LMFDB - Embedded newform 8008.2.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.336827$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.336827 q^{3} +3.95783 q^{5} +1.00000 q^{7} -2.88655 q^{9} +O(q^{10})$	$q+0.336827 q^{3} +3.95783 q^{5} +1.00000 q^{7} -2.88655 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.33310 q^{15} -0.794965 q^{17} +4.35914 q^{19} +0.336827 q^{21} -1.29578 q^{23} +10.6644 q^{25} -1.98275 q^{27} +3.76838 q^{29} +6.69646 q^{31} +0.336827 q^{33} +3.95783 q^{35} -2.72266 q^{37} +0.336827 q^{39} -3.50398 q^{41} -1.63697 q^{43} -11.4245 q^{45} -0.830732 q^{47} +1.00000 q^{49} -0.267766 q^{51} +10.8838 q^{53} +3.95783 q^{55} +1.46828 q^{57} +5.39313 q^{59} +3.96062 q^{61} -2.88655 q^{63} +3.95783 q^{65} -11.6208 q^{67} -0.436456 q^{69} -4.19845 q^{71} +5.43742 q^{73} +3.59206 q^{75} +1.00000 q^{77} -9.39088 q^{79} +7.99180 q^{81} +5.82445 q^{83} -3.14633 q^{85} +1.26929 q^{87} -5.81460 q^{89} +1.00000 q^{91} +2.25555 q^{93} +17.2527 q^{95} -3.83277 q^{97} -2.88655 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * 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
$3$	0.336827	0.194467	0.0972336	−	0.995262i	$-0.469001\pi$
$3$	0.336827	0.194467	0.0972336	+	0.995262i	$0.469001\pi$
$4$	0	0
$5$	3.95783	1.76999	0.884997	−	0.465597i	$-0.154160\pi$
$5$	3.95783	1.76999	0.884997	+	0.465597i	$0.154160\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.88655	−0.962182
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.33310	0.344206
$16$	0	0
$17$	−0.794965	−0.192807	−0.0964036	−	0.995342i	$-0.530734\pi$
$17$	−0.794965	−0.192807	−0.0964036	+	0.995342i	$0.530734\pi$
$18$	0	0
$19$	4.35914	1.00006	0.500028	−	0.866009i	$-0.333323\pi$
$19$	4.35914	1.00006	0.500028	+	0.866009i	$0.333323\pi$
$20$	0	0
$21$	0.336827	0.0735017
$22$	0	0
$23$	−1.29578	−0.270190	−0.135095	−	0.990833i	$-0.543134\pi$
$23$	−1.29578	−0.270190	−0.135095	+	0.990833i	$0.543134\pi$
$24$	0	0
$25$	10.6644	2.13288
$26$	0	0
$27$	−1.98275	−0.381580
$28$	0	0
$29$	3.76838	0.699771	0.349886	−	0.936792i	$-0.386220\pi$
$29$	3.76838	0.699771	0.349886	+	0.936792i	$0.386220\pi$
$30$	0	0
$31$	6.69646	1.20272	0.601360	−	0.798978i	$-0.294626\pi$
$31$	6.69646	1.20272	0.601360	+	0.798978i	$0.294626\pi$
$32$	0	0
$33$	0.336827	0.0586341
$34$	0	0
$35$	3.95783	0.668995
$36$	0	0
$37$	−2.72266	−0.447603	−0.223801	−	0.974635i	$-0.571847\pi$
$37$	−2.72266	−0.447603	−0.223801	+	0.974635i	$0.571847\pi$
$38$	0	0
$39$	0.336827	0.0539355
$40$	0	0
$41$	−3.50398	−0.547229	−0.273615	−	0.961839i	$-0.588219\pi$
$41$	−3.50398	−0.547229	−0.273615	+	0.961839i	$0.588219\pi$
$42$	0	0
$43$	−1.63697	−0.249635	−0.124818	−	0.992180i	$-0.539835\pi$
$43$	−1.63697	−0.249635	−0.124818	+	0.992180i	$0.539835\pi$
$44$	0	0
$45$	−11.4245	−1.70306
$46$	0	0
$47$	−0.830732	−0.121175	−0.0605873	−	0.998163i	$-0.519297\pi$
$47$	−0.830732	−0.121175	−0.0605873	+	0.998163i	$0.519297\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.267766	−0.0374947
$52$	0	0
$53$	10.8838	1.49500	0.747502	−	0.664259i	$-0.231253\pi$
$53$	10.8838	1.49500	0.747502	+	0.664259i	$0.231253\pi$
$54$	0	0
$55$	3.95783	0.533673
$56$	0	0
$57$	1.46828	0.194478
$58$	0	0
$59$	5.39313	0.702126	0.351063	−	0.936352i	$-0.385820\pi$
$59$	5.39313	0.702126	0.351063	+	0.936352i	$0.385820\pi$
$60$	0	0
$61$	3.96062	0.507105	0.253553	−	0.967322i	$-0.418401\pi$
$61$	3.96062	0.507105	0.253553	+	0.967322i	$0.418401\pi$
$62$	0	0
$63$	−2.88655	−0.363671
$64$	0	0
$65$	3.95783	0.490908
$66$	0	0
$67$	−11.6208	−1.41970	−0.709850	−	0.704352i	$-0.751237\pi$
$67$	−11.6208	−1.41970	−0.709850	+	0.704352i	$0.751237\pi$
$68$	0	0
$69$	−0.436456	−0.0525431
$70$	0	0
$71$	−4.19845	−0.498265	−0.249132	−	0.968469i	$-0.580145\pi$
$71$	−4.19845	−0.498265	−0.249132	+	0.968469i	$0.580145\pi$
$72$	0	0
$73$	5.43742	0.636402	0.318201	−	0.948023i	$-0.396921\pi$
$73$	5.43742	0.636402	0.318201	+	0.948023i	$0.396921\pi$
$74$	0	0
$75$	3.59206	0.414775
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−9.39088	−1.05656	−0.528278	−	0.849071i	$-0.677162\pi$
$79$	−9.39088	−1.05656	−0.528278	+	0.849071i	$0.677162\pi$
$80$	0	0
$81$	7.99180	0.887978
$82$	0	0
$83$	5.82445	0.639316	0.319658	−	0.947533i	$-0.396432\pi$
$83$	5.82445	0.639316	0.319658	+	0.947533i	$0.396432\pi$
$84$	0	0
$85$	−3.14633	−0.341268
$86$	0	0
$87$	1.26929	0.136083
$88$	0	0
$89$	−5.81460	−0.616346	−0.308173	−	0.951330i	$-0.599718\pi$
