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Properties

Label 675.2.u.d
Level $675$
Weight $2$
Character orbit 675.u
Analytic conductor $5.390$
Analytic rank $0$
Dimension $84$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(49,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([14, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.u (of order \(18\), degree \(6\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 12 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 12 q^{6} + 6 q^{9} - 12 q^{11} + 54 q^{14} - 24 q^{16} + 48 q^{19} + 24 q^{21} + 6 q^{24} - 36 q^{26} - 24 q^{31} - 102 q^{34} - 72 q^{36} + 60 q^{39} + 66 q^{41} - 24 q^{44} - 60 q^{46} - 36 q^{49} - 78 q^{51} - 60 q^{54} - 48 q^{56} + 54 q^{59} - 12 q^{61} + 126 q^{64} - 60 q^{66} - 102 q^{69} + 24 q^{71} + 144 q^{74} + 276 q^{76} + 36 q^{79} + 90 q^{81} + 102 q^{84} + 36 q^{86} - 18 q^{89} - 138 q^{91} + 66 q^{94} + 306 q^{96} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.75919 − 2.09652i −1.40194 + 1.01713i −0.953350 + 5.40672i 0 4.59871 + 1.14987i −2.48743 + 0.438601i 8.27211 − 4.77590i 0.930888 − 2.85192i 0
49.2 −1.69563 − 2.02078i 0.112798 − 1.72837i −0.861072 + 4.88338i 0 −3.68392 + 2.70275i −0.0335646 + 0.00591834i 6.75926 − 3.90246i −2.97455 − 0.389915i 0
49.3 −1.19967 − 1.42971i 1.58534 + 0.697646i −0.257569 + 1.46075i 0 −0.904448 − 3.10352i 3.42349 − 0.603654i −0.835177 + 0.482190i 2.02658 + 2.21201i 0
49.4 −0.949646 − 1.13174i −0.0588713 − 1.73105i −0.0317206 + 0.179896i 0 −1.90320 + 1.71051i −3.64163 + 0.642118i −2.32519 + 1.34245i −2.99307 + 0.203818i 0
49.5 −0.708387 − 0.844223i −0.290600 + 1.70750i 0.136396 − 0.773542i 0 1.64737 − 0.964239i 2.94069 − 0.518523i −2.65848 + 1.53487i −2.83110 − 0.992398i 0
49.6 −0.659865 − 0.786397i −0.795743 + 1.53844i 0.164299 − 0.931783i 0 1.73491 − 0.389393i −4.04068 + 0.712481i −2.61923 + 1.51222i −1.73359 − 2.44840i 0
49.7 −0.504838 − 0.601643i −1.44576 − 0.953823i 0.240184 − 1.36215i 0 0.156014 + 1.35136i 1.40161 − 0.247142i −2.30112 + 1.32855i 1.18044 + 2.75800i 0
49.8 0.504838 + 0.601643i 1.44576 + 0.953823i 0.240184 − 1.36215i 0 0.156014 + 1.35136i −1.40161 + 0.247142i 2.30112 − 1.32855i 1.18044 + 2.75800i 0
49.9 0.659865 + 0.786397i 0.795743 − 1.53844i 0.164299 − 0.931783i 0 1.73491 − 0.389393i 4.04068 − 0.712481i 2.61923 − 1.51222i −1.73359 − 2.44840i 0
49.10 0.708387 + 0.844223i 0.290600 − 1.70750i 0.136396 − 0.773542i 0 1.64737 − 0.964239i −2.94069 + 0.518523i 2.65848 − 1.53487i −2.83110 − 0.992398i 0
49.11 0.949646 + 1.13174i 0.0588713 + 1.73105i −0.0317206 + 0.179896i 0 −1.90320 + 1.71051i 3.64163 − 0.642118i 2.32519 − 1.34245i −2.99307 + 0.203818i 0
49.12 1.19967 + 1.42971i −1.58534 − 0.697646i −0.257569 + 1.46075i 0 −0.904448 − 3.10352i −3.42349 + 0.603654i 0.835177 − 0.482190i 2.02658 + 2.21201i 0
49.13 1.69563 + 2.02078i −0.112798 + 1.72837i −0.861072 + 4.88338i 0 −3.68392 + 2.70275i 0.0335646 − 0.00591834i −6.75926 + 3.90246i −2.97455 − 0.389915i 0
49.14 1.75919 + 2.09652i 1.40194 − 1.01713i −0.953350 + 5.40672i 0 4.59871 + 1.14987i 2.48743 − 0.438601i −8.27211 + 4.77590i 0.930888 − 2.85192i 0
124.1 −1.75919 + 2.09652i −1.40194 − 1.01713i −0.953350 − 5.40672i 0 4.59871 − 1.14987i −2.48743 − 0.438601i 8.27211 + 4.77590i 0.930888 + 2.85192i 0
124.2 −1.69563 + 2.02078i 0.112798 + 1.72837i −0.861072 − 4.88338i 0 −3.68392 − 2.70275i −0.0335646 − 0.00591834i 6.75926 + 3.90246i −2.97455 + 0.389915i 0
124.3 −1.19967 + 1.42971i 1.58534 − 0.697646i −0.257569 − 1.46075i 0 −0.904448 + 3.10352i 3.42349 + 0.603654i −0.835177 − 0.482190i 2.02658 − 2.21201i 0
124.4 −0.949646 + 1.13174i −0.0588713 + 1.73105i −0.0317206 − 0.179896i 0 −1.90320 − 1.71051i −3.64163 − 0.642118i −2.32519 − 1.34245i −2.99307 − 0.203818i 0
124.5 −0.708387 + 0.844223i −0.290600 − 1.70750i 0.136396 + 0.773542i 0 1.64737 + 0.964239i 2.94069 + 0.518523i −2.65848 − 1.53487i −2.83110 + 0.992398i 0
124.6 −0.659865 + 0.786397i −0.795743 − 1.53844i 0.164299 + 0.931783i 0 1.73491 + 0.389393i −4.04068 − 0.712481i −2.61923 − 1.51222i −1.73359 + 2.44840i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.e even 9 1 inner
135.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.u.d 84
5.b even 2 1 inner 675.2.u.d 84
5.c odd 4 1 135.2.k.b ✓ 42
5.c odd 4 1 675.2.l.e 42
15.e even 4 1 405.2.k.b 42
27.e even 9 1 inner 675.2.u.d 84
135.p even 18 1 inner 675.2.u.d 84
135.q even 36 1 405.2.k.b 42
135.q even 36 1 3645.2.a.k 21
135.r odd 36 1 135.2.k.b ✓ 42
135.r odd 36 1 675.2.l.e 42
135.r odd 36 1 3645.2.a.l 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.b ✓ 42 5.c odd 4 1
135.2.k.b ✓ 42 135.r odd 36 1
405.2.k.b 42 15.e even 4 1
405.2.k.b 42 135.q even 36 1
675.2.l.e 42 5.c odd 4 1
675.2.l.e 42 135.r odd 36 1
675.2.u.d 84 1.a even 1 1 trivial
675.2.u.d 84 5.b even 2 1 inner
675.2.u.d 84 27.e even 9 1 inner
675.2.u.d 84 135.p even 18 1 inner
3645.2.a.k 21 135.q even 36 1
3645.2.a.l 21 135.r odd 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} + 6 T_{2}^{80} - 755 T_{2}^{78} - 261 T_{2}^{76} - 1530 T_{2}^{74} + 397962 T_{2}^{72} + \cdots + 855036081 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display