# Properties

 Label 36.6.h.a Level 36 Weight 6 Character orbit 36.h Analytic conductor 5.774 Analytic rank 0 Dimension 56 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.77381751327$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 3q^{2} - q^{4} - 6q^{5} - 27q^{6} + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 3q^{2} - q^{4} - 6q^{5} - 27q^{6} + 18q^{9} - 68q^{10} - 486q^{12} - 2q^{13} - 1518q^{14} - q^{16} + 1992q^{18} - 1242q^{20} + 330q^{21} + 63q^{22} + 2235q^{24} + 12498q^{25} - 2052q^{28} - 11946q^{29} - 6882q^{30} - 7233q^{32} + 17040q^{33} + 6361q^{34} + 6399q^{36} - 8q^{37} - 14877q^{38} - 1526q^{40} - 43536q^{41} + 18564q^{42} - 42q^{45} - 26880q^{46} - 5931q^{48} + 38414q^{49} + 38631q^{50} + 24988q^{52} + 37587q^{54} + 21186q^{56} - 90786q^{57} - 3314q^{58} + 60930q^{60} - 2q^{61} - 106342q^{64} + 35970q^{65} - 47838q^{66} + 31413q^{68} - 1854q^{69} + 10524q^{70} - 130941q^{72} + 53620q^{73} - 20406q^{74} + 26193q^{76} + 26178q^{77} - 96684q^{78} - 14790q^{81} - 151286q^{82} + 141630q^{84} + 6248q^{85} + 279237q^{86} - 122541q^{88} + 235278q^{90} + 435804q^{92} + 71838q^{93} + 63480q^{94} - 37476q^{96} - 58148q^{97} + O(q^{100})$$

## 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 −5.58478 + 0.900118i −10.9673 + 11.0778i 30.3796 10.0539i −78.5398 + 45.3450i 51.2786 71.7391i 69.6792 + 40.2293i −160.614 + 83.4942i −2.43645 242.988i 397.812 323.937i
11.2 −5.50860 + 1.28660i −13.2152 8.26786i 28.6893 14.1747i 38.3603 22.1474i 83.4349 + 28.5416i −102.940 59.4323i −139.801 + 114.994i 106.285 + 218.523i −182.817 + 171.355i
11.3 −5.49437 1.34609i 13.5204 + 7.75878i 28.3761 + 14.7918i −27.3241 + 15.7756i −63.8421 60.8292i 2.81745 + 1.62665i −135.998 119.468i 122.603 + 209.804i 171.364 49.8961i
11.4 −5.35087 1.83526i 6.54818 14.1464i 25.2636 + 19.6405i 36.0599 20.8192i −61.0009 + 63.6781i 117.288 + 67.7164i −99.1369 151.459i −157.243 185.267i −231.160 + 45.2214i
11.5 −4.78727 + 3.01364i 13.9499 6.95697i 13.8359 28.8542i −0.143132 + 0.0826372i −45.8163 + 75.3450i −192.634 111.217i 20.7202 + 179.830i 146.201 194.098i 0.436172 0.826955i
11.6 −4.76722 + 3.04525i 2.08778 + 15.4480i 13.4528 29.0348i 84.3255 48.6853i −56.9960 67.2863i 87.9661 + 50.7872i 24.2857 + 179.383i −234.282 + 64.5040i −253.739 + 488.886i
11.7 −4.26482 3.71636i −6.54818 + 14.1464i 4.37736 + 31.6992i 36.0599 20.8192i 80.5000 35.9966i −117.288 67.7164i 99.1369 151.459i −157.243 185.267i −231.160 45.2214i
11.8 −3.91293 4.08522i −13.5204 7.75878i −1.37798 + 31.9703i −27.3241 + 15.7756i 21.2081 + 85.5933i −2.81745 1.62665i 135.998 119.468i 122.603 + 209.804i 171.364 + 49.8961i
