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Properties

Label 275.2.bm.c
Level $275$
Weight $2$
Character orbit 275.bm
Analytic conductor $2.196$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,2,Mod(7,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.bm (of order \(20\), degree \(8\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 20 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 20 q^{6} + 44 q^{11} - 16 q^{16} - 120 q^{26} + 8 q^{31} - 228 q^{36} - 180 q^{41} + 40 q^{46} - 60 q^{51} + 240 q^{56} + 160 q^{61} + 460 q^{66} + 8 q^{71} - 140 q^{81} - 40 q^{86} - 200 q^{91} - 240 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.45848 + 0.389385i 1.42041 + 2.78770i 3.99039 − 1.29656i 0 −4.57753 − 6.30043i −1.01159 − 0.515430i −4.86977 + 2.48127i −3.99039 + 5.49230i 0
7.2 −2.06640 + 0.327285i −1.18788 − 2.33135i 2.26078 − 0.734571i 0 3.21766 + 4.42872i 2.55575 + 1.30222i −0.702999 + 0.358196i −2.26078 + 3.11169i 0
7.3 −1.67541 + 0.265359i −0.954428 − 1.87317i 0.834479 − 0.271139i 0 2.09612 + 2.88507i −1.62781 − 0.829412i 1.69668 − 0.864500i −0.834479 + 1.14856i 0
7.4 −0.404529 + 0.0640711i 0.0853641 + 0.167537i −1.74257 + 0.566197i 0 −0.0452665 − 0.0623040i −3.17198 − 1.61620i 1.39851 − 0.712575i 1.74257 − 2.39845i 0
7.5 0.404529 − 0.0640711i −0.0853641 − 0.167537i −1.74257 + 0.566197i 0 −0.0452665 − 0.0623040i 3.17198 + 1.61620i −1.39851 + 0.712575i 1.74257 − 2.39845i 0
7.6 1.67541 − 0.265359i 0.954428 + 1.87317i 0.834479 − 0.271139i 0 2.09612 + 2.88507i 1.62781 + 0.829412i −1.69668 + 0.864500i −0.834479 + 1.14856i 0
7.7 2.06640 − 0.327285i 1.18788 + 2.33135i 2.26078 − 0.734571i 0 3.21766 + 4.42872i −2.55575 − 1.30222i 0.702999 − 0.358196i −2.26078 + 3.11169i 0
7.8 2.45848 − 0.389385i −1.42041 − 2.78770i 3.99039 − 1.29656i 0 −4.57753 − 6.30043i 1.01159 + 0.515430i 4.86977 − 2.48127i −3.99039 + 5.49230i 0
18.1 −0.389385 − 2.45848i 2.78770 − 1.42041i −3.99039 + 1.29656i 0 −4.57753 − 6.30043i 0.515430 − 1.01159i 2.48127 + 4.86977i 3.99039 − 5.49230i 0
18.2 −0.327285 − 2.06640i −2.33135 + 1.18788i −2.26078 + 0.734571i 0 3.21766 + 4.42872i −1.30222 + 2.55575i 0.358196 + 0.702999i 2.26078 − 3.11169i 0
18.3 −0.265359 − 1.67541i −1.87317 + 0.954428i −0.834479 + 0.271139i 0 2.09612 + 2.88507i 0.829412 − 1.62781i −0.864500 − 1.69668i 0.834479 − 1.14856i 0
18.4 −0.0640711 − 0.404529i 0.167537 − 0.0853641i 1.74257 − 0.566197i 0 −0.0452665 − 0.0623040i 1.61620 − 3.17198i −0.712575 − 1.39851i −1.74257 + 2.39845i 0
18.5 0.0640711 + 0.404529i −0.167537 + 0.0853641i 1.74257 − 0.566197i 0 −0.0452665 − 0.0623040i −1.61620 + 3.17198i 0.712575 + 1.39851i −1.74257 + 2.39845i 0
18.6 0.265359 + 1.67541i 1.87317 − 0.954428i −0.834479 + 0.271139i 0 2.09612 + 2.88507i −0.829412 + 1.62781i 0.864500 + 1.69668i 0.834479 − 1.14856i 0
18.7 0.327285 + 2.06640i 2.33135 − 1.18788i −2.26078 + 0.734571i 0 3.21766 + 4.42872i 1.30222 − 2.55575i −0.358196 − 0.702999i 2.26078 − 3.11169i 0
18.8 0.389385 + 2.45848i −2.78770 + 1.42041i −3.99039 + 1.29656i 0 −4.57753 − 6.30043i −0.515430 + 1.01159i −2.48127 − 4.86977i 3.99039 − 5.49230i 0
57.1 −1.18170 − 2.31921i −0.0345558 + 0.218177i −2.80676 + 3.86318i 0 0.546832 − 0.177677i 3.07699 − 0.487347i 7.13454 + 1.13000i 2.80676 + 0.911972i 0
57.2 −0.847550 − 1.66341i −0.225724 + 1.42517i −0.873024 + 1.20161i 0 2.56195 − 0.832427i −0.0507901 + 0.00804437i −0.949101 − 0.150323i 0.873024 + 0.283663i 0
57.3 −0.463887 − 0.910430i 0.296434 − 1.87161i 0.561879 − 0.773360i 0 −1.84148 + 0.598334i −4.66765 + 0.739283i −2.98318 − 0.472489i −0.561879 − 0.182565i 0
57.4 −0.126357 − 0.247990i −0.320144 + 2.02131i 1.13004 − 1.55536i 0 0.541718 − 0.176015i 1.53987 − 0.243891i −1.07830 − 0.170786i −1.13004 − 0.367171i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
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
11.d odd 10 1 inner
55.h odd 10 1 inner
55.l even 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.bm.c ✓ 64
5.b even 2 1 inner 275.2.bm.c ✓ 64
5.c odd 4 2 inner 275.2.bm.c ✓ 64
11.d odd 10 1 inner 275.2.bm.c ✓ 64
55.h odd 10 1 inner 275.2.bm.c ✓ 64
55.l even 20 2 inner 275.2.bm.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.bm.c ✓ 64 1.a even 1 1 trivial
275.2.bm.c ✓ 64 5.b even 2 1 inner
275.2.bm.c ✓ 64 5.c odd 4 2 inner
275.2.bm.c ✓ 64 11.d odd 10 1 inner
275.2.bm.c ✓ 64 55.h odd 10 1 inner
275.2.bm.c ✓ 64 55.l even 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 70 T_{2}^{60} + 3685 T_{2}^{56} - 183530 T_{2}^{52} + 7828150 T_{2}^{48} - 186237550 T_{2}^{44} + \cdots + 390625 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display