LMFDB - Embedded newform 6008.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.962699 q^{3} +2.73078 q^{5} -4.42659 q^{7} -2.07321 q^{9} +O(q^{10})$	$q-0.962699 q^{3} +2.73078 q^{5} -4.42659 q^{7} -2.07321 q^{9} -2.59005 q^{11} +2.44381 q^{13} -2.62892 q^{15} +0.359015 q^{17} +4.11543 q^{19} +4.26147 q^{21} +0.0487888 q^{23} +2.45715 q^{25} +4.88398 q^{27} +1.62661 q^{29} -1.82266 q^{31} +2.49344 q^{33} -12.0880 q^{35} +7.04333 q^{37} -2.35265 q^{39} -2.33305 q^{41} -0.598973 q^{43} -5.66148 q^{45} +12.9710 q^{47} +12.5947 q^{49} -0.345624 q^{51} +3.71721 q^{53} -7.07287 q^{55} -3.96192 q^{57} +0.779197 q^{59} -12.1921 q^{61} +9.17724 q^{63} +6.67350 q^{65} -5.58624 q^{67} -0.0469689 q^{69} -14.3849 q^{71} +10.0286 q^{73} -2.36550 q^{75} +11.4651 q^{77} -5.23078 q^{79} +1.51783 q^{81} -11.6419 q^{83} +0.980391 q^{85} -1.56594 q^{87} +13.4868 q^{89} -10.8177 q^{91} +1.75468 q^{93} +11.2383 q^{95} -6.27432 q^{97} +5.36973 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.962699	−0.555815	−0.277907	−	0.960608i	$-0.589641\pi$
$3$	−0.962699	−0.555815	−0.277907	+	0.960608i	$0.589641\pi$
$4$	0	0
$5$	2.73078	1.22124	0.610621	−	0.791923i	$-0.290920\pi$
$5$	2.73078	1.22124	0.610621	+	0.791923i	$0.290920\pi$
$6$	0	0
$7$	−4.42659	−1.67309	−0.836546	−	0.547896i	$-0.815429\pi$
$7$	−4.42659	−1.67309	−0.836546	+	0.547896i	$0.815429\pi$
$8$	0	0
$9$	−2.07321	−0.691070
$10$	0	0
$11$	−2.59005	−0.780931	−0.390465	−	0.920618i	$-0.627686\pi$
$11$	−2.59005	−0.780931	−0.390465	+	0.920618i	$0.627686\pi$
$12$	0	0
$13$	2.44381	0.677791	0.338895	−	0.940824i	$-0.389947\pi$
$13$	2.44381	0.677791	0.338895	+	0.940824i	$0.389947\pi$
$14$	0	0
$15$	−2.62892	−0.678784
$16$	0	0
$17$	0.359015	0.0870740	0.0435370	−	0.999052i	$-0.486137\pi$
$17$	0.359015	0.0870740	0.0435370	+	0.999052i	$0.486137\pi$
$18$	0	0
$19$	4.11543	0.944145	0.472072	−	0.881560i	$-0.343506\pi$
$19$	4.11543	0.944145	0.472072	+	0.881560i	$0.343506\pi$
$20$	0	0
$21$	4.26147	0.929930
$22$	0	0
$23$	0.0487888	0.0101732	0.00508658	−	0.999987i	$-0.498381\pi$
$23$	0.0487888	0.0101732	0.00508658	+	0.999987i	$0.498381\pi$
$24$	0	0
$25$	2.45715	0.491431
$26$	0	0
$27$	4.88398	0.939922
$28$	0	0
$29$	1.62661	0.302054	0.151027	−	0.988530i	$-0.451742\pi$
$29$	1.62661	0.302054	0.151027	+	0.988530i	$0.451742\pi$
$30$	0	0
$31$	−1.82266	−0.327360	−0.163680	−	0.986513i	$-0.552336\pi$
$31$	−1.82266	−0.327360	−0.163680	+	0.986513i	$0.552336\pi$
$32$	0	0
$33$	2.49344	0.434053
$34$	0	0
$35$	−12.0880	−2.04325
$36$	0	0
$37$	7.04333	1.15792	0.578958	−	0.815357i	$-0.303459\pi$
$37$	7.04333	1.15792	0.578958	+	0.815357i	$0.303459\pi$
$38$	0	0
$39$	−2.35265	−0.376726
$40$	0	0
$41$	−2.33305	−0.364361	−0.182181	−	0.983265i	$-0.558316\pi$
$41$	−2.33305	−0.364361	−0.182181	+	0.983265i	$0.558316\pi$
$42$	0	0
$43$	−0.598973	−0.0913425	−0.0456712	−	0.998957i	$-0.514543\pi$
$43$	−0.598973	−0.0913425	−0.0456712	+	0.998957i	$0.514543\pi$
$44$	0	0
$45$	−5.66148	−0.843963
$46$	0	0
$47$	12.9710	1.89202	0.946010	−	0.324136i	$-0.105074\pi$
$47$	12.9710	1.89202	0.946010	+	0.324136i	$0.105074\pi$
$48$	0	0
$49$	12.5947	1.79924
$50$	0	0
$51$	−0.345624	−0.0483970
$52$	0	0
$53$	3.71721	0.510598	0.255299	−	0.966862i	$-0.417826\pi$
$53$	3.71721	0.510598	0.255299	+	0.966862i	$0.417826\pi$
$54$	0	0
$55$	−7.07287	−0.953705
$56$	0	0
$57$	−3.96192	−0.524770
$58$	0	0
$59$	0.779197	0.101443	0.0507214	−	0.998713i	$-0.483848\pi$
$59$	0.779197	0.101443	0.0507214	+	0.998713i	$0.483848\pi$
$60$	0	0
$61$	−12.1921	−1.56104	−0.780522	−	0.625128i	$-0.785047\pi$
$61$	−12.1921	−1.56104	−0.780522	+	0.625128i	$0.785047\pi$
$62$	0	0
$63$	9.17724	1.15622
$64$	0	0
$65$	6.67350	0.827746
$66$	0	0
$67$	−5.58624	−0.682468	−0.341234	−	0.939978i	$-0.610845\pi$
$67$	−5.58624	−0.682468	−0.341234	+	0.939978i	$0.610845\pi$
$68$	0	0
$69$	−0.0469689	−0.00565439
$70$	0	0
$71$	−14.3849	−1.70717	−0.853585	−	0.520953i	$-0.825577\pi$
$71$	−14.3849	−1.70717	−0.853585	+	0.520953i	$0.825577\pi$
$72$	0	0
$73$	10.0286	1.17376	0.586881	−	0.809673i	$-0.300356\pi$
$73$	10.0286	1.17376	0.586881	+	0.809673i	$0.300356\pi$
$74$	0	0
$75$	−2.36550	−0.273144
$76$	0	0
$77$	11.4651	1.30657
$78$	0	0
$79$	−5.23078	−0.588509	−0.294254	−	0.955727i	$-0.595071\pi$
$79$	−5.23078	−0.588509	−0.294254	+	0.955727i	$0.595071\pi$
$80$	0	0
$81$	1.51783	0.168647
$82$	0	0
$83$	−11.6419	−1.27786	−0.638930	−	0.769265i	$-0.720622\pi$
$83$	−11.6419	−1.27786	−0.638930	+	0.769265i	$0.720622\pi$
