Properties

Label 69.6.a.c
Level $69$
Weight $6$
Character orbit 69.a
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 39 x^{2} - 30 x + 20\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} -9 q^{3} + ( -12 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{4} + ( -29 + 9 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( -9 - 9 \beta_{1} ) q^{6} + ( 14 + 34 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{7} + ( 17 - 17 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} -9 q^{3} + ( -12 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{4} + ( -29 + 9 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( -9 - 9 \beta_{1} ) q^{6} + ( 14 + 34 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{7} + ( 17 - 17 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{8} + 81 q^{9} + ( 155 + \beta_{1} - 11 \beta_{2} + 13 \beta_{3} ) q^{10} + ( 19 + 23 \beta_{1} + 22 \beta_{2} - 19 \beta_{3} ) q^{11} + ( 108 - 36 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{12} + ( 330 + 76 \beta_{1} - 22 \beta_{2} - 12 \beta_{3} ) q^{13} + ( 671 + 77 \beta_{1} - 35 \beta_{2} + 27 \beta_{3} ) q^{14} + ( 261 - 81 \beta_{1} - 27 \beta_{2} - 9 \beta_{3} ) q^{15} + ( 32 - 108 \beta_{1} + 55 \beta_{2} - 47 \beta_{3} ) q^{16} + ( 124 + 158 \beta_{1} + 109 \beta_{2} + 8 \beta_{3} ) q^{17} + ( 81 + 81 \beta_{1} ) q^{18} + ( 725 + 13 \beta_{1} + 19 \beta_{2} + 101 \beta_{3} ) q^{19} + ( 921 + 59 \beta_{1} - 73 \beta_{2} - 29 \beta_{3} ) q^{20} + ( -126 - 306 \beta_{1} + 27 \beta_{2} + 36 \beta_{3} ) q^{21} + ( 762 - 206 \beta_{1} - 64 \beta_{2} + 26 \beta_{3} ) q^{22} + 529 q^{23} + ( -153 + 153 \beta_{1} + 18 \beta_{2} - 36 \beta_{3} ) q^{24} + ( 1361 - 430 \beta_{1} - 250 \beta_{2} - 10 \beta_{3} ) q^{25} + ( 1716 + 480 \beta_{1} - 66 \beta_{2} + 42 \beta_{3} ) q^{26} -729 q^{27} + ( 1225 + 243 \beta_{1} + 81 \beta_{2} + 197 \beta_{3} ) q^{28} + ( -1298 - 592 \beta_{1} - 6 \beta_{2} - 316 \beta_{3} ) q^{29} + ( -1395 - 9 \beta_{1} + 99 \beta_{2} - 117 \beta_{3} ) q^{30} + ( -176 - 936 \beta_{1} + 542 \beta_{2} - 292 \beta_{3} ) q^{31} + ( -1803 - 477 \beta_{1} + 70 \beta_{2} - 228 \beta_{3} ) q^{32} + ( -171 - 207 \beta_{1} - 198 \beta_{2} + 171 \beta_{3} ) q^{33} + ( 3825 + 367 \beta_{1} - 259 \beta_{2} + 275 \beta_{3} ) q^{34} + ( 2736 - 424 \beta_{1} - 412 \beta_{2} + 724 \beta_{3} ) q^{35} + ( -972 + 324 \beta_{1} - 81 \beta_{2} + 81 \beta_{3} ) q^{36} + ( 931 - 595 \beta_{1} + 188 \beta_{2} - 477 \beta_{3} ) q^{37} + ( 297 + 1919 \beta_{1} + 69 \beta_{2} + 133 \beta_{3} ) q^{38} + ( -2970 - 684 \beta_{1} + 198 \beta_{2} + 108 \beta_{3} ) q^{39} + ( -3197 + 937 \beta_{1} + 337 \beta_{2} - 459 \beta_{3} ) q^{40} + ( -4180 + 1170 \beta_{1} + 472 \beta_{2} + 134 \beta_{3} ) q^{41} + ( -6039 - 693 \beta_{1} + 315 \beta_{2} - 243 \beta_{3} ) q^{42} + ( 6319 + 951 \beta_{1} - 61 \beta_{2} - 181 \beta_{3} ) q^{43} + ( -4416 - 88 \beta_{1} - 408 \beta_{2} + 364 \beta_{3} ) q^{44} + ( -2349 + 729 \beta_{1} + 243 \beta_{2} + 81 \beta_{3} ) q^{45} + ( 529 + 529 \beta_{1} ) q^{46} + ( -6844 + 1852 \beta_{1} - 668 \beta_{2} - 488 \beta_{3} ) q^{47} + ( -288 + 972 \beta_{1} - 495 \beta_{2} + 423 \beta_{3} ) q^{48} + ( 13355 + 554 \beta_{1} - 1150 \beta_{2} + 366 \beta_{3} ) q^{49} + ( -8479 + 701 \beta_{1} + 670 \beta_{2} - 690 \beta_{3} ) q^{50} + ( -1116 - 1422 \beta_{1} - 981 \beta_{2} - 72 \beta_{3} ) q^{51} + ( -522 + 1426 \beta_{1} + 332 \beta_{2} + 840 \beta_{3} ) q^{52} + ( -8101 + 2273 \beta_{1} - 199 \beta_{2} - 27 \beta_{3} ) q^{53} + ( -729 - 729 \beta_{1} ) q^{54} + ( 8392 - 4016 \beta_{1} - 668 \beta_{2} + 1512 \beta_{3} ) q^{55} + ( -16639 + 1611 \beta_{1} + 993 \beta_{2} - 343 \beta_{3} ) q^{56} + ( -6525 - 117 \beta_{1} - 171 \beta_{2} - 909 \beta_{3} ) q^{57} + ( -10060 - 6848 \beta_{1} + 282 \beta_{2} - 914 \beta_{3} ) q^{58} + ( -14266 - 2710 \beta_{1} + 2036 \beta_{2} + 162 \beta_{3} ) q^{59} + ( -8289 - 531 \beta_{1} + 657 \beta_{2} + 261 \beta_{3} ) q^{60} + ( 9137 + 51 \beta_{1} + 352 \beta_{2} - 1683 \beta_{3} ) q^{61} + ( -11830 - 8114 \beta_{1} + 102 \beta_{2} - 686 \beta_{3} ) q^{62} + ( 1134 + 2754 \beta_{1} - 243 \beta_{2} - 324 \beta_{3} ) q^{63} + ( -9576 - 2724 \beta_{1} - 1581 \beta_{2} + 869 \beta_{3} ) q^{64} + ( -11922 + 2490 \beta_{1} + 174 \beta_{2} + 2226 \beta_{3} ) q^{65} + ( -6858 + 1854 \beta_{1} + 576 \beta_{2} - 234 \beta_{3} ) q^{66} + ( 19641 + 3033 \beta_{1} + 2553 \beta_{2} + 877 \beta_{3} ) q^{67} + ( 2817 + 3947 \beta_{1} - 3321 \beta_{2} + 127 \beta_{3} ) q^{68} -4761 q^{69} + ( -13996 + 11388 \beta_{1} + 1560 \beta_{2} - 112 \beta_{3} ) q^{70} + ( 6386 - 8430 \beta_{1} + 556 \beta_{2} - 1042 \beta_{3} ) q^{71} + ( 1377 - 1377 \beta_{1} - 162 \beta_{2} + 324 \beta_{3} ) q^{72} + ( 30086 + 1284 \beta_{1} - 1084 \beta_{2} - 756 \beta_{3} ) q^{73} + ( -5242 - 7142 \beta_{1} - 70 \beta_{2} - 884 \beta_{3} ) q^{74} + ( -12249 + 3870 \beta_{1} + 2250 \beta_{2} + 90 \beta_{3} ) q^{75} + ( 12977 + 7027 \beta_{1} - 2463 \beta_{2} - 1111 \beta_{3} ) q^{76} + ( 37690 - 7694 \beta_{1} - 4368 \beta_{2} + 390 \beta_{3} ) q^{77} + ( -15444 - 4320 \beta_{1} + 594 \beta_{2} - 378 \beta_{3} ) q^{78} + ( 23538 + 3390 \beta_{1} + 1991 \beta_{2} + 460 \beta_{3} ) q^{79} + ( -8835 - 8793 \beta_{1} + 603 \beta_{2} + 1743 \beta_{3} ) q^{80} + 6561 q^{81} + ( 20282 - 478 \beta_{1} - 1508 \beta_{2} + 1776 \beta_{3} ) q^{82} + ( 8305 - 3539 \beta_{1} + 2314 \beta_{2} - 1881 \beta_{3} ) q^{83} + ( -11025 - 2187 \beta_{1} - 729 \beta_{2} - 1773 \beta_{3} ) q^{84} + ( 87118 - 7214 \beta_{1} - 3326 \beta_{2} + 798 \beta_{3} ) q^{85} + ( 25409 + 7183 \beta_{1} - 1071 \beta_{2} + 709 \beta_{3} ) q^{86} + ( 11682 + 5328 \beta_{1} + 54 \beta_{2} + 2844 \beta_{3} ) q^{87} + ( -36240 + 7504 \beta_{1} + 2908 \beta_{2} - 964 \beta_{3} ) q^{88} + ( -4846 - 12740 \beta_{1} + 4271 \beta_{2} + 1102 \beta_{3} ) q^{89} + ( 12555 + 81 \beta_{1} - 891 \beta_{2} + 1053 \beta_{3} ) q^{90} + ( 82428 + 11324 \beta_{1} - 2234 \beta_{2} - 1848 \beta_{3} ) q^{91} + ( -6348 + 2116 \beta_{1} - 529 \beta_{2} + 529 \beta_{3} ) q^{92} + ( 1584 + 8424 \beta_{1} - 4878 \beta_{2} + 2628 \beta_{3} ) q^{93} + ( 27572 - 5140 \beta_{1} - 1672 \beta_{2} + 696 \beta_{3} ) q^{94} + ( 29464 + 17640 \beta_{1} + 2232 \beta_{2} - 6404 \beta_{3} ) q^{95} + ( 16227 + 4293 \beta_{1} - 630 \beta_{2} + 2052 \beta_{3} ) q^{96} + ( 7724 - 11646 \beta_{1} + 7158 \beta_{2} - 1182 \beta_{3} ) q^{97} + ( 12903 + 22859 \beta_{1} + 962 \beta_{2} - 230 \beta_{3} ) q^{98} + ( 1539 + 1863 \beta_{1} + 1782 \beta_{2} - 1539 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 36q^{3} - 46q^{4} - 122q^{5} - 36q^{6} + 62q^{7} + 72q^{8} + 324q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 36q^{3} - 46q^{4} - 122q^{5} - 36q^{6} + 62q^{7} + 72q^{8} + 324q^{9} + 642q^{10} + 32q^{11} + 414q^{12} + 1364q^{13} + 2754q^{14} + 1098q^{15} + 18q^{16} + 278q^{17} + 324q^{18} + 2862q^{19} + 3830q^{20} - 558q^{21} + 3176q^{22} + 2116q^{23} - 648q^{24} + 5944q^{25} + 6996q^{26} - 2916q^{27} + 4738q^{28} - 5180q^{29} - 5778q^{30} - 1788q^{31} - 7352q^{32} - 288q^{33} + 15818q^{34} + 11768q^{35} - 3726q^{36} + 3348q^{37} + 1050q^{38} - 12276q^{39} - 13462q^{40} - 17664q^{41} - 24786q^{42} + 25398q^{43} - 16848q^{44} - 9882q^{45} + 2116q^{46} - 26040q^{47} - 162q^{48} + 55720q^{49} - 35256q^{50} - 2502q^{51} - 2752q^{52} - 32006q^{53} - 2916q^{54} + 34904q^{55} - 68542q^{56} - 25758q^{57} - 40804q^{58} - 61136q^{59} - 34470q^{60} + 35844q^{61} - 47524q^{62} + 5022q^{63} - 35142q^{64} - 48036q^{65} - 28584q^{66} + 73458q^{67} + 17910q^{68} - 19044q^{69} - 59104q^{70} + 24432q^{71} + 5832q^{72} + 122512q^{73} - 20828q^{74} - 53496q^{75} + 56834q^{76} + 159496q^{77} - 62964q^{78} + 90170q^{79} - 36546q^{80} + 26244q^{81} + 84144q^{82} + 28592q^{83} - 42642q^{84} + 355124q^{85} + 103778q^{86} + 46620q^{87} - 150776q^{88} - 27926q^{89} + 52002q^{90} + 334180q^{91} - 24334q^{92} + 16092q^{93} + 113632q^{94} + 113392q^{95} + 66168q^{96} + 16580q^{97} + 49688q^{98} + 2592q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 39 x^{2} - 30 x + 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 36 \nu - 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 40 \nu - 42 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + 2 \beta_{1} + 19\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 38 \beta_{1} + 23\)

