LMFDB - Embedded newform 3381.2.a.bb.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.62622704.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 5x^{4} + 13x^{3} + 9x^{2} - 5x - 2 $ x^6 - 3x^5 - 5x^4 + 13x^3 + 9x^2 - 5*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.52648$ 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)

$f(q)$	$=$	$q+1.52648 q^{2} +1.00000 q^{3} +0.330154 q^{4} +0.362203 q^{5} +1.52648 q^{6} -2.54899 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.52648 q^{2} +1.00000 q^{3} +0.330154 q^{4} +0.362203 q^{5} +1.52648 q^{6} -2.54899 q^{8} +1.00000 q^{9} +0.552896 q^{10} -1.14818 q^{11} +0.330154 q^{12} -3.55853 q^{13} +0.362203 q^{15} -4.55131 q^{16} -4.81035 q^{17} +1.52648 q^{18} +4.61068 q^{19} +0.119582 q^{20} -1.75268 q^{22} -1.00000 q^{23} -2.54899 q^{24} -4.86881 q^{25} -5.43204 q^{26} +1.00000 q^{27} +1.29122 q^{29} +0.552896 q^{30} -9.28398 q^{31} -1.84951 q^{32} -1.14818 q^{33} -7.34292 q^{34} +0.330154 q^{36} -9.08983 q^{37} +7.03813 q^{38} -3.55853 q^{39} -0.923252 q^{40} -6.09893 q^{41} +6.31350 q^{43} -0.379076 q^{44} +0.362203 q^{45} -1.52648 q^{46} -0.0368774 q^{47} -4.55131 q^{48} -7.43216 q^{50} -4.81035 q^{51} -1.17486 q^{52} +8.82952 q^{53} +1.52648 q^{54} -0.415874 q^{55} +4.61068 q^{57} +1.97102 q^{58} -3.94636 q^{59} +0.119582 q^{60} +2.77141 q^{61} -14.1718 q^{62} +6.27937 q^{64} -1.28891 q^{65} -1.75268 q^{66} +11.7895 q^{67} -1.58815 q^{68} -1.00000 q^{69} -9.56084 q^{71} -2.54899 q^{72} +0.157279 q^{73} -13.8755 q^{74} -4.86881 q^{75} +1.52223 q^{76} -5.43204 q^{78} +8.32568 q^{79} -1.64849 q^{80} +1.00000 q^{81} -9.30993 q^{82} -16.3582 q^{83} -1.74232 q^{85} +9.63746 q^{86} +1.29122 q^{87} +2.92670 q^{88} -1.02159 q^{89} +0.552896 q^{90} -0.330154 q^{92} -9.28398 q^{93} -0.0562928 q^{94} +1.67000 q^{95} -1.84951 q^{96} -7.41172 q^{97} -1.14818 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 6 q^{3} + 7 q^{4} - 2 q^{5} - 3 q^{6} - 9 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 3 * q^2 + 6 * q^3 + 7 * q^4 - 2 * q^5 - 3 * q^6 - 9 * q^8 + 6 * q^9	$ 6 q - 3 q^{2} + 6 q^{3} + 7 q^{4} - 2 q^{5} - 3 q^{6} - 9 q^{8} + 6 q^{9} - 8 q^{10} - 3 q^{11} + 7 q^{12} - 2 q^{15} + 5 q^{16} - 8 q^{17} - 3 q^{18} - 19 q^{19} + 24 q^{22} - 6 q^{23} - 9 q^{24} - 20 q^{26} + 6 q^{27} - 12 q^{29} - 8 q^{30} + 2 q^{31} - 41 q^{32} - 3 q^{33} - 2 q^{34} + 7 q^{36} - 10 q^{37} + 20 q^{38} - 22 q^{40} - 3 q^{41} - 2 q^{43} - 14 q^{44} - 2 q^{45} + 3 q^{46} - 7 q^{47} + 5 q^{48} + 17 q^{50} - 8 q^{51} + 44 q^{52} - 5 q^{53} - 3 q^{54} - 24 q^{55} - 19 q^{57} - 12 q^{58} - 21 q^{59} - 29 q^{61} - 10 q^{62} + 59 q^{64} - 18 q^{65} + 24 q^{66} + 16 q^{67} - 54 q^{68} - 6 q^{69} - 6 q^{71} - 9 q^{72} + 8 q^{73} - 16 q^{74} - 40 q^{76} - 20 q^{78} - 12 q^{79} + 78 q^{80} + 6 q^{81} - 44 q^{82} - 24 q^{83} + 10 q^{85} + 12 q^{86} - 12 q^{87} + 28 q^{88} - 18 q^{89} - 8 q^{90} - 7 q^{92} + 2 q^{93} + 8 q^{94} + 6 q^{95} - 41 q^{96} - 22 q^{97} - 3 q^{99}+O(q^{100}) $ 6 * q - 3 * q^2 + 6 * q^3 + 7 * q^4 - 2 * q^5 - 3 * q^6 - 9 * q^8 + 6 * q^9 - 8 * q^10 - 3 * q^11 + 7 * q^12 - 2 * q^15 + 5 * q^16 - 8 * q^17 - 3 * q^18 - 19 * q^19 + 24 * q^22 - 6 * q^23 - 9 * q^24 - 20 * q^26 + 6 * q^27 - 12 * q^29 - 8 * q^30 + 2 * q^31 - 41 * q^32 - 3 * q^33 - 2 * q^34 + 7 * q^36 - 10 * q^37 + 20 * q^38 - 22 * q^40 - 3 * q^41 - 2 * q^43 - 14 * q^44 - 2 * q^45 + 3 * q^46 - 7 * q^47 + 5 * q^48 + 17 * q^50 - 8 * q^51 + 44 * q^52 - 5 * q^53 - 3 * q^54 - 24 * q^55 - 19 * q^57 - 12 * q^58 - 21 * q^59 - 29 * q^61 - 10 * q^62 + 59 * q^64 - 18 * q^65 + 24 * q^66 + 16 * q^67 - 54 * q^68 - 6 * q^69 - 6 * q^71 - 9 * q^72 + 8 * q^73 - 16 * q^74 - 40 * q^76 - 20 * q^78 - 12 * q^79 + 78 * q^80 + 6 * q^81 - 44 * q^82 - 24 * q^83 + 10 * q^85 + 12 * q^86 - 12 * q^87 + 28 * q^88 - 18 * q^89 - 8 * q^90 - 7 * q^92 + 2 * q^93 + 8 * q^94 + 6 * q^95 - 41 * q^96 - 22 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.52648	1.07939	0.539694	−	0.841862i	$-0.318540\pi$
$2$	1.52648	1.07939	0.539694	+	0.841862i	$0.318540\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.330154	0.165077
$5$	0.362203	0.161982	0.0809909	−	0.996715i	$-0.474192\pi$
$5$	0.362203	0.161982	0.0809909	+	0.996715i	$0.474192\pi$
$6$	1.52648	0.623185
$7$	0	0
$7$	0	0
$8$	−2.54899	−0.901205
$9$	1.00000	0.333333
$10$	0.552896	0.174841
$11$	−1.14818	−0.346189	−0.173095	−	0.984905i	$-0.555377\pi$
$11$	−1.14818	−0.346189	−0.173095	+	0.984905i	$0.555377\pi$
$12$	0.330154	0.0953071
$13$	−3.55853	−0.986959	−0.493480	−	0.869757i	$-0.664275\pi$
$13$	−3.55853	−0.986959	−0.493480	+	0.869757i	$0.664275\pi$
$14$	0	0
$15$	0.362203	0.0935203
$16$	−4.55131	−1.13783
$17$	−4.81035	−1.16668	−0.583340	−	0.812228i	$-0.698255\pi$
$17$	−4.81035	−1.16668	−0.583340	+	0.812228i	$0.698255\pi$
$18$	1.52648	0.359796
$19$	4.61068	1.05776	0.528882	−	0.848696i	$-0.322612\pi$
$19$	4.61068	1.05776	0.528882	+	0.848696i	$0.322612\pi$
$20$	0.119582	0.0267395
$21$	0	0
$22$	−1.75268	−0.373672
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.54899	−0.520311
$25$	−4.86881	−0.973762
$26$	−5.43204	−1.06531
$27$	1.00000	0.192450
$28$	0	0
$29$	1.29122	0.239773	0.119887	−	0.992788i	$-0.461747\pi$
$29$	1.29122	0.239773	0.119887	+	0.992788i	$0.461747\pi$
$30$	0.552896	0.100945
$31$	−9.28398	−1.66745	−0.833726	−	0.552178i	$-0.813797\pi$
$31$	−9.28398	−1.66745	−0.833726	+	0.552178i	$0.813797\pi$
$32$	−1.84951	−0.326950
$33$	−1.14818	−0.199872
$34$	−7.34292	−1.25930
$35$	0	0
$36$	0.330154	0.0550256
$37$	−9.08983	−1.49436	−0.747180	−	0.664622i	$-0.768593\pi$
$37$	−9.08983	−1.49436	−0.747180	+	0.664622i	$0.768593\pi$
$38$	7.03813	1.14174
$39$	−3.55853	−0.569821
$40$	−0.923252	−0.145979
$41$	−6.09893	−0.952493	−0.476247	−	0.879312i	$-0.658003\pi$
$41$	−6.09893	−0.952493	−0.476247	+	0.879312i	$0.658003\pi$
