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Properties

Label 1183.1.n.b.867.2
Level $1183$
Weight $1$
Character 1183.867
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(146,1183)] 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("1183.146"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) 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.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590393909945\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.64827.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.1655595487.1

Embedding invariants

Embedding label 867.2
Root \(-0.623490 + 1.07992i\) of defining polynomial
Character \(\chi\) \(=\) 1183.867
Dual form 1183.1.n.b.146.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.222521 - 0.385418i) q^{2} +(0.400969 + 0.694498i) q^{4} +(-0.500000 - 0.866025i) q^{7} +0.801938 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.900969 - 1.56052i) q^{11} -0.445042 q^{14} +(-0.222521 + 0.385418i) q^{16} -0.445042 q^{18} +(-0.400969 - 0.694498i) q^{22} +(-0.623490 + 1.07992i) q^{23} +1.00000 q^{25} +(0.400969 - 0.694498i) q^{28} +(0.222521 - 0.385418i) q^{29} +(0.500000 + 0.866025i) q^{32} +(0.400969 - 0.694498i) q^{36} +(-0.623490 + 1.07992i) q^{37} +(0.222521 + 0.385418i) q^{43} +1.44504 q^{44} +(0.277479 + 0.480608i) q^{46} +(-0.500000 + 0.866025i) q^{49} +(0.222521 - 0.385418i) q^{50} +1.24698 q^{53} +(-0.400969 - 0.694498i) q^{56} +(-0.0990311 - 0.171527i) q^{58} +(-0.500000 + 0.866025i) q^{63} +(-0.623490 + 1.07992i) q^{67} +(-0.623490 - 1.07992i) q^{71} +(-0.400969 - 0.694498i) q^{72} +(0.277479 + 0.480608i) q^{74} -1.80194 q^{77} -1.80194 q^{79} +(-0.500000 + 0.866025i) q^{81} +0.198062 q^{86} +(0.722521 - 1.25144i) q^{88} -1.00000 q^{92} +(0.222521 + 0.385418i) q^{98} -1.80194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 2 q^{4} - 3 q^{7} - 4 q^{8} - 3 q^{9} + q^{11} - 2 q^{14} - q^{16} - 2 q^{18} + 2 q^{22} + q^{23} + 6 q^{25} - 2 q^{28} + q^{29} + 3 q^{32} - 2 q^{36} + q^{37} + q^{43} + 8 q^{44} + 2 q^{46}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.222521 − 0.385418i 0.222521 − 0.385418i −0.733052 − 0.680173i \(-0.761905\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(3\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0.400969 + 0.694498i 0.400969 + 0.694498i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(8\) 0.801938 0.801938
\(9\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0.900969 − 1.56052i 0.900969 − 1.56052i 0.0747301 − 0.997204i \(-0.476190\pi\)
0.826239 − 0.563320i \(-0.190476\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −0.445042 −0.445042
\(15\) 0 0
\(16\) −0.222521 + 0.385418i −0.222521 + 0.385418i
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) −0.445042 −0.445042
\(19\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.400969 − 0.694498i −0.400969 − 0.694498i
\(23\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0.400969 − 0.694498i 0.400969 − 0.694498i
\(29\) 0.222521 − 0.385418i 0.222521 − 0.385418i −0.733052 − 0.680173i \(-0.761905\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.400969 − 0.694498i 0.400969 − 0.694498i
\(37\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0.222521 + 0.385418i 0.222521 + 0.385418i 0.955573 − 0.294755i \(-0.0952381\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(44\) 1.44504 1.44504
\(45\) 0 0
\(46\) 0.277479 + 0.480608i 0.277479 + 0.480608i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0.222521 − 0.385418i 0.222521 − 0.385418i
\(51\) 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
\(56\) −0.400969 − 0.694498i −0.400969 − 0.694498i
\(57\) 0 0
\(58\) −0.0990311 − 0.171527i −0.0990311 − 0.171527i
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.623490 + 1.07992i −0.623490 + 1.07992i 0.365341 + 0.930874i \(0.380952\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.623490 − 1.07992i −0.623490 − 1.07992i −0.988831 − 0.149042i \(-0.952381\pi\)
0.365341 − 0.930874i \(-0.380952\pi\)
\(72\) −0.400969 − 0.694498i −0.400969 − 0.694498i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0.277479 + 0.480608i 0.277479 + 0.480608i
\(75\) 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
\(81\) −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
\(86\) 0.198062 0.198062
\(87\) 0 0
\(88\) 0.722521 − 1.25144i 0.722521 − 1.25144i
\(89\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 0.222521 + 0.385418i 0.222521 + 0.385418i
\(99\) −1.80194 −1.80194
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.n.b.867.2 6
7.6 odd 2 CM 1183.1.n.b.867.2 6
13.2 odd 12 1183.1.t.a.699.4 12
13.3 even 3 inner 1183.1.n.b.146.2 6
13.4 even 6 1183.1.d.b.846.2 yes 3
13.5 odd 4 1183.1.t.a.1161.3 12
13.6 odd 12 1183.1.b.a.1182.4 6
13.7 odd 12 1183.1.b.a.1182.3 6
13.8 odd 4 1183.1.t.a.1161.4 12
13.9 even 3 1183.1.d.a.846.2 ✓ 3
13.10 even 6 1183.1.n.a.146.2 6
13.11 odd 12 1183.1.t.a.699.3 12
13.12 even 2 1183.1.n.a.867.2 6
91.6 even 12 1183.1.b.a.1182.4 6
91.20 even 12 1183.1.b.a.1182.3 6
91.34 even 4 1183.1.t.a.1161.4 12
91.41 even 12 1183.1.t.a.699.4 12
91.48 odd 6 1183.1.d.a.846.2 ✓ 3
91.55 odd 6 inner 1183.1.n.b.146.2 6
91.62 odd 6 1183.1.n.a.146.2 6
91.69 odd 6 1183.1.d.b.846.2 yes 3
91.76 even 12 1183.1.t.a.699.3 12
91.83 even 4 1183.1.t.a.1161.3 12
91.90 odd 2 1183.1.n.a.867.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1183.1.b.a.1182.3 6 13.7 odd 12
1183.1.b.a.1182.3 6 91.20 even 12
1183.1.b.a.1182.4 6 13.6 odd 12
1183.1.b.a.1182.4 6 91.6 even 12
1183.1.d.a.846.2 ✓ 3 13.9 even 3
1183.1.d.a.846.2 ✓ 3 91.48 odd 6
1183.1.d.b.846.2 yes 3 13.4 even 6
1183.1.d.b.846.2 yes 3 91.69 odd 6
1183.1.n.a.146.2 6 13.10 even 6
1183.1.n.a.146.2 6 91.62 odd 6
1183.1.n.a.867.2 6 13.12 even 2
1183.1.n.a.867.2 6 91.90 odd 2
1183.1.n.b.146.2 6 13.3 even 3 inner
1183.1.n.b.146.2 6 91.55 odd 6 inner
1183.1.n.b.867.2 6 1.1 even 1 trivial
1183.1.n.b.867.2 6 7.6 odd 2 CM
1183.1.t.a.699.3 12 13.11 odd 12
1183.1.t.a.699.3 12 91.76 even 12
1183.1.t.a.699.4 12 13.2 odd 12
1183.1.t.a.699.4 12 91.41 even 12
1183.1.t.a.1161.3 12 13.5 odd 4
1183.1.t.a.1161.3 12 91.83 even 4
1183.1.t.a.1161.4 12 13.8 odd 4
1183.1.t.a.1161.4 12 91.34 even 4