LMFDB - Embedded newform 6008.2.a.b.1.22

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			1.22
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)

$f(q)$	$=$	$q-0.447974 q^{3} +2.48474 q^{5} +3.20909 q^{7} -2.79932 q^{9} +O(q^{10})$	$q-0.447974 q^{3} +2.48474 q^{5} +3.20909 q^{7} -2.79932 q^{9} -0.332333 q^{11} -4.86212 q^{13} -1.11310 q^{15} -4.61386 q^{17} -7.39401 q^{19} -1.43759 q^{21} +5.99857 q^{23} +1.17393 q^{25} +2.59794 q^{27} +8.23188 q^{29} +7.16322 q^{31} +0.148877 q^{33} +7.97375 q^{35} -5.58586 q^{37} +2.17810 q^{39} -3.18131 q^{41} +3.29032 q^{43} -6.95558 q^{45} -9.99363 q^{47} +3.29825 q^{49} +2.06689 q^{51} -2.53894 q^{53} -0.825761 q^{55} +3.31233 q^{57} -5.30087 q^{59} +8.55552 q^{61} -8.98326 q^{63} -12.0811 q^{65} -5.87268 q^{67} -2.68721 q^{69} -13.6528 q^{71} -2.82965 q^{73} -0.525891 q^{75} -1.06649 q^{77} -16.1660 q^{79} +7.23415 q^{81} +13.0448 q^{83} -11.4642 q^{85} -3.68767 q^{87} +6.60233 q^{89} -15.6030 q^{91} -3.20894 q^{93} -18.3722 q^{95} +12.8090 q^{97} +0.930306 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.447974	−0.258638	−0.129319	−	0.991603i	$-0.541279\pi$
$3$	−0.447974	−0.258638	−0.129319	+	0.991603i	$0.541279\pi$
$4$	0	0
$5$	2.48474	1.11121	0.555605	−	0.831447i	$-0.312487\pi$
$5$	2.48474	1.11121	0.555605	+	0.831447i	$0.312487\pi$
$6$	0	0
$7$	3.20909	1.21292	0.606461	−	0.795113i	$-0.292589\pi$
$7$	3.20909	1.21292	0.606461	+	0.795113i	$0.292589\pi$
$8$	0	0
$9$	−2.79932	−0.933106
$10$	0	0
$11$	−0.332333	−0.100202	−0.0501011	−	0.998744i	$-0.515954\pi$
$11$	−0.332333	−0.100202	−0.0501011	+	0.998744i	$0.515954\pi$
$12$	0	0
$13$	−4.86212	−1.34851	−0.674254	−	0.738499i	$-0.735535\pi$
$13$	−4.86212	−1.34851	−0.674254	+	0.738499i	$0.735535\pi$
$14$	0	0
$15$	−1.11310	−0.287401
$16$	0	0
$17$	−4.61386	−1.11903	−0.559513	−	0.828822i	$-0.689012\pi$
$17$	−4.61386	−1.11903	−0.559513	+	0.828822i	$0.689012\pi$
$18$	0	0
$19$	−7.39401	−1.69630	−0.848151	−	0.529754i	$-0.822284\pi$
$19$	−7.39401	−1.69630	−0.848151	+	0.529754i	$0.822284\pi$
$20$	0	0
$21$	−1.43759	−0.313708
$22$	0	0
$23$	5.99857	1.25079	0.625395	−	0.780309i	$-0.284938\pi$
$23$	5.99857	1.25079	0.625395	+	0.780309i	$0.284938\pi$
$24$	0	0
$25$	1.17393	0.234786
$26$	0	0
$27$	2.59794	0.499975
$28$	0	0
$29$	8.23188	1.52862	0.764310	−	0.644848i	$-0.223080\pi$
$29$	8.23188	1.52862	0.764310	+	0.644848i	$0.223080\pi$
$30$	0	0
$31$	7.16322	1.28655	0.643276	−	0.765634i	$-0.277575\pi$
$31$	7.16322	1.28655	0.643276	+	0.765634i	$0.277575\pi$
$32$	0	0
$33$	0.148877	0.0259161
$34$	0	0
$35$	7.97375	1.34781
$36$	0	0
$37$	−5.58586	−0.918309	−0.459155	−	0.888356i	$-0.651848\pi$
$37$	−5.58586	−0.918309	−0.459155	+	0.888356i	$0.651848\pi$
$38$	0	0
$39$	2.17810	0.348776
$40$	0	0
$41$	−3.18131	−0.496837	−0.248418	−	0.968653i	$-0.579911\pi$
$41$	−3.18131	−0.496837	−0.248418	+	0.968653i	$0.579911\pi$
$42$	0	0
$43$	3.29032	0.501769	0.250884	−	0.968017i	$-0.419279\pi$
$43$	3.29032	0.501769	0.250884	+	0.968017i	$0.419279\pi$
$44$	0	0
$45$	−6.95558	−1.03688
$46$	0	0
$47$	−9.99363	−1.45772	−0.728861	−	0.684662i	$-0.759950\pi$
$47$	−9.99363	−1.45772	−0.728861	+	0.684662i	$0.759950\pi$
$48$	0	0
$49$	3.29825	0.471179
$50$	0	0
$51$	2.06689	0.289423
$52$	0	0
$53$	−2.53894	−0.348750	−0.174375	−	0.984679i	$-0.555790\pi$
$53$	−2.53894	−0.348750	−0.174375	+	0.984679i	$0.555790\pi$
$54$	0	0
$55$	−0.825761	−0.111346
$56$	0	0
$57$	3.31233	0.438728
$58$	0	0
$59$	−5.30087	−0.690115	−0.345057	−	0.938582i	$-0.612141\pi$
$59$	−5.30087	−0.690115	−0.345057	+	0.938582i	$0.612141\pi$
$60$	0	0
$61$	8.55552	1.09542	0.547711	−	0.836668i	$-0.315499\pi$
$61$	8.55552	1.09542	0.547711	+	0.836668i	$0.315499\pi$
$62$	0	0
$63$	−8.98326	−1.13178
$64$	0	0
$65$	−12.0811	−1.49848
$66$	0	0
$67$	−5.87268	−0.717462	−0.358731	−	0.933441i	$-0.616790\pi$
$67$	−5.87268	−0.717462	−0.358731	+	0.933441i	$0.616790\pi$
$68$	0	0
$69$	−2.68721	−0.323502
$70$	0	0
$71$	−13.6528	−1.62029	−0.810147	−	0.586226i	$-0.800613\pi$
$71$	−13.6528	−1.62029	−0.810147	+	0.586226i	$0.800613\pi$
$72$	0	0
$73$	−2.82965	−0.331185	−0.165593	−	0.986194i	$-0.552954\pi$
$73$	−2.82965	−0.331185	−0.165593	+	0.986194i	$0.552954\pi$
$74$	0	0
$75$	−0.525891	−0.0607246
$76$	0	0
$77$	−1.06649	−0.121537
$78$	0	0
$79$	−16.1660	−1.81881	−0.909407	−	0.415907i	$-0.863464\pi$
$79$	−16.1660	−1.81881	−0.909407	+	0.415907i	$0.863464\pi$
$80$	0	0
$81$	7.23415	0.803794
$82$	0	0