$42$	0	0
$43$	6.31350	0.962800	0.481400	−	0.876501i	$-0.340128\pi$
$43$	6.31350	0.962800	0.481400	+	0.876501i	$0.340128\pi$
$44$	−0.379076	−0.0571478
$45$	0.362203	0.0539940
$46$	−1.52648	−0.225068
$47$	−0.0368774	−0.00537912	−0.00268956	−	0.999996i	$-0.500856\pi$
$47$	−0.0368774	−0.00537912	−0.00268956	+	0.999996i	$0.500856\pi$
$48$	−4.55131	−0.656924
$49$	0	0
$50$	−7.43216	−1.05107
$51$	−4.81035	−0.673583
$52$	−1.17486	−0.162924
$53$	8.82952	1.21283	0.606414	−	0.795149i	$-0.292607\pi$
$53$	8.82952	1.21283	0.606414	+	0.795149i	$0.292607\pi$
$54$	1.52648	0.207728
$55$	−0.415874	−0.0560764
$56$	0	0
$57$	4.61068	0.610700
$58$	1.97102	0.258808
$59$	−3.94636	−0.513772	−0.256886	−	0.966442i	$-0.582696\pi$
$59$	−3.94636	−0.513772	−0.256886	+	0.966442i	$0.582696\pi$
$60$	0.119582	0.0154380
$61$	2.77141	0.354843	0.177422	−	0.984135i	$-0.443224\pi$
$61$	2.77141	0.354843	0.177422	+	0.984135i	$0.443224\pi$
$62$	−14.1718	−1.79983
$63$	0	0
$64$	6.27937	0.784921
$65$	−1.28891	−0.159870
$66$	−1.75268	−0.215740
$67$	11.7895	1.44032	0.720161	−	0.693807i	$-0.244068\pi$
$67$	11.7895	1.44032	0.720161	+	0.693807i	$0.244068\pi$
$68$	−1.58815	−0.192592
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−9.56084	−1.13466	−0.567332	−	0.823489i	$-0.692024\pi$
$71$	−9.56084	−1.13466	−0.567332	+	0.823489i	$0.692024\pi$
$72$	−2.54899	−0.300402
$73$	0.157279	0.0184081	0.00920405	−	0.999958i	$-0.497070\pi$
$73$	0.157279	0.0184081	0.00920405	+	0.999958i	$0.497070\pi$
$74$	−13.8755	−1.61299
$75$	−4.86881	−0.562202
$76$	1.52223	0.174612
$77$	0	0
$78$	−5.43204	−0.615058
$79$	8.32568	0.936712	0.468356	−	0.883540i	$-0.344847\pi$
$79$	8.32568	0.936712	0.468356	+	0.883540i	$0.344847\pi$
$80$	−1.64849	−0.184307
$81$	1.00000	0.111111
$82$	−9.30993	−1.02811
$83$	−16.3582	−1.79554	−0.897772	−	0.440460i	$-0.854815\pi$
$83$	−16.3582	−1.79554	−0.897772	+	0.440460i	$0.854815\pi$
$84$	0	0
$85$	−1.74232	−0.188981
$86$	9.63746	1.03923
$87$	1.29122	0.138433
$88$	2.92670	0.311988
$89$	−1.02159	−0.108289	−0.0541444	−	0.998533i	$-0.517243\pi$
$89$	−1.02159	−0.108289	−0.0541444	+	0.998533i	$0.517243\pi$
$90$	0.552896	0.0582804
$91$	0	0
$92$	−0.330154	−0.0344209
$93$	−9.28398	−0.962704
$94$	−0.0562928	−0.00580616
$95$	1.67000	0.171338
$96$	−1.84951	−0.188765
$97$	−7.41172	−0.752546	−0.376273	−	0.926509i	$-0.622795\pi$
$97$	−7.41172	−0.752546	−0.376273	+	0.926509i	$0.622795\pi$
$98$	0	0
$99$	−1.14818	−0.115396
$100$	−1.60746	−0.160746
$101$	0.484421	0.0482017	0.0241009	−	0.999710i	$-0.492328\pi$
$101$	0.484421	0.0482017	0.0241009	+	0.999710i	$0.492328\pi$
$102$	−7.34292	−0.727057
$103$	−15.3646	−1.51392	−0.756961	−	0.653460i	$-0.773317\pi$
$103$	−15.3646	−1.51392	−0.756961	+	0.653460i	$0.773317\pi$
$104$	9.07068	0.889453
$105$	0	0
$106$	13.4781	1.30911
$107$	7.81576	0.755578	0.377789	−	0.925892i	$-0.376684\pi$
$107$	7.81576	0.755578	0.377789	+	0.925892i	$0.376684\pi$
$108$	0.330154	0.0317690
$109$	−5.14991	−0.493272	−0.246636	−	0.969108i	$-0.579325\pi$
$109$	−5.14991	−0.493272	−0.246636	+	0.969108i	$0.579325\pi$
$110$	−0.634825	−0.0605281
$111$	−9.08983	−0.862769
$112$	0	0
$113$	−9.04584	−0.850961	−0.425480	−	0.904968i	$-0.639895\pi$
$113$	−9.04584	−0.850961	−0.425480	+	0.904968i	$0.639895\pi$
$114$	7.03813	0.659182
$115$	−0.362203	−0.0337756
$116$	0.426300	0.0395810
$117$	−3.55853	−0.328986
$118$	−6.02405	−0.554559
$119$	0	0
$120$	−0.923252	−0.0842810
$121$	−9.68168	−0.880153
$122$	4.23052	0.383013
$123$	−6.09893	−0.549922
$124$	−3.06514	−0.275258
$125$	−3.57451	−0.319714
$126$	0	0
$127$	15.3531	1.36237	0.681186	−	0.732111i	$-0.261465\pi$
$127$	15.3531	1.36237	0.681186	+	0.732111i	$0.261465\pi$
$128$	13.2844	1.17418
$129$	6.31350	0.555873
$130$	−1.96750	−0.172561
$131$	11.3557	0.992151	0.496075	−	0.868280i	$-0.334774\pi$
$131$	11.3557	0.992151	0.496075	+	0.868280i	$0.334774\pi$
$132$	−0.379076	−0.0329943
$133$	0	0
$134$	17.9966	1.55467
$135$	0.362203	0.0311734
$136$	12.2615	1.05142
$137$	−9.72371	−0.830753	−0.415376	−	0.909650i	$-0.636350\pi$
$137$	−9.72371	−0.830753	−0.415376	+	0.909650i	$0.636350\pi$
$138$	−1.52648	−0.129943
$139$	−18.5445	−1.57293	−0.786463	−	0.617638i	$-0.788090\pi$
$139$	−18.5445	−1.57293	−0.786463	+	0.617638i	$0.788090\pi$
$140$	0	0
$141$	−0.0368774	−0.00310564
$142$	−14.5945	−1.22474
$143$	4.08584	0.341675
$144$	−4.55131	−0.379275
$145$	0.467682	0.0388389
$146$	0.240084	0.0198695
$147$	0	0
$148$	−3.00104	−0.246684
$149$	15.5066	1.27035	0.635175	−	0.772368i	$-0.280928\pi$
$149$	15.5066	1.27035	0.635175	+	0.772368i	$0.280928\pi$
$150$	−7.43216	−0.606833
$151$	17.6314	1.43482	0.717411	−	0.696651i	$-0.245327\pi$
$151$	17.6314	1.43482	0.717411	+	0.696651i	$0.245327\pi$
$152$	−11.7526	−0.953262
$153$	−4.81035	−0.388894
$154$	0	0
$155$	−3.36268	−0.270097
$156$	−1.17486	−0.0940643
$157$	−6.37508	−0.508786	−0.254393	−	0.967101i	$-0.581876\pi$
$157$	−6.37508	−0.508786	−0.254393	+	0.967101i	$0.581876\pi$
$158$	12.7090	1.01107
$159$	8.82952	0.700227
$160$	−0.669897	−0.0529600
$161$	0	0
$162$	1.52648	0.119932
$163$	14.3869	1.12687	0.563435	−	0.826160i	$-0.309480\pi$
$163$	14.3869	1.12687	0.563435	+	0.826160i	$0.309480\pi$
$164$	−2.01359	−0.157235
$165$	−0.415874	−0.0323757
$166$	−24.9705	−1.93809
$167$	−2.05019	−0.158649	−0.0793243	−	0.996849i	$-0.525276\pi$
$167$	−2.05019	−0.158649	−0.0793243	+	0.996849i	$0.525276\pi$
$168$	0	0
$169$	−0.336843	−0.0259110
$170$	−2.65962	−0.203984
$171$	4.61068	0.352588
$172$	2.08443	0.158936
$173$	18.8742	1.43498	0.717490	−	0.696569i	$-0.245291\pi$
$173$	18.8742	1.43498	0.717490	+	0.696569i	$0.245291\pi$
$174$	1.97102	0.149423
$175$	0	0
$176$	5.22572	0.393903
$177$	−3.94636	−0.296626
$178$	−1.55945	−0.116886
$179$	15.4735	1.15655	0.578274	−	0.815843i	$-0.303727\pi$
$179$	15.4735	1.15655	0.578274	+	0.815843i	$0.303727\pi$
