LMFDB - Embedded newform 6032.2.a.be.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 159x^9 - 827x^8 - 362x^7 + 3029x^6 - 142x^5 - 4316x^4 + 208x^3 + 2228x^2 + 376x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.07047$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.07047 q^{3} -1.28770 q^{5} -4.21640 q^{7} +1.28687 q^{9} +O(q^{10})$	$q+2.07047 q^{3} -1.28770 q^{5} -4.21640 q^{7} +1.28687 q^{9} -6.44307 q^{11} -1.00000 q^{13} -2.66615 q^{15} +7.60944 q^{17} -4.67826 q^{19} -8.72996 q^{21} +6.94581 q^{23} -3.34183 q^{25} -3.54700 q^{27} +1.00000 q^{29} -7.98467 q^{31} -13.3402 q^{33} +5.42946 q^{35} +3.04399 q^{37} -2.07047 q^{39} +6.59888 q^{41} +8.31730 q^{43} -1.65709 q^{45} -0.728070 q^{47} +10.7781 q^{49} +15.7551 q^{51} +1.44109 q^{53} +8.29673 q^{55} -9.68623 q^{57} +14.0002 q^{59} -2.99431 q^{61} -5.42594 q^{63} +1.28770 q^{65} +13.5568 q^{67} +14.3811 q^{69} -0.514661 q^{71} -4.71562 q^{73} -6.91918 q^{75} +27.1666 q^{77} -5.02425 q^{79} -11.2046 q^{81} +6.53628 q^{83} -9.79867 q^{85} +2.07047 q^{87} +9.98400 q^{89} +4.21640 q^{91} -16.5321 q^{93} +6.02420 q^{95} -16.3973 q^{97} -8.29136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * 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
$3$	2.07047	1.19539	0.597695	−	0.801724i	$-0.296084\pi$
$3$	2.07047	1.19539	0.597695	+	0.801724i	$0.296084\pi$
$4$	0	0
$5$	−1.28770	−0.575877	−0.287938	−	0.957649i	$-0.592970\pi$
$5$	−1.28770	−0.575877	−0.287938	+	0.957649i	$0.592970\pi$
$6$	0	0
$7$	−4.21640	−1.59365	−0.796825	−	0.604210i	$-0.793489\pi$
$7$	−4.21640	−1.59365	−0.796825	+	0.604210i	$0.793489\pi$
$8$	0	0
$9$	1.28687	0.428955
$10$	0	0
$11$	−6.44307	−1.94266	−0.971329	−	0.237741i	$-0.923593\pi$
$11$	−6.44307	−1.94266	−0.971329	+	0.237741i	$0.923593\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.66615	−0.688397
$16$	0	0
$17$	7.60944	1.84556	0.922780	−	0.385327i	$-0.125911\pi$
$17$	7.60944	1.84556	0.922780	+	0.385327i	$0.125911\pi$
$18$	0	0
$19$	−4.67826	−1.07327	−0.536634	−	0.843815i	$-0.680304\pi$
$19$	−4.67826	−1.07327	−0.536634	+	0.843815i	$0.680304\pi$
$20$	0	0
$21$	−8.72996	−1.90503
$22$	0	0
$23$	6.94581	1.44830	0.724151	−	0.689642i	$-0.242232\pi$
$23$	6.94581	1.44830	0.724151	+	0.689642i	$0.242232\pi$
$24$	0	0
$25$	−3.34183	−0.668366
$26$	0	0
$27$	−3.54700	−0.682621
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.98467	−1.43409	−0.717044	−	0.697028i	$-0.754505\pi$
$31$	−7.98467	−1.43409	−0.717044	+	0.697028i	$0.754505\pi$
$32$	0	0
$33$	−13.3402	−2.32223
$34$	0	0
$35$	5.42946	0.917746
$36$	0	0
$37$	3.04399	0.500429	0.250214	−	0.968190i	$-0.419499\pi$
$37$	3.04399	0.500429	0.250214	+	0.968190i	$0.419499\pi$
$38$	0	0
$39$	−2.07047	−0.331541
$40$	0	0
$41$	6.59888	1.03057	0.515286	−	0.857018i	$-0.327686\pi$
$41$	6.59888	1.03057	0.515286	+	0.857018i	$0.327686\pi$
$42$	0	0
$43$	8.31730	1.26838	0.634188	−	0.773179i	$-0.281334\pi$
$43$	8.31730	1.26838	0.634188	+	0.773179i	$0.281334\pi$
$44$	0	0
$45$	−1.65709	−0.247025
$46$	0	0
$47$	−0.728070	−0.106200	−0.0530999	−	0.998589i	$-0.516910\pi$
$47$	−0.728070	−0.106200	−0.0530999	+	0.998589i	$0.516910\pi$
$48$	0	0
$49$	10.7781	1.53972
$50$	0	0
$51$	15.7551	2.20616
$52$	0	0
$53$	1.44109	0.197948	0.0989742	−	0.995090i	$-0.468444\pi$
$53$	1.44109	0.197948	0.0989742	+	0.995090i	$0.468444\pi$
$54$	0	0
$55$	8.29673	1.11873
$56$	0	0
$57$	−9.68623	−1.28297
$58$	0	0
$59$	14.0002	1.82267	0.911337	−	0.411661i	$-0.135051\pi$
$59$	14.0002	1.82267	0.911337	+	0.411661i	$0.135051\pi$
$60$	0	0
$61$	−2.99431	−0.383382	−0.191691	−	0.981455i	$-0.561397\pi$
$61$	−2.99431	−0.383382	−0.191691	+	0.981455i	$0.561397\pi$
$62$	0	0
$63$	−5.42594	−0.683605
$64$	0	0
$65$	1.28770	0.159719
$66$	0	0
$67$	13.5568	1.65623	0.828113	−	0.560561i	$-0.189414\pi$
$67$	13.5568	1.65623	0.828113	+	0.560561i	$0.189414\pi$
$68$	0	0
$69$	14.3811	1.73128
$70$	0	0
$71$	−0.514661	−0.0610790	−0.0305395	−	0.999534i	$-0.509723\pi$
$71$	−0.514661	−0.0610790	−0.0305395	+	0.999534i	$0.509723\pi$
$72$	0	0
$73$	−4.71562	−0.551922	−0.275961	−	0.961169i	$-0.588996\pi$
$73$	−4.71562	−0.551922	−0.275961	+	0.961169i	$0.588996\pi$
$74$	0	0
$75$	−6.91918	−0.798958
$76$	0	0
$77$	27.1666	3.09592
$78$	0	0
$79$	−5.02425	−0.565273	−0.282636	−	0.959227i	$-0.591209\pi$
$79$	−5.02425	−0.565273	−0.282636	+	0.959227i	$0.591209\pi$
$80$	0	0
$81$	−11.2046	−1.24495
$82$	0	0
$83$	6.53628	0.717450	0.358725	−	0.933443i	$-0.383212\pi$
