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Properties

Label 7200.2.m.d
Level $7200$
Weight $2$
Character orbit 7200.m
Analytic conductor $57.492$
Analytic rank $0$
Dimension $12$
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] = [7200,2,Mod(3599,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.3599");
 
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.m (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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.426337261060096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 360)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{7} + (\beta_{11} + \beta_{2}) q^{11} + (\beta_{4} - \beta_1) q^{13} + ( - \beta_{3} - \beta_1) q^{17} + (\beta_{10} + 3) q^{19} - \beta_{7} q^{23} + (\beta_{10} - \beta_{5} - 2) q^{29} + (\beta_{9} + \beta_{2}) q^{31} - \beta_{3} q^{37} + (\beta_{11} - \beta_{9} + 2 \beta_{2}) q^{41} + (\beta_{8} + \beta_{7} - 2 \beta_{6}) q^{43} + ( - 2 \beta_{8} + \beta_{7} + \beta_{6}) q^{47} - 5 q^{49} + (\beta_{8} + \beta_{7}) q^{53} + (\beta_{9} - 2 \beta_{2}) q^{59} + ( - 2 \beta_{11} - 2 \beta_{2}) q^{61} + ( - \beta_{8} + \beta_{7}) q^{67} + ( - \beta_{10} + \beta_{5} - 8) q^{71} + (2 \beta_{8} - 2 \beta_{7} + \beta_{6}) q^{73} + ( - \beta_{8} + \beta_{7} + \beta_{6}) q^{77} + (2 \beta_{11} + \beta_{9} - 3 \beta_{2}) q^{79} + 2 \beta_{3} q^{83} + ( - \beta_{11} + \beta_{9} + 2 \beta_{2}) q^{89} + (\beta_{10} - \beta_{5} + 2) q^{91} + ( - 3 \beta_{8} + \beta_{7} + \beta_{6}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 32 q^{19} - 24 q^{29} - 60 q^{49} - 96 q^{71} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - 4\nu^{10} - \nu^{9} - 2\nu^{8} + 7\nu^{7} + 20\nu^{6} - \nu^{5} - 18\nu^{4} - 18\nu^{3} + 16\nu^{2} + 72\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + \nu^{9} - 2\nu^{7} - 3\nu^{6} + 5\nu^{5} + 4\nu^{4} + 2\nu^{3} - 8\nu^{2} - 24\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{11} + 2 \nu^{10} + 3 \nu^{9} + 6 \nu^{8} + 19 \nu^{7} - 30 \nu^{6} - 37 \nu^{5} + \cdots + 104 \nu ) / 80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 6 \nu^{10} + 16 \nu^{9} + 12 \nu^{8} + 3 \nu^{7} - 50 \nu^{6} - 64 \nu^{5} + 48 \nu^{4} + \cdots - 320 ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 8 \nu^{8} + 3 \nu^{7} + 36 \nu^{5} + 28 \nu^{4} - 52 \nu^{3} + \cdots + 120 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{11} - \nu^{10} + 4\nu^{8} + 5\nu^{7} - \nu^{6} - 12\nu^{5} - 8\nu^{4} - 12\nu^{3} + 36\nu^{2} + 16\nu + 64 ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 4 \nu^{8} - 15 \nu^{7} - 16 \nu^{6} + 8 \nu^{5} + 32 \nu^{4} + 28 \nu^{3} + \cdots + 64 ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7 \nu^{11} - 4 \nu^{10} - 20 \nu^{9} - 4 \nu^{8} + 5 \nu^{7} + 16 \nu^{6} + 52 \nu^{5} - 72 \nu^{4} + \cdots + 576 ) / 80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{11} + 9 \nu^{10} + 25 \nu^{9} - 16 \nu^{8} - 40 \nu^{7} - 51 \nu^{6} + 13 \nu^{5} + 172 \nu^{4} + \cdots - 416 ) / 80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9 \nu^{11} - 4 \nu^{10} + 4 \nu^{9} + 28 \nu^{8} + 27 \nu^{7} - 40 \nu^{6} - 36 \nu^{5} + 32 \nu^{4} + \cdots - 80 ) / 80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9 \nu^{11} - 8 \nu^{10} + 12 \nu^{8} + 35 \nu^{7} - 28 \nu^{6} - 56 \nu^{5} - 64 \nu^{4} + 84 \nu^{3} + \cdots + 32 ) / 80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{3} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - 4 \beta_{6} - \beta_{5} - 3 \beta_{4} + \cdots + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + 2\beta_{10} + \beta_{9} + \beta_{5} - 5\beta_{2} + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} + 5 \beta_{10} + \beta_{9} + 5 \beta_{8} - \beta_{7} - 8 \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{8} + 3\beta_{7} + 5\beta_{6} - 4\beta_{4} - \beta_{3} + 13\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5 \beta_{11} + 7 \beta_{10} - \beta_{9} - 7 \beta_{8} - 9 \beta_{7} - 4 \beta_{6} + 9 \beta_{5} + \cdots + 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -10\beta_{11} + 10\beta_{10} - 5\beta_{9} - 5\beta_{5} + 9\beta_{2} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( \beta_{11} - 5 \beta_{10} + 7 \beta_{9} - 5 \beta_{8} - 7 \beta_{7} + 32 \beta_{6} - 7 \beta_{5} + \cdots + 52 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -16\beta_{8} + \beta_{7} + 11\beta_{6} - 24\beta_{4} + 13\beta_{3} - 25\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 21 \beta_{11} + 9 \beta_{10} - 23 \beta_{9} - 9 \beta_{8} + \beta_{7} - 100 \beta_{6} - \beta_{5} + \cdots + 208 ) / 8 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3599.1
