0.537087\pi$
$557$	−5.48710	−0.232496	−0.116248	+	0.993220i	$0.537087\pi$
$558$	0	0
$559$	32.4444	1.37225
$560$	0	0
$561$	2.48314	0.104838
$562$	0	0
$563$	−12.2386	−0.515795	−0.257897	−	0.966172i	$-0.583030\pi$
$563$	−12.2386	−0.515795	−0.257897	+	0.966172i	$0.583030\pi$
$564$	0	0
$565$	−8.27001	−0.347922
$566$	0	0
$567$	43.0364	1.80736
$568$	0	0
$569$	36.6658	1.53711	0.768556	−	0.639783i	$-0.220976\pi$
$569$	36.6658	1.53711	0.768556	+	0.639783i	$0.220976\pi$
$570$	0	0
$571$	19.5692	0.818945	0.409473	−	0.912322i	$-0.365713\pi$
$571$	19.5692	0.818945	0.409473	+	0.912322i	$0.365713\pi$
$572$	0	0
$573$	−38.2573	−1.59822
$574$	0	0
$575$	−33.1058	−1.38061
$576$	0	0
$577$	−8.86315	−0.368978	−0.184489	−	0.982835i	$-0.559063\pi$
$577$	−8.86315	−0.368978	−0.184489	+	0.982835i	$0.559063\pi$
$578$	0	0
$579$	−29.4280	−1.22299
$580$	0	0
$581$	44.0626	1.82803
$582$	0	0
$583$	3.66440	0.151764
$584$	0	0
$585$	−6.13403	−0.253611
$586$	0	0
$587$	43.7234	1.80466	0.902329	−	0.431048i	$-0.141856\pi$
$587$	43.7234	1.80466	0.902329	+	0.431048i	$0.141856\pi$
$588$	0	0
$589$	−32.5426	−1.34090
$590$	0	0
$591$	−51.8420	−2.13250
$592$	0	0
$593$	−32.8150	−1.34755	−0.673776	−	0.738936i	$-0.735329\pi$
$593$	−32.8150	−1.34755	−0.673776	+	0.738936i	$0.735329\pi$
$594$	0	0
$595$	3.70769	0.152000
$596$	0	0
$597$	−13.7775	−0.563876
$598$	0	0
$599$	44.9971	1.83853	0.919266	−	0.393636i	$-0.128783\pi$
$599$	44.9971	1.83853	0.919266	+	0.393636i	$0.128783\pi$
$600$	0	0
$601$	9.41613	0.384092	0.192046	−	0.981386i	$-0.438488\pi$
$601$	9.41613	0.384092	0.192046	+	0.981386i	$0.438488\pi$
$602$	0	0
$603$	−6.07572	−0.247423
$604$	0	0
$605$	−9.05225	−0.368026
$606$	0	0
$607$	−3.24776	−0.131823	−0.0659113	−	0.997825i	$-0.520995\pi$
$607$	−3.24776	−0.131823	−0.0659113	+	0.997825i	$0.520995\pi$
$608$	0	0
$609$	−43.5579	−1.76505
$610$	0	0
$611$	3.40166	0.137616
$612$	0	0
$613$	−22.2148	−0.897246	−0.448623	−	0.893721i	$-0.648085\pi$
$613$	−22.2148	−0.897246	−0.448623	+	0.893721i	$0.648085\pi$
$614$	0	0
$615$	−19.6735	−0.793313
$616$	0	0
$617$	−0.777777	−0.0313121	−0.0156561	−	0.999877i	$-0.504984\pi$
$617$	−0.777777	−0.0313121	−0.0156561	+	0.999877i	$0.504984\pi$
$618$	0	0
$619$	−25.6471	−1.03084	−0.515422	−	0.856937i	$-0.672365\pi$
$619$	−25.6471	−1.03084	−0.515422	+	0.856937i	$0.672365\pi$
$620$	0	0
$621$	−13.7910	−0.553415
$622$	0	0
$623$	20.8555	0.835557
$624$	0	0
$625$	12.9860	0.519438
$626$	0	0
$627$	−16.1397	−0.644557
$628$	0	0
$629$	−10.2510	−0.408734
$630$	0	0
$631$	32.8846	1.30911	0.654557	−	0.756013i	$-0.272855\pi$
