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Properties

Label 1870.2.h.a
Level $1870$
Weight $2$
Character orbit 1870.h
Analytic conductor $14.932$
Analytic rank $0$
Dimension $46$
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] = [1870,2,Mod(1189,1870)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1870, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1870.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1870 = 2 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1870.h (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: \(14.9320251780\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 8 q^{3} - 46 q^{4} - 8 q^{7} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 8 q^{3} - 46 q^{4} - 8 q^{7} + 50 q^{9} + 8 q^{12} - 12 q^{15} + 46 q^{16} + 24 q^{21} + 46 q^{22} + 28 q^{23} + 4 q^{25} - 20 q^{26} - 20 q^{27} + 8 q^{28} + 12 q^{30} + 4 q^{35} - 50 q^{36} + 12 q^{37} + 60 q^{45} - 8 q^{48} + 46 q^{49} - 4 q^{50} - 24 q^{51} - 24 q^{57} + 16 q^{58} + 44 q^{59} + 12 q^{60} - 28 q^{62} - 40 q^{63} - 46 q^{64} + 16 q^{65} - 8 q^{66} + 56 q^{69} - 28 q^{70} - 28 q^{73} - 60 q^{75} + 16 q^{78} + 78 q^{81} + 4 q^{82} - 24 q^{84} - 38 q^{85} + 28 q^{86} - 46 q^{88} - 72 q^{89} + 14 q^{90} - 28 q^{92} - 32 q^{94} + 4 q^{95} + 128 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1 − 1.00000i 1.09054 −1.00000 −2.13991 + 0.648681i − 1.09054i −5.24098 1.00000i −1.81073 0.648681 + 2.13991i
1189.2 1.00000i 1.09054 −1.00000 −2.13991 − 0.648681i 1.09054i −5.24098 − 1.00000i −1.81073 0.648681 − 2.13991i
1189.3 − 1.00000i −2.65875 −1.00000 2.23245 − 0.127198i 2.65875i 0.872967 1.00000i 4.06897 −0.127198 − 2.23245i
1189.4 1.00000i −2.65875 −1.00000 2.23245 + 0.127198i − 2.65875i 0.872967 − 1.00000i 4.06897 −0.127198 + 2.23245i
1189.5 − 1.00000i −2.33459 −1.00000 0.563416 − 2.16392i 2.33459i −5.02816 1.00000i 2.45030 −2.16392 − 0.563416i
1189.6 1.00000i −2.33459 −1.00000 0.563416 + 2.16392i − 2.33459i −5.02816 − 1.00000i 2.45030 −2.16392 + 0.563416i
1189.7 − 1.00000i 2.14927 −1.00000 −0.742958 + 2.10903i − 2.14927i −2.63015 1.00000i 1.61937 2.10903 + 0.742958i
1189.8 1.00000i 2.14927 −1.00000 −0.742958 − 2.10903i 2.14927i −2.63015 − 1.00000i 1.61937 2.10903 − 0.742958i
1189.9 − 1.00000i −0.127852 −1.00000 2.23321 + 0.113073i 0.127852i −4.09681 1.00000i −2.98365 0.113073 − 2.23321i
1189.10 1.00000i −0.127852 −1.00000 2.23321 − 0.113073i − 0.127852i −4.09681 − 1.00000i −2.98365 0.113073 + 2.23321i
1189.11 − 1.00000i −1.91274 −1.00000 −1.34457 + 1.78666i 1.91274i 1.18254 1.00000i 0.658563 1.78666 + 1.34457i
1189.12 1.00000i −1.91274 −1.00000 −1.34457 − 1.78666i − 1.91274i 1.18254 − 1.00000i 0.658563 1.78666 − 1.34457i
1189.13 − 1.00000i 3.41269 −1.00000 0.476326 + 2.18475i − 3.41269i −0.887407 1.00000i 8.64643 2.18475 − 0.476326i
1189.14 1.00000i 3.41269 −1.00000 0.476326 − 2.18475i 3.41269i −0.887407 − 1.00000i 8.64643 2.18475 + 0.476326i
1189.15 − 1.00000i 1.33449 −1.00000 −1.46482 − 1.68946i − 1.33449i −1.70075 1.00000i −1.21913 −1.68946 + 1.46482i
1189.16 1.00000i 1.33449 −1.00000 −1.46482 + 1.68946i 1.33449i −1.70075 − 1.00000i −1.21913 −1.68946 − 1.46482i
1189.17 − 1.00000i −0.223994 −1.00000 0.946650 − 2.02580i 0.223994i 1.39091 1.00000i −2.94983 −2.02580 − 0.946650i
1189.18 1.00000i −0.223994 −1.00000 0.946650 + 2.02580i − 0.223994i 1.39091 − 1.00000i −2.94983 −2.02580 + 0.946650i
1189.19 − 1.00000i −1.22936 −1.00000 2.17597 + 0.514936i 1.22936i −1.17521 1.00000i −1.48868 0.514936 − 2.17597i
1189.20 1.00000i −1.22936 −1.00000 2.17597 − 0.514936i − 1.22936i −1.17521 − 1.00000i −1.48868 0.514936 + 2.17597i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1189.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1870.2.h.a ✓ 46
5.b even 2 1 1870.2.h.b yes 46
17.b even 2 1 1870.2.h.b yes 46
85.c even 2 1 inner 1870.2.h.a ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1870.2.h.a ✓ 46 1.a even 1 1 trivial
1870.2.h.a ✓ 46 85.c even 2 1 inner
1870.2.h.b yes 46 5.b even 2 1
1870.2.h.b yes 46 17.b even 2 1