LMFDB - Embedded newform 6042.2.a.bd.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 35 x^{9} + 77 x^{8} + 394 x^{7} - 994 x^{6} - 1477 x^{5} + 4683 x^{4} + 563 x^{3} + \cdots + 1534 $ x^11 - 2x^10 - 35x^9 + 77x^8 + 394x^7 - 994x^6 - 1477x^5 + 4683x^4 + 563x^3 - 6194x^2 + 1258x + 1534
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.67126$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.67126 q^{5} +1.00000 q^{6} -4.35376 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.67126 q^{5} +1.00000 q^{6} -4.35376 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.67126 q^{10} -1.84428 q^{11} -1.00000 q^{12} -2.26847 q^{13} +4.35376 q^{14} -1.67126 q^{15} +1.00000 q^{16} +7.36934 q^{17} -1.00000 q^{18} +1.00000 q^{19} +1.67126 q^{20} +4.35376 q^{21} +1.84428 q^{22} -0.930417 q^{23} +1.00000 q^{24} -2.20688 q^{25} +2.26847 q^{26} -1.00000 q^{27} -4.35376 q^{28} +2.96149 q^{29} +1.67126 q^{30} -2.29246 q^{31} -1.00000 q^{32} +1.84428 q^{33} -7.36934 q^{34} -7.27627 q^{35} +1.00000 q^{36} -4.06931 q^{37} -1.00000 q^{38} +2.26847 q^{39} -1.67126 q^{40} +3.35450 q^{41} -4.35376 q^{42} -1.54804 q^{43} -1.84428 q^{44} +1.67126 q^{45} +0.930417 q^{46} +7.25706 q^{47} -1.00000 q^{48} +11.9552 q^{49} +2.20688 q^{50} -7.36934 q^{51} -2.26847 q^{52} +1.00000 q^{53} +1.00000 q^{54} -3.08228 q^{55} +4.35376 q^{56} -1.00000 q^{57} -2.96149 q^{58} +4.26529 q^{59} -1.67126 q^{60} +13.4812 q^{61} +2.29246 q^{62} -4.35376 q^{63} +1.00000 q^{64} -3.79121 q^{65} -1.84428 q^{66} -2.24091 q^{67} +7.36934 q^{68} +0.930417 q^{69} +7.27627 q^{70} -6.88372 q^{71} -1.00000 q^{72} -10.3950 q^{73} +4.06931 q^{74} +2.20688 q^{75} +1.00000 q^{76} +8.02955 q^{77} -2.26847 q^{78} -3.40825 q^{79} +1.67126 q^{80} +1.00000 q^{81} -3.35450 q^{82} +10.7760 q^{83} +4.35376 q^{84} +12.3161 q^{85} +1.54804 q^{86} -2.96149 q^{87} +1.84428 q^{88} -3.49401 q^{89} -1.67126 q^{90} +9.87636 q^{91} -0.930417 q^{92} +2.29246 q^{93} -7.25706 q^{94} +1.67126 q^{95} +1.00000 q^{96} +8.06610 q^{97} -11.9552 q^{98} -1.84428 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9}+O(q^{10}) $ 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9	\( 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9} - 2 q^{10} - 6 q^{11} - 11 q^{12} - 7 q^{13} + 2 q^{14} - 2 q^{15} + 11 q^{16} - 16 q^{17} - 11 q^{18} + 11 q^{19} + 2 q^{20} + 2 q^{21} + 6 q^{22} - 11 q^{23} + 11 q^{24} + 19 q^{25} + 7 q^{26} - 11 q^{27} - 2 q^{28} - 7 q^{29} + 2 q^{30} + 12 q^{31} - 11 q^{32} + 6 q^{33} + 16 q^{34} - 5 q^{35} + 11 q^{36} + 3 q^{37} - 11 q^{38} + 7 q^{39} - 2 q^{40} + 11 q^{41} - 2 q^{42} + 7 q^{43} - 6 q^{44} + 2 q^{45} + 11 q^{46} - 37 q^{47} - 11 q^{48} + 23 q^{49} - 19 q^{50} + 16 q^{51} - 7 q^{52} + 11 q^{53} + 11 q^{54} - 11 q^{55} + 2 q^{56} - 11 q^{57} + 7 q^{58} - 16 q^{59} - 2 q^{60} + 24 q^{61} - 12 q^{62} - 2 q^{63} + 11 q^{64} - 7 q^{65} - 6 q^{66} - 28 q^{67} - 16 q^{68} + 11 q^{69} + 5 q^{70} + 4 q^{71} - 11 q^{72} - 7 q^{73} - 3 q^{74} - 19 q^{75} + 11 q^{76} - 13 q^{77} - 7 q^{78} + 5 q^{79} + 2 q^{80} + 11 q^{81} - 11 q^{82} - 11 q^{83} + 2 q^{84} - 11 q^{85} - 7 q^{86} + 7 q^{87} + 6 q^{88} - 10 q^{89} - 2 q^{90} - 24 q^{91} - 11 q^{92} - 12 q^{93} + 37 q^{94} + 2 q^{95} + 11 q^{96} - 24 q^{97} - 23 q^{98} - 6 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9 - 2 * q^10 - 6 * q^11 - 11 * q^12 - 7 * q^13 + 2 * q^14 - 2 * q^15 + 11 * q^16 - 16 * q^17 - 11 * q^18 + 11 * q^19 + 2 * q^20 + 2 * q^21 + 6 * q^22 - 11 * q^23 + 11 * q^24 + 19 * q^25 + 7 * q^26 - 11 * q^27 - 2 * q^28 - 7 * q^29 + 2 * q^30 + 12 * q^31 - 11 * q^32 + 6 * q^33 + 16 * q^34 - 5 * q^35 + 11 * q^36 + 3 * q^37 - 11 * q^38 + 7 * q^39 - 2 * q^40 + 11 * q^41 - 2 * q^42 + 7 * q^43 - 6 * q^44 + 2 * q^45 + 11 * q^46 - 37 * q^47 - 11 * q^48 + 23 * q^49 - 19 * q^50 + 16 * q^51 - 7 * q^52 + 11 * q^53 + 11 * q^54 - 11 * q^55 + 2 * q^56 - 11 * q^57 + 7 * q^58 - 16 * q^59 - 2 * q^60 + 24 * q^61 - 12 * q^62 - 2 * q^63 + 11 * q^64 - 7 * q^65 - 6 * q^66 - 28 * q^67 - 16 * q^68 + 11 * q^69 + 5 * q^70 + 4 * q^71 - 11 * q^72 - 7 * q^73 - 3 * q^74 - 19 * q^75 + 11 * q^76 - 13 * q^77 - 7 * q^78 + 5 * q^79 + 2 * q^80 + 11 * q^81 - 11 * q^82 - 11 * q^83 + 2 * q^84 - 11 * q^85 - 7 * q^86 + 7 * q^87 + 6 * q^88 - 10 * q^89 - 2 * q^90 - 24 * q^91 - 11 * q^92 - 12 * q^93 + 37 * q^94 + 2 * q^95 + 11 * q^96 - 24 * q^97 - 23 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.67126	0.747411	0.373706	−	0.927547i	$-0.378087\pi$
$5$	1.67126	0.747411	0.373706	+	0.927547i	$0.378087\pi$
$6$	1.00000	0.408248
$7$	−4.35376	−1.64557	−0.822783	−	0.568356i	$-0.807580\pi$
$7$	−4.35376	−1.64557	−0.822783	+	0.568356i	$0.807580\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.67126	−0.528500
$11$	−1.84428	−0.556072	−0.278036	−	0.960571i	$-0.589683\pi$
$11$	−1.84428	−0.556072	−0.278036	+	0.960571i	$0.589683\pi$
$12$	−1.00000	−0.288675
$13$	−2.26847	−0.629160	−0.314580	−	0.949231i	$-0.601864\pi$
$13$	−2.26847	−0.629160	−0.314580	+	0.949231i	$0.601864\pi$
$14$	4.35376	1.16359
$15$	−1.67126	−0.431518
$16$	1.00000	0.250000
$17$	7.36934	1.78733	0.893664	−	0.448737i	$-0.148126\pi$
$17$	7.36934	1.78733	0.893664	+	0.448737i	$0.148126\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.67126	0.373706
$21$	4.35376	0.950068
$22$	1.84428	0.393202
$23$	−0.930417	−0.194005	−0.0970027	−	0.995284i	$-0.530926\pi$
$23$	−0.930417	−0.194005	−0.0970027	+	0.995284i	$0.530926\pi$
$24$	1.00000	0.204124
$25$	−2.20688	−0.441376
$26$	2.26847	0.444883
$27$	−1.00000	−0.192450
$28$	−4.35376	−0.822783
$29$	2.96149	0.549934	0.274967	−	0.961454i	$-0.411333\pi$
$29$	2.96149	0.549934	0.274967	+	0.961454i	$0.411333\pi$
$30$	1.67126	0.305129
$31$	−2.29246	−0.411739	−0.205869	−	0.978579i	$-0.566002\pi$
$31$	−2.29246	−0.411739	−0.205869	+	0.978579i	$0.566002\pi$
$32$	−1.00000	−0.176777
$33$	1.84428	0.321048
$34$	−7.36934	−1.26383
$35$	−7.27627	−1.22991
$36$	1.00000	0.166667
$37$	−4.06931	−0.668990	−0.334495	−	0.942398i	$-0.608566\pi$
$37$	−4.06931	−0.668990	−0.334495	+	0.942398i	$0.608566\pi$
$38$	−1.00000	−0.162221
$39$	2.26847	0.363246
$40$	−1.67126	−0.264250
