Properties

Label 275.2.z.c
Level $275$
Weight $2$
Character orbit 275.z
Analytic conductor $2.196$
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] = [275,2,Mod(49,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.z (of order \(10\), degree \(4\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} - 6 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} - 6 q^{6} - 16 q^{9} - 10 q^{11} - 6 q^{14} - 8 q^{16} + 26 q^{19} + 20 q^{21} + 86 q^{24} - 68 q^{26} + 22 q^{29} - 20 q^{31} - 40 q^{34} + 6 q^{36} - 6 q^{39} + 50 q^{41} + 2 q^{44} + 80 q^{46} - 32 q^{49} + 70 q^{51} - 120 q^{54} - 40 q^{56} - 28 q^{59} + 32 q^{61} - 80 q^{64} - 110 q^{66} - 70 q^{69} - 92 q^{71} + 14 q^{74} - 124 q^{76} + 78 q^{79} - 86 q^{81} + 108 q^{84} + 4 q^{86} + 44 q^{89} - 68 q^{91} + 80 q^{94} + 216 q^{96} + 20 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −2.63006 + 0.854560i 0.964848 1.32800i 4.56893 3.31953i 0 −1.40276 + 4.31724i 1.11557 + 1.53545i −5.92892 + 8.16046i 0.0944009 + 0.290536i 0
49.2 −1.59020 + 0.516686i −1.82798 + 2.51599i 0.643729 0.467696i 0 1.60686 4.94542i −1.81624 2.49985i 1.18359 1.62907i −2.06168 6.34519i 0
49.3 −0.911888 + 0.296290i 0.528620 0.727583i −0.874283 + 0.635204i 0 −0.266466 + 0.820099i −0.447684 0.616184i 1.73620 2.38967i 0.677113 + 2.08394i 0
49.4 −0.132580 + 0.0430779i 1.75502 2.41558i −1.60231 + 1.16415i 0 −0.128623 + 0.395861i −2.09815 2.88786i 0.326164 0.448926i −1.82787 5.62561i 0
49.5 0.132580 0.0430779i −1.75502 + 2.41558i −1.60231 + 1.16415i 0 −0.128623 + 0.395861i 2.09815 + 2.88786i −0.326164 + 0.448926i −1.82787 5.62561i 0
49.6 0.911888 0.296290i −0.528620 + 0.727583i −0.874283 + 0.635204i 0 −0.266466 + 0.820099i 0.447684 + 0.616184i −1.73620 + 2.38967i 0.677113 + 2.08394i 0
49.7 1.59020 0.516686i 1.82798 2.51599i 0.643729 0.467696i 0 1.60686 4.94542i 1.81624 + 2.49985i −1.18359 + 1.62907i −2.06168 6.34519i 0
49.8 2.63006 0.854560i −0.964848 + 1.32800i 4.56893 3.31953i 0 −1.40276 + 4.31724i −1.11557 1.53545i 5.92892 8.16046i 0.0944009 + 0.290536i 0
124.1 −1.50618 2.07308i 1.70486 0.553942i −1.41105 + 4.34277i 0 −3.71620 2.69998i 3.61485 + 1.17454i 6.25410 2.03208i 0.172643 0.125433i 0
124.2 −1.34829 1.85576i 0.0403304 0.0131041i −1.00792 + 3.10207i 0 −0.0786950 0.0571753i −1.08390 0.352180i 2.75251 0.894346i −2.42560 + 1.76230i 0
124.3 −0.776830 1.06921i −1.92724 + 0.626199i 0.0782786 0.240917i 0 2.16668 + 1.57419i 0.331213 + 0.107618i −2.83228 + 0.920262i 0.895086 0.650318i 0
124.4 −0.122427 0.168506i −1.80155 + 0.585361i 0.604628 1.86085i 0 0.319195 + 0.231909i 1.22573 + 0.398265i −0.783769 + 0.254662i 0.475901 0.345762i 0
124.5 0.122427 + 0.168506i 1.80155 0.585361i 0.604628 1.86085i 0 0.319195 + 0.231909i −1.22573 0.398265i 0.783769 0.254662i 0.475901 0.345762i 0
124.6 0.776830 + 1.06921i 1.92724 0.626199i 0.0782786 0.240917i 0 2.16668 + 1.57419i −0.331213 0.107618i 2.83228 0.920262i 0.895086 0.650318i 0
124.7 1.34829 + 1.85576i −0.0403304 + 0.0131041i −1.00792 + 3.10207i 0 −0.0786950 0.0571753i 1.08390 + 0.352180i −2.75251 + 0.894346i −2.42560 + 1.76230i 0
124.8 1.50618 + 2.07308i −1.70486 + 0.553942i −1.41105 + 4.34277i 0 −3.71620 2.69998i −3.61485 1.17454i −6.25410 + 2.03208i 0.172643 0.125433i 0
174.1 −2.63006 0.854560i 0.964848 + 1.32800i 4.56893 + 3.31953i 0 −1.40276 4.31724i 1.11557 1.53545i −5.92892 8.16046i 0.0944009 0.290536i 0
174.2 −1.59020 0.516686i −1.82798 2.51599i 0.643729 + 0.467696i 0 1.60686 + 4.94542i −1.81624 + 2.49985i 1.18359 + 1.62907i −2.06168 + 6.34519i 0
174.3 −0.911888 0.296290i 0.528620 + 0.727583i −0.874283 0.635204i 0 −0.266466 0.820099i −0.447684 + 0.616184i 1.73620 + 2.38967i 0.677113 2.08394i 0
174.4 −0.132580 0.0430779i 1.75502 + 2.41558i −1.60231 1.16415i 0 −0.128623 0.395861i −2.09815 + 2.88786i 0.326164 + 0.448926i −1.82787 + 5.62561i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.z.c 32
5.b even 2 1 inner 275.2.z.c 32
5.c odd 4 1 275.2.h.c 16
5.c odd 4 1 275.2.h.e yes 16
11.c even 5 1 inner 275.2.z.c 32
55.j even 10 1 inner 275.2.z.c 32
55.k odd 20 1 275.2.h.c 16
55.k odd 20 1 275.2.h.e yes 16
55.k odd 20 1 3025.2.a.bi 8
55.k odd 20 1 3025.2.a.bn 8
55.l even 20 1 3025.2.a.bj 8
55.l even 20 1 3025.2.a.bm 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.h.c 16 5.c odd 4 1
275.2.h.c 16 55.k odd 20 1
275.2.h.e yes 16 5.c odd 4 1
275.2.h.e yes 16 55.k odd 20 1
275.2.z.c 32 1.a even 1 1 trivial
275.2.z.c 32 5.b even 2 1 inner
275.2.z.c 32 11.c even 5 1 inner
275.2.z.c 32 55.j even 10 1 inner
3025.2.a.bi 8 55.k odd 20 1
3025.2.a.bj 8 55.l even 20 1
3025.2.a.bm 8 55.l even 20 1
3025.2.a.bn 8 55.k odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 10 T_{2}^{30} + 89 T_{2}^{28} - 694 T_{2}^{26} + 5502 T_{2}^{24} - 20142 T_{2}^{22} + 113853 T_{2}^{20} - 411465 T_{2}^{18} + 961802 T_{2}^{16} - 1330529 T_{2}^{14} + 2642317 T_{2}^{12} - 2806839 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display