Properties

Label 76.8.a.b
Level $76$
Weight $8$
Character orbit 76.a
Self dual yes
Analytic conductor $23.741$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 8376 x^{4} + 135458 x^{3} + 16275767 x^{2} - 280013424 x - 6276171312\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 7 - \beta_{1} ) q^{3} + ( 47 - \beta_{1} - \beta_{2} ) q^{5} + ( -262 + 5 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( 662 - 34 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( 7 - \beta_{1} ) q^{3} + ( 47 - \beta_{1} - \beta_{2} ) q^{5} + ( -262 + 5 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( 662 - 34 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{9} + ( 1346 - 52 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{11} + ( 221 - 52 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{13} + ( 4540 - 217 \beta_{1} - 3 \beta_{2} - 13 \beta_{3} + 7 \beta_{4} - 9 \beta_{5} ) q^{15} + ( 3647 - 46 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} - 7 \beta_{4} - 19 \beta_{5} ) q^{17} + 6859 q^{19} + ( -14601 + 533 \beta_{1} + 81 \beta_{2} - 31 \beta_{3} - 44 \beta_{4} + 67 \beta_{5} ) q^{21} + ( 16649 + 494 \beta_{1} - 38 \beta_{2} + 12 \beta_{3} - \beta_{4} + 26 \beta_{5} ) q^{23} + ( 62269 + 12 \beta_{1} - 93 \beta_{2} + 123 \beta_{3} + 42 \beta_{4} - 108 \beta_{5} ) q^{25} + ( 89359 - 514 \beta_{1} - 261 \beta_{2} - 33 \beta_{3} + 111 \beta_{4} - 99 \beta_{5} ) q^{27} + ( -10315 + 1379 \beta_{1} + 143 \beta_{2} - 165 \beta_{3} - 90 \beta_{4} + 171 \beta_{5} ) q^{29} + ( 66842 + 1281 \beta_{1} - 133 \beta_{2} - 65 \beta_{3} + 15 \beta_{4} + 99 \beta_{5} ) q^{31} + ( 153504 - 2033 \beta_{1} - 27 \beta_{2} - 27 \beta_{3} + 111 \beta_{4} - 241 \beta_{5} ) q^{33} + ( 75982 + 3832 \beta_{1} + 163 \beta_{2} + 315 \beta_{3} - 174 \beta_{4} + 42 \beta_{5} ) q^{35} + ( 135038 - 1059 \beta_{1} + 385 \beta_{2} + 95 \beta_{3} - 15 \beta_{4} + 105 \beta_{5} ) q^{37} + ( 125967 + 1624 \beta_{1} + 168 \beta_{2} + 26 \beta_{3} - 161 \beta_{4} - 64 \beta_{5} ) q^{39} + ( 93658 - 2693 \beta_{1} + 609 \beta_{2} - 327 \beta_{3} - 193 \beta_{4} + 53 \beta_{5} ) q^{41} + ( 203132 + 96 \beta_{1} + 95 \beta_{2} - 59 \beta_{3} + 36 \beta_{4} + 528 \beta_{5} ) q^{43} + ( 544345 - 15535 \beta_{1} - 63 \beta_{2} + 68 \beta_{3} + 664 \beta_{4} - 474 \beta_{5} ) q^{45} + ( 320025 - 333 \beta_{1} - 985 \beta_{2} - 114 \beta_{3} - 124 \beta_{4} - 982 \beta_{5} ) q^{47} + ( 343482 - 7868 \beta_{1} - 676 \beta_{2} + 68 \beta_{3} + 247 \beta_{4} + 303 \beta_{5} ) q^{49} + ( 150417 + 991 \beta_{1} + 882 \beta_{2} - 54 \beta_{3} - 6 \beta_{4} + 1154 \beta_{5} ) q^{51} + ( 84255 + 2955 \beta_{1} + 127 \beta_{2} + 1089 \beta_{3} - 464 \beta_{4} + 337 \beta_{5} ) q^{53} + ( 303243 - 123 \beta_{1} - 2925 \beta_{2} - 1596 \beta_{3} + 264 \beta_{4} - 1158 \beta_{5} ) q^{55} + ( 48013 - 6859 \beta_{1} ) q^{57} + ( 213157 + 11722 \beta_{1} - 