LMFDB - Embedded newform 3381.2.a.bg.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 100x^{3} - 17x^{2} - 28x - 2 $ x^10 - 4x^9 - 6x^8 + 36x^7 + x^6 - 100x^5 + 26x^4 + 100x^3 - 17x^2 - 28x - 2
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			1.5
Root			$0.795395$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.795395 q^{2} -1.00000 q^{3} -1.36735 q^{4} -1.31903 q^{5} +0.795395 q^{6} +2.67837 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.795395 q^{2} -1.00000 q^{3} -1.36735 q^{4} -1.31903 q^{5} +0.795395 q^{6} +2.67837 q^{8} +1.00000 q^{9} +1.04915 q^{10} +4.89739 q^{11} +1.36735 q^{12} +5.51611 q^{13} +1.31903 q^{15} +0.604329 q^{16} +3.91876 q^{17} -0.795395 q^{18} +4.67871 q^{19} +1.80357 q^{20} -3.89536 q^{22} +1.00000 q^{23} -2.67837 q^{24} -3.26017 q^{25} -4.38749 q^{26} -1.00000 q^{27} +6.49718 q^{29} -1.04915 q^{30} +6.46887 q^{31} -5.83742 q^{32} -4.89739 q^{33} -3.11696 q^{34} -1.36735 q^{36} -9.60783 q^{37} -3.72142 q^{38} -5.51611 q^{39} -3.53284 q^{40} +8.54564 q^{41} +3.92282 q^{43} -6.69643 q^{44} -1.31903 q^{45} -0.795395 q^{46} -8.34179 q^{47} -0.604329 q^{48} +2.59312 q^{50} -3.91876 q^{51} -7.54244 q^{52} -5.18739 q^{53} +0.795395 q^{54} -6.45979 q^{55} -4.67871 q^{57} -5.16783 q^{58} -1.03999 q^{59} -1.80357 q^{60} +10.5284 q^{61} -5.14531 q^{62} +3.43440 q^{64} -7.27590 q^{65} +3.89536 q^{66} -11.2479 q^{67} -5.35830 q^{68} -1.00000 q^{69} +3.62821 q^{71} +2.67837 q^{72} +16.2125 q^{73} +7.64202 q^{74} +3.26017 q^{75} -6.39742 q^{76} +4.38749 q^{78} +11.0959 q^{79} -0.797126 q^{80} +1.00000 q^{81} -6.79716 q^{82} -3.53762 q^{83} -5.16895 q^{85} -3.12019 q^{86} -6.49718 q^{87} +13.1170 q^{88} -16.7543 q^{89} +1.04915 q^{90} -1.36735 q^{92} -6.46887 q^{93} +6.63502 q^{94} -6.17134 q^{95} +5.83742 q^{96} -4.67418 q^{97} +4.89739 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} - 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q - 4 * q^2 - 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 12 * q^8 + 10 * q^9	$ 10 q - 4 q^{2} - 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 12 q^{8} + 10 q^{9} + 8 q^{10} - 2 q^{11} - 8 q^{12} + 16 q^{13} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 4 q^{18} + 26 q^{19} - 8 q^{22} + 10 q^{23} + 12 q^{24} + 14 q^{25} - 12 q^{26} - 10 q^{27} - 16 q^{29} - 8 q^{30} + 20 q^{31} - 8 q^{32} + 2 q^{33} - 4 q^{34} + 8 q^{36} + 8 q^{37} - 8 q^{38} - 16 q^{39} - 12 q^{40} + 22 q^{41} - 4 q^{43} - 24 q^{44} + 4 q^{45} - 4 q^{46} + 6 q^{47} - 4 q^{48} - 48 q^{50} - 12 q^{51} + 24 q^{52} - 30 q^{53} + 4 q^{54} + 48 q^{55} - 26 q^{57} + 24 q^{58} + 42 q^{59} + 14 q^{61} + 40 q^{62} + 8 q^{64} - 44 q^{65} + 8 q^{66} + 8 q^{68} - 10 q^{69} + 8 q^{71} - 12 q^{72} + 24 q^{73} + 8 q^{74} - 14 q^{75} + 32 q^{76} + 12 q^{78} + 32 q^{79} + 28 q^{80} + 10 q^{81} - 64 q^{82} + 28 q^{83} - 4 q^{85} - 4 q^{86} + 16 q^{87} + 20 q^{88} + 8 q^{90} + 8 q^{92} - 20 q^{93} + 8 q^{94} - 16 q^{95} + 8 q^{96} - 12 q^{97} - 2 q^{99}+O(q^{100}) $ 10 * q - 4 * q^2 - 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 12 * q^8 + 10 * q^9 + 8 * q^10 - 2 * q^11 - 8 * q^12 + 16 * q^13 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 4 * q^18 + 26 * q^19 - 8 * q^22 + 10 * q^23 + 12 * q^24 + 14 * q^25 - 12 * q^26 - 10 * q^27 - 16 * q^29 - 8 * q^30 + 20 * q^31 - 8 * q^32 + 2 * q^33 - 4 * q^34 + 8 * q^36 + 8 * q^37 - 8 * q^38 - 16 * q^39 - 12 * q^40 + 22 * q^41 - 4 * q^43 - 24 * q^44 + 4 * q^45 - 4 * q^46 + 6 * q^47 - 4 * q^48 - 48 * q^50 - 12 * q^51 + 24 * q^52 - 30 * q^53 + 4 * q^54 + 48 * q^55 - 26 * q^57 + 24 * q^58 + 42 * q^59 + 14 * q^61 + 40 * q^62 + 8 * q^64 - 44 * q^65 + 8 * q^66 + 8 * q^68 - 10 * q^69 + 8 * q^71 - 12 * q^72 + 24 * q^73 + 8 * q^74 - 14 * q^75 + 32 * q^76 + 12 * q^78 + 32 * q^79 + 28 * q^80 + 10 * q^81 - 64 * q^82 + 28 * q^83 - 4 * q^85 - 4 * q^86 + 16 * q^87 + 20 * q^88 + 8 * q^90 + 8 * q^92 - 20 * q^93 + 8 * q^94 - 16 * q^95 + 8 * q^96 - 12 * q^97 - 2 * 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 $

$2$	−0.795395	−0.562429	−0.281215	−	0.959645i	$-0.590737\pi$
$2$	−0.795395	−0.562429	−0.281215	+	0.959645i	$0.590737\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.36735	−0.683673
$5$	−1.31903	−0.589887	−0.294943	−	0.955515i	$-0.595301\pi$
$5$	−1.31903	−0.589887	−0.294943	+	0.955515i	$0.595301\pi$
$6$	0.795395	0.324719
$7$	0	0
$7$	0	0
$8$	2.67837	0.946947
$9$	1.00000	0.333333
$10$	1.04915	0.331770
$11$	4.89739	1.47662	0.738310	−	0.674462i	$-0.235624\pi$
$11$	4.89739	1.47662	0.738310	+	0.674462i	$0.235624\pi$
$12$	1.36735	0.394719
$13$	5.51611	1.52989	0.764947	−	0.644093i	$-0.222765\pi$
$13$	5.51611	1.52989	0.764947	+	0.644093i	$0.222765\pi$
$14$	0	0
$15$	1.31903	0.340571
$16$	0.604329	0.151082
$17$	3.91876	0.950439	0.475219	−	0.879867i	$-0.342369\pi$
$17$	3.91876	0.950439	0.475219	+	0.879867i	$0.342369\pi$
$18$	−0.795395	−0.187476
$19$	4.67871	1.07337	0.536685	−	0.843783i	$-0.319676\pi$
$19$	4.67871	1.07337	0.536685	+	0.843783i	$0.319676\pi$
$20$	1.80357	0.403290
$21$	0	0
$22$	−3.89536	−0.830494
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.67837	−0.546720
$25$	−3.26017	−0.652034
$26$	−4.38749	−0.860458
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.49718	1.20650	0.603248	−	0.797554i	$-0.293873\pi$
$29$	6.49718	1.20650	0.603248	+	0.797554i	$0.293873\pi$
$30$	−1.04915	−0.191547
$31$	6.46887	1.16184	0.580922	−	0.813959i	$-0.302692\pi$
$31$	6.46887	1.16184	0.580922	+	0.813959i	$0.302692\pi$
$32$	−5.83742	−1.03192
$33$	−4.89739	−0.852527
$34$	−3.11696	−0.534555
$35$	0	0
$36$	−1.36735	−0.227891
$37$	−9.60783	−1.57952	−0.789758	−	0.613418i	$-0.789794\pi$
$37$	−9.60783	−1.57952	−0.789758	+	0.613418i	$0.789794\pi$
$38$	−3.72142	−0.603695
$39$	−5.51611	−0.883285
$40$	−3.53284	−0.558592
