Properties

Label 74.6.a.e
Level $74$
Weight $6$
Character orbit 74.a
Self dual yes
Analytic conductor $11.868$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 870 x^{3} - 2235 x^{2} + 121361 x + 481504\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( 4 - \beta_{1} ) q^{3} + 16 q^{4} + ( 19 + \beta_{2} ) q^{5} + ( 16 - 4 \beta_{1} ) q^{6} + ( 34 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + 64 q^{8} + ( 121 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{9} +O(q^{10})\) \( q + 4 q^{2} + ( 4 - \beta_{1} ) q^{3} + 16 q^{4} + ( 19 + \beta_{2} ) q^{5} + ( 16 - 4 \beta_{1} ) q^{6} + ( 34 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{7} + 64 q^{8} + ( 121 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{9} + ( 76 + 4 \beta_{2} ) q^{10} + ( 177 + 11 \beta_{1} - 6 \beta_{2} - \beta_{3} - 5 \beta_{4} ) q^{11} + ( 64 - 16 \beta_{1} ) q^{12} + ( 276 + 6 \beta_{1} - \beta_{2} + 3 \beta_{3} + 9 \beta_{4} ) q^{13} + ( 136 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} ) q^{14} + ( -3 - 8 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} ) q^{15} + 256 q^{16} + ( 402 + 46 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 10 \beta_{4} ) q^{17} + ( 484 - 12 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 8 \beta_{4} ) q^{18} + ( 468 + 38 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} + 16 \beta_{4} ) q^{19} + ( 304 + 16 \beta_{2} ) q^{20} + ( 565 + 27 \beta_{1} + 20 \beta_{2} + 15 \beta_{3} - 10 \beta_{4} ) q^{21} + ( 708 + 44 \beta_{1} - 24 \beta_{2} - 4 \beta_{3} - 20 \beta_{4} ) q^{22} + ( 396 + 66 \beta_{1} - 11 \beta_{2} - 15 \beta_{3} - 21 \beta_{4} ) q^{23} + ( 256 - 64 \beta_{1} ) q^{24} + ( -75 - 6 \beta_{1} + 35 \beta_{2} - 9 \beta_{3} + 27 \beta_{4} ) q^{25} + ( 1104 + 24 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} + 36 \beta_{4} ) q^{26} + ( 551 - 91 \beta_{1} - 67 \beta_{2} + 18 \beta_{3} + 14 \beta_{4} ) q^{27} + ( 544 - 16 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} + 16 \beta_{4} ) q^{28} + ( 604 - 14 \beta_{1} + 31 \beta_{2} + 43 \beta_{3} - 7 \beta_{4} ) q^{29} + ( -12 - 32 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} + 12 \beta_{4} ) q^{30} + ( -1165 - 146 \beta_{1} - 95 \beta_{2} - 44 \beta_{3} - 34 \beta_{4} ) q^{31} + 1024 q^{32} + ( -2176 - 231 \beta_{1} - 20 \beta_{2} - 30 \beta_{3} - 49 \beta_{4} ) q^{33} + ( 1608 + 184 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 40 \beta_{4} ) q^{34} + ( -2134 - 230 \beta_{1} + 94 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} ) q^{35} + ( 1936 - 48 \beta_{1} - 16 \beta_{2} + 48 \beta_{3} - 32 \beta_{4} ) q^{36} -1369 