Properties

Label 441.3.b.c
Level $441$
Weight $3$
Character orbit 441.b
Analytic conductor $12.016$
Analytic rank $0$
Dimension $6$
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] = [441,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.417489408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 20x^{4} + 93x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 63)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 3) q^{4} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 3) q^{4} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{4} + 2 \beta_{2} - 3) q^{10} + (2 \beta_{3} - \beta_1) q^{11} + ( - \beta_{4} + 2 \beta_{2} + 4) q^{13} + (2 \beta_{4} - \beta_{2} + 7) q^{16} + ( - \beta_{5} - 7 \beta_1) q^{17} + ( - \beta_{4} + 2 \beta_{2} + 4) q^{19} + ( - 7 \beta_{3} - 7 \beta_1) q^{20} + ( - 2 \beta_{4} + \beta_{2} + 3) q^{22} + ( - \beta_{5} - 4 \beta_{3} + 5 \beta_1) q^{23} + ( - 2 \beta_{4} - 3 \beta_{2} - 16) q^{25} + (4 \beta_{5} - 7 \beta_{3}) q^{26} + ( - 4 \beta_{5} - \beta_{3} + \beta_1) q^{29} + ( - 3 \beta_{4} - \beta_{2} + 5) q^{31} + ( - \beta_{5} + 7 \beta_{3} - 2 \beta_1) q^{32} + ( - \beta_{4} - 5 \beta_{2} + 46) q^{34} + ( - \beta_{4} - 4 \beta_{2} + 26) q^{37} + (4 \beta_{5} - 7 \beta_{3}) q^{38} + (3 \beta_{4} - 6 \beta_{2} + 51) q^{40} + ( - \beta_{5} - 7 \beta_{3} + 7 \beta_1) q^{41} + (3 \beta_{4} + 8) q^{43} + (5 \beta_{5} - 3 \beta_{3}) q^{44} + (3 \beta_{4} + 3 \beta_{2} - 30) q^{46} + ( - 2 \beta_{5} - 7 \beta_{3} - 14 \beta_1) q^{47} + (\beta_{5} - 7 \beta_{3} - 3 \beta_1) q^{50} + (7 \beta_{4} - 7 \beta_{2} + 42) q^{52} + (4 \beta_{5} + 9 \beta_{3} - 5 \beta_1) q^{53} + (3 \beta_{4} + 8 \beta_{2} - 5) q^{55} + ( - 3 \beta_{4} + 8 \beta_{2} - 17) q^{58} + ( - \beta_{5} + 7 \beta_{3} + 14 \beta_1) q^{59} + ( - 5 \beta_{4} + 3 \beta_{2} - 22) q^{61} + (5 \beta_{5} - 14 \beta_{3} + 14 \beta_1) q^{62} + (3 \beta_{2} + 25) q^{64} + ( - 6 \beta_{5} - 7 \beta_{3} - 28 \beta_1) q^{65} + (2 \beta_{4} + 3 \beta_{2} - 2) q^{67} + ( - 7 \beta_{5} + 35 \beta_1) q^{68} + (12 \beta_{5} + 3 \beta_{3} + 24 \beta_1) q^{71} + (4 \beta_{4} + 13 \beta_{2} - 86) q^{73} + ( - 2 \beta_{5} - \beta_{3} + 40 \beta_1) q^{74} + (7 \beta_{4} - 7 \beta_{2} + 42) q^{76} + ( - 5 \beta_{4} + 17 \beta_{2} - 11) q^{79} + ( - 12 \beta_{5} - 7 \beta_{3} + 35 \beta_1) q^{80} + (6 \beta_{4} + 2 \beta_{2} - 38) q^{82} + ( - 6 \beta_{5} + 14 \beta_{3} - 21 \beta_1) q^{83} + (5 \beta_{4} - 17 \beta_{2} - 20) q^{85} + ( - 6 \beta_{5} + 15 \beta_{3} + 2 \beta_1) q^{86} + ( - 9 \beta_{2} + 33) q^{88} + (2 \beta_{5} + 14 \beta_{3}) q^{89} + ( - 7 \beta_{5} - 4 \beta_{3} - 25 \beta_1) q^{92} + (5 \beta_{4} - 17 \beta_{2} + 106) q^{94} + ( - 6 \beta_{5} - 7 \beta_{3} - 28 \beta_1) q^{95} + ( - 2 \beta_{4} - 17 \beta_{2} - 41) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 16 q^{10} + 26 q^{13} + 44 q^{16} + 26 q^{19} + 16 q^{22} - 106 q^{25} + 22 q^{31} + 264 q^{34} + 146 q^{37} + 300 q^{40} + 54 q^{43} - 168 q^{46} + 252 q^{52} - 8 q^{55} - 92 q^{58} - 136 q^{61} + 156 q^{64} - 2 q^{67} - 482 q^{73} + 252 q^{76} - 42 q^{79} - 212 q^{82} - 144 q^{85} + 180 q^{88} + 612 q^{94} - 284 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 20x^{4} + 93x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 17\nu^{3} + 60\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 13\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 23\nu^{3} + 120\nu ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} - 13\beta_{2} + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{5} + 23\beta_{3} + 110\beta_1 \) 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
