# Properties

 Label 804.2.j.a Level 804 Weight 2 Character orbit 804.j Analytic conductor 6.420 Analytic rank 0 Dimension 68 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.j (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$34$$ over $$\Q(\zeta_{6})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$68q - 68q^{3} - 2q^{4} + 4q^{7} - 6q^{8} + 68q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q - 68q^{3} - 2q^{4} + 4q^{7} - 6q^{8} + 68q^{9} + 18q^{10} + 2q^{12} + 6q^{13} + 10q^{14} - 2q^{16} - 36q^{20} - 4q^{21} - 22q^{22} + 6q^{24} - 68q^{25} - q^{26} - 68q^{27} + q^{28} - 8q^{29} - 18q^{30} + 2q^{31} + 15q^{32} - 2q^{36} + 12q^{37} - 22q^{38} - 6q^{39} + 18q^{40} - 10q^{42} - 4q^{43} - 31q^{44} + 32q^{46} + 2q^{48} - 46q^{49} - 9q^{50} - 28q^{52} - 11q^{56} + 4q^{58} + 36q^{60} + 6q^{61} - 34q^{62} + 4q^{63} + 16q^{64} + 22q^{66} - 18q^{67} + 34q^{68} + 56q^{70} - 36q^{71} - 6q^{72} + 6q^{73} - 53q^{74} + 68q^{75} + 14q^{76} - 4q^{77} + q^{78} + 6q^{79} + 55q^{80} + 68q^{81} - 26q^{82} + 12q^{83} - q^{84} - 21q^{86} + 8q^{87} - 50q^{88} + 18q^{90} + 10q^{92} - 2q^{93} - 16q^{94} + 20q^{95} - 15q^{96} + 18q^{97} - 70q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1 −1.39800 + 0.213516i −1.00000 1.90882 0.596993i 1.98108i 1.39800 0.213516i −0.674999 + 1.16913i −2.54107 + 1.24216i 1.00000 0.422992 + 2.76955i
499.2 −1.39611 + 0.225574i −1.00000 1.89823 0.629851i 3.70048i 1.39611 0.225574i 0.353219 0.611794i −2.50806 + 1.30753i 1.00000 −0.834732 5.16627i
499.3 −1.38203 0.299971i −1.00000 1.82004 + 0.829139i 2.35066i 1.38203 + 0.299971i 1.38486 2.39864i −2.26663 1.69186i 1.00000 −0.705129 + 3.24869i
499.4 −1.35015 0.420832i −1.00000 1.64580 + 1.13637i 2.13147i 1.35015 + 0.420832i −1.61812 + 2.80266i −1.74385 2.22688i 1.00000 0.896992 2.87780i
499.5 −1.32201 + 0.502280i −1.00000 1.49543 1.32804i 0.0336703i 1.32201 0.502280i 0.290483 0.503131i −1.30993 + 2.50681i 1.00000 0.0169119 + 0.0445126i
499.6 −1.24739 0.666348i −1.00000 1.11196 + 1.66239i 1.77165i 1.24739 + 0.666348i 0.297950 0.516065i −0.279318 2.81460i 1.00000 1.18054 2.20994i
499.7 −1.14695 + 0.827342i −1.00000 0.631011 1.89785i 0.235101i 1.14695 0.827342i 1.44937 2.51039i 0.846427 + 2.69881i 1.00000 −0.194509 0.269650i
499.8 −1.09483 + 0.895180i −1.00000 0.397307 1.96014i 3.14328i 1.09483 0.895180i −2.02612 + 3.50935i 1.31969 + 2.50168i 1.00000 2.81380 + 3.44136i
499.9 −0.925825 1.06904i −1.00000 −0.285696 + 1.97949i 2.40613i 0.925825 + 1.06904i −0.814738 + 1.41117i 2.38066 1.52724i 1.00000 −2.57226 + 2.22766i
499.10 −0.890131 1.09894i −1.00000 −0.415333 + 1.95640i 0.854908i 0.890131 + 1.09894i 2.31397 4.00791i 2.51966 1.28503i 1.00000 −0.939492 + 0.760981i
499.11 −0.727656 + 1.21265i −1.00000 −0.941033 1.76478i 4.20985i 0.727656 1.21265i 2.11068 3.65581i 2.82481 + 0.143013i 1.00000 5.10506 + 3.06332i
499.12 −0.717914 + 1.21844i −1.00000 −0.969198 1.74947i 0.567637i 0.717914 1.21844i −1.03281 + 1.78888i 2.82743 + 0.0750601i 1.00000 −0.691633 0.407515i
499.13 −0.699111 1.22933i −1.00000 −1.02249 + 1.71887i 3.18642i 0.699111 + 1.22933i −2.40111 + 4.15884i 2.82789 + 0.0552894i 1.00000 3.91715 2.22766i
499.14 −0.419658 1.35051i −1.00000 −1.64778 + 1.13351i 3.95644i 0.419658 + 1.35051i 1.34216 2.32469i 2.22232 + 1.74966i 1.00000 5.34323 1.66035i
499.15 −0.411062 + 1.35315i −1.00000 −1.66206 1.11246i 0.264773i 0.411062 1.35315i −0.135503 + 0.234697i 2.18854 1.79173i 1.00000 −0.358279 0.108838i
499.16 −0.125591 1.40863i −1.00000 −1.96845 + 0.353823i 1.31012i 0.125591 + 1.40863i −1.48755 + 2.57651i 0.745625 + 2.72838i 1.00000 −1.84547 + 0.164540i
499.17 0.00630957 1.41420i −1.00000 −1.99992 0.0178460i 0.235538i −0.00630957 + 1.41420i 0.386363 0.669200i −0.0378564 + 2.82817i 1.00000 0.333097 + 0.00148614i
499.18 0.278040 + 1.38661i −1.00000 −1.84539 + 0.771067i 3.05041i −0.278040 1.38661i −1.90650 + 3.30215i −1.58226 2.34445i 1.00000 −4.22974 + 0.848137i
499.19 0.284787 + 1.38524i −1.00000 −1.83779 + 0.788997i 2.76718i −0.284787 1.38524i 2.08271 3.60736i −1.61633 2.32109i 1.00000 −3.83321 + 0.788055i
499.20 0.325475 + 1.37625i −1.00000 −1.78813 + 0.895869i 4.16950i −0.325475 1.37625i −0.853071 + 1.47756i −1.81493 2.16934i 1.00000 5.73827 1.35706i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 775.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{7}^{68} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.