Properties

Label 76.8.a.a
Level $76$
Weight $8$
Character orbit 76.a
Self dual yes
Analytic conductor $23.741$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7412619368\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5014 x^{3} + 113222 x^{2} - 625803 x + 567036\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta_{1} ) q^{3} + ( -56 + \beta_{1} - \beta_{2} ) q^{5} + ( 84 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{7} + ( 762 + 18 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta_{1} ) q^{3} + ( -56 + \beta_{1} - \beta_{2} ) q^{5} + ( 84 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{7} + ( 762 + 18 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{9} + ( -518 + 55 \beta_{1} + \beta_{2} + 8 \beta_{4} ) q^{11} + ( -118 + 64 \beta_{1} + 6 \beta_{2} - 11 \beta_{3} - 18 \beta_{4} ) q^{13} + ( -4146 + 111 \beta_{1} - 13 \beta_{2} - 19 \beta_{3} + 5 \beta_{4} ) q^{15} + ( -5439 + 153 \beta_{1} - 66 \beta_{2} + 27 \beta_{3} + 15 \beta_{4} ) q^{17} -6859 q^{19} + ( -11982 + 5 \beta_{1} - 73 \beta_{2} + 50 \beta_{3} + 11 \beta_{4} ) q^{21} + ( -13466 - 428 \beta_{1} + 80 \beta_{2} + 39 \beta_{3} + 18 \beta_{4} ) q^{23} + ( -21955 - 505 \beta_{1} + 105 \beta_{2} - 40 \beta_{3} - 90 \beta_{4} ) q^{25} + ( -36057 - 1220 \beta_{1} + 353 \beta_{2} - 199 \beta_{3} - 121 \beta_{4} ) q^{27} + ( -74458 - 247 \beta_{1} - 179 \beta_{2} + 60 \beta_{3} + 207 \beta_{4} ) q^{29} + ( -54578 - 1751 \beta_{1} + 477 \beta_{2} - 35 \beta_{3} - 87 \beta_{4} ) q^{31} + ( -158436 + 863 \beta_{1} - 611 \beta_{2} - 149 \beta_{3} + 283 \beta_{4} ) q^{33} + ( -121704 - 1101 \beta_{1} - 257 \beta_{2} + 384 \beta_{3} + 160 \beta_{4} ) q^{35} + ( -112160 + 1813 \beta_{1} - 147 \beta_{2} + 265 \beta_{3} - 183 \beta_{4} ) q^{37} + ( -192990 + 62 \beta_{1} - 290 \beta_{2} - 125 \beta_{3} - 284 \beta_{4} ) q^{39} + ( -191224 + 3593 \beta_{1} - 471 \beta_{2} - 405 \beta_{3} - 859 \beta_{4} ) q^{41} + ( -165152 + 255 \beta_{1} + 291 \beta_{2} + 558 \beta_{3} - 30 \beta_{4} ) q^{43} + ( -230970 + 9085 \beta_{1} - 55 \beta_{2} - 40 \beta_{3} + 650 \beta_{4} ) q^{45} + ( -361646 + 1183 \beta_{1} - 1627 \beta_{2} + 1032 \beta_{3} + 1566 \beta_{4} ) q^{47} + ( -171688 + 6237 \beta_{1} + 2646 \beta_{2} - 1995 \beta_{3} - 651 \beta_{4} ) q^{49} + ( -492453 - 9 \beta_{1} + 6 \beta_{2} - 2370 \beta_{3} + 366 \beta_{4} ) q^{51} + ( 98746 + 3337 \beta_{1} + 2205 \beta_{2} + 990 \beta_{3} - 395 \beta_{4} ) q^{53} + ( 92758 - 11253 \beta_{1} + 2853 \beta_{2} + 2040 \beta_{3} + 90 \beta_{4} ) q^{55} + ( 20577 + 6859 \beta_{1} ) q^{57} + ( -80631 - 22938 \beta_{1} - 1651 \beta_{2} - 885 \beta_{3} - 4807 \beta_{4} ) q^{59} + ( 379142 + 1003 \beta_{1} - 369 \beta_{2} + 4240 \beta_{3} + 3432 \beta_{4} ) q^{61} + ( -204702 - 10913 \beta_{1} - 4043 \beta_{2} - 356 \beta_{3} - 284 \beta_{4} ) q^{63} + ( 354674 - 12424 \beta_{1} - 4098 \beta_{2} + 696 \beta_{3} - 1210 \beta_{4} ) q^{65} + ( -213195 - 17411 \beta_{1} + 1422 \beta_{2} - 4448 \beta_{3} + 2652 \beta_{4} ) q^{67} + ( 1454322 + 2490 \beta_{1} + 4814 \beta_{2} + 2225 \beta_{3} - 928 \beta_{4} ) q^{69} + ( -937688 - 15608 \beta_{1} - 15496 \beta_{2} - 1800 \beta_{3} + 5418 \beta_{4} ) q^{71} + ( 842379 - 10643 \beta_{1} + 1344 \beta_{2} - 4595 \beta_{3} + 2721 \beta_{4} ) q^{73} + ( 1644975 + 23190 \beta_{1} + 4355 \beta_{2} + 4055 \beta_{3} - 2815 \beta_{4} ) q^{75} + ( 1720008 + 3495 \beta_{1} + 3487 \beta_{2} - 5418 \beta_{3} - 3608 \beta_{4} ) q^{77} + ( 1909172 - 17539 \beta_{1} + 11547 \beta_{2} - 2365 \beta_{3} - 8721 \beta_{4} ) q^{79} + ( 2279169 + 45952 \beta_{1} - 11136 \beta_{2} + 4980 \beta_{3} - 372 \beta_{4} ) q^{81} + ( 2363280 + 21564 \beta_{1} + 8796 \beta_{2} + 11898 \beta_{3} + 3720 \beta_{4} ) q^{83} + ( 3741216 - 10251 \beta_{1} + 17013 \beta_{2} - 2256 \beta_{3} - 3960 \beta_{4} ) q^{85} + ( 781422 + 100338 \beta_{1} - 4778 \beta_{2} - 353 \beta_{3} + 5392 \beta_{4} ) q^{87} + ( 547314 - 25143 \beta_{1} - 17633 \beta_{2} - 8355 \beta_{3} + 6757 \beta_{4} ) q^{89} + ( 1486047 + 27226 \beta_{1} + 19347 \beta_{2} - 4505 \beta_{3} + 3729 \beta_{4} ) q^{91} + ( 5923098 + 47058 \beta_{1} + 13792 \beta_{2} + 15370 \beta_{3} - 5552 \beta_{4} ) q^{93} + ( 384104 - 6859 \beta_{1} + 6859 \beta_{2} ) q^{95} + ( 242902 - 58604 \beta_{1} - 50124 \beta_{2} + 6406 \beta_{3} + 9426 \beta_{4} ) q^{97} + ( -1928322 + 167567 \beta_{1} - 34287 \beta_{2} - 3990 \beta_{3} - 3300 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} + O(q^{10}) \) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} - 2662q^{11} - 602q^{13} - 20800q^{15} - 27366q^{17} - 34295q^{19} - 59964q^{21} - 67096q^{23} - 109115q^{25} - 178778q^{27} - 372398q^{29} - 271372q^{31} - 792700q^{33} - 608250q^{35} - 562630q^{37} - 963904q^{39} - 956714q^{41} - 827362q^{43} - 1165100q^{45} - 1812982q^{47} - 862031q^{49} - 2458254q^{51} + 486998q^{53} + 467930q^{55} + 96026q^{57} - 367182q^{59} + 1879732q^{61} - 1007274q^{63} + 1790920q^{65} - 1046394q^{67} + 7261712q^{69} - 4664572q^{71} + 4224942q^{73} + 8194850q^{75} + 8611110q^{77} + 9574024q^{79} + 11351813q^{81} + 11754804q^{83} + 18711750q^{85} + 3801472q^{87} + 2782542q^{89} + 7385214q^{91} + 29535004q^{93} + 1920520q^{95} + 1291574q^{97} - 9760310q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5014 x^{3} + 113222 x^{2} - 625803 x + 567036\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 302 \nu^{4} - 759 \nu^{3} - 1458172 \nu^{2} + 37319919 \nu - 275123626 \)\()/3056594\)
\(\beta_{2}\)\(=\)\((\)\( -440 \nu^{4} + 11227 \nu^{3} + 2418003 \nu^{2} - 101285025 \nu + 456154575 \)\()/1528297\)
\(\beta_{3}\)\(=\)\((\)\( -3831 \nu^{4} + 24810 \nu^{3} + 19701959 \nu^{2} - 484183104 \nu + 2008817956 \)\()/3056594\)
\(\beta_{4}\)\(=\)\((\)\( 15023 \nu^{4} + 78637 \nu^{3} - 74510443 \nu^{2} + 1156525641 \nu - 2210497044 \)\()/3056594\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 7 \beta_{3} - 11 \beta_{2} + 7 \beta_{1} + 36\)\()/84\)
\(\nu^{2}\)\(=\)\((\)\(-13 \beta_{4} - 35 \beta_{3} + 101 \beta_{2} + 497 \beta_{1} + 28190\)\()/14\)
\(\nu^{3}\)\(=\)\((\)\(2299 \beta_{4} + 12845 \beta_{3} - 18625 \beta_{2} - 5691 \beta_{1} - 1732420\)\()/28\)
\(\nu^{4}\)\(=\)\((\)\(-40238 \beta_{4} - 148512 \beta_{3} + 345409 \beta_{2} + 1195040 \beta_{1} + 72973885\)\()/7\)

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
57.9656
−79.8251
15.8518
6.88490
1.12286
0 −84.8970 0 72.4872 0 −210.006 0 5020.49 0
1.2 0 −36.5155 0 −266.507 0 1627.87 0 −853.617 0
1.3 0 8.09048 0 276.339 0 −720.800 0 −2121.54 0
1.4 0 25.4201 0 −3.35619 0 −173.262 0 −1540.82 0
1.5 0 73.9019 0 −358.964 0 −109.800 0 3274.48 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 76.8.a.a 5
