LMFDB - Embedded newform 6004.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22742 q^{3} +2.43630 q^{5} -2.42654 q^{7} +7.41621 q^{9} +O(q^{10})$	$q-3.22742 q^{3} +2.43630 q^{5} -2.42654 q^{7} +7.41621 q^{9} +4.13185 q^{11} -6.00645 q^{13} -7.86295 q^{15} +2.79598 q^{17} -1.00000 q^{19} +7.83146 q^{21} +0.561526 q^{23} +0.935550 q^{25} -14.2529 q^{27} +6.38727 q^{29} -2.28821 q^{31} -13.3352 q^{33} -5.91178 q^{35} +8.62184 q^{37} +19.3853 q^{39} -5.14036 q^{41} +0.407117 q^{43} +18.0681 q^{45} +1.84353 q^{47} -1.11190 q^{49} -9.02380 q^{51} +1.53824 q^{53} +10.0664 q^{55} +3.22742 q^{57} +2.20768 q^{59} +10.7785 q^{61} -17.9957 q^{63} -14.6335 q^{65} -9.26213 q^{67} -1.81228 q^{69} -2.66414 q^{71} +10.8888 q^{73} -3.01941 q^{75} -10.0261 q^{77} -1.00000 q^{79} +23.7516 q^{81} +3.29227 q^{83} +6.81185 q^{85} -20.6144 q^{87} +7.49553 q^{89} +14.5749 q^{91} +7.38499 q^{93} -2.43630 q^{95} -10.0965 q^{97} +30.6427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−3.22742	−1.86335	−0.931675	−	0.363294i	$-0.881652\pi$
$3$	−3.22742	−1.86335	−0.931675	+	0.363294i	$0.881652\pi$
$4$	0	0
$5$	2.43630	1.08955	0.544773	−	0.838584i	$-0.316616\pi$
$5$	2.43630	1.08955	0.544773	+	0.838584i	$0.316616\pi$
$6$	0	0
$7$	−2.42654	−0.917146	−0.458573	−	0.888657i	$-0.651639\pi$
$7$	−2.42654	−0.917146	−0.458573	+	0.888657i	$0.651639\pi$
$8$	0	0
$9$	7.41621	2.47207
$10$	0	0
$11$	4.13185	1.24580	0.622900	−	0.782302i	$-0.285954\pi$
$11$	4.13185	1.24580	0.622900	+	0.782302i	$0.285954\pi$
$12$	0	0
$13$	−6.00645	−1.66589	−0.832945	−	0.553356i	$-0.813347\pi$
$13$	−6.00645	−1.66589	−0.832945	+	0.553356i	$0.813347\pi$
$14$	0	0
$15$	−7.86295	−2.03020
$16$	0	0
$17$	2.79598	0.678125	0.339063	−	0.940764i	$-0.389890\pi$
$17$	2.79598	0.678125	0.339063	+	0.940764i	$0.389890\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.83146	1.70896
$22$	0	0
$23$	0.561526	0.117086	0.0585432	−	0.998285i	$-0.481354\pi$
$23$	0.561526	0.117086	0.0585432	+	0.998285i	$0.481354\pi$
$24$	0	0
$25$	0.935550	0.187110
$26$	0	0
$27$	−14.2529	−2.74298
$28$	0	0
$29$	6.38727	1.18609	0.593043	−	0.805170i	$-0.297926\pi$
$29$	6.38727	1.18609	0.593043	+	0.805170i	$0.297926\pi$
$30$	0	0
$31$	−2.28821	−0.410974	−0.205487	−	0.978660i	$-0.565878\pi$
$31$	−2.28821	−0.410974	−0.205487	+	0.978660i	$0.565878\pi$
$32$	0	0
$33$	−13.3352	−2.32136
$34$	0	0
$35$	−5.91178	−0.999273
$36$	0	0
$37$	8.62184	1.41742	0.708711	−	0.705499i	$-0.249277\pi$
$37$	8.62184	1.41742	0.708711	+	0.705499i	$0.249277\pi$
$38$	0	0
$39$	19.3853	3.10414
$40$	0	0
$41$	−5.14036	−0.802790	−0.401395	−	0.915905i	$-0.631475\pi$
$41$	−5.14036	−0.802790	−0.401395	+	0.915905i	$0.631475\pi$
$42$	0	0
$43$	0.407117	0.0620848	0.0310424	−	0.999518i	$-0.490117\pi$
$43$	0.407117	0.0620848	0.0310424	+	0.999518i	$0.490117\pi$
$44$	0	0
$45$	18.0681	2.69343
$46$	0	0
$47$	1.84353	0.268906	0.134453	−	0.990920i	$-0.457072\pi$
$47$	1.84353	0.268906	0.134453	+	0.990920i	$0.457072\pi$
$48$	0	0
$49$	−1.11190	−0.158842
$50$	0	0
$51$	−9.02380	−1.26358
$52$	0	0
$53$	1.53824	0.211294	0.105647	−	0.994404i	$-0.466309\pi$
$53$	1.53824	0.211294	0.105647	+	0.994404i	$0.466309\pi$
$54$	0	0
$55$	10.0664	1.35736
$56$	0	0
$57$	3.22742	0.427482
$58$	0	0
$59$	2.20768	0.287415	0.143708	−	0.989620i	$-0.454097\pi$
$59$	2.20768	0.287415	0.143708	+	0.989620i	$0.454097\pi$
$60$	0	0
$61$	10.7785	1.38005	0.690024	−	0.723786i	$-0.257600\pi$
$61$	10.7785	1.38005	0.690024	+	0.723786i	$0.257600\pi$
$62$	0	0
$63$	−17.9957	−2.26725
$64$	0	0
$65$	−14.6335	−1.81506
$66$	0	0
$67$	−9.26213	−1.13155	−0.565775	−	0.824560i	$-0.691423\pi$
$67$	−9.26213	−1.13155	−0.565775	+	0.824560i	$0.691423\pi$
$68$	0	0
$69$	−1.81228	−0.218173
$70$	0	0
$71$	−2.66414	−0.316176	−0.158088	−	0.987425i	$-0.550533\pi$
$71$	−2.66414	−0.316176	−0.158088	+	0.987425i	$0.550533\pi$
$72$	0	0
$73$	10.8888	1.27443	0.637217	−	0.770684i	$-0.280086\pi$
$73$	10.8888	1.27443	0.637217	+	0.770684i	$0.280086\pi$
$74$	0	0
$75$	−3.01941	−0.348651
$76$	0	0
$77$	−10.0261	−1.14258
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	23.7516	2.63906
$82$	0	0
$83$	3.29227	0.361373	0.180687	−	0.983541i	$-0.442168\pi$
$83$	3.29227	0.361373	0.180687	+	0.983541i	$0.442168\pi$
$84$	0	0
$85$	6.81185	0.738849
$86$	0	0
$87$	−20.6144	−2.21009
$88$	0	0
$89$	7.49553	0.794524	0.397262	−	0.917705i	$-0.369960\pi$
$89$	7.49553	0.794524	0.397262	+	0.917705i	$0.369960\pi$
