LMFDB - Embedded newform 8008.2.a.r.1.5

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 $ x^9 - 4x^8 - 9x^7 + 36x^6 + 23x^5 - 89x^4 - 20x^3 + 51x^2 + 18x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0694951$ 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)

$f(q)$	$=$	$q+1.06950 q^{3} +0.381633 q^{5} -1.00000 q^{7} -1.85618 q^{9} +O(q^{10})$	$q+1.06950 q^{3} +0.381633 q^{5} -1.00000 q^{7} -1.85618 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.408155 q^{15} +7.47860 q^{17} +1.51254 q^{19} -1.06950 q^{21} -3.96682 q^{23} -4.85436 q^{25} -5.19366 q^{27} +2.36137 q^{29} -3.61930 q^{31} -1.06950 q^{33} -0.381633 q^{35} -4.82850 q^{37} -1.06950 q^{39} -2.43493 q^{41} +12.2806 q^{43} -0.708380 q^{45} +8.39193 q^{47} +1.00000 q^{49} +7.99833 q^{51} -5.64794 q^{53} -0.381633 q^{55} +1.61766 q^{57} +7.25793 q^{59} +7.70112 q^{61} +1.85618 q^{63} -0.381633 q^{65} +8.61532 q^{67} -4.24250 q^{69} +6.79647 q^{71} +11.3501 q^{73} -5.19171 q^{75} +1.00000 q^{77} +2.58414 q^{79} +0.0139460 q^{81} -1.86936 q^{83} +2.85408 q^{85} +2.52547 q^{87} +4.79957 q^{89} +1.00000 q^{91} -3.87083 q^{93} +0.577236 q^{95} +11.1990 q^{97} +1.85618 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * 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
$3$	1.06950	0.617473	0.308737	−	0.951148i	$-0.400094\pi$
$3$	1.06950	0.617473	0.308737	+	0.951148i	$0.400094\pi$
$4$	0	0
$5$	0.381633	0.170672	0.0853358	−	0.996352i	$-0.472804\pi$
$5$	0.381633	0.170672	0.0853358	+	0.996352i	$0.472804\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.85618	−0.618727
$10$	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
$15$	0.408155	0.105385
$16$	0	0
$17$	7.47860	1.81383	0.906914	−	0.421316i	$-0.138432\pi$
$17$	7.47860	1.81383	0.906914	+	0.421316i	$0.138432\pi$
$18$	0	0
$19$	1.51254	0.347001	0.173501	−	0.984834i	$-0.444492\pi$
$19$	1.51254	0.347001	0.173501	+	0.984834i	$0.444492\pi$
$20$	0	0
$21$	−1.06950	−0.233383
$22$	0	0
$23$	−3.96682	−0.827140	−0.413570	−	0.910472i	$-0.635718\pi$
$23$	−3.96682	−0.827140	−0.413570	+	0.910472i	$0.635718\pi$
$24$	0	0
$25$	−4.85436	−0.970871
$26$	0	0
$27$	−5.19366	−0.999521
$28$	0	0
$29$	2.36137	0.438495	0.219248	−	0.975669i	$-0.429640\pi$
$29$	2.36137	0.438495	0.219248	+	0.975669i	$0.429640\pi$
$30$	0	0
$31$	−3.61930	−0.650046	−0.325023	−	0.945706i	$-0.605372\pi$
$31$	−3.61930	−0.650046	−0.325023	+	0.945706i	$0.605372\pi$
$32$	0	0
$33$	−1.06950	−0.186175
$34$	0	0
$35$	−0.381633	−0.0645078
$36$	0	0
$37$	−4.82850	−0.793800	−0.396900	−	0.917862i	$-0.629914\pi$
$37$	−4.82850	−0.793800	−0.396900	+	0.917862i	$0.629914\pi$
$38$	0	0
$39$	−1.06950	−0.171256
$40$	0	0
$41$	−2.43493	−0.380272	−0.190136	−	0.981758i	$-0.560893\pi$
$41$	−2.43493	−0.380272	−0.190136	+	0.981758i	$0.560893\pi$
$42$	0	0
$43$	12.2806	1.87277	0.936384	−	0.350977i	$-0.114150\pi$
$43$	12.2806	1.87277	0.936384	+	0.350977i	$0.114150\pi$
$44$	0	0
$45$	−0.708380	−0.105599
$46$	0	0
$47$	8.39193	1.22409	0.612044	−	0.790824i	$-0.290347\pi$
$47$	8.39193	1.22409	0.612044	+	0.790824i	$0.290347\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.99833	1.11999
$52$	0	0
$53$	−5.64794	−0.775804	−0.387902	−	0.921701i	$-0.626800\pi$
$53$	−5.64794	−0.775804	−0.387902	+	0.921701i	$0.626800\pi$
$54$	0	0
$55$	−0.381633	−0.0514594
$56$	0	0
$57$	1.61766	0.214264
$58$	0	0
$59$	7.25793	0.944903	0.472451	−	0.881357i	$-0.343369\pi$
$59$	7.25793	0.944903	0.472451	+	0.881357i	$0.343369\pi$
$60$	0	0
$61$	7.70112	0.986027	0.493014	−	0.870022i	$-0.335895\pi$
$61$	7.70112	0.986027	0.493014	+	0.870022i	$0.335895\pi$
$62$	0	0
$63$	1.85618	0.233857
$64$	0	0
$65$	−0.381633	−0.0473358
$66$	0	0
$67$	8.61532	1.05253	0.526265	−	0.850321i	$-0.323592\pi$
$67$	8.61532	1.05253	0.526265	+	0.850321i	$0.323592\pi$
$68$	0	0
$69$	−4.24250	−0.510737
$70$	0	0
$71$	6.79647	0.806592	0.403296	−	0.915070i	$-0.367864\pi$
$71$	6.79647	0.806592	0.403296	+	0.915070i	$0.367864\pi$
$72$	0	0
$73$	11.3501	1.32842	0.664212	−	0.747544i	$-0.268767\pi$
$73$	11.3501	1.32842	0.664212	+	0.747544i	$0.268767\pi$
$74$	0	0
$75$	−5.19171	−0.599487
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	2.58414	0.290738	0.145369	−	0.989377i	$-0.453563\pi$
$79$	2.58414	0.290738	0.145369	+	0.989377i	$0.453563\pi$
$80$	0	0
$81$	0.0139460	0.00154956
$82$	0	0
$83$	−1.86936	−0.205189	−0.102595	−	0.994723i	$-0.532714\pi$
$83$	−1.86936	−0.205189	−0.102595	+	0.994723i	$0.532714\pi$
$84$	0	0
