LMFDB - Embedded newform 6004.2.a.e.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22506 q^{3} -3.09603 q^{5} -0.873879 q^{7} +1.95088 q^{9} +O(q^{10})$	$q+2.22506 q^{3} -3.09603 q^{5} -0.873879 q^{7} +1.95088 q^{9} -6.13971 q^{11} -6.06842 q^{13} -6.88885 q^{15} +3.12561 q^{17} +1.00000 q^{19} -1.94443 q^{21} -2.75685 q^{23} +4.58542 q^{25} -2.33436 q^{27} +8.05387 q^{29} +6.25590 q^{31} -13.6612 q^{33} +2.70556 q^{35} +9.84640 q^{37} -13.5026 q^{39} +8.30989 q^{41} -11.1777 q^{43} -6.03998 q^{45} +2.54321 q^{47} -6.23633 q^{49} +6.95466 q^{51} -5.31982 q^{53} +19.0087 q^{55} +2.22506 q^{57} +5.54567 q^{59} +12.4143 q^{61} -1.70483 q^{63} +18.7880 q^{65} -15.1857 q^{67} -6.13414 q^{69} -9.47528 q^{71} +1.94076 q^{73} +10.2028 q^{75} +5.36537 q^{77} +1.00000 q^{79} -11.0467 q^{81} +3.56701 q^{83} -9.67698 q^{85} +17.9203 q^{87} +5.57267 q^{89} +5.30307 q^{91} +13.9197 q^{93} -3.09603 q^{95} +10.4884 q^{97} -11.9778 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	2.22506	1.28464	0.642319	−	0.766438i	$-0.277973\pi$
$3$	2.22506	1.28464	0.642319	+	0.766438i	$0.277973\pi$
$4$	0	0
$5$	−3.09603	−1.38459	−0.692294	−	0.721616i	$-0.743400\pi$
$5$	−3.09603	−1.38459	−0.692294	+	0.721616i	$0.743400\pi$
$6$	0	0
$7$	−0.873879	−0.330295	−0.165148	−	0.986269i	$-0.552810\pi$
$7$	−0.873879	−0.330295	−0.165148	+	0.986269i	$0.552810\pi$
$8$	0	0
$9$	1.95088	0.650293
$10$	0	0
$11$	−6.13971	−1.85119	−0.925596	−	0.378513i	$-0.876436\pi$
$11$	−6.13971	−1.85119	−0.925596	+	0.378513i	$0.876436\pi$
$12$	0	0
$13$	−6.06842	−1.68308	−0.841539	−	0.540196i	$-0.818350\pi$
$13$	−6.06842	−1.68308	−0.841539	+	0.540196i	$0.818350\pi$
$14$	0	0
$15$	−6.88885	−1.77869
$16$	0	0
$17$	3.12561	0.758071	0.379036	−	0.925382i	$-0.376256\pi$
$17$	3.12561	0.758071	0.379036	+	0.925382i	$0.376256\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.94443	−0.424310
$22$	0	0
$23$	−2.75685	−0.574842	−0.287421	−	0.957804i	$-0.592798\pi$
$23$	−2.75685	−0.574842	−0.287421	+	0.957804i	$0.592798\pi$
$24$	0	0
$25$	4.58542	0.917083
$26$	0	0
$27$	−2.33436	−0.449247
$28$	0	0
$29$	8.05387	1.49557	0.747783	−	0.663943i	$-0.231118\pi$
$29$	8.05387	1.49557	0.747783	+	0.663943i	$0.231118\pi$
$30$	0	0
$31$	6.25590	1.12359	0.561797	−	0.827275i	$-0.310110\pi$
$31$	6.25590	1.12359	0.561797	+	0.827275i	$0.310110\pi$
$32$	0	0
$33$	−13.6612	−2.37811
$34$	0	0
$35$	2.70556	0.457323
$36$	0	0
$37$	9.84640	1.61874	0.809369	−	0.587300i	$-0.199809\pi$
$37$	9.84640	1.61874	0.809369	+	0.587300i	$0.199809\pi$
$38$	0	0
$39$	−13.5026	−2.16214
$40$	0	0
$41$	8.30989	1.29779	0.648893	−	0.760880i	$-0.275232\pi$
$41$	8.30989	1.29779	0.648893	+	0.760880i	$0.275232\pi$
$42$	0	0
$43$	−11.1777	−1.70459	−0.852293	−	0.523064i	$-0.824789\pi$
$43$	−11.1777	−1.70459	−0.852293	+	0.523064i	$0.824789\pi$
$44$	0	0
$45$	−6.03998	−0.900387
$46$	0	0
$47$	2.54321	0.370965	0.185483	−	0.982648i	$-0.440615\pi$
$47$	2.54321	0.370965	0.185483	+	0.982648i	$0.440615\pi$
$48$	0	0
$49$	−6.23633	−0.890905
$50$	0	0
$51$	6.95466	0.973847
$52$	0	0
$53$	−5.31982	−0.730733	−0.365367	−	0.930864i	$-0.619056\pi$
$53$	−5.31982	−0.730733	−0.365367	+	0.930864i	$0.619056\pi$
$54$	0	0
$55$	19.0087	2.56314
$56$	0	0
$57$	2.22506	0.294716
$58$	0	0
$59$	5.54567	0.721984	0.360992	−	0.932569i	$-0.382438\pi$
$59$	5.54567	0.721984	0.360992	+	0.932569i	$0.382438\pi$
$60$	0	0
$61$	12.4143	1.58949	0.794744	−	0.606944i	$-0.207605\pi$
$61$	12.4143	1.58949	0.794744	+	0.606944i	$0.207605\pi$
$62$	0	0
$63$	−1.70483	−0.214789
$64$	0	0
$65$	18.7880	2.33037
$66$	0	0
$67$	−15.1857	−1.85522	−0.927611	−	0.373547i	$-0.878142\pi$
$67$	−15.1857	−1.85522	−0.927611	+	0.373547i	$0.878142\pi$
$68$	0	0
$69$	−6.13414	−0.738463
$70$	0	0
$71$	−9.47528	−1.12451	−0.562254	−	0.826964i	$-0.690066\pi$
$71$	−9.47528	−1.12451	−0.562254	+	0.826964i	$0.690066\pi$
$72$	0	0
$73$	1.94076	0.227149	0.113575	−	0.993529i	$-0.463770\pi$
$73$	1.94076	0.227149	0.113575	+	0.993529i	$0.463770\pi$
$74$	0	0
$75$	10.2028	1.17812
$76$	0	0
$77$	5.36537	0.611440
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−11.0467	−1.22741
$82$	0	0
$83$	3.56701	0.391530	0.195765	−	0.980651i	$-0.437281\pi$
$83$	3.56701	0.391530	0.195765	+	0.980651i	$0.437281\pi$
$84$	0	0
$85$	−9.67698	−1.04962
$86$	0	0
$87$	17.9203	1.92126
$88$	0	0
$89$	5.57267	0.590702	0.295351	−	0.955389i	$-0.404563\pi$
$89$	5.57267	0.590702	0.295351	+	0.955389i	$0.404563\pi$