$84$	0	0
$85$	0.980391	0.106338
$86$	0	0
$87$	−1.56594	−0.167886
$88$	0	0
$89$	13.4868	1.42959	0.714797	−	0.699332i	$-0.246519\pi$
$89$	13.4868	1.42959	0.714797	+	0.699332i	$0.246519\pi$
$90$	0	0
$91$	−10.8177	−1.13401
$92$	0	0
$93$	1.75468	0.181951
$94$	0	0
$95$	11.2383	1.15303
$96$	0	0
$97$	−6.27432	−0.637061	−0.318530	−	0.947913i	$-0.603189\pi$
$97$	−6.27432	−0.637061	−0.318530	+	0.947913i	$0.603189\pi$
$98$	0	0
$99$	5.36973	0.539678
$100$	0	0
$101$	−5.38530	−0.535858	−0.267929	−	0.963439i	$-0.586339\pi$
$101$	−5.38530	−0.535858	−0.267929	+	0.963439i	$0.586339\pi$
$102$	0	0
$103$	−13.9396	−1.37351	−0.686754	−	0.726890i	$-0.740965\pi$
$103$	−13.9396	−1.37351	−0.686754	+	0.726890i	$0.740965\pi$
$104$	0	0
$105$	11.6371	1.13567
$106$	0	0
$107$	−6.61587	−0.639581	−0.319790	−	0.947488i	$-0.603612\pi$
$107$	−6.61587	−0.639581	−0.319790	+	0.947488i	$0.603612\pi$
$108$	0	0
$109$	−11.7060	−1.12123	−0.560614	−	0.828077i	$-0.689435\pi$
$109$	−11.7060	−1.12123	−0.560614	+	0.828077i	$0.689435\pi$
$110$	0	0
$111$	−6.78061	−0.643587
$112$	0	0
$113$	−16.8266	−1.58291	−0.791455	−	0.611228i	$-0.790676\pi$
$113$	−16.8266	−1.58291	−0.791455	+	0.611228i	$0.790676\pi$
$114$	0	0
$115$	0.133231	0.0124239
$116$	0	0
$117$	−5.06653	−0.468401
$118$	0	0
$119$	−1.58921	−0.145683
$120$	0	0
$121$	−4.29162	−0.390147
$122$	0	0
$123$	2.24603	0.202517
$124$	0	0
$125$	−6.94395	−0.621086
$126$	0	0
$127$	−10.3619	−0.919466	−0.459733	−	0.888057i	$-0.652055\pi$
$127$	−10.3619	−0.919466	−0.459733	+	0.888057i	$0.652055\pi$
$128$	0	0
$129$	0.576631	0.0507695
$130$	0	0
$131$	−10.7168	−0.936327	−0.468163	−	0.883642i	$-0.655084\pi$
$131$	−10.7168	−0.936327	−0.468163	+	0.883642i	$0.655084\pi$
$132$	0	0
$133$	−18.2173	−1.57964
$134$	0	0
$135$	13.3371	1.14787
$136$	0	0
$137$	8.26324	0.705976	0.352988	−	0.935628i	$-0.385166\pi$
$137$	8.26324	0.705976	0.352988	+	0.935628i	$0.385166\pi$
$138$	0	0
$139$	−3.86367	−0.327712	−0.163856	−	0.986484i	$-0.552393\pi$
$139$	−3.86367	−0.327712	−0.163856	+	0.986484i	$0.552393\pi$
$140$	0	0
$141$	−12.4872	−1.05161
$142$	0	0
$143$	−6.32960	−0.529308
$144$	0	0
$145$	4.44192	0.368881
$146$	0	0
$147$	−12.1249	−1.00004
$148$	0	0
$149$	20.5012	1.67952	0.839761	−	0.542957i	$-0.182695\pi$
$149$	20.5012	1.67952	0.839761	+	0.542957i	$0.182695\pi$
$150$	0	0
$151$	5.70679	0.464412	0.232206	−	0.972667i	$-0.425406\pi$
$151$	5.70679	0.464412	0.232206	+	0.972667i	$0.425406\pi$
$152$	0	0
$153$	−0.744314	−0.0601742
$154$	0	0
$155$	−4.97729	−0.399785
$156$	0	0
$157$	9.28446	0.740980	0.370490	−	0.928836i	$-0.379190\pi$
$157$	9.28446	0.740980	0.370490	+	0.928836i	$0.379190\pi$
$158$	0	0
$159$	−3.57856	−0.283798
$160$	0	0
$161$	−0.215968	−0.0170206
$162$	0	0
$163$	−0.709508	−0.0555729	−0.0277865	−	0.999614i	$-0.508846\pi$
$163$	−0.709508	−0.0555729	−0.0277865	+	0.999614i	$0.508846\pi$
$164$	0	0
$165$	6.80905	0.530083
$166$	0	0
$167$	1.30141	0.100706	0.0503531	−	0.998731i	$-0.483965\pi$
$167$	1.30141	0.100706	0.0503531	+	0.998731i	$0.483965\pi$
$168$	0	0
$169$	−7.02780	−0.540600
$170$	0	0
$171$	−8.53215	−0.652470
$172$	0	0
$173$	−14.5407	−1.10551	−0.552753	−	0.833345i	$-0.686423\pi$
$173$	−14.5407	−1.10551	−0.552753	+	0.833345i	$0.686423\pi$
$174$	0	0
$175$	−10.8768	−0.822209
$176$	0	0
$177$	−0.750132	−0.0563834
$178$	0	0
$179$	−25.4508	−1.90228	−0.951139	−	0.308762i	$-0.900085\pi$
$179$	−25.4508	−1.90228	−0.951139	+	0.308762i	$0.900085\pi$
$180$	0	0
$181$	−18.1748	−1.35092	−0.675461	−	0.737396i	$-0.736055\pi$
$181$	−18.1748	−1.35092	−0.675461	+	0.737396i	$0.736055\pi$
$182$	0	0
$183$	11.7374	0.867652
$184$	0	0
$185$	19.2338	1.41410
$186$	0	0
$187$	−0.929869	−0.0679988
$188$	0	0
$189$	−21.6193	−1.57258
$190$	0	0
$191$	−18.9402	−1.37047	−0.685233	−	0.728324i	$-0.740300\pi$
$191$	−18.9402	−1.37047	−0.685233	+	0.728324i	$0.740300\pi$
$192$	0	0
$193$	−3.96192	−0.285185	−0.142592	−	0.989781i	$-0.545544\pi$
$193$	−3.96192	−0.285185	−0.142592	+	0.989781i	$0.545544\pi$
$194$	0	0
$195$	−6.42458	−0.460073
$196$	0	0
$197$	−1.44725	−0.103112	−0.0515562	−	0.998670i	$-0.516418\pi$
$197$	−1.44725	−0.103112	−0.0515562	+	0.998670i	$0.516418\pi$
$198$	0	0
$199$	11.3615	0.805393	0.402696	−	0.915334i	$-0.368073\pi$
$199$	11.3615	0.805393	0.402696	+	0.915334i	$0.368073\pi$
$200$	0	0
$201$	5.37787	0.379326
$202$	0	0
$203$	−7.20034	−0.505365
$204$	0	0
$205$	−6.37104	−0.444973
$206$	0	0
$207$	−0.101149	−0.00703036
$208$	0	0
