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Properties

Label 990.2.o.a
Level $990$
Weight $2$
Character orbit 990.o
Analytic conductor $7.905$
Analytic rank $0$
Dimension $144$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(329,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("990.329"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(72\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 72 q^{4} - 6 q^{5} + 12 q^{9} + 12 q^{11} - 2 q^{15} - 72 q^{16} - 6 q^{20} + 6 q^{25} + 12 q^{31} + 12 q^{45} - 72 q^{49} + 12 q^{55} + 60 q^{59} - 4 q^{60} - 144 q^{64} - 36 q^{66} - 8 q^{69}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
329.1 −0.866025 + 0.500000i −1.72505 − 0.155544i 0.500000 − 0.866025i −2.11767 + 0.717978i 1.57171 − 0.727821i 0.0167009 + 0.0289269i 1.00000i 2.95161 + 0.536644i 1.47496 − 1.68062i
329.2 −0.866025 + 0.500000i −1.72406 + 0.166190i 0.500000 − 0.866025i 1.97778 + 1.04326i 1.40998 − 1.00595i 1.12523 + 1.94895i 1.00000i 2.94476 − 0.573043i −2.23444 + 0.0853978i
329.3 −0.866025 + 0.500000i −1.67359 − 0.446217i 0.500000 − 0.866025i −0.339852 − 2.21009i 1.67248 − 0.450358i 1.06306 + 1.84128i 1.00000i 2.60178 + 1.49356i 1.39937 + 1.74407i
329.4 −0.866025 + 0.500000i −1.62901 + 0.588492i 0.500000 − 0.866025i 1.65730 − 1.50112i 1.11652 − 1.32415i −0.753042 − 1.30431i 1.00000i 2.30735 − 1.91732i −0.684705 + 2.12866i
329.5 −0.866025 + 0.500000i −1.56736 − 0.737142i 0.500000 − 0.866025i −0.558209 − 2.16527i 1.72595 − 0.145297i −1.97281 − 3.41701i 1.00000i 1.91324 + 2.31074i 1.56606 + 1.59608i
329.6 −0.866025 + 0.500000i −1.49496 − 0.874693i 0.500000 − 0.866025i 0.235152 + 2.22367i 1.73202 + 0.0100253i −0.109626 − 0.189878i 1.00000i 1.46982 + 2.61527i −1.31548 − 1.80818i
329.7 −0.866025 + 0.500000i −1.46936 + 0.917053i 0.500000 − 0.866025i 0.434042 + 2.19354i 0.813975 − 1.52887i 0.730956 + 1.26605i 1.00000i 1.31803 − 2.69496i −1.47266 − 1.68264i
329.8 −0.866025 + 0.500000i −1.41927 + 0.992805i 0.500000 − 0.866025i −0.425673 − 2.19518i 0.732725 − 1.56943i 1.49028 + 2.58123i 1.00000i 1.02868 − 2.81812i 1.46623 + 1.68824i
329.9 −0.866025 + 0.500000i −1.40449 + 1.01361i 0.500000 − 0.866025i −2.22159 − 0.254024i 0.709522 − 1.58006i −2.11537 − 3.66393i 1.00000i 0.945199 − 2.84721i 2.05097 − 0.890805i
329.10 −0.866025 + 0.500000i −1.13835 + 1.30543i 0.500000 − 0.866025i 1.78534 + 1.34631i 0.333128 − 1.69971i −2.44430 − 4.23366i 1.00000i −0.408300 − 2.97209i −2.21931 − 0.273267i
329.11 −0.866025 + 0.500000i −1.11889 − 1.32215i 0.500000 − 0.866025i 2.15002 + 0.614355i 1.63006 + 0.585572i −1.04399 − 1.80824i 1.00000i −0.496170 + 2.95868i −2.16915 + 0.542961i
329.12 −0.866025 + 0.500000i −1.03091 − 1.39184i 0.500000 − 0.866025i −1.73198 + 1.41430i 1.58871 + 0.689918i 2.43971 + 4.22570i 1.00000i −0.874459 + 2.86972i 0.792789 − 2.09081i
329.13 −0.866025 + 0.500000i −0.782108 − 1.54542i 0.500000 − 0.866025i 1.55321 − 1.60859i 1.45003 + 0.947315i −0.781159 − 1.35301i 1.00000i −1.77662 + 2.41736i −0.540822 + 2.16968i
329.14 −0.866025 + 0.500000i −0.690097 + 1.58864i 0.500000 − 0.866025i −1.04557 + 1.97656i −0.196677 − 1.72085i 0.998734 + 1.72986i 1.00000i −2.04753 − 2.19263i −0.0827910 − 2.23453i
329.15 −0.866025 + 0.500000i −0.648796 + 1.60595i 0.500000 − 0.866025i −2.04128 − 0.912789i −0.241100 − 1.71519i −0.226164 − 0.391728i 1.00000i −2.15813 − 2.08386i 2.22419 − 0.230141i
329.16 −0.866025 + 0.500000i −0.468711 − 1.66743i 0.500000 − 0.866025i −1.39844 + 1.74481i 1.23963 + 1.20968i −1.50667 − 2.60963i 1.00000i −2.56062 + 1.56308i 0.338674 − 2.21027i
329.17 −0.866025 + 0.500000i −0.353989 − 1.69549i 0.500000 − 0.866025i −1.45625 − 1.69686i 1.15431 + 1.29134i 1.23806 + 2.14438i 1.00000i −2.74938 + 1.20037i 2.10958 + 0.741395i
329.18 −0.866025 + 0.500000i −0.212993 + 1.71890i 0.500000 − 0.866025i 2.22842 − 0.184830i −0.674995 − 1.59511i 1.85041 + 3.20500i 1.00000i −2.90927 − 0.732230i −1.83745 + 1.27428i
329.19 −0.866025 + 0.500000i 0.212993 − 1.71890i 0.500000 − 0.866025i 0.954141 + 2.02228i 0.674995 + 1.59511i 1.85041 + 3.20500i 1.00000i −2.90927 − 0.732230i −1.83745 − 1.27428i
329.20 −0.866025 + 0.500000i 0.353989 + 1.69549i 0.500000 − 0.866025i −2.19765 − 0.412724i −1.15431 − 1.29134i 1.23806 + 2.14438i 1.00000i −2.74938 + 1.20037i 2.10958 − 0.741395i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 329.72
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
9.d odd 6 1 inner
11.b odd 2 1 inner
45.h odd 6 1 inner
55.d odd 2 1 inner
99.g even 6 1 inner
495.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.o.a ✓ 144
5.b even 2 1 inner 990.2.o.a ✓ 144
9.d odd 6 1 inner 990.2.o.a ✓ 144
11.b odd 2 1 inner 990.2.o.a ✓ 144
45.h odd 6 1 inner 990.2.o.a ✓ 144
55.d odd 2 1 inner 990.2.o.a ✓ 144
99.g even 6 1 inner 990.2.o.a ✓ 144
495.r even 6 1 inner 990.2.o.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.o.a ✓ 144 1.a even 1 1 trivial
990.2.o.a ✓ 144 5.b even 2 1 inner
990.2.o.a ✓ 144 9.d odd 6 1 inner
990.2.o.a ✓ 144 11.b odd 2 1 inner
990.2.o.a ✓ 144 45.h odd 6 1 inner
990.2.o.a ✓ 144 55.d odd 2 1 inner
990.2.o.a ✓ 144 99.g even 6 1 inner
990.2.o.a ✓ 144 495.r even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(990, [\chi])\).