LMFDB - Embedded newform 3380.2.f.i.3041.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.7
Root			$1.20036 - 0.747754i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.i.3041.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82684 q^{3} -1.00000i q^{5} +2.09479i q^{7} +4.99102 q^{9} +O(q^{10})$	$q+2.82684 q^{3} -1.00000i q^{5} +2.09479i q^{7} +4.99102 q^{9} +1.73205i q^{11} -2.82684i q^{15} +3.62828 q^{17} -1.06939i q^{19} +5.92163i q^{21} +7.81785 q^{23} -1.00000 q^{25} +5.62828 q^{27} -0.526914 q^{29} -5.84325i q^{31} +4.89623i q^{33} +2.09479 q^{35} +9.74846i q^{37} -4.26795i q^{41} -9.34477 q^{43} -4.99102i q^{45} +3.46410i q^{47} +2.61186 q^{49} +10.2566 q^{51} +12.5939 q^{53} +1.73205 q^{55} -3.02299i q^{57} +1.40370i q^{59} -11.1088 q^{61} +10.4551i q^{63} -10.8334i q^{67} +22.0998 q^{69} +14.1692i q^{71} -2.64469i q^{73} -2.82684 q^{75} -3.62828 q^{77} +13.5729 q^{79} +0.937188 q^{81} +15.7925i q^{83} -3.62828i q^{85} -1.48950 q^{87} -5.52451i q^{89} -16.5179i q^{93} -1.06939 q^{95} +15.1946i q^{97} +8.64469i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 8 * q^9	$ 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) $ 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	2.09479i	0.791755i	0.918303	+	0.395878i	$0.129560\pi$
$7$	2.09479i	0.791755i	−0.918303	+	0.395878i	$0.870440\pi$
$8$	0	0
$9$	4.99102	1.66367
$10$	0	0
$11$	1.73205i	0.522233i	0.965307	+	0.261116i	$0.0840907\pi$
$11$	1.73205i	0.522233i	−0.965307	+	0.261116i	$0.915909\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.82684i	− 0.729887i
$16$	0	0
$17$	3.62828	0.879987	0.439993	−	0.898001i	$-0.354981\pi$
$17$	3.62828	0.879987	0.439993	+	0.898001i	$0.354981\pi$
$18$	0	0
$19$	− 1.06939i	− 0.245335i	−0.992448	−	0.122667i	$-0.960855\pi$
$19$	− 1.06939i	− 0.245335i	0.992448	−	0.122667i	$-0.0391448\pi$
$20$	0	0
$21$	5.92163i	1.29220i
$22$	0	0
$23$	7.81785	1.63014	0.815068	−	0.579366i	$-0.196700\pi$
$23$	7.81785	1.63014	0.815068	+	0.579366i	$0.196700\pi$
$24$	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
$31$	− 5.84325i	− 1.04948i	−0.851263	−	0.524740i	$-0.824163\pi$
$31$	− 5.84325i	− 1.04948i	0.851263	−	0.524740i	$-0.175837\pi$
$32$	0	0
$33$	4.89623i	0.852324i
$34$	0	0
$35$	2.09479	0.354084
$36$	0	0
$37$	9.74846i	1.60264i	0.598238	+	0.801319i	$0.295868\pi$
$37$	9.74846i	1.60264i	−0.598238	+	0.801319i	$0.704132\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.26795i	− 0.666542i	−0.942831	−	0.333271i	$-0.891848\pi$
$41$	− 4.26795i	− 0.666542i	0.942831	−	0.333271i	$-0.108152\pi$
$42$	0	0
$43$	−9.34477	−1.42506	−0.712532	−	0.701640i	$-0.752452\pi$
$43$	−9.34477	−1.42506	−0.712532	+	0.701640i	$0.752452\pi$
$44$	0	0
$45$	− 4.99102i	− 0.744017i
$46$	0	0
$47$	3.46410i	0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$	3.46410i	0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	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
$57$	− 3.02299i	− 0.400405i
$58$	0	0
$59$	1.40370i	0.182746i	0.995817	+	0.0913729i	$0.0291255\pi$
$59$	1.40370i	0.182746i	−0.995817	+	0.0913729i	$0.970874\pi$
$60$	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
$63$	10.4551i	1.31722i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.8334i	− 1.32351i	−0.749719	−	0.661756i	$-0.769811\pi$
$67$	− 10.8334i	− 1.32351i	0.749719	−	0.661756i	$-0.230189\pi$
$68$	0	0
$69$	22.0998	2.66050
$70$	0	0
$71$	14.1692i	1.68157i	0.541366	+	0.840787i	$0.317907\pi$
$71$	14.1692i	1.68157i	−0.541366	+	0.840787i	$0.682093\pi$
$72$	0	0
$73$	− 2.64469i	− 0.309538i	−0.987951	−	0.154769i	$-0.950537\pi$
$73$	− 2.64469i	− 0.309538i	0.987951	−	0.154769i	$-0.0494633\pi$
$74$	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
$83$	15.7925i	1.73345i	0.498789	+	0.866724i	$0.333778\pi$
$83$	15.7925i	1.73345i	−0.498789	+	0.866724i	$0.666222\pi$
$84$	0	0
$85$	− 3.62828i	− 0.393542i
$86$	0	0
$87$	−1.48950	−0.159691
$88$	0	0
$89$	− 5.52451i	− 0.585596i	−0.956174	−	0.292798i	$-0.905414\pi$
