LMFDB - Embedded newform 4014.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 $ x^7 - 18x^5 - 8x^4 + 51x^3 + 47x^2 - 2*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.369925$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.69148 q^{5} +2.17419 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.69148 q^{5} +2.17419 q^{7} -1.00000 q^{8} +2.69148 q^{10} +2.79991 q^{11} +5.60736 q^{13} -2.17419 q^{14} +1.00000 q^{16} +4.05158 q^{17} +3.43987 q^{19} -2.69148 q^{20} -2.79991 q^{22} +0.907338 q^{23} +2.24408 q^{25} -5.60736 q^{26} +2.17419 q^{28} -3.53306 q^{29} +2.49443 q^{31} -1.00000 q^{32} -4.05158 q^{34} -5.85180 q^{35} +4.44606 q^{37} -3.43987 q^{38} +2.69148 q^{40} +10.8990 q^{41} -7.91186 q^{43} +2.79991 q^{44} -0.907338 q^{46} +13.4073 q^{47} -2.27289 q^{49} -2.24408 q^{50} +5.60736 q^{52} -13.1947 q^{53} -7.53592 q^{55} -2.17419 q^{56} +3.53306 q^{58} -11.3491 q^{59} +0.466941 q^{61} -2.49443 q^{62} +1.00000 q^{64} -15.0921 q^{65} +6.52324 q^{67} +4.05158 q^{68} +5.85180 q^{70} -2.84960 q^{71} +4.29281 q^{73} -4.44606 q^{74} +3.43987 q^{76} +6.08755 q^{77} -7.97329 q^{79} -2.69148 q^{80} -10.8990 q^{82} +1.80391 q^{83} -10.9048 q^{85} +7.91186 q^{86} -2.79991 q^{88} -10.9572 q^{89} +12.1915 q^{91} +0.907338 q^{92} -13.4073 q^{94} -9.25835 q^{95} +16.3978 q^{97} +2.27289 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29} + 11 q^{31} - 7 q^{32} + 16 q^{34} + 17 q^{37} - 2 q^{38} + 6 q^{40} - 18 q^{41} - q^{43} + q^{44} + 8 q^{46} - 5 q^{47} + 24 q^{49} - 19 q^{50} + 8 q^{52} - 6 q^{53} + 21 q^{55} - 3 q^{56} + 4 q^{58} + 7 q^{59} + 24 q^{61} - 11 q^{62} + 7 q^{64} - 11 q^{65} - 4 q^{67} - 16 q^{68} + 7 q^{71} + 28 q^{73} - 17 q^{74} + 2 q^{76} + 2 q^{77} - 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} + q^{86} - q^{88} - 5 q^{89} + 2 q^{91} - 8 q^{92} + 5 q^{94} + 14 q^{95} + 23 q^{97} - 24 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8 + 6 * q^10 + q^11 + 8 * q^13 - 3 * q^14 + 7 * q^16 - 16 * q^17 + 2 * q^19 - 6 * q^20 - q^22 - 8 * q^23 + 19 * q^25 - 8 * q^26 + 3 * q^28 - 4 * q^29 + 11 * q^31 - 7 * q^32 + 16 * q^34 + 17 * q^37 - 2 * q^38 + 6 * q^40 - 18 * q^41 - q^43 + q^44 + 8 * q^46 - 5 * q^47 + 24 * q^49 - 19 * q^50 + 8 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^56 + 4 * q^58 + 7 * q^59 + 24 * q^61 - 11 * q^62 + 7 * q^64 - 11 * q^65 - 4 * q^67 - 16 * q^68 + 7 * q^71 + 28 * q^73 - 17 * q^74 + 2 * q^76 + 2 * q^77 - 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 + q^86 - q^88 - 5 * q^89 + 2 * q^91 - 8 * q^92 + 5 * q^94 + 14 * q^95 + 23 * q^97 - 24 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.69148	−1.20367	−0.601834	−	0.798621i	$-0.705563\pi$
$5$	−2.69148	−1.20367	−0.601834	+	0.798621i	$0.705563\pi$
$6$	0	0
$7$	2.17419	0.821767	0.410884	−	0.911688i	$-0.365220\pi$
$7$	2.17419	0.821767	0.410884	+	0.911688i	$0.365220\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.69148	0.851122
$11$	2.79991	0.844206	0.422103	−	0.906548i	$-0.361292\pi$
$11$	2.79991	0.844206	0.422103	+	0.906548i	$0.361292\pi$
$12$	0	0
$13$	5.60736	1.55520	0.777601	−	0.628758i	$-0.216436\pi$
$13$	5.60736	1.55520	0.777601	+	0.628758i	$0.216436\pi$
$14$	−2.17419	−0.581077
$15$	0	0
$16$	1.00000	0.250000
$17$	4.05158	0.982654	0.491327	−	0.870975i	$-0.336512\pi$
$17$	4.05158	0.982654	0.491327	+	0.870975i	$0.336512\pi$
$18$	0	0
$19$	3.43987	0.789160	0.394580	−	0.918861i	$-0.370890\pi$
$19$	3.43987	0.789160	0.394580	+	0.918861i	$0.370890\pi$
$20$	−2.69148	−0.601834
$21$	0	0
$22$	−2.79991	−0.596944
$23$	0.907338	0.189193	0.0945965	−	0.995516i	$-0.469844\pi$
$23$	0.907338	0.189193	0.0945965	+	0.995516i	$0.469844\pi$
$24$	0	0
$25$	2.24408	0.448816
$26$	−5.60736	−1.09969
$27$	0	0
$28$	2.17419	0.410884
$29$	−3.53306	−0.656073	−0.328036	−	0.944665i	$-0.606387\pi$
$29$	−3.53306	−0.656073	−0.328036	+	0.944665i	$0.606387\pi$
$30$	0	0
$31$	2.49443	0.448012	0.224006	−	0.974588i	$-0.428086\pi$
$31$	2.49443	0.448012	0.224006	+	0.974588i	$0.428086\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.05158	−0.694841
$35$	−5.85180	−0.989135
$36$	0	0
$37$	4.44606	0.730928	0.365464	−	0.930826i	$-0.380910\pi$
$37$	4.44606	0.730928	0.365464	+	0.930826i	$0.380910\pi$
$38$	−3.43987	−0.558021
$39$	0	0
$40$	2.69148	0.425561
$41$	10.8990	1.70214	0.851071	−	0.525050i	$-0.175953\pi$
$41$	10.8990	1.70214	0.851071	+	0.525050i	$0.175953\pi$
$42$	0	0
$43$	−7.91186	−1.20655	−0.603274	−	0.797534i	$-0.706137\pi$
$43$	−7.91186	−1.20655	−0.603274	+	0.797534i	$0.706137\pi$
$44$	2.79991	0.422103
$45$	0	0
$46$	−0.907338	−0.133780
$47$	13.4073	1.95566	0.977829	−	0.209406i	$-0.0671531\pi$
$47$	13.4073	1.95566	0.977829	+	0.209406i	$0.0671531\pi$
$48$	0	0
$49$	−2.27289	−0.324698
$50$	−2.24408	−0.317361
$51$	0	0
$52$	5.60736	0.777601
$53$	−13.1947	−1.81243	−0.906213	−	0.422822i	$-0.861039\pi$
$53$	−13.1947	−1.81243	−0.906213	+	0.422822i	$0.861039\pi$
$54$	0	0
$55$	−7.53592	−1.01614
$56$	−2.17419	−0.290539
$57$	0	0
$58$	3.53306	0.463913
$59$	−11.3491	−1.47752	−0.738762	−	0.673966i	$-0.764589\pi$
$59$	−11.3491	−1.47752	−0.738762	+	0.673966i	$0.764589\pi$
