Properties

Label 375.3.h.a
Level $375$
Weight $3$
Character orbit 375.h
Analytic conductor $10.218$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [375,3,Mod(74,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.74");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 375.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2180099135\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 5 q^{3} - 38 q^{4} + 5 q^{6} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 5 q^{3} - 38 q^{4} + 5 q^{6} - 13 q^{9} + 45 q^{12} + 10 q^{13} + 22 q^{16} - 36 q^{19} + 54 q^{21} + 50 q^{22} - 20 q^{24} - 100 q^{27} - 270 q^{28} - 126 q^{31} - 20 q^{33} + 210 q^{34} - 213 q^{36} - 110 q^{37} - 191 q^{39} + 175 q^{42} - 210 q^{46} - 150 q^{48} - 224 q^{49} - 60 q^{51} + 320 q^{52} + 320 q^{54} + 70 q^{58} + 294 q^{61} - 795 q^{63} + 362 q^{64} - 470 q^{66} + 260 q^{67} + 335 q^{69} - 215 q^{72} + 150 q^{73} - 16 q^{76} + 1295 q^{78} - 346 q^{79} + 507 q^{81} - 456 q^{84} + 430 q^{87} + 1710 q^{88} + 538 q^{91} - 560 q^{94} + 740 q^{96} + 150 q^{97} + 60 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −2.91455 + 2.11754i 1.30764 2.70002i 2.77453 8.53912i 0 1.90623 + 10.6383i 9.61929i 5.54243 + 17.0578i −5.58018 7.06128i 0
74.2 −2.79240 + 2.02880i 1.49624 + 2.60025i 2.44541 7.52621i 0 −9.45346 4.22537i 4.79411i 4.17417 + 12.8468i −4.52255 + 7.78116i 0
74.3 −2.48516 + 1.80557i −2.15485 2.08725i 1.67984 5.17002i 0 9.12382 + 1.29641i 8.51768i 1.36320 + 4.19549i 0.286751 + 8.99543i 0
74.4 −2.05424 + 1.49250i −2.06574 + 2.17548i 0.756307 2.32767i 0 0.996648 7.55208i 6.97211i −1.21820 3.74924i −0.465411 8.98796i 0
74.5 −1.88806 + 1.37175i 2.67611 + 1.35589i 0.446980 1.37566i 0 −6.91259 + 1.11096i 12.5544i −1.84155 5.66769i 5.32311 + 7.25703i 0
74.6 −1.27392 + 0.925557i 1.60087 2.53717i −0.469852 + 1.44606i 0 0.308913 + 4.71384i 3.45468i −2.68623 8.26736i −3.87443 8.12335i 0
74.7 −0.949573 + 0.689905i −1.65555 + 2.50183i −0.810348 + 2.49399i 0 −0.153954 3.51784i 4.52613i −2.40195 7.39246i −3.51828 8.28382i 0
74.8 −0.680034 + 0.494074i 2.98817 0.266209i −1.01773 + 3.13225i 0 −1.90053 + 1.65741i 9.76040i −1.89447 5.83059i 8.85827 1.59096i 0
74.9 −0.624639 + 0.453826i −2.28023 1.94950i −1.05185 + 3.23727i 0 2.30905 + 0.182906i 1.71117i −1.76649 5.43671i 1.39889 + 8.89062i 0
74.10 0.624639 0.453826i −2.99063 + 0.236895i −1.05185 + 3.23727i 0 −1.76055 + 1.50520i 1.71117i 1.76649 + 5.43671i 8.88776 1.41693i 0
74.11 0.680034 0.494074i 2.26100 + 1.97177i −1.01773 + 3.13225i 0 2.51176 + 0.223767i 9.76040i 1.89447 + 5.83059i 1.22427 + 8.91634i 0
74.12 0.949573 0.689905i 0.131165 2.99713i −0.810348 + 2.49399i 0 −1.94319 2.93649i 4.52613i 2.40195 + 7.39246i −8.96559 0.786239i 0
74.13 1.27392 0.925557i −0.196179 + 2.99358i −0.469852 + 1.44606i 0 2.52081 + 3.99515i 3.45468i 2.68623 + 8.26736i −8.92303 1.17455i 0
74.14 1.88806 1.37175i 2.96199 + 0.476037i 0.446980 1.37566i 0 6.24541 3.16433i 12.5544i 1.84155 + 5.66769i 8.54678 + 2.82003i 0
74.15 2.05424 1.49250i −0.392507 2.97421i 0.756307 2.32767i 0 −5.24530 5.52394i 6.97211i 1.21820 + 3.74924i −8.69188 + 2.33480i 0
74.16 2.48516 1.80557i −2.97017 + 0.422034i 1.67984 5.17002i 0 −6.61931 + 6.41167i 8.51768i −1.36320 4.19549i 8.64377 2.50702i 0
74.17 2.79240 2.02880i 2.73887 1.22418i 2.44541 7.52621i 0 5.16441 8.97500i 4.79411i −4.17417 12.8468i 6.00278 6.70572i 0
74.18 2.91455 2.11754i −0.529130 + 2.95297i 2.77453 8.53912i 0 4.71086 + 9.72702i 9.61929i −5.54243 17.0578i −8.44004 3.12501i 0
149.1 −1.16846 + 3.59614i −2.98146 + 0.333051i −8.33088 6.05274i 0 2.28600 11.1109i 3.47419i 19.2645 13.9965i 8.77815 1.98595i 0
149.2 −1.10327 + 3.39551i 2.43927 1.74641i −7.07623 5.14118i 0 3.23878 + 10.2093i 0.132726i 13.7104 9.96116i 2.90011 8.51994i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.3.h.a 72
3.b odd 2 1 inner 375.3.h.a 72
5.b even 2 1 75.3.h.a 72
5.c odd 4 2 375.3.j.b 144
15.d odd 2 1 75.3.h.a 72
15.e even 4 2 375.3.j.b 144
25.d even 5 1 75.3.h.a 72
25.e even 10 1 inner 375.3.h.a 72
25.f odd 20 2 375.3.j.b 144
75.h odd 10 1 inner 375.3.h.a 72
75.j odd 10 1 75.3.h.a 72
75.l even 20 2 375.3.j.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.h.a 72 5.b even 2 1
75.3.h.a 72 15.d odd 2 1
75.3.h.a 72 25.d even 5 1
75.3.h.a 72 75.j odd 10 1
375.3.h.a 72 1.a even 1 1 trivial
375.3.h.a 72 3.b odd 2 1 inner
375.3.h.a 72 25.e even 10 1 inner
375.3.h.a 72 75.h odd 10 1 inner
375.3.j.b 144 5.c odd 4 2
375.3.j.b 144 15.e even 4 2
375.3.j.b 144 25.f odd 20 2
375.3.j.b 144 75.l even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 55 T_{2}^{70} + 1735 T_{2}^{68} + 41465 T_{2}^{66} + 847535 T_{2}^{64} + 14877956 T_{2}^{62} + 227671100 T_{2}^{60} + 3100547455 T_{2}^{58} + 38147838725 T_{2}^{56} + \cdots + 16\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display