LMFDB - Embedded newform 6012.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 $ x^10 - 4x^9 - 26x^8 + 82x^7 + 211x^6 - 340x^5 - 593x^4 + 192x^3 + 423x^2 + 126*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.617293$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.860385 q^{5} -0.617293 q^{7} +O(q^{10})$	$q-0.860385 q^{5} -0.617293 q^{7} -4.88516 q^{11} -4.79718 q^{13} -2.56620 q^{17} +0.574616 q^{19} +1.04502 q^{23} -4.25974 q^{25} +3.36951 q^{29} -2.95507 q^{31} +0.531109 q^{35} -6.11849 q^{37} +9.79812 q^{41} -5.38076 q^{43} +6.46311 q^{47} -6.61895 q^{49} +2.44729 q^{53} +4.20311 q^{55} -1.74970 q^{59} +10.6921 q^{61} +4.12742 q^{65} +7.38118 q^{67} -7.40854 q^{71} +11.5558 q^{73} +3.01557 q^{77} -0.281783 q^{79} +8.62608 q^{83} +2.20792 q^{85} -12.7820 q^{89} +2.96126 q^{91} -0.494391 q^{95} -9.14519 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) $ 10 * q + 6 * q^5 + 4 * q^7	$ 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) $ 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−0.860385	−0.384776	−0.192388	−	0.981319i	$-0.561623\pi$
$5$	−0.860385	−0.384776	−0.192388	+	0.981319i	$0.561623\pi$
$6$	0	0
$7$	−0.617293	−0.233315	−0.116657	−	0.993172i	$-0.537218\pi$
$7$	−0.617293	−0.233315	−0.116657	+	0.993172i	$0.537218\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.88516	−1.47293	−0.736465	−	0.676475i	$-0.763507\pi$
$11$	−4.88516	−1.47293	−0.736465	+	0.676475i	$0.763507\pi$
$12$	0	0
$13$	−4.79718	−1.33050	−0.665249	−	0.746621i	$-0.731675\pi$
$13$	−4.79718	−1.33050	−0.665249	+	0.746621i	$0.731675\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.56620	−0.622395	−0.311198	−	0.950345i	$-0.600730\pi$
$17$	−2.56620	−0.622395	−0.311198	+	0.950345i	$0.600730\pi$
$18$	0	0
$19$	0.574616	0.131826	0.0659130	−	0.997825i	$-0.479004\pi$
$19$	0.574616	0.131826	0.0659130	+	0.997825i	$0.479004\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.04502	0.217902	0.108951	−	0.994047i	$-0.465251\pi$
$23$	1.04502	0.217902	0.108951	+	0.994047i	$0.465251\pi$
$24$	0	0
$25$	−4.25974	−0.851948
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.36951	0.625702	0.312851	−	0.949802i	$-0.398716\pi$
$29$	3.36951	0.625702	0.312851	+	0.949802i	$0.398716\pi$
$30$	0	0
$31$	−2.95507	−0.530745	−0.265373	−	0.964146i	$-0.585495\pi$
$31$	−2.95507	−0.530745	−0.265373	+	0.964146i	$0.585495\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.531109	0.0897738
$36$	0	0
$37$	−6.11849	−1.00587	−0.502937	−	0.864323i	$-0.667747\pi$
$37$	−6.11849	−1.00587	−0.502937	+	0.864323i	$0.667747\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.79812	1.53021	0.765104	−	0.643906i	$-0.222687\pi$
$41$	9.79812	1.53021	0.765104	+	0.643906i	$0.222687\pi$
$42$	0	0
$43$	−5.38076	−0.820558	−0.410279	−	0.911960i	$-0.634569\pi$
$43$	−5.38076	−0.820558	−0.410279	+	0.911960i	$0.634569\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.46311	0.942741	0.471370	−	0.881935i	$-0.343759\pi$
$47$	6.46311	0.942741	0.471370	+	0.881935i	$0.343759\pi$
$48$	0	0
$49$	−6.61895	−0.945564
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.44729	0.336161	0.168080	−	0.985773i	$-0.446243\pi$
$53$	2.44729	0.336161	0.168080	+	0.985773i	$0.446243\pi$
$54$	0	0
$55$	4.20311	0.566748
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.74970	−0.227792	−0.113896	−	0.993493i	$-0.536333\pi$
$59$	−1.74970	−0.227792	−0.113896	+	0.993493i	$0.536333\pi$
$60$	0	0
$61$	10.6921	1.36899	0.684494	−	0.729018i	$-0.260023\pi$
$61$	10.6921	1.36899	0.684494	+	0.729018i	$0.260023\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.12742	0.511944
$66$	0	0
$67$	7.38118	0.901754	0.450877	−	0.892586i	$-0.351111\pi$
$67$	7.38118	0.901754	0.450877	+	0.892586i	$0.351111\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.40854	−0.879233	−0.439616	−	0.898186i	$-0.644886\pi$
$71$	−7.40854	−0.879233	−0.439616	+	0.898186i	$0.644886\pi$
$72$	0	0
$73$	11.5558	1.35250	0.676249	−	0.736673i	$-0.263604\pi$
$73$	11.5558	1.35250	0.676249	+	0.736673i	$0.263604\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.01557	0.343656
$78$	0	0
$79$	−0.281783	−0.0317031	−0.0158516	−	0.999874i	$-0.505046\pi$
$79$	−0.281783	−0.0317031	−0.0158516	+	0.999874i	$0.505046\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.62608	0.946835	0.473418	−	0.880838i	$-0.343020\pi$
$83$	8.62608	0.946835	0.473418	+	0.880838i	$0.343020\pi$
