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Properties

Label 152.1.g.a.37.1
Level $152$
Weight $1$
Character 152.37
Self dual yes
Analytic conductor $0.076$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -152
Inner twists $2$

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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] = [152,1,Mod(37,152)] 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("152.37"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 152.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-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: yes
Analytic conductor: \(0.0758578819202\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.184832.1
Stark unit: Root of $x^{6} - 2x^{5} - 5x^{4} - 20x^{3} - 5x^{2} - 2x + 1$

Embedding invariants

Embedding label 37.1
Character \(\chi\) \(=\) 152.37

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +1.00000 q^{29} -1.00000 q^{32} +1.00000 q^{34} -2.00000 q^{37} +1.00000 q^{38} +1.00000 q^{39} +1.00000 q^{42} +1.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} -1.00000 q^{58} +1.00000 q^{59} +1.00000 q^{64} +1.00000 q^{67} -1.00000 q^{68} -1.00000 q^{69} -1.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -1.00000 q^{78} -1.00000 q^{81} -1.00000 q^{84} +1.00000 q^{87} -1.00000 q^{91} -1.00000 q^{92} -2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000
\(3\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.00000 −1.00000
\(7\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 1.00000
\(13\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 1.00000 1.00000
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) −1.00000 −1.00000
\(25\) 1.00000 1.00000
\(26\) −1.00000 −1.00000
\(27\) −1.00000 −1.00000
\(28\) −1.00000 −1.00000
\(29\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(38\) 1.00000 1.00000
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 1.00000 1.00000
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 1.00000
\(47\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(48\) 1.00000 1.00000
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) −1.00000 −1.00000
\(52\) 1.00000 1.00000
\(53\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) −1.00000 −1.00000
\(58\) −1.00000 −1.00000
\(59\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) −1.00000 −1.00000
\(69\) −1.00000 −1.00000
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 2.00000 2.00000
\(75\) 1.00000 1.00000
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) −1.00000 −1.00000
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) −1.00000 −1.00000
\(93\) 0 0
\(94\) −2.00000 −2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 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 152.1.g.a.37.1 ✓ 1
3.2 odd 2 1368.1.i.b.37.1 1
4.3 odd 2 608.1.g.a.113.1 1
5.2 odd 4 3800.1.b.a.949.1 2
5.3 odd 4 3800.1.b.a.949.2 2
5.4 even 2 3800.1.o.b.1101.1 1
8.3 odd 2 608.1.g.b.113.1 1
8.5 even 2 152.1.g.b.37.1 yes 1
19.2 odd 18 2888.1.s.a.2789.1 6
19.3 odd 18 2888.1.s.a.333.1 6
19.4 even 9 2888.1.s.b.1029.1 6
19.5 even 9 2888.1.s.b.2293.1 6
19.6 even 9 2888.1.s.b.477.1 6
19.7 even 3 2888.1.l.b.293.1 2
19.8 odd 6 2888.1.l.a.69.1 2
