LMFDB - Embedded newform 6008.2.a.b.1.4

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			1.4
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.18197 q^{3} -2.41204 q^{5} +2.14400 q^{7} +7.12495 q^{9} +O(q^{10})$	$q-3.18197 q^{3} -2.41204 q^{5} +2.14400 q^{7} +7.12495 q^{9} +4.27932 q^{11} +6.04339 q^{13} +7.67503 q^{15} -4.10147 q^{17} -5.68811 q^{19} -6.82215 q^{21} -7.79317 q^{23} +0.817913 q^{25} -13.1255 q^{27} +4.70794 q^{29} +2.59911 q^{31} -13.6167 q^{33} -5.17141 q^{35} -9.09774 q^{37} -19.2299 q^{39} +0.788241 q^{41} +8.65874 q^{43} -17.1856 q^{45} -3.21353 q^{47} -2.40326 q^{49} +13.0508 q^{51} +14.2957 q^{53} -10.3219 q^{55} +18.0994 q^{57} +1.45990 q^{59} -3.31142 q^{61} +15.2759 q^{63} -14.5769 q^{65} -8.27604 q^{67} +24.7976 q^{69} +6.59994 q^{71} -8.46722 q^{73} -2.60258 q^{75} +9.17487 q^{77} -5.68149 q^{79} +20.3900 q^{81} -2.17621 q^{83} +9.89289 q^{85} -14.9805 q^{87} -3.16790 q^{89} +12.9570 q^{91} -8.27031 q^{93} +13.7199 q^{95} +13.7492 q^{97} +30.4899 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.18197	−1.83711	−0.918556	−	0.395290i	$-0.870644\pi$
$3$	−3.18197	−1.83711	−0.918556	+	0.395290i	$0.870644\pi$
$4$	0	0
$5$	−2.41204	−1.07869	−0.539347	−	0.842083i	$-0.681329\pi$
$5$	−2.41204	−1.07869	−0.539347	+	0.842083i	$0.681329\pi$
$6$	0	0
$7$	2.14400	0.810356	0.405178	−	0.914238i	$-0.367209\pi$
$7$	2.14400	0.810356	0.405178	+	0.914238i	$0.367209\pi$
$8$	0	0
$9$	7.12495	2.37498
$10$	0	0
$11$	4.27932	1.29026	0.645132	−	0.764071i	$-0.276802\pi$
$11$	4.27932	1.29026	0.645132	+	0.764071i	$0.276802\pi$
$12$	0	0
$13$	6.04339	1.67613	0.838067	−	0.545567i	$-0.183686\pi$
$13$	6.04339	1.67613	0.838067	+	0.545567i	$0.183686\pi$
$14$	0	0
$15$	7.67503	1.98168
$16$	0	0
$17$	−4.10147	−0.994753	−0.497376	−	0.867535i	$-0.665703\pi$
$17$	−4.10147	−0.994753	−0.497376	+	0.867535i	$0.665703\pi$
$18$	0	0
$19$	−5.68811	−1.30494	−0.652471	−	0.757814i	$-0.726268\pi$
$19$	−5.68811	−1.30494	−0.652471	+	0.757814i	$0.726268\pi$
$20$	0	0
$21$	−6.82215	−1.48872
$22$	0	0
$23$	−7.79317	−1.62499	−0.812494	−	0.582969i	$-0.801891\pi$
$23$	−7.79317	−1.62499	−0.812494	+	0.582969i	$0.801891\pi$
$24$	0	0
$25$	0.817913	0.163583
$26$	0	0
$27$	−13.1255	−2.52600
$28$	0	0
$29$	4.70794	0.874243	0.437121	−	0.899402i	$-0.355998\pi$
$29$	4.70794	0.874243	0.437121	+	0.899402i	$0.355998\pi$
$30$	0	0
$31$	2.59911	0.466815	0.233407	−	0.972379i	$-0.425012\pi$
$31$	2.59911	0.466815	0.233407	+	0.972379i	$0.425012\pi$
$32$	0	0
$33$	−13.6167	−2.37036
$34$	0	0
$35$	−5.17141	−0.874127
$36$	0	0
$37$	−9.09774	−1.49566	−0.747829	−	0.663891i	$-0.768904\pi$
$37$	−9.09774	−1.49566	−0.747829	+	0.663891i	$0.768904\pi$
$38$	0	0
$39$	−19.2299	−3.07925
$40$	0	0
$41$	0.788241	0.123103	0.0615513	−	0.998104i	$-0.480395\pi$
$41$	0.788241	0.123103	0.0615513	+	0.998104i	$0.480395\pi$
$42$	0	0
$43$	8.65874	1.32045	0.660223	−	0.751070i	$-0.270462\pi$
$43$	8.65874	1.32045	0.660223	+	0.751070i	$0.270462\pi$
$44$	0	0
$45$	−17.1856	−2.56188
$46$	0	0
$47$	−3.21353	−0.468741	−0.234371	−	0.972147i	$-0.575303\pi$
$47$	−3.21353	−0.468741	−0.234371	+	0.972147i	$0.575303\pi$
$48$	0	0
$49$	−2.40326	−0.343322
$50$	0	0
$51$	13.0508	1.82747
$52$	0	0
$53$	14.2957	1.96367	0.981836	−	0.189732i	$-0.0607619\pi$
$53$	14.2957	1.96367	0.981836	+	0.189732i	$0.0607619\pi$
$54$	0	0
$55$	−10.3219	−1.39180
$56$	0	0
$57$	18.0994	2.39733
$58$	0	0
$59$	1.45990	0.190062	0.0950312	−	0.995474i	$-0.469705\pi$
$59$	1.45990	0.190062	0.0950312	+	0.995474i	$0.469705\pi$
$60$	0	0
$61$	−3.31142	−0.423984	−0.211992	−	0.977271i	$-0.567995\pi$
$61$	−3.31142	−0.423984	−0.211992	+	0.977271i	$0.567995\pi$
$62$	0	0
$63$	15.2759	1.92458
$64$	0	0
$65$	−14.5769	−1.80804
$66$	0	0
$67$	−8.27604	−1.01108	−0.505540	−	0.862803i	$-0.668707\pi$
$67$	−8.27604	−1.01108	−0.505540	+	0.862803i	$0.668707\pi$
$68$	0	0
$69$	24.7976	2.98529
$70$	0	0
$71$	6.59994	0.783268	0.391634	−	0.920121i	$-0.371910\pi$
$71$	6.59994	0.783268	0.391634	+	0.920121i	$0.371910\pi$
$72$	0	0
$73$	−8.46722	−0.991013	−0.495507	−	0.868604i	$-0.665018\pi$
$73$	−8.46722	−0.991013	−0.495507	+	0.868604i	$0.665018\pi$
$74$	0	0
$75$	−2.60258	−0.300520
$76$	0	0
$77$	9.17487	1.04557
$78$	0	0
$79$	−5.68149	−0.639218	−0.319609	−	0.947550i	$-0.603552\pi$
$79$	−5.68149	−0.639218	−0.319609	+	0.947550i	$0.603552\pi$
$80$	0	0
$81$	20.3900	2.26556
$82$	0	0
$83$	−2.17621	−0.238870	−0.119435	−	0.992842i	$-0.538108\pi$