$84$	0	0
$85$	2.20792	0.239483
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.7820	−1.35488	−0.677442	−	0.735576i	$-0.736912\pi$
$89$	−12.7820	−1.35488	−0.677442	+	0.735576i	$0.736912\pi$
$90$	0	0
$91$	2.96126	0.310425
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.494391	−0.0507234
$96$	0	0
$97$	−9.14519	−0.928553	−0.464277	−	0.885690i	$-0.653686\pi$
$97$	−9.14519	−0.928553	−0.464277	+	0.885690i	$0.653686\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.1710	1.01205	0.506025	−	0.862519i	$-0.331115\pi$
$101$	10.1710	1.01205	0.506025	+	0.862519i	$0.331115\pi$
$102$	0	0
$103$	1.20675	0.118905	0.0594524	−	0.998231i	$-0.481065\pi$
$103$	1.20675	0.118905	0.0594524	+	0.998231i	$0.481065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.43408	0.525332	0.262666	−	0.964887i	$-0.415398\pi$
$107$	5.43408	0.525332	0.262666	+	0.964887i	$0.415398\pi$
$108$	0	0
$109$	−0.0703012	−0.00673363	−0.00336682	−	0.999994i	$-0.501072\pi$
$109$	−0.0703012	−0.00673363	−0.00336682	+	0.999994i	$0.501072\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.26390	0.401114	0.200557	−	0.979682i	$-0.435725\pi$
$113$	4.26390	0.401114	0.200557	+	0.979682i	$0.435725\pi$
$114$	0	0
$115$	−0.899121	−0.0838435
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.58410	0.145214
$120$	0	0
$121$	12.8648	1.16952
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.96694	0.712584
$126$	0	0
$127$	−1.01569	−0.0901279	−0.0450640	−	0.998984i	$-0.514349\pi$
$127$	−1.01569	−0.0901279	−0.0450640	+	0.998984i	$0.514349\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.0962	1.49371	0.746853	−	0.664989i	$-0.231564\pi$
$131$	17.0962	1.49371	0.746853	+	0.664989i	$0.231564\pi$
$132$	0	0
$133$	−0.354706	−0.0307569
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.4807	1.32261	0.661304	−	0.750118i	$-0.270003\pi$
$137$	15.4807	1.32261	0.661304	+	0.750118i	$0.270003\pi$
$138$	0	0
$139$	15.3455	1.30159	0.650795	−	0.759253i	$-0.274436\pi$
$139$	15.3455	1.30159	0.650795	+	0.759253i	$0.274436\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	23.4350	1.95973
$144$	0	0
$145$	−2.89907	−0.240755
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.58506	−0.211777	−0.105888	−	0.994378i	$-0.533769\pi$
$149$	−2.58506	−0.211777	−0.105888	+	0.994378i	$0.533769\pi$
$150$	0	0
$151$	−18.6361	−1.51659	−0.758293	−	0.651914i	$-0.773966\pi$
$151$	−18.6361	−1.51659	−0.758293	+	0.651914i	$0.773966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.54249	0.204218
$156$	0	0
$157$	−1.20249	−0.0959692	−0.0479846	−	0.998848i	$-0.515280\pi$
$157$	−1.20249	−0.0959692	−0.0479846	+	0.998848i	$0.515280\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.645084	−0.0508398
$162$	0	0
$163$	−4.89558	−0.383452	−0.191726	−	0.981449i	$-0.561408\pi$
$163$	−4.89558	−0.383452	−0.191726	+	0.981449i	$0.561408\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	10.0130	0.770227
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.92713	−0.450631	−0.225316	−	0.974286i	$-0.572341\pi$
$173$	−5.92713	−0.450631	−0.225316	+	0.974286i	$0.572341\pi$
$174$	0	0
$175$	2.62950	0.198772
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.71178	0.277432	0.138716	−	0.990332i	$-0.455703\pi$
$179$	3.71178	0.277432	0.138716	+	0.990332i	$0.455703\pi$
$180$	0	0
$181$	−5.55697	−0.413046	−0.206523	−	0.978442i	$-0.566215\pi$
$181$	−5.55697	−0.413046	−0.206523	+	0.978442i	$0.566215\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.26425	0.387036
$186$	0	0
$187$	12.5363	0.916745
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.74173	−0.415457	−0.207729	−	0.978186i	$-0.566607\pi$
$191$	−5.74173	−0.415457	−0.207729	+	0.978186i	$0.566607\pi$
$192$	0	0
$193$	−6.02016	−0.433341	−0.216670	−	0.976245i	$-0.569520\pi$
$193$	−6.02016	−0.433341	−0.216670	+	0.976245i	$0.569520\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.7101	−0.834307	−0.417154	−	0.908836i	$-0.636972\pi$
$197$	−11.7101	−0.834307	−0.417154	+	0.908836i	$0.636972\pi$
$198$	0	0
$199$	−8.82823	−0.625817	−0.312908	−	0.949783i	$-0.601303\pi$
$199$	−8.82823	−0.625817	−0.312908	+	0.949783i	$0.601303\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.07997	−0.145986
$204$	0	0
$205$	−8.43015	−0.588787
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.80709	−0.194170
$210$	0	0
$211$	17.8879	1.23145	0.615727	−	0.787959i	$-0.288862\pi$
