Properties

Label 1100.2.cb.c
Level $1100$
Weight $2$
Character orbit 1100.cb
Analytic conductor $8.784$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,2,Mod(49,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.cb (of order \(10\), degree \(4\), 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: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.45212176000000000000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{13} + \cdots + 2 \beta_1) q^{3}+ \cdots + (\beta_{12} - \beta_{10} + 2 \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{13} + \cdots + 2 \beta_1) q^{3}+ \cdots + ( - \beta_{14} + 2 \beta_{12} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} + 10 q^{11} + 14 q^{19} + 56 q^{21} - 6 q^{29} - 4 q^{31} + 34 q^{39} - 24 q^{41} - 42 q^{49} + 42 q^{51} + 18 q^{59} - 68 q^{61} + 2 q^{69} - 52 q^{71} - 38 q^{79} + 24 q^{81} + 16 q^{89} - 36 q^{91} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + \nu^{13} - 14\nu^{11} - 44\nu^{9} + \nu^{7} + 108\nu^{5} + 288\nu^{3} + 192\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - \nu^{12} + 14\nu^{10} + 44\nu^{8} - \nu^{6} - 108\nu^{4} - 288\nu^{2} - 192 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 9\nu^{12} + 10\nu^{10} - 12\nu^{8} - 63\nu^{6} - 76\nu^{4} - 48\nu^{2} - 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} + 3\nu^{11} + 4\nu^{9} - 8\nu^{7} - 23\nu^{5} - 18\nu^{3} - 8\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{14} + 9\nu^{12} + 26\nu^{10} + 4\nu^{8} - 95\nu^{6} - 204\nu^{4} - 32\nu^{2} + 128 ) / 192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} - 5\nu^{13} + 10\nu^{11} + 100\nu^{9} + 175\nu^{7} - 112\nu^{5} - 720\nu^{3} - 960\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{14} - 5\nu^{12} + 10\nu^{10} + 100\nu^{8} + 175\nu^{6} - 112\nu^{4} - 720\nu^{2} - 960 ) / 192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{15} - 9\nu^{13} - 42\nu^{11} - 20\nu^{9} + 127\nu^{7} + 332\nu^{5} + 208\nu^{3} - 128\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} - 4\nu^{10} + 8\nu^{8} + 23\nu^{6} + 18\nu^{4} + 8\nu^{2} ) / 48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} + 5\nu^{13} + 6\nu^{11} - 20\nu^{9} - 79\nu^{7} - 80\nu^{5} + 96\nu^{3} + 320\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5\nu^{14} + 9\nu^{12} - 18\nu^{10} - 100\nu^{8} - 139\nu^{6} + 96\nu^{4} + 512\nu^{2} + 512 ) / 192 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + 9\nu^{13} + 10\nu^{11} - 12\nu^{9} - 63\nu^{7} - 76\nu^{5} - 48\nu^{3} - 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -5\nu^{14} - 13\nu^{12} + 14\nu^{10} + 124\nu^{8} + 187\nu^{6} - 132\nu^{4} - 688\nu^{2} - 640 ) / 96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5\nu^{15} + 15\nu^{13} + 4\nu^{11} - 72\nu^{9} - 147\nu^{7} - 26\nu^{5} + 216\nu^{3} + 224\nu ) / 192 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{12} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + 2\beta_{5} + 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{14} - 3\beta_{12} + \beta_{8} - \beta_{6} - \beta_{4} + 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 3\beta_{11} - 2\beta_{7} + \beta_{5} - 3\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - \beta_{10} + \beta_{6} - 5\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{15} + \beta_{13} - \beta_{11} - 3\beta_{9} + 2\beta_{7} + 10\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -13\beta_{14} - 11\beta_{12} + 12\beta_{10} + 13\beta_{8} + 5\beta_{6} - 7\beta_{4} + 12\beta_{3} + 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3\beta_{15} - 9\beta_{13} + \beta_{11} + 9\beta_{9} + 18\beta_{5} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -12\beta_{12} - 15\beta_{10} - 6\beta_{8} - 12\beta_{4} - 6\beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3\beta_{15} - 27\beta_{11} - 27\beta_{9} - 12\beta_{7} + 12\beta_{2} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -28\beta_{14} - 31\beta_{12} + 31\beta_{10} + 31\beta_{8} + 7\beta_{6} + 23\beta_{4} + 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -6\beta_{15} + 26\beta_{13} + 21\beta_{9} + 6\beta_{7} + 35\beta_{5} + 35\beta_{2} + 26\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -41\beta_{14} - 9\beta_{12} + 61\beta_{8} + 32\beta_{6} - 103\beta_{4} + 41\beta_{3} + 41 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 73\beta_{15} - 135\beta_{13} - 9\beta_{11} + 40\beta_{7} + 73\beta_{5} - 9\beta_{2} + 9\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(\beta_{12}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.41395 + 0.0272949i
