LMFDB - Embedded newform 3344.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 $ x^6 - 2x^5 - 12x^4 + 28x^3 + 16x^2 - 60*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.64714$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64714 q^{3} +1.84900 q^{5} -3.60453 q^{7} -0.286920 q^{9} +O(q^{10})$	$q-1.64714 q^{3} +1.84900 q^{5} -3.60453 q^{7} -0.286920 q^{9} -1.00000 q^{11} -4.60764 q^{13} -3.04556 q^{15} +4.53875 q^{17} -1.00000 q^{19} +5.93718 q^{21} -7.43020 q^{23} -1.58121 q^{25} +5.41403 q^{27} -1.20922 q^{29} +8.16972 q^{31} +1.64714 q^{33} -6.66477 q^{35} -2.67467 q^{37} +7.58945 q^{39} -10.6575 q^{41} -7.58121 q^{43} -0.530515 q^{45} +7.21528 q^{47} +5.99263 q^{49} -7.47598 q^{51} +9.86813 q^{53} -1.84900 q^{55} +1.64714 q^{57} +7.61190 q^{59} +13.0950 q^{61} +1.03421 q^{63} -8.51952 q^{65} -9.37878 q^{67} +12.2386 q^{69} +10.9414 q^{71} +8.17013 q^{73} +2.60447 q^{75} +3.60453 q^{77} -14.1049 q^{79} -8.05692 q^{81} +15.3819 q^{83} +8.39215 q^{85} +1.99176 q^{87} +6.58945 q^{89} +16.6084 q^{91} -13.4567 q^{93} -1.84900 q^{95} -12.8914 q^{97} +0.286920 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * 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
$3$	−1.64714	−0.950978	−0.475489	−	0.879722i	$-0.657729\pi$
$3$	−1.64714	−0.950978	−0.475489	+	0.879722i	$0.657729\pi$
$4$	0	0
$5$	1.84900	0.826897	0.413449	−	0.910527i	$-0.364324\pi$
$5$	1.84900	0.826897	0.413449	+	0.910527i	$0.364324\pi$
$6$	0	0
$7$	−3.60453	−1.36238	−0.681192	−	0.732105i	$-0.738538\pi$
$7$	−3.60453	−1.36238	−0.681192	+	0.732105i	$0.738538\pi$
$8$	0	0
$9$	−0.286920	−0.0956400
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.60764	−1.27793	−0.638965	−	0.769236i	$-0.720637\pi$
$13$	−4.60764	−1.27793	−0.638965	+	0.769236i	$0.720637\pi$
$14$	0	0
$15$	−3.04556	−0.786361
$16$	0	0
$17$	4.53875	1.10081	0.550405	−	0.834898i	$-0.314473\pi$
$17$	4.53875	1.10081	0.550405	+	0.834898i	$0.314473\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.93718	1.29560
$22$	0	0
$23$	−7.43020	−1.54930	−0.774652	−	0.632387i	$-0.782075\pi$
$23$	−7.43020	−1.54930	−0.774652	+	0.632387i	$0.782075\pi$
$24$	0	0
$25$	−1.58121	−0.316241
$26$	0	0
$27$	5.41403	1.04193
$28$	0	0
$29$	−1.20922	−0.224547	−0.112273	−	0.993677i	$-0.535813\pi$
$29$	−1.20922	−0.224547	−0.112273	+	0.993677i	$0.535813\pi$
$30$	0	0
$31$	8.16972	1.46733	0.733663	−	0.679514i	$-0.237809\pi$
$31$	8.16972	1.46733	0.733663	+	0.679514i	$0.237809\pi$
$32$	0	0
$33$	1.64714	0.286731
$34$	0	0
$35$	−6.66477	−1.12655
$36$	0	0
$37$	−2.67467	−0.439713	−0.219857	−	0.975532i	$-0.570559\pi$
$37$	−2.67467	−0.439713	−0.219857	+	0.975532i	$0.570559\pi$
$38$	0	0
$39$	7.58945	1.21528
$40$	0	0
$41$	−10.6575	−1.66442	−0.832208	−	0.554464i	$-0.812923\pi$
$41$	−10.6575	−1.66442	−0.832208	+	0.554464i	$0.812923\pi$
$42$	0	0
$43$	−7.58121	−1.15612	−0.578062	−	0.815993i	$-0.696191\pi$
$43$	−7.58121	−1.15612	−0.578062	+	0.815993i	$0.696191\pi$
$44$	0	0
$45$	−0.530515	−0.0790844
$46$	0	0
$47$	7.21528	1.05246	0.526229	−	0.850343i	$-0.323606\pi$
$47$	7.21528	1.05246	0.526229	+	0.850343i	$0.323606\pi$
$48$	0	0
$49$	5.99263	0.856091
$50$	0	0
$51$	−7.47598	−1.04685
$52$	0	0
$53$	9.86813	1.35549	0.677746	−	0.735296i	$-0.262957\pi$
$53$	9.86813	1.35549	0.677746	+	0.735296i	$0.262957\pi$
$54$	0	0
$55$	−1.84900	−0.249319
$56$	0	0
$57$	1.64714	0.218169
$58$	0	0
$59$	7.61190	0.990984	0.495492	−	0.868612i	$-0.334988\pi$
$59$	7.61190	0.990984	0.495492	+	0.868612i	$0.334988\pi$
$60$	0	0
$61$	13.0950	1.67664	0.838320	−	0.545179i	$-0.183538\pi$
$61$	13.0950	1.67664	0.838320	+	0.545179i	$0.183538\pi$
$62$	0	0
$63$	1.03421	0.130298
$64$	0	0
$65$	−8.51952	−1.05672
$66$	0	0
$67$	−9.37878	−1.14580	−0.572900	−	0.819625i	$-0.694182\pi$
$67$	−9.37878	−1.14580	−0.572900	+	0.819625i	$0.694182\pi$
$68$	0	0
$69$	12.2386	1.47336
$70$	0	0
$71$	10.9414	1.29851	0.649254	−	0.760571i	$-0.275081\pi$
$71$	10.9414	1.29851	0.649254	+	0.760571i	$0.275081\pi$
$72$	0	0
$73$	8.17013	0.956241	0.478121	−	0.878294i	$-0.341318\pi$
$73$	8.17013	0.956241	0.478121	+	0.878294i	$0.341318\pi$
$74$	0	0
$75$	2.60447	0.300739
$76$	0	0
$77$	3.60453	0.410774
$78$	0	0
$79$	−14.1049	−1.58692	−0.793461	−	0.608621i	$-0.791723\pi$
$79$	−14.1049	−1.58692	−0.793461	+	0.608621i	$0.791723\pi$
$80$	0	0
$81$	−8.05692	−0.895213
$82$	0	0
$83$	15.3819	1.68838	0.844191	−	0.536042i	$-0.180081\pi$
$83$	15.3819	1.68838	0.844191	+	0.536042i	$0.180081\pi$
$84$	0	0
