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Properties

Label 900.2.bq.a
Level $900$
Weight $2$
Character orbit 900.bq
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(169,900)] 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("900.169"); 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, 27])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bq (of order \(30\), 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 \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 35 q^{15} - 12 q^{21} - 12 q^{25} - 15 q^{27} + 12 q^{29} - 6 q^{31} - 25 q^{33} + 34 q^{35} + 8 q^{39} - 16 q^{41} - 69 q^{45} + 25 q^{47} + 120 q^{49} - 2 q^{51}+ \cdots - 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 0 −1.69178 − 0.371316i 0 −2.20641 − 0.362991i 0 1.22444 + 0.706932i 0 2.72425 + 1.25637i 0
169.2 0 −1.68537 + 0.399405i 0 −0.129921 + 2.23229i 0 2.62185 + 1.51373i 0 2.68095 − 1.34629i 0
169.3 0 −1.67753 + 0.431145i 0 −0.383154 − 2.20300i 0 −3.46041 − 1.99787i 0 2.62823 − 1.44652i 0
169.4 0 −1.54179 + 0.789230i 0 1.92812 − 1.13241i 0 3.46038 + 1.99785i 0 1.75423 − 2.43365i 0
169.5 0 −1.48544 − 0.890766i 0 2.21953 − 0.271422i 0 −0.920289 − 0.531329i 0 1.41307 + 2.64636i 0
169.6 0 −1.44729 − 0.951493i 0 −1.23418 + 1.86462i 0 −3.93849 − 2.27389i 0 1.18932 + 2.75418i 0
169.7 0 −1.42125 − 0.989977i 0 0.147145 − 2.23122i 0 2.29699 + 1.32617i 0 1.03989 + 2.81400i 0
169.8 0 −1.38548 + 1.03944i 0 1.17355 + 1.90336i 0 −1.37169 − 0.791945i 0 0.839112 − 2.88026i 0
169.9 0 −1.14511 − 1.29951i 0 1.16657 + 1.90765i 0 0.233273 + 0.134680i 0 −0.377463 + 2.97616i 0
169.10 0 −1.04456 + 1.38163i 0 −2.23594 − 0.0237652i 0 −0.0756016 − 0.0436486i 0 −0.817781 − 2.88639i 0
169.11 0 −0.711027 − 1.57938i 0 −1.41435 + 1.73194i 0 3.40643 + 1.96670i 0 −1.98888 + 2.24596i 0
169.12 0 −0.494354 + 1.66000i 0 2.23121 + 0.147318i 0 −3.85003 − 2.22282i 0 −2.51123 − 1.64126i 0
169.13 0 −0.286387 − 1.70821i 0 1.28846 − 1.82753i 0 −2.61918 − 1.51218i 0 −2.83597 + 0.978417i 0
169.14 0 −0.0466535 + 1.73142i 0 −0.283641 − 2.21801i 0 0.423988 + 0.244790i 0 −2.99565 − 0.161554i 0
169.15 0 −0.0348683 + 1.73170i 0 2.12968 − 0.681505i 0 1.77046 + 1.02218i 0 −2.99757 − 0.120763i 0
169.16 0 0.00533247 − 1.73204i 0 1.77841 + 1.35546i 0 1.40979 + 0.813945i 0 −2.99994 − 0.0184721i 0
169.17 0 0.398144 − 1.68567i 0 −1.99592 + 1.00812i 0 −1.56304 − 0.902424i 0 −2.68296 − 1.34228i 0
169.18 0 0.499953 + 1.65833i 0 −1.04126 + 1.97883i 0 −1.99836 − 1.15376i 0 −2.50009 + 1.65817i 0
169.19 0 0.677813 − 1.59392i 0 0.241444 − 2.22299i 0 0.158252 + 0.0913667i 0 −2.08114 − 2.16075i 0
169.20 0 0.809585 − 1.53120i 0 −1.53030 − 1.63040i 0 3.10904 + 1.79501i 0 −1.68915 − 2.47927i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.30
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.e even 10 1 inner
225.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bq.a ✓ 240
9.c even 3 1 inner 900.2.bq.a ✓ 240
25.e even 10 1 inner 900.2.bq.a ✓ 240
225.u even 30 1 inner 900.2.bq.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.bq.a ✓ 240 1.a even 1 1 trivial
900.2.bq.a ✓ 240 9.c even 3 1 inner
900.2.bq.a ✓ 240 25.e even 10 1 inner
900.2.bq.a ✓ 240 225.u even 30 1 inner

Hecke kernels

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