LMFDB - Embedded newform 6008.2.a.b.1.13

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			1.13
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)

$f(q)$	$=$	$q-2.00709 q^{3} +0.0492234 q^{5} +3.84191 q^{7} +1.02840 q^{9} +O(q^{10})$	$q-2.00709 q^{3} +0.0492234 q^{5} +3.84191 q^{7} +1.02840 q^{9} -1.21458 q^{11} +1.27285 q^{13} -0.0987957 q^{15} -0.819483 q^{17} -1.24303 q^{19} -7.71105 q^{21} +1.37611 q^{23} -4.99758 q^{25} +3.95718 q^{27} +0.881623 q^{29} -1.56310 q^{31} +2.43776 q^{33} +0.189112 q^{35} +5.05616 q^{37} -2.55473 q^{39} -5.34858 q^{41} -9.30023 q^{43} +0.0506213 q^{45} -3.16855 q^{47} +7.76027 q^{49} +1.64477 q^{51} +3.52915 q^{53} -0.0597856 q^{55} +2.49486 q^{57} +9.34369 q^{59} -5.45413 q^{61} +3.95101 q^{63} +0.0626542 q^{65} -5.39897 q^{67} -2.76196 q^{69} -15.1439 q^{71} -14.3312 q^{73} +10.0306 q^{75} -4.66629 q^{77} -13.4366 q^{79} -11.0276 q^{81} -7.25318 q^{83} -0.0403378 q^{85} -1.76949 q^{87} +1.79541 q^{89} +4.89018 q^{91} +3.13727 q^{93} -0.0611860 q^{95} +15.1851 q^{97} -1.24907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	−2.00709	−1.15879	−0.579396	−	0.815046i	$-0.696712\pi$
$3$	−2.00709	−1.15879	−0.579396	+	0.815046i	$0.696712\pi$
$4$	0	0
$5$	0.0492234	0.0220134	0.0110067	−	0.999939i	$-0.496496\pi$
$5$	0.0492234	0.0220134	0.0110067	+	0.999939i	$0.496496\pi$
$6$	0	0
$7$	3.84191	1.45211	0.726053	−	0.687639i	$-0.241353\pi$
$7$	3.84191	1.45211	0.726053	+	0.687639i	$0.241353\pi$
$8$	0	0
$9$	1.02840	0.342799
$10$	0	0
$11$	−1.21458	−0.366208	−0.183104	−	0.983094i	$-0.558615\pi$
$11$	−1.21458	−0.366208	−0.183104	+	0.983094i	$0.558615\pi$
$12$	0	0
$13$	1.27285	0.353026	0.176513	−	0.984298i	$-0.443518\pi$
$13$	1.27285	0.353026	0.176513	+	0.984298i	$0.443518\pi$
$14$	0	0
$15$	−0.0987957	−0.0255089
$16$	0	0
$17$	−0.819483	−0.198754	−0.0993770	−	0.995050i	$-0.531685\pi$
$17$	−0.819483	−0.198754	−0.0993770	+	0.995050i	$0.531685\pi$
$18$	0	0
$19$	−1.24303	−0.285170	−0.142585	−	0.989783i	$-0.545541\pi$
$19$	−1.24303	−0.285170	−0.142585	+	0.989783i	$0.545541\pi$
$20$	0	0
$21$	−7.71105	−1.68269
$22$	0	0
$23$	1.37611	0.286938	0.143469	−	0.989655i	$-0.454174\pi$
$23$	1.37611	0.286938	0.143469	+	0.989655i	$0.454174\pi$
$24$	0	0
$25$	−4.99758	−0.999515
$26$	0	0
$27$	3.95718	0.761559
$28$	0	0
$29$	0.881623	0.163713	0.0818566	−	0.996644i	$-0.473915\pi$
$29$	0.881623	0.163713	0.0818566	+	0.996644i	$0.473915\pi$
$30$	0	0
$31$	−1.56310	−0.280741	−0.140370	−	0.990099i	$-0.544829\pi$
$31$	−1.56310	−0.280741	−0.140370	+	0.990099i	$0.544829\pi$
$32$	0	0
$33$	2.43776	0.424359
$34$	0	0
$35$	0.189112	0.0319658
$36$	0	0
$37$	5.05616	0.831228	0.415614	−	0.909541i	$-0.363567\pi$
$37$	5.05616	0.831228	0.415614	+	0.909541i	$0.363567\pi$
$38$	0	0
$39$	−2.55473	−0.409083
$40$	0	0
$41$	−5.34858	−0.835307	−0.417654	−	0.908606i	$-0.637147\pi$
$41$	−5.34858	−0.835307	−0.417654	+	0.908606i	$0.637147\pi$
$42$	0	0
$43$	−9.30023	−1.41827	−0.709136	−	0.705072i	$-0.750915\pi$
$43$	−9.30023	−1.41827	−0.709136	+	0.705072i	$0.750915\pi$
$44$	0	0
$45$	0.0506213	0.00754618
$46$	0	0
$47$	−3.16855	−0.462181	−0.231090	−	0.972932i	$-0.574229\pi$
$47$	−3.16855	−0.462181	−0.231090	+	0.972932i	$0.574229\pi$
$48$	0	0
$49$	7.76027	1.10861
$50$	0	0
$51$	1.64477	0.230314
$52$	0	0
$53$	3.52915	0.484767	0.242383	−	0.970181i	$-0.422071\pi$
$53$	3.52915	0.484767	0.242383	+	0.970181i	$0.422071\pi$
$54$	0	0
$55$	−0.0597856	−0.00806149
$56$	0	0
$57$	2.49486	0.330452
$58$	0	0
$59$	9.34369	1.21644	0.608222	−	0.793767i	$-0.291883\pi$
$59$	9.34369	1.21644	0.608222	+	0.793767i	$0.291883\pi$
$60$	0	0
$61$	−5.45413	−0.698329	−0.349165	−	0.937061i	$-0.613535\pi$
$61$	−5.45413	−0.698329	−0.349165	+	0.937061i	$0.613535\pi$
$62$	0	0
$63$	3.95101	0.497781
$64$	0	0
$65$	0.0626542	0.00777129
$66$	0	0
$67$	−5.39897	−0.659590	−0.329795	−	0.944053i	$-0.606980\pi$
$67$	−5.39897	−0.659590	−0.329795	+	0.944053i	$0.606980\pi$
$68$	0	0
$69$	−2.76196	−0.332501
$70$	0	0
$71$	−15.1439	−1.79725	−0.898625	−	0.438718i	$-0.855433\pi$
$71$	−15.1439	−1.79725	−0.898625	+	0.438718i	$0.855433\pi$
$72$	0	0
$73$	−14.3312	−1.67733	−0.838667	−	0.544644i	$-0.816665\pi$
$73$	−14.3312	−1.67733	−0.838667	+	0.544644i	$0.816665\pi$
$74$	0	0
$75$	10.0306	1.15823
$76$	0	0
$77$	−4.66629	−0.531773
$78$	0	0
$79$	−13.4366	−1.51174	−0.755868	−	0.654725i	$-0.772785\pi$
$79$	−13.4366	−1.51174	−0.755868	+	0.654725i	$0.772785\pi$
$80$	0	0
$81$	−11.0276	−1.22529
$82$	0	0
