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Properties

Label 726.2.i.a
Level $726$
Weight $2$
Character orbit 726.i
Analytic conductor $5.797$
Analytic rank $0$
Dimension $50$
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: \(50\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 5 q^{2} + 50 q^{3} - 5 q^{4} + 10 q^{5} - 5 q^{6} + 9 q^{7} - 5 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 5 q^{2} + 50 q^{3} - 5 q^{4} + 10 q^{5} - 5 q^{6} + 9 q^{7} - 5 q^{8} + 50 q^{9} - 12 q^{10} - q^{11} - 5 q^{12} - 13 q^{13} - 2 q^{14} + 10 q^{15} - 5 q^{16} + 6 q^{17} - 5 q^{18} + 18 q^{19} + 10 q^{20} + 9 q^{21} - q^{22} + 6 q^{23} - 5 q^{24} + 13 q^{25} + 9 q^{26} + 50 q^{27} + 9 q^{28} + 16 q^{29} - 12 q^{30} - 20 q^{31} - 5 q^{32} - q^{33} + 6 q^{34} - 19 q^{35} - 5 q^{36} - 9 q^{37} - 4 q^{38} - 13 q^{39} - 12 q^{40} + 13 q^{41} - 2 q^{42} - 50 q^{43} - q^{44} + 10 q^{45} - 5 q^{46} + 24 q^{47} - 5 q^{48} + 24 q^{49} + 13 q^{50} + 6 q^{51} + 9 q^{52} - 31 q^{53} - 5 q^{54} + 37 q^{55} - 2 q^{56} + 18 q^{57} - 28 q^{58} - 11 q^{59} + 10 q^{60} + 28 q^{61} + 24 q^{62} + 9 q^{63} - 5 q^{64} - q^{66} - 40 q^{67} + 17 q^{68} + 6 q^{69} + 14 q^{70} - 5 q^{72} + 15 q^{73} + 24 q^{74} + 13 q^{75} - 26 q^{76} - 31 q^{77} + 9 q^{78} + 30 q^{79} + 10 q^{80} + 50 q^{81} + 24 q^{82} + 62 q^{83} + 9 q^{84} - 57 q^{85} + 38 q^{86} + 16 q^{87} - q^{88} + 42 q^{89} - 12 q^{90} + 42 q^{91} - 5 q^{92} - 20 q^{93} - 174 q^{94} + 55 q^{95} - 5 q^{96} - 11 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) \) 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 −1.70382 − 1.96631i −0.959493 + 0.281733i 0.487486 + 1.06745i −0.654861 + 0.755750i 1.00000 2.18878 + 1.40664i
67.2 −0.959493 + 0.281733i 1.00000 0.841254 − 0.540641i 0.308222 + 0.355707i −0.959493 + 0.281733i 0.915456 + 2.00457i −0.654861 + 0.755750i 1.00000 −0.395951 − 0.254462i
67.3 −0.959493 + 0.281733i 1.00000 0.841254 − 0.540641i 0.940143 + 1.08498i −0.959493 + 0.281733i −1.93867 − 4.24509i −0.654861 + 0.755750i 1.00000 −1.20774 − 0.776165i
67.4 −0.959493 + 0.281733i 1.00000 0.841254 − 0.540641i 1.33017 + 1.53510i −0.959493 + 0.281733i −0.0989826 − 0.216742i −0.654861 + 0.755750i 1.00000 −1.70878 − 1.09816i
67.5 −0.959493 + 0.281733i 1.00000 0.841254 − 0.540641i 2.26584 + 2.61491i −0.959493 + 0.281733i 0.847958 + 1.85677i −0.654861 + 0.755750i 1.00000 −2.91076 − 1.87063i
133.1 0.841254 − 0.540641i 1.00000 0.415415 − 0.909632i −0.442082 + 3.07474i 0.841254 − 0.540641i −1.58201 + 1.82573i −0.142315 − 0.989821i 1.00000 1.29043 + 2.82565i
133.2 0.841254 − 0.540641i 1.00000 0.415415 − 0.909632i −0.367165 + 2.55369i 0.841254 − 0.540641i 1.59538 − 1.84117i −0.142315 − 0.989821i 1.00000 1.07175 + 2.34680i
