LMFDB - Embedded newform 6008.2.a.c.1.12

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:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.66100 q^{3} -3.23151 q^{5} +4.31694 q^{7} -0.241072 q^{9} +O(q^{10})$	$q-1.66100 q^{3} -3.23151 q^{5} +4.31694 q^{7} -0.241072 q^{9} -0.366901 q^{11} -0.733107 q^{13} +5.36755 q^{15} +1.89322 q^{17} +0.903188 q^{19} -7.17045 q^{21} -4.65025 q^{23} +5.44266 q^{25} +5.38343 q^{27} +6.97988 q^{29} -10.4769 q^{31} +0.609423 q^{33} -13.9502 q^{35} -8.90802 q^{37} +1.21769 q^{39} +5.85492 q^{41} -8.62204 q^{43} +0.779028 q^{45} +6.39241 q^{47} +11.6360 q^{49} -3.14465 q^{51} +7.70361 q^{53} +1.18564 q^{55} -1.50020 q^{57} +2.27013 q^{59} -9.01693 q^{61} -1.04069 q^{63} +2.36904 q^{65} +1.58414 q^{67} +7.72407 q^{69} +14.3922 q^{71} +7.66000 q^{73} -9.04028 q^{75} -1.58389 q^{77} +5.91255 q^{79} -8.21867 q^{81} -16.8028 q^{83} -6.11798 q^{85} -11.5936 q^{87} +8.32265 q^{89} -3.16478 q^{91} +17.4021 q^{93} -2.91866 q^{95} +14.2747 q^{97} +0.0884496 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * 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
$3$	−1.66100	−0.958980	−0.479490	−	0.877547i	$-0.659178\pi$
$3$	−1.66100	−0.958980	−0.479490	+	0.877547i	$0.659178\pi$
$4$	0	0
$5$	−3.23151	−1.44518	−0.722588	−	0.691279i	$-0.757048\pi$
$5$	−3.23151	−1.44518	−0.722588	+	0.691279i	$0.757048\pi$
$6$	0	0
$7$	4.31694	1.63165	0.815825	−	0.578299i	$-0.196283\pi$
$7$	4.31694	1.63165	0.815825	+	0.578299i	$0.196283\pi$
$8$	0	0
$9$	−0.241072	−0.0803574
$10$	0	0
$11$	−0.366901	−0.110625	−0.0553124	−	0.998469i	$-0.517615\pi$
$11$	−0.366901	−0.110625	−0.0553124	+	0.998469i	$0.517615\pi$
$12$	0	0
$13$	−0.733107	−0.203327	−0.101664	−	0.994819i	$-0.532417\pi$
$13$	−0.733107	−0.203327	−0.101664	+	0.994819i	$0.532417\pi$
$14$	0	0
$15$	5.36755	1.38589
$16$	0	0
$17$	1.89322	0.459174	0.229587	−	0.973288i	$-0.426262\pi$
$17$	1.89322	0.459174	0.229587	+	0.973288i	$0.426262\pi$
$18$	0	0
$19$	0.903188	0.207206	0.103603	−	0.994619i	$-0.466963\pi$
$19$	0.903188	0.207206	0.103603	+	0.994619i	$0.466963\pi$
$20$	0	0
$21$	−7.17045	−1.56472
$22$	0	0
$23$	−4.65025	−0.969644	−0.484822	−	0.874613i	$-0.661116\pi$
$23$	−4.65025	−0.969644	−0.484822	+	0.874613i	$0.661116\pi$
$24$	0	0
$25$	5.44266	1.08853
$26$	0	0
$27$	5.38343	1.03604
$28$	0	0
$29$	6.97988	1.29613	0.648066	−	0.761584i	$-0.275578\pi$
$29$	6.97988	1.29613	0.648066	+	0.761584i	$0.275578\pi$
$30$	0	0
$31$	−10.4769	−1.88170	−0.940849	−	0.338826i	$-0.889970\pi$
$31$	−10.4769	−1.88170	−0.940849	+	0.338826i	$0.889970\pi$
$32$	0	0
$33$	0.609423	0.106087
$34$	0	0
$35$	−13.9502	−2.35802
$36$	0	0
$37$	−8.90802	−1.46447	−0.732235	−	0.681052i	$-0.761523\pi$
$37$	−8.90802	−1.46447	−0.732235	+	0.681052i	$0.761523\pi$
$38$	0	0
$39$	1.21769	0.194987
$40$	0	0
$41$	5.85492	0.914384	0.457192	−	0.889368i	$-0.348855\pi$
$41$	5.85492	0.914384	0.457192	+	0.889368i	$0.348855\pi$
$42$	0	0
$43$	−8.62204	−1.31485	−0.657424	−	0.753521i	$-0.728354\pi$
$43$	−8.62204	−1.31485	−0.657424	+	0.753521i	$0.728354\pi$
$44$	0	0
$45$	0.779028	0.116131
$46$	0	0
$47$	6.39241	0.932429	0.466215	−	0.884672i	$-0.345617\pi$
$47$	6.39241	0.932429	0.466215	+	0.884672i	$0.345617\pi$
$48$	0	0
$49$	11.6360	1.66228
$50$	0	0
$51$	−3.14465	−0.440339
$52$	0	0
$53$	7.70361	1.05817	0.529086	−	0.848568i	$-0.322535\pi$
$53$	7.70361	1.05817	0.529086	+	0.848568i	$0.322535\pi$
$54$	0	0
$55$	1.18564	0.159872
$56$	0	0
$57$	−1.50020	−0.198706
$58$	0	0
$59$	2.27013	0.295545	0.147773	−	0.989021i	$-0.452790\pi$
$59$	2.27013	0.295545	0.147773	+	0.989021i	$0.452790\pi$
$60$	0	0
$61$	−9.01693	−1.15450	−0.577250	−	0.816568i	$-0.695874\pi$
$61$	−9.01693	−1.15450	−0.577250	+	0.816568i	$0.695874\pi$
$62$	0	0
$63$	−1.04069	−0.131115
$64$	0	0
$65$	2.36904	0.293844
$66$	0	0
$67$	1.58414	0.193533	0.0967667	−	0.995307i	$-0.469150\pi$
$67$	1.58414	0.193533	0.0967667	+	0.995307i	$0.469150\pi$
$68$	0	0
$69$	7.72407	0.929869
$70$	0	0
$71$	14.3922	1.70804	0.854018	−	0.520243i	$-0.174159\pi$
$71$	14.3922	1.70804	0.854018	+	0.520243i	$0.174159\pi$
$72$	0	0
$73$	7.66000	0.896535	0.448267	−	0.893900i	$-0.352041\pi$
$73$	7.66000	0.896535	0.448267	+	0.893900i	$0.352041\pi$
$74$	0	0
$75$	−9.04028	−1.04388
$76$	0	0
$77$	−1.58389	−0.180501
$78$	0	0
$79$	5.91255	0.665214	0.332607	−	0.943066i	$-0.392072\pi$
$79$	5.91255	0.665214	0.332607	+	0.943066i	$0.392072\pi$
$80$	0	0
$81$	−8.21867	−0.913185
$82$	0	0
