Properties

Label 74.6.a.c
Level $74$
Weight $6$
Character orbit 74.a
Self dual yes
Analytic conductor $11.868$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 74.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8684026662\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.324233.1
Defining polynomial: \(x^{3} - 77 x - 140\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} + ( -3 + \beta_{1} - \beta_{2} ) q^{3} + 16 q^{4} -14 q^{5} + ( 12 - 4 \beta_{1} + 4 \beta_{2} ) q^{6} + ( 31 - 7 \beta_{1} + 9 \beta_{2} ) q^{7} -64 q^{8} + ( 66 - 20 \beta_{1} + 5 \beta_{2} ) q^{9} +O(q^{10})\) \( q -4 q^{2} + ( -3 + \beta_{1} - \beta_{2} ) q^{3} + 16 q^{4} -14 q^{5} + ( 12 - 4 \beta_{1} + 4 \beta_{2} ) q^{6} + ( 31 - 7 \beta_{1} + 9 \beta_{2} ) q^{7} -64 q^{8} + ( 66 - 20 \beta_{1} + 5 \beta_{2} ) q^{9} + 56 q^{10} + ( 107 + 11 \beta_{1} + 17 \beta_{2} ) q^{11} + ( -48 + 16 \beta_{1} - 16 \beta_{2} ) q^{12} + ( -274 + 22 \beta_{1} - 16 \beta_{2} ) q^{13} + ( -124 + 28 \beta_{1} - 36 \beta_{2} ) q^{14} + ( 42 - 14 \beta_{1} + 14 \beta_{2} ) q^{15} + 256 q^{16} + ( 80 - 106 \beta_{1} - 6 \beta_{2} ) q^{17} + ( -264 + 80 \beta_{1} - 20 \beta_{2} ) q^{18} + ( -1492 + 24 \beta_{1} - 34 \beta_{2} ) q^{19} -224 q^{20} + ( -2553 + 158 \beta_{1} - 29 \beta_{2} ) q^{21} + ( -428 - 44 \beta_{1} - 68 \beta_{2} ) q^{22} + ( -934 + 18 \beta_{1} + 96 \beta_{2} ) q^{23} + ( 192 - 64 \beta_{1} + 64 \beta_{2} ) q^{24} -2929 q^{25} + ( 1096 - 88 \beta_{1} + 64 \beta_{2} ) q^{26} + ( -2769 + 103 \beta_{1} + 17 \beta_{2} ) q^{27} + ( 496 - 112 \beta_{1} + 144 \beta_{2} ) q^{28} + ( -598 - 330 \beta_{1} - 4 \beta_{2} ) q^{29} + ( -168 + 56 \beta_{1} - 56 \beta_{2} ) q^{30} + ( -4948 - 40 \beta_{1} - 238 \beta_{2} ) q^{31} -1024 q^{32} + ( -2061 + 32 \beta_{1} + 139 \beta_{2} ) q^{33} + ( -320 + 424 \beta_{1} + 24 \beta_{2} ) q^{34} + ( -434 + 98 \beta_{1} - 126 \beta_{2} ) q^{35} + ( 1056 - 320 \beta_{1} + 80 \beta_{2} ) q^{36} -1369 q^{37} + ( 5968 - 96 \beta_{1} + 136 \beta_{2} ) q^{38} + ( 6342 - 624 \beta_{1} + 366 \beta_{2} ) q^{39} + 896 q^{40} + ( 2715 + 656 \beta_{1} + 101 \beta_{2} ) q^{41} + ( 10212 - 632 \beta_{1} + 116 \beta_{2} ) q^{42} + ( -2758 + 816 \beta_{1} - 124 \beta_{2} ) q^{43} + ( 1712 + 176 \beta_{1} + 272 \beta_{2} ) q^{44} + ( -924 + 280 \beta_{1} - 70 \beta_{2} ) q^{45} + ( 3736 - 72 \beta_{1} - 384 \beta_{2} ) q^{46} + ( 6327 - 407 \beta_{1} - 807 \beta_{2} ) q^{47} + ( -768 + 256 \beta_{1} - 256 \beta_{2} ) q^{48} + ( 4966 - 1200 \beta_{1} + 149 \beta_{2} ) q^{49} + 11716 q^{50} + ( -11880 + 1434 \beta_{1} - 1188 \beta_{2} ) q^{51} + ( -4384 + 352 \beta_{1} - 256 \beta_{2} ) q^{52} + ( 12941 + 742 \beta_{1} + 495 \beta_{2} ) q^{53} + ( 11076 - 412 \beta_{1} - 68 \beta_{2} ) q^{54} + ( -1498 - 154 \beta_{1} - 238 \beta_{2} ) q^{55} + ( -1984 + 448 \beta_{1} - 576 \beta_{2} ) q^{56} + ( 13476 - 1940 \beta_{1} + 1460 \beta_{2} ) q^{57} + ( 2392 + 1320 \beta_{1} + 16 \beta_{2} ) q^{58} + ( -5868 + 2386 \beta_{1} - 542 \beta_{2} ) q^{59} + ( 672 - 224 \beta_{1} + 224 \beta_{2} ) q^{60} + ( -11180 - 2 \beta_{1} - 650 \beta_{2} ) q^{61} + ( 19792 + 160 \beta_{1} + 952 \beta_{2} ) q^{62} + ( 24306 - 3022 \beta_{1} + 1714 \beta_{2} ) q^{63} + 4096 q^{64} + ( 3836 - 308 \beta_{1} + 224 \beta_{2} ) q^{65} + ( 8244 - 128 \beta_{1} - 556 \beta_{2} ) q^{66} + ( 1984 + 372 \beta_{1} + 412 \beta_{2} ) q^{67} + ( 1280 - 1696 \beta_{1} - 96 \beta_{2} ) q^{68} + ( -12318 - 784 \beta_{1} + 1882 \beta_{2} ) q^{69} + ( 1736 - 392 \beta_{1} + 504 \beta_{2} ) q^{70} + ( -5671 - 433 \beta_{1} - 2121 \beta_{2} ) q^{71} + ( -4224 + 1280 \beta_{1} - 320 \beta_{2} ) q^{72} + ( -437 - 3172 \beta_{1} + 1181 \beta_{2} ) q^{73} + 5476 q^{74} + ( 8787 - 2929 \beta_{1} + 2929 \beta_{2} ) q^{75} + ( -23872 + 384 \beta_{1} - 544 \beta_{2} ) q^{76} + ( 26545 + 422 \beta_{1} - 1075 \beta_{2} ) q^{77} + ( -25368 + 2496 \beta_{1} - 1464 \beta_{2} ) q^{78} + ( -42804 + 4670 \beta_{1} + 1546 \beta_{2} ) q^{79} -3584 q^{80} + ( 1569 + 820 \beta_{1} + 2720 \beta_{2} ) q^{81} + ( -10860 - 2624 \beta_{1} - 404 \beta_{2} ) q^{82} + ( -48735 + 1945 \beta_{1} - 6209 \beta_{2} ) q^{83} + ( -40848 + 2528 \beta_{1} - 464 \beta_{2} ) q^{84} + ( -1120 + 1484 \beta_{1} + 84 \beta_{2} ) q^{85} + ( 11032 - 3264 \beta_{1} + 496 \beta_{2} ) q^{86} + ( -37086 + 3676 \beta_{1} - 2734 \beta_{2} ) q^{87} + ( -6848 - 704 \beta_{1} - 1088 \beta_{2} ) q^{88} + ( -19880 - 336 \beta_{1} - 1034 \beta_{2} ) q^{89} + ( 3696 - 1120 \beta_{1} + 280 \beta_{2} ) q^{90} + ( -51838 + 4860 \beta_{1} - 2942 \beta_{2} ) q^{91} + ( -14944 + 288 \beta_{1} + 1536 \beta_{2} ) q^{92} + ( 52884 - 5380 \beta_{1} + 2644 \beta_{2} ) q^{93} + ( -25308 + 1628 \beta_{1} + 3228 \beta_{2} ) q^{94} + ( 20888 - 336 \beta_{1} + 476 \beta_{2} ) q^{95} + ( 3072 - 1024 \beta_{1} + 1024 \beta_{2} ) q^{96} + ( 8816 - 4550 \beta_{1} - 3366 \beta_{2} ) q^{97} + ( -19864 + 4800 \beta_{1} - 596 \beta_{2} ) q^{98} + ( -40998 - 4594 \beta_{1} - 638 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 12q^{2} - 8q^{3} + 48q^{4} - 42q^{5} + 32q^{6} + 84q^{7} - 192q^{8} + 193q^{9} + O(q^{10}) \) \( 3q - 12q^{2} - 8q^{3} + 48q^{4} - 42q^{5} + 32q^{6} + 84q^{7} - 192q^{8} + 193q^{9} + 168q^{10} + 304q^{11} - 128q^{12} - 806q^{13} - 336q^{14} + 112q^{15} + 768q^{16} + 246q^{17} - 772q^{18} - 4442q^{19} - 672q^{20} - 7630q^{21} - 1216q^{22} - 2898q^{23} + 512q^{24} - 8787q^{25} + 3224q^{26} - 8324q^{27} + 1344q^{28} - 1790q^{29} - 448q^{30} - 14606q^{31} - 3072q^{32} - 6322q^{33} - 984q^{34} - 1176q^{35} + 3088q^{36} - 4107q^{37} + 17768q^{38} + 18660q^{39} + 2688q^{40} + 8044q^{41} + 30520q^{42} - 8150q^{43} + 4864q^{44} - 2702q^{45} + 11592q^{46} + 19788q^{47} - 2048q^{48} + 14749q^{49} + 35148q^{50} - 34452q^{51} - 12896q^{52} + 38328q^{53} + 33296q^{54} - 4256q^{55} - 5376q^{56} + 38968q^{57} + 7160q^{58} - 17062q^{59} + 1792q^{60} - 32890q^{61} + 