LMFDB - Embedded newform 6042.2.a.bb.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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 16x^{7} + 76x^{6} + 30x^{5} - 366x^{4} + 300x^{3} + 101x^{2} - 106x + 16 $ x^9 - 4x^8 - 16x^7 + 76x^6 + 30x^5 - 366x^4 + 300x^3 + 101x^2 - 106x + 16
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			$-2.88457$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.64105 q^{5} -1.00000 q^{6} -3.88457 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.64105 q^{5} -1.00000 q^{6} -3.88457 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.64105 q^{10} +0.455198 q^{11} -1.00000 q^{12} +6.27753 q^{13} -3.88457 q^{14} -1.64105 q^{15} +1.00000 q^{16} -7.54529 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.64105 q^{20} +3.88457 q^{21} +0.455198 q^{22} +7.87030 q^{23} -1.00000 q^{24} -2.30696 q^{25} +6.27753 q^{26} -1.00000 q^{27} -3.88457 q^{28} -8.82914 q^{29} -1.64105 q^{30} -9.74498 q^{31} +1.00000 q^{32} -0.455198 q^{33} -7.54529 q^{34} -6.37477 q^{35} +1.00000 q^{36} +1.01478 q^{37} -1.00000 q^{38} -6.27753 q^{39} +1.64105 q^{40} -10.0728 q^{41} +3.88457 q^{42} +8.44779 q^{43} +0.455198 q^{44} +1.64105 q^{45} +7.87030 q^{46} -5.98098 q^{47} -1.00000 q^{48} +8.08987 q^{49} -2.30696 q^{50} +7.54529 q^{51} +6.27753 q^{52} +1.00000 q^{53} -1.00000 q^{54} +0.747002 q^{55} -3.88457 q^{56} +1.00000 q^{57} -8.82914 q^{58} +6.68275 q^{59} -1.64105 q^{60} -7.04256 q^{61} -9.74498 q^{62} -3.88457 q^{63} +1.00000 q^{64} +10.3017 q^{65} -0.455198 q^{66} +4.81283 q^{67} -7.54529 q^{68} -7.87030 q^{69} -6.37477 q^{70} +6.55760 q^{71} +1.00000 q^{72} -6.32429 q^{73} +1.01478 q^{74} +2.30696 q^{75} -1.00000 q^{76} -1.76825 q^{77} -6.27753 q^{78} +6.24584 q^{79} +1.64105 q^{80} +1.00000 q^{81} -10.0728 q^{82} -6.59700 q^{83} +3.88457 q^{84} -12.3822 q^{85} +8.44779 q^{86} +8.82914 q^{87} +0.455198 q^{88} -12.0090 q^{89} +1.64105 q^{90} -24.3855 q^{91} +7.87030 q^{92} +9.74498 q^{93} -5.98098 q^{94} -1.64105 q^{95} -1.00000 q^{96} -0.429380 q^{97} +8.08987 q^{98} +0.455198 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20} + 5 q^{21} + 3 q^{22} - 9 q^{24} + 8 q^{25} - 4 q^{26} - 9 q^{27} - 5 q^{28} - 2 q^{29} + 5 q^{30} - 10 q^{31} + 9 q^{32} - 3 q^{33} - 16 q^{34} - q^{35} + 9 q^{36} - 14 q^{37} - 9 q^{38} + 4 q^{39} - 5 q^{40} - 21 q^{41} + 5 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} - 8 q^{47} - 9 q^{48} - 14 q^{49} + 8 q^{50} + 16 q^{51} - 4 q^{52} + 9 q^{53} - 9 q^{54} - 14 q^{55} - 5 q^{56} + 9 q^{57} - 2 q^{58} - 12 q^{59} + 5 q^{60} - 13 q^{61} - 10 q^{62} - 5 q^{63} + 9 q^{64} - 13 q^{65} - 3 q^{66} + 22 q^{67} - 16 q^{68} - q^{70} - 13 q^{71} + 9 q^{72} - 17 q^{73} - 14 q^{74} - 8 q^{75} - 9 q^{76} - 25 q^{77} + 4 q^{78} - 5 q^{80} + 9 q^{81} - 21 q^{82} - 24 q^{83} + 5 q^{84} - 16 q^{85} - 4 q^{86} + 2 q^{87} + 3 q^{88} - 23 q^{89} - 5 q^{90} - 11 q^{91} + 10 q^{93} - 8 q^{94} + 5 q^{95} - 9 q^{96} - 29 q^{97} - 14 q^{98} + 3 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9 - 5 * q^10 + 3 * q^11 - 9 * q^12 - 4 * q^13 - 5 * q^14 + 5 * q^15 + 9 * q^16 - 16 * q^17 + 9 * q^18 - 9 * q^19 - 5 * q^20 + 5 * q^21 + 3 * q^22 - 9 * q^24 + 8 * q^25 - 4 * q^26 - 9 * q^27 - 5 * q^28 - 2 * q^29 + 5 * q^30 - 10 * q^31 + 9 * q^32 - 3 * q^33 - 16 * q^34 - q^35 + 9 * q^36 - 14 * q^37 - 9 * q^38 + 4 * q^39 - 5 * q^40 - 21 * q^41 + 5 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 - 8 * q^47 - 9 * q^48 - 14 * q^49 + 8 * q^50 + 16 * q^51 - 4 * q^52 + 9 * q^53 - 9 * q^54 - 14 * q^55 - 5 * q^56 + 9 * q^57 - 2 * q^58 - 12 * q^59 + 5 * q^60 - 13 * q^61 - 10 * q^62 - 5 * q^63 + 9 * q^64 - 13 * q^65 - 3 * q^66 + 22 * q^67 - 16 * q^68 - q^70 - 13 * q^71 + 9 * q^72 - 17 * q^73 - 14 * q^74 - 8 * q^75 - 9 * q^76 - 25 * q^77 + 4 * q^78 - 5 * q^80 + 9 * q^81 - 21 * q^82 - 24 * q^83 + 5 * q^84 - 16 * q^85 - 4 * q^86 + 2 * q^87 + 3 * q^88 - 23 * q^89 - 5 * q^90 - 11 * q^91 + 10 * q^93 - 8 * q^94 + 5 * q^95 - 9 * q^96 - 29 * q^97 - 14 * q^98 + 3 * 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
$5$	1.64105	0.733900	0.366950	−	0.930241i	$-0.380402\pi$
$5$	1.64105	0.733900	0.366950	+	0.930241i	$0.380402\pi$
$6$	−1.00000	−0.408248
$7$	−3.88457	−1.46823	−0.734114	−	0.679026i	$-0.762402\pi$
$7$	−3.88457	−1.46823	−0.734114	+	0.679026i	$0.762402\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.64105	0.518945
$11$	0.455198	0.137247	0.0686236	−	0.997643i	$-0.478139\pi$
$11$	0.455198	0.137247	0.0686236	+	0.997643i	$0.478139\pi$
$12$	−1.00000	−0.288675
$13$	6.27753	1.74107	0.870537	−	0.492103i	$-0.163772\pi$
$13$	6.27753	1.74107	0.870537	+	0.492103i	$0.163772\pi$
$14$	−3.88457	−1.03819
$15$	−1.64105	−0.423717
$16$	1.00000	0.250000
$17$	−7.54529	−1.83000	−0.915001	−	0.403451i	$-0.867810\pi$
$17$	−7.54529	−1.83000	−0.915001	+	0.403451i	$0.867810\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.64105	0.366950
$21$	3.88457	0.847682
$22$	0.455198	0.0970484
$23$	7.87030	1.64107	0.820535	−	0.571596i	$-0.193676\pi$
$23$	7.87030	1.64107	0.820535	+	0.571596i	$0.193676\pi$
$24$	−1.00000	−0.204124
$25$	−2.30696	−0.461392
$26$	6.27753	1.23113
$27$	−1.00000	−0.192450
$28$	−3.88457	−0.734114
$29$	−8.82914	−1.63953	−0.819765	−	0.572700i	$-0.805896\pi$
$29$	−8.82914	−1.63953	−0.819765	+	0.572700i	$0.805896\pi$
$30$	−1.64105	−0.299613
$31$	−9.74498	−1.75025	−0.875125	−	0.483897i	$-0.839221\pi$
$31$	−9.74498	−1.75025	−0.875125	+	0.483897i	$0.839221\pi$
$32$	1.00000	0.176777
$33$	−0.455198	−0.0792397
$34$	−7.54529	−1.29401
$35$	−6.37477	−1.07753
$36$	1.00000	0.166667
$37$	1.01478	0.166828	0.0834141	−	0.996515i	$-0.473418\pi$
$37$	1.01478	0.166828	0.0834141	+	0.996515i	$0.473418\pi$
$38$	−1.00000	−0.162221
$39$	−6.27753	−1.00521
$40$	1.64105	0.259473
$41$	−10.0728	−1.57311	−0.786554	−	0.617521i	$-0.788137\pi$
$41$	−10.0728	−1.57311	−0.786554	+	0.617521i	$0.788137\pi$
$42$	3.88457	0.599402
$43$	8.44779	1.28828	0.644138	−	0.764909i	$-0.277216\pi$
$43$	8.44779	1.28828	0.644138	+	0.764909i	$0.277216\pi$
