# Properties

 Label 324.2.p.a Level $324$ Weight $2$ Character orbit 324.p Analytic conductor $2.587$ Analytic rank $0$ Dimension $936$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 324.p (of order $$54$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$936$$ Relative dimension: $$52$$ over $$\Q(\zeta_{54})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$936q - 18q^{2} - 18q^{4} - 36q^{5} - 18q^{6} - 18q^{8} - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$936q - 18q^{2} - 18q^{4} - 36q^{5} - 18q^{6} - 18q^{8} - 36q^{9} - 18q^{10} - 18q^{12} - 36q^{13} - 18q^{14} - 18q^{16} - 36q^{17} - 18q^{18} - 18q^{20} - 36q^{21} - 18q^{22} - 18q^{24} - 36q^{25} - 27q^{26} - 9q^{28} - 36q^{29} - 18q^{30} - 18q^{32} - 36q^{33} - 18q^{34} - 18q^{36} - 36q^{37} - 18q^{38} - 18q^{40} - 36q^{41} - 63q^{42} - 90q^{44} - 36q^{45} - 18q^{46} - 117q^{48} - 36q^{49} - 135q^{50} - 18q^{52} - 54q^{53} - 144q^{54} - 144q^{56} - 36q^{57} - 18q^{58} - 135q^{60} - 36q^{61} - 117q^{62} - 18q^{64} - 36q^{65} - 90q^{66} - 63q^{68} - 36q^{69} - 18q^{70} - 18q^{72} - 36q^{73} - 18q^{74} - 18q^{76} - 36q^{77} + 9q^{78} - 36q^{81} - 36q^{82} - 45q^{84} - 36q^{85} - 18q^{86} - 18q^{88} - 54q^{89} + 45q^{90} + 72q^{92} - 144q^{93} - 18q^{94} + 99q^{96} - 36q^{97} + 153q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41337 + 0.0488211i −1.16649 + 1.28035i 1.99523 0.138005i 0.190968 1.63383i 1.58618 1.86656i 1.05815 + 2.10695i −2.81327 + 0.292461i −0.278579 2.98704i −0.190143 + 2.31854i
11.2 −1.41238 + 0.0718931i −1.73203 0.00891691i 1.98966 0.203082i −0.493585 + 4.22289i 2.44693 0.111927i −0.262610 0.522900i −2.79557 + 0.429872i 2.99984 + 0.0308887i 0.393535 5.99983i
11.3 −1.41031 0.105029i −0.330952 1.70014i 1.97794 + 0.296246i 0.00377236 0.0322746i 0.288181 + 2.43248i −0.446224 0.888506i −2.75839 0.625539i −2.78094 + 1.12533i −0.00870995 + 0.0451209i
11.4 −1.36952 0.352718i 1.60792 + 0.643879i 1.75118 + 0.966111i −0.221174 + 1.89226i −1.97498 1.44895i 0.167545 + 0.333610i −2.05751 1.94078i 2.17084 + 2.07062i 0.970339 2.51349i
11.5 −1.31425 + 0.522253i 1.25530 + 1.19341i 1.45450 1.37274i 0.241988 2.07034i −2.27303 0.912860i 0.716180 + 1.42603i −1.19466 + 2.56374i 0.151532 + 2.99617i 0.763210 + 2.84733i
11.6 −1.31106 0.530208i 1.52671 0.818025i 1.43776 + 1.39027i 0.412015 3.52501i −2.43533 + 0.263006i −0.153752 0.306145i −1.14785 2.58504i 1.66167 2.49777i −2.40917 + 4.40305i
11.7 −1.30907 + 0.535100i 1.04386 1.38215i 1.42734 1.40097i −0.309185 + 2.64525i −0.626899 + 2.36791i 2.01774 + 4.01764i −1.11883 + 2.59773i −0.820704 2.88556i −1.01073 3.62826i
11.8 −1.28696 + 0.586296i −1.69911 0.336182i 1.31251 1.50908i 0.366702 3.13733i 2.38379 0.563531i −1.91508 3.81323i −0.804381 + 2.71164i 2.77396 + 1.14242i 1.36748 + 4.25261i
11.9 −1.24875 0.663801i −1.42257 0.988080i 1.11874 + 1.65784i 0.180183 1.54156i 1.12054 + 2.17816i 1.41761 + 2.82269i −0.296543 2.81284i 1.04740 + 2.81122i −1.24830 + 1.80542i
