LMFDB - Embedded newform 3380.2.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.49551$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82684 q^{3} +1.00000 q^{5} +2.09479 q^{7} +4.99102 q^{9} +O(q^{10})$	$q+2.82684 q^{3} +1.00000 q^{5} +2.09479 q^{7} +4.99102 q^{9} +1.73205 q^{11} +2.82684 q^{15} -3.62828 q^{17} +1.06939 q^{19} +5.92163 q^{21} -7.81785 q^{23} +1.00000 q^{25} +5.62828 q^{27} -0.526914 q^{29} +5.84325 q^{31} +4.89623 q^{33} +2.09479 q^{35} +9.74846 q^{37} +4.26795 q^{41} +9.34477 q^{43} +4.99102 q^{45} +3.46410 q^{47} -2.61186 q^{49} -10.2566 q^{51} +12.5939 q^{53} +1.73205 q^{55} +3.02299 q^{57} +1.40370 q^{59} -11.1088 q^{61} +10.4551 q^{63} +10.8334 q^{67} -22.0998 q^{69} -14.1692 q^{71} -2.64469 q^{73} +2.82684 q^{75} +3.62828 q^{77} +13.5729 q^{79} +0.937188 q^{81} -15.7925 q^{83} -3.62828 q^{85} -1.48950 q^{87} -5.52451 q^{89} +16.5179 q^{93} +1.06939 q^{95} -15.1946 q^{97} +8.64469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9	$ 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9} + 2 q^{15} + 6 q^{17} + 12 q^{21} - 6 q^{23} + 4 q^{25} + 2 q^{27} + 6 q^{33} + 6 q^{35} + 18 q^{37} + 24 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{49} + 12 q^{53} - 18 q^{57} + 12 q^{59} + 4 q^{61} + 12 q^{63} + 18 q^{67} - 24 q^{69} + 12 q^{71} + 24 q^{73} + 2 q^{75} - 6 q^{77} - 8 q^{79} - 8 q^{81} - 36 q^{83} + 6 q^{85} + 6 q^{87} + 12 q^{89} + 48 q^{93} + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9 + 2 * q^15 + 6 * q^17 + 12 * q^21 - 6 * q^23 + 4 * q^25 + 2 * q^27 + 6 * q^33 + 6 * q^35 + 18 * q^37 + 24 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^49 + 12 * q^53 - 18 * q^57 + 12 * q^59 + 4 * q^61 + 12 * q^63 + 18 * q^67 - 24 * q^69 + 12 * q^71 + 24 * q^73 + 2 * q^75 - 6 * q^77 - 8 * q^79 - 8 * q^81 - 36 * q^83 + 6 * q^85 + 6 * q^87 + 12 * q^89 + 48 * q^93 + 6 * 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
$3$	2.82684	1.63208	0.816038	−	0.577998i	$-0.196166\pi$
$3$	2.82684	1.63208	0.816038	+	0.577998i	$0.196166\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.09479	0.791755	0.395878	−	0.918303i	$-0.370440\pi$
$7$	2.09479	0.791755	0.395878	+	0.918303i	$0.370440\pi$
$8$	0	0
$9$	4.99102	1.66367
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.82684	0.729887
$16$	0	0
$17$	−3.62828	−0.879987	−0.439993	−	0.898001i	$-0.645019\pi$
$17$	−3.62828	−0.879987	−0.439993	+	0.898001i	$0.645019\pi$
$18$	0	0
$19$	1.06939	0.245335	0.122667	−	0.992448i	$-0.460855\pi$
$19$	1.06939	0.245335	0.122667	+	0.992448i	$0.460855\pi$
$20$	0	0
$21$	5.92163	1.29220
$22$	0	0
$23$	−7.81785	−1.63014	−0.815068	−	0.579366i	$-0.803300\pi$
$23$	−7.81785	−1.63014	−0.815068	+	0.579366i	$0.803300\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.62828	1.08316
$28$	0	0
$29$	−0.526914	−0.0978454	−0.0489227	−	0.998803i	$-0.515579\pi$
$29$	−0.526914	−0.0978454	−0.0489227	+	0.998803i	$0.515579\pi$
$30$	0	0
$31$	5.84325	1.04948	0.524740	−	0.851263i	$-0.324163\pi$
$31$	5.84325	1.04948	0.524740	+	0.851263i	$0.324163\pi$
$32$	0	0
$33$	4.89623	0.852324
$34$	0	0
$35$	2.09479	0.354084
$36$	0	0
$37$	9.74846	1.60264	0.801319	−	0.598238i	$-0.204132\pi$
$37$	9.74846	1.60264	0.801319	+	0.598238i	$0.204132\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.26795	0.666542	0.333271	−	0.942831i	$-0.391848\pi$
$41$	4.26795	0.666542	0.333271	+	0.942831i	$0.391848\pi$
$42$	0	0
$43$	9.34477	1.42506	0.712532	−	0.701640i	$-0.247548\pi$
$43$	9.34477	1.42506	0.712532	+	0.701640i	$0.247548\pi$
$44$	0	0
$45$	4.99102	0.744017
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−2.61186	−0.373124
$50$	0	0
$51$	−10.2566	−1.43621
$52$	0	0
$53$	12.5939	1.72990	0.864952	−	0.501854i	$-0.167349\pi$
$53$	12.5939	1.72990	0.864952	+	0.501854i	$0.167349\pi$
$54$	0	0
$55$	1.73205	0.233550
$56$	0	0
$57$	3.02299	0.400405
$58$	0	0
$59$	1.40370	0.182746	0.0913729	−	0.995817i	$-0.470874\pi$
$59$	1.40370	0.182746	0.0913729	+	0.995817i	$0.470874\pi$
$60$	0	0
$61$	−11.1088	−1.42234	−0.711168	−	0.703022i	$-0.751833\pi$
$61$	−11.1088	−1.42234	−0.711168	+	0.703022i	$0.751833\pi$
$62$	0	0
$63$	10.4551	1.31722
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8334	1.32351	0.661756	−	0.749719i	$-0.269811\pi$
$67$	10.8334	1.32351	0.661756	+	0.749719i	$0.269811\pi$
$68$	0	0
$69$	−22.0998	−2.66050
$70$	0	0
$71$	−14.1692	−1.68157	−0.840787	−	0.541366i	$-0.817907\pi$
$71$	−14.1692	−1.68157	−0.840787	+	0.541366i	$0.817907\pi$
$72$	0	0
$73$	−2.64469	−0.309538	−0.154769	−	0.987951i	$-0.549463\pi$
$73$	−2.64469	−0.309538	−0.154769	+	0.987951i	$0.549463\pi$
$74$	0	0
$75$	2.82684	0.326415
$76$	0	0
$77$	3.62828	0.413481
$78$	0	0
$79$	13.5729	1.52707	0.763535	−	0.645766i	$-0.223462\pi$