$44$	0.455198	0.0686236
$45$	1.64105	0.244633
$46$	7.87030	1.16041
$47$	−5.98098	−0.872416	−0.436208	−	0.899846i	$-0.643679\pi$
$47$	−5.98098	−0.872416	−0.436208	+	0.899846i	$0.643679\pi$
$48$	−1.00000	−0.144338
$49$	8.08987	1.15570
$50$	−2.30696	−0.326253
$51$	7.54529	1.05655
$52$	6.27753	0.870537
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	0.747002	0.100726
$56$	−3.88457	−0.519097
$57$	1.00000	0.132453
$58$	−8.82914	−1.15932
$59$	6.68275	0.870020	0.435010	−	0.900426i	$-0.356745\pi$
$59$	6.68275	0.870020	0.435010	+	0.900426i	$0.356745\pi$
$60$	−1.64105	−0.211859
$61$	−7.04256	−0.901708	−0.450854	−	0.892598i	$-0.648880\pi$
$61$	−7.04256	−0.901708	−0.450854	+	0.892598i	$0.648880\pi$
$62$	−9.74498	−1.23761
$63$	−3.88457	−0.489410
$64$	1.00000	0.125000
$65$	10.3017	1.27777
$66$	−0.455198	−0.0560309
$67$	4.81283	0.587981	0.293991	−	0.955808i	$-0.405017\pi$
$67$	4.81283	0.587981	0.293991	+	0.955808i	$0.405017\pi$
$68$	−7.54529	−0.915001
$69$	−7.87030	−0.947472
$70$	−6.37477	−0.761930
$71$	6.55760	0.778244	0.389122	−	0.921186i	$-0.372778\pi$
$71$	6.55760	0.778244	0.389122	+	0.921186i	$0.372778\pi$
$72$	1.00000	0.117851
$73$	−6.32429	−0.740202	−0.370101	−	0.928992i	$-0.620677\pi$
$73$	−6.32429	−0.740202	−0.370101	+	0.928992i	$0.620677\pi$
$74$	1.01478	0.117965
$75$	2.30696	0.266385
$76$	−1.00000	−0.114708
$77$	−1.76825	−0.201510
$78$	−6.27753	−0.710791
$79$	6.24584	0.702712	0.351356	−	0.936242i	$-0.385721\pi$
$79$	6.24584	0.702712	0.351356	+	0.936242i	$0.385721\pi$
$80$	1.64105	0.183475
$81$	1.00000	0.111111
$82$	−10.0728	−1.11236
$83$	−6.59700	−0.724115	−0.362057	−	0.932156i	$-0.617926\pi$
$83$	−6.59700	−0.724115	−0.362057	+	0.932156i	$0.617926\pi$
$84$	3.88457	0.423841
$85$	−12.3822	−1.34304
$86$	8.44779	0.910949
$87$	8.82914	0.946583
$88$	0.455198	0.0485242
$89$	−12.0090	−1.27295	−0.636474	−	0.771299i	$-0.719608\pi$
$89$	−12.0090	−1.27295	−0.636474	+	0.771299i	$0.719608\pi$
$90$	1.64105	0.172982
$91$	−24.3855	−2.55630
$92$	7.87030	0.820535
$93$	9.74498	1.01051
$94$	−5.98098	−0.616891
$95$	−1.64105	−0.168368
$96$	−1.00000	−0.102062
$97$	−0.429380	−0.0435969	−0.0217985	−	0.999762i	$-0.506939\pi$
$97$	−0.429380	−0.0435969	−0.0217985	+	0.999762i	$0.506939\pi$
$98$	8.08987	0.817200
$99$	0.455198	0.0457491
$100$	−2.30696	−0.230696
$101$	2.17007	0.215930	0.107965	−	0.994155i	$-0.465567\pi$
$101$	2.17007	0.215930	0.107965	+	0.994155i	$0.465567\pi$
$102$	7.54529	0.747095
$103$	15.0801	1.48589	0.742944	−	0.669354i	$-0.233429\pi$
$103$	15.0801	1.48589	0.742944	+	0.669354i	$0.233429\pi$
$104$	6.27753	0.615563
$105$	6.37477	0.622114
$106$	1.00000	0.0971286
$107$	2.87972	0.278393	0.139196	−	0.990265i	$-0.455548\pi$
$107$	2.87972	0.278393	0.139196	+	0.990265i	$0.455548\pi$
$108$	−1.00000	−0.0962250
$109$	−7.21565	−0.691133	−0.345567	−	0.938394i	$-0.612313\pi$
$109$	−7.21565	−0.691133	−0.345567	+	0.938394i	$0.612313\pi$
$110$	0.747002	0.0712238
$111$	−1.01478	−0.0963183
$112$	−3.88457	−0.367057
$113$	6.67886	0.628294	0.314147	−	0.949374i	$-0.398281\pi$
$113$	6.67886	0.628294	0.314147	+	0.949374i	$0.398281\pi$
$114$	1.00000	0.0936586
$115$	12.9155	1.20438
$116$	−8.82914	−0.819765
$117$	6.27753	0.580358
$118$	6.68275	0.615197
$119$	29.3102	2.68686
$120$	−1.64105	−0.149807
$121$	−10.7928	−0.981163
$122$	−7.04256	−0.637604
$123$	10.0728	0.908235
$124$	−9.74498	−0.875125
$125$	−11.9911	−1.07251
$126$	−3.88457	−0.346065
$127$	2.62300	0.232754	0.116377	−	0.993205i	$-0.462872\pi$
$127$	2.62300	0.232754	0.116377	+	0.993205i	$0.462872\pi$
$128$	1.00000	0.0883883
$129$	−8.44779	−0.743787
$130$	10.3017	0.903522
$131$	−7.45205	−0.651088	−0.325544	−	0.945527i	$-0.605547\pi$
$131$	−7.45205	−0.651088	−0.325544	+	0.945527i	$0.605547\pi$
$132$	−0.455198	−0.0396199
$133$	3.88457	0.336835
$134$	4.81283	0.415766
$135$	−1.64105	−0.141239
$136$	−7.54529	−0.647003
$137$	1.99064	0.170072	0.0850360	−	0.996378i	$-0.472899\pi$
$137$	1.99064	0.170072	0.0850360	+	0.996378i	$0.472899\pi$
$138$	−7.87030	−0.669964
$139$	9.62307	0.816218	0.408109	−	0.912933i	$-0.366188\pi$
$139$	9.62307	0.816218	0.408109	+	0.912933i	$0.366188\pi$
$140$	−6.37477	−0.538766
$141$	5.98098	0.503689
$142$	6.55760	0.550302
$143$	2.85752	0.238958
$144$	1.00000	0.0833333
$145$	−14.4891	−1.20325
$146$	−6.32429	−0.523402
$147$	−8.08987	−0.667241
$148$	1.01478	0.0834141
$149$	−9.77775	−0.801024	−0.400512	−	0.916291i	$-0.631168\pi$
$149$	−9.77775	−0.801024	−0.400512	+	0.916291i	$0.631168\pi$
$150$	2.30696	0.188362
$151$	−14.6849	−1.19504	−0.597520	−	0.801854i	$-0.703847\pi$
$151$	−14.6849	−1.19504	−0.597520	+	0.801854i	$0.703847\pi$
$152$	−1.00000	−0.0811107
$153$	−7.54529	−0.610001
$154$	−1.76825	−0.142489
$155$	−15.9920	−1.28451
$156$	−6.27753	−0.502605
$157$	−19.5948	−1.56383	−0.781917	−	0.623383i	$-0.785758\pi$
$157$	−19.5948	−1.56383	−0.781917	+	0.623383i	$0.785758\pi$
$158$	6.24584	0.496893
$159$	−1.00000	−0.0793052
$160$	1.64105	0.129736
$161$	−30.5727	−2.40947
$162$	1.00000	0.0785674
$163$	−17.9262	−1.40409	−0.702045	−	0.712133i	$-0.747729\pi$
$163$	−17.9262	−1.40409	−0.702045	+	0.712133i	$0.747729\pi$
$164$	−10.0728	−0.786554
$165$	−0.747002	−0.0581540
$166$	−6.59700	−0.512027
$167$	−16.5612	−1.28154	−0.640771	−	0.767732i	$-0.721385\pi$
$167$	−16.5612	−1.28154	−0.640771	+	0.767732i	$0.721385\pi$
$168$	3.88457	0.299701
$169$	26.4074	2.03134
$170$	−12.3822	−0.949671
$171$	−1.00000	−0.0764719
$172$	8.44779	0.644138
$173$	−12.3055	−0.935572	−0.467786	−	0.883842i	$-0.654948\pi$
$173$	−12.3055	−0.935572	−0.467786	+	0.883842i	$0.654948\pi$
$174$	8.82914	0.669336
$175$	8.96153	0.677428
$176$	0.455198	0.0343118
$177$	−6.68275	−0.502306
$178$	−12.0090	−0.900110
$179$	−5.25246	−0.392588	−0.196294	−	0.980545i	$-0.562891\pi$
$179$	−5.25246	−0.392588	−0.196294	+	0.980545i	$0.562891\pi$
$180$	1.64105	0.122317
$181$	8.38098	0.622953	0.311477	−	0.950254i	$-0.399176\pi$
$181$	8.38098	0.622953	0.311477	+	0.950254i	$0.399176\pi$
$182$	−24.3855	−1.80757
$183$	7.04256	0.520601
