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Properties

Label 630.2.z.c
Level $630$
Weight $2$
Character orbit 630.z
Analytic conductor $5.031$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

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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: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 22 q^{4} + 2 q^{5} + 2 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 22 q^{4} + 2 q^{5} + 2 q^{6} + 8 q^{9} - 6 q^{11} + 22 q^{14} - 2 q^{15} - 22 q^{16} - 40 q^{19} - 2 q^{20} + 2 q^{21} + 4 q^{24} - 6 q^{25} + 12 q^{26} + 8 q^{29} + 10 q^{30} - 12 q^{31} + 2 q^{34} + 10 q^{36} - 22 q^{39} - 8 q^{41} - 12 q^{44} - 14 q^{45} + 32 q^{46} + 22 q^{49} - 4 q^{50} - 44 q^{51} - 38 q^{54} + 36 q^{55} - 22 q^{56} - 8 q^{59} - 16 q^{60} - 44 q^{61} - 44 q^{64} + 30 q^{65} - 22 q^{66} + 24 q^{69} - 2 q^{70} + 20 q^{71} + 28 q^{74} + 52 q^{75} - 20 q^{76} + 66 q^{79} - 4 q^{80} - 8 q^{81} + 4 q^{84} - 4 q^{85} - 32 q^{86} + 80 q^{89} + 26 q^{90} + 12 q^{91} + 46 q^{94} + 30 q^{95} + 2 q^{96} - 106 q^{99}+O(q^{100}) \) 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 −1.62095 − 0.610356i 0.500000 + 0.866025i −1.05657 − 1.97070i 1.09860 + 1.33906i −0.866025 − 0.500000i − 1.00000i 2.25493 + 1.97871i −0.0703298 + 2.23496i
169.2 −0.866025 − 0.500000i −1.58428 + 0.700045i 0.500000 + 0.866025i −1.44629 + 1.70536i 1.72205 + 0.185882i −0.866025 − 0.500000i − 1.00000i 2.01987 − 2.21813i 2.10520 − 0.753742i
169.3 −0.866025 − 0.500000i −1.46538 − 0.923390i 0.500000 + 0.866025i 2.07585 − 0.831160i 0.807365 + 1.53237i −0.866025 − 0.500000i − 1.00000i 1.29470 + 2.70624i −2.21332 − 0.318122i
169.4 −0.866025 − 0.500000i −0.974304 + 1.43204i 0.500000 + 0.866025i 1.13975 + 1.92379i 1.55979 − 0.753029i −0.866025 − 0.500000i − 1.00000i −1.10146 − 2.79048i −0.0251517 − 2.23593i
169.5 −0.866025 − 0.500000i −0.351901 − 1.69593i 0.500000 + 0.866025i 0.634232 + 2.14424i −0.543208 + 1.64467i −0.866025 − 0.500000i − 1.00000i −2.75233 + 1.19360i 0.522857 − 2.17408i
169.6 −0.866025 − 0.500000i −0.298822 + 1.70608i 0.500000 + 0.866025i 0.302898 − 2.21546i 1.11183 − 1.32810i −0.866025 − 0.500000i − 1.00000i −2.82141 − 1.01963i −1.37005 + 1.76719i
169.7 −0.866025 − 0.500000i 0.374939 + 1.69098i 0.500000 + 0.866025i 2.00883 + 0.982150i 0.520784 − 1.65190i −0.866025 − 0.500000i − 1.00000i −2.71884 + 1.26803i −1.24862 − 1.85498i
169.8 −0.866025 − 0.500000i 1.19578 − 1.25304i 0.500000 + 0.866025i −1.51889 − 1.64103i −1.66210 + 0.487276i −0.866025 − 0.500000i − 1.00000i −0.140224 − 2.99672i 0.494886 + 2.18062i
169.9 −0.866025 − 0.500000i 1.43915 − 0.963770i 0.500000 + 0.866025i −1.55403 + 1.60779i −1.72822 + 0.115075i −0.866025 − 0.500000i − 1.00000i 1.14230 − 2.77402i 2.14973 − 0.615370i
