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Properties

Label 946.2.f.h
Level $946$
Weight $2$
Character orbit 946.f
Analytic conductor $7.554$
Analytic rank $0$
Dimension $40$
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] = [946,2,Mod(345,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.345");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.f (of order \(5\), degree \(4\), 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: \(7.55384803121\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{2} - q^{3} - 10 q^{4} - 5 q^{5} + q^{6} + q^{7} + 10 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{2} - q^{3} - 10 q^{4} - 5 q^{5} + q^{6} + q^{7} + 10 q^{8} - 11 q^{9} - 5 q^{11} + 4 q^{12} + 5 q^{13} - q^{14} - 11 q^{15} - 10 q^{16} + 3 q^{17} + 11 q^{18} + 3 q^{19} - 5 q^{20} - 44 q^{21} + 22 q^{23} + q^{24} - 9 q^{25} - 20 q^{26} - 16 q^{27} + 11 q^{28} - 5 q^{29} + 11 q^{30} - 17 q^{31} - 40 q^{32} + 56 q^{33} + 42 q^{34} + 12 q^{35} - 11 q^{36} + 29 q^{37} - 13 q^{38} - 34 q^{39} - 5 q^{40} - 12 q^{41} - 26 q^{42} - 40 q^{43} + 10 q^{44} - 8 q^{45} - 2 q^{46} - 23 q^{47} - q^{48} - 3 q^{49} + 19 q^{50} - 51 q^{51} + 20 q^{52} - 8 q^{53} - 64 q^{54} + 11 q^{55} + 24 q^{56} + 30 q^{57} + 5 q^{58} + 26 q^{59} + 4 q^{60} - 19 q^{61} - 23 q^{62} + 13 q^{63} - 10 q^{64} - 12 q^{65} - 6 q^{66} + 10 q^{67} + 3 q^{68} + 87 q^{69} + 8 q^{70} + 23 q^{71} + 11 q^{72} - 56 q^{73} - 29 q^{74} - 90 q^{75} - 32 q^{76} + 13 q^{77} + 94 q^{78} + 47 q^{79} + 5 q^{80} - 66 q^{81} + 2 q^{82} - 34 q^{83} - 4 q^{84} - 7 q^{85} - 10 q^{86} + 24 q^{87} + 5 q^{88} + 60 q^{89} - 77 q^{90} - 28 q^{91} - 13 q^{92} + 12 q^{93} - 17 q^{94} - 34 q^{95} + q^{96} + q^{97} - 12 q^{98} + 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
345.1 0.809017 + 0.587785i −0.894288 + 2.75234i 0.309017 + 0.951057i 0.579041 − 0.420698i −2.34128 + 1.70104i 0.182797 + 0.562591i −0.309017 + 0.951057i −4.34855 − 3.15941i 0.715734
345.2 0.809017 + 0.587785i −0.770441 + 2.37117i 0.309017 + 0.951057i 1.63163 − 1.18545i −2.01704 + 1.46546i 0.508794 + 1.56591i −0.309017 + 0.951057i −2.60183 − 1.89034i 2.01681
345.3 0.809017 + 0.587785i −0.414607 + 1.27603i 0.309017 + 0.951057i −3.41058 + 2.47793i −1.08545 + 0.788629i −0.257277 − 0.791818i −0.309017 + 0.951057i 0.970700 + 0.705255i −4.21571
345.4 0.809017 + 0.587785i −0.330423 + 1.01694i 0.309017 + 0.951057i 1.56972 − 1.14047i −0.865060 + 0.628503i −0.0850874 − 0.261872i −0.309017 + 0.951057i 1.50207 + 1.09132i 1.94029
345.5 0.809017 + 0.587785i −0.0600300 + 0.184753i 0.309017 + 0.951057i −1.70424 + 1.23820i −0.157161 + 0.114184i −1.47139 − 4.52846i −0.309017 + 0.951057i 2.39652 + 1.74117i −2.10656
345.6 0.809017 + 0.587785i 0.0127882 − 0.0393581i 0.309017 + 0.951057i −1.38092 + 1.00330i 0.0334800 − 0.0243247i 1.20412 + 3.70589i −0.309017 + 0.951057i 2.42567 + 1.76235i −1.70691
