↑

Properties

Label 912.1.bl.a.353.1
Level $912$
Weight $1$
Character 912.353
Analytic conductor $0.455$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,1,Mod(353,912)] 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("912.353"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 912.bl (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.455147291521\)
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 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of \(\Q(\sqrt[3]{19})\)
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

Embedding invariants

Embedding label 353.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 912.353
Dual form 912.1.bl.a.881.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +1.00000 q^{7} +(-0.500000 - 0.866025i) q^{9} +(0.500000 + 0.866025i) q^{13} -1.00000 q^{19} +(0.500000 - 0.866025i) q^{21} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +1.00000 q^{31} -1.00000 q^{37} +1.00000 q^{39} +(-0.500000 + 0.866025i) q^{43} +(-0.500000 + 0.866025i) q^{57} +(0.500000 + 0.866025i) q^{61} +(-0.500000 - 0.866025i) q^{63} +(-0.500000 - 0.866025i) q^{67} +(0.500000 - 0.866025i) q^{73} -1.00000 q^{75} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(0.500000 + 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{93} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{7} - q^{9} + q^{13} - 2 q^{19} + q^{21} - q^{25} - 2 q^{27} + 2 q^{31} - 2 q^{37} + 2 q^{39} - q^{43} - q^{57} + q^{61} - q^{63} - q^{67} + q^{73} - 2 q^{75} - q^{79} - q^{81}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(1\)