q^{37} + ( 1872 + 152 \beta_{1} + 64 \beta_{2} + 8 \beta_{3} + 64 \beta_{4} ) q^{38} + ( -1847 - 410 \beta_{1} + 19 \beta_{2} + 90 \beta_{3} + 120 \beta_{4} ) q^{39} + ( 1216 + 64 \beta_{2} ) q^{40} + ( -2961 + 215 \beta_{1} + 89 \beta_{2} - 46 \beta_{3} - 71 \beta_{4} ) q^{41} + ( 2260 + 108 \beta_{1} + 80 \beta_{2} + 60 \beta_{3} - 40 \beta_{4} ) q^{42} + ( -1350 - 16 \beta_{1} - 98 \beta_{2} + 38 \beta_{3} + 130 \beta_{4} ) q^{43} + ( 2832 + 176 \beta_{1} - 96 \beta_{2} - 16 \beta_{3} - 80 \beta_{4} ) q^{44} + ( -1517 + 624 \beta_{1} - 82 \beta_{2} - 21 \beta_{3} - 103 \beta_{4} ) q^{45} + ( 1584 + 264 \beta_{1} - 44 \beta_{2} - 60 \beta_{3} - 84 \beta_{4} ) q^{46} + ( -3844 + 227 \beta_{1} + 99 \beta_{2} + 5 \beta_{3} - 41 \beta_{4} ) q^{47} + ( 1024 - 256 \beta_{1} ) q^{48} + ( -2548 + 713 \beta_{1} - 188 \beta_{2} - 67 \beta_{3} + 64 \beta_{4} ) q^{49} + ( -300 - 24 \beta_{1} + 140 \beta_{2} - 36 \beta_{3} + 108 \beta_{4} ) q^{50} + ( -13618 - 380 \beta_{1} + 12 \beta_{2} - 258 \beta_{3} - 8 \beta_{4} ) q^{51} + ( 4416 + 96 \beta_{1} - 16 \beta_{2} + 48 \beta_{3} + 144 \beta_{4} ) q^{52} + ( -4783 + 551 \beta_{1} + 130 \beta_{2} + 11 \beta_{3} + 190 \beta_{4} ) q^{53} + ( 2204 - 364 \beta_{1} - 268 \beta_{2} + 72 \beta_{3} + 56 \beta_{4} ) q^{54} + ( -6358 + 148 \beta_{1} - 137 \beta_{2} + 103 \beta_{3} - 433 \beta_{4} ) q^{55} + ( 2176 - 64 \beta_{1} - 64 \beta_{2} - 64 \beta_{3} + 64 \beta_{4} ) q^{56} + ( -14024 - 328 \beta_{1} + 150 \beta_{2} - 102 \beta_{3} + 278 \beta_{4} ) q^{57} + ( 2416 - 56 \beta_{1} + 124 \beta_{2} + 172 \beta_{3} - 28 \beta_{4} ) q^{58} + ( -3464 - 410 \beta_{1} + 12 \beta_{2} + 406 \beta_{3} - 226 \beta_{4} ) q^{59} + ( -48 - 128 \beta_{1} + 32 \beta_{2} - 144 \beta_{3} + 48 \beta_{4} ) q^{60} + ( -3727 + 290 \beta_{1} - 191 \beta_{2} + 302 \beta_{3} - 254 \beta_{4} ) q^{61} + ( -4660 - 584 \beta_{1} - 380 \beta_{2} - 176 \beta_{3} - 136 \beta_{4} ) q^{62} + ( -17398 - 1154 \beta_{1} - 18 \beta_{3} - 14 \beta_{4} ) q^{63} + 4096 q^{64} + ( -8047 + 234 \beta_{1} + 674 \beta_{2} + 159 \beta_{3} + 387 \beta_{4} ) q^{65} + ( -8704 - 924 \beta_{1} - 80 \beta_{2} - 120 \beta_{3} - 196 \beta_{4} ) q^{66} + ( -16955 - 560 \beta_{1} + 343 \beta_{2} - 254 \beta_{3} + 446 \beta_{4} ) q^{67} + ( 6432 + 736 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} - 160 \beta_{4} ) q^{68} + ( -17677 + 194 \beta_{1} + 137 \beta_{2} - 378 \beta_{3} - 264 \beta_{4} ) q^{69} + ( -8536 - 920 \beta_{1} + 376 \beta_{2} + 16 \beta_{3} + 80 \beta_{4} ) q^{70} + ( 1548 + 601 \beta_{1} - 685 \beta_{2} + 37 \beta_{3} - 493 \beta_{4} ) q^{71} + ( 7744 - 192 \beta_{1} - 