$41$	8.54564	1.33461	0.667303	−	0.744787i	$-0.267449\pi$
$41$	8.54564	1.33461	0.667303	+	0.744787i	$0.267449\pi$
$42$	0	0
$43$	3.92282	0.598224	0.299112	−	0.954218i	$-0.403310\pi$
$43$	3.92282	0.598224	0.299112	+	0.954218i	$0.403310\pi$
$44$	−6.69643	−1.00953
$45$	−1.31903	−0.196629
$46$	−0.795395	−0.117275
$47$	−8.34179	−1.21678	−0.608388	−	0.793640i	$-0.708183\pi$
$47$	−8.34179	−1.21678	−0.608388	+	0.793640i	$0.708183\pi$
$48$	−0.604329	−0.0872273
$49$	0	0
$50$	2.59312	0.366723
$51$	−3.91876	−0.548736
$52$	−7.54244	−1.04595
$53$	−5.18739	−0.712543	−0.356271	−	0.934383i	$-0.615952\pi$
$53$	−5.18739	−0.712543	−0.356271	+	0.934383i	$0.615952\pi$
$54$	0.795395	0.108240
$55$	−6.45979	−0.871038
$56$	0	0
$57$	−4.67871	−0.619710
$58$	−5.16783	−0.678569
$59$	−1.03999	−0.135395	−0.0676974	−	0.997706i	$-0.521565\pi$
$59$	−1.03999	−0.135395	−0.0676974	+	0.997706i	$0.521565\pi$
$60$	−1.80357	−0.232839
$61$	10.5284	1.34802	0.674012	−	0.738720i	$-0.264570\pi$
$61$	10.5284	1.34802	0.674012	+	0.738720i	$0.264570\pi$
$62$	−5.14531	−0.653455
$63$	0	0
$64$	3.43440	0.429300
$65$	−7.27590	−0.902465
$66$	3.89536	0.479486
$67$	−11.2479	−1.37416	−0.687078	−	0.726584i	$-0.741107\pi$
$67$	−11.2479	−1.37416	−0.687078	+	0.726584i	$0.741107\pi$
$68$	−5.35830	−0.649790
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.62821	0.430589	0.215295	−	0.976549i	$-0.430929\pi$
$71$	3.62821	0.430589	0.215295	+	0.976549i	$0.430929\pi$
$72$	2.67837	0.315649
$73$	16.2125	1.89753	0.948765	−	0.315983i	$-0.102334\pi$
$73$	16.2125	1.89753	0.948765	+	0.315983i	$0.102334\pi$
$74$	7.64202	0.888367
$75$	3.26017	0.376452
$76$	−6.39742	−0.733834
$77$	0	0
$78$	4.38749	0.496786
$79$	11.0959	1.24839	0.624193	−	0.781270i	$-0.285428\pi$
$79$	11.0959	1.24839	0.624193	+	0.781270i	$0.285428\pi$
$80$	−0.797126	−0.0891214
$81$	1.00000	0.111111
$82$	−6.79716	−0.750621
$83$	−3.53762	−0.388305	−0.194152	−	0.980971i	$-0.562196\pi$
$83$	−3.53762	−0.388305	−0.194152	+	0.980971i	$0.562196\pi$
$84$	0	0
$85$	−5.16895	−0.560651
$86$	−3.12019	−0.336459
$87$	−6.49718	−0.696571
$88$	13.1170	1.39828
$89$	−16.7543	−1.77595	−0.887974	−	0.459894i	$-0.847888\pi$
$89$	−16.7543	−1.77595	−0.887974	+	0.459894i	$0.847888\pi$
$90$	1.04915	0.110590
$91$	0	0
$92$	−1.36735	−0.142556
$93$	−6.46887	−0.670791
$94$	6.63502	0.684350
$95$	−6.17134	−0.633167
$96$	5.83742	0.595779
$97$	−4.67418	−0.474591	−0.237296	−	0.971437i	$-0.576261\pi$
$97$	−4.67418	−0.474591	−0.237296	+	0.971437i	$0.576261\pi$
$98$	0	0
$99$	4.89739	0.492207
$100$	4.45778	0.445778
$101$	5.20722	0.518138	0.259069	−	0.965859i	$-0.416584\pi$
$101$	5.20722	0.518138	0.259069	+	0.965859i	$0.416584\pi$
$102$	3.11696	0.308625
$103$	−10.3751	−1.02229	−0.511144	−	0.859495i	$-0.670778\pi$
$103$	−10.3751	−1.02229	−0.511144	+	0.859495i	$0.670778\pi$
$104$	14.7742	1.44873
$105$	0	0
$106$	4.12602	0.400755
$107$	6.52924	0.631205	0.315603	−	0.948891i	$-0.397793\pi$
$107$	6.52924	0.631205	0.315603	+	0.948891i	$0.397793\pi$
$108$	1.36735	0.131573
$109$	−8.55728	−0.819639	−0.409819	−	0.912167i	$-0.634408\pi$
$109$	−8.55728	−0.819639	−0.409819	+	0.912167i	$0.634408\pi$
$110$	5.13809	0.489897
$111$	9.60783	0.911935
$112$	0	0
$113$	−7.74311	−0.728410	−0.364205	−	0.931319i	$-0.618659\pi$
$113$	−7.74311	−0.728410	−0.364205	+	0.931319i	$0.618659\pi$
$114$	3.72142	0.348543
$115$	−1.31903	−0.123000
$116$	−8.88389	−0.824849
$117$	5.51611	0.509965
$118$	0.827200	0.0761500
$119$	0	0
$120$	3.53284	0.322503
$121$	12.9845	1.18041
$122$	−8.37424	−0.758168
$123$	−8.54564	−0.770535
$124$	−8.84519	−0.794321
$125$	10.8954	0.974513
$126$	0	0
$127$	13.5910	1.20601	0.603005	−	0.797737i	$-0.293970\pi$
$127$	13.5910	1.20601	0.603005	+	0.797737i	$0.293970\pi$
$128$	8.94314	0.790469
$129$	−3.92282	−0.345385
$130$	5.78722	0.507573
$131$	20.0681	1.75336	0.876678	−	0.481077i	$-0.159754\pi$
$131$	20.0681	1.75336	0.876678	+	0.481077i	$0.159754\pi$
$132$	6.69643	0.582850
$133$	0	0
$134$	8.94656	0.772865
$135$	1.31903	0.113524
$136$	10.4959	0.900015
$137$	16.0942	1.37502	0.687508	−	0.726176i	$-0.258704\pi$
$137$	16.0942	1.37502	0.687508	+	0.726176i	$0.258704\pi$
$138$	0.795395	0.0677085
$139$	7.69854	0.652981	0.326491	−	0.945200i	$-0.394134\pi$
$139$	7.69854	0.652981	0.326491	+	0.945200i	$0.394134\pi$
$140$	0	0
$141$	8.34179	0.702506
$142$	−2.88586	−0.242176
$143$	27.0146	2.25907
$144$	0.604329	0.0503607
$145$	−8.56995	−0.711696
$146$	−12.8953	−1.06723
$147$	0	0
$148$	13.1372	1.07987
$149$	−15.8229	−1.29626	−0.648130	−	0.761530i	$-0.724449\pi$
$149$	−15.8229	−1.29626	−0.648130	+	0.761530i	$0.724449\pi$
$150$	−2.59312	−0.211728
$151$	−16.8398	−1.37040	−0.685202	−	0.728353i	$-0.740286\pi$
$151$	−16.8398	−1.37040	−0.685202	+	0.728353i	$0.740286\pi$
$152$	12.5313	1.01642
$153$	3.91876	0.316813
$154$	0	0
$155$	−8.53261	−0.685356
$156$	7.54244	0.603878
$157$	0.0325284	0.00259605	0.00129802	−	0.999999i	$-0.499587\pi$
$157$	0.0325284	0.00259605	0.00129802	+	0.999999i	$0.499587\pi$
$158$	−8.82563	−0.702129
$159$	5.18739	0.411387
$160$	7.69972	0.608716
$161$	0	0
$162$	−0.795395	−0.0624922
$163$	14.5128	1.13673	0.568364	−	0.822777i	$-0.307576\pi$
$163$	14.5128	1.13673	0.568364	+	0.822777i	$0.307576\pi$
$164$	−11.6849	−0.912434
$165$	6.45979	0.502894
$166$	2.81381	0.218394
$167$	−13.0534	−1.01011	−0.505053	−	0.863088i	$-0.668527\pi$
$167$	−13.0534	−1.01011	−0.505053	+	0.863088i	$0.668527\pi$
$168$	0	0
$169$	17.4275	1.34058
$170$	4.11136	0.315327
$171$	4.67871	0.357790
$172$	−5.36385	−0.408990
$173$	−6.89352	−0.524104	−0.262052	−	0.965054i	$-0.584399\pi$
$173$	−6.89352	−0.524104	−0.262052	+	0.965054i	$0.584399\pi$
$174$	5.16783	0.391772
$175$	0	0
$176$	2.95964	0.223091
$177$	1.03999	0.0781702
$178$	13.3263	0.998845
$179$	−21.5060	−1.60743	−0.803717	−	0.595012i	$-0.797147\pi$