$83$	−16.8028	−1.84435	−0.922174	−	0.386774i	$-0.873589\pi$
$83$	−16.8028	−1.84435	−0.922174	+	0.386774i	$0.873589\pi$
$84$	0	0
$85$	−6.11798	−0.663588
$86$	0	0
$87$	−11.5936	−1.24296
$88$	0	0
$89$	8.32265	0.882199	0.441099	−	0.897458i	$-0.354589\pi$
$89$	8.32265	0.882199	0.441099	+	0.897458i	$0.354589\pi$
$90$	0	0
$91$	−3.16478	−0.331759
$92$	0	0
$93$	17.4021	1.80451
$94$	0	0
$95$	−2.91866	−0.299449
$96$	0	0
$97$	14.2747	1.44938	0.724689	−	0.689076i	$-0.241984\pi$
$97$	14.2747	1.44938	0.724689	+	0.689076i	$0.241984\pi$
$98$	0	0
$99$	0.0884496	0.00888952
$100$	0	0
$101$	−16.4808	−1.63990	−0.819949	−	0.572437i	$-0.805998\pi$
$101$	−16.4808	−1.63990	−0.819949	+	0.572437i	$0.805998\pi$
$102$	0	0
$103$	7.95245	0.783579	0.391789	−	0.920055i	$-0.371856\pi$
$103$	7.95245	0.783579	0.391789	+	0.920055i	$0.371856\pi$
$104$	0	0
$105$	23.1714	2.26130
$106$	0	0
$107$	14.6956	1.42068	0.710338	−	0.703861i	$-0.248542\pi$
$107$	14.6956	1.42068	0.710338	+	0.703861i	$0.248542\pi$
$108$	0	0
$109$	4.41794	0.423162	0.211581	−	0.977360i	$-0.432139\pi$
$109$	4.41794	0.423162	0.211581	+	0.977360i	$0.432139\pi$
$110$	0	0
$111$	14.7962	1.40440
$112$	0	0
$113$	−8.55532	−0.804817	−0.402408	−	0.915460i	$-0.631827\pi$
$113$	−8.55532	−0.804817	−0.402408	+	0.915460i	$0.631827\pi$
$114$	0	0
$115$	15.0273	1.40131
$116$	0	0
$117$	0.176732	0.0163389
$118$	0	0
$119$	8.17294	0.749212
$120$	0	0
$121$	−10.8654	−0.987762
$122$	0	0
$123$	−9.72503	−0.876876
$124$	0	0
$125$	−1.43047	−0.127945
$126$	0	0
$127$	−3.15506	−0.279966	−0.139983	−	0.990154i	$-0.544705\pi$
$127$	−3.15506	−0.279966	−0.139983	+	0.990154i	$0.544705\pi$
$128$	0	0
$129$	14.3212	1.26091
$130$	0	0
$131$	2.95336	0.258036	0.129018	−	0.991642i	$-0.458817\pi$
$131$	2.95336	0.258036	0.129018	+	0.991642i	$0.458817\pi$
$132$	0	0
$133$	3.89901	0.338087
$134$	0	0
$135$	−17.3966	−1.49726
$136$	0	0
$137$	1.29292	0.110461	0.0552306	−	0.998474i	$-0.482411\pi$
$137$	1.29292	0.110461	0.0552306	+	0.998474i	$0.482411\pi$
$138$	0	0
$139$	4.45835	0.378152	0.189076	−	0.981962i	$-0.439451\pi$
$139$	4.45835	0.378152	0.189076	+	0.981962i	$0.439451\pi$
$140$	0	0
$141$	−10.6178	−0.894181
$142$	0	0
$143$	0.268978	0.0224930
$144$	0	0
$145$	−22.5556	−1.87314
$146$	0	0
$147$	−19.3274	−1.59410
$148$	0	0
$149$	−3.01787	−0.247233	−0.123617	−	0.992330i	$-0.539449\pi$
$149$	−3.01787	−0.247233	−0.123617	+	0.992330i	$0.539449\pi$
$150$	0	0
$151$	−14.5985	−1.18801	−0.594005	−	0.804462i	$-0.702454\pi$
$151$	−14.5985	−1.18801	−0.594005	+	0.804462i	$0.702454\pi$
$152$	0	0
$153$	−0.456404	−0.0368981
$154$	0	0
$155$	33.8561	2.71938
$156$	0	0
$157$	−24.4581	−1.95197	−0.975984	−	0.217841i	$-0.930099\pi$
$157$	−24.4581	−1.95197	−0.975984	+	0.217841i	$0.930099\pi$
$158$	0	0
$159$	−12.7957	−1.01477
$160$	0	0
$161$	−20.0749	−1.58212
$162$	0	0
$163$	7.92398	0.620654	0.310327	−	0.950630i	$-0.399561\pi$
$163$	7.92398	0.620654	0.310327	+	0.950630i	$0.399561\pi$
$164$	0	0
$165$	−1.96936	−0.153314
$166$	0	0
$167$	−6.10514	−0.472430	−0.236215	−	0.971701i	$-0.575907\pi$
$167$	−6.10514	−0.472430	−0.236215	+	0.971701i	$0.575907\pi$
$168$	0	0
$169$	−12.4626	−0.958658
$170$	0	0
$171$	−0.217734	−0.0166505
$172$	0	0
$173$	−25.0332	−1.90324	−0.951620	−	0.307277i	$-0.900582\pi$
$173$	−25.0332	−1.90324	−0.951620	+	0.307277i	$0.900582\pi$
$174$	0	0
$175$	23.4957	1.77610
$176$	0	0
$177$	−3.77068	−0.283422
$178$	0	0
$179$	−14.0321	−1.04881	−0.524406	−	0.851468i	$-0.675713\pi$
$179$	−14.0321	−1.04881	−0.524406	+	0.851468i	$0.675713\pi$
$180$	0	0
$181$	−20.1736	−1.49949	−0.749747	−	0.661724i	$-0.769825\pi$
$181$	−20.1736	−1.49949	−0.749747	+	0.661724i	$0.769825\pi$
$182$	0	0
$183$	14.9771	1.10714
$184$	0	0
$185$	28.7864	2.11642
$186$	0	0
$187$	−0.694626	−0.0507961
$188$	0	0
$189$	23.2399	1.69046
$190$	0	0
$191$	−15.1646	−1.09727	−0.548637	−	0.836061i	$-0.684853\pi$
$191$	−15.1646	−1.09727	−0.548637	+	0.836061i	$0.684853\pi$
$192$	0	0
$193$	−14.4188	−1.03789	−0.518945	−	0.854808i	$-0.673675\pi$
$193$	−14.4188	−1.03789	−0.518945	+	0.854808i	$0.673675\pi$
$194$	0	0
$195$	−3.93499	−0.281790
$196$	0	0
$197$	19.7343	1.40601	0.703004	−	0.711186i	$-0.251842\pi$
$197$	19.7343	1.40601	0.703004	+	0.711186i	$0.251842\pi$
$198$	0	0
$199$	−14.5102	−1.02860	−0.514299	−	0.857611i	$-0.671948\pi$
$199$	−14.5102	−1.02860	−0.514299	+	0.857611i	$0.671948\pi$
$200$	0	0
$201$	−2.63126	−0.185595
$202$	0	0
$203$	30.1317	2.11483
$204$	0	0
$205$	−18.9202	−1.32145