$631$	32.8846	1.30911	0.654557	+	0.756013i	$0.272855\pi$
$632$	0	0
$633$	−5.31806	−0.211374
$634$	0	0
$635$	−11.9439	−0.473980
$636$	0	0
$637$	−27.2588	−1.08003
$638$	0	0
$639$	−22.8802	−0.905128
$640$	0	0
$641$	22.5827	0.891962	0.445981	−	0.895043i	$-0.352855\pi$
$641$	22.5827	0.891962	0.445981	+	0.895043i	$0.352855\pi$
$642$	0	0
$643$	32.3591	1.27612	0.638059	−	0.769988i	$-0.279738\pi$
$643$	32.3591	1.27612	0.638059	+	0.769988i	$0.279738\pi$
$644$	0	0
$645$	−23.0873	−0.909060
$646$	0	0
$647$	35.3915	1.39138	0.695691	−	0.718341i	$-0.255098\pi$
$647$	35.3915	1.39138	0.695691	+	0.718341i	$0.255098\pi$
$648$	0	0
$649$	−1.08425	−0.0425606
$650$	0	0
$651$	46.1405	1.80839
$652$	0	0
$653$	−7.06841	−0.276608	−0.138304	−	0.990390i	$-0.544165\pi$
$653$	−7.06841	−0.276608	−0.138304	+	0.990390i	$0.544165\pi$
$654$	0	0
$655$	2.35083	0.0918545
$656$	0	0
$657$	1.65566	0.0645936
$658$	0	0
$659$	48.6615	1.89558	0.947792	−	0.318890i	$-0.103310\pi$
$659$	48.6615	1.89558	0.947792	+	0.318890i	$0.103310\pi$
$660$	0	0
$661$	−23.6847	−0.921230	−0.460615	−	0.887600i	$-0.652371\pi$
$661$	−23.6847	−0.921230	−0.460615	+	0.887600i	$0.652371\pi$
$662$	0	0
$663$	6.79139	0.263756
$664$	0	0
$665$	−24.0989	−0.934516
$666$	0	0
$667$	37.6958	1.45959
$668$	0	0
$669$	52.1957	2.01800
$670$	0	0
$671$	12.7073	0.490559
$672$	0	0
$673$	−0.277788	−0.0107079	−0.00535396	−	0.999986i	$-0.501704\pi$
$673$	−0.277788	−0.0107079	−0.00535396	+	0.999986i	$0.501704\pi$
$674$	0	0
$675$	7.17796	0.276280
$676$	0	0
$677$	13.7580	0.528763	0.264381	−	0.964418i	$-0.414832\pi$
$677$	13.7580	0.528763	0.264381	+	0.964418i	$0.414832\pi$
$678$	0	0
$679$	−69.8468	−2.68047
$680$	0	0
$681$	−12.3748	−0.474205
$682$	0	0
$683$	40.1348	1.53571	0.767857	−	0.640621i	$-0.221323\pi$
$683$	40.1348	1.53571	0.767857	+	0.640621i	$0.221323\pi$
$684$	0	0
$685$	−3.58443	−0.136954
$686$	0	0
$687$	46.4593	1.77253
$688$	0	0
$689$	10.0222	0.381814
$690$	0	0
$691$	28.9902	1.10284	0.551419	−	0.834228i	$-0.314086\pi$
$691$	28.9902	1.10284	0.551419	+	0.834228i	$0.314086\pi$
$692$	0	0
$693$	9.79466	0.372068
$694$	0	0
$695$	16.4342	0.623386
$696$	0	0
$697$	9.32311	0.353138
$698$	0	0
$699$	−21.1249	−0.799019
$700$	0	0
$701$	−40.2534	−1.52035	−0.760174	−	0.649719i	$-0.774887\pi$
$701$	−40.2534	−1.52035	−0.760174	+	0.649719i	$0.774887\pi$
$702$	0	0
$703$	66.6284	2.51294
$704$	0	0
$705$	−2.42060	−0.0911652
$706$	0	0
$707$	−30.1604	−1.13430
$708$	0	0
$709$	24.7488	0.929461	0.464731	−	0.885452i	$-0.346151\pi$