$211$	17.8879	1.23145	0.615727	+	0.787959i	$0.288862\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.62952	0.315731
$216$	0	0
$217$	1.82414	0.123831
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.3105	0.828096
$222$	0	0
$223$	−0.573280	−0.0383897	−0.0191948	−	0.999816i	$-0.506110\pi$
$223$	−0.573280	−0.0383897	−0.0191948	+	0.999816i	$0.506110\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.69805	0.378193	0.189096	−	0.981959i	$-0.439444\pi$
$227$	5.69805	0.378193	0.189096	+	0.981959i	$0.439444\pi$
$228$	0	0
$229$	9.30463	0.614867	0.307434	−	0.951569i	$-0.400530\pi$
$229$	9.30463	0.614867	0.307434	+	0.951569i	$0.400530\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.6059	−1.08789	−0.543943	−	0.839122i	$-0.683069\pi$
$233$	−16.6059	−1.08789	−0.543943	+	0.839122i	$0.683069\pi$
$234$	0	0
$235$	−5.56076	−0.362744
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.4323	1.51571	0.757855	−	0.652423i	$-0.226247\pi$
$239$	23.4323	1.51571	0.757855	+	0.652423i	$0.226247\pi$
$240$	0	0
$241$	−10.8268	−0.697414	−0.348707	−	0.937232i	$-0.613379\pi$
$241$	−10.8268	−0.697414	−0.348707	+	0.937232i	$0.613379\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.69484	0.363830
$246$	0	0
$247$	−2.75654	−0.175394
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.21581	−0.139861	−0.0699305	−	0.997552i	$-0.522278\pi$
$251$	−2.21581	−0.139861	−0.0699305	+	0.997552i	$0.522278\pi$
$252$	0	0
$253$	−5.10510	−0.320955
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.5453	−0.782555	−0.391277	−	0.920273i	$-0.627967\pi$
$257$	−12.5453	−0.782555	−0.391277	+	0.920273i	$0.627967\pi$
$258$	0	0
$259$	3.77690	0.234685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.2161	1.86321	0.931604	−	0.363476i	$-0.118410\pi$
$263$	30.2161	1.86321	0.931604	+	0.363476i	$0.118410\pi$
$264$	0	0
$265$	−2.10561	−0.129347
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.11364	0.250813	0.125407	−	0.992105i	$-0.459976\pi$
$269$	4.11364	0.250813	0.125407	+	0.992105i	$0.459976\pi$
$270$	0	0
$271$	−19.2634	−1.17017	−0.585084	−	0.810973i	$-0.698938\pi$
$271$	−19.2634	−1.17017	−0.585084	+	0.810973i	$0.698938\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.8095	1.25486
$276$	0	0
$277$	5.03762	0.302681	0.151340	−	0.988482i	$-0.451641\pi$
$277$	5.03762	0.302681	0.151340	+	0.988482i	$0.451641\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−21.5736	−1.28697	−0.643486	−	0.765458i	$-0.722512\pi$
$281$	−21.5736	−1.28697	−0.643486	+	0.765458i	$0.722512\pi$
$282$	0	0
$283$	−14.3098	−0.850629	−0.425314	−	0.905046i	$-0.639836\pi$
$283$	−14.3098	−0.850629	−0.425314	+	0.905046i	$0.639836\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.04831	−0.357020
$288$	0	0
$289$	−10.4146	−0.612624
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.4005	1.30865	0.654326	−	0.756212i	$-0.272952\pi$
$293$	22.4005	1.30865	0.654326	+	0.756212i	$0.272952\pi$
$294$	0	0
$295$	1.50542	0.0876487
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.01316	−0.289919
$300$	0	0
$301$	3.32150	0.191448
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.19936	−0.526754
$306$	0	0
$307$	25.9729	1.48235	0.741177	−	0.671309i	$-0.234268\pi$
$307$	25.9729	1.48235	0.741177	+	0.671309i	$0.234268\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.14792	−0.518731	−0.259366	−	0.965779i	$-0.583513\pi$
$311$	−9.14792	−0.518731	−0.259366	+	0.965779i	$0.583513\pi$
$312$	0	0
$313$	−5.04602	−0.285218	−0.142609	−	0.989779i	$-0.545549\pi$
$313$	−5.04602	−0.285218	−0.142609	+	0.989779i	$0.545549\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.1865	1.07762	0.538811	−	0.842427i	$-0.318874\pi$
$317$	19.1865	1.07762	0.538811	+	0.842427i	$0.318874\pi$
$318$	0	0
$319$	−16.4606	−0.921616
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.47458	−0.0820479
$324$	0	0
$325$	20.4347	1.13352
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.98963	−0.219955
$330$	0	0
$331$	−29.0306	−1.59567	−0.797834	−	0.602878i	$-0.794021\pi$
$331$	−29.0306	−1.59567	−0.797834	+	0.602878i	$0.794021\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.35065	−0.346973
$336$	0	0
$337$	6.26605	0.341334	0.170667	−	0.985329i	$-0.445408\pi$
$337$	6.26605	0.341334	0.170667	+	0.985329i	$0.445408\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.4360	0.781751
$342$	0	0
$343$	8.40688	0.453929