141 \beta_{2} + 651 \beta_{3} + 209 \beta_{4} - 697 \beta_{5} ) q^{59} + ( 92723 + 4843 \beta_{1} - 3229 \beta_{2} + 1058 \beta_{3} + 532 \beta_{4} + 492 \beta_{5} ) q^{61} + ( -1209611 + 39599 \beta_{1} + 5571 \beta_{2} + 460 \beta_{3} - 2044 \beta_{4} + 2220 \beta_{5} ) q^{63} + ( -1035068 + 8272 \beta_{1} - 38 \beta_{2} - 2994 \beta_{3} - 636 \beta_{4} + 54 \beta_{5} ) q^{65} + ( -13057 + 20179 \beta_{1} + 142 \beta_{2} + 1758 \beta_{3} + 340 \beta_{4} - 2616 \beta_{5} ) q^{67} + ( -1229577 - 2012 \beta_{1} + 2916 \beta_{2} - 198 \beta_{3} - 717 \beta_{4} - 508 \beta_{5} ) q^{69} + ( -242236 + 9992 \beta_{1} + 3444 \beta_{2} + 246 \beta_{3} + 1684 \beta_{4} + 838 \beta_{5} ) q^{71} + ( -1864249 + 15252 \beta_{1} + 382 \beta_{2} - 16 \beta_{3} - 1089 \beta_{4} + 987 \beta_{5} ) q^{73} + ( 675139 - 60880 \beta_{1} + 4545 \beta_{2} - 771 \beta_{3} + 4641 \beta_{4} - 309 \beta_{5} ) q^{75} + ( -2932317 + 42087 \beta_{1} - 2113 \beta_{2} - 1398 \beta_{3} - 2866 \beta_{4} + 716 \beta_{5} ) q^{77} + ( -1068724 + 15445 \beta_{1} - 5115 \beta_{2} + 25 \beta_{3} - 695 \beta_{4} - 1395 \beta_{5} ) q^{79} + ( 1072103 - 117862 \beta_{1} - 6564 \beta_{2} + 1898 \beta_{3} + 6472 \beta_{4} - 5248 \beta_{5} ) q^{81} + ( -517244 - 45434 \beta_{1} + 9308 \beta_{2} - 42 \beta_{3} + 2686 \beta_{4} + 5164 \beta_{5} ) q^{83} + ( -3248511 + 13071 \beta_{1} - 14307 \beta_{2} - 1716 \beta_{3} - 2352 \beta_{4} + 1560 \beta_{5} ) q^{85} + ( -4352219 + 60914 \beta_{1} + 780 \beta_{2} + 1736 \beta_{3} - 9689 \beta_{4} + 3528 \beta_{5} ) q^{87} + ( -2421858 + 17253 \beta_{1} + 2831 \beta_{2} + 9039 \beta_{3} - 1759 \beta_{4} - 1783 \beta_{5} ) q^{89} + ( -1604921 - 50760 \beta_{1} - 3455 \beta_{2} - 5575 \beta_{3} - 2055 \beta_{4} + 795 \beta_{5} ) q^{91} + ( -3033262 - 64756 \beta_{1} - 240 \beta_{2} + 234 \beta_{3} - 1686 \beta_{4} - 3868 \beta_{5} ) q^{93} + ( 322373 - 6859 \beta_{1} - 6859 \beta_{2} ) q^{95} + ( -3557830 - 25110 \beta_{1} + 168 \beta_{2} - 9084 \beta_{3} + 2514 \beta_{4} + 4998 \beta_{5} ) q^{97} + ( 4064986 - 163550 \beta_{1} - 21975 \beta_{2} - 5047 \beta_{3} + 6832 \beta_{4} - 7190 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 40q^{3} + 279q^{5} - 1565q^{7} + 3900q^{9} + O(q^{10}) \) \( 6q + 40q^{3} + 279q^{5} - 1565q^{7} + 3900q^{9} + 7983q^{11} + 1250q^{13} + 26790q^{15} + 21735q^{17} + 41154q^{19} - 86346q^{21} + 100920q^{23} + 373305q^{25} + 534790q^{27} - 58656q^{29} + 403808q^{31} + 916430q^{33} + 463497q^{35} + 808780q^{37} + 758704q^{39} + 556944q^{41} + 1220735q^{43} + 3234843q^{45} + 1915305q^{47} + 2045883q^{49} + 908816q^{51} + 511650q^{53} + 1813341q^{55} + 274360q^{57} + 1300572q^{59} + 565335q^{61} - 7170325q^{63} - 6195012q^{65} - 45010q^{67} - 7381528q^{69} - 1424106q^{71} - 11153825q^{73} + 3941974q^{75} - 17515425q^{77} - 6392144q^{79} + 6187530q^{81} - 3164160q^{83} - 19479255q^{85} - 25999500q^{87} - 14502678q^{89} - 9736226q^{91} - 18344300q^{93} + 1913661q^{95} - 21377010q^{97} + 24032935q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 8376 x^{4} + 135458 x^{3} + 16275767 x^{2} - 280013424 x - 6276171312\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -23 \nu^{5} + 1337 \nu^{4} + 331031 \nu^{3} - 3798869 \nu^{2} - 655626372 \nu - 394340616 \)\()/12622824\)