$83$	−7.25318	−0.796140	−0.398070	−	0.917355i	$-0.630320\pi$
$83$	−7.25318	−0.796140	−0.398070	+	0.917355i	$0.630320\pi$
$84$	0	0
$85$	−0.0403378	−0.00437525
$86$	0	0
$87$	−1.76949	−0.189710
$88$	0	0
$89$	1.79541	0.190313	0.0951565	−	0.995462i	$-0.469665\pi$
$89$	1.79541	0.190313	0.0951565	+	0.995462i	$0.469665\pi$
$90$	0	0
$91$	4.89018	0.512630
$92$	0	0
$93$	3.13727	0.325320
$94$	0	0
$95$	−0.0611860	−0.00627755
$96$	0	0
$97$	15.1851	1.54181	0.770907	−	0.636948i	$-0.219803\pi$
$97$	15.1851	1.54181	0.770907	+	0.636948i	$0.219803\pi$
$98$	0	0
$99$	−1.24907	−0.125536
$100$	0	0
$101$	14.9487	1.48746	0.743728	−	0.668483i	$-0.233056\pi$
$101$	14.9487	1.48746	0.743728	+	0.668483i	$0.233056\pi$
$102$	0	0
$103$	17.2962	1.70424	0.852122	−	0.523343i	$-0.175315\pi$
$103$	17.2962	1.70424	0.852122	+	0.523343i	$0.175315\pi$
$104$	0	0
$105$	−0.379564	−0.0370417
$106$	0	0
$107$	3.86798	0.373932	0.186966	−	0.982366i	$-0.440135\pi$
$107$	3.86798	0.373932	0.186966	+	0.982366i	$0.440135\pi$
$108$	0	0
$109$	9.28036	0.888897	0.444449	−	0.895804i	$-0.353400\pi$
$109$	9.28036	0.888897	0.444449	+	0.895804i	$0.353400\pi$
$110$	0	0
$111$	−10.1482	−0.963221
$112$	0	0
$113$	10.0199	0.942594	0.471297	−	0.881975i	$-0.343786\pi$
$113$	10.0199	0.942594	0.471297	+	0.881975i	$0.343786\pi$
$114$	0	0
$115$	0.0677367	0.00631648
$116$	0	0
$117$	1.30900	0.121017
$118$	0	0
$119$	−3.14838	−0.288612
$120$	0	0
$121$	−9.52481	−0.865891
$122$	0	0
$123$	10.7351	0.967947
$124$	0	0
$125$	−0.492115	−0.0440161
$126$	0	0
$127$	−15.0908	−1.33909	−0.669546	−	0.742770i	$-0.733511\pi$
$127$	−15.0908	−1.33909	−0.669546	+	0.742770i	$0.733511\pi$
$128$	0	0
$129$	18.6664	1.64348
$130$	0	0
$131$	−1.30772	−0.114256	−0.0571281	−	0.998367i	$-0.518194\pi$
$131$	−1.30772	−0.114256	−0.0571281	+	0.998367i	$0.518194\pi$
$132$	0	0
$133$	−4.77559	−0.414096
$134$	0	0
$135$	0.194786	0.0167645
$136$	0	0
$137$	21.3694	1.82571	0.912855	−	0.408285i	$-0.133873\pi$
$137$	21.3694	1.82571	0.912855	+	0.408285i	$0.133873\pi$
$138$	0	0
$139$	−6.42998	−0.545383	−0.272692	−	0.962101i	$-0.587914\pi$
$139$	−6.42998	−0.545383	−0.272692	+	0.962101i	$0.587914\pi$
$140$	0	0
$141$	6.35956	0.535571
$142$	0	0
$143$	−1.54598	−0.129281
$144$	0	0
$145$	0.0433965	0.00360388
$146$	0	0
$147$	−15.5755	−1.28465
$148$	0	0
$149$	12.3057	1.00812	0.504062	−	0.863668i	$-0.331838\pi$
$149$	12.3057	1.00812	0.504062	+	0.863668i	$0.331838\pi$
$150$	0	0
$151$	−2.95247	−0.240269	−0.120134	−	0.992758i	$-0.538333\pi$
$151$	−2.95247	−0.240269	−0.120134	+	0.992758i	$0.538333\pi$
$152$	0	0
$153$	−0.842755	−0.0681327
$154$	0	0
$155$	−0.0769411	−0.00618005
$156$	0	0
$157$	10.5631	0.843025	0.421512	−	0.906823i	$-0.361499\pi$
$157$	10.5631	0.843025	0.421512	+	0.906823i	$0.361499\pi$
$158$	0	0
$159$	−7.08332	−0.561744
$160$	0	0
$161$	5.28687	0.416664
$162$	0	0
$163$	19.7136	1.54409	0.772044	−	0.635570i	$-0.219235\pi$
$163$	19.7136	1.54409	0.772044	+	0.635570i	$0.219235\pi$
$164$	0	0
$165$	0.119995	0.00934159
$166$	0	0
$167$	−16.9375	−1.31066	−0.655331	−	0.755342i	$-0.727471\pi$
$167$	−16.9375	−1.31066	−0.655331	+	0.755342i	$0.727471\pi$
$168$	0	0
$169$	−11.3798	−0.875373
$170$	0	0
$171$	−1.27833	−0.0977560
$172$	0	0
$173$	−23.1723	−1.76175	−0.880877	−	0.473346i	$-0.843046\pi$
$173$	−23.1723	−1.76175	−0.880877	+	0.473346i	$0.843046\pi$
$174$	0	0
$175$	−19.2002	−1.45140
$176$	0	0
$177$	−18.7536	−1.40961
$178$	0	0
$179$	1.05850	0.0791160	0.0395580	−	0.999217i	$-0.487405\pi$
$179$	1.05850	0.0791160	0.0395580	+	0.999217i	$0.487405\pi$
$180$	0	0
$181$	5.93329	0.441018	0.220509	−	0.975385i	$-0.429228\pi$
$181$	5.93329	0.441018	0.220509	+	0.975385i	$0.429228\pi$
$182$	0	0
$183$	10.9469	0.809219
$184$	0	0
$185$	0.248882	0.0182982
$186$	0	0
$187$	0.995325	0.0727853
$188$	0	0
$189$	15.2031	1.10586
$190$	0	0
$191$	−2.85605	−0.206657	−0.103328	−	0.994647i	$-0.532949\pi$
$191$	−2.85605	−0.206657	−0.103328	+	0.994647i	$0.532949\pi$
$192$	0	0
$193$	3.42983	0.246885	0.123442	−	0.992352i	$-0.460607\pi$
$193$	3.42983	0.246885	0.123442	+	0.992352i	$0.460607\pi$
$194$	0	0
$195$	−0.125752	−0.00900531
$196$	0	0
$197$	1.67085	0.119043	0.0595216	−	0.998227i	$-0.481042\pi$
$197$	1.67085	0.119043	0.0595216	+	0.998227i	$0.481042\pi$
$198$	0	0
$199$	−11.0992	−0.786801	−0.393401	−	0.919367i	$-0.628701\pi$
$199$	−11.0992	−0.786801	−0.393401	+	0.919367i	$0.628701\pi$
$200$	0	0
$201$	10.8362	0.764327
$202$	0	0
$203$	3.38711	0.237729
$204$	0	0
$205$	−0.263275	−0.0183879
