# Properties

 Label 378.2.bf.a Level $378$ Weight $2$ Character orbit 378.bf 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.bf (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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q + 18 q^{6} - 12 q^{9}+O(q^{10})$$ 144 * q + 18 * q^6 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$144 q + 18 q^{6} - 12 q^{9} + 12 q^{11} + 6 q^{14} + 12 q^{15} + 6 q^{21} - 6 q^{23} - 6 q^{29} - 18 q^{30} - 54 q^{35} + 6 q^{36} - 6 q^{39} + 24 q^{42} - 54 q^{47} + 18 q^{49} + 12 q^{50} + 18 q^{51} + 90 q^{53} - 54 q^{54} - 12 q^{56} - 6 q^{57} - 90 q^{59} - 36 q^{60} - 24 q^{63} + 72 q^{64} - 84 q^{65} - 36 q^{69} - 18 q^{70} - 72 q^{71} + 12 q^{72} + 18 q^{74} - 90 q^{75} - 78 q^{77} - 60 q^{78} + 36 q^{79} - 6 q^{84} - 72 q^{85} + 24 q^{86} + 90 q^{87} - 18 q^{91} - 42 q^{92} - 12 q^{93} + 78 q^{95} - 36 q^{98} - 108 q^{99}+O(q^{100})$$ 144 * q + 18 * q^6 - 12 * q^9 + 12 * q^11 + 6 * q^14 + 12 * q^15 + 6 * q^21 - 6 * q^23 - 6 * q^29 - 18 * q^30 - 54 * q^35 + 6 * q^36 - 6 * q^39 + 24 * q^42 - 54 * q^47 + 18 * q^49 + 12 * q^50 + 18 * q^51 + 90 * q^53 - 54 * q^54 - 12 * q^56 - 6 * q^57 - 90 * q^59 - 36 * q^60 - 24 * q^63 + 72 * q^64 - 84 * q^65 - 36 * q^69 - 18 * q^70 - 72 * q^71 + 12 * q^72 + 18 * q^74 - 90 * q^75 - 78 * q^77 - 60 * q^78 + 36 * q^79 - 6 * q^84 - 72 * q^85 + 24 * q^86 + 90 * q^87 - 18 * q^91 - 42 * q^92 - 12 * q^93 + 78 * q^95 - 36 * q^98 - 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.984808 + 0.173648i −1.69494 0.356630i 0.939693 0.342020i 0.295406 1.67533i 1.73112 + 0.0568892i 2.46278 0.966803i −0.866025 + 0.500000i 2.74563 + 1.20893i 1.70117i
5.2 −0.984808 + 0.173648i −1.68531 + 0.399650i 0.939693 0.342020i −0.662712 + 3.75843i 1.59031 0.686230i −0.925433 + 2.47862i −0.866025 + 0.500000i 2.68056 1.34707i 3.81641i
5.3 −0.984808 + 0.173648i −1.31364 1.12887i 0.939693 0.342020i −0.00506474 + 0.0287236i 1.48971 + 0.883607i −2.31258 + 1.28530i −0.866025 + 0.500000i 0.451313 + 2.96586i 0.0291667i
5.4 −0.984808 + 0.173648i −0.906167 + 1.47610i 0.939693 0.342020i 0.392637 2.22675i 0.636079 1.61103i 1.14836 + 2.38354i −0.866025 + 0.500000i −1.35772 2.67518i 2.26111i
5.5 −0.984808 + 0.173648i −0.755599 + 1.55855i 0.939693 0.342020i −0.0946220 + 0.536628i 0.473481 1.66608i −2.19872 1.47161i −0.866025 + 0.500000i −1.85814 2.35527i 0.544906i
5.6 −0.984808 + 0.173648i −0.542374 1.64494i 0.939693 0.342020i 0.381353 2.16276i 0.819775 + 1.52577i −1.53538 2.15467i −0.866025 + 0.500000i −2.41166 + 1.78435i 2.19613i
5.7 −0.984808 + 0.173648i −0.507755 1.65595i 0.939693 0.342020i −0.693451 + 3.93276i 0.787595 + 1.54263i 1.74878 1.98539i −0.866025 + 0.500000i −2.48437 + 1.68164i 3.99342i