$41$	3.35450	0.523884	0.261942	−	0.965084i	$-0.415637\pi$
$41$	3.35450	0.523884	0.261942	+	0.965084i	$0.415637\pi$
$42$	−4.35376	−0.671799
$43$	−1.54804	−0.236074	−0.118037	−	0.993009i	$-0.537660\pi$
$43$	−1.54804	−0.236074	−0.118037	+	0.993009i	$0.537660\pi$
$44$	−1.84428	−0.278036
$45$	1.67126	0.249137
$46$	0.930417	0.137183
$47$	7.25706	1.05855	0.529276	−	0.848450i	$-0.322464\pi$
$47$	7.25706	1.05855	0.529276	+	0.848450i	$0.322464\pi$
$48$	−1.00000	−0.144338
$49$	11.9552	1.70789
$50$	2.20688	0.312100
$51$	−7.36934	−1.03191
$52$	−2.26847	−0.314580
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	−3.08228	−0.415614
$56$	4.35376	0.581795
$57$	−1.00000	−0.132453
$58$	−2.96149	−0.388862
$59$	4.26529	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$59$	4.26529	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$60$	−1.67126	−0.215759
$61$	13.4812	1.72610	0.863048	−	0.505123i	$-0.168553\pi$
$61$	13.4812	1.72610	0.863048	+	0.505123i	$0.168553\pi$
$62$	2.29246	0.291143
$63$	−4.35376	−0.548522
$64$	1.00000	0.125000
$65$	−3.79121	−0.470241
$66$	−1.84428	−0.227015
$67$	−2.24091	−0.273771	−0.136886	−	0.990587i	$-0.543709\pi$
$67$	−2.24091	−0.273771	−0.136886	+	0.990587i	$0.543709\pi$
$68$	7.36934	0.893664
$69$	0.930417	0.112009
$70$	7.27627	0.869681
$71$	−6.88372	−0.816948	−0.408474	−	0.912770i	$-0.633939\pi$
$71$	−6.88372	−0.816948	−0.408474	+	0.912770i	$0.633939\pi$
$72$	−1.00000	−0.117851
$73$	−10.3950	−1.21665	−0.608324	−	0.793689i	$-0.708158\pi$
$73$	−10.3950	−1.21665	−0.608324	+	0.793689i	$0.708158\pi$
$74$	4.06931	0.473047
$75$	2.20688	0.254829
$76$	1.00000	0.114708
$77$	8.02955	0.915052
$78$	−2.26847	−0.256854
$79$	−3.40825	−0.383458	−0.191729	−	0.981448i	$-0.561410\pi$
$79$	−3.40825	−0.383458	−0.191729	+	0.981448i	$0.561410\pi$
$80$	1.67126	0.186853
$81$	1.00000	0.111111
$82$	−3.35450	−0.370442
$83$	10.7760	1.18282	0.591412	−	0.806370i	$-0.298571\pi$
$83$	10.7760	1.18282	0.591412	+	0.806370i	$0.298571\pi$
$84$	4.35376	0.475034
$85$	12.3161	1.33587
$86$	1.54804	0.166929
$87$	−2.96149	−0.317505
$88$	1.84428	0.196601
$89$	−3.49401	−0.370364	−0.185182	−	0.982704i	$-0.559287\pi$
$89$	−3.49401	−0.370364	−0.185182	+	0.982704i	$0.559287\pi$
$90$	−1.67126	−0.176167
$91$	9.87636	1.03532
$92$	−0.930417	−0.0970027
$93$	2.29246	0.237718
$94$	−7.25706	−0.748509
$95$	1.67126	0.171468
$96$	1.00000	0.102062
$97$	8.06610	0.818989	0.409494	−	0.912313i	$-0.365705\pi$
$97$	8.06610	0.818989	0.409494	+	0.912313i	$0.365705\pi$
$98$	−11.9552	−1.20766
$99$	−1.84428	−0.185357
$100$	−2.20688	−0.220688
$101$	−1.19330	−0.118738	−0.0593691	−	0.998236i	$-0.518909\pi$
$101$	−1.19330	−0.118738	−0.0593691	+	0.998236i	$0.518909\pi$
$102$	7.36934	0.729673
$103$	−5.63508	−0.555241	−0.277620	−	0.960691i	$-0.589546\pi$
$103$	−5.63508	−0.555241	−0.277620	+	0.960691i	$0.589546\pi$
$104$	2.26847	0.222442
$105$	7.27627	0.710092
$106$	−1.00000	−0.0971286
$107$	−6.76209	−0.653716	−0.326858	−	0.945074i	$-0.605990\pi$
$107$	−6.76209	−0.653716	−0.326858	+	0.945074i	$0.605990\pi$
$108$	−1.00000	−0.0962250
$109$	8.51282	0.815381	0.407690	−	0.913120i	$-0.366334\pi$
$109$	8.51282	0.815381	0.407690	+	0.913120i	$0.366334\pi$
$110$	3.08228	0.293884
$111$	4.06931	0.386241
$112$	−4.35376	−0.411391
$113$	−11.1310	−1.04711	−0.523556	−	0.851991i	$-0.675395\pi$
$113$	−11.1310	−1.04711	−0.523556	+	0.851991i	$0.675395\pi$
$114$	1.00000	0.0936586
$115$	−1.55497	−0.145002
$116$	2.96149	0.274967
$117$	−2.26847	−0.209720
$118$	−4.26529	−0.392652
$119$	−32.0843	−2.94116
$120$	1.67126	0.152565
$121$	−7.59863	−0.690784
$122$	−13.4812	−1.22053
$123$	−3.35450	−0.302465
$124$	−2.29246	−0.205869
$125$	−12.0446	−1.07730
$126$	4.35376	0.387864
$127$	1.75055	0.155336	0.0776680	−	0.996979i	$-0.475253\pi$
$127$	1.75055	0.155336	0.0776680	+	0.996979i	$0.475253\pi$
$128$	−1.00000	−0.0883883
$129$	1.54804	0.136297
$130$	3.79121	0.332511
$131$	−6.41181	−0.560203	−0.280101	−	0.959970i	$-0.590368\pi$
$131$	−6.41181	−0.560203	−0.280101	+	0.959970i	$0.590368\pi$
$132$	1.84428	0.160524
$133$	−4.35376	−0.377519
$134$	2.24091	0.193585
$135$	−1.67126	−0.143839
$136$	−7.36934	−0.631916
$137$	14.3623	1.22705	0.613526	−	0.789675i	$-0.289751\pi$
$137$	14.3623	1.22705	0.613526	+	0.789675i	$0.289751\pi$
$138$	−0.930417	−0.0792024
$139$	0.688554	0.0584024	0.0292012	−	0.999574i	$-0.490704\pi$
$139$	0.688554	0.0584024	0.0292012	+	0.999574i	$0.490704\pi$
$140$	−7.27627	−0.614957
$141$	−7.25706	−0.611155
$142$	6.88372	0.577669
$143$	4.18369	0.349858
$144$	1.00000	0.0833333
$145$	4.94942	0.411027
$146$	10.3950	0.860300
$147$	−11.9552	−0.986049
$148$	−4.06931	−0.334495
$149$	9.30506	0.762300	0.381150	−	0.924513i	$-0.375528\pi$
$149$	9.30506	0.762300	0.381150	+	0.924513i	$0.375528\pi$
$150$	−2.20688	−0.180191
$151$	2.42618	0.197440	0.0987199	−	0.995115i	$-0.468525\pi$
$151$	2.42618	0.197440	0.0987199	+	0.995115i	$0.468525\pi$
$152$	−1.00000	−0.0811107
$153$	7.36934	0.595776
$154$	−8.02955	−0.647040
$155$	−3.83131	−0.307738
$156$	2.26847	0.181623
$157$	−3.75931	−0.300025	−0.150013	−	0.988684i	$-0.547931\pi$
$157$	−3.75931	−0.300025	−0.150013	+	0.988684i	$0.547931\pi$
$158$	3.40825	0.271146
$159$	−1.00000	−0.0793052
$160$	−1.67126	−0.132125
$161$	4.05081	0.319249
$162$	−1.00000	−0.0785674
$163$	−20.9931	−1.64431	−0.822154	−	0.569266i	$-0.807228\pi$
$163$	−20.9931	−1.64431	−0.822154	+	0.569266i	$0.807228\pi$
$164$	3.35450	0.261942
$165$	3.08228	0.239955
$166$	−10.7760	−0.836382
$167$	−0.857116	−0.0663257	−0.0331628	−	0.999450i	$-0.510558\pi$
$167$	−0.857116	−0.0663257	−0.0331628	+	0.999450i	$0.510558\pi$
$168$	−4.35376	−0.335900
$169$	−7.85405	−0.604158
$170$	−12.3161	−0.944602
$171$	1.00000	0.0764719
$172$	−1.54804	−0.118037
$173$	−3.89944	−0.296469	−0.148235	−	0.988952i	$-0.547359\pi$
$173$	−3.89944	−0.296469	−0.148235	+	0.988952i	$0.547359\pi$
$174$	2.96149	0.224510
$175$	9.60822	0.726313
$176$	−1.84428	−0.139018
$177$	−4.26529	−0.320599
$178$	3.49401	0.261887
$179$	−17.1902	−1.28485	−0.642427	−	0.766347i	$-0.722073\pi$
$179$	−17.1902	−1.28485	−0.642427	+	0.766347i	$0.722073\pi$
$180$	1.67126	0.124569
