LMFDB - Embedded newform 1216.2.a.w.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,2,Mod(1,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,-1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.70980888579$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.329727$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +0.844614 q^{5} -5.06562 q^{7} -0.561553 q^{9} -2.84461 q^{11} -2.90210 q^{13} +1.31891 q^{15} +7.72508 q^{17} -1.00000 q^{19} -7.91023 q^{21} +3.91023 q^{23} -4.28663 q^{25} -5.56155 q^{27} -2.90210 q^{29} -7.00814 q^{31} -4.44201 q^{33} -4.27849 q^{35} -9.00814 q^{37} -4.53178 q^{39} -6.81233 q^{41} -5.96772 q^{43} -0.474295 q^{45} -1.04042 q^{47} +18.6605 q^{49} +12.0631 q^{51} +9.03334 q^{53} -2.40260 q^{55} -1.56155 q^{57} -0.749220 q^{59} -2.27849 q^{61} +2.84461 q^{63} -2.45115 q^{65} -10.3739 q^{67} +6.10604 q^{69} +2.13124 q^{71} +4.60197 q^{73} -6.69380 q^{75} +14.4097 q^{77} +3.88503 q^{79} -7.00000 q^{81} -8.44201 q^{83} +6.52470 q^{85} -4.53178 q^{87} +16.0667 q^{89} +14.7009 q^{91} -10.9436 q^{93} -0.844614 q^{95} -3.68923 q^{97} +1.59740 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37}+ \cdots - 19 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * q^99

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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	0.844614	0.377723	0.188861	−	0.982004i	$-0.439520\pi$
$5$	0.844614	0.377723	0.188861	+	0.982004i	$0.439520\pi$
$6$	0	0
$7$	−5.06562	−1.91462	−0.957312	−	0.289056i	$-0.906659\pi$
$7$	−5.06562	−1.91462	−0.957312	+	0.289056i	$0.906659\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−2.84461	−0.857683	−0.428842	−	0.903380i	$-0.641078\pi$
$11$	−2.84461	−0.857683	−0.428842	+	0.903380i	$0.641078\pi$
$12$	0	0
$13$	−2.90210	−0.804897	−0.402449	−	0.915443i	$-0.631841\pi$
$13$	−2.90210	−0.804897	−0.402449	+	0.915443i	$0.631841\pi$
$14$	0	0
$15$	1.31891	0.340541
$16$	0	0
$17$	7.72508	1.87361	0.936803	−	0.349857i	$-0.113770\pi$
$17$	7.72508	1.87361	0.936803	+	0.349857i	$0.113770\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−7.91023	−1.72615
$22$	0	0
$23$	3.91023	0.815340	0.407670	−	0.913129i	$-0.366341\pi$
$23$	3.91023	0.815340	0.407670	+	0.913129i	$0.366341\pi$
$24$	0	0
$25$	−4.28663	−0.857326
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−2.90210	−0.538906	−0.269453	−	0.963014i	$-0.586843\pi$
$29$	−2.90210	−0.538906	−0.269453	+	0.963014i	$0.586843\pi$
$30$	0	0
$31$	−7.00814	−1.25870	−0.629349	−	0.777123i	$-0.716678\pi$
$31$	−7.00814	−1.25870	−0.629349	+	0.777123i	$0.716678\pi$
$32$	0	0
$33$	−4.44201	−0.773255
$34$	0	0
$35$	−4.27849	−0.723197
$36$	0	0
$37$	−9.00814	−1.48093	−0.740464	−	0.672096i	$-0.765394\pi$
$37$	−9.00814	−1.48093	−0.740464	+	0.672096i	$0.765394\pi$
$38$	0	0
$39$	−4.53178	−0.725666
$40$	0	0
$41$	−6.81233	−1.06391	−0.531954	−	0.846773i	$-0.678542\pi$
$41$	−6.81233	−1.06391	−0.531954	+	0.846773i	$0.678542\pi$
$42$	0	0
$43$	−5.96772	−0.910069	−0.455034	−	0.890474i	$-0.650373\pi$
$43$	−5.96772	−0.910069	−0.455034	+	0.890474i	$0.650373\pi$
$44$	0	0
$45$	−0.474295	−0.0707037
$46$	0	0
$47$	−1.04042	−0.151760	−0.0758802	−	0.997117i	$-0.524177\pi$
$47$	−1.04042	−0.151760	−0.0758802	+	0.997117i	$0.524177\pi$
$48$	0	0
$49$	18.6605	2.66579
$50$	0	0
$51$	12.0631	1.68917
$52$	0	0
$53$	9.03334	1.24082	0.620412	−	0.784276i	$-0.286965\pi$
$53$	9.03334	1.24082	0.620412	+	0.784276i	$0.286965\pi$
$54$	0	0
$55$	−2.40260	−0.323966
$56$	0	0
$57$	−1.56155	−0.206833
$58$	0	0
$59$	−0.749220	−0.0975401	−0.0487701	−	0.998810i	$-0.515530\pi$
$59$	−0.749220	−0.0975401	−0.0487701	+	0.998810i	$0.515530\pi$
$60$	0	0
$61$	−2.27849	−0.291731	−0.145866	−	0.989304i	$-0.546597\pi$
$61$	−2.27849	−0.291731	−0.145866	+	0.989304i	$0.546597\pi$
$62$	0	0
$63$	2.84461	0.358388
$64$	0	0
$65$	−2.45115	−0.304028
$66$	0	0
$67$	−10.3739	−1.26737	−0.633686	−	0.773590i	$-0.718459\pi$
$67$	−10.3739	−1.26737	−0.633686	+	0.773590i	$0.718459\pi$
$68$	0	0
$69$	6.10604	0.735081
$70$	0	0
$71$	2.13124	0.252932	0.126466	−	0.991971i	$-0.459637\pi$
$71$	2.13124	0.252932	0.126466	+	0.991971i	$0.459637\pi$
$72$	0	0
$73$	4.60197	0.538620	0.269310	−	0.963054i	$-0.413204\pi$
$73$	4.60197	0.538620	0.269310	+	0.963054i	$0.413204\pi$
$74$	0	0
$75$	−6.69380	−0.772933
$76$	0	0
$77$	14.4097	1.64214
$78$	0	0
$79$	3.88503	0.437100	0.218550	−	0.975826i	$-0.429867\pi$
$79$	3.88503	0.437100	0.218550	+	0.975826i	$0.429867\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−8.44201	−0.926631	−0.463316	−	0.886193i	$-0.653340\pi$
