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Properties

Label 1104.6.a.t
Level $1104$
Weight $6$
Character orbit 1104.a
Self dual yes
Analytic conductor $177.064$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1104,6,Mod(1,1104)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

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: yes
Analytic conductor: \(177.063737074\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 1267x^{4} - 6846x^{3} + 387078x^{2} + 4232368x + 11363808 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + (\beta_{5} - 15) q^{5} + ( - \beta_{3} + \beta_1 - 8) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + (\beta_{5} - 15) q^{5} + ( - \beta_{3} + \beta_1 - 8) q^{7} + 81 q^{9} + (\beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots - 105) q^{11}+ \cdots + (81 \beta_{4} - 81 \beta_{3} + \cdots - 8505) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} - 90 q^{5} - 50 q^{7} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 54 q^{3} - 90 q^{5} - 50 q^{7} + 486 q^{9} - 632 q^{11} - 196 q^{13} - 810 q^{15} + 466 q^{17} - 518 q^{19} - 450 q^{21} + 3174 q^{23} + 186 q^{25} + 4374 q^{27} - 2016 q^{29} + 9540 q^{31} - 5688 q^{33} + 7028 q^{35} - 2664 q^{37} - 1764 q^{39} - 23196 q^{41} + 10034 q^{43} - 7290 q^{45} + 8036 q^{47} - 24386 q^{49} + 4194 q^{51} - 18198 q^{53} - 12088 q^{55} - 4662 q^{57} - 9140 q^{59} - 43992 q^{61} - 4050 q^{63} + 9764 q^{65} - 31418 q^{67} + 28566 q^{69} - 79296 q^{71} + 7708 q^{73} + 1674 q^{75} + 47928 q^{77} - 120438 q^{79} + 39366 q^{81} - 81288 q^{83} + 73248 q^{85} - 18144 q^{87} + 77750 q^{89} + 29716 q^{91} + 85860 q^{93} - 469252 q^{95} - 83192 q^{97} - 51192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 1267x^{4} - 6846x^{3} + 387078x^{2} + 4232368x + 11363808 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13315\nu^{5} + 266159\nu^{4} + 10528589\nu^{3} - 120073610\nu^{2} - 2126496346\nu - 3913524120 ) / 40141872 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16249\nu^{5} - 14285\nu^{4} - 22908935\nu^{3} - 53664658\nu^{2} + 7724908942\nu + 41607337224 ) / 40141872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3295\nu^{5} + 18131\nu^{4} + 3957089\nu^{3} + 5282590\nu^{2} - 1212497194\nu - 7640621760 ) / 6690312 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4037\nu^{5} - 91249\nu^{4} - 3175099\nu^{3} + 41324470\nu^{2} + 694449758\nu + 2011004736 ) / 6690312 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 5\beta_{4} + 4\beta_{3} - 8\beta_{2} + 6\beta _1 + 1692 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -65\beta_{5} - 29\beta_{4} - 40\beta_{3} - 124\beta_{2} + 655\beta _1 + 16445 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4040\beta_{5} + 2284\beta_{4} + 1800\beta_{3} - 8544\beta_{2} + 8569\beta _1 + 1132669 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -105101\beta_{5} - 22365\beta_{4} - 31720\beta_{3} - 208756\beta_{2} + 475404\beta _1 + 19051262 ) / 4 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−21.6157
−4.92062
−6.71632
31.0252
24.3146
−21.0872
0 9.00000 0 −103.992 0 −123.613 0 81.0000 0
1.2 0 9.00000 0 −28.8121 0 −152.512 0 81.0000 0
1.3 0 9.00000 0 −25.1547 0 113.776 0 81.0000 0
1.4 0 9.00000 0 −12.6999 0 131.872 0 81.0000 0
1.5 0 9.00000 0 0.0507387 0 51.0255 0 81.0000 0
1.6 0 9.00000 0 80.6075 0 −70.5485 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.6.a.t 6
4.b odd 2 1 552.6.a.c ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.6.a.c ✓ 6 4.b odd 2 1
1104.6.a.t 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 90T_{5}^{5} - 5418T_{5}^{4} - 516380T_{5}^{3} - 11578992T_{5}^{2} - 76566960T_{5} + 3914784 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1104))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 90 T^{5} + \cdots + 3914784 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 1018230787584 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 758010921984000 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 85\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 37\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 13\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 36\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 34\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 49\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 32\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 17\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 54\!\cdots\!52 \) Copy content Toggle raw display
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