# Properties

 Label 378.2.ba.a Level $378$ Weight $2$ Character orbit 378.ba Analytic conductor $3.018$ Analytic rank $0$ Dimension $144$ 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.ba (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$24$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 18 q^{6} + 24 q^{9}+O(q^{10})$$ 144 * q - 18 * q^6 + 24 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 18 q^{6} + 24 q^{9} - 24 q^{11} + 6 q^{14} + 12 q^{15} - 30 q^{21} - 42 q^{23} - 6 q^{29} - 18 q^{30} + 6 q^{36} + 48 q^{39} - 48 q^{42} + 18 q^{45} - 54 q^{47} - 36 q^{49} + 12 q^{50} - 36 q^{51} - 90 q^{53} + 6 q^{56} - 6 q^{57} + 18 q^{60} + 54 q^{61} - 24 q^{63} + 72 q^{64} + 78 q^{65} - 72 q^{66} - 54 q^{68} + 36 q^{69} - 18 q^{70} - 72 q^{71} + 12 q^{72} - 36 q^{74} - 6 q^{77} - 60 q^{78} + 36 q^{79} - 6 q^{84} - 72 q^{85} + 24 q^{86} + 36 q^{91} - 42 q^{92} + 96 q^{93} - 66 q^{95} - 108 q^{99}+O(q^{100})$$ 144 * q - 18 * q^6 + 24 * q^9 - 24 * q^11 + 6 * q^14 + 12 * q^15 - 30 * q^21 - 42 * q^23 - 6 * q^29 - 18 * q^30 + 6 * q^36 + 48 * q^39 - 48 * q^42 + 18 * q^45 - 54 * q^47 - 36 * q^49 + 12 * q^50 - 36 * q^51 - 90 * q^53 + 6 * q^56 - 6 * q^57 + 18 * q^60 + 54 * q^61 - 24 * q^63 + 72 * q^64 + 78 * q^65 - 72 * q^66 - 54 * q^68 + 36 * q^69 - 18 * q^70 - 72 * q^71 + 12 * q^72 - 36 * q^74 - 6 * q^77 - 60 * q^78 + 36 * q^79 - 6 * q^84 - 72 * q^85 + 24 * q^86 + 36 * q^91 - 42 * q^92 + 96 * q^93 - 66 * q^95 - 108 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −0.984808 0.173648i −1.69591 + 0.351999i 0.939693 + 0.342020i −2.70582 0.984837i 1.73127 0.0521605i 2.49997 0.866117i −0.866025 0.500000i 2.75219 1.19391i 2.49370 + 1.43974i
47.2 −0.984808 0.173648i −1.65689 0.504694i 0.939693 + 0.342020i 1.24189 + 0.452012i 1.54408 + 0.784742i −2.58194 0.577558i −0.866025 0.500000i 2.49057 + 1.67244i −1.14453 0.660797i
47.3 −0.984808 0.173648i −1.45666 + 0.937089i 0.939693 + 0.342020i 3.19574 + 1.16316i 1.59726 0.669906i 0.0714494 + 2.64479i −0.866025 0.500000i 1.24373 2.73004i −2.94521 1.70042i
47.4 −0.984808 0.173648i −1.17037 1.27680i 0.939693 + 0.342020i 1.68087 + 0.611788i 0.930879 + 1.46064i 1.98134 1.75337i −0.866025 0.500000i −0.260448 + 2.98867i −1.54910 0.894374i
47.5 −0.984808 0.173648i −0.235548 1.71596i 0.939693 + 0.342020i −1.87429 0.682186i −0.0660042 + 1.73079i −0.317208 + 2.62667i −0.866025 0.500000i −2.88903 + 0.808380i 1.72736 + 0.997290i
47.6 −0.984808 0.173648i 0.206907 + 1.71965i 0.939693 + 0.342020i −2.47224 0.899821i 0.0948504 1.72945i −1.11322 + 2.40015i −0.866025 0.500000i −2.91438 + 0.711614i 2.27843 + 1.31545i
47.7 −0.984808 0.173648i 0.363371 + 1.69351i 0.939693 + 0.342020i −0.772614 0.281208i −0.0637767 1.73088i −1.54265 2.14948i −0.866025 0.500000i −2.73592 + 1.23074i 0.712045 + 0.411099i