$206$	0	0
$207$	1.41518	0.0983622
$208$	0	0
$209$	1.50975	0.104432
$210$	0	0
$211$	−24.4236	−1.68139	−0.840697	−	0.541506i	$-0.817854\pi$
$211$	−24.4236	−1.68139	−0.840697	+	0.541506i	$0.817854\pi$
$212$	0	0
$213$	30.3951	2.08264
$214$	0	0
$215$	−0.457789	−0.0312210
$216$	0	0
$217$	−6.00528	−0.407665
$218$	0	0
$219$	28.7639	1.94368
$220$	0	0
$221$	−1.04308	−0.0701652
$222$	0	0
$223$	−5.66896	−0.379622	−0.189811	−	0.981821i	$-0.560787\pi$
$223$	−5.66896	−0.379622	−0.189811	+	0.981821i	$0.560787\pi$
$224$	0	0
$225$	−5.13950	−0.342633
$226$	0	0
$227$	27.5843	1.83083	0.915416	−	0.402510i	$-0.131862\pi$
$227$	27.5843	1.83083	0.915416	+	0.402510i	$0.131862\pi$
$228$	0	0
$229$	−21.2397	−1.40356	−0.701781	−	0.712393i	$-0.747611\pi$
$229$	−21.2397	−1.40356	−0.701781	+	0.712393i	$0.747611\pi$
$230$	0	0
$231$	9.36565	0.616214
$232$	0	0
$233$	−21.5364	−1.41089	−0.705447	−	0.708763i	$-0.749254\pi$
$233$	−21.5364	−1.41089	−0.705447	+	0.708763i	$0.749254\pi$
$234$	0	0
$235$	−0.155967	−0.0101742
$236$	0	0
$237$	26.9684	1.75179
$238$	0	0
$239$	0.920812	0.0595624	0.0297812	−	0.999556i	$-0.490519\pi$
$239$	0.920812	0.0595624	0.0297812	+	0.999556i	$0.490519\pi$
$240$	0	0
$241$	−21.1325	−1.36126	−0.680632	−	0.732625i	$-0.738295\pi$
$241$	−21.1325	−1.36126	−0.680632	+	0.732625i	$0.738295\pi$
$242$	0	0
$243$	10.2618	0.658295
$244$	0	0
$245$	0.381987	0.0244042
$246$	0	0
$247$	−1.58219	−0.100672
$248$	0	0
$249$	14.5578	0.922561
$250$	0	0
$251$	−23.7687	−1.50027	−0.750133	−	0.661287i	$-0.770010\pi$
$251$	−23.7687	−1.50027	−0.750133	+	0.661287i	$0.770010\pi$
$252$	0	0
$253$	−1.67139	−0.105079
$254$	0	0
$255$	0.0809614	0.00507000
$256$	0	0
$257$	6.80865	0.424712	0.212356	−	0.977192i	$-0.431886\pi$
$257$	6.80865	0.424712	0.212356	+	0.977192i	$0.431886\pi$
$258$	0	0
$259$	19.4253	1.20703
$260$	0	0
$261$	0.906659	0.0561208
$262$	0	0
$263$	−10.9060	−0.672491	−0.336246	−	0.941774i	$-0.609157\pi$
$263$	−10.9060	−0.672491	−0.336246	+	0.941774i	$0.609157\pi$
$264$	0	0
$265$	0.173717	0.0106714
$266$	0	0
$267$	−3.60354	−0.220533
$268$	0	0
$269$	−12.3222	−0.751298	−0.375649	−	0.926762i	$-0.622580\pi$
$269$	−12.3222	−0.751298	−0.375649	+	0.926762i	$0.622580\pi$
$270$	0	0
$271$	−4.05699	−0.246445	−0.123222	−	0.992379i	$-0.539323\pi$
$271$	−4.05699	−0.246445	−0.123222	+	0.992379i	$0.539323\pi$
$272$	0	0
$273$	−9.81502	−0.594032
$274$	0	0
$275$	6.06994	0.366031
$276$	0	0
$277$	−22.6107	−1.35855	−0.679273	−	0.733886i	$-0.737705\pi$
$277$	−22.6107	−1.35855	−0.679273	+	0.733886i	$0.737705\pi$
$278$	0	0
$279$	−1.60749	−0.0962378
$280$	0	0
$281$	0.573502	0.0342122	0.0171061	−	0.999854i	$-0.494555\pi$
$281$	0.573502	0.0342122	0.0171061	+	0.999854i	$0.494555\pi$
$282$	0	0
$283$	−19.0894	−1.13474	−0.567372	−	0.823461i	$-0.692040\pi$
$283$	−19.0894	−1.13474	−0.567372	+	0.823461i	$0.692040\pi$
$284$	0	0
$285$	0.122806	0.00727438
$286$	0	0
$287$	−20.5487	−1.21295
$288$	0	0
$289$	−16.3284	−0.960497
$290$	0	0
$291$	−30.4778	−1.78664
$292$	0	0
$293$	19.7114	1.15156	0.575778	−	0.817606i	$-0.304699\pi$
$293$	19.7114	1.15156	0.575778	+	0.817606i	$0.304699\pi$
$294$	0	0
$295$	0.459928	0.0267781
$296$	0	0
$297$	−4.80629	−0.278889
$298$	0	0
$299$	1.75158	0.101296
$300$	0	0
$301$	−35.7307	−2.05948
$302$	0	0
$303$	−30.0034	−1.72365
$304$	0	0
$305$	−0.268471	−0.0153726
$306$	0	0
$307$	−12.9458	−0.738854	−0.369427	−	0.929260i	$-0.620446\pi$
$307$	−12.9458	−0.738854	−0.369427	+	0.929260i	$0.620446\pi$
$308$	0	0
$309$	−34.7150	−1.97487
$310$	0	0
$311$	−21.1415	−1.19882	−0.599412	−	0.800441i	$-0.704599\pi$
$311$	−21.1415	−1.19882	−0.599412	+	0.800441i	$0.704599\pi$
$312$	0	0
$313$	0.237249	0.0134101	0.00670504	−	0.999978i	$-0.497866\pi$
$313$	0.237249	0.0134101	0.00670504	+	0.999978i	$0.497866\pi$
$314$	0	0
$315$	0.194482	0.0109578
$316$	0	0
$317$	−24.5299	−1.37774	−0.688868	−	0.724887i	$-0.741892\pi$
$317$	−24.5299	−1.37774	−0.688868	+	0.724887i	$0.741892\pi$
$318$	0	0
$319$	−1.07080	−0.0599532
$320$	0	0
$321$	−7.76338	−0.433309
$322$	0	0
$323$	1.01864	0.0566786
$324$	0	0
$325$	−6.36118	−0.352855
$326$	0	0
$327$	−18.6265	−1.03005
$328$	0	0
$329$	−12.1733	−0.671135
$330$	0	0
$331$	−16.9064	−0.929258	−0.464629	−	0.885505i	$-0.653812\pi$
$331$	−16.9064	−0.929258	−0.464629	+	0.885505i	$0.653812\pi$
$332$	0	0
$333$	5.19975	0.284945
$334$	0	0
$335$	−0.265756	−0.0145198
$336$	0	0
$337$	28.4680	1.55075	0.775377	−	0.631499i	$-0.217560\pi$
$337$	28.4680	1.55075	0.775377	+	0.631499i	$0.217560\pi$
