Properties

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

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Newspace parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.bs (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.

\(\operatorname{Tr}(f)(q) = \) \( 840q - 3q^{2} - 3q^{4} - 6q^{5} - 12q^{6} - 18q^{8} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 840q - 3q^{2} - 3q^{4} - 6q^{5} - 12q^{6} - 18q^{8} - 6q^{9} - 6q^{10} - 3q^{12} - 24q^{13} - 21q^{14} - 3q^{16} - 3q^{18} - 12q^{20} - 12q^{21} - 12q^{22} - 15q^{24} - 6q^{25} - 12q^{28} - 12q^{29} + 33q^{30} + 27q^{32} - 6q^{33} - 6q^{34} - 42q^{36} + 6q^{37} - 9q^{38} + 12q^{40} - 24q^{41} - 21q^{42} - 9q^{44} - 30q^{45} + 3q^{46} - 78q^{48} - 12q^{49} - 27q^{50} - 3q^{52} + 39q^{54} - 27q^{56} - 6q^{57} - 3q^{58} + 27q^{60} - 6q^{61} - 117q^{62} - 6q^{64} + 18q^{65} - 3q^{66} + 60q^{68} - 48q^{69} + 24q^{70} - 63q^{72} + 6q^{73} - 75q^{74} + 12q^{77} + 6q^{78} + 18q^{81} - 6q^{82} + 132q^{84} + 6q^{85} - 81q^{86} + 45q^{88} - 39q^{90} + 48q^{92} - 6q^{93} - 3q^{94} - 111q^{96} - 24q^{97} - 9q^{98} + O(q^{100}) \)