$90$	0	0
$91$	14.5749	1.52787
$92$	0	0
$93$	7.38499	0.765788
$94$	0	0
$95$	−2.43630	−0.249959
$96$	0	0
$97$	−10.0965	−1.02515	−0.512573	−	0.858644i	$-0.671308\pi$
$97$	−10.0965	−1.02515	−0.512573	+	0.858644i	$0.671308\pi$
$98$	0	0
$99$	30.6427	3.07970
$100$	0	0
$101$	−0.277735	−0.0276357	−0.0138178	−	0.999905i	$-0.504398\pi$
$101$	−0.277735	−0.0276357	−0.0138178	+	0.999905i	$0.504398\pi$
$102$	0	0
$103$	−12.5928	−1.24080	−0.620400	−	0.784285i	$-0.713030\pi$
$103$	−12.5928	−1.24080	−0.620400	+	0.784285i	$0.713030\pi$
$104$	0	0
$105$	19.0798	1.86199
$106$	0	0
$107$	−12.0963	−1.16939	−0.584697	−	0.811252i	$-0.698787\pi$
$107$	−12.0963	−1.16939	−0.584697	+	0.811252i	$0.698787\pi$
$108$	0	0
$109$	−3.27078	−0.313284	−0.156642	−	0.987655i	$-0.550067\pi$
$109$	−3.27078	−0.313284	−0.156642	+	0.987655i	$0.550067\pi$
$110$	0	0
$111$	−27.8263	−2.64115
$112$	0	0
$113$	11.6378	1.09479	0.547395	−	0.836875i	$-0.315620\pi$
$113$	11.6378	1.09479	0.547395	+	0.836875i	$0.315620\pi$
$114$	0	0
$115$	1.36805	0.127571
$116$	0	0
$117$	−44.5451	−4.11820
$118$	0	0
$119$	−6.78457	−0.621940
$120$	0	0
$121$	6.07219	0.552017
$122$	0	0
$123$	16.5901	1.49588
$124$	0	0
$125$	−9.90221	−0.885681
$126$	0	0
$127$	−2.44656	−0.217097	−0.108548	−	0.994091i	$-0.534620\pi$
$127$	−2.44656	−0.217097	−0.108548	+	0.994091i	$0.534620\pi$
$128$	0	0
$129$	−1.31394	−0.115686
$130$	0	0
$131$	−18.2085	−1.59089	−0.795444	−	0.606027i	$-0.792762\pi$
$131$	−18.2085	−1.59089	−0.795444	+	0.606027i	$0.792762\pi$
$132$	0	0
$133$	2.42654	0.210408
$134$	0	0
$135$	−34.7244	−2.98860
$136$	0	0
$137$	4.41445	0.377152	0.188576	−	0.982059i	$-0.439613\pi$
$137$	4.41445	0.377152	0.188576	+	0.982059i	$0.439613\pi$
$138$	0	0
$139$	−0.757392	−0.0642411	−0.0321206	−	0.999484i	$-0.510226\pi$
$139$	−0.757392	−0.0642411	−0.0321206	+	0.999484i	$0.510226\pi$
$140$	0	0
$141$	−5.94982	−0.501066
$142$	0	0
$143$	−24.8178	−2.07537
$144$	0	0
$145$	15.5613	1.29230
$146$	0	0
$147$	3.58855	0.295979
$148$	0	0
$149$	−4.54307	−0.372183	−0.186091	−	0.982532i	$-0.559582\pi$
$149$	−4.54307	−0.372183	−0.186091	+	0.982532i	$0.559582\pi$
$150$	0	0
$151$	17.0856	1.39041	0.695205	−	0.718812i	$-0.255314\pi$
$151$	17.0856	1.39041	0.695205	+	0.718812i	$0.255314\pi$
$152$	0	0
$153$	20.7356	1.67637
$154$	0	0
$155$	−5.57475	−0.447775
$156$	0	0
$157$	3.34688	0.267110	0.133555	−	0.991041i	$-0.457361\pi$
$157$	3.34688	0.267110	0.133555	+	0.991041i	$0.457361\pi$
$158$	0	0
$159$	−4.96454	−0.393714
$160$	0	0
$161$	−1.36257	−0.107385
$162$	0	0
$163$	1.94673	0.152480	0.0762398	−	0.997090i	$-0.475709\pi$
$163$	1.94673	0.152480	0.0762398	+	0.997090i	$0.475709\pi$
$164$	0	0
$165$	−32.4885	−2.52923
$166$	0	0
$167$	−1.50916	−0.116783	−0.0583913	−	0.998294i	$-0.518597\pi$
$167$	−1.50916	−0.116783	−0.0583913	+	0.998294i	$0.518597\pi$
$168$	0	0
$169$	23.0775	1.77519
$170$	0	0
$171$	−7.41621	−0.567132
$172$	0	0
$173$	4.41996	0.336043	0.168022	−	0.985783i	$-0.446262\pi$
$173$	4.41996	0.336043	0.168022	+	0.985783i	$0.446262\pi$
$174$	0	0
$175$	−2.27015	−0.171607
$176$	0	0
$177$	−7.12510	−0.535555
$178$	0	0
$179$	23.4947	1.75608	0.878038	−	0.478591i	$-0.158852\pi$
$179$	23.4947	1.75608	0.878038	+	0.478591i	$0.158852\pi$
$180$	0	0
$181$	−1.98233	−0.147345	−0.0736727	−	0.997282i	$-0.523472\pi$
$181$	−1.98233	−0.147345	−0.0736727	+	0.997282i	$0.523472\pi$
$182$	0	0
$183$	−34.7868	−2.57151
$184$	0	0
$185$	21.0054	1.54435
$186$	0	0
$187$	11.5526	0.844808
$188$	0	0
$189$	34.5854	2.51572
$190$	0	0
$191$	−5.36219	−0.387995	−0.193997	−	0.981002i	$-0.562145\pi$
$191$	−5.36219	−0.387995	−0.193997	+	0.981002i	$0.562145\pi$
$192$	0	0
$193$	1.83305	0.131946	0.0659729	−	0.997821i	$-0.478985\pi$
$193$	1.83305	0.131946	0.0659729	+	0.997821i	$0.478985\pi$
$194$	0	0
$195$	47.2284	3.38210
$196$	0	0
$197$	−1.04215	−0.0742499	−0.0371249	−	0.999311i	$-0.511820\pi$
$197$	−1.04215	−0.0742499	−0.0371249	+	0.999311i	$0.511820\pi$
$198$	0	0
$199$	9.89605	0.701512	0.350756	−	0.936467i	$-0.385925\pi$
$199$	9.89605	0.701512	0.350756	+	0.936467i	$0.385925\pi$
$200$	0	0
$201$	29.8927	2.10847
$202$	0	0
$203$	−15.4990	−1.08782
$204$	0	0
$205$	−12.5235	−0.874676
$206$	0	0
$207$	4.16440	0.289446
$208$	0	0
$209$	−4.13185	−0.285806
$210$	0	0
$211$	26.7103	1.83882	0.919408	−	0.393306i	$-0.128669\pi$
$211$	26.7103	1.83882	0.919408	+	0.393306i	$0.128669\pi$
$212$	0	0
$213$	8.59830	0.589146
