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Properties

Label 320.5.p.a
Level $320$
Weight $5$
Character orbit 320.p
Analytic conductor $33.078$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,5,Mod(193,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.193"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-20,0,-40,0,-84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (10 i - 10) q^{3} + ( - 15 i - 20) q^{5} + ( - 42 i - 42) q^{7} - 119 i q^{9} - 184 q^{11} + ( - 117 i + 117) q^{13} + ( - 50 i + 350) q^{15} + ( - 129 i - 129) q^{17} - 212 i q^{19} + 840 q^{21} + (406 i - 406) q^{23} + \cdots + 21896 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{3} - 40 q^{5} - 84 q^{7} - 368 q^{11} + 234 q^{13} + 700 q^{15} - 258 q^{17} + 1680 q^{21} - 812 q^{23} + 350 q^{25} + 760 q^{27} + 824 q^{31} + 3680 q^{33} + 420 q^{35} - 266 q^{37} - 3632 q^{41}+ \cdots + 19666 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
− 1.00000i
1.00000i
0 −10.0000 − 10.0000i 0 −20.0000 + 15.0000i 0 −42.0000 + 42.0000i 0 119.000i 0
257.1 0 −10.0000 + 10.0000i 0 −20.0000 − 15.0000i 0 −42.0000 − 42.0000i 0 − 119.000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.5.p.a 2
4.b odd 2 1 320.5.p.j 2
5.c odd 4 1 inner 320.5.p.a 2
8.b even 2 1 40.5.l.a ✓ 2
8.d odd 2 1 80.5.p.a 2
20.e even 4 1 320.5.p.j 2
24.h odd 2 1 360.5.v.a 2
40.e odd 2 1 400.5.p.d 2
40.f even 2 1 200.5.l.a 2
40.i odd 4 1 40.5.l.a ✓ 2
40.i odd 4 1 200.5.l.a 2
40.k even 4 1 80.5.p.a 2
40.k even 4 1 400.5.p.d 2
120.w even 4 1 360.5.v.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.5.l.a ✓ 2 8.b even 2 1
40.5.l.a ✓ 2 40.i odd 4 1
80.5.p.a 2 8.d odd 2 1
80.5.p.a 2 40.k even 4 1
200.5.l.a 2 40.f even 2 1
200.5.l.a 2 40.i odd 4 1
320.5.p.a 2 1.a even 1 1 trivial
320.5.p.a 2 5.c odd 4 1 inner
320.5.p.j 2 4.b odd 2 1
320.5.p.j 2 20.e even 4 1
360.5.v.a 2 24.h odd 2 1
360.5.v.a 2 120.w even 4 1
400.5.p.d 2 40.e odd 2 1
400.5.p.d 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} + 20T_{3} + 200 \) Copy content Toggle raw display
\( T_{13}^{2} - 234T_{13} + 27378 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 20T + 200 \) Copy content Toggle raw display
$5$ \( T^{2} + 40T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} + 84T + 3528 \) Copy content Toggle raw display
$11$ \( (T + 184)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 234T + 27378 \) Copy content Toggle raw display
$17$ \( T^{2} + 258T + 33282 \) Copy content Toggle raw display
$19$ \( T^{2} + 44944 \) Copy content Toggle raw display
$23$ \( T^{2} + 812T + 329672 \) Copy content Toggle raw display
$29$ \( T^{2} + 1401856 \) Copy content Toggle raw display
$31$ \( (T - 412)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 266T + 35378 \) Copy content Toggle raw display
$41$ \( (T + 1816)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2084 T + 2171528 \) Copy content Toggle raw display
$47$ \( T^{2} + 2956 T + 4368968 \) Copy content Toggle raw display
$53$ \( T^{2} + 4602 T + 10589202 \) Copy content Toggle raw display
$59$ \( T^{2} + 4376464 \) Copy content Toggle raw display
$61$ \( (T + 1848)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 10252 T + 52551752 \) Copy content Toggle raw display
$71$ \( (T + 980)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10542 T + 55566882 \) Copy content Toggle raw display
$79$ \( T^{2} + 24601600 \) Copy content Toggle raw display
$83$ \( T^{2} + 2828 T + 3998792 \) Copy content Toggle raw display
$89$ \( T^{2} + 78287104 \) Copy content Toggle raw display
$97$ \( T^{2} - 19666 T + 193375778 \) Copy content Toggle raw display
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