LMFDB - Embedded newform 3332.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.113481.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 9x^{2} + 3x + 12 $ x^4 - 2x^3 - 9x^2 + 3*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.90611$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.90611 q^{3} -3.90611 q^{5} +5.44549 q^{9} +O(q^{10})$	$q-2.90611 q^{3} -3.90611 q^{5} +5.44549 q^{9} +2.70537 q^{11} +1.75526 q^{13} +11.3516 q^{15} -1.00000 q^{17} +3.74011 q^{19} -8.40148 q^{23} +10.2577 q^{25} -7.10685 q^{27} -8.96308 q^{29} -7.25771 q^{31} -7.86211 q^{33} +6.20074 q^{37} -5.10097 q^{39} -10.7623 q^{41} +0.849144 q^{43} -21.2707 q^{45} -0.661368 q^{47} +2.90611 q^{51} -3.29463 q^{53} -10.5675 q^{55} -10.8692 q^{57} +3.24474 q^{59} -2.39560 q^{61} -6.85623 q^{65} +0.395600 q^{67} +24.4156 q^{69} -5.99074 q^{71} +13.6462 q^{73} -29.8100 q^{75} -6.84108 q^{79} +4.31685 q^{81} -0.0591443 q^{83} +3.90611 q^{85} +26.0477 q^{87} -12.5523 q^{89} +21.0917 q^{93} -14.6093 q^{95} +8.01296 q^{97} +14.7321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 2 * q^5 + 10 * q^9	$ 4 q + 2 q^{3} - 2 q^{5} + 10 q^{9} + 11 q^{11} + 5 q^{13} + 20 q^{15} - 4 q^{17} + 3 q^{19} - 6 q^{23} + 2 q^{25} - q^{27} + 3 q^{29} + 10 q^{31} - 2 q^{33} + 11 q^{37} - 11 q^{39} - 18 q^{41} + 15 q^{43} - 5 q^{45} + 13 q^{47} - 2 q^{51} - 13 q^{53} - 13 q^{55} + 9 q^{57} + 15 q^{59} - 16 q^{65} - 8 q^{67} + 24 q^{69} + 4 q^{71} + 29 q^{73} - 41 q^{75} - 6 q^{79} + 4 q^{81} - 26 q^{83} + 2 q^{85} + 51 q^{87} - 11 q^{89} + 47 q^{93} + 6 q^{95} - 9 q^{97} + 14 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^5 + 10 * q^9 + 11 * q^11 + 5 * q^13 + 20 * q^15 - 4 * q^17 + 3 * q^19 - 6 * q^23 + 2 * q^25 - q^27 + 3 * q^29 + 10 * q^31 - 2 * q^33 + 11 * q^37 - 11 * q^39 - 18 * q^41 + 15 * q^43 - 5 * q^45 + 13 * q^47 - 2 * q^51 - 13 * q^53 - 13 * q^55 + 9 * q^57 + 15 * q^59 - 16 * q^65 - 8 * q^67 + 24 * q^69 + 4 * q^71 + 29 * q^73 - 41 * q^75 - 6 * q^79 + 4 * q^81 - 26 * q^83 + 2 * q^85 + 51 * q^87 - 11 * q^89 + 47 * q^93 + 6 * q^95 - 9 * q^97 + 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.90611	−1.67784	−0.838922	−	0.544251i	$-0.816814\pi$
$3$	−2.90611	−1.67784	−0.838922	+	0.544251i	$0.816814\pi$
$4$	0	0
$5$	−3.90611	−1.74687	−0.873433	−	0.486944i	$-0.838112\pi$
$5$	−3.90611	−1.74687	−0.873433	+	0.486944i	$0.838112\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.44549	1.81516
$10$	0	0
$11$	2.70537	0.815700	0.407850	−	0.913049i	$-0.366279\pi$
$11$	2.70537	0.815700	0.407850	+	0.913049i	$0.366279\pi$
$12$	0	0
$13$	1.75526	0.486820	0.243410	−	0.969923i	$-0.421734\pi$
$13$	1.75526	0.486820	0.243410	+	0.969923i	$0.421734\pi$
$14$	0	0
$15$	11.3516	2.93097
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.74011	0.858041	0.429021	−	0.903295i	$-0.358859\pi$
$19$	3.74011	0.858041	0.429021	+	0.903295i	$0.358859\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.40148	−1.75183	−0.875915	−	0.482465i	$-0.839741\pi$
$23$	−8.40148	−1.75183	−0.875915	+	0.482465i	$0.839741\pi$
$24$	0	0
$25$	10.2577	2.05154
$26$	0	0
$27$	−7.10685	−1.36771
$28$	0	0
$29$	−8.96308	−1.66440	−0.832201	−	0.554474i	$-0.812919\pi$
$29$	−8.96308	−1.66440	−0.832201	+	0.554474i	$0.812919\pi$
$30$	0	0
$31$	−7.25771	−1.30352	−0.651761	−	0.758424i	$-0.725970\pi$
$31$	−7.25771	−1.30352	−0.651761	+	0.758424i	$0.725970\pi$
$32$	0	0
$33$	−7.86211	−1.36862
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.20074	1.01940	0.509698	−	0.860354i	$-0.329757\pi$
$37$	6.20074	1.01940	0.509698	+	0.860354i	$0.329757\pi$
$38$	0	0
$39$	−5.10097	−0.816809
$40$	0	0
$41$	−10.7623	−1.68079	−0.840397	−	0.541971i	$-0.817678\pi$
$41$	−10.7623	−1.68079	−0.840397	+	0.541971i	$0.817678\pi$
$42$	0	0
$43$	0.849144	0.129493	0.0647467	−	0.997902i	$-0.479376\pi$
$43$	0.849144	0.129493	0.0647467	+	0.997902i	$0.479376\pi$
$44$	0	0
$45$	−21.2707	−3.17084
$46$	0	0
$47$	−0.661368	−0.0964704	−0.0482352	−	0.998836i	$-0.515360\pi$
$47$	−0.661368	−0.0964704	−0.0482352	+	0.998836i	$0.515360\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.90611	0.406937
$52$	0	0
$53$	−3.29463	−0.452552	−0.226276	−	0.974063i	$-0.572655\pi$
$53$	−3.29463	−0.452552	−0.226276	+	0.974063i	$0.572655\pi$
$54$	0	0
$55$	−10.5675	−1.42492
$56$	0	0
$57$	−10.8692	−1.43966
$58$	0	0
$59$	3.24474	0.422430	0.211215	−	0.977440i	$-0.432258\pi$
$59$	3.24474	0.422430	0.211215	+	0.977440i	$0.432258\pi$
$60$	0	0
$61$	−2.39560	−0.306725	−0.153363	−	0.988170i	$-0.549010\pi$
$61$	−2.39560	−0.306725	−0.153363	+	0.988170i	$0.549010\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.85623	−0.850410
$66$	0	0
$67$	0.395600	0.0483302	0.0241651	−	0.999708i	$-0.492307\pi$
