LMFDB - Embedded newform 8024.2.a.y.1.11

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:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-0.823534 q^{3} +3.63980 q^{5} +3.42292 q^{7} -2.32179 q^{9} +O(q^{10})$	$q-0.823534 q^{3} +3.63980 q^{5} +3.42292 q^{7} -2.32179 q^{9} -1.67000 q^{11} +1.71525 q^{13} -2.99750 q^{15} -1.00000 q^{17} -8.21569 q^{19} -2.81889 q^{21} +1.40727 q^{23} +8.24811 q^{25} +4.38268 q^{27} -9.95781 q^{29} -6.20012 q^{31} +1.37530 q^{33} +12.4587 q^{35} -1.91074 q^{37} -1.41257 q^{39} +4.14541 q^{41} -7.78851 q^{43} -8.45085 q^{45} +3.59899 q^{47} +4.71635 q^{49} +0.823534 q^{51} -9.21150 q^{53} -6.07845 q^{55} +6.76590 q^{57} -1.00000 q^{59} -9.69562 q^{61} -7.94730 q^{63} +6.24317 q^{65} -7.29905 q^{67} -1.15893 q^{69} -13.5788 q^{71} +4.25258 q^{73} -6.79260 q^{75} -5.71627 q^{77} +15.8841 q^{79} +3.35609 q^{81} -7.87799 q^{83} -3.63980 q^{85} +8.20059 q^{87} +6.32519 q^{89} +5.87116 q^{91} +5.10601 q^{93} -29.9034 q^{95} +12.2886 q^{97} +3.87739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	−0.823534	−0.475468	−0.237734	−	0.971330i	$-0.576405\pi$
$3$	−0.823534	−0.475468	−0.237734	+	0.971330i	$0.576405\pi$
$4$	0	0
$5$	3.63980	1.62777	0.813883	−	0.581029i	$-0.197350\pi$
$5$	3.63980	1.62777	0.813883	+	0.581029i	$0.197350\pi$
$6$	0	0
$7$	3.42292	1.29374	0.646870	−	0.762600i	$-0.276078\pi$
$7$	3.42292	1.29374	0.646870	+	0.762600i	$0.276078\pi$
$8$	0	0
$9$	−2.32179	−0.773931
$10$	0	0
$11$	−1.67000	−0.503524	−0.251762	−	0.967789i	$-0.581010\pi$
$11$	−1.67000	−0.503524	−0.251762	+	0.967789i	$0.581010\pi$
$12$	0	0
$13$	1.71525	0.475725	0.237863	−	0.971299i	$-0.423553\pi$
$13$	1.71525	0.475725	0.237863	+	0.971299i	$0.423553\pi$
$14$	0	0
$15$	−2.99750	−0.773950
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−8.21569	−1.88481	−0.942405	−	0.334475i	$-0.891441\pi$
$19$	−8.21569	−1.88481	−0.942405	+	0.334475i	$0.891441\pi$
$20$	0	0
$21$	−2.81889	−0.615132
$22$	0	0
$23$	1.40727	0.293436	0.146718	−	0.989178i	$-0.453129\pi$
$23$	1.40727	0.293436	0.146718	+	0.989178i	$0.453129\pi$
$24$	0	0
$25$	8.24811	1.64962
$26$	0	0
$27$	4.38268	0.843447
$28$	0	0
$29$	−9.95781	−1.84912	−0.924559	−	0.381039i	$-0.875566\pi$
$29$	−9.95781	−1.84912	−0.924559	+	0.381039i	$0.875566\pi$
$30$	0	0
$31$	−6.20012	−1.11357	−0.556787	−	0.830655i	$-0.687966\pi$
$31$	−6.20012	−1.11357	−0.556787	+	0.830655i	$0.687966\pi$
$32$	0	0
$33$	1.37530	0.239409
$34$	0	0
$35$	12.4587	2.10591
$36$	0	0
$37$	−1.91074	−0.314124	−0.157062	−	0.987589i	$-0.550202\pi$
$37$	−1.91074	−0.314124	−0.157062	+	0.987589i	$0.550202\pi$
$38$	0	0
$39$	−1.41257	−0.226192
$40$	0	0
$41$	4.14541	0.647405	0.323702	−	0.946159i	$-0.395072\pi$
$41$	4.14541	0.647405	0.323702	+	0.946159i	$0.395072\pi$
$42$	0	0
$43$	−7.78851	−1.18774	−0.593868	−	0.804563i	$-0.702400\pi$
$43$	−7.78851	−1.18774	−0.593868	+	0.804563i	$0.702400\pi$
$44$	0	0
$45$	−8.45085	−1.25978
$46$	0	0
$47$	3.59899	0.524967	0.262483	−	0.964936i	$-0.415458\pi$
$47$	3.59899	0.524967	0.262483	+	0.964936i	$0.415458\pi$
$48$	0	0
$49$	4.71635	0.673765
$50$	0	0
$51$	0.823534	0.115318
$52$	0	0
$53$	−9.21150	−1.26530	−0.632649	−	0.774439i	$-0.718032\pi$
$53$	−9.21150	−1.26530	−0.632649	+	0.774439i	$0.718032\pi$
$54$	0	0
$55$	−6.07845	−0.819619
$56$	0	0
$57$	6.76590	0.896166
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−9.69562	−1.24140	−0.620698	−	0.784050i	$-0.713151\pi$
$61$	−9.69562	−1.24140	−0.620698	+	0.784050i	$0.713151\pi$
$62$	0	0
$63$	−7.94730	−1.00127
$64$	0	0
$65$	6.24317	0.774370
$66$	0	0
$67$	−7.29905	−0.891721	−0.445861	−	0.895102i	$-0.647102\pi$
$67$	−7.29905	−0.891721	−0.445861	+	0.895102i	$0.647102\pi$
$68$	0	0
$69$	−1.15893	−0.139519
$70$	0	0
$71$	−13.5788	−1.61151	−0.805756	−	0.592248i	$-0.798241\pi$
$71$	−13.5788	−1.61151	−0.805756	+	0.592248i	$0.798241\pi$
$72$	0	0
$73$	4.25258	0.497727	0.248863	−	0.968539i	$-0.419943\pi$
$73$	4.25258	0.497727	0.248863	+	0.968539i	$0.419943\pi$
$74$	0	0
$75$	−6.79260	−0.784342
$76$	0	0
$77$	−5.71627	−0.651429
$78$	0	0
$79$	15.8841	1.78710	0.893552	−	0.448960i	$-0.148205\pi$
$79$	15.8841	1.78710	0.893552	+	0.448960i	$0.148205\pi$
$80$	0	0
$81$	3.35609	0.372899
$82$	0	0
$83$	−7.87799	−0.864722	−0.432361	−	0.901701i	$-0.642319\pi$
$83$	−7.87799	−0.864722	−0.432361	+	0.901701i	$0.642319\pi$
$84$	0	0
$85$	−3.63980	−0.394791
$86$	0	0
$87$	8.20059	0.879196
$88$	0	0
$89$	6.32519	0.670469	0.335235	−	0.942135i	$-0.391184\pi$
