# Properties

 Label 378.2.v.a Level $378$ Weight $2$ Character orbit 378.v Analytic conductor $3.018$ Analytic rank $0$ Dimension $72$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.v (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$12$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 3 q^{6} - 6 q^{7} + 36 q^{8}+O(q^{10})$$ 72 * q + 3 * q^6 - 6 * q^7 + 36 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$72 q + 3 q^{6} - 6 q^{7} + 36 q^{8} + 6 q^{10} + 12 q^{11} + 12 q^{13} - 3 q^{14} - 6 q^{15} - 12 q^{17} + 18 q^{19} + 6 q^{22} + 12 q^{23} - 6 q^{25} - 18 q^{26} + 6 q^{27} + 3 q^{29} + 15 q^{30} - 9 q^{31} - 18 q^{33} - 9 q^{34} - 18 q^{35} - 3 q^{36} - 48 q^{39} - 6 q^{41} + 36 q^{42} - 24 q^{43} - 45 q^{45} - 27 q^{47} + 6 q^{48} + 48 q^{49} + 6 q^{50} + 15 q^{51} - 6 q^{52} - 15 q^{53} - 63 q^{54} - 72 q^{55} - 3 q^{56} + 57 q^{57} - 3 q^{58} - 30 q^{59} + 21 q^{60} + 9 q^{61} - 24 q^{62} - 21 q^{63} - 36 q^{64} + 45 q^{65} + 6 q^{67} + 9 q^{68} - 39 q^{69} + 15 q^{70} + 12 q^{71} - 60 q^{73} - 18 q^{74} + 42 q^{75} + 3 q^{77} + 6 q^{78} - 27 q^{79} - 6 q^{80} + 12 q^{81} + 33 q^{82} + 18 q^{83} - 21 q^{84} + 51 q^{85} - 12 q^{86} - 117 q^{87} + 6 q^{88} - 36 q^{89} - 69 q^{90} - 3 q^{91} + 12 q^{92} - 30 q^{93} - 27 q^{94} + 3 q^{95} + 48 q^{97} - 12 q^{98} + 90 q^{99}+O(q^{100})$$ 72 * q + 3 * q^6 - 6 * q^7 + 36 * q^8 + 6 * q^10 + 12 * q^11 + 12 * q^13 - 3 * q^14 - 6 * q^15 - 12 * q^17 + 18 * q^19 + 6 * q^22 + 12 * q^23 - 6 * q^25 - 18 * q^26 + 6 * q^27 + 3 * q^29 + 15 * q^30 - 9 * q^31 - 18 * q^33 - 9 * q^34 - 18 * q^35 - 3 * q^36 - 48 * q^39 - 6 * q^41 + 36 * q^42 - 24 * q^43 - 45 * q^45 - 27 * q^47 + 6 * q^48 + 48 * q^49 + 6 * q^50 + 15 * q^51 - 6 * q^52 - 15 * q^53 - 63 * q^54 - 72 * q^55 - 3 * q^56 + 57 * q^57 - 3 * q^58 - 30 * q^59 + 21 * q^60 + 9 * q^61 - 24 * q^62 - 21 * q^63 - 36 * q^64 + 45 * q^65 + 6 * q^67 + 9 * q^68 - 39 * q^69 + 15 * q^70 + 12 * q^71 - 60 * q^73 - 18 * q^74 + 42 * q^75 + 3 * q^77 + 6 * q^78 - 27 * q^79 - 6 * q^80 + 12 * q^81 + 33 * q^82 + 18 * q^83 - 21 * q^84 + 51 * q^85 - 12 * q^86 - 117 * q^87 + 6 * q^88 - 36 * q^89 - 69 * q^90 - 3 * q^91 + 12 * q^92 - 30 * q^93 - 27 * q^94 + 3 * q^95 + 48 * q^97 - 12 * q^98 + 90 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −0.766044 + 0.642788i −1.71753 0.223844i 0.173648 0.984808i 0.0598945 0.339678i 1.45959 0.932530i −1.66417 + 2.05683i 0.500000 + 0.866025i 2.89979 + 0.768916i 0.172459 + 0.298708i
67.2 −0.766044 + 0.642788i −1.42403 0.985977i 0.173648 0.984808i −0.327106 + 1.85511i 1.72464 0.160044i −1.01788 2.44211i 0.500000 + 0.866025i 1.05570 + 2.80811i −0.941864 1.63136i
67.3 −0.766044 + 0.642788i −1.38102 + 1.04536i 0.173648 0.984808i −0.716530 + 4.06364i 0.385982 1.68850i 2.64332 + 0.113332i 0.500000 + 0.866025i 0.814451 2.88733i −2.06317 3.57351i
67.4 −0.766044 + 0.642788i −0.921674 1.46646i 0.173648 0.984808i 0.622926 3.53279i 1.64867 + 0.530936i 2.63270 + 0.262455i 0.500000 + 0.866025i −1.30103 + 2.70320i 1.79365 + 3.10668i
67.5 −0.766044 + 0.642788i −0.829129 + 1.52071i 0.173648 0.984808i 0.684323 3.88099i −0.342341 1.69788i −1.85424 + 1.88727i 0.500000 + 0.866025i −1.62509 2.52172i 1.97043 + 3.41288i
67.6 −0.766044 + 0.642788i −0.402311 + 1.68468i 0.173648 0.984808i 0.256458 1.45445i −0.774703 1.54914i 2.04878 1.67407i 0.500000 + 0.866025i −2.67629 1.35553i 0.738441 + 1.27902i
