# Properties

 Label 225.4.e.g Level $225$ Weight $4$ Character orbit 225.e Analytic conductor $13.275$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{3})$$ Twist minimal: no (minimal twist has level 45) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 54 q^{4} - 12 q^{6} + 18 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32 q - 54 q^{4} - 12 q^{6} + 18 q^{9} + 90 q^{11} + 102 q^{14} - 146 q^{16} + 8 q^{19} + 30 q^{21} - 462 q^{24} - 936 q^{26} + 516 q^{29} - 38 q^{31} - 212 q^{34} + 864 q^{36} - 330 q^{39} + 576 q^{41} - 3288 q^{44} - 580 q^{46} + 4 q^{49} + 1260 q^{51} + 3726 q^{54} + 2430 q^{56} + 2202 q^{59} - 20 q^{61} - 644 q^{64} - 5052 q^{66} - 1452 q^{69} - 5904 q^{71} + 4080 q^{74} + 396 q^{76} + 218 q^{79} + 198 q^{81} - 4662 q^{84} + 6108 q^{86} - 8148 q^{89} - 1884 q^{91} + 1078 q^{94} + 11874 q^{96} + 1602 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1 −2.52718 4.37720i 0.559681 5.16592i −8.77324 + 15.1957i 0 −24.0267 + 10.6054i 10.5059 + 18.1967i 48.2513 −26.3735 5.78253i 0
76.2 −2.51186 4.35066i −4.79730 + 1.99648i −8.61884 + 14.9283i 0 20.7361 + 15.8565i −2.69099 4.66092i 46.4074 19.0281 19.1554i 0
76.3 −2.03813 3.53014i 4.38850 + 2.78227i −4.30793 + 7.46156i 0 0.877481 21.1627i 6.67148 + 11.5553i 2.51043 11.5179 + 24.4200i 0
76.4 −1.54370 2.67377i 4.77188 2.05649i −0.766022 + 1.32679i 0 −12.8649 9.58430i −15.6602 27.1243i −19.9692 18.5417 19.6266i 0
76.5 −1.33753 2.31667i −0.694136 + 5.14958i 0.422017 0.730954i 0 12.8583 5.27964i 7.70766 + 13.3501i −23.6584 −26.0363 7.14902i 0
76.6 −1.23192 2.13376i −4.97325 1.50557i 0.964722 1.67095i 0 2.91415 + 12.4665i −9.62007 16.6624i −24.4647 22.4665 + 14.9752i 0
76.7 −0.392666 0.680118i −3.69592 3.65242i 3.69163 6.39408i 0 −1.03281 + 3.94785i 10.4568 + 18.1117i −12.0810 0.319654 + 26.9981i 0
76.8 −0.236995 0.410487i 2.45316 4.58061i 3.88767 6.73364i 0 −2.46167 + 0.0785933i 4.10329 + 7.10710i −7.47735 −14.9641 22.4739i 0
76.9 0.236995 + 0.410487i −2.45316 + 4.58061i 3.88767 6.73364i 0 −2.46167 + 0.0785933i −4.10329 7.10710i 7.47735 −14.9641 22.4739i 0
76.10 0.392666 + 0.680118i 3.69592 + 3.65242i 3.69163 6.39408i 0 −1.03281 + 3.94785i −10.4568 18.1117i 12.0810 0.319654 + 26.9981i 0
76.11 1.23192 + 2.13376i 4.97325 + 1.50557i 0.964722 1.67095i 0 2.91415 + 12.4665i 9.62007 + 16.6624i 24.4647 22.4665 + 14.9752i 0
76.12 1.33753 + 2.31667i 0.694136 5.14958i 0.422017 0.730954i 0 12.8583 5.27964i −7.70766 13.3501i 23.6584 −26.0363 7.14902i 0
76.13 1.54370 + 2.67377i −4.77188 + 2.05649i −0.766022 + 1.32679i 0 −12.8649 9.58430i 15.6602 + 27.1243i 19.9692 18.5417 19.6266i 0
76.14 2.03813 + 3.53014i −4.38850 2.78227i −4.30793 + 7.46156i 0 0.877481 21.1627i −6.67148 11.5553i −2.51043 11.5179 + 24.4200i 0
76.15 2.51186 + 4.35066i 4.79730 1.99648i −8.61884 + 14.9283i 0 20.7361 + 15.8565i 2.69099 + 4.66092i −46.4074 19.0281 19.1554i 0
76.16 2.52718 + 4.37720i −0.559681 + 5.16592i −8.77324 + 15.1957i 0 −24.0267 + 10.6054i −10.5059 18.1967i −48.2513 −26.3735 5.78253i 0
151.1 −2.52718 + 4.37720i 0.559681 + 5.16592i −8.77324 15.1957i 0 −24.0267 10.6054i 10.5059 18.1967i 48.2513 −26.3735 + 5.78253i 0
151.2 −2.51186 + 4.35066i −4.79730 1.99648i −8.61884 14.9283i 0 20.7361 15.8565i −2.69099 + 4.66092i 46.4074 19.0281 + 19.1554i 0
151.3 −2.03813 + 3.53014i 4.38850 2.78227i −4.30793 7.46156i 0 0.877481 + 21.1627i 6.67148 11.5553i 2.51043 11.5179 24.4200i 0
151.4 −1.54370 + 2.67377i 4.77188 + 2.05649i −0.766022 1.32679i 0 −12.8649 + 9.58430i −15.6602 + 27.1243i −19.9692 18.5417 + 19.6266i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.e.g 32
5.b even 2 1 inner 225.4.e.g 32
5.c odd 4 2 45.4.j.a 32
9.c even 3 1 inner 225.4.e.g 32
9.c even 3 1 2025.4.a.bk 16
9.d odd 6 1 2025.4.a.bl 16
15.e even 4 2 135.4.j.a 32
45.h odd 6 1 2025.4.a.bl 16
45.j even 6 1 inner 225.4.e.g 32
45.j even 6 1 2025.4.a.bk 16
45.k odd 12 2 45.4.j.a 32
45.k odd 12 2 405.4.b.e 16
45.l even 12 2 135.4.j.a 32
45.l even 12 2 405.4.b.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.j.a 32 5.c odd 4 2
45.4.j.a 32 45.k odd 12 2
135.4.j.a 32 15.e even 4 2
135.4.j.a 32 45.l even 12 2
225.4.e.g 32 1.a even 1 1 trivial
225.4.e.g 32 5.b even 2 1 inner
225.4.e.g 32 9.c even 3 1 inner
225.4.e.g 32 45.j even 6 1 inner
405.4.b.e 16 45.k odd 12 2
405.4.b.f 16 45.l even 12 2
2025.4.a.bk 16 9.c even 3 1
2025.4.a.bk 16 45.j even 6 1
2025.4.a.bl 16 9.d odd 6 1
2025.4.a.bl 16 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$43\!\cdots\!85$$$$T_{2}^{12} +$$$$14\!\cdots\!04$$$$T_{2}^{10} +$$$$30\!\cdots\!52$$$$T_{2}^{8} +$$$$23\!\cdots\!92$$$$T_{2}^{6} +$$$$12\!\cdots\!40$$$$T_{2}^{4} +$$$$25\!\cdots\!24$$$$T_{2}^{2} + 377801998336$$">$$T_{2}^{32} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.