$709$	24.7488	0.929461	0.464731	+	0.885452i	$0.346151\pi$
$710$	0	0
$711$	−11.7813	−0.441833
$712$	0	0
$713$	−39.9308	−1.49542
$714$	0	0
$715$	2.96258	0.110794
$716$	0	0
$717$	10.2498	0.382785
$718$	0	0
$719$	12.7932	0.477106	0.238553	−	0.971129i	$-0.423327\pi$
$719$	12.7932	0.477106	0.238553	+	0.971129i	$0.423327\pi$
$720$	0	0
$721$	−20.9888	−0.781664
$722$	0	0
$723$	−42.5416	−1.58214
$724$	0	0
$725$	−19.6199	−0.728666
$726$	0	0
$727$	8.28101	0.307126	0.153563	−	0.988139i	$-0.450925\pi$
$727$	8.28101	0.307126	0.153563	+	0.988139i	$0.450925\pi$
$728$	0	0
$729$	−12.1292	−0.449231
$730$	0	0
$731$	10.9408	0.404662
$732$	0	0
$733$	4.68295	0.172969	0.0864844	−	0.996253i	$-0.472437\pi$
$733$	4.68295	0.172969	0.0864844	+	0.996253i	$0.472437\pi$
$734$	0	0
$735$	19.3972	0.715478
$736$	0	0
$737$	2.93442	0.108091
$738$	0	0
$739$	−43.5650	−1.60257	−0.801283	−	0.598285i	$-0.795849\pi$
$739$	−43.5650	−1.60257	−0.801283	+	0.598285i	$0.795849\pi$
$740$	0	0
$741$	−44.1421	−1.62160
$742$	0	0
$743$	43.7559	1.60525	0.802623	−	0.596486i	$-0.203437\pi$
$743$	43.7559	1.60525	0.802623	+	0.596486i	$0.203437\pi$
$744$	0	0
$745$	−6.36951	−0.233361
$746$	0	0
$747$	−24.5824	−0.899422
$748$	0	0
$749$	5.48719	0.200497
$750$	0	0
$751$	7.83302	0.285831	0.142915	−	0.989735i	$-0.454352\pi$
$751$	7.83302	0.285831	0.142915	+	0.989735i	$0.454352\pi$
$752$	0	0
$753$	15.7319	0.573301
$754$	0	0
$755$	7.10620	0.258621
$756$	0	0
$757$	1.97402	0.0717468	0.0358734	−	0.999356i	$-0.488579\pi$
$757$	1.97402	0.0717468	0.0358734	+	0.999356i	$0.488579\pi$
$758$	0	0
$759$	−19.8039	−0.718836
$760$	0	0
$761$	24.9875	0.905796	0.452898	−	0.891562i	$-0.350390\pi$
$761$	24.9875	0.905796	0.452898	+	0.891562i	$0.350390\pi$
$762$	0	0
$763$	49.4459	1.79006
$764$	0	0
$765$	−2.06851	−0.0747871
$766$	0	0
$767$	−2.96543	−0.107076
$768$	0	0
$769$	13.1782	0.475217	0.237608	−	0.971361i	$-0.423636\pi$
$769$	13.1782	0.475217	0.237608	+	0.971361i	$0.423636\pi$
$770$	0	0
$771$	−59.2582	−2.13413
$772$	0	0
$773$	−13.4141	−0.482471	−0.241235	−	0.970467i	$-0.577553\pi$
$773$	−13.4141	−0.482471	−0.241235	+	0.970467i	$0.577553\pi$
$774$	0	0
$775$	20.7832	0.746555
$776$	0	0
$777$	−94.4689	−3.38905
$778$	0	0
$779$	−60.5975	−2.17113
$780$	0	0
$781$	11.0506	0.395420
$782$	0	0
$783$	−8.17317	−0.292085
$784$	0	0
$785$	10.4268	0.372147
$786$	0	0
$787$	25.9052	0.923421	0.461711	−	0.887031i	$-0.347236\pi$
$787$	25.9052	0.923421	0.461711	+	0.887031i	$0.347236\pi$
$788$	0	0
$789$	−10.3631	−0.368936
$790$	0	0
$791$	36.1167	1.28416
$792$	0	0
$793$	34.7545	1.23417
$794$	0	0
