LMFDB - Embedded newform 6008.2.a.e.1.45

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			1.45
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)

$f(q)$	$=$	$q+2.93661 q^{3} +3.50463 q^{5} +3.59612 q^{7} +5.62368 q^{9} +O(q^{10})$	$q+2.93661 q^{3} +3.50463 q^{5} +3.59612 q^{7} +5.62368 q^{9} +2.69390 q^{11} -2.54493 q^{13} +10.2917 q^{15} +2.23138 q^{17} -7.67199 q^{19} +10.5604 q^{21} -1.01256 q^{23} +7.28243 q^{25} +7.70474 q^{27} +5.38015 q^{29} -2.95587 q^{31} +7.91094 q^{33} +12.6031 q^{35} -2.73071 q^{37} -7.47347 q^{39} +3.58852 q^{41} -11.3625 q^{43} +19.7089 q^{45} -1.21067 q^{47} +5.93210 q^{49} +6.55270 q^{51} -11.6872 q^{53} +9.44112 q^{55} -22.5296 q^{57} -3.51792 q^{59} -1.42170 q^{61} +20.2235 q^{63} -8.91905 q^{65} -1.46876 q^{67} -2.97351 q^{69} -8.41321 q^{71} -11.1092 q^{73} +21.3857 q^{75} +9.68759 q^{77} +7.93478 q^{79} +5.75476 q^{81} +10.7358 q^{83} +7.82016 q^{85} +15.7994 q^{87} +14.2270 q^{89} -9.15189 q^{91} -8.68024 q^{93} -26.8875 q^{95} -7.71302 q^{97} +15.1496 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.93661	1.69545	0.847727	−	0.530434i	$-0.177971\pi$
$3$	2.93661	1.69545	0.847727	+	0.530434i	$0.177971\pi$
$4$	0	0
$5$	3.50463	1.56732	0.783659	−	0.621191i	$-0.213351\pi$
$5$	3.50463	1.56732	0.783659	+	0.621191i	$0.213351\pi$
$6$	0	0
$7$	3.59612	1.35921	0.679603	−	0.733580i	$-0.262152\pi$
$7$	3.59612	1.35921	0.679603	+	0.733580i	$0.262152\pi$
$8$	0	0
$9$	5.62368	1.87456
$10$	0	0
$11$	2.69390	0.812241	0.406121	−	0.913819i	$-0.366881\pi$
$11$	2.69390	0.812241	0.406121	+	0.913819i	$0.366881\pi$
$12$	0	0
$13$	−2.54493	−0.705837	−0.352919	−	0.935654i	$-0.614811\pi$
$13$	−2.54493	−0.705837	−0.352919	+	0.935654i	$0.614811\pi$
$14$	0	0
$15$	10.2917	2.65731
$16$	0	0
$17$	2.23138	0.541189	0.270595	−	0.962693i	$-0.412780\pi$
$17$	2.23138	0.541189	0.270595	+	0.962693i	$0.412780\pi$
$18$	0	0
$19$	−7.67199	−1.76007	−0.880037	−	0.474905i	$-0.842482\pi$
$19$	−7.67199	−1.76007	−0.880037	+	0.474905i	$0.842482\pi$
$20$	0	0
$21$	10.5604	2.30447
$22$	0	0
$23$	−1.01256	−0.211134	−0.105567	−	0.994412i	$-0.533666\pi$
$23$	−1.01256	−0.211134	−0.105567	+	0.994412i	$0.533666\pi$
$24$	0	0
$25$	7.28243	1.45649
$26$	0	0
$27$	7.70474	1.48278
$28$	0	0
$29$	5.38015	0.999070	0.499535	−	0.866294i	$-0.333504\pi$
$29$	5.38015	0.999070	0.499535	+	0.866294i	$0.333504\pi$
$30$	0	0
$31$	−2.95587	−0.530890	−0.265445	−	0.964126i	$-0.585519\pi$
$31$	−2.95587	−0.530890	−0.265445	+	0.964126i	$0.585519\pi$
$32$	0	0
$33$	7.91094	1.37712
$34$	0	0
$35$	12.6031	2.13031
$36$	0	0
$37$	−2.73071	−0.448926	−0.224463	−	0.974483i	$-0.572063\pi$
$37$	−2.73071	−0.448926	−0.224463	+	0.974483i	$0.572063\pi$
$38$	0	0
$39$	−7.47347	−1.19671
$40$	0	0
$41$	3.58852	0.560432	0.280216	−	0.959937i	$-0.409594\pi$
$41$	3.58852	0.560432	0.280216	+	0.959937i	$0.409594\pi$
$42$	0	0
$43$	−11.3625	−1.73277	−0.866385	−	0.499377i	$-0.833562\pi$
$43$	−11.3625	−1.73277	−0.866385	+	0.499377i	$0.833562\pi$
$44$	0	0
$45$	19.7089	2.93803
$46$	0	0
$47$	−1.21067	−0.176594	−0.0882968	−	0.996094i	$-0.528142\pi$
$47$	−1.21067	−0.176594	−0.0882968	+	0.996094i	$0.528142\pi$
$48$	0	0
$49$	5.93210	0.847443
$50$	0	0
$51$	6.55270	0.917561
$52$	0	0
$53$	−11.6872	−1.60535	−0.802677	−	0.596414i	$-0.796592\pi$
$53$	−11.6872	−1.60535	−0.802677	+	0.596414i	$0.796592\pi$
$54$	0	0
$55$	9.44112	1.27304
$56$	0	0
$57$	−22.5296	−2.98412
$58$	0	0
$59$	−3.51792	−0.457994	−0.228997	−	0.973427i	$-0.573545\pi$
$59$	−3.51792	−0.457994	−0.228997	+	0.973427i	$0.573545\pi$
$60$	0	0
$61$	−1.42170	−0.182030	−0.0910152	−	0.995850i	$-0.529011\pi$
$61$	−1.42170	−0.182030	−0.0910152	+	0.995850i	$0.529011\pi$
$62$	0	0
$63$	20.2235	2.54792
$64$	0	0
$65$	−8.91905	−1.10627
$66$	0	0
$67$	−1.46876	−0.179438	−0.0897188	−	0.995967i	$-0.528597\pi$
$67$	−1.46876	−0.179438	−0.0897188	+	0.995967i	$0.528597\pi$
$68$	0	0
$69$	−2.97351	−0.357968
$70$	0	0
$71$	−8.41321	−0.998464	−0.499232	−	0.866468i	$-0.666384\pi$
$71$	−8.41321	−0.998464	−0.499232	+	0.866468i	$0.666384\pi$
$72$	0	0
$73$	−11.1092	−1.30023	−0.650114	−	0.759837i	$-0.725279\pi$
$73$	−11.1092	−1.30023	−0.650114	+	0.759837i	$0.725279\pi$
$74$	0	0
$75$	21.3857	2.46941
$76$	0	0
$77$	9.68759	1.10400
$78$	0	0
$79$	7.93478	0.892732	0.446366	−	0.894850i	$-0.352718\pi$
$79$	7.93478	0.892732	0.446366	+	0.894850i	$0.352718\pi$
$80$	0	0
$81$	5.75476	0.639418
$82$	0	0
$83$	10.7358	1.17840	0.589201	−	0.807986i	$-0.299443\pi$