$85$	2.85408	0.309569
$86$	0	0
$87$	2.52547	0.270759
$88$	0	0
$89$	4.79957	0.508753	0.254377	−	0.967105i	$-0.418130\pi$
$89$	4.79957	0.508753	0.254377	+	0.967105i	$0.418130\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−3.87083	−0.401386
$94$	0	0
$95$	0.577236	0.0592232
$96$	0	0
$97$	11.1990	1.13709	0.568543	−	0.822653i	$-0.307507\pi$
$97$	11.1990	1.13709	0.568543	+	0.822653i	$0.307507\pi$
$98$	0	0
$99$	1.85618	0.186553
$100$	0	0
$101$	−8.60262	−0.855992	−0.427996	−	0.903781i	$-0.640780\pi$
$101$	−8.60262	−0.855992	−0.427996	+	0.903781i	$0.640780\pi$
$102$	0	0
$103$	−4.43557	−0.437049	−0.218525	−	0.975831i	$-0.570124\pi$
$103$	−4.43557	−0.437049	−0.218525	+	0.975831i	$0.570124\pi$
$104$	0	0
$105$	−0.408155	−0.0398318
$106$	0	0
$107$	−4.15796	−0.401965	−0.200983	−	0.979595i	$-0.564413\pi$
$107$	−4.15796	−0.401965	−0.200983	+	0.979595i	$0.564413\pi$
$108$	0	0
$109$	−14.2843	−1.36819	−0.684096	−	0.729392i	$-0.739803\pi$
$109$	−14.2843	−1.36819	−0.684096	+	0.729392i	$0.739803\pi$
$110$	0	0
$111$	−5.16406	−0.490150
$112$	0	0
$113$	−3.11747	−0.293267	−0.146634	−	0.989191i	$-0.546844\pi$
$113$	−3.11747	−0.293267	−0.146634	+	0.989191i	$0.546844\pi$
$114$	0	0
$115$	−1.51387	−0.141169
$116$	0	0
$117$	1.85618	0.171604
$118$	0	0
$119$	−7.47860	−0.685562
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.60415	−0.234808
$124$	0	0
$125$	−3.76075	−0.336372
$126$	0	0
$127$	4.63861	0.411610	0.205805	−	0.978593i	$-0.434019\pi$
$127$	4.63861	0.411610	0.205805	+	0.978593i	$0.434019\pi$
$128$	0	0
$129$	13.1340	1.15638
$130$	0	0
$131$	8.82475	0.771022	0.385511	−	0.922703i	$-0.374025\pi$
$131$	8.82475	0.771022	0.385511	+	0.922703i	$0.374025\pi$
$132$	0	0
$133$	−1.51254	−0.131154
$134$	0	0
$135$	−1.98207	−0.170590
$136$	0	0
$137$	−7.94651	−0.678916	−0.339458	−	0.940621i	$-0.610244\pi$
$137$	−7.94651	−0.678916	−0.339458	+	0.940621i	$0.610244\pi$
$138$	0	0
$139$	7.95980	0.675142	0.337571	−	0.941300i	$-0.390395\pi$
$139$	7.95980	0.675142	0.337571	+	0.941300i	$0.390395\pi$
$140$	0	0
$141$	8.97513	0.755842
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.901177	0.0748386
$146$	0	0
$147$	1.06950	0.0882105
$148$	0	0
$149$	7.65337	0.626988	0.313494	−	0.949590i	$-0.398500\pi$
$149$	7.65337	0.626988	0.313494	+	0.949590i	$0.398500\pi$
$150$	0	0
$151$	0.0530390	0.00431625	0.00215812	−	0.999998i	$-0.499313\pi$
$151$	0.0530390	0.00431625	0.00215812	+	0.999998i	$0.499313\pi$
$152$	0	0
$153$	−13.8816	−1.12226
$154$	0	0
$155$	−1.38125	−0.110944
$156$	0	0
$157$	8.30719	0.662986	0.331493	−	0.943458i	$-0.392448\pi$
$157$	8.30719	0.662986	0.331493	+	0.943458i	$0.392448\pi$
$158$	0	0
$159$	−6.04044	−0.479038
$160$	0	0
$161$	3.96682	0.312630
$162$	0	0
$163$	−21.0907	−1.65195	−0.825975	−	0.563707i	$-0.809375\pi$
$163$	−21.0907	−1.65195	−0.825975	+	0.563707i	$0.809375\pi$
$164$	0	0
$165$	−0.408155	−0.0317748
$166$	0	0
$167$	10.3189	0.798497	0.399248	−	0.916843i	$-0.369271\pi$
$167$	10.3189	0.798497	0.399248	+	0.916843i	$0.369271\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.80755	−0.214699
$172$	0	0
$173$	17.5045	1.33084	0.665421	−	0.746468i	$-0.268252\pi$
$173$	17.5045	1.33084	0.665421	+	0.746468i	$0.268252\pi$
$174$	0	0
$175$	4.85436	0.366955
$176$	0	0
$177$	7.76233	0.583452
$178$	0	0
$179$	−6.69431	−0.500356	−0.250178	−	0.968200i	$-0.580489\pi$
$179$	−6.69431	−0.500356	−0.250178	+	0.968200i	$0.580489\pi$
$180$	0	0
$181$	24.7060	1.83638	0.918190	−	0.396141i	$-0.129651\pi$
$181$	24.7060	1.83638	0.918190	+	0.396141i	$0.129651\pi$
$182$	0	0
$183$	8.23631	0.608846
$184$	0	0
$185$	−1.84272	−0.135479
$186$	0	0
$187$	−7.47860	−0.546890
$188$	0	0
$189$	5.19366	0.377783
$190$	0	0
$191$	−0.555042	−0.0401615	−0.0200807	−	0.999798i	$-0.506392\pi$
$191$	−0.555042	−0.0401615	−0.0200807	+	0.999798i	$0.506392\pi$
$192$	0	0
$193$	17.4789	1.25816	0.629079	−	0.777341i	$-0.283432\pi$
$193$	17.4789	1.25816	0.629079	+	0.777341i	$0.283432\pi$
$194$	0	0
$195$	−0.408155	−0.0292286
$196$	0	0
$197$	0.453490	0.0323098	0.0161549	−	0.999870i	$-0.494858\pi$
$197$	0.453490	0.0323098	0.0161549	+	0.999870i	$0.494858\pi$
$198$	0	0
$199$	2.95026	0.209138	0.104569	−	0.994518i	$-0.466654\pi$
$199$	2.95026	0.209138	0.104569	+	0.994518i	$0.466654\pi$
$200$	0	0
$201$	9.21405	0.649909
$202$	0	0
$203$	−2.36137	−0.165736
$204$	0	0
$205$	−0.929250	−0.0649016
$206$	0	0
$207$	7.36314	0.511774
$208$	0	0
$209$	−1.51254	−0.104625
$210$	0	0
$211$	−5.92231	−0.407709	−0.203854	−	0.979001i	$-0.565347\pi$
