# Properties

 Label 144.3.v.a Level $144$ Weight $3$ Character orbit 144.v Analytic conductor $3.924$ Analytic rank $0$ Dimension $184$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.v (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 8q^{8} - 8q^{10} - 2q^{11} + 56q^{12} - 2q^{13} + 14q^{14} - 2q^{16} - 16q^{17} + 38q^{18} - 8q^{19} - 44q^{20} + 14q^{21} - 2q^{22} - 4q^{23} + 120q^{24} - 104q^{26} - 52q^{27} + 56q^{28} - 2q^{29} - 130q^{30} - 182q^{32} - 8q^{33} - 10q^{34} + 92q^{35} - 2q^{36} - 8q^{37} - 254q^{38} + 184q^{39} - 2q^{40} - 252q^{42} - 2q^{43} - 140q^{44} - 54q^{45} + 176q^{46} + 162q^{48} - 480q^{49} - 96q^{50} - 120q^{51} - 2q^{52} - 8q^{53} + 94q^{54} - 16q^{55} + 260q^{56} + 88q^{58} + 142q^{59} - 434q^{60} - 2q^{61} - 636q^{62} + 244q^{64} - 4q^{65} - 100q^{66} - 2q^{67} - 112q^{68} + 14q^{69} - 100q^{70} - 16q^{71} + 98q^{72} + 82q^{74} - 296q^{75} + 154q^{76} + 194q^{77} + 228q^{78} + 592q^{80} - 8q^{81} - 420q^{82} + 238q^{83} - 22q^{84} - 52q^{85} - 170q^{86} - 456q^{87} - 26q^{88} + 808q^{90} + 188q^{91} + 176q^{92} + 26q^{93} - 18q^{94} - 202q^{96} - 4q^{97} + 408q^{98} - 38q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.99771 + 0.0956780i −2.20012 + 2.03947i 3.98169 0.382274i 0.981821 3.66421i 4.20008 4.28478i −1.53303 + 2.65528i −7.91769 + 1.14463i 0.681097 8.97419i −1.61081 + 7.41396i
43.2 −1.96237 + 0.386118i 1.43023 2.63713i 3.70183 1.51541i −1.66775 + 6.22412i −1.78840 + 5.72727i −3.37227 + 5.84094i −6.67924 + 4.40315i −4.90890 7.54339i 0.869504 12.8580i
43.3 −1.95734 0.410895i −2.99626 0.149794i 3.66233 + 1.60852i −1.99641 + 7.45071i 5.80313 + 1.52435i 3.29099 5.70016i −6.50748 4.65325i 8.95512 + 0.897641i 6.96911 13.7632i
43.4 −1.94172 + 0.479310i −1.68621 2.48127i 3.54052 1.86137i 1.25155 4.67084i 4.46343 + 4.00971i 1.88587 3.26643i −5.98252 + 5.31125i −3.31343 + 8.36787i −0.191372 + 9.66933i
43.5 −1.92787 0.532286i 2.18726 + 2.05326i 3.43334 + 2.05235i −0.931824 + 3.47761i −3.12383 5.12267i −4.96512 + 8.59983i −5.52659 5.78418i 0.568222 + 8.98204i 3.64752 6.20838i
43.6 −1.91763 0.568063i 2.62502 1.45234i 3.35461 + 2.17867i 2.50565 9.35123i −5.85883 + 1.29387i −1.43574 + 2.48677i −5.19527 6.08351i 4.78143 7.62483i −10.1170 + 16.5088i
43.7 −1.83642 + 0.792194i 2.99507 + 0.171846i 2.74486 2.90960i −0.222103 + 0.828901i −5.63634 + 2.05710i 5.23163 9.06144i −2.73574 + 7.51769i 8.94094 + 1.02938i −0.248776 1.69816i
43.8 −1.81345 0.843436i 0.915091 + 2.85703i 2.57723 + 3.05906i 0.136028 0.507662i 0.750245 5.95291i 6.49628 11.2519i −2.09357 7.72120i −7.32522 + 5.22888i −0.674860 + 0.805891i
43.9 −1.62235 + 1.16960i −0.520377 + 2.95452i 1.26405 3.79502i −1.70362 + 6.35798i −2.61139 5.40191i −0.581649 + 1.00744i 2.38794 + 7.63530i −8.45841 3.07493i −4.67246 12.3074i
