Properties

Label 360.2.bk.a
Level $360$
Weight $2$
Character orbit 360.bk
Analytic conductor $2.875$
Analytic rank $0$
Dimension $136$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(68\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136q - 2q^{4} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 136q - 2q^{4} - 8q^{9} - 10q^{14} + 2q^{15} - 2q^{16} - 2q^{20} - 38q^{24} - 2q^{25} + 16q^{26} - 12q^{30} - 4q^{31} + 8q^{34} - 24q^{36} + 4q^{39} - 6q^{40} - 4q^{41} - 56q^{44} - 28q^{46} + 40q^{49} - 12q^{50} + 10q^{54} - 28q^{55} + 26q^{56} - 58q^{60} - 20q^{64} + 18q^{65} + 16q^{66} - 6q^{70} - 128q^{71} + 36q^{74} + 12q^{76} - 4q^{79} - 36q^{80} - 16q^{81} - 110q^{84} + 44q^{86} - 32q^{89} + 32q^{90} - 34q^{94} + 8q^{95} - 2q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 −1.40884 + 0.123126i 0.777229 + 1.54787i 1.96968 0.346930i 1.94405 + 1.10484i −1.28558 2.08502i −2.92425 + 1.68832i −2.73226 + 0.731288i −1.79183 + 2.40611i −2.87489 1.31719i
229.2 −1.40262 0.180707i −1.01615 1.40266i 1.93469 + 0.506927i −1.43541 + 1.71453i 1.17180 + 2.15102i 2.42894 1.40235i −2.62203 1.06064i −0.934882 + 2.85061i 2.32316 2.14545i
229.3 −1.39716 + 0.218938i −1.67702 + 0.433134i 1.90413 0.611784i −1.45294 1.69970i 2.24824 0.972322i −0.435632 + 0.251512i −2.52644 + 1.27165i 2.62479 1.45275i 2.40212 + 2.05665i
229.4 −1.39375 + 0.239699i 1.69556 + 0.353674i 1.88509 0.668161i 0.971012 2.01423i −2.44796 0.0865103i −1.11197 + 0.641993i −2.46719 + 1.38310i 2.74983 + 1.19935i −0.870541 + 3.04009i
229.5 −1.38109 + 0.304279i 1.07030 1.36178i 1.81483 0.840474i −1.96832 + 1.06099i −1.06383 + 2.20642i −1.60442 + 0.926310i −2.25071 + 1.71299i −0.708903 2.91504i 2.39560 2.06425i
229.6 −1.37959 0.311008i 1.28537 1.16096i 1.80655 + 0.858129i 2.15709 + 0.589033i −2.13435 + 1.20190i 2.73484 1.57896i −2.22541 1.74572i 0.304332 2.98452i −2.79271 1.48350i
229.7 −1.35366 0.409391i −0.0938520 1.72951i 1.66480 + 1.10835i 0.465822 2.18701i −0.581001 + 2.37959i −3.18214 + 1.83721i −1.79982 2.18189i −2.98238 + 0.324635i −1.52591 + 2.76977i
229.8 −1.34429 0.439174i −0.277804 + 1.70963i 1.61425 + 1.18076i −2.16653 + 0.553294i 1.12427 2.17624i −0.442721 + 0.255605i −1.65147 2.29622i −2.84565 0.949882i 3.15545 + 0.207696i
229.9 −1.30878 + 0.535802i −0.883424 + 1.48982i 1.42583 1.40250i 1.08655 + 1.95433i 0.357964 2.42319i 4.23238 2.44357i −1.11465 + 2.59953i −1.43912 2.63229i −2.46920 1.97562i
229.10 −1.25986 0.642465i 1.66988 + 0.459910i 1.17448 + 1.61883i −1.65905 1.49919i −1.80833 1.65226i 1.92958 1.11405i −0.439633 2.79405i 2.57697 + 1.53598i 1.12698 + 2.95464i
