LMFDB - Embedded newform 1338.2.a.j.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1338,2,Mod(1,1338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1338.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1338 = 2 \cdot 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1338.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.6839837904$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.369925$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.69148 q^{5} -1.00000 q^{6} +2.17419 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.69148 q^{10} -2.79991 q^{11} -1.00000 q^{12} +5.60736 q^{13} +2.17419 q^{14} -2.69148 q^{15} +1.00000 q^{16} -4.05158 q^{17} +1.00000 q^{18} +3.43987 q^{19} +2.69148 q^{20} -2.17419 q^{21} -2.79991 q^{22} -0.907338 q^{23} -1.00000 q^{24} +2.24408 q^{25} +5.60736 q^{26} -1.00000 q^{27} +2.17419 q^{28} +3.53306 q^{29} -2.69148 q^{30} +2.49443 q^{31} +1.00000 q^{32} +2.79991 q^{33} -4.05158 q^{34} +5.85180 q^{35} +1.00000 q^{36} +4.44606 q^{37} +3.43987 q^{38} -5.60736 q^{39} +2.69148 q^{40} -10.8990 q^{41} -2.17419 q^{42} -7.91186 q^{43} -2.79991 q^{44} +2.69148 q^{45} -0.907338 q^{46} -13.4073 q^{47} -1.00000 q^{48} -2.27289 q^{49} +2.24408 q^{50} +4.05158 q^{51} +5.60736 q^{52} +13.1947 q^{53} -1.00000 q^{54} -7.53592 q^{55} +2.17419 q^{56} -3.43987 q^{57} +3.53306 q^{58} +11.3491 q^{59} -2.69148 q^{60} +0.466941 q^{61} +2.49443 q^{62} +2.17419 q^{63} +1.00000 q^{64} +15.0921 q^{65} +2.79991 q^{66} +6.52324 q^{67} -4.05158 q^{68} +0.907338 q^{69} +5.85180 q^{70} +2.84960 q^{71} +1.00000 q^{72} +4.29281 q^{73} +4.44606 q^{74} -2.24408 q^{75} +3.43987 q^{76} -6.08755 q^{77} -5.60736 q^{78} -7.97329 q^{79} +2.69148 q^{80} +1.00000 q^{81} -10.8990 q^{82} -1.80391 q^{83} -2.17419 q^{84} -10.9048 q^{85} -7.91186 q^{86} -3.53306 q^{87} -2.79991 q^{88} +10.9572 q^{89} +2.69148 q^{90} +12.1915 q^{91} -0.907338 q^{92} -2.49443 q^{93} -13.4073 q^{94} +9.25835 q^{95} -1.00000 q^{96} +16.3978 q^{97} -2.27289 q^{98} -2.79991 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} - 7 q^{3} + 7 q^{4} + 6 q^{5} - 7 q^{6} + 3 q^{7} + 7 q^{8} + 7 q^{9} + 6 q^{10} - q^{11} - 7 q^{12} + 8 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} + 16 q^{17} + 7 q^{18} + 2 q^{19} + 6 q^{20}+ \cdots - q^{99}+O(q^{100}) $ 7 * q + 7 * q^2 - 7 * q^3 + 7 * q^4 + 6 * q^5 - 7 * q^6 + 3 * q^7 + 7 * q^8 + 7 * q^9 + 6 * q^10 - q^11 - 7 * q^12 + 8 * q^13 + 3 * q^14 - 6 * q^15 + 7 * q^16 + 16 * q^17 + 7 * q^18 + 2 * q^19 + 6 * q^20 - 3 * q^21 - q^22 + 8 * q^23 - 7 * q^24 + 19 * q^25 + 8 * q^26 - 7 * q^27 + 3 * q^28 + 4 * q^29 - 6 * q^30 + 11 * q^31 + 7 * q^32 + q^33 + 16 * q^34 + 7 * q^36 + 17 * q^37 + 2 * q^38 - 8 * q^39 + 6 * q^40 + 18 * q^41 - 3 * q^42 - q^43 - q^44 + 6 * q^45 + 8 * q^46 + 5 * q^47 - 7 * q^48 + 24 * q^49 + 19 * q^50 - 16 * q^51 + 8 * q^52 + 6 * q^53 - 7 * q^54 + 21 * q^55 + 3 * q^56 - 2 * q^57 + 4 * q^58 - 7 * q^59 - 6 * q^60 + 24 * q^61 + 11 * q^62 + 3 * q^63 + 7 * q^64 + 11 * q^65 + q^66 - 4 * q^67 + 16 * q^68 - 8 * q^69 - 7 * q^71 + 7 * q^72 + 28 * q^73 + 17 * q^74 - 19 * q^75 + 2 * q^76 - 2 * q^77 - 8 * q^78 + 6 * q^80 + 7 * q^81 + 18 * q^82 - 2 * q^83 - 3 * q^84 + 4 * q^85 - q^86 - 4 * q^87 - q^88 + 5 * q^89 + 6 * q^90 + 2 * q^91 + 8 * q^92 - 11 * q^93 + 5 * q^94 - 14 * q^95 - 7 * q^96 + 23 * q^97 + 24 * q^98 - q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.69148	1.20367	0.601834	−	0.798621i	$-0.294437\pi$
$5$	2.69148	1.20367	0.601834	+	0.798621i	$0.294437\pi$
$6$	−1.00000	−0.408248
$7$	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
$9$	1.00000	0.333333
$10$	2.69148	0.851122
$11$	−2.79991	−0.844206	−0.422103	−	0.906548i	$-0.638708\pi$
$11$	−2.79991	−0.844206	−0.422103	+	0.906548i	$0.638708\pi$
$12$	−1.00000	−0.288675
$13$	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
$15$	−2.69148	−0.694938
$16$	1.00000	0.250000
$17$	−4.05158	−0.982654	−0.491327	−	0.870975i	$-0.663488\pi$
$17$	−4.05158	−0.982654	−0.491327	+	0.870975i	$0.663488\pi$
$18$	1.00000	0.235702
$19$	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
$21$	−2.17419	−0.474448
$22$	−2.79991	−0.596944
$23$	−0.907338	−0.189193	−0.0945965	−	0.995516i	$-0.530156\pi$
$23$	−0.907338	−0.189193	−0.0945965	+	0.995516i	$0.530156\pi$
$24$	−1.00000	−0.204124
$25$	2.24408	0.448816
$26$	5.60736	1.09969
$27$	−1.00000	−0.192450
$28$	2.17419	0.410884
$29$	3.53306	0.656073	0.328036	−	0.944665i	$-0.393613\pi$
$29$	3.53306	0.656073	0.328036	+	0.944665i	$0.393613\pi$
$30$	−2.69148	−0.491395
$31$	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
$33$	2.79991	0.487402
$34$	−4.05158	−0.694841
$35$	5.85180	0.989135
$36$	1.00000	0.166667
$37$	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
$39$	−5.60736	−0.897896
$40$	2.69148	0.425561
$41$	−10.8990	−1.70214	−0.851071	−	0.525050i	$-0.824047\pi$
$41$	−10.8990	−1.70214	−0.851071	+	0.525050i	$0.824047\pi$
$42$	−2.17419	−0.335485
$43$	−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
