LMFDB - Embedded newform 4080.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4080,2,Mod(1,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.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:	$32.5789640247$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	4080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -3.48929 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -3.48929 q^{7} +1.00000 q^{9} +4.17513 q^{11} +0.292731 q^{13} +1.00000 q^{15} -1.00000 q^{17} -5.78202 q^{19} -3.48929 q^{21} -6.39312 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.78202 q^{29} -6.39312 q^{31} +4.17513 q^{33} -3.48929 q^{35} +2.17513 q^{37} +0.292731 q^{39} -10.8610 q^{41} +3.66442 q^{43} +1.00000 q^{45} -11.1537 q^{47} +5.17513 q^{49} -1.00000 q^{51} -3.19656 q^{53} +4.17513 q^{55} -5.78202 q^{57} -6.29273 q^{59} +11.3717 q^{61} -3.48929 q^{63} +0.292731 q^{65} +11.6644 q^{67} -6.39312 q^{69} +7.07896 q^{71} +5.48929 q^{73} +1.00000 q^{75} -14.5682 q^{77} -5.02142 q^{79} +1.00000 q^{81} +0.585462 q^{83} -1.00000 q^{85} -3.78202 q^{87} +13.7648 q^{89} -1.02142 q^{91} -6.39312 q^{93} -5.78202 q^{95} -13.6644 q^{97} +4.17513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 7 q^{11} - 2 q^{13} + 3 q^{15} - 3 q^{17} - 7 q^{19} - 3 q^{21} - 10 q^{23} + 3 q^{25} + 3 q^{27} - q^{29} - 10 q^{31} - 7 q^{33} - 3 q^{35} - 13 q^{37} - 2 q^{39} - q^{41} - 16 q^{43} + 3 q^{45} + q^{47} - 4 q^{49} - 3 q^{51} - 5 q^{53} - 7 q^{55} - 7 q^{57} - 16 q^{59} + 10 q^{61} - 3 q^{63} - 2 q^{65} + 8 q^{67} - 10 q^{69} + 9 q^{73} + 3 q^{75} - 15 q^{77} - 30 q^{79} + 3 q^{81} - 4 q^{83} - 3 q^{85} - q^{87} + 8 q^{89} - 18 q^{91} - 10 q^{93} - 7 q^{95} - 14 q^{97} - 7 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 7 * q^11 - 2 * q^13 + 3 * q^15 - 3 * q^17 - 7 * q^19 - 3 * q^21 - 10 * q^23 + 3 * q^25 + 3 * q^27 - q^29 - 10 * q^31 - 7 * q^33 - 3 * q^35 - 13 * q^37 - 2 * q^39 - q^41 - 16 * q^43 + 3 * q^45 + q^47 - 4 * q^49 - 3 * q^51 - 5 * q^53 - 7 * q^55 - 7 * q^57 - 16 * q^59 + 10 * q^61 - 3 * q^63 - 2 * q^65 + 8 * q^67 - 10 * q^69 + 9 * q^73 + 3 * q^75 - 15 * q^77 - 30 * q^79 + 3 * q^81 - 4 * q^83 - 3 * q^85 - q^87 + 8 * q^89 - 18 * q^91 - 10 * q^93 - 7 * q^95 - 14 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.48929	−1.31883	−0.659414	−	0.751780i	$-0.729195\pi$
$7$	−3.48929	−1.31883	−0.659414	+	0.751780i	$0.729195\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.17513	1.25885	0.629425	−	0.777061i	$-0.283290\pi$
$11$	4.17513	1.25885	0.629425	+	0.777061i	$0.283290\pi$
$12$	0	0
$13$	0.292731	0.0811890	0.0405945	−	0.999176i	$-0.487075\pi$
$13$	0.292731	0.0811890	0.0405945	+	0.999176i	$0.487075\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.78202	−1.32649	−0.663243	−	0.748404i	$-0.730820\pi$
$19$	−5.78202	−1.32649	−0.663243	+	0.748404i	$0.730820\pi$
$20$	0	0
$21$	−3.48929	−0.761425
$22$	0	0
$23$	−6.39312	−1.33306	−0.666528	−	0.745480i	$-0.732220\pi$
$23$	−6.39312	−1.33306	−0.666528	+	0.745480i	$0.732220\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.78202	−0.702303	−0.351152	−	0.936319i	$-0.614210\pi$
$29$	−3.78202	−0.702303	−0.351152	+	0.936319i	$0.614210\pi$
$30$	0	0
$31$	−6.39312	−1.14824	−0.574119	−	0.818772i	$-0.694655\pi$
$31$	−6.39312	−1.14824	−0.574119	+	0.818772i	$0.694655\pi$
$32$	0	0
$33$	4.17513	0.726798
$34$	0	0
$35$	−3.48929	−0.589797
$36$	0	0
$37$	2.17513	0.357590	0.178795	−	0.983886i	$-0.442780\pi$
$37$	2.17513	0.357590	0.178795	+	0.983886i	$0.442780\pi$
$38$	0	0
$39$	0.292731	0.0468745
$40$	0	0
$41$	−10.8610	−1.69620	−0.848100	−	0.529836i	$-0.822253\pi$
$41$	−10.8610	−1.69620	−0.848100	+	0.529836i	$0.822253\pi$
$42$	0	0
$43$	3.66442	0.558819	0.279410	−	0.960172i	$-0.409861\pi$
$43$	3.66442	0.558819	0.279410	+	0.960172i	$0.409861\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−11.1537	−1.62694	−0.813468	−	0.581610i	$-0.802423\pi$
$47$	−11.1537	−1.62694	−0.813468	+	0.581610i	$0.802423\pi$
$48$	0	0
$49$	5.17513	0.739305
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−3.19656	−0.439081	−0.219540	−	0.975603i	$-0.570456\pi$
$53$	−3.19656	−0.439081	−0.219540	+	0.975603i	$0.570456\pi$
$54$	0	0
$55$	4.17513	0.562975
$56$	0	0
$57$	−5.78202	−0.765847
$58$	0	0
$59$	−6.29273	−0.819244	−0.409622	−	0.912255i	$-0.634339\pi$
$59$	−6.29273	−0.819244	−0.409622	+	0.912255i	$0.634339\pi$
$60$	0	0
$61$	11.3717	1.45600	0.727998	−	0.685579i	$-0.240451\pi$
$61$	11.3717	1.45600	0.727998	+	0.685579i	$0.240451\pi$
$62$	0	0
$63$	−3.48929	−0.439609
$64$	0	0
$65$	0.292731	0.0363088
$66$	0	0
$67$	11.6644	1.42504	0.712518	−	0.701654i	$-0.247555\pi$
$67$	11.6644	1.42504	0.712518	+	0.701654i	$0.247555\pi$
$68$	0	0