$180$	0.119582	0.00891315
$181$	−18.0888	−1.34453	−0.672266	−	0.740310i	$-0.734679\pi$
$181$	−18.0888	−1.34453	−0.672266	+	0.740310i	$0.734679\pi$
$182$	0	0
$183$	2.77141	0.204869
$184$	2.54899	0.187914
$185$	−3.29236	−0.242059
$186$	−14.1718	−1.03913
$187$	5.52314	0.403892
$188$	−0.0121752	−0.000887968 0
$189$	0	0
$190$	2.54923	0.184941
$191$	−11.3386	−0.820436	−0.410218	−	0.911988i	$-0.634547\pi$
$191$	−11.3386	−0.820436	−0.410218	+	0.911988i	$0.634547\pi$
$192$	6.27937	0.453174
$193$	−9.25207	−0.665979	−0.332989	−	0.942931i	$-0.608057\pi$
$193$	−9.25207	−0.665979	−0.332989	+	0.942931i	$0.608057\pi$
$194$	−11.3139	−0.812289
$195$	−1.28891	−0.0923007
$196$	0	0
$197$	−16.7108	−1.19059	−0.595296	−	0.803506i	$-0.702965\pi$
$197$	−16.7108	−1.19059	−0.595296	+	0.803506i	$0.702965\pi$
$198$	−1.75268	−0.124557
$199$	13.4796	0.955546	0.477773	−	0.878483i	$-0.341444\pi$
$199$	13.4796	0.955546	0.477773	+	0.878483i	$0.341444\pi$
$200$	12.4106	0.877559
$201$	11.7895	0.831570
$202$	0.739461	0.0520283
$203$	0	0
$204$	−1.58815	−0.111193
$205$	−2.20905	−0.154287
$206$	−23.4539	−1.63411
$207$	−1.00000	−0.0695048
$208$	16.1960	1.12299
$209$	−5.29389	−0.366186
$210$	0	0
$211$	2.29341	0.157885	0.0789426	−	0.996879i	$-0.474846\pi$
$211$	2.29341	0.157885	0.0789426	+	0.996879i	$0.474846\pi$
$212$	2.91510	0.200210
$213$	−9.56084	−0.655098
$214$	11.9306	0.815561
$215$	2.28677	0.155956
$216$	−2.54899	−0.173437
$217$	0	0
$218$	−7.86126	−0.532432
$219$	0.157279	0.0106279
$220$	−0.137302	−0.00925691
$221$	17.1178	1.15147
$222$	−13.8755	−0.931262
$223$	19.3317	1.29454	0.647272	−	0.762259i	$-0.275910\pi$
$223$	19.3317	1.29454	0.647272	+	0.762259i	$0.275910\pi$
$224$	0	0
$225$	−4.86881	−0.324587
$226$	−13.8083	−0.918516
$227$	4.17014	0.276782	0.138391	−	0.990378i	$-0.455807\pi$
$227$	4.17014	0.276782	0.138391	+	0.990378i	$0.455807\pi$
$228$	1.52223	0.100812
$229$	20.4191	1.34933	0.674667	−	0.738123i	$-0.264287\pi$
$229$	20.4191	1.34933	0.674667	+	0.738123i	$0.264287\pi$
$230$	−0.552896	−0.0364569
$231$	0	0
$232$	−3.29131	−0.216085
$233$	−6.36384	−0.416909	−0.208455	−	0.978032i	$-0.566843\pi$
$233$	−6.36384	−0.416909	−0.208455	+	0.978032i	$0.566843\pi$
$234$	−5.43204	−0.355104
$235$	−0.0133571	−0.000871320 0
$236$	−1.30290	−0.0848118
$237$	8.32568	0.540811
$238$	0	0
$239$	−0.437573	−0.0283043	−0.0141521	−	0.999900i	$-0.504505\pi$
$239$	−0.437573	−0.0283043	−0.0141521	+	0.999900i	$0.504505\pi$
$240$	−1.64849	−0.106410
$241$	21.3225	1.37350	0.686750	−	0.726893i	$-0.259037\pi$
$241$	21.3225	1.37350	0.686750	+	0.726893i	$0.259037\pi$
$242$	−14.7789	−0.950026
$243$	1.00000	0.0641500
$244$	0.914992	0.0585764
$245$	0	0
$246$	−9.30993	−0.593579
$247$	−16.4073	−1.04397
$248$	23.6648	1.50272
$249$	−16.3582	−1.03666
$250$	−5.45643	−0.345095
$251$	−19.2965	−1.21798	−0.608992	−	0.793176i	$-0.708426\pi$
$251$	−19.2965	−1.21798	−0.608992	+	0.793176i	$0.708426\pi$
$252$	0	0
$253$	1.14818	0.0721855
$254$	23.4363	1.47053
$255$	−1.74232	−0.109108
$256$	7.71965	0.482478
$257$	−31.7543	−1.98078	−0.990389	−	0.138307i	$-0.955834\pi$
$257$	−31.7543	−1.98078	−0.990389	+	0.138307i	$0.955834\pi$
$258$	9.63746	0.600002
$259$	0	0
$260$	−0.425538	−0.0263908
$261$	1.29122	0.0799244
$262$	17.3343	1.07091
$263$	−12.5428	−0.773421	−0.386711	−	0.922201i	$-0.626389\pi$
$263$	−12.5428	−0.773421	−0.386711	+	0.922201i	$0.626389\pi$
$264$	2.92670	0.180126
$265$	3.19808	0.196456
$266$	0	0
$267$	−1.02159	−0.0625206
$268$	3.89236	0.237764
$269$	8.77914	0.535273	0.267637	−	0.963520i	$-0.413757\pi$
$269$	8.77914	0.535273	0.267637	+	0.963520i	$0.413757\pi$
$270$	0.552896	0.0336482
$271$	−19.1142	−1.16110	−0.580552	−	0.814223i	$-0.697163\pi$
$271$	−19.1142	−1.16110	−0.580552	+	0.814223i	$0.697163\pi$
$272$	21.8934	1.32748
$273$	0	0
$274$	−14.8431	−0.896704
$275$	5.59027	0.337106
$276$	−0.330154	−0.0198729
$277$	−0.388387	−0.0233359	−0.0116680	−	0.999932i	$-0.503714\pi$
$277$	−0.388387	−0.0233359	−0.0116680	+	0.999932i	$0.503714\pi$
$278$	−28.3079	−1.69780
$279$	−9.28398	−0.555817
$280$	0	0
$281$	−17.6703	−1.05412	−0.527062	−	0.849827i	$-0.676706\pi$
$281$	−17.6703	−1.05412	−0.527062	+	0.849827i	$0.676706\pi$
$282$	−0.0562928	−0.00335219
$283$	−11.7855	−0.700576	−0.350288	−	0.936642i	$-0.613916\pi$
$283$	−11.7855	−0.700576	−0.350288	+	0.936642i	$0.613916\pi$
$284$	−3.15655	−0.187307
$285$	1.67000	0.0989223
$286$	6.23696	0.368799
$287$	0	0
$288$	−1.84951	−0.108983
$289$	6.13945	0.361144
$290$	0.713910	0.0419222
$291$	−7.41172	−0.434483
$292$	0.0519262	0.00303875
$293$	6.25525	0.365436	0.182718	−	0.983165i	$-0.441510\pi$
$293$	6.25525	0.365436	0.182718	+	0.983165i	$0.441510\pi$
$294$	0	0
$295$	−1.42938	−0.0832217
$296$	23.1699	1.34672
$297$	−1.14818	−0.0666242
$298$	23.6706	1.37120
$299$	3.55853	0.205795
$300$	−1.60746	−0.0928065
$301$	0	0
$302$	26.9140	1.54873
$303$	0.484421	0.0278293
$304$	−20.9846	−1.20355
$305$	1.00381	0.0574781
$306$	−7.34292	−0.419767
$307$	22.6051	1.29014	0.645072	−	0.764122i	$-0.276828\pi$
$307$	22.6051	1.29014	0.645072	+	0.764122i	$0.276828\pi$
$308$	0	0
$309$	−15.3646	−0.874063
$310$	−5.13308	−0.291539
$311$	18.2540	1.03509	0.517546	−	0.855656i	$-0.326846\pi$
$311$	18.2540	1.03509	0.517546	+	0.855656i	$0.326846\pi$
$312$	9.07068	0.513526
$313$	3.19524	0.180606	0.0903029	−	0.995914i	$-0.471216\pi$
$313$	3.19524	0.180606	0.0903029	+	0.995914i	$0.471216\pi$
$314$	−9.73145	−0.549178
$315$	0	0
$316$	2.74875	0.154629
$317$	28.0472	1.57529	0.787645	−	0.616130i	$-0.211300\pi$
$317$	28.0472	1.57529	0.787645	+	0.616130i	$0.211300\pi$
$318$	13.4781	0.755816
$319$	−1.48255	−0.0830069
$320$	2.27440	0.127143
$321$	7.81576	0.436233
$322$	0	0
$323$	−22.1790	−1.23407
$324$	0.330154	0.0183419
$325$	17.3258	0.961063
$326$	21.9614	1.21633
$327$	−5.14991	−0.284791
