Properties

Label 224.2.u.a
Level $224$
Weight $2$
Character orbit 224.u
Analytic conductor $1.789$
Analytic rank $1$
Dimension $4$
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] = [224,2,Mod(29,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.u (of order \(8\), degree \(4\), 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: \(1.78864900528\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{2} - 1) q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} - 1) q^{3} - 2 \zeta_{8}^{2} q^{4} + ( - 2 \zeta_{8} - 2) q^{5} + (2 \zeta_{8} + 2) q^{6} + \zeta_{8} q^{7} + (2 \zeta_{8}^{2} + 2) q^{8} + (\zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{2} - 1) q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} - 1) q^{3} - 2 \zeta_{8}^{2} q^{4} + ( - 2 \zeta_{8} - 2) q^{5} + (2 \zeta_{8} + 2) q^{6} + \zeta_{8} q^{7} + (2 \zeta_{8}^{2} + 2) q^{8} + (\zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{9} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{10} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 3 \zeta_{8} - 3) q^{11} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{12} + ( - 2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{13} + (\zeta_{8}^{3} - \zeta_{8}) q^{14} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4 \zeta_{8}) q^{15} - 4 q^{16} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8}) q^{18} + (\zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3 \zeta_{8} - 1) q^{19} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2}) q^{20} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} + 1) q^{21} + (\zeta_{8}^{3} - 5 \zeta_{8}^{2} - 5 \zeta_{8} + 1) q^{22} + (\zeta_{8}^{2} - 1) q^{23} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2}) q^{24} + (4 \zeta_{8}^{2} + 3 \zeta_{8} + 4) q^{25} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8} + 2) q^{26} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{27} - 2 \zeta_{8}^{3} q^{28} + ( - 4 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3 \zeta_{8} - 4) q^{29} + ( - 4 \zeta_{8}^{2} - 8 \zeta_{8} - 4) q^{30} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 6) q^{31} + ( - 4 \zeta_{8}^{2} + 4) q^{32} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} + 10) q^{33} + ( - 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{35} + (4 \zeta_{8}^{2} + 2 \zeta_{8} + 4) q^{36} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 4 \zeta_{8} - 4) q^{37} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 4 \zeta_{8} + 4) q^{38} - 4 \zeta_{8} q^{39} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4 \zeta_{8} - 4) q^{40} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{41} + (2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{42} + (\zeta_{8}^{3} + \zeta_{8}^{2} + 4 \zeta_{8} - 4) q^{43} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{44} + ( - 6 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8} + 6) q^{45} - 2 \zeta_{8}^{2} q^{46} + (2 \zeta_{8}^{3} + 6 \zeta_{8}^{2} + 2 \zeta_{8}) q^{47} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{48} + \zeta_{8}^{2} q^{49} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 8) q^{50} + ( - 4 \zeta_{8}^{3} - 4) q^{52} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 5 \zeta_{8} + 5) q^{53} + (4 \zeta_{8}^{2} - 4 \zeta_{8}) q^{54} + ( - 8 \zeta_{8}^{3} - 10 \zeta_{8}^{2} + 10) q^{55} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{56} + (2 \zeta_{8}^{2} - 4 \zeta_{8} + 2) q^{57} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 7 \zeta_{8} + 7) q^{58} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 5 \zeta_{8} - 5) q^{59} + ( - 8 \zeta_{8}^{3} + 8 \zeta_{8} + 8) q^{60} + ( - 4 \zeta_{8}^{3} - 4) q^{61} + ( - 4 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 6) q^{62} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{63} + 8 \zeta_{8}^{2} q^{64} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{65} + (8 \zeta_{8}^{3} + 10 \zeta_{8}^{2} - 10) q^{66} + (4 \zeta_{8}^{3} + 5 \zeta_{8}^{2} + 5 \zeta_{8} + 4) q^{67} + (2 \zeta_{8} + 2) q^{69} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{70} - 6 \zeta_{8} q^{71} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 8) q^{72} + (4 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 6) q^{73} + ( - \zeta_{8}^{3} - 7 \zeta_{8}^{2} + 7 \zeta_{8} + 1) q^{74} + ( - 11 \zeta_{8}^{3} - 11 \zeta_{8}^{2} - 3 \zeta_{8} + 3) q^{75} + ( - 6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8} - 