11.10 −1.22866 + 0.700279i 1.64246 0.549842i 1.01922 1.72081i −0.0995817 + 0.851976i −1.63298 + 1.82575i −2.18215 4.34501i −0.0472259 + 2.82803i 2.39535 1.80619i −0.474269 1.11652i
11.11 −1.18334 0.774405i −1.00564 + 1.41021i 0.800594 + 1.83277i −0.0544195 + 0.465589i 2.28209 0.889987i −1.91763 3.81831i 0.471930 2.78878i −0.977379 2.83632i 0.424951 0.508808i
11.12 −1.09593 0.893834i 0.138112 + 1.72654i 0.402121 + 1.95916i −0.283509 + 2.42558i 1.39188 2.01561i 1.43973 + 2.86675i 1.31047 2.50653i −2.96185 + 0.476909i 2.47877 2.40485i
11.13 −1.05140 + 0.945816i −0.445336 + 1.67382i 0.210865 1.98885i −0.155074 + 1.32674i −1.11490 2.18105i 0.202244 + 0.402700i 1.65939 + 2.29051i −2.60335 1.49083i −1.09181 1.54160i
11.14 −0.969113 + 1.02996i 0.112425 1.72840i −0.121640 1.99630i 0.460520 3.94000i 1.67123 + 1.79081i 0.896764 + 1.78560i 2.17399 + 1.80935i −2.97472 0.388631i 3.61175 + 4.29262i
11.15 −0.956045 1.04210i −0.138112 1.72654i −0.171958 + 1.99259i −0.283509 + 2.42558i −1.66719 + 1.79457i −1.43973 2.86675i 2.24089 1.72581i −2.96185 + 0.476909i 2.79875 2.02351i
11.16 −0.870676 + 1.11442i −0.828844 1.52086i −0.483846 1.94059i −0.291150 + 2.49095i 2.41653 + 0.400499i −0.659322 1.31282i 2.58390 + 1.15042i −1.62603 + 2.52111i −2.52246 2.49327i
11.17 −0.841900 1.13631i 1.00564 1.41021i −0.582409 + 1.91332i −0.0544195 + 0.465589i −2.44908 + 0.0445349i 1.91763 + 3.81831i 2.66446 0.949027i −0.977379 2.83632i 0.574870 0.330141i
11.18 −0.735286 1.20804i 1.42257 + 0.988080i −0.918708 + 1.77651i 0.180183 1.54156i 0.147643 2.44504i −1.41761 2.82269i 2.82160 0.196407i 1.04740 + 2.81122i −1.99475 + 0.915823i
11.19 −0.607055 + 1.27730i −1.65046 + 0.525329i −1.26297 1.55078i −0.0392046 + 0.335417i 0.330922 2.42703i −0.385340 0.767275i 2.74749 0.671779i 2.44806 1.73407i −0.404627 0.253692i
11.20 −0.605543 1.27801i −1.52671 + 0.818025i −1.26664 + 1.54778i 0.412015 3.52501i 1.96993 + 1.45580i 0.153752 + 0.306145i 2.74509 + 0.681528i 1.66167 2.49777i −4.75450 + 1.60799i
See next 80 embeddings (of 936 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.52 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
81.h odd 54 1 inner
324.p even 54 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.p.a 936
3.b odd 2 1 972.2.p.a 936
4.b odd 2 1 inner 324.2.p.a 936
12.b even 2 1 972.2.p.a 936
81.g even 27 1 972.2.p.a 936
81.h odd 54 1 inner 324.2.p.a 936
324.n odd 54 1 972.2.p.a 936
324.p even 54 1 inner 324.2.p.a 936

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.p.a 936 1.a even 1 1 trivial
324.2.p.a 936 4.b odd 2 1 inner
324.2.p.a 936 81.h odd 54 1 inner
324.2.p.a 936 324.p even 54 1 inner
972.2.p.a 936 3.b odd 2 1
972.2.p.a 936 12.b even 2 1
972.2.p.a 936 81.g even 27 1
972.2.p.a 936 324.n odd 54 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(324, [\chi])$$.