Properties

Label 304.8.a.d
Level $304$
Weight $8$
Character orbit 304.a
Self dual yes
Analytic conductor $94.965$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(94.9650477472\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{633})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 36 - 3 \beta ) q^{3} + ( 93 - 31 \beta ) q^{5} + ( 1133 - 28 \beta ) q^{7} + ( 531 - 207 \beta ) q^{9} +O(q^{10})\) \( q + ( 36 - 3 \beta ) q^{3} + ( 93 - 31 \beta ) q^{5} + ( 1133 - 28 \beta ) q^{7} + ( 531 - 207 \beta ) q^{9} + ( 1723 - 151 \beta ) q^{11} + ( -6938 + 449 \beta ) q^{13} + ( 18042 - 1302 \beta ) q^{15} + ( -16023 - 210 \beta ) q^{17} + 6859 q^{19} + ( 54060 - 4323 \beta ) q^{21} + ( 43362 - 4199 \beta ) q^{23} + ( 82362 - 4805 \beta ) q^{25} + ( 38502 - 1863 \beta ) q^{27} + ( -4788 - 3173 \beta ) q^{29} + ( -132756 + 6568 \beta ) q^{31} + ( 133602 - 10152 \beta ) q^{33} + ( 242513 - 36859 \beta ) q^{35} + ( -70326 - 8608 \beta ) q^{37} + ( -462594 + 35631 \beta ) q^{39} + ( 143726 + 51678 \beta ) q^{41} + ( 62579 - 41289 \beta ) q^{43} + ( 1063269 - 29295 \beta ) q^{45} + ( -725295 - 20435 \beta ) q^{47} + ( 584018 - 62664 \beta ) q^{49} + ( -477288 + 41139 \beta ) q^{51} + ( -457730 - 30183 \beta ) q^{53} + ( 899837 - 62775 \beta ) q^{55} + ( 246924 - 20577 \beta ) q^{57} + ( 427288 + 114433 \beta ) q^{59} + ( -791651 + 76547 \beta ) q^{61} + ( 1517391 - 243603 \beta ) q^{63} + ( -2844436 + 242916 \beta ) q^{65} + ( 868450 + 111319 \beta ) q^{67} + ( 3551358 - 268653 \beta ) q^{69} + ( 1562970 + 291244 \beta ) q^{71} + ( -1151887 - 196048 \beta ) q^{73} + ( 5242602 - 405651 \beta ) q^{75} + ( 2620183 - 215099 \beta ) q^{77} + ( -1422876 + 208826 \beta ) q^{79} + ( 1107837 + 275724 \beta ) q^{81} + ( 4970568 + 118218 \beta ) q^{83} + ( -461559 + 483693 \beta ) q^{85} + ( 1331634 - 90345 \beta ) q^{87} + ( -1981580 + 457000 \beta ) q^{89} + ( -9847130 + 690409 \beta ) q^{91} + ( -7892448 + 615012 \beta ) q^{93} + ( 637887 - 212629 \beta ) q^{95} + ( 3338292 - 783058 \beta ) q^{97} + ( 5853519 - 405585 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + O(q^{10}) \) \( 2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + 3295q^{11} - 13427q^{13} + 34782q^{15} - 32256q^{17} + 13718q^{19} + 103797q^{21} + 82525q^{23} + 159919q^{25} + 75141q^{27} - 12749q^{29} - 258944q^{31} + 257052q^{33} + 448167q^{35} - 149260q^{37} - 889557q^{39} + 339130q^{41} + 83869q^{43} + 2097243q^{45} - 1471025q^{47} + 1105372q^{49} - 913437q^{51} - 945643q^{53} + 1736899q^{55} + 473271q^{57} + 969009q^{59} - 1506755q^{61} + 2791179q^{63} - 5445956q^{65} + 1848219q^{67} + 6834063q^{69} + 3417184q^{71} - 2499822q^{73} + 10079553q^{75} + 5025267q^{77} - 2636926q^{79} + 2491398q^{81} + 10059354q^{83} - 439425q^{85} + 2572923q^{87} - 3506160q^{89} - 19003851q^{91} - 15169884q^{93} + 1063145q^{95} + 5893526q^{97} + 11301453q^{99} + O(q^{100}) \)

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
13.0797
−12.0797
0 −3.23924 0 −312.472 0 766.767 0 −2176.51 0
1.2 0 72.2392 0 467.472 0 1471.23 0 3031.51 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 304.8.a.d 2
4.b odd 2 1 38.8.a.d 2
12.b even 2 1 342.8.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.d 2 4.b odd 2 1
304.8.a.d 2 1.a even 1 1 trivial
342.8.a.g 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 69 T_{3} - 234 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -234 - 69 T + T^{2} \)
$5$ \( -146072 - 155 T + T^{2} \)
$7$ \( 1128093 - 2238 T + T^{2} \)
$11$ \( -894002 - 3295 T + T^{2} \)
$13$ \( 13167724 + 13427 T + T^{2} \)
$17$ \( 253133559 + 32256 T + T^{2} \)
$19$ \( ( -6859 + T )^{2} \)
$23$ \( -1087606952 - 82525 T + T^{2} \)
$29$ \( -1552615514 + 12749 T + T^{2} \)
$31$ \( 9936311536 + 258944 T + T^{2} \)
$37$ \( -6156318428 + 149260 T + T^{2} \)
$41$ \( -393872642768 - 339130 T + T^{2} \)
$43$ \( -268023173408 - 83869 T + T^{2} \)
$47$ \( 474895142800 + 1471025 T + T^{2} \)
$53$ \( 79392286228 + 945643 T + T^{2} \)
$59$ \( -1837525132614 - 969009 T + T^{2} \)
$61$ \( -359679230318 + 1506755 T + T^{2} \)
$67$ \( -1107042934188 - 1848219 T + T^{2} \)
$71$ \( -10503963815108 - 3417184 T + T^{2} \)
$73$ \( -4520032488687 + 2499822 T + T^{2} \)
$79$ \( -5162668519808 + 2636926 T + T^{2} \)
$83$ \( 23086028557656 - 10059354 T + T^{2} \)
$89$ \( -29977064763600 + 3506160 T + T^{2} \)
$97$ \( -88352296135184 - 5893526 T + T^{2} \)
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