LMFDB - Embedded newform 6008.2.a.b.1.25

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			1.25
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)

$f(q)$	$=$	$q-0.107352 q^{3} +0.451859 q^{5} +2.31721 q^{7} -2.98848 q^{9} +O(q^{10})$	$q-0.107352 q^{3} +0.451859 q^{5} +2.31721 q^{7} -2.98848 q^{9} +2.84297 q^{11} +2.31366 q^{13} -0.0485078 q^{15} +0.174832 q^{17} -2.11240 q^{19} -0.248756 q^{21} +0.730513 q^{23} -4.79582 q^{25} +0.642873 q^{27} -7.10282 q^{29} -10.1070 q^{31} -0.305197 q^{33} +1.04705 q^{35} -2.63181 q^{37} -0.248376 q^{39} -7.51725 q^{41} -0.351209 q^{43} -1.35037 q^{45} -6.18043 q^{47} -1.63054 q^{49} -0.0187686 q^{51} -0.440982 q^{53} +1.28462 q^{55} +0.226770 q^{57} -1.57828 q^{59} -4.03536 q^{61} -6.92493 q^{63} +1.04545 q^{65} -4.68956 q^{67} -0.0784218 q^{69} +14.4991 q^{71} +3.78102 q^{73} +0.514840 q^{75} +6.58775 q^{77} +16.4620 q^{79} +8.89641 q^{81} -7.15521 q^{83} +0.0789996 q^{85} +0.762500 q^{87} +3.87479 q^{89} +5.36124 q^{91} +1.08500 q^{93} -0.954508 q^{95} +11.6388 q^{97} -8.49614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.107352	−0.0619795	−0.0309898	−	0.999520i	$-0.509866\pi$
$3$	−0.107352	−0.0619795	−0.0309898	+	0.999520i	$0.509866\pi$
$4$	0	0
$5$	0.451859	0.202077	0.101039	−	0.994882i	$-0.467783\pi$
$5$	0.451859	0.202077	0.101039	+	0.994882i	$0.467783\pi$
$6$	0	0
$7$	2.31721	0.875823	0.437912	−	0.899018i	$-0.355718\pi$
$7$	2.31721	0.875823	0.437912	+	0.899018i	$0.355718\pi$
$8$	0	0
$9$	−2.98848	−0.996159
$10$	0	0
$11$	2.84297	0.857187	0.428593	−	0.903498i	$-0.359009\pi$
$11$	2.84297	0.857187	0.428593	+	0.903498i	$0.359009\pi$
$12$	0	0
$13$	2.31366	0.641695	0.320847	−	0.947131i	$-0.396032\pi$
$13$	2.31366	0.641695	0.320847	+	0.947131i	$0.396032\pi$
$14$	0	0
$15$	−0.0485078	−0.0125247
$16$	0	0
$17$	0.174832	0.0424031	0.0212016	−	0.999775i	$-0.493251\pi$
$17$	0.174832	0.0424031	0.0212016	+	0.999775i	$0.493251\pi$
$18$	0	0
$19$	−2.11240	−0.484619	−0.242309	−	0.970199i	$-0.577905\pi$
$19$	−2.11240	−0.484619	−0.242309	+	0.970199i	$0.577905\pi$
$20$	0	0
$21$	−0.248756	−0.0542831
$22$	0	0
$23$	0.730513	0.152323	0.0761613	−	0.997096i	$-0.475734\pi$
$23$	0.730513	0.152323	0.0761613	+	0.997096i	$0.475734\pi$
$24$	0	0
$25$	−4.79582	−0.959165
$26$	0	0
$27$	0.642873	0.123721
$28$	0	0
$29$	−7.10282	−1.31896	−0.659480	−	0.751722i	$-0.729224\pi$
$29$	−7.10282	−1.31896	−0.659480	+	0.751722i	$0.729224\pi$
$30$	0	0
$31$	−10.1070	−1.81526	−0.907632	−	0.419767i	$-0.862112\pi$
$31$	−10.1070	−1.81526	−0.907632	+	0.419767i	$0.862112\pi$
$32$	0	0
$33$	−0.305197	−0.0531280
$34$	0	0
$35$	1.04705	0.176984
$36$	0	0
$37$	−2.63181	−0.432667	−0.216333	−	0.976320i	$-0.569410\pi$
$37$	−2.63181	−0.432667	−0.216333	+	0.976320i	$0.569410\pi$
$38$	0	0
$39$	−0.248376	−0.0397719
$40$	0	0
$41$	−7.51725	−1.17400	−0.586998	−	0.809588i	$-0.699691\pi$
$41$	−7.51725	−1.17400	−0.586998	+	0.809588i	$0.699691\pi$
$42$	0	0
$43$	−0.351209	−0.0535589	−0.0267795	−	0.999641i	$-0.508525\pi$
$43$	−0.351209	−0.0535589	−0.0267795	+	0.999641i	$0.508525\pi$
$44$	0	0
$45$	−1.35037	−0.201301
$46$	0	0
$47$	−6.18043	−0.901508	−0.450754	−	0.892648i	$-0.648845\pi$
$47$	−6.18043	−0.901508	−0.450754	+	0.892648i	$0.648845\pi$
$48$	0	0
$49$	−1.63054	−0.232934
$50$	0	0
$51$	−0.0187686	−0.00262812
$52$	0	0
$53$	−0.440982	−0.0605735	−0.0302867	−	0.999541i	$-0.509642\pi$
$53$	−0.440982	−0.0605735	−0.0302867	+	0.999541i	$0.509642\pi$
$54$	0	0
$55$	1.28462	0.173218
$56$	0	0
$57$	0.226770	0.0300364
$58$	0	0
$59$	−1.57828	−0.205474	−0.102737	−	0.994709i	$-0.532760\pi$
$59$	−1.57828	−0.205474	−0.102737	+	0.994709i	$0.532760\pi$
$60$	0	0
$61$	−4.03536	−0.516674	−0.258337	−	0.966055i	$-0.583175\pi$
$61$	−4.03536	−0.516674	−0.258337	+	0.966055i	$0.583175\pi$
$62$	0	0
$63$	−6.92493	−0.872459
$64$	0	0
$65$	1.04545	0.129672
$66$	0	0
$67$	−4.68956	−0.572921	−0.286461	−	0.958092i	$-0.592479\pi$
$67$	−4.68956	−0.572921	−0.286461	+	0.958092i	$0.592479\pi$
$68$	0	0
$69$	−0.0784218	−0.00944088
$70$	0	0
$71$	14.4991	1.72073	0.860363	−	0.509682i	$-0.170237\pi$
$71$	14.4991	1.72073	0.860363	+	0.509682i	$0.170237\pi$
$72$	0	0
$73$	3.78102	0.442535	0.221268	−	0.975213i	$-0.428981\pi$
$73$	3.78102	0.442535	0.221268	+	0.975213i	$0.428981\pi$
$74$	0	0
$75$	0.514840	0.0594486
$76$	0	0
$77$	6.58775	0.750744
$78$	0	0
$79$	16.4620	1.85211	0.926057	−	0.377383i	$-0.123176\pi$
$79$	16.4620	1.85211	0.926057	+	0.377383i	$0.123176\pi$
$80$	0	0
$81$	8.89641	0.988490
$82$	0	0
