LMFDB - Embedded newform 8024.2.a.bc.1.2

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			1.2
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)

$f(q)$	$=$	$q-3.36646 q^{3} +0.242538 q^{5} +4.57037 q^{7} +8.33308 q^{9} +O(q^{10})$	$q-3.36646 q^{3} +0.242538 q^{5} +4.57037 q^{7} +8.33308 q^{9} -1.90941 q^{11} -6.85930 q^{13} -0.816495 q^{15} -1.00000 q^{17} -1.47614 q^{19} -15.3860 q^{21} -8.75041 q^{23} -4.94118 q^{25} -17.9536 q^{27} -8.58154 q^{29} +2.36730 q^{31} +6.42797 q^{33} +1.10849 q^{35} +1.79260 q^{37} +23.0916 q^{39} +0.885886 q^{41} -1.60052 q^{43} +2.02109 q^{45} -7.99453 q^{47} +13.8883 q^{49} +3.36646 q^{51} +0.915755 q^{53} -0.463105 q^{55} +4.96936 q^{57} -1.00000 q^{59} +8.00420 q^{61} +38.0853 q^{63} -1.66364 q^{65} +11.3806 q^{67} +29.4579 q^{69} -9.24925 q^{71} +3.30440 q^{73} +16.6343 q^{75} -8.72672 q^{77} -9.38524 q^{79} +35.4410 q^{81} -4.48048 q^{83} -0.242538 q^{85} +28.8894 q^{87} +17.7828 q^{89} -31.3496 q^{91} -7.96945 q^{93} -0.358019 q^{95} +11.5876 q^{97} -15.9113 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.36646	−1.94363	−0.971815	−	0.235746i	$-0.924247\pi$
$3$	−3.36646	−1.94363	−0.971815	+	0.235746i	$0.924247\pi$
$4$	0	0
$5$	0.242538	0.108466	0.0542331	−	0.998528i	$-0.482729\pi$
$5$	0.242538	0.108466	0.0542331	+	0.998528i	$0.482729\pi$
$6$	0	0
$7$	4.57037	1.72744	0.863719	−	0.503973i	$-0.168129\pi$
$7$	4.57037	1.72744	0.863719	+	0.503973i	$0.168129\pi$
$8$	0	0
$9$	8.33308	2.77769
$10$	0	0
$11$	−1.90941	−0.575709	−0.287855	−	0.957674i	$-0.592942\pi$
$11$	−1.90941	−0.575709	−0.287855	+	0.957674i	$0.592942\pi$
$12$	0	0
$13$	−6.85930	−1.90243	−0.951214	−	0.308532i	$-0.900162\pi$
$13$	−6.85930	−1.90243	−0.951214	+	0.308532i	$0.900162\pi$
$14$	0	0
$15$	−0.816495	−0.210818
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.47614	−0.338649	−0.169325	−	0.985560i	$-0.554159\pi$
$19$	−1.47614	−0.338649	−0.169325	+	0.985560i	$0.554159\pi$
$20$	0	0
$21$	−15.3860	−3.35750
$22$	0	0
$23$	−8.75041	−1.82459	−0.912293	−	0.409538i	$-0.865690\pi$
$23$	−8.75041	−1.82459	−0.912293	+	0.409538i	$0.865690\pi$
$24$	0	0
$25$	−4.94118	−0.988235
$26$	0	0
$27$	−17.9536	−3.45518
$28$	0	0
$29$	−8.58154	−1.59355	−0.796776	−	0.604275i	$-0.793463\pi$
$29$	−8.58154	−1.59355	−0.796776	+	0.604275i	$0.793463\pi$
$30$	0	0
$31$	2.36730	0.425180	0.212590	−	0.977141i	$-0.431810\pi$
$31$	2.36730	0.425180	0.212590	+	0.977141i	$0.431810\pi$
$32$	0	0
$33$	6.42797	1.11897
$34$	0	0
$35$	1.10849	0.187369
$36$	0	0
$37$	1.79260	0.294702	0.147351	−	0.989084i	$-0.452925\pi$
$37$	1.79260	0.294702	0.147351	+	0.989084i	$0.452925\pi$
$38$	0	0
$39$	23.0916	3.69761
$40$	0	0
$41$	0.885886	0.138352	0.0691761	−	0.997604i	$-0.477963\pi$
$41$	0.885886	0.138352	0.0691761	+	0.997604i	$0.477963\pi$
$42$	0	0
$43$	−1.60052	−0.244077	−0.122039	−	0.992525i	$-0.538943\pi$
$43$	−1.60052	−0.244077	−0.122039	+	0.992525i	$0.538943\pi$
$44$	0	0
$45$	2.02109	0.301286
$46$	0	0
$47$	−7.99453	−1.16612	−0.583061	−	0.812428i	$-0.698145\pi$
$47$	−7.99453	−1.16612	−0.583061	+	0.812428i	$0.698145\pi$
$48$	0	0
$49$	13.8883	1.98404
$50$	0	0
$51$	3.36646	0.471399
$52$	0	0
$53$	0.915755	0.125789	0.0628943	−	0.998020i	$-0.479967\pi$
$53$	0.915755	0.125789	0.0628943	+	0.998020i	$0.479967\pi$
$54$	0	0
$55$	−0.463105	−0.0624450
$56$	0	0
$57$	4.96936	0.658208
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	8.00420	1.02483	0.512416	−	0.858737i	$-0.328750\pi$
$61$	8.00420	1.02483	0.512416	+	0.858737i	$0.328750\pi$
$62$	0	0
$63$	38.0853	4.79830
$64$	0	0
$65$	−1.66364	−0.206349
$66$	0	0
$67$	11.3806	1.39036	0.695182	−	0.718834i	$-0.255324\pi$
$67$	11.3806	1.39036	0.695182	+	0.718834i	$0.255324\pi$
$68$	0	0
$69$	29.4579	3.54632
$70$	0	0
$71$	−9.24925	−1.09768	−0.548842	−	0.835926i	$-0.684931\pi$
$71$	−9.24925	−1.09768	−0.548842	+	0.835926i	$0.684931\pi$
$72$	0	0
$73$	3.30440	0.386751	0.193376	−	0.981125i	$-0.438056\pi$
$73$	3.30440	0.386751	0.193376	+	0.981125i	$0.438056\pi$
$74$	0	0
$75$	16.6343	1.92076
$76$	0	0
$77$	−8.72672	−0.994502
$78$	0	0
$79$	−9.38524	−1.05592	−0.527961	−	0.849269i	$-0.677043\pi$
$79$	−9.38524	−1.05592	−0.527961	+	0.849269i	$0.677043\pi$
$80$	0	0
$81$	35.4410	3.93789
$82$	0	0
$83$	−4.48048	−0.491797	−0.245898	−	0.969296i	$-0.579083\pi$
$83$	−4.48048	−0.491797	−0.245898	+	0.969296i	$0.579083\pi$
$84$	0	0
$85$	−0.242538	−0.0263069
$86$	0	0
$87$	28.8894	3.09727
$88$	0	0
