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Properties

Label 448.6.b.b
Level $448$
Weight $6$
Character orbit 448.b
Analytic conductor $71.852$
Analytic rank $0$
Dimension $10$
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] = [448,6,Mod(225,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.225");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.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: \(71.8519512762\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 2 x^{8} - 168 x^{7} + 104662 x^{6} - 325916 x^{5} + 456620 x^{4} + 25173960 x^{3} + \cdots + 235228050 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{25} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{7} q^{5} + 49 q^{7} + (\beta_{2} - 91) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{7} q^{5} + 49 q^{7} + (\beta_{2} - 91) q^{9} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{11}+ \cdots + ( - 134 \beta_{9} - 45 \beta_{8} + \cdots - 769 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 490 q^{7} - 906 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 490 q^{7} - 906 q^{9} + 656 q^{15} - 1996 q^{17} + 448 q^{23} - 4430 q^{25} - 10896 q^{31} + 5384 q^{33} - 26576 q^{39} + 868 q^{41} - 47536 q^{47} + 24010 q^{49} - 42544 q^{55} + 15608 q^{57} - 44394 q^{63} + 62144 q^{65} - 176272 q^{71} - 64476 q^{73} - 223376 q^{79} - 172206 q^{81} - 552992 q^{87} + 107140 q^{89} - 653200 q^{95} - 372364 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 2 x^{8} - 168 x^{7} + 104662 x^{6} - 325916 x^{5} + 456620 x^{4} + 25173960 x^{3} + \cdots + 235228050 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16795145490112 \nu^{9} + 50053085465291 \nu^{8} + \cdots + 12\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5487598161689 \nu^{9} + 456231627466432 \nu^{8} - 222139316797720 \nu^{7} + \cdots + 10\!\cdots\!40 ) / 32\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33590290980224 \nu^{9} + 100106170930582 \nu^{8} + \cdots + 98\!\cdots\!55 ) / 48\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!08 \nu^{9} + \cdots - 11\!\cdots\!20 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59\!\cdots\!64 \nu^{9} + \cdots + 36\!\cdots\!80 ) / 40\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 75\!\cdots\!92 \nu^{9} + \cdots + 27\!\cdots\!60 ) / 35\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!48 \nu^{9} + \cdots + 53\!\cdots\!80 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!70 \nu^{9} + \cdots + 86\!\cdots\!30 ) / 19\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38\!\cdots\!60 \nu^{9} + \cdots + 14\!\cdots\!40 ) / 23\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - 8\beta _1 + 3 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{7} - 83\beta_{6} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{9} - 2 \beta_{8} + 90 \beta_{7} + 126 \beta_{6} - \beta_{5} - 48 \beta_{4} - 109 \beta_{3} + \cdots + 407 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -49\beta_{5} + 105\beta_{4} + 105\beta_{3} + 664\beta_{2} - 167189 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5664 \beta_{9} + 2056 \beta_{8} - 73368 \beta_{7} - 165984 \beta_{6} - 1028 \beta_{5} - 39516 \beta_{4} + \cdots + 934939 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -104083\beta_{9} + 40796\beta_{8} - 8239\beta_{7} + 5790089\beta_{6} - 740194\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1075074 \beta_{9} + 342846 \beta_{8} - 12098982 \beta_{7} - 37624902 \beta_{6} + 171423 \beta_{5} + \cdots - 240330561 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6735183\beta_{5} - 19116279\beta_{4} - 28258461\beta_{3} - 64441960\beta_{2} + 13598707657 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 765649680 \beta_{9} - 191479544 \beta_{8} + 7522155576 \beta_{7} + 29505120624 \beta_{6} + \cdots - 199254080321 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
−12.6670 − 12.6670i
11.7650 − 11.7650i
8.43461 − 8.43461i
−6.84758 − 6.84758i
0.314993 − 0.314993i
0.314993 + 0.314993i
−6.84758 + 6.84758i
8.43461 + 8.43461i
11.7650 + 11.7650i
−12.6670 + 12.6670i
0 − 25.3340i 0 35.3154i 0 49.0000 0 −398.814 0
225.2 0 − 23.5300i 0 65.6520i 0 49.0000 0 −310.661 0
225.3 0 − 16.8692i 0 − 55.3642i 0 49.0000 0 −41.5709 0
225.4 0 − 13.6952i 0 − 83.8281i 0 49.0000 0 55.4425 0
225.5 0 − 0.629986i 0 − 46.8005i 0 49.0000 0 242.603 0
225.6 0 0.629986i 0 46.8005i 0 49.0000 0 242.603 0
225.7 0 13.6952i 0 83.8281i 0 49.0000 0 55.4425 0
225.8 0 16.8692i 0 55.3642i 0 49.0000 0 −41.5709 0
225.9 0 23.5300i 0 − 65.6520i 0 49.0000 0 −310.661 0
225.10 0 25.3340i 0 − 35.3154i 0 49.0000 0 −398.814 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.6.b.b yes 10
4.b odd 2 1 448.6.b.a ✓ 10
8.b even 2 1 inner 448.6.b.b yes 10
8.d odd 2 1 448.6.b.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.6.b.a ✓ 10 4.b odd 2 1
448.6.b.a ✓ 10 8.d odd 2 1
448.6.b.b yes 10 1.a even 1 1 trivial
448.6.b.b yes 10 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{10} + 1668T_{3}^{8} + 973800T_{3}^{6} + 231962272T_{3}^{4} + 19057996944T_{3}^{2} + 7527297600 \) Copy content Toggle raw display
\( T_{23}^{5} - 224T_{23}^{4} - 10015320T_{23}^{3} - 10462919040T_{23}^{2} + 3191279106960T_{23} + 2464801784524800 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 7527297600 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( (T - 49)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 85362245858016)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 35\!\cdots\!60)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 80\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 31\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 28\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 39\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 40\!\cdots\!40)^{2} \) Copy content Toggle raw display
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