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Properties

Label 725.2.bd.c
Level $725$
Weight $2$
Character orbit 725.bd
Analytic conductor $5.789$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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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] = [725,2,Mod(43,725)] 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("725.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([21, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.bd (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240] 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(20\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 40 q^{4} - 56 q^{9} - 12 q^{14} - 80 q^{16} + 4 q^{19} - 28 q^{21} + 12 q^{29} - 16 q^{31} + 8 q^{34} - 104 q^{36} - 100 q^{39} + 28 q^{41} - 264 q^{44} - 100 q^{46} + 168 q^{49} - 112 q^{51} + 168 q^{54}+ \cdots + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.16243 − 1.72448i −0.539817 + 2.36509i 1.25723 + 5.50829i 0 5.24588 − 4.18345i −1.28727 − 2.04868i 4.38015 − 9.09547i −2.59936 − 1.25179i 0
43.2 −1.88099 − 1.50004i −0.247674 + 1.08513i 0.842965 + 3.69327i 0 2.09361 − 1.66960i −0.665536 − 1.05919i 1.86670 − 3.87625i 1.58674 + 0.764134i 0
43.3 −1.87958 − 1.49891i 0.662907 − 2.90439i 0.841030 + 3.68479i 0 −5.59941 + 4.46538i 1.59856 + 2.54409i 1.85623 − 3.85450i −5.29311 − 2.54903i 0
43.4 −1.47630 − 1.17731i 0.104891 − 0.459558i 0.348360 + 1.52627i 0 −0.695892 + 0.554956i −1.14738 − 1.82605i −0.355966 + 0.739170i 2.50272 + 1.20524i 0
43.5 −1.21576 − 0.969534i 0.491176 − 2.15198i 0.0930275 + 0.407580i 0 −2.68357 + 2.14008i −1.68596 − 2.68319i −1.06732 + 2.21632i −1.68687 − 0.812355i 0
43.6 −1.04744 − 0.835306i 0.321965 − 1.41062i −0.0456466 − 0.199991i 0 −1.51554 + 1.20860i 1.77909 + 2.83140i −1.28181 + 2.66171i 0.816717 + 0.393310i 0
43.7 −0.882991 − 0.704162i −0.531870 + 2.33027i −0.161213 − 0.706319i 0 2.11053 − 1.68309i 2.07932 + 3.30922i −1.33506 + 2.77228i −2.44438 − 1.17715i 0
43.8 −0.624636 − 0.498130i −0.662747 + 2.90368i −0.303006 − 1.32756i 0 1.86039 − 1.48361i −0.0612624 − 0.0974985i −1.16532 + 2.41981i −5.28923 − 2.54716i 0
43.9 −0.366257 − 0.292081i −0.00192371 + 0.00842832i −0.396208 − 1.73590i 0 0.00316632 − 0.00252506i 0.422156 + 0.671858i −0.768424 + 1.59565i 2.70284 + 1.30162i 0
43.10 −0.0410962 − 0.0327731i −0.470287 + 2.06046i −0.444427 − 1.94716i 0 0.0868547 − 0.0692644i −1.98211 − 3.15451i −0.0911637 + 0.189303i −1.32142 − 0.636365i 0
43.11 0.0410962 + 0.0327731i 0.470287 − 2.06046i −0.444427 − 1.94716i 0 0.0868547 − 0.0692644i 1.98211 + 3.15451i 0.0911637 − 0.189303i −1.32142 − 0.636365i 0
43.12 0.366257 + 0.292081i 0.00192371 − 0.00842832i −0.396208 − 1.73590i 0 0.00316632 − 0.00252506i −0.422156 − 0.671858i 0.768424 − 1.59565i 2.70284 + 1.30162i 0
43.13 0.624636 + 0.498130i 0.662747 − 2.90368i −0.303006 − 1.32756i 0 1.86039 − 1.48361i 0.0612624 + 0.0974985i 1.16532 − 2.41981i −5.28923 − 2.54716i 0
43.14 0.882991 + 0.704162i 0.531870 − 2.33027i −0.161213 − 0.706319i 0 2.11053 − 1.68309i −2.07932 − 3.30922i 1.33506 − 2.77228i −2.44438 − 1.17715i 0
43.15 1.04744 + 0.835306i −0.321965 + 1.41062i −0.0456466 − 0.199991i 0 −1.51554 + 1.20860i −1.77909 − 2.83140i 1.28181 − 2.66171i 0.816717 + 0.393310i 0
43.16 1.21576 + 0.969534i −0.491176 + 2.15198i 0.0930275 + 0.407580i 0 −2.68357 + 2.14008i 1.68596 + 2.68319i 1.06732 − 2.21632i −1.68687 − 0.812355i 0
43.17 1.47630 + 1.17731i −0.104891 + 0.459558i 0.348360 + 1.52627i 0 −0.695892 + 0.554956i 1.14738 + 1.82605i 0.355966 − 0.739170i 2.50272 + 1.20524i 0
43.18 1.87958 + 1.49891i −0.662907 + 2.90439i 0.841030 + 3.68479i 0 −5.59941 + 4.46538i −1.59856 − 2.54409i −1.85623 + 3.85450i −5.29311 − 2.54903i 0
43.19 1.88099 + 1.50004i 0.247674 − 1.08513i 0.842965 + 3.69327i 0 2.09361 − 1.66960i 0.665536 + 1.05919i −1.86670 + 3.87625i 1.58674 + 0.764134i 0
43.20 2.16243 + 1.72448i 0.539817 − 2.36509i 1.25723 + 5.50829i 0 5.24588 − 4.18345i 1.28727 + 2.04868i −4.38015 + 9.09547i −2.59936 − 1.25179i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.20
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
145.o even 28 1 inner
145.t even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.bd.c yes 240
5.b even 2 1 inner 725.2.bd.c yes 240
5.c odd 4 2 725.2.y.c ✓ 240
29.f odd 28 1 725.2.y.c ✓ 240
145.o even 28 1 inner 725.2.bd.c yes 240
145.s odd 28 1 725.2.y.c ✓ 240
145.t even 28 1 inner 725.2.bd.c yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.y.c ✓ 240 5.c odd 4 2
725.2.y.c ✓ 240 29.f odd 28 1
725.2.y.c ✓ 240 145.s odd 28 1
725.2.bd.c yes 240 1.a even 1 1 trivial
725.2.bd.c yes 240 5.b even 2 1 inner
725.2.bd.c yes 240 145.o even 28 1 inner
725.2.bd.c yes 240 145.t even 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{240} - 60 T_{2}^{238} + 1960 T_{2}^{236} - 45960 T_{2}^{234} + 863454 T_{2}^{232} + \cdots + 90\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display