$214$	0	0
$215$	0.991859	0.0676443
$216$	0	0
$217$	5.55243	0.376923
$218$	0	0
$219$	−35.1426	−2.37472
$220$	0	0
$221$	−16.7939	−1.12968
$222$	0	0
$223$	13.4870	0.903156	0.451578	−	0.892232i	$-0.350861\pi$
$223$	13.4870	0.903156	0.451578	+	0.892232i	$0.350861\pi$
$224$	0	0
$225$	6.93823	0.462549
$226$	0	0
$227$	−27.1562	−1.80242	−0.901211	−	0.433381i	$-0.857321\pi$
$227$	−27.1562	−1.80242	−0.901211	+	0.433381i	$0.857321\pi$
$228$	0	0
$229$	−26.5325	−1.75331	−0.876657	−	0.481116i	$-0.840232\pi$
$229$	−26.5325	−1.75331	−0.876657	+	0.481116i	$0.840232\pi$
$230$	0	0
$231$	32.3584	2.12903
$232$	0	0
$233$	8.12698	0.532416	0.266208	−	0.963916i	$-0.414229\pi$
$233$	8.12698	0.532416	0.266208	+	0.963916i	$0.414229\pi$
$234$	0	0
$235$	4.49138	0.292985
$236$	0	0
$237$	3.22742	0.209643
$238$	0	0
$239$	−5.67917	−0.367355	−0.183677	−	0.982987i	$-0.558800\pi$
$239$	−5.67917	−0.367355	−0.183677	+	0.982987i	$0.558800\pi$
$240$	0	0
$241$	20.5178	1.32167	0.660835	−	0.750531i	$-0.270202\pi$
$241$	20.5178	1.32167	0.660835	+	0.750531i	$0.270202\pi$
$242$	0	0
$243$	−33.8973	−2.17451
$244$	0	0
$245$	−2.70891	−0.173066
$246$	0	0
$247$	6.00645	0.382181
$248$	0	0
$249$	−10.6255	−0.673365
$250$	0	0
$251$	3.57112	0.225407	0.112704	−	0.993629i	$-0.464049\pi$
$251$	3.57112	0.225407	0.112704	+	0.993629i	$0.464049\pi$
$252$	0	0
$253$	2.32014	0.145866
$254$	0	0
$255$	−21.9847	−1.37673
$256$	0	0
$257$	−10.2190	−0.637446	−0.318723	−	0.947848i	$-0.603254\pi$
$257$	−10.2190	−0.637446	−0.318723	+	0.947848i	$0.603254\pi$
$258$	0	0
$259$	−20.9213	−1.29998
$260$	0	0
$261$	47.3694	2.93209
$262$	0	0
$263$	3.25540	0.200736	0.100368	−	0.994950i	$-0.467998\pi$
$263$	3.25540	0.200736	0.100368	+	0.994950i	$0.467998\pi$
$264$	0	0
$265$	3.74761	0.230214
$266$	0	0
$267$	−24.1912	−1.48048
$268$	0	0
$269$	−2.26071	−0.137838	−0.0689190	−	0.997622i	$-0.521955\pi$
$269$	−2.26071	−0.137838	−0.0689190	+	0.997622i	$0.521955\pi$
$270$	0	0
$271$	20.5642	1.24919	0.624594	−	0.780950i	$-0.285264\pi$
$271$	20.5642	1.24919	0.624594	+	0.780950i	$0.285264\pi$
$272$	0	0
$273$	−47.0393	−2.84695
$274$	0	0
$275$	3.86555	0.233102
$276$	0	0
$277$	17.8153	1.07042	0.535210	−	0.844719i	$-0.320233\pi$
$277$	17.8153	1.07042	0.535210	+	0.844719i	$0.320233\pi$
$278$	0	0
$279$	−16.9698	−1.01596
$280$	0	0
$281$	−25.3604	−1.51287	−0.756436	−	0.654067i	$-0.773061\pi$
$281$	−25.3604	−1.51287	−0.756436	+	0.654067i	$0.773061\pi$
$282$	0	0
$283$	−23.2694	−1.38322	−0.691611	−	0.722270i	$-0.743099\pi$
$283$	−23.2694	−1.38322	−0.691611	+	0.722270i	$0.743099\pi$
$284$	0	0
$285$	7.86295	0.465761
$286$	0	0
$287$	12.4733	0.736276
$288$	0	0
$289$	−9.18248	−0.540146
$290$	0	0
$291$	32.5857	1.91020
$292$	0	0
$293$	18.5633	1.08448	0.542240	−	0.840224i	$-0.317576\pi$
$293$	18.5633	1.08448	0.542240	+	0.840224i	$0.317576\pi$
$294$	0	0
$295$	5.37857	0.313152
$296$	0	0
$297$	−58.8911	−3.41721
$298$	0	0
$299$	−3.37278	−0.195053
$300$	0	0
$301$	−0.987887	−0.0569409
$302$	0	0
$303$	0.896367	0.0514949
$304$	0	0
$305$	26.2597	1.50363
$306$	0	0
$307$	28.7269	1.63953	0.819764	−	0.572701i	$-0.194105\pi$
$307$	28.7269	1.63953	0.819764	+	0.572701i	$0.194105\pi$
$308$	0	0
$309$	40.6420	2.31205
$310$	0	0
$311$	32.4842	1.84201	0.921005	−	0.389551i	$-0.127370\pi$
$311$	32.4842	1.84201	0.921005	+	0.389551i	$0.127370\pi$
$312$	0	0
$313$	11.8857	0.671818	0.335909	−	0.941894i	$-0.390957\pi$
$313$	11.8857	0.671818	0.335909	+	0.941894i	$0.390957\pi$
$314$	0	0
$315$	−43.8430	−2.47027
$316$	0	0
$317$	−12.8132	−0.719658	−0.359829	−	0.933018i	$-0.617165\pi$
$317$	−12.8132	−0.719658	−0.359829	+	0.933018i	$0.617165\pi$
$318$	0	0
$319$	26.3913	1.47763
$320$	0	0
$321$	39.0398	2.17899
$322$	0	0
$323$	−2.79598	−0.155573
$324$	0	0
$325$	−5.61934	−0.311705
$326$	0	0
$327$	10.5562	0.583758
$328$	0	0
$329$	−4.47339	−0.246626
$330$	0	0
$331$	23.6501	1.29993	0.649963	−	0.759965i	$-0.274784\pi$
$331$	23.6501	1.29993	0.649963	+	0.759965i	$0.274784\pi$
$332$	0	0
$333$	63.9414	3.50397
$334$	0	0
$335$	−22.5653	−1.23287
$336$	0	0
$337$	6.32644	0.344623	0.172312	−	0.985043i	$-0.444876\pi$
$337$	6.32644	0.344623	0.172312	+	0.985043i	$0.444876\pi$
$338$	0	0
$339$	−37.5599	−2.03997
$340$	0	0
$341$	−9.45453	−0.511991
$342$	0	0
$343$	19.6839	1.06283
$344$	0	0
$345$	−4.41525	−0.237709
$346$	0	0
$347$	−31.6636	−1.69979	−0.849897	−	0.526949i	$-0.823336\pi$
$347$	−31.6636	−1.69979	−0.849897	+	0.526949i	$0.823336\pi$
