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Properties

Label 630.2.bf.e
Level $630$
Weight $2$
Character orbit 630.bf
Analytic conductor $5.031$
Analytic rank $0$
Dimension $32$
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(209,630)]
 
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([1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.209");
 
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.bf (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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} - 6 q^{9} - 24 q^{11} - 16 q^{15} - 16 q^{16} + 6 q^{18} + 2 q^{21} + 24 q^{22} - 24 q^{23} - 58 q^{25} - 36 q^{29} - 10 q^{30} - 16 q^{32} - 48 q^{35} - 66 q^{39} + 2 q^{42} - 54 q^{43} + 48 q^{46} + 32 q^{49} + 50 q^{50} - 26 q^{51} - 24 q^{53} + 110 q^{57} + 36 q^{58} + 26 q^{60} + 20 q^{63} + 32 q^{64} - 90 q^{65} - 66 q^{67} + 36 q^{70} - 6 q^{72} + 12 q^{74} + 18 q^{77} + 60 q^{78} + 34 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{85} + 54 q^{86} - 24 q^{88} + 16 q^{91} - 24 q^{92} + 28 q^{93} + 12 q^{95} - 64 q^{98} + 58 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 −0.500000 + 0.866025i −1.71915 − 0.210974i −0.500000 − 0.866025i −1.22443 − 1.87103i 1.04229 − 1.38334i 2.07602 + 1.64016i 1.00000 2.91098 + 0.725392i 2.23258 − 0.124872i
209.2 −0.500000 + 0.866025i −1.65659 − 0.505669i −0.500000 − 0.866025i −0.150780 + 2.23098i 1.26622 − 1.18182i −2.28974 + 1.32556i 1.00000 2.48860 + 1.67538i −1.85669 − 1.24607i
209.3 −0.500000 + 0.866025i −1.42161 − 0.989450i −0.500000 − 0.866025i 1.07274 − 1.96195i 1.56770 − 0.736429i −0.858483 − 2.50260i 1.00000 1.04198 + 2.81323i 1.16272 + 1.90999i
209.4 −0.500000 + 0.866025i −1.33289 + 1.10608i −0.500000 − 0.866025i 0.693615 − 2.12577i −0.291452 − 1.70735i −2.50039 − 0.864900i 1.00000 0.553169 − 2.94856i 1.49416 + 1.66357i
209.5 −0.500000 + 0.866025i −0.979752 + 1.42832i −0.500000 − 0.866025i −0.452503 + 2.18980i −0.747081 − 1.56265i 2.48907 + 0.896960i 1.00000 −1.08017 − 2.79879i −1.67017 − 1.48678i
209.6 −0.500000 + 0.866025i −0.777586 + 1.54770i −0.500000 − 0.866025i 1.55878 − 1.60318i −0.951551 − 1.44726i 2.64557 − 0.0312377i 1.00000 −1.79072 − 2.40693i 0.609005 + 2.15154i
209.7 −0.500000 + 0.866025i −0.341957 − 1.69796i −0.500000 − 0.866025i 2.23011 − 0.163074i 1.64145 + 0.552836i −2.62496 + 0.331028i 1.00000 −2.76613 + 1.16126i −0.973830 + 2.01287i
209.8 −0.500000 + 0.866025i −0.266741 − 1.71139i −0.500000 − 0.866025i −1.41003 − 1.73546i 1.61548 + 0.624689i 1.74972 − 1.98456i 1.00000 −2.85770 + 0.912995i 2.20797 − 0.353395i
209.9 −0.500000 + 0.866025i 0.266741 + 1.71139i −0.500000 − 0.866025i 1.41003 + 1.73546i −1.61548 − 0.624689i −0.843817 + 2.50758i 1.00000 −2.85770 + 0.912995i −2.20797 + 0.353395i
209.10 −0.500000 + 0.866025i 0.341957 + 1.69796i −0.500000 − 0.866025i −2.23011 + 0.163074i −1.64145 − 0.552836i −1.02580 − 2.43880i 1.00000 −2.76613 + 1.16126i 0.973830 − 2.01287i
209.11 −0.500000 + 0.866025i 0.777586 − 1.54770i −0.500000 − 0.866025i −1.55878 + 1.60318i 0.951551 + 1.44726i 1.29573 + 2.30675i 1.00000 −1.79072 − 2.40693i −0.609005 − 2.15154i