$206$	0	0
$207$	1.12105	0.0779181
$208$	0	0
$209$	−0.331381	−0.0229221
$210$	0	0
$211$	18.1867	1.25203	0.626013	−	0.779813i	$-0.284686\pi$
$211$	18.1867	1.25203	0.626013	+	0.779813i	$0.284686\pi$
$212$	0	0
$213$	−23.9054	−1.63797
$214$	0	0
$215$	27.8622	1.90019
$216$	0	0
$217$	−45.2280	−3.07027
$218$	0	0
$219$	−12.7233	−0.859759
$220$	0	0
$221$	−1.38794	−0.0933627
$222$	0	0
$223$	21.7768	1.45828	0.729141	−	0.684364i	$-0.239920\pi$
$223$	21.7768	1.45828	0.729141	+	0.684364i	$0.239920\pi$
$224$	0	0
$225$	−1.31208	−0.0874717
$226$	0	0
$227$	12.4840	0.828593	0.414296	−	0.910142i	$-0.364028\pi$
$227$	12.4840	0.828593	0.414296	+	0.910142i	$0.364028\pi$
$228$	0	0
$229$	24.3660	1.61015	0.805077	−	0.593170i	$-0.202124\pi$
$229$	24.3660	1.61015	0.805077	+	0.593170i	$0.202124\pi$
$230$	0	0
$231$	2.63084	0.173097
$232$	0	0
$233$	13.1451	0.861164	0.430582	−	0.902552i	$-0.358308\pi$
$233$	13.1451	0.861164	0.430582	+	0.902552i	$0.358308\pi$
$234$	0	0
$235$	−20.6572	−1.34752
$236$	0	0
$237$	−9.82076	−0.637927
$238$	0	0
$239$	15.4433	0.998943	0.499471	−	0.866330i	$-0.333528\pi$
$239$	15.4433	0.998943	0.499471	+	0.866330i	$0.333528\pi$
$240$	0	0
$241$	10.3111	0.664197	0.332098	−	0.943245i	$-0.392243\pi$
$241$	10.3111	0.664197	0.332098	+	0.943245i	$0.392243\pi$
$242$	0	0
$243$	−2.49906	−0.160315
$244$	0	0
$245$	−37.6018	−2.40229
$246$	0	0
$247$	−0.662134	−0.0421306
$248$	0	0
$249$	27.9095	1.76869
$250$	0	0
$251$	−6.32966	−0.399525	−0.199762	−	0.979844i	$-0.564017\pi$
$251$	−6.32966	−0.399525	−0.199762	+	0.979844i	$0.564017\pi$
$252$	0	0
$253$	1.70618	0.107267
$254$	0	0
$255$	10.1620	0.636367
$256$	0	0
$257$	−8.71283	−0.543491	−0.271746	−	0.962369i	$-0.587601\pi$
$257$	−8.71283	−0.543491	−0.271746	+	0.962369i	$0.587601\pi$
$258$	0	0
$259$	−38.4554	−2.38950
$260$	0	0
$261$	−1.68266	−0.104154
$262$	0	0
$263$	−29.9839	−1.84888	−0.924442	−	0.381323i	$-0.875469\pi$
$263$	−29.9839	−1.84888	−0.924442	+	0.381323i	$0.875469\pi$
$264$	0	0
$265$	−24.8943	−1.52924
$266$	0	0
$267$	−13.8239	−0.846011
$268$	0	0
$269$	−17.3932	−1.06048	−0.530240	−	0.847847i	$-0.677898\pi$
$269$	−17.3932	−1.06048	−0.530240	+	0.847847i	$0.677898\pi$
$270$	0	0
$271$	17.4977	1.06291	0.531454	−	0.847087i	$-0.321646\pi$
$271$	17.4977	1.06291	0.531454	+	0.847087i	$0.321646\pi$
$272$	0	0
$273$	5.25671	0.318150
$274$	0	0
$275$	−1.99692	−0.120419
$276$	0	0
$277$	−7.23213	−0.434537	−0.217268	−	0.976112i	$-0.569715\pi$
$277$	−7.23213	−0.434537	−0.217268	+	0.976112i	$0.569715\pi$
$278$	0	0
$279$	2.52568	0.151208
$280$	0	0
$281$	19.8933	1.18674	0.593368	−	0.804931i	$-0.297798\pi$
$281$	19.8933	1.18674	0.593368	+	0.804931i	$0.297798\pi$
$282$	0	0
$283$	−8.57642	−0.509815	−0.254908	−	0.966965i	$-0.582045\pi$
$283$	−8.57642	−0.509815	−0.254908	+	0.966965i	$0.582045\pi$
$284$	0	0
$285$	4.84791	0.287165
$286$	0	0
$287$	25.2753	1.49196
$288$	0	0
$289$	−13.4157	−0.789159
$290$	0	0
$291$	−23.7103	−1.38992
$292$	0	0
$293$	−4.83370	−0.282388	−0.141194	−	0.989982i	$-0.545094\pi$
$293$	−4.83370	−0.282388	−0.141194	+	0.989982i	$0.545094\pi$
$294$	0	0
$295$	−7.33593	−0.427115
$296$	0	0
$297$	−1.97518	−0.114612
$298$	0	0
$299$	3.40913	0.197155
$300$	0	0
$301$	−37.2208	−2.14537
$302$	0	0
$303$	27.3746	1.57263
$304$	0	0
$305$	29.1383	1.66845
$306$	0	0
$307$	−25.9041	−1.47842	−0.739212	−	0.673472i	$-0.764802\pi$
$307$	−25.9041	−1.47842	−0.739212	+	0.673472i	$0.764802\pi$
$308$	0	0
$309$	−13.2090	−0.751436
$310$	0	0
$311$	−3.99010	−0.226258	−0.113129	−	0.993580i	$-0.536087\pi$
$311$	−3.99010	−0.226258	−0.113129	+	0.993580i	$0.536087\pi$
$312$	0	0
$313$	−29.8169	−1.68535	−0.842677	−	0.538420i	$-0.819022\pi$
$313$	−29.8169	−1.68535	−0.842677	+	0.538420i	$0.819022\pi$
$314$	0	0
$315$	3.36302	0.189484
$316$	0	0
$317$	24.2133	1.35996	0.679978	−	0.733232i	$-0.261989\pi$
$317$	24.2133	1.35996	0.679978	+	0.733232i	$0.261989\pi$
$318$	0	0
$319$	−2.56093	−0.143384
$320$	0	0
$321$	−24.4094	−1.36240
$322$	0	0
$323$	1.70994	0.0951435
$324$	0	0
$325$	−3.99006	−0.221328
$326$	0	0
$327$	−7.33821	−0.405804
$328$	0	0
$329$	27.5957	1.52140
$330$	0	0
$331$	−27.3778	−1.50482	−0.752410	−	0.658695i	$-0.771109\pi$
$331$	−27.3778	−1.50482	−0.752410	+	0.658695i	$0.771109\pi$
$332$	0	0
$333$	2.14748	0.117681
$334$	0	0
$335$	−5.11916	−0.279690
$336$	0	0
$337$	−1.91838	−0.104501	−0.0522503	−	0.998634i	$-0.516639\pi$
$337$	−1.91838	−0.104501	−0.0522503	+	0.998634i	$0.516639\pi$
