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Properties

Label 576.7.b.d
Level $576$
Weight $7$
Character orbit 576.b
Analytic conductor $132.511$
Analytic rank $0$
Dimension $8$
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] = [576,7,Mod(415,576)]
 
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, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.415");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.b (of order \(2\), degree \(1\), 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: 8.0.14637786276096.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 81x^{6} + 4880x^{4} + 136161x^{2} + 2825761 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 192)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + (\beta_{5} - 3 \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + (\beta_{5} - 3 \beta_{3}) q^{7} + ( - 5 \beta_{6} - \beta_1) q^{11} + (21 \beta_{4} + 3 \beta_{2}) q^{13} + ( - 3 \beta_{7} - 3510) q^{17} + ( - 15 \beta_{6} - 6 \beta_1) q^{19} + ( - 30 \beta_{5} - 264 \beta_{3}) q^{23} + ( - 4 \beta_{7} + 8029) q^{25} + (61 \beta_{4} + 62 \beta_{2}) q^{29} + ( - 61 \beta_{5} + 1302 \beta_{3}) q^{31} + (67 \beta_{6} + 182 \beta_1) q^{35} + (603 \beta_{4} - 165 \beta_{2}) q^{37} + ( - 27 \beta_{7} - 1458) q^{41} + (357 \beta_{6} - 120 \beta_1) q^{43} + ( - 102 \beta_{5} + 1581 \beta_{3}) q^{47} + ( - 26 \beta_{7} - 43491) q^{49} + ( - 441 \beta_{4} + 770 \beta_{2}) q^{53} + (176 \beta_{5} - 7507 \beta_{3}) q^{55} + (264 \beta_{6} + 1445 \beta_1) q^{59} + (1065 \beta_{4} - 1803 \beta_{2}) q^{61} + (69 \beta_{7} + 158976) q^{65} + ( - 396 \beta_{6} - 2049 \beta_1) q^{67} + (702 \beta_{5} + 12933 \beta_{3}) q^{71} + (214 \beta_{7} + 91834) q^{73} + ( - 4428 \beta_{4} + 2956 \beta_{2}) q^{77} + (63 \beta_{5} + 20530 \beta_{3}) q^{79} + ( - 1603 \beta_{6} + 2505 \beta_1) q^{83} + (7482 \beta_{4} - 1380 \beta_{2}) q^{85} + (474 \beta_{7} + 164322) q^{89} + ( - 879 \beta_{6} - 4131 \beta_1) q^{91} + (636 \beta_{5} - 23952 \beta_{3}) q^{95} + ( - 370 \beta_{7} + 375086) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 28080 q^{17} + 64232 q^{25} - 11664 q^{41} - 347928 q^{49} + 1271808 q^{65} + 734672 q^{73} + 1314576 q^{89} + 3000688 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 81x^{6} + 4880x^{4} + 136161x^{2} + 2825761 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\nu^{7} - 43200\nu^{5} - 1683720\nu^{3} - 68942853\nu ) / 689210 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1641\nu^{6} - 204960\nu^{4} - 16601760\nu^{2} - 600989439 ) / 2050820 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 648\nu^{7} + 39040\nu^{5} + 1521584\nu^{3} + 22606088\nu ) / 4204181 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1203\nu^{6} + 11712\nu^{4} + 948672\nu^{2} - 45381957 ) / 410164 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1629\nu^{7} - 143716\nu^{5} - 9385094\nu^{3} - 430579745\nu ) / 4204181 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 483\nu^{7} + 34080\nu^{5} + 2155320\nu^{3} + 24614883\nu ) / 689210 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 144\nu^{6} + 11664\nu^{4} + 460656\nu^{2} + 9803592 ) / 1681 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -18\beta_{6} - 24\beta_{5} + 21\beta_{3} + 2\beta_1 ) / 1728 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 134\beta_{4} - 170\beta_{2} - 17496 ) / 864 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 360\beta_{6} - 1647\beta_{3} + 40\beta_1 ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 243\beta_{7} - 8666\beta_{4} + 5750\beta_{2} - 690984 ) / 864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9354\beta_{6} + 12472\beta_{5} + 76927\beta_{3} - 11974\beta_1 ) / 576 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8540\beta_{4} + 2440\beta_{2} + 1659933 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -354234\beta_{6} - 472312\beta_{5} + 4014713\beta_{3} + 572906\beta_1 ) / 576 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
2.75877 + 5.77834i
−2.75877 − 5.77834i
−3.62480 + 5.27834i
3.62480 − 5.27834i
3.62480 + 5.27834i
−3.62480 − 5.27834i
−2.75877 + 5.77834i
2.75877 − 5.77834i
0 0 0 − 118.172i 0 − 450.040i 0 0 0
415.2 0 0 0 − 118.172i 0 450.040i 0 0 0
415.3 0 0 0 − 35.0337i 0 − 346.040i 0 0 0
415.4 0 0 0 − 35.0337i 0 346.040i 0 0 0
415.5 0 0 0 35.0337i 0 − 346.040i 0 0 0
415.6 0 0 0 35.0337i 0 346.040i 0 0 0
415.7 0 0 0 118.172i 0 − 450.040i 0 0 0
415.8 0 0 0 118.172i 0 450.040i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 15192T_{5}^{2} + 17139600 \) 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
$5$ \( (T^{4} + 15192 T^{2} + 17139600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 322280 T^{2} + 24252455824)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 5476947922944)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 10172629575936)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 7020 T - 10494684)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 418927889481984)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 21\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2916 T - 1845871740)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 97\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 40\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 183668 T - 107658277340)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 328644 T - 542546548092)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 750172 T - 206348707004)^{4} \) Copy content Toggle raw display
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