$89$	17.7828	1.88498	0.942489	−	0.334237i	$-0.108479\pi$
$89$	17.7828	1.88498	0.942489	+	0.334237i	$0.108479\pi$
$90$	0	0
$91$	−31.3496	−3.28633
$92$	0	0
$93$	−7.96945	−0.826393
$94$	0	0
$95$	−0.358019	−0.0367320
$96$	0	0
$97$	11.5876	1.17655	0.588273	−	0.808663i	$-0.299808\pi$
$97$	11.5876	1.17655	0.588273	+	0.808663i	$0.299808\pi$
$98$	0	0
$99$	−15.9113	−1.59914
$100$	0	0
$101$	−5.08770	−0.506245	−0.253123	−	0.967434i	$-0.581458\pi$
$101$	−5.08770	−0.506245	−0.253123	+	0.967434i	$0.581458\pi$
$102$	0	0
$103$	−0.363125	−0.0357798	−0.0178899	−	0.999840i	$-0.505695\pi$
$103$	−0.363125	−0.0357798	−0.0178899	+	0.999840i	$0.505695\pi$
$104$	0	0
$105$	−3.73169	−0.364175
$106$	0	0
$107$	−11.7902	−1.13980	−0.569901	−	0.821713i	$-0.693019\pi$
$107$	−11.7902	−1.13980	−0.569901	+	0.821713i	$0.693019\pi$
$108$	0	0
$109$	11.1927	1.07207	0.536035	−	0.844196i	$-0.319922\pi$
$109$	11.1927	1.07207	0.536035	+	0.844196i	$0.319922\pi$
$110$	0	0
$111$	−6.03474	−0.572792
$112$	0	0
$113$	18.3221	1.72360	0.861799	−	0.507251i	$-0.169338\pi$
$113$	18.3221	1.72360	0.861799	+	0.507251i	$0.169338\pi$
$114$	0	0
$115$	−2.12230	−0.197906
$116$	0	0
$117$	−57.1591	−5.28436
$118$	0	0
$119$	−4.57037	−0.418965
$120$	0	0
$121$	−7.35415	−0.668559
$122$	0	0
$123$	−2.98230	−0.268905
$124$	0	0
$125$	−2.41111	−0.215656
$126$	0	0
$127$	−8.53162	−0.757059	−0.378530	−	0.925589i	$-0.623570\pi$
$127$	−8.53162	−0.757059	−0.378530	+	0.925589i	$0.623570\pi$
$128$	0	0
$129$	5.38810	0.474396
$130$	0	0
$131$	−4.78593	−0.418148	−0.209074	−	0.977900i	$-0.567045\pi$
$131$	−4.78593	−0.418148	−0.209074	+	0.977900i	$0.567045\pi$
$132$	0	0
$133$	−6.74650	−0.584996
$134$	0	0
$135$	−4.35444	−0.374770
$136$	0	0
$137$	1.11407	0.0951813	0.0475907	−	0.998867i	$-0.484846\pi$
$137$	1.11407	0.0951813	0.0475907	+	0.998867i	$0.484846\pi$
$138$	0	0
$139$	14.6682	1.24414	0.622069	−	0.782963i	$-0.286292\pi$
$139$	14.6682	1.24414	0.622069	+	0.782963i	$0.286292\pi$
$140$	0	0
$141$	26.9133	2.26651
$142$	0	0
$143$	13.0972	1.09525
$144$	0	0
$145$	−2.08135	−0.172846
$146$	0	0
$147$	−46.7545	−3.85625
$148$	0	0
$149$	5.99554	0.491174	0.245587	−	0.969375i	$-0.421019\pi$
$149$	5.99554	0.491174	0.245587	+	0.969375i	$0.421019\pi$
$150$	0	0
$151$	−0.782487	−0.0636779	−0.0318389	−	0.999493i	$-0.510136\pi$
$151$	−0.782487	−0.0636779	−0.0318389	+	0.999493i	$0.510136\pi$
$152$	0	0
$153$	−8.33308	−0.673690
$154$	0	0
$155$	0.574161	0.0461177
$156$	0	0
$157$	−6.29623	−0.502494	−0.251247	−	0.967923i	$-0.580841\pi$
$157$	−6.29623	−0.502494	−0.251247	+	0.967923i	$0.580841\pi$
$158$	0	0
$159$	−3.08286	−0.244486
$160$	0	0
$161$	−39.9926	−3.15186
$162$	0	0
$163$	6.72976	0.527115	0.263558	−	0.964644i	$-0.415104\pi$
$163$	6.72976	0.527115	0.263558	+	0.964644i	$0.415104\pi$
$164$	0	0
$165$	1.55903	0.121370
$166$	0	0
$167$	−6.12363	−0.473861	−0.236930	−	0.971527i	$-0.576141\pi$
$167$	−6.12363	−0.473861	−0.236930	+	0.971527i	$0.576141\pi$
$168$	0	0
$169$	34.0500	2.61923
$170$	0	0
$171$	−12.3008	−0.940664
$172$	0	0
$173$	7.19446	0.546985	0.273492	−	0.961874i	$-0.411821\pi$
$173$	7.19446	0.546985	0.273492	+	0.961874i	$0.411821\pi$
$174$	0	0
$175$	−22.5830	−1.70712
$176$	0	0
$177$	3.36646	0.253039
$178$	0	0
$179$	21.1319	1.57947	0.789736	−	0.613446i	$-0.210217\pi$
$179$	21.1319	1.57947	0.789736	+	0.613446i	$0.210217\pi$
$180$	0	0
$181$	24.4385	1.81650	0.908250	−	0.418427i	$-0.137419\pi$
$181$	24.4385	1.81650	0.908250	+	0.418427i	$0.137419\pi$
$182$	0	0
$183$	−26.9459	−1.99189
$184$	0	0
$185$	0.434774	0.0319652
$186$	0	0
$187$	1.90941	0.139630
$188$	0	0
$189$	−82.0548	−5.96861
$190$	0	0
$191$	1.96662	0.142300	0.0711499	−	0.997466i	$-0.477333\pi$
$191$	1.96662	0.142300	0.0711499	+	0.997466i	$0.477333\pi$
$192$	0	0
$193$	11.3299	0.815547	0.407773	−	0.913083i	$-0.366305\pi$
$193$	11.3299	0.815547	0.407773	+	0.913083i	$0.366305\pi$
$194$	0	0
$195$	5.60059	0.401066
$196$	0	0
$197$	−17.5392	−1.24962	−0.624810	−	0.780777i	$-0.714823\pi$
$197$	−17.5392	−1.24962	−0.624810	+	0.780777i	$0.714823\pi$
$198$	0	0
$199$	−8.96724	−0.635671	−0.317835	−	0.948146i	$-0.602956\pi$
$199$	−8.96724	−0.635671	−0.317835	+	0.948146i	$0.602956\pi$
$200$	0	0
$201$	−38.3124	−2.70235
$202$	0	0
$203$	−39.2208	−2.75276
$204$	0	0
$205$	0.214861	0.0150065
$206$	0	0
$207$	−72.9179	−5.06814
$208$	0	0
$209$	2.81855	0.194963
$210$	0	0
$211$	22.0033	1.51477	0.757385	−	0.652968i	$-0.226476\pi$
$211$	22.0033	1.51477	0.757385	+	0.652968i	$0.226476\pi$