64 \beta_{2} + 192 \beta_{3} - 128 \beta_{4} ) q^{72} + ( -2118 + 349 \beta_{1} - 229 \beta_{2} - 137 \beta_{3} + 458 \beta_{4} ) q^{73} -5476 q^{74} + ( -2363 + 1249 \beta_{1} + 397 \beta_{2} - 108 \beta_{3} + 192 \beta_{4} ) q^{75} + ( 7488 + 608 \beta_{1} + 256 \beta_{2} + 32 \beta_{3} + 256 \beta_{4} ) q^{76} + ( 3255 + 185 \beta_{1} - 986 \beta_{2} - 445 \beta_{3} + 112 \beta_{4} ) q^{77} + ( -7388 - 1640 \beta_{1} + 76 \beta_{2} + 360 \beta_{3} + 480 \beta_{4} ) q^{78} + ( 9742 + 180 \beta_{1} + 677 \beta_{2} + 567 \beta_{3} - 669 \beta_{4} ) q^{79} + ( 4864 + 256 \beta_{2} ) q^{80} + ( 7230 - 1490 \beta_{1} - 172 \beta_{2} + 381 \beta_{3} + 449 \beta_{4} ) q^{81} + ( -11844 + 860 \beta_{1} + 356 \beta_{2} - 184 \beta_{3} - 284 \beta_{4} ) q^{82} + ( 5236 + 289 \beta_{1} + 237 \beta_{2} - 749 \beta_{3} + 263 \beta_{4} ) q^{83} + ( 9040 + 432 \beta_{1} + 320 \beta_{2} + 240 \beta_{3} - 160 \beta_{4} ) q^{84} + ( 27738 + 404 \beta_{1} + 46 \beta_{2} + 296 \beta_{3} - 560 \beta_{4} ) q^{85} + ( -5400 - 64 \beta_{1} - 392 \beta_{2} + 152 \beta_{3} + 520 \beta_{4} ) q^{86} + ( 1945 - 3014 \beta_{1} - 689 \beta_{2} - 42 \beta_{3} + 568 \beta_{4} ) q^{87} + ( 11328 + 704 \beta_{1} - 384 \beta_{2} - 64 \beta_{3} - 320 \beta_{4} ) q^{88} + ( 46614 - 262 \beta_{1} - 1700 \beta_{2} + 140 \beta_{3} - 1118 \beta_{4} ) q^{89} + ( -6068 + 2496 \beta_{1} - 328 \beta_{2} - 84 \beta_{3} - 412 \beta_{4} ) q^{90} + ( 29144 + 370 \beta_{1} - 734 \beta_{2} - 542 \beta_{3} + 740 \beta_{4} ) q^{91} + ( 6336 + 1056 \beta_{1} - 176 \beta_{2} - 240 \beta_{3} - 336 \beta_{4} ) q^{92} + ( 59825 + 2766 \beta_{1} + 130 \beta_{2} + 723 \beta_{3} - 1421 \beta_{4} ) q^{93} + ( -15376 + 908 \beta_{1} + 396 \beta_{2} + 20 \beta_{3} - 164 \beta_{4} ) q^{94} + ( 38274 - 92 \beta_{1} + 1598 \beta_{2} + 352 \beta_{3} + 1076 \beta_{4} ) q^{95} + ( 4096 - 1024 \beta_{1} ) q^{96} + ( 49974 + 1174 \beta_{1} + 810 \beta_{2} - 542 \beta_{3} + 434 \beta_{4} ) q^{97} + ( -10192 + 2852 \beta_{1} - 752 \beta_{2} - 268 \beta_{3} + 256 \beta_{4} ) q^{98} + ( 36475 + 1012 \beta_{1} + 1324 \beta_{2} + 495 \beta_{3} - 89 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 20q^{2} + 19q^{3} + 80q^{4} + 95q^{5} + 76q^{6} + 170q^{7} + 320q^{8} + 598q^{9} + O(q^{10}) \) \( 5q + 20q^{2} + 19q^{3} + 80q^{4} + 95q^{5} + 76q^{6} + 170q^{7} + 320q^{8} + 598q^{9} + 380q^{10} + 903q^{11} + 304q^{12} + 1371q^{13} + 680q^{14} - 8q^{15} + 1280q^{16} + 2070q^{17} + 2392q^{18} + 2358q^{19} + 1520q^{20} + 2832q^{21} + 3612q^{22} + 2097q^{23} + 1216q^{24} - 390q^{25} + 5484q^{26} + 2614q^{27} + 2720q^{28} + 2927q^{29} - 32q^{30} - 