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Properties

Label 444.2.bb.a
Level $444$
Weight $2$
Character orbit 444.bb
Analytic conductor $3.545$
Analytic rank $0$
Dimension $18$
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] = [444,2,Mod(25,444)] 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("444.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18] 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: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 48 x^{16} + 954 x^{14} + 10172 x^{12} + 63093 x^{10} + 231255 x^{8} + 486910 x^{6} + \cdots + 36963 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1) q^{3} - \beta_{3} q^{5} + (\beta_{16} + \beta_{13} + \cdots - \beta_1) q^{7} - \beta_{6} q^{9} + ( - \beta_{16} + \beta_{15} - \beta_{13} + \cdots - 1) q^{11} + (\beta_{17} + \beta_{15} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{15} - \beta_{11} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{11} + 18 q^{13} + 15 q^{17} - 12 q^{19} + 6 q^{25} - 9 q^{27} + 27 q^{29} - 6 q^{33} - 12 q^{35} - 9 q^{37} + 18 q^{39} + 33 q^{41} + 21 q^{47} + 30 q^{49} - 9 q^{51} - 18 q^{53} - 51 q^{55}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 48 x^{16} + 954 x^{14} + 10172 x^{12} + 63093 x^{10} + 231255 x^{8} + 486910 x^{6} + \cdots + 36963 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 422945 \nu^{17} - 8436 \nu^{16} - 24823059 \nu^{15} - 35133165 \nu^{14} + \cdots + 75598877448 ) / 49576951422 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 422945 \nu^{17} + 8436 \nu^{16} - 24823059 \nu^{15} + 35133165 \nu^{14} + \cdots - 75598877448 ) / 49576951422 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2812 \nu^{17} + 1507233 \nu^{16} + 11711055 \nu^{15} + 44096746 \nu^{14} + 412074957 \nu^{13} + \cdots - 5211105345 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2812 \nu^{17} - 1507233 \nu^{16} + 11711055 \nu^{15} - 44096746 \nu^{14} + 412074957 \nu^{13} + \cdots + 5211105345 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1683344 \nu^{17} - 12132744 \nu^{16} - 64159071 \nu^{15} - 517050765 \nu^{14} + \cdots - 179109727983 ) / 49576951422 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1683344 \nu^{17} + 12132744 \nu^{16} - 64159071 \nu^{15} + 517050765 \nu^{14} + \cdots + 179109727983 ) / 49576951422 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1887500 \nu^{17} + 6157267 \nu^{16} - 72055791 \nu^{15} + 282177827 \nu^{14} + \cdots + 85423902921 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{17} + 214 \nu^{15} + 3675 \nu^{13} + 32373 \nu^{11} + 155349 \nu^{9} + 399564 \nu^{7} + \cdots - 9879 ) / 19758 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4047060 \nu^{17} - 4039914 \nu^{16} - 184061310 \nu^{15} - 188908579 \nu^{14} + \cdots - 25951586769 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3581437 \nu^{17} - 10690214 \nu^{16} - 162374237 \nu^{15} - 461312285 \nu^{14} + \cdots - 35094685518 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3581437 \nu^{17} + 11056264 \nu^{16} + 162374237 \nu^{15} + 454549000 \nu^{14} + \cdots + 102959484453 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4044248 \nu^{17} - 5547147 \nu^{16} + 172350255 \nu^{15} - 233005325 \nu^{14} + \cdots - 20740481424 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3581437 \nu^{17} + 10690214 \nu^{16} - 162374237 \nu^{15} + 461312285 \nu^{14} + \cdots + 51620335992 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3581437 \nu^{17} + 11056264 \nu^{16} - 162374237 \nu^{15} + 454549000 \nu^{14} + \cdots + 119485134927 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10204327 \nu^{16} + 466239137 \nu^{14} + 8606448832 \nu^{12} + 82007392210 \nu^{10} + \cdots + 119927519766 ) / 8262825237 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10021070 \nu^{17} - 4650034 \nu^{16} + 451024081 \nu^{15} - 238081081 \nu^{14} + \cdots - 140211959688 ) / 16525650474 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 33548288 \nu^{17} + 5987796 \nu^{16} - 1467079329 \nu^{15} + 152642847 \nu^{14} + \cdots - 420096123360 ) / 49576951422 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{16} - \beta_{15} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8 \beta_{14} - 8 \beta_{13} - 8 \beta_{11} - 8 \beta_{10} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{17} + 10 \beta_{16} + 9 \beta_{15} + \beta_{13} + 9 \beta_{10} + 10 \beta_{8} - 9 \beta_{6} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{15} - 75 \beta_{14} + 72 \beta_{13} - 4 \beta_{12} + 75 \beta_{11} + 72 \beta_{10} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 98 \beta_{17} - 98 \beta_{16} - 83 \beta_{15} - \beta_{14} - 17 \beta_{13} + 10 \beta_{12} + \cdots - 387 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{17} - 2 \beta_{16} + 70 \beta_{15} + 751 \beta_{14} - 689 \beta_{13} + 82 \beta_{12} + \cdots - 93 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 973 \beta_{17} + 973 \beta_{16} + 787 \beta_{15} + 33 \beta_{14} + 219 \beta_{13} - 225 \beta_{12} + \cdots + 3875 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 33 \beta_{17} + 33 \beta_{16} - 936 \beta_{15} - 7763 \beta_{14} + 6821 \beta_{13} - 1212 \beta_{12} + \cdots + 1320 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9826 \beta_{17} - 9826 \beta_{16} - 7629 \beta_{15} - 627 \beta_{14} - 2608 \beta_{13} + 3486 \beta_{12} + \cdots - 40004 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 411 \beta_{17} - 411 \beta_{16} + 11425 \beta_{15} + 81744 \beta_{14} - 69225 \beta_{13} + 15886 \beta_{12} + \cdots - 16952 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 100823 \beta_{17} + 100823 \beta_{16} + 75467 \beta_{15} + 9577 \beta_{14} + 30200 \beta_{13} + \cdots + 420735 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4844 \beta_{17} + 4844 \beta_{16} - 133729 \beta_{15} - 871429 \beta_{14} + 716738 \beta_{13} + \cdots + 207081 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1048657 \beta_{17} - 1048657 \beta_{16} - 760273 \beta_{15} - 131334 \beta_{14} - 345618 \beta_{13} + \cdots - 4481150 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 57234 \beta_{17} - 57234 \beta_{16} + 1529544 \beta_{15} + 9371459 \beta_{14} - 7541351 \beta_{13} + \cdots - 2457732 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11028073 \beta_{17} + 11028073 \beta_{16} + 7780134 \beta_{15} + 1696611 \beta_{14} + 3932797 \beta_{13} + \cdots + 48160658 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 684858 \beta_{17} + 684858 \beta_{16} - 17246902 \beta_{15} - 101425140 \beta_{14} + 80360769 \beta_{13} + \cdots + 28651001 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.91345i
0.880292i
− 2.79375i
− 1.81967i
− 1.51171i
3.33138i
1.81967i
1.51171i
− 3.33138i
− 2.81975i
− 0.479259i
3.29901i
2.81975i
0.479259i
− 3.29901i
− 1.91345i
− 0.880292i
2.79375i
0 0.766044 + 0.642788i 0 −0.654440 − 1.79806i 0 −0.283425 + 0.103158i 0 0.173648 + 0.984808i 0
25.2 0 0.766044 + 0.642788i 0 −0.301078 − 0.827204i 0 4.88956 − 1.77965i 0 0.173648 + 0.984808i 0
25.3 0 0.766044 + 0.642788i 0 0.955517 + 2.62526i 0 −1.61341 + 0.587231i 0 0.173648 + 0.984808i 0
169.1 0 −0.939693 + 0.342020i 0 −1.79202 − 0.315982i 0 −0.352471 + 1.99897i 0 0.766044 − 0.642788i 0
