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Properties

Label 1890.2.r.a
Level $1890$
Weight $2$
Character orbit 1890.r
Analytic conductor $15.092$
Analytic rank $0$
Dimension $48$
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] = [1890,2,Mod(89,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.r (of order \(6\), degree \(2\), not 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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{2} - 24 q^{4} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{2} - 24 q^{4} + 48 q^{8} - 3 q^{14} - 24 q^{16} - 6 q^{22} - 6 q^{23} + 3 q^{28} + 3 q^{29} - 24 q^{32} - 12 q^{35} - 3 q^{41} + 6 q^{44} + 3 q^{46} - 6 q^{49} + 18 q^{50} - 42 q^{55} - 9 q^{61} + 48 q^{64} + 21 q^{65} + 33 q^{67} + 12 q^{70} - 18 q^{73} + 6 q^{77} - 3 q^{82} - 9 q^{83} + 33 q^{85} - 33 q^{89} + 3 q^{92} - 33 q^{95} - 24 q^{97} + 3 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.23138 + 0.144779i 0 −2.37108 − 1.17388i 1.00000 0 1.24107 + 1.86004i
89.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.20770 + 0.355056i 0 −0.692019 − 2.55365i 1.00000 0 1.41134 + 1.73440i
89.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.19316 − 0.435950i 0 2.64572 − 0.0128895i 1.00000 0 0.719036 + 2.11731i
89.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.07866 − 0.824119i 0 −0.837534 + 2.50969i 1.00000 0 0.325622 + 2.21223i
89.5 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.96632 − 1.06470i 0 0.733422 − 2.54206i 1.00000 0 0.0611015 + 2.23523i
89.6 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.76507 + 1.37278i 0 −1.05699 + 2.42544i 1.00000 0 2.07140 + 0.842204i
89.7 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.29544 − 1.82259i 0 1.77597 + 1.96110i 1.00000 0 −0.930691 + 2.03318i
89.8 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.28213 + 1.83198i 0 2.37211 + 1.17179i 1.00000 0 2.22760 + 0.194363i
89.9 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.782489 + 2.09469i 0 −2.24311 − 1.40302i 1.00000 0 2.20530 − 0.369687i
89.10 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.619562 − 2.14852i 0 −2.06156 + 1.65830i 1.00000 0 −1.55089 + 1.61082i
89.11 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.390088 − 2.20178i 0 0.160573 − 2.64087i 1.00000 0 −1.71175 + 1.43872i
89.12 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.0986685 + 2.23389i 0 1.62563 − 2.08742i 1.00000 0 1.98394 − 1.03150i
89.13 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.00346114 − 2.23607i 0 −2.57793 − 0.595221i 1.00000 0 −1.93822 + 1.11504i
89.14 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.438968 + 2.19256i 0 0.272771 + 2.63165i 1.00000 0 1.67933 − 1.47644i
89.15 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.717414 − 2.11786i 0 0.647443 − 2.56531i 1.00000 0 −2.19282 + 0.437630i
89.16 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.00280 + 1.99860i 0 2.55747 − 0.677764i 1.00000 0 1.22943 − 1.86775i
89.17 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.26026 − 1.84709i 0 2.59865 − 0.497024i 1.00000 0 −2.22976 − 0.167870i
89.18 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.60670 + 1.55516i 0 −1.24799 + 2.33292i 1.00000 0 0.543458 − 2.16902i
89.19 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.76448 + 1.37353i 0 −2.04479 + 1.67894i 1.00000 0 0.307273 − 2.21486i
89.20 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.83728 − 1.27452i 0 −2.46948 + 0.949557i 1.00000 0 −2.02241 − 0.953871i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
315.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.r.a 48
3.b odd 2 1 630.2.r.b yes 48
5.b even 2 1 1890.2.r.b 48
7.d odd 6 1 1890.2.bi.b 48
9.c even 3 1 630.2.bi.b yes 48
9.d odd 6 1 1890.2.bi.a 48
15.d odd 2 1 630.2.r.a ✓ 48
21.g even 6 1 630.2.bi.a yes 48
35.i odd 6 1 1890.2.bi.a 48
45.h odd 6 1 1890.2.bi.b 48
45.j even 6 1 630.2.bi.a yes 48
63.k odd 6 1 630.2.r.a ✓ 48
63.s even 6 1 1890.2.r.b 48
105.p even 6 1 630.2.bi.b yes 48
315.u even 6 1 inner 1890.2.r.a 48
315.bn odd 6 1 630.2.r.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.r.a ✓ 48 15.d odd 2 1
630.2.r.a ✓ 48 63.k odd 6 1
630.2.r.b yes 48 3.b odd 2 1
630.2.r.b yes 48 315.bn odd 6 1
630.2.bi.a yes 48 21.g even 6 1
630.2.bi.a yes 48 45.j even 6 1
630.2.bi.b yes 48 9.c even 3 1
630.2.bi.b yes 48 105.p even 6 1
1890.2.r.a 48 1.a even 1 1 trivial
1890.2.r.a 48 315.u even 6 1 inner
1890.2.r.b 48 5.b even 2 1
1890.2.r.b 48 63.s even 6 1
1890.2.bi.a 48 9.d odd 6 1
1890.2.bi.a 48 35.i odd 6 1
1890.2.bi.b 48 7.d odd 6 1
1890.2.bi.b 48 45.h odd 6 1