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Properties

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

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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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 74x^{6} + 1469x^{4} - 8828x^{2} + 2500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} + 8) q^{4} + \beta_1 q^{5} + ( - 2 \beta_{4} - 11 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} + 8) q^{4} + \beta_1 q^{5} + ( - 2 \beta_{4} - 11 \beta_{2}) q^{8} + ( - 2 \beta_{7} - 3 \beta_{5}) q^{10} + (\beta_{4} - 7 \beta_{2}) q^{11} + (3 \beta_{7} + 2 \beta_{5}) q^{13} + ( - 9 \beta_{3} + 96) q^{16} + ( - 2 \beta_{6} + \beta_1) q^{17} + ( - 2 \beta_{7} + 6 \beta_{5}) q^{19} + (5 \beta_{6} + 14 \beta_1) q^{20} + ( - 4 \beta_{3} + 120) q^{22} + (3 \beta_{4} + 7 \beta_{2}) q^{23} + ( - 4 \beta_{3} + 65) q^{25} + ( - 5 \beta_{6} - 18 \beta_1) q^{26} + (9 \beta_{4} - 13 \beta_{2}) q^{29} + (10 \beta_{7} - 2 \beta_{5}) q^{31} + ( - 2 \beta_{4} - 107 \beta_{2}) q^{32} + (18 \beta_{7} - \beta_{5}) q^{34} + (24 \beta_{3} + 20) q^{37} + ( - 4 \beta_{6} - 32 \beta_1) q^{38} + ( - 62 \beta_{7} - 23 \beta_{5}) q^{40} + (10 \beta_{6} + \beta_1) q^{41} + (20 \beta_{3} + 280) q^{43} + ( - 16 \beta_{4} - 108 \beta_{2}) q^{44} + (16 \beta_{3} - 88) q^{46} + (2 \beta_{6} + 30 \beta_1) q^{47} + ( - 8 \beta_{4} - 109 \beta_{2}) q^{50} + (62 \beta_{7} + 43 \beta_{5}) q^{52} + ( - 14 \beta_{4} - 34 \beta_{2}) q^{53} + (6 \beta_{7} - 26 \beta_{5}) q^{55} + (14 \beta_{3} + 280) q^{58} + ( - 10 \beta_{6} - 2 \beta_1) q^{59} + ( - 55 \beta_{7} + 22 \beta_{5}) q^{61} + ( - 8 \beta_{6} - 8 \beta_1) q^{62} + ( - 41 \beta_{3} + 928) q^{64} + (17 \beta_{4} + 191 \beta_{2}) q^{65} + (40 \beta_{3} + 20) q^{67} + ( - \beta_{6} - 38 \beta_1) q^{68} + ( - 9 \beta_{4} + 163 \beta_{2}) q^{71} + (23 \beta_{7} - 28 \beta_{5}) q^{73} + (48 \beta_{4} + 244 \beta_{2}) q^{74} + (120 \beta_{7} + 52 \beta_{5}) q^{76} + ( - 28 \beta_{3} + 660) q^{79} + (45 \beta_{6} + 150 \beta_1) q^{80} + ( - 102 \beta_{7} - 13 \beta_{5}) q^{82} + ( - 16 \beta_{6} + 52 \beta_1) q^{83} + (64 \beta_{3} + 390) q^{85} + (40 \beta_{4} - 60 \beta_{2}) q^{86} + ( - 124 \beta_{3} + 640) q^{88} + (10 \beta_{6} + 7 \beta_1) q^{89} + (8 \beta_{4} + 208 \beta_{2}) q^{92} + ( - 80 \beta_{7} - 92 \beta_{5}) q^{94} + ( - 26 \beta_{4} + 342 \beta_{2}) q^{95} + ( - 57 \beta_{7} - 28 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 68 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 68 q^{4} + 804 q^{16} + 976 q^{22} + 536 q^{25} + 64 q^{37} + 2160 q^{43} - 768 q^{46} + 2184 q^{58} + 7588 q^{64} + 5392 q^{79} + 2864 q^{85} + 5616 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 74x^{6} + 1469x^{4} - 8828x^{2} + 2500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 148\nu^{4} + 4633\nu^{2} - 18202 ) / 1416 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{7} - 862\nu^{5} + 13147\nu^{3} - 67414\nu ) / 17700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} - 119\nu^{4} + 1124\nu^{2} + 1828 ) / 177 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 148\nu^{5} - 6049\nu^{3} + 60682\nu ) / 1416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -28\nu^{7} + 2197\nu^{5} - 46357\nu^{3} + 246634\nu ) / 8850 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{6} + 1814\nu^{4} - 22847\nu^{2} + 37214 ) / 708 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 91\nu^{7} - 6034\nu^{5} + 92029\nu^{3} - 347998\nu ) / 17700 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - 7\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 7\beta_{3} - 4\beta _1 + 126 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 44\beta_{7} + 21\beta_{5} - 14\beta_{4} - 231\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -92\beta_{6} - 315\beta_{3} - 296\beta _1 + 4284 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2034\beta_{7} + 1295\beta_{5} - 742\beta_{4} - 9373\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4350\beta_{6} - 14189\beta_{3} - 15364\beta _1 + 177688 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 95558\beta_{7} + 64631\beta_{5} - 35042\beta_{4} - 414659\beta_{2} ) / 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−6.81412