−0.462894 + 1.33631i
0.462894 1.33631i
1.41395 0.0272949i
−1.41395 0.0272949i
−0.462894 1.33631i
0.462894 + 1.33631i
1.41395 + 0.0272949i
−0.132563 + 1.40799i
−0.720348 + 1.21700i
0.720348 1.21700i
0.132563 1.40799i
−0.132563 1.40799i
−0.720348 1.21700i
0.720348 + 1.21700i
0.132563 + 1.40799i
0 −1.59076 + 2.18949i 0 0 0 −2.21063 3.04267i 0 −1.33631 4.11275i 0
49.2 0 −1.00297 + 1.38048i 0 0 0 −1.98612 2.73366i 0 0.0272949 + 0.0840051i 0
49.3 0 1.00297 1.38048i 0 0 0 1.98612 + 2.73366i 0 0.0272949 + 0.0840051i 0
49.4 0 1.59076 2.18949i 0 0 0 2.21063 + 3.04267i 0 −1.33631 4.11275i 0
449.1 0 −1.59076 2.18949i 0 0 0 −2.21063 + 3.04267i 0 −1.33631 + 4.11275i 0
449.2 0 −1.00297 1.38048i 0 0 0 −1.98612 + 2.73366i 0 0.0272949 0.0840051i 0
449.3 0 1.00297 + 1.38048i 0 0 0 1.98612 2.73366i 0 0.0272949 0.0840051i 0
449.4 0 1.59076 + 2.18949i 0 0 0 2.21063 3.04267i 0 −1.33631 + 4.11275i 0
949.1 0 −2.01846 + 0.655837i 0 0 0 −0.291365 0.0946704i 0 1.21700 0.884205i 0
949.2 0 −1.06740 + 0.346820i 0 0 0 2.19853 + 0.714347i 0 −1.40799 + 1.02296i 0
949.3 0 1.06740 0.346820i 0 0 0 −2.19853 0.714347i 0 −1.40799 + 1.02296i 0
949.4 0 2.01846 0.655837i 0 0 0 0.291365 + 0.0946704i 0 1.21700 0.884205i 0
1049.1 0 −2.01846 0.655837i 0 0 0 −0.291365 + 0.0946704i 0 1.21700 + 0.884205i 0
1049.2 0 −1.06740 0.346820i 0 0 0 2.19853 0.714347i 0 −1.40799 1.02296i 0
1049.3 0 1.06740 + 0.346820i 0 0 0 −2.19853 + 0.714347i 0 −1.40799 1.02296i 0
1049.4 0 2.01846 + 0.655837i 0 0 0 0.291365 0.0946704i 0 1.21700 + 0.884205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.2.cb.c 16
5.b even 2 1 inner 1100.2.cb.c 16
5.c odd 4 1 220.2.m.a 8
5.c odd 4 1 1100.2.n.c 8
11.c even 5 1 inner 1100.2.cb.c 16
15.e even 4 1 1980.2.z.c 8
20.e even 4 1 880.2.bo.f 8
55.j even 10 1 inner 1100.2.cb.c 16
55.k odd 20 1 220.2.m.a 8
55.k odd 20 1 1100.2.n.c 8
55.k odd 20 1 2420.2.a.n 4
55.l even 20 1 2420.2.a.m 4
165.v even 20 1 1980.2.z.c 8
220.v even 20 1 880.2.bo.f 8
220.v even 20 1 9680.2.a.ck 4
220.w odd 20 1 9680.2.a.cl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.m.a 8 5.c odd 4 1
220.2.m.a 8 55.k odd 20 1
880.2.bo.f 8 20.e even 4 1
880.2.bo.f 8 220.v even 20 1
1100.2.n.c 8 5.c odd 4 1
1100.2.n.c 8 55.k odd 20 1
1100.2.cb.c 16 1.a even 1 1 trivial
1100.2.cb.c 16 5.b even 2 1 inner
1100.2.cb.c 16 11.c even 5 1 inner
1100.2.cb.c 16 55.j even 10 1 inner
1980.2.z.c 8 15.e even 4 1
1980.2.z.c 8 165.v even 20 1
2420.2.a.m 4 55.l even 20 1
2420.2.a.n 4 55.k odd 20 1
9680.2.a.ck 4 220.v even 20 1
9680.2.a.cl 4 220.w odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 3T_{3}^{14} + 48T_{3}^{12} - 341T_{3}^{10} + 1475T_{3}^{8} - 2801T_{3}^{6} + 11828T_{3}^{4} - 19723T_{3}^{2} + 14641 \) acting on \(S_{2}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 3 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 7 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$11$ \( (T^{8} - 5 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 2 T^{14} + \cdots + 62742241 \) Copy content Toggle raw display
$17$ \( T^{16} - 7 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$19$ \( (T^{8} - 7 T^{7} + 23 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 149 T^{6} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 3 T^{7} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + \cdots + 2313441)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1078193566321 \) Copy content Toggle raw display
$41$ \( (T^{8} + 12 T^{7} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 95 T^{6} + \cdots + 21025)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 104060401 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3373402561 \) Copy content Toggle raw display
$59$ \( (T^{8} - 9 T^{7} + \cdots + 1185921)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 34 T^{7} + \cdots + 273207841)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 421 T^{6} + \cdots + 776161)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 26 T^{7} + \cdots + 43546801)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 411651843201 \) Copy content Toggle raw display
$79$ \( (T^{8} + 19 T^{7} + \cdots + 2128681)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 35387281535841 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + \cdots + 881)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 926138694321 \) Copy content Toggle raw display
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