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Properties

Label 945.2.bo.b
Level $945$
Weight $2$
Character orbit 945.bo
Analytic conductor $7.546$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(289,945)]
 
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([4, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bo (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(42\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 315)
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 44 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 44 q^{4} + 6 q^{5} + 6 q^{10} + 24 q^{11} + 10 q^{14} - 36 q^{16} + 8 q^{19} + 10 q^{20} + 10 q^{25} + 40 q^{26} + 10 q^{29} - 6 q^{31} - 12 q^{34} - 4 q^{35} - 8 q^{40} + 30 q^{41} + 4 q^{44} + 4 q^{46} + 8 q^{49} - 42 q^{50} - 54 q^{55} - 48 q^{56} - 42 q^{59} + 22 q^{61} - 28 q^{64} - 8 q^{65} - 26 q^{70} + 4 q^{71} + 108 q^{74} + 24 q^{76} + 24 q^{79} + 9 q^{80} + q^{85} + 92 q^{86} - 46 q^{89} - 44 q^{91} - 8 q^{94} + 25 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −2.35408 + 1.35913i 0 2.69445 − 4.66693i −0.317086 − 2.21347i 0 −1.81310 − 1.92683i 9.21190i 0 3.75483 + 4.77972i
289.2 −2.26777 + 1.30930i 0 2.42851 − 4.20630i −2.23375 + 0.101685i 0 0.957411 + 2.46645i 7.48135i 0 4.93250 − 3.15524i
289.3 −2.26367 + 1.30693i 0 2.41613 − 4.18486i 1.53331 + 1.62756i 0 2.64512 + 0.0579738i 7.40313i 0 −5.59802 − 1.68034i
289.4 −2.10397 + 1.21472i 0 1.95111 − 3.37943i 2.23603 + 0.0133001i 0 −1.65650 + 2.06301i 4.62136i 0 −4.72068 + 2.68818i
289.5 −2.05276 + 1.18516i 0 1.80922 − 3.13366i 0.218973 + 2.22532i 0 −1.63234 + 2.08218i 3.83621i 0 −3.08686 − 4.30853i
289.6 −1.90125 + 1.09768i 0 1.40982 − 2.44189i 1.91379 − 1.15647i 0 2.51786 − 0.812625i 1.79943i 0 −2.36914 + 4.29947i
289.7 −1.84157 + 1.06323i 0 1.26093 − 2.18399i −1.02419 − 1.98772i 0 −0.441480 − 2.60866i 1.10970i 0 3.99953 + 2.57158i
289.8 −1.75373 + 1.01252i 0 1.05039 − 1.81932i 0.846932 + 2.06947i 0 −1.36749 − 2.26495i 0.204070i 0 −3.58067 − 2.77176i
289.9 −1.56714 + 0.904786i 0 0.637277 − 1.10380i 0.0315838 − 2.23584i 0 2.37354 − 1.16889i − 1.31275i 0 1.97347 + 3.53245i
289.10 −1.56370 + 0.902800i 0 0.630096 − 1.09136i −1.94086 − 1.11043i 0 −0.480299 + 2.60179i − 1.33580i 0 4.03741 − 0.0158424i
289.11 −1.42338 + 0.821790i 0 0.350678 − 0.607392i −2.06509 + 0.857547i 0 −2.61685 − 0.390007i − 2.13443i 0 2.23469 − 2.91769i
289.12 −1.40440 + 0.810833i 0 0.314901 − 0.545425i −1.75547 + 1.38504i 0 1.09563 − 2.40824i − 2.22200i 0 1.34235 − 3.36855i
289.13 −1.17913 + 0.680771i 0 −0.0731024 + 0.126617i 2.23088 + 0.152304i 0 −2.52908 − 0.777026i − 2.92215i 0 −2.73418 + 1.33913i
289.14 −0.941848 + 0.543776i 0 −0.408615 + 0.707741i 1.28643 − 1.82896i 0 −2.64561 − 0.0270009i − 3.06389i 0 −0.217080 + 2.42213i
289.15 −0.931541 + 0.537825i 0 −0.421488 + 0.730038i 2.11877 − 0.714718i 0 1.08220 + 2.41430i − 3.05805i 0 −1.58933 + 1.80532i
289.16 −0.755931 + 0.436437i 0 −0.619046 + 1.07222i 1.55427 + 1.60756i 0 2.14141 + 1.55382i − 2.82644i 0 −1.87652 − 0.536861i
289.17 −0.627169 + 0.362096i 0 −0.737773 + 1.27786i −0.851637 − 2.06754i 0 0.722021 + 2.54533i − 2.51696i 0 1.28277 + 0.988321i
289.18 −0.582215 + 0.336142i 0 −0.774017 + 1.34064i 1.25057 + 1.85366i 0 0.680768 − 2.55667i − 2.38529i 0 −1.35120 − 0.658862i
289.19 −0.248464 + 0.143451i 0 −0.958844 + 1.66077i −2.21083 + 0.335020i 0 2.25543 − 1.38312i − 1.12399i 0 0.501253 − 0.400386i
289.20 −0.223223 + 0.128878i 0 −0.966781 + 1.67451i −1.91223 − 1.15903i 0 −2.07085 − 1.64669i − 1.01390i 0 0.576229 + 0.0122786i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.g even 3 1 inner
315.bo even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bo.b 84
3.b odd 2 1 315.2.bo.b yes 84
5.b even 2 1 inner 945.2.bo.b 84
7.c even 3 1 945.2.r.b 84
9.c even 3 1 945.2.r.b 84
9.d odd 6 1 315.2.r.b ✓ 84
15.d odd 2 1 315.2.bo.b yes 84
21.h odd 6 1 315.2.r.b ✓ 84
35.j even 6 1 945.2.r.b 84
45.h odd 6 1 315.2.r.b ✓ 84
45.j even 6 1 945.2.r.b 84
63.g even 3 1 inner 945.2.bo.b 84
63.n odd 6 1 315.2.bo.b yes 84
105.o odd 6 1 315.2.r.b ✓ 84
315.v odd 6 1 315.2.bo.b yes 84
315.bo even 6 1 inner 945.2.bo.b 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.b ✓ 84 9.d odd 6 1
315.2.r.b ✓ 84 21.h odd 6 1
315.2.r.b ✓ 84 45.h odd 6 1
315.2.r.b ✓ 84 105.o odd 6 1
315.2.bo.b yes 84 3.b odd 2 1
315.2.bo.b yes 84 15.d odd 2 1
315.2.bo.b yes 84 63.n odd 6 1
315.2.bo.b yes 84 315.v odd 6 1
945.2.r.b 84 7.c even 3 1
945.2.r.b 84 9.c even 3 1
945.2.r.b 84 35.j even 6 1
945.2.r.b 84 45.j even 6 1
945.2.bo.b 84 1.a even 1 1 trivial
945.2.bo.b 84 5.b even 2 1 inner
945.2.bo.b 84 63.g even 3 1 inner
945.2.bo.b 84 315.bo even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} - 64 T_{2}^{82} + 2207 T_{2}^{80} - 52654 T_{2}^{78} + 962504 T_{2}^{76} - 14220262 T_{2}^{74} + \cdots + 5764801 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display