$83$	−7.15521	−0.785386	−0.392693	−	0.919670i	$-0.628457\pi$
$83$	−7.15521	−0.785386	−0.392693	+	0.919670i	$0.628457\pi$
$84$	0	0
$85$	0.0789996	0.00856871
$86$	0	0
$87$	0.762500	0.0817486
$88$	0	0
$89$	3.87479	0.410726	0.205363	−	0.978686i	$-0.434162\pi$
$89$	3.87479	0.410726	0.205363	+	0.978686i	$0.434162\pi$
$90$	0	0
$91$	5.36124	0.562011
$92$	0	0
$93$	1.08500	0.112509
$94$	0	0
$95$	−0.954508	−0.0979305
$96$	0	0
$97$	11.6388	1.18174	0.590872	−	0.806765i	$-0.298784\pi$
$97$	11.6388	1.18174	0.590872	+	0.806765i	$0.298784\pi$
$98$	0	0
$99$	−8.49614	−0.853894
$100$	0	0
$101$	−10.0335	−0.998371	−0.499186	−	0.866495i	$-0.666367\pi$
$101$	−10.0335	−0.998371	−0.499186	+	0.866495i	$0.666367\pi$
$102$	0	0
$103$	−7.62007	−0.750828	−0.375414	−	0.926857i	$-0.622499\pi$
$103$	−7.62007	−0.750828	−0.375414	+	0.926857i	$0.622499\pi$
$104$	0	0
$105$	−0.112403	−0.0109694
$106$	0	0
$107$	−14.7831	−1.42914	−0.714568	−	0.699566i	$-0.753377\pi$
$107$	−14.7831	−1.42914	−0.714568	+	0.699566i	$0.753377\pi$
$108$	0	0
$109$	−11.2344	−1.07606	−0.538029	−	0.842926i	$-0.680831\pi$
$109$	−11.2344	−1.07606	−0.538029	+	0.842926i	$0.680831\pi$
$110$	0	0
$111$	0.282529	0.0268165
$112$	0	0
$113$	1.39916	0.131622	0.0658112	−	0.997832i	$-0.479037\pi$
$113$	1.39916	0.131622	0.0658112	+	0.997832i	$0.479037\pi$
$114$	0	0
$115$	0.330089	0.0307809
$116$	0	0
$117$	−6.91432	−0.639230
$118$	0	0
$119$	0.405124	0.0371376
$120$	0	0
$121$	−2.91754	−0.265231
$122$	0	0
$123$	0.806989	0.0727638
$124$	0	0
$125$	−4.42633	−0.395903
$126$	0	0
$127$	12.6770	1.12491	0.562453	−	0.826829i	$-0.309858\pi$
$127$	12.6770	1.12491	0.562453	+	0.826829i	$0.309858\pi$
$128$	0	0
$129$	0.0377029	0.00331956
$130$	0	0
$131$	−15.5557	−1.35911	−0.679553	−	0.733627i	$-0.737826\pi$
$131$	−15.5557	−1.35911	−0.679553	+	0.733627i	$0.737826\pi$
$132$	0	0
$133$	−4.89488	−0.424440
$134$	0	0
$135$	0.290488	0.0250012
$136$	0	0
$137$	4.15472	0.354962	0.177481	−	0.984124i	$-0.443205\pi$
$137$	4.15472	0.354962	0.177481	+	0.984124i	$0.443205\pi$
$138$	0	0
$139$	10.8863	0.923362	0.461681	−	0.887046i	$-0.347246\pi$
$139$	10.8863	0.923362	0.461681	+	0.887046i	$0.347246\pi$
$140$	0	0
$141$	0.663479	0.0558750
$142$	0	0
$143$	6.57767	0.550052
$144$	0	0
$145$	−3.20947	−0.266532
$146$	0	0
$147$	0.175041	0.0144371
$148$	0	0
$149$	11.5547	0.946599	0.473300	−	0.880901i	$-0.343063\pi$
$149$	11.5547	0.946599	0.473300	+	0.880901i	$0.343063\pi$
$150$	0	0
$151$	−0.593480	−0.0482967	−0.0241484	−	0.999708i	$-0.507687\pi$
$151$	−0.593480	−0.0482967	−0.0241484	+	0.999708i	$0.507687\pi$
$152$	0	0
$153$	−0.522483	−0.0422402
$154$	0	0
$155$	−4.56692	−0.366824
$156$	0	0
$157$	−6.89377	−0.550183	−0.275091	−	0.961418i	$-0.588708\pi$
$157$	−6.89377	−0.550183	−0.275091	+	0.961418i	$0.588708\pi$
$158$	0	0
$159$	0.0473401	0.00375432
$160$	0	0
$161$	1.69275	0.133408
$162$	0	0
$163$	12.7980	1.00242	0.501208	−	0.865327i	$-0.332889\pi$
$163$	12.7980	1.00242	0.501208	+	0.865327i	$0.332889\pi$
$164$	0	0
$165$	−0.137906	−0.0107360
$166$	0	0
$167$	−12.3617	−0.956576	−0.478288	−	0.878203i	$-0.658742\pi$
$167$	−12.3617	−0.956576	−0.478288	+	0.878203i	$0.658742\pi$
$168$	0	0
$169$	−7.64697	−0.588228
$170$	0	0
$171$	6.31287	0.482757
$172$	0	0
$173$	17.9487	1.36462	0.682308	−	0.731065i	$-0.260976\pi$
$173$	17.9487	1.36462	0.682308	+	0.731065i	$0.260976\pi$
$174$	0	0
$175$	−11.1129	−0.840059
$176$	0	0
$177$	0.169431	0.0127352
$178$	0	0
$179$	−15.5568	−1.16277	−0.581386	−	0.813628i	$-0.697489\pi$
$179$	−15.5568	−1.16277	−0.581386	+	0.813628i	$0.697489\pi$
$180$	0	0
$181$	−7.18764	−0.534253	−0.267127	−	0.963661i	$-0.586074\pi$
$181$	−7.18764	−0.534253	−0.267127	+	0.963661i	$0.586074\pi$
$182$	0	0
$183$	0.433202	0.0320232
$184$	0	0
$185$	−1.18921	−0.0874322
$186$	0	0
$187$	0.497043	0.0363474
$188$	0	0
$189$	1.48967	0.108358
$190$	0	0
$191$	−10.6830	−0.772993	−0.386496	−	0.922291i	$-0.626315\pi$
$191$	−10.6830	−0.772993	−0.386496	+	0.922291i	$0.626315\pi$
$192$	0	0
$193$	23.2083	1.67057	0.835284	−	0.549818i	$-0.185303\pi$
$193$	23.2083	1.67057	0.835284	+	0.549818i	$0.185303\pi$
$194$	0	0
$195$	−0.112231	−0.00803701
$196$	0	0
$197$	6.22567	0.443561	0.221780	−	0.975097i	$-0.428813\pi$
$197$	6.22567	0.443561	0.221780	+	0.975097i	$0.428813\pi$
$198$	0	0
$199$	−20.4044	−1.44643	−0.723216	−	0.690622i	$-0.757337\pi$
$199$	−20.4044	−1.44643	−0.723216	+	0.690622i	$0.757337\pi$
$200$	0	0
$201$	0.503433	0.0355094
$202$	0	0
$203$	−16.4587	−1.15518
$204$	0	0
$205$	−3.39673	−0.237238
