Properties

Label 690.2.i.c
Level $690$
Weight $2$
Character orbit 690.i
Analytic conductor $5.510$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2} + ( -1 + \zeta_{24}^{3} + \zeta_{24}^{6} ) q^{3} -\zeta_{24}^{6} q^{4} + ( -\zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( -1 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{3} q^{8} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{2} + ( -1 + \zeta_{24}^{3} + \zeta_{24}^{6} ) q^{3} -\zeta_{24}^{6} q^{4} + ( -\zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( -1 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{3} q^{8} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{9} + ( -1 - \zeta_{24}^{3} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{10} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{11} + ( 1 + \zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{12} + ( -1 + 2 \zeta_{24}^{3} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{13} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{14} + ( -2 + \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{15} - q^{16} + ( 3 + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{17} + ( 2 + \zeta_{24}^{3} - 2 \zeta_{24}^{6} ) q^{18} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( 1 + \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{20} + ( 2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{21} + ( 4 + 4 \zeta_{24}^{6} ) q^{22} -\zeta_{24}^{3} q^{23} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{24} + ( 1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{25} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{26} + ( -1 + 5 \zeta_{24} - 5 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{27} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{28} + ( -4 \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{29} + ( 2 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{30} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{32} + ( 4 + 8 \zeta_{24}^{3} - 4 \zeta_{24}^{6} ) q^{33} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{34} + ( 2 + 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{35} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{36} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{37} + ( -1 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{38} + ( \zeta_{24} + 4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{39} + ( -1 + \zeta_{24} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{40} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{41} + ( -2 + 2 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{42} + ( -2 - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{43} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{44} + ( 1 - \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{45} + q^{46} + ( 2 - 2 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{47} + ( 1 - \zeta_{24}^{3} - \zeta_{24}^{6} ) q^{48} + 5 \zeta_{24}^{6} q^{49} + ( 2 - \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{50} + ( -6 + 3 \zeta_{24} + 3 \zeta_{24}^{3} + 12 \zeta_{24}^{4} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{51} + ( 1 - 2 \zeta_{24} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{52} + ( -3 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{53} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{54} + ( 8 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{55} + ( 2 - 4 \zeta_{24}^{4} ) q^{56} + ( 1 + 2 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{57} + ( 4 + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{58} + ( \zeta_{24} + 8 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{59} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{60} + ( -4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{61} + ( 2 + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{62} + ( -4 + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{63} + \zeta_{24}^{6} q^{64} + ( -3 + 6 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{65} + ( -8 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{66} + ( -6 + 4 \zeta_{24} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{67} + ( 3 - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{68} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{69} + ( 6 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{70} + ( 6 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{71} + ( -2 + \zeta_{24} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{72} + ( 3 + 4 \zeta_{24}^{3} - 3 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{73} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{74} + ( -3 - 4 \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{75} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{76} + ( -8 - 16 \zeta_{24}^{2} + 16 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{77} + ( 3 - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{78} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{79} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{80} + ( 7 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{81} + ( 6 + 6 \zeta_{24}^{6} ) q^{82} + 2 \zeta_{24}^{3} q^{83} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{84} + ( 9 + 6 \zeta_{24} + 6 \zeta_{24}^{5} - 9 \zeta_{24}^{6} ) q^{85} + ( 4 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{86} + ( -2 + 6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 10 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{87} + ( 4 - 4 \zeta_{24}^{6} ) q^{88} + ( 4 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{89} + ( 1 + \zeta_{24} + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{90} + ( -12 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{91} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{92} + ( 2 - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{93} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{94} + ( 3 \zeta_{24} + 4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{95} + ( 1 - \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{96} + ( 2 + 2 \zeta_{24} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{97} -5 \zeta_{24}^{3} q^{98} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 16 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} - 8q^{6} + O(q^{10}) \) \( 8q - 8q^{3} - 8q^{6} - 8q^{10} + 8q^{12} - 8q^{13} - 8q^{15} - 8q^{16} + 16q^{18} + 32q^{22} + 8q^{25} - 8q^{27} + 16q^{30} + 32q^{33} - 8q^{36} - 8q^{40} - 16q^{43} + 8q^{46} + 8q^{48} + 8q^{52} + 64q^{55} + 32q^{58} + 8q^{60} - 32q^{61} - 64q^{66} - 48q^{67} + 48q^{70} - 16q^{72} + 24q^{73} - 8q^{75} + 8q^{78} + 56q^{81} + 48q^{82} + 72q^{85} - 32q^{87} + 32q^{88} - 8q^{90} - 96q^{91} + 8q^{96} + 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(\zeta_{24}^{3}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.258819 0.965926i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
0.258819 0.965926i
−0.965926 + 0.258819i
−0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i −1.73205 + 1.41421i −1.00000 1.41421i 2.44949 + 2.44949i 0.707107 + 0.707107i −2.82843 1.00000i 0.224745 2.22474i
47.2 −0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i 1.73205 + 1.41421i −1.00000 1.41421i −2.44949 2.44949i 0.707107 + 0.707107i −2.82843 1.00000i −2.22474 + 0.224745i
47.3 0.707107 0.707107i −1.70711 + 0.292893i 1.00000i −1.73205 1.41421i −1.00000 + 1.41421i −2.44949 2.44949i −0.707107 0.707107i 2.82843 1.00000i −2.22474 + 0.224745i
47.4 0.707107 0.707107i −1.70711 + 0.292893i 1.00000i 1.73205 1.41421i −1.00000 + 1.41421i 2.44949 + 2.44949i −0.707107 0.707107i 2.82843 1.00000i 0.224745 2.22474i
323.1 −0.707107 0.707107i −0.292893 1.70711i 1.00000i −1.73205 1.41421i −1.00000 + 1.41421i 2.44949 2.44949i 0.707107 0.707107i −2.82843 + 1.00000i 0.224745 + 2.22474i
323.2 −0.707107 0.707107i −0.292893 1.70711i 1.00000i 1.73205 1.41421i −1.00000 + 1.41421i −2.44949 + 2.44949i 0.707107 0.707107i −2.82843 + 1.00000i −2.22474 0.224745i
323.3 0.707107 + 0.707107i −1.70711 0.292893i 1.00000i −1.73205 + 1.41421i −1.00000 1.41421i −2.44949 + 2.44949i −0.707107 + 0.707107i 2.82843 + 1.00000i −2.22474 0.224745i
323.4 0.707107 + 0.707107i −1.70711 0.292893i 1.00000i 1.73205 + 1.41421i −1.00000 1.41421i 2.44949 2.44949i −0.707107 + 0.707107i 2.82843 + 1.00000i 0.224745 + 2.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.4
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.i.c 8
3.b odd 2 1 inner 690.2.i.c 8
5.c odd 4 1 inner 690.2.i.c 8
15.e even 4 1 inner 690.2.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.i.c 8 1.a even 1 1 trivial
690.2.i.c 8 3.b odd 2 1 inner
690.2.i.c 8 5.c odd 4 1 inner
690.2.i.c 8 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 144 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$5$ \( ( 25 - 2 T^{2} + T^{4} )^{2} \)
$7$ \( ( 144 + T^{4} )^{2} \)
$11$ \( ( 32 + T^{2} )^{4} \)
$13$ \( ( 100 - 40 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$17$ \( ( 2916 + T^{4} )^{2} \)
$19$ \( ( 6 + T^{2} )^{4} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( 400 - 88 T^{2} + T^{4} )^{2} \)
$31$ \( ( -24 + T^{2} )^{4} \)
$37$ \( ( 144 + T^{4} )^{2} \)
$41$ \( ( 72 + T^{2} )^{4} \)
$43$ \( ( 1600 - 320 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$47$ \( 160000 + 2336 T^{4} + T^{8} \)
$53$ \( 104976 + 31752 T^{4} + T^{8} \)
$59$ \( ( 2116 - 100 T^{2} + T^{4} )^{2} \)
$61$ \( ( -8 + 8 T + T^{2} )^{4} \)
$67$ \( ( 576 + 576 T + 288 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$71$ \( ( 8100 + 252 T^{2} + T^{4} )^{2} \)
$73$ \( ( 900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$79$ \( ( 4 + 20 T^{2} + T^{4} )^{2} \)
$83$ \( ( 16 + T^{4} )^{2} \)
$89$ \( ( 400 - 88 T^{2} + T^{4} )^{2} \)
$97$ \( ( 16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
show more
show less