Properties

Label 240.2.y.d
Level $240$
Weight $2$
Character orbit 240.y
Analytic conductor $1.916$
Analytic rank $0$
Dimension $6$
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] = [240,2,Mod(163,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.y (of order \(4\), degree \(2\), 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: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + q^{3} + ( - \beta_{5} - \beta_1) q^{4} + (2 \beta_1 + 1) q^{5} + \beta_{3} q^{6} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + q^{3} + ( - \beta_{5} - \beta_1) q^{4} + (2 \beta_1 + 1) q^{5} + \beta_{3} q^{6} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{8} + q^{9} + (\beta_{3} + 2 \beta_{2}) q^{10} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_1) q^{12} + (2 \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{14} + (2 \beta_1 + 1) q^{15} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{16} + ( - 2 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{17} + \beta_{3} q^{18} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 2) q^{19} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 + 2) q^{20} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{21} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{22} + (\beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 2) q^{23} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{24} + (4 \beta_1 - 3) q^{25} + q^{27} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{28} + ( - \beta_1 - 1) q^{29} + (\beta_{3} + 2 \beta_{2}) q^{30} + (2 \beta_{4} - 2 \beta_{2}) q^{31} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{32} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{33} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{34} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - 3 \beta_{2}) q^{35} + ( - \beta_{5} - \beta_1) q^{36} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{37} + (2 \beta_{5} - \beta_{3} + \beta_{2} + 6 \beta_1) q^{38} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + 4 \beta_1 - 3) q^{40} - 4 \beta_1 q^{41} + (2 \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{42} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{43} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{44} + (2 \beta_1 + 1) q^{45} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} - 6 \beta_1) q^{46} + (\beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{47} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{48} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + \beta_1 - 2) q^{49} + ( - 3 \beta_{3} + 4 \beta_{2}) q^{50} + ( - 2 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{51} + ( - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{53} + \beta_{3} q^{54} + (3 \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2}) q^{55} + (\beta_{5} + \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{56} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 2) q^{57} + ( - \beta_{3} - \beta_{2}) q^{58} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} + 5 \beta_{2} - 4) q^{59} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 + 2) q^{60} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_1 - 3) q^{61} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 2) q^{62} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{63} + ( - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 5) q^{64} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{66} + ( - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 6 \beta_1 - 2) q^{67} + (\beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_1 + 7) q^{68} + (\beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 2) q^{69} + (2 \beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 4) q^{70} + ( - 4 \beta_{5} + 4 \beta_{3} - 4) q^{71} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{72} + ( - 4 \beta_{4} - 4 \beta_{2} - 3 \beta_1 + 1) q^{73} + ( - 4 \beta_{5} + 4 \beta_{2} - 4 \beta_1 - 8) q^{74} + (4 \beta_1 - 3) q^{75} + (\beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 1) q^{76} + ( - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{77} + (4 \beta_{5} - 4 \beta_{3} + 4) q^{79} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 5) q^{80} + q^{81} - 4 \beta_{2} q^{82} + ( - 4 \beta_{5} + 4 \beta_{3} - 8) q^{83} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{84} + (4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{85} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 6 \beta_1 + 6) q^{86} + ( - \beta_1 - 1) q^{87} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 1) q^{88} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{89} + (\beta_{3} + 2 \beta_{2}) q^{90} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 1) q^{92} + (2 \beta_{4} - 2 \beta_{2}) q^{93} + (2 \beta_{5} - 4 \beta_{4} + \beta_{3} - \beta_{2} + 6 \beta_1 - 4) q^{94} + (\beta_{5} - 3 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 2) q^{95} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{96} + ( - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 3) q^{97} + ( - 4 \beta_{4} + 3 \beta_{2} + 8 \beta_1 + 4) q^{98} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 2 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 2 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{8} + 6 q^{9} + 4 