# Properties

 Label 156.2.p.b Level $156$ Weight $2$ Character orbit 156.p Analytic conductor $1.246$ Analytic rank $0$ Dimension $40$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 156.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.24566627153$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 6 q^{4} + 2 q^{6} - 10 q^{9}+O(q^{10})$$ 40 * q + 6 * q^4 + 2 * q^6 - 10 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 6 q^{4} + 2 q^{6} - 10 q^{9} + 6 q^{10} - 16 q^{12} + 16 q^{13} + 18 q^{16} - 20 q^{18} - 60 q^{21} + 20 q^{22} - 20 q^{24} - 32 q^{25} - 24 q^{28} - 26 q^{30} - 2 q^{33} - 28 q^{34} + 6 q^{36} + 16 q^{37} + 20 q^{40} - 4 q^{42} + 28 q^{45} + 48 q^{46} - 24 q^{48} - 28 q^{49} + 12 q^{52} - 4 q^{54} + 44 q^{57} - 50 q^{58} - 12 q^{60} - 12 q^{61} + 12 q^{64} + 24 q^{66} - 6 q^{69} + 48 q^{72} - 16 q^{73} + 30 q^{78} - 14 q^{81} - 2 q^{82} - 6 q^{84} + 52 q^{85} - 12 q^{88} + 156 q^{90} + 40 q^{93} - 8 q^{94} + 128 q^{96} - 56 q^{97}+O(q^{100})$$ 40 * q + 6 * q^4 + 2 * q^6 - 10 * q^9 + 6 * q^10 - 16 * q^12 + 16 * q^13 + 18 * q^16 - 20 * q^18 - 60 * q^21 + 20 * q^22 - 20 * q^24 - 32 * q^25 - 24 * q^28 - 26 * q^30 - 2 * q^33 - 28 * q^34 + 6 * q^36 + 16 * q^37 + 20 * q^40 - 4 * q^42 + 28 * q^45 + 48 * q^46 - 24 * q^48 - 28 * q^49 + 12 * q^52 - 4 * q^54 + 44 * q^57 - 50 * q^58 - 12 * q^60 - 12 * q^61 + 12 * q^64 + 24 * q^66 - 6 * q^69 + 48 * q^72 - 16 * q^73 + 30 * q^78 - 14 * q^81 - 2 * q^82 - 6 * q^84 + 52 * q^85 - 12 * q^88 + 156 * q^90 + 40 * q^93 - 8 * q^94 + 128 * q^96 - 56 * q^97

## 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 −1.40167 0.187933i −1.29017 1.15562i 1.92936 + 0.526840i 1.98084i 1.59121 + 1.86227i 2.42516 1.40017i −2.60532 1.10105i 0.329066 + 2.98190i 0.372266 2.77649i
35.2 −1.39117 + 0.254263i 0.564863 1.63735i 1.87070 0.707445i 3.40345i −0.369501 + 2.42146i −2.05768 + 1.18800i −2.42258 + 1.45983i −2.36186 1.84976i 0.865372 + 4.73478i
35.3 −1.36641 0.364576i −1.10270 + 1.33569i 1.73417 + 0.996323i 0.171572i 1.99370 1.42308i −2.51891 + 1.45429i −2.00636 1.99362i −0.568109 2.94572i 0.0625510 0.234438i
35.4 −1.32222 + 0.501741i 1.71957 0.207570i 1.49651 1.32682i 3.37008i −2.16949 + 1.13723i −1.77448 + 1.02450i −1.31299 + 2.50520i 2.91383 0.713863i −1.69091 4.45597i
35.5 −1.06855 + 0.926389i −1.63287 + 0.577710i 0.283607 1.97979i 1.45117i 1.20962 2.12998i 1.17936 0.680902i 1.53101 + 2.37824i 2.33250 1.88665i 1.34435 + 1.55065i
35.6 −0.998939 1.00106i 1.10270 1.33569i −0.00424367 + 2.00000i 0.171572i −2.43863 + 0.230399i 2.51891 1.45429i 2.00636 1.99362i −0.568109 2.94572i 0.171754 0.171390i
35.7 −0.863590 1.11992i 1.29017 + 1.15562i −0.508424 + 1.93430i 1.98084i 0.180026 2.44287i −2.42516 + 1.40017i 2.60532 1.10105i 0.329066 + 2.98190i 2.21838 1.71064i
35.8 −0.475386 1.33192i −0.564863 + 1.63735i −1.54802 + 1.26635i 3.40345i 2.44935 0.0260236i 2.05768 1.18800i 2.42258 + 1.45983i −2.36186 1.84976i −4.53312 + 1.61795i
