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Properties

Label 576.7.g.r
Level $576$
Weight $7$
Character orbit 576.g
Analytic conductor $132.511$
Analytic rank $0$
Dimension $8$
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] = [576,7,Mod(127,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.127");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.g (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: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.19752615936.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} + 195x^{4} - 976x^{2} + 3721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{41}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 96)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 14) q^{5} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 14) q^{5} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{7} + ( - 15 \beta_{5} - \beta_{4} + \cdots + 4 \beta_{2}) q^{11}+ \cdots + ( - 46 \beta_{7} - 128 \beta_{6} + \cdots - 925118) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 112 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 112 q^{5} + 6448 q^{13} + 13872 q^{17} + 39768 q^{25} + 45872 q^{29} + 87024 q^{37} - 119888 q^{41} - 721272 q^{49} - 51024 q^{53} + 369136 q^{61} - 445024 q^{65} - 378864 q^{73} + 2347904 q^{77} + 5663136 q^{85} - 2565264 q^{89} - 7400944 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 195x^{4} - 976x^{2} + 3721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 352\nu^{7} - 732\nu^{6} - 28080\nu^{5} + 258960\nu^{3} - 2056432\nu - 427488 ) / 11895 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 384\nu^{6} - 4680\nu^{4} + 74880\nu^{2} - 232044 ) / 3965 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9472 \nu^{7} + 1684 \nu^{6} + 115440 \nu^{5} - 25480 \nu^{4} - 895440 \nu^{3} + 249080 \nu^{2} + \cdots - 866444 ) / 11895 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22528 \nu^{7} + 384 \nu^{6} + 274560 \nu^{5} - 4680 \nu^{4} - 2870400 \nu^{3} + 74880 \nu^{2} + \cdots - 232044 ) / 11895 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -64\nu^{6} + 1024\nu^{4} - 8576\nu^{2} + 31232 ) / 183 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4736\nu^{7} - 11712\nu^{6} - 37440\nu^{5} - 162240\nu^{3} + 2338496\nu - 6839808 ) / 11895 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 352\nu^{7} + 34404\nu^{6} - 28080\nu^{5} + 258960\nu^{3} - 2056432\nu + 20091936 ) / 11895 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + 12\beta_{6} + 5\beta_{5} - 10\beta_{4} + 16\beta_{3} - 2\beta_{2} - 51\beta_1 ) / 9216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 27\beta_{5} + 128\beta_{2} + \beta _1 + 4608 ) / 1152 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 55\beta_{5} - 74\beta_{4} + 176\beta_{3} - 34\beta_{2} ) / 4608 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 27\beta_{5} + 67\beta_{2} - \beta _1 - 2412 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -345\beta_{7} - 996\beta_{6} + 575\beta_{5} - 574\beta_{4} + 1840\beta_{3} - 422\beta_{2} - 279\beta_1 ) / 9216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 65\beta_{7} - 65\beta _1 - 112128 ) / 192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3507 \beta_{7} - 9348 \beta_{6} - 5845 \beta_{5} + 4670 \beta_{4} - 18704 \beta_{3} + \cdots - 15261 \beta_1 ) / 9216 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−2.16817 − 1.25179i
−2.16817 + 1.25179i
2.70167 + 1.55981i
2.70167 − 1.55981i
−2.70167 + 1.55981i
−2.70167 − 1.55981i
2.16817 − 1.25179i
2.16817 + 1.25179i
0 0 0 −154.769 0 − 78.8120i 0 0 0
127.2 0 0 0 −154.769 0 78.8120i 0 0 0
127.3 0 0 0 −70.0730 0 − 522.087i 0 0 0
127.4 0 0 0 −70.0730 0 522.087i 0 0 0
127.5 0 0 0 56.5038 0 − 641.933i 0 0 0
127.6 0 0 0 56.5038 0 641.933i 0 0 0
127.7 0 0 0 224.338 0 − 374.658i 0 0 0
127.8 0 0 0 224.338 0 374.658i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.g.r 8
3.b odd 2 1 192.7.g.f 8
4.b odd 2 1 inner 576.7.g.r 8
8.b even 2 1 288.7.g.f 8
8.d odd 2 1 288.7.g.f 8
12.b even 2 1 192.7.g.f 8
24.f even 2 1 96.7.g.b ✓ 8
24.h odd 2 1 96.7.g.b ✓ 8
48.i odd 4 1 768.7.b.f 8
48.i odd 4 1 768.7.b.g 8
48.k even 4 1 768.7.b.f 8
48.k even 4 1 768.7.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.7.g.b ✓ 8 24.f even 2 1
96.7.g.b ✓ 8 24.h odd 2 1
192.7.g.f 8 3.b odd 2 1
192.7.g.f 8 12.b even 2 1
288.7.g.f 8 8.b even 2 1
288.7.g.f 8 8.d odd 2 1
576.7.g.r 8 1.a even 1 1 trivial
576.7.g.r 8 4.b odd 2 1 inner
768.7.b.f 8 48.i odd 4 1
768.7.b.f 8 48.k even 4 1
768.7.b.g 8 48.i odd 4 1
768.7.b.g 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 56T_{5}^{3} - 39624T_{5}^{2} - 195680T_{5} + 137472400 \) acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 56 T^{3} + \cdots + 137472400)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 97\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 4205886954256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 65249673849072)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 50\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 28\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 23\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 85\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 70\!\cdots\!08)^{2} \) Copy content Toggle raw display
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