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Properties

Label 924.2.bu.b
Level $924$
Weight $2$
Character orbit 924.bu
Analytic conductor $7.378$
Analytic rank $0$
Dimension $192$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [924,2,Mod(223,924)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("924.223"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(924, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 0, 5, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 924.bu (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192,2,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.37817714677\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 2 q^{2} + 48 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{8} - 48 q^{9} + 2 q^{12} + 13 q^{14} + 6 q^{16} - 3 q^{18} + 6 q^{20} + 24 q^{22} + 3 q^{24} + 56 q^{25} - 18 q^{26} + 48 q^{27} + 16 q^{29} + 12 q^{31}+ \cdots + 50 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 −1.41179 − 0.0828155i −0.309017 − 0.951057i 1.98628 + 0.233836i 1.42882 − 1.96660i 0.357504 + 1.36828i −0.147880 − 2.64162i −2.78484 − 0.494621i −0.809017 + 0.587785i −2.18005 + 2.65809i
223.2 −1.40373 + 0.171848i −0.309017 − 0.951057i 1.94094 − 0.482458i 1.31503 − 1.80998i 0.597215 + 1.28193i −0.260969 + 2.63285i −2.64165 + 1.01079i −0.809017 + 0.587785i −1.53491 + 2.76672i
223.3 −1.39300 − 0.244004i −0.309017 − 0.951057i 1.88092 + 0.679798i −2.37164 + 3.26428i 0.198400 + 1.40023i 1.26943 − 2.32132i −2.45426 − 1.40591i −0.809017 + 0.587785i 4.10020 − 3.96846i
223.4 −1.37326 − 0.337888i −0.309017 − 0.951057i 1.77166 + 0.928015i 0.104179 − 0.143390i 0.103008 + 1.41046i 2.64485 − 0.0688624i −2.11938 − 1.87303i −0.809017 + 0.587785i −0.191514 + 0.161710i
223.5 −1.35559 + 0.402946i −0.309017 − 0.951057i 1.67527 − 1.09246i 0.147952 − 0.203638i 0.802126 + 1.16473i 2.30140 + 1.30521i −1.83078 + 2.15598i −0.809017 + 0.587785i −0.118507 + 0.335667i
223.6 −1.33390 − 0.469785i −0.309017 − 0.951057i 1.55860 + 1.25330i 0.181677 − 0.250056i −0.0345930 + 1.41379i −1.96680 + 1.76966i −1.49025 − 2.40399i −0.809017 + 0.587785i −0.359812 + 0.248202i
223.7 −1.28527 − 0.589979i −0.309017 − 0.951057i 1.30385 + 1.51657i −1.77459 + 2.44251i −0.163932 + 1.40468i −2.24812 + 1.39497i −0.781060 − 2.71845i −0.809017 + 0.587785i 3.72185 − 2.09232i
223.8 −1.20316 + 0.743245i −0.309017 − 0.951057i 0.895173 − 1.78848i 0.698827 − 0.961853i 1.07866 + 0.914595i 0.350754 − 2.62240i 0.252248 + 2.81716i −0.809017 + 0.587785i −0.125906 + 1.67666i
223.9 −1.17589 + 0.785678i −0.309017 − 0.951057i 0.765420 − 1.84774i 2.25375 − 3.10202i 1.11059 + 0.875547i −2.63609 + 0.225872i 0.551680 + 2.77410i −0.809017 + 0.587785i −0.212965 + 5.41835i
223.10 −1.16658 − 0.799429i −0.309017 − 0.951057i 0.721826 + 1.86520i 0.710256 − 0.977584i −0.399809 + 1.35652i −1.93475 − 1.80465i 0.649026 − 2.75296i −0.809017 + 0.587785i −1.61008 + 0.572632i