$212$	0	0
$213$	31.1373	2.13349
$214$	0	0
$215$	−0.388187	−0.0264741
$216$	0	0
$217$	10.8195	0.734473
$218$	0	0
$219$	−11.1242	−0.751701
$220$	0	0
$221$	6.85930	0.461407
$222$	0	0
$223$	−7.18444	−0.481106	−0.240553	−	0.970636i	$-0.577329\pi$
$223$	−7.18444	−0.481106	−0.240553	+	0.970636i	$0.577329\pi$
$224$	0	0
$225$	−41.1752	−2.74502
$226$	0	0
$227$	0.103684	0.00688173	0.00344086	−	0.999994i	$-0.498905\pi$
$227$	0.103684	0.00688173	0.00344086	+	0.999994i	$0.498905\pi$
$228$	0	0
$229$	−17.6659	−1.16740	−0.583699	−	0.811970i	$-0.698395\pi$
$229$	−17.6659	−1.16740	−0.583699	+	0.811970i	$0.698395\pi$
$230$	0	0
$231$	29.3782	1.93294
$232$	0	0
$233$	−16.3916	−1.07385	−0.536926	−	0.843630i	$-0.680414\pi$
$233$	−16.3916	−1.07385	−0.536926	+	0.843630i	$0.680414\pi$
$234$	0	0
$235$	−1.93898	−0.126485
$236$	0	0
$237$	31.5951	2.05232
$238$	0	0
$239$	9.01996	0.583452	0.291726	−	0.956502i	$-0.405770\pi$
$239$	9.01996	0.583452	0.291726	+	0.956502i	$0.405770\pi$
$240$	0	0
$241$	20.7184	1.33459	0.667294	−	0.744794i	$-0.267452\pi$
$241$	20.7184	1.33459	0.667294	+	0.744794i	$0.267452\pi$
$242$	0	0
$243$	−65.4501	−4.19863
$244$	0	0
$245$	3.36844	0.215202
$246$	0	0
$247$	10.1253	0.644256
$248$	0	0
$249$	15.0834	0.955871
$250$	0	0
$251$	19.9112	1.25678	0.628391	−	0.777897i	$-0.283714\pi$
$251$	19.9112	1.25678	0.628391	+	0.777897i	$0.283714\pi$
$252$	0	0
$253$	16.7081	1.05043
$254$	0	0
$255$	0.816495	0.0511309
$256$	0	0
$257$	25.2742	1.57656	0.788282	−	0.615314i	$-0.210971\pi$
$257$	25.2742	1.57656	0.788282	+	0.615314i	$0.210971\pi$
$258$	0	0
$259$	8.19287	0.509080
$260$	0	0
$261$	−71.5107	−4.42640
$262$	0	0
$263$	19.6853	1.21385	0.606925	−	0.794759i	$-0.292403\pi$
$263$	19.6853	1.21385	0.606925	+	0.794759i	$0.292403\pi$
$264$	0	0
$265$	0.222105	0.0136438
$266$	0	0
$267$	−59.8653	−3.66370
$268$	0	0
$269$	−17.9960	−1.09724	−0.548618	−	0.836073i	$-0.684846\pi$
$269$	−17.9960	−1.09724	−0.548618	+	0.836073i	$0.684846\pi$
$270$	0	0
$271$	−10.2765	−0.624255	−0.312128	−	0.950040i	$-0.601042\pi$
$271$	−10.2765	−0.624255	−0.312128	+	0.950040i	$0.601042\pi$
$272$	0	0
$273$	105.537	6.38740
$274$	0	0
$275$	9.43474	0.568936
$276$	0	0
$277$	−21.0021	−1.26190	−0.630948	−	0.775825i	$-0.717334\pi$
$277$	−21.0021	−1.26190	−0.630948	+	0.775825i	$0.717334\pi$
$278$	0	0
$279$	19.7270	1.18102
$280$	0	0
$281$	28.0043	1.67059	0.835297	−	0.549798i	$-0.185295\pi$
$281$	28.0043	1.67059	0.835297	+	0.549798i	$0.185295\pi$
$282$	0	0
$283$	−19.0726	−1.13375	−0.566873	−	0.823805i	$-0.691847\pi$
$283$	−19.0726	−1.13375	−0.566873	+	0.823805i	$0.691847\pi$
$284$	0	0
$285$	1.20526	0.0713934
$286$	0	0
$287$	4.04883	0.238995
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−39.0093	−2.28677
$292$	0	0
$293$	−4.96240	−0.289906	−0.144953	−	0.989439i	$-0.546303\pi$
$293$	−4.96240	−0.289906	−0.144953	+	0.989439i	$0.546303\pi$
$294$	0	0
$295$	−0.242538	−0.0141211
$296$	0	0
$297$	34.2809	1.98918
$298$	0	0
$299$	60.0217	3.47114
$300$	0	0
$301$	−7.31498	−0.421629
$302$	0	0
$303$	17.1276	0.983954
$304$	0	0
$305$	1.94132	0.111160
$306$	0	0
$307$	28.2625	1.61303	0.806514	−	0.591215i	$-0.201352\pi$
$307$	28.2625	1.61303	0.806514	+	0.591215i	$0.201352\pi$
$308$	0	0
$309$	1.22245	0.0695426
$310$	0	0
$311$	−1.13210	−0.0641953	−0.0320976	−	0.999485i	$-0.510219\pi$
$311$	−1.13210	−0.0641953	−0.0320976	+	0.999485i	$0.510219\pi$
$312$	0	0
$313$	2.37169	0.134056	0.0670280	−	0.997751i	$-0.478648\pi$
$313$	2.37169	0.134056	0.0670280	+	0.997751i	$0.478648\pi$
$314$	0	0
$315$	9.23713	0.520453
$316$	0	0
$317$	30.6734	1.72279	0.861395	−	0.507936i	$-0.169591\pi$
$317$	30.6734	1.72279	0.861395	+	0.507936i	$0.169591\pi$
$318$	0	0
$319$	16.3857	0.917422
$320$	0	0
$321$	39.6913	2.21535
$322$	0	0
$323$	1.47614	0.0821345
$324$	0	0
$325$	33.8930	1.88005
$326$	0	0
$327$	−37.6799	−2.08371
$328$	0	0
$329$	−36.5380	−2.01440
$330$	0	0
$331$	−18.3573	−1.00901	−0.504504	−	0.863409i	$-0.668325\pi$
$331$	−18.3573	−1.00901	−0.504504	+	0.863409i	$0.668325\pi$
$332$	0	0
$333$	14.9379	0.818593
$334$	0	0
$335$	2.76023	0.150807
$336$	0	0
$337$	−7.59692	−0.413830	−0.206915	−	0.978359i	$-0.566342\pi$
$337$	−7.59692	−0.413830	−0.206915	+	0.978359i	$0.566342\pi$
$338$	0	0
$339$	−61.6807	−3.35003
$340$	0	0
$341$	−4.52016	−0.244780
$342$	0	0
$343$	31.4821	1.69987
$344$	0	0
$345$	7.14466	0.384656
$346$	0	0
$347$	21.6998	1.16491	0.582453	−	0.812864i	$-0.302093\pi$