$338$	0	0
$339$	−20.1108	−1.09227
$340$	0	0
$341$	1.89850	0.102810
$342$	0	0
$343$	2.92087	0.157712
$344$	0	0
$345$	−0.135953	−0.00731948
$346$	0	0
$347$	29.2531	1.57039	0.785194	−	0.619250i	$-0.212563\pi$
$347$	29.2531	1.57039	0.785194	+	0.619250i	$0.212563\pi$
$348$	0	0
$349$	29.7647	1.59327	0.796633	−	0.604463i	$-0.206612\pi$
$349$	29.7647	1.59327	0.796633	+	0.604463i	$0.206612\pi$
$350$	0	0
$351$	5.03690	0.268850
$352$	0	0
$353$	11.1388	0.592858	0.296429	−	0.955055i	$-0.404204\pi$
$353$	11.1388	0.592858	0.296429	+	0.955055i	$0.404204\pi$
$354$	0	0
$355$	−0.745434	−0.0395635
$356$	0	0
$357$	6.31907	0.334441
$358$	0	0
$359$	11.6789	0.616390	0.308195	−	0.951323i	$-0.400275\pi$
$359$	11.6789	0.616390	0.308195	+	0.951323i	$0.400275\pi$
$360$	0	0
$361$	−17.4549	−0.918678
$362$	0	0
$363$	19.1171	1.00339
$364$	0	0
$365$	−0.705428	−0.0369238
$366$	0	0
$367$	−34.3649	−1.79384	−0.896918	−	0.442197i	$-0.854199\pi$
$367$	−34.3649	−1.79384	−0.896918	+	0.442197i	$0.854199\pi$
$368$	0	0
$369$	−5.50047	−0.286343
$370$	0	0
$371$	13.5587	0.703932
$372$	0	0
$373$	20.2744	1.04977	0.524885	−	0.851173i	$-0.324108\pi$
$373$	20.2744	1.04977	0.524885	+	0.851173i	$0.324108\pi$
$374$	0	0
$375$	0.987718	0.0510055
$376$	0	0
$377$	1.12218	0.0577950
$378$	0	0
$379$	−8.24662	−0.423601	−0.211800	−	0.977313i	$-0.567933\pi$
$379$	−8.24662	−0.423601	−0.211800	+	0.977313i	$0.567933\pi$
$380$	0	0
$381$	30.2886	1.55173
$382$	0	0
$383$	−22.0089	−1.12460	−0.562302	−	0.826932i	$-0.690084\pi$
$383$	−22.0089	−1.12460	−0.562302	+	0.826932i	$0.690084\pi$
$384$	0	0
$385$	−0.229691	−0.0117061
$386$	0	0
$387$	−9.56434	−0.486183
$388$	0	0
$389$	−4.20479	−0.213191	−0.106596	−	0.994302i	$-0.533995\pi$
$389$	−4.20479	−0.213191	−0.106596	+	0.994302i	$0.533995\pi$
$390$	0	0
$391$	−1.12770	−0.0570300
$392$	0	0
$393$	2.62471	0.132399
$394$	0	0
$395$	−0.661395	−0.0332784
$396$	0	0
$397$	−19.6563	−0.986523	−0.493261	−	0.869881i	$-0.664195\pi$
$397$	−19.6563	−0.986523	−0.493261	+	0.869881i	$0.664195\pi$
$398$	0	0
$399$	9.58503	0.479852
$400$	0	0
$401$	9.95152	0.496955	0.248478	−	0.968638i	$-0.420070\pi$
$401$	9.95152	0.496955	0.248478	+	0.968638i	$0.420070\pi$
$402$	0	0
$403$	−1.98959	−0.0991087
$404$	0	0
$405$	−0.542816	−0.0269727
$406$	0	0
$407$	−6.14109	−0.304403
$408$	0	0
$409$	19.6345	0.970864	0.485432	−	0.874274i	$-0.338662\pi$
$409$	19.6345	0.970864	0.485432	+	0.874274i	$0.338662\pi$
$410$	0	0
$411$	−42.8902	−2.11562
$412$	0	0
$413$	35.8976	1.76641
$414$	0	0
$415$	−0.357027	−0.0175257
$416$	0	0
$417$	12.9055	0.631986
$418$	0	0
$419$	8.43531	0.412092	0.206046	−	0.978542i	$-0.433940\pi$
$419$	8.43531	0.412092	0.206046	+	0.978542i	$0.433940\pi$
$420$	0	0
$421$	0.170507	0.00831001	0.00415501	−	0.999991i	$-0.498677\pi$
$421$	0.170507	0.00831001	0.00415501	+	0.999991i	$0.498677\pi$
$422$	0	0
$423$	−3.25853	−0.158435
$424$	0	0
$425$	4.09543	0.198658
$426$	0	0
$427$	−20.9543	−1.01405
$428$	0	0
$429$	3.10291	0.149810
$430$	0	0
$431$	2.79223	0.134497	0.0672484	−	0.997736i	$-0.478578\pi$
$431$	2.79223	0.134497	0.0672484	+	0.997736i	$0.478578\pi$
$432$	0	0
$433$	7.63595	0.366960	0.183480	−	0.983023i	$-0.441264\pi$
$433$	7.63595	0.366960	0.183480	+	0.983023i	$0.441264\pi$
$434$	0	0
$435$	−0.0871005	−0.00417615
$436$	0	0
$437$	−1.71054	−0.0818260
$438$	0	0
$439$	−32.3147	−1.54230	−0.771149	−	0.636655i	$-0.780318\pi$
$439$	−32.3147	−1.54230	−0.771149	+	0.636655i	$0.780318\pi$
$440$	0	0
$441$	7.98064	0.380031
$442$	0	0
$443$	−40.5594	−1.92703	−0.963517	−	0.267646i	$-0.913754\pi$
$443$	−40.5594	−1.92703	−0.963517	+	0.267646i	$0.913754\pi$
$444$	0	0
$445$	0.0883762	0.00418943
$446$	0	0
$447$	−24.6986	−1.16821
$448$	0	0
$449$	−13.9801	−0.659760	−0.329880	−	0.944023i	$-0.607008\pi$
$449$	−13.9801	−0.659760	−0.329880	+	0.944023i	$0.607008\pi$
$450$	0	0
$451$	6.49625	0.305896
$452$	0	0
$453$	5.92587	0.278422
$454$	0	0
$455$	0.240712	0.0112847
$456$	0	0
$457$	1.60020	0.0748541	0.0374270	−	0.999299i	$-0.488084\pi$
$457$	1.60020	0.0748541	0.0374270	+	0.999299i	$0.488084\pi$
$458$	0	0
$459$	−3.24284	−0.151363
$460$	0	0
$461$	14.9778	0.697587	0.348793	−	0.937200i	$-0.386591\pi$
$461$	14.9778	0.697587	0.348793	+	0.937200i	$0.386591\pi$
$462$	0	0
$463$	3.03428	0.141015	0.0705075	−	0.997511i	$-0.477538\pi$
$463$	3.03428	0.141015	0.0705075	+	0.997511i	$0.477538\pi$
$464$	0	0
$465$	0.154427	0.00716140
$466$	0	0
$467$	0.611036	0.0282754	0.0141377	−	0.999900i	$-0.495500\pi$
