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Properties

Label 441.4.e.q
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,4,Mod(226,441)] 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("441.226"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-3,0,-17,6,0,0,174,0,66,-6,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{4} + (2 \beta_{3} + 2 \beta_1 + 2) q^{5} + (5 \beta_{2} + 41) q^{8} + ( - 6 \beta_{3} + 6 \beta_{2} - 30 \beta_1) q^{10}+ \cdots + (276 \beta_{2} + 266) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 17 q^{4} + 6 q^{5} + 174 q^{8} + 66 q^{10} - 6 q^{11} + 32 q^{13} - 137 q^{16} - 6 q^{17} - 64 q^{19} - 444 q^{20} - 552 q^{22} + 6 q^{23} + 118 q^{25} + 318 q^{26} + 504 q^{29} - 40 q^{31}+ \cdots + 1616 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
2.13746 − 0.656712i
−1.63746 + 1.52274i
2.13746 + 0.656712i
−1.63746 − 1.52274i
−2.63746 + 4.56821i 0 −9.91238 − 17.1687i 5.27492 − 9.13642i 0 0 62.3746 0 27.8248 + 48.1939i
226.2 1.13746 − 1.97014i 0 1.41238 + 2.44631i −2.27492 + 3.94027i 0 0 24.6254 0 5.17525 + 8.96379i
361.1 −2.63746 − 4.56821i 0 −9.91238 + 17.1687i 5.27492 + 9.13642i 0 0 62.3746 0 27.8248 − 48.1939i
361.2 1.13746 + 1.97014i 0 1.41238 − 2.44631i −2.27492 − 3.94027i 0 0 24.6254 0 5.17525 − 8.96379i
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.q 4
3.b odd 2 1 147.4.e.l 4
7.b odd 2 1 441.4.e.p 4
7.c even 3 1 63.4.a.e 2
7.c even 3 1 inner 441.4.e.q 4
7.d odd 6 1 441.4.a.r 2
7.d odd 6 1 441.4.e.p 4
21.c even 2 1 147.4.e.m 4
21.g even 6 1 147.4.a.i 2
21.g even 6 1 147.4.e.m 4
21.h odd 6 1 21.4.a.c ✓ 2
21.h odd 6 1 147.4.e.l 4
28.g odd 6 1 1008.4.a.ba 2
35.j even 6 1 1575.4.a.p 2
84.j odd 6 1 2352.4.a.bz 2
84.n even 6 1 336.4.a.m 2
105.o odd 6 1 525.4.a.n 2
105.x even 12 2 525.4.d.g 4
168.s odd 6 1 1344.4.a.bg 2
168.v even 6 1 1344.4.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c ✓ 2 21.h odd 6 1
63.4.a.e 2 7.c even 3 1
147.4.a.i 2 21.g even 6 1
147.4.e.l 4 3.b odd 2 1
147.4.e.l 4 21.h odd 6 1
147.4.e.m 4 21.c even 2 1
147.4.e.m 4 21.g even 6 1
336.4.a.m 2 84.n even 6 1
441.4.a.r 2 7.d odd 6 1
441.4.e.p 4 7.b odd 2 1
441.4.e.p 4 7.d odd 6 1
441.4.e.q 4 1.a even 1 1 trivial
441.4.e.q 4 7.c even 3 1 inner
525.4.a.n 2 105.o odd 6 1
525.4.d.g 4 105.x even 12 2
1008.4.a.ba 2 28.g odd 6 1
1344.4.a.bg 2 168.s odd 6 1
1344.4.a.bo 2 168.v even 6 1
1575.4.a.p 2 35.j even 6 1
2352.4.a.bz 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{3} + 21T_{2}^{2} - 36T_{2} + 144 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{3} + 84T_{5}^{2} + 288T_{5} + 2304 \) Copy content Toggle raw display
\( T_{13}^{2} - 16T_{13} - 1988 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 2005056 \) Copy content Toggle raw display
$13$ \( (T^{2} - 16 T - 1988)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( T^{4} + 64 T^{3} + \cdots + 51609856 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 271063296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 252 T + 7668)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 5398134784 \) Copy content Toggle raw display
$37$ \( T^{4} - 248 T^{3} + \cdots + 9560464 \) Copy content Toggle raw display
$41$ \( (T^{2} - 450 T + 37800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 376 T + 2512)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 4337012736 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 92705634576 \) Copy content Toggle raw display
$59$ \( T^{4} - 804 T^{3} + \cdots + 908660736 \) Copy content Toggle raw display
$61$ \( T^{4} - 428 T^{3} + \cdots + 788261776 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 25836061696 \) Copy content Toggle raw display
$71$ \( (T^{2} + 954 T + 214704)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 81364139536 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 7126061056 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1944 T + 813456)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 64438807104 \) Copy content Toggle raw display
$97$ \( (T^{2} - 808 T - 922292)^{2} \) Copy content Toggle raw display
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