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Properties

Label 900.2.bg.a
Level $900$
Weight $2$
Character orbit 900.bg
Analytic conductor $7.187$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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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] = [900,2,Mod(61,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.61"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 20, 24])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bg (of order \(15\), degree \(8\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 2 q^{3} - 4 q^{5} - 2 q^{9} + 4 q^{11} + 33 q^{15} - 12 q^{17} + 12 q^{21} - 12 q^{23} - 12 q^{25} + 13 q^{27} + 12 q^{29} + 6 q^{31} + 31 q^{33} + 14 q^{35} - 12 q^{37} + 8 q^{39} + 16 q^{41} + 19 q^{45}+ \cdots - 66 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 0 −1.67673 + 0.434241i 0 −1.00789 + 1.99604i 0 0.516228 − 0.894133i 0 2.62287 − 1.45621i 0
61.2 0 −1.64289 + 0.548549i 0 1.37194 − 1.76573i 0 −2.37185 + 4.10816i 0 2.39819 − 1.80241i 0
61.3 0 −1.62661 − 0.595101i 0 2.22506 + 0.221639i 0 1.44975 − 2.51104i 0 2.29171 + 1.93599i 0
61.4 0 −1.60007 − 0.663151i 0 1.06657 + 1.96530i 0 −2.15417 + 3.73113i 0 2.12046 + 2.12218i 0
61.5 0 −1.59842 + 0.667132i 0 −0.502718 − 2.17882i 0 1.58299 − 2.74182i 0 2.10987 − 2.13271i 0
61.6 0 −1.58608 − 0.695944i 0 −2.23484 + 0.0742255i 0 −0.867218 + 1.50207i 0 2.03132 + 2.20765i 0
61.7 0 −1.27651 + 1.17069i 0 2.23269 − 0.122868i 0 0.279123 − 0.483455i 0 0.258969 − 2.98880i 0
61.8 0 −1.26899 + 1.17885i 0 −1.96299 − 1.07083i 0 −0.910263 + 1.57662i 0 0.220648 − 2.99187i 0
61.9 0 −0.893308 − 1.48391i 0 −2.05489 − 0.881726i 0 2.49276 − 4.31759i 0 −1.40400 + 2.65118i 0
61.10 0 −0.878097 − 1.49297i 0 0.0507636 − 2.23549i 0 −0.545715 + 0.945206i 0 −1.45789 + 2.62194i 0
61.11 0 −0.874214 + 1.49524i 0 0.535572 + 2.17098i 0 1.54954 − 2.68389i 0 −1.47150 − 2.61432i 0
61.12 0 −0.671690 − 1.59651i 0 0.0838407 + 2.23450i 0 −0.905890 + 1.56905i 0 −2.09766 + 2.14471i 0
61.13 0 −0.637479 + 1.61047i 0 −1.19376 + 1.89075i 0 −2.07969 + 3.60213i 0 −2.18724 − 2.05328i 0
61.14 0 −0.631875 − 1.61268i 0 0.308404 + 2.21470i 0 1.91374 − 3.31469i 0 −2.20147 + 2.03802i 0
61.15 0 −0.312229 − 1.70368i 0 2.01459 − 0.970277i 0 −0.856597 + 1.48367i 0 −2.80503 + 1.06387i 0
61.16 0 −0.175370 + 1.72315i 0 −1.95088 − 1.09274i 0 1.38079 − 2.39159i 0 −2.93849 − 0.604379i 0
61.17 0 0.0513649 + 1.73129i 0 2.13415 + 0.667390i 0 −0.801301 + 1.38789i 0 −2.99472 + 0.177855i 0
61.18 0 0.179429 + 1.72273i 0 0.694082 − 2.12562i 0 −0.0881730 + 0.152720i 0 −2.93561 + 0.618215i 0
61.19 0 0.579526 − 1.63222i 0 −2.01757 − 0.964056i 0 −1.83528 + 3.17879i 0 −2.32830 − 1.89183i 0
61.20 0 0.626630 − 1.61472i 0 −1.66045 + 1.49764i 0 0.0756354 − 0.131004i 0 −2.21467 − 2.02367i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.d even 5 1 inner
225.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bg.a ✓ 240
9.c even 3 1 inner 900.2.bg.a ✓ 240
25.d even 5 1 inner 900.2.bg.a ✓ 240
225.q even 15 1 inner 900.2.bg.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.bg.a ✓ 240 1.a even 1 1 trivial
900.2.bg.a ✓ 240 9.c even 3 1 inner
900.2.bg.a ✓ 240 25.d even 5 1 inner
900.2.bg.a ✓ 240 225.q even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(900, [\chi])\).