5849q^{31} + 5120q^{32} - 11002q^{33} + 8280q^{34} - 10928q^{35} + 9568q^{36} - 6845q^{37} + 9432q^{38} - 9945q^{39} + 6080q^{40} - 14427q^{41} + 11328q^{42} - 6972q^{43} + 14448q^{44} - 6816q^{45} + 8388q^{46} - 18962q^{47} + 4864q^{48} - 11957q^{49} - 1560q^{50} - 67946q^{51} + 21936q^{52} - 23576q^{53} + 10456q^{54} - 31415q^{55} + 10880q^{56} - 70522q^{57} + 11708q^{58} - 18316q^{59} - 128q^{60} - 18695q^{61} - 23396q^{62} - 88094q^{63} + 20480q^{64} - 40706q^{65} - 44008q^{66} - 85273q^{67} + 33120q^{68} - 87171q^{69} - 43712q^{70} + 8760q^{71} + 38272q^{72} - 10425q^{73} - 27380q^{74} - 10542q^{75} + 37728q^{76} + 17238q^{77} - 39780q^{78} + 48425q^{79} + 24320q^{80} + 33449q^{81} - 57708q^{82} + 27704q^{83} + 45312q^{84} + 139062q^{85} - 27888q^{86} + 6227q^{87} + 57792q^{88} + 233646q^{89} - 27264q^{90} + 146434q^{91} + 33552q^{92} + 301866q^{93} - 75848q^{94} + 189498q^{95} + 19456q^{96} + 251694q^{97} - 47828q^{98} + 182486q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 870 x^{3} - 2235 x^{2} + 121361 x + 481504\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 26 \nu^{4} + 287 \nu^{3} - 23978 \nu^{2} - 192751 \nu + 2440911 \)\()/41713\)
\(\beta_{3}\)\(=\)\((\)\( 148 \nu^{4} - 1575 \nu^{3} - 75525 \nu^{2} + 487896 \nu - 1722289 \)\()/125139\)
\(\beta_{4}\)\(=\)\((\)\( 61 \nu^{4} - 931 \nu^{3} - 46630 \nu^{2} + 444606 \nu + 5176462 \)\()/41713\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 5 \beta_{1} + 348\)
\(\nu^{3}\)\(=\)\(-38 \beta_{4} + 18 \beta_{3} + 55 \beta_{2} + 589 \beta_{1} + 1745\)
\(\nu^{4}\)\(=\)\(-1425 \beta_{4} + 2568 \beta_{3} + 75 \beta_{2} + 5523 \beta_{1} + 207793\)

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
28.3792
13.4504
−4.15018
−12.6478
−24.0316
4.00000 −24.3792 16.0000 44.9778 −97.5168 −145.948 64.0000 351.346 179.911
1.2 4.00000 −9.45038 16.0000 −51.4877 −37.8015 212.238 64.0000 −153.690 −205.951
1.3 4.00000 8.15018 16.0000 86.4865 32.6007 72.3887 64.0000 −176.574 345.946
1.4 4.00000 16.6478 16.0000 46.0360 66.5913 16.5452 64.0000 34.1497 184.144
1.5 4.00000 28.0316 16.0000 −31.0127 112.126 14.7764 64.0000 542.770 −124.051
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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.e 5
3.b odd 2 1 666.6.a.l 5
4.b odd 2 1 592.6.a.e 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.a.e 5 1.a even 1 1 trivial
592.6.a.e 5 4.b odd 2 1
666.6.a.l 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 19 T_{3}^{4} - 726 T_{3}^{3} + 12131 T_{3}^{2} + 62745 T_{3} - 876276 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(74))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{5} \)
