# Properties

 Label 354.6.c.b Level 354 Weight 6 Character orbit 354.c Analytic conductor 56.776 Analytic rank 0 Dimension 50 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 354.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$56.7758722138$$ Analytic rank: $$0$$ Dimension: $$50$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$50q + 200q^{2} - 13q^{3} + 800q^{4} - 52q^{6} + 38q^{7} + 3200q^{8} + 51q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$50q + 200q^{2} - 13q^{3} + 800q^{4} - 52q^{6} + 38q^{7} + 3200q^{8} + 51q^{9} + 652q^{11} - 208q^{12} + 152q^{14} - 2107q^{15} + 12800q^{16} + 204q^{18} - 894q^{19} - 3801q^{21} + 2608q^{22} - 2456q^{23} - 832q^{24} - 23956q^{25} + 9890q^{27} + 608q^{28} - 8428q^{30} + 51200q^{32} + 9744q^{33} + 816q^{36} - 3576q^{38} - 1388q^{39} - 15204q^{42} + 10432q^{44} + 33067q^{45} - 9824q^{46} + 27144q^{47} - 3328q^{48} + 85768q^{49} - 95824q^{50} + 3338q^{51} + 39560q^{54} + 2432q^{56} - 63969q^{57} + 23840q^{59} - 33712q^{60} + 94781q^{63} + 204800q^{64} + 9400q^{65} + 38976q^{66} + 115930q^{69} + 3264q^{72} + 24248q^{75} - 14304q^{76} + 150240q^{77} - 5552q^{78} - 68658q^{79} + 47095q^{81} + 175140q^{83} - 60816q^{84} - 138524q^{85} + 138261q^{87} + 41728q^{88} - 104908q^{89} + 132268q^{90} - 39296q^{92} + 91204q^{93} + 108576q^{94} - 13312q^{96} + 343072q^{98} + 111406q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
353.1 4.00000 −15.5824 0.434789i 16.0000 38.7472i −62.3296 1.73916i −41.9790 64.0000 242.622 + 13.5501i 154.989i
353.2 4.00000 −15.5824 + 0.434789i 16.0000 38.7472i −62.3296 + 1.73916i −41.9790 64.0000 242.622 13.5501i 154.989i
353.3 4.00000 −15.5391 1.23947i 16.0000 14.2214i −62.1564 4.95787i 129.028 64.0000 239.927 + 38.5204i 56.8857i
353.4 4.00000 −15.5391 + 1.23947i 16.0000 14.2214i −62.1564 + 4.95787i 129.028 64.0000 239.927 38.5204i 56.8857i
353.5 4.00000 −14.5480 5.59965i 16.0000 83.1853i −58.1920 22.3986i 242.629 64.0000 180.288 + 162.927i 332.741i
353.6 4.00000 −14.5480 + 5.59965i 16.0000 83.1853i −58.1920 + 22.3986i 242.629 64.0000 180.288 162.927i 332.741i
353.7 4.00000 −13.3335 8.07581i 16.0000 83.4718i −53.3339 32.3032i −123.882 64.0000 112.563 + 215.357i 333.887i
353.8 4.00000 −13.3335 + 8.07581i 16.0000 83.4718i −53.3339 + 32.3032i −123.882 64.0000 112.563 215.357i 333.887i
353.9 4.00000 −12.9095 8.73750i 16.0000 27.1818i −51.6381 34.9500i −249.363 64.0000 90.3121 + 225.594i 108.727i
353.10 4.00000 −12.9095 + 8.73750i 16.0000 27.1818i −51.6381 + 34.9500i −249.363 64.0000 90.3121 225.594i 108.727i
353.11 4.00000 −11.6908 10.3115i 16.0000 106.082i −46.7630 41.2458i 71.9340 64.0000 30.3476 + 241.098i 424.326i
353.12 4.00000 −11.6908 + 10.3115i 16.0000 106.082i −46.7630 + 41.2458i 71.9340 64.0000 30.3476 241.098i 424.326i
353.13 4.00000 −11.5231 10.4984i 16.0000 67.9871i −46.0925 41.9938i −61.6810 64.0000 22.5654 + 241.950i 271.948i
353.14 4.00000 −11.5231 + 10.4984i 16.0000 67.9871i −46.0925 + 41.9938i −61.6810 64.0000 22.5654 241.950i 271.948i
353.15 4.00000 −9.75722 12.1572i 16.0000 43.2791i −39.0289 48.6287i −5.13563 64.0000 −52.5934 + 237.240i 173.116i
353.16 4.00000 −9.75722 + 12.1572i 16.0000 43.2791i −39.0289 + 48.6287i −5.13563 64.0000 −52.5934 237.240i 173.116i
353.17 4.00000 −9.68192 12.2172i 16.0000 48.6180i −38.7277 48.8689i 149.149 64.0000 −55.5209 + 236.572i 194.472i
353.18 4.00000 −9.68192 + 12.2172i 16.0000 48.6180i −38.7277 + 48.8689i 149.149 64.0000 −55.5209 236.572i 194.472i
353.19 4.00000 −4.24050 15.0006i 16.0000 2.81364i −16.9620 60.0024i −187.063 64.0000 −207.036 + 127.220i 11.2546i
353.20 4.00000 −4.24050 + 15.0006i 16.0000 2.81364i −16.9620 + 60.0024i −187.063 64.0000 −207.036 127.220i 11.2546i
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.50 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.