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Properties

Label 2312.1.j.c.1483.4
Level $2312$
Weight $1$
Character 2312.1483
Analytic conductor $1.154$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -8
Inner twists $8$

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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] = [2312,1,Mod(251,2312)] 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("2312.251"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2312.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.15383830921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.1680747204608.3

Embedding invariants

Embedding label 1483.4
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 2312.1483
Dual form 2312.1.j.c.251.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +(1.30656 - 1.30656i) q^{3} -1.00000 q^{4} +(-1.30656 - 1.30656i) q^{6} +1.00000i q^{8} -2.41421i q^{9} +(-0.541196 - 0.541196i) q^{11} +(-1.30656 + 1.30656i) q^{12} +1.00000 q^{16} -2.41421 q^{18} +(-0.541196 + 0.541196i) q^{22} +(1.30656 + 1.30656i) q^{24} -1.00000i q^{25} +(-1.84776 - 1.84776i) q^{27} -1.00000i q^{32} -1.41421 q^{33} +2.41421i q^{36} +(-0.541196 - 0.541196i) q^{41} +1.41421i q^{43} +(0.541196 + 0.541196i) q^{44} +(1.30656 - 1.30656i) q^{48} +1.00000i q^{49} -1.00000 q^{50} +(-1.84776 + 1.84776i) q^{54} +1.41421i q^{59} -1.00000 q^{64} +1.41421i q^{66} +1.41421 q^{67} +2.41421 q^{72} +(0.541196 - 0.541196i) q^{73} +(-1.30656 - 1.30656i) q^{75} -2.41421 q^{81} +(-0.541196 + 0.541196i) q^{82} +1.41421i q^{83} +1.41421 q^{86} +(0.541196 - 0.541196i) q^{88} +1.41421 q^{89} +(-1.30656 - 1.30656i) q^{96} +(1.30656 - 1.30656i) q^{97} +1.00000 q^{98} +(-1.30656 + 1.30656i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{16} - 8 q^{18} - 8 q^{50} - 8 q^{64} + 8 q^{72} - 8 q^{81} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 1.00000i
\(3\) 1.30656 − 1.30656i 1.30656 − 1.30656i 0.382683 − 0.923880i \(-0.375000\pi\)
0.923880 − 0.382683i \(-0.125000\pi\)
\(4\) −1.00000 −1.00000
\(5\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) −1.30656 − 1.30656i −1.30656 − 1.30656i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 1.00000i 1.00000i
\(9\) − 2.41421i − 2.41421i
\(10\) 0 0
\(11\) −0.541196 − 0.541196i −0.541196 − 0.541196i 0.382683 − 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0
\(18\) −2.41421 −2.41421
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(25\) − 1.00000i − 1.00000i
\(26\) 0 0
\(27\) −1.84776 − 1.84776i −1.84776 − 1.84776i
\(28\) 0 0
\(29\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) −1.41421 −1.41421
\(34\) 0 0
\(35\) 0 0
\(36\) 2.41421i 2.41421i
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.541196 − 0.541196i −0.541196 − 0.541196i 0.382683 − 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(42\) 0 0
\(43\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.30656 − 1.30656i 1.30656 − 1.30656i
\(49\) 1.00000i 1.00000i
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.84776 + 1.84776i −1.84776 + 1.84776i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 1.41421i 1.41421i
\(67\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 2.41421 2.41421
\(73\) 0.541196 − 0.541196i 0.541196 − 0.541196i −0.382683 − 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(74\) 0 0
\(75\) −1.30656 − 1.30656i −1.30656 − 1.30656i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(80\) 0 0
\(81\) −2.41421 −2.41421
\(82\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(83\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.41421 1.41421
\(87\) 0 0
\(88\) 0.541196 − 0.541196i 0.541196 − 0.541196i
\(89\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.30656 − 1.30656i −1.30656 − 1.30656i
\(97\) 1.30656 − 1.30656i 1.30656 − 1.30656i 0.382683 − 0.923880i \(-0.375000\pi\)
