LMFDB - Embedded newform 6042.2.a.p.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} +1.00000 q^{6} +1.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} +1.00000 q^{6} +1.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{10} -4.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.56155 q^{14} -1.56155 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -1.56155 q^{20} +1.56155 q^{21} -4.00000 q^{22} -0.438447 q^{23} +1.00000 q^{24} -2.56155 q^{25} -4.00000 q^{26} +1.00000 q^{27} +1.56155 q^{28} +6.68466 q^{29} -1.56155 q^{30} -0.438447 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -2.43845 q^{35} +1.00000 q^{36} +7.12311 q^{37} +1.00000 q^{38} -4.00000 q^{39} -1.56155 q^{40} -5.12311 q^{41} +1.56155 q^{42} -9.56155 q^{43} -4.00000 q^{44} -1.56155 q^{45} -0.438447 q^{46} +8.24621 q^{47} +1.00000 q^{48} -4.56155 q^{49} -2.56155 q^{50} -2.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +6.24621 q^{55} +1.56155 q^{56} +1.00000 q^{57} +6.68466 q^{58} -7.80776 q^{59} -1.56155 q^{60} -13.1231 q^{61} -0.438447 q^{62} +1.56155 q^{63} +1.00000 q^{64} +6.24621 q^{65} -4.00000 q^{66} -0.684658 q^{67} -2.00000 q^{68} -0.438447 q^{69} -2.43845 q^{70} -6.24621 q^{71} +1.00000 q^{72} -10.0000 q^{73} +7.12311 q^{74} -2.56155 q^{75} +1.00000 q^{76} -6.24621 q^{77} -4.00000 q^{78} -13.1231 q^{79} -1.56155 q^{80} +1.00000 q^{81} -5.12311 q^{82} +4.00000 q^{83} +1.56155 q^{84} +3.12311 q^{85} -9.56155 q^{86} +6.68466 q^{87} -4.00000 q^{88} -8.43845 q^{89} -1.56155 q^{90} -6.24621 q^{91} -0.438447 q^{92} -0.438447 q^{93} +8.24621 q^{94} -1.56155 q^{95} +1.00000 q^{96} -8.24621 q^{97} -4.56155 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} - 8 q^{13} - q^{14} + q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} + q^{20} - q^{21} - 8 q^{22} - 5 q^{23} + 2 q^{24} - q^{25} - 8 q^{26} + 2 q^{27} - q^{28} + q^{29} + q^{30} - 5 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 9 q^{35} + 2 q^{36} + 6 q^{37} + 2 q^{38} - 8 q^{39} + q^{40} - 2 q^{41} - q^{42} - 15 q^{43} - 8 q^{44} + q^{45} - 5 q^{46} + 2 q^{48} - 5 q^{49} - q^{50} - 4 q^{51} - 8 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} - q^{56} + 2 q^{57} + q^{58} + 5 q^{59} + q^{60} - 18 q^{61} - 5 q^{62} - q^{63} + 2 q^{64} - 4 q^{65} - 8 q^{66} + 11 q^{67} - 4 q^{68} - 5 q^{69} - 9 q^{70} + 4 q^{71} + 2 q^{72} - 20 q^{73} + 6 q^{74} - q^{75} + 2 q^{76} + 4 q^{77} - 8 q^{78} - 18 q^{79} + q^{80} + 2 q^{81} - 2 q^{82} + 8 q^{83} - q^{84} - 2 q^{85} - 15 q^{86} + q^{87} - 8 q^{88} - 21 q^{89} + q^{90} + 4 q^{91} - 5 q^{92} - 5 q^{93} + q^{95} + 2 q^{96} - 5 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 + q^10 - 8 * q^11 + 2 * q^12 - 8 * q^13 - q^14 + q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 + q^20 - q^21 - 8 * q^22 - 5 * q^23 + 2 * q^24 - q^25 - 8 * q^26 + 2 * q^27 - q^28 + q^29 + q^30 - 5 * q^31 + 2 * q^32 - 8 * q^33 - 4 * q^34 - 9 * q^35 + 2 * q^36 + 6 * q^37 + 2 * q^38 - 8 * q^39 + q^40 - 2 * q^41 - q^42 - 15 * q^43 - 8 * q^44 + q^45 - 5 * q^46 + 2 * q^48 - 5 * q^49 - q^50 - 4 * q^51 - 8 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 - q^56 + 2 * q^57 + q^58 + 5 * q^59 + q^60 - 18 * q^61 - 5 * q^62 - q^63 + 2 * q^64 - 4 * q^65 - 8 * q^66 + 11 * q^67 - 4 * q^68 - 5 * q^69 - 9 * q^70 + 4 * q^71 + 2 * q^72 - 20 * q^73 + 6 * q^74 - q^75 + 2 * q^76 + 4 * q^77 - 8 * q^78 - 18 * q^79 + q^80 + 2 * q^81 - 2 * q^82 + 8 * q^83 - q^84 - 2 * q^85 - 15 * q^86 + q^87 - 8 * q^88 - 21 * q^89 + q^90 + 4 * q^91 - 5 * q^92 - 5 * q^93 + q^95 + 2 * q^96 - 5 * q^98 - 8 * 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
$5$	−1.56155	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$6$	1.00000	0.408248
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.56155	−0.493806
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.56155	0.417343
$15$	−1.56155	−0.403191
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.56155	−0.349174
$21$	1.56155	0.340759
$22$	−4.00000	−0.852803
$23$	−0.438447	−0.0914226	−0.0457113	−	0.998955i	$-0.514555\pi$
$23$	−0.438447	−0.0914226	−0.0457113	+	0.998955i	$0.514555\pi$
$24$	1.00000	0.204124
$25$	−2.56155	−0.512311
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	1.56155	0.295106
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	−1.56155	−0.285099
$31$	−0.438447	−0.0787474	−0.0393737	−	0.999225i	$-0.512536\pi$
$31$	−0.438447	−0.0787474	−0.0393737	+	0.999225i	$0.512536\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	−2.43845	−0.412173
$36$	1.00000	0.166667
$37$	7.12311	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$38$	1.00000	0.162221
$39$	−4.00000	−0.640513
$40$	−1.56155	−0.246903
$41$	−5.12311	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$41$	−5.12311	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$42$	1.56155	0.240953
$43$	−9.56155	−1.45812	−0.729062	−	0.684448i	$-0.760043\pi$
$43$	−9.56155	−1.45812	−0.729062	+	0.684448i	$0.760043\pi$
$44$	−4.00000	−0.603023
$45$	−1.56155	−0.232783
$46$	−0.438447	−0.0646455
$47$	8.24621	1.20283	0.601417	−	0.798935i	$-0.294603\pi$
$47$	8.24621	1.20283	0.601417	+	0.798935i	$0.294603\pi$
$48$	1.00000	0.144338
$49$	−4.56155	−0.651650
$50$	−2.56155	−0.362258
$51$	−2.00000	−0.280056
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	6.24621	0.842239
$56$	1.56155	0.208671
$57$	1.00000	0.132453
$58$	6.68466	0.877739
$59$	−7.80776	−1.01648	−0.508242	−	0.861214i	$-0.669705\pi$
$59$	−7.80776	−1.01648	−0.508242	+	0.861214i	$0.669705\pi$
$60$	−1.56155	−0.201596
$61$	−13.1231	−1.68024	−0.840121	−	0.542399i	$-0.817516\pi$
$61$	−13.1231	−1.68024	−0.840121	+	0.542399i	$0.817516\pi$
$62$	−0.438447	−0.0556828
$63$	1.56155	0.196737
$64$	1.00000	0.125000
$65$	6.24621	0.774747
$66$	−4.00000	−0.492366
$67$	−0.684658	−0.0836443	−0.0418222	−	0.999125i	$-0.513316\pi$