$328$	15.5461	0.858392
$329$	0	0
$330$	−0.634825	−0.0349459
$331$	−22.1074	−1.21513	−0.607567	−	0.794269i	$-0.707854\pi$
$331$	−22.1074	−1.21513	−0.607567	+	0.794269i	$0.707854\pi$
$332$	−5.40072	−0.296403
$333$	−9.08983	−0.498120
$334$	−3.12959	−0.171243
$335$	4.27020	0.233306
$336$	0	0
$337$	6.18599	0.336972	0.168486	−	0.985704i	$-0.446112\pi$
$337$	6.18599	0.336972	0.168486	+	0.985704i	$0.446112\pi$
$338$	−0.514186	−0.0279680
$339$	−9.04584	−0.491302
$340$	−0.575233	−0.0311964
$341$	10.6597	0.577254
$342$	7.03813	0.380579
$343$	0	0
$344$	−16.0931	−0.867681
$345$	−0.362203	−0.0195003
$346$	28.8112	1.54890
$347$	30.6463	1.64518	0.822591	−	0.568634i	$-0.192528\pi$
$347$	30.6463	1.64518	0.822591	+	0.568634i	$0.192528\pi$
$348$	0.426300	0.0228521
$349$	−14.4898	−0.775623	−0.387812	−	0.921739i	$-0.626769\pi$
$349$	−14.4898	−0.775623	−0.387812	+	0.921739i	$0.626769\pi$
$350$	0	0
$351$	−3.55853	−0.189940
$352$	2.12357	0.113187
$353$	19.2273	1.02337	0.511683	−	0.859174i	$-0.329022\pi$
$353$	19.2273	1.02337	0.511683	+	0.859174i	$0.329022\pi$
$354$	−6.02405	−0.320175
$355$	−3.46296	−0.183795
$356$	−0.337283	−0.0178760
$357$	0	0
$358$	23.6201	1.24836
$359$	−9.38911	−0.495538	−0.247769	−	0.968819i	$-0.579697\pi$
$359$	−9.38911	−0.495538	−0.247769	+	0.968819i	$0.579697\pi$
$360$	−0.923252	−0.0486597
$361$	2.25839	0.118863
$362$	−27.6123	−1.45127
$363$	−9.68168	−0.508157
$364$	0	0
$365$	0.0569668	0.00298178
$366$	4.23052	0.221133
$367$	−11.5737	−0.604142	−0.302071	−	0.953285i	$-0.597678\pi$
$367$	−11.5737	−0.604142	−0.302071	+	0.953285i	$0.597678\pi$
$368$	4.55131	0.237253
$369$	−6.09893	−0.317498
$370$	−5.02574	−0.261276
$371$	0	0
$372$	−3.06514	−0.158920
$373$	3.71521	0.192366	0.0961830	−	0.995364i	$-0.469337\pi$
$373$	3.71521	0.192366	0.0961830	+	0.995364i	$0.469337\pi$
$374$	8.43099	0.435956
$375$	−3.57451	−0.184587
$376$	0.0940003	0.00484769
$377$	−4.59484	−0.236646
$378$	0	0
$379$	14.6927	0.754715	0.377358	−	0.926068i	$-0.376833\pi$
$379$	14.6927	0.754715	0.377358	+	0.926068i	$0.376833\pi$
$380$	0.551357	0.0282840
$381$	15.3531	0.786565
$382$	−17.3083	−0.885568
$383$	5.25211	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$383$	5.25211	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$384$	13.2844	0.677915
$385$	0	0
$386$	−14.1231	−0.718849
$387$	6.31350	0.320933
$388$	−2.44701	−0.124228
$389$	−16.2374	−0.823268	−0.411634	−	0.911349i	$-0.635042\pi$
$389$	−16.2374	−0.823268	−0.411634	+	0.911349i	$0.635042\pi$
$390$	−1.96750	−0.0996282
$391$	4.81035	0.243270
$392$	0	0
$393$	11.3557	0.572818
$394$	−25.5087	−1.28511
$395$	3.01558	0.151730
$396$	−0.379076	−0.0190493
$397$	8.53398	0.428308	0.214154	−	0.976800i	$-0.431301\pi$
$397$	8.53398	0.428308	0.214154	+	0.976800i	$0.431301\pi$
$398$	20.5764	1.03140
$399$	0	0
$400$	22.1594	1.10797
$401$	−4.19425	−0.209451	−0.104725	−	0.994501i	$-0.533396\pi$
$401$	−4.19425	−0.209451	−0.104725	+	0.994501i	$0.533396\pi$
$402$	17.9966	0.897586
$403$	33.0373	1.64571
$404$	0.159933	0.00795698
$405$	0.362203	0.0179980
$406$	0	0
$407$	10.4368	0.517331
$408$	12.2615	0.607037
$409$	−17.6741	−0.873929	−0.436965	−	0.899479i	$-0.643946\pi$
$409$	−17.6741	−0.873929	−0.436965	+	0.899479i	$0.643946\pi$
$410$	−3.37208	−0.166535
$411$	−9.72371	−0.479635
$412$	−5.07269	−0.249913
$413$	0	0
$414$	−1.52648	−0.0750226
$415$	−5.92498	−0.290846
$416$	6.58154	0.322686
$417$	−18.5445	−0.908129
$418$	−8.08104	−0.395257
$419$	−17.1268	−0.836699	−0.418350	−	0.908286i	$-0.637391\pi$
$419$	−17.1268	−0.836699	−0.418350	+	0.908286i	$0.637391\pi$
$420$	0	0
$421$	32.8574	1.60137	0.800685	−	0.599086i	$-0.204469\pi$
$421$	32.8574	1.60137	0.800685	+	0.599086i	$0.204469\pi$
$422$	3.50086	0.170419
$423$	−0.0368774	−0.00179304
$424$	−22.5064	−1.09301
$425$	23.4207	1.13607
$426$	−14.5945	−0.707104
$427$	0	0
$428$	2.58040	0.124728
$429$	4.08584	0.197266
$430$	3.49071	0.168337
$431$	−18.6306	−0.897405	−0.448702	−	0.893681i	$-0.648114\pi$
$431$	−18.6306	−0.897405	−0.448702	+	0.893681i	$0.648114\pi$
$432$	−4.55131	−0.218975
$433$	−18.2241	−0.875794	−0.437897	−	0.899025i	$-0.644277\pi$
$433$	−18.2241	−0.875794	−0.437897	+	0.899025i	$0.644277\pi$
$434$	0	0
$435$	0.467682	0.0224237
$436$	−1.70026	−0.0814278
$437$	−4.61068	−0.220559
$438$	0.240084	0.0114716
$439$	−22.4033	−1.06925	−0.534626	−	0.845089i	$-0.679548\pi$
$439$	−22.4033	−1.06925	−0.534626	+	0.845089i	$0.679548\pi$
$440$	1.06006	0.0505364
$441$	0	0
$442$	26.1300	1.24288
$443$	−19.0957	−0.907263	−0.453632	−	0.891189i	$-0.649872\pi$
$443$	−19.0957	−0.907263	−0.453632	+	0.891189i	$0.649872\pi$
$444$	−3.00104	−0.142423
$445$	−0.370024	−0.0175408
$446$	29.5095	1.39731
$447$	15.5066	0.733437
$448$	0	0
$449$	−22.4684	−1.06035	−0.530174	−	0.847889i	$-0.677873\pi$
$449$	−22.4684	−1.06035	−0.530174	+	0.847889i	$0.677873\pi$
$450$	−7.43216	−0.350355
$451$	7.00267	0.329743
$452$	−2.98652	−0.140474
$453$	17.6314	0.828394
$454$	6.36565	0.298755
$455$	0	0
$456$	−11.7526	−0.550366
$457$	−11.5263	−0.539177	−0.269588	−	0.962976i	$-0.586888\pi$
$457$	−11.5263	−0.539177	−0.269588	+	0.962976i	$0.586888\pi$
$458$	31.1695	1.45645
$459$	−4.81035	−0.224528
$460$	−0.119582	−0.00557556
$461$	28.3690	1.32128	0.660638	−	0.750705i	$-0.270286\pi$
$461$	28.3690	1.32128	0.660638	+	0.750705i	$0.270286\pi$
$462$	0	0
$463$	−4.83032	−0.224484	−0.112242	−	0.993681i	$-0.535803\pi$
$463$	−4.83032	−0.224484	−0.112242	+	0.993681i	$0.535803\pi$
$464$	−5.87673	−0.272820
$465$	−3.36268	−0.155941
$466$	−9.71430	−0.450006
$467$	−2.02413	−0.0936656	−0.0468328	−	0.998903i	$-0.514913\pi$
$467$	−2.02413	−0.0936656	−0.0468328	+	0.998903i	$0.514913\pi$
$468$	−1.17486	−0.0543080
$469$	0	0
$470$	−0.0203894	−0.000940492 0
$471$	−6.37508	−0.293748
$472$	10.0592	0.463014
$473$	−7.24904	−0.333311
$474$	12.7090	0.583744
$475$	−22.4485	−1.03001
