Properties

Label 441.5.b.c
Level $441$
Weight $5$
Character orbit 441.b
Analytic conductor $45.586$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,5,Mod(197,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.197");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 441.b (of order \(2\), degree \(1\), 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: \(45.5861537200\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{93})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 41x^{2} + 42x + 627 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 14 q^{4} + \beta_{3} q^{5} + 30 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 14 q^{4} + \beta_{3} q^{5} + 30 \beta_1 q^{8} - \beta_{2} q^{10} + 106 \beta_1 q^{11} + 5 \beta_{2} q^{13} + 164 q^{16} - 5 \beta_{3} q^{17} + 10 \beta_{2} q^{19} + 14 \beta_{3} q^{20} - 212 q^{22} + 74 \beta_1 q^{23} - 1049 q^{25} + 10 \beta_{3} q^{26} + 509 \beta_1 q^{29} - 10 \beta_{2} q^{31} + 644 \beta_1 q^{32} + 5 \beta_{2} q^{34} - 750 q^{37} + 20 \beta_{3} q^{38} - 30 \beta_{2} q^{40} - 45 \beta_{3} q^{41} + 2820 q^{43} + 1484 \beta_1 q^{44} - 148 q^{46} - 40 \beta_{3} q^{47} - 1049 \beta_1 q^{50} + 70 \beta_{2} q^{52} + 1139 \beta_1 q^{53} - 106 \beta_{2} q^{55} - 1018 q^{58} - 40 \beta_{3} q^{59} - 20 \beta_{2} q^{61} - 20 \beta_{3} q^{62} + 1336 q^{64} + 8370 \beta_1 q^{65} - 3272 q^{67} - 70 \beta_{3} q^{68} - 5654 \beta_1 q^{71} - 115 \beta_{2} q^{73} - 750 \beta_1 q^{74} + 140 \beta_{2} q^{76} - 8668 q^{79} + 164 \beta_{3} q^{80} + 45 \beta_{2} q^{82} + 80 \beta_{3} q^{83} + 8370 q^{85} + 2820 \beta_1 q^{86} - 6360 q^{88} + 355 \beta_{3} q^{89} + 1036 \beta_1 q^{92} + 40 \beta_{2} q^{94} + 16740 \beta_1 q^{95} + 115 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 56 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 56 q^{4} + 656 q^{16} - 848 q^{22} - 4196 q^{25} - 3000 q^{37} + 11280 q^{43} - 592 q^{46} - 4072 q^{58} + 5344 q^{64} - 13088 q^{67} - 34672 q^{79} + 33480 q^{85} - 25440 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 41x^{2} + 42x + 627 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 33\nu + 17 ) / 101 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -24\nu^{3} + 36\nu^{2} + 1608\nu - 810 ) / 101 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} - 3\nu - 63 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 12\beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + \beta_{2} + 12\beta _1 + 258 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + 137\beta _1 + 64 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
5.32183 1.41421i
−4.32183 1.41421i
−4.32183 + 1.41421i
5.32183 + 1.41421i
1.41421i 0 14.0000 40.9145i 0 0 42.4264i 0 −57.8619
197.2 1.41421i 0 14.0000 40.9145i 0 0 42.4264i 0 57.8619
197.3 1.41421i 0 14.0000 40.9145i 0 0 42.4264i 0 57.8619
197.4 1.41421i 0 14.0000 40.9145i 0 0 42.4264i 0 −57.8619
\(n\): 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.5.b.c 4
3.b odd 2 1 inner 441.5.b.c 4
7.b odd 2 1 inner 441.5.b.c 4
21.c even 2 1 inner 441.5.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.5.b.c 4 1.a even 1 1 trivial
441.5.b.c 4 3.b odd 2 1 inner
441.5.b.c 4 7.b odd 2 1 inner
441.5.b.c 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 83700 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1674)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 22472)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 83700)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 41850)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 334800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 10952)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 518162)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 334800)^{2} \) Copy content Toggle raw display
$37$ \( (T + 750)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3389850)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2820)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2678400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2594642)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2678400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 1339200)^{2} \) Copy content Toggle raw display
$67$ \( (T + 3272)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 63935432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 44277300)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8668)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10713600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 210965850)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 44277300)^{2} \) Copy content Toggle raw display
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