Properties

Label 380.2.k.d
Level $380$
Weight $2$
Character orbit 380.k
Analytic conductor $3.034$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52q + 2q^{2} + 2q^{3} + 4q^{5} - 4q^{6} + 4q^{7} - 4q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 52q + 2q^{2} + 2q^{3} + 4q^{5} - 4q^{6} + 4q^{7} - 4q^{8} + 2q^{10} + 22q^{12} - 2q^{13} - 2q^{15} + 16q^{16} - 20q^{17} - 2q^{18} - 52q^{19} - 12q^{20} - 16q^{21} - 36q^{22} - 20q^{23} - 16q^{25} + 8q^{27} - 24q^{28} + 40q^{30} + 2q^{32} + 8q^{33} + 20q^{34} - 12q^{35} - 4q^{36} + 10q^{37} - 2q^{38} + 64q^{39} - 36q^{40} - 4q^{41} - 60q^{42} + 28q^{43} - 8q^{44} + 12q^{45} - 8q^{46} + 4q^{47} - 2q^{48} + 46q^{50} + 74q^{52} - 2q^{53} + 24q^{54} + 12q^{56} - 2q^{57} - 20q^{58} - 28q^{59} - 110q^{60} - 4q^{61} - 32q^{62} - 44q^{63} - 24q^{64} + 10q^{65} + 36q^{66} - 6q^{67} + 28q^{68} + 124q^{70} + 124q^{72} + 8q^{73} + 88q^{74} - 2q^{75} + 12q^{77} - 40q^{78} + 52q^{79} - 120q^{80} - 24q^{81} - 80q^{82} + 76q^{83} - 40q^{84} + 12q^{85} - 8q^{86} - 12q^{87} + 20q^{88} + 78q^{90} + 8q^{92} + 24q^{93} + 32q^{94} - 4q^{95} - 4q^{96} - 10q^{97} - 122q^{98} - 128q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
267.1 −1.38393 0.291109i −1.18694 1.18694i 1.83051 + 0.805748i 1.27222 1.83887i 1.29711 + 1.98816i 1.53689 1.53689i −2.29873 1.64798i 0.182360i −2.29598 + 2.17451i
267.2 −1.36260 0.378581i 1.58108 + 1.58108i 1.71335 + 1.03171i −1.74843 1.39392i −1.55581 2.75294i 1.94368 1.94368i −1.94403 2.05445i 1.99961i 1.85470 + 2.56127i
267.3 −1.34889 0.424847i −2.38733 2.38733i 1.63901 + 1.14614i −2.23504 0.0679064i 2.20600 + 4.23450i −0.160827 + 0.160827i −1.72391 2.24235i 8.39872i 2.98597 + 1.04115i
267.4 −1.34528 + 0.436150i 1.71881 + 1.71881i 1.61955 1.17349i −0.923046 + 2.03666i −3.06193 1.56262i 0.323011 0.323011i −1.66692 + 2.28503i 2.90860i 0.353464 3.14246i
267.5 −1.23831 + 0.683080i 0.494605 + 0.494605i 1.06680 1.69172i 1.31079 1.81158i −0.950327 0.274617i −0.327577 + 0.327577i −0.165443 + 2.82358i 2.51073i −0.385706 + 3.13867i
267.6 −1.14458 0.830627i 1.85040 + 1.85040i 0.620117 + 1.90143i 1.13686 + 1.92550i −0.580936 3.65493i −1.96775 + 1.96775i 0.869611 2.69143i 3.84798i 0.298148 3.14819i
267.7 −1.07073 0.923868i −1.14885 1.14885i 0.292934 + 1.97843i 1.95382 1.08746i 0.168724 + 2.29149i −2.70883 + 2.70883i 1.51416 2.38900i 0.360306i −3.09669 0.640694i
267.8 −0.685298 + 1.23708i −1.29571 1.29571i −1.06073 1.69554i 0.171865 + 2.22945i 2.49084 0.714946i −0.654476 + 0.654476i 2.82443 0.150262i 0.357708i −2.87579 1.31523i
267.9 −0.615749 + 1.27313i 1.93808 + 1.93808i −1.24171 1.56785i −0.168423 2.22972i −3.66079 + 1.27405i 2.91797 2.91797i 2.76066 0.615446i 4.51229i 2.94242 + 1.15852i
267.10 −0.287770 1.38463i −1.15541 1.15541i −1.83438 + 0.796907i −0.283261 2.21805i −1.26732 + 1.93230i 3.26325 3.26325i 1.63130 + 2.31060i 0.330062i −2.98966 + 1.03050i
