Properties

Label 29.6.a.a
Level $29$
Weight $6$
Character orbit 29.a
Self dual yes
Analytic conductor $4.651$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65113077458\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
Defining polynomial: \(x^{4} - 34 x^{2} - 27 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -7 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{4} + ( -15 - 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{5} + ( -48 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{6} + ( -58 - 4 \beta_{2} - 12 \beta_{3} ) q^{7} + ( -126 + 14 \beta_{1} - 5 \beta_{2} ) q^{8} + ( -70 - 8 \beta_{1} - 20 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -7 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{4} + ( -15 - 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{5} + ( -48 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{6} + ( -58 - 4 \beta_{2} - 12 \beta_{3} ) q^{7} + ( -126 + 14 \beta_{1} - 5 \beta_{2} ) q^{8} + ( -70 - 8 \beta_{1} - 20 \beta_{2} ) q^{9} + ( -184 + 22 \beta_{1} - 17 \beta_{2} + 26 \beta_{3} ) q^{10} + ( -41 - 21 \beta_{1} - 29 \beta_{2} - 20 \beta_{3} ) q^{11} + ( 10 - 16 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{12} + ( -85 - 10 \beta_{1} - 38 \beta_{2} + 60 \beta_{3} ) q^{13} + ( 160 - 24 \beta_{1} + 66 \beta_{2} - 64 \beta_{3} ) q^{14} + ( 193 - 49 \beta_{1} - 29 \beta_{2} - 80 \beta_{3} ) q^{15} + ( -90 + 81 \beta_{1} + 33 \beta_{2} + 27 \beta_{3} ) q^{16} + ( 44 + 14 \beta_{1} + 50 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 792 - 60 \beta_{1} + 146 \beta_{2} - 20 \beta_{3} ) q^{18} + ( -540 + 168 \beta_{1} - 80 \beta_{2} + 116 \beta_{3} ) q^{19} + ( 698 + 39 \beta_{1} + 169 \beta_{2} - 15 \beta_{3} ) q^{20} + ( 358 - 10 \beta_{1} - 38 \beta_{2} + 220 \beta_{3} ) q^{21} + ( 1320 - 107 \beta_{1} + 103 \beta_{2} - 129 \beta_{3} ) q^{22} + ( -368 + 146 \beta_{1} + 14 \beta_{2} - 140 \beta_{3} ) q^{23} + ( 1706 - 83 \beta_{1} - 213 \beta_{2} + 19 \beta_{3} ) q^{24} + ( 316 - 140 \beta_{1} - 220 \beta_{2} - 280 \beta_{3} ) q^{25} + ( 1312 - 54 \beta_{1} + 305 \beta_{2} + 262 \beta_{3} ) q^{26} + ( 623 - 269 \beta_{1} - 149 \beta_{2} + 12 \beta_{3} ) q^{27} + ( 76 + 134 \beta_{1} - 498 \beta_{2} + 130 \beta_{3} ) q^{28} -841 q^{29} + ( 1832 - 167 \beta_{1} - 275 \beta_{2} - 429 \beta_{3} ) q^{30} + ( -4849 + 485 \beta_{1} - 319 \beta_{2} - 92 \beta_{3} ) q^{31} + ( 1722 - 322 \beta_{1} + 355 \beta_{2} + 168 \beta_{3} ) q^{32} + ( -2461 + 64 \beta_{1} + 272 \beta_{2} + 368 \beta_{3} ) q^{33} + ( -1888 + 146 \beta_{1} - 256 \beta_{2} + 30 \beta_{3} ) q^{34} + ( -5626 + 564 \beta_{1} + 304 \beta_{2} + 220 \beta_{3} ) q^{35} + ( -1844 + 674 \beta_{1} - 1082 \beta_{2} + 46 \beta_{3} ) q^{36} + ( -2712 - 712 \beta_{1} + 544 \beta_{2} + 40 \beta_{3} ) q^{37} + ( 136 - 124 \beta_{1} + 1560 \beta_{2} + 500 \beta_{3} ) q^{38} + ( -2709 - 171 \beta_{1} + 185 \beta_{2} - 1052 \beta_{3} ) q^{39} + ( -374 - 212 \beta_{1} - 897 \beta_{2} - 738 \beta_{3} ) q^{40} + ( -682 - 366 \beta_{1} + 1950 \beta_{2} - 804 \beta_{3} ) q^{41} + ( 992 + 106 \beta_{1} + 22 \beta_{2} + 1062 \beta_{3} ) q^{42} + ( -4969 - 341 \beta_{1} + 1603 \beta_{2} + 772 \beta_{3} ) q^{43} + ( -434 + 852 \beta_{1} - 1357 \beta_{2} + 98 \beta_{3} ) q^{44} + ( -1862 + 908 \beta_{1} - 728 \beta_{2} + 448 \beta_{3} ) q^{45} + ( -2240 - 98 \beta_{1} + 596 \beta_{2} - 686 \beta_{3} ) q^{46} + ( 5979 - 1675 \beta_{1} + 449 \beta_{2} + 72 \beta_{3} ) q^{47} + ( 8046 - 108 \beta_{1} - 903 \beta_{2} - 438 \beta_{3} ) q^{48} + ( 3133 - 1152 \beta_{1} - 1424 \beta_{2} + 1040 \beta_{3} ) q^{49} + ( 10000 - 940 \beta_{1} + 84 \beta_{2} - 1620 \beta_{3} ) q^{50} + ( 3204 - 66 \beta_{1} - 318 \beta_{2} + 36 \beta_{3} ) q^{51} + ( -7418 + 1497 \beta_{1} - 1521 \beta_{2} - 305 \beta_{3} ) q^{52} + ( 2529 + 1898 \beta_{1} - 2258 \beta_{2} + 628 \beta_{3} ) q^{53} + ( 8808 - 435 \beta_{1} - 673 \beta_{2} - 89 \beta_{3} ) q^{54} + ( -12305 + 2181 \beta_{1} + 545 \beta_{2} + 1716 \beta_{3} ) q^{55} + ( 9676 - 596 \beta_{1} + 834 \beta_{2} + 2200 \beta_{3} ) q^{56} + ( 11316 - 312 \beta_{1} - 1484 \beta_{2} - 1840 \beta_{3} ) q^{57} + 841 \beta_{2} q^{58} + ( -2780 + 3098 \beta_{1} - 2466 \beta_{2} - 140 \beta_{3} ) q^{59} + ( 6370 + 314 \beta_{1} - 459 \beta_{2} + 140 \beta_{3} ) q^{60} + ( 11164 - 3142 \beta_{1} + 3582 \beta_{2} - 2396 \beta_{3} ) q^{61} + ( 4240 - 1049 \beta_{1} + 7807 \beta_{2} - 779 \beta_{3} ) q^{62} + ( 6364 - 208 \beta_{1} + 1600 \beta_{2} - 1016 \beta_{3} ) q^{63} + ( -5018 - 1359 \beta_{1} - 5351 \beta_{2} + 331 \beta_{3} ) q^{64} + ( 24043 - 384 \beta_{1} - 4716 \beta_{2} - 832 \beta_{3} ) q^{65} + ( -10880 + 1184 \beta_{1} + 1661 \beta_{2} + 2112 \beta_{3} ) q^{66} + ( -2476 - 4428 \beta_{1} + 4676 \beta_{2} - 1032 \beta_{3} ) q^{67} + ( 5192 - 1186 \beta_{1} + 2036 \beta_{2} + 22 \beta_{3} ) q^{68} + ( 16024 + 302 \beta_{1} - 1606 \beta_{2} + 2652 \beta_{3} ) q^{69} + ( -18672 + 1132 \beta_{1} + 6018 \beta_{2} + 1404 \beta_{3} ) q^{70} + ( -12234 + 3250 \beta_{1} - 3598 \beta_{2} - 96 \beta_{3} ) q^{71} + ( 1916 - 1280 \beta_{1} + 4650 \beta_{2} - 212 \beta_{3} ) q^{72} + ( -19500 + 3784 \beta_{1} - 2200 \beta_{2} - 1504 \beta_{3} ) q^{73} + ( -8608 + 1672 \beta_{1} - 2104 \beta_{2} + 744 \beta_{3} ) q^{74} + ( -20052 + 1536 \beta_{1} + 2536 \beta_{2} + 5120 \beta_{3} ) q^{75} + ( -35024 - 196 \beta_{1} - 5248 \beta_{2} + 348 \beta_{3} ) q^{76} + ( 31226 - 1742 \beta_{1} - 1122 \beta_{2} - 1876 \beta_{3} ) q^{77} + ( -1792 - 497 \beta_{1} + 219 \beta_{2} - 5075 \beta_{3} ) q^{78} + ( -25087 - 1457 \beta_{1} - 5901 \beta_{2} + 2864 \beta_{3} ) q^{79} + ( 12606 - 4677 \beta_{1} - 1923 \beta_{2} - 4107 \beta_{3} ) q^{80} + ( -15091 + 2440 \beta_{1} + 8380 \beta_{2} - 336 \beta_{3} ) q^{81} + ( -59568 + 5046 \beta_{1} - 10970 \beta_{2} - 2070 \beta_{3} ) q^{82} + ( 15168 - 5038 \beta_{1} + 2566 \beta_{2} - 1108 \beta_{3} ) q^{83} + ( -15812 + 1448 \beta_{1} + 1494 \beta_{2} - 1708 \beta_{3} ) q^{84} + ( 5276 - 1642 \beta_{1} + 1298 \beta_{2} - 1372 \beta_{3} ) q^{85} + ( -51272 + 5581 \beta_{1} - 3297 \beta_{2} + 5463 \beta_{3} ) q^{86} + ( 5887 - 841 \beta_{1} - 841 \beta_{2} ) q^{87} + ( -8226 - 549 \beta_{1} + 6577 \beta_{2} + 3261 \beta_{3} ) q^{88} + ( 29082 - 842 \beta_{1} - 7382 \beta_{2} + 4380 \beta_{3} ) q^{89} + ( 11144 - 1736 \beta_{1} + 8674 \beta_{2} + 1512 \beta_{3} ) q^{90} + ( -69790 + 4708 \beta_{1} + 12448 \beta_{2} - 5132 \beta_{3} ) q^{91} + ( -5744 - 3570 \beta_{1} - 2168 \beta_{2} + 1646 \beta_{3} ) q^{92} + ( 56595 - 2646 \beta_{1} - 7134 \beta_{2} + 2460 \beta_{3} ) q^{93} + ( 8040 + 1419 \beta_{1} - 13177 \beta_{2} + 809 \beta_{3} ) q^{94} + ( 32932 - 7528 \beta_{1} - 12232 \beta_{2} - 7812 \beta_{3} ) q^{95} + ( -21502 - 491 \beta_{1} + 2523 \beta_{2} - 3701 \beta_{3} ) q^{96} + ( -8790 + 3962 \beta_{1} + 7150 \beta_{2} + 7180 \beta_{3} ) q^{97} + ( 62464 - 3232 \beta_{1} + 1571 \beta_{2} + 3776 \beta_{3} ) q^{98} + ( 40694 + 850 \beta_{1} + 2714 \beta_{2} - 1972 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9} + O(q^{10}) \) \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9} - 788 q^{10} - 124 q^{11} + 20 q^{12} - 460 q^{13} + 768 q^{14} + 932 q^{15} - 414 q^{16} + 184 q^{17} + 3208 q^{18} - 2392 q^{19} + 2822 q^{20} + 992 q^{21} + 5538 q^{22} - 1192 q^{23} + 6786 q^{24} + 1824 q^{25} + 4724 q^{26} + 2468 q^{27} + 44 q^{28} - 3364 q^{29} + 8186 q^{30} - 19212 q^{31} + 6552 q^{32} - 10580 q^{33} - 7612 q^{34} - 22944 q^{35} - 7468 q^{36} - 10928 q^{37} - 456 q^{38} - 8732 q^{39} - 20 q^{40} - 1120 q^{41} + 1844 q^{42} - 21420 q^{43} - 1932 q^{44} - 8344 q^{45} - 7588 q^{46} + 23772 q^{47} + 33060 q^{48} + 10452 q^{49} + 43240 q^{50} + 12744 q^{51} - 29062 q^{52} + 8860 q^{53} + 35410 q^{54} - 52652 q^{55} + 34304 q^{56} + 48944 q^{57} - 10840 q^{59} + 25200 q^{60} + 49448 q^{61} + 18518 q^{62} + 27488 q^{63} - 20734 q^{64} + 97836 q^{65} - 47744 q^{66} - 7840 q^{67} + 20724 q^{68} + 58792 q^{69} - 77496 q^{70} - 48744 q^{71} + 8088 q^{72} - 74992 q^{73} - 35920 q^{74} - 90448 q^{75} - 140792 q^{76} + 128656 q^{77} + 2982 q^{78} - 106076 q^{79} + 58638 q^{80} - 59692 q^{81} - 234132 q^{82} + 62888 q^{83} - 59832 q^{84} + 23848 q^{85} - 216014 q^{86} + 23548 q^{87} - 39426 q^{88} + 107568 q^{89} + 41552 q^{90} - 268896 q^{91} - 26268 q^{92} + 221460 q^{93} + 30542 q^{94} + 147352 q^{95} - 78606 q^{96} - 49520 q^{97} + 242304 q^{98} + 166720 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 34 x^{2} - 27 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 3 \nu - 17 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 3 \nu - 17 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} - \nu^{2} - 67 \nu - 25 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{2} + 3 \beta_{1} + 34\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} - 32 \beta_{2} + 35 \beta_{1} + 42\)\()/2\)

