Properties

Label 690.2.i.b
Level $690$
Weight $2$
Character orbit 690.i
Analytic conductor $5.510$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(1\)

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
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i 1.70711 0.292893i 1.00000i 0.707107 2.12132i −1.00000 + 1.41421i 3.00000 + 3.00000i 0.707107 + 0.707107i 2.82843 1.00000i 1.00000 + 2.00000i
47.2 0.707107 0.707107i 0.292893 1.70711i 1.00000i −0.707107 + 2.12132i −1.00000 1.41421i 3.00000 + 3.00000i −0.707107 0.707107i −2.82843 1.00000i 1.00000 + 2.00000i
323.1 −0.707107 0.707107i 1.70711 + 0.292893i 1.00000i 0.707107 + 2.12132i −1.00000 1.41421i 3.00000 3.00000i 0.707107 0.707107i 2.82843 + 1.00000i 1.00000 2.00000i
323.2 0.707107 + 0.707107i 0.292893 + 1.70711i 1.00000i −0.707107 2.12132i −1.00000 + 1.41421i 3.00000 3.00000i −0.707107 + 0.707107i −2.82843 + 1.00000i 1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 inner 690.2.i.b 4
5.c odd 4 1 inner 690.2.i.b 4
15.e even 4 1 inner 690.2.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.i.b 4 1.a even 1 1 trivial
690.2.i.b 4 3.b odd 2 1 inner
690.2.i.b 4 5.c odd 4 1 inner
690.2.i.b 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 6 T_{7} + 18 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( 9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( ( 18 - 6 T + T^{2} )^{2} \)
$11$ \( ( 18 + T^{2} )^{2} \)
$13$ \( ( 8 - 4 T + T^{2} )^{2} \)
$17$ \( 1296 + T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( 1 + T^{4} \)
$29$ \( ( -50 + T^{2} )^{2} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( ( 72 - 12 T + T^{2} )^{2} \)
$41$ \( ( 32 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( 20736 + T^{4} \)
$53$ \( 256 + T^{4} \)
$59$ \( ( -2 + T^{2} )^{2} \)
$61$ \( ( 10 + T )^{4} \)
$67$ \( ( 200 + 20 T + T^{2} )^{2} \)
$71$ \( ( 32 + T^{2} )^{2} \)
$73$ \( ( 2 + 2 T + T^{2} )^{2} \)
$79$ \( ( 196 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( -128 + T^{2} )^{2} \)
$97$ \( ( 18 + 6 T + T^{2} )^{2} \)
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