$67$	−0.684658	−0.0836443	−0.0418222	+	0.999125i	$0.513316\pi$
$68$	−2.00000	−0.242536
$69$	−0.438447	−0.0527828
$70$	−2.43845	−0.291450
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	7.12311	0.828044
$75$	−2.56155	−0.295783
$76$	1.00000	0.114708
$77$	−6.24621	−0.711822
$78$	−4.00000	−0.452911
$79$	−13.1231	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$79$	−13.1231	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$80$	−1.56155	−0.174587
$81$	1.00000	0.111111
$82$	−5.12311	−0.565752
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	1.56155	0.170379
$85$	3.12311	0.338748
$86$	−9.56155	−1.03105
$87$	6.68466	0.716671
$88$	−4.00000	−0.426401
$89$	−8.43845	−0.894474	−0.447237	−	0.894416i	$-0.647592\pi$
$89$	−8.43845	−0.894474	−0.447237	+	0.894416i	$0.647592\pi$
$90$	−1.56155	−0.164602
$91$	−6.24621	−0.654781
$92$	−0.438447	−0.0457113
$93$	−0.438447	−0.0454649
$94$	8.24621	0.850532
$95$	−1.56155	−0.160212
$96$	1.00000	0.102062
$97$	−8.24621	−0.837276	−0.418638	−	0.908153i	$-0.637492\pi$
$97$	−8.24621	−0.837276	−0.418638	+	0.908153i	$0.637492\pi$
$98$	−4.56155	−0.460786
$99$	−4.00000	−0.402015
$100$	−2.56155	−0.256155
$101$	4.68466	0.466141	0.233070	−	0.972460i	$-0.425123\pi$
$101$	4.68466	0.466141	0.233070	+	0.972460i	$0.425123\pi$
$102$	−2.00000	−0.198030
$103$	5.31534	0.523736	0.261868	−	0.965104i	$-0.415661\pi$
$103$	5.31534	0.523736	0.261868	+	0.965104i	$0.415661\pi$
$104$	−4.00000	−0.392232
$105$	−2.43845	−0.237968
$106$	1.00000	0.0971286
$107$	−4.68466	−0.452883	−0.226442	−	0.974025i	$-0.572709\pi$
$107$	−4.68466	−0.452883	−0.226442	+	0.974025i	$0.572709\pi$
$108$	1.00000	0.0962250
$109$	−3.31534	−0.317552	−0.158776	−	0.987315i	$-0.550755\pi$
$109$	−3.31534	−0.317552	−0.158776	+	0.987315i	$0.550755\pi$
$110$	6.24621	0.595553
$111$	7.12311	0.676095
$112$	1.56155	0.147553
$113$	0.438447	0.0412456	0.0206228	−	0.999787i	$-0.493435\pi$
$113$	0.438447	0.0412456	0.0206228	+	0.999787i	$0.493435\pi$
$114$	1.00000	0.0936586
$115$	0.684658	0.0638447
$116$	6.68466	0.620655
$117$	−4.00000	−0.369800
$118$	−7.80776	−0.718763
$119$	−3.12311	−0.286295
$120$	−1.56155	−0.142550
$121$	5.00000	0.454545
$122$	−13.1231	−1.18811
$123$	−5.12311	−0.461935
$124$	−0.438447	−0.0393737
$125$	11.8078	1.05612
$126$	1.56155	0.139114
$127$	−9.80776	−0.870298	−0.435149	−	0.900358i	$-0.643304\pi$
$127$	−9.80776	−0.870298	−0.435149	+	0.900358i	$0.643304\pi$
$128$	1.00000	0.0883883
$129$	−9.56155	−0.841848
$130$	6.24621	0.547829
$131$	12.4924	1.09147	0.545734	−	0.837958i	$-0.316251\pi$
$131$	12.4924	1.09147	0.545734	+	0.837958i	$0.316251\pi$
$132$	−4.00000	−0.348155
$133$	1.56155	0.135404
$134$	−0.684658	−0.0591455
$135$	−1.56155	−0.134397
$136$	−2.00000	−0.171499
$137$	−0.438447	−0.0374591	−0.0187295	−	0.999825i	$-0.505962\pi$
$137$	−0.438447	−0.0374591	−0.0187295	+	0.999825i	$0.505962\pi$
$138$	−0.438447	−0.0373231
$139$	−21.3693	−1.81252	−0.906261	−	0.422719i	$-0.861076\pi$
$139$	−21.3693	−1.81252	−0.906261	+	0.422719i	$0.861076\pi$
$140$	−2.43845	−0.206086
$141$	8.24621	0.694456
$142$	−6.24621	−0.524170
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	−10.4384	−0.866866
$146$	−10.0000	−0.827606
$147$	−4.56155	−0.376231
$148$	7.12311	0.585516
$149$	−18.2462	−1.49479	−0.747394	−	0.664381i	$-0.768695\pi$
$149$	−18.2462	−1.49479	−0.747394	+	0.664381i	$0.768695\pi$
$150$	−2.56155	−0.209150
$151$	−5.80776	−0.472629	−0.236315	−	0.971677i	$-0.575940\pi$
$151$	−5.80776	−0.472629	−0.236315	+	0.971677i	$0.575940\pi$
$152$	1.00000	0.0811107
$153$	−2.00000	−0.161690
$154$	−6.24621	−0.503334
$155$	0.684658	0.0549931
$156$	−4.00000	−0.320256
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−13.1231	−1.04402
$159$	1.00000	0.0793052
$160$	−1.56155	−0.123452
$161$	−0.684658	−0.0539586
$162$	1.00000	0.0785674
$163$	−6.43845	−0.504298	−0.252149	−	0.967688i	$-0.581137\pi$
$163$	−6.43845	−0.504298	−0.252149	+	0.967688i	$0.581137\pi$
$164$	−5.12311	−0.400047
$165$	6.24621	0.486267
$166$	4.00000	0.310460
$167$	−0.876894	−0.0678561	−0.0339281	−	0.999424i	$-0.510802\pi$
$167$	−0.876894	−0.0678561	−0.0339281	+	0.999424i	$0.510802\pi$
$168$	1.56155	0.120476
$169$	3.00000	0.230769
$170$	3.12311	0.239531
$171$	1.00000	0.0764719
$172$	−9.56155	−0.729062
$173$	3.36932	0.256164	0.128082	−	0.991764i	$-0.459118\pi$
$173$	3.36932	0.256164	0.128082	+	0.991764i	$0.459118\pi$
$174$	6.68466	0.506763
$175$	−4.00000	−0.302372
$176$	−4.00000	−0.301511
$177$	−7.80776	−0.586867
$178$	−8.43845	−0.632488
$179$	8.87689	0.663490	0.331745	−	0.943369i	$-0.392363\pi$
$179$	8.87689	0.663490	0.331745	+	0.943369i	$0.392363\pi$
$180$	−1.56155	−0.116391
$181$	6.43845	0.478566	0.239283	−	0.970950i	$-0.423088\pi$
$181$	6.43845	0.478566	0.239283	+	0.970950i	$0.423088\pi$
$182$	−6.24621	−0.463000
$183$	−13.1231	−0.970088
$184$	−0.438447	−0.0323228
$185$	−11.1231	−0.817787
$186$	−0.438447	−0.0321485
$187$	8.00000	0.585018
$188$	8.24621	0.601417
$189$	1.56155	0.113586
$190$	−1.56155	−0.113287
$191$	19.3693	1.40151	0.700757	−	0.713400i	$-0.252846\pi$
$191$	19.3693	1.40151	0.700757	+	0.713400i	$0.252846\pi$
$192$	1.00000	0.0721688
$193$	24.2462	1.74528	0.872640	−	0.488364i	$-0.162406\pi$
$193$	24.2462	1.74528	0.872640	+	0.488364i	$0.162406\pi$
$194$	−8.24621	−0.592043
$195$	6.24621	0.447300
$196$	−4.56155	−0.325825
$197$	2.24621	0.160036	0.0800180	−	0.996793i	$-0.474502\pi$
$197$	2.24621	0.160036	0.0800180	+	0.996793i	$0.474502\pi$
$198$	−4.00000	−0.284268
$199$	−17.5616	−1.24491	−0.622453	−	0.782657i	$-0.713864\pi$
$199$	−17.5616	−1.24491	−0.622453	+	0.782657i	$0.713864\pi$
$200$	−2.56155	−0.181129
$201$	−0.684658	−0.0482921
$202$	4.68466	0.329611
$203$	10.4384	0.732635
$204$	−2.00000	−0.140028
$205$	8.00000	0.558744
$206$	5.31534	0.370337
