# Properties

 Label 156.3.f.a Level $156$ Weight $3$ Character orbit 156.f Analytic conductor $4.251$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 156.f (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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} + 8 q^{4} - 12 q^{6} - 32 q^{8} - 72 q^{9}+O(q^{10})$$ 24 * q + 4 * q^2 + 8 * q^4 - 12 * q^6 - 32 * q^8 - 72 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} + 8 q^{4} - 12 q^{6} - 32 q^{8} - 72 q^{9} - 12 q^{10} + 12 q^{12} + 32 q^{14} + 4 q^{16} - 12 q^{18} + 84 q^{20} + 28 q^{22} - 36 q^{24} + 104 q^{25} - 96 q^{28} + 64 q^{29} - 12 q^{30} + 44 q^{32} + 48 q^{33} + 40 q^{34} - 24 q^{36} - 192 q^{37} - 104 q^{38} + 220 q^{40} - 220 q^{44} - 104 q^{46} - 144 q^{48} - 248 q^{49} + 100 q^{50} - 52 q^{52} + 336 q^{53} + 36 q^{54} + 168 q^{56} - 16 q^{58} + 60 q^{60} + 16 q^{61} + 152 q^{62} - 16 q^{64} - 132 q^{66} + 400 q^{68} - 192 q^{69} + 208 q^{70} + 96 q^{72} + 112 q^{73} - 104 q^{74} - 264 q^{76} - 272 q^{77} - 300 q^{80} + 216 q^{81} - 4 q^{82} + 96 q^{84} + 64 q^{85} + 288 q^{86} - 492 q^{88} + 36 q^{90} + 328 q^{92} - 96 q^{93} - 884 q^{94} + 72 q^{96} - 80 q^{97} - 572 q^{98}+O(q^{100})$$ 24 * q + 4 * q^2 + 8 * q^4 - 12 * q^6 - 32 * q^8 - 72 * q^9 - 12 * q^10 + 12 * q^12 + 32 * q^14 + 4 * q^16 - 12 * q^18 + 84 * q^20 + 28 * q^22 - 36 * q^24 + 104 * q^25 - 96 * q^28 + 64 * q^29 - 12 * q^30 + 44 * q^32 + 48 * q^33 + 40 * q^34 - 24 * q^36 - 192 * q^37 - 104 * q^38 + 220 * q^40 - 220 * q^44 - 104 * q^46 - 144 * q^48 - 248 * q^49 + 100 * q^50 - 52 * q^52 + 336 * q^53 + 36 * q^54 + 168 * q^56 - 16 * q^58 + 60 * q^60 + 16 * q^61 + 152 * q^62 - 16 * q^64 - 132 * q^66 + 400 * q^68 - 192 * q^69 + 208 * q^70 + 96 * q^72 + 112 * q^73 - 104 * q^74 - 264 * q^76 - 272 * q^77 - 300 * q^80 + 216 * q^81 - 4 * q^82 + 96 * q^84 + 64 * q^85 + 288 * q^86 - 492 * q^88 + 36 * q^90 + 328 * q^92 - 96 * q^93 - 884 * q^94 + 72 * q^96 - 80 * q^97 - 572 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1 −1.99963 0.0384798i 1.73205i 3.99704 + 0.153891i −4.98564 0.0666489 3.46346i 3.34189i −7.98668 0.461529i −3.00000 9.96944 + 0.191846i
79.2 −1.99963 + 0.0384798i 1.73205i 3.99704 0.153891i −4.98564 0.0666489 + 3.46346i 3.34189i −7.98668 + 0.461529i −3.00000 9.96944 0.191846i
79.3 −1.89675 0.634294i 1.73205i 3.19534 + 2.40620i 6.10663 1.09863 3.28527i 5.33206i −4.53454 6.59075i −3.00000 −11.5828 3.87340i
79.4 −1.89675 + 0.634294i 1.73205i 3.19534 2.40620i 6.10663 1.09863 + 3.28527i 5.33206i −4.53454 + 6.59075i −3.00000 −11.5828 + 3.87340i
79.5 −1.66943 1.10137i 1.73205i 1.57397 + 3.67731i −0.251667 −1.90763 + 2.89153i 2.33352i 1.42246 7.87252i −3.00000 0.420139 + 0.277178i
