LMFDB - Embedded newform 6008.2.a.b.1.34

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			1.34
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63667 q^{3} +2.82337 q^{5} +2.24738 q^{7} -0.321314 q^{9} +O(q^{10})$	$q+1.63667 q^{3} +2.82337 q^{5} +2.24738 q^{7} -0.321314 q^{9} -3.70116 q^{11} -6.16300 q^{13} +4.62093 q^{15} -5.15994 q^{17} -0.846675 q^{19} +3.67821 q^{21} -4.10789 q^{23} +2.97143 q^{25} -5.43589 q^{27} -1.81243 q^{29} -4.67573 q^{31} -6.05758 q^{33} +6.34518 q^{35} -2.39820 q^{37} -10.0868 q^{39} +2.12496 q^{41} +3.93377 q^{43} -0.907189 q^{45} +10.7385 q^{47} -1.94930 q^{49} -8.44512 q^{51} +6.85969 q^{53} -10.4498 q^{55} -1.38573 q^{57} -12.6324 q^{59} -11.6644 q^{61} -0.722113 q^{63} -17.4004 q^{65} +13.2410 q^{67} -6.72326 q^{69} -5.49181 q^{71} +9.21478 q^{73} +4.86325 q^{75} -8.31791 q^{77} -1.24182 q^{79} -7.93282 q^{81} -8.53084 q^{83} -14.5684 q^{85} -2.96636 q^{87} -5.06679 q^{89} -13.8506 q^{91} -7.65263 q^{93} -2.39048 q^{95} +3.83936 q^{97} +1.18924 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.63667	0.944931	0.472466	−	0.881349i	$-0.343364\pi$
$3$	1.63667	0.944931	0.472466	+	0.881349i	$0.343364\pi$
$4$	0	0
$5$	2.82337	1.26265	0.631325	−	0.775518i	$-0.282511\pi$
$5$	2.82337	1.26265	0.631325	+	0.775518i	$0.282511\pi$
$6$	0	0
$7$	2.24738	0.849428	0.424714	−	0.905328i	$-0.360375\pi$
$7$	2.24738	0.849428	0.424714	+	0.905328i	$0.360375\pi$
$8$	0	0
$9$	−0.321314	−0.107105
$10$	0	0
$11$	−3.70116	−1.11594	−0.557971	−	0.829860i	$-0.688420\pi$
$11$	−3.70116	−1.11594	−0.557971	+	0.829860i	$0.688420\pi$
$12$	0	0
$13$	−6.16300	−1.70931	−0.854654	−	0.519197i	$-0.826231\pi$
$13$	−6.16300	−1.70931	−0.854654	+	0.519197i	$0.826231\pi$
$14$	0	0
$15$	4.62093	1.19312
$16$	0	0
$17$	−5.15994	−1.25147	−0.625735	−	0.780036i	$-0.715201\pi$
$17$	−5.15994	−1.25147	−0.625735	+	0.780036i	$0.715201\pi$
$18$	0	0
$19$	−0.846675	−0.194241	−0.0971203	−	0.995273i	$-0.530963\pi$
$19$	−0.846675	−0.194241	−0.0971203	+	0.995273i	$0.530963\pi$
$20$	0	0
$21$	3.67821	0.802651
$22$	0	0
$23$	−4.10789	−0.856554	−0.428277	−	0.903647i	$-0.640879\pi$
$23$	−4.10789	−0.856554	−0.428277	+	0.903647i	$0.640879\pi$
$24$	0	0
$25$	2.97143	0.594286
$26$	0	0
$27$	−5.43589	−1.04614
$28$	0	0
$29$	−1.81243	−0.336561	−0.168280	−	0.985739i	$-0.553821\pi$
$29$	−1.81243	−0.336561	−0.168280	+	0.985739i	$0.553821\pi$
$30$	0	0
$31$	−4.67573	−0.839787	−0.419893	−	0.907573i	$-0.637933\pi$
$31$	−4.67573	−0.839787	−0.419893	+	0.907573i	$0.637933\pi$
$32$	0	0
$33$	−6.05758	−1.05449
$34$	0	0
$35$	6.34518	1.07253
$36$	0	0
$37$	−2.39820	−0.394262	−0.197131	−	0.980377i	$-0.563162\pi$
$37$	−2.39820	−0.394262	−0.197131	+	0.980377i	$0.563162\pi$
$38$	0	0
$39$	−10.0868	−1.61518
$40$	0	0
$41$	2.12496	0.331864	0.165932	−	0.986137i	$-0.446937\pi$
$41$	2.12496	0.331864	0.165932	+	0.986137i	$0.446937\pi$
$42$	0	0
$43$	3.93377	0.599894	0.299947	−	0.953956i	$-0.403031\pi$
$43$	3.93377	0.599894	0.299947	+	0.953956i	$0.403031\pi$
$44$	0	0
$45$	−0.907189	−0.135236
$46$	0	0
$47$	10.7385	1.56637	0.783187	−	0.621786i	$-0.213592\pi$
$47$	10.7385	1.56637	0.783187	+	0.621786i	$0.213592\pi$
$48$	0	0
$49$	−1.94930	−0.278472
$50$	0	0
$51$	−8.44512	−1.18255
$52$	0	0
$53$	6.85969	0.942251	0.471126	−	0.882066i	$-0.343848\pi$
$53$	6.85969	0.942251	0.471126	+	0.882066i	$0.343848\pi$
$54$	0	0
$55$	−10.4498	−1.40905
$56$	0	0
$57$	−1.38573	−0.183544
$58$	0	0
$59$	−12.6324	−1.64460	−0.822302	−	0.569051i	$-0.807311\pi$
$59$	−12.6324	−1.64460	−0.822302	+	0.569051i	$0.807311\pi$
$60$	0	0
$61$	−11.6644	−1.49347	−0.746736	−	0.665121i	$-0.768380\pi$
$61$	−11.6644	−1.49347	−0.746736	+	0.665121i	$0.768380\pi$
$62$	0	0
$63$	−0.722113	−0.0909777
$64$	0	0
$65$	−17.4004	−2.15826
$66$	0	0
$67$	13.2410	1.61765	0.808824	−	0.588051i	$-0.200104\pi$
$67$	13.2410	1.61765	0.808824	+	0.588051i	$0.200104\pi$
$68$	0	0
$69$	−6.72326	−0.809385
$70$	0	0
$71$	−5.49181	−0.651758	−0.325879	−	0.945412i	$-0.605660\pi$
$71$	−5.49181	−0.651758	−0.325879	+	0.945412i	$0.605660\pi$
$72$	0	0
$73$	9.21478	1.07851	0.539254	−	0.842143i	$-0.318706\pi$
$73$	9.21478	1.07851	0.539254	+	0.842143i	$0.318706\pi$
$74$	0	0
$75$	4.86325	0.561560
$76$	0	0
$77$	−8.31791	−0.947913
$78$	0	0
$79$	−1.24182	−0.139716	−0.0698581	−	0.997557i	$-0.522255\pi$
$79$	−1.24182	−0.139716	−0.0698581	+	0.997557i	$0.522255\pi$
$80$	0	0
$81$	−7.93282	−0.881424
$82$	0	0
$83$	−8.53084	−0.936381	−0.468191	−	0.883627i	$-0.655094\pi$