$179$	−21.5060	−1.60743	−0.803717	+	0.595012i	$0.797147\pi$
$180$	1.80357	0.134430
$181$	4.99998	0.371645	0.185823	−	0.982583i	$-0.440505\pi$
$181$	4.99998	0.371645	0.185823	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	−10.5284	−0.778282
$184$	2.67837	0.197452
$185$	12.6730	0.931736
$186$	5.14531	0.377272
$187$	19.1917	1.40344
$188$	11.4061	0.831877
$189$	0	0
$190$	4.90866	0.356112
$191$	−12.3877	−0.896341	−0.448171	−	0.893948i	$-0.647924\pi$
$191$	−12.3877	−0.896341	−0.448171	+	0.893948i	$0.647924\pi$
$192$	−3.43440	−0.247857
$193$	−3.88085	−0.279349	−0.139675	−	0.990197i	$-0.544606\pi$
$193$	−3.88085	−0.279349	−0.139675	+	0.990197i	$0.544606\pi$
$194$	3.71782	0.266924
$195$	7.27590	0.521038
$196$	0	0
$197$	−15.6628	−1.11593	−0.557963	−	0.829866i	$-0.688417\pi$
$197$	−15.6628	−1.11593	−0.557963	+	0.829866i	$0.688417\pi$
$198$	−3.89536	−0.276831
$199$	14.9452	1.05944	0.529719	−	0.848173i	$-0.322298\pi$
$199$	14.9452	1.05944	0.529719	+	0.848173i	$0.322298\pi$
$200$	−8.73194	−0.617442
$201$	11.2479	0.793369
$202$	−4.14180	−0.291416
$203$	0	0
$204$	5.35830	0.375156
$205$	−11.2719	−0.787266
$206$	8.25231	0.574965
$207$	1.00000	0.0695048
$208$	3.33355	0.231140
$209$	22.9135	1.58496
$210$	0	0
$211$	−23.5341	−1.62015	−0.810076	−	0.586324i	$-0.800574\pi$
$211$	−23.5341	−1.62015	−0.810076	+	0.586324i	$0.800574\pi$
$212$	7.09296	0.487146
$213$	−3.62821	−0.248601
$214$	−5.19333	−0.355008
$215$	−5.17430	−0.352884
$216$	−2.67837	−0.182240
$217$	0	0
$218$	6.80642	0.460989
$219$	−16.2125	−1.09554
$220$	8.83277	0.595505
$221$	21.6163	1.45407
$222$	−7.64202	−0.512899
$223$	5.59390	0.374595	0.187298	−	0.982303i	$-0.440027\pi$
$223$	5.59390	0.374595	0.187298	+	0.982303i	$0.440027\pi$
$224$	0	0
$225$	−3.26017	−0.217345
$226$	6.15883	0.409679
$227$	−15.8433	−1.05156	−0.525778	−	0.850622i	$-0.676226\pi$
$227$	−15.8433	−1.05156	−0.525778	+	0.850622i	$0.676226\pi$
$228$	6.39742	0.423679
$229$	12.8109	0.846569	0.423285	−	0.905997i	$-0.360877\pi$
$229$	12.8109	0.846569	0.423285	+	0.905997i	$0.360877\pi$
$230$	1.04915	0.0691787
$231$	0	0
$232$	17.4019	1.14249
$233$	−10.5405	−0.690533	−0.345267	−	0.938505i	$-0.612211\pi$
$233$	−10.5405	−0.690533	−0.345267	+	0.938505i	$0.612211\pi$
$234$	−4.38749	−0.286819
$235$	11.0030	0.717760
$236$	1.42202	0.0925657
$237$	−11.0959	−0.720756
$238$	0	0
$239$	22.0075	1.42354	0.711772	−	0.702410i	$-0.247893\pi$
$239$	22.0075	1.42354	0.711772	+	0.702410i	$0.247893\pi$
$240$	0.797126	0.0514542
$241$	1.60171	0.103175	0.0515876	−	0.998668i	$-0.483572\pi$
$241$	1.60171	0.103175	0.0515876	+	0.998668i	$0.483572\pi$
$242$	−10.3278	−0.663895
$243$	−1.00000	−0.0641500
$244$	−14.3960	−0.921608
$245$	0	0
$246$	6.79716	0.433371
$247$	25.8083	1.64214
$248$	17.3260	1.10020
$249$	3.53762	0.224188
$250$	−8.66614	−0.548095
$251$	−12.5579	−0.792648	−0.396324	−	0.918111i	$-0.629714\pi$
$251$	−12.5579	−0.792648	−0.396324	+	0.918111i	$0.629714\pi$
$252$	0	0
$253$	4.89739	0.307896
$254$	−10.8103	−0.678296
$255$	5.16895	0.323692
$256$	−13.9821	−0.873883
$257$	21.1974	1.32225	0.661127	−	0.750274i	$-0.270078\pi$
$257$	21.1974	1.32225	0.661127	+	0.750274i	$0.270078\pi$
$258$	3.12019	0.194255
$259$	0	0
$260$	9.94868	0.616991
$261$	6.49718	0.402165
$262$	−15.9621	−0.986139
$263$	−11.6953	−0.721160	−0.360580	−	0.932728i	$-0.617421\pi$
$263$	−11.6953	−0.721160	−0.360580	+	0.932728i	$0.617421\pi$
$264$	−13.1170	−0.807298
$265$	6.84230	0.420319
$266$	0	0
$267$	16.7543	1.02534
$268$	15.3798	0.939473
$269$	−13.6435	−0.831860	−0.415930	−	0.909397i	$-0.636544\pi$
$269$	−13.6435	−0.831860	−0.415930	+	0.909397i	$0.636544\pi$
$270$	−1.04915	−0.0638491
$271$	−2.16490	−0.131509	−0.0657543	−	0.997836i	$-0.520945\pi$
$271$	−2.16490	−0.131509	−0.0657543	+	0.997836i	$0.520945\pi$
$272$	2.36822	0.143594
$273$	0	0
$274$	−12.8012	−0.773350
$275$	−15.9663	−0.962806
$276$	1.36735	0.0823046
$277$	29.5308	1.77433	0.887166	−	0.461451i	$-0.152671\pi$
$277$	29.5308	1.77433	0.887166	+	0.461451i	$0.152671\pi$
$278$	−6.12338	−0.367256
$279$	6.46887	0.387281
$280$	0	0
$281$	−27.3139	−1.62941	−0.814705	−	0.579876i	$-0.803101\pi$
$281$	−27.3139	−1.62941	−0.814705	+	0.579876i	$0.803101\pi$
$282$	−6.63502	−0.395110
$283$	19.3681	1.15131	0.575657	−	0.817691i	$-0.304746\pi$
$283$	19.3681	1.15131	0.575657	+	0.817691i	$0.304746\pi$
$284$	−4.96102	−0.294382
$285$	6.17134	0.365559
$286$	−21.4873	−1.27057
$287$	0	0
$288$	−5.83742	−0.343973
$289$	−1.64332	−0.0966661
$290$	6.81650	0.400279
$291$	4.67418	0.274005
$292$	−22.1681	−1.29729
$293$	−4.51942	−0.264027	−0.132014	−	0.991248i	$-0.542144\pi$
$293$	−4.51942	−0.264027	−0.132014	+	0.991248i	$0.542144\pi$
$294$	0	0
$295$	1.37177	0.0798675
$296$	−25.7333	−1.49572
$297$	−4.89739	−0.284176
$298$	12.5854	0.729055
$299$	5.51611	0.319005
$300$	−4.45778	−0.257370
$301$	0	0
$302$	13.3943	0.770756
$303$	−5.20722	−0.299147
$304$	2.82748	0.162167
$305$	−13.8872	−0.795181
$306$	−3.11696	−0.178185
$307$	−31.6101	−1.80409	−0.902043	−	0.431646i	$-0.857933\pi$
$307$	−31.6101	−1.80409	−0.902043	+	0.431646i	$0.857933\pi$
$308$	0	0
$309$	10.3751	0.590219
$310$	6.78680	0.385464
$311$	−5.03731	−0.285640	−0.142820	−	0.989749i	$-0.545617\pi$
$311$	−5.03731	−0.285640	−0.142820	+	0.989749i	$0.545617\pi$
$312$	−14.7742	−0.836425
$313$	32.1635	1.81799	0.908994	−	0.416809i	$-0.136852\pi$
$313$	32.1635	1.81799	0.908994	+	0.416809i	$0.136852\pi$
$314$	−0.0258729	−0.00146009
$315$	0	0
$316$	−15.1719	−0.853488
$317$	−5.74052	−0.322420	−0.161210	−	0.986920i	$-0.551540\pi$
$317$	−5.74052	−0.322420	−0.161210	+	0.986920i	$0.551540\pi$
$318$	−4.12602	−0.231376
$319$	31.8192	1.78154
$320$	−4.53007	−0.253238
$321$	−6.52924	−0.364427
$322$	0	0
$323$	18.3347	1.02017
$324$	−1.36735	−0.0759637
$325$	−17.9835	−0.997543
$326$	−11.5434	−0.639329
$327$	8.55728	0.473219