$184$	7.87030	0.580206
$185$	1.66530	0.122435
$186$	9.74498	0.714537
$187$	−3.43460	−0.251163
$188$	−5.98098	−0.436208
$189$	3.88457	0.282561
$190$	−1.64105	−0.119054
$191$	−6.20812	−0.449204	−0.224602	−	0.974451i	$-0.572108\pi$
$191$	−6.20812	−0.449204	−0.224602	+	0.974451i	$0.572108\pi$
$192$	−1.00000	−0.0721688
$193$	−21.4495	−1.54397	−0.771985	−	0.635641i	$-0.780736\pi$
$193$	−21.4495	−1.54397	−0.771985	+	0.635641i	$0.780736\pi$
$194$	−0.429380	−0.0308277
$195$	−10.3017	−0.737723
$196$	8.08987	0.577848
$197$	10.0505	0.716067	0.358034	−	0.933709i	$-0.383447\pi$
$197$	10.0505	0.716067	0.358034	+	0.933709i	$0.383447\pi$
$198$	0.455198	0.0323495
$199$	−0.241147	−0.0170945	−0.00854723	−	0.999963i	$-0.502721\pi$
$199$	−0.241147	−0.0170945	−0.00854723	+	0.999963i	$0.502721\pi$
$200$	−2.30696	−0.163127
$201$	−4.81283	−0.339471
$202$	2.17007	0.152686
$203$	34.2974	2.40721
$204$	7.54529	0.528276
$205$	−16.5300	−1.15450
$206$	15.0801	1.05068
$207$	7.87030	0.547023
$208$	6.27753	0.435269
$209$	−0.455198	−0.0314867
$210$	6.37477	0.439901
$211$	−28.9130	−1.99045	−0.995227	−	0.0975857i	$-0.968888\pi$
$211$	−28.9130	−1.99045	−0.995227	+	0.0975857i	$0.968888\pi$
$212$	1.00000	0.0686803
$213$	−6.55760	−0.449320
$214$	2.87972	0.196853
$215$	13.8632	0.945465
$216$	−1.00000	−0.0680414
$217$	37.8550	2.56977
$218$	−7.21565	−0.488705
$219$	6.32429	0.427356
$220$	0.747002	0.0503628
$221$	−47.3658	−3.18617
$222$	−1.01478	−0.0681073
$223$	−19.5613	−1.30992	−0.654960	−	0.755663i	$-0.727315\pi$
$223$	−19.5613	−1.30992	−0.654960	+	0.755663i	$0.727315\pi$
$224$	−3.88457	−0.259549
$225$	−2.30696	−0.153797
$226$	6.67886	0.444271
$227$	−18.5754	−1.23289	−0.616447	−	0.787396i	$-0.711429\pi$
$227$	−18.5754	−1.23289	−0.616447	+	0.787396i	$0.711429\pi$
$228$	1.00000	0.0662266
$229$	−8.75285	−0.578405	−0.289202	−	0.957268i	$-0.593390\pi$
$229$	−8.75285	−0.578405	−0.289202	+	0.957268i	$0.593390\pi$
$230$	12.9155	0.851626
$231$	1.76825	0.116342
$232$	−8.82914	−0.579662
$233$	−20.1814	−1.32213	−0.661064	−	0.750329i	$-0.729895\pi$
$233$	−20.1814	−1.32213	−0.661064	+	0.750329i	$0.729895\pi$
$234$	6.27753	0.410375
$235$	−9.81508	−0.640265
$236$	6.68275	0.435010
$237$	−6.24584	−0.405711
$238$	29.3102	1.89990
$239$	20.7366	1.34134	0.670671	−	0.741755i	$-0.266006\pi$
$239$	20.7366	1.34134	0.670671	+	0.741755i	$0.266006\pi$
$240$	−1.64105	−0.105929
$241$	−4.48792	−0.289092	−0.144546	−	0.989498i	$-0.546172\pi$
$241$	−4.48792	−0.289092	−0.144546	+	0.989498i	$0.546172\pi$
$242$	−10.7928	−0.693787
$243$	−1.00000	−0.0641500
$244$	−7.04256	−0.450854
$245$	13.2759	0.848164
$246$	10.0728	0.642219
$247$	−6.27753	−0.399430
$248$	−9.74498	−0.618807
$249$	6.59700	0.418068
$250$	−11.9911	−0.758382
$251$	24.3302	1.53571	0.767854	−	0.640625i	$-0.221325\pi$
$251$	24.3302	1.53571	0.767854	+	0.640625i	$0.221325\pi$
$252$	−3.88457	−0.244705
$253$	3.58254	0.225232
$254$	2.62300	0.164582
$255$	12.3822	0.775403
$256$	1.00000	0.0625000
$257$	7.41352	0.462443	0.231221	−	0.972901i	$-0.425728\pi$
$257$	7.41352	0.462443	0.231221	+	0.972901i	$0.425728\pi$
$258$	−8.44779	−0.525937
$259$	−3.94197	−0.244942
$260$	10.3017	0.638887
$261$	−8.82914	−0.546510
$262$	−7.45205	−0.460389
$263$	13.0571	0.805137	0.402569	−	0.915390i	$-0.368117\pi$
$263$	13.0571	0.805137	0.402569	+	0.915390i	$0.368117\pi$
$264$	−0.455198	−0.0280155
$265$	1.64105	0.100809
$266$	3.88457	0.238178
$267$	12.0090	0.734936
$268$	4.81283	0.293991
$269$	−6.73563	−0.410678	−0.205339	−	0.978691i	$-0.565830\pi$
$269$	−6.73563	−0.410678	−0.205339	+	0.978691i	$0.565830\pi$
$270$	−1.64105	−0.0998711
$271$	−11.8951	−0.722577	−0.361288	−	0.932454i	$-0.617663\pi$
$271$	−11.8951	−0.722577	−0.361288	+	0.932454i	$0.617663\pi$
$272$	−7.54529	−0.457501
$273$	24.3855	1.47588
$274$	1.99064	0.120259
$275$	−1.05012	−0.0633247
$276$	−7.87030	−0.473736
$277$	15.1771	0.911905	0.455952	−	0.890004i	$-0.349299\pi$
$277$	15.1771	0.911905	0.455952	+	0.890004i	$0.349299\pi$
$278$	9.62307	0.577153
$279$	−9.74498	−0.583417
$280$	−6.37477	−0.380965
$281$	25.3816	1.51414	0.757071	−	0.653333i	$-0.226630\pi$
$281$	25.3816	1.51414	0.757071	+	0.653333i	$0.226630\pi$
$282$	5.98098	0.356162
$283$	−19.7617	−1.17471	−0.587355	−	0.809330i	$-0.699831\pi$
$283$	−19.7617	−1.17471	−0.587355	+	0.809330i	$0.699831\pi$
$284$	6.55760	0.389122
$285$	1.64105	0.0972074
$286$	2.85752	0.168969
$287$	39.1285	2.30968
$288$	1.00000	0.0589256
$289$	39.9314	2.34891
$290$	−14.4891	−0.850827
$291$	0.429380	0.0251707
$292$	−6.32429	−0.370101
$293$	−18.2532	−1.06636	−0.533181	−	0.846001i	$-0.679004\pi$
$293$	−18.2532	−1.06636	−0.533181	+	0.846001i	$0.679004\pi$
$294$	−8.08987	−0.471811
$295$	10.9667	0.638507
$296$	1.01478	0.0589827
$297$	−0.455198	−0.0264132
$298$	−9.77775	−0.566410
$299$	49.4060	2.85723
$300$	2.30696	0.133192
$301$	−32.8160	−1.89148
$302$	−14.6849	−0.845021
$303$	−2.17007	−0.124667
$304$	−1.00000	−0.0573539
$305$	−11.5572	−0.661763
$306$	−7.54529	−0.431336
$307$	−0.135600	−0.00773910	−0.00386955	−	0.999993i	$-0.501232\pi$
$307$	−0.135600	−0.00773910	−0.00386955	+	0.999993i	$0.501232\pi$
$308$	−1.76825	−0.100755
$309$	−15.0801	−0.857878
$310$	−15.9920	−0.908284
$311$	−12.9458	−0.734088	−0.367044	−	0.930204i	$-0.619630\pi$
$311$	−12.9458	−0.734088	−0.367044	+	0.930204i	$0.619630\pi$
$312$	−6.27753	−0.355395
$313$	12.5352	0.708532	0.354266	−	0.935145i	$-0.384731\pi$
$313$	12.5352	0.708532	0.354266	+	0.935145i	$0.384731\pi$
$314$	−19.5948	−1.10580
$315$	−6.37477	−0.359177
$316$	6.24584	0.351356
$317$	16.4744	0.925293	0.462647	−	0.886543i	$-0.346900\pi$
$317$	16.4744	0.925293	0.462647	+	0.886543i	$0.346900\pi$
$318$	−1.00000	−0.0560772
$319$	−4.01900	−0.225021
$320$	1.64105	0.0917374
$321$	−2.87972	−0.160730
$322$	−30.5727	−1.70375
$323$	7.54529	0.419831
$324$	1.00000	0.0555556
$325$	−14.4820	−0.803317
$326$	−17.9262	−0.992841
$327$	7.21565	0.399026
$328$	−10.0728	−0.556178
$329$	23.2335	1.28091
$330$	−0.747002	−0.0411211
$331$	33.5886	1.84620	0.923098	−	0.384564i	$-0.125648\pi$
