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Properties

Label 225.4.e.e
Level $225$
Weight $4$
Character orbit 225.e
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(76,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.76");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.e (of order \(3\), degree \(2\), 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: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - q^{3} - 48 q^{4} - 13 q^{6} + 6 q^{7} + 90 q^{8} - 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} - q^{3} - 48 q^{4} - 13 q^{6} + 6 q^{7} + 90 q^{8} - 61 q^{9} - 29 q^{11} - 77 q^{12} + 24 q^{13} + 69 q^{14} - 192 q^{16} + 158 q^{17} + 125 q^{18} - 150 q^{19} - 60 q^{21} - 18 q^{22} - 318 q^{23} + 342 q^{24} - 308 q^{26} - 394 q^{27} - 192 q^{28} - 106 q^{29} - 60 q^{31} - 914 q^{32} - 80 q^{33} + 108 q^{34} + 1303 q^{36} + 168 q^{37} - 640 q^{38} - 410 q^{39} + 353 q^{41} + 1521 q^{42} - 426 q^{43} + 1142 q^{44} + 540 q^{46} - 1210 q^{47} + 2680 q^{48} - 666 q^{49} - 1369 q^{51} - 75 q^{52} + 896 q^{53} - 2128 q^{54} + 570 q^{56} + 1544 q^{57} + 594 q^{58} - 482 q^{59} - 402 q^{61} + 5088 q^{62} - 1038 q^{63} + 1950 q^{64} + 2041 q^{66} - 201 q^{67} - 3437 q^{68} + 2856 q^{69} - 1888 q^{71} - 5493 q^{72} + 906 q^{73} - 10 q^{74} + 462 q^{76} - 2652 q^{77} - 4589 q^{78} - 258 q^{79} + 3071 q^{81} - 1746 q^{82} - 3012 q^{83} - 2703 q^{84} - 1952 q^{86} + 2708 q^{87} - 216 q^{88} - 1476 q^{89} - 1236 q^{91} - 5232 q^{92} + 3024 q^{93} - 63 q^{94} - 10424 q^{96} - 318 q^{97} + 15022 q^{98} - 1697 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −2.77209 − 4.80141i 0.320819 + 5.18624i −11.3690 + 19.6917i 0 24.0119 − 15.9171i 12.8563 + 22.2678i 81.7101 −26.7942 + 3.32769i 0
76.2 −2.39440 − 4.14722i 1.75824 − 4.88964i −7.46632 + 12.9320i 0 −24.4884 + 4.41595i −15.3349 − 26.5609i 33.1990 −20.8172 − 17.1943i 0
76.3 −1.81989 − 3.15215i −5.18137 − 0.391700i −2.62402 + 4.54493i 0 8.19483 + 17.0453i −1.86027 − 3.22208i −10.0166 26.6931 + 4.05909i 0
76.4 −1.48830 − 2.57780i 5.14295 − 0.741698i −0.430049 + 0.744867i 0 −9.56618 − 12.1536i 12.4826 + 21.6204i −21.2526 25.8998 − 7.62902i 0
76.5 −1.07290 − 1.85832i 2.66146 + 4.46280i 1.69777 − 2.94063i 0 5.43782 − 9.73398i −11.6282 − 20.1406i −24.4526 −12.8333 + 23.7552i 0
76.6 −0.694136 − 1.20228i −0.900797 − 5.11748i 3.03635 − 5.25911i 0 −5.52736 + 4.63523i 13.2166 + 22.8919i −19.5367 −25.3771 + 9.21962i 0
76.7 0.238017 + 0.412258i −3.09012 + 4.17746i 3.88670 − 6.73195i 0 −2.45769 − 0.279619i 6.34045 + 10.9820i 7.50869 −7.90234 − 25.8177i 0