\(\beta_{3}\)\(=\)\((\)\( 51 \nu^{5} + 2117 \nu^{4} - 357983 \nu^{3} - 9662021 \nu^{2} + 548036696 \nu + 11111121348 \)\()/2103804\)
\(\beta_{4}\)\(=\)\((\)\( -269 \nu^{5} - 12312 \nu^{4} + 1628074 \nu^{3} + 41365824 \nu^{2} - 2202338405 \nu - 33289998684 \)\()/2103804\)
\(\beta_{5}\)\(=\)\((\)\( -617 \nu^{5} - 30195 \nu^{4} + 3641083 \nu^{3} + 101646951 \nu^{2} - 4929275990 \nu - 76626597048 \)\()/4207608\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 20 \beta_{1} + 2800\)
\(\nu^{3}\)\(=\)\(78 \beta_{5} - 90 \beta_{4} + 12 \beta_{3} + 198 \beta_{2} + 4321 \beta_{1} - 60834\)
\(\nu^{4}\)\(=\)\(-9331 \beta_{5} + 10219 \beta_{4} - 4033 \beta_{3} - 19821 \beta_{2} - 212696 \beta_{1} + 12452470\)
\(\nu^{5}\)\(=\)\(745380 \beta_{5} - 866472 \beta_{4} + 103440 \beta_{3} + 1644228 \beta_{2} + 24624409 \beta_{1} - 631311360\)

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
52.7878
49.5941
44.4724
−13.7370
−46.2258
−84.8914
0 −45.7878 0 −327.289 0 −7.23734 0 −90.4805 0
1.2 0 −42.5941 0 51.7645 0 −1212.43 0 −372.745 0
1.3 0 −37.4724 0 534.866 0 1051.36 0 −782.818 0
1.4 0 20.7370 0 −501.411 0 −1008.02 0 −1756.98 0
1.5 0 53.2258 0 88.7692 0 1094.09 0 645.987 0
1.6 0 91.8914 0 432.300 0 −1482.77 0 6257.03 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.b 6
4.b odd 2 1 304.8.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.a.b 6 1.a even 1 1 trivial
304.8.a.i 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 40 T_{3}^{5} - 7711 T_{3}^{4} + 93190 T_{3}^{3} + 16686996 T_{3}^{2} + 43655400 T_{3} - 7412317344 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(76))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( -7412317344 + 43655400 T + 16686996 T^{2} + 93190 T^{3} - 7711 T^{4} - 40 T^{5} + T^{6} \)
$5$ \( 174360864384000 - 5181395099400 T + 31455689670 T^{2} + 89339211 T^{3} - 382107 T^{4} - 279 T^{5} + T^{6} \)
$7$ \( 15086336025908584 + 2093958739148105 T + 1278964317269 T^{2} - 3630360070 T^{3} - 2268958 T^{4} + 1565 T^{5} + T^{6} \)
$11$ \( -47929083669332301456 + 385525041126730164 T - 880155543756408 T^{2} + 432916267239 T^{3} - 37767781 T^{4} - 7983 T^{5} + T^{6} \)
$13$ \( -\)\(15\!\cdots\!56\)\( + 589363572143005920 T + 3312548747774008 T^{2} - 6544577750 T^{3} - 122286729 T^{4} - 1250 T^{5} + T^{6} \)
$17$ \( \)\(81\!\cdots\!86\)\( - \)\(85\!\cdots\!55\)\( T - 231907545730575987 T^{2} + 36745972939470 T^{3} - 1164308584 T^{4} - 21735 T^{5} + T^{6} \)
$19$ \( ( -6859 + T )^{6} \)