$67$	0.395600	0.0483302	0.0241651	+	0.999708i	$0.492307\pi$
$68$	0	0
$69$	24.4156	2.93930
$70$	0	0
$71$	−5.99074	−0.710970	−0.355485	−	0.934682i	$-0.615684\pi$
$71$	−5.99074	−0.710970	−0.355485	+	0.934682i	$0.615684\pi$
$72$	0	0
$73$	13.6462	1.59717	0.798585	−	0.601882i	$-0.205582\pi$
$73$	13.6462	1.59717	0.798585	+	0.601882i	$0.205582\pi$
$74$	0	0
$75$	−29.8100	−3.44217
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.84108	−0.769682	−0.384841	−	0.922983i	$-0.625744\pi$
$79$	−6.84108	−0.769682	−0.384841	+	0.922983i	$0.625744\pi$
$80$	0	0
$81$	4.31685	0.479650
$82$	0	0
$83$	−0.0591443	−0.00649193	−0.00324596	−	0.999995i	$-0.501033\pi$
$83$	−0.0591443	−0.00649193	−0.00324596	+	0.999995i	$0.501033\pi$
$84$	0	0
$85$	3.90611	0.423677
$86$	0	0
$87$	26.0477	2.79261
$88$	0	0
$89$	−12.5523	−1.33055	−0.665273	−	0.746601i	$-0.731685\pi$
$89$	−12.5523	−1.33055	−0.665273	+	0.746601i	$0.731685\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	21.0917	2.18711
$94$	0	0
$95$	−14.6093	−1.49888
$96$	0	0
$97$	8.01296	0.813593	0.406797	−	0.913519i	$-0.366646\pi$
$97$	8.01296	0.813593	0.406797	+	0.913519i	$0.366646\pi$
$98$	0	0
$99$	14.7321	1.48063
$100$	0	0
$101$	11.0570	1.10021	0.550105	−	0.835096i	$-0.314588\pi$
$101$	11.0570	1.10021	0.550105	+	0.835096i	$0.314588\pi$
$102$	0	0
$103$	−5.93497	−0.584790	−0.292395	−	0.956298i	$-0.594452\pi$
$103$	−5.93497	−0.584790	−0.292395	+	0.956298i	$0.594452\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.2728	1.28313	0.641567	−	0.767067i	$-0.278284\pi$
$107$	13.2728	1.28313	0.641567	+	0.767067i	$0.278284\pi$
$108$	0	0
$109$	4.03474	0.386458	0.193229	−	0.981154i	$-0.438104\pi$
$109$	4.03474	0.386458	0.193229	+	0.981154i	$0.438104\pi$
$110$	0	0
$111$	−18.0200	−1.71039
$112$	0	0
$113$	15.9061	1.49632	0.748161	−	0.663518i	$-0.230937\pi$
$113$	15.9061	1.49632	0.748161	+	0.663518i	$0.230937\pi$
$114$	0	0
$115$	32.8171	3.06021
$116$	0	0
$117$	9.55822	0.883658
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.68097	−0.334634
$122$	0	0
$123$	31.2766	2.82011
$124$	0	0
$125$	−20.5372	−1.83690
$126$	0	0
$127$	−16.6382	−1.47640	−0.738199	−	0.674583i	$-0.764324\pi$
$127$	−16.6382	−1.47640	−0.738199	+	0.674583i	$0.764324\pi$
$128$	0	0
$129$	−2.46771	−0.217270
$130$	0	0
$131$	0.561597	0.0490669	0.0245335	−	0.999699i	$-0.492190\pi$
$131$	0.561597	0.0490669	0.0245335	+	0.999699i	$0.492190\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	27.7602	2.38921
$136$	0	0
$137$	−0.747196	−0.0638373	−0.0319186	−	0.999490i	$-0.510162\pi$
$137$	−0.747196	−0.0638373	−0.0319186	+	0.999490i	$0.510162\pi$
$138$	0	0
$139$	8.02222	0.680436	0.340218	−	0.940347i	$-0.389499\pi$
$139$	8.02222	0.680436	0.340218	+	0.940347i	$0.389499\pi$
$140$	0	0
$141$	1.92201	0.161862
$142$	0	0
$143$	4.74862	0.397099
$144$	0	0
$145$	35.0108	2.90749
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.61148	0.541634	0.270817	−	0.962631i	$-0.412706\pi$
$149$	6.61148	0.541634	0.270817	+	0.962631i	$0.412706\pi$
$150$	0	0
$151$	−17.5897	−1.43143	−0.715715	−	0.698393i	$-0.753899\pi$
$151$	−17.5897	−1.43143	−0.715715	+	0.698393i	$0.753899\pi$
$152$	0	0
$153$	−5.44549	−0.440241
$154$	0	0
$155$	28.3494	2.27708
$156$	0	0
$157$	1.17896	0.0940914	0.0470457	−	0.998893i	$-0.485019\pi$
$157$	1.17896	0.0940914	0.0470457	+	0.998893i	$0.485019\pi$
$158$	0	0
$159$	9.57456	0.759312
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.56748	−0.201101	−0.100550	−	0.994932i	$-0.532060\pi$
$163$	−2.56748	−0.201101	−0.100550	+	0.994932i	$0.532060\pi$
$164$	0	0
$165$	30.7103	2.39079
$166$	0	0
$167$	13.1010	1.01378	0.506892	−	0.862010i	$-0.330794\pi$
$167$	13.1010	1.01378	0.506892	+	0.862010i	$0.330794\pi$
$168$	0	0
$169$	−9.91908	−0.763006
$170$	0	0
$171$	20.3667	1.55748
$172$	0	0
$173$	6.50245	0.494372	0.247186	−	0.968968i	$-0.420494\pi$
$173$	6.50245	0.494372	0.247186	+	0.968968i	$0.420494\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.42959	−0.708771
$178$	0	0
$179$	−20.1198	−1.50383	−0.751913	−	0.659262i	$-0.770869\pi$
$179$	−20.1198	−1.50383	−0.751913	+	0.659262i	$0.770869\pi$
$180$	0	0
$181$	−7.40812	−0.550641	−0.275321	−	0.961352i	$-0.588784\pi$
$181$	−7.40812	−0.550641	−0.275321	+	0.961352i	$0.588784\pi$
$182$	0	0
$183$	6.96188	0.514637
$184$	0	0
$185$	−24.2208	−1.78075
$186$	0	0
$187$	−2.70537	−0.197836
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.144218	0.0104352	0.00521761	−	0.999986i	$-0.498339\pi$
$191$	0.144218	0.0104352	0.00521761	+	0.999986i	$0.498339\pi$
$192$	0	0
$193$	18.4677	1.32933	0.664667	−	0.747139i	$-0.268573\pi$
$193$	18.4677	1.32933	0.664667	+	0.747139i	$0.268573\pi$
