Properties

Label 69.6.a.e
Level $69$
Weight $6$
Character orbit 69.a
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 113 x^{3} - 257 x^{2} + 1404 x + 2197\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{2} ) q^{2} + 9 q^{3} + ( 24 - \beta_{2} + \beta_{3} ) q^{4} + ( 18 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( 18 - 9 \beta_{2} ) q^{6} + ( 52 - 5 \beta_{1} + 5 \beta_{2} - \beta_{4} ) q^{7} + ( 60 + 10 \beta_{1} - 22 \beta_{2} - \beta_{4} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + ( 2 - \beta_{2} ) q^{2} + 9 q^{3} + ( 24 - \beta_{2} + \beta_{3} ) q^{4} + ( 18 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( 18 - 9 \beta_{2} ) q^{6} + ( 52 - 5 \beta_{1} + 5 \beta_{2} - \beta_{4} ) q^{7} + ( 60 + 10 \beta_{1} - 22 \beta_{2} - \beta_{4} ) q^{8} + 81 q^{9} + ( -12 + 22 \beta_{1} - 52 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} ) q^{10} + ( 212 + 14 \beta_{2} - 5 \beta_{3} - 6 \beta_{4} ) q^{11} + ( 216 - 9 \beta_{2} + 9 \beta_{3} ) q^{12} + ( -182 - 22 \beta_{1} - 33 \beta_{2} - 10 \beta_{3} + \beta_{4} ) q^{13} + ( -48 - 12 \beta_{1} - 65 \beta_{2} - 11 \beta_{3} - 13 \beta_{4} ) q^{14} + ( 162 - 27 \beta_{1} + 27 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} ) q^{15} + ( 244 - 12 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{16} + ( 478 + 13 \beta_{1} + 71 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} ) q^{17} + ( 162 - 81 \beta_{2} ) q^{18} + ( 400 + 35 \beta_{1} + 53 \beta_{2} - 9 \beta_{3} + 23 \beta_{4} ) q^{19} + ( 1480 + 84 \beta_{1} + 188 \beta_{2} + 52 \beta_{3} + 28 \beta_{4} ) q^{20} + ( 468 - 45 \beta_{1} + 45 \beta_{2} - 9 \beta_{4} ) q^{21} + ( -496 - 122 \beta_{1} - 89 \beta_{2} - 15 \beta_{3} - 43 \beta_{4} ) q^{22} -529 q^{23} + ( 540 + 90 \beta_{1} - 198 \beta_{2} - 9 \beta_{4} ) q^{24} + ( 2275 + 106 \beta_{1} + 341 \beta_{2} + 24 \beta_{3} + 11 \beta_{4} ) q^{25} + ( 1652 - 88 \beta_{1} + 506 \beta_{2} + 22 \beta_{3} - 4 \beta_{4} ) q^{26} + 729 q^{27} + ( 1488 - 106 \beta_{1} + 243 \beta_{2} + 51 \beta_{3} - 73 \beta_{4} ) q^{28} + ( 198 + 90 \beta_{1} + 242 \beta_{2} - 96 \beta_{3} - 26 \beta_{4} ) q^{29} + ( -108 + 198 \beta_{1} - 468 \beta_{2} - 54 \beta_{3} + 36 \beta_{4} ) q^{30} + ( -1344 + 50 \beta_{1} - 316 \beta_{2} - 36 \beta_{3} + 20 \beta_{4} ) q^{31} + ( -1196 - 306 \beta_{1} + 484 \beta_{2} - 10 \beta_{3} + 37 \beta_{4} ) q^{32} + ( 1908 + 126 \beta_{2} - 45 \beta_{3} - 54 \beta_{4} ) q^{33} + ( -2796 + 140 \beta_{1} - 769 \beta_{2} - 61 \beta_{3} + 45 \beta_{4} ) q^{34} + ( 828 - 220 \beta_{1} + 979 \beta_{2} + 2 \beta_{3} + 21 \beta_{4} ) q^{35} + ( 1944 - 81 \beta_{2} + 81 \beta_{3} ) q^{36} + ( -1442 + 114 \beta_{1} - 530 \beta_{2} - 9 \beta_{3} - 30 \beta_{4} ) q^{37} + ( -2736 + 186 \beta_{1} - 70 \beta_{2} + 14 \beta_{3} + 228 \beta_{4} ) q^{38} + ( -1638 - 198 \beta_{1} - 297 \beta_{2} - 90 \beta_{3} + 9 \beta_{4} ) q^{39} + ( -6864 + 152 \beta_{1} - 1448 \beta_{2} + 64 \beta_{3} + 128 \beta_{4} ) q^{40} + ( 3754 - 132 \beta_{1} + 1036 \beta_{2} + 34 \beta_{3} - 204 \beta_{4} ) q^{41} + ( -432 - 108 \beta_{1} - 585 \beta_{2} - 99 \beta_{3} - 117 \beta_{4} ) q^{42} + ( -784 + 135 \beta_{1} - 295 \beta_{2} + 361 \beta_{3} + 19 \beta_{4} ) q^{43} + ( -1096 - 666 \beta_{1} + 351 \beta_{2} + 99 \beta_{3} - 259 \beta_{4} ) q^{44} + ( 1458 - 243 \beta_{1} + 243 \beta_{2} + 81 \beta_{3} + 81 \beta_{4} ) q^{45} + ( -1058 + 529 \beta_{2} ) q^{46} + ( 540 + 1116 \beta_{1} - 921 \beta_{2} - 102 \beta_{3} + 161 \beta_{4} ) q^{47} + ( 2196 - 108 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} + 18 \beta_{4} ) q^{48} + ( -5571 - 338 \beta_{1} - 127 \beta_{2} + 116 \beta_{3} - 65 \beta_{4} ) q^{49} + ( -15018 + 372 \beta_{1} - 3221 \beta_{2} - 248 \beta_{3} + 170 \beta_{4} ) q^{50} + ( 4302 + 117 \beta_{1} + 639 \beta_{2} + 72 \beta_{3} + 45 \beta_{4} ) q^{51} + ( -14592 + 876 \beta_{1} - 1884 \beta_{2} - 300 \beta_{3} - 174 \beta_{4} ) q^{52} + ( -1846 + 235 \beta_{1} - 1461 \beta_{2} - 55 \beta_{3} - 335 \beta_{4} ) q^{53} + ( 1458 - 729 \beta_{2} ) q^{54} + ( -16160 - 1932 \beta_{1} + 2006 \beta_{2} + 144 \beta_{3} + 138 \beta_{4} ) q^{55} + ( -5232 + 18 \beta_{1} - 1557 \beta_{2} - 121 \beta_{3} - 325 \beta_{4} ) q^{56} + ( 3600 + 315 \beta_{1} + 477 \beta_{2} - 81 \beta_{3} + 207 \beta_{4} ) q^{57} + ( -16964 - 1272 \beta_{1} + 2548 \beta_{2} - 82 \beta_{3} - 22 \beta_{4} ) q^{58} + ( 14872 + 1416 \beta_{1} + 419 \beta_{2} + 324 \beta_{3} + 261 \beta_{4} ) q^{59} + ( 13320 + 756 \beta_{1} + 1692 \beta_{2} + 468 \beta_{3} + 252 \beta_{4} ) q^{60} + ( -9650 + 242 \beta_{1} + 1860 \beta_{2} + 133 \beta_{3} + 44 \beta_{4} ) q^{61} + ( 11920 - 120 \beta_{1} + 2886 \beta_{2} + 422 \beta_{3} + 246 \beta_{4} ) q^{62} + ( 4212 - 405 \beta_{1} + 405 \beta_{2} - 81 \beta_{4} ) q^{63} + ( -27820 + 728 \beta_{1} + 795 \beta_{2} - 711 \beta_{3} - 64 \beta_{4} ) q^{64} + ( -3700 + 762 \beta_{1} - 3140 \beta_{2} - 1202 \beta_{3} - 196 \beta_{4} ) q^{65} + ( -4464 - 1098 \beta_{1} - 801 \beta_{2} - 135 \beta_{3} - 387 \beta_{4} ) q^{66} + ( -2152 - 563 \beta_{1} + 4643 \beta_{2} + 47 \beta_{3} - 31 \beta_{4} ) q^{67} + ( 14816 - 486 \beta_{1} + 3459 \beta_{2} + 759 \beta_{3} + 401 \beta_{4} ) q^{68} -4761 q^{69} + ( -43672 + 272 \beta_{1} - 2026 \beta_{2} - 1180 \beta_{3} - 54 \beta_{4} ) q^{70} + ( 6648 - 4352 \beta_{1} + 2272 \beta_{2} - 58 \beta_{3} - 64 \beta_{4} ) q^{71} + ( 4860 + 810 \beta_{1} - 1782 \beta_{2} - 81 \beta_{4} ) q^{72} + ( 7914 + 1156 \beta_{1} + 3552 \beta_{2} - 88 \beta_{3} + 64 \beta_{4} ) q^{73} + ( 21364 - 450 \beta_{1} + 2275 \beta_{2} + 623 \beta_{3} - 117 \beta_{4} ) q^{74} + ( 20475 + 954 \beta_{1} + 3069 \beta_{2} + 216 \beta_{3} + 99 \beta_{4} ) q^{75} + ( -16024 + 1756 \beta_{1} + 1546 \beta_{2} + 758 \beta_{3} + 1260 \beta_{4} ) q^{76} + ( 32264 - 740 \beta_{1} - 2342 \beta_{2} + 330 \beta_{3} - 186 \beta_{4} ) q^{77} + ( 14868 - 792 \beta_{1} + 4554 \beta_{2} + 198 \beta_{3} - 36 \beta_{4} ) q^{78} + ( 11836 + 4233 \beta_{1} - 