58424q^{62} + 71204q^{63} + 12288q^{64} + 11284q^{65} + 25288q^{66} + 5540q^{67} + 3936q^{68} - 38836q^{69} + 4704q^{70} - 14892q^{71} - 12352q^{72} - 2492q^{73} + 16428q^{74} + 23432q^{75} - 71072q^{76} + 80710q^{77} - 74640q^{78} - 129958q^{79} - 10752q^{80} + 1987q^{81} - 32176q^{82} - 139996q^{83} - 122080q^{84} - 3444q^{85} + 32600q^{86} - 108524q^{87} - 19456q^{88} - 58606q^{89} + 10808q^{90} - 152572q^{91} - 46368q^{92} + 156008q^{93} - 79152q^{94} + 62188q^{95} + 8192q^{96} + 29814q^{97} - 58996q^{98} - 122356q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 77 x - 140\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 52 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{2} + \beta_{1} + 104\)\()/2\)

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
−7.66368
9.57214
−1.90845
−4.00000 −25.5252 16.0000 −14.0000 102.101 203.072 −64.0000 408.537 56.0000
1.2 −4.00000 1.11745 16.0000 −14.0000 −4.46979 32.2315 −64.0000 −241.751 56.0000
1.3 −4.00000 16.4078 16.0000 −14.0000 −65.6311 −151.304 −64.0000 26.2148 56.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(37\) \(1\)

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 74.6.a.c 3
3.b odd 2 1 666.6.a.i 3
4.b odd 2 1 592.6.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.a.c 3 1.a even 1 1 trivial
592.6.a.c 3 4.b odd 2 1
666.6.a.i 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 8 T_{3}^{2} - 429 T_{3} + 468 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(74))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{3} \)
$3$ \( 468 - 429 T + 8 T^{2} + T^{3} \)
$5$ \( ( 14 + T )^{3} \)
$7$ \( 990332 - 29057 T - 84 T^{2} + T^{3} \)
$11$ \( 11483660 - 174213 T - 304 T^{2} + T^{3} \)
$13$ \( -922776 + 56572 T + 806 T^{2} + T^{3} \)
$17$ \( 2114271552 - 3624392 T - 246 T^{2} + T^{3} \)
$19$ \( 2579057312 + 6144736 T + 4442 T^{2} + T^{3} \)
$23$ \( -1430959880 - 1523172 T + 2898 T^{2} + T^{3} \)
$29$ \( 23321731784 - 32830820 T + 1790 T^{2} + T^{3} \)
$31$ \( -41137100320 + 44956992 T + 14606 T^{2} + T^{3} \)
$37$ \( ( 1369 + T )^{3} \)
$41$ \( -237018444942 - 132764347 T - 8044 T^{2} + T^{3} \)
$43$ \( -531931362712 - 162306404 T + 8150 T^{2} + T^{3} \)
$47$ \( 2444798953316 - 273792353 T - 19788 T^{2} + T^{3} \)
$53$ \( 246191549806 + 122353801 T - 38328 T^{2} + T^{3} \)
$59$ \( -3503222325504 - 1432364728 T + 17062 T^{2} + T^{3} \)
$61$ \( -1303290680000 + 187716168 T + 32890 T^{2} + T^{3} \)
$67$ \( -103615970816 - 142472096 T - 5540 T^{2} + T^{3} \)
$71$ \( -29653094181780 - 2065060337 T + 14892 T^{2} + T^{3} \)
$73$ \( -38779212138142 - 2669954147 T + 2492 T^{2} + T^{3} \)
$79$ \( -696052558500480 - 3983883432 T + 129958 T^{2} + T^{3} \)
$83$ \( -1137118842312648 - 7161797773 T + 139996 T^{2} + T^{3} \)
$89$ \( -5026391398400 + 581128960 T + 58606 T^{2} + T^{3} \)
$97$ \( 730628639852224 - 14780346968 T - 29814 T^{2} + T^{3} \)
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