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Properties

Label 875.2.bb.b
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 = [288,2] 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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} + 6 q^{3} + 10 q^{4} - 10 q^{7} + 64 q^{8} + 10 q^{9} - 6 q^{11} - 6 q^{12} + 20 q^{14} - 30 q^{16} - 12 q^{17} - 14 q^{18} + 30 q^{19} - 12 q^{21} - 8 q^{22} + 30 q^{23} - 48 q^{26} - 58 q^{28}+ \cdots + 62 q^{98}+O(q^{100}) \) 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 −2.43619 − 0.935166i −0.773359 + 0.0405301i 3.57420 + 3.21822i 0 1.92195 + 0.624480i −0.425622 − 2.61129i −3.32847 − 6.53248i −2.38712 + 0.250897i 0
82.2 −2.32523 − 0.892573i −1.91879 + 0.100560i 3.12372 + 2.81261i 0 4.55139 + 1.47884i −0.589885 + 2.57915i −2.49145 − 4.88974i 0.688086 − 0.0723207i 0
82.3 −2.09413 − 0.803861i 2.97123 − 0.155716i 2.25290 + 2.02852i 0 −6.34732 − 2.06237i 1.16740 + 2.37427i −1.05050 − 2.06173i 5.82041 − 0.611750i 0
82.4 −1.78057 − 0.683496i 1.92325 − 0.100793i 1.21696 + 1.09576i 0 −3.49337 − 1.13506i −2.62152 − 0.357251i 0.313803 + 0.615873i 0.705160 − 0.0741153i 0
82.5 −1.67005 − 0.641073i 0.719894 − 0.0377280i 0.891813 + 0.802992i 0 −1.22645 − 0.398497i 1.50450 − 2.17635i 0.649660 + 1.27503i −2.46674 + 0.259265i 0
82.6 −1.30359 − 0.500401i −3.22509 + 0.169020i −0.0373446 − 0.0336253i 0 4.28878 + 1.39351i 2.37511 + 1.16570i 1.29970 + 2.55081i 7.38909 − 0.776624i 0
82.7 −0.957274 − 0.367463i −1.16281 + 0.0609404i −0.704945 − 0.634736i 0 1.13552 + 0.368954i 2.47964 − 0.922709i 1.37261 + 2.69390i −1.63515 + 0.171861i 0
82.8 −0.766047 − 0.294058i 0.140222 − 0.00734873i −0.985932 − 0.887737i 0 −0.109578 − 0.0356039i −1.23154 + 2.34165i 1.23927 + 2.43220i −2.96396 + 0.311524i 0
82.9 −0.252007 − 0.0967363i −1.95874 + 0.102653i −1.43214 − 1.28950i 0 0.503546 + 0.163612i −2.64466 − 0.0759508i 0.481263 + 0.944532i 0.842568 − 0.0885574i 0
82.10 −0.00521563 − 0.00200209i 2.07789 − 0.108898i −1.48627 − 1.33824i 0 −0.0110555 − 0.00359216i −1.62930 − 2.08455i 0.0101452 + 0.0199110i 1.32220 − 0.138969i 0
82.11 0.278809 + 0.107025i 3.11927 − 0.163474i −1.42001 − 1.27858i 0 0.887175 + 0.288261i 2.59603 − 0.510505i −0.530235 − 1.04064i 6.71954 − 0.706252i 0
82.12 0.596349 + 0.228917i −0.0491972 + 0.00257832i −1.18306 − 1.06523i 0 −0.0299289 − 0.00972449i 1.82620 + 1.91442i −1.04166 − 2.04438i −2.98115 + 0.313332i 0
82.13 0.782618 + 0.300419i −2.02474 + 0.106112i −0.964050 − 0.868035i 0 −1.61648 − 0.525226i −0.522392 − 2.59367i −1.25487 − 2.46282i 1.10477 − 0.116116i 0
82.14 1.47859 + 0.567578i 0.936069 − 0.0490573i 0.377800 + 0.340173i 0 1.41191 + 0.458757i −0.0623449 − 2.64502i −1.07251 − 2.10492i −2.10975 + 0.221743i 0
82.15 1.81626 + 0.697196i −2.40177 + 0.125871i 1.32642 + 1.19431i 0 −4.44999 − 1.44589i −2.13721 + 1.55960i −0.190005 − 0.372906i 2.76909 − 0.291043i 0
82.16 2.07887 + 0.798002i 2.97775 − 0.156057i 2.19859 + 1.97962i 0 6.31487 + 2.05183i −1.33917 + 2.28180i 0.968973 + 1.90172i 5.85905 − 0.615811i 0
82.17 2.18349 + 0.838165i 1.19738 − 0.0627519i 2.57884 + 2.32200i 0 2.66706 + 0.866582i 2.62853 − 0.301393i 1.56104 + 3.06372i −1.55379 + 0.163310i 0
82.18 2.53114 + 0.971615i −0.977922 + 0.0512507i 3.97637 + 3.58034i 0 −2.52506 − 0.820441i −2.54155 + 0.735207i 4.12431 + 8.09441i −2.02986 + 0.213347i 0
143.1 −0.131292 − 2.50521i −1.20007 − 0.971802i −4.26979 + 0.448773i 0 −2.27701 + 3.13403i 0.377880 − 2.61863i 0.899985 + 5.68228i −0.127953 − 0.601973i 0
143.2 −0.115594 − 2.20567i −2.04764 − 1.65814i −2.86259 + 0.300870i 0 −3.42063 + 4.70809i −2.43764 + 1.02854i 0.303488 + 1.91615i 0.819640 + 3.85610i 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:
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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.b 288
5.b even 2 1 875.2.bb.a 288
5.c odd 4 1 175.2.x.a ✓ 288
5.c odd 4 1 875.2.bb.c 288
7.d odd 6 1 inner 875.2.bb.b 288
25.d even 5 1 175.2.x.a ✓ 288
25.e even 10 1 875.2.bb.c 288
25.f odd 20 1 875.2.bb.a 288
25.f odd 20 1 inner 875.2.bb.b 288
35.i odd 6 1 875.2.bb.a 288
35.k even 12 1 175.2.x.a ✓ 288
35.k even 12 1 875.2.bb.c 288
175.u odd 30 1 875.2.bb.c 288
175.v odd 30 1 175.2.x.a ✓ 288
175.x even 60 1 875.2.bb.a 288
175.x even 60 1 inner 875.2.bb.b 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.x.a ✓ 288 5.c odd 4 1
175.2.x.a ✓ 288 25.d even 5 1
175.2.x.a ✓ 288 35.k even 12 1
175.2.x.a ✓ 288 175.v odd 30 1
875.2.bb.a 288 5.b even 2 1
875.2.bb.a 288 25.f odd 20 1
875.2.bb.a 288 35.i odd 6 1
875.2.bb.a 288 175.x even 60 1
875.2.bb.b 288 1.a even 1 1 trivial
875.2.bb.b 288 7.d odd 6 1 inner
875.2.bb.b 288 25.f odd 20 1 inner
875.2.bb.b 288 175.x even 60 1 inner
875.2.bb.c 288 5.c odd 4 1
875.2.bb.c 288 25.e even 10 1
875.2.bb.c 288 35.k even 12 1
875.2.bb.c 288 175.u odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2 T_{2}^{287} - 3 T_{2}^{286} - 18 T_{2}^{285} + 184 T_{2}^{284} - 184 T_{2}^{283} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display