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Properties

Label 441.3.r.b
Level $441$
Weight $3$
Character orbit 441.r
Analytic conductor $12.016$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,3,Mod(50,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.50");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.r (of order \(6\), degree \(2\), 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.63369648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{2} + 3 \beta_{2} q^{3} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{4} + (\beta_{4} + 2 \beta_{2} + 4) q^{5} + (3 \beta_{5} + 3 \beta_{4} + 3) q^{6} + ( - 3 \beta_{5} - 3 \beta_{4} + \cdots - 2) q^{8}+ \cdots + ( - 9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 1) q^{2} + 3 \beta_{2} q^{3} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{4} + (\beta_{4} + 2 \beta_{2} + 4) q^{5} + (3 \beta_{5} + 3 \beta_{4} + 3) q^{6} + ( - 3 \beta_{5} - 3 \beta_{4} + \cdots - 2) q^{8}+ \cdots + ( - 18 \beta_{4} + 9 \beta_{3} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 9 q^{3} + 13 q^{4} + 15 q^{5} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 9 q^{3} + 13 q^{4} + 15 q^{5} - 27 q^{9} + 38 q^{10} + 9 q^{11} + 39 q^{12} + 11 q^{13} - 45 q^{15} - 47 q^{16} + 27 q^{18} + 38 q^{19} + 45 q^{20} + 65 q^{22} - 15 q^{23} - 63 q^{24} - 26 q^{25} + 162 q^{27} - 51 q^{29} - 57 q^{30} + 46 q^{31} + 291 q^{32} + 93 q^{34} - 234 q^{36} - 14 q^{37} + 21 q^{38} + 33 q^{39} + 57 q^{40} - 27 q^{41} - 99 q^{43} + 114 q^{46} + 156 q^{47} + 282 q^{48} + 294 q^{50} - 99 q^{51} + 63 q^{52} - 81 q^{54} + 166 q^{55} - 57 q^{57} - 7 q^{58} - 144 q^{59} - 22 q^{61} - 138 q^{64} + 147 q^{65} + 195 q^{66} + 98 q^{67} + 567 q^{68} + 45 q^{69} + 189 q^{72} + 202 q^{73} + 411 q^{74} + 156 q^{75} - 99 q^{76} + 243 q^{78} - 90 q^{79} - 243 q^{81} - 302 q^{82} + 99 q^{83} - 159 q^{85} - 249 q^{86} - 495 q^{88} - 171 q^{90} + 147 q^{92} + 138 q^{93} - 444 q^{94} + 135 q^{95} - 873 q^{96} - 161 q^{97} - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 24\nu^{4} + 288\nu^{3} - 236\nu^{2} - 66\nu + 2241 ) / 1449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44\nu^{5} - 45\nu^{4} + 540\nu^{3} + 604\nu^{2} + 5310\nu + 36 ) / 1449 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 3\nu^{4} + 36\nu^{3} + 16\nu^{2} + 354\nu + 99 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{5} - 30\nu^{4} + 843\nu^{3} + 2281\nu^{2} + 9819\nu + 5337 ) / 1449 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\nu^{5} - 62\nu^{4} + 422\nu^{3} + 249\nu^{2} + 3130\nu - 1335 ) / 483 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta_{3} - 7\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} + 13\beta _1 - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -24\beta_{5} - 12\beta_{4} - 19\beta_{3} + 94\beta_{2} + 19\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{5} - 104\beta_{4} - 187\beta_{3} + 571\beta_{2} + 519 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
−1.34153 + 2.32360i
1.98253 − 3.43384i
−0.140998 + 0.244215i
−1.34153 − 2.32360i
1.98253 + 3.43384i
−0.140998 − 0.244215i
−3.28247 + 1.89513i −1.50000 + 2.59808i 5.18306 − 8.97732i −0.282467 − 0.163082i − 11.3708i 0 24.1293i −4.50000 − 7.79423i 1.23625
50.2 −0.895777 + 0.517177i −1.50000 + 2.59808i −1.46506 + 2.53755i 2.10422 + 1.21487i − 3.10306i 0 − 7.16819i −4.50000 − 7.79423i −2.51322