$206$	0	0
$207$	−2.18312	−0.151737
$208$	0	0
$209$	−6.00549	−0.415409
$210$	0	0
$211$	−1.20727	−0.0831119	−0.0415559	−	0.999136i	$-0.513231\pi$
$211$	−1.20727	−0.0831119	−0.0415559	+	0.999136i	$0.513231\pi$
$212$	0	0
$213$	−1.55650	−0.106650
$214$	0	0
$215$	−0.158697	−0.0108230
$216$	0	0
$217$	−23.4200	−1.58985
$218$	0	0
$219$	−0.405899	−0.0274281
$220$	0	0
$221$	0.404503	0.0272098
$222$	0	0
$223$	−27.0513	−1.81149	−0.905746	−	0.423821i	$-0.860688\pi$
$223$	−27.0513	−1.81149	−0.905746	+	0.423821i	$0.860688\pi$
$224$	0	0
$225$	14.3322	0.955480
$226$	0	0
$227$	15.3831	1.02101	0.510505	−	0.859875i	$-0.329458\pi$
$227$	15.3831	1.02101	0.510505	+	0.859875i	$0.329458\pi$
$228$	0	0
$229$	−26.6244	−1.75939	−0.879696	−	0.475536i	$-0.842254\pi$
$229$	−26.6244	−1.75939	−0.879696	+	0.475536i	$0.842254\pi$
$230$	0	0
$231$	−0.707206	−0.0465308
$232$	0	0
$233$	−11.2611	−0.737738	−0.368869	−	0.929481i	$-0.620255\pi$
$233$	−11.2611	−0.737738	−0.368869	+	0.929481i	$0.620255\pi$
$234$	0	0
$235$	−2.79268	−0.182174
$236$	0	0
$237$	−1.76722	−0.114793
$238$	0	0
$239$	−19.5825	−1.26669	−0.633344	−	0.773870i	$-0.718318\pi$
$239$	−19.5825	−1.26669	−0.633344	+	0.773870i	$0.718318\pi$
$240$	0	0
$241$	−12.9928	−0.836940	−0.418470	−	0.908231i	$-0.637434\pi$
$241$	−12.9928	−0.836940	−0.418470	+	0.908231i	$0.637434\pi$
$242$	0	0
$243$	−2.88366	−0.184987
$244$	0	0
$245$	−0.736772	−0.0470706
$246$	0	0
$247$	−4.88739	−0.310977
$248$	0	0
$249$	0.768124	0.0486779
$250$	0	0
$251$	24.9178	1.57280	0.786398	−	0.617720i	$-0.211943\pi$
$251$	24.9178	1.57280	0.786398	+	0.617720i	$0.211943\pi$
$252$	0	0
$253$	2.07682	0.130569
$254$	0	0
$255$	−0.00848074	−0.000531084 0
$256$	0	0
$257$	8.45169	0.527202	0.263601	−	0.964632i	$-0.415090\pi$
$257$	8.45169	0.527202	0.263601	+	0.964632i	$0.415090\pi$
$258$	0	0
$259$	−6.09846	−0.378940
$260$	0	0
$261$	21.2266	1.31389
$262$	0	0
$263$	5.45332	0.336266	0.168133	−	0.985764i	$-0.446226\pi$
$263$	5.45332	0.336266	0.168133	+	0.985764i	$0.446226\pi$
$264$	0	0
$265$	−0.199261	−0.0122405
$266$	0	0
$267$	−0.415965	−0.0254566
$268$	0	0
$269$	17.1104	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$269$	17.1104	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$270$	0	0
$271$	−19.8965	−1.20863	−0.604314	−	0.796747i	$-0.706553\pi$
$271$	−19.8965	−1.20863	−0.604314	+	0.796747i	$0.706553\pi$
$272$	0	0
$273$	−0.575539	−0.0348332
$274$	0	0
$275$	−13.6344	−0.822183
$276$	0	0
$277$	16.4773	0.990027	0.495013	−	0.868885i	$-0.335163\pi$
$277$	16.4773	0.990027	0.495013	+	0.868885i	$0.335163\pi$
$278$	0	0
$279$	30.2044	1.80829
$280$	0	0
$281$	14.4031	0.859216	0.429608	−	0.903016i	$-0.358652\pi$
$281$	14.4031	0.859216	0.429608	+	0.903016i	$0.358652\pi$
$282$	0	0
$283$	24.5725	1.46068	0.730342	−	0.683081i	$-0.239361\pi$
$283$	24.5725	1.46068	0.730342	+	0.683081i	$0.239361\pi$
$284$	0	0
$285$	0.102468	0.00606968
$286$	0	0
$287$	−17.4190	−1.02821
$288$	0	0
$289$	−16.9694	−0.998202
$290$	0	0
$291$	−1.24945	−0.0732439
$292$	0	0
$293$	−5.42243	−0.316782	−0.158391	−	0.987377i	$-0.550631\pi$
$293$	−5.42243	−0.316782	−0.158391	+	0.987377i	$0.550631\pi$
$294$	0	0
$295$	−0.713159	−0.0415217
$296$	0	0
$297$	1.82767	0.106052
$298$	0	0
$299$	1.69016	0.0977445
$300$	0	0
$301$	−0.813826	−0.0469082
$302$	0	0
$303$	1.07711	0.0618786
$304$	0	0
$305$	−1.82341	−0.104408
$306$	0	0
$307$	−23.9641	−1.36771	−0.683853	−	0.729620i	$-0.739697\pi$
$307$	−23.9641	−1.36771	−0.683853	+	0.729620i	$0.739697\pi$
$308$	0	0
$309$	0.818027	0.0465360
$310$	0	0
$311$	−14.0880	−0.798857	−0.399429	−	0.916764i	$-0.630791\pi$
$311$	−14.0880	−0.798857	−0.399429	+	0.916764i	$0.630791\pi$
$312$	0	0
$313$	25.6192	1.44808	0.724042	−	0.689755i	$-0.242282\pi$
$313$	25.6192	1.44808	0.724042	+	0.689755i	$0.242282\pi$
$314$	0	0
$315$	−3.12909	−0.176304
$316$	0	0
$317$	23.5791	1.32433	0.662166	−	0.749357i	$-0.269637\pi$
$317$	23.5791	1.32433	0.662166	+	0.749357i	$0.269637\pi$
$318$	0	0
$319$	−20.1931	−1.13060
$320$	0	0
$321$	1.58699	0.0885771
$322$	0	0
$323$	−0.369317	−0.0205493
$324$	0	0
$325$	−11.0959	−0.615491
$326$	0	0
$327$	1.20603	0.0666936
$328$	0	0
$329$	−14.3213	−0.789562
$330$	0	0
$331$	−6.32751	−0.347791	−0.173896	−	0.984764i	$-0.555636\pi$
$331$	−6.32751	−0.347791	−0.173896	+	0.984764i	$0.555636\pi$
$332$	0	0
$333$	7.86510	0.431005
$334$	0	0
$335$	−2.11902	−0.115774
$336$	0	0
$337$	28.7099	1.56393	0.781965	−	0.623322i	$-0.214217\pi$
$337$	28.7099	1.56393	0.781965	+	0.623322i	$0.214217\pi$