$89$	−5.81460	−0.616346	−0.308173	+	0.951330i	$0.599718\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.25555	0.233890
$94$	0	0
$95$	17.2527	1.77009
$96$	0	0
$97$	−3.83277	−0.389159	−0.194580	−	0.980887i	$-0.562334\pi$
$97$	−3.83277	−0.389159	−0.194580	+	0.980887i	$0.562334\pi$
$98$	0	0
$99$	−2.88655	−0.290109
$100$	0	0
$101$	−9.80841	−0.975973	−0.487986	−	0.872851i	$-0.662268\pi$
$101$	−9.80841	−0.975973	−0.487986	+	0.872851i	$0.662268\pi$
$102$	0	0
$103$	10.7941	1.06358	0.531789	−	0.846877i	$-0.321520\pi$
$103$	10.7941	1.06358	0.531789	+	0.846877i	$0.321520\pi$
$104$	0	0
$105$	1.33310	0.130098
$106$	0	0
$107$	0.389486	0.0376530	0.0188265	−	0.999823i	$-0.494007\pi$
$107$	0.389486	0.0376530	0.0188265	+	0.999823i	$0.494007\pi$
$108$	0	0
$109$	9.34534	0.895121	0.447561	−	0.894254i	$-0.352293\pi$
$109$	9.34534	0.895121	0.447561	+	0.894254i	$0.352293\pi$
$110$	0	0
$111$	−0.917067	−0.0870441
$112$	0	0
$113$	0.726344	0.0683287	0.0341644	−	0.999416i	$-0.489123\pi$
$113$	0.726344	0.0683287	0.0341644	+	0.999416i	$0.489123\pi$
$114$	0	0
$115$	−5.12849	−0.478234
$116$	0	0
$117$	−2.88655	−0.266861
$118$	0	0
$119$	−0.794965	−0.0728743
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.18023	−0.106418
$124$	0	0
$125$	22.4187	2.00519
$126$	0	0
$127$	−12.6242	−1.12022	−0.560109	−	0.828419i	$-0.689241\pi$
$127$	−12.6242	−1.12022	−0.560109	+	0.828419i	$0.689241\pi$
$128$	0	0
$129$	−0.551375	−0.0485459
$130$	0	0
$131$	−5.34083	−0.466630	−0.233315	−	0.972401i	$-0.574957\pi$
$131$	−5.34083	−0.466630	−0.233315	+	0.972401i	$0.574957\pi$
$132$	0	0
$133$	4.35914	0.377986
$134$	0	0
$135$	−7.84738	−0.675395
$136$	0	0
$137$	9.15955	0.782553	0.391277	−	0.920273i	$-0.372034\pi$
$137$	9.15955	0.782553	0.391277	+	0.920273i	$0.372034\pi$
$138$	0	0
$139$	10.4462	0.886036	0.443018	−	0.896513i	$-0.353908\pi$
$139$	10.4462	0.886036	0.443018	+	0.896513i	$0.353908\pi$
$140$	0	0
$141$	−0.279813	−0.0235645
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	14.9146	1.23859
$146$	0	0
$147$	0.336827	0.0277810
$148$	0	0
$149$	17.1601	1.40581	0.702904	−	0.711285i	$-0.251886\pi$
$149$	17.1601	1.40581	0.702904	+	0.711285i	$0.251886\pi$
$150$	0	0
$151$	3.73563	0.304001	0.152001	−	0.988380i	$-0.451428\pi$
$151$	3.73563	0.304001	0.152001	+	0.988380i	$0.451428\pi$
$152$	0	0
$153$	2.29470	0.185516
$154$	0	0
$155$	26.5034	2.12881
$156$	0	0
$157$	−10.0475	−0.801877	−0.400938	−	0.916105i	$-0.631316\pi$
$157$	−10.0475	−0.801877	−0.400938	+	0.916105i	$0.631316\pi$
$158$	0	0
$159$	3.66596	0.290729
$160$	0	0
$161$	−1.29578	−0.102122
$162$	0	0
$163$	7.17571	0.562045	0.281023	−	0.959701i	$-0.409326\pi$
$163$	7.17571	0.562045	0.281023	+	0.959701i	$0.409326\pi$
$164$	0	0
$165$	1.33310	0.103782
$166$	0	0
$167$	−11.6901	−0.904604	−0.452302	−	0.891865i	$-0.649397\pi$
$167$	−11.6901	−0.904604	−0.452302	+	0.891865i	$0.649397\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−12.5829	−0.962237
$172$	0	0
$173$	23.0440	1.75200	0.876001	−	0.482310i	$-0.160202\pi$
$173$	23.0440	1.75200	0.876001	+	0.482310i	$0.160202\pi$
$174$	0	0
$175$	10.6644	0.806152
$176$	0	0
$177$	1.81655	0.136540
$178$	0	0
$179$	1.57678	0.117854	0.0589271	−	0.998262i	$-0.481232\pi$
$179$	1.57678	0.117854	0.0589271	+	0.998262i	$0.481232\pi$
$180$	0	0
$181$	−0.394998	−0.0293600	−0.0146800	−	0.999892i	$-0.504673\pi$
$181$	−0.394998	−0.0293600	−0.0146800	+	0.999892i	$0.504673\pi$
$182$	0	0
$183$	1.33404	0.0986154
$184$	0	0
$185$	−10.7758	−0.792254
$186$	0	0
$187$	−0.794965	−0.0581336
$188$	0	0
$189$	−1.98275	−0.144224
$190$	0	0
$191$	−10.3478	−0.748742	−0.374371	−	0.927279i	$-0.622141\pi$
$191$	−10.3478	−0.748742	−0.374371	+	0.927279i	$0.622141\pi$
$192$	0	0
$193$	18.9024	1.36063	0.680313	−	0.732922i	$-0.261844\pi$
$193$	18.9024	1.36063	0.680313	+	0.732922i	$0.261844\pi$
$194$	0	0
$195$	1.33310	0.0954655
$196$	0	0
$197$	23.7800	1.69426	0.847128	−	0.531389i	$-0.178330\pi$
$197$	23.7800	1.69426	0.847128	+	0.531389i	$0.178330\pi$
$198$	0	0
$199$	4.10297	0.290852	0.145426	−	0.989369i	$-0.453545\pi$
$199$	4.10297	0.290852	0.145426	+	0.989369i	$0.453545\pi$
$200$	0	0
$201$	−3.91419	−0.276085
$202$	0	0
$203$	3.76838	0.264489
$204$	0	0
$205$	−13.8681	−0.968592
$206$	0	0
$207$	3.74034	0.259972
$208$	0	0
$209$	4.35914	0.301528
$210$	0	0
$211$	−2.40373	−0.165479	−0.0827397	−	0.996571i	$-0.526367\pi$
$211$	−2.40373	−0.165479	−0.0827397	+	0.996571i	$0.526367\pi$
$212$	0	0
$213$	−1.41415	−0.0968962
