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Properties

Label 825.2.cw.b
Level $825$
Weight $2$
Character orbit 825.cw
Analytic conductor $6.588$
Analytic rank $0$
Dimension $96$
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] = [825,2,Mod(7,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 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("825.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.cw (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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 165)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 20 q^{7} + 8 q^{11} + 16 q^{12} + 8 q^{16} + 20 q^{17} + 32 q^{22} - 32 q^{23} + 60 q^{28} + 16 q^{31} + 16 q^{33} + 24 q^{36} - 8 q^{37} - 56 q^{38} - 120 q^{41} - 12 q^{42} - 200 q^{46} - 60 q^{47} - 48 q^{48} + 40 q^{51} - 40 q^{52} - 36 q^{53} - 80 q^{56} - 40 q^{57} - 44 q^{58} + 40 q^{61} - 80 q^{62} - 20 q^{63} + 56 q^{66} + 48 q^{67} - 80 q^{68} + 32 q^{71} + 60 q^{73} + 24 q^{77} + 96 q^{78} + 24 q^{81} - 32 q^{82} + 200 q^{83} - 80 q^{86} + 144 q^{88} + 56 q^{91} - 20 q^{92} + 72 q^{93} - 32 q^{97}+O(q^{100}) \) 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 −2.46605 + 0.390584i −0.453990 − 0.891007i 4.02673 − 1.30836i 0 1.46758 + 2.01994i 3.26542 + 1.66382i −4.96977 + 2.53222i −0.587785 + 0.809017i 0
7.2 −2.13992 + 0.338930i 0.453990 + 0.891007i 2.56227 − 0.832532i 0 −1.27349 − 1.75281i 0.770101 + 0.392386i −1.33999 + 0.682757i −0.587785 + 0.809017i 0
7.3 −1.86197 + 0.294907i 0.453990 + 0.891007i 1.47784 − 0.480180i 0 −1.10808 − 1.52514i 0.414728 + 0.211314i 0.749326 − 0.381801i −0.587785 + 0.809017i 0
7.4 −1.73704 + 0.275120i −0.453990 − 0.891007i 1.03949 − 0.337751i 0 1.03373 + 1.42281i −3.16835 − 1.61435i 1.42129 − 0.724185i −0.587785 + 0.809017i 0
7.5 −0.779339 + 0.123435i −0.453990 − 0.891007i −1.30998 + 0.425638i 0 0.463794 + 0.638358i −0.804274 − 0.409798i 2.37448 − 1.20986i −0.587785 + 0.809017i 0
7.6 −0.139396 + 0.0220781i 0.453990 + 0.891007i −1.88317 + 0.611879i 0 −0.0829560 − 0.114179i 1.56314 + 0.796462i 0.500497 − 0.255016i −0.587785 + 0.809017i 0
7.7 0.329725 − 0.0522232i 0.453990 + 0.891007i −1.79612 + 0.583595i 0 0.196223 + 0.270078i −2.61195 − 1.33085i −1.15665 + 0.589341i −0.587785 + 0.809017i 0
7.8 1.12702 − 0.178502i −0.453990 − 0.891007i −0.663811 + 0.215685i 0 −0.670701 − 0.923140i −1.36271 − 0.694338i −2.74302 + 1.39764i −0.587785 + 0.809017i 0
7.9 1.27330 − 0.201671i −0.453990 − 0.891007i −0.321488 + 0.104458i 0 −0.757757 − 1.04296i 4.03315 + 2.05499i −2.68561 + 1.36839i −0.587785 + 0.809017i 0
