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Properties

Label 441.2.p.b
Level $441$
Weight $2$
Character orbit 441.p
Analytic conductor $3.521$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(80,441)] 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("441.80"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - 2 \beta_{4} + \beta_{3} + 2) q^{4} + (\beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_1) q^{8} + (\beta_{7} + \beta_{5}) q^{11} + (2 \beta_{6} - 3 \beta_{4}) q^{16} + ( - \beta_{3} + 5) q^{22}+ \cdots + ( - 3 \beta_{7} + 15 \beta_{5} + \cdots - 15 \beta_1) q^{92}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 258\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 440\nu^{3} + 576\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{7} + 110\nu^{5} - 715\nu^{3} + 936\nu ) / 165 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 4\beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 6\beta_{5} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 23\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} - 39\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{3} - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -55\beta_{2} - 258\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−2.23256 + 1.28897i
−1.00781 + 0.581861i
1.00781 − 0.581861i
2.23256 − 1.28897i
−2.23256 − 1.28897i
−1.00781 − 0.581861i
1.00781 + 0.581861i
2.23256 + 1.28897i
−2.23256 + 1.28897i 0 2.32288 − 4.02334i 0 0 0 6.82058i 0 0
80.2 −1.00781 + 0.581861i 0 −0.322876 + 0.559237i 0 0 0 − 3.07892i 0 0
80.3 1.00781 − 0.581861i 0 −0.322876 + 0.559237i 0 0 0 3.07892i 0 0
80.4 2.23256 − 1.28897i 0 2.32288 − 4.02334i 0 0 0 − 6.82058i 0 0
215.1 −2.23256 − 1.28897i 0 2.32288 + 4.02334i 0 0 0 − 6.82058i 0 0
215.2 −1.00781 − 0.581861i 0 −0.322876 − 0.559237i 0 0 0 3.07892i 0 0
215.3 1.00781 + 0.581861i 0 −0.322876 − 0.559237i 0 0 0 − 3.07892i 0 0
215.4 2.23256 + 1.28897i 0 2.32288 + 4.02334i 0 0 0 6.82058i 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 80.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.p.b 8
3.b odd 2 1 inner 441.2.p.b 8
7.b odd 2 1 CM 441.2.p.b 8
7.c even 3 1 63.2.c.a ✓ 4
7.c even 3 1 inner 441.2.p.b 8
7.d odd 6 1 63.2.c.a ✓ 4
7.d odd 6 1 inner 441.2.p.b 8
21.c even 2 1 inner 441.2.p.b 8
21.g even 6 1 63.2.c.a ✓ 4
21.g even 6 1 inner 441.2.p.b 8
21.h odd 6 1 63.2.c.a ✓ 4
21.h odd 6 1 inner 441.2.p.b 8
28.f even 6 1 1008.2.k.a 4
28.g odd 6 1 1008.2.k.a 4
35.i odd 6 1 1575.2.b.a 4
35.j even 6 1 1575.2.b.a 4
35.k even 12 2 1575.2.g.d 8
35.l odd 12 2 1575.2.g.d 8
56.j odd 6 1 4032.2.k.c 4
56.k odd 6 1 4032.2.k.b 4
56.m even 6 1 4032.2.k.b 4
56.p even 6 1 4032.2.k.c 4
63.g even 3 1 567.2.o.f 8
63.h even 3 1 567.2.o.f 8
63.i even 6 1 567.2.o.f 8
63.j odd 6 1 567.2.o.f 8
63.k odd 6 1 567.2.o.f 8
63.n odd 6 1 567.2.o.f 8
63.s even 6 1 567.2.o.f 8
63.t odd 6 1 567.2.o.f 8
84.j odd 6 1 1008.2.k.a 4
84.n even 6 1 1008.2.k.a 4
105.o odd 6 1 1575.2.b.a 4
105.p even 6 1 1575.2.b.a 4
105.w odd 12 2 1575.2.g.d 8
105.x even 12 2 1575.2.g.d 8
168.s odd 6 1 4032.2.k.c 4
168.v even 6 1 4032.2.k.b 4
168.ba even 6 1 4032.2.k.c 4
168.be odd 6 1 4032.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a ✓ 4 7.c even 3 1
63.2.c.a ✓ 4 7.d odd 6 1
63.2.c.a ✓ 4 21.g even 6 1
63.2.c.a ✓ 4 21.h odd 6 1
441.2.p.b 8 1.a even 1 1 trivial
441.2.p.b 8 3.b odd 2 1 inner
441.2.p.b 8 7.b odd 2 1 CM
441.2.p.b 8 7.c even 3 1 inner
441.2.p.b 8 7.d odd 6 1 inner
441.2.p.b 8 21.c even 2 1 inner
441.2.p.b 8 21.g even 6 1 inner
441.2.p.b 8 21.h odd 6 1 inner
567.2.o.f 8 63.g even 3 1
567.2.o.f 8 63.h even 3 1
567.2.o.f 8 63.i even 6 1
567.2.o.f 8 63.j odd 6 1
567.2.o.f 8 63.k odd 6 1
567.2.o.f 8 63.n odd 6 1
567.2.o.f 8 63.s even 6 1
567.2.o.f 8 63.t odd 6 1
1008.2.k.a 4 28.f even 6 1
1008.2.k.a 4 28.g odd 6 1
1008.2.k.a 4 84.j odd 6 1
1008.2.k.a 4 84.n even 6 1
1575.2.b.a 4 35.i odd 6 1
1575.2.b.a 4 35.j even 6 1
1575.2.b.a 4 105.o odd 6 1
1575.2.b.a 4 105.p even 6 1
1575.2.g.d 8 35.k even 12 2
1575.2.g.d 8 35.l odd 12 2
1575.2.g.d 8 105.w odd 12 2
1575.2.g.d 8 105.x even 12 2
4032.2.k.b 4 56.k odd 6 1
4032.2.k.b 4 56.m even 6 1
4032.2.k.b 4 168.v even 6 1
4032.2.k.b 4 168.be odd 6 1
4032.2.k.c 4 56.j odd 6 1
4032.2.k.c 4 56.p even 6 1
4032.2.k.c 4 168.s odd 6 1
4032.2.k.c 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 8T_{2}^{6} + 55T_{2}^{4} - 72T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 8 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 44 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 92 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 112 T^{2} + 12544)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 212 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 284 T^{2} + 12996)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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