$83$	6.53628	0.717450	0.358725	+	0.933443i	$0.383212\pi$
$84$	0	0
$85$	−9.79867	−1.06281
$86$	0	0
$87$	2.07047	0.221978
$88$	0	0
$89$	9.98400	1.05830	0.529151	−	0.848528i	$-0.322511\pi$
$89$	9.98400	1.05830	0.529151	+	0.848528i	$0.322511\pi$
$90$	0	0
$91$	4.21640	0.441999
$92$	0	0
$93$	−16.5321	−1.71429
$94$	0	0
$95$	6.02420	0.618070
$96$	0	0
$97$	−16.3973	−1.66489	−0.832445	−	0.554108i	$-0.813059\pi$
$97$	−16.3973	−1.66489	−0.832445	+	0.554108i	$0.813059\pi$
$98$	0	0
$99$	−8.29136	−0.833313
$100$	0	0
$101$	−1.65308	−0.164487	−0.0822437	−	0.996612i	$-0.526209\pi$
$101$	−1.65308	−0.164487	−0.0822437	+	0.996612i	$0.526209\pi$
$102$	0	0
$103$	−14.8809	−1.46626	−0.733130	−	0.680089i	$-0.761941\pi$
$103$	−14.8809	−1.46626	−0.733130	+	0.680089i	$0.761941\pi$
$104$	0	0
$105$	11.2416	1.09706
$106$	0	0
$107$	−12.3414	−1.19309	−0.596546	−	0.802579i	$-0.703461\pi$
$107$	−12.3414	−1.19309	−0.596546	+	0.802579i	$0.703461\pi$
$108$	0	0
$109$	9.94995	0.953033	0.476516	−	0.879166i	$-0.341899\pi$
$109$	9.94995	0.953033	0.476516	+	0.879166i	$0.341899\pi$
$110$	0	0
$111$	6.30251	0.598207
$112$	0	0
$113$	15.7937	1.48574	0.742871	−	0.669435i	$-0.233464\pi$
$113$	15.7937	1.48574	0.742871	+	0.669435i	$0.233464\pi$
$114$	0	0
$115$	−8.94411	−0.834043
$116$	0	0
$117$	−1.28687	−0.118971
$118$	0	0
$119$	−32.0845	−2.94118
$120$	0	0
$121$	30.5131	2.77392
$122$	0	0
$123$	13.6628	1.23193
$124$	0	0
$125$	10.7418	0.960773
$126$	0	0
$127$	−5.25917	−0.466675	−0.233338	−	0.972396i	$-0.574965\pi$
$127$	−5.25917	−0.466675	−0.233338	+	0.972396i	$0.574965\pi$
$128$	0	0
$129$	17.2208	1.51620
$130$	0	0
$131$	7.20253	0.629288	0.314644	−	0.949210i	$-0.398115\pi$
$131$	7.20253	0.629288	0.314644	+	0.949210i	$0.398115\pi$
$132$	0	0
$133$	19.7255	1.71041
$134$	0	0
$135$	4.56747	0.393105
$136$	0	0
$137$	−5.41868	−0.462949	−0.231475	−	0.972841i	$-0.574355\pi$
$137$	−5.41868	−0.462949	−0.231475	+	0.972841i	$0.574355\pi$
$138$	0	0
$139$	0.464978	0.0394390	0.0197195	−	0.999806i	$-0.493723\pi$
$139$	0.464978	0.0394390	0.0197195	+	0.999806i	$0.493723\pi$
$140$	0	0
$141$	−1.50745	−0.126950
$142$	0	0
$143$	6.44307	0.538796
$144$	0	0
$145$	−1.28770	−0.106938
$146$	0	0
$147$	22.3157	1.84057
$148$	0	0
$149$	12.3032	1.00791	0.503957	−	0.863729i	$-0.331877\pi$
$149$	12.3032	1.00791	0.503957	+	0.863729i	$0.331877\pi$
$150$	0	0
$151$	−6.96094	−0.566473	−0.283237	−	0.959050i	$-0.591408\pi$
$151$	−6.96094	−0.566473	−0.283237	+	0.959050i	$0.591408\pi$
$152$	0	0
$153$	9.79232	0.791662
$154$	0	0
$155$	10.2818	0.825858
$156$	0	0
$157$	14.1639	1.13040	0.565200	−	0.824954i	$-0.308799\pi$
$157$	14.1639	1.13040	0.565200	+	0.824954i	$0.308799\pi$
$158$	0	0
$159$	2.98373	0.236625
$160$	0	0
$161$	−29.2863	−2.30809
$162$	0	0
$163$	12.9724	1.01608	0.508038	−	0.861334i	$-0.330371\pi$
$163$	12.9724	1.01608	0.508038	+	0.861334i	$0.330371\pi$
$164$	0	0
$165$	17.1782	1.33732
$166$	0	0
$167$	−2.62115	−0.202831	−0.101415	−	0.994844i	$-0.532337\pi$
$167$	−2.62115	−0.202831	−0.101415	+	0.994844i	$0.532337\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.02030	−0.460383
$172$	0	0
$173$	−13.9459	−1.06029	−0.530145	−	0.847907i	$-0.677863\pi$
$173$	−13.9459	−1.06029	−0.530145	+	0.847907i	$0.677863\pi$
$174$	0	0
$175$	14.0905	1.06514
$176$	0	0
$177$	28.9871	2.17880
$178$	0	0
$179$	6.57398	0.491362	0.245681	−	0.969351i	$-0.420988\pi$
$179$	6.57398	0.491362	0.245681	+	0.969351i	$0.420988\pi$
$180$	0	0
$181$	−18.7642	−1.39473	−0.697364	−	0.716717i	$-0.745644\pi$
$181$	−18.7642	−1.39473	−0.697364	+	0.716717i	$0.745644\pi$
$182$	0	0
$183$	−6.19963	−0.458290
$184$	0	0
$185$	−3.91974	−0.288185
$186$	0	0
$187$	−49.0281	−3.58529
$188$	0	0
$189$	14.9556	1.08786
$190$	0	0
$191$	−3.95382	−0.286088	−0.143044	−	0.989716i	$-0.545689\pi$
$191$	−3.95382	−0.286088	−0.143044	+	0.989716i	$0.545689\pi$
$192$	0	0
$193$	21.6189	1.55616	0.778082	−	0.628162i	$-0.216193\pi$
$193$	21.6189	1.55616	0.778082	+	0.628162i	$0.216193\pi$
$194$	0	0
$195$	2.66615	0.190927
$196$	0	0
$197$	10.0498	0.716020	0.358010	−	0.933718i	$-0.383455\pi$
$197$	10.0498	0.716020	0.358010	+	0.933718i	$0.383455\pi$
$198$	0	0
$199$	18.3964	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$199$	18.3964	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$200$	0	0
$201$	28.0690	1.97984
$202$	0	0
$203$	−4.21640	−0.295934
$204$	0	0
$205$	−8.49737	−0.593482
$206$	0	0
$207$	8.93832	0.621256
$208$	0	0
$209$	30.1424	2.08499
