LMFDB - Embedded newform 6004.2.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+1.84886 q^{3} +0.871764 q^{5} -3.24231 q^{7} +0.418286 q^{9} +O(q^{10})$	$q+1.84886 q^{3} +0.871764 q^{5} -3.24231 q^{7} +0.418286 q^{9} +2.03683 q^{11} -0.211070 q^{13} +1.61177 q^{15} +6.97206 q^{17} +1.00000 q^{19} -5.99458 q^{21} +2.72804 q^{23} -4.24003 q^{25} -4.77323 q^{27} -2.83001 q^{29} -6.88989 q^{31} +3.76582 q^{33} -2.82653 q^{35} +6.72730 q^{37} -0.390239 q^{39} -3.53456 q^{41} +10.4324 q^{43} +0.364647 q^{45} +7.54260 q^{47} +3.51259 q^{49} +12.8904 q^{51} +12.8766 q^{53} +1.77564 q^{55} +1.84886 q^{57} +6.09771 q^{59} +9.61862 q^{61} -1.35621 q^{63} -0.184003 q^{65} -1.50251 q^{67} +5.04376 q^{69} +0.683995 q^{71} +3.52069 q^{73} -7.83922 q^{75} -6.60405 q^{77} +1.00000 q^{79} -10.0799 q^{81} -4.54782 q^{83} +6.07799 q^{85} -5.23229 q^{87} -7.61940 q^{89} +0.684355 q^{91} -12.7385 q^{93} +0.871764 q^{95} +18.9825 q^{97} +0.851979 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.84886	1.06744	0.533720	−	0.845661i	$-0.320794\pi$
$3$	1.84886	1.06744	0.533720	+	0.845661i	$0.320794\pi$
$4$	0	0
$5$	0.871764	0.389865	0.194932	−	0.980817i	$-0.437551\pi$
$5$	0.871764	0.389865	0.194932	+	0.980817i	$0.437551\pi$
$6$	0	0
$7$	−3.24231	−1.22548	−0.612739	−	0.790285i	$-0.709932\pi$
$7$	−3.24231	−1.22548	−0.612739	+	0.790285i	$0.709932\pi$
$8$	0	0
$9$	0.418286	0.139429
$10$	0	0
$11$	2.03683	0.614128	0.307064	−	0.951689i	$-0.400653\pi$
$11$	2.03683	0.614128	0.307064	+	0.951689i	$0.400653\pi$
$12$	0	0
$13$	−0.211070	−0.0585403	−0.0292701	−	0.999572i	$-0.509318\pi$
$13$	−0.211070	−0.0585403	−0.0292701	+	0.999572i	$0.509318\pi$
$14$	0	0
$15$	1.61177	0.416157
$16$	0	0
$17$	6.97206	1.69097	0.845486	−	0.533998i	$-0.179311\pi$
$17$	6.97206	1.69097	0.845486	+	0.533998i	$0.179311\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.99458	−1.30813
$22$	0	0
$23$	2.72804	0.568835	0.284418	−	0.958700i	$-0.408200\pi$
$23$	2.72804	0.568835	0.284418	+	0.958700i	$0.408200\pi$
$24$	0	0
$25$	−4.24003	−0.848005
$26$	0	0
$27$	−4.77323	−0.918608
$28$	0	0
$29$	−2.83001	−0.525519	−0.262760	−	0.964861i	$-0.584633\pi$
$29$	−2.83001	−0.525519	−0.262760	+	0.964861i	$0.584633\pi$
$30$	0	0
$31$	−6.88989	−1.23746	−0.618731	−	0.785603i	$-0.712353\pi$
$31$	−6.88989	−1.23746	−0.618731	+	0.785603i	$0.712353\pi$
$32$	0	0
$33$	3.76582	0.655545
$34$	0	0
$35$	−2.82653	−0.477771
$36$	0	0
$37$	6.72730	1.10596	0.552981	−	0.833194i	$-0.313490\pi$
$37$	6.72730	1.10596	0.552981	+	0.833194i	$0.313490\pi$
$38$	0	0
$39$	−0.390239	−0.0624883
$40$	0	0
$41$	−3.53456	−0.552006	−0.276003	−	0.961157i	$-0.589010\pi$
$41$	−3.53456	−0.552006	−0.276003	+	0.961157i	$0.589010\pi$
$42$	0	0
$43$	10.4324	1.59092	0.795461	−	0.606005i	$-0.207229\pi$
$43$	10.4324	1.59092	0.795461	+	0.606005i	$0.207229\pi$
$44$	0	0
$45$	0.364647	0.0543583
$46$	0	0
$47$	7.54260	1.10020	0.550101	−	0.835098i	$-0.314589\pi$
$47$	7.54260	1.10020	0.550101	+	0.835098i	$0.314589\pi$
$48$	0	0
$49$	3.51259	0.501799
$50$	0	0
$51$	12.8904	1.80501
$52$	0	0
$53$	12.8766	1.76874	0.884368	−	0.466790i	$-0.154590\pi$
$53$	12.8766	1.76874	0.884368	+	0.466790i	$0.154590\pi$
$54$	0	0
$55$	1.77564	0.239427
$56$	0	0
$57$	1.84886	0.244888
$58$	0	0
$59$	6.09771	0.793855	0.396927	−	0.917850i	$-0.370076\pi$
$59$	6.09771	0.793855	0.396927	+	0.917850i	$0.370076\pi$
$60$	0	0
$61$	9.61862	1.23154	0.615769	−	0.787927i	$-0.288845\pi$
$61$	9.61862	1.23154	0.615769	+	0.787927i	$0.288845\pi$
$62$	0	0
$63$	−1.35621	−0.170867
$64$	0	0
$65$	−0.184003	−0.0228228
$66$	0	0
$67$	−1.50251	−0.183561	−0.0917804	−	0.995779i	$-0.529256\pi$
$67$	−1.50251	−0.183561	−0.0917804	+	0.995779i	$0.529256\pi$
$68$	0	0
$69$	5.04376	0.607198
$70$	0	0
$71$	0.683995	0.0811753	0.0405876	−	0.999176i	$-0.487077\pi$
$71$	0.683995	0.0811753	0.0405876	+	0.999176i	$0.487077\pi$
$72$	0	0
$73$	3.52069	0.412066	0.206033	−	0.978545i	$-0.433945\pi$
$73$	3.52069	0.412066	0.206033	+	0.978545i	$0.433945\pi$
$74$	0	0
$75$	−7.83922	−0.905195
$76$	0	0
$77$	−6.60405	−0.752601
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−10.0799	−1.11999
$82$	0	0
$83$	−4.54782	−0.499188	−0.249594	−	0.968351i	$-0.580297\pi$
$83$	−4.54782	−0.499188	−0.249594	+	0.968351i	$0.580297\pi$
$84$	0	0
$85$	6.07799	0.659250
$86$	0	0
$87$	−5.23229	−0.560961
$88$	0	0
$89$	−7.61940	−0.807654	−0.403827	−	0.914835i	$-0.632320\pi$
$89$	−7.61940	−0.807654	−0.403827	+	0.914835i	$0.632320\pi$
