Properties

Label 225.4.m.a
Level $225$
Weight $4$
Character orbit 225.m
Analytic conductor $13.275$
Analytic rank $0$
Dimension $24$
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(19,225)]
 
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, 9]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.m (of order \(10\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 25)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5 q^{2} + 13 q^{4} - 15 q^{5} + 110 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5 q^{2} + 13 q^{4} - 15 q^{5} + 110 q^{8} - 5 q^{10} - 83 q^{11} - 5 q^{13} - 31 q^{14} + 89 q^{16} + 165 q^{17} - 115 q^{19} - 395 q^{20} + 30 q^{22} - 75 q^{23} + 595 q^{25} + 822 q^{26} + 675 q^{28} - 125 q^{29} + 633 q^{31} - 779 q^{34} - 190 q^{35} - 1510 q^{37} - 255 q^{38} - 2470 q^{40} + 117 q^{41} - 516 q^{44} + 1233 q^{46} + 95 q^{47} + 1148 q^{49} - 2165 q^{50} + 1740 q^{52} - 2580 q^{53} - 315 q^{55} - 3160 q^{56} - 4230 q^{58} + 1905 q^{59} - 567 q^{61} + 6880 q^{62} - 3612 q^{64} + 3170 q^{65} + 4195 q^{67} + 3630 q^{70} - 2473 q^{71} - 845 q^{73} - 3596 q^{74} - 3280 q^{76} - 3870 q^{77} + 775 q^{79} + 3670 q^{80} + 4625 q^{83} - 1460 q^{85} + 3897 q^{86} - 1650 q^{88} + 2410 q^{89} - 382 q^{91} - 4090 q^{92} + 5401 q^{94} - 3545 q^{95} + 5185 q^{97} - 5180 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.47870 + 1.13030i 0 4.35165 3.16166i 11.1800 0.0808329i 0 8.69522i 5.63517 7.75614i 0 −38.8007 + 12.9180i
19.2 −3.33666 + 1.08415i 0 3.48577 2.53256i −10.7075 3.21691i 0 25.7483i 7.61219 10.4773i 0 39.2150 0.874818i
19.3 −0.331937 + 0.107853i 0 −6.37359 + 4.63068i −10.0178 + 4.96433i 0 14.4107i 3.25738 4.48340i 0 2.78985 2.72829i
19.4 1.48841 0.483614i 0 −4.49065 + 3.26265i 5.33345 + 9.82621i 0 1.18261i −12.4652 + 17.1569i 0 12.6905 + 12.0461i
19.5 2.51328 0.816614i 0 −0.822422 + 0.597525i 5.31827 9.83443i 0 26.3705i −14.0054 + 19.2767i 0 5.33537 29.0596i
19.6 4.95462 1.60985i 0 15.4845 11.2501i −11.0057 1.96864i 0 24.5811i 34.1117 46.9507i 0 −57.6981 + 7.96365i
64.1 −2.81526 3.87487i 0 −4.61680 + 14.2091i −11.1717 + 0.439337i 0 14.5499i 31.6143 10.2721i 0 33.1536 + 42.0521i
64.2 −1.81265 2.49489i 0 −0.466674 + 1.43628i 10.8670 + 2.62823i 0 5.10302i −19.0341 + 6.18456i 0 −13.1409 31.8762i
64.3 0.217515 + 0.299383i 0 2.42982 7.47821i −7.78705 8.02258i 0 0.707538i 5.58294 1.81401i 0 0.708030 4.07634i
64.4 0.442260 + 0.608718i 0 2.29719 7.07003i −1.00498 + 11.1351i 0 18.3105i 11.0443 3.58852i 0 −7.22259 + 4.31284i
64.5 2.23725 + 3.07931i 0 −2.00472 + 6.16988i 3.48187 + 10.6243i 0 4.54748i 5.47554 1.77911i 0 −24.9258 + 34.4910i
64.6 2.42187 + 3.33341i 0 −2.77407 + 8.53772i 8.01402 7.79587i 0 26.4674i −3.82887 + 1.24408i 0 45.3957 + 7.83348i
109.1 −2.81526 + 3.87487i 0 −4.61680 14.2091i −11.1717 0.439337i 0 14.5499i 31.6143 + 10.2721i 0 33.1536 42.0521i
109.2 −1.81265 + 2.49489i 0 −0.466674 1.43628i 10.8670 2.62823i 0 5.10302i −19.0341 6.18456i 0 −13.1409 + 31.8762i
109.3 0.217515 0.299383i 0 2.42982 + 7.47821i −7.78705 + 8.02258i 0 0.707538i 5.58294 + 1.81401i 0 0.708030 + 4.07634i
109.4 0.442260 0.608718i 0 2.29719 + 7.07003i −1.00498 11.1351i 0 18.3105i 11.0443 + 3.58852i 0 −7.22259 4.31284i
109.5 2.23725 3.07931i 0 −2.00472 6.16988i 3.48187 10.6243i 0 4.54748i 5.47554 + 1.77911i 0 −24.9258 34.4910i
109.6 2.42187 3.33341i 0 −2.77407 8.53772i 8.01402 + 7.79587i 0 26.4674i −3.82887 1.24408i 0 45.3957 7.83348i
154.1 −3.47870 1.13030i 0 4.35165 + 3.16166i 11.1800 + 0.0808329i 0 8.69522i 5.63517 + 7.75614i 0 −38.8007 12.9180i
154.2 −3.33666 1.08415i 0 3.48577 + 2.53256i −10.7075 + 3.21691i 0 25.7483i 7.61219 + 10.4773i 0 39.2150 + 0.874818i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.m.a 24
3.b odd 2 1 25.4.e.a 24
15.d odd 2 1 125.4.e.a 24
15.e even 4 2 125.4.d.b 48
25.e even 10 1 inner 225.4.m.a 24
75.h odd 10 1 25.4.e.a 24
75.j odd 10 1 125.4.e.a 24
75.l even 20 2 125.4.d.b 48
75.l even 20 2 625.4.a.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.e.a 24 3.b odd 2 1
25.4.e.a 24 75.h odd 10 1
125.4.d.b 48 15.e even 4 2
125.4.d.b 48 75.l even 20 2
125.4.e.a 24 15.d odd 2 1
125.4.e.a 24 75.j odd 10 1
225.4.m.a 24 1.a even 1 1 trivial
225.4.m.a 24 25.e even 10 1 inner
625.4.a.g 24 75.l even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 5 T_{2}^{23} - 18 T_{2}^{22} + 95 T_{2}^{21} + 571 T_{2}^{20} - 2445 T_{2}^{19} - 11935 T_{2}^{18} + 72075 T_{2}^{17} + 190000 T_{2}^{16} - 1217055 T_{2}^{15} - 720663 T_{2}^{14} + 13173050 T_{2}^{13} + \cdots + 38738176 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display