−1.35818 − 0.394157i
1.41127 + 0.0912546i
−0.760198 + 1.19252i
−0.760198 − 1.19252i
1.41127 − 0.0912546i
−1.35818 + 0.394157i
−0.394157 + 1.35818i
−0.0912546 + 1.41127i
1.19252 + 0.760198i
1.19252 − 0.760198i
−0.0912546 − 1.41127i
−0.394157 − 1.35818i
0 0 0 0 0 −1.41421 0 0 0
3599.2 0 0 0 0 0 −1.41421 0 0 0
3599.3 0 0 0 0 0 −1.41421 0 0 0
3599.4 0 0 0 0 0 −1.41421 0 0 0
3599.5 0 0 0 0 0 −1.41421 0 0 0
3599.6 0 0 0 0 0 −1.41421 0 0 0
3599.7 0 0 0 0 0 1.41421 0 0 0
3599.8 0 0 0 0 0 1.41421 0 0 0
3599.9 0 0 0 0 0 1.41421 0 0 0
3599.10 0 0 0 0 0 1.41421 0 0 0
3599.11 0 0 0 0 0 1.41421 0 0 0
3599.12 0 0 0 0 0 1.41421 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3599.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
24.f even 2 1 inner
120.m 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.m.d 12
3.b odd 2 1 7200.2.m.e 12
4.b odd 2 1 1800.2.m.d 12
5.b even 2 1 inner 7200.2.m.d 12
5.c odd 4 1 1440.2.b.c 6
5.c odd 4 1 7200.2.b.d 6
8.b even 2 1 1800.2.m.e 12
8.d odd 2 1 7200.2.m.e 12
12.b even 2 1 1800.2.m.e 12
15.d odd 2 1 7200.2.m.e 12
15.e even 4 1 1440.2.b.d 6
15.e even 4 1 7200.2.b.e 6
20.d odd 2 1 1800.2.m.d 12
20.e even 4 1 360.2.b.c ✓ 6
20.e even 4 1 1800.2.b.e 6
24.f even 2 1 inner 7200.2.m.d 12
24.h odd 2 1 1800.2.m.d 12
40.e odd 2 1 7200.2.m.e 12
40.f even 2 1 1800.2.m.e 12
40.i odd 4 1 360.2.b.d yes 6
40.i odd 4 1 1800.2.b.d 6
40.k even 4 1 1440.2.b.d 6
40.k even 4 1 7200.2.b.e 6
60.h even 2 1 1800.2.m.e 12
60.l odd 4 1 360.2.b.d yes 6
60.l odd 4 1 1800.2.b.d 6
120.i odd 2 1 1800.2.m.d 12
120.m even 2 1 inner 7200.2.m.d 12
120.q odd 4 1 1440.2.b.c 6
120.q odd 4 1 7200.2.b.d 6
120.w even 4 1 360.2.b.c ✓ 6
120.w even 4 1 1800.2.b.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.b.c ✓ 6 20.e even 4 1
360.2.b.c ✓ 6 120.w even 4 1
360.2.b.d yes 6 40.i odd 4 1
360.2.b.d yes 6 60.l odd 4 1
1440.2.b.c 6 5.c odd 4 1
1440.2.b.c 6 120.q odd 4 1
1440.2.b.d 6 15.e even 4 1
1440.2.b.d 6 40.k even 4 1
1800.2.b.d 6 40.i odd 4 1
1800.2.b.d 6 60.l odd 4 1
1800.2.b.e 6 20.e even 4 1
1800.2.b.e 6 120.w even 4 1
1800.2.m.d 12 4.b odd 2 1
1800.2.m.d 12 20.d odd 2 1
1800.2.m.d 12 24.h odd 2 1
1800.2.m.d 12 120.i odd 2 1
1800.2.m.e 12 8.b even 2 1
1800.2.m.e 12 12.b even 2 1
1800.2.m.e 12 40.f even 2 1
1800.2.m.e 12 60.h even 2 1
7200.2.b.d 6 5.c odd 4 1
7200.2.b.d 6 120.q odd 4 1
7200.2.b.e 6 15.e even 4 1
7200.2.b.e 6 40.k even 4 1
7200.2.m.d 12 1.a even 1 1 trivial
7200.2.m.d 12 5.b even 2 1 inner
7200.2.m.d 12 24.f even 2 1 inner
7200.2.m.d 12 120.m even 2 1 inner
7200.2.m.e 12 3.b odd 2 1
7200.2.m.e 12 8.d odd 2 1
7200.2.m.e 12 15.d odd 2 1
7200.2.m.e 12 40.e odd 2 1

Hecke kernels

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

\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{29}^{3} + 6T_{29}^{2} - 28T_{29} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} + 46 T^{4} + 220 T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 46 T^{4} + \cdots - 2312)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 80 T^{4} + \cdots - 15488)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 8 T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 68 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 28 T - 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 80 T^{4} + \cdots + 15488)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 78 T^{4} + \cdots - 5000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 134 T^{4} + \cdots + 49928)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 176 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 200 T^{4} + \cdots + 123904)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 160 T^{4} + \cdots + 123904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 110 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 184 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 80 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 24 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 332 T^{4} + \cdots + 678976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 336 T^{4} + \cdots + 881792)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 312 T^{4} + \cdots - 320000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 118 T^{4} + \cdots + 200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 412 T^{4} + \cdots + 1459264)^{2} \) Copy content Toggle raw display
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