Properties

Label 240.7.c.b.209.1
Level $240$
Weight $7$
Character 240.209
Self dual yes
Analytic conductor $55.213$
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] = [240,7,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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,27,0,125,0,0,0,729] 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: \(55.2129800688\)
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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +125.000 q^{5} +729.000 q^{9} +3375.00 q^{15} +9394.00 q^{17} -13178.0 q^{19} +14654.0 q^{23} +15625.0 q^{25} +19683.0 q^{27} +5758.00 q^{31} +91125.0 q^{45} -90034.0 q^{47} +117649. q^{49} +253638. q^{51} +88666.0 q^{53} -355806. q^{57} -325798. q^{61} +395658. q^{69} +421875. q^{75} +893662. q^{79} +531441. q^{81} -469546. q^{83} +1.17425e6 q^{85} +155466. q^{93} -1.64725e6 q^{95} +O(q^{100})\)

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
\(3\) 27.0000 1.00000
\(4\) 0 0
\(5\) 125.000 1.00000
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 729.000 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
\(15\) 3375.00 1.00000
\(16\) 0 0
\(17\) 9394.00 1.91207 0.956035 − 0.293253i \(-0.0947377\pi\)
0.956035 + 0.293253i \(0.0947377\pi\)
\(18\) 0 0
\(19\) −13178.0 −1.92127 −0.960636 − 0.277811i \(-0.910391\pi\)
−0.960636 + 0.277811i \(0.910391\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14654.0 1.20441 0.602203 − 0.798343i \(-0.294290\pi\)
0.602203 + 0.798343i \(0.294290\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 19683.0 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 5758.00 0.193280 0.0966399 − 0.995319i \(-0.469190\pi\)
0.0966399 + 0.995319i \(0.469190\pi\)
\(32\) 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
\(45\) 91125.0 1.00000
\(46\) 0 0
\(47\) −90034.0 −0.867187 −0.433594 − 0.901108i \(-0.642755\pi\)
−0.433594 + 0.901108i \(0.642755\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 253638. 1.91207
\(52\) 0 0
\(53\) 88666.0 0.595565 0.297783 − 0.954634i \(-0.403753\pi\)
0.297783 + 0.954634i \(0.403753\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −355806. −1.92127
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −325798. −1.43535 −0.717677 − 0.696376i \(-0.754794\pi\)
−0.717677 + 0.696376i \(0.754794\pi\)
\(62\) 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
\(69\) 395658. 1.20441
\(70\) 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
\(75\) 421875. 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 893662. 1.81256 0.906279 − 0.422680i \(-0.138910\pi\)
0.906279 + 0.422680i \(0.138910\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) −469546. −0.821190 −0.410595 − 0.911818i \(-0.634679\pi\)
−0.410595 + 0.911818i \(0.634679\pi\)
\(84\) 0 0
\(85\) 1.17425e6 1.91207
\(86\) 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
\(93\) 155466. 0.193280
\(94\) 0 0
\(95\) −1.64725e6 −1.92127
\(96\) 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 240.7.c.b.209.1 1
3.2 odd 2 240.7.c.a.209.1 1
4.3 odd 2 15.7.d.a.14.1 ✓ 1
5.4 even 2 240.7.c.a.209.1 1
12.11 even 2 15.7.d.b.14.1 yes 1
15.14 odd 2 CM 240.7.c.b.209.1 1
20.3 even 4 75.7.c.b.26.2 2
20.7 even 4 75.7.c.b.26.1 2
20.19 odd 2 15.7.d.b.14.1 yes 1
60.23 odd 4 75.7.c.b.26.1 2
60.47 odd 4 75.7.c.b.26.2 2
60.59 even 2 15.7.d.a.14.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.7.d.a.14.1 ✓ 1 4.3 odd 2
15.7.d.a.14.1 ✓ 1 60.59 even 2
15.7.d.b.14.1 yes 1 12.11 even 2
15.7.d.b.14.1 yes 1 20.19 odd 2
75.7.c.b.26.1 2 20.7 even 4
75.7.c.b.26.1 2 60.23 odd 4
75.7.c.b.26.2 2 20.3 even 4
75.7.c.b.26.2 2 60.47 odd 4
240.7.c.a.209.1 1 3.2 odd 2
240.7.c.a.209.1 1 5.4 even 2
240.7.c.b.209.1 1 1.1 even 1 trivial
240.7.c.b.209.1 1 15.14 odd 2 CM