$90$	0	0
$91$	5.30307	0.555913
$92$	0	0
$93$	13.9197	1.44341
$94$	0	0
$95$	−3.09603	−0.317646
$96$	0	0
$97$	10.4884	1.06493	0.532466	−	0.846451i	$-0.321265\pi$
$97$	10.4884	1.06493	0.532466	+	0.846451i	$0.321265\pi$
$98$	0	0
$99$	−11.9778	−1.20382
$100$	0	0
$101$	9.45537	0.940844	0.470422	−	0.882442i	$-0.344102\pi$
$101$	9.45537	0.940844	0.470422	+	0.882442i	$0.344102\pi$
$102$	0	0
$103$	8.70697	0.857923	0.428962	−	0.903323i	$-0.358880\pi$
$103$	8.70697	0.857923	0.428962	+	0.903323i	$0.358880\pi$
$104$	0	0
$105$	6.02002	0.587494
$106$	0	0
$107$	2.46787	0.238578	0.119289	−	0.992860i	$-0.461938\pi$
$107$	2.46787	0.238578	0.119289	+	0.992860i	$0.461938\pi$
$108$	0	0
$109$	5.20742	0.498781	0.249390	−	0.968403i	$-0.419770\pi$
$109$	5.20742	0.498781	0.249390	+	0.968403i	$0.419770\pi$
$110$	0	0
$111$	21.9088	2.07949
$112$	0	0
$113$	5.05149	0.475204	0.237602	−	0.971363i	$-0.423639\pi$
$113$	5.05149	0.475204	0.237602	+	0.971363i	$0.423639\pi$
$114$	0	0
$115$	8.53528	0.795919
$116$	0	0
$117$	−11.8388	−1.09449
$118$	0	0
$119$	−2.73140	−0.250387
$120$	0	0
$121$	26.6960	2.42691
$122$	0	0
$123$	18.4900	1.66718
$124$	0	0
$125$	1.28356	0.114805
$126$	0	0
$127$	−14.2864	−1.26772	−0.633858	−	0.773449i	$-0.718530\pi$
$127$	−14.2864	−1.26772	−0.633858	+	0.773449i	$0.718530\pi$
$128$	0	0
$129$	−24.8711	−2.18978
$130$	0	0
$131$	5.08111	0.443939	0.221969	−	0.975054i	$-0.428751\pi$
$131$	5.08111	0.443939	0.221969	+	0.975054i	$0.428751\pi$
$132$	0	0
$133$	−0.873879	−0.0757750
$134$	0	0
$135$	7.22724	0.622022
$136$	0	0
$137$	−3.87749	−0.331276	−0.165638	−	0.986187i	$-0.552968\pi$
$137$	−3.87749	−0.331276	−0.165638	+	0.986187i	$0.552968\pi$
$138$	0	0
$139$	−7.75992	−0.658188	−0.329094	−	0.944297i	$-0.606743\pi$
$139$	−7.75992	−0.658188	−0.329094	+	0.944297i	$0.606743\pi$
$140$	0	0
$141$	5.65879	0.476556
$142$	0	0
$143$	37.2584	3.11570
$144$	0	0
$145$	−24.9350	−2.07074
$146$	0	0
$147$	−13.8762	−1.14449
$148$	0	0
$149$	11.5577	0.946846	0.473423	−	0.880835i	$-0.343018\pi$
$149$	11.5577	0.946846	0.473423	+	0.880835i	$0.343018\pi$
$150$	0	0
$151$	8.20750	0.667917	0.333959	−	0.942588i	$-0.391615\pi$
$151$	8.20750	0.667917	0.333959	+	0.942588i	$0.391615\pi$
$152$	0	0
$153$	6.09768	0.492968
$154$	0	0
$155$	−19.3685	−1.55571
$156$	0	0
$157$	12.4556	0.994067	0.497034	−	0.867731i	$-0.334423\pi$
$157$	12.4556	0.994067	0.497034	+	0.867731i	$0.334423\pi$
$158$	0	0
$159$	−11.8369	−0.938727
$160$	0	0
$161$	2.40915	0.189868
$162$	0	0
$163$	2.47686	0.194003	0.0970015	−	0.995284i	$-0.469075\pi$
$163$	2.47686	0.194003	0.0970015	+	0.995284i	$0.469075\pi$
$164$	0	0
$165$	42.2955	3.29270
$166$	0	0
$167$	21.5643	1.66870	0.834349	−	0.551237i	$-0.185844\pi$
$167$	21.5643	1.66870	0.834349	+	0.551237i	$0.185844\pi$
$168$	0	0
$169$	23.8258	1.83275
$170$	0	0
$171$	1.95088	0.149187
$172$	0	0
$173$	−2.56565	−0.195063	−0.0975313	−	0.995232i	$-0.531095\pi$
$173$	−2.56565	−0.195063	−0.0975313	+	0.995232i	$0.531095\pi$
$174$	0	0
$175$	−4.00710	−0.302908
$176$	0	0
$177$	12.3394	0.927488
$178$	0	0
$179$	18.4509	1.37908	0.689541	−	0.724246i	$-0.257812\pi$
$179$	18.4509	1.37908	0.689541	+	0.724246i	$0.257812\pi$
$180$	0	0
$181$	24.0449	1.78725	0.893623	−	0.448819i	$-0.148155\pi$
$181$	24.0449	1.78725	0.893623	+	0.448819i	$0.148155\pi$
$182$	0	0
$183$	27.6225	2.04192
$184$	0	0
$185$	−30.4848	−2.24129
$186$	0	0
$187$	−19.1903	−1.40334
$188$	0	0
$189$	2.03995	0.148384
$190$	0	0
$191$	−8.21352	−0.594309	−0.297155	−	0.954829i	$-0.596038\pi$
$191$	−8.21352	−0.594309	−0.297155	+	0.954829i	$0.596038\pi$
$192$	0	0
$193$	−25.3226	−1.82276	−0.911379	−	0.411567i	$-0.864982\pi$
$193$	−25.3226	−1.82276	−0.911379	+	0.411567i	$0.864982\pi$
$194$	0	0
$195$	41.8045	2.99368
$196$	0	0
$197$	−24.4551	−1.74235	−0.871175	−	0.490972i	$-0.836642\pi$
$197$	−24.4551	−1.74235	−0.871175	+	0.490972i	$0.836642\pi$
$198$	0	0
$199$	−18.0083	−1.27657	−0.638287	−	0.769799i	$-0.720357\pi$
$199$	−18.0083	−1.27657	−0.638287	+	0.769799i	$0.720357\pi$
$200$	0	0
$201$	−33.7889	−2.38329
$202$	0	0
$203$	−7.03811	−0.493978
$204$	0	0
$205$	−25.7277	−1.79690
$206$	0	0
$207$	−5.37827	−0.373816
$208$	0	0
$209$	−6.13971	−0.424693
$210$	0	0
$211$	−16.4938	−1.13548	−0.567739	−	0.823209i	$-0.692182\pi$
$211$	−16.4938	−1.13548	−0.567739	+	0.823209i	$0.692182\pi$
$212$	0	0
$213$	−21.0830	−1.44459
$214$	0	0
$215$	34.6066	2.36015
$216$	0	0
