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Properties

Label 117.2.t.b.103.1
Level $117$
Weight $2$
Character 117.103
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [117,2,Mod(25,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3,-2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 103.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 117.103
Dual form 117.2.t.b.25.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(3.00000 + 1.73205i) q^{5} +(3.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-3.00000 + 1.73205i) q^{11} +(3.00000 - 1.73205i) q^{12} +(3.50000 + 0.866025i) q^{13} +(-3.00000 - 5.19615i) q^{15} +(-2.00000 - 3.46410i) q^{16} -3.00000 q^{17} -3.46410i q^{19} +(-6.00000 + 3.46410i) q^{20} -6.00000 q^{21} +(-1.50000 + 2.59808i) q^{23} +(3.50000 + 6.06218i) q^{25} -5.19615i q^{27} +6.92820i q^{28} +(-3.00000 - 5.19615i) q^{29} +(-3.00000 - 1.73205i) q^{31} +6.00000 q^{33} +12.0000 q^{35} -6.00000 q^{36} -6.92820i q^{37} +(-4.50000 - 4.33013i) q^{39} +(6.00000 + 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} -6.92820i q^{44} +10.3923i q^{45} +(6.00000 - 3.46410i) q^{47} +6.92820i q^{48} +(2.50000 - 4.33013i) q^{49} +(4.50000 + 2.59808i) q^{51} +(-5.00000 + 5.19615i) q^{52} -9.00000 q^{53} -12.0000 q^{55} +(-3.00000 + 5.19615i) q^{57} +(-3.00000 - 1.73205i) q^{59} +12.0000 q^{60} +(-3.50000 - 6.06218i) q^{61} +(9.00000 + 5.19615i) q^{63} +8.00000 q^{64} +(9.00000 + 8.66025i) q^{65} +(3.00000 - 5.19615i) q^{68} +(4.50000 - 2.59808i) q^{69} +10.3923i q^{73} -12.1244i q^{75} +(6.00000 + 3.46410i) q^{76} +(-6.00000 + 10.3923i) q^{77} +(-0.500000 - 0.866025i) q^{79} -13.8564i q^{80} +(-4.50000 + 7.79423i) q^{81} +(-6.00000 + 3.46410i) q^{83} +(6.00000 - 10.3923i) q^{84} +(-9.00000 - 5.19615i) q^{85} +10.3923i q^{87} -3.46410i q^{89} +(12.0000 - 3.46410i) q^{91} +(-3.00000 - 5.19615i) q^{92} +(3.00000 + 5.19615i) q^{93} +(6.00000 - 10.3923i) q^{95} +(-6.00000 + 3.46410i) q^{97} +(-9.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{12} + 7 q^{13} - 6 q^{15} - 4 q^{16} - 6 q^{17} - 12 q^{20} - 12 q^{21} - 3 q^{23} + 7 q^{25} - 6 q^{29} - 6 q^{31} + 12 q^{33} + 24 q^{35}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) −1.50000 − 0.866025i −0.866025 − 0.500000i
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 3.00000 + 1.73205i 1.34164 + 0.774597i 0.987048 − 0.160424i \(-0.0512862\pi\)
0.354593 + 0.935021i \(0.384620\pi\)
\(6\) 0 0
\(7\) 3.00000 − 1.73205i 1.13389 − 0.654654i 0.188982 − 0.981981i \(-0.439481\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) −3.00000 + 1.73205i −0.904534 + 0.522233i −0.878668 − 0.477432i \(-0.841568\pi\)
−0.0258656 + 0.999665i \(0.508234\pi\)
\(12\) 3.00000 − 1.73205i 0.866025 − 0.500000i
\(13\) 3.50000 + 0.866025i 0.970725 + 0.240192i
\(14\) 0 0
\(15\) −3.00000 − 5.19615i −0.774597 − 1.34164i
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 − 0.397360i \(-0.869927\pi\)
0.917663 − 0.397360i \(-0.130073\pi\)
\(20\) −6.00000 + 3.46410i −1.34164 + 0.774597i
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) −1.50000 + 2.59808i −0.312772 + 0.541736i −0.978961 − 0.204046i \(-0.934591\pi\)
0.666190 + 0.745782i \(0.267924\pi\)
\(24\) 0 0
\(25\) 3.50000 + 6.06218i 0.700000 + 1.21244i
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 6.92820i 1.30931i
\(29\) −3.00000 − 5.19615i −0.557086 − 0.964901i −0.997738 − 0.0672232i \(-0.978586\pi\)
0.440652 − 0.897678i \(-0.354747\pi\)
\(30\) 0 0
\(31\) −3.00000 − 1.73205i −0.538816 − 0.311086i 0.205783 − 0.978598i \(-0.434026\pi\)
−0.744599 + 0.667512i \(0.767359\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) −6.00000 −1.00000
