# Properties

 Label 819.2.ew.a Level $819$ Weight $2$ Character orbit 819.ew Analytic conductor $6.540$ Analytic rank $0$ Dimension $336$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.ew (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$336$$ Relative dimension: $$84$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$336 q + 24 q^{6} - 36 q^{8}+O(q^{10})$$ 336 * q + 24 * q^6 - 36 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$336 q + 24 q^{6} - 36 q^{8} + 16 q^{15} - 336 q^{16} + 20 q^{18} + 8 q^{21} + 8 q^{24} + 96 q^{26} + 24 q^{27} - 96 q^{30} - 72 q^{32} - 36 q^{33} - 12 q^{36} - 72 q^{38} - 16 q^{39} - 48 q^{41} - 28 q^{45} + 60 q^{47} - 48 q^{48} + 132 q^{50} - 36 q^{52} - 108 q^{54} + 68 q^{57} - 72 q^{58} - 44 q^{60} - 36 q^{62} - 8 q^{63} - 72 q^{65} - 20 q^{66} - 72 q^{69} + 48 q^{71} - 104 q^{72} + 12 q^{74} + 104 q^{78} - 12 q^{79} + 96 q^{80} - 68 q^{81} - 120 q^{83} - 12 q^{84} + 36 q^{85} + 48 q^{86} - 24 q^{87} - 60 q^{89} - 72 q^{92} - 172 q^{93} - 48 q^{94} - 152 q^{96} + 48 q^{97}+O(q^{100})$$ 336 * q + 24 * q^6 - 36 * q^8 + 16 * q^15 - 336 * q^16 + 20 * q^18 + 8 * q^21 + 8 * q^24 + 96 * q^26 + 24 * q^27 - 96 * q^30 - 72 * q^32 - 36 * q^33 - 12 * q^36 - 72 * q^38 - 16 * q^39 - 48 * q^41 - 28 * q^45 + 60 * q^47 - 48 * q^48 + 132 * q^50 - 36 * q^52 - 108 * q^54 + 68 * q^57 - 72 * q^58 - 44 * q^60 - 36 * q^62 - 8 * q^63 - 72 * q^65 - 20 * q^66 - 72 * q^69 + 48 * q^71 - 104 * q^72 + 12 * q^74 + 104 * q^78 - 12 * q^79 + 96 * q^80 - 68 * q^81 - 120 * q^83 - 12 * q^84 + 36 * q^85 + 48 * q^86 - 24 * q^87 - 60 * q^89 - 72 * q^92 - 172 * q^93 - 48 * q^94 - 152 * q^96 + 48 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
176.1 −1.95788 + 1.95788i −1.00105 + 1.41347i 5.66657i −0.576104 2.15005i −0.807482 4.72733i −0.258819 0.965926i 7.17868 + 7.17868i −0.995814 2.82990i 5.33747 + 3.08159i
176.2 −1.91264 + 1.91264i −1.50144 + 0.863523i 5.31637i 0.990693 + 3.69732i 1.22011 4.52332i 0.258819 + 0.965926i 6.34303 + 6.34303i 1.50865 2.59306i −8.96647 5.17679i
176.3 −1.87295 + 1.87295i 0.832889 1.51865i 5.01588i 0.0815777 + 0.304452i 1.28440 + 4.40431i 0.258819 + 0.965926i 5.64860 + 5.64860i −1.61259 2.52973i −0.723014 0.417432i
176.4 −1.86338 + 1.86338i 0.258684 1.71262i 4.94436i −0.423179 1.57933i 2.70924 + 3.67329i −0.258819 0.965926i 5.48647 + 5.48647i −2.86617 0.886057i 3.73143 + 2.15434i
176.5 −1.83201 + 1.83201i −1.42869 0.979200i 4.71250i −0.819999 3.06028i 4.41128 0.823478i 0.258819 + 0.965926i 4.96932 + 4.96932i 1.08234 + 2.79795i 7.10869 + 4.10421i
176.6 −1.80918 + 1.80918i −0.649457 1.60568i 4.54630i 0.921409 + 3.43874i 4.07996 + 1.72998i −0.258819 0.965926i 4.60672 + 4.60672i −2.15641 + 2.08564i −7.88832 4.55433i
176.7 −1.80571 + 1.80571i 0.527557 + 1.64975i 4.52115i −0.366549 1.36798i −3.93158 2.02636i 0.258819 + 0.965926i 4.55245 + 4.55245i −2.44337 + 1.74068i 3.13205 + 1.80829i
