# Properties

 Label 225.4.e.f Level $225$ Weight $4$ Character orbit 225.e Analytic conductor $13.275$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# 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: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} + q^{3} - 48 q^{4} - 13 q^{6} - 6 q^{7} - 90 q^{8} - 61 q^{9}+O(q^{10})$$ 24 * q + 4 * q^2 + q^3 - 48 * q^4 - 13 * q^6 - 6 * q^7 - 90 * q^8 - 61 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} + q^{3} - 48 q^{4} - 13 q^{6} - 6 q^{7} - 90 q^{8} - 61 q^{9} - 29 q^{11} + 77 q^{12} - 24 q^{13} + 69 q^{14} - 192 q^{16} - 158 q^{17} - 125 q^{18} - 150 q^{19} - 60 q^{21} + 18 q^{22} + 318 q^{23} + 342 q^{24} - 308 q^{26} + 394 q^{27} + 192 q^{28} - 106 q^{29} - 60 q^{31} + 914 q^{32} + 80 q^{33} + 108 q^{34} + 1303 q^{36} - 168 q^{37} + 640 q^{38} - 410 q^{39} + 353 q^{41} - 1521 q^{42} + 426 q^{43} + 1142 q^{44} + 540 q^{46} + 1210 q^{47} - 2680 q^{48} - 666 q^{49} - 1369 q^{51} + 75 q^{52} - 896 q^{53} - 2128 q^{54} + 570 q^{56} - 1544 q^{57} - 594 q^{58} - 482 q^{59} - 402 q^{61} - 5088 q^{62} + 1038 q^{63} + 1950 q^{64} + 2041 q^{66} + 201 q^{67} + 3437 q^{68} + 2856 q^{69} - 1888 q^{71} + 5493 q^{72} - 906 q^{73} - 10 q^{74} + 462 q^{76} + 2652 q^{77} + 4589 q^{78} - 258 q^{79} + 3071 q^{81} + 1746 q^{82} + 3012 q^{83} - 2703 q^{84} - 1952 q^{86} - 2708 q^{87} + 216 q^{88} - 1476 q^{89} - 1236 q^{91} + 5232 q^{92} - 3024 q^{93} - 63 q^{94} - 10424 q^{96} + 318 q^{97} - 15022 q^{98} - 1697 q^{99}+O(q^{100})$$ 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

## 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 −2.42567 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
76.2 −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
76.3 −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
76.4 −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
76.5 −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
76.6 −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
76.7 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
76.8 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
76.9 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
76.10 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
76.11 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
76.12 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
151.1 −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
151.2 −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
151.3 −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
151.4 −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
151.5 −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
151.6 −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
151.7 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
151.8 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
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f yes 24
5.b even 2 1 225.4.e.e 24
5.c odd 4 2 225.4.k.e 48
9.c even 3 1 inner 225.4.e.f yes 24
9.c even 3 1 2025.4.a.bf 12
9.d odd 6 1 2025.4.a.bj 12
45.h odd 6 1 2025.4.a.be 12
45.j even 6 1 225.4.e.e 24
45.j even 6 1 2025.4.a.bi 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 5.b even 2 1
225.4.e.e 24 45.j even 6 1
225.4.e.f yes 24 1.a even 1 1 trivial
225.4.e.f yes 24 9.c even 3 1 inner
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 45.h odd 6 1
2025.4.a.bf 12 9.c even 3 1
2025.4.a.bi 12 45.j even 6 1
2025.4.a.bj 12 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 4 T_{2}^{23} + 80 T_{2}^{22} - 226 T_{2}^{21} + 3608 T_{2}^{20} - 8938 T_{2}^{19} + 102821 T_{2}^{18} - 206125 T_{2}^{17} + 2044613 T_{2}^{16} - 3598036 T_{2}^{15} + 28276085 T_{2}^{14} + \cdots + 5330168064$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.