Properties

Label 224.6.a.i
Level $224$
Weight $6$
Character orbit 224.a
Self dual yes
Analytic conductor $35.926$
Analytic rank $0$
Dimension $5$
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] = [224,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(35.9259756381\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} + ( - \beta_{3} + 7) q^{5} + 49 q^{7} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 127) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{3} + ( - \beta_{3} + 7) q^{5} + 49 q^{7} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 127) q^{9}+ \cdots + (183 \beta_{4} - 1212 \beta_{3} + \cdots - 52253) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33} + 1764 q^{35} + 19642 q^{37} + 10872 q^{39} + 23398 q^{41} + 22044 q^{43} + 49476 q^{45} + 16004 q^{47} + 12005 q^{49} - 45676 q^{51} + 54246 q^{53} + 53456 q^{55} + 109556 q^{57} - 74366 q^{59} + 68316 q^{61} + 31213 q^{63} + 152568 q^{65} - 26560 q^{67} + 214720 q^{69} + 93072 q^{71} + 136098 q^{73} - 124510 q^{75} + 5684 q^{77} + 96080 q^{79} + 104801 q^{81} - 145894 q^{83} + 117352 q^{85} + 168876 q^{87} + 188554 q^{89} + 1960 q^{91} + 86296 q^{93} + 74736 q^{95} - 88146 q^{97} - 260236 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56\nu^{4} - 64\nu^{3} - 8772\nu^{2} - 20218\nu - 3528 ) / 633 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\nu^{4} - 32\nu^{3} - 5652\nu^{2} - 8210\nu + 114075 ) / 633 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36\nu^{4} + 200\nu^{3} - 8352\nu^{2} - 46908\nu + 206833 ) / 211 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} + \beta_{2} + 3\beta _1 + 366 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{4} - 45\beta_{3} + 9\beta_{2} + 630\beta _1 + 1298 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{4} - 339\beta_{3} + 207\beta_{2} + 1552\beta _1 + 58325 ) / 4 \) 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
−11.8926
−8.98949
2.24605
3.97668
14.6594
0 −25.7852 0 −34.4729 0 49.0000 0 421.876 0
1.2 0 −19.9790 0 106.158 0 49.0000 0 156.159 0
1.3 0 2.49210 0 −99.5911 0 49.0000 0 −236.789 0
1.4 0 5.95336 0 11.6830 0 49.0000 0 −207.557 0
1.5 0 27.3187 0 52.2232 0 49.0000 0 503.312 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 224.6.a.i 5
4.b odd 2 1 224.6.a.j yes 5
8.b even 2 1 448.6.a.bf 5
8.d odd 2 1 448.6.a.be 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.i 5 1.a even 1 1 trivial
224.6.a.j yes 5 4.b odd 2 1
448.6.a.be 5 8.d odd 2 1
448.6.a.bf 5 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 10T_{3}^{4} - 876T_{3}^{3} - 7592T_{3}^{2} + 107952T_{3} - 208800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 10 T^{4} + \cdots - 208800 \) Copy content Toggle raw display
$5$ \( T^{5} - 36 T^{4} + \cdots - 222365696 \) Copy content Toggle raw display
$7$ \( (T - 49)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 183286596608 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots + 247266196544 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 353511151373280 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots + 71\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 38\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 77\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 62\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 30\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 26\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 64\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
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