LMFDB - Embedded newform 6008.2.a.e.1.46

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.46
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00349 q^{3} +3.90143 q^{5} -4.05459 q^{7} +6.02097 q^{9} +O(q^{10})$	$q+3.00349 q^{3} +3.90143 q^{5} -4.05459 q^{7} +6.02097 q^{9} -1.15784 q^{11} +3.06308 q^{13} +11.7179 q^{15} +2.39254 q^{17} -5.08326 q^{19} -12.1779 q^{21} +5.68251 q^{23} +10.2211 q^{25} +9.07346 q^{27} -6.19683 q^{29} +1.20190 q^{31} -3.47758 q^{33} -15.8187 q^{35} -0.702594 q^{37} +9.19995 q^{39} +3.20524 q^{41} +0.852294 q^{43} +23.4904 q^{45} +9.40596 q^{47} +9.43969 q^{49} +7.18597 q^{51} +2.44430 q^{53} -4.51724 q^{55} -15.2676 q^{57} +8.28158 q^{59} +10.0969 q^{61} -24.4126 q^{63} +11.9504 q^{65} +13.8704 q^{67} +17.0674 q^{69} -0.0237590 q^{71} +1.97219 q^{73} +30.6991 q^{75} +4.69458 q^{77} -11.3522 q^{79} +9.18918 q^{81} -14.2999 q^{83} +9.33431 q^{85} -18.6121 q^{87} -9.64695 q^{89} -12.4195 q^{91} +3.60989 q^{93} -19.8320 q^{95} -4.93677 q^{97} -6.97134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	3.00349	1.73407	0.867034	−	0.498249i	$-0.166024\pi$
$3$	3.00349	1.73407	0.867034	+	0.498249i	$0.166024\pi$
$4$	0	0
$5$	3.90143	1.74477	0.872385	−	0.488819i	$-0.162572\pi$
$5$	3.90143	1.74477	0.872385	+	0.488819i	$0.162572\pi$
$6$	0	0
$7$	−4.05459	−1.53249	−0.766245	−	0.642548i	$-0.777877\pi$
$7$	−4.05459	−1.53249	−0.766245	+	0.642548i	$0.777877\pi$
$8$	0	0
$9$	6.02097	2.00699
$10$	0	0
$11$	−1.15784	−0.349103	−0.174551	−	0.984648i	$-0.555848\pi$
$11$	−1.15784	−0.349103	−0.174551	+	0.984648i	$0.555848\pi$
$12$	0	0
$13$	3.06308	0.849547	0.424773	−	0.905300i	$-0.360354\pi$
$13$	3.06308	0.849547	0.424773	+	0.905300i	$0.360354\pi$
$14$	0	0
$15$	11.7179	3.02555
$16$	0	0
$17$	2.39254	0.580276	0.290138	−	0.956985i	$-0.406299\pi$
$17$	2.39254	0.580276	0.290138	+	0.956985i	$0.406299\pi$
$18$	0	0
$19$	−5.08326	−1.16618	−0.583090	−	0.812407i	$-0.698157\pi$
$19$	−5.08326	−1.16618	−0.583090	+	0.812407i	$0.698157\pi$
$20$	0	0
$21$	−12.1779	−2.65744
$22$	0	0
$23$	5.68251	1.18489	0.592443	−	0.805613i	$-0.298164\pi$
$23$	5.68251	1.18489	0.592443	+	0.805613i	$0.298164\pi$
$24$	0	0
$25$	10.2211	2.04422
$26$	0	0
$27$	9.07346	1.74619
$28$	0	0
$29$	−6.19683	−1.15072	−0.575361	−	0.817900i	$-0.695138\pi$
$29$	−6.19683	−1.15072	−0.575361	+	0.817900i	$0.695138\pi$
$30$	0	0
$31$	1.20190	0.215867	0.107933	−	0.994158i	$-0.465577\pi$
$31$	1.20190	0.215867	0.107933	+	0.994158i	$0.465577\pi$
$32$	0	0
$33$	−3.47758	−0.605368
$34$	0	0
$35$	−15.8187	−2.67384
$36$	0	0
$37$	−0.702594	−0.115506	−0.0577529	−	0.998331i	$-0.518394\pi$
$37$	−0.702594	−0.115506	−0.0577529	+	0.998331i	$0.518394\pi$
$38$	0	0
$39$	9.19995	1.47317
$40$	0	0
$41$	3.20524	0.500575	0.250287	−	0.968172i	$-0.419475\pi$
$41$	3.20524	0.500575	0.250287	+	0.968172i	$0.419475\pi$
$42$	0	0
$43$	0.852294	0.129974	0.0649868	−	0.997886i	$-0.479299\pi$
$43$	0.852294	0.129974	0.0649868	+	0.997886i	$0.479299\pi$
$44$	0	0
$45$	23.4904	3.50174
$46$	0	0
$47$	9.40596	1.37200	0.686000	−	0.727601i	$-0.259365\pi$
$47$	9.40596	1.37200	0.686000	+	0.727601i	$0.259365\pi$
$48$	0	0
$49$	9.43969	1.34853
$50$	0	0
$51$	7.18597	1.00624
$52$	0	0
$53$	2.44430	0.335750	0.167875	−	0.985808i	$-0.446309\pi$
$53$	2.44430	0.335750	0.167875	+	0.985808i	$0.446309\pi$
$54$	0	0
$55$	−4.51724	−0.609105
$56$	0	0
$57$	−15.2676	−2.02224
$58$	0	0
$59$	8.28158	1.07817	0.539085	−	0.842251i	$-0.318770\pi$
$59$	8.28158	1.07817	0.539085	+	0.842251i	$0.318770\pi$
$60$	0	0
$61$	10.0969	1.29278	0.646389	−	0.763008i	$-0.276278\pi$
$61$	10.0969	1.29278	0.646389	+	0.763008i	$0.276278\pi$
$62$	0	0
$63$	−24.4126	−3.07569
$64$	0	0
$65$	11.9504	1.48226
$66$	0	0
$67$	13.8704	1.69454	0.847271	−	0.531161i	$-0.178244\pi$
$67$	13.8704	1.69454	0.847271	+	0.531161i	$0.178244\pi$
$68$	0	0
$69$	17.0674	2.05467
$70$	0	0
$71$	−0.0237590	−0.00281967	−0.00140984	−	0.999999i	$-0.500449\pi$
$71$	−0.0237590	−0.00281967	−0.00140984	+	0.999999i	$0.500449\pi$
$72$	0	0
$73$	1.97219	0.230828	0.115414	−	0.993317i	$-0.463181\pi$
$73$	1.97219	0.230828	0.115414	+	0.993317i	$0.463181\pi$
$74$	0	0
$75$	30.6991	3.54482
$76$	0	0
$77$	4.69458	0.534997
$78$	0	0
$79$	−11.3522	−1.27722	−0.638610	−	0.769531i	$-0.720490\pi$
$79$	−11.3522	−1.27722	−0.638610	+	0.769531i	$0.720490\pi$
$80$	0	0
$81$	9.18918	1.02102
$82$	0	0
$83$	−14.2999	−1.56962	−0.784808	−	0.619740i	$-0.787238\pi$
