Properties

Label 76.3.l.a
Level $76$
Weight $3$
Character orbit 76.l
Analytic conductor $2.071$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 108q - 6q^{2} - 12q^{5} + 12q^{6} - 3q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 108q - 6q^{2} - 12q^{5} + 12q^{6} - 3q^{8} - 9q^{10} - 3q^{12} - 36q^{13} - 63q^{14} - 48q^{16} - 12q^{17} - 12q^{18} + 18q^{20} + 6q^{21} - 18q^{22} + 72q^{24} - 12q^{25} + 69q^{26} - 216q^{28} - 12q^{29} - 270q^{30} - 261q^{32} - 6q^{33} - 120q^{34} - 165q^{36} - 24q^{37} + 240q^{38} + 330q^{40} - 168q^{41} + 153q^{42} + 57q^{44} - 6q^{45} + 132q^{46} + 549q^{48} + 120q^{49} + 114q^{50} + 249q^{52} - 36q^{53} + 51q^{54} - 306q^{56} - 12q^{57} - 84q^{58} + 576q^{60} - 276q^{61} + 432q^{62} + 207q^{64} - 126q^{65} + 648q^{66} + 234q^{68} - 294q^{69} + 459q^{70} + 498q^{72} + 276q^{73} + 459q^{74} - 582q^{76} - 468q^{77} - 903q^{78} + 57q^{80} - 270q^{81} - 321q^{82} - 621q^{84} + 900q^{85} - 456q^{86} - 699q^{88} + 348q^{89} - 1566q^{90} - 348q^{92} + 366q^{93} + 162q^{94} - 726q^{96} + 96q^{97} - 1659q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.96172 0.389448i −1.94869 + 0.343607i 3.69666 + 1.52797i 1.01207 0.849231i 3.95660 + 0.0848543i −7.34996 4.24350i −6.65673 4.43710i −4.77790 + 1.73901i −2.31613 + 1.27180i
23.2 −1.92177 + 0.553897i 4.90961 0.865697i 3.38640 2.12892i 2.37531 1.99312i −8.95563 + 4.38309i −8.41114 4.85617i −5.32867 + 5.96702i 14.8976 5.42229i −3.46081 + 5.14599i
23.3 −1.77891 + 0.914055i 0.408304 0.0719951i 2.32901 3.25203i −5.90028 + 4.95092i −0.660528 + 0.501285i 6.78454 + 3.91705i −1.17055 + 7.91390i −8.29570 + 3.01939i 5.97063 14.2004i
23.4 −1.75309 0.962633i 1.94869 0.343607i 2.14668 + 3.37517i 1.01207 0.849231i −3.74701 1.27350i 7.34996 + 4.24350i −0.514277 7.98345i −4.77790 + 1.73901i −2.59176 + 0.514526i
23.5 −1.57322 + 1.23490i −4.66468 + 0.822509i 0.950039 3.88554i 2.08183 1.74687i 6.32285 7.05441i −0.551331 0.318311i 3.30364 + 7.28601i 12.6255 4.59531i −1.11798 + 5.31906i
23.6 −1.11612 1.65960i −4.90961 + 0.865697i −1.50854 + 3.70463i 2.37531 1.99312i 6.91644 + 7.18176i 8.41114 + 4.85617i 7.83192 1.63125i 14.8976 5.42229i −5.95892 1.71749i
23.7 −0.831743 + 1.81885i 1.75796 0.309976i −2.61641 3.02563i 6.16490 5.17297i −0.898372 + 3.45528i 6.99959 + 4.04121i 7.67933 2.24230i −5.46290 + 1.98833i 4.28122 + 15.5156i
23.8 −0.775177 1.84366i −0.408304 + 0.0719951i −2.79820 + 2.85833i −5.90028 + 4.95092i 0.449243 + 0.696968i −6.78454 3.91705i 7.43891 + 2.94323i −8.29570 + 3.01939i 13.7016 + 7.04030i
