LMFDB - Embedded newform 6422.2.a.bo.1.13

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			1.13
Root			$2.61859$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +2.61859 q^{3} +1.00000 q^{4} -3.36169 q^{5} -2.61859 q^{6} -2.21822 q^{7} -1.00000 q^{8} +3.85700 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.61859 q^{3} +1.00000 q^{4} -3.36169 q^{5} -2.61859 q^{6} -2.21822 q^{7} -1.00000 q^{8} +3.85700 q^{9} +3.36169 q^{10} +1.60410 q^{11} +2.61859 q^{12} +2.21822 q^{14} -8.80287 q^{15} +1.00000 q^{16} -0.647275 q^{17} -3.85700 q^{18} +1.00000 q^{19} -3.36169 q^{20} -5.80860 q^{21} -1.60410 q^{22} +6.07154 q^{23} -2.61859 q^{24} +6.30094 q^{25} +2.24414 q^{27} -2.21822 q^{28} -3.73238 q^{29} +8.80287 q^{30} -7.57907 q^{31} -1.00000 q^{32} +4.20047 q^{33} +0.647275 q^{34} +7.45695 q^{35} +3.85700 q^{36} +9.85900 q^{37} -1.00000 q^{38} +3.36169 q^{40} +1.46711 q^{41} +5.80860 q^{42} -3.17837 q^{43} +1.60410 q^{44} -12.9660 q^{45} -6.07154 q^{46} +2.45793 q^{47} +2.61859 q^{48} -2.07951 q^{49} -6.30094 q^{50} -1.69495 q^{51} -8.54484 q^{53} -2.24414 q^{54} -5.39247 q^{55} +2.21822 q^{56} +2.61859 q^{57} +3.73238 q^{58} +11.1316 q^{59} -8.80287 q^{60} +2.85301 q^{61} +7.57907 q^{62} -8.55567 q^{63} +1.00000 q^{64} -4.20047 q^{66} +9.50862 q^{67} -0.647275 q^{68} +15.8988 q^{69} -7.45695 q^{70} -7.26865 q^{71} -3.85700 q^{72} -0.556551 q^{73} -9.85900 q^{74} +16.4996 q^{75} +1.00000 q^{76} -3.55823 q^{77} -15.2397 q^{79} -3.36169 q^{80} -5.69454 q^{81} -1.46711 q^{82} -1.35072 q^{83} -5.80860 q^{84} +2.17594 q^{85} +3.17837 q^{86} -9.77355 q^{87} -1.60410 q^{88} +2.33354 q^{89} +12.9660 q^{90} +6.07154 q^{92} -19.8464 q^{93} -2.45793 q^{94} -3.36169 q^{95} -2.61859 q^{96} -16.5432 q^{97} +2.07951 q^{98} +6.18700 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.61859	1.51184	0.755921	−	0.654663i	$-0.227189\pi$
$3$	2.61859	1.51184	0.755921	+	0.654663i	$0.227189\pi$
$4$	1.00000	0.500000
$5$	−3.36169	−1.50339	−0.751696	−	0.659510i	$-0.770764\pi$
$5$	−3.36169	−1.50339	−0.751696	+	0.659510i	$0.770764\pi$
$6$	−2.61859	−1.06903
$7$	−2.21822	−0.838407	−0.419204	−	0.907892i	$-0.637691\pi$
$7$	−2.21822	−0.838407	−0.419204	+	0.907892i	$0.637691\pi$
$8$	−1.00000	−0.353553
$9$	3.85700	1.28567
$10$	3.36169	1.06306
$11$	1.60410	0.483653	0.241827	−	0.970320i	$-0.422254\pi$
$11$	1.60410	0.483653	0.241827	+	0.970320i	$0.422254\pi$
$12$	2.61859	0.755921
$13$	0	0
$13$	0	0
$14$	2.21822	0.592843
$15$	−8.80287	−2.27289
$16$	1.00000	0.250000
$17$	−0.647275	−0.156987	−0.0784936	−	0.996915i	$-0.525011\pi$
$17$	−0.647275	−0.156987	−0.0784936	+	0.996915i	$0.525011\pi$
$18$	−3.85700	−0.909104
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.36169	−0.751696
$21$	−5.80860	−1.26754
$22$	−1.60410	−0.341994
$23$	6.07154	1.26600	0.633001	−	0.774151i	$-0.281823\pi$
$23$	6.07154	1.26600	0.633001	+	0.774151i	$0.281823\pi$
$24$	−2.61859	−0.534517
$25$	6.30094	1.26019
$26$	0	0
$27$	2.24414	0.431884
$28$	−2.21822	−0.419204
$29$	−3.73238	−0.693085	−0.346542	−	0.938034i	$-0.612644\pi$
$29$	−3.73238	−0.693085	−0.346542	+	0.938034i	$0.612644\pi$
$30$	8.80287	1.60718
$31$	−7.57907	−1.36124	−0.680620	−	0.732636i	$-0.738290\pi$
$31$	−7.57907	−1.36124	−0.680620	+	0.732636i	$0.738290\pi$
$32$	−1.00000	−0.176777
$33$	4.20047	0.731207
$34$	0.647275	0.111007
$35$	7.45695	1.26045
$36$	3.85700	0.642834
$37$	9.85900	1.62081	0.810405	−	0.585870i	$-0.199247\pi$
$37$	9.85900	1.62081	0.810405	+	0.585870i	$0.199247\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.36169	0.531529
$41$	1.46711	0.229123	0.114562	−	0.993416i	$-0.463454\pi$
$41$	1.46711	0.229123	0.114562	+	0.993416i	$0.463454\pi$
$42$	5.80860	0.896286
$43$	−3.17837	−0.484697	−0.242348	−	0.970189i	$-0.577918\pi$
$43$	−3.17837	−0.484697	−0.242348	+	0.970189i	$0.577918\pi$
$44$	1.60410	0.241827
$45$	−12.9660	−1.93286
$46$	−6.07154	−0.895199
$47$	2.45793	0.358525	0.179263	−	0.983801i	$-0.442629\pi$
$47$	2.45793	0.358525	0.179263	+	0.983801i	$0.442629\pi$
$48$	2.61859	0.377961
$49$	−2.07951	−0.297073
$50$	−6.30094	−0.891088
$51$	−1.69495	−0.237340
$52$	0	0
$53$	−8.54484	−1.17372	−0.586862	−	0.809687i	$-0.699637\pi$
$53$	−8.54484	−1.17372	−0.586862	+	0.809687i	$0.699637\pi$
$54$	−2.24414	−0.305388
$55$	−5.39247	−0.727120
$56$	2.21822	0.296422
$57$	2.61859	0.346840
$58$	3.73238	0.490085
$59$	11.1316	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$59$	11.1316	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$60$	−8.80287	−1.13645
$61$	2.85301	0.365291	0.182646	−	0.983179i	$-0.441534\pi$
$61$	2.85301	0.365291	0.182646	+	0.983179i	$0.441534\pi$
$62$	7.57907	0.962542
$63$	−8.55567	−1.07791
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.20047	−0.517042
$67$	9.50862	1.16166	0.580831	−	0.814024i	$-0.302728\pi$
$67$	9.50862	1.16166	0.580831	+	0.814024i	$0.302728\pi$
$68$	−0.647275	−0.0784936
$69$	15.8988	1.91400
$70$	−7.45695	−0.891276
$71$	−7.26865	−0.862630	−0.431315	−	0.902201i	$-0.641950\pi$
$71$	−7.26865	−0.862630	−0.431315	+	0.902201i	$0.641950\pi$
$72$	−3.85700	−0.454552
$73$	−0.556551	−0.0651393	−0.0325697	−	0.999469i	$-0.510369\pi$
$73$	−0.556551	−0.0651393	−0.0325697	+	0.999469i	$0.510369\pi$
$74$	−9.85900	−1.14609
$75$	16.4996	1.90521
$76$	1.00000	0.114708
$77$	−3.55823	−0.405498
$78$	0	0
$79$	−15.2397	−1.71460	−0.857299	−	0.514819i	$-0.827859\pi$
