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Properties

Label 588.2.ba.b
Level $588$
Weight $2$
Character orbit 588.ba
Analytic conductor $4.695$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,2,Mod(103,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 0, 29]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.ba (of order \(42\), degree \(12\), 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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{3} - 2 q^{7} - 6 q^{8} + 28 q^{9} - 23 q^{10} - 6 q^{11} - 30 q^{14} - 12 q^{16} + 6 q^{19} + 25 q^{20} + 4 q^{21} + 6 q^{22} + 15 q^{24} - 26 q^{25} - 12 q^{26} - 56 q^{27} + 36 q^{28} - 13 q^{30} - 2 q^{31} - 25 q^{32} + 6 q^{33} + 68 q^{34} + 12 q^{35} + 16 q^{37} + 82 q^{38} - 8 q^{39} - 19 q^{40} - 9 q^{42} - 11 q^{44} + 10 q^{46} - 4 q^{47} - 8 q^{48} - 4 q^{49} - 114 q^{50} - 8 q^{52} - 4 q^{53} - 41 q^{56} - 12 q^{57} - 33 q^{58} + 10 q^{59} + 17 q^{60} + 2 q^{61} + 16 q^{62} + 12 q^{63} + 84 q^{64} - 4 q^{65} + 15 q^{66} - 42 q^{67} + 10 q^{68} - 38 q^{70} - 28 q^{71} + 33 q^{72} + 18 q^{73} + 2 q^{74} - 54 q^{75} - 7 q^{76} - 8 q^{77} - 6 q^{78} + 6 q^{79} - 14 q^{80} + 28 q^{81} - 87 q^{82} - 10 q^{83} - 14 q^{84} + 24 q^{85} + 126 q^{86} - 244 q^{88} - 20 q^{90} + 34 q^{91} + 14 q^{92} - 2 q^{93} - 184 q^{94} + 24 q^{95} - 20 q^{96} - 122 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −1.41341 + 0.0477030i 0.826239 + 0.563320i 1.99545 − 0.134848i −0.793563 − 0.0594693i −1.19469 − 0.756787i −0.251406 − 2.63378i −2.81395 + 0.285784i 0.365341 + 0.930874i 1.12447 + 0.0461991i
103.2 −1.38479 − 0.286956i 0.826239 + 0.563320i 1.83531 + 0.794750i 0.961592 + 0.0720614i −0.982523 − 1.01718i −2.50588 + 0.848875i −2.31347 − 1.62722i 0.365341 + 0.930874i −1.31093 − 0.375725i
103.3 −1.34609 − 0.433645i 0.826239 + 0.563320i 1.62390 + 1.16745i 1.33423 + 0.0999870i −0.867909 − 1.11657i 2.38917 + 1.13661i −1.67966 − 2.27569i 0.365341 + 0.930874i −1.75264 − 0.713175i
103.4 −1.31353 + 0.524063i 0.826239 + 0.563320i 1.45072 − 1.37674i 3.32240 + 0.248980i −1.38050 − 0.306936i 2.08944 + 1.62303i −1.18406 + 2.56866i 0.365341 + 0.930874i −4.49455 + 1.41411i
103.5 −1.13745 − 0.840366i 0.826239 + 0.563320i 0.587569 + 1.91174i −4.04500 − 0.303131i −0.466407 − 1.33509i 2.59666 − 0.507311i 0.938236 − 2.66828i 0.365341 + 0.930874i 4.34623 + 3.74408i
103.6 −1.12991 + 0.850479i 0.826239 + 0.563320i 0.553371 − 1.92192i −1.47341 − 0.110417i −1.41266 + 0.0662004i −2.51658 − 0.816579i 1.00930 + 2.64222i 0.365341 + 0.930874i 1.75872 − 1.12834i
103.7 −0.984808 + 1.01496i 0.826239 + 0.563320i −0.0603079 − 1.99909i −2.59884 − 0.194756i −1.38544 + 0.283842i 2.28001 − 1.34222i 2.08840 + 1.90751i 0.365341 + 0.930874i 2.75703 − 2.44594i
103.8 −0.945130 − 1.05201i 0.826239 + 0.563320i −0.213459 + 1.98858i 4.20665 + 0.315245i −0.188284 − 1.40162i −2.58503 − 0.563570i 2.29375 − 1.65490i 0.365341 + 0.930874i −3.64419 − 4.72340i