$85$	8.39215	0.910256
$86$	0	0
$87$	1.99176	0.213539
$88$	0	0
$89$	6.58945	0.698480	0.349240	−	0.937033i	$-0.386440\pi$
$89$	6.58945	0.698480	0.349240	+	0.937033i	$0.386440\pi$
$90$	0	0
$91$	16.6084	1.74103
$92$	0	0
$93$	−13.4567	−1.39539
$94$	0	0
$95$	−1.84900	−0.189703
$96$	0	0
$97$	−12.8914	−1.30893	−0.654464	−	0.756093i	$-0.727106\pi$
$97$	−12.8914	−1.30893	−0.654464	+	0.756093i	$0.727106\pi$
$98$	0	0
$99$	0.286920	0.0288365
$100$	0	0
$101$	0.942465	0.0937787	0.0468894	−	0.998900i	$-0.485069\pi$
$101$	0.942465	0.0937787	0.0468894	+	0.998900i	$0.485069\pi$
$102$	0	0
$103$	10.5386	1.03840	0.519199	−	0.854653i	$-0.326230\pi$
$103$	10.5386	1.03840	0.519199	+	0.854653i	$0.326230\pi$
$104$	0	0
$105$	10.9778	1.07133
$106$	0	0
$107$	−7.04516	−0.681081	−0.340540	−	0.940230i	$-0.610610\pi$
$107$	−7.04516	−0.681081	−0.340540	+	0.940230i	$0.610610\pi$
$108$	0	0
$109$	−10.2076	−0.977708	−0.488854	−	0.872366i	$-0.662585\pi$
$109$	−10.2076	−0.977708	−0.488854	+	0.872366i	$0.662585\pi$
$110$	0	0
$111$	4.40557	0.418158
$112$	0	0
$113$	3.83490	0.360757	0.180378	−	0.983597i	$-0.442268\pi$
$113$	3.83490	0.360757	0.180378	+	0.983597i	$0.442268\pi$
$114$	0	0
$115$	−13.7384	−1.28112
$116$	0	0
$117$	1.32202	0.122221
$118$	0	0
$119$	−16.3601	−1.49973
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	17.5544	1.58282
$124$	0	0
$125$	−12.1686	−1.08840
$126$	0	0
$127$	−10.6913	−0.948696	−0.474348	−	0.880337i	$-0.657316\pi$
$127$	−10.6913	−0.948696	−0.474348	+	0.880337i	$0.657316\pi$
$128$	0	0
$129$	12.4873	1.09945
$130$	0	0
$131$	10.2811	0.898260	0.449130	−	0.893466i	$-0.351734\pi$
$131$	10.2811	0.898260	0.449130	+	0.893466i	$0.351734\pi$
$132$	0	0
$133$	3.60453	0.312552
$134$	0	0
$135$	10.0105	0.861569
$136$	0	0
$137$	5.15505	0.440425	0.220213	−	0.975452i	$-0.429325\pi$
$137$	5.15505	0.440425	0.220213	+	0.975452i	$0.429325\pi$
$138$	0	0
$139$	−2.23710	−0.189749	−0.0948743	−	0.995489i	$-0.530245\pi$
$139$	−2.23710	−0.189749	−0.0948743	+	0.995489i	$0.530245\pi$
$140$	0	0
$141$	−11.8846	−1.00086
$142$	0	0
$143$	4.60764	0.385310
$144$	0	0
$145$	−2.23585	−0.185677
$146$	0	0
$147$	−9.87073	−0.814124
$148$	0	0
$149$	−13.7042	−1.12269	−0.561347	−	0.827581i	$-0.689717\pi$
$149$	−13.7042	−1.12269	−0.561347	+	0.827581i	$0.689717\pi$
$150$	0	0
$151$	16.4420	1.33803	0.669014	−	0.743250i	$-0.266717\pi$
$151$	16.4420	1.33803	0.669014	+	0.743250i	$0.266717\pi$
$152$	0	0
$153$	−1.30226	−0.105281
$154$	0	0
$155$	15.1058	1.21333
$156$	0	0
$157$	−21.4827	−1.71451	−0.857254	−	0.514894i	$-0.827831\pi$
$157$	−21.4827	−1.71451	−0.857254	+	0.514894i	$0.827831\pi$
$158$	0	0
$159$	−16.2542	−1.28904
$160$	0	0
$161$	26.7824	2.11075
$162$	0	0
$163$	−17.1301	−1.34173	−0.670865	−	0.741580i	$-0.734077\pi$
$163$	−17.1301	−1.34173	−0.670865	+	0.741580i	$0.734077\pi$
$164$	0	0
$165$	3.04556	0.237097
$166$	0	0
$167$	18.8782	1.46084	0.730419	−	0.682999i	$-0.239325\pi$
$167$	18.8782	1.46084	0.730419	+	0.682999i	$0.239325\pi$
$168$	0	0
$169$	8.23037	0.633105
$170$	0	0
$171$	0.286920	0.0219413
$172$	0	0
$173$	9.79779	0.744912	0.372456	−	0.928050i	$-0.378516\pi$
$173$	9.79779	0.744912	0.372456	+	0.928050i	$0.378516\pi$
$174$	0	0
$175$	5.69950	0.430842
$176$	0	0
$177$	−12.5379	−0.942405
$178$	0	0
$179$	9.54923	0.713743	0.356872	−	0.934153i	$-0.383843\pi$
$179$	9.54923	0.713743	0.356872	+	0.934153i	$0.383843\pi$
$180$	0	0
$181$	20.9124	1.55441	0.777203	−	0.629249i	$-0.216638\pi$
$181$	20.9124	1.55441	0.777203	+	0.629249i	$0.216638\pi$
$182$	0	0
$183$	−21.5693	−1.59445
$184$	0	0
$185$	−4.94546	−0.363598
$186$	0	0
$187$	−4.53875	−0.331907
$188$	0	0
$189$	−19.5150	−1.41951
$190$	0	0
$191$	23.5827	1.70639	0.853193	−	0.521596i	$-0.174663\pi$
$191$	23.5827	1.70639	0.853193	+	0.521596i	$0.174663\pi$
$192$	0	0
$193$	16.8780	1.21491	0.607454	−	0.794355i	$-0.292191\pi$
$193$	16.8780	1.21491	0.607454	+	0.794355i	$0.292191\pi$
$194$	0	0
$195$	14.0329	1.00491
$196$	0	0
$197$	16.2427	1.15724	0.578621	−	0.815597i	$-0.303591\pi$
$197$	16.2427	1.15724	0.578621	+	0.815597i	$0.303591\pi$
$198$	0	0
$199$	22.0695	1.56446	0.782232	−	0.622988i	$-0.214081\pi$
$199$	22.0695	1.56446	0.782232	+	0.622988i	$0.214081\pi$
$200$	0	0
$201$	15.4482	1.08963
$202$	0	0
$203$	4.35867	0.305919
$204$	0	0
$205$	−19.7056	−1.37630
$206$	0	0
$207$	2.13187	0.148175
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	1.06376	0.0732323	0.0366162	−	0.999329i	$-0.488342\pi$
$211$	1.06376	0.0732323	0.0366162	+	0.999329i	$0.488342\pi$