$331$	33.5886	1.84620	0.923098	+	0.384564i	$0.125648\pi$
$332$	−6.59700	−0.362057
$333$	1.01478	0.0556094
$334$	−16.5612	−0.906188
$335$	7.89810	0.431519
$336$	3.88457	0.211921
$337$	−25.2827	−1.37724	−0.688618	−	0.725124i	$-0.741782\pi$
$337$	−25.2827	−1.37724	−0.688618	+	0.725124i	$0.741782\pi$
$338$	26.4074	1.43637
$339$	−6.67886	−0.362746
$340$	−12.3822	−0.671519
$341$	−4.43589	−0.240217
$342$	−1.00000	−0.0540738
$343$	−4.23367	−0.228597
$344$	8.44779	0.455474
$345$	−12.9155	−0.695350
$346$	−12.3055	−0.661549
$347$	−8.12594	−0.436224	−0.218112	−	0.975924i	$-0.569990\pi$
$347$	−8.12594	−0.436224	−0.218112	+	0.975924i	$0.569990\pi$
$348$	8.82914	0.473292
$349$	27.0402	1.44743	0.723714	−	0.690100i	$-0.242434\pi$
$349$	27.0402	1.44743	0.723714	+	0.690100i	$0.242434\pi$
$350$	8.96153	0.479014
$351$	−6.27753	−0.335070
$352$	0.455198	0.0242621
$353$	−11.6746	−0.621377	−0.310688	−	0.950512i	$-0.600560\pi$
$353$	−11.6746	−0.621377	−0.310688	+	0.950512i	$0.600560\pi$
$354$	−6.68275	−0.355184
$355$	10.7613	0.571153
$356$	−12.0090	−0.636474
$357$	−29.3102	−1.55126
$358$	−5.25246	−0.277601
$359$	22.4127	1.18290	0.591449	−	0.806343i	$-0.298556\pi$
$359$	22.4127	1.18290	0.591449	+	0.806343i	$0.298556\pi$
$360$	1.64105	0.0864909
$361$	1.00000	0.0526316
$362$	8.38098	0.440495
$363$	10.7928	0.566475
$364$	−24.3855	−1.27815
$365$	−10.3785	−0.543234
$366$	7.04256	0.368121
$367$	−1.25478	−0.0654988	−0.0327494	−	0.999464i	$-0.510426\pi$
$367$	−1.25478	−0.0654988	−0.0327494	+	0.999464i	$0.510426\pi$
$368$	7.87030	0.410268
$369$	−10.0728	−0.524370
$370$	1.66530	0.0865747
$371$	−3.88457	−0.201677
$372$	9.74498	0.505254
$373$	−12.6148	−0.653171	−0.326586	−	0.945168i	$-0.605898\pi$
$373$	−12.6148	−0.653171	−0.326586	+	0.945168i	$0.605898\pi$
$374$	−3.43460	−0.177599
$375$	11.9911	0.619217
$376$	−5.98098	−0.308445
$377$	−55.4252	−2.85454
$378$	3.88457	0.199801
$379$	−13.2521	−0.680712	−0.340356	−	0.940297i	$-0.610548\pi$
$379$	−13.2521	−0.680712	−0.340356	+	0.940297i	$0.610548\pi$
$380$	−1.64105	−0.0841840
$381$	−2.62300	−0.134380
$382$	−6.20812	−0.317635
$383$	25.2253	1.28895	0.644476	−	0.764625i	$-0.277076\pi$
$383$	25.2253	1.28895	0.644476	+	0.764625i	$0.277076\pi$
$384$	−1.00000	−0.0510310
$385$	−2.90178	−0.147888
$386$	−21.4495	−1.09175
$387$	8.44779	0.429425
$388$	−0.429380	−0.0217985
$389$	2.96573	0.150369	0.0751843	−	0.997170i	$-0.476045\pi$
$389$	2.96573	0.150369	0.0751843	+	0.997170i	$0.476045\pi$
$390$	−10.3017	−0.521649
$391$	−59.3837	−3.00316
$392$	8.08987	0.408600
$393$	7.45205	0.375906
$394$	10.0505	0.506336
$395$	10.2497	0.515720
$396$	0.455198	0.0228745
$397$	−0.210589	−0.0105692	−0.00528458	−	0.999986i	$-0.501682\pi$
$397$	−0.210589	−0.0105692	−0.00528458	+	0.999986i	$0.501682\pi$
$398$	−0.241147	−0.0120876
$399$	−3.88457	−0.194472
$400$	−2.30696	−0.115348
$401$	−19.6442	−0.980985	−0.490492	−	0.871445i	$-0.663183\pi$
$401$	−19.6442	−0.980985	−0.490492	+	0.871445i	$0.663183\pi$
$402$	−4.81283	−0.240042
$403$	−61.1744	−3.04732
$404$	2.17007	0.107965
$405$	1.64105	0.0815444
$406$	34.2974	1.70215
$407$	0.461924	0.0228967
$408$	7.54529	0.373548
$409$	−9.81228	−0.485186	−0.242593	−	0.970128i	$-0.577998\pi$
$409$	−9.81228	−0.485186	−0.242593	+	0.970128i	$0.577998\pi$
$410$	−16.5300	−0.816357
$411$	−1.99064	−0.0981911
$412$	15.0801	0.742944
$413$	−25.9596	−1.27739
$414$	7.87030	0.386804
$415$	−10.8260	−0.531428
$416$	6.27753	0.307781
$417$	−9.62307	−0.471244
$418$	−0.455198	−0.0222644
$419$	−14.4027	−0.703619	−0.351809	−	0.936072i	$-0.614433\pi$
$419$	−14.4027	−0.703619	−0.351809	+	0.936072i	$0.614433\pi$
$420$	6.37477	0.311057
$421$	8.01126	0.390445	0.195222	−	0.980759i	$-0.437457\pi$
$421$	8.01126	0.390445	0.195222	+	0.980759i	$0.437457\pi$
$422$	−28.9130	−1.40746
$423$	−5.98098	−0.290805
$424$	1.00000	0.0485643
$425$	17.4067	0.844347
$426$	−6.55760	−0.317717
$427$	27.3573	1.32391
$428$	2.87972	0.139196
$429$	−2.85752	−0.137962
$430$	13.8632	0.668545
$431$	31.4180	1.51335	0.756675	−	0.653791i	$-0.226822\pi$
$431$	31.4180	1.51335	0.756675	+	0.653791i	$0.226822\pi$
$432$	−1.00000	−0.0481125
$433$	5.39232	0.259138	0.129569	−	0.991570i	$-0.458641\pi$
$433$	5.39232	0.259138	0.129569	+	0.991570i	$0.458641\pi$
$434$	37.8550	1.81710
$435$	14.4891	0.694697
$436$	−7.21565	−0.345567
$437$	−7.87030	−0.376487
$438$	6.32429	0.302186
$439$	1.56435	0.0746624	0.0373312	−	0.999303i	$-0.488114\pi$
$439$	1.56435	0.0746624	0.0373312	+	0.999303i	$0.488114\pi$
$440$	0.747002	0.0356119
$441$	8.08987	0.385232
$442$	−47.3658	−2.25296
$443$	4.07636	0.193674	0.0968368	−	0.995300i	$-0.469128\pi$
$443$	4.07636	0.193674	0.0968368	+	0.995300i	$0.469128\pi$
$444$	−1.01478	−0.0481592
$445$	−19.7073	−0.934215
$446$	−19.5613	−0.926254
$447$	9.77775	0.462472
$448$	−3.88457	−0.183529
$449$	13.1011	0.618277	0.309139	−	0.951017i	$-0.399959\pi$
$449$	13.1011	0.618277	0.309139	+	0.951017i	$0.399959\pi$
$450$	−2.30696	−0.108751
$451$	−4.58512	−0.215905
$452$	6.67886	0.314147
$453$	14.6849	0.689957
$454$	−18.5754	−0.871788
$455$	−40.0178	−1.87606
$456$	1.00000	0.0468293
$457$	7.86376	0.367851	0.183926	−	0.982940i	$-0.441119\pi$
$457$	7.86376	0.367851	0.183926	+	0.982940i	$0.441119\pi$
$458$	−8.75285	−0.408994
$459$	7.54529	0.352184
$460$	12.9155	0.602190
$461$	−12.3032	−0.573018	−0.286509	−	0.958078i	$-0.592495\pi$
$461$	−12.3032	−0.573018	−0.286509	+	0.958078i	$0.592495\pi$
$462$	1.76825	0.0822662
$463$	40.3764	1.87645	0.938226	−	0.346022i	$-0.112468\pi$
$463$	40.3764	1.87645	0.938226	+	0.346022i	$0.112468\pi$
$464$	−8.82914	−0.409883
$465$	15.9920	0.741611
$466$	−20.1814	−0.934886
$467$	−20.0150	−0.926185	−0.463093	−	0.886310i	$-0.653260\pi$
$467$	−20.0150	−0.926185	−0.463093	+	0.886310i	$0.653260\pi$
$468$	6.27753	0.290179
$469$	−18.6958	−0.863291
$470$	−9.81508	−0.452736
$471$	19.5948	0.902880
$472$	6.68275	0.307599
$473$	3.84541	0.176812
$474$	−6.24584	−0.286881
$475$	2.30696	0.105850
$476$	29.3102	1.34343
$477$	1.00000	0.0457869
