LMFDB - Embedded newform 8008.2.a.l.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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 24x^4 - 10x^3 - 18x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.20852$ 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)

$f(q)$	$=$	$q+0.425730 q^{3} +2.69163 q^{5} +1.00000 q^{7} -2.81875 q^{9} +O(q^{10})$	$q+0.425730 q^{3} +2.69163 q^{5} +1.00000 q^{7} -2.81875 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.14591 q^{15} -3.40165 q^{17} -6.46699 q^{19} +0.425730 q^{21} +1.01391 q^{23} +2.24487 q^{25} -2.47722 q^{27} -4.96288 q^{29} +4.33115 q^{31} +0.425730 q^{33} +2.69163 q^{35} -8.50935 q^{37} +0.425730 q^{39} +8.23368 q^{41} -9.54987 q^{43} -7.58704 q^{45} -0.151885 q^{47} +1.00000 q^{49} -1.44818 q^{51} +0.201775 q^{53} +2.69163 q^{55} -2.75319 q^{57} -0.501293 q^{59} -13.5811 q^{61} -2.81875 q^{63} +2.69163 q^{65} -13.8501 q^{67} +0.431652 q^{69} +6.45440 q^{71} -6.05652 q^{73} +0.955707 q^{75} +1.00000 q^{77} +7.84192 q^{79} +7.40164 q^{81} -12.2613 q^{83} -9.15599 q^{85} -2.11285 q^{87} +17.3175 q^{89} +1.00000 q^{91} +1.84390 q^{93} -17.4067 q^{95} +5.62520 q^{97} -2.81875 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * 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
$3$	0.425730	0.245795	0.122898	−	0.992419i	$-0.460781\pi$
$3$	0.425730	0.245795	0.122898	+	0.992419i	$0.460781\pi$
$4$	0	0
$5$	2.69163	1.20373	0.601867	−	0.798597i	$-0.294424\pi$
$5$	2.69163	1.20373	0.601867	+	0.798597i	$0.294424\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.81875	−0.939585
$10$	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
$15$	1.14591	0.295872
$16$	0	0
$17$	−3.40165	−0.825022	−0.412511	−	0.910953i	$-0.635348\pi$
$17$	−3.40165	−0.825022	−0.412511	+	0.910953i	$0.635348\pi$
$18$	0	0
$19$	−6.46699	−1.48363	−0.741815	−	0.670605i	$-0.766035\pi$
$19$	−6.46699	−1.48363	−0.741815	+	0.670605i	$0.766035\pi$
$20$	0	0
$21$	0.425730	0.0929018
$22$	0	0
$23$	1.01391	0.211415	0.105707	−	0.994397i	$-0.466289\pi$
$23$	1.01391	0.211415	0.105707	+	0.994397i	$0.466289\pi$
$24$	0	0
$25$	2.24487	0.448974
$26$	0	0
$27$	−2.47722	−0.476740
$28$	0	0
$29$	−4.96288	−0.921584	−0.460792	−	0.887508i	$-0.652435\pi$
$29$	−4.96288	−0.921584	−0.460792	+	0.887508i	$0.652435\pi$
$30$	0	0
$31$	4.33115	0.777897	0.388949	−	0.921259i	$-0.372838\pi$
$31$	4.33115	0.777897	0.388949	+	0.921259i	$0.372838\pi$
$32$	0	0
$33$	0.425730	0.0741100
$34$	0	0
$35$	2.69163	0.454968
$36$	0	0
$37$	−8.50935	−1.39893	−0.699464	−	0.714668i	$-0.746578\pi$
$37$	−8.50935	−1.39893	−0.699464	+	0.714668i	$0.746578\pi$
$38$	0	0
$39$	0.425730	0.0681713
$40$	0	0
$41$	8.23368	1.28588	0.642942	−	0.765915i	$-0.277713\pi$
$41$	8.23368	1.28588	0.642942	+	0.765915i	$0.277713\pi$
$42$	0	0
$43$	−9.54987	−1.45634	−0.728171	−	0.685396i	$-0.759629\pi$
$43$	−9.54987	−1.45634	−0.728171	+	0.685396i	$0.759629\pi$
$44$	0	0
$45$	−7.58704	−1.13101
$46$	0	0
$47$	−0.151885	−0.0221547	−0.0110773	−	0.999939i	$-0.503526\pi$
$47$	−0.151885	−0.0221547	−0.0110773	+	0.999939i	$0.503526\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.44818	−0.202786
$52$	0	0
$53$	0.201775	0.0277159	0.0138579	−	0.999904i	$-0.495589\pi$
$53$	0.201775	0.0277159	0.0138579	+	0.999904i	$0.495589\pi$
$54$	0	0
$55$	2.69163	0.362939
$56$	0	0
$57$	−2.75319	−0.364669
$58$	0	0
$59$	−0.501293	−0.0652628	−0.0326314	−	0.999467i	$-0.510389\pi$
$59$	−0.501293	−0.0652628	−0.0326314	+	0.999467i	$0.510389\pi$
$60$	0	0
$61$	−13.5811	−1.73889	−0.869443	−	0.494034i	$-0.835522\pi$
$61$	−13.5811	−1.73889	−0.869443	+	0.494034i	$0.835522\pi$
$62$	0	0
$63$	−2.81875	−0.355130
$64$	0	0
$65$	2.69163	0.333856
$66$	0	0
$67$	−13.8501	−1.69206	−0.846031	−	0.533133i	$-0.821015\pi$
$67$	−13.8501	−1.69206	−0.846031	+	0.533133i	$0.821015\pi$
$68$	0	0
$69$	0.431652	0.0519648
$70$	0	0
$71$	6.45440	0.765996	0.382998	−	0.923749i	$-0.374892\pi$
$71$	6.45440	0.765996	0.382998	+	0.923749i	$0.374892\pi$
$72$	0	0
$73$	−6.05652	−0.708862	−0.354431	−	0.935082i	$-0.615325\pi$
$73$	−6.05652	−0.708862	−0.354431	+	0.935082i	$0.615325\pi$
$74$	0	0
$75$	0.955707	0.110356
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.84192	0.882285	0.441142	−	0.897437i	$-0.354573\pi$
$79$	7.84192	0.882285	0.441142	+	0.897437i	$0.354573\pi$
$80$	0	0
$81$	7.40164	0.822404
$82$	0	0
$83$	−12.2613	−1.34585	−0.672924	−	0.739712i	$-0.734962\pi$
$83$	−12.2613	−1.34585	−0.672924	+	0.739712i	$0.734962\pi$
$84$	0	0
$85$	−9.15599	−0.993106
