LMFDB - Embedded newform 6008.2.a.b.1.37

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			1.37
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)

$f(q)$	$=$	$q+1.77382 q^{3} -3.34355 q^{5} +0.838649 q^{7} +0.146429 q^{9} +O(q^{10})$	$q+1.77382 q^{3} -3.34355 q^{5} +0.838649 q^{7} +0.146429 q^{9} -0.297273 q^{11} -1.29535 q^{13} -5.93085 q^{15} +2.69073 q^{17} -2.59218 q^{19} +1.48761 q^{21} +5.00797 q^{23} +6.17933 q^{25} -5.06171 q^{27} +4.26847 q^{29} -1.16112 q^{31} -0.527309 q^{33} -2.80407 q^{35} +6.31269 q^{37} -2.29772 q^{39} +0.0977895 q^{41} +7.74167 q^{43} -0.489594 q^{45} -4.23944 q^{47} -6.29667 q^{49} +4.77286 q^{51} -3.41633 q^{53} +0.993949 q^{55} -4.59806 q^{57} +5.04471 q^{59} -11.7533 q^{61} +0.122803 q^{63} +4.33108 q^{65} -4.02906 q^{67} +8.88323 q^{69} -2.77177 q^{71} -12.8527 q^{73} +10.9610 q^{75} -0.249308 q^{77} +9.39050 q^{79} -9.41785 q^{81} -15.8476 q^{83} -8.99658 q^{85} +7.57148 q^{87} -11.7826 q^{89} -1.08635 q^{91} -2.05962 q^{93} +8.66710 q^{95} -8.20439 q^{97} -0.0435295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.77382	1.02411	0.512057	−	0.858951i	$-0.328884\pi$
$3$	1.77382	1.02411	0.512057	+	0.858951i	$0.328884\pi$
$4$	0	0
$5$	−3.34355	−1.49528	−0.747641	−	0.664103i	$-0.768813\pi$
$5$	−3.34355	−1.49528	−0.747641	+	0.664103i	$0.768813\pi$
$6$	0	0
$7$	0.838649	0.316980	0.158490	−	0.987361i	$-0.449338\pi$
$7$	0.838649	0.316980	0.158490	+	0.987361i	$0.449338\pi$
$8$	0	0
$9$	0.146429	0.0488097
$10$	0	0
$11$	−0.297273	−0.0896313	−0.0448157	−	0.998995i	$-0.514270\pi$
$11$	−0.297273	−0.0896313	−0.0448157	+	0.998995i	$0.514270\pi$
$12$	0	0
$13$	−1.29535	−0.359267	−0.179633	−	0.983734i	$-0.557491\pi$
$13$	−1.29535	−0.359267	−0.179633	+	0.983734i	$0.557491\pi$
$14$	0	0
$15$	−5.93085	−1.53134
$16$	0	0
$17$	2.69073	0.652597	0.326299	−	0.945267i	$-0.394199\pi$
$17$	2.69073	0.652597	0.326299	+	0.945267i	$0.394199\pi$
$18$	0	0
$19$	−2.59218	−0.594688	−0.297344	−	0.954770i	$-0.596101\pi$
$19$	−2.59218	−0.594688	−0.297344	+	0.954770i	$0.596101\pi$
$20$	0	0
$21$	1.48761	0.324623
$22$	0	0
$23$	5.00797	1.04423	0.522117	−	0.852874i	$-0.325142\pi$
$23$	5.00797	1.04423	0.522117	+	0.852874i	$0.325142\pi$
$24$	0	0
$25$	6.17933	1.23587
$26$	0	0
$27$	−5.06171	−0.974127
$28$	0	0
$29$	4.26847	0.792635	0.396317	−	0.918114i	$-0.370288\pi$
$29$	4.26847	0.792635	0.396317	+	0.918114i	$0.370288\pi$
$30$	0	0
$31$	−1.16112	−0.208544	−0.104272	−	0.994549i	$-0.533251\pi$
$31$	−1.16112	−0.208544	−0.104272	+	0.994549i	$0.533251\pi$
$32$	0	0
$33$	−0.527309	−0.0917927
$34$	0	0
$35$	−2.80407	−0.473974
$36$	0	0
$37$	6.31269	1.03780	0.518900	−	0.854835i	$-0.326342\pi$
$37$	6.31269	1.03780	0.518900	+	0.854835i	$0.326342\pi$
$38$	0	0
$39$	−2.29772	−0.367930
$40$	0	0
$41$	0.0977895	0.0152722	0.00763608	−	0.999971i	$-0.497569\pi$
$41$	0.0977895	0.0152722	0.00763608	+	0.999971i	$0.497569\pi$
$42$	0	0
$43$	7.74167	1.18059	0.590297	−	0.807186i	$-0.299011\pi$
$43$	7.74167	1.18059	0.590297	+	0.807186i	$0.299011\pi$
$44$	0	0
$45$	−0.489594	−0.0729843
$46$	0	0
$47$	−4.23944	−0.618385	−0.309193	−	0.950999i	$-0.600059\pi$
$47$	−4.23944	−0.618385	−0.309193	+	0.950999i	$0.600059\pi$
$48$	0	0
$49$	−6.29667	−0.899524
$50$	0	0
$51$	4.77286	0.668334
$52$	0	0
$53$	−3.41633	−0.469269	−0.234634	−	0.972084i	$-0.575389\pi$
$53$	−3.41633	−0.469269	−0.234634	+	0.972084i	$0.575389\pi$
$54$	0	0
$55$	0.993949	0.134024
$56$	0	0
$57$	−4.59806	−0.609028
$58$	0	0
$59$	5.04471	0.656765	0.328382	−	0.944545i	$-0.393497\pi$
$59$	5.04471	0.656765	0.328382	+	0.944545i	$0.393497\pi$
$60$	0	0
$61$	−11.7533	−1.50485	−0.752426	−	0.658677i	$-0.771117\pi$
$61$	−11.7533	−1.50485	−0.752426	+	0.658677i	$0.771117\pi$
$62$	0	0
$63$	0.122803	0.0154717
$64$	0	0
$65$	4.33108	0.537205
$66$	0	0
$67$	−4.02906	−0.492228	−0.246114	−	0.969241i	$-0.579154\pi$
$67$	−4.02906	−0.492228	−0.246114	+	0.969241i	$0.579154\pi$
$68$	0	0
$69$	8.88323	1.06942
$70$	0	0
$71$	−2.77177	−0.328949	−0.164475	−	0.986381i	$-0.552593\pi$
$71$	−2.77177	−0.328949	−0.164475	+	0.986381i	$0.552593\pi$
$72$	0	0
$73$	−12.8527	−1.50429	−0.752146	−	0.658997i	$-0.770981\pi$
$73$	−12.8527	−1.50429	−0.752146	+	0.658997i	$0.770981\pi$
$74$	0	0
$75$	10.9610	1.26567
$76$	0	0
$77$	−0.249308	−0.0284113
$78$	0	0
$79$	9.39050	1.05651	0.528257	−	0.849085i	$-0.322846\pi$
$79$	9.39050	1.05651	0.528257	+	0.849085i	$0.322846\pi$
$80$	0	0
$81$	−9.41785	−1.04643
$82$	0	0
