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Properties

Label 441.4.e.s
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(226,441)]
 
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([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//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);
 
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

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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 19x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 11 \beta_{2} q^{4} - 2 \beta_1 q^{5} + 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 11 \beta_{2} q^{4} - 2 \beta_1 q^{5} + 3 \beta_{3} q^{8} - 38 \beta_{2} q^{10} + (10 \beta_{3} + 10 \beta_1) q^{11} - 82 q^{13} + (31 \beta_{2} + 31) q^{16} + ( - 18 \beta_{3} - 18 \beta_1) q^{17} + ( - 20 \beta_{2} - 20) q^{19} - 22 \beta_{3} q^{20} - 190 q^{22} - 30 \beta_1 q^{23} - 49 \beta_{2} q^{25} - 82 \beta_1 q^{26} + 56 \beta_{3} q^{29} - 156 \beta_{2} q^{31} + (55 \beta_{3} + 55 \beta_1) q^{32} + 342 q^{34} + ( - 186 \beta_{2} - 186) q^{37} + ( - 20 \beta_{3} - 20 \beta_1) q^{38} + (114 \beta_{2} + 114) q^{40} + 38 \beta_{3} q^{41} + 164 q^{43} - 110 \beta_1 q^{44} - 570 \beta_{2} q^{46} - 108 \beta_1 q^{47} - 49 \beta_{3} q^{50} - 902 \beta_{2} q^{52} + (36 \beta_{3} + 36 \beta_1) q^{53} + 380 q^{55} + ( - 1064 \beta_{2} - 1064) q^{58} + (36 \beta_{3} + 36 \beta_1) q^{59} + (790 \beta_{2} + 790) q^{61} - 156 \beta_{3} q^{62} - 797 q^{64} + 164 \beta_1 q^{65} - 44 \beta_{2} q^{67} + 198 \beta_1 q^{68} + 102 \beta_{3} q^{71} - 126 \beta_{2} q^{73} + ( - 186 \beta_{3} - 186 \beta_1) q^{74} + 220 q^{76} + (712 \beta_{2} + 712) q^{79} + ( - 62 \beta_{3} - 62 \beta_1) q^{80} + ( - 722 \beta_{2} - 722) q^{82} + 336 \beta_{3} q^{83} - 684 q^{85} + 164 \beta_1 q^{86} - 570 \beta_{2} q^{88} + 334 \beta_1 q^{89} - 330 \beta_{3} q^{92} - 2052 \beta_{2} q^{94} + (40 \beta_{3} + 40 \beta_1) q^{95} - 798 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 22 q^{4} + 76 q^{10} - 328 q^{13} + 62 q^{16} - 40 q^{19} - 760 q^{22} + 98 q^{25} + 312 q^{31} + 1368 q^{34} - 372 q^{37} + 228 q^{40} + 656 q^{43} + 1140 q^{46} + 1804 q^{52} + 1520 q^{55} - 2128 q^{58} + 1580 q^{61} - 3188 q^{64} + 88 q^{67} + 252 q^{73} + 880 q^{76} + 1424 q^{79} - 1444 q^{82} - 2736 q^{85} + 1140 q^{88} + 4104 q^{94} - 3192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 19x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 19 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 19\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−2.17945 + 3.77492i
2.17945 − 3.77492i
−2.17945 − 3.77492i
2.17945 + 3.77492i
−2.17945 + 3.77492i 0 −5.50000 − 9.52628i 4.35890 − 7.54983i 0 0 13.0767 0 19.0000 + 32.9090i
226.2 2.17945 − 3.77492i 0 −5.50000 − 9.52628i −4.35890 + 7.54983i 0 0 −13.0767 0 19.0000 + 32.9090i
361.1 −2.17945 − 3.77492i 0 −5.50000 + 9.52628i 4.35890 + 7.54983i 0 0 13.0767 0 19.0000 − 32.9090i
361.2 2.17945 + 3.77492i 0 −5.50000 + 9.52628i −4.35890 − 7.54983i 0 0 −13.0767 0 19.0000 − 32.9090i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 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 441.4.e.s 4
3.b odd 2 1 inner 441.4.e.s 4
7.b odd 2 1 441.4.e.r 4
7.c even 3 1 441.4.a.q 2
7.c even 3 1 inner 441.4.e.s 4
7.d odd 6 1 63.4.a.d ✓ 2
7.d odd 6 1 441.4.e.r 4
21.c even 2 1 441.4.e.r 4
21.g even 6 1 63.4.a.d ✓ 2
21.g even 6 1 441.4.e.r 4
21.h odd 6 1 441.4.a.q 2
21.h odd 6 1 inner 441.4.e.s 4
28.f even 6 1 1008.4.a.be 2
35.i odd 6 1 1575.4.a.t 2
84.j odd 6 1 1008.4.a.be 2
105.p even 6 1 1575.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d ✓ 2 7.d odd 6 1
63.4.a.d ✓ 2 21.g even 6 1
441.4.a.q 2 7.c even 3 1
441.4.a.q 2 21.h odd 6 1
441.4.e.r 4 7.b odd 2 1
441.4.e.r 4 7.d odd 6 1
441.4.e.r 4 21.c even 2 1
441.4.e.r 4 21.g even 6 1
441.4.e.s 4 1.a even 1 1 trivial
441.4.e.s 4 3.b odd 2 1 inner
441.4.e.s 4 7.c even 3 1 inner
441.4.e.s 4 21.h odd 6 1 inner
1008.4.a.be 2 28.f even 6 1
1008.4.a.be 2 84.j odd 6 1
1575.4.a.t 2 35.i odd 6 1
1575.4.a.t 2 105.p even 6 1

Hecke kernels

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

\( T_{2}^{4} + 19T_{2}^{2} + 361 \) Copy content Toggle raw display
\( T_{5}^{4} + 76T_{5}^{2} + 5776 \) Copy content Toggle raw display
\( T_{13} + 82 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 19T^{2} + 361 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 76T^{2} + 5776 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 1900 T^{2} + 3610000 \) Copy content Toggle raw display
$13$ \( (T + 82)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 6156 T^{2} + 37896336 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 17100 T^{2} + 292410000 \) Copy content Toggle raw display
$29$ \( (T^{2} - 59584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 156 T + 24336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 186 T + 34596)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 27436)^{2} \) Copy content Toggle raw display
$43$ \( (T - 164)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 49113651456 \) Copy content Toggle raw display
$53$ \( T^{4} + 24624 T^{2} + 606341376 \) Copy content Toggle raw display
$59$ \( T^{4} + 24624 T^{2} + 606341376 \) Copy content Toggle raw display
$61$ \( (T^{2} - 790 T + 624100)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 44 T + 1936)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 197676)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 126 T + 15876)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 712 T + 506944)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2145024)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 4492551550096 \) Copy content Toggle raw display
$97$ \( (T + 798)^{4} \) Copy content Toggle raw display
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