LMFDB - Embedded newform 6004.2.a.g.1.14

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:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q-0.338332 q^{3} +0.413944 q^{5} -1.61145 q^{7} -2.88553 q^{9} +O(q^{10})$	$q-0.338332 q^{3} +0.413944 q^{5} -1.61145 q^{7} -2.88553 q^{9} +1.62423 q^{11} +1.42248 q^{13} -0.140051 q^{15} +0.598456 q^{17} +1.00000 q^{19} +0.545204 q^{21} -0.815594 q^{23} -4.82865 q^{25} +1.99126 q^{27} +7.29709 q^{29} -4.96239 q^{31} -0.549530 q^{33} -0.667050 q^{35} +5.22799 q^{37} -0.481270 q^{39} +2.10875 q^{41} -5.53106 q^{43} -1.19445 q^{45} +5.12276 q^{47} -4.40324 q^{49} -0.202477 q^{51} -5.32257 q^{53} +0.672341 q^{55} -0.338332 q^{57} +11.5040 q^{59} -12.2460 q^{61} +4.64988 q^{63} +0.588827 q^{65} -4.41080 q^{67} +0.275942 q^{69} +13.9441 q^{71} -6.82902 q^{73} +1.63369 q^{75} -2.61736 q^{77} -1.00000 q^{79} +7.98289 q^{81} -11.3773 q^{83} +0.247727 q^{85} -2.46884 q^{87} +0.614336 q^{89} -2.29225 q^{91} +1.67893 q^{93} +0.413944 q^{95} -2.40472 q^{97} -4.68677 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	−0.338332	−0.195336	−0.0976680	−	0.995219i	$-0.531138\pi$
$3$	−0.338332	−0.195336	−0.0976680	+	0.995219i	$0.531138\pi$
$4$	0	0
$5$	0.413944	0.185122	0.0925608	−	0.995707i	$-0.470495\pi$
$5$	0.413944	0.185122	0.0925608	+	0.995707i	$0.470495\pi$
$6$	0	0
$7$	−1.61145	−0.609070	−0.304535	−	0.952501i	$-0.598501\pi$
$7$	−1.61145	−0.609070	−0.304535	+	0.952501i	$0.598501\pi$
$8$	0	0
$9$	−2.88553	−0.961844
$10$	0	0
$11$	1.62423	0.489724	0.244862	−	0.969558i	$-0.421257\pi$
$11$	1.62423	0.489724	0.244862	+	0.969558i	$0.421257\pi$
$12$	0	0
$13$	1.42248	0.394525	0.197262	−	0.980351i	$-0.436795\pi$
$13$	1.42248	0.394525	0.197262	+	0.980351i	$0.436795\pi$
$14$	0	0
$15$	−0.140051	−0.0361609
$16$	0	0
$17$	0.598456	0.145147	0.0725734	−	0.997363i	$-0.476879\pi$
$17$	0.598456	0.145147	0.0725734	+	0.997363i	$0.476879\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.545204	0.118973
$22$	0	0
$23$	−0.815594	−0.170063	−0.0850315	−	0.996378i	$-0.527099\pi$
$23$	−0.815594	−0.170063	−0.0850315	+	0.996378i	$0.527099\pi$
$24$	0	0
$25$	−4.82865	−0.965730
$26$	0	0
$27$	1.99126	0.383219
$28$	0	0
$29$	7.29709	1.35504	0.677518	−	0.735506i	$-0.263056\pi$
$29$	7.29709	1.35504	0.677518	+	0.735506i	$0.263056\pi$
$30$	0	0
$31$	−4.96239	−0.891271	−0.445635	−	0.895215i	$-0.647022\pi$
$31$	−4.96239	−0.891271	−0.445635	+	0.895215i	$0.647022\pi$
$32$	0	0
$33$	−0.549530	−0.0956608
$34$	0	0
$35$	−0.667050	−0.112752
$36$	0	0
$37$	5.22799	0.859476	0.429738	−	0.902954i	$-0.358606\pi$
$37$	5.22799	0.859476	0.429738	+	0.902954i	$0.358606\pi$
$38$	0	0
$39$	−0.481270	−0.0770649
$40$	0	0
$41$	2.10875	0.329332	0.164666	−	0.986349i	$-0.447345\pi$
$41$	2.10875	0.329332	0.164666	+	0.986349i	$0.447345\pi$
$42$	0	0
$43$	−5.53106	−0.843478	−0.421739	−	0.906717i	$-0.638580\pi$
$43$	−5.53106	−0.843478	−0.421739	+	0.906717i	$0.638580\pi$
$44$	0	0
$45$	−1.19445	−0.178058
$46$	0	0
$47$	5.12276	0.747231	0.373616	−	0.927584i	$-0.378118\pi$
$47$	5.12276	0.747231	0.373616	+	0.927584i	$0.378118\pi$
$48$	0	0
$49$	−4.40324	−0.629034
$50$	0	0
$51$	−0.202477	−0.0283524
$52$	0	0
$53$	−5.32257	−0.731111	−0.365556	−	0.930790i	$-0.619121\pi$
$53$	−5.32257	−0.731111	−0.365556	+	0.930790i	$0.619121\pi$
$54$	0	0
$55$	0.672341	0.0906585
$56$	0	0
$57$	−0.338332	−0.0448132
$58$	0	0
$59$	11.5040	1.49769	0.748846	−	0.662744i	$-0.230608\pi$
$59$	11.5040	1.49769	0.748846	+	0.662744i	$0.230608\pi$
$60$	0	0
$61$	−12.2460	−1.56794	−0.783968	−	0.620802i	$-0.786807\pi$
$61$	−12.2460	−1.56794	−0.783968	+	0.620802i	$0.786807\pi$
$62$	0	0
$63$	4.64988	0.585830
$64$	0	0
$65$	0.588827	0.0730350
$66$	0	0
$67$	−4.41080	−0.538865	−0.269433	−	0.963019i	$-0.586836\pi$
$67$	−4.41080	−0.538865	−0.269433	+	0.963019i	$0.586836\pi$
$68$	0	0
$69$	0.275942	0.0332195
$70$	0	0
$71$	13.9441	1.65487	0.827433	−	0.561565i	$-0.189801\pi$
$71$	13.9441	1.65487	0.827433	+	0.561565i	$0.189801\pi$
$72$	0	0
$73$	−6.82902	−0.799276	−0.399638	−	0.916673i	$-0.630864\pi$
$73$	−6.82902	−0.799276	−0.399638	+	0.916673i	$0.630864\pi$
$74$	0	0
$75$	1.63369	0.188642
$76$	0	0
$77$	−2.61736	−0.298276
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	7.98289	0.886987
$82$	0	0
$83$	−11.3773	−1.24882	−0.624410	−	0.781097i	$-0.714661\pi$
$83$	−11.3773	−1.24882	−0.624410	+	0.781097i	$0.714661\pi$
$84$	0	0
$85$	0.247727	0.0268698
$86$	0	0
$87$	−2.46884	−0.264687
$88$	0	0
