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Properties

Label 945.2.bf.c
Level $945$
Weight $2$
Character orbit 945.bf
Analytic conductor $7.546$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(109,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bf (of order \(6\), degree \(2\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 36 q^{4} - 8 q^{10} - 36 q^{16} + 20 q^{19} + 20 q^{25} + 32 q^{31} - 40 q^{34} - 4 q^{40} + 72 q^{46} - 24 q^{49} - 80 q^{55} + 60 q^{61} - 80 q^{64} - 24 q^{70} + 64 q^{76} + 64 q^{79} - 16 q^{85} - 28 q^{91} - 84 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −2.34033 + 1.35119i 0 2.65144 − 4.59243i 0.445344 − 2.19127i 0 2.32921 + 1.25491i 8.92564i 0 1.91857 + 5.73005i
109.2 −2.34033 + 1.35119i 0 2.65144 − 4.59243i 2.12037 + 0.709957i 0 −2.32921 − 1.25491i 8.92564i 0 −5.92165 + 1.20349i
109.3 −2.11287 + 1.21987i 0 1.97616 − 3.42280i −2.10749 + 0.747319i 0 −2.64034 − 0.169153i 4.76312i 0 3.54123 − 4.14985i
109.4 −2.11287 + 1.21987i 0 1.97616 − 3.42280i −1.70094 + 1.45148i 0 2.64034 + 0.169153i 4.76312i 0 1.82326 − 5.14172i
109.5 −1.72237 + 0.994411i 0 0.977708 − 1.69344i −0.889701 − 2.05145i 0 −1.60275 + 2.10504i − 0.0886711i 0 3.57238 + 2.64862i
109.6 −1.72237 + 0.994411i 0 0.977708 − 1.69344i 1.33175 + 1.79623i 0 1.60275 − 2.10504i − 0.0886711i 0 −4.07996 − 1.76946i
109.7 −1.56578 + 0.904005i 0 0.634449 − 1.09890i 1.04413 − 1.97732i 0 −1.08254 − 2.41415i − 1.32184i 0 0.152617 + 4.03995i
109.8 −1.56578 + 0.904005i 0 0.634449 − 1.09890i 2.23447 + 0.0844117i 0 1.08254 + 2.41415i − 1.32184i 0 −3.57501 + 1.88780i
109.9 −1.18504 + 0.684185i 0 −0.0637831 + 0.110475i −2.22269 − 0.244243i 0 0.0806857 − 2.64452i − 2.91130i 0 2.80109 − 1.23129i
109.10 −1.18504 + 0.684185i 0 −0.0637831 + 0.110475i −0.899824 + 2.04703i 0 −0.0806857 + 2.64452i − 2.91130i 0 −0.334214 − 3.04146i
109.11 −0.689052 + 0.397824i 0 −0.683472 + 1.18381i 1.79739 − 1.33019i 0 −0.670648 − 2.55934i − 2.67890i 0 −0.709312 + 1.63161i
109.12 −0.689052 + 0.397824i 0 −0.683472 + 1.18381i 2.05067 − 0.891489i 0 0.670648 + 2.55934i − 2.67890i 0 −1.05836 + 1.43009i
109.13 −0.106101 + 0.0612575i 0 −0.992495 + 1.71905i −0.0593437 − 2.23528i 0 2.53110 + 0.770427i − 0.488221i 0 0.143224 + 0.233530i
109.14 −0.106101 + 0.0612575i 0 −0.992495 + 1.71905i 1.90614 + 1.16903i 0 −2.53110 − 0.770427i − 0.488221i 0 −0.273855 − 0.00727049i
109.15 0.106101 − 0.0612575i 0 −0.992495 + 1.71905i −1.90614 − 1.16903i 0 −2.53110 − 0.770427i 0.488221i 0 −0.273855 − 0.00727049i
109.16 0.106101 − 0.0612575i 0 −0.992495 + 1.71905i 0.0593437 + 2.23528i 0 2.53110 + 0.770427i 0.488221i 0 0.143224 + 0.233530i
109.17 0.689052 − 0.397824i 0 −0.683472 + 1.18381i −2.05067 + 0.891489i 0 0.670648 + 2.55934i 2.67890i 0 −1.05836 + 1.43009i
109.18 0.689052 − 0.397824i 0 −0.683472 + 1.18381i −1.79739 + 1.33019i 0 −0.670648 − 2.55934i 2.67890i 0 −0.709312 + 1.63161i
109.19 1.18504 − 0.684185i 0 −0.0637831 + 0.110475i 0.899824 − 2.04703i 0 −0.0806857 + 2.64452i 2.91130i 0 −0.334214 − 3.04146i
109.20 1.18504 − 0.684185i 0 −0.0637831 + 0.110475i 2.22269 + 0.244243i 0 0.0806857 − 2.64452i 2.91130i 0 2.80109 − 1.23129i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bf.c ✓ 56
3.b odd 2 1 inner 945.2.bf.c ✓ 56
5.b even 2 1 inner 945.2.bf.c ✓ 56
7.c even 3 1 inner 945.2.bf.c ✓ 56
15.d odd 2 1 inner 945.2.bf.c ✓ 56
21.h odd 6 1 inner 945.2.bf.c ✓ 56
35.j even 6 1 inner 945.2.bf.c ✓ 56
105.o odd 6 1 inner 945.2.bf.c ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bf.c ✓ 56 1.a even 1 1 trivial
945.2.bf.c ✓ 56 3.b odd 2 1 inner
945.2.bf.c ✓ 56 5.b even 2 1 inner
945.2.bf.c ✓ 56 7.c even 3 1 inner
945.2.bf.c ✓ 56 15.d odd 2 1 inner
945.2.bf.c ✓ 56 21.h odd 6 1 inner
945.2.bf.c ✓ 56 35.j even 6 1 inner
945.2.bf.c ✓ 56 105.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{28} - 23 T_{2}^{26} + 324 T_{2}^{24} - 2927 T_{2}^{22} + 19491 T_{2}^{20} - 94571 T_{2}^{18} + \cdots + 100 \) Copy content Toggle raw display
\( T_{11}^{28} + 69 T_{11}^{26} + 2997 T_{11}^{24} + 76762 T_{11}^{22} + 1404831 T_{11}^{20} + \cdots + 687970713600 \) Copy content Toggle raw display