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Properties

Label 1100.4.b.j
Level $1100$
Weight $4$
Character orbit 1100.b
Analytic conductor $64.902$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,4,Mod(749,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(64.9021010063\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 187x^{8} + 12537x^{6} + 358849x^{4} + 3893659x^{2} + 7371225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3}) q^{7} + (\beta_1 - 12) q^{9} + 11 q^{11} + (\beta_{9} - \beta_{7} + \cdots - 2 \beta_{3}) q^{13} + (\beta_{9} + \beta_{7} + \cdots - 7 \beta_{3}) q^{17}+ \cdots + (11 \beta_1 - 132) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 118 q^{9} + 110 q^{11} - 6 q^{19} + 316 q^{21} - 314 q^{29} + 114 q^{31} - 1036 q^{39} + 148 q^{41} - 1054 q^{49} - 2292 q^{51} - 1384 q^{59} - 1996 q^{61} - 2492 q^{69} - 3290 q^{71} - 2640 q^{79}+ \cdots - 1298 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 187x^{8} + 12537x^{6} + 358849x^{4} + 3893659x^{2} + 7371225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{8} + 2478\nu^{6} + 156878\nu^{4} + 3654376\nu^{2} + 34976596 ) / 834557 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\nu^{8} + 7434\nu^{6} + 470634\nu^{4} + 6790343\nu^{2} - 51966928 ) / 1669114 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1202\nu^{9} + 194909\nu^{7} + 8341704\nu^{5} + 5412728\nu^{3} - 7507274977\nu ) / 4531644510 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5485\nu^{8} + 780408\nu^{6} + 33765948\nu^{4} + 430998241\nu^{2} + 276192954 ) / 15022026 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -17854\nu^{8} - 2883987\nu^{6} - 155997792\nu^{4} - 3085545094\nu^{2} - 12682445409 ) / 45066078 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1202\nu^{9} + 194909\nu^{7} + 8341704\nu^{5} + 5412728\nu^{3} - 2975630467\nu ) / 906328902 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 388739\nu^{9} + 66177288\nu^{7} + 3817507563\nu^{5} + 82260737921\nu^{3} + 419575647051\nu ) / 20392400295 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2432051\nu^{9} + 288175797\nu^{7} + 7581043422\nu^{5} - 100065977821\nu^{3} - 3547368286641\nu ) / 40784800590 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -847102\nu^{9} - 132334119\nu^{7} - 6806205234\nu^{5} - 133301992558\nu^{3} - 904160923353\nu ) / 13594933530 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 5\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{2} + 3\beta _1 - 188 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{9} - 2\beta_{8} - 17\beta_{7} - 49\beta_{6} + 272\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -18\beta_{5} - 15\beta_{4} + 101\beta_{2} - 277\beta _1 + 9964 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 640\beta_{9} + 190\beta_{8} + 1675\beta_{7} + 701\beta_{6} - 16255\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1758\beta_{5} + 1300\beta_{4} - 3752\beta_{2} + 22456\beta _1 - 587123 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -48714\beta_{9} - 16464\beta_{8} - 128874\beta_{7} + 99491\beta_{6} + 1022309\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -27864\beta_{5} - 15786\beta_{4} + 13846\beta_{2} - 345111\beta _1 + 7343602 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3489181\beta_{9} + 1360136\beta_{8} + 9349721\beta_{7} - 14531387\beta_{6} - 66566756\beta_{3} ) / 5 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
− 8.44158i
− 7.30216i
− 4.57551i
− 6.25411i
− 1.53918i
1.53918i
6.25411i
4.57551i
7.30216i
8.44158i
0 − 9.44158i 0 0 0 27.8042i 0 −62.1434 0
749.2 0 − 6.30216i 0 0 0 − 4.78461i 0 −12.7172 0
749.3 0 − 5.57551i 0 0 0 − 20.7640i 0 −4.08629 0
749.4 0 − 5.25411i 0 0 0 − 7.12487i 0 −0.605666 0
749.5 0 − 2.53918i 0 0 0 31.0503i 0 20.5526 0
749.6 0 2.53918i 0 0 0 − 31.0503i 0 20.5526 0
749.7 0 5.25411i 0 0 0 7.12487i 0 −0.605666 0
749.8 0 5.57551i 0 0 0 20.7640i 0 −4.08629 0
749.9 0 6.30216i 0 0 0 4.78461i 0 −12.7172 0
749.10 0 9.44158i 0 0 0 − 27.8042i 0 −62.1434 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.10
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.j 10
5.b even 2 1 inner 1100.4.b.j 10
5.c odd 4 1 1100.4.a.j ✓ 5
5.c odd 4 1 1100.4.a.m yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.4.a.j ✓ 5 5.c odd 4 1
1100.4.a.m yes 5 5.c odd 4 1
1100.4.b.j 10 1.a even 1 1 trivial
1100.4.b.j 10 5.b 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 \(S_{4}^{\mathrm{new}}(1100, [\chi])\):

\( T_{3}^{10} + 194T_{3}^{8} + 13171T_{3}^{6} + 395506T_{3}^{4} + 5091089T_{3}^{2} + 19589476 \) Copy content Toggle raw display
\( T_{7}^{10} + 2242T_{7}^{8} + 1655199T_{7}^{6} + 433936018T_{7}^{4} + 25405989841T_{7}^{2} + 373443210000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 194 T^{8} + \cdots + 19589476 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 373443210000 \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} + \cdots + 6579183679)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$29$ \( (T^{5} + 157 T^{4} + \cdots - 10780618347)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 57 T^{4} + \cdots - 9310638465)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} - 74 T^{4} + \cdots - 63070168860)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 37\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 2722880626524)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 1054285961106)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 90164424394224)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 235259377206772)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 43\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 749849927110839)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
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