$89$	0.614336	0.0651195	0.0325597	−	0.999470i	$-0.489634\pi$
$89$	0.614336	0.0651195	0.0325597	+	0.999470i	$0.489634\pi$
$90$	0	0
$91$	−2.29225	−0.240293
$92$	0	0
$93$	1.67893	0.174097
$94$	0	0
$95$	0.413944	0.0424698
$96$	0	0
$97$	−2.40472	−0.244163	−0.122081	−	0.992520i	$-0.538957\pi$
$97$	−2.40472	−0.244163	−0.122081	+	0.992520i	$0.538957\pi$
$98$	0	0
$99$	−4.68677	−0.471038
$100$	0	0
$101$	−12.5520	−1.24897	−0.624484	−	0.781037i	$-0.714691\pi$
$101$	−12.5520	−1.24897	−0.624484	+	0.781037i	$0.714691\pi$
$102$	0	0
$103$	10.1972	1.00476	0.502381	−	0.864646i	$-0.332457\pi$
$103$	10.1972	1.00476	0.502381	+	0.864646i	$0.332457\pi$
$104$	0	0
$105$	0.225684	0.0220245
$106$	0	0
$107$	0.0950657	0.00919035	0.00459518	−	0.999989i	$-0.498537\pi$
$107$	0.0950657	0.00919035	0.00459518	+	0.999989i	$0.498537\pi$
$108$	0	0
$109$	1.49100	0.142811	0.0714057	−	0.997447i	$-0.477251\pi$
$109$	1.49100	0.142811	0.0714057	+	0.997447i	$0.477251\pi$
$110$	0	0
$111$	−1.76880	−0.167887
$112$	0	0
$113$	2.10963	0.198457	0.0992285	−	0.995065i	$-0.468363\pi$
$113$	2.10963	0.198457	0.0992285	+	0.995065i	$0.468363\pi$
$114$	0	0
$115$	−0.337610	−0.0314823
$116$	0	0
$117$	−4.10461	−0.379471
$118$	0	0
$119$	−0.964381	−0.0884046
$120$	0	0
$121$	−8.36187	−0.760170
$122$	0	0
$123$	−0.713459	−0.0643304
$124$	0	0
$125$	−4.06851	−0.363899
$126$	0	0
$127$	−8.22200	−0.729585	−0.364792	−	0.931089i	$-0.618860\pi$
$127$	−8.22200	−0.729585	−0.364792	+	0.931089i	$0.618860\pi$
$128$	0	0
$129$	1.87133	0.164762
$130$	0	0
$131$	−15.0824	−1.31776	−0.658879	−	0.752249i	$-0.728969\pi$
$131$	−15.0824	−1.31776	−0.658879	+	0.752249i	$0.728969\pi$
$132$	0	0
$133$	−1.61145	−0.139730
$134$	0	0
$135$	0.824272	0.0709421
$136$	0	0
$137$	9.71607	0.830100	0.415050	−	0.909799i	$-0.363764\pi$
$137$	9.71607	0.830100	0.415050	+	0.909799i	$0.363764\pi$
$138$	0	0
$139$	−4.01186	−0.340281	−0.170141	−	0.985420i	$-0.554422\pi$
$139$	−4.01186	−0.340281	−0.170141	+	0.985420i	$0.554422\pi$
$140$	0	0
$141$	−1.73319	−0.145961
$142$	0	0
$143$	2.31044	0.193208
$144$	0	0
$145$	3.02059	0.250846
$146$	0	0
$147$	1.48976	0.122873
$148$	0	0
$149$	−14.7351	−1.20715	−0.603573	−	0.797308i	$-0.706257\pi$
$149$	−14.7351	−1.20715	−0.603573	+	0.797308i	$0.706257\pi$
$150$	0	0
$151$	−14.2681	−1.16112	−0.580561	−	0.814217i	$-0.697167\pi$
$151$	−14.2681	−1.16112	−0.580561	+	0.814217i	$0.697167\pi$
$152$	0	0
$153$	−1.72686	−0.139609
$154$	0	0
$155$	−2.05415	−0.164993
$156$	0	0
$157$	20.7206	1.65368	0.826841	−	0.562436i	$-0.190136\pi$
$157$	20.7206	1.65368	0.826841	+	0.562436i	$0.190136\pi$
$158$	0	0
$159$	1.80080	0.142812
$160$	0	0
$161$	1.31429	0.103580
$162$	0	0
$163$	−1.21351	−0.0950497	−0.0475248	−	0.998870i	$-0.515133\pi$
$163$	−1.21351	−0.0950497	−0.0475248	+	0.998870i	$0.515133\pi$
$164$	0	0
$165$	−0.227475	−0.0177089
$166$	0	0
$167$	10.8525	0.839789	0.419895	−	0.907573i	$-0.362067\pi$
$167$	10.8525	0.839789	0.419895	+	0.907573i	$0.362067\pi$
$168$	0	0
$169$	−10.9766	−0.844350
$170$	0	0
$171$	−2.88553	−0.220662
$172$	0	0
$173$	−15.4950	−1.17806	−0.589031	−	0.808110i	$-0.700490\pi$
$173$	−15.4950	−1.17806	−0.589031	+	0.808110i	$0.700490\pi$
$174$	0	0
$175$	7.78112	0.588197
$176$	0	0
$177$	−3.89217	−0.292553
$178$	0	0
$179$	−9.59731	−0.717336	−0.358668	−	0.933465i	$-0.616769\pi$
$179$	−9.59731	−0.717336	−0.358668	+	0.933465i	$0.616769\pi$
$180$	0	0
$181$	12.6940	0.943538	0.471769	−	0.881722i	$-0.343616\pi$
$181$	12.6940	0.943538	0.471769	+	0.881722i	$0.343616\pi$
$182$	0	0
$183$	4.14320	0.306274
$184$	0	0
$185$	2.16409	0.159107
$186$	0	0
$187$	0.972031	0.0710819
$188$	0	0
$189$	−3.20882	−0.233407
$190$	0	0
$191$	−15.1508	−1.09628	−0.548138	−	0.836388i	$-0.684663\pi$
$191$	−15.1508	−1.09628	−0.548138	+	0.836388i	$0.684663\pi$
$192$	0	0
$193$	−16.2264	−1.16800	−0.584001	−	0.811753i	$-0.698514\pi$
$193$	−16.2264	−1.16800	−0.584001	+	0.811753i	$0.698514\pi$
$194$	0	0
$195$	−0.199219	−0.0142664
$196$	0	0
$197$	−12.6227	−0.899332	−0.449666	−	0.893197i	$-0.648457\pi$
$197$	−12.6227	−0.899332	−0.449666	+	0.893197i	$0.648457\pi$
$198$	0	0
$199$	−13.8752	−0.983589	−0.491794	−	0.870711i	$-0.663659\pi$
$199$	−13.8752	−0.983589	−0.491794	+	0.870711i	$0.663659\pi$
$200$	0	0
$201$	1.49232	0.105260
$202$	0	0
$203$	−11.7589	−0.825312
$204$	0	0
$205$	0.872906	0.0609664
$206$	0	0
$207$	2.35342	0.163574
$208$	0	0
$209$	1.62423	0.112350
$210$	0	0
$211$	−8.91556	−0.613772	−0.306886	−	0.951746i	$-0.599287\pi$
$211$	−8.91556	−0.613772	−0.306886	+	0.951746i	$0.599287\pi$