$89$	− 5.52451i	− 0.585596i	0.956174	−	0.292798i	$-0.0945864\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 16.5179i	− 1.71283i
$94$	0	0
$95$	−1.06939	−0.109717
$96$	0	0
$97$	15.1946i	1.54278i	0.636364	+	0.771389i	$0.280438\pi$
$97$	15.1946i	1.54278i	−0.636364	+	0.771389i	$0.719562\pi$
$98$	0	0
$99$	8.64469i	0.868824i
$100$	0	0
$101$	3.66266	0.364448	0.182224	−	0.983257i	$-0.441670\pi$
$101$	3.66266	0.364448	0.182224	+	0.983257i	$0.441670\pi$
$102$	0	0
$103$	13.7804	1.35783	0.678914	−	0.734218i	$-0.262451\pi$
$103$	13.7804	1.35783	0.678914	+	0.734218i	$0.262451\pi$
$104$	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
$109$	− 9.12979i	− 0.874476i	−0.899346	−	0.437238i	$-0.855957\pi$
$109$	− 9.12979i	− 0.874476i	0.899346	−	0.437238i	$-0.144043\pi$
$110$	0	0
$111$	27.5573i	2.61563i
$112$	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
$115$	− 7.81785i	− 0.729019i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.60047i	0.696734i
$120$	0	0
$121$	8.00000	0.727273
$122$	0	0
$123$	− 12.0648i	− 1.08785i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−0.907620	−0.0805382	−0.0402691	−	0.999189i	$-0.512822\pi$
$127$	−0.907620	−0.0805382	−0.0402691	+	0.999189i	$0.512822\pi$
$128$	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
$135$	− 5.62828i	− 0.484405i
$136$	0	0
$137$	− 12.0648i	− 1.03077i	−0.856960	−	0.515383i	$-0.827650\pi$
$137$	− 12.0648i	− 1.03077i	0.856960	−	0.515383i	$-0.172350\pi$
$138$	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
$141$	9.79246i	0.824674i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.526914i	0.0437578i
$146$	0	0
$147$	7.38332	0.608966
$148$	0	0
$149$	− 20.9886i	− 1.71945i	−0.510754	−	0.859727i	$-0.670634\pi$
$149$	− 20.9886i	− 1.71945i	0.510754	−	0.859727i	$-0.329366\pi$
$150$	0	0
$151$	6.99102i	0.568921i	0.958688	+	0.284460i	$0.0918144\pi$
$151$	6.99102i	0.568921i	−0.958688	+	0.284460i	$0.908186\pi$
$152$	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
$161$	16.3767i	1.29067i
$162$	0	0
$163$	6.37830i	0.499587i	0.968299	+	0.249793i	$0.0803627\pi$
$163$	6.37830i	0.499587i	−0.968299	+	0.249793i	$0.919637\pi$
$164$	0	0
$165$	4.89623	0.381171
$166$	0	0
$167$	− 15.5289i	− 1.20166i	−0.799376	−	0.600831i	$-0.794836\pi$
$167$	− 15.5289i	− 1.20166i	0.799376	−	0.600831i	$-0.205164\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 5.33734i	− 0.408156i
$172$	0	0
$173$	4.77604	0.363116	0.181558	−	0.983380i	$-0.441886\pi$
$173$	4.77604	0.363116	0.181558	+	0.983380i	$0.441886\pi$
$174$	0	0
$175$	− 2.09479i	− 0.158351i
$176$	0	0
$177$	3.96802i	0.298255i
$178$	0	0
$179$	−15.7027	−1.17367	−0.586837	−	0.809705i	$-0.699627\pi$
$179$	−15.7027	−1.17367	−0.586837	+	0.809705i	$0.699627\pi$
$180$	0	0
$181$	10.8851	0.809080	0.404540	−	0.914520i	$-0.367432\pi$
$181$	10.8851	0.809080	0.404540	+	0.914520i	$0.367432\pi$
$182$	0	0
$183$	−31.4028	−2.32136
$184$	0	0
$185$	9.74846	0.716721
$186$	0	0
$187$	6.28436i	0.459558i
$188$	0	0
$189$	11.7900i	0.857600i
$190$	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
$193$	12.7695i	0.919166i	0.888135	+	0.459583i	$0.152001\pi$
$193$	12.7695i	0.919166i	−0.888135	+	0.459583i	$0.847999\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.1079i	− 1.14764i	−0.818980	−	0.573822i	$-0.805460\pi$
$197$	− 16.1079i	− 1.14764i	0.818980	−	0.573822i	$-0.194540\pi$
$198$	0	0
$199$	8.52805	0.604538	0.302269	−	0.953223i	$-0.402256\pi$
$199$	8.52805	0.604538	0.302269	+	0.953223i	$0.402256\pi$
$200$	0	0
$201$	− 30.6243i	− 2.16007i
$202$	0	0
$203$	− 1.10377i	− 0.0774696i
$204$	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
$213$	40.0540i	2.74446i
$214$	0	0
$215$	9.34477i	0.637308i
$216$	0	0
$217$	12.2404	0.830931
$218$	0	0
$219$	− 7.47612i	− 0.505189i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 10.4991i	− 0.703072i	−0.936174	−	0.351536i	$-0.885659\pi$
$223$	− 10.4991i	− 0.703072i	0.936174	−	0.351536i	$-0.114341\pi$
$224$	0	0
$225$	−4.99102	−0.332734
$226$	0	0
