Properties

Label 882.2.bk.a
Level $882$
Weight $2$
Character orbit 882.bk
Analytic conductor $7.043$
Analytic rank $0$
Dimension $672$
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] = [882,2,Mod(5,882)]
 
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, 29]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.bk (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.

\(\operatorname{Tr}(f)(q) = \) \( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 672 q + 112 q^{4} + 14 q^{6} - 2 q^{7} + 10 q^{9} - 6 q^{13} + 6 q^{14} + 6 q^{15} - 112 q^{16} + 18 q^{17} + 8 q^{18} + 12 q^{21} + 18 q^{23} + 56 q^{25} - 12 q^{26} + 48 q^{27} + 2 q^{28} - 6 q^{29} + 4 q^{30} - 30 q^{35} - 10 q^{36} - 26 q^{37} + 4 q^{39} + 6 q^{41} + 14 q^{42} + 2 q^{43} + 58 q^{45} + 78 q^{46} - 90 q^{47} + 8 q^{49} + 24 q^{50} + 6 q^{51} - 22 q^{52} + 12 q^{53} - 18 q^{54} - 42 q^{55} - 6 q^{56} - 28 q^{57} + 78 q^{58} - 60 q^{59} - 6 q^{60} + 98 q^{61} - 36 q^{62} - 36 q^{63} + 112 q^{64} - 24 q^{66} + 28 q^{67} - 18 q^{68} + 14 q^{69} + 18 q^{70} + 126 q^{71} - 8 q^{72} - 18 q^{74} - 102 q^{75} + 42 q^{77} - 8 q^{78} - 8 q^{79} + 26 q^{81} + 30 q^{84} + 80 q^{87} - 186 q^{89} - 102 q^{90} + 12 q^{91} - 18 q^{92} - 224 q^{93} - 6 q^{97} + 24 q^{98} + 194 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.781831 + 0.623490i −1.73071 0.0681459i 0.222521 0.974928i 1.83461 + 1.25081i 1.39561 1.02580i 0.211960 2.63725i 0.433884 + 0.900969i 2.99071 + 0.235881i −2.21422 + 0.165933i
5.2 −0.781831 + 0.623490i −1.69372 + 0.362376i 0.222521 0.974928i −1.25279 0.854138i 1.09826 1.33933i 2.59469 0.517268i 0.433884 + 0.900969i 2.73737 1.22753i 1.51202 0.113310i
5.3 −0.781831 + 0.623490i −1.65319 + 0.516682i 0.222521 0.974928i −2.27516 1.55118i 0.970370 1.43471i −1.77406 1.96283i 0.433884 + 0.900969i 2.46608 1.70835i 2.74593 0.205779i
5.4 −0.781831 + 0.623490i −1.63652 + 0.567289i 0.222521 0.974928i 3.21299 + 2.19058i 0.925780 1.46388i −2.29563 + 1.31532i 0.433884 + 0.900969i 2.35637 1.85675i −3.87782 + 0.290602i
5.5 −0.781831 + 0.623490i −1.54555 0.781835i 0.222521 0.974928i 2.77812 + 1.89409i 1.69583 0.352373i 2.33331 + 1.24726i 0.433884 + 0.900969i 1.77747 + 2.41674i −3.35297 + 0.251270i
5.6 −0.781831 + 0.623490i −1.52352 0.823950i 0.222521 0.974928i −3.67482 2.50545i 1.70486 0.305708i −0.560178 + 2.58577i 0.433884 + 0.900969i 1.64221 + 2.51060i 4.43521 0.332373i
5.7 −0.781831 + 0.623490i −1.34273 + 1.09411i 0.222521 0.974928i 0.806214 + 0.549667i 0.367626 1.69259i 0.824300 + 2.51407i 0.433884 + 0.900969i 0.605860 2.93819i −0.973035 + 0.0729189i
5.8 −0.781831 + 0.623490i −1.23334 1.21609i 0.222521 0.974928i −0.255877 0.174454i 1.72248 + 0.181809i −2.52813 0.780116i 0.433884 + 0.900969i 0.0422313 + 2.99970i 0.308823 0.0231431i