$83$	−8.44201	−0.926631	−0.463316	+	0.886193i	$0.653340\pi$
$84$	0	0
$85$	6.52470	0.707703
$86$	0	0
$87$	−4.53178	−0.485858
$88$	0	0
$89$	16.0667	1.70306	0.851532	−	0.524302i	$-0.175674\pi$
$89$	16.0667	1.70306	0.851532	+	0.524302i	$0.175674\pi$
$90$	0	0
$91$	14.7009	1.54108
$92$	0	0
$93$	−10.9436	−1.13480
$94$	0	0
$95$	−0.844614	−0.0866555
$96$	0	0
$97$	−3.68923	−0.374584	−0.187292	−	0.982304i	$-0.559971\pi$
$97$	−3.68923	−0.374584	−0.187292	+	0.982304i	$0.559971\pi$
$98$	0	0
$99$	1.59740	0.160545
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	−0.812333	−0.0800415	−0.0400208	−	0.999199i	$-0.512742\pi$
$103$	−0.812333	−0.0800415	−0.0400208	+	0.999199i	$0.512742\pi$
$104$	0	0
$105$	−6.68109	−0.652008
$106$	0	0
$107$	−9.88860	−0.955967	−0.477983	−	0.878369i	$-0.658632\pi$
$107$	−9.88860	−0.955967	−0.477983	+	0.878369i	$0.658632\pi$
$108$	0	0
$109$	10.8375	1.03805	0.519024	−	0.854760i	$-0.326296\pi$
$109$	10.8375	1.03805	0.519024	+	0.854760i	$0.326296\pi$
$110$	0	0
$111$	−14.0667	−1.33515
$112$	0	0
$113$	−0.310773	−0.0292351	−0.0146175	−	0.999893i	$-0.504653\pi$
$113$	−0.310773	−0.0292351	−0.0146175	+	0.999893i	$0.504653\pi$
$114$	0	0
$115$	3.30264	0.307972
$116$	0	0
$117$	1.62968	0.150664
$118$	0	0
$119$	−39.1323	−3.58725
$120$	0	0
$121$	−2.90817	−0.264379
$122$	0	0
$123$	−10.6378	−0.959180
$124$	0	0
$125$	−7.84361	−0.701554
$126$	0	0
$127$	18.6882	1.65831	0.829156	−	0.559017i	$-0.188822\pi$
$127$	18.6882	1.65831	0.829156	+	0.559017i	$0.188822\pi$
$128$	0	0
$129$	−9.31891	−0.820484
$130$	0	0
$131$	16.6055	1.45083	0.725416	−	0.688310i	$-0.241647\pi$
$131$	16.6055	1.45083	0.725416	+	0.688310i	$0.241647\pi$
$132$	0	0
$133$	5.06562	0.439245
$134$	0	0
$135$	−4.69736	−0.404285
$136$	0	0
$137$	−2.91274	−0.248852	−0.124426	−	0.992229i	$-0.539709\pi$
$137$	−2.91274	−0.248852	−0.124426	+	0.992229i	$0.539709\pi$
$138$	0	0
$139$	0.278492	0.0236214	0.0118107	−	0.999930i	$-0.496240\pi$
$139$	0.278492	0.0236214	0.0118107	+	0.999930i	$0.496240\pi$
$140$	0	0
$141$	−1.62467	−0.136822
$142$	0	0
$143$	8.25535	0.690347
$144$	0	0
$145$	−2.45115	−0.203557
$146$	0	0
$147$	29.1394	2.40338
$148$	0	0
$149$	8.08269	0.662160	0.331080	−	0.943603i	$-0.392587\pi$
$149$	8.08269	0.662160	0.331080	+	0.943603i	$0.392587\pi$
$150$	0	0
$151$	15.0585	1.22545	0.612723	−	0.790297i	$-0.290074\pi$
$151$	15.0585	1.22545	0.612723	+	0.790297i	$0.290074\pi$
$152$	0	0
$153$	−4.33804	−0.350710
$154$	0	0
$155$	−5.91917	−0.475439
$156$	0	0
$157$	−10.9273	−0.872094	−0.436047	−	0.899924i	$-0.643622\pi$
$157$	−10.9273	−0.872094	−0.436047	+	0.899924i	$0.643622\pi$
$158$	0	0
$159$	14.1060	1.11868
$160$	0	0
$161$	−19.8078	−1.56107
$162$	0	0
$163$	−0.697363	−0.0546217	−0.0273108	−	0.999627i	$-0.508694\pi$
$163$	−0.697363	−0.0546217	−0.0273108	+	0.999627i	$0.508694\pi$
$164$	0	0
$165$	−3.75179	−0.292076
$166$	0	0
$167$	6.24621	0.483346	0.241673	−	0.970358i	$-0.422304\pi$
$167$	6.24621	0.483346	0.241673	+	0.970358i	$0.422304\pi$
$168$	0	0
$169$	−4.57782	−0.352140
$170$	0	0
$171$	0.561553	0.0429430
$172$	0	0
$173$	−6.69736	−0.509191	−0.254596	−	0.967048i	$-0.581942\pi$
$173$	−6.69736	−0.509191	−0.254596	+	0.967048i	$0.581942\pi$
$174$	0	0
$175$	21.7144	1.64146
$176$	0	0
$177$	−1.16995	−0.0879386
$178$	0	0
$179$	−18.2462	−1.36379	−0.681893	−	0.731452i	$-0.738843\pi$
$179$	−18.2462	−1.36379	−0.681893	+	0.731452i	$0.738843\pi$
$180$	0	0
$181$	−17.5651	−1.30561	−0.652803	−	0.757528i	$-0.726407\pi$
$181$	−17.5651	−1.30561	−0.652803	+	0.757528i	$0.726407\pi$
$182$	0	0
$183$	−3.55799	−0.263014
$184$	0	0
$185$	−7.60839	−0.559380
$186$	0	0
$187$	−21.9749	−1.60696
$188$	0	0
$189$	28.1727	2.04926
$190$	0	0
$191$	7.19686	0.520747	0.260373	−	0.965508i	$-0.416154\pi$
$191$	7.19686	0.520747	0.260373	+	0.965508i	$0.416154\pi$
$192$	0	0
$193$	−2.92730	−0.210712	−0.105356	−	0.994435i	$-0.533598\pi$
$193$	−2.92730	−0.210712	−0.105356	+	0.994435i	$0.533598\pi$
$194$	0	0
$195$	−3.82760	−0.274100
$196$	0	0
$197$	11.1394	0.793648	0.396824	−	0.917895i	$-0.370112\pi$
$197$	11.1394	0.793648	0.396824	+	0.917895i	$0.370112\pi$
$198$	0	0
$199$	5.57220	0.395003	0.197501	−	0.980303i	$-0.436717\pi$
$199$	5.57220	0.395003	0.197501	+	0.980303i	$0.436717\pi$
$200$	0	0
$201$	−16.1994	−1.14262
$202$	0	0
$203$	14.7009	1.03180
$204$	0	0
$205$	−5.75379	−0.401862
$206$	0	0
$207$	−2.19580	−0.152619
$208$	0	0
$209$	2.84461	0.196766
$210$	0	0