$227$	− 15.5858i	− 1.03446i	−0.855845	−	0.517232i	$-0.826963\pi$
$227$	− 15.5858i	− 1.03446i	0.855845	−	0.517232i	$-0.173037\pi$
$228$	0	0
$229$	− 19.2714i	− 1.27349i	−0.771074	−	0.636745i	$-0.780280\pi$
$229$	− 19.2714i	− 1.27349i	0.771074	−	0.636745i	$-0.219720\pi$
$230$	0	0
$231$	−10.2566	−0.674832
$232$	0	0
$233$	2.48794	0.162991	0.0814953	−	0.996674i	$-0.474030\pi$
$233$	2.48794	0.162991	0.0814953	+	0.996674i	$0.474030\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	0	0
$237$	38.3684	2.49229
$238$	0	0
$239$	− 16.7775i	− 1.08525i	−0.839976	−	0.542624i	$-0.817431\pi$
$239$	− 16.7775i	− 1.08525i	0.839976	−	0.542624i	$-0.182569\pi$
$240$	0	0
$241$	− 29.0794i	− 1.87317i	−0.350439	−	0.936585i	$-0.613968\pi$
$241$	− 29.0794i	− 1.87317i	0.350439	−	0.936585i	$-0.386032\pi$
$242$	0	0
$243$	−14.2356	−0.913211
$244$	0	0
$245$	− 2.61186i	− 0.166866i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	44.6427i	2.82912i
$250$	0	0
$251$	−19.1208	−1.20689	−0.603447	−	0.797403i	$-0.706207\pi$
$251$	−19.1208	−1.20689	−0.603447	+	0.797403i	$0.706207\pi$
$252$	0	0
$253$	13.5409i	0.851310i
$254$	0	0
$255$	− 10.2566i	− 0.642291i
$256$	0	0
$257$	−13.7091	−0.855148	−0.427574	−	0.903980i	$-0.640632\pi$
$257$	−13.7091	−0.855148	−0.427574	+	0.903980i	$0.640632\pi$
$258$	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
$265$	− 12.5939i	− 0.773637i
$266$	0	0
$267$	− 15.6169i	− 0.955738i
$268$	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
$271$	1.95161i	0.118552i	0.998242	+	0.0592760i	$0.0188792\pi$
$271$	1.95161i	0.118552i	−0.998242	+	0.0592760i	$0.981121\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.73205i	− 0.104447i
$276$	0	0
$277$	−12.4805	−0.749881	−0.374941	−	0.927049i	$-0.622337\pi$
$277$	−12.4805	−0.749881	−0.374941	+	0.927049i	$0.622337\pi$
$278$	0	0
$279$	− 29.1638i	− 1.74599i
$280$	0	0
$281$	2.29553i	0.136940i	0.997653	+	0.0684698i	$0.0218117\pi$
$281$	2.29553i	0.136940i	−0.997653	+	0.0684698i	$0.978188\pi$
$282$	0	0
$283$	−14.9297	−0.887477	−0.443739	−	0.896156i	$-0.646348\pi$
$283$	−14.9297	−0.887477	−0.443739	+	0.896156i	$0.646348\pi$
$284$	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
$291$	42.9527i	2.51793i
$292$	0	0
$293$	− 1.43213i	− 0.0836657i	−0.999125	−	0.0418329i	$-0.986680\pi$
$293$	− 1.43213i	− 0.0836657i	0.999125	−	0.0418329i	$-0.0133197\pi$
$294$	0	0
$295$	1.40370	0.0817264
$296$	0	0
$297$	9.74846i	0.565663i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 19.5753i	− 1.12830i
$302$	0	0
$303$	10.3538	0.594808
$304$	0	0
$305$	11.1088i	0.636088i
$306$	0	0
$307$	− 17.3833i	− 0.992118i	−0.868289	−	0.496059i	$-0.834780\pi$
$307$	− 17.3833i	− 0.992118i	0.868289	−	0.496059i	$-0.165220\pi$
$308$	0	0
$309$	38.9551	2.21608
$310$	0	0
$311$	−20.2164	−1.14637	−0.573185	−	0.819426i	$-0.694292\pi$
$311$	−20.2164	−1.14637	−0.573185	+	0.819426i	$0.694292\pi$
$312$	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
$317$	23.6177i	1.32650i	0.748396	+	0.663252i	$0.230824\pi$
$317$	23.6177i	1.32650i	−0.748396	+	0.663252i	$0.769176\pi$
$318$	0	0
$319$	− 0.912641i	− 0.0510981i
$320$	0	0
$321$	9.15079	0.510748
$322$	0	0
$323$	− 3.88004i	− 0.215891i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 25.8085i	− 1.42721i
$328$	0	0
$329$	−7.25656	−0.400067
$330$	0	0
$331$	14.8798i	0.817869i	0.912564	+	0.408934i	$0.134099\pi$
$331$	14.8798i	0.817869i	−0.912564	+	0.408934i	$0.865901\pi$
$332$	0	0
$333$	48.6547i	2.66626i
$334$	0	0
$335$	−10.8334	−0.591893
$336$	0	0
$337$	−29.1906	−1.59012	−0.795058	−	0.606534i	$-0.792559\pi$
$337$	−29.1906	−1.59012	−0.795058	+	0.606534i	$0.792559\pi$
$338$	0	0
$339$	−30.9641	−1.68174
$340$	0	0
$341$	10.1208	0.548073
$342$	0	0
$343$	20.1348i	1.08718i
$344$	0	0
$345$	− 22.0998i	− 1.18981i
$346$	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
$349$	− 12.8815i	− 0.689532i	−0.938689	−	0.344766i	$-0.887958\pi$
$349$	− 12.8815i	− 0.689532i	0.938689	−	0.344766i	$-0.112042\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.2167i	0.863130i	0.902082	+	0.431565i	$0.142038\pi$
