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Properties

Label 882.2.bj.a
Level $882$
Weight $2$
Character orbit 882.bj
Analytic conductor $7.043$
Analytic rank $0$
Dimension $672$
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] = [882,2,Mod(41,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([35, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.bj (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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(672\)
Relative dimension: \(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 56 q^{4} - 28 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 672 q - 56 q^{4} - 28 q^{6} - 2 q^{7} + 10 q^{9} + 6 q^{14} - 12 q^{15} + 56 q^{16} + 8 q^{18} - 18 q^{21} - 90 q^{23} + 56 q^{25} - 42 q^{27} - 4 q^{28} - 30 q^{29} + 28 q^{30} + 20 q^{36} - 20 q^{37} + 52 q^{39} - 142 q^{42} - 4 q^{43} - 14 q^{45} + 60 q^{46} + 126 q^{47} + 8 q^{49} - 48 q^{50} - 12 q^{51} + 14 q^{52} + 84 q^{55} + 6 q^{56} + 56 q^{57} - 30 q^{58} - 24 q^{60} + 98 q^{61} - 24 q^{63} + 112 q^{64} - 84 q^{65} + 28 q^{67} + 56 q^{69} - 8 q^{72} - 36 q^{74} - 42 q^{75} - 78 q^{77} + 16 q^{78} - 8 q^{79} - 22 q^{81} + 24 q^{84} + 56 q^{87} + 42 q^{90} - 24 q^{91} + 36 q^{92} - 194 q^{93} + 36 q^{95} - 388 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.930874 − 0.365341i −1.72742 + 0.126511i 0.733052 + 0.680173i −0.121526 + 1.62165i 1.65423 + 0.513334i 0.109858 − 2.64347i −0.433884 − 0.900969i 2.96799 − 0.437075i 0.705578 − 1.46515i
41.2 −0.930874 − 0.365341i −1.70201 − 0.321175i 0.733052 + 0.680173i 0.280353 − 3.74105i 1.46702 + 0.920788i −2.35509 + 1.20563i −0.433884 − 0.900969i 2.79369 + 1.09329i −1.62773 + 3.38002i
41.3 −0.930874 − 0.365341i −1.61964 − 0.613814i 0.733052 + 0.680173i 0.157894 − 2.10695i 1.28343 + 1.16310i 2.64331 − 0.113717i −0.433884 − 0.900969i 2.24646 + 1.98832i −0.916737 + 1.90362i
41.4 −0.930874 − 0.365341i −1.60605 + 0.648546i 0.733052 + 0.680173i −0.0940075 + 1.25444i 1.73197 − 0.0169594i −2.57110 − 0.624050i −0.433884 − 0.900969i 2.15878 − 2.08319i 0.545809 − 1.13338i
41.5 −0.930874 − 0.365341i −1.56662 + 0.738707i 0.733052 + 0.680173i 0.000265561 − 0.00354366i 1.72821 − 0.115290i 1.54355 + 2.14883i −0.433884 − 0.900969i 1.90863 − 2.31455i −0.00154185 + 0.00320168i
41.6 −0.930874 − 0.365341i −1.37774 − 1.04968i 0.733052 + 0.680173i −0.249578 + 3.33038i 0.899017 + 1.48046i −0.0799807 + 2.64454i −0.433884 − 0.900969i 0.796362 + 2.89237i 1.44905 − 3.00898i
41.7 −0.930874 − 0.365341i −1.27448 − 1.17290i 0.733052 + 0.680173i 0.139930 − 1.86724i 0.757873 + 1.55744i −0.987094 − 2.45472i −0.433884 − 0.900969i 0.248608 + 2.98968i −0.812438 + 1.68704i
41.8 −0.930874 − 0.365341i −1.10194 − 1.33631i 0.733052 + 0.680173i −0.0832983 + 1.11154i 0.537563 + 1.64652i 2.61990 + 0.368923i −0.433884 − 0.900969i −0.571436 + 2.94507i 0.483631 − 1.00427i