$214$	0	0
$215$	−6.47884	−0.441853
$216$	0	0
$217$	6.69646	0.454585
$218$	0	0
$219$	1.83147	0.123759
$220$	0	0
$221$	−0.794965	−0.0534751
$222$	0	0
$223$	−5.79837	−0.388288	−0.194144	−	0.980973i	$-0.562193\pi$
$223$	−5.79837	−0.388288	−0.194144	+	0.980973i	$0.562193\pi$
$224$	0	0
$225$	−30.7833	−2.05222
$226$	0	0
$227$	25.5209	1.69388	0.846941	−	0.531687i	$-0.178442\pi$
$227$	25.5209	1.69388	0.846941	+	0.531687i	$0.178442\pi$
$228$	0	0
$229$	5.07831	0.335584	0.167792	−	0.985822i	$-0.446336\pi$
$229$	5.07831	0.335584	0.167792	+	0.985822i	$0.446336\pi$
$230$	0	0
$231$	0.336827	0.0221616
$232$	0	0
$233$	1.98167	0.129823	0.0649116	−	0.997891i	$-0.479323\pi$
$233$	1.98167	0.129823	0.0649116	+	0.997891i	$0.479323\pi$
$234$	0	0
$235$	−3.28789	−0.214478
$236$	0	0
$237$	−3.16310	−0.205466
$238$	0	0
$239$	−1.56668	−0.101340	−0.0506700	−	0.998715i	$-0.516136\pi$
$239$	−1.56668	−0.101340	−0.0506700	+	0.998715i	$0.516136\pi$
$240$	0	0
$241$	−11.1274	−0.716781	−0.358390	−	0.933572i	$-0.616674\pi$
$241$	−11.1274	−0.716781	−0.358390	+	0.933572i	$0.616674\pi$
$242$	0	0
$243$	8.64010	0.554263
$244$	0	0
$245$	3.95783	0.252856
$246$	0	0
$247$	4.35914	0.277366
$248$	0	0
$249$	1.96183	0.124326
$250$	0	0
$251$	9.58852	0.605222	0.302611	−	0.953114i	$-0.402142\pi$
$251$	9.58852	0.605222	0.302611	+	0.953114i	$0.402142\pi$
$252$	0	0
$253$	−1.29578	−0.0814653
$254$	0	0
$255$	−1.05977	−0.0663654
$256$	0	0
$257$	15.6356	0.975320	0.487660	−	0.873034i	$-0.337851\pi$
$257$	15.6356	0.975320	0.487660	+	0.873034i	$0.337851\pi$
$258$	0	0
$259$	−2.72266	−0.169178
$260$	0	0
$261$	−10.8776	−0.673307
$262$	0	0
$263$	−24.8823	−1.53431	−0.767154	−	0.641463i	$-0.778328\pi$
$263$	−24.8823	−1.53431	−0.767154	+	0.641463i	$0.778328\pi$
$264$	0	0
$265$	43.0762	2.64615
$266$	0	0
$267$	−1.95852	−0.119859
$268$	0	0
$269$	−27.4775	−1.67533	−0.837666	−	0.546183i	$-0.816080\pi$
$269$	−27.4775	−1.67533	−0.837666	+	0.546183i	$0.816080\pi$
$270$	0	0
$271$	−21.5418	−1.30857	−0.654287	−	0.756246i	$-0.727031\pi$
$271$	−21.5418	−1.30857	−0.654287	+	0.756246i	$0.727031\pi$
$272$	0	0
$273$	0.336827	0.0203857
$274$	0	0
$275$	10.6644	0.643087
$276$	0	0
$277$	9.56893	0.574941	0.287471	−	0.957789i	$-0.407186\pi$
$277$	9.56893	0.574941	0.287471	+	0.957789i	$0.407186\pi$
$278$	0	0
$279$	−19.3296	−1.15724
$280$	0	0
$281$	0.724053	0.0431934	0.0215967	−	0.999767i	$-0.493125\pi$
$281$	0.724053	0.0431934	0.0215967	+	0.999767i	$0.493125\pi$
$282$	0	0
$283$	−18.1779	−1.08057	−0.540283	−	0.841483i	$-0.681683\pi$
$283$	−18.1779	−1.08057	−0.540283	+	0.841483i	$0.681683\pi$
$284$	0	0
$285$	5.81119	0.344225
$286$	0	0
$287$	−3.50398	−0.206833
$288$	0	0
$289$	−16.3680	−0.962825
$290$	0	0
$291$	−1.29098	−0.0756787
$292$	0	0
$293$	−16.1604	−0.944101	−0.472050	−	0.881572i	$-0.656486\pi$
$293$	−16.1604	−0.944101	−0.472050	+	0.881572i	$0.656486\pi$
$294$	0	0
$295$	21.3451	1.24276
$296$	0	0
$297$	−1.98275	−0.115051
$298$	0	0
$299$	−1.29578	−0.0749372
$300$	0	0
$301$	−1.63697	−0.0943533
$302$	0	0
$303$	−3.30374	−0.189795
$304$	0	0
$305$	15.6754	0.897573
$306$	0	0
$307$	−21.7379	−1.24065	−0.620324	−	0.784346i	$-0.712999\pi$
$307$	−21.7379	−1.24065	−0.620324	+	0.784346i	$0.712999\pi$
$308$	0	0
$309$	3.63576	0.206831
$310$	0	0
$311$	−26.8927	−1.52495	−0.762473	−	0.647020i	$-0.776015\pi$
$311$	−26.8927	−1.52495	−0.762473	+	0.647020i	$0.776015\pi$
$312$	0	0
$313$	−28.2812	−1.59855	−0.799275	−	0.600966i	$-0.794783\pi$
$313$	−28.2812	−1.59855	−0.799275	+	0.600966i	$0.794783\pi$
$314$	0	0
$315$	−11.4245	−0.643695
$316$	0	0
$317$	20.7591	1.16595	0.582974	−	0.812491i	$-0.301889\pi$
$317$	20.7591	1.16595	0.582974	+	0.812491i	$0.301889\pi$
$318$	0	0
$319$	3.76838	0.210989
$320$	0	0
$321$	0.131189	0.00732227
$322$	0	0
$323$	−3.46537	−0.192818
$324$	0	0
$325$	10.6644	0.591554
$326$	0	0
$327$	3.14776	0.174072
$328$	0	0
$329$	−0.830732	−0.0457997
$330$	0	0
$331$	−3.86300	−0.212330	−0.106165	−	0.994349i	$-0.533857\pi$
$331$	−3.86300	−0.212330	−0.106165	+	0.994349i	$0.533857\pi$
$332$	0	0
$333$	7.85909	0.430676
$334$	0	0
$335$	−45.9929	−2.51286
$336$	0	0
$337$	8.75219	0.476762	0.238381	−	0.971172i	$-0.423383\pi$
$337$	8.75219	0.476762	0.238381	+	0.971172i	$0.423383\pi$
$338$	0	0
$339$	0.244653	0.0132877
$340$	0	0
$341$	6.69646	0.362634
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.72742	−0.0930009
$346$	0	0
$347$	11.6853	0.627300	0.313650	−	0.949539i	$-0.398448\pi$
