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Properties

Label 1008.5.f.n
Level $1008$
Weight $5$
Character orbit 1008.f
Analytic conductor $104.197$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} - 4 x^{15} + 618 x^{14} - 752 x^{13} + 245625 x^{12} - 165924 x^{11} + 55954930 x^{10} + \cdots + 64\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + ( - \beta_{4} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + ( - \beta_{4} + 3) q^{7} + ( - \beta_{9} + 18) q^{11} + (\beta_{6} - \beta_{4} - \beta_{3} + \cdots - 1) q^{13}+ \cdots + ( - 2 \beta_{10} - 30 \beta_{8} + \cdots - 16) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 56 q^{7} + 288 q^{11} - 2880 q^{23} - 3648 q^{25} + 384 q^{29} - 1680 q^{35} + 496 q^{37} + 6160 q^{43} - 7264 q^{49} + 10656 q^{53} - 9696 q^{65} - 13328 q^{67} - 2208 q^{71} + 9408 q^{77} + 12368 q^{79} + 47344 q^{85} - 33216 q^{91} + 35808 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 618 x^{14} - 752 x^{13} + 245625 x^{12} - 165924 x^{11} + 55954930 x^{10} + \cdots + 64\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\!\cdots\!00 \nu^{15} + \cdots + 65\!\cdots\!70 ) / 20\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97\!\cdots\!22 \nu^{15} + \cdots - 31\!\cdots\!80 ) / 12\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!30 \nu^{15} + \cdots + 18\!\cdots\!17 ) / 77\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!59 \nu^{15} + \cdots + 40\!\cdots\!03 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28\!\cdots\!63 \nu^{15} + \cdots - 18\!\cdots\!29 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!93 \nu^{15} + \cdots - 37\!\cdots\!54 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 61\!\cdots\!17 \nu^{15} + \cdots - 50\!\cdots\!85 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!91 \nu^{15} + \cdots + 35\!\cdots\!87 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 48\!\cdots\!59 \nu^{15} + \cdots + 30\!\cdots\!57 ) / 21\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16\!\cdots\!39 \nu^{15} + \cdots - 99\!\cdots\!18 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60\!\cdots\!14 \nu^{15} + \cdots + 76\!\cdots\!93 ) / 19\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!09 \nu^{15} + \cdots + 55\!\cdots\!81 ) / 55\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 34\!\cdots\!27 \nu^{15} + \cdots + 96\!\cdots\!38 ) / 70\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!11 \nu^{15} + \cdots - 18\!\cdots\!91 ) / 54\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30\!\cdots\!08 \nu^{15} + \cdots + 10\!\cdots\!87 ) / 54\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 21 \beta_{15} - 21 \beta_{14} - 14 \beta_{12} + 26 \beta_{11} + 5 \beta_{10} - 21 \beta_{9} + \cdots + 251 ) / 1344 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 105 \beta_{15} + 105 \beta_{14} + 28 \beta_{13} + 14 \beta_{12} - 128 \beta_{11} + 103 \beta_{10} + \cdots - 102887 ) / 1344 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 707 \beta_{15} + 903 \beta_{14} + 35 \beta_{13} + 616 \beta_{12} - 218 \beta_{11} - 218 \beta_{10} + \cdots - 35634 ) / 112 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 44835 \beta_{15} + 47187 \beta_{14} + 19264 \beta_{13} + 16338 \beta_{12} + 64266 \beta_{11} + \cdots - 20662149 ) / 1344 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1163043 \beta_{15} - 1524075 \beta_{14} - 355194 \beta_{13} - 1022630 \beta_{12} + 342050 \beta_{11} + \cdots + 93192359 ) / 1344 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2704625 \beta_{15} - 2907681 \beta_{14} - 1373050 \beta_{13} - 1327410 \beta_{12} - 527291 \beta_{11} + \cdots + 822762499 ) / 112 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 50336589 \beta_{15} - 65240205 \beta_{14} - 24172700 \beta_{13} - 