Properties

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

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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: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{5}\)\(=\)\( \nu^{6} + 6 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} + 13 \nu^{3} \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} + 29 \nu^{3} \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 3 \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{6}\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{4} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{5} - 9 \beta_{3}\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 29 \beta_{6}\)\()/2\)

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(1\)

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
1.14412 + 1.14412i
−0.437016 0.437016i
−1.14412 1.14412i
0.437016 + 0.437016i
−0.437016 + 0.437016i
1.14412 1.14412i
0.437016 0.437016i
−1.14412 + 1.14412i
−0.707107 + 0.707107i −1.70711 + 0.292893i 1.00000i 2.23607i 1.00000 1.41421i 0 0.707107 + 0.707107i 2.82843 1.00000i 1.58114 + 1.58114i
47.2 −0.707107 + 0.707107i −1.70711 + 0.292893i 1.00000i 2.23607i 1.00000 1.41421i 0 0.707107 + 0.707107i 2.82843 1.00000i −1.58114 1.58114i
47.3 0.707107 0.707107i −0.292893 + 1.70711i 1.00000i 2.23607i 1.00000 + 1.41421i 0 −0.707107 0.707107i −2.82843 1.00000i −1.58114 1.58114i
47.4 0.707107 0.707107i −0.292893 + 1.70711i 1.00000i 2.23607i 1.00000 + 1.41421i 0 −0.707107 0.707107i −2.82843 1.00000i 1.58114 + 1.58114i
323.1 −0.707107 0.707107i −1.70711 0.292893i 1.00000i 2.23607i 1.00000 + 1.41421i 0 0.707107 0.707107i 2.82843 + 1.00000i −1.58114 + 1.58114i
323.2 −0.707107 0.707107i −1.70711 0.292893i 1.00000i 2.23607i 1.00000 + 1.41421i 0 0.707107 0.707107i 2.82843 + 1.00000i 1.58114 1.58114i
323.3 0.707107 + 0.707107i −0.292893 1.70711i 1.00000i 2.23607i 1.00000 1.41421i 0 −0.707107 + 0.707107i −2.82843 + 1.00000i 1.58114 1.58114i
323.4 0.707107 + 0.707107i −0.292893 1.70711i 1.00000i 2.23607i 1.00000 1.41421i 0 −0.707107 + 0.707107i −2.82843 + 1.00000i −1.58114 + 1.58114i
\(n\): 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.d 8
3.b odd 2 1 inner 690.2.i.d 8
5.c odd 4 1 inner 690.2.i.d 8
15.e even 4 1 inner 690.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.i.d 8 1.a even 1 1 trivial
690.2.i.d 8 3.b odd 2 1 inner
690.2.i.d 8 5.c odd 4 1 inner
690.2.i.d 8 15.e even 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$5$ \( ( 5 + T^{2} )^{4} \)
$7$ \( T^{8} \)
$11$ \( ( 8 + T^{2} )^{4} \)
$13$ \( ( 324 + 72 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$17$ \( 1296 + 712 T^{4} + T^{8} \)
$19$ \( ( 36 + 28 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( -20 + T^{2} )^{4} \)
$31$ \( ( -24 - 8 T + T^{2} )^{4} \)
$37$ \( ( 144 + 96 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$41$ \( ( 5184 + 176 T^{2} + T^{4} )^{2} \)
$43$ \( ( 2704 + 1248 T + 288 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$47$ \( 256 + 23072 T^{4} + T^{8} \)
$53$ \( ( 8100 + T^{4} )^{2} \)
$59$ \( ( -98 + T^{2} )^{4} \)
$61$ \( ( -8 + T )^{8} \)
$67$ \( ( 144 + 96 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$71$ \( ( 324 + 44 T^{2} + T^{4} )^{2} \)
$73$ \( ( 6084 - 312 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$79$ \( ( 10 + T^{2} )^{4} \)
$83$ \( 1679616 + 5152 T^{4} + T^{8} \)
$89$ \( ( 64 - 304 T^{2} + T^{4} )^{2} \)
$97$ \( ( 144 - 96 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
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