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Properties

Label 630.2.bl.a
Level $630$
Weight $2$
Character orbit 630.bl
Analytic conductor $5.031$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(41,630)]
 
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([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.41");
 
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.bl (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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 16 q^{4} - 16 q^{5} - 4 q^{6} - 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 16 q^{4} - 16 q^{5} - 4 q^{6} - 2 q^{7} - 6 q^{9} + 6 q^{11} - 2 q^{12} - 6 q^{13} + 2 q^{15} - 16 q^{16} - 12 q^{17} + 8 q^{18} + 16 q^{20} - 20 q^{21} + 36 q^{23} - 2 q^{24} - 16 q^{25} + 8 q^{27} - 4 q^{28} + 6 q^{29} + 2 q^{30} + 48 q^{31} - 14 q^{33} + 4 q^{35} - 6 q^{36} + 8 q^{37} + 6 q^{39} + 6 q^{41} + 24 q^{42} - 4 q^{43} + 6 q^{45} - 12 q^{46} + 6 q^{47} + 2 q^{48} - 28 q^{49} - 20 q^{51} - 6 q^{52} - 14 q^{54} + 6 q^{56} - 36 q^{57} - 2 q^{60} + 6 q^{61} - 10 q^{63} - 32 q^{64} + 6 q^{65} - 20 q^{66} + 28 q^{67} - 6 q^{68} - 14 q^{69} - 6 q^{70} - 8 q^{72} - 36 q^{74} + 2 q^{75} - 6 q^{77} + 32 q^{78} + 10 q^{79} + 32 q^{80} - 10 q^{81} + 6 q^{83} - 16 q^{84} + 6 q^{85} + 24 q^{86} + 28 q^{87} - 60 q^{89} + 8 q^{90} - 24 q^{91} + 36 q^{92} + 4 q^{93} + 42 q^{94} + 2 q^{96} + 90 q^{97} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.866025 + 0.500000i −1.72836 + 0.112970i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.44032 − 0.962017i 2.55222 − 0.697255i 1.00000i 2.97448 − 0.390508i − 1.00000i
41.2 −0.866025 + 0.500000i −0.998657 + 1.41516i 0.500000 − 0.866025i −0.500000 + 0.866025i 0.157281 − 1.72489i −2.23931 + 1.40907i 1.00000i −1.00537 − 2.82652i − 1.00000i
41.3 −0.866025 + 0.500000i −0.628600 − 1.61396i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.35136 + 1.08343i 1.22724 − 2.34390i 1.00000i −2.20972 + 2.02907i − 1.00000i
41.4 −0.866025 + 0.500000i −0.325592 + 1.70117i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.568616 − 1.63606i 2.64304 + 0.119814i 1.00000i −2.78798 − 1.10778i − 1.00000i
41.5 −0.866025 + 0.500000i −0.258130 − 1.71271i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.07990 + 1.35418i −2.17621 + 1.50470i 1.00000i −2.86674 + 0.884201i − 1.00000i
41.6 −0.866025 + 0.500000i 1.02449 − 1.39657i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.188953 + 1.72171i −0.672634 − 2.55882i 1.00000i −0.900821 − 2.86156i − 1.00000i
41.7 −0.866025 + 0.500000i 1.23327 + 1.21616i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.67612 − 0.436589i 0.0425252 + 2.64541i 1.00000i 0.0419137 + 2.99971i − 1.00000i
41.8 −0.866025 + 0.500000i 1.54760 + 0.777773i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.72915 + 0.100230i −1.01084 − 2.44504i 1.00000i 1.79014 + 2.40736i − 1.00000i
41.9 0.866025 − 0.500000i −1.67705 + 0.433031i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.23585 + 1.21354i 0.943282 + 2.47189i − 1.00000i 2.62497 − 1.45243i 1.00000i
41.10 0.866025 − 0.500000i −1.62688 − 0.594351i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.70610 + 0.298718i −2.13969 + 1.55619i − 1.00000i 2.29349 + 1.93388i 1.00000i
41.11 0.866025 − 0.500000i −1.17115 + 1.27609i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.376205 + 1.69070i 1.00174 − 2.44878i − 1.00000i −0.256800 − 2.98899i 1.00000i
41.12 0.866025 − 0.500000i −0.954824 − 1.44510i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.54945 − 0.774080i −1.54699 − 2.14635i − 1.00000i −1.17662 + 2.75963i 1.00000i
41.13 0.866025 − 0.500000i −0.150193 + 1.72553i 0.500000 − 0.866025i −0.500000 + 0.866025i 0.732692 + 1.56945i −2.63140 − 0.275220i − 1.00000i −2.95488 − 0.518324i 1.00000i
41.14 0.866025 − 0.500000i 0.600052 − 1.62479i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.292734 − 1.70713i 2.32919 + 1.25494i − 1.00000i −2.27988 − 1.94991i 1.00000i
41.15 0.866025 − 0.500000i 1.49242 − 0.879021i 0.500000 − 0.866025i −0.500000 + 0.866025i 0.852964 − 1.50747i 1.15259 − 2.38150i − 1.00000i 1.45464 − 2.62374i 1.00000i
41.16 0.866025 − 0.500000i 1.62160 + 0.608614i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.70865 − 0.283725i −0.474749 + 2.60281i − 1.00000i 2.25918 + 1.97386i 1.00000i
461.1 −0.866025 − 0.500000i −1.72836 − 0.112970i 0.500000 + 0.866025i −0.500000 − 0.866025i 1.44032 + 0.962017i 2.55222 + 0.697255i − 1.00000i 2.97448 + 0.390508i 1.00000i
461.2 −0.866025 − 0.500000i −0.998657 − 1.41516i 0.500000 + 0.866025i −0.500000 − 0.866025i 0.157281 + 1.72489i −2.23931 − 1.40907i − 1.00000i −1.00537 + 2.82652i 1.00000i
461.3 −0.866025 − 0.500000i −0.628600 + 1.61396i 0.500000 + 0.866025i −0.500000 − 0.866025i 1.35136 − 1.08343i 1.22724 + 2.34390i − 1.00000i −2.20972 − 2.02907i 1.00000i
461.4 −0.866025 − 0.500000i −0.325592 − 1.70117i 0.500000 + 0.866025i −0.500000 − 0.866025i −0.568616 + 1.63606i 2.64304 − 0.119814i − 1.00000i −2.78798 + 1.10778i 1.00000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o 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.bl.a ✓ 32
3.b odd 2 1 1890.2.bl.b 32
7.b odd 2 1 630.2.bl.b yes 32
9.c even 3 1 1890.2.bl.a 32
9.d odd 6 1 630.2.bl.b yes 32
21.c even 2 1 1890.2.bl.a 32
63.l odd 6 1 1890.2.bl.b 32
63.o even 6 1 inner 630.2.bl.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bl.a ✓ 32 1.a even 1 1 trivial
630.2.bl.a ✓ 32 63.o even 6 1 inner
630.2.bl.b yes 32 7.b odd 2 1
630.2.bl.b yes 32 9.d odd 6 1
1890.2.bl.a 32 9.c even 3 1
1890.2.bl.a 32 21.c even 2 1
1890.2.bl.b 32 3.b odd 2 1
1890.2.bl.b 32 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{32} + 6 T_{13}^{31} - 105 T_{13}^{30} - 702 T_{13}^{29} + 7017 T_{13}^{28} + \cdots + 22\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display