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Properties

Label 320.3.h.g
Level $320$
Weight $3$
Character orbit 320.h
Analytic conductor $8.719$
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] = [320,3,Mod(319,320)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.h (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: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1827904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{3} - 2) q^{7} + ( - \beta_{4} - \beta_{3} + 3 \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + (\beta_{3} + \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{3} - 2) q^{7} + ( - \beta_{4} - \beta_{3} + 3 \beta_1 + 4) q^{9} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \cdots - 2) q^{11}+ \cdots + ( - 10 \beta_{5} + 4 \beta_{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 2 q^{5} - 12 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 2 q^{5} - 12 q^{7} + 18 q^{9} + 36 q^{15} - 8 q^{21} + 68 q^{23} - 10 q^{25} + 184 q^{27} - 44 q^{29} - 108 q^{35} - 68 q^{41} - 76 q^{43} + 6 q^{45} - 268 q^{47} - 62 q^{49} + 288 q^{55} + 100 q^{61} + 172 q^{63} + 308 q^{67} + 184 q^{69} - 284 q^{75} + 238 q^{81} - 204 q^{83} + 32 q^{85} - 584 q^{87} + 76 q^{89} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 14x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} + 10\nu^{2} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} - 40\nu^{3} - 76\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} + 2\nu^{4} + 50\nu^{3} + 20\nu^{2} + 64\nu + 23 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{5} + 4\nu^{4} - 50\nu^{3} + 30\nu^{2} - 64\nu + 21 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -12\nu^{5} - 100\nu^{3} - 88\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - 3\beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{5} + 7\beta_{4} - 7\beta_{3} - 6\beta_{2} - 7\beta _1 - 7 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{4} - 5\beta_{3} + 25\beta _1 + 79 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{5} - 51\beta_{4} + 51\beta_{3} + 50\beta_{2} + 51\beta _1 + 51 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
1.37720i
− 1.37720i
− 0.273891i
0.273891i
− 2.65109i
2.65109i
0 −2.75441 0 −4.30219 − 2.54778i 0 3.84997 0 −1.41325 0
319.2 0 −2.75441 0 −4.30219 + 2.54778i 0 3.84997 0 −1.41325 0
319.3 0 −0.547781 0 3.75441 − 3.30219i 0 −10.0566 0 −8.69994 0
319.4 0 −0.547781 0 3.75441 + 3.30219i 0 −10.0566 0 −8.69994 0
319.5 0 5.30219 0 1.54778 − 4.75441i 0 0.206625 0 19.1132 0
319.6 0 5.30219 0 1.54778 + 4.75441i 0 0.206625 0 19.1132 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 320.3.h.g 6
4.b odd 2 1 320.3.h.f 6
5.b even 2 1 320.3.h.f 6
5.c odd 4 1 1600.3.b.v 6
5.c odd 4 1 1600.3.b.w 6
8.b even 2 1 160.3.h.a ✓ 6
8.d odd 2 1 160.3.h.b yes 6
16.e even 4 1 1280.3.e.g 6
16.e even 4 1 1280.3.e.i 6
16.f odd 4 1 1280.3.e.f 6
16.f odd 4 1 1280.3.e.h 6
20.d odd 2 1 inner 320.3.h.g 6
20.e even 4 1 1600.3.b.v 6
20.e even 4 1 1600.3.b.w 6
24.f even 2 1 1440.3.j.b 6
24.h odd 2 1 1440.3.j.a 6
40.e odd 2 1 160.3.h.a ✓ 6
40.f even 2 1 160.3.h.b yes 6
40.i odd 4 1 800.3.b.h 6
40.i odd 4 1 800.3.b.i 6
40.k even 4 1 800.3.b.h 6
40.k even 4 1 800.3.b.i 6
80.k odd 4 1 1280.3.e.g 6
80.k odd 4 1 1280.3.e.i 6
80.q even 4 1 1280.3.e.f 6
80.q even 4 1 1280.3.e.h 6
120.i odd 2 1 1440.3.j.b 6
120.m even 2 1 1440.3.j.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a ✓ 6 8.b even 2 1
160.3.h.a ✓ 6 40.e odd 2 1
160.3.h.b yes 6 8.d odd 2 1
160.3.h.b yes 6 40.f even 2 1
320.3.h.f 6 4.b odd 2 1
320.3.h.f 6 5.b even 2 1
320.3.h.g 6 1.a even 1 1 trivial
320.3.h.g 6 20.d odd 2 1 inner
800.3.b.h 6 40.i odd 4 1
800.3.b.h 6 40.k even 4 1
800.3.b.i 6 40.i odd 4 1
800.3.b.i 6 40.k even 4 1
1280.3.e.f 6 16.f odd 4 1
1280.3.e.f 6 80.q even 4 1
1280.3.e.g 6 16.e even 4 1
1280.3.e.g 6 80.k odd 4 1
1280.3.e.h 6 16.f odd 4 1
1280.3.e.h 6 80.q even 4 1
1280.3.e.i 6 16.e even 4 1
1280.3.e.i 6 80.k odd 4 1
1440.3.j.a 6 24.h odd 2 1
1440.3.j.a 6 120.m even 2 1
1440.3.j.b 6 24.f even 2 1
1440.3.j.b 6 120.i odd 2 1
1600.3.b.v 6 5.c odd 4 1
1600.3.b.v 6 20.e even 4 1
1600.3.b.w 6 5.c odd 4 1
1600.3.b.w 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 2T_{3}^{2} - 16T_{3} - 8 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 2 T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{3} + 6 T^{2} - 40 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 560 T^{4} + \cdots + 2560000 \) Copy content Toggle raw display
$13$ \( T^{6} + 416 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$17$ \( T^{6} + 1088 T^{4} + \cdots + 6553600 \) Copy content Toggle raw display
$19$ \( T^{6} + 1712 T^{4} + \cdots + 11505664 \) Copy content Toggle raw display
$23$ \( (T^{3} - 34 T^{2} + \cdots + 9160)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 22 T^{2} + \cdots + 2120)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 2560 T^{4} + \cdots + 419430400 \) Copy content Toggle raw display
$37$ \( T^{6} + 5408 T^{4} + \cdots + 432640000 \) Copy content Toggle raw display
$41$ \( (T^{3} + 34 T^{2} + \cdots - 5000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 38 T^{2} + \cdots - 6280)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 134 T^{2} + \cdots + 22216)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 50319462400 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 25416011776 \) Copy content Toggle raw display
$61$ \( (T^{3} - 50 T^{2} + \cdots + 81544)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 154 T^{2} + \cdots + 24760)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 14231535616 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 3114532864 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 37060870144 \) Copy content Toggle raw display
$83$ \( (T^{3} + 102 T^{2} + \cdots - 483400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 38 T^{2} + \cdots + 155000)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 7488 T^{4} + \cdots + 44302336 \) Copy content Toggle raw display
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