$209$	−10.6592	−0.737312
$210$	0	0
$211$	−1.52224	−0.104796	−0.0523978	−	0.998626i	$-0.516686\pi$
$211$	−1.52224	−0.104796	−0.0523978	+	0.998626i	$0.516686\pi$
$212$	0	0
$213$	13.8483	0.948871
$214$	0	0
$215$	−1.63566	−0.111551
$216$	0	0
$217$	8.06817	0.547703
$218$	0	0
$219$	−9.65455	−0.652394
$220$	0	0
$221$	0.877365	0.0590179
$222$	0	0
$223$	−23.3186	−1.56153	−0.780763	−	0.624827i	$-0.785169\pi$
$223$	−23.3186	−1.56153	−0.780763	+	0.624827i	$0.785169\pi$
$224$	0	0
$225$	−5.09419	−0.339613
$226$	0	0
$227$	−1.70296	−0.113030	−0.0565148	−	0.998402i	$-0.517999\pi$
$227$	−1.70296	−0.113030	−0.0565148	+	0.998402i	$0.517999\pi$
$228$	0	0
$229$	19.9024	1.31519	0.657593	−	0.753374i	$-0.271575\pi$
$229$	19.9024	1.31519	0.657593	+	0.753374i	$0.271575\pi$
$230$	0	0
$231$	−11.0374	−0.726211
$232$	0	0
$233$	17.7608	1.16355	0.581774	−	0.813351i	$-0.302359\pi$
$233$	17.7608	1.16355	0.581774	+	0.813351i	$0.302359\pi$
$234$	0	0
$235$	35.4210	2.31061
$236$	0	0
$237$	5.03567	0.327102
$238$	0	0
$239$	5.89213	0.381130	0.190565	−	0.981675i	$-0.438968\pi$
$239$	5.89213	0.381130	0.190565	+	0.981675i	$0.438968\pi$
$240$	0	0
$241$	28.6483	1.84540	0.922698	−	0.385523i	$-0.125979\pi$
$241$	28.6483	1.84540	0.922698	+	0.385523i	$0.125979\pi$
$242$	0	0
$243$	−16.1131	−1.03366
$244$	0	0
$245$	34.3932	2.19730
$246$	0	0
$247$	10.0573	0.639932
$248$	0	0
$249$	11.2076	0.710253
$250$	0	0
$251$	11.9917	0.756908	0.378454	−	0.925620i	$-0.376456\pi$
$251$	11.9917	0.756908	0.378454	+	0.925620i	$0.376456\pi$
$252$	0	0
$253$	−0.126366	−0.00794454
$254$	0	0
$255$	−0.943822	−0.0591044
$256$	0	0
$257$	−19.4754	−1.21484	−0.607422	−	0.794379i	$-0.707796\pi$
$257$	−19.4754	−1.21484	−0.607422	+	0.794379i	$0.707796\pi$
$258$	0	0
$259$	−31.1779	−1.93730
$260$	0	0
$261$	−3.37231	−0.208741
$262$	0	0
$263$	12.5815	0.775811	0.387906	−	0.921699i	$-0.373199\pi$
$263$	12.5815	0.775811	0.387906	+	0.921699i	$0.373199\pi$
$264$	0	0
$265$	10.1509	0.623564
$266$	0	0
$267$	−12.9837	−0.794590
$268$	0	0
$269$	−26.5638	−1.61962	−0.809811	−	0.586691i	$-0.800430\pi$
$269$	−26.5638	−1.61962	−0.809811	+	0.586691i	$0.800430\pi$
$270$	0	0
$271$	25.9338	1.57537	0.787683	−	0.616081i	$-0.211281\pi$
$271$	25.9338	1.57537	0.787683	+	0.616081i	$0.211281\pi$
$272$	0	0
$273$	10.4142	0.630297
$274$	0	0
$275$	−6.36416	−0.383773
$276$	0	0
$277$	−3.12691	−0.187878	−0.0939388	−	0.995578i	$-0.529946\pi$
$277$	−3.12691	−0.187878	−0.0939388	+	0.995578i	$0.529946\pi$
$278$	0	0
$279$	3.77876	0.226229
$280$	0	0
$281$	14.9313	0.890729	0.445364	−	0.895349i	$-0.353074\pi$
$281$	14.9313	0.890729	0.445364	+	0.895349i	$0.353074\pi$
$282$	0	0
$283$	19.7259	1.17258	0.586291	−	0.810100i	$-0.300587\pi$
$283$	19.7259	1.17258	0.586291	+	0.810100i	$0.300587\pi$
$284$	0	0
$285$	−10.8191	−0.640870
$286$	0	0
$287$	10.3274	0.609610
$288$	0	0
$289$	−16.8711	−0.992418
$290$	0	0
$291$	6.04028	0.354088
$292$	0	0
$293$	−13.9695	−0.816107	−0.408053	−	0.912958i	$-0.633792\pi$
$293$	−13.9695	−0.816107	−0.408053	+	0.912958i	$0.633792\pi$
$294$	0	0
$295$	2.12781	0.123886
$296$	0	0
$297$	−12.6498	−0.734014
$298$	0	0
$299$	0.119230	0.00689527
$300$	0	0
$301$	2.65140	0.152824
$302$	0	0
$303$	5.18443	0.297838
$304$	0	0
$305$	−33.2941	−1.90641
$306$	0	0
$307$	17.7964	1.01569	0.507847	−	0.861447i	$-0.330441\pi$
$307$	17.7964	1.01569	0.507847	+	0.861447i	$0.330441\pi$
$308$	0	0
$309$	13.4196	0.763416
$310$	0	0
$311$	25.9023	1.46878	0.734391	−	0.678727i	$-0.237468\pi$
$311$	25.9023	1.46878	0.734391	+	0.678727i	$0.237468\pi$
$312$	0	0
$313$	11.6576	0.658925	0.329463	−	0.944169i	$-0.393132\pi$
$313$	11.6576	0.658925	0.329463	+	0.944169i	$0.393132\pi$
$314$	0	0
$315$	25.0610	1.41203
$316$	0	0
$317$	−24.5162	−1.37697	−0.688484	−	0.725251i	$-0.741724\pi$
$317$	−24.5162	−1.37697	−0.688484	+	0.725251i	$0.741724\pi$
$318$	0	0
$319$	−4.21301	−0.235883
$320$	0	0
$321$	6.36910	0.355488
$322$	0	0
$323$	1.47750	0.0822104
$324$	0	0
$325$	6.00481	0.333087
$326$	0	0
$327$	11.2693	0.623195
$328$	0	0
$329$	−57.4174	−3.16553
$330$	0	0
$331$	−0.560641	−0.0308156	−0.0154078	−	0.999881i	$-0.504905\pi$
$331$	−0.560641	−0.0308156	−0.0154078	+	0.999881i	$0.504905\pi$
$332$	0	0
$333$	−14.6023	−0.800201
$334$	0	0
$335$	−15.2548	−0.833458
$336$	0	0
$337$	−1.26878	−0.0691147	−0.0345573	−	0.999403i	$-0.511002\pi$
$337$	−1.26878	−0.0691147	−0.0345573	+	0.999403i	$0.511002\pi$
$338$	0	0
$339$	16.1989	0.879805
$340$	0	0
$341$	4.72080	0.255645
