# Properties

 Label 69.6.a.b Level $69$ Weight $6$ Character orbit 69.a Self dual yes Analytic conductor $11.066$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## 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: $$3$$ Coefficient field: 3.3.5333.1 Defining polynomial: $$x^{3} - x^{2} - 11 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 + \beta_{2} ) q^{2} + 9 q^{3} + ( 9 - 4 \beta_{1} - 9 \beta_{2} ) q^{4} + ( -17 + 11 \beta_{1} + 6 \beta_{2} ) q^{5} + ( -27 + 9 \beta_{2} ) q^{6} + ( -44 - 26 \beta_{1} - 8 \beta_{2} ) q^{7} + ( -171 + 32 \beta_{1} + 35 \beta_{2} ) q^{8} + 81 q^{9} +O(q^{10})$$ $$q + ( -3 + \beta_{2} ) q^{2} + 9 q^{3} + ( 9 - 4 \beta_{1} - 9 \beta_{2} ) q^{4} + ( -17 + 11 \beta_{1} + 6 \beta_{2} ) q^{5} + ( -27 + 9 \beta_{2} ) q^{6} + ( -44 - 26 \beta_{1} - 8 \beta_{2} ) q^{7} + ( -171 + 32 \beta_{1} + 35 \beta_{2} ) q^{8} + 81 q^{9} + ( 111 - 13 \beta_{1} - 64 \beta_{2} ) q^{10} + ( -95 + 49 \beta_{1} - 42 \beta_{2} ) q^{11} + ( 81 - 36 \beta_{1} - 81 \beta_{2} ) q^{12} + ( -264 - 66 \beta_{1} - 132 \beta_{2} ) q^{13} + ( 188 + 6 \beta_{1} + 30 \beta_{2} ) q^{14} + ( -153 + 99 \beta_{1} + 54 \beta_{2} ) q^{15} + ( 961 + 20 \beta_{1} - 125 \beta_{2} ) q^{16} + ( -896 - 68 \beta_{1} + 72 \beta_{2} ) q^{17} + ( -243 + 81 \beta_{2} ) q^{18} + ( -993 - 143 \beta_{1} - 10 \beta_{2} ) q^{19} + ( -1681 - 109 \beta_{1} + 316 \beta_{2} ) q^{20} + ( -396 - 234 \beta_{1} - 72 \beta_{2} ) q^{21} + ( -1647 + 217 \beta_{1} + 108 \beta_{2} ) q^{22} + 529 q^{23} + ( -1539 + 288 \beta_{1} + 315 \beta_{2} ) q^{24} + ( 241 + 10 \beta_{1} + 40 \beta_{2} ) q^{25} + ( -2640 + 462 \beta_{1} + 594 \beta_{2} ) q^{26} + 729 q^{27} + ( 1732 + 718 \beta_{1} + 258 \beta_{2} ) q^{28} + ( -5608 - 254 \beta_{1} + 200 \beta_{2} ) q^{29} + ( 999 - 117 \beta_{1} - 576 \beta_{2} ) q^{30} + ( -5158 - 158 \beta_{1} + 560 \beta_{2} ) q^{31} + ( -1651 - 504 \beta_{1} + 571 \beta_{2} ) q^{32} + ( -855 + 441 \beta_{1} - 378 \beta_{2} ) q^{33} + ( 5808 - 356 \beta_{1} - 1260 \beta_{2} ) q^{34} + ( -6154 - 826 \beta_{1} - 884 \beta_{2} ) q^{35} + ( 729 - 324 \beta_{1} - 729 \beta_{2} ) q^{36} + ( 5133 - 105 \beta_{1} + 370 \beta_{2} ) q^{37} + ( 4375 - 103 \beta_{1} - 790 \beta_{2} ) q^{38} + ( -2376 - 594 \beta_{1} - 1188 \beta_{2} ) q^{39} + ( 12911 - 957 \beta_{1} - 1420 \beta_{2} ) q^{40} + ( 4150 + 1048 \beta_{1} + 1204 \beta_{2} ) q^{41} + ( 1692 + 54 \beta_{1} + 270 \beta_{2} ) q^{42} + ( 593 + 163 \beta_{1} + 1538 \beta_{2} ) q^{43} + ( 8833 - 1783 \beta_{1} - 1168 \beta_{2} ) q^{44} + ( -1377 + 891 \beta_{1} + 486 \beta_{2} ) q^{45} + ( -1587 + 529 \beta_{2} ) q^{46} + ( 10548 + 2000 \beta_{1} + 284 \beta_{2} ) q^{47} + ( 8649 + 180 \beta_{1} - 1125 \beta_{2} ) q^{48} + ( 1789 + 3696 \beta_{1} + 2800 \beta_{2} ) q^{49} + ( 437 - 150 \beta_{1} - 9 \beta_{2} ) q^{50} + ( -8064 - 612 \beta_{1} + 648 \beta_{2} ) q^{51} + ( 29832 + 198 \beta_{1} - 2442 \beta_{2} ) q^{52} + ( -13877 + 5619 \beta_{1} + 3166 \beta_{2} ) q^{53} + ( -2187 + 729 \beta_{2} ) q^{54} + ( 11198 - 1542 \beta_{1} + 3224 \beta_{2} ) q^{55} + ( -11572 - 506 \beta_{1} - 1494 \beta_{2} ) q^{56} + ( -8937 - 1287 \beta_{1} - 90 \beta_{2} ) q^{57} + ( 26272 - 1054 \beta_{1} - 6554 \beta_{2} ) q^{58} + ( -10904 - 4164 \beta_{1} - 752 \beta_{2} ) q^{59} + ( -15129 - 981 \beta_{1} + 2844 \beta_{2} ) q^{60} + ( 18847 + 1545 \beta_{1} - 986 \beta_{2} ) q^{61} + ( 35290 - 2398 \beta_{1} - 8360 \beta_{2} ) q^{62} + ( -3564 - 2106 \beta_{1} - 648 \beta_{2} ) q^{63} + ( -1479 - 3428 \beta_{1} - 573 \beta_{2} ) q^{64} + ( -19734 - 6006 \beta_{1} + 1980 \beta_{2} ) q^{65} + ( -14823 + 1953 \beta_{1} + 972 \beta_{2} ) q^{66} + ( 13323 - 9559 \beta_{1} - 6138 \beta_{2} ) q^{67} + ( -24800 + 6860 \beta_{1} + 11420 \beta_{2} ) q^{68} + 4761 q^{69} + ( 86 + 2710 \beta_{1} - 24 \beta_{2} ) q^{70} + ( 9832 + 4708 \beta_{1} - 1364 \beta_{2} ) q^{71} + ( -13851 + 2592 \beta_{1} + 2835 \beta_{2} ) q^{72} + ( -30240 + 8490 \beta_{1} + 7808 \beta_{2} ) q^{73} + ( -2299 - 1585 \beta_{1} + 3018 \beta_{2} ) q^{74} + ( 2169 + 90 \beta_{1} + 360 \beta_{2} ) q^{75} + ( -5393 + 7633 \beta_{1} + 9538 \beta_{2} ) q^{76} + ( -30414 + 1770 \beta_{1} - 4196 \beta_{2} ) q^{77} + ( -23760 + 4158 \beta_{1} + 5346 \beta_{2} ) q^{78} + ( -19652 - 9526 \beta_{1} + 32 \beta_{2} ) q^{79} + ( -18897 + 8211 \beta_{1} + 12276 \beta_{2} ) q^{80} + 6561 q^{81} + ( 13502 - 3768 \beta_{1} - 4122 \beta_{2} ) q^{82} + ( -31417 - 6321 \beta_{1} - 16006 \beta_{2} ) q^{83} + ( 15588 + 6462 \beta_{1} + 2322 \beta_{2} ) q^{84} + ( 2756 - 8892 \beta_{1} - 11272 \beta_{2} ) q^{85} + ( 45481 - 5989 \beta_{1} - 8798 \beta_{2} ) q^{86} + ( -50472 - 2286 \beta_{1} + 1800 \beta_{2} ) q^{87} + ( 10225 - 4055 \beta_{1} + 14168 \beta_{2} ) q^{88} + ( 35534 + 1622 \beta_{1} - 8868 \beta_{2} ) q^{89} + ( 8991 - 1053 \beta_{1} - 5184 \beta_{2} ) q^{90} + ( 47652 + 21384 \beta_{1} + 7656 \beta_{2} ) q^{91} + ( 4761 - 2116 \beta_{1} - 4761 \beta_{2} ) q^{92} + ( -46422 - 1422 \beta_{1} + 5040 \beta_{2} ) q^{93} + ( -46556 + 864 \beta_{1} + 6844 \beta_{2} ) q^{94} + ( -19040 - 12124 \beta_{1} - 10932 \beta_{2} ) q^{95} + ( -14859 - 4536 \beta_{1} + 5139 \beta_{2} ) q^{96} + ( -39412 + 7014 \beta_{1} - 10068 \beta_{2} ) q^{97} + ( 39881 - 7504 \beta_{1} - 18707 \beta_{2} ) q^{98} + ( -7695 + 3969 \beta_{1} - 3402 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9} + O(q^{10})$$ $$3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9} + 282 q^{10} - 376 q^{11} + 198 q^{12} - 858 q^{13} + 588 q^{14} - 504 q^{15} + 2738 q^{16} - 2548 q^{17} - 648 q^{18} - 2846 q^{19} - 4618 q^{20} - 1026 q^{21} - 5050 q^{22} + 1587 q^{23} - 4590 q^{24} + 753 q^{25} - 7788 q^{26} + 2187 q^{27} + 4736 q^{28} - 16370 q^{29} + 2538 q^{30} - 14756 q^{31} - 3878 q^{32} - 3384 q^{33} + 16520 q^{34} - 18520 q^{35} + 1782 q^{36} + 15874 q^{37} + 12438 q^{38} - 7722 q^{39} + 38270 q^{40} + 12606 q^{41} + 