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Properties

Label 1008.5.f.m
Level $1008$
Weight $5$
Character orbit 1008.f
Analytic conductor $104.197$
Analytic rank $0$
Dimension $16$
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] = [1008,5,Mod(433,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.433");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.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: \(104.196922789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 782 x^{14} + 236807 x^{12} + 33890658 x^{10} + 2190442889 x^{8} + 46986787344 x^{6} + \cdots + 4066062536704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 504)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{8} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{8} + 3) q^{7} + \beta_{11} q^{11} + \beta_{9} q^{13} + \beta_{3} q^{17} + ( - \beta_{14} - \beta_{9} - \beta_{8} + \cdots - 1) q^{19}+ \cdots + (20 \beta_{14} + 10 \beta_{12} + \cdots - 26) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 56 q^{7} - 3648 q^{25} + 496 q^{37} - 3440 q^{43} + 6080 q^{49} - 13328 q^{67} + 24080 q^{79} - 1232 q^{85} + 1248 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 782 x^{14} + 236807 x^{12} + 33890658 x^{10} + 2190442889 x^{8} + 46986787344 x^{6} + \cdots + 4066062536704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 18\!\cdots\!65 \nu^{14} + \cdots + 81\!\cdots\!88 ) / 39\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 217523149137419 \nu^{14} + \cdots + 71\!\cdots\!96 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!49 \nu^{14} + \cdots + 36\!\cdots\!96 ) / 82\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!27 \nu^{14} + \cdots + 97\!\cdots\!20 ) / 55\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!69 \nu^{14} + \cdots - 75\!\cdots\!60 ) / 19\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44\!\cdots\!48 \nu^{15} + \cdots - 86\!\cdots\!72 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 52\!\cdots\!51 \nu^{15} + \cdots + 80\!\cdots\!40 ) / 40\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52\!\cdots\!51 \nu^{15} + \cdots + 76\!\cdots\!04 ) / 40\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40\!\cdots\!45 \nu^{15} + \cdots + 45\!\cdots\!60 \nu ) / 29\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!93 \nu^{15} + \cdots - 54\!\cdots\!00 \nu ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22\!\cdots\!97 \nu^{15} + \cdots + 42\!\cdots\!16 \nu ) / 56\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 97\!\cdots\!39 \nu^{15} + \cdots - 99\!\cdots\!40 \nu ) / 20\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!67 \nu^{15} + \cdots - 34\!\cdots\!96 \nu ) / 18\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 57\!\cdots\!67 \nu^{15} + \cdots - 41\!\cdots\!36 \nu ) / 51\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65\!\cdots\!91 \nu^{15} + \cdots - 12\!\cdots\!24 \nu ) / 56\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{14} - \beta_{12} - 3\beta_{10} - 9\beta_{8} + 9\beta_{7} - 9 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 108\beta_{8} + 108\beta_{7} - \beta_{5} + 3\beta_{4} - 8\beta_{3} - 34\beta _1 - 9384 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 304 \beta_{14} - 90 \beta_{13} + 208 \beta_{12} + 54 \beta_{11} + 831 \beta_{10} - 208 \beta_{9} + \cdots + 1824 ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 104 \beta_{10} - 20820 \beta_{8} - 20820 \beta_{7} - 208 \beta_{6} + 464 \beta_{5} - 597 \beta_{4} + \cdots + 1654920 ) / 96 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3360 \beta_{15} - 55200 \beta_{14} + 29790 \beta_{13} - 32675 \beta_{12} - 19890 \beta_{11} + \cdots - 331611 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21629 \beta_{10} + 1832988 \beta_{8} + 1832988 \beta_{7} + 43258 \beta_{6} - 71001 \beta_{5} + \cdots - 146007120 ) / 48 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1651104 \beta_{15} + 9761232 \beta_{14} - 7920192 \beta_{13} + 4701773 \beta_{12} + 7217952 \beta_{11} + \cdots + 58004133 ) / 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4616808 \beta_{10} - 205141012 \beta_{8} - 205141012 \beta_{7} - 9233616 \beta_{6} + \cdots + 16449057080 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 572257728 \beta_{15} - 1633760720 \beta_{14} + 1924160526 \beta_{13} - 616395586 \beta_{12} + \cdots - 9605601666 ) / 96 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3901887984 \beta_{10} + 96523415196 \beta_{8} + 96523415196 \beta_{7} + 7803775968 \beta_{6} + \cdots - 7788890685624 ) / 96 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 167622270816 \beta_{15} + 250183218112 \beta_{14} - 442784522070 \beta_{13} + 68803185991 \beta_{12} + \cdots + 1455703674495 ) / 96 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 252135724061 \beta_{10} - 3349079500104 \beta_{8} - 3349079500104 \beta_{7} - 504271448122 \beta_{6} + \cdots + 272091060616284 ) / 24 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 44176385259552 \beta_{15} - 32147528969040 \beta_{14} + 97966301134884 \beta_{13} + \cdots - 184788033633051 ) / 96 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 244222060043374 \beta_{10} + \cdots - 11\!\cdots\!88 ) / 96 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 10\!\cdots\!00 \beta_{15} + \cdots + 13\!\cdots\!32 ) / 96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−1.41421 − 12.9601i
1.41421 + 12.9601i
−1.41421 + 4.19551i
1.41421 − 4.19551i
1.41421 + 0.905600i
−1.41421 − 0.905600i
1.41421 − 14.5813i
−1.41421 + 14.5813i
−1.41421 − 14.5813i
1.41421 + 14.5813i
−1.41421 + 0.905600i
1.41421 − 0.905600i
1.41421 + 4.19551i
−1.41421 − 4.19551i
1.41421 − 12.9601i
−1.41421 + 12.9601i
0 0 0 − 46.5942i 0 40.0649 − 28.2100i 0 0 0
433.2 0 0 0 − 46.5942i 0 40.0649 + 28.2100i 0 0 0
433.3 0 0 0 − 32.8383i 0 −48.8714 − 3.54787i 0 0 0
433.4 0 0 0 − 32.8383i 0 −48.8714 + 3.54787i 0 0 0
433.5 0 0 0 − 12.4484i 0 −14.1733 − 46.9054i 0 0 0
433.6 0 0 0 − 12.4484i 0 −14.1733 + 46.9054i 0 0 0
433.7 0 0 0 − 2.76873i 0 36.9798 − 32.1480i 0 0 0
433.8 0 0 0 − 2.76873i 0 36.9798 + 32.1480i 0 0 0
433.9 0 0 0 2.76873i 0 36.9798 − 32.1480i 0 0 0
433.10 0 0 0 2.76873i 0 36.9798 + 32.1480i 0 0 0
433.11 0 0 0 12.4484i 0 −14.1733 − 46.9054i 0 0 0
433.12 0 0 0 12.4484i 0 −14.1733 + 46.9054i 0 0 0
433.13 0 0 0 32.8383i 0 −48.8714 − 3.54787i 0 0 0
433.14 0 0 0 32.8383i 0 −48.8714 + 3.54787i 0 0 0
433.15 0 0 0 46.5942i 0 40.0649 − 28.2100i 0 0 0
433.16 0 0 0 46.5942i 0 40.0649 + 28.2100i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.5.f.m 16
3.b odd 2 1 inner 1008.5.f.m 16
4.b odd 2 1 504.5.f.c ✓ 16
7.b odd 2 1 inner 1008.5.f.m 16
12.b even 2 1 504.5.f.c ✓ 16
21.c even 2 1 inner 1008.5.f.m 16
28.d even 2 1 504.5.f.c ✓ 16
84.h odd 2 1 504.5.f.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.5.f.c ✓ 16 4.b odd 2 1
504.5.f.c ✓ 16 12.b even 2 1
504.5.f.c ✓ 16 28.d even 2 1
504.5.f.c ✓ 16 84.h odd 2 1
1008.5.f.m 16 1.a even 1 1 trivial
1008.5.f.m 16 3.b odd 2 1 inner
1008.5.f.m 16 7.b odd 2 1 inner
1008.5.f.m 16 21.c 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 \(S_{5}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{8} + 3412T_{5}^{6} + 2870756T_{5}^{4} + 384594944T_{5}^{2} + 2781085696 \) Copy content Toggle raw display
\( T_{11}^{8} - 55092T_{11}^{6} + 1002203492T_{11}^{4} - 6577525400448T_{11}^{2} + 11016910604406784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 3412 T^{6} + \cdots + 2781085696)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 33232930569601)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 34\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 92\!\cdots\!96)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 3937379860480)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 860 T^{3} + \cdots + 514650919744)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 80\!\cdots\!84)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 20284999338176)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 19\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 12197756851456)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 38\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 54\!\cdots\!36)^{2} \) Copy content Toggle raw display
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