# Properties

 Label 69.4.c.b Level $69$ Weight $4$ Character orbit 69.c Analytic conductor $4.071$ Analytic rank $0$ Dimension $4$ CM discriminant -23 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 69.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.07113179040$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-23})$$ Defining polynomial: $$x^{4} - x^{3} - 5 x^{2} - 6 x + 36$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{3} + ( -4 + 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{4} + ( 4 + \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{6} + ( 9 + 8 \beta_{2} + 10 \beta_{3} ) q^{8} + ( 10 - 8 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} - \beta_{3} ) q^{2} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{3} + ( -4 + 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{4} + ( 4 + \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{6} + ( 9 + 8 \beta_{2} + 10 \beta_{3} ) q^{8} + ( 10 - 8 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{9} + ( 57 - 15 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{12} + ( -35 + 24 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} ) q^{13} + ( 75 - 6 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{16} + ( -7 - 7 \beta_{1} - 26 \beta_{2} - 31 \beta_{3} ) q^{18} + ( 23 + 46 \beta_{3} ) q^{23} + ( -25 - 4 \beta_{1} - 47 \beta_{2} - 48 \beta_{3} ) q^{24} -125 q^{25} + ( 87 + 83 \beta_{2} + 91 \beta_{3} ) q^{26} + ( -141 - 22 \beta_{3} ) q^{27} + ( -61 - 94 \beta_{2} - 28 \beta_{3} ) q^{29} + ( -173 - 12 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{31} + ( -16 - 23 \beta_{2} - 9 \beta_{3} ) q^{32} + ( -267 + 93 \beta_{1} + 33 \beta_{2} - 21 \beta_{3} ) q^{36} + ( 247 - 98 \beta_{1} + 5 \beta_{2} - 48 \beta_{3} ) q^{39} + ( -103 - 142 \beta_{2} - 64 \beta_{3} ) q^{41} + ( 253 - 138 \beta_{1} - 23 \beta_{2} - 69 \beta_{3} ) q^{46} + ( 59 - 16 \beta_{2} + 134 \beta_{3} ) q^{47} + ( -128 + 157 \beta_{1} + 65 \beta_{2} + 12 \beta_{3} ) q^{48} + 343 q^{49} + ( 125 + 125 \beta_{2} + 125 \beta_{3} ) q^{50} + ( 760 - 330 \beta_{1} - 55 \beta_{2} - 165 \beta_{3} ) q^{52} + ( 9 + 66 \beta_{1} + 141 \beta_{2} + 163 \beta_{3} ) q^{54} + ( -765 + 366 \beta_{1} + 61 \beta_{2} + 183 \beta_{3} ) q^{58} + ( -170 - 340 \beta_{3} ) q^{59} + ( 147 + 149 \beta_{2} + 145 \beta_{3} ) q^{62} + ( 401 + 48 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{64} + ( 161 + 92 \beta_{1} - 161 \beta_{2} ) q^{69} + ( 173 + 392 \beta_{2} - 46 \beta_{3} ) q^{71} + ( 130 + 94 \beta_{1} + 245 \beta_{2} + 286 \beta_{3} ) q^{72} + ( -605 + 96 \beta_{1} + 16 \beta_{2} + 48 \beta_{3} ) q^{73} + ( 125 - 250 \beta_{1} - 125 \beta_{2} ) q^{75} + ( -304 - 67 \beta_{1} - 443 \beta_{2} - 498 \beta_{3} ) q^{78} + ( 53 - 304 \beta_{1} - 53 \beta_{2} ) q^{81} + ( -1275 + 618 \beta_{1} + 103 \beta_{2} + 309 \beta_{3} ) q^{82} + ( 607 + 226 \beta_{1} + 239 \beta_{2} + 564 \beta_{3} ) q^{87} + ( -368 - 529 \beta_{2} - 207 \beta_{3} ) q^{92} + ( 67 - 332 \beta_{1} - 193 \beta_{2} + 24 \beta_{3} ) q^{93} + ( 633 - 354 \beta_{1} - 59 \beta_{2} - 177 \beta_{3} ) q^{94} + ( 141 + 51 \beta_{1} + 66 \beta_{2} + 138 \beta_{3} ) q^{96} + ( -343 - 343 \beta_{2} - 343 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 18q^{4} - 5q^{6} + 38q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 18q^{4} - 5q^{6} + 38q^{9} + 225q^{12} - 148q^{13} + 302q^{16} + 79q^{18} + 86q^{24} - 500q^{25} - 520q^{27} - 688q^{31} - 999q^{36} + 976q^{39} + 1058q^{46} - 509q^{48} + 1372q^{49} + 3150q^{52} - 506q^{54} - 3182q^{58} + 1588q^{64} + 1058q^{69} - 448q^{72} - 2452q^{73} + 500q^{75} + 599q^{78} + 14q^{81} - 5306q^{82} + 1048q^{87} + 274q^{93} + 2650q^{94} + 207q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5 x^{2} - 6 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu^{2} - 5 \nu - 26$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 6$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$-5 \beta_{3} + 6$$

## 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
 −1.82666 − 1.63197i 2.32666 + 0.765945i 2.32666 − 0.765945i −1.82666 + 1.63197i
4.99599i −5.15331 0.665865i −16.9599 0 −3.32666 + 25.7459i 0 44.7638i 26.1132 + 6.86282i 0
68.2 0.200160i 3.15331 + 4.12997i 7.95994 0 0.826656 0.631168i 0 3.19455i −7.11325 + 26.0461i 0
68.3 0.200160i 3.15331 4.12997i 7.95994 0 0.826656 + 0.631168i 0 3.19455i −7.11325 26.0461i 0
68.4 4.99599i −5.15331 + 0.665865i −16.9599 0 −3.32666 25.7459i 0 44.7638i 26.1132 6.86282i 0
 $$n$$: e.g. 2-40 or 990-1000 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 69.4.c.b 4
3.b odd 2 1 inner 69.4.c.b 4
23.b odd 2 1 CM 69.4.c.b 4
69.c even 2 1 inner 69.4.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.c.b 4 1.a even 1 1 trivial
69.4.c.b 4 3.b odd 2 1 inner
69.4.c.b 4 23.b odd 2 1 CM
69.4.c.b 4 69.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 25 T_{2}^{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(69, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 25 T^{2} + T^{4}$$
$3$ $$729 + 108 T - 11 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( -1115 + 74 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 12167 + T^{2} )^{2}$$
$29$ $$3039868225 + 128302 T^{2} + T^{4}$$
$31$ $$( 28963 + 344 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$12668628025 + 319318 T^{2} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$10306107361 + 209950 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 664700 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$1050758254225 + 2098798 T^{2} + T^{4}$$
$73$ $$( 336025 + 1226 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$