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Properties

Label 735.2.s.n
Level $735$
Weight $2$
Character orbit 735.s
Analytic conductor $5.869$
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] = [735,2,Mod(521,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.s (of order \(6\), degree \(2\), not 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: \(5.86900454856\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} + 16 q^{4} - 16 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} + 16 q^{4} - 16 q^{5} + 4 q^{9} - 16 q^{12} - 8 q^{15} - 48 q^{16} - 24 q^{17} - 32 q^{20} + 64 q^{22} + 40 q^{24} - 16 q^{25} + 16 q^{26} - 32 q^{27} - 36 q^{33} + 32 q^{36} - 8 q^{38} + 12 q^{39} + 32 q^{41} - 32 q^{43} + 4 q^{45} + 16 q^{46} - 8 q^{47} - 80 q^{48} - 4 q^{51} + 40 q^{54} - 32 q^{57} + 48 q^{58} - 16 q^{59} - 16 q^{60} + 32 q^{62} - 160 q^{64} + 72 q^{66} - 48 q^{67} + 88 q^{68} + 56 q^{72} + 4 q^{75} - 48 q^{78} + 40 q^{79} - 48 q^{80} + 20 q^{81} + 192 q^{83} + 48 q^{85} - 12 q^{87} + 32 q^{88} + 32 q^{89} - 48 q^{93} - 8 q^{96} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 −2.35644 + 1.36049i −1.58737 − 0.693003i 2.70188 − 4.67979i −0.500000 − 0.866025i 4.68337 − 0.526585i 0 9.26157i 2.03949 + 2.20011i 2.35644 + 1.36049i
521.2 −2.30502 + 1.33080i 1.57985 − 0.709980i 2.54208 − 4.40301i −0.500000 − 0.866025i −2.69674 + 3.73899i 0 8.20884i 1.99186 − 2.24333i 2.30502 + 1.33080i
521.3 −1.90025 + 1.09711i −0.228904 − 1.71686i 1.40729 − 2.43750i −0.500000 − 0.866025i 2.31855 + 3.01132i 0 1.78737i −2.89521 + 0.785991i 1.90025 + 1.09711i
521.4 −1.31494 + 0.759178i −0.00282885 + 1.73205i 0.152704 − 0.264491i −0.500000 − 0.866025i −1.31121 − 2.27968i 0 − 2.57300i −2.99998 − 0.00979942i 1.31494 + 0.759178i
521.5 −0.927479 + 0.535480i 1.72786 + 0.120447i −0.426521 + 0.738757i −0.500000 − 0.866025i −1.66705 + 0.813522i 0 − 3.05550i 2.97098 + 0.416231i 0.927479 + 0.535480i
521.6 −0.901314 + 0.520374i 1.07950 + 1.35451i −0.458422 + 0.794010i −0.500000 − 0.866025i −1.67782 − 0.659093i 0 − 3.03570i −0.669372 + 2.92437i 0.901314 + 0.520374i
521.7 −0.291194 + 0.168121i 0.790472 − 1.54115i −0.943471 + 1.63414i −0.500000 − 0.866025i 0.0289195 + 0.581669i 0 − 1.30695i −1.75031 − 2.43648i 0.291194 + 0.168121i
521.8 −0.191543 + 0.110588i −1.72997 + 0.0848672i −0.975541 + 1.68969i −0.500000 − 0.866025i 0.321979 − 0.207569i 0 − 0.873880i 2.98560 − 0.293635i 0.191543 + 0.110588i
521.9 0.191543 − 0.110588i 0.791488 + 1.54063i −0.975541 + 1.68969i −0.500000 − 0.866025i 0.321979 + 0.207569i 0 0.873880i −1.74709 + 2.43878i −0.191543 − 0.110588i
521.10 0.291194 − 0.168121i 0.939442 − 1.45515i −0.943471 + 1.63414i −0.500000 − 0.866025i 0.0289195 − 0.581669i 0 1.30695i −1.23490 − 2.73405i −0.291194 − 0.168121i
521.11 0.901314 − 0.520374i −1.71279 − 0.257619i −0.458422 + 0.794010i −0.500000 − 0.866025i −1.67782 + 0.659093i 0 3.03570i 2.86727 + 0.882492i −0.901314 − 0.520374i
