Properties

Label 690.2.d.d
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: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{5} + 25 \nu^{4} + 10 \nu^{3} - 4 \nu^{2} - 121 \nu + 323 \)\()/121\)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{5} - 27 \nu^{4} - 35 \nu^{3} + 14 \nu^{2} - 121 \nu - 223 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{4}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1573 \nu + 574 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( 75 \nu^{5} + 30 \nu^{4} + 12 \nu^{3} - 392 \nu^{2} + 1573 \nu - 774 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} - \beta_{4} - \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + \beta_{4} - 6 \beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 5 \beta_{1} + 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{5} - 9 \beta_{4} - 5 \beta_{2} + 5 \beta_{1} - 30\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(25 \beta_{5} + 29 \beta_{4} - 32 \beta_{3} + 29 \beta_{2} + 25 \beta_{1} + 32\)\()/2\)

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
−1.75233 + 1.75233i
0.432320 0.432320i
1.32001 1.32001i
−1.75233 1.75233i
0.432320 + 0.432320i
1.32001 + 1.32001i
1.00000i 1.00000i −1.00000 −1.75233 + 1.38900i −1.00000 2.00000i 1.00000i −1.00000 1.38900 + 1.75233i
139.2 1.00000i 1.00000i −1.00000 0.432320 2.19388i −1.00000 2.00000i 1.00000i −1.00000 −2.19388 0.432320i
139.3 1.00000i 1.00000i −1.00000 1.32001 + 1.80487i −1.00000 2.00000i 1.00000i −1.00000 1.80487 1.32001i
139.4 1.00000i 1.00000i −1.00000 −1.75233 1.38900i −1.00000 2.00000i 1.00000i −1.00000 1.38900 1.75233i
139.5 1.00000i 1.00000i −1.00000 0.432320 + 2.19388i −1.00000 2.00000i 1.00000i −1.00000 −2.19388 + 0.432320i
139.6 1.00000i 1.00000i −1.00000 1.32001 1.80487i −1.00000 2.00000i 1.00000i −1.00000 1.80487 + 1.32001i
\(n\): 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.d 6
3.b odd 2 1 2070.2.d.d 6
5.b even 2 1 inner 690.2.d.d 6
5.c odd 4 1 3450.2.a.bq 3
5.c odd 4 1 3450.2.a.br 3
15.d odd 2 1 2070.2.d.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.d.d 6 1.a even 1 1 trivial
690.2.d.d 6 5.b even 2 1 inner
2070.2.d.d 6 3.b odd 2 1
2070.2.d.d 6 15.d odd 2 1
3450.2.a.bq 3 5.c odd 4 1
3450.2.a.br 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 \(S_{2}^{\mathrm{new}}(690, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11}^{3} - 10 T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 + 25 T^{2} + 8 T^{3} + 5 T^{4} + T^{6} \)
$7$ \( ( 4 + T^{2} )^{3} \)
$11$ \( ( -8 - 10 T + T^{3} )^{2} \)
$13$ \( 4096 + 912 T^{2} + 56 T^{4} + T^{6} \)
$17$ \( 256 + 512 T^{2} + 52 T^{4} + T^{6} \)
$19$ \( ( -232 - 62 T + 4 T^{2} + T^{3} )^{2} \)
$23$ \( ( 1 + T^{2} )^{3} \)
$29$ \( ( -2 + T )^{6} \)
$31$ \( ( 80 - 12 T - 8 T^{2} + T^{3} )^{2} \)
$37$ \( 26896 + 3700 T^{2} + 128 T^{4} + T^{6} \)
$41$ \( ( 16 - 24 T + 2 T^{2} + T^{3} )^{2} \)
$43$ \( 16 + 52 T^{2} + 32 T^{4} + T^{6} \)
$47$ \( 4096 + 912 T^{2} + 56 T^{4} + T^{6} \)
$53$ \( 16 + 308 T^{2} + 168 T^{4} + T^{6} \)
$59$ \( ( 848 - 88 T - 10 T^{2} + T^{3} )^{2} \)
$61$ \( ( 284 - 46 T - 10 T^{2} + T^{3} )^{2} \)
$67$ \( 80656 + 7796 T^{2} + 192 T^{4} + T^{6} \)
$71$ \( ( -512 - 64 T + 10 T^{2} + T^{3} )^{2} \)
$73$ \( 262144 + 23312 T^{2} + 328 T^{4} + T^{6} \)
$79$ \( ( 352 - 92 T - 4 T^{2} + T^{3} )^{2} \)
$83$ \( 3748096 + 87268 T^{2} + 548 T^{4} + T^{6} \)
$89$ \( ( 512 + 8 T - 16 T^{2} + T^{3} )^{2} \)
$97$ \( 7744 + 7472 T^{2} + 364 T^{4} + T^{6} \)
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