# Properties

 Label 756.2.bt.a Level 756 Weight 2 Character orbit 756.bt Analytic conductor 6.037 Analytic rank 0 Dimension 840 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.bt (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$840$$ Relative dimension: $$140$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$840q - 3q^{2} - 3q^{4} - 18q^{5} - 6q^{8} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$840q - 3q^{2} - 3q^{4} - 18q^{5} - 6q^{8} - 6q^{9} - 9q^{12} - 21q^{14} - 3q^{16} - 3q^{18} - 12q^{21} - 12q^{22} - 9q^{24} - 6q^{25} - 18q^{26} - 12q^{28} - 36q^{29} - 39q^{30} + 27q^{32} - 18q^{33} + 18q^{34} + 18q^{36} + 6q^{37} - 99q^{38} + 36q^{40} + 9q^{42} + 3q^{44} - 18q^{45} + 3q^{46} - 12q^{49} + 3q^{50} - 9q^{52} - 12q^{53} - 135q^{54} + 15q^{56} - 42q^{57} - 3q^{58} - 33q^{60} - 18q^{61} - 99q^{62} - 6q^{64} + 18q^{65} - 9q^{66} - 54q^{68} + 72q^{69} - 36q^{70} - 111q^{72} - 18q^{73} + 93q^{74} - 36q^{76} - 36q^{77} + 6q^{78} - 18q^{80} - 30q^{81} - 18q^{82} + 84q^{84} + 6q^{85} + 135q^{86} - 51q^{88} + 81q^{90} + 48q^{92} - 6q^{93} - 9q^{94} - 9q^{96} - 117q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1 −1.41414 + 0.0147396i −0.326297 + 1.70104i 1.99957 0.0416875i −1.34679 0.237475i 0.436356 2.41031i 1.86287 + 1.87875i −2.82704 + 0.0884245i −2.78706 1.11009i 1.90804 + 0.315971i
103.2 −1.41407 + 0.0203430i 1.63923 0.559403i 1.99917 0.0575327i −1.49077 0.262863i −2.30660 + 0.824380i 1.66611 2.05525i −2.82579 + 0.122024i 2.37414 1.83398i 2.11340 + 0.341379i
103.3 −1.41387 + 0.0311086i 0.755283 1.55870i 1.99806 0.0879671i 3.35278 + 0.591186i −1.01938 + 2.22730i 2.13149 + 1.56741i −2.82227 + 0.186531i −1.85910 2.35452i −4.75879 0.731561i
103.4 −1.41356 + 0.0430901i −1.54969 + 0.773606i 1.99629 0.121821i −0.175857 0.0310083i 2.15724 1.16031i −2.42990 + 1.04671i −2.81662 + 0.258221i 1.80307 2.39770i 0.249920 + 0.0362544i
103.5 −1.41121 + 0.0920983i −0.696355 1.58590i 1.98304 0.259940i 0.824097 + 0.145310i 1.12876 + 2.17391i −2.46179 + 0.969313i −2.77454 + 0.549465i −2.03018 + 2.20870i −1.17636 0.129166i
103.6 −1.40532 + 0.158328i 1.69106 0.374584i 1.94986 0.445005i 3.97187 + 0.700348i −2.31718 + 0.794154i −2.15958 1.52847i −2.66973 + 0.934094i 2.71937 1.26689i −5.69265 0.355355i
103.7 −1.39958 + 0.202941i −1.70851 0.284574i 1.91763 0.568062i 2.54587 + 0.448906i 2.44895 + 0.0515557i 2.36953 1.17699i −2.56859 + 1.18421i 2.83804 + 0.972395i −3.65425 0.111617i
103.8 −1.39646 + 0.223410i 0.897457 + 1.48141i 1.90018 0.623965i −1.37715 0.242829i −1.58422 1.86822i −1.27459 + 2.31850i −2.51411 + 1.29586i −1.38914 + 2.65900i 1.97738 + 0.0314303i
103.9 −1.39221 + 0.248524i −1.49487 0.874854i 1.87647 0.691994i −2.90630 0.512459i 2.29859 + 0.846466i −1.25872 2.32715i −2.44046 + 1.42975i 1.46926 + 2.61558i 4.17352 0.00883825i