223.11 −1.11059 + 0.875547i −0.309017 − 0.951057i 0.466835 − 1.94475i −2.25375 + 3.10202i 1.17589 + 0.785678i 1.99988 + 1.73219i 1.18426 + 2.56857i −0.809017 + 0.587785i −0.212965 − 5.41835i
223.12 −1.07866 + 0.914595i −0.309017 − 0.951057i 0.327033 − 1.97308i −0.698827 + 0.961853i 1.20316 + 0.743245i 1.25764 − 2.32773i 1.45181 + 2.42740i −0.809017 + 0.587785i −0.125906 − 1.67666i
223.13 −1.00628 − 0.993684i −0.309017 − 0.951057i 0.0251849 + 1.99984i 2.30531 − 3.17299i −0.634093 + 1.26409i 1.04848 + 2.42913i 1.96187 − 2.03742i −0.809017 + 0.587785i −5.47273 + 0.902154i
223.14 −0.844478 − 1.13440i −0.309017 − 0.951057i −0.573713 + 1.91595i −1.05544 + 1.45268i −0.817918 + 1.15369i 1.87322 + 1.86843i 2.65793 − 0.967156i −0.809017 + 0.587785i 2.53921 − 0.0294755i
223.15 −0.802126 + 1.16473i −0.309017 − 0.951057i −0.713188 − 1.86852i −0.147952 + 0.203638i 1.35559 + 0.402946i −2.62905 − 0.296794i 2.74839 + 0.668117i −0.809017 + 0.587785i −0.118507 − 0.335667i
223.16 −0.636345 − 1.26296i −0.309017 − 0.951057i −1.19013 + 1.60736i 0.381488 − 0.525073i −1.00450 + 0.995476i 2.39317 − 1.12816i 2.78736 + 0.480250i −0.809017 + 0.587785i −0.905904 − 0.147676i
223.17 −0.597215 + 1.28193i −0.309017 − 0.951057i −1.28667 − 1.53117i −1.31503 + 1.80998i 1.40373 + 0.171848i −1.33642 + 2.28341i 2.73127 − 0.734977i −0.809017 + 0.587785i −1.53491 − 2.76672i
223.18 −0.529112 − 1.31150i −0.309017 − 0.951057i −1.44008 + 1.38786i −1.72588 + 2.37547i −1.08381 + 0.908492i −2.24077 − 1.40675i 2.58215 + 1.15433i −0.809017 + 0.587785i 4.02862 + 1.00661i
223.19 −0.479623 − 1.33040i −0.309017 − 0.951057i −1.53992 + 1.27618i 0.515967 − 0.710168i −1.11707 + 0.867264i −1.77909 + 1.95828i 2.43641 + 1.43663i −0.809017 + 0.587785i −1.19228 − 0.345829i
223.20 −0.357504 + 1.36828i −0.309017 − 0.951057i −1.74438 − 0.978331i −1.42882 + 1.96660i 1.41179 − 0.0828155i 1.67234 − 2.05019i 1.96225 − 2.03705i −0.809017 + 0.587785i −2.18005 − 2.65809i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
28.d even 2 1 inner
308.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 924.2.bu.b yes 192
4.b odd 2 1 924.2.bu.a ✓ 192
7.b odd 2 1 924.2.bu.a ✓ 192
11.c even 5 1 inner 924.2.bu.b yes 192
28.d even 2 1 inner 924.2.bu.b yes 192
44.h odd 10 1 924.2.bu.a ✓ 192
77.j odd 10 1 924.2.bu.a ✓ 192
308.t even 10 1 inner 924.2.bu.b yes 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
924.2.bu.a ✓ 192 4.b odd 2 1
924.2.bu.a ✓ 192 7.b odd 2 1
924.2.bu.a ✓ 192 44.h odd 10 1
924.2.bu.a ✓ 192 77.j odd 10 1
924.2.bu.b yes 192 1.a even 1 1 trivial
924.2.bu.b yes 192 11.c even 5 1 inner
924.2.bu.b yes 192 28.d even 2 1 inner
924.2.bu.b yes 192 308.t even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{96} + 251 T_{19}^{94} - 228 T_{19}^{93} + 36138 T_{19}^{92} - 54636 T_{19}^{91} + \cdots + 12\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(924, [\chi])\). Copy content Toggle raw display