$83$	10.7358	1.17840	0.589201	+	0.807986i	$0.299443\pi$
$84$	0	0
$85$	7.82016	0.848216
$86$	0	0
$87$	15.7994	1.69388
$88$	0	0
$89$	14.2270	1.50806	0.754029	−	0.656841i	$-0.228108\pi$
$89$	14.2270	1.50806	0.754029	+	0.656841i	$0.228108\pi$
$90$	0	0
$91$	−9.15189	−0.959378
$92$	0	0
$93$	−8.68024	−0.900099
$94$	0	0
$95$	−26.8875	−2.75860
$96$	0	0
$97$	−7.71302	−0.783138	−0.391569	−	0.920149i	$-0.628068\pi$
$97$	−7.71302	−0.783138	−0.391569	+	0.920149i	$0.628068\pi$
$98$	0	0
$99$	15.1496	1.52260
$100$	0	0
$101$	−3.18955	−0.317372	−0.158686	−	0.987329i	$-0.550726\pi$
$101$	−3.18955	−0.317372	−0.158686	+	0.987329i	$0.550726\pi$
$102$	0	0
$103$	14.1045	1.38976	0.694878	−	0.719128i	$-0.255459\pi$
$103$	14.1045	1.38976	0.694878	+	0.719128i	$0.255459\pi$
$104$	0	0
$105$	37.0103	3.61184
$106$	0	0
$107$	15.3289	1.48190	0.740948	−	0.671562i	$-0.234376\pi$
$107$	15.3289	1.48190	0.740948	+	0.671562i	$0.234376\pi$
$108$	0	0
$109$	−1.80717	−0.173096	−0.0865478	−	0.996248i	$-0.527584\pi$
$109$	−1.80717	−0.173096	−0.0865478	+	0.996248i	$0.527584\pi$
$110$	0	0
$111$	−8.01903	−0.761133
$112$	0	0
$113$	9.98652	0.939453	0.469726	−	0.882812i	$-0.344353\pi$
$113$	9.98652	0.939453	0.469726	+	0.882812i	$0.344353\pi$
$114$	0	0
$115$	−3.54866	−0.330914
$116$	0	0
$117$	−14.3119	−1.32313
$118$	0	0
$119$	8.02432	0.735588
$120$	0	0
$121$	−3.74290	−0.340264
$122$	0	0
$123$	10.5381	0.950186
$124$	0	0
$125$	7.99909	0.715460
$126$	0	0
$127$	−8.22182	−0.729568	−0.364784	−	0.931092i	$-0.618857\pi$
$127$	−8.22182	−0.729568	−0.364784	+	0.931092i	$0.618857\pi$
$128$	0	0
$129$	−33.3673	−2.93783
$130$	0	0
$131$	2.39359	0.209129	0.104564	−	0.994518i	$-0.466655\pi$
$131$	2.39359	0.209129	0.104564	+	0.994518i	$0.466655\pi$
$132$	0	0
$133$	−27.5894	−2.39230
$134$	0	0
$135$	27.0023	2.32398
$136$	0	0
$137$	−1.26489	−0.108067	−0.0540336	−	0.998539i	$-0.517208\pi$
$137$	−1.26489	−0.108067	−0.0540336	+	0.998539i	$0.517208\pi$
$138$	0	0
$139$	−13.3386	−1.13137	−0.565684	−	0.824622i	$-0.691388\pi$
$139$	−13.3386	−1.13137	−0.565684	+	0.824622i	$0.691388\pi$
$140$	0	0
$141$	−3.55525	−0.299406
$142$	0	0
$143$	−6.85579	−0.573310
$144$	0	0
$145$	18.8555	1.56586
$146$	0	0
$147$	17.4203	1.43680
$148$	0	0
$149$	10.2709	0.841428	0.420714	−	0.907193i	$-0.361780\pi$
$149$	10.2709	0.841428	0.420714	+	0.907193i	$0.361780\pi$
$150$	0	0
$151$	−19.1035	−1.55462	−0.777310	−	0.629118i	$-0.783416\pi$
$151$	−19.1035	−1.55462	−0.777310	+	0.629118i	$0.783416\pi$
$152$	0	0
$153$	12.5486	1.01449
$154$	0	0
$155$	−10.3592	−0.832074
$156$	0	0
$157$	−11.2678	−0.899268	−0.449634	−	0.893213i	$-0.648446\pi$
$157$	−11.2678	−0.899268	−0.449634	+	0.893213i	$0.648446\pi$
$158$	0	0
$159$	−34.3206	−2.72180
$160$	0	0
$161$	−3.64130	−0.286975
$162$	0	0
$163$	−2.86346	−0.224283	−0.112142	−	0.993692i	$-0.535771\pi$
$163$	−2.86346	−0.224283	−0.112142	+	0.993692i	$0.535771\pi$
$164$	0	0
$165$	27.7249	2.15838
$166$	0	0
$167$	−14.8434	−1.14861	−0.574307	−	0.818640i	$-0.694728\pi$
$167$	−14.8434	−1.14861	−0.574307	+	0.818640i	$0.694728\pi$
$168$	0	0
$169$	−6.52332	−0.501794
$170$	0	0
$171$	−43.1448	−3.29937
$172$	0	0
$173$	18.4449	1.40234	0.701171	−	0.712993i	$-0.252661\pi$
$173$	18.4449	1.40234	0.701171	+	0.712993i	$0.252661\pi$
$174$	0	0
$175$	26.1885	1.97967
$176$	0	0
$177$	−10.3308	−0.776508
$178$	0	0
$179$	16.9032	1.26340	0.631701	−	0.775212i	$-0.282357\pi$
$179$	16.9032	1.26340	0.631701	+	0.775212i	$0.282357\pi$
$180$	0	0
$181$	14.9920	1.11434	0.557171	−	0.830397i	$-0.311887\pi$
$181$	14.9920	1.11434	0.557171	+	0.830397i	$0.311887\pi$
$182$	0	0
$183$	−4.17499	−0.308624
$184$	0	0
$185$	−9.57013	−0.703610
$186$	0	0
$187$	6.01112	0.439576
$188$	0	0
$189$	27.7072	2.01540
$190$	0	0
$191$	−0.588291	−0.0425673	−0.0212836	−	0.999773i	$-0.506775\pi$
$191$	−0.588291	−0.0425673	−0.0212836	+	0.999773i	$0.506775\pi$
$192$	0	0
$193$	−5.57048	−0.400972	−0.200486	−	0.979697i	$-0.564252\pi$
$193$	−5.57048	−0.400972	−0.200486	+	0.979697i	$0.564252\pi$
$194$	0	0
$195$	−26.1918	−1.87563
$196$	0	0
$197$	26.2860	1.87280	0.936401	−	0.350931i	$-0.114135\pi$
$197$	26.2860	1.87280	0.936401	+	0.350931i	$0.114135\pi$
$198$	0	0
$199$	−2.30553	−0.163435	−0.0817175	−	0.996656i	$-0.526041\pi$
$199$	−2.30553	−0.163435	−0.0817175	+	0.996656i	$0.526041\pi$
$200$	0	0
$201$	−4.31318	−0.304228
$202$	0	0
$203$	19.3477	1.35794
$204$	0	0
$205$	12.5764	0.878376
$206$	0	0
$207$	−5.69434	−0.395784