$211$	−5.92231	−0.407709	−0.203854	+	0.979001i	$0.565347\pi$
$212$	0	0
$213$	7.26879	0.498049
$214$	0	0
$215$	4.68667	0.319628
$216$	0	0
$217$	3.61930	0.245694
$218$	0	0
$219$	12.1388	0.820266
$220$	0	0
$221$	−7.47860	−0.503065
$222$	0	0
$223$	2.96888	0.198811	0.0994056	−	0.995047i	$-0.468306\pi$
$223$	2.96888	0.198811	0.0994056	+	0.995047i	$0.468306\pi$
$224$	0	0
$225$	9.01056	0.600704
$226$	0	0
$227$	0.916725	0.0608452	0.0304226	−	0.999537i	$-0.490315\pi$
$227$	0.916725	0.0608452	0.0304226	+	0.999537i	$0.490315\pi$
$228$	0	0
$229$	13.4048	0.885811	0.442905	−	0.896568i	$-0.353948\pi$
$229$	13.4048	0.885811	0.442905	+	0.896568i	$0.353948\pi$
$230$	0	0
$231$	1.06950	0.0703676
$232$	0	0
$233$	−17.1466	−1.12331	−0.561656	−	0.827371i	$-0.689835\pi$
$233$	−17.1466	−1.12331	−0.561656	+	0.827371i	$0.689835\pi$
$234$	0	0
$235$	3.20264	0.208917
$236$	0	0
$237$	2.76372	0.179523
$238$	0	0
$239$	16.3367	1.05674	0.528368	−	0.849016i	$-0.322804\pi$
$239$	16.3367	1.05674	0.528368	+	0.849016i	$0.322804\pi$
$240$	0	0
$241$	9.51797	0.613106	0.306553	−	0.951854i	$-0.400824\pi$
$241$	9.51797	0.613106	0.306553	+	0.951854i	$0.400824\pi$
$242$	0	0
$243$	15.5959	1.00048
$244$	0	0
$245$	0.381633	0.0243817
$246$	0	0
$247$	−1.51254	−0.0962408
$248$	0	0
$249$	−1.99927	−0.126699
$250$	0	0
$251$	22.2428	1.40395	0.701977	−	0.712200i	$-0.252301\pi$
$251$	22.2428	1.40395	0.701977	+	0.712200i	$0.252301\pi$
$252$	0	0
$253$	3.96682	0.249392
$254$	0	0
$255$	3.05243	0.191150
$256$	0	0
$257$	−17.2337	−1.07501	−0.537505	−	0.843260i	$-0.680633\pi$
$257$	−17.2337	−1.07501	−0.537505	+	0.843260i	$0.680633\pi$
$258$	0	0
$259$	4.82850	0.300028
$260$	0	0
$261$	−4.38313	−0.271309
$262$	0	0
$263$	−4.26978	−0.263286	−0.131643	−	0.991297i	$-0.542025\pi$
$263$	−4.26978	−0.263286	−0.131643	+	0.991297i	$0.542025\pi$
$264$	0	0
$265$	−2.15544	−0.132408
$266$	0	0
$267$	5.13311	0.314141
$268$	0	0
$269$	−23.0420	−1.40489	−0.702447	−	0.711736i	$-0.747909\pi$
$269$	−23.0420	−1.40489	−0.702447	+	0.711736i	$0.747909\pi$
$270$	0	0
$271$	24.4492	1.48518	0.742590	−	0.669746i	$-0.233597\pi$
$271$	24.4492	1.48518	0.742590	+	0.669746i	$0.233597\pi$
$272$	0	0
$273$	1.06950	0.0647288
$274$	0	0
$275$	4.85436	0.292729
$276$	0	0
$277$	−5.99918	−0.360456	−0.180228	−	0.983625i	$-0.557683\pi$
$277$	−5.99918	−0.360456	−0.180228	+	0.983625i	$0.557683\pi$
$278$	0	0
$279$	6.71808	0.402201
$280$	0	0
$281$	9.98663	0.595752	0.297876	−	0.954605i	$-0.403722\pi$
$281$	9.98663	0.595752	0.297876	+	0.954605i	$0.403722\pi$
$282$	0	0
$283$	31.1390	1.85102	0.925510	−	0.378723i	$-0.123637\pi$
$283$	31.1390	1.85102	0.925510	+	0.378723i	$0.123637\pi$
$284$	0	0
$285$	0.617352	0.0365687
$286$	0	0
$287$	2.43493	0.143729
$288$	0	0
$289$	38.9295	2.28997
$290$	0	0
$291$	11.9773	0.702120
$292$	0	0
$293$	6.28129	0.366957	0.183478	−	0.983024i	$-0.441264\pi$
$293$	6.28129	0.366957	0.183478	+	0.983024i	$0.441264\pi$
$294$	0	0
$295$	2.76987	0.161268
$296$	0	0
$297$	5.19366	0.301367
$298$	0	0
$299$	3.96682	0.229407
$300$	0	0
$301$	−12.2806	−0.707840
$302$	0	0
$303$	−9.20045	−0.528552
$304$	0	0
$305$	2.93900	0.168287
$306$	0	0
$307$	17.3864	0.992294	0.496147	−	0.868238i	$-0.334748\pi$
$307$	17.3864	0.992294	0.496147	+	0.868238i	$0.334748\pi$
$308$	0	0
$309$	−4.74382	−0.269866
$310$	0	0
$311$	−22.1882	−1.25818	−0.629090	−	0.777332i	$-0.716572\pi$
$311$	−22.1882	−1.25818	−0.629090	+	0.777332i	$0.716572\pi$
$312$	0	0
$313$	7.02028	0.396810	0.198405	−	0.980120i	$-0.436424\pi$
$313$	7.02028	0.396810	0.198405	+	0.980120i	$0.436424\pi$
$314$	0	0
$315$	0.708380	0.0399127
$316$	0	0
$317$	3.73600	0.209835	0.104917	−	0.994481i	$-0.466542\pi$
$317$	3.73600	0.209835	0.104917	+	0.994481i	$0.466542\pi$
$318$	0	0
$319$	−2.36137	−0.132211
$320$	0	0
$321$	−4.44692	−0.248203
$322$	0	0
$323$	11.3117	0.629400
$324$	0	0
$325$	4.85436	0.269271
$326$	0	0
$327$	−15.2770	−0.844822
$328$	0	0
$329$	−8.39193	−0.462662
$330$	0	0
$331$	−9.34839	−0.513834	−0.256917	−	0.966433i	$-0.582707\pi$
$331$	−9.34839	−0.513834	−0.256917	+	0.966433i	$0.582707\pi$
$332$	0	0
$333$	8.96256	0.491145
$334$	0	0
$335$	3.28789	0.179637
$336$	0	0
$337$	−32.0943	−1.74829	−0.874143	−	0.485668i	$-0.838576\pi$
$337$	−32.0943	−1.74829	−0.874143	+	0.485668i	$0.838576\pi$
$338$	0	0
$339$	−3.33412	−0.181085
$340$	0	0
$341$	3.61930	0.195996
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.61908	−0.0871683
$346$	0	0
