Properties

Label 224.4.b.a
Level $224$
Weight $4$
Character orbit 224.b
Analytic conductor $13.216$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,4,Mod(113,224)]
 
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, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.113");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.b (of order \(2\), degree \(1\), 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: \(13.2164278413\)
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: \( x^{8} - 2x^{7} - 6x^{6} + 12x^{5} + 96x^{3} - 384x^{2} - 1024x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + 7 q^{7} + ( - \beta_{6} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + 7 q^{7} + ( - \beta_{6} - 4) q^{9} + (\beta_{7} + 2 \beta_1) q^{11} + (\beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{13} + ( - \beta_{5} + 7) q^{15} + ( - 2 \beta_{6} - \beta_{5} + \beta_{2} - 17) q^{17} + (2 \beta_{7} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{19} + 7 \beta_1 q^{21} + ( - \beta_{6} - 3 \beta_{5} - \beta_{2} - 34) q^{23} + (2 \beta_{6} - 4 \beta_{5} + 3 \beta_{2} - 5) q^{25} + ( - 4 \beta_{7} - 11 \beta_{4} + 5 \beta_{3} + \beta_1) q^{27} + ( - \beta_{7} - 15 \beta_{4} + 10 \beta_{3} - 10 \beta_1) q^{29} + ( - 5 \beta_{6} + \beta_{5} - 4 \beta_{2} + 34) q^{31} + ( - 7 \beta_{6} - 2 \beta_{5} - 3 \beta_{2} - 63) q^{33} + 7 \beta_{3} q^{35} + ( - 5 \beta_{7} - 13 \beta_{4} - 4 \beta_{3} + 4 \beta_1) q^{37} + ( - 5 \beta_{6} - 3 \beta_{5} + \beta_{2} - 86) q^{39} + (\beta_{6} + \beta_{5} - 6 \beta_{2} + 86) q^{41} + ( - 5 \beta_{7} + 4 \beta_{4} + 6 \beta_{3} + 44 \beta_1) q^{43} + ( - 2 \beta_{7} + \beta_{4} + 18 \beta_{3} + 17 \beta_1) q^{45} + (3 \beta_{6} - 9 \beta_{5} + 12 \beta_{2}) q^{47} + 49 q^{49} + ( - 6 \beta_{7} - 28 \beta_{4} + 6 \beta_{3} - 52 \beta_1) q^{51} + (17 \beta_{7} + \beta_{4} - 22 \beta_{3} - 4 \beta_1) q^{53} + (2 \beta_{6} + 6 \beta_{5} + 2 \beta_{2} + 12) q^{55} + ( - 2 \beta_{6} - 2 \beta_{5} - 9 \beta_{2} + 88) q^{57} + (12 \beta_{7} + 11 \beta_{4} + 7 \beta_{3} + 46 \beta_1) q^{59} + (12 \beta_{7} + 26 \beta_{4} + 17 \beta_{3} + 18 \beta_1) q^{61} + ( - 7 \beta_{6} - 28) q^{63} + (5 \beta_{6} - 12 \beta_{5} + 6 \beta_{2} - 277) q^{65} + (5 \beta_{7} + 34 \beta_{4} + 8 \beta_{3} - 52 \beta_1) q^{67} + ( - 14 \beta_{7} - \beta_{4} - 27 \beta_{3} - 25 \beta_1) q^{69} + ( - 15 \beta_{6} + 7 \beta_{2} - 153) q^{71} + ( - \beta_{6} + 10 \beta_{5} + 19 \beta_{2} + 189) q^{73} + (12 \beta_{7} + 5 \beta_{4} - 31 \beta_{3} + 76 \beta_1) q^{75} + (7 \beta_{7} + 14 \beta_1) q^{77} + (7 \beta_{6} + 8 \beta_{5} - 23 \beta_{2} + 65) q^{79} + (14 \beta_{6} + 14 \beta_{5} + \beta_{2} - 23) q^{81} + ( - 10 \beta_{7} + 54 \beta_{4} - 42 \beta_{3} - 63 \beta_1) q^{83} + (2 \beta_{7} + 12 \beta_{4} - 24 \beta_{3} - 6 \beta_1) q^{85} + (45 \beta_{6} + 7 \beta_{5} - 12 \beta_{2} + 486) q^{87} + ( - 9 \beta_{6} + 4 \beta_{5} + 3 \beta_{2} + 239) q^{89} + (7 \beta_{4} + 14 \beta_{3} + 21 \beta_1) q^{91} + ( - 34 \beta_{7} - 28 \beta_{4} + 14 \beta_{3} - 82 \beta_1) q^{93} + (3 \beta_{6} + 17 \beta_{5} + 7 \beta_{2} - 34) q^{95} + (32 \beta_{6} + 39 \beta_{5} - 7 \beta_{2} - 275) q^{97} + ( - 17 \beta_{7} - 54 \beta_{4} + 2 \beta_{3} - 140 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 56 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 56 q^{7} - 36 q^{9} + 60 q^{15} - 144 q^{17} - 260 q^{23} - 28 q^{25} + 264 q^{31} - 512 q^{33} - 700 q^{39} + 712 q^{41} + 392 q^{49} + 72 q^{55} + 740 q^{57} - 252 q^{63} - 2172 q^{65} - 1312 q^{71} + 1392 q^{73} + 608 q^{79} - 188 q^{81} + 4088 q^{87} + 1848 q^{89} - 356 q^{95} - 2200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 6x^{6} + 12x^{5} + 96x^{3} - 384x^{2} - 1024x + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} + 2\nu^{6} - 2\nu^{5} - 12\nu^{4} + 32\nu^{3} + 288\nu^{2} - 640\nu - 3072 ) / 512 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 6\nu^{6} - 18\nu^{5} - 4\nu^{4} + 272\nu^{3} - 800\nu^{2} + 1024\nu - 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 6\nu^{6} + 18\nu^{5} + 28\nu^{4} - 128\nu^{3} - 672\nu^{2} + 384\nu + 3072 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{5} + 16\nu^{4} - 24\nu^{3} + 64\nu - 1024 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 10\nu^{6} - 14\nu^{5} - 92\nu^{4} - 80\nu^{3} - 224\nu^{2} + 2560\nu - 2816 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - 2\nu^{6} + 38\nu^{5} + 44\nu^{4} + 