$347$	11.6853	0.627300	0.313650	+	0.949539i	$0.398448\pi$
$348$	0	0
$349$	10.2569	0.549037	0.274518	−	0.961582i	$-0.411482\pi$
$349$	10.2569	0.549037	0.274518	+	0.961582i	$0.411482\pi$
$350$	0	0
$351$	−1.98275	−0.105831
$352$	0	0
$353$	12.0121	0.639337	0.319669	−	0.947529i	$-0.396428\pi$
$353$	12.0121	0.639337	0.319669	+	0.947529i	$0.396428\pi$
$354$	0	0
$355$	−16.6167	−0.881925
$356$	0	0
$357$	−0.267766	−0.0141717
$358$	0	0
$359$	−10.4699	−0.552579	−0.276289	−	0.961074i	$-0.589105\pi$
$359$	−10.4699	−0.552579	−0.276289	+	0.961074i	$0.589105\pi$
$360$	0	0
$361$	0.00214236	0.000112756 0
$362$	0	0
$363$	0.336827	0.0176788
$364$	0	0
$365$	21.5204	1.12643
$366$	0	0
$367$	−13.4555	−0.702372	−0.351186	−	0.936306i	$-0.614222\pi$
$367$	−13.4555	−0.702372	−0.351186	+	0.936306i	$0.614222\pi$
$368$	0	0
$369$	10.1144	0.526534
$370$	0	0
$371$	10.8838	0.565059
$372$	0	0
$373$	5.20738	0.269628	0.134814	−	0.990871i	$-0.456956\pi$
$373$	5.20738	0.269628	0.134814	+	0.990871i	$0.456956\pi$
$374$	0	0
$375$	7.55122	0.389943
$376$	0	0
$377$	3.76838	0.194082
$378$	0	0
$379$	−2.66278	−0.136778	−0.0683888	−	0.997659i	$-0.521786\pi$
$379$	−2.66278	−0.136778	−0.0683888	+	0.997659i	$0.521786\pi$
$380$	0	0
$381$	−4.25218	−0.217846
$382$	0	0
$383$	−26.3811	−1.34801	−0.674005	−	0.738727i	$-0.735427\pi$
$383$	−26.3811	−1.34801	−0.674005	+	0.738727i	$0.735427\pi$
$384$	0	0
$385$	3.95783	0.201710
$386$	0	0
$387$	4.72519	0.240195
$388$	0	0
$389$	−9.61217	−0.487356	−0.243678	−	0.969856i	$-0.578354\pi$
$389$	−9.61217	−0.487356	−0.243678	+	0.969856i	$0.578354\pi$
$390$	0	0
$391$	1.03010	0.0520946
$392$	0	0
$393$	−1.79894	−0.0907443
$394$	0	0
$395$	−37.1675	−1.87010
$396$	0	0
$397$	4.45767	0.223724	0.111862	−	0.993724i	$-0.464319\pi$
$397$	4.45767	0.223724	0.111862	+	0.993724i	$0.464319\pi$
$398$	0	0
$399$	1.46828	0.0735059
$400$	0	0
$401$	37.1092	1.85314	0.926572	−	0.376118i	$-0.122741\pi$
$401$	37.1092	1.85314	0.926572	+	0.376118i	$0.122741\pi$
$402$	0	0
$403$	6.69646	0.333574
$404$	0	0
$405$	31.6302	1.57171
$406$	0	0
$407$	−2.72266	−0.134957
$408$	0	0
$409$	18.8731	0.933216	0.466608	−	0.884464i	$-0.345476\pi$
$409$	18.8731	0.933216	0.466608	+	0.884464i	$0.345476\pi$
$410$	0	0
$411$	3.08518	0.152181
$412$	0	0
$413$	5.39313	0.265379
$414$	0	0
$415$	23.0522	1.13159
$416$	0	0
$417$	3.51857	0.172305
$418$	0	0
$419$	10.7169	0.523555	0.261777	−	0.965128i	$-0.415691\pi$
$419$	10.7169	0.523555	0.261777	+	0.965128i	$0.415691\pi$
$420$	0	0
$421$	25.9174	1.26314	0.631569	−	0.775319i	$-0.282411\pi$
$421$	25.9174	1.26314	0.631569	+	0.775319i	$0.282411\pi$
$422$	0	0
$423$	2.39795	0.116592
$424$	0	0
$425$	−8.47782	−0.411234
$426$	0	0
$427$	3.96062	0.191668
$428$	0	0
$429$	0.336827	0.0162622
$430$	0	0
$431$	17.0872	0.823061	0.411531	−	0.911396i	$-0.364994\pi$
$431$	17.0872	0.823061	0.411531	+	0.911396i	$0.364994\pi$
$432$	0	0
$433$	−1.54115	−0.0740632	−0.0370316	−	0.999314i	$-0.511790\pi$
$433$	−1.54115	−0.0740632	−0.0370316	+	0.999314i	$0.511790\pi$
$434$	0	0
$435$	5.02364	0.240865
$436$	0	0
$437$	−5.64851	−0.270205
$438$	0	0
$439$	17.3830	0.829647	0.414824	−	0.909902i	$-0.363843\pi$
$439$	17.3830	0.829647	0.414824	+	0.909902i	$0.363843\pi$
$440$	0	0
$441$	−2.88655	−0.137455
$442$	0	0
$443$	−13.3658	−0.635029	−0.317514	−	0.948253i	$-0.602848\pi$
$443$	−13.3658	−0.635029	−0.317514	+	0.948253i	$0.602848\pi$
$444$	0	0
$445$	−23.0132	−1.09093
$446$	0	0
$447$	5.77998	0.273384
$448$	0	0
$449$	−2.01774	−0.0952232	−0.0476116	−	0.998866i	$-0.515161\pi$
$449$	−2.01774	−0.0952232	−0.0476116	+	0.998866i	$0.515161\pi$
$450$	0	0
$451$	−3.50398	−0.164996
$452$	0	0
$453$	1.25826	0.0591183
$454$	0	0
$455$	3.95783	0.185546
$456$	0	0
$457$	1.76238	0.0824409	0.0412204	−	0.999150i	$-0.486875\pi$
$457$	1.76238	0.0824409	0.0412204	+	0.999150i	$0.486875\pi$
$458$	0	0
$459$	1.57622	0.0735715
$460$	0	0
$461$	8.69135	0.404797	0.202398	−	0.979303i	$-0.435126\pi$
$461$	8.69135	0.404797	0.202398	+	0.979303i	$0.435126\pi$
$462$	0	0
$463$	−23.0868	−1.07294	−0.536468	−	0.843921i	$-0.680242\pi$
$463$	−23.0868	−1.07294	−0.536468	+	0.843921i	$0.680242\pi$
$464$	0	0
$465$	8.92707	0.413983
$466$	0	0
$467$	0.453519	0.0209864	0.0104932	−	0.999945i	$-0.496660\pi$
$467$	0.453519	0.0209864	0.0104932	+	0.999945i	$0.496660\pi$
$468$	0	0
$469$	−11.6208	−0.536596
$470$	0	0
$471$	−3.38427	−0.155939
$472$	0	0
$473$	−1.63697	−0.0752679
$474$	0	0
$475$	46.4876	2.13300
$476$	0	0