$342$	0	0
$343$	−24.7653	−1.33720
$344$	0	0
$345$	−0.128262	−0.00690538
$346$	0	0
$347$	−10.4243	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$347$	−10.4243	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$348$	0	0
$349$	11.2503	0.602216	0.301108	−	0.953590i	$-0.402643\pi$
$349$	11.2503	0.602216	0.301108	+	0.953590i	$0.402643\pi$
$350$	0	0
$351$	11.9355	0.637070
$352$	0	0
$353$	15.5508	0.827686	0.413843	−	0.910348i	$-0.364186\pi$
$353$	15.5508	0.827686	0.413843	+	0.910348i	$0.364186\pi$
$354$	0	0
$355$	−39.2819	−2.08487
$356$	0	0
$357$	1.52993	0.0809727
$358$	0	0
$359$	−29.9534	−1.58088	−0.790440	−	0.612540i	$-0.790148\pi$
$359$	−29.9534	−1.58088	−0.790440	+	0.612540i	$0.790148\pi$
$360$	0	0
$361$	−2.06323	−0.108591
$362$	0	0
$363$	4.13154	0.216849
$364$	0	0
$365$	27.3860	1.43345
$366$	0	0
$367$	−5.19537	−0.271196	−0.135598	−	0.990764i	$-0.543296\pi$
$367$	−5.19537	−0.271196	−0.135598	+	0.990764i	$0.543296\pi$
$368$	0	0
$369$	4.83690	0.251799
$370$	0	0
$371$	−16.4546	−0.854278
$372$	0	0
$373$	−25.3415	−1.31213	−0.656066	−	0.754704i	$-0.727781\pi$
$373$	−25.3415	−1.31213	−0.656066	+	0.754704i	$0.727781\pi$
$374$	0	0
$375$	6.68494	0.345209
$376$	0	0
$377$	3.97513	0.204729
$378$	0	0
$379$	5.82446	0.299182	0.149591	−	0.988748i	$-0.452204\pi$
$379$	5.82446	0.299182	0.149591	+	0.988748i	$0.452204\pi$
$380$	0	0
$381$	9.97535	0.511053
$382$	0	0
$383$	25.4681	1.30136	0.650678	−	0.759353i	$-0.274485\pi$
$383$	25.4681	1.30136	0.650678	+	0.759353i	$0.274485\pi$
$384$	0	0
$385$	31.3087	1.59564
$386$	0	0
$387$	1.24180	0.0631240
$388$	0	0
$389$	−30.9788	−1.57069	−0.785343	−	0.619060i	$-0.787514\pi$
$389$	−30.9788	−1.57069	−0.785343	+	0.619060i	$0.787514\pi$
$390$	0	0
$391$	0.0175159	0.000885818 0
$392$	0	0
$393$	10.3170	0.520424
$394$	0	0
$395$	−14.2841	−0.718711
$396$	0	0
$397$	−17.2785	−0.867186	−0.433593	−	0.901109i	$-0.642754\pi$
$397$	−17.2785	−0.867186	−0.433593	+	0.901109i	$0.642754\pi$
$398$	0	0
$399$	17.5378	0.877988
$400$	0	0
$401$	−34.6417	−1.72992	−0.864962	−	0.501837i	$-0.832658\pi$
$401$	−34.6417	−1.72992	−0.864962	+	0.501837i	$0.832658\pi$
$402$	0	0
$403$	−4.45424	−0.221881
$404$	0	0
$405$	4.14485	0.205959
$406$	0	0
$407$	−18.2426	−0.904253
$408$	0	0
$409$	−9.92750	−0.490883	−0.245442	−	0.969411i	$-0.578933\pi$
$409$	−9.92750	−0.490883	−0.245442	+	0.969411i	$0.578933\pi$
$410$	0	0
$411$	−7.95501	−0.392392
$412$	0	0
$413$	−3.44918	−0.169723
$414$	0	0
$415$	−31.7913	−1.56057
$416$	0	0
$417$	3.71955	0.182147
$418$	0	0
$419$	23.4441	1.14532	0.572660	−	0.819793i	$-0.305912\pi$
$419$	23.4441	1.14532	0.572660	+	0.819793i	$0.305912\pi$
$420$	0	0
$421$	18.9664	0.924364	0.462182	−	0.886785i	$-0.347067\pi$
$421$	18.9664	0.924364	0.462182	+	0.886785i	$0.347067\pi$
$422$	0	0
$423$	−26.8917	−1.30752
$424$	0	0
$425$	0.882156	0.0427908
$426$	0	0
$427$	53.9696	2.61177
$428$	0	0
$429$	6.09350	0.294197
$430$	0	0
$431$	9.35894	0.450804	0.225402	−	0.974266i	$-0.427630\pi$
$431$	9.35894	0.450804	0.225402	+	0.974266i	$0.427630\pi$
$432$	0	0
$433$	8.91181	0.428274	0.214137	−	0.976804i	$-0.431306\pi$
$433$	8.91181	0.428274	0.214137	+	0.976804i	$0.431306\pi$
$434$	0	0
$435$	−4.27623	−0.205030
$436$	0	0
$437$	0.200787	0.00960493
$438$	0	0
$439$	−2.80387	−0.133821	−0.0669107	−	0.997759i	$-0.521314\pi$
$439$	−2.80387	−0.133821	−0.0669107	+	0.997759i	$0.521314\pi$
$440$	0	0
$441$	−26.1114	−1.24340
$442$	0	0
$443$	−31.4842	−1.49586	−0.747931	−	0.663777i	$-0.768952\pi$
$443$	−31.4842	−1.49586	−0.747931	+	0.663777i	$0.768952\pi$
$444$	0	0
$445$	36.8294	1.74588
$446$	0	0
$447$	−19.7365	−0.933503
$448$	0	0
$449$	2.79800	0.132046	0.0660230	−	0.997818i	$-0.478969\pi$
$449$	2.79800	0.132046	0.0660230	+	0.997818i	$0.478969\pi$
$450$	0	0
$451$	6.04273	0.284541
$452$	0	0
$453$	−5.49392	−0.258127
$454$	0	0
$455$	−29.5408	−1.38490
$456$	0	0
$457$	−29.6301	−1.38604	−0.693020	−	0.720919i	$-0.743720\pi$
$457$	−29.6301	−1.38604	−0.693020	+	0.720919i	$0.743720\pi$
$458$	0	0
$459$	1.75342	0.0818427
$460$	0	0
$461$	28.4306	1.32415	0.662074	−	0.749439i	$-0.269677\pi$
$461$	28.4306	1.32415	0.662074	+	0.749439i	$0.269677\pi$
$462$	0	0
$463$	−9.60304	−0.446291	−0.223146	−	0.974785i	$-0.571633\pi$
$463$	−9.60304	−0.446291	−0.223146	+	0.974785i	$0.571633\pi$
$464$	0	0
$465$	4.79163	0.222207
$466$	0	0
$467$	−24.5788	−1.13737	−0.568685	−	0.822555i	$-0.692548\pi$
$467$	−24.5788	−1.13737	−0.568685	+	0.822555i	$0.692548\pi$
