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Properties

Label 1890.2.bi.b
Level $1890$
Weight $2$
Character orbit 1890.bi
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(719,1890)]
 
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([5, 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("1890.719");
 
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.bi (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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} + 48 q^{4} - 3 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} + 48 q^{4} - 3 q^{7} + 48 q^{8} - 6 q^{11} - 3 q^{14} + 48 q^{16} - 6 q^{22} + 3 q^{23} + 18 q^{25} - 3 q^{28} + 3 q^{29} + 48 q^{32} + 18 q^{35} + 3 q^{41} - 6 q^{44} + 3 q^{46} + 3 q^{49} + 18 q^{50} + 42 q^{55} - 3 q^{56} + 3 q^{58} + 48 q^{64} + 12 q^{65} + 18 q^{70} + 18 q^{73} - 12 q^{77} + 3 q^{82} + 9 q^{83} + 33 q^{85} - 6 q^{88} + 33 q^{89} + 3 q^{92} + 24 q^{97} + 3 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1 1.00000 0 1.00000 −2.22613 − 0.210587i 0 2.58635 − 0.557485i 1.00000 0 −2.22613 − 0.210587i
719.2 1.00000 0 1.00000 −2.17045 + 0.537704i 0 0.405354 + 2.61451i 1.00000 0 −2.17045 + 0.537704i
719.3 1.00000 0 1.00000 −2.10184 + 0.763064i 0 −2.20678 − 1.45950i 1.00000 0 −2.10184 + 0.763064i
719.4 1.00000 0 1.00000 −1.93476 + 1.12103i 0 −1.80444 + 1.93494i 1.00000 0 −1.93476 + 1.12103i
719.5 1.00000 0 1.00000 −1.90522 − 1.17053i 0 −1.83478 − 1.90619i 1.00000 0 −1.90522 − 1.17053i
719.6 1.00000 0 1.00000 −1.75304 − 1.38811i 0 1.75469 + 1.98017i 1.00000 0 −1.75304 − 1.38811i
719.7 1.00000 0 1.00000 −1.47541 + 1.68023i 0 −1.89790 − 1.84336i 1.00000 0 −1.47541 + 1.68023i
719.8 1.00000 0 1.00000 −1.47412 − 1.68136i 0 1.31170 − 2.29771i 1.00000 0 −1.47412 − 1.68136i
719.9 1.00000 0 1.00000 −0.990306 − 2.00482i 0 −2.20215 + 1.46647i 1.00000 0 −0.990306 − 2.00482i
719.10 1.00000 0 1.00000 −0.969500 + 2.01496i 0 0.868888 − 2.49901i 1.00000 0 −0.969500 + 2.01496i
719.11 1.00000 0 1.00000 −0.796362 − 2.08945i 0 −2.55753 − 0.677517i 1.00000 0 −0.796362 − 2.08945i
719.12 1.00000 0 1.00000 −0.185127 + 2.22839i 0 −0.412401 + 2.61341i 1.00000 0 −0.185127 + 2.22839i
719.13 1.00000 0 1.00000 −0.0426305 + 2.23566i 0 2.42207 − 1.06471i 1.00000 0 −0.0426305 + 2.23566i
719.14 1.00000 0 1.00000 0.306329 − 2.21499i 0 1.57200 + 2.12810i 1.00000 0 0.306329 − 2.21499i
719.15 1.00000 0 1.00000 0.945479 − 2.02634i 0 2.20085 − 1.46842i 1.00000 0 0.945479 − 2.02634i
719.16 1.00000 0 1.00000 1.37960 + 1.75975i 0 1.94702 − 1.79140i 1.00000 0 1.37960 + 1.75975i
719.17 1.00000 0 1.00000 1.42281 − 1.72500i 0 −2.33660 + 1.24108i 1.00000 0 1.42281 − 1.72500i
719.18 1.00000 0 1.00000 1.71718 + 1.43224i 0 −2.63972 + 0.178501i 1.00000 0 1.71718 + 1.43224i
719.19 1.00000 0 1.00000 1.79580 + 1.33234i 0 −2.61687 − 0.389830i 1.00000 0 1.79580 + 1.33234i
719.20 1.00000 0 1.00000 1.88527 − 1.20239i 0 −0.994947 − 2.45155i 1.00000 0 1.88527 − 1.20239i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.24
Significant digits:
Format:

Inner twists

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