↑

Properties

Label 630.2.z.b
Level $630$
Weight $2$
Character orbit 630.z
Analytic conductor $5.031$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(169,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 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("630.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.z (of order \(6\), degree \(2\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{9} - 4 q^{10} + 4 q^{11} - 12 q^{14} + 28 q^{15} - 12 q^{16} + 32 q^{19} + 2 q^{20} - 4 q^{21} - 4 q^{24} - 16 q^{26} + 20 q^{29} - 26 q^{30} + 36 q^{31} + 4 q^{35} + 4 q^{36} + 24 q^{39} - 2 q^{40} - 20 q^{41} + 8 q^{44} - 22 q^{45} - 32 q^{46} + 12 q^{49} - 68 q^{51} - 16 q^{54} + 28 q^{55} + 12 q^{56} + 26 q^{60} + 16 q^{61} - 24 q^{64} + 16 q^{65} + 32 q^{66} - 40 q^{69} - 2 q^{70} + 24 q^{71} + 44 q^{74} - 28 q^{75} + 16 q^{76} - 68 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{84} + 26 q^{85} - 24 q^{86} + 40 q^{89} + 22 q^{90} + 16 q^{91} - 48 q^{94} - 50 q^{95} - 8 q^{96} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 −0.866025 − 0.500000i −1.70090 − 0.327034i 0.500000 + 0.866025i 1.08556 + 1.95488i 1.30950 + 1.13367i 0.866025 + 0.500000i − 1.00000i 2.78610 + 1.11250i 0.0373163 − 2.23576i
169.2 −0.866025 − 0.500000i −1.21114 + 1.23820i 0.500000 + 0.866025i −2.13204 − 0.674085i 1.66798 − 0.466738i 0.866025 + 0.500000i − 1.00000i −0.0662639 − 2.99927i 1.50936 + 1.64980i
169.3 −0.866025 − 0.500000i −1.13367 − 1.30950i 0.500000 + 0.866025i −1.91756 + 1.15020i 0.327034 + 1.70090i 0.866025 + 0.500000i − 1.00000i −0.429595 + 2.96908i 2.23576 − 0.0373163i
169.4 −0.866025 − 0.500000i 0.241007 + 1.71520i 0.500000 + 0.866025i 0.868341 − 2.06058i 0.648883 − 1.60591i 0.866025 + 0.500000i − 1.00000i −2.88383 + 0.826751i −1.78229 + 1.35034i
169.5 −0.866025 − 0.500000i 0.466738 − 1.66798i 0.500000 + 0.866025i 2.18345 + 0.482247i −1.23820 + 1.21114i 0.866025 + 0.500000i − 1.00000i −2.56431 − 1.55702i −1.64980 − 1.50936i
169.6 −0.866025 − 0.500000i 1.60591 − 0.648883i 0.500000 + 0.866025i 0.278284 − 2.21868i −1.71520 − 0.241007i 0.866025 + 0.500000i − 1.00000i 2.15790 − 2.08410i −1.35034 + 1.78229i
169.7 0.866025 + 0.500000i −1.60591 + 0.648883i 0.500000 + 0.866025i −2.06058 − 0.868341i −1.71520 − 0.241007i −0.866025 − 0.500000i 1.00000i 2.15790 − 2.08410i −1.35034 − 1.78229i
169.8 0.866025 + 0.500000i −0.466738 + 1.66798i 0.500000 + 0.866025i −0.674085 + 2.13204i −1.23820 + 1.21114i −0.866025 − 0.500000i 1.00000i −2.56431 − 1.55702i −1.64980 + 1.50936i
169.9 0.866025 + 0.500000i −0.241007 − 1.71520i 0.500000 + 0.866025i −2.21868 − 0.278284i 0.648883 − 1.60591i −0.866025 − 0.500000i 1.00000i −2.88383 + 0.826751i −1.78229 − 1.35034i
