LMFDB - Embedded newform 6422.2.a.bo.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 $ x^15 - 31x^13 - 4x^12 + 373x^11 + 85x^10 - 2208x^9 - 636x^8 + 6791x^7 + 2005x^6 - 10779x^5 - 2515x^4 + 8205x^3 + 686x^2 - 2352*x + 392
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			1.10
Root			$1.01084$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.01084 q^{3} +1.00000 q^{4} -0.455213 q^{5} -1.01084 q^{6} +0.356053 q^{7} -1.00000 q^{8} -1.97821 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.01084 q^{3} +1.00000 q^{4} -0.455213 q^{5} -1.01084 q^{6} +0.356053 q^{7} -1.00000 q^{8} -1.97821 q^{9} +0.455213 q^{10} +1.21707 q^{11} +1.01084 q^{12} -0.356053 q^{14} -0.460146 q^{15} +1.00000 q^{16} +1.03275 q^{17} +1.97821 q^{18} +1.00000 q^{19} -0.455213 q^{20} +0.359912 q^{21} -1.21707 q^{22} +8.71949 q^{23} -1.01084 q^{24} -4.79278 q^{25} -5.03216 q^{27} +0.356053 q^{28} -3.10642 q^{29} +0.460146 q^{30} -1.63814 q^{31} -1.00000 q^{32} +1.23026 q^{33} -1.03275 q^{34} -0.162080 q^{35} -1.97821 q^{36} -3.69898 q^{37} -1.00000 q^{38} +0.455213 q^{40} -11.0089 q^{41} -0.359912 q^{42} +3.76438 q^{43} +1.21707 q^{44} +0.900504 q^{45} -8.71949 q^{46} -10.6023 q^{47} +1.01084 q^{48} -6.87323 q^{49} +4.79278 q^{50} +1.04394 q^{51} +13.1351 q^{53} +5.03216 q^{54} -0.554026 q^{55} -0.356053 q^{56} +1.01084 q^{57} +3.10642 q^{58} -12.0179 q^{59} -0.460146 q^{60} +11.9519 q^{61} +1.63814 q^{62} -0.704346 q^{63} +1.00000 q^{64} -1.23026 q^{66} -2.69986 q^{67} +1.03275 q^{68} +8.81399 q^{69} +0.162080 q^{70} -1.14762 q^{71} +1.97821 q^{72} -2.88020 q^{73} +3.69898 q^{74} -4.84473 q^{75} +1.00000 q^{76} +0.433342 q^{77} -4.40436 q^{79} -0.455213 q^{80} +0.847922 q^{81} +11.0089 q^{82} +13.4531 q^{83} +0.359912 q^{84} -0.470120 q^{85} -3.76438 q^{86} -3.14009 q^{87} -1.21707 q^{88} -14.4690 q^{89} -0.900504 q^{90} +8.71949 q^{92} -1.65589 q^{93} +10.6023 q^{94} -0.455213 q^{95} -1.01084 q^{96} +5.39891 q^{97} +6.87323 q^{98} -2.40762 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.01084	0.583608	0.291804	−	0.956478i	$-0.405745\pi$
$3$	1.01084	0.583608	0.291804	+	0.956478i	$0.405745\pi$
$4$	1.00000	0.500000
$5$	−0.455213	−0.203577	−0.101789	−	0.994806i	$-0.532457\pi$
$5$	−0.455213	−0.203577	−0.101789	+	0.994806i	$0.532457\pi$
$6$	−1.01084	−0.412673
$7$	0.356053	0.134575	0.0672877	−	0.997734i	$-0.478565\pi$
$7$	0.356053	0.134575	0.0672877	+	0.997734i	$0.478565\pi$
$8$	−1.00000	−0.353553
$9$	−1.97821	−0.659402
$10$	0.455213	0.143951
$11$	1.21707	0.366961	0.183481	−	0.983023i	$-0.441264\pi$
$11$	1.21707	0.366961	0.183481	+	0.983023i	$0.441264\pi$
$12$	1.01084	0.291804
$13$	0	0
$13$	0	0
$14$	−0.356053	−0.0951591
$15$	−0.460146	−0.118809
$16$	1.00000	0.250000
$17$	1.03275	0.250478	0.125239	−	0.992127i	$-0.460030\pi$
$17$	1.03275	0.250478	0.125239	+	0.992127i	$0.460030\pi$
$18$	1.97821	0.466268
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.455213	−0.101789
$21$	0.359912	0.0785392
$22$	−1.21707	−0.259481
$23$	8.71949	1.81814	0.909070	−	0.416644i	$-0.136794\pi$
$23$	8.71949	1.81814	0.909070	+	0.416644i	$0.136794\pi$
$24$	−1.01084	−0.206336
$25$	−4.79278	−0.958556
$26$	0	0
$27$	−5.03216	−0.968440
$28$	0.356053	0.0672877
$29$	−3.10642	−0.576848	−0.288424	−	0.957503i	$-0.593131\pi$
$29$	−3.10642	−0.576848	−0.288424	+	0.957503i	$0.593131\pi$
$30$	0.460146	0.0840108
$31$	−1.63814	−0.294218	−0.147109	−	0.989120i	$-0.546997\pi$
$31$	−1.63814	−0.294218	−0.147109	+	0.989120i	$0.546997\pi$
$32$	−1.00000	−0.176777
$33$	1.23026	0.214161
$34$	−1.03275	−0.177115
$35$	−0.162080	−0.0273965
$36$	−1.97821	−0.329701
$37$	−3.69898	−0.608108	−0.304054	−	0.952655i	$-0.598340\pi$
$37$	−3.69898	−0.608108	−0.304054	+	0.952655i	$0.598340\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0.455213	0.0719754
$41$	−11.0089	−1.71929	−0.859647	−	0.510888i	$-0.829317\pi$
$41$	−11.0089	−1.71929	−0.859647	+	0.510888i	$0.829317\pi$
$42$	−0.359912	−0.0555356
$43$	3.76438	0.574062	0.287031	−	0.957921i	$-0.407332\pi$
$43$	3.76438	0.574062	0.287031	+	0.957921i	$0.407332\pi$
$44$	1.21707	0.183481
$45$	0.900504	0.134239
$46$	−8.71949	−1.28562
$47$	−10.6023	−1.54650	−0.773249	−	0.634102i	$-0.781370\pi$
$47$	−10.6023	−1.54650	−0.773249	+	0.634102i	$0.781370\pi$
$48$	1.01084	0.145902
$49$	−6.87323	−0.981889
$50$	4.79278	0.677802
$51$	1.04394	0.146181
$52$	0	0
$53$	13.1351	1.80425	0.902125	−	0.431475i	$-0.142007\pi$
$53$	13.1351	1.80425	0.902125	+	0.431475i	$0.142007\pi$
$54$	5.03216	0.684790
$55$	−0.554026	−0.0747049
$56$	−0.356053	−0.0475796
$57$	1.01084	0.133889
$58$	3.10642	0.407893
$59$	−12.0179	−1.56460	−0.782299	−	0.622903i	$-0.785953\pi$
$59$	−12.0179	−1.56460	−0.782299	+	0.622903i	$0.785953\pi$
$60$	−0.460146	−0.0594046
$61$	11.9519	1.53029	0.765144	−	0.643859i	$-0.222668\pi$
$61$	11.9519	1.53029	0.765144	+	0.643859i	$0.222668\pi$
$62$	1.63814	0.208044
$63$	−0.704346	−0.0887393
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.23026	−0.151435
$67$	−2.69986	−0.329840	−0.164920	−	0.986307i	$-0.552737\pi$
$67$	−2.69986	−0.329840	−0.164920	+	0.986307i	$0.552737\pi$
$68$	1.03275	0.125239
$69$	8.81399	1.06108
$70$	0.162080	0.0193722
$71$	−1.14762	−0.136197	−0.0680985	−	0.997679i	$-0.521693\pi$
$71$	−1.14762	−0.136197	−0.0680985	+	0.997679i	$0.521693\pi$
$72$	1.97821	0.233134
$73$	−2.88020	−0.337102	−0.168551	−	0.985693i	$-0.553909\pi$
$73$	−2.88020	−0.337102	−0.168551	+	0.985693i	$0.553909\pi$
$74$	3.69898	0.429997