$353$	16.2167i	0.863130i	−0.902082	+	0.431565i	$0.857962\pi$
$354$	0	0
$355$	14.1692	0.752023
$356$	0	0
$357$	21.4853i	1.13712i
$358$	0	0
$359$	− 9.19261i	− 0.485167i	−0.970131	−	0.242584i	$-0.922005\pi$
$359$	− 9.19261i	− 0.485167i	0.970131	−	0.242584i	$-0.0779949\pi$
$360$	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
$369$	− 21.3014i	− 1.10891i
$370$	0	0
$371$	26.3815i	1.36966i
$372$	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
$375$	2.82684i	0.145977i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8.02872i	0.412407i	0.978509	+	0.206204i	$0.0661110\pi$
$379$	8.02872i	0.412407i	−0.978509	+	0.206204i	$0.933889\pi$
$380$	0	0
$381$	−2.56569	−0.131445
$382$	0	0
$383$	30.1425i	1.54021i	0.637918	+	0.770104i	$0.279796\pi$
$383$	30.1425i	1.54021i	−0.637918	+	0.770104i	$0.720204\pi$
$384$	0	0
$385$	3.62828i	0.184914i
$386$	0	0
$387$	−46.6399	−2.37084
$388$	0	0
$389$	−35.8264	−1.81647	−0.908236	−	0.418459i	$-0.862570\pi$
$389$	−35.8264	−1.81647	−0.908236	+	0.418459i	$0.862570\pi$
$390$	0	0
$391$	28.3654	1.43450
$392$	0	0
$393$	−37.2086	−1.87693
$394$	0	0
$395$	− 13.5729i	− 0.682926i
$396$	0	0
$397$	15.3830i	0.772052i	0.922488	+	0.386026i	$0.126153\pi$
$397$	15.3830i	0.772052i	−0.922488	+	0.386026i	$0.873847\pi$
$398$	0	0
$399$	6.33252	0.317023
$400$	0	0
$401$	− 9.77689i	− 0.488235i	−0.969746	−	0.244117i	$-0.921502\pi$
$401$	− 9.77689i	− 0.488235i	0.969746	−	0.244117i	$-0.0784982\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 0.937188i	− 0.0465692i
$406$	0	0
$407$	−16.8848	−0.836950
$408$	0	0
$409$	− 1.00658i	− 0.0497720i	−0.999690	−	0.0248860i	$-0.992078\pi$
$409$	− 1.00658i	− 0.0497720i	0.999690	−	0.0248860i	$-0.00792228\pi$
$410$	0	0
$411$	− 34.1052i	− 1.68229i
$412$	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
$421$	− 14.2955i	− 0.696721i	−0.937361	−	0.348361i	$-0.886738\pi$
$421$	− 14.2955i	− 0.696721i	0.937361	−	0.348361i	$-0.113262\pi$
$422$	0	0
$423$	17.2894i	0.840639i
$424$	0	0
$425$	−3.62828	−0.175997
$426$	0	0
$427$	− 23.2706i	− 1.12614i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.1471i	0.585107i	0.956249	+	0.292554i	$0.0945050\pi$
$431$	12.1471i	0.585107i	−0.956249	+	0.292554i	$0.905495\pi$
$432$	0	0
$433$	0.209020	0.0100448	0.00502242	−	0.999987i	$-0.498401\pi$
$433$	0.209020	0.0100448	0.00502242	+	0.999987i	$0.498401\pi$
$434$	0	0
$435$	1.48950i	0.0714161i
$436$	0	0
$437$	− 8.36033i	− 0.399929i
$438$	0	0
$439$	27.7207	1.32303	0.661517	−	0.749930i	$-0.269913\pi$
$439$	27.7207	1.32303	0.661517	+	0.749930i	$0.269913\pi$
$440$	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
$447$	− 59.3314i	− 2.80628i
$448$	0	0
$449$	− 7.13220i	− 0.336589i	−0.985737	−	0.168295i	$-0.946174\pi$
$449$	− 7.13220i	− 0.336589i	0.985737	−	0.168295i	$-0.0538260\pi$
$450$	0	0
$451$	7.39230	0.348090
$452$	0	0
$453$	19.7625i	0.928522i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 31.6808i	− 1.48197i	−0.671523	−	0.740984i	$-0.734360\pi$
$457$	− 31.6808i	− 1.48197i	0.671523	−	0.740984i	$-0.265640\pi$
$458$	0	0
$459$	20.4210	0.953169
$460$	0	0
$461$	11.8529i	0.552043i	0.961151	+	0.276021i	$0.0890161\pi$
$461$	11.8529i	0.552043i	−0.961151	+	0.276021i	$0.910984\pi$
$462$	0	0
$463$	12.7655i	0.593263i	0.954992	+	0.296632i	$0.0958633\pi$
$463$	12.7655i	0.593263i	−0.954992	+	0.296632i	$0.904137\pi$
$464$	0	0
$465$	−16.5179	−0.766001
$466$	0	0
$467$	31.3402	1.45025	0.725125	−	0.688617i	$-0.241782\pi$
$467$	31.3402	1.45025	0.725125	+	0.688617i	$0.241782\pi$
$468$	0	0
$469$	22.6937	1.04790
$470$	0	0
$471$	10.5939	0.488141
$472$	0	0
$473$	− 16.1856i	− 0.744215i
$474$	0	0
$475$	1.06939i	0.0490669i
$476$	0	0
$477$	62.8563	2.87799
$478$	0	0
$479$	16.1261i	0.736818i	0.929664	+	0.368409i	$0.120097\pi$
$479$	16.1261i	0.736818i	−0.929664	+	0.368409i	$0.879903\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	46.2944i	2.10647i
$484$	0	0
$485$	15.1946	0.689951
$486$	0	0
$487$	5.59001i	0.253308i	0.991947	+	0.126654i	$0.0404237\pi$