345.7 0.809017 + 0.587785i 0.230972 − 0.710858i 0.309017 + 0.951057i 1.43419 − 1.04200i 0.604692 − 0.439334i −1.36949 − 4.21486i −0.309017 + 0.951057i 1.97508 + 1.43498i 1.77275
345.8 0.809017 + 0.587785i 0.716858 − 2.20626i 0.309017 + 0.951057i −2.37704 + 1.72702i 1.87676 − 1.36355i −0.0358425 − 0.110312i −0.309017 + 0.951057i −1.92666 − 1.39980i −2.93818
345.9 0.809017 + 0.587785i 0.817338 − 2.51551i 0.309017 + 0.951057i −1.52797 + 1.11014i 2.13982 − 1.55467i −1.03935 − 3.19878i −0.309017 + 0.951057i −3.23269 − 2.34869i −1.88868
345.10 0.809017 + 0.587785i 1.00085 − 3.08030i 0.309017 + 0.951057i 3.37715 − 2.45365i 2.62026 − 1.90373i −0.182364 − 0.561259i −0.309017 + 0.951057i −6.05949 − 4.40248i 4.17439
603.1 −0.309017 + 0.951057i −2.74566 + 1.99484i −0.809017 − 0.587785i 0.251402 + 0.773735i −1.04875 − 3.22772i 3.62743 + 2.63548i 0.809017 − 0.587785i 2.63221 − 8.10112i −0.813553
603.2 −0.309017 + 0.951057i −2.36025 + 1.71482i −0.809017 − 0.587785i 0.311819 + 0.959681i −0.901533 − 2.77463i −2.26259 − 1.64387i 0.809017 − 0.587785i 1.70310 − 5.24161i −1.00907
603.3 −0.309017 + 0.951057i −1.14961 + 0.835240i −0.809017 − 0.587785i −0.111631 − 0.343565i −0.439112 − 1.35145i −1.39326 − 1.01226i 0.809017 − 0.587785i −0.303076 + 0.932771i 0.361245
603.4 −0.309017 + 0.951057i −0.676651 + 0.491615i −0.809017 − 0.587785i 1.14669 + 3.52914i −0.258458 − 0.795450i 1.25750 + 0.913628i 0.809017 − 0.587785i −0.710881 + 2.18787i −3.71076
603.5 −0.309017 + 0.951057i −0.200599 + 0.145744i −0.809017 − 0.587785i −0.994520 − 3.06082i −0.0766221 − 0.235819i 0.913989 + 0.664052i 0.809017 − 0.587785i −0.908052 + 2.79470i 3.21833
603.6 −0.309017 + 0.951057i −0.0862097 + 0.0626350i −0.809017 − 0.587785i −0.737598 − 2.27009i −0.0329292 − 0.101346i −2.67212 − 1.94141i 0.809017 − 0.587785i −0.923542 + 2.84237i 2.38692
603.7 −0.309017 + 0.951057i 0.707602 − 0.514103i −0.809017 − 0.587785i −1.18479 − 3.64640i 0.270280 + 0.831837i 3.11061 + 2.25999i 0.809017 − 0.587785i −0.690652 + 2.12561i 3.83405
603.8 −0.309017 + 0.951057i 1.31730 − 0.957072i −0.809017 − 0.587785i 0.742785 + 2.28606i 0.503162 + 1.54857i 2.13225 + 1.54917i 0.809017 − 0.587785i −0.107768 + 0.331676i −2.40370
603.9 −0.309017 + 0.951057i 2.03763 − 1.48043i −0.809017 − 0.587785i −0.733937 − 2.25882i 0.778306 + 2.39538i 0.258520 + 0.187826i 0.809017 − 0.587785i 1.03323 − 3.17996i 2.37507
603.10 −0.309017 + 0.951057i 2.34742 − 1.70550i −0.809017 − 0.587785i 0.618796 + 1.90446i 0.896637 + 2.75956i −1.92725 − 1.40023i 0.809017 − 0.587785i 1.67461 − 5.15392i −2.00247
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 345.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.f.h ✓ 40
11.c even 5 1 inner 946.2.f.h ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.f.h ✓ 40 1.a even 1 1 trivial
946.2.f.h ✓ 40 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + T_{3}^{39} + 21 T_{3}^{38} + 28 T_{3}^{37} + 314 T_{3}^{36} + 215 T_{3}^{35} + 3852 T_{3}^{34} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display