Properties

Label 69.6.a.d
Level $69$
Weight $6$
Character orbit 69.a
Self dual yes
Analytic conductor $11.066$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 75 x^{2} - 42 x + 736\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} -9 q^{3} + ( 7 - \beta_{1} + \beta_{3} ) q^{4} + ( 5 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} + ( -9 + 9 \beta_{1} ) q^{6} + ( -18 + 12 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{7} + ( 17 + 9 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} -9 q^{3} + ( 7 - \beta_{1} + \beta_{3} ) q^{4} + ( 5 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{5} + ( -9 + 9 \beta_{1} ) q^{6} + ( -18 + 12 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{7} + ( 17 + 9 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{8} + 81 q^{9} + ( -124 + 24 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{10} + ( -261 + 41 \beta_{1} + 3 \beta_{2} + 13 \beta_{3} ) q^{11} + ( -63 + 9 \beta_{1} - 9 \beta_{3} ) q^{12} + ( -88 - 22 \beta_{1} + 22 \beta_{2} ) q^{13} + ( -450 + 46 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} ) q^{14} + ( -45 - 27 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{15} + ( -513 - 5 \beta_{1} - 16 \beta_{2} - 19 \beta_{3} ) q^{16} + ( 22 - 66 \beta_{1} - 34 \beta_{2} + 43 \beta_{3} ) q^{17} + ( 81 - 81 \beta_{1} ) q^{18} + ( -1605 + 43 \beta_{1} - 19 \beta_{2} - 8 \beta_{3} ) q^{19} + ( -1306 + 158 \beta_{1} + 28 \beta_{2} - 20 \beta_{3} ) q^{20} + ( 162 - 108 \beta_{1} + 36 \beta_{2} + 9 \beta_{3} ) q^{21} + ( -1788 + 44 \beta_{1} - 46 \beta_{2} - 43 \beta_{3} ) q^{22} -529 q^{23} + ( -153 - 81 \beta_{1} + 36 \beta_{2} - 18 \beta_{3} ) q^{24} + ( 321 - 450 \beta_{1} - 10 \beta_{2} + 20 \beta_{3} ) q^{25} + ( 594 + 22 \beta_{1} + 44 \beta_{2} - 88 \beta_{3} ) q^{26} -729 q^{27} + ( -1566 - 34 \beta_{1} + 92 \beta_{2} + 13 \beta_{3} ) q^{28} + ( 932 + 2 \beta_{1} + 10 \beta_{2} - 120 \beta_{3} ) q^{29} + ( 1116 - 216 \beta_{1} - 90 \beta_{2} + 90 \beta_{3} ) q^{30} + ( 818 - 770 \beta_{1} + 14 \beta_{2} + 76 \beta_{3} ) q^{31} + ( -831 + 577 \beta_{1} + 172 \beta_{2} + 2 \beta_{3} ) q^{32} + ( 2349 - 369 \beta_{1} - 27 \beta_{2} - 117 \beta_{3} ) q^{33} + ( 2940 - 608 \beta_{1} - 240 \beta_{2} + 279 \beta_{3} ) q^{34} + ( 230 + 258 \beta_{1} - 282 \beta_{2} + 90 \beta_{3} ) q^{35} + ( 567 - 81 \beta_{1} + 81 \beta_{3} ) q^{36} + ( -4267 - 11 \beta_{1} + 231 \beta_{2} - 33 \beta_{3} ) q^{37} + ( -3138 + 1790 \beta_{1} - 6 \beta_{2} + 44 \beta_{3} ) q^{38} + ( 792 + 198 \beta_{1} - 198 \beta_{2} ) q^{39} + ( -3618 + 774 \beta_{1} - 184 \beta_{2} + 2 \beta_{3} ) q^{40} + ( 4822 - 248 \beta_{1} + 300 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 4050 - 414 \beta_{1} + 36 \beta_{2} - 63 \beta_{3} ) q^{42} + ( -6207 + 365 \beta_{1} + 451 \beta_{2} + 58 \beta_{3} ) q^{43} + ( 5042 + 1302 \beta_{1} - 16 \beta_{2} - 273 \beta_{3} ) q^{44} + ( 405 + 243 \beta_{1} + 81 \beta_{2} - 162 \beta_{3} ) q^{45} + ( -529 + 529 \beta_{1} ) q^{46} + ( 4392 - 348 \beta_{1} - 424 \beta_{2} + 12 \beta_{3} ) q^{47} + ( 4617 + 45 \beta_{1} + 144 \beta_{2} + 171 \beta_{3} ) q^{48} + ( 1677 - 456 \beta_{1} + 40 \beta_{2} - 662 \beta_{3} ) q^{49} + ( 17571 - 611 \beta_{1} - 100 \beta_{2} + 520 \beta_{3} ) q^{50} + ( -198 + 594 \beta_{1} + 306 \beta_{2} - 387 \beta_{3} ) q^{51} + ( 1914 + 1386 \beta_{1} - 264 \beta_{2} - 330 \beta_{3} ) q^{52} + ( -6723 + 1519 \beta_{1} - 203 \beta_{2} + 16 \beta_{3} ) q^{53} + ( -729 + 729 \beta_{1} ) q^{54} + ( -9942 - 646 \beta_{1} - 366 \beta_{2} + 906 \beta_{3} ) q^{55} + ( 13534 - 390 \beta_{1} + 260 \beta_{2} - 637 \beta_{3} ) q^{56} + ( 14445 - 387 \beta_{1} + 171 \beta_{2} + 72 \beta_{3} ) q^{57} + ( 306 + 958 \beta_{1} + 500 \beta_{2} - 172 \beta_{3} ) q^{58} + ( -3420 - 1792 \beta_{1} + 288 \beta_{2} + 132 \beta_{3} ) q^{59} + ( 11754 - 1422 \beta_{1} - 252 \beta_{2} + 180 \beta_{3} ) q^{60} + ( -3849 - 661 \beta_{1} - 1087 \beta_{2} + 233 \beta_{3} ) q^{61} + ( 30284 - 2076 \beta_{1} - 276 \beta_{2} + 776 \beta_{3} ) q^{62} + ( -1458 + 972 \beta_{1} - 324 \beta_{2} - 81 \beta_{3} ) q^{63} + ( -7537 + 443 \beta_{1} + 848 \beta_{2} - 827 \beta_{3} ) q^{64} + ( 10142 - 3542 \beta_{1} + 902 \beta_{2} + 220 \beta_{3} ) q^{65} + ( 16092 - 396 \beta_{1} + 414 \beta_{2} + 387 \beta_{3} ) q^{66} + ( -2845 - 2001 \beta_{1} + 497 \beta_{2} - 638 \beta_{3} ) q^{67} + ( 28136 - 4572 \beta_{1} - 508 \beta_{2} + 711 \beta_{3} ) q^{68} + 4761 q^{69} + ( -7240 - 824 \beta_{1} - 924 \beta_{2} + 1242 \beta_{3} ) q^{70} + ( -14824 - 3940 \beta_{1} - 1408 \beta_{2} + 508 \beta_{3} ) q^{71} + ( 1377 + 729 \beta_{1} - 324 \beta_{2} + 162 \beta_{3} ) q^{72} + ( -27872 + 1990 \beta_{1} - 362 \beta_{2} + 754 \beta_{3} ) q^{73} + ( -5598 + 4102 \beta_{1} + 594 \beta_{2} - 1177 \beta_{3} ) q^{74} + ( -2889 + 4050 \beta_{1} + 90 \beta_{2} - 180 \beta_{3} ) q^{75} + ( -19580 + 1076 \beta_{1} + 420 \beta_{2} - 1460 \beta_{3} ) q^{76} + ( 330 - 6514 \beta_{1} + 2538 \beta_{2} + 518 \beta_{3} ) q^{77} + ( -5346 - 198 \beta_{1} - 396 \beta_{2} + 792 \beta_{3} ) q^{78} + ( -60686 + 2484 \beta_{1} - 116 \beta_{2} - 879 \beta_{3} ) q^{79} + ( 10058 - 918 \beta_{1} - 1272 \beta_{2} + 788 \beta_{3} ) q^{80} + 6561 q^{81} + ( 12162 - 5786 \beta_{1} + 584 \beta_{2} - 1248 \beta_{3} ) q^{82} + ( -24283 - 2161 \beta_{1} + 629 \beta_{2} - 2561 \beta_{3} ) q^{83} + ( 14094 + 306 \beta_{1} - 828 \beta_{2} - 117 \beta_{3} ) q^{84} + ( -81260 + 12220 \beta_{1} - 92 \beta_{2} - 826 \beta_{3} ) q^{85} + ( -23002 + 3926 \beta_{1} + 670 \beta_{2} - 2562 \beta_{3} ) q^{86} + ( -8388 - 18 \beta_{1} - 90 \beta_{2} + 1080 \beta_{3} ) q^{87} + ( 11802 - 2034 \beta_{1} + 2532 \beta_{2} - 119 \beta_{3} ) q^{88} + ( -29176 + 4520 \beta_{1} - 2260 \beta_{2} - 2301 \beta_{3} ) q^{89} + ( -10044 + 1944 \beta_{1} + 810 \beta_{2} - 810 \beta_{3} ) q^{90} + ( -73744 - 396 \beta_{1} - 132 \beta_{2} + 3410 \beta_{3} ) q^{91} + ( -3703 + 529 \beta_{1} - 529 \beta_{3} ) q^{92} + ( -7362 + 6930 \beta_{1} - 126 \beta_{2} - 684 \beta_{3} ) q^{93} + ( 20632 - 3312 \beta_{1} - 896 \beta_{2} + 2480 \beta_{3} ) q^{94} + ( -4372 - 3416 \beta_{1} - 2772 \beta_{2} + 3348 \beta_{3} ) q^{95} + ( 7479 - 5193 \beta_{1} - 1548 \beta_{2} - 18 \beta_{3} ) q^{96} + ( -12408 + 9542 \beta_{1} - 1646 \beta_{2} + 3368 \beta_{3} ) q^{97} + ( 16077 + 8795 \beta_{1} + 2728 \beta_{2} - 406 \beta_{3} ) q^{98} + ( -21141 + 3321 \beta_{1} + 243 \beta_{2} + 1053 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 36q^{3} + 26q^{4} + 22q^{5} - 36q^{6} - 62q^{7} + 72q^{8} + 324q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 36q^{3} + 26q^{4} + 22q^{5} - 36q^{6} - 62q^{7} + 72q^{8} + 324q^{9} - 496q^{10} - 1076q^{11} - 234q^{12} - 396q^{13} - 1806q^{14} - 198q^{15} - 1982q^{16} + 70q^{17} + 324q^{18} - 6366q^{19} - 5240q^{20} + 558q^{21} - 6974q^{22} - 2116q^{23} - 648q^{24} + 1264q^{25} + 2464q^{26} - 2916q^{27} - 6474q^{28} + 3948q^{29} + 4464q^{30} + 3092q^{31} - 3672q^{32} + 9684q^{33} + 11682q^{34} + 1304q^{35} + 2106q^{36} - 17464q^{37} - 12628q^{38} + 3564q^{39} - 14108q^{40} + 18680q^{41} + 16254q^{42} - 25846q^{43} + 20746q^{44} + 1782q^{45} - 2116q^{46} + 18392q^{47} + 17838q^{48} + 7952q^{49} + 69444q^{50} - 630q^{51} + 8844q^{52} - 26518q^{53} - 2916q^{54} - 40848q^{55} + 54890q^{56} + 57294q^{57} + 568q^{58} - 14520q^{59} + 47160q^{60} - 13688q^{61} + 120136q^{62} - 5022q^{63} - 30190q^{64} + 38324q^{65} + 62766q^{66} - 11098q^{67} + 112138q^{68} + 19044q^{69} - 29596q^{70} - 57496q^{71} + 5832q^{72} - 112272q^{73} - 21226q^{74} - 11376q^{75} - 76240q^{76} - 4792q^{77} - 22176q^{78} - 240754q^{79} + 41200q^{80} + 26244q^{81} + 49976q^{82} - 93268q^{83} + 58266q^{84} - 323204q^{85} - 88224q^{86} - 35532q^{87} + 42382q^{88} - 107582q^{89} - 40176q^{90} - 301532q^{91} - 13754q^{92} - 27828q^{93} + 79360q^{94} - 18640q^{95} + 33048q^{96} - 53076q^{97} + 59664q^{98} - 87156q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 75 x^{2} - 42 x + 736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 54 \nu + 4 \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 38 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 38\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{2} + 55 \beta_{1} + 34\)

