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Properties

Label 927.2.e.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 11 q^{2} - q^{3} - 51 q^{4} - 12 q^{5} - 4 q^{6} + 60 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q - 11 q^{2} - q^{3} - 51 q^{4} - 12 q^{5} - 4 q^{6} + 60 q^{8} - 3 q^{9} - 13 q^{11} + 12 q^{12} - 16 q^{14} + 5 q^{15} - 51 q^{16} + 92 q^{17} - 9 q^{18} - 6 q^{19} - 30 q^{20} + 11 q^{21} + 3 q^{22} - 35 q^{23} + 9 q^{24} - 51 q^{25} + 56 q^{26} - 13 q^{27} - 43 q^{29} - 32 q^{30} - 61 q^{32} + 19 q^{33} - 3 q^{34} + 70 q^{35} - 42 q^{36} - 18 q^{37} - 31 q^{38} - 30 q^{39} - 6 q^{40} - 47 q^{41} + 101 q^{42} + 3 q^{43} + 56 q^{44} - 20 q^{45} + 132 q^{46} - 27 q^{47} - 12 q^{48} - 51 q^{49} - 65 q^{50} + 42 q^{51} + 9 q^{52} + 140 q^{53} - 33 q^{54} - 34 q^{56} - 33 q^{57} + 9 q^{58} - 31 q^{59} + 155 q^{60} + 108 q^{62} - 36 q^{63} - 48 q^{64} - 84 q^{65} - 56 q^{66} + 3 q^{67} - 89 q^{68} + 79 q^{69} + 66 q^{71} - 67 q^{72} + 6 q^{73} - 58 q^{74} - 47 q^{75} + 3 q^{76} - 57 q^{77} + 90 q^{78} + 154 q^{80} - 11 q^{81} - 63 q^{83} - 143 q^{84} - 12 q^{85} - 7 q^{86} + 47 q^{87} + 9 q^{88} + 100 q^{89} - 121 q^{90} - 60 q^{91} - 105 q^{92} - 35 q^{93} + 9 q^{94} - 68 q^{95} + 164 q^{96} - 3 q^{97} + 228 q^{98} - 18 q^{99}+O(q^{100}) \) 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 −1.35907 − 2.35397i −0.622359 − 1.61638i −2.69413 + 4.66636i 1.84259 − 3.19147i −2.95908 + 3.66178i −1.06278 − 1.84078i 9.20972 −2.22534 + 2.01193i −10.0168
310.2 −1.35348 − 2.34430i 1.25353 − 1.19527i −2.66383 + 4.61389i −0.349170 + 0.604780i −4.49870 − 1.32087i 0.108442 + 0.187827i 9.00785 0.142662 − 2.99661i 1.89038
310.3 −1.33383 − 2.31026i −1.64631 − 0.538189i −2.55821 + 4.43095i −1.90070 + 3.29211i 0.952547 + 4.52128i −0.0568686 − 0.0984994i 8.31357 2.42070 + 1.77206i 10.1409
310.4 −1.32546 − 2.29577i −0.352722 + 1.69576i −2.51371 + 4.35388i −0.985377 + 1.70672i 4.36059 − 1.43790i −0.677991 − 1.17432i 8.02549 −2.75117 − 1.19626i 5.22433
310.5 −1.31443 − 2.27665i 1.28320 + 1.16336i −2.45544 + 4.25294i −1.49491 + 2.58926i 0.961892 − 4.45055i 2.59980 + 4.50298i 7.65226 0.293198 + 2.98564i 7.85980
310.6 −1.22205 − 2.11666i −1.72041 + 0.200498i −1.98684 + 3.44130i 0.378402 − 0.655411i 2.52682 + 3.39650i −2.10958 − 3.65390i 4.82387 2.91960 − 0.689876i −1.84971
310.7 −1.20586 − 2.08861i 1.73108 − 0.0579992i −1.90818 + 3.30507i 1.31723 − 2.28151i −2.20857 − 3.54560i −0.0784380 − 0.135859i 4.38056 2.99327 − 0.200803i −6.35357
310.8 −1.17229 − 2.03046i −0.0878207 − 1.72982i −1.74852 + 3.02852i −1.56640 + 2.71309i −3.40939 + 2.20617i −2.39362 − 4.14587i 3.50990 −2.98458 + 0.303829i 7.34510
