# Properties

 Label 225.4.h.a Level $225$ Weight $4$ Character orbit 225.h Analytic conductor $13.275$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## 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: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: no (minimal twist has level 75) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 30 q^{4} + 15 q^{5} - 54 q^{7} + 63 q^{8}+O(q^{10})$$ 28 * q - 30 * q^4 + 15 * q^5 - 54 * q^7 + 63 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 30 q^{4} + 15 q^{5} - 54 q^{7} + 63 q^{8} + 165 q^{10} - 19 q^{11} + 4 q^{13} + 24 q^{14} - 66 q^{16} - 208 q^{17} + 42 q^{19} - 295 q^{20} - 89 q^{22} - 32 q^{23} + 95 q^{25} - 206 q^{26} - 482 q^{28} + 716 q^{29} + 637 q^{31} + 844 q^{32} - 90 q^{34} - 430 q^{35} + 216 q^{37} - 2314 q^{38} - 500 q^{40} + 38 q^{41} - 1392 q^{43} - 603 q^{44} + 1622 q^{46} + 536 q^{47} + 162 q^{49} + 2265 q^{50} - 1922 q^{52} - 1672 q^{53} - 1000 q^{55} - 3000 q^{56} - 827 q^{58} - 973 q^{59} - 2712 q^{61} - 1057 q^{62} + 4439 q^{64} + 4360 q^{65} + 2768 q^{67} + 1370 q^{68} + 3230 q^{70} + 1074 q^{71} - 1018 q^{73} + 1414 q^{74} - 11408 q^{76} - 1607 q^{77} - 1820 q^{79} + 1290 q^{80} + 1772 q^{82} - 4045 q^{83} + 1850 q^{85} + 3986 q^{86} + 2407 q^{88} - 4542 q^{89} + 4412 q^{91} + 1089 q^{92} + 5137 q^{94} + 720 q^{95} - 5977 q^{97} + 10689 q^{98}+O(q^{100})$$ 28 * q - 30 * q^4 + 15 * q^5 - 54 * q^7 + 63 * q^8 + 165 * q^10 - 19 * q^11 + 4 * q^13 + 24 * q^14 - 66 * q^16 - 208 * q^17 + 42 * q^19 - 295 * q^20 - 89 * q^22 - 32 * q^23 + 95 * q^25 - 206 * q^26 - 482 * q^28 + 716 * q^29 + 637 * q^31 + 844 * q^32 - 90 * q^34 - 430 * q^35 + 216 * q^37 - 2314 * q^38 - 500 * q^40 + 38 * q^41 - 1392 * q^43 - 603 * q^44 + 1622 * q^46 + 536 * q^47 + 162 * q^49 + 2265 * q^50 - 1922 * q^52 - 1672 * q^53 - 1000 * q^55 - 3000 * q^56 - 827 * q^58 - 973 * q^59 - 2712 * q^61 - 1057 * q^62 + 4439 * q^64 + 4360 * q^65 + 2768 * q^67 + 1370 * q^68 + 3230 * q^70 + 1074 * q^71 - 1018 * q^73 + 1414 * q^74 - 11408 * q^76 - 1607 * q^77 - 1820 * q^79 + 1290 * q^80 + 1772 * q^82 - 4045 * q^83 + 1850 * q^85 + 3986 * q^86 + 2407 * q^88 - 4542 * q^89 + 4412 * q^91 + 1089 * q^92 + 5137 * q^94 + 720 * q^95 - 5977 * q^97 + 10689 * q^98

## 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 −1.48860 + 4.58143i 0 −12.3014 8.93752i −5.89885 + 9.49755i 0 1.13492 28.0809 20.4020i 0 −34.7314 41.1632i
46.2 −1.31310 + 4.04131i 0 −8.13578 5.91099i 8.20823 7.59111i 0 −28.2853 7.06929 5.13614i 0 19.8998 + 43.1398i
46.3 −0.536885 + 1.65236i 0 4.03008 + 2.92802i −8.44668 7.32487i 0 7.66213 −18.2465 + 13.2569i 0 16.6382 10.0244i