$86$	0	0
$87$	−2.11285	−0.226521
$88$	0	0
$89$	17.3175	1.83565	0.917824	−	0.396988i	$-0.129944\pi$
$89$	17.3175	1.83565	0.917824	+	0.396988i	$0.129944\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.84390	0.191203
$94$	0	0
$95$	−17.4067	−1.78589
$96$	0	0
$97$	5.62520	0.571152	0.285576	−	0.958356i	$-0.407815\pi$
$97$	5.62520	0.571152	0.285576	+	0.958356i	$0.407815\pi$
$98$	0	0
$99$	−2.81875	−0.283295
$100$	0	0
$101$	3.60565	0.358775	0.179388	−	0.983778i	$-0.442588\pi$
$101$	3.60565	0.358775	0.179388	+	0.983778i	$0.442588\pi$
$102$	0	0
$103$	−15.8512	−1.56186	−0.780930	−	0.624618i	$-0.785255\pi$
$103$	−15.8512	−1.56186	−0.780930	+	0.624618i	$0.785255\pi$
$104$	0	0
$105$	1.14591	0.111829
$106$	0	0
$107$	2.64124	0.255338	0.127669	−	0.991817i	$-0.459250\pi$
$107$	2.64124	0.255338	0.127669	+	0.991817i	$0.459250\pi$
$108$	0	0
$109$	−8.18079	−0.783577	−0.391789	−	0.920055i	$-0.628144\pi$
$109$	−8.18079	−0.783577	−0.391789	+	0.920055i	$0.628144\pi$
$110$	0	0
$111$	−3.62268	−0.343850
$112$	0	0
$113$	4.56119	0.429081	0.214540	−	0.976715i	$-0.431175\pi$
$113$	4.56119	0.429081	0.214540	+	0.976715i	$0.431175\pi$
$114$	0	0
$115$	2.72907	0.254487
$116$	0	0
$117$	−2.81875	−0.260594
$118$	0	0
$119$	−3.40165	−0.311829
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.50532	0.316064
$124$	0	0
$125$	−7.41579	−0.663289
$126$	0	0
$127$	−1.12365	−0.0997081	−0.0498541	−	0.998757i	$-0.515876\pi$
$127$	−1.12365	−0.0997081	−0.0498541	+	0.998757i	$0.515876\pi$
$128$	0	0
$129$	−4.06566	−0.357961
$130$	0	0
$131$	−16.1156	−1.40803	−0.704015	−	0.710185i	$-0.748611\pi$
$131$	−16.1156	−1.40803	−0.704015	+	0.710185i	$0.748611\pi$
$132$	0	0
$133$	−6.46699	−0.560759
$134$	0	0
$135$	−6.66775	−0.573868
$136$	0	0
$137$	−18.1208	−1.54816	−0.774080	−	0.633087i	$-0.781787\pi$
$137$	−18.1208	−1.54816	−0.774080	+	0.633087i	$0.781787\pi$
$138$	0	0
$139$	12.2435	1.03848	0.519240	−	0.854628i	$-0.326215\pi$
$139$	12.2435	1.03848	0.519240	+	0.854628i	$0.326215\pi$
$140$	0	0
$141$	−0.0646619	−0.00544551
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−13.3582	−1.10934
$146$	0	0
$147$	0.425730	0.0351136
$148$	0	0
$149$	−14.2339	−1.16609	−0.583044	−	0.812441i	$-0.698138\pi$
$149$	−14.2339	−1.16609	−0.583044	+	0.812441i	$0.698138\pi$
$150$	0	0
$151$	−12.1025	−0.984890	−0.492445	−	0.870344i	$-0.663897\pi$
$151$	−12.1025	−0.984890	−0.492445	+	0.870344i	$0.663897\pi$
$152$	0	0
$153$	9.58842	0.775178
$154$	0	0
$155$	11.6578	0.936381
$156$	0	0
$157$	−3.97611	−0.317328	−0.158664	−	0.987333i	$-0.550719\pi$
$157$	−3.97611	−0.317328	−0.158664	+	0.987333i	$0.550719\pi$
$158$	0	0
$159$	0.0859014	0.00681243
$160$	0	0
$161$	1.01391	0.0799073
$162$	0	0
$163$	−3.18314	−0.249323	−0.124661	−	0.992199i	$-0.539784\pi$
$163$	−3.18314	−0.249323	−0.124661	+	0.992199i	$0.539784\pi$
$164$	0	0
$165$	1.14591	0.0892087
$166$	0	0
$167$	−3.92114	−0.303427	−0.151714	−	0.988425i	$-0.548479\pi$
$167$	−3.92114	−0.303427	−0.151714	+	0.988425i	$0.548479\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.2289	1.39400
$172$	0	0
$173$	2.99166	0.227452	0.113726	−	0.993512i	$-0.463721\pi$
$173$	2.99166	0.227452	0.113726	+	0.993512i	$0.463721\pi$
$174$	0	0
$175$	2.24487	0.169696
$176$	0	0
$177$	−0.213415	−0.0160413
$178$	0	0
$179$	14.3524	1.07275	0.536376	−	0.843979i	$-0.319793\pi$
$179$	14.3524	1.07275	0.536376	+	0.843979i	$0.319793\pi$
$180$	0	0
$181$	21.9900	1.63450	0.817251	−	0.576281i	$-0.195497\pi$
$181$	21.9900	1.63450	0.817251	+	0.576281i	$0.195497\pi$
$182$	0	0
$183$	−5.78189	−0.427410
$184$	0	0
$185$	−22.9040	−1.68394
$186$	0	0
$187$	−3.40165	−0.248753
$188$	0	0
$189$	−2.47722	−0.180191
$190$	0	0
$191$	2.33211	0.168745	0.0843727	−	0.996434i	$-0.473111\pi$
$191$	2.33211	0.168745	0.0843727	+	0.996434i	$0.473111\pi$
$192$	0	0
$193$	15.3017	1.10144	0.550720	−	0.834690i	$-0.314353\pi$
$193$	15.3017	1.10144	0.550720	+	0.834690i	$0.314353\pi$
$194$	0	0
$195$	1.14591	0.0820600
$196$	0	0
$197$	13.5681	0.966686	0.483343	−	0.875431i	$-0.339422\pi$
$197$	13.5681	0.966686	0.483343	+	0.875431i	$0.339422\pi$
$198$	0	0
$199$	−13.1216	−0.930163	−0.465081	−	0.885268i	$-0.653975\pi$
$199$	−13.1216	−0.930163	−0.465081	+	0.885268i	$0.653975\pi$
$200$	0	0
$201$	−5.89641	−0.415901
$202$	0	0
$203$	−4.96288	−0.348326
$204$	0	0
$205$	22.1620	1.54786
$206$	0	0
$207$	−2.85796	−0.198642
$208$	0	0
$209$	−6.46699	−0.447331
$210$	0	0
$211$	13.1706	0.906703	0.453351	−	0.891332i	$-0.350228\pi$
$211$	13.1706	0.906703	0.453351	+	0.891332i	$0.350228\pi$