41.9 −0.930874 − 0.365341i −0.884580 + 1.48913i 0.733052 + 0.680173i 0.0893189 − 1.19188i 1.36747 − 1.06302i 2.22662 + 1.42904i −0.433884 − 0.900969i −1.43504 − 2.63452i −0.518586 + 1.07686i
41.10 −0.930874 − 0.365341i −0.852620 + 1.50766i 0.733052 + 0.680173i −0.303428 + 4.04896i 1.34449 − 1.09194i −2.21880 + 1.44116i −0.433884 − 0.900969i −1.54608 − 2.57092i 1.76170 − 3.65822i
41.11 −0.930874 − 0.365341i −0.830646 − 1.51988i 0.733052 + 0.680173i −0.0709640 + 0.946948i 0.217953 + 1.71828i −2.55055 + 0.703332i −0.433884 − 0.900969i −1.62005 + 2.52496i 0.412018 − 0.855563i
41.12 −0.930874 − 0.365341i −0.616235 + 1.61872i 0.733052 + 0.680173i 0.295819 − 3.94743i 1.16502 − 1.28169i −2.17832 + 1.50165i −0.433884 − 0.900969i −2.24051 − 1.99502i −1.71753 + 3.56648i
41.13 −0.930874 − 0.365341i −0.484260 + 1.66298i 0.733052 + 0.680173i −0.225510 + 3.00922i 1.05834 − 1.37110i 1.50363 − 2.17695i −0.433884 − 0.900969i −2.53099 − 1.61063i 1.30931 − 2.71881i
41.14 −0.930874 − 0.365341i −0.176138 − 1.72307i 0.733052 + 0.680173i 0.171092 − 2.28306i −0.465546 + 1.66831i −0.657295 + 2.56280i −0.433884 − 0.900969i −2.93795 + 0.606998i −0.993360 + 2.06273i
41.15 −0.930874 − 0.365341i 0.0460028 + 1.73144i 0.733052 + 0.680173i 0.0946357 − 1.26283i 0.589743 − 1.62856i −1.07631 − 2.41693i −0.433884 − 0.900969i −2.99577 + 0.159302i −0.549456 + 1.14096i
41.16 −0.930874 − 0.365341i 0.469949 − 1.66708i 0.733052 + 0.680173i 0.0730220 − 0.974410i −1.04652 + 1.38015i 2.36652 − 1.18305i −0.433884 − 0.900969i −2.55830 − 1.56688i −0.423966 + 0.880375i
41.17 −0.930874 − 0.365341i 0.505244 − 1.65672i 0.733052 + 0.680173i 0.100443 − 1.34032i −1.07559 + 1.35761i −2.44546 − 1.00980i −0.433884 − 0.900969i −2.48946 − 1.67410i −0.583172 + 1.21097i
41.18 −0.930874 − 0.365341i 0.684299 + 1.59114i 0.733052 + 0.680173i −0.158151 + 2.11038i −0.0556866 − 1.73116i 1.06320 + 2.42273i −0.433884 − 0.900969i −2.06347 + 2.17764i 0.918227 − 1.90672i
41.19 −0.930874 − 0.365341i 0.896330 + 1.48209i 0.733052 + 0.680173i 0.112988 − 1.50772i −0.292902 − 1.70711i 2.12914 − 1.57059i −0.433884 − 0.900969i −1.39318 + 2.65689i −0.656009 + 1.36222i
41.20 −0.930874 − 0.365341i 1.17951 − 1.26837i 0.733052 + 0.680173i −0.259180 + 3.45852i −1.56136 + 0.749767i −2.33472 + 1.24462i −0.433884 − 0.900969i −0.217514 − 2.99210i 1.50480 − 3.12476i
See next 80 embeddings (of 672 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
49.f odd 14 1 inner
441.bh even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.bj.a ✓ 672
9.d odd 6 1 inner 882.2.bj.a ✓ 672
49.f odd 14 1 inner 882.2.bj.a ✓ 672
441.bh even 42 1 inner 882.2.bj.a ✓ 672
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.bj.a ✓ 672 1.a even 1 1 trivial
882.2.bj.a ✓ 672 9.d odd 6 1 inner
882.2.bj.a ✓ 672 49.f odd 14 1 inner
882.2.bj.a ✓ 672 441.bh even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(882, [\chi])\).