$478$	20.7366	0.948472
$479$	7.30106	0.333594	0.166797	−	0.985991i	$-0.446658\pi$
$479$	7.30106	0.333594	0.166797	+	0.985991i	$0.446658\pi$
$480$	−1.64105	−0.0749033
$481$	6.37029	0.290460
$482$	−4.48792	−0.204419
$483$	30.5727	1.39111
$484$	−10.7928	−0.490582
$485$	−0.704634	−0.0319958
$486$	−1.00000	−0.0453609
$487$	16.2556	0.736611	0.368305	−	0.929705i	$-0.379938\pi$
$487$	16.2556	0.736611	0.368305	+	0.929705i	$0.379938\pi$
$488$	−7.04256	−0.318802
$489$	17.9262	0.810651
$490$	13.2759	0.599743
$491$	−33.8947	−1.52965	−0.764824	−	0.644239i	$-0.777174\pi$
$491$	−33.8947	−1.52965	−0.764824	+	0.644239i	$0.777174\pi$
$492$	10.0728	0.454117
$493$	66.6185	3.00034
$494$	−6.27753	−0.282440
$495$	0.747002	0.0335752
$496$	−9.74498	−0.437563
$497$	−25.4735	−1.14264
$498$	6.59700	0.295619
$499$	19.0922	0.854685	0.427343	−	0.904090i	$-0.359450\pi$
$499$	19.0922	0.854685	0.427343	+	0.904090i	$0.359450\pi$
$500$	−11.9911	−0.536257
$501$	16.5612	0.739899
$502$	24.3302	1.08591
$503$	33.1166	1.47660	0.738298	−	0.674475i	$-0.235630\pi$
$503$	33.1166	1.47660	0.738298	+	0.674475i	$0.235630\pi$
$504$	−3.88457	−0.173032
$505$	3.56120	0.158471
$506$	3.58254	0.159263
$507$	−26.4074	−1.17279
$508$	2.62300	0.116377
$509$	19.3444	0.857425	0.428713	−	0.903441i	$-0.358967\pi$
$509$	19.3444	0.857425	0.428713	+	0.903441i	$0.358967\pi$
$510$	12.3822	0.548293
$511$	24.5671	1.08679
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	7.41352	0.326996
$515$	24.7472	1.09049
$516$	−8.44779	−0.371893
$517$	−2.72253	−0.119737
$518$	−3.94197	−0.173200
$519$	12.3055	0.540153
$520$	10.3017	0.451761
$521$	−12.0027	−0.525846	−0.262923	−	0.964817i	$-0.584686\pi$
$521$	−12.0027	−0.525846	−0.262923	+	0.964817i	$0.584686\pi$
$522$	−8.82914	−0.386441
$523$	28.0488	1.22649	0.613245	−	0.789893i	$-0.289864\pi$
$523$	28.0488	1.22649	0.613245	+	0.789893i	$0.289864\pi$
$524$	−7.45205	−0.325544
$525$	−8.96153	−0.391113
$526$	13.0571	0.569318
$527$	73.5287	3.20296
$528$	−0.455198	−0.0198099
$529$	38.9416	1.69311
$530$	1.64105	0.0712826
$531$	6.68275	0.290007
$532$	3.88457	0.168417
$533$	−63.2324	−2.73890
$534$	12.0090	0.519678
$535$	4.72575	0.204312
$536$	4.81283	0.207883
$537$	5.25246	0.226661
$538$	−6.73563	−0.290394
$539$	3.68249	0.158616
$540$	−1.64105	−0.0706195
$541$	21.2807	0.914928	0.457464	−	0.889228i	$-0.348758\pi$
$541$	21.2807	0.914928	0.457464	+	0.889228i	$0.348758\pi$
$542$	−11.8951	−0.510939
$543$	−8.38098	−0.359662
$544$	−7.54529	−0.323502
$545$	−11.8412	−0.507223
$546$	24.3855	1.04360
$547$	16.7688	0.716982	0.358491	−	0.933533i	$-0.383291\pi$
$547$	16.7688	0.716982	0.358491	+	0.933533i	$0.383291\pi$
$548$	1.99064	0.0850360
$549$	−7.04256	−0.300569
$550$	−1.05012	−0.0447773
$551$	8.82914	0.376134
$552$	−7.87030	−0.334982
$553$	−24.2624	−1.03174
$554$	15.1771	0.644814
$555$	−1.66530	−0.0706880
$556$	9.62307	0.408109
$557$	−21.7473	−0.921461	−0.460730	−	0.887540i	$-0.652412\pi$
$557$	−21.7473	−0.921461	−0.460730	+	0.887540i	$0.652412\pi$
$558$	−9.74498	−0.412538
$559$	53.0313	2.24298
$560$	−6.37477	−0.269383
$561$	3.43460	0.145009
$562$	25.3816	1.07066
$563$	−29.5220	−1.24420	−0.622102	−	0.782936i	$-0.713721\pi$
$563$	−29.5220	−1.24420	−0.622102	+	0.782936i	$0.713721\pi$
$564$	5.98098	0.251845
$565$	10.9603	0.461105
$566$	−19.7617	−0.830645
$567$	−3.88457	−0.163137
$568$	6.55760	0.275151
$569$	−45.0668	−1.88930	−0.944649	−	0.328082i	$-0.893598\pi$
$569$	−45.0668	−1.88930	−0.944649	+	0.328082i	$0.893598\pi$
$570$	1.64105	0.0687360
$571$	−11.5247	−0.482292	−0.241146	−	0.970489i	$-0.577523\pi$
$571$	−11.5247	−0.482292	−0.241146	+	0.970489i	$0.577523\pi$
$572$	2.85752	0.119479
$573$	6.20812	0.259348
$574$	39.1285	1.63319
$575$	−18.1564	−0.757176
$576$	1.00000	0.0416667
$577$	−16.6511	−0.693193	−0.346596	−	0.938014i	$-0.612663\pi$
$577$	−16.6511	−0.693193	−0.346596	+	0.938014i	$0.612663\pi$
$578$	39.9314	1.66093
$579$	21.4495	0.891411
$580$	−14.4891	−0.601625
$581$	25.6265	1.06317
$582$	0.429380	0.0177984
$583$	0.455198	0.0188524
$584$	−6.32429	−0.261701
$585$	10.3017	0.425925
$586$	−18.2532	−0.754032
$587$	−29.3234	−1.21031	−0.605153	−	0.796110i	$-0.706888\pi$
$587$	−29.3234	−1.21031	−0.605153	+	0.796110i	$0.706888\pi$
$588$	−8.08987	−0.333621
$589$	9.74498	0.401535
$590$	10.9667	0.451493
$591$	−10.0505	−0.413422
$592$	1.01478	0.0417071
$593$	31.1207	1.27797	0.638987	−	0.769217i	$-0.279354\pi$
$593$	31.1207	1.27797	0.638987	+	0.769217i	$0.279354\pi$
$594$	−0.455198	−0.0186770
$595$	48.0995	1.97189
$596$	−9.77775	−0.400512
$597$	0.241147	0.00986949
$598$	49.4060	2.02036
$599$	−38.7339	−1.58262	−0.791312	−	0.611412i	$-0.790602\pi$
$599$	−38.7339	−1.58262	−0.791312	+	0.611412i	$0.790602\pi$
$600$	2.30696	0.0941811
$601$	12.0265	0.490571	0.245286	−	0.969451i	$-0.421118\pi$
$601$	12.0265	0.490571	0.245286	+	0.969451i	$0.421118\pi$
$602$	−32.8160	−1.33748
$603$	4.81283	0.195994
$604$	−14.6849	−0.597520
$605$	−17.7115	−0.720075
$606$	−2.17007	−0.0881532
$607$	2.90501	0.117911	0.0589553	−	0.998261i	$-0.481223\pi$
$607$	2.90501	0.117911	0.0589553	+	0.998261i	$0.481223\pi$
$608$	−1.00000	−0.0405554
$609$	−34.2974	−1.38980
$610$	−11.5572	−0.467937
$611$	−37.5458	−1.51894
$612$	−7.54529	−0.305000
$613$	18.1752	0.734088	0.367044	−	0.930203i	$-0.380370\pi$
$613$	18.1752	0.734088	0.367044	+	0.930203i	$0.380370\pi$
$614$	−0.135600	−0.00547237
$615$	16.5300	0.666553
$616$	−1.76825	−0.0712447
$617$	−27.9002	−1.12322	−0.561609	−	0.827403i	$-0.689818\pi$
$617$	−27.9002	−1.12322	−0.561609	+	0.827403i	$0.689818\pi$
$618$	−15.0801	−0.606611
$619$	17.3612	0.697807	0.348903	−	0.937159i	$-0.386554\pi$
$619$	17.3612	0.697807	0.348903	+	0.937159i	$0.386554\pi$
$620$	−15.9920	−0.642254
$621$	−7.87030	−0.315824
$622$	−12.9458	−0.519079
$623$	46.6496	1.86898
$624$	−6.27753	−0.251302
$625$	−8.14316	−0.325726
$626$	12.5352	0.501008
$627$	0.455198	0.0181788
$628$	−19.5948	−0.781917
$629$	−7.65678	−0.305296
$630$	−6.37477	−0.253977
$631$	−12.3070	−0.489934	−0.244967	−	0.969531i	$-0.578777\pi$
