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Properties

Label 875.2.bb.c
Level $875$
Weight $2$
Character orbit 875.bb
Analytic conductor $6.987$
Analytic rank $0$
Dimension $288$
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] = [875,2,Mod(82,875)] 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("875.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([27, 50])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.bb (of order \(60\), degree \(16\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [288,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{60})\)
Twist minimal: no (minimal twist has level 175)
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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 8 q^{2} + 24 q^{3} - 10 q^{4} + 10 q^{7} + 36 q^{8} - 10 q^{9} - 6 q^{11} + 36 q^{12} - 20 q^{14} - 30 q^{16} + 42 q^{17} + 14 q^{18} - 30 q^{19} - 12 q^{21} - 32 q^{22} + 40 q^{23} - 48 q^{26}+ \cdots - 222 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −2.48322 − 0.953219i 1.59480 − 0.0835800i 3.77146 + 3.39584i 0 −4.03991 − 1.31265i −0.777975 + 2.52879i −3.71326 − 7.28769i −0.447157 + 0.0469981i 0
82.2 −2.14795 − 0.824519i 0.448786 − 0.0235199i 2.44755 + 2.20378i 0 −0.983361 − 0.319513i −2.20882 − 1.45640i −1.35109 − 2.65167i −2.78271 + 0.292475i 0
82.3 −1.86573 − 0.716188i −0.228491 + 0.0119747i 1.48175 + 1.33417i 0 0.434879 + 0.141301i 2.53073 + 0.771626i 0.00554342 + 0.0108796i −2.93150 + 0.308113i 0
82.4 −1.46963 − 0.564138i −2.21961 + 0.116325i 0.355272 + 0.319888i 0 3.32763 + 1.08121i −0.803682 + 2.52073i 1.08768 + 2.13468i 1.92958 − 0.202807i 0
82.5 −1.26154 − 0.484261i 2.64206 − 0.138464i −0.129310 − 0.116431i 0 −3.40012 − 1.10477i 1.71479 − 2.01482i 1.33370 + 2.61753i 3.97774 − 0.418077i 0
82.6 −1.09256 − 0.419394i −1.42745 + 0.0748097i −0.468495 − 0.421835i 0 1.59095 + 0.516932i −0.744971 − 2.53870i 1.39754 + 2.74284i −0.951539 + 0.100011i 0
82.7 −0.947352 − 0.363654i 3.05010 − 0.159849i −0.721058 − 0.649244i 0 −2.94765 − 0.957749i −2.27212 + 1.35553i 1.36837 + 2.68558i 6.29399 − 0.661525i 0
82.8 −0.653633 − 0.250906i 1.07442 − 0.0563079i −1.12201 − 1.01026i 0 −0.716404 − 0.232774i 2.24299 + 1.40321i 1.11561 + 2.18951i −1.83236 + 0.192589i 0
82.9 −0.343451 − 0.131839i −2.87847 + 0.150854i −1.38571 − 1.24770i 0 1.00850 + 0.327683i 0.714065 − 2.54757i 0.645463 + 1.26679i 5.27927 − 0.554874i 0
82.10 0.195749 + 0.0751409i −0.218665 + 0.0114597i −1.45362 − 1.30884i 0 −0.0436645 − 0.0141875i −2.12696 + 1.57354i −0.376577 − 0.739075i −2.93588 + 0.308574i 0
82.11 0.411843 + 0.158092i 1.36161 − 0.0713588i −1.34167 − 1.20804i 0 0.572049 + 0.185870i −2.62799 − 0.306079i −0.762125 − 1.49575i −1.13469 + 0.119260i 0