$212$	0	0
$213$	2.74783	0.188278
$214$	0	0
$215$	−25.7047	−1.75305
$216$	0	0
$217$	4.33115	0.294018
$218$	0	0
$219$	−2.57844	−0.174235
$220$	0	0
$221$	−3.40165	−0.228820
$222$	0	0
$223$	−15.4505	−1.03464	−0.517320	−	0.855792i	$-0.673070\pi$
$223$	−15.4505	−1.03464	−0.517320	+	0.855792i	$0.673070\pi$
$224$	0	0
$225$	−6.32774	−0.421849
$226$	0	0
$227$	11.0454	0.733106	0.366553	−	0.930397i	$-0.380538\pi$
$227$	11.0454	0.733106	0.366553	+	0.930397i	$0.380538\pi$
$228$	0	0
$229$	−5.49385	−0.363044	−0.181522	−	0.983387i	$-0.558102\pi$
$229$	−5.49385	−0.363044	−0.181522	+	0.983387i	$0.558102\pi$
$230$	0	0
$231$	0.425730	0.0280109
$232$	0	0
$233$	−25.9316	−1.69884	−0.849418	−	0.527720i	$-0.823047\pi$
$233$	−25.9316	−1.69884	−0.849418	+	0.527720i	$0.823047\pi$
$234$	0	0
$235$	−0.408818	−0.0266683
$236$	0	0
$237$	3.33854	0.216861
$238$	0	0
$239$	−13.2093	−0.854440	−0.427220	−	0.904148i	$-0.640507\pi$
$239$	−13.2093	−0.854440	−0.427220	+	0.904148i	$0.640507\pi$
$240$	0	0
$241$	26.5275	1.70878	0.854392	−	0.519629i	$-0.173930\pi$
$241$	26.5275	1.70878	0.854392	+	0.519629i	$0.173930\pi$
$242$	0	0
$243$	10.5827	0.678883
$244$	0	0
$245$	2.69163	0.171962
$246$	0	0
$247$	−6.46699	−0.411485
$248$	0	0
$249$	−5.21998	−0.330803
$250$	0	0
$251$	−23.6058	−1.48999	−0.744994	−	0.667071i	$-0.767548\pi$
$251$	−23.6058	−1.48999	−0.744994	+	0.667071i	$0.767548\pi$
$252$	0	0
$253$	1.01391	0.0637440
$254$	0	0
$255$	−3.89797	−0.244101
$256$	0	0
$257$	27.0220	1.68559	0.842794	−	0.538237i	$-0.180909\pi$
$257$	27.0220	1.68559	0.842794	+	0.538237i	$0.180909\pi$
$258$	0	0
$259$	−8.50935	−0.528745
$260$	0	0
$261$	13.9891	0.865906
$262$	0	0
$263$	−11.8495	−0.730673	−0.365336	−	0.930876i	$-0.619046\pi$
$263$	−11.8495	−0.730673	−0.365336	+	0.930876i	$0.619046\pi$
$264$	0	0
$265$	0.543103	0.0333625
$266$	0	0
$267$	7.37256	0.451193
$268$	0	0
$269$	26.0162	1.58623	0.793117	−	0.609069i	$-0.208457\pi$
$269$	26.0162	1.58623	0.793117	+	0.609069i	$0.208457\pi$
$270$	0	0
$271$	25.4819	1.54792	0.773958	−	0.633237i	$-0.218274\pi$
$271$	25.4819	1.54792	0.773958	+	0.633237i	$0.218274\pi$
$272$	0	0
$273$	0.425730	0.0257663
$274$	0	0
$275$	2.24487	0.135371
$276$	0	0
$277$	−1.33004	−0.0799142	−0.0399571	−	0.999201i	$-0.512722\pi$
$277$	−1.33004	−0.0799142	−0.0399571	+	0.999201i	$0.512722\pi$
$278$	0	0
$279$	−12.2084	−0.730900
$280$	0	0
$281$	17.2202	1.02727	0.513636	−	0.858008i	$-0.328298\pi$
$281$	17.2202	1.02727	0.513636	+	0.858008i	$0.328298\pi$
$282$	0	0
$283$	3.04258	0.180863	0.0904314	−	0.995903i	$-0.471175\pi$
$283$	3.04258	0.180863	0.0904314	+	0.995903i	$0.471175\pi$
$284$	0	0
$285$	−7.41057	−0.438964
$286$	0	0
$287$	8.23368	0.486019
$288$	0	0
$289$	−5.42876	−0.319339
$290$	0	0
$291$	2.39481	0.140386
$292$	0	0
$293$	−5.86083	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$293$	−5.86083	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$294$	0	0
$295$	−1.34930	−0.0785590
$296$	0	0
$297$	−2.47722	−0.143743
$298$	0	0
$299$	1.01391	0.0586360
$300$	0	0
$301$	−9.54987	−0.550445
$302$	0	0
$303$	1.53503	0.0881852
$304$	0	0
$305$	−36.5554	−2.09315
$306$	0	0
$307$	−28.9905	−1.65458	−0.827288	−	0.561778i	$-0.810118\pi$
$307$	−28.9905	−1.65458	−0.827288	+	0.561778i	$0.810118\pi$
$308$	0	0
$309$	−6.74831	−0.383898
$310$	0	0
$311$	9.57713	0.543069	0.271535	−	0.962429i	$-0.412469\pi$
$311$	9.57713	0.543069	0.271535	+	0.962429i	$0.412469\pi$
$312$	0	0
$313$	−0.160075	−0.00904797	−0.00452398	−	0.999990i	$-0.501440\pi$
$313$	−0.160075	−0.00904797	−0.00452398	+	0.999990i	$0.501440\pi$
$314$	0	0
$315$	−7.58704	−0.427481
$316$	0	0
$317$	−13.3427	−0.749401	−0.374701	−	0.927146i	$-0.622255\pi$
$317$	−13.3427	−0.749401	−0.374701	+	0.927146i	$0.622255\pi$
$318$	0	0
$319$	−4.96288	−0.277868
$320$	0	0
$321$	1.12445	0.0627609
$322$	0	0
$323$	21.9985	1.22403
$324$	0	0
$325$	2.24487	0.124523
$326$	0	0
$327$	−3.48280	−0.192599
$328$	0	0
$329$	−0.151885	−0.00837369
$330$	0	0
$331$	−9.40817	−0.517120	−0.258560	−	0.965995i	$-0.583248\pi$
$331$	−9.40817	−0.517120	−0.258560	+	0.965995i	$0.583248\pi$
$332$	0	0
$333$	23.9858	1.31441
$334$	0	0
$335$	−37.2794	−2.03679
$336$	0	0
$337$	14.0075	0.763038	0.381519	−	0.924361i	$-0.375401\pi$
$337$	14.0075	0.763038	0.381519	+	0.924361i	$0.375401\pi$
$338$	0	0
$339$	1.94183	0.105466
$340$	0	0
$341$	4.33115	0.234545
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.16185	0.0625517
$346$	0	0
$347$	−8.36270	−0.448933	−0.224467	−	0.974482i	$-0.572064\pi$