$212$	0	0
$213$	−4.71775	−0.323255
$214$	0	0
$215$	−2.28955	−0.156146
$216$	0	0
$217$	7.99663	0.542846
$218$	0	0
$219$	2.31047	0.156127
$220$	0	0
$221$	0.851291	0.0572641
$222$	0	0
$223$	−0.639245	−0.0428070	−0.0214035	−	0.999771i	$-0.506813\pi$
$223$	−0.639245	−0.0428070	−0.0214035	+	0.999771i	$0.506813\pi$
$224$	0	0
$225$	13.9332	0.928881
$226$	0	0
$227$	−6.83330	−0.453542	−0.226771	−	0.973948i	$-0.572817\pi$
$227$	−6.83330	−0.453542	−0.226771	+	0.973948i	$0.572817\pi$
$228$	0	0
$229$	−8.01872	−0.529892	−0.264946	−	0.964263i	$-0.585354\pi$
$229$	−8.01872	−0.529892	−0.264946	+	0.964263i	$0.585354\pi$
$230$	0	0
$231$	0.885538	0.0582641
$232$	0	0
$233$	23.0368	1.50919	0.754594	−	0.656192i	$-0.227834\pi$
$233$	23.0368	1.50919	0.754594	+	0.656192i	$0.227834\pi$
$234$	0	0
$235$	2.12054	0.138329
$236$	0	0
$237$	0.338332	0.0219770
$238$	0	0
$239$	10.2334	0.661941	0.330971	−	0.943641i	$-0.392624\pi$
$239$	10.2334	0.661941	0.330971	+	0.943641i	$0.392624\pi$
$240$	0	0
$241$	13.6100	0.876699	0.438349	−	0.898805i	$-0.355563\pi$
$241$	13.6100	0.876699	0.438349	+	0.898805i	$0.355563\pi$
$242$	0	0
$243$	−8.67466	−0.556480
$244$	0	0
$245$	−1.82269	−0.116448
$246$	0	0
$247$	1.42248	0.0905102
$248$	0	0
$249$	3.84930	0.243940
$250$	0	0
$251$	−21.9818	−1.38748	−0.693738	−	0.720227i	$-0.744037\pi$
$251$	−21.9818	−1.38748	−0.693738	+	0.720227i	$0.744037\pi$
$252$	0	0
$253$	−1.32471	−0.0832840
$254$	0	0
$255$	−0.0838141	−0.00524864
$256$	0	0
$257$	−30.9121	−1.92825	−0.964123	−	0.265457i	$-0.914477\pi$
$257$	−30.9121	−1.92825	−0.964123	+	0.265457i	$0.914477\pi$
$258$	0	0
$259$	−8.42463	−0.523481
$260$	0	0
$261$	−21.0560	−1.30333
$262$	0	0
$263$	3.15646	0.194636	0.0973179	−	0.995253i	$-0.468974\pi$
$263$	3.15646	0.194636	0.0973179	+	0.995253i	$0.468974\pi$
$264$	0	0
$265$	−2.20325	−0.135344
$266$	0	0
$267$	−0.207849	−0.0127202
$268$	0	0
$269$	−21.8955	−1.33499	−0.667497	−	0.744613i	$-0.732634\pi$
$269$	−21.8955	−1.33499	−0.667497	+	0.744613i	$0.732634\pi$
$270$	0	0
$271$	−7.39092	−0.448966	−0.224483	−	0.974478i	$-0.572069\pi$
$271$	−7.39092	−0.448966	−0.224483	+	0.974478i	$0.572069\pi$
$272$	0	0
$273$	0.775542	0.0469380
$274$	0	0
$275$	−7.84285	−0.472941
$276$	0	0
$277$	−25.4178	−1.52721	−0.763605	−	0.645684i	$-0.776572\pi$
$277$	−25.4178	−1.52721	−0.763605	+	0.645684i	$0.776572\pi$
$278$	0	0
$279$	14.3191	0.857263
$280$	0	0
$281$	−10.3840	−0.619457	−0.309729	−	0.950825i	$-0.600238\pi$
$281$	−10.3840	−0.619457	−0.309729	+	0.950825i	$0.600238\pi$
$282$	0	0
$283$	16.9184	1.00570	0.502849	−	0.864375i	$-0.332285\pi$
$283$	16.9184	1.00570	0.502849	+	0.864375i	$0.332285\pi$
$284$	0	0
$285$	−0.140051	−0.00829588
$286$	0	0
$287$	−3.39815	−0.200586
$288$	0	0
$289$	−16.6419	−0.978932
$290$	0	0
$291$	0.813595	0.0476938
$292$	0	0
$293$	−6.66743	−0.389515	−0.194758	−	0.980851i	$-0.562392\pi$
$293$	−6.66743	−0.389515	−0.194758	+	0.980851i	$0.562392\pi$
$294$	0	0
$295$	4.76201	0.277255
$296$	0	0
$297$	3.23427	0.187672
$298$	0	0
$299$	−1.16017	−0.0670941
$300$	0	0
$301$	8.91301	0.513737
$302$	0	0
$303$	4.24674	0.243969
$304$	0	0
$305$	−5.06915	−0.290259
$306$	0	0
$307$	13.4905	0.769945	0.384973	−	0.922928i	$-0.374211\pi$
$307$	13.4905	0.769945	0.384973	+	0.922928i	$0.374211\pi$
$308$	0	0
$309$	−3.45005	−0.196266
$310$	0	0
$311$	27.7482	1.57346	0.786728	−	0.617300i	$-0.211773\pi$
$311$	27.7482	1.57346	0.786728	+	0.617300i	$0.211773\pi$
$312$	0	0
$313$	30.8571	1.74415	0.872074	−	0.489375i	$-0.162775\pi$
$313$	30.8571	1.74415	0.872074	+	0.489375i	$0.162775\pi$
$314$	0	0
$315$	1.92479	0.108450
$316$	0	0
$317$	22.6357	1.27135	0.635674	−	0.771957i	$-0.280722\pi$
$317$	22.6357	1.27135	0.635674	+	0.771957i	$0.280722\pi$
$318$	0	0
$319$	11.8522	0.663594
$320$	0	0
$321$	−0.0321638	−0.00179521
$322$	0	0
$323$	0.598456	0.0332990
$324$	0	0
$325$	−6.86866	−0.381004
$326$	0	0
$327$	−0.504452	−0.0278962
$328$	0	0
$329$	−8.25506	−0.455116
$330$	0	0
$331$	10.4160	0.572514	0.286257	−	0.958153i	$-0.407589\pi$
$331$	10.4160	0.572514	0.286257	+	0.958153i	$0.407589\pi$
$332$	0	0
$333$	−15.0855	−0.826681
$334$	0	0
$335$	−1.82583	−0.0997555
$336$	0	0
$337$	−24.6952	−1.34523	−0.672615	−	0.739992i	$-0.734829\pi$
$337$	−24.6952	−1.34523	−0.672615	+	0.739992i	$0.734829\pi$
$338$	0	0
$339$	−0.713754	−0.0387658
$340$	0	0
$341$	−8.06006	−0.436477
$342$	0	0
$343$	18.3757	0.992196
$344$	0	0
$345$	0.114224	0.00614964
$346$	0	0