$338$	0	0
$339$	14.2104	0.771803
$340$	0	0
$341$	3.84397	0.208162
$342$	0	0
$343$	20.0133	1.08061
$344$	0	0
$345$	−24.9604	−1.34382
$346$	0	0
$347$	−21.0001	−1.12735	−0.563673	−	0.825998i	$-0.690612\pi$
$347$	−21.0001	−1.12735	−0.563673	+	0.825998i	$0.690612\pi$
$348$	0	0
$349$	14.6346	0.783374	0.391687	−	0.920099i	$-0.371892\pi$
$349$	14.6346	0.783374	0.391687	+	0.920099i	$0.371892\pi$
$350$	0	0
$351$	−3.94663	−0.210656
$352$	0	0
$353$	−12.1225	−0.645215	−0.322608	−	0.946533i	$-0.604559\pi$
$353$	−12.1225	−0.645215	−0.322608	+	0.946533i	$0.604559\pi$
$354$	0	0
$355$	−46.5085	−2.46841
$356$	0	0
$357$	−13.5753	−0.718479
$358$	0	0
$359$	5.91583	0.312226	0.156113	−	0.987739i	$-0.450104\pi$
$359$	5.91583	0.312226	0.156113	+	0.987739i	$0.450104\pi$
$360$	0	0
$361$	−18.1843	−0.957066
$362$	0	0
$363$	18.0474	0.947244
$364$	0	0
$365$	−24.7534	−1.29565
$366$	0	0
$367$	0.0862769	0.00450362	0.00225181	−	0.999997i	$-0.499283\pi$
$367$	0.0862769	0.00450362	0.00225181	+	0.999997i	$0.499283\pi$
$368$	0	0
$369$	−1.41146	−0.0734776
$370$	0	0
$371$	33.2560	1.72657
$372$	0	0
$373$	−12.5790	−0.651314	−0.325657	−	0.945488i	$-0.605586\pi$
$373$	−12.5790	−0.651314	−0.325657	+	0.945488i	$0.605586\pi$
$374$	0	0
$375$	2.37602	0.122697
$376$	0	0
$377$	−5.11700	−0.263539
$378$	0	0
$379$	6.93654	0.356306	0.178153	−	0.984003i	$-0.442988\pi$
$379$	6.93654	0.356306	0.178153	+	0.984003i	$0.442988\pi$
$380$	0	0
$381$	5.24056	0.268482
$382$	0	0
$383$	−12.9763	−0.663058	−0.331529	−	0.943445i	$-0.607564\pi$
$383$	−12.9763	−0.663058	−0.331529	+	0.943445i	$0.607564\pi$
$384$	0	0
$385$	5.11836	0.260856
$386$	0	0
$387$	2.07853	0.105658
$388$	0	0
$389$	−37.6300	−1.90791	−0.953957	−	0.299942i	$-0.903033\pi$
$389$	−37.6300	−1.90791	−0.953957	+	0.299942i	$0.903033\pi$
$390$	0	0
$391$	−8.80397	−0.445236
$392$	0	0
$393$	−4.90553	−0.247451
$394$	0	0
$395$	−19.1065	−0.961351
$396$	0	0
$397$	−2.48449	−0.124693	−0.0623464	−	0.998055i	$-0.519858\pi$
$397$	−2.48449	−0.124693	−0.0623464	+	0.998055i	$0.519858\pi$
$398$	0	0
$399$	−6.47627	−0.324219
$400$	0	0
$401$	−28.4879	−1.42262	−0.711310	−	0.702878i	$-0.751898\pi$
$401$	−28.4879	−1.42262	−0.711310	+	0.702878i	$0.751898\pi$
$402$	0	0
$403$	7.68066	0.382601
$404$	0	0
$405$	26.5587	1.31971
$406$	0	0
$407$	3.26836	0.162007
$408$	0	0
$409$	−21.8429	−1.08006	−0.540031	−	0.841645i	$-0.681588\pi$
$409$	−21.8429	−1.08006	−0.540031	+	0.841645i	$0.681588\pi$
$410$	0	0
$411$	−2.14754	−0.105930
$412$	0	0
$413$	9.80000	0.482226
$414$	0	0
$415$	54.2985	2.66541
$416$	0	0
$417$	−7.40532	−0.362640
$418$	0	0
$419$	−33.3697	−1.63021	−0.815107	−	0.579310i	$-0.803322\pi$
$419$	−33.3697	−1.63021	−0.815107	+	0.579310i	$0.803322\pi$
$420$	0	0
$421$	−23.5965	−1.15002	−0.575011	−	0.818146i	$-0.695002\pi$
$421$	−23.5965	−1.15002	−0.575011	+	0.818146i	$0.695002\pi$
$422$	0	0
$423$	−1.54103	−0.0749276
$424$	0	0
$425$	10.3042	0.499826
$426$	0	0
$427$	−38.9255	−1.88374
$428$	0	0
$429$	−0.446773	−0.0215704
$430$	0	0
$431$	14.4874	0.697832	0.348916	−	0.937154i	$-0.386550\pi$
$431$	14.4874	0.697832	0.348916	+	0.937154i	$0.386550\pi$
$432$	0	0
$433$	−5.72718	−0.275230	−0.137615	−	0.990486i	$-0.543944\pi$
$433$	−5.72718	−0.275230	−0.137615	+	0.990486i	$0.543944\pi$
$434$	0	0
$435$	37.4648	1.79630
$436$	0	0
$437$	−4.20005	−0.200916
$438$	0	0
$439$	−34.7021	−1.65624	−0.828121	−	0.560550i	$-0.810590\pi$
$439$	−34.7021	−1.65624	−0.828121	+	0.560550i	$0.810590\pi$
$440$	0	0
$441$	−2.80511	−0.133577
$442$	0	0
$443$	23.0225	1.09383	0.546917	−	0.837187i	$-0.315801\pi$
$443$	23.0225	1.09383	0.546917	+	0.837187i	$0.315801\pi$
$444$	0	0
$445$	−26.8947	−1.27493
$446$	0	0
$447$	5.01268	0.237092
$448$	0	0
$449$	10.8813	0.513520	0.256760	−	0.966475i	$-0.417345\pi$
$449$	10.8813	0.513520	0.256760	+	0.966475i	$0.417345\pi$
$450$	0	0
$451$	−2.14817	−0.101154
$452$	0	0
$453$	24.2481	1.13928
$454$	0	0
$455$	10.2270	0.479450
$456$	0	0
$457$	−2.14390	−0.100287	−0.0501437	−	0.998742i	$-0.515968\pi$
$457$	−2.14390	−0.100287	−0.0501437	+	0.998742i	$0.515968\pi$
$458$	0	0
$459$	10.1920	0.475724
$460$	0	0
$461$	−31.4114	−1.46297	−0.731486	−	0.681856i	$-0.761173\pi$
$461$	−31.4114	−1.46297	−0.731486	+	0.681856i	$0.761173\pi$
$462$	0	0
$463$	−10.1094	−0.469825	−0.234912	−	0.972017i	$-0.575480\pi$
$463$	−10.1094	−0.469825	−0.234912	+	0.972017i	$0.575480\pi$
$464$	0	0
$465$	−56.2350	−2.60784
$466$	0	0
$467$	−35.7154	−1.65271	−0.826357	−	0.563147i	$-0.809590\pi$