$83$	−8.53084	−0.936381	−0.468191	+	0.883627i	$0.655094\pi$
$84$	0	0
$85$	−14.5684	−1.58017
$86$	0	0
$87$	−2.96636	−0.318027
$88$	0	0
$89$	−5.06679	−0.537079	−0.268539	−	0.963269i	$-0.586541\pi$
$89$	−5.06679	−0.537079	−0.268539	+	0.963269i	$0.586541\pi$
$90$	0	0
$91$	−13.8506	−1.45194
$92$	0	0
$93$	−7.65263	−0.793541
$94$	0	0
$95$	−2.39048	−0.245258
$96$	0	0
$97$	3.83936	0.389828	0.194914	−	0.980820i	$-0.437557\pi$
$97$	3.83936	0.389828	0.194914	+	0.980820i	$0.437557\pi$
$98$	0	0
$99$	1.18924	0.119523
$100$	0	0
$101$	2.79839	0.278450	0.139225	−	0.990261i	$-0.455539\pi$
$101$	2.79839	0.278450	0.139225	+	0.990261i	$0.455539\pi$
$102$	0	0
$103$	6.60990	0.651293	0.325646	−	0.945492i	$-0.394418\pi$
$103$	6.60990	0.651293	0.325646	+	0.945492i	$0.394418\pi$
$104$	0	0
$105$	10.3850	1.01347
$106$	0	0
$107$	17.5247	1.69418	0.847091	−	0.531448i	$-0.178352\pi$
$107$	17.5247	1.69418	0.847091	+	0.531448i	$0.178352\pi$
$108$	0	0
$109$	0.585236	0.0560554	0.0280277	−	0.999607i	$-0.491077\pi$
$109$	0.585236	0.0560554	0.0280277	+	0.999607i	$0.491077\pi$
$110$	0	0
$111$	−3.92506	−0.372550
$112$	0	0
$113$	−4.42019	−0.415817	−0.207908	−	0.978148i	$-0.566666\pi$
$113$	−4.42019	−0.415817	−0.207908	+	0.978148i	$0.566666\pi$
$114$	0	0
$115$	−11.5981	−1.08153
$116$	0	0
$117$	1.98026	0.183075
$118$	0	0
$119$	−11.5963	−1.06303
$120$	0	0
$121$	2.69861	0.245328
$122$	0	0
$123$	3.47786	0.313588
$124$	0	0
$125$	−5.72741	−0.512275
$126$	0	0
$127$	−11.5338	−1.02346	−0.511729	−	0.859147i	$-0.670995\pi$
$127$	−11.5338	−1.02346	−0.511729	+	0.859147i	$0.670995\pi$
$128$	0	0
$129$	6.43828	0.566859
$130$	0	0
$131$	3.20736	0.280228	0.140114	−	0.990135i	$-0.455253\pi$
$131$	3.20736	0.280228	0.140114	+	0.990135i	$0.455253\pi$
$132$	0	0
$133$	−1.90280	−0.164993
$134$	0	0
$135$	−15.3475	−1.32091
$136$	0	0
$137$	−3.04503	−0.260155	−0.130077	−	0.991504i	$-0.541523\pi$
$137$	−3.04503	−0.260155	−0.130077	+	0.991504i	$0.541523\pi$
$138$	0	0
$139$	−15.8028	−1.34037	−0.670186	−	0.742193i	$-0.733786\pi$
$139$	−15.8028	−1.34037	−0.670186	+	0.742193i	$0.733786\pi$
$140$	0	0
$141$	17.5754	1.48012
$142$	0	0
$143$	22.8103	1.90749
$144$	0	0
$145$	−5.11718	−0.424958
$146$	0	0
$147$	−3.19036	−0.263137
$148$	0	0
$149$	16.0924	1.31834	0.659170	−	0.751994i	$-0.270908\pi$
$149$	16.0924	1.31834	0.659170	+	0.751994i	$0.270908\pi$
$150$	0	0
$151$	−7.50521	−0.610765	−0.305383	−	0.952230i	$-0.598784\pi$
$151$	−7.50521	−0.610765	−0.305383	+	0.952230i	$0.598784\pi$
$152$	0	0
$153$	1.65796	0.134038
$154$	0	0
$155$	−13.2013	−1.06036
$156$	0	0
$157$	−10.9383	−0.872967	−0.436484	−	0.899712i	$-0.643776\pi$
$157$	−10.9383	−0.872967	−0.436484	+	0.899712i	$0.643776\pi$
$158$	0	0
$159$	11.2270	0.890363
$160$	0	0
$161$	−9.23197	−0.727581
$162$	0	0
$163$	8.47019	0.663437	0.331718	−	0.943378i	$-0.392372\pi$
$163$	8.47019	0.663437	0.331718	+	0.943378i	$0.392372\pi$
$164$	0	0
$165$	−17.1028	−1.33145
$166$	0	0
$167$	−21.2987	−1.64815	−0.824073	−	0.566484i	$-0.808303\pi$
$167$	−21.2987	−1.64815	−0.824073	+	0.566484i	$0.808303\pi$
$168$	0	0
$169$	24.9826	1.92174
$170$	0	0
$171$	0.272048	0.0208041
$172$	0	0
$173$	20.8485	1.58508	0.792542	−	0.609817i	$-0.208757\pi$
$173$	20.8485	1.58508	0.792542	+	0.609817i	$0.208757\pi$
$174$	0	0
$175$	6.67792	0.504803
$176$	0	0
$177$	−20.6751	−1.55404
$178$	0	0
$179$	11.1161	0.830859	0.415430	−	0.909625i	$-0.363631\pi$
$179$	11.1161	0.830859	0.415430	+	0.909625i	$0.363631\pi$
$180$	0	0
$181$	−2.05779	−0.152955	−0.0764773	−	0.997071i	$-0.524367\pi$
$181$	−2.05779	−0.152955	−0.0764773	+	0.997071i	$0.524367\pi$
$182$	0	0
$183$	−19.0907	−1.41123
$184$	0	0
$185$	−6.77101	−0.497815
$186$	0	0
$187$	19.0978	1.39657
$188$	0	0
$189$	−12.2165	−0.888619
$190$	0	0
$191$	−14.9637	−1.08274	−0.541368	−	0.840786i	$-0.682093\pi$
$191$	−14.9637	−1.08274	−0.541368	+	0.840786i	$0.682093\pi$
$192$	0	0
$193$	23.4561	1.68841	0.844204	−	0.536022i	$-0.180074\pi$
$193$	23.4561	1.68841	0.844204	+	0.536022i	$0.180074\pi$
$194$	0	0
$195$	−28.4788	−2.03941
$196$	0	0
$197$	−2.89264	−0.206092	−0.103046	−	0.994677i	$-0.532859\pi$
$197$	−2.89264	−0.206092	−0.103046	+	0.994677i	$0.532859\pi$
$198$	0	0
$199$	−24.8205	−1.75948	−0.879739	−	0.475458i	$-0.842282\pi$
$199$	−24.8205	−1.75948	−0.879739	+	0.475458i	$0.842282\pi$
$200$	0	0
$201$	21.6712	1.52857
$202$	0	0
$203$	−4.07322	−0.285884
$204$	0	0
$205$	5.99956	0.419028
$206$	0	0
$207$	1.31992	0.0917409
