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Properties

Label 441.2.u.b
Level $441$
Weight $2$
Character orbit 441.u
Analytic conductor $3.521$
Analytic rank $0$
Dimension $12$
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] = [441,2,Mod(64,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.u (of order \(7\), degree \(6\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{9} + \cdots - \beta_{2}) q^{2}+ \cdots + (\beta_{10} + \beta_{5} - \beta_{4} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{9} + \cdots - \beta_{2}) q^{2}+ \cdots + (\beta_{11} - 9 \beta_{10} - \beta_{9} + \cdots + 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 2 q^{4} + 7 q^{5} - 7 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 2 q^{4} + 7 q^{5} - 7 q^{7} - 6 q^{8} + 14 q^{10} + 8 q^{11} + 7 q^{13} + 28 q^{14} + 16 q^{16} - 14 q^{19} - 7 q^{20} + 13 q^{22} + 2 q^{23} + 3 q^{25} + 7 q^{26} + 14 q^{28} + 11 q^{29} - 14 q^{31} + 24 q^{32} - 42 q^{34} - 21 q^{35} - 30 q^{37} - 21 q^{38} - 21 q^{41} + 17 q^{43} + 6 q^{44} - 16 q^{46} + 21 q^{47} + 7 q^{49} + 46 q^{50} - 7 q^{52} - 6 q^{53} - 28 q^{55} + 14 q^{56} - 32 q^{58} - 14 q^{59} - 7 q^{61} - 56 q^{62} + 14 q^{64} - 14 q^{65} + 48 q^{67} - 56 q^{68} + 21 q^{70} + 39 q^{71} + 42 q^{73} - 61 q^{74} - 28 q^{76} - 21 q^{77} - 16 q^{79} - 42 q^{80} + 28 q^{82} + 7 q^{83} + 28 q^{85} - 17 q^{86} - 11 q^{88} + 14 q^{89} - 21 q^{91} - 16 q^{92} - 49 q^{94} + 7 q^{95} - 28 q^{97} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{21}^{2} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{21}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{21}^{5} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{21}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{21}^{8} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{21}^{9} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{21}^{11} + \zeta_{21} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \zeta_{21}^{8} + \zeta_{21}^{7} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{21}^{10} - \zeta_{21}^{8} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{21}^{11} + \zeta_{21}^{9} - \zeta_{21}^{8} + \zeta_{21}^{6} - \zeta_{21}^{4} + \zeta_{21}^{3} - \zeta_{21} + 1 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{21}^{11} + \zeta_{21}^{10} - \zeta_{21}^{8} + \zeta_{21}^{7} - \zeta_{21}^{5} + \zeta_{21}^{3} - \zeta_{21}^{2} + \zeta_{21} + 1 \) Copy content Toggle raw display

Display \(\zeta_{21}^j\) in terms of \(\beta_i\)