$83$	−14.2999	−1.56962	−0.784808	+	0.619740i	$0.787238\pi$
$84$	0	0
$85$	9.33431	1.01245
$86$	0	0
$87$	−18.6121	−1.99543
$88$	0	0
$89$	−9.64695	−1.02257	−0.511287	−	0.859410i	$-0.670831\pi$
$89$	−9.64695	−1.02257	−0.511287	+	0.859410i	$0.670831\pi$
$90$	0	0
$91$	−12.4195	−1.30192
$92$	0	0
$93$	3.60989	0.374328
$94$	0	0
$95$	−19.8320	−2.03472
$96$	0	0
$97$	−4.93677	−0.501253	−0.250626	−	0.968084i	$-0.580637\pi$
$97$	−4.93677	−0.501253	−0.250626	+	0.968084i	$0.580637\pi$
$98$	0	0
$99$	−6.97134	−0.700646
$100$	0	0
$101$	18.7732	1.86800	0.933999	−	0.357275i	$-0.116294\pi$
$101$	18.7732	1.86800	0.933999	+	0.357275i	$0.116294\pi$
$102$	0	0
$103$	−0.107261	−0.0105688	−0.00528439	−	0.999986i	$-0.501682\pi$
$103$	−0.107261	−0.0105688	−0.00528439	+	0.999986i	$0.501682\pi$
$104$	0	0
$105$	−47.5113	−4.63663
$106$	0	0
$107$	−16.8588	−1.62980	−0.814899	−	0.579603i	$-0.803208\pi$
$107$	−16.8588	−1.62980	−0.814899	+	0.579603i	$0.803208\pi$
$108$	0	0
$109$	−6.81198	−0.652470	−0.326235	−	0.945289i	$-0.605780\pi$
$109$	−6.81198	−0.652470	−0.326235	+	0.945289i	$0.605780\pi$
$110$	0	0
$111$	−2.11024	−0.200295
$112$	0	0
$113$	14.8033	1.39257	0.696287	−	0.717763i	$-0.254834\pi$
$113$	14.8033	1.39257	0.696287	+	0.717763i	$0.254834\pi$
$114$	0	0
$115$	22.1699	2.06735
$116$	0	0
$117$	18.4427	1.70503
$118$	0	0
$119$	−9.70076	−0.889267
$120$	0	0
$121$	−9.65940	−0.878127
$122$	0	0
$123$	9.62692	0.868030
$124$	0	0
$125$	20.3698	1.82193
$126$	0	0
$127$	−5.70889	−0.506582	−0.253291	−	0.967390i	$-0.581513\pi$
$127$	−5.70889	−0.506582	−0.253291	+	0.967390i	$0.581513\pi$
$128$	0	0
$129$	2.55986	0.225383
$130$	0	0
$131$	2.52028	0.220198	0.110099	−	0.993921i	$-0.464883\pi$
$131$	2.52028	0.220198	0.110099	+	0.993921i	$0.464883\pi$
$132$	0	0
$133$	20.6105	1.78716
$134$	0	0
$135$	35.3994	3.04670
$136$	0	0
$137$	18.5337	1.58344	0.791720	−	0.610884i	$-0.209186\pi$
$137$	18.5337	1.58344	0.791720	+	0.610884i	$0.209186\pi$
$138$	0	0
$139$	−1.88645	−0.160006	−0.0800032	−	0.996795i	$-0.525493\pi$
$139$	−1.88645	−0.160006	−0.0800032	+	0.996795i	$0.525493\pi$
$140$	0	0
$141$	28.2507	2.37914
$142$	0	0
$143$	−3.54657	−0.296579
$144$	0	0
$145$	−24.1765	−2.00775
$146$	0	0
$147$	28.3521	2.33844
$148$	0	0
$149$	14.5348	1.19074	0.595369	−	0.803452i	$-0.297006\pi$
$149$	14.5348	1.19074	0.595369	+	0.803452i	$0.297006\pi$
$150$	0	0
$151$	5.11314	0.416102	0.208051	−	0.978118i	$-0.433288\pi$
$151$	5.11314	0.416102	0.208051	+	0.978118i	$0.433288\pi$
$152$	0	0
$153$	14.4054	1.16461
$154$	0	0
$155$	4.68911	0.376638
$156$	0	0
$157$	−19.6197	−1.56582	−0.782911	−	0.622134i	$-0.786266\pi$
$157$	−19.6197	−1.56582	−0.782911	+	0.622134i	$0.786266\pi$
$158$	0	0
$159$	7.34143	0.582213
$160$	0	0
$161$	−23.0402	−1.81583
$162$	0	0
$163$	−15.5788	−1.22023	−0.610113	−	0.792314i	$-0.708876\pi$
$163$	−15.5788	−1.22023	−0.610113	+	0.792314i	$0.708876\pi$
$164$	0	0
$165$	−13.5675	−1.05623
$166$	0	0
$167$	−9.61496	−0.744028	−0.372014	−	0.928227i	$-0.621333\pi$
$167$	−9.61496	−0.744028	−0.372014	+	0.928227i	$0.621333\pi$
$168$	0	0
$169$	−3.61752	−0.278271
$170$	0	0
$171$	−30.6062	−2.34051
$172$	0	0
$173$	−15.8050	−1.20163	−0.600815	−	0.799388i	$-0.705157\pi$
$173$	−15.8050	−1.20163	−0.600815	+	0.799388i	$0.705157\pi$
$174$	0	0
$175$	−41.4424	−3.13275
$176$	0	0
$177$	24.8737	1.86962
$178$	0	0
$179$	−8.66479	−0.647637	−0.323818	−	0.946119i	$-0.604967\pi$
$179$	−8.66479	−0.647637	−0.323818	+	0.946119i	$0.604967\pi$
$180$	0	0
$181$	15.5265	1.15407	0.577036	−	0.816719i	$-0.304209\pi$
$181$	15.5265	1.15407	0.577036	+	0.816719i	$0.304209\pi$
$182$	0	0
$183$	30.3260	2.24177
$184$	0	0
$185$	−2.74112	−0.201531
$186$	0	0
$187$	−2.77019	−0.202576
$188$	0	0
$189$	−36.7892	−2.67602
$190$	0	0
$191$	−11.8356	−0.856397	−0.428198	−	0.903685i	$-0.640852\pi$
$191$	−11.8356	−0.856397	−0.428198	+	0.903685i	$0.640852\pi$
$192$	0	0
$193$	−11.3978	−0.820433	−0.410216	−	0.911988i	$-0.634547\pi$
$193$	−11.3978	−0.820433	−0.410216	+	0.911988i	$0.634547\pi$
$194$	0	0
$195$	35.8929	2.57035
$196$	0	0
$197$	13.5156	0.962948	0.481474	−	0.876460i	$-0.340102\pi$
$197$	13.5156	0.962948	0.481474	+	0.876460i	$0.340102\pi$
$198$	0	0
$199$	−17.0080	−1.20567	−0.602834	−	0.797867i	$-0.705962\pi$
$199$	−17.0080	−1.20567	−0.602834	+	0.797867i	$0.705962\pi$
$200$	0	0
$201$	41.6597	2.93845
$202$	0	0
$203$	25.1256	1.76347
$204$	0	0
$205$	12.5050	0.873388
$206$	0	0
$207$	34.2142	2.37805
$208$	0	0
$209$	5.88563	0.407117
