Properties

Label 650.2.d.b
Level $650$
Weight $2$
Character orbit 650.d
Analytic conductor $5.190$
Analytic rank $0$
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} + i q^{6} + 3 i q^{7} -i q^{8} -2 q^{9} +O(q^{10})\) \( q + i q^{2} + q^{3} - q^{4} + i q^{6} + 3 i q^{7} -i q^{8} -2 q^{9} - q^{12} + ( -2 + 3 i ) q^{13} -3 q^{14} + q^{16} -3 q^{17} -2 i q^{18} + 6 i q^{19} + 3 i q^{21} + 6 q^{23} -i q^{24} + ( -3 - 2 i ) q^{26} -5 q^{27} -3 i q^{28} + i q^{32} -3 i q^{34} + 2 q^{36} + 3 i q^{37} -6 q^{38} + ( -2 + 3 i ) q^{39} -3 q^{42} + q^{43} + 6 i q^{46} + 3 i q^{47} + q^{48} -2 q^{49} -3 q^{51} + ( 2 - 3 i ) q^{52} + 6 q^{53} -5 i q^{54} + 3 q^{56} + 6 i q^{57} + 6 i q^{59} -8 q^{61} -6 i q^{63} - q^{64} -12 i q^{67} + 3 q^{68} + 6 q^{69} -15 i q^{71} + 2 i q^{72} + 6 i q^{73} -3 q^{74} -6 i q^{76} + ( -3 - 2 i ) q^{78} + 10 q^{79} + q^{81} + 6 i q^{83} -3 i q^{84} + i q^{86} + 6 i q^{89} + ( -9 - 6 i ) q^{91} -6 q^{92} -3 q^{94} + i q^{96} -12 i q^{97} -2 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} - 4q^{9} - 2q^{12} - 4q^{13} - 6q^{14} + 2q^{16} - 6q^{17} + 12q^{23} - 6q^{26} - 10q^{27} + 4q^{36} - 12q^{38} - 4q^{39} - 6q^{42} + 2q^{43} + 2q^{48} - 4q^{49} - 6q^{51} + 4q^{52} + 12q^{53} + 6q^{56} - 16q^{61} - 2q^{64} + 6q^{68} + 12q^{69} - 6q^{74} - 6q^{78} + 20q^{79} + 2q^{81} - 18q^{91} - 12q^{92} - 6q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 0 1.00000i 3.00000i 1.00000i −2.00000 0
51.2 1.00000i 1.00000 −1.00000 0 1.00000i 3.00000i 1.00000i −2.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.d.b 2
5.b even 2 1 26.2.b.a 2
5.c odd 4 1 650.2.c.a 2
5.c odd 4 1 650.2.c.d 2
13.b even 2 1 inner 650.2.d.b 2
13.d odd 4 1 8450.2.a.h 1
13.d odd 4 1 8450.2.a.u 1
15.d odd 2 1 234.2.b.b 2
20.d odd 2 1 208.2.f.a 2
35.c odd 2 1 1274.2.d.c 2
35.i odd 6 2 1274.2.n.c 4
35.j even 6 2 1274.2.n.d 4
40.e odd 2 1 832.2.f.b 2
40.f even 2 1 832.2.f.d 2
60.h even 2 1 1872.2.c.f 2
65.d even 2 1 26.2.b.a 2
65.g odd 4 1 338.2.a.b 1
65.g odd 4 1 338.2.a.d 1
65.h odd 4 1 650.2.c.a 2
65.h odd 4 1 650.2.c.d 2
65.l even 6 2 338.2.e.c 4
65.n even 6 2 338.2.e.c 4
65.s odd 12 2 338.2.c.b 2
65.s odd 12 2 338.2.c.f 2
195.e odd 2 1 234.2.b.b 2
195.n even 4 1 3042.2.a.g 1
195.n even 4 1 3042.2.a.j 1
260.g odd 2 1 208.2.f.a 2
260.u even 4 1 2704.2.a.j 1
260.u even 4 1 2704.2.a.k 1
455.h odd 2 1 1274.2.d.c 2
455.bf odd 6 2 1274.2.n.c 4
455.bh even 6 2 1274.2.n.d 4
520.b odd 2 1 832.2.f.b 2
520.p even 2 1 832.2.f.d 2
780.d even 2 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 5.b even 2 1
26.2.b.a 2 65.d even 2 1
208.2.f.a 2 20.d odd 2 1
208.2.f.a 2 260.g odd 2 1
234.2.b.b 2 15.d odd 2 1
234.2.b.b 2 195.e odd 2 1
338.2.a.b 1 65.g odd 4 1
338.2.a.d 1 65.g odd 4 1
338.2.c.b 2 65.s odd 12 2
338.2.c.f 2 65.s odd 12 2
338.2.e.c 4 65.l even 6 2
338.2.e.c 4 65.n even 6 2
650.2.c.a 2 5.c odd 4 1
650.2.c.a 2 65.h odd 4 1
650.2.c.d 2 5.c odd 4 1
650.2.c.d 2 65.h odd 4 1
650.2.d.b 2 1.a even 1 1 trivial
650.2.d.b 2 13.b even 2 1 inner
832.2.f.b 2 40.e odd 2 1
832.2.f.b 2 520.b odd 2 1
832.2.f.d 2 40.f even 2 1
832.2.f.d 2 520.p even 2 1
1274.2.d.c 2 35.c odd 2 1
1274.2.d.c 2 455.h odd 2 1
1274.2.n.c 4 35.i odd 6 2
1274.2.n.c 4 455.bf odd 6 2
1274.2.n.d 4 35.j even 6 2
1274.2.n.d 4 455.bh even 6 2
1872.2.c.f 2 60.h even 2 1
1872.2.c.f 2 780.d even 2 1
2704.2.a.j 1 260.u even 4 1
2704.2.a.k 1 260.u even 4 1
3042.2.a.g 1 195.n even 4 1
3042.2.a.j 1 195.n even 4 1
8450.2.a.h 1 13.d odd 4 1
8450.2.a.u 1 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 4 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 9 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 225 + T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( 144 + T^{2} \)
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