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Properties

Label 927.2.e.b
Level $927$
Weight $2$
Character orbit 927.e
Analytic conductor $7.402$
Analytic rank $0$
Dimension $102$
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] = [927,2,Mod(310,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(6))
 
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("927.310");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.e (of order \(3\), degree \(2\), 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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(102\)
Relative dimension: \(51\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 102 q + 9 q^{2} - q^{3} - 51 q^{4} + 12 q^{5} - 4 q^{6} - 60 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q + 9 q^{2} - q^{3} - 51 q^{4} + 12 q^{5} - 4 q^{6} - 60 q^{8} - 3 q^{9} + 15 q^{11} + 12 q^{12} + 8 q^{14} + 5 q^{15} - 51 q^{16} - 100 q^{17} - 9 q^{18} - 6 q^{19} + 30 q^{20} + 11 q^{21} + 3 q^{22} + 37 q^{23} + 9 q^{24} - 51 q^{25} - 24 q^{26} - 7 q^{27} + 37 q^{29} + 16 q^{30} + 79 q^{32} - 21 q^{33} - 3 q^{34} - 90 q^{35} + 2 q^{36} + 18 q^{37} + 33 q^{38} + 4 q^{39} - 6 q^{40} + 41 q^{41} - 83 q^{42} + 3 q^{43} - 112 q^{44} + 24 q^{45} - 108 q^{46} + 37 q^{47} + 42 q^{48} - 51 q^{49} + 35 q^{50} - 18 q^{51} + 9 q^{52} - 132 q^{53} + 101 q^{54} + 38 q^{56} + 3 q^{57} + 9 q^{58} + 17 q^{59} - 149 q^{60} - 116 q^{62} + 4 q^{63} + 240 q^{64} + 100 q^{65} + 72 q^{66} + 3 q^{67} + 103 q^{68} - 21 q^{69} - 46 q^{71} + 57 q^{72} - 42 q^{73} + 42 q^{74} + 27 q^{75} + 3 q^{76} + 119 q^{77} - 158 q^{78} - 110 q^{80} + 21 q^{81} + 49 q^{83} + 99 q^{84} - 12 q^{85} + 77 q^{86} - 61 q^{87} + 9 q^{88} - 116 q^{89} + 65 q^{90} + 60 q^{91} + 111 q^{92} - 3 q^{93} + 9 q^{94} + 52 q^{95} - 160 q^{96} - 3 q^{97} - 52 q^{98} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
310.1 −1.35398 − 2.34516i 1.43374 + 0.971797i −2.66652 + 4.61854i 0.902869 − 1.56382i 0.337765 − 4.67814i −1.07932 − 1.86944i 9.02572 1.11122 + 2.78661i −4.88986
310.2 −1.29558 − 2.24402i −1.62972 + 0.586523i −2.35708 + 4.08258i 0.621877 − 1.07712i 3.42761 + 2.89723i 1.24743 + 2.16062i 7.03284 2.31198 − 1.91174i −3.22278
310.3 −1.27631 − 2.21063i −0.0129279 − 1.73200i −2.25792 + 3.91083i −0.663690 + 1.14955i −3.81231 + 2.23915i 1.81017 + 3.13531i 6.42197 −2.99967 + 0.0447825i 3.38829
310.4 −1.09390 − 1.89470i −1.12846 − 1.31399i −1.39325 + 2.41318i −0.243017 + 0.420917i −1.25518 + 3.57548i −0.247173 − 0.428117i 1.72072 −0.453143 + 2.96558i 1.06335
310.5 −1.08773 − 1.88400i −0.226907 + 1.71712i −1.36630 + 2.36650i −1.03475 + 1.79224i 3.48187 − 1.44027i −0.292007 − 0.505772i 1.59373 −2.89703 − 0.779255i 4.50211
310.6 −1.04910 − 1.81710i −1.33202 − 1.10713i −1.20124 + 2.08061i 1.16588 − 2.01936i −0.614340 + 3.58190i 2.17732 + 3.77122i 0.844484 0.548535 + 2.94943i −4.89252
