# Properties

 Label 38.3.f.a Level $38$ Weight $3$ Character orbit 38.f Analytic conductor $1.035$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 38.f (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.03542500457$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{3} + 12q^{6} - 18q^{7} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{3} + 12q^{6} - 18q^{7} + 6q^{9} + 30q^{11} - 36q^{12} - 90q^{13} - 48q^{14} - 114q^{15} + 18q^{17} - 12q^{19} + 24q^{20} + 90q^{21} + 84q^{22} + 120q^{23} - 24q^{24} + 252q^{25} + 48q^{26} + 126q^{27} + 72q^{28} - 210q^{29} - 108q^{31} - 132q^{33} - 24q^{34} - 66q^{35} - 12q^{36} + 84q^{38} + 120q^{39} + 54q^{41} + 72q^{42} + 90q^{43} - 48q^{44} - 144q^{45} - 360q^{46} - 246q^{47} - 48q^{48} + 54q^{49} - 432q^{50} - 342q^{51} + 36q^{52} - 174q^{53} - 42q^{55} - 12q^{57} + 48q^{58} + 228q^{59} + 132q^{60} + 12q^{61} + 204q^{62} + 174q^{63} + 96q^{64} + 630q^{65} + 696q^{66} + 72q^{67} - 48q^{68} + 702q^{69} + 528q^{70} + 432q^{71} + 96q^{72} - 144q^{74} + 72q^{76} - 144q^{77} - 708q^{78} - 246q^{79} - 642q^{81} - 384q^{82} - 126q^{83} - 540q^{84} - 684q^{85} - 12q^{86} - 324q^{87} - 12q^{89} - 336q^{90} + 372q^{91} - 132q^{92} - 168q^{93} - 570q^{95} + 72q^{97} + 384q^{98} - 204q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −0.909039 + 1.08335i −1.73025 4.75383i −0.347296 1.96962i 0.678914 3.85031i 6.72293 + 2.44695i 1.77574 + 3.07567i 2.44949 + 1.41421i −12.7107 + 10.6656i 3.55408 + 4.23559i
3.2 −0.909039 + 1.08335i 0.836494 + 2.29825i −0.347296 1.96962i −0.852562 + 4.83512i −3.25021 1.18298i 0.583107 + 1.00997i 2.44949 + 1.41421i 2.31218 1.94015i −4.46312 5.31894i
3.3 0.909039 1.08335i −0.830875 2.28281i −0.347296 1.96962i 0.0233547 0.132451i −3.22838 1.17503i 4.79341 + 8.30244i −2.44949 1.41421i 2.37354 1.99163i −0.122261 0.145705i
3.4 0.909039 1.08335i 1.41923 + 3.89929i −0.347296 1.96962i −0.197003 + 1.11726i 5.51443 + 2.00709i −5.55599 9.62326i −2.44949 1.41421i −6.29588 + 5.28287i 1.03130 + 1.22906i
13.1 −0.909039 1.08335i −1.73025 + 4.75383i −0.347296 + 1.96962i 0.678914 + 3.85031i 6.72293 2.44695i 1.77574 3.07567i 2.44949 1.41421i −12.7107 10.6656i 3.55408 4.23559i
13.2 −0.909039 1.08335i 0.836494 2.29825i −0.347296 + 1.96962i −0.852562 4.83512i −3.25021 + 1.18298i 0.583107 1.00997i 2.44949 1.41421i 2.31218 + 1.94015i −4.46312 + 5.31894i
13.3 0.909039 + 1.08335i −0.830875 + 2.28281i −0.347296 + 1.96962i 0.0233547 + 0.132451i −3.22838 + 1.17503i 4.79341 8.30244i −2.44949 + 1.41421i 2.37354 + 1.99163i −0.122261 + 0.145705i
13.4 0.909039 + 1.08335i 1.41923 3.89929i −0.347296 + 1.96962i −0.197003 1.11726i 5.51443 2.00709i −5.55599 + 9.62326i −2.44949 + 1.41421i −6.29588 5.28287i 1.03130 1.22906i