$468$	0	0
$469$	24.7280	1.14183
$470$	0	0
$471$	−8.93814	−0.411848
$472$	0	0
$473$	1.55137	0.0713322
$474$	0	0
$475$	10.1122	0.463982
$476$	0	0
$477$	−7.70656	−0.352859
$478$	0	0
$479$	22.4632	1.02637	0.513186	−	0.858278i	$-0.328465\pi$
$479$	22.4632	1.02637	0.513186	+	0.858278i	$0.328465\pi$
$480$	0	0
$481$	17.2126	0.784825
$482$	0	0
$483$	0.207912	0.00946032
$484$	0	0
$485$	−17.1338	−0.778005
$486$	0	0
$487$	23.4568	1.06293	0.531465	−	0.847081i	$-0.321642\pi$
$487$	23.4568	1.06293	0.531465	+	0.847081i	$0.321642\pi$
$488$	0	0
$489$	0.683043	0.0308883
$490$	0	0
$491$	18.3499	0.828119	0.414059	−	0.910250i	$-0.364111\pi$
$491$	18.3499	0.828119	0.414059	+	0.910250i	$0.364111\pi$
$492$	0	0
$493$	0.583978	0.0263011
$494$	0	0
$495$	14.6635	0.659077
$496$	0	0
$497$	63.6759	2.85625
$498$	0	0
$499$	28.9004	1.29376	0.646880	−	0.762592i	$-0.276074\pi$
$499$	28.9004	1.29376	0.646880	+	0.762592i	$0.276074\pi$
$500$	0	0
$501$	−1.25287	−0.0559740
$502$	0	0
$503$	−24.5369	−1.09404	−0.547022	−	0.837118i	$-0.684239\pi$
$503$	−24.5369	−1.09404	−0.547022	+	0.837118i	$0.684239\pi$
$504$	0	0
$505$	−14.7061	−0.654412
$506$	0	0
$507$	6.76566	0.300473
$508$	0	0
$509$	28.2975	1.25426	0.627132	−	0.778913i	$-0.284228\pi$
$509$	28.2975	1.25426	0.627132	+	0.778913i	$0.284228\pi$
$510$	0	0
$511$	−44.3926	−1.96381
$512$	0	0
$513$	20.0997	0.887422
$514$	0	0
$515$	−38.0659	−1.67739
$516$	0	0
$517$	−33.5957	−1.47754
$518$	0	0
$519$	13.9983	0.614457
$520$	0	0
$521$	−34.1186	−1.49476	−0.747381	−	0.664395i	$-0.768689\pi$
$521$	−34.1186	−1.49476	−0.747381	+	0.664395i	$0.768689\pi$
$522$	0	0
$523$	36.0230	1.57518	0.787588	−	0.616202i	$-0.211330\pi$
$523$	36.0230	1.57518	0.787588	+	0.616202i	$0.211330\pi$
$524$	0	0
$525$	10.4711	0.456996
$526$	0	0
$527$	−0.654364	−0.0285045
$528$	0	0
$529$	−22.9976	−0.999897
$530$	0	0
$531$	−1.61544	−0.0701040
$532$	0	0
$533$	−5.70153	−0.246961
$534$	0	0
$535$	−18.0665	−0.781082
$536$	0	0
$537$	24.5014	1.05731
$538$	0	0
$539$	−32.6209	−1.40508
$540$	0	0
$541$	−5.10069	−0.219296	−0.109648	−	0.993970i	$-0.534972\pi$
$541$	−5.10069	−0.219296	−0.109648	+	0.993970i	$0.534972\pi$
$542$	0	0
$543$	17.4969	0.750863
$544$	0	0
$545$	−31.9664	−1.36929
$546$	0	0
$547$	−13.3585	−0.571169	−0.285584	−	0.958354i	$-0.592188\pi$
$547$	−13.3585	−0.571169	−0.285584	+	0.958354i	$0.592188\pi$
$548$	0	0
$549$	25.2769	1.07879
$550$	0	0
$551$	6.69421	0.285183
$552$	0	0
$553$	23.1545	0.984630
$554$	0	0
$555$	−18.5164	−0.785975
$556$	0	0
$557$	−35.8853	−1.52051	−0.760255	−	0.649625i	$-0.774926\pi$
$557$	−35.8853	−1.52051	−0.760255	+	0.649625i	$0.774926\pi$
$558$	0	0
$559$	−1.46377	−0.0619111
$560$	0	0
$561$	0.895185	0.0377947
$562$	0	0
$563$	−12.3579	−0.520822	−0.260411	−	0.965498i	$-0.583858\pi$
$563$	−12.3579	−0.520822	−0.260411	+	0.965498i	$0.583858\pi$
$564$	0	0
$565$	−45.9496	−1.93311
$566$	0	0
$567$	−6.71879	−0.282163
$568$	0	0
$569$	−15.4665	−0.648389	−0.324194	−	0.945990i	$-0.605093\pi$
$569$	−15.4665	−0.648389	−0.324194	+	0.945990i	$0.605093\pi$
$570$	0	0
$571$	13.4153	0.561412	0.280706	−	0.959794i	$-0.409431\pi$
$571$	13.4153	0.561412	0.280706	+	0.959794i	$0.409431\pi$
$572$	0	0
$573$	18.2337	0.761725
$574$	0	0
$575$	0.119881	0.00499940
$576$	0	0
$577$	−9.17308	−0.381880	−0.190940	−	0.981602i	$-0.561154\pi$
$577$	−9.17308	−0.381880	−0.190940	+	0.981602i	$0.561154\pi$
$578$	0	0
$579$	3.81413	0.158510
$580$	0	0
$581$	51.5337	2.13798
$582$	0	0
$583$	−9.62778	−0.398742
$584$	0	0
$585$	−13.8356	−0.572030
$586$	0	0
$587$	35.6720	1.47234	0.736170	−	0.676796i	$-0.236632\pi$
$587$	35.6720	1.47234	0.736170	+	0.676796i	$0.236632\pi$
$588$	0	0
$589$	−7.50104	−0.309075
$590$	0	0
$591$	1.39327	0.0573114
$592$	0	0
$593$	14.7902	0.607361	0.303681	−	0.952774i	$-0.401784\pi$
$593$	14.7902	0.607361	0.303681	+	0.952774i	$0.401784\pi$
$594$	0	0
$595$	−4.33979	−0.177914
$596$	0	0
$597$	−10.9377	−0.447649
$598$	0	0
$599$	−11.7824	−0.481416	−0.240708	−	0.970598i	$-0.577380\pi$
$599$	−11.7824	−0.481416	−0.240708	+	0.970598i	$0.577380\pi$
$600$	0	0
$601$	21.8868	0.892783	0.446391	−	0.894838i	$-0.352709\pi$
$601$	21.8868	0.892783	0.446391	+	0.894838i	$0.352709\pi$
$602$	0	0
$603$	11.5814	0.471633
$604$	0	0
$605$	−11.7195	−0.476464
$606$	0	0
$607$	−38.8295	−1.57604	−0.788020	−	0.615650i	$-0.788894\pi$
$607$	−38.8295	−1.57604	−0.788020	+	0.615650i	$0.788894\pi$
$608$	0	0
$609$	6.93176	0.280889
$610$	0	0
$611$	31.6987	1.28239
$612$	0	0