$476$	0	0
$477$	8.82952	0.404276
$478$	−0.667949	−0.0305513
$479$	−37.4692	−1.71201	−0.856007	−	0.516964i	$-0.827062\pi$
$479$	−37.4692	−1.71201	−0.856007	+	0.516964i	$0.827062\pi$
$480$	−0.669897	−0.0305765
$481$	32.3465	1.47487
$482$	32.5484	1.48254
$483$	0	0
$484$	−3.19644	−0.145293
$485$	−2.68454	−0.121899
$486$	1.52648	0.0692427
$487$	24.5261	1.11138	0.555691	−	0.831389i	$-0.312454\pi$
$487$	24.5261	1.11138	0.555691	+	0.831389i	$0.312454\pi$
$488$	−7.06431	−0.319786
$489$	14.3869	0.650599
$490$	0	0
$491$	1.79882	0.0811798	0.0405899	−	0.999176i	$-0.487076\pi$
$491$	1.79882	0.0811798	0.0405899	+	0.999176i	$0.487076\pi$
$492$	−2.01359	−0.0907794
$493$	−6.21121	−0.279739
$494$	−25.0454	−1.12685
$495$	−0.415874	−0.0186921
$496$	42.2542	1.89727
$497$	0	0
$498$	−24.9705	−1.11896
$499$	7.21026	0.322776	0.161388	−	0.986891i	$-0.448403\pi$
$499$	7.21026	0.322776	0.161388	+	0.986891i	$0.448403\pi$
$500$	−1.18014	−0.0527773
$501$	−2.05019	−0.0915958
$502$	−29.4558	−1.31468
$503$	33.8658	1.51000	0.755000	−	0.655725i	$-0.227637\pi$
$503$	33.8658	1.51000	0.755000	+	0.655725i	$0.227637\pi$
$504$	0	0
$505$	0.175459	0.00780780
$506$	1.75268	0.0779161
$507$	−0.336843	−0.0149597
$508$	5.06890	0.224896
$509$	−23.4274	−1.03840	−0.519200	−	0.854653i	$-0.673770\pi$
$509$	−23.4274	−1.03840	−0.519200	+	0.854653i	$0.673770\pi$
$510$	−2.65962	−0.117770
$511$	0	0
$512$	−14.7848	−0.653403
$513$	4.61068	0.203567
$514$	−48.4724	−2.13803
$515$	−5.56511	−0.245228
$516$	2.08443	0.0917617
$517$	0.0423419	0.00186219
$518$	0	0
$519$	18.8742	0.828486
$520$	3.28542	0.144075
$521$	7.31051	0.320279	0.160140	−	0.987094i	$-0.448806\pi$
$521$	7.31051	0.320279	0.160140	+	0.987094i	$0.448806\pi$
$522$	1.97102	0.0862694
$523$	26.4075	1.15472	0.577360	−	0.816490i	$-0.304083\pi$
$523$	26.4075	1.15472	0.577360	+	0.816490i	$0.304083\pi$
$524$	3.74912	0.163781
$525$	0	0
$526$	−19.1464	−0.834821
$527$	44.6592	1.94538
$528$	5.22572	0.227420
$529$	1.00000	0.0434783
$530$	4.88181	0.212052
$531$	−3.94636	−0.171257
$532$	0	0
$533$	21.7033	0.940072
$534$	−1.55945	−0.0674839
$535$	2.83089	0.122390
$536$	−30.0515	−1.29803
$537$	15.4735	0.667733
$538$	13.4012	0.577767
$539$	0	0
$540$	0.119582	0.00514601
$541$	30.9483	1.33057	0.665286	−	0.746589i	$-0.268310\pi$
$541$	30.9483	1.33057	0.665286	+	0.746589i	$0.268310\pi$
$542$	−29.1775	−1.25328
$543$	−18.0888	−0.776265
$544$	8.89678	0.381446
$545$	−1.86531	−0.0799012
$546$	0	0
$547$	−15.3008	−0.654215	−0.327108	−	0.944987i	$-0.606074\pi$
$547$	−15.3008	−0.654215	−0.327108	+	0.944987i	$0.606074\pi$
$548$	−3.21032	−0.137138
$549$	2.77141	0.118281
$550$	8.53346	0.363868
$551$	5.95340	0.253623
$552$	2.54899	0.108492
$553$	0	0
$554$	−0.592866	−0.0251885
$555$	−3.29236	−0.139753
$556$	−6.12254	−0.259653
$557$	22.9275	0.971471	0.485735	−	0.874106i	$-0.338552\pi$
$557$	22.9275	0.971471	0.485735	+	0.874106i	$0.338552\pi$
$558$	−14.1718	−0.599942
$559$	−22.4668	−0.950245
$560$	0	0
$561$	5.52314	0.233187
$562$	−26.9735	−1.13781
$563$	−20.6641	−0.870887	−0.435444	−	0.900216i	$-0.643408\pi$
$563$	−20.6641	−0.870887	−0.435444	+	0.900216i	$0.643408\pi$
$564$	−0.0121752	−0.000512669 0
$565$	−3.27643	−0.137840
$566$	−17.9904	−0.756193
$567$	0	0
$568$	24.3705	1.02256
$569$	−13.1030	−0.549305	−0.274653	−	0.961544i	$-0.588563\pi$
$569$	−13.1030	−0.549305	−0.274653	+	0.961544i	$0.588563\pi$
$570$	2.54923	0.106775
$571$	3.84337	0.160840	0.0804200	−	0.996761i	$-0.474374\pi$
$571$	3.84337	0.160840	0.0804200	+	0.996761i	$0.474374\pi$
$572$	1.34895	0.0564026
$573$	−11.3386	−0.473679
$574$	0	0
$575$	4.86881	0.203043
$576$	6.27937	0.261640
$577$	0.416558	0.0173415	0.00867077	−	0.999962i	$-0.497240\pi$
$577$	0.416558	0.0173415	0.00867077	+	0.999962i	$0.497240\pi$
$578$	9.37177	0.389814
$579$	−9.25207	−0.384503
$580$	0.154407	0.00641140
$581$	0	0
$582$	−11.3139	−0.468975
$583$	−10.1379	−0.419868
$584$	−0.400903	−0.0165895
$585$	−1.28891	−0.0532899
$586$	9.54855	0.394447
$587$	−14.5968	−0.602474	−0.301237	−	0.953549i	$-0.597400\pi$
$587$	−14.5968	−0.602474	−0.301237	+	0.953549i	$0.597400\pi$
$588$	0	0
$589$	−42.8055	−1.76377
$590$	−2.18193	−0.0898285
$591$	−16.7108	−0.687389
$592$	41.3706	1.70032
$593$	14.8949	0.611658	0.305829	−	0.952086i	$-0.401066\pi$
$593$	14.8949	0.611658	0.305829	+	0.952086i	$0.401066\pi$
$594$	−1.75268	−0.0719133
$595$	0	0
$596$	5.11956	0.209705
$597$	13.4796	0.551685
$598$	5.43204	0.222133
$599$	−26.7160	−1.09159	−0.545794	−	0.837920i	$-0.683772\pi$
$599$	−26.7160	−1.09159	−0.545794	+	0.837920i	$0.683772\pi$
$600$	12.4106	0.506659
$601$	1.43227	0.0584234	0.0292117	−	0.999573i	$-0.490700\pi$
$601$	1.43227	0.0584234	0.0292117	+	0.999573i	$0.490700\pi$
$602$	0	0
$603$	11.7895	0.480107
$604$	5.82106	0.236856
$605$	−3.50673	−0.142569
$606$	0.739461	0.0300386
$607$	−10.1424	−0.411665	−0.205833	−	0.978587i	$-0.565990\pi$
$607$	−10.1424	−0.411665	−0.205833	+	0.978587i	$0.565990\pi$
$608$	−8.52749	−0.345836
$609$	0	0
$610$	1.53230	0.0620412
$611$	0.131229	0.00530897
$612$	−1.58815	−0.0641973
$613$	19.7885	0.799250	0.399625	−	0.916679i	$-0.369140\pi$
$613$	19.7885	0.799250	0.399625	+	0.916679i	$0.369140\pi$
$614$	34.5064	1.39256
$615$	−2.20905	−0.0890775
$616$	0	0
$617$	14.9335	0.601201	0.300601	−	0.953750i	$-0.402813\pi$
$617$	14.9335	0.601201	0.300601	+	0.953750i	$0.402813\pi$
$618$	−23.4539	−0.943453
$619$	13.6719	0.549521	0.274761	−	0.961513i	$-0.411401\pi$
$619$	13.6719	0.549521	0.274761	+	0.961513i	$0.411401\pi$
$620$	−1.11020	−0.0445868
$621$	−1.00000	−0.0401286
$622$	27.8645	1.11726
$623$	0	0
$624$	16.1960	0.648358
$625$	23.0494	0.921974
$626$	4.87748	0.194943
$627$	−5.29389	−0.211418
$628$	−2.10475	−0.0839888
$629$	43.7253	1.74344
$630$	0	0
$631$	−12.7959	−0.509397	−0.254698	−	0.967021i	$-0.581976\pi$