$347$	−22.6736	−1.21718	−0.608590	−	0.793485i	$-0.708265\pi$
$347$	−22.6736	−1.21718	−0.608590	+	0.793485i	$0.708265\pi$
$348$	0	0
$349$	0.284829	0.0152466	0.00762328	−	0.999971i	$-0.497573\pi$
$349$	0.284829	0.0152466	0.00762328	+	0.999971i	$0.497573\pi$
$350$	0	0
$351$	2.83253	0.151189
$352$	0	0
$353$	−11.6885	−0.622115	−0.311057	−	0.950391i	$-0.600683\pi$
$353$	−11.6885	−0.622115	−0.311057	+	0.950391i	$0.600683\pi$
$354$	0	0
$355$	5.77210	0.306351
$356$	0	0
$357$	0.326281	0.0172686
$358$	0	0
$359$	13.3124	0.702604	0.351302	−	0.936262i	$-0.385739\pi$
$359$	13.3124	0.702604	0.351302	+	0.936262i	$0.385739\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.82909	0.148489
$364$	0	0
$365$	−2.82683	−0.147963
$366$	0	0
$367$	−23.7394	−1.23919	−0.619594	−	0.784922i	$-0.712703\pi$
$367$	−23.7394	−1.23919	−0.619594	+	0.784922i	$0.712703\pi$
$368$	0	0
$369$	−6.08487	−0.316766
$370$	0	0
$371$	8.57704	0.445298
$372$	0	0
$373$	−21.4212	−1.10915	−0.554574	−	0.832134i	$-0.687119\pi$
$373$	−21.4212	−1.10915	−0.554574	+	0.832134i	$0.687119\pi$
$374$	0	0
$375$	1.37651	0.0710826
$376$	0	0
$377$	10.3800	0.534595
$378$	0	0
$379$	19.8946	1.02192	0.510960	−	0.859605i	$-0.329290\pi$
$379$	19.8946	1.02192	0.510960	+	0.859605i	$0.329290\pi$
$380$	0	0
$381$	2.78177	0.142514
$382$	0	0
$383$	25.0286	1.27890	0.639451	−	0.768832i	$-0.279162\pi$
$383$	25.0286	1.27890	0.639451	+	0.768832i	$0.279162\pi$
$384$	0	0
$385$	−1.08344	−0.0552174
$386$	0	0
$387$	15.9600	0.811294
$388$	0	0
$389$	22.8639	1.15924	0.579622	−	0.814885i	$-0.303200\pi$
$389$	22.8639	1.15924	0.579622	+	0.814885i	$0.303200\pi$
$390$	0	0
$391$	−0.488097	−0.0246841
$392$	0	0
$393$	5.10287	0.257406
$394$	0	0
$395$	−0.413944	−0.0208278
$396$	0	0
$397$	−22.0635	−1.10733	−0.553667	−	0.832738i	$-0.686772\pi$
$397$	−22.0635	−1.10733	−0.553667	+	0.832738i	$0.686772\pi$
$398$	0	0
$399$	0.545204	0.0272944
$400$	0	0
$401$	−18.7571	−0.936685	−0.468342	−	0.883547i	$-0.655149\pi$
$401$	−18.7571	−0.936685	−0.468342	+	0.883547i	$0.655149\pi$
$402$	0	0
$403$	−7.05889	−0.351629
$404$	0	0
$405$	3.30447	0.164200
$406$	0	0
$407$	8.49146	0.420906
$408$	0	0
$409$	28.6473	1.41652	0.708259	−	0.705953i	$-0.249481\pi$
$409$	28.6473	1.41652	0.708259	+	0.705953i	$0.249481\pi$
$410$	0	0
$411$	−3.28726	−0.162149
$412$	0	0
$413$	−18.5381	−0.912200
$414$	0	0
$415$	−4.70957	−0.231184
$416$	0	0
$417$	1.35734	0.0664692
$418$	0	0
$419$	17.8953	0.874242	0.437121	−	0.899403i	$-0.355998\pi$
$419$	17.8953	0.874242	0.437121	+	0.899403i	$0.355998\pi$
$420$	0	0
$421$	−27.9131	−1.36040	−0.680201	−	0.733026i	$-0.738107\pi$
$421$	−27.9131	−1.36040	−0.680201	+	0.733026i	$0.738107\pi$
$422$	0	0
$423$	−14.7819	−0.718720
$424$	0	0
$425$	−2.88973	−0.140173
$426$	0	0
$427$	19.7337	0.954983
$428$	0	0
$429$	−0.781695	−0.0377406
$430$	0	0
$431$	20.2859	0.977136	0.488568	−	0.872526i	$-0.337519\pi$
$431$	20.2859	0.977136	0.488568	+	0.872526i	$0.337519\pi$
$432$	0	0
$433$	−5.31006	−0.255185	−0.127593	−	0.991827i	$-0.540725\pi$
$433$	−5.31006	−0.255185	−0.127593	+	0.991827i	$0.540725\pi$
$434$	0	0
$435$	−1.02196	−0.0489993
$436$	0	0
$437$	−0.815594	−0.0390151
$438$	0	0
$439$	−32.0477	−1.52955	−0.764777	−	0.644295i	$-0.777151\pi$
$439$	−32.0477	−1.52955	−0.764777	+	0.644295i	$0.777151\pi$
$440$	0	0
$441$	12.7057	0.605032
$442$	0	0
$443$	−12.7242	−0.604543	−0.302271	−	0.953222i	$-0.597745\pi$
$443$	−12.7242	−0.604543	−0.302271	+	0.953222i	$0.597745\pi$
$444$	0	0
$445$	0.254301	0.0120550
$446$	0	0
$447$	4.98535	0.235799
$448$	0	0
$449$	27.6838	1.30648	0.653240	−	0.757151i	$-0.273409\pi$
$449$	27.6838	1.30648	0.653240	+	0.757151i	$0.273409\pi$
$450$	0	0
$451$	3.42510	0.161282
$452$	0	0
$453$	4.82736	0.226809
$454$	0	0
$455$	−0.948865	−0.0444835
$456$	0	0
$457$	−38.5568	−1.80361	−0.901806	−	0.432141i	$-0.857758\pi$
$457$	−38.5568	−1.80361	−0.901806	+	0.432141i	$0.857758\pi$
$458$	0	0
$459$	1.19168	0.0556230
$460$	0	0
$461$	−20.1377	−0.937905	−0.468953	−	0.883223i	$-0.655369\pi$
$461$	−20.1377	−0.937905	−0.468953	+	0.883223i	$0.655369\pi$
$462$	0	0
$463$	−3.29768	−0.153256	−0.0766280	−	0.997060i	$-0.524415\pi$
$463$	−3.29768	−0.153256	−0.0766280	+	0.997060i	$0.524415\pi$
$464$	0	0
$465$	0.694985	0.0322292
$466$	0	0
$467$	−20.3525	−0.941803	−0.470902	−	0.882186i	$-0.656071\pi$
$467$	−20.3525	−0.941803	−0.470902	+	0.882186i	$0.656071\pi$
$468$	0	0
$469$	7.10778	0.328207
$470$	0	0
$471$	−7.01043	−0.323024
$472$	0	0
$473$	−8.98372	−0.413072
$474$	0	0
$475$	−4.82865	−0.221554