4569 \beta_{2} + 624 \beta_{3} - 251 \beta_{4} ) q^{79} + ( 13632 - 512 \beta_{1} + 848 \beta_{2} + 216 \beta_{4} ) q^{80} + 6561 q^{81} + ( -44828 - 2108 \beta_{1} - 6588 \beta_{2} - 1406 \beta_{3} - 1798 \beta_{4} ) q^{82} + ( 21124 - 1080 \beta_{1} - 7684 \beta_{2} + 781 \beta_{3} + 332 \beta_{4} ) q^{83} + ( 13392 - 954 \beta_{1} + 2187 \beta_{2} + 459 \beta_{3} - 657 \beta_{4} ) q^{84} + ( 27752 - 2490 \beta_{1} + 4987 \beta_{2} + 1568 \beta_{3} + 421 \beta_{4} ) q^{85} + ( 19424 + 3838 \beta_{1} - 9920 \beta_{2} + 88 \beta_{3} - 74 \beta_{4} ) q^{86} + ( 1782 + 810 \beta_{1} + 2178 \beta_{2} - 864 \beta_{3} - 234 \beta_{4} ) q^{87} + ( 10680 + 1786 \beta_{1} - 919 \beta_{2} - 895 \beta_{3} - 1461 \beta_{4} ) q^{88} + ( 59454 + 2883 \beta_{1} - 8269 \beta_{2} - 558 \beta_{3} + 161 \beta_{4} ) q^{89} + ( -972 + 1782 \beta_{1} - 4212 \beta_{2} - 486 \beta_{3} + 324 \beta_{4} ) q^{90} + ( -3168 - 38 \beta_{1} - 5602 \beta_{2} - 1152 \beta_{3} + 186 \beta_{4} ) q^{91} + ( -12696 + 529 \beta_{2} - 529 \beta_{3} ) q^{92} + ( -12096 + 450 \beta_{1} - 2844 \beta_{2} - 324 \beta_{3} + 180 \beta_{4} ) q^{93} + ( 21672 + 912 \beta_{1} + 5142 \beta_{2} + 2300 \beta_{3} + 2506 \beta_{4} ) q^{94} + ( 42764 - 3664 \beta_{1} - 4831 \beta_{2} + 1130 \beta_{3} + 519 \beta_{4} ) q^{95} + ( -10764 - 2754 \beta_{1} + 4356 \beta_{2} - 90 \beta_{3} + 333 \beta_{4} ) q^{96} + ( 20826 + 1122 \beta_{1} - 7270 \beta_{2} - 958 \beta_{3} - 242 \beta_{4} ) q^{97} + ( 5578 + 380 \beta_{1} + 1569 \beta_{2} - 392 \beta_{3} - 974 \beta_{4} ) q^{98} + ( 17172 + 1134 \beta_{2} - 405 \beta_{3} - 486 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} + O(q^{10}) \) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 113 x^{3} - 257 x^{2} + 1404 x + 2197\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{4} - 13 \nu^{3} - 700 \nu^{2} - 499 \nu + 5941 \)\()/468\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 13 \nu^{3} - 56 \nu^{2} - 419 \nu + 4511 \)\()/117\)
\(\beta_{4}\)\(=\)\((\)\( 23 \nu^{4} - 221 \nu^{3} - 1676 \nu^{2} + 11821 \nu + 18005 \)\()/468\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} - 4 \beta_{3} + \beta_{2} + 6 \beta_{1} + 180\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{4} - 7 \beta_{3} + 19 \beta_{2} + 86 \beta_{1} + 298\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-63 \beta_{4} - 213 \beta_{3} + 219 \beta_{2} + 531 \beta_{1} + 7856\)\()/2\)

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
−1.42196
11.0973
−7.90234
−5.17654
3.40352
−9.32729 9.00000 54.9984 99.7124 −83.9456 125.983 −214.513 81.0000 −930.047
1.2 −3.54286 9.00000 −19.4482 −92.1306 −31.8857 −8.98894 182.273 81.0000 326.406
1.3 2.24792 9.00000 −26.9469 53.3906 20.2313 89.8688 −132.508 81.0000 120.018
1.4 9.27315 9.00000 53.9914 −37.4928 83.4584 154.850 203.929 81.0000 −347.676
1.5 9.34908 9.00000 55.4053 70.5203 84.1417 −89.7132 218.818 81.0000 659.300
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)