50.3 2.67824 − 1.54629i −1.50000 + 2.59808i 2.78200 − 4.81856i 5.67824 + 3.27834i 9.27771i 0 − 4.83675i −4.50000 − 7.79423i 20.2770
344.1 −3.28247 − 1.89513i −1.50000 − 2.59808i 5.18306 + 8.97732i −0.282467 + 0.163082i 11.3708i 0 − 24.1293i −4.50000 + 7.79423i 1.23625
344.2 −0.895777 − 0.517177i −1.50000 − 2.59808i −1.46506 − 2.53755i 2.10422 − 1.21487i 3.10306i 0 7.16819i −4.50000 + 7.79423i −2.51322
344.3 2.67824 + 1.54629i −1.50000 − 2.59808i 2.78200 + 4.81856i 5.67824 − 3.27834i − 9.27771i 0 4.83675i −4.50000 + 7.79423i 20.2770
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.r.b 6
7.b odd 2 1 441.3.r.c 6
7.c even 3 1 63.3.j.a ✓ 6
7.c even 3 1 63.3.n.a yes 6
7.d odd 6 1 441.3.j.c 6
7.d odd 6 1 441.3.n.c 6
9.d odd 6 1 inner 441.3.r.b 6
21.h odd 6 1 189.3.j.a 6
21.h odd 6 1 189.3.n.a 6
63.g even 3 1 189.3.j.a 6
63.h even 3 1 189.3.n.a 6
63.i even 6 1 441.3.n.c 6
63.j odd 6 1 63.3.n.a yes 6
63.n odd 6 1 63.3.j.a ✓ 6
63.o even 6 1 441.3.r.c 6
63.s even 6 1 441.3.j.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.a ✓ 6 7.c even 3 1
63.3.j.a ✓ 6 63.n odd 6 1
63.3.n.a yes 6 7.c even 3 1
63.3.n.a yes 6 63.j odd 6 1
189.3.j.a 6 21.h odd 6 1
189.3.j.a 6 63.g even 3 1
189.3.n.a 6 21.h odd 6 1
189.3.n.a 6 63.h even 3 1
441.3.j.c 6 7.d odd 6 1
441.3.j.c 6 63.s even 6 1
441.3.n.c 6 7.d odd 6 1
441.3.n.c 6 63.i even 6 1
441.3.r.b 6 1.a even 1 1 trivial
441.3.r.b 6 9.d odd 6 1 inner
441.3.r.c 6 7.b odd 2 1
441.3.r.c 6 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{6} + 3T_{2}^{5} - 8T_{2}^{4} - 33T_{2}^{3} + 100T_{2}^{2} + 231T_{2} + 147 \) Copy content Toggle raw display
\( T_{5}^{6} - 15T_{5}^{5} + 88T_{5}^{4} - 195T_{5}^{3} + 124T_{5}^{2} + 117T_{5} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} + \cdots + 147 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 15 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 9 T^{5} + \cdots + 1982907 \) Copy content Toggle raw display
$13$ \( T^{6} - 11 T^{5} + \cdots + 499849 \) Copy content Toggle raw display
$17$ \( T^{6} + 1593 T^{4} + \cdots + 31434507 \) Copy content Toggle raw display
$19$ \( (T^{3} - 19 T^{2} + \cdots + 463)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 15 T^{5} + \cdots + 128547 \) Copy content Toggle raw display
$29$ \( T^{6} + 51 T^{5} + \cdots + 694083 \) Copy content Toggle raw display
$31$ \( T^{6} - 46 T^{5} + \cdots + 21455424 \) Copy content Toggle raw display
$37$ \( (T^{3} + 7 T^{2} + \cdots - 23751)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 1387137027 \) Copy content Toggle raw display
$43$ \( T^{6} + 99 T^{5} + \cdots + 432265681 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 29274835968 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 5141134827 \) Copy content Toggle raw display
$59$ \( T^{6} + 144 T^{5} + \cdots + 813189888 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 25823204416 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 44264793664 \) Copy content Toggle raw display
$71$ \( T^{6} + 5568 T^{4} + \cdots + 109734912 \) Copy content Toggle raw display
$73$ \( (T^{3} - 101 T^{2} + \cdots + 808473)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 90 T^{5} + \cdots + 279290944 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 165842832483 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 15857178627 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 36908941689 \) Copy content Toggle raw display
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