$89$	6.32519	0.670469	0.335235	+	0.942135i	$0.391184\pi$
$90$	0	0
$91$	5.87116	0.615465
$92$	0	0
$93$	5.10601	0.529469
$94$	0	0
$95$	−29.9034	−3.06803
$96$	0	0
$97$	12.2886	1.24772	0.623861	−	0.781535i	$-0.285563\pi$
$97$	12.2886	1.24772	0.623861	+	0.781535i	$0.285563\pi$
$98$	0	0
$99$	3.87739	0.389692
$100$	0	0
$101$	−0.139659	−0.0138966	−0.00694828	−	0.999976i	$-0.502212\pi$
$101$	−0.139659	−0.0138966	−0.00694828	+	0.999976i	$0.502212\pi$
$102$	0	0
$103$	−17.4212	−1.71656	−0.858282	−	0.513178i	$-0.828468\pi$
$103$	−17.4212	−1.71656	−0.858282	+	0.513178i	$0.828468\pi$
$104$	0	0
$105$	−10.2602	−1.00129
$106$	0	0
$107$	−11.8435	−1.14496	−0.572479	−	0.819919i	$-0.694018\pi$
$107$	−11.8435	−1.14496	−0.572479	+	0.819919i	$0.694018\pi$
$108$	0	0
$109$	5.88639	0.563814	0.281907	−	0.959442i	$-0.409033\pi$
$109$	5.88639	0.563814	0.281907	+	0.959442i	$0.409033\pi$
$110$	0	0
$111$	1.57356	0.149356
$112$	0	0
$113$	−0.374802	−0.0352584	−0.0176292	−	0.999845i	$-0.505612\pi$
$113$	−0.374802	−0.0352584	−0.0176292	+	0.999845i	$0.505612\pi$
$114$	0	0
$115$	5.12218	0.477645
$116$	0	0
$117$	−3.98246	−0.368178
$118$	0	0
$119$	−3.42292	−0.313778
$120$	0	0
$121$	−8.21110	−0.746464
$122$	0	0
$123$	−3.41389	−0.307820
$124$	0	0
$125$	11.8225	1.05743
$126$	0	0
$127$	−8.86284	−0.786450	−0.393225	−	0.919442i	$-0.628641\pi$
$127$	−8.86284	−0.786450	−0.393225	+	0.919442i	$0.628641\pi$
$128$	0	0
$129$	6.41410	0.564730
$130$	0	0
$131$	−21.3583	−1.86609	−0.933043	−	0.359764i	$-0.882857\pi$
$131$	−21.3583	−1.86609	−0.933043	+	0.359764i	$0.882857\pi$
$132$	0	0
$133$	−28.1216	−2.43845
$134$	0	0
$135$	15.9520	1.37293
$136$	0	0
$137$	−11.7961	−1.00781	−0.503903	−	0.863760i	$-0.668103\pi$
$137$	−11.7961	−1.00781	−0.503903	+	0.863760i	$0.668103\pi$
$138$	0	0
$139$	−7.38315	−0.626230	−0.313115	−	0.949715i	$-0.601373\pi$
$139$	−7.38315	−0.626230	−0.313115	+	0.949715i	$0.601373\pi$
$140$	0	0
$141$	−2.96389	−0.249605
$142$	0	0
$143$	−2.86447	−0.239539
$144$	0	0
$145$	−36.2444	−3.00993
$146$	0	0
$147$	−3.88408	−0.320353
$148$	0	0
$149$	17.2514	1.41329	0.706645	−	0.707568i	$-0.250208\pi$
$149$	17.2514	1.41329	0.706645	+	0.707568i	$0.250208\pi$
$150$	0	0
$151$	−10.7179	−0.872209	−0.436104	−	0.899896i	$-0.643642\pi$
$151$	−10.7179	−0.872209	−0.436104	+	0.899896i	$0.643642\pi$
$152$	0	0
$153$	2.32179	0.187706
$154$	0	0
$155$	−22.5672	−1.81264
$156$	0	0
$157$	10.8328	0.864551	0.432276	−	0.901741i	$-0.357711\pi$
$157$	10.8328	0.864551	0.432276	+	0.901741i	$0.357711\pi$
$158$	0	0
$159$	7.58599	0.601608
$160$	0	0
$161$	4.81697	0.379630
$162$	0	0
$163$	13.6522	1.06932	0.534661	−	0.845067i	$-0.320439\pi$
$163$	13.6522	1.06932	0.534661	+	0.845067i	$0.320439\pi$
$164$	0	0
$165$	5.00581	0.389702
$166$	0	0
$167$	18.6055	1.43974	0.719869	−	0.694110i	$-0.244202\pi$
$167$	18.6055	1.43974	0.719869	+	0.694110i	$0.244202\pi$
$168$	0	0
$169$	−10.0579	−0.773685
$170$	0	0
$171$	19.0751	1.45871
$172$	0	0
$173$	9.08486	0.690709	0.345354	−	0.938472i	$-0.387759\pi$
$173$	9.08486	0.690709	0.345354	+	0.938472i	$0.387759\pi$
$174$	0	0
$175$	28.2326	2.13418
$176$	0	0
$177$	0.823534	0.0619006
$178$	0	0
$179$	−8.47002	−0.633079	−0.316539	−	0.948579i	$-0.602521\pi$
$179$	−8.47002	−0.633079	−0.316539	+	0.948579i	$0.602521\pi$
$180$	0	0
$181$	−10.8643	−0.807537	−0.403769	−	0.914861i	$-0.632300\pi$
$181$	−10.8643	−0.807537	−0.403769	+	0.914861i	$0.632300\pi$
$182$	0	0
$183$	7.98467	0.590244
$184$	0	0
$185$	−6.95470	−0.511320
$186$	0	0
$187$	1.67000	0.122122
$188$	0	0
$189$	15.0015	1.09120
$190$	0	0
$191$	9.82479	0.710897	0.355448	−	0.934696i	$-0.384328\pi$
$191$	9.82479	0.710897	0.355448	+	0.934696i	$0.384328\pi$
$192$	0	0
$193$	3.28013	0.236109	0.118054	−	0.993007i	$-0.462334\pi$
$193$	3.28013	0.236109	0.118054	+	0.993007i	$0.462334\pi$
$194$	0	0
$195$	−5.14146	−0.368188
$196$	0	0
$197$	11.3655	0.809761	0.404880	−	0.914370i	$-0.367313\pi$
$197$	11.3655	0.809761	0.404880	+	0.914370i	$0.367313\pi$
$198$	0	0
$199$	17.8798	1.26746	0.633731	−	0.773554i	$-0.281523\pi$
$199$	17.8798	1.26746	0.633731	+	0.773554i	$0.281523\pi$
$200$	0	0
$201$	6.01102	0.423985
$202$	0	0
$203$	−34.0847	−2.39228
$204$	0	0
$205$	15.0885	1.05382
$206$	0	0
$207$	−3.26739	−0.227099
$208$	0	0
$209$	13.7202	0.949046
$210$	0	0
$211$	12.3055	0.847146	0.423573	−	0.905862i	$-0.360776\pi$
$211$	12.3055	0.847146	0.423573	+	0.905862i	$0.360776\pi$
$212$	0	0
