LMFDB - Embedded newform 6008.2.a.b.1.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.36
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67289 q^{3} +3.24936 q^{5} -2.55182 q^{7} -0.201452 q^{9} +O(q^{10})$	$q+1.67289 q^{3} +3.24936 q^{5} -2.55182 q^{7} -0.201452 q^{9} -3.86634 q^{11} +4.77285 q^{13} +5.43581 q^{15} -7.34637 q^{17} +5.13856 q^{19} -4.26891 q^{21} -4.94464 q^{23} +5.55832 q^{25} -5.35566 q^{27} -0.579676 q^{29} -6.31968 q^{31} -6.46794 q^{33} -8.29178 q^{35} -7.52260 q^{37} +7.98443 q^{39} -7.67286 q^{41} -6.46398 q^{43} -0.654589 q^{45} -12.1946 q^{47} -0.488212 q^{49} -12.2896 q^{51} -2.15145 q^{53} -12.5631 q^{55} +8.59623 q^{57} +5.57924 q^{59} +2.81215 q^{61} +0.514069 q^{63} +15.5087 q^{65} +15.2590 q^{67} -8.27182 q^{69} +12.4025 q^{71} -1.40887 q^{73} +9.29844 q^{75} +9.86619 q^{77} -12.3648 q^{79} -8.35506 q^{81} +1.68284 q^{83} -23.8710 q^{85} -0.969732 q^{87} +15.0207 q^{89} -12.1794 q^{91} -10.5721 q^{93} +16.6970 q^{95} +5.00023 q^{97} +0.778880 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.67289	0.965841	0.482921	−	0.875664i	$-0.339576\pi$
$3$	1.67289	0.965841	0.482921	+	0.875664i	$0.339576\pi$
$4$	0	0
$5$	3.24936	1.45316	0.726578	−	0.687084i	$-0.241109\pi$
$5$	3.24936	1.45316	0.726578	+	0.687084i	$0.241109\pi$
$6$	0	0
$7$	−2.55182	−0.964497	−0.482249	−	0.876034i	$-0.660180\pi$
$7$	−2.55182	−0.964497	−0.482249	+	0.876034i	$0.660180\pi$
$8$	0	0
$9$	−0.201452	−0.0671506
$10$	0	0
$11$	−3.86634	−1.16574	−0.582872	−	0.812564i	$-0.698071\pi$
$11$	−3.86634	−1.16574	−0.582872	+	0.812564i	$0.698071\pi$
$12$	0	0
$13$	4.77285	1.32375	0.661875	−	0.749614i	$-0.269761\pi$
$13$	4.77285	1.32375	0.661875	+	0.749614i	$0.269761\pi$
$14$	0	0
$15$	5.43581	1.40352
$16$	0	0
$17$	−7.34637	−1.78176	−0.890878	−	0.454243i	$-0.849910\pi$
$17$	−7.34637	−1.78176	−0.890878	+	0.454243i	$0.849910\pi$
$18$	0	0
$19$	5.13856	1.17887	0.589433	−	0.807817i	$-0.299351\pi$
$19$	5.13856	1.17887	0.589433	+	0.807817i	$0.299351\pi$
$20$	0	0
$21$	−4.26891	−0.931551
$22$	0	0
$23$	−4.94464	−1.03103	−0.515514	−	0.856881i	$-0.672399\pi$
$23$	−4.94464	−1.03103	−0.515514	+	0.856881i	$0.672399\pi$
$24$	0	0
$25$	5.55832	1.11166
$26$	0	0
$27$	−5.35566	−1.03070
$28$	0	0
$29$	−0.579676	−0.107643	−0.0538216	−	0.998551i	$-0.517140\pi$
$29$	−0.579676	−0.107643	−0.0538216	+	0.998551i	$0.517140\pi$
$30$	0	0
$31$	−6.31968	−1.13505	−0.567524	−	0.823357i	$-0.692099\pi$
$31$	−6.31968	−1.13505	−0.567524	+	0.823357i	$0.692099\pi$
$32$	0	0
$33$	−6.46794	−1.12592
$34$	0	0
$35$	−8.29178	−1.40157
$36$	0	0
$37$	−7.52260	−1.23671	−0.618354	−	0.785900i	$-0.712200\pi$
$37$	−7.52260	−1.23671	−0.618354	+	0.785900i	$0.712200\pi$
$38$	0	0
$39$	7.98443	1.27853
$40$	0	0
$41$	−7.67286	−1.19830	−0.599150	−	0.800637i	$-0.704495\pi$
$41$	−7.67286	−1.19830	−0.599150	+	0.800637i	$0.704495\pi$
$42$	0	0
$43$	−6.46398	−0.985747	−0.492874	−	0.870101i	$-0.664054\pi$
$43$	−6.46398	−0.985747	−0.492874	+	0.870101i	$0.664054\pi$
$44$	0	0
$45$	−0.654589	−0.0975804
$46$	0	0
$47$	−12.1946	−1.77876	−0.889380	−	0.457169i	$-0.848863\pi$
$47$	−12.1946	−1.77876	−0.889380	+	0.457169i	$0.848863\pi$
$48$	0	0
$49$	−0.488212	−0.0697446
$50$	0	0
$51$	−12.2896	−1.72089
$52$	0	0
$53$	−2.15145	−0.295524	−0.147762	−	0.989023i	$-0.547207\pi$
$53$	−2.15145	−0.295524	−0.147762	+	0.989023i	$0.547207\pi$
$54$	0	0
$55$	−12.5631	−1.69401
$56$	0	0
$57$	8.59623	1.13860
$58$	0	0
$59$	5.57924	0.726356	0.363178	−	0.931720i	$-0.381692\pi$
$59$	5.57924	0.726356	0.363178	+	0.931720i	$0.381692\pi$
$60$	0	0
$61$	2.81215	0.360058	0.180029	−	0.983661i	$-0.442381\pi$
$61$	2.81215	0.360058	0.180029	+	0.983661i	$0.442381\pi$
$62$	0	0
$63$	0.514069	0.0647666
$64$	0	0
$65$	15.5087	1.92362
$66$	0	0
$67$	15.2590	1.86419	0.932094	−	0.362215i	$-0.117980\pi$
$67$	15.2590	1.86419	0.932094	+	0.362215i	$0.117980\pi$
$68$	0	0
$69$	−8.27182	−0.995810
$70$	0	0
$71$	12.4025	1.47191	0.735953	−	0.677033i	$-0.236735\pi$
$71$	12.4025	1.47191	0.735953	+	0.677033i	$0.236735\pi$
$72$	0	0
$73$	−1.40887	−0.164895	−0.0824477	−	0.996595i	$-0.526274\pi$
$73$	−1.40887	−0.164895	−0.0824477	+	0.996595i	$0.526274\pi$
$74$	0	0
$75$	9.29844	1.07369
$76$	0	0
$77$	9.86619	1.12436
$78$	0	0
$79$	−12.3648	−1.39115	−0.695574	−	0.718455i	$-0.744850\pi$
$79$	−12.3648	−1.39115	−0.695574	+	0.718455i	$0.744850\pi$
$80$	0	0
$81$	−8.35506	−0.928340
$82$	0	0
$83$	1.68284	0.184716	0.0923579	−	0.995726i	$-0.470560\pi$