$208$	0	0
$209$	3.13368	0.216761
$210$	0	0
$211$	−10.9749	−0.755543	−0.377772	−	0.925899i	$-0.623310\pi$
$211$	−10.9749	−0.755543	−0.377772	+	0.925899i	$0.623310\pi$
$212$	0	0
$213$	−8.98827	−0.615866
$214$	0	0
$215$	11.1065	0.757456
$216$	0	0
$217$	−10.5081	−0.713339
$218$	0	0
$219$	15.0816	1.01912
$220$	0	0
$221$	31.8007	2.13915
$222$	0	0
$223$	22.7524	1.52361	0.761807	−	0.647804i	$-0.224312\pi$
$223$	22.7524	1.52361	0.761807	+	0.647804i	$0.224312\pi$
$224$	0	0
$225$	−0.954762	−0.0636508
$226$	0	0
$227$	15.2523	1.01233	0.506165	−	0.862437i	$-0.331063\pi$
$227$	15.2523	1.01233	0.506165	+	0.862437i	$0.331063\pi$
$228$	0	0
$229$	−4.67557	−0.308971	−0.154485	−	0.987995i	$-0.549372\pi$
$229$	−4.67557	−0.308971	−0.154485	+	0.987995i	$0.549372\pi$
$230$	0	0
$231$	−13.6137	−0.895713
$232$	0	0
$233$	−2.21870	−0.145352	−0.0726760	−	0.997356i	$-0.523154\pi$
$233$	−2.21870	−0.145352	−0.0726760	+	0.997356i	$0.523154\pi$
$234$	0	0
$235$	30.3189	1.97778
$236$	0	0
$237$	−2.03246	−0.132022
$238$	0	0
$239$	6.75297	0.436813	0.218407	−	0.975858i	$-0.429914\pi$
$239$	6.75297	0.436813	0.218407	+	0.975858i	$0.429914\pi$
$240$	0	0
$241$	8.00501	0.515648	0.257824	−	0.966192i	$-0.416995\pi$
$241$	8.00501	0.515648	0.257824	+	0.966192i	$0.416995\pi$
$242$	0	0
$243$	3.32428	0.213253
$244$	0	0
$245$	−5.50360	−0.351612
$246$	0	0
$247$	5.21806	0.332017
$248$	0	0
$249$	−13.9622	−0.884816
$250$	0	0
$251$	3.48447	0.219938	0.109969	−	0.993935i	$-0.464925\pi$
$251$	3.48447	0.219938	0.109969	+	0.993935i	$0.464925\pi$
$252$	0	0
$253$	15.2040	0.955865
$254$	0	0
$255$	−23.8437	−1.49315
$256$	0	0
$257$	−7.75768	−0.483911	−0.241955	−	0.970287i	$-0.577789\pi$
$257$	−7.75768	−0.483911	−0.241955	+	0.970287i	$0.577789\pi$
$258$	0	0
$259$	−5.38966	−0.334897
$260$	0	0
$261$	0.582360	0.0360472
$262$	0	0
$263$	13.5499	0.835521	0.417760	−	0.908557i	$-0.362815\pi$
$263$	13.5499	0.835521	0.417760	+	0.908557i	$0.362815\pi$
$264$	0	0
$265$	19.3675	1.18973
$266$	0	0
$267$	−8.29266	−0.507503
$268$	0	0
$269$	7.18871	0.438303	0.219152	−	0.975691i	$-0.429671\pi$
$269$	7.18871	0.438303	0.219152	+	0.975691i	$0.429671\pi$
$270$	0	0
$271$	−9.70848	−0.589748	−0.294874	−	0.955536i	$-0.595278\pi$
$271$	−9.70848	−0.589748	−0.294874	+	0.955536i	$0.595278\pi$
$272$	0	0
$273$	−22.6688	−1.37198
$274$	0	0
$275$	−10.9977	−0.663189
$276$	0	0
$277$	25.3127	1.52089	0.760446	−	0.649401i	$-0.224980\pi$
$277$	25.3127	1.52089	0.760446	+	0.649401i	$0.224980\pi$
$278$	0	0
$279$	1.50238	0.0899450
$280$	0	0
$281$	13.7441	0.819902	0.409951	−	0.912107i	$-0.365546\pi$
$281$	13.7441	0.819902	0.409951	+	0.912107i	$0.365546\pi$
$282$	0	0
$283$	−7.60434	−0.452031	−0.226016	−	0.974124i	$-0.572570\pi$
$283$	−7.60434	−0.452031	−0.226016	+	0.974124i	$0.572570\pi$
$284$	0	0
$285$	−3.91242	−0.231752
$286$	0	0
$287$	4.77559	0.281894
$288$	0	0
$289$	9.62503	0.566178
$290$	0	0
$291$	6.28377	0.368361
$292$	0	0
$293$	22.1781	1.29566	0.647828	−	0.761787i	$-0.275678\pi$
$293$	22.1781	1.29566	0.647828	+	0.761787i	$0.275678\pi$
$294$	0	0
$295$	−35.6661	−2.07656
$296$	0	0
$297$	20.1191	1.16743
$298$	0	0
$299$	25.3169	1.46412
$300$	0	0
$301$	8.84065	0.509567
$302$	0	0
$303$	4.58003	0.263116
$304$	0	0
$305$	−32.9329	−1.88573
$306$	0	0
$307$	−4.69321	−0.267856	−0.133928	−	0.990991i	$-0.542759\pi$
$307$	−4.69321	−0.267856	−0.133928	+	0.990991i	$0.542759\pi$
$308$	0	0
$309$	10.8182	0.615427
$310$	0	0
$311$	9.00452	0.510599	0.255300	−	0.966862i	$-0.417826\pi$
$311$	9.00452	0.510599	0.255300	+	0.966862i	$0.417826\pi$
$312$	0	0
$313$	2.12786	0.120274	0.0601368	−	0.998190i	$-0.480846\pi$
$313$	2.12786	0.120274	0.0601368	+	0.998190i	$0.480846\pi$
$314$	0	0
$315$	−2.03879	−0.114873
$316$	0	0
$317$	−5.97888	−0.335807	−0.167904	−	0.985803i	$-0.553700\pi$
$317$	−5.97888	−0.335807	−0.167904	+	0.985803i	$0.553700\pi$
$318$	0	0
$319$	6.70812	0.375582
$320$	0	0
$321$	28.6822	1.60089
$322$	0	0
$323$	4.36879	0.243086
$324$	0	0
$325$	−18.3129	−1.01582
$326$	0	0
$327$	0.957837	0.0529685
$328$	0	0
$329$	24.1335	1.33052
$330$	0	0
$331$	7.61128	0.418354	0.209177	−	0.977878i	$-0.432922\pi$
$331$	7.61128	0.418354	0.209177	+	0.977878i	$0.432922\pi$
$332$	0	0
$333$	0.770575	0.0422272
$334$	0	0
$335$	37.3843	2.04252
$336$	0	0
$337$	26.8154	1.46073	0.730365	−	0.683057i	$-0.239350\pi$
$337$	26.8154	1.46073	0.730365	+	0.683057i	$0.239350\pi$
$338$	0	0
$339$	−7.23439	−0.392918
$340$	0	0
$341$	17.3057	0.937154