$338$	0	0
$339$	−0.150203	−0.00815789
$340$	0	0
$341$	−28.7338	−1.55602
$342$	0	0
$343$	−19.9988	−1.07983
$344$	0	0
$345$	−0.0354356	−0.00190779
$346$	0	0
$347$	−5.17786	−0.277962	−0.138981	−	0.990295i	$-0.544383\pi$
$347$	−5.17786	−0.277962	−0.138981	+	0.990295i	$0.544383\pi$
$348$	0	0
$349$	7.66160	0.410116	0.205058	−	0.978750i	$-0.434262\pi$
$349$	7.66160	0.410116	0.205058	+	0.978750i	$0.434262\pi$
$350$	0	0
$351$	1.48739	0.0793911
$352$	0	0
$353$	17.0257	0.906187	0.453093	−	0.891463i	$-0.350320\pi$
$353$	17.0257	0.906187	0.453093	+	0.891463i	$0.350320\pi$
$354$	0	0
$355$	6.55154	0.347720
$356$	0	0
$357$	−0.0434907	−0.00230177
$358$	0	0
$359$	−36.1540	−1.90813	−0.954067	−	0.299593i	$-0.903149\pi$
$359$	−36.1540	−1.90813	−0.954067	+	0.299593i	$0.903149\pi$
$360$	0	0
$361$	−14.5378	−0.765145
$362$	0	0
$363$	0.313203	0.0164389
$364$	0	0
$365$	1.70849	0.0894263
$366$	0	0
$367$	21.8084	1.13839	0.569193	−	0.822204i	$-0.307256\pi$
$367$	21.8084	1.13839	0.569193	+	0.822204i	$0.307256\pi$
$368$	0	0
$369$	22.4651	1.16949
$370$	0	0
$371$	−1.02185	−0.0530517
$372$	0	0
$373$	20.6769	1.07061	0.535305	−	0.844659i	$-0.320197\pi$
$373$	20.6769	1.07061	0.535305	+	0.844659i	$0.320197\pi$
$374$	0	0
$375$	0.475174	0.0245379
$376$	0	0
$377$	−16.4335	−0.846370
$378$	0	0
$379$	−26.0304	−1.33709	−0.668546	−	0.743671i	$-0.733083\pi$
$379$	−26.0304	−1.33709	−0.668546	+	0.743671i	$0.733083\pi$
$380$	0	0
$381$	−1.36090	−0.0697211
$382$	0	0
$383$	−11.7677	−0.601300	−0.300650	−	0.953734i	$-0.597204\pi$
$383$	−11.7677	−0.601300	−0.300650	+	0.953734i	$0.597204\pi$
$384$	0	0
$385$	2.97673	0.151708
$386$	0	0
$387$	1.04958	0.0533532
$388$	0	0
$389$	−27.5703	−1.39787	−0.698936	−	0.715184i	$-0.746343\pi$
$389$	−27.5703	−1.39787	−0.698936	+	0.715184i	$0.746343\pi$
$390$	0	0
$391$	0.127717	0.00645895
$392$	0	0
$393$	1.66993	0.0842367
$394$	0	0
$395$	7.43848	0.374270
$396$	0	0
$397$	1.42649	0.0715933	0.0357967	−	0.999359i	$-0.488603\pi$
$397$	1.42649	0.0715933	0.0357967	+	0.999359i	$0.488603\pi$
$398$	0	0
$399$	0.525474	0.0263066
$400$	0	0
$401$	13.9135	0.694806	0.347403	−	0.937716i	$-0.387064\pi$
$401$	13.9135	0.694806	0.347403	+	0.937716i	$0.387064\pi$
$402$	0	0
$403$	−23.3841	−1.16484
$404$	0	0
$405$	4.01992	0.199752
$406$	0	0
$407$	−7.48215	−0.370876
$408$	0	0
$409$	−31.0654	−1.53608	−0.768042	−	0.640400i	$-0.778769\pi$
$409$	−31.0654	−1.53608	−0.768042	+	0.640400i	$0.778769\pi$
$410$	0	0
$411$	−0.446017	−0.0220004
$412$	0	0
$413$	−3.65720	−0.179959
$414$	0	0
$415$	−3.23314	−0.158709
$416$	0	0
$417$	−1.16866	−0.0572296
$418$	0	0
$419$	21.8128	1.06562	0.532812	−	0.846233i	$-0.321135\pi$
$419$	21.8128	1.06562	0.532812	+	0.846233i	$0.321135\pi$
$420$	0	0
$421$	18.1058	0.882421	0.441210	−	0.897404i	$-0.354549\pi$
$421$	18.1058	0.882421	0.441210	+	0.897404i	$0.354549\pi$
$422$	0	0
$423$	18.4701	0.898045
$424$	0	0
$425$	−0.838466	−0.0406716
$426$	0	0
$427$	−9.35077	−0.452515
$428$	0	0
$429$	−0.706124	−0.0340920
$430$	0	0
$431$	11.1394	0.536567	0.268283	−	0.963340i	$-0.413544\pi$
$431$	11.1394	0.536567	0.268283	+	0.963340i	$0.413544\pi$
$432$	0	0
$433$	−1.34326	−0.0645531	−0.0322766	−	0.999479i	$-0.510276\pi$
$433$	−1.34326	−0.0645531	−0.0322766	+	0.999479i	$0.510276\pi$
$434$	0	0
$435$	0.344542	0.0165195
$436$	0	0
$437$	−1.54314	−0.0738183
$438$	0	0
$439$	−24.9691	−1.19171	−0.595855	−	0.803092i	$-0.703187\pi$
$439$	−24.9691	−1.19171	−0.595855	+	0.803092i	$0.703187\pi$
$440$	0	0
$441$	4.87282	0.232039
$442$	0	0
$443$	−0.639008	−0.0303602	−0.0151801	−	0.999885i	$-0.504832\pi$
$443$	−0.639008	−0.0303602	−0.0151801	+	0.999885i	$0.504832\pi$
$444$	0	0
$445$	1.75086	0.0829985
$446$	0	0
$447$	−1.24042	−0.0586698
$448$	0	0
$449$	27.6952	1.30702	0.653508	−	0.756920i	$-0.273297\pi$
$449$	27.6952	1.30702	0.653508	+	0.756920i	$0.273297\pi$
$450$	0	0
$451$	−21.3713	−1.00633
$452$	0	0
$453$	0.0637111	0.00299341
$454$	0	0
$455$	2.42252	0.113570
$456$	0	0
$457$	−20.8213	−0.973979	−0.486989	−	0.873408i	$-0.661905\pi$
$457$	−20.8213	−0.973979	−0.486989	+	0.873408i	$0.661905\pi$
$458$	0	0
$459$	0.112395	0.00524615
$460$	0	0
$461$	29.6918	1.38289	0.691443	−	0.722431i	$-0.256975\pi$
$461$	29.6918	1.38289	0.691443	+	0.722431i	$0.256975\pi$
$462$	0	0
$463$	−17.3241	−0.805118	−0.402559	−	0.915394i	$-0.631879\pi$
$463$	−17.3241	−0.805118	−0.402559	+	0.915394i	$0.631879\pi$
$464$	0	0
$465$	0.490266	0.0227356
$466$	0	0
$467$	14.0705	0.651104	0.325552	−	0.945524i	$-0.394450\pi$