229.11 −1.25228 + 0.657113i −1.28051 1.16631i 1.13640 1.64578i 2.01086 0.977977i 2.36996 + 0.619105i 1.13221 0.653679i −0.341633 + 2.80772i 0.279433 + 2.98696i −1.87552 + 2.54606i
229.12 −1.19676 0.753499i −0.761001 + 1.55592i 0.864479 + 1.80352i 1.70599 1.44555i 2.08312 1.28865i −1.65398 + 0.954927i 0.324373 2.80977i −1.84175 2.36811i −3.13088 + 0.444516i
229.13 −1.19522 + 0.755949i 1.28051 + 1.16631i 0.857083 1.80704i −2.01086 + 0.977977i −2.41216 0.425992i 1.13221 0.653679i 0.341633 + 2.80772i 0.279433 + 2.98696i 1.66411 2.68900i
229.14 −1.11841 + 0.865540i 0.883424 1.48982i 0.501683 1.93606i −1.08655 1.95433i 0.301467 + 2.43087i 4.23238 2.44357i 1.11465 + 2.59953i −1.43912 2.63229i 2.90676 + 1.24528i
229.15 −1.07641 0.917248i 1.70505 0.304648i 0.317314 + 1.97467i −0.596890 + 2.15493i −2.11477 1.23603i −4.00243 + 2.31080i 1.46970 2.41661i 2.81438 1.03888i 2.61910 1.77209i
229.16 −1.04987 0.947511i −1.58768 + 0.692297i 0.204446 + 1.98952i 0.371573 + 2.20498i 2.32281 + 0.777522i 1.08980 0.629197i 1.67045 2.28245i 2.04145 2.19829i 1.69914 2.66701i
229.17 −1.00550 0.994471i −1.61972 0.613615i 0.0220563 + 1.99988i −0.521343 2.17444i 1.01840 + 2.22775i 2.84021 1.63979i 1.96664 2.03281i 2.24695 + 1.98776i −1.63821 + 2.70486i
229.18 −0.954059 + 1.04392i −1.07030 + 1.36178i −0.179543 1.99192i 1.96832 1.06099i −0.400461 2.41653i −1.60442 + 0.926310i 2.25071 + 1.71299i −0.708903 2.91504i −0.770302 + 3.06702i
229.19 −0.904461 + 1.08718i −1.69556 0.353674i −0.363900 1.96662i −0.971012 + 2.01423i 1.91807 1.52348i −1.11197 + 0.641993i 2.46719 + 1.38310i 2.74983 + 1.19935i −1.31158 2.87746i
229.20 −0.888188 + 1.10051i 1.67702 0.433134i −0.422246 1.95492i 1.45294 + 1.69970i −1.01284 + 2.23028i −0.435632 + 0.251512i 2.52644 + 1.27165i 2.62479 1.45275i −3.16102 + 0.0893258i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
40.f even 2 1 inner
45.j even 6 1 inner
72.n even 6 1 inner
360.bk even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.bk.a 136
5.b even 2 1 inner 360.2.bk.a 136
8.b even 2 1 inner 360.2.bk.a 136
9.c even 3 1 inner 360.2.bk.a 136
40.f even 2 1 inner 360.2.bk.a 136
45.j even 6 1 inner 360.2.bk.a 136
72.n even 6 1 inner 360.2.bk.a 136
360.bk even 6 1 inner 360.2.bk.a 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bk.a 136 1.a even 1 1 trivial
360.2.bk.a 136 5.b even 2 1 inner
360.2.bk.a 136 8.b even 2 1 inner
360.2.bk.a 136 9.c even 3 1 inner
360.2.bk.a 136 40.f even 2 1 inner
360.2.bk.a 136 45.j even 6 1 inner
360.2.bk.a 136 72.n even 6 1 inner
360.2.bk.a 136 360.bk even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(360, [\chi])\).