$213$	11.1826	0.766221
$214$	0	0
$215$	−28.3486	−1.93336
$216$	0	0
$217$	−21.2225	−1.44068
$218$	0	0
$219$	−3.50214	−0.236653
$220$	0	0
$221$	−1.71525	−0.115380
$222$	0	0
$223$	7.01903	0.470029	0.235015	−	0.971992i	$-0.424486\pi$
$223$	7.01903	0.470029	0.235015	+	0.971992i	$0.424486\pi$
$224$	0	0
$225$	−19.1504	−1.27669
$226$	0	0
$227$	−12.0704	−0.801140	−0.400570	−	0.916266i	$-0.631188\pi$
$227$	−12.0704	−0.801140	−0.400570	+	0.916266i	$0.631188\pi$
$228$	0	0
$229$	14.8254	0.979689	0.489845	−	0.871810i	$-0.337053\pi$
$229$	14.8254	0.979689	0.489845	+	0.871810i	$0.337053\pi$
$230$	0	0
$231$	4.70754	0.309733
$232$	0	0
$233$	−7.32389	−0.479804	−0.239902	−	0.970797i	$-0.577115\pi$
$233$	−7.32389	−0.479804	−0.239902	+	0.970797i	$0.577115\pi$
$234$	0	0
$235$	13.0996	0.854523
$236$	0	0
$237$	−13.0811	−0.849710
$238$	0	0
$239$	−8.14583	−0.526910	−0.263455	−	0.964672i	$-0.584862\pi$
$239$	−8.14583	−0.526910	−0.263455	+	0.964672i	$0.584862\pi$
$240$	0	0
$241$	0.717882	0.0462428	0.0231214	−	0.999733i	$-0.492640\pi$
$241$	0.717882	0.0462428	0.0231214	+	0.999733i	$0.492640\pi$
$242$	0	0
$243$	−15.9119	−1.02075
$244$	0	0
$245$	17.1666	1.09673
$246$	0	0
$247$	−14.0920	−0.896652
$248$	0	0
$249$	6.48779	0.411147
$250$	0	0
$251$	31.0093	1.95729	0.978645	−	0.205557i	$-0.0659007\pi$
$251$	31.0093	1.95729	0.978645	+	0.205557i	$0.0659007\pi$
$252$	0	0
$253$	−2.35014	−0.147752
$254$	0	0
$255$	2.99750	0.187710
$256$	0	0
$257$	22.3229	1.39247	0.696233	−	0.717816i	$-0.254858\pi$
$257$	22.3229	1.39247	0.696233	+	0.717816i	$0.254858\pi$
$258$	0	0
$259$	−6.54030	−0.406394
$260$	0	0
$261$	23.1199	1.43109
$262$	0	0
$263$	22.6184	1.39471	0.697355	−	0.716726i	$-0.254360\pi$
$263$	22.6184	1.39471	0.697355	+	0.716726i	$0.254360\pi$
$264$	0	0
$265$	−33.5280	−2.05961
$266$	0	0
$267$	−5.20901	−0.318786
$268$	0	0
$269$	8.03588	0.489956	0.244978	−	0.969529i	$-0.421219\pi$
$269$	8.03588	0.489956	0.244978	+	0.969529i	$0.421219\pi$
$270$	0	0
$271$	2.90056	0.176196	0.0880982	−	0.996112i	$-0.471921\pi$
$271$	2.90056	0.176196	0.0880982	+	0.996112i	$0.471921\pi$
$272$	0	0
$273$	−4.83510	−0.292634
$274$	0	0
$275$	−13.7743	−0.830624
$276$	0	0
$277$	7.72804	0.464333	0.232167	−	0.972676i	$-0.425419\pi$
$277$	7.72804	0.464333	0.232167	+	0.972676i	$0.425419\pi$
$278$	0	0
$279$	14.3954	0.861830
$280$	0	0
$281$	5.90146	0.352051	0.176026	−	0.984386i	$-0.443676\pi$
$281$	5.90146	0.352051	0.176026	+	0.984386i	$0.443676\pi$
$282$	0	0
$283$	30.2948	1.80084	0.900419	−	0.435024i	$-0.143260\pi$
$283$	30.2948	1.80084	0.900419	+	0.435024i	$0.143260\pi$
$284$	0	0
$285$	24.6265	1.45875
$286$	0	0
$287$	14.1894	0.837573
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.1201	−0.593252
$292$	0	0
$293$	20.1311	1.17607	0.588036	−	0.808835i	$-0.299901\pi$
$293$	20.1311	1.17607	0.588036	+	0.808835i	$0.299901\pi$
$294$	0	0
$295$	−3.63980	−0.211917
$296$	0	0
$297$	−7.31906	−0.424695
$298$	0	0
$299$	2.41382	0.139595
$300$	0	0
$301$	−26.6594	−1.53662
$302$	0	0
$303$	0.115014	0.00660736
$304$	0	0
$305$	−35.2901	−2.02070
$306$	0	0
$307$	−10.4030	−0.593729	−0.296864	−	0.954920i	$-0.595941\pi$
$307$	−10.4030	−0.593729	−0.296864	+	0.954920i	$0.595941\pi$
$308$	0	0
$309$	14.3470	0.816171
$310$	0	0
$311$	−10.0866	−0.571959	−0.285979	−	0.958236i	$-0.592319\pi$
$311$	−10.0866	−0.571959	−0.285979	+	0.958236i	$0.592319\pi$
$312$	0	0
$313$	−13.7156	−0.775254	−0.387627	−	0.921816i	$-0.626705\pi$
$313$	−13.7156	−0.775254	−0.387627	+	0.921816i	$0.626705\pi$
$314$	0	0
$315$	−28.9265	−1.62983
$316$	0	0
$317$	−32.6090	−1.83150	−0.915751	−	0.401745i	$-0.868404\pi$
$317$	−32.6090	−1.83150	−0.915751	+	0.401745i	$0.868404\pi$
$318$	0	0
$319$	16.6295	0.931074
$320$	0	0
$321$	9.75356	0.544390
$322$	0	0
$323$	8.21569	0.457133
$324$	0	0
$325$	14.1476	0.784767
$326$	0	0
$327$	−4.84764	−0.268075
$328$	0	0
$329$	12.3190	0.679171
$330$	0	0
$331$	−2.91630	−0.160294	−0.0801472	−	0.996783i	$-0.525539\pi$
$331$	−2.91630	−0.160294	−0.0801472	+	0.996783i	$0.525539\pi$
$332$	0	0
$333$	4.43634	0.243110
$334$	0	0
$335$	−26.5671	−1.45151
$336$	0	0
$337$	−7.45047	−0.405853	−0.202926	−	0.979194i	$-0.565045\pi$
$337$	−7.45047	−0.405853	−0.202926	+	0.979194i	$0.565045\pi$
$338$	0	0
$339$	0.308662	0.0167642
$340$	0	0
$341$	10.3542	0.560711
$342$	0	0
$343$	−7.81673	−0.422064
$344$	0	0
$345$	−4.21829	−0.227105
$346$	0	0
$347$	1.08658	0.0583307	0.0291653	−	0.999575i	$-0.490715\pi$
