# Properties

 Label 156.2.w.c Level $156$ Weight $2$ Character orbit 156.w Analytic conductor $1.246$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 156.w (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.24566627153$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{8} + 12 q^{9}+O(q^{10})$$ 24 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^8 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{8} + 12 q^{9} + 12 q^{10} + 4 q^{11} - 4 q^{12} - 22 q^{14} - 4 q^{15} + 12 q^{16} - 2 q^{18} - 22 q^{19} - 4 q^{20} + 10 q^{21} - 34 q^{22} + 2 q^{24} - 4 q^{26} - 14 q^{28} + 8 q^{29} - 16 q^{30} - 2 q^{31} - 34 q^{32} + 4 q^{33} + 20 q^{34} + 24 q^{35} - 4 q^{36} - 12 q^{37} - 12 q^{38} - 6 q^{39} + 28 q^{40} - 36 q^{41} + 6 q^{42} + 10 q^{43} + 20 q^{44} + 4 q^{45} - 2 q^{46} + 20 q^{47} + 32 q^{48} - 54 q^{49} - 24 q^{50} + 36 q^{51} + 4 q^{52} - 36 q^{53} - 4 q^{54} - 24 q^{55} + 70 q^{56} - 32 q^{57} + 48 q^{58} - 36 q^{59} - 28 q^{60} - 2 q^{61} + 64 q^{62} + 8 q^{63} - 8 q^{64} + 16 q^{65} + 20 q^{66} - 16 q^{67} - 10 q^{68} + 12 q^{69} + 36 q^{71} - 2 q^{72} - 40 q^{73} - 30 q^{74} - 8 q^{75} + 58 q^{76} + 24 q^{78} - 48 q^{80} - 12 q^{81} - 30 q^{82} - 24 q^{83} - 26 q^{84} + 18 q^{85} + 30 q^{86} + 6 q^{87} - 78 q^{88} + 66 q^{89} + 12 q^{90} + 42 q^{91} + 36 q^{92} - 10 q^{93} + 6 q^{94} - 28 q^{95} + 8 q^{96} + 4 q^{97} + 44 q^{98} - 4 q^{99}+O(q^{100})$$ 24 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^8 + 12 * q^9 + 12 * q^10 + 4 * q^11 - 4 * q^12 - 22 * q^14 - 4 * q^15 + 12 * q^16 - 2 * q^18 - 22 * q^19 - 4 * q^20 + 10 * q^21 - 34 * q^22 + 2 * q^24 - 4 * q^26 - 14 * q^28 + 8 * q^29 - 16 * q^30 - 2 * q^31 - 34 * q^32 + 4 * q^33 + 20 * q^34 + 24 * q^35 - 4 * q^36 - 12 * q^37 - 12 * q^38 - 6 * q^39 + 28 * q^40 - 36 * q^41 + 6 * q^42 + 10 * q^43 + 20 * q^44 + 4 * q^45 - 2 * q^46 + 20 * q^47 + 32 * q^48 - 54 * q^49 - 24 * q^50 + 36 * q^51 + 4 * q^52 - 36 * q^53 - 4 * q^54 - 24 * q^55 + 70 * q^56 - 32 * q^57 + 48 * q^58 - 36 * q^59 - 28 * q^60 - 2 * q^61 + 64 * q^62 + 8 * q^63 - 8 * q^64 + 16 * q^65 + 20 * q^66 - 16 * q^67 - 10 * q^68 + 12 * q^69 + 36 * q^71 - 2 * q^72 - 40 * q^73 - 30 * q^74 - 8 * q^75 + 58 * q^76 + 24 * q^78 - 48 * q^80 - 12 * q^81 - 30 * q^82 - 24 * q^83 - 26 * q^84 + 18 * q^85 + 30 * q^86 + 6 * q^87 - 78 * q^88 + 66 * q^89 + 12 * q^90 + 42 * q^91 + 36 * q^92 - 10 * q^93 + 6 * q^94 - 28 * q^95 + 8 * q^96 + 4 * q^97 + 44 * q^98 - 4 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.39918 + 0.205665i 0.866025 + 0.500000i 1.91540 0.575524i −1.90811 1.90811i −1.31456 0.521479i 1.18978 4.44032i −2.56163 + 1.19919i 0.500000 + 0.866025i 3.06221 + 2.27735i