$347$	−8.36270	−0.448933	−0.224467	+	0.974482i	$0.572064\pi$
$348$	0	0
$349$	−5.87392	−0.314424	−0.157212	−	0.987565i	$-0.550251\pi$
$349$	−5.87392	−0.314424	−0.157212	+	0.987565i	$0.550251\pi$
$350$	0	0
$351$	−2.47722	−0.132224
$352$	0	0
$353$	14.4027	0.766579	0.383290	−	0.923628i	$-0.374791\pi$
$353$	14.4027	0.766579	0.383290	+	0.923628i	$0.374791\pi$
$354$	0	0
$355$	17.3729	0.922055
$356$	0	0
$357$	−1.44818	−0.0766460
$358$	0	0
$359$	25.2654	1.33346	0.666728	−	0.745301i	$-0.267694\pi$
$359$	25.2654	1.33346	0.666728	+	0.745301i	$0.267694\pi$
$360$	0	0
$361$	22.8220	1.20116
$362$	0	0
$363$	0.425730	0.0223450
$364$	0	0
$365$	−16.3019	−0.853281
$366$	0	0
$367$	−11.2483	−0.587158	−0.293579	−	0.955935i	$-0.594846\pi$
$367$	−11.2483	−0.587158	−0.293579	+	0.955935i	$0.594846\pi$
$368$	0	0
$369$	−23.2087	−1.20820
$370$	0	0
$371$	0.201775	0.0104756
$372$	0	0
$373$	−1.52952	−0.0791953	−0.0395977	−	0.999216i	$-0.512608\pi$
$373$	−1.52952	−0.0791953	−0.0395977	+	0.999216i	$0.512608\pi$
$374$	0	0
$375$	−3.15712	−0.163033
$376$	0	0
$377$	−4.96288	−0.255601
$378$	0	0
$379$	−18.9744	−0.974651	−0.487326	−	0.873220i	$-0.662027\pi$
$379$	−18.9744	−0.974651	−0.487326	+	0.873220i	$0.662027\pi$
$380$	0	0
$381$	−0.478373	−0.0245078
$382$	0	0
$383$	9.66876	0.494051	0.247025	−	0.969009i	$-0.420547\pi$
$383$	9.66876	0.494051	0.247025	+	0.969009i	$0.420547\pi$
$384$	0	0
$385$	2.69163	0.137178
$386$	0	0
$387$	26.9187	1.36836
$388$	0	0
$389$	−17.5901	−0.891852	−0.445926	−	0.895070i	$-0.647126\pi$
$389$	−17.5901	−0.891852	−0.445926	+	0.895070i	$0.647126\pi$
$390$	0	0
$391$	−3.44897	−0.174422
$392$	0	0
$393$	−6.86091	−0.346087
$394$	0	0
$395$	21.1075	1.06204
$396$	0	0
$397$	−2.92599	−0.146851	−0.0734255	−	0.997301i	$-0.523393\pi$
$397$	−2.92599	−0.146851	−0.0734255	+	0.997301i	$0.523393\pi$
$398$	0	0
$399$	−2.75319	−0.137832
$400$	0	0
$401$	13.3880	0.668567	0.334283	−	0.942473i	$-0.391506\pi$
$401$	13.3880	0.668567	0.334283	+	0.942473i	$0.391506\pi$
$402$	0	0
$403$	4.33115	0.215750
$404$	0	0
$405$	19.9225	0.989955
$406$	0	0
$407$	−8.50935	−0.421793
$408$	0	0
$409$	−7.49657	−0.370681	−0.185341	−	0.982674i	$-0.559339\pi$
$409$	−7.49657	−0.370681	−0.185341	+	0.982674i	$0.559339\pi$
$410$	0	0
$411$	−7.71454	−0.380530
$412$	0	0
$413$	−0.501293	−0.0246670
$414$	0	0
$415$	−33.0028	−1.62004
$416$	0	0
$417$	5.21242	0.255253
$418$	0	0
$419$	−12.8405	−0.627300	−0.313650	−	0.949539i	$-0.601552\pi$
$419$	−12.8405	−0.627300	−0.313650	+	0.949539i	$0.601552\pi$
$420$	0	0
$421$	18.7097	0.911854	0.455927	−	0.890017i	$-0.349308\pi$
$421$	18.7097	0.911854	0.455927	+	0.890017i	$0.349308\pi$
$422$	0	0
$423$	0.428126	0.0208162
$424$	0	0
$425$	−7.63626	−0.370413
$426$	0	0
$427$	−13.5811	−0.657237
$428$	0	0
$429$	0.425730	0.0205544
$430$	0	0
$431$	−13.4006	−0.645482	−0.322741	−	0.946487i	$-0.604604\pi$
$431$	−13.4006	−0.645482	−0.322741	+	0.946487i	$0.604604\pi$
$432$	0	0
$433$	33.3000	1.60030	0.800148	−	0.599802i	$-0.204754\pi$
$433$	33.3000	1.60030	0.800148	+	0.599802i	$0.204754\pi$
$434$	0	0
$435$	−5.68700	−0.272671
$436$	0	0
$437$	−6.55695	−0.313661
$438$	0	0
$439$	−35.5307	−1.69579	−0.847894	−	0.530166i	$-0.822130\pi$
$439$	−35.5307	−1.69579	−0.847894	+	0.530166i	$0.822130\pi$
$440$	0	0
$441$	−2.81875	−0.134226
$442$	0	0
$443$	7.92872	0.376705	0.188352	−	0.982101i	$-0.439685\pi$
$443$	7.92872	0.376705	0.188352	+	0.982101i	$0.439685\pi$
$444$	0	0
$445$	46.6122	2.20963
$446$	0	0
$447$	−6.05979	−0.286618
$448$	0	0
$449$	−29.0647	−1.37165	−0.685823	−	0.727768i	$-0.740558\pi$
$449$	−29.0647	−1.37165	−0.685823	+	0.727768i	$0.740558\pi$
$450$	0	0
$451$	8.23368	0.387709
$452$	0	0
$453$	−5.15241	−0.242081
$454$	0	0
$455$	2.69163	0.126186
$456$	0	0
$457$	−6.11293	−0.285951	−0.142975	−	0.989726i	$-0.545667\pi$
$457$	−6.11293	−0.285951	−0.142975	+	0.989726i	$0.545667\pi$
$458$	0	0
$459$	8.42663	0.393321
$460$	0	0
$461$	20.5919	0.959060	0.479530	−	0.877525i	$-0.340807\pi$
$461$	20.5919	0.959060	0.479530	+	0.877525i	$0.340807\pi$
$462$	0	0
$463$	−9.75312	−0.453266	−0.226633	−	0.973980i	$-0.572772\pi$
$463$	−9.75312	−0.453266	−0.226633	+	0.973980i	$0.572772\pi$
$464$	0	0
$465$	4.96309	0.230158
$466$	0	0
$467$	12.1156	0.560642	0.280321	−	0.959906i	$-0.409559\pi$
$467$	12.1156	0.560642	0.280321	+	0.959906i	$0.409559\pi$
$468$	0	0
$469$	−13.8501	−0.639540
$470$	0	0
$471$	−1.69275	−0.0779977
$472$	0	0
$473$	−9.54987	−0.439103
$474$	0	0
$475$	−14.5176	−0.666111
$476$	0	0
$477$	−0.568753	−0.0260414
$478$	0	0