$348$	0	0
$349$	15.2973	0.818845	0.409423	−	0.912345i	$-0.365730\pi$
$349$	15.2973	0.818845	0.409423	+	0.912345i	$0.365730\pi$
$350$	0	0
$351$	85.6097	4.56951
$352$	0	0
$353$	8.56113	0.455663	0.227832	−	0.973701i	$-0.426836\pi$
$353$	8.56113	0.455663	0.227832	+	0.973701i	$0.426836\pi$
$354$	0	0
$355$	−6.49065	−0.344488
$356$	0	0
$357$	21.8966	1.15889
$358$	0	0
$359$	35.5625	1.87691	0.938457	−	0.345395i	$-0.112255\pi$
$359$	35.5625	1.87691	0.938457	+	0.345395i	$0.112255\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−19.5975	−1.02860
$364$	0	0
$365$	26.5283	1.38855
$366$	0	0
$367$	16.2433	0.847891	0.423945	−	0.905688i	$-0.360645\pi$
$367$	16.2433	0.847891	0.423945	+	0.905688i	$0.360645\pi$
$368$	0	0
$369$	−38.1220	−1.98455
$370$	0	0
$371$	−3.73260	−0.193787
$372$	0	0
$373$	−27.5573	−1.42686	−0.713432	−	0.700724i	$-0.752860\pi$
$373$	−27.5573	−1.42686	−0.713432	+	0.700724i	$0.752860\pi$
$374$	0	0
$375$	31.9586	1.65033
$376$	0	0
$377$	−38.3649	−1.97589
$378$	0	0
$379$	10.2541	0.526718	0.263359	−	0.964698i	$-0.415170\pi$
$379$	10.2541	0.526718	0.263359	+	0.964698i	$0.415170\pi$
$380$	0	0
$381$	7.89606	0.404527
$382$	0	0
$383$	−9.53209	−0.487067	−0.243533	−	0.969893i	$-0.578307\pi$
$383$	−9.53209	−0.487067	−0.243533	+	0.969893i	$0.578307\pi$
$384$	0	0
$385$	−24.4266	−1.24489
$386$	0	0
$387$	3.01927	0.153478
$388$	0	0
$389$	10.4211	0.528369	0.264185	−	0.964472i	$-0.414897\pi$
$389$	10.4211	0.528369	0.264185	+	0.964472i	$0.414897\pi$
$390$	0	0
$391$	1.57002	0.0793992
$392$	0	0
$393$	58.7666	2.96438
$394$	0	0
$395$	−2.43630	−0.122583
$396$	0	0
$397$	19.7583	0.991643	0.495821	−	0.868425i	$-0.334867\pi$
$397$	19.7583	0.991643	0.495821	+	0.868425i	$0.334867\pi$
$398$	0	0
$399$	−7.83146	−0.392063
$400$	0	0
$401$	23.8482	1.19092	0.595460	−	0.803385i	$-0.296970\pi$
$401$	23.8482	1.19092	0.595460	+	0.803385i	$0.296970\pi$
$402$	0	0
$403$	13.7440	0.684638
$404$	0	0
$405$	57.8659	2.87538
$406$	0	0
$407$	35.6242	1.76582
$408$	0	0
$409$	35.4696	1.75386	0.876930	−	0.480618i	$-0.159588\pi$
$409$	35.4696	1.75386	0.876930	+	0.480618i	$0.159588\pi$
$410$	0	0
$411$	−14.2473	−0.702765
$412$	0	0
$413$	−5.35703	−0.263602
$414$	0	0
$415$	8.02095	0.393733
$416$	0	0
$417$	2.44442	0.119704
$418$	0	0
$419$	13.3868	0.653988	0.326994	−	0.945026i	$-0.393964\pi$
$419$	13.3868	0.653988	0.326994	+	0.945026i	$0.393964\pi$
$420$	0	0
$421$	13.1976	0.643212	0.321606	−	0.946874i	$-0.395777\pi$
$421$	13.1976	0.643212	0.321606	+	0.946874i	$0.395777\pi$
$422$	0	0
$423$	13.6720	0.664754
$424$	0	0
$425$	2.61578	0.126884
$426$	0	0
$427$	−26.1545	−1.26571
$428$	0	0
$429$	80.0972	3.86713
$430$	0	0
$431$	5.50527	0.265179	0.132590	−	0.991171i	$-0.457671\pi$
$431$	5.50527	0.265179	0.132590	+	0.991171i	$0.457671\pi$
$432$	0	0
$433$	7.55832	0.363229	0.181615	−	0.983370i	$-0.441868\pi$
$433$	7.55832	0.363229	0.181615	+	0.983370i	$0.441868\pi$
$434$	0	0
$435$	−50.2228	−2.40800
$436$	0	0
$437$	−0.561526	−0.0268614
$438$	0	0
$439$	−22.6438	−1.08073	−0.540364	−	0.841431i	$-0.681714\pi$
$439$	−22.6438	−1.08073	−0.540364	+	0.841431i	$0.681714\pi$
$440$	0	0
$441$	−8.24606	−0.392670
$442$	0	0
$443$	34.3538	1.63220	0.816099	−	0.577912i	$-0.196132\pi$
$443$	34.3538	1.63220	0.816099	+	0.577912i	$0.196132\pi$
$444$	0	0
$445$	18.2613	0.865671
$446$	0	0
$447$	14.6624	0.693507
$448$	0	0
$449$	30.0122	1.41636	0.708182	−	0.706030i	$-0.249516\pi$
$449$	30.0122	1.41636	0.708182	+	0.706030i	$0.249516\pi$
$450$	0	0
$451$	−21.2392	−1.00012
$452$	0	0
$453$	−55.1424	−2.59082
$454$	0	0
$455$	35.5088	1.66468
$456$	0	0
$457$	39.4655	1.84612	0.923059	−	0.384659i	$-0.125681\pi$
$457$	39.4655	1.84612	0.923059	+	0.384659i	$0.125681\pi$
$458$	0	0
$459$	−39.8510	−1.86009
$460$	0	0
$461$	16.7465	0.779964	0.389982	−	0.920822i	$-0.372481\pi$
$461$	16.7465	0.779964	0.389982	+	0.920822i	$0.372481\pi$
$462$	0	0
$463$	1.16793	0.0542784	0.0271392	−	0.999632i	$-0.491360\pi$
$463$	1.16793	0.0542784	0.0271392	+	0.999632i	$0.491360\pi$
$464$	0	0
$465$	17.9921	0.834361
$466$	0	0
$467$	13.4682	0.623235	0.311618	−	0.950208i	$-0.399129\pi$
$467$	13.4682	0.623235	0.311618	+	0.950208i	$0.399129\pi$
$468$	0	0
$469$	22.4749	1.03780
$470$	0	0
$471$	−10.8018	−0.497719
$472$	0	0
$473$	1.68215	0.0773452
$474$	0	0
$475$	−0.935550	−0.0429260
$476$	0	0
$477$	11.4079	0.522333
$478$	0	0
$479$	−16.3855	−0.748672	−0.374336	−	0.927293i	$-0.622129\pi$