$75$	−4.84473	−0.559421
$76$	1.00000	0.114708
$77$	0.433342	0.0493839
$78$	0	0
$79$	−4.40436	−0.495529	−0.247765	−	0.968820i	$-0.579696\pi$
$79$	−4.40436	−0.495529	−0.247765	+	0.968820i	$0.579696\pi$
$80$	−0.455213	−0.0508943
$81$	0.847922	0.0942135
$82$	11.0089	1.21572
$83$	13.4531	1.47667	0.738333	−	0.674436i	$-0.235613\pi$
$83$	13.4531	1.47667	0.738333	+	0.674436i	$0.235613\pi$
$84$	0.359912	0.0392696
$85$	−0.470120	−0.0509917
$86$	−3.76438	−0.405923
$87$	−3.14009	−0.336653
$88$	−1.21707	−0.129740
$89$	−14.4690	−1.53371	−0.766854	−	0.641821i	$-0.778179\pi$
$89$	−14.4690	−1.53371	−0.766854	+	0.641821i	$0.778179\pi$
$90$	−0.900504	−0.0949215
$91$	0	0
$92$	8.71949	0.909070
$93$	−1.65589	−0.171708
$94$	10.6023	1.09354
$95$	−0.455213	−0.0467038
$96$	−1.01084	−0.103168
$97$	5.39891	0.548176	0.274088	−	0.961705i	$-0.411624\pi$
$97$	5.39891	0.548176	0.274088	+	0.961705i	$0.411624\pi$
$98$	6.87323	0.694301
$99$	−2.40762	−0.241975
$100$	−4.79278	−0.479278
$101$	17.7590	1.76709	0.883545	−	0.468347i	$-0.155150\pi$
$101$	17.7590	1.76709	0.883545	+	0.468347i	$0.155150\pi$
$102$	−1.04394	−0.103366
$103$	1.59833	0.157488	0.0787441	−	0.996895i	$-0.474909\pi$
$103$	1.59833	0.157488	0.0787441	+	0.996895i	$0.474909\pi$
$104$	0	0
$105$	−0.163836	−0.0159888
$106$	−13.1351	−1.27580
$107$	−10.4313	−1.00843	−0.504216	−	0.863578i	$-0.668218\pi$
$107$	−10.4313	−1.00843	−0.504216	+	0.863578i	$0.668218\pi$
$108$	−5.03216	−0.484220
$109$	15.3958	1.47465	0.737325	−	0.675539i	$-0.236089\pi$
$109$	15.3958	1.47465	0.737325	+	0.675539i	$0.236089\pi$
$110$	0.554026	0.0528244
$111$	−3.73907	−0.354897
$112$	0.356053	0.0336438
$113$	−2.16339	−0.203514	−0.101757	−	0.994809i	$-0.532446\pi$
$113$	−2.16339	−0.203514	−0.101757	+	0.994809i	$0.532446\pi$
$114$	−1.01084	−0.0946736
$115$	−3.96922	−0.370132
$116$	−3.10642	−0.288424
$117$	0	0
$118$	12.0179	1.10634
$119$	0.367713	0.0337082
$120$	0.460146	0.0420054
$121$	−9.51874	−0.865340
$122$	−11.9519	−1.08208
$123$	−11.1282	−1.00339
$124$	−1.63814	−0.147109
$125$	4.45780	0.398717
$126$	0.704346	0.0627482
$127$	−0.628624	−0.0557814	−0.0278907	−	0.999611i	$-0.508879\pi$
$127$	−0.628624	−0.0557814	−0.0278907	+	0.999611i	$0.508879\pi$
$128$	−1.00000	−0.0883883
$129$	3.80517	0.335027
$130$	0	0
$131$	−12.2759	−1.07255	−0.536277	−	0.844042i	$-0.680170\pi$
$131$	−12.2759	−1.07255	−0.536277	+	0.844042i	$0.680170\pi$
$132$	1.23026	0.107081
$133$	0.356053	0.0308737
$134$	2.69986	0.233232
$135$	2.29070	0.197152
$136$	−1.03275	−0.0885575
$137$	2.02764	0.173233	0.0866164	−	0.996242i	$-0.472395\pi$
$137$	2.02764	0.173233	0.0866164	+	0.996242i	$0.472395\pi$
$138$	−8.81399	−0.750297
$139$	−5.90936	−0.501225	−0.250612	−	0.968087i	$-0.580632\pi$
$139$	−5.90936	−0.501225	−0.250612	+	0.968087i	$0.580632\pi$
$140$	−0.162080	−0.0136982
$141$	−10.7172	−0.902548
$142$	1.14762	0.0963058
$143$	0	0
$144$	−1.97821	−0.164851
$145$	1.41408	0.117433
$146$	2.88020	0.238367
$147$	−6.94772	−0.573038
$148$	−3.69898	−0.304054
$149$	−0.329395	−0.0269851	−0.0134926	−	0.999909i	$-0.504295\pi$
$149$	−0.329395	−0.0269851	−0.0134926	+	0.999909i	$0.504295\pi$
$150$	4.84473	0.395570
$151$	−12.8042	−1.04199	−0.520994	−	0.853561i	$-0.674438\pi$
$151$	−12.8042	−1.04199	−0.520994	+	0.853561i	$0.674438\pi$
$152$	−1.00000	−0.0811107
$153$	−2.04299	−0.165166
$154$	−0.433342	−0.0349197
$155$	0.745700	0.0598961
$156$	0	0
$157$	−3.98723	−0.318216	−0.159108	−	0.987261i	$-0.550862\pi$
$157$	−3.98723	−0.318216	−0.159108	+	0.987261i	$0.550862\pi$
$158$	4.40436	0.350392
$159$	13.2775	1.05297
$160$	0.455213	0.0359877
$161$	3.10460	0.244677
$162$	−0.847922	−0.0666190
$163$	1.51863	0.118948	0.0594741	−	0.998230i	$-0.481058\pi$
$163$	1.51863	0.118948	0.0594741	+	0.998230i	$0.481058\pi$
$164$	−11.0089	−0.859647
$165$	−0.560031	−0.0435984
$166$	−13.4531	−1.04416
$167$	−14.5657	−1.12713	−0.563566	−	0.826071i	$-0.690571\pi$
$167$	−14.5657	−1.12713	−0.563566	+	0.826071i	$0.690571\pi$
$168$	−0.359912	−0.0277678
$169$	0	0
$170$	0.470120	0.0360566
$171$	−1.97821	−0.151277
$172$	3.76438	0.287031
$173$	23.7698	1.80718	0.903591	−	0.428397i	$-0.140922\pi$
$173$	23.7698	1.80718	0.903591	+	0.428397i	$0.140922\pi$
$174$	3.14009	0.238050
$175$	−1.70648	−0.128998
$176$	1.21707	0.0917403
$177$	−12.1482	−0.913111
$178$	14.4690	1.08450
$179$	−24.0274	−1.79589	−0.897945	−	0.440107i	$-0.854941\pi$
$179$	−24.0274	−1.79589	−0.897945	+	0.440107i	$0.854941\pi$
$180$	0.900504	0.0671196
$181$	−17.1006	−1.27107	−0.635537	−	0.772070i	$-0.719221\pi$
$181$	−17.1006	−1.27107	−0.635537	+	0.772070i	$0.719221\pi$
$182$	0	0
$183$	12.0815	0.893088
$184$	−8.71949	−0.642809
$185$	1.68382	0.123797
$186$	1.65589	0.121416
$187$	1.25693	0.0919158
$188$	−10.6023	−0.773249
$189$	−1.79172	−0.130328
$190$	0.455213	0.0330246
$191$	2.23298	0.161573	0.0807864	−	0.996731i	$-0.474257\pi$
$191$	2.23298	0.161573	0.0807864	+	0.996731i	$0.474257\pi$
$192$	1.01084	0.0729509
$193$	−5.42750	−0.390680	−0.195340	−	0.980736i	$-0.562581\pi$
$193$	−5.42750	−0.390680	−0.195340	+	0.980736i	$0.562581\pi$
$194$	−5.39891	−0.387619
$195$	0	0
$196$	−6.87323	−0.490945
$197$	−19.1142	−1.36183	−0.680916	−	0.732361i	$-0.738418\pi$
$197$	−19.1142	−1.36183	−0.680916	+	0.732361i	$0.738418\pi$
$198$	2.40762	0.171102
$199$	−6.50523	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$199$	−6.50523	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$200$	4.79278	0.338901
$201$	−2.72912	−0.192497