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.84603	0.260149	0.130074	−	0.991504i	$-0.458478\pi$
$347$	4.84603	0.260149	0.130074	+	0.991504i	$0.458478\pi$
$348$	0	0
$349$	17.4662	0.934944	0.467472	−	0.884008i	$-0.345165\pi$
$349$	17.4662	0.934944	0.467472	+	0.884008i	$0.345165\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.1109	0.963943	0.481972	−	0.876187i	$-0.339921\pi$
$353$	18.1109	0.963943	0.481972	+	0.876187i	$0.339921\pi$
$354$	0	0
$355$	6.37420	0.338307
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.5072	1.29344	0.646721	−	0.762726i	$-0.276140\pi$
$359$	24.5072	1.29344	0.646721	+	0.762726i	$0.276140\pi$
$360$	0	0
$361$	−18.6698	−0.982622
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.94239	−0.520408
$366$	0	0
$367$	13.8832	0.724699	0.362350	−	0.932042i	$-0.381975\pi$
$367$	13.8832	0.724699	0.362350	+	0.932042i	$0.381975\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.51069	−0.0784312
$372$	0	0
$373$	−25.1842	−1.30399	−0.651994	−	0.758224i	$-0.726067\pi$
$373$	−25.1842	−1.30399	−0.651994	+	0.758224i	$0.726067\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.1642	−0.832496
$378$	0	0
$379$	29.8997	1.53584	0.767922	−	0.640543i	$-0.221291\pi$
$379$	29.8997	1.53584	0.767922	+	0.640543i	$0.221291\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.8239	1.16625	0.583125	−	0.812383i	$-0.301830\pi$
$383$	22.8239	1.16625	0.583125	+	0.812383i	$0.301830\pi$
$384$	0	0
$385$	−2.59455	−0.132231
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.6600	−0.844697	−0.422348	−	0.906434i	$-0.638794\pi$
$389$	−16.6600	−0.844697	−0.422348	+	0.906434i	$0.638794\pi$
$390$	0	0
$391$	−2.68174	−0.135621
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.242442	0.0121986
$396$	0	0
$397$	−11.0317	−0.553667	−0.276833	−	0.960918i	$-0.589285\pi$
$397$	−11.0317	−0.553667	−0.276833	+	0.960918i	$0.589285\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.2937	−0.813671	−0.406835	−	0.913502i	$-0.633368\pi$
$401$	−16.2937	−0.813671	−0.406835	+	0.913502i	$0.633368\pi$
$402$	0	0
$403$	14.1760	0.706156
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.8898	1.48158
$408$	0	0
$409$	22.9352	1.13407	0.567037	−	0.823692i	$-0.308089\pi$
$409$	22.9352	1.13407	0.567037	+	0.823692i	$0.308089\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.08008	0.0531471
$414$	0	0
$415$	−7.42174	−0.364319
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.8728	0.824290	0.412145	−	0.911118i	$-0.364780\pi$
$419$	16.8728	0.824290	0.412145	+	0.911118i	$0.364780\pi$
$420$	0	0
$421$	15.9695	0.778305	0.389152	−	0.921173i	$-0.372768\pi$
$421$	15.9695	0.778305	0.389152	+	0.921173i	$0.372768\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.9313	0.530248
$426$	0	0
$427$	−6.60018	−0.319405
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.9498	−0.816442	−0.408221	−	0.912883i	$-0.633851\pi$
$431$	−16.9498	−0.816442	−0.408221	+	0.912883i	$0.633851\pi$
$432$	0	0
$433$	24.5432	1.17947	0.589736	−	0.807596i	$-0.299232\pi$
$433$	24.5432	1.17947	0.589736	+	0.807596i	$0.299232\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.600486	0.0287252
$438$	0	0
$439$	−24.7644	−1.18194	−0.590971	−	0.806693i	$-0.701255\pi$
$439$	−24.7644	−1.18194	−0.590971	+	0.806693i	$0.701255\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.6243	0.742332	0.371166	−	0.928566i	$-0.378958\pi$
$443$	15.6243	0.742332	0.371166	+	0.928566i	$0.378958\pi$
$444$	0	0
$445$	10.9974	0.521327
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.8934	−1.03321	−0.516607	−	0.856223i	$-0.672805\pi$
$449$	−21.8934	−1.03321	−0.516607	+	0.856223i	$0.672805\pi$
$450$	0	0
$451$	−47.8654	−2.25389
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.54783	−0.119444
$456$	0	0
$457$	31.2345	1.46109	0.730545	−	0.682864i	$-0.239266\pi$
$457$	31.2345	1.46109	0.730545	+	0.682864i	$0.239266\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.9995	1.30407	0.652034	−	0.758190i	$-0.273916\pi$
$461$	27.9995	1.30407	0.652034	+	0.758190i	$0.273916\pi$
$462$	0	0
$463$	22.2414	1.03365	0.516823	−	0.856092i	$-0.327114\pi$
$463$	22.2414	1.03365	0.516823	+	0.856092i	$0.327114\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.39721	0.249753	0.124877	−	0.992172i	$-0.460146\pi$
$467$	5.39721	0.249753	0.124877	+	0.992172i	$0.460146\pi$
$468$	0	0
$469$	−4.55634	−0.210392
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26.2859	1.20863
$474$	0	0