76.8 0.669825 + 1.16017i −4.26770 − 2.96424i 3.10267 − 5.37398i 0 0.580411 − 6.93678i −11.1645 − 19.3375i 19.0302 9.42656 + 25.3010i 0
76.9 0.832171 + 1.44136i 4.88113 − 1.78172i 2.61498 − 4.52928i 0 6.63004 + 5.55278i −5.28358 − 9.15142i 22.0192 20.6509 − 17.3936i 0
76.10 1.90831 + 3.30529i 2.77901 + 4.39057i −3.28331 + 5.68685i 0 −9.20891 + 17.5640i 4.99402 + 8.64990i 5.47071 −11.5542 + 24.4029i 0
76.11 2.16773 + 3.75461i 0.193913 − 5.19253i −5.39807 + 9.34974i 0 19.9163 − 10.5279i 6.50075 + 11.2596i −12.1226 −26.9248 − 2.01380i 0
76.12 2.42567 + 4.20138i −4.79753 + 1.99591i −7.76772 + 13.4541i 0 −20.0228 − 15.3148i −8.11924 − 14.0629i −36.5569 19.0327 − 19.1509i 0
151.1 −2.77209 + 4.80141i 0.320819 − 5.18624i −11.3690 − 19.6917i 0 24.0119 + 15.9171i 12.8563 − 22.2678i 81.7101 −26.7942 − 3.32769i 0
151.2 −2.39440 + 4.14722i 1.75824 + 4.88964i −7.46632 − 12.9320i 0 −24.4884 − 4.41595i −15.3349 + 26.5609i 33.1990 −20.8172 + 17.1943i 0
151.3 −1.81989 + 3.15215i −5.18137 + 0.391700i −2.62402 − 4.54493i 0 8.19483 − 17.0453i −1.86027 + 3.22208i −10.0166 26.6931 − 4.05909i 0
151.4 −1.48830 + 2.57780i 5.14295 + 0.741698i −0.430049 − 0.744867i 0 −9.56618 + 12.1536i 12.4826 − 21.6204i −21.2526 25.8998 + 7.62902i 0
151.5 −1.07290 + 1.85832i 2.66146 − 4.46280i 1.69777 + 2.94063i 0 5.43782 + 9.73398i −11.6282 + 20.1406i −24.4526 −12.8333 − 23.7552i 0
151.6 −0.694136 + 1.20228i −0.900797 + 5.11748i 3.03635 + 5.25911i 0 −5.52736 − 4.63523i 13.2166 − 22.8919i −19.5367 −25.3771 − 9.21962i 0
151.7 0.238017 − 0.412258i −3.09012 − 4.17746i 3.88670 + 6.73195i 0 −2.45769 + 0.279619i 6.34045 − 10.9820i 7.50869 −7.90234 + 25.8177i 0
151.8 0.669825 − 1.16017i −4.26770 + 2.96424i 3.10267 + 5.37398i 0 0.580411 + 6.93678i −11.1645 + 19.3375i 19.0302 9.42656 − 25.3010i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.e.e ✓ 24
5.b even 2 1 225.4.e.f yes 24
5.c odd 4 2 225.4.k.e 48
9.c even 3 1 inner 225.4.e.e ✓ 24
9.c even 3 1 2025.4.a.bi 12
9.d odd 6 1 2025.4.a.be 12
45.h odd 6 1 2025.4.a.bj 12
45.j even 6 1 225.4.e.f yes 24
45.j even 6 1 2025.4.a.bf 12
45.k odd 12 2 225.4.k.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.e.e ✓ 24 1.a even 1 1 trivial
225.4.e.e ✓ 24 9.c even 3 1 inner
225.4.e.f yes 24 5.b even 2 1
225.4.e.f yes 24 45.j even 6 1
225.4.k.e 48 5.c odd 4 2
225.4.k.e 48 45.k odd 12 2
2025.4.a.be 12 9.d odd 6 1
2025.4.a.bf 12 45.j even 6 1
2025.4.a.bi 12 9.c even 3 1
2025.4.a.bj 12 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 4 T_{2}^{23} + 80 T_{2}^{22} + 226 T_{2}^{21} + 3608 T_{2}^{20} + 8938 T_{2}^{19} + \cdots + 5330168064 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display