$202$	−17.7590	−1.24952
$203$	−1.10605	−0.0776296
$204$	1.04394	0.0730906
$205$	5.01137	0.350009
$206$	−1.59833	−0.111361
$207$	−17.2490	−1.19889
$208$	0	0
$209$	1.21707	0.0841866
$210$	0.163836	0.0113058
$211$	2.62576	0.180765	0.0903823	−	0.995907i	$-0.471191\pi$
$211$	2.62576	0.180765	0.0903823	+	0.995907i	$0.471191\pi$
$212$	13.1351	0.902125
$213$	−1.16005	−0.0794856
$214$	10.4313	0.713069
$215$	−1.71359	−0.116866
$216$	5.03216	0.342395
$217$	−0.583263	−0.0395945
$218$	−15.3958	−1.04273
$219$	−2.91141	−0.196735
$220$	−0.554026	−0.0373525
$221$	0	0
$222$	3.73907	0.250950
$223$	−15.9495	−1.06806	−0.534028	−	0.845467i	$-0.679322\pi$
$223$	−15.9495	−1.06806	−0.534028	+	0.845467i	$0.679322\pi$
$224$	−0.356053	−0.0237898
$225$	9.48111	0.632074
$226$	2.16339	0.143906
$227$	12.3280	0.818239	0.409120	−	0.912481i	$-0.365836\pi$
$227$	12.3280	0.818239	0.409120	+	0.912481i	$0.365836\pi$
$228$	1.01084	0.0669444
$229$	−9.93342	−0.656419	−0.328209	−	0.944605i	$-0.606445\pi$
$229$	−9.93342	−0.656419	−0.328209	+	0.944605i	$0.606445\pi$
$230$	3.96922	0.261723
$231$	0.438039	0.0288208
$232$	3.10642	0.203947
$233$	3.90623	0.255905	0.127953	−	0.991780i	$-0.459159\pi$
$233$	3.90623	0.255905	0.127953	+	0.991780i	$0.459159\pi$
$234$	0	0
$235$	4.82628	0.314832
$236$	−12.0179	−0.782299
$237$	−4.45210	−0.289195
$238$	−0.367713	−0.0238353
$239$	20.5380	1.32849	0.664246	−	0.747514i	$-0.268753\pi$
$239$	20.5380	1.32849	0.664246	+	0.747514i	$0.268753\pi$
$240$	−0.460146	−0.0297023
$241$	−1.41961	−0.0914452	−0.0457226	−	0.998954i	$-0.514559\pi$
$241$	−1.41961	−0.0914452	−0.0457226	+	0.998954i	$0.514559\pi$
$242$	9.51874	0.611887
$243$	15.9536	1.02342
$244$	11.9519	0.765144
$245$	3.12878	0.199890
$246$	11.1282	0.709506
$247$	0	0
$248$	1.63814	0.104022
$249$	13.5989	0.861794
$250$	−4.45780	−0.281936
$251$	13.2240	0.834690	0.417345	−	0.908748i	$-0.362961\pi$
$251$	13.2240	0.834690	0.417345	+	0.908748i	$0.362961\pi$
$252$	−0.704346	−0.0443696
$253$	10.6122	0.667186
$254$	0.628624	0.0394434
$255$	−0.475216	−0.0297592
$256$	1.00000	0.0625000
$257$	−16.8982	−1.05408	−0.527041	−	0.849840i	$-0.676699\pi$
$257$	−16.8982	−1.05408	−0.527041	+	0.849840i	$0.676699\pi$
$258$	−3.80517	−0.236900
$259$	−1.31703	−0.0818364
$260$	0	0
$261$	6.14515	0.380375
$262$	12.2759	0.758410
$263$	−20.5694	−1.26836	−0.634181	−	0.773184i	$-0.718663\pi$
$263$	−20.5694	−1.26836	−0.634181	+	0.773184i	$0.718663\pi$
$264$	−1.23026	−0.0757174
$265$	−5.97928	−0.367304
$266$	−0.356053	−0.0218310
$267$	−14.6258	−0.895084
$268$	−2.69986	−0.164920
$269$	−14.3269	−0.873529	−0.436764	−	0.899576i	$-0.643876\pi$
$269$	−14.3269	−0.873529	−0.436764	+	0.899576i	$0.643876\pi$
$270$	−2.29070	−0.139408
$271$	−16.5580	−1.00583	−0.502914	−	0.864337i	$-0.667739\pi$
$271$	−16.5580	−1.00583	−0.502914	+	0.864337i	$0.667739\pi$
$272$	1.03275	0.0626196
$273$	0	0
$274$	−2.02764	−0.122494
$275$	−5.83316	−0.351753
$276$	8.81399	0.530540
$277$	5.16478	0.310322	0.155161	−	0.987889i	$-0.450410\pi$
$277$	5.16478	0.310322	0.155161	+	0.987889i	$0.450410\pi$
$278$	5.90936	0.354420
$279$	3.24057	0.194008
$280$	0.162080	0.00968612
$281$	−14.3906	−0.858473	−0.429237	−	0.903192i	$-0.641217\pi$
$281$	−14.3906	−0.858473	−0.429237	+	0.903192i	$0.641217\pi$
$282$	10.7172	0.638198
$283$	−15.3559	−0.912814	−0.456407	−	0.889771i	$-0.650864\pi$
$283$	−15.3559	−0.912814	−0.456407	+	0.889771i	$0.650864\pi$
$284$	−1.14762	−0.0680985
$285$	−0.460146	−0.0272567
$286$	0	0
$287$	−3.91974	−0.231375
$288$	1.97821	0.116567
$289$	−15.9334	−0.937261
$290$	−1.41408	−0.0830378
$291$	5.45742	0.319920
$292$	−2.88020	−0.168551
$293$	14.8401	0.866968	0.433484	−	0.901161i	$-0.357284\pi$
$293$	14.8401	0.866968	0.433484	+	0.901161i	$0.357284\pi$
$294$	6.94772	0.405199
$295$	5.47070	0.318516
$296$	3.69898	0.214999
$297$	−6.12450	−0.355380
$298$	0.329395	0.0190814
$299$	0	0
$300$	−4.84473	−0.279710
$301$	1.34032	0.0772546
$302$	12.8042	0.736796
$303$	17.9515	1.03129
$304$	1.00000	0.0573539
$305$	−5.44067	−0.311532
$306$	2.04299	0.116790
$307$	11.2907	0.644397	0.322198	−	0.946672i	$-0.395578\pi$
$307$	11.2907	0.644397	0.322198	+	0.946672i	$0.395578\pi$
$308$	0.433342	0.0246920
$309$	1.61565	0.0919113
$310$	−0.745700	−0.0423529
$311$	−13.7991	−0.782472	−0.391236	−	0.920290i	$-0.627952\pi$
$311$	−13.7991	−0.782472	−0.391236	+	0.920290i	$0.627952\pi$
$312$	0	0
$313$	−20.9614	−1.18481	−0.592404	−	0.805641i	$-0.701821\pi$
$313$	−20.9614	−1.18481	−0.592404	+	0.805641i	$0.701821\pi$
$314$	3.98723	0.225013
$315$	0.320627	0.0180653
$316$	−4.40436	−0.247765
$317$	11.2576	0.632287	0.316144	−	0.948711i	$-0.397612\pi$
$317$	11.2576	0.632287	0.316144	+	0.948711i	$0.397612\pi$
$318$	−13.2775	−0.744565
$319$	−3.78074	−0.211681
$320$	−0.455213	−0.0254472
$321$	−10.5444	−0.588529
$322$	−3.10460	−0.173013
$323$	1.03275	0.0574637
$324$	0.847922	0.0471068
$325$	0	0
$326$	−1.51863	−0.0841090
$327$	15.5627	0.860617
$328$	11.0089	0.607862
$329$	−3.77497	−0.208121
$330$	0.560031	0.0308287
$331$	−23.9433	−1.31604	−0.658022	−	0.752998i	$-0.728607\pi$
$331$	−23.9433	−1.31604	−0.658022	+	0.752998i	$0.728607\pi$
$332$	13.4531	0.738333
$333$	7.31734	0.400988
$334$	14.5657	0.797002
$335$	1.22901	0.0671479
$336$	0.359912	0.0196348
$337$	−26.9760	−1.46947	−0.734737	−	0.678352i	$-0.762695\pi$
$337$	−26.9760	−1.46947	−0.734737	+	0.678352i	$0.762695\pi$
$338$	0	0
$339$	−2.18683	−0.118772
$340$	−0.470120	−0.0254959
$341$	−1.99373	−0.107967