6) q^{76} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3 \zeta_{8} - 2) q^{77} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{78} + ( - 7 \zeta_{8}^{3} - 7 \zeta_{8}) q^{79} + (8 \zeta_{8} + 8) q^{80} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 2 \zeta_{8}) q^{81} + ( - 2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{82} + (\zeta_{8}^{3} - 7 \zeta_{8}^{2} + 7 \zeta_{8} - 1) q^{83} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{84} + (3 \zeta_{8}^{3} - 5 \zeta_{8}^{2} - 5 \zeta_{8} + 3) q^{86} + (14 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 6) q^{87} + (10 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} - 10) q^{88} + ( - 8 \zeta_{8}^{2} + 4 \zeta_{8} - 8) q^{89} + (10 \zeta_{8}^{3} + 10 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{90} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2}) q^{91} + (2 \zeta_{8}^{2} + 2) q^{92} + (6 \zeta_{8}^{3} + 10 \zeta_{8}^{2} + 10 \zeta_{8} + 6) q^{93} + ( - 6 \zeta_{8}^{2} - 4 \zeta_{8} - 6) q^{94} + (4 \zeta_{8}^{3} - 4 \zeta_{8} + 4) q^{95} + ( - 8 \zeta_{8} - 8) q^{96} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 4) q^{97} + ( - \zeta_{8}^{2} - 1) q^{98} + ( - \zeta_{8}^{3} - 12 \zeta_{8}^{2} - 12 \zeta_{8} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} - 8 q^{5} + 8 q^{6} + 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} - 8 q^{5} + 8 q^{6} + 8 q^{8} - 8 q^{9} + 8 q^{10} - 12 q^{11} - 8 q^{12} - 16 q^{16} - 4 q^{19} + 4 q^{21} + 4 q^{22} - 4 q^{23} + 16 q^{25} + 8 q^{26} + 8 q^{27} - 16 q^{29} - 16 q^{30} - 24 q^{31} + 16 q^{32} + 40 q^{33} + 16 q^{36} - 16 q^{37} + 16 q^{38} - 16 q^{40} - 8 q^{41} - 16 q^{43} + 16 q^{44} + 24 q^{45} + 16 q^{48} - 32 q^{50} - 16 q^{52} + 20 q^{53} + 40 q^{55} + 8 q^{57} + 28 q^{58} - 20 q^{59} + 32 q^{60} - 16 q^{61} + 24 q^{62} - 4 q^{63} - 40 q^{66} + 16 q^{67} + 8 q^{69} + 8 q^{70} - 32 q^{72} + 24 q^{73} + 4 q^{74} + 12 q^{75} - 24 q^{76} - 8 q^{77} + 32 q^{80} - 4 q^{83} - 8 q^{84} + 12 q^{86} - 24 q^{87} - 40 q^{88} - 32 q^{89} - 8 q^{90} + 8 q^{92} + 24 q^{93} - 24 q^{94} + 16 q^{95} - 32 q^{96} + 16 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.00000 1.00000i −1.00000 0.414214i 2.00000i −0.585786 1.41421i 0.585786 + 1.41421i −0.707107 + 0.707107i 2.00000 2.00000i −1.29289 1.29289i −0.828427 + 2.00000i
85.1 −1.00000 + 1.00000i −1.00000 + 0.414214i 2.00000i −0.585786 + 1.41421i 0.585786 1.41421i −0.707107 0.707107i 2.00000 + 2.00000i −1.29289 + 1.29289i −0.828427 2.00000i
141.1 −1.00000 1.00000i −1.00000 + 2.41421i 2.00000i −3.41421 + 1.41421i 3.41421 1.41421i 0.707107 0.707107i 2.00000 2.00000i −2.70711 2.70711i 4.82843 + 2.00000i
197.1 −1.00000 + 1.00000i −1.00000 2.41421i 2.00000i −3.41421 1.41421i 3.41421 + 1.41421i 0.707107 + 0.707107i 2.00000 + 2.00000i −2.70711 + 2.70711i 4.82843 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.u.a 4
4.b odd 2 1 896.2.u.a 4
32.g even 8 1 inner 224.2.u.a 4
32.h odd 8 1 896.2.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.a 4 1.a even 1 1 trivial
224.2.u.a 4 32.g even 8 1 inner
896.2.u.a 4 4.b odd 2 1
896.2.u.a 4 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{3} + 12T_{3}^{2} + 16T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 12 T^{2} + 16 T + 8 \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 24 T^{2} + 32 T + 32 \) Copy content Toggle raw display
$7$ \( T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + 86 T^{2} + \cdots + 578 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + 36 T^{2} - 32 T + 8 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 16 T^{3} + 162 T^{2} + \cdots + 1922 \) Copy content Toggle raw display
$31$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 16 T^{3} + 162 T^{2} + \cdots + 1922 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 16 T^{3} + 114 T^{2} + \cdots + 1058 \) Copy content Toggle raw display
$47$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} + 118 T^{2} - 12 T + 2 \) Copy content Toggle raw display
$59$ \( T^{4} + 20 T^{3} + 300 T^{2} + \cdots + 5000 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + 226 T^{2} + \cdots + 1922 \) Copy content Toggle raw display
$71$ \( T^{4} + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$79$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + 132 T^{2} + \cdots + 2312 \) Copy content Toggle raw display
$89$ \( T^{4} + 32 T^{3} + 512 T^{2} + \cdots + 12544 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
show more
show less