$181$	−5.23625	−0.389207	−0.194604	−	0.980882i	$-0.562342\pi$
$181$	−5.23625	−0.389207	−0.194604	+	0.980882i	$0.562342\pi$
$182$	−9.87636	−0.732085
$183$	−13.4812	−0.996562
$184$	0.930417	0.0685913
$185$	−6.80088	−0.500011
$186$	−2.29246	−0.168092
$187$	−13.5911	−0.993882
$188$	7.25706	0.529276
$189$	4.35376	0.316689
$190$	−1.67126	−0.121246
$191$	10.0993	0.730761	0.365380	−	0.930858i	$-0.380939\pi$
$191$	10.0993	0.730761	0.365380	+	0.930858i	$0.380939\pi$
$192$	−1.00000	−0.0721688
$193$	−2.76539	−0.199057	−0.0995285	−	0.995035i	$-0.531733\pi$
$193$	−2.76539	−0.199057	−0.0995285	+	0.995035i	$0.531733\pi$
$194$	−8.06610	−0.579113
$195$	3.79121	0.271494
$196$	11.9552	0.853943
$197$	−23.0428	−1.64173	−0.820866	−	0.571121i	$-0.806509\pi$
$197$	−23.0428	−1.64173	−0.820866	+	0.571121i	$0.806509\pi$
$198$	1.84428	0.131067
$199$	−13.5793	−0.962609	−0.481304	−	0.876553i	$-0.659837\pi$
$199$	−13.5793	−0.962609	−0.481304	+	0.876553i	$0.659837\pi$
$200$	2.20688	0.156050
$201$	2.24091	0.158062
$202$	1.19330	0.0839606
$203$	−12.8936	−0.904953
$204$	−7.36934	−0.515957
$205$	5.60624	0.391557
$206$	5.63508	0.392614
$207$	−0.930417	−0.0646685
$208$	−2.26847	−0.157290
$209$	−1.84428	−0.127572
$210$	−7.27627	−0.502111
$211$	−5.83845	−0.401935	−0.200968	−	0.979598i	$-0.564409\pi$
$211$	−5.83845	−0.401935	−0.200968	+	0.979598i	$0.564409\pi$
$212$	1.00000	0.0686803
$213$	6.88372	0.471665
$214$	6.76209	0.462247
$215$	−2.58718	−0.176444
$216$	1.00000	0.0680414
$217$	9.98084	0.677543
$218$	−8.51282	−0.576561
$219$	10.3950	0.702432
$220$	−3.08228	−0.207807
$221$	−16.7171	−1.12452
$222$	−4.06931	−0.273114
$223$	4.38022	0.293321	0.146661	−	0.989187i	$-0.453148\pi$
$223$	4.38022	0.293321	0.146661	+	0.989187i	$0.453148\pi$
$224$	4.35376	0.290898
$225$	−2.20688	−0.147125
$226$	11.1310	0.740421
$227$	−13.7028	−0.909487	−0.454743	−	0.890622i	$-0.650269\pi$
$227$	−13.7028	−0.909487	−0.454743	+	0.890622i	$0.650269\pi$
$228$	−1.00000	−0.0662266
$229$	15.1797	1.00310	0.501550	−	0.865129i	$-0.332763\pi$
$229$	15.1797	1.00310	0.501550	+	0.865129i	$0.332763\pi$
$230$	1.55497	0.102532
$231$	−8.02955	−0.528306
$232$	−2.96149	−0.194431
$233$	−17.1277	−1.12207	−0.561036	−	0.827791i	$-0.689597\pi$
$233$	−17.1277	−1.12207	−0.561036	+	0.827791i	$0.689597\pi$
$234$	2.26847	0.148294
$235$	12.1285	0.791173
$236$	4.26529	0.277647
$237$	3.40825	0.221390
$238$	32.0843	2.07972
$239$	−0.951639	−0.0615564	−0.0307782	−	0.999526i	$-0.509799\pi$
$239$	−0.951639	−0.0615564	−0.0307782	+	0.999526i	$0.509799\pi$
$240$	−1.67126	−0.107880
$241$	24.2146	1.55980	0.779900	−	0.625904i	$-0.215270\pi$
$241$	24.2146	1.55980	0.779900	+	0.625904i	$0.215270\pi$
$242$	7.59863	0.488458
$243$	−1.00000	−0.0641500
$244$	13.4812	0.863048
$245$	19.9803	1.27649
$246$	3.35450	0.213875
$247$	−2.26847	−0.144339
$248$	2.29246	0.145572
$249$	−10.7760	−0.682903
$250$	12.0446	0.761767
$251$	17.2008	1.08571	0.542853	−	0.839827i	$-0.317344\pi$
$251$	17.2008	1.08571	0.542853	+	0.839827i	$0.317344\pi$
$252$	−4.35376	−0.274261
$253$	1.71595	0.107881
$254$	−1.75055	−0.109839
$255$	−12.3161	−0.771264
$256$	1.00000	0.0625000
$257$	6.23625	0.389007	0.194503	−	0.980902i	$-0.437691\pi$
$257$	6.23625	0.389007	0.194503	+	0.980902i	$0.437691\pi$
$258$	−1.54804	−0.0963768
$259$	17.7168	1.10087
$260$	−3.79121	−0.235121
$261$	2.96149	0.183311
$262$	6.41181	0.396123
$263$	−6.16923	−0.380411	−0.190206	−	0.981744i	$-0.560915\pi$
$263$	−6.16923	−0.380411	−0.190206	+	0.981744i	$0.560915\pi$
$264$	−1.84428	−0.113508
$265$	1.67126	0.102665
$266$	4.35376	0.266946
$267$	3.49401	0.213830
$268$	−2.24091	−0.136886
$269$	5.48625	0.334503	0.167251	−	0.985914i	$-0.446511\pi$
$269$	5.48625	0.334503	0.167251	+	0.985914i	$0.446511\pi$
$270$	1.67126	0.101710
$271$	22.9242	1.39255	0.696273	−	0.717777i	$-0.254840\pi$
$271$	22.9242	1.39255	0.696273	+	0.717777i	$0.254840\pi$
$272$	7.36934	0.446832
$273$	−9.87636	−0.597745
$274$	−14.3623	−0.867657
$275$	4.07011	0.245437
$276$	0.930417	0.0560045
$277$	−4.76154	−0.286093	−0.143047	−	0.989716i	$-0.545690\pi$
$277$	−4.76154	−0.286093	−0.143047	+	0.989716i	$0.545690\pi$
$278$	−0.688554	−0.0412967
$279$	−2.29246	−0.137246
$280$	7.27627	0.434840
$281$	1.20090	0.0716397	0.0358198	−	0.999358i	$-0.488596\pi$
$281$	1.20090	0.0716397	0.0358198	+	0.999358i	$0.488596\pi$
$282$	7.25706	0.432152
$283$	−4.46304	−0.265300	−0.132650	−	0.991163i	$-0.542349\pi$
$283$	−4.46304	−0.265300	−0.132650	+	0.991163i	$0.542349\pi$
$284$	−6.88372	−0.408474
$285$	−1.67126	−0.0989971
$286$	−4.18369	−0.247387
$287$	−14.6047	−0.862086
$288$	−1.00000	−0.0589256
$289$	37.3072	2.19454
$290$	−4.94942	−0.290640
$291$	−8.06610	−0.472843
$292$	−10.3950	−0.608324
$293$	14.9470	0.873212	0.436606	−	0.899653i	$-0.356180\pi$
$293$	14.9470	0.873212	0.436606	+	0.899653i	$0.356180\pi$
$294$	11.9552	0.697242
$295$	7.12843	0.415033
$296$	4.06931	0.236524
$297$	1.84428	0.107016
$298$	−9.30506	−0.539028
$299$	2.11062	0.122060
$300$	2.20688	0.127414
$301$	6.73979	0.388475
$302$	−2.42618	−0.139611
$303$	1.19330	0.0685536
$304$	1.00000	0.0573539
$305$	22.5307	1.29010
$306$	−7.36934	−0.421277
$307$	−13.0288	−0.743595	−0.371798	−	0.928314i	$-0.621258\pi$
$307$	−13.0288	−0.743595	−0.371798	+	0.928314i	$0.621258\pi$
$308$	8.02955	0.457526
$309$	5.63508	0.320568
$310$	3.83131	0.217604
$311$	−19.8082	−1.12322	−0.561611	−	0.827402i	$-0.689818\pi$
$311$	−19.8082	−1.12322	−0.561611	+	0.827402i	$0.689818\pi$
$312$	−2.26847	−0.128427
$313$	−8.09870	−0.457765	−0.228883	−	0.973454i	$-0.573507\pi$
$313$	−8.09870	−0.457765	−0.228883	+	0.973454i	$0.573507\pi$
$314$	3.75931	0.212150
$315$	−7.27627	−0.409972
$316$	−3.40825	−0.191729
$317$	−11.0613	−0.621264	−0.310632	−	0.950530i	$-0.600541\pi$
$317$	−11.0613	−0.621264	−0.310632	+	0.950530i	$0.600541\pi$
$318$	1.00000	0.0560772
$319$	−5.46181	−0.305803
$320$	1.67126	0.0934264
$321$	6.76209	0.377423
$322$	−4.05081	−0.225743
$323$	7.36934	0.410041
$324$	1.00000	0.0555556
$325$	5.00624	0.277696
$326$	20.9931	1.16270
$327$	−8.51282	−0.470760
$328$	−3.35450	−0.185221
$329$	−31.5955	−1.74192
$330$	−3.08228	−0.169674
$331$	−3.44441	−0.189322	−0.0946609	−	0.995510i	$-0.530177\pi$