$479$	−6.87439	−0.314099	−0.157050	−	0.987591i	$-0.550198\pi$
$479$	−6.87439	−0.314099	−0.157050	+	0.987591i	$0.550198\pi$
$480$	0	0
$481$	−8.50935	−0.387993
$482$	0	0
$483$	0.431652	0.0196408
$484$	0	0
$485$	15.1409	0.687515
$486$	0	0
$487$	−1.44629	−0.0655379	−0.0327689	−	0.999463i	$-0.510433\pi$
$487$	−1.44629	−0.0655379	−0.0327689	+	0.999463i	$0.510433\pi$
$488$	0	0
$489$	−1.35516	−0.0612823
$490$	0	0
$491$	19.9376	0.899770	0.449885	−	0.893087i	$-0.351465\pi$
$491$	19.9376	0.899770	0.449885	+	0.893087i	$0.351465\pi$
$492$	0	0
$493$	16.8820	0.760327
$494$	0	0
$495$	−7.58704	−0.341012
$496$	0	0
$497$	6.45440	0.289519
$498$	0	0
$499$	−6.70733	−0.300261	−0.150131	−	0.988666i	$-0.547969\pi$
$499$	−6.70733	−0.300261	−0.150131	+	0.988666i	$0.547969\pi$
$500$	0	0
$501$	−1.66935	−0.0745809
$502$	0	0
$503$	−27.6634	−1.23345	−0.616726	−	0.787178i	$-0.711541\pi$
$503$	−27.6634	−1.23345	−0.616726	+	0.787178i	$0.711541\pi$
$504$	0	0
$505$	9.70507	0.431870
$506$	0	0
$507$	0.425730	0.0189073
$508$	0	0
$509$	−12.7919	−0.566992	−0.283496	−	0.958973i	$-0.591494\pi$
$509$	−12.7919	−0.566992	−0.283496	+	0.958973i	$0.591494\pi$
$510$	0	0
$511$	−6.05652	−0.267925
$512$	0	0
$513$	16.0201	0.707306
$514$	0	0
$515$	−42.6654	−1.88006
$516$	0	0
$517$	−0.151885	−0.00667989
$518$	0	0
$519$	1.27364	0.0559065
$520$	0	0
$521$	22.8624	1.00162	0.500810	−	0.865557i	$-0.333036\pi$
$521$	22.8624	1.00162	0.500810	+	0.865557i	$0.333036\pi$
$522$	0	0
$523$	−36.9767	−1.61688	−0.808439	−	0.588580i	$-0.799687\pi$
$523$	−36.9767	−1.61688	−0.808439	+	0.588580i	$0.799687\pi$
$524$	0	0
$525$	0.955707	0.0417105
$526$	0	0
$527$	−14.7331	−0.641782
$528$	0	0
$529$	−21.9720	−0.955304
$530$	0	0
$531$	1.41302	0.0613200
$532$	0	0
$533$	8.23368	0.356640
$534$	0	0
$535$	7.10923	0.307359
$536$	0	0
$537$	6.11026	0.263677
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−46.2520	−1.98853	−0.994264	−	0.106952i	$-0.965891\pi$
$541$	−46.2520	−1.98853	−0.994264	+	0.106952i	$0.965891\pi$
$542$	0	0
$543$	9.36179	0.401753
$544$	0	0
$545$	−22.0196	−0.943218
$546$	0	0
$547$	33.1710	1.41829	0.709146	−	0.705062i	$-0.249081\pi$
$547$	33.1710	1.41829	0.709146	+	0.705062i	$0.249081\pi$
$548$	0	0
$549$	38.2819	1.63383
$550$	0	0
$551$	32.0949	1.36729
$552$	0	0
$553$	7.84192	0.333472
$554$	0	0
$555$	−9.75091	−0.413903
$556$	0	0
$557$	−10.1961	−0.432022	−0.216011	−	0.976391i	$-0.569305\pi$
$557$	−10.1961	−0.432022	−0.216011	+	0.976391i	$0.569305\pi$
$558$	0	0
$559$	−9.54987	−0.403916
$560$	0	0
$561$	−1.44818	−0.0611424
$562$	0	0
$563$	0.698443	0.0294359	0.0147179	−	0.999892i	$-0.495315\pi$
$563$	0.698443	0.0294359	0.0147179	+	0.999892i	$0.495315\pi$
$564$	0	0
$565$	12.2770	0.516499
$566$	0	0
$567$	7.40164	0.310840
$568$	0	0
$569$	9.10551	0.381723	0.190861	−	0.981617i	$-0.438872\pi$
$569$	9.10551	0.381723	0.190861	+	0.981617i	$0.438872\pi$
$570$	0	0
$571$	−8.82178	−0.369180	−0.184590	−	0.982816i	$-0.559096\pi$
$571$	−8.82178	−0.369180	−0.184590	+	0.982816i	$0.559096\pi$
$572$	0	0
$573$	0.992847	0.0414768
$574$	0	0
$575$	2.27610	0.0949198
$576$	0	0
$577$	46.7707	1.94709	0.973544	−	0.228500i	$-0.0733821\pi$
$577$	46.7707	1.94709	0.973544	+	0.228500i	$0.0733821\pi$
$578$	0	0
$579$	6.51438	0.270729
$580$	0	0
$581$	−12.2613	−0.508683
$582$	0	0
$583$	0.201775	0.00835665
$584$	0	0
$585$	−7.58704	−0.313686
$586$	0	0
$587$	18.4409	0.761138	0.380569	−	0.924753i	$-0.375728\pi$
$587$	18.4409	0.761138	0.380569	+	0.924753i	$0.375728\pi$
$588$	0	0
$589$	−28.0095	−1.15411
$590$	0	0
$591$	5.77634	0.237607
$592$	0	0
$593$	−1.71774	−0.0705391	−0.0352695	−	0.999378i	$-0.511229\pi$
$593$	−1.71774	−0.0705391	−0.0352695	+	0.999378i	$0.511229\pi$
$594$	0	0
$595$	−9.15599	−0.375359
$596$	0	0
$597$	−5.58624	−0.228629
$598$	0	0
$599$	39.6847	1.62147	0.810736	−	0.585412i	$-0.199067\pi$
$599$	39.6847	1.62147	0.810736	+	0.585412i	$0.199067\pi$
$600$	0	0
$601$	37.9499	1.54801	0.774004	−	0.633180i	$-0.218251\pi$
$601$	37.9499	1.54801	0.774004	+	0.633180i	$0.218251\pi$
$602$	0	0
$603$	39.0401	1.58984
$604$	0	0
$605$	2.69163	0.109430
$606$	0	0
$607$	11.5000	0.466770	0.233385	−	0.972384i	$-0.425020\pi$
$607$	11.5000	0.466770	0.233385	+	0.972384i	$0.425020\pi$
$608$	0	0
$609$	−2.11285	−0.0856168
$610$	0	0
$611$	−0.151885	−0.00614461
$612$	0	0
$613$	−11.6110	−0.468965	−0.234483	−	0.972120i	$-0.575340\pi$
$613$	−11.6110	−0.468965	−0.234483	+	0.972120i	$0.575340\pi$
$614$	0	0
$615$	9.43503	0.380457
$616$	0	0
$617$	−30.5105	−1.22831	−0.614153	−	0.789187i	$-0.710502\pi$