$217$	−5.46691	−0.371118
$218$	0	0
$219$	4.31831	0.291804
$220$	0	0
$221$	−18.9675	−1.27589
$222$	0	0
$223$	−18.9562	−1.26940	−0.634699	−	0.772759i	$-0.718876\pi$
$223$	−18.9562	−1.26940	−0.634699	+	0.772759i	$0.718876\pi$
$224$	0	0
$225$	8.94559	0.596373
$226$	0	0
$227$	−21.1904	−1.40646	−0.703228	−	0.710964i	$-0.748259\pi$
$227$	−21.1904	−1.40646	−0.703228	+	0.710964i	$0.748259\pi$
$228$	0	0
$229$	−15.3608	−1.01507	−0.507534	−	0.861632i	$-0.669443\pi$
$229$	−15.3608	−1.01507	−0.507534	+	0.861632i	$0.669443\pi$
$230$	0	0
$231$	11.9382	0.785479
$232$	0	0
$233$	−12.4770	−0.817393	−0.408696	−	0.912670i	$-0.634016\pi$
$233$	−12.4770	−0.817393	−0.408696	+	0.912670i	$0.634016\pi$
$234$	0	0
$235$	−7.87386	−0.513634
$236$	0	0
$237$	2.22506	0.144533
$238$	0	0
$239$	3.10784	0.201029	0.100515	−	0.994936i	$-0.467951\pi$
$239$	3.10784	0.201029	0.100515	+	0.994936i	$0.467951\pi$
$240$	0	0
$241$	17.4390	1.12335	0.561674	−	0.827359i	$-0.310158\pi$
$241$	17.4390	1.12335	0.561674	+	0.827359i	$0.310158\pi$
$242$	0	0
$243$	−17.5765	−1.12753
$244$	0	0
$245$	19.3079	1.23354
$246$	0	0
$247$	−6.06842	−0.386125
$248$	0	0
$249$	7.93680	0.502974
$250$	0	0
$251$	31.3295	1.97750	0.988752	−	0.149567i	$-0.0477878\pi$
$251$	31.3295	1.97750	0.988752	+	0.149567i	$0.0477878\pi$
$252$	0	0
$253$	16.9262	1.06414
$254$	0	0
$255$	−21.5318	−1.34838
$256$	0	0
$257$	−4.14850	−0.258776	−0.129388	−	0.991594i	$-0.541301\pi$
$257$	−4.14850	−0.258776	−0.129388	+	0.991594i	$0.541301\pi$
$258$	0	0
$259$	−8.60457	−0.534662
$260$	0	0
$261$	15.7121	0.972556
$262$	0	0
$263$	−1.89218	−0.116677	−0.0583383	−	0.998297i	$-0.518580\pi$
$263$	−1.89218	−0.116677	−0.0583383	+	0.998297i	$0.518580\pi$
$264$	0	0
$265$	16.4703	1.01176
$266$	0	0
$267$	12.3995	0.758838
$268$	0	0
$269$	14.8593	0.905986	0.452993	−	0.891514i	$-0.350356\pi$
$269$	14.8593	0.905986	0.452993	+	0.891514i	$0.350356\pi$
$270$	0	0
$271$	29.9015	1.81639	0.908195	−	0.418548i	$-0.137461\pi$
$271$	29.9015	1.81639	0.908195	+	0.418548i	$0.137461\pi$
$272$	0	0
$273$	11.7996	0.714146
$274$	0	0
$275$	−28.1531	−1.69770
$276$	0	0
$277$	−30.3673	−1.82460	−0.912299	−	0.409526i	$-0.865694\pi$
$277$	−30.3673	−1.82460	−0.912299	+	0.409526i	$0.865694\pi$
$278$	0	0
$279$	12.2045	0.730665
$280$	0	0
$281$	23.5311	1.40375	0.701874	−	0.712301i	$-0.252347\pi$
$281$	23.5311	1.40375	0.701874	+	0.712301i	$0.252347\pi$
$282$	0	0
$283$	−16.9213	−1.00587	−0.502935	−	0.864324i	$-0.667746\pi$
$283$	−16.9213	−1.00587	−0.502935	+	0.864324i	$0.667746\pi$
$284$	0	0
$285$	−6.88885	−0.408060
$286$	0	0
$287$	−7.26184	−0.428653
$288$	0	0
$289$	−7.23057	−0.425328
$290$	0	0
$291$	23.3372	1.36805
$292$	0	0
$293$	8.29711	0.484722	0.242361	−	0.970186i	$-0.422078\pi$
$293$	8.29711	0.484722	0.242361	+	0.970186i	$0.422078\pi$
$294$	0	0
$295$	−17.1696	−0.999651
$296$	0	0
$297$	14.3323	0.831643
$298$	0	0
$299$	16.7297	0.967504
$300$	0	0
$301$	9.76798	0.563017
$302$	0	0
$303$	21.0387	1.20864
$304$	0	0
$305$	−38.4351	−2.20079
$306$	0	0
$307$	17.5586	1.00212	0.501060	−	0.865413i	$-0.332944\pi$
$307$	17.5586	1.00212	0.501060	+	0.865413i	$0.332944\pi$
$308$	0	0
$309$	19.3735	1.10212
$310$	0	0
$311$	25.9064	1.46902	0.734508	−	0.678600i	$-0.237413\pi$
$311$	25.9064	1.46902	0.734508	+	0.678600i	$0.237413\pi$
$312$	0	0
$313$	2.69710	0.152449	0.0762246	−	0.997091i	$-0.475713\pi$
$313$	2.69710	0.152449	0.0762246	+	0.997091i	$0.475713\pi$
$314$	0	0
$315$	5.27822	0.297394
$316$	0	0
$317$	−18.3335	−1.02971	−0.514857	−	0.857276i	$-0.672155\pi$
$317$	−18.3335	−1.02971	−0.514857	+	0.857276i	$0.672155\pi$
$318$	0	0
$319$	−49.4484	−2.76858
$320$	0	0
$321$	5.49116	0.306487
$322$	0	0
$323$	3.12561	0.173913
$324$	0	0
$325$	−27.8263	−1.54352
$326$	0	0
$327$	11.5868	0.640752
$328$	0	0
$329$	−2.22246	−0.122528
$330$	0	0
$331$	17.3003	0.950912	0.475456	−	0.879739i	$-0.342283\pi$
$331$	17.3003	0.950912	0.475456	+	0.879739i	$0.342283\pi$
$332$	0	0
$333$	19.2091	1.05265
$334$	0	0
$335$	47.0153	2.56872
$336$	0	0
$337$	−24.9241	−1.35770	−0.678850	−	0.734277i	$-0.737521\pi$
$337$	−24.9241	−1.35770	−0.678850	+	0.734277i	$0.737521\pi$
$338$	0	0
$339$	11.2399	0.610465
$340$	0	0
$341$	−38.4094	−2.07999
$342$	0	0
$343$	11.5670	0.624557
$344$	0	0
$345$	18.9915	1.02247
$346$	0	0
$347$	−19.9282	−1.06980	−0.534901	−	0.844915i	$-0.679651\pi$
$347$	−19.9282	−1.06980	−0.534901	+	0.844915i	$0.679651\pi$
$348$	0	0