$328$	22.8884	1.26380
$329$	0	0
$330$	−5.13809	−0.282842
$331$	18.4175	1.01232	0.506160	−	0.862440i	$-0.331065\pi$
$331$	18.4175	1.01232	0.506160	+	0.862440i	$0.331065\pi$
$332$	4.83716	0.265474
$333$	−9.60783	−0.526506
$334$	10.3826	0.568113
$335$	14.8363	0.810596
$336$	0	0
$337$	30.8244	1.67911	0.839557	−	0.543271i	$-0.182815\pi$
$337$	30.8244	1.67911	0.839557	+	0.543271i	$0.182815\pi$
$338$	−13.8618	−0.753981
$339$	7.74311	0.420548
$340$	7.06774	0.383302
$341$	31.6806	1.71560
$342$	−3.72142	−0.201232
$343$	0	0
$344$	10.5068	0.566487
$345$	1.31903	0.0710140
$346$	5.48307	0.294772
$347$	15.7805	0.847139	0.423570	−	0.905863i	$-0.360777\pi$
$347$	15.7805	0.847139	0.423570	+	0.905863i	$0.360777\pi$
$348$	8.88389	0.476227
$349$	8.33122	0.445960	0.222980	−	0.974823i	$-0.428422\pi$
$349$	8.33122	0.445960	0.222980	+	0.974823i	$0.428422\pi$
$350$	0	0
$351$	−5.51611	−0.294428
$352$	−28.5882	−1.52375
$353$	−17.1771	−0.914246	−0.457123	−	0.889403i	$-0.651120\pi$
$353$	−17.1771	−0.914246	−0.457123	+	0.889403i	$0.651120\pi$
$354$	−0.827200	−0.0439652
$355$	−4.78570	−0.253999
$356$	22.9089	1.21417
$357$	0	0
$358$	17.1058	0.904068
$359$	−9.36781	−0.494414	−0.247207	−	0.968963i	$-0.579513\pi$
$359$	−9.36781	−0.494414	−0.247207	+	0.968963i	$0.579513\pi$
$360$	−3.53284	−0.186197
$361$	2.89034	0.152123
$362$	−3.97696	−0.209024
$363$	−12.9845	−0.681507
$364$	0	0
$365$	−21.3847	−1.11933
$366$	8.37424	0.437729
$367$	10.8026	0.563892	0.281946	−	0.959430i	$-0.409020\pi$
$367$	10.8026	0.563892	0.281946	+	0.959430i	$0.409020\pi$
$368$	0.604329	0.0315028
$369$	8.54564	0.444868
$370$	−10.0800	−0.524036
$371$	0	0
$372$	8.84519	0.458602
$373$	−28.7847	−1.49042	−0.745208	−	0.666832i	$-0.767650\pi$
$373$	−28.7847	−1.49042	−0.745208	+	0.666832i	$0.767650\pi$
$374$	−15.2650	−0.789334
$375$	−10.8954	−0.562635
$376$	−22.3424	−1.15222
$377$	35.8392	1.84581
$378$	0	0
$379$	−2.41325	−0.123960	−0.0619802	−	0.998077i	$-0.519742\pi$
$379$	−2.41325	−0.123960	−0.0619802	+	0.998077i	$0.519742\pi$
$380$	8.43837	0.432879
$381$	−13.5910	−0.696290
$382$	9.85310	0.504129
$383$	−6.14458	−0.313973	−0.156987	−	0.987601i	$-0.550178\pi$
$383$	−6.14458	−0.313973	−0.156987	+	0.987601i	$0.550178\pi$
$384$	−8.94314	−0.456378
$385$	0	0
$386$	3.08681	0.157114
$387$	3.92282	0.199408
$388$	6.39123	0.324465
$389$	−1.64382	−0.0833448	−0.0416724	−	0.999131i	$-0.513269\pi$
$389$	−1.64382	−0.0833448	−0.0416724	+	0.999131i	$0.513269\pi$
$390$	−5.78722	−0.293047
$391$	3.91876	0.198180
$392$	0	0
$393$	−20.0681	−1.01230
$394$	12.4581	0.627629
$395$	−14.6358	−0.736406
$396$	−6.69643	−0.336508
$397$	−19.4988	−0.978615	−0.489308	−	0.872111i	$-0.662750\pi$
$397$	−19.4988	−0.978615	−0.489308	+	0.872111i	$0.662750\pi$
$398$	−11.8873	−0.595859
$399$	0	0
$400$	−1.97021	−0.0985107
$401$	−5.44197	−0.271759	−0.135879	−	0.990725i	$-0.543386\pi$
$401$	−5.44197	−0.271759	−0.135879	+	0.990725i	$0.543386\pi$
$402$	−8.94656	−0.446214
$403$	35.6830	1.77750
$404$	−7.12008	−0.354237
$405$	−1.31903	−0.0655430
$406$	0	0
$407$	−47.0533	−2.33235
$408$	−10.4959	−0.519624
$409$	5.48304	0.271119	0.135559	−	0.990769i	$-0.456717\pi$
$409$	5.48304	0.271119	0.135559	+	0.990769i	$0.456717\pi$
$410$	8.96564	0.442781
$411$	−16.0942	−0.793866
$412$	14.1864	0.698912
$413$	0	0
$414$	−0.795395	−0.0390915
$415$	4.66622	0.229056
$416$	−32.1999	−1.57873
$417$	−7.69854	−0.376999
$418$	−18.2253	−0.891428
$419$	25.2750	1.23476	0.617382	−	0.786663i	$-0.288193\pi$
$419$	25.2750	1.23476	0.617382	+	0.786663i	$0.288193\pi$
$420$	0	0
$421$	−8.05217	−0.392439	−0.196219	−	0.980560i	$-0.562867\pi$
$421$	−8.05217	−0.392439	−0.196219	+	0.980560i	$0.562867\pi$
$422$	18.7189	0.911222
$423$	−8.34179	−0.405592
$424$	−13.8938	−0.674740
$425$	−12.7758	−0.619718
$426$	2.88586	0.139820
$427$	0	0
$428$	−8.92773	−0.431538
$429$	−27.0146	−1.30428
$430$	4.11561	0.198473
$431$	−0.356660	−0.0171797	−0.00858985	−	0.999963i	$-0.502734\pi$
$431$	−0.356660	−0.0171797	−0.00858985	+	0.999963i	$0.502734\pi$
$432$	−0.604329	−0.0290758
$433$	−30.0466	−1.44395	−0.721975	−	0.691919i	$-0.756765\pi$
$433$	−30.0466	−1.44395	−0.721975	+	0.691919i	$0.756765\pi$
$434$	0	0
$435$	8.56995	0.410898
$436$	11.7008	0.560365
$437$	4.67871	0.223813
$438$	12.8953	0.616163
$439$	28.3861	1.35479	0.677397	−	0.735618i	$-0.263108\pi$
$439$	28.3861	1.35479	0.677397	+	0.735618i	$0.263108\pi$
$440$	−17.3017	−0.824827
$441$	0	0
$442$	−17.1935	−0.817813
$443$	32.8789	1.56213	0.781063	−	0.624453i	$-0.214678\pi$
$443$	32.8789	1.56213	0.781063	+	0.624453i	$0.214678\pi$
$444$	−13.1372	−0.623465
$445$	22.0993	1.04761
$446$	−4.44936	−0.210683
$447$	15.8229	0.748396
$448$	0	0
$449$	−35.3812	−1.66974	−0.834872	−	0.550444i	$-0.814458\pi$
$449$	−35.3812	−1.66974	−0.834872	+	0.550444i	$0.814458\pi$
$450$	2.59312	0.122241
$451$	41.8514	1.97070
$452$	10.5875	0.497995
$453$	16.8398	0.791203
$454$	12.6017	0.591426
$455$	0	0
$456$	−12.5313	−0.586833
$457$	9.46688	0.442842	0.221421	−	0.975178i	$-0.428931\pi$
$457$	9.46688	0.442842	0.221421	+	0.975178i	$0.428931\pi$
$458$	−10.1897	−0.476135
$459$	−3.91876	−0.182912
$460$	1.80357	0.0840917
$461$	−3.99489	−0.186061	−0.0930303	−	0.995663i	$-0.529655\pi$
$461$	−3.99489	−0.186061	−0.0930303	+	0.995663i	$0.529655\pi$
$462$	0	0
$463$	−4.77992	−0.222142	−0.111071	−	0.993812i	$-0.535428\pi$
$463$	−4.77992	−0.222142	−0.111071	+	0.993812i	$0.535428\pi$
$464$	3.92643	0.182280
$465$	8.53261	0.395690
$466$	8.38389	0.388376
$467$	−12.8331	−0.593847	−0.296923	−	0.954901i	$-0.595961\pi$
$467$	−12.8331	−0.593847	−0.296923	+	0.954901i	$0.595961\pi$
$468$	−7.54244	−0.348649
$469$	0	0
$470$	−8.75177	−0.403689
$471$	−0.0325284	−0.00149883
$472$	−2.78547	−0.128212
$473$	19.2116	0.883349
$474$	8.82563	0.405374
$475$	−15.2534	−0.699873
$476$	0	0
$477$	−5.18739	−0.237514