$207$	−0.438447	−0.0304742
$208$	−4.00000	−0.277350
$209$	−4.00000	−0.276686
$210$	−2.43845	−0.168269
$211$	11.1231	0.765746	0.382873	−	0.923801i	$-0.374935\pi$
$211$	11.1231	0.765746	0.382873	+	0.923801i	$0.374935\pi$
$212$	1.00000	0.0686803
$213$	−6.24621	−0.427983
$214$	−4.68466	−0.320237
$215$	14.9309	1.01828
$216$	1.00000	0.0680414
$217$	−0.684658	−0.0464776
$218$	−3.31534	−0.224543
$219$	−10.0000	−0.675737
$220$	6.24621	0.421119
$221$	8.00000	0.538138
$222$	7.12311	0.478072
$223$	12.2462	0.820067	0.410033	−	0.912070i	$-0.365517\pi$
$223$	12.2462	0.820067	0.410033	+	0.912070i	$0.365517\pi$
$224$	1.56155	0.104336
$225$	−2.56155	−0.170770
$226$	0.438447	0.0291651
$227$	15.8078	1.04920	0.524599	−	0.851349i	$-0.324215\pi$
$227$	15.8078	1.04920	0.524599	+	0.851349i	$0.324215\pi$
$228$	1.00000	0.0662266
$229$	−16.9309	−1.11882	−0.559412	−	0.828890i	$-0.688973\pi$
$229$	−16.9309	−1.11882	−0.559412	+	0.828890i	$0.688973\pi$
$230$	0.684658	0.0451450
$231$	−6.24621	−0.410971
$232$	6.68466	0.438869
$233$	−0.246211	−0.0161298	−0.00806492	−	0.999967i	$-0.502567\pi$
$233$	−0.246211	−0.0161298	−0.00806492	+	0.999967i	$0.502567\pi$
$234$	−4.00000	−0.261488
$235$	−12.8769	−0.839996
$236$	−7.80776	−0.508242
$237$	−13.1231	−0.852437
$238$	−3.12311	−0.202441
$239$	15.3693	0.994158	0.497079	−	0.867705i	$-0.334406\pi$
$239$	15.3693	0.994158	0.497079	+	0.867705i	$0.334406\pi$
$240$	−1.56155	−0.100798
$241$	−0.246211	−0.0158599	−0.00792993	−	0.999969i	$-0.502524\pi$
$241$	−0.246211	−0.0158599	−0.00792993	+	0.999969i	$0.502524\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−13.1231	−0.840121
$245$	7.12311	0.455079
$246$	−5.12311	−0.326637
$247$	−4.00000	−0.254514
$248$	−0.438447	−0.0278414
$249$	4.00000	0.253490
$250$	11.8078	0.746789
$251$	−16.4924	−1.04099	−0.520496	−	0.853864i	$-0.674253\pi$
$251$	−16.4924	−1.04099	−0.520496	+	0.853864i	$0.674253\pi$
$252$	1.56155	0.0983686
$253$	1.75379	0.110260
$254$	−9.80776	−0.615394
$255$	3.12311	0.195576
$256$	1.00000	0.0625000
$257$	13.1231	0.818597	0.409298	−	0.912401i	$-0.365773\pi$
$257$	13.1231	0.818597	0.409298	+	0.912401i	$0.365773\pi$
$258$	−9.56155	−0.595276
$259$	11.1231	0.691156
$260$	6.24621	0.387374
$261$	6.68466	0.413770
$262$	12.4924	0.771784
$263$	−16.9309	−1.04400	−0.522001	−	0.852945i	$-0.674814\pi$
$263$	−16.9309	−1.04400	−0.522001	+	0.852945i	$0.674814\pi$
$264$	−4.00000	−0.246183
$265$	−1.56155	−0.0959254
$266$	1.56155	0.0957449
$267$	−8.43845	−0.516425
$268$	−0.684658	−0.0418222
$269$	1.31534	0.0801978	0.0400989	−	0.999196i	$-0.487233\pi$
$269$	1.31534	0.0801978	0.0400989	+	0.999196i	$0.487233\pi$
$270$	−1.56155	−0.0950331
$271$	9.56155	0.580823	0.290411	−	0.956902i	$-0.406208\pi$
$271$	9.56155	0.580823	0.290411	+	0.956902i	$0.406208\pi$
$272$	−2.00000	−0.121268
$273$	−6.24621	−0.378038
$274$	−0.438447	−0.0264876
$275$	10.2462	0.617870
$276$	−0.438447	−0.0263914
$277$	13.6155	0.818078	0.409039	−	0.912517i	$-0.365864\pi$
$277$	13.6155	0.818078	0.409039	+	0.912517i	$0.365864\pi$
$278$	−21.3693	−1.28165
$279$	−0.438447	−0.0262491
$280$	−2.43845	−0.145725
$281$	28.9309	1.72587	0.862935	−	0.505314i	$-0.168623\pi$
$281$	28.9309	1.72587	0.862935	+	0.505314i	$0.168623\pi$
$282$	8.24621	0.491055
$283$	−22.7386	−1.35167	−0.675836	−	0.737052i	$-0.736217\pi$
$283$	−22.7386	−1.35167	−0.675836	+	0.737052i	$0.736217\pi$
$284$	−6.24621	−0.370644
$285$	−1.56155	−0.0924984
$286$	16.0000	0.946100
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−10.4384	−0.612967
$291$	−8.24621	−0.483401
$292$	−10.0000	−0.585206
$293$	−7.75379	−0.452981	−0.226491	−	0.974013i	$-0.572725\pi$
$293$	−7.75379	−0.452981	−0.226491	+	0.974013i	$0.572725\pi$
$294$	−4.56155	−0.266035
$295$	12.1922	0.709859
$296$	7.12311	0.414022
$297$	−4.00000	−0.232104
$298$	−18.2462	−1.05697
$299$	1.75379	0.101424
$300$	−2.56155	−0.147891
$301$	−14.9309	−0.860601
$302$	−5.80776	−0.334199
$303$	4.68466	0.269127
$304$	1.00000	0.0573539
$305$	20.4924	1.17339
$306$	−2.00000	−0.114332
$307$	−10.2462	−0.584782	−0.292391	−	0.956299i	$-0.594451\pi$
$307$	−10.2462	−0.584782	−0.292391	+	0.956299i	$0.594451\pi$
$308$	−6.24621	−0.355911
$309$	5.31534	0.302379
$310$	0.684658	0.0388860
$311$	−1.12311	−0.0636855	−0.0318427	−	0.999493i	$-0.510138\pi$
$311$	−1.12311	−0.0636855	−0.0318427	+	0.999493i	$0.510138\pi$
$312$	−4.00000	−0.226455
$313$	9.12311	0.515668	0.257834	−	0.966189i	$-0.416991\pi$
$313$	9.12311	0.515668	0.257834	+	0.966189i	$0.416991\pi$
$314$	18.0000	1.01580
$315$	−2.43845	−0.137391
$316$	−13.1231	−0.738232
$317$	14.4924	0.813976	0.406988	−	0.913434i	$-0.366579\pi$
$317$	14.4924	0.813976	0.406988	+	0.913434i	$0.366579\pi$
$318$	1.00000	0.0560772
$319$	−26.7386	−1.49708
$320$	−1.56155	−0.0872935
$321$	−4.68466	−0.261472
$322$	−0.684658	−0.0381545
$323$	−2.00000	−0.111283
$324$	1.00000	0.0555556
$325$	10.2462	0.568358
$326$	−6.43845	−0.356593
$327$	−3.31534	−0.183339
$328$	−5.12311	−0.282876
$329$	12.8769	0.709926
$330$	6.24621	0.343843
$331$	11.6155	0.638447	0.319224	−	0.947679i	$-0.396578\pi$
$331$	11.6155	0.638447	0.319224	+	0.947679i	$0.396578\pi$
$332$	4.00000	0.219529
$333$	7.12311	0.390344
$334$	−0.876894	−0.0479815
$335$	1.06913	0.0584128
$336$	1.56155	0.0851897
$337$	1.80776	0.0984752	0.0492376	−	0.998787i	$-0.484321\pi$
$337$	1.80776	0.0984752	0.0492376	+	0.998787i	$0.484321\pi$
$338$	3.00000	0.163178
$339$	0.438447	0.0238132
$340$	3.12311	0.169374
$341$	1.75379	0.0949730
$342$	1.00000	0.0540738
$343$	−18.0540	−0.974823
$344$	−9.56155	−0.515524
$345$	0.684658	0.0368608
$346$	3.36932	0.181136
$347$	−2.24621	−0.120583	−0.0602915	−	0.998181i	$-0.519203\pi$
$347$	−2.24621	−0.120583	−0.0602915	+	0.998181i	$0.519203\pi$
$348$	6.68466	0.358335
$349$	−20.2462	−1.08375	−0.541877	−	0.840458i	$-0.682286\pi$