$347$	21.6998	1.16491	0.582453	+	0.812864i	$0.302093\pi$
$348$	0	0
$349$	−10.4404	−0.558863	−0.279432	−	0.960166i	$-0.590146\pi$
$349$	−10.4404	−0.558863	−0.279432	+	0.960166i	$0.590146\pi$
$350$	0	0
$351$	123.149	6.57323
$352$	0	0
$353$	10.3189	0.549221	0.274611	−	0.961556i	$-0.411451\pi$
$353$	10.3189	0.549221	0.274611	+	0.961556i	$0.411451\pi$
$354$	0	0
$355$	−2.24329	−0.119062
$356$	0	0
$357$	15.3860	0.814313
$358$	0	0
$359$	27.4583	1.44919	0.724596	−	0.689174i	$-0.242026\pi$
$359$	27.4583	1.44919	0.724596	+	0.689174i	$0.242026\pi$
$360$	0	0
$361$	−16.8210	−0.885317
$362$	0	0
$363$	24.7575	1.29943
$364$	0	0
$365$	0.801443	0.0419494
$366$	0	0
$367$	−16.1122	−0.841052	−0.420526	−	0.907280i	$-0.638155\pi$
$367$	−16.1122	−0.841052	−0.420526	+	0.907280i	$0.638155\pi$
$368$	0	0
$369$	7.38216	0.384300
$370$	0	0
$371$	4.18534	0.217292
$372$	0	0
$373$	−14.5213	−0.751885	−0.375942	−	0.926643i	$-0.622681\pi$
$373$	−14.5213	−0.751885	−0.375942	+	0.926643i	$0.622681\pi$
$374$	0	0
$375$	8.11692	0.419156
$376$	0	0
$377$	58.8633	3.03162
$378$	0	0
$379$	−12.5797	−0.646174	−0.323087	−	0.946369i	$-0.604721\pi$
$379$	−12.5797	−0.646174	−0.323087	+	0.946369i	$0.604721\pi$
$380$	0	0
$381$	28.7214	1.47144
$382$	0	0
$383$	10.8021	0.551960	0.275980	−	0.961163i	$-0.410998\pi$
$383$	10.8021	0.551960	0.275980	+	0.961163i	$0.410998\pi$
$384$	0	0
$385$	−2.11656	−0.107870
$386$	0	0
$387$	−13.3373	−0.677972
$388$	0	0
$389$	−24.7178	−1.25324	−0.626621	−	0.779324i	$-0.715562\pi$
$389$	−24.7178	−1.25324	−0.626621	+	0.779324i	$0.715562\pi$
$390$	0	0
$391$	8.75041	0.442527
$392$	0	0
$393$	16.1116	0.812725
$394$	0	0
$395$	−2.27628	−0.114532
$396$	0	0
$397$	−18.6460	−0.935814	−0.467907	−	0.883778i	$-0.654992\pi$
$397$	−18.6460	−0.935814	−0.467907	+	0.883778i	$0.654992\pi$
$398$	0	0
$399$	22.7118	1.13701
$400$	0	0
$401$	4.42106	0.220777	0.110389	−	0.993889i	$-0.464790\pi$
$401$	4.42106	0.220777	0.110389	+	0.993889i	$0.464790\pi$
$402$	0	0
$403$	−16.2381	−0.808875
$404$	0	0
$405$	8.59579	0.427128
$406$	0	0
$407$	−3.42282	−0.169663
$408$	0	0
$409$	6.35858	0.314412	0.157206	−	0.987566i	$-0.449751\pi$
$409$	6.35858	0.314412	0.157206	+	0.987566i	$0.449751\pi$
$410$	0	0
$411$	−3.75047	−0.184997
$412$	0	0
$413$	−4.57037	−0.224893
$414$	0	0
$415$	−1.08669	−0.0533433
$416$	0	0
$417$	−49.3799	−2.41814
$418$	0	0
$419$	16.7441	0.818001	0.409000	−	0.912534i	$-0.365877\pi$
$419$	16.7441	0.818001	0.409000	+	0.912534i	$0.365877\pi$
$420$	0	0
$421$	−14.2347	−0.693759	−0.346879	−	0.937910i	$-0.612759\pi$
$421$	−14.2347	−0.693759	−0.346879	+	0.937910i	$0.612759\pi$
$422$	0	0
$423$	−66.6191	−3.23913
$424$	0	0
$425$	4.94118	0.239682
$426$	0	0
$427$	36.5822	1.77034
$428$	0	0
$429$	−44.0914	−2.12875
$430$	0	0
$431$	−28.9224	−1.39314	−0.696571	−	0.717487i	$-0.745292\pi$
$431$	−28.9224	−1.39314	−0.696571	+	0.717487i	$0.745292\pi$
$432$	0	0
$433$	27.3246	1.31314	0.656569	−	0.754266i	$-0.272007\pi$
$433$	27.3246	1.31314	0.656569	+	0.754266i	$0.272007\pi$
$434$	0	0
$435$	7.00678	0.335949
$436$	0	0
$437$	12.9168	0.617895
$438$	0	0
$439$	15.6378	0.746352	0.373176	−	0.927761i	$-0.378269\pi$
$439$	15.6378	0.746352	0.373176	+	0.927761i	$0.378269\pi$
$440$	0	0
$441$	115.732	5.51107
$442$	0	0
$443$	15.1302	0.718858	0.359429	−	0.933172i	$-0.382971\pi$
$443$	15.1302	0.718858	0.359429	+	0.933172i	$0.382971\pi$
$444$	0	0
$445$	4.31301	0.204456
$446$	0	0
$447$	−20.1838	−0.954660
$448$	0	0
$449$	−38.4897	−1.81644	−0.908222	−	0.418490i	$-0.862560\pi$
$449$	−38.4897	−1.81644	−0.908222	+	0.418490i	$0.862560\pi$
$450$	0	0
$451$	−1.69152	−0.0796506
$452$	0	0
$453$	2.63421	0.123766
$454$	0	0
$455$	−7.60345	−0.356455
$456$	0	0
$457$	−19.5418	−0.914125	−0.457063	−	0.889435i	$-0.651098\pi$
$457$	−19.5418	−0.914125	−0.457063	+	0.889435i	$0.651098\pi$
$458$	0	0
$459$	17.9536	0.838004
$460$	0	0
$461$	−23.0405	−1.07310	−0.536552	−	0.843867i	$-0.680274\pi$
$461$	−23.0405	−1.07310	−0.536552	+	0.843867i	$0.680274\pi$
$462$	0	0
$463$	−22.4924	−1.04531	−0.522655	−	0.852545i	$-0.675058\pi$
$463$	−22.4924	−1.04531	−0.522655	+	0.852545i	$0.675058\pi$
$464$	0	0
$465$	−1.93289	−0.0896357
$466$	0	0
$467$	−30.6142	−1.41665	−0.708327	−	0.705884i	$-0.750550\pi$
$467$	−30.6142	−1.41665	−0.708327	+	0.705884i	$0.750550\pi$
$468$	0	0
$469$	52.0137	2.40177
$470$	0	0
$471$	21.1960	0.976662
$472$	0	0
$473$	3.05606	0.140518
$474$	0	0
$475$	7.29385	0.334665
$476$	0	0
$477$	7.63106	0.349402