$60$	0	0
$61$	0.466941	0.0597857	0.0298928	−	0.999553i	$-0.490483\pi$
$61$	0.466941	0.0597857	0.0298928	+	0.999553i	$0.490483\pi$
$62$	−2.49443	−0.316792
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.0921	−1.87195
$66$	0	0
$67$	6.52324	0.796940	0.398470	−	0.917181i	$-0.369541\pi$
$67$	6.52324	0.796940	0.398470	+	0.917181i	$0.369541\pi$
$68$	4.05158	0.491327
$69$	0	0
$70$	5.85180	0.699424
$71$	−2.84960	−0.338186	−0.169093	−	0.985600i	$-0.554084\pi$
$71$	−2.84960	−0.338186	−0.169093	+	0.985600i	$0.554084\pi$
$72$	0	0
$73$	4.29281	0.502435	0.251218	−	0.967931i	$-0.419169\pi$
$73$	4.29281	0.502435	0.251218	+	0.967931i	$0.419169\pi$
$74$	−4.44606	−0.516844
$75$	0	0
$76$	3.43987	0.394580
$77$	6.08755	0.693741
$78$	0	0
$79$	−7.97329	−0.897065	−0.448533	−	0.893766i	$-0.648053\pi$
$79$	−7.97329	−0.897065	−0.448533	+	0.893766i	$0.648053\pi$
$80$	−2.69148	−0.300917
$81$	0	0
$82$	−10.8990	−1.20360
$83$	1.80391	0.198005	0.0990024	−	0.995087i	$-0.468435\pi$
$83$	1.80391	0.198005	0.0990024	+	0.995087i	$0.468435\pi$
$84$	0	0
$85$	−10.9048	−1.18279
$86$	7.91186	0.853158
$87$	0	0
$88$	−2.79991	−0.298472
$89$	−10.9572	−1.16146	−0.580730	−	0.814096i	$-0.697233\pi$
$89$	−10.9572	−1.16146	−0.580730	+	0.814096i	$0.697233\pi$
$90$	0	0
$91$	12.1915	1.27801
$92$	0.907338	0.0945965
$93$	0	0
$94$	−13.4073	−1.38286
$95$	−9.25835	−0.949887
$96$	0	0
$97$	16.3978	1.66495	0.832473	−	0.554066i	$-0.186924\pi$
$97$	16.3978	1.66495	0.832473	+	0.554066i	$0.186924\pi$
$98$	2.27289	0.229596
$99$	0	0
$100$	2.24408	0.224408
$101$	−3.53689	−0.351933	−0.175967	−	0.984396i	$-0.556305\pi$
$101$	−3.53689	−0.351933	−0.175967	+	0.984396i	$0.556305\pi$
$102$	0	0
$103$	−5.03869	−0.496477	−0.248239	−	0.968699i	$-0.579852\pi$
$103$	−5.03869	−0.496477	−0.248239	+	0.968699i	$0.579852\pi$
$104$	−5.60736	−0.549847
$105$	0	0
$106$	13.1947	1.28158
$107$	8.32832	0.805129	0.402565	−	0.915392i	$-0.368119\pi$
$107$	8.32832	0.805129	0.402565	+	0.915392i	$0.368119\pi$
$108$	0	0
$109$	2.72745	0.261242	0.130621	−	0.991432i	$-0.458303\pi$
$109$	2.72745	0.261242	0.130621	+	0.991432i	$0.458303\pi$
$110$	7.53592	0.718522
$111$	0	0
$112$	2.17419	0.205442
$113$	−13.6313	−1.28233	−0.641165	−	0.767403i	$-0.721548\pi$
$113$	−13.6313	−1.28233	−0.641165	+	0.767403i	$0.721548\pi$
$114$	0	0
$115$	−2.44208	−0.227725
$116$	−3.53306	−0.328036
$117$	0	0
$118$	11.3491	1.04477
$119$	8.80892	0.807513
$120$	0	0
$121$	−3.16049	−0.287317
$122$	−0.466941	−0.0422749
$123$	0	0
$124$	2.49443	0.224006
$125$	7.41751	0.663443
$126$	0	0
$127$	−13.6128	−1.20794	−0.603969	−	0.797008i	$-0.706415\pi$
$127$	−13.6128	−1.20794	−0.603969	+	0.797008i	$0.706415\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	15.0921	1.32367
$131$	17.7178	1.54801	0.774004	−	0.633181i	$-0.218251\pi$
$131$	17.7178	1.54801	0.774004	+	0.633181i	$0.218251\pi$
$132$	0	0
$133$	7.47894	0.648506
$134$	−6.52324	−0.563522
$135$	0	0
$136$	−4.05158	−0.347420
$137$	15.2395	1.30200	0.650998	−	0.759079i	$-0.274351\pi$
$137$	15.2395	1.30200	0.650998	+	0.759079i	$0.274351\pi$
$138$	0	0
$139$	−4.87274	−0.413300	−0.206650	−	0.978415i	$-0.566256\pi$
$139$	−4.87274	−0.413300	−0.206650	+	0.978415i	$0.566256\pi$
$140$	−5.85180	−0.494567
$141$	0	0
$142$	2.84960	0.239133
$143$	15.7001	1.31291
$144$	0	0
$145$	9.50917	0.789693
$146$	−4.29281	−0.355275
$147$	0	0
$148$	4.44606	0.365464
$149$	14.5888	1.19516	0.597579	−	0.801810i	$-0.296129\pi$
$149$	14.5888	1.19516	0.597579	+	0.801810i	$0.296129\pi$
$150$	0	0
$151$	12.6330	1.02806	0.514028	−	0.857773i	$-0.328153\pi$
$151$	12.6330	1.02806	0.514028	+	0.857773i	$0.328153\pi$
$152$	−3.43987	−0.279010
$153$	0	0
$154$	−6.08755	−0.490549
$155$	−6.71371	−0.539258
$156$	0	0
$157$	−12.0602	−0.962507	−0.481254	−	0.876581i	$-0.659818\pi$
$157$	−12.0602	−0.962507	−0.481254	+	0.876581i	$0.659818\pi$
$158$	7.97329	0.634321
$159$	0	0
$160$	2.69148	0.212780
$161$	1.97273	0.155473
$162$	0	0
$163$	0.156053	0.0122230	0.00611149	−	0.999981i	$-0.498055\pi$
$163$	0.156053	0.0122230	0.00611149	+	0.999981i	$0.498055\pi$
$164$	10.8990	0.851071
$165$	0	0
$166$	−1.80391	−0.140011
$167$	−1.39325	−0.107813	−0.0539065	−	0.998546i	$-0.517167\pi$
$167$	−1.39325	−0.107813	−0.0539065	+	0.998546i	$0.517167\pi$
$168$	0	0
$169$	18.4425	1.41865
$170$	10.9048	0.836358
$171$	0	0
$172$	−7.91186	−0.603274
$173$	16.6528	1.26609	0.633043	−	0.774117i	$-0.281806\pi$
$173$	16.6528	1.26609	0.633043	+	0.774117i	$0.281806\pi$
$174$	0	0
$175$	4.87906	0.368822
$176$	2.79991	0.211051
$177$	0	0
$178$	10.9572	0.821277
$179$	12.9554	0.968331	0.484165	−	0.874977i	$-0.339123\pi$
$179$	12.9554	0.968331	0.484165	+	0.874977i	$0.339123\pi$
$180$	0	0
$181$	−8.76887	−0.651785	−0.325892	−	0.945407i	$-0.605665\pi$
$181$	−8.76887	−0.651785	−0.325892	+	0.945407i	$0.605665\pi$
$182$	−12.1915	−0.903692
$183$	0	0
$184$	−0.907338	−0.0668898
$185$	−11.9665	−0.879794
$186$	0	0
$187$	11.3441	0.829562
$188$	13.4073	0.977829
$189$	0	0
$190$	9.25835	0.671672
$191$	−2.78928	−0.201825	−0.100912	−	0.994895i	$-0.532176\pi$
$191$	−2.78928	−0.201825	−0.100912	+	0.994895i	$0.532176\pi$
