Properties

Label 441.4.e.x
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 19x^{6} + 319x^{4} + 798x^{2} + 1764 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 63)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{6} - 2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{6} - 2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{8} + (5 \beta_{6} + 8 \beta_{2} + 8) q^{10} + ( - 3 \beta_{5} + \beta_1) q^{11} + (\beta_{4} - 25) q^{13} + ( - 5 \beta_{6} + 5 \beta_{4} - 28 \beta_{2}) q^{16} + ( - 2 \beta_{5} + 26 \beta_1) q^{17} + ( - 11 \beta_{6} + 11 \beta_{4} - 61 \beta_{2}) q^{19} + ( - 3 \beta_{7} + 3 \beta_{5} + 25 \beta_{3}) q^{20} + (11 \beta_{4} - 16) q^{22} + (2 \beta_{7} + 46 \beta_{3} + 46 \beta_1) q^{23} + (17 \beta_{6} - 83 \beta_{2} - 83) q^{25} + (\beta_{7} - 20 \beta_{3} - 20 \beta_1) q^{26} + (\beta_{7} - \beta_{5} + 45 \beta_{3}) q^{29} + ( - 20 \beta_{6} + 45 \beta_{2} + 45) q^{31} + (13 \beta_{5} + 45 \beta_1) q^{32} + ( - 18 \beta_{4} - 264) q^{34} + (25 \beta_{6} - 25 \beta_{4} - 81 \beta_{2}) q^{37} + (11 \beta_{5} + 116 \beta_1) q^{38} + (27 \beta_{6} - 27 \beta_{4} - 192 \beta_{2}) q^{40} + (16 \beta_{7} - 16 \beta_{5} + 72 \beta_{3}) q^{41} + ( - 27 \beta_{4} - 223) q^{43} + ( - 13 \beta_{7} + 47 \beta_{3} + 47 \beta_1) q^{44} + ( - 54 \beta_{6} - 456 \beta_{2} - 456) q^{46} + (16 \beta_{7} + 44 \beta_{3} + 44 \beta_1) q^{47} + (17 \beta_{7} - 17 \beta_{5} + 2 \beta_{3}) q^{50} + (24 \beta_{6} + 2 \beta_{2} + 2) q^{52} + ( - 7 \beta_{5} + \beta_1) q^{53} + (71 \beta_{4} - 592) q^{55} + ( - 49 \beta_{6} + 49 \beta_{4} - 448 \beta_{2}) q^{58} + ( - 17 \beta_{5} - 73 \beta_1) q^{59} + (46 \beta_{6} - 46 \beta_{4} - 310 \beta_{2}) q^{61} + ( - 20 \beta_{7} + 20 \beta_{5} - 55 \beta_{3}) q^{62} + ( - 57 \beta_{4} - 200) q^{64} + (30 \beta_{7} + 2 \beta_{3} + 2 \beta_1) q^{65} + (19 \beta_{6} - 463 \beta_{2} - 463) q^{67} + ( - 34 \beta_{7} - 146 \beta_{3} - 146 \beta_1) q^{68} + ( - 36 \beta_{7} + 36 \beta_{5}) q^{71} + (7 \beta_{6} + 441 \beta_{2} + 441) q^{73} + ( - 25 \beta_{5} - 44 \beta_1) q^{74} + ( - 72 \beta_{4} - 650) q^{76} + (42 \beta_{6} - 42 \beta_{4} + 23 \beta_{2}) q^{79} + ( - 3 \beta_{5} - 143 \beta_1) q^{80} + ( - 136 \beta_{6} + 136 \beta_{4} - 688 \beta_{2}) q^{82} + ( - 21 \beta_{7} + 21 \beta_{5} - 189 \beta_{3}) q^{83} + (174 \beta_{4} - 192) q^{85} + ( - 27 \beta_{7} - 358 \beta_{3} - 358 \beta_1) q^{86} + (93 \beta_{6} - 624 \beta_{2} - 624) q^{88} + ( - 22 \beta_{7} + 314 \beta_{3} + 314 \beta_1) q^{89} + ( - 38 \beta_{7} + 38 \beta_{5} - 358 \beta_{3}) q^{92} + ( - 108 \beta_{6} - 408 \beta_{2} - 408) q^{94} + ( - 6 \beta_{5} - 314 \beta_1) q^{95} + ( - 91 \beta_{4} - 798) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} + 22 q^{10} - 204 q^{13} + 102 q^{16} + 222 q^{19} - 172 q^{22} - 366 q^{25} + 220 q^{31} - 2040 q^{34} + 374 q^{37} + 822 q^{40} - 1676 q^{43} - 1716 q^{46} - 40 q^{52} - 5020 q^{55} + 1694 q^{58} + 1332 q^{61} - 1372 q^{64} - 1890 q^{67} + 1750 q^{73} - 4912 q^{76} - 8 q^{79} + 2480 q^{82} - 2232 q^{85} - 2682 q^{88} - 1416 q^{94} - 6020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 19x^{6} + 319x^{4} + 798x^{2} + 1764 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\nu^{6} + 319\nu^{4} + 6061\nu^{2} + 1764 ) / 13398 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\nu^{7} + 319\nu^{5} + 6061\nu^{3} + 1764\nu ) / 13398 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 2392 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3987\nu ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -74\nu^{6} - 1595\nu^{4} - 23606\nu^{2} - 59052 ) / 6699 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -81\nu^{7} - 1595\nu^{5} - 25839\nu^{3} - 64638\nu ) / 4466 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -19\beta_{6} - 148\beta_{2} - 148 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{7} - 243\beta_{3} - 243\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 319\beta_{4} + 2392 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 319\beta_{5} + 3987\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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
2.02770 + 3.51207i
0.799027 + 1.38396i
−0.799027 1.38396i
−2.02770 3.51207i
2.02770 3.51207i
0.799027 1.38396i
−0.799027 + 1.38396i
−2.02770 + 3.51207i
−2.02770 + 3.51207i 0 −4.22311 7.31464i 4.96020 8.59131i 0 0 1.80961 0 20.1156 + 34.8412i
226.2 −0.799027 + 1.38396i 0 2.72311 + 4.71657i −9.14584 + 15.8411i 0 0 −21.4878 0 −14.6156 25.3149i
226.3 0.799027 1.38396i 0 2.72311 + 4.71657i 9.14584 15.8411i 0 0 21.4878 0 −14.6156 25.3149i
226.4 2.02770 3.51207i 0 −4.22311 7.31464i −4.96020 + 8.59131i 0 0 −1.80961 0 20.1156 + 34.8412i
361.1 −2.02770 3.51207i 0 −4.22311 + 7.31464i 4.96020 + 8.59131i 0 0 1.80961 0 20.1156 34.8412i
361.2 −0.799027 1.38396i 0 2.72311 4.71657i −9.14584 15.8411i 0 0 −21.4878 0 −14.6156 + 25.3149i
361.3 0.799027 + 1.38396i 0 2.72311 4.71657i 9.14584 + 15.8411i 0 0 21.4878 0 −14.6156 + 25.3149i
361.4 2.02770 + 3.51207i 0 −4.22311 + 7.31464i −4.96020 8.59131i 0 0 −1.80961 0 20.1156 34.8412i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.x 8
3.b odd 2 1 inner 441.4.e.x 8
7.b odd 2 1 63.4.e.d 8
7.c even 3 1 441.4.a.v 4
7.c even 3 1 inner 441.4.e.x 8
7.d odd 6 1 63.4.e.d 8
7.d odd 6 1 441.4.a.w 4
21.c even 2 1 63.4.e.d 8
21.g even 6 1 63.4.e.d 8
21.g even 6 1 441.4.a.w 4
21.h odd 6 1 441.4.a.v 4
21.h odd 6 1 inner 441.4.e.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.e.d 8 7.b odd 2 1
63.4.e.d 8 7.d odd 6 1
63.4.e.d 8 21.c even 2 1
63.4.e.d 8 21.g even 6 1
441.4.a.v 4 7.c even 3 1
441.4.a.v 4 21.h odd 6 1
441.4.a.w 4 7.d odd 6 1
441.4.a.w 4 21.g even 6 1
441.4.e.x 8 1.a even 1 1 trivial
441.4.e.x 8 3.b odd 2 1 inner
441.4.e.x 8 7.c even 3 1 inner
441.4.e.x 8 21.h odd 6 1 inner

