Properties

Label 672.2.bs.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 192q + 16q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 192q + 16q^{4} + 8q^{10} + 16q^{16} + 16q^{22} + 24q^{27} + 44q^{30} - 104q^{36} - 32q^{37} + 24q^{39} + 32q^{46} - 32q^{48} - 80q^{52} - 32q^{55} - 72q^{58} - 28q^{60} + 80q^{61} + 64q^{64} + 76q^{66} - 32q^{67} - 56q^{69} - 24q^{70} + 60q^{72} - 80q^{75} - 56q^{76} - 76q^{78} - 32q^{79} + 120q^{82} - 56q^{87} + 40q^{88} + 60q^{90} + 80q^{93} - 120q^{94} - 32q^{96} - 64q^{99} + O(q^{100}) \)

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 −1.41421 0.00182353i 1.66131 + 0.489956i 1.99999 + 0.00515773i −0.328767 + 0.793714i −2.34855 0.695932i 0.707107 0.707107i −2.82841 0.0109412i 2.51989 + 1.62794i 0.466394 1.12188i
155.2 −1.41292 0.0604352i 0.257532 + 1.71280i 1.99270 + 0.170780i 0.740432 1.78756i −0.260359 2.43561i 0.707107 0.707107i −2.80520 0.361728i −2.86735 + 0.882200i −1.15420 + 2.48093i
155.3 −1.41064 + 0.100442i −1.73160 0.0395123i 1.97982 0.283377i 1.24365 3.00244i 2.44664 0.118189i 0.707107 0.707107i −2.76436 + 0.598602i 2.99688 + 0.136839i −1.45278 + 4.36029i
155.4 −1.38411 0.290248i −0.626099 1.61493i 1.83151 + 0.803469i 1.22449 2.95618i 0.397859 + 2.41696i 0.707107 0.707107i −2.30181 1.64368i −2.21600 + 2.02221i −2.55285 + 3.73627i
155.5 −1.36323 0.376307i −0.709246 + 1.58018i 1.71679 + 1.02599i −0.958822 + 2.31480i 1.56150 1.88725i 0.707107 0.707107i −1.95428 2.04469i −1.99394 2.24147i 2.17817 2.79479i
155.6 −1.34336 0.442023i −1.14969 1.29546i 1.60923 + 1.18759i −1.30330 + 3.14645i 0.971830 + 2.24845i 0.707107 0.707107i −1.63683 2.30668i −0.356409 + 2.97875i 3.14161 3.65073i
155.7 −1.31896 + 0.510228i 0.712074 1.57891i 1.47933 1.34595i −0.745289 + 1.79929i −0.133596 + 2.44584i 0.707107 0.707107i −1.26445 + 2.53005i −1.98590 2.24860i 0.0649625 2.75346i
155.8 −1.24057 + 0.678963i 0.613308 1.61983i 1.07802 1.68460i 1.04531 2.52359i 0.338955 + 2.42592i 0.707107 0.707107i −0.193574 + 2.82180i −2.24771 1.98691i 0.416652 + 3.84041i
155.9 −1.19021 + 0.763800i −1.51796 + 0.834142i 0.833219 1.81817i −1.24406 + 3.00343i 1.16958 2.15223i 0.707107 0.707107i 0.397011 + 2.80043i 1.60842 2.53239i −0.813320 4.52493i
155.10 −1.16411 0.803019i 0.712483 1.57872i 0.710322 + 1.86961i −0.879803 + 2.12403i −2.09716 + 1.26568i 0.707107 0.707107i 0.674435 2.74684i −1.98474 2.24963i 2.72983 1.76612i
155.11 −1.14196 + 0.834221i −1.41654 0.996701i 0.608152 1.90530i 0.0343602 0.0829528i 2.44910 0.0435137i 0.707107 0.707107i 0.894951 + 2.68311i 1.01317 + 2.82373i 0.0299630 + 0.123393i
155.12 −1.12590 0.855778i 1.58528 0.697781i 0.535287 + 1.92704i 0.618840 1.49401i −2.38200 0.571016i 0.707107 0.707107i 1.04644 2.62773i 2.02620 2.21235i −1.97529 + 1.15251i
155.13 −1.09166 + 0.899047i 0.650725 + 1.60517i 0.383428 1.96290i −0.365005 + 0.881200i −2.15349 1.16726i 0.707107 0.707107i 1.34617 + 2.48753i −2.15311 + 2.08904i −0.393780 1.29012i
155.14 −0.770079 1.18616i −0.845496 + 1.51167i −0.813957 + 1.82688i −0.484034 + 1.16856i 2.44418 0.161208i 0.707107 0.707107i 2.79378 0.441355i −1.57027 2.55622i 1.75885 0.325742i
155.15 −0.768409 1.18724i −1.35115 1.08370i −0.819096 + 1.82458i 0.0800577 0.193276i −0.248380 + 2.43686i 0.707107 0.707107i 2.79562 0.429554i 0.651199 + 2.92847i −0.290983 + 0.0534673i
155.16 −0.684676 + 1.23742i 1.72238 0.182818i −1.06244 1.69447i 0.622207 1.50214i −0.953046 + 2.25648i 0.707107 0.707107i 2.82420 0.154520i 2.93315 0.629763i 1.43277 + 1.79841i
155.17 −0.658473 + 1.25156i −1.56926 + 0.733092i −1.13283 1.64824i 0.544247 1.31393i 0.115802 2.44675i 0.707107 0.707107i 2.80882 0.332485i 1.92515 2.30082i 1.28610 + 1.54635i
155.18 −0.584571 1.28774i 1.72479 + 0.158475i −1.31655 + 1.50555i −1.55272 + 3.74859i −0.804185 2.31372i 0.707107 0.707107i 2.70838 + 0.815281i 2.94977 + 0.546669i 5.73489 0.191818i
155.19 −0.540503 + 1.30685i 1.42862 0.979314i −1.41571 1.41271i −1.40353 + 3.38841i 0.507644 + 2.39631i 0.707107 0.707107i 2.61140 1.08655i 1.08189 2.79813i −3.66954 3.66565i
155.20 −0.503558 1.32153i 0.984099 + 1.42532i −1.49286 + 1.33093i 0.483125 1.16637i 1.38805 2.01825i 0.707107 0.707107i 2.51060 + 1.30265i −1.06310 + 2.80532i −1.78467 0.0511282i
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.b 192
3.b odd 2 1 inner 672.2.bs.b 192
32.h odd 8 1 inner 672.2.bs.b 192
96.o even 8 1 inner 672.2.bs.b 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bs.b 192 1.a even 1 1 trivial
672.2.bs.b 192 3.b odd 2 1 inner
672.2.bs.b 192 32.h odd 8 1 inner
672.2.bs.b 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