$349$	−0.873828	−0.0467750	−0.0233875	−	0.999726i	$-0.507445\pi$
$349$	−0.873828	−0.0467750	−0.0233875	+	0.999726i	$0.507445\pi$
$350$	0	0
$351$	14.1659	0.756118
$352$	0	0
$353$	6.05881	0.322478	0.161239	−	0.986915i	$-0.448451\pi$
$353$	6.05881	0.322478	0.161239	+	0.986915i	$0.448451\pi$
$354$	0	0
$355$	29.3358	1.55698
$356$	0	0
$357$	−6.07753	−0.321657
$358$	0	0
$359$	−15.0678	−0.795246	−0.397623	−	0.917549i	$-0.630165\pi$
$359$	−15.0678	−0.795246	−0.397623	+	0.917549i	$0.630165\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	59.4002	3.11770
$364$	0	0
$365$	−6.00866	−0.314508
$366$	0	0
$367$	21.2960	1.11164	0.555820	−	0.831302i	$-0.312404\pi$
$367$	21.2960	1.11164	0.555820	+	0.831302i	$0.312404\pi$
$368$	0	0
$369$	16.2116	0.843941
$370$	0	0
$371$	4.64888	0.241358
$372$	0	0
$373$	7.26468	0.376151	0.188075	−	0.982155i	$-0.439775\pi$
$373$	7.26468	0.376151	0.188075	+	0.982155i	$0.439775\pi$
$374$	0	0
$375$	2.85600	0.147483
$376$	0	0
$377$	−48.8743	−2.51715
$378$	0	0
$379$	−11.1270	−0.571555	−0.285777	−	0.958296i	$-0.592252\pi$
$379$	−11.1270	−0.571555	−0.285777	+	0.958296i	$0.592252\pi$
$380$	0	0
$381$	−31.7881	−1.62856
$382$	0	0
$383$	19.5773	1.00035	0.500176	−	0.865924i	$-0.333269\pi$
$383$	19.5773	1.00035	0.500176	+	0.865924i	$0.333269\pi$
$384$	0	0
$385$	−16.6113	−0.846593
$386$	0	0
$387$	−21.8064	−1.10848
$388$	0	0
$389$	15.9409	0.808234	0.404117	−	0.914707i	$-0.367579\pi$
$389$	15.9409	0.808234	0.404117	+	0.914707i	$0.367579\pi$
$390$	0	0
$391$	−8.61682	−0.435771
$392$	0	0
$393$	11.3058	0.570300
$394$	0	0
$395$	−3.09603	−0.155778
$396$	0	0
$397$	12.9616	0.650523	0.325261	−	0.945624i	$-0.394548\pi$
$397$	12.9616	0.650523	0.325261	+	0.945624i	$0.394548\pi$
$398$	0	0
$399$	−1.94443	−0.0973433
$400$	0	0
$401$	23.5638	1.17672	0.588361	−	0.808598i	$-0.299773\pi$
$401$	23.5638	1.17672	0.588361	+	0.808598i	$0.299773\pi$
$402$	0	0
$403$	−37.9635	−1.89110
$404$	0	0
$405$	34.2010	1.69946
$406$	0	0
$407$	−60.4541	−2.99660
$408$	0	0
$409$	−10.2677	−0.507703	−0.253852	−	0.967243i	$-0.581697\pi$
$409$	−10.2677	−0.507703	−0.253852	+	0.967243i	$0.581697\pi$
$410$	0	0
$411$	−8.62763	−0.425570
$412$	0	0
$413$	−4.84624	−0.238468
$414$	0	0
$415$	−11.0436	−0.542108
$416$	0	0
$417$	−17.2663	−0.845532
$418$	0	0
$419$	−27.7098	−1.35371	−0.676856	−	0.736115i	$-0.736658\pi$
$419$	−27.7098	−1.35371	−0.676856	+	0.736115i	$0.736658\pi$
$420$	0	0
$421$	−12.1413	−0.591729	−0.295865	−	0.955230i	$-0.595608\pi$
$421$	−12.1413	−0.591729	−0.295865	+	0.955230i	$0.595608\pi$
$422$	0	0
$423$	4.96149	0.241236
$424$	0	0
$425$	14.3322	0.695215
$426$	0	0
$427$	−10.8486	−0.525001
$428$	0	0
$429$	82.9020	4.00255
$430$	0	0
$431$	39.5929	1.90712	0.953561	−	0.301199i	$-0.0973870\pi$
$431$	39.5929	1.90712	0.953561	+	0.301199i	$0.0973870\pi$
$432$	0	0
$433$	−11.3268	−0.544332	−0.272166	−	0.962250i	$-0.587740\pi$
$433$	−11.3268	−0.544332	−0.272166	+	0.962250i	$0.587740\pi$
$434$	0	0
$435$	−55.4819	−2.66015
$436$	0	0
$437$	−2.75685	−0.131878
$438$	0	0
$439$	4.44240	0.212024	0.106012	−	0.994365i	$-0.466192\pi$
$439$	4.44240	0.212024	0.106012	+	0.994365i	$0.466192\pi$
$440$	0	0
$441$	−12.1663	−0.579349
$442$	0	0
$443$	−14.4595	−0.686993	−0.343497	−	0.939154i	$-0.611611\pi$
$443$	−14.4595	−0.686993	−0.343497	+	0.939154i	$0.611611\pi$
$444$	0	0
$445$	−17.2532	−0.817879
$446$	0	0
$447$	25.7166	1.21635
$448$	0	0
$449$	11.4457	0.540154	0.270077	−	0.962839i	$-0.412951\pi$
$449$	11.4457	0.540154	0.270077	+	0.962839i	$0.412951\pi$
$450$	0	0
$451$	−51.0203	−2.40245
$452$	0	0
$453$	18.2622	0.858031
$454$	0	0
$455$	−16.4185	−0.769710
$456$	0	0
$457$	31.9567	1.49487	0.747436	−	0.664334i	$-0.231285\pi$
$457$	31.9567	1.49487	0.747436	+	0.664334i	$0.231285\pi$
$458$	0	0
$459$	−7.29628	−0.340561
$460$	0	0
$461$	17.4735	0.813822	0.406911	−	0.913468i	$-0.366606\pi$
$461$	17.4735	0.813822	0.406911	+	0.913468i	$0.366606\pi$
$462$	0	0
$463$	11.5694	0.537677	0.268839	−	0.963185i	$-0.413360\pi$
$463$	11.5694	0.537677	0.268839	+	0.963185i	$0.413360\pi$
$464$	0	0
$465$	−43.0960	−1.99853
$466$	0	0
$467$	−1.84092	−0.0851876	−0.0425938	−	0.999092i	$-0.513562\pi$
$467$	−1.84092	−0.0851876	−0.0425938	+	0.999092i	$0.513562\pi$
$468$	0	0
$469$	13.2704	0.612771
$470$	0	0
$471$	27.7145	1.27702
$472$	0	0
$473$	68.6280	3.15552
$474$	0	0
$475$	4.58542	0.210393
$476$	0	0
$477$	−10.3783	−0.475190
$478$	0	0