$479$	−16.3855	−0.748672	−0.374336	+	0.927293i	$0.622129\pi$
$480$	0	0
$481$	−51.7867	−2.36127
$482$	0	0
$483$	4.39757	0.200096
$484$	0	0
$485$	−24.5981	−1.11694
$486$	0	0
$487$	−38.9603	−1.76546	−0.882730	−	0.469881i	$-0.844297\pi$
$487$	−38.9603	−1.76546	−0.882730	+	0.469881i	$0.844297\pi$
$488$	0	0
$489$	−6.28291	−0.284123
$490$	0	0
$491$	36.1576	1.63177	0.815884	−	0.578216i	$-0.196251\pi$
$491$	36.1576	1.63177	0.815884	+	0.578216i	$0.196251\pi$
$492$	0	0
$493$	17.8587	0.804316
$494$	0	0
$495$	74.6547	3.35548
$496$	0	0
$497$	6.46465	0.289979
$498$	0	0
$499$	−7.61814	−0.341035	−0.170517	−	0.985355i	$-0.554544\pi$
$499$	−7.61814	−0.341035	−0.170517	+	0.985355i	$0.554544\pi$
$500$	0	0
$501$	4.87070	0.217607
$502$	0	0
$503$	0.875550	0.0390389	0.0195194	−	0.999809i	$-0.493786\pi$
$503$	0.875550	0.0390389	0.0195194	+	0.999809i	$0.493786\pi$
$504$	0	0
$505$	−0.676646	−0.0301103
$506$	0	0
$507$	−74.4806	−3.30780
$508$	0	0
$509$	−14.9472	−0.662525	−0.331262	−	0.943539i	$-0.607474\pi$
$509$	−14.9472	−0.662525	−0.331262	+	0.943539i	$0.607474\pi$
$510$	0	0
$511$	−26.4220	−1.16884
$512$	0	0
$513$	14.2529	0.629283
$514$	0	0
$515$	−30.6797	−1.35191
$516$	0	0
$517$	7.61717	0.335003
$518$	0	0
$519$	−14.2651	−0.626166
$520$	0	0
$521$	38.0220	1.66577	0.832887	−	0.553443i	$-0.186686\pi$
$521$	38.0220	1.66577	0.832887	+	0.553443i	$0.186686\pi$
$522$	0	0
$523$	26.7170	1.16825	0.584126	−	0.811663i	$-0.301437\pi$
$523$	26.7170	1.16825	0.584126	+	0.811663i	$0.301437\pi$
$524$	0	0
$525$	7.32672	0.319764
$526$	0	0
$527$	−6.39779	−0.278692
$528$	0	0
$529$	−22.6847	−0.986291
$530$	0	0
$531$	16.3726	0.710511
$532$	0	0
$533$	30.8754	1.33736
$534$	0	0
$535$	−29.4702	−1.27411
$536$	0	0
$537$	−75.8271	−3.27218
$538$	0	0
$539$	−4.59419	−0.197886
$540$	0	0
$541$	10.5526	0.453691	0.226846	−	0.973931i	$-0.427159\pi$
$541$	10.5526	0.453691	0.226846	+	0.973931i	$0.427159\pi$
$542$	0	0
$543$	6.39780	0.274556
$544$	0	0
$545$	−7.96861	−0.341338
$546$	0	0
$547$	44.5552	1.90504	0.952522	−	0.304469i	$-0.0984792\pi$
$547$	44.5552	1.90504	0.952522	+	0.304469i	$0.0984792\pi$
$548$	0	0
$549$	79.9358	3.41158
$550$	0	0
$551$	−6.38727	−0.272107
$552$	0	0
$553$	2.42654	0.103187
$554$	0	0
$555$	−67.7931	−2.87766
$556$	0	0
$557$	21.5940	0.914966	0.457483	−	0.889218i	$-0.348751\pi$
$557$	21.5940	0.914966	0.457483	+	0.889218i	$0.348751\pi$
$558$	0	0
$559$	−2.44533	−0.103427
$560$	0	0
$561$	−37.2850	−1.57417
$562$	0	0
$563$	8.42790	0.355193	0.177597	−	0.984103i	$-0.443168\pi$
$563$	8.42790	0.355193	0.177597	+	0.984103i	$0.443168\pi$
$564$	0	0
$565$	28.3531	1.19282
$566$	0	0
$567$	−57.6341	−2.42041
$568$	0	0
$569$	0.737890	0.0309339	0.0154670	−	0.999880i	$-0.495077\pi$
$569$	0.737890	0.0309339	0.0154670	+	0.999880i	$0.495077\pi$
$570$	0	0
$571$	17.9191	0.749889	0.374944	−	0.927047i	$-0.377662\pi$
$571$	17.9191	0.749889	0.374944	+	0.927047i	$0.377662\pi$
$572$	0	0
$573$	17.3060	0.722970
$574$	0	0
$575$	0.525336	0.0219080
$576$	0	0
$577$	−2.72124	−0.113286	−0.0566432	−	0.998394i	$-0.518040\pi$
$577$	−2.72124	−0.113286	−0.0566432	+	0.998394i	$0.518040\pi$
$578$	0	0
$579$	−5.91601	−0.245861
$580$	0	0
$581$	−7.98882	−0.331432
$582$	0	0
$583$	6.35578	0.263229
$584$	0	0
$585$	−108.525	−4.48697
$586$	0	0
$587$	−42.3525	−1.74807	−0.874037	−	0.485859i	$-0.838507\pi$
$587$	−42.3525	−1.74807	−0.874037	+	0.485859i	$0.838507\pi$
$588$	0	0
$589$	2.28821	0.0942839
$590$	0	0
$591$	3.36344	0.138353
$592$	0	0
$593$	26.6661	1.09504	0.547522	−	0.836791i	$-0.315571\pi$
$593$	26.6661	1.09504	0.547522	+	0.836791i	$0.315571\pi$
$594$	0	0
$595$	−16.5292	−0.677632
$596$	0	0
$597$	−31.9387	−1.30716
$598$	0	0
$599$	18.1982	0.743558	0.371779	−	0.928321i	$-0.378748\pi$
$599$	18.1982	0.743558	0.371779	+	0.928321i	$0.378748\pi$
$600$	0	0
$601$	−31.9838	−1.30465	−0.652323	−	0.757941i	$-0.726205\pi$
$601$	−31.9838	−1.30465	−0.652323	+	0.757941i	$0.726205\pi$
$602$	0	0
$603$	−68.6899	−2.79727
$604$	0	0
$605$	14.7937	0.601448
$606$	0	0
$607$	−33.2304	−1.34878	−0.674390	−	0.738375i	$-0.735593\pi$
$607$	−33.2304	−1.34878	−0.674390	+	0.738375i	$0.735593\pi$
$608$	0	0
$609$	50.0217	2.02698
$610$	0	0
$611$	−11.0731	−0.447968
$612$	0	0
$613$	−20.6479	−0.833961	−0.416980	−	0.908916i	$-0.636912\pi$
$613$	−20.6479	−0.833961	−0.416980	+	0.908916i	$0.636912\pi$
$614$	0	0
$615$	40.4184	1.62983
$616$	0	0
$617$	−24.6281	−0.991489	−0.495745	−	0.868468i	$-0.665105\pi$