7.2 −1.31582 0.518286i 0.866025 + 0.500000i 1.46276 + 1.36394i 0.713774 + 0.713774i −0.880389 1.10676i −0.872392 + 3.25581i −1.21781 2.55283i 0.500000 + 0.866025i −0.569258 1.30914i
7.3 −0.400093 1.35644i 0.866025 + 0.500000i −1.67985 + 1.08540i 1.23779 + 1.23779i 0.331729 1.37476i 1.04473 3.89897i 2.14438 + 1.84435i 0.500000 + 0.866025i 1.18375 2.17421i
7.4 −0.101912 + 1.41054i 0.866025 + 0.500000i −1.97923 0.287501i 2.43250 + 2.43250i −0.793527 + 1.17060i 0.522652 1.95057i 0.607239 2.76247i 0.500000 + 0.866025i −3.67903 + 3.18323i
7.5 0.806088 + 1.16199i 0.866025 + 0.500000i −0.700446 + 1.87333i −0.922964 0.922964i 0.117097 + 1.40936i −0.757268 + 2.82616i −2.74142 + 0.696159i 0.500000 + 0.866025i 0.328486 1.81647i
7.6 1.41092 + 0.0965326i 0.866025 + 0.500000i 1.98136 + 0.272399i −1.91901 1.91901i 1.17362 + 0.789057i 0.104554 0.390202i 2.76924 + 0.575598i 0.500000 + 0.866025i −2.52232 2.89281i
19.1 −1.41136 + 0.0898367i −0.866025 0.500000i 1.98386 0.253583i −1.54484 + 1.54484i 1.26719 + 0.627878i −1.07009 0.286731i −2.77715 + 0.536120i 0.500000 + 0.866025i 2.04154 2.31911i
19.2 −1.15700 0.813242i −0.866025 0.500000i 0.677275 + 1.88183i 1.41026 1.41026i 0.595366 + 1.28279i 0.844432 + 0.226265i 0.746782 2.72806i 0.500000 + 0.866025i −2.77854 + 0.484781i
19.3 −0.737204 + 1.20687i −0.866025 0.500000i −0.913059 1.77942i 0.218120 0.218120i 1.24187 0.676576i 3.40856 + 0.913320i 2.82063 + 0.209852i 0.500000 + 0.866025i 0.102443 + 0.424042i
19.4 0.0144886 1.41414i −0.866025 0.500000i −1.99958 0.0409778i −2.26006 + 2.26006i −0.719617 + 1.21744i −3.65215 0.978592i −0.0869195 + 2.82709i 0.500000 + 0.866025i 3.16330 + 3.22879i
19.5 1.12047 0.862866i −0.866025 0.500000i 0.510925 1.93364i 0.756294 0.756294i −1.40179 + 0.187026i −0.936058 0.250816i −1.09599 2.60745i 0.500000 + 0.866025i 0.194828 1.49999i
19.6 1.17059 + 0.793543i −0.866025 0.500000i 0.740580 + 1.85783i 2.78626 2.78626i −0.616993 1.27253i −0.826736 0.221523i −0.607351 + 2.76245i 0.500000 + 0.866025i 5.47259 1.05056i
67.1 −1.39918 0.205665i 0.866025 0.500000i 1.91540 + 0.575524i −1.90811 + 1.90811i −1.31456 + 0.521479i 1.18978 + 4.44032i −2.56163 1.19919i 0.500000 0.866025i 3.06221 2.27735i
67.2 −1.31582 + 0.518286i 0.866025 0.500000i 1.46276 1.36394i 0.713774 0.713774i −0.880389 + 1.10676i −0.872392 3.25581i −1.21781 + 2.55283i 0.500000 0.866025i −0.569258 + 1.30914i
67.3 −0.400093 + 1.35644i 0.866025 0.500000i −1.67985 1.08540i 1.23779 1.23779i 0.331729 + 1.37476i 1.04473 + 3.89897i 2.14438 1.84435i 0.500000 0.866025i 1.18375 + 2.17421i