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
−5.34807
6.17343
−1.10057
0.275208
−9.21534 0.734546 52.9225 41.6400 −6.76909 −90.0205 −192.808 −242.460 −383.727
1.2 −0.863638 7.07413 −31.2541 −44.4758 −6.10949 −36.7447 54.6287 −192.957 38.4110
1.3 4.16235 −17.5258 −14.6748 32.5670 −72.9487 −220.793 −194.277 64.1549 135.555
1.4 5.91663 −18.2828 3.00648 −97.7313 −108.173 139.558 −171.544 91.2623 −578.240
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

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 29.6.a.a 4
3.b odd 2 1 261.6.a.a 4
4.b odd 2 1 464.6.a.i 4
5.b even 2 1 725.6.a.a 4
29.b even 2 1 841.6.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.a 4 1.a even 1 1 trivial
261.6.a.a 4 3.b odd 2 1
464.6.a.i 4 4.b odd 2 1
725.6.a.a 4 5.b even 2 1
841.6.a.a 4 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 69 T_{2}^{2} + 168 T_{2} + 196 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 196 + 168 T - 69 T^{2} + T^{4} \)
$3$ \( 1665 - 2316 T + 46 T^{2} + 28 T^{3} + T^{4} \)
$5$ \( 5894489 - 129708 T - 4850 T^{2} + 68 T^{3} + T^{4} \)
$7$ \( -101924272 - 3637376 T - 17208 T^{2} + 208 T^{3} + T^{4} \)
$11$ \( -3717303119 - 62500012 T - 219970 T^{2} + 124 T^{3} + T^{4} \)
$13$ \( -116053863479 - 631235044 T - 825826 T^{2} + 460 T^{3} + T^{4} \)
$17$ \( 11464717824 + 14747712 T - 225472 T^{2} - 184 T^{3} + T^{4} \)
$19$ \( 2620094791680 - 4969666944 T - 2930512 T^{2} + 2392 T^{3} + T^{4} \)
$23$ \( 2033080361984 + 2037166656 T - 7363712 T^{2} + 1192 T^{3} + T^{4} \)
$29$ \( ( 841 + T )^{4} \)
$31$ \( -421127233952247 + 84080514516 T + 106781782 T^{2} + 19212 T^{3} + T^{4} \)
$37$ \( 16079593861120 - 102820006912 T - 18294784 T^{2} + 10928 T^{3} + T^{4} \)
$41$ \( 1019972529416656 - 263561233280 T - 338485368 T^{2} + 1120 T^{3} + T^{4} \)
$43$ \( 4775546353139537 - 2087625753308 T - 153019426 T^{2} + 21420 T^{3} + T^{4} \)
$47$ \( -35698126979657687 + 6017632887324 T - 111991658 T^{2} - 23772 T^{3} + T^{4} \)
$53$ \( 16318164064396153 + 3132903897524 T - 533663890 T^{2} - 8860 T^{3} + T^{4} \)
$59$ \( -43504164239687680 - 18906134931392 T - 1157799648 T^{2} + 10840 T^{3} + T^{4} \)
$61$ \( 904039178556583424 + 46768750609216 T - 1321063392 T^{2} - 49448 T^{3} + T^{4} \)
$67$ \( 205602700687246592 - 2185142259200 T - 2682598752 T^{2} + 7840 T^{3} + T^{4} \)
$71$ \( -188937352325855216 - 33729660413856 T - 668565640 T^{2} + 48744 T^{3} + T^{4} \)
$73$ \( -403244927683116800 - 84099984892160 T - 324259744 T^{2} + 74992 T^{3} + T^{4} \)
$79$ \( -5341429243214578375 - 284943152993980 T - 597387258 T^{2} + 106076 T^{3} + T^{4} \)
$83$ \( -1885332073450789888 + 138280248509248 T - 1340665728 T^{2} - 62888 T^{3} + T^{4} \)
$89$ \( -182566548896698160 + 81190152298944 T - 3628346168 T^{2} - 107568 T^{3} + T^{4} \)
$97$ \( -12990327399638977328 + 864813553737920 T - 17061626872 T^{2} + 49520 T^{3} + T^{4} \)
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