521.12 0.927479 − 0.535480i −0.968239 − 1.43615i −0.426521 + 0.738757i −0.500000 − 0.866025i −1.66705 − 0.813522i 0 3.05550i −1.12503 + 2.78106i −0.927479 − 0.535480i
521.13 1.31494 − 0.759178i −1.49858 + 0.868474i 0.152704 − 0.264491i −0.500000 − 0.866025i −1.31121 + 2.27968i 0 2.57300i 1.49151 − 2.60296i −1.31494 − 0.759178i
521.14 1.90025 − 1.09711i 1.60130 − 0.660193i 1.40729 − 2.43750i −0.500000 − 0.866025i 2.31855 − 3.01132i 0 − 1.78737i 2.12829 − 2.11433i −1.90025 − 1.09711i
521.15 2.30502 − 1.33080i −0.175064 − 1.72318i 2.54208 − 4.40301i −0.500000 − 0.866025i −2.69674 − 3.73899i 0 − 8.20884i −2.93871 + 0.603334i −2.30502 − 1.33080i
521.16 2.35644 − 1.36049i 1.39384 + 1.02820i 2.70188 − 4.67979i −0.500000 − 0.866025i 4.68337 + 0.526585i 0 − 9.26157i 0.885601 + 2.86631i −2.35644 − 1.36049i
656.1 −2.35644 − 1.36049i −1.58737 + 0.693003i 2.70188 + 4.67979i −0.500000 + 0.866025i 4.68337 + 0.526585i 0 − 9.26157i 2.03949 − 2.20011i 2.35644 − 1.36049i
656.2 −2.30502 − 1.33080i 1.57985 + 0.709980i 2.54208 + 4.40301i −0.500000 + 0.866025i −2.69674 − 3.73899i 0 − 8.20884i 1.99186 + 2.24333i 2.30502 − 1.33080i
656.3 −1.90025 − 1.09711i −0.228904 + 1.71686i 1.40729 + 2.43750i −0.500000 + 0.866025i 2.31855 − 3.01132i 0 − 1.78737i −2.89521 − 0.785991i 1.90025 − 1.09711i
656.4 −1.31494 − 0.759178i −0.00282885 − 1.73205i 0.152704 + 0.264491i −0.500000 + 0.866025i −1.31121 + 2.27968i 0 2.57300i −2.99998 + 0.00979942i 1.31494 − 0.759178i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
21.c even 2 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.s.n 32
3.b odd 2 1 735.2.s.m 32
7.b odd 2 1 735.2.s.m 32
7.c even 3 1 735.2.b.e ✓ 16
7.c even 3 1 inner 735.2.s.n 32
7.d odd 6 1 735.2.b.f yes 16
7.d odd 6 1 735.2.s.m 32
21.c even 2 1 inner 735.2.s.n 32
21.g even 6 1 735.2.b.e ✓ 16
21.g even 6 1 inner 735.2.s.n 32
21.h odd 6 1 735.2.b.f yes 16
21.h odd 6 1 735.2.s.m 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.b.e ✓ 16 7.c even 3 1
735.2.b.e ✓ 16 21.g even 6 1
735.2.b.f yes 16 7.d odd 6 1
735.2.b.f yes 16 21.h odd 6 1
735.2.s.m 32 3.b odd 2 1
735.2.s.m 32 7.b odd 2 1
735.2.s.m 32 7.d odd 6 1
735.2.s.m 32 21.h odd 6 1
735.2.s.n 32 1.a even 1 1 trivial
735.2.s.n 32 7.c even 3 1 inner
735.2.s.n 32 21.c even 2 1 inner
735.2.s.n 32 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{32} - 24 T_{2}^{30} + 356 T_{2}^{28} - 3344 T_{2}^{26} + 23035 T_{2}^{24} - 112864 T_{2}^{22} + \cdots + 16 \) Copy content Toggle raw display
\( T_{13}^{16} + 140 T_{13}^{14} + 7618 T_{13}^{12} + 208500 T_{13}^{10} + 3077473 T_{13}^{8} + \cdots + 66846976 \) Copy content Toggle raw display
\( T_{17}^{16} + 12 T_{17}^{15} + 138 T_{17}^{14} + 736 T_{17}^{13} + 4897 T_{17}^{12} + 19776 T_{17}^{11} + \cdots + 2676496 \) Copy content Toggle raw display