# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 72 q^{4} - 62 q^{6} - 34 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28 q + 72 q^{4} - 62 q^{6} - 34 q^{9} + 46 q^{11} + 42 q^{14} - 648 q^{16} - 1116 q^{19} + 360 q^{21} - 96 q^{24} + 1432 q^{26} + 592 q^{29} - 488 q^{31} + 250 q^{34} - 4798 q^{36} - 1268 q^{39} - 94 q^{41} + 220 q^{44} + 2868 q^{46} + 2450 q^{49} + 3034 q^{51} + 8132 q^{54} - 1962 q^{56} + 170 q^{59} - 1656 q^{61} - 8944 q^{64} + 9860 q^{66} + 1644 q^{69} - 1312 q^{71} + 2632 q^{74} - 5578 q^{76} + 4220 q^{79} - 4334 q^{81} - 11550 q^{84} - 5138 q^{86} - 12192 q^{89} + 13352 q^{91} - 1034 q^{94} - 1186 q^{96} - 4640 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 −4.66357 + 2.69252i 3.19791 4.09553i 10.4993 18.1853i 0 −3.88642 + 27.7102i −10.8861 + 6.28510i 69.9976i −6.54673 26.1943i 0
49.2 −4.60336 + 2.65775i −0.153351 + 5.19389i 10.1273 17.5410i 0 −13.0981 24.3169i −11.6339 + 6.71686i 65.1396i −26.9530 1.59298i 0
49.3 −3.69081 + 2.13089i 2.76977 4.39640i 5.08138 8.80120i 0 −0.854448 + 22.1284i 26.6423 15.3820i 9.21718i −11.6567 24.3541i 0
49.4 −2.63422 + 1.52087i 3.29398 + 4.01867i 0.626094 1.08443i 0 −14.7890 5.57636i 11.8751 6.85611i 20.5251i −5.29939 + 26.4748i 0
49.5 −1.90044 + 1.09722i −5.19206 0.206141i −1.59221 + 2.75778i 0 10.0934 5.30509i 2.39547 1.38302i 24.5436i 26.9150 + 2.14059i 0
49.6 −1.35984 + 0.785104i −3.43441 3.89934i −2.76722 + 4.79297i 0 7.73164 + 2.60611i −29.6525 + 17.1199i 21.2519i −3.40971 + 26.7838i 0
49.7 −0.195072 + 0.112625i 4.76710 2.06755i −3.97463 + 6.88426i 0 −0.697071 + 0.940215i −27.0148 + 15.5970i 3.59257i 18.4505 19.7124i 0
49.8 0.195072 0.112625i −4.76710 + 2.06755i −3.97463 + 6.88426i 0 −0.697071 + 0.940215i 27.0148 15.5970i 3.59257i 18.4505 19.7124i 0
49.9 1.35984 0.785104i 3.43441 + 3.89934i −2.76722 + 4.79297i 0 7.73164 + 2.60611i 29.6525 17.1199i 21.2519i −3.40971 + 26.7838i 0
49.10 1.90044 1.09722i 5.19206 + 0.206141i −1.59221 + 2.75778i 0 10.0934 5.30509i −2.39547 + 1.38302i 24.5436i 26.9150 + 2.14059i 0
49.11 2.63422 1.52087i −3.29398 4.01867i 0.626094 1.08443i 0 −14.7890 5.57636i −11.8751 + 6.85611i 20.5251i −5.29939 + 26.4748i 0
49.12 3.69081 2.13089i −2.76977 + 4.39640i 5.08138 8.80120i 0 −0.854448 + 22.1284i −26.6423 + 15.3820i 9.21718i −11.6567 24.3541i 0
49.13 4.60336 2.65775i 0.153351 5.19389i 10.1273 17.5410i 0 −13.0981 24.3169i 11.6339 6.71686i 65.1396i −26.9530 1.59298i 0
49.14 4.66357 2.69252i −3.19791 + 4.09553i 10.4993 18.1853i 0 −3.88642 + 27.7102i 10.8861 6.28510i 69.9976i −6.54673 26.1943i 0
124.1 −4.66357 2.69252i 3.19791 + 4.09553i 10.4993 + 18.1853i 0 −3.88642 27.7102i −10.8861 6.28510i 69.9976i −6.54673 + 26.1943i 0
124.2 −4.60336 2.65775i −0.153351 5.19389i 10.1273 + 17.5410i 0 −13.0981 + 24.3169i −11.6339 6.71686i 65.1396i −26.9530 + 1.59298i 0
124.3 −3.69081 2.13089i 2.76977 + 4.39640i 5.08138 + 8.80120i 0 −0.854448 22.1284i 26.6423 + 15.3820i 9.21718i −11.6567 + 24.3541i 0
124.4 −2.63422 1.52087i 3.29398 4.01867i 0.626094 + 1.08443i 0 −14.7890 + 5.57636i 11.8751 + 6.85611i 20.5251i −5.29939 26.4748i 0
124.5 −1.90044 1.09722i −5.19206 + 0.206141i −1.59221 2.75778i 0 10.0934 + 5.30509i 2.39547 + 1.38302i 24.5436i 26.9150 2.14059i 0
124.6 −1.35984 0.785104i −3.43441 + 3.89934i −2.76722 4.79297i 0 7.73164 2.60611i −29.6525 17.1199i 21.2519i −3.40971 26.7838i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.14 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.k.d 28
5.b even 2 1 inner 225.4.k.d 28
5.c odd 4 1 45.4.e.c 14
5.c odd 4 1 225.4.e.d 14
9.c even 3 1 inner 225.4.k.d 28
15.e even 4 1 135.4.e.c 14
45.j even 6 1 inner 225.4.k.d 28
45.k odd 12 1 45.4.e.c 14
45.k odd 12 1 225.4.e.d 14
45.k odd 12 1 405.4.a.m 7
45.k odd 12 1 2025.4.a.bb 7
45.l even 12 1 135.4.e.c 14
45.l even 12 1 405.4.a.n 7
45.l even 12 1 2025.4.a.ba 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 5.c odd 4 1
45.4.e.c 14 45.k odd 12 1
135.4.e.c 14 15.e even 4 1
135.4.e.c 14 45.l even 12 1
225.4.e.d 14 5.c odd 4 1
225.4.e.d 14 45.k odd 12 1
225.4.k.d 28 1.a even 1 1 trivial
225.4.k.d 28 5.b even 2 1 inner
225.4.k.d 28 9.c even 3 1 inner
225.4.k.d 28 45.j even 6 1 inner
405.4.a.m 7 45.k odd 12 1
405.4.a.n 7 45.l even 12 1
2025.4.a.ba 7 45.l even 12 1
2025.4.a.bb 7 45.k odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!04$$$$T_{2}^{8} -$$$$23\!\cdots\!16$$$$T_{2}^{6} +$$$$27\!\cdots\!44$$$$T_{2}^{4} - 141416202240 T_{2}^{2} + 6879707136$$">$$T_{2}^{28} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.