15.1 −0.483690 + 1.32893i −5.23364 + 0.922832i −1.53209 1.28558i −1.06384 + 0.892670i 1.30508 7.40149i −4.39759 + 7.61684i 2.44949 1.41421i 18.0821 6.58136i −0.671723 1.84554i
15.2 −0.483690 + 1.32893i 3.64826 0.643286i −1.53209 1.28558i 0.297798 0.249882i −0.909744 + 5.15941i −2.96431 + 5.13434i 2.44949 1.41421i 4.43872 1.61556i 0.188033 + 0.516617i
15.3 0.483690 1.32893i −0.173101 + 0.0305223i −1.53209 1.28558i 6.04513 5.07246i −0.0431650 + 0.244801i −3.82954 + 6.63297i −2.44949 + 1.41421i −8.42820 + 3.06761i −3.81697 10.4870i
15.4 0.483690 1.32893i 3.82266 0.674039i −1.53209 1.28558i −6.81117 + 5.71525i 0.953235 5.40606i 2.55329 4.42243i −2.44949 + 1.41421i 5.70119 2.07506i 4.30065 + 11.8160i
21.1 −1.39273 0.245576i −2.33610 + 2.78405i 1.87939 + 0.684040i −7.04907 + 2.56565i 3.93724 3.30374i 3.79852 + 6.57923i −2.44949 1.41421i −0.730761 4.14435i 10.4475 1.84218i
21.2 −1.39273 0.245576i 0.272417 0.324654i 1.87939 + 0.684040i 7.98876 2.90767i −0.459130 + 0.385256i −2.58545 4.47813i −2.44949 1.41421i 1.53164 + 8.68639i −11.8402 + 2.08775i
21.3 1.39273 + 0.245576i −3.75536 + 4.47547i 1.87939 + 0.684040i 5.30025 1.92913i −6.32926 + 5.31088i −0.990297 1.71524i 2.44949 + 1.41421i −4.36422 24.7507i 7.85555 1.38515i
21.4 1.39273 + 0.245576i 1.06027 1.26358i 1.87939 + 0.684040i −4.36056 + 1.58711i 1.78697 1.49945i −2.18088 3.77740i 2.44949 + 1.41421i 1.09037 + 6.18380i −6.46283 + 1.13957i
29.1 −1.39273 + 0.245576i −2.33610 2.78405i 1.87939 0.684040i −7.04907 2.56565i 3.93724 + 3.30374i 3.79852 6.57923i −2.44949 + 1.41421i −0.730761 + 4.14435i 10.4475 + 1.84218i
29.2 −1.39273 + 0.245576i 0.272417 + 0.324654i 1.87939 0.684040i 7.98876 + 2.90767i −0.459130 0.385256i −2.58545 + 4.47813i −2.44949 + 1.41421i 1.53164 8.68639i −11.8402 2.08775i
29.3 1.39273 0.245576i −3.75536 4.47547i 1.87939 0.684040i 5.30025 + 1.92913i −6.32926 5.31088i −0.990297 + 1.71524i 2.44949 1.41421i −4.36422 + 24.7507i 7.85555 + 1.38515i
29.4 1.39273 0.245576i 1.06027 + 1.26358i 1.87939 0.684040i −4.36056 1.58711i 1.78697 + 1.49945i −2.18088 + 3.77740i 2.44949 1.41421i 1.09037 6.18380i −6.46283 1.13957i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 33.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.3.f.a 24
3.b odd 2 1 342.3.z.b 24
4.b odd 2 1 304.3.z.c 24
19.e even 9 1 722.3.b.f 24
19.f odd 18 1 inner 38.3.f.a 24
19.f odd 18 1 722.3.b.f 24
57.j even 18 1 342.3.z.b 24
76.k even 18 1 304.3.z.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.f.a 24 1.a even 1 1 trivial
38.3.f.a 24 19.f odd 18 1 inner
304.3.z.c 24 4.b odd 2 1
304.3.z.c 24 76.k even 18 1
342.3.z.b 24 3.b odd 2 1
342.3.z.b 24 57.j even 18 1
722.3.b.f 24 19.e even 9 1
722.3.b.f 24 19.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(38, [\chi])$$.