169.2 0 −0.939693 + 0.342020i 0 −1.48875 − 0.262507i 0 0.180286 − 1.02245i 0 0.766044 − 0.642788i 0
169.3 0 −0.939693 + 0.342020i 0 3.28077 + 0.578489i 0 0.417285 − 2.36654i 0 0.766044 − 0.642788i 0
289.1 0 −0.939693 − 0.342020i 0 −1.79202 + 0.315982i 0 −0.352471 − 1.99897i 0 0.766044 + 0.642788i 0
289.2 0 −0.939693 − 0.342020i 0 −1.48875 + 0.262507i 0 0.180286 + 1.02245i 0 0.766044 + 0.642788i 0
289.3 0 −0.939693 − 0.342020i 0 3.28077 − 0.578489i 0 0.417285 + 2.36654i 0 0.766044 + 0.642788i 0
337.1 0 0.173648 + 0.984808i 0 −1.81250 − 2.16006i 0 0.309819 − 0.259969i 0 −0.939693 + 0.342020i 0
337.2 0 0.173648 + 0.984808i 0 −0.308062 − 0.367134i 0 −2.54489 + 2.13542i 0 −0.939693 + 0.342020i 0
337.3 0 0.173648 + 0.984808i 0 2.12056 + 2.52719i 0 −1.00275 + 0.841410i 0 −0.939693 + 0.342020i 0
361.1 0 0.173648 − 0.984808i 0 −1.81250 + 2.16006i 0 0.309819 + 0.259969i 0 −0.939693 − 0.342020i 0
361.2 0 0.173648 − 0.984808i 0 −0.308062 + 0.367134i 0 −2.54489 − 2.13542i 0 −0.939693 − 0.342020i 0
361.3 0 0.173648 − 0.984808i 0 2.12056 − 2.52719i 0 −1.00275 − 0.841410i 0 −0.939693 − 0.342020i 0
373.1 0 0.766044 − 0.642788i 0 −0.654440 + 1.79806i 0 −0.283425 − 0.103158i 0 0.173648 − 0.984808i 0
373.2 0 0.766044 − 0.642788i 0 −0.301078 + 0.827204i 0 4.88956 + 1.77965i 0 0.173648 − 0.984808i 0
373.3 0 0.766044 − 0.642788i 0 0.955517 − 2.62526i 0 −1.61341 − 0.587231i 0 0.173648 − 0.984808i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 25.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.bb.a ✓ 18
3.b odd 2 1 1332.2.ct.d 18
37.h even 18 1 inner 444.2.bb.a ✓ 18
111.n odd 18 1 1332.2.ct.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.bb.a ✓ 18 1.a even 1 1 trivial
444.2.bb.a ✓ 18 37.h even 18 1 inner
1332.2.ct.d 18 3.b odd 2 1
1332.2.ct.d 18 111.n odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 3 T_{5}^{16} + 30 T_{5}^{14} - 126 T_{5}^{13} - 703 T_{5}^{12} - 630 T_{5}^{11} + \cdots + 36963 \) acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} - 3 T^{16} + \cdots + 36963 \) Copy content Toggle raw display
$7$ \( T^{18} - 15 T^{16} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{18} + 6 T^{17} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( T^{18} - 18 T^{17} + \cdots + 34334067 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 220694787 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 4824992448 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 5717539008 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 1561154328387 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 2099859310272 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 129961739795077 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 792422212761 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 397330916487168 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 30559756935744 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 32982555321 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 45\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 253534353738363 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 467669970496 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 81252362304 \) Copy content Toggle raw display
$73$ \( (T^{9} + 12 T^{8} + \cdots - 3546379)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 24\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 166131238464 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 28\!\cdots\!43 \) Copy content Toggle raw display
$97$ \( T^{18} + 18 T^{17} + \cdots + 42887883 \) Copy content Toggle raw display
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