$349$	−20.2462	−1.08375	−0.541877	+	0.840458i	$0.682286\pi$
$350$	−4.00000	−0.213809
$351$	−4.00000	−0.213504
$352$	−4.00000	−0.213201
$353$	−20.7386	−1.10381	−0.551903	−	0.833908i	$-0.686098\pi$
$353$	−20.7386	−1.10381	−0.551903	+	0.833908i	$0.686098\pi$
$354$	−7.80776	−0.414978
$355$	9.75379	0.517677
$356$	−8.43845	−0.447237
$357$	−3.12311	−0.165292
$358$	8.87689	0.469158
$359$	−1.80776	−0.0954101	−0.0477051	−	0.998861i	$-0.515191\pi$
$359$	−1.80776	−0.0954101	−0.0477051	+	0.998861i	$0.515191\pi$
$360$	−1.56155	−0.0823011
$361$	1.00000	0.0526316
$362$	6.43845	0.338397
$363$	5.00000	0.262432
$364$	−6.24621	−0.327390
$365$	15.6155	0.817354
$366$	−13.1231	−0.685956
$367$	22.7386	1.18695	0.593474	−	0.804854i	$-0.297756\pi$
$367$	22.7386	1.18695	0.593474	+	0.804854i	$0.297756\pi$
$368$	−0.438447	−0.0228556
$369$	−5.12311	−0.266698
$370$	−11.1231	−0.578263
$371$	1.56155	0.0810718
$372$	−0.438447	−0.0227324
$373$	18.0540	0.934799	0.467400	−	0.884046i	$-0.345191\pi$
$373$	18.0540	0.934799	0.467400	+	0.884046i	$0.345191\pi$
$374$	8.00000	0.413670
$375$	11.8078	0.609750
$376$	8.24621	0.425266
$377$	−26.7386	−1.37711
$378$	1.56155	0.0803176
$379$	−10.7386	−0.551607	−0.275803	−	0.961214i	$-0.588944\pi$
$379$	−10.7386	−0.551607	−0.275803	+	0.961214i	$0.588944\pi$
$380$	−1.56155	−0.0801060
$381$	−9.80776	−0.502467
$382$	19.3693	0.991020
$383$	−6.63068	−0.338812	−0.169406	−	0.985546i	$-0.554185\pi$
$383$	−6.63068	−0.338812	−0.169406	+	0.985546i	$0.554185\pi$
$384$	1.00000	0.0510310
$385$	9.75379	0.497099
$386$	24.2462	1.23410
$387$	−9.56155	−0.486041
$388$	−8.24621	−0.418638
$389$	6.05398	0.306949	0.153474	−	0.988153i	$-0.450954\pi$
$389$	6.05398	0.306949	0.153474	+	0.988153i	$0.450954\pi$
$390$	6.24621	0.316289
$391$	0.876894	0.0443465
$392$	−4.56155	−0.230393
$393$	12.4924	0.630159
$394$	2.24621	0.113162
$395$	20.4924	1.03109
$396$	−4.00000	−0.201008
$397$	11.3693	0.570610	0.285305	−	0.958437i	$-0.407905\pi$
$397$	11.3693	0.570610	0.285305	+	0.958437i	$0.407905\pi$
$398$	−17.5616	−0.880281
$399$	1.56155	0.0781754
$400$	−2.56155	−0.128078
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−0.684658	−0.0341477
$403$	1.75379	0.0873624
$404$	4.68466	0.233070
$405$	−1.56155	−0.0775942
$406$	10.4384	0.518051
$407$	−28.4924	−1.41232
$408$	−2.00000	−0.0990148
$409$	−13.1231	−0.648896	−0.324448	−	0.945904i	$-0.605178\pi$
$409$	−13.1231	−0.648896	−0.324448	+	0.945904i	$0.605178\pi$
$410$	8.00000	0.395092
$411$	−0.438447	−0.0216270
$412$	5.31534	0.261868
$413$	−12.1922	−0.599941
$414$	−0.438447	−0.0215485
$415$	−6.24621	−0.306614
$416$	−4.00000	−0.196116
$417$	−21.3693	−1.04646
$418$	−4.00000	−0.195646
$419$	−6.24621	−0.305147	−0.152574	−	0.988292i	$-0.548756\pi$
$419$	−6.24621	−0.305147	−0.152574	+	0.988292i	$0.548756\pi$
$420$	−2.43845	−0.118984
$421$	1.06913	0.0521062	0.0260531	−	0.999661i	$-0.491706\pi$
$421$	1.06913	0.0521062	0.0260531	+	0.999661i	$0.491706\pi$
$422$	11.1231	0.541464
$423$	8.24621	0.400945
$424$	1.00000	0.0485643
$425$	5.12311	0.248507
$426$	−6.24621	−0.302630
$427$	−20.4924	−0.991698
$428$	−4.68466	−0.226442
$429$	16.0000	0.772487
$430$	14.9309	0.720030
$431$	−11.3153	−0.545041	−0.272520	−	0.962150i	$-0.587857\pi$
$431$	−11.3153	−0.545041	−0.272520	+	0.962150i	$0.587857\pi$
$432$	1.00000	0.0481125
$433$	16.7386	0.804407	0.402204	−	0.915550i	$-0.368244\pi$
$433$	16.7386	0.804407	0.402204	+	0.915550i	$0.368244\pi$
$434$	−0.684658	−0.0328647
$435$	−10.4384	−0.500485
$436$	−3.31534	−0.158776
$437$	−0.438447	−0.0209738
$438$	−10.0000	−0.477818
$439$	−3.75379	−0.179159	−0.0895793	−	0.995980i	$-0.528552\pi$
$439$	−3.75379	−0.179159	−0.0895793	+	0.995980i	$0.528552\pi$
$440$	6.24621	0.297776
$441$	−4.56155	−0.217217
$442$	8.00000	0.380521
$443$	21.5616	1.02442	0.512210	−	0.858860i	$-0.328827\pi$
$443$	21.5616	1.02442	0.512210	+	0.858860i	$0.328827\pi$
$444$	7.12311	0.338048
$445$	13.1771	0.624654
$446$	12.2462	0.579875
$447$	−18.2462	−0.863016
$448$	1.56155	0.0737764
$449$	−7.75379	−0.365924	−0.182962	−	0.983120i	$-0.558569\pi$
$449$	−7.75379	−0.365924	−0.182962	+	0.983120i	$0.558569\pi$
$450$	−2.56155	−0.120753
$451$	20.4924	0.964950
$452$	0.438447	0.0206228
$453$	−5.80776	−0.272873
$454$	15.8078	0.741895
$455$	9.75379	0.457265
$456$	1.00000	0.0468293
$457$	−1.12311	−0.0525367	−0.0262683	−	0.999655i	$-0.508362\pi$
$457$	−1.12311	−0.0525367	−0.0262683	+	0.999655i	$0.508362\pi$
$458$	−16.9309	−0.791128
$459$	−2.00000	−0.0933520
$460$	0.684658	0.0319224
$461$	2.63068	0.122523	0.0612616	−	0.998122i	$-0.480488\pi$
$461$	2.63068	0.122523	0.0612616	+	0.998122i	$0.480488\pi$
$462$	−6.24621	−0.290600
$463$	8.49242	0.394676	0.197338	−	0.980335i	$-0.436770\pi$
$463$	8.49242	0.394676	0.197338	+	0.980335i	$0.436770\pi$
$464$	6.68466	0.310327
$465$	0.684658	0.0317503
$466$	−0.246211	−0.0114055
$467$	4.87689	0.225676	0.112838	−	0.993613i	$-0.464006\pi$
$467$	4.87689	0.225676	0.112838	+	0.993613i	$0.464006\pi$
$468$	−4.00000	−0.184900
$469$	−1.06913	−0.0493679
$470$	−12.8769	−0.593967
$471$	18.0000	0.829396
$472$	−7.80776	−0.359381
$473$	38.2462	1.75856
$474$	−13.1231	−0.602764
$475$	−2.56155	−0.117532
$476$	−3.12311	−0.143147
$477$	1.00000	0.0457869
$478$	15.3693	0.702976
$479$	−16.0540	−0.733525	−0.366762	−	0.930315i	$-0.619534\pi$
$479$	−16.0540	−0.733525	−0.366762	+	0.930315i	$0.619534\pi$
$480$	−1.56155	−0.0712748
$481$	−28.4924	−1.29914
$482$	−0.246211	−0.0112146
$483$	−0.684658	−0.0311530
$484$	5.00000	0.227273
$485$	12.8769	0.584710
$486$	1.00000	0.0453609
$487$	−34.4924	−1.56300	−0.781500	−	0.623905i	$-0.785545\pi$
$487$	−34.4924	−1.56300	−0.781500	+	0.623905i	$0.785545\pi$
$488$	−13.1231	−0.594055
$489$	−6.43845	−0.291157
$490$	7.12311	0.321789
$491$	4.68466	0.211416	0.105708	−	0.994397i	$-0.466289\pi$
$491$	4.68466	0.211416	0.105708	+	0.994397i	$0.466289\pi$