$347$	1.08658	0.0583307	0.0291653	+	0.999575i	$0.490715\pi$
$348$	0	0
$349$	29.8659	1.59869	0.799344	−	0.600874i	$-0.205181\pi$
$349$	29.8659	1.59869	0.799344	+	0.600874i	$0.205181\pi$
$350$	0	0
$351$	7.51740	0.401249
$352$	0	0
$353$	30.3790	1.61691	0.808454	−	0.588559i	$-0.200305\pi$
$353$	30.3790	1.61691	0.808454	+	0.588559i	$0.200305\pi$
$354$	0	0
$355$	−49.4242	−2.62316
$356$	0	0
$357$	2.81889	0.149191
$358$	0	0
$359$	28.7554	1.51765	0.758826	−	0.651293i	$-0.225773\pi$
$359$	28.7554	1.51765	0.758826	+	0.651293i	$0.225773\pi$
$360$	0	0
$361$	48.4976	2.55251
$362$	0	0
$363$	6.76212	0.354919
$364$	0	0
$365$	15.4785	0.810183
$366$	0	0
$367$	−10.1650	−0.530609	−0.265305	−	0.964165i	$-0.585473\pi$
$367$	−10.1650	−0.530609	−0.265305	+	0.964165i	$0.585473\pi$
$368$	0	0
$369$	−9.62478	−0.501046
$370$	0	0
$371$	−31.5302	−1.63697
$372$	0	0
$373$	−12.3167	−0.637733	−0.318866	−	0.947800i	$-0.603302\pi$
$373$	−12.3167	−0.637733	−0.318866	+	0.947800i	$0.603302\pi$
$374$	0	0
$375$	−9.73620	−0.502775
$376$	0	0
$377$	−17.0801	−0.879672
$378$	0	0
$379$	−21.7104	−1.11519	−0.557593	−	0.830114i	$-0.688275\pi$
$379$	−21.7104	−1.11519	−0.557593	+	0.830114i	$0.688275\pi$
$380$	0	0
$381$	7.29885	0.373932
$382$	0	0
$383$	−25.1847	−1.28688	−0.643438	−	0.765498i	$-0.722493\pi$
$383$	−25.1847	−1.28688	−0.643438	+	0.765498i	$0.722493\pi$
$384$	0	0
$385$	−20.8060	−1.06037
$386$	0	0
$387$	18.0833	0.919225
$388$	0	0
$389$	19.3852	0.982870	0.491435	−	0.870914i	$-0.336472\pi$
$389$	19.3852	0.982870	0.491435	+	0.870914i	$0.336472\pi$
$390$	0	0
$391$	−1.40727	−0.0711687
$392$	0	0
$393$	17.5893	0.887264
$394$	0	0
$395$	57.8150	2.90899
$396$	0	0
$397$	−8.35493	−0.419322	−0.209661	−	0.977774i	$-0.567236\pi$
$397$	−8.35493	−0.419322	−0.209661	+	0.977774i	$0.567236\pi$
$398$	0	0
$399$	23.1591	1.15941
$400$	0	0
$401$	−14.7173	−0.734946	−0.367473	−	0.930034i	$-0.619777\pi$
$401$	−14.7173	−0.734946	−0.367473	+	0.930034i	$0.619777\pi$
$402$	0	0
$403$	−10.6348	−0.529756
$404$	0	0
$405$	12.2155	0.606992
$406$	0	0
$407$	3.19093	0.158169
$408$	0	0
$409$	−15.5965	−0.771195	−0.385597	−	0.922667i	$-0.626005\pi$
$409$	−15.5965	−0.771195	−0.385597	+	0.922667i	$0.626005\pi$
$410$	0	0
$411$	9.71446	0.479179
$412$	0	0
$413$	−3.42292	−0.168431
$414$	0	0
$415$	−28.6743	−1.40757
$416$	0	0
$417$	6.08027	0.297752
$418$	0	0
$419$	−23.7769	−1.16158	−0.580789	−	0.814054i	$-0.697256\pi$
$419$	−23.7769	−1.16158	−0.580789	+	0.814054i	$0.697256\pi$
$420$	0	0
$421$	5.13183	0.250110	0.125055	−	0.992150i	$-0.460089\pi$
$421$	5.13183	0.250110	0.125055	+	0.992150i	$0.460089\pi$
$422$	0	0
$423$	−8.35611	−0.406288
$424$	0	0
$425$	−8.24811	−0.400092
$426$	0	0
$427$	−33.1873	−1.60604
$428$	0	0
$429$	2.35899	0.113893
$430$	0	0
$431$	−15.5724	−0.750094	−0.375047	−	0.927006i	$-0.622374\pi$
$431$	−15.5724	−0.750094	−0.375047	+	0.927006i	$0.622374\pi$
$432$	0	0
$433$	3.35744	0.161348	0.0806742	−	0.996741i	$-0.474293\pi$
$433$	3.35744	0.161348	0.0806742	+	0.996741i	$0.474293\pi$
$434$	0	0
$435$	29.8485	1.43113
$436$	0	0
$437$	−11.5617	−0.553071
$438$	0	0
$439$	13.2109	0.630521	0.315261	−	0.949005i	$-0.397908\pi$
$439$	13.2109	0.630521	0.315261	+	0.949005i	$0.397908\pi$
$440$	0	0
$441$	−10.9504	−0.521447
$442$	0	0
$443$	−15.8529	−0.753195	−0.376598	−	0.926377i	$-0.622906\pi$
$443$	−15.8529	−0.753195	−0.376598	+	0.926377i	$0.622906\pi$
$444$	0	0
$445$	23.0224	1.09137
$446$	0	0
$447$	−14.2071	−0.671974
$448$	0	0
$449$	34.2691	1.61726	0.808630	−	0.588317i	$-0.200209\pi$
$449$	34.2691	1.61726	0.808630	+	0.588317i	$0.200209\pi$
$450$	0	0
$451$	−6.92283	−0.325983
$452$	0	0
$453$	8.82654	0.414707
$454$	0	0
$455$	21.3698	1.00183
$456$	0	0
$457$	−19.3054	−0.903068	−0.451534	−	0.892254i	$-0.649123\pi$
$457$	−19.3054	−0.903068	−0.451534	+	0.892254i	$0.649123\pi$
$458$	0	0
$459$	−4.38268	−0.204566
$460$	0	0
$461$	−16.8151	−0.783157	−0.391579	−	0.920145i	$-0.628071\pi$
$461$	−16.8151	−0.783157	−0.391579	+	0.920145i	$0.628071\pi$
$462$	0	0
$463$	−23.5431	−1.09414	−0.547070	−	0.837087i	$-0.684257\pi$
$463$	−23.5431	−1.09414	−0.547070	+	0.837087i	$0.684257\pi$
$464$	0	0
$465$	18.5848	0.861851
$466$	0	0
$467$	−20.8183	−0.963356	−0.481678	−	0.876348i	$-0.659972\pi$
$467$	−20.8183	−0.963356	−0.481678	+	0.876348i	$0.659972\pi$
$468$	0	0
$469$	−24.9841	−1.15366
$470$	0	0
$471$	−8.92118	−0.411066
$472$	0	0
$473$	13.0068	0.598053
$474$	0	0
$475$	−67.7640	−3.10922
$476$	0	0
$477$	21.3872	0.979252