$208$	0	0
$209$	−20.6676	−1.42960
$210$	0	0
$211$	−13.5911	−0.935651	−0.467825	−	0.883821i	$-0.654962\pi$
$211$	−13.5911	−0.935651	−0.467825	+	0.883821i	$0.654962\pi$
$212$	0	0
$213$	−24.7063	−1.69285
$214$	0	0
$215$	−39.8215	−2.71580
$216$	0	0
$217$	−10.6297	−0.721589
$218$	0	0
$219$	−32.6233	−2.20448
$220$	0	0
$221$	−5.67871	−0.381991
$222$	0	0
$223$	−5.85854	−0.392317	−0.196158	−	0.980572i	$-0.562847\pi$
$223$	−5.85854	−0.392317	−0.196158	+	0.980572i	$0.562847\pi$
$224$	0	0
$225$	40.9541	2.73027
$226$	0	0
$227$	16.9583	1.12556	0.562781	−	0.826606i	$-0.309731\pi$
$227$	16.9583	1.12556	0.562781	+	0.826606i	$0.309731\pi$
$228$	0	0
$229$	9.70965	0.641631	0.320816	−	0.947142i	$-0.396043\pi$
$229$	9.70965	0.641631	0.320816	+	0.947142i	$0.396043\pi$
$230$	0	0
$231$	28.4487	1.87179
$232$	0	0
$233$	−29.4730	−1.93084	−0.965420	−	0.260701i	$-0.916046\pi$
$233$	−29.4730	−1.93084	−0.965420	+	0.260701i	$0.916046\pi$
$234$	0	0
$235$	−4.24293	−0.276779
$236$	0	0
$237$	23.3014	1.51359
$238$	0	0
$239$	8.18187	0.529241	0.264621	−	0.964353i	$-0.414753\pi$
$239$	8.18187	0.529241	0.264621	+	0.964353i	$0.414753\pi$
$240$	0	0
$241$	−19.2420	−1.23948	−0.619742	−	0.784806i	$-0.712763\pi$
$241$	−19.2420	−1.23948	−0.619742	+	0.784806i	$0.712763\pi$
$242$	0	0
$243$	−6.21471	−0.398674
$244$	0	0
$245$	20.7898	1.32821
$246$	0	0
$247$	19.5247	1.24233
$248$	0	0
$249$	31.5267	1.99793
$250$	0	0
$251$	−2.42671	−0.153173	−0.0765864	−	0.997063i	$-0.524402\pi$
$251$	−2.42671	−0.153173	−0.0765864	+	0.997063i	$0.524402\pi$
$252$	0	0
$253$	−2.72775	−0.171492
$254$	0	0
$255$	22.9648	1.43811
$256$	0	0
$257$	−15.5109	−0.967544	−0.483772	−	0.875194i	$-0.660734\pi$
$257$	−15.5109	−0.967544	−0.483772	+	0.875194i	$0.660734\pi$
$258$	0	0
$259$	−9.81997	−0.610183
$260$	0	0
$261$	30.2563	1.87282
$262$	0	0
$263$	−1.35072	−0.0832891	−0.0416446	−	0.999132i	$-0.513260\pi$
$263$	−1.35072	−0.0832891	−0.0416446	+	0.999132i	$0.513260\pi$
$264$	0	0
$265$	−40.9591	−2.51610
$266$	0	0
$267$	41.7791	2.55684
$268$	0	0
$269$	−5.53476	−0.337460	−0.168730	−	0.985662i	$-0.553967\pi$
$269$	−5.53476	−0.337460	−0.168730	+	0.985662i	$0.553967\pi$
$270$	0	0
$271$	−3.07472	−0.186776	−0.0933881	−	0.995630i	$-0.529770\pi$
$271$	−3.07472	−0.186776	−0.0933881	+	0.995630i	$0.529770\pi$
$272$	0	0
$273$	−26.8755	−1.62658
$274$	0	0
$275$	19.6181	1.18302
$276$	0	0
$277$	0.965160	0.0579908	0.0289954	−	0.999580i	$-0.490769\pi$
$277$	0.965160	0.0579908	0.0289954	+	0.999580i	$0.490769\pi$
$278$	0	0
$279$	−16.6229	−0.995186
$280$	0	0
$281$	11.1746	0.666623	0.333311	−	0.942817i	$-0.391834\pi$
$281$	11.1746	0.666623	0.333311	+	0.942817i	$0.391834\pi$
$282$	0	0
$283$	1.17440	0.0698110	0.0349055	−	0.999391i	$-0.488887\pi$
$283$	1.17440	0.0698110	0.0349055	+	0.999391i	$0.488887\pi$
$284$	0	0
$285$	−78.9580	−4.67707
$286$	0	0
$287$	12.9047	0.761743
$288$	0	0
$289$	−12.0209	−0.707114
$290$	0	0
$291$	−22.6501	−1.32777
$292$	0	0
$293$	−7.19135	−0.420123	−0.210062	−	0.977688i	$-0.567366\pi$
$293$	−7.19135	−0.420123	−0.210062	+	0.977688i	$0.567366\pi$
$294$	0	0
$295$	−12.3290	−0.717823
$296$	0	0
$297$	20.7558	1.20437
$298$	0	0
$299$	2.57691	0.149026
$300$	0	0
$301$	−40.8611	−2.35519
$302$	0	0
$303$	−9.36647	−0.538090
$304$	0	0
$305$	−4.98254	−0.285300
$306$	0	0
$307$	−11.3913	−0.650138	−0.325069	−	0.945690i	$-0.605388\pi$
$307$	−11.3913	−0.650138	−0.325069	+	0.945690i	$0.605388\pi$
$308$	0	0
$309$	41.4193	2.35626
$310$	0	0
$311$	−22.8473	−1.29555	−0.647776	−	0.761831i	$-0.724301\pi$
$311$	−22.8473	−1.29555	−0.647776	+	0.761831i	$0.724301\pi$
$312$	0	0
$313$	21.9858	1.24271	0.621356	−	0.783528i	$-0.286582\pi$
$313$	21.9858	1.24271	0.621356	+	0.783528i	$0.286582\pi$
$314$	0	0
$315$	70.8757	3.99340
$316$	0	0
$317$	19.3667	1.08774	0.543871	−	0.839169i	$-0.316958\pi$
$317$	19.3667	1.08774	0.543871	+	0.839169i	$0.316958\pi$
$318$	0	0
$319$	14.4936	0.811486
$320$	0	0
$321$	45.0149	2.51249
$322$	0	0
$323$	−17.1191	−0.952533
$324$	0	0
$325$	−18.5333	−1.02804
$326$	0	0
$327$	−5.30696	−0.293476
$328$	0	0
$329$	−4.35370	−0.240027
$330$	0	0
$331$	26.0381	1.43118	0.715592	−	0.698519i	$-0.246157\pi$
$331$	26.0381	1.43118	0.715592	+	0.698519i	$0.246157\pi$
$332$	0	0
$333$	−15.3567	−0.841539
$334$	0	0
$335$	−5.14746	−0.281236
$336$	0	0
$337$	10.4605	0.569820	0.284910	−	0.958554i	$-0.408036\pi$
$337$	10.4605	0.569820	0.284910	+	0.958554i	$0.408036\pi$
$338$	0	0
$339$	29.3265	1.59280