−3.98569
−0.545636
−3.37406
0.545636
3.37406
6.81412
3.98569
−5.39991 0 21.1590 −15.5768 0 0 −71.0573 0 84.1131
1.2 −5.39991 0 21.1590 15.5768 0 0 −71.0573 0 −84.1131
1.3 −1.95985 0 −4.15899 −11.8897 0 0 23.8298 0 23.3019
1.4 −1.95985 0 −4.15899 11.8897 0 0 23.8298 0 −23.3019
1.5 1.95985 0 −4.15899 −11.8897 0 0 −23.8298 0 −23.3019
1.6 1.95985 0 −4.15899 11.8897 0 0 −23.8298 0 23.3019
1.7 5.39991 0 21.1590 −15.5768 0 0 71.0573 0 −84.1131
1.8 5.39991 0 21.1590 15.5768 0 0 71.0573 0 84.1131
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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
7.b odd 2 1 inner
21.c even 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.x ✓ 8
3.b odd 2 1 inner 441.4.a.x ✓ 8
7.b odd 2 1 inner 441.4.a.x ✓ 8
7.c even 3 2 441.4.e.z 16
7.d odd 6 2 441.4.e.z 16
21.c even 2 1 inner 441.4.a.x ✓ 8
21.g even 6 2 441.4.e.z 16
21.h odd 6 2 441.4.e.z 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.x ✓ 8 1.a even 1 1 trivial
441.4.a.x ✓ 8 3.b odd 2 1 inner
441.4.a.x ✓ 8 7.b odd 2 1 inner
441.4.a.x ✓ 8 21.c even 2 1 inner
441.4.e.z 16 7.c even 3 2
441.4.e.z 16 7.d odd 6 2
441.4.e.z 16 21.g even 6 2
441.4.e.z 16 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{4} - 33T_{2}^{2} + 112 \) Copy content Toggle raw display
\( T_{5}^{4} - 384T_{5}^{2} + 34300 \) Copy content Toggle raw display
\( T_{13}^{4} - 5268T_{13}^{2} + 4900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 33 T^{2} + 112)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 384 T^{2} + 34300)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2348 T^{2} + 1355200)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 5268 T^{2} + 4900)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 10496 T^{2} + 8134588)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 23080 T^{2} + 133079296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6012 T^{2} + 9033472)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 53088 T^{2} + 16601200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 19464 T^{2} + 51380224)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T - 92240)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 233264 T^{2} + 10038958300)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 540 T + 8800)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 335128 T^{2} + 19149958912)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 133376 T^{2} + 4433997568)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 231096 T^{2} + 10756480000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 745844 T^{2} + 3928531684)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 256400)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 959388 T^{2} + 187904819200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 535668 T^{2} + 55088784100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 1348 T + 328640)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1919232 T^{2} + 351934815232)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 231776 T^{2} + 13398437500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1382388 T^{2} + 35588822500)^{2} \) Copy content Toggle raw display
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