$617$	−24.6281	−0.991489	−0.495745	+	0.868468i	$0.665105\pi$
$618$	0	0
$619$	−41.2667	−1.65865	−0.829324	−	0.558767i	$-0.811274\pi$
$619$	−41.2667	−1.65865	−0.829324	+	0.558767i	$0.811274\pi$
$620$	0	0
$621$	−8.00341	−0.321166
$622$	0	0
$623$	−18.1882	−0.728695
$624$	0	0
$625$	−28.8025	−1.15210
$626$	0	0
$627$	13.3352	0.532556
$628$	0	0
$629$	24.1065	0.961190
$630$	0	0
$631$	43.2655	1.72237	0.861186	−	0.508290i	$-0.169722\pi$
$631$	43.2655	1.72237	0.861186	+	0.508290i	$0.169722\pi$
$632$	0	0
$633$	−86.2054	−3.42636
$634$	0	0
$635$	−5.96055	−0.236537
$636$	0	0
$637$	6.67855	0.264614
$638$	0	0
$639$	−19.7578	−0.781608
$640$	0	0
$641$	−28.8993	−1.14145	−0.570727	−	0.821140i	$-0.693339\pi$
$641$	−28.8993	−1.14145	−0.570727	+	0.821140i	$0.693339\pi$
$642$	0	0
$643$	−49.0956	−1.93614	−0.968071	−	0.250678i	$-0.919347\pi$
$643$	−49.0956	−1.93614	−0.968071	+	0.250678i	$0.919347\pi$
$644$	0	0
$645$	−3.20114	−0.126045
$646$	0	0
$647$	5.66014	0.222523	0.111262	−	0.993791i	$-0.464511\pi$
$647$	5.66014	0.222523	0.111262	+	0.993791i	$0.464511\pi$
$648$	0	0
$649$	9.12180	0.358062
$650$	0	0
$651$	−17.9200	−0.702340
$652$	0	0
$653$	−24.4574	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$653$	−24.4574	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$654$	0	0
$655$	−44.3615	−1.73335
$656$	0	0
$657$	80.7534	3.15049
$658$	0	0
$659$	45.1491	1.75876	0.879379	−	0.476122i	$-0.157958\pi$
$659$	45.1491	1.75876	0.879379	+	0.476122i	$0.157958\pi$
$660$	0	0
$661$	−43.1678	−1.67903	−0.839517	−	0.543334i	$-0.817162\pi$
$661$	−43.1678	−1.67903	−0.839517	+	0.543334i	$0.817162\pi$
$662$	0	0
$663$	54.2010	2.10499
$664$	0	0
$665$	5.91178	0.229249
$666$	0	0
$667$	3.58662	0.138875
$668$	0	0
$669$	−43.5282	−1.68290
$670$	0	0
$671$	44.5352	1.71926
$672$	0	0
$673$	29.9023	1.15265	0.576324	−	0.817221i	$-0.304487\pi$
$673$	29.9023	1.15265	0.576324	+	0.817221i	$0.304487\pi$
$674$	0	0
$675$	−13.3343	−0.513239
$676$	0	0
$677$	−38.0161	−1.46108	−0.730539	−	0.682871i	$-0.760731\pi$
$677$	−38.0161	−1.46108	−0.730539	+	0.682871i	$0.760731\pi$
$678$	0	0
$679$	24.4996	0.940209
$680$	0	0
$681$	87.6444	3.35854
$682$	0	0
$683$	−10.5664	−0.404313	−0.202156	−	0.979353i	$-0.564795\pi$
$683$	−10.5664	−0.404313	−0.202156	+	0.979353i	$0.564795\pi$
$684$	0	0
$685$	10.7549	0.410924
$686$	0	0
$687$	85.6313	3.26704
$688$	0	0
$689$	−9.23937	−0.351992
$690$	0	0
$691$	−19.5266	−0.742828	−0.371414	−	0.928467i	$-0.621127\pi$
$691$	−19.5266	−0.742828	−0.371414	+	0.928467i	$0.621127\pi$
$692$	0	0
$693$	−74.3557	−2.82454
$694$	0	0
$695$	−1.84523	−0.0699936
$696$	0	0
$697$	−14.3724	−0.544392
$698$	0	0
$699$	−26.2291	−0.992077
$700$	0	0
$701$	−24.1165	−0.910869	−0.455434	−	0.890269i	$-0.650516\pi$
$701$	−24.1165	−0.910869	−0.455434	+	0.890269i	$0.650516\pi$
$702$	0	0
$703$	−8.62184	−0.325179
$704$	0	0
$705$	−14.4955	−0.545934
$706$	0	0
$707$	0.673936	0.0253460
$708$	0	0
$709$	20.8227	0.782012	0.391006	−	0.920388i	$-0.372127\pi$
$709$	20.8227	0.782012	0.391006	+	0.920388i	$0.372127\pi$
$710$	0	0
$711$	−7.41621	−0.278130
$712$	0	0
$713$	−1.28489	−0.0481195
$714$	0	0
$715$	−60.4635	−2.26121
$716$	0	0
$717$	18.3290	0.684510
$718$	0	0
$719$	18.1349	0.676319	0.338159	−	0.941089i	$-0.390196\pi$
$719$	18.1349	0.676319	0.338159	+	0.941089i	$0.390196\pi$
$720$	0	0
$721$	30.5568	1.13800
$722$	0	0
$723$	−66.2196	−2.46273
$724$	0	0
$725$	5.97561	0.221929
$726$	0	0
$727$	11.4870	0.426029	0.213015	−	0.977049i	$-0.431672\pi$
$727$	11.4870	0.426029	0.213015	+	0.977049i	$0.431672\pi$
$728$	0	0
$729$	38.1460	1.41281
$730$	0	0
$731$	1.13829	0.0421013
$732$	0	0
$733$	21.0626	0.777963	0.388982	−	0.921246i	$-0.372827\pi$
$733$	21.0626	0.777963	0.388982	+	0.921246i	$0.372827\pi$
$734$	0	0
$735$	8.74278	0.322482
$736$	0	0
$737$	−38.2697	−1.40968
$738$	0	0
$739$	−5.16663	−0.190058	−0.0950288	−	0.995475i	$-0.530294\pi$
$739$	−5.16663	−0.190058	−0.0950288	+	0.995475i	$0.530294\pi$
$740$	0	0
$741$	−19.3853	−0.712138
$742$	0	0
$743$	8.85118	0.324718	0.162359	−	0.986732i	$-0.448090\pi$
$743$	8.85118	0.324718	0.162359	+	0.986732i	$0.448090\pi$
$744$	0	0
$745$	−11.0683	−0.405510
$746$	0	0
$747$	24.4162	0.893340
$748$	0	0
$749$	29.3522	1.07251
$750$	0	0
$751$	49.8397	1.81868	0.909339	−	0.416056i	$-0.136588\pi$
$751$	49.8397	1.81868	0.909339	+	0.416056i	$0.136588\pi$
$752$	0	0
$753$	−11.5255	−0.420012
$754$	0	0