\(\zeta_{21}\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + 3\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{2}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 3\beta_{5} - \beta_{3} + \beta_{2} + 6\beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{21}^{4}\)\(=\) \( ( \beta_{11} - 7 \beta_{10} + \beta_{9} - \beta_{8} - 6 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 7 \beta_{4} + \cdots + 6 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{5}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 3\beta_{5} + 6\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{21}^{7}\)\(=\) \( ( \beta_{11} + \beta_{9} + 6\beta_{8} + \beta_{7} - 4\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{8}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + 4\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{9}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{21}^{10}\)\(=\) \( ( \beta_{11} - 6\beta_{9} - \beta_{8} + \beta_{7} - 4\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7 \) Copy content Toggle raw display
\(\zeta_{21}^{11}\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} + 6\beta_{7} - 3\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{21}^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.955573 − 0.294755i
−0.733052 − 0.680173i
0.826239 − 0.563320i
0.0747301 + 0.997204i
0.365341 + 0.930874i
−0.988831 − 0.149042i
0.365341 − 0.930874i
−0.988831 + 0.149042i
0.826239 + 0.563320i
0.0747301 − 0.997204i
0.955573 + 0.294755i
−0.733052 + 0.680173i
−0.367711 + 1.61105i 0 −0.658322 − 0.317031i 2.42463 − 3.04039i 0 −1.62586 − 2.08724i −1.30778 + 1.63991i 0 4.00665 + 5.02418i
64.2 −0.0332580 + 0.145713i 0 1.78181 + 0.858075i −1.36967 + 1.71752i 0 −2.64558 + 0.0302261i −0.370666 + 0.464800i 0 −0.204712 − 0.256700i
127.1 −0.658322 − 0.317031i 0 −0.914101 − 1.14625i 0.0575591 + 0.252183i 0 −2.16885 + 1.51528i 0.563561 + 2.46912i 0 0.0420574 − 0.184265i
127.2 1.78181 + 0.858075i 0 1.19158 + 1.49419i −0.359497 − 1.57506i 0 1.95991 − 1.77729i −0.0391023 − 0.171318i 0 0.710963 − 3.11493i
190.1 −0.914101 + 1.14625i 0 −0.0332580 − 0.145713i 0.830509 + 0.399952i 0 −0.938402 + 2.47374i −2.44440 − 1.17716i 0 −1.21761 + 0.586371i
190.2 1.19158 − 1.49419i 0 −0.367711 − 1.61105i 1.91647 + 0.922924i 0 1.91879 + 1.82161i 0.598393 + 0.288171i 0 3.66265 − 1.76384i
253.1 −0.914101 − 1.14625i 0 −0.0332580 + 0.145713i 0.830509 − 0.399952i 0 −0.938402 − 2.47374i −2.44440 + 1.17716i 0 −1.21761 − 0.586371i
253.2 1.19158 + 1.49419i 0 −0.367711 + 1.61105i 1.91647 − 0.922924i 0 1.91879 − 1.82161i 0.598393 − 0.288171i 0 3.66265 + 1.76384i
316.1 −0.658322 + 0.317031i 0 −0.914101 + 1.14625i 0.0575591 − 0.252183i 0 −2.16885 − 1.51528i 0.563561 − 2.46912i 0 0.0420574 + 0.184265i
316.2 1.78181 − 0.858075i 0 1.19158 − 1.49419i −0.359497 + 1.57506i 0 1.95991 + 1.77729i −0.0391023 + 0.171318i 0 0.710963 + 3.11493i
379.1 −0.367711 − 1.61105i 0 −0.658322 + 0.317031i 2.42463 + 3.04039i 0 −1.62586 + 2.08724i −1.30778 − 1.63991i 0 4.00665 − 5.02418i
379.2 −0.0332580 − 0.145713i 0 1.78181 − 0.858075i −1.36967 − 1.71752i 0 −2.64558 − 0.0302261i −0.370666 − 0.464800i 0 −0.204712 + 0.256700i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.u.b 12
3.b odd 2 1 49.2.e.b ✓ 12
12.b even 2 1 784.2.u.b 12
21.c even 2 1 343.2.e.b 12
21.g even 6 1 343.2.g.a 12
21.g even 6 1 343.2.g.c 12
21.h odd 6 1 343.2.g.b 12
21.h odd 6 1 343.2.g.d 12
49.e even 7 1 inner 441.2.u.b 12
147.k even 14 1 343.2.e.b 12
147.k even 14 1 2401.2.a.d 6
147.l odd 14 1 49.2.e.b ✓ 12
147.l odd 14 1 2401.2.a.c 6
147.n odd 42 1 343.2.g.b 12
147.n odd 42 1 343.2.g.d 12
147.o even 42 1 343.2.g.a 12
147.o even 42 1 343.2.g.c 12
588.u even 14 1 784.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.b ✓ 12 3.b odd 2 1
49.2.e.b ✓ 12 147.l odd 14 1
343.2.e.b 12 21.c even 2 1
343.2.e.b 12 147.k even 14 1
343.2.g.a 12 21.g even 6 1
343.2.g.a 12 147.o even 42 1
343.2.g.b 12 21.h odd 6 1
343.2.g.b 12 147.n odd 42 1
343.2.g.c 12 21.g even 6 1
343.2.g.c 12 147.o even 42 1
343.2.g.d 12 21.h odd 6 1
343.2.g.d 12 147.n odd 42 1
441.2.u.b 12 1.a even 1 1 trivial
441.2.u.b 12 49.e even 7 1 inner
784.2.u.b 12 12.b even 2 1
784.2.u.b 12 588.u even 14 1
2401.2.a.c 6 147.l odd 14 1
2401.2.a.d 6 147.k even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 2 T_{2}^{11} + 3 T_{2}^{10} - 4 T_{2}^{9} + 12 T_{2}^{8} - 6 T_{2}^{7} + 7 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 7 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{12} + 7 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 8 T^{11} + \cdots + 1849 \) Copy content Toggle raw display
$13$ \( T^{12} - 7 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{12} + 35 T^{10} + \cdots + 3087049 \) Copy content Toggle raw display
$19$ \( (T^{6} + 7 T^{5} + \cdots + 2107)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} - 11 T^{11} + \cdots + 1681 \) Copy content Toggle raw display
$31$ \( (T^{6} + 7 T^{5} + \cdots - 8183)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 295118041 \) Copy content Toggle raw display
$41$ \( T^{12} + 21 T^{11} + \cdots + 10413529 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 200307409 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 3021810841 \) Copy content Toggle raw display
$53$ \( T^{12} + 6 T^{11} + \cdots + 2985984 \) Copy content Toggle raw display
$59$ \( T^{12} + 14 T^{11} + \cdots + 54125449 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 261242569 \) Copy content Toggle raw display
$67$ \( (T^{6} - 24 T^{5} + \cdots - 293)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 312925003609 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 460917961 \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 21629)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 118178641 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 89755965649 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 18571)^{2} \) Copy content Toggle raw display
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