43.10 −1.59468 1.20706i −2.00451 2.23203i 1.08601 + 3.84975i 0.218413 0.815129i 0.502363 + 5.97893i −4.45068 + 7.70881i 2.91503 7.45001i −0.963885 + 8.94824i −1.33221 + 1.03623i
43.11 −1.48296 1.34195i 1.38752 2.65985i 0.398351 + 3.98012i −0.631342 + 2.35620i −5.62701 + 2.08248i 3.43083 5.94237i 4.75037 6.43692i −5.14959 7.38117i 4.09816 2.64693i
43.12 −1.35588 + 1.47023i −2.99965 + 0.0461027i −0.323163 3.98692i 0.803565 2.99895i 3.99939 4.47268i −0.723340 + 1.25286i 6.29987 + 4.93068i 8.99575 0.276584i 3.31960 + 5.24765i
43.13 −1.23592 1.57242i −1.52445 + 2.58381i −0.945018 + 3.88676i 0.892831 3.33209i 5.94692 0.796299i −1.63564 + 2.83300i 7.27960 3.31775i −4.35212 7.87776i −6.34292 + 2.71428i
43.14 −1.23219 + 1.57534i 0.874159 2.86982i −0.963395 3.88225i 0.661418 2.46845i 3.44380 + 4.91327i −2.72334 + 4.71696i 7.30296 + 3.26601i −7.47169 5.01735i 3.07365 + 4.08356i
43.15 −1.17470 + 1.61867i 2.85433 + 0.923476i −1.24016 3.80289i 0.872094 3.25470i −4.84778 + 3.53540i −5.18069 + 8.97322i 7.61243 + 2.45986i 7.29438 + 5.27181i 4.24382 + 5.23492i
43.16 −0.933784 1.76863i 2.99980 + 0.0348191i −2.25609 + 3.30303i −1.49557 + 5.58156i −2.73958 5.33804i −1.11666 + 1.93410i 7.94855 + 0.905871i 8.99758 + 0.208901i 11.2682 2.56685i
43.17 −0.915081 + 1.77838i −1.51894 2.58705i −2.32525 3.25472i −1.91133 + 7.13319i 5.99070 0.333879i 5.45754 9.45274i 7.91591 1.15685i −4.38567 + 7.85913i −10.9365 9.92651i
43.18 −0.809359 + 1.82892i −0.155921 + 2.99595i −2.68988 2.96050i 2.03358 7.58942i −5.35314 2.70996i 4.60505 7.97618i 7.59159 2.52345i −8.95138 0.934260i 12.2345 + 9.86181i
43.19 −0.661532 1.88743i 2.43372 + 1.75414i −3.12475 + 2.49718i 1.97813 7.38249i 1.70082 5.75389i 0.0635994 0.110157i 6.78037 + 4.24577i 2.84600 + 8.53817i −15.2425 + 1.15017i
43.20 −0.536163 1.92679i −2.98561 + 0.293508i −3.42506 + 2.06615i −0.686399 + 2.56168i 2.16630 + 5.59528i 1.45323 2.51707i 5.81743 + 5.49158i 8.82771 1.75260i 5.30384 0.0509280i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
16.f odd 4 1 inner
144.v odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.v.a 184
3.b odd 2 1 432.3.w.a 184
9.c even 3 1 inner 144.3.v.a 184
9.d odd 6 1 432.3.w.a 184
16.f odd 4 1 inner 144.3.v.a 184
48.k even 4 1 432.3.w.a 184
144.u even 12 1 432.3.w.a 184
144.v odd 12 1 inner 144.3.v.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.v.a 184 1.a even 1 1 trivial
144.3.v.a 184 9.c even 3 1 inner
144.3.v.a 184 16.f odd 4 1 inner
144.3.v.a 184 144.v odd 12 1 inner
432.3.w.a 184 3.b odd 2 1
432.3.w.a 184 9.d odd 6 1
432.3.w.a 184 48.k even 4 1
432.3.w.a 184 144.u even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database