$79$	−15.2397	−1.71460	−0.857299	+	0.514819i	$0.827859\pi$
$80$	−3.36169	−0.375848
$81$	−5.69454	−0.632727
$82$	−1.46711	−0.162015
$83$	−1.35072	−0.148261	−0.0741305	−	0.997249i	$-0.523618\pi$
$83$	−1.35072	−0.148261	−0.0741305	+	0.997249i	$0.523618\pi$
$84$	−5.80860	−0.633770
$85$	2.17594	0.236013
$86$	3.17837	0.342732
$87$	−9.77355	−1.04784
$88$	−1.60410	−0.170997
$89$	2.33354	0.247354	0.123677	−	0.992323i	$-0.460531\pi$
$89$	2.33354	0.247354	0.123677	+	0.992323i	$0.460531\pi$
$90$	12.9660	1.36674
$91$	0	0
$92$	6.07154	0.633001
$93$	−19.8464	−2.05798
$94$	−2.45793	−0.253516
$95$	−3.36169	−0.344902
$96$	−2.61859	−0.267259
$97$	−16.5432	−1.67971	−0.839854	−	0.542812i	$-0.817359\pi$
$97$	−16.5432	−1.67971	−0.839854	+	0.542812i	$0.817359\pi$
$98$	2.07951	0.210063
$99$	6.18700	0.621817
$100$	6.30094	0.630094
$101$	5.16007	0.513446	0.256723	−	0.966485i	$-0.417357\pi$
$101$	5.16007	0.513446	0.256723	+	0.966485i	$0.417357\pi$
$102$	1.69495	0.167825
$103$	−10.1163	−0.996793	−0.498396	−	0.866949i	$-0.666078\pi$
$103$	−10.1163	−0.996793	−0.498396	+	0.866949i	$0.666078\pi$
$104$	0	0
$105$	19.5267	1.90561
$106$	8.54484	0.829948
$107$	−18.3478	−1.77375	−0.886873	−	0.462014i	$-0.847127\pi$
$107$	−18.3478	−1.77375	−0.886873	+	0.462014i	$0.847127\pi$
$108$	2.24414	0.215942
$109$	−15.2774	−1.46331	−0.731654	−	0.681676i	$-0.761251\pi$
$109$	−15.2774	−1.46331	−0.731654	+	0.681676i	$0.761251\pi$
$110$	5.39247	0.514152
$111$	25.8167	2.45041
$112$	−2.21822	−0.209602
$113$	4.32441	0.406807	0.203403	−	0.979095i	$-0.434800\pi$
$113$	4.32441	0.406807	0.203403	+	0.979095i	$0.434800\pi$
$114$	−2.61859	−0.245253
$115$	−20.4106	−1.90330
$116$	−3.73238	−0.346542
$117$	0	0
$118$	−11.1316	−1.02475
$119$	1.43580	0.131619
$120$	8.80287	0.803589
$121$	−8.42688	−0.766080
$122$	−2.85301	−0.258300
$123$	3.84174	0.346398
$124$	−7.57907	−0.680620
$125$	−4.37335	−0.391165
$126$	8.55567	0.762199
$127$	6.58824	0.584612	0.292306	−	0.956325i	$-0.405577\pi$
$127$	6.58824	0.584612	0.292306	+	0.956325i	$0.405577\pi$
$128$	−1.00000	−0.0883883
$129$	−8.32284	−0.732785
$130$	0	0
$131$	21.0794	1.84172	0.920858	−	0.389899i	$-0.127490\pi$
$131$	21.0794	1.84172	0.920858	+	0.389899i	$0.127490\pi$
$132$	4.20047	0.365604
$133$	−2.21822	−0.192344
$134$	−9.50862	−0.821420
$135$	−7.54408	−0.649291
$136$	0.647275	0.0555034
$137$	−18.2556	−1.55968	−0.779841	−	0.625977i	$-0.784700\pi$
$137$	−18.2556	−1.55968	−0.779841	+	0.625977i	$0.784700\pi$
$138$	−15.8988	−1.35340
$139$	1.23467	0.104724	0.0523619	−	0.998628i	$-0.483325\pi$
$139$	1.23467	0.104724	0.0523619	+	0.998628i	$0.483325\pi$
$140$	7.45695	0.630227
$141$	6.43629	0.542034
$142$	7.26865	0.609972
$143$	0	0
$144$	3.85700	0.321417
$145$	12.5471	1.04198
$146$	0.556551	0.0460605
$147$	−5.44539	−0.449128
$148$	9.85900	0.810405
$149$	1.78363	0.146121	0.0730604	−	0.997328i	$-0.476723\pi$
$149$	1.78363	0.146121	0.0730604	+	0.997328i	$0.476723\pi$
$150$	−16.4996	−1.34718
$151$	−13.9267	−1.13334	−0.566671	−	0.823944i	$-0.691769\pi$
$151$	−13.9267	−1.13334	−0.566671	+	0.823944i	$0.691769\pi$
$152$	−1.00000	−0.0811107
$153$	−2.49654	−0.201833
$154$	3.55823	0.286731
$155$	25.4784	2.04648
$156$	0	0
$157$	20.2798	1.61850	0.809252	−	0.587461i	$-0.199873\pi$
$157$	20.2798	1.61850	0.809252	+	0.587461i	$0.199873\pi$
$158$	15.2397	1.21240
$159$	−22.3754	−1.77448
$160$	3.36169	0.265765
$161$	−13.4680	−1.06143
$162$	5.69454	0.447405
$163$	18.2909	1.43266	0.716328	−	0.697763i	$-0.245821\pi$
$163$	18.2909	1.43266	0.716328	+	0.697763i	$0.245821\pi$
$164$	1.46711	0.114562
$165$	−14.1207	−1.09929
$166$	1.35072	0.104836
$167$	−4.93553	−0.381923	−0.190961	−	0.981598i	$-0.561160\pi$
$167$	−4.93553	−0.381923	−0.190961	+	0.981598i	$0.561160\pi$
$168$	5.80860	0.448143
$169$	0	0
$170$	−2.17594	−0.166887
$171$	3.85700	0.294952
$172$	−3.17837	−0.242348
$173$	3.42041	0.260049	0.130024	−	0.991511i	$-0.458494\pi$
$173$	3.42041	0.260049	0.130024	+	0.991511i	$0.458494\pi$
$174$	9.77355	0.740931
$175$	−13.9769	−1.05655
$176$	1.60410	0.120913
$177$	29.1491	2.19098
$178$	−2.33354	−0.174906
$179$	−20.6266	−1.54171	−0.770853	−	0.637013i	$-0.780170\pi$
$179$	−20.6266	−1.54171	−0.770853	+	0.637013i	$0.780170\pi$
$180$	−12.9660	−0.966431
$181$	2.89422	0.215125	0.107563	−	0.994198i	$-0.465695\pi$
$181$	2.89422	0.215125	0.107563	+	0.994198i	$0.465695\pi$
$182$	0	0
$183$	7.47087	0.552263
$184$	−6.07154	−0.447600
$185$	−33.1429	−2.43671
$186$	19.8464	1.45521
$187$	−1.03829	−0.0759273
$188$	2.45793	0.179263
$189$	−4.97798	−0.362095
$190$	3.36169	0.243882
$191$	10.5320	0.762066	0.381033	−	0.924561i	$-0.375568\pi$
$191$	10.5320	0.762066	0.381033	+	0.924561i	$0.375568\pi$
$192$	2.61859	0.188980
$193$	−0.427246	−0.0307539	−0.0153769	−	0.999882i	$-0.504895\pi$
$193$	−0.427246	−0.0307539	−0.0153769	+	0.999882i	$0.504895\pi$
$194$	16.5432	1.18773
$195$	0	0
$196$	−2.07951	−0.148537
$197$	−1.10019	−0.0783856	−0.0391928	−	0.999232i	$-0.512479\pi$
$197$	−1.10019	−0.0783856	−0.0391928	+	0.999232i	$0.512479\pi$
$198$	−6.18700	−0.439691
$199$	−16.4906	−1.16899	−0.584495	−	0.811397i	$-0.698707\pi$
$199$	−16.4906	−1.16899	−0.584495	+	0.811397i	$0.698707\pi$
$200$	−6.30094	−0.445544
$201$	24.8992	1.75625
$202$	−5.16007	−0.363061
$203$	8.27922	0.581087
$204$	−1.69495	−0.118670
$205$	−4.93195	−0.344462
$206$	10.1163	0.704839
$207$	23.4179	1.62766
$208$	0	0