Hecke kernels

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

\( T_{2}^{8} + 19T_{2}^{6} + 319T_{2}^{4} + 798T_{2}^{2} + 1764 \) Copy content Toggle raw display
\( T_{5}^{8} + 433T_{5}^{6} + 154561T_{5}^{4} + 14257824T_{5}^{2} + 1084253184 \) Copy content Toggle raw display
\( T_{13}^{2} + 51T_{13} + 602 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 19 T^{6} + 319 T^{4} + \cdots + 1764 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 433 T^{6} + \cdots + 1084253184 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 3937 T^{6} + \cdots + 473520144384 \) Copy content Toggle raw display
$13$ \( (T^{2} + 51 T + 602)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 15396 T^{6} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{4} - 111 T^{3} + 15079 T^{2} + \cdots + 7606564)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 40452 T^{6} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} - 38185 T^{2} + 96018048)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 110 T^{3} + 28375 T^{2} + \cdots + 264875625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 187 T^{3} + 56383 T^{2} + \cdots + 458559396)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 190144 T^{2} + \cdots + 6145155072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 419 T + 8716)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 135600 T^{6} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 737093829878784 \) Copy content Toggle raw display
$59$ \( T^{8} + 205665 T^{6} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{4} - 666 T^{3} + 434764 T^{2} + \cdots + 77299264)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 945 T^{3} + \cdots + 42369282244)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 557280 T^{2} + \cdots + 22856214528)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 875 T^{3} + \cdots + 35736877764)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + 85125 T^{2} + \cdots + 7243541881)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 804825 T^{2} + \cdots + 10585989792)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 2191972 T^{6} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1505 T + 166698)^{4} \) Copy content Toggle raw display
show more
show less