$79$	13.5729	1.52707	0.763535	+	0.645766i	$0.223462\pi$
$80$	0	0
$81$	0.937188	0.104132
$82$	0	0
$83$	−15.7925	−1.73345	−0.866724	−	0.498789i	$-0.833778\pi$
$83$	−15.7925	−1.73345	−0.866724	+	0.498789i	$0.833778\pi$
$84$	0	0
$85$	−3.62828	−0.393542
$86$	0	0
$87$	−1.48950	−0.159691
$88$	0	0
$89$	−5.52451	−0.585596	−0.292798	−	0.956174i	$-0.594586\pi$
$89$	−5.52451	−0.585596	−0.292798	+	0.956174i	$0.594586\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	16.5179	1.71283
$94$	0	0
$95$	1.06939	0.109717
$96$	0	0
$97$	−15.1946	−1.54278	−0.771389	−	0.636364i	$-0.780438\pi$
$97$	−15.1946	−1.54278	−0.771389	+	0.636364i	$0.780438\pi$
$98$	0	0
$99$	8.64469	0.868824
$100$	0	0
$101$	−3.66266	−0.364448	−0.182224	−	0.983257i	$-0.558330\pi$
$101$	−3.66266	−0.364448	−0.182224	+	0.983257i	$0.558330\pi$
$102$	0	0
$103$	−13.7804	−1.35783	−0.678914	−	0.734218i	$-0.737549\pi$
$103$	−13.7804	−1.35783	−0.678914	+	0.734218i	$0.737549\pi$
$104$	0	0
$105$	5.92163	0.577892
$106$	0	0
$107$	3.23711	0.312943	0.156472	−	0.987682i	$-0.449988\pi$
$107$	3.23711	0.312943	0.156472	+	0.987682i	$0.449988\pi$
$108$	0	0
$109$	9.12979	0.874476	0.437238	−	0.899346i	$-0.355957\pi$
$109$	9.12979	0.874476	0.437238	+	0.899346i	$0.355957\pi$
$110$	0	0
$111$	27.5573	2.61563
$112$	0	0
$113$	−10.9536	−1.03043	−0.515214	−	0.857062i	$-0.672288\pi$
$113$	−10.9536	−1.03043	−0.515214	+	0.857062i	$0.672288\pi$
$114$	0	0
$115$	−7.81785	−0.729019
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.60047	−0.696734
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	12.0648	1.08785
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.907620	0.0805382	0.0402691	−	0.999189i	$-0.487178\pi$
$127$	0.907620	0.0805382	0.0402691	+	0.999189i	$0.487178\pi$
$128$	0	0
$129$	26.4161	2.32581
$130$	0	0
$131$	−13.1626	−1.15002	−0.575012	−	0.818145i	$-0.695003\pi$
$131$	−13.1626	−1.15002	−0.575012	+	0.818145i	$0.695003\pi$
$132$	0	0
$133$	2.24014	0.194245
$134$	0	0
$135$	5.62828	0.484405
$136$	0	0
$137$	−12.0648	−1.03077	−0.515383	−	0.856960i	$-0.672350\pi$
$137$	−12.0648	−1.03077	−0.515383	+	0.856960i	$0.672350\pi$
$138$	0	0
$139$	−5.61186	−0.475992	−0.237996	−	0.971266i	$-0.576491\pi$
$139$	−5.61186	−0.475992	−0.237996	+	0.971266i	$0.576491\pi$
$140$	0	0
$141$	9.79246	0.824674
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.526914	−0.0437578
$146$	0	0
$147$	−7.38332	−0.608966
$148$	0	0
$149$	20.9886	1.71945	0.859727	−	0.510754i	$-0.170634\pi$
$149$	20.9886	1.71945	0.859727	+	0.510754i	$0.170634\pi$
$150$	0	0
$151$	6.99102	0.568921	0.284460	−	0.958688i	$-0.408186\pi$
$151$	6.99102	0.568921	0.284460	+	0.958688i	$0.408186\pi$
$152$	0	0
$153$	−18.1088	−1.46401
$154$	0	0
$155$	5.84325	0.469341
$156$	0	0
$157$	3.74761	0.299092	0.149546	−	0.988755i	$-0.452219\pi$
$157$	3.74761	0.299092	0.149546	+	0.988755i	$0.452219\pi$
$158$	0	0
$159$	35.6009	2.82334
$160$	0	0
$161$	−16.3767	−1.29067
$162$	0	0
$163$	6.37830	0.499587	0.249793	−	0.968299i	$-0.419637\pi$
$163$	6.37830	0.499587	0.249793	+	0.968299i	$0.419637\pi$
$164$	0	0
$165$	4.89623	0.381171
$166$	0	0
$167$	−15.5289	−1.20166	−0.600831	−	0.799376i	$-0.705164\pi$
$167$	−15.5289	−1.20166	−0.600831	+	0.799376i	$0.705164\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	5.33734	0.408156
$172$	0	0
$173$	−4.77604	−0.363116	−0.181558	−	0.983380i	$-0.558114\pi$
$173$	−4.77604	−0.363116	−0.181558	+	0.983380i	$0.558114\pi$
$174$	0	0
$175$	2.09479	0.158351
$176$	0	0
$177$	3.96802	0.298255
$178$	0	0
$179$	15.7027	1.17367	0.586837	−	0.809705i	$-0.300373\pi$
$179$	15.7027	1.17367	0.586837	+	0.809705i	$0.300373\pi$
$180$	0	0
$181$	−10.8851	−0.809080	−0.404540	−	0.914520i	$-0.632568\pi$
$181$	−10.8851	−0.809080	−0.404540	+	0.914520i	$0.632568\pi$
$182$	0	0
$183$	−31.4028	−2.32136
$184$	0	0
$185$	9.74846	0.716721
$186$	0	0
$187$	−6.28436	−0.459558
$188$	0	0
$189$	11.7900	0.857600
$190$	0	0
$191$	−5.95819	−0.431119	−0.215560	−	0.976491i	$-0.569158\pi$
$191$	−5.95819	−0.431119	−0.215560	+	0.976491i	$0.569158\pi$
$192$	0	0
$193$	12.7695	0.919166	0.459583	−	0.888135i	$-0.347999\pi$
$193$	12.7695	0.919166	0.459583	+	0.888135i	$0.347999\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.1079	1.14764	0.573822	−	0.818980i	$-0.305460\pi$
$197$	16.1079	1.14764	0.573822	+	0.818980i	$0.305460\pi$
$198$	0	0
$199$	−8.52805	−0.604538	−0.302269	−	0.953223i	$-0.597744\pi$
$199$	−8.52805	−0.604538	−0.302269	+	0.953223i	$0.597744\pi$
$200$	0	0
$201$	30.6243	2.16007
$202$	0	0
$203$	−1.10377	−0.0774696
$204$	0	0
$205$	4.26795	0.298087
$206$	0	0
$207$	−39.0190	−2.71201
$208$	0	0
$209$	1.85224	0.128122
$210$	0	0
$211$	−4.18059	−0.287804	−0.143902	−	0.989592i	$-0.545965\pi$
$211$	−4.18059	−0.287804	−0.143902	+	0.989592i	$0.545965\pi$