$83$	13.0448	1.43185	0.715927	−	0.698175i	$-0.246004\pi$
$83$	13.0448	1.43185	0.715927	+	0.698175i	$0.246004\pi$
$84$	0	0
$85$	−11.4642	−1.24347
$86$	0	0
$87$	−3.68767	−0.395359
$88$	0	0
$89$	6.60233	0.699845	0.349923	−	0.936779i	$-0.386208\pi$
$89$	6.60233	0.699845	0.349923	+	0.936779i	$0.386208\pi$
$90$	0	0
$91$	−15.6030	−1.63564
$92$	0	0
$93$	−3.20894	−0.332751
$94$	0	0
$95$	−18.3722	−1.88495
$96$	0	0
$97$	12.8090	1.30055	0.650277	−	0.759697i	$-0.274653\pi$
$97$	12.8090	1.30055	0.650277	+	0.759697i	$0.274653\pi$
$98$	0	0
$99$	0.930306	0.0934993
$100$	0	0
$101$	−15.8727	−1.57940	−0.789699	−	0.613495i	$-0.789763\pi$
$101$	−15.8727	−1.57940	−0.789699	+	0.613495i	$0.789763\pi$
$102$	0	0
$103$	−6.96888	−0.686664	−0.343332	−	0.939214i	$-0.611556\pi$
$103$	−6.96888	−0.686664	−0.343332	+	0.939214i	$0.611556\pi$
$104$	0	0
$105$	−3.57203	−0.348595
$106$	0	0
$107$	−20.1653	−1.94945	−0.974725	−	0.223408i	$-0.928282\pi$
$107$	−20.1653	−1.94945	−0.974725	+	0.223408i	$0.928282\pi$
$108$	0	0
$109$	4.23463	0.405604	0.202802	−	0.979220i	$-0.434995\pi$
$109$	4.23463	0.405604	0.202802	+	0.979220i	$0.434995\pi$
$110$	0	0
$111$	2.50232	0.237510
$112$	0	0
$113$	−13.7102	−1.28974	−0.644871	−	0.764291i	$-0.723089\pi$
$113$	−13.7102	−1.28974	−0.644871	+	0.764291i	$0.723089\pi$
$114$	0	0
$115$	14.9049	1.38989
$116$	0	0
$117$	13.6106	1.25830
$118$	0	0
$119$	−14.8063	−1.35729
$120$	0	0
$121$	−10.8896	−0.989960
$122$	0	0
$123$	1.42514	0.128501
$124$	0	0
$125$	−9.50679	−0.850313
$126$	0	0
$127$	6.35585	0.563990	0.281995	−	0.959416i	$-0.409004\pi$
$127$	6.35585	0.563990	0.281995	+	0.959416i	$0.409004\pi$
$128$	0	0
$129$	−1.47398	−0.129777
$130$	0	0
$131$	−7.70378	−0.673082	−0.336541	−	0.941669i	$-0.609257\pi$
$131$	−7.70378	−0.673082	−0.336541	+	0.941669i	$0.609257\pi$
$132$	0	0
$133$	−23.7280	−2.05748
$134$	0	0
$135$	6.45522	0.555577
$136$	0	0
$137$	−7.49056	−0.639962	−0.319981	−	0.947424i	$-0.603677\pi$
$137$	−7.49056	−0.639962	−0.319981	+	0.947424i	$0.603677\pi$
$138$	0	0
$139$	6.23746	0.529054	0.264527	−	0.964378i	$-0.414784\pi$
$139$	6.23746	0.529054	0.264527	+	0.964378i	$0.414784\pi$
$140$	0	0
$141$	4.47689	0.377022
$142$	0	0
$143$	1.61584	0.135124
$144$	0	0
$145$	20.4541	1.69862
$146$	0	0
$147$	−1.47753	−0.121865
$148$	0	0
$149$	5.46677	0.447855	0.223928	−	0.974606i	$-0.428112\pi$
$149$	5.46677	0.447855	0.223928	+	0.974606i	$0.428112\pi$
$150$	0	0
$151$	5.74213	0.467288	0.233644	−	0.972322i	$-0.424935\pi$
$151$	5.74213	0.467288	0.233644	+	0.972322i	$0.424935\pi$
$152$	0	0
$153$	12.9157	1.04417
$154$	0	0
$155$	17.7987	1.42963
$156$	0	0
$157$	5.05111	0.403122	0.201561	−	0.979476i	$-0.435399\pi$
$157$	5.05111	0.403122	0.201561	+	0.979476i	$0.435399\pi$
$158$	0	0
$159$	1.13738	0.0901999
$160$	0	0
$161$	19.2500	1.51711
$162$	0	0
$163$	2.48335	0.194511	0.0972554	−	0.995259i	$-0.468994\pi$
$163$	2.48335	0.194511	0.0972554	+	0.995259i	$0.468994\pi$
$164$	0	0
$165$	0.369920	0.0287982
$166$	0	0
$167$	−13.6806	−1.05864	−0.529318	−	0.848423i	$-0.677552\pi$
$167$	−13.6806	−1.05864	−0.529318	+	0.848423i	$0.677552\pi$
$168$	0	0
$169$	10.6402	0.818476
$170$	0	0
$171$	20.6982	1.58283
$172$	0	0
$173$	2.89798	0.220330	0.110165	−	0.993913i	$-0.464862\pi$
$173$	2.89798	0.220330	0.110165	+	0.993913i	$0.464862\pi$
$174$	0	0
$175$	3.76725	0.284777
$176$	0	0
$177$	2.37465	0.178490
$178$	0	0
$179$	−6.37568	−0.476541	−0.238271	−	0.971199i	$-0.576580\pi$
$179$	−6.37568	−0.476541	−0.238271	+	0.971199i	$0.576580\pi$
$180$	0	0
$181$	0.982308	0.0730144	0.0365072	−	0.999333i	$-0.488377\pi$
$181$	0.982308	0.0730144	0.0365072	+	0.999333i	$0.488377\pi$
$182$	0	0
$183$	−3.83265	−0.283318
$184$	0	0
$185$	−13.8794	−1.02043
$186$	0	0
$187$	1.53334	0.112129
$188$	0	0
$189$	8.33704	0.606430
$190$	0	0
$191$	−12.9651	−0.938121	−0.469060	−	0.883166i	$-0.655407\pi$
$191$	−12.9651	−0.938121	−0.469060	+	0.883166i	$0.655407\pi$
$192$	0	0
$193$	−12.8917	−0.927965	−0.463983	−	0.885844i	$-0.653580\pi$
$193$	−12.8917	−0.927965	−0.463983	+	0.885844i	$0.653580\pi$
$194$	0	0
$195$	5.41202	0.387563
$196$	0	0
$197$	24.5782	1.75113	0.875564	−	0.483103i	$-0.160490\pi$
$197$	24.5782	1.75113	0.875564	+	0.483103i	$0.160490\pi$
$198$	0	0
$199$	−22.1065	−1.56709	−0.783543	−	0.621338i	$-0.786590\pi$
$199$	−22.1065	−1.56709	−0.783543	+	0.621338i	$0.786590\pi$
$200$	0	0
$201$	2.63081	0.185563
$202$	0	0
$203$	26.4168	1.85410
$204$	0	0
$205$	−7.90472	−0.552090
$206$	0	0
$207$	−16.7919	−1.16712