$347$	13.6259	0.731477	0.365739	−	0.930718i	$-0.380816\pi$
$347$	13.6259	0.731477	0.365739	+	0.930718i	$0.380816\pi$
$348$	0	0
$349$	−8.83840	−0.473108	−0.236554	−	0.971618i	$-0.576018\pi$
$349$	−8.83840	−0.473108	−0.236554	+	0.971618i	$0.576018\pi$
$350$	0	0
$351$	5.19366	0.277217
$352$	0	0
$353$	−11.4696	−0.610467	−0.305234	−	0.952277i	$-0.598735\pi$
$353$	−11.4696	−0.610467	−0.305234	+	0.952277i	$0.598735\pi$
$354$	0	0
$355$	2.59376	0.137662
$356$	0	0
$357$	−7.99833	−0.423317
$358$	0	0
$359$	−2.58585	−0.136476	−0.0682380	−	0.997669i	$-0.521738\pi$
$359$	−2.58585	−0.136476	−0.0682380	+	0.997669i	$0.521738\pi$
$360$	0	0
$361$	−16.7122	−0.879590
$362$	0	0
$363$	1.06950	0.0561339
$364$	0	0
$365$	4.33156	0.226724
$366$	0	0
$367$	−3.60552	−0.188207	−0.0941033	−	0.995562i	$-0.529998\pi$
$367$	−3.60552	−0.188207	−0.0941033	+	0.995562i	$0.529998\pi$
$368$	0	0
$369$	4.51967	0.235285
$370$	0	0
$371$	5.64794	0.293226
$372$	0	0
$373$	−29.7885	−1.54239	−0.771196	−	0.636598i	$-0.780341\pi$
$373$	−29.7885	−1.54239	−0.771196	+	0.636598i	$0.780341\pi$
$374$	0	0
$375$	−4.02210	−0.207701
$376$	0	0
$377$	−2.36137	−0.121617
$378$	0	0
$379$	−0.376384	−0.0193335	−0.00966677	−	0.999953i	$-0.503077\pi$
$379$	−0.376384	−0.0193335	−0.00966677	+	0.999953i	$0.503077\pi$
$380$	0	0
$381$	4.96097	0.254158
$382$	0	0
$383$	24.1009	1.23150	0.615749	−	0.787942i	$-0.288854\pi$
$383$	24.1009	1.23150	0.615749	+	0.787942i	$0.288854\pi$
$384$	0	0
$385$	0.381633	0.0194498
$386$	0	0
$387$	−22.7949	−1.15873
$388$	0	0
$389$	−17.2981	−0.877047	−0.438523	−	0.898720i	$-0.644498\pi$
$389$	−17.2981	−0.877047	−0.438523	+	0.898720i	$0.644498\pi$
$390$	0	0
$391$	−29.6663	−1.50029
$392$	0	0
$393$	9.43803	0.476086
$394$	0	0
$395$	0.986193	0.0496208
$396$	0	0
$397$	−12.4302	−0.623854	−0.311927	−	0.950106i	$-0.600974\pi$
$397$	−12.4302	−0.623854	−0.311927	+	0.950106i	$0.600974\pi$
$398$	0	0
$399$	−1.61766	−0.0809841
$400$	0	0
$401$	12.7929	0.638847	0.319424	−	0.947612i	$-0.396511\pi$
$401$	12.7929	0.638847	0.319424	+	0.947612i	$0.396511\pi$
$402$	0	0
$403$	3.61930	0.180290
$404$	0	0
$405$	0.00532227	0.000264466 0
$406$	0	0
$407$	4.82850	0.239340
$408$	0	0
$409$	−18.0386	−0.891953	−0.445977	−	0.895045i	$-0.647144\pi$
$409$	−18.0386	−0.891953	−0.445977	+	0.895045i	$0.647144\pi$
$410$	0	0
$411$	−8.49875	−0.419212
$412$	0	0
$413$	−7.25793	−0.357140
$414$	0	0
$415$	−0.713411	−0.0350200
$416$	0	0
$417$	8.51297	0.416882
$418$	0	0
$419$	28.4290	1.38885	0.694424	−	0.719566i	$-0.255659\pi$
$419$	28.4290	1.38885	0.694424	+	0.719566i	$0.255659\pi$
$420$	0	0
$421$	2.51816	0.122727	0.0613637	−	0.998115i	$-0.480455\pi$
$421$	2.51816	0.122727	0.0613637	+	0.998115i	$0.480455\pi$
$422$	0	0
$423$	−15.5769	−0.757376
$424$	0	0
$425$	−36.3038	−1.76099
$426$	0	0
$427$	−7.70112	−0.372683
$428$	0	0
$429$	1.06950	0.0516357
$430$	0	0
$431$	−0.391524	−0.0188591	−0.00942953	−	0.999956i	$-0.503002\pi$
$431$	−0.391524	−0.0188591	−0.00942953	+	0.999956i	$0.503002\pi$
$432$	0	0
$433$	0.457527	0.0219873	0.0109937	−	0.999940i	$-0.496501\pi$
$433$	0.457527	0.0219873	0.0109937	+	0.999940i	$0.496501\pi$
$434$	0	0
$435$	0.963804	0.0462109
$436$	0	0
$437$	−5.99999	−0.287018
$438$	0	0
$439$	−7.69049	−0.367047	−0.183523	−	0.983015i	$-0.558750\pi$
$439$	−7.69049	−0.367047	−0.183523	+	0.983015i	$0.558750\pi$
$440$	0	0
$441$	−1.85618	−0.0883895
$442$	0	0
$443$	26.5492	1.26139	0.630696	−	0.776030i	$-0.282770\pi$
$443$	26.5492	1.26139	0.630696	+	0.776030i	$0.282770\pi$
$444$	0	0
$445$	1.83167	0.0868297
$446$	0	0
$447$	8.18524	0.387149
$448$	0	0
$449$	14.0401	0.662593	0.331297	−	0.943527i	$-0.392514\pi$
$449$	14.0401	0.662593	0.331297	+	0.943527i	$0.392514\pi$
$450$	0	0
$451$	2.43493	0.114656
$452$	0	0
$453$	0.0567249	0.00266517
$454$	0	0
$455$	0.381633	0.0178912
$456$	0	0
$457$	12.7009	0.594123	0.297062	−	0.954858i	$-0.403993\pi$
$457$	12.7009	0.594123	0.297062	+	0.954858i	$0.403993\pi$
$458$	0	0
$459$	−38.8413	−1.81296
$460$	0	0
$461$	18.7474	0.873152	0.436576	−	0.899667i	$-0.356191\pi$
$461$	18.7474	0.873152	0.436576	+	0.899667i	$0.356191\pi$
$462$	0	0
$463$	−20.3333	−0.944969	−0.472484	−	0.881339i	$-0.656643\pi$
$463$	−20.3333	−0.944969	−0.472484	+	0.881339i	$0.656643\pi$
$464$	0	0
$465$	−1.47724	−0.0685052
$466$	0	0
$467$	6.67553	0.308906	0.154453	−	0.988000i	$-0.450638\pi$
$467$	6.67553	0.308906	0.154453	+	0.988000i	$0.450638\pi$
$468$	0	0
$469$	−8.61532	−0.397819
$470$	0	0
$471$	8.88450	0.409376
$472$	0	0