$212$	0	0
$213$	−40.0540	−2.74446
$214$	0	0
$215$	9.34477	0.637308
$216$	0	0
$217$	12.2404	0.830931
$218$	0	0
$219$	−7.47612	−0.505189
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.4991	0.703072	0.351536	−	0.936174i	$-0.385659\pi$
$223$	10.4991	0.703072	0.351536	+	0.936174i	$0.385659\pi$
$224$	0	0
$225$	4.99102	0.332734
$226$	0	0
$227$	15.5858	1.03446	0.517232	−	0.855845i	$-0.326963\pi$
$227$	15.5858	1.03446	0.517232	+	0.855845i	$0.326963\pi$
$228$	0	0
$229$	−19.2714	−1.27349	−0.636745	−	0.771074i	$-0.719720\pi$
$229$	−19.2714	−1.27349	−0.636745	+	0.771074i	$0.719720\pi$
$230$	0	0
$231$	10.2566	0.674832
$232$	0	0
$233$	−2.48794	−0.162991	−0.0814953	−	0.996674i	$-0.525970\pi$
$233$	−2.48794	−0.162991	−0.0814953	+	0.996674i	$0.525970\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	0	0
$237$	38.3684	2.49229
$238$	0	0
$239$	16.7775	1.08525	0.542624	−	0.839976i	$-0.317431\pi$
$239$	16.7775	1.08525	0.542624	+	0.839976i	$0.317431\pi$
$240$	0	0
$241$	−29.0794	−1.87317	−0.936585	−	0.350439i	$-0.886032\pi$
$241$	−29.0794	−1.87317	−0.936585	+	0.350439i	$0.886032\pi$
$242$	0	0
$243$	−14.2356	−0.913211
$244$	0	0
$245$	−2.61186	−0.166866
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−44.6427	−2.82912
$250$	0	0
$251$	19.1208	1.20689	0.603447	−	0.797403i	$-0.293793\pi$
$251$	19.1208	1.20689	0.603447	+	0.797403i	$0.293793\pi$
$252$	0	0
$253$	−13.5409	−0.851310
$254$	0	0
$255$	−10.2566	−0.642291
$256$	0	0
$257$	13.7091	0.855148	0.427574	−	0.903980i	$-0.359368\pi$
$257$	13.7091	0.855148	0.427574	+	0.903980i	$0.359368\pi$
$258$	0	0
$259$	20.4210	1.26890
$260$	0	0
$261$	−2.62983	−0.162783
$262$	0	0
$263$	−11.9135	−0.734617	−0.367309	−	0.930099i	$-0.619721\pi$
$263$	−11.9135	−0.734617	−0.367309	+	0.930099i	$0.619721\pi$
$264$	0	0
$265$	12.5939	0.773637
$266$	0	0
$267$	−15.6169	−0.955738
$268$	0	0
$269$	−20.9312	−1.27620	−0.638100	−	0.769954i	$-0.720279\pi$
$269$	−20.9312	−1.27620	−0.638100	+	0.769954i	$0.720279\pi$
$270$	0	0
$271$	1.95161	0.118552	0.0592760	−	0.998242i	$-0.481121\pi$
$271$	1.95161	0.118552	0.0592760	+	0.998242i	$0.481121\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.73205	0.104447
$276$	0	0
$277$	12.4805	0.749881	0.374941	−	0.927049i	$-0.377663\pi$
$277$	12.4805	0.749881	0.374941	+	0.927049i	$0.377663\pi$
$278$	0	0
$279$	29.1638	1.74599
$280$	0	0
$281$	2.29553	0.136940	0.0684698	−	0.997653i	$-0.478188\pi$
$281$	2.29553	0.136940	0.0684698	+	0.997653i	$0.478188\pi$
$282$	0	0
$283$	14.9297	0.887477	0.443739	−	0.896156i	$-0.353652\pi$
$283$	14.9297	0.887477	0.443739	+	0.896156i	$0.353652\pi$
$284$	0	0
$285$	3.02299	0.179067
$286$	0	0
$287$	8.94045	0.527738
$288$	0	0
$289$	−3.83559	−0.225623
$290$	0	0
$291$	−42.9527	−2.51793
$292$	0	0
$293$	−1.43213	−0.0836657	−0.0418329	−	0.999125i	$-0.513320\pi$
$293$	−1.43213	−0.0836657	−0.0418329	+	0.999125i	$0.513320\pi$
$294$	0	0
$295$	1.40370	0.0817264
$296$	0	0
$297$	9.74846	0.565663
$298$	0	0
$299$	0	0
$300$	0	0
$301$	19.5753	1.12830
$302$	0	0
$303$	−10.3538	−0.594808
$304$	0	0
$305$	−11.1088	−0.636088
$306$	0	0
$307$	−17.3833	−0.992118	−0.496059	−	0.868289i	$-0.665220\pi$
$307$	−17.3833	−0.992118	−0.496059	+	0.868289i	$0.665220\pi$
$308$	0	0
$309$	−38.9551	−2.21608
$310$	0	0
$311$	20.2164	1.14637	0.573185	−	0.819426i	$-0.305708\pi$
$311$	20.2164	1.14637	0.573185	+	0.819426i	$0.305708\pi$
$312$	0	0
$313$	−4.86425	−0.274944	−0.137472	−	0.990506i	$-0.543898\pi$
$313$	−4.86425	−0.274944	−0.137472	+	0.990506i	$0.543898\pi$
$314$	0	0
$315$	10.4551	0.589079
$316$	0	0
$317$	−23.6177	−1.32650	−0.663252	−	0.748396i	$-0.730824\pi$
$317$	−23.6177	−1.32650	−0.663252	+	0.748396i	$0.730824\pi$
$318$	0	0
$319$	−0.912641	−0.0510981
$320$	0	0
$321$	9.15079	0.510748
$322$	0	0
$323$	−3.88004	−0.215891
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.8085	1.42721
$328$	0	0
$329$	7.25656	0.400067
$330$	0	0
$331$	−14.8798	−0.817869	−0.408934	−	0.912564i	$-0.634099\pi$
$331$	−14.8798	−0.817869	−0.408934	+	0.912564i	$0.634099\pi$
$332$	0	0
$333$	48.6547	2.66626
$334$	0	0
$335$	10.8334	0.591893
$336$	0	0
$337$	29.1906	1.59012	0.795058	−	0.606534i	$-0.207441\pi$
$337$	29.1906	1.59012	0.795058	+	0.606534i	$0.207441\pi$
$338$	0	0
$339$	−30.9641	−1.68174
$340$	0	0
$341$	10.1208	0.548073
$342$	0	0
$343$	−20.1348	−1.08718
$344$	0	0
$345$	−22.0998	−1.18981
$346$	0	0
$347$	−24.2250	−1.30047	−0.650234	−	0.759734i	$-0.725329\pi$
$347$	−24.2250	−1.30047	−0.650234	+	0.759734i	$0.725329\pi$
$348$	0	0
$349$	−12.8815	−0.689532	−0.344766	−	0.938689i	$-0.612042\pi$
$349$	−12.8815	−0.689532	−0.344766	+	0.938689i	$0.612042\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.2167	−0.863130	−0.431565	−	0.902082i	$-0.642038\pi$