79.6 −1.66943 + 1.10137i 1.73205i 1.57397 3.67731i −0.251667 −1.90763 2.89153i 2.33352i 1.42246 + 7.87252i −3.00000 0.420139 0.277178i
79.7 −1.25145 1.56009i 1.73205i −0.867741 + 3.90474i 6.81265 −2.70215 + 2.16758i 11.2335i 7.17767 3.53285i −3.00000 −8.52570 10.6283i
79.8 −1.25145 + 1.56009i 1.73205i −0.867741 3.90474i 6.81265 −2.70215 2.16758i 11.2335i 7.17767 + 3.53285i −3.00000 −8.52570 + 10.6283i
79.9 −0.253155 1.98391i 1.73205i −3.87182 + 1.00448i −5.54446 −3.43624 + 0.438478i 3.35464i 2.97297 + 7.42708i −3.00000 1.40361 + 10.9997i
79.10 −0.253155 + 1.98391i 1.73205i −3.87182 1.00448i −5.54446 −3.43624 0.438478i 3.35464i 2.97297 7.42708i −3.00000 1.40361 10.9997i
79.11 0.337137 1.97138i 1.73205i −3.77268 1.32925i −3.28562 3.41453 + 0.583939i 11.8734i −3.89237 + 6.98924i −3.00000 −1.10771 + 6.47720i
79.12 0.337137 + 1.97138i 1.73205i −3.77268 + 1.32925i −3.28562 3.41453 0.583939i 11.8734i −3.89237 6.98924i −3.00000 −1.10771 6.47720i
79.13 0.805717 1.83052i 1.73205i −2.70164 2.94977i 5.48615 −3.17056 1.39554i 6.35783i −7.57638 + 2.56874i −3.00000 4.42028 10.0425i
79.14 0.805717 + 1.83052i 1.73205i −2.70164 + 2.94977i 5.48615 −3.17056 + 1.39554i 6.35783i −7.57638 2.56874i −3.00000 4.42028 + 10.0425i
79.15 0.969763 1.74916i 1.73205i −2.11912 3.39254i −4.25416 3.02963 + 1.67968i 10.6496i −7.98914 + 0.416717i −3.00000 −4.12553 + 7.44120i
79.16 0.969763 + 1.74916i 1.73205i −2.11912 + 3.39254i −4.25416 3.02963 1.67968i 10.6496i −7.98914 0.416717i −3.00000 −4.12553 7.44120i
79.17 1.41412 1.41431i 1.73205i −0.000546494 4.00000i −9.54425 −2.44966 2.44932i 4.66845i −5.65801 5.65569i −3.00000 −13.4967 + 13.4985i
79.18 1.41412 + 1.41431i 1.73205i −0.000546494 4.00000i −9.54425 −2.44966 + 2.44932i 4.66845i −5.65801 + 5.65569i −3.00000 −13.4967 13.4985i
79.19 1.69411 1.06301i 1.73205i 1.74001 3.60171i 5.51671 1.84119 + 2.93428i 1.01261i −0.880886 7.95135i −3.00000 9.34592 5.86433i
79.20 1.69411 + 1.06301i 1.73205i 1.74001 + 3.60171i 5.51671 1.84119 2.93428i 1.01261i −0.880886 + 7.95135i −3.00000 9.34592 + 5.86433i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.24 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.f.a 24
3.b odd 2 1 468.3.f.b 24
4.b odd 2 1 inner 156.3.f.a 24
8.b even 2 1 2496.3.k.e 24
8.d odd 2 1 2496.3.k.e 24
12.b even 2 1 468.3.f.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.f.a 24 1.a even 1 1 trivial
156.3.f.a 24 4.b odd 2 1 inner
468.3.f.b 24 3.b odd 2 1
468.3.f.b 24 12.b even 2 1
2496.3.k.e 24 8.b even 2 1
2496.3.k.e 24 8.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(156, [\chi])$$.