$212$	0	0
$213$	−18.0221	−1.23485
$214$	0	0
$215$	−14.0176	−0.955995
$216$	0	0
$217$	−29.4480	−1.99906
$218$	0	0
$219$	−13.4574	−0.909365
$220$	0	0
$221$	−20.9130	−1.40676
$222$	0	0
$223$	3.85252	0.257984	0.128992	−	0.991646i	$-0.458826\pi$
$223$	3.85252	0.257984	0.128992	+	0.991646i	$0.458826\pi$
$224$	0	0
$225$	0.453680	0.0302453
$226$	0	0
$227$	1.38224	0.0917428	0.0458714	−	0.998947i	$-0.485394\pi$
$227$	1.38224	0.0917428	0.0458714	+	0.998947i	$0.485394\pi$
$228$	0	0
$229$	−4.79798	−0.317060	−0.158530	−	0.987354i	$-0.550675\pi$
$229$	−4.79798	−0.317060	−0.158530	+	0.987354i	$0.550675\pi$
$230$	0	0
$231$	−5.93718	−0.390637
$232$	0	0
$233$	6.77649	0.443943	0.221971	−	0.975053i	$-0.428751\pi$
$233$	6.77649	0.443943	0.221971	+	0.975053i	$0.428751\pi$
$234$	0	0
$235$	13.3410	0.870274
$236$	0	0
$237$	23.2327	1.50913
$238$	0	0
$239$	17.8035	1.15161	0.575807	−	0.817586i	$-0.304688\pi$
$239$	17.8035	1.15161	0.575807	+	0.817586i	$0.304688\pi$
$240$	0	0
$241$	16.9019	1.08875	0.544373	−	0.838843i	$-0.316768\pi$
$241$	16.9019	1.08875	0.544373	+	0.838843i	$0.316768\pi$
$242$	0	0
$243$	−2.97119	−0.190602
$244$	0	0
$245$	11.0804	0.707899
$246$	0	0
$247$	4.60764	0.293177
$248$	0	0
$249$	−25.3362	−1.60561
$250$	0	0
$251$	23.7083	1.49645	0.748226	−	0.663444i	$-0.230906\pi$
$251$	23.7083	1.49645	0.748226	+	0.663444i	$0.230906\pi$
$252$	0	0
$253$	7.43020	0.467133
$254$	0	0
$255$	−13.8231	−0.865634
$256$	0	0
$257$	1.23002	0.0767263	0.0383632	−	0.999264i	$-0.487786\pi$
$257$	1.23002	0.0767263	0.0383632	+	0.999264i	$0.487786\pi$
$258$	0	0
$259$	9.64094	0.599059
$260$	0	0
$261$	0.346950	0.0214756
$262$	0	0
$263$	−8.29315	−0.511377	−0.255689	−	0.966759i	$-0.582302\pi$
$263$	−8.29315	−0.511377	−0.255689	+	0.966759i	$0.582302\pi$
$264$	0	0
$265$	18.2461	1.12085
$266$	0	0
$267$	−10.8538	−0.664239
$268$	0	0
$269$	20.8457	1.27098	0.635492	−	0.772108i	$-0.280797\pi$
$269$	20.8457	1.27098	0.635492	+	0.772108i	$0.280797\pi$
$270$	0	0
$271$	−8.06428	−0.489870	−0.244935	−	0.969539i	$-0.578767\pi$
$271$	−8.06428	−0.489870	−0.244935	+	0.969539i	$0.578767\pi$
$272$	0	0
$273$	−27.3564	−1.65568
$274$	0	0
$275$	1.58121	0.0953503
$276$	0	0
$277$	−8.94531	−0.537472	−0.268736	−	0.963214i	$-0.586606\pi$
$277$	−8.94531	−0.537472	−0.268736	+	0.963214i	$0.586606\pi$
$278$	0	0
$279$	−2.34406	−0.140335
$280$	0	0
$281$	−5.65424	−0.337304	−0.168652	−	0.985676i	$-0.553941\pi$
$281$	−5.65424	−0.337304	−0.168652	+	0.985676i	$0.553941\pi$
$282$	0	0
$283$	−28.2519	−1.67940	−0.839700	−	0.543051i	$-0.817269\pi$
$283$	−28.2519	−1.67940	−0.839700	+	0.543051i	$0.817269\pi$
$284$	0	0
$285$	3.04556	0.180404
$286$	0	0
$287$	38.4151	2.26757
$288$	0	0
$289$	3.60029	0.211782
$290$	0	0
$291$	21.2341	1.24476
$292$	0	0
$293$	3.67042	0.214428	0.107214	−	0.994236i	$-0.465807\pi$
$293$	3.67042	0.214428	0.107214	+	0.994236i	$0.465807\pi$
$294$	0	0
$295$	14.0744	0.819442
$296$	0	0
$297$	−5.41403	−0.314154
$298$	0	0
$299$	34.2357	1.97990
$300$	0	0
$301$	27.3267	1.57508
$302$	0	0
$303$	−1.55237	−0.0891816
$304$	0	0
$305$	24.2126	1.38641
$306$	0	0
$307$	−9.06344	−0.517278	−0.258639	−	0.965974i	$-0.583274\pi$
$307$	−9.06344	−0.517278	−0.258639	+	0.965974i	$0.583274\pi$
$308$	0	0
$309$	−17.3586	−0.987494
$310$	0	0
$311$	15.7300	0.891968	0.445984	−	0.895041i	$-0.352854\pi$
$311$	15.7300	0.891968	0.445984	+	0.895041i	$0.352854\pi$
$312$	0	0
$313$	−4.58157	−0.258966	−0.129483	−	0.991582i	$-0.541332\pi$
$313$	−4.58157	−0.258966	−0.129483	+	0.991582i	$0.541332\pi$
$314$	0	0
$315$	1.91226	0.107743
$316$	0	0
$317$	−5.37416	−0.301843	−0.150921	−	0.988546i	$-0.548224\pi$
$317$	−5.37416	−0.301843	−0.150921	+	0.988546i	$0.548224\pi$
$318$	0	0
$319$	1.20922	0.0677034
$320$	0	0
$321$	11.6044	0.647693
$322$	0	0
$323$	−4.53875	−0.252543
$324$	0	0
$325$	7.28563	0.404134
$326$	0	0
$327$	16.8133	0.929779
$328$	0	0
$329$	−26.0077	−1.43385
$330$	0	0
$331$	−14.7794	−0.812349	−0.406174	−	0.913796i	$-0.633137\pi$
$331$	−14.7794	−0.812349	−0.406174	+	0.913796i	$0.633137\pi$
$332$	0	0
$333$	0.767417	0.0420542
$334$	0	0
$335$	−17.3413	−0.947459
$336$	0	0
$337$	12.3056	0.670331	0.335165	−	0.942159i	$-0.391208\pi$
$337$	12.3056	0.670331	0.335165	+	0.942159i	$0.391208\pi$
$338$	0	0
$339$	−6.31663	−0.343072
$340$	0	0
$341$	−8.16972	−0.442415
$342$	0	0
$343$	3.63108	0.196060
$344$	0	0
$345$	22.6292	1.21831
$346$	0	0
$347$	−16.6446	−0.893530	−0.446765	−	0.894651i	$-0.647424\pi$