$478$	−17.5046	−0.800643
$479$	0.398589	0.0182120	0.00910600	−	0.999959i	$-0.497101\pi$
$479$	0.398589	0.0182120	0.00910600	+	0.999959i	$0.497101\pi$
$480$	−7.69972	−0.351442
$481$	−52.9979	−2.41650
$482$	−1.27399	−0.0580288
$483$	0	0
$484$	−17.7543	−0.807012
$485$	6.16537	0.279955
$486$	0.795395	0.0360799
$487$	2.59910	0.117777	0.0588883	−	0.998265i	$-0.481244\pi$
$487$	2.59910	0.117777	0.0588883	+	0.998265i	$0.481244\pi$
$488$	28.1990	1.27651
$489$	−14.5128	−0.656290
$490$	0	0
$491$	6.27020	0.282970	0.141485	−	0.989940i	$-0.454812\pi$
$491$	6.27020	0.282970	0.141485	+	0.989940i	$0.454812\pi$
$492$	11.6849	0.526794
$493$	25.4609	1.14670
$494$	−20.5278	−0.923590
$495$	−6.45979	−0.290346
$496$	3.90932	0.175534
$497$	0	0
$498$	−2.81381	−0.126090
$499$	5.47089	0.244911	0.122455	−	0.992474i	$-0.460923\pi$
$499$	5.47089	0.244911	0.122455	+	0.992474i	$0.460923\pi$
$500$	−14.8978	−0.666248
$501$	13.0534	0.583185
$502$	9.98850	0.445809
$503$	−12.6490	−0.563993	−0.281996	−	0.959415i	$-0.590997\pi$
$503$	−12.6490	−0.563993	−0.281996	+	0.959415i	$0.590997\pi$
$504$	0	0
$505$	−6.86847	−0.305643
$506$	−3.89536	−0.173170
$507$	−17.4275	−0.773983
$508$	−18.5837	−0.824517
$509$	−8.13556	−0.360602	−0.180301	−	0.983611i	$-0.557707\pi$
$509$	−8.13556	−0.360602	−0.180301	+	0.983611i	$0.557707\pi$
$510$	−4.11136	−0.182054
$511$	0	0
$512$	−6.76496	−0.298972
$513$	−4.67871	−0.206570
$514$	−16.8603	−0.743675
$515$	13.6850	0.603035
$516$	5.36385	0.236130
$517$	−40.8530	−1.79671
$518$	0	0
$519$	6.89352	0.302592
$520$	−19.4876	−0.854586
$521$	3.95156	0.173121	0.0865606	−	0.996247i	$-0.472412\pi$
$521$	3.95156	0.173121	0.0865606	+	0.996247i	$0.472412\pi$
$522$	−5.16783	−0.226190
$523$	39.1771	1.71310	0.856548	−	0.516068i	$-0.172605\pi$
$523$	39.1771	1.71310	0.856548	+	0.516068i	$0.172605\pi$
$524$	−27.4400	−1.19872
$525$	0	0
$526$	9.30235	0.405602
$527$	25.3500	1.10426
$528$	−2.95964	−0.128802
$529$	1.00000	0.0434783
$530$	−5.44234	−0.236400
$531$	−1.03999	−0.0451316
$532$	0	0
$533$	47.1387	2.04181
$534$	−13.3263	−0.576684
$535$	−8.61224	−0.372340
$536$	−30.1262	−1.30125
$537$	21.5060	0.928052
$538$	10.8520	0.467863
$539$	0	0
$540$	−1.80357	−0.0776131
$541$	26.5703	1.14235	0.571174	−	0.820829i	$-0.306488\pi$
$541$	26.5703	1.14235	0.571174	+	0.820829i	$0.306488\pi$
$542$	1.72195	0.0739643
$543$	−4.99998	−0.214570
$544$	−22.8755	−0.980777
$545$	11.2873	0.483494
$546$	0	0
$547$	−14.2706	−0.610169	−0.305084	−	0.952325i	$-0.598685\pi$
$547$	−14.2706	−0.610169	−0.305084	+	0.952325i	$0.598685\pi$
$548$	−22.0063	−0.940062
$549$	10.5284	0.449341
$550$	12.6995	0.541510
$551$	30.3984	1.29502
$552$	−2.67837	−0.113999
$553$	0	0
$554$	−23.4886	−0.997936
$555$	−12.6730	−0.537938
$556$	−10.5266	−0.446426
$557$	−3.08610	−0.130762	−0.0653812	−	0.997860i	$-0.520826\pi$
$557$	−3.08610	−0.130762	−0.0653812	+	0.997860i	$0.520826\pi$
$558$	−5.14531	−0.217818
$559$	21.6387	0.915220
$560$	0	0
$561$	−19.1917	−0.810274
$562$	21.7253	0.916428
$563$	37.3769	1.57525	0.787625	−	0.616155i	$-0.211310\pi$
$563$	37.3769	1.57525	0.787625	+	0.616155i	$0.211310\pi$
$564$	−11.4061	−0.480284
$565$	10.2134	0.429680
$566$	−15.4053	−0.647533
$567$	0	0
$568$	9.71769	0.407745
$569$	37.0665	1.55391	0.776954	−	0.629557i	$-0.216763\pi$
$569$	37.0665	1.55391	0.776954	+	0.629557i	$0.216763\pi$
$570$	−4.90866	−0.205601
$571$	37.4228	1.56609	0.783047	−	0.621962i	$-0.213664\pi$
$571$	37.4228	1.56609	0.783047	+	0.621962i	$0.213664\pi$
$572$	−36.9383	−1.54447
$573$	12.3877	0.517503
$574$	0	0
$575$	−3.26017	−0.135958
$576$	3.43440	0.143100
$577$	13.3089	0.554057	0.277028	−	0.960862i	$-0.410650\pi$
$577$	13.3089	0.554057	0.277028	+	0.960862i	$0.410650\pi$
$578$	1.30709	0.0543678
$579$	3.88085	0.161282
$580$	11.7181	0.486567
$581$	0	0
$582$	−3.71782	−0.154109
$583$	−25.4047	−1.05215
$584$	43.4231	1.79686
$585$	−7.27590	−0.300822
$586$	3.59472	0.148497
$587$	−8.54186	−0.352560	−0.176280	−	0.984340i	$-0.556406\pi$
$587$	−8.54186	−0.352560	−0.176280	+	0.984340i	$0.556406\pi$
$588$	0	0
$589$	30.2660	1.24709
$590$	−1.09110	−0.0449198
$591$	15.6628	0.644280
$592$	−5.80629	−0.238637
$593$	−12.8396	−0.527261	−0.263630	−	0.964624i	$-0.584920\pi$
$593$	−12.8396	−0.527261	−0.263630	+	0.964624i	$0.584920\pi$
$594$	3.89536	0.159829
$595$	0	0
$596$	21.6353	0.886218
$597$	−14.9452	−0.611666
$598$	−4.38749	−0.179418
$599$	31.1203	1.27154	0.635771	−	0.771877i	$-0.280682\pi$
$599$	31.1203	1.27154	0.635771	+	0.771877i	$0.280682\pi$
$600$	8.73194	0.356480
$601$	−14.6600	−0.597993	−0.298996	−	0.954254i	$-0.596652\pi$
$601$	−14.6600	−0.597993	−0.298996	+	0.954254i	$0.596652\pi$
$602$	0	0
$603$	−11.2479	−0.458052
$604$	23.0259	0.936909
$605$	−17.1268	−0.696305
$606$	4.14180	0.168249
$607$	−15.9205	−0.646195	−0.323097	−	0.946366i	$-0.604724\pi$
$607$	−15.9205	−0.646195	−0.323097	+	0.946366i	$0.604724\pi$
$608$	−27.3116	−1.10763
$609$	0	0
$610$	11.0458	0.447233
$611$	−46.0143	−1.86154
$612$	−5.35830	−0.216597
$613$	−18.5970	−0.751127	−0.375563	−	0.926797i	$-0.622551\pi$
$613$	−18.5970	−0.751127	−0.375563	+	0.926797i	$0.622551\pi$
$614$	25.1426	1.01467
$615$	11.2719	0.454528
$616$	0	0
$617$	−15.7884	−0.635618	−0.317809	−	0.948155i	$-0.602947\pi$
$617$	−15.7884	−0.635618	−0.317809	+	0.948155i	$0.602947\pi$
$618$	−8.25231	−0.331956
$619$	2.54302	0.102213	0.0511064	−	0.998693i	$-0.483725\pi$
$619$	2.54302	0.102213	0.0511064	+	0.998693i	$0.483725\pi$
$620$	11.6670	0.468560
$621$	−1.00000	−0.0401286
$622$	4.00665	0.160652
$623$	0	0
$624$	−3.33355	−0.133449
$625$	1.92954	0.0771818
$626$	−25.5827	−1.02249
$627$	−22.9135	−0.915077
$628$	−0.0444775	−0.00177485
$629$	−37.6508	−1.50123
$630$	0	0
$631$	14.7377	0.586700	0.293350	−	0.956005i	$-0.405230\pi$
$631$	14.7377	0.586700	0.293350	+	0.956005i	$0.405230\pi$
$632$	29.7189	1.18216