$342$	1.97821	0.106969
$343$	−4.93960	−0.266713
$344$	−3.76438	−0.202962
$345$	−4.01224	−0.216012
$346$	−23.7698	−1.27787
$347$	28.2555	1.51684	0.758418	−	0.651769i	$-0.225973\pi$
$347$	28.2555	1.51684	0.758418	+	0.651769i	$0.225973\pi$
$348$	−3.14009	−0.168327
$349$	−5.63737	−0.301762	−0.150881	−	0.988552i	$-0.548211\pi$
$349$	−5.63737	−0.301762	−0.150881	+	0.988552i	$0.548211\pi$
$350$	1.70648	0.0912154
$351$	0	0
$352$	−1.21707	−0.0648702
$353$	35.4087	1.88461	0.942307	−	0.334749i	$-0.108652\pi$
$353$	35.4087	1.88461	0.942307	+	0.334749i	$0.108652\pi$
$354$	12.1482	0.645667
$355$	0.522409	0.0277266
$356$	−14.4690	−0.766854
$357$	0.371699	0.0196724
$358$	24.0274	1.26989
$359$	−10.7395	−0.566809	−0.283405	−	0.959000i	$-0.591464\pi$
$359$	−10.7395	−0.566809	−0.283405	+	0.959000i	$0.591464\pi$
$360$	−0.900504	−0.0474608
$361$	1.00000	0.0526316
$362$	17.1006	0.898786
$363$	−9.62190	−0.505019
$364$	0	0
$365$	1.31110	0.0686262
$366$	−12.0815	−0.631509
$367$	−24.4885	−1.27829	−0.639144	−	0.769087i	$-0.720711\pi$
$367$	−24.4885	−1.27829	−0.639144	+	0.769087i	$0.720711\pi$
$368$	8.71949	0.454535
$369$	21.7778	1.13371
$370$	−1.68382	−0.0875377
$371$	4.67680	0.242808
$372$	−1.65589	−0.0858539
$373$	−15.8314	−0.819721	−0.409860	−	0.912148i	$-0.634423\pi$
$373$	−15.8314	−0.819721	−0.409860	+	0.912148i	$0.634423\pi$
$374$	−1.25693	−0.0649943
$375$	4.50611	0.232695
$376$	10.6023	0.546770
$377$	0	0
$378$	1.79172	0.0921559
$379$	−7.94916	−0.408321	−0.204161	−	0.978937i	$-0.565446\pi$
$379$	−7.94916	−0.408321	−0.204161	+	0.978937i	$0.565446\pi$
$380$	−0.455213	−0.0233519
$381$	−0.635437	−0.0325544
$382$	−2.23298	−0.114249
$383$	−7.88922	−0.403120	−0.201560	−	0.979476i	$-0.564601\pi$
$383$	−7.88922	−0.403120	−0.201560	+	0.979476i	$0.564601\pi$
$384$	−1.01084	−0.0515841
$385$	−0.197263	−0.0100534
$386$	5.42750	0.276253
$387$	−7.44672	−0.378538
$388$	5.39891	0.274088
$389$	−28.5085	−1.44544	−0.722718	−	0.691143i	$-0.757107\pi$
$389$	−28.5085	−1.44544	−0.722718	+	0.691143i	$0.757107\pi$
$390$	0	0
$391$	9.00505	0.455405
$392$	6.87323	0.347150
$393$	−12.4090	−0.625951
$394$	19.1142	0.962961
$395$	2.00492	0.100878
$396$	−2.40762	−0.120987
$397$	−14.2501	−0.715190	−0.357595	−	0.933877i	$-0.616403\pi$
$397$	−14.2501	−0.715190	−0.357595	+	0.933877i	$0.616403\pi$
$398$	6.50523	0.326078
$399$	0.359912	0.0180181
$400$	−4.79278	−0.239639
$401$	−7.63991	−0.381519	−0.190759	−	0.981637i	$-0.561095\pi$
$401$	−7.63991	−0.381519	−0.190759	+	0.981637i	$0.561095\pi$
$402$	2.72912	0.136116
$403$	0	0
$404$	17.7590	0.883545
$405$	−0.385985	−0.0191797
$406$	1.10605	0.0548924
$407$	−4.50192	−0.223152
$408$	−1.04394	−0.0516828
$409$	−10.5196	−0.520159	−0.260079	−	0.965587i	$-0.583749\pi$
$409$	−10.5196	−0.520159	−0.260079	+	0.965587i	$0.583749\pi$
$410$	−5.01137	−0.247494
$411$	2.04961	0.101100
$412$	1.59833	0.0787441
$413$	−4.27901	−0.210556
$414$	17.2490	0.847740
$415$	−6.12401	−0.300616
$416$	0	0
$417$	−5.97340	−0.292519
$418$	−1.21707	−0.0595289
$419$	23.7711	1.16129	0.580647	−	0.814156i	$-0.302800\pi$
$419$	23.7711	1.16129	0.580647	+	0.814156i	$0.302800\pi$
$420$	−0.163836	−0.00799440
$421$	7.09093	0.345591	0.172795	−	0.984958i	$-0.444720\pi$
$421$	7.09093	0.345591	0.172795	+	0.984958i	$0.444720\pi$
$422$	−2.62576	−0.127820
$423$	20.9735	1.01976
$424$	−13.1351	−0.637899
$425$	−4.94974	−0.240098
$426$	1.16005	0.0562048
$427$	4.25552	0.205939
$428$	−10.4313	−0.504216
$429$	0	0
$430$	1.71359	0.0826367
$431$	19.6994	0.948886	0.474443	−	0.880286i	$-0.342649\pi$
$431$	19.6994	0.948886	0.474443	+	0.880286i	$0.342649\pi$
$432$	−5.03216	−0.242110
$433$	17.5617	0.843959	0.421980	−	0.906605i	$-0.361335\pi$
$433$	17.5617	0.843959	0.421980	+	0.906605i	$0.361335\pi$
$434$	0.583263	0.0279975
$435$	1.42941	0.0685349
$436$	15.3958	0.737325
$437$	8.71949	0.417110
$438$	2.91141	0.139113
$439$	28.1401	1.34305	0.671526	−	0.740981i	$-0.265639\pi$
$439$	28.1401	1.34305	0.671526	+	0.740981i	$0.265639\pi$
$440$	0.554026	0.0264122
$441$	13.5967	0.647460
$442$	0	0
$443$	−11.6857	−0.555202	−0.277601	−	0.960696i	$-0.589539\pi$
$443$	−11.6857	−0.555202	−0.277601	+	0.960696i	$0.589539\pi$
$444$	−3.73907	−0.177448
$445$	6.58646	0.312228
$446$	15.9495	0.755230
$447$	−0.332965	−0.0157487
$448$	0.356053	0.0168219
$449$	21.2368	1.00222	0.501112	−	0.865382i	$-0.332924\pi$
$449$	21.2368	1.00222	0.501112	+	0.865382i	$0.332924\pi$
$450$	−9.48111	−0.446944
$451$	−13.3986	−0.630914
$452$	−2.16339	−0.101757
$453$	−12.9429	−0.608112
$454$	−12.3280	−0.578582
$455$	0	0
$456$	−1.01084	−0.0473368
$457$	−18.8960	−0.883916	−0.441958	−	0.897036i	$-0.645716\pi$
$457$	−18.8960	−0.883916	−0.441958	+	0.897036i	$0.645716\pi$
$458$	9.93342	0.464158
$459$	−5.19696	−0.242573
$460$	−3.96922	−0.185066
$461$	1.39732	0.0650798	0.0325399	−	0.999470i	$-0.489640\pi$
$461$	1.39732	0.0650798	0.0325399	+	0.999470i	$0.489640\pi$
$462$	−0.438039	−0.0203794
$463$	16.6035	0.771630	0.385815	−	0.922576i	$-0.373920\pi$
$463$	16.6035	0.771630	0.385815	+	0.922576i	$0.373920\pi$
$464$	−3.10642	−0.144212
$465$	0.753782	0.0349558
$466$	−3.90623	−0.180952
$467$	8.81724	0.408013	0.204007	−	0.978969i	$-0.434604\pi$
$467$	8.81724	0.408013	0.204007	+	0.978969i	$0.434604\pi$
$468$	0	0
$469$	−0.961291	−0.0443883
$470$	−4.82628	−0.222620
$471$	−4.03045	−0.185713
$472$	12.0179	0.553169
$473$	4.58152	0.210658
$474$	4.45210	0.204492
$475$	−4.79278	−0.219908
$476$	0.367713	0.0168541