$347$	−16.6446	−0.893530	−0.446765	+	0.894651i	$0.647424\pi$
$348$	0	0
$349$	−12.8958	−0.690297	−0.345149	−	0.938548i	$-0.612171\pi$
$349$	−12.8958	−0.690297	−0.345149	+	0.938548i	$0.612171\pi$
$350$	0	0
$351$	−24.9459	−1.33151
$352$	0	0
$353$	9.50692	0.506003	0.253001	−	0.967466i	$-0.418582\pi$
$353$	9.50692	0.506003	0.253001	+	0.967466i	$0.418582\pi$
$354$	0	0
$355$	20.2307	1.07373
$356$	0	0
$357$	26.9474	1.42621
$358$	0	0
$359$	−19.5140	−1.02991	−0.514954	−	0.857218i	$-0.672191\pi$
$359$	−19.5140	−1.02991	−0.514954	+	0.857218i	$0.672191\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.64714	−0.0864526
$364$	0	0
$365$	15.1066	0.790713
$366$	0	0
$367$	26.2094	1.36812	0.684060	−	0.729425i	$-0.260212\pi$
$367$	26.2094	1.36812	0.684060	+	0.729425i	$0.260212\pi$
$368$	0	0
$369$	3.05784	0.159185
$370$	0	0
$371$	−35.5700	−1.84670
$372$	0	0
$373$	15.3119	0.792821	0.396410	−	0.918073i	$-0.370256\pi$
$373$	15.3119	0.792821	0.396410	+	0.918073i	$0.370256\pi$
$374$	0	0
$375$	20.0435	1.03504
$376$	0	0
$377$	5.57166	0.286955
$378$	0	0
$379$	11.9993	0.616361	0.308180	−	0.951328i	$-0.400280\pi$
$379$	11.9993	0.616361	0.308180	+	0.951328i	$0.400280\pi$
$380$	0	0
$381$	17.6100	0.902190
$382$	0	0
$383$	−27.5695	−1.40874	−0.704369	−	0.709834i	$-0.748770\pi$
$383$	−27.5695	−1.40874	−0.704369	+	0.709834i	$0.748770\pi$
$384$	0	0
$385$	6.66477	0.339668
$386$	0	0
$387$	2.17520	0.110572
$388$	0	0
$389$	−26.3579	−1.33640	−0.668198	−	0.743983i	$-0.732934\pi$
$389$	−26.3579	−1.33640	−0.668198	+	0.743983i	$0.732934\pi$
$390$	0	0
$391$	−33.7239	−1.70549
$392$	0	0
$393$	−16.9344	−0.854226
$394$	0	0
$395$	−26.0799	−1.31222
$396$	0	0
$397$	−14.6384	−0.734683	−0.367341	−	0.930086i	$-0.619732\pi$
$397$	−14.6384	−0.734683	−0.367341	+	0.930086i	$0.619732\pi$
$398$	0	0
$399$	−5.93718	−0.297231
$400$	0	0
$401$	0.884350	0.0441623	0.0220812	−	0.999756i	$-0.492971\pi$
$401$	0.884350	0.0441623	0.0220812	+	0.999756i	$0.492971\pi$
$402$	0	0
$403$	−37.6432	−1.87514
$404$	0	0
$405$	−14.8972	−0.740249
$406$	0	0
$407$	2.67467	0.132579
$408$	0	0
$409$	21.6864	1.07233	0.536163	−	0.844115i	$-0.319873\pi$
$409$	21.6864	1.07233	0.536163	+	0.844115i	$0.319873\pi$
$410$	0	0
$411$	−8.49110	−0.418835
$412$	0	0
$413$	−27.4373	−1.35010
$414$	0	0
$415$	28.4411	1.39612
$416$	0	0
$417$	3.68483	0.180447
$418$	0	0
$419$	−29.6191	−1.44699	−0.723494	−	0.690331i	$-0.757465\pi$
$419$	−29.6191	−1.44699	−0.723494	+	0.690331i	$0.757465\pi$
$420$	0	0
$421$	33.1830	1.61724	0.808621	−	0.588329i	$-0.200214\pi$
$421$	33.1830	1.61724	0.808621	+	0.588329i	$0.200214\pi$
$422$	0	0
$423$	−2.07021	−0.100657
$424$	0	0
$425$	−7.17671	−0.348121
$426$	0	0
$427$	−47.2012	−2.28423
$428$	0	0
$429$	−7.58945	−0.366422
$430$	0	0
$431$	5.37951	0.259122	0.129561	−	0.991571i	$-0.458643\pi$
$431$	5.37951	0.259122	0.129561	+	0.991571i	$0.458643\pi$
$432$	0	0
$433$	32.5638	1.56491	0.782457	−	0.622704i	$-0.213966\pi$
$433$	32.5638	1.56491	0.782457	+	0.622704i	$0.213966\pi$
$434$	0	0
$435$	3.68276	0.176575
$436$	0	0
$437$	7.43020	0.355435
$438$	0	0
$439$	−25.6281	−1.22316	−0.611580	−	0.791182i	$-0.709466\pi$
$439$	−25.6281	−1.22316	−0.611580	+	0.791182i	$0.709466\pi$
$440$	0	0
$441$	−1.71941	−0.0818765
$442$	0	0
$443$	−9.87249	−0.469056	−0.234528	−	0.972109i	$-0.575355\pi$
$443$	−9.87249	−0.469056	−0.234528	+	0.972109i	$0.575355\pi$
$444$	0	0
$445$	12.1839	0.577571
$446$	0	0
$447$	22.5728	1.06766
$448$	0	0
$449$	−1.43358	−0.0676550	−0.0338275	−	0.999428i	$-0.510770\pi$
$449$	−1.43358	−0.0676550	−0.0338275	+	0.999428i	$0.510770\pi$
$450$	0	0
$451$	10.6575	0.501840
$452$	0	0
$453$	−27.0823	−1.27244
$454$	0	0
$455$	30.7089	1.43965
$456$	0	0
$457$	−1.81542	−0.0849217	−0.0424609	−	0.999098i	$-0.513520\pi$
$457$	−1.81542	−0.0849217	−0.0424609	+	0.999098i	$0.513520\pi$
$458$	0	0
$459$	24.5729	1.14697
$460$	0	0
$461$	−30.3391	−1.41303	−0.706517	−	0.707696i	$-0.749735\pi$
$461$	−30.3391	−1.41303	−0.706517	+	0.707696i	$0.749735\pi$
$462$	0	0
$463$	24.1414	1.12195	0.560974	−	0.827833i	$-0.310427\pi$
$463$	24.1414	1.12195	0.560974	+	0.827833i	$0.310427\pi$
$464$	0	0
$465$	−24.8814	−1.15385
$466$	0	0
$467$	−13.7532	−0.636421	−0.318210	−	0.948020i	$-0.603082\pi$
$467$	−13.7532	−0.636421	−0.318210	+	0.948020i	$0.603082\pi$
$468$	0	0
$469$	33.8061	1.56102
$470$	0	0
$471$	35.3851	1.63046
$472$	0	0
$473$	7.58121	0.348584
$474$	0	0
$475$	1.58121	0.0725507
$476$	0	0
$477$	−2.83136	−0.129639
$478$	0	0
$479$	19.1851	0.876588	0.438294	−	0.898832i	$-0.355583\pi$