$83$	1.68284	0.184716	0.0923579	+	0.995726i	$0.470560\pi$
$84$	0	0
$85$	−23.8710	−2.58917
$86$	0	0
$87$	−0.969732	−0.103966
$88$	0	0
$89$	15.0207	1.59219	0.796095	−	0.605172i	$-0.206896\pi$
$89$	15.0207	1.59219	0.796095	+	0.605172i	$0.206896\pi$
$90$	0	0
$91$	−12.1794	−1.27675
$92$	0	0
$93$	−10.5721	−1.09628
$94$	0	0
$95$	16.6970	1.71308
$96$	0	0
$97$	5.00023	0.507697	0.253848	−	0.967244i	$-0.418304\pi$
$97$	5.00023	0.507697	0.253848	+	0.967244i	$0.418304\pi$
$98$	0	0
$99$	0.778880	0.0782804
$100$	0	0
$101$	16.4546	1.63730	0.818649	−	0.574295i	$-0.194724\pi$
$101$	16.4546	1.63730	0.818649	+	0.574295i	$0.194724\pi$
$102$	0	0
$103$	−14.7144	−1.44985	−0.724926	−	0.688827i	$-0.758126\pi$
$103$	−14.7144	−1.44985	−0.724926	+	0.688827i	$0.758126\pi$
$104$	0	0
$105$	−13.8712	−1.35369
$106$	0	0
$107$	−16.9464	−1.63827	−0.819136	−	0.573600i	$-0.805547\pi$
$107$	−16.9464	−1.63827	−0.819136	+	0.573600i	$0.805547\pi$
$108$	0	0
$109$	11.9858	1.14803	0.574015	−	0.818845i	$-0.305385\pi$
$109$	11.9858	1.14803	0.574015	+	0.818845i	$0.305385\pi$
$110$	0	0
$111$	−12.5845	−1.19446
$112$	0	0
$113$	12.1742	1.14525	0.572625	−	0.819817i	$-0.305925\pi$
$113$	12.1742	1.14525	0.572625	+	0.819817i	$0.305925\pi$
$114$	0	0
$115$	−16.0669	−1.49825
$116$	0	0
$117$	−0.961499	−0.0888906
$118$	0	0
$119$	18.7466	1.71850
$120$	0	0
$121$	3.94855	0.358959
$122$	0	0
$123$	−12.8358	−1.15737
$124$	0	0
$125$	1.81419	0.162266
$126$	0	0
$127$	2.93088	0.260073	0.130037	−	0.991509i	$-0.458490\pi$
$127$	2.93088	0.260073	0.130037	+	0.991509i	$0.458490\pi$
$128$	0	0
$129$	−10.8135	−0.952076
$130$	0	0
$131$	8.44438	0.737789	0.368895	−	0.929471i	$-0.379736\pi$
$131$	8.44438	0.737789	0.368895	+	0.929471i	$0.379736\pi$
$132$	0	0
$133$	−13.1127	−1.13701
$134$	0	0
$135$	−17.4025	−1.49777
$136$	0	0
$137$	7.31259	0.624756	0.312378	−	0.949958i	$-0.398874\pi$
$137$	7.31259	0.624756	0.312378	+	0.949958i	$0.398874\pi$
$138$	0	0
$139$	−19.1762	−1.62650	−0.813251	−	0.581913i	$-0.802305\pi$
$139$	−19.1762	−1.62650	−0.813251	+	0.581913i	$0.802305\pi$
$140$	0	0
$141$	−20.4001	−1.71800
$142$	0	0
$143$	−18.4534	−1.54315
$144$	0	0
$145$	−1.88357	−0.156422
$146$	0	0
$147$	−0.816723	−0.0673622
$148$	0	0
$149$	17.5058	1.43413	0.717067	−	0.697004i	$-0.245484\pi$
$149$	17.5058	1.43413	0.717067	+	0.697004i	$0.245484\pi$
$150$	0	0
$151$	−15.4411	−1.25658	−0.628291	−	0.777979i	$-0.716245\pi$
$151$	−15.4411	−1.25658	−0.628291	+	0.777979i	$0.716245\pi$
$152$	0	0
$153$	1.47994	0.119646
$154$	0	0
$155$	−20.5349	−1.64940
$156$	0	0
$157$	−6.55771	−0.523362	−0.261681	−	0.965154i	$-0.584277\pi$
$157$	−6.55771	−0.523362	−0.261681	+	0.965154i	$0.584277\pi$
$158$	0	0
$159$	−3.59913	−0.285429
$160$	0	0
$161$	12.6178	0.994425
$162$	0	0
$163$	−9.81113	−0.768467	−0.384234	−	0.923236i	$-0.625534\pi$
$163$	−9.81113	−0.768467	−0.384234	+	0.923236i	$0.625534\pi$
$164$	0	0
$165$	−21.0166	−1.63614
$166$	0	0
$167$	2.47178	0.191272	0.0956359	−	0.995416i	$-0.469512\pi$
$167$	2.47178	0.191272	0.0956359	+	0.995416i	$0.469512\pi$
$168$	0	0
$169$	9.78007	0.752313
$170$	0	0
$171$	−1.03517	−0.0791616
$172$	0	0
$173$	0.144954	0.0110206	0.00551031	−	0.999985i	$-0.498246\pi$
$173$	0.144954	0.0110206	0.00551031	+	0.999985i	$0.498246\pi$
$174$	0	0
$175$	−14.1838	−1.07220
$176$	0	0
$177$	9.33344	0.701544
$178$	0	0
$179$	0.265766	0.0198643	0.00993214	−	0.999951i	$-0.496838\pi$
$179$	0.265766	0.0198643	0.00993214	+	0.999951i	$0.496838\pi$
$180$	0	0
$181$	−16.2296	−1.20634	−0.603169	−	0.797613i	$-0.706096\pi$
$181$	−16.2296	−1.20634	−0.603169	+	0.797613i	$0.706096\pi$
$182$	0	0
$183$	4.70440	0.347759
$184$	0	0
$185$	−24.4436	−1.79713
$186$	0	0
$187$	28.4035	2.07707
$188$	0	0
$189$	13.6667	0.994106
$190$	0	0
$191$	16.1429	1.16806	0.584030	−	0.811732i	$-0.301475\pi$
$191$	16.1429	1.16806	0.584030	+	0.811732i	$0.301475\pi$
$192$	0	0
$193$	5.60406	0.403389	0.201694	−	0.979448i	$-0.435355\pi$
$193$	5.60406	0.403389	0.201694	+	0.979448i	$0.435355\pi$
$194$	0	0
$195$	25.9443	1.85791
$196$	0	0
$197$	18.7095	1.33300	0.666498	−	0.745507i	$-0.267793\pi$
$197$	18.7095	1.33300	0.666498	+	0.745507i	$0.267793\pi$
$198$	0	0
$199$	8.35159	0.592029	0.296014	−	0.955183i	$-0.404342\pi$
$199$	8.35159	0.592029	0.296014	+	0.955183i	$0.404342\pi$
$200$	0	0
$201$	25.5266	1.80051
$202$	0	0
$203$	1.47923	0.103822
$204$	0	0
$205$	−24.9319	−1.74132
$206$	0	0
$207$	0.996107	0.0692342