$331$	−3.44441	−0.189322	−0.0946609	+	0.995510i	$0.530177\pi$
$332$	10.7760	0.591412
$333$	−4.06931	−0.222997
$334$	0.857116	0.0468993
$335$	−3.74515	−0.204620
$336$	4.35376	0.237517
$337$	−12.4548	−0.678456	−0.339228	−	0.940704i	$-0.610166\pi$
$337$	−12.4548	−0.678456	−0.339228	+	0.940704i	$0.610166\pi$
$338$	7.85405	0.427204
$339$	11.1310	0.604551
$340$	12.3161	0.667935
$341$	4.22795	0.228956
$342$	−1.00000	−0.0540738
$343$	−21.5738	−1.16487
$344$	1.54804	0.0834647
$345$	1.55497	0.0837168
$346$	3.89944	0.209635
$347$	−8.63910	−0.463771	−0.231886	−	0.972743i	$-0.574490\pi$
$347$	−8.63910	−0.463771	−0.231886	+	0.972743i	$0.574490\pi$
$348$	−2.96149	−0.158752
$349$	−11.9465	−0.639481	−0.319741	−	0.947505i	$-0.603596\pi$
$349$	−11.9465	−0.639481	−0.319741	+	0.947505i	$0.603596\pi$
$350$	−9.60822	−0.513581
$351$	2.26847	0.121082
$352$	1.84428	0.0983005
$353$	−32.9804	−1.75537	−0.877685	−	0.479238i	$-0.840913\pi$
$353$	−32.9804	−1.75537	−0.877685	+	0.479238i	$0.840913\pi$
$354$	4.26529	0.226698
$355$	−11.5045	−0.610596
$356$	−3.49401	−0.185182
$357$	32.0843	1.69808
$358$	17.1902	0.908529
$359$	4.90302	0.258772	0.129386	−	0.991594i	$-0.458699\pi$
$359$	4.90302	0.258772	0.129386	+	0.991594i	$0.458699\pi$
$360$	−1.67126	−0.0880833
$361$	1.00000	0.0526316
$362$	5.23625	0.275211
$363$	7.59863	0.398825
$364$	9.87636	0.517662
$365$	−17.3729	−0.909337
$366$	13.4812	0.704675
$367$	19.7191	1.02933	0.514664	−	0.857392i	$-0.327917\pi$
$367$	19.7191	1.02933	0.514664	+	0.857392i	$0.327917\pi$
$368$	−0.930417	−0.0485013
$369$	3.35450	0.174628
$370$	6.80088	0.353561
$371$	−4.35376	−0.226036
$372$	2.29246	0.118859
$373$	−16.6864	−0.863990	−0.431995	−	0.901876i	$-0.642190\pi$
$373$	−16.6864	−0.863990	−0.431995	+	0.901876i	$0.642190\pi$
$374$	13.5911	0.702781
$375$	12.0446	0.621980
$376$	−7.25706	−0.374254
$377$	−6.71804	−0.345997
$378$	−4.35376	−0.223933
$379$	−31.1137	−1.59821	−0.799103	−	0.601194i	$-0.794692\pi$
$379$	−31.1137	−1.59821	−0.799103	+	0.601194i	$0.794692\pi$
$380$	1.67126	0.0857340
$381$	−1.75055	−0.0896833
$382$	−10.0993	−0.516726
$383$	−0.807985	−0.0412861	−0.0206430	−	0.999787i	$-0.506571\pi$
$383$	−0.807985	−0.0412861	−0.0206430	+	0.999787i	$0.506571\pi$
$384$	1.00000	0.0510310
$385$	13.4195	0.683921
$386$	2.76539	0.140755
$387$	−1.54804	−0.0786913
$388$	8.06610	0.409494
$389$	−7.71004	−0.390914	−0.195457	−	0.980712i	$-0.562619\pi$
$389$	−7.71004	−0.390914	−0.195457	+	0.980712i	$0.562619\pi$
$390$	−3.79121	−0.191975
$391$	−6.85656	−0.346751
$392$	−11.9552	−0.603829
$393$	6.41181	0.323433
$394$	23.0428	1.16088
$395$	−5.69609	−0.286601
$396$	−1.84428	−0.0926786
$397$	−21.9940	−1.10385	−0.551923	−	0.833895i	$-0.686106\pi$
$397$	−21.9940	−1.10385	−0.551923	+	0.833895i	$0.686106\pi$
$398$	13.5793	0.680667
$399$	4.35376	0.217960
$400$	−2.20688	−0.110344
$401$	−25.6669	−1.28174	−0.640871	−	0.767649i	$-0.721427\pi$
$401$	−25.6669	−1.28174	−0.640871	+	0.767649i	$0.721427\pi$
$402$	−2.24091	−0.111767
$403$	5.20038	0.259050
$404$	−1.19330	−0.0593691
$405$	1.67126	0.0830457
$406$	12.8936	0.639898
$407$	7.50494	0.372006
$408$	7.36934	0.364837
$409$	18.5846	0.918948	0.459474	−	0.888191i	$-0.348038\pi$
$409$	18.5846	0.918948	0.459474	+	0.888191i	$0.348038\pi$
$410$	−5.60624	−0.276873
$411$	−14.3623	−0.708439
$412$	−5.63508	−0.277620
$413$	−18.5701	−0.913773
$414$	0.930417	0.0457275
$415$	18.0096	0.884056
$416$	2.26847	0.111221
$417$	−0.688554	−0.0337186
$418$	1.84428	0.0902067
$419$	−8.30302	−0.405629	−0.202814	−	0.979217i	$-0.565009\pi$
$419$	−8.30302	−0.405629	−0.202814	+	0.979217i	$0.565009\pi$
$420$	7.27627	0.355046
$421$	25.7217	1.25360	0.626800	−	0.779180i	$-0.284364\pi$
$421$	25.7217	1.25360	0.626800	+	0.779180i	$0.284364\pi$
$422$	5.83845	0.284211
$423$	7.25706	0.352850
$424$	−1.00000	−0.0485643
$425$	−16.2633	−0.788884
$426$	−6.88372	−0.333518
$427$	−58.6940	−2.84040
$428$	−6.76209	−0.326858
$429$	−4.18369	−0.201991
$430$	2.58718	0.124765
$431$	−20.6895	−0.996580	−0.498290	−	0.867010i	$-0.666038\pi$
$431$	−20.6895	−0.996580	−0.498290	+	0.867010i	$0.666038\pi$
$432$	−1.00000	−0.0481125
$433$	−2.40163	−0.115415	−0.0577074	−	0.998334i	$-0.518379\pi$
$433$	−2.40163	−0.115415	−0.0577074	+	0.998334i	$0.518379\pi$
$434$	−9.98084	−0.479095
$435$	−4.94942	−0.237307
$436$	8.51282	0.407690
$437$	−0.930417	−0.0445079
$438$	−10.3950	−0.496695
$439$	−34.6821	−1.65528	−0.827642	−	0.561256i	$-0.810318\pi$
$439$	−34.6821	−1.65528	−0.827642	+	0.561256i	$0.810318\pi$
$440$	3.08228	0.146942
$441$	11.9552	0.569295
$442$	16.7171	0.795152
$443$	−23.2346	−1.10391	−0.551955	−	0.833874i	$-0.686118\pi$
$443$	−23.2346	−1.10391	−0.551955	+	0.833874i	$0.686118\pi$
$444$	4.06931	0.193121
$445$	−5.83941	−0.276814
$446$	−4.38022	−0.207409
$447$	−9.30506	−0.440114
$448$	−4.35376	−0.205696
$449$	−33.9240	−1.60097	−0.800486	−	0.599352i	$-0.795425\pi$
$449$	−33.9240	−1.60097	−0.800486	+	0.599352i	$0.795425\pi$
$450$	2.20688	0.104033
$451$	−6.18663	−0.291317
$452$	−11.1310	−0.523556
$453$	−2.42618	−0.113992
$454$	13.7028	0.643104
$455$	16.5060	0.773813
$456$	1.00000	0.0468293
$457$	16.0379	0.750222	0.375111	−	0.926980i	$-0.377605\pi$
$457$	16.0379	0.750222	0.375111	+	0.926980i	$0.377605\pi$
$458$	−15.1797	−0.709299
$459$	−7.36934	−0.343971
$460$	−1.55497	−0.0725009
$461$	8.13576	0.378920	0.189460	−	0.981888i	$-0.439326\pi$
$461$	8.13576	0.378920	0.189460	+	0.981888i	$0.439326\pi$
$462$	8.02955	0.373569
$463$	0.830441	0.0385938	0.0192969	−	0.999814i	$-0.493857\pi$
$463$	0.830441	0.0385938	0.0192969	+	0.999814i	$0.493857\pi$
$464$	2.96149	0.137484
$465$	3.83131	0.177673
$466$	17.1277	0.793425
$467$	31.0739	1.43793	0.718963	−	0.695048i	$-0.244617\pi$
$467$	31.0739	1.43793	0.718963	+	0.695048i	$0.244617\pi$
$468$	−2.26847	−0.104860
$469$	9.75639	0.450508
$470$	−12.1285	−0.559444
$471$	3.75931	0.173220
$472$	−4.26529	−0.196326
$473$	2.85502	0.131274
$474$	−3.40825	−0.156546
$475$	−2.20688	−0.101259
$476$	−32.0843	−1.47058
$477$	1.00000	0.0457869
$478$	0.951639	0.0435270
$479$	−1.54324	−0.0705124	−0.0352562	−	0.999378i	$-0.511225\pi$