$467$	14.0705	0.651104	0.325552	+	0.945524i	$0.394450\pi$
$468$	0	0
$469$	−10.8667	−0.501778
$470$	0	0
$471$	0.740058	0.0341001
$472$	0	0
$473$	−0.998477	−0.0459100
$474$	0	0
$475$	10.1307	0.464829
$476$	0	0
$477$	1.31786	0.0603408
$478$	0	0
$479$	−9.28506	−0.424245	−0.212123	−	0.977243i	$-0.568038\pi$
$479$	−9.28506	−0.424245	−0.212123	+	0.977243i	$0.568038\pi$
$480$	0	0
$481$	−6.08912	−0.277640
$482$	0	0
$483$	−0.181720	−0.00826854
$484$	0	0
$485$	5.25910	0.238804
$486$	0	0
$487$	25.0038	1.13303	0.566516	−	0.824051i	$-0.308291\pi$
$487$	25.0038	1.13303	0.566516	+	0.824051i	$0.308291\pi$
$488$	0	0
$489$	−1.37389	−0.0621292
$490$	0	0
$491$	−12.4337	−0.561124	−0.280562	−	0.959836i	$-0.590521\pi$
$491$	−12.4337	−0.561124	−0.280562	+	0.959836i	$0.590521\pi$
$492$	0	0
$493$	−1.24180	−0.0559280
$494$	0	0
$495$	−3.83905	−0.172553
$496$	0	0
$497$	33.5974	1.50705
$498$	0	0
$499$	−20.1592	−0.902451	−0.451226	−	0.892410i	$-0.649013\pi$
$499$	−20.1592	−0.902451	−0.451226	+	0.892410i	$0.649013\pi$
$500$	0	0
$501$	1.32705	0.0592881
$502$	0	0
$503$	−16.6026	−0.740275	−0.370137	−	0.928977i	$-0.620689\pi$
$503$	−16.6026	−0.740275	−0.370137	+	0.928977i	$0.620689\pi$
$504$	0	0
$505$	−4.53373	−0.201748
$506$	0	0
$507$	0.820915	0.0364581
$508$	0	0
$509$	−29.9133	−1.32588	−0.662941	−	0.748672i	$-0.730692\pi$
$509$	−29.9133	−1.32588	−0.662941	+	0.748672i	$0.730692\pi$
$510$	0	0
$511$	8.76142	0.387583
$512$	0	0
$513$	−1.35801	−0.0599575
$514$	0	0
$515$	−3.44320	−0.151725
$516$	0	0
$517$	−17.5707	−0.772761
$518$	0	0
$519$	−1.92682	−0.0845782
$520$	0	0
$521$	−1.12510	−0.0492913	−0.0246457	−	0.999696i	$-0.507846\pi$
$521$	−1.12510	−0.0492913	−0.0246457	+	0.999696i	$0.507846\pi$
$522$	0	0
$523$	30.1807	1.31971	0.659855	−	0.751393i	$-0.270618\pi$
$523$	30.1807	1.31971	0.659855	+	0.751393i	$0.270618\pi$
$524$	0	0
$525$	1.19299	0.0520664
$526$	0	0
$527$	−1.76702	−0.0769728
$528$	0	0
$529$	−22.4664	−0.976798
$530$	0	0
$531$	4.71665	0.204685
$532$	0	0
$533$	−17.3924	−0.753347
$534$	0	0
$535$	−6.67987	−0.288796
$536$	0	0
$537$	1.67005	0.0720680
$538$	0	0
$539$	−4.63556	−0.199668
$540$	0	0
$541$	−6.11773	−0.263022	−0.131511	−	0.991315i	$-0.541983\pi$
$541$	−6.11773	−0.263022	−0.131511	+	0.991315i	$0.541983\pi$
$542$	0	0
$543$	0.771605	0.0331128
$544$	0	0
$545$	−5.07635	−0.217447
$546$	0	0
$547$	−27.0335	−1.15587	−0.577935	−	0.816083i	$-0.696141\pi$
$547$	−27.0335	−1.15587	−0.577935	+	0.816083i	$0.696141\pi$
$548$	0	0
$549$	12.0596	0.514690
$550$	0	0
$551$	15.0040	0.639193
$552$	0	0
$553$	38.1458	1.62212
$554$	0	0
$555$	0.127663	0.00541901
$556$	0	0
$557$	−5.99360	−0.253957	−0.126979	−	0.991905i	$-0.540528\pi$
$557$	−5.99360	−0.253957	−0.126979	+	0.991905i	$0.540528\pi$
$558$	0	0
$559$	−0.812580	−0.0343685
$560$	0	0
$561$	−0.0533584	−0.00225279
$562$	0	0
$563$	1.04268	0.0439436	0.0219718	−	0.999759i	$-0.493006\pi$
$563$	1.04268	0.0439436	0.0219718	+	0.999759i	$0.493006\pi$
$564$	0	0
$565$	0.632225	0.0265979
$566$	0	0
$567$	20.6149	0.865743
$568$	0	0
$569$	11.2345	0.470974	0.235487	−	0.971877i	$-0.424331\pi$
$569$	11.2345	0.470974	0.235487	+	0.971877i	$0.424331\pi$
$570$	0	0
$571$	−18.8623	−0.789361	−0.394681	−	0.918818i	$-0.629145\pi$
$571$	−18.8623	−0.789361	−0.394681	+	0.918818i	$0.629145\pi$
$572$	0	0
$573$	1.14683	0.0479097
$574$	0	0
$575$	−3.50341	−0.146102
$576$	0	0
$577$	−44.6470	−1.85868	−0.929339	−	0.369227i	$-0.879622\pi$
$577$	−44.6470	−1.85868	−0.929339	+	0.369227i	$0.879622\pi$
$578$	0	0
$579$	−2.49145	−0.103541
$580$	0	0
$581$	−16.5801	−0.687860
$582$	0	0
$583$	−1.25370	−0.0519228
$584$	0	0
$585$	−3.12430	−0.129174
$586$	0	0
$587$	−8.91920	−0.368135	−0.184067	−	0.982914i	$-0.558926\pi$
$587$	−8.91920	−0.368135	−0.184067	+	0.982914i	$0.558926\pi$
$588$	0	0
$589$	21.3500	0.879711
$590$	0	0
$591$	−0.668336	−0.0274917
$592$	0	0
$593$	4.02432	0.165259	0.0826294	−	0.996580i	$-0.473668\pi$
$593$	4.02432	0.165259	0.0826294	+	0.996580i	$0.473668\pi$
$594$	0	0
$595$	0.183059	0.00750467
$596$	0	0
$597$	2.19045	0.0896491
$598$	0	0
$599$	−1.32840	−0.0542769	−0.0271385	−	0.999632i	$-0.508640\pi$
$599$	−1.32840	−0.0542769	−0.0271385	+	0.999632i	$0.508640\pi$
$600$	0	0
$601$	−14.2283	−0.580383	−0.290191	−	0.956969i	$-0.593719\pi$
$601$	−14.2283	−0.580383	−0.290191	+	0.956969i	$0.593719\pi$
$602$	0	0
$603$	14.0146	0.570721
$604$	0	0
$605$	−1.31832	−0.0535972
$606$	0	0
$607$	0.892481	0.0362247	0.0181123	−	0.999836i	$-0.494234\pi$
$607$	0.892481	0.0362247	0.0181123	+	0.999836i	$0.494234\pi$