7.10 1.44976 − 0.229620i 0.453990 + 0.891007i 0.146976 − 0.0477555i 0 0.862772 + 1.18750i 3.00927 + 1.53330i −2.41359 + 1.22978i −0.587785 + 0.809017i 0
7.11 2.36180 − 0.374072i 0.453990 + 0.891007i 3.53603 − 1.14893i 0 1.40553 + 1.93455i 0.732807 + 0.373384i 3.66039 − 1.86506i −0.587785 + 0.809017i 0
7.12 2.58211 − 0.408966i −0.453990 − 0.891007i 4.59791 − 1.49395i 0 −1.53664 − 2.11501i 1.91487 + 0.975676i 6.60261 − 3.36420i −0.587785 + 0.809017i 0
118.1 −2.46605 − 0.390584i −0.453990 + 0.891007i 4.02673 + 1.30836i 0 1.46758 − 2.01994i 3.26542 − 1.66382i −4.96977 − 2.53222i −0.587785 − 0.809017i 0
118.2 −2.13992 − 0.338930i 0.453990 − 0.891007i 2.56227 + 0.832532i 0 −1.27349 + 1.75281i 0.770101 − 0.392386i −1.33999 − 0.682757i −0.587785 − 0.809017i 0
118.3 −1.86197 − 0.294907i 0.453990 − 0.891007i 1.47784 + 0.480180i 0 −1.10808 + 1.52514i 0.414728 − 0.211314i 0.749326 + 0.381801i −0.587785 − 0.809017i 0
118.4 −1.73704 − 0.275120i −0.453990 + 0.891007i 1.03949 + 0.337751i 0 1.03373 − 1.42281i −3.16835 + 1.61435i 1.42129 + 0.724185i −0.587785 − 0.809017i 0
118.5 −0.779339 − 0.123435i −0.453990 + 0.891007i −1.30998 − 0.425638i 0 0.463794 − 0.638358i −0.804274 + 0.409798i 2.37448 + 1.20986i −0.587785 − 0.809017i 0
118.6 −0.139396 − 0.0220781i 0.453990 − 0.891007i −1.88317 − 0.611879i 0 −0.0829560 + 0.114179i 1.56314 − 0.796462i 0.500497 + 0.255016i −0.587785 − 0.809017i 0
118.7 0.329725 + 0.0522232i 0.453990 − 0.891007i −1.79612 − 0.583595i 0 0.196223 − 0.270078i −2.61195 + 1.33085i −1.15665 − 0.589341i −0.587785 − 0.809017i 0
118.8 1.12702 + 0.178502i −0.453990 + 0.891007i −0.663811 − 0.215685i 0 −0.670701 + 0.923140i −1.36271 + 0.694338i −2.74302 − 1.39764i −0.587785 − 0.809017i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.cw.b 96
5.b even 2 1 165.2.w.a ✓ 96
5.c odd 4 1 165.2.w.a ✓ 96
5.c odd 4 1 inner 825.2.cw.b 96
11.d odd 10 1 inner 825.2.cw.b 96
15.d odd 2 1 495.2.bj.c 96
15.e even 4 1 495.2.bj.c 96
55.h odd 10 1 165.2.w.a ✓ 96
55.l even 20 1 165.2.w.a ✓ 96
55.l even 20 1 inner 825.2.cw.b 96
165.r even 10 1 495.2.bj.c 96
165.u odd 20 1 495.2.bj.c 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.w.a ✓ 96 5.b even 2 1
165.2.w.a ✓ 96 5.c odd 4 1
165.2.w.a ✓ 96 55.h odd 10 1
165.2.w.a ✓ 96 55.l even 20 1
495.2.bj.c 96 15.d odd 2 1
495.2.bj.c 96 15.e even 4 1
495.2.bj.c 96 165.r even 10 1
495.2.bj.c 96 165.u odd 20 1
825.2.cw.b 96 1.a even 1 1 trivial
825.2.cw.b 96 5.c odd 4 1 inner
825.2.cw.b 96 11.d odd 10 1 inner
825.2.cw.b 96 55.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 86 T_{2}^{92} - 180 T_{2}^{89} + 4943 T_{2}^{88} - 20 T_{2}^{87} + 15800 T_{2}^{85} + \cdots + 43046721 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display