$476$	0	0
$477$	15.3584	0.703215
$478$	0	0
$479$	−2.45347	−0.112102	−0.0560509	−	0.998428i	$-0.517851\pi$
$479$	−2.45347	−0.112102	−0.0560509	+	0.998428i	$0.517851\pi$
$480$	0	0
$481$	7.43670	0.339084
$482$	0	0
$483$	−0.444665	−0.0202330
$484$	0	0
$485$	−0.995422	−0.0451998
$486$	0	0
$487$	33.9858	1.54004	0.770021	−	0.638019i	$-0.220246\pi$
$487$	33.9858	1.54004	0.770021	+	0.638019i	$0.220246\pi$
$488$	0	0
$489$	0.410570	0.0185666
$490$	0	0
$491$	29.1688	1.31637	0.658184	−	0.752857i	$-0.271325\pi$
$491$	29.1688	1.31637	0.658184	+	0.752857i	$0.271325\pi$
$492$	0	0
$493$	4.36699	0.196679
$494$	0	0
$495$	−1.94006	−0.0871993
$496$	0	0
$497$	−22.4703	−1.00793
$498$	0	0
$499$	17.0035	0.761181	0.380591	−	0.924744i	$-0.375721\pi$
$499$	17.0035	0.761181	0.380591	+	0.924744i	$0.375721\pi$
$500$	0	0
$501$	−3.67174	−0.164041
$502$	0	0
$503$	−0.648168	−0.0289004	−0.0144502	−	0.999896i	$-0.504600\pi$
$503$	−0.648168	−0.0289004	−0.0144502	+	0.999896i	$0.504600\pi$
$504$	0	0
$505$	−5.19582	−0.231211
$506$	0	0
$507$	3.71372	0.164932
$508$	0	0
$509$	−25.1086	−1.11292	−0.556461	−	0.830874i	$-0.687841\pi$
$509$	−25.1086	−1.11292	−0.556461	+	0.830874i	$0.687841\pi$
$510$	0	0
$511$	11.0046	0.486815
$512$	0	0
$513$	1.99126	0.0879164
$514$	0	0
$515$	4.22108	0.186003
$516$	0	0
$517$	8.32055	0.365937
$518$	0	0
$519$	5.24245	0.230118
$520$	0	0
$521$	−12.6135	−0.552609	−0.276305	−	0.961070i	$-0.589110\pi$
$521$	−12.6135	−0.552609	−0.276305	+	0.961070i	$0.589110\pi$
$522$	0	0
$523$	14.7001	0.642789	0.321394	−	0.946945i	$-0.395848\pi$
$523$	14.7001	0.642789	0.321394	+	0.946945i	$0.395848\pi$
$524$	0	0
$525$	−2.63260	−0.114896
$526$	0	0
$527$	−2.96977	−0.129365
$528$	0	0
$529$	−22.3348	−0.971079
$530$	0	0
$531$	−33.1951	−1.44055
$532$	0	0
$533$	2.99966	0.129930
$534$	0	0
$535$	0.0393519	0.00170133
$536$	0	0
$537$	3.24708	0.140122
$538$	0	0
$539$	−7.15187	−0.308053
$540$	0	0
$541$	−8.11654	−0.348957	−0.174479	−	0.984661i	$-0.555824\pi$
$541$	−8.11654	−0.348957	−0.174479	+	0.984661i	$0.555824\pi$
$542$	0	0
$543$	−4.29479	−0.184307
$544$	0	0
$545$	0.617189	0.0264375
$546$	0	0
$547$	−6.71606	−0.287158	−0.143579	−	0.989639i	$-0.545861\pi$
$547$	−6.71606	−0.287158	−0.143579	+	0.989639i	$0.545861\pi$
$548$	0	0
$549$	35.3361	1.50811
$550$	0	0
$551$	7.29709	0.310867
$552$	0	0
$553$	1.61145	0.0685257
$554$	0	0
$555$	−0.732183	−0.0310794
$556$	0	0
$557$	5.80162	0.245823	0.122911	−	0.992418i	$-0.460777\pi$
$557$	5.80162	0.245823	0.122911	+	0.992418i	$0.460777\pi$
$558$	0	0
$559$	−7.86782	−0.332773
$560$	0	0
$561$	−0.328869	−0.0138849
$562$	0	0
$563$	−6.42477	−0.270772	−0.135386	−	0.990793i	$-0.543227\pi$
$563$	−6.42477	−0.270772	−0.135386	+	0.990793i	$0.543227\pi$
$564$	0	0
$565$	0.873268	0.0367386
$566$	0	0
$567$	−12.8640	−0.540237
$568$	0	0
$569$	−15.4100	−0.646023	−0.323011	−	0.946395i	$-0.604695\pi$
$569$	−15.4100	−0.646023	−0.323011	+	0.946395i	$0.604695\pi$
$570$	0	0
$571$	−23.3203	−0.975923	−0.487962	−	0.872865i	$-0.662259\pi$
$571$	−23.3203	−0.975923	−0.487962	+	0.872865i	$0.662259\pi$
$572$	0	0
$573$	5.12601	0.214142
$574$	0	0
$575$	3.93822	0.164235
$576$	0	0
$577$	−10.5312	−0.438418	−0.219209	−	0.975678i	$-0.570348\pi$
$577$	−10.5312	−0.438418	−0.219209	+	0.975678i	$0.570348\pi$
$578$	0	0
$579$	5.48991	0.228153
$580$	0	0
$581$	18.3339	0.760619
$582$	0	0
$583$	−8.64509	−0.358043
$584$	0	0
$585$	−1.69908	−0.0702483
$586$	0	0
$587$	28.5141	1.17690	0.588451	−	0.808533i	$-0.299738\pi$
$587$	28.5141	1.17690	0.588451	+	0.808533i	$0.299738\pi$
$588$	0	0
$589$	−4.96239	−0.204472
$590$	0	0
$591$	4.27067	0.175672
$592$	0	0
$593$	−18.1479	−0.745246	−0.372623	−	0.927983i	$-0.621542\pi$
$593$	−18.1479	−0.745246	−0.372623	+	0.927983i	$0.621542\pi$
$594$	0	0
$595$	−0.399200	−0.0163656
$596$	0	0
$597$	4.69443	0.192130
$598$	0	0
$599$	−22.2881	−0.910665	−0.455333	−	0.890321i	$-0.650480\pi$
$599$	−22.2881	−0.910665	−0.455333	+	0.890321i	$0.650480\pi$
$600$	0	0
$601$	−5.30668	−0.216464	−0.108232	−	0.994126i	$-0.534519\pi$
$601$	−5.30668	−0.216464	−0.108232	+	0.994126i	$0.534519\pi$
$602$	0	0
$603$	12.7275	0.518304
$604$	0	0
$605$	−3.46135	−0.140724
$606$	0	0
$607$	−29.7063	−1.20574	−0.602871	−	0.797838i	$-0.705977\pi$
$607$	−29.7063	−1.20574	−0.602871	+	0.797838i	$0.705977\pi$
$608$	0	0
$609$	3.97841	0.161213
$610$	0	0
$611$	7.28702	0.294801
$612$	0	0
$613$	20.1642	0.814426	0.407213	−	0.913333i	$-0.366501\pi$
$613$	20.1642	0.814426	0.407213	+	0.913333i	$0.366501\pi$
