# Properties

 Label 69.8.c.a Level $69$ Weight $8$ Character orbit 69.c Analytic conductor $21.555$ Analytic rank $0$ Dimension $6$ CM discriminant -23 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 69.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.5545667584$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8869743.1 Defining polynomial: $$x^{6} - 3 x^{3} + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{2} + ( 8 \beta_{1} - 23 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{3} + ( -128 - 15 \beta_{1} + 71 \beta_{2} - 28 \beta_{3} + 28 \beta_{4} + 28 \beta_{5} ) q^{4} + ( -510 - 58 \beta_{1} - 161 \beta_{3} + 95 \beta_{4} + 124 \beta_{5} ) q^{6} + ( 433 + 256 \beta_{1} + 768 \beta_{2} - 384 \beta_{3} - 689 \beta_{4} + 177 \beta_{5} ) q^{8} + ( -145 \beta_{1} - 767 \beta_{3} + 456 \beta_{4} + 456 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -2 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{2} + ( 8 \beta_{1} - 23 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{3} + ( -128 - 15 \beta_{1} + 71 \beta_{2} - 28 \beta_{3} + 28 \beta_{4} + 28 \beta_{5} ) q^{4} + ( -510 - 58 \beta_{1} - 161 \beta_{3} + 95 \beta_{4} + 124 \beta_{5} ) q^{6} + ( 433 + 256 \beta_{1} + 768 \beta_{2} - 384 \beta_{3} - 689 \beta_{4} + 177 \beta_{5} ) q^{8} + ( -145 \beta_{1} - 767 \beta_{3} + 456 \beta_{4} + 456 \beta_{5} ) q^{9} + ( -2571 + 817 \beta_{1} + 2944 \beta_{2} - 1356 \beta_{3} - 2268 \beta_{4} - 265 \beta_{5} ) q^{12} + ( -4241 \beta_{1} + 3455 \beta_{2} + 393 \beta_{3} - 393 \beta_{4} - 393 \beta_{5} ) q^{13} + ( 16384 - 6928 \beta_{1} - 6062 \beta_{2} + 6495 \beta_{3} - 6495 \beta_{4} - 6495 \beta_{5} ) q^{16} + ( -7521 - 4747 \beta_{1} + 11005 \beta_{2} - 4747 \beta_{3} - 6882 \beta_{4} + 2135 \beta_{5} ) q^{18} + ( 12167 - 12167 \beta_{4} + 12167 \beta_{5} ) q^{23} + ( 65280 - 9030 \beta_{1} - 31609 \beta_{2} + 4154 \beta_{3} - 20387 \beta_{4} - 24099 \beta_{5} ) q^{24} -78125 q^{25} + ( -10529 + 14213 \beta_{1} + 42639 \beta_{2} + 41002 \beta_{3} - 3684 \beta_{4} - 24742 \beta_{5} ) q^{26} + ( -17709 - 19837 \beta_{4} + 19837 \beta_{5} ) q^{27} + ( -8793 \beta_{1} - 26379 \beta_{2} + 85121 \beta_{3} + 8793 \beta_{4} + 8793 \beta_{5} ) q^{29} + ( 84347 \beta_{1} - 13897 \beta_{2} - 35225 \beta_{3} + 35225 \beta_{4} + 35225 \beta_{5} ) q^{31} + ( -55424 - 17753 \beta_{1} - 53259 \beta_{2} + 172334 \beta_{3} + 73177 \beta_{4} - 37671 \beta_{5} ) q^{32} + ( 255054 - 48077 \beta_{1} + 9058 \beta_{2} + 31539 \beta_{3} - 76476 \beta_{4} - 106897 \beta_{5} ) q^{36} + ( -377907 + 5321 \beta_{1} + 147109 \beta_{3} - 64384 \beta_{4} - 88046 \beta_{5} ) q^{39} + ( 71801 \beta_{1} + 215403 \beta_{2} - 157205 \beta_{3} - 71801 \beta_{4} - 71801 \beta_{5} ) q^{41} + ( -194672 \beta_{1} - 170338 \beta_{2} + 182505 \beta_{3} - 182505 \beta_{4} - 182505 \beta_{5} ) q^{46} + ( -88735 \beta_{1} - 266205 \beta_{2} + 