$209$	1.60410	0.110958
$210$	−19.5267	−1.34747
$211$	−15.6615	−1.07818	−0.539090	−	0.842248i	$-0.681232\pi$
$211$	−15.6615	−1.07818	−0.539090	+	0.842248i	$0.681232\pi$
$212$	−8.54484	−0.586862
$213$	−19.0336	−1.30416
$214$	18.3478	1.25423
$215$	10.6847	0.728689
$216$	−2.24414	−0.152694
$217$	16.8120	1.14127
$218$	15.2774	1.03472
$219$	−1.45738	−0.0984804
$220$	−5.39247	−0.363560
$221$	0	0
$222$	−25.8167	−1.73270
$223$	−26.2934	−1.76074	−0.880368	−	0.474292i	$-0.842704\pi$
$223$	−26.2934	−1.76074	−0.880368	+	0.474292i	$0.842704\pi$
$224$	2.21822	0.148211
$225$	24.3027	1.62018
$226$	−4.32441	−0.287656
$227$	8.84941	0.587356	0.293678	−	0.955904i	$-0.405121\pi$
$227$	8.84941	0.587356	0.293678	+	0.955904i	$0.405121\pi$
$228$	2.61859	0.173420
$229$	−9.57906	−0.633002	−0.316501	−	0.948592i	$-0.602508\pi$
$229$	−9.57906	−0.633002	−0.316501	+	0.948592i	$0.602508\pi$
$230$	20.4106	1.34584
$231$	−9.31754	−0.613049
$232$	3.73238	0.245042
$233$	10.1733	0.666472	0.333236	−	0.942843i	$-0.391859\pi$
$233$	10.1733	0.666472	0.333236	+	0.942843i	$0.391859\pi$
$234$	0	0
$235$	−8.26277	−0.539004
$236$	11.1316	0.724605
$237$	−39.9064	−2.59220
$238$	−1.43580	−0.0930688
$239$	−17.5454	−1.13492	−0.567459	−	0.823402i	$-0.692074\pi$
$239$	−17.5454	−1.13492	−0.567459	+	0.823402i	$0.692074\pi$
$240$	−8.80287	−0.568223
$241$	−28.2062	−1.81692	−0.908460	−	0.417972i	$-0.862741\pi$
$241$	−28.2062	−1.81692	−0.908460	+	0.417972i	$0.862741\pi$
$242$	8.42688	0.541700
$243$	−21.6441	−1.38847
$244$	2.85301	0.182646
$245$	6.99067	0.446618
$246$	−3.84174	−0.244941
$247$	0	0
$248$	7.57907	0.481271
$249$	−3.53699	−0.224147
$250$	4.37335	0.276595
$251$	12.2703	0.774493	0.387247	−	0.921976i	$-0.373426\pi$
$251$	12.2703	0.774493	0.387247	+	0.921976i	$0.373426\pi$
$252$	−8.55567	−0.538956
$253$	9.73932	0.612306
$254$	−6.58824	−0.413383
$255$	5.69788	0.356815
$256$	1.00000	0.0625000
$257$	−18.0070	−1.12325	−0.561623	−	0.827393i	$-0.689823\pi$
$257$	−18.0070	−1.12325	−0.561623	+	0.827393i	$0.689823\pi$
$258$	8.32284	0.518157
$259$	−21.8694	−1.35890
$260$	0	0
$261$	−14.3958	−0.891077
$262$	−21.0794	−1.30229
$263$	−11.9794	−0.738678	−0.369339	−	0.929295i	$-0.620416\pi$
$263$	−11.9794	−0.738678	−0.369339	+	0.929295i	$0.620416\pi$
$264$	−4.20047	−0.258521
$265$	28.7251	1.76457
$266$	2.21822	0.136008
$267$	6.11057	0.373961
$268$	9.50862	0.580831
$269$	−30.9659	−1.88802	−0.944012	−	0.329911i	$-0.892981\pi$
$269$	−30.9659	−1.88802	−0.944012	+	0.329911i	$0.892981\pi$
$270$	7.54408	0.459118
$271$	−22.9462	−1.39388	−0.696940	−	0.717129i	$-0.745456\pi$
$271$	−22.9462	−1.39388	−0.696940	+	0.717129i	$0.745456\pi$
$272$	−0.647275	−0.0392468
$273$	0	0
$274$	18.2556	1.10286
$275$	10.1073	0.609494
$276$	15.8988	0.956998
$277$	−29.1550	−1.75175	−0.875876	−	0.482536i	$-0.839716\pi$
$277$	−29.1550	−1.75175	−0.875876	+	0.482536i	$0.839716\pi$
$278$	−1.23467	−0.0740508
$279$	−29.2325	−1.75010
$280$	−7.45695	−0.445638
$281$	20.4344	1.21901	0.609507	−	0.792781i	$-0.291368\pi$
$281$	20.4344	1.21901	0.609507	+	0.792781i	$0.291368\pi$
$282$	−6.43629	−0.383276
$283$	−4.26150	−0.253320	−0.126660	−	0.991946i	$-0.540426\pi$
$283$	−4.26150	−0.253320	−0.126660	+	0.991946i	$0.540426\pi$
$284$	−7.26865	−0.431315
$285$	−8.80287	−0.521437
$286$	0	0
$287$	−3.25436	−0.192099
$288$	−3.85700	−0.227276
$289$	−16.5810	−0.975355
$290$	−12.5471	−0.736790
$291$	−43.3198	−2.53945
$292$	−0.556551	−0.0325697
$293$	−25.8746	−1.51161	−0.755806	−	0.654795i	$-0.772755\pi$
$293$	−25.8746	−1.51161	−0.755806	+	0.654795i	$0.772755\pi$
$294$	5.44539	0.317582
$295$	−37.4209	−2.17873
$296$	−9.85900	−0.573043
$297$	3.59981	0.208882
$298$	−1.78363	−0.103323
$299$	0	0
$300$	16.4996	0.952603
$301$	7.05031	0.406373
$302$	13.9267	0.801394
$303$	13.5121	0.776249
$304$	1.00000	0.0573539
$305$	−9.59094	−0.549176
$306$	2.49654	0.142718
$307$	−24.2794	−1.38570	−0.692850	−	0.721081i	$-0.743645\pi$
$307$	−24.2794	−1.38570	−0.692850	+	0.721081i	$0.743645\pi$
$308$	−3.55823	−0.202749
$309$	−26.4905	−1.50699
$310$	−25.4784	−1.44708
$311$	11.1369	0.631516	0.315758	−	0.948840i	$-0.397741\pi$
$311$	11.1369	0.631516	0.315758	+	0.948840i	$0.397741\pi$
$312$	0	0
$313$	13.4830	0.762106	0.381053	−	0.924553i	$-0.375562\pi$
$313$	13.4830	0.762106	0.381053	+	0.924553i	$0.375562\pi$
$314$	−20.2798	−1.14446
$315$	28.7615	1.62053
$316$	−15.2397	−0.857299
$317$	−3.73035	−0.209517	−0.104759	−	0.994498i	$-0.533407\pi$
$317$	−3.73035	−0.209517	−0.104759	+	0.994498i	$0.533407\pi$
$318$	22.3754	1.25475
$319$	−5.98709	−0.335213
$320$	−3.36169	−0.187924
$321$	−48.0452	−2.68162
$322$	13.4680	0.750541
$323$	−0.647275	−0.0360153
$324$	−5.69454	−0.316363
$325$	0	0
$326$	−18.2909	−1.01304
$327$	−40.0052	−2.21229
$328$	−1.46711	−0.0810074
$329$	−5.45221	−0.300590
$330$	14.1207	0.777316
$331$	35.3031	1.94044	0.970218	−	0.242233i	$-0.0778797\pi$
$331$	35.3031	1.94044	0.970218	+	0.242233i	$0.0778797\pi$
$332$	−1.35072	−0.0741305
$333$	38.0262	2.08382
$334$	4.93553	0.270060
$335$	−31.9650	−1.74643
$336$	−5.80860	−0.316885
$337$	9.75302	0.531281	0.265641	−	0.964072i	$-0.414417\pi$
$337$	9.75302	0.531281	0.265641	+	0.964072i	$0.414417\pi$
$338$	0	0
$339$	11.3239	0.615027
$340$	2.17594	0.118007
$341$	−12.1575	−0.658368
$342$	−3.85700	−0.208563
$343$	20.1403	1.08748
$344$	3.17837	0.171366
$345$	−53.4470	−2.87749
$346$	−3.42041	−0.183882