$194$	0	0
$195$	19.9250	1.42686
$196$	0	0
$197$	9.90611	0.705781	0.352891	−	0.935665i	$-0.385199\pi$
$197$	9.90611	0.705781	0.352891	+	0.935665i	$0.385199\pi$
$198$	0	0
$199$	−13.9919	−0.991862	−0.495931	−	0.868362i	$-0.665173\pi$
$199$	−13.9919	−0.991862	−0.495931	+	0.868362i	$0.665173\pi$
$200$	0	0
$201$	−1.14966	−0.0810905
$202$	0	0
$203$	0	0
$204$	0	0
$205$	42.0389	2.93612
$206$	0	0
$207$	−45.7501	−3.17986
$208$	0	0
$209$	10.1184	0.699904
$210$	0	0
$211$	−8.91275	−0.613579	−0.306789	−	0.951777i	$-0.599255\pi$
$211$	−8.91275	−0.613579	−0.306789	+	0.951777i	$0.599255\pi$
$212$	0	0
$213$	17.4098	1.19290
$214$	0	0
$215$	−3.31685	−0.226207
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−39.6575	−2.67980
$220$	0	0
$221$	−1.75526	−0.118071
$222$	0	0
$223$	12.1416	0.813061	0.406531	−	0.913637i	$-0.366738\pi$
$223$	12.1416	0.813061	0.406531	+	0.913637i	$0.366738\pi$
$224$	0	0
$225$	55.8582	3.72388
$226$	0	0
$227$	−4.15086	−0.275502	−0.137751	−	0.990467i	$-0.543987\pi$
$227$	−4.15086	−0.275502	−0.137751	+	0.990467i	$0.543987\pi$
$228$	0	0
$229$	−2.97189	−0.196388	−0.0981941	−	0.995167i	$-0.531307\pi$
$229$	−2.97189	−0.196388	−0.0981941	+	0.995167i	$0.531307\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.92125	−0.322402	−0.161201	−	0.986922i	$-0.551537\pi$
$233$	−4.92125	−0.322402	−0.161201	+	0.986922i	$0.551537\pi$
$234$	0	0
$235$	2.58338	0.168521
$236$	0	0
$237$	19.8810	1.29141
$238$	0	0
$239$	8.97604	0.580612	0.290306	−	0.956934i	$-0.406243\pi$
$239$	8.97604	0.580612	0.290306	+	0.956934i	$0.406243\pi$
$240$	0	0
$241$	13.9853	0.900873	0.450436	−	0.892809i	$-0.351268\pi$
$241$	13.9853	0.900873	0.450436	+	0.892809i	$0.351268\pi$
$242$	0	0
$243$	8.77530	0.562936
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.56486	0.417712
$248$	0	0
$249$	0.171880	0.0108924
$250$	0	0
$251$	−17.3054	−1.09231	−0.546154	−	0.837685i	$-0.683909\pi$
$251$	−17.3054	−1.09231	−0.546154	+	0.837685i	$0.683909\pi$
$252$	0	0
$253$	−22.7291	−1.42897
$254$	0	0
$255$	−11.3516	−0.710865
$256$	0	0
$257$	23.6729	1.47668	0.738338	−	0.674431i	$-0.235611\pi$
$257$	23.6729	1.47668	0.738338	+	0.674431i	$0.235611\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−48.8083	−3.02116
$262$	0	0
$263$	−13.5176	−0.833531	−0.416765	−	0.909014i	$-0.636836\pi$
$263$	−13.5176	−0.833531	−0.416765	+	0.909014i	$0.636836\pi$
$264$	0	0
$265$	12.8692	0.790548
$266$	0	0
$267$	36.4785	2.23245
$268$	0	0
$269$	29.8311	1.81883	0.909416	−	0.415887i	$-0.136529\pi$
$269$	29.8311	1.81883	0.909416	+	0.415887i	$0.136529\pi$
$270$	0	0
$271$	−4.05914	−0.246575	−0.123288	−	0.992371i	$-0.539344\pi$
$271$	−4.05914	−0.246575	−0.123288	+	0.992371i	$0.539344\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.7509	1.67344
$276$	0	0
$277$	−9.55942	−0.574370	−0.287185	−	0.957875i	$-0.592719\pi$
$277$	−9.55942	−0.574370	−0.287185	+	0.957875i	$0.592719\pi$
$278$	0	0
$279$	−39.5217	−2.36610
$280$	0	0
$281$	28.1408	1.67874	0.839371	−	0.543559i	$-0.182924\pi$
$281$	28.1408	1.67874	0.839371	+	0.543559i	$0.182924\pi$
$282$	0	0
$283$	−3.61954	−0.215159	−0.107580	−	0.994196i	$-0.534310\pi$
$283$	−3.61954	−0.215159	−0.107580	+	0.994196i	$0.534310\pi$
$284$	0	0
$285$	42.4563	2.51489
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−23.2866	−1.36508
$292$	0	0
$293$	−3.70755	−0.216597	−0.108299	−	0.994118i	$-0.534540\pi$
$293$	−3.70755	−0.216597	−0.108299	+	0.994118i	$0.534540\pi$
$294$	0	0
$295$	−12.6743	−0.737928
$296$	0	0
$297$	−19.2267	−1.11564
$298$	0	0
$299$	−14.7468	−0.852827
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−32.1328	−1.84598
$304$	0	0
$305$	9.35748	0.535808
$306$	0	0
$307$	−6.02005	−0.343582	−0.171791	−	0.985133i	$-0.554955\pi$
$307$	−6.02005	−0.343582	−0.171791	+	0.985133i	$0.554955\pi$
$308$	0	0
$309$	17.2477	0.981187
$310$	0	0
$311$	2.07167	0.117473	0.0587367	−	0.998274i	$-0.481293\pi$
$311$	2.07167	0.117473	0.0587367	+	0.998274i	$0.481293\pi$
$312$	0	0
$313$	4.31021	0.243628	0.121814	−	0.992553i	$-0.461129\pi$
$313$	4.31021	0.243628	0.121814	+	0.992553i	$0.461129\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.5225	1.82665	0.913323	−	0.407236i	$-0.133508\pi$
$317$	32.5225	1.82665	0.913323	+	0.407236i	$0.133508\pi$
$318$	0	0
$319$	−24.2484	−1.35765
$320$	0	0
$321$	−38.5724	−2.15290
$322$	0	0
$323$	−3.74011	−0.208106
$324$	0	0
$325$	18.0049	0.998732
$326$	0	0
$327$	−11.7254	−0.648417
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.5025	1.45671	0.728353	−	0.685202i	$-0.240286\pi$
$331$	26.5025	1.45671	0.728353	+	0.685202i	$0.240286\pi$
$332$	0	0
$333$	33.7660	1.85037
$334$	0	0
$335$	−1.54526	−0.0844264
$336$	0	0