$83$	−15.8476	−1.73950	−0.869750	−	0.493493i	$-0.835720\pi$
$83$	−15.8476	−1.73950	−0.869750	+	0.493493i	$0.835720\pi$
$84$	0	0
$85$	−8.99658	−0.975816
$86$	0	0
$87$	7.57148	0.811748
$88$	0	0
$89$	−11.7826	−1.24895	−0.624477	−	0.781043i	$-0.714688\pi$
$89$	−11.7826	−1.24895	−0.624477	+	0.781043i	$0.714688\pi$
$90$	0	0
$91$	−1.08635	−0.113880
$92$	0	0
$93$	−2.05962	−0.213573
$94$	0	0
$95$	8.66710	0.889225
$96$	0	0
$97$	−8.20439	−0.833029	−0.416515	−	0.909129i	$-0.636749\pi$
$97$	−8.20439	−0.833029	−0.416515	+	0.909129i	$0.636749\pi$
$98$	0	0
$99$	−0.0435295	−0.00437488
$100$	0	0
$101$	−4.74409	−0.472054	−0.236027	−	0.971746i	$-0.575845\pi$
$101$	−4.74409	−0.472054	−0.236027	+	0.971746i	$0.575845\pi$
$102$	0	0
$103$	−1.44010	−0.141898	−0.0709488	−	0.997480i	$-0.522603\pi$
$103$	−1.44010	−0.141898	−0.0709488	+	0.997480i	$0.522603\pi$
$104$	0	0
$105$	−4.97390	−0.485403
$106$	0	0
$107$	−5.75172	−0.556039	−0.278020	−	0.960575i	$-0.589678\pi$
$107$	−5.75172	−0.556039	−0.278020	+	0.960575i	$0.589678\pi$
$108$	0	0
$109$	6.93535	0.664287	0.332143	−	0.943229i	$-0.392228\pi$
$109$	6.93535	0.664287	0.332143	+	0.943229i	$0.392228\pi$
$110$	0	0
$111$	11.1976	1.06283
$112$	0	0
$113$	−18.6893	−1.75814	−0.879071	−	0.476691i	$-0.841836\pi$
$113$	−18.6893	−1.75814	−0.879071	+	0.476691i	$0.841836\pi$
$114$	0	0
$115$	−16.7444	−1.56142
$116$	0	0
$117$	−0.189678	−0.0175357
$118$	0	0
$119$	2.25658	0.206860
$120$	0	0
$121$	−10.9116	−0.991966
$122$	0	0
$123$	0.173461	0.0156404
$124$	0	0
$125$	−3.94316	−0.352687
$126$	0	0
$127$	−10.9674	−0.973196	−0.486598	−	0.873626i	$-0.661762\pi$
$127$	−10.9674	−0.973196	−0.486598	+	0.873626i	$0.661762\pi$
$128$	0	0
$129$	13.7323	1.20906
$130$	0	0
$131$	18.3875	1.60653	0.803263	−	0.595625i	$-0.203096\pi$
$131$	18.3875	1.60653	0.803263	+	0.595625i	$0.203096\pi$
$132$	0	0
$133$	−2.17393	−0.188504
$134$	0	0
$135$	16.9241	1.45659
$136$	0	0
$137$	−13.2031	−1.12801	−0.564007	−	0.825770i	$-0.690741\pi$
$137$	−13.2031	−1.12801	−0.564007	+	0.825770i	$0.690741\pi$
$138$	0	0
$139$	−17.2135	−1.46003	−0.730014	−	0.683432i	$-0.760487\pi$
$139$	−17.2135	−1.46003	−0.730014	+	0.683432i	$0.760487\pi$
$140$	0	0
$141$	−7.51999	−0.633297
$142$	0	0
$143$	0.385075	0.0322016
$144$	0	0
$145$	−14.2718	−1.18521
$146$	0	0
$147$	−11.1691	−0.921215
$148$	0	0
$149$	−1.64240	−0.134551	−0.0672753	−	0.997734i	$-0.521431\pi$
$149$	−1.64240	−0.134551	−0.0672753	+	0.997734i	$0.521431\pi$
$150$	0	0
$151$	−16.4309	−1.33713	−0.668564	−	0.743655i	$-0.733091\pi$
$151$	−16.4309	−1.33713	−0.668564	+	0.743655i	$0.733091\pi$
$152$	0	0
$153$	0.394001	0.0318531
$154$	0	0
$155$	3.88228	0.311832
$156$	0	0
$157$	18.2844	1.45925	0.729627	−	0.683845i	$-0.239694\pi$
$157$	18.2844	1.45925	0.729627	+	0.683845i	$0.239694\pi$
$158$	0	0
$159$	−6.05994	−0.480585
$160$	0	0
$161$	4.19993	0.331001
$162$	0	0
$163$	10.1749	0.796960	0.398480	−	0.917177i	$-0.369538\pi$
$163$	10.1749	0.796960	0.398480	+	0.917177i	$0.369538\pi$
$164$	0	0
$165$	1.76308	0.137256
$166$	0	0
$167$	−3.99023	−0.308773	−0.154387	−	0.988010i	$-0.549340\pi$
$167$	−3.99023	−0.308773	−0.154387	+	0.988010i	$0.549340\pi$
$168$	0	0
$169$	−11.3221	−0.870927
$170$	0	0
$171$	−0.379571	−0.0290266
$172$	0	0
$173$	−11.6567	−0.886241	−0.443121	−	0.896462i	$-0.646129\pi$
$173$	−11.6567	−0.886241	−0.443121	+	0.896462i	$0.646129\pi$
$174$	0	0
$175$	5.18229	0.391744
$176$	0	0
$177$	8.94839	0.672602
$178$	0	0
$179$	12.1000	0.904399	0.452199	−	0.891917i	$-0.350640\pi$
$179$	12.1000	0.904399	0.452199	+	0.891917i	$0.350640\pi$
$180$	0	0
$181$	−5.13687	−0.381820	−0.190910	−	0.981608i	$-0.561144\pi$
$181$	−5.13687	−0.381820	−0.190910	+	0.981608i	$0.561144\pi$
$182$	0	0
$183$	−20.8482	−1.54114
$184$	0	0
$185$	−21.1068	−1.55180
$186$	0	0
$187$	−0.799881	−0.0584931
$188$	0	0
$189$	−4.24500	−0.308779
$190$	0	0
$191$	13.2013	0.955213	0.477607	−	0.878574i	$-0.341504\pi$
$191$	13.2013	0.955213	0.477607	+	0.878574i	$0.341504\pi$
$192$	0	0
$193$	15.0529	1.08353	0.541765	−	0.840530i	$-0.317756\pi$
$193$	15.0529	1.08353	0.541765	+	0.840530i	$0.317756\pi$
$194$	0	0
$195$	7.68256	0.550159
$196$	0	0
$197$	27.6452	1.96964	0.984821	−	0.173572i	$-0.0555310\pi$
$197$	27.6452	1.96964	0.984821	+	0.173572i	$0.0555310\pi$
$198$	0	0
$199$	−4.08758	−0.289761	−0.144880	−	0.989449i	$-0.546280\pi$
$199$	−4.08758	−0.289761	−0.144880	+	0.989449i	$0.546280\pi$
$200$	0	0
$201$	−7.14681	−0.504097
$202$	0	0
$203$	3.57975	0.251249
$204$	0	0
$205$	−0.326964	−0.0228362