$90$	0	0
$91$	0.684355	0.0717399
$92$	0	0
$93$	−12.7385	−1.32092
$94$	0	0
$95$	0.871764	0.0894411
$96$	0	0
$97$	18.9825	1.92738	0.963690	−	0.267025i	$-0.0860407\pi$
$97$	18.9825	1.92738	0.963690	+	0.267025i	$0.0860407\pi$
$98$	0	0
$99$	0.851979	0.0856271
$100$	0	0
$101$	3.72328	0.370480	0.185240	−	0.982693i	$-0.440694\pi$
$101$	3.72328	0.370480	0.185240	+	0.982693i	$0.440694\pi$
$102$	0	0
$103$	16.0213	1.57862	0.789312	−	0.613993i	$-0.210438\pi$
$103$	16.0213	1.57862	0.789312	+	0.613993i	$0.210438\pi$
$104$	0	0
$105$	−5.22586	−0.509992
$106$	0	0
$107$	−16.5924	−1.60405	−0.802024	−	0.597292i	$-0.796243\pi$
$107$	−16.5924	−1.60405	−0.802024	+	0.597292i	$0.796243\pi$
$108$	0	0
$109$	5.94284	0.569221	0.284610	−	0.958643i	$-0.408136\pi$
$109$	5.94284	0.569221	0.284610	+	0.958643i	$0.408136\pi$
$110$	0	0
$111$	12.4378	1.18055
$112$	0	0
$113$	−15.4102	−1.44967	−0.724836	−	0.688921i	$-0.758085\pi$
$113$	−15.4102	−1.44967	−0.724836	+	0.688921i	$0.758085\pi$
$114$	0	0
$115$	2.37821	0.221769
$116$	0	0
$117$	−0.0882877	−0.00816220
$118$	0	0
$119$	−22.6056	−2.07225
$120$	0	0
$121$	−6.85131	−0.622846
$122$	0	0
$123$	−6.53492	−0.589234
$124$	0	0
$125$	−8.05512	−0.720472
$126$	0	0
$127$	−4.11569	−0.365209	−0.182604	−	0.983186i	$-0.558453\pi$
$127$	−4.11569	−0.365209	−0.182604	+	0.983186i	$0.558453\pi$
$128$	0	0
$129$	19.2880	1.69821
$130$	0	0
$131$	−16.7707	−1.46526	−0.732629	−	0.680628i	$-0.761707\pi$
$131$	−16.7707	−1.46526	−0.732629	+	0.680628i	$0.761707\pi$
$132$	0	0
$133$	−3.24231	−0.281144
$134$	0	0
$135$	−4.16113	−0.358133
$136$	0	0
$137$	3.04912	0.260504	0.130252	−	0.991481i	$-0.458421\pi$
$137$	3.04912	0.260504	0.130252	+	0.991481i	$0.458421\pi$
$138$	0	0
$139$	21.2024	1.79836	0.899181	−	0.437577i	$-0.144163\pi$
$139$	21.2024	1.79836	0.899181	+	0.437577i	$0.144163\pi$
$140$	0	0
$141$	13.9452	1.17440
$142$	0	0
$143$	−0.429914	−0.0359512
$144$	0	0
$145$	−2.46710	−0.204881
$146$	0	0
$147$	6.49429	0.535640
$148$	0	0
$149$	16.7176	1.36956	0.684781	−	0.728749i	$-0.259898\pi$
$149$	16.7176	1.36956	0.684781	+	0.728749i	$0.259898\pi$
$150$	0	0
$151$	13.8593	1.12786	0.563928	−	0.825824i	$-0.309290\pi$
$151$	13.8593	1.12786	0.563928	+	0.825824i	$0.309290\pi$
$152$	0	0
$153$	2.91631	0.235770
$154$	0	0
$155$	−6.00636	−0.482443
$156$	0	0
$157$	18.5532	1.48071	0.740353	−	0.672219i	$-0.234659\pi$
$157$	18.5532	1.48071	0.740353	+	0.672219i	$0.234659\pi$
$158$	0	0
$159$	23.8070	1.88802
$160$	0	0
$161$	−8.84515	−0.697096
$162$	0	0
$163$	−7.02815	−0.550487	−0.275243	−	0.961375i	$-0.588758\pi$
$163$	−7.02815	−0.550487	−0.275243	+	0.961375i	$0.588758\pi$
$164$	0	0
$165$	3.28291	0.255574
$166$	0	0
$167$	14.8525	1.14932	0.574660	−	0.818392i	$-0.305134\pi$
$167$	14.8525	1.14932	0.574660	+	0.818392i	$0.305134\pi$
$168$	0	0
$169$	−12.9554	−0.996573
$170$	0	0
$171$	0.418286	0.0319871
$172$	0	0
$173$	17.8265	1.35532	0.677660	−	0.735375i	$-0.262994\pi$
$173$	17.8265	1.35532	0.677660	+	0.735375i	$0.262994\pi$
$174$	0	0
$175$	13.7475	1.03921
$176$	0	0
$177$	11.2738	0.847393
$178$	0	0
$179$	6.19221	0.462827	0.231414	−	0.972855i	$-0.425665\pi$
$179$	6.19221	0.462827	0.231414	+	0.972855i	$0.425665\pi$
$180$	0	0
$181$	0.556338	0.0413523	0.0206761	−	0.999786i	$-0.493418\pi$
$181$	0.556338	0.0413523	0.0206761	+	0.999786i	$0.493418\pi$
$182$	0	0
$183$	17.7835	1.31459
$184$	0	0
$185$	5.86462	0.431175
$186$	0	0
$187$	14.2009	1.03847
$188$	0	0
$189$	15.4763	1.12574
$190$	0	0
$191$	3.36632	0.243579	0.121789	−	0.992556i	$-0.461137\pi$
$191$	3.36632	0.243579	0.121789	+	0.992556i	$0.461137\pi$
$192$	0	0
$193$	8.57711	0.617394	0.308697	−	0.951160i	$-0.400107\pi$
$193$	8.57711	0.617394	0.308697	+	0.951160i	$0.400107\pi$
$194$	0	0
$195$	−0.340196	−0.0243620
$196$	0	0
$197$	19.7374	1.40623	0.703116	−	0.711075i	$-0.251791\pi$
$197$	19.7374	1.40623	0.703116	+	0.711075i	$0.251791\pi$
$198$	0	0
$199$	−19.0171	−1.34809	−0.674043	−	0.738692i	$-0.735444\pi$
$199$	−19.0171	−1.34809	−0.674043	+	0.738692i	$0.735444\pi$
$200$	0	0
$201$	−2.77793	−0.195940
$202$	0	0
$203$	9.17577	0.644013
$204$	0	0
$205$	−3.08131	−0.215208
$206$	0	0
$207$	1.14110	0.0793120
$208$	0	0
$209$	2.03683	0.140891
$210$	0	0
$211$	−0.289572	−0.0199350	−0.00996748	−	0.999950i	$-0.503173\pi$
$211$	−0.289572	−0.0199350	−0.00996748	+	0.999950i	$0.503173\pi$
$212$	0	0
$213$	1.26461	0.0866498
$214$	0	0
$215$	9.09457	0.620244
$216$	0	0
$217$	22.3392	1.51648