$477$	−25.9840	−1.18973
$478$	−20.5380	−0.939385
$479$	−30.2392	−1.38167	−0.690833	−	0.723014i	$-0.742756\pi$
$479$	−30.2392	−1.38167	−0.690833	+	0.723014i	$0.742756\pi$
$480$	0.460146	0.0210027
$481$	0	0
$482$	1.41961	0.0646615
$483$	3.13825	0.142795
$484$	−9.51874	−0.432670
$485$	−2.45765	−0.111596
$486$	−15.9536	−0.723670
$487$	−38.0794	−1.72554	−0.862771	−	0.505595i	$-0.831273\pi$
$487$	−38.0794	−1.72554	−0.862771	+	0.505595i	$0.831273\pi$
$488$	−11.9519	−0.541039
$489$	1.53509	0.0694190
$490$	−3.12878	−0.141344
$491$	27.7528	1.25247	0.626234	−	0.779635i	$-0.284596\pi$
$491$	27.7528	1.25247	0.626234	+	0.779635i	$0.284596\pi$
$492$	−11.1282	−0.501697
$493$	−3.20816	−0.144488
$494$	0	0
$495$	1.09598	0.0492606
$496$	−1.63814	−0.0735545
$497$	−0.408612	−0.0183288
$498$	−13.5989	−0.609380
$499$	19.1496	0.857254	0.428627	−	0.903482i	$-0.358997\pi$
$499$	19.1496	0.857254	0.428627	+	0.903482i	$0.358997\pi$
$500$	4.45780	0.199359
$501$	−14.7236	−0.657802
$502$	−13.2240	−0.590215
$503$	29.8016	1.32879	0.664393	−	0.747383i	$-0.268690\pi$
$503$	29.8016	1.32879	0.664393	+	0.747383i	$0.268690\pi$
$504$	0.704346	0.0313741
$505$	−8.08413	−0.359739
$506$	−10.6122	−0.471772
$507$	0	0
$508$	−0.628624	−0.0278907
$509$	−3.39359	−0.150418	−0.0752091	−	0.997168i	$-0.523962\pi$
$509$	−3.39359	−0.150418	−0.0752091	+	0.997168i	$0.523962\pi$
$510$	0.475216	0.0210429
$511$	−1.02550	−0.0453656
$512$	−1.00000	−0.0441942
$513$	−5.03216	−0.222175
$514$	16.8982	0.745349
$515$	−0.727580	−0.0320610
$516$	3.80517	0.167513
$517$	−12.9037	−0.567505
$518$	1.31703	0.0578671
$519$	24.0274	1.05468
$520$	0	0
$521$	27.9107	1.22279	0.611396	−	0.791325i	$-0.290608\pi$
$521$	27.9107	1.22279	0.611396	+	0.791325i	$0.290608\pi$
$522$	−6.14515	−0.268966
$523$	−16.2195	−0.709230	−0.354615	−	0.935012i	$-0.615388\pi$
$523$	−16.2195	−0.709230	−0.354615	+	0.935012i	$0.615388\pi$
$524$	−12.2759	−0.536277
$525$	−1.72498	−0.0752842
$526$	20.5694	0.896868
$527$	−1.69178	−0.0736953
$528$	1.23026	0.0535403
$529$	53.0295	2.30563
$530$	5.97928	0.259723
$531$	23.7739	1.03170
$532$	0.356053	0.0154369
$533$	0	0
$534$	14.6258	0.632920
$535$	4.74846	0.205294
$536$	2.69986	0.116616
$537$	−24.2878	−1.04810
$538$	14.3269	0.617678
$539$	−8.36521	−0.360315
$540$	2.29070	0.0985761
$541$	−29.3698	−1.26271	−0.631353	−	0.775495i	$-0.717500\pi$
$541$	−29.3698	−1.26271	−0.631353	+	0.775495i	$0.717500\pi$
$542$	16.5580	0.711228
$543$	−17.2859	−0.741809
$544$	−1.03275	−0.0442788
$545$	−7.00836	−0.300205
$546$	0	0
$547$	−32.3204	−1.38192	−0.690961	−	0.722892i	$-0.742812\pi$
$547$	−32.3204	−1.38192	−0.690961	+	0.722892i	$0.742812\pi$
$548$	2.02764	0.0866164
$549$	−23.6434	−1.00908
$550$	5.83316	0.248727
$551$	−3.10642	−0.132338
$552$	−8.81399	−0.375148
$553$	−1.56819	−0.0666860
$554$	−5.16478	−0.219431
$555$	1.70207	0.0722489
$556$	−5.90936	−0.250612
$557$	−15.2330	−0.645442	−0.322721	−	0.946494i	$-0.604597\pi$
$557$	−15.2330	−0.645442	−0.322721	+	0.946494i	$0.604597\pi$
$558$	−3.24057	−0.137184
$559$	0	0
$560$	−0.162080	−0.00684912
$561$	1.27055	0.0536428
$562$	14.3906	0.607032
$563$	−7.42136	−0.312773	−0.156387	−	0.987696i	$-0.549985\pi$
$563$	−7.42136	−0.312773	−0.156387	+	0.987696i	$0.549985\pi$
$564$	−10.7172	−0.451274
$565$	0.984801	0.0414309
$566$	15.3559	0.645457
$567$	0.301905	0.0126788
$568$	1.14762	0.0481529
$569$	−12.6785	−0.531511	−0.265755	−	0.964041i	$-0.585621\pi$
$569$	−12.6785	−0.531511	−0.265755	+	0.964041i	$0.585621\pi$
$570$	0.460146	0.0192734
$571$	−27.8889	−1.16711	−0.583557	−	0.812072i	$-0.698340\pi$
$571$	−27.8889	−1.16711	−0.583557	+	0.812072i	$0.698340\pi$
$572$	0	0
$573$	2.25718	0.0942952
$574$	3.91974	0.163607
$575$	−41.7906	−1.74279
$576$	−1.97821	−0.0824253
$577$	−20.1429	−0.838561	−0.419280	−	0.907857i	$-0.637718\pi$
$577$	−20.1429	−0.838561	−0.419280	+	0.907857i	$0.637718\pi$
$578$	15.9334	0.662743
$579$	−5.48633	−0.228004
$580$	1.41408	0.0587166
$581$	4.79001	0.198723
$582$	−5.45742	−0.226217
$583$	15.9864	0.662089
$584$	2.88020	0.119183
$585$	0	0
$586$	−14.8401	−0.613039
$587$	48.0974	1.98519	0.992595	−	0.121470i	$-0.0387609\pi$
$587$	48.0974	1.98519	0.992595	+	0.121470i	$0.0387609\pi$
$588$	−6.94772	−0.286519
$589$	−1.63814	−0.0674982
$590$	−5.47070	−0.225225
$591$	−19.3214	−0.794776
$592$	−3.69898	−0.152027
$593$	12.5637	0.515930	0.257965	−	0.966154i	$-0.416948\pi$
$593$	12.5637	0.515930	0.257965	+	0.966154i	$0.416948\pi$
$594$	6.12450	0.251291
$595$	−0.167388	−0.00686223
$596$	−0.329395	−0.0134926
$597$	−6.57574	−0.269127
$598$	0	0
$599$	32.6000	1.33200	0.666000	−	0.745952i	$-0.268005\pi$
$599$	32.6000	1.33200	0.666000	+	0.745952i	$0.268005\pi$
$600$	4.84473	0.197785
$601$	−3.91344	−0.159633	−0.0798163	−	0.996810i	$-0.525433\pi$
$601$	−3.91344	−0.159633	−0.0798163	+	0.996810i	$0.525433\pi$
$602$	−1.34032	−0.0546273
$603$	5.34087	0.217497
$604$	−12.8042	−0.520994
$605$	4.33305	0.176163
$606$	−17.9515	−0.729230
$607$	1.27107	0.0515911	0.0257955	−	0.999667i	$-0.491788\pi$
$607$	1.27107	0.0515911	0.0257955	+	0.999667i	$0.491788\pi$
$608$	−1.00000	−0.0405554
$609$	−1.11804	−0.0453052
$610$	5.44067	0.220286
$611$	0	0
$612$	−2.04299	−0.0825830
$613$	−18.9291	−0.764539	−0.382269	−	0.924051i	$-0.624857\pi$
$613$	−18.9291	−0.764539	−0.382269	+	0.924051i	$0.624857\pi$
$614$	−11.2907	−0.455657
$615$	5.06568	0.204268
$616$	−0.433342	−0.0174598
$617$	30.4713	1.22673	0.613364	−	0.789801i	$-0.289816\pi$
$617$	30.4713	1.22673	0.613364	+	0.789801i	$0.289816\pi$