$479$	5.03853	0.230216	0.115108	−	0.993353i	$-0.463279\pi$
$479$	5.03853	0.230216	0.115108	+	0.993353i	$0.463279\pi$
$480$	0	0
$481$	−59.7521	−2.72446
$482$	0	0
$483$	5.36050	0.243911
$484$	0	0
$485$	−32.4723	−1.47449
$486$	0	0
$487$	33.5355	1.51964	0.759818	−	0.650135i	$-0.225288\pi$
$487$	33.5355	1.51964	0.759818	+	0.650135i	$0.225288\pi$
$488$	0	0
$489$	5.51116	0.249223
$490$	0	0
$491$	4.05782	0.183127	0.0915635	−	0.995799i	$-0.470814\pi$
$491$	4.05782	0.183127	0.0915635	+	0.995799i	$0.470814\pi$
$492$	0	0
$493$	25.1732	1.13375
$494$	0	0
$495$	37.0837	1.66679
$496$	0	0
$497$	8.28025	0.371420
$498$	0	0
$499$	−23.1694	−1.03721	−0.518603	−	0.855015i	$-0.673548\pi$
$499$	−23.1694	−1.03721	−0.518603	+	0.855015i	$0.673548\pi$
$500$	0	0
$501$	47.9818	2.14367
$502$	0	0
$503$	−8.56255	−0.381785	−0.190893	−	0.981611i	$-0.561138\pi$
$503$	−8.56255	−0.381785	−0.190893	+	0.981611i	$0.561138\pi$
$504$	0	0
$505$	−29.2741	−1.30268
$506$	0	0
$507$	53.0137	2.35442
$508$	0	0
$509$	1.68747	0.0747957	0.0373979	−	0.999300i	$-0.488093\pi$
$509$	1.68747	0.0747957	0.0373979	+	0.999300i	$0.488093\pi$
$510$	0	0
$511$	−1.69599	−0.0750263
$512$	0	0
$513$	−2.33436	−0.103064
$514$	0	0
$515$	−26.9571	−1.18787
$516$	0	0
$517$	−15.6146	−0.686728
$518$	0	0
$519$	−5.70871	−0.250585
$520$	0	0
$521$	−36.7097	−1.60828	−0.804140	−	0.594440i	$-0.797374\pi$
$521$	−36.7097	−1.60828	−0.804140	+	0.594440i	$0.797374\pi$
$522$	0	0
$523$	−3.98172	−0.174109	−0.0870543	−	0.996204i	$-0.527745\pi$
$523$	−3.98172	−0.174109	−0.0870543	+	0.996204i	$0.527745\pi$
$524$	0	0
$525$	−8.91603	−0.389127
$526$	0	0
$527$	19.5535	0.851764
$528$	0	0
$529$	−15.3998	−0.669557
$530$	0	0
$531$	10.8189	0.469501
$532$	0	0
$533$	−50.4279	−2.18428
$534$	0	0
$535$	−7.64062	−0.330333
$536$	0	0
$537$	41.0542	1.77162
$538$	0	0
$539$	38.2893	1.64924
$540$	0	0
$541$	−7.02837	−0.302173	−0.151087	−	0.988521i	$-0.548277\pi$
$541$	−7.02837	−0.302173	−0.151087	+	0.988521i	$0.548277\pi$
$542$	0	0
$543$	53.5013	2.29596
$544$	0	0
$545$	−16.1223	−0.690606
$546$	0	0
$547$	−7.81564	−0.334172	−0.167086	−	0.985942i	$-0.553436\pi$
$547$	−7.81564	−0.334172	−0.167086	+	0.985942i	$0.553436\pi$
$548$	0	0
$549$	24.2188	1.03363
$550$	0	0
$551$	8.05387	0.343106
$552$	0	0
$553$	−0.873879	−0.0371611
$554$	0	0
$555$	−67.8304	−2.87924
$556$	0	0
$557$	−2.63364	−0.111591	−0.0557954	−	0.998442i	$-0.517769\pi$
$557$	−2.63364	−0.111591	−0.0557954	+	0.998442i	$0.517769\pi$
$558$	0	0
$559$	67.8312	2.86895
$560$	0	0
$561$	−42.6996	−1.80278
$562$	0	0
$563$	1.40876	0.0593722	0.0296861	−	0.999559i	$-0.490549\pi$
$563$	1.40876	0.0593722	0.0296861	+	0.999559i	$0.490549\pi$
$564$	0	0
$565$	−15.6396	−0.657962
$566$	0	0
$567$	9.65349	0.405409
$568$	0	0
$569$	8.49794	0.356252	0.178126	−	0.984008i	$-0.442997\pi$
$569$	8.49794	0.356252	0.178126	+	0.984008i	$0.442997\pi$
$570$	0	0
$571$	−23.0595	−0.965010	−0.482505	−	0.875893i	$-0.660273\pi$
$571$	−23.0595	−0.965010	−0.482505	+	0.875893i	$0.660273\pi$
$572$	0	0
$573$	−18.2755	−0.763472
$574$	0	0
$575$	−12.6413	−0.527178
$576$	0	0
$577$	44.3936	1.84813	0.924064	−	0.382237i	$-0.124846\pi$
$577$	44.3936	1.84813	0.924064	+	0.382237i	$0.124846\pi$
$578$	0	0
$579$	−56.3442	−2.34158
$580$	0	0
$581$	−3.11714	−0.129321
$582$	0	0
$583$	32.6621	1.35273
$584$	0	0
$585$	36.6532	1.51542
$586$	0	0
$587$	−33.6150	−1.38744	−0.693719	−	0.720246i	$-0.744029\pi$
$587$	−33.6150	−1.38744	−0.693719	+	0.720246i	$0.744029\pi$
$588$	0	0
$589$	6.25590	0.257770
$590$	0	0
$591$	−54.4139	−2.23829
$592$	0	0
$593$	34.8254	1.43011	0.715054	−	0.699070i	$-0.246402\pi$
$593$	34.8254	1.43011	0.715054	+	0.699070i	$0.246402\pi$
$594$	0	0
$595$	8.45652	0.346683
$596$	0	0
$597$	−40.0695	−1.63993
$598$	0	0
$599$	41.1260	1.68036	0.840181	−	0.542306i	$-0.182449\pi$
$599$	41.1260	1.68036	0.840181	+	0.542306i	$0.182449\pi$
$600$	0	0
$601$	−44.0950	−1.79867	−0.899337	−	0.437257i	$-0.855950\pi$
$601$	−44.0950	−1.79867	−0.899337	+	0.437257i	$0.855950\pi$
$602$	0	0
$603$	−29.6254	−1.20644
$604$	0	0
$605$	−82.6518	−3.36027
$606$	0	0
$607$	2.05947	0.0835912	0.0417956	−	0.999126i	$-0.486692\pi$
$607$	2.05947	0.0835912	0.0417956	+	0.999126i	$0.486692\pi$
$608$	0	0
$609$	−15.6602	−0.634583
$610$	0	0
$611$	−15.4333	−0.624363
$612$	0	0
$613$	−19.0967	−0.771307	−0.385654	−	0.922644i	$-0.626024\pi$
$613$	−19.0967	−0.771307	−0.385654	+	0.922644i	$0.626024\pi$
$614$	0	0
$615$	−57.2455	−2.30836