$467$	−35.7154	−1.65271	−0.826357	+	0.563147i	$0.809590\pi$
$468$	0	0
$469$	6.83864	0.315779
$470$	0	0
$471$	40.6249	1.87190
$472$	0	0
$473$	3.16343	0.145455
$474$	0	0
$475$	4.91575	0.225550
$476$	0	0
$477$	−1.85713	−0.0850320
$478$	0	0
$479$	24.5293	1.12077	0.560386	−	0.828232i	$-0.310653\pi$
$479$	24.5293	1.12077	0.560386	+	0.828232i	$0.310653\pi$
$480$	0	0
$481$	6.53053	0.297767
$482$	0	0
$483$	33.3444	1.51722
$484$	0	0
$485$	−46.1289	−2.09460
$486$	0	0
$487$	24.0779	1.09107	0.545536	−	0.838087i	$-0.316326\pi$
$487$	24.0779	1.09107	0.545536	+	0.838087i	$0.316326\pi$
$488$	0	0
$489$	−13.1617	−0.595195
$490$	0	0
$491$	32.2055	1.45341	0.726707	−	0.686947i	$-0.241050\pi$
$491$	32.2055	1.45341	0.726707	+	0.686947i	$0.241050\pi$
$492$	0	0
$493$	13.2145	0.595151
$494$	0	0
$495$	−0.285826	−0.0128469
$496$	0	0
$497$	62.1302	2.78692
$498$	0	0
$499$	−27.1615	−1.21592	−0.607958	−	0.793969i	$-0.708011\pi$
$499$	−27.1615	−1.21592	−0.607958	+	0.793969i	$0.708011\pi$
$500$	0	0
$501$	10.1407	0.453051
$502$	0	0
$503$	−5.66417	−0.252553	−0.126276	−	0.991995i	$-0.540303\pi$
$503$	−5.66417	−0.252553	−0.126276	+	0.991995i	$0.540303\pi$
$504$	0	0
$505$	53.2578	2.36994
$506$	0	0
$507$	20.7003	0.919334
$508$	0	0
$509$	13.7928	0.611355	0.305678	−	0.952135i	$-0.401117\pi$
$509$	13.7928	0.611355	0.305678	+	0.952135i	$0.401117\pi$
$510$	0	0
$511$	33.0678	1.46283
$512$	0	0
$513$	4.86225	0.214674
$514$	0	0
$515$	−25.6984	−1.13241
$516$	0	0
$517$	−2.34538	−0.103150
$518$	0	0
$519$	41.5802	1.82517
$520$	0	0
$521$	12.6852	0.555750	0.277875	−	0.960617i	$-0.410370\pi$
$521$	12.6852	0.555750	0.277875	+	0.960617i	$0.410370\pi$
$522$	0	0
$523$	15.1712	0.663391	0.331695	−	0.943387i	$-0.392379\pi$
$523$	15.1712	0.663391	0.331695	+	0.943387i	$0.392379\pi$
$524$	0	0
$525$	−39.0263	−1.70325
$526$	0	0
$527$	−19.8350	−0.864028
$528$	0	0
$529$	−1.37518	−0.0597903
$530$	0	0
$531$	−0.547264	−0.0237492
$532$	0	0
$533$	−4.29228	−0.185919
$534$	0	0
$535$	−47.4890	−2.05313
$536$	0	0
$537$	23.3074	1.00579
$538$	0	0
$539$	−4.26925	−0.183890
$540$	0	0
$541$	14.1091	0.606597	0.303298	−	0.952896i	$-0.401912\pi$
$541$	14.1091	0.606597	0.303298	+	0.952896i	$0.401912\pi$
$542$	0	0
$543$	33.5084	1.43799
$544$	0	0
$545$	−14.2766	−0.611544
$546$	0	0
$547$	3.01938	0.129099	0.0645496	−	0.997914i	$-0.479439\pi$
$547$	3.01938	0.129099	0.0645496	+	0.997914i	$0.479439\pi$
$548$	0	0
$549$	2.17373	0.0927726
$550$	0	0
$551$	6.30415	0.268566
$552$	0	0
$553$	25.5241	1.08540
$554$	0	0
$555$	−47.8142	−2.02960
$556$	0	0
$557$	−11.4107	−0.483487	−0.241744	−	0.970340i	$-0.577719\pi$
$557$	−11.4107	−0.483487	−0.241744	+	0.970340i	$0.577719\pi$
$558$	0	0
$559$	6.32088	0.267345
$560$	0	0
$561$	1.15377	0.0487124
$562$	0	0
$563$	−38.6377	−1.62838	−0.814192	−	0.580596i	$-0.802820\pi$
$563$	−38.6377	−1.62838	−0.814192	+	0.580596i	$0.802820\pi$
$564$	0	0
$565$	27.6466	1.16310
$566$	0	0
$567$	−35.4795	−1.49000
$568$	0	0
$569$	−1.89230	−0.0793293	−0.0396647	−	0.999213i	$-0.512629\pi$
$569$	−1.89230	−0.0793293	−0.0396647	+	0.999213i	$0.512629\pi$
$570$	0	0
$571$	−16.7255	−0.699942	−0.349971	−	0.936761i	$-0.613809\pi$
$571$	−16.7255	−0.699942	−0.349971	+	0.936761i	$0.613809\pi$
$572$	0	0
$573$	25.1885	1.05226
$574$	0	0
$575$	−25.3097	−1.05549
$576$	0	0
$577$	−6.74516	−0.280805	−0.140402	−	0.990095i	$-0.544840\pi$
$577$	−6.74516	−0.280805	−0.140402	+	0.990095i	$0.544840\pi$
$578$	0	0
$579$	23.9497	0.995315
$580$	0	0
$581$	−72.5368	−3.00933
$582$	0	0
$583$	−2.82646	−0.117060
$584$	0	0
$585$	−0.571111	−0.0236125
$586$	0	0
$587$	15.0455	0.620996	0.310498	−	0.950574i	$-0.399504\pi$
$587$	15.0455	0.620996	0.310498	+	0.950574i	$0.399504\pi$
$588$	0	0
$589$	−9.46257	−0.389899
$590$	0	0
$591$	−32.7787	−1.34833
$592$	0	0
$593$	−24.1405	−0.991331	−0.495665	−	0.868514i	$-0.665076\pi$
$593$	−24.1405	−0.991331	−0.495665	+	0.868514i	$0.665076\pi$
$594$	0	0
$595$	−26.4109	−1.08274
$596$	0	0
$597$	24.1014	0.986405
$598$	0	0
$599$	23.2959	0.951844	0.475922	−	0.879487i	$-0.342114\pi$
$599$	23.2959	0.951844	0.475922	+	0.879487i	$0.342114\pi$
$600$	0	0
$601$	−43.3135	−1.76680	−0.883398	−	0.468624i	$-0.844750\pi$
$601$	−43.3135	−1.76680	−0.883398	+	0.468624i	$0.844750\pi$
$602$	0	0
$603$	−0.381892	−0.0155518
$604$	0	0
$605$	35.1116	1.42749
$606$	0	0
$607$	−8.27451	−0.335852	−0.167926	−	0.985800i	$-0.553707\pi$
$607$	−8.27451	−0.335852	−0.167926	+	0.985800i	$0.553707\pi$