$478$	0	0
$479$	−29.0694	−1.32821	−0.664107	−	0.747637i	$-0.731188\pi$
$479$	−29.0694	−1.32821	−0.664107	+	0.747637i	$0.731188\pi$
$480$	0	0
$481$	−3.27740	−0.149437
$482$	0	0
$483$	−3.96694	−0.180502
$484$	0	0
$485$	44.7282	2.03100
$486$	0	0
$487$	−12.2714	−0.556068	−0.278034	−	0.960571i	$-0.589683\pi$
$487$	−12.2714	−0.556068	−0.278034	+	0.960571i	$0.589683\pi$
$488$	0	0
$489$	−11.2430	−0.508428
$490$	0	0
$491$	19.8158	0.894277	0.447138	−	0.894465i	$-0.352443\pi$
$491$	19.8158	0.894277	0.447138	+	0.894465i	$0.352443\pi$
$492$	0	0
$493$	9.95781	0.448477
$494$	0	0
$495$	14.1129	0.634328
$496$	0	0
$497$	−46.4792	−2.08488
$498$	0	0
$499$	−6.43165	−0.287920	−0.143960	−	0.989584i	$-0.545984\pi$
$499$	−6.43165	−0.287920	−0.143960	+	0.989584i	$0.545984\pi$
$500$	0	0
$501$	−15.3223	−0.684549
$502$	0	0
$503$	10.2693	0.457886	0.228943	−	0.973440i	$-0.426473\pi$
$503$	10.2693	0.457886	0.228943	+	0.973440i	$0.426473\pi$
$504$	0	0
$505$	−0.508329	−0.0226203
$506$	0	0
$507$	8.28303	0.367862
$508$	0	0
$509$	15.6208	0.692380	0.346190	−	0.938164i	$-0.387475\pi$
$509$	15.6208	0.692380	0.346190	+	0.938164i	$0.387475\pi$
$510$	0	0
$511$	14.5562	0.643929
$512$	0	0
$513$	−36.0067	−1.58974
$514$	0	0
$515$	−63.4097	−2.79416
$516$	0	0
$517$	−6.01031	−0.264333
$518$	0	0
$519$	−7.48169	−0.328410
$520$	0	0
$521$	5.55808	0.243504	0.121752	−	0.992561i	$-0.461149\pi$
$521$	5.55808	0.243504	0.121752	+	0.992561i	$0.461149\pi$
$522$	0	0
$523$	4.22156	0.184596	0.0922980	−	0.995731i	$-0.470579\pi$
$523$	4.22156	0.184596	0.0922980	+	0.995731i	$0.470579\pi$
$524$	0	0
$525$	−23.2505	−1.01474
$526$	0	0
$527$	6.20012	0.270082
$528$	0	0
$529$	−21.0196	−0.913895
$530$	0	0
$531$	2.32179	0.100757
$532$	0	0
$533$	7.11043	0.307987
$534$	0	0
$535$	−43.1081	−1.86372
$536$	0	0
$537$	6.97535	0.301009
$538$	0	0
$539$	−7.87630	−0.339256
$540$	0	0
$541$	−39.0422	−1.67856	−0.839278	−	0.543703i	$-0.817022\pi$
$541$	−39.0422	−1.67856	−0.839278	+	0.543703i	$0.817022\pi$
$542$	0	0
$543$	8.94712	0.383958
$544$	0	0
$545$	21.4253	0.917757
$546$	0	0
$547$	−27.8232	−1.18964	−0.594818	−	0.803861i	$-0.702776\pi$
$547$	−27.8232	−1.18964	−0.594818	+	0.803861i	$0.702776\pi$
$548$	0	0
$549$	22.5112	0.960755
$550$	0	0
$551$	81.8103	3.48523
$552$	0	0
$553$	54.3700	2.31205
$554$	0	0
$555$	5.72743	0.243116
$556$	0	0
$557$	−35.7360	−1.51418	−0.757092	−	0.653308i	$-0.773381\pi$
$557$	−35.7360	−1.51418	−0.757092	+	0.653308i	$0.773381\pi$
$558$	0	0
$559$	−13.3593	−0.565036
$560$	0	0
$561$	−1.37530	−0.0580652
$562$	0	0
$563$	−36.5933	−1.54222	−0.771112	−	0.636700i	$-0.780299\pi$
$563$	−36.5933	−1.54222	−0.771112	+	0.636700i	$0.780299\pi$
$564$	0	0
$565$	−1.36420	−0.0573924
$566$	0	0
$567$	11.4876	0.482435
$568$	0	0
$569$	38.6467	1.62016	0.810078	−	0.586323i	$-0.199425\pi$
$569$	38.6467	1.62016	0.810078	+	0.586323i	$0.199425\pi$
$570$	0	0
$571$	23.8657	0.998747	0.499374	−	0.866387i	$-0.333564\pi$
$571$	23.8657	0.998747	0.499374	+	0.866387i	$0.333564\pi$
$572$	0	0
$573$	−8.09105	−0.338008
$574$	0	0
$575$	11.6073	0.484059
$576$	0	0
$577$	8.29767	0.345437	0.172718	−	0.984971i	$-0.444745\pi$
$577$	8.29767	0.345437	0.172718	+	0.984971i	$0.444745\pi$
$578$	0	0
$579$	−2.70130	−0.112262
$580$	0	0
$581$	−26.9657	−1.11873
$582$	0	0
$583$	15.3832	0.637107
$584$	0	0
$585$	−14.4953	−0.599308
$586$	0	0
$587$	−1.22441	−0.0505367	−0.0252684	−	0.999681i	$-0.508044\pi$
$587$	−1.22441	−0.0505367	−0.0252684	+	0.999681i	$0.508044\pi$
$588$	0	0
$589$	50.9383	2.09888
$590$	0	0
$591$	−9.35990	−0.385015
$592$	0	0
$593$	28.8137	1.18324	0.591618	−	0.806219i	$-0.298490\pi$
$593$	28.8137	1.18324	0.591618	+	0.806219i	$0.298490\pi$
$594$	0	0
$595$	−12.4587	−0.510758
$596$	0	0
$597$	−14.7246	−0.602637
$598$	0	0
$599$	35.4720	1.44935	0.724673	−	0.689093i	$-0.241991\pi$
$599$	35.4720	1.44935	0.724673	+	0.689093i	$0.241991\pi$
$600$	0	0
$601$	−25.8542	−1.05461	−0.527307	−	0.849675i	$-0.676798\pi$
$601$	−25.8542	−1.05461	−0.527307	+	0.849675i	$0.676798\pi$
$602$	0	0
$603$	16.9469	0.690130
$604$	0	0
$605$	−29.8867	−1.21507
$606$	0	0
$607$	−22.6106	−0.917737	−0.458869	−	0.888504i	$-0.651745\pi$
$607$	−22.6106	−0.917737	−0.458869	+	0.888504i	$0.651745\pi$
$608$	0	0
$609$	28.0699	1.13745
$610$	0	0
$611$	6.17318	0.249740
$612$	0	0
$613$	20.9462	0.846009	0.423005	−	0.906128i	$-0.360975\pi$
$613$	20.9462	0.846009	0.423005	+	0.906128i	$0.360975\pi$
$614$	0	0
$615$	−12.4259	−0.501059