\(37\) − 6.92820i − 1.13899i −0.821995 − 0.569495i \(-0.807139\pi\)
0.821995 − 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) −4.50000 − 4.33013i −0.720577 − 0.693375i
\(40\) 0 0
\(41\) 6.00000 + 3.46410i 0.937043 + 0.541002i 0.889032 − 0.457845i \(-0.151379\pi\)
0.0480106 + 0.998847i \(0.484712\pi\)
\(42\) 0 0
\(43\) 0.500000 + 0.866025i 0.0762493 + 0.132068i 0.901629 − 0.432511i \(-0.142372\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) − 6.92820i − 1.04447i
\(45\) 10.3923i 1.54919i
\(46\) 0 0
\(47\) 6.00000 − 3.46410i 0.875190 − 0.505291i 0.00612051 − 0.999981i \(-0.498052\pi\)
0.869069 + 0.494690i \(0.164718\pi\)
\(48\) 6.92820i 1.00000i
\(49\) 2.50000 − 4.33013i 0.357143 − 0.618590i
\(50\) 0 0
\(51\) 4.50000 + 2.59808i 0.630126 + 0.363803i
\(52\) −5.00000 + 5.19615i −0.693375 + 0.720577i
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) −3.00000 + 5.19615i −0.397360 + 0.688247i
\(58\) 0 0
\(59\) −3.00000 − 1.73205i −0.390567 − 0.225494i 0.291839 − 0.956467i \(-0.405733\pi\)
−0.682406 + 0.730974i \(0.739066\pi\)
\(60\) 12.0000 1.54919
\(61\) −3.50000 − 6.06218i −0.448129 − 0.776182i 0.550135 − 0.835076i \(-0.314576\pi\)
−0.998264 + 0.0588933i \(0.981243\pi\)
\(62\) 0 0
\(63\) 9.00000 + 5.19615i 1.13389 + 0.654654i
\(64\) 8.00000 1.00000
\(65\) 9.00000 + 8.66025i 1.11631 + 1.07417i
\(66\) 0 0
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 3.00000 − 5.19615i 0.363803 − 0.630126i
\(69\) 4.50000 − 2.59808i 0.541736 − 0.312772i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 10.3923i 1.21633i 0.793812 + 0.608164i \(0.208094\pi\)
−0.793812 + 0.608164i \(0.791906\pi\)
\(74\) 0 0
\(75\) − 12.1244i − 1.40000i
\(76\) 6.00000 + 3.46410i 0.688247 + 0.397360i
\(77\) −6.00000 + 10.3923i −0.683763 + 1.18431i
\(78\) 0 0
\(79\) −0.500000 − 0.866025i −0.0562544 − 0.0974355i 0.836527 − 0.547926i \(-0.184582\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) − 13.8564i − 1.54919i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −6.00000 + 3.46410i −0.658586 + 0.380235i −0.791738 − 0.610861i \(-0.790823\pi\)
0.133152 + 0.991096i \(0.457490\pi\)
\(84\) 6.00000 − 10.3923i 0.654654 − 1.13389i
\(85\) −9.00000 − 5.19615i −0.976187 − 0.563602i
\(86\) 0 0
\(87\) 10.3923i 1.11417i
\(88\) 0 0
\(89\) − 3.46410i − 0.367194i −0.983002 − 0.183597i \(-0.941226\pi\)
0.983002 − 0.183597i \(-0.0587741\pi\)
\(90\) 0 0
\(91\) 12.0000 − 3.46410i 1.25794 − 0.363137i
\(92\) −3.00000 − 5.19615i −0.312772 − 0.541736i
\(93\) 3.00000 + 5.19615i 0.311086 + 0.538816i
\(94\) 0 0
\(95\) 6.00000 − 10.3923i 0.615587 − 1.06623i
\(96\) 0 0
\(97\) −6.00000 + 3.46410i −0.609208 + 0.351726i −0.772655 − 0.634826i \(-0.781072\pi\)
0.163448 + 0.986552i \(0.447739\pi\)
\(98\) 0 0
\(99\) −9.00000 − 5.19615i −0.904534 − 0.522233i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 117.2.t.b.103.1 yes 2
3.2 odd 2 351.2.t.a.64.1 2
9.2 odd 6 351.2.t.b.181.1 2
9.4 even 3 1053.2.b.e.649.1 2
9.5 odd 6 1053.2.b.f.649.2 2
9.7 even 3 117.2.t.a.25.1 ✓ 2
13.12 even 2 117.2.t.a.103.1 yes 2
39.38 odd 2 351.2.t.b.64.1 2
117.25 even 6 inner 117.2.t.b.25.1 yes 2
117.38 odd 6 351.2.t.a.181.1 2
117.77 odd 6 1053.2.b.f.649.1 2
117.103 even 6 1053.2.b.e.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.a.25.1 ✓ 2 9.7 even 3
117.2.t.a.103.1 yes 2 13.12 even 2
117.2.t.b.25.1 yes 2 117.25 even 6 inner
117.2.t.b.103.1 yes 2 1.1 even 1 trivial
351.2.t.a.64.1 2 3.2 odd 2
351.2.t.a.181.1 2 117.38 odd 6
351.2.t.b.64.1 2 39.38 odd 2
351.2.t.b.181.1 2 9.2 odd 6
1053.2.b.e.649.1 2 9.4 even 3
1053.2.b.e.649.2 2 117.103 even 6
1053.2.b.f.649.1 2 117.77 odd 6
1053.2.b.f.649.2 2 9.5 odd 6