$477$	−31.4166	−1.43847
$478$	0	0
$479$	−37.6634	−1.72088	−0.860442	−	0.509549i	$-0.829812\pi$
$479$	−37.6634	−1.72088	−0.860442	+	0.509549i	$0.829812\pi$
$480$	0	0
$481$	−2.72266	−0.124143
$482$	0	0
$483$	−0.436456	−0.0198594
$484$	0	0
$485$	−15.1694	−0.688809
$486$	0	0
$487$	5.63786	0.255476	0.127738	−	0.991808i	$-0.459228\pi$
$487$	5.63786	0.255476	0.127738	+	0.991808i	$0.459228\pi$
$488$	0	0
$489$	2.41698	0.109299
$490$	0	0
$491$	4.31545	0.194753	0.0973767	−	0.995248i	$-0.468955\pi$
$491$	4.31545	0.194753	0.0973767	+	0.995248i	$0.468955\pi$
$492$	0	0
$493$	−2.99573	−0.134921
$494$	0	0
$495$	−11.4245	−0.513491
$496$	0	0
$497$	−4.19845	−0.188326
$498$	0	0
$499$	25.5577	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$499$	25.5577	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$500$	0	0
$501$	−3.93753	−0.175916
$502$	0	0
$503$	5.06786	0.225965	0.112982	−	0.993597i	$-0.463960\pi$
$503$	5.06786	0.225965	0.112982	+	0.993597i	$0.463960\pi$
$504$	0	0
$505$	−38.8200	−1.72747
$506$	0	0
$507$	0.336827	0.0149590
$508$	0	0
$509$	−17.6900	−0.784097	−0.392048	−	0.919945i	$-0.628233\pi$
$509$	−17.6900	−0.784097	−0.392048	+	0.919945i	$0.628233\pi$
$510$	0	0
$511$	5.43742	0.240538
$512$	0	0
$513$	−8.64309	−0.381602
$514$	0	0
$515$	42.7214	1.88253
$516$	0	0
$517$	−0.830732	−0.0365355
$518$	0	0
$519$	7.76184	0.340707
$520$	0	0
$521$	−36.5334	−1.60056	−0.800278	−	0.599630i	$-0.795315\pi$
$521$	−36.5334	−1.60056	−0.800278	+	0.599630i	$0.795315\pi$
$522$	0	0
$523$	15.9604	0.697899	0.348950	−	0.937141i	$-0.386538\pi$
$523$	15.9604	0.697899	0.348950	+	0.937141i	$0.386538\pi$
$524$	0	0
$525$	3.59206	0.156770
$526$	0	0
$527$	−5.32345	−0.231893
$528$	0	0
$529$	−21.3209	−0.926997
$530$	0	0
$531$	−15.5675	−0.675573
$532$	0	0
$533$	−3.50398	−0.151774
$534$	0	0
$535$	1.54152	0.0666456
$536$	0	0
$537$	0.531103	0.0229188
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	5.47164	0.235244	0.117622	−	0.993058i	$-0.462473\pi$
$541$	5.47164	0.235244	0.117622	+	0.993058i	$0.462473\pi$
$542$	0	0
$543$	−0.133046	−0.00570956
$544$	0	0
$545$	36.9872	1.58436
$546$	0	0
$547$	28.9482	1.23774	0.618868	−	0.785495i	$-0.287592\pi$
$547$	28.9482	1.23774	0.618868	+	0.785495i	$0.287592\pi$
$548$	0	0
$549$	−11.4325	−0.487928
$550$	0	0
$551$	16.4269	0.699810
$552$	0	0
$553$	−9.39088	−0.399341
$554$	0	0
$555$	−3.62959	−0.154068
$556$	0	0
$557$	−19.1903	−0.813120	−0.406560	−	0.913624i	$-0.633272\pi$
$557$	−19.1903	−0.813120	−0.406560	+	0.913624i	$0.633272\pi$
$558$	0	0
$559$	−1.63697	−0.0692364
$560$	0	0
$561$	−0.267766	−0.0113051
$562$	0	0
$563$	−24.8877	−1.04889	−0.524446	−	0.851444i	$-0.675728\pi$
$563$	−24.8877	−1.04889	−0.524446	+	0.851444i	$0.675728\pi$
$564$	0	0
$565$	2.87475	0.120941
$566$	0	0
$567$	7.99180	0.335624
$568$	0	0
$569$	−36.7788	−1.54185	−0.770924	−	0.636927i	$-0.780205\pi$
$569$	−36.7788	−1.54185	−0.770924	+	0.636927i	$0.780205\pi$
$570$	0	0
$571$	−11.3683	−0.475747	−0.237874	−	0.971296i	$-0.576450\pi$
$571$	−11.3683	−0.475747	−0.237874	+	0.971296i	$0.576450\pi$
$572$	0	0
$573$	−3.48543	−0.145606
$574$	0	0
$575$	−13.8188	−0.576282
$576$	0	0
$577$	−13.4864	−0.561448	−0.280724	−	0.959789i	$-0.590575\pi$
$577$	−13.4864	−0.561448	−0.280724	+	0.959789i	$0.590575\pi$
$578$	0	0
$579$	6.36685	0.264597
$580$	0	0
$581$	5.82445	0.241639
$582$	0	0
$583$	10.8838	0.450761
$584$	0	0
$585$	−11.4245	−0.472343
$586$	0	0
$587$	−9.35731	−0.386218	−0.193109	−	0.981177i	$-0.561857\pi$
$587$	−9.35731	−0.386218	−0.193109	+	0.981177i	$0.561857\pi$
$588$	0	0
$589$	29.1908	1.20279
$590$	0	0
$591$	8.00976	0.329477
$592$	0	0
$593$	28.6023	1.17456	0.587278	−	0.809386i	$-0.300200\pi$
$593$	28.6023	1.17456	0.587278	+	0.809386i	$0.300200\pi$
$594$	0	0
$595$	−3.14633	−0.128987
$596$	0	0
$597$	1.38199	0.0565611
$598$	0	0
$599$	−44.6891	−1.82595	−0.912974	−	0.408017i	$-0.866220\pi$
$599$	−44.6891	−1.82595	−0.912974	+	0.408017i	$0.866220\pi$
$600$	0	0
$601$	−2.21302	−0.0902709	−0.0451355	−	0.998981i	$-0.514372\pi$
$601$	−2.21302	−0.0902709	−0.0451355	+	0.998981i	$0.514372\pi$
$602$	0	0
$603$	33.5439	1.36601
$604$	0	0
$605$	3.95783	0.160909
$606$	0	0
$607$	−10.5592	−0.428584	−0.214292	−	0.976770i	$-0.568744\pi$
$607$	−10.5592	−0.428584	−0.214292	+	0.976770i	$0.568744\pi$
$608$	0	0
$609$	1.26929	0.0514344
$610$	0	0
$611$	−0.830732	−0.0336078
$612$	0	0
$613$	18.2277	0.736208	0.368104	−	0.929785i	$-0.380007\pi$
