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Properties

Label 1960.2.q.x
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $0$
Dimension $8$
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] = [1960,2,Mod(361,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.q (of order \(3\), degree \(2\), 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: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.21913473024.16
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 11x^{6} - 2x^{5} + 51x^{4} + 162x^{2} + 112x + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \cdots - 1) q^{9}+ \cdots + ( - 2 \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 4 q^{5} - 6 q^{9} - 2 q^{11} + 20 q^{13} + 4 q^{15} - 6 q^{17} + 4 q^{23} - 4 q^{25} + 28 q^{27} - 4 q^{29} + 12 q^{31} - 18 q^{33} - 14 q^{39} + 24 q^{41} - 16 q^{43} - 6 q^{45} + 2 q^{47} - 2 q^{51} + 4 q^{53} + 4 q^{55} - 16 q^{57} - 8 q^{59} - 20 q^{61} - 10 q^{65} + 8 q^{67} + 48 q^{69} + 8 q^{71} - 16 q^{73} - 2 q^{75} - 22 q^{79} + 20 q^{81} + 72 q^{83} + 12 q^{85} + 18 q^{87} - 40 q^{89} + 32 q^{93} + 52 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 11x^{6} - 2x^{5} + 51x^{4} + 162x^{2} + 112x + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 329\nu^{7} + 154\nu^{6} + 2209\nu^{5} + 4794\nu^{4} + 28611\nu^{3} + 20492\nu^{2} + 19740\nu + 169702 ) / 84134 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3056 \nu^{7} - 8415 \nu^{6} + 32538 \nu^{5} - 21575 \nu^{4} + 122298 \nu^{3} - 200277 \nu^{2} + \cdots + 204092 ) / 588938 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{7} - 68\nu^{6} + 141\nu^{5} + 306\nu^{4} - 353\nu^{3} + 1308\nu^{2} + 1260\nu + 8108 ) / 3658 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} + 45\nu^{6} - 174\nu^{5} + 712\nu^{4} - 654\nu^{3} + 1071\nu^{2} - 2878\nu + 2058 ) / 1829 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3270 \nu^{7} + 11895 \nu^{6} - 45994 \nu^{5} + 84562 \nu^{4} - 172874 \nu^{3} + 283101 \nu^{2} + \cdots + 543998 ) / 294469 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1064\nu^{7} + 397\nu^{6} - 7144\nu^{5} - 15504\nu^{4} - 25401\nu^{3} - 66272\nu^{2} - 63840\nu - 138894 ) / 42067 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} - 4\beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{4} + 5\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{6} - 2\beta_{5} + 22\beta_{3} - 9\beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{7} + 17\beta_{6} - 11\beta_{5} - 17\beta_{4} + 40\beta_{3} - 31\beta_{2} - 31\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -28\beta_{7} - 75\beta_{4} - 71\beta_{2} + 217 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -183\beta_{6} + 103\beta_{5} - 340\beta_{3} + 217\beta _1 + 340 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.43998 − 2.49412i
1.09199 − 1.89138i
−0.591990 + 1.02536i
−0.939980 + 1.62809i
1.43998 + 2.49412i
1.09199 + 1.89138i
−0.591990 − 1.02536i
−0.939980 − 1.62809i
0 −1.43998 + 2.49412i 0 −0.500000 − 0.866025i 0 0 0 −2.64709 − 4.58489i 0
361.2 0 −1.09199 + 1.89138i 0 −0.500000 − 0.866025i 0 0 0 −0.884883 − 1.53266i 0
361.3 0 0.591990 − 1.02536i 0 −0.500000 − 0.866025i 0 0 0 0.799096 + 1.38408i 0
361.4 0 0.939980 − 1.62809i 0 −0.500000 − 0.866025i 0 0 0 −0.267126 − 0.462676i 0
961.1 0 −1.43998 − 2.49412i 0 −0.500000 + 0.866025i 0 0 0 −2.64709 + 4.58489i 0
961.2 0 −1.09199 − 1.89138i 0 −0.500000 + 0.866025i 0 0 0 −0.884883 + 1.53266i 0
961.3 0 0.591990 + 1.02536i 0 −0.500000 + 0.866025i 0 0 0 0.799096 − 1.38408i 0
961.4 0 0.939980 + 1.62809i 0 −0.500000 + 0.866025i 0 0 0 −0.267126 + 0.462676i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.x 8
7.b odd 2 1 1960.2.q.y 8
7.c even 3 1 1960.2.a.y yes 4
7.c even 3 1 inner 1960.2.q.x 8
7.d odd 6 1 1960.2.a.x ✓ 4
7.d odd 6 1 1960.2.q.y 8
28.f even 6 1 3920.2.a.ce 4
28.g odd 6 1 3920.2.a.cd 4
35.i odd 6 1 9800.2.a.cs 4
35.j even 6 1 9800.2.a.cl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.x ✓ 4 7.d odd 6 1
1960.2.a.y yes 4 7.c even 3 1
1960.2.q.x 8 1.a even 1 1 trivial
1960.2.q.x 8 7.c even 3 1 inner
1960.2.q.y 8 7.b odd 2 1
1960.2.q.y 8 7.d odd 6 1
3920.2.a.cd 4 28.g odd 6 1
3920.2.a.ce 4 28.f even 6 1
9800.2.a.cl 4 35.j even 6 1
9800.2.a.cs 4 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 11T_{3}^{6} + 2T_{3}^{5} + 51T_{3}^{4} + 162T_{3}^{2} - 112T_{3} + 196 \) Copy content Toggle raw display
\( T_{11}^{8} + 2T_{11}^{7} + 15T_{11}^{6} + 18T_{11}^{5} + 165T_{11}^{4} + 236T_{11}^{3} + 356T_{11}^{2} + 80T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 10T_{13}^{3} + 19T_{13}^{2} + 52T_{13} - 124 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{4} - 10 T^{3} + \cdots - 124)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 188356 \) Copy content Toggle raw display
$19$ \( T^{8} + 26 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + \cdots + 188)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$37$ \( T^{8} + 42 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + \cdots - 4228)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + \cdots + 752)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 2 T^{7} + \cdots + 3136 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} + \cdots + 322624 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} + \cdots + 28558336 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots + 5456896 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots + 10976)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots + 201412864 \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 3136 \) Copy content Toggle raw display
$83$ \( (T^{4} - 36 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 40 T^{7} + \cdots + 29246464 \) Copy content Toggle raw display
$97$ \( (T^{4} - 26 T^{3} + \cdots - 1022)^{2} \) Copy content Toggle raw display
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