Properties

 Label 324.2.l.a Level $324$ Weight $2$ Character orbit 324.l Analytic conductor $2.587$ Analytic rank $0$ Dimension $96$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 324.l (of order $$18$$, degree $$6$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$16$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 108) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 6 q^{2} - 6 q^{4} + 12 q^{5} + 9 q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96 q + 6 q^{2} - 6 q^{4} + 12 q^{5} + 9 q^{8} - 3 q^{10} - 12 q^{13} + 21 q^{14} - 6 q^{16} + 18 q^{17} + 27 q^{20} - 6 q^{22} - 12 q^{25} - 12 q^{28} + 24 q^{29} - 24 q^{32} - 12 q^{34} - 6 q^{37} - 18 q^{38} - 21 q^{40} + 42 q^{41} - 63 q^{44} - 3 q^{46} - 12 q^{49} - 87 q^{50} - 33 q^{52} - 99 q^{56} - 33 q^{58} - 12 q^{61} - 90 q^{62} - 3 q^{64} - 12 q^{65} - 51 q^{68} - 21 q^{70} - 6 q^{73} - 21 q^{74} - 18 q^{76} - 12 q^{77} - 12 q^{82} - 42 q^{85} + 30 q^{86} + 18 q^{88} + 123 q^{92} + 21 q^{94} - 30 q^{97} + 180 q^{98} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.40284 0.179019i 0 1.93590 + 0.502269i 0.847966 2.32977i 0 −4.59553 0.810316i −2.62584 1.05116i 0 −1.60663 + 3.11648i
35.2 −1.37675 + 0.323366i 0 1.79087 0.890387i −0.470103 + 1.29160i 0 1.57428 + 0.277589i −2.17765 + 1.80495i 0 0.229554 1.93022i
35.3 −1.18497 + 0.771906i 0 0.808323 1.82938i 0.197421 0.542409i 0 −1.70706 0.301001i 0.454264 + 2.79171i 0 0.184750 + 0.795131i
35.4 −1.09968 0.889210i 0 0.418610 + 1.95570i −0.605021 + 1.66228i 0 −0.748045 0.131901i 1.27869 2.52289i 0 2.14345 1.28999i
35.5 −0.608544 1.27659i 0 −1.25935 + 1.55372i −0.420820 + 1.15619i 0 −1.81474 0.319988i 2.74983 + 0.662160i 0 1.73207 0.166382i
35.6 −0.557089 1.29987i 0 −1.37930 + 1.44828i 1.27150 3.49343i 0 1.63150 + 0.287677i 2.65097 + 0.986086i 0 −5.24933 + 0.293367i
35.7 −0.554410 + 1.30101i 0 −1.38526 1.44259i 0.197421 0.542409i 0 1.70706 + 0.301001i 2.64482 1.00245i 0 0.596228 + 0.557564i
35.8 −0.0793838 + 1.41198i 0 −1.98740 0.224177i −0.470103 + 1.29160i 0 −1.57428 0.277589i 0.474302 2.78838i 0 −1.78640 0.766310i
35.9 0.376137 1.36328i 0 −1.71704 1.02556i −1.29726 + 3.56419i 0 −2.58045 0.455003i −2.04396 + 1.95505i 0 4.37103 + 3.10915i
35.10 0.419899 + 1.35044i 0 −1.64737 + 1.13410i 0.847966 2.32977i 0 4.59553 + 0.810316i −2.22326 1.74846i 0 3.50227 + 0.166858i
35.11 0.805437 1.16244i 0 −0.702543 1.87255i 0.710267 1.95144i 0 3.83975 + 0.677051i −2.74258 0.691554i 0 −1.69636 2.39741i
35.12 1.00492 0.995056i 0 0.0197253 1.99990i 0.710267 1.95144i 0 −3.83975 0.677051i −1.97019 2.02937i 0 −1.22804 2.66780i
