# Properties

 Label 36.4.h.b Level 36 Weight 4 Character orbit 36.h Analytic conductor 2.124 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 36.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.12406876021$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{4} - 72q^{5} + 60q^{6} - 84q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{4} - 72q^{5} + 60q^{6} - 84q^{9} + 96q^{10} + 216q^{12} - 216q^{13} - 36q^{14} - 72q^{16} + 276q^{18} - 540q^{20} + 384q^{21} - 192q^{22} - 168q^{24} + 252q^{25} - 672q^{28} + 576q^{29} + 660q^{30} + 360q^{32} - 1236q^{33} - 660q^{34} + 276q^{36} + 1248q^{37} - 144q^{38} + 636q^{40} + 1116q^{41} - 288q^{42} - 1296q^{45} + 960q^{46} - 288q^{48} + 348q^{49} - 648q^{50} + 132q^{52} - 2616q^{54} - 1692q^{56} - 1668q^{57} + 516q^{58} - 192q^{60} - 264q^{61} + 960q^{64} - 2592q^{65} + 1068q^{66} + 5688q^{68} + 1608q^{69} + 564q^{70} + 4224q^{72} - 4776q^{73} + 5652q^{74} - 600q^{76} + 648q^{77} + 3660q^{78} + 3948q^{81} - 4104q^{82} - 4872q^{84} + 720q^{85} - 9540q^{86} + 1956q^{88} - 7656q^{90} - 7416q^{92} + 5400q^{93} - 1188q^{94} - 5640q^{96} + 588q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.81402 0.285097i −0.828122 5.12974i 7.83744 + 1.60454i −14.4924 + 8.36717i 0.867881 + 14.6713i −16.7175 9.65186i −21.5973 6.74964i −25.6284 + 8.49610i 43.1673 19.4137i
11.2 −2.52436 + 1.27578i 5.18398 + 0.355390i 4.74478 6.44105i −1.23846 + 0.715028i −13.5396 + 5.71649i 23.8818 + 13.7882i −3.76017 + 22.3128i 26.7474 + 3.68468i 2.21411 3.38499i
11.3 −1.93145 + 2.06627i −2.72340 + 4.42528i −0.538974 7.98182i −4.71466 + 2.72201i −3.88372 14.1745i −20.9358 12.0873i 17.5336 + 14.3029i −12.1662 24.1036i 3.48173 14.9992i
11.4 −1.65391 2.29447i 0.828122 + 5.12974i −2.52915 + 7.58969i −14.4924 + 8.36717i 10.4004 10.3842i 16.7175 + 9.65186i 21.5973 6.74964i −25.6284 + 8.49610i 43.1673 + 19.4137i
11.5 −1.38284 + 2.46734i −1.42987 4.99554i −4.17551 6.82387i 14.6499 8.45813i 14.3030 + 3.38006i −3.08966 1.78382i 22.6108 0.866066i −22.9109 + 14.2860i 0.610574 + 47.8425i
11.6 −0.157323 2.82405i −5.18398 0.355390i −7.95050 + 0.888573i −1.23846 + 0.715028i −0.188082 + 14.6957i −23.8818 13.7882i 3.76017 + 22.3128i 26.7474 + 3.68468i 2.21411 + 3.38499i
11.7 0.664105 + 2.74936i −5.00415 1.39947i −7.11793 + 3.65173i −14.2911 + 8.25096i 0.524375 14.6876i 19.2620 + 11.1209i −14.7670 17.1446i 23.0829 + 14.0063i −32.1756 33.8118i
11.8 0.823719 2.70582i 2.72340 4.42528i −6.64298 4.45768i −4.71466 + 2.72201i −9.73072 11.0142i 20.9358 + 12.0873i −17.5336 + 14.3029i −12.1662 24.1036i 3.48173 + 14.9992i
11.9 1.44536 2.43124i 1.42987 + 4.99554i −3.82189 7.02803i 14.6499 8.45813i 14.2121 + 3.74398i 3.08966 + 1.78382i −22.6108 0.866066i −22.9109 + 14.2860i 0.610574 47.8425i