$347$	28.8899	1.55089	0.775446	−	0.631414i	$-0.217525\pi$
$347$	28.8899	1.55089	0.775446	+	0.631414i	$0.217525\pi$
$348$	−9.77355	−0.523918
$349$	−16.4727	−0.881764	−0.440882	−	0.897565i	$-0.645334\pi$
$349$	−16.4727	−0.881764	−0.440882	+	0.897565i	$0.645334\pi$
$350$	13.9769	0.747094
$351$	0	0
$352$	−1.60410	−0.0854986
$353$	1.40560	0.0748124	0.0374062	−	0.999300i	$-0.488090\pi$
$353$	1.40560	0.0748124	0.0374062	+	0.999300i	$0.488090\pi$
$354$	−29.1491	−1.54926
$355$	24.4349	1.29687
$356$	2.33354	0.123677
$357$	3.75976	0.198987
$358$	20.6266	1.09015
$359$	5.92174	0.312538	0.156269	−	0.987715i	$-0.450053\pi$
$359$	5.92174	0.312538	0.156269	+	0.987715i	$0.450053\pi$
$360$	12.9660	0.683370
$361$	1.00000	0.0526316
$362$	−2.89422	−0.152117
$363$	−22.0665	−1.15819
$364$	0	0
$365$	1.87095	0.0979299
$366$	−7.47087	−0.390509
$367$	35.6819	1.86258	0.931290	−	0.364278i	$-0.118684\pi$
$367$	35.6819	1.86258	0.931290	+	0.364278i	$0.118684\pi$
$368$	6.07154	0.316501
$369$	5.65863	0.294576
$370$	33.1429	1.72302
$371$	18.9543	0.984058
$372$	−19.8464	−1.02899
$373$	−1.99326	−0.103207	−0.0516035	−	0.998668i	$-0.516433\pi$
$373$	−1.99326	−0.103207	−0.0516035	+	0.998668i	$0.516433\pi$
$374$	1.03829	0.0536887
$375$	−11.4520	−0.591379
$376$	−2.45793	−0.126758
$377$	0	0
$378$	4.97798	0.256040
$379$	18.0685	0.928114	0.464057	−	0.885805i	$-0.346393\pi$
$379$	18.0685	0.928114	0.464057	+	0.885805i	$0.346393\pi$
$380$	−3.36169	−0.172451
$381$	17.2519	0.883841
$382$	−10.5320	−0.538862
$383$	−23.6300	−1.20744	−0.603718	−	0.797198i	$-0.706315\pi$
$383$	−23.6300	−1.20744	−0.603718	+	0.797198i	$0.706315\pi$
$384$	−2.61859	−0.133629
$385$	11.9617	0.609623
$386$	0.427246	0.0217463
$387$	−12.2590	−0.623159
$388$	−16.5432	−0.839854
$389$	6.26339	0.317566	0.158783	−	0.987313i	$-0.449243\pi$
$389$	6.26339	0.317566	0.158783	+	0.987313i	$0.449243\pi$
$390$	0	0
$391$	−3.92995	−0.198746
$392$	2.07951	0.105031
$393$	55.1983	2.78438
$394$	1.10019	0.0554270
$395$	51.2310	2.57771
$396$	6.18700	0.310908
$397$	2.86780	0.143931	0.0719653	−	0.997407i	$-0.477073\pi$
$397$	2.86780	0.143931	0.0719653	+	0.997407i	$0.477073\pi$
$398$	16.4906	0.826601
$399$	−5.80860	−0.290794
$400$	6.30094	0.315047
$401$	27.2599	1.36130	0.680648	−	0.732610i	$-0.261698\pi$
$401$	27.2599	1.36130	0.680648	+	0.732610i	$0.261698\pi$
$402$	−24.8992	−1.24186
$403$	0	0
$404$	5.16007	0.256723
$405$	19.1433	0.951236
$406$	−8.27922	−0.410891
$407$	15.8148	0.783909
$408$	1.69495	0.0839123
$409$	11.9721	0.591980	0.295990	−	0.955191i	$-0.404350\pi$
$409$	11.9721	0.591980	0.295990	+	0.955191i	$0.404350\pi$
$410$	4.93195	0.243572
$411$	−47.8039	−2.35799
$412$	−10.1163	−0.498396
$413$	−24.6923	−1.21503
$414$	−23.4179	−1.15093
$415$	4.54071	0.222895
$416$	0	0
$417$	3.23310	0.158326
$418$	−1.60410	−0.0784589
$419$	7.91833	0.386836	0.193418	−	0.981116i	$-0.438043\pi$
$419$	7.91833	0.386836	0.193418	+	0.981116i	$0.438043\pi$
$420$	19.5267	0.952805
$421$	1.41209	0.0688210	0.0344105	−	0.999408i	$-0.489045\pi$
$421$	1.41209	0.0688210	0.0344105	+	0.999408i	$0.489045\pi$
$422$	15.6615	0.762388
$423$	9.48022	0.460944
$424$	8.54484	0.414974
$425$	−4.07844	−0.197833
$426$	19.0336	0.922181
$427$	−6.32861	−0.306263
$428$	−18.3478	−0.886873
$429$	0	0
$430$	−10.6847	−0.515261
$431$	−22.2579	−1.07213	−0.536063	−	0.844178i	$-0.680089\pi$
$431$	−22.2579	−1.07213	−0.536063	+	0.844178i	$0.680089\pi$
$432$	2.24414	0.107971
$433$	0.00720591	0.000346294 0	0.000173147	−	1.00000i	$-0.499945\pi$
$433$	0.00720591	0.000346294 0	0.000173147		1.00000i	$0.499945\pi$
$434$	−16.8120	−0.807002
$435$	32.8556	1.57531
$436$	−15.2774	−0.731654
$437$	6.07154	0.290441
$438$	1.45738	0.0696361
$439$	−23.3107	−1.11256	−0.556279	−	0.830996i	$-0.687771\pi$
$439$	−23.3107	−1.11256	−0.556279	+	0.830996i	$0.687771\pi$
$440$	5.39247	0.257076
$441$	−8.02069	−0.381938
$442$	0	0
$443$	−1.09078	−0.0518244	−0.0259122	−	0.999664i	$-0.508249\pi$
$443$	−1.09078	−0.0518244	−0.0259122	+	0.999664i	$0.508249\pi$
$444$	25.8167	1.22520
$445$	−7.84462	−0.371871
$446$	26.2934	1.24503
$447$	4.67059	0.220911
$448$	−2.21822	−0.104801
$449$	13.7588	0.649319	0.324659	−	0.945831i	$-0.394750\pi$
$449$	13.7588	0.649319	0.324659	+	0.945831i	$0.394750\pi$
$450$	−24.3027	−1.14564
$451$	2.35338	0.110816
$452$	4.32441	0.203403
$453$	−36.4684	−1.71343
$454$	−8.84941	−0.415323
$455$	0	0
$456$	−2.61859	−0.122627
$457$	−3.86418	−0.180759	−0.0903794	−	0.995907i	$-0.528808\pi$
$457$	−3.86418	−0.180759	−0.0903794	+	0.995907i	$0.528808\pi$
$458$	9.57906	0.447600
$459$	−1.45257	−0.0678003
$460$	−20.4106	−0.951649
$461$	25.6865	1.19634	0.598170	−	0.801369i	$-0.295895\pi$
$461$	25.6865	1.19634	0.598170	+	0.801369i	$0.295895\pi$
$462$	9.31754	0.433491
$463$	−24.4845	−1.13789	−0.568946	−	0.822375i	$-0.692649\pi$
$463$	−24.4845	−1.13789	−0.568946	+	0.822375i	$0.692649\pi$
$464$	−3.73238	−0.173271
$465$	66.7176	3.09395
$466$	−10.1733	−0.471267
$467$	−32.9015	−1.52250	−0.761251	−	0.648458i	$-0.775414\pi$
$467$	−32.9015	−1.52250	−0.761251	+	0.648458i	$0.775414\pi$
$468$	0	0
$469$	−21.0922	−0.973947
$470$	8.26277	0.381133
$471$	53.1044	2.44692
$472$	−11.1316	−0.512373
$473$	−5.09841	−0.234425
$474$	39.9064	1.83296
$475$	6.30094	0.289107
$476$	1.43580	0.0658096
$477$	−32.9575	−1.50902
$478$	17.5454	0.802508
$479$	−10.5388	−0.481529	−0.240764	−	0.970584i	$-0.577398\pi$