$353$	−16.2167	−0.863130	−0.431565	+	0.902082i	$0.642038\pi$
$354$	0	0
$355$	−14.1692	−0.752023
$356$	0	0
$357$	−21.4853	−1.13712
$358$	0	0
$359$	−9.19261	−0.485167	−0.242584	−	0.970131i	$-0.577995\pi$
$359$	−9.19261	−0.485167	−0.242584	+	0.970131i	$0.577995\pi$
$360$	0	0
$361$	−17.8564	−0.939811
$362$	0	0
$363$	−22.6147	−1.18696
$364$	0	0
$365$	−2.64469	−0.138430
$366$	0	0
$367$	−4.60132	−0.240187	−0.120094	−	0.992763i	$-0.538319\pi$
$367$	−4.60132	−0.240187	−0.120094	+	0.992763i	$0.538319\pi$
$368$	0	0
$369$	21.3014	1.10891
$370$	0	0
$371$	26.3815	1.36966
$372$	0	0
$373$	−20.4925	−1.06106	−0.530532	−	0.847665i	$-0.678008\pi$
$373$	−20.4925	−1.06106	−0.530532	+	0.847665i	$0.678008\pi$
$374$	0	0
$375$	2.82684	0.145977
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−8.02872	−0.412407	−0.206204	−	0.978509i	$-0.566111\pi$
$379$	−8.02872	−0.412407	−0.206204	+	0.978509i	$0.566111\pi$
$380$	0	0
$381$	2.56569	0.131445
$382$	0	0
$383$	−30.1425	−1.54021	−0.770104	−	0.637918i	$-0.779796\pi$
$383$	−30.1425	−1.54021	−0.770104	+	0.637918i	$0.779796\pi$
$384$	0	0
$385$	3.62828	0.184914
$386$	0	0
$387$	46.6399	2.37084
$388$	0	0
$389$	35.8264	1.81647	0.908236	−	0.418459i	$-0.137430\pi$
$389$	35.8264	1.81647	0.908236	+	0.418459i	$0.137430\pi$
$390$	0	0
$391$	28.3654	1.43450
$392$	0	0
$393$	−37.2086	−1.87693
$394$	0	0
$395$	13.5729	0.682926
$396$	0	0
$397$	15.3830	0.772052	0.386026	−	0.922488i	$-0.373847\pi$
$397$	15.3830	0.772052	0.386026	+	0.922488i	$0.373847\pi$
$398$	0	0
$399$	6.33252	0.317023
$400$	0	0
$401$	−9.77689	−0.488235	−0.244117	−	0.969746i	$-0.578498\pi$
$401$	−9.77689	−0.488235	−0.244117	+	0.969746i	$0.578498\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.937188	0.0465692
$406$	0	0
$407$	16.8848	0.836950
$408$	0	0
$409$	1.00658	0.0497720	0.0248860	−	0.999690i	$-0.492078\pi$
$409$	1.00658	0.0497720	0.0248860	+	0.999690i	$0.492078\pi$
$410$	0	0
$411$	−34.1052	−1.68229
$412$	0	0
$413$	2.94045	0.144690
$414$	0	0
$415$	−15.7925	−0.775221
$416$	0	0
$417$	−15.8638	−0.776855
$418$	0	0
$419$	9.70269	0.474007	0.237004	−	0.971509i	$-0.423835\pi$
$419$	9.70269	0.474007	0.237004	+	0.971509i	$0.423835\pi$
$420$	0	0
$421$	14.2955	0.696721	0.348361	−	0.937361i	$-0.386738\pi$
$421$	14.2955	0.696721	0.348361	+	0.937361i	$0.386738\pi$
$422$	0	0
$423$	17.2894	0.840639
$424$	0	0
$425$	−3.62828	−0.175997
$426$	0	0
$427$	−23.2706	−1.12614
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.1471	−0.585107	−0.292554	−	0.956249i	$-0.594505\pi$
$431$	−12.1471	−0.585107	−0.292554	+	0.956249i	$0.594505\pi$
$432$	0	0
$433$	−0.209020	−0.0100448	−0.00502242	−	0.999987i	$-0.501599\pi$
$433$	−0.209020	−0.0100448	−0.00502242	+	0.999987i	$0.501599\pi$
$434$	0	0
$435$	−1.48950	−0.0714161
$436$	0	0
$437$	−8.36033	−0.399929
$438$	0	0
$439$	−27.7207	−1.32303	−0.661517	−	0.749930i	$-0.730087\pi$
$439$	−27.7207	−1.32303	−0.661517	+	0.749930i	$0.730087\pi$
$440$	0	0
$441$	−13.0359	−0.620755
$442$	0	0
$443$	−7.98798	−0.379521	−0.189760	−	0.981830i	$-0.560771\pi$
$443$	−7.98798	−0.379521	−0.189760	+	0.981830i	$0.560771\pi$
$444$	0	0
$445$	−5.52451	−0.261887
$446$	0	0
$447$	59.3314	2.80628
$448$	0	0
$449$	−7.13220	−0.336589	−0.168295	−	0.985737i	$-0.553826\pi$
$449$	−7.13220	−0.336589	−0.168295	+	0.985737i	$0.553826\pi$
$450$	0	0
$451$	7.39230	0.348090
$452$	0	0
$453$	19.7625	0.928522
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.6808	1.48197	0.740984	−	0.671523i	$-0.234360\pi$
$457$	31.6808	1.48197	0.740984	+	0.671523i	$0.234360\pi$
$458$	0	0
$459$	−20.4210	−0.953169
$460$	0	0
$461$	−11.8529	−0.552043	−0.276021	−	0.961151i	$-0.589016\pi$
$461$	−11.8529	−0.552043	−0.276021	+	0.961151i	$0.589016\pi$
$462$	0	0
$463$	12.7655	0.593263	0.296632	−	0.954992i	$-0.404137\pi$
$463$	12.7655	0.593263	0.296632	+	0.954992i	$0.404137\pi$
$464$	0	0
$465$	16.5179	0.766001
$466$	0	0
$467$	−31.3402	−1.45025	−0.725125	−	0.688617i	$-0.758218\pi$
$467$	−31.3402	−1.45025	−0.725125	+	0.688617i	$0.758218\pi$
$468$	0	0
$469$	22.6937	1.04790
$470$	0	0
$471$	10.5939	0.488141
$472$	0	0
$473$	16.1856	0.744215
$474$	0	0
$475$	1.06939	0.0490669
$476$	0	0
$477$	62.8563	2.87799
$478$	0	0
$479$	16.1261	0.736818	0.368409	−	0.929664i	$-0.379903\pi$
$479$	16.1261	0.736818	0.368409	+	0.929664i	$0.379903\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−46.2944	−2.10647
$484$	0	0
$485$	−15.1946	−0.689951
$486$	0	0
$487$	−5.59001	−0.253308	−0.126654	−	0.991947i	$-0.540424\pi$
$487$	−5.59001	−0.253308	−0.126654	+	0.991947i	$0.540424\pi$
$488$	0	0
$489$	18.0304	0.815364
$490$	0	0
$491$	−18.2954	−0.825662	−0.412831	−	0.910808i	$-0.635460\pi$
$491$	−18.2954	−0.825662	−0.412831	+	0.910808i	$0.635460\pi$
$492$	0	0
$493$	1.91179	0.0861027