$206$	0	0
$207$	0.733314	0.0509688
$208$	0	0
$209$	0.770587	0.0533026
$210$	0	0
$211$	8.92644	0.614522	0.307261	−	0.951625i	$-0.400588\pi$
$211$	8.92644	0.614522	0.307261	+	0.951625i	$0.400588\pi$
$212$	0	0
$213$	−4.91662	−0.336881
$214$	0	0
$215$	−25.8847	−1.76532
$216$	0	0
$217$	−0.973777	−0.0661043
$218$	0	0
$219$	−22.7983	−1.54057
$220$	0	0
$221$	−3.48545	−0.234456
$222$	0	0
$223$	8.07168	0.540519	0.270260	−	0.962787i	$-0.412890\pi$
$223$	8.07168	0.540519	0.270260	+	0.962787i	$0.412890\pi$
$224$	0	0
$225$	0.904835	0.0603223
$226$	0	0
$227$	−10.8324	−0.718970	−0.359485	−	0.933151i	$-0.617048\pi$
$227$	−10.8324	−0.718970	−0.359485	+	0.933151i	$0.617048\pi$
$228$	0	0
$229$	−4.06858	−0.268860	−0.134430	−	0.990923i	$-0.542920\pi$
$229$	−4.06858	−0.268860	−0.134430	+	0.990923i	$0.542920\pi$
$230$	0	0
$231$	−0.442227	−0.0290964
$232$	0	0
$233$	4.37960	0.286917	0.143459	−	0.989656i	$-0.454178\pi$
$233$	4.37960	0.286917	0.143459	+	0.989656i	$0.454178\pi$
$234$	0	0
$235$	14.1748	0.924660
$236$	0	0
$237$	16.6570	1.08199
$238$	0	0
$239$	−22.4530	−1.45236	−0.726182	−	0.687502i	$-0.758707\pi$
$239$	−22.4530	−1.45236	−0.726182	+	0.687502i	$0.758707\pi$
$240$	0	0
$241$	−1.50796	−0.0971362	−0.0485681	−	0.998820i	$-0.515466\pi$
$241$	−1.50796	−0.0971362	−0.0485681	+	0.998820i	$0.515466\pi$
$242$	0	0
$243$	−1.52040	−0.0975337
$244$	0	0
$245$	21.0532	1.34504
$246$	0	0
$247$	3.35780	0.213652
$248$	0	0
$249$	−28.1108	−1.78145
$250$	0	0
$251$	−10.5663	−0.666939	−0.333469	−	0.942761i	$-0.608219\pi$
$251$	−10.5663	−0.666939	−0.333469	+	0.942761i	$0.608219\pi$
$252$	0	0
$253$	−1.48874	−0.0935961
$254$	0	0
$255$	−15.9583	−0.999347
$256$	0	0
$257$	−6.34538	−0.395814	−0.197907	−	0.980221i	$-0.563414\pi$
$257$	−6.34538	−0.395814	−0.197907	+	0.980221i	$0.563414\pi$
$258$	0	0
$259$	5.29413	0.328961
$260$	0	0
$261$	0.625029	0.0386883
$262$	0	0
$263$	25.1599	1.55142	0.775711	−	0.631088i	$-0.217391\pi$
$263$	25.1599	1.55142	0.775711	+	0.631088i	$0.217391\pi$
$264$	0	0
$265$	11.4227	0.701689
$266$	0	0
$267$	−20.9002	−1.27907
$268$	0	0
$269$	16.8199	1.02553	0.512765	−	0.858529i	$-0.328621\pi$
$269$	16.8199	1.02553	0.512765	+	0.858529i	$0.328621\pi$
$270$	0	0
$271$	−14.7714	−0.897298	−0.448649	−	0.893708i	$-0.648095\pi$
$271$	−14.7714	−0.897298	−0.448649	+	0.893708i	$0.648095\pi$
$272$	0	0
$273$	−1.92698	−0.116626
$274$	0	0
$275$	−1.83695	−0.110772
$276$	0	0
$277$	−16.3951	−0.985086	−0.492543	−	0.870288i	$-0.663933\pi$
$277$	−16.3951	−0.985086	−0.492543	+	0.870288i	$0.663933\pi$
$278$	0	0
$279$	−0.170023	−0.0101790
$280$	0	0
$281$	2.11245	0.126018	0.0630092	−	0.998013i	$-0.479930\pi$
$281$	2.11245	0.126018	0.0630092	+	0.998013i	$0.479930\pi$
$282$	0	0
$283$	22.9049	1.36156	0.680778	−	0.732490i	$-0.261642\pi$
$283$	22.9049	1.36156	0.680778	+	0.732490i	$0.261642\pi$
$284$	0	0
$285$	15.3738	0.910668
$286$	0	0
$287$	0.0820111	0.00484096
$288$	0	0
$289$	−9.75999	−0.574117
$290$	0	0
$291$	−14.5531	−0.853117
$292$	0	0
$293$	−29.4136	−1.71836	−0.859181	−	0.511672i	$-0.829026\pi$
$293$	−29.4136	−1.71836	−0.859181	+	0.511672i	$0.829026\pi$
$294$	0	0
$295$	−16.8672	−0.982048
$296$	0	0
$297$	1.50471	0.0873123
$298$	0	0
$299$	−6.48710	−0.375159
$300$	0	0
$301$	6.49255	0.374224
$302$	0	0
$303$	−8.41514	−0.483437
$304$	0	0
$305$	39.2977	2.25018
$306$	0	0
$307$	8.96347	0.511572	0.255786	−	0.966733i	$-0.417666\pi$
$307$	8.96347	0.511572	0.255786	+	0.966733i	$0.417666\pi$
$308$	0	0
$309$	−2.55448	−0.145319
$310$	0	0
$311$	−4.27774	−0.242568	−0.121284	−	0.992618i	$-0.538701\pi$
$311$	−4.27774	−0.242568	−0.121284	+	0.992618i	$0.538701\pi$
$312$	0	0
$313$	1.98402	0.112143	0.0560716	−	0.998427i	$-0.482142\pi$
$313$	1.98402	0.112143	0.0560716	+	0.998427i	$0.482142\pi$
$314$	0	0
$315$	−0.410597	−0.0231345
$316$	0	0
$317$	21.1918	1.19025	0.595126	−	0.803632i	$-0.297102\pi$
$317$	21.1918	1.19025	0.595126	+	0.803632i	$0.297102\pi$
$318$	0	0
$319$	−1.26890	−0.0710449
$320$	0	0
$321$	−10.2025	−0.569448
$322$	0	0
$323$	−6.97486	−0.388091
$324$	0	0
$325$	−8.00443	−0.444006
$326$	0	0
$327$	12.3021	0.680305
$328$	0	0
$329$	−3.55540	−0.196015
$330$	0	0
$331$	−23.2369	−1.27721	−0.638607	−	0.769533i	$-0.720489\pi$
$331$	−23.2369	−1.27721	−0.638607	+	0.769533i	$0.720489\pi$
$332$	0	0
$333$	0.924363	0.0506548
$334$	0	0
$335$	13.4714	0.736019
$336$	0	0
$337$	−21.4801	−1.17010	−0.585049	−	0.810998i	$-0.698925\pi$
$337$	−21.4801	−1.17010	−0.585049	+	0.810998i	$0.698925\pi$
$338$	0	0