$83$	−2.17621	−0.238870	−0.119435	+	0.992842i	$0.538108\pi$
$84$	0	0
$85$	9.89289	1.07303
$86$	0	0
$87$	−14.9805	−1.60608
$88$	0	0
$89$	−3.16790	−0.335797	−0.167898	−	0.985804i	$-0.553698\pi$
$89$	−3.16790	−0.335797	−0.167898	+	0.985804i	$0.553698\pi$
$90$	0	0
$91$	12.9570	1.35827
$92$	0	0
$93$	−8.27031	−0.857591
$94$	0	0
$95$	13.7199	1.40763
$96$	0	0
$97$	13.7492	1.39602	0.698011	−	0.716087i	$-0.254069\pi$
$97$	13.7492	1.39602	0.698011	+	0.716087i	$0.254069\pi$
$98$	0	0
$99$	30.4899	3.06435
$100$	0	0
$101$	7.96474	0.792522	0.396261	−	0.918138i	$-0.370308\pi$
$101$	7.96474	0.792522	0.396261	+	0.918138i	$0.370308\pi$
$102$	0	0
$103$	−16.7528	−1.65070	−0.825352	−	0.564619i	$-0.809023\pi$
$103$	−16.7528	−1.65070	−0.825352	+	0.564619i	$0.809023\pi$
$104$	0	0
$105$	16.4553	1.60587
$106$	0	0
$107$	5.45238	0.527102	0.263551	−	0.964645i	$-0.415106\pi$
$107$	5.45238	0.527102	0.263551	+	0.964645i	$0.415106\pi$
$108$	0	0
$109$	−0.704006	−0.0674315	−0.0337157	−	0.999431i	$-0.510734\pi$
$109$	−0.704006	−0.0674315	−0.0337157	+	0.999431i	$0.510734\pi$
$110$	0	0
$111$	28.9487	2.74769
$112$	0	0
$113$	9.31870	0.876630	0.438315	−	0.898821i	$-0.355575\pi$
$113$	9.31870	0.876630	0.438315	+	0.898821i	$0.355575\pi$
$114$	0	0
$115$	18.7974	1.75287
$116$	0	0
$117$	43.0588	3.98079
$118$	0	0
$119$	−8.79356	−0.806104
$120$	0	0
$121$	7.31260	0.664782
$122$	0	0
$123$	−2.50816	−0.226153
$124$	0	0
$125$	10.0873	0.902239
$126$	0	0
$127$	−21.0967	−1.87203	−0.936014	−	0.351963i	$-0.885514\pi$
$127$	−21.0967	−1.87203	−0.936014	+	0.351963i	$0.885514\pi$
$128$	0	0
$129$	−27.5519	−2.42581
$130$	0	0
$131$	3.12482	0.273017	0.136508	−	0.990639i	$-0.456412\pi$
$131$	3.12482	0.273017	0.136508	+	0.990639i	$0.456412\pi$
$132$	0	0
$133$	−12.1953	−1.05747
$134$	0	0
$135$	31.6591	2.72478
$136$	0	0
$137$	−1.15467	−0.0986504	−0.0493252	−	0.998783i	$-0.515707\pi$
$137$	−1.15467	−0.0986504	−0.0493252	+	0.998783i	$0.515707\pi$
$138$	0	0
$139$	2.06879	0.175472	0.0877362	−	0.996144i	$-0.472037\pi$
$139$	2.06879	0.175472	0.0877362	+	0.996144i	$0.472037\pi$
$140$	0	0
$141$	10.2254	0.861130
$142$	0	0
$143$	25.8616	2.16266
$144$	0	0
$145$	−11.3557	−0.943041
$146$	0	0
$147$	7.64710	0.630722
$148$	0	0
$149$	−3.45818	−0.283305	−0.141653	−	0.989916i	$-0.545242\pi$
$149$	−3.45818	−0.283305	−0.141653	+	0.989916i	$0.545242\pi$
$150$	0	0
$151$	0.585657	0.0476601	0.0238301	−	0.999716i	$-0.492414\pi$
$151$	0.585657	0.0476601	0.0238301	+	0.999716i	$0.492414\pi$
$152$	0	0
$153$	−29.2228	−2.36252
$154$	0	0
$155$	−6.26916	−0.503551
$156$	0	0
$157$	−18.9851	−1.51518	−0.757588	−	0.652733i	$-0.773623\pi$
$157$	−18.9851	−1.51518	−0.757588	+	0.652733i	$0.773623\pi$
$158$	0	0
$159$	−45.4887	−3.60749
$160$	0	0
$161$	−16.7086	−1.31682
$162$	0	0
$163$	−12.1185	−0.949191	−0.474595	−	0.880204i	$-0.657406\pi$
$163$	−12.1185	−0.949191	−0.474595	+	0.880204i	$0.657406\pi$
$164$	0	0
$165$	32.8439	2.55690
$166$	0	0
$167$	−25.0534	−1.93869	−0.969346	−	0.245701i	$-0.920982\pi$
$167$	−25.0534	−1.93869	−0.969346	+	0.245701i	$0.920982\pi$
$168$	0	0
$169$	23.5226	1.80943
$170$	0	0
$171$	−40.5275	−3.09921
$172$	0	0
$173$	19.1817	1.45836	0.729179	−	0.684323i	$-0.239902\pi$
$173$	19.1817	1.45836	0.729179	+	0.684323i	$0.239902\pi$
$174$	0	0
$175$	1.75361	0.132560
$176$	0	0
$177$	−4.64535	−0.349166
$178$	0	0
$179$	14.2807	1.06739	0.533694	−	0.845677i	$-0.320803\pi$
$179$	14.2807	1.06739	0.533694	+	0.845677i	$0.320803\pi$
$180$	0	0
$181$	−14.6368	−1.08795	−0.543973	−	0.839103i	$-0.683081\pi$
$181$	−14.6368	−1.08795	−0.543973	+	0.839103i	$0.683081\pi$
$182$	0	0
$183$	10.5368	0.778906
$184$	0	0
$185$	21.9441	1.61336
$186$	0	0
$187$	−17.5515	−1.28349
$188$	0	0
$189$	−28.1410	−2.04696
$190$	0	0
$191$	−4.68767	−0.339188	−0.169594	−	0.985514i	$-0.554246\pi$
$191$	−4.68767	−0.339188	−0.169594	+	0.985514i	$0.554246\pi$
$192$	0	0
$193$	14.2675	1.02700	0.513500	−	0.858090i	$-0.328349\pi$
$193$	14.2675	1.02700	0.513500	+	0.858090i	$0.328349\pi$
$194$	0	0
$195$	46.3832	3.32157
$196$	0	0
$197$	23.5457	1.67756	0.838780	−	0.544470i	$-0.183269\pi$
$197$	23.5457	1.67756	0.838780	+	0.544470i	$0.183269\pi$
$198$	0	0
$199$	0.553056	0.0392051	0.0196026	−	0.999808i	$-0.493760\pi$
$199$	0.553056	0.0392051	0.0196026	+	0.999808i	$0.493760\pi$
$200$	0	0
$201$	26.3341	1.85747
$202$	0	0
$203$	10.0938	0.708448
$204$	0	0
$205$	−1.90127	−0.132790
$206$	0	0