$218$	0	0
$219$	6.50927	0.439856
$220$	0	0
$221$	−1.47159	−0.0989900
$222$	0	0
$223$	13.4645	0.901648	0.450824	−	0.892613i	$-0.351130\pi$
$223$	13.4645	0.901648	0.450824	+	0.892613i	$0.351130\pi$
$224$	0	0
$225$	−1.77354	−0.118236
$226$	0	0
$227$	−12.5665	−0.834069	−0.417034	−	0.908891i	$-0.636931\pi$
$227$	−12.5665	−0.834069	−0.417034	+	0.908891i	$0.636931\pi$
$228$	0	0
$229$	−9.22437	−0.609564	−0.304782	−	0.952422i	$-0.598584\pi$
$229$	−9.22437	−0.609564	−0.304782	+	0.952422i	$0.598584\pi$
$230$	0	0
$231$	−12.2100	−0.803357
$232$	0	0
$233$	−18.0151	−1.18021	−0.590104	−	0.807327i	$-0.700913\pi$
$233$	−18.0151	−1.18021	−0.590104	+	0.807327i	$0.700913\pi$
$234$	0	0
$235$	6.57537	0.428930
$236$	0	0
$237$	1.84886	0.120096
$238$	0	0
$239$	−17.7934	−1.15096	−0.575478	−	0.817817i	$-0.695184\pi$
$239$	−17.7934	−1.15096	−0.575478	+	0.817817i	$0.695184\pi$
$240$	0	0
$241$	−17.5566	−1.13092	−0.565459	−	0.824776i	$-0.691301\pi$
$241$	−17.5566	−1.13092	−0.565459	+	0.824776i	$0.691301\pi$
$242$	0	0
$243$	−4.31663	−0.276912
$244$	0	0
$245$	3.06215	0.195634
$246$	0	0
$247$	−0.211070	−0.0134301
$248$	0	0
$249$	−8.40829	−0.532854
$250$	0	0
$251$	−3.98676	−0.251642	−0.125821	−	0.992053i	$-0.540157\pi$
$251$	−3.98676	−0.251642	−0.125821	+	0.992053i	$0.540157\pi$
$252$	0	0
$253$	5.55656	0.349338
$254$	0	0
$255$	11.2374	0.703710
$256$	0	0
$257$	15.9209	0.993116	0.496558	−	0.868003i	$-0.334597\pi$
$257$	15.9209	0.993116	0.496558	+	0.868003i	$0.334597\pi$
$258$	0	0
$259$	−21.8120	−1.35533
$260$	0	0
$261$	−1.18375	−0.0732725
$262$	0	0
$263$	15.6323	0.963931	0.481966	−	0.876190i	$-0.339923\pi$
$263$	15.6323	0.963931	0.481966	+	0.876190i	$0.339923\pi$
$264$	0	0
$265$	11.2254	0.689568
$266$	0	0
$267$	−14.0872	−0.862123
$268$	0	0
$269$	12.9098	0.787125	0.393563	−	0.919298i	$-0.371242\pi$
$269$	12.9098	0.787125	0.393563	+	0.919298i	$0.371242\pi$
$270$	0	0
$271$	−14.9403	−0.907560	−0.453780	−	0.891114i	$-0.649925\pi$
$271$	−14.9403	−0.907560	−0.453780	+	0.891114i	$0.649925\pi$
$272$	0	0
$273$	1.26528	0.0765781
$274$	0	0
$275$	−8.63623	−0.520784
$276$	0	0
$277$	−21.1146	−1.26865	−0.634326	−	0.773066i	$-0.718722\pi$
$277$	−21.1146	−1.26865	−0.634326	+	0.773066i	$0.718722\pi$
$278$	0	0
$279$	−2.88195	−0.172538
$280$	0	0
$281$	−7.50474	−0.447695	−0.223848	−	0.974624i	$-0.571862\pi$
$281$	−7.50474	−0.447695	−0.223848	+	0.974624i	$0.571862\pi$
$282$	0	0
$283$	−7.92172	−0.470897	−0.235449	−	0.971887i	$-0.575656\pi$
$283$	−7.92172	−0.470897	−0.235449	+	0.971887i	$0.575656\pi$
$284$	0	0
$285$	1.61177	0.0954730
$286$	0	0
$287$	11.4602	0.676472
$288$	0	0
$289$	31.6096	1.85939
$290$	0	0
$291$	35.0960	2.05736
$292$	0	0
$293$	−1.15659	−0.0675684	−0.0337842	−	0.999429i	$-0.510756\pi$
$293$	−1.15659	−0.0675684	−0.0337842	+	0.999429i	$0.510756\pi$
$294$	0	0
$295$	5.31577	0.309496
$296$	0	0
$297$	−9.72227	−0.564143
$298$	0	0
$299$	−0.575807	−0.0332998
$300$	0	0
$301$	−33.8250	−1.94964
$302$	0	0
$303$	6.88383	0.395465
$304$	0	0
$305$	8.38517	0.480133
$306$	0	0
$307$	3.21159	0.183295	0.0916476	−	0.995791i	$-0.470787\pi$
$307$	3.21159	0.183295	0.0916476	+	0.995791i	$0.470787\pi$
$308$	0	0
$309$	29.6211	1.68509
$310$	0	0
$311$	−14.9656	−0.848623	−0.424312	−	0.905516i	$-0.639484\pi$
$311$	−14.9656	−0.848623	−0.424312	+	0.905516i	$0.639484\pi$
$312$	0	0
$313$	2.86367	0.161864	0.0809321	−	0.996720i	$-0.474210\pi$
$313$	2.86367	0.161864	0.0809321	+	0.996720i	$0.474210\pi$
$314$	0	0
$315$	−1.18230	−0.0666150
$316$	0	0
$317$	27.6047	1.55044	0.775218	−	0.631694i	$-0.217640\pi$
$317$	27.6047	1.55044	0.775218	+	0.631694i	$0.217640\pi$
$318$	0	0
$319$	−5.76425	−0.322736
$320$	0	0
$321$	−30.6770	−1.71223
$322$	0	0
$323$	6.97206	0.387936
$324$	0	0
$325$	0.894943	0.0496425
$326$	0	0
$327$	10.9875	0.607609
$328$	0	0
$329$	−24.4555	−1.34827
$330$	0	0
$331$	−7.45602	−0.409820	−0.204910	−	0.978781i	$-0.565690\pi$
$331$	−7.45602	−0.409820	−0.204910	+	0.978781i	$0.565690\pi$
$332$	0	0
$333$	2.81394	0.154203
$334$	0	0
$335$	−1.30983	−0.0715639
$336$	0	0
$337$	−0.801574	−0.0436645	−0.0218323	−	0.999762i	$-0.506950\pi$
$337$	−0.801574	−0.0436645	−0.0218323	+	0.999762i	$0.506950\pi$
$338$	0	0
$339$	−28.4914	−1.54744
$340$	0	0
$341$	−14.0336	−0.759960
$342$	0	0
$343$	11.3073	0.610535
$344$	0	0
$345$	4.39697	0.236725
$346$	0	0
$347$	−14.9190	−0.800896	−0.400448	−	0.916319i	$-0.631146\pi$
$347$	−14.9190	−0.800896	−0.400448	+	0.916319i	$0.631146\pi$