$487$	5.59001i	0.253308i	−0.991947	+	0.126654i	$0.959576\pi$
$488$	0	0
$489$	18.0304i	0.815364i
$490$	0	0
$491$	18.2954	0.825662	0.412831	−	0.910808i	$-0.364540\pi$
$491$	18.2954	0.825662	0.412831	+	0.910808i	$0.364540\pi$
$492$	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
$499$	− 0.553868i	− 0.0247945i	−0.999923	−	0.0123973i	$-0.996054\pi$
$499$	− 0.553868i	− 0.0247945i	0.999923	−	0.0123973i	$-0.00394627\pi$
$500$	0	0
$501$	− 43.8977i	− 1.96120i
$502$	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
$505$	− 3.66266i	− 0.162986i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.92295i	0.306855i	0.988160	+	0.153427i	$0.0490311\pi$
$509$	6.92295i	0.306855i	−0.988160	+	0.153427i	$0.950969\pi$
$510$	0	0
$511$	5.54007	0.245078
$512$	0	0
$513$	− 6.01882i	− 0.265737i
$514$	0	0
$515$	− 13.7804i	− 0.607239i
$516$	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
$525$	− 5.92163i	− 0.258441i
$526$	0	0
$527$	− 21.2009i	− 0.923528i
$528$	0	0
$529$	38.1188	1.65734
$530$	0	0
$531$	7.00587i	0.304029i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 3.23711i	− 0.139953i
$536$	0	0
$537$	−44.3890	−1.91553
$538$	0	0
$539$	4.52388i	0.194857i
$540$	0	0
$541$	31.8881i	1.37098i	0.728084	+	0.685488i	$0.240411\pi$
$541$	31.8881i	1.37098i	−0.728084	+	0.685488i	$0.759589\pi$
$542$	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
$551$	0.563476i	0.0240049i
$552$	0	0
$553$	28.4323i	1.20907i
$554$	0	0
$555$	27.5573	1.16974
$556$	0	0
$557$	26.3064i	1.11464i	0.830298	+	0.557319i	$0.188170\pi$
$557$	26.3064i	1.11464i	−0.830298	+	0.557319i	$0.811830\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	17.7649i	0.750034i
$562$	0	0
$563$	5.78984	0.244013	0.122006	−	0.992529i	$-0.461067\pi$
$563$	5.78984	0.244013	0.122006	+	0.992529i	$0.461067\pi$
$564$	0	0
$565$	10.9536i	0.460821i
$566$	0	0
$567$	1.96321i	0.0824471i
$568$	0	0
$569$	−23.6368	−0.990905	−0.495452	−	0.868635i	$-0.664998\pi$
$569$	−23.6368	−0.990905	−0.495452	+	0.868635i	$0.664998\pi$
$570$	0	0
$571$	−11.4641	−0.479758	−0.239879	−	0.970803i	$-0.577108\pi$
$571$	−11.4641	−0.479758	−0.239879	+	0.970803i	$0.577108\pi$
$572$	0	0
$573$	−16.8428	−0.703620
$574$	0	0
$575$	−7.81785	−0.326027
$576$	0	0
$577$	− 34.2415i	− 1.42549i	−0.701422	−	0.712746i	$-0.747451\pi$
$577$	− 34.2415i	− 1.42549i	0.701422	−	0.712746i	$-0.252549\pi$
$578$	0	0
$579$	36.0972i	1.50015i
$580$	0	0
$581$	−33.0818	−1.37247
$582$	0	0
$583$	21.8133i	0.903413i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.63599i	− 0.232622i	−0.993213	−	0.116311i	$-0.962893\pi$
$587$	− 5.63599i	− 0.232622i	0.993213	−	0.116311i	$-0.0371070\pi$
$588$	0	0
$589$	−6.24871	−0.257474
$590$	0	0
$591$	− 45.5345i	− 1.87304i
$592$	0	0
$593$	− 15.3014i	− 0.628353i	−0.949365	−	0.314177i	$-0.898272\pi$
$593$	− 15.3014i	− 0.628353i	0.949365	−	0.314177i	$-0.101728\pi$
$594$	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
$603$	− 54.0697i	− 2.20189i
$604$	0	0
$605$	− 8.00000i	− 0.325246i
$606$	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
$609$	− 3.12019i	− 0.126436i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 16.6455i	− 0.672307i	−0.941807	−	0.336154i	$-0.890874\pi$
$613$	− 16.6455i	− 0.672307i	0.941807	−	0.336154i	$-0.109126\pi$
$614$	0	0
$615$	−12.0648	−0.486500
$616$	0	0
$617$	1.61564i	0.0650432i	0.999471	+	0.0325216i	$0.0103538\pi$
$617$	1.61564i	0.0650432i	−0.999471	+	0.0325216i	$0.989646\pi$
$618$	0	0
$619$	23.3922i	0.940213i	0.882610	+	0.470106i	$0.155784\pi$
$619$	23.3922i	0.940213i	−0.882610	+	0.470106i	$0.844216\pi$
$620$	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
$629$	35.3701i	1.41030i
$630$	0	0
$631$	− 3.20381i	− 0.127542i	−0.997965	−	0.0637708i	$-0.979687\pi$
$631$	− 3.20381i	− 0.127542i	0.997965	−	0.0637708i	$-0.0203127\pi$
$632$	0	0
$633$	−11.8179	−0.469718
$634$	0	0
$635$	0.907620i	0.0360178i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	70.7187i	2.79759i
$640$	0	0
