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Properties

Label 525.1.p.a.74.2
Level $525$
Weight $1$
Character 525.74
Analytic conductor $0.262$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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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] = [525,1,Mod(74,525)] 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("525.74"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 525.p (of order \(6\), degree \(2\), 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: \(0.262009131632\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of \(\Q(\sqrt[3]{35})\)

Embedding invariants

Embedding label 74.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 525.74
Dual form 525.1.p.a.149.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.866025 - 0.500000i) q^{7} +(0.500000 + 0.866025i) q^{9} +(0.866025 - 0.500000i) q^{12} +1.00000i q^{13} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{19} +(-0.500000 - 0.866025i) q^{21} +1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} +(-1.00000 + 1.73205i) q^{31} +1.00000 q^{36} +(-0.866025 + 0.500000i) q^{37} +(-0.500000 + 0.866025i) q^{39} -2.00000i q^{43} -1.00000i q^{48} +(0.500000 + 0.866025i) q^{49} +(0.866025 + 0.500000i) q^{52} -1.00000i q^{57} +(0.500000 + 0.866025i) q^{61} -1.00000i q^{63} -1.00000 q^{64} +(0.866025 + 0.500000i) q^{67} +(-0.866025 - 0.500000i) q^{73} -1.00000 q^{76} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} -1.00000 q^{84} +(0.500000 - 0.866025i) q^{91} +(-1.73205 + 1.00000i) q^{93} -1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{16} - 2 q^{19} - 2 q^{21} - 4 q^{31} + 4 q^{36} - 2 q^{39} + 2 q^{49} + 2 q^{61} - 4 q^{64} - 4 q^{76} - 2 q^{79} - 2 q^{81} - 4 q^{84} + 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(4\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(13\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.73205i −1.00000 + 1.73205i −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
\(36\) 1.00000 1.00000
\(37\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) − 1.00000i − 1.00000i
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(53\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 1.00000i
\(58\) 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 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 1.00000i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i \(-0.166667\pi\)
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
\(73\) −0.866025 − 0.500000i −0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 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 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(92\) 0 0
\(93\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(98\) 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 525.1.p.a.74.2 4
3.2 odd 2 CM 525.1.p.a.74.2 4
5.2 odd 4 525.1.u.b.326.1 yes 2
5.3 odd 4 525.1.u.a.326.1 ✓ 2
5.4 even 2 inner 525.1.p.a.74.1 4
7.2 even 3 inner 525.1.p.a.149.1 4
7.3 odd 6 3675.1.f.a.2549.2 2