$795$	−7.13172	−0.252936
$796$	0	0
$797$	−50.6783	−1.79512	−0.897559	−	0.440894i	$-0.854662\pi$
$797$	−50.6783	−1.79512	−0.897559	+	0.440894i	$0.854662\pi$
$798$	0	0
$799$	1.14710	0.0405816
$800$	0	0
$801$	−11.6352	−0.411110
$802$	0	0
$803$	−0.799643	−0.0282188
$804$	0	0
$805$	−29.5701	−1.04221
$806$	0	0
$807$	−21.4148	−0.753836
$808$	0	0
$809$	29.5863	1.04020	0.520100	−	0.854105i	$-0.325895\pi$
$809$	29.5863	1.04020	0.520100	+	0.854105i	$0.325895\pi$
$810$	0	0
$811$	21.1321	0.742047	0.371024	−	0.928623i	$-0.379007\pi$
$811$	21.1321	0.742047	0.371024	+	0.928623i	$0.379007\pi$
$812$	0	0
$813$	−0.391013	−0.0137134
$814$	0	0
$815$	−21.2143	−0.743104
$816$	0	0
$817$	−71.1123	−2.48791
$818$	0	0
$819$	26.7884	0.936064
$820$	0	0
$821$	−0.433367	−0.0151246	−0.00756231	−	0.999971i	$-0.502407\pi$
$821$	−0.433367	−0.0151246	−0.00756231	+	0.999971i	$0.502407\pi$
$822$	0	0
$823$	27.8437	0.970572	0.485286	−	0.874356i	$-0.338716\pi$
$823$	27.8437	0.970572	0.485286	+	0.874356i	$0.338716\pi$
$824$	0	0
$825$	10.3075	0.358862
$826$	0	0
$827$	−52.6823	−1.83194	−0.915972	−	0.401242i	$-0.868579\pi$
$827$	−52.6823	−1.83194	−0.915972	+	0.401242i	$0.868579\pi$
$828$	0	0
$829$	−22.9192	−0.796015	−0.398008	−	0.917382i	$-0.630298\pi$
$829$	−22.9192	−0.796015	−0.398008	+	0.917382i	$0.630298\pi$
$830$	0	0
$831$	−49.0711	−1.70226
$832$	0	0
$833$	−9.19218	−0.318490
$834$	0	0
$835$	18.7445	0.648679
$836$	0	0
$837$	8.65777	0.299256
$838$	0	0
$839$	−9.57501	−0.330566	−0.165283	−	0.986246i	$-0.552854\pi$
$839$	−9.57501	−0.330566	−0.165283	+	0.986246i	$0.552854\pi$
$840$	0	0
$841$	−6.65980	−0.229648
$842$	0	0
$843$	15.7796	0.543480
$844$	0	0
$845$	−3.87561	−0.133325
$846$	0	0
$847$	39.5329	1.35837
$848$	0	0
$849$	−40.3210	−1.38381
$850$	0	0
$851$	81.7552	2.80253
$852$	0	0
$853$	−7.45704	−0.255324	−0.127662	−	0.991818i	$-0.540747\pi$
$853$	−7.45704	−0.255324	−0.127662	+	0.991818i	$0.540747\pi$
$854$	0	0
$855$	13.4447	0.459799
$856$	0	0
$857$	34.8690	1.19110	0.595552	−	0.803317i	$-0.296933\pi$
$857$	34.8690	1.19110	0.595552	+	0.803317i	$0.296933\pi$
$858$	0	0
$859$	0.395760	0.0135031	0.00675157	−	0.999977i	$-0.497851\pi$
$859$	0.395760	0.0135031	0.00675157	+	0.999977i	$0.497851\pi$
$860$	0	0
$861$	85.9180	2.92808
$862$	0	0
$863$	−17.5546	−0.597567	−0.298783	−	0.954321i	$-0.596581\pi$
$863$	−17.5546	−0.597567	−0.298783	+	0.954321i	$0.596581\pi$
$864$	0	0
$865$	2.78833	0.0948061
$866$	0	0
$867$	2.29019	0.0777788
$868$	0	0
$869$	5.69005	0.193022
$870$	0	0
$871$	8.02564	0.271938
$872$	0	0