$467$	0.611036	0.0282754	0.0141377	+	0.999900i	$0.495500\pi$
$468$	0	0
$469$	−20.7424	−0.957793
$470$	0	0
$471$	−21.2010	−0.976891
$472$	0	0
$473$	11.2958	0.519383
$474$	0	0
$475$	6.21212	0.285031
$476$	0	0
$477$	3.62937	0.166178
$478$	0	0
$479$	19.6418	0.897457	0.448728	−	0.893668i	$-0.351877\pi$
$479$	19.6418	0.897457	0.448728	+	0.893668i	$0.351877\pi$
$480$	0	0
$481$	6.43575	0.293445
$482$	0	0
$483$	−10.6112	−0.482827
$484$	0	0
$485$	0.747463	0.0339405
$486$	0	0
$487$	−21.8782	−0.991394	−0.495697	−	0.868496i	$-0.665087\pi$
$487$	−21.8782	−0.991394	−0.495697	+	0.868496i	$0.665087\pi$
$488$	0	0
$489$	−39.5669	−1.78928
$490$	0	0
$491$	26.3666	1.18991	0.594954	−	0.803759i	$-0.297170\pi$
$491$	26.3666	1.18991	0.594954	+	0.803759i	$0.297170\pi$
$492$	0	0
$493$	−0.722475	−0.0325386
$494$	0	0
$495$	−0.0614834	−0.00276347
$496$	0	0
$497$	−58.1815	−2.60979
$498$	0	0
$499$	−5.58422	−0.249984	−0.124992	−	0.992158i	$-0.539891\pi$
$499$	−5.58422	−0.249984	−0.124992	+	0.992158i	$0.539891\pi$
$500$	0	0
$501$	33.9950	1.51878
$502$	0	0
$503$	−31.0425	−1.38412	−0.692059	−	0.721841i	$-0.743296\pi$
$503$	−31.0425	−1.38412	−0.692059	+	0.721841i	$0.743296\pi$
$504$	0	0
$505$	0.735828	0.0327439
$506$	0	0
$507$	22.8403	1.01438
$508$	0	0
$509$	−18.4287	−0.816836	−0.408418	−	0.912795i	$-0.633919\pi$
$509$	−18.4287	−0.816836	−0.408418	+	0.912795i	$0.633919\pi$
$510$	0	0
$511$	−55.0590	−2.43567
$512$	0	0
$513$	−4.91887	−0.217173
$514$	0	0
$515$	0.851378	0.0375162
$516$	0	0
$517$	3.84844	0.169254
$518$	0	0
$519$	46.5087	2.04151
$520$	0	0
$521$	28.0249	1.22780	0.613898	−	0.789386i	$-0.289601\pi$
$521$	28.0249	1.22780	0.613898	+	0.789386i	$0.289601\pi$
$522$	0	0
$523$	−43.0613	−1.88294	−0.941469	−	0.337100i	$-0.890554\pi$
$523$	−43.0613	−1.88294	−0.941469	+	0.337100i	$0.890554\pi$
$524$	0	0
$525$	38.5365	1.68187
$526$	0	0
$527$	1.28093	0.0557983
$528$	0	0
$529$	−21.1063	−0.917667
$530$	0	0
$531$	9.60903	0.416996
$532$	0	0
$533$	−6.80795	−0.294885
$534$	0	0
$535$	0.190395	0.00823151
$536$	0	0
$537$	−2.12450	−0.0916790
$538$	0	0
$539$	−9.42543	−0.405982
$540$	0	0
$541$	6.99953	0.300933	0.150467	−	0.988615i	$-0.451922\pi$
$541$	6.99953	0.300933	0.150467	+	0.988615i	$0.451922\pi$
$542$	0	0
$543$	−11.9086	−0.511048
$544$	0	0
$545$	0.456811	0.0195676
$546$	0	0
$547$	35.6994	1.52640	0.763199	−	0.646164i	$-0.223628\pi$
$547$	35.6994	1.52640	0.763199	+	0.646164i	$0.223628\pi$
$548$	0	0
$549$	−5.60901	−0.239387
$550$	0	0
$551$	−1.09588	−0.0466860
$552$	0	0
$553$	−51.6222	−2.19520
$554$	0	0
$555$	−0.499527	−0.0212038
$556$	0	0
$557$	−23.9989	−1.01686	−0.508432	−	0.861102i	$-0.669775\pi$
$557$	−23.9989	−1.01686	−0.508432	+	0.861102i	$0.669775\pi$
$558$	0	0
$559$	−11.8378	−0.500687
$560$	0	0
$561$	−1.99770	−0.0843431
$562$	0	0
$563$	37.5164	1.58113	0.790564	−	0.612379i	$-0.209788\pi$
$563$	37.5164	1.58113	0.790564	+	0.612379i	$0.209788\pi$
$564$	0	0
$565$	0.493214	0.0207497
$566$	0	0
$567$	−42.3670	−1.77925
$568$	0	0
$569$	−44.8656	−1.88086	−0.940431	−	0.339984i	$-0.889578\pi$
$569$	−44.8656	−1.88086	−0.940431	+	0.339984i	$0.889578\pi$
$570$	0	0
$571$	−10.1072	−0.422973	−0.211486	−	0.977381i	$-0.567830\pi$
$571$	−10.1072	−0.422973	−0.211486	+	0.977381i	$0.567830\pi$
$572$	0	0
$573$	5.73235	0.239472
$574$	0	0
$575$	−6.87720	−0.286799
$576$	0	0
$577$	29.1503	1.21354	0.606771	−	0.794877i	$-0.292465\pi$
$577$	29.1503	1.21354	0.606771	+	0.794877i	$0.292465\pi$
$578$	0	0
$579$	−6.88397	−0.286088
$580$	0	0
$581$	−27.8661	−1.15608
$582$	0	0
$583$	−4.28642	−0.177526
$584$	0	0
$585$	0.0644334	0.00266399
$586$	0	0
$587$	4.09267	0.168922	0.0844612	−	0.996427i	$-0.473083\pi$
$587$	4.09267	0.168922	0.0844612	+	0.996427i	$0.473083\pi$
$588$	0	0
$589$	1.94297	0.0800587
$590$	0	0
$591$	−3.35354	−0.137946
$592$	0	0
$593$	2.14775	0.0881977	0.0440988	−	0.999027i	$-0.485958\pi$
$593$	2.14775	0.0881977	0.0440988	+	0.999027i	$0.485958\pi$
$594$	0	0
$595$	−0.154974	−0.00635332
$596$	0	0
$597$	22.2771	0.911739
$598$	0	0
$599$	4.88974	0.199789	0.0998946	−	0.994998i	$-0.468149\pi$
$599$	4.88974	0.199789	0.0998946	+	0.994998i	$0.468149\pi$
$600$	0	0
$601$	45.0994	1.83964	0.919821	−	0.392337i	$-0.128333\pi$
$601$	45.0994	1.83964	0.919821	+	0.392337i	$0.128333\pi$
$602$	0	0
$603$	−5.55229	−0.226107
$604$	0	0
$605$	−0.468844	−0.0190612
$606$	0	0
$607$	−1.56212	−0.0634043	−0.0317022	−	0.999497i	$-0.510093\pi$
$607$	−1.56212	−0.0634043	−0.0317022	+	0.999497i	$0.510093\pi$