$608$	0	0
$609$	−50.0489	−2.02808
$610$	0	0
$611$	−4.68632	−0.189588
$612$	0	0
$613$	−10.6889	−0.431722	−0.215861	−	0.976424i	$-0.569256\pi$
$613$	−10.6889	−0.431722	−0.215861	+	0.976424i	$0.569256\pi$
$614$	0	0
$615$	31.4265	1.26724
$616$	0	0
$617$	8.90394	0.358459	0.179230	−	0.983807i	$-0.442640\pi$
$617$	8.90394	0.358459	0.179230	+	0.983807i	$0.442640\pi$
$618$	0	0
$619$	−13.4370	−0.540078	−0.270039	−	0.962849i	$-0.587037\pi$
$619$	−13.4370	−0.540078	−0.270039	+	0.962849i	$0.587037\pi$
$620$	0	0
$621$	−25.0343	−1.00459
$622$	0	0
$623$	35.9284	1.43944
$624$	0	0
$625$	−22.5907	−0.903629
$626$	0	0
$627$	0.550424	0.0219818
$628$	0	0
$629$	−16.8649	−0.672447
$630$	0	0
$631$	−41.0373	−1.63367	−0.816834	−	0.576873i	$-0.804273\pi$
$631$	−41.0373	−1.63367	−0.816834	+	0.576873i	$0.804273\pi$
$632$	0	0
$633$	−30.2082	−1.20067
$634$	0	0
$635$	10.1956	0.404601
$636$	0	0
$637$	−8.53042	−0.337988
$638$	0	0
$639$	−3.46955	−0.137253
$640$	0	0
$641$	−3.55229	−0.140307	−0.0701534	−	0.997536i	$-0.522349\pi$
$641$	−3.55229	−0.140307	−0.0701534	+	0.997536i	$0.522349\pi$
$642$	0	0
$643$	7.22054	0.284750	0.142375	−	0.989813i	$-0.454526\pi$
$643$	7.22054	0.284750	0.142375	+	0.989813i	$0.454526\pi$
$644$	0	0
$645$	−46.2792	−1.82224
$646$	0	0
$647$	28.0927	1.10444	0.552219	−	0.833699i	$-0.313781\pi$
$647$	28.0927	1.10444	0.552219	+	0.833699i	$0.313781\pi$
$648$	0	0
$649$	−0.832911	−0.0326946
$650$	0	0
$651$	75.1237	2.94433
$652$	0	0
$653$	−17.0580	−0.667530	−0.333765	−	0.942656i	$-0.608319\pi$
$653$	−17.0580	−0.667530	−0.333765	+	0.942656i	$0.608319\pi$
$654$	0	0
$655$	−9.54381	−0.372908
$656$	0	0
$657$	−1.84661	−0.0720432
$658$	0	0
$659$	48.1405	1.87529	0.937643	−	0.347600i	$-0.113003\pi$
$659$	48.1405	1.87529	0.937643	+	0.347600i	$0.113003\pi$
$660$	0	0
$661$	−21.1547	−0.822823	−0.411411	−	0.911450i	$-0.634964\pi$
$661$	−21.1547	−0.822823	−0.411411	+	0.911450i	$0.634964\pi$
$662$	0	0
$663$	2.30537	0.0895330
$664$	0	0
$665$	−12.5997	−0.488595
$666$	0	0
$667$	−32.4582	−1.25679
$668$	0	0
$669$	−36.1713	−1.39846
$670$	0	0
$671$	3.30832	0.127716
$672$	0	0
$673$	28.8738	1.11301	0.556503	−	0.830846i	$-0.312143\pi$
$673$	28.8738	1.11301	0.556503	+	0.830846i	$0.312143\pi$
$674$	0	0
$675$	29.3002	1.12776
$676$	0	0
$677$	−16.0659	−0.617462	−0.308731	−	0.951149i	$-0.599904\pi$
$677$	−16.0659	−0.617462	−0.308731	+	0.951149i	$0.599904\pi$
$678$	0	0
$679$	61.6231	2.36488
$680$	0	0
$681$	−20.7360	−0.794604
$682$	0	0
$683$	22.3159	0.853893	0.426946	−	0.904277i	$-0.359589\pi$
$683$	22.3159	0.853893	0.426946	+	0.904277i	$0.359589\pi$
$684$	0	0
$685$	−4.17807	−0.159636
$686$	0	0
$687$	−40.4721	−1.54411
$688$	0	0
$689$	−5.64757	−0.215155
$690$	0	0
$691$	34.4159	1.30924	0.654621	−	0.755958i	$-0.272828\pi$
$691$	34.4159	1.30924	0.654621	+	0.755958i	$0.272828\pi$
$692$	0	0
$693$	0.381832	0.0145046
$694$	0	0
$695$	−14.4072	−0.546496
$696$	0	0
$697$	11.0847	0.419862
$698$	0	0
$699$	−21.8340	−0.825839
$700$	0	0
$701$	19.3120	0.729403	0.364701	−	0.931125i	$-0.381171\pi$
$701$	19.3120	0.729403	0.364701	+	0.931125i	$0.381171\pi$
$702$	0	0
$703$	−8.04562	−0.303446
$704$	0	0
$705$	34.3116	1.29225
$706$	0	0
$707$	−71.1465	−2.67574
$708$	0	0
$709$	−47.1982	−1.77257	−0.886283	−	0.463145i	$-0.846721\pi$
$709$	−47.1982	−1.77257	−0.886283	+	0.463145i	$0.846721\pi$
$710$	0	0
$711$	−1.42535	−0.0534549
$712$	0	0
$713$	48.7200	1.82458
$714$	0	0
$715$	−0.869205	−0.0325064
$716$	0	0
$717$	−25.6513	−0.957966
$718$	0	0
$719$	44.4900	1.65920	0.829598	−	0.558361i	$-0.188569\pi$
$719$	44.4900	1.65920	0.829598	+	0.558361i	$0.188569\pi$
$720$	0	0
$721$	34.3303	1.27853
$722$	0	0
$723$	−17.1268	−0.636951
$724$	0	0
$725$	37.9892	1.41088
$726$	0	0
$727$	16.5953	0.615486	0.307743	−	0.951470i	$-0.400426\pi$
$727$	16.5953	0.615486	0.307743	+	0.951470i	$0.400426\pi$
$728$	0	0
$729$	28.8069	1.06692
$730$	0	0
$731$	−16.3235	−0.603745
$732$	0	0
$733$	33.9335	1.25336	0.626681	−	0.779276i	$-0.284413\pi$
$733$	33.9335	1.25336	0.626681	+	0.779276i	$0.284413\pi$
$734$	0	0
$735$	62.4567	2.30375
$736$	0	0
$737$	−0.581222	−0.0214096
$738$	0	0
$739$	−6.83431	−0.251404	−0.125702	−	0.992068i	$-0.540118\pi$
$739$	−6.83431	−0.251404	−0.125702	+	0.992068i	$0.540118\pi$
$740$	0	0
$741$	1.09981	0.0404024
$742$	0	0
$743$	11.6149	0.426108	0.213054	−	0.977040i	$-0.431659\pi$
$743$	11.6149	0.426108	0.213054	+	0.977040i	$0.431659\pi$
$744$	0	0
$745$	9.75226	0.357295
$746$	0	0
