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Properties

Label 444.2.y.a
Level $444$
Weight $2$
Character orbit 444.y
Analytic conductor $3.545$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(103,444)] 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("444.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,-4] 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.3468738816.6
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{4} - 1) q^{2} + \beta_{6} q^{3} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{4} + (\beta_{6} + \beta_{4} - \beta_{3} + \cdots + 1) q^{5} + ( - \beta_{2} + 1) q^{6} + (\beta_{6} - \beta_{5} + \beta_1 + 2) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{3} + 8 q^{6} + 12 q^{7} - 16 q^{8} - 4 q^{9} + 12 q^{10} - 4 q^{11} - 12 q^{13} - 4 q^{14} - 6 q^{15} + 16 q^{16} - 4 q^{18} - 6 q^{19} - 12 q^{21} - 16 q^{22} + 16 q^{23} + 8 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 503\nu^{7} - 3756\nu^{6} + 6055\nu^{5} + 3917\nu^{4} - 5566\nu^{3} - 287375\nu^{2} + 15024\nu + 24068 ) / 697554 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 179 \nu^{7} + 3753 \nu^{6} - 13144 \nu^{5} + 24892 \nu^{4} + 28855 \nu^{3} + 95837 \nu^{2} + \cdots + 479290 ) / 126828 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14009 \nu^{7} + 13080 \nu^{6} + 74744 \nu^{5} - 561878 \nu^{4} - 708257 \nu^{3} + \cdots - 4434056 ) / 2790216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 679 \nu^{7} - 4671 \nu^{6} + 14078 \nu^{5} - 16838 \nu^{4} + 23395 \nu^{3} - 97837 \nu^{2} + \cdots - 308198 ) / 126828 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{7} - 94\nu^{6} + 204\nu^{5} + 274\nu^{4} + 825\nu^{3} - 880\nu^{2} + 2950\nu - 132 ) / 3432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 7\nu^{6} - 8\nu^{5} - 50\nu^{4} - 40\nu^{3} + 125\nu^{2} - 106\nu - 550 ) / 156 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - 2\beta_{6} + \beta_{5} - 5\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} - 6\beta_{6} + 7\beta_{5} + 6\beta_{4} + 7\beta_{3} - 7\beta_{2} - 2\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -15\beta_{7} + 3\beta_{5} + 34\beta_{4} + 18\beta_{3} - 17\beta_{2} - 18\beta _1 - 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 96\beta_{6} - 35\beta_{5} + 96\beta_{4} + 35\beta_{3} + 27\beta_{2} - 74\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 205\beta_{7} + 498\beta_{6} - 267\beta_{5} - 62\beta_{3} + 475\beta_{2} - 62\beta _1 + 311 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1032\beta_{7} + 1354\beta_{6} - 947\beta_{5} - 1354\beta_{4} - 947\beta_{3} + 1807\beta_{2} + 516\beta _1 + 2754 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{2} - \beta_{4}\)

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
1.10829 + 1.10829i
−1.47432 − 1.47432i
−1.27467 − 1.27467i
2.64070 + 2.64070i
1.10829 − 1.10829i
−1.47432 + 1.47432i
−1.27467 + 1.27467i
2.64070 − 2.64070i
0.366025 − 1.36603i −0.500000 + 0.866025i −1.73205 − 1.00000i −0.0396377 − 0.147930i 1.00000 + 1.00000i 4.12225 + 2.37998i −2.00000 + 2.00000i −0.500000 − 0.866025i −0.216584
103.2 0.366025 − 1.36603i −0.500000 + 0.866025i −1.73205 − 1.00000i 0.905663 + 3.37998i 1.00000 + 1.00000i −1.98827 − 1.14793i −2.00000 + 2.00000i −0.500000 − 0.866025i 4.94864
199.1 −1.36603 + 0.366025i −0.500000 − 0.866025i 1.73205 − 1.00000i −3.10726 − 0.832588i 1.00000 + 1.00000i 0.691890 − 0.399463i −2.00000 + 2.00000i −0.500000 + 0.866025i 4.54935
199.2 −1.36603 + 0.366025i −0.500000 − 0.866025i 1.73205 − 1.00000i 2.24124 + 0.600537i 1.00000 + 1.00000i 3.17414 − 1.83259i −2.00000 + 2.00000i −0.500000 + 0.866025i −3.28140
319.1 0.366025 + 1.36603i −0.500000 − 0.866025i −1.73205 + 1.00000i −0.0396377 + 0.147930i 1.00000 − 1.00000i 4.12225 − 2.37998i −2.00000 − 2.00000i −0.500000 + 0.866025i −0.216584
319.2 0.366025 + 1.36603i −0.500000 − 0.866025i −1.73205 + 1.00000i 0.905663 − 3.37998i 1.00000 − 1.00000i −1.98827 + 1.14793i −2.00000 − 2.00000i −0.500000 + 0.866025i 4.94864
415.1 −1.36603 − 0.366025i −0.500000 + 0.866025i 1.73205 + 1.00000i −3.10726 + 0.832588i 1.00000 − 1.00000i 0.691890 + 0.399463i −2.00000 − 2.00000i −0.500000 − 0.866025i 4.54935
415.2 −1.36603 − 0.366025i −0.500000 + 0.866025i 1.73205 + 1.00000i 2.24124 − 0.600537i 1.00000 − 1.00000i 3.17414 + 1.83259i −2.00000 − 2.00000i −0.500000 − 0.866025i −3.28140
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 103.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
148.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.y.a ✓ 8
4.b odd 2 1 444.2.y.b yes 8
37.g odd 12 1 444.2.y.b yes 8
148.l even 12 1 inner 444.2.y.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.y.a ✓ 8 1.a even 1 1 trivial
444.2.y.a ✓ 8 148.l even 12 1 inner
444.2.y.b yes 8 4.b odd 2 1
444.2.y.b yes 8 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):

\( T_{5}^{8} - 3T_{5}^{6} + 30T_{5}^{5} - 67T_{5}^{4} - 264T_{5}^{3} + 660T_{5}^{2} + 48T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 12T_{7}^{7} + 51T_{7}^{6} - 36T_{7}^{5} - 247T_{7}^{4} + 216T_{7}^{3} + 1632T_{7}^{2} - 2304T_{7} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 34 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{8} + 27 T^{6} + \cdots + 39204 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 173056 \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + \cdots + 640000 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} + \cdots + 43264 \) Copy content Toggle raw display
$37$ \( T^{8} - 87 T^{6} + \cdots + 1874161 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 5476 \) Copy content Toggle raw display
$43$ \( T^{8} - 22 T^{7} + \cdots + 495616 \) Copy content Toggle raw display
$47$ \( T^{8} + 184 T^{6} + \cdots + 123904 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$59$ \( T^{8} + 34 T^{7} + \cdots + 2560000 \) Copy content Toggle raw display
$61$ \( T^{8} + 32 T^{7} + \cdots + 702244 \) Copy content Toggle raw display
$67$ \( T^{8} + 28 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{8} + 18 T^{7} + \cdots + 262144 \) Copy content Toggle raw display
$73$ \( T^{8} + 282 T^{6} + \cdots + 3701776 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + \cdots + 350464 \) Copy content Toggle raw display
$83$ \( T^{8} - 18 T^{7} + \cdots + 5161984 \) Copy content Toggle raw display
$89$ \( T^{8} + 20 T^{7} + \cdots + 81396484 \) Copy content Toggle raw display
$97$ \( T^{8} - 1272 T^{5} + \cdots + 466489 \) Copy content Toggle raw display
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