169.10 −0.866025 − 0.500000i 1.56197 + 0.748506i 0.500000 + 0.866025i −2.22745 − 0.196142i −0.978449 − 1.42921i −0.866025 − 0.500000i − 1.00000i 1.87948 + 2.33828i 1.83096 + 1.28359i
169.11 −0.866025 − 0.500000i 1.72380 + 0.168833i 0.500000 + 0.866025i 2.14168 − 0.642820i −1.40844 − 1.00811i −0.866025 − 0.500000i − 1.00000i 2.94299 + 0.582068i −2.17616 − 0.514141i
169.12 0.866025 + 0.500000i −1.72380 − 0.168833i 0.500000 + 0.866025i −1.62754 + 1.53334i −1.40844 − 1.00811i 0.866025 + 0.500000i 1.00000i 2.94299 + 0.582068i −2.17616 + 0.514141i
169.13 0.866025 + 0.500000i −1.56197 − 0.748506i 0.500000 + 0.866025i 0.943861 − 2.02710i −0.978449 − 1.42921i 0.866025 + 0.500000i 1.00000i 1.87948 + 2.33828i 1.83096 − 1.28359i
169.14 0.866025 + 0.500000i −1.43915 + 0.963770i 0.500000 + 0.866025i 2.16940 − 0.541937i −1.72822 + 0.115075i 0.866025 + 0.500000i 1.00000i 1.14230 − 2.77402i 2.14973 + 0.615370i
169.15 0.866025 + 0.500000i −1.19578 + 1.25304i 0.500000 + 0.866025i −0.661724 − 2.13591i −1.66210 + 0.487276i 0.866025 + 0.500000i 1.00000i −0.140224 − 2.99672i 0.494886 − 2.18062i
169.16 0.866025 + 0.500000i −0.374939 − 1.69098i 0.500000 + 0.866025i −0.153846 + 2.23077i 0.520784 − 1.65190i 0.866025 + 0.500000i 1.00000i −2.71884 + 1.26803i −1.24862 + 1.85498i
169.17 0.866025 + 0.500000i 0.298822 − 1.70608i 0.500000 + 0.866025i −2.07009 − 0.845411i 1.11183 − 1.32810i 0.866025 + 0.500000i 1.00000i −2.82141 − 1.01963i −1.37005 − 1.76719i
169.18 0.866025 + 0.500000i 0.351901 + 1.69593i 0.500000 + 0.866025i 1.53985 + 1.62138i −0.543208 + 1.64467i 0.866025 + 0.500000i 1.00000i −2.75233 + 1.19360i 0.522857 + 2.17408i
169.19 0.866025 + 0.500000i 0.974304 − 1.43204i 0.500000 + 0.866025i 1.09618 + 1.94895i 1.55979 − 0.753029i 0.866025 + 0.500000i 1.00000i −1.10146 − 2.79048i −0.0251517 + 2.23593i
169.20 0.866025 + 0.500000i 1.46538 + 0.923390i 0.500000 + 0.866025i −1.75773 + 1.38216i 0.807365 + 1.53237i 0.866025 + 0.500000i 1.00000i 1.29470 + 2.70624i −2.21332 + 0.318122i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.22
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.c ✓ 44
3.b odd 2 1 1890.2.z.c 44
5.b even 2 1 inner 630.2.z.c ✓ 44
9.c even 3 1 inner 630.2.z.c ✓ 44
9.d odd 6 1 1890.2.z.c 44
15.d odd 2 1 1890.2.z.c 44
45.h odd 6 1 1890.2.z.c 44
45.j even 6 1 inner 630.2.z.c ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.z.c ✓ 44 1.a even 1 1 trivial
630.2.z.c ✓ 44 5.b even 2 1 inner
630.2.z.c ✓ 44 9.c even 3 1 inner
630.2.z.c ✓ 44 45.j even 6 1 inner
1890.2.z.c 44 3.b odd 2 1
1890.2.z.c 44 9.d odd 6 1
1890.2.z.c 44 15.d odd 2 1
1890.2.z.c 44 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{22} + 3 T_{11}^{21} + 90 T_{11}^{20} + 235 T_{11}^{19} + 5169 T_{11}^{18} + 12930 T_{11}^{17} + \cdots + 20736 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display