Properties

Label 684.3.be.a
Level $684$
Weight $3$
Character orbit 684.be
Analytic conductor $18.638$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + q^{7} + 4 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + q^{7} + 4 q^{9} + 18 q^{11} + 10 q^{13} - 11 q^{15} + 9 q^{17} + 20 q^{19} - 30 q^{21} + 200 q^{25} + 25 q^{27} - 27 q^{29} - 8 q^{31} + 23 q^{33} + 22 q^{37} + 39 q^{39} - 54 q^{41} + 88 q^{43} - 196 q^{45} + 198 q^{47} - 267 q^{49} - 56 q^{51} + 36 q^{53} + 78 q^{57} + 171 q^{59} + 7 q^{61} + 82 q^{63} - 144 q^{65} + 154 q^{67} + 44 q^{69} + 135 q^{71} + 43 q^{73} + 69 q^{75} + 216 q^{77} + 34 q^{79} - 44 q^{81} - 171 q^{83} - 244 q^{87} - 216 q^{89} + 122 q^{91} - 104 q^{93} - 216 q^{95} + 16 q^{97} - 305 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
425.1 0 −2.99940 0.0599360i 0 1.93690 + 1.11827i 0 1.30590 2.26189i 0 8.99282 + 0.359544i 0
425.2 0 −2.95244 0.532061i 0 −2.56758 1.48239i 0 −4.97804 + 8.62221i 0 8.43382 + 3.14176i 0
425.3 0 −2.83801 + 0.972476i 0 4.92443 + 2.84312i 0 −1.27053 + 2.20061i 0 7.10858 5.51979i 0
425.4 0 −2.82991 0.995790i 0 −7.96394 4.59798i 0 6.18838 10.7186i 0 7.01680 + 5.63600i 0
425.5 0 −2.75973 + 1.17639i 0 0.349882 + 0.202004i 0 5.14359 8.90897i 0 6.23221 6.49304i 0
425.6 0 −2.69852 + 1.31072i 0 −7.80899 4.50852i 0 −2.96809 + 5.14088i 0 5.56402 7.07401i 0
425.7 0 −2.66330 1.38089i 0 7.06805 + 4.08074i 0 −1.62444 + 2.81362i 0 5.18629 + 7.35543i 0
425.8 0 −2.59273 1.50922i 0 −3.90584 2.25504i 0 1.48435 2.57096i 0 4.44449 + 7.82601i 0
425.9 0 −2.37911 + 1.82752i 0 5.32501 + 3.07440i 0 −6.67945 + 11.5692i 0 2.32035 8.69574i 0
425.10 0 −2.18435 + 2.05636i 0 −3.91903 2.26265i 0 −1.50207 + 2.60166i 0 0.542755 8.98362i 0
425.11 0 −1.89926 2.32224i 0 1.60071 + 0.924171i 0 2.46982 4.27785i 0 −1.78559 + 8.82109i 0
425.12 0 −1.76470 2.42608i 0 2.53393 + 1.46297i 0 −2.71227 + 4.69779i 0 −2.77170 + 8.56257i 0
425.13 0 −1.61894 + 2.52567i 0 6.09418 + 3.51848i 0 2.95479 5.11784i 0 −3.75805 8.17784i 0
425.14 0 −1.44827 2.62727i 0 −4.85266 2.80168i 0 −5.05578 + 8.75686i 0 −4.80506 + 7.60996i 0
425.15 0 −1.18332 2.75677i 0 3.94117 + 2.27544i 0 5.85150 10.1351i 0 −6.19951 + 6.52427i 0
425.16 0 −1.01102 + 2.82451i 0 −1.73205 0.999998i 0 −0.0735988 + 0.127477i 0 −6.95566 5.71129i 0
425.17 0 −0.759041 + 2.90239i 0 −4.68550 2.70517i 0 0.913116 1.58156i 0 −7.84771 4.40606i 0
425.18 0 −0.456660 + 2.96504i 0 5.47473 + 3.16084i 0 2.00114 3.46607i 0 −8.58292 2.70803i 0
425.19 0 −0.391198 2.97438i 0 −5.10431 2.94698i 0 −0.351537 + 0.608880i 0 −8.69393 + 2.32715i 0
425.20 0 0.0979727 2.99840i 0 5.49889 + 3.17478i 0 −3.73312 + 6.46596i 0 −8.98080 0.587522i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 581.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.be.a yes 80
3.b odd 2 1 2052.3.be.a 80
9.c even 3 1 2052.3.m.a 80
9.d odd 6 1 684.3.m.a 80
19.c even 3 1 684.3.m.a 80
57.h odd 6 1 2052.3.m.a 80
171.h even 3 1 2052.3.be.a 80
171.j odd 6 1 inner 684.3.be.a yes 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.m.a 80 9.d odd 6 1
684.3.m.a 80 19.c even 3 1
684.3.be.a yes 80 1.a even 1 1 trivial
684.3.be.a yes 80 171.j odd 6 1 inner
2052.3.m.a 80 9.c even 3 1
2052.3.m.a 80 57.h odd 6 1
2052.3.be.a 80 3.b odd 2 1
2052.3.be.a 80 171.h even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).