80\nu^{3} - 160\nu^{2} + 1024\nu + 5888 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 14\nu^{6} - 42\nu^{5} + 12\nu^{4} + 16\nu^{3} + 736\nu^{2} + 512\nu - 9216 ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{3} - 2\beta _1 + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{4} - 2\beta_{3} - \beta_{2} + 15 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{6} - \beta_{5} - 2\beta_{4} - 6\beta_{3} + 4\beta_{2} + 10\beta _1 + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} - 4\beta_{5} + 13\beta_{4} + 2\beta_{3} + \beta_{2} - 8\beta _1 + 41 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{7} + 17\beta_{6} + 5\beta_{5} + 2\beta_{4} + 14\beta_{3} - 4\beta_{2} + 78\beta _1 - 232 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{7} + 5\beta_{6} - 25\beta_{4} + 62\beta_{3} + 3\beta_{2} + 80\beta _1 + 411 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -32\beta_{7} + 3\beta_{6} + 23\beta_{5} + 126\beta_{4} - 22\beta_{3} + 12\beta_{2} + 218\beta _1 + 1504 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
2.79819 0.412490i
−2.76031 0.616979i
−0.948616 + 2.66461i
1.91074 + 2.08544i
1.91074 2.08544i
−0.948616 2.66461i
−2.76031 + 0.616979i
2.79819 + 0.412490i
0 8.87222i 0 1.57179i 0 7.00000 0 −51.7163 0
113.2 0 5.24924i 0 9.85277i 0 7.00000 0 −0.554549 0
113.3 0 3.82805i 0 2.56837i 0 7.00000 0 12.3460 0
113.4 0 2.25282i 0 20.1954i 0 7.00000 0 21.9248 0
113.5 0 2.25282i 0 20.1954i 0 7.00000 0 21.9248 0
113.6 0 3.82805i 0 2.56837i 0 7.00000 0 12.3460 0
113.7 0 5.24924i 0 9.85277i 0 7.00000 0 −0.554549 0
113.8 0 8.87222i 0 1.57179i 0 7.00000 0 −51.7163 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.b.a 8
3.b odd 2 1 2016.4.c.a 8
4.b odd 2 1 56.4.b.a 8
8.b even 2 1 inner 224.4.b.a 8
8.d odd 2 1 56.4.b.a 8
12.b even 2 1 504.4.c.a 8
16.e even 4 2 1792.4.a.u 8
16.f odd 4 2 1792.4.a.w 8
24.f even 2 1 504.4.c.a 8
24.h odd 2 1 2016.4.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.b.a 8 4.b odd 2 1
56.4.b.a 8 8.d odd 2 1
224.4.b.a 8 1.a even 1 1 trivial
224.4.b.a 8 8.b even 2 1 inner
504.4.c.a 8 12.b even 2 1
504.4.c.a 8 24.f even 2 1
1792.4.a.u 8 16.e even 4 2
1792.4.a.w 8 16.f odd 4 2
2016.4.c.a 8 3.b odd 2 1
2016.4.c.a 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 126T_{3}^{6} + 4340T_{3}^{4} + 50696T_{3}^{2} + 161312 \) acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 126 T^{6} + 4340 T^{4} + \cdots + 161312 \) Copy content Toggle raw display
$5$ \( T^{8} + 514 T^{6} + 44188 T^{4} + \cdots + 645248 \) Copy content Toggle raw display
$7$ \( (T - 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 3908 T^{6} + \cdots + 223860787200 \) Copy content Toggle raw display
$13$ \( T^{8} + 4162 T^{6} + \cdots + 8031006848 \) Copy content Toggle raw display
$17$ \( (T^{4} + 72 T^{3} - 5384 T^{2} + \cdots - 1875824)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 20958 T^{6} + \cdots + 61589268512 \) Copy content Toggle raw display
$23$ \( (T^{4} + 130 T^{3} - 19220 T^{2} + \cdots - 85969984)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 157048 T^{6} + \cdots + 60\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( (T^{4} - 132 T^{3} - 64720 T^{2} + \cdots + 793521152)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 182136 T^{6} + \cdots + 64\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( (T^{4} - 356 T^{3} - 28784 T^{2} + \cdots - 1202210128)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 361156 T^{6} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( (T^{4} - 381312 T^{2} + \cdots + 12485394432)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 1194896 T^{6} + \cdots + 31\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{8} + 851214 T^{6} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{8} + 1109282 T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + 1033628 T^{6} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( (T^{4} + 656 T^{3} + \cdots + 31205572608)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 696 T^{3} + \cdots - 78638316912)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 304 T^{3} + \cdots - 117180334080)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 2480782 T^{6} + \cdots + 49\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( (T^{4} - 924 T^{3} + 45920 T^{2} + \cdots + 3549530480)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1100 T^{3} + \cdots + 2140769108912)^{2} \) Copy content Toggle raw display
show more
show less