$614$	0	0
$615$	−0.295332	−0.0119089
$616$	0	0
$617$	−44.9895	−1.81121	−0.905605	−	0.424123i	$-0.860582\pi$
$617$	−44.9895	−1.81121	−0.905605	+	0.424123i	$0.860582\pi$
$618$	0	0
$619$	−13.6631	−0.549166	−0.274583	−	0.961563i	$-0.588540\pi$
$619$	−13.6631	−0.549166	−0.274583	+	0.961563i	$0.588540\pi$
$620$	0	0
$621$	−1.62406	−0.0651714
$622$	0	0
$623$	−0.989970	−0.0396623
$624$	0	0
$625$	22.4591	0.898365
$626$	0	0
$627$	−0.549530	−0.0219461
$628$	0	0
$629$	3.12872	0.124750
$630$	0	0
$631$	−27.3683	−1.08951	−0.544756	−	0.838594i	$-0.683378\pi$
$631$	−27.3683	−1.08951	−0.544756	+	0.838594i	$0.683378\pi$
$632$	0	0
$633$	3.01642	0.119892
$634$	0	0
$635$	−3.40345	−0.135062
$636$	0	0
$637$	−6.26351	−0.248169
$638$	0	0
$639$	−40.2363	−1.59172
$640$	0	0
$641$	47.0443	1.85814	0.929070	−	0.369905i	$-0.120610\pi$
$641$	47.0443	1.85814	0.929070	+	0.369905i	$0.120610\pi$
$642$	0	0
$643$	19.9980	0.788644	0.394322	−	0.918972i	$-0.370979\pi$
$643$	19.9980	0.788644	0.394322	+	0.918972i	$0.370979\pi$
$644$	0	0
$645$	0.774628	0.0305009
$646$	0	0
$647$	−35.3485	−1.38969	−0.694847	−	0.719158i	$-0.744528\pi$
$647$	−35.3485	−1.38969	−0.694847	+	0.719158i	$0.744528\pi$
$648$	0	0
$649$	18.6852	0.733457
$650$	0	0
$651$	−2.70552	−0.106038
$652$	0	0
$653$	26.1725	1.02421	0.512104	−	0.858923i	$-0.328866\pi$
$653$	26.1725	1.02421	0.512104	+	0.858923i	$0.328866\pi$
$654$	0	0
$655$	−6.24329	−0.243945
$656$	0	0
$657$	19.7053	0.768778
$658$	0	0
$659$	32.9819	1.28479	0.642396	−	0.766373i	$-0.277941\pi$
$659$	32.9819	1.28479	0.642396	+	0.766373i	$0.277941\pi$
$660$	0	0
$661$	−4.59821	−0.178850	−0.0894249	−	0.995994i	$-0.528503\pi$
$661$	−4.59821	−0.178850	−0.0894249	+	0.995994i	$0.528503\pi$
$662$	0	0
$663$	−0.288019	−0.0111857
$664$	0	0
$665$	−0.667050	−0.0258671
$666$	0	0
$667$	−5.95146	−0.230442
$668$	0	0
$669$	0.216277	0.00836175
$670$	0	0
$671$	−19.8903	−0.767856
$672$	0	0
$673$	21.6841	0.835861	0.417930	−	0.908479i	$-0.362756\pi$
$673$	21.6841	0.835861	0.417930	+	0.908479i	$0.362756\pi$
$674$	0	0
$675$	−9.61512	−0.370086
$676$	0	0
$677$	−37.1049	−1.42606	−0.713029	−	0.701135i	$-0.752677\pi$
$677$	−37.1049	−1.42606	−0.713029	+	0.701135i	$0.752677\pi$
$678$	0	0
$679$	3.87509	0.148712
$680$	0	0
$681$	2.31192	0.0885931
$682$	0	0
$683$	−12.7002	−0.485960	−0.242980	−	0.970031i	$-0.578125\pi$
$683$	−12.7002	−0.485960	−0.242980	+	0.970031i	$0.578125\pi$
$684$	0	0
$685$	4.02191	0.153669
$686$	0	0
$687$	2.71299	0.103507
$688$	0	0
$689$	−7.57125	−0.288442
$690$	0	0
$691$	30.1820	1.14818	0.574089	−	0.818793i	$-0.305356\pi$
$691$	30.1820	1.14818	0.574089	+	0.818793i	$0.305356\pi$
$692$	0	0
$693$	7.55249	0.286895
$694$	0	0
$695$	−1.66069	−0.0629934
$696$	0	0
$697$	1.26200	0.0478015
$698$	0	0
$699$	−7.79408	−0.294799
$700$	0	0
$701$	48.0683	1.81552	0.907758	−	0.419495i	$-0.137793\pi$
$701$	48.0683	1.81552	0.907758	+	0.419495i	$0.137793\pi$
$702$	0	0
$703$	5.22799	0.197177
$704$	0	0
$705$	−0.717446	−0.0270206
$706$	0	0
$707$	20.2269	0.760710
$708$	0	0
$709$	−14.7547	−0.554126	−0.277063	−	0.960852i	$-0.589361\pi$
$709$	−14.7547	−0.554126	−0.277063	+	0.960852i	$0.589361\pi$
$710$	0	0
$711$	2.88553	0.108216
$712$	0	0
$713$	4.04729	0.151572
$714$	0	0
$715$	0.956392	0.0357670
$716$	0	0
$717$	−3.46227	−0.129301
$718$	0	0
$719$	−27.8461	−1.03848	−0.519242	−	0.854627i	$-0.673786\pi$
$719$	−27.8461	−1.03848	−0.519242	+	0.854627i	$0.673786\pi$
$720$	0	0
$721$	−16.4323	−0.611971
$722$	0	0
$723$	−4.60471	−0.171251
$724$	0	0
$725$	−35.2351	−1.30860
$726$	0	0
$727$	−36.4582	−1.35216	−0.676081	−	0.736828i	$-0.736323\pi$
$727$	−36.4582	−1.35216	−0.676081	+	0.736828i	$0.736323\pi$
$728$	0	0
$729$	−21.0137	−0.778287
$730$	0	0
$731$	−3.31009	−0.122428
$732$	0	0
$733$	−5.98155	−0.220933	−0.110467	−	0.993880i	$-0.535235\pi$
$733$	−5.98155	−0.220933	−0.110467	+	0.993880i	$0.535235\pi$
$734$	0	0
$735$	0.616676	0.0227464
$736$	0	0
$737$	−7.16416	−0.263895
$738$	0	0
$739$	−12.6232	−0.464351	−0.232175	−	0.972674i	$-0.574584\pi$
$739$	−12.6232	−0.464351	−0.232175	+	0.972674i	$0.574584\pi$
$740$	0	0
$741$	−0.481270	−0.0176799
$742$	0	0
$743$	22.5942	0.828901	0.414450	−	0.910072i	$-0.363974\pi$
$743$	22.5942	0.828901	0.414450	+	0.910072i	$0.363974\pi$
$744$	0	0
$745$	−6.09950	−0.223469
$746$	0	0
$747$	32.8295	1.20117
$748$	0	0
$749$	−0.153193	−0.00559757
$750$	0	0
$751$	33.6398	1.22753	0.613767	−	0.789487i	$-0.289653\pi$
$751$	33.6398	1.22753	0.613767	+	0.789487i	$0.289653\pi$
