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Properties

Label 225.4.h.c
Level $225$
Weight $4$
Character orbit 225.h
Analytic conductor $13.275$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(46,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.46");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), degree \(4\), 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} - 18 q^{4} - 15 q^{5} + 58 q^{7} + 111 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} - 18 q^{4} - 15 q^{5} + 58 q^{7} + 111 q^{8} - 155 q^{10} - 65 q^{11} - 100 q^{13} + 108 q^{14} + 46 q^{16} - 72 q^{17} + 146 q^{19} - 265 q^{20} - 901 q^{22} + 464 q^{23} + 95 q^{25} + 114 q^{26} - 66 q^{28} - 372 q^{29} + 149 q^{31} - 2968 q^{32} + 734 q^{34} - 650 q^{35} - 72 q^{37} - 568 q^{38} + 1080 q^{40} + 1306 q^{41} + 928 q^{43} + 2297 q^{44} - 186 q^{46} + 1416 q^{47} + 498 q^{49} + 2315 q^{50} - 2018 q^{52} - 56 q^{53} - 1520 q^{55} + 300 q^{56} - 979 q^{58} - 419 q^{59} + 1292 q^{61} - 501 q^{62} + 259 q^{64} - 1000 q^{65} + 1772 q^{67} - 1218 q^{68} - 5030 q^{70} - 2506 q^{71} - 2234 q^{73} - 1882 q^{74} + 2576 q^{76} + 999 q^{77} + 1500 q^{79} - 730 q^{80} + 3956 q^{82} + 953 q^{83} + 4370 q^{85} + 10 q^{86} + 8439 q^{88} + 774 q^{89} - 5896 q^{91} - 2663 q^{92} - 7295 q^{94} + 5340 q^{95} - 3753 q^{97} + 9855 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.72597 + 5.31199i 0 −18.7661 − 13.6344i 9.86834 + 5.25509i 0 14.5931 68.6662 − 49.8889i 0 −44.9474 + 43.3504i
46.2 −1.11464 + 3.43050i 0 −4.05377 − 2.94523i −10.2207 + 4.53191i 0 −7.04529 −8.72315 + 6.33774i 0 −4.15441 − 40.1134i
46.3 −0.792217 + 2.43819i 0 1.15496 + 0.839130i 5.97279 − 9.45123i 0 25.1018 −19.5533 + 14.2063i 0 18.3122 + 22.0502i
46.4 0.00365961 − 0.0112631i 0 6.47202 + 4.70220i −2.23120 − 10.9554i 0 −26.1506 0.153295 − 0.111375i 0 −0.131558 − 0.0149624i
46.5 0.180146 − 0.554432i 0 6.19719 + 4.50252i −1.75128 + 11.0423i 0 13.5827 7.38577 − 5.36608i 0 5.80674 + 2.96020i
46.6 1.09475 − 3.36931i 0 −3.68162 − 2.67485i 10.9568 + 2.22453i 0 −15.6925 9.88596 − 7.18257i 0 19.4901 − 34.4815i
46.7 1.11819 − 3.44145i 0 −4.12106 − 2.99413i −7.95953 − 7.85149i 0 15.7010 8.50748 − 6.18105i 0 −35.9208 + 18.6128i
91.1 −3.67736 − 2.67176i 0 3.91255 + 12.0416i −6.73234 − 8.92612i 0 9.62995 6.54736 − 20.1507i 0 0.908811 + 50.8118i
91.2 −2.30514 − 1.67478i 0 0.0366450 + 0.112782i −2.98333 + 10.7750i 0 6.69745 −6.93948 + 21.3575i 0 24.9227 − 19.8414i
91.3 −1.20373 − 0.874560i 0 −1.78803 − 5.50298i 10.4360 − 4.01110i 0 −27.0465 −6.33866 + 19.5084i 0 −16.0701 − 4.29868i
91.4 1.04899 + 0.762136i 0 −1.95261 − 6.00951i −10.7157 + 3.18962i 0 −7.54047 5.73722 − 17.6574i 0 −13.6716 − 4.82095i
91.5 2.01897 + 1.46687i 0 −0.547592 − 1.68532i 10.1178 + 4.75718i 0 10.8656 7.53599 − 23.1934i 0 13.4493 + 24.4461i
91.6 3.10085 + 2.25290i 0 2.06758 + 6.36335i −3.45281 − 10.6338i 0 −20.3108 1.55062 − 4.77231i 0 13.2503 − 40.7527i
91.7 4.25349 + 3.09034i 0 6.06983 + 18.6810i −8.80488 + 6.89015i 0 36.6146 −18.9152 + 58.2151i 0 −58.7444 + 2.09710i
136.1 −3.67736 + 2.67176i 0 3.91255 − 12.0416i −6.73234 + 8.92612i 0 9.62995 6.54736 + 20.1507i 0 0.908811 − 50.8118i
136.2 −2.30514 + 1.67478i 0 0.0366450 − 0.112782i −2.98333 − 10.7750i 0 6.69745 −6.93948 − 21.3575i 0 24.9227 + 19.8414i
136.3 −1.20373 + 0.874560i 0 −1.78803 + 5.50298i 10.4360 + 4.01110i 0 −27.0465 −6.33866 − 19.5084i 0 −16.0701 + 4.29868i
136.4 1.04899 − 0.762136i 0 −1.95261 + 6.00951i −10.7157 − 3.18962i 0 −7.54047 5.73722 + 17.6574i 0 −13.6716 + 4.82095i
136.5 2.01897 − 1.46687i 0 −0.547592 + 1.68532i 10.1178 − 4.75718i 0 10.8656 7.53599 + 23.1934i 0 13.4493 − 24.4461i
136.6 3.10085 − 2.25290i 0 2.06758 − 6.36335i −3.45281 + 10.6338i 0 −20.3108 1.55062 + 4.77231i 0 13.2503 + 40.7527i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.h.c 28
3.b odd 2 1 75.4.g.a ✓ 28
25.d even 5 1 inner 225.4.h.c 28
75.h odd 10 1 1875.4.a.e 14
75.j odd 10 1 75.4.g.a ✓ 28
75.j odd 10 1 1875.4.a.h 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.a ✓ 28 3.b odd 2 1
75.4.g.a ✓ 28 75.j odd 10 1
225.4.h.c 28 1.a even 1 1 trivial
225.4.h.c 28 25.d even 5 1 inner
1875.4.a.e 14 75.h odd 10 1
1875.4.a.h 14 75.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 4 T_{2}^{27} + 45 T_{2}^{26} - 217 T_{2}^{25} + 1605 T_{2}^{24} - 4173 T_{2}^{23} + \cdots + 32993536 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display