$211$	1.23451	0.0849871	0.0424935	−	0.999097i	$-0.486470\pi$
$211$	1.23451	0.0849871	0.0424935	+	0.999097i	$0.486470\pi$
$212$	0	0
$213$	3.32805	0.228034
$214$	0	0
$215$	−5.04042	−0.343754
$216$	0	0
$217$	35.5006	2.40993
$218$	0	0
$219$	7.18622	0.485600
$220$	0	0
$221$	−22.4189	−1.50806
$222$	0	0
$223$	−5.76092	−0.385780	−0.192890	−	0.981220i	$-0.561786\pi$
$223$	−5.76092	−0.385780	−0.192890	+	0.981220i	$0.561786\pi$
$224$	0	0
$225$	2.40717	0.160478
$226$	0	0
$227$	11.1862	0.742455	0.371228	−	0.928542i	$-0.378937\pi$
$227$	11.1862	0.742455	0.371228	+	0.928542i	$0.378937\pi$
$228$	0	0
$229$	−25.3624	−1.67600	−0.837999	−	0.545672i	$-0.816274\pi$
$229$	−25.3624	−1.67600	−0.837999	+	0.545672i	$0.816274\pi$
$230$	0	0
$231$	22.5016	1.48049
$232$	0	0
$233$	8.33804	0.546243	0.273122	−	0.961980i	$-0.411944\pi$
$233$	8.33804	0.546243	0.273122	+	0.961980i	$0.411944\pi$
$234$	0	0
$235$	−0.878750	−0.0573233
$236$	0	0
$237$	6.06668	0.394073
$238$	0	0
$239$	−16.4441	−1.06368	−0.531839	−	0.846845i	$-0.678499\pi$
$239$	−16.4441	−1.06368	−0.531839	+	0.846845i	$0.678499\pi$
$240$	0	0
$241$	−12.8286	−0.826363	−0.413182	−	0.910649i	$-0.635583\pi$
$241$	−12.8286	−0.826363	−0.413182	+	0.910649i	$0.635583\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	15.7609	1.00693
$246$	0	0
$247$	2.90210	0.184656
$248$	0	0
$249$	−13.1827	−0.835417
$250$	0	0
$251$	−17.8527	−1.12686	−0.563428	−	0.826165i	$-0.690518\pi$
$251$	−17.8527	−1.12686	−0.563428	+	0.826165i	$0.690518\pi$
$252$	0	0
$253$	−11.1231	−0.699304
$254$	0	0
$255$	10.1887	0.638039
$256$	0	0
$257$	1.49342	0.0931572	0.0465786	−	0.998915i	$-0.485168\pi$
$257$	1.49342	0.0931572	0.0465786	+	0.998915i	$0.485168\pi$
$258$	0	0
$259$	45.6318	2.83542
$260$	0	0
$261$	1.62968	0.100875
$262$	0	0
$263$	0.483433	0.0298097	0.0149049	−	0.999889i	$-0.495255\pi$
$263$	0.483433	0.0298097	0.0149049	+	0.999889i	$0.495255\pi$
$264$	0	0
$265$	7.62968	0.468688
$266$	0	0
$267$	25.0890	1.53542
$268$	0	0
$269$	−8.45115	−0.515276	−0.257638	−	0.966242i	$-0.582944\pi$
$269$	−8.45115	−0.515276	−0.257638	+	0.966242i	$0.582944\pi$
$270$	0	0
$271$	−16.5822	−1.00730	−0.503648	−	0.863909i	$-0.668009\pi$
$271$	−16.5822	−1.00730	−0.503648	+	0.863909i	$0.668009\pi$
$272$	0	0
$273$	22.9563	1.38938
$274$	0	0
$275$	12.1938	0.735314
$276$	0	0
$277$	−15.7124	−0.944065	−0.472032	−	0.881581i	$-0.656479\pi$
$277$	−15.7124	−0.944065	−0.472032	+	0.881581i	$0.656479\pi$
$278$	0	0
$279$	3.93544	0.235609
$280$	0	0
$281$	3.13938	0.187280	0.0936398	−	0.995606i	$-0.470150\pi$
$281$	3.13938	0.187280	0.0936398	+	0.995606i	$0.470150\pi$
$282$	0	0
$283$	26.3884	1.56863	0.784315	−	0.620363i	$-0.213015\pi$
$283$	26.3884	1.56863	0.784315	+	0.620363i	$0.213015\pi$
$284$	0	0
$285$	−1.31891	−0.0781254
$286$	0	0
$287$	34.5087	2.03698
$288$	0	0
$289$	42.6768	2.51040
$290$	0	0
$291$	−5.76092	−0.337711
$292$	0	0
$293$	−13.5903	−0.793955	−0.396978	−	0.917828i	$-0.629941\pi$
$293$	−13.5903	−0.793955	−0.396978	+	0.917828i	$0.629941\pi$
$294$	0	0
$295$	−0.632801	−0.0368431
$296$	0	0
$297$	15.8205	0.917997
$298$	0	0
$299$	−11.3479	−0.656265
$300$	0	0
$301$	30.2302	1.74244
$302$	0	0
$303$	25.3693	1.45743
$304$	0	0
$305$	−1.92445	−0.110193
$306$	0	0
$307$	3.44302	0.196503	0.0982516	−	0.995162i	$-0.468675\pi$
$307$	3.44302	0.196503	0.0982516	+	0.995162i	$0.468675\pi$
$308$	0	0
$309$	−1.26850	−0.0721625
$310$	0	0
$311$	−29.4935	−1.67242	−0.836211	−	0.548408i	$-0.815234\pi$
$311$	−29.4935	−1.67242	−0.836211	+	0.548408i	$0.815234\pi$
$312$	0	0
$313$	−10.0631	−0.568801	−0.284400	−	0.958706i	$-0.591794\pi$
$313$	−10.0631	−0.568801	−0.284400	+	0.958706i	$0.591794\pi$
$314$	0	0
$315$	2.40260	0.135371
$316$	0	0
$317$	−4.14931	−0.233049	−0.116524	−	0.993188i	$-0.537175\pi$
$317$	−4.14931	−0.233049	−0.116524	+	0.993188i	$0.537175\pi$
$318$	0	0
$319$	8.25535	0.462211
$320$	0	0
$321$	−15.4416	−0.861864
$322$	0	0
$323$	−7.72508	−0.429835
$324$	0	0
$325$	12.4402	0.690059
$326$	0	0
$327$	16.9234	0.935865
$328$	0	0
$329$	5.27036	0.290564
$330$	0	0
$331$	−25.8886	−1.42297	−0.711483	−	0.702703i	$-0.751976\pi$
$331$	−25.8886	−1.42297	−0.711483	+	0.702703i	$0.751976\pi$
$332$	0	0
$333$	5.05854	0.277207
$334$	0	0
$335$	−8.76192	−0.478715
$336$	0	0
$337$	31.5097	1.71644	0.858221	−	0.513280i	$-0.171570\pi$
$337$	31.5097	1.71644	0.858221	+	0.513280i	$0.171570\pi$
$338$	0	0
$339$	−0.485288	−0.0263572
$340$	0	0
$341$	19.9354	1.07956
$342$	0	0
$343$	−59.0677	−3.18936
$344$	0	0
$345$	5.15724	0.277657
$346$	0	0
