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Properties

Label 726.2.i.c
Level $726$
Weight $2$
Character orbit 726.i
Analytic conductor $5.797$
Analytic rank $0$
Dimension $60$
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] = [726,2,Mod(67,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.i (of order \(11\), degree \(10\), 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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{2} - 60 q^{3} - 6 q^{4} - 2 q^{5} + 6 q^{6} + 5 q^{7} - 6 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{2} - 60 q^{3} - 6 q^{4} - 2 q^{5} + 6 q^{6} + 5 q^{7} - 6 q^{8} + 60 q^{9} - 13 q^{10} - 10 q^{11} + 6 q^{12} - 17 q^{13} - 6 q^{14} + 2 q^{15} - 6 q^{16} + 10 q^{17} - 6 q^{18} - 4 q^{19} - 2 q^{20} - 5 q^{21} - 10 q^{22} - 26 q^{23} + 6 q^{24} - 2 q^{25} + 5 q^{26} - 60 q^{27} + 5 q^{28} - 6 q^{29} + 13 q^{30} - 52 q^{31} - 6 q^{32} + 10 q^{33} + 10 q^{34} + 63 q^{35} - 6 q^{36} - 47 q^{37} + 18 q^{38} + 17 q^{39} - 2 q^{40} + 27 q^{41} + 6 q^{42} + 84 q^{43} + q^{44} - 2 q^{45} + 7 q^{46} + 10 q^{47} + 6 q^{48} - 33 q^{49} - 2 q^{50} - 10 q^{51} + 5 q^{52} + 42 q^{53} + 6 q^{54} + 13 q^{55} - 6 q^{56} + 4 q^{57} + 16 q^{58} - 11 q^{59} + 2 q^{60} - 24 q^{61} + 14 q^{62} + 5 q^{63} - 6 q^{64} - 80 q^{65} + 10 q^{66} - 2 q^{67} - q^{68} + 26 q^{69} - 14 q^{70} - 30 q^{71} - 6 q^{72} - 51 q^{73} + 8 q^{74} + 2 q^{75} - 4 q^{76} + 73 q^{77} - 5 q^{78} - 30 q^{79} - 2 q^{80} + 60 q^{81} - 6 q^{82} + 4 q^{83} - 5 q^{84} + 45 q^{85} - 4 q^{86} + 6 q^{87} + 12 q^{88} - 44 q^{89} - 13 q^{90} - 90 q^{91} + 7 q^{92} + 52 q^{93} - 78 q^{94} - 65 q^{95} + 6 q^{96} - 41 q^{97} + 33 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −0.959493 + 0.281733i −1.00000 0.841254 − 0.540641i −2.56949 − 2.96535i 0.959493 − 0.281733i 1.95930 + 4.29026i −0.654861 + 0.755750i 1.00000 3.30085 + 2.12132i
67.2 −0.959493 + 0.281733i −1.00000 0.841254 − 0.540641i −0.806572 − 0.930834i 0.959493 − 0.281733i −0.184135 − 0.403198i −0.654861 + 0.755750i 1.00000 1.03615 + 0.665891i
67.3 −0.959493 + 0.281733i −1.00000 0.841254 − 0.540641i −0.0837911 − 0.0967001i 0.959493 − 0.281733i −0.653344 − 1.43062i −0.654861 + 0.755750i 1.00000 0.107641 + 0.0691764i
67.4 −0.959493 + 0.281733i −1.00000 0.841254 − 0.540641i 0.727009 + 0.839013i 0.959493 − 0.281733i 0.246193 + 0.539087i −0.654861 + 0.755750i 1.00000 −0.933937 − 0.600205i
67.5 −0.959493 + 0.281733i −1.00000 0.841254 − 0.540641i 1.77550 + 2.04904i 0.959493 − 0.281733i 2.17193 + 4.75586i −0.654861 + 0.755750i 1.00000 −2.28087 − 1.46582i
67.6 −0.959493 + 0.281733i −1.00000 0.841254 − 0.540641i 2.43194 + 2.80660i 0.959493 − 0.281733i −1.66503 − 3.64591i −0.654861 + 0.755750i 1.00000 −3.12414 − 2.00776i