$494$	0	0
$495$	8.64469	0.388550
$496$	0	0
$497$	−29.6815	−1.33140
$498$	0	0
$499$	0.553868	0.0247945	0.0123973	−	0.999923i	$-0.496054\pi$
$499$	0.553868	0.0247945	0.0123973	+	0.999923i	$0.496054\pi$
$500$	0	0
$501$	−43.8977	−1.96120
$502$	0	0
$503$	14.6832	0.654694	0.327347	−	0.944904i	$-0.393845\pi$
$503$	14.6832	0.654694	0.327347	+	0.944904i	$0.393845\pi$
$504$	0	0
$505$	−3.66266	−0.162986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.92295	−0.306855	−0.153427	−	0.988160i	$-0.549031\pi$
$509$	−6.92295	−0.306855	−0.153427	+	0.988160i	$0.549031\pi$
$510$	0	0
$511$	−5.54007	−0.245078
$512$	0	0
$513$	6.01882	0.265737
$514$	0	0
$515$	−13.7804	−0.607239
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	−13.5011	−0.592632
$520$	0	0
$521$	−18.3551	−0.804152	−0.402076	−	0.915606i	$-0.631711\pi$
$521$	−18.3551	−0.804152	−0.402076	+	0.915606i	$0.631711\pi$
$522$	0	0
$523$	−18.2713	−0.798947	−0.399473	−	0.916745i	$-0.630807\pi$
$523$	−18.2713	−0.798947	−0.399473	+	0.916745i	$0.630807\pi$
$524$	0	0
$525$	5.92163	0.258441
$526$	0	0
$527$	−21.2009	−0.923528
$528$	0	0
$529$	38.1188	1.65734
$530$	0	0
$531$	7.00587	0.304029
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3.23711	0.139953
$536$	0	0
$537$	44.3890	1.91553
$538$	0	0
$539$	−4.52388	−0.194857
$540$	0	0
$541$	31.8881	1.37098	0.685488	−	0.728084i	$-0.259589\pi$
$541$	31.8881	1.37098	0.685488	+	0.728084i	$0.259589\pi$
$542$	0	0
$543$	−30.7703	−1.32048
$544$	0	0
$545$	9.12979	0.391077
$546$	0	0
$547$	44.7966	1.91537	0.957683	−	0.287826i	$-0.0929325\pi$
$547$	44.7966	1.91537	0.957683	+	0.287826i	$0.0929325\pi$
$548$	0	0
$549$	−55.4442	−2.36630
$550$	0	0
$551$	−0.563476	−0.0240049
$552$	0	0
$553$	28.4323	1.20907
$554$	0	0
$555$	27.5573	1.16974
$556$	0	0
$557$	26.3064	1.11464	0.557319	−	0.830298i	$-0.311830\pi$
$557$	26.3064	1.11464	0.557319	+	0.830298i	$0.311830\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−17.7649	−0.750034
$562$	0	0
$563$	−5.78984	−0.244013	−0.122006	−	0.992529i	$-0.538933\pi$
$563$	−5.78984	−0.244013	−0.122006	+	0.992529i	$0.538933\pi$
$564$	0	0
$565$	−10.9536	−0.460821
$566$	0	0
$567$	1.96321	0.0824471
$568$	0	0
$569$	23.6368	0.990905	0.495452	−	0.868635i	$-0.335002\pi$
$569$	23.6368	0.990905	0.495452	+	0.868635i	$0.335002\pi$
$570$	0	0
$571$	11.4641	0.479758	0.239879	−	0.970803i	$-0.422892\pi$
$571$	11.4641	0.479758	0.239879	+	0.970803i	$0.422892\pi$
$572$	0	0
$573$	−16.8428	−0.703620
$574$	0	0
$575$	−7.81785	−0.326027
$576$	0	0
$577$	34.2415	1.42549	0.712746	−	0.701422i	$-0.247451\pi$
$577$	34.2415	1.42549	0.712746	+	0.701422i	$0.247451\pi$
$578$	0	0
$579$	36.0972	1.50015
$580$	0	0
$581$	−33.0818	−1.37247
$582$	0	0
$583$	21.8133	0.903413
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.63599	0.232622	0.116311	−	0.993213i	$-0.462893\pi$
$587$	5.63599	0.232622	0.116311	+	0.993213i	$0.462893\pi$
$588$	0	0
$589$	6.24871	0.257474
$590$	0	0
$591$	45.5345	1.87304
$592$	0	0
$593$	−15.3014	−0.628353	−0.314177	−	0.949365i	$-0.601728\pi$
$593$	−15.3014	−0.628353	−0.314177	+	0.949365i	$0.601728\pi$
$594$	0	0
$595$	−7.60047	−0.311589
$596$	0	0
$597$	−24.1074	−0.986651
$598$	0	0
$599$	9.82414	0.401404	0.200702	−	0.979652i	$-0.435678\pi$
$599$	9.82414	0.401404	0.200702	+	0.979652i	$0.435678\pi$
$600$	0	0
$601$	−46.1889	−1.88409	−0.942043	−	0.335492i	$-0.891097\pi$
$601$	−46.1889	−1.88409	−0.942043	+	0.335492i	$0.891097\pi$
$602$	0	0
$603$	54.0697	2.20189
$604$	0	0
$605$	−8.00000	−0.325246
$606$	0	0
$607$	24.6964	1.00240	0.501198	−	0.865333i	$-0.332893\pi$
$607$	24.6964	1.00240	0.501198	+	0.865333i	$0.332893\pi$
$608$	0	0
$609$	−3.12019	−0.126436
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.6455	0.672307	0.336154	−	0.941807i	$-0.390874\pi$
$613$	16.6455	0.672307	0.336154	+	0.941807i	$0.390874\pi$
$614$	0	0
$615$	12.0648	0.486500
$616$	0	0
$617$	−1.61564	−0.0650432	−0.0325216	−	0.999471i	$-0.510354\pi$
$617$	−1.61564	−0.0650432	−0.0325216	+	0.999471i	$0.510354\pi$
$618$	0	0
$619$	23.3922	0.940213	0.470106	−	0.882610i	$-0.344216\pi$
$619$	23.3922	0.940213	0.470106	+	0.882610i	$0.344216\pi$
$620$	0	0
$621$	−44.0011	−1.76570
$622$	0	0
$623$	−11.5727	−0.463649
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.23597	0.209105
$628$	0	0
$629$	−35.3701	−1.41030
$630$	0	0
$631$	−3.20381	−0.127542	−0.0637708	−	0.997965i	$-0.520313\pi$
$631$	−3.20381	−0.127542	−0.0637708	+	0.997965i	$0.520313\pi$
$632$	0	0
$633$	−11.8179	−0.469718
$634$	0	0
$635$	0.907620	0.0360178
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−70.7187	−2.79759
$640$	0	0
$641$	−19.9759	−0.789000	−0.394500	−	0.918896i	$-0.629082\pi$
$641$	−19.9759	−0.789000	−0.394500	+	0.918896i	$0.629082\pi$
$642$	0	0