$631$	−12.3070	−0.489934	−0.244967	+	0.969531i	$0.578777\pi$
$632$	6.24584	0.248446
$633$	28.9130	1.14919
$634$	16.4744	0.654281
$635$	4.30448	0.170818
$636$	−1.00000	−0.0396526
$637$	50.7844	2.01215
$638$	−4.01900	−0.159114
$639$	6.55760	0.259415
$640$	1.64105	0.0648682
$641$	30.3965	1.20059	0.600294	−	0.799779i	$-0.295050\pi$
$641$	30.3965	1.20059	0.600294	+	0.799779i	$0.295050\pi$
$642$	−2.87972	−0.113653
$643$	21.9378	0.865142	0.432571	−	0.901600i	$-0.357606\pi$
$643$	21.9378	0.865142	0.432571	+	0.901600i	$0.357606\pi$
$644$	−30.5727	−1.20473
$645$	−13.8632	−0.545865
$646$	7.54529	0.296866
$647$	−40.1075	−1.57679	−0.788395	−	0.615170i	$-0.789087\pi$
$647$	−40.1075	−1.57679	−0.788395	+	0.615170i	$0.789087\pi$
$648$	1.00000	0.0392837
$649$	3.04197	0.119408
$650$	−14.4820	−0.568031
$651$	−37.8550	−1.48366
$652$	−17.9262	−0.702045
$653$	22.4997	0.880481	0.440240	−	0.897880i	$-0.354893\pi$
$653$	22.4997	0.880481	0.440240	+	0.897880i	$0.354893\pi$
$654$	7.21565	0.282154
$655$	−12.2292	−0.477833
$656$	−10.0728	−0.393277
$657$	−6.32429	−0.246734
$658$	23.2335	0.905737
$659$	−48.2006	−1.87763	−0.938814	−	0.344425i	$-0.888074\pi$
$659$	−48.2006	−1.87763	−0.938814	+	0.344425i	$0.888074\pi$
$660$	−0.747002	−0.0290770
$661$	−7.91057	−0.307685	−0.153843	−	0.988095i	$-0.549165\pi$
$661$	−7.91057	−0.307685	−0.153843	+	0.988095i	$0.549165\pi$
$662$	33.5886	1.30546
$663$	47.3658	1.83954
$664$	−6.59700	−0.256013
$665$	6.37477	0.247203
$666$	1.01478	0.0393218
$667$	−69.4880	−2.69058
$668$	−16.5612	−0.640771
$669$	19.5613	0.756283
$670$	7.89810	0.305130
$671$	−3.20576	−0.123757
$672$	3.88457	0.149850
$673$	−18.5337	−0.714422	−0.357211	−	0.934024i	$-0.616272\pi$
$673$	−18.5337	−0.714422	−0.357211	+	0.934024i	$0.616272\pi$
$674$	−25.2827	−0.973853
$675$	2.30696	0.0887948
$676$	26.4074	1.01567
$677$	−27.7226	−1.06547	−0.532733	−	0.846283i	$-0.678835\pi$
$677$	−27.7226	−1.06547	−0.532733	+	0.846283i	$0.678835\pi$
$678$	−6.67886	−0.256500
$679$	1.66796	0.0640103
$680$	−12.3822	−0.474836
$681$	18.5754	0.711812
$682$	−4.43589	−0.169859
$683$	−2.25540	−0.0863003	−0.0431502	−	0.999069i	$-0.513739\pi$
$683$	−2.25540	−0.0863003	−0.0431502	+	0.999069i	$0.513739\pi$
$684$	−1.00000	−0.0382360
$685$	3.26674	0.124816
$686$	−4.23367	−0.161642
$687$	8.75285	0.333942
$688$	8.44779	0.322069
$689$	6.27753	0.239155
$690$	−12.9155	−0.491686
$691$	28.2227	1.07364	0.536822	−	0.843696i	$-0.319625\pi$
$691$	28.2227	1.07364	0.536822	+	0.843696i	$0.319625\pi$
$692$	−12.3055	−0.467786
$693$	−1.76825	−0.0671701
$694$	−8.12594	−0.308457
$695$	15.7919	0.599022
$696$	8.82914	0.334668
$697$	76.0023	2.87879
$698$	27.0402	1.02349
$699$	20.1814	0.763331
$700$	8.96153	0.338714
$701$	−12.4352	−0.469670	−0.234835	−	0.972035i	$-0.575455\pi$
$701$	−12.4352	−0.469670	−0.234835	+	0.972035i	$0.575455\pi$
$702$	−6.27753	−0.236930
$703$	−1.01478	−0.0382730
$704$	0.455198	0.0171559
$705$	9.81508	0.369657
$706$	−11.6746	−0.439380
$707$	−8.42979	−0.317035
$708$	−6.68275	−0.251153
$709$	5.23661	0.196665	0.0983325	−	0.995154i	$-0.468649\pi$
$709$	5.23661	0.196665	0.0983325	+	0.995154i	$0.468649\pi$
$710$	10.7613	0.403866
$711$	6.24584	0.234237
$712$	−12.0090	−0.450055
$713$	−76.6959	−2.87228
$714$	−29.3102	−1.09691
$715$	4.68933	0.175371
$716$	−5.25246	−0.196294
$717$	−20.7366	−0.774424
$718$	22.4127	0.836435
$719$	28.4931	1.06261	0.531307	−	0.847180i	$-0.321701\pi$
$719$	28.4931	1.06261	0.531307	+	0.847180i	$0.321701\pi$
$720$	1.64105	0.0611583
$721$	−58.5797	−2.18162
$722$	1.00000	0.0372161
$723$	4.48792	0.166907
$724$	8.38098	0.311477
$725$	20.3685	0.756465
$726$	10.7928	0.400558
$727$	31.8467	1.18113	0.590563	−	0.806991i	$-0.298906\pi$
$727$	31.8467	1.18113	0.590563	+	0.806991i	$0.298906\pi$
$728$	−24.3855	−0.903787
$729$	1.00000	0.0370370
$730$	−10.3785	−0.384124
$731$	−63.7411	−2.35755
$732$	7.04256	0.260301
$733$	33.2244	1.22717	0.613585	−	0.789629i	$-0.289727\pi$
$733$	33.2244	1.22717	0.613585	+	0.789629i	$0.289727\pi$
$734$	−1.25478	−0.0463146
$735$	−13.2759	−0.489688
$736$	7.87030	0.290103
$737$	2.19079	0.0806988
$738$	−10.0728	−0.370785
$739$	10.6162	0.390524	0.195262	−	0.980751i	$-0.437444\pi$
$739$	10.6162	0.390524	0.195262	+	0.980751i	$0.437444\pi$
$740$	1.66530	0.0612176
$741$	6.27753	0.230611
$742$	−3.88457	−0.142607
$743$	46.2940	1.69836	0.849181	−	0.528102i	$-0.177096\pi$
$743$	46.2940	1.69836	0.849181	+	0.528102i	$0.177096\pi$
$744$	9.74498	0.357268
$745$	−16.0458	−0.587871
$746$	−12.6148	−0.461862
$747$	−6.59700	−0.241372
$748$	−3.43460	−0.125581
$749$	−11.1865	−0.408744
$750$	11.9911	0.437852
$751$	−21.6442	−0.789806	−0.394903	−	0.918723i	$-0.629222\pi$
$751$	−21.6442	−0.789806	−0.394903	+	0.918723i	$0.629222\pi$
$752$	−5.98098	−0.218104
$753$	−24.3302	−0.886642
$754$	−55.4252	−2.01847
$755$	−24.0986	−0.877040
$756$	3.88457	0.141280
$757$	−12.2432	−0.444986	−0.222493	−	0.974934i	$-0.571420\pi$
$757$	−12.2432	−0.444986	−0.222493	+	0.974934i	$0.571420\pi$
$758$	−13.2521	−0.481336
$759$	−3.58254	−0.130038
$760$	−1.64105	−0.0595271
$761$	28.3999	1.02949	0.514747	−	0.857342i	$-0.327886\pi$
$761$	28.3999	1.02949	0.514747	+	0.857342i	$0.327886\pi$
$762$	−2.62300	−0.0950213
$763$	28.0297	1.01474
$764$	−6.20812	−0.224602
$765$	−12.3822	−0.447679
$766$	25.2253	0.911426
$767$	41.9512	1.51477
$768$	−1.00000	−0.0360844
$769$	0.585196	0.0211027	0.0105513	−	0.999944i	$-0.496641\pi$
$769$	0.585196	0.0211027	0.0105513	+	0.999944i	$0.496641\pi$
$770$	−2.90178	−0.104573
$771$	−7.41352	−0.266991
$772$	−21.4495	−0.771985
$773$	−18.0261	−0.648355	−0.324178	−	0.945996i	$-0.605088\pi$
$773$	−18.0261	−0.648355	−0.324178	+	0.945996i	$0.605088\pi$
$774$	8.44779	0.303650
$775$	22.4813	0.807551
$776$	−0.429380	−0.0154138
$777$	3.94197	0.141417
$778$	2.96573	0.106327
$779$	10.0728	0.360896
$780$	−10.3017	−0.368861
$781$	2.98500	0.106812
$782$	−59.3837	−2.12356
$783$	8.82914	0.315528
$784$	8.08987	0.288924
$785$	−32.1560	−1.14770
$786$	7.45205	0.265806
$787$	−33.0081	−1.17661	−0.588307	−	0.808638i	$-0.700205\pi$