$339$	−33.1514	−1.80054
$340$	0	0
$341$	0.345172	0.0186921
$342$	0	0
$343$	−11.1512	−0.602110
$344$	0	0
$345$	−29.7015	−1.59908
$346$	0	0
$347$	32.3974	1.73919	0.869593	−	0.493770i	$-0.164381\pi$
$347$	32.3974	1.73919	0.869593	+	0.493770i	$0.164381\pi$
$348$	0	0
$349$	−8.97027	−0.480167	−0.240084	−	0.970752i	$-0.577175\pi$
$349$	−8.97027	−0.480167	−0.240084	+	0.970752i	$0.577175\pi$
$350$	0	0
$351$	6.55672	0.349972
$352$	0	0
$353$	5.98482	0.318540	0.159270	−	0.987235i	$-0.449086\pi$
$353$	5.98482	0.318540	0.159270	+	0.987235i	$0.449086\pi$
$354$	0	0
$355$	9.26757	0.491872
$356$	0	0
$357$	4.00275	0.211848
$358$	0	0
$359$	1.69118	0.0892571	0.0446285	−	0.999004i	$-0.485790\pi$
$359$	1.69118	0.0892571	0.0446285	+	0.999004i	$0.485790\pi$
$360$	0	0
$361$	−12.2806	−0.646347
$362$	0	0
$363$	−19.3552	−1.01589
$364$	0	0
$365$	42.9736	2.24934
$366$	0	0
$367$	15.1243	0.789481	0.394741	−	0.918793i	$-0.370834\pi$
$367$	15.1243	0.789481	0.394741	+	0.918793i	$0.370834\pi$
$368$	0	0
$369$	0.0143192	0.000745430 0
$370$	0	0
$371$	−2.86510	−0.148749
$372$	0	0
$373$	1.18552	0.0613839	0.0306919	−	0.999529i	$-0.490229\pi$
$373$	1.18552	0.0613839	0.0306919	+	0.999529i	$0.490229\pi$
$374$	0	0
$375$	−6.99445	−0.361192
$376$	0	0
$377$	−5.52918	−0.284767
$378$	0	0
$379$	−28.7848	−1.47857	−0.739287	−	0.673390i	$-0.764837\pi$
$379$	−28.7848	−1.47857	−0.739287	+	0.673390i	$0.764837\pi$
$380$	0	0
$381$	−19.4541	−0.996664
$382$	0	0
$383$	−26.3351	−1.34566	−0.672829	−	0.739798i	$-0.734921\pi$
$383$	−26.3351	−1.34566	−0.672829	+	0.739798i	$0.734921\pi$
$384$	0	0
$385$	0.833574	0.0424829
$386$	0	0
$387$	1.13361	0.0576245
$388$	0	0
$389$	−1.72684	−0.0875544	−0.0437772	−	0.999041i	$-0.513939\pi$
$389$	−1.72684	−0.0875544	−0.0437772	+	0.999041i	$0.513939\pi$
$390$	0	0
$391$	13.4751	0.681465
$392$	0	0
$393$	32.6161	1.64527
$394$	0	0
$395$	−31.3976	−1.57978
$396$	0	0
$397$	−16.7181	−0.839057	−0.419528	−	0.907742i	$-0.637804\pi$
$397$	−16.7181	−0.839057	−0.419528	+	0.907742i	$0.637804\pi$
$398$	0	0
$399$	−3.85616	−0.193049
$400$	0	0
$401$	22.2971	1.11346	0.556731	−	0.830693i	$-0.312055\pi$
$401$	22.2971	1.11346	0.556731	+	0.830693i	$0.312055\pi$
$402$	0	0
$403$	1.50407	0.0749230
$404$	0	0
$405$	31.4890	1.56470
$406$	0	0
$407$	−1.87659	−0.0930194
$408$	0	0
$409$	−37.7115	−1.86471	−0.932356	−	0.361541i	$-0.882251\pi$
$409$	−37.7115	−1.86471	−0.932356	+	0.361541i	$0.882251\pi$
$410$	0	0
$411$	−23.4198	−1.15521
$412$	0	0
$413$	4.23074	0.208181
$414$	0	0
$415$	52.9873	2.60104
$416$	0	0
$417$	−30.5336	−1.49524
$418$	0	0
$419$	−35.6033	−1.73934	−0.869668	−	0.493638i	$-0.835667\pi$
$419$	−35.6033	−1.73934	−0.869668	+	0.493638i	$0.835667\pi$
$420$	0	0
$421$	18.4505	0.899223	0.449611	−	0.893224i	$-0.351562\pi$
$421$	18.4505	0.899223	0.449611	+	0.893224i	$0.351562\pi$
$422$	0	0
$423$	−0.620777	−0.0301832
$424$	0	0
$425$	16.6269	0.806523
$426$	0	0
$427$	−9.85687	−0.477008
$428$	0	0
$429$	0.683052	0.0329781
$430$	0	0
$431$	−15.3236	−0.738114	−0.369057	−	0.929407i	$-0.620319\pi$
$431$	−15.3236	−0.738114	−0.369057	+	0.929407i	$0.620319\pi$
$432$	0	0
$433$	−25.2338	−1.21266	−0.606329	−	0.795214i	$-0.707359\pi$
$433$	−25.2338	−1.21266	−0.606329	+	0.795214i	$0.707359\pi$
$434$	0	0
$435$	−25.3156	−1.21379
$436$	0	0
$437$	−12.9816	−0.620994
$438$	0	0
$439$	−10.0447	−0.479406	−0.239703	−	0.970846i	$-0.577050\pi$
$439$	−10.0447	−0.479406	−0.239703	+	0.970846i	$0.577050\pi$
$440$	0	0
$441$	−0.922016	−0.0439055
$442$	0	0
$443$	26.8947	1.27780	0.638902	−	0.769288i	$-0.279389\pi$
$443$	26.8947	1.27780	0.638902	+	0.769288i	$0.279389\pi$
$444$	0	0
$445$	39.3957	1.86754
$446$	0	0
$447$	−2.91332	−0.137795
$448$	0	0
$449$	27.2235	1.28476	0.642379	−	0.766387i	$-0.277948\pi$
$449$	27.2235	1.28476	0.642379	+	0.766387i	$0.277948\pi$
$450$	0	0
$451$	−0.0290702	−0.00136886
$452$	0	0
$453$	−29.1454	−1.36937
$454$	0	0
$455$	3.63226	0.170283
$456$	0	0
$457$	−15.9248	−0.744929	−0.372465	−	0.928046i	$-0.621487\pi$
$457$	−15.9248	−0.744929	−0.372465	+	0.928046i	$0.621487\pi$
$458$	0	0
$459$	−13.6197	−0.635713
$460$	0	0
$461$	30.5077	1.42088	0.710442	−	0.703756i	$-0.248495\pi$
$461$	30.5077	1.42088	0.710442	+	0.703756i	$0.248495\pi$
$462$	0	0
$463$	−6.96729	−0.323797	−0.161899	−	0.986807i	$-0.551762\pi$
$463$	−6.96729	−0.323797	−0.161899	+	0.986807i	$0.551762\pi$
$464$	0	0
$465$	6.88646	0.319352
$466$	0	0
$467$	−14.5795	−0.674660	−0.337330	−	0.941386i	$-0.609524\pi$
