Properties

Label 441.6.a.w
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

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: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 18) q^{4} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{5} + (2 \beta_{3} - \beta_{2} + 27 \beta_1 - 38) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 18) q^{4} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{5} + (2 \beta_{3} - \beta_{2} + 27 \beta_1 - 38) q^{8} + ( - \beta_{2} - 23 \beta_1 + 76) q^{10} + ( - 3 \beta_{3} - 9 \beta_{2} + \cdots - 107) q^{11}+ \cdots + ( - 669 \beta_{3} - 863 \beta_{2} + \cdots - 47705) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 89\nu + 52 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 91\beta _1 + 46 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.22385
−1.74818
1.79080
10.1812
−10.2239 0 72.5272 −23.7528 0 0 −414.344 0 242.845
1.2 −2.74818 0 −24.4475 −58.3673 0 0 155.128 0 160.404
1.3 0.790805 0 −31.3746 104.192 0 0 −50.1170 0 82.3953
1.4 9.18123 0 52.2950 −22.0716 0 0 186.333 0 −202.644
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.w 4
3.b odd 2 1 147.6.a.m 4
7.b odd 2 1 441.6.a.v 4
7.c even 3 2 63.6.e.e 8
21.c even 2 1 147.6.a.l 4
21.g even 6 2 147.6.e.o 8
21.h odd 6 2 21.6.e.c 8
84.n even 6 2 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 21.h odd 6 2
63.6.e.e 8 7.c even 3 2
147.6.a.l 4 21.c even 2 1
147.6.a.m 4 3.b odd 2 1
147.6.e.o 8 21.g even 6 2
336.6.q.j 8 84.n even 6 2
441.6.a.v 4 7.b odd 2 1
441.6.a.w 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{4} + 3T_{2}^{3} - 94T_{2}^{2} - 186T_{2} + 204 \) Copy content Toggle raw display
\( T_{5}^{4} - 7657T_{5}^{2} - 302700T_{5} - 3188244 \) Copy content Toggle raw display
\( T_{13}^{4} - 462T_{13}^{3} - 1148423T_{13}^{2} + 515112852T_{13} + 149501563456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + \cdots + 204 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7657 T^{2} + \cdots - 3188244 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 1682132124 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 149501563456 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 50104147968 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 7391138416576 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 3007939608576 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 408027025117872 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 86716089209547 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 50\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 991662745581932 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 270685655359056 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 85\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 16\!\cdots\!59 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 47\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
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