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Properties

Label 900.2.bt.a
Level $900$
Weight $2$
Character orbit 900.bt
Analytic conductor $7.187$
Analytic rank $0$
Dimension $480$
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(77,900)] 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("900.77"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([0, 50, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bt (of order \(60\), degree \(16\), 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: \(480\)
Relative dimension: \(30\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q + 2 q^{3} - 10 q^{15} + 24 q^{23} + 24 q^{25} + 26 q^{27} + 26 q^{33} - 12 q^{37} + 40 q^{39} + 90 q^{45} + 42 q^{47} - 12 q^{55} + 94 q^{57} + 30 q^{59} + 56 q^{63} + 24 q^{65} - 6 q^{67} + 70 q^{69}+ \cdots + 18 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 0 −1.73118 − 0.0550599i 0 0.490512 − 2.18160i 0 −0.208136 + 0.776775i 0 2.99394 + 0.190637i 0
77.2 0 −1.71252 − 0.259372i 0 1.42106 + 1.72643i 0 −0.690144 + 2.57565i 0 2.86545 + 0.888359i 0
77.3 0 −1.69504 + 0.356139i 0 1.61164 + 1.55003i 0 0.149305 − 0.557215i 0 2.74633 − 1.20734i 0
77.4 0 −1.52917 − 0.813405i 0 −1.40476 + 1.73973i 0 1.06949 − 3.99140i 0 1.67675 + 2.48767i 0
77.5 0 −1.51474 + 0.839976i 0 −2.21405 + 0.312996i 0 −1.19563 + 4.46214i 0 1.58888 − 2.54469i 0
77.6 0 −1.43603 − 0.968403i 0 −2.05046 − 0.891969i 0 0.199875 − 0.745944i 0 1.12439 + 2.78132i 0
77.7 0 −1.21535 + 1.23407i 0 −1.29947 + 1.81972i 0 0.569362 − 2.12489i 0 −0.0458376 − 2.99965i 0
77.8 0 −1.21271 + 1.23666i 0 2.16904 − 0.543379i 0 0.155273 − 0.579487i 0 −0.0586764 − 2.99943i 0
77.9 0 −1.20723 + 1.24202i 0 −1.27582 − 1.83638i 0 0.910740 − 3.39893i 0 −0.0852039 − 2.99879i 0
77.10 0 −1.16258 − 1.28390i 0 0.843055 − 2.07105i 0 −1.06196 + 3.96329i 0 −0.296812 + 2.98528i 0
77.11 0 −0.861206 − 1.50277i 0 2.21604 − 0.298606i 0 −0.253773 + 0.947095i 0 −1.51665 + 2.58839i 0
77.12 0 −0.853917 − 1.50693i 0 0.726094 + 2.11490i 0 0.669419 − 2.49830i 0 −1.54165 + 2.57358i 0
77.13 0 −0.478671 + 1.66459i 0 1.71117 − 1.43940i 0 −0.290896 + 1.08564i 0 −2.54175 − 1.59359i 0
77.14 0 −0.244653 − 1.71469i 0 −0.0557366 − 2.23537i 0 0.921281 − 3.43827i 0 −2.88029 + 0.839006i 0
77.15 0 −0.218627 − 1.71820i 0 −2.23563 − 0.0442100i 0 −0.426153 + 1.59043i 0 −2.90440 + 0.751288i 0
77.16 0 −0.0932360 + 1.72954i 0 −0.308380 − 2.21470i 0 −0.633695 + 2.36498i 0 −2.98261 − 0.322511i 0
77.17 0 −0.0663451 + 1.73078i 0 0.194165 + 2.22762i 0 −0.917725 + 3.42499i 0 −2.99120 − 0.229658i 0
77.18 0 0.297923 − 1.70624i 0 −0.490393 + 2.18163i 0 −0.736506 + 2.74868i 0 −2.82248 − 1.01665i 0
77.19 0 0.429867 + 1.67786i 0 2.23540 − 0.0547012i 0 1.31646 − 4.91308i 0 −2.63043 + 1.44251i 0
77.20 0 0.543843 + 1.64446i 0 −1.93015 − 1.12894i 0 −0.204965 + 0.764939i 0 −2.40847 + 1.78865i 0
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
25.f odd 20 1 inner
225.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bt.a ✓ 480
9.d odd 6 1 inner 900.2.bt.a ✓ 480
25.f odd 20 1 inner 900.2.bt.a ✓ 480
225.w even 60 1 inner 900.2.bt.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.bt.a ✓ 480 1.a even 1 1 trivial
900.2.bt.a ✓ 480 9.d odd 6 1 inner
900.2.bt.a ✓ 480 25.f odd 20 1 inner
900.2.bt.a ✓ 480 225.w even 60 1 inner

Hecke kernels

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