11.9 −3.75408 + 4.23165i −1.51698 15.5145i −3.81374 31.7719i −52.1793 + 30.1257i 71.3467 + 51.8233i 185.316 + 106.992i 148.765 + 103.136i −238.398 + 47.0702i 68.4038 333.899i
11.10 −2.01287 5.28662i 10.9673 11.0778i −23.8967 + 21.2825i −78.5398 + 45.3450i −80.6400 35.6818i −69.6792 40.2293i 160.614 + 83.4942i −2.43645 242.988i 397.812 + 323.937i
11.11 −1.90579 + 5.32616i 9.73802 + 12.1726i −24.7359 20.3011i −51.8636 + 29.9435i −83.3916 + 28.6679i −33.8827 19.5622i 155.268 93.0578i −53.3420 + 237.073i −60.6425 333.300i
11.12 −1.78216 + 5.36879i −15.4584 + 2.00950i −25.6478 19.1360i 12.8631 7.42651i 16.7607 86.5741i −15.2989 8.83282i 148.446 103.595i 234.924 62.1273i 16.9473 + 82.2944i
11.13 −1.64008 5.41389i 13.2152 + 8.26786i −26.6203 + 17.7584i 38.3603 22.1474i 23.0872 85.1057i 102.940 + 59.4323i 139.801 + 114.994i 106.285 + 218.523i −182.817 171.355i
11.14 −0.0499164 + 5.65663i 14.1974 6.43687i −31.9950 0.564717i 70.1957 40.5275i 35.7024 + 80.6309i 89.9985 + 51.9607i 4.79147 180.956i 160.133 182.774i 225.745 + 399.094i
11.15 0.216257 5.65272i −13.9499 + 6.95697i −31.9065 2.44488i −0.143132 + 0.0826372i 36.3090 + 80.3595i 192.634 + 111.217i −20.7202 + 179.830i 146.201 194.098i 0.436172 + 0.826955i
11.16 0.253656 5.65116i −2.08778 15.4480i −31.8713 2.86690i 84.3255 48.6853i −87.8289 + 7.87989i −87.9661 50.7872i −24.2857 + 179.383i −234.282 + 64.5040i −253.739 488.886i
11.17 1.72594 + 5.38713i −3.10718 15.2756i −26.0423 + 18.5957i −20.1454 + 11.6310i 76.9290 43.1036i −156.832 90.5473i −145.125 108.198i −223.691 + 94.9283i −97.4272 88.4515i
11.18 1.78768 5.36696i 1.51698 + 15.5145i −25.6084 19.1888i −52.1793 + 30.1257i 85.9773 + 19.5933i −185.316 106.992i −148.765 + 103.136i −238.398 + 47.0702i 68.4038 + 333.899i
11.19 2.66122 + 4.99179i −5.80725 + 14.4664i −17.8359 + 26.5684i 3.68052 2.12495i −87.6674 + 9.50956i 13.9322 + 8.04377i −180.089 18.3284i −175.552 168.020i 20.4019 + 12.7174i
11.20 3.65969 4.31354i −9.73802 12.1726i −5.21330 31.5725i −51.8636 + 29.9435i −88.1450 2.54244i 33.8827 + 19.5622i −155.268 93.0578i −53.3420 + 237.073i −60.6425 + 333.300i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.28 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 36.6.h.a 56
3.b odd 2 1 108.6.h.a 56
4.b odd 2 1 inner 36.6.h.a 56
9.c even 3 1 108.6.h.a 56
9.d odd 6 1 inner 36.6.h.a 56
12.b even 2 1 108.6.h.a 56
36.f odd 6 1 108.6.h.a 56
36.h even 6 1 inner 36.6.h.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.h.a 56 1.a even 1 1 trivial
36.6.h.a 56 4.b odd 2 1 inner
36.6.h.a 56 9.d odd 6 1 inner
36.6.h.a 56 36.h even 6 1 inner
108.6.h.a 56 3.b odd 2 1
108.6.h.a 56 9.c even 3 1
108.6.h.a 56 12.b even 2 1
108.6.h.a 56 36.f odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database