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Properties

Label 225.3.o.c
Level $225$
Weight $3$
Character orbit 225.o
Analytic conductor $6.131$
Analytic rank $0$
Dimension $64$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,3,Mod(7,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 36 q^{6} + 128 q^{16} - 180 q^{21} - 432 q^{26} + 64 q^{31} - 612 q^{36} + 432 q^{41} + 240 q^{46} + 1080 q^{51} + 1080 q^{56} - 220 q^{61} - 2484 q^{66} - 1728 q^{71} + 392 q^{76} - 24 q^{81} + 1296 q^{86}+ \cdots + 4848 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.77829 + 1.01239i 0.609203 − 2.93749i 9.78643 − 5.65020i 0 0.672142 + 11.7154i −3.12611 + 0.837640i −20.1921 + 20.1921i −8.25774 − 3.57906i 0
7.2 −3.09389 + 0.829004i −2.93358 + 0.627778i 5.42078 − 3.12969i 0 8.55573 − 4.37422i −1.55916 + 0.417775i −5.11721 + 5.11721i 8.21179 − 3.68327i 0
7.3 −2.83272 + 0.759025i 2.79842 − 1.08112i 3.98409 − 2.30022i 0 −7.10655 + 5.18660i −9.40715 + 2.52064i −1.24511 + 1.24511i 6.66234 − 6.05088i 0
7.4 −2.55363 + 0.684243i −1.96512 + 2.26679i 2.58873 − 1.49461i 0 3.46714 − 7.13315i −5.37715 + 1.44080i 1.88955 − 1.88955i −1.27665 − 8.90899i 0
7.5 −1.86913 + 0.500832i 1.04848 + 2.81082i −0.221279 + 0.127756i 0 −3.36750 − 4.72867i 12.1355 − 3.25169i 5.82282 − 5.82282i −6.80136 + 5.89419i 0
7.6 −1.22704 + 0.328785i −2.24709 − 1.98761i −2.06657 + 1.19313i 0 3.41077 + 1.70007i 9.94163 − 2.66385i 5.73651 − 5.73651i 1.09881 + 8.93267i 0
7.7 −0.944333 + 0.253033i 2.74905 − 1.20114i −2.63636 + 1.52210i 0 −2.29209 + 1.82988i −2.68352 + 0.719047i 4.86966 − 4.86966i 6.11450 − 6.60400i 0
7.8 −0.707555 + 0.189589i −0.870705 − 2.87087i −2.99941 + 1.73171i 0 1.16036 + 1.86622i −0.741289 + 0.198628i 3.86580 − 3.86580i −7.48375 + 4.99935i 0
7.9 0.707555 − 0.189589i 0.870705 + 2.87087i −2.99941 + 1.73171i 0 1.16036 + 1.86622i 0.741289 − 0.198628i −3.86580 + 3.86580i −7.48375 + 4.99935i 0
7.10 0.944333 − 0.253033i −2.74905 + 1.20114i −2.63636 + 1.52210i 0 −2.29209 + 1.82988i 2.68352 − 0.719047i −4.86966 + 4.86966i 6.11450 − 6.60400i 0
7.11 1.22704 − 0.328785i 2.24709 + 1.98761i −2.06657 + 1.19313i 0 3.41077 + 1.70007i −9.94163 + 2.66385i −5.73651 + 5.73651i 1.09881 + 8.93267i 0
7.12 1.86913 − 0.500832i −1.04848 − 2.81082i −0.221279 + 0.127756i 0 −3.36750 − 4.72867i −12.1355 + 3.25169i −5.82282 + 5.82282i −6.80136 + 5.89419i 0
7.13 2.55363 − 0.684243i 1.96512 − 2.26679i 2.58873 − 1.49461i 0 3.46714 − 7.13315i 5.37715 − 1.44080i −1.88955 + 1.88955i −1.27665 − 8.90899i 0
7.14 2.83272 − 0.759025i −2.79842 + 1.08112i 3.98409 − 2.30022i 0 −7.10655 + 5.18660i 9.40715 − 2.52064i 1.24511 − 1.24511i 6.66234 − 6.05088i 0
7.15 3.09389 − 0.829004i 2.93358 − 0.627778i 5.42078 − 3.12969i 0 8.55573 − 4.37422i 1.55916 − 0.417775i 5.11721 − 5.11721i 8.21179 − 3.68327i 0
7.16 3.77829 − 1.01239i −0.609203 + 2.93749i 9.78643 − 5.65020i 0 0.672142 + 11.7154i 3.12611 − 0.837640i 20.1921 − 20.1921i −8.25774 − 3.57906i 0
43.1 −1.01239 − 3.77829i −2.93749 − 0.609203i −9.78643 + 5.65020i 0 0.672142 + 11.7154i −0.837640 − 3.12611i 20.1921 + 20.1921i 8.25774 + 3.57906i 0
43.2 −0.829004 − 3.09389i 0.627778 + 2.93358i −5.42078 + 3.12969i 0 8.55573 − 4.37422i −0.417775 − 1.55916i 5.11721 + 5.11721i −8.21179 + 3.68327i 0
43.3 −0.759025 − 2.83272i −1.08112 − 2.79842i −3.98409 + 2.30022i 0 −7.10655 + 5.18660i −2.52064 − 9.40715i 1.24511 + 1.24511i −6.66234 + 6.05088i 0
43.4 −0.684243 − 2.55363i 2.26679 + 1.96512i −2.58873 + 1.49461i 0 3.46714 − 7.13315i −1.44080 − 5.37715i −1.88955 − 1.88955i 1.27665 + 8.90899i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
9.c even 3 1 inner
45.j even 6 1 inner
45.k odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.o.c ✓ 64
5.b even 2 1 inner 225.3.o.c ✓ 64
5.c odd 4 2 inner 225.3.o.c ✓ 64
9.c even 3 1 inner 225.3.o.c ✓ 64
45.j even 6 1 inner 225.3.o.c ✓ 64
45.k odd 12 2 inner 225.3.o.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.o.c ✓ 64 1.a even 1 1 trivial
225.3.o.c ✓ 64 5.b even 2 1 inner
225.3.o.c ✓ 64 5.c odd 4 2 inner
225.3.o.c ✓ 64 9.c even 3 1 inner
225.3.o.c ✓ 64 45.j even 6 1 inner
225.3.o.c ✓ 64 45.k odd 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 480 T_{2}^{60} + 152172 T_{2}^{56} - 26498718 T_{2}^{52} + 3298617270 T_{2}^{48} + \cdots + 73\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display