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Properties

Label 927.2.n.a
Level $927$
Weight $2$
Character orbit 927.n
Analytic conductor $7.402$
Analytic rank $0$
Dimension $204$
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] = [927,2,Mod(308,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.308");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.n (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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(102\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 6 q^{2} + 98 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q - 6 q^{2} + 98 q^{4} - 14 q^{9} - 4 q^{13} + 12 q^{14} - 6 q^{15} - 94 q^{16} - 26 q^{18} - 12 q^{19} - 6 q^{23} - 102 q^{25} + 16 q^{28} - 42 q^{29} + 56 q^{30} + 18 q^{32} + 6 q^{33} + 2 q^{34} - 48 q^{36} + 42 q^{38} + 6 q^{41} - 40 q^{46} - 78 q^{49} - 54 q^{50} + 26 q^{52} + 48 q^{56} - 22 q^{58} - 42 q^{59} + 18 q^{60} - 28 q^{61} + 52 q^{63} - 176 q^{64} - 24 q^{66} + 66 q^{68} + 18 q^{72} - 26 q^{76} + 20 q^{79} + 42 q^{81} - 16 q^{82} - 126 q^{83} + 88 q^{91} - 78 q^{92} - 86 q^{93} - 10 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
308.1 −2.38576 − 1.37742i −0.0848428 + 1.72997i 2.79458 + 4.84035i 1.77625 + 3.07655i 2.58531 − 4.01044i −1.18557 + 2.05348i − 9.88756i −2.98560 − 0.293551i − 9.78656i
308.2 −2.38576 − 1.37742i 0.0848428 − 1.72997i 2.79458 + 4.84035i −1.77625 − 3.07655i −2.58531 + 4.01044i −1.18557 + 2.05348i − 9.88756i −2.98560 − 0.293551i 9.78656i
308.3 −2.33273 − 1.34680i −1.59334 − 0.679175i 2.62774 + 4.55138i 1.21354 + 2.10191i 2.80211 + 3.73024i 1.24513 − 2.15664i − 8.76897i 2.07744 + 2.16431i − 6.53758i
308.4 −2.33273 − 1.34680i 1.59334 + 0.679175i 2.62774 + 4.55138i −1.21354 − 2.10191i −2.80211 − 3.73024i 1.24513 − 2.15664i − 8.76897i 2.07744 + 2.16431i 6.53758i
308.5 −2.30910 − 1.33316i −1.21101 + 1.23832i 2.55463 + 4.42475i −0.798011 − 1.38220i 4.44724 − 1.24494i −0.599739 + 1.03878i − 8.29028i −0.0668936 − 2.99925i 4.25550i
308.6 −2.30910 − 1.33316i 1.21101 − 1.23832i 2.55463 + 4.42475i 0.798011 + 1.38220i −4.44724 + 1.24494i −0.599739 + 1.03878i − 8.29028i −0.0668936 − 2.99925i − 4.25550i
308.7 −2.19716 − 1.26853i −1.48629 − 0.889347i 2.21834 + 3.84227i −1.15731 − 2.00452i 2.13745 + 3.83944i 1.13885 − 1.97255i − 6.18199i 1.41812 + 2.64366i 5.87234i
308.8 −2.19716 − 1.26853i 1.48629 + 0.889347i 2.21834 + 3.84227i 1.15731 + 2.00452i −2.13745 − 3.83944i 1.13885 − 1.97255i − 6.18199i 1.41812 + 2.64366i − 5.87234i
308.9 −2.05646 − 1.18730i −0.144800 − 1.72599i 1.81936 + 3.15122i 0.580200 + 1.00494i −1.75149 + 3.72135i 1.35112 − 2.34020i − 3.89128i −2.95807 + 0.499846i − 2.75548i
308.10 −2.05646 − 1.18730i 0.144800 + 1.72599i 1.81936 + 3.15122i −0.580200 − 1.00494i 1.75149 − 3.72135i 1.35112 − 2.34020i − 3.89128i −2.95807 + 0.499846i 2.75548i
308.11 −1.96035 − 1.13181i −1.71885 − 0.213418i 1.56198 + 2.70544i −1.24094 − 2.14937i 3.12800 + 2.36379i −2.31576 + 4.01101i − 2.54423i 2.90891 + 0.733668i 5.61801i
308.12 −1.96035 − 1.13181i 1.71885 + 0.213418i 1.56198 + 2.70544i 1.24094 + 2.14937i −3.12800 − 2.36379i −2.31576 + 4.01101i − 2.54423i 2.90891 + 0.733668i − 5.61801i
308.13 −1.92782 − 1.11303i −1.69196 + 0.370517i 1.47765 + 2.55937i 1.29770 + 2.24769i 3.67418 + 1.16890i −0.836598 + 1.44903i − 2.12657i 2.72543 − 1.25380i − 5.77751i
308.14 −1.92782 − 1.11303i 1.69196 − 0.370517i 1.47765 + 2.55937i −1.29770 − 2.24769i −3.67418 − 1.16890i −0.836598 + 1.44903i − 2.12657i 2.72543 − 1.25380i 5.77751i
308.15 −1.79225 − 1.03475i −0.753714 − 1.55946i 1.14143 + 1.97702i −1.12511 − 1.94874i −0.262816 + 3.57485i 0.306817 − 0.531422i − 0.585395i −1.86383 + 2.35077i 4.65684i
308.16 −1.79225 − 1.03475i 0.753714 + 1.55946i 1.14143 + 1.97702i 1.12511 + 1.94874i 0.262816 − 3.57485i 0.306817 − 0.531422i − 0.585395i −1.86383 + 2.35077i − 4.65684i
308.17 −1.76568 − 1.01942i −1.24006 + 1.20923i 1.07842 + 1.86788i 1.01273 + 1.75409i 3.42227 − 0.870975i 2.38124 − 4.12442i − 0.319777i 0.0755195 − 2.99905i − 4.12956i
308.18 −1.76568 − 1.01942i 1.24006 − 1.20923i 1.07842 + 1.86788i −1.01273 − 1.75409i −3.42227 + 0.870975i 2.38124 − 4.12442i − 0.319777i 0.0755195 − 2.99905i 4.12956i
308.19 −1.73339 − 1.00077i −1.68677 − 0.393434i 1.00310 + 1.73742i −0.212408 − 0.367901i 2.53010 + 2.37006i −0.957553 + 1.65853i − 0.0124109i 2.69042 + 1.32727i 0.850290i
308.20 −1.73339 − 1.00077i 1.68677 + 0.393434i 1.00310 + 1.73742i 0.212408 + 0.367901i −2.53010 − 2.37006i −0.957553 + 1.65853i − 0.0124109i 2.69042 + 1.32727i − 0.850290i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 308.102
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
103.b odd 2 1 inner
927.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.n.a ✓ 204
9.d odd 6 1 inner 927.2.n.a ✓ 204
103.b odd 2 1 inner 927.2.n.a ✓ 204
927.n even 6 1 inner 927.2.n.a ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.n.a ✓ 204 1.a even 1 1 trivial
927.2.n.a ✓ 204 9.d odd 6 1 inner
927.2.n.a ✓ 204 103.b odd 2 1 inner
927.2.n.a ✓ 204 927.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(927, [\chi])\).