# Properties

 Label 230.4.j.a Level $230$ Weight $4$ Character orbit 230.j Analytic conductor $13.570$ Analytic rank $0$ Dimension $360$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 230.j (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.5704393013$$ Analytic rank: $$0$$ Dimension: $$360$$ Relative dimension: $$36$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$360q + 144q^{4} + 32q^{5} + 8q^{6} + 224q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$360q + 144q^{4} + 32q^{5} + 8q^{6} + 224q^{9} + 96q^{11} + 208q^{14} - 752q^{15} - 576q^{16} + 48q^{19} + 224q^{20} - 272q^{21} - 32q^{24} + 904q^{25} - 152q^{26} - 20q^{29} - 1272q^{30} + 160q^{31} + 128q^{34} - 364q^{35} - 896q^{36} - 1476q^{39} + 216q^{41} - 384q^{44} - 820q^{45} + 416q^{46} + 3814q^{49} + 1000q^{50} + 296q^{51} - 2772q^{54} - 4272q^{55} + 1104q^{56} - 6250q^{59} - 72q^{60} + 96q^{61} + 2304q^{64} + 8708q^{65} - 6144q^{66} + 14176q^{69} + 6136q^{70} - 11144q^{71} + 4944q^{74} + 6722q^{75} - 192q^{76} - 2680q^{79} - 896q^{80} + 4820q^{81} - 2696q^{84} - 7290q^{85} + 2900q^{86} + 880q^{89} - 1104q^{90} + 5272q^{91} + 864q^{94} + 7456q^{95} + 128q^{96} + 23448q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −0.563465 1.91899i −7.34353 + 6.36321i −3.36501 + 2.16256i −7.26047 8.50209i 16.3487 + 10.5067i −18.7911 8.58160i 6.04600 + 5.23889i 9.59456 66.7316i −12.2244 + 18.7234i
9.2 −0.563465 1.91899i −6.85739 + 5.94196i −3.36501 + 2.16256i 11.1057 1.29006i 15.2664 + 9.81115i 9.56919 + 4.37010i 6.04600 + 5.23889i 7.87438 54.7675i −8.73326 20.5847i
9.3 −0.563465 1.91899i −6.12185 + 5.30461i −3.36501 + 2.16256i −10.8097 + 2.85485i 13.6289 + 8.75878i 32.7383 + 14.9511i 6.04600 + 5.23889i 5.49562 38.2229i 11.5693 + 19.1351i
9.4 −0.563465 1.91899i −4.91203 + 4.25629i −3.36501 + 2.16256i 6.66177 + 8.97891i 10.9355 + 7.02783i −6.13831 2.80327i 6.04600 + 5.23889i 2.16945 15.0889i 13.4767 17.8432i
9.5 −0.563465 1.91899i −4.12395 + 3.57342i −3.36501 + 2.16256i 9.20079 6.35181i 9.18104 + 5.90030i −12.3150 5.62406i 6.04600 + 5.23889i 0.395105 2.74801i −17.3734 14.0772i
9.6 −0.563465 1.91899i −3.19029 + 2.76440i −3.36501 + 2.16256i −9.29353 + 6.21533i 7.10247 + 4.56448i −18.3244 8.36846i 6.04600 + 5.23889i −1.30647 + 9.08669i 17.1637 + 14.3320i
9.7 −0.563465 1.91899i −3.14614 + 2.72615i −3.36501 + 2.16256i −3.36640 10.6615i 7.00418 + 4.50131i 17.7210 + 8.09292i 6.04600 + 5.23889i −1.37617 + 9.57148i −18.5624 + 12.4675i
9.8 −0.563465 1.91899i −2.50934 + 2.17435i −3.36501 + 2.16256i 0.326731 + 11.1756i 5.58647 + 3.59021i 12.0046 + 5.48233i 6.04600 + 5.23889i −2.27354 + 15.8128i 21.2617 6.92403i