47.8 −0.984808 0.173648i 0.679714 1.59311i 0.939693 + 0.342020i 1.56056 + 0.567996i −0.946028 + 1.45087i −2.50129 0.862294i −0.866025 0.500000i −2.07598 2.16571i −1.43822 0.830355i
47.9 −0.984808 0.173648i 1.48209 0.896330i 0.939693 + 0.342020i −2.03467 0.740558i −1.61522 + 0.625351i 2.23500 + 1.41591i −0.866025 0.500000i 1.39318 2.65689i 1.87516 + 1.08262i
47.10 −0.984808 0.173648i 1.50439 0.858369i 0.939693 + 0.342020i 2.90863 + 1.05865i −1.63059 + 0.584093i 2.64574 0.00779754i −0.866025 0.500000i 1.52640 2.58265i −2.68061 1.54765i
47.11 −0.984808 0.173648i 1.63664 + 0.566931i 0.939693 + 0.342020i 1.63454 + 0.594924i −1.51333 0.842517i −2.03053 + 1.69616i −0.866025 0.500000i 2.35718 + 1.85572i −1.50640 0.869721i
47.12 −0.984808 0.173648i 1.66909 + 0.462748i 0.939693 + 0.342020i −2.36261 0.859919i −1.56338 0.745552i 0.311323 2.62737i −0.866025 0.500000i 2.57173 + 1.54474i 2.17739 + 1.25712i
47.13 0.984808 + 0.173648i −1.69863 0.338627i 0.939693 + 0.342020i −0.525051 0.191103i −1.61402 0.628446i −0.0974999 2.64395i 0.866025 + 0.500000i 2.77066 + 1.15040i −0.483890 0.279374i
47.14 0.984808 + 0.173648i −1.63576 0.569458i 0.939693 + 0.342020i −3.08742 1.12373i −1.51203 0.844853i −0.0850968 + 2.64438i 0.866025 + 0.500000i 2.35144 + 1.86299i −2.84538 1.64278i
47.15 0.984808 + 0.173648i −1.49340 0.877362i 0.939693 + 0.342020i 3.97042 + 1.44511i −1.31836 1.12336i −1.73420 + 1.99814i 0.866025 + 0.500000i 1.46047 + 2.62050i 3.65916 + 2.11262i
47.16 0.984808 + 0.173648i −1.40287 + 1.01585i 0.939693 + 0.342020i 2.68457 + 0.977104i −1.55796 + 0.756808i 1.78067 1.95684i 0.866025 + 0.500000i 0.936109 2.85021i 2.47411 + 1.42843i
47.17 0.984808 + 0.173648i −0.611133 + 1.62065i 0.939693 + 0.342020i −4.17040 1.51790i −0.883272 + 1.48991i −1.63543 2.07976i 0.866025 + 0.500000i −2.25303 1.98087i −3.84346 2.21902i
47.18 0.984808 + 0.173648i −0.505130 + 1.65676i 0.939693 + 0.342020i −0.342190 0.124547i −0.785149 + 1.54387i 2.40698 + 1.09838i 0.866025 + 0.500000i −2.48969 1.67375i −0.315364 0.182076i
47.19 0.984808 + 0.173648i −0.0904059 1.72969i 0.939693 + 0.342020i 1.22429 + 0.445607i 0.211325 1.71911i 2.52854 + 0.778769i 0.866025 + 0.500000i −2.98365 + 0.312748i 1.12832 + 0.651433i
47.20 0.984808 + 0.173648i 0.692095 + 1.58777i 0.939693 + 0.342020i 0.235782 + 0.0858176i 0.405868 + 1.68383i −2.16551 + 1.52006i 0.866025 + 0.500000i −2.04201 + 2.19777i 0.217298 + 0.125457i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.24 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.bd even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.ba.a 144
7.d odd 6 1 378.2.bf.a yes 144
27.f odd 18 1 378.2.bf.a yes 144
189.bd even 18 1 inner 378.2.ba.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.ba.a 144 1.a even 1 1 trivial
378.2.ba.a 144 189.bd even 18 1 inner
378.2.bf.a yes 144 7.d odd 6 1
378.2.bf.a yes 144 27.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(378, [\chi])$$.