$45$	2.69148	0.401223
$46$	−0.907338	−0.133780
$47$	−13.4073	−1.95566	−0.977829	−	0.209406i	$-0.932847\pi$
$47$	−13.4073	−1.95566	−0.977829	+	0.209406i	$0.932847\pi$
$48$	−1.00000	−0.144338
$49$	−2.27289	−0.324698
$50$	2.24408	0.317361
$51$	4.05158	0.567335
$52$	5.60736	0.777601
$53$	13.1947	1.81243	0.906213	−	0.422822i	$-0.138961\pi$
$53$	13.1947	1.81243	0.906213	+	0.422822i	$0.138961\pi$
$54$	−1.00000	−0.136083
$55$	−7.53592	−1.01614
$56$	2.17419	0.290539
$57$	−3.43987	−0.455622
$58$	3.53306	0.463913
$59$	11.3491	1.47752	0.738762	−	0.673966i	$-0.235411\pi$
$59$	11.3491	1.47752	0.738762	+	0.673966i	$0.235411\pi$
$60$	−2.69148	−0.347469
$61$	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
$63$	2.17419	0.273922
$64$	1.00000	0.125000
$65$	15.0921	1.87195
$66$	2.79991	0.344646
$67$	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
$69$	0.907338	0.109231
$70$	5.85180	0.699424
$71$	2.84960	0.338186	0.169093	−	0.985600i	$-0.445916\pi$
$71$	2.84960	0.338186	0.169093	+	0.985600i	$0.445916\pi$
$72$	1.00000	0.117851
$73$	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
$75$	−2.24408	−0.259124
$76$	3.43987	0.394580
$77$	−6.08755	−0.693741
$78$	−5.60736	−0.634908
$79$	−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
$81$	1.00000	0.111111
$82$	−10.8990	−1.20360
$83$	−1.80391	−0.198005	−0.0990024	−	0.995087i	$-0.531565\pi$
$83$	−1.80391	−0.198005	−0.0990024	+	0.995087i	$0.531565\pi$
$84$	−2.17419	−0.237224
$85$	−10.9048	−1.18279
$86$	−7.91186	−0.853158
$87$	−3.53306	−0.378784
$88$	−2.79991	−0.298472
$89$	10.9572	1.16146	0.580730	−	0.814096i	$-0.302767\pi$
$89$	10.9572	1.16146	0.580730	+	0.814096i	$0.302767\pi$
$90$	2.69148	0.283707
$91$	12.1915	1.27801
$92$	−0.907338	−0.0945965
$93$	−2.49443	−0.258660
$94$	−13.4073	−1.38286
$95$	9.25835	0.949887
$96$	−1.00000	−0.102062
$97$	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
$99$	−2.79991	−0.281402
$100$	2.24408	0.224408
$101$	3.53689	0.351933	0.175967	−	0.984396i	$-0.443695\pi$
$101$	3.53689	0.351933	0.175967	+	0.984396i	$0.443695\pi$
$102$	4.05158	0.401167
$103$	−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
$105$	−5.85180	−0.571077
$106$	13.1947	1.28158
$107$	−8.32832	−0.805129	−0.402565	−	0.915392i	$-0.631881\pi$
$107$	−8.32832	−0.805129	−0.402565	+	0.915392i	$0.631881\pi$
$108$	−1.00000	−0.0962250
$109$	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
$111$	−4.44606	−0.422001
$112$	2.17419	0.205442
$113$	13.6313	1.28233	0.641165	−	0.767403i	$-0.278452\pi$
$113$	13.6313	1.28233	0.641165	+	0.767403i	$0.278452\pi$
$114$	−3.43987	−0.322173
$115$	−2.44208	−0.227725
$116$	3.53306	0.328036
$117$	5.60736	0.518401
$118$	11.3491	1.04477
$119$	−8.80892	−0.807513
$120$	−2.69148	−0.245698
$121$	−3.16049	−0.287317
$122$	0.466941	0.0422749
$123$	10.8990	0.982733
$124$	2.49443	0.224006
$125$	−7.41751	−0.663443
$126$	2.17419	0.193692
$127$	−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
$129$	7.91186	0.696601
$130$	15.0921	1.32367
$131$	−17.7178	−1.54801	−0.774004	−	0.633181i	$-0.781749\pi$
$131$	−17.7178	−1.54801	−0.774004	+	0.633181i	$0.781749\pi$
$132$	2.79991	0.243701
$133$	7.47894	0.648506
$134$	6.52324	0.563522
$135$	−2.69148	−0.231646
$136$	−4.05158	−0.347420
$137$	−15.2395	−1.30200	−0.650998	−	0.759079i	$-0.725649\pi$
$137$	−15.2395	−1.30200	−0.650998	+	0.759079i	$0.725649\pi$
$138$	0.907338	0.0772377
$139$	−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
$141$	13.4073	1.12910
$142$	2.84960	0.239133
$143$	−15.7001	−1.31291
$144$	1.00000	0.0833333
$145$	9.50917	0.789693
$146$	4.29281	0.355275
$147$	2.27289	0.187465
$148$	4.44606	0.365464
$149$	−14.5888	−1.19516	−0.597579	−	0.801810i	$-0.703871\pi$
$149$	−14.5888	−1.19516	−0.597579	+	0.801810i	$0.703871\pi$
$150$	−2.24408	−0.183228
$151$	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
$153$	−4.05158	−0.327551
$154$	−6.08755	−0.490549
$155$	6.71371	0.539258
$156$	−5.60736	−0.448948
$157$	−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
$159$	−13.1947	−1.04640
$160$	2.69148	0.212780
$161$	−1.97273	−0.155473
$162$	1.00000	0.0785674
$163$	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
$165$	7.53592	0.586670
$166$	−1.80391	−0.140011
$167$	1.39325	0.107813	0.0539065	−	0.998546i	$-0.482833\pi$
$167$	1.39325	0.107813	0.0539065	+	0.998546i	$0.482833\pi$
$168$	−2.17419	−0.167743
$169$	18.4425	1.41865
$170$	−10.9048	−0.836358
$171$	3.43987	0.263053
$172$	−7.91186	−0.603274
$173$	−16.6528	−1.26609	−0.633043	−	0.774117i	$-0.718194\pi$
$173$	−16.6528	−1.26609	−0.633043	+	0.774117i	$0.718194\pi$
$174$	−3.53306	−0.267841
$175$	4.87906	0.368822
$176$	−2.79991	−0.211051
$177$	−11.3491	−0.853049
$178$	10.9572	0.821277
$179$	−12.9554	−0.968331	−0.484165	−	0.874977i	$-0.660877\pi$
$179$	−12.9554	−0.968331	−0.484165	+	0.874977i	$0.660877\pi$
$180$	2.69148	0.200611
$181$	−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
$183$	−0.466941	−0.0345173
$184$	−0.907338	−0.0668898
$185$	11.9665	0.879794
