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Properties

Label 448.6.f.c
Level $448$
Weight $6$
Character orbit 448.f
Analytic conductor $71.852$
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] = [448,6,Mod(447,448)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.447");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.f (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: \(71.8519512762\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 100x^{10} + 7635x^{8} - 236404x^{6} + 5588425x^{4} - 113520x^{2} + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 112)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{6} q^{5} + (\beta_{8} + 2 \beta_{4} + \beta_{3}) q^{7} + ( - \beta_1 + 36) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{6} q^{5} + (\beta_{8} + 2 \beta_{4} + \beta_{3}) q^{7} + ( - \beta_1 + 36) q^{9} + ( - \beta_{5} - 7 \beta_{3}) q^{11} + ( - \beta_{11} + \beta_{6}) q^{13} + (\beta_{8} + \beta_{7} + \cdots + 10 \beta_{3}) q^{15}+ \cdots + ( - 90 \beta_{8} + \cdots - 6913 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 428 q^{9} + 6608 q^{21} - 3636 q^{25} + 14136 q^{29} + 21528 q^{37} + 5964 q^{49} + 49656 q^{53} + 90512 q^{57} + 30768 q^{65} + 94248 q^{77} - 67300 q^{81} - 272256 q^{85} - 437824 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 100x^{10} + 7635x^{8} - 236404x^{6} + 5588425x^{4} - 113520x^{2} + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16720 \nu^{10} - 1276572 \nu^{8} + 74716260 \nu^{6} - 934384660 \nu^{4} + 18980544 \nu^{2} + 173202679125 ) / 3981252135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61520 \nu^{10} - 4697052 \nu^{8} + 335869908 \nu^{6} - 3437999060 \nu^{4} + \cdots + 6539323166229 ) / 3981252135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 240757 \nu^{10} + 24069300 \nu^{8} - 1837691055 \nu^{6} + 56878610164 \nu^{4} + \cdots + 13673462040 ) / 1706250915 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69768689 \nu^{11} + 6978111124 \nu^{9} - 532809117667 \nu^{7} + 16503397096708 \nu^{5} + \cdots + 15846199997664 \nu ) / 127400068320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 147035881 \nu^{10} - 14712628884 \nu^{8} + 1123036215147 \nu^{6} - 34781354490772 \nu^{4} + \cdots - 8347425828984 ) / 47775025620 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 588499505 \nu^{11} + 58837882164 \nu^{9} - 4491999303075 \nu^{7} + 139032443196260 \nu^{5} + \cdots + 57332044800 \nu ) / 382200204960 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1618839827 \nu^{11} + 424516148 \nu^{10} + 161918920508 \nu^{9} - 42444563280 \nu^{8} + \cdots - 24104455322976 ) / 382200204960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1618839827 \nu^{11} + 424516148 \nu^{10} - 161918920508 \nu^{9} - 42444563280 \nu^{8} + \cdots - 24104455322976 ) / 382200204960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1873357651 \nu^{11} - 187296692892 \nu^{9} + 14299283501865 \nu^{7} - 442578946942252 \nu^{5} + \cdots - 182503848960 \nu ) / 127400068320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 66398091 \nu^{11} - 6641140924 \nu^{9} + 507081442897 \nu^{7} - 15707096554412 \nu^{5} + \cdots - 15081471221664 \nu ) / 3981252135 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 88252340 \nu^{11} + 8823483944 \nu^{9} - 673627499100 \nu^{7} + 20849530617680 \nu^{5} + \cdots + 8597606400 \nu ) / 3981252135 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} + 9\beta_{6} + 32\beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{8} - 12\beta_{7} - 189\beta_{3} - \beta_{2} + \beta _1 + 1600 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{11} + 35\beta_{9} + 507\beta_{6} ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -744\beta_{8} - 744\beta_{7} + 84\beta_{5} - 9879\beta_{3} + 55\beta_{2} - 139\beta _1 - 84292 ) / 96 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1248 \beta_{11} - 3523 \beta_{10} + 1195 \beta_{9} + 3744 \beta_{8} - 3744 \beta_{7} + \cdots - 49376 \beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1045\beta_{2} - 3845\beta _1 - 1549168 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 96420 \beta_{11} - 212813 \beta_{10} - 36773 \beta_{9} + 289260 \beta_{8} - 289260 \beta_{7} + \cdots - 2006536 \beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2843016 \beta_{8} + 2843016 \beta_{7} - 641340 \beta_{5} + 30765537 \beta_{3} + 183473 \beta_{2} + \cdots - 265476620 ) / 96 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6691056\beta_{11} - 852805\beta_{9} - 104470701\beta_{6} ) / 96 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 175500012 \beta_{8} + 175500012 \beta_{7} - 44272032 \beta_{5} + 1797082797 \beta_{3} + \cdots + 15560361056 ) / 96 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 441132204 \beta_{11} + 792090059 \beta_{10} + 1630285 \beta_{9} - 1323396612 \beta_{8} + \cdots + 3650800408 \beta_{4} ) / 192 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
447.1
−5.36148 + 3.09545i
−5.36148 − 3.09545i
−0.123430 − 0.0712626i
−0.123430 + 0.0712626i
−6.79995 + 3.92595i
−6.79995 − 3.92595i
6.79995 + 3.92595i
6.79995 − 3.92595i
0.123430 − 0.0712626i
0.123430 + 0.0712626i
5.36148 + 3.09545i
5.36148 − 3.09545i
0 −24.4951 0 − 31.8640i 0 −124.098 + 37.5068i 0 357.011 0
447.2 0 −24.4951 0 31.8640i 0 −124.098 − 37.5068i 0 357.011 0
447.3 0 −15.3459 0 − 24.8695i 0 88.0080 + 95.1924i 0 −7.50467 0
447.4 0 −15.3459 0 24.8695i 0 88.0080 − 95.1924i 0 −7.50467 0
447.5 0 −0.702317 0 − 93.0064i 0 53.0128 + 118.307i 0 −242.507 0
447.6 0 −0.702317 0 93.0064i 0 53.0128 − 118.307i 0 −242.507 0
447.7 0 0.702317 0 − 93.0064i 0 −53.0128 − 118.307i 0 −242.507 0
447.8 0 0.702317 0 93.0064i 0 −53.0128 + 118.307i 0 −242.507 0
447.9 0 15.3459 0 − 24.8695i 0 −88.0080 − 95.1924i 0 −7.50467 0
447.10 0 15.3459 0 24.8695i 0 −88.0080 + 95.1924i 0 −7.50467 0
447.11 0 24.4951 0 − 31.8640i 0 124.098 − 37.5068i 0 357.011 0
447.12 0 24.4951 0 31.8640i 0 124.098 + 37.5068i 0 357.011 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 447.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.f.c 12
4.b odd 2 1 inner 448.6.f.c 12
7.b odd 2 1 inner 448.6.f.c 12
8.b even 2 1 112.6.f.b ✓ 12
8.d odd 2 1 112.6.f.b ✓ 12
24.f even 2 1 1008.6.b.h 12
24.h odd 2 1 1008.6.b.h 12
28.d even 2 1 inner 448.6.f.c 12
56.e even 2 1 112.6.f.b ✓ 12
56.h odd 2 1 112.6.f.b ✓ 12
168.e odd 2 1 1008.6.b.h 12
168.i even 2 1 1008.6.b.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.6.f.b ✓ 12 8.b even 2 1
112.6.f.b ✓ 12 8.d odd 2 1
112.6.f.b ✓ 12 56.e even 2 1
112.6.f.b ✓ 12 56.h odd 2 1
448.6.f.c 12 1.a even 1 1 trivial
448.6.f.c 12 4.b odd 2 1 inner
448.6.f.c 12 7.b odd 2 1 inner
448.6.f.c 12 28.d even 2 1 inner
1008.6.b.h 12 24.f even 2 1
1008.6.b.h 12 24.h odd 2 1
1008.6.b.h 12 168.e odd 2 1
1008.6.b.h 12 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 836T_{3}^{4} + 141712T_{3}^{2} - 69696 \) acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 836 T^{4} + \cdots - 69696)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 10284 T^{4} + \cdots + 5432018112)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 689944777622208)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 847854984892608)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 41\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 46\!\cdots\!12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 3534 T^{2} + \cdots + 110122734168)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 94\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 5382 T^{2} + \cdots + 344383920376)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 58\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 99\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 12414 T^{2} + \cdots - 27889524072)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 81\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 49\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 58\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 88\!\cdots\!48)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 55\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 35\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 13\!\cdots\!28)^{2} \) Copy content Toggle raw display
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