$475$	−2.44771	−0.112309
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.5826	0.986133	0.493067	−	0.869992i	$-0.335876\pi$
$479$	21.5826	0.986133	0.493067	+	0.869992i	$0.335876\pi$
$480$	0	0
$481$	29.3515	1.33831
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.86838	0.357285
$486$	0	0
$487$	−4.26564	−0.193294	−0.0966472	−	0.995319i	$-0.530812\pi$
$487$	−4.26564	−0.193294	−0.0966472	+	0.995319i	$0.530812\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.30191	0.419789	0.209895	−	0.977724i	$-0.432688\pi$
$491$	9.30191	0.419789	0.209895	+	0.977724i	$0.432688\pi$
$492$	0	0
$493$	−8.64685	−0.389434
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.57324	0.205138
$498$	0	0
$499$	7.04042	0.315172	0.157586	−	0.987505i	$-0.449629\pi$
$499$	7.04042	0.315172	0.157586	+	0.987505i	$0.449629\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.7163	0.700754	0.350377	−	0.936609i	$-0.386053\pi$
$503$	15.7163	0.700754	0.350377	+	0.936609i	$0.386053\pi$
$504$	0	0
$505$	−8.75094	−0.389412
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.08468	−0.225374	−0.112687	−	0.993631i	$-0.535946\pi$
$509$	−5.08468	−0.225374	−0.112687	+	0.993631i	$0.535946\pi$
$510$	0	0
$511$	−7.13328	−0.315558
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.03827	−0.0457517
$516$	0	0
$517$	−31.5733	−1.38859
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.3961	−1.46311	−0.731554	−	0.681784i	$-0.761205\pi$
$521$	−33.3961	−1.46311	−0.731554	+	0.681784i	$0.761205\pi$
$522$	0	0
$523$	−29.6396	−1.29605	−0.648024	−	0.761620i	$-0.724404\pi$
$523$	−29.6396	−1.29605	−0.648024	+	0.761620i	$0.724404\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.58329	0.330333
$528$	0	0
$529$	−21.9079	−0.952519
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−47.0034	−2.03594
$534$	0	0
$535$	−4.67540	−0.202135
$536$	0	0
$537$	0	0
$538$	0	0
$539$	32.3346	1.39275
$540$	0	0
$541$	2.64530	0.113730	0.0568652	−	0.998382i	$-0.481889\pi$
$541$	2.64530	0.113730	0.0568652	+	0.998382i	$0.481889\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.0604861	0.00259094
$546$	0	0
$547$	25.3974	1.08592	0.542958	−	0.839760i	$-0.317304\pi$
$547$	25.3974	1.08592	0.542958	+	0.839760i	$0.317304\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.93617	0.0824838
$552$	0	0
$553$	0.173943	0.00739680
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.0196	1.10249	0.551243	−	0.834345i	$-0.314154\pi$
$557$	26.0196	1.10249	0.551243	+	0.834345i	$0.314154\pi$
$558$	0	0
$559$	25.8125	1.09175
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.2797	−0.475381	−0.237691	−	0.971341i	$-0.576390\pi$
$563$	−11.2797	−0.475381	−0.237691	+	0.971341i	$0.576390\pi$
$564$	0	0
$565$	−3.66860	−0.154339
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.61130	−0.319082	−0.159541	−	0.987191i	$-0.551001\pi$
$569$	−7.61130	−0.319082	−0.159541	+	0.987191i	$0.551001\pi$
$570$	0	0
$571$	24.3791	1.02023	0.510116	−	0.860106i	$-0.329602\pi$
$571$	24.3791	1.02023	0.510116	+	0.860106i	$0.329602\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.45152	−0.185641
$576$	0	0
$577$	0.323021	0.0134476	0.00672378	−	0.999977i	$-0.497860\pi$
$577$	0.323021	0.0134476	0.00672378	+	0.999977i	$0.497860\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.32481	−0.220911
$582$	0	0
$583$	−11.9554	−0.495142
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.7236	0.525161	0.262581	−	0.964910i	$-0.415426\pi$
$587$	12.7236	0.525161	0.262581	+	0.964910i	$0.415426\pi$
$588$	0	0
$589$	−1.69803	−0.0699660
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.8781	−1.06269	−0.531344	−	0.847156i	$-0.678313\pi$
$593$	−25.8781	−1.06269	−0.531344	+	0.847156i	$0.678313\pi$
$594$	0	0
$595$	−1.36293	−0.0558748
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.3367	−0.422346	−0.211173	−	0.977449i	$-0.567728\pi$
$599$	−10.3367	−0.422346	−0.211173	+	0.977449i	$0.567728\pi$
$600$	0	0
$601$	40.1789	1.63893	0.819466	−	0.573128i	$-0.194270\pi$
$601$	40.1789	1.63893	0.819466	+	0.573128i	$0.194270\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.0686	−0.450005
$606$	0	0
$607$	−15.7229	−0.638171	−0.319085	−	0.947726i	$-0.603376\pi$
$607$	−15.7229	−0.638171	−0.319085	+	0.947726i	$0.603376\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.0047	−1.25432
$612$	0	0
$613$	−28.1808	−1.13821	−0.569105	−	0.822265i	$-0.692710\pi$
