Properties

Label 171.1.c.a.37.1
Level $171$
Weight $1$
Character 171.37
Self dual yes
Analytic conductor $0.085$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -19, 57
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] = [171,1,Mod(37,171)] 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("171.37"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 171.c (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.0853401171602\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\)
Artin image: $D_4$
Artin field: Galois closure of \(\Q(\sqrt{30 +2 \sqrt{-3}})\)
Stark unit: Root of $x^{4} - 7x^{3} - 7x + 1$

Embedding invariants

Embedding label 37.1
Character \(\chi\) \(=\) 171.37

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{4} -2.00000 q^{7} +1.00000 q^{16} -1.00000 q^{19} -1.00000 q^{25} -2.00000 q^{28} +2.00000 q^{43} +3.00000 q^{49} -2.00000 q^{61} +1.00000 q^{64} +2.00000 q^{73} -1.00000 q^{76} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1\)

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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 1.00000 1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.00000 −2.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 3.00000 3.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 171.1.c.a.37.1 ✓ 1
3.2 odd 2 CM 171.1.c.a.37.1 ✓ 1
4.3 odd 2 2736.1.o.a.721.1 1
9.2 odd 6 1539.1.o.b.1405.1 2
9.4 even 3 1539.1.o.b.379.1 2
9.5 odd 6 1539.1.o.b.379.1 2
9.7 even 3 1539.1.o.b.1405.1 2
12.11 even 2 2736.1.o.a.721.1 1
19.2 odd 18 3249.1.ba.c.262.1 6
19.3 odd 18 3249.1.ba.c.694.1 6
19.4 even 9 3249.1.ba.c.307.1 6
19.5 even 9 3249.1.ba.c.127.1 6
19.6 even 9 3249.1.ba.c.838.1 6
19.7 even 3 3249.1.p.a.2098.1 2
19.8 odd 6 3249.1.p.a.1513.1 2
19.9 even 9 3249.1.ba.c.3187.1 6
19.10 odd 18 3249.1.ba.c.3187.1 6
19.11 even 3 3249.1.p.a.1513.1 2
19.12 odd 6 3249.1.p.a.2098.1 2
19.13 odd 18 3249.1.ba.c.838.1 6
19.14 odd 18 3249.1.ba.c.127.1 6
19.15 odd 18 3249.1.ba.c.307.1 6
19.16 even 9 3249.1.ba.c.694.1 6
19.17 even 9 3249.1.ba.c.262.1 6
19.18 odd 2 CM 171.1.c.a.37.1 ✓ 1
57.2 even 18 3249.1.ba.c.262.1 6
57.5 odd 18 3249.1.ba.c.127.1 6
57.8 even 6 3249.1.p.a.1513.1 2
57.11 odd 6 3249.1.p.a.1513.1 2
57.14 even 18 3249.1.ba.c.127.1 6
57.17 odd 18 3249.1.ba.c.262.1 6
57.23 odd 18 3249.1.ba.c.307.1 6
57.26 odd 6 3249.1.p.a.2098.1 2
57.29 even 18 3249.1.ba.c.3187.1 6
57.32 even 18 3249.1.ba.c.838.1 6
57.35 odd 18 3249.1.ba.c.694.1 6
57.41 even 18 3249.1.ba.c.694.1 6
57.44 odd 18 3249.1.ba.c.838.1 6
57.47 odd 18 3249.1.ba.c.3187.1 6
57.50 even 6 3249.1.p.a.2098.1 2
57.53 even 18 3249.1.ba.c.307.1 6
57.56 even 2 RM 171.1.c.a.37.1 ✓ 1
76.75 even 2 2736.1.o.a.721.1 1
171.56 even 6 1539.1.o.b.1405.1 2
171.94 odd 6 1539.1.o.b.379.1 2
171.113 even 6 1539.1.o.b.379.1 2
171.151 odd 6 1539.1.o.b.1405.1 2
228.227 odd 2 2736.1.o.a.721.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.1.c.a.37.1 ✓ 1 1.1 even 1 trivial
171.1.c.a.37.1 ✓ 1 3.2 odd 2 CM
171.1.c.a.37.1 ✓ 1 19.18 odd 2 CM
171.1.c.a.37.1 ✓ 1 57.56 even 2 RM
1539.1.o.b.379.1 2 9.4 even 3
1539.1.o.b.379.1 2 9.5 odd 6
1539.1.o.b.379.1 2 171.94 odd 6
1539.1.o.b.379.1 2 171.113 even 6
1539.1.o.b.1405.1 2 9.2 odd 6
1539.1.o.b.1405.1 2 9.7 even 3
1539.1.o.b.1405.1 2 171.56 even 6
1539.1.o.b.1405.1 2 171.151 odd 6
2736.1.o.a.721.1 1 4.3 odd 2
2736.1.o.a.721.1 1 12.11 even 2
2736.1.o.a.721.1 1 76.75 even 2
2736.1.o.a.721.1 1 228.227 odd 2
3249.1.p.a.1513.1 2 19.8 odd 6
3249.1.p.a.1513.1 2 19.11 even 3
3249.1.p.a.1513.1 2 57.8 even 6
3249.1.p.a.1513.1 2 57.11 odd 6
3249.1.p.a.2098.1 2 19.7 even 3
3249.1.p.a.2098.1 2 19.12 odd 6
3249.1.p.a.2098.1 2 57.26 odd 6
3249.1.p.a.2098.1 2 57.50 even 6
3249.1.ba.c.127.1 6 19.5 even 9
3249.1.ba.c.127.1 6 19.14 odd 18
3249.1.ba.c.127.1 6 57.5 odd 18
3249.1.ba.c.127.1 6 57.14 even 18
3249.1.ba.c.262.1 6 19.2 odd 18
3249.1.ba.c.262.1 6 19.17 even 9
3249.1.ba.c.262.1 6 57.2 even 18
3249.1.ba.c.262.1 6 57.17 odd 18
3249.1.ba.c.307.1 6 19.4 even 9
3249.1.ba.c.307.1 6 19.15 odd 18
3249.1.ba.c.307.1 6 57.23 odd 18
3249.1.ba.c.307.1 6 57.53 even 18
3249.1.ba.c.694.1 6 19.3 odd 18
3249.1.ba.c.694.1 6 19.16 even 9
3249.1.ba.c.694.1 6 57.35 odd 18
3249.1.ba.c.694.1 6 57.41 even 18
3249.1.ba.c.838.1 6 19.6 even 9
3249.1.ba.c.838.1 6 19.13 odd 18
3249.1.ba.c.838.1 6 57.32 even 18
3249.1.ba.c.838.1 6 57.44 odd 18
3249.1.ba.c.3187.1 6 19.9 even 9
3249.1.ba.c.3187.1 6 19.10 odd 18
3249.1.ba.c.3187.1 6 57.29 even 18
3249.1.ba.c.3187.1 6 57.47 odd 18