$479$	19.1851	0.876588	0.438294	+	0.898832i	$0.355583\pi$
$480$	0	0
$481$	12.3239	0.561923
$482$	0	0
$483$	−44.1144	−2.00728
$484$	0	0
$485$	−23.8363	−1.08235
$486$	0	0
$487$	−14.3820	−0.651711	−0.325855	−	0.945420i	$-0.605652\pi$
$487$	−14.3820	−0.651711	−0.325855	+	0.945420i	$0.605652\pi$
$488$	0	0
$489$	28.2157	1.27596
$490$	0	0
$491$	−32.7834	−1.47950	−0.739748	−	0.672885i	$-0.765055\pi$
$491$	−32.7834	−1.47950	−0.739748	+	0.672885i	$0.765055\pi$
$492$	0	0
$493$	−5.48836	−0.247183
$494$	0	0
$495$	0.530515	0.0238449
$496$	0	0
$497$	−39.4387	−1.76907
$498$	0	0
$499$	7.89797	0.353562	0.176781	−	0.984250i	$-0.443432\pi$
$499$	7.89797	0.353562	0.176781	+	0.984250i	$0.443432\pi$
$500$	0	0
$501$	−31.0951	−1.38923
$502$	0	0
$503$	2.50370	0.111634	0.0558172	−	0.998441i	$-0.482224\pi$
$503$	2.50370	0.111634	0.0558172	+	0.998441i	$0.482224\pi$
$504$	0	0
$505$	1.74262	0.0775454
$506$	0	0
$507$	−13.5566	−0.602069
$508$	0	0
$509$	28.5076	1.26358	0.631788	−	0.775142i	$-0.282321\pi$
$509$	28.5076	1.26358	0.631788	+	0.775142i	$0.282321\pi$
$510$	0	0
$511$	−29.4495	−1.30277
$512$	0	0
$513$	−5.41403	−0.239035
$514$	0	0
$515$	19.4858	0.858649
$516$	0	0
$517$	−7.21528	−0.317328
$518$	0	0
$519$	−16.1384	−0.708396
$520$	0	0
$521$	33.6574	1.47456	0.737279	−	0.675588i	$-0.236110\pi$
$521$	33.6574	1.47456	0.737279	+	0.675588i	$0.236110\pi$
$522$	0	0
$523$	−17.2318	−0.753492	−0.376746	−	0.926317i	$-0.622957\pi$
$523$	−17.2318	−0.753492	−0.376746	+	0.926317i	$0.622957\pi$
$524$	0	0
$525$	−9.38790	−0.409721
$526$	0	0
$527$	37.0804	1.61525
$528$	0	0
$529$	32.2079	1.40034
$530$	0	0
$531$	−2.18400	−0.0947777
$532$	0	0
$533$	49.1058	2.12701
$534$	0	0
$535$	−13.0265	−0.563184
$536$	0	0
$537$	−15.7290	−0.678754
$538$	0	0
$539$	−5.99263	−0.258121
$540$	0	0
$541$	41.6025	1.78863	0.894315	−	0.447438i	$-0.147663\pi$
$541$	41.6025	1.78863	0.894315	+	0.447438i	$0.147663\pi$
$542$	0	0
$543$	−34.4457	−1.47821
$544$	0	0
$545$	−18.8738	−0.808464
$546$	0	0
$547$	−0.872294	−0.0372966	−0.0186483	−	0.999826i	$-0.505936\pi$
$547$	−0.872294	−0.0372966	−0.0186483	+	0.999826i	$0.505936\pi$
$548$	0	0
$549$	−3.75721	−0.160354
$550$	0	0
$551$	1.20922	0.0515146
$552$	0	0
$553$	50.8414	2.16200
$554$	0	0
$555$	8.14589	0.345774
$556$	0	0
$557$	−9.76642	−0.413816	−0.206908	−	0.978360i	$-0.566340\pi$
$557$	−9.76642	−0.413816	−0.206908	+	0.978360i	$0.566340\pi$
$558$	0	0
$559$	34.9315	1.47744
$560$	0	0
$561$	7.47598	0.315636
$562$	0	0
$563$	−11.0277	−0.464762	−0.232381	−	0.972625i	$-0.574652\pi$
$563$	−11.0277	−0.464762	−0.232381	+	0.972625i	$0.574652\pi$
$564$	0	0
$565$	7.09072	0.298309
$566$	0	0
$567$	29.0414	1.21962
$568$	0	0
$569$	5.57307	0.233635	0.116818	−	0.993153i	$-0.462731\pi$
$569$	5.57307	0.233635	0.116818	+	0.993153i	$0.462731\pi$
$570$	0	0
$571$	8.86690	0.371068	0.185534	−	0.982638i	$-0.440598\pi$
$571$	8.86690	0.371068	0.185534	+	0.982638i	$0.440598\pi$
$572$	0	0
$573$	−38.8441	−1.62274
$574$	0	0
$575$	11.7487	0.489954
$576$	0	0
$577$	−31.3329	−1.30440	−0.652202	−	0.758045i	$-0.726155\pi$
$577$	−31.3329	−1.30440	−0.652202	+	0.758045i	$0.726155\pi$
$578$	0	0
$579$	−27.8005	−1.15535
$580$	0	0
$581$	−55.4445	−2.30022
$582$	0	0
$583$	−9.86813	−0.408696
$584$	0	0
$585$	2.44442	0.101064
$586$	0	0
$587$	19.2704	0.795376	0.397688	−	0.917521i	$-0.369813\pi$
$587$	19.2704	0.795376	0.397688	+	0.917521i	$0.369813\pi$
$588$	0	0
$589$	−8.16972	−0.336627
$590$	0	0
$591$	−26.7540	−1.10051
$592$	0	0
$593$	−29.5232	−1.21237	−0.606186	−	0.795323i	$-0.707301\pi$
$593$	−29.5232	−1.21237	−0.606186	+	0.795323i	$0.707301\pi$
$594$	0	0
$595$	−30.2497	−1.24012
$596$	0	0
$597$	−36.3516	−1.48777
$598$	0	0
$599$	−12.5386	−0.512313	−0.256157	−	0.966635i	$-0.582456\pi$
$599$	−12.5386	−0.512313	−0.256157	+	0.966635i	$0.582456\pi$
$600$	0	0
$601$	5.36819	0.218973	0.109486	−	0.993988i	$-0.465079\pi$
$601$	5.36819	0.218973	0.109486	+	0.993988i	$0.465079\pi$
$602$	0	0
$603$	2.69096	0.109584
$604$	0	0
$605$	1.84900	0.0751725
$606$	0	0
$607$	47.5693	1.93078	0.965389	−	0.260814i	$-0.0839910\pi$
$607$	47.5693	1.93078	0.965389	+	0.260814i	$0.0839910\pi$
$608$	0	0
$609$	−7.17936	−0.290922
$610$	0	0
$611$	−33.2455	−1.34497
$612$	0	0
$613$	12.0498	0.486688	0.243344	−	0.969940i	$-0.421756\pi$
$613$	12.0498	0.486688	0.243344	+	0.969940i	$0.421756\pi$
$614$	0	0
$615$	32.4580	1.30883
$616$	0	0
$617$	−2.65285	−0.106800	−0.0534000	−	0.998573i	$-0.517006\pi$
