Properties

Label 105.4.u.a
Level $105$
Weight $4$
Character orbit 105.u
Analytic conductor $6.195$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96q + 48q^{5} - 4q^{7} + 168q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 96q + 48q^{5} - 4q^{7} + 168q^{8} - 72q^{10} + 56q^{11} - 168q^{15} + 880q^{16} + 96q^{21} + 624q^{22} - 64q^{23} + 152q^{25} + 912q^{26} + 628q^{28} + 48q^{30} - 528q^{31} - 672q^{32} - 108q^{33} - 1616q^{35} - 3456q^{36} - 552q^{37} - 4044q^{38} + 2052q^{40} - 1068q^{42} + 720q^{43} + 2560q^{46} + 240q^{47} - 3528q^{50} + 336q^{51} + 4644q^{52} + 1728q^{53} + 4896q^{56} + 1392q^{57} + 512q^{58} + 420q^{60} - 2592q^{61} - 288q^{63} - 824q^{65} - 4104q^{66} - 3784q^{67} - 8844q^{68} - 2256q^{70} - 128q^{71} - 756q^{72} - 3312q^{73} - 1872q^{75} - 6600q^{77} - 1200q^{78} + 12108q^{80} + 3888q^{81} + 6084q^{82} + 4768q^{85} - 1088q^{86} + 5652q^{87} + 4844q^{88} + 10128q^{91} - 2152q^{92} + 1368q^{93} - 7872q^{95} + 20184q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −1.41620 + 5.28532i −2.89778 + 0.776457i −19.0007 10.9701i 11.0616 1.62545i 16.4153i 0.587638 18.5109i 53.9362 53.9362i 7.79423 4.50000i −7.07431 + 60.7657i
52.2 −1.31873 + 4.92156i 2.89778 0.776457i −15.5545 8.98042i −0.535886 + 11.1675i 15.2855i −13.3869 + 12.7980i 35.8873 35.8873i 7.79423 4.50000i −54.2548 17.3643i
52.3 −1.24190 + 4.63484i −2.89778 + 0.776457i −13.0112 7.51204i −10.1233 4.74540i 14.3950i −3.69659 + 18.1476i 23.8323 23.8323i 7.79423 4.50000i 34.5663 41.0266i
52.4 −1.17018 + 4.36716i 2.89778 0.776457i −10.7746 6.22070i −6.73223 8.92620i 13.5637i −1.57380 18.4533i 14.1991 14.1991i 7.79423 4.50000i 46.8601 18.9555i
52.5 −1.14346 + 4.26746i 2.89778 0.776457i −9.97548 5.75935i 9.24082 6.29343i 13.2540i 15.1987 + 10.5830i 10.9924 10.9924i 7.79423 4.50000i 16.2904 + 46.6311i
52.6 −0.871352 + 3.25193i −2.89778 + 0.776457i −2.88758 1.66715i 6.94823 + 8.75911i 10.0999i 10.8772 + 14.9896i −11.1071 + 11.1071i 7.79423 4.50000i −34.5384 + 14.9629i
52.7 −0.624851 + 2.33198i −2.89778 + 0.776457i 1.88053 + 1.08572i −9.58456 + 5.75641i 7.24272i −5.69491 17.6229i −17.3639 + 17.3639i 7.79423 4.50000i −7.43490 25.9479i
52.8 −0.529124 + 1.97472i 2.89778 0.776457i 3.30867 + 1.91026i −11.1089 + 1.26224i 6.13313i −16.3965 + 8.61137i −17.0877 + 17.0877i 7.79423 4.50000i 3.38539 22.6047i
52.9 −0.514613 + 1.92056i −2.89778 + 0.776457i 3.50448 + 2.02331i 5.44490 9.76489i 5.96493i 14.3387 11.7218i −16.9369 + 16.9369i 7.79423 4.50000i 15.9520 + 15.4824i
52.10 −0.465010 + 1.73544i 2.89778 0.776457i 4.13268 + 2.38600i 2.32023 + 10.9369i 5.38999i 13.1952 12.9956i −16.2260 + 16.2260i 7.79423 4.50000i −20.0594 1.05916i
52.11 −0.121182 + 0.452258i 2.89778 0.776457i 6.73835 + 3.89039i 10.5131 3.80453i 1.40463i −5.91256 + 17.5511i −5.22463 + 5.22463i 7.79423 4.50000i 0.446628 + 5.21568i
52.12 0.0422777 0.157783i −2.89778 + 0.776457i 6.90510 + 3.98666i 9.25781 + 6.26841i 0.490046i −18.5183 0.270750i 1.84500 1.84500i 7.79423 4.50000i 1.38044 1.19571i
52.13 0.121038 0.451720i −2.89778 + 0.776457i 6.73880 + 3.89065i −10.1913 4.59752i 1.40296i 16.4236 + 8.55956i 5.21859 5.21859i 7.79423 4.50000i −3.31032 + 4.04714i
52.14 0.131114 0.489325i 2.89778 0.776457i 6.70596 + 3.87169i 1.00965 11.1347i 1.51976i −6.72481 17.2562i 5.63944 5.63944i 7.79423 4.50000i −5.31609 1.95396i
52.15 0.366039 1.36607i −2.89778 + 0.776457i 5.19603 + 2.99993i −4.85914 10.0692i 4.24279i −18.5166 0.368999i 14.0004 14.0004i 7.79423 4.50000i −15.5339 + 2.95223i
52.16 0.503088 1.87755i 2.89778 0.776457i 3.65611 + 2.11085i −11.1743 + 0.367232i 5.83135i 18.4739 1.30903i 16.7983 16.7983i 7.79423 4.50000i −4.93216 + 21.1651i
52.17 0.662502 2.47249i 2.89778 0.776457i 1.25390 + 0.723938i 4.16719 + 10.3747i 7.67914i 6.62391 + 17.2952i 17.1006 17.1006i 7.79423 4.50000i 28.4121 3.43007i
52.18 0.791608 2.95432i −2.89778 + 0.776457i −1.17317 0.677329i 2.51558 + 10.8937i 9.17562i 5.89275 17.5578i 14.3720 14.3720i 7.79423 4.50000i 34.1747 + 1.19167i
52.19 0.815920 3.04505i −2.89778 + 0.776457i −1.67842 0.969037i 10.1281 4.73506i 9.45741i 13.1472 + 13.0442i 13.5128 13.5128i 7.79423 4.50000i −6.15476 34.7042i
52.20 1.00662 3.75676i 2.89778 0.776457i −6.17173 3.56325i 10.7872 + 2.93879i 11.6678i −14.4985 11.5236i 2.40224 2.40224i 7.79423 4.50000i 21.8989 37.5666i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.u.a 96
5.c odd 4 1 inner 105.4.u.a 96
7.d odd 6 1 inner 105.4.u.a 96
35.k even 12 1 inner 105.4.u.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.u.a 96 1.a even 1 1 trivial
105.4.u.a 96 5.c odd 4 1 inner
105.4.u.a 96 7.d odd 6 1 inner
105.4.u.a 96 35.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(105, [\chi])\).