$186$	−2.49443	−0.182900
$187$	11.3441	0.829562
$188$	−13.4073	−0.977829
$189$	−2.17419	−0.158149
$190$	9.25835	0.671672
$191$	2.78928	0.201825	0.100912	−	0.994895i	$-0.467824\pi$
$191$	2.78928	0.201825	0.100912	+	0.994895i	$0.467824\pi$
$192$	−1.00000	−0.0721688
$193$	−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
$195$	−15.0921	−1.08077
$196$	−2.27289	−0.162349
$197$	20.5331	1.46293	0.731463	−	0.681881i	$-0.238838\pi$
$197$	20.5331	1.46293	0.731463	+	0.681881i	$0.238838\pi$
$198$	−2.79991	−0.198981
$199$	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
$201$	−6.52324	−0.460114
$202$	3.53689	0.248855
$203$	7.68155	0.539139
$204$	4.05158	0.283668
$205$	−29.3346	−2.04881
$206$	−5.03869	−0.351062
$207$	−0.907338	−0.0630643
$208$	5.60736	0.388800
$209$	−9.63134	−0.666214
$210$	−5.85180	−0.403813
$211$	−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
$213$	−2.84960	−0.195252
$214$	−8.32832	−0.569312
$215$	−21.2946	−1.45228
$216$	−1.00000	−0.0680414
$217$	5.42336	0.368162
$218$	2.72745	0.184726
$219$	−4.29281	−0.290081
$220$	−7.53592	−0.508072
$221$	−22.7187	−1.52822
$222$	−4.44606	−0.298400
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	2.17419	0.145269
$225$	2.24408	0.149605
$226$	13.6313	0.906744
$227$	−19.8177	−1.31535	−0.657675	−	0.753302i	$-0.728460\pi$
$227$	−19.8177	−1.31535	−0.657675	+	0.753302i	$0.728460\pi$
$228$	−3.43987	−0.227811
$229$	−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
$231$	6.08755	0.400531
$232$	3.53306	0.231957
$233$	8.41472	0.551266	0.275633	−	0.961263i	$-0.411112\pi$
$233$	8.41472	0.551266	0.275633	+	0.961263i	$0.411112\pi$
$234$	5.60736	0.366565
$235$	−36.0855	−2.35396
$236$	11.3491	0.738762
$237$	7.97329	0.517921
$238$	−8.80892	−0.570998
$239$	−28.9489	−1.87255	−0.936273	−	0.351272i	$-0.885749\pi$
$239$	−28.9489	−1.87255	−0.936273	+	0.351272i	$0.885749\pi$
$240$	−2.69148	−0.173734
$241$	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
$243$	−1.00000	−0.0641500
$244$	0.466941	0.0298928
$245$	−6.11744	−0.390829
$246$	10.8990	0.694897
$247$	19.2886	1.22730
$248$	2.49443	0.158396
$249$	1.80391	0.114318
$250$	−7.41751	−0.469125
$251$	14.7729	0.932456	0.466228	−	0.884665i	$-0.345613\pi$
$251$	14.7729	0.932456	0.466228	+	0.884665i	$0.345613\pi$
$252$	2.17419	0.136961
$253$	2.54047	0.159718
$254$	−13.6128	−0.854140
$255$	10.9048	0.682883
$256$	1.00000	0.0625000
$257$	−20.3220	−1.26765	−0.633827	−	0.773475i	$-0.718517\pi$
$257$	−20.3220	−1.26765	−0.633827	+	0.773475i	$0.718517\pi$
$258$	7.91186	0.492571
$259$	9.66659	0.600653
$260$	15.0921	0.935973
$261$	3.53306	0.218691
$262$	−17.7178	−1.09461
$263$	−13.9440	−0.859826	−0.429913	−	0.902870i	$-0.641456\pi$
$263$	−13.9440	−0.859826	−0.429913	+	0.902870i	$0.641456\pi$
$264$	2.79991	0.172323
$265$	35.5132	2.18156
$266$	7.47894	0.458563
$267$	−10.9572	−0.670570
$268$	6.52324	0.398470
$269$	−1.33483	−0.0813862	−0.0406931	−	0.999172i	$-0.512957\pi$
$269$	−1.33483	−0.0813862	−0.0406931	+	0.999172i	$0.512957\pi$
$270$	−2.69148	−0.163798
$271$	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
$273$	−12.1915	−0.737862
$274$	−15.2395	−0.920650
$275$	−6.28323	−0.378893
$276$	0.907338	0.0546153
$277$	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
$279$	2.49443	0.149337
$280$	5.85180	0.349712
$281$	8.16492	0.487078	0.243539	−	0.969891i	$-0.421692\pi$
$281$	8.16492	0.487078	0.243539	+	0.969891i	$0.421692\pi$
$282$	13.4073	0.798394
$283$	−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
$285$	−9.25835	−0.548417
$286$	−15.7001	−0.928368
$287$	−23.6966	−1.39877
$288$	1.00000	0.0589256
$289$	−0.584665	−0.0343921
$290$	9.50917	0.558398
$291$	−16.3978	−0.961257
$292$	4.29281	0.251218
$293$	−22.2675	−1.30088	−0.650439	−	0.759558i	$-0.725415\pi$
$293$	−22.2675	−1.30088	−0.650439	+	0.759558i	$0.725415\pi$
$294$	2.27289	0.132558
$295$	30.5459	1.77845
$296$	4.44606	0.258422
$297$	2.79991	0.162467
$298$	−14.5888	−0.845105
$299$	−5.08777	−0.294233
$300$	−2.24408	−0.129562
$301$	−17.2019	−0.991502
$302$	12.6330	0.726946
$303$	−3.53689	−0.203189
$304$	3.43987	0.197290
$305$	1.25676	0.0719621
$306$	−4.05158	−0.231614
$307$	−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
$309$	5.03869	0.286641
$310$	6.71371	0.381313
$311$	5.19327	0.294483	0.147242	−	0.989101i	$-0.452960\pi$
$311$	5.19327	0.294483	0.147242	+	0.989101i	$0.452960\pi$
$312$	−5.60736	−0.317454
$313$	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
$315$	5.85180	0.329712
$316$	−7.97329	−0.448533
$317$	16.7624	0.941470	0.470735	−	0.882275i	$-0.343989\pi$
$317$	16.7624	0.941470	0.470735	+	0.882275i	$0.343989\pi$
$318$	−13.1947	−0.739920
$319$	−9.89226	−0.553860
$320$	2.69148	0.150458
$321$	8.32832	0.464842
$322$	−1.97273	−0.109936
$323$	−13.9369	−0.775471
$324$	1.00000	0.0555556
$325$	12.5834	0.697999
$326$	0.156053	0.00864295
$327$	−2.72745	−0.150828
$328$	−10.8990	−0.601798
$329$	−29.1501	−1.60710
$330$	7.53592	0.414839
$331$	−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