$613$	−8.98728	−0.362993	−0.181496	−	0.983392i	$-0.558094\pi$
$613$	−8.98728	−0.362993	−0.181496	+	0.983392i	$0.558094\pi$
$614$	0	0
$615$	6.13340	0.247323
$616$	0	0
$617$	−27.9125	−1.12371	−0.561857	−	0.827234i	$-0.689913\pi$
$617$	−27.9125	−1.12371	−0.561857	+	0.827234i	$0.689913\pi$
$618$	0	0
$619$	−30.7965	−1.23782	−0.618909	−	0.785463i	$-0.712425\pi$
$619$	−30.7965	−1.23782	−0.618909	+	0.785463i	$0.712425\pi$
$620$	0	0
$621$	0.238283	0.00956197
$622$	0	0
$623$	−59.7003	−2.39184
$624$	0	0
$625$	−31.2482	−1.24993
$626$	0	0
$627$	10.2616	0.409809
$628$	0	0
$629$	2.52866	0.100824
$630$	0	0
$631$	20.0024	0.796282	0.398141	−	0.917324i	$-0.369655\pi$
$631$	20.0024	0.796282	0.398141	+	0.917324i	$0.369655\pi$
$632$	0	0
$633$	1.46546	0.0582469
$634$	0	0
$635$	−28.2959	−1.12289
$636$	0	0
$637$	30.7790	1.21951
$638$	0	0
$639$	29.8229	1.17977
$640$	0	0
$641$	−2.52474	−0.0997212	−0.0498606	−	0.998756i	$-0.515878\pi$
$641$	−2.52474	−0.0997212	−0.0498606	+	0.998756i	$0.515878\pi$
$642$	0	0
$643$	−26.6427	−1.05069	−0.525343	−	0.850890i	$-0.676063\pi$
$643$	−26.6427	−1.05069	−0.525343	+	0.850890i	$0.676063\pi$
$644$	0	0
$645$	1.57465	0.0620018
$646$	0	0
$647$	21.1454	0.831312	0.415656	−	0.909522i	$-0.363552\pi$
$647$	21.1454	0.831312	0.415656	+	0.909522i	$0.363552\pi$
$648$	0	0
$649$	−2.01816	−0.0792198
$650$	0	0
$651$	−7.76723	−0.304422
$652$	0	0
$653$	−28.1936	−1.10330	−0.551650	−	0.834076i	$-0.686002\pi$
$653$	−28.1936	−1.10330	−0.551650	+	0.834076i	$0.686002\pi$
$654$	0	0
$655$	−29.2651	−1.14348
$656$	0	0
$657$	−20.7914	−0.811151
$658$	0	0
$659$	−11.2653	−0.438835	−0.219418	−	0.975631i	$-0.570416\pi$
$659$	−11.2653	−0.438835	−0.219418	+	0.975631i	$0.570416\pi$
$660$	0	0
$661$	−6.65340	−0.258787	−0.129394	−	0.991593i	$-0.541303\pi$
$661$	−6.65340	−0.258787	−0.129394	+	0.991593i	$0.541303\pi$
$662$	0	0
$663$	−0.844639	−0.0328030
$664$	0	0
$665$	−49.7474	−1.92912
$666$	0	0
$667$	0.0793604	0.00307285
$668$	0	0
$669$	22.4488	0.867919
$670$	0	0
$671$	31.5783	1.21907
$672$	0	0
$673$	−41.0956	−1.58412	−0.792059	−	0.610444i	$-0.790991\pi$
$673$	−41.0956	−1.58412	−0.792059	+	0.610444i	$0.790991\pi$
$674$	0	0
$675$	12.0007	0.461906
$676$	0	0
$677$	−42.8256	−1.64592	−0.822960	−	0.568099i	$-0.807679\pi$
$677$	−42.8256	−1.64592	−0.822960	+	0.568099i	$0.807679\pi$
$678$	0	0
$679$	27.7738	1.06586
$680$	0	0
$681$	1.63944	0.0628235
$682$	0	0
$683$	−4.31978	−0.165292	−0.0826459	−	0.996579i	$-0.526337\pi$
$683$	−4.31978	−0.165292	−0.0826459	+	0.996579i	$0.526337\pi$
$684$	0	0
$685$	22.5651	0.862167
$686$	0	0
$687$	−19.1600	−0.731000
$688$	0	0
$689$	9.08415	0.346079
$690$	0	0
$691$	20.9671	0.797626	0.398813	−	0.917032i	$-0.369422\pi$
$691$	20.9671	0.797626	0.398813	+	0.917032i	$0.369422\pi$
$692$	0	0
$693$	−23.7696	−0.902931
$694$	0	0
$695$	−10.5508	−0.400215
$696$	0	0
$697$	−0.837600	−0.0317264
$698$	0	0
$699$	−17.0983	−0.646717
$700$	0	0
$701$	7.59617	0.286903	0.143452	−	0.989657i	$-0.454180\pi$
$701$	7.59617	0.286903	0.143452	+	0.989657i	$0.454180\pi$
$702$	0	0
$703$	28.9863	1.09324
$704$	0	0
$705$	−34.0998	−1.28427
$706$	0	0
$707$	23.8385	0.896539
$708$	0	0
$709$	−13.0690	−0.490816	−0.245408	−	0.969420i	$-0.578922\pi$
$709$	−13.0690	−0.490816	−0.245408	+	0.969420i	$0.578922\pi$
$710$	0	0
$711$	10.8445	0.406701
$712$	0	0
$713$	−0.0889255	−0.00333028
$714$	0	0
$715$	−17.2847	−0.646412
$716$	0	0
$717$	−5.67235	−0.211838
$718$	0	0
$719$	−26.7304	−0.996876	−0.498438	−	0.866925i	$-0.666093\pi$
$719$	−26.7304	−0.996876	−0.498438	+	0.866925i	$0.666093\pi$
$720$	0	0
$721$	61.7048	2.29801
$722$	0	0
$723$	−27.5797	−1.02570
$724$	0	0
$725$	3.99683	0.148439
$726$	0	0
$727$	−43.7784	−1.62365	−0.811826	−	0.583899i	$-0.801526\pi$
$727$	−43.7784	−1.62365	−0.811826	+	0.583899i	$0.801526\pi$
$728$	0	0
$729$	10.9586	0.405875
$730$	0	0
$731$	−0.215040	−0.00795356
$732$	0	0
$733$	5.89844	0.217864	0.108932	−	0.994049i	$-0.465257\pi$
$733$	5.89844	0.217864	0.108932	+	0.994049i	$0.465257\pi$
$734$	0	0
$735$	−33.1104	−1.22129
$736$	0	0
$737$	14.4687	0.532960
$738$	0	0
$739$	11.2819	0.415010	0.207505	−	0.978234i	$-0.433466\pi$
$739$	11.2819	0.415010	0.207505	+	0.978234i	$0.433466\pi$
$740$	0	0
$741$	−9.68218	−0.355684
$742$	0	0
$743$	2.51753	0.0923591	0.0461796	−	0.998933i	$-0.485295\pi$
$743$	2.51753	0.0923591	0.0461796	+	0.998933i	$0.485295\pi$
$744$	0	0
$745$	55.9842	2.05110
$746$	0	0
$747$	24.1360	0.883090
$748$	0	0
$749$	29.2857	1.07008