$207$	−55.5259	−3.85932
$208$	0	0
$209$	−24.3413	−1.68372
$210$	0	0
$211$	8.30437	0.571696	0.285848	−	0.958275i	$-0.407725\pi$
$211$	8.30437	0.571696	0.285848	+	0.958275i	$0.407725\pi$
$212$	0	0
$213$	−21.0008	−1.43895
$214$	0	0
$215$	−20.8852	−1.42436
$216$	0	0
$217$	5.57251	0.378286
$218$	0	0
$219$	26.9425	1.82060
$220$	0	0
$221$	−24.7868	−1.66734
$222$	0	0
$223$	−24.7570	−1.65785	−0.828927	−	0.559357i	$-0.811048\pi$
$223$	−24.7570	−1.65785	−0.828927	+	0.559357i	$0.811048\pi$
$224$	0	0
$225$	5.82759	0.388506
$226$	0	0
$227$	−17.3506	−1.15160	−0.575801	−	0.817590i	$-0.695310\pi$
$227$	−17.3506	−1.15160	−0.575801	+	0.817590i	$0.695310\pi$
$228$	0	0
$229$	6.99497	0.462241	0.231120	−	0.972925i	$-0.425761\pi$
$229$	6.99497	0.462241	0.231120	+	0.972925i	$0.425761\pi$
$230$	0	0
$231$	−29.1942	−1.92084
$232$	0	0
$233$	13.2735	0.869574	0.434787	−	0.900533i	$-0.356824\pi$
$233$	13.2735	0.869574	0.434787	+	0.900533i	$0.356824\pi$
$234$	0	0
$235$	7.75114	0.505629
$236$	0	0
$237$	18.0784	1.17432
$238$	0	0
$239$	1.51469	0.0979772	0.0489886	−	0.998799i	$-0.484400\pi$
$239$	1.51469	0.0979772	0.0489886	+	0.998799i	$0.484400\pi$
$240$	0	0
$241$	−12.7844	−0.823517	−0.411758	−	0.911293i	$-0.635085\pi$
$241$	−12.7844	−0.823517	−0.411758	+	0.911293i	$0.635085\pi$
$242$	0	0
$243$	−25.5041	−1.63609
$244$	0	0
$245$	5.79674	0.370340
$246$	0	0
$247$	−34.3755	−2.18726
$248$	0	0
$249$	6.92463	0.438831
$250$	0	0
$251$	−5.90243	−0.372558	−0.186279	−	0.982497i	$-0.559643\pi$
$251$	−5.90243	−0.372558	−0.186279	+	0.982497i	$0.559643\pi$
$252$	0	0
$253$	−33.3495	−2.09666
$254$	0	0
$255$	−31.4789	−1.97129
$256$	0	0
$257$	8.31994	0.518984	0.259492	−	0.965745i	$-0.416445\pi$
$257$	8.31994	0.518984	0.259492	+	0.965745i	$0.416445\pi$
$258$	0	0
$259$	−19.5056	−1.21202
$260$	0	0
$261$	33.5438	2.07631
$262$	0	0
$263$	3.88818	0.239755	0.119878	−	0.992789i	$-0.461750\pi$
$263$	3.88818	0.239755	0.119878	+	0.992789i	$0.461750\pi$
$264$	0	0
$265$	−34.4818	−2.11820
$266$	0	0
$267$	10.0802	0.616896
$268$	0	0
$269$	20.6433	1.25864	0.629322	−	0.777145i	$-0.283333\pi$
$269$	20.6433	1.25864	0.629322	+	0.777145i	$0.283333\pi$
$270$	0	0
$271$	12.4758	0.757850	0.378925	−	0.925427i	$-0.376294\pi$
$271$	12.4758	0.757850	0.378925	+	0.925427i	$0.376294\pi$
$272$	0	0
$273$	−41.2289	−2.49529
$274$	0	0
$275$	3.50011	0.211065
$276$	0	0
$277$	−11.8192	−0.710148	−0.355074	−	0.934838i	$-0.615544\pi$
$277$	−11.8192	−0.710148	−0.355074	+	0.934838i	$0.615544\pi$
$278$	0	0
$279$	18.5186	1.10868
$280$	0	0
$281$	−10.9486	−0.653140	−0.326570	−	0.945173i	$-0.605893\pi$
$281$	−10.9486	−0.653140	−0.326570	+	0.945173i	$0.605893\pi$
$282$	0	0
$283$	−24.5751	−1.46084	−0.730420	−	0.682999i	$-0.760676\pi$
$283$	−24.5751	−1.46084	−0.730420	+	0.682999i	$0.760676\pi$
$284$	0	0
$285$	−43.6564	−2.58598
$286$	0	0
$287$	1.68999	0.0997570
$288$	0	0
$289$	−0.177939	−0.0104670
$290$	0	0
$291$	−43.7496	−2.56465
$292$	0	0
$293$	4.22642	0.246910	0.123455	−	0.992350i	$-0.460603\pi$
$293$	4.22642	0.246910	0.123455	+	0.992350i	$0.460603\pi$
$294$	0	0
$295$	−3.52132	−0.205019
$296$	0	0
$297$	−56.1681	−3.25920
$298$	0	0
$299$	−47.0972	−2.72370
$300$	0	0
$301$	18.5644	1.07003
$302$	0	0
$303$	−25.3436	−1.45595
$304$	0	0
$305$	7.98726	0.457349
$306$	0	0
$307$	28.4382	1.62305	0.811527	−	0.584315i	$-0.198637\pi$
$307$	28.4382	1.62305	0.811527	+	0.584315i	$0.198637\pi$
$308$	0	0
$309$	53.3070	3.03253
$310$	0	0
$311$	−24.2743	−1.37647	−0.688236	−	0.725487i	$-0.741615\pi$
$311$	−24.2743	−1.37647	−0.688236	+	0.725487i	$0.741615\pi$
$312$	0	0
$313$	−13.1400	−0.742717	−0.371359	−	0.928490i	$-0.621108\pi$
$313$	−13.1400	−0.742717	−0.371359	+	0.928490i	$0.621108\pi$
$314$	0	0
$315$	−36.8460	−2.07604
$316$	0	0
$317$	−27.3949	−1.53865	−0.769324	−	0.638859i	$-0.779407\pi$
$317$	−27.3949	−1.53865	−0.769324	+	0.638859i	$0.779407\pi$
$318$	0	0
$319$	20.1468	1.12800
$320$	0	0
$321$	−17.3493	−0.968345
$322$	0	0
$323$	23.3296	1.29809
$324$	0	0
$325$	4.94297	0.274187
$326$	0	0
$327$	2.24013	0.123879
$328$	0	0
$329$	−6.88981	−0.379847
$330$	0	0
$331$	−10.0742	−0.553727	−0.276864	−	0.960909i	$-0.589295\pi$
$331$	−10.0742	−0.553727	−0.276864	+	0.960909i	$0.589295\pi$
$332$	0	0
$333$	−64.8209	−3.55216
$334$	0	0
$335$	19.9621	1.09065
$336$	0	0
$337$	34.3371	1.87046	0.935230	−	0.354041i	$-0.115193\pi$
$337$	34.3371	1.87046	0.935230	+	0.354041i	$0.115193\pi$
$338$	0	0
$339$	−29.6519	−1.61047