$747$	4.05069	0.148207
$748$	0	0
$749$	63.4400	2.31805
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	10.5136	0.383136
$754$	0	0
$755$	47.1752	1.71688
$756$	0	0
$757$	38.3127	1.39250	0.696250	−	0.717800i	$-0.254851\pi$
$757$	38.3127	1.39250	0.696250	+	0.717800i	$0.254851\pi$
$758$	0	0
$759$	−2.83397	−0.102867
$760$	0	0
$761$	29.0411	1.05274	0.526370	−	0.850256i	$-0.323553\pi$
$761$	29.0411	1.05274	0.526370	+	0.850256i	$0.323553\pi$
$762$	0	0
$763$	19.0720	0.690453
$764$	0	0
$765$	1.47487	0.0533242
$766$	0	0
$767$	−1.66425	−0.0600924
$768$	0	0
$769$	−14.5786	−0.525719	−0.262859	−	0.964834i	$-0.584665\pi$
$769$	−14.5786	−0.525719	−0.262859	+	0.964834i	$0.584665\pi$
$770$	0	0
$771$	14.4720	0.521197
$772$	0	0
$773$	26.2165	0.942941	0.471470	−	0.881882i	$-0.343723\pi$
$773$	26.2165	0.942941	0.471470	+	0.881882i	$0.343723\pi$
$774$	0	0
$775$	−57.0220	−2.04829
$776$	0	0
$777$	63.8745	2.29148
$778$	0	0
$779$	5.28809	0.189466
$780$	0	0
$781$	−5.28050	−0.188951
$782$	0	0
$783$	37.5757	1.34285
$784$	0	0
$785$	79.0366	2.82094
$786$	0	0
$787$	−29.6767	−1.05786	−0.528931	−	0.848665i	$-0.677407\pi$
$787$	−29.6767	−1.05786	−0.528931	+	0.848665i	$0.677407\pi$
$788$	0	0
$789$	49.8032	1.77304
$790$	0	0
$791$	−36.9328	−1.31318
$792$	0	0
$793$	6.61037	0.234741
$794$	0	0
$795$	41.3495	1.46651
$796$	0	0
$797$	7.26734	0.257422	0.128711	−	0.991682i	$-0.458916\pi$
$797$	7.26734	0.257422	0.128711	+	0.991682i	$0.458916\pi$
$798$	0	0
$799$	12.1023	0.428148
$800$	0	0
$801$	−2.00636	−0.0708912
$802$	0	0
$803$	−2.81046	−0.0991790
$804$	0	0
$805$	64.8721	2.28644
$806$	0	0
$807$	28.8901	1.01698
$808$	0	0
$809$	−37.9697	−1.33494	−0.667472	−	0.744635i	$-0.732624\pi$
$809$	−37.9697	−1.33494	−0.667472	+	0.744635i	$0.732624\pi$
$810$	0	0
$811$	2.60733	0.0915558	0.0457779	−	0.998952i	$-0.485423\pi$
$811$	2.60733	0.0915558	0.0457779	+	0.998952i	$0.485423\pi$
$812$	0	0
$813$	−29.0637	−1.01931
$814$	0	0
$815$	−25.6064	−0.896954
$816$	0	0
$817$	−7.78732	−0.272444
$818$	0	0
$819$	0.762941	0.0266593
$820$	0	0
$821$	−7.68740	−0.268292	−0.134146	−	0.990962i	$-0.542829\pi$
$821$	−7.68740	−0.268292	−0.134146	+	0.990962i	$0.542829\pi$
$822$	0	0
$823$	−38.9510	−1.35775	−0.678873	−	0.734256i	$-0.737531\pi$
$823$	−38.9510	−1.35775	−0.678873	+	0.734256i	$0.737531\pi$
$824$	0	0
$825$	3.31689	0.115479
$826$	0	0
$827$	49.4667	1.72013	0.860063	−	0.510188i	$-0.170424\pi$
$827$	49.4667	1.72013	0.860063	+	0.510188i	$0.170424\pi$
$828$	0	0
$829$	28.2433	0.980930	0.490465	−	0.871461i	$-0.336827\pi$
$829$	28.2433	0.980930	0.490465	+	0.871461i	$0.336827\pi$
$830$	0	0
$831$	12.0126	0.416712
$832$	0	0
$833$	22.0295	0.763278
$834$	0	0
$835$	19.7288	0.682745
$836$	0	0
$837$	−56.4014	−1.94952
$838$	0	0
$839$	−54.4674	−1.88042	−0.940211	−	0.340592i	$-0.889373\pi$
$839$	−54.4674	−1.88042	−0.940211	+	0.340592i	$0.889373\pi$
$840$	0	0
$841$	19.7188	0.679958
$842$	0	0
$843$	−33.0429	−1.13806
$844$	0	0
$845$	40.2729	1.38543
$846$	0	0
$847$	−46.9052	−1.61168
$848$	0	0
$849$	14.2455	0.488903
$850$	0	0
$851$	41.4245	1.42001
$852$	0	0
$853$	31.9379	1.09353	0.546767	−	0.837285i	$-0.315858\pi$
$853$	31.9379	1.09353	0.546767	+	0.837285i	$0.315858\pi$
$854$	0	0
$855$	0.703609	0.0240629
$856$	0	0
$857$	−19.7813	−0.675715	−0.337857	−	0.941197i	$-0.609702\pi$
$857$	−19.7813	−0.675715	−0.337857	+	0.941197i	$0.609702\pi$
$858$	0	0
$859$	19.6258	0.669624	0.334812	−	0.942285i	$-0.391327\pi$
$859$	19.6258	0.669624	0.334812	+	0.942285i	$0.391327\pi$
$860$	0	0
$861$	−41.9824	−1.43076
$862$	0	0
$863$	24.3669	0.829458	0.414729	−	0.909945i	$-0.363876\pi$
$863$	24.3669	0.829458	0.414729	+	0.909945i	$0.363876\pi$
$864$	0	0
$865$	80.8951	2.75052
$866$	0	0
$867$	22.2835	0.756788
$868$	0	0
$869$	−2.16932	−0.0735891
$870$	0	0
$871$	−1.16134	−0.0393506
$872$	0	0
$873$	−3.44124	−0.116468
$874$	0	0
$875$	−6.17527	−0.208762
$876$	0	0
$877$	−10.4726	−0.353635	−0.176818	−	0.984244i	$-0.556580\pi$
$877$	−10.4726	−0.353635	−0.176818	+	0.984244i	$0.556580\pi$
$878$	0	0
$879$	8.02878	0.270804
$880$	0	0
$881$	−2.55601	−0.0861143	−0.0430571	−	0.999073i	$-0.513710\pi$
$881$	−2.55601	−0.0861143	−0.0430571	+	0.999073i	$0.513710\pi$
$882$	0	0
$883$	25.9102	0.871948	0.435974	−	0.899959i	$-0.356404\pi$
$883$	25.9102	0.871948	0.435974	+	0.899959i	$0.356404\pi$
$884$	0	0
$885$	12.1850	0.409594
$886$	0	0
$887$	21.2438	0.713295	0.356648	−	0.934239i	$-0.383920\pi$
$887$	21.2438	0.713295	0.356648	+	0.934239i	$0.383920\pi$