$69$	−6.39312	−0.769641
$70$	0	0
$71$	7.07896	0.840118	0.420059	−	0.907497i	$-0.362009\pi$
$71$	7.07896	0.840118	0.420059	+	0.907497i	$0.362009\pi$
$72$	0	0
$73$	5.48929	0.642473	0.321236	−	0.946999i	$-0.395902\pi$
$73$	5.48929	0.642473	0.321236	+	0.946999i	$0.395902\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−14.5682	−1.66021
$78$	0	0
$79$	−5.02142	−0.564954	−0.282477	−	0.959274i	$-0.591156\pi$
$79$	−5.02142	−0.564954	−0.282477	+	0.959274i	$0.591156\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.585462	0.0642628	0.0321314	−	0.999484i	$-0.489770\pi$
$83$	0.585462	0.0642628	0.0321314	+	0.999484i	$0.489770\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−3.78202	−0.405475
$88$	0	0
$89$	13.7648	1.45907	0.729533	−	0.683945i	$-0.239737\pi$
$89$	13.7648	1.45907	0.729533	+	0.683945i	$0.239737\pi$
$90$	0	0
$91$	−1.02142	−0.107074
$92$	0	0
$93$	−6.39312	−0.662935
$94$	0	0
$95$	−5.78202	−0.593223
$96$	0	0
$97$	−13.6644	−1.38741	−0.693706	−	0.720258i	$-0.744023\pi$
$97$	−13.6644	−1.38741	−0.693706	+	0.720258i	$0.744023\pi$
$98$	0	0
$99$	4.17513	0.419617
$100$	0	0
$101$	9.66442	0.961646	0.480823	−	0.876818i	$-0.340338\pi$
$101$	9.66442	0.961646	0.480823	+	0.876818i	$0.340338\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	−3.48929	−0.340520
$106$	0	0
$107$	0.585462	0.0565987	0.0282994	−	0.999599i	$-0.490991\pi$
$107$	0.585462	0.0565987	0.0282994	+	0.999599i	$0.490991\pi$
$108$	0	0
$109$	−2.58546	−0.247642	−0.123821	−	0.992305i	$-0.539515\pi$
$109$	−2.58546	−0.247642	−0.123821	+	0.992305i	$0.539515\pi$
$110$	0	0
$111$	2.17513	0.206455
$112$	0	0
$113$	15.3717	1.44605	0.723024	−	0.690823i	$-0.242752\pi$
$113$	15.3717	1.44605	0.723024	+	0.690823i	$0.242752\pi$
$114$	0	0
$115$	−6.39312	−0.596161
$116$	0	0
$117$	0.292731	0.0270630
$118$	0	0
$119$	3.48929	0.319863
$120$	0	0
$121$	6.43175	0.584705
$122$	0	0
$123$	−10.8610	−0.979302
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.3503	−1.80579	−0.902897	−	0.429857i	$-0.858564\pi$
$127$	−20.3503	−1.80579	−0.902897	+	0.429857i	$0.858564\pi$
$128$	0	0
$129$	3.66442	0.322634
$130$	0	0
$131$	−3.41454	−0.298330	−0.149165	−	0.988812i	$-0.547658\pi$
$131$	−3.41454	−0.298330	−0.149165	+	0.988812i	$0.547658\pi$
$132$	0	0
$133$	20.1751	1.74941
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−20.1323	−1.72002	−0.860009	−	0.510279i	$-0.829542\pi$
$137$	−20.1323	−1.72002	−0.860009	+	0.510279i	$0.829542\pi$
$138$	0	0
$139$	7.76481	0.658602	0.329301	−	0.944225i	$-0.393187\pi$
$139$	7.76481	0.658602	0.329301	+	0.944225i	$0.393187\pi$
$140$	0	0
$141$	−11.1537	−0.939312
$142$	0	0
$143$	1.22219	0.102205
$144$	0	0
$145$	−3.78202	−0.314080
$146$	0	0
$147$	5.17513	0.426838
$148$	0	0
$149$	9.66442	0.791740	0.395870	−	0.918306i	$-0.370443\pi$
$149$	9.66442	0.791740	0.395870	+	0.918306i	$0.370443\pi$
$150$	0	0
$151$	−19.1537	−1.55871	−0.779353	−	0.626585i	$-0.784452\pi$
$151$	−19.1537	−1.55871	−0.779353	+	0.626585i	$0.784452\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−6.39312	−0.513507
$156$	0	0
$157$	−16.0575	−1.28153	−0.640766	−	0.767737i	$-0.721383\pi$
$157$	−16.0575	−1.28153	−0.640766	+	0.767737i	$0.721383\pi$
$158$	0	0
$159$	−3.19656	−0.253504
$160$	0	0
$161$	22.3074	1.75807
$162$	0	0
$163$	−18.7178	−1.46609	−0.733044	−	0.680182i	$-0.761901\pi$
$163$	−18.7178	−1.46609	−0.733044	+	0.680182i	$0.761901\pi$
$164$	0	0
$165$	4.17513	0.325034
$166$	0	0
$167$	−21.7220	−1.68090	−0.840448	−	0.541892i	$-0.817708\pi$
$167$	−21.7220	−1.68090	−0.840448	+	0.541892i	$0.817708\pi$
$168$	0	0
$169$	−12.9143	−0.993408
$170$	0	0
$171$	−5.78202	−0.442162
$172$	0	0
$173$	0.393115	0.0298880	0.0149440	−	0.999888i	$-0.495243\pi$
$173$	0.393115	0.0298880	0.0149440	+	0.999888i	$0.495243\pi$
$174$	0	0
$175$	−3.48929	−0.263765
$176$	0	0
$177$	−6.29273	−0.472991
$178$	0	0
$179$	−18.0575	−1.34968	−0.674842	−	0.737962i	$-0.735788\pi$
$179$	−18.0575	−1.34968	−0.674842	+	0.737962i	$0.735788\pi$
$180$	0	0
$181$	25.3288	1.88268	0.941339	−	0.337462i	$-0.109568\pi$
$181$	25.3288	1.88268	0.941339	+	0.337462i	$0.109568\pi$
$182$	0	0
$183$	11.3717	0.840620
$184$	0	0
$185$	2.17513	0.159919
$186$	0	0
$187$	−4.17513	−0.305316
$188$	0	0
$189$	−3.48929	−0.253808
$190$	0	0
$191$	−15.3288	−1.10916	−0.554578	−	0.832132i	$-0.687120\pi$
$191$	−15.3288	−1.10916	−0.554578	+	0.832132i	$0.687120\pi$
$192$	0	0
$193$	−24.4078	−1.75691	−0.878456	−	0.477823i	$-0.841426\pi$
$193$	−24.4078	−1.75691	−0.878456	+	0.477823i	$0.841426\pi$
$194$	0	0
$195$	0.292731	0.0209629
$196$	0	0
$197$	−13.5640	−0.966398	−0.483199	−	0.875511i	$-0.660525\pi$
$197$	−13.5640	−0.966398	−0.483199	+	0.875511i	$0.660525\pi$
$198$	0	0
$199$	−7.56404	−0.536201	−0.268100	−	0.963391i	$-0.586396\pi$
