Properties

Label 3600.1.cc.a.751.1
Level $3600$
Weight $1$
Character 3600.751
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -20
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] = [3600,1,Mod(751,3600)] 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("3600.751"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.cc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-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: \(1.79663404548\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 720)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.10497600.1

Embedding invariants

Embedding label 751.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3600.751
Dual form 3600.1.cc.a.1951.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(-1.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} -1.73205i q^{21} +(-1.50000 - 0.866025i) q^{23} +1.00000 q^{27} +(-0.500000 - 0.866025i) q^{29} +(-0.500000 + 0.866025i) q^{41} +(1.50000 - 0.866025i) q^{47} +(1.00000 - 1.73205i) q^{49} +(-0.500000 - 0.866025i) q^{61} +(1.50000 + 0.866025i) q^{63} +(-1.50000 - 0.866025i) q^{67} +(1.50000 - 0.866025i) q^{69} +(-0.500000 + 0.866025i) q^{81} +(1.50000 - 0.866025i) q^{83} +1.00000 q^{87} +1.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 3 q^{7} - q^{9} - 3 q^{23} + 2 q^{27} - q^{29} - q^{41} + 3 q^{47} + 2 q^{49} - q^{61} + 3 q^{63} - 3 q^{67} + 3 q^{69} - q^{81} + 3 q^{83} + 2 q^{87} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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
\(3\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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
\(21\) − 1.73205i − 1.73205i
\(22\) 0 0
\(23\) −1.50000 − 0.866025i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.50000 − 0.866025i 1.50000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(48\) 0 0
\(49\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.50000 − 0.866025i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 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 0
\(83\) 1.50000 − 0.866025i 1.50000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) 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
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(98\) 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 3600.1.cc.a.751.1 2
4.3 odd 2 3600.1.cc.b.751.1 2
5.2 odd 4 720.1.bu.a.319.2 yes 4
5.3 odd 4 720.1.bu.a.319.1 yes 4
5.4 even 2 3600.1.cc.b.751.1 2
9.7 even 3 3600.1.cc.b.1951.1 2
15.2 even 4 2160.1.bu.a.1279.1 4
15.8 even 4 2160.1.bu.a.1279.2 4
20.3 even 4 720.1.bu.a.319.2 yes 4
20.7 even 4 720.1.bu.a.319.1 yes 4
20.19 odd 2 CM 3600.1.cc.a.751.1 2
36.7 odd 6 inner 3600.1.cc.a.1951.1 2
40.3 even 4 2880.1.bu.c.319.1 4
40.13 odd 4 2880.1.bu.c.319.2 4
40.27 even 4 2880.1.bu.c.319.2 4
40.37 odd 4 2880.1.bu.c.319.1 4
45.2 even 12 2160.1.bu.a.559.1 4
45.7 odd 12 720.1.bu.a.79.2 yes 4
45.34 even 6 inner 3600.1.cc.a.1951.1 2
45.38 even 12 2160.1.bu.a.559.2 4
45.43 odd 12 720.1.bu.a.79.1 ✓ 4
60.23 odd 4 2160.1.bu.a.1279.1 4
60.47 odd 4 2160.1.bu.a.1279.2 4
180.7 even 12 720.1.bu.a.79.1 ✓ 4
180.43 even 12 720.1.bu.a.79.2 yes 4
180.47 odd 12 2160.1.bu.a.559.2 4
180.79 odd 6 3600.1.cc.b.1951.1 2
180.83 odd 12 2160.1.bu.a.559.1 4
360.43 even 12 2880.1.bu.c.2239.1 4
360.133 odd 12 2880.1.bu.c.2239.2 4
360.187 even 12 2880.1.bu.c.2239.2 4
360.277 odd 12 2880.1.bu.c.2239.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.1.bu.a.79.1 ✓ 4 45.43 odd 12
720.1.bu.a.79.1 ✓ 4 180.7 even 12
720.1.bu.a.79.2 yes 4 45.7 odd 12
720.1.bu.a.79.2 yes 4 180.43 even 12
720.1.bu.a.319.1 yes 4 5.3 odd 4
720.1.bu.a.319.1 yes 4 20.7 even 4
720.1.bu.a.319.2 yes 4 5.2 odd 4
720.1.bu.a.319.2 yes 4 20.3 even 4
2160.1.bu.a.559.1 4 45.2 even 12
2160.1.bu.a.559.1 4 180.83 odd 12
2160.1.bu.a.559.2 4 45.38 even 12
2160.1.bu.a.559.2 4 180.47 odd 12
2160.1.bu.a.1279.1 4 15.2 even 4
2160.1.bu.a.1279.1 4 60.23 odd 4
2160.1.bu.a.1279.2 4 15.8 even 4
2160.1.bu.a.1279.2 4 60.47 odd 4
2880.1.bu.c.319.1 4 40.3 even 4
2880.1.bu.c.319.1 4 40.37 odd 4
2880.1.bu.c.319.2 4 40.13 odd 4
2880.1.bu.c.319.2 4 40.27 even 4
2880.1.bu.c.2239.1 4 360.43 even 12
2880.1.bu.c.2239.1 4 360.277 odd 12
2880.1.bu.c.2239.2 4 360.133 odd 12
2880.1.bu.c.2239.2 4 360.187 even 12
3600.1.cc.a.751.1 2 1.1 even 1 trivial
3600.1.cc.a.751.1 2 20.19 odd 2 CM
3600.1.cc.a.1951.1 2 36.7 odd 6 inner
3600.1.cc.a.1951.1 2 45.34 even 6 inner
3600.1.cc.b.751.1 2 4.3 odd 2
3600.1.cc.b.751.1 2 5.4 even 2
3600.1.cc.b.1951.1 2 9.7 even 3
3600.1.cc.b.1951.1 2 180.79 odd 6