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
8.33314
3.04157
−3.86863
−7.50608
−7.33314 −9.00000 21.7749 −0.408582 65.9983 −4.34307 74.9818 81.0000 2.99619
1.2 −2.04157 −9.00000 −27.8320 42.3660 18.3742 191.647 122.151 81.0000 −86.4934
1.3 4.86863 −9.00000 −8.29644 66.7344 −43.8177 −185.299 −196.188 81.0000 324.905
1.4 8.50608 −9.00000 40.3535 −86.6918 −76.5548 −64.0051 71.0553 81.0000 −737.408
\(n\): 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.d 4
3.b odd 2 1 207.6.a.e 4
4.b odd 2 1 1104.6.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.d 4 1.a even 1 1 trivial
207.6.a.e 4 3.b odd 2 1
1104.6.a.o 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4 T_{2}^{3} - 69 T_{2}^{2} + 188 T_{2} + 620 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 620 + 188 T - 69 T^{2} - 4 T^{3} + T^{4} \)
$3$ \( ( 9 + T )^{4} \)
$5$ \( 100144 + 242392 T - 6640 T^{2} - 22 T^{3} + T^{4} \)
$7$ \( -9871616 - 2428952 T - 35668 T^{2} + 62 T^{3} + T^{4} \)
$11$ \( -45159083072 - 201957832 T + 72632 T^{2} + 1076 T^{3} + T^{4} \)
$13$ \( 16813724400 - 169111536 T - 700832 T^{2} + 396 T^{3} + T^{4} \)
$17$ \( -95458629376 - 1618545376 T - 4039272 T^{2} - 70 T^{3} + T^{4} \)
$19$ \( 3908943190016 + 13265159200 T + 14381900 T^{2} + 6366 T^{3} + T^{4} \)
$23$ \( ( 529 + T )^{4} \)
$29$ \( 61820529282864 + 31954993392 T - 12443696 T^{2} - 3948 T^{3} + T^{4} \)
$31$ \( 378047008189440 + 10631956992 T - 50204560 T^{2} - 3092 T^{3} + T^{4} \)
$37$ \( -684323468629888 - 448799300360 T + 34985088 T^{2} + 17464 T^{3} + T^{4} \)
$41$ \( -1264675422828464 + 874280152544 T - 9058840 T^{2} - 18680 T^{3} + T^{4} \)
$43$ \( 13132961579165184 - 2905674582960 T - 84206564 T^{2} + 25846 T^{3} + T^{4} \)
$47$ \( 12164338199022592 + 2264418253312 T - 155883760 T^{2} - 18392 T^{3} + T^{4} \)
$53$ \( 4003378120659632 - 1687966063688 T + 35844360 T^{2} + 26518 T^{3} + T^{4} \)
$59$ \( 23251251284928256 - 1703086899072 T - 319101312 T^{2} + 14520 T^{3} + T^{4} \)
$61$ \( 609195340524114656 - 4835992564696 T - 1747623024 T^{2} + 13688 T^{3} + T^{4} \)
$67$ \( -73643463180406528 - 18173151537008 T - 986825700 T^{2} + 11098 T^{3} + T^{4} \)
$71$ \( 1217198076112510976 - 160479941003520 T - 3157729520 T^{2} + 57496 T^{3} + T^{4} \)
$73$ \( -14479128931571888 + 33631197210816 T + 3648345384 T^{2} + 112272 T^{3} + T^{4} \)
$79$ \( 7536932540782966080 + 685119845220504 T + 20172607868 T^{2} + 240754 T^{3} + T^{4} \)
$83$ \( 12198224737513093952 - 375266370364536 T - 5396841896 T^{2} + 93268 T^{3} + T^{4} \)
$89$ \( -75077688976055350528 - 2210856074146336 T - 13698324048 T^{2} + 107582 T^{3} + T^{4} \)
$97$ \( 20933686069973108720 - 64038776661904 T - 21394056160 T^{2} + 53076 T^{3} + T^{4} \)
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