$787$	−33.0081	−1.17661	−0.588307	+	0.808638i	$0.700205\pi$
$788$	10.0505	0.358034
$789$	−13.0571	−0.464846
$790$	10.2497	0.364669
$791$	−25.9445	−0.922480
$792$	0.455198	0.0161747
$793$	−44.2099	−1.56994
$794$	−0.210589	−0.00747352
$795$	−1.64105	−0.0582020
$796$	−0.241147	−0.00854723
$797$	12.2924	0.435418	0.217709	−	0.976014i	$-0.430142\pi$
$797$	12.2924	0.435418	0.217709	+	0.976014i	$0.430142\pi$
$798$	−3.88457	−0.137512
$799$	45.1282	1.59652
$800$	−2.30696	−0.0815633
$801$	−12.0090	−0.424316
$802$	−19.6442	−0.693661
$803$	−2.87880	−0.101591
$804$	−4.81283	−0.169736
$805$	−50.1713	−1.76831
$806$	−61.1744	−2.15478
$807$	6.73563	0.237105
$808$	2.17007	0.0763429
$809$	−30.3470	−1.06695	−0.533473	−	0.845817i	$-0.679113\pi$
$809$	−30.3470	−1.06695	−0.533473	+	0.845817i	$0.679113\pi$
$810$	1.64105	0.0576606
$811$	45.2566	1.58917	0.794587	−	0.607150i	$-0.207687\pi$
$811$	45.2566	1.58917	0.794587	+	0.607150i	$0.207687\pi$
$812$	34.2974	1.20360
$813$	11.8951	0.417180
$814$	0.461924	0.0161904
$815$	−29.4178	−1.03046
$816$	7.54529	0.264138
$817$	−8.44779	−0.295551
$818$	−9.81228	−0.343078
$819$	−24.3855	−0.852098
$820$	−16.5300	−0.577252
$821$	−18.5295	−0.646686	−0.323343	−	0.946282i	$-0.604807\pi$
$821$	−18.5295	−0.646686	−0.323343	+	0.946282i	$0.604807\pi$
$822$	−1.99064	−0.0694316
$823$	−44.5962	−1.55452	−0.777262	−	0.629177i	$-0.783392\pi$
$823$	−44.5962	−1.55452	−0.777262	+	0.629177i	$0.783392\pi$
$824$	15.0801	0.525341
$825$	1.05012	0.0365605
$826$	−25.9596	−0.903250
$827$	−4.87450	−0.169503	−0.0847515	−	0.996402i	$-0.527010\pi$
$827$	−4.87450	−0.169503	−0.0847515	+	0.996402i	$0.527010\pi$
$828$	7.87030	0.273512
$829$	47.6101	1.65357	0.826783	−	0.562521i	$-0.190169\pi$
$829$	47.6101	1.65357	0.826783	+	0.562521i	$0.190169\pi$
$830$	−10.8260	−0.375776
$831$	−15.1771	−0.526488
$832$	6.27753	0.217634
$833$	−61.0404	−2.11493
$834$	−9.62307	−0.333220
$835$	−27.1777	−0.940524
$836$	−0.455198	−0.0157433
$837$	9.74498	0.336836
$838$	−14.4027	−0.497534
$839$	1.88565	0.0650997	0.0325499	−	0.999470i	$-0.489637\pi$
$839$	1.88565	0.0650997	0.0325499	+	0.999470i	$0.489637\pi$
$840$	6.37477	0.219950
$841$	48.9537	1.68806
$842$	8.01126	0.276086
$843$	−25.3816	−0.874190
$844$	−28.9130	−0.995227
$845$	43.3359	1.49080
$846$	−5.98098	−0.205630
$847$	41.9253	1.44057
$848$	1.00000	0.0343401
$849$	19.7617	0.678219
$850$	17.4067	0.597044
$851$	7.98659	0.273777
$852$	−6.55760	−0.224660
$853$	3.31068	0.113355	0.0566777	−	0.998393i	$-0.481949\pi$
$853$	3.31068	0.113355	0.0566777	+	0.998393i	$0.481949\pi$
$854$	27.3573	0.936148
$855$	−1.64105	−0.0561227
$856$	2.87972	0.0984266
$857$	28.4190	0.970776	0.485388	−	0.874299i	$-0.338678\pi$
$857$	28.4190	0.970776	0.485388	+	0.874299i	$0.338678\pi$
$858$	−2.85752	−0.0975540
$859$	34.1290	1.16446	0.582232	−	0.813022i	$-0.302179\pi$
$859$	34.1290	1.16446	0.582232	+	0.813022i	$0.302179\pi$
$860$	13.8632	0.472733
$861$	−39.1285	−1.33350
$862$	31.4180	1.07010
$863$	6.32630	0.215350	0.107675	−	0.994186i	$-0.465659\pi$
$863$	6.32630	0.215350	0.107675	+	0.994186i	$0.465659\pi$
$864$	−1.00000	−0.0340207
$865$	−20.1940	−0.686616
$866$	5.39232	0.183238
$867$	−39.9314	−1.35614
$868$	37.8550	1.28488
$869$	2.84309	0.0964453
$870$	14.4891	0.491225
$871$	30.2127	1.02372
$872$	−7.21565	−0.244353
$873$	−0.429380	−0.0145323
$874$	−7.87030	−0.266217
$875$	46.5802	1.57470
$876$	6.32429	0.213678
$877$	−40.8895	−1.38074	−0.690371	−	0.723456i	$-0.742553\pi$
$877$	−40.8895	−1.38074	−0.690371	+	0.723456i	$0.742553\pi$
$878$	1.56435	0.0527943
$879$	18.2532	0.615665
$880$	0.747002	0.0251814
$881$	47.6873	1.60663	0.803313	−	0.595557i	$-0.203069\pi$
$881$	47.6873	1.60663	0.803313	+	0.595557i	$0.203069\pi$
$882$	8.08987	0.272400
$883$	−30.2713	−1.01871	−0.509356	−	0.860556i	$-0.670116\pi$
$883$	−30.2713	−1.01871	−0.509356	+	0.860556i	$0.670116\pi$
$884$	−47.3658	−1.59308
$885$	−10.9667	−0.368642
$886$	4.07636	0.136948
$887$	18.9345	0.635758	0.317879	−	0.948131i	$-0.397029\pi$
$887$	18.9345	0.635758	0.317879	+	0.948131i	$0.397029\pi$
$888$	−1.01478	−0.0340537
$889$	−10.1892	−0.341736
$890$	−19.7073	−0.660590
$891$	0.455198	0.0152497
$892$	−19.5613	−0.654960
$893$	5.98098	0.200146
$894$	9.77775	0.327017
$895$	−8.61955	−0.288120
$896$	−3.88457	−0.129774
$897$	−49.4060	−1.64962
$898$	13.1011	0.437188
$899$	86.0398	2.86959
$900$	−2.30696	−0.0768986
$901$	−7.54529	−0.251370
$902$	−4.58512	−0.152668
$903$	32.8160	1.09205
$904$	6.67886	0.222136
$905$	13.7536	0.457185
$906$	14.6849	0.487873
$907$	−28.8061	−0.956491	−0.478246	−	0.878226i	$-0.658727\pi$
$907$	−28.8061	−0.956491	−0.478246	+	0.878226i	$0.658727\pi$
$908$	−18.5754	−0.616447
$909$	2.17007	0.0719768
$910$	−40.0178	−1.32658
$911$	30.9497	1.02541	0.512705	−	0.858565i	$-0.328644\pi$
$911$	30.9497	1.02541	0.512705	+	0.858565i	$0.328644\pi$
$912$	1.00000	0.0331133
$913$	−3.00294	−0.0993828
$914$	7.86376	0.260110
$915$	11.5572	0.382069
$916$	−8.75285	−0.289202
$917$	28.9480	0.955947
$918$	7.54529	0.249032
$919$	−46.8611	−1.54581	−0.772903	−	0.634524i	$-0.781196\pi$
$919$	−46.8611	−1.54581	−0.772903	+	0.634524i	$0.781196\pi$
$920$	12.9155	0.425813
$921$	0.135600	0.00446817
$922$	−12.3032	−0.405185
$923$	41.1656	1.35498
$924$	1.76825	0.0581710
$925$	−2.34105	−0.0769731
$926$	40.3764	1.32685
$927$	15.0801	0.495296
$928$	−8.82914	−0.289831
$929$	−18.9009	−0.620117	−0.310059	−	0.950717i	$-0.600349\pi$
$929$	−18.9009	−0.620117	−0.310059	+	0.950717i	$0.600349\pi$
$930$	15.9920	0.524398
$931$	−8.08987	−0.265135
$932$	−20.1814	−0.661064
$933$	12.9458	0.423826
$934$	−20.0150	−0.654912
$935$	−5.63634	−0.184328
$936$	6.27753	0.205188
$937$	33.4036	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$937$	33.4036	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$938$	−18.6958	−0.610439
$939$	−12.5352	−0.409071
$940$	−9.81508	−0.320133
$941$	22.7615	0.742004	0.371002	−	0.928632i	$-0.379014\pi$
$941$	22.7615	0.742004	0.371002	+	0.928632i	$0.379014\pi$