$333$	4.44606	0.243643
$334$	1.39325	0.0762353
$335$	17.5572	0.959251
$336$	−2.17419	−0.118612
$337$	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
$339$	−13.6313	−0.740353
$340$	−10.9048	−0.591394
$341$	−6.98418	−0.378214
$342$	3.43987	0.186007
$343$	−20.1610	−1.08859
$344$	−7.91186	−0.426579
$345$	2.44208	0.131477
$346$	−16.6528	−0.895258
$347$	20.1398	1.08116	0.540580	−	0.841293i	$-0.318205\pi$
$347$	20.1398	1.08116	0.540580	+	0.841293i	$0.318205\pi$
$348$	−3.53306	−0.189392
$349$	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
$351$	−5.60736	−0.299299
$352$	−2.79991	−0.149236
$353$	26.7829	1.42551	0.712755	−	0.701413i	$-0.247447\pi$
$353$	26.7829	1.42551	0.712755	+	0.701413i	$0.247447\pi$
$354$	−11.3491	−0.603197
$355$	7.66966	0.407063
$356$	10.9572	0.580730
$357$	8.80892	0.466218
$358$	−12.9554	−0.684713
$359$	9.32994	0.492415	0.246208	−	0.969217i	$-0.420815\pi$
$359$	9.32994	0.492415	0.246208	+	0.969217i	$0.420815\pi$
$360$	2.69148	0.141854
$361$	−7.16729	−0.377226
$362$	−8.76887	−0.460881
$363$	3.16049	0.165882
$364$	12.1915	0.639007
$365$	11.5540	0.604765
$366$	−0.466941	−0.0244074
$367$	−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
$369$	−10.8990	−0.567381
$370$	11.9665	0.622108
$371$	28.6877	1.48939
$372$	−2.49443	−0.129330
$373$	−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
$375$	7.41751	0.383039
$376$	−13.4073	−0.691429
$377$	19.8111	1.02033
$378$	−2.17419	−0.111828
$379$	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
$381$	13.6128	0.697403
$382$	2.78928	0.142712
$383$	−21.8176	−1.11483	−0.557414	−	0.830235i	$-0.688206\pi$
$383$	−21.8176	−1.11483	−0.557414	+	0.830235i	$0.688206\pi$
$384$	−1.00000	−0.0510310
$385$	−16.3845	−0.835033
$386$	−11.6058	−0.590718
$387$	−7.91186	−0.402183
$388$	16.3978	0.832473
$389$	−16.2181	−0.822290	−0.411145	−	0.911570i	$-0.634871\pi$
$389$	−16.2181	−0.822290	−0.411145	+	0.911570i	$0.634871\pi$
$390$	−15.0921	−0.764219
$391$	3.67616	0.185911
$392$	−2.27289	−0.114798
$393$	17.7178	0.893743
$394$	20.5331	1.03444
$395$	−21.4600	−1.07977
$396$	−2.79991	−0.140701
$397$	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
$399$	−7.47894	−0.374415
$400$	2.24408	0.112204
$401$	−33.0190	−1.64889	−0.824446	−	0.565940i	$-0.808513\pi$
$401$	−33.0190	−1.64889	−0.824446	+	0.565940i	$0.808513\pi$
$402$	−6.52324	−0.325349
$403$	13.9871	0.696749
$404$	3.53689	0.175967
$405$	2.69148	0.133741
$406$	7.68155	0.381229
$407$	−12.4486	−0.617053
$408$	4.05158	0.200583
$409$	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
$411$	15.2395	0.751708
$412$	−5.03869	−0.248239
$413$	24.6751	1.21418
$414$	−0.907338	−0.0445932
$415$	−4.85519	−0.238332
$416$	5.60736	0.274923
$417$	4.87274	0.238619
$418$	−9.63134	−0.471084
$419$	−8.61709	−0.420972	−0.210486	−	0.977597i	$-0.567505\pi$
$419$	−8.61709	−0.420972	−0.210486	+	0.977597i	$0.567505\pi$
$420$	−5.85180	−0.285539
$421$	−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
$423$	−13.4073	−0.651886
$424$	13.1947	0.640789
$425$	−9.09208	−0.441030
$426$	−2.84960	−0.138064
$427$	1.01522	0.0491299
$428$	−8.32832	−0.402565
$429$	15.7001	0.758009
$430$	−21.2946	−1.02692
$431$	7.17551	0.345632	0.172816	−	0.984954i	$-0.444713\pi$
$431$	7.17551	0.345632	0.172816	+	0.984954i	$0.444713\pi$
$432$	−1.00000	−0.0481125
$433$	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
$435$	−9.50917	−0.455930
$436$	2.72745	0.130621
$437$	−3.12112	−0.149304
$438$	−4.29281	−0.205118
$439$	−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
$441$	−2.27289	−0.108233
$442$	−22.7187	−1.08062
$443$	−23.6850	−1.12531	−0.562654	−	0.826693i	$-0.690220\pi$
$443$	−23.6850	−1.12531	−0.562654	+	0.826693i	$0.690220\pi$
$444$	−4.44606	−0.211001
$445$	29.4911	1.39801
$446$	−1.00000	−0.0473514
$447$	14.5888	0.690025
$448$	2.17419	0.102721
$449$	26.0448	1.22913	0.614565	−	0.788866i	$-0.289332\pi$
$449$	26.0448	1.22913	0.614565	+	0.788866i	$0.289332\pi$
$450$	2.24408	0.105787
$451$	30.5163	1.43696
$452$	13.6313	0.641165
$453$	−12.6330	−0.593549
$454$	−19.8177	−0.930093
$455$	32.8131	1.53830
$456$	−3.43987	−0.161087
$457$	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
$459$	4.05158	0.189112
$460$	−2.44208	−0.113863
$461$	−1.34885	−0.0628221	−0.0314110	−	0.999507i	$-0.510000\pi$
$461$	−1.34885	−0.0628221	−0.0314110	+	0.999507i	$0.510000\pi$
$462$	6.08755	0.283218
$463$	−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
$465$	−6.71371	−0.311341
$466$	8.41472	0.389804
$467$	30.1602	1.39565	0.697824	−	0.716269i	$-0.254152\pi$
$467$	30.1602	1.39565	0.697824	+	0.716269i	$0.254152\pi$
$468$	5.60736	0.259200
$469$	14.1828	0.654899
$470$	−36.0855	−1.66450
$471$	12.0602	0.555704
$472$	11.3491	0.522384
$473$	22.1525	1.01857
$474$	7.97329	0.366225
$475$	7.71934	0.354188
$476$	−8.80892	−0.403756
$477$	13.1947	0.604142
$478$	−28.9489	−1.32409
$479$	10.4392	0.476979	0.238490	−	0.971145i	$-0.423348\pi$
$479$	10.4392	0.476979	0.238490	+	0.971145i	$0.423348\pi$