46.4 0.177563 0.546483i 0 6.20502 + 4.50821i 9.45304 5.96993i 0 −2.67744 7.28438 5.29241i 0 −1.58395 6.22597i
46.5 0.722966 2.22506i 0 2.04392 + 1.48499i 1.87995 + 11.0212i 0 −32.9322 19.9239 14.4756i 0 25.8819 + 3.78491i
46.6 0.907834 2.79403i 0 −0.510282 0.370741i −10.7234 + 3.16365i 0 18.9115 17.5148 12.7253i 0 −0.895733 + 32.8335i
46.7 1.53022 4.70953i 0 −13.3660 9.71093i 9.83672 + 5.31403i 0 28.2766 −34.1374 + 24.8023i 0 40.0789 38.1947i
91.1 −3.87763 2.81726i 0 4.62691 + 14.2402i −7.51887 + 8.27445i 0 0.140520 10.3279 31.7859i 0 52.4667 10.9026i
91.2 −3.25026 2.36146i 0 2.51561 + 7.74226i 6.00716 9.42943i 0 1.75849 0.174670 0.537580i 0 −41.7920 + 16.4625i
91.3 −0.772797 0.561470i 0 −2.19017 6.74065i −11.0970 + 1.36218i 0 12.4836 −4.45357 + 13.7067i 0 9.34059 + 5.17797i
91.4 −0.109191 0.0793317i 0 −2.46651 7.59113i 6.22327 + 9.28822i 0 −17.3099 −0.666555 + 2.05145i 0 0.0573272 1.50789i
91.5 1.08389 + 0.787491i 0 −1.91746 5.90135i 3.83217 10.5031i 0 −12.2101 5.88101 18.0999i 0 12.4247 8.36635i
91.6 3.16070 + 2.29638i 0 2.24451 + 6.90789i 11.0852 + 1.45535i 0 22.0918 0.889297 2.73697i 0 31.6950 + 30.0558i
91.7 3.76530 + 2.73565i 0 4.22155 + 12.9926i −5.34090 + 9.82216i 0 −26.0445 −8.14206 + 25.0587i 0 −46.9800 + 22.3725i
136.1 −3.87763 + 2.81726i 0 4.62691 14.2402i −7.51887 8.27445i 0 0.140520 10.3279 + 31.7859i 0 52.4667 + 10.9026i
136.2 −3.25026 + 2.36146i 0 2.51561 7.74226i 6.00716 + 9.42943i 0 1.75849 0.174670 + 0.537580i 0 −41.7920 16.4625i
136.3 −0.772797 + 0.561470i 0 −2.19017 + 6.74065i −11.0970 1.36218i 0 12.4836 −4.45357 13.7067i 0 9.34059 5.17797i
136.4 −0.109191 + 0.0793317i 0 −2.46651 + 7.59113i 6.22327 9.28822i 0 −17.3099 −0.666555 2.05145i 0 0.0573272 + 1.50789i
136.5 1.08389 0.787491i 0 −1.91746 + 5.90135i 3.83217 + 10.5031i 0 −12.2101 5.88101 + 18.0999i 0 12.4247 + 8.36635i
136.6 3.16070 2.29638i 0 2.24451 6.90789i 11.0852 1.45535i 0 22.0918 0.889297 + 2.73697i 0 31.6950 30.0558i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.h.a 28
3.b odd 2 1 75.4.g.b 28
25.d even 5 1 inner 225.4.h.a 28
75.h odd 10 1 1875.4.a.f 14
75.j odd 10 1 75.4.g.b 28
75.j odd 10 1 1875.4.a.g 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.b 28 3.b odd 2 1
75.4.g.b 28 75.j odd 10 1
225.4.h.a 28 1.a even 1 1 trivial
225.4.h.a 28 25.d even 5 1 inner
1875.4.a.f 14 75.h odd 10 1
1875.4.a.g 14 75.j odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 43 T_{2}^{26} - 21 T_{2}^{25} + 1285 T_{2}^{24} - 803 T_{2}^{23} + 33580 T_{2}^{22} - 26209 T_{2}^{21} + 852818 T_{2}^{20} - 655690 T_{2}^{19} + 13441499 T_{2}^{18} - 14783120 T_{2}^{17} + \cdots + 1769380096$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.