$337$	7.76898	0.423203	0.211602	−	0.977356i	$-0.432132\pi$
$337$	7.76898	0.423203	0.211602	+	0.977356i	$0.432132\pi$
$338$	0	0
$339$	−46.2249	−2.51059
$340$	0	0
$341$	−19.6348	−1.06328
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−95.3702	−5.13456
$346$	0	0
$347$	12.2714	0.658765	0.329382	−	0.944197i	$-0.393159\pi$
$347$	12.2714	0.658765	0.329382	+	0.944197i	$0.393159\pi$
$348$	0	0
$349$	18.8344	1.00818	0.504092	−	0.863650i	$-0.331827\pi$
$349$	18.8344	1.00818	0.504092	+	0.863650i	$0.331827\pi$
$350$	0	0
$351$	−12.4743	−0.665831
$352$	0	0
$353$	4.01514	0.213704	0.106852	−	0.994275i	$-0.465923\pi$
$353$	4.01514	0.213704	0.106852	+	0.994275i	$0.465923\pi$
$354$	0	0
$355$	23.4005	1.24197
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.2907	1.54591	0.772953	−	0.634464i	$-0.218779\pi$
$359$	29.2907	1.54591	0.772953	+	0.634464i	$0.218779\pi$
$360$	0	0
$361$	−5.01154	−0.263765
$362$	0	0
$363$	10.6973	0.561463
$364$	0	0
$365$	−53.3037	−2.79004
$366$	0	0
$367$	−24.0384	−1.25480	−0.627398	−	0.778698i	$-0.715880\pi$
$367$	−24.0384	−1.25480	−0.627398	+	0.778698i	$0.715880\pi$
$368$	0	0
$369$	−58.6062	−3.05091
$370$	0	0
$371$	0	0
$372$	0	0
$373$	24.1002	1.24786	0.623931	−	0.781480i	$-0.285535\pi$
$373$	24.1002	1.24786	0.623931	+	0.781480i	$0.285535\pi$
$374$	0	0
$375$	59.6834	3.08204
$376$	0	0
$377$	−15.7325	−0.810265
$378$	0	0
$379$	11.1993	0.575270	0.287635	−	0.957740i	$-0.407131\pi$
$379$	11.1993	0.575270	0.287635	+	0.957740i	$0.407131\pi$
$380$	0	0
$381$	48.3524	2.47717
$382$	0	0
$383$	−11.8496	−0.605486	−0.302743	−	0.953072i	$-0.597902\pi$
$383$	−11.8496	−0.605486	−0.302743	+	0.953072i	$0.597902\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.62400	0.235051
$388$	0	0
$389$	−13.7623	−0.697778	−0.348889	−	0.937164i	$-0.613441\pi$
$389$	−13.7623	−0.697778	−0.348889	+	0.937164i	$0.613441\pi$
$390$	0	0
$391$	8.40148	0.424881
$392$	0	0
$393$	−1.63206	−0.0823267
$394$	0	0
$395$	26.7220	1.34453
$396$	0	0
$397$	−37.2751	−1.87079	−0.935393	−	0.353611i	$-0.884954\pi$
$397$	−37.2751	−1.87079	−0.935393	+	0.353611i	$0.884954\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.4596	−0.871893	−0.435947	−	0.899973i	$-0.643586\pi$
$401$	−17.4596	−0.871893	−0.435947	+	0.899973i	$0.643586\pi$
$402$	0	0
$403$	−12.7391	−0.634582
$404$	0	0
$405$	−16.8621	−0.837885
$406$	0	0
$407$	16.7753	0.831521
$408$	0	0
$409$	3.69491	0.182702	0.0913508	−	0.995819i	$-0.470882\pi$
$409$	3.69491	0.182702	0.0913508	+	0.995819i	$0.470882\pi$
$410$	0	0
$411$	2.17144	0.107109
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.231024	0.0113405
$416$	0	0
$417$	−23.3135	−1.14167
$418$	0	0
$419$	23.8680	1.16603	0.583014	−	0.812462i	$-0.301873\pi$
$419$	23.8680	1.16603	0.583014	+	0.812462i	$0.301873\pi$
$420$	0	0
$421$	24.4530	1.19177	0.595883	−	0.803071i	$-0.296802\pi$
$421$	24.4530	1.19177	0.595883	+	0.803071i	$0.296802\pi$
$422$	0	0
$423$	−3.60147	−0.175109
$424$	0	0
$425$	−10.2577	−0.497572
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.8000	−0.666271
$430$	0	0
$431$	−37.9325	−1.82714	−0.913572	−	0.406678i	$-0.866687\pi$
$431$	−37.9325	−1.82714	−0.913572	+	0.406678i	$0.866687\pi$
$432$	0	0
$433$	0.232223	0.0111599	0.00557997	−	0.999984i	$-0.498224\pi$
$433$	0.232223	0.0111599	0.00557997	+	0.999984i	$0.498224\pi$
$434$	0	0
$435$	−101.745	−4.87831
$436$	0	0
$437$	−31.4225	−1.50314
$438$	0	0
$439$	28.5594	1.36307	0.681533	−	0.731787i	$-0.261313\pi$
$439$	28.5594	1.36307	0.681533	+	0.731787i	$0.261313\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.0883	−0.669357	−0.334679	−	0.942332i	$-0.608628\pi$
$443$	−14.0883	−0.669357	−0.334679	+	0.942332i	$0.608628\pi$
$444$	0	0
$445$	49.0308	2.32428
$446$	0	0
$447$	−19.2137	−0.908777
$448$	0	0
$449$	17.1188	0.807888	0.403944	−	0.914784i	$-0.367639\pi$
$449$	17.1188	0.807888	0.403944	+	0.914784i	$0.367639\pi$
$450$	0	0
$451$	−29.1161	−1.37102
$452$	0	0
$453$	51.1176	2.40172
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.8411	1.48946	0.744732	−	0.667364i	$-0.232578\pi$
$457$	31.8411	1.48946	0.744732	+	0.667364i	$0.232578\pi$
$458$	0	0
$459$	7.10685	0.331719
$460$	0	0
$461$	−6.41662	−0.298852	−0.149426	−	0.988773i	$-0.547743\pi$
$461$	−6.41662	−0.298852	−0.149426	+	0.988773i	$0.547743\pi$
$462$	0	0
$463$	31.2907	1.45420	0.727102	−	0.686530i	$-0.240867\pi$
$463$	31.2907	1.45420	0.727102	+	0.686530i	$0.240867\pi$
$464$	0	0
$465$	−82.3866	−3.82059
$466$	0	0
$467$	−4.29583	−0.198787	−0.0993936	−	0.995048i	$-0.531690\pi$
$467$	−4.29583	−0.198787	−0.0993936	+	0.995048i	$0.531690\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.42619	−0.157871
$472$	0	0
$473$	2.29725	0.105628
$474$	0	0
$475$	38.3650	1.76031
