Properties

Label 240.5.c.a.209.1
Level $240$
Weight $5$
Character 240.209
Self dual yes
Analytic conductor $24.809$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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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] = [240,5,Mod(209,240)] 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("240.209"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-9,0,25,0,0,0,81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.8087911401\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 209.1
Character \(\chi\) \(=\) 240.209

$q$-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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +25.0000 q^{5} +81.0000 q^{9} -225.000 q^{15} -382.000 q^{17} +238.000 q^{19} -98.0000 q^{23} +625.000 q^{25} -729.000 q^{27} +1918.00 q^{31} +2025.00 q^{45} +4222.00 q^{47} +2401.00 q^{49} +3438.00 q^{51} +1778.00 q^{53} -2142.00 q^{57} +6482.00 q^{61} +882.000 q^{69} -5625.00 q^{75} +2878.00 q^{79} +6561.00 q^{81} -9938.00 q^{83} -9550.00 q^{85} -17262.0 q^{93} +5950.00 q^{95} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) −9.00000 −1.00000
\(4\) 0 0
\(5\) 25.0000 1.00000
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −225.000 −1.00000
\(16\) 0 0
\(17\) −382.000 −1.32180 −0.660900 − 0.750474i \(-0.729825\pi\)
−0.660900 + 0.750474i \(0.729825\pi\)
\(18\) 0 0
\(19\) 238.000 0.659280 0.329640 − 0.944107i \(-0.393073\pi\)
0.329640 + 0.944107i \(0.393073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −98.0000 −0.185255 −0.0926276 − 0.995701i \(-0.529527\pi\)
−0.0926276 + 0.995701i \(0.529527\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) −729.000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1918.00 1.99584 0.997919 − 0.0644826i \(-0.0205397\pi\)
0.997919 + 0.0644826i \(0.0205397\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 2025.00 1.00000
\(46\) 0 0
\(47\) 4222.00 1.91127 0.955636 − 0.294550i \(-0.0951698\pi\)
0.955636 + 0.294550i \(0.0951698\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 3438.00 1.32180
\(52\) 0 0
\(53\) 1778.00 0.632965 0.316483 − 0.948598i \(-0.397498\pi\)
0.316483 + 0.948598i \(0.397498\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2142.00 −0.659280
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 6482.00 1.74200 0.871002 − 0.491279i \(-0.163470\pi\)
0.871002 + 0.491279i \(0.163470\pi\)
\(62\) 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
\(69\) 882.000 0.185255
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −5625.00 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2878.00 0.461144 0.230572 − 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) −9938.00 −1.44259 −0.721295 − 0.692628i \(-0.756453\pi\)
−0.721295 + 0.692628i \(0.756453\pi\)
\(84\) 0 0
\(85\) −9550.00 −1.32180
\(86\) 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
\(93\) −17262.0 −1.99584
\(94\) 0 0
\(95\) 5950.00 0.659280
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 240.5.c.a.209.1 1
3.2 odd 2 240.5.c.b.209.1 1
4.3 odd 2 15.5.d.a.14.1 ✓ 1
5.4 even 2 240.5.c.b.209.1 1
12.11 even 2 15.5.d.b.14.1 yes 1
15.14 odd 2 CM 240.5.c.a.209.1 1
20.3 even 4 75.5.c.e.26.2 2
20.7 even 4 75.5.c.e.26.1 2
20.19 odd 2 15.5.d.b.14.1 yes 1
60.23 odd 4 75.5.c.e.26.1 2
60.47 odd 4 75.5.c.e.26.2 2
60.59 even 2 15.5.d.a.14.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.5.d.a.14.1 ✓ 1 4.3 odd 2
15.5.d.a.14.1 ✓ 1 60.59 even 2
15.5.d.b.14.1 yes 1 12.11 even 2
15.5.d.b.14.1 yes 1 20.19 odd 2
75.5.c.e.26.1 2 20.7 even 4
75.5.c.e.26.1 2 60.23 odd 4
75.5.c.e.26.2 2 20.3 even 4
75.5.c.e.26.2 2 60.47 odd 4
240.5.c.a.209.1 1 1.1 even 1 trivial
240.5.c.a.209.1 1 15.14 odd 2 CM
240.5.c.b.209.1 1 3.2 odd 2
240.5.c.b.209.1 1 5.4 even 2