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Properties

Label 152.1.k.a.83.1
Level $152$
Weight $1$
Character 152.83
Analytic conductor $0.076$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
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] = [152,1,Mod(11,152)] 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("152.11"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) 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.k (of order \(6\), degree \(2\), 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: \(0.0758578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2888.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.184832.1

Embedding invariants

Embedding label 83.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 152.83
Dual form 152.1.k.a.11.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{6} +1.00000 q^{8} -1.00000 q^{11} -1.00000 q^{12} +(-0.500000 - 0.866025i) q^{16} +(-1.00000 - 1.73205i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(0.500000 + 0.866025i) q^{22} +(0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +(-0.500000 + 0.866025i) q^{32} +(-0.500000 - 0.866025i) q^{33} +(-1.00000 + 1.73205i) q^{34} +1.00000 q^{38} +(0.500000 + 0.866025i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(0.500000 - 0.866025i) q^{44} +(0.500000 - 0.866025i) q^{48} +1.00000 q^{49} +1.00000 q^{50} +(1.00000 - 1.73205i) q^{51} +(-0.500000 - 0.866025i) q^{54} -1.00000 q^{57} +(0.500000 + 0.866025i) q^{59} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{66} +(0.500000 - 0.866025i) q^{67} +2.00000 q^{68} +(0.500000 + 0.866025i) q^{73} -1.00000 q^{75} +(-0.500000 - 0.866025i) q^{76} +(0.500000 + 0.866025i) q^{81} +(0.500000 - 0.866025i) q^{82} -1.00000 q^{83} +(-1.00000 + 1.73205i) q^{86} -1.00000 q^{88} +(-1.00000 + 1.73205i) q^{89} -1.00000 q^{96} +(0.500000 + 0.866025i) q^{97} +(-0.500000 - 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{8} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - q^{19} + q^{22} + q^{24} - q^{25} + 2 q^{27} - q^{32} - q^{33} - 2 q^{34} + 2 q^{38} + q^{41} - 2 q^{43}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(3\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −1.00000 −1.00000
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(18\) 0 0
\(19\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(20\) 0 0
\(21\) 0 0
\(22\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(33\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(34\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.00000 1.00000
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(44\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(49\) 1.00000 1.00000
\(50\) 1.00000 1.00000
\(51\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(52\) 0 0
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −1.00000
\(58\) 0 0
\(59\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(67\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(68\) 2.00000 2.00000
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) −1.00000 −1.00000
\(76\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(83\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(87\) 0 0
\(88\) −1.00000 −1.00000
\(89\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(99\) 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.k.a.83.1 yes 2
3.2 odd 2 1368.1.bz.a.235.1 2
4.3 odd 2 608.1.o.a.463.1 2
5.2 odd 4 3800.1.bn.b.1299.2 4
5.3 odd 4 3800.1.bn.b.1299.1 4
5.4 even 2 3800.1.bd.c.1451.1 2
8.3 odd 2 CM 152.1.k.a.83.1 yes 2