$210$	0	0
$211$	15.1575	1.04348	0.521741	−	0.853104i	$-0.325283\pi$
$211$	15.1575	1.04348	0.521741	+	0.853104i	$0.325283\pi$
$212$	0	0
$213$	−1.06559	−0.0730132
$214$	0	0
$215$	−10.7102	−0.730428
$216$	0	0
$217$	33.6666	2.28544
$218$	0	0
$219$	−9.76358	−0.659762
$220$	0	0
$221$	−7.60944	−0.511866
$222$	0	0
$223$	−4.69968	−0.314714	−0.157357	−	0.987542i	$-0.550297\pi$
$223$	−4.69968	−0.314714	−0.157357	+	0.987542i	$0.550297\pi$
$224$	0	0
$225$	−4.30049	−0.286699
$226$	0	0
$227$	17.2192	1.14288	0.571439	−	0.820645i	$-0.306385\pi$
$227$	17.2192	1.14288	0.571439	+	0.820645i	$0.306385\pi$
$228$	0	0
$229$	−3.65585	−0.241586	−0.120793	−	0.992678i	$-0.538544\pi$
$229$	−3.65585	−0.241586	−0.120793	+	0.992678i	$0.538544\pi$
$230$	0	0
$231$	56.2477	3.70083
$232$	0	0
$233$	16.1725	1.05950	0.529748	−	0.848155i	$-0.322286\pi$
$233$	16.1725	1.05950	0.529748	+	0.848155i	$0.322286\pi$
$234$	0	0
$235$	0.937535	0.0611580
$236$	0	0
$237$	−10.4026	−0.675721
$238$	0	0
$239$	−19.4995	−1.26132	−0.630658	−	0.776061i	$-0.717215\pi$
$239$	−19.4995	−1.26132	−0.630658	+	0.776061i	$0.717215\pi$
$240$	0	0
$241$	−13.7687	−0.886920	−0.443460	−	0.896294i	$-0.646249\pi$
$241$	−13.7687	−0.886920	−0.443460	+	0.896294i	$0.646249\pi$
$242$	0	0
$243$	−12.5578	−0.805582
$244$	0	0
$245$	−13.8789	−0.886690
$246$	0	0
$247$	4.67826	0.297671
$248$	0	0
$249$	13.5332	0.857632
$250$	0	0
$251$	−20.7749	−1.31130	−0.655651	−	0.755064i	$-0.727606\pi$
$251$	−20.7749	−1.31130	−0.655651	+	0.755064i	$0.727606\pi$
$252$	0	0
$253$	−44.7523	−2.81355
$254$	0	0
$255$	−20.2879	−1.27048
$256$	0	0
$257$	−5.09273	−0.317675	−0.158838	−	0.987305i	$-0.550775\pi$
$257$	−5.09273	−0.317675	−0.158838	+	0.987305i	$0.550775\pi$
$258$	0	0
$259$	−12.8347	−0.797509
$260$	0	0
$261$	1.28687	0.0796549
$262$	0	0
$263$	−11.9096	−0.734376	−0.367188	−	0.930147i	$-0.619679\pi$
$263$	−11.9096	−0.734376	−0.367188	+	0.930147i	$0.619679\pi$
$264$	0	0
$265$	−1.85568	−0.113994
$266$	0	0
$267$	20.6716	1.26508
$268$	0	0
$269$	25.6772	1.56557	0.782785	−	0.622293i	$-0.213799\pi$
$269$	25.6772	1.56557	0.782785	+	0.622293i	$0.213799\pi$
$270$	0	0
$271$	2.37626	0.144347	0.0721737	−	0.997392i	$-0.477006\pi$
$271$	2.37626	0.144347	0.0721737	+	0.997392i	$0.477006\pi$
$272$	0	0
$273$	8.72996	0.528361
$274$	0	0
$275$	21.5316	1.29841
$276$	0	0
$277$	−27.8863	−1.67553	−0.837764	−	0.546033i	$-0.816137\pi$
$277$	−27.8863	−1.67553	−0.837764	+	0.546033i	$0.816137\pi$
$278$	0	0
$279$	−10.2752	−0.615159
$280$	0	0
$281$	2.19072	0.130687	0.0653437	−	0.997863i	$-0.479186\pi$
$281$	2.19072	0.130687	0.0653437	+	0.997863i	$0.479186\pi$
$282$	0	0
$283$	8.24449	0.490084	0.245042	−	0.969512i	$-0.421198\pi$
$283$	8.24449	0.490084	0.245042	+	0.969512i	$0.421198\pi$
$284$	0	0
$285$	12.4729	0.738834
$286$	0	0
$287$	−27.8235	−1.64237
$288$	0	0
$289$	40.9036	2.40609
$290$	0	0
$291$	−33.9501	−1.99019
$292$	0	0
$293$	0.550270	0.0321471	0.0160736	−	0.999871i	$-0.494883\pi$
$293$	0.550270	0.0321471	0.0160736	+	0.999871i	$0.494883\pi$
$294$	0	0
$295$	−18.0281	−1.04964
$296$	0	0
$297$	22.8536	1.32610
$298$	0	0
$299$	−6.94581	−0.401686
$300$	0	0
$301$	−35.0691	−2.02135
$302$	0	0
$303$	−3.42265	−0.196626
$304$	0	0
$305$	3.85577	0.220780
$306$	0	0
$307$	21.2947	1.21535	0.607677	−	0.794184i	$-0.292101\pi$
$307$	21.2947	1.21535	0.607677	+	0.794184i	$0.292101\pi$
$308$	0	0
$309$	−30.8105	−1.75275
$310$	0	0
$311$	9.11842	0.517058	0.258529	−	0.966003i	$-0.416762\pi$
$311$	9.11842	0.517058	0.258529	+	0.966003i	$0.416762\pi$
$312$	0	0
$313$	−0.513399	−0.0290191	−0.0145095	−	0.999895i	$-0.504619\pi$
$313$	−0.513399	−0.0290191	−0.0145095	+	0.999895i	$0.504619\pi$
$314$	0	0
$315$	6.98698	0.393672
$316$	0	0
$317$	−7.22825	−0.405979	−0.202989	−	0.979181i	$-0.565066\pi$
$317$	−7.22825	−0.405979	−0.202989	+	0.979181i	$0.565066\pi$
$318$	0	0
$319$	−6.44307	−0.360742
$320$	0	0
$321$	−25.5526	−1.42621
$322$	0	0
$323$	−35.5990	−1.98078
$324$	0	0
$325$	3.34183	0.185371
$326$	0	0
$327$	20.6011	1.13924
$328$	0	0
$329$	3.06984	0.169246
$330$	0	0
$331$	−15.8129	−0.869157	−0.434578	−	0.900634i	$-0.643103\pi$
$331$	−15.8129	−0.869157	−0.434578	+	0.900634i	$0.643103\pi$
$332$	0	0
$333$	3.91721	0.214662
$334$	0	0
$335$	−17.4571	−0.953782
$336$	0	0
$337$	−5.06628	−0.275978	−0.137989	−	0.990434i	$-0.544064\pi$
$337$	−5.06628	−0.275978	−0.137989	+	0.990434i	$0.544064\pi$
$338$	0	0
$339$	32.7004	1.77604
$340$	0	0
$341$	51.4457	2.78594
$342$	0	0