$210$	0	0
$211$	9.71123	0.668549	0.334274	−	0.942476i	$-0.391509\pi$
$211$	9.71123	0.668549	0.334274	+	0.942476i	$0.391509\pi$
$212$	0	0
$213$	−0.0713600	−0.00488951
$214$	0	0
$215$	3.32516	0.226774
$216$	0	0
$217$	−4.87319	−0.330814
$218$	0	0
$219$	5.92347	0.400271
$220$	0	0
$221$	7.32855	0.492971
$222$	0	0
$223$	20.4988	1.37270	0.686352	−	0.727269i	$-0.259211\pi$
$223$	20.4988	1.37270	0.686352	+	0.727269i	$0.259211\pi$
$224$	0	0
$225$	61.5411	4.10274
$226$	0	0
$227$	0.250591	0.0166323	0.00831614	−	0.999965i	$-0.497353\pi$
$227$	0.250591	0.0166323	0.00831614	+	0.999965i	$0.497353\pi$
$228$	0	0
$229$	−22.1239	−1.46199	−0.730994	−	0.682384i	$-0.760943\pi$
$229$	−22.1239	−1.46199	−0.730994	+	0.682384i	$0.760943\pi$
$230$	0	0
$231$	14.1001	0.927721
$232$	0	0
$233$	−6.70985	−0.439577	−0.219788	−	0.975548i	$-0.570537\pi$
$233$	−6.70985	−0.439577	−0.219788	+	0.975548i	$0.570537\pi$
$234$	0	0
$235$	36.6967	2.39383
$236$	0	0
$237$	−34.0962	−2.21479
$238$	0	0
$239$	−10.8665	−0.702895	−0.351447	−	0.936208i	$-0.614310\pi$
$239$	−10.8665	−0.702895	−0.351447	+	0.936208i	$0.614310\pi$
$240$	0	0
$241$	−1.17637	−0.0757767	−0.0378884	−	0.999282i	$-0.512063\pi$
$241$	−1.17637	−0.0757767	−0.0378884	+	0.999282i	$0.512063\pi$
$242$	0	0
$243$	0.379235	0.0243279
$244$	0	0
$245$	36.8283	2.35287
$246$	0	0
$247$	−15.5705	−0.990725
$248$	0	0
$249$	−42.9496	−2.72182
$250$	0	0
$251$	−19.9124	−1.25686	−0.628431	−	0.777865i	$-0.716302\pi$
$251$	−19.9124	−1.25686	−0.628431	+	0.777865i	$0.716302\pi$
$252$	0	0
$253$	−6.57946	−0.413647
$254$	0	0
$255$	28.0355	1.75565
$256$	0	0
$257$	−14.7530	−0.920264	−0.460132	−	0.887850i	$-0.652198\pi$
$257$	−14.7530	−0.920264	−0.460132	+	0.887850i	$0.652198\pi$
$258$	0	0
$259$	2.84873	0.177011
$260$	0	0
$261$	−37.3109	−2.30949
$262$	0	0
$263$	23.8677	1.47174	0.735872	−	0.677120i	$-0.236772\pi$
$263$	23.8677	1.47174	0.735872	+	0.677120i	$0.236772\pi$
$264$	0	0
$265$	9.53625	0.585807
$266$	0	0
$267$	−28.9745	−1.77321
$268$	0	0
$269$	8.84139	0.539069	0.269535	−	0.962991i	$-0.413130\pi$
$269$	8.84139	0.539069	0.269535	+	0.962991i	$0.413130\pi$
$270$	0	0
$271$	7.77312	0.472184	0.236092	−	0.971731i	$-0.424133\pi$
$271$	7.77312	0.472184	0.236092	+	0.971731i	$0.424133\pi$
$272$	0	0
$273$	−37.3020	−2.25762
$274$	0	0
$275$	−11.8345	−0.713645
$276$	0	0
$277$	−16.1444	−0.970026	−0.485013	−	0.874507i	$-0.661185\pi$
$277$	−16.1444	−0.970026	−0.485013	+	0.874507i	$0.661185\pi$
$278$	0	0
$279$	7.23658	0.433243
$280$	0	0
$281$	32.4431	1.93539	0.967697	−	0.252117i	$-0.0811268\pi$
$281$	32.4431	1.93539	0.967697	+	0.252117i	$0.0811268\pi$
$282$	0	0
$283$	−21.4402	−1.27449	−0.637245	−	0.770661i	$-0.719926\pi$
$283$	−21.4402	−1.27449	−0.637245	+	0.770661i	$0.719926\pi$
$284$	0	0
$285$	−59.5652	−3.52834
$286$	0	0
$287$	−12.9959	−0.767126
$288$	0	0
$289$	−11.2758	−0.663280
$290$	0	0
$291$	−14.8275	−0.869206
$292$	0	0
$293$	14.2796	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$293$	14.2796	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$294$	0	0
$295$	32.3100	1.88116
$296$	0	0
$297$	−10.5057	−0.609600
$298$	0	0
$299$	17.4060	1.00662
$300$	0	0
$301$	−3.45570	−0.199183
$302$	0	0
$303$	56.3850	3.23924
$304$	0	0
$305$	39.3924	2.25560
$306$	0	0
$307$	8.01951	0.457698	0.228849	−	0.973462i	$-0.426504\pi$
$307$	8.01951	0.457698	0.228849	+	0.973462i	$0.426504\pi$
$308$	0	0
$309$	−0.322159	−0.0183270
$310$	0	0
$311$	23.7129	1.34464	0.672318	−	0.740263i	$-0.265299\pi$
$311$	23.7129	1.34464	0.672318	+	0.740263i	$0.265299\pi$
$312$	0	0
$313$	−27.6785	−1.56448	−0.782242	−	0.622975i	$-0.785924\pi$
$313$	−27.6785	−1.56448	−0.782242	+	0.622975i	$0.785924\pi$
$314$	0	0
$315$	−95.2438	−5.36638
$316$	0	0
$317$	−21.0060	−1.17981	−0.589907	−	0.807471i	$-0.700836\pi$
$317$	−21.0060	−1.17981	−0.589907	+	0.807471i	$0.700836\pi$
$318$	0	0
$319$	7.17496	0.401721
$320$	0	0
$321$	−50.6352	−2.82618
$322$	0	0
$323$	−12.1619	−0.676707
$324$	0	0
$325$	31.3081	1.73666
$326$	0	0
$327$	−20.4597	−1.13143
$328$	0	0
$329$	−38.1373	−2.10258
$330$	0	0
$331$	−5.87277	−0.322796	−0.161398	−	0.986889i	$-0.551600\pi$
$331$	−5.87277	−0.322796	−0.161398	+	0.986889i	$0.551600\pi$
$332$	0	0
$333$	−4.23030	−0.231819
$334$	0	0
$335$	54.1144	2.95659
$336$	0	0
$337$	−25.0951	−1.36702	−0.683508	−	0.729943i	$-0.739547\pi$
$337$	−25.0951	−1.36702	−0.683508	+	0.729943i	$0.739547\pi$
$338$	0	0
$339$	44.4615	2.41482
$340$	0	0