$208$	0	0
$209$	−19.8674	−1.37426
$210$	0	0
$211$	25.7042	1.76955	0.884777	−	0.466015i	$-0.154311\pi$
$211$	25.7042	1.76955	0.884777	+	0.466015i	$0.154311\pi$
$212$	0	0
$213$	20.7480	1.42163
$214$	0	0
$215$	−21.0038	−1.43245
$216$	0	0
$217$	16.1267	1.09475
$218$	0	0
$219$	−2.35687	−0.159263
$220$	0	0
$221$	−35.0631	−2.35860
$222$	0	0
$223$	−24.7287	−1.65596	−0.827979	−	0.560759i	$-0.810509\pi$
$223$	−24.7287	−1.65596	−0.827979	+	0.560759i	$0.810509\pi$
$224$	0	0
$225$	−1.11973	−0.0746490
$226$	0	0
$227$	−11.8321	−0.785326	−0.392663	−	0.919682i	$-0.628446\pi$
$227$	−11.8321	−0.785326	−0.392663	+	0.919682i	$0.628446\pi$
$228$	0	0
$229$	−5.19219	−0.343110	−0.171555	−	0.985175i	$-0.554879\pi$
$229$	−5.19219	−0.343110	−0.171555	+	0.985175i	$0.554879\pi$
$230$	0	0
$231$	16.5050	1.08595
$232$	0	0
$233$	2.17352	0.142392	0.0711959	−	0.997462i	$-0.477318\pi$
$233$	2.17352	0.142392	0.0711959	+	0.997462i	$0.477318\pi$
$234$	0	0
$235$	−39.6245	−2.58482
$236$	0	0
$237$	−20.6849	−1.34363
$238$	0	0
$239$	12.0873	0.781860	0.390930	−	0.920420i	$-0.372153\pi$
$239$	12.0873	0.781860	0.390930	+	0.920420i	$0.372153\pi$
$240$	0	0
$241$	−11.0591	−0.712378	−0.356189	−	0.934414i	$-0.615924\pi$
$241$	−11.0591	−0.712378	−0.356189	+	0.934414i	$0.615924\pi$
$242$	0	0
$243$	2.08993	0.134069
$244$	0	0
$245$	−1.58638	−0.101350
$246$	0	0
$247$	24.5256	1.56052
$248$	0	0
$249$	2.81520	0.178406
$250$	0	0
$251$	14.2693	0.900669	0.450335	−	0.892860i	$-0.351305\pi$
$251$	14.2693	0.900669	0.450335	+	0.892860i	$0.351305\pi$
$252$	0	0
$253$	19.1176	1.20192
$254$	0	0
$255$	−39.9334	−2.50073
$256$	0	0
$257$	−7.83668	−0.488839	−0.244419	−	0.969670i	$-0.578597\pi$
$257$	−7.83668	−0.488839	−0.244419	+	0.969670i	$0.578597\pi$
$258$	0	0
$259$	19.1963	1.19280
$260$	0	0
$261$	0.116777	0.00722830
$262$	0	0
$263$	−18.3948	−1.13427	−0.567135	−	0.823625i	$-0.691948\pi$
$263$	−18.3948	−1.13427	−0.567135	+	0.823625i	$0.691948\pi$
$264$	0	0
$265$	−6.99082	−0.429443
$266$	0	0
$267$	25.1279	1.53780
$268$	0	0
$269$	−5.11081	−0.311612	−0.155806	−	0.987788i	$-0.549797\pi$
$269$	−5.11081	−0.311612	−0.155806	+	0.987788i	$0.549797\pi$
$270$	0	0
$271$	19.2724	1.17072	0.585358	−	0.810775i	$-0.300954\pi$
$271$	19.2724	1.17072	0.585358	+	0.810775i	$0.300954\pi$
$272$	0	0
$273$	−20.3748	−1.23314
$274$	0	0
$275$	−21.4903	−1.29592
$276$	0	0
$277$	−32.5523	−1.95588	−0.977939	−	0.208890i	$-0.933015\pi$
$277$	−32.5523	−1.95588	−0.977939	+	0.208890i	$0.933015\pi$
$278$	0	0
$279$	1.27311	0.0762192
$280$	0	0
$281$	−10.5395	−0.628737	−0.314368	−	0.949301i	$-0.601793\pi$
$281$	−10.5395	−0.628737	−0.314368	+	0.949301i	$0.601793\pi$
$282$	0	0
$283$	−18.0435	−1.07257	−0.536287	−	0.844036i	$-0.680173\pi$
$283$	−18.0435	−1.07257	−0.536287	+	0.844036i	$0.680173\pi$
$284$	0	0
$285$	27.9322	1.65456
$286$	0	0
$287$	19.5798	1.15576
$288$	0	0
$289$	36.9691	2.17465
$290$	0	0
$291$	8.36482	0.490354
$292$	0	0
$293$	−29.1173	−1.70105	−0.850524	−	0.525936i	$-0.823715\pi$
$293$	−29.1173	−1.70105	−0.850524	+	0.525936i	$0.823715\pi$
$294$	0	0
$295$	18.1290	1.05551
$296$	0	0
$297$	20.7068	1.20153
$298$	0	0
$299$	−23.6000	−1.36482
$300$	0	0
$301$	16.4949	0.950751
$302$	0	0
$303$	27.5267	1.58137
$304$	0	0
$305$	9.13767	0.523221
$306$	0	0
$307$	−1.88567	−0.107621	−0.0538105	−	0.998551i	$-0.517137\pi$
$307$	−1.88567	−0.107621	−0.0538105	+	0.998551i	$0.517137\pi$
$308$	0	0
$309$	−24.6155	−1.40033
$310$	0	0
$311$	−26.4793	−1.50150	−0.750750	−	0.660586i	$-0.770308\pi$
$311$	−26.4793	−1.50150	−0.750750	+	0.660586i	$0.770308\pi$
$312$	0	0
$313$	−17.9790	−1.01623	−0.508116	−	0.861289i	$-0.669658\pi$
$313$	−17.9790	−1.01623	−0.508116	+	0.861289i	$0.669658\pi$
$314$	0	0
$315$	1.67039	0.0941160
$316$	0	0
$317$	17.4857	0.982094	0.491047	−	0.871133i	$-0.336614\pi$
$317$	17.4857	0.982094	0.491047	+	0.871133i	$0.336614\pi$
$318$	0	0
$319$	2.24122	0.125484
$320$	0	0
$321$	−28.3494	−1.58231
$322$	0	0
$323$	−37.7498	−2.10045
$324$	0	0
$325$	26.5290	1.47157
$326$	0	0
$327$	20.0509	1.10881
$328$	0	0
$329$	31.1183	1.71561
$330$	0	0
$331$	−22.1630	−1.21819	−0.609093	−	0.793099i	$-0.708466\pi$
$331$	−22.1630	−1.21819	−0.609093	+	0.793099i	$0.708466\pi$
$332$	0	0
$333$	1.51544	0.0830457
$334$	0	0
$335$	49.5821	2.70896
$336$	0	0
$337$	13.2710	0.722917	0.361459	−	0.932388i	$-0.382279\pi$
$337$	13.2710	0.722917	0.361459	+	0.932388i	$0.382279\pi$
$338$	0	0
$339$	20.3660	1.10613
