# Properties

 Label 378.2.z.a Level 378 Weight 2 Character orbit 378.z Analytic conductor 3.018 Analytic rank 0 Dimension 144 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 378.z (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$24$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q - 12q^{9} + 12q^{11} + 6q^{14} + 12q^{15} + 24q^{21} - 60q^{23} + 12q^{29} - 72q^{30} + 54q^{35} - 12q^{36} + 48q^{39} + 24q^{42} + 18q^{49} - 60q^{50} - 36q^{51} + 6q^{56} - 60q^{57} - 36q^{60} - 78q^{63} + 72q^{64} - 120q^{65} + 36q^{70} - 72q^{71} - 24q^{72} - 36q^{74} + 66q^{77} - 60q^{78} - 72q^{79} + 12q^{84} - 72q^{85} - 48q^{86} - 18q^{91} + 12q^{92} + 24q^{93} - 120q^{95} + 36q^{98} + 144q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 −0.984808 + 0.173648i −1.71625 + 0.233448i 0.939693 0.342020i −0.0848896 0.0712308i 1.64964 0.527925i −1.74235 1.99104i −0.866025 + 0.500000i 2.89100 0.801309i 0.0959690 + 0.0554077i
41.2 −0.984808 + 0.173648i −1.56441 0.743380i 0.939693 0.342020i −0.941136 0.789707i 1.66973 + 0.460429i 2.62387 + 0.339554i −0.866025 + 0.500000i 1.89477 + 2.32590i 1.06397 + 0.614283i
41.3 −0.984808 + 0.173648i −1.34465 + 1.09174i 0.939693 0.342020i −2.60134 2.18278i 1.13465 1.30865i −0.615646 + 2.57313i −0.866025 + 0.500000i 0.616190 2.93604i 2.94085 + 1.69790i
41.4 −0.984808 + 0.173648i −0.826027 1.52239i 0.939693 0.342020i 1.99843 + 1.67688i 1.07784 + 1.35583i 0.187552 + 2.63910i −0.866025 + 0.500000i −1.63536 + 2.51507i −2.25926 1.30438i
41.5 −0.984808 + 0.173648i −0.754516 + 1.55907i 0.939693 0.342020i 0.486827 + 0.408497i 0.472323 1.66641i −0.719601 2.54601i −0.866025 + 0.500000i −1.86141 2.35269i −0.550366 0.317754i
41.6 −0.984808 + 0.173648i −0.0283469 1.73182i 0.939693 0.342020i −2.71537 2.27847i 0.328643 + 1.70059i 0.173696 2.64004i −0.866025 + 0.500000i −2.99839 + 0.0981834i 3.06977 + 1.77233i
41.7 −0.984808 + 0.173648i 0.0283469 + 1.73182i 0.939693 0.342020i 2.71537 + 2.27847i −0.328643 1.70059i −1.56393 + 2.13404i −0.866025 + 0.500000i −2.99839 + 0.0981834i −3.06977 1.77233i
41.8 −0.984808 + 0.173648i 0.754516 1.55907i 0.939693 0.342020i −0.486827 0.408497i −0.472323 + 1.66641i −2.18779 + 1.48781i −0.866025 + 0.500000i −1.86141 2.35269i 0.550366 + 0.317754i
41.9 −0.984808 + 0.173648i 0.826027 + 1.52239i 0.939693 0.342020i −1.99843 1.67688i −1.07784 1.35583i 1.84005 1.90111i −0.866025 + 0.500000i −1.63536 + 2.51507i 2.25926 + 1.30438i
41.10 −0.984808 + 0.173648i 1.34465 1.09174i 0.939693 0.342020i 2.60134 + 2.18278i −1.13465 + 1.30865i 1.18236 2.36686i −0.866025 + 0.500000i 0.616190 2.93604i −2.94085 1.69790i
41.11 −0.984808 + 0.173648i 1.56441 + 0.743380i 0.939693 0.342020i 0.941136 + 0.789707i −1.66973 0.460429i 2.22826 + 1.42648i −0.866025 + 0.500000i 1.89477 + 2.32590i −1.06397 0.614283i
41.12 −0.984808 + 0.173648i 1.71625 0.233448i 0.939693 0.342020i 0.0848896 + 0.0712308i −1.64964 + 0.527925i −2.61453 + 0.405264i −0.866025 + 0.500000i 2.89100 0.801309i −0.0959690 0.0554077i
41.13 0.984808 0.173648i −1.64560 0.540384i 0.939693 0.342020i 0.881233 + 0.739442i −1.71443 0.246420i −0.336437 + 2.62427i 0.866025 0.500000i 2.41597 + 1.77851i 0.996248 + 0.575184i
41.14 0.984808 0.173648i −1.47964 + 0.900370i 0.939693 0.342020i −1.23395 1.03541i −1.30081 + 1.14363i 1.54951 2.14453i 0.866025 0.500000i 1.37867 2.66445i −1.39500 0.805406i
41.15 0.984808 0.173648i −1.41919 + 0.992922i 0.939693 0.342020i 3.28093 + 2.75302i −1.22521 + 1.22428i −2.27542 1.34999i 0.866025 0.500000i 1.02821 2.81829i 3.70914 + 2.14147i
41.16 0.984808 0.173648i −1.33409 1.10463i 0.939693 0.342020i −1.22837 1.03073i −1.50564 0.856190i −2.38077 1.15409i 0.866025 0.500000i 0.559569 + 2.94735i −1.38870 0.801764i
41.17 0.984808 0.173648i −0.419652 1.68044i 0.939693 0.342020i 1.95067 + 1.63681i −0.705083 1.58204i 2.43278 1.04000i 0.866025 0.500000i −2.64778 + 1.41040i 2.20527 + 1.27321i
41.18 0.984808 0.173648i −0.0750633 1.73042i 0.939693 0.342020i −2.14704 1.80158i −0.374408 1.69110i 2.53928 + 0.743006i 0.866025 0.500000i −2.98873 + 0.259783i −2.42726 1.40138i
41.19 0.984808 0.173648i 0.0750633 + 1.73042i 0.939693 0.342020i 2.14704 + 1.80158i 0.374408 + 1.69110i 2.42280 + 1.06304i 0.866025 0.500000i −2.98873 + 0.259783i 2.42726 + 1.40138i
41.20 0.984808 0.173648i 0.419652 + 1.68044i 0.939693 0.342020i −1.95067 1.63681i 0.705083 + 1.58204i 1.19511 + 2.36045i 0.866025 0.500000i −2.64778 + 1.41040i −2.20527 1.27321i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 335.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
27.f odd 18 1 inner
189.be 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.z.a 144
7.b odd 2 1 inner 378.2.z.a 144
27.f odd 18 1 inner 378.2.z.a 144
189.be even 18 1 inner 378.2.z.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.z.a 144 1.a even 1 1 trivial
378.2.z.a 144 7.b odd 2 1 inner
378.2.z.a 144 27.f odd 18 1 inner
378.2.z.a 144 189.be even 18 1 inner

## Hecke kernels

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

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database