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Properties

Label 400.1.b.a.351.1
Level $400$
Weight $1$
Character 400.351
Self dual yes
Analytic conductor $0.200$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
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] = [400,1,Mod(351,400)] 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("400.351"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 400.b (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $D_4$
Artin field: Galois closure of \(\Q(\sqrt[4]{5})\)
Stark unit: Root of $x^{4} - 24x^{3} - 34x^{2} - 24x + 1$

Embedding invariants

Embedding label 351.1
Character \(\chi\) \(=\) 400.351

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{9} -2.00000 q^{29} -2.00000 q^{41} +1.00000 q^{49} -2.00000 q^{61} +1.00000 q^{81} -2.00000 q^{89} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 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
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 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
\(41\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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 − 1.00000i \(-0.5\pi\)
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
\(61\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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 400.1.b.a.351.1 1
3.2 odd 2 3600.1.e.a.3151.1 1
4.3 odd 2 CM 400.1.b.a.351.1 1
5.2 odd 4 80.1.h.a.79.1 ✓ 1
5.3 odd 4 80.1.h.a.79.1 ✓ 1
5.4 even 2 RM 400.1.b.a.351.1 1
8.3 odd 2 1600.1.b.a.1151.1 1
8.5 even 2 1600.1.b.a.1151.1 1
12.11 even 2 3600.1.e.a.3151.1 1
15.2 even 4 720.1.j.a.559.1 1
15.8 even 4 720.1.j.a.559.1 1
15.14 odd 2 3600.1.e.a.3151.1 1
20.3 even 4 80.1.h.a.79.1 ✓ 1
20.7 even 4 80.1.h.a.79.1 ✓ 1
20.19 odd 2 CM 400.1.b.a.351.1 1
35.2 odd 12 3920.1.bt.b.1439.1 2
35.3 even 12 3920.1.bt.a.79.1 2
35.12 even 12 3920.1.bt.a.1439.1 2
35.13 even 4 3920.1.j.a.3039.1 1
35.17 even 12 3920.1.bt.a.79.1 2
35.18 odd 12 3920.1.bt.b.79.1 2
35.23 odd 12 3920.1.bt.b.1439.1 2
35.27 even 4 3920.1.j.a.3039.1 1
35.32 odd 12 3920.1.bt.b.79.1 2
35.33 even 12 3920.1.bt.a.1439.1 2
40.3 even 4 320.1.h.a.319.1 1
40.13 odd 4 320.1.h.a.319.1 1
40.19 odd 2 1600.1.b.a.1151.1 1
40.27 even 4 320.1.h.a.319.1 1
40.29 even 2 1600.1.b.a.1151.1 1
40.37 odd 4 320.1.h.a.319.1 1
60.23 odd 4 720.1.j.a.559.1 1
60.47 odd 4 720.1.j.a.559.1 1
60.59 even 2 3600.1.e.a.3151.1 1
80.3 even 4 1280.1.e.a.639.2 2
80.13 odd 4 1280.1.e.a.639.2 2
80.27 even 4 1280.1.e.a.639.1 2
80.37 odd 4 1280.1.e.a.639.1 2
80.43 even 4 1280.1.e.a.639.1 2
80.53 odd 4 1280.1.e.a.639.1 2
80.67 even 4 1280.1.e.a.639.2 2
80.77 odd 4 1280.1.e.a.639.2 2
120.53 even 4 2880.1.j.a.1279.1 1
120.77 even 4 2880.1.j.a.1279.1 1
120.83 odd 4 2880.1.j.a.1279.1 1
120.107 odd 4 2880.1.j.a.1279.1 1
140.3 odd 12 3920.1.bt.a.79.1 2
140.23 even 12 3920.1.bt.b.1439.1 2
140.27 odd 4 3920.1.j.a.3039.1 1
140.47 odd 12 3920.1.bt.a.1439.1 2
140.67 even 12 3920.1.bt.b.79.1 2
140.83 odd 4 3920.1.j.a.3039.1 1
140.87 odd 12 3920.1.bt.a.79.1 2
140.103 odd 12 3920.1.bt.a.1439.1 2
140.107 even 12 3920.1.bt.b.1439.1 2
140.123 even 12 3920.1.bt.b.79.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.1.h.a.79.1 ✓ 1 5.2 odd 4
80.1.h.a.79.1 ✓ 1 5.3 odd 4
80.1.h.a.79.1 ✓ 1 20.3 even 4
80.1.h.a.79.1 ✓ 1 20.7 even 4
320.1.h.a.319.1 1 40.3 even 4
320.1.h.a.319.1 1 40.13 odd 4
320.1.h.a.319.1 1 40.27 even 4
320.1.h.a.319.1 1 40.37 odd 4
400.1.b.a.351.1 1 1.1 even 1 trivial
400.1.b.a.351.1 1 4.3 odd 2 CM
400.1.b.a.351.1 1 5.4 even 2 RM
400.1.b.a.351.1 1 20.19 odd 2 CM
720.1.j.a.559.1 1 15.2 even 4
720.1.j.a.559.1 1 15.8 even 4
720.1.j.a.559.1 1 60.23 odd 4
720.1.j.a.559.1 1 60.47 odd 4
1280.1.e.a.639.1 2 80.27 even 4
1280.1.e.a.639.1 2 80.37 odd 4
1280.1.e.a.639.1 2 80.43 even 4
1280.1.e.a.639.1 2 80.53 odd 4
1280.1.e.a.639.2 2 80.3 even 4
1280.1.e.a.639.2 2 80.13 odd 4
1280.1.e.a.639.2 2 80.67 even 4
1280.1.e.a.639.2 2 80.77 odd 4
1600.1.b.a.1151.1 1 8.3 odd 2
1600.1.b.a.1151.1 1 8.5 even 2
1600.1.b.a.1151.1 1 40.19 odd 2
1600.1.b.a.1151.1 1 40.29 even 2
2880.1.j.a.1279.1 1 120.53 even 4
2880.1.j.a.1279.1 1 120.77 even 4
2880.1.j.a.1279.1 1 120.83 odd 4
2880.1.j.a.1279.1 1 120.107 odd 4
3600.1.e.a.3151.1 1 3.2 odd 2
3600.1.e.a.3151.1 1 12.11 even 2
3600.1.e.a.3151.1 1 15.14 odd 2
3600.1.e.a.3151.1 1 60.59 even 2
3920.1.j.a.3039.1 1 35.13 even 4
3920.1.j.a.3039.1 1 35.27 even 4
3920.1.j.a.3039.1 1 140.27 odd 4
3920.1.j.a.3039.1 1 140.83 odd 4
3920.1.bt.a.79.1 2 35.3 even 12
3920.1.bt.a.79.1 2 35.17 even 12
3920.1.bt.a.79.1 2 140.3 odd 12
3920.1.bt.a.79.1 2 140.87 odd 12
3920.1.bt.a.1439.1 2 35.12 even 12
3920.1.bt.a.1439.1 2 35.33 even 12
3920.1.bt.a.1439.1 2 140.47 odd 12
3920.1.bt.a.1439.1 2 140.103 odd 12
3920.1.bt.b.79.1 2 35.18 odd 12
3920.1.bt.b.79.1 2 35.32 odd 12
3920.1.bt.b.79.1 2 140.67 even 12
3920.1.bt.b.79.1 2 140.123 even 12
3920.1.bt.b.1439.1 2 35.2 odd 12
3920.1.bt.b.1439.1 2 35.23 odd 12
3920.1.bt.b.1439.1 2 140.23 even 12
3920.1.bt.b.1439.1 2 140.107 even 12