$618$	−1.61565	−0.0649911
$619$	18.6147	0.748187	0.374093	−	0.927391i	$-0.377954\pi$
$619$	18.6147	0.748187	0.374093	+	0.927391i	$0.377954\pi$
$620$	0.745700	0.0299480
$621$	−43.8779	−1.76076
$622$	13.7991	0.553292
$623$	−5.15172	−0.206399
$624$	0	0
$625$	21.9347	0.877387
$626$	20.9614	0.837785
$627$	1.23026	0.0491320
$628$	−3.98723	−0.159108
$629$	−3.82012	−0.152318
$630$	−0.320627	−0.0127741
$631$	−40.5022	−1.61237	−0.806184	−	0.591665i	$-0.798471\pi$
$631$	−40.5022	−1.61237	−0.806184	+	0.591665i	$0.798471\pi$
$632$	4.40436	0.175196
$633$	2.65422	0.105496
$634$	−11.2576	−0.447095
$635$	0.286158	0.0113558
$636$	13.2775	0.526487
$637$	0	0
$638$	3.78074	0.149681
$639$	2.27022	0.0898086
$640$	0.455213	0.0179939
$641$	27.6768	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$641$	27.6768	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$642$	10.5444	0.416153
$643$	−3.99847	−0.157684	−0.0788422	−	0.996887i	$-0.525122\pi$
$643$	−3.99847	−0.157684	−0.0788422	+	0.996887i	$0.525122\pi$
$644$	3.10460	0.122338
$645$	−1.73216	−0.0682039
$646$	−1.03275	−0.0406330
$647$	30.3738	1.19412	0.597058	−	0.802198i	$-0.296336\pi$
$647$	30.3738	1.19412	0.597058	+	0.802198i	$0.296336\pi$
$648$	−0.847922	−0.0333095
$649$	−14.6267	−0.574146
$650$	0	0
$651$	−0.589585	−0.0231076
$652$	1.51863	0.0594741
$653$	−46.4149	−1.81636	−0.908179	−	0.418583i	$-0.862527\pi$
$653$	−46.4149	−1.81636	−0.908179	+	0.418583i	$0.862527\pi$
$654$	−15.5627	−0.608548
$655$	5.58816	0.218348
$656$	−11.0089	−0.429824
$657$	5.69762	0.222285
$658$	3.77497	0.147163
$659$	−29.1809	−1.13673	−0.568364	−	0.822777i	$-0.692423\pi$
$659$	−29.1809	−1.13673	−0.568364	+	0.822777i	$0.692423\pi$
$660$	−0.560031	−0.0217992
$661$	47.9597	1.86542	0.932708	−	0.360632i	$-0.117439\pi$
$661$	47.9597	1.86542	0.932708	+	0.360632i	$0.117439\pi$
$662$	23.9433	0.930584
$663$	0	0
$664$	−13.4531	−0.522081
$665$	−0.162080	−0.00628518
$666$	−7.31734	−0.283541
$667$	−27.0864	−1.04879
$668$	−14.5657	−0.563566
$669$	−16.1223	−0.623326
$670$	−1.22901	−0.0474807
$671$	14.5464	0.561556
$672$	−0.359912	−0.0138839
$673$	22.0030	0.848153	0.424076	−	0.905626i	$-0.360599\pi$
$673$	22.0030	0.848153	0.424076	+	0.905626i	$0.360599\pi$
$674$	26.9760	1.03908
$675$	24.1180	0.928304
$676$	0	0
$677$	1.63871	0.0629808	0.0314904	−	0.999504i	$-0.489975\pi$
$677$	1.63871	0.0629808	0.0314904	+	0.999504i	$0.489975\pi$
$678$	2.18683	0.0839848
$679$	1.92230	0.0737710
$680$	0.470120	0.0180283
$681$	12.4616	0.477530
$682$	1.99373	0.0763439
$683$	−11.5324	−0.441275	−0.220638	−	0.975356i	$-0.570814\pi$
$683$	−11.5324	−0.441275	−0.220638	+	0.975356i	$0.570814\pi$
$684$	−1.97821	−0.0756386
$685$	−0.923006	−0.0352662
$686$	4.93960	0.188595
$687$	−10.0411	−0.383091
$688$	3.76438	0.143516
$689$	0	0
$690$	4.01224	0.152743
$691$	−4.19182	−0.159464	−0.0797322	−	0.996816i	$-0.525407\pi$
$691$	−4.19182	−0.159464	−0.0797322	+	0.996816i	$0.525407\pi$
$692$	23.7698	0.903591
$693$	−0.857240	−0.0325639
$694$	−28.2555	−1.07256
$695$	2.69001	0.102038
$696$	3.14009	0.119025
$697$	−11.3694	−0.430646
$698$	5.63737	0.213378
$699$	3.94856	0.149348
$700$	−1.70648	−0.0644990
$701$	14.9078	0.563061	0.281531	−	0.959552i	$-0.409158\pi$
$701$	14.9078	0.563061	0.281531	+	0.959552i	$0.409158\pi$
$702$	0	0
$703$	−3.69898	−0.139510
$704$	1.21707	0.0458701
$705$	4.87859	0.183738
$706$	−35.4087	−1.33262
$707$	6.32315	0.237807
$708$	−12.1482	−0.456556
$709$	−24.9595	−0.937373	−0.468687	−	0.883365i	$-0.655273\pi$
$709$	−24.9595	−0.937373	−0.468687	+	0.883365i	$0.655273\pi$
$710$	−0.522409	−0.0196057
$711$	8.71274	0.326753
$712$	14.4690	0.542248
$713$	−14.2837	−0.534929
$714$	−0.371699	−0.0139105
$715$	0	0
$716$	−24.0274	−0.897945
$717$	20.7606	0.775317
$718$	10.7395	0.400795
$719$	−9.68202	−0.361078	−0.180539	−	0.983568i	$-0.557784\pi$
$719$	−9.68202	−0.361078	−0.180539	+	0.983568i	$0.557784\pi$
$720$	0.900504	0.0335598
$721$	0.569090	0.0211940
$722$	−1.00000	−0.0372161
$723$	−1.43500	−0.0533681
$724$	−17.1006	−0.635537
$725$	14.8884	0.552942
$726$	9.62190	0.357102
$727$	17.8810	0.663168	0.331584	−	0.943426i	$-0.392417\pi$
$727$	17.8810	0.663168	0.331584	+	0.943426i	$0.392417\pi$
$728$	0	0
$729$	13.5827	0.503064
$730$	−1.31110	−0.0485260
$731$	3.88766	0.143790
$732$	12.0815	0.446544
$733$	−44.1328	−1.63008	−0.815041	−	0.579403i	$-0.803286\pi$
$733$	−44.1328	−1.63008	−0.815041	+	0.579403i	$0.803286\pi$
$734$	24.4885	0.903887
$735$	3.16269	0.116658
$736$	−8.71949	−0.321405
$737$	−3.28592	−0.121038
$738$	−21.7778	−0.801652
$739$	−40.5905	−1.49315	−0.746573	−	0.665303i	$-0.768302\pi$
$739$	−40.5905	−1.49315	−0.746573	+	0.665303i	$0.768302\pi$
$740$	1.68382	0.0618985
$741$	0	0
$742$	−4.67680	−0.171691
$743$	5.23568	0.192078	0.0960392	−	0.995378i	$-0.469383\pi$
$743$	5.23568	0.192078	0.0960392	+	0.995378i	$0.469383\pi$
$744$	1.65589	0.0607079
$745$	0.149945	0.00549356
$746$	15.8314	0.579630
$747$	−26.6130	−0.973717
$748$	1.25693	0.0459579
$749$	−3.71410	−0.135710
$750$	−4.50611	−0.164540
$751$	25.6642	0.936499	0.468249	−	0.883596i	$-0.344885\pi$
$751$	25.6642	0.936499	0.468249	+	0.883596i	$0.344885\pi$
$752$	−10.6023	−0.386625
$753$	13.3673	0.487132
$754$	0	0
$755$	5.82861	0.212125
$756$	−1.79172	−0.0651641
$757$	40.9853	1.48963	0.744817	−	0.667268i	$-0.232537\pi$
$757$	40.9853	1.48963	0.744817	+	0.667268i	$0.232537\pi$
$758$	7.94916	0.288727
$759$	10.7273	0.389375
$760$	0.455213	0.0165123