$342$	0	0
$343$	−20.1124	−1.08597
$344$	0	0
$345$	−18.9823	−1.02197
$346$	0	0
$347$	−16.5979	−0.891021	−0.445511	−	0.895277i	$-0.646978\pi$
$347$	−16.5979	−0.891021	−0.445511	+	0.895277i	$0.646978\pi$
$348$	0	0
$349$	−1.73924	−0.0930993	−0.0465497	−	0.998916i	$-0.514823\pi$
$349$	−1.73924	−0.0930993	−0.0465497	+	0.998916i	$0.514823\pi$
$350$	0	0
$351$	33.5014	1.78817
$352$	0	0
$353$	−16.3115	−0.868176	−0.434088	−	0.900871i	$-0.642929\pi$
$353$	−16.3115	−0.868176	−0.434088	+	0.900871i	$0.642929\pi$
$354$	0	0
$355$	−15.5054	−0.822942
$356$	0	0
$357$	−18.9794	−1.00449
$358$	0	0
$359$	−3.90195	−0.205937	−0.102968	−	0.994685i	$-0.532834\pi$
$359$	−3.90195	−0.205937	−0.102968	+	0.994685i	$0.532834\pi$
$360$	0	0
$361$	−18.2831	−0.962271
$362$	0	0
$363$	4.41673	0.231818
$364$	0	0
$365$	26.0168	1.36178
$366$	0	0
$367$	−20.4779	−1.06894	−0.534468	−	0.845189i	$-0.679488\pi$
$367$	−20.4779	−1.06894	−0.534468	+	0.845189i	$0.679488\pi$
$368$	0	0
$369$	−0.682780	−0.0355441
$370$	0	0
$371$	15.4163	0.800375
$372$	0	0
$373$	−16.7146	−0.865450	−0.432725	−	0.901526i	$-0.642448\pi$
$373$	−16.7146	−0.865450	−0.432725	+	0.901526i	$0.642448\pi$
$374$	0	0
$375$	−9.37387	−0.484065
$376$	0	0
$377$	11.1700	0.575286
$378$	0	0
$379$	−10.8625	−0.557968	−0.278984	−	0.960296i	$-0.589998\pi$
$379$	−10.8625	−0.557968	−0.278984	+	0.960296i	$0.589998\pi$
$380$	0	0
$381$	−18.8770	−0.967097
$382$	0	0
$383$	−29.8421	−1.52486	−0.762430	−	0.647071i	$-0.775994\pi$
$383$	−29.8421	−1.52486	−0.762430	+	0.647071i	$0.775994\pi$
$384$	0	0
$385$	−23.4845	−1.19688
$386$	0	0
$387$	−1.26397	−0.0642514
$388$	0	0
$389$	−0.257173	−0.0130392	−0.00651960	−	0.999979i	$-0.502075\pi$
$389$	−0.257173	−0.0130392	−0.00651960	+	0.999979i	$0.502075\pi$
$390$	0	0
$391$	21.1965	1.07195
$392$	0	0
$393$	5.24939	0.264797
$394$	0	0
$395$	−3.50613	−0.176413
$396$	0	0
$397$	−20.5449	−1.03112	−0.515559	−	0.856854i	$-0.672416\pi$
$397$	−20.5449	−1.03112	−0.515559	+	0.856854i	$0.672416\pi$
$398$	0	0
$399$	−3.11425	−0.155907
$400$	0	0
$401$	−14.9742	−0.747775	−0.373888	−	0.927474i	$-0.621975\pi$
$401$	−14.9742	−0.747775	−0.373888	+	0.927474i	$0.621975\pi$
$402$	0	0
$403$	28.8166	1.43545
$404$	0	0
$405$	−22.3973	−1.11293
$406$	0	0
$407$	8.87613	0.439973
$408$	0	0
$409$	−38.6274	−1.91000	−0.955001	−	0.296602i	$-0.904147\pi$
$409$	−38.6274	−1.91000	−0.955001	+	0.296602i	$0.904147\pi$
$410$	0	0
$411$	−4.98371	−0.245828
$412$	0	0
$413$	−28.3899	−1.39697
$414$	0	0
$415$	−24.0857	−1.18232
$416$	0	0
$417$	−25.8639	−1.26656
$418$	0	0
$419$	−7.25831	−0.354591	−0.177296	−	0.984158i	$-0.556735\pi$
$419$	−7.25831	−0.354591	−0.177296	+	0.984158i	$0.556735\pi$
$420$	0	0
$421$	21.1192	1.02929	0.514643	−	0.857405i	$-0.327925\pi$
$421$	21.1192	1.02929	0.514643	+	0.857405i	$0.327925\pi$
$422$	0	0
$423$	−3.45044	−0.167766
$424$	0	0
$425$	−15.3324	−0.743731
$426$	0	0
$427$	−26.2143	−1.26860
$428$	0	0
$429$	37.3329	1.80245
$430$	0	0
$431$	4.86169	0.234179	0.117090	−	0.993121i	$-0.462643\pi$
$431$	4.86169	0.234179	0.117090	+	0.993121i	$0.462643\pi$
$432$	0	0
$433$	−7.17595	−0.344854	−0.172427	−	0.985022i	$-0.555161\pi$
$433$	−7.17595	−0.344854	−0.172427	+	0.985022i	$0.555161\pi$
$434$	0	0
$435$	−8.37513	−0.401557
$436$	0	0
$437$	3.47805	0.166378
$438$	0	0
$439$	−27.2703	−1.30154	−0.650770	−	0.759275i	$-0.725554\pi$
$439$	−27.2703	−1.30154	−0.650770	+	0.759275i	$0.725554\pi$
$440$	0	0
$441$	0.626337	0.0298256
$442$	0	0
$443$	−0.147529	−0.00700931	−0.00350466	−	0.999994i	$-0.501116\pi$
$443$	−0.147529	−0.00700931	−0.00350466	+	0.999994i	$0.501116\pi$
$444$	0	0
$445$	−14.3054	−0.678143
$446$	0	0
$447$	26.3379	1.24574
$448$	0	0
$449$	19.1787	0.905096	0.452548	−	0.891740i	$-0.350515\pi$
$449$	19.1787	0.905096	0.452548	+	0.891740i	$0.350515\pi$
$450$	0	0
$451$	−7.86484	−0.370341
$452$	0	0
$453$	−12.2835	−0.577131
$454$	0	0
$455$	−39.1053	−1.83329
$456$	0	0
$457$	21.4999	1.00572	0.502861	−	0.864367i	$-0.332281\pi$
$457$	21.4999	1.00572	0.502861	+	0.864367i	$0.332281\pi$
$458$	0	0
$459$	28.0489	1.30921
$460$	0	0
$461$	−34.9411	−1.62737	−0.813685	−	0.581306i	$-0.802542\pi$
$461$	−34.9411	−1.62737	−0.813685	+	0.581306i	$0.802542\pi$
$462$	0	0
$463$	−3.18896	−0.148204	−0.0741018	−	0.997251i	$-0.523609\pi$
$463$	−3.18896	−0.148204	−0.0741018	+	0.997251i	$0.523609\pi$
$464$	0	0
$465$	−21.6062	−1.00196
$466$	0	0
$467$	−20.0791	−0.929149	−0.464575	−	0.885534i	$-0.653793\pi$