$208$	0	0
$209$	2.45727	0.169973
$210$	0	0
$211$	18.9156	1.30221	0.651103	−	0.758989i	$-0.274307\pi$
$211$	18.9156	1.30221	0.651103	+	0.758989i	$0.274307\pi$
$212$	0	0
$213$	6.11612	0.419070
$214$	0	0
$215$	8.17559	0.557570
$216$	0	0
$217$	22.9874	1.56049
$218$	0	0
$219$	1.26761	0.0856571
$220$	0	0
$221$	22.4331	1.50902
$222$	0	0
$223$	−9.30269	−0.622954	−0.311477	−	0.950254i	$-0.600824\pi$
$223$	−9.30269	−0.622954	−0.311477	+	0.950254i	$0.600824\pi$
$224$	0	0
$225$	−3.28621	−0.219080
$226$	0	0
$227$	−10.5996	−0.703521	−0.351760	−	0.936090i	$-0.614417\pi$
$227$	−10.5996	−0.703521	−0.351760	+	0.936090i	$0.614417\pi$
$228$	0	0
$229$	10.5095	0.694484	0.347242	−	0.937775i	$-0.387118\pi$
$229$	10.5095	0.694484	0.347242	+	0.937775i	$0.387118\pi$
$230$	0	0
$231$	0.477758	0.0314342
$232$	0	0
$233$	15.1138	0.990138	0.495069	−	0.868854i	$-0.335143\pi$
$233$	15.1138	0.990138	0.495069	+	0.868854i	$0.335143\pi$
$234$	0	0
$235$	−24.8316	−1.61983
$236$	0	0
$237$	7.24194	0.470415
$238$	0	0
$239$	9.41488	0.608998	0.304499	−	0.952513i	$-0.401511\pi$
$239$	9.41488	0.608998	0.304499	+	0.952513i	$0.401511\pi$
$240$	0	0
$241$	25.2390	1.62578	0.812892	−	0.582414i	$-0.197892\pi$
$241$	25.2390	1.62578	0.812892	+	0.582414i	$0.197892\pi$
$242$	0	0
$243$	−11.0345	−0.707866
$244$	0	0
$245$	8.19529	0.523578
$246$	0	0
$247$	35.9506	2.28748
$248$	0	0
$249$	−5.84374	−0.370332
$250$	0	0
$251$	20.1843	1.27402	0.637012	−	0.770854i	$-0.280170\pi$
$251$	20.1843	1.27402	0.637012	+	0.770854i	$0.280170\pi$
$252$	0	0
$253$	−1.99352	−0.125332
$254$	0	0
$255$	5.13568	0.321609
$256$	0	0
$257$	−14.7762	−0.921711	−0.460856	−	0.887475i	$-0.652457\pi$
$257$	−14.7762	−0.921711	−0.460856	+	0.887475i	$0.652457\pi$
$258$	0	0
$259$	−17.9255	−1.11384
$260$	0	0
$261$	−23.0436	−1.42637
$262$	0	0
$263$	−22.3298	−1.37692	−0.688458	−	0.725276i	$-0.741712\pi$
$263$	−22.3298	−1.37692	−0.688458	+	0.725276i	$0.741712\pi$
$264$	0	0
$265$	−6.30859	−0.387534
$266$	0	0
$267$	−2.95767	−0.181007
$268$	0	0
$269$	2.74236	0.167205	0.0836023	−	0.996499i	$-0.473357\pi$
$269$	2.74236	0.167205	0.0836023	+	0.996499i	$0.473357\pi$
$270$	0	0
$271$	−21.3192	−1.29505	−0.647525	−	0.762044i	$-0.724196\pi$
$271$	−21.3192	−1.29505	−0.647525	+	0.762044i	$0.724196\pi$
$272$	0	0
$273$	6.98973	0.423037
$274$	0	0
$275$	−0.390136	−0.0235261
$276$	0	0
$277$	−0.644053	−0.0386974	−0.0193487	−	0.999813i	$-0.506159\pi$
$277$	−0.644053	−0.0386974	−0.0193487	+	0.999813i	$0.506159\pi$
$278$	0	0
$279$	−20.0521	−1.20049
$280$	0	0
$281$	−7.38031	−0.440272	−0.220136	−	0.975469i	$-0.570650\pi$
$281$	−7.38031	−0.440272	−0.220136	+	0.975469i	$0.570650\pi$
$282$	0	0
$283$	−7.22725	−0.429615	−0.214808	−	0.976656i	$-0.568912\pi$
$283$	−7.22725	−0.429615	−0.214808	+	0.976656i	$0.568912\pi$
$284$	0	0
$285$	8.23027	0.487519
$286$	0	0
$287$	−10.2091	−0.602624
$288$	0	0
$289$	4.28771	0.252218
$290$	0	0
$291$	−5.73809	−0.336373
$292$	0	0
$293$	−32.9084	−1.92253	−0.961265	−	0.275626i	$-0.911115\pi$
$293$	−32.9084	−1.92253	−0.961265	+	0.275626i	$0.911115\pi$
$294$	0	0
$295$	−13.1713	−0.766862
$296$	0	0
$297$	−0.863383	−0.0500986
$298$	0	0
$299$	−29.1658	−1.68670
$300$	0	0
$301$	10.5589	0.608606
$302$	0	0
$303$	7.11058	0.408492
$304$	0	0
$305$	21.2582	1.21724
$306$	0	0
$307$	0.389674	0.0222398	0.0111199	−	0.999938i	$-0.496460\pi$
$307$	0.389674	0.0222398	0.0111199	+	0.999938i	$0.496460\pi$
$308$	0	0
$309$	3.12188	0.177598
$310$	0	0
$311$	0.0938662	0.00532267	0.00266133	−	0.999996i	$-0.499153\pi$
$311$	0.0938662	0.00532267	0.00266133	+	0.999996i	$0.499153\pi$
$312$	0	0
$313$	9.20154	0.520102	0.260051	−	0.965595i	$-0.416261\pi$
$313$	9.20154	0.520102	0.260051	+	0.965595i	$0.416261\pi$
$314$	0	0
$315$	−22.3211	−1.25765
$316$	0	0
$317$	0.962046	0.0540339	0.0270170	−	0.999635i	$-0.491399\pi$
$317$	0.962046	0.0540339	0.0270170	+	0.999635i	$0.491399\pi$
$318$	0	0
$319$	−2.73572	−0.153171
$320$	0	0
$321$	9.03352	0.504202
$322$	0	0
$323$	34.1150	1.89821
$324$	0	0
$325$	−5.70779	−0.316611
$326$	0	0
$327$	−1.89700	−0.104905
$328$	0	0
$329$	−32.0705	−1.76810
$330$	0	0
$331$	−32.2189	−1.77091	−0.885455	−	0.464725i	$-0.846153\pi$
$331$	−32.2189	−1.77091	−0.885455	+	0.464725i	$0.846153\pi$
$332$	0	0
$333$	15.6366	0.856880
$334$	0	0
$335$	−14.5921	−0.797250
$336$	0	0
$337$	−28.5432	−1.55485	−0.777425	−	0.628976i	$-0.783474\pi$
$337$	−28.5432	−1.55485	−0.777425	+	0.628976i	$0.783474\pi$
$338$	0	0