$340$	0	0
$341$	24.4340	1.32318
$342$	0	0
$343$	19.1086	1.03177
$344$	0	0
$345$	−26.8781	−1.44707
$346$	0	0
$347$	5.00182	0.268512	0.134256	−	0.990947i	$-0.457136\pi$
$347$	5.00182	0.268512	0.134256	+	0.990947i	$0.457136\pi$
$348$	0	0
$349$	−13.6624	−0.731334	−0.365667	−	0.930746i	$-0.619159\pi$
$349$	−13.6624	−0.731334	−0.365667	+	0.930746i	$0.619159\pi$
$350$	0	0
$351$	−25.5618	−1.36439
$352$	0	0
$353$	0.383286	0.0204002	0.0102001	−	0.999948i	$-0.496753\pi$
$353$	0.383286	0.0204002	0.0102001	+	0.999948i	$0.496753\pi$
$354$	0	0
$355$	40.3001	2.13891
$356$	0	0
$357$	31.3609	1.65980
$358$	0	0
$359$	−2.89299	−0.152686	−0.0763430	−	0.997082i	$-0.524324\pi$
$359$	−2.89299	−0.152686	−0.0763430	+	0.997082i	$0.524324\pi$
$360$	0	0
$361$	7.40480	0.389726
$362$	0	0
$363$	6.60547	0.346698
$364$	0	0
$365$	−4.57791	−0.239619
$366$	0	0
$367$	3.36998	0.175912	0.0879558	−	0.996124i	$-0.471967\pi$
$367$	3.36998	0.175912	0.0879558	+	0.996124i	$0.471967\pi$
$368$	0	0
$369$	1.54571	0.0804666
$370$	0	0
$371$	5.49011	0.285032
$372$	0	0
$373$	−7.43334	−0.384884	−0.192442	−	0.981308i	$-0.561641\pi$
$373$	−7.43334	−0.384884	−0.192442	+	0.981308i	$0.561641\pi$
$374$	0	0
$375$	3.03494	0.156724
$376$	0	0
$377$	−2.76671	−0.142493
$378$	0	0
$379$	−3.45621	−0.177534	−0.0887669	−	0.996052i	$-0.528293\pi$
$379$	−3.45621	−0.177534	−0.0887669	+	0.996052i	$0.528293\pi$
$380$	0	0
$381$	4.90303	0.251190
$382$	0	0
$383$	17.0652	0.871990	0.435995	−	0.899949i	$-0.356397\pi$
$383$	17.0652	0.871990	0.435995	+	0.899949i	$0.356397\pi$
$384$	0	0
$385$	32.0588	1.63387
$386$	0	0
$387$	1.30218	0.0661935
$388$	0	0
$389$	26.2792	1.33241	0.666204	−	0.745769i	$-0.267918\pi$
$389$	26.2792	1.33241	0.666204	+	0.745769i	$0.267918\pi$
$390$	0	0
$391$	36.3251	1.83704
$392$	0	0
$393$	14.1265	0.712587
$394$	0	0
$395$	−40.1776	−2.02156
$396$	0	0
$397$	−2.25105	−0.112977	−0.0564884	−	0.998403i	$-0.517990\pi$
$397$	−2.25105	−0.112977	−0.0564884	+	0.998403i	$0.517990\pi$
$398$	0	0
$399$	−21.9360	−1.09817
$400$	0	0
$401$	−2.97404	−0.148517	−0.0742583	−	0.997239i	$-0.523659\pi$
$401$	−2.97404	−0.148517	−0.0742583	+	0.997239i	$0.523659\pi$
$402$	0	0
$403$	−30.1629	−1.50252
$404$	0	0
$405$	−27.1486	−1.34902
$406$	0	0
$407$	29.0849	1.44168
$408$	0	0
$409$	17.3041	0.855631	0.427816	−	0.903866i	$-0.359283\pi$
$409$	17.3041	0.855631	0.427816	+	0.903866i	$0.359283\pi$
$410$	0	0
$411$	12.2331	0.603415
$412$	0	0
$413$	−14.2372	−0.700568
$414$	0	0
$415$	5.46815	0.268421
$416$	0	0
$417$	−32.0796	−1.57094
$418$	0	0
$419$	−25.2643	−1.23424	−0.617120	−	0.786869i	$-0.711701\pi$
$419$	−25.2643	−1.23424	−0.617120	+	0.786869i	$0.711701\pi$
$420$	0	0
$421$	29.5621	1.44077	0.720385	−	0.693574i	$-0.243965\pi$
$421$	29.5621	1.44077	0.720385	+	0.693574i	$0.243965\pi$
$422$	0	0
$423$	2.45662	0.119445
$424$	0	0
$425$	−40.8335	−1.98072
$426$	0	0
$427$	−7.17609	−0.347275
$428$	0	0
$429$	−30.8705	−1.49044
$430$	0	0
$431$	41.0232	1.97602	0.988009	−	0.154399i	$-0.0493440\pi$
$431$	41.0232	1.97602	0.988009	+	0.154399i	$0.0493440\pi$
$432$	0	0
$433$	−37.3663	−1.79571	−0.897855	−	0.440291i	$-0.854875\pi$
$433$	−37.3663	−1.79571	−0.897855	+	0.440291i	$0.854875\pi$
$434$	0	0
$435$	−3.15101	−0.151079
$436$	0	0
$437$	−25.4083	−1.21545
$438$	0	0
$439$	−13.7981	−0.658547	−0.329273	−	0.944235i	$-0.606804\pi$
$439$	−13.7981	−0.658547	−0.329273	+	0.944235i	$0.606804\pi$
$440$	0	0
$441$	0.0983512	0.00468339
$442$	0	0
$443$	−6.29465	−0.299068	−0.149534	−	0.988757i	$-0.547777\pi$
$443$	−6.29465	−0.299068	−0.149534	+	0.988757i	$0.547777\pi$
$444$	0	0
$445$	48.8076	2.31370
$446$	0	0
$447$	29.2853	1.38515
$448$	0	0
$449$	13.6628	0.644786	0.322393	−	0.946606i	$-0.395513\pi$
$449$	13.6628	0.644786	0.322393	+	0.946606i	$0.395513\pi$
$450$	0	0
$451$	29.6659	1.39691
$452$	0	0
$453$	−25.8313	−1.21366
$454$	0	0
$455$	−39.5754	−1.85532
$456$	0	0
$457$	−11.2587	−0.526661	−0.263330	−	0.964706i	$-0.584821\pi$
$457$	−11.2587	−0.526661	−0.263330	+	0.964706i	$0.584821\pi$
$458$	0	0
$459$	39.3447	1.83645
$460$	0	0
$461$	−27.9452	−1.30154	−0.650769	−	0.759276i	$-0.725553\pi$
$461$	−27.9452	−1.30154	−0.650769	+	0.759276i	$0.725553\pi$
$462$	0	0
$463$	19.8264	0.921413	0.460707	−	0.887552i	$-0.347596\pi$
$463$	19.8264	0.921413	0.460707	+	0.887552i	$0.347596\pi$
$464$	0	0
$465$	−34.3526	−1.59306
$466$	0	0
$467$	13.6220	0.630352	0.315176	−	0.949033i	$-0.397936\pi$
$467$	13.6220	0.630352	0.315176	+	0.949033i	$0.397936\pi$