$341$	−1.39161	−0.0753598
$342$	0	0
$343$	−9.89195	−0.534115
$344$	0	0
$345$	66.5871	3.58493
$346$	0	0
$347$	−28.6888	−1.54010	−0.770048	−	0.637986i	$-0.779768\pi$
$347$	−28.6888	−1.54010	−0.770048	+	0.637986i	$0.779768\pi$
$348$	0	0
$349$	−0.227950	−0.0122019	−0.00610093	−	0.999981i	$-0.501942\pi$
$349$	−0.227950	−0.0122019	−0.00610093	+	0.999981i	$0.501942\pi$
$350$	0	0
$351$	27.7928	1.48347
$352$	0	0
$353$	3.91068	0.208144	0.104072	−	0.994570i	$-0.466813\pi$
$353$	3.91068	0.208144	0.104072	+	0.994570i	$0.466813\pi$
$354$	0	0
$355$	−0.0926939	−0.00491968
$356$	0	0
$357$	−29.1362	−1.54205
$358$	0	0
$359$	19.2834	1.01774	0.508870	−	0.860843i	$-0.330063\pi$
$359$	19.2834	1.01774	0.508870	+	0.860843i	$0.330063\pi$
$360$	0	0
$361$	6.83958	0.359978
$362$	0	0
$363$	−29.0119	−1.52273
$364$	0	0
$365$	7.69437	0.402742
$366$	0	0
$367$	−27.9097	−1.45687	−0.728437	−	0.685113i	$-0.759753\pi$
$367$	−27.9097	−1.45687	−0.728437	+	0.685113i	$0.759753\pi$
$368$	0	0
$369$	19.2987	1.00465
$370$	0	0
$371$	−9.91062	−0.514534
$372$	0	0
$373$	−4.58737	−0.237525	−0.118762	−	0.992923i	$-0.537893\pi$
$373$	−4.58737	−0.237525	−0.118762	+	0.992923i	$0.537893\pi$
$374$	0	0
$375$	61.1806	3.15935
$376$	0	0
$377$	−18.9814	−0.977592
$378$	0	0
$379$	−2.24248	−0.115189	−0.0575943	−	0.998340i	$-0.518343\pi$
$379$	−2.24248	−0.115189	−0.0575943	+	0.998340i	$0.518343\pi$
$380$	0	0
$381$	−17.1466	−0.878447
$382$	0	0
$383$	−34.1039	−1.74263	−0.871315	−	0.490724i	$-0.836732\pi$
$383$	−34.1039	−1.74263	−0.871315	+	0.490724i	$0.836732\pi$
$384$	0	0
$385$	18.3156	0.933447
$386$	0	0
$387$	5.13164	0.260856
$388$	0	0
$389$	23.2078	1.17668	0.588342	−	0.808612i	$-0.299781\pi$
$389$	23.2078	1.17668	0.588342	+	0.808612i	$0.299781\pi$
$390$	0	0
$391$	13.5956	0.687561
$392$	0	0
$393$	7.56965	0.381838
$394$	0	0
$395$	−44.2897	−2.22846
$396$	0	0
$397$	18.2947	0.918182	0.459091	−	0.888389i	$-0.348175\pi$
$397$	18.2947	0.918182	0.459091	+	0.888389i	$0.348175\pi$
$398$	0	0
$399$	61.9036	3.09906
$400$	0	0
$401$	−20.1942	−1.00845	−0.504224	−	0.863573i	$-0.668222\pi$
$401$	−20.1942	−1.00845	−0.504224	+	0.863573i	$0.668222\pi$
$402$	0	0
$403$	3.68151	0.183389
$404$	0	0
$405$	35.8509	1.78144
$406$	0	0
$407$	0.813494	0.0403234
$408$	0	0
$409$	−6.90809	−0.341583	−0.170791	−	0.985307i	$-0.554632\pi$
$409$	−6.90809	−0.341583	−0.170791	+	0.985307i	$0.554632\pi$
$410$	0	0
$411$	55.6658	2.74579
$412$	0	0
$413$	−33.5784	−1.65229
$414$	0	0
$415$	−55.7899	−2.73862
$416$	0	0
$417$	−5.66594	−0.277462
$418$	0	0
$419$	32.8613	1.60538	0.802689	−	0.596397i	$-0.203402\pi$
$419$	32.8613	1.60538	0.802689	+	0.596397i	$0.203402\pi$
$420$	0	0
$421$	3.32397	0.162000	0.0810002	−	0.996714i	$-0.474189\pi$
$421$	3.32397	0.162000	0.0810002	+	0.996714i	$0.474189\pi$
$422$	0	0
$423$	56.6330	2.75359
$424$	0	0
$425$	24.4544	1.18621
$426$	0	0
$427$	−40.9389	−1.98117
$428$	0	0
$429$	−10.6521	−0.514288
$430$	0	0
$431$	−0.563527	−0.0271442	−0.0135721	−	0.999908i	$-0.504320\pi$
$431$	−0.563527	−0.0271442	−0.0135721	+	0.999908i	$0.504320\pi$
$432$	0	0
$433$	−34.6983	−1.66749	−0.833746	−	0.552148i	$-0.813808\pi$
$433$	−34.6983	−1.66749	−0.833746	+	0.552148i	$0.813808\pi$
$434$	0	0
$435$	−72.6138	−3.48157
$436$	0	0
$437$	−28.8857	−1.38179
$438$	0	0
$439$	−8.95916	−0.427597	−0.213799	−	0.976878i	$-0.568584\pi$
$439$	−8.95916	−0.427597	−0.213799	+	0.976878i	$0.568584\pi$
$440$	0	0
$441$	56.8361	2.70648
$442$	0	0
$443$	−37.2674	−1.77063	−0.885314	−	0.464993i	$-0.846057\pi$
$443$	−37.2674	−1.77063	−0.885314	+	0.464993i	$0.846057\pi$
$444$	0	0
$445$	−37.6369	−1.78416
$446$	0	0
$447$	43.6552	2.06482
$448$	0	0
$449$	23.1091	1.09058	0.545292	−	0.838246i	$-0.316419\pi$
$449$	23.1091	1.09058	0.545292	+	0.838246i	$0.316419\pi$
$450$	0	0
$451$	−3.71117	−0.174752
$452$	0	0
$453$	15.3573	0.721549
$454$	0	0
$455$	−48.4539	−2.27156
$456$	0	0
$457$	−4.86020	−0.227350	−0.113675	−	0.993518i	$-0.536262\pi$
$457$	−4.86020	−0.227350	−0.113675	+	0.993518i	$0.536262\pi$
$458$	0	0
$459$	21.7086	1.01327
$460$	0	0
$461$	18.9215	0.881261	0.440630	−	0.897689i	$-0.354755\pi$
$461$	18.9215	0.881261	0.440630	+	0.897689i	$0.354755\pi$
$462$	0	0
$463$	−1.98053	−0.0920429	−0.0460215	−	0.998940i	$-0.514654\pi$
$463$	−1.98053	−0.0920429	−0.0460215	+	0.998940i	$0.514654\pi$
$464$	0	0
$465$	14.0837	0.653116
$466$	0	0
$467$	0.206755	0.00956749	0.00478375	−	0.999989i	$-0.498477\pi$