$617$	−30.5105	−1.22831	−0.614153	+	0.789187i	$0.710502\pi$
$618$	0	0
$619$	−7.47562	−0.300470	−0.150235	−	0.988650i	$-0.548003\pi$
$619$	−7.47562	−0.300470	−0.150235	+	0.988650i	$0.548003\pi$
$620$	0	0
$621$	−2.51167	−0.100790
$622$	0	0
$623$	17.3175	0.693810
$624$	0	0
$625$	−31.1849	−1.24740
$626$	0	0
$627$	−2.75319	−0.109952
$628$	0	0
$629$	28.9458	1.15415
$630$	0	0
$631$	6.39646	0.254639	0.127319	−	0.991862i	$-0.459363\pi$
$631$	6.39646	0.254639	0.127319	+	0.991862i	$0.459363\pi$
$632$	0	0
$633$	5.60712	0.222863
$634$	0	0
$635$	−3.02446	−0.120022
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−18.1934	−0.719718
$640$	0	0
$641$	19.0755	0.753436	0.376718	−	0.926328i	$-0.377053\pi$
$641$	19.0755	0.753436	0.376718	+	0.926328i	$0.377053\pi$
$642$	0	0
$643$	35.8924	1.41546	0.707728	−	0.706485i	$-0.249720\pi$
$643$	35.8924	1.41546	0.707728	+	0.706485i	$0.249720\pi$
$644$	0	0
$645$	−10.9433	−0.430890
$646$	0	0
$647$	−43.6975	−1.71793	−0.858963	−	0.512037i	$-0.828891\pi$
$647$	−43.6975	−1.71793	−0.858963	+	0.512037i	$0.828891\pi$
$648$	0	0
$649$	−0.501293	−0.0196775
$650$	0	0
$651$	1.84390	0.0722681
$652$	0	0
$653$	13.0917	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$653$	13.0917	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$654$	0	0
$655$	−43.3773	−1.69489
$656$	0	0
$657$	17.0718	0.666036
$658$	0	0
$659$	−44.3874	−1.72909	−0.864544	−	0.502557i	$-0.832393\pi$
$659$	−44.3874	−1.72909	−0.864544	+	0.502557i	$0.832393\pi$
$660$	0	0
$661$	5.41866	0.210761	0.105381	−	0.994432i	$-0.466394\pi$
$661$	5.41866	0.210761	0.105381	+	0.994432i	$0.466394\pi$
$662$	0	0
$663$	−1.44818	−0.0562428
$664$	0	0
$665$	−17.4067	−0.675005
$666$	0	0
$667$	−5.03192	−0.194837
$668$	0	0
$669$	−6.57772	−0.254309
$670$	0	0
$671$	−13.5811	−0.524294
$672$	0	0
$673$	−8.12896	−0.313348	−0.156674	−	0.987650i	$-0.550077\pi$
$673$	−8.12896	−0.313348	−0.156674	+	0.987650i	$0.550077\pi$
$674$	0	0
$675$	−5.56103	−0.214044
$676$	0	0
$677$	31.5105	1.21105	0.605524	−	0.795827i	$-0.292963\pi$
$677$	31.5105	1.21105	0.605524	+	0.795827i	$0.292963\pi$
$678$	0	0
$679$	5.62520	0.215875
$680$	0	0
$681$	4.70234	0.180194
$682$	0	0
$683$	7.34236	0.280948	0.140474	−	0.990084i	$-0.455137\pi$
$683$	7.34236	0.280948	0.140474	+	0.990084i	$0.455137\pi$
$684$	0	0
$685$	−48.7744	−1.86357
$686$	0	0
$687$	−2.33890	−0.0892344
$688$	0	0
$689$	0.201775	0.00768700
$690$	0	0
$691$	−10.8230	−0.411728	−0.205864	−	0.978581i	$-0.566000\pi$
$691$	−10.8230	−0.411728	−0.205864	+	0.978581i	$0.566000\pi$
$692$	0	0
$693$	−2.81875	−0.107076
$694$	0	0
$695$	32.9550	1.25005
$696$	0	0
$697$	−28.0081	−1.06088
$698$	0	0
$699$	−11.0399	−0.417566
$700$	0	0
$701$	19.9165	0.752236	0.376118	−	0.926572i	$-0.377259\pi$
$701$	19.9165	0.752236	0.376118	+	0.926572i	$0.377259\pi$
$702$	0	0
$703$	55.0299	2.07549
$704$	0	0
$705$	−0.174046	−0.00655495
$706$	0	0
$707$	3.60565	0.135604
$708$	0	0
$709$	−32.5187	−1.22126	−0.610632	−	0.791914i	$-0.709085\pi$
$709$	−32.5187	−1.22126	−0.610632	+	0.791914i	$0.709085\pi$
$710$	0	0
$711$	−22.1044	−0.828981
$712$	0	0
$713$	4.39140	0.164459
$714$	0	0
$715$	2.69163	0.100661
$716$	0	0
$717$	−5.62360	−0.210017
$718$	0	0
$719$	7.81940	0.291614	0.145807	−	0.989313i	$-0.453422\pi$
$719$	7.81940	0.291614	0.145807	+	0.989313i	$0.453422\pi$
$720$	0	0
$721$	−15.8512	−0.590328
$722$	0	0
$723$	11.2935	0.420011
$724$	0	0
$725$	−11.1410	−0.413767
$726$	0	0
$727$	38.3647	1.42287	0.711434	−	0.702753i	$-0.248046\pi$
$727$	38.3647	1.42287	0.711434	+	0.702753i	$0.248046\pi$
$728$	0	0
$729$	−17.6995	−0.655538
$730$	0	0
$731$	32.4853	1.20151
$732$	0	0
$733$	−34.9747	−1.29182	−0.645909	−	0.763414i	$-0.723522\pi$
$733$	−34.9747	−1.29182	−0.645909	+	0.763414i	$0.723522\pi$
$734$	0	0
$735$	1.14591	0.0422674
$736$	0	0
$737$	−13.8501	−0.510176
$738$	0	0
$739$	−41.2887	−1.51883	−0.759414	−	0.650608i	$-0.774514\pi$
$739$	−41.2887	−1.51883	−0.759414	+	0.650608i	$0.774514\pi$
$740$	0	0
$741$	−2.75319	−0.101141
$742$	0	0
$743$	−4.82663	−0.177072	−0.0885359	−	0.996073i	$-0.528219\pi$
$743$	−4.82663	−0.177072	−0.0885359	+	0.996073i	$0.528219\pi$
$744$	0	0
$745$	−38.3124	−1.40366
$746$	0	0
$747$	34.5615	1.26454
$748$	0	0
$749$	2.64124	0.0965087
$750$	0	0
$751$	12.1130	0.442010	0.221005	−	0.975273i	$-0.429066\pi$
$751$	12.1130	0.442010	0.221005	+	0.975273i	$0.429066\pi$
$752$	0	0
$753$	−10.0497	−0.366232
$754$	0	0
$755$	−32.5755	−1.18555
$756$	0	0
$757$	14.4253	0.524296	0.262148	−	0.965028i	$-0.415569\pi$
