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Properties

Label 3600.1.da.a.2207.2
Level $3600$
Weight $1$
Character 3600.2207
Analytic conductor $1.797$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -20
Inner twists $16$

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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(1343,3600)] 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("3600.1343"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 2, 9])) 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.da (of order \(12\), degree \(4\), 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.157464000.2

Embedding invariants

Embedding label 2207.2
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 3600.2207
Dual form 3600.1.da.a.2543.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{3} +(0.258819 + 0.965926i) q^{7} +(-0.866025 - 0.500000i) q^{9} +1.00000 q^{21} +(1.67303 + 0.448288i) q^{23} +(-0.707107 + 0.707107i) q^{27} +(0.866025 - 1.50000i) q^{29} +(1.50000 - 0.866025i) q^{41} +(-1.93185 + 0.517638i) q^{43} +(1.67303 - 0.448288i) q^{47} +(0.500000 - 0.866025i) q^{61} +(0.258819 - 0.965926i) q^{63} +(0.965926 + 0.258819i) q^{67} +(0.866025 - 1.50000i) q^{69} +(0.500000 + 0.866025i) q^{81} +(0.448288 + 1.67303i) q^{83} +(-1.22474 - 1.22474i) q^{87} -1.73205 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{21} + 12 q^{41} + 4 q^{61} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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
\(3\) 0.258819 − 0.965926i 0.258819 − 0.965926i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 1.00000
\(22\) 0 0
\(23\) 1.67303 + 0.448288i 1.67303 + 0.448288i 0.965926 − 0.258819i \(-0.0833333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 0 0
\(29\) 0.866025 − 1.50000i 0.866025 − 1.50000i − 1.00000i \(-0.5\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.50000 − 0.866025i 1.50000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(42\) 0 0
\(43\) −1.93185 + 0.517638i −1.93185 + 0.517638i −0.965926 + 0.258819i \(0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.67303 − 0.448288i 1.67303 − 0.448288i 0.707107 − 0.707107i \(-0.250000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(62\) 0 0
\(63\) 0.258819 − 0.965926i 0.258819 − 0.965926i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 − 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(68\) 0 0
\(69\) 0.866025 − 1.50000i 0.866025 − 1.50000i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0.448288 + 1.67303i 0.448288 + 1.67303i 0.707107 + 0.707107i \(0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.22474 − 1.22474i −1.22474 − 1.22474i
\(88\) 0 0
\(89\) −1.73205 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(98\) 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.da.a.2207.2 yes 8
4.3 odd 2 inner 3600.1.da.a.2207.1 yes 8
5.2 odd 4 inner 3600.1.da.a.1343.1 ✓ 8
5.3 odd 4 inner 3600.1.da.a.1343.2 yes 8
5.4 even 2 inner 3600.1.da.a.2207.1 yes 8
9.5 odd 6 inner 3600.1.da.a.3407.1 yes 8
20.3 even 4 inner 3600.1.da.a.1343.1 ✓ 8
20.7 even 4 inner 3600.1.da.a.1343.2 yes 8
20.19 odd 2 CM 3600.1.da.a.2207.2 yes 8
36.23 even 6 inner 3600.1.da.a.3407.2 yes 8
45.14 odd 6 inner 3600.1.da.a.3407.2 yes 8
45.23 even 12 inner 3600.1.da.a.2543.1 yes 8
45.32 even 12 inner 3600.1.da.a.2543.2 yes 8
180.23 odd 12 inner 3600.1.da.a.2543.2 yes 8
180.59 even 6 inner 3600.1.da.a.3407.1 yes 8
180.167 odd 12 inner 3600.1.da.a.2543.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3600.1.da.a.1343.1 ✓ 8 5.2 odd 4 inner
3600.1.da.a.1343.1 ✓ 8 20.3 even 4 inner
3600.1.da.a.1343.2 yes 8 5.3 odd 4 inner
3600.1.da.a.1343.2 yes 8 20.7 even 4 inner
3600.1.da.a.2207.1 yes 8 4.3 odd 2 inner
3600.1.da.a.2207.1 yes 8 5.4 even 2 inner
3600.1.da.a.2207.2 yes 8 1.1 even 1 trivial
3600.1.da.a.2207.2 yes 8 20.19 odd 2 CM
3600.1.da.a.2543.1 yes 8 45.23 even 12 inner
3600.1.da.a.2543.1 yes 8 180.167 odd 12 inner
3600.1.da.a.2543.2 yes 8 45.32 even 12 inner
3600.1.da.a.2543.2 yes 8 180.23 odd 12 inner
3600.1.da.a.3407.1 yes 8 9.5 odd 6 inner
3600.1.da.a.3407.1 yes 8 180.59 even 6 inner
3600.1.da.a.3407.2 yes 8 36.23 even 6 inner
3600.1.da.a.3407.2 yes 8 45.14 odd 6 inner