$478$	0	0
$479$	7.31237	0.334111	0.167055	−	0.985948i	$-0.446574\pi$
$479$	7.31237	0.334111	0.167055	+	0.985948i	$0.446574\pi$
$480$	0	0
$481$	−12.2960	−0.560650
$482$	0	0
$483$	134.634	6.12605
$484$	0	0
$485$	2.81044	0.127615
$486$	0	0
$487$	−25.1393	−1.13917	−0.569585	−	0.821933i	$-0.692896\pi$
$487$	−25.1393	−1.13917	−0.569585	+	0.821933i	$0.692896\pi$
$488$	0	0
$489$	−22.6555	−1.02452
$490$	0	0
$491$	3.80867	0.171883	0.0859413	−	0.996300i	$-0.472610\pi$
$491$	3.80867	0.171883	0.0859413	+	0.996300i	$0.472610\pi$
$492$	0	0
$493$	8.58154	0.386493
$494$	0	0
$495$	−3.85909	−0.173453
$496$	0	0
$497$	−42.2725	−1.89618
$498$	0	0
$499$	2.76233	0.123659	0.0618295	−	0.998087i	$-0.480306\pi$
$499$	2.76233	0.123659	0.0618295	+	0.998087i	$0.480306\pi$
$500$	0	0
$501$	20.6150	0.921010
$502$	0	0
$503$	1.98390	0.0884579	0.0442290	−	0.999021i	$-0.485917\pi$
$503$	1.98390	0.0884579	0.0442290	+	0.999021i	$0.485917\pi$
$504$	0	0
$505$	−1.23396	−0.0549105
$506$	0	0
$507$	−114.628	−5.09082
$508$	0	0
$509$	10.6004	0.469853	0.234927	−	0.972013i	$-0.424515\pi$
$509$	10.6004	0.469853	0.234927	+	0.972013i	$0.424515\pi$
$510$	0	0
$511$	15.1024	0.668089
$512$	0	0
$513$	26.5020	1.17009
$514$	0	0
$515$	−0.0880716	−0.00388090
$516$	0	0
$517$	15.2649	0.671348
$518$	0	0
$519$	−24.2199	−1.06314
$520$	0	0
$521$	−17.4694	−0.765347	−0.382674	−	0.923884i	$-0.624997\pi$
$521$	−17.4694	−0.765347	−0.382674	+	0.923884i	$0.624997\pi$
$522$	0	0
$523$	38.2144	1.67100	0.835499	−	0.549492i	$-0.185179\pi$
$523$	38.2144	1.67100	0.835499	+	0.549492i	$0.185179\pi$
$524$	0	0
$525$	76.0249	3.31800
$526$	0	0
$527$	−2.36730	−0.103121
$528$	0	0
$529$	53.5696	2.32911
$530$	0	0
$531$	−8.33308	−0.361625
$532$	0	0
$533$	−6.07656	−0.263205
$534$	0	0
$535$	−2.85957	−0.123630
$536$	0	0
$537$	−71.1398	−3.06991
$538$	0	0
$539$	−26.5185	−1.14223
$540$	0	0
$541$	39.9986	1.71967	0.859837	−	0.510568i	$-0.170565\pi$
$541$	39.9986	1.71967	0.859837	+	0.510568i	$0.170565\pi$
$542$	0	0
$543$	−82.2714	−3.53060
$544$	0	0
$545$	2.71466	0.116283
$546$	0	0
$547$	−22.0637	−0.943374	−0.471687	−	0.881766i	$-0.656355\pi$
$547$	−22.0637	−0.943374	−0.471687	+	0.881766i	$0.656355\pi$
$548$	0	0
$549$	66.6997	2.84667
$550$	0	0
$551$	12.6675	0.539655
$552$	0	0
$553$	−42.8940	−1.82404
$554$	0	0
$555$	−1.46365	−0.0621286
$556$	0	0
$557$	6.78180	0.287354	0.143677	−	0.989625i	$-0.454107\pi$
$557$	6.78180	0.287354	0.143677	+	0.989625i	$0.454107\pi$
$558$	0	0
$559$	10.9785	0.464340
$560$	0	0
$561$	−6.42797	−0.271389
$562$	0	0
$563$	−25.9570	−1.09396	−0.546978	−	0.837147i	$-0.684222\pi$
$563$	−25.9570	−1.09396	−0.546978	+	0.837147i	$0.684222\pi$
$564$	0	0
$565$	4.44380	0.186952
$566$	0	0
$567$	161.979	6.80247
$568$	0	0
$569$	−3.08392	−0.129285	−0.0646424	−	0.997908i	$-0.520591\pi$
$569$	−3.08392	−0.129285	−0.0646424	+	0.997908i	$0.520591\pi$
$570$	0	0
$571$	−42.2139	−1.76660	−0.883299	−	0.468810i	$-0.844683\pi$
$571$	−42.2139	−1.76660	−0.883299	+	0.468810i	$0.844683\pi$
$572$	0	0
$573$	−6.62057	−0.276578
$574$	0	0
$575$	43.2373	1.80312
$576$	0	0
$577$	24.2623	1.01005	0.505027	−	0.863104i	$-0.331483\pi$
$577$	24.2623	1.01005	0.505027	+	0.863104i	$0.331483\pi$
$578$	0	0
$579$	−38.1418	−1.58512
$580$	0	0
$581$	−20.4775	−0.849549
$582$	0	0
$583$	−1.74855	−0.0724177
$584$	0	0
$585$	−13.8633	−0.573175
$586$	0	0
$587$	11.2644	0.464930	0.232465	−	0.972605i	$-0.425321\pi$
$587$	11.2644	0.464930	0.232465	+	0.972605i	$0.425321\pi$
$588$	0	0
$589$	−3.49447	−0.143987
$590$	0	0
$591$	59.0453	2.42880
$592$	0	0
$593$	−18.2768	−0.750539	−0.375269	−	0.926916i	$-0.622450\pi$
$593$	−18.2768	−0.750539	−0.375269	+	0.926916i	$0.622450\pi$
$594$	0	0
$595$	−1.10849	−0.0454436
$596$	0	0
$597$	30.1879	1.23551
$598$	0	0
$599$	31.5811	1.29037	0.645185	−	0.764026i	$-0.276780\pi$
$599$	31.5811	1.29037	0.645185	+	0.764026i	$0.276780\pi$
$600$	0	0
$601$	40.5270	1.65313	0.826566	−	0.562840i	$-0.190291\pi$
$601$	40.5270	1.65313	0.826566	+	0.562840i	$0.190291\pi$
$602$	0	0
$603$	94.8356	3.86201
$604$	0	0
$605$	−1.78366	−0.0725160
$606$	0	0
$607$	15.0479	0.610775	0.305388	−	0.952228i	$-0.401214\pi$
$607$	15.0479	0.610775	0.305388	+	0.952228i	$0.401214\pi$
$608$	0	0
$609$	132.036	5.35035
$610$	0	0
$611$	54.8369	2.21846
$612$	0	0
$613$	39.4306	1.59259	0.796294	−	0.604910i	$-0.206791\pi$
$613$	39.4306	1.59259	0.796294	+	0.604910i	$0.206791\pi$
$614$	0	0
$615$	−0.723322	−0.0291671
$616$	0	0