$479$	−1.54324	−0.0705124	−0.0352562	+	0.999378i	$0.511225\pi$
$480$	1.67126	0.0762824
$481$	9.23110	0.420902
$482$	−24.2146	−1.10295
$483$	−4.05081	−0.184318
$484$	−7.59863	−0.345392
$485$	13.4806	0.612122
$486$	1.00000	0.0453609
$487$	−12.3374	−0.559061	−0.279530	−	0.960137i	$-0.590179\pi$
$487$	−12.3374	−0.559061	−0.279530	+	0.960137i	$0.590179\pi$
$488$	−13.4812	−0.610267
$489$	20.9931	0.949341
$490$	−19.9803	−0.902617
$491$	−37.9905	−1.71449	−0.857243	−	0.514912i	$-0.827824\pi$
$491$	−37.9905	−1.71449	−0.857243	+	0.514912i	$0.827824\pi$
$492$	−3.35450	−0.151232
$493$	21.8242	0.982912
$494$	2.26847	0.102063
$495$	−3.08228	−0.138538
$496$	−2.29246	−0.102935
$497$	29.9701	1.34434
$498$	10.7760	0.482886
$499$	27.0505	1.21095	0.605473	−	0.795866i	$-0.292984\pi$
$499$	27.0505	1.21095	0.605473	+	0.795866i	$0.292984\pi$
$500$	−12.0446	−0.538651
$501$	0.857116	0.0382931
$502$	−17.2008	−0.767711
$503$	−7.93359	−0.353741	−0.176871	−	0.984234i	$-0.556597\pi$
$503$	−7.93359	−0.353741	−0.176871	+	0.984234i	$0.556597\pi$
$504$	4.35376	0.193932
$505$	−1.99433	−0.0887463
$506$	−1.71595	−0.0762833
$507$	7.85405	0.348811
$508$	1.75055	0.0776680
$509$	28.2179	1.25074	0.625369	−	0.780329i	$-0.284949\pi$
$509$	28.2179	1.25074	0.625369	+	0.780329i	$0.284949\pi$
$510$	12.3161	0.545366
$511$	45.2575	2.00208
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	−6.23625	−0.275069
$515$	−9.41769	−0.414993
$516$	1.54804	0.0681487
$517$	−13.3841	−0.588630
$518$	−17.7168	−0.778430
$519$	3.89944	0.171167
$520$	3.79121	0.166255
$521$	−11.4727	−0.502627	−0.251314	−	0.967906i	$-0.580863\pi$
$521$	−11.4727	−0.502627	−0.251314	+	0.967906i	$0.580863\pi$
$522$	−2.96149	−0.129621
$523$	1.74288	0.0762108	0.0381054	−	0.999274i	$-0.487868\pi$
$523$	1.74288	0.0762108	0.0381054	+	0.999274i	$0.487868\pi$
$524$	−6.41181	−0.280101
$525$	−9.60822	−0.419337
$526$	6.16923	0.268991
$527$	−16.8940	−0.735912
$528$	1.84428	0.0802620
$529$	−22.1343	−0.962362
$530$	−1.67126	−0.0725950
$531$	4.26529	0.185098
$532$	−4.35376	−0.188759
$533$	−7.60957	−0.329607
$534$	−3.49401	−0.151200
$535$	−11.3012	−0.488595
$536$	2.24091	0.0967927
$537$	17.1902	0.741811
$538$	−5.48625	−0.236529
$539$	−22.0488	−0.949707
$540$	−1.67126	−0.0719197
$541$	−4.55556	−0.195859	−0.0979294	−	0.995193i	$-0.531222\pi$
$541$	−4.55556	−0.195859	−0.0979294	+	0.995193i	$0.531222\pi$
$542$	−22.9242	−0.984679
$543$	5.23625	0.224709
$544$	−7.36934	−0.315958
$545$	14.2272	0.609425
$546$	9.87636	0.422669
$547$	−34.8358	−1.48947	−0.744735	−	0.667360i	$-0.767424\pi$
$547$	−34.8358	−1.48947	−0.744735	+	0.667360i	$0.767424\pi$
$548$	14.3623	0.613526
$549$	13.4812	0.575365
$550$	−4.07011	−0.173550
$551$	2.96149	0.126164
$552$	−0.930417	−0.0396012
$553$	14.8387	0.631006
$554$	4.76154	0.202299
$555$	6.80088	0.288681
$556$	0.688554	0.0292012
$557$	−25.2631	−1.07043	−0.535217	−	0.844715i	$-0.679770\pi$
$557$	−25.2631	−1.07043	−0.535217	+	0.844715i	$0.679770\pi$
$558$	2.29246	0.0970478
$559$	3.51168	0.148528
$560$	−7.27627	−0.307479
$561$	13.5911	0.573818
$562$	−1.20090	−0.0506569
$563$	−40.3987	−1.70260	−0.851301	−	0.524678i	$-0.824186\pi$
$563$	−40.3987	−1.70260	−0.851301	+	0.524678i	$0.824186\pi$
$564$	−7.25706	−0.305577
$565$	−18.6028	−0.782624
$566$	4.46304	0.187596
$567$	−4.35376	−0.182841
$568$	6.88372	0.288835
$569$	−5.42652	−0.227492	−0.113746	−	0.993510i	$-0.536285\pi$
$569$	−5.42652	−0.227492	−0.113746	+	0.993510i	$0.536285\pi$
$570$	1.67126	0.0700015
$571$	−34.6255	−1.44903	−0.724515	−	0.689259i	$-0.757936\pi$
$571$	−34.6255	−1.44903	−0.724515	+	0.689259i	$0.757936\pi$
$572$	4.18369	0.174929
$573$	−10.0993	−0.421905
$574$	14.6047	0.609587
$575$	2.05332	0.0856293
$576$	1.00000	0.0416667
$577$	−5.65721	−0.235513	−0.117756	−	0.993043i	$-0.537570\pi$
$577$	−5.65721	−0.235513	−0.117756	+	0.993043i	$0.537570\pi$
$578$	−37.3072	−1.55177
$579$	2.76539	0.114926
$580$	4.94942	0.205514
$581$	−46.9162	−1.94641
$582$	8.06610	0.334351
$583$	−1.84428	−0.0763823
$584$	10.3950	0.430150
$585$	−3.79121	−0.156747
$586$	−14.9470	−0.617454
$587$	38.0167	1.56912	0.784559	−	0.620055i	$-0.212890\pi$
$587$	38.0167	1.56912	0.784559	+	0.620055i	$0.212890\pi$
$588$	−11.9552	−0.493024
$589$	−2.29246	−0.0944594
$590$	−7.12843	−0.293473
$591$	23.0428	0.947854
$592$	−4.06931	−0.167247
$593$	1.89245	0.0777136	0.0388568	−	0.999245i	$-0.487628\pi$
$593$	1.89245	0.0777136	0.0388568	+	0.999245i	$0.487628\pi$
$594$	−1.84428	−0.0756718
$595$	−53.6213	−2.19826
$596$	9.30506	0.381150
$597$	13.5793	0.555763
$598$	−2.11062	−0.0863098
$599$	−29.9760	−1.22478	−0.612392	−	0.790554i	$-0.709793\pi$
$599$	−29.9760	−1.22478	−0.612392	+	0.790554i	$0.709793\pi$
$600$	−2.20688	−0.0900955
$601$	13.1615	0.536868	0.268434	−	0.963298i	$-0.413494\pi$
$601$	13.1615	0.536868	0.268434	+	0.963298i	$0.413494\pi$
$602$	−6.73979	−0.274693
$603$	−2.24091	−0.0912570
$604$	2.42618	0.0987199
$605$	−12.6993	−0.516300
$606$	−1.19330	−0.0484747
$607$	18.4430	0.748577	0.374288	−	0.927312i	$-0.377887\pi$
$607$	18.4430	0.748577	0.374288	+	0.927312i	$0.377887\pi$
$608$	−1.00000	−0.0405554
$609$	12.8936	0.522475
$610$	−22.5307	−0.912241
$611$	−16.4624	−0.665998
$612$	7.36934	0.297888
$613$	20.3842	0.823308	0.411654	−	0.911340i	$-0.364951\pi$
$613$	20.3842	0.823308	0.411654	+	0.911340i	$0.364951\pi$
$614$	13.0288	0.525801
$615$	−5.60624	−0.226066
$616$	−8.02955	−0.323520
$617$	6.21890	0.250364	0.125182	−	0.992134i	$-0.460049\pi$
$617$	6.21890	0.250364	0.125182	+	0.992134i	$0.460049\pi$
$618$	−5.63508	−0.226676
$619$	12.7080	0.510779	0.255389	−	0.966838i	$-0.417796\pi$
$619$	12.7080	0.510779	0.255389	+	0.966838i	$0.417796\pi$
$620$	−3.83131	−0.153869
$621$	0.930417	0.0373363
$622$	19.8082	0.794237
$623$	15.2121	0.609458
$624$	2.26847	0.0908114
$625$	−9.09527	−0.363811
$626$	8.09870	0.323689
$627$	1.84428	0.0736535
$628$	−3.75931	−0.150013
$629$	−29.9881	−1.19570
$630$	7.27627	0.289894
$631$	21.0959	0.839815	0.419908	−	0.907567i	$-0.362062\pi$
$631$	21.0959	0.839815	0.419908	+	0.907567i	$0.362062\pi$
$632$	3.40825	0.135573
$633$	5.83845	0.232057
$634$	11.0613	0.439300
$635$	2.92563	0.116100