$761$	38.3926	1.39173	0.695865	−	0.718173i	$-0.255021\pi$
$761$	38.3926	1.39173	0.695865	+	0.718173i	$0.255021\pi$
$762$	0.635437	0.0230195
$763$	5.48172	0.198451
$764$	2.23298	0.0807864
$765$	0.929995	0.0336241
$766$	7.88922	0.285049
$767$	0	0
$768$	1.01084	0.0364755
$769$	−12.7679	−0.460421	−0.230210	−	0.973141i	$-0.573941\pi$
$769$	−12.7679	−0.460421	−0.230210	+	0.973141i	$0.573941\pi$
$770$	0.197263	0.00710886
$771$	−17.0814	−0.615171
$772$	−5.42750	−0.195340
$773$	16.4578	0.591945	0.295972	−	0.955197i	$-0.404356\pi$
$773$	16.4578	0.591945	0.295972	+	0.955197i	$0.404356\pi$
$774$	7.44672	0.267667
$775$	7.85123	0.282024
$776$	−5.39891	−0.193809
$777$	−1.33131	−0.0477603
$778$	28.5085	1.02208
$779$	−11.0089	−0.394433
$780$	0	0
$781$	−1.39673	−0.0499790
$782$	−9.00505	−0.322020
$783$	15.6320	0.558643
$784$	−6.87323	−0.245472
$785$	1.81504	0.0647815
$786$	12.4090	0.442614
$787$	16.8153	0.599401	0.299700	−	0.954033i	$-0.403113\pi$
$787$	16.8153	0.599401	0.299700	+	0.954033i	$0.403113\pi$
$788$	−19.1142	−0.680916
$789$	−20.7923	−0.740226
$790$	−2.00492	−0.0713319
$791$	−0.770280	−0.0273880
$792$	2.40762	0.0855511
$793$	0	0
$794$	14.2501	0.505716
$795$	−6.04408	−0.214361
$796$	−6.50523	−0.230572
$797$	1.03972	0.0368287	0.0184143	−	0.999830i	$-0.494138\pi$
$797$	1.03972	0.0368287	0.0184143	+	0.999830i	$0.494138\pi$
$798$	−0.359912	−0.0127407
$799$	−10.9495	−0.387365
$800$	4.79278	0.169450
$801$	28.6226	1.01133
$802$	7.63991	0.269775
$803$	−3.50541	−0.123703
$804$	−2.72912	−0.0962485
$805$	−1.41325	−0.0498106
$806$	0	0
$807$	−14.4822	−0.509798
$808$	−17.7590	−0.624760
$809$	−30.5997	−1.07583	−0.537914	−	0.842999i	$-0.680788\pi$
$809$	−30.5997	−1.07583	−0.537914	+	0.842999i	$0.680788\pi$
$810$	0.385985	0.0135621
$811$	−19.7819	−0.694636	−0.347318	−	0.937747i	$-0.612907\pi$
$811$	−19.7819	−0.694636	−0.347318	+	0.937747i	$0.612907\pi$
$812$	−1.10605	−0.0388148
$813$	−16.7375	−0.587009
$814$	4.50192	0.157792
$815$	−0.691299	−0.0242151
$816$	1.04394	0.0365453
$817$	3.76438	0.131699
$818$	10.5196	0.367808
$819$	0	0
$820$	5.01137	0.175005
$821$	−5.90990	−0.206257	−0.103129	−	0.994668i	$-0.532885\pi$
$821$	−5.90990	−0.206257	−0.103129	+	0.994668i	$0.532885\pi$
$822$	−2.04961	−0.0714884
$823$	−31.2302	−1.08862	−0.544308	−	0.838885i	$-0.683208\pi$
$823$	−31.2302	−1.08862	−0.544308	+	0.838885i	$0.683208\pi$
$824$	−1.59833	−0.0556805
$825$	−5.89638	−0.205286
$826$	4.27901	0.148886
$827$	−35.9202	−1.24907	−0.624534	−	0.780998i	$-0.714711\pi$
$827$	−35.9202	−1.24907	−0.624534	+	0.780998i	$0.714711\pi$
$828$	−17.2490	−0.599443
$829$	−18.3647	−0.637833	−0.318916	−	0.947783i	$-0.603319\pi$
$829$	−18.3647	−0.637833	−0.318916	+	0.947783i	$0.603319\pi$
$830$	6.12401	0.212567
$831$	5.22076	0.181106
$832$	0	0
$833$	−7.09832	−0.245942
$834$	5.97340	0.206842
$835$	6.63051	0.229458
$836$	1.21707	0.0420933
$837$	8.24336	0.284932
$838$	−23.7711	−0.821158
$839$	52.9320	1.82741	0.913707	−	0.406373i	$-0.133207\pi$
$839$	52.9320	1.82741	0.913707	+	0.406373i	$0.133207\pi$
$840$	0.163836	0.00565289
$841$	−19.3501	−0.667246
$842$	−7.09093	−0.244370
$843$	−14.5466	−0.501012
$844$	2.62576	0.0903823
$845$	0	0
$846$	−20.9735	−0.721082
$847$	−3.38917	−0.116453
$848$	13.1351	0.451062
$849$	−15.5223	−0.532725
$850$	4.94974	0.169775
$851$	−32.2532	−1.10563
$852$	−1.16005	−0.0397428
$853$	28.7683	0.985008	0.492504	−	0.870310i	$-0.336082\pi$
$853$	28.7683	0.985008	0.492504	+	0.870310i	$0.336082\pi$
$854$	−4.25552	−0.145621
$855$	0.900504	0.0307966
$856$	10.4313	0.356535
$857$	0.568499	0.0194196	0.00970978	−	0.999953i	$-0.496909\pi$
$857$	0.568499	0.0194196	0.00970978	+	0.999953i	$0.496909\pi$
$858$	0	0
$859$	−9.80150	−0.334423	−0.167211	−	0.985921i	$-0.553476\pi$
$859$	−9.80150	−0.334423	−0.167211	+	0.985921i	$0.553476\pi$
$860$	−1.71359	−0.0584330
$861$	−3.96222	−0.135032
$862$	−19.6994	−0.670964
$863$	−53.8305	−1.83241	−0.916205	−	0.400710i	$-0.868764\pi$
$863$	−53.8305	−1.83241	−0.916205	+	0.400710i	$0.868764\pi$
$864$	5.03216	0.171198
$865$	−10.8203	−0.367901
$866$	−17.5617	−0.596769
$867$	−16.1061	−0.546992
$868$	−0.583263	−0.0197972
$869$	−5.36042	−0.181840
$870$	−1.42941	−0.0484615
$871$	0	0
$872$	−15.3958	−0.521367
$873$	−10.6802	−0.361468
$874$	−8.71949	−0.294941
$875$	1.58721	0.0536575
$876$	−2.91141	−0.0983675
$877$	38.0794	1.28585	0.642926	−	0.765928i	$-0.277720\pi$
$877$	38.0794	1.28585	0.642926	+	0.765928i	$0.277720\pi$
$878$	−28.1401	−0.949682
$879$	15.0009	0.505969
$880$	−0.554026	−0.0186762
$881$	53.7104	1.80955	0.904775	−	0.425890i	$-0.140039\pi$
$881$	53.7104	1.80955	0.904775	+	0.425890i	$0.140039\pi$
$882$	−13.5967	−0.457823
$883$	−30.9959	−1.04310	−0.521548	−	0.853222i	$-0.674645\pi$
$883$	−30.9959	−1.04310	−0.521548	+	0.853222i	$0.674645\pi$
$884$	0	0
$885$	5.52999	0.185889
$886$	11.6857	0.392587
$887$	−10.8899	−0.365648	−0.182824	−	0.983146i	$-0.558524\pi$
$887$	−10.8899	−0.365648	−0.182824	+	0.983146i	$0.558524\pi$
$888$	3.73907	0.125475
$889$	−0.223824	−0.00750680
$890$	−6.58646	−0.220779
$891$	1.03198	0.0345727
$892$	−15.9495	−0.534028
$893$	−10.6023	−0.354791
$894$	0.332965	0.0111360
$895$	10.9376	0.365602
$896$	−0.356053	−0.0118949
$897$	0	0
$898$	−21.2368	−0.708680
$899$	5.08875	0.169719
$900$	9.48111	0.316037
$901$	13.5653	0.451926
$902$	13.3986	0.446124
$903$	1.35484	0.0450864
$904$	2.16339	0.0719532
$905$	7.78439	0.258762
$906$	12.9429	0.430000
$907$	17.2997	0.574428	0.287214	−	0.957866i	$-0.407271\pi$