$755$	41.6257	1.51491
$756$	0	0
$757$	−4.37221	−0.158911	−0.0794553	−	0.996838i	$-0.525318\pi$
$757$	−4.37221	−0.158911	−0.0794553	+	0.996838i	$0.525318\pi$
$758$	0	0
$759$	−7.48807	−0.271800
$760$	0	0
$761$	23.7911	0.862426	0.431213	−	0.902250i	$-0.358086\pi$
$761$	23.7911	0.862426	0.431213	+	0.902250i	$0.358086\pi$
$762$	0	0
$763$	7.93669	0.287328
$764$	0	0
$765$	50.5181	1.82649
$766$	0	0
$767$	−13.2603	−0.478803
$768$	0	0
$769$	−18.9720	−0.684147	−0.342074	−	0.939673i	$-0.611129\pi$
$769$	−18.9720	−0.684147	−0.342074	+	0.939673i	$0.611129\pi$
$770$	0	0
$771$	32.9811	1.18779
$772$	0	0
$773$	3.97878	0.143107	0.0715534	−	0.997437i	$-0.477204\pi$
$773$	3.97878	0.143107	0.0715534	+	0.997437i	$0.477204\pi$
$774$	0	0
$775$	−2.14073	−0.0768973
$776$	0	0
$777$	67.5216	2.42232
$778$	0	0
$779$	5.14036	0.184173
$780$	0	0
$781$	−11.0078	−0.393892
$782$	0	0
$783$	−91.0375	−3.25341
$784$	0	0
$785$	8.15399	0.291029
$786$	0	0
$787$	−1.67512	−0.0597116	−0.0298558	−	0.999554i	$-0.509505\pi$
$787$	−1.67512	−0.0597116	−0.0298558	+	0.999554i	$0.509505\pi$
$788$	0	0
$789$	−10.5065	−0.374042
$790$	0	0
$791$	−28.2395	−1.00408
$792$	0	0
$793$	−64.7407	−2.29901
$794$	0	0
$795$	−12.0951	−0.428969
$796$	0	0
$797$	30.4464	1.07847	0.539234	−	0.842156i	$-0.318714\pi$
$797$	30.4464	1.07847	0.539234	+	0.842156i	$0.318714\pi$
$798$	0	0
$799$	5.15447	0.182352
$800$	0	0
$801$	55.5884	1.96412
$802$	0	0
$803$	44.9908	1.58769
$804$	0	0
$805$	−3.31962	−0.117001
$806$	0	0
$807$	7.29626	0.256841
$808$	0	0
$809$	8.49109	0.298531	0.149265	−	0.988797i	$-0.452309\pi$
$809$	8.49109	0.298531	0.149265	+	0.988797i	$0.452309\pi$
$810$	0	0
$811$	−5.19152	−0.182299	−0.0911495	−	0.995837i	$-0.529054\pi$
$811$	−5.19152	−0.182299	−0.0911495	+	0.995837i	$0.529054\pi$
$812$	0	0
$813$	−66.3693	−2.32767
$814$	0	0
$815$	4.74281	0.166134
$816$	0	0
$817$	−0.407117	−0.0142432
$818$	0	0
$819$	108.091	3.77699
$820$	0	0
$821$	−37.6349	−1.31347	−0.656733	−	0.754123i	$-0.728062\pi$
$821$	−37.6349	−1.31347	−0.656733	+	0.754123i	$0.728062\pi$
$822$	0	0
$823$	27.1127	0.945089	0.472544	−	0.881307i	$-0.343336\pi$
$823$	27.1127	0.945089	0.472544	+	0.881307i	$0.343336\pi$
$824$	0	0
$825$	−12.4757	−0.434350
$826$	0	0
$827$	−12.1079	−0.421035	−0.210517	−	0.977590i	$-0.567515\pi$
$827$	−12.1079	−0.421035	−0.210517	+	0.977590i	$0.567515\pi$
$828$	0	0
$829$	4.64789	0.161428	0.0807140	−	0.996737i	$-0.474280\pi$
$829$	4.64789	0.161428	0.0807140	+	0.996737i	$0.474280\pi$
$830$	0	0
$831$	−57.4975	−1.99456
$832$	0	0
$833$	−3.10884	−0.107715
$834$	0	0
$835$	−3.67678	−0.127240
$836$	0	0
$837$	32.6137	1.12729
$838$	0	0
$839$	36.3070	1.25346	0.626729	−	0.779237i	$-0.284393\pi$
$839$	36.3070	1.25346	0.626729	+	0.779237i	$0.284393\pi$
$840$	0	0
$841$	11.7973	0.406802
$842$	0	0
$843$	81.8485	2.81901
$844$	0	0
$845$	56.2236	1.93415
$846$	0	0
$847$	−14.7344	−0.506280
$848$	0	0
$849$	75.1000	2.57743
$850$	0	0
$851$	4.84139	0.165961
$852$	0	0
$853$	−29.0258	−0.993826	−0.496913	−	0.867800i	$-0.665533\pi$
$853$	−29.0258	−0.993826	−0.496913	+	0.867800i	$0.665533\pi$
$854$	0	0
$855$	−18.0681	−0.617916
$856$	0	0
$857$	−55.2142	−1.88608	−0.943040	−	0.332678i	$-0.892048\pi$
$857$	−55.2142	−1.88608	−0.943040	+	0.332678i	$0.892048\pi$
$858$	0	0
$859$	46.8935	1.59999	0.799993	−	0.600009i	$-0.204836\pi$
$859$	46.8935	1.59999	0.799993	+	0.600009i	$0.204836\pi$
$860$	0	0
$861$	−40.2565	−1.37194
$862$	0	0
$863$	−11.2197	−0.381922	−0.190961	−	0.981598i	$-0.561160\pi$
$863$	−11.2197	−0.381922	−0.190961	+	0.981598i	$0.561160\pi$
$864$	0	0
$865$	10.7683	0.366135
$866$	0	0
$867$	29.6357	1.00648
$868$	0	0
$869$	−4.13185	−0.140163
$870$	0	0
$871$	55.6325	1.88504
$872$	0	0
$873$	−74.8779	−2.53423
$874$	0	0
$875$	24.0281	0.812299
$876$	0	0
$877$	30.7936	1.03983	0.519914	−	0.854219i	$-0.325964\pi$
$877$	30.7936	1.03983	0.519914	+	0.854219i	$0.325964\pi$
$878$	0	0
$879$	−59.9115	−2.02076
$880$	0	0
$881$	−24.9879	−0.841865	−0.420933	−	0.907092i	$-0.638297\pi$
$881$	−24.9879	−0.841865	−0.420933	+	0.907092i	$0.638297\pi$
$882$	0	0
$883$	−30.9113	−1.04025	−0.520125	−	0.854090i	$-0.674115\pi$
$883$	−30.9113	−1.04025	−0.520125	+	0.854090i	$0.674115\pi$
$884$	0	0
$885$	−17.3589	−0.583512
$886$	0	0
$887$	−10.6985	−0.359221	−0.179611	−	0.983738i	$-0.557484\pi$
$887$	−10.6985	−0.359221	−0.179611	+	0.983738i	$0.557484\pi$
$888$	0	0
$889$	5.93667	0.199110
$890$	0	0
