# Properties

 Label 354.7.d.a Level 354 Weight 7 Character orbit 354.d Analytic conductor 81.439 Analytic rank 0 Dimension 60 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 354.d (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$81.4391456014$$ Analytic rank: $$0$$ Dimension: $$60$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60q - 1920q^{4} + 408q^{7} + 14580q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q - 1920q^{4} + 408q^{7} + 14580q^{9} + 4536q^{15} + 61440q^{16} - 15840q^{17} - 5616q^{19} - 17472q^{22} + 226260q^{25} - 34048q^{26} - 13056q^{28} - 75392q^{29} + 278000q^{35} - 466560q^{36} + 67376q^{41} + 209856q^{46} + 269100q^{49} - 206064q^{51} + 490000q^{53} - 373248q^{57} - 863472q^{59} - 145152q^{60} - 155072q^{62} + 99144q^{63} - 1966080q^{64} - 404352q^{66} + 506880q^{68} - 2041856q^{71} - 2146176q^{74} + 808704q^{75} + 179712q^{76} + 228096q^{78} + 670248q^{79} + 3542940q^{81} + 873408q^{85} + 1832576q^{86} - 2568024q^{87} + 559104q^{88} + 1049472q^{94} - 245856q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1 5.65685i −15.5885 −32.0000 −92.5158 88.1816i 505.471 181.019i 243.000 523.348i
235.2 5.65685i −15.5885 −32.0000 −92.5158 88.1816i 505.471 181.019i 243.000 523.348i
235.3 5.65685i 15.5885 −32.0000 117.083 88.1816i 626.230 181.019i 243.000 662.324i
235.4 5.65685i 15.5885 −32.0000 117.083 88.1816i 626.230 181.019i 243.000 662.324i
235.5 5.65685i 15.5885 −32.0000 −229.516 88.1816i −374.873 181.019i 243.000 1298.34i
235.6 5.65685i 15.5885 −32.0000 −229.516 88.1816i −374.873 181.019i 243.000 1298.34i
235.7 5.65685i −15.5885 −32.0000 133.407 88.1816i −65.3612 181.019i 243.000 754.665i
235.8 5.65685i −15.5885 −32.0000 133.407 88.1816i −65.3612 181.019i 243.000 754.665i
235.9 5.65685i −15.5885 −32.0000 204.428 88.1816i 398.882 181.019i 243.000 1156.42i
235.10 5.65685i −15.5885 −32.0000 204.428 88.1816i 398.882 181.019i 243.000 1156.42i
235.11 5.65685i −15.5885 −32.0000 −30.8626 88.1816i −9.02297 181.019i 243.000 174.585i
235.12 5.65685i −15.5885 −32.0000 −30.8626 88.1816i −9.02297 181.019i 243.000 174.585i
235.13 5.65685i −15.5885 −32.0000 151.165 88.1816i −215.009 181.019i 243.000 855.120i
235.14 5.65685i −15.5885 −32.0000 151.165 88.1816i −215.009 181.019i 243.000 855.120i
235.15 5.65685i −15.5885 −32.0000 65.2792 88.1816i −409.923 181.019i 243.000 369.275i
235.16 5.65685i −15.5885 −32.0000 65.2792 88.1816i −409.923 181.019i 243.000 369.275i
235.17 5.65685i 15.5885 −32.0000 −16.5609 88.1816i 303.476 181.019i 243.000 93.6824i
235.18 5.65685i 15.5885 −32.0000 −16.5609 88.1816i 303.476 181.019i 243.000 93.6824i
235.19 5.65685i −15.5885 −32.0000 −96.2981 88.1816i −484.887 181.019i 243.000 544.744i
235.20 5.65685i −15.5885 −32.0000 −96.2981 88.1816i −484.887 181.019i 243.000 544.744i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 235.60 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

There are no other newforms in $$S_{7}^{\mathrm{new}}(354, [\chi])$$.