$636$	−1.00000	−0.0396526
$637$	−27.1200	−1.07453
$638$	5.46181	0.216235
$639$	−6.88372	−0.272316
$640$	−1.67126	−0.0660625
$641$	−4.46655	−0.176418	−0.0882090	−	0.996102i	$-0.528114\pi$
$641$	−4.46655	−0.176418	−0.0882090	+	0.996102i	$0.528114\pi$
$642$	−6.76209	−0.266878
$643$	10.6249	0.419006	0.209503	−	0.977808i	$-0.432815\pi$
$643$	10.6249	0.419006	0.209503	+	0.977808i	$0.432815\pi$
$644$	4.05081	0.159624
$645$	2.58718	0.101870
$646$	−7.36934	−0.289943
$647$	40.9846	1.61127	0.805636	−	0.592411i	$-0.201824\pi$
$647$	40.9846	1.61127	0.805636	+	0.592411i	$0.201824\pi$
$648$	−1.00000	−0.0392837
$649$	−7.86640	−0.308783
$650$	−5.00624	−0.196361
$651$	−9.98084	−0.391180
$652$	−20.9931	−0.822154
$653$	−24.4840	−0.958131	−0.479066	−	0.877779i	$-0.659024\pi$
$653$	−24.4840	−0.958131	−0.479066	+	0.877779i	$0.659024\pi$
$654$	8.51282	0.332878
$655$	−10.7158	−0.418702
$656$	3.35450	0.130971
$657$	−10.3950	−0.405550
$658$	31.5955	1.23172
$659$	−28.4957	−1.11003	−0.555017	−	0.831839i	$-0.687288\pi$
$659$	−28.4957	−1.11003	−0.555017	+	0.831839i	$0.687288\pi$
$660$	3.08228	0.119978
$661$	−25.9141	−1.00794	−0.503971	−	0.863721i	$-0.668128\pi$
$661$	−25.9141	−1.00794	−0.503971	+	0.863721i	$0.668128\pi$
$662$	3.44441	0.133871
$663$	16.7171	0.649239
$664$	−10.7760	−0.418191
$665$	−7.27627	−0.282162
$666$	4.06931	0.157682
$667$	−2.75542	−0.106690
$668$	−0.857116	−0.0331628
$669$	−4.38022	−0.169349
$670$	3.74515	0.144688
$671$	−24.8632	−0.959833
$672$	−4.35376	−0.167950
$673$	−35.2090	−1.35721	−0.678604	−	0.734504i	$-0.737415\pi$
$673$	−35.2090	−1.35721	−0.678604	+	0.734504i	$0.737415\pi$
$674$	12.4548	0.479741
$675$	2.20688	0.0849429
$676$	−7.85405	−0.302079
$677$	−4.94331	−0.189987	−0.0949933	−	0.995478i	$-0.530283\pi$
$677$	−4.94331	−0.189987	−0.0949933	+	0.995478i	$0.530283\pi$
$678$	−11.1310	−0.427482
$679$	−35.1179	−1.34770
$680$	−12.3161	−0.472301
$681$	13.7028	0.525093
$682$	−4.22795	−0.161897
$683$	9.11502	0.348777	0.174388	−	0.984677i	$-0.444205\pi$
$683$	9.11502	0.348777	0.174388	+	0.984677i	$0.444205\pi$
$684$	1.00000	0.0382360
$685$	24.0031	0.917113
$686$	21.5738	0.823690
$687$	−15.1797	−0.579140
$688$	−1.54804	−0.0590185
$689$	−2.26847	−0.0864218
$690$	−1.55497	−0.0591967
$691$	25.0947	0.954646	0.477323	−	0.878728i	$-0.341607\pi$
$691$	25.0947	0.954646	0.477323	+	0.878728i	$0.341607\pi$
$692$	−3.89944	−0.148235
$693$	8.02955	0.305017
$694$	8.63910	0.327936
$695$	1.15076	0.0436506
$696$	2.96149	0.112255
$697$	24.7204	0.936353
$698$	11.9465	0.452181
$699$	17.1277	0.647829
$700$	9.60822	0.363157
$701$	−1.53209	−0.0578661	−0.0289330	−	0.999581i	$-0.509211\pi$
$701$	−1.53209	−0.0578661	−0.0289330	+	0.999581i	$0.509211\pi$
$702$	−2.26847	−0.0856178
$703$	−4.06931	−0.153477
$704$	−1.84428	−0.0695089
$705$	−12.1285	−0.456784
$706$	32.9804	1.24123
$707$	5.19536	0.195392
$708$	−4.26529	−0.160300
$709$	45.9882	1.72712	0.863562	−	0.504243i	$-0.168228\pi$
$709$	45.9882	1.72712	0.863562	+	0.504243i	$0.168228\pi$
$710$	11.5045	0.431757
$711$	−3.40825	−0.127819
$712$	3.49401	0.130943
$713$	2.13295	0.0798795
$714$	−32.0843	−1.20073
$715$	6.99205	0.261488
$716$	−17.1902	−0.642427
$717$	0.951639	0.0355396
$718$	−4.90302	−0.182979
$719$	−22.7701	−0.849180	−0.424590	−	0.905386i	$-0.639582\pi$
$719$	−22.7701	−0.849180	−0.424590	+	0.905386i	$0.639582\pi$
$720$	1.67126	0.0622843
$721$	24.5338	0.913685
$722$	−1.00000	−0.0372161
$723$	−24.2146	−0.900551
$724$	−5.23625	−0.194604
$725$	−6.53565	−0.242728
$726$	−7.59863	−0.282012
$727$	−7.91433	−0.293526	−0.146763	−	0.989172i	$-0.546886\pi$
$727$	−7.91433	−0.293526	−0.146763	+	0.989172i	$0.546886\pi$
$728$	−9.87636	−0.366042
$729$	1.00000	0.0370370
$730$	17.3729	0.642998
$731$	−11.4080	−0.421941
$732$	−13.4812	−0.498281
$733$	−18.5153	−0.683878	−0.341939	−	0.939722i	$-0.611084\pi$
$733$	−18.5153	−0.683878	−0.341939	+	0.939722i	$0.611084\pi$
$734$	−19.7191	−0.727845
$735$	−19.9803	−0.736984
$736$	0.930417	0.0342956
$737$	4.13287	0.152236
$738$	−3.35450	−0.123481
$739$	−6.94899	−0.255623	−0.127811	−	0.991798i	$-0.540795\pi$
$739$	−6.94899	−0.255623	−0.127811	+	0.991798i	$0.540795\pi$
$740$	−6.80088	−0.250005
$741$	2.26847	0.0833343
$742$	4.35376	0.159831
$743$	−11.9353	−0.437864	−0.218932	−	0.975740i	$-0.570257\pi$
$743$	−11.9353	−0.437864	−0.218932	+	0.975740i	$0.570257\pi$
$744$	−2.29246	−0.0840458
$745$	15.5512	0.569752
$746$	16.6864	0.610933
$747$	10.7760	0.394274
$748$	−13.5911	−0.496941
$749$	29.4405	1.07573
$750$	−12.0446	−0.439806
$751$	6.12593	0.223538	0.111769	−	0.993734i	$-0.464348\pi$
$751$	6.12593	0.223538	0.111769	+	0.993734i	$0.464348\pi$
$752$	7.25706	0.264638
$753$	−17.2008	−0.626833
$754$	6.71804	0.244657
$755$	4.05479	0.147569
$756$	4.35376	0.158345
$757$	31.5568	1.14695	0.573475	−	0.819223i	$-0.305595\pi$
$757$	31.5568	1.14695	0.573475	+	0.819223i	$0.305595\pi$
$758$	31.1137	1.13010
$759$	−1.71595	−0.0622850
$760$	−1.67126	−0.0606231
$761$	9.08690	0.329400	0.164700	−	0.986344i	$-0.447334\pi$
$761$	9.08690	0.329400	0.164700	+	0.986344i	$0.447334\pi$
$762$	1.75055	0.0634157
$763$	−37.0628	−1.34176
$764$	10.0993	0.365380
$765$	12.3161	0.445290
$766$	0.807985	0.0291937
$767$	−9.67569	−0.349369
$768$	−1.00000	−0.0360844
$769$	−46.1659	−1.66478	−0.832392	−	0.554187i	$-0.813029\pi$
$769$	−46.1659	−1.66478	−0.832392	+	0.554187i	$0.813029\pi$
$770$	−13.4195	−0.483605
$771$	−6.23625	−0.224593
$772$	−2.76539	−0.0995285
$773$	−5.07611	−0.182575	−0.0912874	−	0.995825i	$-0.529098\pi$
$773$	−5.07611	−0.182575	−0.0912874	+	0.995825i	$0.529098\pi$
$774$	1.54804	0.0556432
$775$	5.05920	0.181732
$776$	−8.06610	−0.289556
$777$	−17.7168	−0.635586
$778$	7.71004	0.276418
$779$	3.35450	0.120187
$780$	3.79121	0.135747
$781$	12.6955	0.454281
$782$	6.85656	0.245190
$783$	−2.96149	−0.105835
$784$	11.9552	0.426972
$785$	−6.28279	−0.224242
$786$	−6.41181	−0.228702
$787$	−25.0447	−0.892746	−0.446373	−	0.894847i	$-0.647285\pi$
$787$	−25.0447	−0.892746	−0.446373	+	0.894847i	$0.647285\pi$
$788$	−23.0428	−0.820866
$789$	6.16923	0.219630
$790$	5.69609	0.202658
$791$	48.4615	1.72309
$792$	1.84428	0.0655337
$793$	−30.5818	−1.08599