$631$	−12.7959	−0.509397	−0.254698	+	0.967021i	$0.581976\pi$
$632$	−21.2221	−0.844170
$633$	2.29341	0.0911550
$634$	42.8136	1.70035
$635$	5.56095	0.220680
$636$	2.91510	0.115591
$637$	0	0
$638$	−2.26309	−0.0895966
$639$	−9.56084	−0.378221
$640$	4.81163	0.190196
$641$	−36.4415	−1.43935	−0.719677	−	0.694309i	$-0.755710\pi$
$641$	−36.4415	−1.43935	−0.719677	+	0.694309i	$0.755710\pi$
$642$	11.9306	0.470865
$643$	46.2850	1.82530	0.912652	−	0.408738i	$-0.134031\pi$
$643$	46.2850	1.82530	0.912652	+	0.408738i	$0.134031\pi$
$644$	0	0
$645$	2.28677	0.0900414
$646$	−33.8559	−1.33204
$647$	−2.48710	−0.0977779	−0.0488890	−	0.998804i	$-0.515568\pi$
$647$	−2.48710	−0.0977779	−0.0488890	+	0.998804i	$0.515568\pi$
$648$	−2.54899	−0.100134
$649$	4.53113	0.177862
$650$	26.4476	1.03736
$651$	0	0
$652$	4.74989	0.186020
$653$	−32.5227	−1.27271	−0.636356	−	0.771396i	$-0.719559\pi$
$653$	−32.5227	−1.27271	−0.636356	+	0.771396i	$0.719559\pi$
$654$	−7.86126	−0.307400
$655$	4.11306	0.160710
$656$	27.7581	1.08377
$657$	0.157279	0.00613603
$658$	0	0
$659$	−25.6593	−0.999545	−0.499773	−	0.866157i	$-0.666583\pi$
$659$	−25.6593	−0.999545	−0.499773	+	0.866157i	$0.666583\pi$
$660$	−0.137302	−0.00534448
$661$	15.8031	0.614671	0.307335	−	0.951601i	$-0.400563\pi$
$661$	15.8031	0.614671	0.307335	+	0.951601i	$0.400563\pi$
$662$	−33.7466	−1.31160
$663$	17.1178	0.664800
$664$	41.6969	1.61815
$665$	0	0
$666$	−13.8755	−0.537664
$667$	−1.29122	−0.0499962
$668$	−0.676878	−0.0261892
$669$	19.3317	0.747405
$670$	6.51840	0.251828
$671$	−3.18208	−0.122843
$672$	0	0
$673$	28.4301	1.09590	0.547950	−	0.836511i	$-0.315409\pi$
$673$	28.4301	1.09590	0.547950	+	0.836511i	$0.315409\pi$
$674$	9.44281	0.363724
$675$	−4.86881	−0.187401
$676$	−0.111210	−0.00427731
$677$	−32.7622	−1.25915	−0.629577	−	0.776938i	$-0.716772\pi$
$677$	−32.7622	−1.25915	−0.629577	+	0.776938i	$0.716772\pi$
$678$	−13.8083	−0.530306
$679$	0	0
$680$	4.44116	0.170311
$681$	4.17014	0.159800
$682$	16.2718	0.623081
$683$	−10.1105	−0.386868	−0.193434	−	0.981113i	$-0.561963\pi$
$683$	−10.1105	−0.386868	−0.193434	+	0.981113i	$0.561963\pi$
$684$	1.52223	0.0582041
$685$	−3.52195	−0.134567
$686$	0	0
$687$	20.4191	0.779038
$688$	−28.7347	−1.09550
$689$	−31.4201	−1.19701
$690$	−0.552896	−0.0210484
$691$	51.3417	1.95313	0.976565	−	0.215223i	$-0.0690479\pi$
$691$	51.3417	1.95313	0.976565	+	0.215223i	$0.0690479\pi$
$692$	6.23139	0.236882
$693$	0	0
$694$	46.7811	1.77579
$695$	−6.71687	−0.254785
$696$	−3.29131	−0.124757
$697$	29.3380	1.11126
$698$	−22.1185	−0.837198
$699$	−6.36384	−0.240703
$700$	0	0
$701$	−10.7533	−0.406145	−0.203073	−	0.979164i	$-0.565093\pi$
$701$	−10.7533	−0.406145	−0.203073	+	0.979164i	$0.565093\pi$
$702$	−5.43204	−0.205019
$703$	−41.9103	−1.58068
$704$	−7.20984	−0.271731
$705$	−0.0133571	−0.000503057 0
$706$	29.3502	1.10461
$707$	0	0
$708$	−1.30290	−0.0489661
$709$	15.5488	0.583949	0.291975	−	0.956426i	$-0.405688\pi$
$709$	15.5488	0.583949	0.291975	+	0.956426i	$0.405688\pi$
$710$	−5.28615	−0.198386
$711$	8.32568	0.312237
$712$	2.60404	0.0975905
$713$	9.28398	0.347688
$714$	0	0
$715$	1.47990	0.0553451
$716$	5.10865	0.190919
$717$	−0.437573	−0.0163415
$718$	−14.3323	−0.534878
$719$	27.1655	1.01310	0.506552	−	0.862209i	$-0.330920\pi$
$719$	27.1655	1.01310	0.506552	+	0.862209i	$0.330920\pi$
$720$	−1.64849	−0.0614358
$721$	0	0
$722$	3.44740	0.128299
$723$	21.3225	0.792991
$724$	−5.97209	−0.221951
$725$	−6.28669	−0.233482
$726$	−14.7789	−0.548498
$727$	−14.8781	−0.551798	−0.275899	−	0.961187i	$-0.588976\pi$
$727$	−14.8781	−0.551798	−0.275899	+	0.961187i	$0.588976\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.0869589	0.00321849
$731$	−30.3701	−1.12328
$732$	0.914992	0.0338191
$733$	17.4530	0.644642	0.322321	−	0.946630i	$-0.395537\pi$
$733$	17.4530	0.644642	0.322321	+	0.946630i	$0.395537\pi$
$734$	−17.6671	−0.652103
$735$	0	0
$736$	1.84951	0.0681738
$737$	−13.5365	−0.498624
$738$	−9.30993	−0.342703
$739$	−8.59955	−0.316340	−0.158170	−	0.987412i	$-0.550559\pi$
$739$	−8.59955	−0.316340	−0.158170	+	0.987412i	$0.550559\pi$
$740$	−1.08699	−0.0399584
$741$	−16.4073	−0.602736
$742$	0	0
$743$	2.61075	0.0957790	0.0478895	−	0.998853i	$-0.484750\pi$
$743$	2.61075	0.0957790	0.0478895	+	0.998853i	$0.484750\pi$
$744$	23.6648	0.867594
$745$	5.61653	0.205774
$746$	5.67120	0.207638
$747$	−16.3582	−0.598515
$748$	1.82349	0.0666733
$749$	0	0
$750$	−5.45643	−0.199241
$751$	23.1435	0.844517	0.422259	−	0.906475i	$-0.361237\pi$
$751$	23.1435	0.844517	0.422259	+	0.906475i	$0.361237\pi$
$752$	0.167840	0.00612051
$753$	−19.2965	−0.703204
$754$	−7.01395	−0.255433
$755$	6.38613	0.232415
$756$	0	0
$757$	52.5217	1.90893	0.954466	−	0.298319i	$-0.0964259\pi$
$757$	52.5217	1.90893	0.954466	+	0.298319i	$0.0964259\pi$
$758$	22.4282	0.814630
$759$	1.14818	0.0416763
$760$	−4.25682	−0.154411
$761$	50.5578	1.83272	0.916359	−	0.400358i	$-0.131114\pi$
$761$	50.5578	1.83272	0.916359	+	0.400358i	$0.131114\pi$
$762$	23.4363	0.849009
$763$	0	0
$764$	−3.74350	−0.135435
$765$	−1.74232	−0.0629937
$766$	8.01726	0.289676
$767$	14.0432	0.507072
$768$	7.71965	0.278559
$769$	−31.1463	−1.12316	−0.561581	−	0.827421i	$-0.689807\pi$
$769$	−31.1463	−1.12316	−0.561581	+	0.827421i	$0.689807\pi$
$770$	0	0
$771$	−31.7543	−1.14360
$772$	−3.05461	−0.109938
$773$	−10.3788	−0.373301	−0.186650	−	0.982426i	$-0.559763\pi$
$773$	−10.3788	−0.373301	−0.186650	+	0.982426i	$0.559763\pi$
$774$	9.63746	0.346411
$775$	45.2019	1.62370
$776$	18.8924	0.678199
$777$	0	0
$778$	−24.7861	−0.888625
$779$	−28.1202	−1.00751
$780$	−0.425538	−0.0152367
$781$	10.9776	0.392808
$782$	7.34292	0.262582
$783$	1.29122	0.0461444
$784$	0	0
$785$	−2.30907	−0.0824142
$786$	17.3343	0.618293
$787$	−32.7493	−1.16739	−0.583694	−	0.811974i	$-0.698393\pi$
$787$	−32.7493	−1.16739	−0.583694	+	0.811974i	$0.698393\pi$