$340$	0	0
$341$	−7.96282	−0.431211
$342$	0	0
$343$	−3.84031	−0.207357
$344$	0	0
$345$	−10.4210	−0.561050
$346$	0	0
$347$	−8.17453	−0.438832	−0.219416	−	0.975631i	$-0.570415\pi$
$347$	−8.17453	−0.438832	−0.219416	+	0.975631i	$0.570415\pi$
$348$	0	0
$349$	−6.22532	−0.333234	−0.166617	−	0.986022i	$-0.553284\pi$
$349$	−6.22532	−0.333234	−0.166617	+	0.986022i	$0.553284\pi$
$350$	0	0
$351$	−19.6080	−1.04660
$352$	0	0
$353$	33.8340	1.80080	0.900402	−	0.435060i	$-0.143273\pi$
$353$	33.8340	1.80080	0.900402	+	0.435060i	$0.143273\pi$
$354$	0	0
$355$	−29.4852	−1.56491
$356$	0	0
$357$	23.5643	1.24716
$358$	0	0
$359$	4.15995	0.219554	0.109777	−	0.993956i	$-0.464986\pi$
$359$	4.15995	0.219554	0.109777	+	0.993956i	$0.464986\pi$
$360$	0	0
$361$	39.8594	2.09786
$362$	0	0
$363$	−10.9915	−0.576902
$364$	0	0
$365$	−38.9335	−2.03787
$366$	0	0
$367$	29.0208	1.51488	0.757438	−	0.652907i	$-0.226451\pi$
$367$	29.0208	1.51488	0.757438	+	0.652907i	$0.226451\pi$
$368$	0	0
$369$	20.1807	1.05056
$370$	0	0
$371$	−42.0284	−2.18201
$372$	0	0
$373$	27.5934	1.42873	0.714366	−	0.699773i	$-0.246715\pi$
$373$	27.5934	1.42873	0.714366	+	0.699773i	$0.246715\pi$
$374$	0	0
$375$	23.4902	1.21303
$376$	0	0
$377$	−13.6921	−0.705180
$378$	0	0
$379$	24.1132	1.23861	0.619306	−	0.785150i	$-0.287414\pi$
$379$	24.1132	1.23861	0.619306	+	0.785150i	$0.287414\pi$
$380$	0	0
$381$	−24.1443	−1.23695
$382$	0	0
$383$	−9.50697	−0.485783	−0.242892	−	0.970053i	$-0.578096\pi$
$383$	−9.50697	−0.485783	−0.242892	+	0.970053i	$0.578096\pi$
$384$	0	0
$385$	33.9514	1.73033
$386$	0	0
$387$	−63.8993	−3.24818
$388$	0	0
$389$	3.43073	0.173945	0.0869724	−	0.996211i	$-0.472281\pi$
$389$	3.43073	0.173945	0.0869724	+	0.996211i	$0.472281\pi$
$390$	0	0
$391$	−2.25942	−0.114264
$392$	0	0
$393$	7.02904	0.354568
$394$	0	0
$395$	27.8085	1.39920
$396$	0	0
$397$	38.4129	1.92789	0.963944	−	0.266105i	$-0.0857367\pi$
$397$	38.4129	1.92789	0.963944	+	0.266105i	$0.0857367\pi$
$398$	0	0
$399$	−81.0193	−4.05604
$400$	0	0
$401$	5.64454	0.281875	0.140938	−	0.990018i	$-0.454988\pi$
$401$	5.64454	0.281875	0.140938	+	0.990018i	$0.454988\pi$
$402$	0	0
$403$	7.52249	0.374722
$404$	0	0
$405$	20.1683	1.00217
$406$	0	0
$407$	−7.35626	−0.364636
$408$	0	0
$409$	−8.46598	−0.418616	−0.209308	−	0.977850i	$-0.567121\pi$
$409$	−8.46598	−0.418616	−0.209308	+	0.977850i	$0.567121\pi$
$410$	0	0
$411$	−3.71450	−0.183223
$412$	0	0
$413$	−12.6509	−0.622509
$414$	0	0
$415$	37.6249	1.84693
$416$	0	0
$417$	−39.1704	−1.91818
$418$	0	0
$419$	0.309716	0.0151306	0.00756532	−	0.999971i	$-0.497592\pi$
$419$	0.309716	0.0151306	0.00756532	+	0.999971i	$0.497592\pi$
$420$	0	0
$421$	18.2903	0.891417	0.445709	−	0.895178i	$-0.352952\pi$
$421$	18.2903	0.891417	0.445709	+	0.895178i	$0.352952\pi$
$422$	0	0
$423$	−6.80840	−0.331036
$424$	0	0
$425$	16.2499	0.788235
$426$	0	0
$427$	−5.11262	−0.247417
$428$	0	0
$429$	−20.1328	−0.972020
$430$	0	0
$431$	35.2770	1.69923	0.849616	−	0.527402i	$-0.176834\pi$
$431$	35.2770	1.69923	0.849616	+	0.527402i	$0.176834\pi$
$432$	0	0
$433$	−14.6692	−0.704957	−0.352478	−	0.935820i	$-0.614661\pi$
$433$	−14.6692	−0.704957	−0.352478	+	0.935820i	$0.614661\pi$
$434$	0	0
$435$	55.3711	2.65484
$436$	0	0
$437$	7.76838	0.371612
$438$	0	0
$439$	12.1285	0.578862	0.289431	−	0.957199i	$-0.406534\pi$
$439$	12.1285	0.578862	0.289431	+	0.957199i	$0.406534\pi$
$440$	0	0
$441$	33.3602	1.58858
$442$	0	0
$443$	−15.5872	−0.740568	−0.370284	−	0.928919i	$-0.620740\pi$
$443$	−15.5872	−0.740568	−0.370284	+	0.928919i	$0.620740\pi$
$444$	0	0
$445$	49.8603	2.36361
$446$	0	0
$447$	30.1618	1.42660
$448$	0	0
$449$	−20.3759	−0.961597	−0.480799	−	0.876831i	$-0.659653\pi$
$449$	−20.3759	−0.961597	−0.480799	+	0.876831i	$0.659653\pi$
$450$	0	0
$451$	9.66710	0.455206
$452$	0	0
$453$	−56.0995	−2.63578
$454$	0	0
$455$	−32.0740	−1.50365
$456$	0	0
$457$	30.3165	1.41815	0.709073	−	0.705136i	$-0.249114\pi$
$457$	30.3165	1.41815	0.709073	+	0.705136i	$0.249114\pi$
$458$	0	0
$459$	17.1922	0.802463
$460$	0	0
$461$	−38.7107	−1.80294	−0.901468	−	0.432846i	$-0.857509\pi$
$461$	−38.7107	−1.80294	−0.901468	+	0.432846i	$0.857509\pi$
$462$	0	0
$463$	8.39874	0.390323	0.195161	−	0.980771i	$-0.437477\pi$
$463$	8.39874	0.390323	0.195161	+	0.980771i	$0.437477\pi$
$464$	0	0
$465$	−30.4210	−1.41074
$466$	0	0
$467$	1.32431	0.0612819	0.0306410	−	0.999530i	$-0.490245\pi$
$467$	1.32431	0.0612819	0.0306410	+	0.999530i	$0.490245\pi$