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
\(3\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(4\) 0 0
\(5\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000
\(20\) 0 0
\(21\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(74\) 0 0
\(75\) −1.00000 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(92\) 0 0
\(93\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) 0 0
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 912.1.bl.a.353.1 2
3.2 odd 2 CM 912.1.bl.a.353.1 2
4.3 odd 2 57.1.h.a.11.1 ✓ 2
8.3 odd 2 3648.1.bl.b.2177.1 2
8.5 even 2 3648.1.bl.a.2177.1 2
12.11 even 2 57.1.h.a.11.1 ✓ 2
19.7 even 3 inner 912.1.bl.a.881.1 2
20.3 even 4 1425.1.o.a.524.1 4
20.7 even 4 1425.1.o.a.524.2 4
20.19 odd 2 1425.1.t.a.1151.1 2
24.5 odd 2 3648.1.bl.a.2177.1 2
24.11 even 2 3648.1.bl.b.2177.1 2
28.3 even 6 2793.1.bi.a.1892.1 2
28.11 odd 6 2793.1.bi.b.1892.1 2
28.19 even 6 2793.1.n.b.410.1 2
28.23 odd 6 2793.1.n.a.410.1 2
28.27 even 2 2793.1.bf.a.638.1 2
36.7 odd 6 1539.1.j.a.296.1 2
36.11 even 6 1539.1.j.a.296.1 2
36.23 even 6 1539.1.n.a.1322.1 2
36.31 odd 6 1539.1.n.a.1322.1 2
57.26 odd 6 inner 912.1.bl.a.881.1 2
60.23 odd 4 1425.1.o.a.524.1 4
60.47 odd 4 1425.1.o.a.524.2 4
60.59 even 2 1425.1.t.a.1151.1 2
76.3 even 18 1083.1.l.b.821.1 6
76.7 odd 6 57.1.h.a.26.1 yes 2
76.11 odd 6 1083.1.b.b.362.1 1
76.15 even 18 1083.1.l.b.62.1 6
76.23 odd 18 1083.1.l.a.62.1 6
76.27 even 6 1083.1.b.a.362.1 1
76.31 even 6 1083.1.h.a.653.1 2
76.35 odd 18 1083.1.l.a.821.1 6
76.43 odd 18 1083.1.l.a.389.1 6
76.47 odd 18 1083.1.l.a.245.1 6
76.51 even 18 1083.1.l.b.776.1 6
76.55 odd 18 1083.1.l.a.956.1 6
76.59 even 18 1083.1.l.b.956.1 6
76.63 odd 18 1083.1.l.a.776.1 6
76.67 even 18 1083.1.l.b.245.1 6
76.71 even 18 1083.1.l.b.389.1 6
76.75 even 2 1083.1.h.a.68.1 2
84.11 even 6 2793.1.bi.b.1892.1 2
84.23 even 6 2793.1.n.a.410.1 2
84.47 odd 6 2793.1.n.b.410.1 2
84.59 odd 6 2793.1.bi.a.1892.1 2
84.83 odd 2 2793.1.bf.a.638.1 2
152.45 even 6 3648.1.bl.a.1793.1 2
152.83 odd 6 3648.1.bl.b.1793.1 2
228.11 even 6 1083.1.b.b.362.1 1
228.23 even 18 1083.1.l.a.62.1 6
228.35 even 18 1083.1.l.a.821.1 6
228.47 even 18 1083.1.l.a.245.1 6
228.59 odd 18 1083.1.l.b.956.1 6
228.71 odd 18 1083.1.l.b.389.1 6
228.83 even 6 57.1.h.a.26.1 yes 2
228.107 odd 6 1083.1.h.a.653.1 2
228.119 even 18 1083.1.l.a.389.1 6
228.131 even 18 1083.1.l.a.956.1 6
228.143 odd 18 1083.1.l.b.245.1 6
228.155 odd 18 1083.1.l.b.821.1 6
228.167 odd 18 1083.1.l.b.62.1 6
228.179 odd 6 1083.1.b.a.362.1 1
228.203 odd 18 1083.1.l.b.776.1 6
228.215 even 18 1083.1.l.a.776.1 6
228.227 odd 2 1083.1.h.a.68.1 2
380.7 even 12 1425.1.o.a.824.1 4
380.83 even 12 1425.1.o.a.824.2 4
380.159 odd 6 1425.1.t.a.26.1 2
456.83 even 6 3648.1.bl.b.1793.1 2
456.197 odd 6 3648.1.bl.a.1793.1 2
532.83 even 6 2793.1.bf.a.197.1 2
532.159 even 6 2793.1.bi.a.2762.1 2
532.235 odd 6 2793.1.n.a.1451.1 2
532.311 even 6 2793.1.n.b.1451.1 2
532.387 odd 6 2793.1.bi.b.2762.1 2
684.7 odd 6 1539.1.n.a.539.1 2
684.83 even 6 1539.1.n.a.539.1 2
684.311 even 6 1539.1.j.a.26.1 2
684.463 odd 6 1539.1.j.a.26.1 2
1140.83 odd 12 1425.1.o.a.824.2 4
1140.539 even 6 1425.1.t.a.26.1 2
1140.767 odd 12 1425.1.o.a.824.1 4
