Properties

Label 432.2.bg.a
Level 432
Weight 2
Character orbit 432.bg
Analytic conductor 3.450
Analytic rank 0
Dimension 840
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 432.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(840\)
Relative dimension: \(70\) over \(\Q(\zeta_{36})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840q - 12q^{2} - 12q^{3} - 12q^{4} - 12q^{5} - 12q^{6} - 6q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 840q - 12q^{2} - 12q^{3} - 12q^{4} - 12q^{5} - 12q^{6} - 6q^{8} - 6q^{10} - 12q^{11} - 30q^{12} - 12q^{13} - 12q^{14} - 24q^{15} - 12q^{16} - 12q^{17} - 12q^{18} - 6q^{19} - 54q^{20} - 12q^{21} - 12q^{22} + 18q^{24} - 24q^{26} - 12q^{27} - 24q^{28} - 12q^{29} - 12q^{30} - 24q^{31} - 12q^{32} - 24q^{33} - 6q^{35} - 12q^{36} - 6q^{37} - 12q^{38} - 12q^{40} - 72q^{42} - 12q^{43} - 6q^{44} - 12q^{45} - 6q^{46} - 24q^{47} - 12q^{48} - 24q^{49} - 168q^{50} + 6q^{51} - 12q^{52} - 24q^{53} + 156q^{54} - 54q^{56} + 42q^{58} - 48q^{59} - 72q^{60} - 12q^{61} + 60q^{62} - 24q^{63} - 6q^{64} - 24q^{65} - 198q^{66} - 12q^{67} - 66q^{68} - 12q^{69} - 54q^{70} - 144q^{72} - 96q^{74} - 96q^{75} - 12q^{76} - 12q^{77} - 90q^{78} - 24q^{79} - 204q^{80} - 24q^{81} - 24q^{82} + 48q^{83} + 36q^{84} + 18q^{85} - 72q^{86} - 12q^{88} - 66q^{90} - 6q^{91} + 102q^{92} - 12q^{93} - 12q^{94} - 24q^{95} - 120q^{96} - 24q^{97} - 6q^{98} - 12q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.41419 + 0.00816009i 0.757020 1.55786i 1.99987 0.0230798i 2.92668 0.256052i −1.05786 + 2.20928i 1.42579 3.91732i −2.82800 + 0.0489584i −1.85384 2.35866i −4.13680 + 0.385988i
13.2 −1.41151 0.0874761i 0.0727719 + 1.73052i 1.98470 + 0.246946i −0.723668 + 0.0633127i 0.0486614 2.44901i 0.398828 1.09577i −2.77981 0.522179i −2.98941 + 0.251867i 1.02700 0.0260626i
13.3 −1.40392 0.170309i −1.60100 0.660910i 1.94199 + 0.478202i 0.199885 0.0174877i 2.13512 + 1.20053i −0.807683 + 2.21909i −2.64496 1.00210i 2.12640 + 2.11623i −0.283601 0.00949098i
13.4 −1.39069 0.256893i −0.791539 1.54061i 1.86801 + 0.714515i −2.39012 + 0.209108i 0.705011 + 2.34584i 0.307660 0.845290i −2.41426 1.47355i −1.74693 + 2.43890i 3.37762 + 0.323202i
13.5 −1.37047 0.349003i 1.72507 + 0.155359i 1.75639 + 0.956599i 1.88606 0.165009i −2.30994 0.814971i −0.627115 + 1.72298i −2.07323 1.92398i 2.95173 + 0.536012i −2.64238 0.432100i
13.6 −1.36057 + 0.385801i −1.47892 + 0.901550i 1.70231 1.04982i 2.53117 0.221449i 1.66436 1.79719i −0.599847 + 1.64807i −1.91110 + 2.08511i 1.37441 2.66664i −3.35841 + 1.27783i
13.7 −1.35642 + 0.400143i 1.26410 1.18408i 1.67977 1.08553i −2.21064 + 0.193406i −1.24086 + 2.11193i −0.170100 + 0.467346i −1.84412 + 2.14458i 0.195921 2.99360i 2.92117 1.14691i
13.8 −1.31559 + 0.518866i −1.54765 + 0.777673i 1.46156 1.36523i −4.10207 + 0.358885i 1.63257 1.82612i 0.659902 1.81306i −1.21444 + 2.55443i 1.79045 2.40713i 5.21044 2.60057i