$343$	−15.9298	−0.860130
$344$	0	0
$345$	−18.5186	−0.997005
$346$	0	0
$347$	−5.61575	−0.301469	−0.150735	−	0.988574i	$-0.548164\pi$
$347$	−5.61575	−0.301469	−0.150735	+	0.988574i	$0.548164\pi$
$348$	0	0
$349$	−2.64198	−0.141422	−0.0707111	−	0.997497i	$-0.522527\pi$
$349$	−2.64198	−0.141422	−0.0707111	+	0.997497i	$0.522527\pi$
$350$	0	0
$351$	3.54700	0.189325
$352$	0	0
$353$	−26.9302	−1.43335	−0.716674	−	0.697409i	$-0.754336\pi$
$353$	−26.9302	−1.43335	−0.716674	+	0.697409i	$0.754336\pi$
$354$	0	0
$355$	0.662729	0.0351740
$356$	0	0
$357$	−66.4301	−3.51585
$358$	0	0
$359$	15.2003	0.802244	0.401122	−	0.916025i	$-0.368620\pi$
$359$	15.2003	0.802244	0.401122	+	0.916025i	$0.368620\pi$
$360$	0	0
$361$	2.88616	0.151903
$362$	0	0
$363$	63.1766	3.31591
$364$	0	0
$365$	6.07231	0.317839
$366$	0	0
$367$	−26.6810	−1.39274	−0.696369	−	0.717684i	$-0.745202\pi$
$367$	−26.6810	−1.39274	−0.696369	+	0.717684i	$0.745202\pi$
$368$	0	0
$369$	8.49187	0.442069
$370$	0	0
$371$	−6.07620	−0.315461
$372$	0	0
$373$	1.22761	0.0635634	0.0317817	−	0.999495i	$-0.489882\pi$
$373$	1.22761	0.0635634	0.0317817	+	0.999495i	$0.489882\pi$
$374$	0	0
$375$	22.2406	1.14850
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	30.9674	1.59069	0.795344	−	0.606158i	$-0.207290\pi$
$379$	30.9674	1.59069	0.795344	+	0.606158i	$0.207290\pi$
$380$	0	0
$381$	−10.8890	−0.557859
$382$	0	0
$383$	−9.15598	−0.467849	−0.233924	−	0.972255i	$-0.575157\pi$
$383$	−9.15598	−0.467849	−0.233924	+	0.972255i	$0.575157\pi$
$384$	0	0
$385$	−34.9824	−1.78287
$386$	0	0
$387$	10.7032	0.544077
$388$	0	0
$389$	26.6970	1.35359	0.676796	−	0.736171i	$-0.263368\pi$
$389$	26.6970	1.35359	0.676796	+	0.736171i	$0.263368\pi$
$390$	0	0
$391$	52.8537	2.67293
$392$	0	0
$393$	14.9127	0.752244
$394$	0	0
$395$	6.46973	0.325527
$396$	0	0
$397$	19.3877	0.973043	0.486521	−	0.873669i	$-0.338266\pi$
$397$	19.3877	0.973043	0.486521	+	0.873669i	$0.338266\pi$
$398$	0	0
$399$	40.8411	2.04461
$400$	0	0
$401$	6.75414	0.337286	0.168643	−	0.985677i	$-0.446062\pi$
$401$	6.75414	0.337286	0.168643	+	0.985677i	$0.446062\pi$
$402$	0	0
$403$	7.98467	0.397745
$404$	0	0
$405$	14.4281	0.716939
$406$	0	0
$407$	−19.6126	−0.972162
$408$	0	0
$409$	−26.3180	−1.30134	−0.650669	−	0.759361i	$-0.725512\pi$
$409$	−26.3180	−1.30134	−0.650669	+	0.759361i	$0.725512\pi$
$410$	0	0
$411$	−11.2192	−0.553405
$412$	0	0
$413$	−59.0306	−2.90471
$414$	0	0
$415$	−8.41677	−0.413163
$416$	0	0
$417$	0.962726	0.0471449
$418$	0	0
$419$	−27.7816	−1.35722	−0.678610	−	0.734499i	$-0.737417\pi$
$419$	−27.7816	−1.35722	−0.678610	+	0.734499i	$0.737417\pi$
$420$	0	0
$421$	6.70790	0.326923	0.163462	−	0.986550i	$-0.447734\pi$
$421$	6.70790	0.326923	0.163462	+	0.986550i	$0.447734\pi$
$422$	0	0
$423$	−0.936927	−0.0455550
$424$	0	0
$425$	−25.4295	−1.23351
$426$	0	0
$427$	12.6252	0.610976
$428$	0	0
$429$	13.3402	0.644071
$430$	0	0
$431$	17.1143	0.824369	0.412185	−	0.911100i	$-0.364766\pi$
$431$	17.1143	0.824369	0.412185	+	0.911100i	$0.364766\pi$
$432$	0	0
$433$	−22.7010	−1.09094	−0.545470	−	0.838131i	$-0.683649\pi$
$433$	−22.7010	−1.09094	−0.545470	+	0.838131i	$0.683649\pi$
$434$	0	0
$435$	−2.66615	−0.127832
$436$	0	0
$437$	−32.4943	−1.55441
$438$	0	0
$439$	36.1435	1.72503	0.862517	−	0.506028i	$-0.168887\pi$
$439$	36.1435	1.72503	0.862517	+	0.506028i	$0.168887\pi$
$440$	0	0
$441$	13.8699	0.660472
$442$	0	0
$443$	0.795656	0.0378028	0.0189014	−	0.999821i	$-0.493983\pi$
$443$	0.795656	0.0378028	0.0189014	+	0.999821i	$0.493983\pi$
$444$	0	0
$445$	−12.8564	−0.609451
$446$	0	0
$447$	25.4734	1.20485
$448$	0	0
$449$	8.18283	0.386172	0.193086	−	0.981182i	$-0.438150\pi$
$449$	8.18283	0.386172	0.193086	+	0.981182i	$0.438150\pi$
$450$	0	0
$451$	−42.5170	−2.00205
$452$	0	0
$453$	−14.4125	−0.677156
$454$	0	0
$455$	−5.42946	−0.254537
$456$	0	0
$457$	21.3722	0.999752	0.499876	−	0.866097i	$-0.333379\pi$
$457$	21.3722	0.999752	0.499876	+	0.866097i	$0.333379\pi$
$458$	0	0
$459$	−26.9907	−1.25982
$460$	0	0
$461$	39.1582	1.82378	0.911889	−	0.410437i	$-0.134624\pi$
$461$	39.1582	1.82378	0.911889	+	0.410437i	$0.134624\pi$
$462$	0	0
$463$	−2.97351	−0.138191	−0.0690954	−	0.997610i	$-0.522011\pi$
$463$	−2.97351	−0.138191	−0.0690954	+	0.997610i	$0.522011\pi$
$464$	0	0
$465$	21.2883	0.987222
$466$	0	0
$467$	−23.9921	−1.11022	−0.555111	−	0.831776i	$-0.687324\pi$
$467$	−23.9921	−1.11022	−0.555111	+	0.831776i	$0.687324\pi$
$468$	0	0
$469$	−57.1609	−2.63945
$470$	0	0