1596.83 odd 6 2793.1.bf.a.197.1 2
1596.311 odd 6 2793.1.n.b.1451.1 2
1596.767 even 6 2793.1.n.a.1451.1 2
1596.1223 odd 6 2793.1.bi.a.2762.1 2
1596.1451 even 6 2793.1.bi.b.2762.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.1.h.a.11.1 ✓ 2 4.3 odd 2
57.1.h.a.11.1 ✓ 2 12.11 even 2
57.1.h.a.26.1 yes 2 76.7 odd 6
57.1.h.a.26.1 yes 2 228.83 even 6
912.1.bl.a.353.1 2 1.1 even 1 trivial
912.1.bl.a.353.1 2 3.2 odd 2 CM
912.1.bl.a.881.1 2 19.7 even 3 inner
912.1.bl.a.881.1 2 57.26 odd 6 inner
1083.1.b.a.362.1 1 76.27 even 6
1083.1.b.a.362.1 1 228.179 odd 6
1083.1.b.b.362.1 1 76.11 odd 6
1083.1.b.b.362.1 1 228.11 even 6
1083.1.h.a.68.1 2 76.75 even 2
1083.1.h.a.68.1 2 228.227 odd 2
1083.1.h.a.653.1 2 76.31 even 6
1083.1.h.a.653.1 2 228.107 odd 6
1083.1.l.a.62.1 6 76.23 odd 18
1083.1.l.a.62.1 6 228.23 even 18
1083.1.l.a.245.1 6 76.47 odd 18
1083.1.l.a.245.1 6 228.47 even 18
1083.1.l.a.389.1 6 76.43 odd 18
1083.1.l.a.389.1 6 228.119 even 18
1083.1.l.a.776.1 6 76.63 odd 18
1083.1.l.a.776.1 6 228.215 even 18
1083.1.l.a.821.1 6 76.35 odd 18
1083.1.l.a.821.1 6 228.35 even 18
1083.1.l.a.956.1 6 76.55 odd 18
1083.1.l.a.956.1 6 228.131 even 18
1083.1.l.b.62.1 6 76.15 even 18
1083.1.l.b.62.1 6 228.167 odd 18
1083.1.l.b.245.1 6 76.67 even 18
1083.1.l.b.245.1 6 228.143 odd 18
1083.1.l.b.389.1 6 76.71 even 18
1083.1.l.b.389.1 6 228.71 odd 18
1083.1.l.b.776.1 6 76.51 even 18
1083.1.l.b.776.1 6 228.203 odd 18
1083.1.l.b.821.1 6 76.3 even 18
1083.1.l.b.821.1 6 228.155 odd 18
1083.1.l.b.956.1 6 76.59 even 18
1083.1.l.b.956.1 6 228.59 odd 18
1425.1.o.a.524.1 4 20.3 even 4
1425.1.o.a.524.1 4 60.23 odd 4
1425.1.o.a.524.2 4 20.7 even 4
1425.1.o.a.524.2 4 60.47 odd 4
1425.1.o.a.824.1 4 380.7 even 12
1425.1.o.a.824.1 4 1140.767 odd 12
1425.1.o.a.824.2 4 380.83 even 12
1425.1.o.a.824.2 4 1140.83 odd 12
1425.1.t.a.26.1 2 380.159 odd 6
1425.1.t.a.26.1 2 1140.539 even 6
1425.1.t.a.1151.1 2 20.19 odd 2
1425.1.t.a.1151.1 2 60.59 even 2
1539.1.j.a.26.1 2 684.311 even 6
1539.1.j.a.26.1 2 684.463 odd 6
1539.1.j.a.296.1 2 36.7 odd 6
1539.1.j.a.296.1 2 36.11 even 6
1539.1.n.a.539.1 2 684.7 odd 6
1539.1.n.a.539.1 2 684.83 even 6
1539.1.n.a.1322.1 2 36.23 even 6
1539.1.n.a.1322.1 2 36.31 odd 6
2793.1.n.a.410.1 2 28.23 odd 6
2793.1.n.a.410.1 2 84.23 even 6
2793.1.n.a.1451.1 2 532.235 odd 6
2793.1.n.a.1451.1 2 1596.767 even 6
2793.1.n.b.410.1 2 28.19 even 6
2793.1.n.b.410.1 2 84.47 odd 6
2793.1.n.b.1451.1 2 532.311 even 6
2793.1.n.b.1451.1 2 1596.311 odd 6
2793.1.bf.a.197.1 2 532.83 even 6
2793.1.bf.a.197.1 2 1596.83 odd 6
2793.1.bf.a.638.1 2 28.27 even 2
2793.1.bf.a.638.1 2 84.83 odd 2
2793.1.bi.a.1892.1 2 28.3 even 6
2793.1.bi.a.1892.1 2 84.59 odd 6
2793.1.bi.a.2762.1 2 532.159 even 6
2793.1.bi.a.2762.1 2 1596.1223 odd 6
2793.1.bi.b.1892.1 2 28.11 odd 6
2793.1.bi.b.1892.1 2 84.11 even 6
2793.1.bi.b.2762.1 2 532.387 odd 6
2793.1.bi.b.2762.1 2 1596.1451 even 6
3648.1.bl.a.1793.1 2 152.45 even 6
3648.1.bl.a.1793.1 2 456.197 odd 6
3648.1.bl.a.2177.1 2 8.5 even 2
3648.1.bl.a.2177.1 2 24.5 odd 2
3648.1.bl.b.1793.1 2 152.83 odd 6
3648.1.bl.b.1793.1 2 456.83 even 6
3648.1.bl.b.2177.1 2 8.3 odd 2
3648.1.bl.b.2177.1 2 24.11 even 2