$348$	0	0
$349$	−0.998196	−0.0534322	−0.0267161	−	0.999643i	$-0.508505\pi$
$349$	−0.998196	−0.0534322	−0.0267161	+	0.999643i	$0.508505\pi$
$350$	0	0
$351$	1.00749	0.0537756
$352$	0	0
$353$	22.0318	1.17263	0.586317	−	0.810082i	$-0.300577\pi$
$353$	22.0318	1.17263	0.586317	+	0.810082i	$0.300577\pi$
$354$	0	0
$355$	0.596282	0.0316474
$356$	0	0
$357$	−41.7946	−2.21200
$358$	0	0
$359$	−16.4987	−0.870769	−0.435384	−	0.900245i	$-0.643387\pi$
$359$	−16.4987	−0.870769	−0.435384	+	0.900245i	$0.643387\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.6671	−0.664851
$364$	0	0
$365$	3.06922	0.160650
$366$	0	0
$367$	−18.7536	−0.978928	−0.489464	−	0.872023i	$-0.662808\pi$
$367$	−18.7536	−0.978928	−0.489464	+	0.872023i	$0.662808\pi$
$368$	0	0
$369$	−1.47846	−0.0769655
$370$	0	0
$371$	−41.7500	−2.16755
$372$	0	0
$373$	20.4058	1.05657	0.528287	−	0.849066i	$-0.322834\pi$
$373$	20.4058	1.05657	0.528287	+	0.849066i	$0.322834\pi$
$374$	0	0
$375$	−14.8928	−0.769061
$376$	0	0
$377$	0.597330	0.0307641
$378$	0	0
$379$	33.2918	1.71008	0.855042	−	0.518558i	$-0.173531\pi$
$379$	33.2918	1.71008	0.855042	+	0.518558i	$0.173531\pi$
$380$	0	0
$381$	−7.60934	−0.389839
$382$	0	0
$383$	−21.8380	−1.11587	−0.557936	−	0.829884i	$-0.688407\pi$
$383$	−21.8380	−1.11587	−0.557936	+	0.829884i	$0.688407\pi$
$384$	0	0
$385$	−5.75717	−0.293413
$386$	0	0
$387$	4.36372	0.221820
$388$	0	0
$389$	−12.5145	−0.634508	−0.317254	−	0.948341i	$-0.602761\pi$
$389$	−12.5145	−0.634508	−0.317254	+	0.948341i	$0.602761\pi$
$390$	0	0
$391$	19.0200	0.961885
$392$	0	0
$393$	−31.0066	−1.56408
$394$	0	0
$395$	0.871764	0.0438632
$396$	0	0
$397$	7.47970	0.375395	0.187698	−	0.982227i	$-0.439897\pi$
$397$	7.47970	0.375395	0.187698	+	0.982227i	$0.439897\pi$
$398$	0	0
$399$	−5.99458	−0.300105
$400$	0	0
$401$	17.3262	0.865229	0.432614	−	0.901579i	$-0.357591\pi$
$401$	17.3262	0.865229	0.432614	+	0.901579i	$0.357591\pi$
$402$	0	0
$403$	1.45425	0.0724413
$404$	0	0
$405$	−8.78729	−0.436644
$406$	0	0
$407$	13.7024	0.679202
$408$	0	0
$409$	−28.2449	−1.39662	−0.698311	−	0.715794i	$-0.746065\pi$
$409$	−28.2449	−1.39662	−0.698311	+	0.715794i	$0.746065\pi$
$410$	0	0
$411$	5.63741	0.278073
$412$	0	0
$413$	−19.7707	−0.972852
$414$	0	0
$415$	−3.96463	−0.194616
$416$	0	0
$417$	39.2002	1.91964
$418$	0	0
$419$	31.8746	1.55717	0.778587	−	0.627537i	$-0.215937\pi$
$419$	31.8746	1.55717	0.778587	+	0.627537i	$0.215937\pi$
$420$	0	0
$421$	7.84594	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$421$	7.84594	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$422$	0	0
$423$	3.15496	0.153400
$424$	0	0
$425$	−29.5617	−1.43395
$426$	0	0
$427$	−31.1866	−1.50922
$428$	0	0
$429$	−0.794852	−0.0383758
$430$	0	0
$431$	26.8366	1.29267	0.646336	−	0.763053i	$-0.276301\pi$
$431$	26.8366	1.29267	0.646336	+	0.763053i	$0.276301\pi$
$432$	0	0
$433$	12.1690	0.584806	0.292403	−	0.956295i	$-0.405545\pi$
$433$	12.1690	0.584806	0.292403	+	0.956295i	$0.405545\pi$
$434$	0	0
$435$	−4.56132	−0.218699
$436$	0	0
$437$	2.72804	0.130500
$438$	0	0
$439$	16.6038	0.792458	0.396229	−	0.918152i	$-0.370319\pi$
$439$	16.6038	0.792458	0.396229	+	0.918152i	$0.370319\pi$
$440$	0	0
$441$	1.46927	0.0699651
$442$	0	0
$443$	17.2469	0.819424	0.409712	−	0.912215i	$-0.365629\pi$
$443$	17.2469	0.819424	0.409712	+	0.912215i	$0.365629\pi$
$444$	0	0
$445$	−6.64232	−0.314876
$446$	0	0
$447$	30.9086	1.46192
$448$	0	0
$449$	−10.5670	−0.498690	−0.249345	−	0.968415i	$-0.580215\pi$
$449$	−10.5670	−0.498690	−0.249345	+	0.968415i	$0.580215\pi$
$450$	0	0
$451$	−7.19932	−0.339003
$452$	0	0
$453$	25.6240	1.20392
$454$	0	0
$455$	0.596596	0.0279689
$456$	0	0
$457$	5.23025	0.244661	0.122330	−	0.992489i	$-0.460963\pi$
$457$	5.23025	0.244661	0.122330	+	0.992489i	$0.460963\pi$
$458$	0	0
$459$	−33.2792	−1.55334
$460$	0	0
$461$	7.28860	0.339464	0.169732	−	0.985490i	$-0.445710\pi$
$461$	7.28860	0.339464	0.169732	+	0.985490i	$0.445710\pi$
$462$	0	0
$463$	3.55796	0.165352	0.0826761	−	0.996576i	$-0.473653\pi$
$463$	3.55796	0.165352	0.0826761	+	0.996576i	$0.473653\pi$
$464$	0	0
$465$	−11.1049	−0.514979
$466$	0	0
$467$	−33.3650	−1.54395	−0.771975	−	0.635653i	$-0.780731\pi$
$467$	−33.3650	−1.54395	−0.771975	+	0.635653i	$0.780731\pi$
$468$	0	0
$469$	4.87161	0.224950
$470$	0	0
$471$	34.3023	1.58056
$472$	0	0
$473$	21.2490	0.977030
$474$	0	0
$475$	−4.24003	−0.194546
$476$	0	0
$477$	5.38610	0.246613
$478$	0	0