$643$	−28.5368	−1.12538	−0.562690	−	0.826668i	$-0.690233\pi$
$643$	−28.5368	−1.12538	−0.562690	+	0.826668i	$0.690233\pi$
$644$	0	0
$645$	26.4161	1.04013
$646$	0	0
$647$	6.08651	0.239285	0.119643	−	0.992817i	$-0.461825\pi$
$647$	6.08651	0.239285	0.119643	+	0.992817i	$0.461825\pi$
$648$	0	0
$649$	2.43127	0.0954359
$650$	0	0
$651$	34.6016	1.35614
$652$	0	0
$653$	19.6300	0.768181	0.384090	−	0.923296i	$-0.374515\pi$
$653$	19.6300	0.768181	0.384090	+	0.923296i	$0.374515\pi$
$654$	0	0
$655$	−13.1626	−0.514306
$656$	0	0
$657$	−13.1997	−0.514969
$658$	0	0
$659$	−19.5950	−0.763314	−0.381657	−	0.924304i	$-0.624646\pi$
$659$	−19.5950	−0.763314	−0.381657	+	0.924304i	$0.624646\pi$
$660$	0	0
$661$	34.7259	1.35068	0.675341	−	0.737506i	$-0.263997\pi$
$661$	34.7259	1.35068	0.675341	+	0.737506i	$0.263997\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.24014	0.0868690
$666$	0	0
$667$	4.11933	0.159501
$668$	0	0
$669$	29.6793	1.14747
$670$	0	0
$671$	−19.2410	−0.742790
$672$	0	0
$673$	−34.0103	−1.31100	−0.655500	−	0.755195i	$-0.727542\pi$
$673$	−34.0103	−1.31100	−0.655500	+	0.755195i	$0.727542\pi$
$674$	0	0
$675$	5.62828	0.216633
$676$	0	0
$677$	29.9209	1.14995	0.574977	−	0.818169i	$-0.305011\pi$
$677$	29.9209	1.14995	0.574977	+	0.818169i	$0.305011\pi$
$678$	0	0
$679$	−31.8295	−1.22150
$680$	0	0
$681$	44.0584	1.68832
$682$	0	0
$683$	−27.8573	−1.06593	−0.532964	−	0.846138i	$-0.678922\pi$
$683$	−27.8573	−1.06593	−0.532964	+	0.846138i	$0.678922\pi$
$684$	0	0
$685$	−12.0648	−0.460972
$686$	0	0
$687$	−54.4772	−2.07843
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−36.3211	−1.38172	−0.690860	−	0.722988i	$-0.742768\pi$
$691$	−36.3211	−1.38172	−0.690860	+	0.722988i	$0.742768\pi$
$692$	0	0
$693$	18.1088	0.687896
$694$	0	0
$695$	−5.61186	−0.212870
$696$	0	0
$697$	−15.4853	−0.586548
$698$	0	0
$699$	−7.03302	−0.266013
$700$	0	0
$701$	−2.16156	−0.0816411	−0.0408206	−	0.999166i	$-0.512997\pi$
$701$	−2.16156	−0.0816411	−0.0408206	+	0.999166i	$0.512997\pi$
$702$	0	0
$703$	10.4249	0.393183
$704$	0	0
$705$	9.79246	0.368805
$706$	0	0
$707$	−7.67250	−0.288554
$708$	0	0
$709$	47.6165	1.78827	0.894137	−	0.447793i	$-0.147790\pi$
$709$	47.6165	1.78827	0.894137	+	0.447793i	$0.147790\pi$
$710$	0	0
$711$	67.7425	2.54054
$712$	0	0
$713$	−45.6817	−1.71079
$714$	0	0
$715$	0	0
$716$	0	0
$717$	47.4273	1.77121
$718$	0	0
$719$	−20.2937	−0.756829	−0.378414	−	0.925636i	$-0.623531\pi$
$719$	−20.2937	−0.756829	−0.378414	+	0.925636i	$0.623531\pi$
$720$	0	0
$721$	−28.8671	−1.07507
$722$	0	0
$723$	−82.2029	−3.05716
$724$	0	0
$725$	−0.526914	−0.0195691
$726$	0	0
$727$	−11.0681	−0.410494	−0.205247	−	0.978710i	$-0.565800\pi$
$727$	−11.0681	−0.410494	−0.205247	+	0.978710i	$0.565800\pi$
$728$	0	0
$729$	−43.0532	−1.59456
$730$	0	0
$731$	−33.9054	−1.25404
$732$	0	0
$733$	23.4002	0.864304	0.432152	−	0.901801i	$-0.357754\pi$
$733$	23.4002	0.864304	0.432152	+	0.901801i	$0.357754\pi$
$734$	0	0
$735$	−7.38332	−0.272338
$736$	0	0
$737$	18.7640	0.691182
$738$	0	0
$739$	−25.5902	−0.941349	−0.470675	−	0.882307i	$-0.655989\pi$
$739$	−25.5902	−0.941349	−0.470675	+	0.882307i	$0.655989\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.41424	−0.308688	−0.154344	−	0.988017i	$-0.549326\pi$
$743$	−8.41424	−0.308688	−0.154344	+	0.988017i	$0.549326\pi$
$744$	0	0
$745$	20.9886	0.768963
$746$	0	0
$747$	−78.8204	−2.88389
$748$	0	0
$749$	6.78106	0.247775
$750$	0	0
$751$	40.5190	1.47856	0.739279	−	0.673399i	$-0.235166\pi$
$751$	40.5190	1.47856	0.739279	+	0.673399i	$0.235166\pi$
$752$	0	0
$753$	54.0514	1.96974
$754$	0	0
$755$	6.99102	0.254429
$756$	0	0
$757$	−11.3519	−0.412590	−0.206295	−	0.978490i	$-0.566141\pi$
$757$	−11.3519	−0.412590	−0.206295	+	0.978490i	$0.566141\pi$
$758$	0	0
$759$	−38.2780	−1.38940
$760$	0	0
$761$	6.77075	0.245440	0.122720	−	0.992441i	$-0.460838\pi$
$761$	6.77075	0.245440	0.122720	+	0.992441i	$0.460838\pi$
$762$	0	0
$763$	19.1250	0.692371
$764$	0	0
$765$	−18.1088	−0.654725
$766$	0	0
$767$	0	0
$768$	0	0
$769$	23.1591	0.835138	0.417569	−	0.908645i	$-0.362882\pi$
$769$	23.1591	0.835138	0.417569	+	0.908645i	$0.362882\pi$
$770$	0	0
$771$	38.7533	1.39567
$772$	0	0
$773$	41.9540	1.50898	0.754491	−	0.656311i	$-0.227884\pi$
$773$	41.9540	1.50898	0.754491	+	0.656311i	$0.227884\pi$
$774$	0	0
$775$	5.84325	0.209896
$776$	0	0
$777$	57.7268	2.07094
$778$	0	0
$779$	4.56410	0.163526
$780$	0	0
$781$	−24.5418	−0.878174
$782$	0	0
$783$	−2.96562	−0.105983
$784$	0	0
$785$	3.74761	0.133758
$786$	0	0
$787$	−37.4153	−1.33371	−0.666856	−	0.745187i	$-0.732360\pi$
$787$	−37.4153	−1.33371	−0.666856	+	0.745187i	$0.732360\pi$
$788$	0	0
$789$	−33.6775	−1.19895
$790$	0	0
$791$	−22.9455	−0.815847
$792$	0	0
$793$	0	0
$794$	0	0
$795$	35.6009	1.26263
$796$	0	0
