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Properties

Label 221.2.m.b
Level $221$
Weight $2$
Character orbit 221.m
Analytic conductor $1.765$
Analytic rank $0$
Dimension $12$
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(69,221)] 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("221.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(221, base_ring=CyclotomicField(6)) 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.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.76469388467\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 13x^{10} + 57x^{8} + 104x^{6} + 78x^{4} + 19x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{10} - \beta_{9} - \beta_{7} + \cdots + 1) q^{3} + (\beta_{7} + \beta_{5} - \beta_{2}) q^{4} + (\beta_{7} + \beta_{6}) q^{5} + (2 \beta_{9} + \beta_{7} + \beta_{6} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{11} - 4 \beta_{10} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 4 q^{3} + q^{4} - 12 q^{6} - 4 q^{9} - 3 q^{10} - 24 q^{11} + 20 q^{12} + 7 q^{13} + 10 q^{14} + 9 q^{15} - q^{16} + 6 q^{17} + 15 q^{19} - 18 q^{20} + 11 q^{22} - 12 q^{23} - 42 q^{24}+ \cdots - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 13x^{10} + 57x^{8} + 104x^{6} + 78x^{4} + 19x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 10\nu^{9} + 22\nu^{7} - 22\nu^{5} - 81\nu^{3} - 38\nu - 5 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{10} + 35\nu^{8} + 126\nu^{6} + 159\nu^{4} + 57\nu^{2} + 5\nu + 1 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{10} + 35\nu^{8} + 126\nu^{6} + 159\nu^{4} + 57\nu^{2} - 5\nu + 1 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 3 \nu^{10} - 15 \nu^{9} - 40 \nu^{8} - 82 \nu^{7} - 181 \nu^{6} - 203 \nu^{5} - 334 \nu^{4} + \cdots - 26 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} - 60 \nu^{9} + 25 \nu^{8} - 225 \nu^{7} + 104 \nu^{6} - 295 \nu^{5} + \cdots + 24 ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6 \nu^{11} - \nu^{10} + 75 \nu^{9} - 10 \nu^{8} + 307 \nu^{7} - 22 \nu^{6} + 498 \nu^{5} + 22 \nu^{4} + \cdots + 33 ) / 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6 \nu^{11} + \nu^{10} + 75 \nu^{9} + 10 \nu^{8} + 307 \nu^{7} + 22 \nu^{6} + 498 \nu^{5} - 22 \nu^{4} + \cdots - 33 ) / 10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7 \nu^{11} + 4 \nu^{10} + 90 \nu^{9} + 50 \nu^{8} + 384 \nu^{7} + 203 \nu^{6} + 651 \nu^{5} + 317 \nu^{4} + \cdots + 28 ) / 10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{11} - 25\nu^{9} - 102\nu^{7} - 163\nu^{5} - 96\nu^{3} + \nu^{2} - 13\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8 \nu^{11} + \nu^{10} + 105 \nu^{9} + 10 \nu^{8} + 466 \nu^{7} + 22 \nu^{6} + 854 \nu^{5} - 17 \nu^{4} + \cdots + 2 ) / 10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{11} + 4 \nu^{10} - 90 \nu^{9} + 50 \nu^{8} - 384 \nu^{7} + 203 \nu^{6} - 651 \nu^{5} + \cdots + 28 ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 4\beta_{3} - 5\beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - 7\beta_{9} - \beta_{7} - 6\beta_{6} + 7\beta_{5} + \beta_{4} - 7\beta_{3} + 7\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 6 \beta_{10} - 8 \beta_{9} - 7 \beta_{8} - 14 \beta_{7} - 6 \beta_{6} - 8 \beta_{5} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{11} - 9 \beta_{10} + 43 \beta_{9} - \beta_{8} + 7 \beta_{7} + 36 \beta_{6} - 43 \beta_{5} + \cdots - 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9 \beta_{11} - 33 \beta_{10} + 54 \beta_{9} + 42 \beta_{8} + 89 \beta_{7} + 35 \beta_{6} + 54 \beta_{5} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75 \beta_{11} + 63 \beta_{10} - 261 \beta_{9} + 12 \beta_{8} - 42 \beta_{7} - 219 \beta_{6} + 261 \beta_{5} + \cdots + 418 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 63 \beta_{11} + 186 \beta_{10} - 347 \beta_{9} - 249 \beta_{8} - 556 \beta_{7} - 209 \beta_{6} + \cdots + 121 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 508 \beta_{11} - 410 \beta_{10} + 1591 \beta_{9} - 98 \beta_{8} + 249 \beta_{7} + 1342 \beta_{6} + \cdots - 2454 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 410 \beta_{11} - 1083 \beta_{10} + 2187 \beta_{9} + 1493 \beta_{8} + 3456 \beta_{7} + 1269 \beta_{6} + \cdots - 648 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(105\) \(171\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
