Properties

Label 690.3.b.a
Level $690$
Weight $3$
Character orbit 690.b
Analytic conductor $18.801$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 690.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.8011382409\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88q + 176q^{4} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 88q + 176q^{4} + 8q^{9} + 32q^{10} - 12q^{15} + 352q^{16} - 16q^{19} - 176q^{21} + 72q^{25} - 72q^{30} + 32q^{31} + 160q^{34} + 16q^{36} + 144q^{39} + 64q^{40} + 92q^{45} - 360q^{49} + 48q^{51} - 144q^{54} + 16q^{55} - 24q^{60} + 208q^{61} + 704q^{64} + 512q^{66} + 304q^{70} + 536q^{75} - 32q^{76} + 448q^{79} - 24q^{81} - 352q^{84} - 96q^{85} + 32q^{90} - 64q^{91} + 160q^{94} + 296q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1 −1.41421 −2.99999 0.00596006i 2.00000 1.21044 + 4.85127i 4.24263 + 0.00842880i 6.90074i −2.82843 8.99993 + 0.0357603i −1.71182 6.86073i
599.2 −1.41421 −2.99999 + 0.00596006i 2.00000 1.21044 4.85127i 4.24263 0.00842880i 6.90074i −2.82843 8.99993 0.0357603i −1.71182 + 6.86073i
599.3 −1.41421 −2.90508 0.748659i 2.00000 −4.90392 0.975491i 4.10841 + 1.05876i 3.09484i −2.82843 7.87902 + 4.34983i 6.93519 + 1.37955i
599.4 −1.41421 −2.90508 + 0.748659i 2.00000 −4.90392 + 0.975491i 4.10841 1.05876i 3.09484i −2.82843 7.87902 4.34983i 6.93519 1.37955i
599.5 −1.41421 −2.79936 1.07869i 2.00000 4.72534 + 1.63438i 3.95890 + 1.52550i 7.69411i −2.82843 6.67285 + 6.03929i −6.68264 2.31136i
599.6 −1.41421 −2.79936 + 1.07869i 2.00000 4.72534 1.63438i 3.95890 1.52550i 7.69411i −2.82843 6.67285 6.03929i −6.68264 + 2.31136i
599.7 −1.41421 −2.76001 1.17574i 2.00000 −2.52270 4.31694i 3.90324 + 1.66274i 6.80400i −2.82843 6.23528 + 6.49009i 3.56764 + 6.10507i
599.8 −1.41421 −2.76001 + 1.17574i 2.00000 −2.52270 + 4.31694i 3.90324 1.66274i 6.80400i −2.82843 6.23528 6.49009i 3.56764 6.10507i
599.9 −1.41421 −2.16793 2.07367i 2.00000 4.21296 2.69276i 3.06591 + 2.93261i 3.87132i −2.82843 0.399825 + 8.99111i −5.95803 + 3.80814i
599.10 −1.41421 −2.16793 + 2.07367i 2.00000 4.21296 + 2.69276i 3.06591 2.93261i 3.87132i −2.82843 0.399825 8.99111i −5.95803 3.80814i
599.11 −1.41421 −2.13931 2.10317i 2.00000 −2.73059 + 4.18854i 3.02545 + 2.97433i 4.11953i −2.82843 0.153334 + 8.99869i 3.86164 5.92349i
599.12 −1.41421 −2.13931 + 2.10317i 2.00000 −2.73059 4.18854i 3.02545 2.97433i 4.11953i −2.82843 0.153334 8.99869i 3.86164 + 5.92349i
599.13 −1.41421 −1.52209 2.58520i 2.00000 −4.64244 + 1.85681i 2.15256 + 3.65602i 10.7642i −2.82843 −4.36648 + 7.86981i 6.56540 2.62593i
599.14 −1.41421 −1.52209 + 2.58520i 2.00000 −4.64244 1.85681i 2.15256 3.65602i 10.7642i −2.82843 −4.36648 7.86981i 6.56540 + 2.62593i
599.15 −1.41421 −1.42652 2.63914i 2.00000 −3.84011 3.20212i 2.01740 + 3.73230i 12.7200i −2.82843 −4.93010 + 7.52955i 5.43073 + 4.52848i
599.16 −1.41421 −1.42652 + 2.63914i 2.00000 −3.84011 + 3.20212i 2.01740 3.73230i 12.7200i −2.82843 −4.93010 7.52955i 5.43073 4.52848i
599.17 −1.41421 −1.23483 2.73408i 2.00000 4.77075 1.49664i 1.74631 + 3.86657i 2.54772i −2.82843 −5.95040 + 6.75224i −6.74686 + 2.11656i
599.18 −1.41421 −1.23483 + 2.73408i 2.00000 4.77075 + 1.49664i 1.74631 3.86657i 2.54772i −2.82843 −5.95040 6.75224i −6.74686 2.11656i
599.19 −1.41421 −0.996475 2.82967i 2.00000 3.01455 + 3.98905i 1.40923 + 4.00176i 10.6805i −2.82843 −7.01408 + 5.63939i −4.26321 5.64137i
599.20 −1.41421 −0.996475 + 2.82967i 2.00000 3.01455 3.98905i 1.40923 4.00176i 10.6805i −2.82843 −7.01408 5.63939i −4.26321 + 5.64137i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.3.b.a 88
3.b odd 2 1 inner 690.3.b.a 88
5.b even 2 1 inner 690.3.b.a 88
15.d odd 2 1 inner 690.3.b.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.3.b.a 88 1.a even 1 1 trivial
690.3.b.a 88 3.b odd 2 1 inner
690.3.b.a 88 5.b even 2 1 inner
690.3.b.a 88 15.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(690, [\chi])\).