$340$	0	0
$341$	11.1225	0.602314
$342$	0	0
$343$	−20.1606	−1.08857
$344$	0	0
$345$	−59.8128	−3.22021
$346$	0	0
$347$	−29.6494	−1.59166	−0.795832	−	0.605517i	$-0.792966\pi$
$347$	−29.6494	−1.59166	−0.795832	+	0.605517i	$0.792966\pi$
$348$	0	0
$349$	−8.29858	−0.444213	−0.222106	−	0.975022i	$-0.571293\pi$
$349$	−8.29858	−0.444213	−0.222106	+	0.975022i	$0.571293\pi$
$350$	0	0
$351$	−79.3223	−4.23391
$352$	0	0
$353$	−9.16523	−0.487816	−0.243908	−	0.969798i	$-0.578430\pi$
$353$	−9.16523	−0.487816	−0.243908	+	0.969798i	$0.578430\pi$
$354$	0	0
$355$	−15.9193	−0.844907
$356$	0	0
$357$	27.9809	1.48090
$358$	0	0
$359$	28.6357	1.51134	0.755668	−	0.654954i	$-0.227312\pi$
$359$	28.6357	1.51134	0.755668	+	0.654954i	$0.227312\pi$
$360$	0	0
$361$	13.3546	0.702874
$362$	0	0
$363$	−23.2685	−1.22128
$364$	0	0
$365$	20.4232	1.06900
$366$	0	0
$367$	6.08254	0.317506	0.158753	−	0.987318i	$-0.449253\pi$
$367$	6.08254	0.317506	0.158753	+	0.987318i	$0.449253\pi$
$368$	0	0
$369$	5.61618	0.292367
$370$	0	0
$371$	30.6501	1.59127
$372$	0	0
$373$	−8.63400	−0.447052	−0.223526	−	0.974698i	$-0.571757\pi$
$373$	−8.63400	−0.447052	−0.223526	+	0.974698i	$0.571757\pi$
$374$	0	0
$375$	−32.0976	−1.65751
$376$	0	0
$377$	28.4519	1.46535
$378$	0	0
$379$	−6.95523	−0.357266	−0.178633	−	0.983916i	$-0.557167\pi$
$379$	−6.95523	−0.357266	−0.178633	+	0.983916i	$0.557167\pi$
$380$	0	0
$381$	67.1291	3.43913
$382$	0	0
$383$	35.4097	1.80935	0.904676	−	0.426100i	$-0.140113\pi$
$383$	35.4097	1.80935	0.904676	+	0.426100i	$0.140113\pi$
$384$	0	0
$385$	−22.1301	−1.12786
$386$	0	0
$387$	61.6931	3.13604
$388$	0	0
$389$	−3.62879	−0.183987	−0.0919934	−	0.995760i	$-0.529324\pi$
$389$	−3.62879	−0.183987	−0.0919934	+	0.995760i	$0.529324\pi$
$390$	0	0
$391$	31.9635	1.61646
$392$	0	0
$393$	−9.94309	−0.501563
$394$	0	0
$395$	13.7040	0.689521
$396$	0	0
$397$	8.06758	0.404900	0.202450	−	0.979293i	$-0.435110\pi$
$397$	8.06758	0.404900	0.202450	+	0.979293i	$0.435110\pi$
$398$	0	0
$399$	38.8052	1.94269
$400$	0	0
$401$	−21.9377	−1.09551	−0.547757	−	0.836637i	$-0.684518\pi$
$401$	−21.9377	−1.09551	−0.547757	+	0.836637i	$0.684518\pi$
$402$	0	0
$403$	15.7075	0.782445
$404$	0	0
$405$	−49.1815	−2.44385
$406$	0	0
$407$	−38.9321	−1.92979
$408$	0	0
$409$	−28.2850	−1.39860	−0.699301	−	0.714827i	$-0.746505\pi$
$409$	−28.2850	−1.39860	−0.699301	+	0.714827i	$0.746505\pi$
$410$	0	0
$411$	3.67414	0.181232
$412$	0	0
$413$	3.13002	0.154018
$414$	0	0
$415$	5.24909	0.257668
$416$	0	0
$417$	−6.58283	−0.322363
$418$	0	0
$419$	−19.3503	−0.945323	−0.472661	−	0.881244i	$-0.656707\pi$
$419$	−19.3503	−0.945323	−0.472661	+	0.881244i	$0.656707\pi$
$420$	0	0
$421$	3.28476	0.160089	0.0800447	−	0.996791i	$-0.474494\pi$
$421$	3.28476	0.160089	0.0800447	+	0.996791i	$0.474494\pi$
$422$	0	0
$423$	−22.8962	−1.11325
$424$	0	0
$425$	−3.35465	−0.162724
$426$	0	0
$427$	−7.09969	−0.343578
$428$	0	0
$429$	−82.2909	−3.97304
$430$	0	0
$431$	−30.8826	−1.48756	−0.743780	−	0.668424i	$-0.766969\pi$
$431$	−30.8826	−1.48756	−0.743780	+	0.668424i	$0.766969\pi$
$432$	0	0
$433$	5.34217	0.256728	0.128364	−	0.991727i	$-0.459027\pi$
$433$	5.34217	0.256728	0.128364	+	0.991727i	$0.459027\pi$
$434$	0	0
$435$	36.1336	1.73247
$436$	0	0
$437$	44.3284	2.12052
$438$	0	0
$439$	26.9492	1.28621	0.643107	−	0.765776i	$-0.277645\pi$
$439$	26.9492	1.28621	0.643107	+	0.765776i	$0.277645\pi$
$440$	0	0
$441$	−17.1231	−0.815385
$442$	0	0
$443$	−18.0731	−0.858677	−0.429339	−	0.903144i	$-0.641253\pi$
$443$	−18.0731	−0.858677	−0.429339	+	0.903144i	$0.641253\pi$
$444$	0	0
$445$	7.64108	0.362222
$446$	0	0
$447$	11.0038	0.520464
$448$	0	0
$449$	17.7406	0.837230	0.418615	−	0.908164i	$-0.362516\pi$
$449$	17.7406	0.837230	0.418615	+	0.908164i	$0.362516\pi$
$450$	0	0
$451$	3.37314	0.158835
$452$	0	0
$453$	−1.86354	−0.0875570
$454$	0	0
$455$	−31.2528	−1.46516
$456$	0	0
$457$	−10.5462	−0.493331	−0.246665	−	0.969101i	$-0.579335\pi$
$457$	−10.5462	−0.493331	−0.246665	+	0.969101i	$0.579335\pi$
$458$	0	0
$459$	53.8337	2.51274
$460$	0	0
$461$	33.2430	1.54828	0.774140	−	0.633015i	$-0.218183\pi$
$461$	33.2430	1.54828	0.774140	+	0.633015i	$0.218183\pi$
$462$	0	0
$463$	9.43193	0.438339	0.219169	−	0.975687i	$-0.429665\pi$
$463$	9.43193	0.438339	0.219169	+	0.975687i	$0.429665\pi$
$464$	0	0
$465$	19.9483	0.925079
$466$	0	0
$467$	19.8667	0.919320	0.459660	−	0.888095i	$-0.347971\pi$
$467$	19.8667	0.919320	0.459660	+	0.888095i	$0.347971\pi$
