Properties

Label 608.2.a.h
Level $608$
Weight $2$
Character orbit 608.a
Self dual yes
Analytic conductor $4.855$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( -1 + \beta ) q^{5} + 3 q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( -1 + \beta ) q^{5} + 3 q^{7} + ( 1 + \beta ) q^{9} + ( -1 - \beta ) q^{11} + \beta q^{13} + 4 q^{15} + ( -3 - 2 \beta ) q^{17} + q^{19} + 3 \beta q^{21} + ( 4 - \beta ) q^{23} -\beta q^{25} + ( 4 - \beta ) q^{27} -3 \beta q^{29} + ( 6 - 2 \beta ) q^{31} + ( -4 - 2 \beta ) q^{33} + ( -3 + 3 \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( 4 + \beta ) q^{39} + 4 q^{41} + ( 7 + \beta ) q^{43} + ( 3 + \beta ) q^{45} + ( 1 + 3 \beta ) q^{47} + 2 q^{49} + ( -8 - 5 \beta ) q^{51} + ( -6 - \beta ) q^{53} + ( -3 - \beta ) q^{55} + \beta q^{57} + ( -6 - \beta ) q^{59} + ( 7 - 5 \beta ) q^{61} + ( 3 + 3 \beta ) q^{63} + 4 q^{65} + ( -2 - \beta ) q^{67} + ( -4 + 3 \beta ) q^{69} + ( 2 + 4 \beta ) q^{71} + ( -3 + 4 \beta ) q^{73} + ( -4 - \beta ) q^{75} + ( -3 - 3 \beta ) q^{77} + 10 q^{79} -7 q^{81} + ( -8 + 6 \beta ) q^{83} + ( -5 - 3 \beta ) q^{85} + ( -12 - 3 \beta ) q^{87} + ( -6 + 6 \beta ) q^{89} + 3 \beta q^{91} + ( -8 + 4 \beta ) q^{93} + ( -1 + \beta ) q^{95} + ( 4 - 2 \beta ) q^{97} + ( -5 - 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} + 6 q^{7} + 3 q^{9} + O(q^{10}) \) \( 2 q + q^{3} - q^{5} + 6 q^{7} + 3 q^{9} - 3 q^{11} + q^{13} + 8 q^{15} - 8 q^{17} + 2 q^{19} + 3 q^{21} + 7 q^{23} - q^{25} + 7 q^{27} - 3 q^{29} + 10 q^{31} - 10 q^{33} - 3 q^{35} + 6 q^{37} + 9 q^{39} + 8 q^{41} + 15 q^{43} + 7 q^{45} + 5 q^{47} + 4 q^{49} - 21 q^{51} - 13 q^{53} - 7 q^{55} + q^{57} - 13 q^{59} + 9 q^{61} + 9 q^{63} + 8 q^{65} - 5 q^{67} - 5 q^{69} + 8 q^{71} - 2 q^{73} - 9 q^{75} - 9 q^{77} + 20 q^{79} - 14 q^{81} - 10 q^{83} - 13 q^{85} - 27 q^{87} - 6 q^{89} + 3 q^{91} - 12 q^{93} - q^{95} + 6 q^{97} - 13 q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
0 −1.56155 0 −2.56155 0 3.00000 0 −0.561553 0
1.2 0 2.56155 0 1.56155 0 3.00000 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)

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 608.2.a.h yes 2
3.b odd 2 1 5472.2.a.bf 2
4.b odd 2 1 608.2.a.g 2
8.b even 2 1 1216.2.a.s 2
8.d odd 2 1 1216.2.a.t 2
12.b even 2 1 5472.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.g 2 4.b odd 2 1
608.2.a.h yes 2 1.a even 1 1 trivial
1216.2.a.s 2 8.b even 2 1
1216.2.a.t 2 8.d odd 2 1
5472.2.a.bc 2 12.b even 2 1
5472.2.a.bf 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(608))\):

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{5}^{2} + T_{5} - 4 \)
\( T_{7} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - T + T^{2} \)
$5$ \( -4 + T + T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( -2 + 3 T + T^{2} \)
$13$ \( -4 - T + T^{2} \)
$17$ \( -1 + 8 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 8 - 7 T + T^{2} \)
$29$ \( -36 + 3 T + T^{2} \)
$31$ \( 8 - 10 T + T^{2} \)
$37$ \( -8 - 6 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( 52 - 15 T + T^{2} \)
$47$ \( -32 - 5 T + T^{2} \)
$53$ \( 38 + 13 T + T^{2} \)
$59$ \( 38 + 13 T + T^{2} \)
$61$ \( -86 - 9 T + T^{2} \)
$67$ \( 2 + 5 T + T^{2} \)
$71$ \( -52 - 8 T + T^{2} \)
$73$ \( -67 + 2 T + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( -128 + 10 T + T^{2} \)
$89$ \( -144 + 6 T + T^{2} \)
$97$ \( -8 - 6 T + T^{2} \)
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