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Properties

Label 825.2.cb.a
Level $825$
Weight $2$
Character orbit 825.cb
Analytic conductor $6.588$
Analytic rank $0$
Dimension $240$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(169,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.cb (of order \(10\), degree \(4\), 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: \(240\)
Relative dimension: \(60\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 60 q^{4} + 6 q^{5} - 16 q^{6} + 10 q^{7} + 60 q^{9} + 8 q^{10} + 4 q^{11} - 20 q^{12} - 64 q^{16} - 24 q^{19} - 44 q^{20} + 8 q^{21} - 40 q^{22} + 40 q^{23} - 12 q^{24} + 4 q^{25} + 40 q^{26} + 30 q^{28}+ \cdots - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 −2.63258 − 0.855376i 0.951057 − 0.309017i 4.58076 + 3.32812i −0.694498 + 2.12548i −2.76806 3.66791 − 1.19178i −5.95837 − 8.20099i 0.809017 − 0.587785i 3.64641 − 5.00144i
169.2 −2.60741 − 0.847200i −0.951057 + 0.309017i 4.46282 + 3.24243i 1.30322 + 1.81703i 2.74160 −3.97271 + 1.29081i −5.66648 − 7.79924i 0.809017 − 0.587785i −1.85864 − 5.84185i
169.3 −2.55995 − 0.831779i −0.951057 + 0.309017i 4.24347 + 3.08306i −2.21665 − 0.294050i 2.69169 0.542994 − 0.176430i −5.13438 − 7.06687i 0.809017 − 0.587785i 5.42993 + 2.59652i
169.4 −2.47523 − 0.804252i 0.951057 − 0.309017i 3.86193 + 2.80585i 2.18794 − 0.461453i −2.60261 −0.220156 + 0.0715332i −4.24300 − 5.83999i 0.809017 − 0.587785i −5.78678 − 0.617448i
169.5 −2.45958 − 0.799165i 0.951057 − 0.309017i 3.79282 + 2.75564i −1.27760 − 1.83514i −2.58615 0.283269 − 0.0920396i −4.08631 − 5.62432i 0.809017 − 0.587785i 1.67576 + 5.53468i
169.6 −2.10834 − 0.685041i −0.951057 + 0.309017i 2.35778 + 1.71303i 0.644933 − 2.14104i 2.21684 −3.97500 + 1.29155i −1.19146 − 1.63991i 0.809017 − 0.587785i −2.82644 + 4.07224i
169.7 −2.09650 − 0.681193i 0.951057 − 0.309017i 2.31323 + 1.68066i 2.21589 + 0.299753i −2.20439 −3.07728 + 0.999868i −1.11342 − 1.53249i 0.809017 − 0.587785i −4.44140 − 2.13788i
169.8 −2.08139 − 0.676285i −0.951057 + 0.309017i 2.25679 + 1.63966i −1.84257 + 1.26685i 2.18850 −1.00018 + 0.324978i −1.01565 − 1.39792i 0.809017 − 0.587785i 4.69187 − 1.39072i
169.9 −2.03594 − 0.661516i −0.951057 + 0.309017i 2.08940 + 1.51804i 0.00440980 + 2.23606i 2.14071 4.70550 − 1.52891i −0.733119 − 1.00905i 0.809017 − 0.587785i 1.47021 − 4.55540i
169.10 −1.96058 − 0.637033i 0.951057 − 0.309017i 1.82005 + 1.32234i −2.14188 − 0.642157i −2.06148 1.60997 − 0.523111i −0.302567 − 0.416448i 0.809017 − 0.587785i 3.79025 + 2.62345i
169.11 −1.95539 − 0.635346i 0.951057 − 0.309017i 1.80187 + 1.30913i −2.22400 − 0.231974i −2.05602 −4.36107 + 1.41700i −0.274612 − 0.377970i 0.809017 − 0.587785i 4.20142 + 1.86661i
169.12 −1.95335 − 0.634681i 0.951057 − 0.309017i 1.79471 + 1.30393i 1.33152 − 1.79640i −2.05387 4.36588 − 1.41856i −0.263635 − 0.362863i 0.809017 − 0.587785i −3.74106 + 2.66390i
169.13 −1.81721 − 0.590448i −0.951057 + 0.309017i 1.33560 + 0.970368i 1.53007 + 1.63061i 1.91073 1.82631 − 0.593403i 0.392086 + 0.539660i 0.809017 − 0.587785i −1.81767 − 3.86659i
169.14 −1.75058 − 0.568798i −0.951057 + 0.309017i 1.12297 + 0.815886i 2.12442 − 0.697744i 1.84067 0.161477 − 0.0524671i 0.662061 + 0.911249i 0.809017 − 0.587785i −4.11584 + 0.0130915i
169.15 −1.42855 − 0.464164i −0.951057 + 0.309017i 0.207274 + 0.150593i −1.19097 − 1.89251i 1.50207 4.03713 − 1.31174i 1.53958 + 2.11906i 0.809017 − 0.587785i 0.822929 + 3.25635i
169.16 −1.41304 − 0.459125i 0.951057 − 0.309017i 0.167853 + 0.121952i 1.85059 + 1.25512i −1.48576 1.08303 − 0.351897i 1.56542 + 2.15462i 0.809017 − 0.587785i −2.03870 − 2.62318i
169.17 −1.27409 − 0.413977i −0.951057 + 0.309017i −0.166107 − 0.120683i −0.599520 − 2.15420i 1.33966 −0.114201 + 0.0371061i 1.73654 + 2.39014i 0.809017 − 0.587785i −0.127946 + 2.99283i
169.18 −1.18834 − 0.386117i 0.951057 − 0.309017i −0.354957 − 0.257892i −1.39539 + 1.74725i −1.24950 −1.93521 + 0.628789i 1.79111 + 2.46525i 0.809017 − 0.587785i 2.33284 − 1.53756i
169.19 −1.18185 − 0.384006i 0.951057 − 0.309017i −0.368730 − 0.267898i −0.282910 − 2.21810i −1.24267 −2.77555 + 0.901831i 1.79375 + 2.46889i 0.809017 − 0.587785i −0.517406 + 2.73009i
169.20 −1.11155 − 0.361164i 0.951057 − 0.309017i −0.512931 − 0.372666i −0.391185 + 2.20158i −1.16875 2.43045 − 0.789701i 1.80951 + 2.49057i 0.809017 − 0.587785i 1.22996 − 2.30589i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.bb even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.cb.a yes 240
11.c even 5 1 825.2.by.a ✓ 240
25.e even 10 1 825.2.by.a ✓ 240
275.bb even 10 1 inner 825.2.cb.a yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.by.a ✓ 240 11.c even 5 1
825.2.by.a ✓ 240 25.e even 10 1
825.2.cb.a yes 240 1.a even 1 1 trivial
825.2.cb.a yes 240 275.bb even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(825, [\chi])\).