82.12 0.634788 + 0.243672i −2.14109 + 0.112210i −1.14271 − 1.02890i 0 −1.38648 − 0.450496i 0.628287 + 2.57007i −1.09205 − 2.14326i 1.58813 − 0.166919i 0
82.13 1.32882 + 0.510085i −0.741032 + 0.0388358i 0.0192759 + 0.0173561i 0 −1.00450 − 0.326383i 2.35634 − 1.20318i −1.27562 − 2.50354i −2.43595 + 0.256028i 0
82.14 1.53966 + 0.591020i 2.58725 − 0.135592i 0.534955 + 0.481675i 0 4.06362 + 1.32035i 2.63441 + 0.244749i −0.958472 − 1.88111i 3.69193 − 0.388038i 0
82.15 1.73544 + 0.666174i 1.78758 − 0.0936832i 1.08168 + 0.973949i 0 3.16465 + 1.02826i 0.0904486 + 2.64420i −0.459481 − 0.901783i 0.203110 − 0.0213477i 0
82.16 1.83327 + 0.703726i −0.797776 + 0.0418096i 1.37935 + 1.24198i 0 −1.49196 − 0.484767i −2.02454 − 1.70330i −0.128282 − 0.251768i −2.34887 + 0.246876i 0
82.17 2.43845 + 0.936034i 2.16962 − 0.113705i 3.58360 + 3.22669i 0 5.39694 + 1.75357i −2.32567 − 1.26145i 3.34656 + 6.56798i 1.71075 − 0.179807i 0
82.18 2.43914 + 0.936298i −2.98133 + 0.156245i 3.58646 + 3.22926i 0 −7.41816 − 2.41031i 1.17017 + 2.37291i 3.35206 + 6.57879i 5.88033 − 0.618047i 0
143.1 −0.127170 − 2.42654i 1.78675 + 1.44688i −3.88289 + 0.408108i 0 3.28370 − 4.51962i −1.54352 + 2.14884i 0.723845 + 4.57018i 0.475273 + 2.23598i 0
143.2 −0.126421 − 2.41226i −0.433801 − 0.351285i −3.81395 + 0.400862i 0 −0.792548 + 1.09085i −2.23919 − 1.40926i 0.693391 + 4.37790i −0.558953 − 2.62967i 0
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 82.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 875.2.bb.c 288
5.b even 2 1 175.2.x.a ✓ 288
5.c odd 4 1 875.2.bb.a 288
5.c odd 4 1 875.2.bb.b 288
7.d odd 6 1 inner 875.2.bb.c 288
25.d even 5 1 875.2.bb.a 288
25.e even 10 1 875.2.bb.b 288
25.f odd 20 1 175.2.x.a ✓ 288
25.f odd 20 1 inner 875.2.bb.c 288
35.i odd 6 1 175.2.x.a ✓ 288
35.k even 12 1 875.2.bb.a 288
35.k even 12 1 875.2.bb.b 288
175.u odd 30 1 875.2.bb.b 288
175.v odd 30 1 875.2.bb.a 288
175.x even 60 1 175.2.x.a ✓ 288
175.x even 60 1 inner 875.2.bb.c 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.x.a ✓ 288 5.b even 2 1
175.2.x.a ✓ 288 25.f odd 20 1
175.2.x.a ✓ 288 35.i odd 6 1
175.2.x.a ✓ 288 175.x even 60 1
875.2.bb.a 288 5.c odd 4 1
875.2.bb.a 288 25.d even 5 1
875.2.bb.a 288 35.k even 12 1
875.2.bb.a 288 175.v odd 30 1
875.2.bb.b 288 5.c odd 4 1
875.2.bb.b 288 25.e even 10 1
875.2.bb.b 288 35.k even 12 1
875.2.bb.b 288 175.u odd 30 1
875.2.bb.c 288 1.a even 1 1 trivial
875.2.bb.c 288 7.d odd 6 1 inner
875.2.bb.c 288 25.f odd 20 1 inner
875.2.bb.c 288 175.x even 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 8 T_{2}^{287} + 37 T_{2}^{286} - 132 T_{2}^{285} + 524 T_{2}^{284} - 2076 T_{2}^{283} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display