Properties

Label 441.4.e.m
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 21)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} - 4 \zeta_{6} q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{4} - 4 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{10} + ( - 62 \zeta_{6} + 62) q^{11} - 62 q^{13} + 64 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} - 100 \zeta_{6} q^{19} + 32 q^{20} + 248 q^{22} - 42 \zeta_{6} q^{23} + ( - 109 \zeta_{6} + 109) q^{25} - 248 \zeta_{6} q^{26} + 10 q^{29} + ( - 48 \zeta_{6} + 48) q^{31} + (256 \zeta_{6} - 256) q^{32} + 336 q^{34} + 246 \zeta_{6} q^{37} + ( - 400 \zeta_{6} + 400) q^{38} + 248 q^{41} + 68 q^{43} + 496 \zeta_{6} q^{44} + ( - 168 \zeta_{6} + 168) q^{46} + 324 \zeta_{6} q^{47} + 436 q^{50} + ( - 496 \zeta_{6} + 496) q^{52} + ( - 258 \zeta_{6} + 258) q^{53} - 248 q^{55} + 40 \zeta_{6} q^{58} + ( - 120 \zeta_{6} + 120) q^{59} - 622 \zeta_{6} q^{61} + 192 q^{62} - 512 q^{64} + 248 \zeta_{6} q^{65} + (904 \zeta_{6} - 904) q^{67} + 672 \zeta_{6} q^{68} + 678 q^{71} + ( - 642 \zeta_{6} + 642) q^{73} + (984 \zeta_{6} - 984) q^{74} + 800 q^{76} - 740 \zeta_{6} q^{79} + ( - 256 \zeta_{6} + 256) q^{80} + 992 \zeta_{6} q^{82} - 468 q^{83} - 336 q^{85} + 272 \zeta_{6} q^{86} + 200 \zeta_{6} q^{89} + 336 q^{92} + (1296 \zeta_{6} - 1296) q^{94} + (400 \zeta_{6} - 400) q^{95} - 1266 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 8 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 8 q^{4} - 4 q^{5} + 16 q^{10} + 62 q^{11} - 124 q^{13} + 64 q^{16} + 84 q^{17} - 100 q^{19} + 64 q^{20} + 496 q^{22} - 42 q^{23} + 109 q^{25} - 248 q^{26} + 20 q^{29} + 48 q^{31} - 256 q^{32} + 672 q^{34} + 246 q^{37} + 400 q^{38} + 496 q^{41} + 136 q^{43} + 496 q^{44} + 168 q^{46} + 324 q^{47} + 872 q^{50} + 496 q^{52} + 258 q^{53} - 496 q^{55} + 40 q^{58} + 120 q^{59} - 622 q^{61} + 384 q^{62} - 1024 q^{64} + 248 q^{65} - 904 q^{67} + 672 q^{68} + 1356 q^{71} + 642 q^{73} - 984 q^{74} + 1600 q^{76} - 740 q^{79} + 256 q^{80} + 992 q^{82} - 936 q^{83} - 672 q^{85} + 272 q^{86} + 200 q^{89} + 672 q^{92} - 1296 q^{94} - 400 q^{95} - 2532 q^{97}+O(q^{100}) \) 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 −2.00000 + 3.46410i 0 0 0 0 8.00000 + 13.8564i
361.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i −2.00000 3.46410i 0 0 0 0 8.00000 13.8564i
\(n\): 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.m 2
3.b odd 2 1 147.4.e.c 2
7.b odd 2 1 441.4.e.n 2
7.c even 3 1 63.4.a.a 1
7.c even 3 1 inner 441.4.e.m 2
7.d odd 6 1 441.4.a.b 1
7.d odd 6 1 441.4.e.n 2
21.c even 2 1 147.4.e.b 2
21.g even 6 1 147.4.a.g 1
21.g even 6 1 147.4.e.b 2
21.h odd 6 1 21.4.a.b 1
21.h odd 6 1 147.4.e.c 2
28.g odd 6 1 1008.4.a.m 1
35.j even 6 1 1575.4.a.k 1
84.j odd 6 1 2352.4.a.l 1
84.n even 6 1 336.4.a.h 1
105.o odd 6 1 525.4.a.b 1
105.x even 12 2 525.4.d.b 2
168.s odd 6 1 1344.4.a.w 1
168.v even 6 1 1344.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 21.h odd 6 1
63.4.a.a 1 7.c even 3 1
147.4.a.g 1 21.g even 6 1
147.4.e.b 2 21.c even 2 1
147.4.e.b 2 21.g even 6 1
147.4.e.c 2 3.b odd 2 1
147.4.e.c 2 21.h odd 6 1
336.4.a.h 1 84.n even 6 1
441.4.a.b 1 7.d odd 6 1
441.4.e.m 2 1.a even 1 1 trivial
441.4.e.m 2 7.c even 3 1 inner
441.4.e.n 2 7.b odd 2 1
441.4.e.n 2 7.d odd 6 1
525.4.a.b 1 105.o odd 6 1
525.4.d.b 2 105.x even 12 2
1008.4.a.m 1 28.g odd 6 1
1344.4.a.i 1 168.v even 6 1
1344.4.a.w 1 168.s odd 6 1
1575.4.a.k 1 35.j even 6 1
2352.4.a.l 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
\( T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{13} + 62 \) 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
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 62T + 3844 \) Copy content Toggle raw display
$13$ \( (T + 62)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} + 100T + 10000 \) Copy content Toggle raw display
$23$ \( T^{2} + 42T + 1764 \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 48T + 2304 \) Copy content Toggle raw display
$37$ \( T^{2} - 246T + 60516 \) Copy content Toggle raw display
$41$ \( (T - 248)^{2} \) Copy content Toggle raw display
$43$ \( (T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 324T + 104976 \) Copy content Toggle raw display
$53$ \( T^{2} - 258T + 66564 \) Copy content Toggle raw display
$59$ \( T^{2} - 120T + 14400 \) Copy content Toggle raw display
$61$ \( T^{2} + 622T + 386884 \) Copy content Toggle raw display
$67$ \( T^{2} + 904T + 817216 \) Copy content Toggle raw display
$71$ \( (T - 678)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 642T + 412164 \) Copy content Toggle raw display
$79$ \( T^{2} + 740T + 547600 \) Copy content Toggle raw display
$83$ \( (T + 468)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 200T + 40000 \) Copy content Toggle raw display
$97$ \( (T + 1266)^{2} \) Copy content Toggle raw display
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