$616$	0	0
$617$	18.7733	0.755787	0.377893	−	0.925849i	$-0.376649\pi$
$617$	18.7733	0.755787	0.377893	+	0.925849i	$0.376649\pi$
$618$	0	0
$619$	28.9514	1.16365	0.581827	−	0.813312i	$-0.302338\pi$
$619$	28.9514	1.16365	0.581827	+	0.813312i	$0.302338\pi$
$620$	0	0
$621$	6.43546	0.258246
$622$	0	0
$623$	−4.86984	−0.195106
$624$	0	0
$625$	−26.9010	−1.07604
$626$	0	0
$627$	−13.6612	−0.545576
$628$	0	0
$629$	30.7760	1.22712
$630$	0	0
$631$	37.4711	1.49170	0.745850	−	0.666114i	$-0.232043\pi$
$631$	37.4711	1.49170	0.745850	+	0.666114i	$0.232043\pi$
$632$	0	0
$633$	−36.6996	−1.45868
$634$	0	0
$635$	44.2313	1.75526
$636$	0	0
$637$	37.8447	1.49946
$638$	0	0
$639$	−18.4851	−0.731260
$640$	0	0
$641$	9.27299	0.366261	0.183131	−	0.983089i	$-0.441377\pi$
$641$	9.27299	0.366261	0.183131	+	0.983089i	$0.441377\pi$
$642$	0	0
$643$	−15.0536	−0.593655	−0.296827	−	0.954931i	$-0.595929\pi$
$643$	−15.0536	−0.593655	−0.296827	+	0.954931i	$0.595929\pi$
$644$	0	0
$645$	77.0016	3.03194
$646$	0	0
$647$	28.0674	1.10345	0.551723	−	0.834028i	$-0.313971\pi$
$647$	28.0674	1.10345	0.551723	+	0.834028i	$0.313971\pi$
$648$	0	0
$649$	−34.0488	−1.33653
$650$	0	0
$651$	−12.1642	−0.476752
$652$	0	0
$653$	−7.72136	−0.302160	−0.151080	−	0.988522i	$-0.548275\pi$
$653$	−7.72136	−0.302160	−0.151080	+	0.988522i	$0.548275\pi$
$654$	0	0
$655$	−15.7313	−0.614672
$656$	0	0
$657$	3.78619	0.147713
$658$	0	0
$659$	16.1256	0.628164	0.314082	−	0.949396i	$-0.398303\pi$
$659$	16.1256	0.628164	0.314082	+	0.949396i	$0.398303\pi$
$660$	0	0
$661$	−35.9838	−1.39961	−0.699804	−	0.714335i	$-0.746729\pi$
$661$	−35.9838	−1.39961	−0.699804	+	0.714335i	$0.746729\pi$
$662$	0	0
$663$	−42.2038	−1.63906
$664$	0	0
$665$	2.70556	0.104917
$666$	0	0
$667$	−22.2033	−0.859714
$668$	0	0
$669$	−42.1785	−1.63072
$670$	0	0
$671$	−76.2202	−2.94245
$672$	0	0
$673$	−33.3989	−1.28743	−0.643717	−	0.765263i	$-0.722609\pi$
$673$	−33.3989	−1.28743	−0.643717	+	0.765263i	$0.722609\pi$
$674$	0	0
$675$	−10.7040	−0.411997
$676$	0	0
$677$	−50.6799	−1.94779	−0.973894	−	0.227002i	$-0.927108\pi$
$677$	−50.6799	−1.94779	−0.973894	+	0.227002i	$0.927108\pi$
$678$	0	0
$679$	−9.16557	−0.351742
$680$	0	0
$681$	−47.1498	−1.80679
$682$	0	0
$683$	22.2700	0.852140	0.426070	−	0.904690i	$-0.359898\pi$
$683$	22.2700	0.852140	0.426070	+	0.904690i	$0.359898\pi$
$684$	0	0
$685$	12.0048	0.458681
$686$	0	0
$687$	−34.1786	−1.30399
$688$	0	0
$689$	32.2829	1.22988
$690$	0	0
$691$	0.353890	0.0134626	0.00673131	−	0.999977i	$-0.497857\pi$
$691$	0.353890	0.0134626	0.00673131	+	0.999977i	$0.497857\pi$
$692$	0	0
$693$	10.4672	0.397615
$694$	0	0
$695$	24.0250	0.911318
$696$	0	0
$697$	25.9734	0.983814
$698$	0	0
$699$	−27.7619	−1.05005
$700$	0	0
$701$	19.4645	0.735164	0.367582	−	0.929991i	$-0.380186\pi$
$701$	19.4645	0.735164	0.367582	+	0.929991i	$0.380186\pi$
$702$	0	0
$703$	9.84640	0.371364
$704$	0	0
$705$	−17.5198	−0.659833
$706$	0	0
$707$	−8.26285	−0.310756
$708$	0	0
$709$	30.0714	1.12936	0.564678	−	0.825311i	$-0.309000\pi$
$709$	30.0714	1.12936	0.564678	+	0.825311i	$0.309000\pi$
$710$	0	0
$711$	1.95088	0.0731636
$712$	0	0
$713$	−17.2466	−0.645889
$714$	0	0
$715$	−115.353	−4.31396
$716$	0	0
$717$	6.91512	0.258250
$718$	0	0
$719$	14.5092	0.541102	0.270551	−	0.962706i	$-0.412794\pi$
$719$	14.5092	0.541102	0.270551	+	0.962706i	$0.412794\pi$
$720$	0	0
$721$	−7.60884	−0.283368
$722$	0	0
$723$	38.8028	1.44309
$724$	0	0
$725$	36.9303	1.37156
$726$	0	0
$727$	4.78687	0.177535	0.0887675	−	0.996052i	$-0.471707\pi$
$727$	4.78687	0.177535	0.0887675	+	0.996052i	$0.471707\pi$
$728$	0	0
$729$	−5.96856	−0.221058
$730$	0	0
$731$	−34.9372	−1.29220
$732$	0	0
$733$	−0.734181	−0.0271176	−0.0135588	−	0.999908i	$-0.504316\pi$
$733$	−0.734181	−0.0271176	−0.0135588	+	0.999908i	$0.504316\pi$
$734$	0	0
$735$	42.9612	1.58465
$736$	0	0
$737$	93.2355	3.43437
$738$	0	0
$739$	−24.8814	−0.915276	−0.457638	−	0.889139i	$-0.651304\pi$
$739$	−24.8814	−0.915276	−0.457638	+	0.889139i	$0.651304\pi$
$740$	0	0
$741$	−13.5026	−0.496030
$742$	0	0
$743$	−5.75509	−0.211134	−0.105567	−	0.994412i	$-0.533666\pi$
$743$	−5.75509	−0.211134	−0.105567	+	0.994412i	$0.533666\pi$
$744$	0	0
$745$	−35.7831	−1.31099
$746$	0	0
$747$	6.95880	0.254609
$748$	0	0
$749$	−2.15662	−0.0788013
$750$	0	0
$751$	−50.3559	−1.83751	−0.918757	−	0.394823i	$-0.870806\pi$
$751$	−50.3559	−1.83751	−0.918757	+	0.394823i	$0.870806\pi$
