Properties

Label 76.4.a.a
Level $76$
Weight $4$
Character orbit 76.a
Self dual yes
Analytic conductor $4.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.48414516044\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta ) q^{3} + ( -5 + 5 \beta ) q^{5} + ( -13 - 4 \beta ) q^{7} + ( -15 + 5 \beta ) q^{9} +O(q^{10})\) \( q + ( -2 - \beta ) q^{3} + ( -5 + 5 \beta ) q^{5} + ( -13 - 4 \beta ) q^{7} + ( -15 + 5 \beta ) q^{9} + ( -33 - 5 \beta ) q^{11} + ( -12 - 11 \beta ) q^{13} + ( -30 - 10 \beta ) q^{15} + ( -3 + 6 \beta ) q^{17} -19 q^{19} + ( 58 + 25 \beta ) q^{21} + ( -32 + 59 \beta ) q^{23} + ( 100 - 25 \beta ) q^{25} + ( 44 + 27 \beta ) q^{27} + ( 110 - 65 \beta ) q^{29} + ( -4 - 80 \beta ) q^{31} + ( 106 + 48 \beta ) q^{33} + ( -95 - 65 \beta ) q^{35} + ( 194 - 8 \beta ) q^{37} + ( 112 + 45 \beta ) q^{39} + ( -86 + 30 \beta ) q^{41} + ( 19 + 117 \beta ) q^{43} + ( 275 - 75 \beta ) q^{45} + ( -239 + 23 \beta ) q^{47} + ( -46 + 120 \beta ) q^{49} + ( -42 - 15 \beta ) q^{51} + ( -76 - 123 \beta ) q^{53} + ( -35 - 165 \beta ) q^{55} + ( 38 + 19 \beta ) q^{57} + ( -454 + 35 \beta ) q^{59} + ( 135 + 175 \beta ) q^{61} + ( 35 - 25 \beta ) q^{63} + ( -380 - 60 \beta ) q^{65} + ( 292 + 61 \beta ) q^{67} + ( -408 - 145 \beta ) q^{69} + ( -866 + 20 \beta ) q^{71} + ( -527 + 64 \beta ) q^{73} -25 \beta q^{75} + ( 589 + 217 \beta ) q^{77} + ( -632 - 10 \beta ) q^{79} + ( 101 - 260 \beta ) q^{81} + ( 12 - 114 \beta ) q^{83} + ( 255 - 15 \beta ) q^{85} + ( 300 + 85 \beta ) q^{87} + ( -404 - 80 \beta ) q^{89} + ( 508 + 235 \beta ) q^{91} + ( 648 + 244 \beta ) q^{93} + ( 95 - 95 \beta ) q^{95} + ( 584 - 458 \beta ) q^{97} + ( 295 - 115 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{3} - 5q^{5} - 30q^{7} - 25q^{9} + O(q^{10}) \) \( 2q - 5q^{3} - 5q^{5} - 30q^{7} - 25q^{9} - 71q^{11} - 35q^{13} - 70q^{15} - 38q^{19} + 141q^{21} - 5q^{23} + 175q^{25} + 115q^{27} + 155q^{29} - 88q^{31} + 260q^{33} - 255q^{35} + 380q^{37} + 269q^{39} - 142q^{41} + 155q^{43} + 475q^{45} - 455q^{47} + 28q^{49} - 99q^{51} - 275q^{53} - 235q^{55} + 95q^{57} - 873q^{59} + 445q^{61} + 45q^{63} - 820q^{65} + 645q^{67} - 961q^{69} - 1712q^{71} - 990q^{73} - 25q^{75} + 1395q^{77} - 1274q^{79} - 58q^{81} - 90q^{83} + 495q^{85} + 685q^{87} - 888q^{89} + 1251q^{91} + 1540q^{93} + 95q^{95} + 710q^{97} + 475q^{99} + O(q^{100}) \)

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
3.37228
−2.37228
0 −5.37228 0 11.8614 0 −26.4891 0 1.86141 0
1.2 0 0.372281 0 −16.8614 0 −3.51087 0 −26.8614 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 76.4.a.a 2
3.b odd 2 1 684.4.a.g 2
4.b odd 2 1 304.4.a.f 2
5.b even 2 1 1900.4.a.b 2
5.c odd 4 2 1900.4.c.b 4
8.b even 2 1 1216.4.a.o 2
8.d odd 2 1 1216.4.a.h 2
19.b odd 2 1 1444.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 1.a even 1 1 trivial
304.4.a.f 2 4.b odd 2 1
684.4.a.g 2 3.b odd 2 1
1216.4.a.h 2 8.d odd 2 1
1216.4.a.o 2 8.b even 2 1
1444.4.a.d 2 19.b odd 2 1
1900.4.a.b 2 5.b even 2 1
1900.4.c.b 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5 T_{3} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(76))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + 5 T + T^{2} \)
$5$ \( -200 + 5 T + T^{2} \)
$7$ \( 93 + 30 T + T^{2} \)
$11$ \( 1054 + 71 T + T^{2} \)
$13$ \( -692 + 35 T + T^{2} \)
$17$ \( -297 + T^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( -28712 + 5 T + T^{2} \)
$29$ \( -28850 - 155 T + T^{2} \)
$31$ \( -50864 + 88 T + T^{2} \)
$37$ \( 35572 - 380 T + T^{2} \)
$41$ \( -2384 + 142 T + T^{2} \)
$43$ \( -106928 - 155 T + T^{2} \)
$47$ \( 47392 + 455 T + T^{2} \)
$53$ \( -105908 + 275 T + T^{2} \)
$59$ \( 180426 + 873 T + T^{2} \)
$61$ \( -203150 - 445 T + T^{2} \)
$67$ \( 73308 - 645 T + T^{2} \)
$71$ \( 729436 + 1712 T + T^{2} \)
$73$ \( 211233 + 990 T + T^{2} \)
$79$ \( 404944 + 1274 T + T^{2} \)
$83$ \( -105192 + 90 T + T^{2} \)
$89$ \( 144336 + 888 T + T^{2} \)
$97$ \( -1604528 - 710 T + T^{2} \)
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