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Properties

Label 6.25.b.a
Level $6$
Weight $25$
Character orbit 6.b
Analytic conductor $21.898$
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] = [6,25,Mod(5,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.5");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(21.8980291355\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9921984x^{6} + 31297402621425x^{4} + 35629505313218665424x^{2} + 11190322069687119538557504 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{53}\cdot 3^{32}\cdot 17^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 - 16485) q^{3} - 8388608 q^{4} + ( - \beta_{3} - 155 \beta_{2} + 27689 \beta_1) q^{5} + (\beta_{7} + \beta_{4} + \cdots + 4214784) q^{6}+ \cdots + ( - 107 \beta_{7} + 37 \beta_{6} + \cdots + 36896131737) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 - 16485) q^{3} - 8388608 q^{4} + ( - \beta_{3} - 155 \beta_{2} + 27689 \beta_1) q^{5} + (\beta_{7} + \beta_{4} + \cdots + 4214784) q^{6}+ \cdots + ( - 168291602235474 \beta_{7} + \cdots + 86\!\cdots\!68) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 131880 q^{3} - 67108864 q^{4} + 33718272 q^{6} - 10160794640 q^{7} + 295169053896 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 131880 q^{3} - 67108864 q^{4} + 33718272 q^{6} - 10160794640 q^{7} + 295169053896 q^{9} - 1863369424896 q^{10} + 1106289623040 q^{12} + 50568363679120 q^{13} - 348034956760512 q^{15} + 562949953421312 q^{16} - 514738292981760 q^{18} - 978083631341264 q^{19} + 36\!\cdots\!36 q^{21}+ \cdots + 69\!\cdots\!44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9921984x^{6} + 31297402621425x^{4} + 35629505313218665424x^{2} + 11190322069687119538557504 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3943712992 \nu^{7} + \cdots + 75\!\cdots\!64 \nu ) / 71\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!35 \nu^{7} + \cdots + 68\!\cdots\!08 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35\!\cdots\!77 \nu^{7} + \cdots - 10\!\cdots\!40 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!31 \nu^{7} + \cdots - 24\!\cdots\!48 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32\!\cdots\!35 \nu^{7} + \cdots + 17\!\cdots\!32 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!01 \nu^{7} + \cdots - 15\!\cdots\!08 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\!\cdots\!85 \nu^{7} + \cdots + 66\!\cdots\!56 ) / 60\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 965 \beta_{7} - 618 \beta_{6} - 3784 \beta_{5} - 1776 \beta_{4} + 3651 \beta_{3} + \cdots - 1842900 \beta_1 ) / 2741133312 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 18797 \beta_{7} + 69228 \beta_{6} + 135280 \beta_{5} - 102729 \beta_{4} + 261816 \beta_{3} + \cdots - 399962953875456 ) / 161243136 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3528444415 \beta_{7} + 2240481438 \beta_{6} + 13778333144 \beta_{5} + \cdots - 11\!\cdots\!08 \beta_1 ) / 2741133312 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10151333995 \beta_{7} - 33506446020 \beta_{6} - 34594106480 \beta_{5} + 32535664335 \beta_{4} + \cdots + 90\!\cdots\!44 ) / 10077696 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17\!\cdots\!35 \beta_{7} + \cdots + 91\!\cdots\!00 \beta_1 ) / 2741133312 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10\!\cdots\!23 \beta_{7} + \cdots - 61\!\cdots\!84 ) / 161243136 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 96\!\cdots\!85 \beta_{7} + \cdots - 55\!\cdots\!12 \beta_1 ) / 2741133312 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
− 2234.51i
1694.95i
− 708.500i
1246.64i
2234.51i
− 1694.95i
708.500i
− 1246.64i
− 2896.31i −490828. − 203759.i −8.38861e6 1.24013e8i −5.90150e8 + 1.42159e9i 1.50611e10 2.42960e10i 1.99394e11 + 2.00021e11i 3.59180e11
5.2 − 2896.31i −317945. + 425841.i −8.38861e6 − 6.76938e7i 1.23337e9 + 9.20867e8i −2.41596e10 2.42960e10i −8.02515e10 − 2.70788e11i −1.96062e11
5.3 − 2896.31i 269557. − 458005.i −8.38861e6 − 4.56581e8i −1.32652e9 − 7.80721e8i 1.81935e9 2.42960e10i −1.37107e11 − 2.46917e11i −1.32240e12
5.4 − 2896.31i 473275. + 241744.i −8.38861e6 7.85820e7i 7.00165e8 − 1.37075e9i 2.19870e9 2.42960e10i 1.65549e11 + 2.28823e11i 2.27598e11
5.5 2896.31i −490828. + 203759.i −8.38861e6 − 1.24013e8i −5.90150e8 − 1.42159e9i 1.50611e10 − 2.42960e10i 1.99394e11 − 2.00021e11i 3.59180e11
5.6 2896.31i −317945. − 425841.i −8.38861e6 6.76938e7i 1.23337e9 − 9.20867e8i −2.41596e10 − 2.42960e10i −8.02515e10 + 2.70788e11i −1.96062e11
5.7 2896.31i 269557. + 458005.i −8.38861e6 4.56581e8i −1.32652e9 + 7.80721e8i 1.81935e9 − 2.42960e10i −1.37107e11 + 2.46917e11i −1.32240e12
5.8 2896.31i 473275. − 241744.i −8.38861e6 − 7.85820e7i 7.00165e8 + 1.37075e9i 2.19870e9 − 2.42960e10i 1.65549e11 − 2.28823e11i 2.27598e11
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.25.b.a ✓ 8
3.b odd 2 1 inner 6.25.b.a ✓ 8
4.b odd 2 1 48.25.e.d 8
12.b even 2 1 48.25.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.25.b.a ✓ 8 1.a even 1 1 trivial
6.25.b.a ✓ 8 3.b odd 2 1 inner
48.25.e.d 8 4.b odd 2 1
48.25.e.d 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{25}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8388608)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 63\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots - 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 73\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 54\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 78\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 78\!\cdots\!44)^{2} \) Copy content Toggle raw display
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