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Properties

Label 1100.4.b.h
Level $1100$
Weight $4$
Character orbit 1100.b
Analytic conductor $64.902$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1100,4,Mod(749,1100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1100.749"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1100.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-48,0,66,0,0,0,0,0,0,0,-342] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.9021010063\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1351885824.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 37x^{4} + 384x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 220)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{3} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{7} + (\beta_{5} - 2 \beta_{4} - 9) q^{9} + 11 q^{11} + ( - 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{13} + (5 \beta_{3} + 8 \beta_{2} + 9 \beta_1) q^{17}+ \cdots + (11 \beta_{5} - 22 \beta_{4} - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{9} + 66 q^{11} - 342 q^{19} + 518 q^{21} - 110 q^{29} + 362 q^{31} - 1108 q^{39} + 604 q^{41} + 536 q^{49} + 1666 q^{51} + 84 q^{59} + 698 q^{61} + 516 q^{69} + 1854 q^{71} + 2292 q^{79} - 3066 q^{81}+ \cdots - 528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 37x^{4} + 384x^{2} + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 7\nu^{3} - 186\nu ) / 180 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 27\nu^{3} + 174\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 27\nu^{3} + 134\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 22\nu^{2} + 72 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{3} - 8\beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{5} - 22\beta_{4} + 192 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -156\beta_{3} + 149\beta_{2} + 243\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
1.81626i
− 3.67648i
− 4.49274i
4.49274i
3.67648i
− 1.81626i
0 − 7.06868i 0 0 0 22.8386i 0 −22.9662 0
749.2 0 − 6.86946i 0 0 0 15.2554i 0 −20.1894 0
749.3 0 − 2.80078i 0 0 0 − 2.58315i 0 19.1557 0
749.4 0 2.80078i 0 0 0 2.58315i 0 19.1557 0
749.5 0 6.86946i 0 0 0 − 15.2554i 0 −20.1894 0
749.6 0 7.06868i 0 0 0 − 22.8386i 0 −22.9662 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.4.b.h 6
5.b even 2 1 inner 1100.4.b.h 6
5.c odd 4 1 220.4.a.f ✓ 3
5.c odd 4 1 1100.4.a.i 3
15.e even 4 1 1980.4.a.l 3
20.e even 4 1 880.4.a.w 3
55.e even 4 1 2420.4.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.4.a.f ✓ 3 5.c odd 4 1
880.4.a.w 3 20.e even 4 1
1100.4.a.i 3 5.c odd 4 1
1100.4.b.h 6 1.a even 1 1 trivial
1100.4.b.h 6 5.b even 2 1 inner
1980.4.a.l 3 15.e even 4 1
2420.4.a.i 3 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1100, [\chi])\):

\( T_{3}^{6} + 105T_{3}^{4} + 3120T_{3}^{2} + 18496 \) Copy content Toggle raw display
\( T_{7}^{6} + 761T_{7}^{4} + 126424T_{7}^{2} + 810000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 105 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 761 T^{4} + \cdots + 810000 \) Copy content Toggle raw display
$11$ \( (T - 11)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 4271929600 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 207476606016 \) Copy content Toggle raw display
$19$ \( (T^{3} + 171 T^{2} + \cdots - 46336)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 1099276759296 \) Copy content Toggle raw display
$29$ \( (T^{3} + 55 T^{2} + \cdots - 1502868)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 181 T^{2} + \cdots + 10494720)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{3} - 302 T^{2} + \cdots + 4589880)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 11776976697600 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{3} - 42 T^{2} + \cdots - 46277856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 349 T^{2} + \cdots + 44132796)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 939244983586816 \) Copy content Toggle raw display
$71$ \( (T^{3} - 927 T^{2} + \cdots + 103206528)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{3} - 1146 T^{2} + \cdots - 14051104)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{3} - 875 T^{2} + \cdots - 3833436)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
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