Properties

Label 76.5.c.b
Level $76$
Weight $5$
Character orbit 76.c
Analytic conductor $7.856$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Defining polynomial: \(x^{4} + 269 x^{2} + 17592\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 6 - \beta_{2} ) q^{5} + ( 7 - 2 \beta_{2} ) q^{7} + ( -54 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 6 - \beta_{2} ) q^{5} + ( 7 - 2 \beta_{2} ) q^{7} + ( -54 + \beta_{2} ) q^{9} + ( -48 - \beta_{2} ) q^{11} + ( 4 \beta_{1} - \beta_{3} ) q^{13} + ( 9 \beta_{1} - \beta_{3} ) q^{15} + ( -273 + 8 \beta_{2} ) q^{17} + ( -160 - 10 \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{19} + ( 13 \beta_{1} - 2 \beta_{3} ) q^{21} + ( 285 - 25 \beta_{2} ) q^{23} + ( -91 - 11 \beta_{2} ) q^{25} + ( 24 \beta_{1} + \beta_{3} ) q^{27} + 27 \beta_{1} q^{29} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{31} + ( -45 \beta_{1} - \beta_{3} ) q^{33} + ( 1038 - 17 \beta_{2} ) q^{35} + ( -31 \beta_{1} + \beta_{3} ) q^{37} + ( -633 + 135 \beta_{2} ) q^{39} + ( -81 \beta_{1} + 3 \beta_{3} ) q^{41} + ( 1034 - 15 \beta_{2} ) q^{43} + ( -822 + 59 \beta_{2} ) q^{45} + ( 1284 + 59 \beta_{2} ) q^{47} + ( -360 - 24 \beta_{2} ) q^{49} + ( -297 \beta_{1} + 8 \beta_{3} ) q^{51} + ( 207 \beta_{1} + 6 \beta_{3} ) q^{53} + ( 210 + 43 \beta_{2} ) q^{55} + ( 1443 - 139 \beta_{1} - 141 \beta_{2} - 7 \beta_{3} ) q^{57} + ( -72 \beta_{1} + 15 \beta_{3} ) q^{59} + ( 1618 - 197 \beta_{2} ) q^{61} + ( -1374 + 113 \beta_{2} ) q^{63} + ( 522 \beta_{1} - 6 \beta_{3} ) q^{65} + ( 331 \beta_{1} + 2 \beta_{3} ) q^{67} + ( 360 \beta_{1} - 25 \beta_{3} ) q^{69} + ( -468 \beta_{1} + 6 \beta_{3} ) q^{71} + ( -2395 - 138 \beta_{2} ) q^{73} + ( -58 \beta_{1} - 11 \beta_{3} ) q^{75} + ( 660 + 91 \beta_{2} ) q^{77} + ( -379 \beta_{1} - 23 \beta_{3} ) q^{79} + ( -7521 - 26 \beta_{2} ) q^{81} + ( 1068 - 202 \beta_{2} ) q^{83} + ( -5622 + 313 \beta_{2} ) q^{85} + ( -3645 + 27 \beta_{2} ) q^{87} + ( -711 \beta_{1} - 21 \beta_{3} ) q^{89} + ( 1024 \beta_{1} - 7 \beta_{3} ) q^{91} + ( -126 - 390 \beta_{2} ) q^{93} + ( 2526 - 576 \beta_{1} + 125 \beta_{2} + 12 \beta_{3} ) q^{95} + ( 422 \beta_{1} - 32 \beta_{3} ) q^{97} + ( 2094 + 5 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 22q^{5} + 24q^{7} - 214q^{9} + O(q^{10}) \) \( 4q + 22q^{5} + 24q^{7} - 214q^{9} - 194q^{11} - 1076q^{17} - 654q^{19} + 1090q^{23} - 386q^{25} + 4118q^{35} - 2262q^{39} + 4106q^{43} - 3170q^{45} + 5254q^{47} - 1488q^{49} + 926q^{55} + 5490q^{57} + 6078q^{61} - 5270q^{63} - 9856q^{73} + 2822q^{77} - 30136q^{81} + 3868q^{83} - 21862q^{85} - 14526q^{87} - 1284q^{93} + 10354q^{95} + 8386q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 269 x^{2} + 17592\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 135 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 138 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 135\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 138 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
12.5228i
10.5914i
10.5914i
12.5228i
0 12.5228i 0 27.8215 0 50.6430 0 −75.8215 0
37.2 0 10.5914i 0 −16.8215 0 −38.6430 0 −31.1785 0
37.3 0 10.5914i 0 −16.8215 0 −38.6430 0 −31.1785 0
37.4 0 12.5228i 0 27.8215 0 50.6430 0 −75.8215 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
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.5.c.b 4
3.b odd 2 1 684.5.h.c 4
4.b odd 2 1 304.5.e.d 4
19.b odd 2 1 inner 76.5.c.b 4
57.d even 2 1 684.5.h.c 4
76.d even 2 1 304.5.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.c.b 4 1.a even 1 1 trivial
76.5.c.b 4 19.b odd 2 1 inner
304.5.e.d 4 4.b odd 2 1
304.5.e.d 4 76.d even 2 1
684.5.h.c 4 3.b odd 2 1
684.5.h.c 4 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 269 T_{3}^{2} + 17592 \) acting on \(S_{5}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 17592 + 269 T^{2} + T^{4} \)
$5$ \( ( -468 - 11 T + T^{2} )^{2} \)
$7$ \( ( -1957 - 12 T + T^{2} )^{2} \)
$11$ \( ( 1854 + 97 T + T^{2} )^{2} \)
$13$ \( 4362886368 + 135093 T^{2} + T^{4} \)
$17$ \( ( 40473 + 538 T + T^{2} )^{2} \)
$19$ \( 16983563041 + 85229934 T + 269914 T^{2} + 654 T^{3} + T^{4} \)
$23$ \( ( -237150 - 545 T + T^{2} )^{2} \)
$29$ \( 9349110072 + 196101 T^{2} + T^{4} \)
$31$ \( 325578732768 + 1174572 T^{2} + T^{4} \)
$37$ \( 1170993888 + 392268 T^{2} + T^{4} \)
$41$ \( 4155160032 + 2964780 T^{2} + T^{4} \)
$43$ \( ( 941596 - 2053 T + T^{2} )^{2} \)
$47$ \( ( -9126 - 2627 T + T^{2} )^{2} \)
$53$ \( 20392614892728 + 16082325 T^{2} + T^{4} \)
$59$ \( 219595642144992 + 30841821 T^{2} + T^{4} \)
$61$ \( ( -17027704 - 3039 T + T^{2} )^{2} \)
$67$ \( 221664248921592 + 29920485 T^{2} + T^{4} \)
$71$ \( 581898049763328 + 63918900 T^{2} + T^{4} \)
$73$ \( ( -3417377 + 4928 T + T^{2} )^{2} \)
$79$ \( 48366114070752 + 106635180 T^{2} + T^{4} \)
$83$ \( ( -19395504 - 1934 T + T^{2} )^{2} \)
$89$ \( 2754695866318848 + 191826648 T^{2} + T^{4} \)
$97$ \( 3016771784201088 + 182867412 T^{2} + T^{4} \)
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