$467$	−14.5795	−0.674660	−0.337330	+	0.941386i	$0.609524\pi$
$468$	0	0
$469$	−3.37897	−0.156026
$470$	0	0
$471$	32.4332	1.49444
$472$	0	0
$473$	−2.30139	−0.105818
$474$	0	0
$475$	−16.0180	−0.734955
$476$	0	0
$477$	−0.500250	−0.0229049
$478$	0	0
$479$	−11.8995	−0.543702	−0.271851	−	0.962339i	$-0.587636\pi$
$479$	−11.8995	−0.543702	−0.271851	+	0.962339i	$0.587636\pi$
$480$	0	0
$481$	−8.17717	−0.372847
$482$	0	0
$483$	7.44992	0.338983
$484$	0	0
$485$	27.4318	1.24561
$486$	0	0
$487$	18.1105	0.820663	0.410331	−	0.911936i	$-0.365413\pi$
$487$	18.1105	0.820663	0.410331	+	0.911936i	$0.365413\pi$
$488$	0	0
$489$	18.0484	0.816178
$490$	0	0
$491$	1.41825	0.0640049	0.0320024	−	0.999488i	$-0.489812\pi$
$491$	1.41825	0.0640049	0.0320024	+	0.999488i	$0.489812\pi$
$492$	0	0
$493$	11.4853	0.517271
$494$	0	0
$495$	0.145543	0.00654168
$496$	0	0
$497$	−2.32455	−0.104270
$498$	0	0
$499$	−15.0810	−0.675118	−0.337559	−	0.941304i	$-0.609601\pi$
$499$	−15.0810	−0.675118	−0.337559	+	0.941304i	$0.609601\pi$
$500$	0	0
$501$	−7.07795	−0.316219
$502$	0	0
$503$	28.6085	1.27559	0.637794	−	0.770207i	$-0.279847\pi$
$503$	28.6085	1.27559	0.637794	+	0.770207i	$0.279847\pi$
$504$	0	0
$505$	15.8621	0.705854
$506$	0	0
$507$	−20.0833	−0.891929
$508$	0	0
$509$	21.5883	0.956887	0.478443	−	0.878118i	$-0.341201\pi$
$509$	21.5883	0.956887	0.478443	+	0.878118i	$0.341201\pi$
$510$	0	0
$511$	−10.7789	−0.476830
$512$	0	0
$513$	13.1209	0.579301
$514$	0	0
$515$	4.81506	0.212177
$516$	0	0
$517$	1.26027	0.0554267
$518$	0	0
$519$	−20.6768	−0.907612
$520$	0	0
$521$	29.3717	1.28680	0.643398	−	0.765532i	$-0.277524\pi$
$521$	29.3717	1.28680	0.643398	+	0.765532i	$0.277524\pi$
$522$	0	0
$523$	5.73648	0.250839	0.125419	−	0.992104i	$-0.459972\pi$
$523$	5.73648	0.250839	0.125419	+	0.992104i	$0.459972\pi$
$524$	0	0
$525$	9.19244	0.401191
$526$	0	0
$527$	−3.12427	−0.136095
$528$	0	0
$529$	2.07981	0.0904264
$530$	0	0
$531$	0.738693	0.0320565
$532$	0	0
$533$	−0.126672	−0.00548678
$534$	0	0
$535$	19.2312	0.831435
$536$	0	0
$537$	21.4633	0.926208
$538$	0	0
$539$	1.87183	0.0806255
$540$	0	0
$541$	35.9411	1.54523	0.772614	−	0.634876i	$-0.218949\pi$
$541$	35.9411	1.54523	0.772614	+	0.634876i	$0.218949\pi$
$542$	0	0
$543$	−9.11187	−0.391028
$544$	0	0
$545$	−23.1887	−0.993295
$546$	0	0
$547$	11.0642	0.473073	0.236536	−	0.971623i	$-0.423988\pi$
$547$	11.0642	0.473073	0.236536	+	0.971623i	$0.423988\pi$
$548$	0	0
$549$	−1.72102	−0.0734515
$550$	0	0
$551$	−11.0647	−0.471370
$552$	0	0
$553$	7.87533	0.334893
$554$	0	0
$555$	−37.4396	−1.58922
$556$	0	0
$557$	25.5527	1.08270	0.541352	−	0.840796i	$-0.317913\pi$
$557$	25.5527	1.08270	0.541352	+	0.840796i	$0.317913\pi$
$558$	0	0
$559$	−10.0282	−0.424148
$560$	0	0
$561$	−1.41884	−0.0599036
$562$	0	0
$563$	−33.0701	−1.39374	−0.696869	−	0.717198i	$-0.745424\pi$
$563$	−33.0701	−1.39374	−0.696869	+	0.717198i	$0.745424\pi$
$564$	0	0
$565$	62.4887	2.62892
$566$	0	0
$567$	−7.89827	−0.331696
$568$	0	0
$569$	32.7223	1.37179	0.685895	−	0.727701i	$-0.259411\pi$
$569$	32.7223	1.37179	0.685895	+	0.727701i	$0.259411\pi$
$570$	0	0
$571$	22.8069	0.954437	0.477219	−	0.878785i	$-0.341645\pi$
$571$	22.8069	0.954437	0.477219	+	0.878785i	$0.341645\pi$
$572$	0	0
$573$	23.4167	0.978247
$574$	0	0
$575$	30.9459	1.29053
$576$	0	0
$577$	19.2941	0.803226	0.401613	−	0.915810i	$-0.368450\pi$
$577$	19.2941	0.803226	0.401613	+	0.915810i	$0.368450\pi$
$578$	0	0
$579$	26.7011	1.10966
$580$	0	0
$581$	−13.2906	−0.551386
$582$	0	0
$583$	1.01558	0.0420612
$584$	0	0
$585$	0.634198	0.0262208
$586$	0	0
$587$	3.86822	0.159658	0.0798292	−	0.996809i	$-0.474563\pi$
$587$	3.86822	0.159658	0.0798292	+	0.996809i	$0.474563\pi$
$588$	0	0
$589$	3.00985	0.124019
$590$	0	0
$591$	49.0376	2.01714
$592$	0	0
$593$	17.8742	0.734007	0.367004	−	0.930219i	$-0.380384\pi$
$593$	17.8742	0.734007	0.367004	+	0.930219i	$0.380384\pi$
$594$	0	0
$595$	−7.54498	−0.309314
$596$	0	0
$597$	−7.25062	−0.296748
$598$	0	0
$599$	−10.5041	−0.429184	−0.214592	−	0.976704i	$-0.568842\pi$
$599$	−10.5041	−0.429184	−0.214592	+	0.976704i	$0.568842\pi$
$600$	0	0
$601$	−16.9813	−0.692682	−0.346341	−	0.938109i	$-0.612576\pi$
$601$	−16.9813	−0.692682	−0.346341	+	0.938109i	$0.612576\pi$
$602$	0	0
$603$	−0.589972	−0.0240255
$604$	0	0
$605$	36.4836	1.48327
$606$	0	0
$607$	−14.0223	−0.569149	−0.284574	−	0.958654i	$-0.591852\pi$
$607$	−14.0223	−0.569149	−0.284574	+	0.958654i	$0.591852\pi$
$608$	0	0
$609$	6.34982	0.257308
$610$	0	0
$611$	5.49157	0.222165
