Properties

Label 240.4.f.f
Level $240$
Weight $4$
Character orbit 240.f
Analytic conductor $14.160$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,4,Mod(49,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} - 9 q^{9} + (2 \beta_{3} - \beta_{2} - \beta_1 + 20) q^{11} + (13 \beta_{2} + 3 \beta_1) q^{13} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 14) q^{15} + ( - 17 \beta_{2} + 5 \beta_1) q^{17} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 32) q^{19} + ( - 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 12) q^{21} + (8 \beta_{2} + 4 \beta_1) q^{23} + (6 \beta_{3} + 13 \beta_{2} + 9 \beta_1 + 61) q^{25} + 9 \beta_{2} q^{27} + ( - 14 \beta_{3} + 7 \beta_{2} + 7 \beta_1 + 166) q^{29} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 28) q^{31} + ( - 21 \beta_{2} + 9 \beta_1) q^{33} + ( - 10 \beta_{3} - 55 \beta_{2} + 5 \beta_1 + 80) q^{35} + ( - 51 \beta_{2} + 27 \beta_1) q^{37} + ( - 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 120) q^{39} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 202) q^{41} + ( - 20 \beta_{2} + 48 \beta_1) q^{43} + ( - 9 \beta_{3} + 18 \beta_{2} + 9 \beta_1 - 9) q^{45} + (30 \beta_{2} + 46 \beta_1) q^{47} + (12 \beta_{3} - 6 \beta_{2} - 6 \beta_1 - 41) q^{49} + ( - 10 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 148) q^{51} + (67 \beta_{2} + 41 \beta_1) q^{53} + (24 \beta_{3} - 23 \beta_{2} - 9 \beta_1 + 204) q^{55} + ( - 28 \beta_{2} - 36 \beta_1) q^{57} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 92) q^{59} + (32 \beta_{3} - 16 \beta_{2} - 16 \beta_1 + 154) q^{61} + ( - 9 \beta_{2} - 27 \beta_1) q^{63} + ( - 22 \beta_{3} - 31 \beta_{2} - 43 \beta_1 + 248) q^{65} + (122 \beta_{2} - 30 \beta_1) q^{67} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 76) q^{69} + ( - 60 \beta_{3} + 30 \beta_{2} + 30 \beta_1 + 48) q^{71} + (222 \beta_{2} - 54 \beta_1) q^{73} + ( - 24 \beta_{3} - 52 \beta_{2} + 39 \beta_1 + 156) q^{75} + ( - 102 \beta_{2} + 54 \beta_1) q^{77} + (100 \beta_{3} - 50 \beta_{2} - 50 \beta_1 + 140) q^{79} + 81 q^{81} + (164 \beta_{2} - 104 \beta_1) q^{83} + (2 \beta_{3} - 129 \beta_{2} + 83 \beta_1 - 128) q^{85} + ( - 159 \beta_{2} - 63 \beta_1) q^{87} + (48 \beta_{3} - 24 \beta_{2} - 24 \beta_1 - 582) q^{89} + (84 \beta_{3} - 42 \beta_{2} - 42 \beta_1 - 528) q^{91} + (26 \beta_{2} + 18 \beta_1) q^{93} + (16 \beta_{3} - 132 \beta_{2} - 76 \beta_1 - 704) q^{95} + (128 \beta_{2} - 120 \beta_1) q^{97} + ( - 18 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 180) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{5} - 36 q^{9} + 84 q^{11} - 54 q^{15} + 112 q^{19} + 36 q^{21} + 256 q^{25} + 636 q^{29} - 104 q^{31} + 300 q^{35} + 468 q^{39} - 816 q^{41} - 54 q^{45} - 140 q^{49} - 612 q^{51} + 864 q^{55} - 372 q^{59} + 680 q^{61} + 948 q^{65} + 288 q^{69} + 72 q^{71} + 576 q^{75} + 760 q^{79} + 324 q^{81} - 508 q^{85} - 2232 q^{89} - 1944 q^{91} - 2784 q^{95} - 756 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 31\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} - 33\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 30\nu^{2} - \nu + 320 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - \beta_{2} - \beta _1 - 64 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -31\beta_{2} - 33\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