$480$	−2.69148	−0.122849
$481$	24.9307	1.13674
$482$	2.56905	0.117017
$483$	1.97273	0.0897622
$484$	−3.16049	−0.143658
$485$	44.1344	2.00404
$486$	−1.00000	−0.0453609
$487$	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
$489$	−0.156053	−0.00705694
$490$	−6.11744	−0.276358
$491$	17.6989	0.798739	0.399370	−	0.916790i	$-0.369229\pi$
$491$	17.6989	0.798739	0.399370	+	0.916790i	$0.369229\pi$
$492$	10.8990	0.491366
$493$	−14.3145	−0.644692
$494$	19.2886	0.867835
$495$	−7.53592	−0.338714
$496$	2.49443	0.112003
$497$	6.19559	0.277910
$498$	1.80391	0.0808351
$499$	−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
$501$	−1.39325	−0.0622458
$502$	14.7729	0.659346
$503$	−26.2160	−1.16892	−0.584458	−	0.811424i	$-0.698693\pi$
$503$	−26.2160	−1.16892	−0.584458	+	0.811424i	$0.698693\pi$
$504$	2.17419	0.0968462
$505$	9.51947	0.423611
$506$	2.54047	0.112938
$507$	−18.4425	−0.819059
$508$	−13.6128	−0.603969
$509$	1.95494	0.0866510	0.0433255	−	0.999061i	$-0.486205\pi$
$509$	1.95494	0.0866510	0.0433255	+	0.999061i	$0.486205\pi$
$510$	10.9048	0.482871
$511$	9.33339	0.412885
$512$	1.00000	0.0441942
$513$	−3.43987	−0.151874
$514$	−20.3220	−0.896367
$515$	−13.5616	−0.597593
$516$	7.91186	0.348300
$517$	37.5393	1.65098
$518$	9.66659	0.424726
$519$	16.6528	0.730975
$520$	15.0921	0.661833
$521$	23.9402	1.04884	0.524420	−	0.851459i	$-0.324282\pi$
$521$	23.9402	1.04884	0.524420	+	0.851459i	$0.324282\pi$
$522$	3.53306	0.154638
$523$	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
$525$	−4.87906	−0.212940
$526$	−13.9440	−0.607989
$527$	−10.1064	−0.440241
$528$	2.79991	0.121851
$529$	−22.1767	−0.964206
$530$	35.5132	1.54259
$531$	11.3491	0.492508
$532$	7.47894	0.324253
$533$	−61.1148	−2.64718
$534$	−10.9572	−0.474164
$535$	−22.4155	−0.969108
$536$	6.52324	0.281761
$537$	12.9554	0.559066
$538$	−1.33483	−0.0575487
$539$	6.36389	0.274112
$540$	−2.69148	−0.115823
$541$	−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
$543$	8.76887	0.376308
$544$	−4.05158	−0.173710
$545$	7.34088	0.314449
$546$	−12.1915	−0.521747
$547$	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
$549$	0.466941	0.0199286
$550$	−6.28323	−0.267918
$551$	12.1533	0.517747
$552$	0.907338	0.0386189
$553$	−17.3355	−0.737179
$554$	6.92461	0.294198
$555$	−11.9665	−0.507949
$556$	−4.87274	−0.206650
$557$	33.3538	1.41325	0.706624	−	0.707590i	$-0.250217\pi$
$557$	33.3538	1.41325	0.706624	+	0.707590i	$0.250217\pi$
$558$	2.49443	0.105597
$559$	−44.3647	−1.87643
$560$	5.85180	0.247284
$561$	−11.3441	−0.478948
$562$	8.16492	0.344416
$563$	19.0344	0.802205	0.401102	−	0.916033i	$-0.368627\pi$
$563$	19.0344	0.802205	0.401102	+	0.916033i	$0.368627\pi$
$564$	13.4073	0.564550
$565$	36.6885	1.54350
$566$	−13.9290	−0.585479
$567$	2.17419	0.0913075
$568$	2.84960	0.119567
$569$	−36.6564	−1.53672	−0.768358	−	0.640020i	$-0.778926\pi$
$569$	−36.6564	−1.53672	−0.768358	+	0.640020i	$0.778926\pi$
$570$	−9.25835	−0.387790
$571$	−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
$573$	−2.78928	−0.116524
$574$	−23.6966	−0.989076
$575$	−2.03614	−0.0849128
$576$	1.00000	0.0416667
$577$	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
$579$	11.6058	0.482319
$580$	9.50917	0.394847
$581$	−3.92205	−0.162714
$582$	−16.3978	−0.679711
$583$	−36.9439	−1.53006
$584$	4.29281	0.177638
$585$	15.0921	0.623982
$586$	−22.2675	−0.919860
$587$	−26.5520	−1.09592	−0.547960	−	0.836505i	$-0.684595\pi$
$587$	−26.5520	−1.09592	−0.547960	+	0.836505i	$0.684595\pi$
$588$	2.27289	0.0937323
$589$	8.58051	0.353554
$590$	30.5459	1.25755
$591$	−20.5331	−0.844620
$592$	4.44606	0.182732
$593$	−13.9180	−0.571543	−0.285771	−	0.958298i	$-0.592250\pi$
$593$	−13.9180	−0.571543	−0.285771	+	0.958298i	$0.592250\pi$
$594$	2.79991	0.114882
$595$	−23.7091	−0.971977
$596$	−14.5888	−0.597579
$597$	−17.9919	−0.736359
$598$	−5.08777	−0.208054
$599$	−3.31413	−0.135412	−0.0677059	−	0.997705i	$-0.521568\pi$
$599$	−3.31413	−0.135412	−0.0677059	+	0.997705i	$0.521568\pi$
$600$	−2.24408	−0.0916142
$601$	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
$603$	6.52324	0.265647
$604$	12.6330	0.514028
$605$	−8.50639	−0.345834
$606$	−3.53689	−0.143676
$607$	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
$609$	−7.68155	−0.311272
$610$	1.25676	0.0508849
$611$	−75.1796	−3.04144
$612$	−4.05158	−0.163776
$613$	−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
$615$	29.3346	1.18288
$616$	−6.08755	−0.245274
$617$	6.14776	0.247500	0.123750	−	0.992313i	$-0.460508\pi$
$617$	6.14776	0.247500	0.123750	+	0.992313i	$0.460508\pi$
$618$	5.03869	0.202686
$619$	−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
$621$	0.907338	0.0364102
$622$	5.19327	0.208231
$623$	23.8231	0.954451
$624$	−5.60736	−0.224474
$625$	−31.1845	−1.24738
$626$	1.77635	0.0709971
$627$	9.63134	0.384639
$628$	−12.0602	−0.481254
$629$	−18.0136	−0.718249
$630$	5.85180	0.233141
$631$	−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
$633$	3.08882	0.122769
$634$	16.7624	0.665720
