↑

Properties

Label 85.2.l.a
Level $85$
Weight $2$
Character orbit 85.l
Analytic conductor $0.679$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [85,2,Mod(26,85)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.l (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: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} + 24 q^{12} - 8 q^{15} - 24 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 32 q^{22} - 16 q^{23} - 8 q^{24} + 16 q^{26} + 24 q^{27} + 48 q^{28} - 8 q^{29} + 16 q^{34} - 32 q^{35} - 24 q^{36} + 24 q^{37} + 8 q^{39} + 16 q^{40} + 16 q^{41} - 24 q^{42} + 8 q^{43} + 16 q^{44} + 16 q^{45} + 8 q^{46} + 80 q^{48} + 8 q^{50} - 56 q^{51} - 48 q^{52} + 24 q^{53} - 32 q^{54} + 64 q^{56} + 32 q^{57} - 64 q^{58} + 32 q^{59} + 24 q^{60} + 32 q^{61} - 32 q^{62} - 56 q^{63} + 8 q^{65} + 96 q^{66} + 16 q^{67} - 40 q^{68} + 96 q^{69} - 24 q^{71} - 64 q^{74} - 8 q^{75} - 8 q^{76} + 24 q^{77} - 112 q^{78} - 32 q^{80} - 80 q^{82} - 96 q^{83} - 64 q^{84} - 16 q^{86} - 48 q^{87} - 8 q^{88} - 24 q^{91} + 80 q^{92} + 64 q^{93} + 56 q^{94} - 16 q^{95} - 168 q^{96} - 40 q^{97} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.93083 + 1.93083i −1.42916 − 0.591976i − 5.45623i 0.382683 − 0.923880i 3.90247 − 1.61646i −0.483886 − 1.16820i 6.67340 + 6.67340i −0.429267 − 0.429267i 1.04496 + 2.52275i
26.2 −1.27691 + 1.27691i 0.635552 + 0.263254i − 1.26102i −0.382683 + 0.923880i −1.14770 + 0.475393i 1.66158 + 4.01142i −0.943613 − 0.943613i −1.78670 − 1.78670i −0.691061 − 1.66837i
26.3 −1.09994 + 1.09994i 2.77900 + 1.15110i − 0.419729i 0.382683 − 0.923880i −4.32287 + 1.79059i −1.32205 − 3.19170i −1.73820 − 1.73820i 4.27649 + 4.27649i 0.595282 + 1.43714i
26.4 0.213325 − 0.213325i 0.980249 + 0.406032i 1.90899i −0.382683 + 0.923880i 0.295728 − 0.122495i −0.960473 − 2.31879i 0.833883 + 0.833883i −1.32529 − 1.32529i 0.115451 + 0.278722i
26.5 1.01710 − 1.01710i −0.101541 − 0.0420595i − 0.0689897i 0.382683 − 0.923880i −0.146056 + 0.0604983i −0.265997 − 0.642174i 1.96403 + 1.96403i −2.11278 − 2.11278i −0.550451 − 1.32891i
26.6 1.66305 − 1.66305i −1.44989 − 0.600564i − 3.53144i −0.382683 + 0.923880i −3.41000 + 1.41247i 1.37082 + 3.30945i −2.54686 − 2.54686i −0.379815 − 0.379815i 0.900034 + 2.17287i
36.1 −1.93083 − 1.93083i −1.42916 + 0.591976i 5.45623i 0.382683 + 0.923880i 3.90247 + 1.61646i −0.483886 + 1.16820i 6.67340 − 6.67340i −0.429267 + 0.429267i 1.04496 − 2.52275i
36.2 −1.27691 − 1.27691i 0.635552 − 0.263254i 1.26102i −0.382683 − 0.923880i −1.14770 − 0.475393i 1.66158 − 4.01142i −0.943613 + 0.943613i −1.78670 + 1.78670i −0.691061 + 1.66837i
36.3 −1.09994 − 1.09994i 2.77900 − 1.15110i 0.419729i 0.382683 + 0.923880i −4.32287 − 1.79059i −1.32205 + 3.19170i −1.73820 + 1.73820i 4.27649 − 4.27649i 0.595282 − 1.43714i
36.4 0.213325 + 0.213325i 0.980249 − 0.406032i − 1.90899i −0.382683 − 0.923880i 0.295728 + 0.122495i −0.960473 + 2.31879i 0.833883 − 0.833883i −1.32529 + 1.32529i 0.115451 − 0.278722i