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
−5.76108
−1.23317
0.428783
6.56547
−4.76108 −9.00000 −9.33211 −97.1410 42.8497 −211.220 196.786 81.0000 462.496
1.2 −0.233171 −9.00000 −31.9456 18.8848 2.09853 −97.3695 14.9102 81.0000 −4.40338
1.3 1.42878 −9.00000 −29.9586 −83.8971 −12.8590 175.668 −88.5253 81.0000 −119.871
1.4 7.56547 −9.00000 25.2363 40.1532 −68.0892 194.921 −51.1704 81.0000 303.778
\(n\): e.g. 2-40 or 990-1000
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.c 4
3.b odd 2 1 207.6.a.d 4
4.b odd 2 1 1104.6.a.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.c 4 1.a even 1 1 trivial
207.6.a.d 4 3.b odd 2 1
1104.6.a.n 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4 T_{2}^{3} - 33 T_{2}^{2} + 44 T_{2} + 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 12 + 44 T - 33 T^{2} - 4 T^{3} + T^{4} \)
$3$ \( ( 9 + T )^{4} \)
$5$ \( 6179904 - 343872 T - 1780 T^{2} + 122 T^{3} + T^{4} \)
$7$ \( 704219920 + 2944856 T - 59552 T^{2} - 62 T^{3} + T^{4} \)
$11$ \( 3071914368 + 65732992 T - 325544 T^{2} - 32 T^{3} + T^{4} \)
$13$ \( 8279608752 + 157801104 T + 224112 T^{2} - 1364 T^{3} + T^{4} \)
$17$ \( 4839573197664 + 628668496 T - 4907108 T^{2} - 278 T^{3} + T^{4} \)
$19$ \( -12894399037424 + 17126580168 T - 3614768 T^{2} - 2862 T^{3} + T^{4} \)
$23$ \( ( -529 + T )^{4} \)
$29$ \( -313971545146320 - 292058201520 T - 58940976 T^{2} + 5180 T^{3} + T^{4} \)
$31$ \( 1282101776467200 - 204369792192 T - 109983424 T^{2} + 1788 T^{3} + T^{4} \)
$37$ \( 1778615087096912 - 65675524528 T - 128477752 T^{2} - 3348 T^{3} + T^{4} \)
$41$ \( 2927035685383248 - 973290541376 T - 32677496 T^{2} + 17664 T^{3} + T^{4} \)
$43$ \( -579704020143984 - 184056480792 T + 179431184 T^{2} - 25398 T^{3} + T^{4} \)
$47$ \( 1374692307102720 - 6125720328960 T - 200587184 T^{2} + 26040 T^{3} + T^{4} \)
$53$ \( -8267030656284672 - 1614300349440 T + 182738612 T^{2} + 32006 T^{3} + T^{4} \)
$59$ \( -591985895589135360 - 46012606646400 T + 64325168 T^{2} + 61136 T^{3} + T^{4} \)
$61$ \( 95539705166646480 + 4099603292048 T - 1179441560 T^{2} - 35844 T^{3} + T^{4} \)
$67$ \( 1079662317324715088 + 93118183336760 T - 1172084512 T^{2} - 73458 T^{3} + T^{4} \)
$71$ \( -1316744113740134400 + 120925168230400 T - 2739085664 T^{2} - 24432 T^{3} + T^{4} \)
$73$ \( 255273216714652240 - 67567796624000 T + 4748249032 T^{2} - 122512 T^{3} + T^{4} \)
$79$ \( 230387954096403472 + 43475279098568 T + 1049929504 T^{2} - 90170 T^{3} + T^{4} \)
$83$ \( 1671969331111706496 + 39469728544512 T - 2499878056 T^{2} - 28592 T^{3} + T^{4} \)
$89$ \( 24867785160773497440 - 265198177383760 T - 11401415132 T^{2} + 27926 T^{3} + T^{4} \)
$97$ \( -7418914108619824400 - 673721776750640 T - 15313044576 T^{2} - 16580 T^{3} + T^{4} \)
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