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Properties

Label 7200.2.d.e
Level $7200$
Weight $2$
Character orbit 7200.d
Analytic conductor $57.492$
Analytic rank $1$
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] = [7200,2,Mod(2449,7200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7200.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, -20,0,0,0,0,0,-8,0,0,0,-20,0,8,0,0,0,0,0,6,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(53)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.4922894553\)
Analytic rank: \(1\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{7} - 2 \beta q^{11} - 3 \beta q^{17} - 2 \beta q^{19} - 2 \beta q^{23} + 3 \beta q^{29} - 10 q^{31} - 4 q^{37} - 10 q^{41} + 4 q^{43} + 2 \beta q^{47} + 3 q^{49} + 10 q^{53} + 4 \beta q^{59} + \cdots + 5 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{31} - 8 q^{37} - 20 q^{41} + 8 q^{43} + 6 q^{49} + 20 q^{53} - 24 q^{67} - 8 q^{71} + 16 q^{77} - 28 q^{79} + 28 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
− 1.00000i
1.00000i
0 0 0 0 0 − 2.00000i 0 0 0
2449.2 0 0 0 0 0 2.00000i 0 0 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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.d.e 2
3.b odd 2 1 2400.2.d.a 2
4.b odd 2 1 1800.2.d.c 2
5.b even 2 1 7200.2.d.f 2
5.c odd 4 1 1440.2.k.a 2
5.c odd 4 1 7200.2.k.f 2
8.b even 2 1 7200.2.d.f 2
8.d odd 2 1 1800.2.d.h 2
12.b even 2 1 600.2.d.d 2
15.d odd 2 1 2400.2.d.d 2
15.e even 4 1 480.2.k.a 2
15.e even 4 1 2400.2.k.b 2
20.d odd 2 1 1800.2.d.h 2
20.e even 4 1 360.2.k.b 2
20.e even 4 1 1800.2.k.g 2
24.f even 2 1 600.2.d.a 2
24.h odd 2 1 2400.2.d.d 2
40.e odd 2 1 1800.2.d.c 2
40.f even 2 1 inner 7200.2.d.e 2
40.i odd 4 1 1440.2.k.a 2
40.i odd 4 1 7200.2.k.f 2
40.k even 4 1 360.2.k.b 2
40.k even 4 1 1800.2.k.g 2
60.h even 2 1 600.2.d.a 2
60.l odd 4 1 120.2.k.a ✓ 2
60.l odd 4 1 600.2.k.a 2
120.i odd 2 1 2400.2.d.a 2
120.m even 2 1 600.2.d.d 2
120.q odd 4 1 120.2.k.a ✓ 2
120.q odd 4 1 600.2.k.a 2
120.w even 4 1 480.2.k.a 2
120.w even 4 1 2400.2.k.b 2
240.z odd 4 1 3840.2.a.d 1
240.bb even 4 1 3840.2.a.r 1
240.bd odd 4 1 3840.2.a.w 1
240.bf even 4 1 3840.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.a ✓ 2 60.l odd 4 1
120.2.k.a ✓ 2 120.q odd 4 1
360.2.k.b 2 20.e even 4 1
360.2.k.b 2 40.k even 4 1
480.2.k.a 2 15.e even 4 1
480.2.k.a 2 120.w even 4 1
600.2.d.a 2 24.f even 2 1
600.2.d.a 2 60.h even 2 1
600.2.d.d 2 12.b even 2 1
600.2.d.d 2 120.m even 2 1
600.2.k.a 2 60.l odd 4 1
600.2.k.a 2 120.q odd 4 1
1440.2.k.a 2 5.c odd 4 1
1440.2.k.a 2 40.i odd 4 1
1800.2.d.c 2 4.b odd 2 1
1800.2.d.c 2 40.e odd 2 1
1800.2.d.h 2 8.d odd 2 1
1800.2.d.h 2 20.d odd 2 1
1800.2.k.g 2 20.e even 4 1
1800.2.k.g 2 40.k even 4 1
2400.2.d.a 2 3.b odd 2 1
2400.2.d.a 2 120.i odd 2 1
2400.2.d.d 2 15.d odd 2 1
2400.2.d.d 2 24.h odd 2 1
2400.2.k.b 2 15.e even 4 1
2400.2.k.b 2 120.w even 4 1
3840.2.a.d 1 240.z odd 4 1
3840.2.a.m 1 240.bf even 4 1
3840.2.a.r 1 240.bb even 4 1
3840.2.a.w 1 240.bd odd 4 1
7200.2.d.e 2 1.a even 1 1 trivial
7200.2.d.e 2 40.f even 2 1 inner
7200.2.d.f 2 5.b even 2 1
7200.2.d.f 2 8.b even 2 1
7200.2.k.f 2 5.c odd 4 1
7200.2.k.f 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{37} + 4 \) Copy content Toggle raw display
\( T_{41} + 10 \) Copy content Toggle raw display
\( T_{53} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( (T + 12)^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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