Properties

Label 650.2.b.d
Level $650$
Weight $2$
Character orbit 650.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + i q^{7} - i q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + i q^{7} - i q^{8} + 2 q^{9} + 6 q^{11} - i q^{12} + i q^{13} - q^{14} + q^{16} + 3 i q^{17} + 2 i q^{18} - 2 q^{19} - q^{21} + 6 i q^{22} + q^{24} - q^{26} + 5 i q^{27} - i q^{28} - 6 q^{29} - 4 q^{31} + i q^{32} + 6 i q^{33} - 3 q^{34} - 2 q^{36} + 7 i q^{37} - 2 i q^{38} - q^{39} - i q^{42} - i q^{43} - 6 q^{44} - 3 i q^{47} + i q^{48} + 6 q^{49} - 3 q^{51} - i q^{52} - 5 q^{54} + q^{56} - 2 i q^{57} - 6 i q^{58} + 6 q^{59} + 8 q^{61} - 4 i q^{62} + 2 i q^{63} - q^{64} - 6 q^{66} - 14 i q^{67} - 3 i q^{68} - 3 q^{71} - 2 i q^{72} + 2 i q^{73} - 7 q^{74} + 2 q^{76} + 6 i q^{77} - i q^{78} - 8 q^{79} + q^{81} + 12 i q^{83} + q^{84} + q^{86} - 6 i q^{87} - 6 i q^{88} + 6 q^{89} - q^{91} - 4 i q^{93} + 3 q^{94} - q^{96} + 10 i q^{97} + 6 i q^{98} + 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 12 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 2 q^{21} + 2 q^{24} - 2 q^{26} - 12 q^{29} - 8 q^{31} - 6 q^{34} - 4 q^{36} - 2 q^{39} - 12 q^{44} + 12 q^{49} - 6 q^{51} - 10 q^{54} + 2 q^{56} + 12 q^{59} + 16 q^{61} - 2 q^{64} - 12 q^{66} - 6 q^{71} - 14 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{81} + 2 q^{84} + 2 q^{86} + 12 q^{89} - 2 q^{91} + 6 q^{94} - 2 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 2.00000 0
599.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 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
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 650.2.b.d 2
3.b odd 2 1 5850.2.e.a 2
5.b even 2 1 inner 650.2.b.d 2
5.c odd 4 1 26.2.a.a 1
5.c odd 4 1 650.2.a.j 1
15.d odd 2 1 5850.2.e.a 2
15.e even 4 1 234.2.a.e 1
15.e even 4 1 5850.2.a.p 1
20.e even 4 1 208.2.a.a 1
20.e even 4 1 5200.2.a.x 1
35.f even 4 1 1274.2.a.d 1
35.k even 12 2 1274.2.f.r 2
35.l odd 12 2 1274.2.f.p 2
40.i odd 4 1 832.2.a.d 1
40.k even 4 1 832.2.a.i 1
45.k odd 12 2 2106.2.e.ba 2
45.l even 12 2 2106.2.e.b 2
55.e even 4 1 3146.2.a.n 1
60.l odd 4 1 1872.2.a.q 1
65.f even 4 1 338.2.b.c 2
65.h odd 4 1 338.2.a.f 1
65.h odd 4 1 8450.2.a.c 1
65.k even 4 1 338.2.b.c 2
65.o even 12 2 338.2.e.a 4
65.q odd 12 2 338.2.c.d 2
65.r odd 12 2 338.2.c.a 2
65.t even 12 2 338.2.e.a 4
80.i odd 4 1 3328.2.b.m 2
80.j even 4 1 3328.2.b.j 2
80.s even 4 1 3328.2.b.j 2
80.t odd 4 1 3328.2.b.m 2
85.g odd 4 1 7514.2.a.c 1
95.g even 4 1 9386.2.a.j 1
120.q odd 4 1 7488.2.a.h 1
120.w even 4 1 7488.2.a.g 1
195.j odd 4 1 3042.2.b.a 2
195.s even 4 1 3042.2.a.a 1
195.u odd 4 1 3042.2.b.a 2
260.l odd 4 1 2704.2.f.d 2
260.p even 4 1 2704.2.a.f 1
260.s odd 4 1 2704.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 5.c odd 4 1
208.2.a.a 1 20.e even 4 1
234.2.a.e 1 15.e even 4 1
338.2.a.f 1 65.h odd 4 1
338.2.b.c 2 65.f even 4 1
338.2.b.c 2 65.k even 4 1
338.2.c.a 2 65.r odd 12 2
338.2.c.d 2 65.q odd 12 2
338.2.e.a 4 65.o even 12 2
338.2.e.a 4 65.t even 12 2
650.2.a.j 1 5.c odd 4 1
650.2.b.d 2 1.a even 1 1 trivial
650.2.b.d 2 5.b even 2 1 inner
832.2.a.d 1 40.i odd 4 1
832.2.a.i 1 40.k even 4 1
1274.2.a.d 1 35.f even 4 1
1274.2.f.p 2 35.l odd 12 2
1274.2.f.r 2 35.k even 12 2
1872.2.a.q 1 60.l odd 4 1
2106.2.e.b 2 45.l even 12 2
2106.2.e.ba 2 45.k odd 12 2
2704.2.a.f 1 260.p even 4 1
2704.2.f.d 2 260.l odd 4 1
2704.2.f.d 2 260.s odd 4 1
3042.2.a.a 1 195.s even 4 1
3042.2.b.a 2 195.j odd 4 1
3042.2.b.a 2 195.u odd 4 1
3146.2.a.n 1 55.e even 4 1
3328.2.b.j 2 80.j even 4 1
3328.2.b.j 2 80.s even 4 1
3328.2.b.m 2 80.i odd 4 1
3328.2.b.m 2 80.t odd 4 1
5200.2.a.x 1 20.e even 4 1
5850.2.a.p 1 15.e even 4 1
5850.2.e.a 2 3.b odd 2 1
5850.2.e.a 2 15.d odd 2 1
7488.2.a.g 1 120.w even 4 1
7488.2.a.h 1 120.q odd 4 1
7514.2.a.c 1 85.g odd 4 1
8450.2.a.c 1 65.h odd 4 1
9386.2.a.j 1 95.g even 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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