$617$	−2.65285	−0.106800	−0.0534000	+	0.998573i	$0.517006\pi$
$618$	0	0
$619$	−23.1495	−0.930458	−0.465229	−	0.885190i	$-0.654028\pi$
$619$	−23.1495	−0.930458	−0.465229	+	0.885190i	$0.654028\pi$
$620$	0	0
$621$	−40.2273	−1.61427
$622$	0	0
$623$	−23.7519	−0.951598
$624$	0	0
$625$	−14.5938	−0.583750
$626$	0	0
$627$	−1.64714	−0.0657806
$628$	0	0
$629$	−12.1397	−0.484041
$630$	0	0
$631$	−20.9858	−0.835433	−0.417717	−	0.908577i	$-0.637170\pi$
$631$	−20.9858	−0.835433	−0.417717	+	0.908577i	$0.637170\pi$
$632$	0	0
$633$	−1.75217	−0.0696424
$634$	0	0
$635$	−19.7681	−0.784474
$636$	0	0
$637$	−27.6119	−1.09402
$638$	0	0
$639$	−3.13931	−0.124189
$640$	0	0
$641$	23.7666	0.938724	0.469362	−	0.883006i	$-0.344484\pi$
$641$	23.7666	0.938724	0.469362	+	0.883006i	$0.344484\pi$
$642$	0	0
$643$	−23.8230	−0.939486	−0.469743	−	0.882803i	$-0.655653\pi$
$643$	−23.8230	−0.939486	−0.469743	+	0.882803i	$0.655653\pi$
$644$	0	0
$645$	23.0891	0.909130
$646$	0	0
$647$	−25.4698	−1.00132	−0.500661	−	0.865643i	$-0.666910\pi$
$647$	−25.4698	−1.00132	−0.500661	+	0.865643i	$0.666910\pi$
$648$	0	0
$649$	−7.61190	−0.298793
$650$	0	0
$651$	48.5051	1.90106
$652$	0	0
$653$	−40.5654	−1.58745	−0.793724	−	0.608278i	$-0.791861\pi$
$653$	−40.5654	−1.58745	−0.793724	+	0.608278i	$0.791861\pi$
$654$	0	0
$655$	19.0097	0.742769
$656$	0	0
$657$	−2.34417	−0.0914549
$658$	0	0
$659$	−30.3240	−1.18126	−0.590628	−	0.806944i	$-0.701120\pi$
$659$	−30.3240	−1.18126	−0.590628	+	0.806944i	$0.701120\pi$
$660$	0	0
$661$	−2.02555	−0.0787847	−0.0393923	−	0.999224i	$-0.512542\pi$
$661$	−2.02555	−0.0787847	−0.0393923	+	0.999224i	$0.512542\pi$
$662$	0	0
$663$	34.4466	1.33780
$664$	0	0
$665$	6.66477	0.258449
$666$	0	0
$667$	8.98476	0.347891
$668$	0	0
$669$	−6.34565	−0.245337
$670$	0	0
$671$	−13.0950	−0.505526
$672$	0	0
$673$	−48.5899	−1.87300	−0.936501	−	0.350664i	$-0.885956\pi$
$673$	−48.5899	−1.87300	−0.936501	+	0.350664i	$0.885956\pi$
$674$	0	0
$675$	−8.56069	−0.329501
$676$	0	0
$677$	−16.4471	−0.632115	−0.316057	−	0.948740i	$-0.602359\pi$
$677$	−16.4471	−0.632115	−0.316057	+	0.948740i	$0.602359\pi$
$678$	0	0
$679$	46.4676	1.78326
$680$	0	0
$681$	−2.27675	−0.0872454
$682$	0	0
$683$	5.22449	0.199910	0.0999549	−	0.994992i	$-0.468130\pi$
$683$	5.22449	0.199910	0.0999549	+	0.994992i	$0.468130\pi$
$684$	0	0
$685$	9.53167	0.364186
$686$	0	0
$687$	7.90296	0.301517
$688$	0	0
$689$	−45.4688	−1.73222
$690$	0	0
$691$	−1.93791	−0.0737217	−0.0368608	−	0.999320i	$-0.511736\pi$
$691$	−1.93791	−0.0737217	−0.0368608	+	0.999320i	$0.511736\pi$
$692$	0	0
$693$	−1.03421	−0.0392865
$694$	0	0
$695$	−4.13640	−0.156903
$696$	0	0
$697$	−48.3716	−1.83220
$698$	0	0
$699$	−11.1618	−0.422180
$700$	0	0
$701$	−18.0143	−0.680391	−0.340196	−	0.940355i	$-0.610493\pi$
$701$	−18.0143	−0.680391	−0.340196	+	0.940355i	$0.610493\pi$
$702$	0	0
$703$	2.67467	0.100877
$704$	0	0
$705$	−21.9746	−0.827612
$706$	0	0
$707$	−3.39714	−0.127763
$708$	0	0
$709$	6.68923	0.251219	0.125610	−	0.992080i	$-0.459911\pi$
$709$	6.68923	0.251219	0.125610	+	0.992080i	$0.459911\pi$
$710$	0	0
$711$	4.04697	0.151773
$712$	0	0
$713$	−60.7027	−2.27333
$714$	0	0
$715$	8.51952	0.318612
$716$	0	0
$717$	−29.3249	−1.09516
$718$	0	0
$719$	−35.4030	−1.32031	−0.660155	−	0.751129i	$-0.729510\pi$
$719$	−35.4030	−1.32031	−0.660155	+	0.751129i	$0.729510\pi$
$720$	0	0
$721$	−37.9867	−1.41470
$722$	0	0
$723$	−27.8398	−1.03537
$724$	0	0
$725$	1.91203	0.0710109
$726$	0	0
$727$	−30.7973	−1.14221	−0.571104	−	0.820878i	$-0.693485\pi$
$727$	−30.7973	−1.14221	−0.571104	+	0.820878i	$0.693485\pi$
$728$	0	0
$729$	29.0647	1.07647
$730$	0	0
$731$	−34.4092	−1.27267
$732$	0	0
$733$	50.1274	1.85150	0.925748	−	0.378140i	$-0.123436\pi$
$733$	50.1274	1.85150	0.925748	+	0.378140i	$0.123436\pi$
$734$	0	0
$735$	−18.2510	−0.673197
$736$	0	0
$737$	9.37878	0.345472
$738$	0	0
$739$	−7.52288	−0.276734	−0.138367	−	0.990381i	$-0.544185\pi$
$739$	−7.52288	−0.276734	−0.138367	+	0.990381i	$0.544185\pi$
$740$	0	0
$741$	−7.58945	−0.278805
$742$	0	0
$743$	43.3485	1.59030	0.795151	−	0.606412i	$-0.207392\pi$
$743$	43.3485	1.59030	0.795151	+	0.606412i	$0.207392\pi$
$744$	0	0
$745$	−25.3391	−0.928352
$746$	0	0
$747$	−4.41337	−0.161477
$748$	0	0
$749$	25.3945	0.927894
$750$	0	0
$751$	7.77361	0.283663	0.141832	−	0.989891i	$-0.454701\pi$
$751$	7.77361	0.283663	0.141832	+	0.989891i	$0.454701\pi$
$752$	0	0
$753$	−39.0509	−1.42309
$754$	0	0
$755$	30.4012	1.10641
$756$	0	0
$757$	15.8968	0.577779	0.288890	−	0.957362i	$-0.406714\pi$
