Properties

Label 770.2.bi.a
Level $770$
Weight $2$
Character orbit 770.bi
Analytic conductor $6.148$
Analytic rank $0$
Dimension $384$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bi (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(48\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384q - 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 384q - 4q^{7} + 24q^{15} + 96q^{16} + 16q^{22} + 16q^{23} - 40q^{25} + 4q^{28} + 16q^{30} - 32q^{35} + 80q^{36} - 64q^{37} + 44q^{42} - 80q^{43} - 16q^{46} - 64q^{50} - 136q^{51} + 80q^{53} + 8q^{56} - 112q^{57} - 96q^{58} + 68q^{63} - 272q^{65} + 160q^{67} - 8q^{70} + 16q^{71} + 96q^{77} + 24q^{81} - 80q^{85} - 16q^{88} - 84q^{91} + 16q^{92} + 88q^{93} - 184q^{95} - 32q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −0.453990 + 0.891007i −0.454148 2.86738i −0.587785 0.809017i −1.32097 + 1.80417i 2.76103 + 0.897114i −2.60332 0.471965i 0.987688 0.156434i −5.16244 + 1.67738i −1.00782 1.99607i
27.2 −0.453990 + 0.891007i −0.445742 2.81431i −0.587785 0.809017i 2.06857 + 0.849127i 2.70993 + 0.880509i 1.00237 2.44852i 0.987688 0.156434i −4.86846 + 1.58186i −1.69569 + 1.45761i
27.3 −0.453990 + 0.891007i −0.442024 2.79083i −0.587785 0.809017i 0.523489 2.17393i 2.68732 + 0.873164i −2.62728 + 0.312118i 0.987688 0.156434i −4.74017 + 1.54018i 1.69932 + 1.45337i
27.4 −0.453990 + 0.891007i −0.376983 2.38018i −0.587785 0.809017i 0.805590 + 2.08591i 2.29190 + 0.744683i 0.945612 + 2.47100i 0.987688 0.156434i −2.66995 + 0.867519i −2.22429 0.229198i
27.5 −0.453990 + 0.891007i −0.370844 2.34142i −0.587785 0.809017i −1.72342 1.42472i 2.25458 + 0.732557i 1.14583 + 2.38476i 0.987688 0.156434i −2.49155 + 0.809553i 2.05185 0.888763i
27.6 −0.453990 + 0.891007i −0.355652 2.24550i −0.587785 0.809017i −1.77729 + 1.35692i 2.16222 + 0.702547i 2.50760 0.843757i 0.987688 0.156434i −2.06261 + 0.670181i −0.402154 2.19961i
27.7 −0.453990 + 0.891007i −0.275125 1.73707i −0.587785 0.809017i 2.06245 0.863892i 1.67264 + 0.543475i −0.479663 + 2.60191i 0.987688 0.156434i −0.0885490 + 0.0287713i −0.166598 + 2.22985i
27.8 −0.453990 + 0.891007i −0.205427 1.29702i −0.587785 0.809017i −0.486212 2.18257i 1.24891 + 0.405797i −0.970889 2.46117i 0.987688 0.156434i 1.21312 0.394165i 2.16542 + 0.557647i
27.9 −0.453990 + 0.891007i −0.160862 1.01564i −0.587785 0.809017i 2.16890 0.543949i 0.977976 + 0.317764i 2.64207 0.139461i 0.987688 0.156434i 1.84751 0.600293i −0.499997 + 2.17945i
27.10 −0.453990 + 0.891007i −0.0917129 0.579052i −0.587785 0.809017i −2.22927 0.174197i 0.557576 + 0.181167i −1.87792 + 1.86371i 0.987688 0.156434i 2.52628 0.820838i 1.16728 1.90721i
27.11 −0.453990 + 0.891007i −0.0293280 0.185170i −0.587785 0.809017i −0.0774713 + 2.23473i 0.178302 + 0.0579339i 2.16357 + 1.52281i 0.987688 0.156434i 2.81974 0.916190i −1.95598 1.08357i
27.12 −0.453990 + 0.891007i −0.00505097 0.0318906i −0.587785 0.809017i 2.20552 + 0.368341i 0.0307078 + 0.00997756i −2.58959 + 0.542252i 0.987688 0.156434i 2.85218 0.926729i −1.32948 + 1.79791i
27.13 −0.453990 + 0.891007i 0.00505097 + 0.0318906i −0.587785 0.809017i −2.20552 0.368341i −0.0307078 0.00997756i −2.29528 1.31594i 0.987688 0.156434i 2.85218 0.926729i 1.32948 1.79791i
27.14 −0.453990 + 0.891007i 0.0293280 + 0.185170i −0.587785 0.809017i 0.0774713 2.23473i −0.178302 0.0579339i 2.52825 0.779702i 0.987688 0.156434i 2.81974 0.916190i 1.95598 + 1.08357i
27.15 −0.453990 + 0.891007i 0.0917129 + 0.579052i −0.587785 0.809017i 2.22927 + 0.174197i −0.557576 0.181167i −1.21009 2.35280i 0.987688 0.156434i 2.52628 0.820838i −1.16728 + 1.90721i
27.16 −0.453990 + 0.891007i 0.160862 + 1.01564i −0.587785 0.809017i −2.16890 + 0.543949i −0.977976 0.317764i 2.46967 + 0.949081i 0.987688 0.156434i 1.84751 0.600293i 0.499997 2.17945i
27.17 −0.453990 + 0.891007i 0.205427 + 1.29702i −0.587785 0.809017i 0.486212 + 2.18257i −1.24891 0.405797i −1.68391 + 2.04069i 0.987688 0.156434i 1.21312 0.394165i −2.16542 0.557647i
27.18 −0.453990 + 0.891007i 0.275125 + 1.73707i −0.587785 0.809017i −2.06245 + 0.863892i −1.67264 0.543475i 0.347847 2.62279i 0.987688 0.156434i −0.0885490 + 0.0287713i 0.166598 2.22985i
27.19 −0.453990 + 0.891007i 0.355652 + 2.24550i −0.587785 0.809017i 1.77729 1.35692i −2.16222 0.702547i 2.12414 + 1.57735i 0.987688 0.156434i −2.06261 + 0.670181i 0.402154 + 2.19961i
27.20 −0.453990 + 0.891007i 0.370844 + 2.34142i −0.587785 0.809017i 1.72342 + 1.42472i −2.25458 0.732557i 1.82668 1.91396i 0.987688 0.156434i −2.49155 + 0.809553i −2.05185 + 0.888763i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 713.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
11.c even 5 1 inner
35.f even 4 1 inner
55.k odd 20 1 inner
77.j odd 10 1 inner
385.bk even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bi.a 384
5.c odd 4 1 inner 770.2.bi.a 384
7.b odd 2 1 inner 770.2.bi.a 384
11.c even 5 1 inner 770.2.bi.a 384
35.f even 4 1 inner 770.2.bi.a 384
55.k odd 20 1 inner 770.2.bi.a 384
77.j odd 10 1 inner 770.2.bi.a 384
385.bk even 20 1 inner 770.2.bi.a 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bi.a 384 1.a even 1 1 trivial
770.2.bi.a 384 5.c odd 4 1 inner
770.2.bi.a 384 7.b odd 2 1 inner
770.2.bi.a 384 11.c even 5 1 inner
770.2.bi.a 384 35.f even 4 1 inner
770.2.bi.a 384 55.k odd 20 1 inner
770.2.bi.a 384 77.j odd 10 1 inner
770.2.bi.a 384 385.bk even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(770, [\chi])\).