$788$	−5.51712	−0.196539
$789$	−12.5428	−0.446535
$790$	4.60324	0.163776
$791$	0	0
$792$	2.92670	0.103996
$793$	−9.86216	−0.350216
$794$	13.0270	0.462310
$795$	3.19808	0.113424
$796$	4.45035	0.157738
$797$	−46.9165	−1.66187	−0.830934	−	0.556371i	$-0.812194\pi$
$797$	−46.9165	−1.66187	−0.830934	+	0.556371i	$0.812194\pi$
$798$	0	0
$799$	0.177393	0.00627572
$800$	9.00490	0.318371
$801$	−1.02159	−0.0360963
$802$	−6.40245	−0.226078
$803$	−0.180584	−0.00637269
$804$	3.89236	0.137273
$805$	0	0
$806$	50.4310	1.77636
$807$	8.77914	0.309040
$808$	−1.23479	−0.0434396
$809$	−22.2380	−0.781846	−0.390923	−	0.920423i	$-0.627844\pi$
$809$	−22.2380	−0.781846	−0.390923	+	0.920423i	$0.627844\pi$
$810$	0.552896	0.0194268
$811$	10.3032	0.361794	0.180897	−	0.983502i	$-0.442100\pi$
$811$	10.3032	0.361794	0.180897	+	0.983502i	$0.442100\pi$
$812$	0	0
$813$	−19.1142	−0.670363
$814$	15.9316	0.558401
$815$	5.21098	0.182533
$816$	21.8934	0.766421
$817$	29.1096	1.01841
$818$	−26.9793	−0.943308
$819$	0	0
$820$	−0.729326	−0.0254692
$821$	−8.13312	−0.283848	−0.141924	−	0.989878i	$-0.545329\pi$
$821$	−8.13312	−0.283848	−0.141924	+	0.989878i	$0.545329\pi$
$822$	−14.8431	−0.517712
$823$	−31.4034	−1.09465	−0.547326	−	0.836919i	$-0.684354\pi$
$823$	−31.4034	−1.09465	−0.547326	+	0.836919i	$0.684354\pi$
$824$	39.1643	1.36435
$825$	5.59027	0.194628
$826$	0	0
$827$	−24.1633	−0.840240	−0.420120	−	0.907468i	$-0.638012\pi$
$827$	−24.1633	−0.840240	−0.420120	+	0.907468i	$0.638012\pi$
$828$	−0.330154	−0.0114736
$829$	−41.3530	−1.43625	−0.718124	−	0.695915i	$-0.754999\pi$
$829$	−41.3530	−1.43625	−0.718124	+	0.695915i	$0.754999\pi$
$830$	−9.04439	−0.313935
$831$	−0.388387	−0.0134730
$832$	−22.3453	−0.774685
$833$	0	0
$834$	−28.3079	−0.980223
$835$	−0.742585	−0.0256982
$836$	−1.74780	−0.0604489
$837$	−9.28398	−0.320901
$838$	−26.1438	−0.903123
$839$	−33.7897	−1.16655	−0.583275	−	0.812275i	$-0.698229\pi$
$839$	−33.7897	−1.16655	−0.583275	+	0.812275i	$0.698229\pi$
$840$	0	0
$841$	−27.3328	−0.942509
$842$	50.1562	1.72850
$843$	−17.6703	−0.608599
$844$	0.757179	0.0260632
$845$	−0.122005	−0.00419711
$846$	−0.0562928	−0.00193539
$847$	0	0
$848$	−40.1859	−1.37999
$849$	−11.7855	−0.404478
$850$	35.7513	1.22626
$851$	9.08983	0.311596
$852$	−3.15655	−0.108142
$853$	−13.6538	−0.467496	−0.233748	−	0.972297i	$-0.575099\pi$
$853$	−13.6538	−0.467496	−0.233748	+	0.972297i	$0.575099\pi$
$854$	0	0
$855$	1.67000	0.0571128
$856$	−19.9223	−0.680931
$857$	−43.3590	−1.48111	−0.740557	−	0.671993i	$-0.765438\pi$
$857$	−43.3590	−1.48111	−0.740557	+	0.671993i	$0.765438\pi$
$858$	6.23696	0.212926
$859$	−17.3944	−0.593488	−0.296744	−	0.954957i	$-0.595901\pi$
$859$	−17.3944	−0.593488	−0.296744	+	0.954957i	$0.595901\pi$
$860$	0.754984	0.0257448
$861$	0	0
$862$	−28.4393	−0.968647
$863$	−44.3182	−1.50861	−0.754305	−	0.656524i	$-0.772026\pi$
$863$	−44.3182	−1.50861	−0.754305	+	0.656524i	$0.772026\pi$
$864$	−1.84951	−0.0629215
$865$	6.83629	0.232441
$866$	−27.8188	−0.945321
$867$	6.13945	0.208507
$868$	0	0
$869$	−9.55937	−0.324280
$870$	0.713910	0.0242038
$871$	−41.9535	−1.42154
$872$	13.1271	0.444540
$873$	−7.41172	−0.250849
$874$	−7.03813	−0.238068
$875$	0	0
$876$	0.0519262	0.00175442
$877$	15.1032	0.509999	0.254999	−	0.966941i	$-0.417925\pi$
$877$	15.1032	0.509999	0.254999	+	0.966941i	$0.417925\pi$
$878$	−34.1983	−1.15414
$879$	6.25525	0.210984
$880$	1.89277	0.0638052
$881$	−28.8598	−0.972312	−0.486156	−	0.873872i	$-0.661601\pi$
$881$	−28.8598	−0.972312	−0.486156	+	0.873872i	$0.661601\pi$
$882$	0	0
$883$	−28.3145	−0.952860	−0.476430	−	0.879212i	$-0.658069\pi$
$883$	−28.3145	−0.952860	−0.476430	+	0.879212i	$0.658069\pi$
$884$	5.65150	0.190080
$885$	−1.42938	−0.0480481
$886$	−29.1493	−0.979289
$887$	−6.73068	−0.225994	−0.112997	−	0.993595i	$-0.536045\pi$
$887$	−6.73068	−0.225994	−0.112997	+	0.993595i	$0.536045\pi$
$888$	23.1699	0.777532
$889$	0	0
$890$	−0.564836	−0.0189333
$891$	−1.14818	−0.0384655
$892$	6.38242	0.213699
$893$	−0.170030	−0.00568984
$894$	23.6706	0.791662
$895$	5.60456	0.187340
$896$	0	0
$897$	3.55853	0.118816
$898$	−34.2976	−1.14453
$899$	−11.9876	−0.399810
$900$	−1.60746	−0.0535818
$901$	−42.4731	−1.41498
$902$	10.6895	0.355920
$903$	0	0
$904$	23.0578	0.766890
$905$	−6.55182	−0.217790
$906$	26.9140	0.894158
$907$	39.9950	1.32801	0.664007	−	0.747726i	$-0.268855\pi$
$907$	39.9950	1.32801	0.664007	+	0.747726i	$0.268855\pi$
$908$	1.37679	0.0456903
$909$	0.484421	0.0160672
$910$	0	0
$911$	43.3937	1.43770	0.718849	−	0.695167i	$-0.244669\pi$
$911$	43.3937	1.43770	0.718849	+	0.695167i	$0.244669\pi$
$912$	−20.9846	−0.694870
$913$	18.7821	0.621598
$914$	−17.5947	−0.581981
$915$	1.00381	0.0331850
$916$	6.74145	0.222744
$917$	0	0
$918$	−7.34292	−0.242352
$919$	−13.4176	−0.442607	−0.221304	−	0.975205i	$-0.571031\pi$
$919$	−13.4176	−0.442607	−0.221304	+	0.975205i	$0.571031\pi$
$920$	0.923252	0.0304387
$921$	22.6051	0.744865
$922$	43.3048	1.42617
$923$	34.0226	1.11987
$924$	0	0
$925$	44.2567	1.45515
$926$	−7.37341	−0.242305
$927$	−15.3646	−0.504641
$928$	−2.38812	−0.0783938
$929$	44.7540	1.46833	0.734166	−	0.678970i	$-0.237573\pi$
$929$	44.7540	1.46833	0.734166	+	0.678970i	$0.237573\pi$
$930$	−5.13308	−0.168320
$931$	0	0
$932$	−2.10105	−0.0688220
$933$	18.2540	0.597610
$934$	−3.08980	−0.101101
$935$	2.00050	0.0654233
$936$	9.07068	0.296484
$937$	11.0734	0.361753	0.180877	−	0.983506i	$-0.442107\pi$
$937$	11.0734	0.361753	0.180877	+	0.983506i	$0.442107\pi$
$938$	0	0
$939$	3.19524	0.104273
$940$	−0.00440989	−0.000143835 0
$941$	18.2486	0.594887	0.297443	−	0.954739i	$-0.403866\pi$
$941$	18.2486	0.594887	0.297443	+	0.954739i	$0.403866\pi$
$942$	−9.73145	−0.317068
$943$	6.09893	0.198609
$944$	17.9611	0.584583
$945$	0	0
$946$	−11.0655	−0.359772
$947$	−29.1174	−0.946187	−0.473093	−	0.881012i	$-0.656863\pi$
