Properties

Label 1440.1.bh.b.577.1
Level $1440$
Weight $1$
Character 1440.577
Analytic conductor $0.719$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
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] = [1440,1,Mod(577,1440)] 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("1440.577"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1440.bh (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.718653618192\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Projective image: \(D_{4}\)
Projective field: Galois closure of \(\Q(\sqrt[4]{5})\)
Artin image: $C_4^2:C_2^2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

Embedding invariants

Embedding label 577.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1440.577
Dual form 1440.1.bh.b.1153.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{5} +(-1.00000 + 1.00000i) q^{13} +(1.00000 + 1.00000i) q^{17} -1.00000 q^{25} +(1.00000 + 1.00000i) q^{37} -1.00000i q^{49} +(-1.00000 + 1.00000i) q^{53} +(-1.00000 - 1.00000i) q^{65} +(1.00000 - 1.00000i) q^{73} +(-1.00000 + 1.00000i) q^{85} +(1.00000 + 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{13} + 2 q^{17} - 2 q^{25} + 2 q^{37} - 2 q^{53} - 2 q^{65} + 2 q^{73} - 2 q^{85} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\) \(1\)

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
\(4\) 0 0
\(5\) 1.00000i 1.00000i
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
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 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) − 1.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(54\) 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
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 − 1.00000i −1.00000 − 1.00000i
\(66\) 0 0
\(67\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1.00000 − 1.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(74\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
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 1440.1.bh.b.577.1 2
3.2 odd 2 160.1.p.a.97.1 yes 2
4.3 odd 2 CM 1440.1.bh.b.577.1 2
5.3 odd 4 inner 1440.1.bh.b.1153.1 2
8.3 odd 2 2880.1.bh.b.577.1 2
8.5 even 2 2880.1.bh.b.577.1 2
12.11 even 2 160.1.p.a.97.1 yes 2
15.2 even 4 800.1.p.b.193.1 2
15.8 even 4 160.1.p.a.33.1 ✓ 2
15.14 odd 2 800.1.p.b.257.1 2
20.3 even 4 inner 1440.1.bh.b.1153.1 2
24.5 odd 2 320.1.p.a.257.1 2
24.11 even 2 320.1.p.a.257.1 2
40.3 even 4 2880.1.bh.b.1153.1 2
40.13 odd 4 2880.1.bh.b.1153.1 2
48.5 odd 4 1280.1.m.b.897.1 2
48.11 even 4 1280.1.m.b.897.1 2
48.29 odd 4 1280.1.m.a.897.1 2
48.35 even 4 1280.1.m.a.897.1 2
60.23 odd 4 160.1.p.a.33.1 ✓ 2
60.47 odd 4 800.1.p.b.193.1 2
60.59 even 2 800.1.p.b.257.1 2
120.29 odd 2 1600.1.p.b.257.1 2
120.53 even 4 320.1.p.a.193.1 2
120.59 even 2 1600.1.p.b.257.1 2
120.77 even 4 1600.1.p.b.193.1 2
120.83 odd 4 320.1.p.a.193.1 2
120.107 odd 4 1600.1.p.b.193.1 2
240.53 even 4 1280.1.m.a.1153.1 2
240.83 odd 4 1280.1.m.b.1153.1 2
240.173 even 4 1280.1.m.b.1153.1 2
240.203 odd 4 1280.1.m.a.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.1.p.a.33.1 ✓ 2 15.8 even 4
160.1.p.a.33.1 ✓ 2 60.23 odd 4
160.1.p.a.97.1 yes 2 3.2 odd 2
160.1.p.a.97.1 yes 2 12.11 even 2
320.1.p.a.193.1 2 120.53 even 4
320.1.p.a.193.1 2 120.83 odd 4
320.1.p.a.257.1 2 24.5 odd 2
320.1.p.a.257.1 2 24.11 even 2
800.1.p.b.193.1 2 15.2 even 4
800.1.p.b.193.1 2 60.47 odd 4
800.1.p.b.257.1 2 15.14 odd 2
800.1.p.b.257.1 2 60.59 even 2
1280.1.m.a.897.1 2 48.29 odd 4
1280.1.m.a.897.1 2 48.35 even 4
1280.1.m.a.1153.1 2 240.53 even 4
1280.1.m.a.1153.1 2 240.203 odd 4
1280.1.m.b.897.1 2 48.5 odd 4
1280.1.m.b.897.1 2 48.11 even 4
1280.1.m.b.1153.1 2 240.83 odd 4
1280.1.m.b.1153.1 2 240.173 even 4
1440.1.bh.b.577.1 2 1.1 even 1 trivial
1440.1.bh.b.577.1 2 4.3 odd 2 CM
1440.1.bh.b.1153.1 2 5.3 odd 4 inner
1440.1.bh.b.1153.1 2 20.3 even 4 inner
1600.1.p.b.193.1 2 120.77 even 4
1600.1.p.b.193.1 2 120.107 odd 4
1600.1.p.b.257.1 2 120.29 odd 2
1600.1.p.b.257.1 2 120.59 even 2
2880.1.bh.b.577.1 2 8.3 odd 2
2880.1.bh.b.577.1 2 8.5 even 2
2880.1.bh.b.1153.1 2 40.3 even 4
2880.1.bh.b.1153.1 2 40.13 odd 4