$471$	29.3259	1.35127
$472$	0	0
$473$	−53.5889	−2.46402
$474$	0	0
$475$	15.6340	0.717336
$476$	0	0
$477$	1.85448	0.0849109
$478$	0	0
$479$	−9.02877	−0.412535	−0.206268	−	0.978496i	$-0.566132\pi$
$479$	−9.02877	−0.412535	−0.206268	+	0.978496i	$0.566132\pi$
$480$	0	0
$481$	−3.04399	−0.138794
$482$	0	0
$483$	−60.6366	−2.75906
$484$	0	0
$485$	21.1147	0.958771
$486$	0	0
$487$	18.3769	0.832737	0.416369	−	0.909196i	$-0.363303\pi$
$487$	18.3769	0.832737	0.416369	+	0.909196i	$0.363303\pi$
$488$	0	0
$489$	26.8590	1.21461
$490$	0	0
$491$	19.7655	0.892005	0.446002	−	0.895032i	$-0.352847\pi$
$491$	19.7655	0.892005	0.446002	+	0.895032i	$0.352847\pi$
$492$	0	0
$493$	7.60944	0.342712
$494$	0	0
$495$	10.6768	0.479885
$496$	0	0
$497$	2.17002	0.0973387
$498$	0	0
$499$	18.7116	0.837645	0.418823	−	0.908068i	$-0.362443\pi$
$499$	18.7116	0.837645	0.418823	+	0.908068i	$0.362443\pi$
$500$	0	0
$501$	−5.42703	−0.242462
$502$	0	0
$503$	−13.8977	−0.619666	−0.309833	−	0.950791i	$-0.600273\pi$
$503$	−13.8977	−0.619666	−0.309833	+	0.950791i	$0.600273\pi$
$504$	0	0
$505$	2.12867	0.0947244
$506$	0	0
$507$	2.07047	0.0919530
$508$	0	0
$509$	−38.6042	−1.71110	−0.855551	−	0.517719i	$-0.826781\pi$
$509$	−38.6042	−1.71110	−0.855551	+	0.517719i	$0.826781\pi$
$510$	0	0
$511$	19.8830	0.879571
$512$	0	0
$513$	16.5938	0.732635
$514$	0	0
$515$	19.1621	0.844384
$516$	0	0
$517$	4.69100	0.206310
$518$	0	0
$519$	−28.8747	−1.26746
$520$	0	0
$521$	1.58677	0.0695175	0.0347587	−	0.999396i	$-0.488934\pi$
$521$	1.58677	0.0695175	0.0347587	+	0.999396i	$0.488934\pi$
$522$	0	0
$523$	5.82921	0.254893	0.127447	−	0.991845i	$-0.459322\pi$
$523$	5.82921	0.254893	0.127447	+	0.991845i	$0.459322\pi$
$524$	0	0
$525$	29.1740	1.27326
$526$	0	0
$527$	−60.7588	−2.64670
$528$	0	0
$529$	25.2442	1.09758
$530$	0	0
$531$	18.0164	0.781845
$532$	0	0
$533$	−6.59888	−0.285829
$534$	0	0
$535$	15.8921	0.687074
$536$	0	0
$537$	13.6112	0.587369
$538$	0	0
$539$	−69.4438	−2.99115
$540$	0	0
$541$	35.6797	1.53399	0.766994	−	0.641654i	$-0.221751\pi$
$541$	35.6797	1.53399	0.766994	+	0.641654i	$0.221751\pi$
$542$	0	0
$543$	−38.8507	−1.66724
$544$	0	0
$545$	−12.8125	−0.548829
$546$	0	0
$547$	3.32382	0.142116	0.0710581	−	0.997472i	$-0.477362\pi$
$547$	3.32382	0.142116	0.0710581	+	0.997472i	$0.477362\pi$
$548$	0	0
$549$	−3.85327	−0.164453
$550$	0	0
$551$	−4.67826	−0.199301
$552$	0	0
$553$	21.1843	0.900848
$554$	0	0
$555$	−8.11573	−0.344494
$556$	0	0
$557$	9.89468	0.419251	0.209626	−	0.977782i	$-0.432775\pi$
$557$	9.89468	0.419251	0.209626	+	0.977782i	$0.432775\pi$
$558$	0	0
$559$	−8.31730	−0.351784
$560$	0	0
$561$	−101.511	−4.28582
$562$	0	0
$563$	−0.922340	−0.0388720	−0.0194360	−	0.999811i	$-0.506187\pi$
$563$	−0.922340	−0.0388720	−0.0194360	+	0.999811i	$0.506187\pi$
$564$	0	0
$565$	−20.3375	−0.855604
$566$	0	0
$567$	47.2430	1.98402
$568$	0	0
$569$	−3.74522	−0.157008	−0.0785039	−	0.996914i	$-0.525014\pi$
$569$	−3.74522	−0.157008	−0.0785039	+	0.996914i	$0.525014\pi$
$570$	0	0
$571$	0.100417	0.00420230	0.00210115	−	0.999998i	$-0.499331\pi$
$571$	0.100417	0.00420230	0.00210115	+	0.999998i	$0.499331\pi$
$572$	0	0
$573$	−8.18628	−0.341987
$574$	0	0
$575$	−23.2117	−0.967996
$576$	0	0
$577$	−12.6652	−0.527261	−0.263631	−	0.964624i	$-0.584920\pi$
$577$	−12.6652	−0.527261	−0.263631	+	0.964624i	$0.584920\pi$
$578$	0	0
$579$	44.7614	1.86022
$580$	0	0
$581$	−27.5596	−1.14337
$582$	0	0
$583$	−9.28501	−0.384546
$584$	0	0
$585$	1.65709	0.0685124
$586$	0	0
$587$	30.8626	1.27384	0.636919	−	0.770931i	$-0.280209\pi$
$587$	30.8626	1.27384	0.636919	+	0.770931i	$0.280209\pi$
$588$	0	0
$589$	37.3544	1.53916
$590$	0	0
$591$	20.8079	0.855922
$592$	0	0
$593$	−3.57627	−0.146860	−0.0734299	−	0.997300i	$-0.523395\pi$
$593$	−3.57627	−0.146860	−0.0734299	+	0.997300i	$0.523395\pi$
$594$	0	0
$595$	41.3151	1.69376
$596$	0	0
$597$	38.0894	1.55889
$598$	0	0
$599$	−12.4342	−0.508049	−0.254025	−	0.967198i	$-0.581754\pi$
$599$	−12.4342	−0.508049	−0.254025	+	0.967198i	$0.581754\pi$
$600$	0	0
$601$	−4.21750	−0.172035	−0.0860176	−	0.996294i	$-0.527414\pi$
$601$	−4.21750	−0.172035	−0.0860176	+	0.996294i	$0.527414\pi$
$602$	0	0
$603$	17.4458	0.710447
$604$	0	0
$605$	−39.2917	−1.59743
$606$	0	0
$607$	−13.4402	−0.545520	−0.272760	−	0.962082i	$-0.587937\pi$
$607$	−13.4402	−0.545520	−0.272760	+	0.962082i	$0.587937\pi$
$608$	0	0
$609$	−8.72996	−0.353756
$610$	0	0
$611$	0.728070	0.0294545
$612$	0	0