$891$	98.1379	3.28774
$892$	0	0
$893$	−1.84353	−0.0616912
$894$	0	0
$895$	57.2401	1.91332
$896$	0	0
$897$	10.8854	0.363452
$898$	0	0
$899$	−14.6154	−0.487451
$900$	0	0
$901$	4.30089	0.143284
$902$	0	0
$903$	3.18832	0.106101
$904$	0	0
$905$	−4.82955	−0.160540
$906$	0	0
$907$	2.86204	0.0950326	0.0475163	−	0.998870i	$-0.484869\pi$
$907$	2.86204	0.0950326	0.0475163	+	0.998870i	$0.484869\pi$
$908$	0	0
$909$	−2.05974	−0.0683174
$910$	0	0
$911$	18.9774	0.628751	0.314375	−	0.949299i	$-0.398205\pi$
$911$	18.9774	0.628751	0.314375	+	0.949299i	$0.398205\pi$
$912$	0	0
$913$	13.6032	0.450199
$914$	0	0
$915$	−84.7509	−2.80178
$916$	0	0
$917$	44.1838	1.45908
$918$	0	0
$919$	−6.86987	−0.226616	−0.113308	−	0.993560i	$-0.536145\pi$
$919$	−6.86987	−0.226616	−0.113308	+	0.993560i	$0.536145\pi$
$920$	0	0
$921$	−92.7135	−3.05501
$922$	0	0
$923$	16.0021	0.526714
$924$	0	0
$925$	8.06616	0.265214
$926$	0	0
$927$	−93.3905	−3.06735
$928$	0	0
$929$	−45.2017	−1.48302	−0.741510	−	0.670942i	$-0.765890\pi$
$929$	−45.2017	−1.48302	−0.741510	+	0.670942i	$0.765890\pi$
$930$	0	0
$931$	1.11190	0.0364409
$932$	0	0
$933$	−104.840	−3.43231
$934$	0	0
$935$	28.1455	0.920457
$936$	0	0
$937$	53.5230	1.74852	0.874260	−	0.485457i	$-0.161347\pi$
$937$	53.5230	1.74852	0.874260	+	0.485457i	$0.161347\pi$
$938$	0	0
$939$	−38.3600	−1.25183
$940$	0	0
$941$	−5.83487	−0.190211	−0.0951056	−	0.995467i	$-0.530319\pi$
$941$	−5.83487	−0.190211	−0.0951056	+	0.995467i	$0.530319\pi$
$942$	0	0
$943$	−2.88645	−0.0939957
$944$	0	0
$945$	84.2603	2.74099
$946$	0	0
$947$	−9.80691	−0.318682	−0.159341	−	0.987224i	$-0.550937\pi$
$947$	−9.80691	−0.318682	−0.159341	+	0.987224i	$0.550937\pi$
$948$	0	0
$949$	−65.4029	−2.12307
$950$	0	0
$951$	41.3534	1.34098
$952$	0	0
$953$	−13.6002	−0.440555	−0.220277	−	0.975437i	$-0.570696\pi$
$953$	−13.6002	−0.440555	−0.220277	+	0.975437i	$0.570696\pi$
$954$	0	0
$955$	−13.0639	−0.422738
$956$	0	0
$957$	−85.1756	−2.75333
$958$	0	0
$959$	−10.7118	−0.345903
$960$	0	0
$961$	−25.7641	−0.831100
$962$	0	0
$963$	−89.7087	−2.89082
$964$	0	0
$965$	4.46586	0.143761
$966$	0	0
$967$	−34.1118	−1.09696	−0.548481	−	0.836163i	$-0.684794\pi$
$967$	−34.1118	−1.09696	−0.548481	+	0.836163i	$0.684794\pi$
$968$	0	0
$969$	9.02380	0.289886
$970$	0	0
$971$	−9.77589	−0.313723	−0.156862	−	0.987621i	$-0.550138\pi$
$971$	−9.77589	−0.313723	−0.156862	+	0.987621i	$0.550138\pi$
$972$	0	0
$973$	1.83784	0.0589185
$974$	0	0
$975$	18.1359	0.580815
$976$	0	0
$977$	−42.5203	−1.36034	−0.680172	−	0.733052i	$-0.738095\pi$
$977$	−42.5203	−1.36034	−0.680172	+	0.733052i	$0.738095\pi$
$978$	0	0
$979$	30.9704	0.989818
$980$	0	0
$981$	−24.2568	−0.774461
$982$	0	0
$983$	41.1728	1.31321	0.656604	−	0.754236i	$-0.271992\pi$
$983$	41.1728	1.31321	0.656604	+	0.754236i	$0.271992\pi$
$984$	0	0
$985$	−2.53898	−0.0808986
$986$	0	0
$987$	14.4375	0.459551
$988$	0	0
$989$	0.228607	0.00726928
$990$	0	0
$991$	37.0970	1.17843	0.589213	−	0.807978i	$-0.299438\pi$
$991$	37.0970	1.17843	0.589213	+	0.807978i	$0.299438\pi$
$992$	0	0
$993$	−76.3287	−2.42222
$994$	0	0
$995$	24.1097	0.764330
$996$	0	0
$997$	17.6090	0.557683	0.278841	−	0.960337i	$-0.410050\pi$
$997$	17.6090	0.557683	0.278841	+	0.960337i	$0.410050\pi$
$998$	0	0
$999$	−122.887	−3.88796

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.1	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.1	✓	31	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-3.22742 q^{3} +2.43630 q^{5} -2.42654 q^{7} +7.41621 q^{9} +O(q^{10})\)	\(q-3.22742 q^{3} +2.43630 q^{5} -2.42654 q^{7} +7.41621 q^{9} +4.13185 q^{11} -6.00645 q^{13} -7.86295 q^{15} +2.79598 q^{17} -1.00000 q^{19} +7.83146 q^{21} +0.561526 q^{23} +0.935550 q^{25} -14.2529 q^{27} +6.38727 q^{29} -2.28821 q^{31} -13.3352 q^{33} -5.91178 q^{35} +8.62184 q^{37} +19.3853 q^{39} -5.14036 q^{41} +0.407117 q^{43} +18.0681 q^{45} +1.84353 q^{47} -1.11190 q^{49} -9.02380 q^{51} +1.53824 q^{53} +10.0664 q^{55} +3.22742 q^{57} +2.20768 q^{59} +10.7785 q^{61} -17.9957 q^{63} -14.6335 q^{65} -9.26213 q^{67} -1.81228 q^{69} -2.66414 q^{71} +10.8888 q^{73} -3.01941 q^{75} -10.0261 q^{77} -1.00000 q^{79} +23.7516 q^{81} +3.29227 q^{83} +6.81185 q^{85} -20.6144 q^{87} +7.49553 q^{89} +14.5749 q^{91} +7.38499 q^{93} -2.43630 q^{95} -10.0965 q^{97} +30.6427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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