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Properties

Label 704.2.be.a
Level $704$
Weight $2$
Character orbit 704.be
Analytic conductor $5.621$
Analytic rank $0$
Dimension $176$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,2,Mod(49,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 5, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.be (of order \(20\), degree \(8\), 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: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(22\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 176)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 6 q^{3} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q + 6 q^{3} - 6 q^{5} + 12 q^{11} - 6 q^{13} + 12 q^{15} - 12 q^{17} + 6 q^{19} - 28 q^{21} + 18 q^{27} - 22 q^{29} + 12 q^{31} - 16 q^{33} + 26 q^{35} + 18 q^{37} + 40 q^{43} - 24 q^{45} + 12 q^{47} + 8 q^{49} - 6 q^{51} - 30 q^{53} - 10 q^{59} - 6 q^{61} + 28 q^{63} - 32 q^{65} - 24 q^{67} + 12 q^{69} + 46 q^{75} - 14 q^{77} + 52 q^{79} - 54 q^{83} + 14 q^{85} + 122 q^{91} + 6 q^{93} - 52 q^{95} - 12 q^{97} - 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −3.10742 + 0.492167i 0 −1.27797 + 0.651156i 0 −2.30623 − 3.17425i 0 6.56066 − 2.13169i 0
49.2 0 −2.68580 + 0.425388i 0 0.102082 − 0.0520132i 0 2.36506 + 3.25523i 0 4.17938 − 1.35796i 0
49.3 0 −2.39469 + 0.379282i 0 2.15797 − 1.09954i 0 0.678287 + 0.933582i 0 2.73753 − 0.889476i 0
49.4 0 −2.35196 + 0.372513i 0 −3.24121 + 1.65148i 0 −0.349519 − 0.481071i 0 2.53977 − 0.825220i 0
49.5 0 −2.09875 + 0.332409i 0 −1.30316 + 0.663991i 0 0.801284 + 1.10287i 0 1.44108 − 0.468236i 0
49.6 0 −1.33374 + 0.211244i 0 0.141837 − 0.0722696i 0 −1.45574 − 2.00365i 0 −1.11893 + 0.363561i 0
49.7 0 −1.22266 + 0.193650i 0 2.51383 − 1.28086i 0 −2.24956 − 3.09626i 0 −1.39578 + 0.453517i 0
49.8 0 −0.901291 + 0.142750i 0 3.45027 − 1.75800i 0 1.81367 + 2.49631i 0 −2.06122 + 0.669732i 0
49.9 0 −0.662536 + 0.104935i 0 −0.264470 + 0.134754i 0 0.696271 + 0.958335i 0 −2.42523 + 0.788004i 0
49.10 0 −0.562832 + 0.0891439i 0 1.83674 − 0.935866i 0 −0.677755 − 0.932849i 0 −2.54434 + 0.826705i 0
49.11 0 −0.310032 + 0.0491042i 0 −3.81933 + 1.94605i 0 −1.09214 − 1.50320i 0 −2.75946 + 0.896603i 0
49.12 0 0.417657 − 0.0661504i 0 −0.736011 + 0.375016i 0 −1.33653 − 1.83957i 0 −2.68311 + 0.871795i 0
49.13 0 0.757844 − 0.120031i 0 −0.814347 + 0.414930i 0 2.91240 + 4.00857i 0 −2.29325 + 0.745122i 0
49.14 0 0.925602 − 0.146601i 0 −1.96526 + 1.00135i 0 −0.432076 − 0.594702i 0 −2.01792 + 0.655663i 0
49.15 0 1.17460 − 0.186039i 0 2.66815 − 1.35949i 0 −2.54995 − 3.50970i 0 −1.50808 + 0.490006i 0
49.16 0 1.67818 − 0.265797i 0 −1.78176 + 0.907854i 0 2.03350 + 2.79887i 0 −0.107544 + 0.0349431i 0
49.17 0 1.93030 − 0.305729i 0 1.24626 − 0.635003i 0 1.09820 + 1.51155i 0 0.779411 − 0.253246i 0
49.18 0 1.99895 − 0.316603i 0 −2.43455 + 1.24047i 0 −2.40520 − 3.31048i 0 1.04239 − 0.338694i 0
49.19 0 2.20058 − 0.348538i 0 2.16222 − 1.10171i 0 0.296642 + 0.408292i 0 1.86792 − 0.606923i 0
49.20 0 2.54744 − 0.403475i 0 2.98748 − 1.52220i 0 1.07945 + 1.48574i 0 3.47349 − 1.12860i 0
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
16.e even 4 1 inner
176.w even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.2.be.a 176
4.b odd 2 1 176.2.w.a ✓ 176
11.c even 5 1 inner 704.2.be.a 176
16.e even 4 1 inner 704.2.be.a 176
16.f odd 4 1 176.2.w.a ✓ 176
44.h odd 10 1 176.2.w.a ✓ 176
176.v odd 20 1 176.2.w.a ✓ 176
176.w even 20 1 inner 704.2.be.a 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.w.a ✓ 176 4.b odd 2 1
176.2.w.a ✓ 176 16.f odd 4 1
176.2.w.a ✓ 176 44.h odd 10 1
176.2.w.a ✓ 176 176.v odd 20 1
704.2.be.a 176 1.a even 1 1 trivial
704.2.be.a 176 11.c even 5 1 inner
704.2.be.a 176 16.e even 4 1 inner
704.2.be.a 176 176.w even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(704, [\chi])\).