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Properties

Label 912.2.f.e.113.2
Level $912$
Weight $2$
Character 912.113
Analytic conductor $7.282$
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] = [912,2,Mod(113,912)] 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("912.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
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: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.2
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 912.113
Dual form 912.2.f.e.113.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} -3.46410i q^{11} +1.73205i q^{13} +(3.00000 - 5.19615i) q^{15} -1.73205i q^{17} +(4.00000 + 1.73205i) q^{19} +(-1.50000 - 0.866025i) q^{21} -5.19615i q^{23} -7.00000 q^{25} +5.19615i q^{27} +9.00000 q^{29} -10.3923i q^{31} +(3.00000 - 5.19615i) q^{33} +3.46410i q^{35} -6.92820i q^{37} +(-1.50000 + 2.59808i) q^{39} -2.00000 q^{43} +(9.00000 - 5.19615i) q^{45} +3.46410i q^{47} -6.00000 q^{49} +(1.50000 - 2.59808i) q^{51} -9.00000 q^{53} -12.0000 q^{55} +(4.50000 + 6.06218i) q^{57} +3.00000 q^{59} -8.00000 q^{61} +(-1.50000 - 2.59808i) q^{63} +6.00000 q^{65} +8.66025i q^{67} +(4.50000 - 7.79423i) q^{69} +12.0000 q^{71} +11.0000 q^{73} +(-10.5000 - 6.06218i) q^{75} +3.46410i q^{77} +6.92820i q^{79} +(-4.50000 + 7.79423i) q^{81} +10.3923i q^{83} -6.00000 q^{85} +(13.5000 + 7.79423i) q^{87} +6.00000 q^{89} -1.73205i q^{91} +(9.00000 - 15.5885i) q^{93} +(6.00000 - 13.8564i) q^{95} +13.8564i q^{97} +(9.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{15} + 8 q^{19} - 3 q^{21} - 14 q^{25} + 18 q^{29} + 6 q^{33} - 3 q^{39} - 4 q^{43} + 18 q^{45} - 12 q^{49} + 3 q^{51} - 18 q^{53} - 24 q^{55} + 9 q^{57} + 6 q^{59}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) − 3.46410i − 1.54919i −0.632456 − 0.774597i \(-0.717953\pi\)
0.632456 − 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 − 0.522233i \(-0.825099\pi\)
0.852803 − 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 3.00000 − 5.19615i 0.774597 − 1.34164i
\(16\) 0 0
\(17\) − 1.73205i − 0.420084i −0.977692 − 0.210042i \(-0.932640\pi\)
0.977692 − 0.210042i \(-0.0673601\pi\)
\(18\) 0 0
\(19\) 4.00000 + 1.73205i 0.917663 + 0.397360i
\(20\) 0 0
\(21\) −1.50000 − 0.866025i −0.327327 − 0.188982i
\(22\) 0 0
\(23\) − 5.19615i − 1.08347i −0.840548 − 0.541736i \(-0.817767\pi\)
0.840548 − 0.541736i \(-0.182233\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) − 10.3923i − 1.86651i −0.359211 − 0.933257i \(-0.616954\pi\)
0.359211 − 0.933257i \(-0.383046\pi\)
\(32\) 0 0
\(33\) 3.00000 − 5.19615i 0.522233 − 0.904534i
\(34\) 0 0
\(35\) 3.46410i 0.585540i
\(36\) 0 0
\(37\) − 6.92820i − 1.13899i −0.821995 − 0.569495i \(-0.807139\pi\)
0.821995 − 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) −1.50000 + 2.59808i −0.240192 + 0.416025i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 9.00000 − 5.19615i 1.34164 − 0.774597i
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.50000 − 2.59808i 0.210042 − 0.363803i
\(52\) 0 0
\(53\) −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
\(57\) 4.50000 + 6.06218i 0.596040 + 0.802955i
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −1.50000 − 2.59808i −0.188982 − 0.327327i
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.66025i 1.05802i 0.848616 + 0.529009i \(0.177436\pi\)
−0.848616 + 0.529009i \(0.822564\pi\)
\(68\) 0 0
\(69\) 4.50000 − 7.79423i 0.541736 − 0.938315i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −10.5000 − 6.06218i −1.21244 − 0.700000i
\(76\) 0 0
\(77\) 3.46410i 0.394771i
\(78\) 0 0
\(79\) 6.92820i 0.779484i 0.920924 + 0.389742i \(0.127436\pi\)
−0.920924 + 0.389742i \(0.872564\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 10.3923i 1.14070i 0.821401 + 0.570352i \(0.193193\pi\)
−0.821401 + 0.570352i \(0.806807\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 13.5000 + 7.79423i 1.44735 + 0.835629i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 1.73205i − 0.181568i
\(92\) 0 0
\(93\) 9.00000 − 15.5885i 0.933257 − 1.61645i
\(94\) 0 0
\(95\) 6.00000 − 13.8564i 0.615587 − 1.42164i
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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 912.2.f.e.113.2 2
3.2 odd 2 912.2.f.a.113.2 2
4.3 odd 2 114.2.b.c.113.1 yes 2
12.11 even 2 114.2.b.b.113.1 ✓ 2
19.18 odd 2 912.2.f.a.113.1 2
57.56 even 2 inner 912.2.f.e.113.1 2
76.75 even 2 114.2.b.b.113.2 yes 2
228.227 odd 2 114.2.b.c.113.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.b.b.113.1 ✓ 2 12.11 even 2
114.2.b.b.113.2 yes 2 76.75 even 2
114.2.b.c.113.1 yes 2 4.3 odd 2
114.2.b.c.113.2 yes 2 228.227 odd 2
912.2.f.a.113.1 2 19.18 odd 2
912.2.f.a.113.2 2 3.2 odd 2
912.2.f.e.113.1 2 57.56 even 2 inner
912.2.f.e.113.2 2 1.1 even 1 trivial