$613$	21.4871	0.867855	0.433927	−	0.900948i	$-0.357127\pi$
$613$	21.4871	0.867855	0.433927	+	0.900948i	$0.357127\pi$
$614$	0	0
$615$	−17.5936	−0.709442
$616$	0	0
$617$	−6.22375	−0.250559	−0.125279	−	0.992121i	$-0.539983\pi$
$617$	−6.22375	−0.250559	−0.125279	+	0.992121i	$0.539983\pi$
$618$	0	0
$619$	15.1384	0.608466	0.304233	−	0.952598i	$-0.401600\pi$
$619$	15.1384	0.608466	0.304233	+	0.952598i	$0.401600\pi$
$620$	0	0
$621$	−24.6368	−0.988641
$622$	0	0
$623$	−42.0966	−1.68656
$624$	0	0
$625$	2.87699	0.115080
$626$	0	0
$627$	62.4090	2.49238
$628$	0	0
$629$	23.1631	0.923572
$630$	0	0
$631$	−30.8092	−1.22649	−0.613247	−	0.789891i	$-0.710137\pi$
$631$	−30.8092	−1.22649	−0.613247	+	0.789891i	$0.710137\pi$
$632$	0	0
$633$	31.3831	1.24737
$634$	0	0
$635$	6.77222	0.268747
$636$	0	0
$637$	−10.7781	−0.427042
$638$	0	0
$639$	−0.662299	−0.0262002
$640$	0	0
$641$	25.9117	1.02345	0.511725	−	0.859149i	$-0.329007\pi$
$641$	25.9117	1.02345	0.511725	+	0.859149i	$0.329007\pi$
$642$	0	0
$643$	10.6438	0.419750	0.209875	−	0.977728i	$-0.432694\pi$
$643$	10.6438	0.419750	0.209875	+	0.977728i	$0.432694\pi$
$644$	0	0
$645$	−22.1752	−0.873146
$646$	0	0
$647$	−34.8501	−1.37010	−0.685050	−	0.728496i	$-0.740220\pi$
$647$	−34.8501	−1.37010	−0.685050	+	0.728496i	$0.740220\pi$
$648$	0	0
$649$	−90.2044	−3.54083
$650$	0	0
$651$	69.7058	2.73199
$652$	0	0
$653$	−25.8342	−1.01097	−0.505486	−	0.862835i	$-0.668687\pi$
$653$	−25.8342	−1.01097	−0.505486	+	0.862835i	$0.668687\pi$
$654$	0	0
$655$	−9.27469	−0.362392
$656$	0	0
$657$	−6.06837	−0.236750
$658$	0	0
$659$	2.19545	0.0855225	0.0427612	−	0.999085i	$-0.486385\pi$
$659$	2.19545	0.0855225	0.0427612	+	0.999085i	$0.486385\pi$
$660$	0	0
$661$	−18.7388	−0.728856	−0.364428	−	0.931232i	$-0.618735\pi$
$661$	−18.7388	−0.728856	−0.364428	+	0.931232i	$0.618735\pi$
$662$	0	0
$663$	−15.7551	−0.611879
$664$	0	0
$665$	−25.4004	−0.984987
$666$	0	0
$667$	6.94581	0.268943
$668$	0	0
$669$	−9.73056	−0.376205
$670$	0	0
$671$	19.2925	0.744779
$672$	0	0
$673$	−47.8161	−1.84318	−0.921589	−	0.388168i	$-0.873108\pi$
$673$	−47.8161	−1.84318	−0.921589	+	0.388168i	$0.873108\pi$
$674$	0	0
$675$	11.8535	0.456241
$676$	0	0
$677$	0.154451	0.00593603	0.00296801	−	0.999996i	$-0.499055\pi$
$677$	0.154451	0.00593603	0.00296801	+	0.999996i	$0.499055\pi$
$678$	0	0
$679$	69.1375	2.65325
$680$	0	0
$681$	35.6519	1.36618
$682$	0	0
$683$	−18.8702	−0.722048	−0.361024	−	0.932557i	$-0.617573\pi$
$683$	−18.8702	−0.722048	−0.361024	+	0.932557i	$0.617573\pi$
$684$	0	0
$685$	6.97764	0.266602
$686$	0	0
$687$	−7.56935	−0.288789
$688$	0	0
$689$	−1.44109	−0.0549010
$690$	0	0
$691$	21.0218	0.799707	0.399853	−	0.916579i	$-0.369061\pi$
$691$	21.0218	0.799707	0.399853	+	0.916579i	$0.369061\pi$
$692$	0	0
$693$	34.9597	1.32801
$694$	0	0
$695$	−0.598752	−0.0227120
$696$	0	0
$697$	50.2138	1.90198
$698$	0	0
$699$	33.4848	1.26651
$700$	0	0
$701$	11.6707	0.440796	0.220398	−	0.975410i	$-0.429264\pi$
$701$	11.6707	0.440796	0.220398	+	0.975410i	$0.429264\pi$
$702$	0	0
$703$	−14.2406	−0.537094
$704$	0	0
$705$	1.94114	0.0731076
$706$	0	0
$707$	6.97004	0.262135
$708$	0	0
$709$	0.123865	0.00465184	0.00232592	−	0.999997i	$-0.499260\pi$
$709$	0.123865	0.00465184	0.00232592	+	0.999997i	$0.499260\pi$
$710$	0	0
$711$	−6.46554	−0.242477
$712$	0	0
$713$	−55.4600	−2.07699
$714$	0	0
$715$	−8.29673	−0.310280
$716$	0	0
$717$	−40.3731	−1.50776
$718$	0	0
$719$	−14.2714	−0.532233	−0.266116	−	0.963941i	$-0.585741\pi$
$719$	−14.2714	−0.532233	−0.266116	+	0.963941i	$0.585741\pi$
$720$	0	0
$721$	62.7439	2.33671
$722$	0	0
$723$	−28.5078	−1.06021
$724$	0	0
$725$	−3.34183	−0.124112
$726$	0	0
$727$	42.1049	1.56159	0.780793	−	0.624790i	$-0.214815\pi$
$727$	42.1049	1.56159	0.780793	+	0.624790i	$0.214815\pi$
$728$	0	0
$729$	7.61316	0.281969
$730$	0	0
$731$	63.2900	2.34087
$732$	0	0
$733$	−7.80820	−0.288403	−0.144201	−	0.989548i	$-0.546061\pi$
$733$	−7.80820	−0.288403	−0.144201	+	0.989548i	$0.546061\pi$
$734$	0	0
$735$	−28.7359	−1.05994
$736$	0	0
$737$	−87.3473	−3.21748
$738$	0	0
$739$	5.28014	0.194233	0.0971165	−	0.995273i	$-0.469038\pi$
$739$	5.28014	0.194233	0.0971165	+	0.995273i	$0.469038\pi$
$740$	0	0
$741$	9.68623	0.355832
$742$	0	0
$743$	14.9185	0.547308	0.273654	−	0.961828i	$-0.411768\pi$
$743$	14.9185	0.547308	0.273654	+	0.961828i	$0.411768\pi$
$744$	0	0
$745$	−15.8428	−0.580434
$746$	0	0
$747$	8.41132	0.307754
$748$	0	0
$749$	52.0365	1.90137
$750$	0	0
$751$	−24.6320	−0.898834	−0.449417	−	0.893322i	$-0.648368\pi$
