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Properties

Label 731.2.p.a
Level $731$
Weight $2$
Character orbit 731.p
Analytic conductor $5.837$
Analytic rank $0$
Dimension $384$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [731,2,Mod(16,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.p (of order \(14\), degree \(6\), 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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(64\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9} - 18 q^{13} - 12 q^{15} - 20 q^{16} - 14 q^{17} + 30 q^{18} + 8 q^{19} + 20 q^{21} + 46 q^{25} + 2 q^{26} - 30 q^{30} + 50 q^{32} - 90 q^{33} + 8 q^{34} + 52 q^{35} - 328 q^{36} - 46 q^{38} + 184 q^{42} + 60 q^{43} - 2 q^{47} - 340 q^{49} - 172 q^{50} - 68 q^{51} + 38 q^{52} - 8 q^{53} - 28 q^{55} + 10 q^{59} + 58 q^{60} - 44 q^{64} + 90 q^{66} - 104 q^{67} + 60 q^{68} - 2 q^{69} + 40 q^{70} - 164 q^{72} - 72 q^{76} + 94 q^{77} + 120 q^{81} - 66 q^{83} + 112 q^{84} - 96 q^{85} - 136 q^{86} + 72 q^{87} - 6 q^{89} - 220 q^{93} - 142 q^{94} + 124 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.608972 − 2.66808i −0.149961 − 0.0342275i −4.94587 + 2.38180i −1.75767 + 1.40169i 0.420950i − 1.31934i 5.95414 + 7.46625i −2.68159 − 1.29139i 4.81020 + 3.83601i
16.2 −0.608972 − 2.66808i 0.149961 + 0.0342275i −4.94587 + 2.38180i 1.75767 − 1.40169i − 0.420950i 1.31934i 5.95414 + 7.46625i −2.68159 − 1.29139i −4.81020 − 3.83601i
16.3 −0.594326 − 2.60391i −2.90090 − 0.662112i −4.62520 + 2.22738i −1.09262 + 0.871338i 7.94720i − 2.92823i 5.21824 + 6.54347i 5.27392 + 2.53979i 2.91826 + 2.32724i
16.4 −0.594326 − 2.60391i 2.90090 + 0.662112i −4.62520 + 2.22738i 1.09262 − 0.871338i − 7.94720i 2.92823i 5.21824 + 6.54347i 5.27392 + 2.53979i −2.91826 − 2.32724i
16.5 −0.534495 − 2.34178i −2.00623 − 0.457909i −3.39629 + 1.63557i 1.96143 − 1.56419i 4.94289i − 1.32734i 2.65019 + 3.32324i 1.11237 + 0.535690i −4.71134 − 3.75717i
16.6 −0.534495 − 2.34178i 2.00623 + 0.457909i −3.39629 + 1.63557i −1.96143 + 1.56419i − 4.94289i 1.32734i 2.65019 + 3.32324i 1.11237 + 0.535690i 4.71134 + 3.75717i
16.7 −0.521303 − 2.28398i −1.00960 − 0.230435i −3.14286 + 1.51352i −2.97746 + 2.37445i 2.42604i 3.68726i 2.17391 + 2.72600i −1.73671 − 0.836354i 6.97534 + 5.56265i
16.8 −0.521303 − 2.28398i 1.00960 + 0.230435i −3.14286 + 1.51352i 2.97746 − 2.37445i − 2.42604i − 3.68726i 2.17391 + 2.72600i −1.73671 − 0.836354i −6.97534 − 5.56265i
16.9 −0.501288 − 2.19629i −1.02764 − 0.234553i −2.77045 + 1.33418i 1.62476 − 1.29571i 2.37458i 4.11644i 1.50987 + 1.89332i −1.70187 − 0.819578i −3.66022 − 2.91893i
16.10 −0.501288 − 2.19629i 1.02764 + 0.234553i −2.77045 + 1.33418i −1.62476 + 1.29571i − 2.37458i − 4.11644i 1.50987 + 1.89332i −1.70187 − 0.819578i 3.66022 + 2.91893i
16.11 −0.473651 − 2.07520i −3.20126 − 0.730666i −2.28018 + 1.09808i −0.606394 + 0.483583i 6.98933i 4.01427i 0.704462 + 0.883367i 7.01126 + 3.37644i 1.29075 + 1.02934i
16.12 −0.473651 − 2.07520i 3.20126 + 0.730666i −2.28018 + 1.09808i 0.606394 − 0.483583i − 6.98933i − 4.01427i 0.704462 + 0.883367i 7.01126 + 3.37644i −1.29075 − 1.02934i
16.13 −0.456593 − 2.00046i −2.15206 − 0.491193i −1.99144 + 0.959029i 0.261751 − 0.208739i 4.52939i − 2.19293i 0.269092 + 0.337431i 1.68717 + 0.812500i −0.537089 − 0.428314i
16.14 −0.456593 − 2.00046i 2.15206 + 0.491193i −1.99144 + 0.959029i −0.261751 + 0.208739i − 4.52939i 2.19293i 0.269092 + 0.337431i 1.68717 + 0.812500i 0.537089 + 0.428314i
16.15 −0.359164 − 1.57360i −1.14363 − 0.261026i −0.545279 + 0.262592i −0.961694 + 0.766926i 1.89337i 0.0287002i −1.40365 − 1.76012i −1.46315 − 0.704615i 1.55224 + 1.23787i
16.16 −0.359164 − 1.57360i 1.14363 + 0.261026i −0.545279 + 0.262592i 0.961694 − 0.766926i − 1.89337i − 0.0287002i −1.40365 − 1.76012i −1.46315 − 0.704615i −1.55224 − 1.23787i
16.17 −0.343890 − 1.50668i −2.15737 − 0.492406i −0.349888 + 0.168497i 2.57313 − 2.05201i 3.41980i − 1.34370i −1.55292 − 1.94730i 1.70888 + 0.822954i −3.97659 − 3.17123i
16.18 −0.343890 − 1.50668i 2.15737 + 0.492406i −0.349888 + 0.168497i −2.57313 + 2.05201i − 3.41980i 1.34370i −1.55292 − 1.94730i 1.70888 + 0.822954i 3.97659 + 3.17123i
16.19 −0.332663 − 1.45749i −2.23027 − 0.509045i −0.211677 + 0.101938i −3.41321 + 2.72194i 3.41994i − 3.16066i −1.64521 − 2.06303i 2.01209 + 0.968970i 5.10265 + 4.06923i
16.20 −0.332663 − 1.45749i 2.23027 + 0.509045i −0.211677 + 0.101938i 3.41321 − 2.72194i − 3.41994i 3.16066i −1.64521 − 2.06303i 2.01209 + 0.968970i −5.10265 − 4.06923i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
43.e even 7 1 inner
731.p even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.p.a ✓ 384
17.b even 2 1 inner 731.2.p.a ✓ 384
43.e even 7 1 inner 731.2.p.a ✓ 384
731.p even 14 1 inner 731.2.p.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.p.a ✓ 384 1.a even 1 1 trivial
731.2.p.a ✓ 384 17.b even 2 1 inner
731.2.p.a ✓ 384 43.e even 7 1 inner
731.2.p.a ✓ 384 731.p even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(731, [\chi])\).