$635$	−36.6385	−1.45395
$636$	−13.1947	−0.523202
$637$	−12.7449	−0.504971
$638$	−9.89226	−0.391638
$639$	2.84960	0.112729
$640$	2.69148	0.106390
$641$	−31.0690	−1.22715	−0.613575	−	0.789636i	$-0.710269\pi$
$641$	−31.0690	−1.22715	−0.613575	+	0.789636i	$0.710269\pi$
$642$	8.32832	0.328693
$643$	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
$645$	21.2946	0.838476
$646$	−13.9369	−0.548341
$647$	46.2606	1.81869	0.909346	−	0.416042i	$-0.136583\pi$
$647$	46.2606	1.81869	0.909346	+	0.416042i	$0.136583\pi$
$648$	1.00000	0.0392837
$649$	−31.7764	−1.24733
$650$	12.5834	0.493560
$651$	−5.42336	−0.212558
$652$	0.156053	0.00611149
$653$	−39.3565	−1.54014	−0.770068	−	0.637961i	$-0.779778\pi$
$653$	−39.3565	−1.54014	−0.770068	+	0.637961i	$0.779778\pi$
$654$	−2.72745	−0.106652
$655$	−47.6870	−1.86329
$656$	−10.8990	−0.425536
$657$	4.29281	0.167478
$658$	−29.1501	−1.13639
$659$	−19.6764	−0.766485	−0.383242	−	0.923648i	$-0.625193\pi$
$659$	−19.6764	−0.766485	−0.383242	+	0.923648i	$0.625193\pi$
$660$	7.53592	0.293335
$661$	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
$663$	22.7187	0.882321
$664$	−1.80391	−0.0700053
$665$	20.1294	0.780586
$666$	4.44606	0.172281
$667$	−3.20568	−0.124124
$668$	1.39325	0.0539065
$669$	1.00000	0.0386622
$670$	17.5572	0.678293
$671$	−1.30739	−0.0504714
$672$	−2.17419	−0.0838713
$673$	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
$675$	−2.24408	−0.0863747
$676$	18.4425	0.709326
$677$	21.4801	0.825548	0.412774	−	0.910833i	$-0.364560\pi$
$677$	21.4801	0.825548	0.412774	+	0.910833i	$0.364560\pi$
$678$	−13.6313	−0.523509
$679$	35.6520	1.36820
$680$	−10.9048	−0.418179
$681$	19.8177	0.759418
$682$	−6.98418	−0.267438
$683$	43.7953	1.67578	0.837890	−	0.545839i	$-0.183789\pi$
$683$	43.7953	1.67578	0.837890	+	0.545839i	$0.183789\pi$
$684$	3.43987	0.131527
$685$	−41.0168	−1.56717
$686$	−20.1610	−0.769752
$687$	25.3745	0.968099
$688$	−7.91186	−0.301637
$689$	73.9872	2.81869
$690$	2.44208	0.0929685
$691$	−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
$693$	−6.08755	−0.231247
$694$	20.1398	0.764495
$695$	−13.1149	−0.497476
$696$	−3.53306	−0.133920
$697$	44.1583	1.67262
$698$	30.6684	1.16082
$699$	−8.41472	−0.318274
$700$	4.87906	0.184411
$701$	−45.2816	−1.71026	−0.855132	−	0.518411i	$-0.826524\pi$
$701$	−45.2816	−1.71026	−0.855132	+	0.518411i	$0.826524\pi$
$702$	−5.60736	−0.211636
$703$	15.2939	0.576819
$704$	−2.79991	−0.105526
$705$	36.0855	1.35906
$706$	26.7829	1.00799
$707$	7.68987	0.289207
$708$	−11.3491	−0.426525
$709$	−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
$711$	−7.97329	−0.299022
$712$	10.9572	0.410638
$713$	−2.26329	−0.0847608
$714$	8.80892	0.329666
$715$	−42.2566	−1.58031
$716$	−12.9554	−0.484165
$717$	28.9489	1.08112
$718$	9.32994	0.348190
$719$	25.6525	0.956675	0.478338	−	0.878176i	$-0.341240\pi$
$719$	25.6525	0.956675	0.478338	+	0.878176i	$0.341240\pi$
$720$	2.69148	0.100306
$721$	−10.9551	−0.407989
$722$	−7.16729	−0.266739
$723$	−2.56905	−0.0955440
$724$	−8.76887	−0.325892
$725$	7.92846	0.294456
$726$	3.16049	0.117297
$727$	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
$729$	1.00000	0.0370370
$730$	11.5540	0.427633
$731$	32.0556	1.18562
$732$	−0.466941	−0.0172586
$733$	−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
$735$	6.11744	0.225645
$736$	−0.907338	−0.0334449
$737$	−18.2645	−0.672781
$738$	−10.8990	−0.401199
$739$	−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
$741$	−19.2886	−0.708584
$742$	28.6877	1.05316
$743$	12.7178	0.466572	0.233286	−	0.972408i	$-0.425052\pi$
$743$	12.7178	0.466572	0.233286	+	0.972408i	$0.425052\pi$
$744$	−2.49443	−0.0914501
$745$	−39.2654	−1.43857
$746$	−24.5202	−0.897750
$747$	−1.80391	−0.0660016
$748$	11.3441	0.414781
$749$	−18.1074	−0.661629
$750$	7.41751	0.270849
$751$	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
$753$	−14.7729	−0.538354
$754$	19.8111	0.721479
$755$	34.0014	1.23744
$756$	−2.17419	−0.0790746
$757$	−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
$759$	−2.54047	−0.0922131
$760$	9.25835	0.335836
$761$	8.48297	0.307508	0.153754	−	0.988109i	$-0.450864\pi$
$761$	8.48297	0.307508	0.153754	+	0.988109i	$0.450864\pi$
$762$	13.6128	0.493138
$763$	5.93000	0.214680
$764$	2.78928	0.100912
$765$	−10.9048	−0.394263
$766$	−21.8176	−0.788303
$767$	63.6384	2.29785
$768$	−1.00000	−0.0360844
$769$	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
$771$	20.3220	0.731881
$772$	−11.6058	−0.417701
$773$	30.8916	1.11109	0.555546	−	0.831486i	$-0.312509\pi$
$773$	30.8916	1.11109	0.555546	+	0.831486i	$0.312509\pi$
$774$	−7.91186	−0.284386
$775$	5.59769	0.201075
$776$	16.3978	0.588647
$777$	−9.66659	−0.346787
$778$	−16.2181	−0.581447
$779$	−37.4913	−1.34326
$780$	−15.0921	−0.540384
$781$	−7.97864	−0.285498
$782$	3.67616	0.131459
$783$	−3.53306	−0.126261
$784$	−2.27289	−0.0811746
$785$	−32.4598	−1.15854
$786$	17.7178	0.631971
$787$	−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
$789$	13.9440	0.496421
