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Properties

Label 675.1.d.a.674.2
Level $675$
Weight $1$
Character 675.674
Analytic conductor $0.337$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
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] = [675,1,Mod(674,675)] 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("675.674"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 675.d (of order \(2\), degree \(1\), not 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: \(0.336868883527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.675.1
Artin image: $C_4\times S_3$
Artin field: Galois closure of 12.0.25949267578125.1

Embedding invariants

Embedding label 674.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 675.674
Dual form 675.1.d.a.674.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{4} +1.00000i q^{7} +2.00000i q^{13} +1.00000 q^{16} +1.00000 q^{19} -1.00000i q^{28} -1.00000 q^{31} +1.00000i q^{37} -1.00000i q^{43} -2.00000i q^{52} -1.00000 q^{61} -1.00000 q^{64} -2.00000i q^{67} -1.00000i q^{73} -1.00000 q^{76} +1.00000 q^{79} -2.00000 q^{91} +1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{16} + 2 q^{19} - 2 q^{31} - 2 q^{61} - 2 q^{64} - 2 q^{76} + 2 q^{79} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −1.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) − 1.00000i − 1.00000i
\(29\) 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
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 2.00000i
\(53\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −2.00000 −2.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000i 1.00000i 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 675.1.d.a.674.2 2
3.2 odd 2 CM 675.1.d.a.674.2 2
5.2 odd 4 675.1.c.a.26.1 ✓ 1
5.3 odd 4 675.1.c.b.26.1 yes 1
5.4 even 2 inner 675.1.d.a.674.1 2
9.2 odd 6 2025.1.i.a.1349.2 4
9.4 even 3 2025.1.i.a.674.1 4
9.5 odd 6 2025.1.i.a.674.1 4
9.7 even 3 2025.1.i.a.1349.2 4
15.2 even 4 675.1.c.a.26.1 ✓ 1
15.8 even 4 675.1.c.b.26.1 yes 1
15.14 odd 2 inner 675.1.d.a.674.1 2
45.2 even 12 2025.1.j.b.701.1 2
45.4 even 6 2025.1.i.a.674.2 4
45.7 odd 12 2025.1.j.b.701.1 2
45.13 odd 12 2025.1.j.a.26.1 2
45.14 odd 6 2025.1.i.a.674.2 4
45.22 odd 12 2025.1.j.b.26.1 2
45.23 even 12 2025.1.j.a.26.1 2
45.29 odd 6 2025.1.i.a.1349.1 4
45.32 even 12 2025.1.j.b.26.1 2
45.34 even 6 2025.1.i.a.1349.1 4
45.38 even 12 2025.1.j.a.701.1 2
45.43 odd 12 2025.1.j.a.701.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.1.c.a.26.1 ✓ 1 5.2 odd 4
675.1.c.a.26.1 ✓ 1 15.2 even 4
675.1.c.b.26.1 yes 1 5.3 odd 4
675.1.c.b.26.1 yes 1 15.8 even 4
675.1.d.a.674.1 2 5.4 even 2 inner
675.1.d.a.674.1 2 15.14 odd 2 inner
675.1.d.a.674.2 2 1.1 even 1 trivial
675.1.d.a.674.2 2 3.2 odd 2 CM
2025.1.i.a.674.1 4 9.4 even 3
2025.1.i.a.674.1 4 9.5 odd 6
2025.1.i.a.674.2 4 45.4 even 6
2025.1.i.a.674.2 4 45.14 odd 6
2025.1.i.a.1349.1 4 45.29 odd 6
2025.1.i.a.1349.1 4 45.34 even 6
2025.1.i.a.1349.2 4 9.2 odd 6
2025.1.i.a.1349.2 4 9.7 even 3
2025.1.j.a.26.1 2 45.13 odd 12
2025.1.j.a.26.1 2 45.23 even 12
2025.1.j.a.701.1 2 45.38 even 12
2025.1.j.a.701.1 2 45.43 odd 12
2025.1.j.b.26.1 2 45.22 odd 12
2025.1.j.b.26.1 2 45.32 even 12
2025.1.j.b.701.1 2 45.2 even 12
2025.1.j.b.701.1 2 45.7 odd 12