$752$	0	0
$753$	7.43713	0.271024
$754$	0	0
$755$	−5.90620	−0.214949
$756$	0	0
$757$	−38.5796	−1.40220	−0.701100	−	0.713063i	$-0.747307\pi$
$757$	−38.5796	−1.40220	−0.701100	+	0.713063i	$0.747307\pi$
$758$	0	0
$759$	0.448193	0.0162684
$760$	0	0
$761$	0.701726	0.0254375	0.0127188	−	0.999919i	$-0.495951\pi$
$761$	0.701726	0.0254375	0.0127188	+	0.999919i	$0.495951\pi$
$762$	0	0
$763$	−2.40266	−0.0869822
$764$	0	0
$765$	−0.714825	−0.0258446
$766$	0	0
$767$	16.3642	0.590877
$768$	0	0
$769$	0.0981808	0.00354049	0.00177024	−	0.999998i	$-0.499437\pi$
$769$	0.0981808	0.00354049	0.00177024	+	0.999998i	$0.499437\pi$
$770$	0	0
$771$	10.4586	0.376656
$772$	0	0
$773$	0.324378	0.0116671	0.00583354	−	0.999983i	$-0.498143\pi$
$773$	0.324378	0.0116671	0.00583354	+	0.999983i	$0.498143\pi$
$774$	0	0
$775$	23.9616	0.860727
$776$	0	0
$777$	2.85032	0.102255
$778$	0	0
$779$	2.10875	0.0755539
$780$	0	0
$781$	22.6485	0.810428
$782$	0	0
$783$	14.5304	0.519275
$784$	0	0
$785$	8.57716	0.306132
$786$	0	0
$787$	20.4633	0.729438	0.364719	−	0.931118i	$-0.381165\pi$
$787$	20.4633	0.729438	0.364719	+	0.931118i	$0.381165\pi$
$788$	0	0
$789$	−1.06793	−0.0380194
$790$	0	0
$791$	−3.39955	−0.120874
$792$	0	0
$793$	−17.4196	−0.618589
$794$	0	0
$795$	0.745429	0.0264376
$796$	0	0
$797$	−35.0236	−1.24060	−0.620300	−	0.784365i	$-0.712989\pi$
$797$	−35.0236	−1.24060	−0.620300	+	0.784365i	$0.712989\pi$
$798$	0	0
$799$	3.06575	0.108458
$800$	0	0
$801$	−1.77268	−0.0626347
$802$	0	0
$803$	−11.0919	−0.391425
$804$	0	0
$805$	0.544042	0.0191749
$806$	0	0
$807$	7.40796	0.260772
$808$	0	0
$809$	−31.2774	−1.09965	−0.549827	−	0.835278i	$-0.685306\pi$
$809$	−31.2774	−1.09965	−0.549827	+	0.835278i	$0.685306\pi$
$810$	0	0
$811$	27.2968	0.958521	0.479261	−	0.877673i	$-0.340905\pi$
$811$	27.2968	0.958521	0.479261	+	0.877673i	$0.340905\pi$
$812$	0	0
$813$	2.50058	0.0876993
$814$	0	0
$815$	−0.502327	−0.0175957
$816$	0	0
$817$	−5.53106	−0.193507
$818$	0	0
$819$	6.61436	0.231125
$820$	0	0
$821$	−28.8166	−1.00571	−0.502854	−	0.864371i	$-0.667717\pi$
$821$	−28.8166	−1.00571	−0.502854	+	0.864371i	$0.667717\pi$
$822$	0	0
$823$	32.7428	1.14134	0.570671	−	0.821179i	$-0.306683\pi$
$823$	32.7428	1.14134	0.570671	+	0.821179i	$0.306683\pi$
$824$	0	0
$825$	2.65349	0.0923825
$826$	0	0
$827$	−32.8914	−1.14375	−0.571873	−	0.820342i	$-0.693783\pi$
$827$	−32.8914	−1.14375	−0.571873	+	0.820342i	$0.693783\pi$
$828$	0	0
$829$	−11.3646	−0.394708	−0.197354	−	0.980332i	$-0.563235\pi$
$829$	−11.3646	−0.394708	−0.197354	+	0.980332i	$0.563235\pi$
$830$	0	0
$831$	8.59967	0.298319
$832$	0	0
$833$	−2.63514	−0.0913023
$834$	0	0
$835$	4.49232	0.155463
$836$	0	0
$837$	−9.88142	−0.341552
$838$	0	0
$839$	26.1915	0.904230	0.452115	−	0.891960i	$-0.350670\pi$
$839$	26.1915	0.904230	0.452115	+	0.891960i	$0.350670\pi$
$840$	0	0
$841$	24.2475	0.836122
$842$	0	0
$843$	3.51324	0.121002
$844$	0	0
$845$	−4.54368	−0.156307
$846$	0	0
$847$	13.4747	0.462997
$848$	0	0
$849$	−5.72405	−0.196449
$850$	0	0
$851$	−4.26391	−0.146165
$852$	0	0
$853$	42.6980	1.46195	0.730976	−	0.682403i	$-0.239065\pi$
$853$	42.6980	1.46195	0.730976	+	0.682403i	$0.239065\pi$
$854$	0	0
$855$	−1.19445	−0.0408493
$856$	0	0
$857$	−26.0061	−0.888352	−0.444176	−	0.895940i	$-0.646503\pi$
$857$	−26.0061	−0.888352	−0.444176	+	0.895940i	$0.646503\pi$
$858$	0	0
$859$	−26.4047	−0.900915	−0.450458	−	0.892798i	$-0.648739\pi$
$859$	−26.4047	−0.900915	−0.450458	+	0.892798i	$0.648739\pi$
$860$	0	0
$861$	1.14970	0.0391817
$862$	0	0
$863$	49.3537	1.68002	0.840010	−	0.542570i	$-0.182549\pi$
$863$	49.3537	1.68002	0.840010	+	0.542570i	$0.182549\pi$
$864$	0	0
$865$	−6.41406	−0.218085
$866$	0	0
$867$	5.63047	0.191221
$868$	0	0
$869$	−1.62423	−0.0550983
$870$	0	0
$871$	−6.27427	−0.212596
$872$	0	0
$873$	6.93891	0.234846
$874$	0	0
$875$	6.55620	0.221640
$876$	0	0
$877$	30.2291	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$877$	30.2291	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$878$	0	0
$879$	2.25580	0.0760864
$880$	0	0
$881$	−40.5269	−1.36539	−0.682693	−	0.730706i	$-0.739191\pi$
$881$	−40.5269	−1.36539	−0.682693	+	0.730706i	$0.739191\pi$
$882$	0	0
$883$	56.3441	1.89613	0.948064	−	0.318079i	$-0.103038\pi$
$883$	56.3441	1.89613	0.948064	+	0.318079i	$0.103038\pi$
$884$	0	0
$885$	−1.61114	−0.0541579
$886$	0	0
$887$	−41.9554	−1.40873	−0.704363	−	0.709840i	$-0.748767\pi$
$887$	−41.9554	−1.40873	−0.704363	+	0.709840i	$0.748767\pi$
$888$	0	0
$889$	13.2493	0.444368