$608$	0	0
$609$	1.76687	0.0715973
$610$	0	0
$611$	−14.2994	−0.578493
$612$	0	0
$613$	46.4577	1.87641	0.938204	−	0.346082i	$-0.112488\pi$
$613$	46.4577	1.87641	0.938204	+	0.346082i	$0.112488\pi$
$614$	0	0
$615$	0.364645	0.0147039
$616$	0	0
$617$	3.60787	0.145247	0.0726236	−	0.997359i	$-0.476863\pi$
$617$	3.60787	0.145247	0.0726236	+	0.997359i	$0.476863\pi$
$618$	0	0
$619$	−6.47639	−0.260308	−0.130154	−	0.991494i	$-0.541547\pi$
$619$	−6.47639	−0.260308	−0.130154	+	0.991494i	$0.541547\pi$
$620$	0	0
$621$	0.469627	0.0188455
$622$	0	0
$623$	8.97869	0.359724
$624$	0	0
$625$	21.9790	0.879162
$626$	0	0
$627$	0.644700	0.0257468
$628$	0	0
$629$	−0.460126	−0.0183464
$630$	0	0
$631$	28.0230	1.11558	0.557789	−	0.829983i	$-0.311650\pi$
$631$	28.0230	1.11558	0.557789	+	0.829983i	$0.311650\pi$
$632$	0	0
$633$	0.129602	0.00515123
$634$	0	0
$635$	5.72823	0.227318
$636$	0	0
$637$	−3.77251	−0.149472
$638$	0	0
$639$	−43.3302	−1.71412
$640$	0	0
$641$	−46.6522	−1.84265	−0.921325	−	0.388793i	$-0.872892\pi$
$641$	−46.6522	−1.84265	−0.921325	+	0.388793i	$0.872892\pi$
$642$	0	0
$643$	−42.5101	−1.67643	−0.838217	−	0.545337i	$-0.816402\pi$
$643$	−42.5101	−1.67643	−0.838217	+	0.545337i	$0.816402\pi$
$644$	0	0
$645$	0.0170364	0.000670807 0
$646$	0	0
$647$	6.42299	0.252514	0.126257	−	0.991998i	$-0.459704\pi$
$647$	6.42299	0.252514	0.126257	+	0.991998i	$0.459704\pi$
$648$	0	0
$649$	−4.48699	−0.176130
$650$	0	0
$651$	2.51417	0.0985382
$652$	0	0
$653$	23.8518	0.933394	0.466697	−	0.884417i	$-0.345444\pi$
$653$	23.8518	0.933394	0.466697	+	0.884417i	$0.345444\pi$
$654$	0	0
$655$	−7.02896	−0.274644
$656$	0	0
$657$	−11.2995	−0.440835
$658$	0	0
$659$	27.9511	1.08882	0.544411	−	0.838819i	$-0.316753\pi$
$659$	27.9511	1.08882	0.544411	+	0.838819i	$0.316753\pi$
$660$	0	0
$661$	39.1041	1.52097	0.760486	−	0.649354i	$-0.224961\pi$
$661$	39.1041	1.52097	0.760486	+	0.649354i	$0.224961\pi$
$662$	0	0
$663$	−0.0434241	−0.00168645
$664$	0	0
$665$	−2.21180	−0.0857698
$666$	0	0
$667$	−5.18870	−0.200907
$668$	0	0
$669$	2.90401	0.112275
$670$	0	0
$671$	−11.4724	−0.442886
$672$	0	0
$673$	−46.0585	−1.77542	−0.887712	−	0.460398i	$-0.847707\pi$
$673$	−46.0585	−1.77542	−0.887712	+	0.460398i	$0.847707\pi$
$674$	0	0
$675$	−3.08311	−0.118669
$676$	0	0
$677$	32.9725	1.26723	0.633617	−	0.773647i	$-0.281569\pi$
$677$	32.9725	1.26723	0.633617	+	0.773647i	$0.281569\pi$
$678$	0	0
$679$	26.9696	1.03500
$680$	0	0
$681$	−1.65140	−0.0632817
$682$	0	0
$683$	37.4858	1.43435	0.717176	−	0.696892i	$-0.245434\pi$
$683$	37.4858	1.43435	0.717176	+	0.696892i	$0.245434\pi$
$684$	0	0
$685$	1.87735	0.0717298
$686$	0	0
$687$	2.85818	0.109046
$688$	0	0
$689$	−1.02028	−0.0388697
$690$	0	0
$691$	24.7575	0.941819	0.470910	−	0.882181i	$-0.343926\pi$
$691$	24.7575	0.941819	0.470910	+	0.882181i	$0.343926\pi$
$692$	0	0
$693$	−19.6873	−0.747860
$694$	0	0
$695$	4.91906	0.186591
$696$	0	0
$697$	−1.31426	−0.0497811
$698$	0	0
$699$	1.20890	0.0457247
$700$	0	0
$701$	15.2572	0.576255	0.288127	−	0.957592i	$-0.406967\pi$
$701$	15.2572	0.576255	0.288127	+	0.957592i	$0.406967\pi$
$702$	0	0
$703$	5.55945	0.209679
$704$	0	0
$705$	0.299799	0.0112911
$706$	0	0
$707$	−23.2498	−0.874397
$708$	0	0
$709$	4.80936	0.180619	0.0903097	−	0.995914i	$-0.471214\pi$
$709$	4.80936	0.180619	0.0903097	+	0.995914i	$0.471214\pi$
$710$	0	0
$711$	−49.1961	−1.84500
$712$	0	0
$713$	−7.38327	−0.276506
$714$	0	0
$715$	2.97218	0.111153
$716$	0	0
$717$	2.10222	0.0785087
$718$	0	0
$719$	40.7242	1.51876	0.759378	−	0.650649i	$-0.225503\pi$
$719$	40.7242	1.51876	0.759378	+	0.650649i	$0.225503\pi$
$720$	0	0
$721$	−17.6573	−0.657592
$722$	0	0
$723$	1.39480	0.0518732
$724$	0	0
$725$	34.0639	1.26510
$726$	0	0
$727$	22.3452	0.828738	0.414369	−	0.910109i	$-0.364002\pi$
$727$	22.3452	0.828738	0.414369	+	0.910109i	$0.364002\pi$
$728$	0	0
$729$	−26.3797	−0.977025
$730$	0	0
$731$	−0.0614028	−0.00227106
$732$	0	0
$733$	−42.7743	−1.57990	−0.789952	−	0.613169i	$-0.789895\pi$
$733$	−42.7743	−1.57990	−0.789952	+	0.613169i	$0.789895\pi$
$734$	0	0
$735$	0.0790937	0.00291741
$736$	0	0
$737$	−13.3323	−0.491101
$738$	0	0
$739$	−38.8243	−1.42818	−0.714088	−	0.700056i	$-0.753158\pi$
$739$	−38.8243	−1.42818	−0.714088	+	0.700056i	$0.753158\pi$
$740$	0	0
$741$	0.524670	0.0192742
$742$	0	0
$743$	44.1174	1.61851	0.809254	−	0.587458i	$-0.199871\pi$
$743$	44.1174	1.61851	0.809254	+	0.587458i	$0.199871\pi$
$744$	0	0
$745$	5.22110	0.191286
$746$	0	0
$747$	21.3832	0.782369
$748$	0	0