$617$	−42.2323	−1.70021	−0.850104	−	0.526614i	$-0.823461\pi$
$617$	−42.2323	−1.70021	−0.850104	+	0.526614i	$0.823461\pi$
$618$	0	0
$619$	19.2753	0.774740	0.387370	−	0.921924i	$-0.373384\pi$
$619$	19.2753	0.774740	0.387370	+	0.921924i	$0.373384\pi$
$620$	0	0
$621$	157.102	6.30427
$622$	0	0
$623$	81.2742	3.25618
$624$	0	0
$625$	24.1211	0.964844
$626$	0	0
$627$	−9.48856	−0.378937
$628$	0	0
$629$	−1.79260	−0.0714758
$630$	0	0
$631$	−22.2966	−0.887612	−0.443806	−	0.896123i	$-0.646372\pi$
$631$	−22.2966	−0.887612	−0.443806	+	0.896123i	$0.646372\pi$
$632$	0	0
$633$	−74.0734	−2.94415
$634$	0	0
$635$	−2.06924	−0.0821154
$636$	0	0
$637$	−95.2641	−3.77450
$638$	0	0
$639$	−77.0748	−3.04903
$640$	0	0
$641$	−12.0020	−0.474049	−0.237025	−	0.971504i	$-0.576172\pi$
$641$	−12.0020	−0.474049	−0.237025	+	0.971504i	$0.576172\pi$
$642$	0	0
$643$	42.0803	1.65948	0.829742	−	0.558147i	$-0.188488\pi$
$643$	42.0803	1.65948	0.829742	+	0.558147i	$0.188488\pi$
$644$	0	0
$645$	1.30682	0.0514559
$646$	0	0
$647$	7.85377	0.308764	0.154382	−	0.988011i	$-0.450661\pi$
$647$	7.85377	0.308764	0.154382	+	0.988011i	$0.450661\pi$
$648$	0	0
$649$	1.90941	0.0749510
$650$	0	0
$651$	−36.4233	−1.42754
$652$	0	0
$653$	−14.7463	−0.577066	−0.288533	−	0.957470i	$-0.593168\pi$
$653$	−14.7463	−0.577066	−0.288533	+	0.957470i	$0.593168\pi$
$654$	0	0
$655$	−1.16077	−0.0453550
$656$	0	0
$657$	27.5359	1.07428
$658$	0	0
$659$	−39.9655	−1.55683	−0.778417	−	0.627748i	$-0.783977\pi$
$659$	−39.9655	−1.55683	−0.778417	+	0.627748i	$0.783977\pi$
$660$	0	0
$661$	14.8312	0.576868	0.288434	−	0.957500i	$-0.406865\pi$
$661$	14.8312	0.576868	0.288434	+	0.957500i	$0.406865\pi$
$662$	0	0
$663$	−23.0916	−0.896803
$664$	0	0
$665$	−1.63628	−0.0634523
$666$	0	0
$667$	75.0919	2.90757
$668$	0	0
$669$	24.1862	0.935091
$670$	0	0
$671$	−15.2833	−0.590006
$672$	0	0
$673$	32.9226	1.26907	0.634536	−	0.772893i	$-0.281191\pi$
$673$	32.9226	1.26907	0.634536	+	0.772893i	$0.281191\pi$
$674$	0	0
$675$	88.7121	3.41453
$676$	0	0
$677$	28.4692	1.09416	0.547080	−	0.837081i	$-0.315740\pi$
$677$	28.4692	1.09416	0.547080	+	0.837081i	$0.315740\pi$
$678$	0	0
$679$	52.9598	2.03241
$680$	0	0
$681$	−0.349047	−0.0133755
$682$	0	0
$683$	−30.3592	−1.16166	−0.580831	−	0.814024i	$-0.697272\pi$
$683$	−30.3592	−1.16166	−0.580831	+	0.814024i	$0.697272\pi$
$684$	0	0
$685$	0.270204	0.0103240
$686$	0	0
$687$	59.4717	2.26899
$688$	0	0
$689$	−6.28144	−0.239304
$690$	0	0
$691$	8.73298	0.332218	0.166109	−	0.986107i	$-0.446880\pi$
$691$	8.73298	0.332218	0.166109	+	0.986107i	$0.446880\pi$
$692$	0	0
$693$	−72.7205	−2.76242
$694$	0	0
$695$	3.55758	0.134947
$696$	0	0
$697$	−0.885886	−0.0335553
$698$	0	0
$699$	55.1818	2.08717
$700$	0	0
$701$	−3.20852	−0.121184	−0.0605920	−	0.998163i	$-0.519299\pi$
$701$	−3.20852	−0.121184	−0.0605920	+	0.998163i	$0.519299\pi$
$702$	0	0
$703$	−2.64613	−0.0998007
$704$	0	0
$705$	6.52750	0.245840
$706$	0	0
$707$	−23.2527	−0.874508
$708$	0	0
$709$	15.3407	0.576133	0.288066	−	0.957610i	$-0.406988\pi$
$709$	15.3407	0.576133	0.288066	+	0.957610i	$0.406988\pi$
$710$	0	0
$711$	−78.2080	−2.93303
$712$	0	0
$713$	−20.7149	−0.775778
$714$	0	0
$715$	3.17657	0.118797
$716$	0	0
$717$	−30.3654	−1.13402
$718$	0	0
$719$	11.8536	0.442063	0.221032	−	0.975267i	$-0.429058\pi$
$719$	11.8536	0.442063	0.221032	+	0.975267i	$0.429058\pi$
$720$	0	0
$721$	−1.65962	−0.0618074
$722$	0	0
$723$	−69.7477	−2.59395
$724$	0	0
$725$	42.4029	1.57480
$726$	0	0
$727$	−14.8306	−0.550038	−0.275019	−	0.961439i	$-0.588684\pi$
$727$	−14.8306	−0.550038	−0.275019	+	0.961439i	$0.588684\pi$
$728$	0	0
$729$	114.012	4.22268
$730$	0	0
$731$	1.60052	0.0591974
$732$	0	0
$733$	−3.80199	−0.140430	−0.0702148	−	0.997532i	$-0.522368\pi$
$733$	−3.80199	−0.140430	−0.0702148	+	0.997532i	$0.522368\pi$
$734$	0	0
$735$	−11.3397	−0.418272
$736$	0	0
$737$	−21.7303	−0.800445
$738$	0	0
$739$	28.3563	1.04311	0.521553	−	0.853219i	$-0.325353\pi$
$739$	28.3563	1.04311	0.521553	+	0.853219i	$0.325353\pi$
$740$	0	0
$741$	−34.0864	−1.25219
$742$	0	0
$743$	−2.99448	−0.109857	−0.0549284	−	0.998490i	$-0.517493\pi$
$743$	−2.99448	−0.109857	−0.0549284	+	0.998490i	$0.517493\pi$
$744$	0	0
$745$	1.45415	0.0532758
$746$	0	0
$747$	−37.3362	−1.36606
$748$	0	0
$749$	−53.8857	−1.96894
$750$	0	0
$751$	−17.9538	−0.655142	−0.327571	−	0.944827i	$-0.606230\pi$
$751$	−17.9538	−0.655142	−0.327571	+	0.944827i	$0.606230\pi$
$752$	0	0
$753$	−67.0303	−2.44272
$754$	0	0