11.10 2.15223 + 1.83518i 2.90476 4.30841i 1.26420 + 7.89948i 2.08666 1.20474i 14.1584 3.94193i −2.30362 1.33000i −11.7761 + 19.3216i −10.1248 25.0298i 6.70190 + 1.23654i
11.11 2.66543 + 0.946295i −2.90476 + 4.30841i 6.20905 + 5.04457i 2.08666 1.20474i −11.8195 + 8.73501i 2.30362 + 1.33000i 11.7761 + 19.3216i −10.1248 25.0298i 6.70190 1.23654i
11.12 2.71307 0.799546i 5.00415 + 1.39947i 6.72145 4.33844i −14.2911 + 8.25096i 14.6955 0.204180i −19.2620 11.1209i 14.7670 17.1446i 23.0829 + 14.0063i −32.1756 + 33.8118i
23.1 −2.81402 + 0.285097i −0.828122 + 5.12974i 7.83744 1.60454i −14.4924 8.36717i 0.867881 14.6713i −16.7175 + 9.65186i −21.5973 + 6.74964i −25.6284 8.49610i 43.1673 + 19.4137i
23.2 −2.52436 1.27578i 5.18398 0.355390i 4.74478 + 6.44105i −1.23846 0.715028i −13.5396 5.71649i 23.8818 13.7882i −3.76017 22.3128i 26.7474 3.68468i 2.21411 + 3.38499i
23.3 −1.93145 2.06627i −2.72340 4.42528i −0.538974 + 7.98182i −4.71466 2.72201i −3.88372 + 14.1745i −20.9358 + 12.0873i 17.5336 14.3029i −12.1662 + 24.1036i 3.48173 + 14.9992i
23.4 −1.65391 + 2.29447i 0.828122 5.12974i −2.52915 7.58969i −14.4924 8.36717i 10.4004 + 10.3842i 16.7175 9.65186i 21.5973 + 6.74964i −25.6284 8.49610i 43.1673 19.4137i
23.5 −1.38284 2.46734i −1.42987 + 4.99554i −4.17551 + 6.82387i 14.6499 + 8.45813i 14.3030 3.38006i −3.08966 + 1.78382i 22.6108 + 0.866066i −22.9109 14.2860i 0.610574 47.8425i
23.6 −0.157323 + 2.82405i −5.18398 + 0.355390i −7.95050 0.888573i −1.23846 0.715028i −0.188082 14.6957i −23.8818 + 13.7882i 3.76017 22.3128i 26.7474 3.68468i 2.21411 3.38499i
23.7 0.664105 2.74936i −5.00415 + 1.39947i −7.11793 3.65173i −14.2911 8.25096i 0.524375 + 14.6876i 19.2620 11.1209i −14.7670 + 17.1446i 23.0829 14.0063i −32.1756 + 33.8118i
23.8 0.823719 + 2.70582i 2.72340 + 4.42528i −6.64298 + 4.45768i −4.71466 2.72201i −9.73072 + 11.0142i 20.9358 12.0873i −17.5336 14.3029i −12.1662 + 24.1036i 3.48173 14.9992i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.12 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
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.4.h.b 24
3.b odd 2 1 108.4.h.b 24
4.b odd 2 1 inner 36.4.h.b 24
9.c even 3 1 108.4.h.b 24
9.c even 3 1 324.4.b.c 24
9.d odd 6 1 inner 36.4.h.b 24
9.d odd 6 1 324.4.b.c 24
12.b even 2 1 108.4.h.b 24
36.f odd 6 1 108.4.h.b 24
36.f odd 6 1 324.4.b.c 24
36.h even 6 1 inner 36.4.h.b 24
36.h even 6 1 324.4.b.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.b 24 1.a even 1 1 trivial
36.4.h.b 24 4.b odd 2 1 inner
36.4.h.b 24 9.d odd 6 1 inner
36.4.h.b 24 36.h even 6 1 inner
108.4.h.b 24 3.b odd 2 1
108.4.h.b 24 9.c even 3 1
108.4.h.b 24 12.b even 2 1
108.4.h.b 24 36.f odd 6 1
324.4.b.c 24 9.c even 3 1
324.4.b.c 24 9.d odd 6 1
324.4.b.c 24 36.f odd 6 1
324.4.b.c 24 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database