Properties

Label 320.7.h.e
Level $320$
Weight $7$
Character orbit 320.h
Analytic conductor $73.617$
Analytic rank $0$
Dimension $4$
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] = [320,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.319");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(73.6173067584\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - x^{2} + 2x + 115 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 80)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} + 65) q^{5} + 13 \beta_1 q^{7} - 129 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} + 65) q^{5} + 13 \beta_1 q^{7} - 129 q^{9} + \beta_{3} q^{11} + 14 \beta_{2} q^{13} + (\beta_{3} + 65 \beta_1) q^{15} - 52 \beta_{2} q^{17} - \beta_{3} q^{19} + 7800 q^{21} - 727 \beta_1 q^{23} + ( - 130 \beta_{2} - 7175) q^{25} - 858 \beta_1 q^{27} + 22022 q^{29} + 12 \beta_{3} q^{31} - 600 \beta_{2} q^{33} + (13 \beta_{3} + 845 \beta_1) q^{35} + 598 \beta_{2} q^{37} - 14 \beta_{3} q^{39} - 62842 q^{41} - 651 \beta_1 q^{43} + (129 \beta_{2} - 8385) q^{45} + 5629 \beta_1 q^{47} - 16249 q^{49} + 52 \beta_{3} q^{51} - 1014 \beta_{2} q^{53} + (65 \beta_{3} - 11400 \beta_1) q^{55} + 600 \beta_{2} q^{57} - 79 \beta_{3} q^{59} - 45838 q^{61} - 1677 \beta_1 q^{63} + (910 \beta_{2} + 159600) q^{65} - 5915 \beta_1 q^{67} - 436200 q^{69} + 66 \beta_{3} q^{71} + 1664 \beta_{2} q^{73} + (130 \beta_{3} - 7175 \beta_1) q^{75} - 7800 \beta_{2} q^{77} - 208 \beta_{3} q^{79} - 420759 q^{81} + 17173 \beta_1 q^{83} + ( - 3380 \beta_{2} - 592800) q^{85} + 22022 \beta_1 q^{87} + 197938 q^{89} - 182 \beta_{3} q^{91} - 7200 \beta_{2} q^{93} + ( - 65 \beta_{3} + 11400 \beta_1) q^{95} - 12348 \beta_{2} q^{97} - 129 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 260 q^{5} - 516 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 260 q^{5} - 516 q^{9} + 31200 q^{21} - 28700 q^{25} + 88088 q^{29} - 251368 q^{41} - 33540 q^{45} - 64996 q^{49} - 183352 q^{61} + 638400 q^{65} - 1744800 q^{69} - 1683036 q^{81} - 2371200 q^{85} + 791752 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - x^{2} + 2x + 115 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -20\nu^{3} + 30\nu^{2} + 250\nu - 130 ) / 43 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -10\nu^{2} + 10\nu + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2400\nu^{3} - 3600\nu^{2} + 21600\nu - 10200 ) / 43 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 120\beta _1 + 600 ) / 1200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 120\beta_{2} + 120\beta _1 + 1800 ) / 1200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} - 90\beta_{2} - 450\beta _1 + 1200 ) / 600 \) Copy content Toggle raw display

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
−1.94949 + 2.17945i
−1.94949 2.17945i
2.94949 2.17945i
2.94949 + 2.17945i
0 −24.4949 0 65.0000 106.771i 0 −318.434 0 −129.000 0
319.2 0 −24.4949 0 65.0000 + 106.771i 0 −318.434 0 −129.000 0
319.3 0 24.4949 0 65.0000 106.771i 0 318.434 0 −129.000 0
319.4 0 24.4949 0 65.0000 + 106.771i 0 318.434 0 −129.000 0
\(n\): e.g. 2-40 or 990-1000
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 320.7.h.e 4
4.b odd 2 1 inner 320.7.h.e 4
5.b even 2 1 inner 320.7.h.e 4
8.b even 2 1 80.7.h.b 4
8.d odd 2 1 80.7.h.b 4
20.d odd 2 1 inner 320.7.h.e 4
24.f even 2 1 720.7.j.d 4
24.h odd 2 1 720.7.j.d 4
40.e odd 2 1 80.7.h.b 4
40.f even 2 1 80.7.h.b 4
40.i odd 4 2 400.7.b.g 4
40.k even 4 2 400.7.b.g 4
120.i odd 2 1 720.7.j.d 4
120.m even 2 1 720.7.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.7.h.b 4 8.b even 2 1
80.7.h.b 4 8.d odd 2 1
80.7.h.b 4 40.e odd 2 1
80.7.h.b 4 40.f even 2 1
320.7.h.e 4 1.a even 1 1 trivial
320.7.h.e 4 4.b odd 2 1 inner
320.7.h.e 4 5.b even 2 1 inner
320.7.h.e 4 20.d odd 2 1 inner
400.7.b.g 4 40.i odd 4 2
400.7.b.g 4 40.k even 4 2
720.7.j.d 4 24.f even 2 1
720.7.j.d 4 24.h odd 2 1
720.7.j.d 4 120.i odd 2 1
720.7.j.d 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 600 \) acting on \(S_{7}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 600)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 130 T + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 101400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6840000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2234400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 30825600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6840000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 317117400)^{2} \) Copy content Toggle raw display
$29$ \( (T - 22022)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 984960000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4076685600)^{2} \) Copy content Toggle raw display
$41$ \( (T + 62842)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 254280600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 19011384600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 11721434400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 42688440000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 45838)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 20992335000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 29795040000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 31565414400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 295925760000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 176947157400)^{2} \) Copy content Toggle raw display
$89$ \( (T - 197938)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1738193385600)^{2} \) Copy content Toggle raw display
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