Properties

Label 208.4.a.h
Level $208$
Weight $4$
Character orbit 208.a
Self dual yes
Analytic conductor $12.272$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(12.2723972812\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta - 1) q^{3} + ( - \beta - 1) q^{5} + (11 \beta - 1) q^{7} + (15 \beta + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta - 1) q^{3} + ( - \beta - 1) q^{5} + (11 \beta - 1) q^{7} + (15 \beta + 10) q^{9} + (12 \beta - 46) q^{11} - 13 q^{13} + (7 \beta + 13) q^{15} + (17 \beta + 1) q^{17} + ( - 32 \beta + 58) q^{19} + ( - 41 \beta - 131) q^{21} + ( - 12 \beta - 92) q^{23} + (3 \beta - 120) q^{25} + ( - 9 \beta - 163) q^{27} + ( - 96 \beta + 26) q^{29} + ( - 34 \beta + 60) q^{31} + (90 \beta - 98) q^{33} + ( - 21 \beta - 43) q^{35} + ( - 5 \beta + 107) q^{37} + (39 \beta + 13) q^{39} + ( - 22 \beta - 104) q^{41} + (143 \beta - 215) q^{43} + ( - 40 \beta - 70) q^{45} + ( - 121 \beta - 157) q^{47} + (99 \beta + 142) q^{49} + ( - 71 \beta - 205) q^{51} + ( - 30 \beta - 44) q^{53} + (22 \beta - 2) q^{55} + ( - 46 \beta + 326) q^{57} + (124 \beta + 122) q^{59} + (190 \beta - 624) q^{61} + (260 \beta + 650) q^{63} + (13 \beta + 13) q^{65} + ( - 232 \beta + 82) q^{67} + (324 \beta + 236) q^{69} + ( - 231 \beta + 181) q^{71} + ( - 260 \beta + 358) q^{73} + (348 \beta + 84) q^{75} + ( - 386 \beta + 574) q^{77} + (40 \beta + 484) q^{79} + (120 \beta + 1) q^{81} + ( - 182 \beta - 888) q^{83} + ( - 35 \beta - 69) q^{85} + (306 \beta + 1126) q^{87} + (388 \beta - 554) q^{89} + ( - 143 \beta + 13) q^{91} + ( - 44 \beta + 348) q^{93} + (6 \beta + 70) q^{95} + ( - 508 \beta - 210) q^{97} + ( - 390 \beta + 260) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{3} - 3 q^{5} + 9 q^{7} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{3} - 3 q^{5} + 9 q^{7} + 35 q^{9} - 80 q^{11} - 26 q^{13} + 33 q^{15} + 19 q^{17} + 84 q^{19} - 303 q^{21} - 196 q^{23} - 237 q^{25} - 335 q^{27} - 44 q^{29} + 86 q^{31} - 106 q^{33} - 107 q^{35} + 209 q^{37} + 65 q^{39} - 230 q^{41} - 287 q^{43} - 180 q^{45} - 435 q^{47} + 383 q^{49} - 481 q^{51} - 118 q^{53} + 18 q^{55} + 606 q^{57} + 368 q^{59} - 1058 q^{61} + 1560 q^{63} + 39 q^{65} - 68 q^{67} + 796 q^{69} + 131 q^{71} + 456 q^{73} + 516 q^{75} + 762 q^{77} + 1008 q^{79} + 122 q^{81} - 1958 q^{83} - 173 q^{85} + 2558 q^{87} - 720 q^{89} - 117 q^{91} + 652 q^{93} + 146 q^{95} - 928 q^{97} + 130 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2.56155
−1.56155
0 −8.68466 0 −3.56155 0 27.1771 0 48.4233 0
1.2 0 3.68466 0 0.561553 0 −18.1771 0 −13.4233 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 208.4.a.h 2
3.b odd 2 1 1872.4.a.bb 2
4.b odd 2 1 13.4.a.b 2
8.b even 2 1 832.4.a.z 2
8.d odd 2 1 832.4.a.s 2
12.b even 2 1 117.4.a.d 2
20.d odd 2 1 325.4.a.f 2
20.e even 4 2 325.4.b.e 4
28.d even 2 1 637.4.a.b 2
44.c even 2 1 1573.4.a.b 2
52.b odd 2 1 169.4.a.g 2
52.f even 4 2 169.4.b.f 4
52.i odd 6 2 169.4.c.j 4
52.j odd 6 2 169.4.c.g 4
52.l even 12 4 169.4.e.f 8
156.h even 2 1 1521.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 4.b odd 2 1
117.4.a.d 2 12.b even 2 1
169.4.a.g 2 52.b odd 2 1
169.4.b.f 4 52.f even 4 2
169.4.c.g 4 52.j odd 6 2
169.4.c.j 4 52.i odd 6 2
169.4.e.f 8 52.l even 12 4
208.4.a.h 2 1.a even 1 1 trivial
325.4.a.f 2 20.d odd 2 1
325.4.b.e 4 20.e even 4 2
637.4.a.b 2 28.d even 2 1
832.4.a.s 2 8.d odd 2 1
832.4.a.z 2 8.b even 2 1
1521.4.a.r 2 156.h even 2 1
1573.4.a.b 2 44.c even 2 1
1872.4.a.bb 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} - 32 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(208))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 9T - 494 \) Copy content Toggle raw display
$11$ \( T^{2} + 80T + 988 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} + 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} + 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} - 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} - 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} + 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} + 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} + 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} + 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} - 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} + 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} + 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} - 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} - 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} - 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} + 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} + 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} + 928T - 881476 \) Copy content Toggle raw display
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