$616$	0	0
$617$	−29.7620	−1.19817	−0.599086	−	0.800685i	$-0.704469\pi$
$617$	−29.7620	−1.19817	−0.599086	+	0.800685i	$0.704469\pi$
$618$	0	0
$619$	−25.5098	−1.02533	−0.512663	−	0.858590i	$-0.671341\pi$
$619$	−25.5098	−1.02533	−0.512663	+	0.858590i	$0.671341\pi$
$620$	0	0
$621$	6.16761	0.247498
$622$	0	0
$623$	21.6506	0.867413
$624$	0	0
$625$	1.79080	0.0716321
$626$	0	0
$627$	−11.2990	−0.451241
$628$	0	0
$629$	1.91074	0.0761862
$630$	0	0
$631$	1.66481	0.0662751	0.0331376	−	0.999451i	$-0.489450\pi$
$631$	1.66481	0.0662751	0.0331376	+	0.999451i	$0.489450\pi$
$632$	0	0
$633$	−10.1340	−0.402790
$634$	0	0
$635$	−32.2589	−1.28016
$636$	0	0
$637$	8.08974	0.320527
$638$	0	0
$639$	31.5272	1.24720
$640$	0	0
$641$	−2.10967	−0.0833271	−0.0416636	−	0.999132i	$-0.513266\pi$
$641$	−2.10967	−0.0833271	−0.0416636	+	0.999132i	$0.513266\pi$
$642$	0	0
$643$	0.0991497	0.00391008	0.00195504	−	0.999998i	$-0.499378\pi$
$643$	0.0991497	0.00391008	0.00195504	+	0.999998i	$0.499378\pi$
$644$	0	0
$645$	23.3460	0.919248
$646$	0	0
$647$	−18.7453	−0.736956	−0.368478	−	0.929637i	$-0.620121\pi$
$647$	−18.7453	−0.736956	−0.368478	+	0.929637i	$0.620121\pi$
$648$	0	0
$649$	1.67000	0.0655532
$650$	0	0
$651$	17.4775	0.684995
$652$	0	0
$653$	−30.5985	−1.19741	−0.598707	−	0.800968i	$-0.704318\pi$
$653$	−30.5985	−1.19741	−0.598707	+	0.800968i	$0.704318\pi$
$654$	0	0
$655$	−77.7400	−3.03755
$656$	0	0
$657$	−9.87361	−0.385206
$658$	0	0
$659$	−34.2995	−1.33612	−0.668059	−	0.744108i	$-0.732875\pi$
$659$	−34.2995	−1.33612	−0.668059	+	0.744108i	$0.732875\pi$
$660$	0	0
$661$	12.6750	0.492999	0.246500	−	0.969143i	$-0.420720\pi$
$661$	12.6750	0.492999	0.246500	+	0.969143i	$0.420720\pi$
$662$	0	0
$663$	1.41257	0.0548596
$664$	0	0
$665$	−102.357	−3.96923
$666$	0	0
$667$	−14.0133	−0.542598
$668$	0	0
$669$	−5.78041	−0.223484
$670$	0	0
$671$	16.1917	0.625072
$672$	0	0
$673$	−11.4390	−0.440940	−0.220470	−	0.975394i	$-0.570759\pi$
$673$	−11.4390	−0.440940	−0.220470	+	0.975394i	$0.570759\pi$
$674$	0	0
$675$	36.1488	1.39137
$676$	0	0
$677$	28.5089	1.09569	0.547844	−	0.836581i	$-0.315449\pi$
$677$	28.5089	1.09569	0.547844	+	0.836581i	$0.315449\pi$
$678$	0	0
$679$	42.0630	1.61423
$680$	0	0
$681$	9.94038	0.380916
$682$	0	0
$683$	−26.8553	−1.02759	−0.513795	−	0.857913i	$-0.671761\pi$
$683$	−26.8553	−1.02759	−0.513795	+	0.857913i	$0.671761\pi$
$684$	0	0
$685$	−42.9353	−1.64047
$686$	0	0
$687$	−12.2092	−0.465811
$688$	0	0
$689$	−15.8001	−0.601934
$690$	0	0
$691$	40.7700	1.55096	0.775482	−	0.631370i	$-0.217507\pi$
$691$	40.7700	1.55096	0.775482	+	0.631370i	$0.217507\pi$
$692$	0	0
$693$	13.2720	0.504161
$694$	0	0
$695$	−26.8732	−1.01936
$696$	0	0
$697$	−4.14541	−0.157019
$698$	0	0
$699$	6.03148	0.228131
$700$	0	0
$701$	25.4972	0.963016	0.481508	−	0.876442i	$-0.340089\pi$
$701$	25.4972	0.963016	0.481508	+	0.876442i	$0.340089\pi$
$702$	0	0
$703$	15.6980	0.592063
$704$	0	0
$705$	−10.7880	−0.406298
$706$	0	0
$707$	−0.478040	−0.0179785
$708$	0	0
$709$	14.0923	0.529246	0.264623	−	0.964352i	$-0.414752\pi$
$709$	14.0923	0.529246	0.264623	+	0.964352i	$0.414752\pi$
$710$	0	0
$711$	−36.8796	−1.38309
$712$	0	0
$713$	−8.72525	−0.326763
$714$	0	0
$715$	−10.4261	−0.389913
$716$	0	0
$717$	6.70837	0.250529
$718$	0	0
$719$	36.8851	1.37558	0.687791	−	0.725909i	$-0.258581\pi$
$719$	36.8851	1.37558	0.687791	+	0.725909i	$0.258581\pi$
$720$	0	0
$721$	−59.6314	−2.22079
$722$	0	0
$723$	−0.591200	−0.0219870
$724$	0	0
$725$	−82.1331	−3.05035
$726$	0	0
$727$	−27.3173	−1.01314	−0.506572	−	0.862198i	$-0.669087\pi$
$727$	−27.3173	−1.01314	−0.506572	+	0.862198i	$0.669087\pi$
$728$	0	0
$729$	3.03571	0.112434
$730$	0	0
$731$	7.78851	0.288068
$732$	0	0
$733$	−6.50008	−0.240086	−0.120043	−	0.992769i	$-0.538303\pi$
$733$	−6.50008	−0.240086	−0.120043	+	0.992769i	$0.538303\pi$
$734$	0	0
$735$	−14.1372	−0.521460
$736$	0	0
$737$	12.1894	0.449003
$738$	0	0
$739$	53.5474	1.96977	0.984887	−	0.173201i	$-0.0554109\pi$
$739$	53.5474	1.96977	0.984887	+	0.173201i	$0.0554109\pi$
$740$	0	0
$741$	11.6052	0.426329
$742$	0	0
$743$	−22.3363	−0.819441	−0.409720	−	0.912211i	$-0.634374\pi$
$743$	−22.3363	−0.819441	−0.409720	+	0.912211i	$0.634374\pi$
$744$	0	0
$745$	62.7916	2.30051
$746$	0	0
$747$	18.2911	0.669235
$748$	0	0
$749$	−40.5394	−1.48128
$750$	0	0
$751$	18.5529	0.677004	0.338502	−	0.940966i	$-0.390080\pi$
$751$	18.5529	0.677004	0.338502	+	0.940966i	$0.390080\pi$
$752$	0	0
$753$	−25.5372	−0.930628