$751$	−24.6320	−0.898834	−0.449417	+	0.893322i	$0.648368\pi$
$752$	0	0
$753$	−43.0140	−1.56752
$754$	0	0
$755$	8.96360	0.326219
$756$	0	0
$757$	21.3591	0.776309	0.388154	−	0.921594i	$-0.373113\pi$
$757$	21.3591	0.776309	0.388154	+	0.921594i	$0.373113\pi$
$758$	0	0
$759$	−92.6585	−3.36329
$760$	0	0
$761$	−1.37078	−0.0496909	−0.0248454	−	0.999691i	$-0.507909\pi$
$761$	−1.37078	−0.0496909	−0.0248454	+	0.999691i	$0.507909\pi$
$762$	0	0
$763$	−41.9530	−1.51880
$764$	0	0
$765$	−12.6096	−0.455900
$766$	0	0
$767$	−14.0002	−0.505519
$768$	0	0
$769$	49.4732	1.78405	0.892024	−	0.451987i	$-0.149285\pi$
$769$	49.4732	1.78405	0.892024	+	0.451987i	$0.149285\pi$
$770$	0	0
$771$	−10.5444	−0.379746
$772$	0	0
$773$	28.6076	1.02895	0.514473	−	0.857507i	$-0.327988\pi$
$773$	28.6076	1.02895	0.514473	+	0.857507i	$0.327988\pi$
$774$	0	0
$775$	26.6834	0.958496
$776$	0	0
$777$	−26.5739	−0.953334
$778$	0	0
$779$	−30.8713	−1.10608
$780$	0	0
$781$	3.31600	0.118656
$782$	0	0
$783$	−3.54700	−0.126760
$784$	0	0
$785$	−18.2388	−0.650971
$786$	0	0
$787$	35.9321	1.28084	0.640420	−	0.768025i	$-0.278760\pi$
$787$	35.9321	1.28084	0.640420	+	0.768025i	$0.278760\pi$
$788$	0	0
$789$	−24.6585	−0.877865
$790$	0	0
$791$	−66.5924	−2.36775
$792$	0	0
$793$	2.99431	0.106331
$794$	0	0
$795$	−3.84215	−0.136267
$796$	0	0
$797$	35.3704	1.25288	0.626442	−	0.779468i	$-0.284510\pi$
$797$	35.3704	1.25288	0.626442	+	0.779468i	$0.284510\pi$
$798$	0	0
$799$	−5.54020	−0.195998
$800$	0	0
$801$	12.8481	0.453964
$802$	0	0
$803$	30.3831	1.07220
$804$	0	0
$805$	37.7120	1.32917
$806$	0	0
$807$	53.1641	1.87146
$808$	0	0
$809$	16.9994	0.597667	0.298833	−	0.954305i	$-0.403403\pi$
$809$	16.9994	0.597667	0.298833	+	0.954305i	$0.403403\pi$
$810$	0	0
$811$	−54.0255	−1.89709	−0.948546	−	0.316639i	$-0.897446\pi$
$811$	−54.0255	−1.89709	−0.948546	+	0.316639i	$0.897446\pi$
$812$	0	0
$813$	4.91998	0.172551
$814$	0	0
$815$	−16.7045	−0.585135
$816$	0	0
$817$	−38.9105	−1.36131
$818$	0	0
$819$	5.42594	0.189598
$820$	0	0
$821$	22.5987	0.788700	0.394350	−	0.918960i	$-0.370970\pi$
$821$	22.5987	0.788700	0.394350	+	0.918960i	$0.370970\pi$
$822$	0	0
$823$	51.6910	1.80183	0.900917	−	0.433991i	$-0.142895\pi$
$823$	51.6910	1.80183	0.900917	+	0.433991i	$0.142895\pi$
$824$	0	0
$825$	44.5807	1.55210
$826$	0	0
$827$	−7.67065	−0.266735	−0.133367	−	0.991067i	$-0.542579\pi$
$827$	−7.67065	−0.266735	−0.133367	+	0.991067i	$0.542579\pi$
$828$	0	0
$829$	−36.5413	−1.26913	−0.634565	−	0.772869i	$-0.718821\pi$
$829$	−36.5413	−1.26913	−0.634565	+	0.772869i	$0.718821\pi$
$830$	0	0
$831$	−57.7380	−2.00291
$832$	0	0
$833$	82.0150	2.84165
$834$	0	0
$835$	3.37526	0.116806
$836$	0	0
$837$	28.3216	0.978939
$838$	0	0
$839$	29.5523	1.02026	0.510130	−	0.860097i	$-0.329597\pi$
$839$	29.5523	1.02026	0.510130	+	0.860097i	$0.329597\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	4.53583	0.156222
$844$	0	0
$845$	−1.28770	−0.0442982
$846$	0	0
$847$	−128.656	−4.42066
$848$	0	0
$849$	17.0700	0.585841
$850$	0	0
$851$	21.1430	0.724772
$852$	0	0
$853$	6.59695	0.225875	0.112938	−	0.993602i	$-0.463974\pi$
$853$	6.59695	0.225875	0.112938	+	0.993602i	$0.463974\pi$
$854$	0	0
$855$	7.75233	0.265124
$856$	0	0
$857$	−26.7833	−0.914899	−0.457449	−	0.889236i	$-0.651237\pi$
$857$	−26.7833	−0.914899	−0.457449	+	0.889236i	$0.651237\pi$
$858$	0	0
$859$	−52.9959	−1.80820	−0.904098	−	0.427325i	$-0.859456\pi$
$859$	−52.9959	−1.80820	−0.904098	+	0.427325i	$0.859456\pi$
$860$	0	0
$861$	−57.6079	−1.96327
$862$	0	0
$863$	31.9756	1.08846	0.544231	−	0.838935i	$-0.316821\pi$
$863$	31.9756	1.08846	0.544231	+	0.838935i	$0.316821\pi$
$864$	0	0
$865$	17.9582	0.610597
$866$	0	0
$867$	84.6898	2.87622
$868$	0	0
$869$	32.3716	1.09813
$870$	0	0
$871$	−13.5568	−0.459355
$872$	0	0
$873$	−21.1011	−0.714163
$874$	0	0
$875$	−45.2916	−1.53114
$876$	0	0
$877$	−27.2728	−0.920938	−0.460469	−	0.887676i	$-0.652319\pi$
$877$	−27.2728	−0.920938	−0.460469	+	0.887676i	$0.652319\pi$
$878$	0	0
$879$	1.13932	0.0384283
$880$	0	0
$881$	11.5961	0.390682	0.195341	−	0.980735i	$-0.437419\pi$
$881$	11.5961	0.390682	0.195341	+	0.980735i	$0.437419\pi$
$882$	0	0
$883$	−16.2132	−0.545617	−0.272808	−	0.962068i	$-0.587953\pi$
$883$	−16.2132	−0.545617	−0.272808	+	0.962068i	$0.587953\pi$
$884$	0	0
$885$	−37.3267	−1.25472
$886$	0	0
$887$	27.4548	0.921841	0.460920	−	0.887442i	$-0.347519\pi$
$887$	27.4548	0.921841	0.460920	+	0.887442i	$0.347519\pi$