$873$	38.9673	1.31884
$874$	0	0
$875$	33.9291	1.14701
$876$	0	0
$877$	−11.8943	−0.401642	−0.200821	−	0.979628i	$-0.564361\pi$
$877$	−11.8943	−0.401642	−0.200821	+	0.979628i	$0.564361\pi$
$878$	0	0
$879$	10.5360	0.355371
$880$	0	0
$881$	−51.6519	−1.74020	−0.870099	−	0.492878i	$-0.835945\pi$
$881$	−51.6519	−1.74020	−0.870099	+	0.492878i	$0.835945\pi$
$882$	0	0
$883$	12.5928	0.423780	0.211890	−	0.977294i	$-0.432038\pi$
$883$	12.5928	0.423780	0.211890	+	0.977294i	$0.432038\pi$
$884$	0	0
$885$	2.11019	0.0709332
$886$	0	0
$887$	2.11579	0.0710413	0.0355207	−	0.999369i	$-0.488691\pi$
$887$	2.11579	0.0710413	0.0355207	+	0.999369i	$0.488691\pi$
$888$	0	0
$889$	52.1613	1.74943
$890$	0	0
$891$	11.5961	0.388485
$892$	0	0
$893$	−7.45583	−0.249500
$894$	0	0
$895$	1.81707	0.0607379
$896$	0	0
$897$	−54.1637	−1.80847
$898$	0	0
$899$	−23.6648	−0.789264
$900$	0	0
$901$	3.37966	0.112593
$902$	0	0
$903$	100.826	3.35529
$904$	0	0
$905$	1.64359	0.0546347
$906$	0	0
$907$	21.6072	0.717456	0.358728	−	0.933442i	$-0.383211\pi$
$907$	21.6072	0.717456	0.358728	+	0.933442i	$0.383211\pi$
$908$	0	0
$909$	16.8264	0.558095
$910$	0	0
$911$	−18.4443	−0.611087	−0.305544	−	0.952178i	$-0.598838\pi$
$911$	−18.4443	−0.611087	−0.305544	+	0.952178i	$0.598838\pi$
$912$	0	0
$913$	11.8727	0.392928
$914$	0	0
$915$	−24.7311	−0.817585
$916$	0	0
$917$	−10.2665	−0.339030
$918$	0	0
$919$	−51.5847	−1.70162	−0.850811	−	0.525471i	$-0.823889\pi$
$919$	−51.5847	−1.70162	−0.850811	+	0.525471i	$0.823889\pi$
$920$	0	0
$921$	61.6757	2.03228
$922$	0	0
$923$	30.2233	0.994813
$924$	0	0
$925$	−42.5520	−1.39910
$926$	0	0
$927$	11.7096	0.384593
$928$	0	0
$929$	30.8782	1.01308	0.506540	−	0.862217i	$-0.330924\pi$
$929$	30.8782	1.01308	0.506540	+	0.862217i	$0.330924\pi$
$930$	0	0
$931$	59.7465	1.95811
$932$	0	0
$933$	57.3047	1.87607
$934$	0	0
$935$	0.999036	0.0326720
$936$	0	0
$937$	−37.9104	−1.23848	−0.619239	−	0.785203i	$-0.712559\pi$
$937$	−37.9104	−1.23848	−0.619239	+	0.785203i	$0.712559\pi$
$938$	0	0
$939$	38.3769	1.25238
$940$	0	0
$941$	−3.28126	−0.106966	−0.0534831	−	0.998569i	$-0.517032\pi$
$941$	−3.28126	−0.106966	−0.0534831	+	0.998569i	$0.517032\pi$
$942$	0	0
$943$	−74.3551	−2.42133
$944$	0	0
$945$	6.41136	0.208562
$946$	0	0
$947$	−51.9879	−1.68938	−0.844689	−	0.535257i	$-0.820215\pi$
$947$	−51.9879	−1.68938	−0.844689	+	0.535257i	$0.820215\pi$
$948$	0	0
$949$	−2.18703	−0.0709939
$950$	0	0
$951$	−30.6789	−0.994832
$952$	0	0
$953$	21.2716	0.689053	0.344527	−	0.938777i	$-0.388039\pi$
$953$	21.2716	0.689053	0.344527	+	0.938777i	$0.388039\pi$