$947$	−29.1174	−0.946187	−0.473093	+	0.881012i	$0.656863\pi$
$948$	2.74875	0.0892753
$949$	−0.559682	−0.0181680
$950$	−34.2673	−1.11178
$951$	28.0472	0.909494
$952$	0	0
$953$	−27.1747	−0.880274	−0.440137	−	0.897931i	$-0.645070\pi$
$953$	−27.1747	−0.880274	−0.440137	+	0.897931i	$0.645070\pi$
$954$	13.4781	0.436370
$955$	−4.10689	−0.132896
$956$	−0.144466	−0.00467238
$957$	−1.48255	−0.0479241
$958$	−57.1962	−1.84793
$959$	0	0
$960$	2.27440	0.0734060
$961$	55.1923	1.78040
$962$	49.3764	1.59196
$963$	7.81576	0.251859
$964$	7.03969	0.226733
$965$	−3.35112	−0.107877
$966$	0	0
$967$	−45.0093	−1.44740	−0.723701	−	0.690113i	$-0.757561\pi$
$967$	−45.0093	−1.44740	−0.723701	+	0.690113i	$0.757561\pi$
$968$	24.6786	0.793199
$969$	−22.1790	−0.712492
$970$	−4.09791	−0.131576
$971$	−54.0959	−1.73602	−0.868011	−	0.496545i	$-0.834602\pi$
$971$	−54.0959	−1.73602	−0.868011	+	0.496545i	$0.834602\pi$
$972$	0.330154	0.0105897
$973$	0	0
$974$	37.4386	1.19961
$975$	17.3258	0.554870
$976$	−12.6135	−0.403750
$977$	−14.0548	−0.449654	−0.224827	−	0.974399i	$-0.572182\pi$
$977$	−14.0548	−0.449654	−0.224827	+	0.974399i	$0.572182\pi$
$978$	21.9614	0.702248
$979$	1.17297	0.0374884
$980$	0	0
$981$	−5.14991	−0.164424
$982$	2.74588	0.0876245
$983$	25.9120	0.826465	0.413232	−	0.910626i	$-0.364400\pi$
$983$	25.9120	0.826465	0.413232	+	0.910626i	$0.364400\pi$
$984$	15.5461	0.495593
$985$	−6.05268	−0.192854
$986$	−9.48131	−0.301946
$987$	0	0
$988$	−5.41692	−0.172335
$989$	−6.31350	−0.200758
$990$	−0.634825	−0.0201760
$991$	−12.2782	−0.390028	−0.195014	−	0.980800i	$-0.562475\pi$
$991$	−12.2782	−0.390028	−0.195014	+	0.980800i	$0.562475\pi$
$992$	17.1708	0.545173
$993$	−22.1074	−0.701557
$994$	0	0
$995$	4.88236	0.154781
$996$	−5.40072	−0.171128
$997$	−8.82008	−0.279335	−0.139667	−	0.990198i	$-0.544603\pi$
$997$	−8.82008	−0.279335	−0.139667	+	0.990198i	$0.544603\pi$
$998$	11.0064	0.348400
$999$	−9.08983	−0.287590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bb.1.5	yes	6
7.6	odd	2		3381.2.a.ba.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.ba.1.5	✓	6	7.6	odd	2
3381.2.a.bb.1.5	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+1.52648 q^{2} +1.00000 q^{3} +0.330154 q^{4} +0.362203 q^{5} +1.52648 q^{6} -2.54899 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.52648 q^{2} +1.00000 q^{3} +0.330154 q^{4} +0.362203 q^{5} +1.52648 q^{6} -2.54899 q^{8} +1.00000 q^{9} +0.552896 q^{10} -1.14818 q^{11} +0.330154 q^{12} -3.55853 q^{13} +0.362203 q^{15} -4.55131 q^{16} -4.81035 q^{17} +1.52648 q^{18} +4.61068 q^{19} +0.119582 q^{20} -1.75268 q^{22} -1.00000 q^{23} -2.54899 q^{24} -4.86881 q^{25} -5.43204 q^{26} +1.00000 q^{27} +1.29122 q^{29} +0.552896 q^{30} -9.28398 q^{31} -1.84951 q^{32} -1.14818 q^{33} -7.34292 q^{34} +0.330154 q^{36} -9.08983 q^{37} +7.03813 q^{38} -3.55853 q^{39} -0.923252 q^{40} -6.09893 q^{41} +6.31350 q^{43} -0.379076 q^{44} +0.362203 q^{45} -1.52648 q^{46} -0.0368774 q^{47} -4.55131 q^{48} -7.43216 q^{50} -4.81035 q^{51} -1.17486 q^{52} +8.82952 q^{53} +1.52648 q^{54} -0.415874 q^{55} +4.61068 q^{57} +1.97102 q^{58} -3.94636 q^{59} +0.119582 q^{60} +2.77141 q^{61} -14.1718 q^{62} +6.27937 q^{64} -1.28891 q^{65} -1.75268 q^{66} +11.7895 q^{67} -1.58815 q^{68} -1.00000 q^{69} -9.56084 q^{71} -2.54899 q^{72} +0.157279 q^{73} -13.8755 q^{74} -4.86881 q^{75} +1.52223 q^{76} -5.43204 q^{78} +8.32568 q^{79} -1.64849 q^{80} +1.00000 q^{81} -9.30993 q^{82} -16.3582 q^{83} -1.74232 q^{85} +9.63746 q^{86} +1.29122 q^{87} +2.92670 q^{88} -1.02159 q^{89} +0.552896 q^{90} -0.330154 q^{92} -9.28398 q^{93} -0.0562928 q^{94} +1.67000 q^{95} -1.84951 q^{96} -7.41172 q^{97} -1.14818 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 6 q^{3} + 7 q^{4} - 2 q^{5} - 3 q^{6} - 9 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 3 * q^2 + 6 * q^3 + 7 * q^4 - 2 * q^5 - 3 * q^6 - 9 * q^8 + 6 * q^9	\( 6 q - 3 q^{2} + 6 q^{3} + 7 q^{4} - 2 q^{5} - 3 q^{6} - 9 q^{8} + 6 q^{9} - 8 q^{10} - 3 q^{11} + 7 q^{12} - 2 q^{15} + 5 q^{16} - 8 q^{17} - 3 q^{18} - 19 q^{19} + 24 q^{22} - 6 q^{23} - 9 q^{24} - 20 q^{26} + 6 q^{27} - 12 q^{29} - 8 q^{30} + 2 q^{31} - 41 q^{32} - 3 q^{33} - 2 q^{34} + 7 q^{36} - 10 q^{37} + 20 q^{38} - 22 q^{40} - 3 q^{41} - 2 q^{43} - 14 q^{44} - 2 q^{45} + 3 q^{46} - 7 q^{47} + 5 q^{48} + 17 q^{50} - 8 q^{51} + 44 q^{52} - 5 q^{53} - 3 q^{54} - 24 q^{55} - 19 q^{57} - 12 q^{58} - 21 q^{59} - 29 q^{61} - 10 q^{62} + 59 q^{64} - 18 q^{65} + 24 q^{66} + 16 q^{67} - 54 q^{68} - 6 q^{69} - 6 q^{71} - 9 q^{72} + 8 q^{73} - 16 q^{74} - 40 q^{76} - 20 q^{78} - 12 q^{79} + 78 q^{80} + 6 q^{81} - 44 q^{82} - 24 q^{83} + 10 q^{85} + 12 q^{86} - 12 q^{87} + 28 q^{88} - 18 q^{89} - 8 q^{90} - 7 q^{92} + 2 q^{93} + 8 q^{94} + 6 q^{95} - 41 q^{96} - 22 q^{97} - 3 q^{99}+O(q^{100}) \) 6 * q - 3 * q^2 + 6 * q^3 + 7 * q^4 - 2 * q^5 - 3 * q^6 - 9 * q^8 + 6 * q^9 - 8 * q^10 - 3 * q^11 + 7 * q^12 - 2 * q^15 + 5 * q^16 - 8 * q^17 - 3 * q^18 - 19 * q^19 + 24 * q^22 - 6 * q^23 - 9 * q^24 - 20 * q^26 + 6 * q^27 - 12 * q^29 - 8 * q^30 + 2 * q^31 - 41 * q^32 - 3 * q^33 - 2 * q^34 + 7 * q^36 - 10 * q^37 + 20 * q^38 - 22 * q^40 - 3 * q^41 - 2 * q^43 - 14 * q^44 - 2 * q^45 + 3 * q^46 - 7 * q^47 + 5 * q^48 + 17 * q^50 - 8 * q^51 + 44 * q^52 - 5 * q^53 - 3 * q^54 - 24 * q^55 - 19 * q^57 - 12 * q^58 - 21 * q^59 - 29 * q^61 - 10 * q^62 + 59 * q^64 - 18 * q^65 + 24 * q^66 + 16 * q^67 - 54 * q^68 - 6 * q^69 - 6 * q^71 - 9 * q^72 + 8 * q^73 - 16 * q^74 - 40 * q^76 - 20 * q^78 - 12 * q^79 + 78 * q^80 + 6 * q^81 - 44 * q^82 - 24 * q^83 + 10 * q^85 + 12 * q^86 - 12 * q^87 + 28 * q^88 - 18 * q^89 - 8 * q^90 - 7 * q^92 + 2 * q^93 + 8 * q^94 + 6 * q^95 - 41 * q^96 - 22 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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