19.9 even 9 2888.1.s.b.1021.1 6
19.10 odd 18 2888.1.s.a.1021.1 6
19.11 even 3 2888.1.l.b.69.1 2
19.12 odd 6 2888.1.l.a.293.1 2
19.13 odd 18 2888.1.s.a.477.1 6
19.14 odd 18 2888.1.s.a.2293.1 6
19.15 odd 18 2888.1.s.a.1029.1 6
19.16 even 9 2888.1.s.b.333.1 6
19.17 even 9 2888.1.s.b.2789.1 6
19.18 odd 2 152.1.g.b.37.1 yes 1
24.5 odd 2 1368.1.i.a.37.1 1
40.13 odd 4 3800.1.b.b.949.1 2
40.29 even 2 3800.1.o.a.1101.1 1
40.37 odd 4 3800.1.b.b.949.2 2
57.56 even 2 1368.1.i.a.37.1 1
76.75 even 2 608.1.g.b.113.1 1
95.18 even 4 3800.1.b.b.949.1 2
95.37 even 4 3800.1.b.b.949.2 2
95.94 odd 2 3800.1.o.a.1101.1 1
152.5 even 18 2888.1.s.a.2293.1 6
152.13 odd 18 2888.1.s.b.477.1 6
152.21 odd 18 2888.1.s.b.2789.1 6
152.29 odd 18 2888.1.s.b.1021.1 6
152.37 odd 2 CM 152.1.g.a.37.1 ✓ 1
152.45 even 6 2888.1.l.a.293.1 2
152.53 odd 18 2888.1.s.b.1029.1 6
152.61 even 18 2888.1.s.a.1029.1 6
152.69 odd 6 2888.1.l.b.293.1 2
152.75 even 2 608.1.g.a.113.1 1
152.85 even 18 2888.1.s.a.1021.1 6
152.93 even 18 2888.1.s.a.2789.1 6
152.101 even 18 2888.1.s.a.477.1 6
152.109 odd 18 2888.1.s.b.2293.1 6
152.117 odd 18 2888.1.s.b.333.1 6
152.125 even 6 2888.1.l.a.69.1 2
152.141 odd 6 2888.1.l.b.69.1 2
152.149 even 18 2888.1.s.a.333.1 6
456.341 even 2 1368.1.i.b.37.1 1
760.37 even 4 3800.1.b.a.949.1 2
760.189 odd 2 3800.1.o.b.1101.1 1
760.493 even 4 3800.1.b.a.949.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.g.a.37.1 ✓ 1 1.1 even 1 trivial
152.1.g.a.37.1 ✓ 1 152.37 odd 2 CM
152.1.g.b.37.1 yes 1 8.5 even 2
152.1.g.b.37.1 yes 1 19.18 odd 2
608.1.g.a.113.1 1 4.3 odd 2
608.1.g.a.113.1 1 152.75 even 2
608.1.g.b.113.1 1 8.3 odd 2
608.1.g.b.113.1 1 76.75 even 2
1368.1.i.a.37.1 1 24.5 odd 2
1368.1.i.a.37.1 1 57.56 even 2
1368.1.i.b.37.1 1 3.2 odd 2
1368.1.i.b.37.1 1 456.341 even 2
2888.1.l.a.69.1 2 19.8 odd 6
2888.1.l.a.69.1 2 152.125 even 6
2888.1.l.a.293.1 2 19.12 odd 6
2888.1.l.a.293.1 2 152.45 even 6
2888.1.l.b.69.1 2 19.11 even 3
2888.1.l.b.69.1 2 152.141 odd 6
2888.1.l.b.293.1 2 19.7 even 3
2888.1.l.b.293.1 2 152.69 odd 6
2888.1.s.a.333.1 6 19.3 odd 18
2888.1.s.a.333.1 6 152.149 even 18
2888.1.s.a.477.1 6 19.13 odd 18
2888.1.s.a.477.1 6 152.101 even 18
2888.1.s.a.1021.1 6 19.10 odd 18
2888.1.s.a.1021.1 6 152.85 even 18
2888.1.s.a.1029.1 6 19.15 odd 18
2888.1.s.a.1029.1 6 152.61 even 18
2888.1.s.a.2293.1 6 19.14 odd 18
2888.1.s.a.2293.1 6 152.5 even 18
2888.1.s.a.2789.1 6 19.2 odd 18
2888.1.s.a.2789.1 6 152.93 even 18
2888.1.s.b.333.1 6 19.16 even 9
2888.1.s.b.333.1 6 152.117 odd 18
2888.1.s.b.477.1 6 19.6 even 9
2888.1.s.b.477.1 6 152.13 odd 18
2888.1.s.b.1021.1 6 19.9 even 9
2888.1.s.b.1021.1 6 152.29 odd 18
2888.1.s.b.1029.1 6 19.4 even 9
2888.1.s.b.1029.1 6 152.53 odd 18
2888.1.s.b.2293.1 6 19.5 even 9
2888.1.s.b.2293.1 6 152.109 odd 18
2888.1.s.b.2789.1 6 19.17 even 9
2888.1.s.b.2789.1 6 152.21 odd 18
3800.1.b.a.949.1 2 5.2 odd 4
3800.1.b.a.949.1 2 760.37 even 4
3800.1.b.a.949.2 2 5.3 odd 4
3800.1.b.a.949.2 2 760.493 even 4
3800.1.b.b.949.1 2 40.13 odd 4
3800.1.b.b.949.1 2 95.18 even 4
3800.1.b.b.949.2 2 40.37 odd 4
3800.1.b.b.949.2 2 95.37 even 4
3800.1.o.a.1101.1 1 40.29 even 2
3800.1.o.a.1101.1 1 95.94 odd 2
3800.1.o.b.1101.1 1 5.4 even 2
3800.1.o.b.1101.1 1 760.189 odd 2