$612$	0	0
$613$	−31.9887	−1.29201	−0.646005	−	0.763333i	$-0.723561\pi$
$613$	−31.9887	−1.29201	−0.646005	+	0.763333i	$0.723561\pi$
$614$	0	0
$615$	−0.579975	−0.0233868
$616$	0	0
$617$	22.4234	0.902734	0.451367	−	0.892338i	$-0.350937\pi$
$617$	22.4234	0.902734	0.451367	+	0.892338i	$0.350937\pi$
$618$	0	0
$619$	−25.9749	−1.04402	−0.522009	−	0.852940i	$-0.674817\pi$
$619$	−25.9749	−1.04402	−0.522009	+	0.852940i	$0.674817\pi$
$620$	0	0
$621$	−25.3489	−1.01722
$622$	0	0
$623$	−9.88147	−0.395893
$624$	0	0
$625$	−17.7125	−0.708500
$626$	0	0
$627$	1.36688	0.0545880
$628$	0	0
$629$	16.9857	0.677265
$630$	0	0
$631$	1.28039	0.0509713	0.0254857	−	0.999675i	$-0.491887\pi$
$631$	1.28039	0.0509713	0.0254857	+	0.999675i	$0.491887\pi$
$632$	0	0
$633$	15.8339	0.629340
$634$	0	0
$635$	36.6699	1.45520
$636$	0	0
$637$	8.15642	0.323169
$638$	0	0
$639$	−0.405869	−0.0160559
$640$	0	0
$641$	1.02127	0.0403378	0.0201689	−	0.999797i	$-0.493580\pi$
$641$	1.02127	0.0403378	0.0201689	+	0.999797i	$0.493580\pi$
$642$	0	0
$643$	33.3204	1.31403	0.657014	−	0.753878i	$-0.271819\pi$
$643$	33.3204	1.31403	0.657014	+	0.753878i	$0.271819\pi$
$644$	0	0
$645$	−45.9147	−1.80789
$646$	0	0
$647$	8.19602	0.322219	0.161109	−	0.986937i	$-0.448493\pi$
$647$	8.19602	0.322219	0.161109	+	0.986937i	$0.448493\pi$
$648$	0	0
$649$	−1.49966	−0.0588667
$650$	0	0
$651$	−1.72730	−0.0676983
$652$	0	0
$653$	−15.4705	−0.605408	−0.302704	−	0.953085i	$-0.597889\pi$
$653$	−15.4705	−0.605408	−0.302704	+	0.953085i	$0.597889\pi$
$654$	0	0
$655$	−61.4796	−2.40221
$656$	0	0
$657$	−1.88201	−0.0734241
$658$	0	0
$659$	25.9045	1.00909	0.504547	−	0.863384i	$-0.331659\pi$
$659$	25.9045	1.00909	0.504547	+	0.863384i	$0.331659\pi$
$660$	0	0
$661$	−30.8453	−1.19974	−0.599872	−	0.800096i	$-0.704782\pi$
$661$	−30.8453	−1.19974	−0.599872	+	0.800096i	$0.704782\pi$
$662$	0	0
$663$	−6.18255	−0.240110
$664$	0	0
$665$	7.26865	0.281866
$666$	0	0
$667$	21.3764	0.827697
$668$	0	0
$669$	14.3177	0.553554
$670$	0	0
$671$	3.49394	0.134882
$672$	0	0
$673$	−6.75901	−0.260541	−0.130270	−	0.991479i	$-0.541585\pi$
$673$	−6.75901	−0.260541	−0.130270	+	0.991479i	$0.541585\pi$
$674$	0	0
$675$	−31.2780	−1.20389
$676$	0	0
$677$	−19.4418	−0.747208	−0.373604	−	0.927588i	$-0.621878\pi$
$677$	−19.4418	−0.747208	−0.373604	+	0.927588i	$0.621878\pi$
$678$	0	0
$679$	−6.88060	−0.264053
$680$	0	0
$681$	−19.2147	−0.736307
$682$	0	0
$683$	2.82651	0.108153	0.0540767	−	0.998537i	$-0.482778\pi$
$683$	2.82651	0.108153	0.0540767	+	0.998537i	$0.482778\pi$
$684$	0	0
$685$	44.1451	1.68670
$686$	0	0
$687$	−7.21693	−0.275343
$688$	0	0
$689$	4.42536	0.168593
$690$	0	0
$691$	−13.0757	−0.497421	−0.248711	−	0.968578i	$-0.580007\pi$
$691$	−13.0757	−0.497421	−0.248711	+	0.968578i	$0.580007\pi$
$692$	0	0
$693$	−0.0365060	−0.00138675
$694$	0	0
$695$	57.5542	2.18315
$696$	0	0
$697$	0.263125	0.00996656
$698$	0	0
$699$	7.76861	0.293836
$700$	0	0
$701$	26.0613	0.984321	0.492160	−	0.870505i	$-0.336207\pi$
$701$	26.0613	0.984321	0.492160	+	0.870505i	$0.336207\pi$
$702$	0	0
$703$	−16.3637	−0.617167
$704$	0	0
$705$	25.1435	0.946957
$706$	0	0
$707$	−3.97862	−0.149632
$708$	0	0
$709$	−1.89091	−0.0710144	−0.0355072	−	0.999369i	$-0.511305\pi$
$709$	−1.89091	−0.0710144	−0.0355072	+	0.999369i	$0.511305\pi$
$710$	0	0
$711$	1.37504	0.0515681
$712$	0	0
$713$	−5.81488	−0.217769
$714$	0	0
$715$	−1.28752	−0.0481504
$716$	0	0
$717$	−39.8276	−1.48739
$718$	0	0
$719$	32.8096	1.22359	0.611796	−	0.791016i	$-0.290448\pi$
$719$	32.8096	1.22359	0.611796	+	0.791016i	$0.290448\pi$
$720$	0	0
$721$	−1.20774	−0.0449786
$722$	0	0
$723$	−2.67485	−0.0994786
$724$	0	0
$725$	26.3763	0.979591
$726$	0	0
$727$	−47.1519	−1.74877	−0.874384	−	0.485235i	$-0.838734\pi$
$727$	−47.1519	−1.74877	−0.874384	+	0.485235i	$0.838734\pi$
$728$	0	0
$729$	25.5566	0.946542
$730$	0	0
$731$	20.8307	0.770452
$732$	0	0
$733$	9.47504	0.349968	0.174984	−	0.984571i	$-0.444013\pi$
$733$	9.47504	0.349968	0.174984	+	0.984571i	$0.444013\pi$
$734$	0	0
$735$	37.3446	1.37748
$736$	0	0
$737$	1.19773	0.0441190
$738$	0	0
$739$	−21.5448	−0.792540	−0.396270	−	0.918134i	$-0.629695\pi$
$739$	−21.5448	−0.792540	−0.396270	+	0.918134i	$0.629695\pi$
$740$	0	0
$741$	5.95612	0.218804
$742$	0	0
$743$	25.8284	0.947550	0.473775	−	0.880646i	$-0.342891\pi$
$743$	25.8284	0.947550	0.473775	+	0.880646i	$0.342891\pi$
$744$	0	0
$745$	5.49144	0.201191
$746$	0	0
$747$	−2.32055	−0.0849046
$748$	0	0
$749$	−4.82367	−0.176253