267.11 −0.283752 + 1.38545i 1.20015 + 1.20015i −1.83897 0.786251i −2.06624 + 0.854787i −2.00330 + 1.32221i −1.52124 + 1.52124i 1.61113 2.32471i 0.119262i −0.597969 3.10523i
267.12 −0.282971 1.38561i −0.321015 0.321015i −1.83985 + 0.784178i 1.77712 + 1.35715i −0.353965 + 0.535642i −1.91243 + 1.91243i 1.60719 + 2.32743i 2.79390i 1.37761 2.84643i
267.13 0.00522099 + 1.41420i −0.212700 0.212700i −1.99995 + 0.0147671i 2.19176 + 0.442915i 0.299691 0.301912i 2.65088 2.65088i −0.0313254 2.82825i 2.90952i −0.614930 + 3.10191i
267.14 0.0365245 1.41374i 0.389264 + 0.389264i −1.99733 0.103273i −1.49421 + 1.66354i 0.564536 0.536101i 2.85353 2.85353i −0.218952 + 2.81994i 2.69695i 2.29723 + 2.17318i
267.15 0.433304 1.34620i 0.497882 + 0.497882i −1.62450 1.16662i −2.01079 0.978129i 0.885982 0.454514i −2.91849 + 2.91849i −2.27441 + 1.68139i 2.50423i −2.18804 + 2.28309i
267.16 0.435901 1.34536i 1.82161 + 1.82161i −1.61998 1.17289i 2.11542 + 0.724569i 3.24476 1.65668i 1.36053 1.36053i −2.28411 + 1.66819i 3.63653i 1.89692 2.53016i
267.17 0.647117 + 1.25747i −1.78847 1.78847i −1.16248 + 1.62746i 0.0829302 2.23453i 1.09160 3.40630i −0.904221 + 0.904221i −2.79875 0.408628i 3.39723i 2.86353 1.34172i
267.18 0.707119 + 1.22474i 1.27216 + 1.27216i −0.999965 + 1.73207i 2.08931 0.796745i −0.658494 + 2.45763i −0.691067 + 0.691067i −2.82843 8.52354e-5i 0.236784i 2.45319 + 1.99546i
267.19 0.912296 + 1.08061i −1.34016 1.34016i −0.335432 + 1.97167i −2.21942 0.272360i 0.225567 2.67081i 0.697742 0.697742i −2.43662 + 1.43628i 0.592060i −1.73045 2.64680i
267.20 0.979881 1.01972i −0.582309 0.582309i −0.0796647 1.99841i 1.46731 + 1.68731i −1.16439 + 0.0231993i −0.972933 + 0.972933i −2.11589 1.87697i 2.32183i 3.15837 + 0.157113i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.k.d yes 52
4.b odd 2 1 380.2.k.c 52
5.c odd 4 1 380.2.k.c 52
20.e even 4 1 inner 380.2.k.d yes 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.k.c 52 4.b odd 2 1
380.2.k.c 52 5.c odd 4 1
380.2.k.d yes 52 1.a even 1 1 trivial
380.2.k.d yes 52 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!00\)\( T_{3}^{21} + \)\(23\!\cdots\!96\)\( T_{3}^{20} - \)\(23\!\cdots\!20\)\( T_{3}^{19} + \)\(20\!\cdots\!24\)\( T_{3}^{18} + \)\(34\!\cdots\!96\)\( T_{3}^{17} + \)\(57\!\cdots\!52\)\( T_{3}^{16} - \)\(37\!\cdots\!88\)\( T_{3}^{15} + \)\(19\!\cdots\!40\)\( T_{3}^{14} + \)\(19\!\cdots\!76\)\( T_{3}^{13} + \)\(23\!\cdots\!40\)\( T_{3}^{12} - \)\(14\!\cdots\!60\)\( T_{3}^{11} + 671027970048 T_{3}^{10} + 404471357440 T_{3}^{9} + 178171249664 T_{3}^{8} - 62174060544 T_{3}^{7} + 33952006144 T_{3}^{6} + 21210497024 T_{3}^{5} + 6514049024 T_{3}^{4} + 35782656 T_{3}^{3} + 25690112 T_{3}^{2} + 18350080 T_{3} + 6553600 \)">\(T_{3}^{52} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).