$754$	0	0
$755$	−39.0109	−1.41975
$756$	0	0
$757$	26.2770	0.955054	0.477527	−	0.878617i	$-0.341533\pi$
$757$	26.2770	0.955054	0.477527	+	0.878617i	$0.341533\pi$
$758$	0	0
$759$	1.93542	0.0702513
$760$	0	0
$761$	−36.6124	−1.32720	−0.663600	−	0.748088i	$-0.730972\pi$
$761$	−36.6124	−1.32720	−0.663600	+	0.748088i	$0.730972\pi$
$762$	0	0
$763$	20.1486	0.729429
$764$	0	0
$765$	8.45085	0.305541
$766$	0	0
$767$	−1.71525	−0.0619342
$768$	0	0
$769$	42.6282	1.53721	0.768606	−	0.639723i	$-0.220951\pi$
$769$	42.6282	1.53721	0.768606	+	0.639723i	$0.220951\pi$
$770$	0	0
$771$	−18.3837	−0.662073
$772$	0	0
$773$	4.55031	0.163663	0.0818316	−	0.996646i	$-0.473923\pi$
$773$	4.55031	0.163663	0.0818316	+	0.996646i	$0.473923\pi$
$774$	0	0
$775$	−51.1393	−1.83698
$776$	0	0
$777$	5.38616	0.193227
$778$	0	0
$779$	−34.0574	−1.22023
$780$	0	0
$781$	22.6766	0.811434
$782$	0	0
$783$	−43.6418	−1.55963
$784$	0	0
$785$	39.4292	1.40729
$786$	0	0
$787$	40.8442	1.45594	0.727969	−	0.685610i	$-0.240464\pi$
$787$	40.8442	1.45594	0.727969	+	0.685610i	$0.240464\pi$
$788$	0	0
$789$	−18.6270	−0.663139
$790$	0	0
$791$	−1.28291	−0.0456152
$792$	0	0
$793$	−16.6304	−0.590564
$794$	0	0
$795$	27.6114	0.979277
$796$	0	0
$797$	39.3181	1.39272	0.696359	−	0.717694i	$-0.254802\pi$
$797$	39.3181	1.39272	0.696359	+	0.717694i	$0.254802\pi$
$798$	0	0
$799$	−3.59899	−0.127323
$800$	0	0
$801$	−14.6858	−0.518897
$802$	0	0
$803$	−7.10180	−0.250617
$804$	0	0
$805$	17.5328	0.617949
$806$	0	0
$807$	−6.61782	−0.232958
$808$	0	0
$809$	23.0635	0.810871	0.405435	−	0.914124i	$-0.367120\pi$
$809$	23.0635	0.810871	0.405435	+	0.914124i	$0.367120\pi$
$810$	0	0
$811$	−34.6779	−1.21770	−0.608852	−	0.793284i	$-0.708370\pi$
$811$	−34.6779	−1.21770	−0.608852	+	0.793284i	$0.708370\pi$
$812$	0	0
$813$	−2.38871	−0.0837757
$814$	0	0
$815$	49.6912	1.74061
$816$	0	0
$817$	63.9880	2.23866
$818$	0	0
$819$	−13.6316	−0.476327
$820$	0	0
$821$	−16.0009	−0.558434	−0.279217	−	0.960228i	$-0.590075\pi$
$821$	−16.0009	−0.558434	−0.279217	+	0.960228i	$0.590075\pi$
$822$	0	0
$823$	−19.6078	−0.683485	−0.341742	−	0.939794i	$-0.611017\pi$
$823$	−19.6078	−0.683485	−0.341742	+	0.939794i	$0.611017\pi$
$824$	0	0
$825$	11.3436	0.394935
$826$	0	0
$827$	0.558266	0.0194128	0.00970641	−	0.999953i	$-0.496910\pi$
$827$	0.558266	0.0194128	0.00970641	+	0.999953i	$0.496910\pi$
$828$	0	0
$829$	−42.5804	−1.47888	−0.739439	−	0.673224i	$-0.764909\pi$
$829$	−42.5804	−1.47888	−0.739439	+	0.673224i	$0.764909\pi$
$830$	0	0
$831$	−6.36431	−0.220775
$832$	0	0
$833$	−4.71635	−0.163412
$834$	0	0
$835$	67.7203	2.34356
$836$	0	0
$837$	−27.1731	−0.939241
$838$	0	0
$839$	12.9527	0.447178	0.223589	−	0.974684i	$-0.428223\pi$
$839$	12.9527	0.447178	0.223589	+	0.974684i	$0.428223\pi$
$840$	0	0
$841$	70.1579	2.41924
$842$	0	0
$843$	−4.86005	−0.167389
$844$	0	0
$845$	−36.6087	−1.25938
$846$	0	0
$847$	−28.1059	−0.965731
$848$	0	0
$849$	−24.9488	−0.856240
$850$	0	0
$851$	−2.68893	−0.0921752
$852$	0	0
$853$	40.7420	1.39498	0.697490	−	0.716594i	$-0.254300\pi$
$853$	40.7420	1.39498	0.697490	+	0.716594i	$0.254300\pi$
$854$	0	0
$855$	69.4296	2.37444
$856$	0	0
$857$	−45.3564	−1.54934	−0.774672	−	0.632363i	$-0.782085\pi$
$857$	−45.3564	−1.54934	−0.774672	+	0.632363i	$0.782085\pi$
$858$	0	0
$859$	−3.30053	−0.112613	−0.0563064	−	0.998414i	$-0.517932\pi$
$859$	−3.30053	−0.112613	−0.0563064	+	0.998414i	$0.517932\pi$
$860$	0	0
$861$	−11.6855	−0.398239
$862$	0	0
$863$	−32.4319	−1.10399	−0.551997	−	0.833846i	$-0.686134\pi$
$863$	−32.4319	−1.10399	−0.551997	+	0.833846i	$0.686134\pi$
$864$	0	0
$865$	33.0670	1.12431
$866$	0	0
$867$	−0.823534	−0.0279687
$868$	0	0
$869$	−26.5265	−0.899849
$870$	0	0
$871$	−12.5197	−0.424215
$872$	0	0
$873$	−28.5317	−0.965651
$874$	0	0
$875$	40.4673	1.36804
$876$	0	0
$877$	−28.1805	−0.951587	−0.475794	−	0.879557i	$-0.657839\pi$
$877$	−28.1805	−0.951587	−0.475794	+	0.879557i	$0.657839\pi$
$878$	0	0
$879$	−16.5787	−0.559184
$880$	0	0
$881$	−41.2321	−1.38915	−0.694573	−	0.719423i	$-0.744407\pi$
$881$	−41.2321	−1.38915	−0.694573	+	0.719423i	$0.744407\pi$
$882$	0	0
$883$	−13.8254	−0.465263	−0.232631	−	0.972565i	$-0.574734\pi$
$883$	−13.8254	−0.465263	−0.232631	+	0.972565i	$0.574734\pi$
$884$	0	0
$885$	2.99750	0.100760
$886$	0	0
$887$	−14.1521	−0.475181	−0.237590	−	0.971365i	$-0.576358\pi$
$887$	−14.1521	−0.475181	−0.237590	+	0.971365i	$0.576358\pi$
$888$	0	0
$889$	−30.3368	−1.01746
$890$	0	0