3.70156i
2.70156i
3.70156i
2.70156i
0 3.00000i 0 −8.10469 7.70156i 0 22.2094i 0 −9.00000 0
49.2 0 3.00000i 0 11.1047 1.29844i 0 16.2094i 0 −9.00000 0
49.3 0 3.00000i 0 −8.10469 + 7.70156i 0 22.2094i 0 −9.00000 0
49.4 0 3.00000i 0 11.1047 + 1.29844i 0 16.2094i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.f.f 4
3.b odd 2 1 720.4.f.j 4
4.b odd 2 1 15.4.b.a 4
5.b even 2 1 inner 240.4.f.f 4
5.c odd 4 1 1200.4.a.bn 2
5.c odd 4 1 1200.4.a.bt 2
8.b even 2 1 960.4.f.p 4
8.d odd 2 1 960.4.f.q 4
12.b even 2 1 45.4.b.b 4
15.d odd 2 1 720.4.f.j 4
20.d odd 2 1 15.4.b.a 4
20.e even 4 1 75.4.a.c 2
20.e even 4 1 75.4.a.f 2
40.e odd 2 1 960.4.f.q 4
40.f even 2 1 960.4.f.p 4
60.h even 2 1 45.4.b.b 4
60.l odd 4 1 225.4.a.i 2
60.l odd 4 1 225.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 4.b odd 2 1
15.4.b.a 4 20.d odd 2 1
45.4.b.b 4 12.b even 2 1
45.4.b.b 4 60.h even 2 1
75.4.a.c 2 20.e even 4 1
75.4.a.f 2 20.e even 4 1
225.4.a.i 2 60.l odd 4 1
225.4.a.o 2 60.l odd 4 1
240.4.f.f 4 1.a even 1 1 trivial
240.4.f.f 4 5.b even 2 1 inner
720.4.f.j 4 3.b odd 2 1
720.4.f.j 4 15.d odd 2 1
960.4.f.p 4 8.b even 2 1
960.4.f.p 4 40.f even 2 1
960.4.f.q 4 8.d odd 2 1
960.4.f.q 4 40.e odd 2 1
1200.4.a.bn 2 5.c odd 4 1
1200.4.a.bt 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{4} + 756T_{7}^{2} + 129600 \) Copy content Toggle raw display
\( T_{11}^{2} - 42T_{11} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} - 110 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 756 T^{2} + 129600 \) Copy content Toggle raw display
$11$ \( (T^{2} - 42 T + 72)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 3780 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$17$ \( T^{4} + 7252 T^{2} + \cdots + 2483776 \) Copy content Toggle raw display
$19$ \( (T^{2} - 56 T - 5120)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2464 T^{2} + 6400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 318 T + 7200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 52 T - 800)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 106596 T^{2} + \cdots + 41990400 \) Copy content Toggle raw display
$41$ \( (T^{2} + 408 T + 40140)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 196128 T^{2} + \cdots + 8256266496 \) Copy content Toggle raw display
$47$ \( T^{4} + 189712 T^{2} + \cdots + 6186766336 \) Copy content Toggle raw display
$53$ \( T^{4} + 218644 T^{2} + \cdots + 813390400 \) Copy content Toggle raw display
$59$ \( (T^{2} + 186 T + 8280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 340 T - 65564)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 341712 T^{2} + \cdots + 9419867136 \) Copy content Toggle raw display
$71$ \( (T^{2} - 36 T - 331776)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 1126224 T^{2} + \cdots + 104976000000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 380 T - 886400)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1371040 T^{2} + \cdots + 40558737664 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1116 T + 98820)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1475712 T^{2} + \cdots + 196199387136 \) Copy content Toggle raw display
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