$468$	0	0
$469$	−17.7438	−0.819335
$470$	0	0
$471$	60.4101	2.78355
$472$	0	0
$473$	37.0535	1.70372
$474$	0	0
$475$	−4.65238	−0.213466
$476$	0	0
$477$	101.856	4.66369
$478$	0	0
$479$	−13.7806	−0.629653	−0.314826	−	0.949149i	$-0.601946\pi$
$479$	−13.7806	−0.629653	−0.314826	+	0.949149i	$0.601946\pi$
$480$	0	0
$481$	−54.9812	−2.50693
$482$	0	0
$483$	53.1662	2.41915
$484$	0	0
$485$	−33.1636	−1.50588
$486$	0	0
$487$	0.762405	0.0345479	0.0172739	−	0.999851i	$-0.494501\pi$
$487$	0.762405	0.0345479	0.0172739	+	0.999851i	$0.494501\pi$
$488$	0	0
$489$	38.5606	1.74377
$490$	0	0
$491$	−29.4198	−1.32770	−0.663849	−	0.747867i	$-0.731078\pi$
$491$	−29.4198	−1.32770	−0.663849	+	0.747867i	$0.731078\pi$
$492$	0	0
$493$	−19.3095	−0.869656
$494$	0	0
$495$	−73.5428	−3.30550
$496$	0	0
$497$	14.1503	0.634726
$498$	0	0
$499$	15.4130	0.689980	0.344990	−	0.938606i	$-0.387882\pi$
$499$	15.4130	0.689980	0.344990	+	0.938606i	$0.387882\pi$
$500$	0	0
$501$	79.7193	3.56159
$502$	0	0
$503$	1.92155	0.0856775	0.0428387	−	0.999082i	$-0.486360\pi$
$503$	1.92155	0.0856775	0.0428387	+	0.999082i	$0.486360\pi$
$504$	0	0
$505$	−19.2112	−0.854889
$506$	0	0
$507$	−74.8481	−3.32412
$508$	0	0
$509$	−34.0545	−1.50944	−0.754719	−	0.656048i	$-0.772227\pi$
$509$	−34.0545	−1.50944	−0.754719	+	0.656048i	$0.772227\pi$
$510$	0	0
$511$	−18.1537	−0.803074
$512$	0	0
$513$	74.6591	3.29628
$514$	0	0
$515$	40.4084	1.78061
$516$	0	0
$517$	−13.7517	−0.604800
$518$	0	0
$519$	−61.0357	−2.67917
$520$	0	0
$521$	14.3394	0.628222	0.314111	−	0.949386i	$-0.398294\pi$
$521$	14.3394	0.628222	0.314111	+	0.949386i	$0.398294\pi$
$522$	0	0
$523$	24.7546	1.08244	0.541221	−	0.840880i	$-0.317962\pi$
$523$	24.7546	1.08244	0.541221	+	0.840880i	$0.317962\pi$
$524$	0	0
$525$	−5.57993	−0.243528
$526$	0	0
$527$	−10.6602	−0.464365
$528$	0	0
$529$	37.7335	1.64059
$530$	0	0
$531$	10.4017	0.451395
$532$	0	0
$533$	4.76365	0.206337
$534$	0	0
$535$	−13.1513	−0.568582
$536$	0	0
$537$	−45.4408	−1.96091
$538$	0	0
$539$	−10.2843	−0.442977
$540$	0	0
$541$	10.9539	0.470945	0.235472	−	0.971881i	$-0.424336\pi$
$541$	10.9539	0.470945	0.235472	+	0.971881i	$0.424336\pi$
$542$	0	0
$543$	46.5739	1.99868
$544$	0	0
$545$	1.69809	0.0727380
$546$	0	0
$547$	−14.1730	−0.605992	−0.302996	−	0.952992i	$-0.597987\pi$
$547$	−14.1730	−0.605992	−0.302996	+	0.952992i	$0.597987\pi$
$548$	0	0
$549$	−23.5937	−1.00695
$550$	0	0
$551$	−26.7793	−1.14084
$552$	0	0
$553$	−12.1811	−0.517994
$554$	0	0
$555$	−69.8254	−2.96392
$556$	0	0
$557$	−7.48107	−0.316983	−0.158492	−	0.987360i	$-0.550663\pi$
$557$	−7.48107	−0.316983	−0.158492	+	0.987360i	$0.550663\pi$
$558$	0	0
$559$	52.3282	2.21325
$560$	0	0
$561$	55.8484	2.35792
$562$	0	0
$563$	−24.3726	−1.02718	−0.513592	−	0.858035i	$-0.671685\pi$
$563$	−24.3726	−1.02718	−0.513592	+	0.858035i	$0.671685\pi$
$564$	0	0
$565$	−22.4770	−0.945616
$566$	0	0
$567$	43.7162	1.83591
$568$	0	0
$569$	22.5084	0.943603	0.471802	−	0.881705i	$-0.343604\pi$
$569$	22.5084	0.943603	0.471802	+	0.881705i	$0.343604\pi$
$570$	0	0
$571$	−32.8894	−1.37638	−0.688190	−	0.725531i	$-0.741594\pi$
$571$	−32.8894	−1.37638	−0.688190	+	0.725531i	$0.741594\pi$
$572$	0	0
$573$	14.9160	0.623126
$574$	0	0
$575$	−6.37414	−0.265820
$576$	0	0
$577$	6.95324	0.289467	0.144733	−	0.989471i	$-0.453768\pi$
$577$	6.95324	0.289467	0.144733	+	0.989471i	$0.453768\pi$
$578$	0	0
$579$	−45.3989	−1.88671
$580$	0	0
$581$	−4.66579	−0.193570
$582$	0	0
$583$	61.1761	2.53366
$584$	0	0
$585$	−103.859	−4.29406
$586$	0	0
$587$	−28.0595	−1.15814	−0.579070	−	0.815278i	$-0.696584\pi$
$587$	−28.0595	−1.15814	−0.579070	+	0.815278i	$0.696584\pi$
$588$	0	0
$589$	−14.7841	−0.609166
$590$	0	0
$591$	−74.9217	−3.08187
$592$	0	0
$593$	−22.8304	−0.937532	−0.468766	−	0.883322i	$-0.655301\pi$
$593$	−22.8304	−0.937532	−0.468766	+	0.883322i	$0.655301\pi$
$594$	0	0
$595$	21.2104	0.869541
$596$	0	0
$597$	−1.75981	−0.0720242
$598$	0	0
$599$	26.5132	1.08330	0.541649	−	0.840605i	$-0.317800\pi$
$599$	26.5132	1.08330	0.541649	+	0.840605i	$0.317800\pi$
$600$	0	0
$601$	−9.27038	−0.378147	−0.189073	−	0.981963i	$-0.560548\pi$
$601$	−9.27038	−0.378147	−0.189073	+	0.981963i	$0.560548\pi$
$602$	0	0
$603$	−58.9664	−2.40130
$604$	0	0
$605$	−17.6382	−0.717096
$606$	0	0
$607$	−37.1211	−1.50670	−0.753349	−	0.657621i	$-0.771563\pi$
$607$	−37.1211	−1.50670	−0.753349	+	0.657621i	$0.771563\pi$
$608$	0	0
$609$	−32.1183	−1.30150