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 69.6.a.e 5
3.b odd 2 1 207.6.a.f 5
4.b odd 2 1 1104.6.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.e 5 1.a even 1 1 trivial
207.6.a.f 5 3.b odd 2 1
1104.6.a.r 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 8 T_{2}^{4} - 107 T_{2}^{3} + 770 T_{2}^{2} + 1740 T_{2} - 6440 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6440 + 1740 T + 770 T^{2} - 107 T^{3} - 8 T^{4} + T^{5} \)
$3$ \( ( -9 + T )^{5} \)
$5$ \( -1296820224 + 7019776 T + 941728 T^{2} - 9412 T^{3} - 94 T^{4} + T^{5} \)
$7$ \( -1413831904 - 136960224 T + 2364440 T^{2} + 8964 T^{3} - 272 T^{4} + T^{5} \)
$11$ \( 5578745996288 - 63217955584 T + 182462016 T^{2} + 144912 T^{3} - 1100 T^{4} + T^{5} \)
$13$ \( -2474410088160 - 99589342320 T - 490490640 T^{2} - 365096 T^{3} + 978 T^{4} + T^{5} \)
$17$ \( 13327498013440 - 314262009600 T + 337647680 T^{2} + 1404748 T^{3} - 2522 T^{4} + T^{5} \)
$19$ \( 3010484594372960 - 12155163321120 T + 14349684600 T^{2} - 4228108 T^{3} - 2060 T^{4} + T^{5} \)
$23$ \( ( 529 + T )^{5} \)
$29$ \( 321077683424449440 + 278387272580880 T - 35016763280 T^{2} - 49105544 T^{3} - 1526 T^{4} + T^{5} \)
$31$ \( 16557143540917248 - 11345767296000 T - 31920628544 T^{2} + 437104 T^{3} + 7392 T^{4} + T^{5} \)
$37$ \( 84165275968030880 + 14275410551760 T - 118822183280 T^{2} - 13572728 T^{3} + 8210 T^{4} + T^{5} \)
$41$ \( -\)\(22\!\cdots\!12\)\( - 15270087820165296 T + 7880216624752 T^{2} - 293297688 T^{3} - 21250 T^{4} + T^{5} \)
$43$ \( -\)\(23\!\cdots\!72\)\( + 49174513307987040 T + 533272451576 T^{2} - 447278508 T^{3} + 4548 T^{4} + T^{5} \)
$47$ \( \)\(94\!\cdots\!64\)\( + 193920884674686976 T - 2009810793728 T^{2} - 882704752 T^{3} - 536 T^{4} + T^{5} \)
$53$ \( \)\(77\!\cdots\!36\)\( - 43031658091872448 T - 20148540401360 T^{2} - 974674548 T^{3} + 11482 T^{4} + T^{5} \)
$59$ \( -\)\(10\!\cdots\!88\)\( - 402125524395219968 T + 49842241849920 T^{2} + 514568688 T^{3} - 74676 T^{4} + T^{5} \)
$61$ \( 1194743717850416160 - 125247918659625520 T - 11771098826800 T^{2} + 214301064 T^{3} + 44618 T^{4} + T^{5} \)
$67$ \( -\)\(11\!\cdots\!24\)\( + 650574262058706912 T - 27683158835768 T^{2} - 2709666380 T^{3} + 1412 T^{4} + T^{5} \)
$71$ \( -\)\(42\!\cdots\!28\)\( + 11320178672395489280 T + 281487303913984 T^{2} - 7554678976 T^{3} - 37912 T^{4} + T^{5} \)
$73$ \( \)\(27\!\cdots\!68\)\( + 977910679949935696 T + 66492374421360 T^{2} - 2018921304 T^{3} - 46546 T^{4} + T^{5} \)
$79$ \( -\)\(39\!\cdots\!64\)\( + 2361477741365623584 T + 568163217678280 T^{2} - 8823710876 T^{3} - 50544 T^{4} + T^{5} \)
$83$ \( -\)\(29\!\cdots\!24\)\( + 15328688118239528192 T + 636243818529216 T^{2} - 9243278864 T^{3} - 89588 T^{4} + T^{5} \)
$89$ \( \)\(28\!\cdots\!76\)\( - \)\(10\!\cdots\!56\)\( T + 610285590892496 T^{2} + 19791083228 T^{3} - 280410 T^{4} + T^{5} \)
$97$ \( \)\(55\!\cdots\!56\)\( - 41635589139920255984 T + 1021916488616848 T^{2} - 6254051272 T^{3} - 90074 T^{4} + T^{5} \)
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