$467$	−20.0791	−0.929149	−0.464575	+	0.885534i	$0.653793\pi$
$468$	0	0
$469$	29.7576	1.37408
$470$	0	0
$471$	−17.9023	−0.824894
$472$	0	0
$473$	−14.5595	−0.669447
$474$	0	0
$475$	−2.51583	−0.115434
$476$	0	0
$477$	−2.20411	−0.100919
$478$	0	0
$479$	7.78398	0.355659	0.177830	−	0.984061i	$-0.443092\pi$
$479$	7.78398	0.355659	0.177830	+	0.984061i	$0.443092\pi$
$480$	0	0
$481$	14.7801	0.673915
$482$	0	0
$483$	−15.1097	−0.687514
$484$	0	0
$485$	10.8400	0.492217
$486$	0	0
$487$	33.3208	1.50991	0.754955	−	0.655776i	$-0.227659\pi$
$487$	33.3208	1.50991	0.754955	+	0.655776i	$0.227659\pi$
$488$	0	0
$489$	13.8629	0.626902
$490$	0	0
$491$	16.4851	0.743962	0.371981	−	0.928240i	$-0.378679\pi$
$491$	16.4851	0.743962	0.371981	+	0.928240i	$0.378679\pi$
$492$	0	0
$493$	9.35206	0.421196
$494$	0	0
$495$	3.35765	0.150915
$496$	0	0
$497$	−12.3422	−0.553621
$498$	0	0
$499$	−2.06220	−0.0923167	−0.0461584	−	0.998934i	$-0.514698\pi$
$499$	−2.06220	−0.0923167	−0.0461584	+	0.998934i	$0.514698\pi$
$500$	0	0
$501$	−34.8590	−1.55738
$502$	0	0
$503$	22.6791	1.01121	0.505604	−	0.862765i	$-0.331270\pi$
$503$	22.6791	1.01121	0.505604	+	0.862765i	$0.331270\pi$
$504$	0	0
$505$	7.90089	0.351585
$506$	0	0
$507$	40.8882	1.81591
$508$	0	0
$509$	−5.17625	−0.229433	−0.114717	−	0.993398i	$-0.536596\pi$
$509$	−5.17625	−0.229433	−0.114717	+	0.993398i	$0.536596\pi$
$510$	0	0
$511$	20.7091	0.916116
$512$	0	0
$513$	4.60243	0.203202
$514$	0	0
$515$	18.6622	0.822355
$516$	0	0
$517$	−39.7450	−1.74798
$518$	0	0
$519$	34.1222	1.49780
$520$	0	0
$521$	8.89698	0.389784	0.194892	−	0.980825i	$-0.437564\pi$
$521$	8.89698	0.389784	0.194892	+	0.980825i	$0.437564\pi$
$522$	0	0
$523$	−23.2575	−1.01698	−0.508489	−	0.861068i	$-0.669796\pi$
$523$	−23.2575	−1.01698	−0.508489	+	0.861068i	$0.669796\pi$
$524$	0	0
$525$	10.9295	0.477005
$526$	0	0
$527$	24.1265	1.05097
$528$	0	0
$529$	−6.12525	−0.266315
$530$	0	0
$531$	4.05898	0.176145
$532$	0	0
$533$	−13.0962	−0.567257
$534$	0	0
$535$	49.4789	2.13916
$536$	0	0
$537$	18.1934	0.785105
$538$	0	0
$539$	7.21468	0.310758
$540$	0	0
$541$	−4.51972	−0.194318	−0.0971590	−	0.995269i	$-0.530976\pi$
$541$	−4.51972	−0.194318	−0.0971590	+	0.995269i	$0.530976\pi$
$542$	0	0
$543$	−3.36793	−0.144532
$544$	0	0
$545$	1.65234	0.0707784
$546$	0	0
$547$	−2.81470	−0.120348	−0.0601739	−	0.998188i	$-0.519166\pi$
$547$	−2.81470	−0.120348	−0.0601739	+	0.998188i	$0.519166\pi$
$548$	0	0
$549$	3.74793	0.159958
$550$	0	0
$551$	1.53454	0.0653737
$552$	0	0
$553$	−2.79085	−0.118679
$554$	0	0
$555$	−11.0819	−0.470401
$556$	0	0
$557$	−16.5865	−0.702794	−0.351397	−	0.936227i	$-0.614293\pi$
$557$	−16.5865	−0.702794	−0.351397	+	0.936227i	$0.614293\pi$
$558$	0	0
$559$	−24.2438	−1.02540
$560$	0	0
$561$	31.2568	1.31966
$562$	0	0
$563$	−37.4842	−1.57977	−0.789885	−	0.613255i	$-0.789860\pi$
$563$	−37.4842	−1.57977	−0.789885	+	0.613255i	$0.789860\pi$
$564$	0	0
$565$	−12.4798	−0.525031
$566$	0	0
$567$	−17.8280	−0.748706
$568$	0	0
$569$	−12.5148	−0.524649	−0.262324	−	0.964980i	$-0.584489\pi$
$569$	−12.5148	−0.524649	−0.262324	+	0.964980i	$0.584489\pi$
$570$	0	0
$571$	15.6114	0.653319	0.326659	−	0.945142i	$-0.394077\pi$
$571$	15.6114	0.653319	0.326659	+	0.945142i	$0.394077\pi$
$572$	0	0
$573$	−24.4906	−1.02311
$574$	0	0
$575$	−12.2063	−0.509038
$576$	0	0
$577$	17.9380	0.746769	0.373385	−	0.927677i	$-0.378197\pi$
$577$	17.9380	0.746769	0.373385	+	0.927677i	$0.378197\pi$
$578$	0	0
$579$	38.3899	1.59543
$580$	0	0
$581$	−19.1720	−0.795389
$582$	0	0
$583$	−25.3888	−1.05150
$584$	0	0
$585$	5.59100	0.231160
$586$	0	0
$587$	−27.1614	−1.12107	−0.560536	−	0.828130i	$-0.689405\pi$
$587$	−27.1614	−1.12107	−0.560536	+	0.828130i	$0.689405\pi$
$588$	0	0
$589$	3.95883	0.163121
$590$	0	0
$591$	−4.73430	−0.194743
$592$	0	0
$593$	13.3843	0.549627	0.274813	−	0.961498i	$-0.411384\pi$
$593$	13.3843	0.549627	0.274813	+	0.961498i	$0.411384\pi$
$594$	0	0
$595$	−32.7408	−1.34224
$596$	0	0
$597$	−40.6229	−1.66259
$598$	0	0
$599$	−44.4158	−1.81478	−0.907391	−	0.420288i	$-0.861929\pi$
$599$	−44.4158	−1.81478	−0.907391	+	0.420288i	$0.861929\pi$
$600$	0	0
$601$	19.7666	0.806296	0.403148	−	0.915135i	$-0.367916\pi$
$601$	19.7666	0.806296	0.403148	+	0.915135i	$0.367916\pi$
$602$	0	0
$603$	−4.25452	−0.173258
$604$	0	0
$605$	7.61918	0.309764
$606$	0	0
$607$	−29.6081	−1.20176	−0.600878	−	0.799341i	$-0.705182\pi$
$607$	−29.6081	−1.20176	−0.600878	+	0.799341i	$0.705182\pi$
$608$	0	0
$609$	−6.66652	−0.270141