$339$	6.14179	0.333576
$340$	0	0
$341$	−2.38058	−0.128915
$342$	0	0
$343$	−11.8792	−0.641419
$344$	0	0
$345$	−6.67701	−0.359478
$346$	0	0
$347$	−7.10425	−0.381376	−0.190688	−	0.981651i	$-0.561072\pi$
$347$	−7.10425	−0.381376	−0.190688	+	0.981651i	$0.561072\pi$
$348$	0	0
$349$	−20.2532	−1.08413	−0.542065	−	0.840336i	$-0.682358\pi$
$349$	−20.2532	−1.08413	−0.542065	+	0.840336i	$0.682358\pi$
$350$	0	0
$351$	−12.6315	−0.674220
$352$	0	0
$353$	17.3628	0.924130	0.462065	−	0.886846i	$-0.347109\pi$
$353$	17.3628	0.924130	0.462065	+	0.886846i	$0.347109\pi$
$354$	0	0
$355$	−33.9238	−1.80049
$356$	0	0
$357$	6.63283	0.351047
$358$	0	0
$359$	−2.05041	−0.108217	−0.0541083	−	0.998535i	$-0.517232\pi$
$359$	−2.05041	−0.108217	−0.0541083	+	0.998535i	$0.517232\pi$
$360$	0	0
$361$	35.6714	1.87744
$362$	0	0
$363$	4.87824	0.256041
$364$	0	0
$365$	−7.03094	−0.368016
$366$	0	0
$367$	−6.16599	−0.321862	−0.160931	−	0.986966i	$-0.551450\pi$
$367$	−6.16599	−0.321862	−0.160931	+	0.986966i	$0.551450\pi$
$368$	0	0
$369$	8.90550	0.463602
$370$	0	0
$371$	−8.14767	−0.423006
$372$	0	0
$373$	3.30418	0.171084	0.0855419	−	0.996335i	$-0.472738\pi$
$373$	3.30418	0.171084	0.0855419	+	0.996335i	$0.472738\pi$
$374$	0	0
$375$	4.25879	0.219923
$376$	0	0
$377$	−40.0244	−2.06136
$378$	0	0
$379$	9.91790	0.509448	0.254724	−	0.967014i	$-0.418015\pi$
$379$	9.91790	0.509448	0.254724	+	0.967014i	$0.418015\pi$
$380$	0	0
$381$	−2.84726	−0.145869
$382$	0	0
$383$	26.6968	1.36414	0.682072	−	0.731285i	$-0.261079\pi$
$383$	26.6968	1.36414	0.682072	+	0.731285i	$0.261079\pi$
$384$	0	0
$385$	−2.64994	−0.135053
$386$	0	0
$387$	−9.21065	−0.468204
$388$	0	0
$389$	21.5226	1.09124	0.545620	−	0.838033i	$-0.316294\pi$
$389$	21.5226	1.09124	0.545620	+	0.838033i	$0.316294\pi$
$390$	0	0
$391$	−27.6766	−1.39967
$392$	0	0
$393$	3.45109	0.174085
$394$	0	0
$395$	−40.1682	−2.02108
$396$	0	0
$397$	−26.5816	−1.33409	−0.667047	−	0.745016i	$-0.732442\pi$
$397$	−26.5816	−1.33409	−0.667047	+	0.745016i	$0.732442\pi$
$398$	0	0
$399$	10.6295	0.532143
$400$	0	0
$401$	31.5167	1.57387	0.786934	−	0.617037i	$-0.211667\pi$
$401$	31.5167	1.57387	0.786934	+	0.617037i	$0.211667\pi$
$402$	0	0
$403$	−34.8284	−1.73493
$404$	0	0
$405$	17.9750	0.893183
$406$	0	0
$407$	1.85637	0.0920166
$408$	0	0
$409$	−15.7098	−0.776797	−0.388399	−	0.921491i	$-0.626972\pi$
$409$	−15.7098	−0.776797	−0.388399	+	0.921491i	$0.626972\pi$
$410$	0	0
$411$	3.35558	0.165518
$412$	0	0
$413$	−17.0110	−0.837055
$414$	0	0
$415$	32.4130	1.59109
$416$	0	0
$417$	−2.79422	−0.136834
$418$	0	0
$419$	−14.7265	−0.719436	−0.359718	−	0.933061i	$-0.617127\pi$
$419$	−14.7265	−0.719436	−0.359718	+	0.933061i	$0.617127\pi$
$420$	0	0
$421$	18.5655	0.904828	0.452414	−	0.891808i	$-0.350563\pi$
$421$	18.5655	0.904828	0.452414	+	0.891808i	$0.350563\pi$
$422$	0	0
$423$	27.9754	1.36021
$424$	0	0
$425$	−5.41635	−0.262732
$426$	0	0
$427$	27.4554	1.32866
$428$	0	0
$429$	−0.723856	−0.0349481
$430$	0	0
$431$	8.88531	0.427990	0.213995	−	0.976835i	$-0.431352\pi$
$431$	8.88531	0.427990	0.213995	+	0.976835i	$0.431352\pi$
$432$	0	0
$433$	24.3929	1.17225	0.586123	−	0.810222i	$-0.300653\pi$
$433$	24.3929	1.17225	0.586123	+	0.810222i	$0.300653\pi$
$434$	0	0
$435$	−9.16289	−0.439327
$436$	0	0
$437$	−44.3535	−2.12172
$438$	0	0
$439$	23.7205	1.13212	0.566058	−	0.824365i	$-0.308468\pi$
$439$	23.7205	1.13212	0.566058	+	0.824365i	$0.308468\pi$
$440$	0	0
$441$	−9.23285	−0.439660
$442$	0	0
$443$	35.9712	1.70904	0.854521	−	0.519417i	$-0.173851\pi$
$443$	35.9712	1.70904	0.854521	+	0.519417i	$0.173851\pi$
$444$	0	0
$445$	16.4051	0.777675
$446$	0	0
$447$	−2.44897	−0.115832
$448$	0	0
$449$	8.95175	0.422459	0.211230	−	0.977436i	$-0.432253\pi$
$449$	8.95175	0.422459	0.211230	+	0.977436i	$0.432253\pi$
$450$	0	0
$451$	1.05725	0.0497841
$452$	0	0
$453$	−2.57233	−0.120858
$454$	0	0
$455$	−38.7693	−1.81753
$456$	0	0
$457$	−21.2645	−0.994713	−0.497357	−	0.867546i	$-0.665696\pi$
$457$	−21.2645	−0.994713	−0.497357	+	0.867546i	$0.665696\pi$
$458$	0	0
$459$	−11.9866	−0.559485
$460$	0	0
$461$	2.06005	0.0959462	0.0479731	−	0.998849i	$-0.484724\pi$
$461$	2.06005	0.0959462	0.0479731	+	0.998849i	$0.484724\pi$
$462$	0	0
$463$	4.69056	0.217989	0.108994	−	0.994042i	$-0.465237\pi$
$463$	4.69056	0.217989	0.108994	+	0.994042i	$0.465237\pi$
$464$	0	0
$465$	−7.97338	−0.369756
$466$	0	0
$467$	33.8156	1.56480	0.782401	−	0.622775i	$-0.213995\pi$
$467$	33.8156	1.56480	0.782401	+	0.622775i	$0.213995\pi$