$479$	−28.1510	−1.28625	−0.643126	−	0.765761i	$-0.722363\pi$
$479$	−28.1510	−1.28625	−0.643126	+	0.765761i	$0.722363\pi$
$480$	0	0
$481$	−1.41993	−0.0647433
$482$	0	0
$483$	−16.3535	−0.744108
$484$	0	0
$485$	16.5482	0.751417
$486$	0	0
$487$	−7.38112	−0.334470	−0.167235	−	0.985917i	$-0.553484\pi$
$487$	−7.38112	−0.334470	−0.167235	+	0.985917i	$0.553484\pi$
$488$	0	0
$489$	−12.9941	−0.587612
$490$	0	0
$491$	−17.8181	−0.804119	−0.402059	−	0.915614i	$-0.631705\pi$
$491$	−17.8181	−0.804119	−0.402059	+	0.915614i	$0.631705\pi$
$492$	0	0
$493$	−19.7310	−0.888639
$494$	0	0
$495$	0.742725	0.0333830
$496$	0	0
$497$	−2.21773	−0.0994786
$498$	0	0
$499$	5.75811	0.257768	0.128884	−	0.991660i	$-0.458860\pi$
$499$	5.75811	0.257768	0.128884	+	0.991660i	$0.458860\pi$
$500$	0	0
$501$	27.4602	1.22683
$502$	0	0
$503$	−10.3221	−0.460239	−0.230120	−	0.973162i	$-0.573912\pi$
$503$	−10.3221	−0.460239	−0.230120	+	0.973162i	$0.573912\pi$
$504$	0	0
$505$	3.24582	0.144437
$506$	0	0
$507$	−23.9528	−1.06378
$508$	0	0
$509$	−4.25324	−0.188521	−0.0942607	−	0.995548i	$-0.530049\pi$
$509$	−4.25324	−0.188521	−0.0942607	+	0.995548i	$0.530049\pi$
$510$	0	0
$511$	−11.4152	−0.504978
$512$	0	0
$513$	−4.77323	−0.210743
$514$	0	0
$515$	13.9668	0.615450
$516$	0	0
$517$	15.3630	0.675665
$518$	0	0
$519$	32.9586	1.44672
$520$	0	0
$521$	−7.37174	−0.322962	−0.161481	−	0.986876i	$-0.551627\pi$
$521$	−7.37174	−0.322962	−0.161481	+	0.986876i	$0.551627\pi$
$522$	0	0
$523$	−18.4521	−0.806853	−0.403426	−	0.915012i	$-0.632181\pi$
$523$	−18.4521	−0.806853	−0.403426	+	0.915012i	$0.632181\pi$
$524$	0	0
$525$	25.4172	1.10930
$526$	0	0
$527$	−48.0367	−2.09251
$528$	0	0
$529$	−15.5578	−0.676426
$530$	0	0
$531$	2.55059	0.110686
$532$	0	0
$533$	0.746041	0.0323146
$534$	0	0
$535$	−14.4647	−0.625362
$536$	0	0
$537$	11.4485	0.494041
$538$	0	0
$539$	7.15456	0.308169
$540$	0	0
$541$	−9.82428	−0.422379	−0.211189	−	0.977445i	$-0.567734\pi$
$541$	−9.82428	−0.422379	−0.211189	+	0.977445i	$0.567734\pi$
$542$	0	0
$543$	1.02859	0.0441411
$544$	0	0
$545$	5.18076	0.221919
$546$	0	0
$547$	44.9795	1.92318	0.961592	−	0.274483i	$-0.0885066\pi$
$547$	44.9795	1.92318	0.961592	+	0.274483i	$0.0885066\pi$
$548$	0	0
$549$	4.02334	0.171712
$550$	0	0
$551$	−2.83001	−0.120562
$552$	0	0
$553$	−3.24231	−0.137877
$554$	0	0
$555$	10.8429	0.460254
$556$	0	0
$557$	−17.1820	−0.728024	−0.364012	−	0.931394i	$-0.618593\pi$
$557$	−17.1820	−0.728024	−0.364012	+	0.931394i	$0.618593\pi$
$558$	0	0
$559$	−2.20196	−0.0931330
$560$	0	0
$561$	26.2555	1.10851
$562$	0	0
$563$	−39.5335	−1.66614	−0.833069	−	0.553169i	$-0.813418\pi$
$563$	−39.5335	−1.66614	−0.833069	+	0.553169i	$0.813418\pi$
$564$	0	0
$565$	−13.4341	−0.565176
$566$	0	0
$567$	32.6822	1.37252
$568$	0	0
$569$	−30.4830	−1.27792	−0.638958	−	0.769242i	$-0.720634\pi$
$569$	−30.4830	−1.27792	−0.638958	+	0.769242i	$0.720634\pi$
$570$	0	0
$571$	−12.8704	−0.538610	−0.269305	−	0.963055i	$-0.586794\pi$
$571$	−12.8704	−0.538610	−0.269305	+	0.963055i	$0.586794\pi$
$572$	0	0
$573$	6.22387	0.260006
$574$	0	0
$575$	−11.5670	−0.482375
$576$	0	0
$577$	−6.06503	−0.252490	−0.126245	−	0.991999i	$-0.540293\pi$
$577$	−6.06503	−0.252490	−0.126245	+	0.991999i	$0.540293\pi$
$578$	0	0
$579$	15.8579	0.659031
$580$	0	0
$581$	14.7455	0.611745
$582$	0	0
$583$	26.2275	1.08623
$584$	0	0
$585$	−0.0769660	−0.00318215
$586$	0	0
$587$	35.9282	1.48292	0.741459	−	0.670999i	$-0.234134\pi$
$587$	35.9282	1.48292	0.741459	+	0.670999i	$0.234134\pi$
$588$	0	0
$589$	−6.88989	−0.283893
$590$	0	0
$591$	36.4917	1.50107
$592$	0	0
$593$	20.0102	0.821721	0.410861	−	0.911698i	$-0.365228\pi$
$593$	20.0102	0.821721	0.410861	+	0.911698i	$0.365228\pi$
$594$	0	0
$595$	−19.7067	−0.807898
$596$	0	0
$597$	−35.1600	−1.43900
$598$	0	0
$599$	31.7894	1.29888	0.649440	−	0.760413i	$-0.275004\pi$
$599$	31.7894	1.29888	0.649440	+	0.760413i	$0.275004\pi$
$600$	0	0
$601$	15.6508	0.638411	0.319206	−	0.947685i	$-0.396584\pi$
$601$	15.6508	0.638411	0.319206	+	0.947685i	$0.396584\pi$
$602$	0	0
$603$	−0.628479	−0.0255937
$604$	0	0
$605$	−5.97273	−0.242826
$606$	0	0
$607$	−8.40361	−0.341092	−0.170546	−	0.985350i	$-0.554553\pi$
$607$	−8.40361	−0.341092	−0.170546	+	0.985350i	$0.554553\pi$
$608$	0	0
$609$	16.9647	0.687445
$610$	0	0
$611$	−1.59202	−0.0644061
$612$	0	0
$613$	8.34311	0.336975	0.168488	−	0.985704i	$-0.446112\pi$
$613$	8.34311	0.336975	0.168488	+	0.985704i	$0.446112\pi$
$614$	0	0
$615$	−5.69691	−0.229721
$616$	0	0