$613$	18.2277	0.736208	0.368104	+	0.929785i	$0.380007\pi$
$614$	0	0
$615$	−4.67116	−0.188360
$616$	0	0
$617$	10.1181	0.407338	0.203669	−	0.979040i	$-0.434713\pi$
$617$	10.1181	0.407338	0.203669	+	0.979040i	$0.434713\pi$
$618$	0	0
$619$	27.0093	1.08559	0.542797	−	0.839864i	$-0.317365\pi$
$619$	27.0093	1.08559	0.542797	+	0.839864i	$0.317365\pi$
$620$	0	0
$621$	2.56922	0.103099
$622$	0	0
$623$	−5.81460	−0.232957
$624$	0	0
$625$	35.4073	1.41629
$626$	0	0
$627$	1.46828	0.0586374
$628$	0	0
$629$	2.16442	0.0863011
$630$	0	0
$631$	−10.0289	−0.399244	−0.199622	−	0.979873i	$-0.563971\pi$
$631$	−10.0289	−0.399244	−0.199622	+	0.979873i	$0.563971\pi$
$632$	0	0
$633$	−0.809641	−0.0321803
$634$	0	0
$635$	−49.9645	−1.98278
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	12.1190	0.479421
$640$	0	0
$641$	−28.9240	−1.14243	−0.571214	−	0.820801i	$-0.693527\pi$
$641$	−28.9240	−1.14243	−0.571214	+	0.820801i	$0.693527\pi$
$642$	0	0
$643$	−7.80891	−0.307953	−0.153977	−	0.988074i	$-0.549208\pi$
$643$	−7.80891	−0.307953	−0.153977	+	0.988074i	$0.549208\pi$
$644$	0	0
$645$	−2.18225	−0.0859259
$646$	0	0
$647$	8.12016	0.319237	0.159618	−	0.987179i	$-0.448974\pi$
$647$	8.12016	0.319237	0.159618	+	0.987179i	$0.448974\pi$
$648$	0	0
$649$	5.39313	0.211699
$650$	0	0
$651$	2.25555	0.0884019
$652$	0	0
$653$	3.79190	0.148388	0.0741942	−	0.997244i	$-0.476362\pi$
$653$	3.79190	0.148388	0.0741942	+	0.997244i	$0.476362\pi$
$654$	0	0
$655$	−21.1381	−0.825933
$656$	0	0
$657$	−15.6954	−0.612335
$658$	0	0
$659$	20.2398	0.788431	0.394215	−	0.919018i	$-0.371016\pi$
$659$	20.2398	0.788431	0.394215	+	0.919018i	$0.371016\pi$
$660$	0	0
$661$	−21.7373	−0.845482	−0.422741	−	0.906250i	$-0.638932\pi$
$661$	−21.7373	−0.845482	−0.422741	+	0.906250i	$0.638932\pi$
$662$	0	0
$663$	−0.267766	−0.0103992
$664$	0	0
$665$	17.2527	0.669033
$666$	0	0
$667$	−4.88301	−0.189071
$668$	0	0
$669$	−1.95305	−0.0755093
$670$	0	0
$671$	3.96062	0.152898
$672$	0	0
$673$	30.8592	1.18953	0.594767	−	0.803898i	$-0.297244\pi$
$673$	30.8592	1.18953	0.594767	+	0.803898i	$0.297244\pi$
$674$	0	0
$675$	−21.1448	−0.813864
$676$	0	0
$677$	9.87967	0.379707	0.189853	−	0.981812i	$-0.439199\pi$
$677$	9.87967	0.379707	0.189853	+	0.981812i	$0.439199\pi$
$678$	0	0
$679$	−3.83277	−0.147088
$680$	0	0
$681$	8.59614	0.329405
$682$	0	0
$683$	36.0754	1.38039	0.690193	−	0.723625i	$-0.257525\pi$
$683$	36.0754	1.38039	0.690193	+	0.723625i	$0.257525\pi$
$684$	0	0
$685$	36.2519	1.38511
$686$	0	0
$687$	1.71051	0.0652602
$688$	0	0
$689$	10.8838	0.414640
$690$	0	0
$691$	41.0924	1.56323	0.781615	−	0.623761i	$-0.214396\pi$
$691$	41.0924	1.56323	0.781615	+	0.623761i	$0.214396\pi$
$692$	0	0
$693$	−2.88655	−0.109651
$694$	0	0
$695$	41.3443	1.56828
$696$	0	0
$697$	2.78554	0.105510
$698$	0	0
$699$	0.667479	0.0252464
$700$	0	0
$701$	−34.9458	−1.31989	−0.659943	−	0.751315i	$-0.729420\pi$
$701$	−34.9458	−1.31989	−0.659943	+	0.751315i	$0.729420\pi$
$702$	0	0
$703$	−11.8685	−0.447628
$704$	0	0
$705$	−1.10745	−0.0417090
$706$	0	0
$707$	−9.80841	−0.368883
$708$	0	0
$709$	31.9192	1.19875	0.599376	−	0.800468i	$-0.295415\pi$
$709$	31.9192	1.19875	0.599376	+	0.800468i	$0.295415\pi$
$710$	0	0
$711$	27.1072	1.01660
$712$	0	0
$713$	−8.67717	−0.324963
$714$	0	0
$715$	3.95783	0.148014
$716$	0	0
$717$	−0.527700	−0.0197073
$718$	0	0
$719$	23.0166	0.858373	0.429186	−	0.903216i	$-0.358800\pi$
$719$	23.0166	0.858373	0.429186	+	0.903216i	$0.358800\pi$
$720$	0	0
$721$	10.7941	0.401995
$722$	0	0
$723$	−3.74802	−0.139390
$724$	0	0
$725$	40.1875	1.49253
$726$	0	0
$727$	41.3596	1.53394	0.766971	−	0.641681i	$-0.221763\pi$
$727$	41.3596	1.53394	0.766971	+	0.641681i	$0.221763\pi$
$728$	0	0
$729$	−21.0652	−0.780192
$730$	0	0
$731$	1.30133	0.0481315
$732$	0	0
$733$	−30.1628	−1.11409	−0.557045	−	0.830483i	$-0.688065\pi$
$733$	−30.1628	−1.11409	−0.557045	+	0.830483i	$0.688065\pi$
$734$	0	0
$735$	1.33310	0.0491723
$736$	0	0
$737$	−11.6208	−0.428056
$738$	0	0
$739$	19.5381	0.718720	0.359360	−	0.933199i	$-0.382995\pi$
$739$	19.5381	0.718720	0.359360	+	0.933199i	$0.382995\pi$
$740$	0	0
$741$	1.46828	0.0539386
$742$	0	0
$743$	18.4164	0.675631	0.337815	−	0.941212i	$-0.390312\pi$
$743$	18.4164	0.675631	0.337815	+	0.941212i	$0.390312\pi$
$744$	0	0
$745$	67.9166	2.48827
$746$	0	0
$747$	−16.8125	−0.615139
$748$	0	0
$749$	0.389486	0.0142315
$750$	0	0
$751$	−4.09345	−0.149372	−0.0746861	−	0.997207i	$-0.523795\pi$
$751$	−4.09345	−0.149372	−0.0746861	+	0.997207i	$0.523795\pi$
