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Properties

Label 1183.1.b.a.1182.3
Level $1183$
Weight $1$
Character 1183.1182
Analytic conductor $0.590$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -7
Inner twists $4$

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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] = [1183,1,Mod(1182,1183)] 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("1183.1182"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.b (of order \(2\), degree \(1\), 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: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.1655595487.1

Embedding invariants

Embedding label 1182.3
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 1183.1182
Dual form 1183.1.b.a.1182.4

$q$-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)
 
\(f(q)\) \(=\) \(q-0.445042i q^{2} +0.801938 q^{4} -1.00000i q^{7} -0.801938i q^{8} +1.00000 q^{9} +1.80194i q^{11} -0.445042 q^{14} +0.445042 q^{16} -0.445042i q^{18} +0.801938 q^{22} -1.24698 q^{23} -1.00000 q^{25} -0.801938i q^{28} -0.445042 q^{29} -1.00000i q^{32} +0.801938 q^{36} -1.24698i q^{37} +0.445042 q^{43} +1.44504i q^{44} +0.554958i q^{46} -1.00000 q^{49} +0.445042i q^{50} +1.24698 q^{53} -0.801938 q^{56} +0.198062i q^{58} -1.00000i q^{63} +1.24698i q^{67} +1.24698i q^{71} -0.801938i q^{72} -0.554958 q^{74} +1.80194 q^{77} -1.80194 q^{79} +1.00000 q^{81} -0.198062i q^{86} +1.44504 q^{88} -1.00000 q^{92} +0.445042i q^{98} +1.80194i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} + 6 q^{9} - 2 q^{14} + 2 q^{16} - 4 q^{22} + 2 q^{23} - 6 q^{25} - 2 q^{29} - 4 q^{36} + 2 q^{43} - 6 q^{49} - 2 q^{53} + 4 q^{56} - 4 q^{74} + 2 q^{77} - 2 q^{79} + 6 q^{81} + 8 q^{88} - 6 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.445042i − 0.445042i −0.974928 − 0.222521i \(-0.928571\pi\)
0.974928 − 0.222521i \(-0.0714286\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0.801938 0.801938
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 1.00000i
\(8\) − 0.801938i − 0.801938i
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 1.80194i 1.80194i 0.433884 + 0.900969i \(0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −0.445042 −0.445042
\(15\) 0 0
\(16\) 0.445042 0.445042
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 0.445042i − 0.445042i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.801938 0.801938
\(23\) −1.24698 −1.24698 −0.623490 − 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) − 0.801938i − 0.801938i
\(29\) −0.445042 −0.445042 −0.222521 − 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.801938 0.801938
\(37\) − 1.24698i − 1.24698i −0.781831 − 0.623490i \(-0.785714\pi\)
0.781831 − 0.623490i \(-0.214286\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0.445042 0.445042 0.222521 − 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(44\) 1.44504i 1.44504i
\(45\) 0 0
\(46\) 0.554958i 0.554958i
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0.445042i 0.445042i
\(51\) 0 0
\(52\) 0 0
\(53\) 1.24698 1.24698 0.623490 − 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.801938 −0.801938
\(57\) 0 0
\(58\) 0.198062i 0.198062i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 1.00000i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(72\) − 0.801938i − 0.801938i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −0.554958 −0.554958
\(75\) 0 0
\(76\) 0 0
\(77\) 1.80194 1.80194
\(78\) 0 0
\(79\) −1.80194 −1.80194 −0.900969 − 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 0.198062i − 0.198062i
\(87\) 0 0
\(88\) 1.44504 1.44504
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0.445042i 0.445042i
\(99\) 1.80194i 1.80194i
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 1183.1.b.a.1182.3 6
7.6 odd 2 CM 1183.1.b.a.1182.3 6
13.2 odd 12 1183.1.n.b.867.2 6
13.3 even 3 1183.1.t.a.1161.4 12
13.4 even 6 1183.1.t.a.699.4 12
13.5 odd 4 1183.1.d.a.846.2 ✓ 3
13.6 odd 12 1183.1.n.b.146.2 6
13.7 odd 12 1183.1.n.a.146.2 6
13.8 odd 4 1183.1.d.b.846.2 yes 3
13.9 even 3 1183.1.t.a.699.3 12
13.10 even 6 1183.1.t.a.1161.3 12
13.11 odd 12 1183.1.n.a.867.2 6
13.12 even 2 inner 1183.1.b.a.1182.4 6
91.6 even 12 1183.1.n.b.146.2 6
91.20 even 12 1183.1.n.a.146.2 6
91.34 even 4 1183.1.d.b.846.2 yes 3
91.41 even 12 1183.1.n.b.867.2 6
91.48 odd 6 1183.1.t.a.699.3 12
91.55 odd 6 1183.1.t.a.1161.4 12
91.62 odd 6 1183.1.t.a.1161.3 12
91.69 odd 6 1183.1.t.a.699.4 12
91.76 even 12 1183.1.n.a.867.2 6
91.83 even 4 1183.1.d.a.846.2 ✓ 3
91.90 odd 2 inner 1183.1.b.a.1182.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1183.1.b.a.1182.3 6 1.1 even 1 trivial
1183.1.b.a.1182.3 6 7.6 odd 2 CM
1183.1.b.a.1182.4 6 13.12 even 2 inner
1183.1.b.a.1182.4 6 91.90 odd 2 inner
1183.1.d.a.846.2 ✓ 3 13.5 odd 4
1183.1.d.a.846.2 ✓ 3 91.83 even 4
1183.1.d.b.846.2 yes 3 13.8 odd 4
1183.1.d.b.846.2 yes 3 91.34 even 4
1183.1.n.a.146.2 6 13.7 odd 12
1183.1.n.a.146.2 6 91.20 even 12
1183.1.n.a.867.2 6 13.11 odd 12
1183.1.n.a.867.2 6 91.76 even 12
1183.1.n.b.146.2 6 13.6 odd 12
1183.1.n.b.146.2 6 91.6 even 12
1183.1.n.b.867.2 6 13.2 odd 12
1183.1.n.b.867.2 6 91.41 even 12
1183.1.t.a.699.3 12 13.9 even 3
1183.1.t.a.699.3 12 91.48 odd 6
1183.1.t.a.699.4 12 13.4 even 6
1183.1.t.a.699.4 12 91.69 odd 6
1183.1.t.a.1161.3 12 13.10 even 6
1183.1.t.a.1161.3 12 91.62 odd 6
1183.1.t.a.1161.4 12 13.3 even 3
1183.1.t.a.1161.4 12 91.55 odd 6