$347$	13.9173	0.747120	0.373560	−	0.927606i	$-0.378137\pi$
$347$	13.9173	0.747120	0.373560	+	0.927606i	$0.378137\pi$
$348$	0	0
$349$	−6.34305	−0.339536	−0.169768	−	0.985484i	$-0.554302\pi$
$349$	−6.34305	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$350$	0	0
$351$	16.1402	0.861499
$352$	0	0
$353$	−32.0794	−1.70741	−0.853707	−	0.520754i	$-0.825651\pi$
$353$	−32.0794	−1.70741	−0.853707	+	0.520754i	$0.825651\pi$
$354$	0	0
$355$	1.80008	0.0955381
$356$	0	0
$357$	−61.1072	−3.23413
$358$	0	0
$359$	−1.86670	−0.0985205	−0.0492603	−	0.998786i	$-0.515686\pi$
$359$	−1.86670	−0.0985205	−0.0492603	+	0.998786i	$0.515686\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.54127	−0.238355
$364$	0	0
$365$	3.88689	0.203449
$366$	0	0
$367$	20.8840	1.09014	0.545069	−	0.838391i	$-0.316504\pi$
$367$	20.8840	1.09014	0.545069	+	0.838391i	$0.316504\pi$
$368$	0	0
$369$	3.82548	0.199147
$370$	0	0
$371$	−45.7595	−2.37571
$372$	0	0
$373$	−25.9819	−1.34529	−0.672647	−	0.739964i	$-0.734843\pi$
$373$	−25.9819	−1.34529	−0.672647	+	0.739964i	$0.734843\pi$
$374$	0	0
$375$	−12.2482	−0.632495
$376$	0	0
$377$	8.42218	0.433764
$378$	0	0
$379$	−2.50301	−0.128571	−0.0642855	−	0.997932i	$-0.520477\pi$
$379$	−2.50301	−0.128571	−0.0642855	+	0.997932i	$0.520477\pi$
$380$	0	0
$381$	29.1827	1.49507
$382$	0	0
$383$	−21.8155	−1.11472	−0.557359	−	0.830272i	$-0.688185\pi$
$383$	−21.8155	−1.11472	−0.557359	+	0.830272i	$0.688185\pi$
$384$	0	0
$385$	12.1707	0.620274
$386$	0	0
$387$	3.35119	0.170351
$388$	0	0
$389$	−35.2383	−1.78665	−0.893327	−	0.449407i	$-0.851635\pi$
$389$	−35.2383	−1.78665	−0.893327	+	0.449407i	$0.851635\pi$
$390$	0	0
$391$	30.2069	1.52763
$392$	0	0
$393$	25.9304	1.30802
$394$	0	0
$395$	3.28135	0.165103
$396$	0	0
$397$	27.0263	1.35641	0.678205	−	0.734873i	$-0.262758\pi$
$397$	27.0263	1.35641	0.678205	+	0.734873i	$0.262758\pi$
$398$	0	0
$399$	7.91023	0.396007
$400$	0	0
$401$	−24.8932	−1.24311	−0.621553	−	0.783372i	$-0.713498\pi$
$401$	−24.8932	−1.24311	−0.621553	+	0.783372i	$0.713498\pi$
$402$	0	0
$403$	20.3383	1.01312
$404$	0	0
$405$	−5.91230	−0.293784
$406$	0	0
$407$	25.6247	1.27017
$408$	0	0
$409$	−14.8336	−0.733475	−0.366738	−	0.930324i	$-0.619525\pi$
$409$	−14.8336	−0.733475	−0.366738	+	0.930324i	$0.619525\pi$
$410$	0	0
$411$	−4.54840	−0.224356
$412$	0	0
$413$	3.79526	0.186753
$414$	0	0
$415$	−7.13024	−0.350010
$416$	0	0
$417$	0.434880	0.0212962
$418$	0	0
$419$	6.05955	0.296028	0.148014	−	0.988985i	$-0.452712\pi$
$419$	6.05955	0.296028	0.148014	+	0.988985i	$0.452712\pi$
$420$	0	0
$421$	−13.8670	−0.675834	−0.337917	−	0.941176i	$-0.609722\pi$
$421$	−13.8670	−0.675834	−0.337917	+	0.941176i	$0.609722\pi$
$422$	0	0
$423$	0.584249	0.0284072
$424$	0	0
$425$	−33.1145	−1.60629
$426$	0	0
$427$	11.5420	0.558555
$428$	0	0
$429$	12.8912	0.622391
$430$	0	0
$431$	−11.5580	−0.556729	−0.278364	−	0.960476i	$-0.589792\pi$
$431$	−11.5580	−0.556729	−0.278364	+	0.960476i	$0.589792\pi$
$432$	0	0
$433$	−31.7468	−1.52565	−0.762826	−	0.646604i	$-0.776189\pi$
$433$	−31.7468	−1.52565	−0.762826	+	0.646604i	$0.776189\pi$
$434$	0	0
$435$	−3.82760	−0.183520
$436$	0	0
$437$	−3.91023	−0.187052
$438$	0	0
$439$	−24.8840	−1.18765	−0.593825	−	0.804594i	$-0.702383\pi$
$439$	−24.8840	−1.18765	−0.593825	+	0.804594i	$0.702383\pi$
$440$	0	0
$441$	−10.4789	−0.498994
$442$	0	0
$443$	23.9911	1.13985	0.569926	−	0.821696i	$-0.306972\pi$
$443$	23.9911	1.13985	0.569926	+	0.821696i	$0.306972\pi$
$444$	0	0
$445$	13.5701	0.643286
$446$	0	0
$447$	12.6215	0.596979
$448$	0	0
$449$	32.1384	1.51670	0.758352	−	0.651845i	$-0.226005\pi$
$449$	32.1384	1.51670	0.758352	+	0.651845i	$0.226005\pi$
$450$	0	0
$451$	19.3785	0.912496
$452$	0	0
$453$	23.5147	1.10482
$454$	0	0
$455$	12.4166	0.582099
$456$	0	0
$457$	29.1590	1.36400	0.681999	−	0.731353i	$-0.261111\pi$
$457$	29.1590	1.36400	0.681999	+	0.731353i	$0.261111\pi$
$458$	0	0
$459$	−42.9634	−2.00536
$460$	0	0
$461$	−18.6559	−0.868894	−0.434447	−	0.900697i	$-0.643056\pi$
$461$	−18.6559	−0.868894	−0.434447	+	0.900697i	$0.643056\pi$
$462$	0	0
$463$	22.6005	1.05034	0.525168	−	0.850999i	$-0.324003\pi$
$463$	22.6005	1.05034	0.525168	+	0.850999i	$0.324003\pi$
$464$	0	0
$465$	−9.24309	−0.428638
$466$	0	0
$467$	17.1666	0.794377	0.397189	−	0.917737i	$-0.369986\pi$
$467$	17.1666	0.794377	0.397189	+	0.917737i	$0.369986\pi$
$468$	0	0
$469$	52.5502	2.42654
$470$	0	0
$471$	−17.0636	−0.786247
$472$	0	0
$473$	16.9759	0.780551
$474$	0	0
$475$	4.28663	0.196684
$476$	0	0
$477$	−5.07270	−0.232263