$492$	−5.12311	−0.230967
$493$	−13.3693	−0.602124
$494$	−4.00000	−0.179969
$495$	6.24621	0.280746
$496$	−0.438447	−0.0196869
$497$	−9.75379	−0.437517
$498$	4.00000	0.179244
$499$	22.7386	1.01792	0.508961	−	0.860790i	$-0.330030\pi$
$499$	22.7386	1.01792	0.508961	+	0.860790i	$0.330030\pi$
$500$	11.8078	0.528059
$501$	−0.876894	−0.0391768
$502$	−16.4924	−0.736093
$503$	−12.4384	−0.554603	−0.277301	−	0.960783i	$-0.589440\pi$
$503$	−12.4384	−0.554603	−0.277301	+	0.960783i	$0.589440\pi$
$504$	1.56155	0.0695571
$505$	−7.31534	−0.325528
$506$	1.75379	0.0779654
$507$	3.00000	0.133235
$508$	−9.80776	−0.435149
$509$	−12.2462	−0.542804	−0.271402	−	0.962466i	$-0.587487\pi$
$509$	−12.2462	−0.542804	−0.271402	+	0.962466i	$0.587487\pi$
$510$	3.12311	0.138293
$511$	−15.6155	−0.690790
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	13.1231	0.578835
$515$	−8.30019	−0.365750
$516$	−9.56155	−0.420924
$517$	−32.9848	−1.45067
$518$	11.1231	0.488721
$519$	3.36932	0.147897
$520$	6.24621	0.273914
$521$	−7.75379	−0.339700	−0.169850	−	0.985470i	$-0.554328\pi$
$521$	−7.75379	−0.339700	−0.169850	+	0.985470i	$0.554328\pi$
$522$	6.68466	0.292580
$523$	−2.63068	−0.115032	−0.0575159	−	0.998345i	$-0.518318\pi$
$523$	−2.63068	−0.115032	−0.0575159	+	0.998345i	$0.518318\pi$
$524$	12.4924	0.545734
$525$	−4.00000	−0.174574
$526$	−16.9309	−0.738221
$527$	0.876894	0.0381981
$528$	−4.00000	−0.174078
$529$	−22.8078	−0.991642
$530$	−1.56155	−0.0678295
$531$	−7.80776	−0.338828
$532$	1.56155	0.0677019
$533$	20.4924	0.887625
$534$	−8.43845	−0.365167
$535$	7.31534	0.316270
$536$	−0.684658	−0.0295727
$537$	8.87689	0.383066
$538$	1.31534	0.0567084
$539$	18.2462	0.785920
$540$	−1.56155	−0.0671985
$541$	−35.5616	−1.52891	−0.764455	−	0.644677i	$-0.776992\pi$
$541$	−35.5616	−1.52891	−0.764455	+	0.644677i	$0.776992\pi$
$542$	9.56155	0.410704
$543$	6.43845	0.276300
$544$	−2.00000	−0.0857493
$545$	5.17708	0.221762
$546$	−6.24621	−0.267313
$547$	16.4924	0.705165	0.352583	−	0.935781i	$-0.385304\pi$
$547$	16.4924	0.705165	0.352583	+	0.935781i	$0.385304\pi$
$548$	−0.438447	−0.0187295
$549$	−13.1231	−0.560080
$550$	10.2462	0.436900
$551$	6.68466	0.284776
$552$	−0.438447	−0.0186616
$553$	−20.4924	−0.871426
$554$	13.6155	0.578468
$555$	−11.1231	−0.472150
$556$	−21.3693	−0.906261
$557$	−21.5616	−0.913592	−0.456796	−	0.889571i	$-0.651003\pi$
$557$	−21.5616	−0.913592	−0.456796	+	0.889571i	$0.651003\pi$
$558$	−0.438447	−0.0185609
$559$	38.2462	1.61764
$560$	−2.43845	−0.103043
$561$	8.00000	0.337760
$562$	28.9309	1.22038
$563$	−18.2462	−0.768986	−0.384493	−	0.923128i	$-0.625624\pi$
$563$	−18.2462	−0.768986	−0.384493	+	0.923128i	$0.625624\pi$
$564$	8.24621	0.347228
$565$	−0.684658	−0.0288038
$566$	−22.7386	−0.955776
$567$	1.56155	0.0655791
$568$	−6.24621	−0.262085
$569$	21.1231	0.885527	0.442763	−	0.896639i	$-0.353998\pi$
$569$	21.1231	0.885527	0.442763	+	0.896639i	$0.353998\pi$
$570$	−1.56155	−0.0654062
$571$	19.6155	0.820884	0.410442	−	0.911887i	$-0.365374\pi$
$571$	19.6155	0.820884	0.410442	+	0.911887i	$0.365374\pi$
$572$	16.0000	0.668994
$573$	19.3693	0.809165
$574$	−8.00000	−0.333914
$575$	1.12311	0.0468367
$576$	1.00000	0.0416667
$577$	26.9848	1.12339	0.561697	−	0.827343i	$-0.310149\pi$
$577$	26.9848	1.12339	0.561697	+	0.827343i	$0.310149\pi$
$578$	−13.0000	−0.540729
$579$	24.2462	1.00764
$580$	−10.4384	−0.433433
$581$	6.24621	0.259137
$582$	−8.24621	−0.341816
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	6.24621	0.258249
$586$	−7.75379	−0.320306
$587$	−22.7386	−0.938524	−0.469262	−	0.883059i	$-0.655480\pi$
$587$	−22.7386	−0.938524	−0.469262	+	0.883059i	$0.655480\pi$
$588$	−4.56155	−0.188115
$589$	−0.438447	−0.0180659
$590$	12.1922	0.501946
$591$	2.24621	0.0923968
$592$	7.12311	0.292758
$593$	29.1231	1.19594	0.597971	−	0.801518i	$-0.295974\pi$
$593$	29.1231	1.19594	0.597971	+	0.801518i	$0.295974\pi$
$594$	−4.00000	−0.164122
$595$	4.87689	0.199933
$596$	−18.2462	−0.747394
$597$	−17.5616	−0.718747
$598$	1.75379	0.0717178
$599$	15.5076	0.633622	0.316811	−	0.948489i	$-0.397388\pi$
$599$	15.5076	0.633622	0.316811	+	0.948489i	$0.397388\pi$
$600$	−2.56155	−0.104575
$601$	3.17708	0.129596	0.0647979	−	0.997898i	$-0.479360\pi$
$601$	3.17708	0.129596	0.0647979	+	0.997898i	$0.479360\pi$
$602$	−14.9309	−0.608537
$603$	−0.684658	−0.0278814
$604$	−5.80776	−0.236315
$605$	−7.80776	−0.317431
$606$	4.68466	0.190301
$607$	−21.6155	−0.877347	−0.438674	−	0.898646i	$-0.644552\pi$
$607$	−21.6155	−0.877347	−0.438674	+	0.898646i	$0.644552\pi$
$608$	1.00000	0.0405554
$609$	10.4384	0.422987
$610$	20.4924	0.829714
$611$	−32.9848	−1.33442
$612$	−2.00000	−0.0808452
$613$	−21.6155	−0.873043	−0.436521	−	0.899694i	$-0.643790\pi$
$613$	−21.6155	−0.873043	−0.436521	+	0.899694i	$0.643790\pi$
$614$	−10.2462	−0.413503
$615$	8.00000	0.322591
$616$	−6.24621	−0.251667
$617$	−5.50758	−0.221727	−0.110863	−	0.993836i	$-0.535362\pi$
$617$	−5.50758	−0.221727	−0.110863	+	0.993836i	$0.535362\pi$
$618$	5.31534	0.213814
$619$	15.8078	0.635368	0.317684	−	0.948197i	$-0.397095\pi$
$619$	15.8078	0.635368	0.317684	+	0.948197i	$0.397095\pi$
$620$	0.684658	0.0274965
$621$	−0.438447	−0.0175943
$622$	−1.12311	−0.0450324
$623$	−13.1771	−0.527929
$624$	−4.00000	−0.160128
$625$	−5.63068	−0.225227
$626$	9.12311	0.364633
$627$	−4.00000	−0.159745
$628$	18.0000	0.718278
$629$	−14.2462	−0.568034
$630$	−2.43845	−0.0971501
$631$	29.8617	1.18878	0.594389	−	0.804178i	$-0.297394\pi$
$631$	29.8617	1.18878	0.594389	+	0.804178i	$0.297394\pi$
$632$	−13.1231	−0.522009
$633$	11.1231	0.442104
$634$	14.4924	0.575568
$635$	15.3153	0.607771
$636$	1.00000	0.0396526
$637$	18.2462	0.722941
$638$	−26.7386	−1.05859
$639$	−6.24621	−0.247096
$640$	−1.56155	−0.0617258
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	−4.68466	−0.184889