$468$	0	0
$469$	−38.9383	−1.79801
$470$	0	0
$471$	−10.9703	−0.505485
$472$	0	0
$473$	24.9919	1.14913
$474$	0	0
$475$	28.5618	1.31050
$476$	0	0
$477$	0.433413	0.0198446
$478$	0	0
$479$	−5.39842	−0.246660	−0.123330	−	0.992366i	$-0.539357\pi$
$479$	−5.39842	−0.246660	−0.123330	+	0.992366i	$0.539357\pi$
$480$	0	0
$481$	−35.9042	−1.63709
$482$	0	0
$483$	21.1082	0.960456
$484$	0	0
$485$	16.2475	0.737763
$486$	0	0
$487$	−5.54090	−0.251082	−0.125541	−	0.992088i	$-0.540067\pi$
$487$	−5.54090	−0.251082	−0.125541	+	0.992088i	$0.540067\pi$
$488$	0	0
$489$	−16.4129	−0.742217
$490$	0	0
$491$	−14.4848	−0.653691	−0.326846	−	0.945078i	$-0.605986\pi$
$491$	−14.4848	−0.653691	−0.326846	+	0.945078i	$0.605986\pi$
$492$	0	0
$493$	4.25851	0.191794
$494$	0	0
$495$	2.53086	0.113754
$496$	0	0
$497$	−31.6489	−1.41965
$498$	0	0
$499$	23.7664	1.06393	0.531964	−	0.846767i	$-0.321454\pi$
$499$	23.7664	1.06393	0.531964	+	0.846767i	$0.321454\pi$
$500$	0	0
$501$	4.13500	0.184738
$502$	0	0
$503$	7.53623	0.336024	0.168012	−	0.985785i	$-0.446265\pi$
$503$	7.53623	0.336024	0.168012	+	0.985785i	$0.446265\pi$
$504$	0	0
$505$	53.4670	2.37925
$506$	0	0
$507$	16.3609	0.726615
$508$	0	0
$509$	28.0122	1.24162	0.620810	−	0.783961i	$-0.286804\pi$
$509$	28.0122	1.24162	0.620810	+	0.783961i	$0.286804\pi$
$510$	0	0
$511$	3.59518	0.159041
$512$	0	0
$513$	−27.5204	−1.21506
$514$	0	0
$515$	−47.8123	−2.10686
$516$	0	0
$517$	47.1483	2.07358
$518$	0	0
$519$	0.242491	0.0106442
$520$	0	0
$521$	27.2359	1.19322	0.596612	−	0.802530i	$-0.296513\pi$
$521$	27.2359	1.19322	0.596612	+	0.802530i	$0.296513\pi$
$522$	0	0
$523$	−40.5393	−1.77266	−0.886329	−	0.463056i	$-0.846753\pi$
$523$	−40.5393	−1.77266	−0.886329	+	0.463056i	$0.846753\pi$
$524$	0	0
$525$	−23.7280	−1.03557
$526$	0	0
$527$	46.4267	2.02238
$528$	0	0
$529$	1.44947	0.0630202
$530$	0	0
$531$	−1.12395	−0.0487752
$532$	0	0
$533$	−36.6214	−1.58625
$534$	0	0
$535$	−55.0650	−2.38067
$536$	0	0
$537$	0.444596	0.0191857
$538$	0	0
$539$	1.88759	0.0813043
$540$	0	0
$541$	−8.28092	−0.356025	−0.178012	−	0.984028i	$-0.556967\pi$
$541$	−8.28092	−0.356025	−0.178012	+	0.984028i	$0.556967\pi$
$542$	0	0
$543$	−27.1503	−1.16513
$544$	0	0
$545$	38.9461	1.66827
$546$	0	0
$547$	−24.4071	−1.04357	−0.521787	−	0.853076i	$-0.674734\pi$
$547$	−24.4071	−1.04357	−0.521787	+	0.853076i	$0.674734\pi$
$548$	0	0
$549$	−0.566512	−0.0241781
$550$	0	0
$551$	−2.97870	−0.126897
$552$	0	0
$553$	31.5527	1.34176
$554$	0	0
$555$	−40.8914	−1.73574
$556$	0	0
$557$	16.8420	0.713620	0.356810	−	0.934177i	$-0.383864\pi$
$557$	16.8420	0.713620	0.356810	+	0.934177i	$0.383864\pi$
$558$	0	0
$559$	−30.8516	−1.30488
$560$	0	0
$561$	47.5159	2.00612
$562$	0	0
$563$	−22.0365	−0.928727	−0.464364	−	0.885645i	$-0.653717\pi$
$563$	−22.0365	−0.928727	−0.464364	+	0.885645i	$0.653717\pi$
$564$	0	0
$565$	39.5582	1.66423
$566$	0	0
$567$	21.3206	0.895382
$568$	0	0
$569$	−2.60335	−0.109138	−0.0545690	−	0.998510i	$-0.517378\pi$
$569$	−2.60335	−0.109138	−0.0545690	+	0.998510i	$0.517378\pi$
$570$	0	0
$571$	12.5814	0.526514	0.263257	−	0.964726i	$-0.415203\pi$
$571$	12.5814	0.526514	0.263257	+	0.964726i	$0.415203\pi$
$572$	0	0
$573$	27.0053	1.12816
$574$	0	0
$575$	−27.4839	−1.14616
$576$	0	0
$577$	23.4115	0.974632	0.487316	−	0.873226i	$-0.337976\pi$
$577$	23.4115	0.974632	0.487316	+	0.873226i	$0.337976\pi$
$578$	0	0
$579$	9.37495	0.389610
$580$	0	0
$581$	−4.29431	−0.178158
$582$	0	0
$583$	8.31822	0.344505
$584$	0	0
$585$	−3.12425	−0.129172
$586$	0	0
$587$	−7.26892	−0.300020	−0.150010	−	0.988684i	$-0.547931\pi$
$587$	−7.26892	−0.300020	−0.150010	+	0.988684i	$0.547931\pi$
$588$	0	0
$589$	−32.4741	−1.33807
$590$	0	0
$591$	31.2988	1.28746
$592$	0	0
$593$	−10.6941	−0.439154	−0.219577	−	0.975595i	$-0.570468\pi$
$593$	−10.6941	−0.439154	−0.219577	+	0.975595i	$0.570468\pi$
$594$	0	0
$595$	60.9144	2.49725
$596$	0	0
$597$	13.9713	0.571806
$598$	0	0
$599$	7.20414	0.294353	0.147177	−	0.989110i	$-0.452981\pi$
$599$	7.20414	0.294353	0.147177	+	0.989110i	$0.452981\pi$
$600$	0	0
$601$	−4.23602	−0.172791	−0.0863954	−	0.996261i	$-0.527535\pi$
$601$	−4.23602	−0.172791	−0.0863954	+	0.996261i	$0.527535\pi$
$602$	0	0
$603$	−3.07396	−0.125181
$604$	0	0
$605$	12.8303	0.521624
$606$	0	0
$607$	−9.40480	−0.381729	−0.190865	−	0.981616i	$-0.561129\pi$
$607$	−9.40480	−0.381729	−0.190865	+	0.981616i	$0.561129\pi$
$608$	0	0
$609$	2.47458	0.100275
