Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
1.61803i
0.618034i
1.61803i
0.618034i
1.00000i 1.00000i −1.00000 −2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
139.2 1.00000i 1.00000i −1.00000 2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
139.3 1.00000i 1.00000i −1.00000 −2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
139.4 1.00000i 1.00000i −1.00000 2.23607 1.00000 4.00000i 1.00000i −1.00000 2.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.d.a 4
3.b odd 2 1 2070.2.d.b 4
5.b even 2 1 inner 690.2.d.a 4
5.c odd 4 1 3450.2.a.bc 2
5.c odd 4 1 3450.2.a.bn 2
15.d odd 2 1 2070.2.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.a 4 1.a even 1 1 trivial
690.2.d.a 4 5.b even 2 1 inner
2070.2.d.b 4 3.b odd 2 1
2070.2.d.b 4 15.d odd 2 1
3450.2.a.bc 2 5.c odd 4 1
3450.2.a.bn 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( 256 + 48 T^{2} + T^{4} \)
$17$ \( 256 + 48 T^{2} + T^{4} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -4 + 8 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 16 + 72 T^{2} + T^{4} \)
$41$ \( ( -76 - 4 T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( 4096 + 192 T^{2} + T^{4} \)
$53$ \( ( 80 + T^{2} )^{2} \)
$59$ \( ( -6 + T )^{4} \)
$61$ \( ( -4 - 8 T + T^{2} )^{2} \)
$67$ \( 4096 + 192 T^{2} + T^{4} \)
$71$ \( ( 124 + 24 T + T^{2} )^{2} \)
$73$ \( 4096 + 192 T^{2} + T^{4} \)
$79$ \( ( 44 + 16 T + T^{2} )^{2} \)
$83$ \( 256 + 112 T^{2} + T^{4} \)
$89$ \( ( 124 - 24 T + T^{2} )^{2} \)
$97$ \( ( 180 + T^{2} )^{2} \)
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