$643$	−42.9309	−1.69303	−0.846514	−	0.532366i	$-0.821303\pi$
$643$	−42.9309	−1.69303	−0.846514	+	0.532366i	$0.821303\pi$
$644$	−0.684658	−0.0269793
$645$	14.9309	0.587902
$646$	−2.00000	−0.0786889
$647$	−27.3693	−1.07600	−0.537999	−	0.842945i	$-0.680820\pi$
$647$	−27.3693	−1.07600	−0.537999	+	0.842945i	$0.680820\pi$
$648$	1.00000	0.0392837
$649$	31.2311	1.22593
$650$	10.2462	0.401889
$651$	−0.684658	−0.0268339
$652$	−6.43845	−0.252149
$653$	35.6155	1.39374	0.696872	−	0.717196i	$-0.254575\pi$
$653$	35.6155	1.39374	0.696872	+	0.717196i	$0.254575\pi$
$654$	−3.31534	−0.129640
$655$	−19.5076	−0.762224
$656$	−5.12311	−0.200024
$657$	−10.0000	−0.390137
$658$	12.8769	0.501994
$659$	34.2462	1.33404	0.667021	−	0.745038i	$-0.267569\pi$
$659$	34.2462	1.33404	0.667021	+	0.745038i	$0.267569\pi$
$660$	6.24621	0.243133
$661$	40.8769	1.58993	0.794963	−	0.606657i	$-0.207490\pi$
$661$	40.8769	1.58993	0.794963	+	0.606657i	$0.207490\pi$
$662$	11.6155	0.451450
$663$	8.00000	0.310694
$664$	4.00000	0.155230
$665$	−2.43845	−0.0945589
$666$	7.12311	0.276015
$667$	−2.93087	−0.113484
$668$	−0.876894	−0.0339281
$669$	12.2462	0.473466
$670$	1.06913	0.0413041
$671$	52.4924	2.02645
$672$	1.56155	0.0602382
$673$	10.8769	0.419273	0.209637	−	0.977779i	$-0.432772\pi$
$673$	10.8769	0.419273	0.209637	+	0.977779i	$0.432772\pi$
$674$	1.80776	0.0696325
$675$	−2.56155	−0.0985942
$676$	3.00000	0.115385
$677$	−11.7538	−0.451735	−0.225867	−	0.974158i	$-0.572522\pi$
$677$	−11.7538	−0.451735	−0.225867	+	0.974158i	$0.572522\pi$
$678$	0.438447	0.0168385
$679$	−12.8769	−0.494170
$680$	3.12311	0.119766
$681$	15.8078	0.605755
$682$	1.75379	0.0671560
$683$	6.73863	0.257847	0.128923	−	0.991655i	$-0.458848\pi$
$683$	6.73863	0.257847	0.128923	+	0.991655i	$0.458848\pi$
$684$	1.00000	0.0382360
$685$	0.684658	0.0261595
$686$	−18.0540	−0.689304
$687$	−16.9309	−0.645953
$688$	−9.56155	−0.364531
$689$	−4.00000	−0.152388
$690$	0.684658	0.0260645
$691$	27.6155	1.05054	0.525272	−	0.850934i	$-0.323964\pi$
$691$	27.6155	1.05054	0.525272	+	0.850934i	$0.323964\pi$
$692$	3.36932	0.128082
$693$	−6.24621	−0.237274
$694$	−2.24621	−0.0852650
$695$	33.3693	1.26577
$696$	6.68466	0.253381
$697$	10.2462	0.388103
$698$	−20.2462	−0.766330
$699$	−0.246211	−0.00931256
$700$	−4.00000	−0.151186
$701$	−17.1771	−0.648769	−0.324385	−	0.945925i	$-0.605157\pi$
$701$	−17.1771	−0.648769	−0.324385	+	0.945925i	$0.605157\pi$
$702$	−4.00000	−0.150970
$703$	7.12311	0.268653
$704$	−4.00000	−0.150756
$705$	−12.8769	−0.484972
$706$	−20.7386	−0.780509
$707$	7.31534	0.275122
$708$	−7.80776	−0.293434
$709$	49.1231	1.84486	0.922428	−	0.386168i	$-0.126201\pi$
$709$	49.1231	1.84486	0.922428	+	0.386168i	$0.126201\pi$
$710$	9.75379	0.366053
$711$	−13.1231	−0.492155
$712$	−8.43845	−0.316244
$713$	0.192236	0.00719929
$714$	−3.12311	−0.116879
$715$	−24.9848	−0.934380
$716$	8.87689	0.331745
$717$	15.3693	0.573978
$718$	−1.80776	−0.0674652
$719$	48.0540	1.79211	0.896055	−	0.443942i	$-0.146420\pi$
$719$	48.0540	1.79211	0.896055	+	0.443942i	$0.146420\pi$
$720$	−1.56155	−0.0581956
$721$	8.30019	0.309115
$722$	1.00000	0.0372161
$723$	−0.246211	−0.00915669
$724$	6.43845	0.239283
$725$	−17.1231	−0.635936
$726$	5.00000	0.185567
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	−6.24621	−0.231500
$729$	1.00000	0.0370370
$730$	15.6155	0.577957
$731$	19.1231	0.707294
$732$	−13.1231	−0.485044
$733$	−20.9309	−0.773099	−0.386550	−	0.922269i	$-0.626333\pi$
$733$	−20.9309	−0.773099	−0.386550	+	0.922269i	$0.626333\pi$
$734$	22.7386	0.839298
$735$	7.12311	0.262740
$736$	−0.438447	−0.0161614
$737$	2.73863	0.100879
$738$	−5.12311	−0.188584
$739$	−13.7538	−0.505941	−0.252971	−	0.967474i	$-0.581408\pi$
$739$	−13.7538	−0.505941	−0.252971	+	0.967474i	$0.581408\pi$
$740$	−11.1231	−0.408893
$741$	−4.00000	−0.146944
$742$	1.56155	0.0573264
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	−0.438447	−0.0160743
$745$	28.4924	1.04388
$746$	18.0540	0.661003
$747$	4.00000	0.146352
$748$	8.00000	0.292509
$749$	−7.31534	−0.267297
$750$	11.8078	0.431159
$751$	42.9848	1.56854	0.784270	−	0.620420i	$-0.213038\pi$
$751$	42.9848	1.56854	0.784270	+	0.620420i	$0.213038\pi$
$752$	8.24621	0.300708
$753$	−16.4924	−0.601017
$754$	−26.7386	−0.973764
$755$	9.06913	0.330059
$756$	1.56155	0.0567931
$757$	39.1771	1.42392	0.711958	−	0.702222i	$-0.247809\pi$
$757$	39.1771	1.42392	0.711958	+	0.702222i	$0.247809\pi$
$758$	−10.7386	−0.390045
$759$	1.75379	0.0636585
$760$	−1.56155	−0.0566435
$761$	−24.7386	−0.896775	−0.448387	−	0.893839i	$-0.648001\pi$
$761$	−24.7386	−0.896775	−0.448387	+	0.893839i	$0.648001\pi$
$762$	−9.80776	−0.355298
$763$	−5.17708	−0.187423
$764$	19.3693	0.700757
$765$	3.12311	0.112916
$766$	−6.63068	−0.239576
$767$	31.2311	1.12769
$768$	1.00000	0.0360844
$769$	−23.8617	−0.860476	−0.430238	−	0.902715i	$-0.641570\pi$
$769$	−23.8617	−0.860476	−0.430238	+	0.902715i	$0.641570\pi$
$770$	9.75379	0.351502
$771$	13.1231	0.472617
$772$	24.2462	0.872640
$773$	15.8617	0.570507	0.285254	−	0.958452i	$-0.407922\pi$
$773$	15.8617	0.570507	0.285254	+	0.958452i	$0.407922\pi$
$774$	−9.56155	−0.343683
$775$	1.12311	0.0403431
$776$	−8.24621	−0.296022
$777$	11.1231	0.399039
$778$	6.05398	0.217046
$779$	−5.12311	−0.183554
$780$	6.24621	0.223650
$781$	24.9848	0.894028
$782$	0.876894	0.0313577
$783$	6.68466	0.238890
$784$	−4.56155	−0.162913
$785$	−28.1080	−1.00322
$786$	12.4924	0.445590
$787$	−21.0691	−0.751033	−0.375517	−	0.926816i	$-0.622535\pi$
$787$	−21.0691	−0.751033	−0.375517	+	0.926816i	$0.622535\pi$
$788$	2.24621	0.0800180
$789$	−16.9309	−0.602755
$790$	20.4924	0.729088
$791$	0.684658	0.0243437
$792$	−4.00000	−0.142134
$793$	52.4924	1.86406
$794$	11.3693	0.403482
$795$	−1.56155	−0.0553826
$796$	−17.5616	−0.622453
$797$	−51.4773	−1.82342	−0.911709	−	0.410836i	$-0.865237\pi$