$468$	0	0
$469$	−5.28184	−0.243893
$470$	0	0
$471$	−33.0891	−1.52467
$472$	0	0
$473$	−30.6095	−1.40743
$474$	0	0
$475$	−55.8707	−2.56352
$476$	0	0
$477$	−65.7248	−3.00933
$478$	0	0
$479$	−10.4732	−0.478531	−0.239265	−	0.970954i	$-0.576907\pi$
$479$	−10.4732	−0.478531	−0.239265	+	0.970954i	$0.576907\pi$
$480$	0	0
$481$	6.94947	0.316869
$482$	0	0
$483$	−10.6931	−0.486553
$484$	0	0
$485$	−27.0313	−1.22743
$486$	0	0
$487$	−32.4707	−1.47139	−0.735693	−	0.677315i	$-0.763144\pi$
$487$	−32.4707	−1.47139	−0.735693	+	0.677315i	$0.763144\pi$
$488$	0	0
$489$	−8.40886	−0.380262
$490$	0	0
$491$	13.6299	0.615108	0.307554	−	0.951531i	$-0.400490\pi$
$491$	13.6299	0.615108	0.307554	+	0.951531i	$0.400490\pi$
$492$	0	0
$493$	12.0052	0.540686
$494$	0	0
$495$	53.0939	2.38639
$496$	0	0
$497$	−30.2549	−1.35712
$498$	0	0
$499$	−29.8315	−1.33544	−0.667720	−	0.744413i	$-0.732730\pi$
$499$	−29.8315	−1.33544	−0.667720	+	0.744413i	$0.732730\pi$
$500$	0	0
$501$	−43.5891	−1.94742
$502$	0	0
$503$	19.4135	0.865605	0.432802	−	0.901489i	$-0.357525\pi$
$503$	19.4135	0.865605	0.432802	+	0.901489i	$0.357525\pi$
$504$	0	0
$505$	−11.1782	−0.497423
$506$	0	0
$507$	−19.1565	−0.850768
$508$	0	0
$509$	35.7324	1.58381	0.791905	−	0.610644i	$-0.209089\pi$
$509$	35.7324	1.58381	0.791905	+	0.610644i	$0.209089\pi$
$510$	0	0
$511$	−39.9499	−1.76728
$512$	0	0
$513$	−59.1106	−2.60980
$514$	0	0
$515$	49.4310	2.17819
$516$	0	0
$517$	−3.26141	−0.143437
$518$	0	0
$519$	54.1656	2.37760
$520$	0	0
$521$	27.2806	1.19518	0.597592	−	0.801800i	$-0.296124\pi$
$521$	27.2806	1.19518	0.597592	+	0.801800i	$0.296124\pi$
$522$	0	0
$523$	−6.84286	−0.299217	−0.149609	−	0.988745i	$-0.547801\pi$
$523$	−6.84286	−0.299217	−0.149609	+	0.988745i	$0.547801\pi$
$524$	0	0
$525$	76.9055	3.35643
$526$	0	0
$527$	−6.59567	−0.287312
$528$	0	0
$529$	−21.9747	−0.955422
$530$	0	0
$531$	−19.7837	−0.858538
$532$	0	0
$533$	−9.13253	−0.395574
$534$	0	0
$535$	53.7220	2.32260
$536$	0	0
$537$	49.6380	2.14204
$538$	0	0
$539$	15.9805	0.688328
$540$	0	0
$541$	5.60345	0.240911	0.120455	−	0.992719i	$-0.461564\pi$
$541$	5.60345	0.240911	0.120455	+	0.992719i	$0.461564\pi$
$542$	0	0
$543$	44.0255	1.88932
$544$	0	0
$545$	−6.33347	−0.271296
$546$	0	0
$547$	−26.4025	−1.12889	−0.564445	−	0.825471i	$-0.690910\pi$
$547$	−26.4025	−1.12889	−0.564445	+	0.825471i	$0.690910\pi$
$548$	0	0
$549$	−7.99521	−0.341227
$550$	0	0
$551$	−41.2765	−1.75844
$552$	0	0
$553$	28.5344	1.21341
$554$	0	0
$555$	−28.1038	−1.19294
$556$	0	0
$557$	−5.85798	−0.248211	−0.124105	−	0.992269i	$-0.539606\pi$
$557$	−5.85798	−0.248211	−0.124105	+	0.992269i	$0.539606\pi$
$558$	0	0
$559$	28.9169	1.22305
$560$	0	0
$561$	17.6523	0.745281
$562$	0	0
$563$	15.1862	0.640022	0.320011	−	0.947414i	$-0.396313\pi$
$563$	15.1862	0.640022	0.320011	+	0.947414i	$0.396313\pi$
$564$	0	0
$565$	34.9991	1.47242
$566$	0	0
$567$	20.6948	0.869101
$568$	0	0
$569$	11.0977	0.465238	0.232619	−	0.972568i	$-0.425270\pi$
$569$	11.0977	0.465238	0.232619	+	0.972568i	$0.425270\pi$
$570$	0	0
$571$	−41.6029	−1.74103	−0.870515	−	0.492143i	$-0.836214\pi$
$571$	−41.6029	−1.74103	−0.870515	+	0.492143i	$0.836214\pi$
$572$	0	0
$573$	−1.72758	−0.0721708
$574$	0	0
$575$	−7.37393	−0.307514
$576$	0	0
$577$	32.0854	1.33573	0.667867	−	0.744281i	$-0.267208\pi$
$577$	32.0854	1.33573	0.667867	+	0.744281i	$0.267208\pi$
$578$	0	0
$579$	−16.3583	−0.679829
$580$	0	0
$581$	38.6071	1.60169
$582$	0	0
$583$	−31.4840	−1.30393
$584$	0	0
$585$	−50.1579	−2.07377
$586$	0	0
$587$	42.5966	1.75815	0.879074	−	0.476685i	$-0.158162\pi$
$587$	42.5966	1.75815	0.879074	+	0.476685i	$0.158162\pi$
$588$	0	0
$589$	22.6774	0.934406
$590$	0	0
$591$	77.1919	3.17525
$592$	0	0
$593$	−39.7371	−1.63181	−0.815903	−	0.578189i	$-0.803760\pi$
$593$	−39.7371	−1.63181	−0.815903	+	0.578189i	$0.803760\pi$
$594$	0	0
$595$	28.1223	1.15290
$596$	0	0
$597$	−6.77046	−0.277096
$598$	0	0
$599$	31.5024	1.28715	0.643577	−	0.765381i	$-0.277450\pi$
$599$	31.5024	1.28715	0.643577	+	0.765381i	$0.277450\pi$
$600$	0	0
$601$	−46.6780	−1.90404	−0.952019	−	0.306040i	$-0.900996\pi$
$601$	−46.6780	−1.90404	−0.952019	+	0.306040i	$0.900996\pi$
$602$	0	0
$603$	−8.25984	−0.336367
$604$	0	0
$605$	−13.1175	−0.533302
$606$	0	0
$607$	−4.68374	−0.190107	−0.0950535	−	0.995472i	$-0.530302\pi$
$607$	−4.68374	−0.190107	−0.0950535	+	0.995472i	$0.530302\pi$
$608$	0	0
$609$	56.8166	2.30233
$610$	0	0
$611$	3.08106	0.124646
$612$	0	0