$617$	−5.08927	−0.204886	−0.102443	−	0.994739i	$-0.532666\pi$
$617$	−5.08927	−0.204886	−0.102443	+	0.994739i	$0.532666\pi$
$618$	0	0
$619$	1.83893	0.0739130	0.0369565	−	0.999317i	$-0.488234\pi$
$619$	1.83893	0.0739130	0.0369565	+	0.999317i	$0.488234\pi$
$620$	0	0
$621$	−13.0216	−0.522537
$622$	0	0
$623$	24.7045	0.989763
$624$	0	0
$625$	14.1780	0.567119
$626$	0	0
$627$	3.76582	0.150392
$628$	0	0
$629$	46.9031	1.87015
$630$	0	0
$631$	−21.6410	−0.861514	−0.430757	−	0.902468i	$-0.641753\pi$
$631$	−21.6410	−0.861514	−0.430757	+	0.902468i	$0.641753\pi$
$632$	0	0
$633$	−0.535378	−0.0212794
$634$	0	0
$635$	−3.58791	−0.142382
$636$	0	0
$637$	−0.741403	−0.0293754
$638$	0	0
$639$	0.286106	0.0113182
$640$	0	0
$641$	−42.6045	−1.68278	−0.841389	−	0.540431i	$-0.818261\pi$
$641$	−42.6045	−1.68278	−0.841389	+	0.540431i	$0.818261\pi$
$642$	0	0
$643$	−49.3292	−1.94535	−0.972676	−	0.232167i	$-0.925418\pi$
$643$	−49.3292	−1.94535	−0.972676	+	0.232167i	$0.925418\pi$
$644$	0	0
$645$	16.8146	0.662074
$646$	0	0
$647$	−35.7579	−1.40579	−0.702893	−	0.711295i	$-0.748109\pi$
$647$	−35.7579	−1.40579	−0.702893	+	0.711295i	$0.748109\pi$
$648$	0	0
$649$	12.4200	0.487529
$650$	0	0
$651$	41.3020	1.61875
$652$	0	0
$653$	−15.9641	−0.624724	−0.312362	−	0.949963i	$-0.601120\pi$
$653$	−15.9641	−0.624724	−0.312362	+	0.949963i	$0.601120\pi$
$654$	0	0
$655$	−14.6201	−0.571253
$656$	0	0
$657$	1.47266	0.0574539
$658$	0	0
$659$	42.7059	1.66359	0.831794	−	0.555085i	$-0.187314\pi$
$659$	42.7059	1.66359	0.831794	+	0.555085i	$0.187314\pi$
$660$	0	0
$661$	−23.7929	−0.925435	−0.462718	−	0.886506i	$-0.653126\pi$
$661$	−23.7929	−0.925435	−0.462718	+	0.886506i	$0.653126\pi$
$662$	0	0
$663$	−2.72077	−0.105666
$664$	0	0
$665$	−2.82653	−0.109608
$666$	0	0
$667$	−7.72037	−0.298934
$668$	0	0
$669$	24.8939	0.962456
$670$	0	0
$671$	19.5915	0.756322
$672$	0	0
$673$	−38.9724	−1.50228	−0.751138	−	0.660145i	$-0.770495\pi$
$673$	−38.9724	−1.50228	−0.751138	+	0.660145i	$0.770495\pi$
$674$	0	0
$675$	20.2386	0.778985
$676$	0	0
$677$	−13.9091	−0.534571	−0.267285	−	0.963617i	$-0.586127\pi$
$677$	−13.9091	−0.534571	−0.267285	+	0.963617i	$0.586127\pi$
$678$	0	0
$679$	−61.5471	−2.36196
$680$	0	0
$681$	−23.2337	−0.890319
$682$	0	0
$683$	32.7840	1.25445	0.627223	−	0.778840i	$-0.284192\pi$
$683$	32.7840	1.25445	0.627223	+	0.778840i	$0.284192\pi$
$684$	0	0
$685$	2.65812	0.101561
$686$	0	0
$687$	−17.0546	−0.650673
$688$	0	0
$689$	−2.71786	−0.103542
$690$	0	0
$691$	−1.74582	−0.0664142	−0.0332071	−	0.999448i	$-0.510572\pi$
$691$	−1.74582	−0.0664142	−0.0332071	+	0.999448i	$0.510572\pi$
$692$	0	0
$693$	−2.76238	−0.104934
$694$	0	0
$695$	18.4835	0.701118
$696$	0	0
$697$	−24.6432	−0.933427
$698$	0	0
$699$	−33.3074	−1.25980
$700$	0	0
$701$	5.69283	0.215015	0.107508	−	0.994204i	$-0.465713\pi$
$701$	5.69283	0.215015	0.107508	+	0.994204i	$0.465713\pi$
$702$	0	0
$703$	6.72730	0.253725
$704$	0	0
$705$	12.1569	0.457857
$706$	0	0
$707$	−12.0720	−0.454016
$708$	0	0
$709$	19.4851	0.731780	0.365890	−	0.930658i	$-0.380765\pi$
$709$	19.4851	0.731780	0.365890	+	0.930658i	$0.380765\pi$
$710$	0	0
$711$	0.418286	0.0156870
$712$	0	0
$713$	−18.7959	−0.703912
$714$	0	0
$715$	−0.374784	−0.0140161
$716$	0	0
$717$	−32.8974	−1.22858
$718$	0	0
$719$	29.2926	1.09243	0.546215	−	0.837645i	$-0.316068\pi$
$719$	29.2926	1.09243	0.546215	+	0.837645i	$0.316068\pi$
$720$	0	0
$721$	−51.9460	−1.93457
$722$	0	0
$723$	−32.4597	−1.20719
$724$	0	0
$725$	11.9993	0.445643
$726$	0	0
$727$	−18.5234	−0.686995	−0.343498	−	0.939154i	$-0.611612\pi$
$727$	−18.5234	−0.686995	−0.343498	+	0.939154i	$0.611612\pi$
$728$	0	0
$729$	22.2588	0.824401
$730$	0	0
$731$	72.7351	2.69020
$732$	0	0
$733$	2.82819	0.104461	0.0522307	−	0.998635i	$-0.483367\pi$
$733$	2.82819	0.104461	0.0522307	+	0.998635i	$0.483367\pi$
$734$	0	0
$735$	5.66149	0.208827
$736$	0	0
$737$	−3.06036	−0.112730
$738$	0	0
$739$	26.5395	0.976273	0.488136	−	0.872767i	$-0.337677\pi$
$739$	26.5395	0.976273	0.488136	+	0.872767i	$0.337677\pi$
$740$	0	0
$741$	−0.390239	−0.0143358
$742$	0	0
$743$	−17.8462	−0.654712	−0.327356	−	0.944901i	$-0.606158\pi$
$743$	−17.8462	−0.654712	−0.327356	+	0.944901i	$0.606158\pi$
$744$	0	0
$745$	14.5738	0.533944
$746$	0	0
$747$	−1.90229	−0.0696012
$748$	0	0
$749$	53.7977	1.96573
$750$	0	0
$751$	−10.9111	−0.398152	−0.199076	−	0.979984i	$-0.563794\pi$
$751$	−10.9111	−0.398152	−0.199076	+	0.979984i	$0.563794\pi$
$752$	0	0