$473$	−12.2806	−0.564661
$474$	0	0
$475$	−7.34242	−0.336893
$476$	0	0
$477$	10.4836	0.480011
$478$	0	0
$479$	12.2951	0.561776	0.280888	−	0.959741i	$-0.409371\pi$
$479$	12.2951	0.561776	0.280888	+	0.959741i	$0.409371\pi$
$480$	0	0
$481$	4.82850	0.220161
$482$	0	0
$483$	4.24250	0.193040
$484$	0	0
$485$	4.27391	0.194068
$486$	0	0
$487$	2.31678	0.104983	0.0524917	−	0.998621i	$-0.483284\pi$
$487$	2.31678	0.104983	0.0524917	+	0.998621i	$0.483284\pi$
$488$	0	0
$489$	−22.5564	−1.02003
$490$	0	0
$491$	−18.5478	−0.837052	−0.418526	−	0.908205i	$-0.637453\pi$
$491$	−18.5478	−0.837052	−0.418526	+	0.908205i	$0.637453\pi$
$492$	0	0
$493$	17.6597	0.795355
$494$	0	0
$495$	0.708380	0.0318393
$496$	0	0
$497$	−6.79647	−0.304863
$498$	0	0
$499$	18.3341	0.820745	0.410373	−	0.911918i	$-0.365399\pi$
$499$	18.3341	0.820745	0.410373	+	0.911918i	$0.365399\pi$
$500$	0	0
$501$	11.0360	0.493050
$502$	0	0
$503$	−14.5119	−0.647056	−0.323528	−	0.946219i	$-0.604869\pi$
$503$	−14.5119	−0.647056	−0.323528	+	0.946219i	$0.604869\pi$
$504$	0	0
$505$	−3.28304	−0.146094
$506$	0	0
$507$	1.06950	0.0474979
$508$	0	0
$509$	1.54051	0.0682820	0.0341410	−	0.999417i	$-0.489130\pi$
$509$	1.54051	0.0682820	0.0341410	+	0.999417i	$0.489130\pi$
$510$	0	0
$511$	−11.3501	−0.502097
$512$	0	0
$513$	−7.85563	−0.346835
$514$	0	0
$515$	−1.69276	−0.0745919
$516$	0	0
$517$	−8.39193	−0.369077
$518$	0	0
$519$	18.7210	0.821759
$520$	0	0
$521$	−5.75095	−0.251954	−0.125977	−	0.992033i	$-0.540207\pi$
$521$	−5.75095	−0.251954	−0.125977	+	0.992033i	$0.540207\pi$
$522$	0	0
$523$	−10.4742	−0.458006	−0.229003	−	0.973426i	$-0.573547\pi$
$523$	−10.4742	−0.458006	−0.229003	+	0.973426i	$0.573547\pi$
$524$	0	0
$525$	5.19171	0.226585
$526$	0	0
$527$	−27.0673	−1.17907
$528$	0	0
$529$	−7.26430	−0.315839
$530$	0	0
$531$	−13.4720	−0.584637
$532$	0	0
$533$	2.43493	0.105469
$534$	0	0
$535$	−1.58682	−0.0686040
$536$	0	0
$537$	−7.15953	−0.308957
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−22.4586	−0.965572	−0.482786	−	0.875738i	$-0.660375\pi$
$541$	−22.4586	−0.965572	−0.482786	+	0.875738i	$0.660375\pi$
$542$	0	0
$543$	26.4229	1.13392
$544$	0	0
$545$	−5.45138	−0.233512
$546$	0	0
$547$	−5.46505	−0.233669	−0.116834	−	0.993151i	$-0.537275\pi$
$547$	−5.46505	−0.233669	−0.116834	+	0.993151i	$0.537275\pi$
$548$	0	0
$549$	−14.2947	−0.610082
$550$	0	0
$551$	3.57167	0.152158
$552$	0	0
$553$	−2.58414	−0.109889
$554$	0	0
$555$	−1.97078	−0.0836547
$556$	0	0
$557$	7.96286	0.337397	0.168699	−	0.985668i	$-0.446044\pi$
$557$	7.96286	0.337397	0.168699	+	0.985668i	$0.446044\pi$
$558$	0	0
$559$	−12.2806	−0.519412
$560$	0	0
$561$	−7.99833	−0.337690
$562$	0	0
$563$	−22.3983	−0.943975	−0.471988	−	0.881605i	$-0.656463\pi$
$563$	−22.3983	−0.943975	−0.471988	+	0.881605i	$0.656463\pi$
$564$	0	0
$565$	−1.18973	−0.0500523
$566$	0	0
$567$	−0.0139460	−0.000585678 0
$568$	0	0
$569$	1.97709	0.0828838	0.0414419	−	0.999141i	$-0.486805\pi$
$569$	1.97709	0.0828838	0.0414419	+	0.999141i	$0.486805\pi$
$570$	0	0
$571$	−5.90315	−0.247039	−0.123519	−	0.992342i	$-0.539418\pi$
$571$	−5.90315	−0.247039	−0.123519	+	0.992342i	$0.539418\pi$
$572$	0	0
$573$	−0.593615	−0.0247986
$574$	0	0
$575$	19.2564	0.803047
$576$	0	0
$577$	36.6572	1.52606	0.763029	−	0.646365i	$-0.223712\pi$
$577$	36.6572	1.52606	0.763029	+	0.646365i	$0.223712\pi$
$578$	0	0
$579$	18.6936	0.776879
$580$	0	0
$581$	1.86936	0.0775543
$582$	0	0
$583$	5.64794	0.233914
$584$	0	0
$585$	0.708380	0.0292879
$586$	0	0
$587$	12.2553	0.505829	0.252914	−	0.967489i	$-0.418611\pi$
$587$	12.2553	0.505829	0.252914	+	0.967489i	$0.418611\pi$
$588$	0	0
$589$	−5.47435	−0.225567
$590$	0	0
$591$	0.485005	0.0199504
$592$	0	0
$593$	−15.0540	−0.618196	−0.309098	−	0.951030i	$-0.600027\pi$
$593$	−15.0540	−0.618196	−0.309098	+	0.951030i	$0.600027\pi$
$594$	0	0
$595$	−2.85408	−0.117006
$596$	0	0
$597$	3.15529	0.129137
$598$	0	0
$599$	37.1260	1.51693	0.758463	−	0.651716i	$-0.225950\pi$
$599$	37.1260	1.51693	0.758463	+	0.651716i	$0.225950\pi$
$600$	0	0
$601$	39.8241	1.62446	0.812230	−	0.583337i	$-0.198253\pi$
$601$	39.8241	1.62446	0.812230	+	0.583337i	$0.198253\pi$
$602$	0	0
$603$	−15.9916	−0.651228
$604$	0	0
$605$	0.381633	0.0155156
$606$	0	0
$607$	26.8049	1.08798	0.543989	−	0.839093i	$-0.316913\pi$
$607$	26.8049	1.08798	0.543989	+	0.839093i	$0.316913\pi$
$608$	0	0
$609$	−2.52547	−0.102337
$610$	0	0
$611$	−8.39193	−0.339501
$612$	0	0
$613$	7.04270	0.284452	0.142226	−	0.989834i	$-0.454574\pi$