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−18.7427	−0.683021
$754$	0	0
$755$	54.9375	1.99938
$756$	0	0
$757$	0.222073	0.00807139	0.00403570	−	0.999992i	$-0.498715\pi$
$757$	0.222073	0.00807139	0.00403570	+	0.999992i	$0.498715\pi$
$758$	0	0
$759$	−2.64075	−0.0958531
$760$	0	0
$761$	−45.5022	−1.64945	−0.824726	−	0.565532i	$-0.808671\pi$
$761$	−45.5022	−1.64945	−0.824726	+	0.565532i	$0.808671\pi$
$762$	0	0
$763$	5.81633	0.210565
$764$	0	0
$765$	−1.31736	−0.0476293
$766$	0	0
$767$	−6.53469	−0.235954
$768$	0	0
$769$	27.1900	0.980496	0.490248	−	0.871583i	$-0.336906\pi$
$769$	27.1900	0.980496	0.490248	+	0.871583i	$0.336906\pi$
$770$	0	0
$771$	−11.2555	−0.405359
$772$	0	0
$773$	35.8941	1.29102	0.645511	−	0.763751i	$-0.276645\pi$
$773$	35.8941	1.29102	0.645511	+	0.763751i	$0.276645\pi$
$774$	0	0
$775$	−7.17498	−0.257733
$776$	0	0
$777$	9.39083	0.336894
$778$	0	0
$779$	−0.253488	−0.00908216
$780$	0	0
$781$	0.823975	0.0294841
$782$	0	0
$783$	−21.6058	−0.772127
$784$	0	0
$785$	−61.1348	−2.18200
$786$	0	0
$787$	−41.3722	−1.47476	−0.737380	−	0.675478i	$-0.763937\pi$
$787$	−41.3722	−1.47476	−0.737380	+	0.675478i	$0.763937\pi$
$788$	0	0
$789$	44.6290	1.58883
$790$	0	0
$791$	−15.6738	−0.557295
$792$	0	0
$793$	15.2247	0.540644
$794$	0	0
$795$	20.2617	0.718610
$796$	0	0
$797$	31.9129	1.13041	0.565207	−	0.824949i	$-0.308796\pi$
$797$	31.9129	1.13041	0.565207	+	0.824949i	$0.308796\pi$
$798$	0	0
$799$	−11.4072	−0.403556
$800$	0	0
$801$	−1.72532	−0.0609611
$802$	0	0
$803$	3.82076	0.134832
$804$	0	0
$805$	−14.0427	−0.494940
$806$	0	0
$807$	29.8355	1.05026
$808$	0	0
$809$	−10.8494	−0.381445	−0.190723	−	0.981644i	$-0.561083\pi$
$809$	−10.8494	−0.381445	−0.190723	+	0.981644i	$0.561083\pi$
$810$	0	0
$811$	−19.3176	−0.678331	−0.339166	−	0.940727i	$-0.610145\pi$
$811$	−19.3176	−0.678331	−0.339166	+	0.940727i	$0.610145\pi$
$812$	0	0
$813$	−26.2017	−0.918935
$814$	0	0
$815$	−34.0203	−1.19168
$816$	0	0
$817$	−20.0678	−0.702085
$818$	0	0
$819$	−0.159073	−0.00555847
$820$	0	0
$821$	0.499718	0.0174403	0.00872013	−	0.999962i	$-0.497224\pi$
$821$	0.499718	0.0174403	0.00872013	+	0.999962i	$0.497224\pi$
$822$	0	0
$823$	−13.3239	−0.464440	−0.232220	−	0.972663i	$-0.574599\pi$
$823$	−13.3239	−0.464440	−0.232220	+	0.972663i	$0.574599\pi$
$824$	0	0
$825$	−3.25842	−0.113444
$826$	0	0
$827$	−50.7089	−1.76332	−0.881661	−	0.471883i	$-0.843574\pi$
$827$	−50.7089	−1.76332	−0.881661	+	0.471883i	$0.843574\pi$
$828$	0	0
$829$	−50.2587	−1.74556	−0.872778	−	0.488117i	$-0.837684\pi$
$829$	−50.2587	−1.74556	−0.872778	+	0.488117i	$0.837684\pi$
$830$	0	0
$831$	−29.0819	−1.00884
$832$	0	0
$833$	−16.9426	−0.587027
$834$	0	0
$835$	13.3415	0.461703
$836$	0	0
$837$	5.87728	0.203149
$838$	0	0
$839$	−8.50592	−0.293657	−0.146828	−	0.989162i	$-0.546907\pi$
$839$	−8.50592	−0.293657	−0.146828	+	0.989162i	$0.546907\pi$
$840$	0	0
$841$	−10.7802	−0.371730
$842$	0	0
$843$	3.74711	0.129057
$844$	0	0
$845$	37.8559	1.30228
$846$	0	0
$847$	−9.15103	−0.314433
$848$	0	0
$849$	40.6291	1.39439
$850$	0	0
$851$	31.6138	1.08371
$852$	0	0
$853$	−11.0777	−0.379295	−0.189647	−	0.981852i	$-0.560734\pi$
$853$	−11.0777	−0.379295	−0.189647	+	0.981852i	$0.560734\pi$
$854$	0	0
$855$	1.26912	0.0434029
$856$	0	0
$857$	9.07021	0.309832	0.154916	−	0.987928i	$-0.450489\pi$
$857$	9.07021	0.309832	0.154916	+	0.987928i	$0.450489\pi$
$858$	0	0
$859$	−40.8995	−1.39547	−0.697737	−	0.716354i	$-0.745810\pi$
$859$	−40.8995	−1.39547	−0.697737	+	0.716354i	$0.745810\pi$
$860$	0	0
$861$	0.145473	0.00495770
$862$	0	0
$863$	−14.0328	−0.477682	−0.238841	−	0.971059i	$-0.576767\pi$
$863$	−14.0328	−0.477682	−0.238841	+	0.971059i	$0.576767\pi$
$864$	0	0
$865$	38.9747	1.32518
$866$	0	0
$867$	−17.3124	−0.587961
$868$	0	0
$869$	−2.79154	−0.0946967
$870$	0	0
$871$	5.21906	0.176841
$872$	0	0
$873$	−1.20136	−0.0406600
$874$	0	0
$875$	−3.30693	−0.111795
$876$	0	0
$877$	−24.8320	−0.838518	−0.419259	−	0.907867i	$-0.637710\pi$
$877$	−24.8320	−0.838518	−0.419259	+	0.907867i	$0.637710\pi$
$878$	0	0
$879$	−52.1744	−1.75980
$880$	0	0
$881$	3.31728	0.111762	0.0558811	−	0.998437i	$-0.482203\pi$
$881$	3.31728	0.111762	0.0558811	+	0.998437i	$0.482203\pi$
$882$	0	0
$883$	−5.25743	−0.176927	−0.0884633	−	0.996079i	$-0.528196\pi$
$883$	−5.25743	−0.176927	−0.0884633	+	0.996079i	$0.528196\pi$
$884$	0	0
$885$	−29.9194	−1.00573
$886$	0	0
$887$	−7.16812	−0.240682	−0.120341	−	0.992733i	$-0.538399\pi$
$887$	−7.16812	−0.240682	−0.120341	+	0.992733i	$0.538399\pi$
