Properties

Label 224.4.a.f
Level $224$
Weight $4$
Character orbit 224.a
Self dual yes
Analytic conductor $13.216$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,4,Mod(1,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("224.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.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: \(13.2164278413\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} - 2 \beta_1 - 2) q^{5} - 7 q^{7} + (3 \beta_{2} - \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} - 2 \beta_1 - 2) q^{5} - 7 q^{7} + (3 \beta_{2} - \beta_1 + 5) q^{9} + (7 \beta_{2} + 3 \beta_1) q^{11} + (3 \beta_{2} - 8 \beta_1 - 2) q^{13} + ( - 7 \beta_{2} - 5 \beta_1 - 56) q^{15} + ( - 14 \beta_{2} - 22) q^{17} + ( - 14 \beta_{2} - 3 \beta_1 - 56) q^{19} - 7 \beta_1 q^{21} + (7 \beta_{2} - 11 \beta_1 - 112) q^{23} + (15 \beta_{2} + 23 \beta_1 + 23) q^{25} + ( - 6 \beta_1 - 56) q^{27} + (2 \beta_{2} + 4 \beta_1 + 30) q^{29} + (14 \beta_{2} + 16 \beta_1 - 168) q^{31} + (16 \beta_{2} + 32 \beta_1 + 40) q^{33} + (7 \beta_{2} + 14 \beta_1 + 14) q^{35} + ( - 24 \beta_{2} + 22 \beta_1 + 6) q^{37} + ( - 21 \beta_{2} + 21 \beta_1 - 280) q^{39} + ( - 28 \beta_{2} - 14 \beta_1 - 150) q^{41} + (21 \beta_{2} + \beta_1) q^{43} + (5 \beta_{2} - 32 \beta_1 - 50) q^{45} + (28 \beta_{2} + 62 \beta_1 - 168) q^{47} + 49 q^{49} + ( - 14 \beta_{2} - 92 \beta_1 + 112) q^{51} + (54 \beta_{2} + 66 \beta_1 - 26) q^{53} + ( - 14 \beta_{2} - 78 \beta_1 - 392) q^{55} + ( - 23 \beta_{2} - 123 \beta_1 + 16) q^{57} + ( - 56 \beta_{2} + 71 \beta_1 + 168) q^{59} + ( - 13 \beta_{2} + 2 \beta_1 + 166) q^{61} + ( - 21 \beta_{2} + 7 \beta_1 - 35) q^{63} + (61 \beta_{2} + 17 \beta_1 + 356) q^{65} + ( - 7 \beta_{2} - 21 \beta_1 + 336) q^{67} + ( - 26 \beta_{2} - 66 \beta_1 - 408) q^{69} + (14 \beta_{2} - 58 \beta_1 - 168) q^{71} + (4 \beta_{2} - 104 \beta_1 - 78) q^{73} + (84 \beta_{2} + 75 \beta_1 + 616) q^{75} + ( - 49 \beta_{2} - 21 \beta_1) q^{77} + ( - 28 \beta_{2} - 144 \beta_1 - 56) q^{79} + ( - 99 \beta_{2} - 23 \beta_1 - 327) q^{81} + (42 \beta_{2} + 29 \beta_1 + 1008) q^{83} + (8 \beta_{2} + 170 \beta_1 + 492) q^{85} + (14 \beta_{2} + 36 \beta_1 + 112) q^{87} + (38 \beta_{2} + 34 \beta_1 + 82) q^{89} + ( - 21 \beta_{2} + 56 \beta_1 + 14) q^{91} + (62 \beta_{2} - 114 \beta_1 + 400) q^{93} + (63 \beta_{2} + 253 \beta_1 + 728) q^{95} + (92 \beta_{2} + 142 \beta_1 - 838) q^{97} + ( - 77 \beta_{2} + 7 \beta_1 + 896) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{5} - 21 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{5} - 21 q^{7} + 15 q^{9} - 6 q^{13} - 168 q^{15} - 66 q^{17} - 168 q^{19} - 336 q^{23} + 69 q^{25} - 168 q^{27} + 90 q^{29} - 504 q^{31} + 120 q^{33} + 42 q^{35} + 18 q^{37} - 840 q^{39} - 450 q^{41} - 150 q^{45} - 504 q^{47} + 147 q^{49} + 336 q^{51} - 78 q^{53} - 1176 q^{55} + 48 q^{57} + 504 q^{59} + 498 q^{61} - 105 q^{63} + 1068 q^{65} + 1008 q^{67} - 1224 q^{69} - 504 q^{71} - 234 q^{73} + 1848 q^{75} - 168 q^{79} - 981 q^{81} + 3024 q^{83} + 1476 q^{85} + 336 q^{87} + 246 q^{89} + 42 q^{91} + 1200 q^{93} + 2184 q^{95} - 2514 q^{97} + 2688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} + 4\nu + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 + 8 ) / 2 \) 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