$757$	15.8968	0.577779	0.288890	+	0.957362i	$0.406714\pi$
$758$	0	0
$759$	−12.2386	−0.444233
$760$	0	0
$761$	29.6717	1.07560	0.537798	−	0.843073i	$-0.319256\pi$
$761$	29.6717	1.07560	0.537798	+	0.843073i	$0.319256\pi$
$762$	0	0
$763$	36.7935	1.33201
$764$	0	0
$765$	−2.40788	−0.0870569
$766$	0	0
$767$	−35.0729	−1.26641
$768$	0	0
$769$	−24.8509	−0.896145	−0.448073	−	0.893997i	$-0.647889\pi$
$769$	−24.8509	−0.896145	−0.448073	+	0.893997i	$0.647889\pi$
$770$	0	0
$771$	−2.02601	−0.0729651
$772$	0	0
$773$	10.5033	0.377779	0.188890	−	0.981998i	$-0.439511\pi$
$773$	10.5033	0.377779	0.188890	+	0.981998i	$0.439511\pi$
$774$	0	0
$775$	−12.9180	−0.464029
$776$	0	0
$777$	−15.8800	−0.569692
$778$	0	0
$779$	10.6575	0.381843
$780$	0	0
$781$	−10.9414	−0.391515
$782$	0	0
$783$	−6.54676	−0.233962
$784$	0	0
$785$	−39.7215	−1.41772
$786$	0	0
$787$	−14.5264	−0.517809	−0.258905	−	0.965903i	$-0.583362\pi$
$787$	−14.5264	−0.517809	−0.258905	+	0.965903i	$0.583362\pi$
$788$	0	0
$789$	13.6600	0.486309
$790$	0	0
$791$	−13.8230	−0.491489
$792$	0	0
$793$	−60.3370	−2.14263
$794$	0	0
$795$	−30.0540	−1.06591
$796$	0	0
$797$	30.5101	1.08072	0.540361	−	0.841433i	$-0.318288\pi$
$797$	30.5101	1.08072	0.540361	+	0.841433i	$0.318288\pi$
$798$	0	0
$799$	32.7484	1.15856
$800$	0	0
$801$	−1.89064	−0.0668026
$802$	0	0
$803$	−8.17013	−0.288318
$804$	0	0
$805$	49.5206	1.74537
$806$	0	0
$807$	−34.3358	−1.20868
$808$	0	0
$809$	−3.17703	−0.111698	−0.0558492	−	0.998439i	$-0.517787\pi$
$809$	−3.17703	−0.111698	−0.0558492	+	0.998439i	$0.517787\pi$
$810$	0	0
$811$	−17.0867	−0.599994	−0.299997	−	0.953940i	$-0.596986\pi$
$811$	−17.0867	−0.599994	−0.299997	+	0.953940i	$0.596986\pi$
$812$	0	0
$813$	13.2830	0.465856
$814$	0	0
$815$	−31.6734	−1.10947
$816$	0	0
$817$	7.58121	0.265233
$818$	0	0
$819$	−4.76528	−0.166512
$820$	0	0
$821$	−13.5078	−0.471425	−0.235713	−	0.971823i	$-0.575742\pi$
$821$	−13.5078	−0.471425	−0.235713	+	0.971823i	$0.575742\pi$
$822$	0	0
$823$	−8.83242	−0.307879	−0.153939	−	0.988080i	$-0.549196\pi$
$823$	−8.83242	−0.307879	−0.153939	+	0.988080i	$0.549196\pi$
$824$	0	0
$825$	−2.60447	−0.0906761
$826$	0	0
$827$	−3.46414	−0.120460	−0.0602299	−	0.998185i	$-0.519183\pi$
$827$	−3.46414	−0.120460	−0.0602299	+	0.998185i	$0.519183\pi$
$828$	0	0
$829$	42.5694	1.47850	0.739249	−	0.673433i	$-0.235181\pi$
$829$	42.5694	1.47850	0.739249	+	0.673433i	$0.235181\pi$
$830$	0	0
$831$	14.7342	0.511124
$832$	0	0
$833$	27.1991	0.942393
$834$	0	0
$835$	34.9057	1.20796
$836$	0	0
$837$	44.2311	1.52885
$838$	0	0
$839$	18.5545	0.640573	0.320287	−	0.947321i	$-0.396221\pi$
$839$	18.5545	0.640573	0.320287	+	0.947321i	$0.396221\pi$
$840$	0	0
$841$	−27.5378	−0.949579
$842$	0	0
$843$	9.31335	0.320769
$844$	0	0
$845$	15.2179	0.523513
$846$	0	0
$847$	−3.60453	−0.123853
$848$	0	0
$849$	46.5349	1.59707
$850$	0	0
$851$	19.8734	0.681250
$852$	0	0
$853$	19.0092	0.650863	0.325431	−	0.945566i	$-0.394491\pi$
$853$	19.0092	0.650863	0.325431	+	0.945566i	$0.394491\pi$
$854$	0	0
$855$	0.530515	0.0181432
$856$	0	0
$857$	−8.84122	−0.302010	−0.151005	−	0.988533i	$-0.548251\pi$
$857$	−8.84122	−0.302010	−0.151005	+	0.988533i	$0.548251\pi$
$858$	0	0
$859$	15.6032	0.532375	0.266188	−	0.963921i	$-0.414236\pi$
$859$	15.6032	0.532375	0.266188	+	0.963921i	$0.414236\pi$
$860$	0	0
$861$	−63.2752	−2.15641
$862$	0	0
$863$	27.2438	0.927388	0.463694	−	0.885995i	$-0.346524\pi$
$863$	27.2438	0.927388	0.463694	+	0.885995i	$0.346524\pi$
$864$	0	0
$865$	18.1161	0.615966
$866$	0	0
$867$	−5.93020	−0.201400
$868$	0	0
$869$	14.1049	0.478475
$870$	0	0
$871$	43.2141	1.46425
$872$	0	0
$873$	3.69881	0.125186
$874$	0	0
$875$	43.8622	1.48281
$876$	0	0
$877$	1.09546	0.0369910	0.0184955	−	0.999829i	$-0.494112\pi$
$877$	1.09546	0.0369910	0.0184955	+	0.999829i	$0.494112\pi$
$878$	0	0
$879$	−6.04571	−0.203917
$880$	0	0
$881$	31.6772	1.06723	0.533615	−	0.845727i	$-0.320833\pi$
$881$	31.6772	1.06723	0.533615	+	0.845727i	$0.320833\pi$
$882$	0	0
$883$	−12.3127	−0.414355	−0.207178	−	0.978303i	$-0.566428\pi$
$883$	−12.3127	−0.414355	−0.207178	+	0.978303i	$0.566428\pi$
$884$	0	0
$885$	−23.1825	−0.779272
$886$	0	0
$887$	37.3490	1.25406	0.627029	−	0.778996i	$-0.284271\pi$
$887$	37.3490	1.25406	0.627029	+	0.778996i	$0.284271\pi$
$888$	0	0
$889$	38.5370	1.29249
$890$	0	0
$891$	8.05692	0.269917
$892$	0	0
$893$	−7.21528	−0.241450
$894$	0	0
$895$	17.6565	0.590192
$896$	0	0
$897$	−56.3911	−1.88285
$898$	0	0