5.8 −0.984808 + 0.173648i 0.801094 1.53566i 0.939693 0.342020i 0.703607 3.99035i −0.522259 + 1.65144i 2.53004 + 0.773898i −0.866025 + 0.500000i −1.71650 2.46041i 4.05191i
5.9 −0.984808 + 0.173648i 0.891925 + 1.48475i 0.939693 0.342020i 0.217432 1.23312i −1.13620 1.30731i −2.23659 + 1.41339i −0.866025 + 0.500000i −1.40894 + 2.64856i 1.25214i
5.10 −0.984808 + 0.173648i 1.22171 + 1.22777i 0.939693 0.342020i 0.204833 1.16167i −1.41635 0.996970i 1.70692 2.02149i −0.866025 + 0.500000i −0.0148421 + 2.99996i 1.17959i
5.11 −0.984808 + 0.173648i 1.43254 0.973561i 0.939693 0.342020i −0.404896 + 2.29628i −1.24172 + 1.20753i −2.01803 1.71101i −0.866025 + 0.500000i 1.10436 2.78934i 2.33170i
5.12 −0.984808 + 0.173648i 1.73169 + 0.0354625i 0.939693 0.342020i −0.334523 + 1.89717i −1.71154 + 0.265781i 2.15385 + 1.53653i −0.866025 + 0.500000i 2.99748 + 0.122820i 1.92644i
5.13 0.984808 0.173648i −1.73120 0.0542063i 0.939693 0.342020i −0.139489 + 0.791082i −1.71431 + 0.247237i 1.46739 + 2.20154i 0.866025 0.500000i 2.99412 + 0.187684i 0.803286i
5.14 0.984808 0.173648i −1.68862 + 0.385431i 0.939693 0.342020i 0.136034 0.771489i −1.59604 + 0.672801i −2.64466 + 0.0761338i 0.866025 0.500000i 2.70289 1.30169i 0.783391i
5.15 0.984808 0.173648i −1.15375 + 1.29185i 0.939693 0.342020i 0.485901 2.75568i −0.911891 + 1.47257i 1.15554 2.38007i 0.866025 0.500000i −0.337739 2.98093i 2.79820i
5.16 0.984808 0.173648i −0.926008 1.46373i 0.939693 0.342020i 0.475930 2.69913i −1.16611 1.28069i 2.49768 + 0.872702i 0.866025 0.500000i −1.28502 + 2.71085i 2.74077i
5.17 0.984808 0.173648i −0.424397 + 1.67925i 0.939693 0.342020i −0.661063 + 3.74907i −0.126350 + 1.72744i −2.47370 0.938514i 0.866025 0.500000i −2.63977 1.42534i 3.80691i
5.18 0.984808 0.173648i 0.228430 1.71692i 0.939693 0.342020i 0.0564431 0.320105i −0.0731809 1.73050i −0.543232 2.58938i 0.866025 0.500000i −2.89564 0.784392i 0.325043i
5.19 0.984808 0.173648i 0.261649 + 1.71217i 0.939693 0.342020i −0.00558222 + 0.0316584i 0.554990 + 1.64073i 2.62068 0.363383i 0.866025 0.500000i −2.86308 + 0.895977i 0.0321467i
5.20 0.984808 0.173648i 0.963600 1.43926i 0.939693 0.342020i −0.689038 + 3.90773i 0.699036 1.58472i 0.360258 + 2.62111i 0.866025 0.500000i −1.14295 2.77375i 3.96801i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.24 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.ba 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.bf.a yes 144
7.d odd 6 1 378.2.ba.a 144
27.f odd 18 1 378.2.ba.a 144
189.ba even 18 1 inner 378.2.bf.a yes 144

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

## Hecke kernels

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