366973 \beta_{3} + 88735 \beta_{4} + 88735 \beta_{5} ) q^{47} + ( -371514 - 223122 \beta_{1} - 376832 \beta_{2} + 541123 \beta_{3} + 264716 \beta_{4} - 189501 \beta_{5} ) q^{48} + 823543 q^{49} + ( 156250 \beta_{1} + 468750 \beta_{2} - 234375 \beta_{3} - 156250 \beta_{4} - 156250 \beta_{5} ) q^{50} + ( 946763 + 168464 \beta_{1} + 147406 \beta_{2} - 157935 \beta_{3} + 157935 \beta_{4} + 157935 \beta_{5} ) q^{52} + ( -242300 \beta_{1} - 52442 \beta_{2} + 184917 \beta_{3} - 372647 \beta_{4} - 372647 \beta_{5} ) q^{54} + ( -2132545 + 437573 \beta_{1} + 384083 \beta_{2} - 410828 \beta_{3} + 410828 \beta_{4} + 410828 \beta_{5} ) q^{58} + ( 657862 - 657862 \beta_{4} + 657862 \beta_{5} ) q^{59} + ( 647947 - 90813 \beta_{1} - 272439 \beta_{2} - 1308826 \beta_{3} - 557134 \beta_{4} + 738760 \beta_{5} ) q^{62} + ( -2215095 + 886784 \beta_{1} + 775936 \beta_{2} - 831360 \beta_{3} + 831360 \beta_{4} + 831360 \beta_{5} ) q^{64} + ( -462346 \beta_{1} - 888191 \beta_{2} - 462346 \beta_{3} - 231173 \beta_{4} - 231173 \beta_{5} ) q^{69} + ( -162273 \beta_{1} - 486819 \beta_{2} - 2091637 \beta_{3} + 162273 \beta_{4} + 162273 \beta_{5} ) q^{71} + ( 962688 - 909183 \beta_{1} - 1408640 \beta_{2} + 2106229 \beta_{3} + 889989 \beta_{4} - 264187 \beta_{5} ) q^{72} + ( -567739 \beta_{1} + 1900877 \beta_{2} - 666569 \beta_{3} + 666569 \beta_{4} + 666569 \beta_{5} ) q^{73} + ( -625000 \beta_{1} + 1796875 \beta_{2} - 625000 \beta_{3} - 312500 \beta_{4} - 312500 \beta_{5} ) q^{75} + ( 632883 + 1809423 \beta_{1} + 768617 \beta_{2} - 387718 \beta_{3} + 709042 \beta_{4} - 930441 \beta_{5} ) q^{78} + ( -1054174 \beta_{1} - 584543 \beta_{2} - 1054174 \beta_{3} - 527087 \beta_{4} - 527087 \beta_{5} ) q^{81} + ( 9883577 + 191983 \beta_{1} - 2598439 \beta_{2} + 1203228 \beta_{3} - 1203228 \beta_{4} - 1203228 \beta_{5} ) q^{82} + ( -5694927 + 895907 \beta_{1} - 2937713 \beta_{3} + 1784366 \beta_{4} + 257440 \beta_{5} ) q^{87} + ( -1557376 - 498847 \beta_{1} - 1496541 \beta_{2} + 4842466 \beta_{3} + 2056223 \beta_{4} - 1058529 \beta_{5} ) q^{92} + ( 6364821 + 1593521 \beta_{1} - 3720629 \beta_{3} - 144760 \beta_{4} + 2271868 \beta_{5} ) q^{93} + ( -14632267 + 971581 \beta_{1} + 3383963 \beta_{2} - 2177772 \beta_{3} + 2177772 \beta_{4} + 2177772 \beta_{5} ) q^{94} + ( -11520831 + 3922547 \beta_{1} + 4045952 \beta_{2} - 3839844 \beta_{3} + 4664709 \beta_{4} + 1570924 \beta_{5} ) q^{96} + ( -1647086 \beta_{1} - 4941258 \beta_{2} + 2470629 \beta_{3} + 1647086 \beta_{4} + 1647086 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 768q^{4} - 3147q^{6} + O(q^{10})$$ $$6q - 768q^{4} - 3147q^{6} - 21435q^{12} + 98304q^{16} - 72177q^{18} + 402816q^{24} - 468750q^{25} - 225276q^{27} + 1621587q^{36} - 2196456q^{39} - 866433q^{48} + 4941258q^{49} + 5680578q^{52} - 12795270q^{58} - 13290570q^{64} + 9238656q^{72} + 8715747q^{78} + 59301462q^{82} - 29588784q^{87} + 30939042q^{93} - 