$192$	0	0
$193$	−11.6058	−0.835402	−0.417701	−	0.908585i	$-0.637164\pi$
$193$	−11.6058	−0.835402	−0.417701	+	0.908585i	$0.637164\pi$
$194$	−16.3978	−1.17729
$195$	0	0
$196$	−2.27289	−0.162349
$197$	−20.5331	−1.46293	−0.731463	−	0.681881i	$-0.761162\pi$
$197$	−20.5331	−1.46293	−0.731463	+	0.681881i	$0.761162\pi$
$198$	0	0
$199$	17.9919	1.27541	0.637706	−	0.770280i	$-0.279883\pi$
$199$	17.9919	1.27541	0.637706	+	0.770280i	$0.279883\pi$
$200$	−2.24408	−0.158680
$201$	0	0
$202$	3.53689	0.248855
$203$	−7.68155	−0.539139
$204$	0	0
$205$	−29.3346	−2.04881
$206$	5.03869	0.351062
$207$	0	0
$208$	5.60736	0.388800
$209$	9.63134	0.666214
$210$	0	0
$211$	−3.08882	−0.212643	−0.106321	−	0.994332i	$-0.533907\pi$
$211$	−3.08882	−0.212643	−0.106321	+	0.994332i	$0.533907\pi$
$212$	−13.1947	−0.906213
$213$	0	0
$214$	−8.32832	−0.569312
$215$	21.2946	1.45228
$216$	0	0
$217$	5.42336	0.368162
$218$	−2.72745	−0.184726
$219$	0	0
$220$	−7.53592	−0.508072
$221$	22.7187	1.52822
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−2.17419	−0.145269
$225$	0	0
$226$	13.6313	0.906744
$227$	19.8177	1.31535	0.657675	−	0.753302i	$-0.271540\pi$
$227$	19.8177	1.31535	0.657675	+	0.753302i	$0.271540\pi$
$228$	0	0
$229$	−25.3745	−1.67680	−0.838398	−	0.545058i	$-0.816508\pi$
$229$	−25.3745	−1.67680	−0.838398	+	0.545058i	$0.816508\pi$
$230$	2.44208	0.161026
$231$	0	0
$232$	3.53306	0.231957
$233$	−8.41472	−0.551266	−0.275633	−	0.961263i	$-0.588888\pi$
$233$	−8.41472	−0.551266	−0.275633	+	0.961263i	$0.588888\pi$
$234$	0	0
$235$	−36.0855	−2.35396
$236$	−11.3491	−0.738762
$237$	0	0
$238$	−8.80892	−0.570998
$239$	28.9489	1.87255	0.936273	−	0.351272i	$-0.114251\pi$
$239$	28.9489	1.87255	0.936273	+	0.351272i	$0.114251\pi$
$240$	0	0
$241$	2.56905	0.165487	0.0827435	−	0.996571i	$-0.473632\pi$
$241$	2.56905	0.165487	0.0827435	+	0.996571i	$0.473632\pi$
$242$	3.16049	0.203164
$243$	0	0
$244$	0.466941	0.0298928
$245$	6.11744	0.390829
$246$	0	0
$247$	19.2886	1.22730
$248$	−2.49443	−0.158396
$249$	0	0
$250$	−7.41751	−0.469125
$251$	−14.7729	−0.932456	−0.466228	−	0.884665i	$-0.654387\pi$
$251$	−14.7729	−0.932456	−0.466228	+	0.884665i	$0.654387\pi$
$252$	0	0
$253$	2.54047	0.159718
$254$	13.6128	0.854140
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.3220	1.26765	0.633827	−	0.773475i	$-0.281483\pi$
$257$	20.3220	1.26765	0.633827	+	0.773475i	$0.281483\pi$
$258$	0	0
$259$	9.66659	0.600653
$260$	−15.0921	−0.935973
$261$	0	0
$262$	−17.7178	−1.09461
$263$	13.9440	0.859826	0.429913	−	0.902870i	$-0.358544\pi$
$263$	13.9440	0.859826	0.429913	+	0.902870i	$0.358544\pi$
$264$	0	0
$265$	35.5132	2.18156
$266$	−7.47894	−0.458563
$267$	0	0
$268$	6.52324	0.398470
$269$	1.33483	0.0813862	0.0406931	−	0.999172i	$-0.487043\pi$
$269$	1.33483	0.0813862	0.0406931	+	0.999172i	$0.487043\pi$
$270$	0	0
$271$	18.0818	1.09839	0.549197	−	0.835693i	$-0.314934\pi$
$271$	18.0818	1.09839	0.549197	+	0.835693i	$0.314934\pi$
$272$	4.05158	0.245663
$273$	0	0
$274$	−15.2395	−0.920650
$275$	6.28323	0.378893
$276$	0	0
$277$	6.92461	0.416059	0.208030	−	0.978123i	$-0.433295\pi$
$277$	6.92461	0.416059	0.208030	+	0.978123i	$0.433295\pi$
$278$	4.87274	0.292247
$279$	0	0
$280$	5.85180	0.349712
$281$	−8.16492	−0.487078	−0.243539	−	0.969891i	$-0.578308\pi$
$281$	−8.16492	−0.487078	−0.243539	+	0.969891i	$0.578308\pi$
$282$	0	0
$283$	−13.9290	−0.827993	−0.413996	−	0.910279i	$-0.635867\pi$
$283$	−13.9290	−0.827993	−0.413996	+	0.910279i	$0.635867\pi$
$284$	−2.84960	−0.169093
$285$	0	0
$286$	−15.7001	−0.928368
$287$	23.6966	1.39877
$288$	0	0
$289$	−0.584665	−0.0343921
$290$	−9.50917	−0.558398
$291$	0	0
$292$	4.29281	0.251218
$293$	22.2675	1.30088	0.650439	−	0.759558i	$-0.274585\pi$
$293$	22.2675	1.30088	0.650439	+	0.759558i	$0.274585\pi$
$294$	0	0
$295$	30.5459	1.77845
$296$	−4.44606	−0.258422
$297$	0	0
$298$	−14.5888	−0.845105
$299$	5.08777	0.294233
$300$	0	0
$301$	−17.2019	−0.991502
$302$	−12.6330	−0.726946
$303$	0	0
$304$	3.43987	0.197290
$305$	−1.25676	−0.0719621
$306$	0	0
$307$	−8.05512	−0.459730	−0.229865	−	0.973222i	$-0.573829\pi$
$307$	−8.05512	−0.459730	−0.229865	+	0.973222i	$0.573829\pi$
$308$	6.08755	0.346870
$309$	0	0
$310$	6.71371	0.381313
$311$	−5.19327	−0.294483	−0.147242	−	0.989101i	$-0.547040\pi$
$311$	−5.19327	−0.294483	−0.147242	+	0.989101i	$0.547040\pi$
$312$	0	0
$313$	1.77635	0.100405	0.0502025	−	0.998739i	$-0.484013\pi$
$313$	1.77635	0.100405	0.0502025	+	0.998739i	$0.484013\pi$
$314$	12.0602	0.680595
$315$	0	0
$316$	−7.97329	−0.448533
$317$	−16.7624	−0.941470	−0.470735	−	0.882275i	$-0.656011\pi$
$317$	−16.7624	−0.941470	−0.470735	+	0.882275i	$0.656011\pi$
$318$	0	0
$319$	−9.89226	−0.553860
$320$	−2.69148	−0.150458
$321$	0	0
$322$	−1.97273	−0.109936
$323$	13.9369	0.775471
$324$	0	0
$325$	12.5834	0.697999
$326$	−0.156053	−0.00864295
$327$	0	0
$328$	−10.8990	−0.601798
$329$	29.1501	1.60710
$330$	0	0
$331$	−3.10572	−0.170706	−0.0853530	−	0.996351i	$-0.527202\pi$
$331$	−3.10572	−0.170706	−0.0853530	+	0.996351i	$0.527202\pi$
$332$	1.80391	0.0990024
$333$	0	0
$334$	1.39325	0.0762353
$335$	−17.5572	−0.959251
$336$	0	0