$888$	0	0
$889$	22.1748	0.743718
$890$	0	0
$891$	72.1918	2.41852
$892$	0	0
$893$	3.40610	0.113981
$894$	0	0
$895$	−8.46530	−0.282964
$896$	0	0
$897$	−14.3811	−0.480172
$898$	0	0
$899$	−7.98467	−0.266304
$900$	0	0
$901$	10.9659	0.365326
$902$	0	0
$903$	−72.6097	−2.41630
$904$	0	0
$905$	24.1626	0.803192
$906$	0	0
$907$	−8.09901	−0.268923	−0.134462	−	0.990919i	$-0.542930\pi$
$907$	−8.09901	−0.268923	−0.134462	+	0.990919i	$0.542930\pi$
$908$	0	0
$909$	−2.12729	−0.0705577
$910$	0	0
$911$	9.07732	0.300745	0.150372	−	0.988629i	$-0.451953\pi$
$911$	9.07732	0.300745	0.150372	+	0.988629i	$0.451953\pi$
$912$	0	0
$913$	−42.1137	−1.39376
$914$	0	0
$915$	7.98326	0.263919
$916$	0	0
$917$	−30.3688	−1.00287
$918$	0	0
$919$	33.7629	1.11373	0.556867	−	0.830601i	$-0.312003\pi$
$919$	33.7629	1.11373	0.556867	+	0.830601i	$0.312003\pi$
$920$	0	0
$921$	44.0902	1.45282
$922$	0	0
$923$	0.514661	0.0169403
$924$	0	0
$925$	−10.1725	−0.334470
$926$	0	0
$927$	−19.1497	−0.628959
$928$	0	0
$929$	−45.1393	−1.48097	−0.740486	−	0.672072i	$-0.765405\pi$
$929$	−45.1393	−1.48097	−0.740486	+	0.672072i	$0.765405\pi$
$930$	0	0
$931$	−50.4226	−1.65253
$932$	0	0
$933$	18.8795	0.618086
$934$	0	0
$935$	63.1335	2.06468
$936$	0	0
$937$	9.36617	0.305980	0.152990	−	0.988228i	$-0.451110\pi$
$937$	9.36617	0.305980	0.152990	+	0.988228i	$0.451110\pi$
$938$	0	0
$939$	−1.06298	−0.0346891
$940$	0	0
$941$	19.0902	0.622324	0.311162	−	0.950357i	$-0.399282\pi$
$941$	19.0902	0.622324	0.311162	+	0.950357i	$0.399282\pi$
$942$	0	0
$943$	45.8346	1.49258
$944$	0	0
$945$	−19.2583	−0.626473
$946$	0	0
$947$	44.6745	1.45173	0.725864	−	0.687839i	$-0.241440\pi$
$947$	44.6745	1.45173	0.725864	+	0.687839i	$0.241440\pi$
$948$	0	0
$949$	4.71562	0.153076
$950$	0	0
$951$	−14.9659	−0.485303
$952$	0	0
$953$	−13.1574	−0.426210	−0.213105	−	0.977029i	$-0.568358\pi$
$953$	−13.1574	−0.426210	−0.213105	+	0.977029i	$0.568358\pi$
$954$	0	0
$955$	5.09133	0.164752
$956$	0	0
$957$	−13.3402	−0.431228
$958$	0	0
$959$	22.8474	0.737780
$960$	0	0
$961$	32.7549	1.05661
$962$	0	0
$963$	−15.8818	−0.511783
$964$	0	0
$965$	−27.8387	−0.896159
$966$	0	0
$967$	−22.8856	−0.735952	−0.367976	−	0.929835i	$-0.619949\pi$
$967$	−22.8856	−0.735952	−0.367976	+	0.929835i	$0.619949\pi$
$968$	0	0
$969$	−73.7068	−2.36780
$970$	0	0
$971$	13.5034	0.433344	0.216672	−	0.976244i	$-0.430480\pi$
$971$	13.5034	0.433344	0.216672	+	0.976244i	$0.430480\pi$
$972$	0	0
$973$	−1.96054	−0.0628519
$974$	0	0
$975$	6.91918	0.221591
$976$	0	0
$977$	−27.0534	−0.865516	−0.432758	−	0.901510i	$-0.642460\pi$
$977$	−27.0534	−0.865516	−0.432758	+	0.901510i	$0.642460\pi$
$978$	0	0
$979$	−64.3275	−2.05592
$980$	0	0
$981$	12.8042	0.408808
$982$	0	0
$983$	55.8595	1.78164	0.890821	−	0.454355i	$-0.150130\pi$
$983$	55.8595	1.78164	0.890821	+	0.454355i	$0.150130\pi$
$984$	0	0
$985$	−12.9411	−0.412339
$986$	0	0
$987$	6.35602	0.202314
$988$	0	0
$989$	57.7704	1.83699
$990$	0	0
$991$	36.2913	1.15283	0.576416	−	0.817156i	$-0.304451\pi$
$991$	36.2913	1.15283	0.576416	+	0.817156i	$0.304451\pi$
$992$	0	0
$993$	−32.7402	−1.03898
$994$	0	0
$995$	−23.6891	−0.750995
$996$	0	0
$997$	9.68378	0.306688	0.153344	−	0.988173i	$-0.450996\pi$
$997$	9.68378	0.306688	0.153344	+	0.988173i	$0.450996\pi$
$998$	0	0
$999$	−10.7970	−0.341603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.10		13
4.3	odd	2		3016.2.a.k.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.4	✓	13	4.3	odd	2
6032.2.a.be.1.10		13	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+2.07047 q^{3} -1.28770 q^{5} -4.21640 q^{7} +1.28687 q^{9} +O(q^{10})\)	\(q+2.07047 q^{3} -1.28770 q^{5} -4.21640 q^{7} +1.28687 q^{9} -6.44307 q^{11} -1.00000 q^{13} -2.66615 q^{15} +7.60944 q^{17} -4.67826 q^{19} -8.72996 q^{21} +6.94581 q^{23} -3.34183 q^{25} -3.54700 q^{27} +1.00000 q^{29} -7.98467 q^{31} -13.3402 q^{33} +5.42946 q^{35} +3.04399 q^{37} -2.07047 q^{39} +6.59888 q^{41} +8.31730 q^{43} -1.65709 q^{45} -0.728070 q^{47} +10.7781 q^{49} +15.7551 q^{51} +1.44109 q^{53} +8.29673 q^{55} -9.68623 q^{57} +14.0002 q^{59} -2.99431 q^{61} -5.42594 q^{63} +1.28770 q^{65} +13.5568 q^{67} +14.3811 q^{69} -0.514661 q^{71} -4.71562 q^{73} -6.91918 q^{75} +27.1666 q^{77} -5.02425 q^{79} -11.2046 q^{81} +6.53628 q^{83} -9.79867 q^{85} +2.07047 q^{87} +9.98400 q^{89} +4.21640 q^{91} -16.5321 q^{93} +6.02420 q^{95} -16.3973 q^{97} -8.29136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more