$199$	−7.56404	−0.536201	−0.268100	+	0.963391i	$0.586396\pi$
$200$	0	0
$201$	11.6644	0.822745
$202$	0	0
$203$	13.1966	0.926217
$204$	0	0
$205$	−10.8610	−0.758564
$206$	0	0
$207$	−6.39312	−0.444352
$208$	0	0
$209$	−24.1407	−1.66985
$210$	0	0
$211$	5.17092	0.355981	0.177991	−	0.984032i	$-0.443040\pi$
$211$	5.17092	0.355981	0.177991	+	0.984032i	$0.443040\pi$
$212$	0	0
$213$	7.07896	0.485042
$214$	0	0
$215$	3.66442	0.249912
$216$	0	0
$217$	22.3074	1.51433
$218$	0	0
$219$	5.48929	0.370932
$220$	0	0
$221$	−0.292731	−0.0196912
$222$	0	0
$223$	4.35027	0.291316	0.145658	−	0.989335i	$-0.453470\pi$
$223$	4.35027	0.291316	0.145658	+	0.989335i	$0.453470\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	23.7648	1.57733	0.788663	−	0.614826i	$-0.210774\pi$
$227$	23.7648	1.57733	0.788663	+	0.614826i	$0.210774\pi$
$228$	0	0
$229$	−10.6111	−0.701201	−0.350600	−	0.936525i	$-0.614022\pi$
$229$	−10.6111	−0.701201	−0.350600	+	0.936525i	$0.614022\pi$
$230$	0	0
$231$	−14.5682	−0.958520
$232$	0	0
$233$	−13.9143	−0.911557	−0.455778	−	0.890093i	$-0.650639\pi$
$233$	−13.9143	−0.911557	−0.455778	+	0.890093i	$0.650639\pi$
$234$	0	0
$235$	−11.1537	−0.727588
$236$	0	0
$237$	−5.02142	−0.326176
$238$	0	0
$239$	24.3503	1.57509	0.787544	−	0.616258i	$-0.211352\pi$
$239$	24.3503	1.57509	0.787544	+	0.616258i	$0.211352\pi$
$240$	0	0
$241$	23.9572	1.54322	0.771608	−	0.636098i	$-0.219453\pi$
$241$	23.9572	1.54322	0.771608	+	0.636098i	$0.219453\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.17513	0.330627
$246$	0	0
$247$	−1.69258	−0.107696
$248$	0	0
$249$	0.585462	0.0371021
$250$	0	0
$251$	26.4078	1.66685	0.833423	−	0.552636i	$-0.186378\pi$
$251$	26.4078	1.66685	0.833423	+	0.552636i	$0.186378\pi$
$252$	0	0
$253$	−26.6921	−1.67812
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	−11.1709	−0.696823	−0.348412	−	0.937342i	$-0.613279\pi$
$257$	−11.1709	−0.696823	−0.348412	+	0.937342i	$0.613279\pi$
$258$	0	0
$259$	−7.58967	−0.471599
$260$	0	0
$261$	−3.78202	−0.234101
$262$	0	0
$263$	15.7392	0.970519	0.485260	−	0.874370i	$-0.338725\pi$
$263$	15.7392	0.970519	0.485260	+	0.874370i	$0.338725\pi$
$264$	0	0
$265$	−3.19656	−0.196363
$266$	0	0
$267$	13.7648	0.842393
$268$	0	0
$269$	−2.96137	−0.180558	−0.0902788	−	0.995917i	$-0.528776\pi$
$269$	−2.96137	−0.180558	−0.0902788	+	0.995917i	$0.528776\pi$
$270$	0	0
$271$	3.41454	0.207418	0.103709	−	0.994608i	$-0.466929\pi$
$271$	3.41454	0.207418	0.103709	+	0.994608i	$0.466929\pi$
$272$	0	0
$273$	−1.02142	−0.0618193
$274$	0	0
$275$	4.17513	0.251770
$276$	0	0
$277$	5.32885	0.320179	0.160090	−	0.987102i	$-0.448822\pi$
$277$	5.32885	0.320179	0.160090	+	0.987102i	$0.448822\pi$
$278$	0	0
$279$	−6.39312	−0.382746
$280$	0	0
$281$	22.9357	1.36823	0.684116	−	0.729374i	$-0.260188\pi$
$281$	22.9357	1.36823	0.684116	+	0.729374i	$0.260188\pi$
$282$	0	0
$283$	11.3889	0.677000	0.338500	−	0.940966i	$-0.390080\pi$
$283$	11.3889	0.677000	0.338500	+	0.940966i	$0.390080\pi$
$284$	0	0
$285$	−5.78202	−0.342497
$286$	0	0
$287$	37.8971	2.23699
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−13.6644	−0.801023
$292$	0	0
$293$	15.3461	0.896526	0.448263	−	0.893902i	$-0.352043\pi$
$293$	15.3461	0.896526	0.448263	+	0.893902i	$0.352043\pi$
$294$	0	0
$295$	−6.29273	−0.366377
$296$	0	0
$297$	4.17513	0.242266
$298$	0	0
$299$	−1.87146	−0.108229
$300$	0	0
$301$	−12.7862	−0.736986
$302$	0	0
$303$	9.66442	0.555207
$304$	0	0
$305$	11.3717	0.651141
$306$	0	0
$307$	6.87819	0.392559	0.196280	−	0.980548i	$-0.437114\pi$
$307$	6.87819	0.392559	0.196280	+	0.980548i	$0.437114\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	11.0533	0.626777	0.313388	−	0.949625i	$-0.398536\pi$
$311$	11.0533	0.626777	0.313388	+	0.949625i	$0.398536\pi$
$312$	0	0
$313$	−5.92525	−0.334915	−0.167457	−	0.985879i	$-0.553556\pi$
$313$	−5.92525	−0.334915	−0.167457	+	0.985879i	$0.553556\pi$
$314$	0	0
$315$	−3.48929	−0.196599
$316$	0	0
$317$	−20.7434	−1.16506	−0.582532	−	0.812808i	$-0.697938\pi$
$317$	−20.7434	−1.16506	−0.582532	+	0.812808i	$0.697938\pi$
$318$	0	0
$319$	−15.7904	−0.884095
$320$	0	0
$321$	0.585462	0.0326773
$322$	0	0
$323$	5.78202	0.321720
$324$	0	0
$325$	0.292731	0.0162378
$326$	0	0
$327$	−2.58546	−0.142976
$328$	0	0
$329$	38.9185	2.14565
$330$	0	0
$331$	−4.41033	−0.242414	−0.121207	−	0.992627i	$-0.538676\pi$
$331$	−4.41033	−0.242414	−0.121207	+	0.992627i	$0.538676\pi$
$332$	0	0
$333$	2.17513	0.119197
$334$	0	0
$335$	11.6644	0.637296
$336$	0	0
$337$	11.4464	0.623527	0.311764	−	0.950160i	$-0.399080\pi$
$337$	11.4464	0.623527	0.311764	+	0.950160i	$0.399080\pi$
$338$	0	0
$339$	15.3717	0.834876
$340$	0	0
$341$	−26.6921	−1.44546
$342$	0	0
$343$	6.36748	0.343812