133.3 0.841254 − 0.540641i 1.00000 0.415415 − 0.909632i 0.146857 − 1.02142i 0.841254 − 0.540641i −0.760730 + 0.877929i −0.142315 − 0.989821i 1.00000 −0.428675 − 0.938666i
133.4 0.841254 − 0.540641i 1.00000 0.415415 − 0.909632i 0.227793 − 1.58434i 0.841254 − 0.540641i 1.74494 − 2.01376i −0.142315 − 0.989821i 1.00000 −0.664926 − 1.45598i
133.5 0.841254 − 0.540641i 1.00000 0.415415 − 0.909632i 0.409504 − 2.84816i 0.841254 − 0.540641i 0.625748 − 0.722151i −0.142315 − 0.989821i 1.00000 −1.19534 − 2.61742i
199.1 −0.654861 + 0.755750i 1.00000 −0.142315 − 0.989821i −3.07785 + 1.97801i −0.654861 + 0.755750i −3.79363 − 1.11391i 0.841254 + 0.540641i 1.00000 0.520680 − 3.62141i
199.2 −0.654861 + 0.755750i 1.00000 −0.142315 − 0.989821i −2.18085 + 1.40155i −0.654861 + 0.755750i 1.49844 + 0.439982i 0.841254 + 0.540641i 1.00000 0.368934 − 2.56599i
199.3 −0.654861 + 0.755750i 1.00000 −0.142315 − 0.989821i −1.62366 + 1.04346i −0.654861 + 0.755750i 4.93951 + 1.45037i 0.841254 + 0.540641i 1.00000 0.274674 − 1.91040i
199.4 −0.654861 + 0.755750i 1.00000 −0.142315 − 0.989821i 1.28126 − 0.823415i −0.654861 + 0.755750i 0.788013 + 0.231382i 0.841254 + 0.540641i 1.00000 −0.216751 + 1.50753i
199.5 −0.654861 + 0.755750i 1.00000 −0.142315 − 0.989821i 2.99961 − 1.92773i −0.654861 + 0.755750i 0.601006 + 0.176471i 0.841254 + 0.540641i 1.00000 −0.507443 + 3.52935i
265.1 0.415415 − 0.909632i 1.00000 −0.654861 − 0.755750i −2.11489 − 0.620987i 0.415415 − 0.909632i −0.318340 − 2.21410i −0.959493 + 0.281733i 1.00000 −1.44343 + 1.66580i
265.2 0.415415 − 0.909632i 1.00000 −0.654861 − 0.755750i −1.60283 − 0.470634i 0.415415 − 0.909632i 0.514993 + 3.58186i −0.959493 + 0.281733i 1.00000 −1.09394 + 1.26248i
265.3 0.415415 − 0.909632i 1.00000 −0.654861 − 0.755750i −0.0584334 − 0.0171576i 0.415415 − 0.909632i −0.252323 − 1.75495i −0.959493 + 0.281733i 1.00000 −0.0398812 + 0.0460254i
265.4 0.415415 − 0.909632i 1.00000 −0.654861 − 0.755750i 2.96556 + 0.870768i 0.415415 − 0.909632i 0.576333 + 4.00848i −0.959493 + 0.281733i 1.00000 2.02402 − 2.33584i
265.5 0.415415 − 0.909632i 1.00000 −0.654861 − 0.755750i 3.44495 + 1.01153i 0.415415 − 0.909632i −0.00913382 − 0.0635271i −0.959493 + 0.281733i 1.00000 2.35120 − 2.71343i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.5
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.a ✓ 50
121.e even 11 1 inner 726.2.i.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.i.a ✓ 50 1.a even 1 1 trivial
726.2.i.a ✓ 50 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{50} - 10 T_{5}^{49} + 56 T_{5}^{48} - 204 T_{5}^{47} + 609 T_{5}^{46} - 1988 T_{5}^{45} + \cdots + 37196136769 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display