$790$	−21.4600	−0.763512
$791$	29.6372	1.05378
$792$	−2.79991	−0.0994906
$793$	2.61831	0.0929788
$794$	1.66612	0.0591283
$795$	−35.5132	−1.25952
$796$	17.9919	0.637706
$797$	24.3923	0.864020	0.432010	−	0.901869i	$-0.357805\pi$
$797$	24.3923	0.864020	0.432010	+	0.901869i	$0.357805\pi$
$798$	−7.47894	−0.264752
$799$	54.3209	1.92173
$800$	2.24408	0.0793402
$801$	10.9572	0.387154
$802$	−33.0190	−1.16594
$803$	−12.0195	−0.424159
$804$	−6.52324	−0.230057
$805$	−5.30956	−0.187137
$806$	13.9871	0.492676
$807$	1.33483	0.0469883
$808$	3.53689	0.124427
$809$	−29.6845	−1.04365	−0.521825	−	0.853052i	$-0.674749\pi$
$809$	−29.6845	−1.04365	−0.521825	+	0.853052i	$0.674749\pi$
$810$	2.69148	0.0945691
$811$	−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
$813$	−18.0818	−0.634158
$814$	−12.4486	−0.436323
$815$	0.420013	0.0147124
$816$	4.05158	0.141834
$817$	−27.2158	−0.952160
$818$	33.0353	1.15505
$819$	12.1915	0.426005
$820$	−29.3346	−1.02441
$821$	21.5725	0.752884	0.376442	−	0.926440i	$-0.377147\pi$
$821$	21.5725	0.752884	0.376442	+	0.926440i	$0.377147\pi$
$822$	15.2395	0.531538
$823$	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
$825$	6.28323	0.218754
$826$	24.6751	0.858556
$827$	−6.69089	−0.232665	−0.116333	−	0.993210i	$-0.537114\pi$
$827$	−6.69089	−0.232665	−0.116333	+	0.993210i	$0.537114\pi$
$828$	−0.907338	−0.0315322
$829$	−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
$831$	−6.92461	−0.240212
$832$	5.60736	0.194400
$833$	9.20880	0.319066
$834$	4.87274	0.168729
$835$	3.74991	0.129771
$836$	−9.63134	−0.333107
$837$	−2.49443	−0.0862200
$838$	−8.61709	−0.297672
$839$	−25.5944	−0.883618	−0.441809	−	0.897109i	$-0.645663\pi$
$839$	−25.5944	−0.883618	−0.441809	+	0.897109i	$0.645663\pi$
$840$	−5.85180	−0.201906
$841$	−16.5175	−0.569569
$842$	−13.7662	−0.474413
$843$	−8.16492	−0.281215
$844$	−3.08882	−0.106321
$845$	49.6376	1.70759
$846$	−13.4073	−0.460953
$847$	−6.87150	−0.236108
$848$	13.1947	0.453106
$849$	13.9290	0.478042
$850$	−9.09208	−0.311856
$851$	−4.03408	−0.138286
$852$	−2.84960	−0.0976258
$853$	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
$855$	9.25835	0.316629
$856$	−8.32832	−0.284656
$857$	−2.35315	−0.0803820	−0.0401910	−	0.999192i	$-0.512797\pi$
$857$	−2.35315	−0.0803820	−0.0401910	+	0.999192i	$0.512797\pi$
$858$	15.7001	0.535993
$859$	−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
$861$	23.6966	0.807578
$862$	7.17551	0.244399
$863$	37.5791	1.27921	0.639604	−	0.768704i	$-0.279098\pi$
$863$	37.5791	1.27921	0.639604	+	0.768704i	$0.279098\pi$
$864$	−1.00000	−0.0340207
$865$	−44.8206	−1.52395
$866$	11.4852	0.390283
$867$	0.584665	0.0198563
$868$	5.42336	0.184081
$869$	22.3245	0.757308
$870$	−9.50917	−0.322391
$871$	36.5781	1.23940
$872$	2.72745	0.0923630
$873$	16.3978	0.554982
$874$	−3.12112	−0.105574
$875$	−16.1271	−0.545195
$876$	−4.29281	−0.145041
$877$	−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
$879$	22.2675	0.751063
$880$	−7.53592	−0.254036
$881$	−25.8200	−0.869897	−0.434949	−	0.900455i	$-0.643233\pi$
$881$	−25.8200	−0.869897	−0.434949	+	0.900455i	$0.643233\pi$
$882$	−2.27289	−0.0765321
$883$	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
$885$	−30.5459	−1.02679
$886$	−23.6850	−0.795713
$887$	58.1805	1.95351	0.976754	−	0.214362i	$-0.0687672\pi$
$887$	58.1805	1.95351	0.976754	+	0.214362i	$0.0687672\pi$
$888$	−4.44606	−0.149200
$889$	−29.5968	−0.992643
$890$	29.4911	0.988544
$891$	−2.79991	−0.0938006
$892$	−1.00000	−0.0334825
$893$	−46.1194	−1.54333
$894$	14.5888	0.487921
$895$	−34.8692	−1.16555
$896$	2.17419	0.0726347
$897$	5.08777	0.169876
$898$	26.0448	0.869126
$899$	8.81296	0.293929
$900$	2.24408	0.0748026
$901$	−53.4593	−1.78099
$902$	30.5163	1.01608
$903$	17.2019	0.572444
$904$	13.6313	0.453372
$905$	−23.6012	−0.784532
$906$	−12.6330	−0.419702
$907$	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
$909$	3.53689	0.117311
$910$	32.8131	1.08775
$911$	26.5194	0.878626	0.439313	−	0.898334i	$-0.355222\pi$
$911$	26.5194	0.878626	0.439313	+	0.898334i	$0.355222\pi$
$912$	−3.43987	−0.113906
$913$	5.05079	0.167157
$914$	19.4587	0.643635
$915$	−1.25676	−0.0415473
$916$	−25.3745	−0.838398
$917$	−38.5218	−1.27210
$918$	4.05158	0.133722
$919$	−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
$921$	8.05512	0.265425
$922$	−1.34885	−0.0444219
$923$	15.9788	0.525947
$924$	6.08755	0.200266
$925$	9.97731	0.328052
$926$	−19.2142	−0.631417
$927$	−5.03869	−0.165492
$928$	3.53306	0.115978
$929$	−2.20228	−0.0722544	−0.0361272	−	0.999347i	$-0.511502\pi$
$929$	−2.20228	−0.0722544	−0.0361272	+	0.999347i	$0.511502\pi$
$930$	−6.71371	−0.220151
$931$	−7.81844	−0.256239
$932$	8.41472	0.275633
$933$	−5.19327	−0.170020
$934$	30.1602	0.986872
$935$	30.5324	0.998517
$936$	5.60736	0.183282
$937$	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
$939$	−1.77635	−0.0579689
$940$	−36.0855	−1.17698
$941$	30.5023	0.994345	0.497173	−	0.867652i	$-0.334372\pi$
$941$	30.5023	0.994345	0.497173	+	0.867652i	$0.334372\pi$