$954$	0	0
$955$	−15.3920	−0.498072
$956$	0	0
$957$	−11.7367	−0.379392
$958$	0	0
$959$	15.6539	0.505490
$960$	0	0
$961$	−5.93211	−0.191359
$962$	0	0
$963$	−3.06128	−0.0986485
$964$	0	0
$965$	−11.8397	−0.381133
$966$	0	0
$967$	1.15791	0.0372360	0.0186180	−	0.999827i	$-0.494073\pi$
$967$	1.15791	0.0372360	0.0186180	+	0.999827i	$0.494073\pi$
$968$	0	0
$969$	−14.8855	−0.478192
$970$	0	0
$971$	−31.0851	−0.997570	−0.498785	−	0.866726i	$-0.666220\pi$
$971$	−31.0851	−0.997570	−0.498785	+	0.866726i	$0.666220\pi$
$972$	0	0
$973$	−71.7714	−2.30089
$974$	0	0
$975$	28.1912	0.902840
$976$	0	0
$977$	44.3031	1.41738	0.708691	−	0.705519i	$-0.249286\pi$
$977$	44.3031	1.41738	0.708691	+	0.705519i	$0.249286\pi$
$978$	0	0
$979$	5.61950	0.179600
$980$	0	0
$981$	−27.5857	−0.880744
$982$	0	0
$983$	45.6969	1.45750	0.728752	−	0.684778i	$-0.240101\pi$
$983$	45.6969	1.45750	0.728752	+	0.684778i	$0.240101\pi$
$984$	0	0
$985$	−20.8575	−0.664575
$986$	0	0
$987$	10.5712	0.336486
$988$	0	0
$989$	−87.2571	−2.77461
$990$	0	0
$991$	−35.9304	−1.14137	−0.570684	−	0.821170i	$-0.693322\pi$
$991$	−35.9304	−1.14137	−0.570684	+	0.821170i	$0.693322\pi$
$992$	0	0
$993$	59.6981	1.89446
$994$	0	0
$995$	−5.54308	−0.175727
$996$	0	0
$997$	4.61433	0.146137	0.0730686	−	0.997327i	$-0.476721\pi$
$997$	4.61433	0.146137	0.0730686	+	0.997327i	$0.476721\pi$
$998$	0	0
$999$	−17.7261	−0.560828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.19	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.19	✓	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.29019 q^{3} +0.921405 q^{5} -4.02395 q^{7} +2.24495 q^{9} +O(q^{10})\)	\(q+2.29019 q^{3} +0.921405 q^{5} -4.02395 q^{7} +2.24495 q^{9} -1.08425 q^{11} -2.96543 q^{13} +2.11019 q^{15} -1.00000 q^{17} +6.49971 q^{19} -9.21559 q^{21} +7.97535 q^{23} -4.15101 q^{25} -1.72921 q^{27} +4.72654 q^{29} -5.00678 q^{31} -2.48314 q^{33} -3.70769 q^{35} +10.2510 q^{37} -6.79139 q^{39} -9.32311 q^{41} -10.9408 q^{43} +2.06851 q^{45} -1.14710 q^{47} +9.19218 q^{49} -2.29019 q^{51} -3.37966 q^{53} -0.999036 q^{55} +14.8855 q^{57} +1.00000 q^{59} -11.7199 q^{61} -9.03356 q^{63} -2.73237 q^{65} -2.70640 q^{67} +18.2650 q^{69} -10.1919 q^{71} +0.737506 q^{73} -9.50659 q^{75} +4.36298 q^{77} -5.24791 q^{79} -10.6951 q^{81} -10.9501 q^{83} -0.921405 q^{85} +10.8247 q^{87} -5.18284 q^{89} +11.9328 q^{91} -11.4665 q^{93} +5.98887 q^{95} +17.3578 q^{97} -2.43409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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