$613$	7.04270	0.284452	0.142226	+	0.989834i	$0.454574\pi$
$614$	0	0
$615$	−0.993828	−0.0400750
$616$	0	0
$617$	26.2663	1.05744	0.528721	−	0.848796i	$-0.322672\pi$
$617$	26.2663	1.05744	0.528721	+	0.848796i	$0.322672\pi$
$618$	0	0
$619$	28.1255	1.13046	0.565230	−	0.824933i	$-0.308787\pi$
$619$	28.1255	1.13046	0.565230	+	0.824933i	$0.308787\pi$
$620$	0	0
$621$	20.6023	0.826744
$622$	0	0
$623$	−4.79957	−0.192291
$624$	0	0
$625$	22.8366	0.913462
$626$	0	0
$627$	−1.61766	−0.0646030
$628$	0	0
$629$	−36.1104	−1.43982
$630$	0	0
$631$	23.0013	0.915669	0.457834	−	0.889038i	$-0.348625\pi$
$631$	23.0013	0.915669	0.457834	+	0.889038i	$0.348625\pi$
$632$	0	0
$633$	−6.33388	−0.251749
$634$	0	0
$635$	1.77025	0.0702502
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−12.6155	−0.499060
$640$	0	0
$641$	34.7039	1.37072	0.685361	−	0.728203i	$-0.259644\pi$
$641$	34.7039	1.37072	0.685361	+	0.728203i	$0.259644\pi$
$642$	0	0
$643$	−23.4339	−0.924142	−0.462071	−	0.886843i	$-0.652893\pi$
$643$	−23.4339	−0.924142	−0.462071	+	0.886843i	$0.652893\pi$
$644$	0	0
$645$	5.01237	0.197362
$646$	0	0
$647$	45.5507	1.79078	0.895392	−	0.445279i	$-0.146895\pi$
$647$	45.5507	1.79078	0.895392	+	0.445279i	$0.146895\pi$
$648$	0	0
$649$	−7.25793	−0.284899
$650$	0	0
$651$	3.87083	0.151710
$652$	0	0
$653$	15.6117	0.610931	0.305466	−	0.952203i	$-0.401188\pi$
$653$	15.6117	0.610931	0.305466	+	0.952203i	$0.401188\pi$
$654$	0	0
$655$	3.36782	0.131592
$656$	0	0
$657$	−21.0678	−0.821931
$658$	0	0
$659$	21.5193	0.838272	0.419136	−	0.907924i	$-0.362333\pi$
$659$	21.5193	0.838272	0.419136	+	0.907924i	$0.362333\pi$
$660$	0	0
$661$	−34.7096	−1.35005	−0.675024	−	0.737796i	$-0.735867\pi$
$661$	−34.7096	−1.35005	−0.675024	+	0.737796i	$0.735867\pi$
$662$	0	0
$663$	−7.99833	−0.310629
$664$	0	0
$665$	−0.577236	−0.0223843
$666$	0	0
$667$	−9.36713	−0.362697
$668$	0	0
$669$	3.17521	0.122761
$670$	0	0
$671$	−7.70112	−0.297298
$672$	0	0
$673$	8.71976	0.336122	0.168061	−	0.985777i	$-0.446249\pi$
$673$	8.71976	0.336122	0.168061	+	0.985777i	$0.446249\pi$
$674$	0	0
$675$	25.2119	0.970406
$676$	0	0
$677$	−22.8935	−0.879868	−0.439934	−	0.898030i	$-0.644998\pi$
$677$	−22.8935	−0.879868	−0.439934	+	0.898030i	$0.644998\pi$
$678$	0	0
$679$	−11.1990	−0.429778
$680$	0	0
$681$	0.980433	0.0375703
$682$	0	0
$683$	25.8313	0.988408	0.494204	−	0.869346i	$-0.335460\pi$
$683$	25.8313	0.988408	0.494204	+	0.869346i	$0.335460\pi$
$684$	0	0
$685$	−3.03265	−0.115872
$686$	0	0
$687$	14.3363	0.546965
$688$	0	0
$689$	5.64794	0.215169
$690$	0	0
$691$	−33.1460	−1.26093	−0.630466	−	0.776217i	$-0.717136\pi$
$691$	−33.1460	−1.26093	−0.630466	+	0.776217i	$0.717136\pi$
$692$	0	0
$693$	−1.85618	−0.0705105
$694$	0	0
$695$	3.03773	0.115228
$696$	0	0
$697$	−18.2099	−0.689748
$698$	0	0
$699$	−18.3382	−0.693615
$700$	0	0
$701$	−23.0747	−0.871518	−0.435759	−	0.900063i	$-0.643520\pi$
$701$	−23.0747	−0.871518	−0.435759	+	0.900063i	$0.643520\pi$
$702$	0	0
$703$	−7.30331	−0.275449
$704$	0	0
$705$	3.42521	0.129001
$706$	0	0
$707$	8.60262	0.323535
$708$	0	0
$709$	−33.8444	−1.27105	−0.635527	−	0.772079i	$-0.719217\pi$
$709$	−33.8444	−1.27105	−0.635527	+	0.772079i	$0.719217\pi$
$710$	0	0
$711$	−4.79663	−0.179888
$712$	0	0
$713$	14.3571	0.537679
$714$	0	0
$715$	0.381633	0.0142723
$716$	0	0
$717$	17.4721	0.652506
$718$	0	0
$719$	−31.7708	−1.18485	−0.592425	−	0.805625i	$-0.701829\pi$
$719$	−31.7708	−1.18485	−0.592425	+	0.805625i	$0.701829\pi$
$720$	0	0
$721$	4.43557	0.165189
$722$	0	0
$723$	10.1794	0.378577
$724$	0	0
$725$	−11.4629	−0.425722
$726$	0	0
$727$	−7.26315	−0.269375	−0.134688	−	0.990888i	$-0.543003\pi$
$727$	−7.26315	−0.269375	−0.134688	+	0.990888i	$0.543003\pi$
$728$	0	0
$729$	16.6379	0.616218
$730$	0	0
$731$	91.8415	3.39688
$732$	0	0
$733$	−5.78955	−0.213842	−0.106921	−	0.994268i	$-0.534099\pi$
$733$	−5.78955	−0.213842	−0.106921	+	0.994268i	$0.534099\pi$
$734$	0	0
$735$	0.408155	0.0150550
$736$	0	0
$737$	−8.61532	−0.317349
$738$	0	0
$739$	−7.60355	−0.279701	−0.139851	−	0.990173i	$-0.544662\pi$
$739$	−7.60355	−0.279701	−0.139851	+	0.990173i	$0.544662\pi$
$740$	0	0
$741$	−1.61766	−0.0594261
$742$	0	0
$743$	4.93024	0.180873	0.0904365	−	0.995902i	$-0.471174\pi$
$743$	4.93024	0.180873	0.0904365	+	0.995902i	$0.471174\pi$
$744$	0	0
$745$	2.92078	0.107009
$746$	0	0
$747$	3.46988	0.126956
$748$	0	0
$749$	4.15796	0.151929
$750$	0	0
$751$	39.8313	1.45346	0.726732	−	0.686921i	$-0.241038\pi$