$478$	0	0
$479$	−16.2625	−0.743052	−0.371526	−	0.928423i	$-0.621165\pi$
$479$	−16.2625	−0.743052	−0.371526	+	0.928423i	$0.621165\pi$
$480$	0	0
$481$	26.1425	1.19200
$482$	0	0
$483$	−30.9309	−1.40740
$484$	0	0
$485$	−3.11597	−0.141489
$486$	0	0
$487$	18.2071	0.825041	0.412520	−	0.910948i	$-0.364649\pi$
$487$	18.2071	0.825041	0.412520	+	0.910948i	$0.364649\pi$
$488$	0	0
$489$	−1.08897	−0.0492449
$490$	0	0
$491$	−39.2493	−1.77130	−0.885649	−	0.464356i	$-0.846286\pi$
$491$	−39.2493	−1.77130	−0.885649	+	0.464356i	$0.846286\pi$
$492$	0	0
$493$	−22.4189	−1.00970
$494$	0	0
$495$	1.34919	0.0606414
$496$	0	0
$497$	−10.7961	−0.484270
$498$	0	0
$499$	35.7541	1.60057	0.800286	−	0.599619i	$-0.204681\pi$
$499$	35.7541	1.60057	0.800286	+	0.599619i	$0.204681\pi$
$500$	0	0
$501$	9.75379	0.435767
$502$	0	0
$503$	27.5512	1.22845	0.614223	−	0.789132i	$-0.289470\pi$
$503$	27.5512	1.22845	0.614223	+	0.789132i	$0.289470\pi$
$504$	0	0
$505$	13.7218	0.610611
$506$	0	0
$507$	−7.14851	−0.317477
$508$	0	0
$509$	14.6328	0.648588	0.324294	−	0.945956i	$-0.394873\pi$
$509$	14.6328	0.648588	0.324294	+	0.945956i	$0.394873\pi$
$510$	0	0
$511$	−23.3118	−1.03125
$512$	0	0
$513$	5.56155	0.245549
$514$	0	0
$515$	−0.686107	−0.0302335
$516$	0	0
$517$	2.95958	0.130162
$518$	0	0
$519$	−10.4583	−0.459068
$520$	0	0
$521$	−7.59926	−0.332929	−0.166465	−	0.986047i	$-0.553235\pi$
$521$	−7.59926	−0.332929	−0.166465	+	0.986047i	$0.553235\pi$
$522$	0	0
$523$	−19.8723	−0.868956	−0.434478	−	0.900682i	$-0.643067\pi$
$523$	−19.8723	−0.868956	−0.434478	+	0.900682i	$0.643067\pi$
$524$	0	0
$525$	33.9082	1.47988
$526$	0	0
$527$	−54.1384	−2.35830
$528$	0	0
$529$	−7.71007	−0.335220
$530$	0	0
$531$	0.420727	0.0182580
$532$	0	0
$533$	19.7701	0.856336
$534$	0	0
$535$	−8.35204	−0.361090
$536$	0	0
$537$	−28.4924	−1.22954
$538$	0	0
$539$	−53.0820	−2.28640
$540$	0	0
$541$	−27.6982	−1.19084	−0.595420	−	0.803415i	$-0.703014\pi$
$541$	−27.6982	−1.19084	−0.595420	+	0.803415i	$0.703014\pi$
$542$	0	0
$543$	−27.4289	−1.17709
$544$	0	0
$545$	9.15353	0.392094
$546$	0	0
$547$	−33.8871	−1.44891	−0.724455	−	0.689322i	$-0.757908\pi$
$547$	−33.8871	−1.44891	−0.724455	+	0.689322i	$0.757908\pi$
$548$	0	0
$549$	1.27949	0.0546075
$550$	0	0
$551$	2.90210	0.123634
$552$	0	0
$553$	−19.6801	−0.836883
$554$	0	0
$555$	−11.8809	−0.504316
$556$	0	0
$557$	8.28763	0.351158	0.175579	−	0.984465i	$-0.443820\pi$
$557$	8.28763	0.351158	0.175579	+	0.984465i	$0.443820\pi$
$558$	0	0
$559$	17.3189	0.732512
$560$	0	0
$561$	−34.3149	−1.44878
$562$	0	0
$563$	5.11397	0.215528	0.107764	−	0.994177i	$-0.465631\pi$
$563$	5.11397	0.215528	0.107764	+	0.994177i	$0.465631\pi$
$564$	0	0
$565$	−0.262483	−0.0110427
$566$	0	0
$567$	35.4593	1.48915
$568$	0	0
$569$	1.92830	0.0808387	0.0404194	−	0.999183i	$-0.487131\pi$
$569$	1.92830	0.0808387	0.0404194	+	0.999183i	$0.487131\pi$
$570$	0	0
$571$	23.3785	0.978358	0.489179	−	0.872183i	$-0.337297\pi$
$571$	23.3785	0.978358	0.489179	+	0.872183i	$0.337297\pi$
$572$	0	0
$573$	11.2383	0.469486
$574$	0	0
$575$	−16.7617	−0.699012
$576$	0	0
$577$	−26.8807	−1.11906	−0.559530	−	0.828810i	$-0.689018\pi$
$577$	−26.8807	−1.11906	−0.559530	+	0.828810i	$0.689018\pi$
$578$	0	0
$579$	−4.57114	−0.189970
$580$	0	0
$581$	42.7640	1.77415
$582$	0	0
$583$	−25.6964	−1.06423
$584$	0	0
$585$	1.37645	0.0569093
$586$	0	0
$587$	36.7922	1.51858	0.759288	−	0.650754i	$-0.225547\pi$
$587$	36.7922	1.51858	0.759288	+	0.650754i	$0.225547\pi$
$588$	0	0
$589$	7.00814	0.288765
$590$	0	0
$591$	17.3947	0.715523
$592$	0	0
$593$	−33.9034	−1.39225	−0.696123	−	0.717922i	$-0.745093\pi$
$593$	−33.9034	−1.39225	−0.696123	+	0.717922i	$0.745093\pi$
$594$	0	0
$595$	−33.0517	−1.35499
$596$	0	0
$597$	8.70128	0.356120
$598$	0	0
$599$	−4.06456	−0.166073	−0.0830367	−	0.996546i	$-0.526462\pi$
$599$	−4.06456	−0.166073	−0.0830367	+	0.996546i	$0.526462\pi$
$600$	0	0
$601$	−27.5560	−1.12403	−0.562016	−	0.827126i	$-0.689974\pi$
$601$	−27.5560	−1.12403	−0.562016	+	0.827126i	$0.689974\pi$
$602$	0	0
$603$	5.82548	0.237232
$604$	0	0
$605$	−2.45628	−0.0998621
$606$	0	0
$607$	25.5743	1.03803	0.519014	−	0.854766i	$-0.326299\pi$
$607$	25.5743	1.03803	0.519014	+	0.854766i	$0.326299\pi$
$608$	0	0
$609$	22.9563	0.930235
$610$	0	0
$611$	3.01939	0.122152
$612$	0	0
$613$	31.2383	1.26170	0.630852	−	0.775903i	$-0.282705\pi$
$613$	31.2383	1.26170	0.630852	+	0.775903i	$0.282705\pi$
$614$	0	0
$615$	−8.98485	−0.362304
$616$	0	0
$617$	−5.86189	−0.235991	−0.117995	−	0.993014i	$-0.537647\pi$