5.9 −0.781831 + 0.623490i −1.17019 + 1.27697i 0.222521 0.974928i −0.474915 0.323792i 0.118716 1.72798i −2.64285 0.123929i 0.433884 + 0.900969i −0.261299 2.98860i 0.573185 0.0429542i
5.10 −0.781831 + 0.623490i −0.724077 + 1.57344i 0.222521 0.974928i 2.44331 + 1.66582i −0.414917 1.68162i 0.419163 2.61234i 0.433884 + 0.900969i −1.95142 2.27858i −2.94888 + 0.220988i
5.11 −0.781831 + 0.623490i −0.569639 1.63570i 0.222521 0.974928i 2.43032 + 1.65696i 1.46520 + 0.923677i −1.41335 2.23661i 0.433884 + 0.900969i −2.35102 + 1.86352i −2.93320 + 0.219813i
5.12 −0.781831 + 0.623490i −0.488019 + 1.66188i 0.222521 0.974928i 0.689539 + 0.470120i −0.654615 1.60358i 2.62337 0.343387i 0.433884 + 0.900969i −2.52368 1.62205i −0.832218 + 0.0623661i
5.13 −0.781831 + 0.623490i −0.485664 1.66257i 0.222521 0.974928i 0.154754 + 0.105510i 1.41630 + 0.997041i 0.0786204 + 2.64458i 0.433884 + 0.900969i −2.52826 + 1.61490i −0.186776 + 0.0139969i
5.14 −0.781831 + 0.623490i −0.310768 1.70394i 0.222521 0.974928i −0.786622 0.536309i 1.30536 + 1.13844i 2.58652 0.556718i 0.433884 + 0.900969i −2.80685 + 1.05906i 0.949389 0.0711469i
5.15 −0.781831 + 0.623490i −0.240845 + 1.71522i 0.222521 0.974928i −2.31965 1.58151i −0.881125 1.49118i 2.04849 + 1.67442i 0.433884 + 0.900969i −2.88399 0.826206i 2.79963 0.209803i
5.16 −0.781831 + 0.623490i 0.227043 1.71711i 0.222521 0.974928i −3.47346 2.36816i 0.893088 + 1.48405i 0.805676 2.52010i 0.433884 + 0.900969i −2.89690 0.779714i 4.19218 0.314161i
5.17 −0.781831 + 0.623490i 0.307117 + 1.70461i 0.222521 0.974928i 1.25571 + 0.856126i −1.30292 1.14123i −0.752068 + 2.53661i 0.433884 + 0.900969i −2.81136 + 1.04703i −1.51554 + 0.113574i
5.18 −0.781831 + 0.623490i 0.366631 + 1.69280i 0.222521 0.974928i −2.85326 1.94532i −1.34209 1.09490i −2.64286 0.123709i 0.433884 + 0.900969i −2.73116 + 1.24127i 3.44365 0.258066i
5.19 −0.781831 + 0.623490i 0.760180 1.55632i 0.222521 0.974928i −1.95730 1.33446i 0.376016 + 1.69074i −1.48234 + 2.19150i 0.433884 + 0.900969i −1.84425 2.36616i 2.36230 0.177030i
5.20 −0.781831 + 0.623490i 0.978483 1.42919i 0.222521 0.974928i 2.35227 + 1.60375i 0.126073 + 1.72746i 2.64481 0.0707346i 0.433884 + 0.900969i −1.08514 2.79687i −2.83900 + 0.212754i
See next 80 embeddings (of 672 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.bn 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.bk.a yes 672
9.d odd 6 1 882.2.bc.a 672
49.h odd 42 1 882.2.bc.a 672
441.bn even 42 1 inner 882.2.bk.a yes 672
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.bc.a 672 9.d odd 6 1
882.2.bc.a 672 49.h odd 42 1
882.2.bk.a yes 672 1.a even 1 1 trivial
882.2.bk.a yes 672 441.bn even 42 1 inner

Hecke kernels

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