$476$	0	0
$477$	−17.9409	−0.821455
$478$	0	0
$479$	22.2756	1.01780	0.508899	−	0.860826i	$-0.330053\pi$
$479$	22.2756	1.01780	0.508899	+	0.860826i	$0.330053\pi$
$480$	0	0
$481$	10.8839	0.496263
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−31.2995	−1.42124
$486$	0	0
$487$	38.2198	1.73190	0.865952	−	0.500126i	$-0.166713\pi$
$487$	38.2198	1.73190	0.865952	+	0.500126i	$0.166713\pi$
$488$	0	0
$489$	7.46138	0.337415
$490$	0	0
$491$	−37.8105	−1.70636	−0.853182	−	0.521614i	$-0.825330\pi$
$491$	−37.8105	−1.70636	−0.853182	+	0.521614i	$0.825330\pi$
$492$	0	0
$493$	8.96308	0.403677
$494$	0	0
$495$	−57.5451	−2.58646
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.3672	−0.911760	−0.455880	−	0.890041i	$-0.650675\pi$
$499$	−20.3672	−0.911760	−0.455880	+	0.890041i	$0.650675\pi$
$500$	0	0
$501$	−38.0729	−1.70097
$502$	0	0
$503$	17.1687	0.765516	0.382758	−	0.923849i	$-0.374974\pi$
$503$	17.1687	0.765516	0.382758	+	0.923849i	$0.374974\pi$
$504$	0	0
$505$	−43.1897	−1.92192
$506$	0	0
$507$	28.8259	1.28021
$508$	0	0
$509$	36.3121	1.60950	0.804752	−	0.593611i	$-0.202298\pi$
$509$	36.3121	1.60950	0.804752	+	0.593611i	$0.202298\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−26.5804	−1.17356
$514$	0	0
$515$	23.1827	1.02155
$516$	0	0
$517$	−1.78924	−0.0786909
$518$	0	0
$519$	−18.8969	−0.829480
$520$	0	0
$521$	3.09607	0.135641	0.0678205	−	0.997698i	$-0.478395\pi$
$521$	3.09607	0.135641	0.0678205	+	0.997698i	$0.478395\pi$
$522$	0	0
$523$	−8.56748	−0.374630	−0.187315	−	0.982300i	$-0.559978\pi$
$523$	−8.56748	−0.374630	−0.187315	+	0.982300i	$0.559978\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.25771	0.316151
$528$	0	0
$529$	47.5849	2.06891
$530$	0	0
$531$	17.6692	0.766778
$532$	0	0
$533$	−18.8907	−0.818245
$534$	0	0
$535$	−51.8452	−2.24146
$536$	0	0
$537$	58.4704	2.52319
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0.232223	0.00998405	0.00499203	−	0.999988i	$-0.498411\pi$
$541$	0.232223	0.00998405	0.00499203	+	0.999988i	$0.498411\pi$
$542$	0	0
$543$	21.5288	0.923890
$544$	0	0
$545$	−15.7602	−0.675091
$546$	0	0
$547$	13.7323	0.587150	0.293575	−	0.955936i	$-0.405155\pi$
$547$	13.7323	0.587150	0.293575	+	0.955936i	$0.405155\pi$
$548$	0	0
$549$	−13.0452	−0.556756
$550$	0	0
$551$	−33.5229	−1.42813
$552$	0	0
$553$	0	0
$554$	0	0
$555$	70.3883	2.98782
$556$	0	0
$557$	−5.85885	−0.248247	−0.124124	−	0.992267i	$-0.539612\pi$
$557$	−5.85885	−0.248247	−0.124124	+	0.992267i	$0.539612\pi$
$558$	0	0
$559$	1.49047	0.0630400
$560$	0	0
$561$	7.86211	0.331938
$562$	0	0
$563$	17.0784	0.719770	0.359885	−	0.932997i	$-0.382816\pi$
$563$	17.0784	0.719770	0.359885	+	0.932997i	$0.382816\pi$
$564$	0	0
$565$	−62.1310	−2.61387
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.19268	0.0919220	0.0459610	−	0.998943i	$-0.485365\pi$
$569$	2.19268	0.0919220	0.0459610	+	0.998943i	$0.485365\pi$
$570$	0	0
$571$	15.3863	0.643898	0.321949	−	0.946757i	$-0.395662\pi$
$571$	15.3863	0.643898	0.321949	+	0.946757i	$0.395662\pi$
$572$	0	0
$573$	−0.419112	−0.0175087
$574$	0	0
$575$	−86.1800	−3.59395
$576$	0	0
$577$	−25.3520	−1.05542	−0.527710	−	0.849425i	$-0.676949\pi$
$577$	−25.3520	−1.05542	−0.527710	+	0.849425i	$0.676949\pi$
$578$	0	0
$579$	−53.6692	−2.23042
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8.91319	−0.369147
$584$	0	0
$585$	−37.3355	−1.54363
$586$	0	0
$587$	−43.5074	−1.79574	−0.897870	−	0.440260i	$-0.854886\pi$
$587$	−43.5074	−1.79574	−0.897870	+	0.440260i	$0.854886\pi$
$588$	0	0
$589$	−27.1447	−1.11848
$590$	0	0
$591$	−28.7883	−1.18419
$592$	0	0
$593$	−11.1113	−0.456287	−0.228143	−	0.973628i	$-0.573266\pi$
$593$	−11.1113	−0.456287	−0.228143	+	0.973628i	$0.573266\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.6621	1.66419
$598$	0	0
$599$	−16.4318	−0.671384	−0.335692	−	0.941972i	$-0.608970\pi$
$599$	−16.4318	−0.671384	−0.335692	+	0.941972i	$0.608970\pi$
$600$	0	0
$601$	26.7786	1.09232	0.546160	−	0.837681i	$-0.316089\pi$
$601$	26.7786	1.09232	0.546160	+	0.837681i	$0.316089\pi$
$602$	0	0
$603$	2.15423	0.0877271
$604$	0	0
$605$	14.3783	0.584560
$606$	0	0
$607$	20.5015	0.832129	0.416065	−	0.909335i	$-0.363409\pi$
$607$	20.5015	0.832129	0.416065	+	0.909335i	$0.363409\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.16087	−0.0469638
$612$	0	0
$613$	11.2355	0.453797	0.226898	−	0.973918i	$-0.427141\pi$
$613$	11.2355	0.453797	0.226898	+	0.973918i	$0.427141\pi$
$614$	0	0
$615$	−122.170	−4.92636
$616$	0	0
$617$	4.58218	0.184472	0.0922358	−	0.995737i	$-0.470599\pi$
$617$	4.58218	0.184472	0.0922358	+	0.995737i	$0.470599\pi$
$618$	0	0
$619$	29.1295	1.17081	0.585407	−	0.810739i	$-0.300935\pi$
$619$	29.1295	1.17081	0.585407	+	0.810739i	$0.300935\pi$