$797$	47.2400	1.67333	0.836663	−	0.547719i	$-0.184504\pi$
$797$	47.2400	1.67333	0.836663	+	0.547719i	$0.184504\pi$
$798$	0	0
$799$	−12.5687	−0.444650
$800$	0	0
$801$	−27.5729	−0.974240
$802$	0	0
$803$	−4.58074	−0.161651
$804$	0	0
$805$	−16.3767	−0.577204
$806$	0	0
$807$	−59.1692	−2.08286
$808$	0	0
$809$	−46.4741	−1.63394	−0.816972	−	0.576677i	$-0.804349\pi$
$809$	−46.4741	−1.63394	−0.816972	+	0.576677i	$0.804349\pi$
$810$	0	0
$811$	−11.4041	−0.400453	−0.200227	−	0.979750i	$-0.564168\pi$
$811$	−11.4041	−0.400453	−0.200227	+	0.979750i	$0.564168\pi$
$812$	0	0
$813$	5.51689	0.193486
$814$	0	0
$815$	6.37830	0.223422
$816$	0	0
$817$	9.99319	0.349618
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.5615	0.857202	0.428601	−	0.903494i	$-0.359007\pi$
$821$	24.5615	0.857202	0.428601	+	0.903494i	$0.359007\pi$
$822$	0	0
$823$	2.96448	0.103335	0.0516676	−	0.998664i	$-0.483546\pi$
$823$	2.96448	0.103335	0.0516676	+	0.998664i	$0.483546\pi$
$824$	0	0
$825$	4.89623	0.170465
$826$	0	0
$827$	17.7265	0.616412	0.308206	−	0.951320i	$-0.400271\pi$
$827$	17.7265	0.616412	0.308206	+	0.951320i	$0.400271\pi$
$828$	0	0
$829$	−11.5758	−0.402045	−0.201022	−	0.979587i	$-0.564426\pi$
$829$	−11.5758	−0.402045	−0.201022	+	0.979587i	$0.564426\pi$
$830$	0	0
$831$	35.2804	1.22386
$832$	0	0
$833$	9.47657	0.328344
$834$	0	0
$835$	−15.5289	−0.537400
$836$	0	0
$837$	32.8874	1.13676
$838$	0	0
$839$	−42.8795	−1.48037	−0.740183	−	0.672405i	$-0.765261\pi$
$839$	−42.8795	−1.48037	−0.740183	+	0.672405i	$0.765261\pi$
$840$	0	0
$841$	−28.7224	−0.990426
$842$	0	0
$843$	6.48908	0.223496
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16.7583	−0.575822
$848$	0	0
$849$	42.2038	1.44843
$850$	0	0
$851$	−76.2121	−2.61252
$852$	0	0
$853$	−13.4599	−0.460857	−0.230428	−	0.973089i	$-0.574013\pi$
$853$	−13.4599	−0.460857	−0.230428	+	0.973089i	$0.574013\pi$
$854$	0	0
$855$	5.33734	0.182533
$856$	0	0
$857$	10.5950	0.361919	0.180960	−	0.983491i	$-0.442080\pi$
$857$	10.5950	0.361919	0.180960	+	0.983491i	$0.442080\pi$
$858$	0	0
$859$	−8.75716	−0.298791	−0.149395	−	0.988778i	$-0.547733\pi$
$859$	−8.75716	−0.298791	−0.149395	+	0.988778i	$0.547733\pi$
$860$	0	0
$861$	25.2732	0.861308
$862$	0	0
$863$	−30.8640	−1.05062	−0.525311	−	0.850910i	$-0.676051\pi$
$863$	−30.8640	−1.05062	−0.525311	+	0.850910i	$0.676051\pi$
$864$	0	0
$865$	−4.77604	−0.162390
$866$	0	0
$867$	−10.8426	−0.368234
$868$	0	0
$869$	23.5089	0.797486
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−75.8365	−2.56668
$874$	0	0
$875$	2.09479	0.0708167
$876$	0	0
$877$	20.1037	0.678853	0.339427	−	0.940633i	$-0.389767\pi$
$877$	20.1037	0.678853	0.339427	+	0.940633i	$0.389767\pi$
$878$	0	0
$879$	−4.04839	−0.136549
$880$	0	0
$881$	3.13396	0.105586	0.0527930	−	0.998605i	$-0.483188\pi$
$881$	3.13396	0.105586	0.0527930	+	0.998605i	$0.483188\pi$
$882$	0	0
$883$	34.0429	1.14563	0.572817	−	0.819683i	$-0.305851\pi$
$883$	34.0429	1.14563	0.572817	+	0.819683i	$0.305851\pi$
$884$	0	0
$885$	3.96802	0.133384
$886$	0	0
$887$	50.8069	1.70593	0.852964	−	0.521969i	$-0.174802\pi$
$887$	50.8069	1.70593	0.852964	+	0.521969i	$0.174802\pi$
$888$	0	0
$889$	1.90127	0.0637666
$890$	0	0
$891$	1.62326	0.0543812
$892$	0	0
$893$	3.70447	0.123965
$894$	0	0
$895$	15.7027	0.524883
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.07889	−0.102687
$900$	0	0
$901$	−45.6942	−1.52229
$902$	0	0
$903$	55.3362	1.84147
$904$	0	0
$905$	−10.8851	−0.361832
$906$	0	0
$907$	−50.9605	−1.69212	−0.846058	−	0.533090i	$-0.821031\pi$
$907$	−50.9605	−1.69212	−0.846058	+	0.533090i	$0.821031\pi$
$908$	0	0
$909$	−18.2804	−0.606323
$910$	0	0
$911$	−23.7176	−0.785800	−0.392900	−	0.919581i	$-0.628528\pi$
$911$	−23.7176	−0.785800	−0.392900	+	0.919581i	$0.628528\pi$
$912$	0	0
$913$	−27.3533	−0.905263
$914$	0	0
$915$	−31.4028	−1.03814
$916$	0	0
$917$	−27.5729	−0.910537
$918$	0	0
$919$	−42.3031	−1.39545	−0.697725	−	0.716365i	$-0.745804\pi$
$919$	−42.3031	−1.39545	−0.697725	+	0.716365i	$0.745804\pi$
$920$	0	0
$921$	−49.1398	−1.61921
$922$	0	0
$923$	0	0
$924$	0	0
$925$	9.74846	0.320528
$926$	0	0
$927$	−68.7784	−2.25898
$928$	0	0
$929$	−13.3290	−0.437310	−0.218655	−	0.975802i	$-0.570167\pi$
$929$	−13.3290	−0.437310	−0.218655	+	0.975802i	$0.570167\pi$
$930$	0	0
$931$	−2.79310	−0.0915402
$932$	0	0
$933$	57.1486	1.87096
$934$	0	0
$935$	−6.28436	−0.205521
$936$	0	0
$937$	14.0848	0.460129	0.230065	−	0.973175i	$-0.426106\pi$
$937$	14.0848	0.460129	0.230065	+	0.973175i	$0.426106\pi$
$938$	0	0
$939$	−13.7505	−0.448729
$940$	0	0
$941$	49.0399	1.59866	0.799328	−	0.600895i	$-0.205189\pi$
$941$	49.0399	1.59866	0.799328	+	0.600895i	$0.205189\pi$
$942$	0	0
$943$	−33.3662	−1.08655
$944$	0	0
$945$	11.7900	0.383530
$946$	0	0