$641$	19.9759	0.789000	0.394500	−	0.918896i	$-0.370918\pi$
$641$	19.9759	0.789000	0.394500	+	0.918896i	$0.370918\pi$
$642$	0	0
$643$	28.5368i	1.12538i	0.826668	+	0.562690i	$0.190233\pi$
$643$	28.5368i	1.12538i	−0.826668	+	0.562690i	$0.809767\pi$
$644$	0	0
$645$	26.4161i	1.04013i
$646$	0	0
$647$	−6.08651	−0.239285	−0.119643	−	0.992817i	$-0.538175\pi$
$647$	−6.08651	−0.239285	−0.119643	+	0.992817i	$0.538175\pi$
$648$	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
$655$	13.1626i	0.514306i
$656$	0	0
$657$	− 13.1997i	− 0.514969i
$658$	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
$661$	34.7259i	1.35068i	0.737506	+	0.675341i	$0.236003\pi$
$661$	34.7259i	1.35068i	−0.737506	+	0.675341i	$0.763997\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.24014i	− 0.0868690i
$666$	0	0
$667$	−4.11933	−0.159501
$668$	0	0
$669$	− 29.6793i	− 1.14747i
$670$	0	0
$671$	− 19.2410i	− 0.742790i
$672$	0	0
$673$	34.0103	1.31100	0.655500	−	0.755195i	$-0.272458\pi$
$673$	34.0103	1.31100	0.655500	+	0.755195i	$0.272458\pi$
$674$	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
$681$	− 44.0584i	− 1.68832i
$682$	0	0
$683$	− 27.8573i	− 1.06593i	−0.846138	−	0.532964i	$-0.821078\pi$
$683$	− 27.8573i	− 1.06593i	0.846138	−	0.532964i	$-0.178922\pi$
$684$	0	0
$685$	−12.0648	−0.460972
$686$	0	0
$687$	− 54.4772i	− 2.07843i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	36.3211i	1.38172i	0.722988	+	0.690860i	$0.242768\pi$
$691$	36.3211i	1.38172i	−0.722988	+	0.690860i	$0.757232\pi$
$692$	0	0
$693$	−18.1088	−0.687896
$694$	0	0
$695$	5.61186i	0.212870i
$696$	0	0
$697$	− 15.4853i	− 0.586548i
$698$	0	0
$699$	7.03302	0.266013
$700$	0	0
$701$	2.16156	0.0816411	0.0408206	−	0.999166i	$-0.487003\pi$
$701$	2.16156	0.0816411	0.0408206	+	0.999166i	$0.487003\pi$
$702$	0	0
$703$	10.4249	0.393183
$704$	0	0
$705$	9.79246	0.368805
$706$	0	0
$707$	7.67250i	0.288554i
$708$	0	0
$709$	47.6165i	1.78827i	0.447793	+	0.894137i	$0.352210\pi$
$709$	47.6165i	1.78827i	−0.447793	+	0.894137i	$0.647790\pi$
$710$	0	0
$711$	67.7425	2.54054
$712$	0	0
$713$	− 45.6817i	− 1.71079i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 47.4273i	− 1.77121i
$718$	0	0
$719$	20.2937	0.756829	0.378414	−	0.925636i	$-0.376469\pi$
$719$	20.2937	0.756829	0.378414	+	0.925636i	$0.376469\pi$
$720$	0	0
$721$	28.8671i	1.07507i
$722$	0	0
$723$	− 82.2029i	− 3.05716i
$724$	0	0
$725$	0.526914	0.0195691
$726$	0	0
$727$	11.0681	0.410494	0.205247	−	0.978710i	$-0.434200\pi$
$727$	11.0681	0.410494	0.205247	+	0.978710i	$0.434200\pi$
$728$	0	0
$729$	−43.0532	−1.59456
$730$	0	0
$731$	−33.9054	−1.25404
$732$	0	0
$733$	− 23.4002i	− 0.864304i	−0.901801	−	0.432152i	$-0.857754\pi$
$733$	− 23.4002i	− 0.864304i	0.901801	−	0.432152i	$-0.142246\pi$
$734$	0	0
$735$	− 7.38332i	− 0.272338i
$736$	0	0
$737$	18.7640	0.691182
$738$	0	0
$739$	− 25.5902i	− 0.941349i	−0.882307	−	0.470675i	$-0.844011\pi$
$739$	− 25.5902i	− 0.941349i	0.882307	−	0.470675i	$-0.155989\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.41424i	0.308688i	0.988017	+	0.154344i	$0.0493265\pi$
$743$	8.41424i	0.308688i	−0.988017	+	0.154344i	$0.950674\pi$
$744$	0	0
$745$	−20.9886	−0.768963
$746$	0	0
$747$	78.8204i	2.88389i
$748$	0	0
$749$	6.78106i	0.247775i
$750$	0	0
$751$	−40.5190	−1.47856	−0.739279	−	0.673399i	$-0.764834\pi$
$751$	−40.5190	−1.47856	−0.739279	+	0.673399i	$0.764834\pi$
$752$	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
$759$	38.2780i	1.38940i
$760$	0	0
$761$	6.77075i	0.245440i	0.992441	+	0.122720i	$0.0391616\pi$
$761$	6.77075i	0.245440i	−0.992441	+	0.122720i	$0.960838\pi$
$762$	0	0
$763$	19.1250	0.692371
$764$	0	0
$765$	− 18.1088i	− 0.654725i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 23.1591i	− 0.835138i	−0.908645	−	0.417569i	$-0.862882\pi$
$769$	− 23.1591i	− 0.835138i	0.908645	−	0.417569i	$-0.137118\pi$
$770$	0	0
$771$	−38.7533	−1.39567
$772$	0	0
$773$	− 41.9540i	− 1.50898i	−0.656311	−	0.754491i	$-0.727884\pi$
$773$	− 41.9540i	− 1.50898i	0.656311	−	0.754491i	$-0.272116\pi$