$610$	0	0
$611$	−58.2028	−2.35463
$612$	0	0
$613$	14.1068	0.569768	0.284884	−	0.958562i	$-0.408045\pi$
$613$	14.1068	0.569768	0.284884	+	0.958562i	$0.408045\pi$
$614$	0	0
$615$	−41.7082	−1.68184
$616$	0	0
$617$	24.4239	0.983271	0.491635	−	0.870801i	$-0.336399\pi$
$617$	24.4239	0.983271	0.491635	+	0.870801i	$0.336399\pi$
$618$	0	0
$619$	−18.6508	−0.749640	−0.374820	−	0.927098i	$-0.622295\pi$
$619$	−18.6508	−0.749640	−0.374820	+	0.927098i	$0.622295\pi$
$620$	0	0
$621$	26.4818	1.06268
$622$	0	0
$623$	−38.3301	−1.53566
$624$	0	0
$625$	−21.8967	−0.875866
$626$	0	0
$627$	−33.2359	−1.32731
$628$	0	0
$629$	55.2638	2.20351
$630$	0	0
$631$	−45.6512	−1.81735	−0.908674	−	0.417507i	$-0.862904\pi$
$631$	−45.6512	−1.81735	−0.908674	+	0.417507i	$0.862904\pi$
$632$	0	0
$633$	43.0003	1.70911
$634$	0	0
$635$	9.52347	0.377927
$636$	0	0
$637$	−2.33016	−0.0923244
$638$	0	0
$639$	−2.49850	−0.0988393
$640$	0	0
$641$	−3.68608	−0.145592	−0.0727958	−	0.997347i	$-0.523192\pi$
$641$	−3.68608	−0.145592	−0.0727958	+	0.997347i	$0.523192\pi$
$642$	0	0
$643$	−37.6364	−1.48423	−0.742117	−	0.670270i	$-0.766178\pi$
$643$	−37.6364	−1.48423	−0.742117	+	0.670270i	$0.766178\pi$
$644$	0	0
$645$	−35.1369	−1.38352
$646$	0	0
$647$	−49.2705	−1.93702	−0.968511	−	0.248971i	$-0.919908\pi$
$647$	−49.2705	−1.93702	−0.968511	+	0.248971i	$0.919908\pi$
$648$	0	0
$649$	−21.5712	−0.846745
$650$	0	0
$651$	26.9781	1.05736
$652$	0	0
$653$	9.86159	0.385914	0.192957	−	0.981207i	$-0.438192\pi$
$653$	9.86159	0.385914	0.192957	+	0.981207i	$0.438192\pi$
$654$	0	0
$655$	27.4388	1.07212
$656$	0	0
$657$	0.283819	0.0110728
$658$	0	0
$659$	31.9293	1.24379	0.621895	−	0.783101i	$-0.286363\pi$
$659$	31.9293	1.24379	0.621895	+	0.783101i	$0.286363\pi$
$660$	0	0
$661$	−18.6107	−0.723874	−0.361937	−	0.932203i	$-0.617884\pi$
$661$	−18.6107	−0.723874	−0.361937	+	0.932203i	$0.617884\pi$
$662$	0	0
$663$	−58.6566	−2.27803
$664$	0	0
$665$	−42.6078	−1.65226
$666$	0	0
$667$	2.86629	0.110983
$668$	0	0
$669$	−41.3683	−1.59939
$670$	0	0
$671$	−10.8727	−0.419736
$672$	0	0
$673$	−13.4646	−0.519021	−0.259510	−	0.965740i	$-0.583561\pi$
$673$	−13.4646	−0.519021	−0.259510	+	0.965740i	$0.583561\pi$
$674$	0	0
$675$	−29.7685	−1.14579
$676$	0	0
$677$	−29.3630	−1.12851	−0.564257	−	0.825599i	$-0.690837\pi$
$677$	−29.3630	−1.12851	−0.564257	+	0.825599i	$0.690837\pi$
$678$	0	0
$679$	−12.7597	−0.489672
$680$	0	0
$681$	−19.7938	−0.758500
$682$	0	0
$683$	−2.74895	−0.105186	−0.0525929	−	0.998616i	$-0.516749\pi$
$683$	−2.74895	−0.105186	−0.0525929	+	0.998616i	$0.516749\pi$
$684$	0	0
$685$	23.7612	0.907869
$686$	0	0
$687$	−8.68595	−0.331390
$688$	0	0
$689$	−10.2685	−0.391200
$690$	0	0
$691$	−34.0741	−1.29624	−0.648119	−	0.761539i	$-0.724444\pi$
$691$	−34.0741	−1.29624	−0.648119	+	0.761539i	$0.724444\pi$
$692$	0	0
$693$	−1.98756	−0.0755013
$694$	0	0
$695$	−62.3103	−2.36356
$696$	0	0
$697$	56.3677	2.13508
$698$	0	0
$699$	3.63605	0.137528
$700$	0	0
$701$	11.0186	0.416165	0.208083	−	0.978111i	$-0.433278\pi$
$701$	11.0186	0.416165	0.208083	+	0.978111i	$0.433278\pi$
$702$	0	0
$703$	−38.6553	−1.45791
$704$	0	0
$705$	−66.2873	−2.49652
$706$	0	0
$707$	−41.9893	−1.57917
$708$	0	0
$709$	−47.9570	−1.80106	−0.900531	−	0.434791i	$-0.856822\pi$
$709$	−47.9570	−1.80106	−0.900531	+	0.434791i	$0.856822\pi$
$710$	0	0
$711$	2.49091	0.0934164
$712$	0	0
$713$	31.2485	1.17027
$714$	0	0
$715$	−59.9618	−2.24244
$716$	0	0
$717$	20.2206	0.755153
$718$	0	0
$719$	−11.2994	−0.421397	−0.210698	−	0.977551i	$-0.567574\pi$
$719$	−11.2994	−0.421397	−0.210698	+	0.977551i	$0.567574\pi$
$720$	0	0
$721$	37.5485	1.39838
$722$	0	0
$723$	−18.5006	−0.688044
$724$	0	0
$725$	−3.22203	−0.119663
$726$	0	0
$727$	52.6897	1.95415	0.977076	−	0.212890i	$-0.0682875\pi$
$727$	52.6897	1.95415	0.977076	+	0.212890i	$0.0682875\pi$
$728$	0	0
$729$	28.5614	1.05783
$730$	0	0
$731$	47.4868	1.75636
$732$	0	0
$733$	49.9995	1.84677	0.923386	−	0.383873i	$-0.125410\pi$
$733$	49.9995	1.84677	0.923386	+	0.383873i	$0.125410\pi$
$734$	0	0
$735$	−2.65383	−0.0978879
$736$	0	0
$737$	−58.9966	−2.17317
$738$	0	0
$739$	38.6115	1.42035	0.710174	−	0.704026i	$-0.248616\pi$
$739$	38.6115	1.42035	0.710174	+	0.704026i	$0.248616\pi$
$740$	0	0
$741$	41.0285	1.50722
$742$	0	0
$743$	53.0323	1.94557	0.972784	−	0.231716i	$-0.0744340\pi$
$743$	53.0323	1.94557	0.972784	+	0.231716i	$0.0744340\pi$
$744$	0	0
$745$	56.8827	2.08402
$746$	0	0
