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Properties

Label 221.2.c.d
Level $221$
Weight $2$
Character orbit 221.c
Analytic conductor $1.765$
Analytic rank $0$
Dimension $10$
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] = [221,2,Mod(103,221)] 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("221.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(221, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 221 = 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 221.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.76469388467\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 107x^{6} + 237x^{4} + 188x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{5} q^{5} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_1) q^{6} + (\beta_{7} + \beta_1) q^{7} + (\beta_{9} - \beta_{8} + \cdots - 3 \beta_1) q^{8}+ \cdots + (\beta_{9} - \beta_{8} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} - 16 q^{4} + 16 q^{9} - 10 q^{10} - 10 q^{12} - 24 q^{14} + 44 q^{16} + 10 q^{17} + 14 q^{22} - 36 q^{23} - 14 q^{25} + 16 q^{27} - 12 q^{29} + 80 q^{30} - 6 q^{35} - 116 q^{36} + 2 q^{38}+ \cdots + 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 107x^{6} + 237x^{4} + 188x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 17\nu^{6} + 86\nu^{4} + 115\nu^{2} + 21 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 17\nu^{6} + 90\nu^{4} + 147\nu^{2} + 49 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} + 49\nu^{7} + 236\nu^{5} + 281\nu^{3} - \nu ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} - 49\nu^{7} - 246\nu^{5} - 371\nu^{3} - 119\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} - 17\nu^{7} - 90\nu^{5} - 151\nu^{3} - 73\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{9} - 10\nu^{8} - 49\nu^{7} - 160\nu^{6} - 246\nu^{5} - 760\nu^{4} - 381\nu^{3} - 930\nu^{2} - 189\nu - 80 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{9} - 10\nu^{8} + 49\nu^{7} - 160\nu^{6} + 246\nu^{5} - 760\nu^{4} + 381\nu^{3} - 930\nu^{2} + 189\nu - 80 ) / 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - \beta_{3} - 8\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{9} + 9\beta_{8} - 10\beta_{6} - \beta_{5} + 51\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + \beta_{8} - 10\beta_{4} + 14\beta_{3} + 58\beta_{2} - 175 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 67\beta_{9} - 67\beta_{8} - 6\beta_{7} + 84\beta_{6} + 12\beta_{5} - 375\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -17\beta_{9} - 17\beta_{8} + 84\beta_{4} - 148\beta_{3} - 413\beta_{2} + 1264 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -480\beta_{9} + 480\beta_{8} + 98\beta_{7} - 679\beta_{6} - 114\beta_{5} + 2769\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/221\mathbb{Z}\right)^\times\).

\(n\) \(105\) \(171\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1
− 2.76630i
− 2.59662i
− 1.36079i
− 1.26044i
− 0.405836i
0.405836i
1.26044i
1.36079i
2.59662i
2.76630i
− 2.76630i 2.94150 −5.65243 0.444845i − 8.13708i − 3.46814i 10.1037i 5.65243 1.23058
103.2 − 2.59662i −2.78252 −4.74243 − 3.58446i 7.22515i − 1.13991i 7.12104i 4.74243 −9.30747
103.3 − 1.36079i −1.68871 0.148260 − 1.63814i 2.29797i 0.615063i − 2.92332i −0.148260 −2.22916
103.4 − 1.26044i 1.60894 0.411297 3.67268i − 2.02797i − 1.74356i − 3.03929i −0.411297 4.62919
103.5 − 0.405836i −1.07921 1.83530 1.66784i 0.437984i 4.71745i − 1.55650i −1.83530 0.676869
103.6 0.405836i −1.07921 1.83530 − 1.66784i − 0.437984i − 4.71745i 1.55650i −1.83530 0.676869
103.7 1.26044i 1.60894 0.411297 − 3.67268i 2.02797i 1.74356i 3.03929i −0.411297 4.62919
103.8 1.36079i −1.68871 0.148260 1.63814i − 2.29797i − 0.615063i 2.92332i −0.148260 −2.22916
103.9 2.59662i −2.78252 −4.74243 3.58446i − 7.22515i 1.13991i − 7.12104i 4.74243 −9.30747
103.10 2.76630i 2.94150 −5.65243 − 0.444845i 8.13708i 3.46814i − 10.1037i 5.65243 1.23058
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 221.2.c.d ✓ 10
3.b odd 2 1 1989.2.b.j 10
13.b even 2 1 inner 221.2.c.d ✓ 10
13.d odd 4 2 2873.2.a.r 10
39.d odd 2 1 1989.2.b.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.c.d ✓ 10 1.a even 1 1 trivial
221.2.c.d ✓ 10 13.b even 2 1 inner
1989.2.b.j 10 3.b odd 2 1
1989.2.b.j 10 39.d odd 2 1
2873.2.a.r 10 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 18T_{2}^{8} + 107T_{2}^{6} + 237T_{2}^{4} + 188T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(221, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 18 T^{8} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{5} + T^{4} - 11 T^{3} + \cdots + 24)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + 32 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{10} + 39 T^{8} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{10} + 53 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{10} - 7 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T - 1)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + 69 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$23$ \( (T^{5} + 18 T^{4} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 6 T^{4} + \cdots - 5384)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 164 T^{8} + \cdots + 1817104 \) Copy content Toggle raw display
$37$ \( T^{10} + 32 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{10} + 264 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$43$ \( (T^{5} + 10 T^{4} + \cdots + 1376)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 136 T^{8} + \cdots + 147456 \) Copy content Toggle raw display
$53$ \( (T^{5} - 29 T^{4} + \cdots - 344)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 156 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( (T^{5} + 19 T^{4} + \cdots + 5240)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1312178176 \) Copy content Toggle raw display
$71$ \( T^{10} + 540 T^{8} + \cdots + 35426304 \) Copy content Toggle raw display
$73$ \( T^{10} + 176 T^{8} + \cdots + 2359296 \) Copy content Toggle raw display
$79$ \( (T^{5} - 8 T^{4} + \cdots - 4528)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 172 T^{8} + \cdots + 7139584 \) Copy content Toggle raw display
$89$ \( T^{10} + 497 T^{8} + \cdots + 16000000 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 158155776 \) Copy content Toggle raw display
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