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Properties

Label 3520.1.i.c.769.2
Level $3520$
Weight $1$
Character 3520.769
Analytic conductor $1.757$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -11
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] = [3520,1,Mod(769,3520)] 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("3520.769"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3520.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-1,0,0,0,-4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.75670884447\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.242000.1

Embedding invariants

Embedding label 769.2
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3520.769
Dual form 3520.1.i.c.769.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +(-0.500000 + 0.866025i) q^{5} -2.00000 q^{9} -1.00000 q^{11} +(-1.50000 - 0.866025i) q^{15} +1.73205i q^{23} +(-0.500000 - 0.866025i) q^{25} -1.73205i q^{27} -1.00000 q^{31} -1.73205i q^{33} -1.73205i q^{37} +(1.00000 - 1.73205i) q^{45} -1.00000 q^{49} +(0.500000 - 0.866025i) q^{55} -1.00000 q^{59} +1.73205i q^{67} -3.00000 q^{69} +1.00000 q^{71} +(1.50000 - 0.866025i) q^{75} +1.00000 q^{81} +1.00000 q^{89} -1.73205i q^{93} -1.73205i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 4 q^{9} - 2 q^{11} - 3 q^{15} - q^{25} - 2 q^{31} + 2 q^{45} - 2 q^{49} + q^{55} - 2 q^{59} - 6 q^{69} + 2 q^{71} + 3 q^{75} + 2 q^{81} + 2 q^{89} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.00000 −2.00000
\(10\) 0 0
\(11\) −1.00000 −1.00000
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.50000 − 0.866025i −1.50000 − 0.866025i
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) − 1.73205i − 1.73205i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) − 1.73205i − 1.73205i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) −3.00000 −3.00000
\(70\) 0 0
\(71\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 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
\(89\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 1.73205i − 1.73205i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(98\) 0 0
\(99\) 2.00000 2.00000
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 3520.1.i.c.769.2 2
4.3 odd 2 3520.1.i.d.769.1 2
5.4 even 2 inner 3520.1.i.c.769.1 2
8.3 odd 2 220.1.e.a.109.2 yes 2
8.5 even 2 880.1.i.b.769.1 2
11.10 odd 2 CM 3520.1.i.c.769.2 2
20.19 odd 2 3520.1.i.d.769.2 2
24.11 even 2 1980.1.p.a.109.2 2
40.3 even 4 1100.1.f.b.901.1 2
40.19 odd 2 220.1.e.a.109.1 ✓ 2
40.27 even 4 1100.1.f.b.901.2 2
40.29 even 2 880.1.i.b.769.2 2
44.43 even 2 3520.1.i.d.769.1 2
55.54 odd 2 inner 3520.1.i.c.769.1 2
88.3 odd 10 2420.1.q.a.2169.1 8
88.19 even 10 2420.1.q.a.2169.1 8
88.21 odd 2 880.1.i.b.769.1 2
88.27 odd 10 2420.1.q.a.1449.1 8
88.35 even 10 2420.1.q.a.1129.2 8
88.43 even 2 220.1.e.a.109.2 yes 2
88.51 even 10 2420.1.q.a.1909.2 8
88.59 odd 10 2420.1.q.a.1909.2 8
88.75 odd 10 2420.1.q.a.1129.2 8
88.83 even 10 2420.1.q.a.1449.1 8
120.59 even 2 1980.1.p.a.109.1 2
220.219 even 2 3520.1.i.d.769.2 2
264.131 odd 2 1980.1.p.a.109.2 2
440.19 even 10 2420.1.q.a.2169.2 8
440.43 odd 4 1100.1.f.b.901.1 2
440.59 odd 10 2420.1.q.a.1909.1 8
440.109 odd 2 880.1.i.b.769.2 2
440.139 even 10 2420.1.q.a.1909.1 8
440.179 odd 10 2420.1.q.a.2169.2 8
440.219 even 2 220.1.e.a.109.1 ✓ 2
440.259 even 10 2420.1.q.a.1449.2 8
440.299 even 10 2420.1.q.a.1129.1 8
440.307 odd 4 1100.1.f.b.901.2 2
440.339 odd 10 2420.1.q.a.1129.1 8
440.379 odd 10 2420.1.q.a.1449.2 8
1320.659 odd 2 1980.1.p.a.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.1.e.a.109.1 ✓ 2 40.19 odd 2
220.1.e.a.109.1 ✓ 2 440.219 even 2
220.1.e.a.109.2 yes 2 8.3 odd 2
220.1.e.a.109.2 yes 2 88.43 even 2
880.1.i.b.769.1 2 8.5 even 2
880.1.i.b.769.1 2 88.21 odd 2
880.1.i.b.769.2 2 40.29 even 2
880.1.i.b.769.2 2 440.109 odd 2
1100.1.f.b.901.1 2 40.3 even 4
1100.1.f.b.901.1 2 440.43 odd 4
1100.1.f.b.901.2 2 40.27 even 4
1100.1.f.b.901.2 2 440.307 odd 4
1980.1.p.a.109.1 2 120.59 even 2
1980.1.p.a.109.1 2 1320.659 odd 2
1980.1.p.a.109.2 2 24.11 even 2
1980.1.p.a.109.2 2 264.131 odd 2
2420.1.q.a.1129.1 8 440.299 even 10
2420.1.q.a.1129.1 8 440.339 odd 10
2420.1.q.a.1129.2 8 88.35 even 10
2420.1.q.a.1129.2 8 88.75 odd 10
2420.1.q.a.1449.1 8 88.27 odd 10
2420.1.q.a.1449.1 8 88.83 even 10
2420.1.q.a.1449.2 8 440.259 even 10
2420.1.q.a.1449.2 8 440.379 odd 10
2420.1.q.a.1909.1 8 440.59 odd 10
2420.1.q.a.1909.1 8 440.139 even 10
2420.1.q.a.1909.2 8 88.51 even 10
2420.1.q.a.1909.2 8 88.59 odd 10
2420.1.q.a.2169.1 8 88.3 odd 10
2420.1.q.a.2169.1 8 88.19 even 10
2420.1.q.a.2169.2 8 440.19 even 10
2420.1.q.a.2169.2 8 440.179 odd 10
3520.1.i.c.769.1 2 5.4 even 2 inner
3520.1.i.c.769.1 2 55.54 odd 2 inner
3520.1.i.c.769.2 2 1.1 even 1 trivial
3520.1.i.c.769.2 2 11.10 odd 2 CM
3520.1.i.d.769.1 2 4.3 odd 2
3520.1.i.d.769.1 2 44.43 even 2
3520.1.i.d.769.2 2 20.19 odd 2
3520.1.i.d.769.2 2 220.219 even 2