$891$	−5.60467	−0.187763
$892$	0	0
$893$	−29.5682	−0.989462
$894$	0	0
$895$	−30.8291	−1.03050
$896$	0	0
$897$	−1.98787	−0.0663729
$898$	0	0
$899$	61.7396	2.05913
$900$	0	0
$901$	9.21150	0.306880
$902$	0	0
$903$	21.9549	0.730614
$904$	0	0
$905$	−39.5438	−1.31448
$906$	0	0
$907$	7.85160	0.260708	0.130354	−	0.991468i	$-0.458389\pi$
$907$	7.85160	0.260708	0.130354	+	0.991468i	$0.458389\pi$
$908$	0	0
$909$	0.324258	0.0107550
$910$	0	0
$911$	15.2329	0.504688	0.252344	−	0.967638i	$-0.418799\pi$
$911$	15.2329	0.504688	0.252344	+	0.967638i	$0.418799\pi$
$912$	0	0
$913$	13.1562	0.435408
$914$	0	0
$915$	29.0626	0.960779
$916$	0	0
$917$	−73.1078	−2.41423
$918$	0	0
$919$	−40.1711	−1.32512	−0.662561	−	0.749008i	$-0.730530\pi$
$919$	−40.1711	−1.32512	−0.662561	+	0.749008i	$0.730530\pi$
$920$	0	0
$921$	8.56720	0.282299
$922$	0	0
$923$	−23.2911	−0.766637
$924$	0	0
$925$	−15.7600	−0.518185
$926$	0	0
$927$	40.4484	1.32850
$928$	0	0
$929$	−0.497465	−0.0163213	−0.00816065	−	0.999967i	$-0.502598\pi$
$929$	−0.497465	−0.0163213	−0.00816065	+	0.999967i	$0.502598\pi$
$930$	0	0
$931$	−38.7481	−1.26992
$932$	0	0
$933$	8.30666	0.271948
$934$	0	0
$935$	6.07845	0.198787
$936$	0	0
$937$	−58.7000	−1.91765	−0.958823	−	0.284004i	$-0.908337\pi$
$937$	−58.7000	−1.91765	−0.958823	+	0.284004i	$0.908337\pi$
$938$	0	0
$939$	11.2953	0.368608
$940$	0	0
$941$	−20.7858	−0.677598	−0.338799	−	0.940859i	$-0.610021\pi$
$941$	−20.7858	−0.677598	−0.338799	+	0.940859i	$0.610021\pi$
$942$	0	0
$943$	5.83371	0.189972
$944$	0	0
$945$	54.6025	1.77622
$946$	0	0
$947$	−12.0761	−0.392421	−0.196210	−	0.980562i	$-0.562864\pi$
$947$	−12.0761	−0.392421	−0.196210	+	0.980562i	$0.562864\pi$
$948$	0	0
$949$	7.29425	0.236781
$950$	0	0
$951$	26.8546	0.870820
$952$	0	0
$953$	−60.9157	−1.97325	−0.986627	−	0.162995i	$-0.947885\pi$
$953$	−60.9157	−1.97325	−0.986627	+	0.162995i	$0.947885\pi$
$954$	0	0
$955$	35.7602	1.15717
$956$	0	0
$957$	−13.6950	−0.442696
$958$	0	0
$959$	−40.3769	−1.30384
$960$	0	0
$961$	7.44152	0.240049
$962$	0	0
$963$	27.4982	0.886118
$964$	0	0
$965$	11.9390	0.384330
$966$	0	0
$967$	−6.76303	−0.217484	−0.108742	−	0.994070i	$-0.534682\pi$
$967$	−6.76303	−0.217484	−0.108742	+	0.994070i	$0.534682\pi$
$968$	0	0
$969$	−6.76590	−0.217352
$970$	0	0
$971$	7.29604	0.234141	0.117070	−	0.993124i	$-0.462650\pi$
$971$	7.29604	0.234141	0.117070	+	0.993124i	$0.462650\pi$
$972$	0	0
$973$	−25.2719	−0.810180
$974$	0	0
$975$	−11.6510	−0.373131
$976$	0	0
$977$	25.0345	0.800924	0.400462	−	0.916313i	$-0.368850\pi$
$977$	25.0345	0.800924	0.400462	+	0.916313i	$0.368850\pi$
$978$	0	0
$979$	−10.5631	−0.337597
$980$	0	0
$981$	−13.6670	−0.436353
$982$	0	0
$983$	49.9355	1.59270	0.796348	−	0.604838i	$-0.206762\pi$
$983$	49.9355	1.59270	0.796348	+	0.604838i	$0.206762\pi$
$984$	0	0
$985$	41.3682	1.31810
$986$	0	0
$987$	−10.1452	−0.322924
$988$	0	0
$989$	−10.9605	−0.348525
$990$	0	0
$991$	17.6320	0.560098	0.280049	−	0.959986i	$-0.409649\pi$
$991$	17.6320	0.560098	0.280049	+	0.959986i	$0.409649\pi$
$992$	0	0
$993$	2.40167	0.0762148
$994$	0	0
$995$	65.0786	2.06313
$996$	0	0
$997$	−10.4382	−0.330580	−0.165290	−	0.986245i	$-0.552856\pi$
$997$	−10.4382	−0.330580	−0.165290	+	0.986245i	$0.552856\pi$
$998$	0	0
$999$	−8.37415	−0.264946

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.11	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.11	✓	23	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)

\(f(q)\)	\(=\)	\(q-0.823534 q^{3} +3.63980 q^{5} +3.42292 q^{7} -2.32179 q^{9} +O(q^{10})\)	\(q-0.823534 q^{3} +3.63980 q^{5} +3.42292 q^{7} -2.32179 q^{9} -1.67000 q^{11} +1.71525 q^{13} -2.99750 q^{15} -1.00000 q^{17} -8.21569 q^{19} -2.81889 q^{21} +1.40727 q^{23} +8.24811 q^{25} +4.38268 q^{27} -9.95781 q^{29} -6.20012 q^{31} +1.37530 q^{33} +12.4587 q^{35} -1.91074 q^{37} -1.41257 q^{39} +4.14541 q^{41} -7.78851 q^{43} -8.45085 q^{45} +3.59899 q^{47} +4.71635 q^{49} +0.823534 q^{51} -9.21150 q^{53} -6.07845 q^{55} +6.76590 q^{57} -1.00000 q^{59} -9.69562 q^{61} -7.94730 q^{63} +6.24317 q^{65} -7.29905 q^{67} -1.15893 q^{69} -13.5788 q^{71} +4.25258 q^{73} -6.79260 q^{75} -5.71627 q^{77} +15.8841 q^{79} +3.35609 q^{81} -7.87799 q^{83} -3.63980 q^{85} +8.20059 q^{87} +6.32519 q^{89} +5.87116 q^{91} +5.10601 q^{93} -29.9034 q^{95} +12.2886 q^{97} +3.87739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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