$610$	0	0
$611$	−66.1815	−2.67742
$612$	0	0
$613$	41.6235	1.68116	0.840579	−	0.541689i	$-0.182215\pi$
$613$	41.6235	1.68116	0.840579	+	0.541689i	$0.182215\pi$
$614$	0	0
$615$	9.81930	0.395953
$616$	0	0
$617$	−35.2189	−1.41786	−0.708929	−	0.705280i	$-0.750821\pi$
$617$	−35.2189	−1.41786	−0.708929	+	0.705280i	$0.750821\pi$
$618$	0	0
$619$	−30.3909	−1.22152	−0.610758	−	0.791818i	$-0.709135\pi$
$619$	−30.3909	−1.22152	−0.610758	+	0.791818i	$0.709135\pi$
$620$	0	0
$621$	22.3300	0.896074
$622$	0	0
$623$	−11.3870	−0.456210
$624$	0	0
$625$	−31.0278	−1.24111
$626$	0	0
$627$	5.12880	0.204825
$628$	0	0
$629$	12.3746	0.493407
$630$	0	0
$631$	9.64462	0.383946	0.191973	−	0.981400i	$-0.438511\pi$
$631$	9.64462	0.383946	0.191973	+	0.981400i	$0.438511\pi$
$632$	0	0
$633$	−17.9623	−0.713937
$634$	0	0
$635$	−32.5641	−1.29227
$636$	0	0
$637$	12.0135	0.475994
$638$	0	0
$639$	1.76459	0.0698062
$640$	0	0
$641$	28.4017	1.12180	0.560900	−	0.827884i	$-0.310455\pi$
$641$	28.4017	1.12180	0.560900	+	0.827884i	$0.310455\pi$
$642$	0	0
$643$	−29.8111	−1.17564	−0.587818	−	0.808993i	$-0.700013\pi$
$643$	−29.8111	−1.17564	−0.587818	+	0.808993i	$0.700013\pi$
$644$	0	0
$645$	18.1776	0.715744
$646$	0	0
$647$	13.4399	0.528377	0.264188	−	0.964471i	$-0.414896\pi$
$647$	13.4399	0.528377	0.264188	+	0.964471i	$0.414896\pi$
$648$	0	0
$649$	46.7548	1.83528
$650$	0	0
$651$	−17.1983	−0.674056
$652$	0	0
$653$	28.0259	1.09674	0.548368	−	0.836237i	$-0.315249\pi$
$653$	28.0259	1.09674	0.548368	+	0.836237i	$0.315249\pi$
$654$	0	0
$655$	9.05557	0.353831
$656$	0	0
$657$	−2.96084	−0.115513
$658$	0	0
$659$	−22.3742	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$659$	−22.3742	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$660$	0	0
$661$	38.4604	1.49593	0.747967	−	0.663736i	$-0.231030\pi$
$661$	38.4604	1.49593	0.747967	+	0.663736i	$0.231030\pi$
$662$	0	0
$663$	52.0473	2.02135
$664$	0	0
$665$	−5.37230	−0.208329
$666$	0	0
$667$	7.44528	0.288282
$668$	0	0
$669$	37.2382	1.43971
$670$	0	0
$671$	43.1718	1.66663
$672$	0	0
$673$	−31.1617	−1.20119	−0.600597	−	0.799552i	$-0.705070\pi$
$673$	−31.1617	−1.20119	−0.600597	+	0.799552i	$0.705070\pi$
$674$	0	0
$675$	−16.1524	−0.621705
$676$	0	0
$677$	18.6610	0.717201	0.358601	−	0.933491i	$-0.383254\pi$
$677$	18.6610	0.717201	0.358601	+	0.933491i	$0.383254\pi$
$678$	0	0
$679$	8.62849	0.331131
$680$	0	0
$681$	24.9629	0.956582
$682$	0	0
$683$	−27.0059	−1.03335	−0.516676	−	0.856181i	$-0.672831\pi$
$683$	−27.0059	−1.03335	−0.516676	+	0.856181i	$0.672831\pi$
$684$	0	0
$685$	−8.59726	−0.328484
$686$	0	0
$687$	−7.65237	−0.291956
$688$	0	0
$689$	−42.2763	−1.61060
$690$	0	0
$691$	13.4701	0.512425	0.256213	−	0.966620i	$-0.417525\pi$
$691$	13.4701	0.512425	0.256213	+	0.966620i	$0.417525\pi$
$692$	0	0
$693$	2.67266	0.101526
$694$	0	0
$695$	−44.6171	−1.69242
$696$	0	0
$697$	−10.9647	−0.415318
$698$	0	0
$699$	−3.63128	−0.137348
$700$	0	0
$701$	9.02012	0.340685	0.170343	−	0.985385i	$-0.445513\pi$
$701$	9.02012	0.340685	0.170343	+	0.985385i	$0.445513\pi$
$702$	0	0
$703$	2.03050	0.0765816
$704$	0	0
$705$	49.6219	1.86887
$706$	0	0
$707$	6.28903	0.236523
$708$	0	0
$709$	30.9131	1.16097	0.580483	−	0.814272i	$-0.302864\pi$
$709$	30.9131	1.16097	0.580483	+	0.814272i	$0.302864\pi$
$710$	0	0
$711$	0.399015	0.0149642
$712$	0	0
$713$	19.2074	0.719323
$714$	0	0
$715$	64.4019	2.40849
$716$	0	0
$717$	11.0524	0.412759
$718$	0	0
$719$	−2.57102	−0.0958828	−0.0479414	−	0.998850i	$-0.515266\pi$
$719$	−2.57102	−0.0958828	−0.0479414	+	0.998850i	$0.515266\pi$
$720$	0	0
$721$	14.8549	0.553227
$722$	0	0
$723$	13.1015	0.487252
$724$	0	0
$725$	−5.38552	−0.200013
$726$	0	0
$727$	−37.0959	−1.37581	−0.687906	−	0.725800i	$-0.741470\pi$
$727$	−37.0959	−1.37581	−0.687906	+	0.725800i	$0.741470\pi$
$728$	0	0
$729$	29.2392	1.08293
$730$	0	0
$731$	−20.2980	−0.750749
$732$	0	0
$733$	19.1665	0.707929	0.353965	−	0.935259i	$-0.384833\pi$
$733$	19.1665	0.707929	0.353965	+	0.935259i	$0.384833\pi$
$734$	0	0
$735$	−9.00758	−0.332249
$736$	0	0
$737$	−49.0072	−1.80520
$738$	0	0
$739$	−37.6726	−1.38581	−0.692904	−	0.721029i	$-0.743669\pi$
$739$	−37.6726	−1.38581	−0.692904	+	0.721029i	$0.743669\pi$
$740$	0	0
$741$	8.54023	0.313733
$742$	0	0
$743$	2.92293	0.107232	0.0536159	−	0.998562i	$-0.482925\pi$
$743$	2.92293	0.107232	0.0536159	+	0.998562i	$0.482925\pi$
$744$	0	0
$745$	45.4348	1.66460
$746$	0	0
$747$	2.74108	0.100291
$748$	0	0
$749$	39.3847	1.43909