$613$	19.2593	0.777874	0.388937	−	0.921264i	$-0.372842\pi$
$613$	19.2593	0.777874	0.388937	+	0.921264i	$0.372842\pi$
$614$	0	0
$615$	36.9321	1.48924
$616$	0	0
$617$	35.6116	1.43367	0.716834	−	0.697244i	$-0.245591\pi$
$617$	35.6116	1.43367	0.716834	+	0.697244i	$0.245591\pi$
$618$	0	0
$619$	48.6921	1.95710	0.978550	−	0.206012i	$-0.0660485\pi$
$619$	48.6921	1.95710	0.978550	+	0.206012i	$0.0660485\pi$
$620$	0	0
$621$	−7.80154	−0.313065
$622$	0	0
$623$	51.1620	2.04976
$624$	0	0
$625$	−8.37832	−0.335133
$626$	0	0
$627$	−60.6926	−2.42383
$628$	0	0
$629$	−6.09326	−0.242954
$630$	0	0
$631$	−38.5674	−1.53534	−0.767672	−	0.640843i	$-0.778585\pi$
$631$	−38.5674	−1.53534	−0.767672	+	0.640843i	$0.778585\pi$
$632$	0	0
$633$	−39.9118	−1.58635
$634$	0	0
$635$	−28.8144	−1.14347
$636$	0	0
$637$	−15.0968	−0.598156
$638$	0	0
$639$	−47.3132	−1.87168
$640$	0	0
$641$	49.6921	1.96272	0.981360	−	0.192177i	$-0.0615547\pi$
$641$	49.6921	1.96272	0.981360	+	0.192177i	$0.0615547\pi$
$642$	0	0
$643$	26.3290	1.03832	0.519158	−	0.854678i	$-0.326246\pi$
$643$	26.3290	1.03832	0.519158	+	0.854678i	$0.326246\pi$
$644$	0	0
$645$	−116.940	−4.60451
$646$	0	0
$647$	−32.8724	−1.29235	−0.646174	−	0.763190i	$-0.723632\pi$
$647$	−32.8724	−1.29235	−0.646174	+	0.763190i	$0.723632\pi$
$648$	0	0
$649$	−9.47693	−0.372002
$650$	0	0
$651$	−31.2152	−1.22342
$652$	0	0
$653$	47.5574	1.86107	0.930533	−	0.366208i	$-0.119344\pi$
$653$	47.5574	1.86107	0.930533	+	0.366208i	$0.119344\pi$
$654$	0	0
$655$	8.38865	0.327772
$656$	0	0
$657$	−62.4744	−2.43736
$658$	0	0
$659$	29.5009	1.14919	0.574596	−	0.818437i	$-0.305159\pi$
$659$	29.5009	1.14919	0.574596	+	0.818437i	$0.305159\pi$
$660$	0	0
$661$	37.1619	1.44543	0.722715	−	0.691146i	$-0.242894\pi$
$661$	37.1619	1.44543	0.722715	+	0.691146i	$0.242894\pi$
$662$	0	0
$663$	−16.6762	−0.647649
$664$	0	0
$665$	−96.6906	−3.74950
$666$	0	0
$667$	−5.44775	−0.210938
$668$	0	0
$669$	−17.2043	−0.665155
$670$	0	0
$671$	−3.82993	−0.147853
$672$	0	0
$673$	10.7232	0.413350	0.206675	−	0.978410i	$-0.433736\pi$
$673$	10.7232	0.413350	0.206675	+	0.978410i	$0.433736\pi$
$674$	0	0
$675$	56.1092	2.15965
$676$	0	0
$677$	0.0109611	0.000421268 0	0.000210634	−	1.00000i	$-0.499933\pi$
$677$	0.0109611	0.000421268 0	0.000210634		1.00000i	$0.499933\pi$
$678$	0	0
$679$	−27.7370	−1.06445
$680$	0	0
$681$	49.7999	1.90834
$682$	0	0
$683$	−35.0223	−1.34009	−0.670045	−	0.742320i	$-0.733725\pi$
$683$	−35.0223	−1.34009	−0.670045	+	0.742320i	$0.733725\pi$
$684$	0	0
$685$	−4.43298	−0.169376
$686$	0	0
$687$	28.5135	1.08786
$688$	0	0
$689$	29.7430	1.13312
$690$	0	0
$691$	−10.6743	−0.406068	−0.203034	−	0.979172i	$-0.565080\pi$
$691$	−10.6743	−0.406068	−0.203034	+	0.979172i	$0.565080\pi$
$692$	0	0
$693$	54.4800	2.06952
$694$	0	0
$695$	−46.7470	−1.77321
$696$	0	0
$697$	8.00735	0.303300
$698$	0	0
$699$	−86.5507	−3.27365
$700$	0	0
$701$	−4.72867	−0.178599	−0.0892997	−	0.996005i	$-0.528463\pi$
$701$	−4.72867	−0.178599	−0.0892997	+	0.996005i	$0.528463\pi$
$702$	0	0
$703$	20.9500	0.790143
$704$	0	0
$705$	−12.4598	−0.469265
$706$	0	0
$707$	−11.4700	−0.431375
$708$	0	0
$709$	9.31840	0.349960	0.174980	−	0.984572i	$-0.444014\pi$
$709$	9.31840	0.349960	0.174980	+	0.984572i	$0.444014\pi$
$710$	0	0
$711$	44.6227	1.67348
$712$	0	0
$713$	2.99301	0.112089
$714$	0	0
$715$	−24.0270	−0.898559
$716$	0	0
$717$	24.0270	0.897303
$718$	0	0
$719$	30.7922	1.14836	0.574178	−	0.818730i	$-0.305322\pi$
$719$	30.7922	1.14836	0.574178	+	0.818730i	$0.305322\pi$
$720$	0	0
$721$	50.7214	1.88896
$722$	0	0
$723$	−56.5062	−2.10149
$724$	0	0
$725$	39.1806	1.45513
$726$	0	0
$727$	−9.59936	−0.356021	−0.178010	−	0.984029i	$-0.556966\pi$
$727$	−9.59936	−0.356021	−0.178010	+	0.984029i	$0.556966\pi$
$728$	0	0
$729$	−35.5145	−1.31535
$730$	0	0
$731$	−25.3541	−0.937757
$732$	0	0
$733$	−1.09657	−0.0405028	−0.0202514	−	0.999795i	$-0.506447\pi$
$733$	−1.09657	−0.0405028	−0.0202514	+	0.999795i	$0.506447\pi$
$734$	0	0
$735$	61.0516	2.25192
$736$	0	0
$737$	−3.95669	−0.145747
$738$	0	0
$739$	−51.7130	−1.90229	−0.951147	−	0.308738i	$-0.900093\pi$
$739$	−51.7130	−1.90229	−0.951147	+	0.308738i	$0.900093\pi$
$740$	0	0
$741$	57.3364	2.10630
$742$	0	0
$743$	30.2172	1.10856	0.554280	−	0.832330i	$-0.312994\pi$
$743$	30.2172	1.10856	0.554280	+	0.832330i	$0.312994\pi$
$744$	0	0
$745$	35.9958	1.31879
$746$	0	0
$747$	60.3745	2.20899
$748$	0	0
$749$	55.1244	2.01420
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−7.12632	−0.259697
