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Properties

Label 765.2.be.b
Level $765$
Weight $2$
Character orbit 765.be
Analytic conductor $6.109$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [765,2,Mod(406,765)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(765, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("765.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 765.be (of order \(8\), degree \(4\), 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: \(6.10855575463\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{11} - 24 q^{16} + 8 q^{17} - 8 q^{19} - 32 q^{22} + 16 q^{23} - 16 q^{26} + 48 q^{28} + 8 q^{29} + 16 q^{34} + 32 q^{35} + 24 q^{37} + 16 q^{40} - 16 q^{41} + 8 q^{43} - 16 q^{44} + 8 q^{46} - 8 q^{50} - 48 q^{52} - 24 q^{53} - 64 q^{56} - 64 q^{58} - 32 q^{59} + 32 q^{61} + 32 q^{62} - 8 q^{65} + 16 q^{67} + 40 q^{68} + 24 q^{71} + 64 q^{74} - 8 q^{76} - 24 q^{77} + 32 q^{80} - 80 q^{82} + 96 q^{83} + 16 q^{86} - 8 q^{88} - 24 q^{91} - 80 q^{92} + 56 q^{94} + 16 q^{95} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1 −1.86672 − 1.86672i 0 4.96928i −0.923880 + 0.382683i 0 −3.75274 − 1.55444i 5.54282 − 5.54282i 0 2.43899 + 1.01026i
406.2 −1.09631 − 1.09631i 0 0.403772i 0.923880 − 0.382683i 0 −3.45666 − 1.43180i −1.74995 + 1.74995i 0 −1.43239 − 0.593316i
406.3 −0.680853 − 0.680853i 0 − 1.07288i 0.923880 − 0.382683i 0 2.85906 + 1.18426i −2.09218 + 2.09218i 0 −0.889577 − 0.368475i
406.4 0.254738 + 0.254738i 0 − 1.87022i −0.923880 + 0.382683i 0 0.275980 + 0.114315i 0.985893 − 0.985893i 0 −0.332832 − 0.137863i
406.5 0.528855 + 0.528855i 0 − 1.44062i 0.923880 − 0.382683i 0 2.98655 + 1.23707i 1.81959 − 1.81959i 0 0.690983 + 0.286214i
406.6 1.44607 + 1.44607i 0 2.18224i −0.923880 + 0.382683i 0 1.08781 + 0.450584i −0.263530 + 0.263530i 0 −1.88938 − 0.782608i
451.1 −1.66305 + 1.66305i 0 − 3.53144i 0.382683 − 0.923880i 0 1.37082 + 3.30945i 2.54686 + 2.54686i 0 0.900034 + 2.17287i
451.2 −1.01710 + 1.01710i 0 − 0.0689897i −0.382683 + 0.923880i 0 −0.265997 − 0.642174i −1.96403 − 1.96403i 0 −0.550451 − 1.32891i
451.3 −0.213325 + 0.213325i 0 1.90899i 0.382683 − 0.923880i 0 −0.960473 − 2.31879i −0.833883 − 0.833883i 0 0.115451 + 0.278722i
451.4 1.09994 − 1.09994i 0 − 0.419729i −0.382683 + 0.923880i 0 −1.32205 − 3.19170i 1.73820 + 1.73820i 0 0.595282 + 1.43714i
451.5 1.27691 − 1.27691i 0 − 1.26102i 0.382683 − 0.923880i 0 1.66158 + 4.01142i 0.943613 + 0.943613i 0 −0.691061 − 1.66837i
451.6 1.93083 − 1.93083i 0 − 5.45623i −0.382683 + 0.923880i 0 −0.483886 − 1.16820i −6.67340 − 6.67340i 0 1.04496 + 2.52275i
586.1 −1.86672 + 1.86672i 0 − 4.96928i −0.923880 − 0.382683i 0 −3.75274 + 1.55444i 5.54282 + 5.54282i 0 2.43899 − 1.01026i
586.2 −1.09631 + 1.09631i 0 − 0.403772i 0.923880 + 0.382683i 0 −3.45666 + 1.43180i −1.74995 − 1.74995i 0 −1.43239 + 0.593316i
586.3 −0.680853 + 0.680853i 0 1.07288i 0.923880 + 0.382683i 0 2.85906 − 1.18426i −2.09218 − 2.09218i 0 −0.889577 + 0.368475i
586.4 0.254738 − 0.254738i 0 1.87022i −0.923880 − 0.382683i 0 0.275980 − 0.114315i 0.985893 + 0.985893i 0 −0.332832 + 0.137863i
586.5 0.528855 − 0.528855i 0 1.44062i 0.923880 + 0.382683i 0 2.98655 − 1.23707i 1.81959 + 1.81959i 0 0.690983 − 0.286214i
586.6 1.44607 − 1.44607i 0 − 2.18224i −0.923880 − 0.382683i 0 1.08781 − 0.450584i −0.263530 − 0.263530i 0 −1.88938 + 0.782608i
631.1 −1.66305 − 1.66305i 0 3.53144i 0.382683 + 0.923880i 0 1.37082 − 3.30945i 2.54686 − 2.54686i 0 0.900034 − 2.17287i
631.2 −1.01710 − 1.01710i 0 0.0689897i −0.382683 − 0.923880i 0 −0.265997 + 0.642174i −1.96403 + 1.96403i 0 −0.550451 + 1.32891i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 765.2.be.b 24
3.b odd 2 1 85.2.l.a ✓ 24
15.d odd 2 1 425.2.m.b 24
15.e even 4 1 425.2.n.c 24
15.e even 4 1 425.2.n.f 24
17.d even 8 1 inner 765.2.be.b 24
51.g odd 8 1 85.2.l.a ✓ 24
51.i even 16 1 1445.2.a.p 12
51.i even 16 1 1445.2.a.q 12
51.i even 16 2 1445.2.d.j 24
255.v even 8 1 425.2.n.f 24
255.y odd 8 1 425.2.m.b 24
255.ba even 8 1 425.2.n.c 24
255.be even 16 1 7225.2.a.bq 12
255.be even 16 1 7225.2.a.bs 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.l.a ✓ 24 3.b odd 2 1
85.2.l.a ✓ 24 51.g odd 8 1
425.2.m.b 24 15.d odd 2 1
425.2.m.b 24 255.y odd 8 1
425.2.n.c 24 15.e even 4 1
425.2.n.c 24 255.ba even 8 1
425.2.n.f 24 15.e even 4 1
425.2.n.f 24 255.v even 8 1
765.2.be.b 24 1.a even 1 1 trivial
765.2.be.b 24 17.d even 8 1 inner
1445.2.a.p 12 51.i even 16 1
1445.2.a.q 12 51.i even 16 1
1445.2.d.j 24 51.i even 16 2
7225.2.a.bq 12 255.be even 16 1
7225.2.a.bs 12 255.be even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90 T_{2}^{20} - 8 T_{2}^{17} + 2327 T_{2}^{16} - 128 T_{2}^{15} + 640 T_{2}^{13} + \cdots + 289 \) acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\). Copy content Toggle raw display