$3$ \( -876276 + 62745 T + 12131 T^{2} - 726 T^{3} - 19 T^{4} + T^{5} \)
$5$ \( -285948168 + 1101124 T + 357718 T^{2} - 3105 T^{3} - 95 T^{4} + T^{5} \)
$7$ \( 548191248 - 76631876 T + 3028298 T^{2} - 21589 T^{3} - 170 T^{4} + T^{5} \)
$11$ \( 18936143228 - 61121515741 T + 270912951 T^{2} - 95912 T^{3} - 903 T^{4} + T^{5} \)
$13$ \( 139399737289736 - 848522680956 T + 1589141482 T^{2} - 475237 T^{3} - 1371 T^{4} + T^{5} \)
$17$ \( -230256902474496 - 455144696288 T + 3224748600 T^{2} - 1248604 T^{3} - 2070 T^{4} + T^{5} \)
$19$ \( -2559940045890432 + 1210439819584 T + 6516600024 T^{2} - 2733892 T^{3} - 2358 T^{4} + T^{5} \)
$23$ \( -27522518850477032 + 14285174359332 T + 18889017230 T^{2} - 9701029 T^{3} - 2097 T^{4} + T^{5} \)
$29$ \( -137213477880692984 + 182579896313476 T + 73825214858 T^{2} - 31640941 T^{3} - 2927 T^{4} + T^{5} \)
$31$ \( -3062440129455661920 + 2612397028880512 T - 215525186366 T^{2} - 96086599 T^{3} + 5849 T^{4} + T^{5} \)
$37$ \( ( 1369 + T )^{5} \)
$41$ \( 2521137745653650398 + 4603635504451745 T - 1045665906399 T^{2} - 158089736 T^{3} + 14427 T^{4} + T^{5} \)
$43$ \( 16975009755781642272 + 14386464613567264 T + 129784439880 T^{2} - 340786156 T^{3} + 6972 T^{4} + T^{5} \)
$47$ \( -161262754472188176 - 1123202815101764 T - 1098135417218 T^{2} - 5739541 T^{3} + 18962 T^{4} + T^{5} \)
$53$ \( -\)\(25\!\cdots\!68\)\( + 92804155078928844 T - 5565026986034 T^{2} - 484297371 T^{3} + 23576 T^{4} + T^{5} \)
$59$ \( \)\(33\!\cdots\!80\)\( + 2056426397251058848 T - 54982855504072 T^{2} - 3174027256 T^{3} + 18316 T^{4} + T^{5} \)
$61$ \( \)\(15\!\cdots\!96\)\( + 1039741242979577880 T - 30786259976486 T^{2} - 2236677861 T^{3} + 18695 T^{4} + T^{5} \)
$67$ \( -\)\(17\!\cdots\!00\)\( - 3680236165924244960 T - 173539986889924 T^{2} - 483464455 T^{3} + 85273 T^{4} + T^{5} \)
$71$ \( \)\(17\!\cdots\!32\)\( - 3076836185262627284 T + 188149811659464 T^{2} - 4128497653 T^{3} - 8760 T^{4} + T^{5} \)
$73$ \( \)\(56\!\cdots\!26\)\( + 882755390020301665 T - 44012986599633 T^{2} - 3025163188 T^{3} + 10425 T^{4} + T^{5} \)
$79$ \( -\)\(95\!\cdots\!84\)\( - 3408763657764962952 T + 617424103027782 T^{2} - 11691949893 T^{3} - 48425 T^{4} + T^{5} \)
$83$ \( -\)\(11\!\cdots\!44\)\( + 25043375616207083632 T + 427604472700820 T^{2} - 10908681493 T^{3} - 27704 T^{4} + T^{5} \)
$89$ \( \)\(85\!\cdots\!48\)\( - \)\(36\!\cdots\!08\)\( T + 4339892398887456 T^{2} - 2481334360 T^{3} - 233646 T^{4} + T^{5} \)
$97$ \( \)\(19\!\cdots\!52\)\( - 91408388188363849696 T + 906061423788888 T^{2} + 12795918820 T^{3} - 251694 T^{4} + T^{5} \)
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