$890$	0	0
$891$	12.9661	0.434379
$892$	0	0
$893$	5.12276	0.171427
$894$	0	0
$895$	−3.97275	−0.132794
$896$	0	0
$897$	0.392521	0.0131059
$898$	0	0
$899$	−36.2110	−1.20770
$900$	0	0
$901$	−3.18532	−0.106119
$902$	0	0
$903$	−3.01556	−0.100351
$904$	0	0
$905$	5.25461	0.174669
$906$	0	0
$907$	21.8127	0.724277	0.362139	−	0.932124i	$-0.382047\pi$
$907$	21.8127	0.724277	0.362139	+	0.932124i	$0.382047\pi$
$908$	0	0
$909$	36.2191	1.20131
$910$	0	0
$911$	32.3550	1.07197	0.535984	−	0.844228i	$-0.319941\pi$
$911$	32.3550	1.07197	0.535984	+	0.844228i	$0.319941\pi$
$912$	0	0
$913$	−18.4794	−0.611578
$914$	0	0
$915$	1.71505	0.0566980
$916$	0	0
$917$	24.3046	0.802607
$918$	0	0
$919$	31.8658	1.05115	0.525577	−	0.850746i	$-0.323849\pi$
$919$	31.8658	1.05115	0.525577	+	0.850746i	$0.323849\pi$
$920$	0	0
$921$	−4.56428	−0.150398
$922$	0	0
$923$	19.8353	0.652885
$924$	0	0
$925$	−25.2441	−0.830021
$926$	0	0
$927$	−29.4244	−0.966424
$928$	0	0
$929$	2.30025	0.0754686	0.0377343	−	0.999288i	$-0.487986\pi$
$929$	2.30025	0.0754686	0.0377343	+	0.999288i	$0.487986\pi$
$930$	0	0
$931$	−4.40324	−0.144310
$932$	0	0
$933$	−9.38811	−0.307353
$934$	0	0
$935$	0.402367	0.0131588
$936$	0	0
$937$	−1.41041	−0.0460761	−0.0230381	−	0.999735i	$-0.507334\pi$
$937$	−1.41041	−0.0460761	−0.0230381	+	0.999735i	$0.507334\pi$
$938$	0	0
$939$	−10.4399	−0.340695
$940$	0	0
$941$	21.0629	0.686632	0.343316	−	0.939220i	$-0.388450\pi$
$941$	21.0629	0.686632	0.343316	+	0.939220i	$0.388450\pi$
$942$	0	0
$943$	−1.71989	−0.0560072
$944$	0	0
$945$	−1.32827	−0.0432087
$946$	0	0
$947$	45.2127	1.46922	0.734608	−	0.678492i	$-0.237366\pi$
$947$	45.2127	1.46922	0.734608	+	0.678492i	$0.237366\pi$
$948$	0	0
$949$	−9.71413	−0.315334
$950$	0	0
$951$	−7.65839	−0.248340
$952$	0	0
$953$	3.57380	0.115767	0.0578834	−	0.998323i	$-0.481565\pi$
$953$	3.57380	0.115767	0.0578834	+	0.998323i	$0.481565\pi$
$954$	0	0
$955$	−6.27160	−0.202944
$956$	0	0
$957$	−4.00997	−0.129624
$958$	0	0
$959$	−15.6569	−0.505589
$960$	0	0
$961$	−6.37472	−0.205636
$962$	0	0
$963$	−0.274315	−0.00883968
$964$	0	0
$965$	−6.71683	−0.216222
$966$	0	0
$967$	48.5589	1.56155	0.780775	−	0.624813i	$-0.214825\pi$
$967$	48.5589	1.56155	0.780775	+	0.624813i	$0.214825\pi$
$968$	0	0
$969$	−0.202477	−0.00650449
$970$	0	0
$971$	42.2841	1.35696	0.678481	−	0.734618i	$-0.262639\pi$
$971$	42.2841	1.35696	0.678481	+	0.734618i	$0.262639\pi$
$972$	0	0
$973$	6.46490	0.207255
$974$	0	0
$975$	2.32389	0.0744239
$976$	0	0
$977$	−23.7753	−0.760639	−0.380320	−	0.924855i	$-0.624186\pi$
$977$	−23.7753	−0.760639	−0.380320	+	0.924855i	$0.624186\pi$
$978$	0	0
$979$	0.997823	0.0318906
$980$	0	0
$981$	−4.30231	−0.137362
$982$	0	0
$983$	16.0244	0.511100	0.255550	−	0.966796i	$-0.417743\pi$
$983$	16.0244	0.511100	0.255550	+	0.966796i	$0.417743\pi$
$984$	0	0
$985$	−5.22511	−0.166486
$986$	0	0
$987$	2.79295	0.0889006
$988$	0	0
$989$	4.51110	0.143445
$990$	0	0
$991$	−50.7254	−1.61135	−0.805673	−	0.592360i	$-0.798196\pi$
$991$	−50.7254	−1.61135	−0.805673	+	0.592360i	$0.798196\pi$
$992$	0	0
$993$	−3.52406	−0.111833
$994$	0	0
$995$	−5.74357	−0.182083
$996$	0	0
$997$	11.6654	0.369447	0.184723	−	0.982791i	$-0.440861\pi$
$997$	11.6654	0.369447	0.184723	+	0.982791i	$0.440861\pi$
$998$	0	0
$999$	10.4103	0.329367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.14	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.14	✓	27	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)

\(f(q)\)	\(=\)	\(q-0.338332 q^{3} +0.413944 q^{5} -1.61145 q^{7} -2.88553 q^{9} +O(q^{10})\)	\(q-0.338332 q^{3} +0.413944 q^{5} -1.61145 q^{7} -2.88553 q^{9} +1.62423 q^{11} +1.42248 q^{13} -0.140051 q^{15} +0.598456 q^{17} +1.00000 q^{19} +0.545204 q^{21} -0.815594 q^{23} -4.82865 q^{25} +1.99126 q^{27} +7.29709 q^{29} -4.96239 q^{31} -0.549530 q^{33} -0.667050 q^{35} +5.22799 q^{37} -0.481270 q^{39} +2.10875 q^{41} -5.53106 q^{43} -1.19445 q^{45} +5.12276 q^{47} -4.40324 q^{49} -0.202477 q^{51} -5.32257 q^{53} +0.672341 q^{55} -0.338332 q^{57} +11.5040 q^{59} -12.2460 q^{61} +4.64988 q^{63} +0.588827 q^{65} -4.41080 q^{67} +0.275942 q^{69} +13.9441 q^{71} -6.82902 q^{73} +1.63369 q^{75} -2.61736 q^{77} -1.00000 q^{79} +7.98289 q^{81} -11.3773 q^{83} +0.247727 q^{85} -2.46884 q^{87} +0.614336 q^{89} -2.29225 q^{91} +1.67893 q^{93} +0.413944 q^{95} -2.40472 q^{97} -4.68677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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