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Properties

Label 760.2.p.i
Level $760$
Weight $2$
Character orbit 760.p
Analytic conductor $6.069$
Analytic rank $0$
Dimension $56$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(379,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.p (of order \(2\), degree \(1\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 16 q^{4} - 8 q^{6} + 88 q^{9} + 32 q^{11} + 48 q^{16} - 56 q^{19} + 4 q^{20} - 32 q^{24} + 80 q^{25} + 24 q^{26} + 24 q^{30} + 48 q^{35} - 96 q^{36} + 104 q^{44} - 72 q^{49} + 104 q^{54} + 16 q^{64}+ \cdots - 272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1 −1.39575 − 0.227746i 1.92999 1.89626 + 0.635755i −0.518150 − 2.17521i −2.69379 − 0.439547i −3.45334 −2.50193 − 1.31922i 0.724849 0.227817 + 3.15406i
379.2 −1.39575 − 0.227746i 1.92999 1.89626 + 0.635755i 0.518150 − 2.17521i −2.69379 − 0.439547i 3.45334 −2.50193 − 1.31922i 0.724849 −1.21861 + 2.91805i
379.3 −1.39575 + 0.227746i 1.92999 1.89626 − 0.635755i −0.518150 + 2.17521i −2.69379 + 0.439547i −3.45334 −2.50193 + 1.31922i 0.724849 0.227817 − 3.15406i
379.4 −1.39575 + 0.227746i 1.92999 1.89626 − 0.635755i 0.518150 + 2.17521i −2.69379 + 0.439547i 3.45334 −2.50193 + 1.31922i 0.724849 −1.21861 − 2.91805i
379.5 −1.39359 − 0.240666i −0.258346 1.88416 + 0.670777i −1.58207 + 1.58020i 0.360027 + 0.0621750i 1.26592 −2.46430 − 1.38824i −2.93326 2.58506 − 1.82140i
379.6 −1.39359 − 0.240666i −0.258346 1.88416 + 0.670777i 1.58207 + 1.58020i 0.360027 + 0.0621750i −1.26592 −2.46430 − 1.38824i −2.93326 −1.82445 − 2.58290i
379.7 −1.39359 + 0.240666i −0.258346 1.88416 − 0.670777i −1.58207 − 1.58020i 0.360027 − 0.0621750i 1.26592 −2.46430 + 1.38824i −2.93326 2.58506 + 1.82140i
379.8 −1.39359 + 0.240666i −0.258346 1.88416 − 0.670777i 1.58207 − 1.58020i 0.360027 − 0.0621750i −1.26592 −2.46430 + 1.38824i −2.93326 −1.82445 + 2.58290i
379.9 −1.31112 − 0.530067i −2.23972 1.43806 + 1.38996i −1.59332 − 1.56886i 2.93653 + 1.18720i 2.42178 −1.14869 − 2.58467i 2.01633 1.25743 + 2.90153i
379.10 −1.31112 − 0.530067i −2.23972 1.43806 + 1.38996i 1.59332 − 1.56886i 2.93653 + 1.18720i −2.42178 −1.14869 − 2.58467i 2.01633 −2.92063 + 1.21239i
379.11 −1.31112 + 0.530067i −2.23972 1.43806 − 1.38996i −1.59332 + 1.56886i 2.93653 − 1.18720i 2.42178 −1.14869 + 2.58467i 2.01633 1.25743 − 2.90153i
379.12 −1.31112 + 0.530067i −2.23972 1.43806 − 1.38996i 1.59332 + 1.56886i 2.93653 − 1.18720i −2.42178 −1.14869 + 2.58467i 2.01633 −2.92063 − 1.21239i
379.13 −1.23558 − 0.687996i 2.17214 1.05332 + 1.70015i −2.19004 − 0.451368i −2.68385 − 1.49442i −2.03346 −0.131770 − 2.82536i 1.71818 2.39543 + 2.06444i
379.14 −1.23558 − 0.687996i 2.17214 1.05332 + 1.70015i 2.19004 − 0.451368i −2.68385 − 1.49442i 2.03346 −0.131770 − 2.82536i 1.71818 −3.01651 − 0.949035i
379.15 −1.23558 + 0.687996i 2.17214 1.05332 − 1.70015i −2.19004 + 0.451368i −2.68385 + 1.49442i −2.03346 −0.131770 + 2.82536i 1.71818 2.39543 − 2.06444i
379.16 −1.23558 + 0.687996i 2.17214 1.05332 − 1.70015i 2.19004 + 0.451368i −2.68385 + 1.49442i 2.03346 −0.131770 + 2.82536i 1.71818 −3.01651 + 0.949035i
379.17 −0.757052 − 1.19452i −2.97713 −0.853744 + 1.80862i −2.06882 + 0.848517i 2.25384 + 3.55623i 1.42544 2.80676 − 0.349410i 5.86328 2.57977 + 1.82887i
379.18 −0.757052 − 1.19452i −2.97713 −0.853744 + 1.80862i 2.06882 + 0.848517i 2.25384 + 3.55623i −1.42544 2.80676 − 0.349410i 5.86328 −0.552636 − 3.11361i
379.19 −0.757052 + 1.19452i −2.97713 −0.853744 − 1.80862i −2.06882 − 0.848517i 2.25384 − 3.55623i 1.42544 2.80676 + 0.349410i 5.86328 2.57977 − 1.82887i
379.20 −0.757052 + 1.19452i −2.97713 −0.853744 − 1.80862i 2.06882 − 0.848517i 2.25384 − 3.55623i −1.42544 2.80676 + 0.349410i 5.86328 −0.552636 + 3.11361i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
19.b odd 2 1 inner
40.e odd 2 1 inner
95.d odd 2 1 inner
152.b even 2 1 inner
760.p even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.p.i ✓ 56
5.b even 2 1 inner 760.2.p.i ✓ 56
8.d odd 2 1 inner 760.2.p.i ✓ 56
19.b odd 2 1 inner 760.2.p.i ✓ 56
40.e odd 2 1 inner 760.2.p.i ✓ 56
95.d odd 2 1 inner 760.2.p.i ✓ 56
152.b even 2 1 inner 760.2.p.i ✓ 56
760.p even 2 1 inner 760.2.p.i ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.p.i ✓ 56 1.a even 1 1 trivial
760.2.p.i ✓ 56 5.b even 2 1 inner
760.2.p.i ✓ 56 8.d odd 2 1 inner
760.2.p.i ✓ 56 19.b odd 2 1 inner
760.2.p.i ✓ 56 40.e odd 2 1 inner
760.2.p.i ✓ 56 95.d odd 2 1 inner
760.2.p.i ✓ 56 152.b even 2 1 inner
760.2.p.i ✓ 56 760.p even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3}^{14} - 32T_{3}^{12} + 410T_{3}^{10} - 2684T_{3}^{8} + 9440T_{3}^{6} - 16868T_{3}^{4} + 12112T_{3}^{2} - 736 \) Copy content Toggle raw display
\( T_{7}^{14} - 40T_{7}^{12} + 605T_{7}^{10} - 4394T_{7}^{8} + 16400T_{7}^{6} - 31216T_{7}^{4} + 27968T_{7}^{2} - 8912 \) Copy content Toggle raw display
\( T_{29}^{14} - 204 T_{29}^{12} + 16376 T_{29}^{10} - 657408 T_{29}^{8} + 13968384 T_{29}^{6} + \cdots - 1679163392 \) Copy content Toggle raw display