$633$	23.5341	0.935396
$634$	4.56598	0.181338
$635$	−17.9270	−0.711409
$636$	−7.09296	−0.281254
$637$	0	0
$638$	−25.3089	−1.00199
$639$	3.62821	0.143530
$640$	−11.7962	−0.466287
$641$	8.35215	0.329890	0.164945	−	0.986303i	$-0.447255\pi$
$641$	8.35215	0.329890	0.164945	+	0.986303i	$0.447255\pi$
$642$	5.19333	0.204964
$643$	−30.6463	−1.20857	−0.604286	−	0.796767i	$-0.706542\pi$
$643$	−30.6463	−1.20857	−0.604286	+	0.796767i	$0.706542\pi$
$644$	0	0
$645$	5.17430	0.203738
$646$	−14.5834	−0.573775
$647$	−27.5490	−1.08306	−0.541532	−	0.840680i	$-0.682156\pi$
$647$	−27.5490	−1.08306	−0.541532	+	0.840680i	$0.682156\pi$
$648$	2.67837	0.105216
$649$	−5.09322	−0.199926
$650$	14.3040	0.561048
$651$	0	0
$652$	−19.8440	−0.777151
$653$	−8.41277	−0.329217	−0.164609	−	0.986359i	$-0.552636\pi$
$653$	−8.41277	−0.329217	−0.164609	+	0.986359i	$0.552636\pi$
$654$	−6.80642	−0.266152
$655$	−26.4703	−1.03428
$656$	5.16438	0.201635
$657$	16.2125	0.632510
$658$	0	0
$659$	6.90882	0.269130	0.134565	−	0.990905i	$-0.457036\pi$
$659$	6.90882	0.269130	0.134565	+	0.990905i	$0.457036\pi$
$660$	−8.83277	−0.343815
$661$	23.2257	0.903374	0.451687	−	0.892176i	$-0.350822\pi$
$661$	23.2257	0.903374	0.451687	+	0.892176i	$0.350822\pi$
$662$	−14.6492	−0.569358
$663$	−21.6163	−0.839509
$664$	−9.47507	−0.367704
$665$	0	0
$666$	7.64202	0.296122
$667$	6.49718	0.251572
$668$	17.8486	0.690582
$669$	−5.59390	−0.216273
$670$	−11.8008	−0.455903
$671$	51.5617	1.99052
$672$	0	0
$673$	−13.6248	−0.525196	−0.262598	−	0.964905i	$-0.584579\pi$
$673$	−13.6248	−0.525196	−0.262598	+	0.964905i	$0.584579\pi$
$674$	−24.5176	−0.944383
$675$	3.26017	0.125484
$676$	−23.8295	−0.916518
$677$	−30.7375	−1.18134	−0.590668	−	0.806914i	$-0.701136\pi$
$677$	−30.7375	−1.18134	−0.590668	+	0.806914i	$0.701136\pi$
$678$	−6.15883	−0.236529
$679$	0	0
$680$	−13.8444	−0.530907
$681$	15.8433	0.607116
$682$	−25.1986	−0.964905
$683$	18.4684	0.706674	0.353337	−	0.935496i	$-0.385047\pi$
$683$	18.4684	0.706674	0.353337	+	0.935496i	$0.385047\pi$
$684$	−6.39742	−0.244611
$685$	−21.2286	−0.811104
$686$	0	0
$687$	−12.8109	−0.488767
$688$	2.37067	0.0903810
$689$	−28.6142	−1.09012
$690$	−1.04915	−0.0399404
$691$	−11.7488	−0.446945	−0.223472	−	0.974710i	$-0.571739\pi$
$691$	−11.7488	−0.446945	−0.223472	+	0.974710i	$0.571739\pi$
$692$	9.42583	0.358316
$693$	0	0
$694$	−12.5517	−0.476456
$695$	−10.1546	−0.385185
$696$	−17.4019	−0.659616
$697$	33.4883	1.26846
$698$	−6.62661	−0.250821
$699$	10.5405	0.398679
$700$	0	0
$701$	−27.0080	−1.02008	−0.510038	−	0.860152i	$-0.670369\pi$
$701$	−27.0080	−1.02008	−0.510038	+	0.860152i	$0.670369\pi$
$702$	4.38749	0.165595
$703$	−44.9522	−1.69541
$704$	16.8196	0.633913
$705$	−11.0030	−0.414399
$706$	13.6626	0.514199
$707$	0	0
$708$	−1.42202	−0.0534428
$709$	−13.3674	−0.502025	−0.251012	−	0.967984i	$-0.580764\pi$
$709$	−13.3674	−0.502025	−0.251012	+	0.967984i	$0.580764\pi$
$710$	3.80653	0.142856
$711$	11.0959	0.416129
$712$	−44.8741	−1.68173
$713$	6.46887	0.242261
$714$	0	0
$715$	−35.6330	−1.33260
$716$	29.4062	1.09896
$717$	−22.0075	−0.821884
$718$	7.45111	0.278073
$719$	18.4113	0.686625	0.343312	−	0.939221i	$-0.388451\pi$
$719$	18.4113	0.686625	0.343312	+	0.939221i	$0.388451\pi$
$720$	−0.797126	−0.0297071
$721$	0	0
$722$	−2.29896	−0.0855586
$723$	−1.60171	−0.0595683
$724$	−6.83670	−0.254084
$725$	−21.1819	−0.786676
$726$	10.3278	0.383300
$727$	5.98605	0.222010	0.111005	−	0.993820i	$-0.464593\pi$
$727$	5.98605	0.222010	0.111005	+	0.993820i	$0.464593\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	17.0093	0.629543
$731$	15.3726	0.568575
$732$	14.3960	0.532091
$733$	6.78462	0.250596	0.125298	−	0.992119i	$-0.460011\pi$
$733$	6.78462	0.250596	0.125298	+	0.992119i	$0.460011\pi$
$734$	−8.59236	−0.317150
$735$	0	0
$736$	−5.83742	−0.215170
$737$	−55.0856	−2.02910
$738$	−6.79716	−0.250207
$739$	−43.1092	−1.58580	−0.792899	−	0.609354i	$-0.791429\pi$
$739$	−43.1092	−1.58580	−0.792899	+	0.609354i	$0.791429\pi$
$740$	−17.3284	−0.637003
$741$	−25.8083	−0.948092
$742$	0	0
$743$	8.67528	0.318265	0.159133	−	0.987257i	$-0.449130\pi$
$743$	8.67528	0.318265	0.159133	+	0.987257i	$0.449130\pi$
$744$	−17.3260	−0.635203
$745$	20.8708	0.764646
$746$	22.8952	0.838254
$747$	−3.53762	−0.129435
$748$	−26.2417	−0.959492
$749$	0	0
$750$	8.66614	0.316443
$751$	26.0656	0.951148	0.475574	−	0.879676i	$-0.342240\pi$
$751$	26.0656	0.951148	0.475574	+	0.879676i	$0.342240\pi$
$752$	−5.04118	−0.183833
$753$	12.5579	0.457636
$754$	−28.5063	−1.03814
$755$	22.2122	0.808383
$756$	0	0
$757$	3.00105	0.109075	0.0545375	−	0.998512i	$-0.482632\pi$
$757$	3.00105	0.109075	0.0545375	+	0.998512i	$0.482632\pi$
$758$	1.91949	0.0697190
$759$	−4.89739	−0.177764
$760$	−16.5292	−0.599575
$761$	17.8557	0.647270	0.323635	−	0.946182i	$-0.395095\pi$
$761$	17.8557	0.647270	0.323635	+	0.946182i	$0.395095\pi$
$762$	10.8103	0.391614
$763$	0	0
$764$	16.9382	0.612804
$765$	−5.16895	−0.186884
$766$	4.88737	0.176588
$767$	−5.73668	−0.207140
$768$	13.9821	0.504537
$769$	37.9875	1.36986	0.684931	−	0.728608i	$-0.259832\pi$
$769$	37.9875	1.36986	0.684931	+	0.728608i	$0.259832\pi$
$770$	0	0
$771$	−21.1974	−0.763404
$772$	5.30646	0.190984
$773$	−20.0770	−0.722120	−0.361060	−	0.932543i	$-0.617585\pi$
$773$	−20.0770	−0.722120	−0.361060	+	0.932543i	$0.617585\pi$
$774$	−3.12019	−0.112153
$775$	−21.0896	−0.757561
$776$	−12.5192	−0.449413
$777$	0	0
$778$	1.30748	0.0468755
$779$	39.9826	1.43253
$780$	−9.94868	−0.356220
$781$	17.7688	0.635816
$782$	−3.11696	−0.111462
$783$	−6.49718	−0.232190
$784$	0	0
$785$	−0.0429058	−0.00153137
$786$	15.9621	0.569348
$787$	3.19313	0.113823	0.0569113	−	0.998379i	$-0.481875\pi$
$787$	3.19313	0.113823	0.0569113	+	0.998379i	$0.481875\pi$
$788$	21.4164	0.762928
$789$	11.6953	0.416362
$790$	11.6412	0.414177
$791$	0	0