9.9 −0.563465 1.91899i 0.694762 0.602015i −3.36501 + 2.16256i 4.50572 10.2322i −1.54673 0.994024i −24.3344 11.1132i 6.04600 + 5.23889i −3.72223 + 25.8887i −22.1743 2.88090i
9.10 −0.563465 1.91899i 0.729490 0.632107i −3.36501 + 2.16256i −6.36706 9.19024i −1.62405 1.04371i 6.54228 + 2.98776i 6.04600 + 5.23889i −3.70990 + 25.8029i −14.0483 + 17.3967i
9.11 −0.563465 1.91899i 1.26527 1.09636i −3.36501 + 2.16256i 9.80304 + 5.37592i −2.81684 1.81027i 20.0538 + 9.15826i 6.04600 + 5.23889i −3.44360 + 23.9508i 4.79264 21.8410i
9.12 −0.563465 1.91899i 2.02304 1.75298i −3.36501 + 2.16256i −11.1730 0.405517i −4.50385 2.89445i 8.71258 + 3.97890i 6.04600 + 5.23889i −2.82272 + 19.6325i 5.51741 + 21.6693i
9.13 −0.563465 1.91899i 2.50028 2.16650i −3.36501 + 2.16256i 8.44207 + 7.33017i −5.56631 3.57725i −13.3837 6.11215i 6.04600 + 5.23889i −2.28485 + 15.8915i 9.30968 20.3305i
9.14 −0.563465 1.91899i 4.45429 3.85967i −3.36501 + 2.16256i 7.13469 8.60792i −9.91648 6.37294i 21.1421 + 9.65527i 6.04600 + 5.23889i 1.10120 7.65898i −20.5386 8.84110i
9.15 −0.563465 1.91899i 4.68867 4.06275i −3.36501 + 2.16256i −10.9539 + 2.23870i −10.4383 6.70826i 0.432083 + 0.197326i 6.04600 + 5.23889i 1.63513 11.3726i 10.4682 + 19.7590i
9.16 −0.563465 1.91899i 6.02364 5.21951i −3.36501 + 2.16256i 10.3489 4.23080i −13.4103 8.61826i −6.46040 2.95037i 6.04600 + 5.23889i 5.19840 36.1557i −13.9501 17.4755i
9.17 −0.563465 1.91899i 6.71164 5.81567i −3.36501 + 2.16256i −2.68913 + 10.8521i −14.9420 9.60261i 25.7501 + 11.7597i 6.04600 + 5.23889i 7.38159 51.3401i 22.3403 0.954398i
9.18 −0.563465 1.91899i 7.66317 6.64017i −3.36501 + 2.16256i −7.45417 8.33278i −17.0603 10.9640i −22.8813 10.4496i 6.04600 + 5.23889i 10.7897 75.0443i −11.7903 + 18.9997i
9.19 0.563465 + 1.91899i −7.66317 + 6.64017i −3.36501 + 2.16256i 4.80461 10.0953i −17.0603 10.9640i 22.8813 + 10.4496i −6.04600 5.23889i 10.7897 75.0443i 22.0800 + 3.53161i
9.20 0.563465 + 1.91899i −6.71164 + 5.81567i −3.36501 + 2.16256i 5.63759 + 9.65492i −14.9420 9.60261i −25.7501 11.7597i −6.04600 5.23889i 7.38159 51.3401i −15.3511 + 16.2587i
See next 80 embeddings (of 360 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 219.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.j.a 360
5.b even 2 1 inner 230.4.j.a 360
23.c even 11 1 inner 230.4.j.a 360
115.j even 22 1 inner 230.4.j.a 360

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.j.a 360 1.a even 1 1 trivial
230.4.j.a 360 5.b even 2 1 inner
230.4.j.a 360 23.c even 11 1 inner
230.4.j.a 360 115.j even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(230, [\chi])$$.