$468$	0	0
$469$	−18.8459	−0.870225
$470$	0	0
$471$	−2.26276	−0.104263
$472$	0	0
$473$	−1.09348	−0.0502783
$474$	0	0
$475$	−8.68006	−0.398268
$476$	0	0
$477$	7.10729	0.325421
$478$	0	0
$479$	37.1794	1.69877	0.849385	−	0.527773i	$-0.176973\pi$
$479$	37.1794	1.69877	0.849385	+	0.527773i	$0.176973\pi$
$480$	0	0
$481$	27.1591	1.23835
$482$	0	0
$483$	−8.62348	−0.392382
$484$	0	0
$485$	31.8270	1.44519
$486$	0	0
$487$	10.5261	0.476982	0.238491	−	0.971145i	$-0.423347\pi$
$487$	10.5261	0.476982	0.238491	+	0.971145i	$0.423347\pi$
$488$	0	0
$489$	−1.11248	−0.0503079
$490$	0	0
$491$	39.4399	1.77990	0.889948	−	0.456062i	$-0.150740\pi$
$491$	39.4399	1.77990	0.889948	+	0.456062i	$0.150740\pi$
$492$	0	0
$493$	−37.9807	−1.71057
$494$	0	0
$495$	2.31157	0.103897
$496$	0	0
$497$	−43.8132	−1.96529
$498$	0	0
$499$	3.30276	0.147852	0.0739259	−	0.997264i	$-0.476447\pi$
$499$	3.30276	0.147852	0.0739259	+	0.997264i	$0.476447\pi$
$500$	0	0
$501$	6.12855	0.273804
$502$	0	0
$503$	19.7041	0.878564	0.439282	−	0.898349i	$-0.355233\pi$
$503$	19.7041	0.878564	0.439282	+	0.898349i	$0.355233\pi$
$504$	0	0
$505$	−39.4396	−1.75504
$506$	0	0
$507$	−4.76653	−0.211689
$508$	0	0
$509$	−5.71119	−0.253144	−0.126572	−	0.991957i	$-0.540398\pi$
$509$	−5.71119	−0.253144	−0.126572	+	0.991957i	$0.540398\pi$
$510$	0	0
$511$	−9.08059	−0.401702
$512$	0	0
$513$	−19.2092	−0.848109
$514$	0	0
$515$	−17.3159	−0.763028
$516$	0	0
$517$	3.32121	0.146067
$518$	0	0
$519$	−1.29822	−0.0569856
$520$	0	0
$521$	−33.3375	−1.46054	−0.730272	−	0.683157i	$-0.760607\pi$
$521$	−33.3375	−1.46054	−0.730272	+	0.683157i	$0.760607\pi$
$522$	0	0
$523$	−21.2348	−0.928531	−0.464266	−	0.885696i	$-0.653682\pi$
$523$	−21.2348	−0.928531	−0.464266	+	0.885696i	$0.653682\pi$
$524$	0	0
$525$	−1.68763	−0.0736542
$526$	0	0
$527$	−33.0501	−1.43969
$528$	0	0
$529$	12.9829	0.564474
$530$	0	0
$531$	14.8388	0.643950
$532$	0	0
$533$	15.4679	0.669989
$534$	0	0
$535$	−50.1054	−2.16625
$536$	0	0
$537$	2.85614	0.123252
$538$	0	0
$539$	−1.09612	−0.0472131
$540$	0	0
$541$	−20.4053	−0.877291	−0.438646	−	0.898660i	$-0.644542\pi$
$541$	−20.4053	−0.877291	−0.438646	+	0.898660i	$0.644542\pi$
$542$	0	0
$543$	−0.440049	−0.0188843
$544$	0	0
$545$	10.5219	0.450711
$546$	0	0
$547$	−10.7335	−0.458933	−0.229466	−	0.973317i	$-0.573698\pi$
$547$	−10.7335	−0.458933	−0.229466	+	0.973317i	$0.573698\pi$
$548$	0	0
$549$	−23.9496	−1.02215
$550$	0	0
$551$	−60.8666	−2.59300
$552$	0	0
$553$	−51.8781	−2.20608
$554$	0	0
$555$	6.21761	0.263923
$556$	0	0
$557$	25.1020	1.06361	0.531804	−	0.846868i	$-0.321514\pi$
$557$	25.1020	1.06361	0.531804	+	0.846868i	$0.321514\pi$
$558$	0	0
$559$	−15.9979	−0.676640
$560$	0	0
$561$	−0.686896	−0.0290008
$562$	0	0
$563$	−30.1743	−1.27169	−0.635847	−	0.771815i	$-0.719349\pi$
$563$	−30.1743	−1.27169	−0.635847	+	0.771815i	$0.719349\pi$
$564$	0	0
$565$	−34.0662	−1.43317
$566$	0	0
$567$	23.2150	0.974939
$568$	0	0
$569$	38.8452	1.62848	0.814238	−	0.580531i	$-0.197155\pi$
$569$	38.8452	1.62848	0.814238	+	0.580531i	$0.197155\pi$
$570$	0	0
$571$	0.495491	0.0207357	0.0103678	−	0.999946i	$-0.496700\pi$
$571$	0.495491	0.0207357	0.0103678	+	0.999946i	$0.496700\pi$
$572$	0	0
$573$	5.80802	0.242634
$574$	0	0
$575$	7.04191	0.293668
$576$	0	0
$577$	46.1553	1.92147	0.960735	−	0.277466i	$-0.0894946\pi$
$577$	46.1553	1.92147	0.960735	+	0.277466i	$0.0894946\pi$
$578$	0	0
$579$	5.77515	0.240007
$580$	0	0
$581$	41.8620	1.73673
$582$	0	0
$583$	0.843772	0.0349455
$584$	0	0
$585$	33.8188	1.39824
$586$	0	0
$587$	19.3829	0.800018	0.400009	−	0.916511i	$-0.369007\pi$
$587$	19.3829	0.800018	0.400009	+	0.916511i	$0.369007\pi$
$588$	0	0
$589$	−52.9650	−2.18238
$590$	0	0
$591$	−11.0104	−0.452908
$592$	0	0
$593$	11.7561	0.482764	0.241382	−	0.970430i	$-0.422399\pi$
$593$	11.7561	0.482764	0.241382	+	0.970430i	$0.422399\pi$
$594$	0	0
$595$	−36.7898	−1.50823
$596$	0	0
$597$	9.90312	0.405308
$598$	0	0
$599$	3.54616	0.144892	0.0724461	−	0.997372i	$-0.476919\pi$
$599$	3.54616	0.144892	0.0724461	+	0.997372i	$0.476919\pi$
$600$	0	0
$601$	−11.2053	−0.457074	−0.228537	−	0.973535i	$-0.573394\pi$
$601$	−11.2053	−0.457074	−0.228537	+	0.973535i	$0.573394\pi$
$602$	0	0
$603$	16.4395	0.669468
$604$	0	0
$605$	−27.0577	−1.10005
$606$	0	0
$607$	−18.5933	−0.754681	−0.377340	−	0.926075i	$-0.623161\pi$
$607$	−18.5933	−0.754681	−0.377340	+	0.926075i	$0.623161\pi$
$608$	0	0
$609$	−11.8341	−0.479540
$610$	0	0