$467$	0.206755	0.00956749	0.00478375	+	0.999989i	$0.498477\pi$
$468$	0	0
$469$	−56.2389	−2.59687
$470$	0	0
$471$	−58.9276	−2.71524
$472$	0	0
$473$	−0.986823	−0.0453742
$474$	0	0
$475$	−51.9567	−2.38393
$476$	0	0
$477$	14.7170	0.673847
$478$	0	0
$479$	11.8039	0.539333	0.269667	−	0.962954i	$-0.413086\pi$
$479$	11.8039	0.539333	0.269667	+	0.962954i	$0.413086\pi$
$480$	0	0
$481$	−2.15210	−0.0981275
$482$	0	0
$483$	−69.2012	−3.14876
$484$	0	0
$485$	−19.2604	−0.874571
$486$	0	0
$487$	−12.1579	−0.550925	−0.275462	−	0.961312i	$-0.588831\pi$
$487$	−12.1579	−0.550925	−0.275462	+	0.961312i	$0.588831\pi$
$488$	0	0
$489$	−46.7909	−2.11596
$490$	0	0
$491$	14.6769	0.662357	0.331179	−	0.943568i	$-0.392554\pi$
$491$	14.6769	0.662357	0.331179	+	0.943568i	$0.392554\pi$
$492$	0	0
$493$	−14.8262	−0.667736
$494$	0	0
$495$	−27.1982	−1.22247
$496$	0	0
$497$	0.0963330	0.00432112
$498$	0	0
$499$	2.63326	0.117881	0.0589403	−	0.998262i	$-0.481228\pi$
$499$	2.63326	0.117881	0.0589403	+	0.998262i	$0.481228\pi$
$500$	0	0
$501$	−28.8785	−1.29019
$502$	0	0
$503$	5.48187	0.244425	0.122212	−	0.992504i	$-0.461001\pi$
$503$	5.48187	0.244425	0.122212	+	0.992504i	$0.461001\pi$
$504$	0	0
$505$	73.2421	3.25923
$506$	0	0
$507$	−10.8652	−0.482540
$508$	0	0
$509$	−10.8677	−0.481703	−0.240852	−	0.970562i	$-0.577427\pi$
$509$	−10.8677	−0.481703	−0.240852	+	0.970562i	$0.577427\pi$
$510$	0	0
$511$	−7.99644	−0.353742
$512$	0	0
$513$	−46.1228	−2.03637
$514$	0	0
$515$	−0.418472	−0.0184401
$516$	0	0
$517$	−10.8906	−0.478969
$518$	0	0
$519$	−47.4702	−2.08371
$520$	0	0
$521$	−22.7378	−0.996161	−0.498081	−	0.867131i	$-0.665962\pi$
$521$	−22.7378	−0.996161	−0.498081	+	0.867131i	$0.665962\pi$
$522$	0	0
$523$	−4.29385	−0.187757	−0.0938784	−	0.995584i	$-0.529927\pi$
$523$	−4.29385	−0.187757	−0.0938784	+	0.995584i	$0.529927\pi$
$524$	0	0
$525$	−124.472	−5.43241
$526$	0	0
$527$	2.87558	0.125262
$528$	0	0
$529$	9.29093	0.403953
$530$	0	0
$531$	49.8632	2.16388
$532$	0	0
$533$	9.81793	0.425262
$534$	0	0
$535$	−65.7732	−2.84362
$536$	0	0
$537$	−26.0246	−1.12305
$538$	0	0
$539$	−10.9297	−0.470775
$540$	0	0
$541$	3.58128	0.153971	0.0769855	−	0.997032i	$-0.475470\pi$
$541$	3.58128	0.153971	0.0769855	+	0.997032i	$0.475470\pi$
$542$	0	0
$543$	46.6336	2.00124
$544$	0	0
$545$	−26.5764	−1.13841
$546$	0	0
$547$	18.6549	0.797624	0.398812	−	0.917033i	$-0.369423\pi$
$547$	18.6549	0.797624	0.398812	+	0.917033i	$0.369423\pi$
$548$	0	0
$549$	60.7933	2.59459
$550$	0	0
$551$	31.5001	1.34195
$552$	0	0
$553$	46.0284	1.95733
$554$	0	0
$555$	−8.23293	−0.349468
$556$	0	0
$557$	15.6996	0.665213	0.332607	−	0.943066i	$-0.392072\pi$
$557$	15.6996	0.665213	0.332607	+	0.943066i	$0.392072\pi$
$558$	0	0
$559$	2.61065	0.110419
$560$	0	0
$561$	−8.32024	−0.351281
$562$	0	0
$563$	30.0469	1.26633	0.633163	−	0.774018i	$-0.281756\pi$
$563$	30.0469	1.26633	0.633163	+	0.774018i	$0.281756\pi$
$564$	0	0
$565$	57.7538	2.42972
$566$	0	0
$567$	−37.2583	−1.56470
$568$	0	0
$569$	42.4359	1.77900	0.889502	−	0.456931i	$-0.151051\pi$
$569$	42.4359	1.77900	0.889502	+	0.456931i	$0.151051\pi$
$570$	0	0
$571$	−43.0387	−1.80111	−0.900557	−	0.434738i	$-0.856841\pi$
$571$	−43.0387	−1.80111	−0.900557	+	0.434738i	$0.856841\pi$
$572$	0	0
$573$	−35.5482	−1.48505
$574$	0	0
$575$	58.0816	2.42217
$576$	0	0
$577$	11.4394	0.476226	0.238113	−	0.971237i	$-0.423471\pi$
$577$	11.4394	0.476226	0.238113	+	0.971237i	$0.423471\pi$
$578$	0	0
$579$	−34.2333	−1.42269
$580$	0	0
$581$	57.9801	2.40542
$582$	0	0
$583$	−2.83011	−0.117211
$584$	0	0
$585$	71.9530	2.97489
$586$	0	0
$587$	−41.6737	−1.72006	−0.860029	−	0.510245i	$-0.829555\pi$
$587$	−41.6737	−1.72006	−0.860029	+	0.510245i	$0.829555\pi$
$588$	0	0
$589$	−6.10955	−0.251740
$590$	0	0
$591$	40.5941	1.66982
$592$	0	0
$593$	26.1597	1.07425	0.537126	−	0.843502i	$-0.319510\pi$
$593$	26.1597	1.07425	0.537126	+	0.843502i	$0.319510\pi$
$594$	0	0
$595$	−37.8468	−1.55157
$596$	0	0
$597$	−51.0835	−2.09071
$598$	0	0
$599$	−26.5297	−1.08397	−0.541987	−	0.840387i	$-0.682328\pi$
$599$	−26.5297	−1.08397	−0.541987	+	0.840387i	$0.682328\pi$
$600$	0	0
$601$	−6.94788	−0.283410	−0.141705	−	0.989909i	$-0.545258\pi$
$601$	−6.94788	−0.283410	−0.141705	+	0.989909i	$0.545258\pi$
$602$	0	0
$603$	83.5134	3.40093
$604$	0	0
$605$	−37.6854	−1.53213
$606$	0	0
$607$	8.63243	0.350380	0.175190	−	0.984535i	$-0.443946\pi$
$607$	8.63243	0.350380	0.175190	+	0.984535i	$0.443946\pi$
$608$	0	0
$609$	75.4645	3.05798
$610$	0	0
