# Properties

 Label 432.2.be.b Level $432$ Weight $2$ Character orbit 432.be Analytic conductor $3.450$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.be (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 3 q^{5} + 6 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36 q + 3 q^{5} + 6 q^{9} - 18 q^{11} + 9 q^{15} + 18 q^{21} - 9 q^{25} + 30 q^{29} + 27 q^{31} + 27 q^{33} + 27 q^{35} - 45 q^{39} + 18 q^{41} + 27 q^{45} + 45 q^{47} - 63 q^{51} - 9 q^{57} + 54 q^{59} - 63 q^{63} - 57 q^{65} - 63 q^{69} + 36 q^{71} + 9 q^{73} - 45 q^{75} - 81 q^{77} - 54 q^{81} - 27 q^{83} - 36 q^{85} + 45 q^{87} - 63 q^{89} + 27 q^{91} - 63 q^{93} - 72 q^{95} + 99 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −1.65284 0.517813i 0 −2.52917 + 0.445961i 0 −1.40789 + 1.67786i 0 2.46374 + 1.71172i 0
47.2 0 −1.59623 0.672336i 0 4.18690 0.738263i 0 1.26944 1.51285i 0 2.09593 + 2.14641i 0
47.3 0 −0.939092 + 1.45537i 0 −1.07086 + 0.188822i 0 −0.0466102 + 0.0555478i 0 −1.23621 2.73346i 0
47.4 0 0.721706 + 1.57453i 0 0.976679 0.172215i 0 3.12803 3.72784i 0 −1.95828 + 2.27269i 0
47.5 0 1.36628 1.06456i 0 1.83017 0.322709i 0 −0.441545 + 0.526213i 0 0.733433 2.90896i 0
47.6 0 1.50778 + 0.852402i 0 −2.12767 + 0.375166i 0 −2.50142 + 2.98107i 0 1.54682 + 2.57047i 0
95.1 0 −1.22376 + 1.22573i 0 1.15344 1.37462i 0 −0.0374696 + 0.102947i 0 −0.00480931 3.00000i 0
95.2 0 −1.19574 1.25308i 0 0.311559 0.371302i 0 −0.958275 + 2.63284i 0 −0.140409 + 2.99671i 0
95.3 0 0.249148 + 1.71404i 0 −1.47612 + 1.75917i 0 0.590654 1.62281i 0 −2.87585 + 0.854099i 0
95.4 0 0.536320 1.64692i 0 −1.32159 + 1.57501i 0 1.34711 3.70114i 0 −2.42472 1.76656i 0
95.5 0 1.63553 + 0.570126i 0 2.34697 2.79701i 0 0.550750 1.51317i 0 2.34991 + 1.86492i 0
95.6 0 1.70424 0.309120i 0 −1.45395 + 1.73275i 0 −1.49276 + 4.10134i 0 2.80889 1.05363i 0
191.1 0 −1.22376 1.22573i 0 1.15344 + 1.37462i 0 −0.0374696 0.102947i 0 −0.00480931 + 3.00000i 0
191.2 0 −1.19574 + 1.25308i 0 0.311559 + 0.371302i 0 −0.958275 2.63284i 0 −0.140409 2.99671i 0
191.3 0 0.249148 1.71404i 0 −1.47612 1.75917i 0 0.590654 + 1.62281i 0 −2.87585 0.854099i 0
191.4 0 0.536320 + 1.64692i 0 −1.32159 1.57501i 0 1.34711 + 3.70114i 0 −2.42472 + 1.76656i 0
191.5 0 1.63553 0.570126i 0 2.34697 + 2.79701i 0 0.550750 + 1.51317i 0 2.34991 1.86492i 0
191.6 0 1.70424 + 0.309120i 0 −1.45395 1.73275i 0 −1.49276 4.10134i 0 2.80889 + 1.05363i 0
239.1 0 −1.65284 + 0.517813i 0 −2.52917 0.445961i 0 −1.40789 1.67786i 0 2.46374 1.71172i 0
239.2 0 −1.59623 + 0.672336i 0 4.18690 + 0.738263i 0 1.26944 + 1.51285i 0 2.09593 2.14641i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 383.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 432.2.be.b 36
4.b odd 2 1 432.2.be.c yes 36
27.f odd 18 1 432.2.be.c yes 36
108.l even 18 1 inner 432.2.be.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.be.b 36 1.a even 1 1 trivial
432.2.be.b 36 108.l even 18 1 inner
432.2.be.c yes 36 4.b odd 2 1
432.2.be.c yes 36 27.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{36} - \cdots$$ $$13\!\cdots\!61$$$$T_{7}^{7} - 264127771323 T_{7}^{6} + 785115858294 T_{7}^{5} + 674560187064 T_{7}^{4} + 110454697440 T_{7}^{3} + 14310643248 T_{7}^{2} + 867311712 T_{7} + 34012224$$">$$T_{7}^{36} + \cdots$$