169.10 0.866025 + 0.500000i 1.13367 + 1.30950i 0.500000 + 0.866025i 1.95488 − 1.08556i 0.327034 + 1.70090i −0.866025 − 0.500000i 1.00000i −0.429595 + 2.96908i 2.23576 + 0.0373163i
169.11 0.866025 + 0.500000i 1.21114 − 1.23820i 0.500000 + 0.866025i 0.482247 − 2.18345i 1.66798 − 0.466738i −0.866025 − 0.500000i 1.00000i −0.0662639 − 2.99927i 1.50936 − 1.64980i
169.12 0.866025 + 0.500000i 1.70090 + 0.327034i 0.500000 + 0.866025i 1.15020 + 1.91756i 1.30950 + 1.13367i −0.866025 − 0.500000i 1.00000i 2.78610 + 1.11250i 0.0373163 + 2.23576i
589.1 −0.866025 + 0.500000i −1.70090 + 0.327034i 0.500000 − 0.866025i 1.08556 − 1.95488i 1.30950 − 1.13367i 0.866025 − 0.500000i 1.00000i 2.78610 − 1.11250i 0.0373163 + 2.23576i
589.2 −0.866025 + 0.500000i −1.21114 − 1.23820i 0.500000 − 0.866025i −2.13204 + 0.674085i 1.66798 + 0.466738i 0.866025 − 0.500000i 1.00000i −0.0662639 + 2.99927i 1.50936 − 1.64980i
589.3 −0.866025 + 0.500000i −1.13367 + 1.30950i 0.500000 − 0.866025i −1.91756 − 1.15020i 0.327034 − 1.70090i 0.866025 − 0.500000i 1.00000i −0.429595 − 2.96908i 2.23576 + 0.0373163i
589.4 −0.866025 + 0.500000i 0.241007 − 1.71520i 0.500000 − 0.866025i 0.868341 + 2.06058i 0.648883 + 1.60591i 0.866025 − 0.500000i 1.00000i −2.88383 − 0.826751i −1.78229 − 1.35034i
589.5 −0.866025 + 0.500000i 0.466738 + 1.66798i 0.500000 − 0.866025i 2.18345 − 0.482247i −1.23820 − 1.21114i 0.866025 − 0.500000i 1.00000i −2.56431 + 1.55702i −1.64980 + 1.50936i
589.6 −0.866025 + 0.500000i 1.60591 + 0.648883i 0.500000 − 0.866025i 0.278284 + 2.21868i −1.71520 + 0.241007i 0.866025 − 0.500000i 1.00000i 2.15790 + 2.08410i −1.35034 − 1.78229i
589.7 0.866025 − 0.500000i −1.60591 − 0.648883i 0.500000 − 0.866025i −2.06058 + 0.868341i −1.71520 + 0.241007i −0.866025 + 0.500000i − 1.00000i 2.15790 + 2.08410i −1.35034 + 1.78229i
589.8 0.866025 − 0.500000i −0.466738 − 1.66798i 0.500000 − 0.866025i −0.674085 − 2.13204i −1.23820 − 1.21114i −0.866025 + 0.500000i − 1.00000i −2.56431 + 1.55702i −1.64980 − 1.50936i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.z.b ✓ 24
3.b odd 2 1 1890.2.z.b 24
5.b even 2 1 inner 630.2.z.b ✓ 24
9.c even 3 1 inner 630.2.z.b ✓ 24
9.d odd 6 1 1890.2.z.b 24
15.d odd 2 1 1890.2.z.b 24
45.h odd 6 1 1890.2.z.b 24
45.j even 6 1 inner 630.2.z.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.z.b ✓ 24 1.a even 1 1 trivial
630.2.z.b ✓ 24 5.b even 2 1 inner
630.2.z.b ✓ 24 9.c even 3 1 inner
630.2.z.b ✓ 24 45.j even 6 1 inner
1890.2.z.b 24 3.b odd 2 1
1890.2.z.b 24 9.d odd 6 1
1890.2.z.b 24 15.d odd 2 1
1890.2.z.b 24 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} - 2 T_{11}^{11} + 27 T_{11}^{10} - 98 T_{11}^{9} + 700 T_{11}^{8} - 1788 T_{11}^{7} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display