$794$	21.9940	0.780537
$795$	−1.67126	−0.0592736
$796$	−13.5793	−0.481304
$797$	−10.1142	−0.358263	−0.179132	−	0.983825i	$-0.557329\pi$
$797$	−10.1142	−0.358263	−0.179132	+	0.983825i	$0.557329\pi$
$798$	−4.35376	−0.154121
$799$	53.4798	1.89198
$800$	2.20688	0.0780250
$801$	−3.49401	−0.123455
$802$	25.6669	0.906328
$803$	19.1714	0.676544
$804$	2.24091	0.0790309
$805$	6.76997	0.238610
$806$	−5.20038	−0.183176
$807$	−5.48625	−0.193125
$808$	1.19330	0.0419803
$809$	−28.5189	−1.00267	−0.501336	−	0.865253i	$-0.667158\pi$
$809$	−28.5189	−1.00267	−0.501336	+	0.865253i	$0.667158\pi$
$810$	−1.67126	−0.0587222
$811$	−20.4681	−0.718733	−0.359366	−	0.933197i	$-0.617007\pi$
$811$	−20.4681	−0.718733	−0.359366	+	0.933197i	$0.617007\pi$
$812$	−12.8936	−0.452476
$813$	−22.9242	−0.803987
$814$	−7.50494	−0.263048
$815$	−35.0850	−1.22897
$816$	−7.36934	−0.257979
$817$	−1.54804	−0.0541591
$818$	−18.5846	−0.649795
$819$	9.87636	0.345108
$820$	5.60624	0.195779
$821$	−29.1889	−1.01870	−0.509350	−	0.860560i	$-0.670114\pi$
$821$	−29.1889	−1.01870	−0.509350	+	0.860560i	$0.670114\pi$
$822$	14.3623	0.500942
$823$	36.7434	1.28080	0.640398	−	0.768044i	$-0.278770\pi$
$823$	36.7434	1.28080	0.640398	+	0.768044i	$0.278770\pi$
$824$	5.63508	0.196307
$825$	−4.07011	−0.141703
$826$	18.5701	0.646135
$827$	−12.6802	−0.440933	−0.220466	−	0.975395i	$-0.570758\pi$
$827$	−12.6802	−0.440933	−0.220466	+	0.975395i	$0.570758\pi$
$828$	−0.930417	−0.0323342
$829$	53.7565	1.86704	0.933519	−	0.358527i	$-0.116721\pi$
$829$	53.7565	1.86704	0.933519	+	0.358527i	$0.116721\pi$
$830$	−18.0096	−0.625122
$831$	4.76154	0.165176
$832$	−2.26847	−0.0786450
$833$	88.1020	3.05255
$834$	0.688554	0.0238427
$835$	−1.43247	−0.0495726
$836$	−1.84428	−0.0637858
$837$	2.29246	0.0792392
$838$	8.30302	0.286823
$839$	32.0868	1.10776	0.553879	−	0.832597i	$-0.313147\pi$
$839$	32.0868	1.10776	0.553879	+	0.832597i	$0.313147\pi$
$840$	−7.27627	−0.251055
$841$	−20.2296	−0.697572
$842$	−25.7217	−0.886429
$843$	−1.20090	−0.0413612
$844$	−5.83845	−0.200968
$845$	−13.1262	−0.451554
$846$	−7.25706	−0.249503
$847$	33.0826	1.13673
$848$	1.00000	0.0343401
$849$	4.46304	0.153171
$850$	16.2633	0.557825
$851$	3.78615	0.129788
$852$	6.88372	0.235833
$853$	15.7914	0.540687	0.270343	−	0.962764i	$-0.412863\pi$
$853$	15.7914	0.540687	0.270343	+	0.962764i	$0.412863\pi$
$854$	58.6940	2.00847
$855$	1.67126	0.0571560
$856$	6.76209	0.231123
$857$	−21.8159	−0.745217	−0.372609	−	0.927989i	$-0.621537\pi$
$857$	−21.8159	−0.745217	−0.372609	+	0.927989i	$0.621537\pi$
$858$	4.18369	0.142829
$859$	5.90527	0.201485	0.100743	−	0.994913i	$-0.467878\pi$
$859$	5.90527	0.201485	0.100743	+	0.994913i	$0.467878\pi$
$860$	−2.58718	−0.0882222
$861$	14.6047	0.497726
$862$	20.6895	0.704688
$863$	−22.9104	−0.779880	−0.389940	−	0.920840i	$-0.627504\pi$
$863$	−22.9104	−0.779880	−0.389940	+	0.920840i	$0.627504\pi$
$864$	1.00000	0.0340207
$865$	−6.51700	−0.221584
$866$	2.40163	0.0816106
$867$	−37.3072	−1.26702
$868$	9.98084	0.338772
$869$	6.28577	0.213230
$870$	4.94942	0.167801
$871$	5.08344	0.172246
$872$	−8.51282	−0.288281
$873$	8.06610	0.272996
$874$	0.930417	0.0314718
$875$	52.4392	1.77277
$876$	10.3950	0.351216
$877$	−42.4621	−1.43384	−0.716922	−	0.697153i	$-0.754450\pi$
$877$	−42.4621	−1.43384	−0.716922	+	0.697153i	$0.754450\pi$
$878$	34.6821	1.17046
$879$	−14.9470	−0.504149
$880$	−3.08228	−0.103904
$881$	29.4611	0.992569	0.496285	−	0.868160i	$-0.334697\pi$
$881$	29.4611	0.992569	0.496285	+	0.868160i	$0.334697\pi$
$882$	−11.9552	−0.402553
$883$	29.8017	1.00291	0.501453	−	0.865185i	$-0.332799\pi$
$883$	29.8017	1.00291	0.501453	+	0.865185i	$0.332799\pi$
$884$	−16.7171	−0.562258
$885$	−7.12843	−0.239619
$886$	23.2346	0.780582
$887$	−11.9316	−0.400622	−0.200311	−	0.979732i	$-0.564195\pi$
$887$	−11.9316	−0.400622	−0.200311	+	0.979732i	$0.564195\pi$
$888$	−4.06931	−0.136557
$889$	−7.62146	−0.255616
$890$	5.83941	0.195737
$891$	−1.84428	−0.0617857
$892$	4.38022	0.146661
$893$	7.25706	0.242848
$894$	9.30506	0.311208
$895$	−28.7293	−0.960315
$896$	4.35376	0.145449
$897$	−2.11062	−0.0704716
$898$	33.9240	1.13206
$899$	−6.78910	−0.226429
$900$	−2.20688	−0.0735627
$901$	7.36934	0.245508
$902$	6.18663	0.205992
$903$	−6.73979	−0.224286
$904$	11.1310	0.370210
$905$	−8.75115	−0.290898
$906$	2.42618	0.0806045
$907$	−36.3283	−1.20626	−0.603130	−	0.797643i	$-0.706080\pi$
$907$	−36.3283	−1.20626	−0.603130	+	0.797643i	$0.706080\pi$
$908$	−13.7028	−0.454743
$909$	−1.19330	−0.0395794
$910$	−16.5060	−0.547169
$911$	21.8478	0.723850	0.361925	−	0.932207i	$-0.382120\pi$
$911$	21.8478	0.723850	0.361925	+	0.932207i	$0.382120\pi$
$912$	−1.00000	−0.0331133
$913$	−19.8740	−0.657734
$914$	−16.0379	−0.530487
$915$	−22.5307	−0.744842
$916$	15.1797	0.501550
$917$	27.9155	0.921850
$918$	7.36934	0.243224
$919$	−15.2670	−0.503612	−0.251806	−	0.967778i	$-0.581024\pi$
$919$	−15.2670	−0.503612	−0.251806	+	0.967778i	$0.581024\pi$
$920$	1.55497	0.0512659
$921$	13.0288	0.429315
$922$	−8.13576	−0.267937
$923$	15.6155	0.513991
$924$	−8.02955	−0.264153
$925$	8.98047	0.295276
$926$	−0.830441	−0.0272900
$927$	−5.63508	−0.185080
$928$	−2.96149	−0.0972155
$929$	58.0264	1.90378	0.951892	−	0.306433i	$-0.0991354\pi$
$929$	58.0264	1.90378	0.951892	+	0.306433i	$0.0991354\pi$
$930$	−3.83131	−0.125634
$931$	11.9552	0.391816
$932$	−17.1277	−0.561036
$933$	19.8082	0.648492
$934$	−31.0739	−1.01677
$935$	−22.7144	−0.742839
$936$	2.26847	0.0741472
$937$	−25.9169	−0.846669	−0.423335	−	0.905973i	$-0.639141\pi$
$937$	−25.9169	−0.846669	−0.423335	+	0.905973i	$0.639141\pi$
$938$	−9.75639	−0.318557
$939$	8.09870	0.264291
$940$	12.1285	0.395587
$941$	42.1935	1.37547	0.687735	−	0.725962i	$-0.258605\pi$
$941$	42.1935	1.37547	0.687735	+	0.725962i	$0.258605\pi$
$942$	−3.75931	−0.122485
$943$	−3.12108	−0.101636
$944$	4.26529	0.138823
$945$	7.27627	0.236697
$946$	−2.85502	−0.0928247
$947$	33.8215	1.09905	0.549526	−	0.835477i	$-0.314808\pi$
$947$	33.8215	1.09905	0.549526	+	0.835477i	$0.314808\pi$
$948$	3.40825	0.110695
$949$	23.5808	0.765467
$950$	2.20688	0.0716007
$951$	11.0613	0.358687
$952$	32.0843	1.03986