$611$	28.8112	1.16558
$612$	0	0
$613$	−35.9695	−1.45279	−0.726396	−	0.687276i	$-0.758806\pi$
$613$	−35.9695	−1.45279	−0.726396	+	0.687276i	$0.758806\pi$
$614$	0	0
$615$	37.5587	1.51451
$616$	0	0
$617$	−30.0267	−1.20883	−0.604415	−	0.796670i	$-0.706593\pi$
$617$	−30.0267	−1.20883	−0.604415	+	0.796670i	$0.706593\pi$
$618$	0	0
$619$	18.4891	0.743139	0.371569	−	0.928405i	$-0.378820\pi$
$619$	18.4891	0.743139	0.371569	+	0.928405i	$0.378820\pi$
$620$	0	0
$621$	51.5601	2.06903
$622$	0	0
$623$	39.1144	1.56709
$624$	0	0
$625$	28.3657	1.13463
$626$	0	0
$627$	17.6774	0.705969
$628$	0	0
$629$	−1.68098	−0.0670252
$630$	0	0
$631$	−5.17204	−0.205896	−0.102948	−	0.994687i	$-0.532827\pi$
$631$	−5.17204	−0.205896	−0.102948	+	0.994687i	$0.532827\pi$
$632$	0	0
$633$	29.1676	1.15931
$634$	0	0
$635$	−22.2728	−0.883869
$636$	0	0
$637$	28.9146	1.14564
$638$	0	0
$639$	−0.143052	−0.00565906
$640$	0	0
$641$	−16.9314	−0.668750	−0.334375	−	0.942440i	$-0.608525\pi$
$641$	−16.9314	−0.668750	−0.334375	+	0.942440i	$0.608525\pi$
$642$	0	0
$643$	45.4872	1.79384	0.896920	−	0.442193i	$-0.145799\pi$
$643$	45.4872	1.79384	0.896920	+	0.442193i	$0.145799\pi$
$644$	0	0
$645$	9.98710	0.393242
$646$	0	0
$647$	33.7207	1.32570	0.662848	−	0.748754i	$-0.269348\pi$
$647$	33.7207	1.32570	0.662848	+	0.748754i	$0.269348\pi$
$648$	0	0
$649$	−9.58878	−0.376393
$650$	0	0
$651$	−14.6366	−0.573654
$652$	0	0
$653$	42.3457	1.65711	0.828557	−	0.559904i	$-0.189162\pi$
$653$	42.3457	1.65711	0.828557	+	0.559904i	$0.189162\pi$
$654$	0	0
$655$	9.83270	0.384195
$656$	0	0
$657$	11.8745	0.463269
$658$	0	0
$659$	−49.8143	−1.94049	−0.970244	−	0.242129i	$-0.922154\pi$
$659$	−49.8143	−1.94049	−0.970244	+	0.242129i	$0.922154\pi$
$660$	0	0
$661$	14.5547	0.566114	0.283057	−	0.959103i	$-0.408651\pi$
$661$	14.5547	0.566114	0.283057	+	0.959103i	$0.408651\pi$
$662$	0	0
$663$	22.0112	0.854846
$664$	0	0
$665$	80.4105	3.11819
$666$	0	0
$667$	−35.2135	−1.36347
$668$	0	0
$669$	61.5681	2.38036
$670$	0	0
$671$	−11.6907	−0.451313
$672$	0	0
$673$	−24.8922	−0.959523	−0.479761	−	0.877399i	$-0.659277\pi$
$673$	−24.8922	−0.959523	−0.479761	+	0.877399i	$0.659277\pi$
$674$	0	0
$675$	92.7410	3.56960
$676$	0	0
$677$	30.3177	1.16520	0.582601	−	0.812758i	$-0.302035\pi$
$677$	30.3177	1.16520	0.582601	+	0.812758i	$0.302035\pi$
$678$	0	0
$679$	20.0166	0.768165
$680$	0	0
$681$	0.752647	0.0288415
$682$	0	0
$683$	20.0088	0.765616	0.382808	−	0.923828i	$-0.374957\pi$
$683$	20.0088	0.765616	0.382808	+	0.923828i	$0.374957\pi$
$684$	0	0
$685$	72.3078	2.76274
$686$	0	0
$687$	−66.4490	−2.53519
$688$	0	0
$689$	7.48709	0.285235
$690$	0	0
$691$	37.5242	1.42749	0.713744	−	0.700406i	$-0.246998\pi$
$691$	37.5242	1.42749	0.713744	+	0.700406i	$0.246998\pi$
$692$	0	0
$693$	28.2659	1.07373
$694$	0	0
$695$	−7.35984	−0.279175
$696$	0	0
$697$	7.66867	0.290471
$698$	0	0
$699$	−20.1530	−0.762256
$700$	0	0
$701$	−28.8949	−1.09134	−0.545672	−	0.837999i	$-0.683726\pi$
$701$	−28.8949	−1.09134	−0.545672	+	0.837999i	$0.683726\pi$
$702$	0	0
$703$	3.57147	0.134701
$704$	0	0
$705$	110.218	4.15106
$706$	0	0
$707$	−76.1174	−2.86269
$708$	0	0
$709$	4.28959	0.161099	0.0805495	−	0.996751i	$-0.474332\pi$
$709$	4.28959	0.161099	0.0805495	+	0.996751i	$0.474332\pi$
$710$	0	0
$711$	−68.3511	−2.56337
$712$	0	0
$713$	6.82979	0.255777
$714$	0	0
$715$	−13.8367	−0.517463
$716$	0	0
$717$	−32.6374	−1.21887
$718$	0	0
$719$	−28.0572	−1.04636	−0.523179	−	0.852223i	$-0.675254\pi$
$719$	−28.0572	−1.04636	−0.523179	+	0.852223i	$0.675254\pi$
$720$	0	0
$721$	0.434901	0.0161965
$722$	0	0
$723$	−3.53322	−0.131402
$724$	0	0
$725$	−63.3385	−2.35233
$726$	0	0
$727$	−41.5405	−1.54065	−0.770327	−	0.637649i	$-0.779907\pi$
$727$	−41.5405	−1.54065	−0.770327	+	0.637649i	$0.779907\pi$
$728$	0	0
$729$	−26.4285	−0.978833
$730$	0	0
$731$	2.03915	0.0754206
$732$	0	0
$733$	51.3178	1.89547	0.947733	−	0.319065i	$-0.103369\pi$
$733$	51.3178	1.89547	0.947733	+	0.319065i	$0.103369\pi$
$734$	0	0
$735$	110.613	4.08004
$736$	0	0
$737$	−16.0598	−0.591570
$738$	0	0
$739$	6.61636	0.243387	0.121693	−	0.992568i	$-0.461168\pi$
$739$	6.61636	0.243387	0.121693	+	0.992568i	$0.461168\pi$
$740$	0	0
$741$	−46.7658	−1.71798
$742$	0	0
$743$	46.8632	1.71924	0.859622	−	0.510930i	$-0.170699\pi$
$743$	46.8632	1.71924	0.859622	+	0.510930i	$0.170699\pi$
$744$	0	0
$745$	56.7065	2.07757
$746$	0	0
$747$	−86.0991	−3.15020
$748$	0	0
$749$	68.3554	2.49765