$620$	0	0
$621$	59.7081	2.39600
$622$	0	0
$623$	0	0
$624$	0	0
$625$	28.9320	1.15728
$626$	0	0
$627$	−29.4052	−1.17433
$628$	0	0
$629$	−6.20074	−0.247240
$630$	0	0
$631$	27.2534	1.08494	0.542469	−	0.840076i	$-0.317489\pi$
$631$	27.2534	1.08494	0.542469	+	0.840076i	$0.317489\pi$
$632$	0	0
$633$	25.9014	1.02949
$634$	0	0
$635$	64.9905	2.57907
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−32.6225	−1.29053
$640$	0	0
$641$	29.8540	1.17916	0.589582	−	0.807709i	$-0.299292\pi$
$641$	29.8540	1.17916	0.589582	+	0.807709i	$0.299292\pi$
$642$	0	0
$643$	43.4274	1.71261	0.856305	−	0.516471i	$-0.172754\pi$
$643$	43.4274	1.71261	0.856305	+	0.516471i	$0.172754\pi$
$644$	0	0
$645$	9.63914	0.379541
$646$	0	0
$647$	5.14759	0.202373	0.101186	−	0.994867i	$-0.467736\pi$
$647$	5.14759	0.202373	0.101186	+	0.994867i	$0.467736\pi$
$648$	0	0
$649$	8.77823	0.344576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.8519	0.424667	0.212333	−	0.977197i	$-0.431894\pi$
$653$	10.8519	0.424667	0.212333	+	0.977197i	$0.431894\pi$
$654$	0	0
$655$	−2.19366	−0.0857134
$656$	0	0
$657$	74.3103	2.89912
$658$	0	0
$659$	−11.3748	−0.443099	−0.221550	−	0.975149i	$-0.571111\pi$
$659$	−11.3748	−0.443099	−0.221550	+	0.975149i	$0.571111\pi$
$660$	0	0
$661$	39.1768	1.52380	0.761900	−	0.647694i	$-0.224267\pi$
$661$	39.1768	1.52380	0.761900	+	0.647694i	$0.224267\pi$
$662$	0	0
$663$	5.10097	0.198105
$664$	0	0
$665$	0	0
$666$	0	0
$667$	75.3031	2.91575
$668$	0	0
$669$	−35.2848	−1.36419
$670$	0	0
$671$	−6.48098	−0.250196
$672$	0	0
$673$	26.4303	1.01881	0.509407	−	0.860525i	$-0.329865\pi$
$673$	26.4303	1.01881	0.509407	+	0.860525i	$0.329865\pi$
$674$	0	0
$675$	−72.9000	−2.80592
$676$	0	0
$677$	−6.40268	−0.246075	−0.123038	−	0.992402i	$-0.539264\pi$
$677$	−6.40268	−0.246075	−0.123038	+	0.992402i	$0.539264\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.0629	0.462249
$682$	0	0
$683$	−3.15041	−0.120547	−0.0602736	−	0.998182i	$-0.519197\pi$
$683$	−3.15041	−0.120547	−0.0602736	+	0.998182i	$0.519197\pi$
$684$	0	0
$685$	2.91863	0.111515
$686$	0	0
$687$	8.63666	0.329509
$688$	0	0
$689$	−5.78292	−0.220312
$690$	0	0
$691$	−7.77040	−0.295600	−0.147800	−	0.989017i	$-0.547219\pi$
$691$	−7.77040	−0.295600	−0.147800	+	0.989017i	$0.547219\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−31.3357	−1.18863
$696$	0	0
$697$	10.7623	0.407653
$698$	0	0
$699$	14.3017	0.540940
$700$	0	0
$701$	36.2098	1.36763	0.683813	−	0.729658i	$-0.260321\pi$
$701$	36.2098	1.36763	0.683813	+	0.729658i	$0.260321\pi$
$702$	0	0
$703$	23.1915	0.874683
$704$	0	0
$705$	−7.50758	−0.282752
$706$	0	0
$707$	0	0
$708$	0	0
$709$	13.0699	0.490852	0.245426	−	0.969415i	$-0.421072\pi$
$709$	13.0699	0.490852	0.245426	+	0.969415i	$0.421072\pi$
$710$	0	0
$711$	−37.2530	−1.39710
$712$	0	0
$713$	60.9755	2.28355
$714$	0	0
$715$	−18.5486	−0.693680
$716$	0	0
$717$	−26.0854	−0.974177
$718$	0	0
$719$	−45.0304	−1.67935	−0.839675	−	0.543089i	$-0.817255\pi$
$719$	−45.0304	−1.67935	−0.839675	+	0.543089i	$0.817255\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−40.6429	−1.51152
$724$	0	0
$725$	−91.9407	−3.41459
$726$	0	0
$727$	37.7719	1.40088	0.700442	−	0.713710i	$-0.252986\pi$
$727$	37.7719	1.40088	0.700442	+	0.713710i	$0.252986\pi$
$728$	0	0
$729$	−38.4526	−1.42417
$730$	0	0
$731$	−0.849144	−0.0314067
$732$	0	0
$733$	−27.2323	−1.00585	−0.502925	−	0.864330i	$-0.667743\pi$
$733$	−27.2323	−1.00585	−0.502925	+	0.864330i	$0.667743\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.07024	0.0394229
$738$	0	0
$739$	31.4581	1.15721	0.578603	−	0.815609i	$-0.303598\pi$
$739$	31.4581	1.15721	0.578603	+	0.815609i	$0.303598\pi$
$740$	0	0
$741$	−19.0782	−0.700856
$742$	0	0
$743$	−4.76604	−0.174849	−0.0874246	−	0.996171i	$-0.527864\pi$
$743$	−4.76604	−0.174849	−0.0874246	+	0.996171i	$0.527864\pi$
$744$	0	0
$745$	−25.8252	−0.946162
$746$	0	0
$747$	−0.322069	−0.0117839
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.7113	−1.04769	−0.523844	−	0.851814i	$-0.675503\pi$
$751$	−28.7113	−1.04769	−0.523844	+	0.851814i	$0.675503\pi$
$752$	0	0
$753$	50.2915	1.83272
$754$	0	0
$755$	68.7073	2.50052
$756$	0	0
$757$	−27.3959	−0.995722	−0.497861	−	0.867257i	$-0.665881\pi$
$757$	−27.3959	−0.995722	−0.497861	+	0.867257i	$0.665881\pi$
$758$	0	0
$759$	66.0534	2.39759
$760$	0	0
$761$	−19.5166	−0.707477	−0.353738	−	0.935344i	$-0.615090\pi$
$761$	−19.5166	−0.707477	−0.353738	+	0.935344i	$0.615090\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	21.2707	0.769043
$766$	0	0
$767$	5.69536	0.205647
$768$	0	0
$769$	−47.4834	−1.71230	−0.856148	−	0.516731i	$-0.827149\pi$
$769$	−47.4834	−1.71230	−0.856148	+	0.516731i	$0.827149\pi$