$752$	0	0
$753$	3.22968	0.117696
$754$	0	0
$755$	14.7850	0.538080
$756$	0	0
$757$	30.4385	1.10631	0.553154	−	0.833079i	$-0.313424\pi$
$757$	30.4385	1.10631	0.553154	+	0.833079i	$0.313424\pi$
$758$	0	0
$759$	−0.436456	−0.0158423
$760$	0	0
$761$	−8.47455	−0.307202	−0.153601	−	0.988133i	$-0.549087\pi$
$761$	−8.47455	−0.307202	−0.153601	+	0.988133i	$0.549087\pi$
$762$	0	0
$763$	9.34534	0.338324
$764$	0	0
$765$	9.08204	0.328362
$766$	0	0
$767$	5.39313	0.194735
$768$	0	0
$769$	−8.92893	−0.321985	−0.160993	−	0.986956i	$-0.551470\pi$
$769$	−8.92893	−0.321985	−0.160993	+	0.986956i	$0.551470\pi$
$770$	0	0
$771$	5.26648	0.189668
$772$	0	0
$773$	10.1629	0.365535	0.182767	−	0.983156i	$-0.441495\pi$
$773$	10.1629	0.365535	0.182767	+	0.983156i	$0.441495\pi$
$774$	0	0
$775$	71.4136	2.56525
$776$	0	0
$777$	−0.917067	−0.0328996
$778$	0	0
$779$	−15.2743	−0.547260
$780$	0	0
$781$	−4.19845	−0.150232
$782$	0	0
$783$	−7.47176	−0.267019
$784$	0	0
$785$	−39.7662	−1.41932
$786$	0	0
$787$	9.52384	0.339488	0.169744	−	0.985488i	$-0.445706\pi$
$787$	9.52384	0.339488	0.169744	+	0.985488i	$0.445706\pi$
$788$	0	0
$789$	−8.38103	−0.298373
$790$	0	0
$791$	0.726344	0.0258258
$792$	0	0
$793$	3.96062	0.140646
$794$	0	0
$795$	14.5092	0.514589
$796$	0	0
$797$	30.9068	1.09477	0.547387	−	0.836879i	$-0.315622\pi$
$797$	30.9068	1.09477	0.547387	+	0.836879i	$0.315622\pi$
$798$	0	0
$799$	0.660402	0.0233634
$800$	0	0
$801$	16.7841	0.593038
$802$	0	0
$803$	5.43742	0.191883
$804$	0	0
$805$	−5.12849	−0.180756
$806$	0	0
$807$	−9.25517	−0.325797
$808$	0	0
$809$	15.4957	0.544799	0.272400	−	0.962184i	$-0.412183\pi$
$809$	15.4957	0.544799	0.272400	+	0.962184i	$0.412183\pi$
$810$	0	0
$811$	−39.1802	−1.37580	−0.687901	−	0.725804i	$-0.741468\pi$
$811$	−39.1802	−1.37580	−0.687901	+	0.725804i	$0.741468\pi$
$812$	0	0
$813$	−7.25588	−0.254475
$814$	0	0
$815$	28.4002	0.994817
$816$	0	0
$817$	−7.13578	−0.249649
$818$	0	0
$819$	−2.88655	−0.100864
$820$	0	0
$821$	−32.0666	−1.11913	−0.559566	−	0.828786i	$-0.689032\pi$
$821$	−32.0666	−1.11913	−0.559566	+	0.828786i	$0.689032\pi$
$822$	0	0
$823$	−21.7489	−0.758121	−0.379060	−	0.925372i	$-0.623753\pi$
$823$	−21.7489	−0.758121	−0.379060	+	0.925372i	$0.623753\pi$
$824$	0	0
$825$	3.59206	0.125059
$826$	0	0
$827$	27.8858	0.969685	0.484842	−	0.874602i	$-0.338877\pi$
$827$	27.8858	0.969685	0.484842	+	0.874602i	$0.338877\pi$
$828$	0	0
$829$	−48.1327	−1.67172	−0.835858	−	0.548946i	$-0.815029\pi$
$829$	−48.1327	−1.67172	−0.835858	+	0.548946i	$0.815029\pi$
$830$	0	0
$831$	3.22308	0.111807
$832$	0	0
$833$	−0.794965	−0.0275439
$834$	0	0
$835$	−46.2672	−1.60114
$836$	0	0
$837$	−13.2774	−0.458934
$838$	0	0
$839$	−43.6380	−1.50655	−0.753276	−	0.657705i	$-0.771528\pi$
$839$	−43.6380	−1.50655	−0.753276	+	0.657705i	$0.771528\pi$
$840$	0	0
$841$	−14.7993	−0.510321
$842$	0	0
$843$	0.243881	0.00839970
$844$	0	0
$845$	3.95783	0.136153
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−6.12283	−0.210135
$850$	0	0
$851$	3.52799	0.120938
$852$	0	0
$853$	45.8831	1.57101	0.785504	−	0.618857i	$-0.212404\pi$
$853$	45.8831	1.57101	0.785504	+	0.618857i	$0.212404\pi$
$854$	0	0
$855$	−49.8008	−1.70315
$856$	0	0
$857$	45.3034	1.54753	0.773767	−	0.633471i	$-0.218370\pi$
$857$	45.3034	1.54753	0.773767	+	0.633471i	$0.218370\pi$
$858$	0	0
$859$	48.1844	1.64403	0.822015	−	0.569465i	$-0.192850\pi$
$859$	48.1844	1.64403	0.822015	+	0.569465i	$0.192850\pi$
$860$	0	0
$861$	−1.18023	−0.0402223
$862$	0	0
$863$	−5.67768	−0.193271	−0.0966353	−	0.995320i	$-0.530808\pi$
$863$	−5.67768	−0.193271	−0.0966353	+	0.995320i	$0.530808\pi$
$864$	0	0
$865$	91.2041	3.10103
$866$	0	0
$867$	−5.51320	−0.187238
$868$	0	0
$869$	−9.39088	−0.318564
$870$	0	0
$871$	−11.6208	−0.393754
$872$	0	0
$873$	11.0635	0.374442
$874$	0	0
$875$	22.4187	0.757890
$876$	0	0
$877$	−19.4504	−0.656792	−0.328396	−	0.944540i	$-0.606508\pi$
$877$	−19.4504	−0.656792	−0.328396	+	0.944540i	$0.606508\pi$
$878$	0	0
$879$	−5.44326	−0.183597
$880$	0	0
$881$	29.5322	0.994966	0.497483	−	0.867474i	$-0.334258\pi$
$881$	29.5322	0.994966	0.497483	+	0.867474i	$0.334258\pi$
$882$	0	0
$883$	−38.5116	−1.29602	−0.648009	−	0.761633i	$-0.724398\pi$
$883$	−38.5116	−1.29602	−0.648009	+	0.761633i	$0.724398\pi$
$884$	0	0
$885$	7.18960	0.241676
$886$	0	0
$887$	−22.6506	−0.760533	−0.380267	−	0.924877i	$-0.624168\pi$
$887$	−22.6506	−0.760533	−0.380267	+	0.924877i	$0.624168\pi$
$888$	0	0
