# Properties

 Label 930.2.bk.a Level $930$ Weight $2$ Character orbit 930.bk Analytic conductor $7.426$ Analytic rank $0$ Dimension $512$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.bk (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$512$$ Relative dimension: $$64$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$512q + 8q^{3} - 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$512q + 8q^{3} - 12q^{7} - 8q^{10} - 8q^{12} + 8q^{15} + 128q^{16} + 28q^{22} + 32q^{25} + 44q^{27} + 8q^{28} + 24q^{30} + 8q^{31} + 28q^{33} + 176q^{37} + 24q^{42} - 40q^{45} - 24q^{46} + 12q^{48} - 16q^{51} - 40q^{55} + 40q^{57} + 56q^{58} + 12q^{60} - 32q^{61} - 224q^{63} + 32q^{66} + 48q^{67} + 24q^{70} + 40q^{73} - 32q^{75} + 48q^{76} - 76q^{78} - 96q^{81} + 24q^{85} - 80q^{87} - 72q^{88} + 16q^{90} + 80q^{91} - 168q^{93} - 112q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −0.156434 + 0.987688i −1.73203 0.00940555i −0.951057 0.309017i 0.500643 + 2.17930i 0.280238 1.70923i −0.753670 + 0.384014i 0.453990 0.891007i 2.99982 + 0.0325813i −2.23079 + 0.153561i
47.2 −0.156434 + 0.987688i −1.70896 0.281880i −0.951057 0.309017i −2.22941 0.172430i 0.545749 1.64382i −0.413436 + 0.210656i 0.453990 0.891007i 2.84109 + 0.963442i 0.519064 2.17499i
47.3 −0.156434 + 0.987688i −1.69850 + 0.339248i −0.951057 0.309017i −1.78466 1.34721i −0.0693673 1.73066i −0.684621 + 0.348832i 0.453990 0.891007i 2.76982 1.15243i 1.60981 1.55194i
47.4 −0.156434 + 0.987688i −1.67858 + 0.427044i −0.951057 0.309017i 0.419593 + 2.19635i −0.159198 1.72472i 3.17140 1.61591i 0.453990 0.891007i 2.63527 1.43366i −2.23495 + 0.0708430i
47.5 −0.156434 + 0.987688i −1.52046 + 0.829577i −0.951057 0.309017i 2.03771 0.920729i −0.581511 1.63152i −3.90942 + 1.99195i 0.453990 0.891007i 1.62360 2.52268i 0.590625 + 2.15666i
47.6 −0.156434 + 0.987688i −1.48019 0.899467i −0.951057 0.309017i 0.191858 2.22782i 1.11995 1.32126i 3.04103 1.54948i 0.453990 0.891007i 1.38192 + 2.66276i 2.17038 + 0.538004i
47.7 −0.156434 + 0.987688i −1.22292 1.22657i −0.951057 0.309017i 2.20307 0.382712i 1.40277 1.01599i −0.157227 + 0.0801110i 0.453990 0.891007i −0.00893248 + 2.99999i 0.0333634 + 2.23582i
47.8 −0.156434 + 0.987688i −1.20093 1.24811i −0.951057 0.309017i −1.74392 + 1.39955i 1.42061 0.990898i −2.75830 + 1.40542i 0.453990 0.891007i −0.115534 + 2.99777i −1.10951 1.94139i
47.9 −0.156434 + 0.987688i −1.00125 + 1.41333i −0.951057 0.309017i 1.95036 + 1.09365i −1.23929 1.21002i 1.24759 0.635678i 0.453990 0.891007i −0.994977 2.83020i −1.38529 + 1.75527i
47.10 −0.156434 + 0.987688i −0.999352 + 1.41467i −0.951057 0.309017i −2.01289 + 0.973791i −1.24092 1.20835i −2.22585 + 1.13413i 0.453990 0.891007i −1.00259 2.82751i −0.646917 2.14044i