$344$	0	0
$345$	−6.39312	−0.344194
$346$	0	0
$347$	4.67115	0.250761	0.125380	−	0.992109i	$-0.459985\pi$
$347$	4.67115	0.250761	0.125380	+	0.992109i	$0.459985\pi$
$348$	0	0
$349$	3.54683	0.189857	0.0949287	−	0.995484i	$-0.469738\pi$
$349$	3.54683	0.189857	0.0949287	+	0.995484i	$0.469738\pi$
$350$	0	0
$351$	0.292731	0.0156248
$352$	0	0
$353$	−9.38890	−0.499721	−0.249860	−	0.968282i	$-0.580385\pi$
$353$	−9.38890	−0.499721	−0.249860	+	0.968282i	$0.580385\pi$
$354$	0	0
$355$	7.07896	0.375712
$356$	0	0
$357$	3.48929	0.184673
$358$	0	0
$359$	4.58546	0.242011	0.121006	−	0.992652i	$-0.461388\pi$
$359$	4.58546	0.242011	0.121006	+	0.992652i	$0.461388\pi$
$360$	0	0
$361$	14.4318	0.759566
$362$	0	0
$363$	6.43175	0.337579
$364$	0	0
$365$	5.48929	0.287322
$366$	0	0
$367$	10.4078	0.543283	0.271642	−	0.962398i	$-0.412433\pi$
$367$	10.4078	0.543283	0.271642	+	0.962398i	$0.412433\pi$
$368$	0	0
$369$	−10.8610	−0.565400
$370$	0	0
$371$	11.1537	0.579072
$372$	0	0
$373$	1.31415	0.0680443	0.0340222	−	0.999421i	$-0.489168\pi$
$373$	1.31415	0.0680443	0.0340222	+	0.999421i	$0.489168\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−1.10711	−0.0570193
$378$	0	0
$379$	−11.3288	−0.581924	−0.290962	−	0.956735i	$-0.593975\pi$
$379$	−11.3288	−0.581924	−0.290962	+	0.956735i	$0.593975\pi$
$380$	0	0
$381$	−20.3503	−1.04258
$382$	0	0
$383$	20.5254	1.04880	0.524400	−	0.851472i	$-0.324290\pi$
$383$	20.5254	1.04880	0.524400	+	0.851472i	$0.324290\pi$
$384$	0	0
$385$	−14.5682	−0.742467
$386$	0	0
$387$	3.66442	0.186273
$388$	0	0
$389$	−16.7287	−0.848178	−0.424089	−	0.905620i	$-0.639406\pi$
$389$	−16.7287	−0.848178	−0.424089	+	0.905620i	$0.639406\pi$
$390$	0	0
$391$	6.39312	0.323314
$392$	0	0
$393$	−3.41454	−0.172241
$394$	0	0
$395$	−5.02142	−0.252655
$396$	0	0
$397$	−35.7820	−1.79585	−0.897924	−	0.440150i	$-0.854925\pi$
$397$	−35.7820	−1.79585	−0.897924	+	0.440150i	$0.854925\pi$
$398$	0	0
$399$	20.1751	1.01002
$400$	0	0
$401$	−17.8396	−0.890865	−0.445433	−	0.895316i	$-0.646950\pi$
$401$	−17.8396	−0.890865	−0.445433	+	0.895316i	$0.646950\pi$
$402$	0	0
$403$	−1.87146	−0.0932242
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	9.08148	0.450152
$408$	0	0
$409$	16.5682	0.819247	0.409624	−	0.912255i	$-0.365660\pi$
$409$	16.5682	0.819247	0.409624	+	0.912255i	$0.365660\pi$
$410$	0	0
$411$	−20.1323	−0.993053
$412$	0	0
$413$	21.9572	1.08044
$414$	0	0
$415$	0.585462	0.0287392
$416$	0	0
$417$	7.76481	0.380244
$418$	0	0
$419$	0.410327	0.0200458	0.0100229	−	0.999950i	$-0.496810\pi$
$419$	0.410327	0.0200458	0.0100229	+	0.999950i	$0.496810\pi$
$420$	0	0
$421$	7.78202	0.379272	0.189636	−	0.981854i	$-0.439269\pi$
$421$	7.78202	0.379272	0.189636	+	0.981854i	$0.439269\pi$
$422$	0	0
$423$	−11.1537	−0.542312
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−39.6791	−1.92021
$428$	0	0
$429$	1.22219	0.0590080
$430$	0	0
$431$	−15.6388	−0.753294	−0.376647	−	0.926357i	$-0.622923\pi$
$431$	−15.6388	−0.753294	−0.376647	+	0.926357i	$0.622923\pi$
$432$	0	0
$433$	0.157923	0.00758928	0.00379464	−	0.999993i	$-0.498792\pi$
$433$	0.157923	0.00758928	0.00379464	+	0.999993i	$0.498792\pi$
$434$	0	0
$435$	−3.78202	−0.181334
$436$	0	0
$437$	36.9651	1.76828
$438$	0	0
$439$	16.9357	0.808298	0.404149	−	0.914693i	$-0.367568\pi$
$439$	16.9357	0.808298	0.404149	+	0.914693i	$0.367568\pi$
$440$	0	0
$441$	5.17513	0.246435
$442$	0	0
$443$	−3.32885	−0.158158	−0.0790791	−	0.996868i	$-0.525198\pi$
$443$	−3.32885	−0.158158	−0.0790791	+	0.996868i	$0.525198\pi$
$444$	0	0
$445$	13.7648	0.652514
$446$	0	0
$447$	9.66442	0.457112
$448$	0	0
$449$	−10.9210	−0.515396	−0.257698	−	0.966226i	$-0.582964\pi$
$449$	−10.9210	−0.515396	−0.257698	+	0.966226i	$0.582964\pi$
$450$	0	0
$451$	−45.3461	−2.13526
$452$	0	0
$453$	−19.1537	−0.899920
$454$	0	0
$455$	−1.02142	−0.0478850
$456$	0	0
$457$	−3.80765	−0.178115	−0.0890573	−	0.996027i	$-0.528385\pi$
$457$	−3.80765	−0.178115	−0.0890573	+	0.996027i	$0.528385\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	16.6430	0.775142	0.387571	−	0.921840i	$-0.373314\pi$
$461$	16.6430	0.775142	0.387571	+	0.921840i	$0.373314\pi$
$462$	0	0
$463$	−13.5212	−0.628383	−0.314192	−	0.949360i	$-0.601733\pi$
$463$	−13.5212	−0.628383	−0.314192	+	0.949360i	$0.601733\pi$
$464$	0	0
$465$	−6.39312	−0.296474
$466$	0	0
$467$	15.3545	0.710521	0.355260	−	0.934767i	$-0.384392\pi$
$467$	15.3545	0.710521	0.355260	+	0.934767i	$0.384392\pi$
$468$	0	0
$469$	−40.7005	−1.87938
$470$	0	0
$471$	−16.0575	−0.739892
$472$	0	0
$473$	15.2995	0.703470
$474$	0	0
$475$	−5.78202	−0.265297
$476$	0	0
$477$	−3.19656	−0.146360
$478$	0	0
$479$	26.2927	1.20135	0.600673	−	0.799495i	$-0.294899\pi$
$479$	26.2927	1.20135	0.600673	+	0.799495i	$0.294899\pi$
