Properties

Label 672.2.bs.a
Level 672
Weight 2
Character orbit 672.bs
Analytic conductor 5.366
Analytic rank 0
Dimension 192
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.bs (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{8})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192q - 16q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 192q - 16q^{4} + 8q^{10} + 16q^{16} + 16q^{22} + 24q^{27} - 84q^{30} + 40q^{36} + 32q^{37} + 24q^{39} - 96q^{46} - 72q^{48} + 16q^{52} - 32q^{55} - 72q^{58} - 28q^{60} - 16q^{61} - 64q^{64} - 12q^{66} - 32q^{67} + 56q^{69} - 24q^{70} + 60q^{72} + 80q^{75} - 56q^{76} + 100q^{78} - 32q^{79} - 120q^{82} - 56q^{87} + 40q^{88} + 60q^{90} - 80q^{93} + 216q^{94} + 96q^{96} - 64q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.41004 + 0.108548i −0.206243 1.71973i 1.97643 0.306113i −0.448541 + 1.08287i 0.477484 + 2.40250i −0.707107 + 0.707107i −2.75363 + 0.646170i −2.91493 + 0.709364i 0.514919 1.57559i
155.2 −1.40912 0.119931i 1.40370 1.01471i 1.97123 + 0.337995i 1.13465 2.73929i −2.09967 + 1.26150i −0.707107 + 0.707107i −2.73717 0.712688i 0.940735 2.84869i −1.92738 + 3.72390i
155.3 −1.38036 + 0.307600i 0.475550 + 1.66549i 1.81076 0.849195i −1.65624 + 3.99852i −1.16873 2.15269i −0.707107 + 0.707107i −2.23829 + 1.72918i −2.54770 + 1.58405i 1.05626 6.02883i
155.4 −1.34893 + 0.424720i −1.25748 + 1.19111i 1.63923 1.14583i 0.367345 0.886850i 1.19037 2.14080i −0.707107 + 0.707107i −1.72454 + 2.24186i 0.162531 2.99559i −0.118861 + 1.35232i
155.5 −1.34768 0.428658i 0.151828 + 1.72538i 1.63251 + 1.15539i 0.849301 2.05039i 0.534982 2.39035i −0.707107 + 0.707107i −1.70483 2.25689i −2.95390 + 0.523925i −2.02351 + 2.39922i
155.6 −1.28283 0.595271i −1.60604 + 0.648559i 1.29131 + 1.52726i −0.327082 + 0.789647i 2.44635 + 0.124039i −0.707107 + 0.707107i −0.747392 2.72789i 2.15874 2.08323i 0.889645 0.818280i
155.7 −1.26204 0.638158i 1.63116 + 0.582516i 1.18551 + 1.61077i −0.102034 + 0.246332i −1.68685 1.77610i −0.707107 + 0.707107i −0.468240 2.78940i 2.32135 + 1.90035i 0.285970 0.245768i
155.8 −1.22233 + 0.711266i 1.73195 + 0.0187876i 0.988202 1.73881i −0.345711 + 0.834621i −2.13038 + 1.20891i −0.707107 + 0.707107i 0.0288427 + 2.82828i 2.99929 + 0.0650782i −0.171063 1.26608i
155.9 −1.21431 0.724880i −1.31477 1.12755i 0.949098 + 1.76046i 0.733476 1.77077i 0.779205 + 2.32225i −0.707107 + 0.707107i 0.123621 2.82572i 0.457262 + 2.96495i −2.17426 + 1.61858i
155.10 −1.07981 + 0.913242i 1.37333 1.05544i 0.331980 1.97225i −0.295998 + 0.714603i −0.519067 + 2.39386i −0.707107 + 0.707107i 1.44267 + 2.43284i 0.772089 2.89894i −0.332983 1.04195i
155.11 −0.958529 + 1.03982i −0.183529 + 1.72230i −0.162445 1.99339i 0.468472 1.13099i −1.61496 1.84171i −0.707107 + 0.707107i 2.22847 + 1.74181i −2.93263 0.632184i 0.726982 + 1.57121i
155.12 −0.938001 1.05837i 0.196518 1.72087i −0.240309 + 1.98551i −0.272376 + 0.657573i −2.00565 + 1.40618i −0.707107 + 0.707107i 2.32682 1.60807i −2.92276 0.676364i 0.951446 0.328529i
155.13 −0.891321 + 1.09797i −1.72725 0.128875i −0.411094 1.95729i 1.42904 3.45001i 1.68104 1.78161i −0.707107 + 0.707107i 2.51548 + 1.29321i 2.96678 + 0.445200i 2.51429 + 4.64412i
155.14 −0.868232 1.11632i −1.73050 + 0.0733671i −0.492347 + 1.93845i −1.63009 + 3.93540i 1.58437 + 1.86809i −0.707107 + 0.707107i 2.59141 1.13341i 2.98923 0.253923i 5.80847 1.59713i
155.15 −0.838438 + 1.13887i −0.303955 1.70517i −0.594043 1.90974i −1.38677 + 3.34796i 2.19681 + 1.08352i −0.707107 + 0.707107i 2.67301 + 0.924664i −2.81522 + 1.03659i −2.65017 4.38641i
155.16 −0.669636 1.24563i 1.65765 + 0.502191i −1.10318 + 1.66823i 1.25795 3.03696i −0.484479 2.40110i −0.707107 + 0.707107i 2.81672 + 0.257038i 2.49561 + 1.66491i −4.62529 + 0.466720i
155.17 −0.651302 1.25531i −0.203847 + 1.72001i −1.15161 + 1.63517i −0.577425 + 1.39403i 2.29192 0.864357i −0.707107 + 0.707107i 2.80270 + 0.380642i −2.91689 0.701238i 2.12602 0.183084i
155.18 −0.550610 + 1.30262i 0.864233 1.50103i −1.39366 1.43447i 1.15904 2.79817i 1.47943 + 1.95225i −0.707107 + 0.707107i 2.63594 1.02558i −1.50620 2.59449i 3.00678 + 3.05049i
155.19 −0.422431 + 1.34965i 1.42128 + 0.989935i −1.64310 1.14027i −0.956738 + 2.30977i −1.93646 + 1.50004i −0.707107 + 0.707107i 2.23306 1.73593i 1.04006 + 2.81394i −2.71322 2.26698i
155.20 −0.393886 1.35825i 1.34373 1.09288i −1.68971 + 1.06999i −0.833504 + 2.01226i −2.01369 1.39465i −0.707107 + 0.707107i 2.11888 + 1.87360i 0.611211 2.93708i 3.06146 + 0.339511i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 659.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bs.a 192
3.b odd 2 1 inner 672.2.bs.a 192
32.h odd 8 1 inner 672.2.bs.a 192
96.o even 8 1 inner 672.2.bs.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bs.a 192 1.a even 1 1 trivial
672.2.bs.a 192 3.b odd 2 1 inner
672.2.bs.a 192 32.h odd 8 1 inner
672.2.bs.a 192 96.o even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{192} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database