$753$	−7.37097	−0.268613
$754$	0	0
$755$	12.0821	0.439711
$756$	0	0
$757$	−13.2426	−0.481311	−0.240656	−	0.970611i	$-0.577362\pi$
$757$	−13.2426	−0.481311	−0.240656	+	0.970611i	$0.577362\pi$
$758$	0	0
$759$	10.2733	0.372897
$760$	0	0
$761$	−8.28994	−0.300510	−0.150255	−	0.988647i	$-0.548009\pi$
$761$	−8.28994	−0.300510	−0.150255	+	0.988647i	$0.548009\pi$
$762$	0	0
$763$	−19.2686	−0.697568
$764$	0	0
$765$	2.54234	0.0919184
$766$	0	0
$767$	−1.28704	−0.0464725
$768$	0	0
$769$	−37.5823	−1.35525	−0.677627	−	0.735406i	$-0.736992\pi$
$769$	−37.5823	−1.35525	−0.677627	+	0.735406i	$0.736992\pi$
$770$	0	0
$771$	29.4355	1.06009
$772$	0	0
$773$	−2.84891	−0.102468	−0.0512341	−	0.998687i	$-0.516315\pi$
$773$	−2.84891	−0.102468	−0.0512341	+	0.998687i	$0.516315\pi$
$774$	0	0
$775$	29.2133	1.04937
$776$	0	0
$777$	−40.3274	−1.44674
$778$	0	0
$779$	−3.53456	−0.126639
$780$	0	0
$781$	1.39318	0.0498520
$782$	0	0
$783$	13.5083	0.482747
$784$	0	0
$785$	16.1740	0.577275
$786$	0	0
$787$	−31.3656	−1.11806	−0.559032	−	0.829146i	$-0.688827\pi$
$787$	−31.3656	−1.11806	−0.559032	+	0.829146i	$0.688827\pi$
$788$	0	0
$789$	28.9020	1.02894
$790$	0	0
$791$	49.9648	1.77654
$792$	0	0
$793$	−2.03020	−0.0720946
$794$	0	0
$795$	20.7541	0.736073
$796$	0	0
$797$	26.8421	0.950796	0.475398	−	0.879771i	$-0.342304\pi$
$797$	26.8421	0.950796	0.475398	+	0.879771i	$0.342304\pi$
$798$	0	0
$799$	52.5874	1.86041
$800$	0	0
$801$	−3.18709	−0.112610
$802$	0	0
$803$	7.17107	0.253061
$804$	0	0
$805$	−7.71089	−0.271773
$806$	0	0
$807$	23.8685	0.840209
$808$	0	0
$809$	−48.1550	−1.69304	−0.846520	−	0.532356i	$-0.821307\pi$
$809$	−48.1550	−1.69304	−0.846520	+	0.532356i	$0.821307\pi$
$810$	0	0
$811$	−9.35042	−0.328338	−0.164169	−	0.986432i	$-0.552494\pi$
$811$	−9.35042	−0.328338	−0.164169	+	0.986432i	$0.552494\pi$
$812$	0	0
$813$	−27.6226	−0.968766
$814$	0	0
$815$	−6.12689	−0.214615
$816$	0	0
$817$	10.4324	0.364983
$818$	0	0
$819$	0.286256	0.0100026
$820$	0	0
$821$	−26.2947	−0.917691	−0.458846	−	0.888516i	$-0.651737\pi$
$821$	−26.2947	−0.917691	−0.458846	+	0.888516i	$0.651737\pi$
$822$	0	0
$823$	−36.9392	−1.28762	−0.643809	−	0.765186i	$-0.722647\pi$
$823$	−36.9392	−1.28762	−0.643809	+	0.765186i	$0.722647\pi$
$824$	0	0
$825$	−15.9672	−0.555906
$826$	0	0
$827$	23.0499	0.801522	0.400761	−	0.916183i	$-0.368746\pi$
$827$	23.0499	0.801522	0.400761	+	0.916183i	$0.368746\pi$
$828$	0	0
$829$	−30.2740	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$829$	−30.2740	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$830$	0	0
$831$	−39.0379	−1.35421
$832$	0	0
$833$	24.4900	0.848528
$834$	0	0
$835$	12.9479	0.448079
$836$	0	0
$837$	32.8870	1.13674
$838$	0	0
$839$	−54.0947	−1.86756	−0.933778	−	0.357852i	$-0.883509\pi$
$839$	−54.0947	−1.86756	−0.933778	+	0.357852i	$0.883509\pi$
$840$	0	0
$841$	−20.9911	−0.723829
$842$	0	0
$843$	−13.8752	−0.477888
$844$	0	0
$845$	−11.2941	−0.388529
$846$	0	0
$847$	22.2141	0.763285
$848$	0	0
$849$	−14.6462	−0.502655
$850$	0	0
$851$	18.3523	0.629110
$852$	0	0
$853$	48.7970	1.67078	0.835389	−	0.549659i	$-0.185242\pi$
$853$	48.7970	1.67078	0.835389	+	0.549659i	$0.185242\pi$
$854$	0	0
$855$	0.364647	0.0124707
$856$	0	0
$857$	53.4412	1.82552	0.912759	−	0.408499i	$-0.133948\pi$
$857$	53.4412	1.82552	0.912759	+	0.408499i	$0.133948\pi$
$858$	0	0
$859$	−43.9862	−1.50079	−0.750395	−	0.660990i	$-0.770137\pi$
$859$	−43.9862	−1.50079	−0.750395	+	0.660990i	$0.770137\pi$
$860$	0	0
$861$	21.1882	0.722094
$862$	0	0
$863$	−9.55378	−0.325215	−0.162607	−	0.986691i	$-0.551990\pi$
$863$	−9.55378	−0.325215	−0.162607	+	0.986691i	$0.551990\pi$
$864$	0	0
$865$	15.5405	0.528392
$866$	0	0
$867$	58.4417	1.98478
$868$	0	0
$869$	2.03683	0.0690948
$870$	0	0
$871$	0.317135	0.0107457
$872$	0	0
$873$	7.94011	0.268732
$874$	0	0
$875$	26.1172	0.882924
$876$	0	0
$877$	12.5890	0.425099	0.212550	−	0.977150i	$-0.431823\pi$
$877$	12.5890	0.425099	0.212550	+	0.977150i	$0.431823\pi$
$878$	0	0
$879$	−2.13836	−0.0721253
$880$	0	0
$881$	9.20686	0.310187	0.155093	−	0.987900i	$-0.450432\pi$
$881$	9.20686	0.310187	0.155093	+	0.987900i	$0.450432\pi$
$882$	0	0
$883$	−13.6152	−0.458187	−0.229093	−	0.973404i	$-0.573576\pi$
$883$	−13.6152	−0.458187	−0.229093	+	0.973404i	$0.573576\pi$
$884$	0	0
$885$	9.82812	0.330369
$886$	0	0
$887$	−49.4330	−1.65980	−0.829898	−	0.557915i	$-0.811602\pi$
$887$	−49.4330	−1.65980	−0.829898	+	0.557915i	$0.811602\pi$
$888$	0	0
$889$	13.3444	0.447556