310.7 −1.02963 − 1.78337i −0.884085 + 1.48943i −1.12026 + 1.94035i 2.15639 − 3.73497i 3.56647 + 0.0430939i −2.04086 − 3.53487i 0.495306 −1.43679 − 2.63356i −8.88110
310.8 −0.980253 − 1.69785i 1.71763 + 0.223012i −0.921791 + 1.59659i −0.621421 + 1.07633i −1.30507 − 3.13489i 0.387100 + 0.670477i −0.306659 2.90053 + 0.766107i 2.43660
310.9 −0.971658 − 1.68296i 1.07388 − 1.35896i −0.888238 + 1.53847i 0.599488 − 1.03834i −3.33053 − 0.486850i −1.93724 − 3.35541i −0.434378 −0.693564 − 2.91873i −2.32999
310.10 −0.927863 − 1.60711i 0.595909 − 1.62631i −0.721859 + 1.25030i 1.33943 − 2.31996i −3.16658 + 0.551307i −0.585462 − 1.01405i −1.03231 −2.28979 − 1.93827i −4.97122
310.11 −0.905887 − 1.56904i 1.37587 − 1.05214i −0.641264 + 1.11070i −1.52974 + 2.64958i −2.89723 − 1.20568i 1.68893 + 2.92531i −1.29990 0.786019 − 2.89520i 5.54308
310.12 −0.720799 − 1.24846i 1.14850 + 1.29651i −0.0391030 + 0.0677284i −1.08111 + 1.87254i 0.790801 − 2.36839i −0.402962 − 0.697950i −2.77046 −0.361875 + 2.97809i 3.11706
310.13 −0.660350 − 1.14376i −1.72768 + 0.123017i 0.127876 − 0.221487i 1.21223 − 2.09964i 1.28157 + 1.89481i −0.261459 − 0.452860i −2.97917 2.96973 − 0.425068i −3.20198
310.14 −0.654134 − 1.13299i −1.51652 + 0.836756i 0.144217 − 0.249792i −1.60315 + 2.77674i 1.94005 + 1.17086i −2.41416 − 4.18144i −2.99389 1.59968 − 2.53792i 4.19471
310.15 −0.650452 − 1.12662i 0.478049 − 1.66477i 0.153825 − 0.266433i 0.386155 − 0.668840i −2.18651 + 0.544277i 0.786757 + 1.36270i −3.00203 −2.54294 − 1.59169i −1.00470
310.16 −0.558030 − 0.966537i 1.23124 + 1.21821i 0.377205 − 0.653338i 2.00256 − 3.46854i 0.490375 − 1.86984i −0.888720 − 1.53931i −3.07409 0.0319224 + 2.99983i −4.46996
310.17 −0.441960 − 0.765497i 1.72988 + 0.0867843i 0.609343 − 1.05541i −0.115668 + 0.200343i −0.698103 − 1.36257i 1.49080 + 2.58215i −2.84506 2.98494 + 0.300252i 0.204483
310.18 −0.412533 − 0.714529i −0.0211970 + 1.73192i 0.659632 − 1.14252i −1.94896 + 3.37569i 1.24625 − 0.699330i 1.97903 + 3.42778i −2.73862 −2.99910 − 0.0734230i 3.21604
310.19 −0.381174 − 0.660213i 0.143094 + 1.72613i 0.709412 − 1.22874i 0.288350 − 0.499437i 1.08507 − 0.752429i −0.935249 − 1.61990i −2.60634 −2.95905 + 0.493997i −0.439646
310.20 −0.254701 − 0.441155i −1.42067 + 0.990813i 0.870255 − 1.50733i −0.850572 + 1.47323i 0.798947 + 0.374373i 1.04845 + 1.81596i −1.90542 1.03658 − 2.81523i 0.866566
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 310.51
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.e.b ✓ 102
9.c even 3 1 inner 927.2.e.b ✓ 102
9.c even 3 1 8343.2.a.g 51
9.d odd 6 1 8343.2.a.h 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.e.b ✓ 102 1.a even 1 1 trivial
927.2.e.b ✓ 102 9.c even 3 1 inner
8343.2.a.g 51 9.c even 3 1
8343.2.a.h 51 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{102} - 9 T_{2}^{101} + 117 T_{2}^{100} - 766 T_{2}^{99} + 6093 T_{2}^{98} - 32795 T_{2}^{97} + \cdots + 7891124224 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display