$611$	48.5902	1.96575
$612$	0	0
$613$	42.8757	1.73173	0.865866	−	0.500276i	$-0.166768\pi$
$613$	42.8757	1.73173	0.865866	+	0.500276i	$0.166768\pi$
$614$	0	0
$615$	3.54111	0.142791
$616$	0	0
$617$	−0.797878	−0.0321214	−0.0160607	−	0.999871i	$-0.505112\pi$
$617$	−0.797878	−0.0321214	−0.0160607	+	0.999871i	$0.505112\pi$
$618$	0	0
$619$	−18.2811	−0.734780	−0.367390	−	0.930067i	$-0.619749\pi$
$619$	−18.2811	−0.734780	−0.367390	+	0.930067i	$0.619749\pi$
$620$	0	0
$621$	15.5840	0.625363
$622$	0	0
$623$	21.1875	0.848857
$624$	0	0
$625$	−29.4915	−1.17966
$626$	0	0
$627$	−1.10080	−0.0439615
$628$	0	0
$629$	25.7724	1.02761
$630$	0	0
$631$	−44.7225	−1.78037	−0.890187	−	0.455596i	$-0.849426\pi$
$631$	−44.7225	−1.78037	−0.890187	+	0.455596i	$0.849426\pi$
$632$	0	0
$633$	−8.47372	−0.336800
$634$	0	0
$635$	15.7926	0.626711
$636$	0	0
$637$	−16.0365	−0.635388
$638$	0	0
$639$	38.2187	1.51191
$640$	0	0
$641$	−18.4267	−0.727812	−0.363906	−	0.931436i	$-0.618557\pi$
$641$	−18.4267	−0.727812	−0.363906	+	0.931436i	$0.618557\pi$
$642$	0	0
$643$	5.41011	0.213354	0.106677	−	0.994294i	$-0.465979\pi$
$643$	5.41011	0.213354	0.106677	+	0.994294i	$0.465979\pi$
$644$	0	0
$645$	−3.66245	−0.144209
$646$	0	0
$647$	9.34078	0.367224	0.183612	−	0.982999i	$-0.441221\pi$
$647$	9.34078	0.367224	0.183612	+	0.982999i	$0.441221\pi$
$648$	0	0
$649$	1.76165	0.0691510
$650$	0	0
$651$	−10.2978	−0.403601
$652$	0	0
$653$	7.12144	0.278683	0.139342	−	0.990244i	$-0.455501\pi$
$653$	7.12144	0.278683	0.139342	+	0.990244i	$0.455501\pi$
$654$	0	0
$655$	−19.1419	−0.747935
$656$	0	0
$657$	7.92109	0.309031
$658$	0	0
$659$	−28.3122	−1.10289	−0.551443	−	0.834213i	$-0.685922\pi$
$659$	−28.3122	−1.10289	−0.551443	+	0.834213i	$0.685922\pi$
$660$	0	0
$661$	−12.6088	−0.490425	−0.245213	−	0.969469i	$-0.578858\pi$
$661$	−12.6088	−0.490425	−0.245213	+	0.969469i	$0.578858\pi$
$662$	0	0
$663$	−10.0495	−0.390289
$664$	0	0
$665$	−58.9580	−2.28629
$666$	0	0
$667$	49.3795	1.91198
$668$	0	0
$669$	4.16736	0.161120
$670$	0	0
$671$	−2.84328	−0.109764
$672$	0	0
$673$	27.5804	1.06314	0.531572	−	0.847013i	$-0.321601\pi$
$673$	27.5804	1.06314	0.531572	+	0.847013i	$0.321601\pi$
$674$	0	0
$675$	3.04981	0.117387
$676$	0	0
$677$	14.4276	0.554496	0.277248	−	0.960798i	$-0.410578\pi$
$677$	14.4276	0.554496	0.277248	+	0.960798i	$0.410578\pi$
$678$	0	0
$679$	41.1052	1.57747
$680$	0	0
$681$	4.74835	0.181957
$682$	0	0
$683$	34.8825	1.33474	0.667371	−	0.744726i	$-0.267420\pi$
$683$	34.8825	1.33474	0.667371	+	0.744726i	$0.267420\pi$
$684$	0	0
$685$	−18.6121	−0.711132
$686$	0	0
$687$	−4.70796	−0.179620
$688$	0	0
$689$	12.3446	0.470292
$690$	0	0
$691$	−38.6552	−1.47051	−0.735257	−	0.677789i	$-0.762938\pi$
$691$	−38.6552	−1.47051	−0.735257	+	0.677789i	$0.762938\pi$
$692$	0	0
$693$	2.98544	0.113407
$694$	0	0
$695$	15.4985	0.587890
$696$	0	0
$697$	14.6781	0.555973
$698$	0	0
$699$	−6.77059	−0.256087
$700$	0	0
$701$	3.35833	0.126842	0.0634211	−	0.997987i	$-0.479799\pi$
$701$	3.35833	0.126842	0.0634211	+	0.997987i	$0.479799\pi$
$702$	0	0
$703$	41.3019	1.55773
$704$	0	0
$705$	11.1239	0.418950
$706$	0	0
$707$	−50.9370	−1.91568
$708$	0	0
$709$	−24.7796	−0.930618	−0.465309	−	0.885148i	$-0.654057\pi$
$709$	−24.7796	−0.930618	−0.465309	+	0.885148i	$0.654057\pi$
$710$	0	0
$711$	45.2537	1.69715
$712$	0	0
$713$	42.9691	1.60921
$714$	0	0
$715$	4.01495	0.150151
$716$	0	0
$717$	−4.21762	−0.157510
$718$	0	0
$719$	−50.8983	−1.89819	−0.949094	−	0.314993i	$-0.897998\pi$
$719$	−50.8983	−1.89819	−0.949094	+	0.314993i	$0.897998\pi$
$720$	0	0
$721$	−22.3638	−0.832870
$722$	0	0
$723$	−11.3064	−0.420490
$724$	0	0
$725$	9.66365	0.358899
$726$	0	0
$727$	−36.4972	−1.35361	−0.676804	−	0.736163i	$-0.736636\pi$
$727$	−36.4972	−1.35361	−0.676804	+	0.736163i	$0.736636\pi$
$728$	0	0
$729$	−16.7592	−0.620713
$730$	0	0
$731$	−15.1811	−0.561492
$732$	0	0
$733$	19.8000	0.731331	0.365666	−	0.930746i	$-0.380841\pi$
$733$	19.8000	0.731331	0.365666	+	0.930746i	$0.380841\pi$
$734$	0	0
$735$	−3.67128	−0.135417
$736$	0	0
$737$	1.95168	0.0718912
$738$	0	0
$739$	−23.2545	−0.855430	−0.427715	−	0.903914i	$-0.640681\pi$
$739$	−23.2545	−0.855430	−0.427715	+	0.903914i	$0.640681\pi$
$740$	0	0
$741$	−16.1049	−0.591629
$742$	0	0
$743$	23.5523	0.864052	0.432026	−	0.901861i	$-0.357799\pi$
$743$	23.5523	0.864052	0.432026	+	0.901861i	$0.357799\pi$
$744$	0	0
$745$	13.5835	0.497661
$746$	0	0
$747$	−36.5166	−1.33607
$748$	0	0
$749$	−64.7121	−2.36453