41862662 \beta_{12} + \cdots + 5116850927 ) / 192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5558124705 \beta_{15} + 6083582673 \beta_{14} + 3023497120 \beta_{13} + 3114823782 \beta_{12} + \cdots - 1307212371819 ) / 1344 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 18412340842 \beta_{15} + 23525423614 \beta_{14} + 10490433681 \beta_{13} + 14445151126 \beta_{12} + \cdots - 2175061732863 ) / 112 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1853082656313 \beta_{15} + 2064175611225 \beta_{14} + 1043844757088 \beta_{13} + \cdots - 371463126919447 ) / 1344 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 35135753580543 \beta_{15} - 44285363654343 \beta_{14} - 21436172342186 \beta_{13} + \cdots + 45\!\cdots\!71 ) / 1344 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 25369016472681 \beta_{15} - 28711279282385 \beta_{14} - 14605924732945 \beta_{13} + \cdots + 45\!\cdots\!90 ) / 28 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!17 \beta_{15} + \cdots + 15\!\cdots\!67 ) / 1344 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 28\!\cdots\!47 \beta_{15} + \cdots - 48\!\cdots\!89 ) / 192 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 60\!\cdots\!67 \beta_{15} + \cdots - 86\!\cdots\!62 ) / 112 \) 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
−6.98190 + 12.0930i
−3.34111 − 5.78697i
−4.98217 + 8.62938i
5.74386 + 9.94867i
3.57942 − 6.19974i
−7.32965 − 12.6953i
6.32948 + 10.9630i
8.98207 − 15.5574i
8.98207 + 15.5574i
6.32948 − 10.9630i
−7.32965 + 12.6953i
3.57942 + 6.19974i
5.74386 − 9.94867i
−4.98217 − 8.62938i
−3.34111 + 5.78697i
−6.98190 − 12.0930i
0 0 0 − 43.1110i 0 1.90684 − 48.9629i 0 0 0
433.2 0 0 0 − 41.9284i 0 −6.04414 − 48.6258i 0 0 0
433.3 0 0 0 − 29.6821i 0 −46.2092 + 16.3007i 0 0 0
433.4 0 0 0 − 29.0892i 0 21.4190 + 44.0707i 0 0 0
433.5 0 0 0 − 24.6093i 0 44.8343 + 19.7709i 0 0 0
433.6 0 0 0 − 24.2927i 0 −8.80201 + 48.2030i 0 0 0
433.7 0 0 0 − 16.8429i 0 48.1827 − 8.91215i 0 0 0
433.8 0 0 0 − 0.894313i 0 −27.2875 + 40.6988i 0 0 0
433.9 0 0 0 0.894313i 0 −27.2875 − 40.6988i 0 0 0
433.10 0 0 0 16.8429i 0 48.1827 + 8.91215i 0 0 0
433.11 0 0 0 24.2927i 0 −8.80201 − 48.2030i 0 0 0
433.12 0 0 0 24.6093i 0 44.8343 − 19.7709i 0 0 0
433.13 0 0 0 29.0892i 0 21.4190 − 44.0707i 0 0 0
433.14 0 0 0 29.6821i 0 −46.2092 − 16.3007i 0 0 0
433.15 0 0 0 41.9284i 0 −6.04414 + 48.6258i 0 0 0
433.16 0 0 0 43.1110i 0 1.90684 + 48.9629i 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
7.b odd 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.n 16
3.b odd 2 1 336.5.f.d 16
4.b odd 2 1 504.5.f.b 16
7.b odd 2 1 inner 1008.5.f.n 16
12.b even 2 1 168.5.f.a ✓ 16
21.c even 2 1 336.5.f.d 16
28.d even 2 1 504.5.f.b 16
84.h odd 2 1 168.5.f.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.5.f.a ✓ 16 12.b even 2 1
168.5.f.a ✓ 16 84.h odd 2 1
336.5.f.d 16 3.b odd 2 1
336.5.f.d 16 21.c even 2 1
504.5.f.b 16 4.b odd 2 1
504.5.f.b 16 28.d even 2 1
1008.5.f.n 16 1.a even 1 1 trivial
1008.5.f.n 16 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{16} + 6824 T_{5}^{14} + 18867224 T_{5}^{12} + 27356683424 T_{5}^{10} + 22484151249424 T_{5}^{8} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
\( T_{11}^{8} - 144 T_{11}^{7} - 62436 T_{11}^{6} + 7152528 T_{11}^{5} + 1225941908 T_{11}^{4} + \cdots + 16\!\cdots\!56 \) 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
$5$ \( T^{16} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 25\!\cdots\!48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 23\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 20\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 35\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 81\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
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