310.9 −1.08749 − 1.88360i 0.968274 + 1.43612i −1.36529 + 2.36475i −0.189895 + 0.328907i 1.65208 − 3.38561i −2.33588 − 4.04586i 1.58899 −1.12489 + 2.78112i 0.826037
310.10 −1.08004 − 1.87069i 1.45543 − 0.939001i −1.33298 + 2.30879i 2.05100 − 3.55243i −3.32851 − 1.70849i 2.50952 + 4.34661i 1.43854 1.23655 − 2.73330i −8.86065
310.11 −1.03531 − 1.79321i 0.916445 + 1.46974i −1.14373 + 1.98100i 0.908894 − 1.57425i 1.68674 − 3.16501i 1.53230 + 2.65402i 0.595216 −1.32026 + 2.69387i −3.76395
310.12 −0.973232 − 1.68569i −1.01396 + 1.40424i −0.894360 + 1.54908i 0.926601 − 1.60492i 3.35392 + 0.342568i 0.754294 + 1.30648i −0.411247 −0.943774 − 2.84768i −3.60719
310.13 −0.944192 − 1.63539i −1.65788 − 0.501420i −0.782997 + 1.35619i −1.68661 + 2.92130i 0.745343 + 3.18472i 1.09782 + 1.90149i −0.819571 2.49716 + 1.66259i 6.36994
310.14 −0.910960 − 1.57783i −1.32336 + 1.11745i −0.659696 + 1.14263i −1.19574 + 2.07107i 2.96868 + 1.07009i 0.0682855 + 0.118274i −1.24001 0.502589 − 2.95760i 4.35707
310.15 −0.803406 − 1.39154i 0.846370 − 1.51118i −0.290922 + 0.503892i −1.97534 + 3.42139i −2.78284 + 0.0363311i −0.236068 − 0.408882i −2.27871 −1.56731 − 2.55803i 6.34801
310.16 −0.729184 − 1.26298i 0.0311748 + 1.73177i −0.0634185 + 0.109844i 0.633933 − 1.09800i 2.16447 − 1.30215i 1.61201 + 2.79209i −2.73176 −2.99806 + 0.107975i −1.84901
310.17 −0.711781 − 1.23284i 1.69159 − 0.372174i −0.0132649 + 0.0229755i 0.368626 − 0.638479i −1.66288 − 1.82056i −2.12689 − 3.68388i −2.80936 2.72297 − 1.25913i −1.04952
310.18 −0.711075 − 1.23162i −1.48089 − 0.898317i −0.0112553 + 0.0194948i 1.46579 − 2.53882i −0.0533610 + 2.46266i −0.743481 − 1.28775i −2.81229 1.38605 + 2.66061i −4.16915
310.19 −0.588221 − 1.01883i −0.840497 − 1.51445i 0.307993 − 0.533460i −0.676362 + 1.17149i −1.04857 + 1.74715i −1.09673 − 1.89959i −3.07755 −1.58713 + 2.54579i 1.59140
310.20 −0.551388 − 0.955031i −0.318207 − 1.70257i 0.391943 − 0.678866i −0.483563 + 0.837555i −1.45055 + 1.24267i 1.61081 + 2.79001i −3.07000 −2.79749 + 1.08354i 1.06652
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.a ✓ 102
9.c even 3 1 inner 927.2.e.a ✓ 102
9.c even 3 1 8343.2.a.i 51
9.d odd 6 1 8343.2.a.f 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.e.a ✓ 102 1.a even 1 1 trivial
927.2.e.a ✓ 102 9.c even 3 1 inner
8343.2.a.f 51 9.d odd 6 1
8343.2.a.i 51 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{102} + 11 T_{2}^{101} + 137 T_{2}^{100} + 1014 T_{2}^{99} + 7813 T_{2}^{98} + 45949 T_{2}^{97} + \cdots + 186624 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display