$752$	0	0
$753$	69.7100	2.54037
$754$	0	0
$755$	−25.4107	−0.924790
$756$	0	0
$757$	28.5153	1.03641	0.518203	−	0.855258i	$-0.326601\pi$
$757$	28.5153	1.03641	0.518203	+	0.855258i	$0.326601\pi$
$758$	0	0
$759$	37.6618	1.36704
$760$	0	0
$761$	38.1886	1.38434	0.692168	−	0.721737i	$-0.256656\pi$
$761$	38.1886	1.38434	0.692168	+	0.721737i	$0.256656\pi$
$762$	0	0
$763$	−4.55066	−0.164745
$764$	0	0
$765$	−18.8786	−0.682558
$766$	0	0
$767$	−33.6535	−1.21516
$768$	0	0
$769$	27.0169	0.974255	0.487127	−	0.873331i	$-0.338045\pi$
$769$	27.0169	0.974255	0.487127	+	0.873331i	$0.338045\pi$
$770$	0	0
$771$	−9.23065	−0.332434
$772$	0	0
$773$	−21.5847	−0.776349	−0.388175	−	0.921586i	$-0.626894\pi$
$773$	−21.5847	−0.776349	−0.388175	+	0.921586i	$0.626894\pi$
$774$	0	0
$775$	28.6859	1.03043
$776$	0	0
$777$	−19.1457	−0.686846
$778$	0	0
$779$	8.30989	0.297733
$780$	0	0
$781$	58.1755	2.08168
$782$	0	0
$783$	−18.8006	−0.671879
$784$	0	0
$785$	−38.5630	−1.37637
$786$	0	0
$787$	33.4429	1.19211	0.596056	−	0.802943i	$-0.296734\pi$
$787$	33.4429	1.19211	0.596056	+	0.802943i	$0.296734\pi$
$788$	0	0
$789$	−4.21020	−0.149887
$790$	0	0
$791$	−4.41439	−0.156958
$792$	0	0
$793$	−75.3353	−2.67523
$794$	0	0
$795$	36.6474	1.29975
$796$	0	0
$797$	21.1764	0.750106	0.375053	−	0.927003i	$-0.377625\pi$
$797$	21.1764	0.750106	0.375053	+	0.927003i	$0.377625\pi$
$798$	0	0
$799$	7.94908	0.281218
$800$	0	0
$801$	10.8716	0.384129
$802$	0	0
$803$	−11.9157	−0.420497
$804$	0	0
$805$	−7.45881	−0.262888
$806$	0	0
$807$	33.0627	1.16386
$808$	0	0
$809$	−6.66026	−0.234162	−0.117081	−	0.993122i	$-0.537354\pi$
$809$	−6.66026	−0.234162	−0.117081	+	0.993122i	$0.537354\pi$
$810$	0	0
$811$	−52.6881	−1.85013	−0.925065	−	0.379809i	$-0.875990\pi$
$811$	−52.6881	−1.85013	−0.925065	+	0.379809i	$0.875990\pi$
$812$	0	0
$813$	66.5326	2.33340
$814$	0	0
$815$	−7.66845	−0.268614
$816$	0	0
$817$	−11.1777	−0.391059
$818$	0	0
$819$	10.3456	0.361506
$820$	0	0
$821$	39.0902	1.36426	0.682129	−	0.731232i	$-0.261054\pi$
$821$	39.0902	1.36426	0.682129	+	0.731232i	$0.261054\pi$
$822$	0	0
$823$	36.6091	1.27611	0.638056	−	0.769990i	$-0.279739\pi$
$823$	36.6091	1.27611	0.638056	+	0.769990i	$0.279739\pi$
$824$	0	0
$825$	−62.6423	−2.18093
$826$	0	0
$827$	14.1569	0.492282	0.246141	−	0.969234i	$-0.420837\pi$
$827$	14.1569	0.492282	0.246141	+	0.969234i	$0.420837\pi$
$828$	0	0
$829$	−28.0739	−0.975048	−0.487524	−	0.873110i	$-0.662100\pi$
$829$	−28.0739	−0.975048	−0.487524	+	0.873110i	$0.662100\pi$
$830$	0	0
$831$	−67.5691	−2.34395
$832$	0	0
$833$	−19.4923	−0.675369
$834$	0	0
$835$	−66.7638	−2.31046
$836$	0	0
$837$	−14.6035	−0.504771
$838$	0	0
$839$	40.3692	1.39370	0.696849	−	0.717218i	$-0.254585\pi$
$839$	40.3692	1.39370	0.696849	+	0.717218i	$0.254585\pi$
$840$	0	0
$841$	35.8648	1.23672
$842$	0	0
$843$	52.3580	1.80331
$844$	0	0
$845$	−73.7654	−2.53761
$846$	0	0
$847$	−23.3291	−0.801598
$848$	0	0
$849$	−37.6510	−1.29218
$850$	0	0
$851$	−27.1450	−0.930519
$852$	0	0
$853$	−29.1042	−0.996509	−0.498255	−	0.867031i	$-0.666026\pi$
$853$	−29.1042	−0.996509	−0.498255	+	0.867031i	$0.666026\pi$
$854$	0	0
$855$	−6.03998	−0.206563
$856$	0	0
$857$	33.3263	1.13841	0.569203	−	0.822197i	$-0.307252\pi$
$857$	33.3263	1.13841	0.569203	+	0.822197i	$0.307252\pi$
$858$	0	0
$859$	−17.8510	−0.609068	−0.304534	−	0.952502i	$-0.598501\pi$
$859$	−17.8510	−0.609068	−0.304534	+	0.952502i	$0.598501\pi$
$860$	0	0
$861$	−16.1580	−0.550663
$862$	0	0
$863$	2.78551	0.0948197	0.0474099	−	0.998876i	$-0.484903\pi$
$863$	2.78551	0.0948197	0.0474099	+	0.998876i	$0.484903\pi$
$864$	0	0
$865$	7.94333	0.270081
$866$	0	0
$867$	−16.0884	−0.546392
$868$	0	0
$869$	−6.13971	−0.208275
$870$	0	0
$871$	92.1530	3.12248
$872$	0	0
$873$	20.4615	0.692518
$874$	0	0
$875$	−1.12168	−0.0379197
$876$	0	0
$877$	34.8355	1.17631	0.588155	−	0.808748i	$-0.299854\pi$
$877$	34.8355	1.17631	0.588155	+	0.808748i	$0.299854\pi$
$878$	0	0
$879$	18.4615	0.622692
$880$	0	0
$881$	54.6117	1.83991	0.919957	−	0.392019i	$-0.128223\pi$
$881$	54.6117	1.83991	0.919957	+	0.392019i	$0.128223\pi$
$882$	0	0
$883$	−4.91480	−0.165396	−0.0826981	−	0.996575i	$-0.526354\pi$
$883$	−4.91480	−0.165396	−0.0826981	+	0.996575i	$0.526354\pi$
$884$	0	0
$885$	−38.2033	−1.28419
$886$	0	0
$887$	−47.5332	−1.59601	−0.798004	−	0.602652i	$-0.794111\pi$
$887$	−47.5332	−1.59601	−0.798004	+	0.602652i	$0.794111\pi$
$888$	0	0
$889$	12.4846	0.418721