$749$	−34.2555	−1.25167
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−2.67497	−0.0974812
$754$	0	0
$755$	−0.268169	−0.00975968
$756$	0	0
$757$	−42.2316	−1.53493	−0.767467	−	0.641089i	$-0.778483\pi$
$757$	−42.2316	−1.53493	−0.767467	+	0.641089i	$0.778483\pi$
$758$	0	0
$759$	−0.222951	−0.00809260
$760$	0	0
$761$	1.89838	0.0688161	0.0344080	−	0.999408i	$-0.489045\pi$
$761$	1.89838	0.0688161	0.0344080	+	0.999408i	$0.489045\pi$
$762$	0	0
$763$	−26.0324	−0.942437
$764$	0	0
$765$	−0.236088	−0.00853579
$766$	0	0
$767$	−3.65160	−0.131852
$768$	0	0
$769$	−18.4764	−0.666276	−0.333138	−	0.942878i	$-0.608108\pi$
$769$	−18.4764	−0.666276	−0.333138	+	0.942878i	$0.608108\pi$
$770$	0	0
$771$	−0.907304	−0.0326757
$772$	0	0
$773$	49.9195	1.79548	0.897740	−	0.440525i	$-0.145208\pi$
$773$	49.9195	1.79548	0.897740	+	0.440525i	$0.145208\pi$
$774$	0	0
$775$	48.4712	1.74114
$776$	0	0
$777$	0.654680	0.0234865
$778$	0	0
$779$	15.8795	0.568941
$780$	0	0
$781$	41.2204	1.47498
$782$	0	0
$783$	−4.56621	−0.163183
$784$	0	0
$785$	−3.11501	−0.111179
$786$	0	0
$787$	18.2768	0.651498	0.325749	−	0.945456i	$-0.394384\pi$
$787$	18.2768	0.651498	0.325749	+	0.945456i	$0.394384\pi$
$788$	0	0
$789$	−0.585423	−0.0208416
$790$	0	0
$791$	3.24216	0.115278
$792$	0	0
$793$	−9.33645	−0.331547
$794$	0	0
$795$	0.0213910	0.000758662 0
$796$	0	0
$797$	−42.3265	−1.49928	−0.749640	−	0.661846i	$-0.769773\pi$
$797$	−42.3265	−1.49928	−0.749640	+	0.661846i	$0.769773\pi$
$798$	0	0
$799$	−1.08054	−0.0382267
$800$	0	0
$801$	−11.5797	−0.409149
$802$	0	0
$803$	10.7493	0.379335
$804$	0	0
$805$	0.764885	0.0269587
$806$	0	0
$807$	−1.83683	−0.0646595
$808$	0	0
$809$	19.5609	0.687726	0.343863	−	0.939020i	$-0.388264\pi$
$809$	19.5609	0.687726	0.343863	+	0.939020i	$0.388264\pi$
$810$	0	0
$811$	3.47091	0.121880	0.0609400	−	0.998141i	$-0.480590\pi$
$811$	3.47091	0.121880	0.0609400	+	0.998141i	$0.480590\pi$
$812$	0	0
$813$	2.13592	0.0749101
$814$	0	0
$815$	5.78288	0.202565
$816$	0	0
$817$	0.741896	0.0259557
$818$	0	0
$819$	−16.0219	−0.559852
$820$	0	0
$821$	25.3122	0.883403	0.441702	−	0.897162i	$-0.354375\pi$
$821$	25.3122	0.883403	0.441702	+	0.897162i	$0.354375\pi$
$822$	0	0
$823$	−29.0117	−1.01128	−0.505642	−	0.862743i	$-0.668744\pi$
$823$	−29.0117	−1.01128	−0.505642	+	0.862743i	$0.668744\pi$
$824$	0	0
$825$	1.46367	0.0509585
$826$	0	0
$827$	5.79862	0.201638	0.100819	−	0.994905i	$-0.467854\pi$
$827$	5.79862	0.201638	0.100819	+	0.994905i	$0.467854\pi$
$828$	0	0
$829$	−39.7053	−1.37902	−0.689511	−	0.724275i	$-0.742175\pi$
$829$	−39.7053	−1.37902	−0.689511	+	0.724275i	$0.742175\pi$
$830$	0	0
$831$	−1.76887	−0.0613614
$832$	0	0
$833$	−0.285071	−0.00987711
$834$	0	0
$835$	−5.58573	−0.193302
$836$	0	0
$837$	−6.49749	−0.224586
$838$	0	0
$839$	−32.9306	−1.13689	−0.568446	−	0.822721i	$-0.692455\pi$
$839$	−32.9306	−1.13689	−0.568446	+	0.822721i	$0.692455\pi$
$840$	0	0
$841$	21.4501	0.739657
$842$	0	0
$843$	−1.54619	−0.0532538
$844$	0	0
$845$	−3.45535	−0.118868
$846$	0	0
$847$	−6.76056	−0.232295
$848$	0	0
$849$	−2.63790	−0.0905325
$850$	0	0
$851$	−1.92257	−0.0659049
$852$	0	0
$853$	37.7219	1.29157	0.645786	−	0.763518i	$-0.276530\pi$
$853$	37.7219	1.29157	0.645786	+	0.763518i	$0.276530\pi$
$854$	0	0
$855$	2.85252	0.0975543
$856$	0	0
$857$	−39.5648	−1.35151	−0.675754	−	0.737128i	$-0.736182\pi$
$857$	−39.5648	−1.35151	−0.675754	+	0.737128i	$0.736182\pi$
$858$	0	0
$859$	5.06855	0.172937	0.0864683	−	0.996255i	$-0.472442\pi$
$859$	5.06855	0.172937	0.0864683	+	0.996255i	$0.472442\pi$
$860$	0	0
$861$	1.86996	0.0637282
$862$	0	0
$863$	−20.1956	−0.687467	−0.343734	−	0.939067i	$-0.611692\pi$
$863$	−20.1956	−0.687467	−0.343734	+	0.939067i	$0.611692\pi$
$864$	0	0
$865$	8.11028	0.275758
$866$	0	0
$867$	1.82170	0.0618681
$868$	0	0
$869$	46.8008	1.58761
$870$	0	0
$871$	−10.8501	−0.367641
$872$	0	0
$873$	−34.7823	−1.17720
$874$	0	0
$875$	−10.2567	−0.346741
$876$	0	0
$877$	−29.7360	−1.00411	−0.502057	−	0.864835i	$-0.667423\pi$
$877$	−29.7360	−1.00411	−0.502057	+	0.864835i	$0.667423\pi$
$878$	0	0
$879$	0.582107	0.0196340
$880$	0	0
$881$	49.4222	1.66508	0.832538	−	0.553968i	$-0.186887\pi$
$881$	49.4222	1.66508	0.832538	+	0.553968i	$0.186887\pi$
$882$	0	0
$883$	26.7724	0.900964	0.450482	−	0.892786i	$-0.351252\pi$
$883$	26.7724	0.900964	0.450482	+	0.892786i	$0.351252\pi$
$884$	0	0
$885$	0.0765588	0.00257350
$886$	0	0
$887$	−38.5906	−1.29575	−0.647873	−	0.761748i	$-0.724341\pi$
$887$	−38.5906	−1.29575	−0.647873	+	0.761748i	$0.724341\pi$
