# Properties

 Label 156.3.e.c Level $156$ Weight $3$ Character orbit 156.e Analytic conductor $4.251$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 156.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.25069212402$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{4} - 72 q^{9}+O(q^{10})$$ 24 * q + 8 * q^4 - 72 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{4} - 72 q^{9} + 28 q^{10} + 36 q^{12} + 48 q^{13} - 40 q^{14} + 100 q^{16} + 32 q^{17} + 84 q^{22} - 312 q^{25} - 16 q^{26} - 80 q^{29} + 60 q^{30} - 24 q^{36} + 120 q^{38} - 204 q^{40} - 96 q^{42} - 144 q^{48} + 392 q^{49} + 28 q^{52} - 224 q^{53} + 800 q^{56} - 96 q^{61} - 352 q^{62} - 184 q^{64} - 112 q^{65} + 252 q^{66} - 344 q^{68} + 144 q^{69} + 232 q^{74} - 16 q^{77} - 168 q^{78} + 216 q^{81} + 20 q^{82} - 92 q^{88} - 84 q^{90} - 616 q^{92} - 684 q^{94}+O(q^{100})$$ 24 * q + 8 * q^4 - 72 * q^9 + 28 * q^10 + 36 * q^12 + 48 * q^13 - 40 * q^14 + 100 * q^16 + 32 * q^17 + 84 * q^22 - 312 * q^25 - 16 * q^26 - 80 * q^29 + 60 * q^30 - 24 * q^36 + 120 * q^38 - 204 * q^40 - 96 * q^42 - 144 * q^48 + 392 * q^49 + 28 * q^52 - 224 * q^53 + 800 * q^56 - 96 * q^61 - 352 * q^62 - 184 * q^64 - 112 * q^65 + 252 * q^66 - 344 * q^68 + 144 * q^69 + 232 * q^74 - 16 * q^77 - 168 * q^78 + 216 * q^81 + 20 * q^82 - 92 * q^88 - 84 * q^90 - 616 * q^92 - 684 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1 −1.99484 0.143522i 1.73205i 3.95880 + 0.572609i 8.96077i 0.248588 3.45517i −7.15347 −7.81501 1.71044i −3.00000 1.28607 17.8753i
103.2 −1.99484 + 0.143522i 1.73205i 3.95880 0.572609i 8.96077i 0.248588 + 3.45517i −7.15347 −7.81501 + 1.71044i −3.00000 1.28607 + 17.8753i
103.3 −1.86766 0.715427i 1.73205i 2.97633 + 2.67235i 4.65917i −1.23916 + 3.23489i −2.30148 −3.64691 7.12040i −3.00000 3.33330 8.70176i
103.4 −1.86766 + 0.715427i 1.73205i 2.97633 2.67235i 4.65917i −1.23916 3.23489i −2.30148 −3.64691 + 7.12040i −3.00000 3.33330 + 8.70176i
103.5 −1.86140 0.731561i 1.73205i 2.92964 + 2.72346i 3.15488i 1.26710 3.22404i 3.05810 −3.46086 7.21266i −3.00000 −2.30799 + 5.87250i
103.6 −1.86140 + 0.731561i 1.73205i 2.92964 2.72346i 3.15488i 1.26710 + 3.22404i 3.05810 −3.46086 + 7.21266i −3.00000 −2.30799 5.87250i
103.7 −1.31729 1.50490i 1.73205i −0.529474 + 3.96480i 7.41736i −2.60657 + 2.28162i 13.3843 6.66412 4.42600i −3.00000 −11.1624 + 9.77084i
103.8 −1.31729 + 1.50490i 1.73205i −0.529474 3.96480i 7.41736i −2.60657 2.28162i 13.3843 6.66412 + 4.42600i −3.00000 −11.1624 9.77084i
103.9 −0.570481 1.91691i 1.73205i −3.34910 + 2.18712i 7.79890i −3.32019 + 0.988102i 7.84779 6.10312 + 5.17222i −3.00000 14.9498 4.44912i