$479$	−10.5388	−0.481529	−0.240764	+	0.970584i	$0.577398\pi$
$480$	8.80287	0.401794
$481$	0	0
$482$	28.2062	1.28476
$483$	−35.2671	−1.60471
$484$	−8.42688	−0.383040
$485$	55.6131	2.52526
$486$	21.6441	0.981795
$487$	−27.7469	−1.25733	−0.628666	−	0.777676i	$-0.716399\pi$
$487$	−27.7469	−1.25733	−0.628666	+	0.777676i	$0.716399\pi$
$488$	−2.85301	−0.129150
$489$	47.8964	2.16595
$490$	−6.99067	−0.315806
$491$	12.0826	0.545279	0.272640	−	0.962116i	$-0.412103\pi$
$491$	12.0826	0.545279	0.272640	+	0.962116i	$0.412103\pi$
$492$	3.84174	0.173199
$493$	2.41587	0.108805
$494$	0	0
$495$	−20.7988	−0.934835
$496$	−7.57907	−0.340310
$497$	16.1234	0.723235
$498$	3.53699	0.158496
$499$	−2.09861	−0.0939467	−0.0469733	−	0.998896i	$-0.514958\pi$
$499$	−2.09861	−0.0939467	−0.0469733	+	0.998896i	$0.514958\pi$
$500$	−4.37335	−0.195582
$501$	−12.9241	−0.577407
$502$	−12.2703	−0.547650
$503$	4.16558	0.185734	0.0928670	−	0.995679i	$-0.470397\pi$
$503$	4.16558	0.185734	0.0928670	+	0.995679i	$0.470397\pi$
$504$	8.55567	0.381100
$505$	−17.3465	−0.771910
$506$	−9.73932	−0.432966
$507$	0	0
$508$	6.58824	0.292306
$509$	−34.5004	−1.52920	−0.764602	−	0.644502i	$-0.777065\pi$
$509$	−34.5004	−1.52920	−0.764602	+	0.644502i	$0.777065\pi$
$510$	−5.69788	−0.252306
$511$	1.23455	0.0546133
$512$	−1.00000	−0.0441942
$513$	2.24414	0.0990810
$514$	18.0070	0.794255
$515$	34.0080	1.49857
$516$	−8.32284	−0.366392
$517$	3.94275	0.173402
$518$	21.8694	0.960886
$519$	8.95663	0.393153
$520$	0	0
$521$	6.87521	0.301209	0.150604	−	0.988594i	$-0.451878\pi$
$521$	6.87521	0.301209	0.150604	+	0.988594i	$0.451878\pi$
$522$	14.3958	0.630086
$523$	−18.4605	−0.807220	−0.403610	−	0.914931i	$-0.632245\pi$
$523$	−18.4605	−0.807220	−0.403610	+	0.914931i	$0.632245\pi$
$524$	21.0794	0.920858
$525$	−36.5996	−1.59734
$526$	11.9794	0.522325
$527$	4.90574	0.213697
$528$	4.20047	0.182802
$529$	13.8635	0.602763
$530$	−28.7251	−1.24774
$531$	42.9346	1.86320
$532$	−2.21822	−0.0961719
$533$	0	0
$534$	−6.11057	−0.264430
$535$	61.6794	2.66663
$536$	−9.50862	−0.410710
$537$	−54.0126	−2.33082
$538$	30.9659	1.33503
$539$	−3.33574	−0.143680
$540$	−7.54408	−0.324646
$541$	−46.2457	−1.98826	−0.994129	−	0.108201i	$-0.965491\pi$
$541$	−46.2457	−1.98826	−0.994129	+	0.108201i	$0.965491\pi$
$542$	22.9462	0.985622
$543$	7.57876	0.325236
$544$	0.647275	0.0277517
$545$	51.3578	2.19993
$546$	0	0
$547$	3.96252	0.169425	0.0847126	−	0.996405i	$-0.473003\pi$
$547$	3.96252	0.169425	0.0847126	+	0.996405i	$0.473003\pi$
$548$	−18.2556	−0.779841
$549$	11.0041	0.469643
$550$	−10.1073	−0.430977
$551$	−3.73238	−0.159005
$552$	−15.8988	−0.676700
$553$	33.8049	1.43753
$554$	29.1550	1.23868
$555$	−86.7875	−3.68393
$556$	1.23467	0.0523619
$557$	34.5995	1.46603	0.733014	−	0.680214i	$-0.238113\pi$
$557$	34.5995	1.46603	0.733014	+	0.680214i	$0.238113\pi$
$558$	29.2325	1.23751
$559$	0	0
$560$	7.45695	0.315114
$561$	−2.71886	−0.114790
$562$	−20.4344	−0.861973
$563$	−33.2821	−1.40267	−0.701337	−	0.712830i	$-0.747413\pi$
$563$	−33.2821	−1.40267	−0.701337	+	0.712830i	$0.747413\pi$
$564$	6.43629	0.271017
$565$	−14.5373	−0.611590
$566$	4.26150	0.179124
$567$	12.6317	0.530483
$568$	7.26865	0.304986
$569$	15.1878	0.636704	0.318352	−	0.947973i	$-0.396871\pi$
$569$	15.1878	0.636704	0.318352	+	0.947973i	$0.396871\pi$
$570$	8.80287	0.368712
$571$	−28.4590	−1.19097	−0.595486	−	0.803365i	$-0.703041\pi$
$571$	−28.4590	−1.19097	−0.595486	+	0.803365i	$0.703041\pi$
$572$	0	0
$573$	27.5789	1.15212
$574$	3.25436	0.135834
$575$	38.2564	1.59540
$576$	3.85700	0.160708
$577$	−12.3424	−0.513819	−0.256910	−	0.966435i	$-0.582704\pi$
$577$	−12.3424	−0.513819	−0.256910	+	0.966435i	$0.582704\pi$
$578$	16.5810	0.689680
$579$	−1.11878	−0.0464950
$580$	12.5471	0.520989
$581$	2.99620	0.124303
$582$	43.3198	1.79567
$583$	−13.7067	−0.567675
$584$	0.556551	0.0230302
$585$	0	0
$586$	25.8746	1.06887
$587$	−12.7738	−0.527231	−0.263616	−	0.964628i	$-0.584915\pi$
$587$	−12.7738	−0.527231	−0.263616	+	0.964628i	$0.584915\pi$
$588$	−5.44539	−0.224564
$589$	−7.57907	−0.312290
$590$	37.4209	1.54060
$591$	−2.88095	−0.118507
$592$	9.85900	0.405202
$593$	26.2954	1.07982	0.539911	−	0.841722i	$-0.318458\pi$
$593$	26.2954	1.07982	0.539911	+	0.841722i	$0.318458\pi$
$594$	−3.59981	−0.147702
$595$	−4.82670	−0.197875
$596$	1.78363	0.0730604
$597$	−43.1822	−1.76733
$598$	0	0
$599$	43.7338	1.78691	0.893456	−	0.449150i	$-0.148273\pi$
$599$	43.7338	1.78691	0.893456	+	0.449150i	$0.148273\pi$
$600$	−16.4996	−0.673592
$601$	19.5551	0.797668	0.398834	−	0.917023i	$-0.369415\pi$
$601$	19.5551	0.797668	0.398834	+	0.917023i	$0.369415\pi$
$602$	−7.05031	−0.287349
$603$	36.6748	1.49351
$604$	−13.9267	−0.566671
$605$	28.3285	1.15172
$606$	−13.5121	−0.548891
$607$	12.1769	0.494247	0.247123	−	0.968984i	$-0.420515\pi$
$607$	12.1769	0.494247	0.247123	+	0.968984i	$0.420515\pi$
$608$	−1.00000	−0.0405554
$609$	21.6799	0.878512
$610$	9.59094	0.388326
$611$	0	0
$612$	−2.49654	−0.100917
$613$	25.7852	1.04146	0.520728	−	0.853723i	$-0.325661\pi$
$613$	25.7852	1.04146	0.520728	+	0.853723i	$0.325661\pi$
$614$	24.2794	0.979838
$615$	−12.9147	−0.520773
$616$	3.55823	0.143365
$617$	11.4089	0.459304	0.229652	−	0.973273i	$-0.426241\pi$
$617$	11.4089	0.459304	0.229652	+	0.973273i	$0.426241\pi$
$618$	26.4905	1.06561
$619$	31.3149	1.25865	0.629325	−	0.777142i	$-0.283331\pi$