$608$	0	0
$609$	−6.79823	−0.275478
$610$	0	0
$611$	−4.03310	−0.163162
$612$	0	0
$613$	−18.3643	−0.741727	−0.370864	−	0.928687i	$-0.620938\pi$
$613$	−18.3643	−0.741727	−0.370864	+	0.928687i	$0.620938\pi$
$614$	0	0
$615$	0.528416	0.0213078
$616$	0	0
$617$	10.3920	0.418368	0.209184	−	0.977876i	$-0.432919\pi$
$617$	10.3920	0.418368	0.209184	+	0.977876i	$0.432919\pi$
$618$	0	0
$619$	−17.7341	−0.712792	−0.356396	−	0.934335i	$-0.615995\pi$
$619$	−17.7341	−0.712792	−0.356396	+	0.934335i	$0.615995\pi$
$620$	0	0
$621$	5.44549	0.218520
$622$	0	0
$623$	6.89780	0.276355
$624$	0	0
$625$	24.9637	0.998546
$626$	0	0
$627$	−3.03020	−0.121014
$628$	0	0
$629$	−4.14344	−0.165210
$630$	0	0
$631$	−11.7614	−0.468214	−0.234107	−	0.972211i	$-0.575217\pi$
$631$	−11.7614	−0.468214	−0.234107	+	0.972211i	$0.575217\pi$
$632$	0	0
$633$	49.0204	1.94839
$634$	0	0
$635$	−0.742821	−0.0294780
$636$	0	0
$637$	9.87767	0.391368
$638$	0	0
$639$	−15.5739	−0.616096
$640$	0	0
$641$	18.7298	0.739782	0.369891	−	0.929075i	$-0.379395\pi$
$641$	18.7298	0.739782	0.369891	+	0.929075i	$0.379395\pi$
$642$	0	0
$643$	−3.33285	−0.131435	−0.0657173	−	0.997838i	$-0.520934\pi$
$643$	−3.33285	−0.131435	−0.0657173	+	0.997838i	$0.520934\pi$
$644$	0	0
$645$	0.918823	0.0361786
$646$	0	0
$647$	−23.5822	−0.927113	−0.463556	−	0.886067i	$-0.653427\pi$
$647$	−23.5822	−0.927113	−0.463556	+	0.886067i	$0.653427\pi$
$648$	0	0
$649$	−11.3486	−0.445472
$650$	0	0
$651$	12.0531	0.472399
$652$	0	0
$653$	30.5579	1.19582	0.597912	−	0.801562i	$-0.295997\pi$
$653$	30.5579	1.19582	0.597912	+	0.801562i	$0.295997\pi$
$654$	0	0
$655$	−0.0643706	−0.00251517
$656$	0	0
$657$	−14.7381	−0.574989
$658$	0	0
$659$	3.20100	0.124693	0.0623467	−	0.998055i	$-0.480142\pi$
$659$	3.20100	0.124693	0.0623467	+	0.998055i	$0.480142\pi$
$660$	0	0
$661$	−38.9181	−1.51374	−0.756870	−	0.653566i	$-0.773272\pi$
$661$	−38.9181	−1.51374	−0.756870	+	0.653566i	$0.773272\pi$
$662$	0	0
$663$	2.09355	0.0813069
$664$	0	0
$665$	−0.235071	−0.00911566
$666$	0	0
$667$	1.21321	0.0469755
$668$	0	0
$669$	11.3781	0.439903
$670$	0	0
$671$	6.62445	0.255734
$672$	0	0
$673$	−32.1720	−1.24014	−0.620069	−	0.784547i	$-0.712895\pi$
$673$	−32.1720	−1.24014	−0.620069	+	0.784547i	$0.712895\pi$
$674$	0	0
$675$	−19.7763	−0.761190
$676$	0	0
$677$	7.16382	0.275328	0.137664	−	0.990479i	$-0.456041\pi$
$677$	7.16382	0.275328	0.137664	+	0.990479i	$0.456041\pi$
$678$	0	0
$679$	58.3398	2.23887
$680$	0	0
$681$	−55.3640	−2.12155
$682$	0	0
$683$	31.4172	1.20215	0.601073	−	0.799194i	$-0.294740\pi$
$683$	31.4172	1.20215	0.601073	+	0.799194i	$0.294740\pi$
$684$	0	0
$685$	1.05187	0.0401900
$686$	0	0
$687$	42.6300	1.62644
$688$	0	0
$689$	4.49209	0.171135
$690$	0	0
$691$	−15.3409	−0.583597	−0.291798	−	0.956480i	$-0.594254\pi$
$691$	−15.3409	−0.583597	−0.291798	+	0.956480i	$0.594254\pi$
$692$	0	0
$693$	−4.79880	−0.182291
$694$	0	0
$695$	−0.316505	−0.0120057
$696$	0	0
$697$	4.38307	0.166021
$698$	0	0
$699$	43.2254	1.63493
$700$	0	0
$701$	−45.8809	−1.73290	−0.866450	−	0.499265i	$-0.833604\pi$
$701$	−45.8809	−1.73290	−0.866450	+	0.499265i	$0.833604\pi$
$702$	0	0
$703$	−6.28494	−0.237041
$704$	0	0
$705$	0.313039	0.0117897
$706$	0	0
$707$	57.4317	2.15994
$708$	0	0
$709$	12.2522	0.460141	0.230071	−	0.973174i	$-0.426104\pi$
$709$	12.2522	0.460141	0.230071	+	0.973174i	$0.426104\pi$
$710$	0	0
$711$	−13.8182	−0.518222
$712$	0	0
$713$	−2.15099	−0.0805552
$714$	0	0
$715$	−0.0760982	−0.00284591
$716$	0	0
$717$	−1.84815	−0.0690204
$718$	0	0
$719$	−25.7323	−0.959653	−0.479826	−	0.877363i	$-0.659300\pi$
$719$	−25.7323	−0.959653	−0.479826	+	0.877363i	$0.659300\pi$
$720$	0	0
$721$	66.4504	2.47474
$722$	0	0
$723$	42.4148	1.57742
$724$	0	0
$725$	−4.40598	−0.163634
$726$	0	0
$727$	43.2514	1.60411	0.802053	−	0.597253i	$-0.203741\pi$
$727$	43.2514	1.60411	0.802053	+	0.597253i	$0.203741\pi$
$728$	0	0
$729$	12.4864	0.462461
$730$	0	0
$731$	7.62139	0.281887
$732$	0	0
$733$	−1.44163	−0.0532480	−0.0266240	−	0.999646i	$-0.508476\pi$
$733$	−1.44163	−0.0532480	−0.0266240	+	0.999646i	$0.508476\pi$
$734$	0	0
$735$	−0.766681	−0.0282795
$736$	0	0
$737$	6.55746	0.241547
$738$	0	0
$739$	−1.82506	−0.0671360	−0.0335680	−	0.999436i	$-0.510687\pi$
$739$	−1.82506	−0.0671360	−0.0335680	+	0.999436i	$0.510687\pi$
$740$	0	0
$741$	3.17559	0.116658
$742$	0	0
$743$	15.9357	0.584622	0.292311	−	0.956323i	$-0.405576\pi$
$743$	15.9357	0.584622	0.292311	+	0.956323i	$0.405576\pi$
$744$	0	0
$745$	0.605729	0.0221922
