Properties

Label 225.4.h.d
Level $225$
Weight $4$
Character orbit 225.h
Analytic conductor $13.275$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 72 q^{4} + 20 q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 72 q^{4} + 20 q^{7} + 50 q^{10} + 180 q^{13} - 244 q^{16} - 116 q^{19} - 210 q^{22} + 440 q^{25} + 980 q^{28} + 42 q^{31} + 40 q^{34} - 1170 q^{37} - 3040 q^{40} + 1040 q^{43} + 700 q^{46} + 4188 q^{49} + 3280 q^{52} + 2640 q^{55} - 4530 q^{58} + 88 q^{61} - 3018 q^{64} - 2860 q^{67} - 3720 q^{70} - 5280 q^{73} + 9288 q^{76} - 1144 q^{79} + 2840 q^{82} + 470 q^{85} + 7610 q^{88} - 1180 q^{91} + 2170 q^{94} + 2050 q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.71061 + 5.26471i 0 −18.3188 13.3094i −11.1283 1.07706i 0 −30.2200 65.5791 47.6460i 0 24.7066 56.7450i
46.2 −1.45629 + 4.48200i 0 −11.4954 8.35189i 10.8155 + 2.83289i 0 2.19013 23.6729 17.1993i 0 −28.4475 + 44.3495i
46.3 −1.31991 + 4.06225i 0 −8.28762 6.02131i −4.79821 + 10.0984i 0 33.6094 7.75447 5.63395i 0 −34.6890 32.8205i
46.4 −1.20400 + 3.70554i 0 −5.80928 4.22069i 3.34114 10.6694i 0 6.27975 −2.58263 + 1.87639i 0 35.5133 + 25.2268i
46.5 −0.856692 + 2.63663i 0 0.254251 + 0.184724i −7.75979 8.04895i 0 −7.15708 −18.6477 + 13.5483i 0 27.8698 13.5642i
46.6 −0.626289 + 1.92752i 0 3.14904 + 2.28791i −3.12321 + 10.7352i 0 −16.4425 −19.4994 + 14.1671i 0 −18.7364 12.7434i
46.7 −0.426412 + 1.31236i 0 4.93167 + 3.58307i 10.9704 2.15653i 0 −13.9864 −15.7361 + 11.4329i 0 −1.84776 + 15.3167i
46.8 −0.0492832 + 0.151678i 0 6.45156 + 4.68733i −9.69768 5.56372i 0 29.3448 −2.06112 + 1.49749i 0 1.32183 1.19673i
46.9 0.0492832 0.151678i 0 6.45156 + 4.68733i 9.69768 + 5.56372i 0 29.3448 2.06112 1.49749i 0 1.32183 1.19673i
46.10 0.426412 1.31236i 0 4.93167 + 3.58307i −10.9704 + 2.15653i 0 −13.9864 15.7361 11.4329i 0 −1.84776 + 15.3167i
46.11 0.626289 1.92752i 0 3.14904 + 2.28791i 3.12321 10.7352i 0 −16.4425 19.4994 14.1671i 0 −18.7364 12.7434i
46.12 0.856692 2.63663i 0 0.254251 + 0.184724i 7.75979 + 8.04895i 0 −7.15708 18.6477 13.5483i 0 27.8698 13.5642i
46.13 1.20400 3.70554i 0 −5.80928 4.22069i −3.34114 + 10.6694i 0 6.27975 2.58263 1.87639i 0 35.5133 + 25.2268i
46.14 1.31991 4.06225i 0 −8.28762 6.02131i 4.79821 10.0984i 0 33.6094 −7.75447 + 5.63395i 0 −34.6890 32.8205i
46.15 1.45629 4.48200i 0 −11.4954 8.35189i −10.8155 2.83289i 0 2.19013 −23.6729 + 17.1993i 0 −28.4475 + 44.3495i
46.16 1.71061 5.26471i 0 −18.3188 13.3094i 11.1283 + 1.07706i 0 −30.2200 −65.5791 + 47.6460i 0 24.7066 56.7450i
91.1 −4.45962 3.24011i 0 6.91782 + 21.2908i −7.22670 8.53082i 0 −10.2696 24.5064 75.4228i 0 4.58760 + 61.4595i
91.2 −3.81348 2.77066i 0 4.39397 + 13.5233i −0.520226 + 11.1682i 0 30.0229 9.05902 27.8808i 0 32.9272 41.1485i
91.3 −3.54222 2.57358i 0 3.45191 + 10.6239i 10.9424 + 2.29411i 0 −21.8567 4.28988 13.2029i 0 −32.8565 36.2875i
91.4 −2.68030 1.94735i 0 0.919680 + 2.83048i −2.31093 10.9389i 0 10.7752 −5.14333 + 15.8295i 0 −15.1079 + 33.8197i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.16
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.d even 5 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.h.d 64
3.b odd 2 1 inner 225.4.h.d 64
25.d even 5 1 inner 225.4.h.d 64
75.j odd 10 1 inner 225.4.h.d 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.h.d 64 1.a even 1 1 trivial
225.4.h.d 64 3.b odd 2 1 inner
225.4.h.d 64 25.d even 5 1 inner
225.4.h.d 64 75.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(75\!\cdots\!00\)\( T_{2}^{48} + \)\(15\!\cdots\!00\)\( T_{2}^{46} + \)\(30\!\cdots\!25\)\( T_{2}^{44} + \)\(51\!\cdots\!00\)\( T_{2}^{42} + \)\(78\!\cdots\!50\)\( T_{2}^{40} + \)\(10\!\cdots\!25\)\( T_{2}^{38} + \)\(12\!\cdots\!00\)\( T_{2}^{36} + \)\(14\!\cdots\!25\)\( T_{2}^{34} + \)\(14\!\cdots\!25\)\( T_{2}^{32} + \)\(12\!\cdots\!75\)\( T_{2}^{30} + \)\(95\!\cdots\!75\)\( T_{2}^{28} + \)\(66\!\cdots\!00\)\( T_{2}^{26} + \)\(40\!\cdots\!25\)\( T_{2}^{24} + \)\(20\!\cdots\!00\)\( T_{2}^{22} + \)\(95\!\cdots\!00\)\( T_{2}^{20} + \)\(34\!\cdots\!00\)\( T_{2}^{18} + \)\(85\!\cdots\!00\)\( T_{2}^{16} + \)\(12\!\cdots\!00\)\( T_{2}^{14} + \)\(14\!\cdots\!00\)\( T_{2}^{12} + \)\(15\!\cdots\!00\)\( T_{2}^{10} + \)\(15\!\cdots\!00\)\( T_{2}^{8} - \)\(34\!\cdots\!00\)\( T_{2}^{6} + \)\(21\!\cdots\!00\)\( T_{2}^{4} + \)\(93\!\cdots\!00\)\( T_{2}^{2} + \)\(15\!\cdots\!00\)\( \)">\(T_{2}^{64} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).