7.4 even 3 3675.1.f.b.2549.1 2
7.5 odd 6 3675.1.p.a.2774.2 4
7.6 odd 2 3675.1.p.a.2174.1 4
15.2 even 4 525.1.u.b.326.1 yes 2
15.8 even 4 525.1.u.a.326.1 ✓ 2
15.14 odd 2 inner 525.1.p.a.74.1 4
21.2 odd 6 inner 525.1.p.a.149.1 4
21.5 even 6 3675.1.p.a.2774.2 4
21.11 odd 6 3675.1.f.b.2549.1 2
21.17 even 6 3675.1.f.a.2549.2 2
21.20 even 2 3675.1.p.a.2174.1 4
35.2 odd 12 525.1.u.b.401.1 yes 2
35.3 even 12 3675.1.c.b.1226.1 1
35.4 even 6 3675.1.f.b.2549.2 2
35.9 even 6 inner 525.1.p.a.149.2 4
35.12 even 12 3675.1.u.a.1451.1 2
35.13 even 4 3675.1.u.b.851.1 2
35.17 even 12 3675.1.c.d.1226.1 1
35.18 odd 12 3675.1.c.c.1226.1 1
35.19 odd 6 3675.1.p.a.2774.1 4
35.23 odd 12 525.1.u.a.401.1 yes 2
35.24 odd 6 3675.1.f.a.2549.1 2
35.27 even 4 3675.1.u.a.851.1 2
35.32 odd 12 3675.1.c.a.1226.1 1
35.33 even 12 3675.1.u.b.1451.1 2
35.34 odd 2 3675.1.p.a.2174.2 4
105.2 even 12 525.1.u.b.401.1 yes 2
105.17 odd 12 3675.1.c.d.1226.1 1
105.23 even 12 525.1.u.a.401.1 yes 2
105.32 even 12 3675.1.c.a.1226.1 1
105.38 odd 12 3675.1.c.b.1226.1 1
105.44 odd 6 inner 525.1.p.a.149.2 4
105.47 odd 12 3675.1.u.a.1451.1 2
105.53 even 12 3675.1.c.c.1226.1 1
105.59 even 6 3675.1.f.a.2549.1 2
105.62 odd 4 3675.1.u.a.851.1 2
105.68 odd 12 3675.1.u.b.1451.1 2
105.74 odd 6 3675.1.f.b.2549.2 2
105.83 odd 4 3675.1.u.b.851.1 2
105.89 even 6 3675.1.p.a.2774.1 4
105.104 even 2 3675.1.p.a.2174.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.1.p.a.74.1 4 5.4 even 2 inner
525.1.p.a.74.1 4 15.14 odd 2 inner
525.1.p.a.74.2 4 1.1 even 1 trivial
525.1.p.a.74.2 4 3.2 odd 2 CM
525.1.p.a.149.1 4 7.2 even 3 inner
525.1.p.a.149.1 4 21.2 odd 6 inner
525.1.p.a.149.2 4 35.9 even 6 inner
525.1.p.a.149.2 4 105.44 odd 6 inner
525.1.u.a.326.1 ✓ 2 5.3 odd 4
525.1.u.a.326.1 ✓ 2 15.8 even 4
525.1.u.a.401.1 yes 2 35.23 odd 12
525.1.u.a.401.1 yes 2 105.23 even 12
525.1.u.b.326.1 yes 2 5.2 odd 4
525.1.u.b.326.1 yes 2 15.2 even 4
525.1.u.b.401.1 yes 2 35.2 odd 12
525.1.u.b.401.1 yes 2 105.2 even 12
3675.1.c.a.1226.1 1 35.32 odd 12
3675.1.c.a.1226.1 1 105.32 even 12
3675.1.c.b.1226.1 1 35.3 even 12
3675.1.c.b.1226.1 1 105.38 odd 12
3675.1.c.c.1226.1 1 35.18 odd 12
3675.1.c.c.1226.1 1 105.53 even 12
3675.1.c.d.1226.1 1 35.17 even 12
3675.1.c.d.1226.1 1 105.17 odd 12
3675.1.f.a.2549.1 2 35.24 odd 6
3675.1.f.a.2549.1 2 105.59 even 6
3675.1.f.a.2549.2 2 7.3 odd 6
3675.1.f.a.2549.2 2 21.17 even 6
3675.1.f.b.2549.1 2 7.4 even 3
3675.1.f.b.2549.1 2 21.11 odd 6
3675.1.f.b.2549.2 2 35.4 even 6
3675.1.f.b.2549.2 2 105.74 odd 6
3675.1.p.a.2174.1 4 7.6 odd 2
3675.1.p.a.2174.1 4 21.20 even 2
3675.1.p.a.2174.2 4 35.34 odd 2
3675.1.p.a.2174.2 4 105.104 even 2
3675.1.p.a.2774.1 4 35.19 odd 6
3675.1.p.a.2774.1 4 105.89 even 6
3675.1.p.a.2774.2 4 7.5 odd 6
3675.1.p.a.2774.2 4 21.5 even 6
3675.1.u.a.851.1 2 35.27 even 4
3675.1.u.a.851.1 2 105.62 odd 4
3675.1.u.a.1451.1 2 35.12 even 12
3675.1.u.a.1451.1 2 105.47 odd 12
3675.1.u.b.851.1 2 35.13 even 4
3675.1.u.b.851.1 2 105.83 odd 4
3675.1.u.b.1451.1 2 35.33 even 12
3675.1.u.b.1451.1 2 105.68 odd 12