$890$	0	0
$891$	67.8236	2.27218
$892$	0	0
$893$	2.54321	0.0851053
$894$	0	0
$895$	−57.1245	−1.90946
$896$	0	0
$897$	37.2245	1.24289
$898$	0	0
$899$	50.3842	1.68041
$900$	0	0
$901$	−16.6277	−0.553948
$902$	0	0
$903$	21.7343	0.723273
$904$	0	0
$905$	−74.4439	−2.47460
$906$	0	0
$907$	28.0388	0.931012	0.465506	−	0.885045i	$-0.345872\pi$
$907$	28.0388	0.931012	0.465506	+	0.885045i	$0.345872\pi$
$908$	0	0
$909$	18.4463	0.611824
$910$	0	0
$911$	10.7766	0.357046	0.178523	−	0.983936i	$-0.442868\pi$
$911$	10.7766	0.357046	0.178523	+	0.983936i	$0.442868\pi$
$912$	0	0
$913$	−21.9004	−0.724798
$914$	0	0
$915$	−85.5203	−2.82721
$916$	0	0
$917$	−4.44028	−0.146631
$918$	0	0
$919$	−10.1782	−0.335747	−0.167873	−	0.985809i	$-0.553690\pi$
$919$	−10.1782	−0.335747	−0.167873	+	0.985809i	$0.553690\pi$
$920$	0	0
$921$	39.0688	1.28736
$922$	0	0
$923$	57.5000	1.89264
$924$	0	0
$925$	45.1499	1.48452
$926$	0	0
$927$	16.9862	0.557901
$928$	0	0
$929$	12.8872	0.422815	0.211407	−	0.977398i	$-0.432195\pi$
$929$	12.8872	0.422815	0.211407	+	0.977398i	$0.432195\pi$
$930$	0	0
$931$	−6.23633	−0.204388
$932$	0	0
$933$	57.6431	1.88715
$934$	0	0
$935$	59.4139	1.94304
$936$	0	0
$937$	25.9382	0.847364	0.423682	−	0.905811i	$-0.360737\pi$
$937$	25.9382	0.847364	0.423682	+	0.905811i	$0.360737\pi$
$938$	0	0
$939$	6.00121	0.195842
$940$	0	0
$941$	−12.3337	−0.402066	−0.201033	−	0.979585i	$-0.564430\pi$
$941$	−12.3337	−0.402066	−0.201033	+	0.979585i	$0.564430\pi$
$942$	0	0
$943$	−22.9091	−0.746022
$944$	0	0
$945$	−6.31574	−0.205451
$946$	0	0
$947$	−13.1752	−0.428136	−0.214068	−	0.976819i	$-0.568671\pi$
$947$	−13.1752	−0.428136	−0.214068	+	0.976819i	$0.568671\pi$
$948$	0	0
$949$	−11.7774	−0.382310
$950$	0	0
$951$	−40.7932	−1.32281
$952$	0	0
$953$	29.6992	0.962050	0.481025	−	0.876707i	$-0.340265\pi$
$953$	29.6992	0.962050	0.481025	+	0.876707i	$0.340265\pi$
$954$	0	0
$955$	25.4293	0.822873
$956$	0	0
$957$	−110.026	−3.55662
$958$	0	0
$959$	3.38846	0.109419
$960$	0	0
$961$	8.13634	0.262463
$962$	0	0
$963$	4.81452	0.155146
$964$	0	0
$965$	78.3995	2.52377
$966$	0	0
$967$	28.3692	0.912292	0.456146	−	0.889905i	$-0.349229\pi$
$967$	28.3692	0.912292	0.456146	+	0.889905i	$0.349229\pi$
$968$	0	0
$969$	6.95466	0.223416
$970$	0	0
$971$	−11.8080	−0.378938	−0.189469	−	0.981887i	$-0.560677\pi$
$971$	−11.8080	−0.378938	−0.189469	+	0.981887i	$0.560677\pi$
$972$	0	0
$973$	6.78123	0.217396
$974$	0	0
$975$	−61.9150	−1.98287
$976$	0	0
$977$	9.57074	0.306195	0.153098	−	0.988211i	$-0.451075\pi$
$977$	9.57074	0.306195	0.153098	+	0.988211i	$0.451075\pi$
$978$	0	0
$979$	−34.2146	−1.09350
$980$	0	0
$981$	10.1590	0.324353
$982$	0	0
$983$	−11.1711	−0.356302	−0.178151	−	0.984003i	$-0.557012\pi$
$983$	−11.1711	−0.356302	−0.178151	+	0.984003i	$0.557012\pi$
$984$	0	0
$985$	75.7137	2.41244
$986$	0	0
$987$	−4.94510	−0.157404
$988$	0	0
$989$	30.8153	0.979868
$990$	0	0
$991$	−3.17800	−0.100952	−0.0504762	−	0.998725i	$-0.516074\pi$
$991$	−3.17800	−0.100952	−0.0504762	+	0.998725i	$0.516074\pi$
$992$	0	0
$993$	38.4942	1.22158
$994$	0	0
$995$	55.7542	1.76753
$996$	0	0
$997$	11.7698	0.372754	0.186377	−	0.982478i	$-0.440325\pi$
$997$	11.7698	0.372754	0.186377	+	0.982478i	$0.440325\pi$
$998$	0	0
$999$	−22.9850	−0.727213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.18	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.18	✓	24	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22506 q^{3} -3.09603 q^{5} -0.873879 q^{7} +1.95088 q^{9} +O(q^{10})\)	\(q+2.22506 q^{3} -3.09603 q^{5} -0.873879 q^{7} +1.95088 q^{9} -6.13971 q^{11} -6.06842 q^{13} -6.88885 q^{15} +3.12561 q^{17} +1.00000 q^{19} -1.94443 q^{21} -2.75685 q^{23} +4.58542 q^{25} -2.33436 q^{27} +8.05387 q^{29} +6.25590 q^{31} -13.6612 q^{33} +2.70556 q^{35} +9.84640 q^{37} -13.5026 q^{39} +8.30989 q^{41} -11.1777 q^{43} -6.03998 q^{45} +2.54321 q^{47} -6.23633 q^{49} +6.95466 q^{51} -5.31982 q^{53} +19.0087 q^{55} +2.22506 q^{57} +5.54567 q^{59} +12.4143 q^{61} -1.70483 q^{63} +18.7880 q^{65} -15.1857 q^{67} -6.13414 q^{69} -9.47528 q^{71} +1.94076 q^{73} +10.2028 q^{75} +5.36537 q^{77} +1.00000 q^{79} -11.0467 q^{81} +3.56701 q^{83} -9.67698 q^{85} +17.9203 q^{87} +5.57267 q^{89} +5.30307 q^{91} +13.9197 q^{93} -3.09603 q^{95} +10.4884 q^{97} -11.9778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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