$754$	0	0
$755$	−66.9507	−2.43658
$756$	0	0
$757$	8.78653	0.319352	0.159676	−	0.987169i	$-0.448955\pi$
$757$	8.78653	0.319352	0.159676	+	0.987169i	$0.448955\pi$
$758$	0	0
$759$	−8.01033	−0.290756
$760$	0	0
$761$	15.5283	0.562900	0.281450	−	0.959576i	$-0.409185\pi$
$761$	15.5283	0.562900	0.281450	+	0.959576i	$0.409185\pi$
$762$	0	0
$763$	−6.49881	−0.235273
$764$	0	0
$765$	43.9781	1.59003
$766$	0	0
$767$	8.95287	0.323269
$768$	0	0
$769$	19.3925	0.699311	0.349655	−	0.936878i	$-0.386299\pi$
$769$	19.3925	0.699311	0.349655	+	0.936878i	$0.386299\pi$
$770$	0	0
$771$	−45.5495	−1.64042
$772$	0	0
$773$	−32.4611	−1.16755	−0.583773	−	0.811917i	$-0.698424\pi$
$773$	−32.4611	−1.16755	−0.583773	+	0.811917i	$0.698424\pi$
$774$	0	0
$775$	−21.5259	−0.773234
$776$	0	0
$777$	−28.8374	−1.03454
$778$	0	0
$779$	−27.5310	−0.986402
$780$	0	0
$781$	−22.6643	−0.810994
$782$	0	0
$783$	41.4527	1.48140
$784$	0	0
$785$	−39.4895	−1.40944
$786$	0	0
$787$	−54.2967	−1.93547	−0.967734	−	0.251974i	$-0.918920\pi$
$787$	−54.2967	−1.93547	−0.967734	+	0.251974i	$0.918920\pi$
$788$	0	0
$789$	−3.96655	−0.141213
$790$	0	0
$791$	35.9127	1.27691
$792$	0	0
$793$	3.61814	0.128484
$794$	0	0
$795$	−120.281	−4.26593
$796$	0	0
$797$	44.4918	1.57598	0.787990	−	0.615688i	$-0.211122\pi$
$797$	44.4918	1.57598	0.787990	+	0.615688i	$0.211122\pi$
$798$	0	0
$799$	−2.70146	−0.0955706
$800$	0	0
$801$	80.0081	2.82695
$802$	0	0
$803$	−29.9269	−1.05610
$804$	0	0
$805$	−12.7614	−0.449781
$806$	0	0
$807$	−16.2534	−0.572148
$808$	0	0
$809$	−25.2194	−0.886665	−0.443333	−	0.896357i	$-0.646204\pi$
$809$	−25.2194	−0.886665	−0.443333	+	0.896357i	$0.646204\pi$
$810$	0	0
$811$	−17.9521	−0.630383	−0.315191	−	0.949028i	$-0.602069\pi$
$811$	−17.9521	−0.630383	−0.315191	+	0.949028i	$0.602069\pi$
$812$	0	0
$813$	−9.02927	−0.316670
$814$	0	0
$815$	−10.0354	−0.351523
$816$	0	0
$817$	87.1732	3.04980
$818$	0	0
$819$	−51.4673	−1.79841
$820$	0	0
$821$	25.8491	0.902139	0.451069	−	0.892489i	$-0.351043\pi$
$821$	25.8491	0.902139	0.451069	+	0.892489i	$0.351043\pi$
$822$	0	0
$823$	−0.656538	−0.0228855	−0.0114427	−	0.999935i	$-0.503642\pi$
$823$	−0.656538	−0.0228855	−0.0114427	+	0.999935i	$0.503642\pi$
$824$	0	0
$825$	57.6109	2.00575
$826$	0	0
$827$	−4.37285	−0.152059	−0.0760295	−	0.997106i	$-0.524224\pi$
$827$	−4.37285	−0.152059	−0.0760295	+	0.997106i	$0.524224\pi$
$828$	0	0
$829$	0.371007	0.0128856	0.00644280	−	0.999979i	$-0.497949\pi$
$829$	0.371007	0.0128856	0.00644280	+	0.999979i	$0.497949\pi$
$830$	0	0
$831$	2.83430	0.0983207
$832$	0	0
$833$	13.2368	0.458627
$834$	0	0
$835$	−52.0205	−1.80024
$836$	0	0
$837$	−22.7742	−0.787192
$838$	0	0
$839$	−23.9574	−0.827100	−0.413550	−	0.910481i	$-0.635711\pi$
$839$	−23.9574	−0.827100	−0.413550	+	0.910481i	$0.635711\pi$
$840$	0	0
$841$	−0.0539434	−0.00186012
$842$	0	0
$843$	32.8155	1.13023
$844$	0	0
$845$	−22.8618	−0.786471
$846$	0	0
$847$	−13.4599	−0.462489
$848$	0	0
$849$	3.44876	0.118361
$850$	0	0
$851$	2.76502	0.0947836
$852$	0	0
$853$	33.2099	1.13708	0.568542	−	0.822654i	$-0.307508\pi$
$853$	33.2099	1.13708	0.568542	+	0.822654i	$0.307508\pi$
$854$	0	0
$855$	−151.207	−5.17116
$856$	0	0
$857$	10.7486	0.367165	0.183582	−	0.983004i	$-0.441231\pi$
$857$	10.7486	0.367165	0.183582	+	0.983004i	$0.441231\pi$
$858$	0	0
$859$	16.2265	0.553643	0.276821	−	0.960921i	$-0.410719\pi$
$859$	16.2265	0.553643	0.276821	+	0.960921i	$0.410719\pi$
$860$	0	0
$861$	37.8962	1.29150
$862$	0	0
$863$	39.4940	1.34439	0.672197	−	0.740373i	$-0.265351\pi$
$863$	39.4940	1.34439	0.672197	+	0.740373i	$0.265351\pi$
$864$	0	0
$865$	64.6426	2.19792
$866$	0	0
$867$	−35.3008	−1.19888
$868$	0	0
$869$	21.3755	0.725114
$870$	0	0
$871$	3.73789	0.126654
$872$	0	0
$873$	−43.3756	−1.46804
$874$	0	0
$875$	28.7657	0.972458
$876$	0	0
$877$	−3.12805	−0.105627	−0.0528134	−	0.998604i	$-0.516819\pi$
$877$	−3.12805	−0.105627	−0.0528134	+	0.998604i	$0.516819\pi$
$878$	0	0
$879$	−21.1182	−0.712299
$880$	0	0
$881$	−20.1801	−0.679885	−0.339942	−	0.940446i	$-0.610408\pi$
$881$	−20.1801	−0.679885	−0.339942	+	0.940446i	$0.610408\pi$
$882$	0	0
$883$	−33.9653	−1.14302	−0.571511	−	0.820594i	$-0.693643\pi$
$883$	−33.9653	−1.14302	−0.571511	+	0.820594i	$0.693643\pi$
$884$	0	0
$885$	−36.2055	−1.21704
$886$	0	0
$887$	−20.3478	−0.683212	−0.341606	−	0.939843i	$-0.610971\pi$
$887$	−20.3478	−0.683212	−0.341606	+	0.939843i	$0.610971\pi$
$888$	0	0