$890$	0	0
$891$	−20.5311	−0.687816
$892$	0	0
$893$	7.54260	0.252403
$894$	0	0
$895$	5.39815	0.180440
$896$	0	0
$897$	−1.06459	−0.0355455
$898$	0	0
$899$	19.4985	0.650310
$900$	0	0
$901$	89.7764	2.99088
$902$	0	0
$903$	−62.5377	−2.08113
$904$	0	0
$905$	0.484995	0.0161218
$906$	0	0
$907$	21.5960	0.717085	0.358542	−	0.933513i	$-0.383274\pi$
$907$	21.5960	0.717085	0.358542	+	0.933513i	$0.383274\pi$
$908$	0	0
$909$	1.55740	0.0516556
$910$	0	0
$911$	−52.7015	−1.74608	−0.873039	−	0.487651i	$-0.837854\pi$
$911$	−52.7015	−1.74608	−0.873039	+	0.487651i	$0.837854\pi$
$912$	0	0
$913$	−9.26315	−0.306566
$914$	0	0
$915$	15.5030	0.512514
$916$	0	0
$917$	54.3757	1.79564
$918$	0	0
$919$	3.36043	0.110850	0.0554252	−	0.998463i	$-0.482349\pi$
$919$	3.36043	0.110850	0.0554252	+	0.998463i	$0.482349\pi$
$920$	0	0
$921$	5.93779	0.195657
$922$	0	0
$923$	−0.144371	−0.00475203
$924$	0	0
$925$	−28.5239	−0.937861
$926$	0	0
$927$	6.70148	0.220105
$928$	0	0
$929$	−21.9699	−0.720808	−0.360404	−	0.932796i	$-0.617361\pi$
$929$	−21.9699	−0.720808	−0.360404	+	0.932796i	$0.617361\pi$
$930$	0	0
$931$	3.51259	0.115121
$932$	0	0
$933$	−27.6694	−0.905855
$934$	0	0
$935$	12.3798	0.404864
$936$	0	0
$937$	47.2712	1.54428	0.772142	−	0.635450i	$-0.219185\pi$
$937$	47.2712	1.54428	0.772142	+	0.635450i	$0.219185\pi$
$938$	0	0
$939$	5.29453	0.172780
$940$	0	0
$941$	21.0381	0.685824	0.342912	−	0.939368i	$-0.388587\pi$
$941$	21.0381	0.685824	0.342912	+	0.939368i	$0.388587\pi$
$942$	0	0
$943$	−9.64243	−0.314001
$944$	0	0
$945$	13.4917	0.438885
$946$	0	0
$947$	7.39820	0.240409	0.120205	−	0.992749i	$-0.461645\pi$
$947$	7.39820	0.240409	0.120205	+	0.992749i	$0.461645\pi$
$948$	0	0
$949$	−0.743113	−0.0241225
$950$	0	0
$951$	51.0373	1.65500
$952$	0	0
$953$	−25.9157	−0.839492	−0.419746	−	0.907642i	$-0.637881\pi$
$953$	−25.9157	−0.839492	−0.419746	+	0.907642i	$0.637881\pi$
$954$	0	0
$955$	2.93464	0.0949628
$956$	0	0
$957$	−10.6573	−0.344502
$958$	0	0
$959$	−9.88621	−0.319243
$960$	0	0
$961$	16.4706	0.531311
$962$	0	0
$963$	−6.94037	−0.223650
$964$	0	0
$965$	7.47721	0.240700
$966$	0	0
$967$	−57.7363	−1.85667	−0.928337	−	0.371741i	$-0.878761\pi$
$967$	−57.7363	−1.85667	−0.928337	+	0.371741i	$0.878761\pi$
$968$	0	0
$969$	12.8904	0.414098
$970$	0	0
$971$	−20.2362	−0.649410	−0.324705	−	0.945815i	$-0.605265\pi$
$971$	−20.2362	−0.649410	−0.324705	+	0.945815i	$0.605265\pi$
$972$	0	0
$973$	−68.7447	−2.20385
$974$	0	0
$975$	1.65462	0.0529904
$976$	0	0
$977$	−57.0862	−1.82635	−0.913175	−	0.407568i	$-0.866377\pi$
$977$	−57.0862	−1.82635	−0.913175	+	0.407568i	$0.866377\pi$
$978$	0	0
$979$	−15.5194	−0.496003
$980$	0	0
$981$	2.48581	0.0793658
$982$	0	0
$983$	−14.2741	−0.455272	−0.227636	−	0.973746i	$-0.573100\pi$
$983$	−14.2741	−0.455272	−0.227636	+	0.973746i	$0.573100\pi$
$984$	0	0
$985$	17.2064	0.548241
$986$	0	0
$987$	−45.2147	−1.43920
$988$	0	0
$989$	28.4599	0.904973
$990$	0	0
$991$	−44.3433	−1.40861	−0.704305	−	0.709897i	$-0.748741\pi$
$991$	−44.3433	−1.40861	−0.704305	+	0.709897i	$0.748741\pi$
$992$	0	0
$993$	−13.7851	−0.437458
$994$	0	0
$995$	−16.5784	−0.525571
$996$	0	0
$997$	15.6375	0.495245	0.247623	−	0.968857i	$-0.420351\pi$
$997$	15.6375	0.495245	0.247623	+	0.968857i	$0.420351\pi$
$998$	0	0
$999$	−32.1109	−1.01595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.15	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.15	✓	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.84886 q^{3} +0.871764 q^{5} -3.24231 q^{7} +0.418286 q^{9} +O(q^{10})\)	\(q+1.84886 q^{3} +0.871764 q^{5} -3.24231 q^{7} +0.418286 q^{9} +2.03683 q^{11} -0.211070 q^{13} +1.61177 q^{15} +6.97206 q^{17} +1.00000 q^{19} -5.99458 q^{21} +2.72804 q^{23} -4.24003 q^{25} -4.77323 q^{27} -2.83001 q^{29} -6.88989 q^{31} +3.76582 q^{33} -2.82653 q^{35} +6.72730 q^{37} -0.390239 q^{39} -3.53456 q^{41} +10.4324 q^{43} +0.364647 q^{45} +7.54260 q^{47} +3.51259 q^{49} +12.8904 q^{51} +12.8766 q^{53} +1.77564 q^{55} +1.84886 q^{57} +6.09771 q^{59} +9.61862 q^{61} -1.35621 q^{63} -0.184003 q^{65} -1.50251 q^{67} +5.04376 q^{69} +0.683995 q^{71} +3.52069 q^{73} -7.83922 q^{75} -6.60405 q^{77} +1.00000 q^{79} -10.0799 q^{81} -4.54782 q^{83} +6.07799 q^{85} -5.23229 q^{87} -7.61940 q^{89} +0.684355 q^{91} -12.7385 q^{93} +0.871764 q^{95} +18.9825 q^{97} +0.851979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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