$619$	31.3149	1.25865	0.629325	+	0.777142i	$0.283331\pi$
$620$	25.4784	1.02324
$621$	13.6253	0.546766
$622$	−11.1369	−0.446550
$623$	−5.17629	−0.207384
$624$	0	0
$625$	−16.8029	−0.672114
$626$	−13.4830	−0.538890
$627$	4.20047	0.167750
$628$	20.2798	0.809252
$629$	−6.38148	−0.254446
$630$	−28.7615	−1.14588
$631$	7.28193	0.289889	0.144944	−	0.989440i	$-0.453700\pi$
$631$	7.28193	0.289889	0.144944	+	0.989440i	$0.453700\pi$
$632$	15.2397	0.606202
$633$	−41.0109	−1.63004
$634$	3.73035	0.148151
$635$	−22.1476	−0.878901
$636$	−22.3754	−0.887242
$637$	0	0
$638$	5.98709	0.237031
$639$	−28.0352	−1.10906
$640$	3.36169	0.132882
$641$	29.9567	1.18322	0.591609	−	0.806225i	$-0.298493\pi$
$641$	29.9567	1.18322	0.591609	+	0.806225i	$0.298493\pi$
$642$	48.0452	1.89619
$643$	−20.3947	−0.804289	−0.402145	−	0.915576i	$-0.631735\pi$
$643$	−20.3947	−0.804289	−0.402145	+	0.915576i	$0.631735\pi$
$644$	−13.4680	−0.530713
$645$	27.9788	1.10166
$646$	0.647275	0.0254667
$647$	10.7071	0.420940	0.210470	−	0.977600i	$-0.432501\pi$
$647$	10.7071	0.420940	0.210470	+	0.977600i	$0.432501\pi$
$648$	5.69454	0.223703
$649$	17.8561	0.700915
$650$	0	0
$651$	44.0237	1.72543
$652$	18.2909	0.716328
$653$	−45.9991	−1.80008	−0.900042	−	0.435804i	$-0.856464\pi$
$653$	−45.9991	−1.80008	−0.900042	+	0.435804i	$0.856464\pi$
$654$	40.0052	1.56433
$655$	−70.8623	−2.76882
$656$	1.46711	0.0572808
$657$	−2.14662	−0.0837475
$658$	5.45221	0.212549
$659$	29.0884	1.13312	0.566562	−	0.824019i	$-0.308273\pi$
$659$	29.0884	1.13312	0.566562	+	0.824019i	$0.308273\pi$
$660$	−14.1207	−0.549646
$661$	28.7408	1.11789	0.558943	−	0.829206i	$-0.311207\pi$
$661$	28.7408	1.11789	0.558943	+	0.829206i	$0.311207\pi$
$662$	−35.3031	−1.37210
$663$	0	0
$664$	1.35072	0.0524182
$665$	7.45695	0.289168
$666$	−38.0262	−1.47348
$667$	−22.6613	−0.877447
$668$	−4.93553	−0.190961
$669$	−68.8515	−2.66195
$670$	31.9650	1.23492
$671$	4.57651	0.176674
$672$	5.80860	0.224071
$673$	47.8203	1.84334	0.921669	−	0.387977i	$-0.126826\pi$
$673$	47.8203	1.84334	0.921669	+	0.387977i	$0.126826\pi$
$674$	−9.75302	−0.375672
$675$	14.1402	0.544255
$676$	0	0
$677$	0.988896	0.0380063	0.0190032	−	0.999819i	$-0.493951\pi$
$677$	0.988896	0.0380063	0.0190032	+	0.999819i	$0.493951\pi$
$678$	−11.3239	−0.434890
$679$	36.6964	1.40828
$680$	−2.17594	−0.0834433
$681$	23.1730	0.887990
$682$	12.1575	0.465537
$683$	31.7867	1.21628	0.608142	−	0.793828i	$-0.291915\pi$
$683$	31.7867	1.21628	0.608142	+	0.793828i	$0.291915\pi$
$684$	3.85700	0.147476
$685$	61.3697	2.34481
$686$	−20.1403	−0.768961
$687$	−25.0836	−0.956999
$688$	−3.17837	−0.121174
$689$	0	0
$690$	53.4470	2.03469
$691$	−16.1975	−0.616184	−0.308092	−	0.951357i	$-0.599690\pi$
$691$	−16.1975	−0.616184	−0.308092	+	0.951357i	$0.599690\pi$
$692$	3.42041	0.130024
$693$	−13.7241	−0.521336
$694$	−28.8899	−1.09665
$695$	−4.15059	−0.157441
$696$	9.77355	0.370466
$697$	−0.949621	−0.0359694
$698$	16.4727	0.623502
$699$	26.6396	1.00760
$700$	−13.9769	−0.528275
$701$	21.5679	0.814609	0.407305	−	0.913292i	$-0.366469\pi$
$701$	21.5679	0.814609	0.407305	+	0.913292i	$0.366469\pi$
$702$	0	0
$703$	9.85900	0.371839
$704$	1.60410	0.0604566
$705$	−21.6368	−0.814889
$706$	−1.40560	−0.0529003
$707$	−11.4461	−0.430477
$708$	29.1491	1.09549
$709$	27.1208	1.01854	0.509272	−	0.860606i	$-0.329915\pi$
$709$	27.1208	1.01854	0.509272	+	0.860606i	$0.329915\pi$
$710$	−24.4349	−0.917027
$711$	−58.7795	−2.20440
$712$	−2.33354	−0.0874529
$713$	−46.0166	−1.72333
$714$	−3.75976	−0.140705
$715$	0	0
$716$	−20.6266	−0.770853
$717$	−45.9442	−1.71582
$718$	−5.92174	−0.220997
$719$	21.3044	0.794522	0.397261	−	0.917706i	$-0.369961\pi$
$719$	21.3044	0.794522	0.397261	+	0.917706i	$0.369961\pi$
$720$	−12.9660	−0.483216
$721$	22.4402	0.835718
$722$	−1.00000	−0.0372161
$723$	−73.8604	−2.74690
$724$	2.89422	0.107563
$725$	−23.5175	−0.873417
$726$	22.0665	0.818965
$727$	−1.52945	−0.0567241	−0.0283620	−	0.999598i	$-0.509029\pi$
$727$	−1.52945	−0.0567241	−0.0283620	+	0.999598i	$0.509029\pi$
$728$	0	0
$729$	−39.5933	−1.46642
$730$	−1.87095	−0.0692469
$731$	2.05728	0.0760912
$732$	7.47087	0.276131
$733$	27.1109	1.00136	0.500682	−	0.865631i	$-0.333083\pi$
$733$	27.1109	1.00136	0.500682	+	0.865631i	$0.333083\pi$
$734$	−35.6819	−1.31704
$735$	18.3057	0.675216
$736$	−6.07154	−0.223800
$737$	15.2527	0.561842
$738$	−5.65863	−0.208297
$739$	−41.9153	−1.54188	−0.770939	−	0.636909i	$-0.780213\pi$
$739$	−41.9153	−1.54188	−0.770939	+	0.636909i	$0.780213\pi$
$740$	−33.1429	−1.21836
$741$	0	0
$742$	−18.9543	−0.695834
$743$	6.61979	0.242857	0.121428	−	0.992600i	$-0.461253\pi$
$743$	6.61979	0.242857	0.121428	+	0.992600i	$0.461253\pi$
$744$	19.8464	0.727606
$745$	−5.99601	−0.219677
$746$	1.99326	0.0729784
$747$	−5.20974	−0.190614
$748$	−1.03829	−0.0379637
$749$	40.6993	1.48712
$750$	11.4520	0.418168
$751$	−25.6359	−0.935466	−0.467733	−	0.883870i	$-0.654929\pi$
$751$	−25.6359	−0.935466	−0.467733	+	0.883870i	$0.654929\pi$
$752$	2.45793	0.0896313
$753$	32.1308	1.17091
$754$	0	0
$755$	46.8174	1.70386
$756$	−4.97798	−0.181047
$757$	1.30269	0.0473472	0.0236736	−	0.999720i	$-0.492464\pi$
$757$	1.30269	0.0473472	0.0236736	+	0.999720i	$0.492464\pi$
$758$	−18.0685	−0.656276
$759$	25.5033	0.925710
$760$	3.36169	0.121941
$761$	17.1544	0.621847	0.310923	−	0.950435i	$-0.399362\pi$
$761$	17.1544	0.621847	0.310923	+	0.950435i	$0.399362\pi$