$747$	−0.339011	−0.0124038
$748$	0	0
$749$	43.2442	1.58011
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	23.8709	0.869903
$754$	0	0
$755$	−50.1738	−1.82601
$756$	0	0
$757$	23.0149	0.836489	0.418245	−	0.908334i	$-0.362646\pi$
$757$	23.0149	0.836489	0.418245	+	0.908334i	$0.362646\pi$
$758$	0	0
$759$	31.9816	1.16086
$760$	0	0
$761$	17.1684	0.622355	0.311178	−	0.950352i	$-0.399277\pi$
$761$	17.1684	0.622355	0.311178	+	0.950352i	$0.399277\pi$
$762$	0	0
$763$	−30.5856	−1.10727
$764$	0	0
$765$	4.80885	0.173864
$766$	0	0
$767$	26.6289	0.961513
$768$	0	0
$769$	−28.0034	−1.00983	−0.504915	−	0.863169i	$-0.668476\pi$
$769$	−28.0034	−1.00983	−0.504915	+	0.863169i	$0.668476\pi$
$770$	0	0
$771$	−13.1099	−0.472141
$772$	0	0
$773$	−21.2341	−0.763737	−0.381869	−	0.924217i	$-0.624719\pi$
$773$	−21.2341	−0.763737	−0.381869	+	0.924217i	$0.624719\pi$
$774$	0	0
$775$	−35.1268	−1.26179
$776$	0	0
$777$	32.1133	1.15206
$778$	0	0
$779$	−39.4275	−1.41264
$780$	0	0
$781$	−47.9522	−1.71586
$782$	0	0
$783$	3.10455	0.110948
$784$	0	0
$785$	−21.3083	−0.760527
$786$	0	0
$787$	4.04884	0.144326	0.0721628	−	0.997393i	$-0.477010\pi$
$787$	4.04884	0.144326	0.0721628	+	0.997393i	$0.477010\pi$
$788$	0	0
$789$	−30.7724	−1.09553
$790$	0	0
$791$	−31.0663	−1.10459
$792$	0	0
$793$	13.4219	0.476627
$794$	0	0
$795$	−11.6949	−0.414774
$796$	0	0
$797$	−21.0550	−0.745807	−0.372903	−	0.927870i	$-0.621638\pi$
$797$	−21.0550	−0.745807	−0.372903	+	0.927870i	$0.621638\pi$
$798$	0	0
$799$	89.5857	3.16932
$800$	0	0
$801$	−3.02594	−0.106916
$802$	0	0
$803$	5.44715	0.192226
$804$	0	0
$805$	40.9999	1.44505
$806$	0	0
$807$	−8.54981	−0.300968
$808$	0	0
$809$	40.9035	1.43809	0.719045	−	0.694964i	$-0.244580\pi$
$809$	40.9035	1.43809	0.719045	+	0.694964i	$0.244580\pi$
$810$	0	0
$811$	15.6356	0.549041	0.274521	−	0.961581i	$-0.411481\pi$
$811$	15.6356	0.549041	0.274521	+	0.961581i	$0.411481\pi$
$812$	0	0
$813$	32.2406	1.13073
$814$	0	0
$815$	−31.8799	−1.11670
$816$	0	0
$817$	−33.2155	−1.16206
$818$	0	0
$819$	2.45357	0.0857347
$820$	0	0
$821$	13.7400	0.479529	0.239764	−	0.970831i	$-0.422930\pi$
$821$	13.7400	0.479529	0.239764	+	0.970831i	$0.422930\pi$
$822$	0	0
$823$	9.88944	0.344724	0.172362	−	0.985034i	$-0.444860\pi$
$823$	9.88944	0.344724	0.172362	+	0.985034i	$0.444860\pi$
$824$	0	0
$825$	−35.9509	−1.25165
$826$	0	0
$827$	30.2012	1.05020	0.525099	−	0.851041i	$-0.324028\pi$
$827$	30.2012	1.05020	0.525099	+	0.851041i	$0.324028\pi$
$828$	0	0
$829$	−37.6915	−1.30908	−0.654539	−	0.756028i	$-0.727137\pi$
$829$	−37.6915	−1.30908	−0.654539	+	0.756028i	$0.727137\pi$
$830$	0	0
$831$	−54.4563	−1.88907
$832$	0	0
$833$	3.58659	0.124268
$834$	0	0
$835$	8.03169	0.277948
$836$	0	0
$837$	33.8461	1.16989
$838$	0	0
$839$	−14.4155	−0.497677	−0.248839	−	0.968545i	$-0.580049\pi$
$839$	−14.4155	−0.497677	−0.248839	+	0.968545i	$0.580049\pi$
$840$	0	0
$841$	−28.6640	−0.988413
$842$	0	0
$843$	−17.6315	−0.607260
$844$	0	0
$845$	31.7789	1.09323
$846$	0	0
$847$	−10.0760	−0.346215
$848$	0	0
$849$	−30.1847	−1.03594
$850$	0	0
$851$	37.1966	1.27508
$852$	0	0
$853$	52.6060	1.80119	0.900597	−	0.434654i	$-0.143129\pi$
$853$	52.6060	1.80119	0.900597	+	0.434654i	$0.143129\pi$
$854$	0	0
$855$	−3.36364	−0.115034
$856$	0	0
$857$	0.511833	0.0174839	0.00874195	−	0.999962i	$-0.497217\pi$
$857$	0.511833	0.0174839	0.00874195	+	0.999962i	$0.497217\pi$
$858$	0	0
$859$	−25.0052	−0.853168	−0.426584	−	0.904448i	$-0.640283\pi$
$859$	−25.0052	−0.853168	−0.426584	+	0.904448i	$0.640283\pi$
$860$	0	0
$861$	32.7547	1.11628
$862$	0	0
$863$	31.1650	1.06087	0.530434	−	0.847726i	$-0.322029\pi$
$863$	31.1650	1.06087	0.530434	+	0.847726i	$0.322029\pi$
$864$	0	0
$865$	0.471006	0.0160147
$866$	0	0
$867$	61.8451	2.10037
$868$	0	0
$869$	47.8064	1.62172
$870$	0	0
$871$	72.8291	2.46772
$872$	0	0
$873$	−1.00731	−0.0340921
$874$	0	0
$875$	−4.62950	−0.156506
$876$	0	0
$877$	12.2573	0.413900	0.206950	−	0.978352i	$-0.433646\pi$
$877$	12.2573	0.413900	0.206950	+	0.978352i	$0.433646\pi$
$878$	0	0
$879$	−48.7099	−1.64294
$880$	0	0
$881$	−20.3024	−0.684004	−0.342002	−	0.939699i	$-0.611105\pi$
$881$	−20.3024	−0.684004	−0.342002	+	0.939699i	$0.611105\pi$
$882$	0	0
$883$	−18.8329	−0.633777	−0.316889	−	0.948463i	$-0.602638\pi$
$883$	−18.8329	−0.633777	−0.316889	+	0.948463i	$0.602638\pi$
$884$	0	0
$885$	30.3277	1.01945
$886$	0	0
$887$	−3.70961	−0.124557	−0.0622783	−	0.998059i	$-0.519837\pi$