$23$ \( -\)\(10\!\cdots\!64\)\( - \)\(70\!\cdots\!80\)\( T + 1502913205656366528 T^{2} + 151244309224680 T^{3} - 284311201 T^{4} - 100920 T^{5} + T^{6} \)
$29$ \( -\)\(19\!\cdots\!56\)\( + \)\(76\!\cdots\!44\)\( T + \)\(25\!\cdots\!08\)\( T^{2} - 4494973587859440 T^{3} - 95711902329 T^{4} + 58656 T^{5} + T^{6} \)
$31$ \( -\)\(59\!\cdots\!24\)\( + \)\(76\!\cdots\!80\)\( T - \)\(31\!\cdots\!08\)\( T^{2} + 4120055684069632 T^{3} + 18356065296 T^{4} - 403808 T^{5} + T^{6} \)
$37$ \( \)\(12\!\cdots\!96\)\( + \)\(51\!\cdots\!40\)\( T - \)\(34\!\cdots\!04\)\( T^{2} - 9069720444468640 T^{3} + 176256645884 T^{4} - 808780 T^{5} + T^{6} \)
$41$ \( \)\(23\!\cdots\!84\)\( - \)\(10\!\cdots\!76\)\( T + \)\(21\!\cdots\!84\)\( T^{2} + 174434267004322080 T^{3} - 326056713208 T^{4} - 556944 T^{5} + T^{6} \)
$43$ \( -\)\(15\!\cdots\!44\)\( + \)\(46\!\cdots\!60\)\( T - \)\(46\!\cdots\!40\)\( T^{2} + 142748983288435795 T^{3} + 158935767027 T^{4} - 1220735 T^{5} + T^{6} \)
$47$ \( \)\(10\!\cdots\!56\)\( - \)\(32\!\cdots\!20\)\( T - \)\(66\!\cdots\!96\)\( T^{2} + 2150474405962335525 T^{3} - 384082342837 T^{4} - 1915305 T^{5} + T^{6} \)
$53$ \( -\)\(69\!\cdots\!76\)\( - \)\(18\!\cdots\!40\)\( T + \)\(39\!\cdots\!40\)\( T^{2} + 1036250365853981250 T^{3} - 3740637014161 T^{4} - 511650 T^{5} + T^{6} \)
$59$ \( -\)\(86\!\cdots\!56\)\( - \)\(61\!\cdots\!76\)\( T + \)\(57\!\cdots\!12\)\( T^{2} + 1965619922082623202 T^{3} - 1547972074327 T^{4} - 1300572 T^{5} + T^{6} \)
$61$ \( -\)\(23\!\cdots\!64\)\( - \)\(32\!\cdots\!16\)\( T + \)\(12\!\cdots\!14\)\( T^{2} + 3565037032223823767 T^{3} - 8735718697011 T^{4} - 565335 T^{5} + T^{6} \)
$67$ \( -\)\(13\!\cdots\!76\)\( - \)\(12\!\cdots\!40\)\( T + \)\(85\!\cdots\!60\)\( T^{2} + 1501881992060699320 T^{3} - 16470947063367 T^{4} + 45010 T^{5} + T^{6} \)
$71$ \( \)\(13\!\cdots\!04\)\( + \)\(50\!\cdots\!52\)\( T + \)\(41\!\cdots\!32\)\( T^{2} - 18728657102643413232 T^{3} - 14152150868392 T^{4} + 1424106 T^{5} + T^{6} \)
$73$ \( \)\(96\!\cdots\!26\)\( + \)\(49\!\cdots\!25\)\( T + \)\(96\!\cdots\!29\)\( T^{2} + 92710992204432255410 T^{3} + 46136474610476 T^{4} + 11153825 T^{5} + T^{6} \)
$79$ \( \)\(29\!\cdots\!76\)\( - \)\(35\!\cdots\!56\)\( T - \)\(79\!\cdots\!60\)\( T^{2} - 43513478257211180320 T^{3} + 2824710229340 T^{4} + 6392144 T^{5} + T^{6} \)
$83$ \( -\)\(65\!\cdots\!64\)\( - \)\(17\!\cdots\!40\)\( T + \)\(28\!\cdots\!68\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} - 103202687694196 T^{4} + 3164160 T^{5} + T^{6} \)
$89$ \( \)\(18\!\cdots\!76\)\( + \)\(69\!\cdots\!72\)\( T - \)\(14\!\cdots\!76\)\( T^{2} - \)\(22\!\cdots\!40\)\( T^{3} - 112082030411632 T^{4} + 14502678 T^{5} + T^{6} \)
$97$ \( \)\(48\!\cdots\!44\)\( + \)\(28\!\cdots\!00\)\( T - \)\(15\!\cdots\!40\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} - 45190604396916 T^{4} + 21377010 T^{5} + T^{6} \)
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