$797$	−51.4773	−1.82342	−0.911709	+	0.410836i	$0.865237\pi$
$798$	1.56155	0.0552784
$799$	−16.4924	−0.583460
$800$	−2.56155	−0.0905646
$801$	−8.43845	−0.298158
$802$	−2.00000	−0.0706225
$803$	40.0000	1.41157
$804$	−0.684658	−0.0241460
$805$	1.06913	0.0376819
$806$	1.75379	0.0617746
$807$	1.31534	0.0463022
$808$	4.68466	0.164806
$809$	16.9309	0.595258	0.297629	−	0.954682i	$-0.403804\pi$
$809$	16.9309	0.595258	0.297629	+	0.954682i	$0.403804\pi$
$810$	−1.56155	−0.0548674
$811$	44.9848	1.57963	0.789816	−	0.613344i	$-0.210176\pi$
$811$	44.9848	1.57963	0.789816	+	0.613344i	$0.210176\pi$
$812$	10.4384	0.366318
$813$	9.56155	0.335338
$814$	−28.4924	−0.998659
$815$	10.0540	0.352175
$816$	−2.00000	−0.0700140
$817$	−9.56155	−0.334516
$818$	−13.1231	−0.458839
$819$	−6.24621	−0.218260
$820$	8.00000	0.279372
$821$	−34.5464	−1.20568	−0.602839	−	0.797863i	$-0.705964\pi$
$821$	−34.5464	−1.20568	−0.602839	+	0.797863i	$0.705964\pi$
$822$	−0.438447	−0.0152926
$823$	−24.9848	−0.870917	−0.435458	−	0.900209i	$-0.643414\pi$
$823$	−24.9848	−0.870917	−0.435458	+	0.900209i	$0.643414\pi$
$824$	5.31534	0.185169
$825$	10.2462	0.356727
$826$	−12.1922	−0.424222
$827$	34.2462	1.19086	0.595429	−	0.803408i	$-0.296982\pi$
$827$	34.2462	1.19086	0.595429	+	0.803408i	$0.296982\pi$
$828$	−0.438447	−0.0152371
$829$	−12.6847	−0.440556	−0.220278	−	0.975437i	$-0.570697\pi$
$829$	−12.6847	−0.440556	−0.220278	+	0.975437i	$0.570697\pi$
$830$	−6.24621	−0.216809
$831$	13.6155	0.472317
$832$	−4.00000	−0.138675
$833$	9.12311	0.316097
$834$	−21.3693	−0.739959
$835$	1.36932	0.0473872
$836$	−4.00000	−0.138343
$837$	−0.438447	−0.0151550
$838$	−6.24621	−0.215772
$839$	−5.06913	−0.175006	−0.0875029	−	0.996164i	$-0.527889\pi$
$839$	−5.06913	−0.175006	−0.0875029	+	0.996164i	$0.527889\pi$
$840$	−2.43845	−0.0841344
$841$	15.6847	0.540850
$842$	1.06913	0.0368447
$843$	28.9309	0.996432
$844$	11.1231	0.382873
$845$	−4.68466	−0.161157
$846$	8.24621	0.283511
$847$	7.80776	0.268278
$848$	1.00000	0.0343401
$849$	−22.7386	−0.780388
$850$	5.12311	0.175721
$851$	−3.12311	−0.107059
$852$	−6.24621	−0.213992
$853$	−5.50758	−0.188576	−0.0942879	−	0.995545i	$-0.530057\pi$
$853$	−5.50758	−0.188576	−0.0942879	+	0.995545i	$0.530057\pi$
$854$	−20.4924	−0.701236
$855$	−1.56155	−0.0534040
$856$	−4.68466	−0.160118
$857$	−41.4233	−1.41499	−0.707496	−	0.706717i	$-0.750175\pi$
$857$	−41.4233	−1.41499	−0.707496	+	0.706717i	$0.750175\pi$
$858$	16.0000	0.546231
$859$	36.9848	1.26191	0.630953	−	0.775821i	$-0.282664\pi$
$859$	36.9848	1.26191	0.630953	+	0.775821i	$0.282664\pi$
$860$	14.9309	0.509138
$861$	−8.00000	−0.272639
$862$	−11.3153	−0.385402
$863$	50.9309	1.73371	0.866853	−	0.498563i	$-0.166139\pi$
$863$	50.9309	1.73371	0.866853	+	0.498563i	$0.166139\pi$
$864$	1.00000	0.0340207
$865$	−5.26137	−0.178892
$866$	16.7386	0.568802
$867$	−13.0000	−0.441503
$868$	−0.684658	−0.0232388
$869$	52.4924	1.78068
$870$	−10.4384	−0.353897
$871$	2.73863	0.0927951
$872$	−3.31534	−0.112272
$873$	−8.24621	−0.279092
$874$	−0.438447	−0.0148307
$875$	18.4384	0.623333
$876$	−10.0000	−0.337869
$877$	4.38447	0.148053	0.0740265	−	0.997256i	$-0.476415\pi$
$877$	4.38447	0.148053	0.0740265	+	0.997256i	$0.476415\pi$
$878$	−3.75379	−0.126684
$879$	−7.75379	−0.261529
$880$	6.24621	0.210560
$881$	16.0540	0.540872	0.270436	−	0.962738i	$-0.412832\pi$
$881$	16.0540	0.540872	0.270436	+	0.962738i	$0.412832\pi$
$882$	−4.56155	−0.153595
$883$	−44.9848	−1.51386	−0.756930	−	0.653496i	$-0.773302\pi$
$883$	−44.9848	−1.51386	−0.756930	+	0.653496i	$0.773302\pi$
$884$	8.00000	0.269069
$885$	12.1922	0.409838
$886$	21.5616	0.724375
$887$	−43.6155	−1.46447	−0.732233	−	0.681054i	$-0.761522\pi$
$887$	−43.6155	−1.46447	−0.732233	+	0.681054i	$0.761522\pi$
$888$	7.12311	0.239036
$889$	−15.3153	−0.513660
$890$	13.1771	0.441697
$891$	−4.00000	−0.134005
$892$	12.2462	0.410033
$893$	8.24621	0.275949
$894$	−18.2462	−0.610245
$895$	−13.8617	−0.463347
$896$	1.56155	0.0521678
$897$	1.75379	0.0585573
$898$	−7.75379	−0.258747
$899$	−2.93087	−0.0977500
$900$	−2.56155	−0.0853851
$901$	−2.00000	−0.0666297
$902$	20.4924	0.682323
$903$	−14.9309	−0.496868
$904$	0.438447	0.0145825
$905$	−10.0540	−0.334205
$906$	−5.80776	−0.192950
$907$	28.9848	0.962426	0.481213	−	0.876604i	$-0.340196\pi$
$907$	28.9848	0.962426	0.481213	+	0.876604i	$0.340196\pi$
$908$	15.8078	0.524599
$909$	4.68466	0.155380
$910$	9.75379	0.323335
$911$	−44.3002	−1.46773	−0.733865	−	0.679295i	$-0.762286\pi$
$911$	−44.3002	−1.46773	−0.733865	+	0.679295i	$0.762286\pi$
$912$	1.00000	0.0331133
$913$	−16.0000	−0.529523
$914$	−1.12311	−0.0371490
$915$	20.4924	0.677459
$916$	−16.9309	−0.559412
$917$	19.5076	0.644197
$918$	−2.00000	−0.0660098
$919$	−28.4924	−0.939878	−0.469939	−	0.882699i	$-0.655724\pi$
$919$	−28.4924	−0.939878	−0.469939	+	0.882699i	$0.655724\pi$
$920$	0.684658	0.0225725
$921$	−10.2462	−0.337624
$922$	2.63068	0.0866369
$923$	24.9848	0.822386
$924$	−6.24621	−0.205485
$925$	−18.2462	−0.599932
$926$	8.49242	0.279078
$927$	5.31534	0.174579
$928$	6.68466	0.219435
$929$	40.7386	1.33659	0.668296	−	0.743896i	$-0.267024\pi$
$929$	40.7386	1.33659	0.668296	+	0.743896i	$0.267024\pi$
$930$	0.684658	0.0224508
$931$	−4.56155	−0.149499
$932$	−0.246211	−0.00806492
$933$	−1.12311	−0.0367688
$934$	4.87689	0.159577
$935$	−12.4924	−0.408546
$936$	−4.00000	−0.130744
$937$	27.1771	0.887837	0.443918	−	0.896067i	$-0.353588\pi$
$937$	27.1771	0.887837	0.443918	+	0.896067i	$0.353588\pi$
$938$	−1.06913	−0.0349083
$939$	9.12311	0.297721
$940$	−12.8769	−0.419998
$941$	40.9309	1.33431	0.667154	−	0.744920i	$-0.267512\pi$
$941$	40.9309	1.33431	0.667154	+	0.744920i	$0.267512\pi$
$942$	18.0000	0.586472
$943$	2.24621	0.0731467
$944$	−7.80776	−0.254121
$945$	−2.43845	−0.0793227
$946$	38.2462	1.24349