$770$	0	0
$771$	−68.7961	−2.47763
$772$	0	0
$773$	−53.0183	−1.90694	−0.953468	−	0.301494i	$-0.902515\pi$
$773$	−53.0183	−1.90694	−0.953468	+	0.301494i	$0.902515\pi$
$774$	0	0
$775$	−74.4475	−2.67423
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−40.2524	−1.44219
$780$	0	0
$781$	−16.2072	−0.579938
$782$	0	0
$783$	63.6993	2.27643
$784$	0	0
$785$	−4.60516	−0.164365
$786$	0	0
$787$	−20.9489	−0.746748	−0.373374	−	0.927681i	$-0.621799\pi$
$787$	−20.9489	−0.746748	−0.373374	+	0.927681i	$0.621799\pi$
$788$	0	0
$789$	39.2836	1.39853
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.20489	−0.149320
$794$	0	0
$795$	−37.3993	−1.32642
$796$	0	0
$797$	−5.47805	−0.194043	−0.0970213	−	0.995282i	$-0.530931\pi$
$797$	−5.47805	−0.194043	−0.0970213	+	0.995282i	$0.530931\pi$
$798$	0	0
$799$	0.661368	0.0233975
$800$	0	0
$801$	−68.3536	−2.41515
$802$	0	0
$803$	36.9181	1.30281
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−86.6924	−3.05172
$808$	0	0
$809$	−13.4817	−0.473990	−0.236995	−	0.971511i	$-0.576162\pi$
$809$	−13.4817	−0.473990	−0.236995	+	0.971511i	$0.576162\pi$
$810$	0	0
$811$	−7.90536	−0.277595	−0.138797	−	0.990321i	$-0.544324\pi$
$811$	−7.90536	−0.277595	−0.138797	+	0.990321i	$0.544324\pi$
$812$	0	0
$813$	11.7963	0.413715
$814$	0	0
$815$	10.0289	0.351296
$816$	0	0
$817$	3.17590	0.111111
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.3061	−0.569086	−0.284543	−	0.958663i	$-0.591842\pi$
$821$	−16.3061	−0.569086	−0.284543	+	0.958663i	$0.591842\pi$
$822$	0	0
$823$	−52.2328	−1.82072	−0.910360	−	0.413817i	$-0.864195\pi$
$823$	−52.2328	−1.82072	−0.910360	+	0.413817i	$0.864195\pi$
$824$	0	0
$825$	−80.6472	−2.80778
$826$	0	0
$827$	−12.8255	−0.445986	−0.222993	−	0.974820i	$-0.571583\pi$
$827$	−12.8255	−0.445986	−0.222993	+	0.974820i	$0.571583\pi$
$828$	0	0
$829$	19.9569	0.693131	0.346566	−	0.938026i	$-0.387348\pi$
$829$	19.9569	0.693131	0.346566	+	0.938026i	$0.387348\pi$
$830$	0	0
$831$	27.7807	0.963703
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−51.1739	−1.77094
$836$	0	0
$837$	51.5795	1.78285
$838$	0	0
$839$	13.7299	0.474008	0.237004	−	0.971509i	$-0.423835\pi$
$839$	13.7299	0.474008	0.237004	+	0.971509i	$0.423835\pi$
$840$	0	0
$841$	51.3368	1.77023
$842$	0	0
$843$	−81.7804	−2.81667
$844$	0	0
$845$	38.7450	1.33287
$846$	0	0
$847$	0	0
$848$	0	0
$849$	10.5188	0.361004
$850$	0	0
$851$	−52.0954	−1.78581
$852$	0	0
$853$	27.5350	0.942781	0.471391	−	0.881925i	$-0.343752\pi$
$853$	27.5350	0.942781	0.471391	+	0.881925i	$0.343752\pi$
$854$	0	0
$855$	−79.5548	−2.72072
$856$	0	0
$857$	4.22372	0.144280	0.0721398	−	0.997395i	$-0.477017\pi$
$857$	4.22372	0.144280	0.0721398	+	0.997395i	$0.477017\pi$
$858$	0	0
$859$	−15.4890	−0.528479	−0.264240	−	0.964457i	$-0.585121\pi$
$859$	−15.4890	−0.528479	−0.264240	+	0.964457i	$0.585121\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.58142	0.0538322	0.0269161	−	0.999638i	$-0.491431\pi$
$863$	1.58142	0.0538322	0.0269161	+	0.999638i	$0.491431\pi$
$864$	0	0
$865$	−25.3993	−0.863602
$866$	0	0
$867$	−2.90611	−0.0986967
$868$	0	0
$869$	−18.5077	−0.627830
$870$	0	0
$871$	0.694379	0.0235281
$872$	0	0
$873$	43.6345	1.47680
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−48.4127	−1.63478	−0.817391	−	0.576084i	$-0.804580\pi$
$877$	−48.4127	−1.63478	−0.817391	+	0.576084i	$0.804580\pi$
$878$	0	0
$879$	10.7745	0.363416
$880$	0	0
$881$	−4.03736	−0.136022	−0.0680111	−	0.997685i	$-0.521665\pi$
$881$	−4.03736	−0.136022	−0.0680111	+	0.997685i	$0.521665\pi$
$882$	0	0
$883$	−38.4086	−1.29255	−0.646276	−	0.763104i	$-0.723674\pi$
$883$	−38.4086	−1.29255	−0.646276	+	0.763104i	$0.723674\pi$
$884$	0	0
$885$	36.8330	1.23813
$886$	0	0
$887$	26.7904	0.899535	0.449768	−	0.893146i	$-0.351507\pi$
$887$	26.7904	0.899535	0.449768	+	0.893146i	$0.351507\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	11.6787	0.391251
$892$	0	0
$893$	−2.47359	−0.0827756
$894$	0	0
$895$	78.5903	2.62698
$896$	0	0
$897$	42.8557	1.43091
$898$	0	0
$899$	65.0514	2.16959
$900$	0	0
$901$	3.29463	0.109760
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.9369	0.961897
$906$	0	0
$907$	−0.238106	−0.00790618	−0.00395309	−	0.999992i	$-0.501258\pi$
$907$	−0.238106	−0.00790618	−0.00395309	+	0.999992i	$0.501258\pi$
$908$	0	0
$909$	60.2106	1.99706
$910$	0	0
$911$	−2.74251	−0.0908635	−0.0454318	−	0.998967i	$-0.514466\pi$
$911$	−2.74251	−0.0908635	−0.0454318	+	0.998967i	$0.514466\pi$
$912$	0	0
$913$	−0.160007	−0.00529546
$914$	0	0
$915$	−27.1939	−0.899002
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35.2071	1.16137	0.580687	−	0.814127i	$-0.302784\pi$
$919$	35.2071	1.16137	0.580687	+	0.814127i	$0.302784\pi$