103.10 −0.570481 + 1.91691i 1.73205i −3.34910 2.18712i 7.79890i −3.32019 0.988102i 7.84779 6.10312 5.17222i −3.00000 14.9498 + 4.44912i
103.11 −0.0830862 1.99827i 1.73205i −3.98619 + 0.332058i 0.451006i 3.46111 0.143910i 9.24405 0.994741 + 7.93791i −3.00000 0.901234 0.0374724i
103.12 −0.0830862 + 1.99827i 1.73205i −3.98619 0.332058i 0.451006i 3.46111 + 0.143910i 9.24405 0.994741 7.93791i −3.00000 0.901234 + 0.0374724i
103.13 0.0830862 1.99827i 1.73205i −3.98619 0.332058i 0.451006i −3.46111 0.143910i −9.24405 −0.994741 + 7.93791i −3.00000 0.901234 + 0.0374724i
103.14 0.0830862 + 1.99827i 1.73205i −3.98619 + 0.332058i 0.451006i −3.46111 + 0.143910i −9.24405 −0.994741 7.93791i −3.00000 0.901234 0.0374724i
103.15 0.570481 1.91691i 1.73205i −3.34910 2.18712i 7.79890i 3.32019 + 0.988102i −7.84779 −6.10312 + 5.17222i −3.00000 14.9498 + 4.44912i
103.16 0.570481 + 1.91691i 1.73205i −3.34910 + 2.18712i 7.79890i 3.32019 0.988102i −7.84779 −6.10312 5.17222i −3.00000 14.9498 4.44912i
103.17 1.31729 1.50490i 1.73205i −0.529474 3.96480i 7.41736i 2.60657 + 2.28162i −13.3843 −6.66412 4.42600i −3.00000 −11.1624 9.77084i
103.18 1.31729 + 1.50490i 1.73205i −0.529474 + 3.96480i 7.41736i 2.60657 2.28162i −13.3843 −6.66412 + 4.42600i −3.00000 −11.1624 + 9.77084i
103.19 1.86140 0.731561i 1.73205i 2.92964 2.72346i 3.15488i −1.26710 3.22404i −3.05810 3.46086 7.21266i −3.00000 −2.30799 5.87250i
103.20 1.86140 + 0.731561i 1.73205i 2.92964 + 2.72346i 3.15488i −1.26710 + 3.22404i −3.05810 3.46086 + 7.21266i −3.00000 −2.30799 + 5.87250i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 103.24 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
13.b even 2 1 inner
52.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.e.c 24
3.b odd 2 1 468.3.e.m 24
4.b odd 2 1 inner 156.3.e.c 24
12.b even 2 1 468.3.e.m 24
13.b even 2 1 inner 156.3.e.c 24
39.d odd 2 1 468.3.e.m 24
52.b odd 2 1 inner 156.3.e.c 24
156.h even 2 1 468.3.e.m 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.e.c 24 1.a even 1 1 trivial
156.3.e.c 24 4.b odd 2 1 inner
156.3.e.c 24 13.b even 2 1 inner
156.3.e.c 24 52.b odd 2 1 inner
468.3.e.m 24 3.b odd 2 1
468.3.e.m 24 12.b even 2 1
468.3.e.m 24 39.d odd 2 1
468.3.e.m 24 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(156, [\chi])$$:

 $$T_{5}^{12} + 228T_{5}^{10} + 19120T_{5}^{8} + 715392T_{5}^{6} + 11384576T_{5}^{4} + 60341248T_{5}^{2} + 11808768$$ T5^12 + 228*T5^10 + 19120*T5^8 + 715392*T5^6 + 11384576*T5^4 + 60341248*T5^2 + 11808768 $$T_{7}^{12} - 392T_{7}^{10} + 53872T_{7}^{8} - 3286144T_{7}^{6} + 88136960T_{7}^{4} - 833523712T_{7}^{2} + 2389782528$$ T7^12 - 392*T7^10 + 53872*T7^8 - 3286144*T7^6 + 88136960*T7^4 - 833523712*T7^2 + 2389782528