$762$	−17.2519	−0.624970
$763$	33.8886	1.22685
$764$	10.5320	0.381033
$765$	8.39259	0.303435
$766$	23.6300	0.853786
$767$	0	0
$768$	2.61859	0.0944901
$769$	33.3648	1.20317	0.601583	−	0.798810i	$-0.294537\pi$
$769$	33.3648	1.20317	0.601583	+	0.798810i	$0.294537\pi$
$770$	−11.9617	−0.431068
$771$	−47.1529	−1.69817
$772$	−0.427246	−0.0153769
$773$	22.2774	0.801264	0.400632	−	0.916239i	$-0.368791\pi$
$773$	22.2774	0.801264	0.400632	+	0.916239i	$0.368791\pi$
$774$	12.2590	0.440640
$775$	−47.7552	−1.71542
$776$	16.5432	0.593866
$777$	−57.2669	−2.05444
$778$	−6.26339	−0.224553
$779$	1.46711	0.0525645
$780$	0	0
$781$	−11.6596	−0.417214
$782$	3.92995	0.140535
$783$	−8.37596	−0.299332
$784$	−2.07951	−0.0742683
$785$	−68.1744	−2.43325
$786$	−55.1983	−1.96886
$787$	−40.9833	−1.46090	−0.730448	−	0.682968i	$-0.760689\pi$
$787$	−40.9833	−1.46090	−0.730448	+	0.682968i	$0.760689\pi$
$788$	−1.10019	−0.0391928
$789$	−31.3690	−1.11677
$790$	−51.2310	−1.82272
$791$	−9.59249	−0.341070
$792$	−6.18700	−0.219845
$793$	0	0
$794$	−2.86780	−0.101774
$795$	75.2191	2.66775
$796$	−16.4906	−0.584495
$797$	−10.1382	−0.359113	−0.179557	−	0.983748i	$-0.557466\pi$
$797$	−10.1382	−0.359113	−0.179557	+	0.983748i	$0.557466\pi$
$798$	5.80860	0.205622
$799$	−1.59095	−0.0562839
$800$	−6.30094	−0.222772
$801$	9.00045	0.318015
$802$	−27.2599	−0.962582
$803$	−0.892760	−0.0315048
$804$	24.8992	0.878126
$805$	45.2751	1.59574
$806$	0	0
$807$	−81.0869	−2.85439
$808$	−5.16007	−0.181530
$809$	−9.63800	−0.338854	−0.169427	−	0.985543i	$-0.554192\pi$
$809$	−9.63800	−0.338854	−0.169427	+	0.985543i	$0.554192\pi$
$810$	−19.1433	−0.672626
$811$	−23.4154	−0.822224	−0.411112	−	0.911585i	$-0.634860\pi$
$811$	−23.4154	−0.822224	−0.411112	+	0.911585i	$0.634860\pi$
$812$	8.27922	0.290544
$813$	−60.0866	−2.10733
$814$	−15.8148	−0.554308
$815$	−61.4884	−2.15385
$816$	−1.69495	−0.0593350
$817$	−3.17837	−0.111197
$818$	−11.9721	−0.418593
$819$	0	0
$820$	−4.93195	−0.172231
$821$	13.9796	0.487891	0.243945	−	0.969789i	$-0.421558\pi$
$821$	13.9796	0.487891	0.243945	+	0.969789i	$0.421558\pi$
$822$	47.8039	1.66735
$823$	−34.7641	−1.21180	−0.605901	−	0.795540i	$-0.707187\pi$
$823$	−34.7641	−1.21180	−0.605901	+	0.795540i	$0.707187\pi$
$824$	10.1163	0.352419
$825$	26.4669	0.921459
$826$	24.6923	0.859155
$827$	−27.8294	−0.967723	−0.483861	−	0.875145i	$-0.660766\pi$
$827$	−27.8294	−0.967723	−0.483861	+	0.875145i	$0.660766\pi$
$828$	23.4179	0.813829
$829$	24.6193	0.855064	0.427532	−	0.904000i	$-0.359383\pi$
$829$	24.6193	0.855064	0.427532	+	0.904000i	$0.359383\pi$
$830$	−4.54071	−0.157610
$831$	−76.3448	−2.64837
$832$	0	0
$833$	1.34602	0.0466367
$834$	−3.23310	−0.111953
$835$	16.5917	0.574179
$836$	1.60410	0.0554788
$837$	−17.0085	−0.587898
$838$	−7.91833	−0.273534
$839$	19.9346	0.688218	0.344109	−	0.938930i	$-0.388181\pi$
$839$	19.9346	0.688218	0.344109	+	0.938930i	$0.388181\pi$
$840$	−19.5267	−0.673735
$841$	−15.0694	−0.519633
$842$	−1.41209	−0.0486638
$843$	53.5092	1.84296
$844$	−15.6615	−0.539090
$845$	0	0
$846$	−9.48022	−0.325937
$847$	18.6926	0.642287
$848$	−8.54484	−0.293431
$849$	−11.1591	−0.382979
$850$	4.07844	0.139889
$851$	59.8593	2.05195
$852$	−19.0336	−0.652080
$853$	17.9841	0.615764	0.307882	−	0.951425i	$-0.400380\pi$
$853$	17.9841	0.615764	0.307882	+	0.951425i	$0.400380\pi$
$854$	6.32861	0.216560
$855$	−12.9660	−0.443429
$856$	18.3478	0.627114
$857$	−18.8568	−0.644135	−0.322067	−	0.946717i	$-0.604378\pi$
$857$	−18.8568	−0.644135	−0.322067	+	0.946717i	$0.604378\pi$
$858$	0	0
$859$	27.7429	0.946574	0.473287	−	0.880908i	$-0.343067\pi$
$859$	27.7429	0.946574	0.473287	+	0.880908i	$0.343067\pi$
$860$	10.6847	0.364345
$861$	−8.52182	−0.290423
$862$	22.2579	0.758108
$863$	−3.48952	−0.118785	−0.0593923	−	0.998235i	$-0.518916\pi$
$863$	−3.48952	−0.118785	−0.0593923	+	0.998235i	$0.518916\pi$
$864$	−2.24414	−0.0763471
$865$	−11.4983	−0.390955
$866$	−0.00720591	−0.000244867 0
$867$	−43.4189	−1.47458
$868$	16.8120	0.570637
$869$	−24.4459	−0.829271
$870$	−32.8556	−1.11391
$871$	0	0
$872$	15.2774	0.517358
$873$	−63.8072	−2.15955
$874$	−6.07154	−0.205373
$875$	9.70105	0.327955
$876$	−1.45738	−0.0492402
$877$	−35.1796	−1.18793	−0.593966	−	0.804490i	$-0.702439\pi$
$877$	−35.1796	−1.18793	−0.593966	+	0.804490i	$0.702439\pi$
$878$	23.3107	0.786697
$879$	−67.7550	−2.28532
$880$	−5.39247	−0.181780
$881$	−38.6123	−1.30088	−0.650440	−	0.759558i	$-0.725415\pi$
$881$	−38.6123	−1.30088	−0.650440	+	0.759558i	$0.725415\pi$
$882$	8.02069	0.270071
$883$	−30.1035	−1.01306	−0.506532	−	0.862221i	$-0.669073\pi$
$883$	−30.1035	−1.01306	−0.506532	+	0.862221i	$0.669073\pi$
$884$	0	0
$885$	−97.9900	−3.29390
$886$	1.09078	0.0366454
$887$	12.5568	0.421617	0.210809	−	0.977527i	$-0.432390\pi$
$887$	12.5568	0.421617	0.210809	+	0.977527i	$0.432390\pi$
$888$	−25.8167	−0.866350
$889$	−14.6142	−0.490143
$890$	7.84462	0.262952
$891$	−9.13459	−0.306020
$892$	−26.2934	−0.880368
$893$	2.45793	0.0822513
$894$	−4.67059	−0.156208
$895$	69.3403	2.31779
$896$	2.21822	0.0741054
$897$	0	0
$898$	−13.7588	−0.459138
$899$	28.2879	0.943455
$900$	24.3027	0.810091
$901$	5.53086	0.184260
$902$	−2.35338	−0.0783589
$903$	18.4619	0.614372
$904$	−4.32441	−0.143828
$905$	−9.72945	−0.323418
$906$	36.4684	1.21158
$907$	−8.15451	−0.270766	−0.135383	−	0.990793i	$-0.543227\pi$
$907$	−8.15451	−0.270766	−0.135383	+	0.990793i	$0.543227\pi$
