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Properties

Label 1104.5.c.c
Level $1104$
Weight $5$
Character orbit 1104.c
Analytic conductor $114.120$
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] = [1104,5,Mod(1057,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, 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("1104.1057");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1104.c (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: \(114.120439245\)
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: \( x^{16} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 69)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{10} q^{5} + \beta_{11} q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{10} q^{5} + \beta_{11} q^{7} + 27 q^{9} + (\beta_{12} - \beta_{10} - \beta_{8}) q^{11} + (\beta_{7} - 2 \beta_{6} + \beta_{4} + \cdots + 6) q^{13}+ \cdots + (27 \beta_{12} - 27 \beta_{10} - 27 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 432 q^{9} + 104 q^{13} + 732 q^{23} - 2984 q^{25} - 3528 q^{29} + 400 q^{31} - 912 q^{35} - 2016 q^{39} + 1008 q^{41} + 8664 q^{47} + 7240 q^{49} - 6816 q^{55} - 20112 q^{59} - 10044 q^{69} - 40368 q^{71} - 9568 q^{73} - 7560 q^{75} + 2952 q^{77} + 11664 q^{81} + 42744 q^{85} - 8352 q^{87} - 10008 q^{93} - 33312 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + \cdots + 13\!\cdots\!52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15\!\cdots\!27 \nu^{14} + \cdots - 20\!\cdots\!40 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38\!\cdots\!97 \nu^{14} + \cdots + 35\!\cdots\!30 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 74\!\cdots\!21 \nu^{14} + \cdots + 54\!\cdots\!44 ) / 70\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!96 \nu^{14} + \cdots - 70\!\cdots\!70 ) / 71\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!94 \nu^{14} + \cdots + 20\!\cdots\!82 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!13 \nu^{14} + \cdots + 25\!\cdots\!98 ) / 70\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!65 \nu^{14} + \cdots - 62\!\cdots\!92 ) / 79\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!69 \nu^{15} + \cdots - 23\!\cdots\!72 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!21 \nu^{15} + \cdots - 23\!\cdots\!72 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 68\!\cdots\!23 \nu^{15} + \cdots + 10\!\cdots\!22 \nu ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 69\!\cdots\!26 \nu^{15} + \cdots + 22\!\cdots\!50 \nu ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 87\!\cdots\!72 \nu^{15} + \cdots - 23\!\cdots\!72 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!00 \nu^{15} + \cdots + 23\!\cdots\!72 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 85\!\cdots\!55 \nu^{15} + \cdots - 77\!\cdots\!24 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51\!\cdots\!04 \nu^{15} + \cdots + 57\!\cdots\!46 \nu ) / 72\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - 8\beta_{11} + 7\beta_{10} - 2\beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} - \beta_{2} - 1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 21\beta_{7} + 4\beta_{6} - 45\beta_{5} + 37\beta_{4} + 96\beta_{3} + 47\beta_{2} + 27\beta _1 - 4130 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 426 \beta_{15} - 1024 \beta_{14} - 3632 \beta_{13} - 582 \beta_{12} + 13185 \beta_{11} - 102 \beta_{10} + \cdots + 769 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20567 \beta_{7} - 18110 \beta_{6} + 39345 \beta_{5} - 36240 \beta_{4} - 68518 \beta_{3} + \cdots + 2104766 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 141975 \beta_{15} + 1370418 \beta_{14} + 3389218 \beta_{13} + 455706 \beta_{12} - 9569255 \beta_{11} + \cdots - 315211 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 54577108 \beta_{7} + 61112447 \beta_{6} - 98514942 \beta_{5} + 90295833 \beta_{4} + 195525424 \beta_{3} + \cdots - 4460095310 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8017965 \beta_{15} - 4002931408 \beta_{14} - 9004377080 \beta_{13} - 952949592 \beta_{12} + \cdots + 708788977 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 141281647023 \beta_{7} - 171819542523 \beta_{6} + 248827903245 \beta_{5} - 226826769192 \beta_{4} + \cdots + 10688344374255 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 133141138584 \beta_{15} + 3574305718014 \beta_{14} + 7799201267100 \beta_{13} + 714774643596 \beta_{12} + \cdots - 623229508037 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 362655908564613 \beta_{7} + 455736614449971 \beta_{6} - 632394156823797 \beta_{5} + \cdots - 26\!\cdots\!06 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 492599294360709 \beta_{15} + \cdots + 16\!\cdots\!30 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 92\!\cdots\!84 \beta_{7} + \cdots + 67\!\cdots\!13 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14\!\cdots\!93 \beta_{15} + \cdots - 42\!\cdots\!36 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 23\!\cdots\!47 \beta_{7} + \cdots - 17\!\cdots\!38 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 37\!\cdots\!60 \beta_{15} + \cdots + 10\!\cdots\!03 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(277\) \(415\) \(737\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
23.3661i
32.2151i
25.3025i
19.7839i
− 19.7839i
− 25.3025i
− 32.2151i
− 23.3661i
19.1905i
1.32778i
− 50.5339i
− 7.67355i
7.67355i
50.5339i
− 1.32778i
− 19.1905i
0 −5.19615 0 − 40.4877i 0 40.5328i 0 27.0000 0
1057.2 0 −5.19615 0 − 30.6703i 0 − 37.9803i 0 27.0000 0
1057.3 0 −5.19615 0 − 12.7157i 0 − 72.2771i 0 27.0000 0
1057.4 0 −5.19615 0 − 11.8600i 0 − 49.3562i 0 27.0000 0
1057.5 0 −5.19615 0 11.8600i 0 49.3562i 0 27.0000 0
1057.6 0 −5.19615 0 12.7157i 0 72.2771i 0 27.0000 0
1057.7 0 −5.19615 0 30.6703i 0 37.9803i 0 27.0000 0
1057.8 0 −5.19615 0 40.4877i 0 − 40.5328i 0 27.0000 0
1057.9 0 5.19615 0 − 49.1084i 0 − 11.1740i 0 27.0000 0
1057.10 0 5.19615 0 − 30.5542i 0 6.50343i 0 27.0000 0
1057.11 0 5.19615 0 − 11.5477i 0 67.1686i 0 27.0000 0
1057.12 0 5.19615 0 − 11.4538i 0 12.7987i 0 27.0000 0
1057.13 0 5.19615 0 11.4538i 0 − 12.7987i 0 27.0000 0
1057.14 0 5.19615 0 11.5477i 0 − 67.1686i 0 27.0000 0
1057.15 0 5.19615 0 30.5542i 0 − 6.50343i 0 27.0000 0
1057.16 0 5.19615 0 49.1084i 0 11.1740i 0 27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.5.c.c 16
4.b odd 2 1 69.5.d.a ✓ 16
12.b even 2 1 207.5.d.c 16
23.b odd 2 1 inner 1104.5.c.c 16
92.b even 2 1 69.5.d.a ✓ 16
276.h odd 2 1 207.5.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.5.d.a ✓ 16 4.b odd 2 1
69.5.d.a ✓ 16 92.b even 2 1
207.5.d.c 16 12.b even 2 1
207.5.d.c 16 276.h odd 2 1
1104.5.c.c 16 1.a even 1 1 trivial
1104.5.c.c 16 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 6492 T_{5}^{14} + 15902820 T_{5}^{12} + 18733188384 T_{5}^{10} + 11249509308948 T_{5}^{8} + \cdots + 13\!\cdots\!68 \) acting on \(S_{5}^{\mathrm{new}}(1104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 4394524804672)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 10\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 58\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 66\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 77\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 70\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 75\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 53\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 17\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 37\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
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