67.4 −0.101912 1.41054i 0.866025 0.500000i −1.97923 + 0.287501i 2.43250 2.43250i −0.793527 1.17060i 0.522652 + 1.95057i 0.607239 + 2.76247i 0.500000 0.866025i −3.67903 3.18323i
67.5 0.806088 1.16199i 0.866025 0.500000i −0.700446 1.87333i −0.922964 + 0.922964i 0.117097 1.40936i −0.757268 2.82616i −2.74142 0.696159i 0.500000 0.866025i 0.328486 + 1.81647i
67.6 1.41092 0.0965326i 0.866025 0.500000i 1.98136 0.272399i −1.91901 + 1.91901i 1.17362 0.789057i 0.104554 + 0.390202i 2.76924 0.575598i 0.500000 0.866025i −2.52232 + 2.89281i
115.1 −1.41136 0.0898367i −0.866025 + 0.500000i 1.98386 + 0.253583i −1.54484 1.54484i 1.26719 0.627878i −1.07009 + 0.286731i −2.77715 0.536120i 0.500000 0.866025i 2.04154 + 2.31911i
115.2 −1.15700 + 0.813242i −0.866025 + 0.500000i 0.677275 1.88183i 1.41026 + 1.41026i 0.595366 1.28279i 0.844432 0.226265i 0.746782 + 2.72806i 0.500000 0.866025i −2.77854 0.484781i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 115.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.w.c 24
3.b odd 2 1 468.2.cb.h 24
4.b odd 2 1 156.2.w.d yes 24
12.b even 2 1 468.2.cb.g 24
13.f odd 12 1 156.2.w.d yes 24
39.k even 12 1 468.2.cb.g 24
52.l even 12 1 inner 156.2.w.c 24
156.v odd 12 1 468.2.cb.h 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.w.c 24 1.a even 1 1 trivial
156.2.w.c 24 52.l even 12 1 inner
156.2.w.d yes 24 4.b odd 2 1
156.2.w.d yes 24 13.f odd 12 1
468.2.cb.g 24 12.b even 2 1
468.2.cb.g 24 39.k even 12 1
468.2.cb.h 24 3.b odd 2 1
468.2.cb.h 24 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(156, [\chi])$$:

 $$T_{5}^{24} - 2 T_{5}^{23} + 2 T_{5}^{22} + 14 T_{5}^{21} + 293 T_{5}^{20} - 388 T_{5}^{19} + 288 T_{5}^{18} + 1752 T_{5}^{17} + 28082 T_{5}^{16} - 24348 T_{5}^{15} + 7948 T_{5}^{14} - 14444 T_{5}^{13} + 843186 T_{5}^{12} + \cdots + 1106704$$ T5^24 - 2*T5^23 + 2*T5^22 + 14*T5^21 + 293*T5^20 - 388*T5^19 + 288*T5^18 + 1752*T5^17 + 28082*T5^16 - 24348*T5^15 + 7948*T5^14 - 14444*T5^13 + 843186*T5^12 - 887104*T5^11 + 451016*T5^10 - 1048984*T5^9 + 9279741*T5^8 - 12447762*T5^7 + 8772290*T5^6 - 5765426*T5^5 + 20431209*T5^4 - 29281292*T5^3 + 22498632*T5^2 - 7056816*T5 + 1106704 $$T_{7}^{24} + 2 T_{7}^{23} + 29 T_{7}^{22} + 112 T_{7}^{21} + 209 T_{7}^{20} + 816 T_{7}^{19} - 2504 T_{7}^{18} - 14992 T_{7}^{17} - 32928 T_{7}^{16} - 150840 T_{7}^{15} + 507036 T_{7}^{14} + 2192304 T_{7}^{13} + \cdots + 2560000$$ T7^24 + 2*T7^23 + 29*T7^22 + 112*T7^21 + 209*T7^20 + 816*T7^19 - 2504*T7^18 - 14992*T7^17 - 32928*T7^16 - 150840*T7^15 + 507036*T7^14 + 2192304*T7^13 + 9660252*T7^12 + 24925376*T7^11 + 39267600*T7^10 + 61262176*T7^9 + 60656480*T7^8 - 27307744*T7^7 - 104760176*T7^6 - 45357504*T7^5 + 38536336*T7^4 + 36034560*T7^3 + 11168000*T7^2 + 5888000*T7 + 2560000