209.12 −0.500000 + 0.866025i 0.979752 − 1.42832i −0.500000 − 0.866025i 0.452503 − 2.18980i 0.747081 + 1.56265i 2.02132 + 1.70712i 1.00000 −1.08017 − 2.79879i 1.67017 + 1.48678i
209.13 −0.500000 + 0.866025i 1.33289 − 1.10608i −0.500000 − 0.866025i −0.693615 + 2.12577i 0.291452 + 1.70735i −1.99922 − 1.73295i 1.00000 0.553169 − 2.94856i −1.49416 − 1.66357i
209.14 −0.500000 + 0.866025i 1.42161 + 0.989450i −0.500000 − 0.866025i −1.07274 + 1.96195i −1.56770 + 0.736429i −2.59656 + 0.507832i 1.00000 1.04198 + 2.81323i −1.16272 − 1.90999i
209.15 −0.500000 + 0.866025i 1.65659 + 0.505669i −0.500000 − 0.866025i 0.150780 − 2.23098i −1.26622 + 1.18182i 0.00309790 − 2.64575i 1.00000 2.48860 + 1.67538i 1.85669 + 1.24607i
209.16 −0.500000 + 0.866025i 1.71915 + 0.210974i −0.500000 − 0.866025i 1.22443 + 1.87103i −1.04229 + 1.38334i 2.45843 + 0.977807i 1.00000 2.91098 + 0.725392i −2.23258 + 0.124872i
419.1 −0.500000 − 0.866025i −1.71915 + 0.210974i −0.500000 + 0.866025i −1.22443 + 1.87103i 1.04229 + 1.38334i 2.07602 − 1.64016i 1.00000 2.91098 − 0.725392i 2.23258 + 0.124872i
419.2 −0.500000 − 0.866025i −1.65659 + 0.505669i −0.500000 + 0.866025i −0.150780 − 2.23098i 1.26622 + 1.18182i −2.28974 − 1.32556i 1.00000 2.48860 − 1.67538i −1.85669 + 1.24607i
419.3 −0.500000 − 0.866025i −1.42161 + 0.989450i −0.500000 + 0.866025i 1.07274 + 1.96195i 1.56770 + 0.736429i −0.858483 + 2.50260i 1.00000 1.04198 − 2.81323i 1.16272 − 1.90999i
419.4 −0.500000 − 0.866025i −1.33289 − 1.10608i −0.500000 + 0.866025i 0.693615 + 2.12577i −0.291452 + 1.70735i −2.50039 + 0.864900i 1.00000 0.553169 + 2.94856i 1.49416 − 1.66357i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
45.h odd 6 1 inner
315.z 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.bf.e ✓ 32
3.b odd 2 1 1890.2.bf.f 32
5.b even 2 1 630.2.bf.f yes 32
7.b odd 2 1 inner 630.2.bf.e ✓ 32
9.c even 3 1 1890.2.bf.e 32
9.d odd 6 1 630.2.bf.f yes 32
15.d odd 2 1 1890.2.bf.e 32
21.c even 2 1 1890.2.bf.f 32
35.c odd 2 1 630.2.bf.f yes 32
45.h odd 6 1 inner 630.2.bf.e ✓ 32
45.j even 6 1 1890.2.bf.f 32
63.l odd 6 1 1890.2.bf.e 32
63.o even 6 1 630.2.bf.f yes 32
105.g even 2 1 1890.2.bf.e 32
315.z even 6 1 inner 630.2.bf.e ✓ 32
315.bg odd 6 1 1890.2.bf.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bf.e ✓ 32 1.a even 1 1 trivial
630.2.bf.e ✓ 32 7.b odd 2 1 inner
630.2.bf.e ✓ 32 45.h odd 6 1 inner
630.2.bf.e ✓ 32 315.z even 6 1 inner
630.2.bf.f yes 32 5.b even 2 1
630.2.bf.f yes 32 9.d odd 6 1
630.2.bf.f yes 32 35.c odd 2 1
630.2.bf.f yes 32 63.o even 6 1
1890.2.bf.e 32 9.c even 3 1
1890.2.bf.e 32 15.d odd 2 1
1890.2.bf.e 32 63.l odd 6 1
1890.2.bf.e 32 105.g even 2 1
1890.2.bf.f 32 3.b odd 2 1
1890.2.bf.f 32 21.c even 2 1
1890.2.bf.f 32 45.j even 6 1
1890.2.bf.f 32 315.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{16} + 12 T_{11}^{15} + 7 T_{11}^{14} - 492 T_{11}^{13} - 830 T_{11}^{12} + 21498 T_{11}^{11} + \cdots + 239754256 \) Copy content Toggle raw display
\( T_{13}^{32} + 131 T_{13}^{30} + 10509 T_{13}^{28} + 533244 T_{13}^{26} + 19752852 T_{13}^{24} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
\( T_{23}^{16} + 12 T_{23}^{15} + 177 T_{23}^{14} + 1156 T_{23}^{13} + 11169 T_{23}^{12} + \cdots + 2958924816 \) Copy content Toggle raw display