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Properties

Label 1860.1.be.c.119.1
Level $1860$
Weight $1$
Character 1860.119
Analytic conductor $0.928$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -15
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] = [1860,1,Mod(119,1860)] 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("1860.119"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1860.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.928260923497\)
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: \(D_{6}\)
Projective field: Galois closure of 6.2.412259774400.1

Embedding invariants

Embedding label 119.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1860.119
Dual form 1860.1.be.c.719.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} -1.00000 q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.500000 - 0.866025i) q^{12} +1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 - 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.50000 - 0.866025i) q^{19} +1.00000 q^{20} -2.00000 q^{23} +(0.500000 - 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} +(0.500000 + 0.866025i) q^{30} +1.00000 q^{31} +(0.500000 - 0.866025i) q^{32} -1.73205i q^{34} +1.00000 q^{36} -1.73205i q^{38} +(0.500000 + 0.866025i) q^{40} +(-0.500000 + 0.866025i) q^{45} +(-1.00000 - 1.73205i) q^{46} +1.73205i q^{47} +1.00000 q^{48} +(0.500000 + 0.866025i) q^{49} -1.00000 q^{50} +(1.50000 - 0.866025i) q^{51} +(0.500000 + 0.866025i) q^{54} +(1.50000 - 0.866025i) q^{57} +(-0.500000 + 0.866025i) q^{60} -1.73205i q^{61} +(0.500000 + 0.866025i) q^{62} +1.00000 q^{64} +(1.50000 - 0.866025i) q^{68} +(1.00000 - 1.73205i) q^{69} +(0.500000 + 0.866025i) q^{72} +(-0.500000 - 0.866025i) q^{75} +(1.50000 - 0.866025i) q^{76} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.500000 - 0.866025i) q^{83} +1.73205i q^{85} -1.00000 q^{90} +(1.00000 - 1.73205i) q^{92} +(-0.500000 + 0.866025i) q^{93} +(-1.50000 + 0.866025i) q^{94} +1.73205i q^{95} +(0.500000 + 0.866025i) q^{96} +(-0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{8} - q^{9} + q^{10} - q^{12} + 2 q^{15} - q^{16} - 3 q^{17} + q^{18} - 3 q^{19} + 2 q^{20} - 4 q^{23} + q^{24} - q^{25} + 2 q^{27} + q^{30}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(6\) −1.00000 −1.00000
\(7\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) −1.00000 −1.00000
\(9\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(10\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(13\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 1.00000 1.00000
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) −1.50000 − 0.866025i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(18\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(19\) −1.50000 − 0.866025i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(20\) 1.00000 1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(24\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(31\) 1.00000 1.00000
\(32\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(33\) 0 0
\(34\) − 1.73205i − 1.73205i
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) − 1.73205i − 1.73205i
\(39\) 0 0
\(40\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(41\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(46\) −1.00000 − 1.73205i −1.00000 − 1.73205i
\(47\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 1.00000 1.00000
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) −1.00000 −1.00000
\(51\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(52\) 0 0
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(55\) 0 0
\(56\) 0 0
\(57\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(61\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(62\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(69\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(70\) 0 0
\(71\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(73\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(76\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 1.73205i 1.73205i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −1.00000
\(91\) 0 0
\(92\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(93\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(94\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(95\) 1.73205i 1.73205i
\(96\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −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 1860.1.be.c.119.1 yes 2
3.2 odd 2 1860.1.be.b.119.1 yes 2
4.3 odd 2 1860.1.be.d.119.1 yes 2
5.4 even 2 1860.1.be.b.119.1 yes 2
12.11 even 2 1860.1.be.a.119.1 ✓ 2
15.14 odd 2 CM 1860.1.be.c.119.1 yes 2
20.19 odd 2 1860.1.be.a.119.1 ✓ 2
31.6 odd 6 1860.1.be.d.719.1 yes 2
60.59 even 2 1860.1.be.d.119.1 yes 2
93.68 even 6 1860.1.be.a.719.1 yes 2
124.99 even 6 inner 1860.1.be.c.719.1 yes 2
155.99 odd 6 1860.1.be.a.719.1 yes 2
372.347 odd 6 1860.1.be.b.719.1 yes 2
465.254 even 6 1860.1.be.d.719.1 yes 2
620.99 even 6 1860.1.be.b.719.1 yes 2
1860.719 odd 6 inner 1860.1.be.c.719.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.1.be.a.119.1 ✓ 2 12.11 even 2
1860.1.be.a.119.1 ✓ 2 20.19 odd 2
1860.1.be.a.719.1 yes 2 93.68 even 6
1860.1.be.a.719.1 yes 2 155.99 odd 6
1860.1.be.b.119.1 yes 2 3.2 odd 2
1860.1.be.b.119.1 yes 2 5.4 even 2
1860.1.be.b.719.1 yes 2 372.347 odd 6
1860.1.be.b.719.1 yes 2 620.99 even 6
1860.1.be.c.119.1 yes 2 1.1 even 1 trivial
1860.1.be.c.119.1 yes 2 15.14 odd 2 CM
1860.1.be.c.719.1 yes 2 124.99 even 6 inner
1860.1.be.c.719.1 yes 2 1860.719 odd 6 inner
1860.1.be.d.119.1 yes 2 4.3 odd 2
1860.1.be.d.119.1 yes 2 60.59 even 2
1860.1.be.d.719.1 yes 2 31.6 odd 6
1860.1.be.d.719.1 yes 2 465.254 even 6