13.9 −1.31389 0.523159i 1.58196 + 0.705273i 1.45261 + 1.37475i −3.33491 + 0.291766i −1.70955 1.75427i 1.51056 4.15024i −1.18936 2.56621i 2.00518 + 2.23142i 4.53434 + 1.36134i
13.10 −1.30739 + 0.539193i 1.39818 + 1.02229i 1.41854 1.40987i 3.17016 0.277353i −2.37919 0.582643i 0.294648 0.809538i −1.09439 + 2.60812i 0.909840 + 2.85870i −3.99510 + 2.07194i
13.11 −1.26954 + 0.623105i −0.0426757 1.73152i 1.22348 1.58212i 1.79177 0.156760i 1.13310 + 2.17165i −1.76857 + 4.85911i −0.567437 + 2.77092i −2.99636 + 0.147788i −2.17706 + 1.31548i
13.12 −1.19582 0.754986i −0.692004 + 1.58781i 0.859992 + 1.80566i 2.72931 0.238784i 2.02629 1.37628i 0.552395 1.51769i 0.334850 2.80854i −2.04226 2.19754i −3.44405 1.77505i
13.13 −1.19078 0.762913i −1.38465 + 1.04055i 0.835928 + 1.81693i −2.45604 + 0.214875i 2.44267 0.182707i −1.20765 + 3.31798i 0.390748 2.80131i 0.834499 2.88160i 3.08854 + 1.61787i
13.14 −1.15556 + 0.815280i −1.32179 1.11931i 0.670638 1.88421i 0.762654 0.0667236i 2.43997 + 0.215802i 1.03298 2.83809i 0.761196 + 2.72407i 0.494277 + 2.95900i −0.826895 + 0.698880i
13.15 −1.09334 0.896994i 1.02688 1.39482i 0.390802 + 1.96145i −3.34554 + 0.292697i −2.37388 + 0.603909i −0.981004 + 2.69529i 1.33213 2.49508i −0.891032 2.86462i 3.92037 + 2.68091i
13.16 −1.09259 + 0.897911i 1.51775 + 0.834518i 0.387511 1.96210i −2.33873 + 0.204612i −2.40761 + 0.451022i −0.349949 + 0.961477i 1.33840 + 2.49172i 1.60716 + 2.53319i 2.37155 2.32353i
13.17 −1.07216 0.922209i −1.54249 0.787864i 0.299063 + 1.97751i 4.29797 0.376024i 0.927222 + 2.26721i −0.248889 + 0.683816i 1.50304 2.39601i 1.75854 + 2.43054i −4.95489 3.56047i
13.18 −0.945342 1.05182i 1.02940 + 1.39296i −0.212657 + 1.98866i 1.13926 0.0996725i 0.492003 2.39957i −1.44119 + 3.95964i 2.29275 1.65629i −0.880655 + 2.86783i −1.18183 1.10408i
13.19 −0.881475 1.10589i −0.338579 1.69864i −0.446003 + 1.94964i 0.423518 0.0370531i −1.58006 + 1.87174i 0.916962 2.51933i 2.54923 1.22532i −2.77073 + 1.15024i −0.414298 0.435705i
13.20 −0.854260 1.12705i 1.64279 0.548844i −0.540481 + 1.92559i 0.365664 0.0319915i −2.02195 1.38265i 0.804838 2.21127i 2.63194 1.03580i 2.39754 1.80328i −0.348428 0.384793i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
27.e even 9 1 inner
432.bg even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.bg.a 840
16.e even 4 1 inner 432.2.bg.a 840
27.e even 9 1 inner 432.2.bg.a 840
432.bg even 36 1 inner 432.2.bg.a 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.bg.a 840 1.a even 1 1 trivial
432.2.bg.a 840 16.e even 4 1 inner
432.2.bg.a 840 27.e even 9 1 inner
432.2.bg.a 840 432.bg even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(432, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database