$337$	16.0670	0.875225	0.437613	−	0.899164i	$-0.355824\pi$
$337$	16.0670	0.875225	0.437613	+	0.899164i	$0.355824\pi$
$338$	−18.4425	−1.00314
$339$	0	0
$340$	−10.9048	−0.591394
$341$	6.98418	0.378214
$342$	0	0
$343$	−20.1610	−1.08859
$344$	7.91186	0.426579
$345$	0	0
$346$	−16.6528	−0.895258
$347$	−20.1398	−1.08116	−0.540580	−	0.841293i	$-0.681795\pi$
$347$	−20.1398	−1.08116	−0.540580	+	0.841293i	$0.681795\pi$
$348$	0	0
$349$	30.6684	1.64164	0.820821	−	0.571186i	$-0.193516\pi$
$349$	30.6684	1.64164	0.820821	+	0.571186i	$0.193516\pi$
$350$	−4.87906	−0.260797
$351$	0	0
$352$	−2.79991	−0.149236
$353$	−26.7829	−1.42551	−0.712755	−	0.701413i	$-0.752553\pi$
$353$	−26.7829	−1.42551	−0.712755	+	0.701413i	$0.752553\pi$
$354$	0	0
$355$	7.66966	0.407063
$356$	−10.9572	−0.580730
$357$	0	0
$358$	−12.9554	−0.684713
$359$	−9.32994	−0.492415	−0.246208	−	0.969217i	$-0.579185\pi$
$359$	−9.32994	−0.492415	−0.246208	+	0.969217i	$0.579185\pi$
$360$	0	0
$361$	−7.16729	−0.377226
$362$	8.76887	0.460881
$363$	0	0
$364$	12.1915	0.639007
$365$	−11.5540	−0.604765
$366$	0	0
$367$	−19.3737	−1.01130	−0.505649	−	0.862739i	$-0.668747\pi$
$367$	−19.3737	−1.01130	−0.505649	+	0.862739i	$0.668747\pi$
$368$	0.907338	0.0472983
$369$	0	0
$370$	11.9665	0.622108
$371$	−28.6877	−1.48939
$372$	0	0
$373$	−24.5202	−1.26961	−0.634805	−	0.772672i	$-0.718920\pi$
$373$	−24.5202	−1.26961	−0.634805	+	0.772672i	$0.718920\pi$
$374$	−11.3441	−0.586589
$375$	0	0
$376$	−13.4073	−0.691429
$377$	−19.8111	−1.02033
$378$	0	0
$379$	27.5623	1.41578	0.707889	−	0.706323i	$-0.249647\pi$
$379$	27.5623	1.41578	0.707889	+	0.706323i	$0.249647\pi$
$380$	−9.25835	−0.474943
$381$	0	0
$382$	2.78928	0.142712
$383$	21.8176	1.11483	0.557414	−	0.830235i	$-0.311794\pi$
$383$	21.8176	1.11483	0.557414	+	0.830235i	$0.311794\pi$
$384$	0	0
$385$	−16.3845	−0.835033
$386$	11.6058	0.590718
$387$	0	0
$388$	16.3978	0.832473
$389$	16.2181	0.822290	0.411145	−	0.911570i	$-0.365129\pi$
$389$	16.2181	0.822290	0.411145	+	0.911570i	$0.365129\pi$
$390$	0	0
$391$	3.67616	0.185911
$392$	2.27289	0.114798
$393$	0	0
$394$	20.5331	1.03444
$395$	21.4600	1.07977
$396$	0	0
$397$	1.66612	0.0836200	0.0418100	−	0.999126i	$-0.486688\pi$
$397$	1.66612	0.0836200	0.0418100	+	0.999126i	$0.486688\pi$
$398$	−17.9919	−0.901852
$399$	0	0
$400$	2.24408	0.112204
$401$	33.0190	1.64889	0.824446	−	0.565940i	$-0.191487\pi$
$401$	33.0190	1.64889	0.824446	+	0.565940i	$0.191487\pi$
$402$	0	0
$403$	13.9871	0.696749
$404$	−3.53689	−0.175967
$405$	0	0
$406$	7.68155	0.381229
$407$	12.4486	0.617053
$408$	0	0
$409$	33.0353	1.63349	0.816744	−	0.577000i	$-0.195777\pi$
$409$	33.0353	1.63349	0.816744	+	0.577000i	$0.195777\pi$
$410$	29.3346	1.44873
$411$	0	0
$412$	−5.03869	−0.248239
$413$	−24.6751	−1.21418
$414$	0	0
$415$	−4.85519	−0.238332
$416$	−5.60736	−0.274923
$417$	0	0
$418$	−9.63134	−0.471084
$419$	8.61709	0.420972	0.210486	−	0.977597i	$-0.432495\pi$
$419$	8.61709	0.420972	0.210486	+	0.977597i	$0.432495\pi$
$420$	0	0
$421$	−13.7662	−0.670922	−0.335461	−	0.942054i	$-0.608892\pi$
$421$	−13.7662	−0.670922	−0.335461	+	0.942054i	$0.608892\pi$
$422$	3.08882	0.150361
$423$	0	0
$424$	13.1947	0.640789
$425$	9.09208	0.441030
$426$	0	0
$427$	1.01522	0.0491299
$428$	8.32832	0.402565
$429$	0	0
$430$	−21.2946	−1.02692
$431$	−7.17551	−0.345632	−0.172816	−	0.984954i	$-0.555287\pi$
$431$	−7.17551	−0.345632	−0.172816	+	0.984954i	$0.555287\pi$
$432$	0	0
$433$	11.4852	0.551944	0.275972	−	0.961166i	$-0.411000\pi$
$433$	11.4852	0.551944	0.275972	+	0.961166i	$0.411000\pi$
$434$	−5.42336	−0.260330
$435$	0	0
$436$	2.72745	0.130621
$437$	3.12112	0.149304
$438$	0	0
$439$	−10.6077	−0.506279	−0.253139	−	0.967430i	$-0.581463\pi$
$439$	−10.6077	−0.506279	−0.253139	+	0.967430i	$0.581463\pi$
$440$	7.53592	0.359261
$441$	0	0
$442$	−22.7187	−1.08062
$443$	23.6850	1.12531	0.562654	−	0.826693i	$-0.309780\pi$
$443$	23.6850	1.12531	0.562654	+	0.826693i	$0.309780\pi$
$444$	0	0
$445$	29.4911	1.39801
$446$	1.00000	0.0473514
$447$	0	0
$448$	2.17419	0.102721
$449$	−26.0448	−1.22913	−0.614565	−	0.788866i	$-0.710668\pi$
$449$	−26.0448	−1.22913	−0.614565	+	0.788866i	$0.710668\pi$
$450$	0	0
$451$	30.5163	1.43696
$452$	−13.6313	−0.641165
$453$	0	0
$454$	−19.8177	−0.930093
$455$	−32.8131	−1.53830
$456$	0	0
$457$	19.4587	0.910237	0.455119	−	0.890431i	$-0.349597\pi$
$457$	19.4587	0.910237	0.455119	+	0.890431i	$0.349597\pi$
$458$	25.3745	1.18567
$459$	0	0
$460$	−2.44208	−0.113863
$461$	1.34885	0.0628221	0.0314110	−	0.999507i	$-0.490000\pi$
$461$	1.34885	0.0628221	0.0314110	+	0.999507i	$0.490000\pi$
$462$	0	0
$463$	−19.2142	−0.892958	−0.446479	−	0.894794i	$-0.647322\pi$
$463$	−19.2142	−0.892958	−0.446479	+	0.894794i	$0.647322\pi$
$464$	−3.53306	−0.164018
$465$	0	0
$466$	8.41472	0.389804
$467$	−30.1602	−1.39565	−0.697824	−	0.716269i	$-0.745848\pi$
$467$	−30.1602	−1.39565	−0.697824	+	0.716269i	$0.745848\pi$
$468$	0	0
$469$	14.1828	0.654899
$470$	36.0855	1.66450
$471$	0	0
$472$	11.3491	0.522384
$473$	−22.1525	−1.01857
$474$	0	0
$475$	7.71934	0.354188
$476$	8.80892	0.403756
$477$	0	0
$478$	−28.9489	−1.32409