$613$	−28.1808	−1.13821	−0.569105	+	0.822265i	$0.692710\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.76396	0.232048	0.116024	−	0.993246i	$-0.462985\pi$
$617$	5.76396	0.232048	0.116024	+	0.993246i	$0.462985\pi$
$618$	0	0
$619$	−3.75727	−0.151017	−0.0755087	−	0.997145i	$-0.524058\pi$
$619$	−3.75727	−0.151017	−0.0755087	+	0.997145i	$0.524058\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.89021	0.316114
$624$	0	0
$625$	14.4441	0.577763
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.7013	0.626051
$630$	0	0
$631$	12.4844	0.496997	0.248498	−	0.968632i	$-0.420063\pi$
$631$	12.4844	0.496997	0.248498	+	0.968632i	$0.420063\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.873884	0.0346790
$636$	0	0
$637$	31.7523	1.25807
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.24462	−0.365141	−0.182570	−	0.983193i	$-0.558442\pi$
$641$	−9.24462	−0.365141	−0.182570	+	0.983193i	$0.558442\pi$
$642$	0	0
$643$	−9.03173	−0.356176	−0.178088	−	0.984015i	$-0.556991\pi$
$643$	−9.03173	−0.356176	−0.178088	+	0.984015i	$0.556991\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.5450	0.453882	0.226941	−	0.973908i	$-0.427127\pi$
$647$	11.5450	0.453882	0.226941	+	0.973908i	$0.427127\pi$
$648$	0	0
$649$	8.54757	0.335521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.8602	1.48158	0.740792	−	0.671735i	$-0.234450\pi$
$653$	37.8602	1.48158	0.740792	+	0.671735i	$0.234450\pi$
$654$	0	0
$655$	−14.7093	−0.574742
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.6344	−0.881710	−0.440855	−	0.897578i	$-0.645325\pi$
$659$	−22.6344	−0.881710	−0.440855	+	0.897578i	$0.645325\pi$
$660$	0	0
$661$	−8.03264	−0.312434	−0.156217	−	0.987723i	$-0.549930\pi$
$661$	−8.03264	−0.312434	−0.156217	+	0.987723i	$0.549930\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.305184	0.0118345
$666$	0	0
$667$	3.52121	0.136342
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−52.2328	−2.01643
$672$	0	0
$673$	−31.3230	−1.20741	−0.603706	−	0.797207i	$-0.706310\pi$
$673$	−31.3230	−1.20741	−0.603706	+	0.797207i	$0.706310\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.7812	−1.83638	−0.918191	−	0.396137i	$-0.870350\pi$
$677$	−47.7812	−1.83638	−0.918191	+	0.396137i	$0.870350\pi$
$678$	0	0
$679$	5.64526	0.216645
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.07060	0.194021	0.0970105	−	0.995283i	$-0.469072\pi$
$683$	5.07060	0.194021	0.0970105	+	0.995283i	$0.469072\pi$
$684$	0	0
$685$	−13.3194	−0.508908
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.7401	−0.447262
$690$	0	0
$691$	−38.5299	−1.46575	−0.732874	−	0.680364i	$-0.761822\pi$
$691$	−38.5299	−1.46575	−0.732874	+	0.680364i	$0.761822\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.2031	−0.500820
$696$	0	0
$697$	−25.1440	−0.952395
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.14159	0.345273	0.172636	−	0.984986i	$-0.444771\pi$
$701$	9.14159	0.345273	0.172636	+	0.984986i	$0.444771\pi$
$702$	0	0
$703$	−3.51578	−0.132600
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.27846	−0.236126
$708$	0	0
$709$	41.7141	1.56661	0.783303	−	0.621640i	$-0.213533\pi$
$709$	41.7141	1.56661	0.783303	+	0.621640i	$0.213533\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.08811	−0.115651
$714$	0	0
$715$	−20.1631	−0.754057
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.1731	−0.528566	−0.264283	−	0.964445i	$-0.585135\pi$
$719$	−14.1731	−0.528566	−0.264283	+	0.964445i	$0.585135\pi$
$720$	0	0
$721$	−0.744919	−0.0277422
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−14.3532	−0.533066
$726$	0	0
$727$	−10.0752	−0.373667	−0.186833	−	0.982392i	$-0.559822\pi$
$727$	−10.0752	−0.373667	−0.186833	+	0.982392i	$0.559822\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.8081	0.510712
$732$	0	0
$733$	29.7191	1.09770	0.548849	−	0.835921i	$-0.315066\pi$
$733$	29.7191	1.09770	0.548849	+	0.835921i	$0.315066\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.0582	−1.32822
$738$	0	0
$739$	30.7669	1.13178	0.565890	−	0.824481i	$-0.308533\pi$
$739$	30.7669	1.13178	0.565890	+	0.824481i	$0.308533\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.2156	1.76886	0.884429	−	0.466674i	$-0.154548\pi$
$743$	48.2156	1.76886	0.884429	+	0.466674i	$0.154548\pi$
$744$	0	0
$745$	2.22415	0.0814865
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.35442	−0.122568
$750$	0	0
$751$	1.10668	0.0403834	0.0201917	−	0.999796i	$-0.493572\pi$