8.5 even 2 608.1.o.a.463.1 2
19.2 odd 18 2888.1.u.d.2555.1 6
19.3 odd 18 2888.1.u.d.595.1 6
19.4 even 9 2888.1.u.c.2411.1 6
19.5 even 9 2888.1.u.c.99.1 6
19.6 even 9 2888.1.u.c.1867.1 6
19.7 even 3 2888.1.f.b.723.1 1
19.8 odd 6 2888.1.k.a.2595.1 2
19.9 even 9 2888.1.u.c.1859.1 6
19.10 odd 18 2888.1.u.d.1859.1 6
19.11 even 3 inner 152.1.k.a.11.1 ✓ 2
19.12 odd 6 2888.1.f.a.723.1 1
19.13 odd 18 2888.1.u.d.1867.1 6
19.14 odd 18 2888.1.u.d.99.1 6
19.15 odd 18 2888.1.u.d.2411.1 6
19.16 even 9 2888.1.u.c.595.1 6
19.17 even 9 2888.1.u.c.2555.1 6
19.18 odd 2 2888.1.k.a.2819.1 2
24.11 even 2 1368.1.bz.a.235.1 2
40.3 even 4 3800.1.bn.b.1299.1 4
40.19 odd 2 3800.1.bd.c.1451.1 2
40.27 even 4 3800.1.bn.b.1299.2 4
57.11 odd 6 1368.1.bz.a.163.1 2
76.11 odd 6 608.1.o.a.239.1 2
95.49 even 6 3800.1.bd.c.3051.1 2
95.68 odd 12 3800.1.bn.b.2899.2 4
95.87 odd 12 3800.1.bn.b.2899.1 4
152.3 even 18 2888.1.u.d.595.1 6
152.11 odd 6 inner 152.1.k.a.11.1 ✓ 2
152.27 even 6 2888.1.k.a.2595.1 2
152.35 odd 18 2888.1.u.c.595.1 6
152.43 odd 18 2888.1.u.c.99.1 6
152.51 even 18 2888.1.u.d.1867.1 6
152.59 even 18 2888.1.u.d.2555.1 6
152.67 even 18 2888.1.u.d.1859.1 6
152.75 even 2 2888.1.k.a.2819.1 2
152.83 odd 6 2888.1.f.b.723.1 1
152.91 even 18 2888.1.u.d.2411.1 6
152.99 odd 18 2888.1.u.c.2411.1 6
152.107 even 6 2888.1.f.a.723.1 1
152.123 odd 18 2888.1.u.c.1859.1 6
152.125 even 6 608.1.o.a.239.1 2
152.131 odd 18 2888.1.u.c.2555.1 6
152.139 odd 18 2888.1.u.c.1867.1 6
152.147 even 18 2888.1.u.d.99.1 6
456.11 even 6 1368.1.bz.a.163.1 2
760.163 even 12 3800.1.bn.b.2899.2 4
760.467 even 12 3800.1.bn.b.2899.1 4
760.619 odd 6 3800.1.bd.c.3051.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.k.a.11.1 ✓ 2 19.11 even 3 inner
152.1.k.a.11.1 ✓ 2 152.11 odd 6 inner
152.1.k.a.83.1 yes 2 1.1 even 1 trivial
152.1.k.a.83.1 yes 2 8.3 odd 2 CM
608.1.o.a.239.1 2 76.11 odd 6
608.1.o.a.239.1 2 152.125 even 6
608.1.o.a.463.1 2 4.3 odd 2
608.1.o.a.463.1 2 8.5 even 2
1368.1.bz.a.163.1 2 57.11 odd 6
1368.1.bz.a.163.1 2 456.11 even 6
1368.1.bz.a.235.1 2 3.2 odd 2
1368.1.bz.a.235.1 2 24.11 even 2
2888.1.f.a.723.1 1 19.12 odd 6
2888.1.f.a.723.1 1 152.107 even 6
2888.1.f.b.723.1 1 19.7 even 3
2888.1.f.b.723.1 1 152.83 odd 6
2888.1.k.a.2595.1 2 19.8 odd 6
2888.1.k.a.2595.1 2 152.27 even 6
2888.1.k.a.2819.1 2 19.18 odd 2
2888.1.k.a.2819.1 2 152.75 even 2
2888.1.u.c.99.1 6 19.5 even 9
2888.1.u.c.99.1 6 152.43 odd 18
2888.1.u.c.595.1 6 19.16 even 9
2888.1.u.c.595.1 6 152.35 odd 18
2888.1.u.c.1859.1 6 19.9 even 9
2888.1.u.c.1859.1 6 152.123 odd 18
2888.1.u.c.1867.1 6 19.6 even 9
2888.1.u.c.1867.1 6 152.139 odd 18
2888.1.u.c.2411.1 6 19.4 even 9
2888.1.u.c.2411.1 6 152.99 odd 18
2888.1.u.c.2555.1 6 19.17 even 9
2888.1.u.c.2555.1 6 152.131 odd 18
2888.1.u.d.99.1 6 19.14 odd 18
2888.1.u.d.99.1 6 152.147 even 18
2888.1.u.d.595.1 6 19.3 odd 18
2888.1.u.d.595.1 6 152.3 even 18
2888.1.u.d.1859.1 6 19.10 odd 18
2888.1.u.d.1859.1 6 152.67 even 18
2888.1.u.d.1867.1 6 19.13 odd 18
2888.1.u.d.1867.1 6 152.51 even 18
2888.1.u.d.2411.1 6 19.15 odd 18
2888.1.u.d.2411.1 6 152.91 even 18
2888.1.u.d.2555.1 6 19.2 odd 18
2888.1.u.d.2555.1 6 152.59 even 18
3800.1.bd.c.1451.1 2 5.4 even 2
3800.1.bd.c.1451.1 2 40.19 odd 2
3800.1.bd.c.3051.1 2 95.49 even 6
3800.1.bd.c.3051.1 2 760.619 odd 6
3800.1.bn.b.1299.1 4 5.3 odd 4
3800.1.bn.b.1299.1 4 40.3 even 4
3800.1.bn.b.1299.2 4 5.2 odd 4
3800.1.bn.b.1299.2 4 40.27 even 4
3800.1.bn.b.2899.1 4 95.87 odd 12
3800.1.bn.b.2899.1 4 760.467 even 12
3800.1.bn.b.2899.2 4 95.68 odd 12
3800.1.bn.b.2899.2 4 760.163 even 12