$617$	−5.86189	−0.235991	−0.117995	+	0.993014i	$0.537647\pi$
$618$	0	0
$619$	31.6481	1.27204	0.636022	−	0.771671i	$-0.280579\pi$
$619$	31.6481	1.27204	0.636022	+	0.771671i	$0.280579\pi$
$620$	0	0
$621$	−21.7470	−0.872676
$622$	0	0
$623$	−81.3877	−3.26073
$624$	0	0
$625$	14.8083	0.592333
$626$	0	0
$627$	4.44201	0.177397
$628$	0	0
$629$	−69.5885	−2.77468
$630$	0	0
$631$	−3.86189	−0.153739	−0.0768696	−	0.997041i	$-0.524493\pi$
$631$	−3.86189	−0.153739	−0.0768696	+	0.997041i	$0.524493\pi$
$632$	0	0
$633$	1.92775	0.0766212
$634$	0	0
$635$	15.7843	0.626382
$636$	0	0
$637$	−54.1546	−2.14569
$638$	0	0
$639$	−1.19680	−0.0473449
$640$	0	0
$641$	−14.9919	−0.592143	−0.296072	−	0.955166i	$-0.595677\pi$
$641$	−14.9919	−0.592143	−0.296072	+	0.955166i	$0.595677\pi$
$642$	0	0
$643$	12.4906	0.492580	0.246290	−	0.969196i	$-0.420788\pi$
$643$	12.4906	0.492580	0.246290	+	0.969196i	$0.420788\pi$
$644$	0	0
$645$	−7.87088	−0.309915
$646$	0	0
$647$	−23.5743	−0.926802	−0.463401	−	0.886149i	$-0.653371\pi$
$647$	−23.5743	−0.926802	−0.463401	+	0.886149i	$0.653371\pi$
$648$	0	0
$649$	2.13124	0.0836585
$650$	0	0
$651$	55.4360	2.17271
$652$	0	0
$653$	−30.4693	−1.19236	−0.596178	−	0.802853i	$-0.703315\pi$
$653$	−30.4693	−1.19236	−0.596178	+	0.802853i	$0.703315\pi$
$654$	0	0
$655$	14.0253	0.548012
$656$	0	0
$657$	−2.58425	−0.100821
$658$	0	0
$659$	−39.9049	−1.55447	−0.777237	−	0.629209i	$-0.783379\pi$
$659$	−39.9049	−1.55447	−0.777237	+	0.629209i	$0.783379\pi$
$660$	0	0
$661$	22.8156	0.887422	0.443711	−	0.896170i	$-0.353662\pi$
$661$	22.8156	0.887422	0.443711	+	0.896170i	$0.353662\pi$
$662$	0	0
$663$	−35.0083	−1.35961
$664$	0	0
$665$	4.27849	0.165913
$666$	0	0
$667$	−11.3479	−0.439392
$668$	0	0
$669$	−8.99599	−0.347805
$670$	0	0
$671$	6.48143	0.250213
$672$	0	0
$673$	−37.0265	−1.42727	−0.713634	−	0.700519i	$-0.752952\pi$
$673$	−37.0265	−1.42727	−0.713634	+	0.700519i	$0.752952\pi$
$674$	0	0
$675$	23.8403	0.917614
$676$	0	0
$677$	45.6733	1.75537	0.877683	−	0.479241i	$-0.159088\pi$
$677$	45.6733	1.75537	0.877683	+	0.479241i	$0.159088\pi$
$678$	0	0
$679$	18.6882	0.717188
$680$	0	0
$681$	17.4679	0.669370
$682$	0	0
$683$	−13.9837	−0.535072	−0.267536	−	0.963548i	$-0.586210\pi$
$683$	−13.9837	−0.535072	−0.267536	+	0.963548i	$0.586210\pi$
$684$	0	0
$685$	−2.46014	−0.0939972
$686$	0	0
$687$	−39.6048	−1.51102
$688$	0	0
$689$	−26.2156	−0.998736
$690$	0	0
$691$	0.00185566	7.05926e−5 0	3.52963e−5	−	1.00000i	$-0.499989\pi$
$691$	0.00185566	7.05926e−5 0	3.52963e−5		1.00000i	$0.499989\pi$
$692$	0	0
$693$	−8.09183	−0.307383
$694$	0	0
$695$	0.235218	0.00892233
$696$	0	0
$697$	−52.6258	−1.99334
$698$	0	0
$699$	13.0203	0.492472
$700$	0	0
$701$	22.0071	0.831198	0.415599	−	0.909548i	$-0.363572\pi$
$701$	22.0071	0.831198	0.415599	+	0.909548i	$0.363572\pi$
$702$	0	0
$703$	9.00814	0.339748
$704$	0	0
$705$	−1.37221	−0.0516806
$706$	0	0
$707$	−82.2971	−3.09510
$708$	0	0
$709$	−3.94045	−0.147987	−0.0739934	−	0.997259i	$-0.523574\pi$
$709$	−3.94045	−0.147987	−0.0739934	+	0.997259i	$0.523574\pi$
$710$	0	0
$711$	−2.18165	−0.0818183
$712$	0	0
$713$	−27.4035	−1.02627
$714$	0	0
$715$	6.97258	0.260760
$716$	0	0
$717$	−25.6783	−0.958973
$718$	0	0
$719$	−29.0656	−1.08396	−0.541982	−	0.840390i	$-0.682326\pi$
$719$	−29.0656	−1.08396	−0.541982	+	0.840390i	$0.682326\pi$
$720$	0	0
$721$	4.11497	0.153249
$722$	0	0
$723$	−20.0325	−0.745018
$724$	0	0
$725$	12.4402	0.462018
$726$	0	0
$727$	10.1617	0.376877	0.188439	−	0.982085i	$-0.439657\pi$
$727$	10.1617	0.376877	0.188439	+	0.982085i	$0.439657\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−46.1011	−1.70511
$732$	0	0
$733$	−51.4777	−1.90137	−0.950686	−	0.310156i	$-0.899619\pi$
$733$	−51.4777	−1.90137	−0.950686	+	0.310156i	$0.899619\pi$
$734$	0	0
$735$	24.6115	0.907809
$736$	0	0
$737$	29.5097	1.08700
$738$	0	0
$739$	−33.1412	−1.21912	−0.609560	−	0.792740i	$-0.708654\pi$
$739$	−33.1412	−1.21912	−0.609560	+	0.792740i	$0.708654\pi$
$740$	0	0
$741$	4.53178	0.166479
$742$	0	0
$743$	35.5006	1.30239	0.651195	−	0.758911i	$-0.274268\pi$
$743$	35.5006	1.30239	0.651195	+	0.758911i	$0.274268\pi$
$744$	0	0
$745$	6.82675	0.250113
$746$	0	0
$747$	4.74064	0.173451
$748$	0	0
$749$	50.0919	1.83032
$750$	0	0
$751$	15.2594	0.556822	0.278411	−	0.960462i	$-0.410192\pi$
$751$	15.2594	0.556822	0.278411	+	0.960462i	$0.410192\pi$
$752$	0	0
$753$	−27.8780	−1.01593
$754$	0	0
$755$	12.7187	0.462879
$756$	0	0
$757$	−7.28161	−0.264655	−0.132327	−	0.991206i	$-0.542245\pi$
