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Properties

Label 448.4.f.d
Level $448$
Weight $4$
Character orbit 448.f
Analytic conductor $26.433$
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] = [448,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.447");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(26.4328556826\)
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: \( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 493x^{4} - 464x^{3} - 3172x^{2} + 1072x + 8978 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{7} q^{5} + ( - \beta_{6} + \beta_{3}) q^{7} + (3 \beta_{4} + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{7} q^{5} + ( - \beta_{6} + \beta_{3}) q^{7} + (3 \beta_{4} + 13) q^{9} + (5 \beta_{5} + \beta_{3}) q^{11} - \beta_{2} q^{13} + (14 \beta_{5} - 3 \beta_{3}) q^{15} + (\beta_{7} - \beta_{2}) q^{17} + ( - 2 \beta_{6} + \beta_{5} - \beta_1) q^{19} + ( - 2 \beta_{7} + 17 \beta_{4} + \cdots + 8) q^{21}+ \cdots + ( - 19 \beta_{5} - 5 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 104 q^{9} + 64 q^{21} - 472 q^{25} + 592 q^{29} - 1392 q^{37} + 1480 q^{49} + 1168 q^{53} - 192 q^{57} + 448 q^{65} - 368 q^{77} - 4984 q^{81} - 1024 q^{85} + 2304 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 493x^{4} - 464x^{3} - 3172x^{2} + 1072x + 8978 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2107780 \nu^{7} + 28020178 \nu^{6} - 12013496 \nu^{5} - 856630098 \nu^{4} + \cdots - 26653239984 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16804 \nu^{7} + 349554 \nu^{6} - 183568 \nu^{5} - 9829550 \nu^{4} - 238208 \nu^{3} + \cdots - 699954360 ) / 7201361 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31087480 \nu^{7} + 94134528 \nu^{6} + 982404328 \nu^{5} - 1886510428 \nu^{4} + \cdots - 43521251908 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24200 \nu^{7} + 126280 \nu^{6} + 572168 \nu^{5} - 3253540 \nu^{4} - 5348520 \nu^{3} + \cdots - 120743112 ) / 7201361 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 71287048 \nu^{7} + 411792664 \nu^{6} + 1067149048 \nu^{5} - 6385173864 \nu^{4} + \cdots - 65434616690 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 85892144 \nu^{7} - 485124166 \nu^{6} - 2515029448 \nu^{5} + 14797753670 \nu^{4} + \cdots + 390648894103 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2876 \nu^{7} + 28790 \nu^{6} - 17704 \nu^{5} - 509450 \nu^{4} + 374248 \nu^{3} + 4834142 \nu^{2} + \cdots - 15131816 ) / 379019 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - 4\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + 6\beta_{4} - 2\beta_{3} + \beta_{2} - 6\beta _1 + 76 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} + 6\beta_{6} + 17\beta_{5} + 35\beta_{4} - 41\beta_{3} + 3\beta_{2} - 40\beta _1 + 172 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{7} + 18\beta_{6} + 68\beta_{5} + 47\beta_{4} - 86\beta_{3} + 15\beta_{2} - 62\beta _1 + 330 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -390\beta_{7} + 50\beta_{6} + 825\beta_{5} + 334\beta_{4} - 1168\beta_{3} + 140\beta_{2} - 212\beta _1 + 1764 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1968\beta_{7} - 68\beta_{6} + 4966\beta_{5} - 675\beta_{4} - 5783\beta_{3} + 788\beta_{2} + 280\beta _1 - 4172 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 20447 \beta_{7} - 6446 \beta_{6} + 48871 \beta_{5} - 22678 \beta_{4} - 56830 \beta_{3} + \cdots - 127112 ) / 16 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
447.1
4.98105 + 1.39897i
4.98105 − 1.39897i
2.19234 − 0.736813i
2.19234 + 0.736813i
−2.60656 + 0.736813i
−2.60656 − 0.736813i
−2.56684 − 1.39897i
−2.56684 + 1.39897i
0 −7.54788 0 − 8.74756i 0 −13.8008 − 12.3507i 0 29.9706 0
447.2 0 −7.54788 0 8.74756i 0 −13.8008 + 12.3507i 0 29.9706 0
447.3 0 −4.79890 0 − 17.0728i 0 18.3722 + 2.33686i 0 −3.97056 0
447.4 0 −4.79890 0 17.0728i 0 18.3722 − 2.33686i 0 −3.97056 0
447.5 0 4.79890 0 − 17.0728i 0 −18.3722 − 2.33686i 0 −3.97056 0
447.6 0 4.79890 0 17.0728i 0 −18.3722 + 2.33686i 0 −3.97056 0
447.7 0 7.54788 0 − 8.74756i 0 13.8008 + 12.3507i 0 29.9706 0
447.8 0 7.54788 0 8.74756i 0 13.8008 − 12.3507i 0 29.9706 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 447.8
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 448.4.f.d 8
4.b odd 2 1 inner 448.4.f.d 8
7.b odd 2 1 inner 448.4.f.d 8
8.b even 2 1 28.4.d.b ✓ 8
8.d odd 2 1 28.4.d.b ✓ 8
24.f even 2 1 252.4.b.d 8
24.h odd 2 1 252.4.b.d 8
28.d even 2 1 inner 448.4.f.d 8
56.e even 2 1 28.4.d.b ✓ 8
56.h odd 2 1 28.4.d.b ✓ 8
56.j odd 6 2 196.4.f.c 16
56.k odd 6 2 196.4.f.c 16
56.m even 6 2 196.4.f.c 16
56.p even 6 2 196.4.f.c 16
168.e odd 2 1 252.4.b.d 8
168.i even 2 1 252.4.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.b ✓ 8 8.b even 2 1
28.4.d.b ✓ 8 8.d odd 2 1
28.4.d.b ✓ 8 56.e even 2 1
28.4.d.b ✓ 8 56.h odd 2 1
196.4.f.c 16 56.j odd 6 2
196.4.f.c 16 56.k odd 6 2
196.4.f.c 16 56.m even 6 2
196.4.f.c 16 56.p even 6 2
252.4.b.d 8 24.f even 2 1
252.4.b.d 8 24.h odd 2 1
252.4.b.d 8 168.e odd 2 1
252.4.b.d 8 168.i even 2 1
448.4.f.d 8 1.a even 1 1 trivial
448.4.f.d 8 4.b odd 2 1 inner
448.4.f.d 8 7.b odd 2 1 inner
448.4.f.d 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 80T_{3}^{2} + 1312 \) acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 80 T^{2} + 1312)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 368 T^{2} + 22304)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{4} + 1720 T^{2} + 272)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 7792 T^{2} + 11798816)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 7936 T^{2} + 5709824)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2128 T^{2} + 694048)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 23800 T^{2} + 132140048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 148 T - 4892)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 23040 T^{2} + 108822528)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 348 T + 24004)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 58304 T^{2} + 342946304)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 34360 T^{2} + 141396752)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 103680 T^{2} + 1789861888)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 292 T - 163516)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 570320 T^{2} + 67995188512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 606832 T^{2} + 6445856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 343672 T^{2} + 12017669648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 275576 T^{2} + 9248152592)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 92864 T^{2} + 1798951424)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1466680 T^{2} + 108115639568)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1267920 T^{2} + 340971272992)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1846976 T^{2} + 661026681344)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 1135775350784)^{2} \) Copy content Toggle raw display
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