$888$	0	0
$889$	29.3754	0.985218
$890$	0	0
$891$	25.2922	0.847321
$892$	0	0
$893$	13.0556	0.436888
$894$	0	0
$895$	−7.02948	−0.234970
$896$	0	0
$897$	−0.181442	−0.00605816
$898$	0	0
$899$	71.7879	2.39426
$900$	0	0
$901$	−0.0770979	−0.00256850
$902$	0	0
$903$	0.0873656	0.00290735
$904$	0	0
$905$	−3.24780	−0.107960
$906$	0	0
$907$	−55.5012	−1.84289	−0.921444	−	0.388512i	$-0.872989\pi$
$907$	−55.5012	−1.84289	−0.921444	+	0.388512i	$0.872989\pi$
$908$	0	0
$909$	29.9849	0.994536
$910$	0	0
$911$	6.73076	0.223000	0.111500	−	0.993764i	$-0.464435\pi$
$911$	6.73076	0.223000	0.111500	+	0.993764i	$0.464435\pi$
$912$	0	0
$913$	−20.3420	−0.673223
$914$	0	0
$915$	0.195746	0.00647117
$916$	0	0
$917$	−36.0458	−1.19034
$918$	0	0
$919$	44.2852	1.46084	0.730418	−	0.683001i	$-0.239326\pi$
$919$	44.2852	1.46084	0.730418	+	0.683001i	$0.239326\pi$
$920$	0	0
$921$	2.57259	0.0847698
$922$	0	0
$923$	33.5460	1.10418
$924$	0	0
$925$	12.6217	0.414999
$926$	0	0
$927$	22.7724	0.747944
$928$	0	0
$929$	4.03372	0.132342	0.0661710	−	0.997808i	$-0.478922\pi$
$929$	4.03372	0.132342	0.0661710	+	0.997808i	$0.478922\pi$
$930$	0	0
$931$	3.44435	0.112884
$932$	0	0
$933$	1.51237	0.0495128
$934$	0	0
$935$	0.224593	0.00734498
$936$	0	0
$937$	−14.2626	−0.465939	−0.232969	−	0.972484i	$-0.574844\pi$
$937$	−14.2626	−0.465939	−0.232969	+	0.972484i	$0.574844\pi$
$938$	0	0
$939$	−2.75027	−0.0897516
$940$	0	0
$941$	30.6788	1.00010	0.500050	−	0.865997i	$-0.333315\pi$
$941$	30.6788	1.00010	0.500050	+	0.865997i	$0.333315\pi$
$942$	0	0
$943$	−5.49145	−0.178826
$944$	0	0
$945$	0.673121	0.0218966
$946$	0	0
$947$	−29.0991	−0.945594	−0.472797	−	0.881171i	$-0.656756\pi$
$947$	−29.0991	−0.945594	−0.472797	+	0.881171i	$0.656756\pi$
$948$	0	0
$949$	8.74801	0.283972
$950$	0	0
$951$	−2.53125	−0.0820815
$952$	0	0
$953$	13.7057	0.443972	0.221986	−	0.975050i	$-0.428746\pi$
$953$	13.7057	0.443972	0.221986	+	0.975050i	$0.428746\pi$
$954$	0	0
$955$	−4.82719	−0.156204
$956$	0	0
$957$	2.16776	0.0700738
$958$	0	0
$959$	9.62737	0.310884
$960$	0	0
$961$	71.1506	2.29518
$962$	0	0
$963$	44.1789	1.42365
$964$	0	0
$965$	10.4869	0.337584
$966$	0	0
$967$	19.8636	0.638770	0.319385	−	0.947625i	$-0.396524\pi$
$967$	19.8636	0.638770	0.319385	+	0.947625i	$0.396524\pi$
$968$	0	0
$969$	0.0396468	0.00127364
$970$	0	0
$971$	5.00233	0.160533	0.0802663	−	0.996773i	$-0.474423\pi$
$971$	5.00233	0.160533	0.0802663	+	0.996773i	$0.474423\pi$
$972$	0	0
$973$	25.2258	0.808702
$974$	0	0
$975$	1.19117	0.0381478
$976$	0	0
$977$	−36.1794	−1.15748	−0.578741	−	0.815512i	$-0.696456\pi$
$977$	−36.1794	−1.15748	−0.578741	+	0.815512i	$0.696456\pi$
$978$	0	0
$979$	11.0159	0.352069
$980$	0	0
$981$	33.5737	1.07192
$982$	0	0
$983$	35.3163	1.12641	0.563207	−	0.826316i	$-0.309567\pi$
$983$	35.3163	1.12641	0.563207	+	0.826316i	$0.309567\pi$
$984$	0	0
$985$	2.81312	0.0896336
$986$	0	0
$987$	1.53742	0.0489366
$988$	0	0
$989$	−0.256563	−0.00815823
$990$	0	0
$991$	38.3902	1.21950	0.609752	−	0.792592i	$-0.291269\pi$
$991$	38.3902	1.21950	0.609752	+	0.792592i	$0.291269\pi$
$992$	0	0
$993$	0.679269	0.0215559
$994$	0	0
$995$	−9.21992	−0.292291
$996$	0	0
$997$	24.3654	0.771661	0.385830	−	0.922570i	$-0.373915\pi$
$997$	24.3654	0.771661	0.385830	+	0.922570i	$0.373915\pi$
$998$	0	0
$999$	−1.69192	−0.0535300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.25	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.25	✓	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)

\(f(q)\)	\(=\)	\(q-0.107352 q^{3} +0.451859 q^{5} +2.31721 q^{7} -2.98848 q^{9} +O(q^{10})\)	\(q-0.107352 q^{3} +0.451859 q^{5} +2.31721 q^{7} -2.98848 q^{9} +2.84297 q^{11} +2.31366 q^{13} -0.0485078 q^{15} +0.174832 q^{17} -2.11240 q^{19} -0.248756 q^{21} +0.730513 q^{23} -4.79582 q^{25} +0.642873 q^{27} -7.10282 q^{29} -10.1070 q^{31} -0.305197 q^{33} +1.04705 q^{35} -2.63181 q^{37} -0.248376 q^{39} -7.51725 q^{41} -0.351209 q^{43} -1.35037 q^{45} -6.18043 q^{47} -1.63054 q^{49} -0.0187686 q^{51} -0.440982 q^{53} +1.28462 q^{55} +0.226770 q^{57} -1.57828 q^{59} -4.03536 q^{61} -6.92493 q^{63} +1.04545 q^{65} -4.68956 q^{67} -0.0784218 q^{69} +14.4991 q^{71} +3.78102 q^{73} +0.514840 q^{75} +6.58775 q^{77} +16.4620 q^{79} +8.89641 q^{81} -7.15521 q^{83} +0.0789996 q^{85} +0.762500 q^{87} +3.87479 q^{89} +5.36124 q^{91} +1.08500 q^{93} -0.954508 q^{95} +11.6388 q^{97} -8.49614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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