Properties

Label 770.2.l.c
Level $770$
Weight $2$
Character orbit 770.l
Analytic conductor $6.148$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 40q^{11} + 56q^{15} - 40q^{16} - 8q^{18} - 4q^{21} - 32q^{25} - 24q^{30} + 48q^{35} - 48q^{36} - 8q^{37} - 40q^{42} - 8q^{43} + 32q^{46} + 16q^{50} - 8q^{51} + 56q^{53} + 4q^{56} + 32q^{57} - 8q^{58} + 8q^{60} - 16q^{63} + 8q^{65} - 24q^{67} - 4q^{70} - 24q^{71} - 8q^{72} - 16q^{78} - 24q^{81} + 96q^{85} - 96q^{86} + 64q^{91} + 24q^{93} - 112q^{95} + 16q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
573.1 −0.707107 + 0.707107i −2.03841 + 2.03841i 1.00000i −2.07139 + 0.842227i 2.88275i −2.55056 0.703324i 0.707107 + 0.707107i 5.31025i 0.869149 2.06024i
573.2 −0.707107 + 0.707107i −2.02780 + 2.02780i 1.00000i −0.515374 2.17587i 2.86774i 1.43371 2.22362i 0.707107 + 0.707107i 5.22391i 1.90299 + 1.17414i
573.3 −0.707107 + 0.707107i −0.903547 + 0.903547i 1.00000i −1.41673 1.72999i 1.27781i 1.36406 + 2.26701i 0.707107 + 0.707107i 1.36720i 2.22507 + 0.221508i
573.4 −0.707107 + 0.707107i −0.839228 + 0.839228i 1.00000i 0.690407 2.12681i 1.18685i −2.60331 + 0.472008i 0.707107 + 0.707107i 1.59139i 1.01569 + 1.99208i
573.5 −0.707107 + 0.707107i −0.0715208 + 0.0715208i 1.00000i 1.95282 1.08927i 0.101146i 2.63480 0.240478i 0.707107 + 0.707107i 2.98977i −0.610623 + 2.15108i
573.6 −0.707107 + 0.707107i 0.0715208 0.0715208i 1.00000i −1.95282 + 1.08927i 0.101146i 0.240478 2.63480i 0.707107 + 0.707107i 2.98977i 0.610623 2.15108i
573.7 −0.707107 + 0.707107i 0.839228 0.839228i 1.00000i −0.690407 + 2.12681i 1.18685i −0.472008 + 2.60331i 0.707107 + 0.707107i 1.59139i −1.01569 1.99208i
573.8 −0.707107 + 0.707107i 0.903547 0.903547i 1.00000i 1.41673 + 1.72999i 1.27781i −2.26701 1.36406i 0.707107 + 0.707107i 1.36720i −2.22507 0.221508i
573.9 −0.707107 + 0.707107i 2.02780 2.02780i 1.00000i 0.515374 + 2.17587i 2.86774i 2.22362 1.43371i 0.707107 + 0.707107i 5.22391i −1.90299 1.17414i
573.10 −0.707107 + 0.707107i 2.03841 2.03841i 1.00000i 2.07139 0.842227i 2.88275i 0.703324 + 2.55056i 0.707107 + 0.707107i 5.31025i −0.869149 + 2.06024i
573.11 0.707107 0.707107i −2.23110 + 2.23110i 1.00000i −0.288161 2.21742i 3.15525i 2.06987 + 1.64792i −0.707107 0.707107i 6.95559i −1.77171 1.36419i
573.12 0.707107 0.707107i −1.65110 + 1.65110i 1.00000i 2.15340 + 0.602381i 2.33501i 0.00754262 + 2.64574i −0.707107 0.707107i 2.45228i 1.94863 1.09674i
573.13 0.707107 0.707107i −1.49597 + 1.49597i 1.00000i −0.446287 + 2.19108i 2.11563i −1.17733 2.36936i −0.707107 0.707107i 1.47587i 1.23375 + 1.86490i
573.14 0.707107 0.707107i −0.949748 + 0.949748i 1.00000i −2.23301 + 0.116862i 1.34315i 2.47882 0.924898i −0.707107 0.707107i 1.19596i −1.49634 + 1.66161i
573.15 0.707107 0.707107i −0.602672 + 0.602672i 1.00000i −0.490670 2.18157i 0.852308i −2.60076 + 0.485856i −0.707107 0.707107i 2.27357i −1.88956 1.19565i
573.16 0.707107 0.707107i 0.602672 0.602672i 1.00000i 0.490670 + 2.18157i 0.852308i −0.485856 + 2.60076i −0.707107 0.707107i 2.27357i 1.88956 + 1.19565i
573.17 0.707107 0.707107i 0.949748 0.949748i 1.00000i 2.23301 0.116862i 1.34315i 0.924898 2.47882i −0.707107 0.707107i 1.19596i 1.49634 1.66161i
573.18 0.707107 0.707107i 1.49597 1.49597i 1.00000i 0.446287 2.19108i 2.11563i 2.36936 + 1.17733i −0.707107 0.707107i 1.47587i −1.23375 1.86490i
573.19 0.707107 0.707107i 1.65110 1.65110i 1.00000i −2.15340 0.602381i 2.33501i −2.64574 0.00754262i −0.707107 0.707107i 2.45228i −1.94863 + 1.09674i
573.20 0.707107 0.707107i 2.23110 2.23110i 1.00000i 0.288161 + 2.21742i 3.15525i −1.64792 2.06987i −0.707107 0.707107i 6.95559i 1.77171 + 1.36419i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 727.20
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
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.l.c 40
5.c odd 4 1 inner 770.2.l.c 40
7.b odd 2 1 inner 770.2.l.c 40
35.f even 4 1 inner 770.2.l.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.l.c 40 1.a even 1 1 trivial
770.2.l.c 40 5.c odd 4 1 inner
770.2.l.c 40 7.b odd 2 1 inner
770.2.l.c 40 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).