$889$	−29.5667	−0.991634
$890$	0	0
$891$	15.5028	0.519362
$892$	0	0
$893$	9.28821	0.310818
$894$	0	0
$895$	59.2394	1.98015
$896$	0	0
$897$	7.56737	0.252667
$898$	0	0
$899$	−15.9030	−0.530396
$900$	0	0
$901$	−26.0785	−0.868800
$902$	0	0
$903$	−119.993	−3.99312
$904$	0	0
$905$	52.5412	1.74653
$906$	0	0
$907$	−6.23956	−0.207181	−0.103591	−	0.994620i	$-0.533033\pi$
$907$	−6.23956	−0.207181	−0.103591	+	0.994620i	$0.533033\pi$
$908$	0	0
$909$	−17.9370	−0.594934
$910$	0	0
$911$	5.34308	0.177024	0.0885120	−	0.996075i	$-0.471789\pi$
$911$	5.34308	0.177024	0.0885120	+	0.996075i	$0.471789\pi$
$912$	0	0
$913$	28.9211	0.957147
$914$	0	0
$915$	−14.6318	−0.483712
$916$	0	0
$917$	8.60764	0.284249
$918$	0	0
$919$	9.09577	0.300042	0.150021	−	0.988683i	$-0.452066\pi$
$919$	9.09577	0.300042	0.150021	+	0.988683i	$0.452066\pi$
$920$	0	0
$921$	−33.4519	−1.10228
$922$	0	0
$923$	21.4110	0.704753
$924$	0	0
$925$	−19.8862	−0.653855
$926$	0	0
$927$	79.3191	2.60518
$928$	0	0
$929$	11.7796	0.386477	0.193238	−	0.981152i	$-0.438101\pi$
$929$	11.7796	0.386477	0.193238	+	0.981152i	$0.438101\pi$
$930$	0	0
$931$	−45.5110	−1.49156
$932$	0	0
$933$	−67.0936	−2.19655
$934$	0	0
$935$	21.0667	0.688956
$936$	0	0
$937$	−30.4723	−0.995487	−0.497743	−	0.867324i	$-0.665838\pi$
$937$	−30.4723	−0.995487	−0.497743	+	0.867324i	$0.665838\pi$
$938$	0	0
$939$	64.5638	2.10696
$940$	0	0
$941$	−10.1624	−0.331285	−0.165642	−	0.986186i	$-0.552970\pi$
$941$	−10.1624	−0.331285	−0.165642	+	0.986186i	$0.552970\pi$
$942$	0	0
$943$	−3.63360	−0.118326
$944$	0	0
$945$	97.1034	3.15877
$946$	0	0
$947$	14.7859	0.480476	0.240238	−	0.970714i	$-0.422775\pi$
$947$	14.7859	0.480476	0.240238	+	0.970714i	$0.422775\pi$
$948$	0	0
$949$	28.2720	0.917749
$950$	0	0
$951$	56.8724	1.84421
$952$	0	0
$953$	−23.8716	−0.773277	−0.386639	−	0.922231i	$-0.626364\pi$
$953$	−23.8716	−0.773277	−0.386639	+	0.922231i	$0.626364\pi$
$954$	0	0
$955$	−2.06174	−0.0667164
$956$	0	0
$957$	42.5620	1.37584
$958$	0	0
$959$	−4.54871	−0.146886
$960$	0	0
$961$	−22.2628	−0.718156
$962$	0	0
$963$	86.2046	2.77790
$964$	0	0
$965$	−19.5225	−0.628451
$966$	0	0
$967$	−14.7395	−0.473990	−0.236995	−	0.971511i	$-0.576162\pi$
$967$	−14.7395	−0.473990	−0.236995	+	0.971511i	$0.576162\pi$
$968$	0	0
$969$	−50.2722	−1.61498
$970$	0	0
$971$	−49.4852	−1.58806	−0.794028	−	0.607881i	$-0.792020\pi$
$971$	−49.4852	−1.58806	−0.794028	+	0.607881i	$0.792020\pi$
$972$	0	0
$973$	−47.9674	−1.53776
$974$	0	0
$975$	−54.4251	−1.74300
$976$	0	0
$977$	−50.8293	−1.62617	−0.813087	−	0.582143i	$-0.802215\pi$
$977$	−50.8293	−1.62617	−0.813087	+	0.582143i	$0.802215\pi$
$978$	0	0
$979$	38.3261	1.22491
$980$	0	0
$981$	−10.1630	−0.324478
$982$	0	0
$983$	27.9347	0.890978	0.445489	−	0.895287i	$-0.353030\pi$
$983$	27.9347	0.890978	0.445489	+	0.895287i	$0.353030\pi$
$984$	0	0
$985$	92.1228	2.93528
$986$	0	0
$987$	−12.7851	−0.406955
$988$	0	0
$989$	11.5053	0.365847
$990$	0	0
$991$	33.0473	1.04978	0.524891	−	0.851170i	$-0.324106\pi$
$991$	33.0473	1.04978	0.524891	+	0.851170i	$0.324106\pi$
$992$	0	0
$993$	76.4637	2.42650
$994$	0	0
$995$	−8.08005	−0.256155
$996$	0	0
$997$	32.0785	1.01594	0.507968	−	0.861376i	$-0.330397\pi$
$997$	32.0785	1.01594	0.507968	+	0.861376i	$0.330397\pi$
$998$	0	0
$999$	−21.0394	−0.665657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.45	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.45	✓	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)

\(f(q)\)	\(=\)	\(q+2.93661 q^{3} +3.50463 q^{5} +3.59612 q^{7} +5.62368 q^{9} +O(q^{10})\)	\(q+2.93661 q^{3} +3.50463 q^{5} +3.59612 q^{7} +5.62368 q^{9} +2.69390 q^{11} -2.54493 q^{13} +10.2917 q^{15} +2.23138 q^{17} -7.67199 q^{19} +10.5604 q^{21} -1.01256 q^{23} +7.28243 q^{25} +7.70474 q^{27} +5.38015 q^{29} -2.95587 q^{31} +7.91094 q^{33} +12.6031 q^{35} -2.73071 q^{37} -7.47347 q^{39} +3.58852 q^{41} -11.3625 q^{43} +19.7089 q^{45} -1.21067 q^{47} +5.93210 q^{49} +6.55270 q^{51} -11.6872 q^{53} +9.44112 q^{55} -22.5296 q^{57} -3.51792 q^{59} -1.42170 q^{61} +20.2235 q^{63} -8.91905 q^{65} -1.46876 q^{67} -2.97351 q^{69} -8.41321 q^{71} -11.1092 q^{73} +21.3857 q^{75} +9.68759 q^{77} +7.93478 q^{79} +5.75476 q^{81} +10.7358 q^{83} +7.82016 q^{85} +15.7994 q^{87} +14.2270 q^{89} -9.15189 q^{91} -8.68024 q^{93} -26.8875 q^{95} -7.71302 q^{97} +15.1496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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