Properties

Label 441.4.e.e
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} - 15 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} - 15 q^{8} + (16 \zeta_{6} - 16) q^{10} + (8 \zeta_{6} - 8) q^{11} - 28 q^{13} - 41 \zeta_{6} q^{16} + (54 \zeta_{6} - 54) q^{17} - 110 \zeta_{6} q^{19} - 112 q^{20} + 8 q^{22} + 48 \zeta_{6} q^{23} + (131 \zeta_{6} - 131) q^{25} + 28 \zeta_{6} q^{26} + 110 q^{29} + ( - 12 \zeta_{6} + 12) q^{31} + (161 \zeta_{6} - 161) q^{32} + 54 q^{34} + 246 \zeta_{6} q^{37} + (110 \zeta_{6} - 110) q^{38} + 240 \zeta_{6} q^{40} + 182 q^{41} + 128 q^{43} + 56 \zeta_{6} q^{44} + ( - 48 \zeta_{6} + 48) q^{46} - 324 \zeta_{6} q^{47} + 131 q^{50} + (196 \zeta_{6} - 196) q^{52} + (162 \zeta_{6} - 162) q^{53} + 128 q^{55} - 110 \zeta_{6} q^{58} + (810 \zeta_{6} - 810) q^{59} - 488 \zeta_{6} q^{61} - 12 q^{62} - 167 q^{64} + 448 \zeta_{6} q^{65} + (244 \zeta_{6} - 244) q^{67} + 378 \zeta_{6} q^{68} + 768 q^{71} + (702 \zeta_{6} - 702) q^{73} + ( - 246 \zeta_{6} + 246) q^{74} - 770 q^{76} - 440 \zeta_{6} q^{79} + (656 \zeta_{6} - 656) q^{80} - 182 \zeta_{6} q^{82} - 1302 q^{83} + 864 q^{85} - 128 \zeta_{6} q^{86} + ( - 120 \zeta_{6} + 120) q^{88} - 730 \zeta_{6} q^{89} + 336 q^{92} + (324 \zeta_{6} - 324) q^{94} + (1760 \zeta_{6} - 1760) q^{95} - 294 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 7 q^{4} - 16 q^{5} - 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 7 q^{4} - 16 q^{5} - 30 q^{8} - 16 q^{10} - 8 q^{11} - 56 q^{13} - 41 q^{16} - 54 q^{17} - 110 q^{19} - 224 q^{20} + 16 q^{22} + 48 q^{23} - 131 q^{25} + 28 q^{26} + 220 q^{29} + 12 q^{31} - 161 q^{32} + 108 q^{34} + 246 q^{37} - 110 q^{38} + 240 q^{40} + 364 q^{41} + 256 q^{43} + 56 q^{44} + 48 q^{46} - 324 q^{47} + 262 q^{50} - 196 q^{52} - 162 q^{53} + 256 q^{55} - 110 q^{58} - 810 q^{59} - 488 q^{61} - 24 q^{62} - 334 q^{64} + 448 q^{65} - 244 q^{67} + 378 q^{68} + 1536 q^{71} - 702 q^{73} + 246 q^{74} - 1540 q^{76} - 440 q^{79} - 656 q^{80} - 182 q^{82} - 2604 q^{83} + 1728 q^{85} - 128 q^{86} + 120 q^{88} - 730 q^{89} + 672 q^{92} - 324 q^{94} - 1760 q^{95} - 588 q^{97}+O(q^{100}) \) 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
−0.500000 + 0.866025i 0 3.50000 + 6.06218i −8.00000 + 13.8564i 0 0 −15.0000 0 −8.00000 13.8564i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i −8.00000 13.8564i 0 0 −15.0000 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.e 2
3.b odd 2 1 49.4.c.b 2
7.b odd 2 1 441.4.e.h 2
7.c even 3 1 441.4.a.i 1
7.c even 3 1 inner 441.4.e.e 2
7.d odd 6 1 63.4.a.b 1
7.d odd 6 1 441.4.e.h 2
21.c even 2 1 49.4.c.c 2
21.g even 6 1 7.4.a.a 1
21.g even 6 1 49.4.c.c 2
21.h odd 6 1 49.4.a.b 1
21.h odd 6 1 49.4.c.b 2
28.f even 6 1 1008.4.a.c 1
35.i odd 6 1 1575.4.a.e 1
84.j odd 6 1 112.4.a.f 1
84.n even 6 1 784.4.a.g 1
105.o odd 6 1 1225.4.a.j 1
105.p even 6 1 175.4.a.b 1
105.w odd 12 2 175.4.b.b 2
168.ba even 6 1 448.4.a.i 1
168.be odd 6 1 448.4.a.e 1
231.k odd 6 1 847.4.a.b 1
273.ba even 6 1 1183.4.a.b 1
357.s even 6 1 2023.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 21.g even 6 1
49.4.a.b 1 21.h odd 6 1
49.4.c.b 2 3.b odd 2 1
49.4.c.b 2 21.h odd 6 1
49.4.c.c 2 21.c even 2 1
49.4.c.c 2 21.g even 6 1
63.4.a.b 1 7.d odd 6 1
112.4.a.f 1 84.j odd 6 1
175.4.a.b 1 105.p even 6 1
175.4.b.b 2 105.w odd 12 2
441.4.a.i 1 7.c even 3 1
441.4.e.e 2 1.a even 1 1 trivial
441.4.e.e 2 7.c even 3 1 inner
441.4.e.h 2 7.b odd 2 1
441.4.e.h 2 7.d odd 6 1
448.4.a.e 1 168.be odd 6 1
448.4.a.i 1 168.ba even 6 1
784.4.a.g 1 84.n even 6 1
847.4.a.b 1 231.k odd 6 1
1008.4.a.c 1 28.f even 6 1
1183.4.a.b 1 273.ba even 6 1
1225.4.a.j 1 105.o odd 6 1
1575.4.a.e 1 35.i odd 6 1
2023.4.a.a 1 357.s even 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 16T_{5} + 256 \) Copy content Toggle raw display
\( T_{13} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$13$ \( (T + 28)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 54T + 2916 \) Copy content Toggle raw display
$19$ \( T^{2} + 110T + 12100 \) Copy content Toggle raw display
$23$ \( T^{2} - 48T + 2304 \) Copy content Toggle raw display
$29$ \( (T - 110)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$37$ \( T^{2} - 246T + 60516 \) Copy content Toggle raw display
$41$ \( (T - 182)^{2} \) Copy content Toggle raw display
$43$ \( (T - 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 324T + 104976 \) Copy content Toggle raw display
$53$ \( T^{2} + 162T + 26244 \) Copy content Toggle raw display
$59$ \( T^{2} + 810T + 656100 \) Copy content Toggle raw display
$61$ \( T^{2} + 488T + 238144 \) Copy content Toggle raw display
$67$ \( T^{2} + 244T + 59536 \) Copy content Toggle raw display
$71$ \( (T - 768)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 702T + 492804 \) Copy content Toggle raw display
$79$ \( T^{2} + 440T + 193600 \) Copy content Toggle raw display
$83$ \( (T + 1302)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 730T + 532900 \) Copy content Toggle raw display
$97$ \( (T + 294)^{2} \) Copy content Toggle raw display
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