$947$	−3.12311	−0.101487	−0.0507436	−	0.998712i	$-0.516159\pi$
$947$	−3.12311	−0.101487	−0.0507436	+	0.998712i	$0.516159\pi$
$948$	−13.1231	−0.426219
$949$	40.0000	1.29845
$950$	−2.56155	−0.0831077
$951$	14.4924	0.469949
$952$	−3.12311	−0.101220
$953$	−21.4233	−0.693969	−0.346984	−	0.937871i	$-0.612794\pi$
$953$	−21.4233	−0.693969	−0.346984	+	0.937871i	$0.612794\pi$
$954$	1.00000	0.0323762
$955$	−30.2462	−0.978744
$956$	15.3693	0.497079
$957$	−26.7386	−0.864337
$958$	−16.0540	−0.518680
$959$	−0.684658	−0.0221088
$960$	−1.56155	−0.0503989
$961$	−30.8078	−0.993799
$962$	−28.4924	−0.918633
$963$	−4.68466	−0.150961
$964$	−0.246211	−0.00792993
$965$	−37.8617	−1.21881
$966$	−0.684658	−0.0220285
$967$	−1.26137	−0.0405628	−0.0202814	−	0.999794i	$-0.506456\pi$
$967$	−1.26137	−0.0405628	−0.0202814	+	0.999794i	$0.506456\pi$
$968$	5.00000	0.160706
$969$	−2.00000	−0.0642493
$970$	12.8769	0.413452
$971$	−7.80776	−0.250563	−0.125282	−	0.992121i	$-0.539983\pi$
$971$	−7.80776	−0.250563	−0.125282	+	0.992121i	$0.539983\pi$
$972$	1.00000	0.0320750
$973$	−33.3693	−1.06977
$974$	−34.4924	−1.10521
$975$	10.2462	0.328141
$976$	−13.1231	−0.420060
$977$	−27.7538	−0.887922	−0.443961	−	0.896046i	$-0.646427\pi$
$977$	−27.7538	−0.887922	−0.443961	+	0.896046i	$0.646427\pi$
$978$	−6.43845	−0.205879
$979$	33.7538	1.07878
$980$	7.12311	0.227539
$981$	−3.31534	−0.105851
$982$	4.68466	0.149493
$983$	−5.17708	−0.165123	−0.0825616	−	0.996586i	$-0.526310\pi$
$983$	−5.17708	−0.165123	−0.0825616	+	0.996586i	$0.526310\pi$
$984$	−5.12311	−0.163319
$985$	−3.50758	−0.111761
$986$	−13.3693	−0.425766
$987$	12.8769	0.409876
$988$	−4.00000	−0.127257
$989$	4.19224	0.133305
$990$	6.24621	0.198518
$991$	−42.9848	−1.36546	−0.682729	−	0.730671i	$-0.739207\pi$
$991$	−42.9848	−1.36546	−0.682729	+	0.730671i	$0.739207\pi$
$992$	−0.438447	−0.0139207
$993$	11.6155	0.368608
$994$	−9.75379	−0.309371
$995$	27.4233	0.869377
$996$	4.00000	0.126745
$997$	31.5616	0.999564	0.499782	−	0.866151i	$-0.333413\pi$
$997$	31.5616	0.999564	0.499782	+	0.866151i	$0.333413\pi$
$998$	22.7386	0.719779
$999$	7.12311	0.225365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.p.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.p.1.1	✓	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} +1.00000 q^{6} +1.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} +1.00000 q^{6} +1.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{10} -4.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.56155 q^{14} -1.56155 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -1.56155 q^{20} +1.56155 q^{21} -4.00000 q^{22} -0.438447 q^{23} +1.00000 q^{24} -2.56155 q^{25} -4.00000 q^{26} +1.00000 q^{27} +1.56155 q^{28} +6.68466 q^{29} -1.56155 q^{30} -0.438447 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -2.43845 q^{35} +1.00000 q^{36} +7.12311 q^{37} +1.00000 q^{38} -4.00000 q^{39} -1.56155 q^{40} -5.12311 q^{41} +1.56155 q^{42} -9.56155 q^{43} -4.00000 q^{44} -1.56155 q^{45} -0.438447 q^{46} +8.24621 q^{47} +1.00000 q^{48} -4.56155 q^{49} -2.56155 q^{50} -2.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +6.24621 q^{55} +1.56155 q^{56} +1.00000 q^{57} +6.68466 q^{58} -7.80776 q^{59} -1.56155 q^{60} -13.1231 q^{61} -0.438447 q^{62} +1.56155 q^{63} +1.00000 q^{64} +6.24621 q^{65} -4.00000 q^{66} -0.684658 q^{67} -2.00000 q^{68} -0.438447 q^{69} -2.43845 q^{70} -6.24621 q^{71} +1.00000 q^{72} -10.0000 q^{73} +7.12311 q^{74} -2.56155 q^{75} +1.00000 q^{76} -6.24621 q^{77} -4.00000 q^{78} -13.1231 q^{79} -1.56155 q^{80} +1.00000 q^{81} -5.12311 q^{82} +4.00000 q^{83} +1.56155 q^{84} +3.12311 q^{85} -9.56155 q^{86} +6.68466 q^{87} -4.00000 q^{88} -8.43845 q^{89} -1.56155 q^{90} -6.24621 q^{91} -0.438447 q^{92} -0.438447 q^{93} +8.24621 q^{94} -1.56155 q^{95} +1.00000 q^{96} -8.24621 q^{97} -4.56155 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} - 8 q^{13} - q^{14} + q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} + q^{20} - q^{21} - 8 q^{22} - 5 q^{23} + 2 q^{24} - q^{25} - 8 q^{26} + 2 q^{27} - q^{28} + q^{29} + q^{30} - 5 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 9 q^{35} + 2 q^{36} + 6 q^{37} + 2 q^{38} - 8 q^{39} + q^{40} - 2 q^{41} - q^{42} - 15 q^{43} - 8 q^{44} + q^{45} - 5 q^{46} + 2 q^{48} - 5 q^{49} - q^{50} - 4 q^{51} - 8 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} - q^{56} + 2 q^{57} + q^{58} + 5 q^{59} + q^{60} - 18 q^{61} - 5 q^{62} - q^{63} + 2 q^{64} - 4 q^{65} - 8 q^{66} + 11 q^{67} - 4 q^{68} - 5 q^{69} - 9 q^{70} + 4 q^{71} + 2 q^{72} - 20 q^{73} + 6 q^{74} - q^{75} + 2 q^{76} + 4 q^{77} - 8 q^{78} - 18 q^{79} + q^{80} + 2 q^{81} - 2 q^{82} + 8 q^{83} - q^{84} - 2 q^{85} - 15 q^{86} + q^{87} - 8 q^{88} - 21 q^{89} + q^{90} + 4 q^{91} - 5 q^{92} - 5 q^{93} + q^{95} + 2 q^{96} - 5 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 + q^10 - 8 * q^11 + 2 * q^12 - 8 * q^13 - q^14 + q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 + q^20 - q^21 - 8 * q^22 - 5 * q^23 + 2 * q^24 - q^25 - 8 * q^26 + 2 * q^27 - q^28 + q^29 + q^30 - 5 * q^31 + 2 * q^32 - 8 * q^33 - 4 * q^34 - 9 * q^35 + 2 * q^36 + 6 * q^37 + 2 * q^38 - 8 * q^39 + q^40 - 2 * q^41 - q^42 - 15 * q^43 - 8 * q^44 + q^45 - 5 * q^46 + 2 * q^48 - 5 * q^49 - q^50 - 4 * q^51 - 8 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 - q^56 + 2 * q^57 + q^58 + 5 * q^59 + q^60 - 18 * q^61 - 5 * q^62 - q^63 + 2 * q^64 - 4 * q^65 - 8 * q^66 + 11 * q^67 - 4 * q^68 - 5 * q^69 - 9 * q^70 + 4 * q^71 + 2 * q^72 - 20 * q^73 + 6 * q^74 - q^75 + 2 * q^76 + 4 * q^77 - 8 * q^78 - 18 * q^79 + q^80 + 2 * q^81 - 2 * q^82 + 8 * q^83 - q^84 - 2 * q^85 - 15 * q^86 + q^87 - 8 * q^88 - 21 * q^89 + q^90 + 4 * q^91 - 5 * q^92 - 5 * q^93 + q^95 + 2 * q^96 - 5 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more