$920$	0	0
$921$	17.4949	0.576477
$922$	0	0
$923$	−10.5153	−0.346115
$924$	0	0
$925$	63.6054	2.09133
$926$	0	0
$927$	−32.3188	−1.06149
$928$	0	0
$929$	−45.0703	−1.47871	−0.739354	−	0.673317i	$-0.764869\pi$
$929$	−45.0703	−1.47871	−0.739354	+	0.673317i	$0.764869\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.02049	−0.197102
$934$	0	0
$935$	10.5675	0.345594
$936$	0	0
$937$	−27.6981	−0.904857	−0.452428	−	0.891801i	$-0.649442\pi$
$937$	−27.6981	−0.904857	−0.452428	+	0.891801i	$0.649442\pi$
$938$	0	0
$939$	−12.5260	−0.408769
$940$	0	0
$941$	−12.6846	−0.413505	−0.206753	−	0.978393i	$-0.566290\pi$
$941$	−12.6846	−0.413505	−0.206753	+	0.978393i	$0.566290\pi$
$942$	0	0
$943$	90.4196	2.94447
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.9179	−0.874713	−0.437357	−	0.899288i	$-0.644085\pi$
$947$	−26.9179	−0.874713	−0.437357	+	0.899288i	$0.644085\pi$
$948$	0	0
$949$	23.9526	0.777535
$950$	0	0
$951$	−94.5140	−3.06483
$952$	0	0
$953$	9.33057	0.302247	0.151123	−	0.988515i	$-0.451711\pi$
$953$	9.33057	0.302247	0.151123	+	0.988515i	$0.451711\pi$
$954$	0	0
$955$	−0.563330	−0.0182289
$956$	0	0
$957$	70.4687	2.27793
$958$	0	0
$959$	0	0
$960$	0	0
$961$	21.6743	0.699172
$962$	0	0
$963$	72.2771	2.32910
$964$	0	0
$965$	−72.1369	−2.32217
$966$	0	0
$967$	−35.4929	−1.14137	−0.570687	−	0.821168i	$-0.693323\pi$
$967$	−35.4929	−1.14137	−0.570687	+	0.821168i	$0.693323\pi$
$968$	0	0
$969$	10.8692	0.349169
$970$	0	0
$971$	29.6575	0.951753	0.475877	−	0.879512i	$-0.342131\pi$
$971$	29.6575	0.951753	0.475877	+	0.879512i	$0.342131\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−52.3243	−1.67572
$976$	0	0
$977$	51.8774	1.65970	0.829852	−	0.557984i	$-0.188425\pi$
$977$	51.8774	1.65970	0.829852	+	0.557984i	$0.188425\pi$
$978$	0	0
$979$	−33.9587	−1.08533
$980$	0	0
$981$	21.9711	0.701484
$982$	0	0
$983$	35.8448	1.14327	0.571636	−	0.820508i	$-0.306309\pi$
$983$	35.8448	1.14327	0.571636	+	0.820508i	$0.306309\pi$
$984$	0	0
$985$	−38.6944	−1.23291
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.13407	−0.226850
$990$	0	0
$991$	−54.0350	−1.71648	−0.858239	−	0.513250i	$-0.828441\pi$
$991$	−54.0350	−1.71648	−0.858239	+	0.513250i	$0.828441\pi$
$992$	0	0
$993$	−77.0191	−2.44413
$994$	0	0
$995$	54.6541	1.73265
$996$	0	0
$997$	−16.1976	−0.512983	−0.256491	−	0.966547i	$-0.582567\pi$
$997$	−16.1976	−0.512983	−0.256491	+	0.966547i	$0.582567\pi$
$998$	0	0
$999$	−44.0678	−1.39424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.s.1.1		4
7.2	even	3		476.2.i.e.137.4	✓	8
7.4	even	3		476.2.i.e.205.4	yes	8
7.6	odd	2		3332.2.a.r.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.i.e.137.4	✓	8	7.2	even	3
476.2.i.e.205.4	yes	8	7.4	even	3
3332.2.a.r.1.4		4	7.6	odd	2
3332.2.a.s.1.1		4	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q-2.90611 q^{3} -3.90611 q^{5} +5.44549 q^{9} +O(q^{10})\)	\(q-2.90611 q^{3} -3.90611 q^{5} +5.44549 q^{9} +2.70537 q^{11} +1.75526 q^{13} +11.3516 q^{15} -1.00000 q^{17} +3.74011 q^{19} -8.40148 q^{23} +10.2577 q^{25} -7.10685 q^{27} -8.96308 q^{29} -7.25771 q^{31} -7.86211 q^{33} +6.20074 q^{37} -5.10097 q^{39} -10.7623 q^{41} +0.849144 q^{43} -21.2707 q^{45} -0.661368 q^{47} +2.90611 q^{51} -3.29463 q^{53} -10.5675 q^{55} -10.8692 q^{57} +3.24474 q^{59} -2.39560 q^{61} -6.85623 q^{65} +0.395600 q^{67} +24.4156 q^{69} -5.99074 q^{71} +13.6462 q^{73} -29.8100 q^{75} -6.84108 q^{79} +4.31685 q^{81} -0.0591443 q^{83} +3.90611 q^{85} +26.0477 q^{87} -12.5523 q^{89} +21.0917 q^{93} -14.6093 q^{95} +8.01296 q^{97} +14.7321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 2 * q^5 + 10 * q^9	\( 4 q + 2 q^{3} - 2 q^{5} + 10 q^{9} + 11 q^{11} + 5 q^{13} + 20 q^{15} - 4 q^{17} + 3 q^{19} - 6 q^{23} + 2 q^{25} - q^{27} + 3 q^{29} + 10 q^{31} - 2 q^{33} + 11 q^{37} - 11 q^{39} - 18 q^{41} + 15 q^{43} - 5 q^{45} + 13 q^{47} - 2 q^{51} - 13 q^{53} - 13 q^{55} + 9 q^{57} + 15 q^{59} - 16 q^{65} - 8 q^{67} + 24 q^{69} + 4 q^{71} + 29 q^{73} - 41 q^{75} - 6 q^{79} + 4 q^{81} - 26 q^{83} + 2 q^{85} + 51 q^{87} - 11 q^{89} + 47 q^{93} + 6 q^{95} - 9 q^{97} + 14 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^5 + 10 * q^9 + 11 * q^11 + 5 * q^13 + 20 * q^15 - 4 * q^17 + 3 * q^19 - 6 * q^23 + 2 * q^25 - q^27 + 3 * q^29 + 10 * q^31 - 2 * q^33 + 11 * q^37 - 11 * q^39 - 18 * q^41 + 15 * q^43 - 5 * q^45 + 13 * q^47 - 2 * q^51 - 13 * q^53 - 13 * q^55 + 9 * q^57 + 15 * q^59 - 16 * q^65 - 8 * q^67 + 24 * q^69 + 4 * q^71 + 29 * q^73 - 41 * q^75 - 6 * q^79 + 4 * q^81 - 26 * q^83 + 2 * q^85 + 51 * q^87 - 11 * q^89 + 47 * q^93 + 6 * q^95 - 9 * q^97 + 14 * q^99

Introduction

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