$942$	12.0602	0.392942
$943$	9.88910	0.322033
$944$	11.3491	0.369381
$945$	−5.85180	−0.190359
$946$	22.1525	0.720241
$947$	19.5995	0.636899	0.318450	−	0.947940i	$-0.396838\pi$
$947$	19.5995	0.636899	0.318450	+	0.947940i	$0.396838\pi$
$948$	7.97329	0.258960
$949$	24.0713	0.781388
$950$	7.71934	0.250449
$951$	−16.7624	−0.543558
$952$	−8.80892	−0.285499
$953$	−9.62406	−0.311754	−0.155877	−	0.987776i	$-0.549820\pi$
$953$	−9.62406	−0.311754	−0.155877	+	0.987776i	$0.549820\pi$
$954$	13.1947	0.427193
$955$	7.50729	0.242930
$956$	−28.9489	−0.936273
$957$	9.89226	0.319771
$958$	10.4392	0.337275
$959$	−33.1336	−1.06994
$960$	−2.69148	−0.0868672
$961$	−24.7778	−0.799285
$962$	24.9307	0.803797
$963$	−8.32832	−0.268376
$964$	2.56905	0.0827435
$965$	−31.2367	−1.00555
$966$	1.97273	0.0634714
$967$	−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
$969$	13.9369	0.447719
$970$	44.1344	1.41707
$971$	34.3885	1.10358	0.551789	−	0.833984i	$-0.313945\pi$
$971$	34.3885	1.10358	0.551789	+	0.833984i	$0.313945\pi$
$972$	−1.00000	−0.0320750
$973$	−10.5943	−0.339637
$974$	18.5437	0.594180
$975$	−12.5834	−0.402990
$976$	0.466941	0.0149464
$977$	14.9389	0.477939	0.238970	−	0.971027i	$-0.423190\pi$
$977$	14.9389	0.477939	0.238970	+	0.971027i	$0.423190\pi$
$978$	−0.156053	−0.00499001
$979$	−30.6792	−0.980512
$980$	−6.11744	−0.195414
$981$	2.72745	0.0870807
$982$	17.6989	0.564794
$983$	58.4046	1.86282	0.931409	−	0.363975i	$-0.118581\pi$
$983$	58.4046	1.86282	0.931409	+	0.363975i	$0.118581\pi$
$984$	10.8990	0.347448
$985$	55.2646	1.76088
$986$	−14.3145	−0.455866
$987$	29.1501	0.927857
$988$	19.2886	0.613652
$989$	7.17873	0.228270
$990$	−7.53592	−0.239507
$991$	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
$993$	3.10572	0.0985571
$994$	6.19559	0.196512
$995$	48.4249	1.53517
$996$	1.80391	0.0571591
$997$	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
$999$	−4.44606	−0.140667

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.69148 q^{5} -1.00000 q^{6} +2.17419 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.69148 q^{10} -2.79991 q^{11} -1.00000 q^{12} +5.60736 q^{13} +2.17419 q^{14} -2.69148 q^{15} +1.00000 q^{16} -4.05158 q^{17} +1.00000 q^{18} +3.43987 q^{19} +2.69148 q^{20} -2.17419 q^{21} -2.79991 q^{22} -0.907338 q^{23} -1.00000 q^{24} +2.24408 q^{25} +5.60736 q^{26} -1.00000 q^{27} +2.17419 q^{28} +3.53306 q^{29} -2.69148 q^{30} +2.49443 q^{31} +1.00000 q^{32} +2.79991 q^{33} -4.05158 q^{34} +5.85180 q^{35} +1.00000 q^{36} +4.44606 q^{37} +3.43987 q^{38} -5.60736 q^{39} +2.69148 q^{40} -10.8990 q^{41} -2.17419 q^{42} -7.91186 q^{43} -2.79991 q^{44} +2.69148 q^{45} -0.907338 q^{46} -13.4073 q^{47} -1.00000 q^{48} -2.27289 q^{49} +2.24408 q^{50} +4.05158 q^{51} +5.60736 q^{52} +13.1947 q^{53} -1.00000 q^{54} -7.53592 q^{55} +2.17419 q^{56} -3.43987 q^{57} +3.53306 q^{58} +11.3491 q^{59} -2.69148 q^{60} +0.466941 q^{61} +2.49443 q^{62} +2.17419 q^{63} +1.00000 q^{64} +15.0921 q^{65} +2.79991 q^{66} +6.52324 q^{67} -4.05158 q^{68} +0.907338 q^{69} +5.85180 q^{70} +2.84960 q^{71} +1.00000 q^{72} +4.29281 q^{73} +4.44606 q^{74} -2.24408 q^{75} +3.43987 q^{76} -6.08755 q^{77} -5.60736 q^{78} -7.97329 q^{79} +2.69148 q^{80} +1.00000 q^{81} -10.8990 q^{82} -1.80391 q^{83} -2.17419 q^{84} -10.9048 q^{85} -7.91186 q^{86} -3.53306 q^{87} -2.79991 q^{88} +10.9572 q^{89} +2.69148 q^{90} +12.1915 q^{91} -0.907338 q^{92} -2.49443 q^{93} -13.4073 q^{94} +9.25835 q^{95} -1.00000 q^{96} +16.3978 q^{97} -2.27289 q^{98} -2.79991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} - 7 q^{3} + 7 q^{4} + 6 q^{5} - 7 q^{6} + 3 q^{7} + 7 q^{8} + 7 q^{9} + 6 q^{10} - q^{11} - 7 q^{12} + 8 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} + 16 q^{17} + 7 q^{18} + 2 q^{19} + 6 q^{20}+ \cdots - q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 - 7 * q^3 + 7 * q^4 + 6 * q^5 - 7 * q^6 + 3 * q^7 + 7 * q^8 + 7 * q^9 + 6 * q^10 - q^11 - 7 * q^12 + 8 * q^13 + 3 * q^14 - 6 * q^15 + 7 * q^16 + 16 * q^17 + 7 * q^18 + 2 * q^19 + 6 * q^20 - 3 * q^21 - q^22 + 8 * q^23 - 7 * q^24 + 19 * q^25 + 8 * q^26 - 7 * q^27 + 3 * q^28 + 4 * q^29 - 6 * q^30 + 11 * q^31 + 7 * q^32 + q^33 + 16 * q^34 + 7 * q^36 + 17 * q^37 + 2 * q^38 - 8 * q^39 + 6 * q^40 + 18 * q^41 - 3 * q^42 - q^43 - q^44 + 6 * q^45 + 8 * q^46 + 5 * q^47 - 7 * q^48 + 24 * q^49 + 19 * q^50 - 16 * q^51 + 8 * q^52 + 6 * q^53 - 7 * q^54 + 21 * q^55 + 3 * q^56 - 2 * q^57 + 4 * q^58 - 7 * q^59 - 6 * q^60 + 24 * q^61 + 11 * q^62 + 3 * q^63 + 7 * q^64 + 11 * q^65 + q^66 - 4 * q^67 + 16 * q^68 - 8 * q^69 - 7 * q^71 + 7 * q^72 + 28 * q^73 + 17 * q^74 - 19 * q^75 + 2 * q^76 - 2 * q^77 - 8 * q^78 + 6 * q^80 + 7 * q^81 + 18 * q^82 - 2 * q^83 - 3 * q^84 + 4 * q^85 - q^86 - 4 * q^87 - q^88 + 5 * q^89 + 6 * q^90 + 2 * q^91 + 8 * q^92 - 11 * q^93 + 5 * q^94 - 14 * q^95 - 7 * q^96 + 23 * q^97 + 24 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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