$888$	0	0
$889$	−9.19777	−0.308483
$890$	0	0
$891$	2.79968	0.0937926
$892$	0	0
$893$	10.9894	0.367746
$894$	0	0
$895$	−40.4571	−1.35233
$896$	0	0
$897$	−11.5069	−0.384206
$898$	0	0
$899$	−4.95623	−0.165299
$900$	0	0
$901$	−9.19241	−0.306243
$902$	0	0
$903$	11.5166	0.383248
$904$	0	0
$905$	17.1754	0.570929
$906$	0	0
$907$	41.8851	1.39077	0.695386	−	0.718636i	$-0.255233\pi$
$907$	41.8851	1.39077	0.695386	+	0.718636i	$0.255233\pi$
$908$	0	0
$909$	−0.694673	−0.0230408
$910$	0	0
$911$	47.7780	1.58296	0.791478	−	0.611197i	$-0.209312\pi$
$911$	47.7780	1.58296	0.791478	+	0.611197i	$0.209312\pi$
$912$	0	0
$913$	4.71107	0.155914
$914$	0	0
$915$	69.7069	2.30444
$916$	0	0
$917$	15.4207	0.509236
$918$	0	0
$919$	47.4339	1.56470	0.782350	−	0.622840i	$-0.214021\pi$
$919$	47.4339	1.56470	0.782350	+	0.622840i	$0.214021\pi$
$920$	0	0
$921$	15.8996	0.523909
$922$	0	0
$923$	3.59043	0.118181
$924$	0	0
$925$	39.0082	1.28258
$926$	0	0
$927$	−0.210873	−0.00692598
$928$	0	0
$929$	16.0929	0.527991	0.263996	−	0.964524i	$-0.414960\pi$
$929$	16.0929	0.527991	0.263996	+	0.964524i	$0.414960\pi$
$930$	0	0
$931$	16.3221	0.534936
$932$	0	0
$933$	−7.58793	−0.248418
$934$	0	0
$935$	2.67444	0.0874637
$936$	0	0
$937$	50.8755	1.66203	0.831015	−	0.556251i	$-0.187761\pi$
$937$	50.8755	1.66203	0.831015	+	0.556251i	$0.187761\pi$
$938$	0	0
$939$	3.51929	0.114848
$940$	0	0
$941$	40.6160	1.32404	0.662021	−	0.749485i	$-0.269699\pi$
$941$	40.6160	1.32404	0.662021	+	0.749485i	$0.269699\pi$
$942$	0	0
$943$	0.489727	0.0159477
$944$	0	0
$945$	14.1934	0.461711
$946$	0	0
$947$	54.3798	1.76711	0.883553	−	0.468332i	$-0.155145\pi$
$947$	54.3798	1.76711	0.883553	+	0.468332i	$0.155145\pi$
$948$	0	0
$949$	16.6488	0.540442
$950$	0	0
$951$	37.5905	1.21895
$952$	0	0
$953$	−40.3359	−1.30661	−0.653304	−	0.757096i	$-0.726618\pi$
$953$	−40.3359	−1.30661	−0.653304	+	0.757096i	$0.726618\pi$
$954$	0	0
$955$	−44.1392	−1.42831
$956$	0	0
$957$	−2.25080	−0.0727581
$958$	0	0
$959$	−11.0727	−0.357557
$960$	0	0
$961$	−29.6518	−0.956509
$962$	0	0
$963$	−0.842219	−0.0271401
$964$	0	0
$965$	−50.3301	−1.62018
$966$	0	0
$967$	−39.8958	−1.28296	−0.641481	−	0.767139i	$-0.721680\pi$
$967$	−39.8958	−1.28296	−0.641481	+	0.767139i	$0.721680\pi$
$968$	0	0
$969$	−12.3721	−0.397450
$970$	0	0
$971$	−35.8777	−1.15137	−0.575684	−	0.817672i	$-0.695264\pi$
$971$	−35.8777	−1.15137	−0.575684	+	0.817672i	$0.695264\pi$
$972$	0	0
$973$	−14.4361	−0.462799
$974$	0	0
$975$	−14.1984	−0.454713
$976$	0	0
$977$	−10.6239	−0.339890	−0.169945	−	0.985454i	$-0.554359\pi$
$977$	−10.6239	−0.339890	−0.169945	+	0.985454i	$0.554359\pi$
$978$	0	0
$979$	3.50265	0.111945
$980$	0	0
$981$	1.01554	0.0324237
$982$	0	0
$983$	41.7017	1.33008	0.665039	−	0.746809i	$-0.268415\pi$
$983$	41.7017	1.33008	0.665039	+	0.746809i	$0.268415\pi$
$984$	0	0
$985$	−92.4333	−2.94517
$986$	0	0
$987$	−6.30663	−0.200742
$988$	0	0
$989$	38.7701	1.23282
$990$	0	0
$991$	−43.2323	−1.37332	−0.686660	−	0.726979i	$-0.740924\pi$
$991$	−43.2323	−1.37332	−0.686660	+	0.726979i	$0.740924\pi$
$992$	0	0
$993$	−41.2180	−1.30801
$994$	0	0
$995$	13.6670	0.433274
$996$	0	0
$997$	23.4982	0.744195	0.372098	−	0.928194i	$-0.378639\pi$
$997$	23.4982	0.744195	0.372098	+	0.928194i	$0.378639\pi$
$998$	0	0
$999$	−31.9530	−1.01095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.37	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.37	✓	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)

\(f(q)\)	\(=\)	\(q+1.77382 q^{3} -3.34355 q^{5} +0.838649 q^{7} +0.146429 q^{9} +O(q^{10})\)	\(q+1.77382 q^{3} -3.34355 q^{5} +0.838649 q^{7} +0.146429 q^{9} -0.297273 q^{11} -1.29535 q^{13} -5.93085 q^{15} +2.69073 q^{17} -2.59218 q^{19} +1.48761 q^{21} +5.00797 q^{23} +6.17933 q^{25} -5.06171 q^{27} +4.26847 q^{29} -1.16112 q^{31} -0.527309 q^{33} -2.80407 q^{35} +6.31269 q^{37} -2.29772 q^{39} +0.0977895 q^{41} +7.74167 q^{43} -0.489594 q^{45} -4.23944 q^{47} -6.29667 q^{49} +4.77286 q^{51} -3.41633 q^{53} +0.993949 q^{55} -4.59806 q^{57} +5.04471 q^{59} -11.7533 q^{61} +0.122803 q^{63} +4.33108 q^{65} -4.02906 q^{67} +8.88323 q^{69} -2.77177 q^{71} -12.8527 q^{73} +10.9610 q^{75} -0.249308 q^{77} +9.39050 q^{79} -9.41785 q^{81} -15.8476 q^{83} -8.99658 q^{85} +7.57148 q^{87} -11.7826 q^{89} -1.08635 q^{91} -2.05962 q^{93} +8.66710 q^{95} -8.20439 q^{97} -0.0435295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

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