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−11.5444	−0.420701
$754$	0	0
$755$	15.5840	0.567159
$756$	0	0
$757$	−17.9539	−0.652545	−0.326272	−	0.945276i	$-0.605793\pi$
$757$	−17.9539	−0.652545	−0.326272	+	0.945276i	$0.605793\pi$
$758$	0	0
$759$	0.121652	0.00441569
$760$	0	0
$761$	20.3575	0.737957	0.368979	−	0.929438i	$-0.379708\pi$
$761$	20.3575	0.737957	0.368979	+	0.929438i	$0.379708\pi$
$762$	0	0
$763$	51.8174	1.87592
$764$	0	0
$765$	−2.03256	−0.0734873
$766$	0	0
$767$	1.90421	0.0687570
$768$	0	0
$769$	−23.5643	−0.849750	−0.424875	−	0.905252i	$-0.639682\pi$
$769$	−23.5643	−0.849750	−0.424875	+	0.905252i	$0.639682\pi$
$770$	0	0
$771$	18.7490	0.675228
$772$	0	0
$773$	24.7255	0.889315	0.444658	−	0.895701i	$-0.353325\pi$
$773$	24.7255	0.889315	0.444658	+	0.895701i	$0.353325\pi$
$774$	0	0
$775$	−4.47856	−0.160875
$776$	0	0
$777$	30.0150	1.07678
$778$	0	0
$779$	−9.60150	−0.344010
$780$	0	0
$781$	37.2576	1.33318
$782$	0	0
$783$	7.94433	0.283907
$784$	0	0
$785$	25.3538	0.904916
$786$	0	0
$787$	−15.8678	−0.565627	−0.282814	−	0.959175i	$-0.591268\pi$
$787$	−15.8678	−0.565627	−0.282814	+	0.959175i	$0.591268\pi$
$788$	0	0
$789$	−12.1122	−0.431207
$790$	0	0
$791$	74.4842	2.64835
$792$	0	0
$793$	−29.7953	−1.05806
$794$	0	0
$795$	−9.77225	−0.346586
$796$	0	0
$797$	26.6301	0.943287	0.471643	−	0.881789i	$-0.343661\pi$
$797$	26.6301	0.943287	0.471643	+	0.881789i	$0.343661\pi$
$798$	0	0
$799$	4.65680	0.164746
$800$	0	0
$801$	−27.9609	−0.987950
$802$	0	0
$803$	−25.9747	−0.916627
$804$	0	0
$805$	−0.589760	−0.0207863
$806$	0	0
$807$	25.5729	0.900210
$808$	0	0
$809$	−18.5436	−0.651957	−0.325978	−	0.945377i	$-0.605694\pi$
$809$	−18.5436	−0.651957	−0.325978	+	0.945377i	$0.605694\pi$
$810$	0	0
$811$	−29.6575	−1.04142	−0.520708	−	0.853735i	$-0.674332\pi$
$811$	−29.6575	−1.04142	−0.520708	+	0.853735i	$0.674332\pi$
$812$	0	0
$813$	−24.9664	−0.875611
$814$	0	0
$815$	−1.93751	−0.0678680
$816$	0	0
$817$	−2.46503	−0.0862405
$818$	0	0
$819$	22.4274	0.783678
$820$	0	0
$821$	2.21087	0.0771599	0.0385799	−	0.999256i	$-0.487717\pi$
$821$	2.21087	0.0771599	0.0385799	+	0.999256i	$0.487717\pi$
$822$	0	0
$823$	−33.1404	−1.15520	−0.577601	−	0.816319i	$-0.696011\pi$
$823$	−33.1404	−1.15520	−0.577601	+	0.816319i	$0.696011\pi$
$824$	0	0
$825$	6.12677	0.213307
$826$	0	0
$827$	−15.0829	−0.524485	−0.262243	−	0.965002i	$-0.584462\pi$
$827$	−15.0829	−0.524485	−0.262243	+	0.965002i	$0.584462\pi$
$828$	0	0
$829$	−21.8670	−0.759473	−0.379736	−	0.925095i	$-0.623985\pi$
$829$	−21.8670	−0.759473	−0.379736	+	0.925095i	$0.623985\pi$
$830$	0	0
$831$	3.01027	0.104425
$832$	0	0
$833$	4.52168	0.156667
$834$	0	0
$835$	3.55387	0.122987
$836$	0	0
$837$	−8.90184	−0.307693
$838$	0	0
$839$	−47.4485	−1.63810	−0.819052	−	0.573719i	$-0.805500\pi$
$839$	−47.4485	−1.63810	−0.819052	+	0.573719i	$0.805500\pi$
$840$	0	0
$841$	−26.3541	−0.908763
$842$	0	0
$843$	−14.3744	−0.495080
$844$	0	0
$845$	−19.1914	−0.660203
$846$	0	0
$847$	18.9972	0.652752
$848$	0	0
$849$	−18.9901	−0.651739
$850$	0	0
$851$	0.343635	0.0117797
$852$	0	0
$853$	22.3940	0.766758	0.383379	−	0.923591i	$-0.374760\pi$
$853$	22.3940	0.766758	0.383379	+	0.923591i	$0.374760\pi$
$854$	0	0
$855$	−23.2994	−0.796823
$856$	0	0
$857$	37.4144	1.27805	0.639025	−	0.769186i	$-0.279338\pi$
$857$	37.4144	1.27805	0.639025	+	0.769186i	$0.279338\pi$
$858$	0	0
$859$	49.2153	1.67920	0.839601	−	0.543203i	$-0.182789\pi$
$859$	49.2153	1.67920	0.839601	+	0.543203i	$0.182789\pi$
$860$	0	0
$861$	−9.94223	−0.338830
$862$	0	0
$863$	−15.8303	−0.538871	−0.269435	−	0.963018i	$-0.586837\pi$
$863$	−15.8303	−0.538871	−0.269435	+	0.963018i	$0.586837\pi$
$864$	0	0
$865$	−39.7073	−1.35009
$866$	0	0
$867$	16.2418	0.551601
$868$	0	0
$869$	13.5480	0.459585
$870$	0	0
$871$	−13.6517	−0.462570
$872$	0	0
$873$	13.0080	0.440253
$874$	0	0
$875$	30.7380	1.03913
$876$	0	0
$877$	−12.3618	−0.417430	−0.208715	−	0.977977i	$-0.566928\pi$
$877$	−12.3618	−0.417430	−0.208715	+	0.977977i	$0.566928\pi$
$878$	0	0
$879$	13.4484	0.453604
$880$	0	0
$881$	23.4019	0.788431	0.394215	−	0.919018i	$-0.371016\pi$
$881$	23.4019	0.788431	0.394215	+	0.919018i	$0.371016\pi$
$882$	0	0
$883$	8.79278	0.295901	0.147950	−	0.988995i	$-0.452732\pi$
$883$	8.79278	0.295901	0.147950	+	0.988995i	$0.452732\pi$
$884$	0	0
$885$	−2.04845	−0.0688577
$886$	0	0
$887$	−5.37404	−0.180443	−0.0902213	−	0.995922i	$-0.528757\pi$
$887$	−5.37404	−0.180443	−0.0902213	+	0.995922i	$0.528757\pi$