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 −1.41413 + 0.0149120i 0.154673 + 1.72513i 1.99956 0.0421753i 3.70389 0.653095i −0.244453 2.43726i −2.38269 1.15013i −2.82701 + 0.0894589i −2.95215 + 0.533661i −5.22806 + 0.978797i
11.2 −1.41382 + 0.0335363i −1.23240 1.21704i 1.99775 0.0948283i −1.44666 + 0.255085i 1.78320 + 1.67934i 1.11827 2.39781i −2.82127 + 0.201067i 0.0376115 + 2.99976i 2.03676 0.409160i
11.3 −1.41197 + 0.0796147i −1.59653 + 0.671641i 1.98732 0.224827i 0.781464 0.137793i 2.20078 1.07544i 2.60906 + 0.439114i −2.78814 + 0.475670i 2.09780 2.14459i −1.09243 + 0.256776i
11.4 −1.41190 + 0.0808315i −1.10309 + 1.33536i 1.98693 0.228252i −1.49610 + 0.263803i 1.44951 1.97457i −1.99141 + 1.74193i −2.78690 + 0.482877i −0.566395 2.94605i 2.09102 0.493396i
11.5 −1.41111 + 0.0937031i 1.68184 0.414019i 1.98244 0.264450i −2.78746 + 0.491505i −2.33446 + 0.741819i 2.41910 + 1.07144i −2.77265 + 0.558927i 2.65718 1.39263i 3.88735 0.954760i
11.6 −1.40534 + 0.158152i 0.564160 + 1.63760i 1.94998 0.444515i −3.91223 + 0.689832i −1.05183 2.21216i −0.574615 2.58260i −2.67008 + 0.933089i −2.36345 + 1.84773i 5.38892 1.58818i
11.7 −1.40267 0.180347i 0.0171663 + 1.73197i 1.93495 + 0.505934i 1.75070 0.308695i 0.288276 2.43247i 2.14695 + 1.54616i −2.62285 1.05862i −2.99941 + 0.0594627i −2.51132 + 0.117263i
11.8 −1.39274 0.245534i −1.54124 0.790313i 1.87943 + 0.683928i 1.51908 0.267855i 1.95248 + 1.47912i −1.79492 + 1.94378i −2.44962 1.41399i 1.75081 + 2.43612i −2.18145 6.52995e-5i
11.9 −1.39272 0.245617i 1.54124 + 0.790313i 1.87934 + 0.684153i 1.51908 0.267855i −1.95240 1.47924i 1.79492 1.94378i −2.44936 1.41443i 1.75081 + 2.43612i −2.18145 6.52995e-5i
11.10 −1.38987 + 0.261245i 0.749350 1.56156i 1.86350 0.726195i −2.82395 + 0.497938i −0.633553 + 2.36614i −2.63326 0.256777i −2.40032 + 1.49615i −1.87695 2.34031i 3.79485 1.42981i
11.11 −1.37976 0.310270i −0.0171663 1.73197i 1.80747 + 0.856194i 1.75070 0.308695i −0.513691 + 2.39502i −2.14695 1.54616i −2.22821 1.74214i −2.99941 + 0.0594627i −2.51132 0.117263i
11.12 −1.35727 + 0.397255i −1.72583 + 0.146654i 1.68438 1.07837i 3.07035 0.541386i 2.28416 0.884644i −1.38755 2.25271i −1.85777 + 2.13276i 2.95699 0.506199i −3.95223 + 1.95452i
11.13 −1.35660 + 0.399529i 0.345844 1.69717i 1.68075 1.08401i 0.871331 0.153639i 0.208897 + 2.44057i 0.218427 + 2.63672i −1.84702 + 2.14208i −2.76078 1.17391i −1.12067 + 0.556550i
11.14 −1.35563 + 0.402825i 1.69073 0.376077i 1.67546 1.09216i 3.16927 0.558828i −2.14051 + 1.19089i −1.27230 + 2.31975i −1.83136 + 2.15549i 2.71713 1.27169i −4.07125 + 2.03423i
11.15 −1.32375 0.497675i −0.154673 1.72513i 1.50464 + 1.31760i 3.70389 0.653095i −0.653807 + 2.36062i 2.38269 + 1.15013i −1.33603 2.49299i −2.95215 + 0.533661i −5.22806 0.978797i
11.16 −1.31708 0.515067i 1.23240 + 1.21704i 1.46941 + 1.35677i −1.44666 + 0.255085i −0.996311 2.23771i −1.11827 + 2.39781i −1.23651 2.54383i 0.0376115 + 2.99976i 2.03676 + 0.409160i
11.17 −1.29959 0.557736i 1.59653 0.671641i 1.37786 + 1.44965i 0.781464 0.137793i −2.44943 0.0175830i −2.60906 0.439114i −0.982129 2.65244i 2.09780 2.14459i −1.09243 0.256776i
11.18 −1.29911 0.558856i 1.10309 1.33536i 1.37536 + 1.45203i −1.49610 + 0.263803i −2.17931 + 1.11831i 1.99141 1.74193i −0.975268 2.65497i −0.566395 2.94605i 2.09102 + 0.493396i
11.19 −1.29808 + 0.561242i −1.72841 0.112253i 1.37001 1.45707i −3.58626 + 0.632355i 2.30661 0.824343i 1.09900 + 2.40670i −0.960616 + 2.66030i 2.97480 + 0.388039i 4.30035 2.83361i
11.20 −1.29496 + 0.568405i 1.56875 + 0.734184i 1.35383 1.47212i −1.56831 + 0.276535i −2.44878 0.0590536i 2.42629 + 1.05504i −0.916395 + 2.67586i 1.92195 + 2.30350i 1.87371 1.24953i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 527.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
189.bf odd 18 1 inner
756.bs 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.bs.a 840
4.b odd 2 1 inner 756.2.bs.a 840
7.c even 3 1 756.2.ci.a yes 840
27.f odd 18 1 756.2.ci.a yes 840
28.g odd 6 1 756.2.ci.a yes 840
108.l even 18 1 756.2.ci.a yes 840
189.bf odd 18 1 inner 756.2.bs.a 840
756.bs even 18 1 inner 756.2.bs.a 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bs.a 840 1.a even 1 1 trivial
756.2.bs.a 840 4.b odd 2 1 inner
756.2.bs.a 840 189.bf odd 18 1 inner
756.2.bs.a 840 756.bs even 18 1 inner
756.2.ci.a yes 840 7.c even 3 1
756.2.ci.a yes 840 27.f odd 18 1
756.2.ci.a yes 840 28.g odd 6 1
756.2.ci.a yes 840 108.l even 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