$480$	0	0
$481$	0.636729	0.0290324
$482$	0	0
$483$	22.3074	1.01502
$484$	0	0
$485$	−13.6644	−0.620470
$486$	0	0
$487$	−16.8353	−0.762882	−0.381441	−	0.924393i	$-0.624572\pi$
$487$	−16.8353	−0.762882	−0.381441	+	0.924393i	$0.624572\pi$
$488$	0	0
$489$	−18.7178	−0.846446
$490$	0	0
$491$	−32.0147	−1.44480	−0.722401	−	0.691474i	$-0.756962\pi$
$491$	−32.0147	−1.44480	−0.722401	+	0.691474i	$0.756962\pi$
$492$	0	0
$493$	3.78202	0.170334
$494$	0	0
$495$	4.17513	0.187658
$496$	0	0
$497$	−24.7005	−1.10797
$498$	0	0
$499$	25.4868	1.14094	0.570472	−	0.821317i	$-0.306760\pi$
$499$	25.4868	1.14094	0.570472	+	0.821317i	$0.306760\pi$
$500$	0	0
$501$	−21.7220	−0.970466
$502$	0	0
$503$	24.7350	1.10288	0.551439	−	0.834215i	$-0.314079\pi$
$503$	24.7350	1.10288	0.551439	+	0.834215i	$0.314079\pi$
$504$	0	0
$505$	9.66442	0.430061
$506$	0	0
$507$	−12.9143	−0.573545
$508$	0	0
$509$	6.92104	0.306770	0.153385	−	0.988167i	$-0.450983\pi$
$509$	6.92104	0.306770	0.153385	+	0.988167i	$0.450983\pi$
$510$	0	0
$511$	−19.1537	−0.847310
$512$	0	0
$513$	−5.78202	−0.255282
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	−46.5682	−2.04807
$518$	0	0
$519$	0.393115	0.0172558
$520$	0	0
$521$	−9.83956	−0.431079	−0.215539	−	0.976495i	$-0.569151\pi$
$521$	−9.83956	−0.431079	−0.215539	+	0.976495i	$0.569151\pi$
$522$	0	0
$523$	26.6430	1.16502	0.582509	−	0.812825i	$-0.302071\pi$
$523$	26.6430	1.16502	0.582509	+	0.812825i	$0.302071\pi$
$524$	0	0
$525$	−3.48929	−0.152285
$526$	0	0
$527$	6.39312	0.278488
$528$	0	0
$529$	17.8719	0.777040
$530$	0	0
$531$	−6.29273	−0.273081
$532$	0	0
$533$	−3.17935	−0.137713
$534$	0	0
$535$	0.585462	0.0253117
$536$	0	0
$537$	−18.0575	−0.779240
$538$	0	0
$539$	21.6069	0.930674
$540$	0	0
$541$	−3.28600	−0.141276	−0.0706381	−	0.997502i	$-0.522504\pi$
$541$	−3.28600	−0.141276	−0.0706381	+	0.997502i	$0.522504\pi$
$542$	0	0
$543$	25.3288	1.08696
$544$	0	0
$545$	−2.58546	−0.110749
$546$	0	0
$547$	28.0894	1.20102	0.600509	−	0.799618i	$-0.294965\pi$
$547$	28.0894	1.20102	0.600509	+	0.799618i	$0.294965\pi$
$548$	0	0
$549$	11.3717	0.485332
$550$	0	0
$551$	21.8677	0.931596
$552$	0	0
$553$	17.5212	0.745077
$554$	0	0
$555$	2.17513	0.0923293
$556$	0	0
$557$	−20.1579	−0.854119	−0.427059	−	0.904224i	$-0.640450\pi$
$557$	−20.1579	−0.854119	−0.427059	+	0.904224i	$0.640450\pi$
$558$	0	0
$559$	1.07269	0.0453700
$560$	0	0
$561$	−4.17513	−0.176274
$562$	0	0
$563$	−29.1109	−1.22688	−0.613438	−	0.789743i	$-0.710214\pi$
$563$	−29.1109	−1.22688	−0.613438	+	0.789743i	$0.710214\pi$
$564$	0	0
$565$	15.3717	0.646692
$566$	0	0
$567$	−3.48929	−0.146536
$568$	0	0
$569$	−1.56404	−0.0655679	−0.0327840	−	0.999462i	$-0.510437\pi$
$569$	−1.56404	−0.0655679	−0.0327840	+	0.999462i	$0.510437\pi$
$570$	0	0
$571$	19.6791	0.823545	0.411773	−	0.911287i	$-0.364910\pi$
$571$	19.6791	0.823545	0.411773	+	0.911287i	$0.364910\pi$
$572$	0	0
$573$	−15.3288	−0.640372
$574$	0	0
$575$	−6.39312	−0.266611
$576$	0	0
$577$	−0.192347	−0.00800750	−0.00400375	−	0.999992i	$-0.501274\pi$
$577$	−0.192347	−0.00800750	−0.00400375	+	0.999992i	$0.501274\pi$
$578$	0	0
$579$	−24.4078	−1.01435
$580$	0	0
$581$	−2.04285	−0.0847515
$582$	0	0
$583$	−13.3461	−0.552737
$584$	0	0
$585$	0.292731	0.0121029
$586$	0	0
$587$	−12.7606	−0.526686	−0.263343	−	0.964702i	$-0.584825\pi$
$587$	−12.7606	−0.526686	−0.263343	+	0.964702i	$0.584825\pi$
$588$	0	0
$589$	36.9651	1.52312
$590$	0	0
$591$	−13.5640	−0.557950
$592$	0	0
$593$	23.7820	0.976610	0.488305	−	0.872673i	$-0.337615\pi$
$593$	23.7820	0.976610	0.488305	+	0.872673i	$0.337615\pi$
$594$	0	0
$595$	3.48929	0.143047
$596$	0	0
$597$	−7.56404	−0.309576
$598$	0	0
$599$	44.9357	1.83602	0.918012	−	0.396552i	$-0.129793\pi$
$599$	44.9357	1.83602	0.918012	+	0.396552i	$0.129793\pi$
$600$	0	0
$601$	18.2008	0.742425	0.371212	−	0.928548i	$-0.378942\pi$
$601$	18.2008	0.742425	0.371212	+	0.928548i	$0.378942\pi$
$602$	0	0
$603$	11.6644	0.475012
$604$	0	0
$605$	6.43175	0.261488
$606$	0	0
$607$	−39.8051	−1.61564	−0.807820	−	0.589429i	$-0.799353\pi$
$607$	−39.8051	−1.61564	−0.807820	+	0.589429i	$0.799353\pi$
$608$	0	0
$609$	13.1966	0.534751
$610$	0	0
$611$	−3.26504	−0.132089
$612$	0	0
$613$	−14.6858	−0.593156	−0.296578	−	0.955009i	$-0.595845\pi$
$613$	−14.6858	−0.593156	−0.296578	+	0.955009i	$0.595845\pi$
$614$	0	0
$615$	−10.8610	−0.437957
$616$	0	0
$617$	0.0428457	0.00172490	0.000862452	−	1.00000i	$-0.499725\pi$
$617$	0.0428457	0.00172490	0.000862452		1.00000i	$0.499725\pi$
$618$	0	0
$619$	24.7350	0.994182	0.497091	−	0.867698i	$-0.334401\pi$
$619$	24.7350	0.994182	0.497091	+	0.867698i	$0.334401\pi$
$620$	0	0
$621$	−6.39312	−0.256547
$622$	0	0
$623$	−48.0294	−1.92426