67.7 −0.766044 + 0.642788i 0.0412776 1.73156i 0.173648 0.984808i −0.194594 + 1.10360i 1.08140 + 1.35298i −0.0465876 + 2.64534i 0.500000 + 0.866025i −2.99659 0.142949i −0.560311 0.970488i
67.8 −0.766044 + 0.642788i 0.309985 + 1.70409i 0.173648 0.984808i −0.504047 + 2.85859i −1.33283 1.10615i −2.58705 + 0.554250i 0.500000 + 0.866025i −2.80782 + 1.05648i −1.45134 2.51380i
67.9 −0.766044 + 0.642788i 0.600510 1.62462i 0.173648 0.984808i 0.421605 2.39104i 0.584268 + 1.63053i −1.84906 1.89235i 0.500000 + 0.866025i −2.27878 1.95120i 1.21396 + 2.10265i
67.10 −0.766044 + 0.642788i 1.54135 0.790081i 0.173648 0.984808i −0.272791 + 1.54708i −0.672891 + 1.59600i 2.24230 1.40432i 0.500000 + 0.866025i 1.75154 2.43559i −0.785472 1.36048i
67.11 −0.766044 + 0.642788i 1.59651 + 0.671670i 0.173648 0.984808i −0.153204 + 0.868862i −1.65474 + 0.511691i 0.258184 + 2.63312i 0.500000 + 0.866025i 2.09772 + 2.14466i −0.441133 0.764065i
67.12 −0.766044 + 0.642788i 1.64636 + 0.538065i 0.173648 0.984808i 0.470361 2.66755i −1.60704 + 0.646075i −1.97994 1.75495i 0.500000 + 0.866025i 2.42097 + 1.77169i 1.35435 + 2.34580i
79.1 −0.766044 0.642788i −1.71753 + 0.223844i 0.173648 + 0.984808i 0.0598945 + 0.339678i 1.45959 + 0.932530i −1.66417 2.05683i 0.500000 0.866025i 2.89979 0.768916i 0.172459 0.298708i
79.2 −0.766044 0.642788i −1.42403 + 0.985977i 0.173648 + 0.984808i −0.327106 1.85511i 1.72464 + 0.160044i −1.01788 + 2.44211i 0.500000 0.866025i 1.05570 2.80811i −0.941864 + 1.63136i
79.3 −0.766044 0.642788i −1.38102 1.04536i 0.173648 + 0.984808i −0.716530 4.06364i 0.385982 + 1.68850i 2.64332 0.113332i 0.500000 0.866025i 0.814451 + 2.88733i −2.06317 + 3.57351i
79.4 −0.766044 0.642788i −0.921674 + 1.46646i 0.173648 + 0.984808i 0.622926 + 3.53279i 1.64867 0.530936i 2.63270 0.262455i 0.500000 0.866025i −1.30103 2.70320i 1.79365 3.10668i
79.5 −0.766044 0.642788i −0.829129 1.52071i 0.173648 + 0.984808i 0.684323 + 3.88099i −0.342341 + 1.69788i −1.85424 1.88727i 0.500000 0.866025i −1.62509 + 2.52172i 1.97043 3.41288i
79.6 −0.766044 0.642788i −0.402311 1.68468i 0.173648 + 0.984808i 0.256458 + 1.45445i −0.774703 + 1.54914i 2.04878 + 1.67407i 0.500000 0.866025i −2.67629 + 1.35553i 0.738441 1.27902i
79.7 −0.766044 0.642788i 0.0412776 + 1.73156i 0.173648 + 0.984808i −0.194594 1.10360i 1.08140 1.35298i −0.0465876 2.64534i 0.500000 0.866025i −2.99659 + 0.142949i −0.560311 + 0.970488i
79.8 −0.766044 0.642788i 0.309985 1.70409i 0.173648 + 0.984808i −0.504047 2.85859i −1.33283 + 1.10615i −2.58705 0.554250i 0.500000 0.866025i −2.80782 1.05648i −1.45134 + 2.51380i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.u even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.v.a 72
7.c even 3 1 378.2.w.b yes 72
27.e even 9 1 378.2.w.b yes 72
189.u even 9 1 inner 378.2.v.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.v.a 72 1.a even 1 1 trivial
378.2.v.a 72 189.u even 9 1 inner
378.2.w.b yes 72 7.c even 3 1
378.2.w.b yes 72 27.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{72} + 3 T_{5}^{70} + 41 T_{5}^{69} - 105 T_{5}^{68} + 207 T_{5}^{67} + 6532 T_{5}^{66} - 7476 T_{5}^{65} + 57867 T_{5}^{64} + 137570 T_{5}^{63} - 299070 T_{5}^{62} + 1828671 T_{5}^{61} + 26671486 T_{5}^{60} + \cdots + 2678994081$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.