103.10 −1.36742 0.360769i −1.03034 + 1.39226i 1.73969 + 0.986648i 1.98405 + 0.349842i 1.91120 1.53209i 1.38264 2.25573i −2.02294 1.97679i −0.876779 2.86902i −2.58683 1.19417i
103.11 −1.36642 0.364532i −0.0919920 1.72961i 1.73423 + 0.996212i −2.28806 0.403446i −0.504797 + 2.39691i 2.63964 0.179668i −2.00655 1.99343i −2.98307 + 0.318220i 2.97939 + 1.38535i
103.12 −1.35942 0.389832i 1.54409 + 0.784711i 1.69606 + 1.05989i 0.874454 + 0.154190i −1.79317 1.66869i 2.54554 + 0.721260i −1.89249 2.10202i 1.76846 + 2.42334i −1.12865 0.550499i
103.13 −1.34560 0.435164i −1.53529 0.801804i 1.62126 + 1.17111i −2.59686 0.457897i 1.71696 + 1.74701i 0.683168 + 2.55603i −1.67194 2.28136i 1.71422 + 2.46200i 3.29507 + 1.74621i
103.14 −1.34481 0.437584i 1.72493 0.156906i 1.61704 + 1.17694i −4.40150 0.776103i −2.38837 0.543792i −2.15811 + 1.53054i −1.65961 2.29035i 2.95076 0.541305i 5.57958 + 2.96974i
103.15 −1.33467 + 0.467620i −0.284726 + 1.70849i 1.56266 1.24823i 1.73945 + 0.306711i −0.418909 2.41340i −0.667920 2.56006i −1.50193 + 2.39671i −2.83786 0.972903i −2.46500 + 0.404043i
103.16 −1.31489 + 0.520649i 1.19189 + 1.25674i 1.45785 1.36919i −3.13702 0.553141i −2.22152 1.03191i 0.170788 2.64023i −1.20404 + 2.55935i −0.158782 + 2.99580i 4.41281 0.905969i
103.17 −1.30594 0.542696i 0.375808 + 1.69079i 1.41096 + 1.41746i 3.92095 + 0.691370i 0.426801 2.41202i −1.73044 + 2.00139i −1.07338 2.61684i −2.71754 + 1.27083i −4.74533 3.03077i
103.18 −1.28497 0.590645i −0.974544 1.43187i 1.30228 + 1.51792i 2.54084 + 0.448019i 0.406527 + 2.41552i −0.471692 2.60336i −0.776834 2.71966i −1.10053 + 2.79085i −3.00028 2.07643i
103.19 −1.27998 0.601365i −1.63693 + 0.566086i 1.27672 + 1.53948i −1.53946 0.271447i 2.43567 + 0.259812i −1.93149 1.80814i −0.708397 2.73828i 2.35909 1.85329i 1.80724 + 1.27322i
103.20 −1.27697 + 0.607749i 1.12163 1.31983i 1.26128 1.55215i −2.50783 0.442198i −0.630164 + 2.36704i 1.61158 + 2.09829i −0.667298 + 2.74858i −0.483880 2.96072i 3.47116 0.959460i
See next 80 embeddings (of 840 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 619.140 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
189.x odd 18 1 inner
756.bt even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bt.a 840
4.b odd 2 1 inner 756.2.bt.a 840
7.d odd 6 1 756.2.cd.a yes 840
27.e even 9 1 756.2.cd.a yes 840
28.f even 6 1 756.2.cd.a yes 840
108.j odd 18 1 756.2.cd.a yes 840
189.x odd 18 1 inner 756.2.bt.a 840
756.bt even 18 1 inner 756.2.bt.a 840

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bt.a 840 1.a even 1 1 trivial
756.2.bt.a 840 4.b odd 2 1 inner
756.2.bt.a 840 189.x odd 18 1 inner
756.2.bt.a 840 756.bt even 18 1 inner
756.2.cd.a yes 840 7.d odd 6 1
756.2.cd.a yes 840 27.e even 9 1
756.2.cd.a yes 840 28.f even 6 1
756.2.cd.a yes 840 108.j odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database