$757$	−7.28161	−0.264655	−0.132327	+	0.991206i	$0.542245\pi$
$758$	0	0
$759$	−17.3693	−0.630466
$760$	0	0
$761$	9.47886	0.343609	0.171804	−	0.985131i	$-0.445040\pi$
$761$	9.47886	0.343609	0.171804	+	0.985131i	$0.445040\pi$
$762$	0	0
$763$	−54.8989	−1.98747
$764$	0	0
$765$	−3.66397	−0.132471
$766$	0	0
$767$	2.17431	0.0785098
$768$	0	0
$769$	19.2760	0.695112	0.347556	−	0.937659i	$-0.387012\pi$
$769$	19.2760	0.695112	0.347556	+	0.937659i	$0.387012\pi$
$770$	0	0
$771$	2.33206	0.0839871
$772$	0	0
$773$	−9.19501	−0.330721	−0.165361	−	0.986233i	$-0.552879\pi$
$773$	−9.19501	−0.330721	−0.165361	+	0.986233i	$0.552879\pi$
$774$	0	0
$775$	30.0413	1.07911
$776$	0	0
$777$	71.2565	2.55631
$778$	0	0
$779$	6.81233	0.244077
$780$	0	0
$781$	−6.06256	−0.216935
$782$	0	0
$783$	16.1402	0.576803
$784$	0	0
$785$	−9.22935	−0.329410
$786$	0	0
$787$	−12.6847	−0.452159	−0.226080	−	0.974109i	$-0.572591\pi$
$787$	−12.6847	−0.452159	−0.226080	+	0.974109i	$0.572591\pi$
$788$	0	0
$789$	0.754905	0.0268753
$790$	0	0
$791$	1.57426	0.0559741
$792$	0	0
$793$	6.61241	0.234814
$794$	0	0
$795$	11.9142	0.422551
$796$	0	0
$797$	−27.4591	−0.972651	−0.486325	−	0.873778i	$-0.661663\pi$
$797$	−27.4591	−0.972651	−0.486325	+	0.873778i	$0.661663\pi$
$798$	0	0
$799$	−8.03730	−0.284339
$800$	0	0
$801$	−9.02229	−0.318787
$802$	0	0
$803$	−13.0908	−0.461965
$804$	0	0
$805$	−16.7299	−0.589652
$806$	0	0
$807$	−13.1969	−0.464554
$808$	0	0
$809$	18.2912	0.643084	0.321542	−	0.946895i	$-0.395799\pi$
$809$	18.2912	0.643084	0.321542	+	0.946895i	$0.395799\pi$
$810$	0	0
$811$	−6.28391	−0.220658	−0.110329	−	0.993895i	$-0.535190\pi$
$811$	−6.28391	−0.220658	−0.110329	+	0.993895i	$0.535190\pi$
$812$	0	0
$813$	−25.8940	−0.908141
$814$	0	0
$815$	−0.589002	−0.0206318
$816$	0	0
$817$	5.96772	0.208784
$818$	0	0
$819$	−8.25535	−0.288465
$820$	0	0
$821$	43.9103	1.53248	0.766240	−	0.642555i	$-0.222125\pi$
$821$	43.9103	1.53248	0.766240	+	0.642555i	$0.222125\pi$
$822$	0	0
$823$	−26.2483	−0.914957	−0.457479	−	0.889221i	$-0.651247\pi$
$823$	−26.2483	−0.914957	−0.457479	+	0.889221i	$0.651247\pi$
$824$	0	0
$825$	19.0413	0.662932
$826$	0	0
$827$	34.7726	1.20916	0.604581	−	0.796543i	$-0.293340\pi$
$827$	34.7726	1.20916	0.604581	+	0.796543i	$0.293340\pi$
$828$	0	0
$829$	20.9184	0.726525	0.363263	−	0.931687i	$-0.381663\pi$
$829$	20.9184	0.726525	0.363263	+	0.931687i	$0.381663\pi$
$830$	0	0
$831$	−24.5357	−0.851134
$832$	0	0
$833$	144.154	4.99464
$834$	0	0
$835$	5.27563	0.182571
$836$	0	0
$837$	38.9761	1.34721
$838$	0	0
$839$	41.0061	1.41569	0.707844	−	0.706368i	$-0.249668\pi$
$839$	41.0061	1.41569	0.707844	+	0.706368i	$0.249668\pi$
$840$	0	0
$841$	−20.5778	−0.709580
$842$	0	0
$843$	4.90230	0.168844
$844$	0	0
$845$	−3.86649	−0.133011
$846$	0	0
$847$	14.7317	0.506187
$848$	0	0
$849$	41.2070	1.41422
$850$	0	0
$851$	−35.2239	−1.20746
$852$	0	0
$853$	−11.5056	−0.393943	−0.196972	−	0.980409i	$-0.563111\pi$
$853$	−11.5056	−0.393943	−0.196972	+	0.980409i	$0.563111\pi$
$854$	0	0
$855$	0.474295	0.0162206
$856$	0	0
$857$	14.7478	0.503774	0.251887	−	0.967757i	$-0.418949\pi$
$857$	14.7478	0.503774	0.251887	+	0.967757i	$0.418949\pi$
$858$	0	0
$859$	−36.3402	−1.23991	−0.619955	−	0.784637i	$-0.712849\pi$
$859$	−36.3402	−1.23991	−0.619955	+	0.784637i	$0.712849\pi$
$860$	0	0
$861$	53.8871	1.83647
$862$	0	0
$863$	48.3454	1.64570	0.822849	−	0.568260i	$-0.192383\pi$
$863$	48.3454	1.64570	0.822849	+	0.568260i	$0.192383\pi$
$864$	0	0
$865$	−5.65668	−0.192333
$866$	0	0
$867$	66.6421	2.26328
$868$	0	0
$869$	−11.0514	−0.374893
$870$	0	0
$871$	30.1060	1.02010
$872$	0	0
$873$	2.07170	0.0701163
$874$	0	0
$875$	39.7328	1.34321
$876$	0	0
$877$	4.23226	0.142913	0.0714567	−	0.997444i	$-0.477235\pi$
$877$	4.23226	0.142913	0.0714567	+	0.997444i	$0.477235\pi$
$878$	0	0
$879$	−21.2220	−0.715801
$880$	0	0
$881$	32.6651	1.10051	0.550257	−	0.834995i	$-0.314530\pi$
$881$	32.6651	1.10051	0.550257	+	0.834995i	$0.314530\pi$
$882$	0	0
$883$	45.3320	1.52554	0.762772	−	0.646668i	$-0.223838\pi$
$883$	45.3320	1.52554	0.762772	+	0.646668i	$0.223838\pi$
$884$	0	0
$885$	−0.988153	−0.0332164
$886$	0	0
$887$	−50.5271	−1.69653	−0.848267	−	0.529569i	$-0.822354\pi$
$887$	−50.5271	−1.69653	−0.848267	+	0.529569i	$0.822354\pi$
$888$	0	0
$889$	−94.6675	−3.17504
$890$	0	0
$891$	19.9123	0.667087
$892$	0	0
$893$	1.04042	0.0348162
$894$	0	0
$895$	−15.4110	−0.515133
$896$	0	0
$897$	−17.7203	−0.591664
$898$	0	0
$899$	20.3383	0.678320
$900$	0	0
$901$	69.7832	2.32482
$902$	0	0