$774$	0	0
$775$	5.84325i	0.209896i
$776$	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
$785$	− 3.74761i	− 0.133758i
$786$	0	0
$787$	− 37.4153i	− 1.33371i	−0.745187	−	0.666856i	$-0.767640\pi$
$787$	− 37.4153i	− 1.33371i	0.745187	−	0.666856i	$-0.232360\pi$
$788$	0	0
$789$	−33.6775	−1.19895
$790$	0	0
$791$	− 22.9455i	− 0.815847i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 35.6009i	− 1.26263i
$796$	0	0
$797$	−47.2400	−1.67333	−0.836663	−	0.547719i	$-0.815496\pi$
$797$	−47.2400	−1.67333	−0.836663	+	0.547719i	$0.815496\pi$
$798$	0	0
$799$	12.5687i	0.444650i
$800$	0	0
$801$	− 27.5729i	− 0.974240i
$802$	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
$811$	11.4041i	0.400453i	0.979750	+	0.200227i	$0.0641678\pi$
$811$	11.4041i	0.400453i	−0.979750	+	0.200227i	$0.935832\pi$
$812$	0	0
$813$	5.51689i	0.193486i
$814$	0	0
$815$	6.37830	0.223422
$816$	0	0
$817$	9.99319i	0.349618i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.5615i	− 0.857202i	−0.903494	−	0.428601i	$-0.859007\pi$
$821$	− 24.5615i	− 0.857202i	0.903494	−	0.428601i	$-0.140993\pi$
$822$	0	0
$823$	−2.96448	−0.103335	−0.0516676	−	0.998664i	$-0.516454\pi$
$823$	−2.96448	−0.103335	−0.0516676	+	0.998664i	$0.516454\pi$
$824$	0	0
$825$	− 4.89623i	− 0.170465i
$826$	0	0
$827$	17.7265i	0.616412i	0.951320	+	0.308206i	$0.0997286\pi$
$827$	17.7265i	0.616412i	−0.951320	+	0.308206i	$0.900271\pi$
$828$	0	0
$829$	11.5758	0.402045	0.201022	−	0.979587i	$-0.435574\pi$
$829$	11.5758	0.402045	0.201022	+	0.979587i	$0.435574\pi$
$830$	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
$837$	− 32.8874i	− 1.13676i
$838$	0	0
$839$	− 42.8795i	− 1.48037i	−0.672405	−	0.740183i	$-0.734739\pi$
$839$	− 42.8795i	− 1.48037i	0.672405	−	0.740183i	$-0.265261\pi$
$840$	0	0
$841$	−28.7224	−0.990426
$842$	0	0
$843$	6.48908i	0.223496i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16.7583i	0.575822i
$848$	0	0
$849$	−42.2038	−1.44843
$850$	0	0
$851$	76.2121i	2.61252i
$852$	0	0
$853$	− 13.4599i	− 0.460857i	−0.973089	−	0.230428i	$-0.925987\pi$
$853$	− 13.4599i	− 0.460857i	0.973089	−	0.230428i	$-0.0740127\pi$
$854$	0	0
$855$	−5.33734	−0.182533
$856$	0	0
$857$	−10.5950	−0.361919	−0.180960	−	0.983491i	$-0.557920\pi$
$857$	−10.5950	−0.361919	−0.180960	+	0.983491i	$0.557920\pi$
$858$	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
$863$	30.8640i	1.05062i	0.850910	+	0.525311i	$0.176051\pi$
$863$	30.8640i	1.05062i	−0.850910	+	0.525311i	$0.823949\pi$
$864$	0	0
$865$	− 4.77604i	− 0.162390i
$866$	0	0
$867$	−10.8426	−0.368234
$868$	0	0
$869$	23.5089i	0.797486i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	75.8365i	2.56668i
$874$	0	0
$875$	−2.09479	−0.0708167
$876$	0	0
$877$	− 20.1037i	− 0.678853i	−0.940633	−	0.339427i	$-0.889767\pi$
$877$	− 20.1037i	− 0.678853i	0.940633	−	0.339427i	$-0.110233\pi$
$878$	0	0
$879$	− 4.04839i	− 0.136549i
$880$	0	0
$881$	−3.13396	−0.105586	−0.0527930	−	0.998605i	$-0.516812\pi$
$881$	−3.13396	−0.105586	−0.0527930	+	0.998605i	$0.516812\pi$
$882$	0	0
$883$	−34.0429	−1.14563	−0.572817	−	0.819683i	$-0.694149\pi$
$883$	−34.0429	−1.14563	−0.572817	+	0.819683i	$0.694149\pi$
$884$	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
$889$	− 1.90127i	− 0.0637666i
$890$	0	0
$891$	1.62326i	0.0543812i
$892$	0	0
$893$	3.70447	0.123965
$894$	0	0
$895$	15.7027i	0.524883i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.07889i	0.102687i
$900$	0	0
$901$	45.6942	1.52229
$902$	0	0
$903$	− 55.3362i	− 1.84147i
$904$	0	0
$905$	− 10.8851i	− 0.361832i
$906$	0	0
$907$	50.9605	1.69212	0.846058	−	0.533090i	$-0.178969\pi$
$907$	50.9605	1.69212	0.846058	+	0.533090i	$0.178969\pi$
$908$	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
$915$	31.4028i	1.03814i
$916$	0	0
$917$	− 27.5729i	− 0.910537i
$918$	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
$921$	− 49.1398i	− 1.61921i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 9.74846i	− 0.320528i
$926$	0	0
$927$	68.7784	2.25898
$928$	0	0