$908$	8.84941	0.293678
$909$	19.9024	0.660120
$910$	0	0
$911$	14.1881	0.470072	0.235036	−	0.971987i	$-0.424479\pi$
$911$	14.1881	0.470072	0.235036	+	0.971987i	$0.424479\pi$
$912$	2.61859	0.0867101
$913$	−2.16669	−0.0717069
$914$	3.86418	0.127816
$915$	−25.1147	−0.830267
$916$	−9.57906	−0.316501
$917$	−46.7587	−1.54411
$918$	1.45257	0.0479420
$919$	−27.4048	−0.904001	−0.452000	−	0.892018i	$-0.649289\pi$
$919$	−27.4048	−0.904001	−0.452000	+	0.892018i	$0.649289\pi$
$920$	20.4106	0.672918
$921$	−63.5778	−2.09496
$922$	−25.6865	−0.845940
$923$	0	0
$924$	−9.31754	−0.306525
$925$	62.1210	2.04252
$926$	24.4845	0.804611
$927$	−39.0187	−1.28154
$928$	3.73238	0.122521
$929$	−14.2249	−0.466703	−0.233351	−	0.972392i	$-0.574969\pi$
$929$	−14.2249	−0.466703	−0.233351	+	0.972392i	$0.574969\pi$
$930$	−66.7176	−2.18775
$931$	−2.07951	−0.0681533
$932$	10.1733	0.333236
$933$	29.1630	0.954753
$934$	32.9015	1.07657
$935$	3.49041	0.114149
$936$	0	0
$937$	1.76038	0.0575093	0.0287546	−	0.999587i	$-0.490846\pi$
$937$	1.76038	0.0575093	0.0287546	+	0.999587i	$0.490846\pi$
$938$	21.0922	0.688684
$939$	35.3065	1.15218
$940$	−8.26277	−0.269502
$941$	44.1074	1.43786	0.718929	−	0.695083i	$-0.244632\pi$
$941$	44.1074	1.43786	0.718929	+	0.695083i	$0.244632\pi$
$942$	−53.1044	−1.73024
$943$	8.90758	0.290071
$944$	11.1316	0.362303
$945$	16.7344	0.544370
$946$	5.09841	0.165764
$947$	−18.9882	−0.617033	−0.308517	−	0.951219i	$-0.599833\pi$
$947$	−18.9882	−0.617033	−0.308517	+	0.951219i	$0.599833\pi$
$948$	−39.9064	−1.29610
$949$	0	0
$950$	−6.30094	−0.204429
$951$	−9.76824	−0.316757
$952$	−1.43580	−0.0465344
$953$	−6.72212	−0.217751	−0.108875	−	0.994055i	$-0.534725\pi$
$953$	−6.72212	−0.217751	−0.108875	+	0.994055i	$0.534725\pi$
$954$	32.9575	1.06704
$955$	−35.4052	−1.14568
$956$	−17.5454	−0.567459
$957$	−15.6777	−0.506789
$958$	10.5388	0.340492
$959$	40.4949	1.30765
$960$	−8.80287	−0.284112
$961$	26.4422	0.852975
$962$	0	0
$963$	−70.7674	−2.28045
$964$	−28.2062	−0.908460
$965$	1.43627	0.0462351
$966$	35.2671	1.13470
$967$	−7.59413	−0.244211	−0.122105	−	0.992517i	$-0.538965\pi$
$967$	−7.59413	−0.244211	−0.122105	+	0.992517i	$0.538965\pi$
$968$	8.42688	0.270850
$969$	−1.69495	−0.0544495
$970$	−55.6131	−1.78563
$971$	25.5435	0.819729	0.409864	−	0.912147i	$-0.365576\pi$
$971$	25.5435	0.819729	0.409864	+	0.912147i	$0.365576\pi$
$972$	−21.6441	−0.694234
$973$	−2.73878	−0.0878011
$974$	27.7469	0.889067
$975$	0	0
$976$	2.85301	0.0913228
$977$	6.11695	0.195698	0.0978492	−	0.995201i	$-0.468804\pi$
$977$	6.11695	0.195698	0.0978492	+	0.995201i	$0.468804\pi$
$978$	−47.8964	−1.53156
$979$	3.74321	0.119634
$980$	6.99067	0.223309
$981$	−58.9249	−1.88133
$982$	−12.0826	−0.385571
$983$	35.3472	1.12740	0.563700	−	0.825980i	$-0.309378\pi$
$983$	35.3472	1.12740	0.563700	+	0.825980i	$0.309378\pi$
$984$	−3.84174	−0.122470
$985$	3.69851	0.117844
$986$	−2.41587	−0.0769371
$987$	−14.2771	−0.454445
$988$	0	0
$989$	−19.2976	−0.613627
$990$	20.7988	0.661028
$991$	22.5088	0.715015	0.357507	−	0.933910i	$-0.383627\pi$
$991$	22.5088	0.715015	0.357507	+	0.933910i	$0.383627\pi$
$992$	7.57907	0.240636
$993$	92.4444	2.93363
$994$	−16.1234	−0.511405
$995$	55.4364	1.75745
$996$	−3.53699	−0.112074
$997$	46.9693	1.48753	0.743766	−	0.668440i	$-0.233038\pi$
$997$	46.9693	1.48753	0.743766	+	0.668440i	$0.233038\pi$
$998$	2.09861	0.0664303
$999$	22.1249	0.700002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bo.1.13	✓	15
13.12	even	2		6422.2.a.bq.1.13	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.13	✓	15	1.1	even	1	trivial
6422.2.a.bq.1.13	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.61859 q^{3} +1.00000 q^{4} -3.36169 q^{5} -2.61859 q^{6} -2.21822 q^{7} -1.00000 q^{8} +3.85700 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.61859 q^{3} +1.00000 q^{4} -3.36169 q^{5} -2.61859 q^{6} -2.21822 q^{7} -1.00000 q^{8} +3.85700 q^{9} +3.36169 q^{10} +1.60410 q^{11} +2.61859 q^{12} +2.21822 q^{14} -8.80287 q^{15} +1.00000 q^{16} -0.647275 q^{17} -3.85700 q^{18} +1.00000 q^{19} -3.36169 q^{20} -5.80860 q^{21} -1.60410 q^{22} +6.07154 q^{23} -2.61859 q^{24} +6.30094 q^{25} +2.24414 q^{27} -2.21822 q^{28} -3.73238 q^{29} +8.80287 q^{30} -7.57907 q^{31} -1.00000 q^{32} +4.20047 q^{33} +0.647275 q^{34} +7.45695 q^{35} +3.85700 q^{36} +9.85900 q^{37} -1.00000 q^{38} +3.36169 q^{40} +1.46711 q^{41} +5.80860 q^{42} -3.17837 q^{43} +1.60410 q^{44} -12.9660 q^{45} -6.07154 q^{46} +2.45793 q^{47} +2.61859 q^{48} -2.07951 q^{49} -6.30094 q^{50} -1.69495 q^{51} -8.54484 q^{53} -2.24414 q^{54} -5.39247 q^{55} +2.21822 q^{56} +2.61859 q^{57} +3.73238 q^{58} +11.1316 q^{59} -8.80287 q^{60} +2.85301 q^{61} +7.57907 q^{62} -8.55567 q^{63} +1.00000 q^{64} -4.20047 q^{66} +9.50862 q^{67} -0.647275 q^{68} +15.8988 q^{69} -7.45695 q^{70} -7.26865 q^{71} -3.85700 q^{72} -0.556551 q^{73} -9.85900 q^{74} +16.4996 q^{75} +1.00000 q^{76} -3.55823 q^{77} -15.2397 q^{79} -3.36169 q^{80} -5.69454 q^{81} -1.46711 q^{82} -1.35072 q^{83} -5.80860 q^{84} +2.17594 q^{85} +3.17837 q^{86} -9.77355 q^{87} -1.60410 q^{88} +2.33354 q^{89} +12.9660 q^{90} +6.07154 q^{92} -19.8464 q^{93} -2.45793 q^{94} -3.36169 q^{95} -2.61859 q^{96} -16.5432 q^{97} +2.07951 q^{98} +6.18700 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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