$888$	0	0
$889$	−13.6202	−0.456807
$890$	0	0
$891$	3.01544	0.101021
$892$	0	0
$893$	5.77355	0.193205
$894$	0	0
$895$	45.3450	1.51572
$896$	0	0
$897$	−5.66258	−0.189068
$898$	0	0
$899$	−73.1272	−2.43893
$900$	0	0
$901$	14.5847	0.485886
$902$	0	0
$903$	61.8239	2.05737
$904$	0	0
$905$	65.1913	2.16703
$906$	0	0
$907$	9.29387	0.308598	0.154299	−	0.988024i	$-0.450688\pi$
$907$	9.29387	0.308598	0.154299	+	0.988024i	$0.450688\pi$
$908$	0	0
$909$	3.97306	0.131778
$910$	0	0
$911$	41.1510	1.36339	0.681696	−	0.731635i	$-0.261243\pi$
$911$	41.1510	1.36339	0.681696	+	0.731635i	$0.261243\pi$
$912$	0	0
$913$	6.16497	0.204031
$914$	0	0
$915$	−48.3988	−1.60001
$916$	0	0
$917$	12.7495	0.421025
$918$	0	0
$919$	3.32372	0.109639	0.0548196	−	0.998496i	$-0.482542\pi$
$919$	3.32372	0.109639	0.0548196	+	0.998496i	$0.482542\pi$
$920$	0	0
$921$	43.0268	1.41778
$922$	0	0
$923$	−10.5510	−0.347291
$924$	0	0
$925$	−48.4834	−1.59412
$926$	0	0
$927$	−1.91712	−0.0629664
$928$	0	0
$929$	−49.7671	−1.63280	−0.816402	−	0.577484i	$-0.804035\pi$
$929$	−49.7671	−1.63280	−0.816402	+	0.577484i	$0.804035\pi$
$930$	0	0
$931$	10.5095	0.344434
$932$	0	0
$933$	6.62757	0.216977
$934$	0	0
$935$	2.24469	0.0734093
$936$	0	0
$937$	−5.53677	−0.180878	−0.0904392	−	0.995902i	$-0.528827\pi$
$937$	−5.53677	−0.180878	−0.0904392	+	0.995902i	$0.528827\pi$
$938$	0	0
$939$	49.5260	1.61622
$940$	0	0
$941$	−26.4791	−0.863194	−0.431597	−	0.902066i	$-0.642050\pi$
$941$	−26.4791	−0.863194	−0.431597	+	0.902066i	$0.642050\pi$
$942$	0	0
$943$	−27.2268	−0.886627
$944$	0	0
$945$	−75.1001	−2.44301
$946$	0	0
$947$	−55.3772	−1.79952	−0.899758	−	0.436389i	$-0.856257\pi$
$947$	−55.3772	−1.79952	−0.899758	+	0.436389i	$0.856257\pi$
$948$	0	0
$949$	−5.61560	−0.182290
$950$	0	0
$951$	−40.2184	−1.30417
$952$	0	0
$953$	41.2105	1.33494	0.667469	−	0.744637i	$-0.267378\pi$
$953$	41.2105	1.33494	0.667469	+	0.744637i	$0.267378\pi$
$954$	0	0
$955$	49.0047	1.58575
$956$	0	0
$957$	4.25370	0.137503
$958$	0	0
$959$	5.58144	0.180234
$960$	0	0
$961$	78.7645	2.54079
$962$	0	0
$963$	−3.54270	−0.114162
$964$	0	0
$965$	46.5946	1.49993
$966$	0	0
$967$	−34.1760	−1.09903	−0.549513	−	0.835485i	$-0.685187\pi$
$967$	−34.1760	−1.09903	−0.549513	+	0.835485i	$0.685187\pi$
$968$	0	0
$969$	−2.84021	−0.0912407
$970$	0	0
$971$	22.9449	0.736336	0.368168	−	0.929759i	$-0.379985\pi$
$971$	22.9449	0.736336	0.368168	+	0.929759i	$0.379985\pi$
$972$	0	0
$973$	19.2464	0.617012
$974$	0	0
$975$	6.62749	0.212250
$976$	0	0
$977$	31.1995	0.998160	0.499080	−	0.866556i	$-0.333671\pi$
$977$	31.1995	0.998160	0.499080	+	0.866556i	$0.333671\pi$
$978$	0	0
$979$	−3.05359	−0.0975931
$980$	0	0
$981$	−1.06504	−0.0340042
$982$	0	0
$983$	−24.4137	−0.778677	−0.389339	−	0.921095i	$-0.627296\pi$
$983$	−24.4137	−0.778677	−0.389339	+	0.921095i	$0.627296\pi$
$984$	0	0
$985$	−63.7715	−2.03193
$986$	0	0
$987$	−45.8365	−1.45899
$988$	0	0
$989$	40.0946	1.27494
$990$	0	0
$991$	−20.5381	−0.652415	−0.326207	−	0.945298i	$-0.605771\pi$
$991$	−20.5381	−0.652415	−0.326207	+	0.945298i	$0.605771\pi$
$992$	0	0
$993$	45.4746	1.44309
$994$	0	0
$995$	46.8897	1.48650
$996$	0	0
$997$	−18.9680	−0.600723	−0.300361	−	0.953825i	$-0.597107\pi$
$997$	−18.9680	−0.600723	−0.300361	+	0.953825i	$0.597107\pi$
$998$	0	0
$999$	−47.9557	−1.51725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.c.1.12	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.c.1.12	✓	44	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)

\(f(q)\)	\(=\)	\(q-1.66100 q^{3} -3.23151 q^{5} +4.31694 q^{7} -0.241072 q^{9} +O(q^{10})\)	\(q-1.66100 q^{3} -3.23151 q^{5} +4.31694 q^{7} -0.241072 q^{9} -0.366901 q^{11} -0.733107 q^{13} +5.36755 q^{15} +1.89322 q^{17} +0.903188 q^{19} -7.17045 q^{21} -4.65025 q^{23} +5.44266 q^{25} +5.38343 q^{27} +6.97988 q^{29} -10.4769 q^{31} +0.609423 q^{33} -13.9502 q^{35} -8.90802 q^{37} +1.21769 q^{39} +5.85492 q^{41} -8.62204 q^{43} +0.779028 q^{45} +6.39241 q^{47} +11.6360 q^{49} -3.14465 q^{51} +7.70361 q^{53} +1.18564 q^{55} -1.50020 q^{57} +2.27013 q^{59} -9.01693 q^{61} -1.04069 q^{63} +2.36904 q^{65} +1.58414 q^{67} +7.72407 q^{69} +14.3922 q^{71} +7.66000 q^{73} -9.04028 q^{75} -1.58389 q^{77} +5.91255 q^{79} -8.21867 q^{81} -16.8028 q^{83} -6.11798 q^{85} -11.5936 q^{87} +8.32265 q^{89} -3.16478 q^{91} +17.4021 q^{93} -2.91866 q^{95} +14.2747 q^{97} +0.0884496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more