5292 q^{42} + 3154 q^{43} + 27114 q^{44} - 4536 q^{45} - 4232 q^{46} + 29928 q^{47} + 24642 q^{48} + 4471 q^{49} + 1452 q^{50} - 22932 q^{51} + 86856 q^{52} - 44084 q^{53} - 5832 q^{54} + 38360 q^{55} - 35704 q^{56} - 25614 q^{57} + 73316 q^{58} - 29300 q^{59} - 41562 q^{60} + 54010 q^{61} + 99908 q^{62} - 9234 q^{63} - 1582 q^{64} - 51216 q^{65} - 45450 q^{66} + 43390 q^{67} - 69840 q^{68} + 14283 q^{69} - 2476 q^{70} + 23424 q^{71} - 41310 q^{72} - 91402 q^{73} - 2294 q^{74} + 6777 q^{75} - 14274 q^{76} - 97208 q^{77} - 70092 q^{78} - 49398 q^{79} - 52626 q^{80} + 19683 q^{81} + 40152 q^{82} - 103936 q^{83} + 42624 q^{84} + 5888 q^{85} + 133634 q^{86} - 147330 q^{87} + 48898 q^{88} + 96112 q^{89} + 22842 q^{90} + 129228 q^{91} + 11638 q^{92} - 132804 q^{93} - 133688 q^{94} - 55928 q^{95} - 34902 q^{96} - 135318 q^{97} + 108440 q^{98} - 30456 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 11 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} + \beta_{1} + 15$$$$)/2$$

## 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
 0.714018 3.49331 −3.20733
−10.2042 9.00000 72.1256 −55.5168 −91.8378 2.50462 −409.450 81.0000 566.504
1.2 −1.29009 9.00000 −30.3357 59.1123 −11.6108 −213.331 80.4187 81.0000 −76.2602
1.3 3.49429 9.00000 −19.7900 −59.5955 31.4486 96.8268 −180.969 81.0000 −208.244
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 3
3.b odd 2 1 207.6.a.c 3
4.b odd 2 1 1104.6.a.i 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 8 T_{2}^{2} - 27 T_{2} - 46$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(69))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-46 - 27 T + 8 T^{2} + T^{3}$$
$3$ $$( -9 + T )^{3}$$
$5$ $$-195576 - 3496 T + 56 T^{2} + T^{3}$$
$7$ $$51736 - 20948 T + 114 T^{2} + T^{3}$$
$11$ $$21141352 - 230672 T + 376 T^{2} + T^{3}$$
$13$ $$-368282376 - 439956 T + 858 T^{2} + T^{3}$$
$17$ $$-129112640 + 1504816 T + 2548 T^{2} + T^{3}$$
$19$ $$-4313168 + 1826204 T + 2846 T^{2} + T^{3}$$
$23$ $$( -529 + T )^{3}$$
$29$ $$117835741080 + 82401708 T + 16370 T^{2} + T^{3}$$
$31$ $$16664141952 + 52692624 T + 14756 T^{2} + T^{3}$$
$37$ $$-103469473312 + 75298092 T - 15874 T^{2} + T^{3}$$
$41$ $$213582513784 - 15571268 T - 12606 T^{2} + T^{3}$$
$43$ $$409945701888 - 102023460 T - 3154 T^{2} + T^{3}$$
$47$ $$524672802816 + 136428688 T - 29928 T^{2} + T^{3}$$
$53$ $$-30187666172280 - 544538824 T + 44084 T^{2} + T^{3}$$
$59$ $$-4070512924224 - 399858640 T + 29300 T^{2} + T^{3}$$
$61$ $$-694379910768 + 755208380 T - 54010 T^{2} + T^{3}$$
$67$ $$128918418373088 - 2949665124 T - 43390 T^{2} + T^{3}$$
$71$ $$26245560332032 - 1173004304 T - 23424 T^{2} + T^{3}$$
$73$ $$-119459239092680 - 733701012 T + 91402 T^{2} + T^{3}$$
$79$ $$-93978622829240 - 3312817172 T + 49398 T^{2} + T^{3}$$
$83$ $$-694250483317800 - 6478606576 T + 103936 T^{2} + T^{3}$$
$89$ $$102645237296960 - 1426041824 T - 96112 T^{2} + T^{3}$$
$97$ $$-82905816948920 - 3897620052 T + 135318 T^{2} + T^{3}$$