$746$	0	0
$747$	−7.45916	−0.272916
$748$	0	0
$749$	14.8604	0.542989
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	47.7058	1.73850
$754$	0	0
$755$	−0.145331	−0.00528913
$756$	0	0
$757$	−51.8492	−1.88449	−0.942246	−	0.334923i	$-0.891290\pi$
$757$	−51.8492	−1.88449	−0.942246	+	0.334923i	$0.891290\pi$
$758$	0	0
$759$	3.35462	0.121765
$760$	0	0
$761$	−39.7590	−1.44126	−0.720631	−	0.693318i	$-0.756148\pi$
$761$	−39.7590	−1.44126	−0.720631	+	0.693318i	$0.756148\pi$
$762$	0	0
$763$	35.6543	1.29077
$764$	0	0
$765$	−0.0414833	−0.00149983
$766$	0	0
$767$	11.8931	0.429436
$768$	0	0
$769$	−18.2323	−0.657472	−0.328736	−	0.944422i	$-0.606623\pi$
$769$	−18.2323	−0.657472	−0.328736	+	0.944422i	$0.606623\pi$
$770$	0	0
$771$	−13.6655	−0.492153
$772$	0	0
$773$	32.3971	1.16524	0.582621	−	0.812744i	$-0.302027\pi$
$773$	32.3971	1.16524	0.582621	+	0.812744i	$0.302027\pi$
$774$	0	0
$775$	7.81170	0.280605
$776$	0	0
$777$	−38.9883	−1.39870
$778$	0	0
$779$	6.64842	0.238204
$780$	0	0
$781$	18.3934	0.658168
$782$	0	0
$783$	3.48874	0.124677
$784$	0	0
$785$	0.519951	0.0185578
$786$	0	0
$787$	−27.5333	−0.981457	−0.490728	−	0.871313i	$-0.663269\pi$
$787$	−27.5333	−0.981457	−0.490728	+	0.871313i	$0.663269\pi$
$788$	0	0
$789$	21.8892	0.779278
$790$	0	0
$791$	38.4956	1.36875
$792$	0	0
$793$	−6.94230	−0.246528
$794$	0	0
$795$	−0.348665	−0.0123659
$796$	0	0
$797$	−3.76869	−0.133494	−0.0667469	−	0.997770i	$-0.521262\pi$
$797$	−3.76869	−0.133494	−0.0667469	+	0.997770i	$0.521262\pi$
$798$	0	0
$799$	2.59657	0.0918602
$800$	0	0
$801$	1.84640	0.0652392
$802$	0	0
$803$	17.4063	0.614254
$804$	0	0
$805$	0.260238	0.00917219
$806$	0	0
$807$	24.7317	0.870598
$808$	0	0
$809$	35.8165	1.25924	0.629621	−	0.776902i	$-0.283210\pi$
$809$	35.8165	1.25924	0.629621	+	0.776902i	$0.283210\pi$
$810$	0	0
$811$	−16.1755	−0.567999	−0.284000	−	0.958824i	$-0.591661\pi$
$811$	−16.1755	−0.567999	−0.284000	+	0.958824i	$0.591661\pi$
$812$	0	0
$813$	8.14274	0.285578
$814$	0	0
$815$	0.970370	0.0339906
$816$	0	0
$817$	11.5604	0.404448
$818$	0	0
$819$	5.02906	0.175729
$820$	0	0
$821$	35.8688	1.25183	0.625914	−	0.779892i	$-0.284726\pi$
$821$	35.8688	1.25183	0.625914	+	0.779892i	$0.284726\pi$
$822$	0	0
$823$	13.1572	0.458631	0.229315	−	0.973352i	$-0.426351\pi$
$823$	13.1572	0.458631	0.229315	+	0.973352i	$0.426351\pi$
$824$	0	0
$825$	−12.1829	−0.424154
$826$	0	0
$827$	1.90486	0.0662386	0.0331193	−	0.999451i	$-0.489456\pi$
$827$	1.90486	0.0662386	0.0331193	+	0.999451i	$0.489456\pi$
$828$	0	0
$829$	−10.1600	−0.352872	−0.176436	−	0.984312i	$-0.556457\pi$
$829$	−10.1600	−0.352872	−0.176436	+	0.984312i	$0.556457\pi$
$830$	0	0
$831$	45.3816	1.57427
$832$	0	0
$833$	−6.35941	−0.220340
$834$	0	0
$835$	−0.833721	−0.0288521
$836$	0	0
$837$	−6.18546	−0.213801
$838$	0	0
$839$	−20.8208	−0.718813	−0.359406	−	0.933181i	$-0.617021\pi$
$839$	−20.8208	−0.718813	−0.359406	+	0.933181i	$0.617021\pi$
$840$	0	0
$841$	−28.2227	−0.973198
$842$	0	0
$843$	−1.15107	−0.0396449
$844$	0	0
$845$	−0.560155	−0.0192699
$846$	0	0
$847$	−36.5934	−1.25737
$848$	0	0
$849$	38.3140	1.31493
$850$	0	0
$851$	6.95782	0.238511
$852$	0	0
$853$	−17.7927	−0.609212	−0.304606	−	0.952478i	$-0.598525\pi$
$853$	−17.7927	−0.609212	−0.304606	+	0.952478i	$0.598525\pi$
$854$	0	0
$855$	−0.0629236	−0.00215194
$856$	0	0
$857$	−57.9789	−1.98052	−0.990261	−	0.139221i	$-0.955540\pi$
$857$	−57.9789	−1.98052	−0.990261	+	0.139221i	$0.955540\pi$
$858$	0	0
$859$	15.3446	0.523551	0.261775	−	0.965129i	$-0.415692\pi$
$859$	15.3446	0.523551	0.261775	+	0.965129i	$0.415692\pi$
$860$	0	0
$861$	41.2431	1.40556
$862$	0	0
$863$	−44.0580	−1.49975	−0.749876	−	0.661578i	$-0.769887\pi$
$863$	−44.0580	−1.49975	−0.749876	+	0.661578i	$0.769887\pi$
$864$	0	0
$865$	−1.14062	−0.0387822
$866$	0	0
$867$	32.7726	1.11302
$868$	0	0
$869$	16.3198	0.553610
$870$	0	0
$871$	−6.87210	−0.232852
$872$	0	0
$873$	15.6163	0.528533
$874$	0	0
$875$	−1.89066	−0.0639160
$876$	0	0
$877$	4.08232	0.137850	0.0689251	−	0.997622i	$-0.478043\pi$
$877$	4.08232	0.137850	0.0689251	+	0.997622i	$0.478043\pi$
$878$	0	0
$879$	−39.5626	−1.33441
$880$	0	0
$881$	47.9429	1.61524	0.807619	−	0.589704i	$-0.200756\pi$
$881$	47.9429	1.61524	0.807619	+	0.589704i	$0.200756\pi$
$882$	0	0
$883$	37.5593	1.26397	0.631986	−	0.774980i	$-0.282240\pi$
$883$	37.5593	1.26397	0.631986	+	0.774980i	$0.282240\pi$
$884$	0	0
$885$	−0.923116	−0.0310302
$886$	0	0
$887$	−37.5277	−1.26006	−0.630029	−	0.776572i	$-0.716957\pi$