$479$	−10.4392	−0.476979	−0.238490	−	0.971145i	$-0.576652\pi$
$479$	−10.4392	−0.476979	−0.238490	+	0.971145i	$0.576652\pi$
$480$	0	0
$481$	24.9307	1.13674
$482$	−2.56905	−0.117017
$483$	0	0
$484$	−3.16049	−0.143658
$485$	−44.1344	−2.00404
$486$	0	0
$487$	18.5437	0.840297	0.420149	−	0.907455i	$-0.361978\pi$
$487$	18.5437	0.840297	0.420149	+	0.907455i	$0.361978\pi$
$488$	−0.466941	−0.0211374
$489$	0	0
$490$	−6.11744	−0.276358
$491$	−17.6989	−0.798739	−0.399370	−	0.916790i	$-0.630771\pi$
$491$	−17.6989	−0.798739	−0.399370	+	0.916790i	$0.630771\pi$
$492$	0	0
$493$	−14.3145	−0.644692
$494$	−19.2886	−0.867835
$495$	0	0
$496$	2.49443	0.112003
$497$	−6.19559	−0.277910
$498$	0	0
$499$	−2.78926	−0.124864	−0.0624321	−	0.998049i	$-0.519886\pi$
$499$	−2.78926	−0.124864	−0.0624321	+	0.998049i	$0.519886\pi$
$500$	7.41751	0.331721
$501$	0	0
$502$	14.7729	0.659346
$503$	26.2160	1.16892	0.584458	−	0.811424i	$-0.301307\pi$
$503$	26.2160	1.16892	0.584458	+	0.811424i	$0.301307\pi$
$504$	0	0
$505$	9.51947	0.423611
$506$	−2.54047	−0.112938
$507$	0	0
$508$	−13.6128	−0.603969
$509$	−1.95494	−0.0866510	−0.0433255	−	0.999061i	$-0.513795\pi$
$509$	−1.95494	−0.0866510	−0.0433255	+	0.999061i	$0.513795\pi$
$510$	0	0
$511$	9.33339	0.412885
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.3220	−0.896367
$515$	13.5616	0.597593
$516$	0	0
$517$	37.5393	1.65098
$518$	−9.66659	−0.424726
$519$	0	0
$520$	15.0921	0.661833
$521$	−23.9402	−1.04884	−0.524420	−	0.851459i	$-0.675718\pi$
$521$	−23.9402	−1.04884	−0.524420	+	0.851459i	$0.675718\pi$
$522$	0	0
$523$	33.6405	1.47100	0.735498	−	0.677527i	$-0.236948\pi$
$523$	33.6405	1.47100	0.735498	+	0.677527i	$0.236948\pi$
$524$	17.7178	0.774004
$525$	0	0
$526$	−13.9440	−0.607989
$527$	10.1064	0.440241
$528$	0	0
$529$	−22.1767	−0.964206
$530$	−35.5132	−1.54259
$531$	0	0
$532$	7.47894	0.324253
$533$	61.1148	2.64718
$534$	0	0
$535$	−22.4155	−0.969108
$536$	−6.52324	−0.281761
$537$	0	0
$538$	−1.33483	−0.0575487
$539$	−6.36389	−0.274112
$540$	0	0
$541$	−28.9644	−1.24528	−0.622638	−	0.782510i	$-0.713939\pi$
$541$	−28.9644	−1.24528	−0.622638	+	0.782510i	$0.713939\pi$
$542$	−18.0818	−0.776681
$543$	0	0
$544$	−4.05158	−0.173710
$545$	−7.34088	−0.314449
$546$	0	0
$547$	40.6631	1.73863	0.869314	−	0.494260i	$-0.164561\pi$
$547$	40.6631	1.73863	0.869314	+	0.494260i	$0.164561\pi$
$548$	15.2395	0.650998
$549$	0	0
$550$	−6.28323	−0.267918
$551$	−12.1533	−0.517747
$552$	0	0
$553$	−17.3355	−0.737179
$554$	−6.92461	−0.294198
$555$	0	0
$556$	−4.87274	−0.206650
$557$	−33.3538	−1.41325	−0.706624	−	0.707590i	$-0.749783\pi$
$557$	−33.3538	−1.41325	−0.706624	+	0.707590i	$0.749783\pi$
$558$	0	0
$559$	−44.3647	−1.87643
$560$	−5.85180	−0.247284
$561$	0	0
$562$	8.16492	0.344416
$563$	−19.0344	−0.802205	−0.401102	−	0.916033i	$-0.631373\pi$
$563$	−19.0344	−0.802205	−0.401102	+	0.916033i	$0.631373\pi$
$564$	0	0
$565$	36.6885	1.54350
$566$	13.9290	0.585479
$567$	0	0
$568$	2.84960	0.119567
$569$	36.6564	1.53672	0.768358	−	0.640020i	$-0.221074\pi$
$569$	36.6564	1.53672	0.768358	+	0.640020i	$0.221074\pi$
$570$	0	0
$571$	−27.9181	−1.16833	−0.584167	−	0.811633i	$-0.698579\pi$
$571$	−27.9181	−1.16833	−0.584167	+	0.811633i	$0.698579\pi$
$572$	15.7001	0.656455
$573$	0	0
$574$	−23.6966	−0.989076
$575$	2.03614	0.0849128
$576$	0	0
$577$	17.9442	0.747028	0.373514	−	0.927624i	$-0.378153\pi$
$577$	17.9442	0.747028	0.373514	+	0.927624i	$0.378153\pi$
$578$	0.584665	0.0243189
$579$	0	0
$580$	9.50917	0.394847
$581$	3.92205	0.162714
$582$	0	0
$583$	−36.9439	−1.53006
$584$	−4.29281	−0.177638
$585$	0	0
$586$	−22.2675	−0.919860
$587$	26.5520	1.09592	0.547960	−	0.836505i	$-0.315405\pi$
$587$	26.5520	1.09592	0.547960	+	0.836505i	$0.315405\pi$
$588$	0	0
$589$	8.58051	0.353554
$590$	−30.5459	−1.25755
$591$	0	0
$592$	4.44606	0.182732
$593$	13.9180	0.571543	0.285771	−	0.958298i	$-0.407750\pi$
$593$	13.9180	0.571543	0.285771	+	0.958298i	$0.407750\pi$
$594$	0	0
$595$	−23.7091	−0.971977
$596$	14.5888	0.597579
$597$	0	0
$598$	−5.08777	−0.208054
$599$	3.31413	0.135412	0.0677059	−	0.997705i	$-0.478432\pi$
$599$	3.31413	0.135412	0.0677059	+	0.997705i	$0.478432\pi$
$600$	0	0
$601$	24.4692	0.998118	0.499059	−	0.866568i	$-0.333679\pi$
$601$	24.4692	0.998118	0.499059	+	0.866568i	$0.333679\pi$
$602$	17.2019	0.701098
$603$	0	0
$604$	12.6330	0.514028
$605$	8.50639	0.345834
$606$	0	0
$607$	37.4463	1.51990	0.759950	−	0.649981i	$-0.225223\pi$
$607$	37.4463	1.51990	0.759950	+	0.649981i	$0.225223\pi$
$608$	−3.43987	−0.139505
$609$	0	0
$610$	1.25676	0.0508849
$611$	75.1796	3.04144
$612$	0	0
$613$	−17.8819	−0.722241	−0.361121	−	0.932519i	$-0.617606\pi$
$613$	−17.8819	−0.722241	−0.361121	+	0.932519i	$0.617606\pi$
$614$	8.05512	0.325078
$615$	0	0
$616$	−6.08755	−0.245274
$617$	−6.14776	−0.247500	−0.123750	−	0.992313i	$-0.539492\pi$
$617$	−6.14776	−0.247500	−0.123750	+	0.992313i	$0.539492\pi$
$618$	0	0
$619$	−21.3510	−0.858169	−0.429085	−	0.903264i	$-0.641164\pi$
$619$	−21.3510	−0.858169	−0.429085	+	0.903264i	$0.641164\pi$
$620$	−6.71371	−0.269629
$621$	0	0
$622$	5.19327	0.208231
$623$	−23.8231	−0.954451
$624$	0	0