$610$	0	0
$611$	−19.4206	−0.785673
$612$	0	0
$613$	−15.4695	−0.624806	−0.312403	−	0.949950i	$-0.601134\pi$
$613$	−15.4695	−0.624806	−0.312403	+	0.949950i	$0.601134\pi$
$614$	0	0
$615$	6.04978	0.243951
$616$	0	0
$617$	−35.2283	−1.41824	−0.709120	−	0.705088i	$-0.750908\pi$
$617$	−35.2283	−1.41824	−0.709120	+	0.705088i	$0.750908\pi$
$618$	0	0
$619$	24.2717	0.975561	0.487781	−	0.872966i	$-0.337807\pi$
$619$	24.2717	0.975561	0.487781	+	0.872966i	$0.337807\pi$
$620$	0	0
$621$	102.289	4.10472
$622$	0	0
$623$	−6.79198	−0.272115
$624$	0	0
$625$	−28.4206	−1.13682
$626$	0	0
$627$	77.4532	3.09318
$628$	0	0
$629$	37.3141	1.48781
$630$	0	0
$631$	19.0657	0.758994	0.379497	−	0.925193i	$-0.376097\pi$
$631$	19.0657	0.758994	0.379497	+	0.925193i	$0.376097\pi$
$632$	0	0
$633$	−26.4243	−1.05027
$634$	0	0
$635$	50.8859	2.01935
$636$	0	0
$637$	−14.5238	−0.575455
$638$	0	0
$639$	47.0242	1.86025
$640$	0	0
$641$	−29.2188	−1.15407	−0.577037	−	0.816718i	$-0.695791\pi$
$641$	−29.2188	−1.15407	−0.577037	+	0.816718i	$0.695791\pi$
$642$	0	0
$643$	−3.60188	−0.142044	−0.0710221	−	0.997475i	$-0.522626\pi$
$643$	−3.60188	−0.142044	−0.0710221	+	0.997475i	$0.522626\pi$
$644$	0	0
$645$	66.4561	2.61671
$646$	0	0
$647$	−25.8057	−1.01453	−0.507263	−	0.861791i	$-0.669343\pi$
$647$	−25.8057	−1.01453	−0.507263	+	0.861791i	$0.669343\pi$
$648$	0	0
$649$	6.24737	0.245231
$650$	0	0
$651$	−17.7316	−0.694955
$652$	0	0
$653$	−24.0728	−0.942041	−0.471021	−	0.882122i	$-0.656114\pi$
$653$	−24.0728	−0.942041	−0.471021	+	0.882122i	$0.656114\pi$
$654$	0	0
$655$	−7.53718	−0.294502
$656$	0	0
$657$	−60.3285	−2.35364
$658$	0	0
$659$	35.3043	1.37526	0.687630	−	0.726061i	$-0.258651\pi$
$659$	35.3043	1.37526	0.687630	+	0.726061i	$0.258651\pi$
$660$	0	0
$661$	4.58117	0.178187	0.0890935	−	0.996023i	$-0.471603\pi$
$661$	4.58117	0.178187	0.0890935	+	0.996023i	$0.471603\pi$
$662$	0	0
$663$	78.8709	3.06309
$664$	0	0
$665$	29.4155	1.14069
$666$	0	0
$667$	−36.6898	−1.42063
$668$	0	0
$669$	78.7762	3.04566
$670$	0	0
$671$	−14.1706	−0.547051
$672$	0	0
$673$	22.7644	0.877503	0.438751	−	0.898609i	$-0.355421\pi$
$673$	22.7644	0.877503	0.438751	+	0.898609i	$0.355421\pi$
$674$	0	0
$675$	−10.7355	−0.413209
$676$	0	0
$677$	−32.4053	−1.24544	−0.622718	−	0.782446i	$-0.713972\pi$
$677$	−32.4053	−1.24544	−0.622718	+	0.782446i	$0.713972\pi$
$678$	0	0
$679$	29.4783	1.13128
$680$	0	0
$681$	55.2092	2.11562
$682$	0	0
$683$	−42.4112	−1.62282	−0.811409	−	0.584478i	$-0.801299\pi$
$683$	−42.4112	−1.62282	−0.811409	+	0.584478i	$0.801299\pi$
$684$	0	0
$685$	2.78511	0.106414
$686$	0	0
$687$	−22.2578	−0.849188
$688$	0	0
$689$	86.3948	3.29138
$690$	0	0
$691$	−37.5434	−1.42822	−0.714109	−	0.700035i	$-0.753168\pi$
$691$	−37.5434	−1.42822	−0.714109	+	0.700035i	$0.753168\pi$
$692$	0	0
$693$	65.3705	2.48322
$694$	0	0
$695$	−4.98999	−0.189281
$696$	0	0
$697$	−3.23295	−0.122457
$698$	0	0
$699$	−42.2358	−1.59751
$700$	0	0
$701$	33.8929	1.28012	0.640059	−	0.768325i	$-0.278910\pi$
$701$	33.8929	1.28012	0.640059	+	0.768325i	$0.278910\pi$
$702$	0	0
$703$	51.7489	1.95175
$704$	0	0
$705$	−24.6639	−0.928897
$706$	0	0
$707$	17.0764	0.642225
$708$	0	0
$709$	−1.98557	−0.0745696	−0.0372848	−	0.999305i	$-0.511871\pi$
$709$	−1.98557	−0.0745696	−0.0372848	+	0.999305i	$0.511871\pi$
$710$	0	0
$711$	−40.4803	−1.51813
$712$	0	0
$713$	−20.2553	−0.758569
$714$	0	0
$715$	−62.3791	−2.33285
$716$	0	0
$717$	−4.81970	−0.179995
$718$	0	0
$719$	−29.4203	−1.09719	−0.548596	−	0.836088i	$-0.684838\pi$
$719$	−29.4203	−1.09719	−0.548596	+	0.836088i	$0.684838\pi$
$720$	0	0
$721$	−35.9180	−1.33766
$722$	0	0
$723$	40.6797	1.51289
$724$	0	0
$725$	3.85069	0.143011
$726$	0	0
$727$	−49.4355	−1.83346	−0.916731	−	0.399505i	$-0.869182\pi$
$727$	−49.4355	−1.83346	−0.916731	+	0.399505i	$0.869182\pi$
$728$	0	0
$729$	19.9832	0.740120
$730$	0	0
$731$	−35.5136	−1.31352
$732$	0	0
$733$	−8.18741	−0.302409	−0.151205	−	0.988503i	$-0.548315\pi$
$733$	−8.18741	−0.302409	−0.151205	+	0.988503i	$0.548315\pi$
$734$	0	0
$735$	−18.4451	−0.680357
$736$	0	0
$737$	−35.4158	−1.30456
$738$	0	0
$739$	−28.3164	−1.04163	−0.520817	−	0.853668i	$-0.674373\pi$
$739$	−28.3164	−1.04163	−0.520817	+	0.853668i	$0.674373\pi$
$740$	0	0
$741$	109.382	4.01824
$742$	0	0
$743$	−19.6007	−0.719078	−0.359539	−	0.933130i	$-0.617066\pi$
$743$	−19.6007	−0.719078	−0.359539	+	0.933130i	$0.617066\pi$
$744$	0	0
$745$	8.34126	0.305600
$746$	0	0