$942$	19.5948	0.638433
$943$	−79.2760	−2.58158
$944$	6.68275	0.217505
$945$	6.37477	0.207371
$946$	3.84541	0.125025
$947$	−43.3336	−1.40815	−0.704077	−	0.710124i	$-0.748639\pi$
$947$	−43.3336	−1.40815	−0.704077	+	0.710124i	$0.748639\pi$
$948$	−6.24584	−0.202856
$949$	−39.7009	−1.28875
$950$	2.30696	0.0748476
$951$	−16.4744	−0.534218
$952$	29.3102	0.949949
$953$	−8.28502	−0.268378	−0.134189	−	0.990956i	$-0.542843\pi$
$953$	−8.28502	−0.268378	−0.134189	+	0.990956i	$0.542843\pi$
$954$	1.00000	0.0323762
$955$	−10.1878	−0.329671
$956$	20.7366	0.670671
$957$	4.01900	0.129916
$958$	7.30106	0.235887
$959$	−7.73278	−0.249705
$960$	−1.64105	−0.0529646
$961$	63.9647	2.06338
$962$	6.37029	0.205386
$963$	2.87972	0.0927975
$964$	−4.48792	−0.144546
$965$	−35.1997	−1.13312
$966$	30.5727	0.983661
$967$	−49.8298	−1.60242	−0.801209	−	0.598385i	$-0.795809\pi$
$967$	−49.8298	−1.60242	−0.801209	+	0.598385i	$0.795809\pi$
$968$	−10.7928	−0.346894
$969$	−7.54529	−0.242390
$970$	−0.704634	−0.0226244
$971$	−43.5238	−1.39675	−0.698373	−	0.715734i	$-0.746092\pi$
$971$	−43.5238	−1.39675	−0.698373	+	0.715734i	$0.746092\pi$
$972$	−1.00000	−0.0320750
$973$	−37.3815	−1.19839
$974$	16.2556	0.520862
$975$	14.4820	0.463795
$976$	−7.04256	−0.225427
$977$	23.4922	0.751581	0.375791	−	0.926705i	$-0.377371\pi$
$977$	23.4922	0.751581	0.375791	+	0.926705i	$0.377371\pi$
$978$	17.9262	0.573217
$979$	−5.46645	−0.174708
$980$	13.2759	0.424082
$981$	−7.21565	−0.230378
$982$	−33.8947	−1.08162
$983$	−3.82132	−0.121881	−0.0609406	−	0.998141i	$-0.519410\pi$
$983$	−3.82132	−0.121881	−0.0609406	+	0.998141i	$0.519410\pi$
$984$	10.0728	0.321109
$985$	16.4933	0.525522
$986$	66.6185	2.12156
$987$	−23.2335	−0.739531
$988$	−6.27753	−0.199715
$989$	66.4866	2.11415
$990$	0.747002	0.0237413
$991$	35.2646	1.12022	0.560109	−	0.828419i	$-0.310759\pi$
$991$	35.2646	1.12022	0.560109	+	0.828419i	$0.310759\pi$
$992$	−9.74498	−0.309403
$993$	−33.5886	−1.06590
$994$	−25.4735	−0.807969
$995$	−0.395734	−0.0125456
$996$	6.59700	0.209034
$997$	55.2148	1.74867	0.874335	−	0.485324i	$-0.161298\pi$
$997$	55.2148	1.74867	0.874335	+	0.485324i	$0.161298\pi$
$998$	19.0922	0.604354
$999$	−1.01478	−0.0321061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bb.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bb.1.7	✓	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.64105 q^{5} -1.00000 q^{6} -3.88457 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.64105 q^{5} -1.00000 q^{6} -3.88457 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.64105 q^{10} +0.455198 q^{11} -1.00000 q^{12} +6.27753 q^{13} -3.88457 q^{14} -1.64105 q^{15} +1.00000 q^{16} -7.54529 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.64105 q^{20} +3.88457 q^{21} +0.455198 q^{22} +7.87030 q^{23} -1.00000 q^{24} -2.30696 q^{25} +6.27753 q^{26} -1.00000 q^{27} -3.88457 q^{28} -8.82914 q^{29} -1.64105 q^{30} -9.74498 q^{31} +1.00000 q^{32} -0.455198 q^{33} -7.54529 q^{34} -6.37477 q^{35} +1.00000 q^{36} +1.01478 q^{37} -1.00000 q^{38} -6.27753 q^{39} +1.64105 q^{40} -10.0728 q^{41} +3.88457 q^{42} +8.44779 q^{43} +0.455198 q^{44} +1.64105 q^{45} +7.87030 q^{46} -5.98098 q^{47} -1.00000 q^{48} +8.08987 q^{49} -2.30696 q^{50} +7.54529 q^{51} +6.27753 q^{52} +1.00000 q^{53} -1.00000 q^{54} +0.747002 q^{55} -3.88457 q^{56} +1.00000 q^{57} -8.82914 q^{58} +6.68275 q^{59} -1.64105 q^{60} -7.04256 q^{61} -9.74498 q^{62} -3.88457 q^{63} +1.00000 q^{64} +10.3017 q^{65} -0.455198 q^{66} +4.81283 q^{67} -7.54529 q^{68} -7.87030 q^{69} -6.37477 q^{70} +6.55760 q^{71} +1.00000 q^{72} -6.32429 q^{73} +1.01478 q^{74} +2.30696 q^{75} -1.00000 q^{76} -1.76825 q^{77} -6.27753 q^{78} +6.24584 q^{79} +1.64105 q^{80} +1.00000 q^{81} -10.0728 q^{82} -6.59700 q^{83} +3.88457 q^{84} -12.3822 q^{85} +8.44779 q^{86} +8.82914 q^{87} +0.455198 q^{88} -12.0090 q^{89} +1.64105 q^{90} -24.3855 q^{91} +7.87030 q^{92} +9.74498 q^{93} -5.98098 q^{94} -1.64105 q^{95} -1.00000 q^{96} -0.429380 q^{97} +8.08987 q^{98} +0.455198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20} + 5 q^{21} + 3 q^{22} - 9 q^{24} + 8 q^{25} - 4 q^{26} - 9 q^{27} - 5 q^{28} - 2 q^{29} + 5 q^{30} - 10 q^{31} + 9 q^{32} - 3 q^{33} - 16 q^{34} - q^{35} + 9 q^{36} - 14 q^{37} - 9 q^{38} + 4 q^{39} - 5 q^{40} - 21 q^{41} + 5 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} - 8 q^{47} - 9 q^{48} - 14 q^{49} + 8 q^{50} + 16 q^{51} - 4 q^{52} + 9 q^{53} - 9 q^{54} - 14 q^{55} - 5 q^{56} + 9 q^{57} - 2 q^{58} - 12 q^{59} + 5 q^{60} - 13 q^{61} - 10 q^{62} - 5 q^{63} + 9 q^{64} - 13 q^{65} - 3 q^{66} + 22 q^{67} - 16 q^{68} - q^{70} - 13 q^{71} + 9 q^{72} - 17 q^{73} - 14 q^{74} - 8 q^{75} - 9 q^{76} - 25 q^{77} + 4 q^{78} - 5 q^{80} + 9 q^{81} - 21 q^{82} - 24 q^{83} + 5 q^{84} - 16 q^{85} - 4 q^{86} + 2 q^{87} + 3 q^{88} - 23 q^{89} - 5 q^{90} - 11 q^{91} + 10 q^{93} - 8 q^{94} + 5 q^{95} - 9 q^{96} - 29 q^{97} - 14 q^{98} + 3 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9 - 5 * q^10 + 3 * q^11 - 9 * q^12 - 4 * q^13 - 5 * q^14 + 5 * q^15 + 9 * q^16 - 16 * q^17 + 9 * q^18 - 9 * q^19 - 5 * q^20 + 5 * q^21 + 3 * q^22 - 9 * q^24 + 8 * q^25 - 4 * q^26 - 9 * q^27 - 5 * q^28 - 2 * q^29 + 5 * q^30 - 10 * q^31 + 9 * q^32 - 3 * q^33 - 16 * q^34 - q^35 + 9 * q^36 - 14 * q^37 - 9 * q^38 + 4 * q^39 - 5 * q^40 - 21 * q^41 + 5 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 - 8 * q^47 - 9 * q^48 - 14 * q^49 + 8 * q^50 + 16 * q^51 - 4 * q^52 + 9 * q^53 - 9 * q^54 - 14 * q^55 - 5 * q^56 + 9 * q^57 - 2 * q^58 - 12 * q^59 + 5 * q^60 - 13 * q^61 - 10 * q^62 - 5 * q^63 + 9 * q^64 - 13 * q^65 - 3 * q^66 + 22 * q^67 - 16 * q^68 - q^70 - 13 * q^71 + 9 * q^72 - 17 * q^73 - 14 * q^74 - 8 * q^75 - 9 * q^76 - 25 * q^77 + 4 * q^78 - 5 * q^80 + 9 * q^81 - 21 * q^82 - 24 * q^83 + 5 * q^84 - 16 * q^85 - 4 * q^86 + 2 * q^87 + 3 * q^88 - 23 * q^89 - 5 * q^90 - 11 * q^91 + 10 * q^93 - 8 * q^94 + 5 * q^95 - 9 * q^96 - 29 * q^97 - 14 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more