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−9.04206	−0.329511
$754$	0	0
$755$	14.2677	0.519255
$756$	0	0
$757$	−10.8167	−0.393139	−0.196570	−	0.980490i	$-0.562980\pi$
$757$	−10.8167	−0.393139	−0.196570	+	0.980490i	$0.562980\pi$
$758$	0	0
$759$	0.893047	0.0324156
$760$	0	0
$761$	8.49964	0.308112	0.154056	−	0.988062i	$-0.450766\pi$
$761$	8.49964	0.308112	0.154056	+	0.988062i	$0.450766\pi$
$762$	0	0
$763$	13.5893	0.491965
$764$	0	0
$765$	32.0921	1.16029
$766$	0	0
$767$	25.7735	0.930626
$768$	0	0
$769$	10.8173	0.390082	0.195041	−	0.980795i	$-0.437516\pi$
$769$	10.8173	0.390082	0.195041	+	0.980795i	$0.437516\pi$
$770$	0	0
$771$	6.61934	0.238390
$772$	0	0
$773$	14.9068	0.536159	0.268079	−	0.963397i	$-0.413611\pi$
$773$	14.9068	0.536159	0.268079	+	0.963397i	$0.413611\pi$
$774$	0	0
$775$	8.40913	0.302065
$776$	0	0
$777$	8.03017	0.288081
$778$	0	0
$779$	23.5226	0.842786
$780$	0	0
$781$	4.53729	0.162357
$782$	0	0
$783$	21.3860	0.764272
$784$	0	0
$785$	12.5507	0.447953
$786$	0	0
$787$	−28.3524	−1.01066	−0.505328	−	0.862928i	$-0.668628\pi$
$787$	−28.3524	−1.01066	−0.505328	+	0.862928i	$0.668628\pi$
$788$	0	0
$789$	10.0032	0.356123
$790$	0	0
$791$	−43.9971	−1.56436
$792$	0	0
$793$	−41.5979	−1.47719
$794$	0	0
$795$	2.82609	0.100231
$796$	0	0
$797$	12.6015	0.446369	0.223185	−	0.974776i	$-0.428355\pi$
$797$	12.6015	0.446369	0.223185	+	0.974776i	$0.428355\pi$
$798$	0	0
$799$	46.1092	1.63123
$800$	0	0
$801$	−18.4820	−0.653030
$802$	0	0
$803$	0.940386	0.0331855
$804$	0	0
$805$	47.8311	1.68583
$806$	0	0
$807$	−1.22851	−0.0432455
$808$	0	0
$809$	−14.5487	−0.511505	−0.255752	−	0.966742i	$-0.582323\pi$
$809$	−14.5487	−0.511505	−0.255752	+	0.966742i	$0.582323\pi$
$810$	0	0
$811$	−9.97636	−0.350317	−0.175159	−	0.984540i	$-0.556044\pi$
$811$	−9.97636	−0.350317	−0.175159	+	0.984540i	$0.556044\pi$
$812$	0	0
$813$	9.55045	0.334949
$814$	0	0
$815$	6.17047	0.216142
$816$	0	0
$817$	−24.3287	−0.851152
$818$	0	0
$819$	43.6777	1.52622
$820$	0	0
$821$	−47.6090	−1.66157	−0.830784	−	0.556596i	$-0.812107\pi$
$821$	−47.6090	−1.66157	−0.830784	+	0.556596i	$0.812107\pi$
$822$	0	0
$823$	46.5601	1.62298	0.811491	−	0.584365i	$-0.198656\pi$
$823$	46.5601	1.62298	0.811491	+	0.584365i	$0.198656\pi$
$824$	0	0
$825$	0.174771	0.00608474
$826$	0	0
$827$	31.8537	1.10766	0.553831	−	0.832629i	$-0.313165\pi$
$827$	31.8537	1.10766	0.553831	+	0.832629i	$0.313165\pi$
$828$	0	0
$829$	35.9049	1.24703	0.623515	−	0.781811i	$-0.285704\pi$
$829$	35.9049	1.24703	0.623515	+	0.781811i	$0.285704\pi$
$830$	0	0
$831$	0.288519	0.0100086
$832$	0	0
$833$	−15.2177	−0.527261
$834$	0	0
$835$	−33.9927	−1.17637
$836$	0	0
$837$	18.6097	0.643244
$838$	0	0
$839$	−49.7051	−1.71601	−0.858005	−	0.513641i	$-0.828296\pi$
$839$	−49.7051	−1.71601	−0.858005	+	0.513641i	$0.828296\pi$
$840$	0	0
$841$	38.7638	1.33668
$842$	0	0
$843$	3.30619	0.113871
$844$	0	0
$845$	26.4381	0.909498
$846$	0	0
$847$	−34.9455	−1.20074
$848$	0	0
$849$	3.23762	0.111115
$850$	0	0
$851$	−33.5072	−1.14861
$852$	0	0
$853$	21.6447	0.741100	0.370550	−	0.928812i	$-0.379169\pi$
$853$	21.6447	0.741100	0.370550	+	0.928812i	$0.379169\pi$
$854$	0	0
$855$	51.4296	1.75886
$856$	0	0
$857$	5.20395	0.177764	0.0888818	−	0.996042i	$-0.471671\pi$
$857$	5.20395	0.177764	0.0888818	+	0.996042i	$0.471671\pi$
$858$	0	0
$859$	48.2358	1.64578	0.822891	−	0.568199i	$-0.192360\pi$
$859$	48.2358	1.64578	0.822891	+	0.568199i	$0.192360\pi$
$860$	0	0
$861$	4.57341	0.155861
$862$	0	0
$863$	11.2075	0.381507	0.190754	−	0.981638i	$-0.438907\pi$
$863$	11.2075	0.381507	0.190754	+	0.981638i	$0.438907\pi$
$864$	0	0
$865$	7.20073	0.244832
$866$	0	0
$867$	−1.92078	−0.0652333
$868$	0	0
$869$	5.37249	0.182249
$870$	0	0
$871$	28.5536	0.967503
$872$	0	0
$873$	−35.8564	−1.21356
$874$	0	0
$875$	−30.5081	−1.03136
$876$	0	0
$877$	−26.0904	−0.881012	−0.440506	−	0.897750i	$-0.645201\pi$
$877$	−26.0904	−0.881012	−0.440506	+	0.897750i	$0.645201\pi$
$878$	0	0
$879$	14.7421	0.497239
$880$	0	0
$881$	−9.32074	−0.314024	−0.157012	−	0.987597i	$-0.550186\pi$
$881$	−9.32074	−0.314024	−0.157012	+	0.987597i	$0.550186\pi$
$882$	0	0
$883$	29.1390	0.980605	0.490303	−	0.871552i	$-0.336886\pi$
$883$	29.1390	0.980605	0.490303	+	0.871552i	$0.336886\pi$
$884$	0	0
$885$	5.90039	0.198340
$886$	0	0
$887$	−8.43141	−0.283099	−0.141550	−	0.989931i	$-0.545208\pi$
$887$	−8.43141	−0.283099	−0.141550	+	0.989931i	$0.545208\pi$
$888$	0	0
$889$	20.3965	0.684076