$747$	−15.5054	−0.567311
$748$	0	0
$749$	11.6899	0.427140
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	18.7814	0.684431
$754$	0	0
$755$	−1.41263	−0.0514107
$756$	0	0
$757$	−30.8106	−1.11983	−0.559914	−	0.828551i	$-0.689166\pi$
$757$	−30.8106	−1.11983	−0.559914	+	0.828551i	$0.689166\pi$
$758$	0	0
$759$	106.117	3.85181
$760$	0	0
$761$	−11.0937	−0.402145	−0.201073	−	0.979576i	$-0.564443\pi$
$761$	−11.0937	−0.402145	−0.201073	+	0.979576i	$0.564443\pi$
$762$	0	0
$763$	−1.50939	−0.0546435
$764$	0	0
$765$	70.4863	2.54844
$766$	0	0
$767$	8.82273	0.318570
$768$	0	0
$769$	−35.9512	−1.29643	−0.648216	−	0.761457i	$-0.724485\pi$
$769$	−35.9512	−1.29643	−0.648216	+	0.761457i	$0.724485\pi$
$770$	0	0
$771$	−26.4738	−0.953431
$772$	0	0
$773$	23.3605	0.840219	0.420110	−	0.907473i	$-0.361992\pi$
$773$	23.3605	0.840219	0.420110	+	0.907473i	$0.361992\pi$
$774$	0	0
$775$	2.12585	0.0763628
$776$	0	0
$777$	62.0662	2.22661
$778$	0	0
$779$	−4.48361	−0.160642
$780$	0	0
$781$	28.2432	1.01062
$782$	0	0
$783$	−61.7939	−2.20834
$784$	0	0
$785$	45.7928	1.63441
$786$	0	0
$787$	23.5839	0.840676	0.420338	−	0.907368i	$-0.361912\pi$
$787$	23.5839	0.840676	0.420338	+	0.907368i	$0.361912\pi$
$788$	0	0
$789$	−12.3721	−0.440458
$790$	0	0
$791$	19.9793	0.710383
$792$	0	0
$793$	−20.0122	−0.710654
$794$	0	0
$795$	109.720	3.89138
$796$	0	0
$797$	18.5554	0.657266	0.328633	−	0.944458i	$-0.393412\pi$
$797$	18.5554	0.657266	0.328633	+	0.944458i	$0.393412\pi$
$798$	0	0
$799$	13.1802	0.466282
$800$	0	0
$801$	−22.5711	−0.797511
$802$	0	0
$803$	−36.2340	−1.27867
$804$	0	0
$805$	40.3016	1.42045
$806$	0	0
$807$	−65.6864	−2.31227
$808$	0	0
$809$	33.5383	1.17914	0.589572	−	0.807716i	$-0.299296\pi$
$809$	33.5383	1.17914	0.589572	+	0.807716i	$0.299296\pi$
$810$	0	0
$811$	−2.34171	−0.0822286	−0.0411143	−	0.999154i	$-0.513091\pi$
$811$	−2.34171	−0.0822286	−0.0411143	+	0.999154i	$0.513091\pi$
$812$	0	0
$813$	−39.6976	−1.39226
$814$	0	0
$815$	29.2301	1.02389
$816$	0	0
$817$	−49.2519	−1.72311
$818$	0	0
$819$	92.3182	3.22586
$820$	0	0
$821$	−11.1611	−0.389527	−0.194763	−	0.980850i	$-0.562394\pi$
$821$	−11.1611	−0.389527	−0.194763	+	0.980850i	$0.562394\pi$
$822$	0	0
$823$	−26.3292	−0.917779	−0.458889	−	0.888493i	$-0.651753\pi$
$823$	−26.3292	−0.917779	−0.458889	+	0.888493i	$0.651753\pi$
$824$	0	0
$825$	−11.1373	−0.387750
$826$	0	0
$827$	17.7898	0.618612	0.309306	−	0.950963i	$-0.399903\pi$
$827$	17.7898	0.618612	0.309306	+	0.950963i	$0.399903\pi$
$828$	0	0
$829$	−49.6223	−1.72345	−0.861727	−	0.507371i	$-0.830617\pi$
$829$	−49.6223	−1.72345	−0.861727	+	0.507371i	$0.830617\pi$
$830$	0	0
$831$	37.6084	1.30462
$832$	0	0
$833$	9.85689	0.341521
$834$	0	0
$835$	60.4297	2.09126
$836$	0	0
$837$	−34.1146	−1.17917
$838$	0	0
$839$	−43.3742	−1.49744	−0.748721	−	0.662885i	$-0.769332\pi$
$839$	−43.3742	−1.49744	−0.748721	+	0.662885i	$0.769332\pi$
$840$	0	0
$841$	−6.83528	−0.235699
$842$	0	0
$843$	34.8382	1.19989
$844$	0	0
$845$	−56.7372	−1.95182
$846$	0	0
$847$	15.6782	0.538710
$848$	0	0
$849$	78.1974	2.68373
$850$	0	0
$851$	70.9002	2.43043
$852$	0	0
$853$	−52.0354	−1.78166	−0.890829	−	0.454339i	$-0.849876\pi$
$853$	−52.0354	−1.78166	−0.890829	+	0.454339i	$0.849876\pi$
$854$	0	0
$855$	97.7537	3.34311
$856$	0	0
$857$	−34.9752	−1.19473	−0.597365	−	0.801969i	$-0.703786\pi$
$857$	−34.9752	−1.19473	−0.597365	+	0.801969i	$0.703786\pi$
$858$	0	0
$859$	46.2471	1.57793	0.788965	−	0.614438i	$-0.210617\pi$
$859$	46.2471	1.57793	0.788965	+	0.614438i	$0.210617\pi$
$860$	0	0
$861$	−5.37750	−0.183265
$862$	0	0
$863$	25.4230	0.865408	0.432704	−	0.901536i	$-0.357559\pi$
$863$	25.4230	0.865408	0.432704	+	0.901536i	$0.357559\pi$
$864$	0	0
$865$	−46.2669	−1.57312
$866$	0	0
$867$	0.566196	0.0192290
$868$	0	0
$869$	−24.3129	−0.824760
$870$	0	0
$871$	−50.0153	−1.69471
$872$	0	0
$873$	97.9625	3.31553
$874$	0	0
$875$	21.6273	0.731135
$876$	0	0
$877$	−42.9765	−1.45121	−0.725606	−	0.688110i	$-0.758440\pi$
$877$	−42.9765	−1.45121	−0.725606	+	0.688110i	$0.758440\pi$
$878$	0	0
$879$	−13.4484	−0.453602
$880$	0	0
$881$	−22.8167	−0.768713	−0.384356	−	0.923185i	$-0.625577\pi$
$881$	−22.8167	−0.768713	−0.384356	+	0.923185i	$0.625577\pi$
$882$	0	0
$883$	2.86518	0.0964211	0.0482105	−	0.998837i	$-0.484648\pi$
$883$	2.86518	0.0964211	0.0482105	+	0.998837i	$0.484648\pi$
$884$	0	0
$885$	11.2048	0.376644
$886$	0	0
$887$	−33.4635	−1.12360	−0.561798	−	0.827275i	$-0.689890\pi$