$907$	17.2997	0.574428	0.287214	+	0.957866i	$0.407271\pi$
$908$	12.3280	0.409120
$909$	−35.1310	−1.16522
$910$	0	0
$911$	−47.7385	−1.58165	−0.790824	−	0.612043i	$-0.790348\pi$
$911$	−47.7385	−1.58165	−0.790824	+	0.612043i	$0.790348\pi$
$912$	1.01084	0.0334722
$913$	16.3734	0.541879
$914$	18.8960	0.625023
$915$	−5.49964	−0.181812
$916$	−9.93342	−0.328209
$917$	−4.37089	−0.144339
$918$	5.19696	0.171525
$919$	28.5643	0.942249	0.471125	−	0.882067i	$-0.343848\pi$
$919$	28.5643	0.942249	0.471125	+	0.882067i	$0.343848\pi$
$920$	3.96922	0.130861
$921$	11.4131	0.376075
$922$	−1.39732	−0.0460183
$923$	0	0
$924$	0.438039	0.0144104
$925$	17.7284	0.582906
$926$	−16.6035	−0.545625
$927$	−3.16183	−0.103848
$928$	3.10642	0.101973
$929$	−16.5850	−0.544135	−0.272067	−	0.962278i	$-0.587707\pi$
$929$	−16.5850	−0.544135	−0.272067	+	0.962278i	$0.587707\pi$
$930$	−0.753782	−0.0247175
$931$	−6.87323	−0.225261
$932$	3.90623	0.127953
$933$	−13.9486	−0.456657
$934$	−8.81724	−0.288509
$935$	−0.572170	−0.0187120
$936$	0	0
$937$	−27.8385	−0.909445	−0.454723	−	0.890633i	$-0.650262\pi$
$937$	−27.8385	−0.909445	−0.454723	+	0.890633i	$0.650262\pi$
$938$	0.961291	0.0313873
$939$	−21.1886	−0.691462
$940$	4.82628	0.157416
$941$	15.5122	0.505684	0.252842	−	0.967508i	$-0.418635\pi$
$941$	15.5122	0.505684	0.252842	+	0.967508i	$0.418635\pi$
$942$	4.03045	0.131319
$943$	−95.9916	−3.12592
$944$	−12.0179	−0.391149
$945$	0.815611	0.0265318
$946$	−4.58152	−0.148958
$947$	48.3592	1.57146	0.785731	−	0.618568i	$-0.212287\pi$
$947$	48.3592	1.57146	0.785731	+	0.618568i	$0.212287\pi$
$948$	−4.45210	−0.144597
$949$	0	0
$950$	4.79278	0.155498
$951$	11.3796	0.369008
$952$	−0.367713	−0.0119177
$953$	−49.3073	−1.59722	−0.798610	−	0.601848i	$-0.794431\pi$
$953$	−49.3073	−1.59722	−0.798610	+	0.601848i	$0.794431\pi$
$954$	25.9840	0.841264
$955$	−1.01648	−0.0328926
$956$	20.5380	0.664246
$957$	−3.82172	−0.123539
$958$	30.2392	0.976985
$959$	0.721946	0.0233129
$960$	−0.460146	−0.0148512
$961$	−28.3165	−0.913436
$962$	0	0
$963$	20.6353	0.664962
$964$	−1.41961	−0.0457226
$965$	2.47067	0.0795336
$966$	−3.13825	−0.100971
$967$	31.6982	1.01935	0.509673	−	0.860368i	$-0.329766\pi$
$967$	31.6982	1.01935	0.509673	+	0.860368i	$0.329766\pi$
$968$	9.51874	0.305944
$969$	1.04394	0.0335363
$970$	2.45765	0.0789104
$971$	37.3445	1.19844	0.599221	−	0.800584i	$-0.295477\pi$
$971$	37.3445	1.19844	0.599221	+	0.800584i	$0.295477\pi$
$972$	15.9536	0.511712
$973$	−2.10404	−0.0674525
$974$	38.0794	1.22014
$975$	0	0
$976$	11.9519	0.382572
$977$	−20.7741	−0.664624	−0.332312	−	0.943170i	$-0.607829\pi$
$977$	−20.7741	−0.664624	−0.332312	+	0.943170i	$0.607829\pi$
$978$	−1.53509	−0.0490867
$979$	−17.6098	−0.562811
$980$	3.12878	0.0999452
$981$	−30.4561	−0.972387
$982$	−27.7528	−0.885628
$983$	−24.8423	−0.792346	−0.396173	−	0.918176i	$-0.629662\pi$
$983$	−24.8423	−0.792346	−0.396173	+	0.918176i	$0.629662\pi$
$984$	11.1282	0.354753
$985$	8.70104	0.277238
$986$	3.20816	0.102169
$987$	−3.81588	−0.121461
$988$	0	0
$989$	32.8235	1.04373
$990$	−1.09598	−0.0348325
$991$	−34.3374	−1.09076	−0.545382	−	0.838188i	$-0.683615\pi$
$991$	−34.3374	−1.09076	−0.545382	+	0.838188i	$0.683615\pi$
$992$	1.63814	0.0520109
$993$	−24.2028	−0.768053
$994$	0.408612	0.0129604
$995$	2.96126	0.0938784
$996$	13.5989	0.430897
$997$	−50.2059	−1.59004	−0.795018	−	0.606586i	$-0.792539\pi$
$997$	−50.2059	−1.59004	−0.795018	+	0.606586i	$0.792539\pi$
$998$	−19.1496	−0.606170
$999$	18.6138	0.588916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bo.1.10	✓	15
13.12	even	2		6422.2.a.bq.1.10	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.10	✓	15	1.1	even	1	trivial
6422.2.a.bq.1.10	yes	15	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.01084 q^{3} +1.00000 q^{4} -0.455213 q^{5} -1.01084 q^{6} +0.356053 q^{7} -1.00000 q^{8} -1.97821 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.01084 q^{3} +1.00000 q^{4} -0.455213 q^{5} -1.01084 q^{6} +0.356053 q^{7} -1.00000 q^{8} -1.97821 q^{9} +0.455213 q^{10} +1.21707 q^{11} +1.01084 q^{12} -0.356053 q^{14} -0.460146 q^{15} +1.00000 q^{16} +1.03275 q^{17} +1.97821 q^{18} +1.00000 q^{19} -0.455213 q^{20} +0.359912 q^{21} -1.21707 q^{22} +8.71949 q^{23} -1.01084 q^{24} -4.79278 q^{25} -5.03216 q^{27} +0.356053 q^{28} -3.10642 q^{29} +0.460146 q^{30} -1.63814 q^{31} -1.00000 q^{32} +1.23026 q^{33} -1.03275 q^{34} -0.162080 q^{35} -1.97821 q^{36} -3.69898 q^{37} -1.00000 q^{38} +0.455213 q^{40} -11.0089 q^{41} -0.359912 q^{42} +3.76438 q^{43} +1.21707 q^{44} +0.900504 q^{45} -8.71949 q^{46} -10.6023 q^{47} +1.01084 q^{48} -6.87323 q^{49} +4.79278 q^{50} +1.04394 q^{51} +13.1351 q^{53} +5.03216 q^{54} -0.554026 q^{55} -0.356053 q^{56} +1.01084 q^{57} +3.10642 q^{58} -12.0179 q^{59} -0.460146 q^{60} +11.9519 q^{61} +1.63814 q^{62} -0.704346 q^{63} +1.00000 q^{64} -1.23026 q^{66} -2.69986 q^{67} +1.03275 q^{68} +8.81399 q^{69} +0.162080 q^{70} -1.14762 q^{71} +1.97821 q^{72} -2.88020 q^{73} +3.69898 q^{74} -4.84473 q^{75} +1.00000 q^{76} +0.433342 q^{77} -4.40436 q^{79} -0.455213 q^{80} +0.847922 q^{81} +11.0089 q^{82} +13.4531 q^{83} +0.359912 q^{84} -0.470120 q^{85} -3.76438 q^{86} -3.14009 q^{87} -1.21707 q^{88} -14.4690 q^{89} -0.900504 q^{90} +8.71949 q^{92} -1.65589 q^{93} +10.6023 q^{94} -0.455213 q^{95} -1.01084 q^{96} +5.39891 q^{97} +6.87323 q^{98} -2.40762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more