Properties

Label 208.2.f.a
Level $208$
Weight $2$
Character orbit 208.f
Analytic conductor $1.661$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + q^{3} + 3 i q^{5} + 3 i q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} + 3 i q^{5} + 3 i q^{7} -2 q^{9} + ( 2 - 3 i ) q^{13} + 3 i q^{15} + 3 q^{17} -6 i q^{19} + 3 i q^{21} + 6 q^{23} -4 q^{25} -5 q^{27} -9 q^{35} -3 i q^{37} + ( 2 - 3 i ) q^{39} + q^{43} -6 i q^{45} + 3 i q^{47} -2 q^{49} + 3 q^{51} -6 q^{53} -6 i q^{57} -6 i q^{59} -8 q^{61} -6 i q^{63} + ( 9 + 6 i ) q^{65} -12 i q^{67} + 6 q^{69} + 15 i q^{71} -6 i q^{73} -4 q^{75} -10 q^{79} + q^{81} + 6 i q^{83} + 9 i q^{85} + 6 i q^{89} + ( 9 + 6 i ) q^{91} + 18 q^{95} + 12 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} + 4q^{13} + 6q^{17} + 12q^{23} - 8q^{25} - 10q^{27} - 18q^{35} + 4q^{39} + 2q^{43} - 4q^{49} + 6q^{51} - 12q^{53} - 16q^{61} + 18q^{65} + 12q^{69} - 8q^{75} - 20q^{79} + 2q^{81} + 18q^{91} + 36q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 1.00000 0 3.00000i 0 3.00000i 0 −2.00000 0
129.2 0 1.00000 0 3.00000i 0 3.00000i 0 −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 208.2.f.a 2
3.b odd 2 1 1872.2.c.f 2
4.b odd 2 1 26.2.b.a 2
8.b even 2 1 832.2.f.b 2
8.d odd 2 1 832.2.f.d 2
12.b even 2 1 234.2.b.b 2
13.b even 2 1 inner 208.2.f.a 2
13.d odd 4 1 2704.2.a.j 1
13.d odd 4 1 2704.2.a.k 1
20.d odd 2 1 650.2.d.b 2
20.e even 4 1 650.2.c.a 2
20.e even 4 1 650.2.c.d 2
28.d even 2 1 1274.2.d.c 2
28.f even 6 2 1274.2.n.c 4
28.g odd 6 2 1274.2.n.d 4
39.d odd 2 1 1872.2.c.f 2
52.b odd 2 1 26.2.b.a 2
52.f even 4 1 338.2.a.b 1
52.f even 4 1 338.2.a.d 1
52.i odd 6 2 338.2.e.c 4
52.j odd 6 2 338.2.e.c 4
52.l even 12 2 338.2.c.b 2
52.l even 12 2 338.2.c.f 2
104.e even 2 1 832.2.f.b 2
104.h odd 2 1 832.2.f.d 2
156.h even 2 1 234.2.b.b 2
156.l odd 4 1 3042.2.a.g 1
156.l odd 4 1 3042.2.a.j 1
260.g odd 2 1 650.2.d.b 2
260.p even 4 1 650.2.c.a 2
260.p even 4 1 650.2.c.d 2
260.u even 4 1 8450.2.a.h 1
260.u even 4 1 8450.2.a.u 1
364.h even 2 1 1274.2.d.c 2
364.x even 6 2 1274.2.n.c 4
364.bl odd 6 2 1274.2.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 4.b odd 2 1
26.2.b.a 2 52.b odd 2 1
208.2.f.a 2 1.a even 1 1 trivial
208.2.f.a 2 13.b even 2 1 inner
234.2.b.b 2 12.b even 2 1
234.2.b.b 2 156.h even 2 1
338.2.a.b 1 52.f even 4 1
338.2.a.d 1 52.f even 4 1
338.2.c.b 2 52.l even 12 2
338.2.c.f 2 52.l even 12 2
338.2.e.c 4 52.i odd 6 2
338.2.e.c 4 52.j odd 6 2
650.2.c.a 2 20.e even 4 1
650.2.c.a 2 260.p even 4 1
650.2.c.d 2 20.e even 4 1
650.2.c.d 2 260.p even 4 1
650.2.d.b 2 20.d odd 2 1
650.2.d.b 2 260.g odd 2 1
832.2.f.b 2 8.b even 2 1
832.2.f.b 2 104.e even 2 1
832.2.f.d 2 8.d odd 2 1
832.2.f.d 2 104.h odd 2 1
1274.2.d.c 2 28.d even 2 1
1274.2.d.c 2 364.h even 2 1
1274.2.n.c 4 28.f even 6 2
1274.2.n.c 4 364.x even 6 2
1274.2.n.d 4 28.g odd 6 2
1274.2.n.d 4 364.bl odd 6 2
1872.2.c.f 2 3.b odd 2 1
1872.2.c.f 2 39.d odd 2 1
2704.2.a.j 1 13.d odd 4 1
2704.2.a.k 1 13.d odd 4 1
3042.2.a.g 1 156.l odd 4 1
3042.2.a.j 1 156.l odd 4 1
8450.2.a.h 1 260.u even 4 1
8450.2.a.u 1 260.u even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(208, [\chi])\).

Hecke characteristic polynomials

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