Properties

Label 672.2.bo.a
Level 672
Weight 2
Character orbit 672.bo
Analytic conductor 5.366
Analytic rank 0
Dimension 496
CM no
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(496\)
Relative dimension: \(124\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 496q - 16q^{4} - 8q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 496q - 16q^{4} - 8q^{7} - 8q^{9} - 48q^{16} - 8q^{18} - 4q^{21} - 16q^{22} - 16q^{25} - 8q^{28} - 64q^{30} - 48q^{36} - 16q^{37} - 8q^{39} + 56q^{42} - 16q^{43} - 16q^{46} + 16q^{51} - 8q^{57} - 80q^{58} + 24q^{60} - 8q^{63} - 112q^{64} + 48q^{67} - 8q^{70} - 8q^{72} - 160q^{78} + 128q^{84} + 64q^{85} - 16q^{88} - 56q^{91} + 16q^{93} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 −1.41228 0.0740064i −0.841142 + 1.51409i 1.98905 + 0.209035i −0.918980 2.21861i 1.29998 2.07607i −1.19970 + 2.35812i −2.79361 0.442417i −1.58496 2.54714i 1.13366 + 3.20130i
125.2 −1.41228 0.0740064i 0.841142 1.51409i 1.98905 + 0.209035i 0.918980 + 2.21861i −1.29998 + 2.07607i −2.35812 + 1.19970i −2.79361 0.442417i −1.58496 2.54714i −1.13366 3.20130i
125.3 −1.40652 0.147357i −0.676569 + 1.59444i 1.95657 + 0.414520i 1.32184 + 3.19120i 1.18656 2.14291i −1.01750 2.44227i −2.69087 0.871344i −2.08451 2.15750i −1.38894 4.68326i
125.4 −1.40652 0.147357i 0.676569 1.59444i 1.95657 + 0.414520i −1.32184 3.19120i −1.18656 + 2.14291i 2.44227 + 1.01750i −2.69087 0.871344i −2.08451 2.15750i 1.38894 + 4.68326i
125.5 −1.39741 + 0.217363i −0.679260 1.59330i 1.90551 0.607491i 0.491250 + 1.18598i 1.29553 + 2.07885i 0.112587 + 2.64335i −2.53073 + 1.26310i −2.07721 + 2.16453i −0.944266 1.55052i
125.6 −1.39741 + 0.217363i 0.679260 + 1.59330i 1.90551 0.607491i −0.491250 1.18598i −1.29553 2.07885i −2.64335 0.112587i −2.53073 + 1.26310i −2.07721 + 2.16453i 0.944266 + 1.55052i
125.7 −1.39544 0.229683i −1.71200 0.262778i 1.89449 + 0.641017i −0.321849 0.777011i 2.32863 + 0.759907i −2.46255 0.967393i −2.49641 1.32963i 2.86190 + 0.899751i 0.270653 + 1.15819i
125.8 −1.39544 0.229683i 1.71200 + 0.262778i 1.89449 + 0.641017i 0.321849 + 0.777011i −2.32863 0.759907i 0.967393 + 2.46255i −2.49641 1.32963i 2.86190 + 0.899751i −0.270653 1.15819i
125.9 −1.38553 + 0.283382i −1.72093 + 0.195961i 1.83939 0.785270i 0.478704 + 1.15569i 2.32887 0.759191i 2.64195 0.141859i −2.32600 + 1.60927i 2.92320 0.674470i −0.990761 1.46559i
125.10 −1.38553 + 0.283382i 1.72093 0.195961i 1.83939 0.785270i −0.478704 1.15569i −2.32887 + 0.759191i 0.141859 2.64195i −2.32600 + 1.60927i 2.92320 0.674470i 0.990761 + 1.46559i
125.11 −1.33124 + 0.477276i −1.25546 + 1.19324i 1.54442 1.27074i −1.54647 3.73351i 1.10181 2.18770i 1.60304 2.10482i −1.44950 + 2.42878i 0.152341 2.99613i 3.84065 + 4.23212i
125.12 −1.33124 + 0.477276i 1.25546 1.19324i 1.54442 1.27074i 1.54647 + 3.73351i −1.10181 + 2.18770i 2.10482 1.60304i −1.44950 + 2.42878i 0.152341 2.99613i −3.84065 4.23212i
125.13 −1.31911 0.509861i −0.856827 1.50527i 1.48008 + 1.34512i 0.903130 + 2.18035i 0.362766 + 2.42248i 0.477357 2.60233i −1.26656 2.52899i −1.53170 + 2.57952i −0.0796507 3.33658i
125.14 −1.31911 0.509861i 0.856827 + 1.50527i 1.48008 + 1.34512i −0.903130 2.18035i −0.362766 2.42248i 2.60233 0.477357i −1.26656 2.52899i −1.53170 + 2.57952i 0.0796507 + 3.33658i
125.15 −1.28797 + 0.584070i −0.124696 1.72756i 1.31772 1.50453i −0.799640 1.93050i 1.16962 + 2.15221i −1.72605 2.00519i −0.818439 + 2.70743i −2.96890 + 0.430838i 2.15746 + 2.01938i
125.16 −1.28797 + 0.584070i 0.124696 + 1.72756i 1.31772 1.50453i 0.799640 + 1.93050i −1.16962 2.15221i 2.00519 + 1.72605i −0.818439 + 2.70743i −2.96890 + 0.430838i −2.15746 2.01938i
125.17 −1.26371 0.634854i −0.826142 + 1.52233i 1.19392 + 1.60454i 0.561080 + 1.35457i 2.01046 1.39930i 2.62474 0.332748i −0.490121 2.78564i −1.63498 2.51532i 0.150910 2.06798i
125.18 −1.26371 0.634854i 0.826142 1.52233i 1.19392 + 1.60454i −0.561080 1.35457i −2.01046 + 1.39930i 0.332748 2.62474i −0.490121 2.78564i −1.63498 2.51532i −0.150910 + 2.06798i
125.19 −1.21868 + 0.717517i −1.53560 0.801206i 0.970338 1.74884i −1.29133 3.11756i 2.44628 0.125409i −1.18607 + 2.36500i 0.0722962 + 2.82750i 1.71614 + 2.46067i 3.81062 + 2.87274i
125.20 −1.21868 + 0.717517i 1.53560 + 0.801206i 0.970338 1.74884i 1.29133 + 3.11756i −2.44628 + 0.125409i −2.36500 + 1.18607i 0.0722962 + 2.82750i 1.71614 + 2.46067i −3.81062 2.87274i
See next 80 embeddings (of 496 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 629.124
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner
32.g even 8 1 inner
96.p odd 8 1 inner
224.v odd 8 1 inner
672.bo 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.bo.a 496
3.b odd 2 1 inner 672.2.bo.a 496
7.b odd 2 1 inner 672.2.bo.a 496
21.c even 2 1 inner 672.2.bo.a 496
32.g even 8 1 inner 672.2.bo.a 496
96.p odd 8 1 inner 672.2.bo.a 496
224.v odd 8 1 inner 672.2.bo.a 496
672.bo even 8 1 inner 672.2.bo.a 496
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bo.a 496 1.a even 1 1 trivial
672.2.bo.a 496 3.b odd 2 1 inner
672.2.bo.a 496 7.b odd 2 1 inner
672.2.bo.a 496 21.c even 2 1 inner
672.2.bo.a 496 32.g even 8 1 inner
672.2.bo.a 496 96.p odd 8 1 inner
672.2.bo.a 496 224.v odd 8 1 inner
672.2.bo.a 496 672.bo even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database