Properties

Label 650.2.e.a
Level $650$
Weight $2$
Character orbit 650.e
Analytic conductor $5.190$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.19027613138\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{6} - \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{6} - \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + 2 q^{12} + (3 \zeta_{6} - 4) q^{13} + q^{14} + (\zeta_{6} - 1) q^{16} - 6 \zeta_{6} q^{17} + q^{18} - 5 \zeta_{6} q^{19} + 2 q^{21} - 3 \zeta_{6} q^{22} + (2 \zeta_{6} - 2) q^{24} + ( - 4 \zeta_{6} + 1) q^{26} - 4 q^{27} + (\zeta_{6} - 1) q^{28} - 4 q^{31} - \zeta_{6} q^{32} - 6 \zeta_{6} q^{33} + 6 q^{34} + (\zeta_{6} - 1) q^{36} + ( - 11 \zeta_{6} + 11) q^{37} + 5 q^{38} + ( - 8 \zeta_{6} + 2) q^{39} + (6 \zeta_{6} - 6) q^{41} + (2 \zeta_{6} - 2) q^{42} + 2 \zeta_{6} q^{43} + 3 q^{44} + 3 q^{47} - 2 \zeta_{6} q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + 12 q^{51} + (\zeta_{6} + 3) q^{52} + 9 q^{53} + ( - 4 \zeta_{6} + 4) q^{54} - \zeta_{6} q^{56} + 10 q^{57} - 8 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + (\zeta_{6} - 1) q^{63} + q^{64} + 6 q^{66} + (16 \zeta_{6} - 16) q^{67} + (6 \zeta_{6} - 6) q^{68} - 6 \zeta_{6} q^{71} - \zeta_{6} q^{72} - 14 q^{73} + 11 \zeta_{6} q^{74} + (5 \zeta_{6} - 5) q^{76} + 3 q^{77} + (2 \zeta_{6} + 6) q^{78} - 16 q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 6 \zeta_{6} q^{82} + 6 q^{83} - 2 \zeta_{6} q^{84} - 2 q^{86} + (3 \zeta_{6} - 3) q^{88} + (9 \zeta_{6} - 9) q^{89} + (\zeta_{6} + 3) q^{91} + ( - 8 \zeta_{6} + 8) q^{93} + (3 \zeta_{6} - 3) q^{94} + 2 q^{96} - 10 \zeta_{6} q^{97} + 6 \zeta_{6} q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{7} + 2 q^{8} - q^{9} - 3 q^{11} + 4 q^{12} - 5 q^{13} + 2 q^{14} - q^{16} - 6 q^{17} + 2 q^{18} - 5 q^{19} + 4 q^{21} - 3 q^{22} - 2 q^{24} - 2 q^{26} - 8 q^{27} - q^{28} - 8 q^{31} - q^{32} - 6 q^{33} + 12 q^{34} - q^{36} + 11 q^{37} + 10 q^{38} - 4 q^{39} - 6 q^{41} - 2 q^{42} + 2 q^{43} + 6 q^{44} + 6 q^{47} - 2 q^{48} + 6 q^{49} + 24 q^{51} + 7 q^{52} + 18 q^{53} + 4 q^{54} - q^{56} + 20 q^{57} - 8 q^{61} + 4 q^{62} - q^{63} + 2 q^{64} + 12 q^{66} - 16 q^{67} - 6 q^{68} - 6 q^{71} - q^{72} - 28 q^{73} + 11 q^{74} - 5 q^{76} + 6 q^{77} + 14 q^{78} - 32 q^{79} + 11 q^{81} - 6 q^{82} + 12 q^{83} - 2 q^{84} - 4 q^{86} - 3 q^{88} - 9 q^{89} + 7 q^{91} + 8 q^{93} - 3 q^{94} + 4 q^{96} - 10 q^{97} + 6 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 0.866025i 0 −1.00000 1.73205i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0
601.1 −0.500000 0.866025i −1.00000 1.73205i −0.500000 + 0.866025i 0 −1.00000 + 1.73205i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.e.a 2
5.b even 2 1 130.2.e.b 2
5.c odd 4 2 650.2.o.b 4
13.c even 3 1 inner 650.2.e.a 2
13.c even 3 1 8450.2.a.w 1
13.e even 6 1 8450.2.a.k 1
15.d odd 2 1 1170.2.i.f 2
20.d odd 2 1 1040.2.q.c 2
65.d even 2 1 1690.2.e.e 2
65.g odd 4 2 1690.2.l.i 4
65.l even 6 1 1690.2.a.g 1
65.l even 6 1 1690.2.e.e 2
65.n even 6 1 130.2.e.b 2
65.n even 6 1 1690.2.a.a 1
65.q odd 12 2 650.2.o.b 4
65.s odd 12 2 1690.2.d.a 2
65.s odd 12 2 1690.2.l.i 4
195.x odd 6 1 1170.2.i.f 2
260.v odd 6 1 1040.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 5.b even 2 1
130.2.e.b 2 65.n even 6 1
650.2.e.a 2 1.a even 1 1 trivial
650.2.e.a 2 13.c even 3 1 inner
650.2.o.b 4 5.c odd 4 2
650.2.o.b 4 65.q odd 12 2
1040.2.q.c 2 20.d odd 2 1
1040.2.q.c 2 260.v odd 6 1
1170.2.i.f 2 15.d odd 2 1
1170.2.i.f 2 195.x odd 6 1
1690.2.a.a 1 65.n even 6 1
1690.2.a.g 1 65.l even 6 1
1690.2.d.a 2 65.s odd 12 2
1690.2.e.e 2 65.d even 2 1
1690.2.e.e 2 65.l even 6 1
1690.2.l.i 4 65.g odd 4 2
1690.2.l.i 4 65.s odd 12 2
8450.2.a.k 1 13.e even 6 1
8450.2.a.w 1 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$47$ \( (T - 3)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 14)^{2} \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
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