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	5.70293	0.207826
$754$	0	0
$755$	−21.1900	−0.771183
$756$	0	0
$757$	−2.76853	−0.100624	−0.0503119	−	0.998734i	$-0.516022\pi$
$757$	−2.76853	−0.100624	−0.0503119	+	0.998734i	$0.516022\pi$
$758$	0	0
$759$	24.8839	0.903227
$760$	0	0
$761$	−34.7780	−1.26070	−0.630350	−	0.776311i	$-0.717089\pi$
$761$	−34.7780	−1.26070	−0.630350	+	0.776311i	$0.717089\pi$
$762$	0	0
$763$	1.31524	0.0476151
$764$	0	0
$765$	4.68104	0.169243
$766$	0	0
$767$	77.8538	2.81114
$768$	0	0
$769$	22.4400	0.809208	0.404604	−	0.914492i	$-0.367409\pi$
$769$	22.4400	0.809208	0.404604	+	0.914492i	$0.367409\pi$
$770$	0	0
$771$	−12.6968	−0.457263
$772$	0	0
$773$	−3.47901	−0.125131	−0.0625656	−	0.998041i	$-0.519928\pi$
$773$	−3.47901	−0.125131	−0.0625656	+	0.998041i	$0.519928\pi$
$774$	0	0
$775$	−13.8936	−0.499074
$776$	0	0
$777$	−8.82108	−0.316455
$778$	0	0
$779$	−1.79915	−0.0644614
$780$	0	0
$781$	20.3261	0.727324
$782$	0	0
$783$	9.85220	0.352089
$784$	0	0
$785$	−30.8828	−1.10225
$786$	0	0
$787$	−25.7043	−0.916259	−0.458129	−	0.888886i	$-0.651480\pi$
$787$	−25.7043	−0.916259	−0.458129	+	0.888886i	$0.651480\pi$
$788$	0	0
$789$	22.1767	0.789510
$790$	0	0
$791$	−9.93383	−0.353206
$792$	0	0
$793$	71.8876	2.55280
$794$	0	0
$795$	31.6981	1.12422
$796$	0	0
$797$	−42.2691	−1.49725	−0.748624	−	0.662995i	$-0.769285\pi$
$797$	−42.2691	−1.49725	−0.748624	+	0.662995i	$0.769285\pi$
$798$	0	0
$799$	−55.4102	−1.96027
$800$	0	0
$801$	1.62803	0.0575236
$802$	0	0
$803$	−34.1054	−1.20355
$804$	0	0
$805$	−26.0653	−0.918681
$806$	0	0
$807$	11.7655	0.414167
$808$	0	0
$809$	28.2635	0.993691	0.496846	−	0.867839i	$-0.334492\pi$
$809$	28.2635	0.993691	0.496846	+	0.867839i	$0.334492\pi$
$810$	0	0
$811$	−40.9201	−1.43690	−0.718450	−	0.695579i	$-0.755148\pi$
$811$	−40.9201	−1.43690	−0.718450	+	0.695579i	$0.755148\pi$
$812$	0	0
$813$	−15.8896	−0.557271
$814$	0	0
$815$	23.9145	0.837689
$816$	0	0
$817$	−3.33062	−0.116524
$818$	0	0
$819$	4.45038	0.155509
$820$	0	0
$821$	−46.4399	−1.62076	−0.810381	−	0.585903i	$-0.800740\pi$
$821$	−46.4399	−1.62076	−0.810381	+	0.585903i	$0.800740\pi$
$822$	0	0
$823$	16.4470	0.573306	0.286653	−	0.958034i	$-0.407457\pi$
$823$	16.4470	0.573306	0.286653	+	0.958034i	$0.407457\pi$
$824$	0	0
$825$	−17.9997	−0.626668
$826$	0	0
$827$	9.70175	0.337363	0.168681	−	0.985671i	$-0.446049\pi$
$827$	9.70175	0.337363	0.168681	+	0.985671i	$0.446049\pi$
$828$	0	0
$829$	19.7687	0.686595	0.343298	−	0.939227i	$-0.388456\pi$
$829$	19.7687	0.686595	0.343298	+	0.939227i	$0.388456\pi$
$830$	0	0
$831$	41.4285	1.43714
$832$	0	0
$833$	10.0583	0.348499
$834$	0	0
$835$	−60.1343	−2.08103
$836$	0	0
$837$	25.4168	0.878533
$838$	0	0
$839$	−6.93077	−0.239277	−0.119638	−	0.992818i	$-0.538174\pi$
$839$	−6.93077	−0.239277	−0.119638	+	0.992818i	$0.538174\pi$
$840$	0	0
$841$	−25.7151	−0.886727
$842$	0	0
$843$	22.4945	0.774751
$844$	0	0
$845$	70.5351	2.42648
$846$	0	0
$847$	6.06479	0.208389
$848$	0	0
$849$	−12.4458	−0.427138
$850$	0	0
$851$	9.85154	0.337706
$852$	0	0
$853$	44.4234	1.52103	0.760513	−	0.649322i	$-0.224947\pi$
$853$	44.4234	1.52103	0.760513	+	0.649322i	$0.224947\pi$
$854$	0	0
$855$	0.768094	0.0262683
$856$	0	0
$857$	−41.8400	−1.42923	−0.714614	−	0.699519i	$-0.753398\pi$
$857$	−41.8400	−1.42923	−0.714614	+	0.699519i	$0.753398\pi$
$858$	0	0
$859$	−56.2569	−1.91946	−0.959731	−	0.280922i	$-0.909360\pi$
$859$	−56.2569	−1.91946	−0.959731	+	0.280922i	$0.909360\pi$
$860$	0	0
$861$	7.81607	0.266371
$862$	0	0
$863$	23.1473	0.787942	0.393971	−	0.919123i	$-0.371101\pi$
$863$	23.1473	0.787942	0.393971	+	0.919123i	$0.371101\pi$
$864$	0	0
$865$	58.8632	2.00141
$866$	0	0
$867$	15.7530	0.534999
$868$	0	0
$869$	4.59620	0.155915
$870$	0	0
$871$	−81.6044	−2.76506
$872$	0	0
$873$	−1.23364	−0.0417524
$874$	0	0
$875$	−12.8716	−0.435141
$876$	0	0
$877$	26.5396	0.896179	0.448089	−	0.893989i	$-0.352105\pi$
$877$	26.5396	0.896179	0.448089	+	0.893989i	$0.352105\pi$
$878$	0	0
$879$	36.2982	1.22431
$880$	0	0
$881$	−37.0510	−1.24828	−0.624141	−	0.781312i	$-0.714551\pi$
$881$	−37.0510	−1.24828	−0.624141	+	0.781312i	$0.714551\pi$
$882$	0	0
$883$	−5.17449	−0.174135	−0.0870677	−	0.996202i	$-0.527750\pi$
$883$	−5.17449	−0.174135	−0.0870677	+	0.996202i	$0.527750\pi$
$884$	0	0
$885$	−58.3736	−1.96221
$886$	0	0
$887$	33.5647	1.12699	0.563496	−	0.826119i	$-0.309456\pi$
$887$	33.5647	1.12699	0.563496	+	0.826119i	$0.309456\pi$
$888$	0	0