103.9 −0.847839 − 1.13189i 0.826239 + 0.563320i −0.562339 + 1.91932i −0.786913 − 0.0589710i −0.0629023 − 1.41281i −0.621496 + 2.57172i 2.64922 − 0.990766i 0.365341 + 0.930874i 0.600427 + 0.940695i
103.10 −0.775324 + 1.18274i 0.826239 + 0.563320i −0.797745 − 1.83401i 2.88553 + 0.216241i −1.30686 + 0.540469i −0.232552 − 2.63551i 2.78767 + 0.478430i 0.365341 + 0.930874i −2.49298 + 3.24518i
103.11 −0.504390 − 1.32121i 0.826239 + 0.563320i −1.49118 + 1.33281i −2.91108 − 0.218156i 0.327516 − 1.37577i −2.63369 − 0.252375i 2.51305 + 1.29790i 0.365341 + 0.930874i 1.18009 + 3.95618i
103.12 −0.467447 − 1.33473i 0.826239 + 0.563320i −1.56299 + 1.24783i 2.71904 + 0.203764i 0.365655 − 1.36612i 2.62441 − 0.335390i 2.39612 + 1.50287i 0.365341 + 0.930874i −0.999039 − 3.72443i
103.13 −0.313470 + 1.37903i 0.826239 + 0.563320i −1.80347 − 0.864572i −0.481855 − 0.0361100i −1.03584 + 0.962828i 1.51904 + 2.16623i 1.75761 − 2.21603i 0.365341 + 0.930874i 0.200844 − 0.653175i
103.14 −0.281403 + 1.38593i 0.826239 + 0.563320i −1.84162 − 0.780012i −2.31910 − 0.173792i −1.01323 + 0.986592i −2.40273 + 1.10766i 1.59928 − 2.33287i 0.365341 + 0.930874i 0.893467 − 3.16521i
103.15 0.227338 + 1.39582i 0.826239 + 0.563320i −1.89663 + 0.634646i 1.91992 + 0.143878i −0.598459 + 1.28135i 0.854457 − 2.50398i −1.31703 − 2.50308i 0.365341 + 0.930874i 0.235643 + 2.71258i
103.16 0.307661 − 1.38034i 0.826239 + 0.563320i −1.81069 − 0.849355i −2.57232 − 0.192768i 1.03178 − 0.967181i 0.135490 + 2.64228i −1.72948 + 2.23806i 0.365341 + 0.930874i −1.05749 + 3.49137i
103.17 0.443386 + 1.34291i 0.826239 + 0.563320i −1.60682 + 1.19085i 4.26189 + 0.319384i −0.390146 + 1.35933i −0.230798 + 2.63567i −2.31165 − 1.62981i 0.365341 + 0.930874i 1.46076 + 5.86495i
103.18 0.471094 − 1.33344i 0.826239 + 0.563320i −1.55614 − 1.25635i 0.00499363 0.000374221i 1.14039 − 0.836366i 2.43918 − 1.02488i −2.40836 + 1.48317i 0.365341 + 0.930874i 0.00285147 − 0.00648243i
103.19 0.604797 + 1.27837i 0.826239 + 0.563320i −1.26844 + 1.54630i −1.87993 − 0.140882i −0.220423 + 1.39693i −1.80315 − 1.93615i −2.74389 − 0.686335i 0.365341 + 0.930874i −0.956880 − 2.48845i
103.20 0.898343 − 1.09224i 0.826239 + 0.563320i −0.385959 − 1.96241i −3.46688 − 0.259807i 1.35752 − 0.396393i −2.42772 − 1.05175i −2.49013 − 1.34135i 0.365341 + 0.930874i −3.39822 + 3.55326i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.p even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.ba.b yes 336
4.b odd 2 1 588.2.ba.a ✓ 336
49.h odd 42 1 588.2.ba.a ✓ 336
196.p even 42 1 inner 588.2.ba.b yes 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.ba.a ✓ 336 4.b odd 2 1
588.2.ba.a ✓ 336 49.h odd 42 1
588.2.ba.b yes 336 1.a even 1 1 trivial
588.2.ba.b yes 336 196.p even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{336} + 6 T_{11}^{335} + 189 T_{11}^{334} + 1230 T_{11}^{333} + 16023 T_{11}^{332} + \cdots + 41\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display