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.1407	−0.964087
$628$	0	0
$629$	−2.17513	−0.0867283
$630$	0	0
$631$	6.91852	0.275422	0.137711	−	0.990472i	$-0.456025\pi$
$631$	6.91852	0.275422	0.137711	+	0.990472i	$0.456025\pi$
$632$	0	0
$633$	5.17092	0.205526
$634$	0	0
$635$	−20.3503	−0.807576
$636$	0	0
$637$	1.51492	0.0600234
$638$	0	0
$639$	7.07896	0.280039
$640$	0	0
$641$	−16.6686	−0.658371	−0.329186	−	0.944265i	$-0.606774\pi$
$641$	−16.6686	−0.658371	−0.329186	+	0.944265i	$0.606774\pi$
$642$	0	0
$643$	−43.0680	−1.69844	−0.849218	−	0.528042i	$-0.822926\pi$
$643$	−43.0680	−1.69844	−0.849218	+	0.528042i	$0.822926\pi$
$644$	0	0
$645$	3.66442	0.144287
$646$	0	0
$647$	37.0852	1.45797	0.728985	−	0.684529i	$-0.239992\pi$
$647$	37.0852	1.45797	0.728985	+	0.684529i	$0.239992\pi$
$648$	0	0
$649$	−26.2730	−1.03131
$650$	0	0
$651$	22.3074	0.874297
$652$	0	0
$653$	5.12808	0.200677	0.100339	−	0.994953i	$-0.468007\pi$
$653$	5.12808	0.200677	0.100339	+	0.994953i	$0.468007\pi$
$654$	0	0
$655$	−3.41454	−0.133417
$656$	0	0
$657$	5.48929	0.214158
$658$	0	0
$659$	−27.2285	−1.06067	−0.530335	−	0.847788i	$-0.677934\pi$
$659$	−27.2285	−1.06067	−0.530335	+	0.847788i	$0.677934\pi$
$660$	0	0
$661$	−33.0680	−1.28620	−0.643098	−	0.765783i	$-0.722351\pi$
$661$	−33.0680	−1.28620	−0.643098	+	0.765783i	$0.722351\pi$
$662$	0	0
$663$	−0.292731	−0.0113687
$664$	0	0
$665$	20.1751	0.782358
$666$	0	0
$667$	24.1789	0.936210
$668$	0	0
$669$	4.35027	0.168191
$670$	0	0
$671$	47.4783	1.83288
$672$	0	0
$673$	−18.9210	−0.729352	−0.364676	−	0.931134i	$-0.618820\pi$
$673$	−18.9210	−0.729352	−0.364676	+	0.931134i	$0.618820\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	31.5725	1.21343	0.606714	−	0.794920i	$-0.292487\pi$
$677$	31.5725	1.21343	0.606714	+	0.794920i	$0.292487\pi$
$678$	0	0
$679$	47.6791	1.82976
$680$	0	0
$681$	23.7648	0.910669
$682$	0	0
$683$	43.8286	1.67706	0.838528	−	0.544859i	$-0.183417\pi$
$683$	43.8286	1.67706	0.838528	+	0.544859i	$0.183417\pi$
$684$	0	0
$685$	−20.1323	−0.769215
$686$	0	0
$687$	−10.6111	−0.404839
$688$	0	0
$689$	−0.935731	−0.0356485
$690$	0	0
$691$	38.7434	1.47387	0.736934	−	0.675965i	$-0.236273\pi$
$691$	38.7434	1.47387	0.736934	+	0.675965i	$0.236273\pi$
$692$	0	0
$693$	−14.5682	−0.553402
$694$	0	0
$695$	7.76481	0.294536
$696$	0	0
$697$	10.8610	0.411389
$698$	0	0
$699$	−13.9143	−0.526287
$700$	0	0
$701$	6.60015	0.249284	0.124642	−	0.992202i	$-0.460222\pi$
$701$	6.60015	0.249284	0.124642	+	0.992202i	$0.460222\pi$
$702$	0	0
$703$	−12.5767	−0.474338
$704$	0	0
$705$	−11.1537	−0.420073
$706$	0	0
$707$	−33.7220	−1.26824
$708$	0	0
$709$	−8.24361	−0.309595	−0.154798	−	0.987946i	$-0.549473\pi$
$709$	−8.24361	−0.309595	−0.154798	+	0.987946i	$0.549473\pi$
$710$	0	0
$711$	−5.02142	−0.188318
$712$	0	0
$713$	40.8719	1.53067
$714$	0	0
$715$	1.22219	0.0457074
$716$	0	0
$717$	24.3503	0.909377
$718$	0	0
$719$	−39.3692	−1.46822	−0.734111	−	0.679029i	$-0.762401\pi$
$719$	−39.3692	−1.46822	−0.734111	+	0.679029i	$0.762401\pi$
$720$	0	0
$721$	41.8715	1.55937
$722$	0	0
$723$	23.9572	0.890976
$724$	0	0
$725$	−3.78202	−0.140461
$726$	0	0
$727$	35.8799	1.33071	0.665356	−	0.746527i	$-0.268280\pi$
$727$	35.8799	1.33071	0.665356	+	0.746527i	$0.268280\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.66442	−0.135534
$732$	0	0
$733$	−6.33558	−0.234010	−0.117005	−	0.993131i	$-0.537329\pi$
$733$	−6.33558	−0.234010	−0.117005	+	0.993131i	$0.537329\pi$
$734$	0	0
$735$	5.17513	0.190888
$736$	0	0
$737$	48.7005	1.79391
$738$	0	0
$739$	−41.9865	−1.54450	−0.772250	−	0.635319i	$-0.780869\pi$
$739$	−41.9865	−1.54450	−0.772250	+	0.635319i	$0.780869\pi$
$740$	0	0
$741$	−1.69258	−0.0621783
$742$	0	0
$743$	14.5939	0.535398	0.267699	−	0.963503i	$-0.413737\pi$
$743$	14.5939	0.535398	0.267699	+	0.963503i	$0.413737\pi$
$744$	0	0
$745$	9.66442	0.354077
$746$	0	0
$747$	0.585462	0.0214209
$748$	0	0
$749$	−2.04285	−0.0746440
$750$	0	0
$751$	−4.55104	−0.166070	−0.0830349	−	0.996547i	$-0.526461\pi$
$751$	−4.55104	−0.166070	−0.0830349	+	0.996547i	$0.526461\pi$
$752$	0	0
$753$	26.4078	0.962354
$754$	0	0
$755$	−19.1537	−0.697075
$756$	0	0
$757$	−41.2285	−1.49847	−0.749237	−	0.662302i	$-0.769580\pi$
$757$	−41.2285	−1.49847	−0.749237	+	0.662302i	$0.769580\pi$
$758$	0	0
$759$	−26.6921	−0.968862
$760$	0	0
$761$	−0.743385	−0.0269477	−0.0134738	−	0.999909i	$-0.504289\pi$
$761$	−0.743385	−0.0269477	−0.0134738	+	0.999909i	$0.504289\pi$
$762$	0	0
$763$	9.02142	0.326597
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	−1.84208	−0.0665136
$768$	0	0
$769$	10.0894	0.363835	0.181917	−	0.983314i	$-0.441770\pi$
$769$	10.0894	0.363835	0.181917	+	0.983314i	$0.441770\pi$
$770$	0	0
$771$	−11.1709	−0.402311
$772$	0	0