$887$	−33.4635	−1.12360	−0.561798	+	0.827275i	$0.689890\pi$
$888$	0	0
$889$	−45.2313	−1.51701
$890$	0	0
$891$	87.2555	2.92317
$892$	0	0
$893$	18.2789	0.611680
$894$	0	0
$895$	−34.4455	−1.15139
$896$	0	0
$897$	149.862	5.00374
$898$	0	0
$899$	12.2365	0.408110
$900$	0	0
$901$	−58.6336	−1.95337
$902$	0	0
$903$	−59.0713	−1.96577
$904$	0	0
$905$	35.3045	1.17356
$906$	0	0
$907$	34.5940	1.14867	0.574337	−	0.818619i	$-0.305260\pi$
$907$	34.5940	1.14867	0.574337	+	0.818619i	$0.305260\pi$
$908$	0	0
$909$	56.7484	1.88222
$910$	0	0
$911$	−16.1897	−0.536388	−0.268194	−	0.963365i	$-0.586427\pi$
$911$	−16.1897	−0.536388	−0.268194	+	0.963365i	$0.586427\pi$
$912$	0	0
$913$	−9.31269	−0.308205
$914$	0	0
$915$	−25.4152	−0.840202
$916$	0	0
$917$	6.69962	0.221241
$918$	0	0
$919$	59.5114	1.96310	0.981549	−	0.191209i	$-0.0612407\pi$
$919$	59.5114	1.96310	0.981549	+	0.191209i	$0.0612407\pi$
$920$	0	0
$921$	−90.4896	−2.98173
$922$	0	0
$923$	39.8860	1.31286
$924$	0	0
$925$	−7.44116	−0.244664
$926$	0	0
$927$	−119.363	−3.92039
$928$	0	0
$929$	−39.2807	−1.28876	−0.644379	−	0.764706i	$-0.722884\pi$
$929$	−39.2807	−1.28876	−0.644379	+	0.764706i	$0.722884\pi$
$930$	0	0
$931$	13.6700	0.448016
$932$	0	0
$933$	77.2403	2.52873
$934$	0	0
$935$	42.3349	1.38450
$936$	0	0
$937$	−4.87273	−0.159185	−0.0795925	−	0.996827i	$-0.525362\pi$
$937$	−4.87273	−0.159185	−0.0795925	+	0.996827i	$0.525362\pi$
$938$	0	0
$939$	41.8111	1.36445
$940$	0	0
$941$	14.3248	0.466975	0.233487	−	0.972360i	$-0.424986\pi$
$941$	14.3248	0.466975	0.233487	+	0.972360i	$0.424986\pi$
$942$	0	0
$943$	−6.14290	−0.200040
$944$	0	0
$945$	67.8771	2.20804
$946$	0	0
$947$	12.8887	0.418826	0.209413	−	0.977827i	$-0.432845\pi$
$947$	12.8887	0.418826	0.209413	+	0.977827i	$0.432845\pi$
$948$	0	0
$949$	−51.1707	−1.66107
$950$	0	0
$951$	87.1697	2.82667
$952$	0	0
$953$	−42.1470	−1.36528	−0.682638	−	0.730757i	$-0.739167\pi$
$953$	−42.1470	−1.36528	−0.682638	+	0.730757i	$0.739167\pi$
$954$	0	0
$955$	11.3068	0.365880
$956$	0	0
$957$	−64.1066	−2.07227
$958$	0	0
$959$	−2.47562	−0.0799420
$960$	0	0
$961$	−24.2446	−0.782084
$962$	0	0
$963$	38.8479	1.25186
$964$	0	0
$965$	−34.4138	−1.10782
$966$	0	0
$967$	−18.4349	−0.592828	−0.296414	−	0.955060i	$-0.595791\pi$
$967$	−18.4349	−0.592828	−0.296414	+	0.955060i	$0.595791\pi$
$968$	0	0
$969$	−74.2342	−2.38475
$970$	0	0
$971$	61.1213	1.96147	0.980737	−	0.195330i	$-0.0625779\pi$
$971$	61.1213	1.96147	0.980737	+	0.195330i	$0.0625779\pi$
$972$	0	0
$973$	4.43549	0.142195
$974$	0	0
$975$	−15.7284	−0.503711
$976$	0	0
$977$	5.52698	0.176824	0.0884118	−	0.996084i	$-0.471821\pi$
$977$	5.52698	0.176824	0.0884118	+	0.996084i	$0.471821\pi$
$978$	0	0
$979$	−13.5565	−0.433266
$980$	0	0
$981$	−5.01600	−0.160149
$982$	0	0
$983$	−36.1706	−1.15366	−0.576831	−	0.816864i	$-0.695711\pi$
$983$	−36.1706	−1.15366	−0.576831	+	0.816864i	$0.695711\pi$
$984$	0	0
$985$	−56.7930	−1.80958
$986$	0	0
$987$	21.9232	0.697823
$988$	0	0
$989$	−67.4790	−2.14571
$990$	0	0
$991$	32.1201	1.02033	0.510164	−	0.860077i	$-0.329585\pi$
$991$	32.1201	1.02033	0.510164	+	0.860077i	$0.329585\pi$
$992$	0	0
$993$	32.0558	1.01726
$994$	0	0
$995$	−1.33399	−0.0422904
$996$	0	0
$997$	59.5135	1.88481	0.942406	−	0.334472i	$-0.108558\pi$
$997$	59.5135	1.88481	0.942406	+	0.334472i	$0.108558\pi$
$998$	0	0
$999$	119.412	3.77803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.4	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-3.18197 q^{3} -2.41204 q^{5} +2.14400 q^{7} +7.12495 q^{9} +O(q^{10})\)	\(q-3.18197 q^{3} -2.41204 q^{5} +2.14400 q^{7} +7.12495 q^{9} +4.27932 q^{11} +6.04339 q^{13} +7.67503 q^{15} -4.10147 q^{17} -5.68811 q^{19} -6.82215 q^{21} -7.79317 q^{23} +0.817913 q^{25} -13.1255 q^{27} +4.70794 q^{29} +2.59911 q^{31} -13.6167 q^{33} -5.17141 q^{35} -9.09774 q^{37} -19.2299 q^{39} +0.788241 q^{41} +8.65874 q^{43} -17.1856 q^{45} -3.21353 q^{47} -2.40326 q^{49} +13.0508 q^{51} +14.2957 q^{53} -10.3219 q^{55} +18.0994 q^{57} +1.45990 q^{59} -3.31142 q^{61} +15.2759 q^{63} -14.5769 q^{65} -8.27604 q^{67} +24.7976 q^{69} +6.59994 q^{71} -8.46722 q^{73} -2.60258 q^{75} +9.17487 q^{77} -5.68149 q^{79} +20.3900 q^{81} -2.17621 q^{83} +9.89289 q^{85} -14.9805 q^{87} -3.16790 q^{89} +12.9570 q^{91} -8.27031 q^{93} +13.7199 q^{95} +13.7492 q^{97} +30.4899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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