$751$	1.10668	0.0403834	0.0201917	+	0.999796i	$0.493572\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.0342	0.583546
$756$	0	0
$757$	−13.7486	−0.499703	−0.249851	−	0.968284i	$-0.580382\pi$
$757$	−13.7486	−0.499703	−0.249851	+	0.968284i	$0.580382\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.69161	0.242571	0.121285	−	0.992618i	$-0.461298\pi$
$761$	6.69161	0.242571	0.121285	+	0.992618i	$0.461298\pi$
$762$	0	0
$763$	0.0433964	0.00157106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.39363	0.303077
$768$	0	0
$769$	−7.66449	−0.276389	−0.138194	−	0.990405i	$-0.544130\pi$
$769$	−7.66449	−0.276389	−0.138194	+	0.990405i	$0.544130\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.1410	−0.400713	−0.200356	−	0.979723i	$-0.564210\pi$
$773$	−11.1410	−0.400713	−0.200356	+	0.979723i	$0.564210\pi$
$774$	0	0
$775$	12.5878	0.452167
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.63016	0.201721
$780$	0	0
$781$	36.1919	1.29505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.03460	0.0369266
$786$	0	0
$787$	15.0024	0.534777	0.267388	−	0.963589i	$-0.413839\pi$
$787$	15.0024	0.534777	0.267388	+	0.963589i	$0.413839\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.63208	−0.0935858
$792$	0	0
$793$	−51.2922	−1.82144
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.5641	0.480466	0.240233	−	0.970715i	$-0.422776\pi$
$797$	13.5641	0.480466	0.240233	+	0.970715i	$0.422776\pi$
$798$	0	0
$799$	−16.5856	−0.586758
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−56.4517	−1.99214
$804$	0	0
$805$	0.555021	0.0195619
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.0531	−1.33788	−0.668939	−	0.743318i	$-0.733251\pi$
$809$	−38.0531	−1.33788	−0.668939	+	0.743318i	$0.733251\pi$
$810$	0	0
$811$	−6.27615	−0.220386	−0.110193	−	0.993910i	$-0.535147\pi$
$811$	−6.27615	−0.220386	−0.110193	+	0.993910i	$0.535147\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.21208	0.147543
$816$	0	0
$817$	−3.09187	−0.108171
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.0291	0.978221	0.489110	−	0.872222i	$-0.337322\pi$
$821$	28.0291	0.978221	0.489110	+	0.872222i	$0.337322\pi$
$822$	0	0
$823$	−26.7592	−0.932768	−0.466384	−	0.884582i	$-0.654444\pi$
$823$	−26.7592	−0.932768	−0.466384	+	0.884582i	$0.654444\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.195116	0.00678484	0.00339242	−	0.999994i	$-0.498920\pi$
$827$	0.195116	0.00678484	0.00339242	+	0.999994i	$0.498920\pi$
$828$	0	0
$829$	2.37733	0.0825681	0.0412841	−	0.999147i	$-0.486855\pi$
$829$	2.37733	0.0825681	0.0412841	+	0.999147i	$0.486855\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.9856	0.588515
$834$	0	0
$835$	0.860385	0.0297748
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.2813	−1.42519	−0.712595	−	0.701576i	$-0.752480\pi$
$839$	−41.2813	−1.42519	−0.712595	+	0.701576i	$0.752480\pi$
$840$	0	0
$841$	−17.6464	−0.608497
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.61499	−0.296365
$846$	0	0
$847$	−7.94133	−0.272867
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.39396	−0.219182
$852$	0	0
$853$	13.3988	0.458765	0.229382	−	0.973336i	$-0.426329\pi$
$853$	13.3988	0.458765	0.229382	+	0.973336i	$0.426329\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.9604	−0.511038	−0.255519	−	0.966804i	$-0.582246\pi$
$857$	−14.9604	−0.511038	−0.255519	+	0.966804i	$0.582246\pi$
$858$	0	0
$859$	−24.2340	−0.826852	−0.413426	−	0.910538i	$-0.635668\pi$
$859$	−24.2340	−0.826852	−0.413426	+	0.910538i	$0.635668\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.71511	−0.0583832	−0.0291916	−	0.999574i	$-0.509293\pi$
$863$	−1.71511	−0.0583832	−0.0291916	+	0.999574i	$0.509293\pi$
$864$	0	0
$865$	5.09961	0.173392
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.37656	0.0466965
$870$	0	0
$871$	−35.4088	−1.19978
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.91793	−0.166256
$876$	0	0
$877$	−11.6804	−0.394417	−0.197209	−	0.980362i	$-0.563188\pi$
$877$	−11.6804	−0.394417	−0.197209	+	0.980362i	$0.563188\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.0746	1.04693	0.523465	−	0.852047i	$-0.324639\pi$
$881$	31.0746	1.04693	0.523465	+	0.852047i	$0.324639\pi$
$882$	0	0
$883$	22.6018	0.760612	0.380306	−	0.924861i	$-0.375819\pi$
$883$	22.6018	0.760612	0.380306	+	0.924861i	$0.375819\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.07992	−0.271297	−0.135649	−	0.990757i	$-0.543312\pi$
