Properties

Label 441.4.e.o
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
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] = [441,4,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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} + 18 \zeta_{6} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} + 18 \zeta_{6} q^{5} + (72 \zeta_{6} - 72) q^{10} + (50 \zeta_{6} - 50) q^{11} - 36 q^{13} + 64 \zeta_{6} q^{16} + ( - 126 \zeta_{6} + 126) q^{17} + 72 \zeta_{6} q^{19} - 144 q^{20} - 200 q^{22} + 14 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} - 144 \zeta_{6} q^{26} - 158 q^{29} + ( - 36 \zeta_{6} + 36) q^{31} + (256 \zeta_{6} - 256) q^{32} + 504 q^{34} + 162 \zeta_{6} q^{37} + (288 \zeta_{6} - 288) q^{38} + 270 q^{41} - 324 q^{43} - 400 \zeta_{6} q^{44} + (56 \zeta_{6} - 56) q^{46} - 72 \zeta_{6} q^{47} - 796 q^{50} + ( - 288 \zeta_{6} + 288) q^{52} + (22 \zeta_{6} - 22) q^{53} - 900 q^{55} - 632 \zeta_{6} q^{58} + ( - 468 \zeta_{6} + 468) q^{59} - 792 \zeta_{6} q^{61} + 144 q^{62} - 512 q^{64} - 648 \zeta_{6} q^{65} + (232 \zeta_{6} - 232) q^{67} + 1008 \zeta_{6} q^{68} + 734 q^{71} + (180 \zeta_{6} - 180) q^{73} + (648 \zeta_{6} - 648) q^{74} - 576 q^{76} - 236 \zeta_{6} q^{79} + (1152 \zeta_{6} - 1152) q^{80} + 1080 \zeta_{6} q^{82} - 36 q^{83} + 2268 q^{85} - 1296 \zeta_{6} q^{86} + 234 \zeta_{6} q^{89} - 112 q^{92} + ( - 288 \zeta_{6} + 288) q^{94} + (1296 \zeta_{6} - 1296) q^{95} + 468 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 8 q^{4} + 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 8 q^{4} + 18 q^{5} - 72 q^{10} - 50 q^{11} - 72 q^{13} + 64 q^{16} + 126 q^{17} + 72 q^{19} - 288 q^{20} - 400 q^{22} + 14 q^{23} - 199 q^{25} - 144 q^{26} - 316 q^{29} + 36 q^{31} - 256 q^{32} + 1008 q^{34} + 162 q^{37} - 288 q^{38} + 540 q^{41} - 648 q^{43} - 400 q^{44} - 56 q^{46} - 72 q^{47} - 1592 q^{50} + 288 q^{52} - 22 q^{53} - 1800 q^{55} - 632 q^{58} + 468 q^{59} - 792 q^{61} + 288 q^{62} - 1024 q^{64} - 648 q^{65} - 232 q^{67} + 1008 q^{68} + 1468 q^{71} - 180 q^{73} - 648 q^{74} - 1152 q^{76} - 236 q^{79} - 1152 q^{80} + 1080 q^{82} - 72 q^{83} + 4536 q^{85} - 1296 q^{86} + 234 q^{89} - 224 q^{92} + 288 q^{94} - 1296 q^{95} + 936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
2.00000 3.46410i 0 −4.00000 6.92820i 9.00000 15.5885i 0 0 0 0 −36.0000 62.3538i
361.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i 9.00000 + 15.5885i 0 0 0 0 −36.0000 + 62.3538i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.o 2
3.b odd 2 1 147.4.e.a 2
7.b odd 2 1 441.4.e.l 2
7.c even 3 1 441.4.a.a 1
7.c even 3 1 inner 441.4.e.o 2
7.d odd 6 1 441.4.a.c 1
7.d odd 6 1 441.4.e.l 2
21.c even 2 1 147.4.e.d 2
21.g even 6 1 147.4.a.f 1
21.g even 6 1 147.4.e.d 2
21.h odd 6 1 147.4.a.h yes 1
21.h odd 6 1 147.4.e.a 2
84.j odd 6 1 2352.4.a.t 1
84.n even 6 1 2352.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 21.g even 6 1
147.4.a.h yes 1 21.h odd 6 1
147.4.e.a 2 3.b odd 2 1
147.4.e.a 2 21.h odd 6 1
147.4.e.d 2 21.c even 2 1
147.4.e.d 2 21.g even 6 1
441.4.a.a 1 7.c even 3 1
441.4.a.c 1 7.d odd 6 1
441.4.e.l 2 7.b odd 2 1
441.4.e.l 2 7.d odd 6 1
441.4.e.o 2 1.a even 1 1 trivial
441.4.e.o 2 7.c even 3 1 inner
2352.4.a.s 1 84.n even 6 1
2352.4.a.t 1 84.j odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{2} - 18T_{5} + 324 \) Copy content Toggle raw display
\( T_{13} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 50T + 2500 \) Copy content Toggle raw display
$13$ \( (T + 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 126T + 15876 \) Copy content Toggle raw display
$19$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$23$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$29$ \( (T + 158)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$37$ \( T^{2} - 162T + 26244 \) Copy content Toggle raw display
$41$ \( (T - 270)^{2} \) Copy content Toggle raw display
$43$ \( (T + 324)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 72T + 5184 \) Copy content Toggle raw display
$53$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$59$ \( T^{2} - 468T + 219024 \) Copy content Toggle raw display
$61$ \( T^{2} + 792T + 627264 \) Copy content Toggle raw display
$67$ \( T^{2} + 232T + 53824 \) Copy content Toggle raw display
$71$ \( (T - 734)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 180T + 32400 \) Copy content Toggle raw display
$79$ \( T^{2} + 236T + 55696 \) Copy content Toggle raw display
$83$ \( (T + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 234T + 54756 \) Copy content Toggle raw display
$97$ \( (T - 468)^{2} \) Copy content Toggle raw display
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