$929$	13.3290i	0.437310i	0.975802	+	0.218655i	$0.0701669\pi$
$929$	13.3290i	0.437310i	−0.975802	+	0.218655i	$0.929833\pi$
$930$	0	0
$931$	− 2.79310i	− 0.0915402i
$932$	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
$941$	− 49.0399i	− 1.59866i	−0.600895	−	0.799328i	$-0.705189\pi$
$941$	− 49.0399i	− 1.59866i	0.600895	−	0.799328i	$-0.294811\pi$
$942$	0	0
$943$	− 33.3662i	− 1.08655i
$944$	0	0
$945$	11.7900	0.383530
$946$	0	0
$947$	− 17.9352i	− 0.582816i	−0.956599	−	0.291408i	$-0.905876\pi$
$947$	− 17.9352i	− 0.582816i	0.956599	−	0.291408i	$-0.0941237\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	66.7635i	2.16496i
$952$	0	0
$953$	−44.3830	−1.43771	−0.718853	−	0.695162i	$-0.755333\pi$
$953$	−44.3830	−1.43771	−0.718853	+	0.695162i	$0.755333\pi$
$954$	0	0
$955$	5.95819i	0.192802i
$956$	0	0
$957$	− 2.57989i	− 0.0833960i
$958$	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
$967$	− 24.3595i	− 0.783348i	−0.920104	−	0.391674i	$-0.871896\pi$
$967$	− 24.3595i	− 0.783348i	0.920104	−	0.391674i	$-0.128104\pi$
$968$	0	0
$969$	− 10.9683i	− 0.352351i
$970$	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
$973$	− 11.7557i	− 0.376869i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.3101i	0.393835i	0.980420	+	0.196917i	$0.0630931\pi$
$977$	12.3101i	0.393835i	−0.980420	+	0.196917i	$0.936907\pi$
$978$	0	0
$979$	9.56873	0.305818
$980$	0	0
$981$	− 45.5669i	− 1.45484i
$982$	0	0
$983$	− 2.29060i	− 0.0730589i	−0.999333	−	0.0365295i	$-0.988370\pi$
$983$	− 2.29060i	− 0.0730589i	0.999333	−	0.0365295i	$-0.0116303\pi$
$984$	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
$993$	42.0628i	1.33482i
$994$	0	0
$995$	− 8.52805i	− 0.270357i
$996$	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
$999$	54.8671i	1.73592i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.i.3041.7		8
13.5	odd	4		3380.2.a.p.1.4		4
13.8	odd	4		3380.2.a.q.1.4		4
13.9	even	3		260.2.x.a.101.1	✓	8
13.10	even	6		260.2.x.a.121.1	yes	8
13.12	even	2	inner	3380.2.f.i.3041.8		8
39.23	odd	6		2340.2.dj.d.901.1		8
39.35	odd	6		2340.2.dj.d.361.3		8
52.23	odd	6		1040.2.da.c.641.4		8
52.35	odd	6		1040.2.da.c.881.4		8
65.9	even	6		1300.2.y.b.101.4		8
65.22	odd	12		1300.2.ba.b.49.1		8
65.23	odd	12		1300.2.ba.b.849.1		8
65.48	odd	12		1300.2.ba.c.49.4		8
65.49	even	6		1300.2.y.b.901.4		8
65.62	odd	12		1300.2.ba.c.849.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.1	✓	8	13.9	even	3
260.2.x.a.121.1	yes	8	13.10	even	6
1040.2.da.c.641.4		8	52.23	odd	6
1040.2.da.c.881.4		8	52.35	odd	6
1300.2.y.b.101.4		8	65.9	even	6
1300.2.y.b.901.4		8	65.49	even	6
1300.2.ba.b.49.1		8	65.22	odd	12
1300.2.ba.b.849.1		8	65.23	odd	12
1300.2.ba.c.49.4		8	65.48	odd	12
1300.2.ba.c.849.4		8	65.62	odd	12
2340.2.dj.d.361.3		8	39.35	odd	6
2340.2.dj.d.901.1		8	39.23	odd	6
3380.2.a.p.1.4		4	13.5	odd	4
3380.2.a.q.1.4		4	13.8	odd	4
3380.2.f.i.3041.7		8	1.1	even	1	trivial
3380.2.f.i.3041.8		8	13.12	even	2	inner

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.82684 q^{3} -1.00000i q^{5} +2.09479i q^{7} +4.99102 q^{9} +O(q^{10})\)	\(q+2.82684 q^{3} -1.00000i q^{5} +2.09479i q^{7} +4.99102 q^{9} +1.73205i q^{11} -2.82684i q^{15} +3.62828 q^{17} -1.06939i q^{19} +5.92163i q^{21} +7.81785 q^{23} -1.00000 q^{25} +5.62828 q^{27} -0.526914 q^{29} -5.84325i q^{31} +4.89623i q^{33} +2.09479 q^{35} +9.74846i q^{37} -4.26795i q^{41} -9.34477 q^{43} -4.99102i q^{45} +3.46410i q^{47} +2.61186 q^{49} +10.2566 q^{51} +12.5939 q^{53} +1.73205 q^{55} -3.02299i q^{57} +1.40370i q^{59} -11.1088 q^{61} +10.4551i q^{63} -10.8334i q^{67} +22.0998 q^{69} +14.1692i q^{71} -2.64469i q^{73} -2.82684 q^{75} -3.62828 q^{77} +13.5729 q^{79} +0.937188 q^{81} +15.7925i q^{83} -3.62828i q^{85} -1.48950 q^{87} -5.52451i q^{89} -16.5179i q^{93} -1.06939 q^{95} +15.1946i q^{97} +8.64469i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 8 * q^9	\( 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) \) 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more