$757$	14.4253	0.524296	0.262148	+	0.965028i	$0.415569\pi$
$758$	0	0
$759$	0.431652	0.0156680
$760$	0	0
$761$	−11.9909	−0.434669	−0.217334	−	0.976097i	$-0.569736\pi$
$761$	−11.9909	−0.434669	−0.217334	+	0.976097i	$0.569736\pi$
$762$	0	0
$763$	−8.18079	−0.296164
$764$	0	0
$765$	25.8085	0.933107
$766$	0	0
$767$	−0.501293	−0.0181007
$768$	0	0
$769$	−32.6229	−1.17641	−0.588207	−	0.808711i	$-0.700166\pi$
$769$	−32.6229	−1.17641	−0.588207	+	0.808711i	$0.700166\pi$
$770$	0	0
$771$	11.5041	0.414309
$772$	0	0
$773$	−13.4810	−0.484878	−0.242439	−	0.970167i	$-0.577947\pi$
$773$	−13.4810	−0.484878	−0.242439	+	0.970167i	$0.577947\pi$
$774$	0	0
$775$	9.72286	0.349256
$776$	0	0
$777$	−3.62268	−0.129963
$778$	0	0
$779$	−53.2471	−1.90778
$780$	0	0
$781$	6.45440	0.230957
$782$	0	0
$783$	12.2941	0.439356
$784$	0	0
$785$	−10.7022	−0.381979
$786$	0	0
$787$	8.40855	0.299732	0.149866	−	0.988706i	$-0.452116\pi$
$787$	8.40855	0.299732	0.149866	+	0.988706i	$0.452116\pi$
$788$	0	0
$789$	−5.04469	−0.179596
$790$	0	0
$791$	4.56119	0.162177
$792$	0	0
$793$	−13.5811	−0.482280
$794$	0	0
$795$	0.231215	0.00820034
$796$	0	0
$797$	10.8705	0.385055	0.192527	−	0.981292i	$-0.438332\pi$
$797$	10.8705	0.385055	0.192527	+	0.981292i	$0.438332\pi$
$798$	0	0
$799$	0.516660	0.0182781
$800$	0	0
$801$	−48.8137	−1.72475
$802$	0	0
$803$	−6.05652	−0.213730
$804$	0	0
$805$	2.72907	0.0961871
$806$	0	0
$807$	11.0759	0.389889
$808$	0	0
$809$	19.8144	0.696636	0.348318	−	0.937376i	$-0.386753\pi$
$809$	19.8144	0.696636	0.348318	+	0.937376i	$0.386753\pi$
$810$	0	0
$811$	−47.1288	−1.65492	−0.827459	−	0.561526i	$-0.810214\pi$
$811$	−47.1288	−1.65492	−0.827459	+	0.561526i	$0.810214\pi$
$812$	0	0
$813$	10.8484	0.380470
$814$	0	0
$815$	−8.56783	−0.300118
$816$	0	0
$817$	61.7589	2.16067
$818$	0	0
$819$	−2.81875	−0.0984952
$820$	0	0
$821$	34.6816	1.21039	0.605197	−	0.796075i	$-0.293094\pi$
$821$	34.6816	1.21039	0.605197	+	0.796075i	$0.293094\pi$
$822$	0	0
$823$	−10.3738	−0.361608	−0.180804	−	0.983519i	$-0.557870\pi$
$823$	−10.3738	−0.361608	−0.180804	+	0.983519i	$0.557870\pi$
$824$	0	0
$825$	0.955707	0.0332735
$826$	0	0
$827$	−25.6016	−0.890254	−0.445127	−	0.895467i	$-0.646842\pi$
$827$	−25.6016	−0.890254	−0.445127	+	0.895467i	$0.646842\pi$
$828$	0	0
$829$	6.21541	0.215870	0.107935	−	0.994158i	$-0.465576\pi$
$829$	6.21541	0.215870	0.107935	+	0.994158i	$0.465576\pi$
$830$	0	0
$831$	−0.566237	−0.0196425
$832$	0	0
$833$	−3.40165	−0.117860
$834$	0	0
$835$	−10.5543	−0.365245
$836$	0	0
$837$	−10.7292	−0.370855
$838$	0	0
$839$	−22.7163	−0.784253	−0.392126	−	0.919911i	$-0.628260\pi$
$839$	−22.7163	−0.784253	−0.392126	+	0.919911i	$0.628260\pi$
$840$	0	0
$841$	−4.36980	−0.150683
$842$	0	0
$843$	7.33115	0.252498
$844$	0	0
$845$	2.69163	0.0925949
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	1.29532	0.0444552
$850$	0	0
$851$	−8.62771	−0.295754
$852$	0	0
$853$	−8.76949	−0.300262	−0.150131	−	0.988666i	$-0.547969\pi$
$853$	−8.76949	−0.300262	−0.150131	+	0.988666i	$0.547969\pi$
$854$	0	0
$855$	49.0653	1.67800
$856$	0	0
$857$	26.0564	0.890069	0.445035	−	0.895513i	$-0.353191\pi$
$857$	26.0564	0.890069	0.445035	+	0.895513i	$0.353191\pi$
$858$	0	0
$859$	−0.757066	−0.0258308	−0.0129154	−	0.999917i	$-0.504111\pi$
$859$	−0.757066	−0.0258308	−0.0129154	+	0.999917i	$0.504111\pi$
$860$	0	0
$861$	3.50532	0.119461
$862$	0	0
$863$	−10.4396	−0.355368	−0.177684	−	0.984088i	$-0.556860\pi$
$863$	−10.4396	−0.355368	−0.177684	+	0.984088i	$0.556860\pi$
$864$	0	0
$865$	8.05244	0.273791
$866$	0	0
$867$	−2.31119	−0.0784920
$868$	0	0
$869$	7.84192	0.266019
$870$	0	0
$871$	−13.8501	−0.469294
$872$	0	0
$873$	−15.8561	−0.536646
$874$	0	0
$875$	−7.41579	−0.250699
$876$	0	0
$877$	7.21091	0.243495	0.121748	−	0.992561i	$-0.461150\pi$
$877$	7.21091	0.243495	0.121748	+	0.992561i	$0.461150\pi$
$878$	0	0
$879$	−2.49513	−0.0841586
$880$	0	0
$881$	−32.5200	−1.09563	−0.547813	−	0.836601i	$-0.684539\pi$
$881$	−32.5200	−1.09563	−0.547813	+	0.836601i	$0.684539\pi$
$882$	0	0
$883$	−43.1943	−1.45360	−0.726801	−	0.686848i	$-0.758994\pi$
$883$	−43.1943	−1.45360	−0.726801	+	0.686848i	$0.758994\pi$
$884$	0	0
$885$	−0.574435	−0.0193094
$886$	0	0
$887$	−32.7618	−1.10003	−0.550016	−	0.835154i	$-0.685379\pi$
$887$	−32.7618	−1.10003	−0.550016	+	0.835154i	$0.685379\pi$
$888$	0	0
$889$	−1.12365	−0.0376861
$890$	0	0
$891$	7.40164	0.247964
$892$	0	0
$893$	0.982238	0.0328694
$894$	0	0
$895$	38.6315	1.29131
$896$	0	0
$897$	0.431652	0.0144124
$898$	0	0
$899$	−21.4950	−0.716898