$889$	−12.6242	−0.423403
$890$	0	0
$891$	7.99180	0.267735
$892$	0	0
$893$	−3.62128	−0.121181
$894$	0	0
$895$	6.24063	0.208601
$896$	0	0
$897$	−0.436456	−0.0145728
$898$	0	0
$899$	25.2348	0.841628
$900$	0	0
$901$	−8.65223	−0.288248
$902$	0	0
$903$	−0.551375	−0.0183486
$904$	0	0
$905$	−1.56333	−0.0519670
$906$	0	0
$907$	−8.06694	−0.267858	−0.133929	−	0.990991i	$-0.542759\pi$
$907$	−8.06694	−0.267858	−0.133929	+	0.990991i	$0.542759\pi$
$908$	0	0
$909$	28.3124	0.939064
$910$	0	0
$911$	−24.4917	−0.811446	−0.405723	−	0.913996i	$-0.632980\pi$
$911$	−24.4917	−0.811446	−0.405723	+	0.913996i	$0.632980\pi$
$912$	0	0
$913$	5.82445	0.192761
$914$	0	0
$915$	5.27991	0.174549
$916$	0	0
$917$	−5.34083	−0.176370
$918$	0	0
$919$	29.7753	0.982196	0.491098	−	0.871104i	$-0.336596\pi$
$919$	29.7753	0.982196	0.491098	+	0.871104i	$0.336596\pi$
$920$	0	0
$921$	−7.32192	−0.241265
$922$	0	0
$923$	−4.19845	−0.138194
$924$	0	0
$925$	−29.0355	−0.954683
$926$	0	0
$927$	−31.1578	−1.02336
$928$	0	0
$929$	−53.5723	−1.75765	−0.878826	−	0.477143i	$-0.841672\pi$
$929$	−53.5723	−1.75765	−0.878826	+	0.477143i	$0.841672\pi$
$930$	0	0
$931$	4.35914	0.142865
$932$	0	0
$933$	−9.05820	−0.296552
$934$	0	0
$935$	−3.14633	−0.102896
$936$	0	0
$937$	9.63864	0.314881	0.157440	−	0.987529i	$-0.449676\pi$
$937$	9.63864	0.314881	0.157440	+	0.987529i	$0.449676\pi$
$938$	0	0
$939$	−9.52589	−0.310866
$940$	0	0
$941$	15.3921	0.501768	0.250884	−	0.968017i	$-0.419279\pi$
$941$	15.3921	0.501768	0.250884	+	0.968017i	$0.419279\pi$
$942$	0	0
$943$	4.54040	0.147856
$944$	0	0
$945$	−7.84738	−0.255275
$946$	0	0
$947$	−33.9331	−1.10268	−0.551339	−	0.834282i	$-0.685883\pi$
$947$	−33.9331	−1.10268	−0.551339	+	0.834282i	$0.685883\pi$
$948$	0	0
$949$	5.43742	0.176506
$950$	0	0
$951$	6.99224	0.226739
$952$	0	0
$953$	−23.8145	−0.771427	−0.385713	−	0.922619i	$-0.626045\pi$
$953$	−23.8145	−0.771427	−0.385713	+	0.922619i	$0.626045\pi$
$954$	0	0
$955$	−40.9549	−1.32527
$956$	0	0
$957$	1.26929	0.0410304
$958$	0	0
$959$	9.15955	0.295777
$960$	0	0
$961$	13.8425	0.446534
$962$	0	0
$963$	−1.12427	−0.0362290
$964$	0	0
$965$	74.8125	2.40830
$966$	0	0
$967$	6.45027	0.207427	0.103713	−	0.994607i	$-0.466928\pi$
$967$	6.45027	0.207427	0.103713	+	0.994607i	$0.466928\pi$
$968$	0	0
$969$	−1.16723	−0.0374968
$970$	0	0
$971$	−19.1096	−0.613255	−0.306628	−	0.951830i	$-0.599201\pi$
$971$	−19.1096	−0.613255	−0.306628	+	0.951830i	$0.599201\pi$
$972$	0	0
$973$	10.4462	0.334890
$974$	0	0
$975$	3.59206	0.115038
$976$	0	0
$977$	−2.96552	−0.0948753	−0.0474376	−	0.998874i	$-0.515106\pi$
$977$	−2.96552	−0.0948753	−0.0474376	+	0.998874i	$0.515106\pi$
$978$	0	0
$979$	−5.81460	−0.185835
$980$	0	0
$981$	−26.9758	−0.861270
$982$	0	0
$983$	−10.5040	−0.335025	−0.167512	−	0.985870i	$-0.553573\pi$
$983$	−10.5040	−0.335025	−0.167512	+	0.985870i	$0.553573\pi$
$984$	0	0
$985$	94.1172	2.99882
$986$	0	0
$987$	−0.279813	−0.00890654
$988$	0	0
$989$	2.12116	0.0674489
$990$	0	0
$991$	−5.83406	−0.185325	−0.0926626	−	0.995698i	$-0.529538\pi$
$991$	−5.83406	−0.185325	−0.0926626	+	0.995698i	$0.529538\pi$
$992$	0	0
$993$	−1.30116	−0.0412912
$994$	0	0
$995$	16.2388	0.514806
$996$	0	0
$997$	−46.4157	−1.47000	−0.735000	−	0.678067i	$-0.762818\pi$
$997$	−46.4157	−1.47000	−0.735000	+	0.678067i	$0.762818\pi$
$998$	0	0
$999$	5.39836	0.170796

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.6	✓	12	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.336827 q^{3} +3.95783 q^{5} +1.00000 q^{7} -2.88655 q^{9} +O(q^{10})\)	\(q+0.336827 q^{3} +3.95783 q^{5} +1.00000 q^{7} -2.88655 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.33310 q^{15} -0.794965 q^{17} +4.35914 q^{19} +0.336827 q^{21} -1.29578 q^{23} +10.6644 q^{25} -1.98275 q^{27} +3.76838 q^{29} +6.69646 q^{31} +0.336827 q^{33} +3.95783 q^{35} -2.72266 q^{37} +0.336827 q^{39} -3.50398 q^{41} -1.63697 q^{43} -11.4245 q^{45} -0.830732 q^{47} +1.00000 q^{49} -0.267766 q^{51} +10.8838 q^{53} +3.95783 q^{55} +1.46828 q^{57} +5.39313 q^{59} +3.96062 q^{61} -2.88655 q^{63} +3.95783 q^{65} -11.6208 q^{67} -0.436456 q^{69} -4.19845 q^{71} +5.43742 q^{73} +3.59206 q^{75} +1.00000 q^{77} -9.39088 q^{79} +7.99180 q^{81} +5.82445 q^{83} -3.14633 q^{85} +1.26929 q^{87} -5.81460 q^{89} +1.00000 q^{91} +2.25555 q^{93} +17.2527 q^{95} -3.83277 q^{97} -2.88655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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