$889$	−25.9207	−0.869353
$890$	0	0
$891$	29.3606	0.983619
$892$	0	0
$893$	−9.09204	−0.304253
$894$	0	0
$895$	31.3850	1.04908
$896$	0	0
$897$	41.4354	1.38349
$898$	0	0
$899$	8.47446	0.282639
$900$	0	0
$901$	−35.3956	−1.17920
$902$	0	0
$903$	14.4692	0.481506
$904$	0	0
$905$	−5.80992	−0.193128
$906$	0	0
$907$	0.229378	0.00761639	0.00380819	−	0.999993i	$-0.498788\pi$
$907$	0.229378	0.00761639	0.00380819	+	0.999993i	$0.498788\pi$
$908$	0	0
$909$	−0.899161	−0.0298233
$910$	0	0
$911$	−26.1240	−0.865525	−0.432763	−	0.901508i	$-0.642461\pi$
$911$	−26.1240	−0.865525	−0.432763	+	0.901508i	$0.642461\pi$
$912$	0	0
$913$	31.5740	1.04495
$914$	0	0
$915$	−53.9003	−1.78189
$916$	0	0
$917$	7.20815	0.238034
$918$	0	0
$919$	19.4548	0.641756	0.320878	−	0.947121i	$-0.396022\pi$
$919$	19.4548	0.641756	0.320878	+	0.947121i	$0.396022\pi$
$920$	0	0
$921$	−7.68124	−0.253105
$922$	0	0
$923$	33.8460	1.11405
$924$	0	0
$925$	−7.12608	−0.234304
$926$	0	0
$927$	−2.12385	−0.0697565
$928$	0	0
$929$	21.7263	0.712818	0.356409	−	0.934330i	$-0.384001\pi$
$929$	21.7263	0.712818	0.356409	+	0.934330i	$0.384001\pi$
$930$	0	0
$931$	1.65042	0.0540905
$932$	0	0
$933$	14.7374	0.482481
$934$	0	0
$935$	53.9202	1.76338
$936$	0	0
$937$	−32.2210	−1.05261	−0.526306	−	0.850295i	$-0.676423\pi$
$937$	−32.2210	−1.05261	−0.526306	+	0.850295i	$0.676423\pi$
$938$	0	0
$939$	3.48260	0.113650
$940$	0	0
$941$	14.6698	0.478223	0.239111	−	0.970992i	$-0.423144\pi$
$941$	14.6698	0.478223	0.239111	+	0.970992i	$0.423144\pi$
$942$	0	0
$943$	−8.72912	−0.284259
$944$	0	0
$945$	−34.4917	−1.12202
$946$	0	0
$947$	22.0982	0.718094	0.359047	−	0.933319i	$-0.383102\pi$
$947$	22.0982	0.718094	0.359047	+	0.933319i	$0.383102\pi$
$948$	0	0
$949$	−56.7907	−1.84350
$950$	0	0
$951$	−9.78545	−0.317315
$952$	0	0
$953$	−2.69782	−0.0873911	−0.0436955	−	0.999045i	$-0.513913\pi$
$953$	−2.69782	−0.0873911	−0.0436955	+	0.999045i	$0.513913\pi$
$954$	0	0
$955$	−42.2481	−1.36712
$956$	0	0
$957$	10.9790	0.354900
$958$	0	0
$959$	−6.84333	−0.220983
$960$	0	0
$961$	−9.13751	−0.294758
$962$	0	0
$963$	−5.63094	−0.181455
$964$	0	0
$965$	66.2253	2.13187
$966$	0	0
$967$	5.16391	0.166060	0.0830301	−	0.996547i	$-0.473540\pi$
$967$	5.16391	0.166060	0.0830301	+	0.996547i	$0.473540\pi$
$968$	0	0
$969$	7.15027	0.229700
$970$	0	0
$971$	−3.22182	−0.103393	−0.0516965	−	0.998663i	$-0.516463\pi$
$971$	−3.22182	−0.103393	−0.0516965	+	0.998663i	$0.516463\pi$
$972$	0	0
$973$	−35.5147	−1.13855
$974$	0	0
$975$	−29.9722	−0.959879
$976$	0	0
$977$	44.4817	1.42310	0.711549	−	0.702637i	$-0.247994\pi$
$977$	44.4817	1.42310	0.711549	+	0.702637i	$0.247994\pi$
$978$	0	0
$979$	18.7530	0.599349
$980$	0	0
$981$	−0.188044	−0.00600379
$982$	0	0
$983$	29.2976	0.934448	0.467224	−	0.884139i	$-0.345254\pi$
$983$	29.2976	0.934448	0.467224	+	0.884139i	$0.345254\pi$
$984$	0	0
$985$	−8.16701	−0.260223
$986$	0	0
$987$	39.4986	1.25725
$988$	0	0
$989$	−16.1595	−0.513841
$990$	0	0
$991$	21.0261	0.667915	0.333958	−	0.942588i	$-0.391616\pi$
$991$	21.0261	0.667915	0.333958	+	0.942588i	$0.391616\pi$
$992$	0	0
$993$	12.4571	0.395316
$994$	0	0
$995$	−70.0774	−2.22160
$996$	0	0
$997$	−44.8922	−1.42175	−0.710876	−	0.703317i	$-0.751701\pi$
$997$	−44.8922	−1.42175	−0.710876	+	0.703317i	$0.751701\pi$
$998$	0	0
$999$	13.0364	0.412452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.34	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.34	✓	44	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.63667 q^{3} +2.82337 q^{5} +2.24738 q^{7} -0.321314 q^{9} +O(q^{10})\)	\(q+1.63667 q^{3} +2.82337 q^{5} +2.24738 q^{7} -0.321314 q^{9} -3.70116 q^{11} -6.16300 q^{13} +4.62093 q^{15} -5.15994 q^{17} -0.846675 q^{19} +3.67821 q^{21} -4.10789 q^{23} +2.97143 q^{25} -5.43589 q^{27} -1.81243 q^{29} -4.67573 q^{31} -6.05758 q^{33} +6.34518 q^{35} -2.39820 q^{37} -10.0868 q^{39} +2.12496 q^{41} +3.93377 q^{43} -0.907189 q^{45} +10.7385 q^{47} -1.94930 q^{49} -8.44512 q^{51} +6.85969 q^{53} -10.4498 q^{55} -1.38573 q^{57} -12.6324 q^{59} -11.6644 q^{61} -0.722113 q^{63} -17.4004 q^{65} +13.2410 q^{67} -6.72326 q^{69} -5.49181 q^{71} +9.21478 q^{73} +4.86325 q^{75} -8.31791 q^{77} -1.24182 q^{79} -7.93282 q^{81} -8.53084 q^{83} -14.5684 q^{85} -2.96636 q^{87} -5.06679 q^{89} -13.8506 q^{91} -7.65263 q^{93} -2.39048 q^{95} +3.83936 q^{97} +1.18924 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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