$792$	13.1170	0.466094
$793$	58.0759	2.06234
$794$	15.5092	0.550402
$795$	−6.84230	−0.242672
$796$	−20.4353	−0.724309
$797$	−40.8078	−1.44549	−0.722744	−	0.691116i	$-0.757119\pi$
$797$	−40.8078	−1.44549	−0.722744	+	0.691116i	$0.757119\pi$
$798$	0	0
$799$	−32.6895	−1.15647
$800$	19.0310	0.672847
$801$	−16.7543	−0.591983
$802$	4.32851	0.152845
$803$	79.3990	2.80193
$804$	−15.3798	−0.542405
$805$	0	0
$806$	−28.3821	−0.999718
$807$	13.6435	0.480275
$808$	13.9469	0.490650
$809$	4.42507	0.155577	0.0777886	−	0.996970i	$-0.475214\pi$
$809$	4.42507	0.155577	0.0777886	+	0.996970i	$0.475214\pi$
$810$	1.04915	0.0368633
$811$	23.5604	0.827318	0.413659	−	0.910432i	$-0.364251\pi$
$811$	23.5604	0.827318	0.413659	+	0.910432i	$0.364251\pi$
$812$	0	0
$813$	2.16490	0.0759265
$814$	37.4260	1.31178
$815$	−19.1427	−0.670541
$816$	−2.36822	−0.0829042
$817$	18.3537	0.642116
$818$	−4.36118	−0.152485
$819$	0	0
$820$	15.4126	0.538232
$821$	51.9940	1.81460	0.907301	−	0.420482i	$-0.138139\pi$
$821$	51.9940	1.81460	0.907301	+	0.420482i	$0.138139\pi$
$822$	12.8012	0.446494
$823$	50.8197	1.77146	0.885731	−	0.464199i	$-0.153658\pi$
$823$	50.8197	1.77146	0.885731	+	0.464199i	$0.153658\pi$
$824$	−27.7884	−0.968054
$825$	15.9663	0.555876
$826$	0	0
$827$	−11.5427	−0.401378	−0.200689	−	0.979655i	$-0.564318\pi$
$827$	−11.5427	−0.401378	−0.200689	+	0.979655i	$0.564318\pi$
$828$	−1.36735	−0.0475186
$829$	−49.4730	−1.71827	−0.859135	−	0.511749i	$-0.828998\pi$
$829$	−49.4730	−1.71827	−0.859135	+	0.511749i	$0.828998\pi$
$830$	−3.71149	−0.128828
$831$	−29.5308	−1.02441
$832$	18.9446	0.656784
$833$	0	0
$834$	6.12338	0.212035
$835$	17.2178	0.595848
$836$	−31.3307	−1.08359
$837$	−6.46887	−0.223597
$838$	−20.1036	−0.694468
$839$	−24.7142	−0.853228	−0.426614	−	0.904434i	$-0.640294\pi$
$839$	−24.7142	−0.853228	−0.426614	+	0.904434i	$0.640294\pi$
$840$	0	0
$841$	13.2133	0.455633
$842$	6.40466	0.220719
$843$	27.3139	0.940740
$844$	32.1792	1.10766
$845$	−22.9874	−0.790789
$846$	6.63502	0.228117
$847$	0	0
$848$	−3.13489	−0.107653
$849$	−19.3681	−0.664711
$850$	10.1618	0.348548
$851$	−9.60783	−0.329352
$852$	4.96102	0.169962
$853$	47.0244	1.61009	0.805043	−	0.593216i	$-0.202142\pi$
$853$	47.0244	1.61009	0.805043	+	0.593216i	$0.202142\pi$
$854$	0	0
$855$	−6.17134	−0.211056
$856$	17.4877	0.597718
$857$	3.95329	0.135042	0.0675210	−	0.997718i	$-0.478491\pi$
$857$	3.95329	0.135042	0.0675210	+	0.997718i	$0.478491\pi$
$858$	21.4873	0.733563
$859$	−13.9000	−0.474262	−0.237131	−	0.971478i	$-0.576207\pi$
$859$	−13.9000	−0.474262	−0.237131	+	0.971478i	$0.576207\pi$
$860$	7.07506	0.241258
$861$	0	0
$862$	0.283686	0.00966237
$863$	20.6227	0.702004	0.351002	−	0.936375i	$-0.385841\pi$
$863$	20.6227	0.702004	0.351002	+	0.936375i	$0.385841\pi$
$864$	5.83742	0.198593
$865$	9.09273	0.309162
$866$	23.8990	0.812120
$867$	1.64332	0.0558102
$868$	0	0
$869$	54.3410	1.84339
$870$	−6.81650	−0.231101
$871$	−62.0450	−2.10231
$872$	−22.9196	−0.776155
$873$	−4.67418	−0.158197
$874$	−3.72142	−0.125879
$875$	0	0
$876$	22.1681	0.748991
$877$	−21.6839	−0.732213	−0.366106	−	0.930573i	$-0.619309\pi$
$877$	−21.6839	−0.732213	−0.366106	+	0.930573i	$0.619309\pi$
$878$	−22.5781	−0.761976
$879$	4.51942	0.152436
$880$	−3.90384	−0.131598
$881$	−21.2107	−0.714607	−0.357304	−	0.933988i	$-0.616304\pi$
$881$	−21.2107	−0.714607	−0.357304	+	0.933988i	$0.616304\pi$
$882$	0	0
$883$	−28.0744	−0.944778	−0.472389	−	0.881390i	$-0.656608\pi$
$883$	−28.0744	−0.944778	−0.472389	+	0.881390i	$0.656608\pi$
$884$	−29.5570	−0.994110
$885$	−1.37177	−0.0461115
$886$	−26.1517	−0.878585
$887$	6.97793	0.234296	0.117148	−	0.993114i	$-0.462625\pi$
$887$	6.97793	0.234296	0.117148	+	0.993114i	$0.462625\pi$
$888$	25.7333	0.863554
$889$	0	0
$890$	−17.5777	−0.589206
$891$	4.89739	0.164069
$892$	−7.64880	−0.256101
$893$	−39.0288	−1.30605
$894$	−12.5854	−0.420920
$895$	28.3670	0.948204
$896$	0	0
$897$	−5.51611	−0.184178
$898$	28.1421	0.939113
$899$	42.0294	1.40176
$900$	4.45778	0.148593
$901$	−20.3281	−0.677228
$902$	−33.2884	−1.10838
$903$	0	0
$904$	−20.7389	−0.689766
$905$	−6.59510	−0.219229
$906$	−13.3943	−0.444996
$907$	−36.0155	−1.19587	−0.597937	−	0.801543i	$-0.704013\pi$
$907$	−36.0155	−1.19587	−0.597937	+	0.801543i	$0.704013\pi$
$908$	21.6633	0.718920
$909$	5.20722	0.172713
$910$	0	0
$911$	−36.1014	−1.19609	−0.598046	−	0.801461i	$-0.704056\pi$
$911$	−36.1014	−1.19609	−0.598046	+	0.801461i	$0.704056\pi$
$912$	−2.82748	−0.0936272
$913$	−17.3251	−0.573378
$914$	−7.52991	−0.249067
$915$	13.8872	0.459098
$916$	−17.5170	−0.578777
$917$	0	0
$918$	3.11696	0.102875
$919$	22.7572	0.750691	0.375345	−	0.926885i	$-0.377524\pi$
$919$	22.7572	0.750691	0.375345	+	0.926885i	$0.377524\pi$
$920$	−3.53284	−0.116474
$921$	31.6101	1.04159
$922$	3.17752	0.104646
$923$	20.0136	0.658756
$924$	0	0
$925$	31.3231	1.02990
$926$	3.80192	0.124939
$927$	−10.3751	−0.340763
$928$	−37.9268	−1.24501
$929$	15.8519	0.520084	0.260042	−	0.965597i	$-0.416264\pi$
$929$	15.8519	0.520084	0.260042	+	0.965597i	$0.416264\pi$
$930$	−6.78680	−0.222548
$931$	0	0
$932$	14.4126	0.472099
$933$	5.03731	0.164914
$934$	10.2074	0.333997
$935$	−25.3144	−0.827868
$936$	14.7742	0.482910
$937$	−36.9250	−1.20629	−0.603143	−	0.797633i	$-0.706085\pi$
$937$	−36.9250	−1.20629	−0.603143	+	0.797633i	$0.706085\pi$
$938$	0	0
$939$	−32.1635	−1.04962
$940$	−15.0450	−0.490713
$941$	−33.9743	−1.10753	−0.553765	−	0.832673i	$-0.686809\pi$
$941$	−33.9743	−1.10753	−0.553765	+	0.832673i	$0.686809\pi$
$942$	0.0258729	0.000842985 0
$943$	8.54564	0.278284
$944$	−0.628494	−0.0204557
$945$	0	0
$946$	−15.2808	−0.496822
$947$	−4.47620	−0.145457	−0.0727285	−	0.997352i	$-0.523171\pi$
$947$	−4.47620	−0.145457	−0.0727285	+	0.997352i	$0.523171\pi$
$948$	15.1719	0.492762
$949$	89.4300	2.90302
$950$	12.1325	0.393629