87793602q^{94} - 59843631q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{3} + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 3 \nu^{2} + 4 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{3} - \nu^{2} + 4 \nu - 4$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} - 4 \nu^{3} - \nu^{2} + 4 \nu + 4$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} - \beta_{4} - 2 \beta_{3} + 4 \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 2 \beta_{3} + 12 \beta_{2} + 2 \beta_{1}$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{5} - 7 \beta_{4} + 10 \beta_{3} + 4 \beta_{1}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$47$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 −1.07255 − 0.921756i 1.33454 + 0.467979i −0.261988 − 1.38973i −0.261988 + 1.38973i 1.33454 − 0.467979i −1.07255 + 0.921756i
21.8836i 5.84430 46.3988i −350.893 0 −1015.37 127.894i 0 4877.70i −2118.69 542.337i 0
68.2 15.9248i 37.2603 28.2607i −125.600 0 −450.047 593.365i 0 38.2170i 589.667 2106.01i 0
68.3 5.95879i −43.1046 18.1381i 92.4928 0 −108.081 + 256.851i 0 1313.87i 1529.02 + 1563.67i 0
68.4 5.95879i −43.1046 + 18.1381i 92.4928 0 −108.081 256.851i 0 1313.87i 1529.02 1563.67i 0
68.5 15.9248i 37.2603 + 28.2607i −125.600 0 −450.047 + 593.365i 0 38.2170i 589.667 + 2106.01i 0
68.6 21.8836i 5.84430 + 46.3988i −350.893 0 −1015.37 + 127.894i 0 4877.70i −2118.69 + 542.337i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 68.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
3.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.8.c.a 6
3.b odd 2 1 inner 69.8.c.a 6
23.b odd 2 1 CM 69.8.c.a 6
69.c even 2 1 inner 69.8.c.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.c.a 6 1.a even 1 1 trivial
69.8.c.a 6 3.b odd 2 1 inner
69.8.c.a 6 23.b odd 2 1 CM
69.8.c.a 6 69.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 768 T_{2}^{4} + 147456 T_{2}^{2} + 4312247$$ acting on $$S_{8}^{\mathrm{new}}(69, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4312247 + 147456 T^{2} + 768 T^{4} + T^{6}$$
$3$ $$10460353203 + 75092 T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$( 84809369662 - 188245551 T + T^{3} )^{2}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$( 3404825447 + T^{2} )^{3}$$
$29$ $$19\!\cdots\!92$$$$+$$$$26\!\cdots\!29$$$$T^{2} + 103499257854 T^{4} + T^{6}$$
$31$ $$( -7613427736873064 - 82537842333 T + T^{3} )^{2}$$
$37$ $$T^{6}$$
$41$ $$26\!\cdots\!88$$$$+$$$$34\!\cdots\!49$$$$T^{2} + 1168525643286 T^{4} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$39\!\cdots\!92$$$$+$$$$23\!\cdots\!21$$$$T^{2} + 3039738722778 T^{4} + T^{6}$$
$53$ $$T^{6}$$
$59$ $$( 9953995454012 + T^{2} )^{3}$$
$61$ $$T^{6}$$
$67$ $$T^{6}$$
$71$ $$27\!\cdots\!08$$$$+$$$$74\!\cdots\!29$$$$T^{2} + 54570720950346 T^{4} + T^{6}$$
$73$ $$( -19699186128365185562 - 33142195557291 T + T^{3} )^{2}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$