$625$	−31.1845	−1.24738
$626$	−1.77635	−0.0709971
$627$	0	0
$628$	−12.0602	−0.481254
$629$	18.0136	0.718249
$630$	0	0
$631$	−40.9747	−1.63118	−0.815589	−	0.578632i	$-0.803587\pi$
$631$	−40.9747	−1.63118	−0.815589	+	0.578632i	$0.803587\pi$
$632$	7.97329	0.317161
$633$	0	0
$634$	16.7624	0.665720
$635$	36.6385	1.45395
$636$	0	0
$637$	−12.7449	−0.504971
$638$	9.89226	0.391638
$639$	0	0
$640$	2.69148	0.106390
$641$	31.0690	1.22715	0.613575	−	0.789636i	$-0.289731\pi$
$641$	31.0690	1.22715	0.613575	+	0.789636i	$0.289731\pi$
$642$	0	0
$643$	28.6129	1.12838	0.564192	−	0.825643i	$-0.309188\pi$
$643$	28.6129	1.12838	0.564192	+	0.825643i	$0.309188\pi$
$644$	1.97273	0.0777363
$645$	0	0
$646$	−13.9369	−0.548341
$647$	−46.2606	−1.81869	−0.909346	−	0.416042i	$-0.863417\pi$
$647$	−46.2606	−1.81869	−0.909346	+	0.416042i	$0.863417\pi$
$648$	0	0
$649$	−31.7764	−1.24733
$650$	−12.5834	−0.493560
$651$	0	0
$652$	0.156053	0.00611149
$653$	39.3565	1.54014	0.770068	−	0.637961i	$-0.220222\pi$
$653$	39.3565	1.54014	0.770068	+	0.637961i	$0.220222\pi$
$654$	0	0
$655$	−47.6870	−1.86329
$656$	10.8990	0.425536
$657$	0	0
$658$	−29.1501	−1.13639
$659$	19.6764	0.766485	0.383242	−	0.923648i	$-0.374807\pi$
$659$	19.6764	0.766485	0.383242	+	0.923648i	$0.374807\pi$
$660$	0	0
$661$	38.9951	1.51673	0.758366	−	0.651829i	$-0.225998\pi$
$661$	38.9951	1.51673	0.758366	+	0.651829i	$0.225998\pi$
$662$	3.10572	0.120707
$663$	0	0
$664$	−1.80391	−0.0700053
$665$	−20.1294	−0.780586
$666$	0	0
$667$	−3.20568	−0.124124
$668$	−1.39325	−0.0539065
$669$	0	0
$670$	17.5572	0.678293
$671$	1.30739	0.0504714
$672$	0	0
$673$	21.3800	0.824139	0.412069	−	0.911152i	$-0.364806\pi$
$673$	21.3800	0.824139	0.412069	+	0.911152i	$0.364806\pi$
$674$	−16.0670	−0.618878
$675$	0	0
$676$	18.4425	0.709326
$677$	−21.4801	−0.825548	−0.412774	−	0.910833i	$-0.635440\pi$
$677$	−21.4801	−0.825548	−0.412774	+	0.910833i	$0.635440\pi$
$678$	0	0
$679$	35.6520	1.36820
$680$	10.9048	0.418179
$681$	0	0
$682$	−6.98418	−0.267438
$683$	−43.7953	−1.67578	−0.837890	−	0.545839i	$-0.816211\pi$
$683$	−43.7953	−1.67578	−0.837890	+	0.545839i	$0.816211\pi$
$684$	0	0
$685$	−41.0168	−1.56717
$686$	20.1610	0.769752
$687$	0	0
$688$	−7.91186	−0.301637
$689$	−73.9872	−2.81869
$690$	0	0
$691$	−14.1162	−0.537007	−0.268504	−	0.963279i	$-0.586529\pi$
$691$	−14.1162	−0.537007	−0.268504	+	0.963279i	$0.586529\pi$
$692$	16.6528	0.633043
$693$	0	0
$694$	20.1398	0.764495
$695$	13.1149	0.497476
$696$	0	0
$697$	44.1583	1.67262
$698$	−30.6684	−1.16082
$699$	0	0
$700$	4.87906	0.184411
$701$	45.2816	1.71026	0.855132	−	0.518411i	$-0.173476\pi$
$701$	45.2816	1.71026	0.855132	+	0.518411i	$0.173476\pi$
$702$	0	0
$703$	15.2939	0.576819
$704$	2.79991	0.105526
$705$	0	0
$706$	26.7829	1.00799
$707$	−7.68987	−0.289207
$708$	0	0
$709$	−18.5070	−0.695046	−0.347523	−	0.937671i	$-0.612977\pi$
$709$	−18.5070	−0.695046	−0.347523	+	0.937671i	$0.612977\pi$
$710$	−7.66966	−0.287837
$711$	0	0
$712$	10.9572	0.410638
$713$	2.26329	0.0847608
$714$	0	0
$715$	−42.2566	−1.58031
$716$	12.9554	0.484165
$717$	0	0
$718$	9.32994	0.348190
$719$	−25.6525	−0.956675	−0.478338	−	0.878176i	$-0.658760\pi$
$719$	−25.6525	−0.956675	−0.478338	+	0.878176i	$0.658760\pi$
$720$	0	0
$721$	−10.9551	−0.407989
$722$	7.16729	0.266739
$723$	0	0
$724$	−8.76887	−0.325892
$725$	−7.92846	−0.294456
$726$	0	0
$727$	5.95897	0.221006	0.110503	−	0.993876i	$-0.464754\pi$
$727$	5.95897	0.221006	0.110503	+	0.993876i	$0.464754\pi$
$728$	−12.1915	−0.451846
$729$	0	0
$730$	11.5540	0.427633
$731$	−32.0556	−1.18562
$732$	0	0
$733$	−5.21396	−0.192582	−0.0962910	−	0.995353i	$-0.530698\pi$
$733$	−5.21396	−0.192582	−0.0962910	+	0.995353i	$0.530698\pi$
$734$	19.3737	0.715096
$735$	0	0
$736$	−0.907338	−0.0334449
$737$	18.2645	0.672781
$738$	0	0
$739$	−7.06983	−0.260068	−0.130034	−	0.991510i	$-0.541509\pi$
$739$	−7.06983	−0.260068	−0.130034	+	0.991510i	$0.541509\pi$
$740$	−11.9665	−0.439897
$741$	0	0
$742$	28.6877	1.05316
$743$	−12.7178	−0.466572	−0.233286	−	0.972408i	$-0.574948\pi$
$743$	−12.7178	−0.466572	−0.233286	+	0.972408i	$0.574948\pi$
$744$	0	0
$745$	−39.2654	−1.43857
$746$	24.5202	0.897750
$747$	0	0
$748$	11.3441	0.414781
$749$	18.1074	0.661629
$750$	0	0
$751$	28.0917	1.02508	0.512541	−	0.858663i	$-0.328704\pi$
$751$	28.0917	1.02508	0.512541	+	0.858663i	$0.328704\pi$
$752$	13.4073	0.488914
$753$	0	0
$754$	19.8111	0.721479
$755$	−34.0014	−1.23744
$756$	0	0
$757$	−13.4644	−0.489373	−0.244687	−	0.969602i	$-0.578685\pi$
$757$	−13.4644	−0.489373	−0.244687	+	0.969602i	$0.578685\pi$
$758$	−27.5623	−1.00111
$759$	0	0
$760$	9.25835	0.335836
$761$	−8.48297	−0.307508	−0.153754	−	0.988109i	$-0.549136\pi$
$761$	−8.48297	−0.307508	−0.153754	+	0.988109i	$0.549136\pi$
$762$	0	0
$763$	5.93000	0.214680
$764$	−2.78928	−0.100912
$765$	0	0
$766$	−21.8176	−0.788303
$767$	−63.6384	−2.29785
$768$	0	0
$769$	48.5298	1.75003	0.875015	−	0.484096i	$-0.160851\pi$
$769$	48.5298	1.75003	0.875015	+	0.484096i	$0.160851\pi$
$770$	16.3845	0.590458
$771$	0	0
$772$	−11.6058	−0.417701
$773$	−30.8916	−1.11109	−0.555546	−	0.831486i	$-0.687491\pi$
