Properties

Label 208.4.f.b
Level $208$
Weight $4$
Character orbit 208.f
Analytic conductor $12.272$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + q^{3} + 3 \beta q^{5} - 5 \beta q^{7} - 26 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 3 \beta q^{5} - 5 \beta q^{7} - 26 q^{9} + 16 \beta q^{11} + (13 \beta + 26) q^{13} + 3 \beta q^{15} - 45 q^{17} + 2 \beta q^{19} - 5 \beta q^{21} - 162 q^{23} + 44 q^{25} - 53 q^{27} - 144 q^{29} + 88 \beta q^{31} + 16 \beta q^{33} + 135 q^{35} + 101 \beta q^{37} + (13 \beta + 26) q^{39} + 64 \beta q^{41} + 97 q^{43} - 78 \beta q^{45} - 37 \beta q^{47} + 118 q^{49} - 45 q^{51} - 414 q^{53} - 432 q^{55} + 2 \beta q^{57} - 174 \beta q^{59} + 376 q^{61} + 130 \beta q^{63} + (78 \beta - 351) q^{65} - 12 \beta q^{67} - 162 q^{69} + 119 \beta q^{71} - 366 \beta q^{73} + 44 q^{75} + 720 q^{77} + 830 q^{79} + 649 q^{81} - 146 \beta q^{83} - 135 \beta q^{85} - 144 q^{87} - 146 \beta q^{89} + ( - 130 \beta + 585) q^{91} + 88 \beta q^{93} - 54 q^{95} + 284 \beta q^{97} - 416 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 52 q^{9} + 52 q^{13} - 90 q^{17} - 324 q^{23} + 88 q^{25} - 106 q^{27} - 288 q^{29} + 270 q^{35} + 52 q^{39} + 194 q^{43} + 236 q^{49} - 90 q^{51} - 828 q^{53} - 864 q^{55} + 752 q^{61} - 702 q^{65} - 324 q^{69} + 88 q^{75} + 1440 q^{77} + 1660 q^{79} + 1298 q^{81} - 288 q^{87} + 1170 q^{91} - 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 9.00000i 0 15.0000i 0 −26.0000 0
129.2 0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 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 208.4.f.b 2
4.b odd 2 1 13.4.b.a 2
8.b even 2 1 832.4.f.c 2
8.d odd 2 1 832.4.f.e 2
12.b even 2 1 117.4.b.a 2
13.b even 2 1 inner 208.4.f.b 2
20.d odd 2 1 325.4.c.b 2
20.e even 4 1 325.4.d.a 2
20.e even 4 1 325.4.d.b 2
52.b odd 2 1 13.4.b.a 2
52.f even 4 1 169.4.a.b 1
52.f even 4 1 169.4.a.c 1
52.i odd 6 2 169.4.e.d 4
52.j odd 6 2 169.4.e.d 4
52.l even 12 2 169.4.c.b 2
52.l even 12 2 169.4.c.c 2
104.e even 2 1 832.4.f.c 2
104.h odd 2 1 832.4.f.e 2
156.h even 2 1 117.4.b.a 2
156.l odd 4 1 1521.4.a.d 1
156.l odd 4 1 1521.4.a.i 1
260.g odd 2 1 325.4.c.b 2
260.p even 4 1 325.4.d.a 2
260.p even 4 1 325.4.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 4.b odd 2 1
13.4.b.a 2 52.b odd 2 1
117.4.b.a 2 12.b even 2 1
117.4.b.a 2 156.h even 2 1
169.4.a.b 1 52.f even 4 1
169.4.a.c 1 52.f even 4 1
169.4.c.b 2 52.l even 12 2
169.4.c.c 2 52.l even 12 2
169.4.e.d 4 52.i odd 6 2
169.4.e.d 4 52.j odd 6 2
208.4.f.b 2 1.a even 1 1 trivial
208.4.f.b 2 13.b even 2 1 inner
325.4.c.b 2 20.d odd 2 1
325.4.c.b 2 260.g odd 2 1
325.4.d.a 2 20.e even 4 1
325.4.d.a 2 260.p even 4 1
325.4.d.b 2 20.e even 4 1
325.4.d.b 2 260.p even 4 1
832.4.f.c 2 8.b even 2 1
832.4.f.c 2 104.e even 2 1
832.4.f.e 2 8.d odd 2 1
832.4.f.e 2 104.h odd 2 1
1521.4.a.d 1 156.l odd 4 1
1521.4.a.i 1 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 225 \) Copy content Toggle raw display
$11$ \( T^{2} + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} - 52T + 2197 \) Copy content Toggle raw display
$17$ \( (T + 45)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 162)^{2} \) Copy content Toggle raw display
$29$ \( (T + 144)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 69696 \) Copy content Toggle raw display
$37$ \( T^{2} + 91809 \) Copy content Toggle raw display
$41$ \( T^{2} + 36864 \) Copy content Toggle raw display
$43$ \( (T - 97)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12321 \) Copy content Toggle raw display
$53$ \( (T + 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 272484 \) Copy content Toggle raw display
$61$ \( (T - 376)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{2} + 127449 \) Copy content Toggle raw display
$73$ \( T^{2} + 1205604 \) Copy content Toggle raw display
$79$ \( (T - 830)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 191844 \) Copy content Toggle raw display
$89$ \( T^{2} + 191844 \) Copy content Toggle raw display
$97$ \( T^{2} + 725904 \) Copy content Toggle raw display
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