$887$	−8.07992	−0.271297	−0.135649	+	0.990757i	$0.543312\pi$
$888$	0	0
$889$	0.626978	0.0210282
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.71380	0.124278
$894$	0	0
$895$	−3.19356	−0.106749
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.95712	−0.332089
$900$	0	0
$901$	−6.28024	−0.209225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.78113	0.158930
$906$	0	0
$907$	−30.7161	−1.01991	−0.509955	−	0.860201i	$-0.670338\pi$
$907$	−30.7161	−1.01991	−0.509955	+	0.860201i	$0.670338\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	54.0001	1.78910	0.894551	−	0.446965i	$-0.147495\pi$
$911$	54.0001	1.78910	0.894551	+	0.446965i	$0.147495\pi$
$912$	0	0
$913$	−42.1398	−1.39462
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.5534	−0.348504
$918$	0	0
$919$	46.4540	1.53238	0.766189	−	0.642616i	$-0.222151\pi$
$919$	46.4540	1.53238	0.766189	+	0.642616i	$0.222151\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.5401	1.16982
$924$	0	0
$925$	26.0632	0.856952
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.8423	−1.01190	−0.505952	−	0.862562i	$-0.668859\pi$
$929$	−30.8423	−1.01190	−0.505952	+	0.862562i	$0.668859\pi$
$930$	0	0
$931$	−3.80335	−0.124650
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.7860	−0.352741
$936$	0	0
$937$	−17.6869	−0.577807	−0.288904	−	0.957358i	$-0.593291\pi$
$937$	−17.6869	−0.577807	−0.288904	+	0.957358i	$0.593291\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39.2387	−1.27914	−0.639572	−	0.768731i	$-0.720889\pi$
$941$	−39.2387	−1.27914	−0.639572	+	0.768731i	$0.720889\pi$
$942$	0	0
$943$	10.2393	0.333436
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.1770	0.525681	0.262840	−	0.964839i	$-0.415341\pi$
$947$	16.1770	0.525681	0.262840	+	0.964839i	$0.415341\pi$
$948$	0	0
$949$	−55.4350	−1.79950
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.8349	0.610122	0.305061	−	0.952333i	$-0.401323\pi$
$953$	18.8349	0.610122	0.305061	+	0.952333i	$0.401323\pi$
$954$	0	0
$955$	4.94010	0.159858
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.55614	−0.308584
$960$	0	0
$961$	−22.2676	−0.718309
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.17965	0.166739
$966$	0	0
$967$	7.99687	0.257162	0.128581	−	0.991699i	$-0.458958\pi$
$967$	7.99687	0.257162	0.128581	+	0.991699i	$0.458958\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.112876	0.00362237	0.00181119	−	0.999998i	$-0.499423\pi$
$971$	0.112876	0.00362237	0.00181119	+	0.999998i	$0.499423\pi$
$972$	0	0
$973$	−9.47268	−0.303680
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.0633	0.321952	0.160976	−	0.986958i	$-0.448536\pi$
$977$	10.0633	0.321952	0.160976	+	0.986958i	$0.448536\pi$
$978$	0	0
$979$	62.4419	1.99565
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.15997	−0.100787	−0.0503937	−	0.998729i	$-0.516048\pi$
$983$	−3.15997	−0.100787	−0.0503937	+	0.998729i	$0.516048\pi$
$984$	0	0
$985$	10.0752	0.321021
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.62301	−0.178801
$990$	0	0
$991$	37.6239	1.19516	0.597581	−	0.801809i	$-0.296129\pi$
$991$	37.6239	1.19516	0.597581	+	0.801809i	$0.296129\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.59568	0.240799
$996$	0	0
$997$	50.6895	1.60535	0.802676	−	0.596416i	$-0.203409\pi$
$997$	50.6895	1.60535	0.802676	+	0.596416i	$0.203409\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.k.1.4	yes	10
3.2	odd	2		6012.2.a.j.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.7	✓	10	3.2	odd	2
6012.2.a.k.1.4	yes	10	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-0.860385 q^{5} -0.617293 q^{7} +O(q^{10})\)	\(q-0.860385 q^{5} -0.617293 q^{7} -4.88516 q^{11} -4.79718 q^{13} -2.56620 q^{17} +0.574616 q^{19} +1.04502 q^{23} -4.25974 q^{25} +3.36951 q^{29} -2.95507 q^{31} +0.531109 q^{35} -6.11849 q^{37} +9.79812 q^{41} -5.38076 q^{43} +6.46311 q^{47} -6.61895 q^{49} +2.44729 q^{53} +4.20311 q^{55} -1.74970 q^{59} +10.6921 q^{61} +4.12742 q^{65} +7.38118 q^{67} -7.40854 q^{71} +11.5558 q^{73} +3.01557 q^{77} -0.281783 q^{79} +8.62608 q^{83} +2.20792 q^{85} -12.7820 q^{89} +2.96126 q^{91} -0.494391 q^{95} -9.14519 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \) 10 * q + 6 * q^5 + 4 * q^7	\( 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) \) 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

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