$899$	−9.87900	−0.329483
$900$	0	0
$901$	44.7890	1.49214
$902$	0	0
$903$	−45.0110	−1.49787
$904$	0	0
$905$	38.6670	1.28533
$906$	0	0
$907$	−15.3443	−0.509498	−0.254749	−	0.967007i	$-0.581993\pi$
$907$	−15.3443	−0.509498	−0.254749	+	0.967007i	$0.581993\pi$
$908$	0	0
$909$	−0.270412	−0.00896900
$910$	0	0
$911$	−34.4470	−1.14128	−0.570641	−	0.821200i	$-0.693305\pi$
$911$	−34.4470	−1.14128	−0.570641	+	0.821200i	$0.693305\pi$
$912$	0	0
$913$	−15.3819	−0.509066
$914$	0	0
$915$	−39.8816	−1.31844
$916$	0	0
$917$	−37.0584	−1.22378
$918$	0	0
$919$	10.1740	0.335608	0.167804	−	0.985820i	$-0.446332\pi$
$919$	10.1740	0.335608	0.167804	+	0.985820i	$0.446332\pi$
$920$	0	0
$921$	14.9288	0.491920
$922$	0	0
$923$	−50.4142	−1.65940
$924$	0	0
$925$	4.22921	0.139056
$926$	0	0
$927$	−3.02373	−0.0993124
$928$	0	0
$929$	11.8942	0.390235	0.195117	−	0.980780i	$-0.437491\pi$
$929$	11.8942	0.390235	0.195117	+	0.980780i	$0.437491\pi$
$930$	0	0
$931$	−5.99263	−0.196401
$932$	0	0
$933$	−25.9096	−0.848242
$934$	0	0
$935$	−8.39215	−0.274453
$936$	0	0
$937$	−24.9975	−0.816634	−0.408317	−	0.912840i	$-0.633884\pi$
$937$	−24.9975	−0.816634	−0.408317	+	0.912840i	$0.633884\pi$
$938$	0	0
$939$	7.54650	0.246271
$940$	0	0
$941$	0.255522	0.00832978	0.00416489	−	0.999991i	$-0.498674\pi$
$941$	0.255522	0.00832978	0.00416489	+	0.999991i	$0.498674\pi$
$942$	0	0
$943$	79.1871	2.57869
$944$	0	0
$945$	−36.0832	−1.17379
$946$	0	0
$947$	54.7242	1.77830	0.889149	−	0.457618i	$-0.151297\pi$
$947$	54.7242	1.77830	0.889149	+	0.457618i	$0.151297\pi$
$948$	0	0
$949$	−37.6450	−1.22201
$950$	0	0
$951$	8.85201	0.287046
$952$	0	0
$953$	2.40355	0.0778587	0.0389293	−	0.999242i	$-0.487605\pi$
$953$	2.40355	0.0778587	0.0389293	+	0.999242i	$0.487605\pi$
$954$	0	0
$955$	43.6044	1.41100
$956$	0	0
$957$	−1.99176	−0.0643845
$958$	0	0
$959$	−18.5815	−0.600028
$960$	0	0
$961$	35.7443	1.15304
$962$	0	0
$963$	2.02140	0.0651386
$964$	0	0
$965$	31.2075	1.00460
$966$	0	0
$967$	−38.0910	−1.22492	−0.612462	−	0.790500i	$-0.709821\pi$
$967$	−38.0910	−1.22492	−0.612462	+	0.790500i	$0.709821\pi$
$968$	0	0
$969$	7.47598	0.240163
$970$	0	0
$971$	47.2442	1.51614	0.758069	−	0.652175i	$-0.226143\pi$
$971$	47.2442	1.51614	0.758069	+	0.652175i	$0.226143\pi$
$972$	0	0
$973$	8.06370	0.258510
$974$	0	0
$975$	−12.0005	−0.384323
$976$	0	0
$977$	4.37638	0.140013	0.0700065	−	0.997547i	$-0.477698\pi$
$977$	4.37638	0.140013	0.0700065	+	0.997547i	$0.477698\pi$
$978$	0	0
$979$	−6.58945	−0.210600
$980$	0	0
$981$	2.92875	0.0935080
$982$	0	0
$983$	−38.8315	−1.23853	−0.619266	−	0.785181i	$-0.712570\pi$
$983$	−38.8315	−1.23853	−0.619266	+	0.785181i	$0.712570\pi$
$984$	0	0
$985$	30.0326	0.956919
$986$	0	0
$987$	42.8384	1.36356
$988$	0	0
$989$	56.3299	1.79119
$990$	0	0
$991$	16.7975	0.533591	0.266795	−	0.963753i	$-0.414035\pi$
$991$	16.7975	0.533591	0.266795	+	0.963753i	$0.414035\pi$
$992$	0	0
$993$	24.3438	0.772526
$994$	0	0
$995$	40.8064	1.29365
$996$	0	0
$997$	−41.1556	−1.30341	−0.651706	−	0.758471i	$-0.725946\pi$
$997$	−41.1556	−1.30341	−0.651706	+	0.758471i	$0.725946\pi$
$998$	0	0
$999$	−14.4807	−0.458151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.x.1.2		6
4.3	odd	2		836.2.a.d.1.5	✓	6
12.11	even	2		7524.2.a.r.1.3		6
44.43	even	2		9196.2.a.k.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.d.1.5	✓	6	4.3	odd	2
3344.2.a.x.1.2		6	1.1	even	1	trivial
7524.2.a.r.1.3		6	12.11	even	2
9196.2.a.k.1.5		6	44.43	even	2

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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64714 q^{3} +1.84900 q^{5} -3.60453 q^{7} -0.286920 q^{9} +O(q^{10})\)	\(q-1.64714 q^{3} +1.84900 q^{5} -3.60453 q^{7} -0.286920 q^{9} -1.00000 q^{11} -4.60764 q^{13} -3.04556 q^{15} +4.53875 q^{17} -1.00000 q^{19} +5.93718 q^{21} -7.43020 q^{23} -1.58121 q^{25} +5.41403 q^{27} -1.20922 q^{29} +8.16972 q^{31} +1.64714 q^{33} -6.66477 q^{35} -2.67467 q^{37} +7.58945 q^{39} -10.6575 q^{41} -7.58121 q^{43} -0.530515 q^{45} +7.21528 q^{47} +5.99263 q^{49} -7.47598 q^{51} +9.86813 q^{53} -1.84900 q^{55} +1.64714 q^{57} +7.61190 q^{59} +13.0950 q^{61} +1.03421 q^{63} -8.51952 q^{65} -9.37878 q^{67} +12.2386 q^{69} +10.9414 q^{71} +8.17013 q^{73} +2.60447 q^{75} +3.60453 q^{77} -14.1049 q^{79} -8.05692 q^{81} +15.3819 q^{83} +8.39215 q^{85} +1.99176 q^{87} +6.58945 q^{89} +16.6084 q^{91} -13.4567 q^{93} -1.84900 q^{95} -12.8914 q^{97} +0.286920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more