Properties

Label 690.2.d.c
Level $690$
Weight $2$
Character orbit 690.d
Analytic conductor $5.510$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -7 \nu^{5} + 10 \nu^{4} - 5 \nu^{3} - 30 \nu^{2} - 32 \nu + 13 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -9 \nu^{5} + 3 \nu^{4} + 10 \nu^{3} - 32 \nu^{2} - 74 \nu - 3 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( -10 \nu^{5} + 11 \nu^{4} - 17 \nu^{3} - 10 \nu^{2} - 72 \nu - 11 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( 12 \nu^{5} - 27 \nu^{4} + 25 \nu^{3} + 12 \nu^{2} + 68 \nu - 65 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -19 \nu^{5} + 37 \nu^{4} - 30 \nu^{3} - 42 \nu^{2} - 54 \nu + 55 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{2} - 4 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} - 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{5} - 5 \beta_{4} - 5 \beta_{3} + \beta_{2} - 14\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{5} - 11 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 18 \beta_{1} - 18\)\()/2\)

Character values

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

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

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
0.403032 + 0.403032i
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
1.00000i 1.00000i −1.00000 −1.67513 1.48119i −1.00000 2.96239i 1.00000i −1.00000 −1.48119 + 1.67513i
139.2 1.00000i 1.00000i −1.00000 −0.539189 + 2.17009i −1.00000 4.34017i 1.00000i −1.00000 2.17009 + 0.539189i
139.3 1.00000i 1.00000i −1.00000 2.21432 + 0.311108i −1.00000 0.622216i 1.00000i −1.00000 0.311108 2.21432i
139.4 1.00000i 1.00000i −1.00000 −1.67513 + 1.48119i −1.00000 2.96239i 1.00000i −1.00000 −1.48119 1.67513i
139.5 1.00000i 1.00000i −1.00000 −0.539189 2.17009i −1.00000 4.34017i 1.00000i −1.00000 2.17009 0.539189i
139.6 1.00000i 1.00000i −1.00000 2.21432 0.311108i −1.00000 0.622216i 1.00000i −1.00000 0.311108 + 2.21432i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.6
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.c 6
3.b odd 2 1 2070.2.d.e 6
5.b even 2 1 inner 690.2.d.c 6
5.c odd 4 1 3450.2.a.bo 3
5.c odd 4 1 3450.2.a.bt 3
15.d odd 2 1 2070.2.d.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.c 6 1.a even 1 1 trivial
690.2.d.c 6 5.b even 2 1 inner
2070.2.d.e 6 3.b odd 2 1
2070.2.d.e 6 15.d odd 2 1
3450.2.a.bo 3 5.c odd 4 1
3450.2.a.bt 3 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 \(S_{2}^{\mathrm{new}}(690, [\chi])\):

\( T_{7}^{6} + 28 T_{7}^{4} + 176 T_{7}^{2} + 64 \)
\( T_{11}^{3} - 16 T_{11} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 - 5 T^{2} - 16 T^{3} - T^{4} + T^{6} \)
$7$ \( 64 + 176 T^{2} + 28 T^{4} + T^{6} \)
$11$ \( ( -16 - 16 T + T^{3} )^{2} \)
$13$ \( 256 + 192 T^{2} + 32 T^{4} + T^{6} \)
$17$ \( 64 + 112 T^{2} + 44 T^{4} + T^{6} \)
$19$ \( ( 16 - 8 T - 4 T^{2} + T^{3} )^{2} \)
$23$ \( ( 1 + T^{2} )^{3} \)
$29$ \( ( -344 + 4 T + 14 T^{2} + T^{3} )^{2} \)
$31$ \( ( 4 + T )^{6} \)
$37$ \( 53824 + 9392 T^{2} + 204 T^{4} + T^{6} \)
$41$ \( ( 8 - 52 T + 6 T^{2} + T^{3} )^{2} \)
$43$ \( 64 + 176 T^{2} + 108 T^{4} + T^{6} \)
$47$ \( 65536 + 8192 T^{2} + 192 T^{4} + T^{6} \)
$53$ \( 64 + 560 T^{2} + 60 T^{4} + T^{6} \)
$59$ \( ( -104 - 84 T + 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( -1112 - 68 T + 14 T^{2} + T^{3} )^{2} \)
$67$ \( 53824 + 9392 T^{2} + 204 T^{4} + T^{6} \)
$71$ \( ( -232 - 124 T - 6 T^{2} + T^{3} )^{2} \)
$73$ \( 16384 + 3072 T^{2} + 128 T^{4} + T^{6} \)
$79$ \( ( 592 - 144 T - 4 T^{2} + T^{3} )^{2} \)
$83$ \( 25600 + 6144 T^{2} + 176 T^{4} + T^{6} \)
$89$ \( ( -128 - 32 T + 8 T^{2} + T^{3} )^{2} \)
$97$ \( 61504 + 14640 T^{2} + 252 T^{4} + T^{6} \)
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