0.923880 − 0.382683i \(-0.125000\pi\)
\(98\) 1.00000 1.00000
\(99\) −1.30656 + 1.30656i −1.30656 + 1.30656i
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 2312.1.j.c.1483.4 8
8.3 odd 2 CM 2312.1.j.c.1483.4 8
17.2 even 8 2312.1.e.b.1155.1 4
17.3 odd 16 2312.1.p.d.155.1 4
17.4 even 4 inner 2312.1.j.c.251.1 8
17.5 odd 16 2312.1.p.b.1579.1 4
17.6 odd 16 2312.1.p.b.1555.1 4
17.7 odd 16 2312.1.p.a.179.1 4
17.8 even 8 2312.1.f.c.579.1 4
17.9 even 8 2312.1.f.c.579.4 4
17.10 odd 16 2312.1.p.d.179.1 4
17.11 odd 16 136.1.p.a.59.1 ✓ 4
17.12 odd 16 136.1.p.a.83.1 yes 4
17.13 even 4 inner 2312.1.j.c.251.4 8
17.14 odd 16 2312.1.p.a.155.1 4
17.15 even 8 2312.1.e.b.1155.4 4
17.16 even 2 inner 2312.1.j.c.1483.1 8
51.11 even 16 1224.1.bv.a.739.1 4
51.29 even 16 1224.1.bv.a.1171.1 4
68.11 even 16 544.1.bl.a.399.1 4
68.63 even 16 544.1.bl.a.15.1 4
85.12 even 16 3400.1.br.a.899.1 4
85.28 even 16 3400.1.br.a.2099.1 4
85.29 odd 16 3400.1.ce.a.1851.1 4
85.62 even 16 3400.1.br.b.2099.1 4
85.63 even 16 3400.1.br.b.899.1 4
85.79 odd 16 3400.1.ce.a.3051.1 4
136.3 even 16 2312.1.p.d.155.1 4
136.11 even 16 136.1.p.a.59.1 ✓ 4
136.19 odd 8 2312.1.e.b.1155.1 4
136.27 even 16 2312.1.p.d.179.1 4
136.29 odd 16 544.1.bl.a.15.1 4
136.43 odd 8 2312.1.f.c.579.4 4
136.45 odd 16 544.1.bl.a.399.1 4
136.59 odd 8 2312.1.f.c.579.1 4
136.67 odd 2 inner 2312.1.j.c.1483.1 8
136.75 even 16 2312.1.p.a.179.1 4
136.83 odd 8 2312.1.e.b.1155.4 4
136.91 even 16 2312.1.p.b.1555.1 4
136.99 even 16 2312.1.p.a.155.1 4
136.107 even 16 2312.1.p.b.1579.1 4
136.115 odd 4 inner 2312.1.j.c.251.4 8
136.123 odd 4 inner 2312.1.j.c.251.1 8
136.131 even 16 136.1.p.a.83.1 yes 4
408.11 odd 16 1224.1.bv.a.739.1 4
408.131 odd 16 1224.1.bv.a.1171.1 4
680.147 odd 16 3400.1.br.b.2099.1 4
680.267 odd 16 3400.1.br.a.899.1 4
680.283 odd 16 3400.1.br.a.2099.1 4
680.403 odd 16 3400.1.br.b.899.1 4
680.419 even 16 3400.1.ce.a.3051.1 4
680.539 even 16 3400.1.ce.a.1851.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.1.p.a.59.1 ✓ 4 17.11 odd 16
136.1.p.a.59.1 ✓ 4 136.11 even 16
136.1.p.a.83.1 yes 4 17.12 odd 16
136.1.p.a.83.1 yes 4 136.131 even 16
544.1.bl.a.15.1 4 68.63 even 16
544.1.bl.a.15.1 4 136.29 odd 16
544.1.bl.a.399.1 4 68.11 even 16
544.1.bl.a.399.1 4 136.45 odd 16
1224.1.bv.a.739.1 4 51.11 even 16
1224.1.bv.a.739.1 4 408.11 odd 16
1224.1.bv.a.1171.1 4 51.29 even 16
1224.1.bv.a.1171.1 4 408.131 odd 16
2312.1.e.b.1155.1 4 17.2 even 8
2312.1.e.b.1155.1 4 136.19 odd 8
2312.1.e.b.1155.4 4 17.15 even 8
2312.1.e.b.1155.4 4 136.83 odd 8
2312.1.f.c.579.1 4 17.8 even 8
2312.1.f.c.579.1 4 136.59 odd 8
2312.1.f.c.579.4 4 17.9 even 8
2312.1.f.c.579.4 4 136.43 odd 8
2312.1.j.c.251.1 8 17.4 even 4 inner
2312.1.j.c.251.1 8 136.123 odd 4 inner
2312.1.j.c.251.4 8 17.13 even 4 inner
2312.1.j.c.251.4 8 136.115 odd 4 inner
2312.1.j.c.1483.1 8 17.16 even 2 inner
2312.1.j.c.1483.1 8 136.67 odd 2 inner
2312.1.j.c.1483.4 8 1.1 even 1 trivial
2312.1.j.c.1483.4 8 8.3 odd 2 CM
2312.1.p.a.155.1 4 17.14 odd 16
2312.1.p.a.155.1 4 136.99 even 16
2312.1.p.a.179.1 4 17.7 odd 16
2312.1.p.a.179.1 4 136.75 even 16
2312.1.p.b.1555.1 4 17.6 odd 16
2312.1.p.b.1555.1 4 136.91 even 16
2312.1.p.b.1579.1 4 17.5 odd 16
2312.1.p.b.1579.1 4 136.107 even 16
2312.1.p.d.155.1 4 17.3 odd 16
2312.1.p.d.155.1 4 136.3 even 16
2312.1.p.d.179.1 4 17.10 odd 16
2312.1.p.d.179.1 4 136.27 even 16
3400.1.br.a.899.1 4 85.12 even 16
3400.1.br.a.899.1 4 680.267 odd 16
3400.1.br.a.2099.1 4 85.28 even 16
3400.1.br.a.2099.1 4 680.283 odd 16
3400.1.br.b.899.1 4 85.63 even 16
3400.1.br.b.899.1 4 680.403 odd 16
3400.1.br.b.2099.1 4 85.62 even 16
3400.1.br.b.2099.1 4 680.147 odd 16
3400.1.ce.a.1851.1 4 85.29 odd 16
3400.1.ce.a.1851.1 4 680.539 even 16
3400.1.ce.a.3051.1 4 85.79 odd 16
3400.1.ce.a.3051.1 4 680.419 even 16