$773$	−30.8916	−1.11109	−0.555546	+	0.831486i	$0.687491\pi$
$774$	0	0
$775$	5.59769	0.201075
$776$	−16.3978	−0.588647
$777$	0	0
$778$	−16.2181	−0.581447
$779$	37.4913	1.34326
$780$	0	0
$781$	−7.97864	−0.285498
$782$	−3.67616	−0.131459
$783$	0	0
$784$	−2.27289	−0.0811746
$785$	32.4598	1.15854
$786$	0	0
$787$	−11.2345	−0.400465	−0.200233	−	0.979748i	$-0.564170\pi$
$787$	−11.2345	−0.400465	−0.200233	+	0.979748i	$0.564170\pi$
$788$	−20.5331	−0.731463
$789$	0	0
$790$	−21.4600	−0.763512
$791$	−29.6372	−1.05378
$792$	0	0
$793$	2.61831	0.0929788
$794$	−1.66612	−0.0591283
$795$	0	0
$796$	17.9919	0.637706
$797$	−24.3923	−0.864020	−0.432010	−	0.901869i	$-0.642195\pi$
$797$	−24.3923	−0.864020	−0.432010	+	0.901869i	$0.642195\pi$
$798$	0	0
$799$	54.3209	1.92173
$800$	−2.24408	−0.0793402
$801$	0	0
$802$	−33.0190	−1.16594
$803$	12.0195	0.424159
$804$	0	0
$805$	−5.30956	−0.187137
$806$	−13.9871	−0.492676
$807$	0	0
$808$	3.53689	0.124427
$809$	29.6845	1.04365	0.521825	−	0.853052i	$-0.325251\pi$
$809$	29.6845	1.04365	0.521825	+	0.853052i	$0.325251\pi$
$810$	0	0
$811$	−1.60658	−0.0564146	−0.0282073	−	0.999602i	$-0.508980\pi$
$811$	−1.60658	−0.0564146	−0.0282073	+	0.999602i	$0.508980\pi$
$812$	−7.68155	−0.269570
$813$	0	0
$814$	−12.4486	−0.436323
$815$	−0.420013	−0.0147124
$816$	0	0
$817$	−27.2158	−0.952160
$818$	−33.0353	−1.15505
$819$	0	0
$820$	−29.3346	−1.02441
$821$	−21.5725	−0.752884	−0.376442	−	0.926440i	$-0.622853\pi$
$821$	−21.5725	−0.752884	−0.376442	+	0.926440i	$0.622853\pi$
$822$	0	0
$823$	30.3535	1.05806	0.529029	−	0.848604i	$-0.322556\pi$
$823$	30.3535	1.05806	0.529029	+	0.848604i	$0.322556\pi$
$824$	5.03869	0.175531
$825$	0	0
$826$	24.6751	0.858556
$827$	6.69089	0.232665	0.116333	−	0.993210i	$-0.462886\pi$
$827$	6.69089	0.232665	0.116333	+	0.993210i	$0.462886\pi$
$828$	0	0
$829$	−21.3700	−0.742209	−0.371105	−	0.928591i	$-0.621021\pi$
$829$	−21.3700	−0.742209	−0.371105	+	0.928591i	$0.621021\pi$
$830$	4.85519	0.168526
$831$	0	0
$832$	5.60736	0.194400
$833$	−9.20880	−0.319066
$834$	0	0
$835$	3.74991	0.129771
$836$	9.63134	0.333107
$837$	0	0
$838$	−8.61709	−0.297672
$839$	25.5944	0.883618	0.441809	−	0.897109i	$-0.354337\pi$
$839$	25.5944	0.883618	0.441809	+	0.897109i	$0.354337\pi$
$840$	0	0
$841$	−16.5175	−0.569569
$842$	13.7662	0.474413
$843$	0	0
$844$	−3.08882	−0.106321
$845$	−49.6376	−1.70759
$846$	0	0
$847$	−6.87150	−0.236108
$848$	−13.1947	−0.453106
$849$	0	0
$850$	−9.09208	−0.311856
$851$	4.03408	0.138286
$852$	0	0
$853$	38.1839	1.30739	0.653696	−	0.756757i	$-0.273217\pi$
$853$	38.1839	1.30739	0.653696	+	0.756757i	$0.273217\pi$
$854$	−1.01522	−0.0347401
$855$	0	0
$856$	−8.32832	−0.284656
$857$	2.35315	0.0803820	0.0401910	−	0.999192i	$-0.487203\pi$
$857$	2.35315	0.0803820	0.0401910	+	0.999192i	$0.487203\pi$
$858$	0	0
$859$	−39.4562	−1.34623	−0.673115	−	0.739538i	$-0.735044\pi$
$859$	−39.4562	−1.34623	−0.673115	+	0.739538i	$0.735044\pi$
$860$	21.2946	0.726141
$861$	0	0
$862$	7.17551	0.244399
$863$	−37.5791	−1.27921	−0.639604	−	0.768704i	$-0.720902\pi$
$863$	−37.5791	−1.27921	−0.639604	+	0.768704i	$0.720902\pi$
$864$	0	0
$865$	−44.8206	−1.52395
$866$	−11.4852	−0.390283
$867$	0	0
$868$	5.42336	0.184081
$869$	−22.3245	−0.757308
$870$	0	0
$871$	36.5781	1.23940
$872$	−2.72745	−0.0923630
$873$	0	0
$874$	−3.12112	−0.105574
$875$	16.1271	0.545195
$876$	0	0
$877$	−18.7077	−0.631714	−0.315857	−	0.948807i	$-0.602292\pi$
$877$	−18.7077	−0.631714	−0.315857	+	0.948807i	$0.602292\pi$
$878$	10.6077	0.357993
$879$	0	0
$880$	−7.53592	−0.254036
$881$	25.8200	0.869897	0.434949	−	0.900455i	$-0.356767\pi$
$881$	25.8200	0.869897	0.434949	+	0.900455i	$0.356767\pi$
$882$	0	0
$883$	17.2777	0.581442	0.290721	−	0.956808i	$-0.406105\pi$
$883$	17.2777	0.581442	0.290721	+	0.956808i	$0.406105\pi$
$884$	22.7187	0.764112
$885$	0	0
$886$	−23.6850	−0.795713
$887$	−58.1805	−1.95351	−0.976754	−	0.214362i	$-0.931233\pi$
$887$	−58.1805	−1.95351	−0.976754	+	0.214362i	$0.931233\pi$
$888$	0	0
$889$	−29.5968	−0.992643
$890$	−29.4911	−0.988544
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	46.1194	1.54333
$894$	0	0
$895$	−34.8692	−1.16555
$896$	−2.17419	−0.0726347
$897$	0	0
$898$	26.0448	0.869126
$899$	−8.81296	−0.293929
$900$	0	0
$901$	−53.4593	−1.78099
$902$	−30.5163	−1.01608
$903$	0	0
$904$	13.6313	0.453372
$905$	23.6012	0.784532
$906$	0	0
$907$	13.3120	0.442019	0.221009	−	0.975272i	$-0.429065\pi$
$907$	13.3120	0.442019	0.221009	+	0.975272i	$0.429065\pi$
$908$	19.8177	0.657675
$909$	0	0
$910$	32.8131	1.08775
$911$	−26.5194	−0.878626	−0.439313	−	0.898334i	$-0.644778\pi$
$911$	−26.5194	−0.878626	−0.439313	+	0.898334i	$0.644778\pi$
$912$	0	0
$913$	5.05079	0.167157
$914$	−19.4587	−0.643635
$915$	0	0
$916$	−25.3745	−0.838398
$917$	38.5218	1.27210
$918$	0	0
$919$	−28.1795	−0.929555	−0.464777	−	0.885428i	$-0.653866\pi$
$919$	−28.1795	−0.929555	−0.464777	+	0.885428i	$0.653866\pi$
$920$	2.44208	0.0805131
$921$	0	0
$922$	−1.34885	−0.0444219
$923$	−15.9788	−0.525947
$924$	0	0
$925$	9.97731	0.328052
$926$	19.2142	0.631417
$927$	0	0
$928$	3.53306	0.115978