$903$	47.2061	1.57092
$904$	0	0
$905$	−14.8357	−0.493157
$906$	0	0
$907$	21.9823	0.729910	0.364955	−	0.931025i	$-0.381084\pi$
$907$	21.9823	0.729910	0.364955	+	0.931025i	$0.381084\pi$
$908$	0	0
$909$	−9.12311	−0.302594
$910$	0	0
$911$	−29.9950	−0.993778	−0.496889	−	0.867814i	$-0.665524\pi$
$911$	−29.9950	−0.993778	−0.496889	+	0.867814i	$0.665524\pi$
$912$	0	0
$913$	24.0143	0.794756
$914$	0	0
$915$	−3.00512	−0.0993463
$916$	0	0
$917$	−84.1174	−2.77780
$918$	0	0
$919$	−30.5481	−1.00769	−0.503844	−	0.863795i	$-0.668081\pi$
$919$	−30.5481	−1.00769	−0.503844	+	0.863795i	$0.668081\pi$
$920$	0	0
$921$	5.37645	0.177160
$922$	0	0
$923$	−6.18507	−0.203584
$924$	0	0
$925$	38.6145	1.26964
$926$	0	0
$927$	0.456168	0.0149825
$928$	0	0
$929$	−5.67151	−0.186076	−0.0930380	−	0.995663i	$-0.529658\pi$
$929$	−5.67151	−0.186076	−0.0930380	+	0.995663i	$0.529658\pi$
$930$	0	0
$931$	−18.6605	−0.611574
$932$	0	0
$933$	−46.0556	−1.50779
$934$	0	0
$935$	−18.5603	−0.606985
$936$	0	0
$937$	33.6051	1.09783	0.548915	−	0.835878i	$-0.315041\pi$
$937$	33.6051	1.09783	0.548915	+	0.835878i	$0.315041\pi$
$938$	0	0
$939$	−15.7141	−0.512810
$940$	0	0
$941$	−30.0543	−0.979743	−0.489871	−	0.871795i	$-0.662956\pi$
$941$	−30.0543	−0.979743	−0.489871	+	0.871795i	$0.662956\pi$
$942$	0	0
$943$	−26.6378	−0.867447
$944$	0	0
$945$	23.7951	0.774053
$946$	0	0
$947$	12.3058	0.399883	0.199942	−	0.979808i	$-0.435925\pi$
$947$	12.3058	0.399883	0.199942	+	0.979808i	$0.435925\pi$
$948$	0	0
$949$	−13.3554	−0.433534
$950$	0	0
$951$	−6.47937	−0.210108
$952$	0	0
$953$	30.4674	0.986937	0.493468	−	0.869764i	$-0.335729\pi$
$953$	30.4674	0.986937	0.493468	+	0.869764i	$0.335729\pi$
$954$	0	0
$955$	6.07857	0.196698
$956$	0	0
$957$	12.8912	0.416712
$958$	0	0
$959$	14.7548	0.476459
$960$	0	0
$961$	18.1140	0.584322
$962$	0	0
$963$	5.55297	0.178942
$964$	0	0
$965$	−2.47244	−0.0795906
$966$	0	0
$967$	−5.26560	−0.169330	−0.0846652	−	0.996409i	$-0.526982\pi$
$967$	−5.26560	−0.169330	−0.0846652	+	0.996409i	$0.526982\pi$
$968$	0	0
$969$	−12.0631	−0.387523
$970$	0	0
$971$	54.7895	1.75828	0.879139	−	0.476566i	$-0.158119\pi$
$971$	54.7895	1.75828	0.879139	+	0.476566i	$0.158119\pi$
$972$	0	0
$973$	−1.41074	−0.0452261
$974$	0	0
$975$	19.4261	0.622132
$976$	0	0
$977$	28.9690	0.926800	0.463400	−	0.886149i	$-0.346629\pi$
$977$	28.9690	0.926800	0.463400	+	0.886149i	$0.346629\pi$
$978$	0	0
$979$	−45.7035	−1.46069
$980$	0	0
$981$	−6.08585	−0.194306
$982$	0	0
$983$	35.2997	1.12589	0.562943	−	0.826495i	$-0.309669\pi$
$983$	35.2997	1.12589	0.562943	+	0.826495i	$0.309669\pi$
$984$	0	0
$985$	9.40847	0.299779
$986$	0	0
$987$	8.22994	0.261962
$988$	0	0
$989$	−23.3352	−0.742016
$990$	0	0
$991$	−8.34833	−0.265194	−0.132597	−	0.991170i	$-0.542332\pi$
$991$	−8.34833	−0.265194	−0.132597	+	0.991170i	$0.542332\pi$
$992$	0	0
$993$	−40.4264	−1.28289
$994$	0	0
$995$	4.70635	0.149201
$996$	0	0
$997$	−5.81519	−0.184169	−0.0920845	−	0.995751i	$-0.529353\pi$
$997$	−5.81519	−0.184169	−0.0920845	+	0.995751i	$0.529353\pi$
$998$	0	0
$999$	50.0992	1.58507

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.w.1.4		4
4.3	odd	2		1216.2.a.x.1.2		4
8.3	odd	2		608.2.a.i.1.3	✓	4
8.5	even	2		608.2.a.j.1.1	yes	4
24.5	odd	2		5472.2.a.bs.1.3		4
24.11	even	2		5472.2.a.bt.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.i.1.3	✓	4	8.3	odd	2
608.2.a.j.1.1	yes	4	8.5	even	2
1216.2.a.w.1.4		4	1.1	even	1	trivial
1216.2.a.x.1.2		4	4.3	odd	2
5472.2.a.bs.1.3		4	24.5	odd	2
5472.2.a.bt.1.3		4	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +0.844614 q^{5} -5.06562 q^{7} -0.561553 q^{9} -2.84461 q^{11} -2.90210 q^{13} +1.31891 q^{15} +7.72508 q^{17} -1.00000 q^{19} -7.91023 q^{21} +3.91023 q^{23} -4.28663 q^{25} -5.56155 q^{27} -2.90210 q^{29} -7.00814 q^{31} -4.44201 q^{33} -4.27849 q^{35} -9.00814 q^{37} -4.53178 q^{39} -6.81233 q^{41} -5.96772 q^{43} -0.474295 q^{45} -1.04042 q^{47} +18.6605 q^{49} +12.0631 q^{51} +9.03334 q^{53} -2.40260 q^{55} -1.56155 q^{57} -0.749220 q^{59} -2.27849 q^{61} +2.84461 q^{63} -2.45115 q^{65} -10.3739 q^{67} +6.10604 q^{69} +2.13124 q^{71} +4.60197 q^{73} -6.69380 q^{75} +14.4097 q^{77} +3.88503 q^{79} -7.00000 q^{81} -8.44201 q^{83} +6.52470 q^{85} -4.53178 q^{87} +16.0667 q^{89} +14.7009 q^{91} -10.9436 q^{93} -0.844614 q^{95} -3.68923 q^{97} +1.59740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37}+ \cdots - 19 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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