4.b odd 2 1 304.8.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.a.a 5 1.a even 1 1 trivial
304.8.a.g 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 14 T_{3}^{4} - 7259 T_{3}^{3} - 22536 T_{3}^{2} + 6469524 T_{3} - 47116944 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(76))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -47116944 + 6469524 T - 22536 T^{2} - 7259 T^{3} + 14 T^{4} + T^{5} \)
$5$ \( 6431464000 + 1846348600 T - 21186050 T^{2} - 101555 T^{3} + 280 T^{4} + T^{5} \)
$7$ \( -4687831896336 - 95696771841 T - 645731154 T^{2} - 1542144 T^{3} - 414 T^{4} + T^{5} \)
$11$ \( -1053955338796946000 + 684520373335036 T - 10845737972 T^{2} - 49903451 T^{3} + 2662 T^{4} + T^{5} \)
$13$ \( -5386040659739527712 + 1804473258241184 T + 635364478342 T^{2} - 166234049 T^{3} + 602 T^{4} + T^{5} \)
$17$ \( -74095261080705325026 + 74135586518583129 T - 13922276457924 T^{2} - 634921434 T^{3} + 27366 T^{4} + T^{5} \)
$19$ \( ( 6859 + T )^{5} \)
$23$ \( \)\(17\!\cdots\!64\)\( - 2669335822660692416 T - 171053808154304 T^{2} - 1552248449 T^{3} + 67096 T^{4} + T^{5} \)
$29$ \( -\)\(16\!\cdots\!00\)\( - \)\(40\!\cdots\!00\)\( T - 1372372529110510 T^{2} + 37345252075 T^{3} + 372398 T^{4} + T^{5} \)
$31$ \( \)\(46\!\cdots\!76\)\( - \)\(62\!\cdots\!28\)\( T - 10804547269027648 T^{2} - 24683984048 T^{3} + 271372 T^{4} + T^{5} \)
$37$ \( -\)\(15\!\cdots\!80\)\( - \)\(41\!\cdots\!56\)\( T - 27012641642832880 T^{2} + 28368593944 T^{3} + 562630 T^{4} + T^{5} \)
$41$ \( \)\(13\!\cdots\!60\)\( - \)\(66\!\cdots\!84\)\( T - 187022106794472136 T^{2} - 114941692004 T^{3} + 956714 T^{4} + T^{5} \)
$43$ \( \)\(14\!\cdots\!28\)\( - \)\(59\!\cdots\!36\)\( T - 57541345607077912 T^{2} + 87107828809 T^{3} + 827362 T^{4} + T^{5} \)
$47$ \( \)\(43\!\cdots\!48\)\( - \)\(43\!\cdots\!56\)\( T - 1153899669604238312 T^{2} + 138216878449 T^{3} + 1812982 T^{4} + T^{5} \)
$53$ \( -\)\(10\!\cdots\!52\)\( + \)\(40\!\cdots\!04\)\( T + 349864873041442318 T^{2} - 1244529609401 T^{3} - 486998 T^{4} + T^{5} \)
$59$ \( \)\(51\!\cdots\!76\)\( + \)\(47\!\cdots\!72\)\( T - 10282362311118342888 T^{2} - 14086815378843 T^{3} + 367182 T^{4} + T^{5} \)
$61$ \( -\)\(22\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T + 13666352240866067290 T^{2} - 7964309784815 T^{3} - 1879732 T^{4} + T^{5} \)
$67$ \( \)\(25\!\cdots\!16\)\( + \)\(64\!\cdots\!64\)\( T - 262329580235863116 T^{2} - 20206084455039 T^{3} + 1046394 T^{4} + T^{5} \)
$71$ \( \)\(83\!\cdots\!08\)\( + \)\(18\!\cdots\!08\)\( T - \)\(14\!\cdots\!88\)\( T^{2} - 33451780289792 T^{3} + 4664572 T^{4} + T^{5} \)
$73$ \( -\)\(35\!\cdots\!38\)\( + \)\(36\!\cdots\!89\)\( T + 49111259849504070768 T^{2} - 13340308046754 T^{3} - 4224942 T^{4} + T^{5} \)
$79$ \( \)\(11\!\cdots\!08\)\( - \)\(42\!\cdots\!68\)\( T + \)\(20\!\cdots\!64\)\( T^{2} - 4689316616852 T^{3} - 9574024 T^{4} + T^{5} \)
$83$ \( -\)\(15\!\cdots\!36\)\( - \)\(16\!\cdots\!16\)\( T + \)\(47\!\cdots\!56\)\( T^{2} - 25411639393764 T^{3} - 11754804 T^{4} + T^{5} \)
$89$ \( -\)\(65\!\cdots\!68\)\( + \)\(28\!\cdots\!28\)\( T + \)\(26\!\cdots\!08\)\( T^{2} - 110216148962952 T^{3} - 2782542 T^{4} + T^{5} \)
$97$ \( \)\(75\!\cdots\!84\)\( + \)\(15\!\cdots\!44\)\( T + \)\(64\!\cdots\!64\)\( T^{2} - 297568136505956 T^{3} - 1291574 T^{4} + T^{5} \)
show more
show less