$887$	−3.70961	−0.124557	−0.0622783	+	0.998059i	$0.519837\pi$
$888$	0	0
$889$	−7.47908	−0.250840
$890$	0	0
$891$	32.3035	1.08221
$892$	0	0
$893$	−62.6625	−2.09692
$894$	0	0
$895$	0.863569	0.0288659
$896$	0	0
$897$	−39.4801	−1.31820
$898$	0	0
$899$	3.66337	0.122180
$900$	0	0
$901$	15.8053	0.526552
$902$	0	0
$903$	27.5941	0.918275
$904$	0	0
$905$	−52.7358	−1.75300
$906$	0	0
$907$	−11.0725	−0.367656	−0.183828	−	0.982958i	$-0.558849\pi$
$907$	−11.0725	−0.367656	−0.183828	+	0.982958i	$0.558849\pi$
$908$	0	0
$909$	−3.31482	−0.109945
$910$	0	0
$911$	19.7890	0.655639	0.327820	−	0.944740i	$-0.393686\pi$
$911$	19.7890	0.655639	0.327820	+	0.944740i	$0.393686\pi$
$912$	0	0
$913$	−6.50643	−0.215331
$914$	0	0
$915$	15.2863	0.505349
$916$	0	0
$917$	−21.5486	−0.711596
$918$	0	0
$919$	7.50904	0.247700	0.123850	−	0.992301i	$-0.460476\pi$
$919$	7.50904	0.247700	0.123850	+	0.992301i	$0.460476\pi$
$920$	0	0
$921$	−3.15452	−0.103945
$922$	0	0
$923$	59.1952	1.94843
$924$	0	0
$925$	−41.8130	−1.37480
$926$	0	0
$927$	2.96424	0.0973584
$928$	0	0
$929$	−40.9324	−1.34295	−0.671474	−	0.741028i	$-0.734338\pi$
$929$	−40.9324	−1.34295	−0.671474	+	0.741028i	$0.734338\pi$
$930$	0	0
$931$	−2.50871	−0.0822196
$932$	0	0
$933$	−44.2968	−1.45021
$934$	0	0
$935$	92.2932	3.01831
$936$	0	0
$937$	48.8681	1.59645	0.798226	−	0.602358i	$-0.205772\pi$
$937$	48.8681	1.59645	0.798226	+	0.602358i	$0.205772\pi$
$938$	0	0
$939$	−30.0768	−0.981519
$940$	0	0
$941$	37.7805	1.23161	0.615805	−	0.787899i	$-0.288831\pi$
$941$	37.7805	1.23161	0.615805	+	0.787899i	$0.288831\pi$
$942$	0	0
$943$	37.9396	1.23548
$944$	0	0
$945$	44.4080	1.44459
$946$	0	0
$947$	−36.1188	−1.17370	−0.586852	−	0.809694i	$-0.699633\pi$
$947$	−36.1188	−1.17370	−0.586852	+	0.809694i	$0.699633\pi$
$948$	0	0
$949$	−6.72431	−0.218280
$950$	0	0
$951$	29.2516	0.948547
$952$	0	0
$953$	−48.0232	−1.55562	−0.777811	−	0.628498i	$-0.783670\pi$
$953$	−48.0232	−1.55562	−0.777811	+	0.628498i	$0.783670\pi$
$954$	0	0
$955$	52.4541	1.69737
$956$	0	0
$957$	3.74931	0.121198
$958$	0	0
$959$	−18.6604	−0.602576
$960$	0	0
$961$	8.93836	0.288334
$962$	0	0
$963$	3.41389	0.110011
$964$	0	0
$965$	18.2096	0.586187
$966$	0	0
$967$	50.3577	1.61940	0.809698	−	0.586847i	$-0.199631\pi$
$967$	50.3577	1.61940	0.809698	+	0.586847i	$0.199631\pi$
$968$	0	0
$969$	−63.1510	−2.02870
$970$	0	0
$971$	−11.3491	−0.364210	−0.182105	−	0.983279i	$-0.558291\pi$
$971$	−11.3491	−0.364210	−0.182105	+	0.983279i	$0.558291\pi$
$972$	0	0
$973$	48.9342	1.56876
$974$	0	0
$975$	44.3800	1.42130
$976$	0	0
$977$	−18.5837	−0.594546	−0.297273	−	0.954792i	$-0.596077\pi$
$977$	−18.5837	−0.594546	−0.297273	+	0.954792i	$0.596077\pi$
$978$	0	0
$979$	−58.0750	−1.85609
$980$	0	0
$981$	−2.41456	−0.0770909
$982$	0	0
$983$	35.6133	1.13589	0.567944	−	0.823067i	$-0.307739\pi$
$983$	35.6133	1.13589	0.567944	+	0.823067i	$0.307739\pi$
$984$	0	0
$985$	60.7938	1.93705
$986$	0	0
$987$	52.0574	1.65701
$988$	0	0
$989$	31.9620	1.01633
$990$	0	0
$991$	−5.59651	−0.177779	−0.0888895	−	0.996041i	$-0.528332\pi$
$991$	−5.59651	−0.177779	−0.0888895	+	0.996041i	$0.528332\pi$
$992$	0	0
$993$	−37.0761	−1.17657
$994$	0	0
$995$	27.1373	0.860311
$996$	0	0
$997$	34.7689	1.10114	0.550572	−	0.834788i	$-0.314410\pi$
$997$	34.7689	1.10114	0.550572	+	0.834788i	$0.314410\pi$
$998$	0	0
$999$	40.2885	1.27467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.36	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.36	✓	44	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67289 q^{3} +3.24936 q^{5} -2.55182 q^{7} -0.201452 q^{9} +O(q^{10})\)	\(q+1.67289 q^{3} +3.24936 q^{5} -2.55182 q^{7} -0.201452 q^{9} -3.86634 q^{11} +4.77285 q^{13} +5.43581 q^{15} -7.34637 q^{17} +5.13856 q^{19} -4.26891 q^{21} -4.94464 q^{23} +5.55832 q^{25} -5.35566 q^{27} -0.579676 q^{29} -6.31968 q^{31} -6.46794 q^{33} -8.29178 q^{35} -7.52260 q^{37} +7.98443 q^{39} -7.67286 q^{41} -6.46398 q^{43} -0.654589 q^{45} -12.1946 q^{47} -0.488212 q^{49} -12.2896 q^{51} -2.15145 q^{53} -12.5631 q^{55} +8.59623 q^{57} +5.57924 q^{59} +2.81215 q^{61} +0.514069 q^{63} +15.5087 q^{65} +15.2590 q^{67} -8.27182 q^{69} +12.4025 q^{71} -1.40887 q^{73} +9.29844 q^{75} +9.86619 q^{77} -12.3648 q^{79} -8.35506 q^{81} +1.68284 q^{83} -23.8710 q^{85} -0.969732 q^{87} +15.0207 q^{89} -12.1794 q^{91} -10.5721 q^{93} +16.6970 q^{95} +5.00023 q^{97} +0.778880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more