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−59.8069	−2.17948
$754$	0	0
$755$	19.9486	0.726002
$756$	0	0
$757$	23.4260	0.851432	0.425716	−	0.904857i	$-0.360022\pi$
$757$	23.4260	0.851432	0.425716	+	0.904857i	$0.360022\pi$
$758$	0	0
$759$	−19.7614	−0.717292
$760$	0	0
$761$	−25.8260	−0.936191	−0.468096	−	0.883678i	$-0.655060\pi$
$761$	−25.8260	−0.936191	−0.468096	+	0.883678i	$0.655060\pi$
$762$	0	0
$763$	27.6198	0.999904
$764$	0	0
$765$	56.2016	2.03197
$766$	0	0
$767$	25.3672	0.915956
$768$	0	0
$769$	−30.5921	−1.10318	−0.551590	−	0.834116i	$-0.685979\pi$
$769$	−30.5921	−1.10318	−0.551590	+	0.834116i	$0.685979\pi$
$770$	0	0
$771$	−44.3104	−1.59580
$772$	0	0
$773$	−25.6287	−0.921799	−0.460900	−	0.887452i	$-0.652473\pi$
$773$	−25.6287	−0.921799	−0.460900	+	0.887452i	$0.652473\pi$
$774$	0	0
$775$	12.2847	0.441280
$776$	0	0
$777$	8.55614	0.306950
$778$	0	0
$779$	−16.2931	−0.583761
$780$	0	0
$781$	0.0275092	0.000984357 0
$782$	0	0
$783$	−56.2267	−2.00938
$784$	0	0
$785$	−76.5447	−2.73200
$786$	0	0
$787$	11.9337	0.425392	0.212696	−	0.977118i	$-0.431776\pi$
$787$	11.9337	0.425392	0.212696	+	0.977118i	$0.431776\pi$
$788$	0	0
$789$	71.6864	2.55210
$790$	0	0
$791$	−60.0212	−2.13411
$792$	0	0
$793$	30.9277	1.09828
$794$	0	0
$795$	28.6420	1.01583
$796$	0	0
$797$	−13.0053	−0.460670	−0.230335	−	0.973111i	$-0.573982\pi$
$797$	−13.0053	−0.460670	−0.230335	+	0.973111i	$0.573982\pi$
$798$	0	0
$799$	22.5041	0.796139
$800$	0	0
$801$	−58.0840	−2.05230
$802$	0	0
$803$	−2.28349	−0.0805827
$804$	0	0
$805$	−89.8898	−3.16820
$806$	0	0
$807$	26.5551	0.934782
$808$	0	0
$809$	30.5912	1.07553	0.537765	−	0.843095i	$-0.319269\pi$
$809$	30.5912	1.07553	0.537765	+	0.843095i	$0.319269\pi$
$810$	0	0
$811$	−9.49311	−0.333348	−0.166674	−	0.986012i	$-0.553303\pi$
$811$	−9.49311	−0.333348	−0.166674	+	0.986012i	$0.553303\pi$
$812$	0	0
$813$	23.3465	0.818798
$814$	0	0
$815$	−60.7796	−2.12902
$816$	0	0
$817$	−4.33244	−0.151573
$818$	0	0
$819$	−74.7777	−2.61295
$820$	0	0
$821$	45.9272	1.60287	0.801435	−	0.598081i	$-0.204070\pi$
$821$	45.9272	1.60287	0.801435	+	0.598081i	$0.204070\pi$
$822$	0	0
$823$	−12.7332	−0.443852	−0.221926	−	0.975064i	$-0.571234\pi$
$823$	−12.7332	−0.443852	−0.221926	+	0.975064i	$0.571234\pi$
$824$	0	0
$825$	−35.5447	−1.23751
$826$	0	0
$827$	1.63891	0.0569906	0.0284953	−	0.999594i	$-0.490928\pi$
$827$	1.63891	0.0569906	0.0284953	+	0.999594i	$0.490928\pi$
$828$	0	0
$829$	0.733575	0.0254781	0.0127391	−	0.999919i	$-0.495945\pi$
$829$	0.733575	0.0254781	0.0127391	+	0.999919i	$0.495945\pi$
$830$	0	0
$831$	−48.4897	−1.68209
$832$	0	0
$833$	22.5848	0.782518
$834$	0	0
$835$	−37.5120	−1.29816
$836$	0	0
$837$	10.9054	0.376944
$838$	0	0
$839$	−4.17085	−0.143994	−0.0719968	−	0.997405i	$-0.522937\pi$
$839$	−4.17085	−0.143994	−0.0719968	+	0.997405i	$0.522937\pi$
$840$	0	0
$841$	9.40069	0.324162
$842$	0	0
$843$	97.4427	3.35610
$844$	0	0
$845$	−14.1135	−0.485518
$846$	0	0
$847$	39.1649	1.34572
$848$	0	0
$849$	−64.3956	−2.21005
$850$	0	0
$851$	−3.99250	−0.136861
$852$	0	0
$853$	−30.0380	−1.02848	−0.514240	−	0.857646i	$-0.671926\pi$
$853$	−30.0380	−1.02848	−0.514240	+	0.857646i	$0.671926\pi$
$854$	0	0
$855$	−119.408	−4.08366
$856$	0	0
$857$	−38.5241	−1.31596	−0.657979	−	0.753036i	$-0.728588\pi$
$857$	−38.5241	−1.31596	−0.657979	+	0.753036i	$0.728588\pi$
$858$	0	0
$859$	47.3399	1.61522	0.807608	−	0.589720i	$-0.200762\pi$
$859$	47.3399	1.61522	0.807608	+	0.589720i	$0.200762\pi$
$860$	0	0
$861$	−39.0332	−1.33025
$862$	0	0
$863$	−14.2960	−0.486642	−0.243321	−	0.969946i	$-0.578237\pi$
$863$	−14.2960	−0.486642	−0.243321	+	0.969946i	$0.578237\pi$
$864$	0	0
$865$	−61.6620	−2.09657
$866$	0	0
$867$	−33.8667	−1.15017
$868$	0	0
$869$	13.1441	0.445881
$870$	0	0
$871$	42.4863	1.43959
$872$	0	0
$873$	−29.7241	−1.00601
$874$	0	0
$875$	−82.5912	−2.79209
$876$	0	0
$877$	−0.720461	−0.0243282	−0.0121641	−	0.999926i	$-0.503872\pi$
$877$	−0.720461	−0.0243282	−0.0121641	+	0.999926i	$0.503872\pi$
$878$	0	0
$879$	42.8888	1.44660
$880$	0	0
$881$	44.7151	1.50649	0.753245	−	0.657740i	$-0.228487\pi$
$881$	44.7151	1.50649	0.753245	+	0.657740i	$0.228487\pi$
$882$	0	0
$883$	−29.3774	−0.988629	−0.494314	−	0.869283i	$-0.664581\pi$
$883$	−29.3774	−0.988629	−0.494314	+	0.869283i	$0.664581\pi$
$884$	0	0
$885$	97.0428	3.26206
$886$	0	0
$887$	25.2596	0.848133	0.424067	−	0.905631i	$-0.360602\pi$
$887$	25.2596	0.848133	0.424067	+	0.905631i	$0.360602\pi$
