Properties

Label 208.2.a.d
Level 208
Weight 2
Character orbit 208.a
Self dual yes
Analytic conductor 1.661
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 208.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - q^{5} - q^{7} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} - q^{5} - q^{7} + 6q^{9} + 2q^{11} - q^{13} - 3q^{15} - 3q^{17} - 6q^{19} - 3q^{21} + 4q^{23} - 4q^{25} + 9q^{27} + 2q^{29} - 4q^{31} + 6q^{33} + q^{35} + 3q^{37} - 3q^{39} + 5q^{43} - 6q^{45} - 13q^{47} - 6q^{49} - 9q^{51} + 12q^{53} - 2q^{55} - 18q^{57} + 10q^{59} - 8q^{61} - 6q^{63} + q^{65} + 2q^{67} + 12q^{69} + 5q^{71} - 10q^{73} - 12q^{75} - 2q^{77} + 4q^{79} + 9q^{81} + 3q^{85} + 6q^{87} + 6q^{89} + q^{91} - 12q^{93} + 6q^{95} + 14q^{97} + 12q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 3.00000 0 −1.00000 0 −1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.2.a.d 1
3.b odd 2 1 1872.2.a.m 1
4.b odd 2 1 26.2.a.b 1
5.b even 2 1 5200.2.a.c 1
8.b even 2 1 832.2.a.a 1
8.d odd 2 1 832.2.a.j 1
12.b even 2 1 234.2.a.b 1
13.b even 2 1 2704.2.a.n 1
13.d odd 4 2 2704.2.f.j 2
16.e even 4 2 3328.2.b.k 2
16.f odd 4 2 3328.2.b.g 2
20.d odd 2 1 650.2.a.g 1
20.e even 4 2 650.2.b.a 2
24.f even 2 1 7488.2.a.w 1
24.h odd 2 1 7488.2.a.v 1
28.d even 2 1 1274.2.a.o 1
28.f even 6 2 1274.2.f.a 2
28.g odd 6 2 1274.2.f.l 2
36.f odd 6 2 2106.2.e.h 2
36.h even 6 2 2106.2.e.t 2
44.c even 2 1 3146.2.a.a 1
52.b odd 2 1 338.2.a.a 1
52.f even 4 2 338.2.b.a 2
52.i odd 6 2 338.2.c.g 2
52.j odd 6 2 338.2.c.c 2
52.l even 12 4 338.2.e.d 4
60.h even 2 1 5850.2.a.bn 1
60.l odd 4 2 5850.2.e.v 2
68.d odd 2 1 7514.2.a.i 1
76.d even 2 1 9386.2.a.f 1
156.h even 2 1 3042.2.a.l 1
156.l odd 4 2 3042.2.b.f 2
260.g odd 2 1 8450.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 4.b odd 2 1
208.2.a.d 1 1.a even 1 1 trivial
234.2.a.b 1 12.b even 2 1
338.2.a.a 1 52.b odd 2 1
338.2.b.a 2 52.f even 4 2
338.2.c.c 2 52.j odd 6 2
338.2.c.g 2 52.i odd 6 2
338.2.e.d 4 52.l even 12 4
650.2.a.g 1 20.d odd 2 1
650.2.b.a 2 20.e even 4 2
832.2.a.a 1 8.b even 2 1
832.2.a.j 1 8.d odd 2 1
1274.2.a.o 1 28.d even 2 1
1274.2.f.a 2 28.f even 6 2
1274.2.f.l 2 28.g odd 6 2
1872.2.a.m 1 3.b odd 2 1
2106.2.e.h 2 36.f odd 6 2
2106.2.e.t 2 36.h even 6 2
2704.2.a.n 1 13.b even 2 1
2704.2.f.j 2 13.d odd 4 2
3042.2.a.l 1 156.h even 2 1
3042.2.b.f 2 156.l odd 4 2
3146.2.a.a 1 44.c even 2 1
3328.2.b.g 2 16.f odd 4 2
3328.2.b.k 2 16.e even 4 2
5200.2.a.c 1 5.b even 2 1
5850.2.a.bn 1 60.h even 2 1
5850.2.e.v 2 60.l odd 4 2
7488.2.a.v 1 24.h odd 2 1
7488.2.a.w 1 24.f even 2 1
7514.2.a.i 1 68.d odd 2 1
8450.2.a.y 1 260.g odd 2 1
9386.2.a.f 1 76.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(208))\):

\( T_{3} - 3 \)
\( T_{5} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 + T + 5 T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 - 2 T + 11 T^{2} \)
$13$ \( 1 + T \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 - 3 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 5 T + 43 T^{2} \)
$47$ \( 1 + 13 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 - 10 T + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 - 2 T + 67 T^{2} \)
$71$ \( 1 - 5 T + 71 T^{2} \)
$73$ \( 1 + 10 T + 73 T^{2} \)
$79$ \( 1 - 4 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - 14 T + 97 T^{2} \)
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