133.1 0.841254 − 0.540641i −1.00000 0.415415 − 0.909632i −0.196232 + 1.36482i −0.841254 + 0.540641i 2.02489 − 2.33685i −0.142315 − 0.989821i 1.00000 0.572797 + 1.25425i
133.2 0.841254 − 0.540641i −1.00000 0.415415 − 0.909632i −0.186845 + 1.29954i −0.841254 + 0.540641i −0.229226 + 0.264540i −0.142315 − 0.989821i 1.00000 0.545399 + 1.19426i
133.3 0.841254 − 0.540641i −1.00000 0.415415 − 0.909632i −0.149872 + 1.04238i −0.841254 + 0.540641i −1.49653 + 1.72708i −0.142315 − 0.989821i 1.00000 0.437474 + 0.957934i
133.4 0.841254 − 0.540641i −1.00000 0.415415 − 0.909632i 0.245189 − 1.70533i −0.841254 + 0.540641i −0.219849 + 0.253720i −0.142315 − 0.989821i 1.00000 −0.715703 − 1.56717i
133.5 0.841254 − 0.540641i −1.00000 0.415415 − 0.909632i 0.349688 − 2.43213i −0.841254 + 0.540641i −3.14342 + 3.62770i −0.142315 − 0.989821i 1.00000 −1.02073 − 2.23510i
133.6 0.841254 − 0.540641i −1.00000 0.415415 − 0.909632i 0.602512 − 4.19056i −0.841254 + 0.540641i 2.06802 − 2.38662i −0.142315 − 0.989821i 1.00000 −1.75872 − 3.85107i
199.1 −0.654861 + 0.755750i −1.00000 −0.142315 − 0.989821i −3.38359 + 2.17450i 0.654861 − 0.755750i 0.0704423 + 0.0206837i 0.841254 + 0.540641i 1.00000 0.572402 − 3.98114i
199.2 −0.654861 + 0.755750i −1.00000 −0.142315 − 0.989821i −0.915370 + 0.588273i 0.654861 − 0.755750i −4.12515 − 1.21125i 0.841254 + 0.540641i 1.00000 0.154853 − 1.07703i
199.3 −0.654861 + 0.755750i −1.00000 −0.142315 − 0.989821i 0.650217 − 0.417869i 0.654861 − 0.755750i 3.42744 + 1.00639i 0.841254 + 0.540641i 1.00000 −0.109997 + 0.765047i
199.4 −0.654861 + 0.755750i −1.00000 −0.142315 − 0.989821i 0.679842 − 0.436908i 0.654861 − 0.755750i −0.703516 − 0.206571i 0.841254 + 0.540641i 1.00000 −0.115009 + 0.799904i
199.5 −0.654861 + 0.755750i −1.00000 −0.142315 − 0.989821i 0.886906 − 0.569980i 0.654861 − 0.755750i −1.75005 − 0.513861i 0.841254 + 0.540641i 1.00000 −0.150038 + 1.04354i
199.6 −0.654861 + 0.755750i −1.00000 −0.142315 − 0.989821i 3.71932 − 2.39026i 0.654861 − 0.755750i 3.27620 + 0.961980i 0.841254 + 0.540641i 1.00000 −0.629197 + 4.37616i
265.1 0.415415 − 0.909632i −1.00000 −0.654861 − 0.755750i −3.97541 − 1.16728i −0.415415 + 0.909632i 0.122923 + 0.854951i −0.959493 + 0.281733i 1.00000 −2.71324 + 3.13125i
265.2 0.415415 − 0.909632i −1.00000 −0.654861 − 0.755750i −3.41436 − 1.00255i −0.415415 + 0.909632i −0.408001 − 2.83771i −0.959493 + 0.281733i 1.00000 −2.33032 + 2.68934i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.i.c ✓ 60
121.e even 11 1 inner 726.2.i.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.i.c ✓ 60 1.a even 1 1 trivial
726.2.i.c ✓ 60 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} + 2 T_{5}^{59} + 18 T_{5}^{58} + 93 T_{5}^{57} + 503 T_{5}^{56} + 1766 T_{5}^{55} + \cdots + 52548001 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display