$888$	0	0
$889$	23.1472	0.776332
$890$	0	0
$891$	−10.6396	−0.356441
$892$	0	0
$893$	−47.8130	−1.60000
$894$	0	0
$895$	−33.8050	−1.12998
$896$	0	0
$897$	52.2788	1.74554
$898$	0	0
$899$	−7.44794	−0.248403
$900$	0	0
$901$	5.84808	0.194828
$902$	0	0
$903$	−10.3792	−0.345397
$904$	0	0
$905$	60.5753	2.01359
$906$	0	0
$907$	12.3854	0.411252	0.205626	−	0.978631i	$-0.434077\pi$
$907$	12.3854	0.411252	0.205626	+	0.978631i	$0.434077\pi$
$908$	0	0
$909$	113.033	3.74906
$910$	0	0
$911$	12.0835	0.400345	0.200173	−	0.979761i	$-0.435850\pi$
$911$	12.0835	0.400345	0.200173	+	0.979761i	$0.435850\pi$
$912$	0	0
$913$	16.5570	0.547957
$914$	0	0
$915$	118.315	3.91137
$916$	0	0
$917$	−10.2187	−0.337452
$918$	0	0
$919$	−26.0939	−0.860758	−0.430379	−	0.902648i	$-0.641620\pi$
$919$	−26.0939	−0.860758	−0.430379	+	0.902648i	$0.641620\pi$
$920$	0	0
$921$	24.0866	0.793679
$922$	0	0
$923$	−0.0727758	−0.00239544
$924$	0	0
$925$	−7.18130	−0.236120
$926$	0	0
$927$	−0.645817	−0.0212114
$928$	0	0
$929$	12.5428	0.411517	0.205759	−	0.978603i	$-0.434034\pi$
$929$	12.5428	0.411517	0.205759	+	0.978603i	$0.434034\pi$
$930$	0	0
$931$	−47.9845	−1.57263
$932$	0	0
$933$	71.2215	2.33169
$934$	0	0
$935$	−10.8077	−0.353449
$936$	0	0
$937$	14.8980	0.486697	0.243348	−	0.969939i	$-0.421754\pi$
$937$	14.8980	0.486697	0.243348	+	0.969939i	$0.421754\pi$
$938$	0	0
$939$	−83.1323	−2.71292
$940$	0	0
$941$	26.1199	0.851485	0.425743	−	0.904844i	$-0.360013\pi$
$941$	26.1199	0.851485	0.425743	+	0.904844i	$0.360013\pi$
$942$	0	0
$943$	18.2138	0.593124
$944$	0	0
$945$	−143.530	−4.66904
$946$	0	0
$947$	−38.9973	−1.26724	−0.633620	−	0.773644i	$-0.718432\pi$
$947$	−38.9973	−1.26724	−0.633620	+	0.773644i	$0.718432\pi$
$948$	0	0
$949$	6.04100	0.196099
$950$	0	0
$951$	−63.0914	−2.04588
$952$	0	0
$953$	15.9205	0.515715	0.257858	−	0.966183i	$-0.416983\pi$
$953$	15.9205	0.515715	0.257858	+	0.966183i	$0.416983\pi$
$954$	0	0
$955$	−46.1758	−1.49422
$956$	0	0
$957$	21.5499	0.696611
$958$	0	0
$959$	−75.1465	−2.42661
$960$	0	0
$961$	−29.5554	−0.953402
$962$	0	0
$963$	−101.506	−3.27099
$964$	0	0
$965$	−44.4677	−1.43147
$966$	0	0
$967$	36.4744	1.17294	0.586470	−	0.809971i	$-0.300517\pi$
$967$	36.4744	1.17294	0.586470	+	0.809971i	$0.300517\pi$
$968$	0	0
$969$	−36.5282	−1.17346
$970$	0	0
$971$	19.2365	0.617328	0.308664	−	0.951171i	$-0.400118\pi$
$971$	19.2365	0.617328	0.308664	+	0.951171i	$0.400118\pi$
$972$	0	0
$973$	7.64877	0.245208
$974$	0	0
$975$	94.0338	3.01149
$976$	0	0
$977$	−2.59668	−0.0830751	−0.0415375	−	0.999137i	$-0.513226\pi$
$977$	−2.59668	−0.0830751	−0.0415375	+	0.999137i	$0.513226\pi$
$978$	0	0
$979$	11.1697	0.356984
$980$	0	0
$981$	−41.0148	−1.30950
$982$	0	0
$983$	14.7358	0.469998	0.234999	−	0.971996i	$-0.424491\pi$
$983$	14.7358	0.469998	0.234999	+	0.971996i	$0.424491\pi$
$984$	0	0
$985$	52.7302	1.68012
$986$	0	0
$987$	−114.545	−3.64601
$988$	0	0
$989$	4.84317	0.154004
$990$	0	0
$991$	36.8704	1.17123	0.585614	−	0.810590i	$-0.300854\pi$
$991$	36.8704	1.17123	0.585614	+	0.810590i	$0.300854\pi$
$992$	0	0
$993$	−17.6388	−0.559751
$994$	0	0
$995$	−66.3556	−2.10361
$996$	0	0
$997$	−24.1457	−0.764703	−0.382352	−	0.924017i	$-0.624886\pi$
$997$	−24.1457	−0.764703	−0.382352	+	0.924017i	$0.624886\pi$
$998$	0	0
$999$	−6.37496	−0.201695

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.46	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.46	✓	50	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00349 q^{3} +3.90143 q^{5} -4.05459 q^{7} +6.02097 q^{9} +O(q^{10})\)	\(q+3.00349 q^{3} +3.90143 q^{5} -4.05459 q^{7} +6.02097 q^{9} -1.15784 q^{11} +3.06308 q^{13} +11.7179 q^{15} +2.39254 q^{17} -5.08326 q^{19} -12.1779 q^{21} +5.68251 q^{23} +10.2211 q^{25} +9.07346 q^{27} -6.19683 q^{29} +1.20190 q^{31} -3.47758 q^{33} -15.8187 q^{35} -0.702594 q^{37} +9.19995 q^{39} +3.20524 q^{41} +0.852294 q^{43} +23.4904 q^{45} +9.40596 q^{47} +9.43969 q^{49} +7.18597 q^{51} +2.44430 q^{53} -4.51724 q^{55} -15.2676 q^{57} +8.28158 q^{59} +10.0969 q^{61} -24.4126 q^{63} +11.9504 q^{65} +13.8704 q^{67} +17.0674 q^{69} -0.0237590 q^{71} +1.97219 q^{73} +30.6991 q^{75} +4.69458 q^{77} -11.3522 q^{79} +9.18918 q^{81} -14.2999 q^{83} +9.33431 q^{85} -18.6121 q^{87} -9.64695 q^{89} -12.4195 q^{91} +3.60989 q^{93} -19.8320 q^{95} -4.93677 q^{97} -6.97134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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