23.9 −0.411377 1.95724i 4.66468 0.822509i −3.66154 + 1.61032i 2.08183 1.74687i −3.52879 8.79152i 0.551331 + 0.318311i 4.65805 + 6.50404i 12.6255 4.59531i −4.27545 3.35602i
23.10 0.0616288 + 1.99905i −2.53724 + 0.447383i −3.99240 + 0.246398i −1.56377 + 1.31216i −1.05071 5.04449i −3.64994 2.10730i −0.738609 7.96583i −2.21981 + 0.807946i −2.71944 3.04518i
23.11 0.156355 + 1.99388i 5.33798 0.941229i −3.95111 + 0.623505i −5.57927 + 4.68156i 2.71132 + 10.4961i 3.83135 + 2.21203i −1.86097 7.78054i 19.1508 6.97034i −10.2068 10.3924i
23.12 0.531980 1.92795i −1.75796 + 0.309976i −3.43399 2.05126i 6.16490 5.17297i −0.337581 + 3.55416i −6.99959 4.04121i −5.78155 + 5.52934i −5.46290 + 1.98833i −6.69363 14.6375i
23.13 1.32019 + 1.50237i 2.09930 0.370163i −0.514209 + 3.96681i 3.93098 3.29849i 3.32759 + 2.66523i −3.79547 2.19131i −6.63846 + 4.46440i −4.18719 + 1.52401i 10.1452 + 1.55116i
23.14 1.33218 1.49175i 2.53724 0.447383i −0.450619 3.97454i −1.56377 + 1.31216i 2.71266 4.38091i 3.64994 + 2.10730i −6.52931 4.62257i −2.21981 + 0.807946i −0.125806 + 4.08077i
23.15 1.40142 1.42690i −5.33798 + 0.941229i −0.0720694 3.99935i −5.57927 + 4.68156i −6.13769 + 8.93580i −3.83135 2.21203i −5.80766 5.50192i 19.1508 6.97034i −1.13877 + 14.5219i
23.16 1.57762 + 1.22927i −2.79584 + 0.492981i 0.977798 + 3.87865i −3.69543 + 3.10084i −5.01679 2.65909i 7.95020 + 4.59005i −3.22530 + 7.32103i −0.883563 + 0.321590i −9.64176 + 0.349277i
23.17 1.97702 0.302279i −2.09930 + 0.370163i 3.81725 1.19523i 3.93098 3.29849i −4.03848 + 1.36640i 3.79547 + 2.19131i 7.18552 3.51687i −4.18719 + 1.52401i 6.77459 7.70944i
23.18 1.99869 + 0.0724033i 2.79584 0.492981i 3.98952 + 0.289423i −3.69543 + 3.10084i 5.62370 0.782889i −7.95020 4.59005i 7.95285 + 0.867321i −0.883563 + 0.321590i −7.61053 + 5.93004i
35.1 −1.97401 0.321352i 1.80941 + 4.97132i 3.79347 + 1.26871i 0.0746739 0.423497i −1.97426 10.3949i −3.59587 + 2.07608i −7.08065 3.72348i −14.5457 + 12.2053i −0.283499 + 0.811992i
35.2 −1.94561 0.463236i −0.599610 1.64741i 3.57083 + 1.80256i −1.33437 + 7.56759i 0.403468 + 3.48299i 9.08670 5.24621i −6.11244 5.16121i 4.53996 3.80948i 6.10175 14.1055i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.e even 9 1 inner
76.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.l.a 108
4.b odd 2 1 inner 76.3.l.a 108
19.e even 9 1 inner 76.3.l.a 108
76.l odd 18 1 inner 76.3.l.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.l.a 108 1.a even 1 1 trivial
76.3.l.a 108 4.b odd 2 1 inner
76.3.l.a 108 19.e even 9 1 inner
76.3.l.a 108 76.l odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(76, [\chi])\).