35.9 −0.268000 + 1.38859i 0.316122 + 1.70296i −1.85635 0.744284i 1.45117i −2.44943 0.0174310i −1.17936 + 0.680902i 1.53101 2.37824i −2.80013 + 1.07668i −2.01508 0.388914i
35.10 −0.226588 1.39594i −1.71957 + 0.207570i −1.89732 + 0.632608i 3.37008i 0.679390 + 2.35339i 1.77448 1.02450i 1.31299 + 2.50520i 2.91383 0.713863i 4.70444 0.763619i
35.11 0.226588 + 1.39594i −0.680023 1.59297i −1.89732 + 0.632608i 3.37008i 2.06962 1.31022i 1.77448 1.02450i −1.31299 2.50520i −2.07514 + 2.16652i 4.70444 0.763619i
35.12 0.268000 1.38859i 1.63287 0.577710i −1.85635 0.744284i 1.45117i −0.364592 2.42220i −1.17936 + 0.680902i −1.53101 + 2.37824i 2.33250 1.88665i −2.01508 0.388914i
35.13 0.475386 + 1.33192i 1.13556 1.30786i −1.54802 + 1.26635i 3.40345i 2.28180 + 0.890732i 2.05768 1.18800i −2.42258 1.45983i −0.421012 2.97031i −4.53312 + 1.61795i
35.14 0.863590 + 1.11992i 1.64588 + 0.539506i −0.508424 + 1.93430i 1.98084i 0.817167 + 2.30916i −2.42516 + 1.40017i −2.60532 + 1.10105i 2.41787 + 1.77593i 2.21838 1.71064i
35.15 0.998939 + 1.00106i −0.605388 + 1.62281i −0.00424367 + 2.00000i 0.171572i −2.22927 + 1.01506i 2.51891 1.45429i −2.00636 + 1.99362i −2.26701 1.96486i 0.171754 0.171390i
35.16 1.06855 0.926389i −0.316122 1.70296i 0.283607 1.97979i 1.45117i −1.91539 1.52685i 1.17936 0.680902i −1.53101 2.37824i −2.80013 + 1.07668i 1.34435 + 1.55065i
35.17 1.32222 0.501741i 0.680023 + 1.59297i 1.49651 1.32682i 3.37008i 1.69840 + 1.76506i −1.77448 + 1.02450i 1.31299 2.50520i −2.07514 + 2.16652i −1.69091 4.45597i
35.18 1.36641 + 0.364576i 0.605388 1.62281i 1.73417 + 0.996323i 0.171572i 1.41885 1.99672i −2.51891 + 1.45429i 2.00636 + 1.99362i −2.26701 1.96486i 0.0625510 0.234438i
35.19 1.39117 0.254263i −1.13556 + 1.30786i 1.87070 0.707445i 3.40345i −1.24721 + 2.10819i −2.05768 + 1.18800i 2.42258 1.45983i −0.421012 2.97031i 0.865372 + 4.73478i
35.20 1.40167 + 0.187933i −1.64588 0.539506i 1.92936 + 0.526840i 1.98084i −2.20560 1.06553i 2.42516 1.40017i 2.60532 + 1.10105i 2.41787 + 1.77593i 0.372266 2.77649i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 107.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner
52.j odd 6 1 inner
156.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.p.b 40
3.b odd 2 1 inner 156.2.p.b 40
4.b odd 2 1 inner 156.2.p.b 40
12.b even 2 1 inner 156.2.p.b 40
13.c even 3 1 inner 156.2.p.b 40
39.i odd 6 1 inner 156.2.p.b 40
52.j odd 6 1 inner 156.2.p.b 40
156.p even 6 1 inner 156.2.p.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.p.b 40 1.a even 1 1 trivial
156.2.p.b 40 3.b odd 2 1 inner
156.2.p.b 40 4.b odd 2 1 inner
156.2.p.b 40 12.b even 2 1 inner
156.2.p.b 40 13.c even 3 1 inner
156.2.p.b 40 39.i odd 6 1 inner
156.2.p.b 40 52.j odd 6 1 inner
156.2.p.b 40 156.p even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + 29T_{5}^{8} + 279T_{5}^{6} + 991T_{5}^{4} + 1116T_{5}^{2} + 32$$ acting on $$S_{2}^{\mathrm{new}}(156, [\chi])$$.