$755$	−0.189783	−0.00690690
$756$	0	0
$757$	50.8630	1.84865	0.924323	−	0.381612i	$-0.124631\pi$
$757$	50.8630	1.84865	0.924323	+	0.381612i	$0.124631\pi$
$758$	0	0
$759$	−56.2473	−2.04165
$760$	0	0
$761$	24.0950	0.873442	0.436721	−	0.899597i	$-0.356140\pi$
$761$	24.0950	0.873442	0.436721	+	0.899597i	$0.356140\pi$
$762$	0	0
$763$	51.1550	1.85193
$764$	0	0
$765$	−2.02109	−0.0730726
$766$	0	0
$767$	6.85930	0.247675
$768$	0	0
$769$	−18.5286	−0.668157	−0.334079	−	0.942545i	$-0.608425\pi$
$769$	−18.5286	−0.668157	−0.334079	+	0.942545i	$0.608425\pi$
$770$	0	0
$771$	−85.0848	−3.06426
$772$	0	0
$773$	22.4221	0.806466	0.403233	−	0.915097i	$-0.367886\pi$
$773$	22.4221	0.806466	0.403233	+	0.915097i	$0.367886\pi$
$774$	0	0
$775$	−11.6973	−0.420178
$776$	0	0
$777$	−27.5810	−0.989463
$778$	0	0
$779$	−1.30769	−0.0468528
$780$	0	0
$781$	17.6606	0.631947
$782$	0	0
$783$	154.070	5.50601
$784$	0	0
$785$	−1.52707	−0.0545036
$786$	0	0
$787$	−18.4372	−0.657214	−0.328607	−	0.944467i	$-0.606579\pi$
$787$	−18.4372	−0.657214	−0.328607	+	0.944467i	$0.606579\pi$
$788$	0	0
$789$	−66.2700	−2.35927
$790$	0	0
$791$	83.7388	2.97741
$792$	0	0
$793$	−54.9032	−1.94967
$794$	0	0
$795$	−0.747709	−0.0265185
$796$	0	0
$797$	−47.2674	−1.67430	−0.837149	−	0.546974i	$-0.815779\pi$
$797$	−47.2674	−1.67430	−0.837149	+	0.546974i	$0.815779\pi$
$798$	0	0
$799$	7.99453	0.282826
$800$	0	0
$801$	148.186	5.23589
$802$	0	0
$803$	−6.30947	−0.222656
$804$	0	0
$805$	−9.69972	−0.341870
$806$	0	0
$807$	60.5830	2.13262
$808$	0	0
$809$	−10.4765	−0.368335	−0.184167	−	0.982895i	$-0.558959\pi$
$809$	−10.4765	−0.368335	−0.184167	+	0.982895i	$0.558959\pi$
$810$	0	0
$811$	−25.0113	−0.878265	−0.439133	−	0.898422i	$-0.644714\pi$
$811$	−25.0113	−0.878265	−0.439133	+	0.898422i	$0.644714\pi$
$812$	0	0
$813$	34.5956	1.21332
$814$	0	0
$815$	1.63222	0.0571742
$816$	0	0
$817$	2.36259	0.0826566
$818$	0	0
$819$	−261.239	−9.12841
$820$	0	0
$821$	45.3876	1.58404	0.792019	−	0.610496i	$-0.209030\pi$
$821$	45.3876	1.58404	0.792019	+	0.610496i	$0.209030\pi$
$822$	0	0
$823$	−27.7128	−0.966008	−0.483004	−	0.875618i	$-0.660454\pi$
$823$	−27.7128	−0.966008	−0.483004	+	0.875618i	$0.660454\pi$
$824$	0	0
$825$	−31.7617	−1.10580
$826$	0	0
$827$	−51.6989	−1.79775	−0.898874	−	0.438207i	$-0.855614\pi$
$827$	−51.6989	−1.79775	−0.898874	+	0.438207i	$0.855614\pi$
$828$	0	0
$829$	2.60466	0.0904636	0.0452318	−	0.998977i	$-0.485597\pi$
$829$	2.60466	0.0904636	0.0452318	+	0.998977i	$0.485597\pi$
$830$	0	0
$831$	70.7030	2.45266
$832$	0	0
$833$	−13.8883	−0.481201
$834$	0	0
$835$	−1.48521	−0.0513979
$836$	0	0
$837$	−42.5017	−1.46908
$838$	0	0
$839$	−6.76561	−0.233575	−0.116787	−	0.993157i	$-0.537260\pi$
$839$	−6.76561	−0.233575	−0.116787	+	0.993157i	$0.537260\pi$
$840$	0	0
$841$	44.6428	1.53941
$842$	0	0
$843$	−94.2754	−3.24702
$844$	0	0
$845$	8.25842	0.284098
$846$	0	0
$847$	−33.6112	−1.15489
$848$	0	0
$849$	64.2071	2.20358
$850$	0	0
$851$	−15.6860	−0.537710
$852$	0	0
$853$	22.1990	0.760081	0.380040	−	0.924970i	$-0.375910\pi$
$853$	22.1990	0.760081	0.380040	+	0.924970i	$0.375910\pi$
$854$	0	0
$855$	−2.98340	−0.102030
$856$	0	0
$857$	−14.9912	−0.512089	−0.256045	−	0.966665i	$-0.582419\pi$
$857$	−14.9912	−0.512089	−0.256045	+	0.966665i	$0.582419\pi$
$858$	0	0
$859$	7.02832	0.239803	0.119902	−	0.992786i	$-0.461742\pi$
$859$	7.02832	0.239803	0.119902	+	0.992786i	$0.461742\pi$
$860$	0	0
$861$	−13.6302	−0.464517
$862$	0	0
$863$	−29.1097	−0.990905	−0.495452	−	0.868635i	$-0.664998\pi$
$863$	−29.1097	−0.990905	−0.495452	+	0.868635i	$0.664998\pi$
$864$	0	0
$865$	1.74493	0.0593294
$866$	0	0
$867$	−3.36646	−0.114331
$868$	0	0
$869$	17.9203	0.607904
$870$	0	0
$871$	−78.0631	−2.64507
$872$	0	0
$873$	96.5607	3.26808
$874$	0	0
$875$	−11.0197	−0.372533
$876$	0	0
$877$	15.6573	0.528708	0.264354	−	0.964426i	$-0.414841\pi$
$877$	15.6573	0.528708	0.264354	+	0.964426i	$0.414841\pi$
$878$	0	0
$879$	16.7057	0.563470
$880$	0	0
$881$	−30.4450	−1.02572	−0.512858	−	0.858473i	$-0.671413\pi$
$881$	−30.4450	−1.02572	−0.512858	+	0.858473i	$0.671413\pi$
$882$	0	0
$883$	44.2603	1.48948	0.744739	−	0.667355i	$-0.232574\pi$
$883$	44.2603	1.48948	0.744739	+	0.667355i	$0.232574\pi$
$884$	0	0
$885$	0.816495	0.0274462
$886$	0	0
$887$	−47.9136	−1.60878	−0.804391	−	0.594100i	$-0.797508\pi$
$887$	−47.9136	−1.60878	−0.804391	+	0.594100i	$0.797508\pi$
$888$	0	0
$889$	−38.9927	−1.30777
$890$	0	0
$891$	−67.6716	−2.26708
$892$	0	0
