Properties

Label 441.4.a.v
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $4$
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(1,441)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

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: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.6257832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 42 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8} + ( - 5 \beta_{2} - 8) q^{10} + (3 \beta_{3} - \beta_1) q^{11} + (\beta_{2} - 25) q^{13} + ( - 5 \beta_{2} - 28) q^{16} + (2 \beta_{3} - 26 \beta_1) q^{17} + ( - 11 \beta_{2} - 61) q^{19} + (3 \beta_{3} - 25 \beta_1) q^{20} + (11 \beta_{2} - 16) q^{22} + (2 \beta_{3} + 46 \beta_1) q^{23} + ( - 17 \beta_{2} + 83) q^{25} + (\beta_{3} - 20 \beta_1) q^{26} + ( - \beta_{3} - 45 \beta_1) q^{29} + (20 \beta_{2} - 45) q^{31} + ( - 13 \beta_{3} - 45 \beta_1) q^{32} + ( - 18 \beta_{2} - 264) q^{34} + (25 \beta_{2} - 81) q^{37} + ( - 11 \beta_{3} - 116 \beta_1) q^{38} + (27 \beta_{2} - 192) q^{40} + ( - 16 \beta_{3} - 72 \beta_1) q^{41} + ( - 27 \beta_{2} - 223) q^{43} + ( - 13 \beta_{3} + 47 \beta_1) q^{44} + (54 \beta_{2} + 456) q^{46} + (16 \beta_{3} + 44 \beta_1) q^{47} + ( - 17 \beta_{3} - 2 \beta_1) q^{50} + ( - 24 \beta_{2} - 2) q^{52} + (7 \beta_{3} - \beta_1) q^{53} + (71 \beta_{2} - 592) q^{55} + ( - 49 \beta_{2} - 448) q^{58} + (17 \beta_{3} + 73 \beta_1) q^{59} + (46 \beta_{2} - 310) q^{61} + (20 \beta_{3} + 55 \beta_1) q^{62} + ( - 57 \beta_{2} - 200) q^{64} + (30 \beta_{3} + 2 \beta_1) q^{65} + ( - 19 \beta_{2} + 463) q^{67} + ( - 34 \beta_{3} - 146 \beta_1) q^{68} + 36 \beta_{3} q^{71} + ( - 7 \beta_{2} - 441) q^{73} + (25 \beta_{3} + 44 \beta_1) q^{74} + ( - 72 \beta_{2} - 650) q^{76} + (42 \beta_{2} + 23) q^{79} + (3 \beta_{3} + 143 \beta_1) q^{80} + ( - 136 \beta_{2} - 688) q^{82} + (21 \beta_{3} + 189 \beta_1) q^{83} + (174 \beta_{2} - 192) q^{85} + ( - 27 \beta_{3} - 358 \beta_1) q^{86} + ( - 93 \beta_{2} + 624) q^{88} + ( - 22 \beta_{3} + 314 \beta_1) q^{89} + (38 \beta_{3} + 358 \beta_1) q^{92} + (108 \beta_{2} + 408) q^{94} + (6 \beta_{3} + 314 \beta_1) q^{95} + ( - 91 \beta_{2} - 798) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} - 22 q^{10} - 102 q^{13} - 102 q^{16} - 222 q^{19} - 86 q^{22} + 366 q^{25} - 220 q^{31} - 1020 q^{34} - 374 q^{37} - 822 q^{40} - 838 q^{43} + 1716 q^{46} + 40 q^{52} - 2510 q^{55} - 1694 q^{58} - 1332 q^{61} - 686 q^{64} + 1890 q^{67} - 1750 q^{73} - 2456 q^{76} + 8 q^{79} - 2480 q^{82} - 1116 q^{85} + 2682 q^{88} + 1416 q^{94} - 3010 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 19x^{2} + 42 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 15\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 15\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.05539
−1.59805
1.59805
4.05539
−4.05539 0 8.44622 9.92039 0 0 −1.80961 0 −40.2311
1.2 −1.59805 0 −5.44622 −18.2917 0 0 21.4878 0 29.2311
1.3 1.59805 0 −5.44622 18.2917 0 0 −21.4878 0 29.2311
1.4 4.05539 0 8.44622 −9.92039 0 0 1.80961 0 −40.2311
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{4} - 19T_{2}^{2} + 42 \) Copy content Toggle raw display
\( T_{5}^{4} - 433T_{5}^{2} + 32928 \) Copy content Toggle raw display
\( T_{13}^{2} + 51T_{13} + 602 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 19T^{2} + 42 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 433 T^{2} + 32928 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 3937 T^{2} + 688128 \) Copy content Toggle raw display
$13$ \( (T^{2} + 51 T + 602)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 15396 T^{2} + \cdots + 58084992 \) Copy content Toggle raw display
$19$ \( (T^{2} + 111 T - 2758)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 40452 T^{2} + \cdots + 44731008 \) Copy content Toggle raw display
$29$ \( T^{4} - 38185 T^{2} + \cdots + 96018048 \) Copy content Toggle raw display
$31$ \( (T^{2} + 110 T - 16275)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 187 T - 21414)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 190144 T^{2} + \cdots + 6145155072 \) Copy content Toggle raw display
$43$ \( (T^{2} + 419 T + 8716)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 135600 T^{2} + \cdots + 4556708352 \) Copy content Toggle raw display
$53$ \( T^{4} - 21201 T^{2} + \cdots + 27149472 \) Copy content Toggle raw display
$59$ \( T^{4} - 205665 T^{2} + \cdots + 7681740192 \) Copy content Toggle raw display
$61$ \( (T^{2} + 666 T + 8792)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 945 T + 205838)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 557280 T^{2} + \cdots + 22856214528 \) Copy content Toggle raw display
$73$ \( (T^{2} + 875 T + 189042)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 85109)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 804825 T^{2} + \cdots + 10585989792 \) Copy content Toggle raw display
$89$ \( T^{4} - 2191972 T^{2} + \cdots + 1155567856128 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1505 T + 166698)^{2} \) Copy content Toggle raw display
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