$888$	0	0
$889$	45.8676	1.53835
$890$	0	0
$891$	−3.93126	−0.131702
$892$	0	0
$893$	53.3814	1.78634
$894$	0	0
$895$	−69.5004	−2.32314
$896$	0	0
$897$	−0.114783	−0.00383249
$898$	0	0
$899$	−2.96476	−0.0988804
$900$	0	0
$901$	1.33454	0.0444598
$902$	0	0
$903$	−2.55251	−0.0849421
$904$	0	0
$905$	−49.6314	−1.64980
$906$	0	0
$907$	−36.7558	−1.22046	−0.610228	−	0.792226i	$-0.708922\pi$
$907$	−36.7558	−1.22046	−0.610228	+	0.792226i	$0.708922\pi$
$908$	0	0
$909$	11.1649	0.370315
$910$	0	0
$911$	47.6696	1.57936	0.789682	−	0.613516i	$-0.210246\pi$
$911$	47.6696	1.57936	0.789682	+	0.613516i	$0.210246\pi$
$912$	0	0
$913$	30.1530	0.997920
$914$	0	0
$915$	32.0522	1.05961
$916$	0	0
$917$	47.4386	1.56656
$918$	0	0
$919$	2.25440	0.0743657	0.0371828	−	0.999308i	$-0.488162\pi$
$919$	2.25440	0.0743657	0.0371828	+	0.999308i	$0.488162\pi$
$920$	0	0
$921$	−17.1326	−0.564538
$922$	0	0
$923$	−35.1539	−1.15710
$924$	0	0
$925$	17.3065	0.569036
$926$	0	0
$927$	28.8997	0.949190
$928$	0	0
$929$	−10.6970	−0.350958	−0.175479	−	0.984483i	$-0.556147\pi$
$929$	−10.6970	−0.350958	−0.175479	+	0.984483i	$0.556147\pi$
$930$	0	0
$931$	51.8325	1.69874
$932$	0	0
$933$	−24.9361	−0.816371
$934$	0	0
$935$	−2.53927	−0.0830429
$936$	0	0
$937$	11.0765	0.361853	0.180927	−	0.983497i	$-0.442090\pi$
$937$	11.0765	0.361853	0.180927	+	0.983497i	$0.442090\pi$
$938$	0	0
$939$	−11.2227	−0.366240
$940$	0	0
$941$	−0.169037	−0.00551045	−0.00275523	−	0.999996i	$-0.500877\pi$
$941$	−0.169037	−0.00551045	−0.00275523	+	0.999996i	$0.500877\pi$
$942$	0	0
$943$	−0.113827	−0.00370670
$944$	0	0
$945$	−59.0376	−1.92049
$946$	0	0
$947$	49.8062	1.61848	0.809241	−	0.587476i	$-0.199879\pi$
$947$	49.8062	1.61848	0.809241	+	0.587476i	$0.199879\pi$
$948$	0	0
$949$	24.5080	0.795565
$950$	0	0
$951$	23.6018	0.765339
$952$	0	0
$953$	0.922640	0.0298873	0.0149436	−	0.999888i	$-0.495243\pi$
$953$	0.922640	0.0298873	0.0149436	+	0.999888i	$0.495243\pi$
$954$	0	0
$955$	−51.7216	−1.67367
$956$	0	0
$957$	4.05587	0.131108
$958$	0	0
$959$	−36.5779	−1.18116
$960$	0	0
$961$	−27.6779	−0.892836
$962$	0	0
$963$	13.7161	0.441995
$964$	0	0
$965$	−10.8191	−0.348280
$966$	0	0
$967$	−7.89127	−0.253766	−0.126883	−	0.991918i	$-0.540497\pi$
$967$	−7.89127	−0.253766	−0.126883	+	0.991918i	$0.540497\pi$
$968$	0	0
$969$	−1.42239	−0.0456938
$970$	0	0
$971$	−33.6352	−1.07940	−0.539702	−	0.841856i	$-0.681463\pi$
$971$	−33.6352	−1.07940	−0.539702	+	0.841856i	$0.681463\pi$
$972$	0	0
$973$	17.1028	0.548292
$974$	0	0
$975$	−5.78083	−0.185135
$976$	0	0
$977$	18.7841	0.600957	0.300479	−	0.953789i	$-0.402854\pi$
$977$	18.7841	0.600957	0.300479	+	0.953789i	$0.402854\pi$
$978$	0	0
$979$	−34.9315	−1.11641
$980$	0	0
$981$	24.2689	0.774847
$982$	0	0
$983$	−11.1163	−0.354554	−0.177277	−	0.984161i	$-0.556729\pi$
$983$	−11.1163	−0.354554	−0.177277	+	0.984161i	$0.556729\pi$
$984$	0	0
$985$	−3.95212	−0.125925
$986$	0	0
$987$	55.2757	1.75945
$988$	0	0
$989$	−0.0292231	−0.000929242 0
$990$	0	0
$991$	11.3653	0.361030	0.180515	−	0.983572i	$-0.442224\pi$
$991$	11.3653	0.361030	0.180515	+	0.983572i	$0.442224\pi$
$992$	0	0
$993$	0.539729	0.0171278
$994$	0	0
$995$	31.0256	0.983579
$996$	0	0
$997$	−4.96012	−0.157088	−0.0785442	−	0.996911i	$-0.525027\pi$
$997$	−4.96012	−0.157088	−0.0785442	+	0.996911i	$0.525027\pi$
$998$	0	0
$999$	34.3995	1.08835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.19	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.19	✓	44	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.962699 q^{3} +2.73078 q^{5} -4.42659 q^{7} -2.07321 q^{9} +O(q^{10})\)	\(q-0.962699 q^{3} +2.73078 q^{5} -4.42659 q^{7} -2.07321 q^{9} -2.59005 q^{11} +2.44381 q^{13} -2.62892 q^{15} +0.359015 q^{17} +4.11543 q^{19} +4.26147 q^{21} +0.0487888 q^{23} +2.45715 q^{25} +4.88398 q^{27} +1.62661 q^{29} -1.82266 q^{31} +2.49344 q^{33} -12.0880 q^{35} +7.04333 q^{37} -2.35265 q^{39} -2.33305 q^{41} -0.598973 q^{43} -5.66148 q^{45} +12.9710 q^{47} +12.5947 q^{49} -0.345624 q^{51} +3.71721 q^{53} -7.07287 q^{55} -3.96192 q^{57} +0.779197 q^{59} -12.1921 q^{61} +9.17724 q^{63} +6.67350 q^{65} -5.58624 q^{67} -0.0469689 q^{69} -14.3849 q^{71} +10.0286 q^{73} -2.36550 q^{75} +11.4651 q^{77} -5.23078 q^{79} +1.51783 q^{81} -11.6419 q^{83} +0.980391 q^{85} -1.56594 q^{87} +13.4868 q^{89} -10.8177 q^{91} +1.75468 q^{93} +11.2383 q^{95} -6.27432 q^{97} +5.36973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more