$751$	39.8313	1.45346	0.726732	+	0.686921i	$0.241038\pi$
$752$	0	0
$753$	23.7886	0.866903
$754$	0	0
$755$	0.0202414	0.000736661 0
$756$	0	0
$757$	−46.4806	−1.68937	−0.844683	−	0.535267i	$-0.820211\pi$
$757$	−46.4806	−1.68937	−0.844683	+	0.535267i	$0.820211\pi$
$758$	0	0
$759$	4.24250	0.153993
$760$	0	0
$761$	−28.4674	−1.03194	−0.515971	−	0.856606i	$-0.672569\pi$
$761$	−28.4674	−1.03194	−0.515971	+	0.856606i	$0.672569\pi$
$762$	0	0
$763$	14.2843	0.517128
$764$	0	0
$765$	−5.29769	−0.191539
$766$	0	0
$767$	−7.25793	−0.262069
$768$	0	0
$769$	33.9128	1.22293	0.611464	−	0.791273i	$-0.290581\pi$
$769$	33.9128	1.22293	0.611464	+	0.791273i	$0.290581\pi$
$770$	0	0
$771$	−18.4314	−0.663790
$772$	0	0
$773$	−29.5772	−1.06382	−0.531909	−	0.846801i	$-0.678525\pi$
$773$	−29.5772	−1.06382	−0.531909	+	0.846801i	$0.678525\pi$
$774$	0	0
$775$	17.5694	0.631111
$776$	0	0
$777$	5.16406	0.185259
$778$	0	0
$779$	−3.68293	−0.131955
$780$	0	0
$781$	−6.79647	−0.243197
$782$	0	0
$783$	−12.2641	−0.438285
$784$	0	0
$785$	3.17030	0.113153
$786$	0	0
$787$	14.7714	0.526544	0.263272	−	0.964722i	$-0.415198\pi$
$787$	14.7714	0.526544	0.263272	+	0.964722i	$0.415198\pi$
$788$	0	0
$789$	−4.56651	−0.162572
$790$	0	0
$791$	3.11747	0.110845
$792$	0	0
$793$	−7.70112	−0.273475
$794$	0	0
$795$	−2.30523	−0.0817582
$796$	0	0
$797$	40.0470	1.41854	0.709268	−	0.704939i	$-0.249026\pi$
$797$	40.0470	1.41854	0.709268	+	0.704939i	$0.249026\pi$
$798$	0	0
$799$	62.7599	2.22029
$800$	0	0
$801$	−8.90886	−0.314779
$802$	0	0
$803$	−11.3501	−0.400535
$804$	0	0
$805$	1.51387	0.0533570
$806$	0	0
$807$	−24.6433	−0.867484
$808$	0	0
$809$	6.67208	0.234578	0.117289	−	0.993098i	$-0.462580\pi$
$809$	6.67208	0.234578	0.117289	+	0.993098i	$0.462580\pi$
$810$	0	0
$811$	−30.2132	−1.06093	−0.530464	−	0.847707i	$-0.677982\pi$
$811$	−30.2132	−1.06093	−0.530464	+	0.847707i	$0.677982\pi$
$812$	0	0
$813$	26.1482	0.917059
$814$	0	0
$815$	−8.04890	−0.281941
$816$	0	0
$817$	18.5749	0.649853
$818$	0	0
$819$	−1.85618	−0.0648602
$820$	0	0
$821$	36.2227	1.26418	0.632090	−	0.774895i	$-0.282197\pi$
$821$	36.2227	1.26418	0.632090	+	0.774895i	$0.282197\pi$
$822$	0	0
$823$	−27.8513	−0.970834	−0.485417	−	0.874283i	$-0.661332\pi$
$823$	−27.8513	−0.970834	−0.485417	+	0.874283i	$0.661332\pi$
$824$	0	0
$825$	5.19171	0.180752
$826$	0	0
$827$	2.65057	0.0921695	0.0460848	−	0.998938i	$-0.485326\pi$
$827$	2.65057	0.0921695	0.0460848	+	0.998938i	$0.485326\pi$
$828$	0	0
$829$	3.29483	0.114434	0.0572171	−	0.998362i	$-0.481777\pi$
$829$	3.29483	0.114434	0.0572171	+	0.998362i	$0.481777\pi$
$830$	0	0
$831$	−6.41609	−0.222572
$832$	0	0
$833$	7.47860	0.259118
$834$	0	0
$835$	3.93802	0.136281
$836$	0	0
$837$	18.7974	0.649734
$838$	0	0
$839$	−14.2551	−0.492141	−0.246071	−	0.969252i	$-0.579140\pi$
$839$	−14.2551	−0.492141	−0.246071	+	0.969252i	$0.579140\pi$
$840$	0	0
$841$	−23.4239	−0.807722
$842$	0	0
$843$	10.6807	0.367861
$844$	0	0
$845$	0.381633	0.0131286
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	33.3030	1.14296
$850$	0	0
$851$	19.1538	0.656584
$852$	0	0
$853$	−40.9371	−1.40166	−0.700831	−	0.713328i	$-0.747187\pi$
$853$	−40.9371	−1.40166	−0.700831	+	0.713328i	$0.747187\pi$
$854$	0	0
$855$	−1.07145	−0.0366430
$856$	0	0
$857$	−38.5555	−1.31703	−0.658515	−	0.752568i	$-0.728815\pi$
$857$	−38.5555	−1.31703	−0.658515	+	0.752568i	$0.728815\pi$
$858$	0	0
$859$	−7.61070	−0.259674	−0.129837	−	0.991535i	$-0.541445\pi$
$859$	−7.61070	−0.259674	−0.129837	+	0.991535i	$0.541445\pi$
$860$	0	0
$861$	2.60415	0.0887490
$862$	0	0
$863$	15.8258	0.538717	0.269358	−	0.963040i	$-0.413188\pi$
$863$	15.8258	0.538717	0.269358	+	0.963040i	$0.413188\pi$
$864$	0	0
$865$	6.68029	0.227137
$866$	0	0
$867$	41.6349	1.41400
$868$	0	0
$869$	−2.58414	−0.0876609
$870$	0	0
$871$	−8.61532	−0.291919
$872$	0	0
$873$	−20.7874	−0.703546
$874$	0	0
$875$	3.76075	0.127137
$876$	0	0
$877$	−36.1409	−1.22039	−0.610195	−	0.792251i	$-0.708909\pi$
$877$	−36.1409	−1.22039	−0.610195	+	0.792251i	$0.708909\pi$
$878$	0	0
$879$	6.71781	0.226586
$880$	0	0
$881$	17.8174	0.600285	0.300142	−	0.953894i	$-0.402966\pi$
$881$	17.8174	0.600285	0.300142	+	0.953894i	$0.402966\pi$
$882$	0	0
$883$	11.6997	0.393725	0.196862	−	0.980431i	$-0.436925\pi$
$883$	11.6997	0.393725	0.196862	+	0.980431i	$0.436925\pi$
$884$	0	0
$885$	2.96236	0.0995787
$886$	0	0
$887$	13.9846	0.469556	0.234778	−	0.972049i	$-0.424564\pi$
$887$	13.9846	0.469556	0.234778	+	0.972049i	$0.424564\pi$
$888$	0	0
$889$	−4.63861	−0.155574