47.11 −0.156434 + 0.987688i −0.893454 1.48383i −0.951057 0.309017i −0.275456 2.21904i 1.60532 0.650333i −4.52946 + 2.30788i 0.453990 0.891007i −1.40348 + 2.65146i 2.23481 + 0.0750690i
47.12 −0.156434 + 0.987688i −0.776954 1.54801i −0.951057 0.309017i 0.00309216 + 2.23607i 1.65050 0.525226i 3.71530 1.89304i 0.453990 0.891007i −1.79268 + 2.40547i −2.20902 0.346744i
47.13 −0.156434 + 0.987688i −0.678564 + 1.59360i −0.951057 0.309017i −1.22925 1.86787i −1.46783 0.919503i −0.247778 + 0.126249i 0.453990 0.891007i −2.07910 2.16271i 2.03717 0.921913i
47.14 −0.156434 + 0.987688i −0.343275 + 1.69769i −0.951057 0.309017i 2.03327 0.930487i −1.62309 0.604626i 3.29626 1.67953i 0.453990 0.891007i −2.76433 1.16555i 0.600958 + 2.15380i
47.15 −0.156434 + 0.987688i −0.212558 + 1.71896i −0.951057 0.309017i −1.82187 + 1.29645i −1.66454 0.478845i 1.93565 0.986265i 0.453990 0.891007i −2.90964 0.730756i −0.995488 2.00225i
47.16 −0.156434 + 0.987688i −0.115485 1.72820i −0.951057 0.309017i −2.08414 0.810158i 1.72499 + 0.156287i 2.37194 1.20856i 0.453990 0.891007i −2.97333 + 0.399160i 1.12621 1.93175i
47.17 −0.156434 + 0.987688i 0.207488 1.71958i −0.951057 0.309017i 1.69493 1.45849i 1.66595 + 0.473934i 0.759514 0.386992i 0.453990 0.891007i −2.91390 0.713583i 1.17539 + 1.90223i
47.18 −0.156434 + 0.987688i 0.219461 + 1.71809i −0.951057 0.309017i 0.873512 + 2.05839i −1.73127 0.0520095i −2.14611 + 1.09350i 0.453990 0.891007i −2.90367 + 0.754108i −2.16970 + 0.540754i
47.19 −0.156434 + 0.987688i 0.345889 1.69716i −0.951057 0.309017i −0.410025 + 2.19815i 1.62216 + 0.607126i −0.710855 + 0.362199i 0.453990 0.891007i −2.76072 1.17406i −2.10695 0.748844i
47.20 −0.156434 + 0.987688i 0.690891 1.58829i −0.951057 0.309017i 1.41201 + 1.73384i 1.46066 + 0.930848i −2.17193 + 1.10665i 0.453990 0.891007i −2.04534 2.19467i −1.93339 + 1.12340i
See next 80 embeddings (of 512 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 653.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
31.d even 5 1 inner
93.l odd 10 1 inner
155.s odd 20 1 inner
465.bj even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bk.a 512
3.b odd 2 1 inner 930.2.bk.a 512
5.c odd 4 1 inner 930.2.bk.a 512
15.e even 4 1 inner 930.2.bk.a 512
31.d even 5 1 inner 930.2.bk.a 512
93.l odd 10 1 inner 930.2.bk.a 512
155.s odd 20 1 inner 930.2.bk.a 512
465.bj even 20 1 inner 930.2.bk.a 512

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bk.a 512 1.a even 1 1 trivial
930.2.bk.a 512 3.b odd 2 1 inner
930.2.bk.a 512 5.c odd 4 1 inner
930.2.bk.a 512 15.e even 4 1 inner
930.2.bk.a 512 31.d even 5 1 inner
930.2.bk.a 512 93.l odd 10 1 inner
930.2.bk.a 512 155.s odd 20 1 inner
930.2.bk.a 512 465.bj even 20 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(930, [\chi])$$.