176.8 −1.64739 + 1.64739i 1.05743 + 1.37180i 3.42777i −0.520990 1.94436i −4.00189 0.517890i −0.258819 0.965926i 2.35209 + 2.35209i −0.763679 + 2.90117i 4.06139 + 2.34484i
176.9 −1.63956 + 1.63956i 1.73159 0.0399945i 3.37635i −1.02893 3.84001i −2.77348 + 2.90463i 0.258819 + 0.965926i 2.25661 + 2.25661i 2.99680 0.138508i 7.98294 + 4.60895i
176.10 −1.59695 + 1.59695i 0.0991579 + 1.72921i 3.10050i 0.947622 + 3.53657i −2.91981 2.60311i −0.258819 0.965926i 1.75745 + 1.75745i −2.98034 + 0.342930i −7.16104 4.13443i
176.11 −1.55099 + 1.55099i 1.57977 0.710165i 2.81113i 0.719217 + 2.68415i −1.34874 + 3.55166i −0.258819 0.965926i 1.25806 + 1.25806i 1.99133 2.24379i −5.27859 3.04759i
176.12 −1.49802 + 1.49802i −0.188264 1.72179i 2.48814i 0.487433 + 1.81912i 2.86130 + 2.29725i 0.258819 + 0.965926i 0.731245 + 0.731245i −2.92911 + 0.648302i −3.45527 1.99490i
176.13 −1.47984 + 1.47984i −1.33313 1.10579i 2.37986i 0.507117 + 1.89258i 3.60921 0.336422i 0.258819 + 0.965926i 0.562125 + 0.562125i 0.554455 + 2.94832i −3.55117 2.05027i
176.14 −1.45048 + 1.45048i −0.883457 + 1.48980i 2.20781i 0.260529 + 0.972306i −0.879492 3.44237i −0.258819 0.965926i 0.301429 + 0.301429i −1.43901 2.63235i −1.78821 1.03242i
176.15 −1.39334 + 1.39334i −1.02425 1.39675i 1.88279i −0.512387 1.91225i 3.37328 + 0.519029i −0.258819 0.965926i −0.163307 0.163307i −0.901840 + 2.86124i 3.37835 + 1.95049i
176.16 −1.39126 + 1.39126i 1.52938 + 0.813025i 1.87123i −0.451446 1.68482i −3.25890 + 0.996634i −0.258819 0.965926i −0.179147 0.179147i 1.67798 + 2.48684i 2.97211 + 1.71595i
176.17 −1.37577 + 1.37577i 1.69723 + 0.345561i 1.78551i 0.367577 + 1.37181i −2.81042 + 1.85959i 0.258819 + 0.965926i −0.295090 0.295090i 2.76117 + 1.17299i −2.39301 1.38160i
176.18 −1.37300 + 1.37300i −1.58085 + 0.707758i 1.77025i 0.189170 + 0.705993i 1.19875 3.14225i 0.258819 + 0.965926i −0.315445 0.315445i 1.99816 2.23772i −1.22906 0.709597i
176.19 −1.35386 + 1.35386i 1.29330 1.15212i 1.66586i 0.0412922 + 0.154105i −0.191139 + 3.31075i −0.258819 0.965926i −0.452377 0.452377i 0.345247 2.98007i −0.264540 0.152732i
176.20 −1.25977 + 1.25977i −0.923456 + 1.46534i 1.17405i −0.838983 3.13113i −0.682653 3.00934i 0.258819 + 0.965926i −1.04050 1.04050i −1.29446 2.70636i 5.00144 + 2.88758i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 617.84 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.ew.a 336
9.d odd 6 1 819.2.fy.a yes 336
13.f odd 12 1 819.2.fy.a yes 336
117.x even 12 1 inner 819.2.ew.a 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.ew.a 336 1.a even 1 1 trivial
819.2.ew.a 336 117.x even 12 1 inner
819.2.fy.a yes 336 9.d odd 6 1
819.2.fy.a yes 336 13.f odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(819, [\chi])$$.