$953$	−40.3541	−1.30720	−0.653598	−	0.756842i	$-0.726741\pi$
$953$	−40.3541	−1.30720	−0.653598	+	0.756842i	$0.726741\pi$
$954$	−1.00000	−0.0323762
$955$	16.8786	0.546179
$956$	−0.951639	−0.0307782
$957$	5.46181	0.176555
$958$	1.54324	0.0498598
$959$	−62.5299	−2.01919
$960$	−1.67126	−0.0539398
$961$	−25.7446	−0.830471
$962$	−9.23110	−0.297622
$963$	−6.76209	−0.217905
$964$	24.2146	0.779900
$965$	−4.62169	−0.148777
$966$	4.05081	0.130333
$967$	11.1019	0.357012	0.178506	−	0.983939i	$-0.442874\pi$
$967$	11.1019	0.357012	0.178506	+	0.983939i	$0.442874\pi$
$968$	7.59863	0.244229
$969$	−7.36934	−0.236737
$970$	−13.4806	−0.432835
$971$	−46.6830	−1.49813	−0.749064	−	0.662498i	$-0.769496\pi$
$971$	−46.6830	−1.49813	−0.749064	+	0.662498i	$0.769496\pi$
$972$	−1.00000	−0.0320750
$973$	−2.99780	−0.0961050
$974$	12.3374	0.395316
$975$	−5.00624	−0.160328
$976$	13.4812	0.431524
$977$	−49.8898	−1.59612	−0.798058	−	0.602581i	$-0.794139\pi$
$977$	−49.8898	−1.59612	−0.798058	+	0.602581i	$0.794139\pi$
$978$	−20.9931	−0.671286
$979$	6.44393	0.205949
$980$	19.9803	0.638247
$981$	8.51282	0.271794
$982$	37.9905	1.21232
$983$	−13.2069	−0.421234	−0.210617	−	0.977569i	$-0.567547\pi$
$983$	−13.2069	−0.421234	−0.210617	+	0.977569i	$0.567547\pi$
$984$	3.35450	0.106937
$985$	−38.5106	−1.22705
$986$	−21.8242	−0.695024
$987$	31.5955	1.00570
$988$	−2.26847	−0.0721696
$989$	1.44032	0.0457996
$990$	3.08228	0.0979612
$991$	20.5831	0.653842	0.326921	−	0.945052i	$-0.393989\pi$
$991$	20.5831	0.653842	0.326921	+	0.945052i	$0.393989\pi$
$992$	2.29246	0.0727858
$993$	3.44441	0.109305
$994$	−29.9701	−0.950593
$995$	−22.6945	−0.719465
$996$	−10.7760	−0.341452
$997$	−36.8944	−1.16846	−0.584228	−	0.811589i	$-0.698603\pi$
$997$	−36.8944	−1.16846	−0.584228	+	0.811589i	$0.698603\pi$
$998$	−27.0505	−0.856268
$999$	4.06931	0.128747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bd.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bd.1.7	✓	11	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.67126 q^{5} +1.00000 q^{6} -4.35376 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.67126 q^{5} +1.00000 q^{6} -4.35376 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.67126 q^{10} -1.84428 q^{11} -1.00000 q^{12} -2.26847 q^{13} +4.35376 q^{14} -1.67126 q^{15} +1.00000 q^{16} +7.36934 q^{17} -1.00000 q^{18} +1.00000 q^{19} +1.67126 q^{20} +4.35376 q^{21} +1.84428 q^{22} -0.930417 q^{23} +1.00000 q^{24} -2.20688 q^{25} +2.26847 q^{26} -1.00000 q^{27} -4.35376 q^{28} +2.96149 q^{29} +1.67126 q^{30} -2.29246 q^{31} -1.00000 q^{32} +1.84428 q^{33} -7.36934 q^{34} -7.27627 q^{35} +1.00000 q^{36} -4.06931 q^{37} -1.00000 q^{38} +2.26847 q^{39} -1.67126 q^{40} +3.35450 q^{41} -4.35376 q^{42} -1.54804 q^{43} -1.84428 q^{44} +1.67126 q^{45} +0.930417 q^{46} +7.25706 q^{47} -1.00000 q^{48} +11.9552 q^{49} +2.20688 q^{50} -7.36934 q^{51} -2.26847 q^{52} +1.00000 q^{53} +1.00000 q^{54} -3.08228 q^{55} +4.35376 q^{56} -1.00000 q^{57} -2.96149 q^{58} +4.26529 q^{59} -1.67126 q^{60} +13.4812 q^{61} +2.29246 q^{62} -4.35376 q^{63} +1.00000 q^{64} -3.79121 q^{65} -1.84428 q^{66} -2.24091 q^{67} +7.36934 q^{68} +0.930417 q^{69} +7.27627 q^{70} -6.88372 q^{71} -1.00000 q^{72} -10.3950 q^{73} +4.06931 q^{74} +2.20688 q^{75} +1.00000 q^{76} +8.02955 q^{77} -2.26847 q^{78} -3.40825 q^{79} +1.67126 q^{80} +1.00000 q^{81} -3.35450 q^{82} +10.7760 q^{83} +4.35376 q^{84} +12.3161 q^{85} +1.54804 q^{86} -2.96149 q^{87} +1.84428 q^{88} -3.49401 q^{89} -1.67126 q^{90} +9.87636 q^{91} -0.930417 q^{92} +2.29246 q^{93} -7.25706 q^{94} +1.67126 q^{95} +1.00000 q^{96} +8.06610 q^{97} -11.9552 q^{98} -1.84428 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9}+O(q^{10}) \) 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9	\( 11 q - 11 q^{2} - 11 q^{3} + 11 q^{4} + 2 q^{5} + 11 q^{6} - 2 q^{7} - 11 q^{8} + 11 q^{9} - 2 q^{10} - 6 q^{11} - 11 q^{12} - 7 q^{13} + 2 q^{14} - 2 q^{15} + 11 q^{16} - 16 q^{17} - 11 q^{18} + 11 q^{19} + 2 q^{20} + 2 q^{21} + 6 q^{22} - 11 q^{23} + 11 q^{24} + 19 q^{25} + 7 q^{26} - 11 q^{27} - 2 q^{28} - 7 q^{29} + 2 q^{30} + 12 q^{31} - 11 q^{32} + 6 q^{33} + 16 q^{34} - 5 q^{35} + 11 q^{36} + 3 q^{37} - 11 q^{38} + 7 q^{39} - 2 q^{40} + 11 q^{41} - 2 q^{42} + 7 q^{43} - 6 q^{44} + 2 q^{45} + 11 q^{46} - 37 q^{47} - 11 q^{48} + 23 q^{49} - 19 q^{50} + 16 q^{51} - 7 q^{52} + 11 q^{53} + 11 q^{54} - 11 q^{55} + 2 q^{56} - 11 q^{57} + 7 q^{58} - 16 q^{59} - 2 q^{60} + 24 q^{61} - 12 q^{62} - 2 q^{63} + 11 q^{64} - 7 q^{65} - 6 q^{66} - 28 q^{67} - 16 q^{68} + 11 q^{69} + 5 q^{70} + 4 q^{71} - 11 q^{72} - 7 q^{73} - 3 q^{74} - 19 q^{75} + 11 q^{76} - 13 q^{77} - 7 q^{78} + 5 q^{79} + 2 q^{80} + 11 q^{81} - 11 q^{82} - 11 q^{83} + 2 q^{84} - 11 q^{85} - 7 q^{86} + 7 q^{87} + 6 q^{88} - 10 q^{89} - 2 q^{90} - 24 q^{91} - 11 q^{92} - 12 q^{93} + 37 q^{94} + 2 q^{95} + 11 q^{96} - 24 q^{97} - 23 q^{98} - 6 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 - 11 * q^3 + 11 * q^4 + 2 * q^5 + 11 * q^6 - 2 * q^7 - 11 * q^8 + 11 * q^9 - 2 * q^10 - 6 * q^11 - 11 * q^12 - 7 * q^13 + 2 * q^14 - 2 * q^15 + 11 * q^16 - 16 * q^17 - 11 * q^18 + 11 * q^19 + 2 * q^20 + 2 * q^21 + 6 * q^22 - 11 * q^23 + 11 * q^24 + 19 * q^25 + 7 * q^26 - 11 * q^27 - 2 * q^28 - 7 * q^29 + 2 * q^30 + 12 * q^31 - 11 * q^32 + 6 * q^33 + 16 * q^34 - 5 * q^35 + 11 * q^36 + 3 * q^37 - 11 * q^38 + 7 * q^39 - 2 * q^40 + 11 * q^41 - 2 * q^42 + 7 * q^43 - 6 * q^44 + 2 * q^45 + 11 * q^46 - 37 * q^47 - 11 * q^48 + 23 * q^49 - 19 * q^50 + 16 * q^51 - 7 * q^52 + 11 * q^53 + 11 * q^54 - 11 * q^55 + 2 * q^56 - 11 * q^57 + 7 * q^58 - 16 * q^59 - 2 * q^60 + 24 * q^61 - 12 * q^62 - 2 * q^63 + 11 * q^64 - 7 * q^65 - 6 * q^66 - 28 * q^67 - 16 * q^68 + 11 * q^69 + 5 * q^70 + 4 * q^71 - 11 * q^72 - 7 * q^73 - 3 * q^74 - 19 * q^75 + 11 * q^76 - 13 * q^77 - 7 * q^78 + 5 * q^79 + 2 * q^80 + 11 * q^81 - 11 * q^82 - 11 * q^83 + 2 * q^84 - 11 * q^85 - 7 * q^86 + 7 * q^87 + 6 * q^88 - 10 * q^89 - 2 * q^90 - 24 * q^91 - 11 * q^92 - 12 * q^93 + 37 * q^94 + 2 * q^95 + 11 * q^96 - 24 * q^97 - 23 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more