$893$	11.8010	0.394906
$894$	0	0
$895$	5.12529	0.171319
$896$	0	0
$897$	−202.061	−6.74662
$898$	0	0
$899$	−20.3151	−0.677547
$900$	0	0
$901$	−0.915755	−0.0305082
$902$	0	0
$903$	24.6256	0.819490
$904$	0	0
$905$	5.92726	0.197029
$906$	0	0
$907$	12.3978	0.411662	0.205831	−	0.978588i	$-0.434010\pi$
$907$	12.3978	0.411662	0.205831	+	0.978588i	$0.434010\pi$
$908$	0	0
$909$	−42.3963	−1.40620
$910$	0	0
$911$	27.7259	0.918600	0.459300	−	0.888281i	$-0.348100\pi$
$911$	27.7259	0.918600	0.459300	+	0.888281i	$0.348100\pi$
$912$	0	0
$913$	8.55509	0.283132
$914$	0	0
$915$	−6.53539	−0.216053
$916$	0	0
$917$	−21.8735	−0.722325
$918$	0	0
$919$	−14.8575	−0.490104	−0.245052	−	0.969510i	$-0.578805\pi$
$919$	−14.8575	−0.490104	−0.245052	+	0.969510i	$0.578805\pi$
$920$	0	0
$921$	−95.1448	−3.13513
$922$	0	0
$923$	63.4434	2.08827
$924$	0	0
$925$	−8.85757	−0.291235
$926$	0	0
$927$	−3.02595	−0.0993853
$928$	0	0
$929$	−4.48947	−0.147295	−0.0736473	−	0.997284i	$-0.523464\pi$
$929$	−4.48947	−0.147295	−0.0736473	+	0.997284i	$0.523464\pi$
$930$	0	0
$931$	−20.5010	−0.671895
$932$	0	0
$933$	3.81116	0.124772
$934$	0	0
$935$	0.463105	0.0151451
$936$	0	0
$937$	−20.2420	−0.661277	−0.330638	−	0.943758i	$-0.607264\pi$
$937$	−20.2420	−0.661277	−0.330638	+	0.943758i	$0.607264\pi$
$938$	0	0
$939$	−7.98422	−0.260555
$940$	0	0
$941$	40.4311	1.31801	0.659007	−	0.752136i	$-0.270977\pi$
$941$	40.4311	1.31801	0.659007	+	0.752136i	$0.270977\pi$
$942$	0	0
$943$	−7.75186	−0.252435
$944$	0	0
$945$	−19.9014	−0.647393
$946$	0	0
$947$	−52.7826	−1.71520	−0.857602	−	0.514314i	$-0.828047\pi$
$947$	−52.7826	−1.71520	−0.857602	+	0.514314i	$0.828047\pi$
$948$	0	0
$949$	−22.6659	−0.735766
$950$	0	0
$951$	−103.261	−3.34846
$952$	0	0
$953$	28.0466	0.908517	0.454259	−	0.890870i	$-0.349904\pi$
$953$	28.0466	0.908517	0.454259	+	0.890870i	$0.349904\pi$
$954$	0	0
$955$	0.476980	0.0154347
$956$	0	0
$957$	−55.1618	−1.78313
$958$	0	0
$959$	5.09171	0.164420
$960$	0	0
$961$	−25.3959	−0.819222
$962$	0	0
$963$	−98.2488	−3.16602
$964$	0	0
$965$	2.74794	0.0884593
$966$	0	0
$967$	−16.2655	−0.523062	−0.261531	−	0.965195i	$-0.584227\pi$
$967$	−16.2655	−0.523062	−0.261531	+	0.965195i	$0.584227\pi$
$968$	0	0
$969$	−4.96936	−0.159639
$970$	0	0
$971$	47.9791	1.53972	0.769861	−	0.638212i	$-0.220326\pi$
$971$	47.9791	1.53972	0.769861	+	0.638212i	$0.220326\pi$
$972$	0	0
$973$	67.0390	2.14917
$974$	0	0
$975$	−114.100	−3.65411
$976$	0	0
$977$	−33.7478	−1.07969	−0.539843	−	0.841766i	$-0.681516\pi$
$977$	−33.7478	−1.07969	−0.539843	+	0.841766i	$0.681516\pi$
$978$	0	0
$979$	−33.9548	−1.08520
$980$	0	0
$981$	93.2700	2.97788
$982$	0	0
$983$	−15.4918	−0.494112	−0.247056	−	0.969001i	$-0.579463\pi$
$983$	−15.4918	−0.494112	−0.247056	+	0.969001i	$0.579463\pi$
$984$	0	0
$985$	−4.25393	−0.135542
$986$	0	0
$987$	123.004	3.91526
$988$	0	0
$989$	14.0052	0.445340
$990$	0	0
$991$	36.7948	1.16882	0.584412	−	0.811457i	$-0.301325\pi$
$991$	36.7948	1.16882	0.584412	+	0.811457i	$0.301325\pi$
$992$	0	0
$993$	61.7992	1.96114
$994$	0	0
$995$	−2.17490	−0.0689488
$996$	0	0
$997$	34.4210	1.09012	0.545062	−	0.838396i	$-0.316506\pi$
$997$	34.4210	1.09012	0.545062	+	0.838396i	$0.316506\pi$
$998$	0	0
$999$	−32.1838	−1.01825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.2	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-3.36646 q^{3} +0.242538 q^{5} +4.57037 q^{7} +8.33308 q^{9} +O(q^{10})\)	\(q-3.36646 q^{3} +0.242538 q^{5} +4.57037 q^{7} +8.33308 q^{9} -1.90941 q^{11} -6.85930 q^{13} -0.816495 q^{15} -1.00000 q^{17} -1.47614 q^{19} -15.3860 q^{21} -8.75041 q^{23} -4.94118 q^{25} -17.9536 q^{27} -8.58154 q^{29} +2.36730 q^{31} +6.42797 q^{33} +1.10849 q^{35} +1.79260 q^{37} +23.0916 q^{39} +0.885886 q^{41} -1.60052 q^{43} +2.02109 q^{45} -7.99453 q^{47} +13.8883 q^{49} +3.36646 q^{51} +0.915755 q^{53} -0.463105 q^{55} +4.96936 q^{57} -1.00000 q^{59} +8.00420 q^{61} +38.0853 q^{63} -1.66364 q^{65} +11.3806 q^{67} +29.4579 q^{69} -9.24925 q^{71} +3.30440 q^{73} +16.6343 q^{75} -8.72672 q^{77} -9.38524 q^{79} +35.4410 q^{81} -4.48048 q^{83} -0.242538 q^{85} +28.8894 q^{87} +17.7828 q^{89} -31.3496 q^{91} -7.96945 q^{93} -0.358019 q^{95} +11.5876 q^{97} -15.9113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 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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Database

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