$887$	−37.5277	−1.26006	−0.630029	+	0.776572i	$0.716957\pi$
$888$	0	0
$889$	−57.9775	−1.94450
$890$	0	0
$891$	13.3938	0.448711
$892$	0	0
$893$	3.93859	0.131800
$894$	0	0
$895$	0.0521030	0.00174161
$896$	0	0
$897$	−3.51557	−0.117382
$898$	0	0
$899$	−1.37806	−0.0459610
$900$	0	0
$901$	−2.89208	−0.0963492
$902$	0	0
$903$	71.7145	2.38651
$904$	0	0
$905$	0.292057	0.00970830
$906$	0	0
$907$	12.0360	0.399650	0.199825	−	0.979832i	$-0.435963\pi$
$907$	12.0360	0.399650	0.199825	+	0.979832i	$0.435963\pi$
$908$	0	0
$909$	15.3733	0.509899
$910$	0	0
$911$	59.9162	1.98511	0.992557	−	0.121784i	$-0.0388614\pi$
$911$	59.9162	1.98511	0.992557	+	0.121784i	$0.0388614\pi$
$912$	0	0
$913$	8.80954	0.291553
$914$	0	0
$915$	0.538844	0.0178136
$916$	0	0
$917$	−5.02415	−0.165912
$918$	0	0
$919$	27.0738	0.893082	0.446541	−	0.894763i	$-0.352656\pi$
$919$	27.0738	0.893082	0.446541	+	0.894763i	$0.352656\pi$
$920$	0	0
$921$	25.9833	0.856178
$922$	0	0
$923$	−19.2759	−0.634475
$924$	0	0
$925$	−25.2686	−0.830826
$926$	0	0
$927$	17.7874	0.584214
$928$	0	0
$929$	21.5169	0.705947	0.352973	−	0.935633i	$-0.385171\pi$
$929$	21.5169	0.705947	0.352973	+	0.935633i	$0.385171\pi$
$930$	0	0
$931$	−9.64621	−0.316142
$932$	0	0
$933$	42.4328	1.38919
$934$	0	0
$935$	0.0489933	0.00160225
$936$	0	0
$937$	4.05888	0.132598	0.0662989	−	0.997800i	$-0.478881\pi$
$937$	4.05888	0.132598	0.0662989	+	0.997800i	$0.478881\pi$
$938$	0	0
$939$	−0.476179	−0.0155395
$940$	0	0
$941$	−33.8277	−1.10275	−0.551376	−	0.834257i	$-0.685897\pi$
$941$	−33.8277	−1.10275	−0.551376	+	0.834257i	$0.685897\pi$
$942$	0	0
$943$	−7.36021	−0.239681
$944$	0	0
$945$	0.748349	0.0243438
$946$	0	0
$947$	35.9997	1.16983	0.584917	−	0.811093i	$-0.301127\pi$
$947$	35.9997	1.16983	0.584917	+	0.811093i	$0.301127\pi$
$948$	0	0
$949$	−18.2414	−0.592142
$950$	0	0
$951$	49.2336	1.59651
$952$	0	0
$953$	41.3634	1.33989	0.669946	−	0.742410i	$-0.266317\pi$
$953$	41.3634	1.33989	0.669946	+	0.742410i	$0.266317\pi$
$954$	0	0
$955$	−0.140585	−0.00454922
$956$	0	0
$957$	2.14918	0.0694733
$958$	0	0
$959$	82.0992	2.65112
$960$	0	0
$961$	−28.5567	−0.921185
$962$	0	0
$963$	3.97783	0.128184
$964$	0	0
$965$	0.168828	0.00543477
$966$	0	0
$967$	23.9050	0.768735	0.384367	−	0.923180i	$-0.374420\pi$
$967$	23.9050	0.768735	0.384367	+	0.923180i	$0.374420\pi$
$968$	0	0
$969$	−2.04450	−0.0656787
$970$	0	0
$971$	−49.4606	−1.58727	−0.793633	−	0.608396i	$-0.791813\pi$
$971$	−49.4606	−1.58727	−0.793633	+	0.608396i	$0.791813\pi$
$972$	0	0
$973$	−24.7034	−0.791954
$974$	0	0
$975$	12.7674	0.408885
$976$	0	0
$977$	−33.6360	−1.07611	−0.538056	−	0.842909i	$-0.680841\pi$
$977$	−33.6360	−1.07611	−0.538056	+	0.842909i	$0.680841\pi$
$978$	0	0
$979$	−2.18066	−0.0696942
$980$	0	0
$981$	9.54390	0.304713
$982$	0	0
$983$	−19.1782	−0.611689	−0.305845	−	0.952081i	$-0.598939\pi$
$983$	−19.1782	−0.611689	−0.305845	+	0.952081i	$0.598939\pi$
$984$	0	0
$985$	0.0822450	0.00262054
$986$	0	0
$987$	24.4328	0.777706
$988$	0	0
$989$	−12.7981	−0.406956
$990$	0	0
$991$	43.1950	1.37214	0.686068	−	0.727538i	$-0.259335\pi$
$991$	43.1950	1.37214	0.686068	+	0.727538i	$0.259335\pi$
$992$	0	0
$993$	33.9325	1.07682
$994$	0	0
$995$	−0.546341	−0.0173202
$996$	0	0
$997$	3.94299	0.124876	0.0624380	−	0.998049i	$-0.480112\pi$
$997$	3.94299	0.124876	0.0624380	+	0.998049i	$0.480112\pi$
$998$	0	0
$999$	20.0081	0.633029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.13	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-2.00709 q^{3} +0.0492234 q^{5} +3.84191 q^{7} +1.02840 q^{9} +O(q^{10})\)	\(q-2.00709 q^{3} +0.0492234 q^{5} +3.84191 q^{7} +1.02840 q^{9} -1.21458 q^{11} +1.27285 q^{13} -0.0987957 q^{15} -0.819483 q^{17} -1.24303 q^{19} -7.71105 q^{21} +1.37611 q^{23} -4.99758 q^{25} +3.95718 q^{27} +0.881623 q^{29} -1.56310 q^{31} +2.43776 q^{33} +0.189112 q^{35} +5.05616 q^{37} -2.55473 q^{39} -5.34858 q^{41} -9.30023 q^{43} +0.0506213 q^{45} -3.16855 q^{47} +7.76027 q^{49} +1.64477 q^{51} +3.52915 q^{53} -0.0597856 q^{55} +2.49486 q^{57} +9.34369 q^{59} -5.45413 q^{61} +3.95101 q^{63} +0.0626542 q^{65} -5.39897 q^{67} -2.76196 q^{69} -15.1439 q^{71} -14.3312 q^{73} +10.0306 q^{75} -4.66629 q^{77} -13.4366 q^{79} -11.0276 q^{81} -7.25318 q^{83} -0.0403378 q^{85} -1.76949 q^{87} +1.79541 q^{89} +4.89018 q^{91} +3.13727 q^{93} -0.0611860 q^{95} +15.1851 q^{97} -1.24907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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