$947$	−17.9352	−0.582816	−0.291408	−	0.956599i	$-0.594124\pi$
$947$	−17.9352	−0.582816	−0.291408	+	0.956599i	$0.594124\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−66.7635	−2.16496
$952$	0	0
$953$	44.3830	1.43771	0.718853	−	0.695162i	$-0.244667\pi$
$953$	44.3830	1.43771	0.718853	+	0.695162i	$0.244667\pi$
$954$	0	0
$955$	−5.95819	−0.192802
$956$	0	0
$957$	−2.57989	−0.0833960
$958$	0	0
$959$	−25.2732	−0.816114
$960$	0	0
$961$	3.14359	0.101406
$962$	0	0
$963$	16.1565	0.520635
$964$	0	0
$965$	12.7695	0.411064
$966$	0	0
$967$	24.3595	0.783348	0.391674	−	0.920104i	$-0.371896\pi$
$967$	24.3595	0.783348	0.391674	+	0.920104i	$0.371896\pi$
$968$	0	0
$969$	−10.9683	−0.352351
$970$	0	0
$971$	−1.17472	−0.0376985	−0.0188492	−	0.999822i	$-0.506000\pi$
$971$	−1.17472	−0.0376985	−0.0188492	+	0.999822i	$0.506000\pi$
$972$	0	0
$973$	−11.7557	−0.376869
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.3101	−0.393835	−0.196917	−	0.980420i	$-0.563093\pi$
$977$	−12.3101	−0.393835	−0.196917	+	0.980420i	$0.563093\pi$
$978$	0	0
$979$	−9.56873	−0.305818
$980$	0	0
$981$	45.5669	1.45484
$982$	0	0
$983$	−2.29060	−0.0730589	−0.0365295	−	0.999333i	$-0.511630\pi$
$983$	−2.29060	−0.0730589	−0.0365295	+	0.999333i	$0.511630\pi$
$984$	0	0
$985$	16.1079	0.513242
$986$	0	0
$987$	20.5131	0.652940
$988$	0	0
$989$	−73.0560	−2.32305
$990$	0	0
$991$	21.6998	0.689318	0.344659	−	0.938728i	$-0.387994\pi$
$991$	21.6998	0.689318	0.344659	+	0.938728i	$0.387994\pi$
$992$	0	0
$993$	−42.0628	−1.33482
$994$	0	0
$995$	−8.52805	−0.270357
$996$	0	0
$997$	−19.0625	−0.603716	−0.301858	−	0.953353i	$-0.597607\pi$
$997$	−19.0625	−0.603716	−0.301858	+	0.953353i	$0.597607\pi$
$998$	0	0
$999$	54.8671	1.73592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.q.1.4		4
13.5	odd	4		3380.2.f.i.3041.7		8
13.6	odd	12		260.2.x.a.101.1	✓	8
13.8	odd	4		3380.2.f.i.3041.8		8
13.11	odd	12		260.2.x.a.121.1	yes	8
13.12	even	2		3380.2.a.p.1.4		4
39.11	even	12		2340.2.dj.d.901.1		8
39.32	even	12		2340.2.dj.d.361.3		8
52.11	even	12		1040.2.da.c.641.4		8
52.19	even	12		1040.2.da.c.881.4		8
65.19	odd	12		1300.2.y.b.101.4		8
65.24	odd	12		1300.2.y.b.901.4		8
65.32	even	12		1300.2.ba.b.49.1		8
65.37	even	12		1300.2.ba.c.849.4		8
65.58	even	12		1300.2.ba.c.49.4		8
65.63	even	12		1300.2.ba.b.849.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.1	✓	8	13.6	odd	12
260.2.x.a.121.1	yes	8	13.11	odd	12
1040.2.da.c.641.4		8	52.11	even	12
1040.2.da.c.881.4		8	52.19	even	12
1300.2.y.b.101.4		8	65.19	odd	12
1300.2.y.b.901.4		8	65.24	odd	12
1300.2.ba.b.49.1		8	65.32	even	12
1300.2.ba.b.849.1		8	65.63	even	12
1300.2.ba.c.49.4		8	65.58	even	12
1300.2.ba.c.849.4		8	65.37	even	12
2340.2.dj.d.361.3		8	39.32	even	12
2340.2.dj.d.901.1		8	39.11	even	12
3380.2.a.p.1.4		4	13.12	even	2
3380.2.a.q.1.4		4	1.1	even	1	trivial
3380.2.f.i.3041.7		8	13.5	odd	4
3380.2.f.i.3041.8		8	13.8	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82684 q^{3} +1.00000 q^{5} +2.09479 q^{7} +4.99102 q^{9} +O(q^{10})\)	\(q+2.82684 q^{3} +1.00000 q^{5} +2.09479 q^{7} +4.99102 q^{9} +1.73205 q^{11} +2.82684 q^{15} -3.62828 q^{17} +1.06939 q^{19} +5.92163 q^{21} -7.81785 q^{23} +1.00000 q^{25} +5.62828 q^{27} -0.526914 q^{29} +5.84325 q^{31} +4.89623 q^{33} +2.09479 q^{35} +9.74846 q^{37} +4.26795 q^{41} +9.34477 q^{43} +4.99102 q^{45} +3.46410 q^{47} -2.61186 q^{49} -10.2566 q^{51} +12.5939 q^{53} +1.73205 q^{55} +3.02299 q^{57} +1.40370 q^{59} -11.1088 q^{61} +10.4551 q^{63} +10.8334 q^{67} -22.0998 q^{69} -14.1692 q^{71} -2.64469 q^{73} +2.82684 q^{75} +3.62828 q^{77} +13.5729 q^{79} +0.937188 q^{81} -15.7925 q^{83} -3.62828 q^{85} -1.48950 q^{87} -5.52451 q^{89} +16.5179 q^{93} +1.06939 q^{95} -15.1946 q^{97} +8.64469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9	\( 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 4 q^{9} + 2 q^{15} + 6 q^{17} + 12 q^{21} - 6 q^{23} + 4 q^{25} + 2 q^{27} + 6 q^{33} + 6 q^{35} + 18 q^{37} + 24 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{49} + 12 q^{53} - 18 q^{57} + 12 q^{59} + 4 q^{61} + 12 q^{63} + 18 q^{67} - 24 q^{69} + 12 q^{71} + 24 q^{73} + 2 q^{75} - 6 q^{77} - 8 q^{79} - 8 q^{81} - 36 q^{83} + 6 q^{85} + 6 q^{87} + 12 q^{89} + 48 q^{93} + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7 + 4 * q^9 + 2 * q^15 + 6 * q^17 + 12 * q^21 - 6 * q^23 + 4 * q^25 + 2 * q^27 + 6 * q^33 + 6 * q^35 + 18 * q^37 + 24 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^49 + 12 * q^53 - 18 * q^57 + 12 * q^59 + 4 * q^61 + 12 * q^63 + 18 * q^67 - 24 * q^69 + 12 * q^71 + 24 * q^73 + 2 * q^75 - 6 * q^77 - 8 * q^79 - 8 * q^81 - 36 * q^83 + 6 * q^85 + 6 * q^87 + 12 * q^89 + 48 * q^93 + 6 * q^97

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