− 2.48782i
− 1.35007i
− 0.557160i
− 0.268136i
1.07190i
1.85923i
2.48782i
1.35007i
0.557160i
0.268136i
− 1.07190i
− 1.85923i
−2.15452 − 1.24391i 1.53909 − 2.66579i 2.09463 + 3.62801i 1.33009i −6.63201 + 3.82899i 0.686981 − 0.396628i − 5.44652i −3.23761 − 5.60771i 1.65452 − 2.86571i
69.2 −1.16919 − 0.675034i 0.160577 − 0.278128i −0.0886588 − 0.153562i 0.991347i −0.375492 + 0.216790i −4.13797 + 2.38906i 2.93953i 1.44843 + 2.50875i 0.669193 − 1.15908i
69.3 −0.482515 − 0.278580i −1.01552 + 1.75892i −0.844786 − 1.46321i − 0.0627664i 0.980002 − 0.565805i 2.50288 − 1.44504i 2.05568i −0.562544 − 0.974355i −0.0174855 + 0.0302857i
69.4 −0.232213 − 0.134068i 0.727013 − 1.25922i −0.964052 − 1.66979i − 1.99740i −0.337643 + 0.194938i −0.880982 + 0.508635i 1.05327i 0.442904 + 0.767132i −0.267787 + 0.463821i
69.5 0.928296 + 0.535952i 1.12274 − 1.94464i −0.425511 − 0.737006i 2.66497i 2.08446 − 1.20347i 2.27321 − 1.31244i − 3.05602i −1.02107 − 1.76855i −1.42830 + 2.47388i
69.6 1.61014 + 0.929616i −0.533903 + 0.924748i 0.728373 + 1.26158i 2.26991i −1.71932 + 0.992650i −0.444116 + 0.256410i − 1.01004i 0.929895 + 1.61062i −2.11014 + 3.65487i
205.1 −2.15452 + 1.24391i 1.53909 + 2.66579i 2.09463 − 3.62801i − 1.33009i −6.63201 − 3.82899i 0.686981 + 0.396628i 5.44652i −3.23761 + 5.60771i 1.65452 + 2.86571i
205.2 −1.16919 + 0.675034i 0.160577 + 0.278128i −0.0886588 + 0.153562i − 0.991347i −0.375492 − 0.216790i −4.13797 − 2.38906i − 2.93953i 1.44843 − 2.50875i 0.669193 + 1.15908i
205.3 −0.482515 + 0.278580i −1.01552 − 1.75892i −0.844786 + 1.46321i 0.0627664i 0.980002 + 0.565805i 2.50288 + 1.44504i − 2.05568i −0.562544 + 0.974355i −0.0174855 − 0.0302857i
205.4 −0.232213 + 0.134068i 0.727013 + 1.25922i −0.964052 + 1.66979i 1.99740i −0.337643 − 0.194938i −0.880982 − 0.508635i − 1.05327i 0.442904 − 0.767132i −0.267787 − 0.463821i
205.5 0.928296 − 0.535952i 1.12274 + 1.94464i −0.425511 + 0.737006i − 2.66497i 2.08446 + 1.20347i 2.27321 + 1.31244i 3.05602i −1.02107 + 1.76855i −1.42830 − 2.47388i
205.6 1.61014 − 0.929616i −0.533903 − 0.924748i 0.728373 − 1.26158i − 2.26991i −1.71932 − 0.992650i −0.444116 − 0.256410i 1.01004i 0.929895 − 1.61062i −2.11014 − 3.65487i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 221.2.m.b ✓ 12
13.e even 6 1 inner 221.2.m.b ✓ 12
13.f odd 12 2 2873.2.a.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.m.b ✓ 12 1.a even 1 1 trivial
221.2.m.b ✓ 12 13.e even 6 1 inner
2873.2.a.u 12 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3 T_{2}^{11} - 2 T_{2}^{10} - 15 T_{2}^{9} + 9 T_{2}^{8} + 51 T_{2}^{7} + 17 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(221, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{12} + 19 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} - 20 T^{10} + \cdots + 225 \) Copy content Toggle raw display
$11$ \( T^{12} + 24 T^{11} + \cdots + 765625 \) Copy content Toggle raw display
$13$ \( T^{12} - 7 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$19$ \( T^{12} - 15 T^{11} + \cdots + 5625 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 762129 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 16370434809 \) Copy content Toggle raw display
$31$ \( T^{12} + 87 T^{10} + \cdots + 3272481 \) Copy content Toggle raw display
$37$ \( T^{12} + 18 T^{11} + \cdots + 39125025 \) Copy content Toggle raw display
$41$ \( T^{12} - 3 T^{11} + \cdots + 16129 \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{11} + \cdots + 625 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 749062161 \) Copy content Toggle raw display
$53$ \( (T^{6} - 3 T^{5} + \cdots - 65367)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 3 T^{11} + \cdots + 18769 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 128754409 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 34735640625 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 15006985009 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15430856841 \) Copy content Toggle raw display
$79$ \( (T^{6} + 19 T^{5} + \cdots + 16987)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 120 T^{10} + \cdots + 18769 \) Copy content Toggle raw display
$89$ \( T^{12} + 12 T^{11} + \cdots + 332929 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 939722788449 \) Copy content Toggle raw display
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