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Properties

Label 320.4.c.j
Level $320$
Weight $4$
Character orbit 320.c
Analytic conductor $18.881$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(129,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("320.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.c (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: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.359712057600.22
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{4} - 4) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - 3 \beta_{5} - 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{4} - 4) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - 3 \beta_{5} - 19) q^{9} - \beta_{6} q^{11} + (\beta_{5} - 2 \beta_{4} - 3 \beta_{2}) q^{13} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{15}+ \cdots + (18 \beta_{7} + \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{5} - 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{5} - 152 q^{9} + 272 q^{21} - 408 q^{25} - 624 q^{29} - 192 q^{41} - 400 q^{45} - 2424 q^{49} - 2112 q^{61} + 2176 q^{65} - 3952 q^{69} - 1000 q^{81} + 5376 q^{85} + 80 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{7} + 130\nu^{5} + 572\nu^{3} + 420\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\nu^{7} + 1040\nu^{5} + 5008\nu^{3} + 7248\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} + 148\nu^{5} + 788\nu^{3} + 996\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -64\nu^{7} - 30\nu^{6} - 1040\nu^{5} - 420\nu^{4} - 4468\nu^{3} - 1470\nu^{2} - 3468\nu - 765 ) / 135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 56\nu^{4} - 196\nu^{2} - 102 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\nu^{6} + 80\nu^{4} - 1088\nu^{2} - 3048 ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{6} + 320\nu^{4} + 1552\nu^{2} + 672 ) / 15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} - 4\beta_{4} + \beta_{2} - 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - 4\beta_{5} - 136 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -18\beta_{5} + 36\beta_{4} + \beta_{2} + 56\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{7} + 16\beta_{6} + 44\beta_{5} + 1000 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 38\beta_{5} - 76\beta_{4} + 4\beta_{3} - 11\beta_{2} - 116\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -133\beta_{7} - 126\beta_{6} - 492\beta_{5} - 8152 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1288\beta_{5} + 2576\beta_{4} - 260\beta_{3} + 591\beta_{2} + 4064\beta_1 ) / 32 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
3.03888i
1.24759i
2.47107i
− 0.320221i
0.320221i
− 2.47107i
− 1.24759i
− 3.03888i
0 − 8.57295i 0 0.582576 − 11.1652i 0 22.1403i 0 −46.4955 0
129.2 0 − 8.57295i 0 0.582576 + 11.1652i 0 22.1403i 0 −46.4955 0
129.3 0 − 4.30169i 0 −8.58258 − 7.16515i 0 − 28.3162i 0 8.49545 0
129.4 0 − 4.30169i 0 −8.58258 + 7.16515i 0 − 28.3162i 0 8.49545 0
129.5 0 4.30169i 0 −8.58258 − 7.16515i 0 28.3162i 0 8.49545 0
129.6 0 4.30169i 0 −8.58258 + 7.16515i 0 28.3162i 0 8.49545 0
129.7 0 8.57295i 0 0.582576 − 11.1652i 0 − 22.1403i 0 −46.4955 0
129.8 0 8.57295i 0 0.582576 + 11.1652i 0 − 22.1403i 0 −46.4955 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.c.j 8
4.b odd 2 1 inner 320.4.c.j 8
5.b even 2 1 inner 320.4.c.j 8
5.c odd 4 1 1600.4.a.cu 4
5.c odd 4 1 1600.4.a.cv 4
8.b even 2 1 160.4.c.d ✓ 8
8.d odd 2 1 160.4.c.d ✓ 8
20.d odd 2 1 inner 320.4.c.j 8
20.e even 4 1 1600.4.a.cu 4
20.e even 4 1 1600.4.a.cv 4
24.f even 2 1 1440.4.f.k 8
24.h odd 2 1 1440.4.f.k 8
40.e odd 2 1 160.4.c.d ✓ 8
40.f even 2 1 160.4.c.d ✓ 8
40.i odd 4 1 800.4.a.y 4
40.i odd 4 1 800.4.a.z 4
40.k even 4 1 800.4.a.y 4
40.k even 4 1 800.4.a.z 4
120.i odd 2 1 1440.4.f.k 8
120.m even 2 1 1440.4.f.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.d ✓ 8 8.b even 2 1
160.4.c.d ✓ 8 8.d odd 2 1
160.4.c.d ✓ 8 40.e odd 2 1
160.4.c.d ✓ 8 40.f even 2 1
320.4.c.j 8 1.a even 1 1 trivial
320.4.c.j 8 4.b odd 2 1 inner
320.4.c.j 8 5.b even 2 1 inner
320.4.c.j 8 20.d odd 2 1 inner
800.4.a.y 4 40.i odd 4 1
800.4.a.y 4 40.k even 4 1
800.4.a.z 4 40.i odd 4 1
800.4.a.z 4 40.k even 4 1
1440.4.f.k 8 24.f even 2 1
1440.4.f.k 8 24.h odd 2 1
1440.4.f.k 8 120.i odd 2 1
1440.4.f.k 8 120.m even 2 1
1600.4.a.cu 4 5.c odd 4 1
1600.4.a.cu 4 20.e even 4 1
1600.4.a.cv 4 5.c odd 4 1
1600.4.a.cv 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{4} + 92T_{3}^{2} + 1360 \) Copy content Toggle raw display
\( T_{11}^{4} - 4992T_{11}^{2} + 3133440 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 92 T^{2} + 1360)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 16 T^{3} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1292 T^{2} + 393040)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4992 T^{2} + 3133440)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6080 T^{2} + 5607424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 5376)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 30080 T^{2} + 217600000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 11852 T^{2} + 2514640)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 156 T - 15420)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 23552 T^{2} + 89128960)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 42176 T^{2} + 140185600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 48 T - 13620)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 251708 T^{2} + 57154000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 211052 T^{2} + 3485953360)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 151488 T^{2} + 5693607936)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 313728 T^{2} + 4289679360)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 528 T + 8460)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 388572 T^{2} + 27064708560)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 46592 T^{2} + 272957440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 584448 T^{2} + 1090584576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1215488 T^{2} + 196885872640)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1986972 T^{2} + 739915650000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 20 T - 537500)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1290496)^{4} \) Copy content Toggle raw display
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