$890$	0	0
$891$	−2.40415	−0.0805419
$892$	0	0
$893$	73.8931	2.47274
$894$	0	0
$895$	−15.8419	−0.529537
$896$	0	0
$897$	13.0655	0.436245
$898$	0	0
$899$	58.9668	1.96665
$900$	0	0
$901$	11.7143	0.390260
$902$	0	0
$903$	−4.73013	−0.157409
$904$	0	0
$905$	2.44078	0.0811343
$906$	0	0
$907$	21.4252	0.711411	0.355706	−	0.934598i	$-0.384241\pi$
$907$	21.4252	0.711411	0.355706	+	0.934598i	$0.384241\pi$
$908$	0	0
$909$	44.4329	1.47375
$910$	0	0
$911$	−10.5613	−0.349913	−0.174956	−	0.984576i	$-0.555978\pi$
$911$	−10.5613	−0.349913	−0.174956	+	0.984576i	$0.555978\pi$
$912$	0	0
$913$	−4.33522	−0.143475
$914$	0	0
$915$	−9.52314	−0.314825
$916$	0	0
$917$	−24.7221	−0.816396
$918$	0	0
$919$	−29.1784	−0.962506	−0.481253	−	0.876582i	$-0.659818\pi$
$919$	−29.1784	−0.962506	−0.481253	+	0.876582i	$0.659818\pi$
$920$	0	0
$921$	−0.174564	−0.00575207
$922$	0	0
$923$	66.3817	2.18498
$924$	0	0
$925$	−6.55741	−0.215606
$926$	0	0
$927$	19.5081	0.640731
$928$	0	0
$929$	−57.0488	−1.87171	−0.935855	−	0.352386i	$-0.885371\pi$
$929$	−57.0488	−1.87171	−0.935855	+	0.352386i	$0.885371\pi$
$930$	0	0
$931$	−24.3873	−0.799262
$932$	0	0
$933$	−0.0420496	−0.00137664
$934$	0	0
$935$	3.80995	0.124599
$936$	0	0
$937$	7.79456	0.254637	0.127319	−	0.991862i	$-0.459363\pi$
$937$	7.79456	0.254637	0.127319	+	0.991862i	$0.459363\pi$
$938$	0	0
$939$	−4.12205	−0.134518
$940$	0	0
$941$	45.8670	1.49522	0.747611	−	0.664137i	$-0.231201\pi$
$941$	45.8670	1.49522	0.747611	+	0.664137i	$0.231201\pi$
$942$	0	0
$943$	−19.0833	−0.621438
$944$	0	0
$945$	20.7154	0.673871
$946$	0	0
$947$	−6.76210	−0.219739	−0.109869	−	0.993946i	$-0.535043\pi$
$947$	−6.76210	−0.219739	−0.109869	+	0.993946i	$0.535043\pi$
$948$	0	0
$949$	13.7581	0.446606
$950$	0	0
$951$	−0.430972	−0.0139752
$952$	0	0
$953$	17.4991	0.566852	0.283426	−	0.958994i	$-0.408529\pi$
$953$	17.4991	0.566852	0.283426	+	0.958994i	$0.408529\pi$
$954$	0	0
$955$	−32.2149	−1.04245
$956$	0	0
$957$	1.22553	0.0396159
$958$	0	0
$959$	−24.0379	−0.776223
$960$	0	0
$961$	20.3118	0.655218
$962$	0	0
$963$	56.4490	1.81904
$964$	0	0
$965$	−32.0325	−1.03116
$966$	0	0
$967$	15.2725	0.491129	0.245565	−	0.969380i	$-0.421027\pi$
$967$	15.2725	0.491129	0.245565	+	0.969380i	$0.421027\pi$
$968$	0	0
$969$	−15.2826	−0.490948
$970$	0	0
$971$	47.8574	1.53582	0.767909	−	0.640560i	$-0.221298\pi$
$971$	47.8574	1.53582	0.767909	+	0.640560i	$0.221298\pi$
$972$	0	0
$973$	20.0166	0.641702
$974$	0	0
$975$	2.55694	0.0818877
$976$	0	0
$977$	−46.7857	−1.49681	−0.748403	−	0.663244i	$-0.769179\pi$
$977$	−46.7857	−1.49681	−0.748403	+	0.663244i	$0.769179\pi$
$978$	0	0
$979$	−2.19417	−0.0701260
$980$	0	0
$981$	−11.8541	−0.378471
$982$	0	0
$983$	−14.2822	−0.455530	−0.227765	−	0.973716i	$-0.573142\pi$
$983$	−14.2822	−0.455530	−0.227765	+	0.973716i	$0.573142\pi$
$984$	0	0
$985$	61.0705	1.94587
$986$	0	0
$987$	14.3667	0.457298
$988$	0	0
$989$	19.7372	0.627607
$990$	0	0
$991$	−32.3109	−1.02639	−0.513195	−	0.858272i	$-0.671538\pi$
$991$	−32.3109	−1.02639	−0.513195	+	0.858272i	$0.671538\pi$
$992$	0	0
$993$	14.4332	0.458025
$994$	0	0
$995$	−54.9288	−1.74136
$996$	0	0
$997$	−56.2135	−1.78030	−0.890150	−	0.455669i	$-0.849400\pi$
$997$	−56.2135	−1.78030	−0.890150	+	0.455669i	$0.849400\pi$
$998$	0	0
$999$	−14.5118	−0.459131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.22	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.22	✓	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)

\(f(q)\)	\(=\)	\(q-0.447974 q^{3} +2.48474 q^{5} +3.20909 q^{7} -2.79932 q^{9} +O(q^{10})\)	\(q-0.447974 q^{3} +2.48474 q^{5} +3.20909 q^{7} -2.79932 q^{9} -0.332333 q^{11} -4.86212 q^{13} -1.11310 q^{15} -4.61386 q^{17} -7.39401 q^{19} -1.43759 q^{21} +5.99857 q^{23} +1.17393 q^{25} +2.59794 q^{27} +8.23188 q^{29} +7.16322 q^{31} +0.148877 q^{33} +7.97375 q^{35} -5.58586 q^{37} +2.17810 q^{39} -3.18131 q^{41} +3.29032 q^{43} -6.95558 q^{45} -9.99363 q^{47} +3.29825 q^{49} +2.06689 q^{51} -2.53894 q^{53} -0.825761 q^{55} +3.31233 q^{57} -5.30087 q^{59} +8.55552 q^{61} -8.98326 q^{63} -12.0811 q^{65} -5.87268 q^{67} -2.68721 q^{69} -13.6528 q^{71} -2.82965 q^{73} -0.525891 q^{75} -1.06649 q^{77} -16.1660 q^{79} +7.23415 q^{81} +13.0448 q^{83} -11.4642 q^{85} -3.68767 q^{87} +6.60233 q^{89} -15.6030 q^{91} -3.20894 q^{93} -18.3722 q^{95} +12.8090 q^{97} +0.930306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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