−0.523976
−2.14510
2.66908
0 −7.45090 0 7.54680 0 −7.00000 0 28.5159 0
1.2 0 1.20293 0 5.37748 0 −7.00000 0 −25.5530 0
1.3 0 6.24797 0 −18.9243 0 −7.00000 0 12.0371 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.a.f 3
3.b odd 2 1 2016.4.a.w 3
4.b odd 2 1 224.4.a.g yes 3
7.b odd 2 1 1568.4.a.w 3
8.b even 2 1 448.4.a.v 3
8.d odd 2 1 448.4.a.w 3
12.b even 2 1 2016.4.a.x 3
28.d even 2 1 1568.4.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.a.f 3 1.a even 1 1 trivial
224.4.a.g yes 3 4.b odd 2 1
448.4.a.v 3 8.b even 2 1
448.4.a.w 3 8.d odd 2 1
1568.4.a.w 3 7.b odd 2 1
1568.4.a.x 3 28.d even 2 1
2016.4.a.w 3 3.b odd 2 1
2016.4.a.x 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 48T_{3} + 56 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(224))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 48T + 56 \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} - 204 T + 768 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 3456T + 48832 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 4284 T - 116816 \) Copy content Toggle raw display
$17$ \( T^{3} + 66 T^{2} - 12660 T - 936424 \) Copy content Toggle raw display
$19$ \( T^{3} + 168 T^{2} - 4128 T - 1148952 \) Copy content Toggle raw display
$23$ \( T^{3} + 336 T^{2} + 26448 T - 215936 \) Copy content Toggle raw display
$29$ \( T^{3} - 90 T^{2} + 1836 T - 10616 \) Copy content Toggle raw display
$31$ \( T^{3} + 504 T^{2} + 63648 T + 366016 \) Copy content Toggle raw display
$37$ \( T^{3} - 18 T^{2} - 77268 T + 2845928 \) Copy content Toggle raw display
$41$ \( T^{3} + 450 T^{2} + 11052 T - 7572776 \) Copy content Toggle raw display
$43$ \( T^{3} - 31296 T + 2128448 \) Copy content Toggle raw display
$47$ \( T^{3} + 504 T^{2} + \cdots - 60547648 \) Copy content Toggle raw display
$53$ \( T^{3} + 78 T^{2} - 331476 T - 67903192 \) Copy content Toggle raw display
$59$ \( T^{3} - 504 T^{2} + \cdots + 192556616 \) Copy content Toggle raw display
$61$ \( T^{3} - 498 T^{2} + 69684 T - 2912608 \) Copy content Toggle raw display
$67$ \( T^{3} - 1008 T^{2} + \cdots - 29986432 \) Copy content Toggle raw display
$71$ \( T^{3} + 504 T^{2} + \cdots - 59523072 \) Copy content Toggle raw display
$73$ \( T^{3} + 234 T^{2} + \cdots - 123555144 \) Copy content Toggle raw display
$79$ \( T^{3} + 168 T^{2} + \cdots + 41853952 \) Copy content Toggle raw display
$83$ \( T^{3} - 3024 T^{2} + \cdots - 883723736 \) Copy content Toggle raw display
$89$ \( T^{3} - 246 T^{2} - 108276 T + 3703224 \) Copy content Toggle raw display
$97$ \( T^{3} + 2514 T^{2} + \cdots - 1004302696 \) Copy content Toggle raw display
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