$929$	2.20228	0.0722544	0.0361272	−	0.999347i	$-0.488498\pi$
$929$	2.20228	0.0722544	0.0361272	+	0.999347i	$0.488498\pi$
$930$	0	0
$931$	−7.81844	−0.256239
$932$	−8.41472	−0.275633
$933$	0	0
$934$	30.1602	0.986872
$935$	−30.5324	−0.998517
$936$	0	0
$937$	11.5251	0.376507	0.188254	−	0.982120i	$-0.439717\pi$
$937$	11.5251	0.376507	0.188254	+	0.982120i	$0.439717\pi$
$938$	−14.1828	−0.463084
$939$	0	0
$940$	−36.0855	−1.17698
$941$	−30.5023	−0.994345	−0.497173	−	0.867652i	$-0.665628\pi$
$941$	−30.5023	−0.994345	−0.497173	+	0.867652i	$0.665628\pi$
$942$	0	0
$943$	9.88910	0.322033
$944$	−11.3491	−0.369381
$945$	0	0
$946$	22.1525	0.720241
$947$	−19.5995	−0.636899	−0.318450	−	0.947940i	$-0.603162\pi$
$947$	−19.5995	−0.636899	−0.318450	+	0.947940i	$0.603162\pi$
$948$	0	0
$949$	24.0713	0.781388
$950$	−7.71934	−0.250449
$951$	0	0
$952$	−8.80892	−0.285499
$953$	9.62406	0.311754	0.155877	−	0.987776i	$-0.450180\pi$
$953$	9.62406	0.311754	0.155877	+	0.987776i	$0.450180\pi$
$954$	0	0
$955$	7.50729	0.242930
$956$	28.9489	0.936273
$957$	0	0
$958$	10.4392	0.337275
$959$	33.1336	1.06994
$960$	0	0
$961$	−24.7778	−0.799285
$962$	−24.9307	−0.803797
$963$	0	0
$964$	2.56905	0.0827435
$965$	31.2367	1.00555
$966$	0	0
$967$	−31.2488	−1.00489	−0.502447	−	0.864608i	$-0.667567\pi$
$967$	−31.2488	−1.00489	−0.502447	+	0.864608i	$0.667567\pi$
$968$	3.16049	0.101582
$969$	0	0
$970$	44.1344	1.41707
$971$	−34.3885	−1.10358	−0.551789	−	0.833984i	$-0.686055\pi$
$971$	−34.3885	−1.10358	−0.551789	+	0.833984i	$0.686055\pi$
$972$	0	0
$973$	−10.5943	−0.339637
$974$	−18.5437	−0.594180
$975$	0	0
$976$	0.466941	0.0149464
$977$	−14.9389	−0.477939	−0.238970	−	0.971027i	$-0.576810\pi$
$977$	−14.9389	−0.477939	−0.238970	+	0.971027i	$0.576810\pi$
$978$	0	0
$979$	−30.6792	−0.980512
$980$	6.11744	0.195414
$981$	0	0
$982$	17.6989	0.564794
$983$	−58.4046	−1.86282	−0.931409	−	0.363975i	$-0.881419\pi$
$983$	−58.4046	−1.86282	−0.931409	+	0.363975i	$0.881419\pi$
$984$	0	0
$985$	55.2646	1.76088
$986$	14.3145	0.455866
$987$	0	0
$988$	19.2886	0.613652
$989$	−7.17873	−0.228270
$990$	0	0
$991$	5.49611	0.174590	0.0872949	−	0.996183i	$-0.472178\pi$
$991$	5.49611	0.174590	0.0872949	+	0.996183i	$0.472178\pi$
$992$	−2.49443	−0.0791981
$993$	0	0
$994$	6.19559	0.196512
$995$	−48.4249	−1.53517
$996$	0	0
$997$	34.0412	1.07809	0.539047	−	0.842275i	$-0.318784\pi$
$997$	34.0412	1.07809	0.539047	+	0.842275i	$0.318784\pi$
$998$	2.78926	0.0882924
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.v.1.3		7
3.2	odd	2		1338.2.a.j.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.j.1.5	✓	7	3.2	odd	2
4014.2.a.v.1.3		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.69148 q^{5} +2.17419 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.69148 q^{5} +2.17419 q^{7} -1.00000 q^{8} +2.69148 q^{10} +2.79991 q^{11} +5.60736 q^{13} -2.17419 q^{14} +1.00000 q^{16} +4.05158 q^{17} +3.43987 q^{19} -2.69148 q^{20} -2.79991 q^{22} +0.907338 q^{23} +2.24408 q^{25} -5.60736 q^{26} +2.17419 q^{28} -3.53306 q^{29} +2.49443 q^{31} -1.00000 q^{32} -4.05158 q^{34} -5.85180 q^{35} +4.44606 q^{37} -3.43987 q^{38} +2.69148 q^{40} +10.8990 q^{41} -7.91186 q^{43} +2.79991 q^{44} -0.907338 q^{46} +13.4073 q^{47} -2.27289 q^{49} -2.24408 q^{50} +5.60736 q^{52} -13.1947 q^{53} -7.53592 q^{55} -2.17419 q^{56} +3.53306 q^{58} -11.3491 q^{59} +0.466941 q^{61} -2.49443 q^{62} +1.00000 q^{64} -15.0921 q^{65} +6.52324 q^{67} +4.05158 q^{68} +5.85180 q^{70} -2.84960 q^{71} +4.29281 q^{73} -4.44606 q^{74} +3.43987 q^{76} +6.08755 q^{77} -7.97329 q^{79} -2.69148 q^{80} -10.8990 q^{82} +1.80391 q^{83} -10.9048 q^{85} +7.91186 q^{86} -2.79991 q^{88} -10.9572 q^{89} +12.1915 q^{91} +0.907338 q^{92} -13.4073 q^{94} -9.25835 q^{95} +16.3978 q^{97} +2.27289 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29} + 11 q^{31} - 7 q^{32} + 16 q^{34} + 17 q^{37} - 2 q^{38} + 6 q^{40} - 18 q^{41} - q^{43} + q^{44} + 8 q^{46} - 5 q^{47} + 24 q^{49} - 19 q^{50} + 8 q^{52} - 6 q^{53} + 21 q^{55} - 3 q^{56} + 4 q^{58} + 7 q^{59} + 24 q^{61} - 11 q^{62} + 7 q^{64} - 11 q^{65} - 4 q^{67} - 16 q^{68} + 7 q^{71} + 28 q^{73} - 17 q^{74} + 2 q^{76} + 2 q^{77} - 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} + q^{86} - q^{88} - 5 q^{89} + 2 q^{91} - 8 q^{92} + 5 q^{94} + 14 q^{95} + 23 q^{97} - 24 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8 + 6 * q^10 + q^11 + 8 * q^13 - 3 * q^14 + 7 * q^16 - 16 * q^17 + 2 * q^19 - 6 * q^20 - q^22 - 8 * q^23 + 19 * q^25 - 8 * q^26 + 3 * q^28 - 4 * q^29 + 11 * q^31 - 7 * q^32 + 16 * q^34 + 17 * q^37 - 2 * q^38 + 6 * q^40 - 18 * q^41 - q^43 + q^44 + 8 * q^46 - 5 * q^47 + 24 * q^49 - 19 * q^50 + 8 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^56 + 4 * q^58 + 7 * q^59 + 24 * q^61 - 11 * q^62 + 7 * q^64 - 11 * q^65 - 4 * q^67 - 16 * q^68 + 7 * q^71 + 28 * q^73 - 17 * q^74 + 2 * q^76 + 2 * q^77 - 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 + q^86 - q^88 - 5 * q^89 + 2 * q^91 - 8 * q^92 + 5 * q^94 + 14 * q^95 + 23 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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