$773$	47.8971	1.72274	0.861369	−	0.507979i	$-0.169607\pi$
$773$	47.8971	1.72274	0.861369	+	0.507979i	$0.169607\pi$
$774$	0	0
$775$	−6.39312	−0.229647
$776$	0	0
$777$	−7.58967	−0.272278
$778$	0	0
$779$	62.7984	2.24999
$780$	0	0
$781$	29.5556	1.05758
$782$	0	0
$783$	−3.78202	−0.135158
$784$	0	0
$785$	−16.0575	−0.573118
$786$	0	0
$787$	−25.1966	−0.898160	−0.449080	−	0.893491i	$-0.648248\pi$
$787$	−25.1966	−0.898160	−0.449080	+	0.893491i	$0.648248\pi$
$788$	0	0
$789$	15.7392	0.560329
$790$	0	0
$791$	−53.6363	−1.90709
$792$	0	0
$793$	3.32885	0.118211
$794$	0	0
$795$	−3.19656	−0.113370
$796$	0	0
$797$	9.73917	0.344979	0.172490	−	0.985011i	$-0.444819\pi$
$797$	9.73917	0.344979	0.172490	+	0.985011i	$0.444819\pi$
$798$	0	0
$799$	11.1537	0.394590
$800$	0	0
$801$	13.7648	0.486356
$802$	0	0
$803$	22.9185	0.808777
$804$	0	0
$805$	22.3074	0.786233
$806$	0	0
$807$	−2.96137	−0.104245
$808$	0	0
$809$	12.8782	0.452773	0.226387	−	0.974038i	$-0.427309\pi$
$809$	12.8782	0.452773	0.226387	+	0.974038i	$0.427309\pi$
$810$	0	0
$811$	39.7135	1.39453	0.697266	−	0.716813i	$-0.254400\pi$
$811$	39.7135	1.39453	0.697266	+	0.716813i	$0.254400\pi$
$812$	0	0
$813$	3.41454	0.119753
$814$	0	0
$815$	−18.7178	−0.655654
$816$	0	0
$817$	−21.1878	−0.741266
$818$	0	0
$819$	−1.02142	−0.0356914
$820$	0	0
$821$	−43.6707	−1.52412	−0.762059	−	0.647508i	$-0.775811\pi$
$821$	−43.6707	−1.52412	−0.762059	+	0.647508i	$0.775811\pi$
$822$	0	0
$823$	12.3612	0.430885	0.215442	−	0.976517i	$-0.430881\pi$
$823$	12.3612	0.430885	0.215442	+	0.976517i	$0.430881\pi$
$824$	0	0
$825$	4.17513	0.145360
$826$	0	0
$827$	4.70054	0.163454	0.0817269	−	0.996655i	$-0.473956\pi$
$827$	4.70054	0.163454	0.0817269	+	0.996655i	$0.473956\pi$
$828$	0	0
$829$	2.49602	0.0866903	0.0433452	−	0.999060i	$-0.486198\pi$
$829$	2.49602	0.0866903	0.0433452	+	0.999060i	$0.486198\pi$
$830$	0	0
$831$	5.32885	0.184856
$832$	0	0
$833$	−5.17513	−0.179308
$834$	0	0
$835$	−21.7220	−0.751719
$836$	0	0
$837$	−6.39312	−0.220978
$838$	0	0
$839$	17.0617	0.589037	0.294519	−	0.955646i	$-0.404841\pi$
$839$	17.0617	0.589037	0.294519	+	0.955646i	$0.404841\pi$
$840$	0	0
$841$	−14.6963	−0.506770
$842$	0	0
$843$	22.9357	0.789949
$844$	0	0
$845$	−12.9143	−0.444266
$846$	0	0
$847$	−22.4422	−0.771124
$848$	0	0
$849$	11.3889	0.390866
$850$	0	0
$851$	−13.9059	−0.476688
$852$	0	0
$853$	2.58546	0.0885245	0.0442623	−	0.999020i	$-0.485906\pi$
$853$	2.58546	0.0885245	0.0442623	+	0.999020i	$0.485906\pi$
$854$	0	0
$855$	−5.78202	−0.197741
$856$	0	0
$857$	−8.77781	−0.299844	−0.149922	−	0.988698i	$-0.547902\pi$
$857$	−8.77781	−0.299844	−0.149922	+	0.988698i	$0.547902\pi$
$858$	0	0
$859$	−2.71775	−0.0927285	−0.0463642	−	0.998925i	$-0.514763\pi$
$859$	−2.71775	−0.0927285	−0.0463642	+	0.998925i	$0.514763\pi$
$860$	0	0
$861$	37.8971	1.29153
$862$	0	0
$863$	−53.8458	−1.83293	−0.916467	−	0.400111i	$-0.868972\pi$
$863$	−53.8458	−1.83293	−0.916467	+	0.400111i	$0.868972\pi$
$864$	0	0
$865$	0.393115	0.0133663
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−20.9651	−0.711193
$870$	0	0
$871$	3.41454	0.115697
$872$	0	0
$873$	−13.6644	−0.462471
$874$	0	0
$875$	−3.48929	−0.117959
$876$	0	0
$877$	−49.0680	−1.65691	−0.828455	−	0.560056i	$-0.810780\pi$
$877$	−49.0680	−1.65691	−0.828455	+	0.560056i	$0.810780\pi$
$878$	0	0
$879$	15.3461	0.517610
$880$	0	0
$881$	21.3398	0.718956	0.359478	−	0.933154i	$-0.382955\pi$
$881$	21.3398	0.718956	0.359478	+	0.933154i	$0.382955\pi$
$882$	0	0
$883$	48.4800	1.63148	0.815742	−	0.578416i	$-0.196329\pi$
$883$	48.4800	1.63148	0.815742	+	0.578416i	$0.196329\pi$
$884$	0	0
$885$	−6.29273	−0.211528
$886$	0	0
$887$	30.0428	1.00874	0.504370	−	0.863488i	$-0.331725\pi$
$887$	30.0428	1.00874	0.504370	+	0.863488i	$0.331725\pi$
$888$	0	0
$889$	71.0080	2.38153
$890$	0	0
$891$	4.17513	0.139872
$892$	0	0
$893$	64.4910	2.15811
$894$	0	0
$895$	−18.0575	−0.603597
$896$	0	0
$897$	−1.87146	−0.0624863
$898$	0	0
$899$	24.1789	0.806411
$900$	0	0
$901$	3.19656	0.106493
$902$	0	0
$903$	−12.7862	−0.425499
$904$	0	0
$905$	25.3288	0.841959
$906$	0	0
$907$	31.9744	1.06169	0.530846	−	0.847468i	$-0.321874\pi$
$907$	31.9744	1.06169	0.530846	+	0.847468i	$0.321874\pi$
$908$	0	0
$909$	9.66442	0.320549
$910$	0	0
$911$	−1.06175	−0.0351773	−0.0175887	−	0.999845i	$-0.505599\pi$
$911$	−1.06175	−0.0351773	−0.0175887	+	0.999845i	$0.505599\pi$
$912$	0	0
$913$	2.44438	0.0808973
$914$	0	0
$915$	11.3717	0.375937
$916$	0	0
$917$	11.9143	0.393445
$918$	0	0
$919$	−1.98279	−0.0654061	−0.0327031	−	0.999465i	$-0.510412\pi$
$919$	−1.98279	−0.0654061	−0.0327031	+	0.999465i	$0.510412\pi$
$920$	0	0
$921$	6.87819	0.226644
$922$	0	0
$923$	2.07223	0.0682083
$924$	0	0
$925$	2.17513	0.0715180