3.59131i
2.62683i
0.449730i
0.449730i
2.62683i
3.59131i
3.59131i 0 −8.89752 6.16325i 0 0 17.5885i 0 −22.1342
197.2 2.62683i 0 −2.90022 3.89994i 0 0 2.88892i 0 10.2445
197.3 0.449730i 0 3.79774 8.64898i 0 0 3.50688i 0 3.88970
197.4 0.449730i 0 3.79774 8.64898i 0 0 3.50688i 0 3.88970
197.5 2.62683i 0 −2.90022 3.89994i 0 0 2.88892i 0 10.2445
197.6 3.59131i 0 −8.89752 6.16325i 0 0 17.5885i 0 −22.1342
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.b.c 6
3.b odd 2 1 inner 441.3.b.c 6
7.b odd 2 1 441.3.b.d 6
7.c even 3 2 441.3.q.d 12
7.d odd 6 2 63.3.q.a 12
21.c even 2 1 441.3.b.d 6
21.g even 6 2 63.3.q.a 12
21.h odd 6 2 441.3.q.d 12
28.f even 6 2 1008.3.dc.e 12
84.j odd 6 2 1008.3.dc.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.q.a 12 7.d odd 6 2
63.3.q.a 12 21.g even 6 2
441.3.b.c 6 1.a even 1 1 trivial
441.3.b.c 6 3.b odd 2 1 inner
441.3.b.d 6 7.b odd 2 1
441.3.b.d 6 21.c even 2 1
441.3.q.d 12 7.c even 3 2
441.3.q.d 12 21.h odd 6 2
1008.3.dc.e 12 28.f even 6 2
1008.3.dc.e 12 84.j odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{6} + 20T_{2}^{4} + 93T_{2}^{2} + 18 \) Copy content Toggle raw display
\( T_{13}^{3} - 13T_{13}^{2} - 238T_{13} + 2842 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 20 T^{4} + 93 T^{2} + 18 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 128 T^{4} + 4557 T^{2} + \cdots + 43218 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 212 T^{4} + 12477 T^{2} + \cdots + 190962 \) Copy content Toggle raw display
$13$ \( (T^{3} - 13 T^{2} - 238 T + 2842)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 996 T^{4} + \cdots + 24893568 \) Copy content Toggle raw display
$19$ \( (T^{3} - 13 T^{2} - 238 T + 2842)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 1284 T^{4} + \cdots + 19170432 \) Copy content Toggle raw display
$29$ \( T^{6} + 2126 T^{4} + \cdots + 64706688 \) Copy content Toggle raw display
$31$ \( (T^{3} - 11 T^{2} - 1211 T - 10731)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 73 T^{2} + 994 T + 12894)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3194 T^{4} + \cdots + 6223392 \) Copy content Toggle raw display
$43$ \( (T^{3} - 27 T^{2} - 966 T + 26278)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 7638 T^{4} + \cdots + 2876762952 \) Copy content Toggle raw display
$53$ \( T^{6} + 6270 T^{4} + \cdots + 124314912 \) Copy content Toggle raw display
$59$ \( T^{6} + 8178 T^{4} + \cdots + 56010528 \) Copy content Toggle raw display
$61$ \( (T^{3} + 68 T^{2} - 2170 T - 116816)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + T^{2} - 904 T - 4432)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 26262 T^{4} + \cdots + 95234699592 \) Copy content Toggle raw display
$73$ \( (T^{3} + 241 T^{2} + 10360 T - 532728)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 21 T^{2} - 14811 T + 587701)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 19476 T^{4} + \cdots + 31505922 \) Copy content Toggle raw display
$89$ \( T^{6} + 10760 T^{4} + \cdots + 34696101888 \) Copy content Toggle raw display
$97$ \( (T^{3} + 142 T^{2} - 5495 T - 125832)^{2} \) Copy content Toggle raw display
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