$890$	0	0
$891$	−0.0139460	−0.000467210 0
$892$	0	0
$893$	12.6931	0.424760
$894$	0	0
$895$	−2.55477	−0.0853966
$896$	0	0
$897$	4.24250	0.141653
$898$	0	0
$899$	−8.54651	−0.285042
$900$	0	0
$901$	−42.2387	−1.40718
$902$	0	0
$903$	−13.1340	−0.437072
$904$	0	0
$905$	9.42862	0.313418
$906$	0	0
$907$	39.5347	1.31273	0.656364	−	0.754444i	$-0.272093\pi$
$907$	39.5347	1.31273	0.656364	+	0.754444i	$0.272093\pi$
$908$	0	0
$909$	15.9680	0.529625
$910$	0	0
$911$	6.57391	0.217803	0.108902	−	0.994053i	$-0.465267\pi$
$911$	6.57391	0.217803	0.108902	+	0.994053i	$0.465267\pi$
$912$	0	0
$913$	1.86936	0.0618669
$914$	0	0
$915$	3.14325	0.103913
$916$	0	0
$917$	−8.82475	−0.291419
$918$	0	0
$919$	12.9743	0.427984	0.213992	−	0.976835i	$-0.431353\pi$
$919$	12.9743	0.427984	0.213992	+	0.976835i	$0.431353\pi$
$920$	0	0
$921$	18.5947	0.612715
$922$	0	0
$923$	−6.79647	−0.223708
$924$	0	0
$925$	23.4393	0.770678
$926$	0	0
$927$	8.23321	0.270414
$928$	0	0
$929$	−33.5705	−1.10141	−0.550707	−	0.834699i	$-0.685642\pi$
$929$	−33.5705	−1.10141	−0.550707	+	0.834699i	$0.685642\pi$
$930$	0	0
$931$	1.51254	0.0495716
$932$	0	0
$933$	−23.7302	−0.776893
$934$	0	0
$935$	−2.85408	−0.0933385
$936$	0	0
$937$	−24.5776	−0.802915	−0.401457	−	0.915878i	$-0.631496\pi$
$937$	−24.5776	−0.802915	−0.401457	+	0.915878i	$0.631496\pi$
$938$	0	0
$939$	7.50815	0.245019
$940$	0	0
$941$	24.9785	0.814276	0.407138	−	0.913367i	$-0.366527\pi$
$941$	24.9785	0.814276	0.407138	+	0.913367i	$0.366527\pi$
$942$	0	0
$943$	9.65894	0.314538
$944$	0	0
$945$	1.98207	0.0644769
$946$	0	0
$947$	−29.4794	−0.957951	−0.478975	−	0.877828i	$-0.658992\pi$
$947$	−29.4794	−0.957951	−0.478975	+	0.877828i	$0.658992\pi$
$948$	0	0
$949$	−11.3501	−0.368438
$950$	0	0
$951$	3.99563	0.129567
$952$	0	0
$953$	−32.5745	−1.05519	−0.527595	−	0.849496i	$-0.676906\pi$
$953$	−32.5745	−1.05519	−0.527595	+	0.849496i	$0.676906\pi$
$954$	0	0
$955$	−0.211823	−0.00685442
$956$	0	0
$957$	−2.52547	−0.0816369
$958$	0	0
$959$	7.94651	0.256606
$960$	0	0
$961$	−17.9006	−0.577440
$962$	0	0
$963$	7.71792	0.248707
$964$	0	0
$965$	6.67053	0.214732
$966$	0	0
$967$	−37.8259	−1.21640	−0.608200	−	0.793784i	$-0.708108\pi$
$967$	−37.8259	−1.21640	−0.608200	+	0.793784i	$0.708108\pi$
$968$	0	0
$969$	12.0978	0.388638
$970$	0	0
$971$	−58.5838	−1.88004	−0.940022	−	0.341113i	$-0.889196\pi$
$971$	−58.5838	−1.88004	−0.940022	+	0.341113i	$0.889196\pi$
$972$	0	0
$973$	−7.95980	−0.255180
$974$	0	0
$975$	5.19171	0.166268
$976$	0	0
$977$	34.2495	1.09574	0.547869	−	0.836564i	$-0.315439\pi$
$977$	34.2495	1.09574	0.547869	+	0.836564i	$0.315439\pi$
$978$	0	0
$979$	−4.79957	−0.153395
$980$	0	0
$981$	26.5143	0.846537
$982$	0	0
$983$	−19.3487	−0.617126	−0.308563	−	0.951204i	$-0.599848\pi$
$983$	−19.3487	−0.617126	−0.308563	+	0.951204i	$0.599848\pi$
$984$	0	0
$985$	0.173067	0.00551437
$986$	0	0
$987$	−8.97513	−0.285681
$988$	0	0
$989$	−48.7148	−1.54904
$990$	0	0
$991$	−10.7964	−0.342958	−0.171479	−	0.985188i	$-0.554855\pi$
$991$	−10.7964	−0.342958	−0.171479	+	0.985188i	$0.554855\pi$
$992$	0	0
$993$	−9.99806	−0.317279
$994$	0	0
$995$	1.12592	0.0356940
$996$	0	0
$997$	−39.1398	−1.23957	−0.619785	−	0.784771i	$-0.712780\pi$
$997$	−39.1398	−1.23957	−0.619785	+	0.784771i	$0.712780\pi$
$998$	0	0
$999$	25.0776	0.793420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.r.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.r.1.5	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.06950 q^{3} +0.381633 q^{5} -1.00000 q^{7} -1.85618 q^{9} +O(q^{10})\)	\(q+1.06950 q^{3} +0.381633 q^{5} -1.00000 q^{7} -1.85618 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.408155 q^{15} +7.47860 q^{17} +1.51254 q^{19} -1.06950 q^{21} -3.96682 q^{23} -4.85436 q^{25} -5.19366 q^{27} +2.36137 q^{29} -3.61930 q^{31} -1.06950 q^{33} -0.381633 q^{35} -4.82850 q^{37} -1.06950 q^{39} -2.43493 q^{41} +12.2806 q^{43} -0.708380 q^{45} +8.39193 q^{47} +1.00000 q^{49} +7.99833 q^{51} -5.64794 q^{53} -0.381633 q^{55} +1.61766 q^{57} +7.25793 q^{59} +7.70112 q^{61} +1.85618 q^{63} -0.381633 q^{65} +8.61532 q^{67} -4.24250 q^{69} +6.79647 q^{71} +11.3501 q^{73} -5.19171 q^{75} +1.00000 q^{77} +2.58414 q^{79} +0.0139460 q^{81} -1.86936 q^{83} +2.85408 q^{85} +2.52547 q^{87} +4.79957 q^{89} +1.00000 q^{91} -3.87083 q^{93} +0.577236 q^{95} +11.1990 q^{97} +1.85618 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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