Properties

Label 690.2.x.a
Level $690$
Weight $2$
Character orbit 690.x
Analytic conductor $5.510$
Analytic rank $0$
Dimension $960$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.x (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(960\)
Relative dimension: \(48\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960q + 8q^{3} + 8q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 960q + 8q^{3} + 8q^{6} - 8q^{12} - 16q^{13} + 8q^{15} + 96q^{16} + 72q^{18} + 16q^{22} + 32q^{25} + 8q^{27} - 16q^{31} + 36q^{33} - 8q^{36} + 24q^{37} - 48q^{43} - 16q^{46} - 8q^{48} - 32q^{51} + 16q^{52} - 64q^{55} - 16q^{57} + 8q^{60} - 96q^{61} + 72q^{63} - 144q^{66} + 64q^{67} - 16q^{70} + 16q^{72} + 48q^{73} + 4q^{75} - 24q^{78} - 248q^{81} - 32q^{82} + 64q^{85} - 8q^{87} + 16q^{88} + 40q^{90} - 96q^{91} - 104q^{93} - 8q^{96} - 264q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −0.599278 0.800541i −1.72986 + 0.0871816i −0.281733 + 0.959493i 2.00169 + 0.996620i 1.10646 + 1.33257i −0.163768 0.752829i 0.936950 0.349464i 2.98480 0.301623i −0.401731 2.19968i
77.2 −0.599278 0.800541i −1.72143 0.191520i −0.281733 + 0.959493i −0.512913 + 2.17645i 0.878295 + 1.49285i 0.0972258 + 0.446940i 0.936950 0.349464i 2.92664 + 0.659376i 2.04971 0.893688i
77.3 −0.599278 0.800541i −1.71709 0.227188i −0.281733 + 0.959493i −1.19043 1.89285i 0.847138 + 1.51075i −0.982544 4.51668i 0.936950 0.349464i 2.89677 + 0.780204i −0.801902 + 2.08733i
77.4 −0.599278 0.800541i −1.54741 + 0.778153i −0.281733 + 0.959493i −1.72896 1.41799i 1.55027 + 0.772436i 0.112122 + 0.515415i 0.936950 0.349464i 1.78896 2.40824i −0.0990383 + 2.23387i
77.5 −0.599278 0.800541i −1.39273 1.02971i −0.281733 + 0.959493i 0.737339 2.11100i 0.0103038 + 1.73202i 0.427775 + 1.96645i 0.936950 0.349464i 0.879385 + 2.86822i −2.13182 + 0.674807i
77.6 −0.599278 0.800541i −1.24279 1.20643i −0.281733 + 0.959493i −2.22744 0.196281i −0.221027 + 1.71789i 0.926409 + 4.25863i 0.936950 0.349464i 0.0890332 + 2.99868i 1.17772 + 1.90078i
77.7 −0.599278 0.800541i −1.10274 + 1.33565i −0.281733 + 0.959493i 1.49334 + 1.66431i 1.73009 + 0.0823679i 1.04650 + 4.81067i 0.936950 0.349464i −0.567912 2.94576i 0.437427 2.19287i
77.8 −0.599278 0.800541i −0.987144 + 1.42322i −0.281733 + 0.959493i 1.97219 1.05378i 1.73092 0.0626522i −0.0325198 0.149491i 0.936950 0.349464i −1.05109 2.80984i −2.02549 0.947313i
77.9 −0.599278 0.800541i −0.983174 1.42596i −0.281733 + 0.959493i 2.07963 0.821679i −0.552347 + 1.64162i −0.944966 4.34394i 0.936950 0.349464i −1.06674 + 2.80394i −1.90406 1.17241i
77.10 −0.599278 0.800541i −0.925925 1.46378i −0.281733 + 0.959493i −1.19343 + 1.89096i −0.616933 + 1.61845i −0.549750 2.52716i 0.936950 0.349464i −1.28533 + 2.71071i 2.22899 0.177821i
77.11 −0.599278 0.800541i −0.682476 + 1.59193i −0.281733 + 0.959493i −1.62117 + 1.54007i 1.68339 0.407655i −0.461734 2.12256i 0.936950 0.349464i −2.06845 2.17290i 2.20442 + 0.374883i
77.12 −0.599278 0.800541i −0.264597 1.71172i −0.281733 + 0.959493i 0.462257 + 2.18777i −1.21174 + 1.23762i 0.120371 + 0.553337i 0.936950 0.349464i −2.85998 + 0.905834i 1.47438 1.68114i
77.13 −0.599278 0.800541i −0.0145841 + 1.73199i −0.281733 + 0.959493i −0.631950 2.14491i 1.39527 1.02627i 0.805423 + 3.70247i 0.936950 0.349464i −2.99957 0.0505191i −1.33838 + 1.79130i
77.14 −0.599278 0.800541i 0.114573 1.72826i −0.281733 + 0.959493i 2.16257 0.568575i −1.45220 + 0.943986i 0.773495 + 3.55570i 0.936950 0.349464i −2.97375 0.396023i −1.75115 1.39049i
77.15 −0.599278 0.800541i 0.543497 + 1.64457i −0.281733 + 0.959493i 1.86405 + 1.23504i 0.990841 1.42065i −0.773513 3.55578i 0.936950 0.349464i −2.40922 + 1.78764i −0.128382 2.23238i
77.16 −0.599278 0.800541i 0.807481 1.53231i −0.281733 + 0.959493i −0.793048 2.09071i −1.71058 + 0.271858i −0.161893 0.744209i 0.936950 0.349464i −1.69595 2.47462i −1.19845 + 1.88778i
77.17 −0.599278 0.800541i 0.959317 + 1.44212i −0.281733 + 0.959493i −0.937919 + 2.02985i 0.579579 1.63220i 0.495106 + 2.27597i 0.936950 0.349464i −1.15942 + 2.76690i 2.18706 0.465603i
77.18 −0.599278 0.800541i 1.00245 + 1.41248i −0.281733 + 0.959493i −1.92562 1.13666i 0.530001 1.64897i −1.05526 4.85093i 0.936950 0.349464i −0.990189 + 2.83188i 0.244034 + 2.22271i
77.19 −0.599278 0.800541i 1.03732 1.38707i −0.281733 + 0.959493i 1.58434 + 1.57793i −1.73205 0.000829396i −0.607715 2.79362i 0.936950 0.349464i −0.847954 2.87767i 0.313734 2.21395i
77.20 −0.599278 0.800541i 1.05792 1.37142i −0.281733 + 0.959493i −2.22480 + 0.224149i −1.73187 0.0250473i −0.0551290 0.253424i 0.936950 0.349464i −0.761601 2.90172i 1.51272 + 1.64672i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 683.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
115.k odd 44 1 inner
345.x even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.x.a 960
3.b odd 2 1 inner 690.2.x.a 960
5.c odd 4 1 inner 690.2.x.a 960
15.e even 4 1 inner 690.2.x.a 960
23.c even 11 1 inner 690.2.x.a 960
69.h odd 22 1 inner 690.2.x.a 960
115.k odd 44 1 inner 690.2.x.a 960
345.x even 44 1 inner 690.2.x.a 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.x.a 960 1.a even 1 1 trivial
690.2.x.a 960 3.b odd 2 1 inner
690.2.x.a 960 5.c odd 4 1 inner
690.2.x.a 960 15.e even 4 1 inner
690.2.x.a 960 23.c even 11 1 inner
690.2.x.a 960 69.h odd 22 1 inner
690.2.x.a 960 115.k odd 44 1 inner
690.2.x.a 960 345.x even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(690, [\chi])\).