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{14} + 6 q^{15} + 10 q^{16} - 2 q^{17} - 10 q^{19} + 10 q^{20} - 2 q^{21} - 14 q^{22} + 10 q^{23} + 6 q^{24} - 18 q^{25} + 6 q^{27} - 26 q^{28} - 6 q^{29} + 4 q^{30} + 10 q^{32} - 2 q^{33} - 26 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{38} - 10 q^{40} - 6 q^{42} - 14 q^{44} + 6 q^{45} + 2 q^{46} + 10 q^{47} + 10 q^{48} + 8 q^{50} - 2 q^{51} - 12 q^{53} - 6 q^{55} - 34 q^{56} - 10 q^{57} - 2 q^{58} - 6 q^{59} + 10 q^{60} - 14 q^{61} + 8 q^{62} - 2 q^{63} - 22 q^{64} - 14 q^{66} + 42 q^{68} + 10 q^{69} + 22 q^{70} - 16 q^{71} + 6 q^{72} - 10 q^{73} - 32 q^{74} - 18 q^{75} - 2 q^{76} + 60 q^{77} + 16 q^{79} + 42 q^{80} + 6 q^{81} - 8 q^{82} - 40 q^{83} - 26 q^{84} + 2 q^{85} + 24 q^{86} - 6 q^{87} + 10 q^{88} - 28 q^{89} + 4 q^{90} + 2 q^{92} - 38 q^{94} - 30 q^{95} + 10 q^{96} - 10 q^{97} + 22 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 6\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 3\nu^{3} + 3\nu^{2} - 2\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 6\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 10\nu^{2} - 2\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} + 7\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + 5\beta_{4} + 3\beta_{3} + 3\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.671462 1.24464i
1.40680 0.144584i
0.264658 + 1.38923i
−0.671462 + 1.24464i
1.40680 + 0.144584i
0.264658 1.38923i
−1.24464 0.671462i 1.00000 1.09828 + 1.67146i 1.00000 2.00000i −1.24464 0.671462i 0.146365 0.146365i −0.244644 2.81783i 1.00000 −2.58757 + 1.81783i
163.2 −0.144584 + 1.40680i 1.00000 −1.95819 0.406803i 1.00000 2.00000i −0.144584 + 1.40680i 2.10278 2.10278i 0.855416 2.69597i 1.00000 2.66902 + 1.69597i
163.3 1.38923 + 0.264658i 1.00000 1.85991 + 0.735342i 1.00000 2.00000i 1.38923 + 0.264658i −3.24914 + 3.24914i 2.38923 + 1.51380i 1.00000 1.91855 2.51380i
187.1 −1.24464 + 0.671462i 1.00000 1.09828 1.67146i 1.00000 + 2.00000i −1.24464 + 0.671462i 0.146365 + 0.146365i −0.244644 + 2.81783i 1.00000 −2.58757 1.81783i
187.2 −0.144584 1.40680i 1.00000 −1.95819 + 0.406803i 1.00000 + 2.00000i −0.144584 1.40680i 2.10278 + 2.10278i 0.855416 + 2.69597i 1.00000 2.66902 1.69597i
187.3 1.38923 0.264658i 1.00000 1.85991 0.735342i 1.00000 + 2.00000i 1.38923 0.264658i −3.24914 3.24914i 2.38923 1.51380i 1.00000 1.91855 + 2.51380i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.y.d 6
3.b odd 2 1 720.2.z.e 6
4.b odd 2 1 960.2.y.d 6
5.c odd 4 1 240.2.bc.d yes 6
8.b even 2 1 1920.2.y.g 6
8.d odd 2 1 1920.2.y.h 6
15.e even 4 1 720.2.bd.e 6
16.e even 4 1 960.2.bc.d 6
16.e even 4 1 1920.2.bc.h 6
16.f odd 4 1 240.2.bc.d yes 6
16.f odd 4 1 1920.2.bc.g 6
20.e even 4 1 960.2.bc.d 6
40.i odd 4 1 1920.2.bc.g 6
40.k even 4 1 1920.2.bc.h 6
48.k even 4 1 720.2.bd.e 6
80.i odd 4 1 960.2.y.d 6
80.j even 4 1 1920.2.y.g 6
80.s even 4 1 inner 240.2.y.d 6
80.t odd 4 1 1920.2.y.h 6
240.z odd 4 1 720.2.z.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.d 6 1.a even 1 1 trivial
240.2.y.d 6 80.s even 4 1 inner
240.2.bc.d yes 6 5.c odd 4 1
240.2.bc.d yes 6 16.f odd 4 1
720.2.z.e 6 3.b odd 2 1
720.2.z.e 6 240.z odd 4 1
720.2.bd.e 6 15.e even 4 1
720.2.bd.e 6 48.k even 4 1
960.2.y.d 6 4.b odd 2 1
960.2.y.d 6 80.i odd 4 1
960.2.bc.d 6 16.e even 4 1
960.2.bc.d 6 20.e even 4 1
1920.2.y.g 6 8.b even 2 1
1920.2.y.g 6 80.j even 4 1
1920.2.y.h 6 8.d odd 2 1
1920.2.y.h 6 80.t odd 4 1
1920.2.bc.g 6 16.f odd 4 1
1920.2.bc.g 6 40.i odd 4 1
1920.2.bc.h 6 16.e even 4 1
1920.2.bc.h 6 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{6} + 2T_{7}^{5} + 2T_{7}^{4} - 32T_{7}^{3} + 196T_{7}^{2} - 56T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{6} + 2T_{11}^{5} + 2T_{11}^{4} - 32T_{11}^{3} + 196T_{11}^{2} - 56T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{4} - 2 T^{3} - 2 T^{2} + 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + 2 T^{4} - 64 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$19$ \( T^{6} + 10 T^{5} + 50 T^{4} + \cdots + 14792 \) Copy content Toggle raw display
$23$ \( T^{6} - 10 T^{5} + 50 T^{4} + \cdots + 14792 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 2)^{3} \) Copy content Toggle raw display
$31$ \( T^{6} + 60 T^{4} + 752 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$43$ \( T^{6} + 124 T^{4} + 3056 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$47$ \( T^{6} - 10 T^{5} + 50 T^{4} + \cdots + 412232 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 100 T - 344)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + 18 T^{4} + \cdots + 35912 \) Copy content Toggle raw display
$61$ \( T^{6} + 14 T^{5} + 98 T^{4} + \cdots + 4232 \) Copy content Toggle raw display
$67$ \( T^{6} + 380 T^{4} + 44272 T^{2} + \cdots + 1459264 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} - 96 T - 512)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 10 T^{5} + 50 T^{4} + \cdots + 42632 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} - 96 T + 512)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 20 T^{2} + 16 T - 704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 14 T^{2} - 4 T - 184)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 10 T^{5} + 50 T^{4} + \cdots + 1338248 \) Copy content Toggle raw display
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