36.5 1.01710 + 1.01710i −0.101541 + 0.0420595i 0.0689897i 0.382683 + 0.923880i −0.146056 − 0.0604983i −0.265997 + 0.642174i 1.96403 − 1.96403i −2.11278 + 2.11278i −0.550451 + 1.32891i
36.6 1.66305 + 1.66305i −1.44989 + 0.600564i 3.53144i −0.382683 − 0.923880i −3.41000 − 1.41247i 1.37082 − 3.30945i −2.54686 + 2.54686i −0.379815 + 0.379815i 0.900034 − 2.17287i
66.1 −1.44607 − 1.44607i −1.22292 − 2.95240i 2.18224i 0.923880 − 0.382683i −2.50094 + 6.03781i 1.08781 + 0.450584i 0.263530 − 0.263530i −5.09981 + 5.09981i −1.88938 − 0.782608i
66.2 −0.528855 − 0.528855i 1.17676 + 2.84096i − 1.44062i −0.923880 + 0.382683i 0.880118 − 2.12479i 2.98655 + 1.23707i −1.81959 + 1.81959i −4.56494 + 4.56494i 0.690983 + 0.286214i
66.3 −0.254738 − 0.254738i 0.0207557 + 0.0501087i − 1.87022i 0.923880 − 0.382683i 0.00747733 − 0.0180519i 0.275980 + 0.114315i −0.985893 + 0.985893i 2.11924 − 2.11924i −0.332832 − 0.137863i
66.4 0.680853 + 0.680853i −1.01372 − 2.44733i − 1.07288i −0.923880 + 0.382683i 0.976080 − 2.35647i 2.85906 + 1.18426i 2.09218 − 2.09218i −2.84049 + 2.84049i −0.889577 − 0.368475i
66.5 1.09631 + 1.09631i 0.436412 + 1.05359i 0.403772i −0.923880 + 0.382683i −0.676617 + 1.63350i −3.45666 − 1.43180i 1.74995 − 1.74995i 1.20172 − 1.20172i −1.43239 − 0.593316i
66.6 1.86672 + 1.86672i −0.811501 − 1.95914i 4.96928i 0.923880 − 0.382683i 2.14231 − 5.17200i −3.75274 − 1.55444i −5.54282 + 5.54282i −1.05836 + 1.05836i 2.43899 + 1.01026i
76.1 −1.44607 + 1.44607i −1.22292 + 2.95240i − 2.18224i 0.923880 + 0.382683i −2.50094 − 6.03781i 1.08781 − 0.450584i 0.263530 + 0.263530i −5.09981 − 5.09981i −1.88938 + 0.782608i
76.2 −0.528855 + 0.528855i 1.17676 − 2.84096i 1.44062i −0.923880 − 0.382683i 0.880118 + 2.12479i 2.98655 − 1.23707i −1.81959 − 1.81959i −4.56494 − 4.56494i 0.690983 − 0.286214i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
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 85.2.l.a ✓ 24
3.b odd 2 1 765.2.be.b 24
5.b even 2 1 425.2.m.b 24
5.c odd 4 1 425.2.n.c 24
5.c odd 4 1 425.2.n.f 24
17.d even 8 1 inner 85.2.l.a ✓ 24
17.e odd 16 1 1445.2.a.p 12
17.e odd 16 1 1445.2.a.q 12
17.e odd 16 2 1445.2.d.j 24
51.g odd 8 1 765.2.be.b 24
85.k odd 8 1 425.2.n.c 24
85.m even 8 1 425.2.m.b 24
85.n odd 8 1 425.2.n.f 24
85.p odd 16 1 7225.2.a.bq 12
85.p odd 16 1 7225.2.a.bs 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.l.a ✓ 24 1.a even 1 1 trivial
85.2.l.a ✓ 24 17.d even 8 1 inner
425.2.m.b 24 5.b even 2 1
425.2.m.b 24 85.m even 8 1
425.2.n.c 24 5.c odd 4 1
425.2.n.c 24 85.k odd 8 1
425.2.n.f 24 5.c odd 4 1
425.2.n.f 24 85.n odd 8 1
765.2.be.b 24 3.b odd 2 1
765.2.be.b 24 51.g odd 8 1
1445.2.a.p 12 17.e odd 16 1
1445.2.a.q 12 17.e odd 16 1
1445.2.d.j 24 17.e odd 16 2
7225.2.a.bq 12 85.p odd 16 1
7225.2.a.bs 12 85.p odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(85, [\chi])\).