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	43.3263	1.42149	0.710745	−	0.703449i	$-0.248358\pi$
$929$	43.3263	1.42149	0.710745	+	0.703449i	$0.248358\pi$
$930$	0	0
$931$	−29.9227	−0.980678
$932$	0	0
$933$	11.0533	0.361870
$934$	0	0
$935$	−4.17513	−0.136542
$936$	0	0
$937$	13.4145	0.438234	0.219117	−	0.975699i	$-0.429682\pi$
$937$	13.4145	0.438234	0.219117	+	0.975699i	$0.429682\pi$
$938$	0	0
$939$	−5.92525	−0.193363
$940$	0	0
$941$	24.0722	0.784732	0.392366	−	0.919809i	$-0.371657\pi$
$941$	24.0722	0.784732	0.392366	+	0.919809i	$0.371657\pi$
$942$	0	0
$943$	69.4355	2.26113
$944$	0	0
$945$	−3.48929	−0.113507
$946$	0	0
$947$	21.3717	0.694487	0.347243	−	0.937775i	$-0.387118\pi$
$947$	21.3717	0.694487	0.347243	+	0.937775i	$0.387118\pi$
$948$	0	0
$949$	1.60688	0.0521617
$950$	0	0
$951$	−20.7434	−0.672650
$952$	0	0
$953$	47.6106	1.54226	0.771130	−	0.636678i	$-0.219692\pi$
$953$	47.6106	1.54226	0.771130	+	0.636678i	$0.219692\pi$
$954$	0	0
$955$	−15.3288	−0.496030
$956$	0	0
$957$	−15.7904	−0.510432
$958$	0	0
$959$	70.2474	2.26841
$960$	0	0
$961$	9.87192	0.318449
$962$	0	0
$963$	0.585462	0.0188662
$964$	0	0
$965$	−24.4078	−0.785715
$966$	0	0
$967$	−29.3717	−0.944530	−0.472265	−	0.881457i	$-0.656563\pi$
$967$	−29.3717	−0.944530	−0.472265	+	0.881457i	$0.656563\pi$
$968$	0	0
$969$	5.78202	0.185745
$970$	0	0
$971$	−44.2793	−1.42099	−0.710495	−	0.703703i	$-0.751529\pi$
$971$	−44.2793	−1.42099	−0.710495	+	0.703703i	$0.751529\pi$
$972$	0	0
$973$	−27.0937	−0.868583
$974$	0	0
$975$	0.292731	0.00937489
$976$	0	0
$977$	−15.4868	−0.495466	−0.247733	−	0.968828i	$-0.579686\pi$
$977$	−15.4868	−0.495466	−0.247733	+	0.968828i	$0.579686\pi$
$978$	0	0
$979$	57.4699	1.83675
$980$	0	0
$981$	−2.58546	−0.0825474
$982$	0	0
$983$	38.8929	1.24049	0.620245	−	0.784408i	$-0.287033\pi$
$983$	38.8929	1.24049	0.620245	+	0.784408i	$0.287033\pi$
$984$	0	0
$985$	−13.5640	−0.432186
$986$	0	0
$987$	38.9185	1.23879
$988$	0	0
$989$	−23.4271	−0.744938
$990$	0	0
$991$	−15.5640	−0.494408	−0.247204	−	0.968963i	$-0.579512\pi$
$991$	−15.5640	−0.494408	−0.247204	+	0.968963i	$0.579512\pi$
$992$	0	0
$993$	−4.41033	−0.139958
$994$	0	0
$995$	−7.56404	−0.239796
$996$	0	0
$997$	−24.0466	−0.761563	−0.380782	−	0.924665i	$-0.624345\pi$
$997$	−24.0466	−0.761563	−0.380782	+	0.924665i	$0.624345\pi$
$998$	0	0
$999$	2.17513	0.0688182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.a.bs.1.1		3
4.3	odd	2		2040.2.a.x.1.3	✓	3
12.11	even	2		6120.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.a.x.1.3	✓	3	4.3	odd	2
4080.2.a.bs.1.1		3	1.1	even	1	trivial
6120.2.a.bp.1.3		3	12.11	even	2

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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.48929 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.48929 q^{7} +1.00000 q^{9} +4.17513 q^{11} +0.292731 q^{13} +1.00000 q^{15} -1.00000 q^{17} -5.78202 q^{19} -3.48929 q^{21} -6.39312 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.78202 q^{29} -6.39312 q^{31} +4.17513 q^{33} -3.48929 q^{35} +2.17513 q^{37} +0.292731 q^{39} -10.8610 q^{41} +3.66442 q^{43} +1.00000 q^{45} -11.1537 q^{47} +5.17513 q^{49} -1.00000 q^{51} -3.19656 q^{53} +4.17513 q^{55} -5.78202 q^{57} -6.29273 q^{59} +11.3717 q^{61} -3.48929 q^{63} +0.292731 q^{65} +11.6644 q^{67} -6.39312 q^{69} +7.07896 q^{71} +5.48929 q^{73} +1.00000 q^{75} -14.5682 q^{77} -5.02142 q^{79} +1.00000 q^{81} +0.585462 q^{83} -1.00000 q^{85} -3.78202 q^{87} +13.7648 q^{89} -1.02142 q^{91} -6.39312 q^{93} -5.78202 q^{95} -13.6644 q^{97} +4.17513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 7 q^{11} - 2 q^{13} + 3 q^{15} - 3 q^{17} - 7 q^{19} - 3 q^{21} - 10 q^{23} + 3 q^{25} + 3 q^{27} - q^{29} - 10 q^{31} - 7 q^{33} - 3 q^{35} - 13 q^{37} - 2 q^{39} - q^{41} - 16 q^{43} + 3 q^{45} + q^{47} - 4 q^{49} - 3 q^{51} - 5 q^{53} - 7 q^{55} - 7 q^{57} - 16 q^{59} + 10 q^{61} - 3 q^{63} - 2 q^{65} + 8 q^{67} - 10 q^{69} + 9 q^{73} + 3 q^{75} - 15 q^{77} - 30 q^{79} + 3 q^{81} - 4 q^{83} - 3 q^{85} - q^{87} + 8 q^{89} - 18 q^{91} - 10 q^{93} - 7 q^{95} - 14 q^{97} - 7 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 7 * q^11 - 2 * q^13 + 3 * q^15 - 3 * q^17 - 7 * q^19 - 3 * q^21 - 10 * q^23 + 3 * q^25 + 3 * q^27 - q^29 - 10 * q^31 - 7 * q^33 - 3 * q^35 - 13 * q^37 - 2 * q^39 - q^41 - 16 * q^43 + 3 * q^45 + q^47 - 4 * q^49 - 3 * q^51 - 5 * q^53 - 7 * q^55 - 7 * q^57 - 16 * q^59 + 10 * q^61 - 3 * q^63 - 2 * q^65 + 8 * q^67 - 10 * q^69 + 9 * q^73 + 3 * q^75 - 15 * q^77 - 30 * q^79 + 3 * q^81 - 4 * q^83 - 3 * q^85 - q^87 + 8 * q^89 - 18 * q^91 - 10 * q^93 - 7 * q^95 - 14 * q^97 - 7 * q^99

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