$900$	0	0
$901$	−0.686367	−0.0228662
$902$	0	0
$903$	−4.06566	−0.135297
$904$	0	0
$905$	59.1889	1.96751
$906$	0	0
$907$	25.3170	0.840636	0.420318	−	0.907377i	$-0.361919\pi$
$907$	25.3170	0.840636	0.420318	+	0.907377i	$0.361919\pi$
$908$	0	0
$909$	−10.1634	−0.337100
$910$	0	0
$911$	4.88414	0.161819	0.0809093	−	0.996721i	$-0.474218\pi$
$911$	4.88414	0.161819	0.0809093	+	0.996721i	$0.474218\pi$
$912$	0	0
$913$	−12.2613	−0.405788
$914$	0	0
$915$	−15.5627	−0.514487
$916$	0	0
$917$	−16.1156	−0.532185
$918$	0	0
$919$	28.6244	0.944232	0.472116	−	0.881536i	$-0.343490\pi$
$919$	28.6244	0.944232	0.472116	+	0.881536i	$0.343490\pi$
$920$	0	0
$921$	−12.3421	−0.406687
$922$	0	0
$923$	6.45440	0.212449
$924$	0	0
$925$	−19.1024	−0.628082
$926$	0	0
$927$	44.6805	1.46750
$928$	0	0
$929$	−24.0289	−0.788364	−0.394182	−	0.919032i	$-0.628972\pi$
$929$	−24.0289	−0.788364	−0.394182	+	0.919032i	$0.628972\pi$
$930$	0	0
$931$	−6.46699	−0.211947
$932$	0	0
$933$	4.07727	0.133484
$934$	0	0
$935$	−9.15599	−0.299433
$936$	0	0
$937$	−48.0192	−1.56872	−0.784360	−	0.620306i	$-0.787008\pi$
$937$	−48.0192	−1.56872	−0.784360	+	0.620306i	$0.787008\pi$
$938$	0	0
$939$	−0.0681486	−0.00222395
$940$	0	0
$941$	21.0351	0.685724	0.342862	−	0.939386i	$-0.388604\pi$
$941$	21.0351	0.685724	0.342862	+	0.939386i	$0.388604\pi$
$942$	0	0
$943$	8.34821	0.271855
$944$	0	0
$945$	−6.66775	−0.216902
$946$	0	0
$947$	−2.38693	−0.0775649	−0.0387824	−	0.999248i	$-0.512348\pi$
$947$	−2.38693	−0.0775649	−0.0387824	+	0.999248i	$0.512348\pi$
$948$	0	0
$949$	−6.05652	−0.196603
$950$	0	0
$951$	−5.68039	−0.184199
$952$	0	0
$953$	−55.3782	−1.79388	−0.896939	−	0.442155i	$-0.854214\pi$
$953$	−55.3782	−1.79388	−0.896939	+	0.442155i	$0.854214\pi$
$954$	0	0
$955$	6.27717	0.203124
$956$	0	0
$957$	−2.11285	−0.0682986
$958$	0	0
$959$	−18.1208	−0.585150
$960$	0	0
$961$	−12.2412	−0.394876
$962$	0	0
$963$	−7.44500	−0.239912
$964$	0	0
$965$	41.1865	1.32584
$966$	0	0
$967$	5.86547	0.188621	0.0943105	−	0.995543i	$-0.469935\pi$
$967$	5.86547	0.188621	0.0943105	+	0.995543i	$0.469935\pi$
$968$	0	0
$969$	9.36539	0.300860
$970$	0	0
$971$	4.20741	0.135022	0.0675111	−	0.997719i	$-0.478494\pi$
$971$	4.20741	0.135022	0.0675111	+	0.997719i	$0.478494\pi$
$972$	0	0
$973$	12.2435	0.392509
$974$	0	0
$975$	0.955707	0.0306071
$976$	0	0
$977$	−40.6208	−1.29958	−0.649788	−	0.760116i	$-0.725142\pi$
$977$	−40.6208	−1.29958	−0.649788	+	0.760116i	$0.725142\pi$
$978$	0	0
$979$	17.3175	0.553469
$980$	0	0
$981$	23.0596	0.736237
$982$	0	0
$983$	12.4200	0.396136	0.198068	−	0.980188i	$-0.436533\pi$
$983$	12.4200	0.396136	0.198068	+	0.980188i	$0.436533\pi$
$984$	0	0
$985$	36.5203	1.16363
$986$	0	0
$987$	−0.0646619	−0.00205821
$988$	0	0
$989$	−9.68271	−0.307892
$990$	0	0
$991$	−3.34045	−0.106113	−0.0530564	−	0.998592i	$-0.516896\pi$
$991$	−3.34045	−0.106113	−0.0530564	+	0.998592i	$0.516896\pi$
$992$	0	0
$993$	−4.00534	−0.127105
$994$	0	0
$995$	−35.3184	−1.11967
$996$	0	0
$997$	−9.03347	−0.286093	−0.143046	−	0.989716i	$-0.545690\pi$
$997$	−9.03347	−0.286093	−0.143046	+	0.989716i	$0.545690\pi$
$998$	0	0
$999$	21.0795	0.666925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.l.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.l.1.6	✓	8	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)

\(f(q)\)	\(=\)	\(q+0.425730 q^{3} +2.69163 q^{5} +1.00000 q^{7} -2.81875 q^{9} +O(q^{10})\)	\(q+0.425730 q^{3} +2.69163 q^{5} +1.00000 q^{7} -2.81875 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.14591 q^{15} -3.40165 q^{17} -6.46699 q^{19} +0.425730 q^{21} +1.01391 q^{23} +2.24487 q^{25} -2.47722 q^{27} -4.96288 q^{29} +4.33115 q^{31} +0.425730 q^{33} +2.69163 q^{35} -8.50935 q^{37} +0.425730 q^{39} +8.23368 q^{41} -9.54987 q^{43} -7.58704 q^{45} -0.151885 q^{47} +1.00000 q^{49} -1.44818 q^{51} +0.201775 q^{53} +2.69163 q^{55} -2.75319 q^{57} -0.501293 q^{59} -13.5811 q^{61} -2.81875 q^{63} +2.69163 q^{65} -13.8501 q^{67} +0.431652 q^{69} +6.45440 q^{71} -6.05652 q^{73} +0.955707 q^{75} +1.00000 q^{77} +7.84192 q^{79} +7.40164 q^{81} -12.2613 q^{83} -9.15599 q^{85} -2.11285 q^{87} +17.3175 q^{89} +1.00000 q^{91} +1.84390 q^{93} -17.4067 q^{95} +5.62520 q^{97} -2.81875 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Introduction

L-functions

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Varieties

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Representations

Groups

Database

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