Properties

Label 3920.2.k.c
Level $3920$
Weight $2$
Character orbit 3920.k
Analytic conductor $31.301$
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] = [3920,2,Mod(2351,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3920.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.k (of order \(2\), degree \(1\), 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: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5} + 1) q^{3} + \beta_1 q^{5} + (2 \beta_{7} - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5} + 1) q^{3} + \beta_1 q^{5} + (2 \beta_{7} - \beta_{6} + 2) q^{9} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{15} + (\beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{17} + ( - 3 \beta_{7} + \beta_{5} - 2) q^{19} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{23} - q^{25} + (5 \beta_{7} - 2 \beta_{6} + 1) q^{27} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 1) q^{29} + (2 \beta_{7} - \beta_{5} - 6) q^{31} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{33} + (2 \beta_{7} + \beta_{6} + 3 \beta_{5}) q^{37} + (2 \beta_{4} - \beta_{3} + 5 \beta_1) q^{39} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{41} + ( - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2}) q^{43} + (2 \beta_{4} - \beta_{3} + 2 \beta_1) q^{45} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3) q^{47} + (4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 7 \beta_1) q^{51} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} + 4) q^{53} + (\beta_{6} - 2 \beta_{5} + 1) q^{55} + ( - 7 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} - 4) q^{57} + (\beta_{7} - 4 \beta_{6} - 2) q^{59} + (\beta_{4} - 4 \beta_{3} - \beta_{2}) q^{61} + (\beta_{6} + \beta_{5} - 1) q^{65} + (4 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{4} - 8 \beta_{3} + 4 \beta_1) q^{69} + (4 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{71} + ( - \beta_{4} + 5 \beta_{3} + 3 \beta_{2}) q^{73} + (\beta_{6} + \beta_{5} - 1) q^{75} + ( - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1) q^{79} + (6 \beta_{7} - 5 \beta_{6} - 4 \beta_{5} - 1) q^{81} + (2 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 6) q^{83} + ( - \beta_{7} + 2 \beta_{5} - 3) q^{85} + (5 \beta_{7} - 8 \beta_{6} + 3) q^{87} + ( - 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 8 \beta_1) q^{89} + (5 \beta_{7} + 5 \beta_{6} + 4 \beta_{5} - 4) q^{93} + ( - 3 \beta_{4} + \beta_{2} - 2 \beta_1) q^{95} + ( - 6 \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_1) q^{97} + ( - 2 \beta_{4} + 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 16 q^{9} - 16 q^{19} - 8 q^{25} + 8 q^{27} + 8 q^{29} - 48 q^{31} - 24 q^{47} + 32 q^{53} + 8 q^{55} - 32 q^{57} - 16 q^{59} - 8 q^{65} - 8 q^{75} - 8 q^{81} + 48 q^{83} - 24 q^{85} + 24 q^{87} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\zeta_{16}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( ( -\beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
0.923880 0.382683i
0.923880 + 0.382683i
−0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 + 0.923880i
0.382683 0.923880i
−0.382683 0.923880i
−0.382683 + 0.923880i
0 −2.26197 0 1.00000i 0 0 0 2.11652 0
2351.2 0 −2.26197 0 1.00000i 0 0 0 2.11652 0
2351.3 0 1.43355 0 1.00000i 0 0 0 −0.944947 0
2351.4 0 1.43355 0 1.00000i 0 0 0 −0.944947 0
2351.5 0 1.64885 0 1.00000i 0 0 0 −0.281305 0
2351.6 0 1.64885 0 1.00000i 0 0 0 −0.281305 0
2351.7 0 3.17958 0 1.00000i 0 0 0 7.10973 0
2351.8 0 3.17958 0 1.00000i 0 0 0 7.10973 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2351.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.2.k.c yes 8
4.b odd 2 1 3920.2.k.a 8
7.b odd 2 1 3920.2.k.a 8
28.d even 2 1 inner 3920.2.k.c yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.2.k.a 8 4.b odd 2 1
3920.2.k.a 8 7.b odd 2 1
3920.2.k.c yes 8 1.a even 1 1 trivial
3920.2.k.c yes 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 2T_{3}^{2} + 20T_{3} - 17 \) acting on \(S_{2}^{\mathrm{new}}(3920, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{3} - 2 T^{2} + 20 T - 17)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 44 T^{6} + 262 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{8} + 20 T^{6} + 130 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( T^{8} + 76 T^{6} + 1442 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} - 16 T^{2} - 128 T - 136)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 128 T^{6} + 5628 T^{4} + \cdots + 334084 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 62 T^{2} + 324 T - 383)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 24 T^{3} + 196 T^{2} + 624 T + 578)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 56 T^{2} - 136 T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 88 T^{6} + 1876 T^{4} + \cdots + 24964 \) Copy content Toggle raw display
$43$ \( T^{8} + 176 T^{6} + 8016 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} - 2 T^{2} - 204 T - 289)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 16 T^{3} + 40 T^{2} + 328 T - 1186)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} - 44 T^{2} - 272 T + 578)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 144 T^{6} + 6736 T^{4} + \cdots + 602176 \) Copy content Toggle raw display
$67$ \( T^{8} + 224 T^{6} + 14076 T^{4} + \cdots + 334084 \) Copy content Toggle raw display
$71$ \( T^{8} + 616 T^{6} + \cdots + 74580496 \) Copy content Toggle raw display
$73$ \( T^{8} + 280 T^{6} + 21384 T^{4} + \cdots + 795664 \) Copy content Toggle raw display
$79$ \( T^{8} + 164 T^{6} + 8230 T^{4} + \cdots + 277729 \) Copy content Toggle raw display
$83$ \( (T^{4} - 24 T^{3} + 72 T^{2} + 992 T - 4624)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 448 T^{6} + 44992 T^{4} + \cdots + 295936 \) Copy content Toggle raw display
$97$ \( T^{8} + 500 T^{6} + \cdots + 12271009 \) Copy content Toggle raw display
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