35.13 1.06666 + 0.928568i 0 0.275524 + 1.98093i −0.605021 + 1.66228i 0 0.748045 + 0.131901i −1.54554 + 2.36882i 0 −2.18889 + 1.21129i
35.14 1.27725 0.607153i 0 1.26273 1.55097i −1.29726 + 3.56419i 0 2.58045 + 0.455003i 0.671144 2.74765i 0 0.507086 + 5.33999i
35.15 1.36287 + 0.377622i 0 1.71480 + 1.02930i −0.420820 + 1.15619i 0 1.81474 + 0.319988i 1.94836 + 2.05034i 0 −1.01013 + 1.41682i
35.16 1.37686 + 0.322907i 0 1.79146 + 0.889191i 1.27150 3.49343i 0 −1.63150 0.287677i 2.17946 + 1.80276i 0 2.87873 4.39937i
71.1 −1.39986 0.201007i 0 1.91919 + 0.562762i 2.12601 2.53368i 0 1.10219 3.02825i −2.57347 1.17356i 0 −3.48540 + 3.11945i
71.2 −1.24669 + 0.667664i 0 1.10845 1.66473i 2.12601 2.53368i 0 −1.10219 + 3.02825i −0.270407 + 2.81547i 0 −0.958821 + 4.57817i
71.3 −1.24471 0.671335i 0 1.09862 + 1.67124i −0.546554 + 0.651358i 0 0.348909 0.958618i −0.245503 2.81775i 0 1.11758 0.443832i
71.4 −0.940037 + 1.05657i 0 −0.232661 1.98642i −0.546554 + 0.651358i 0 −0.348909 + 0.958618i 2.31749 + 1.62149i 0 −0.174421 1.18977i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 287.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.l.a 96
3.b odd 2 1 108.2.l.a 96
4.b odd 2 1 inner 324.2.l.a 96
9.c even 3 1 972.2.l.a 96
9.c even 3 1 972.2.l.b 96
9.d odd 6 1 972.2.l.c 96
9.d odd 6 1 972.2.l.d 96
12.b even 2 1 108.2.l.a 96
27.e even 9 1 108.2.l.a 96
27.e even 9 1 972.2.l.c 96
27.e even 9 1 972.2.l.d 96
27.f odd 18 1 inner 324.2.l.a 96
27.f odd 18 1 972.2.l.a 96
27.f odd 18 1 972.2.l.b 96
36.f odd 6 1 972.2.l.a 96
36.f odd 6 1 972.2.l.b 96
36.h even 6 1 972.2.l.c 96
36.h even 6 1 972.2.l.d 96
108.j odd 18 1 108.2.l.a 96
108.j odd 18 1 972.2.l.c 96
108.j odd 18 1 972.2.l.d 96
108.l even 18 1 inner 324.2.l.a 96
108.l even 18 1 972.2.l.a 96
108.l even 18 1 972.2.l.b 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.l.a 96 3.b odd 2 1
108.2.l.a 96 12.b even 2 1
108.2.l.a 96 27.e even 9 1
108.2.l.a 96 108.j odd 18 1
324.2.l.a 96 1.a even 1 1 trivial
324.2.l.a 96 4.b odd 2 1 inner
324.2.l.a 96 27.f odd 18 1 inner
324.2.l.a 96 108.l even 18 1 inner
972.2.l.a 96 9.c even 3 1
972.2.l.a 96 27.f odd 18 1
972.2.l.a 96 36.f odd 6 1
972.2.l.a 96 108.l even 18 1
972.2.l.b 96 9.c even 3 1
972.2.l.b 96 27.f odd 18 1
972.2.l.b 96 36.f odd 6 1
972.2.l.b 96 108.l even 18 1
972.2.l.c 96 9.d odd 6 1
972.2.l.c 96 27.e even 9 1
972.2.l.c 96 36.h even 6 1
972.2.l.c 96 108.j odd 18 1
972.2.l.d 96 9.d odd 6 1
972.2.l.d 96 27.e even 9 1
972.2.l.d 96 36.h even 6 1
972.2.l.d 96 108.j odd 18 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(324, [\chi])$$.