$951$	5.74052	0.186149
$952$	0	0
$953$	−2.38191	−0.0771576	−0.0385788	−	0.999256i	$-0.512283\pi$
$953$	−2.38191	−0.0771576	−0.0385788	+	0.999256i	$0.512283\pi$
$954$	4.12602	0.133585
$955$	16.3397	0.528740
$956$	−30.0918	−0.973239
$957$	−31.8192	−1.02857
$958$	−0.317036	−0.0102430
$959$	0	0
$960$	4.53007	0.146207
$961$	10.8463	0.349881
$962$	42.1543	1.35911
$963$	6.52924	0.210402
$964$	−2.19009	−0.0705382
$965$	5.11894	0.164785
$966$	0	0
$967$	−2.14760	−0.0690623	−0.0345312	−	0.999404i	$-0.510994\pi$
$967$	−2.14760	−0.0690623	−0.0345312	+	0.999404i	$0.510994\pi$
$968$	34.7772	1.11778
$969$	−18.3347	−0.588997
$970$	−4.90391	−0.157455
$971$	9.14640	0.293522	0.146761	−	0.989172i	$-0.453115\pi$
$971$	9.14640	0.293522	0.146761	+	0.989172i	$0.453115\pi$
$972$	1.36735	0.0438577
$973$	0	0
$974$	−2.06731	−0.0662410
$975$	17.9835	0.575932
$976$	6.36262	0.203662
$977$	−27.4555	−0.878379	−0.439190	−	0.898394i	$-0.644734\pi$
$977$	−27.4555	−0.878379	−0.439190	+	0.898394i	$0.644734\pi$
$978$	11.5434	0.369117
$979$	−82.0522	−2.62240
$980$	0	0
$981$	−8.55728	−0.273213
$982$	−4.98729	−0.159151
$983$	−24.5201	−0.782070	−0.391035	−	0.920376i	$-0.627883\pi$
$983$	−24.5201	−0.782070	−0.391035	+	0.920376i	$0.627883\pi$
$984$	−22.8884	−0.729656
$985$	20.6596	0.658269
$986$	−20.2515	−0.644938
$987$	0	0
$988$	−35.2889	−1.12269
$989$	3.92282	0.124738
$990$	5.13809	0.163299
$991$	−2.64828	−0.0841253	−0.0420626	−	0.999115i	$-0.513393\pi$
$991$	−2.64828	−0.0841253	−0.0420626	+	0.999115i	$0.513393\pi$
$992$	−37.7615	−1.19893
$993$	−18.4175	−0.584463
$994$	0	0
$995$	−19.7131	−0.624948
$996$	−4.83716	−0.153271
$997$	−23.9305	−0.757886	−0.378943	−	0.925420i	$-0.623712\pi$
$997$	−23.9305	−0.757886	−0.378943	+	0.925420i	$0.623712\pi$
$998$	−4.35152	−0.137745
$999$	9.60783	0.303978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bg.1.5	✓	10
7.6	odd	2		3381.2.a.bh.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.bg.1.5	✓	10	1.1	even	1	trivial
3381.2.a.bh.1.5	yes	10	7.6	odd	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-0.795395 q^{2} -1.00000 q^{3} -1.36735 q^{4} -1.31903 q^{5} +0.795395 q^{6} +2.67837 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.795395 q^{2} -1.00000 q^{3} -1.36735 q^{4} -1.31903 q^{5} +0.795395 q^{6} +2.67837 q^{8} +1.00000 q^{9} +1.04915 q^{10} +4.89739 q^{11} +1.36735 q^{12} +5.51611 q^{13} +1.31903 q^{15} +0.604329 q^{16} +3.91876 q^{17} -0.795395 q^{18} +4.67871 q^{19} +1.80357 q^{20} -3.89536 q^{22} +1.00000 q^{23} -2.67837 q^{24} -3.26017 q^{25} -4.38749 q^{26} -1.00000 q^{27} +6.49718 q^{29} -1.04915 q^{30} +6.46887 q^{31} -5.83742 q^{32} -4.89739 q^{33} -3.11696 q^{34} -1.36735 q^{36} -9.60783 q^{37} -3.72142 q^{38} -5.51611 q^{39} -3.53284 q^{40} +8.54564 q^{41} +3.92282 q^{43} -6.69643 q^{44} -1.31903 q^{45} -0.795395 q^{46} -8.34179 q^{47} -0.604329 q^{48} +2.59312 q^{50} -3.91876 q^{51} -7.54244 q^{52} -5.18739 q^{53} +0.795395 q^{54} -6.45979 q^{55} -4.67871 q^{57} -5.16783 q^{58} -1.03999 q^{59} -1.80357 q^{60} +10.5284 q^{61} -5.14531 q^{62} +3.43440 q^{64} -7.27590 q^{65} +3.89536 q^{66} -11.2479 q^{67} -5.35830 q^{68} -1.00000 q^{69} +3.62821 q^{71} +2.67837 q^{72} +16.2125 q^{73} +7.64202 q^{74} +3.26017 q^{75} -6.39742 q^{76} +4.38749 q^{78} +11.0959 q^{79} -0.797126 q^{80} +1.00000 q^{81} -6.79716 q^{82} -3.53762 q^{83} -5.16895 q^{85} -3.12019 q^{86} -6.49718 q^{87} +13.1170 q^{88} -16.7543 q^{89} +1.04915 q^{90} -1.36735 q^{92} -6.46887 q^{93} +6.63502 q^{94} -6.17134 q^{95} +5.83742 q^{96} -4.67418 q^{97} +4.89739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} - 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q - 4 * q^2 - 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 12 * q^8 + 10 * q^9	\( 10 q - 4 q^{2} - 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 12 q^{8} + 10 q^{9} + 8 q^{10} - 2 q^{11} - 8 q^{12} + 16 q^{13} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 4 q^{18} + 26 q^{19} - 8 q^{22} + 10 q^{23} + 12 q^{24} + 14 q^{25} - 12 q^{26} - 10 q^{27} - 16 q^{29} - 8 q^{30} + 20 q^{31} - 8 q^{32} + 2 q^{33} - 4 q^{34} + 8 q^{36} + 8 q^{37} - 8 q^{38} - 16 q^{39} - 12 q^{40} + 22 q^{41} - 4 q^{43} - 24 q^{44} + 4 q^{45} - 4 q^{46} + 6 q^{47} - 4 q^{48} - 48 q^{50} - 12 q^{51} + 24 q^{52} - 30 q^{53} + 4 q^{54} + 48 q^{55} - 26 q^{57} + 24 q^{58} + 42 q^{59} + 14 q^{61} + 40 q^{62} + 8 q^{64} - 44 q^{65} + 8 q^{66} + 8 q^{68} - 10 q^{69} + 8 q^{71} - 12 q^{72} + 24 q^{73} + 8 q^{74} - 14 q^{75} + 32 q^{76} + 12 q^{78} + 32 q^{79} + 28 q^{80} + 10 q^{81} - 64 q^{82} + 28 q^{83} - 4 q^{85} - 4 q^{86} + 16 q^{87} + 20 q^{88} + 8 q^{90} + 8 q^{92} - 20 q^{93} + 8 q^{94} - 16 q^{95} + 8 q^{96} - 12 q^{97} - 2 q^{99}+O(q^{100}) \) 10 * q - 4 * q^2 - 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 12 * q^8 + 10 * q^9 + 8 * q^10 - 2 * q^11 - 8 * q^12 + 16 * q^13 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 4 * q^18 + 26 * q^19 - 8 * q^22 + 10 * q^23 + 12 * q^24 + 14 * q^25 - 12 * q^26 - 10 * q^27 - 16 * q^29 - 8 * q^30 + 20 * q^31 - 8 * q^32 + 2 * q^33 - 4 * q^34 + 8 * q^36 + 8 * q^37 - 8 * q^38 - 16 * q^39 - 12 * q^40 + 22 * q^41 - 4 * q^43 - 24 * q^44 + 4 * q^45 - 4 * q^46 + 6 * q^47 - 4 * q^48 - 48 * q^50 - 12 * q^51 + 24 * q^52 - 30 * q^53 + 4 * q^54 + 48 * q^55 - 26 * q^57 + 24 * q^58 + 42 * q^59 + 14 * q^61 + 40 * q^62 + 8 * q^64 - 44 * q^65 + 8 * q^66 + 8 * q^68 - 10 * q^69 + 8 * q^71 - 12 * q^72 + 24 * q^73 + 8 * q^74 - 14 * q^75 + 32 * q^76 + 12 * q^78 + 32 * q^79 + 28 * q^80 + 10 * q^81 - 64 * q^82 + 28 * q^83 - 4 * q^85 - 4 * q^86 + 16 * q^87 + 20 * q^88 + 8 * q^90 + 8 * q^92 - 20 * q^93 + 8 * q^94 - 16 * q^95 + 8 * q^96 - 12 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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