# Properties

 Label 169.2.c.c Level $169$ Weight $2$ Character orbit 169.c Analytic conductor $1.349$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{4} ) q^{2} + ( 1 - \beta_{4} - \beta_{5} ) q^{3} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{7} + ( -1 - \beta_{2} - \beta_{3} ) q^{8} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{4} ) q^{2} + ( 1 - \beta_{4} - \beta_{5} ) q^{3} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{7} + ( -1 - \beta_{2} - \beta_{3} ) q^{8} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{9} + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{10} + ( 3 - \beta_{4} - 3 \beta_{5} ) q^{11} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{12} + ( -1 - 2 \beta_{2} ) q^{14} + ( 1 - \beta_{1} - \beta_{5} ) q^{15} + ( -1 + \beta_{1} + \beta_{5} ) q^{16} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{17} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{18} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{5} ) q^{20} + ( 2 - \beta_{2} + 3 \beta_{3} ) q^{21} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{22} + ( 1 + 2 \beta_{4} - \beta_{5} ) q^{23} + ( -4 + \beta_{1} + 2 \beta_{4} + 4 \beta_{5} ) q^{24} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{25} + ( 2 \beta_{2} + \beta_{3} ) q^{27} + ( -4 - \beta_{1} - \beta_{4} + 4 \beta_{5} ) q^{28} + ( -2 \beta_{1} + 3 \beta_{4} ) q^{29} + ( -\beta_{1} + \beta_{2} ) q^{30} + ( -2 - 2 \beta_{2} - 3 \beta_{3} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{32} + ( -\beta_{1} + \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} ) q^{33} + ( 2 + 5 \beta_{2} - 2 \beta_{3} ) q^{34} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{35} + ( 1 + 4 \beta_{1} - \beta_{5} ) q^{36} + ( -4 - \beta_{1} + \beta_{4} + 4 \beta_{5} ) q^{37} + ( 1 + 6 \beta_{2} - 3 \beta_{3} ) q^{38} + ( -2 + 3 \beta_{2} ) q^{40} + ( 5 - 2 \beta_{1} - 6 \beta_{4} - 5 \beta_{5} ) q^{41} + ( -3 + \beta_{4} + 3 \beta_{5} ) q^{42} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{43} + ( 3 - 5 \beta_{2} + 4 \beta_{3} ) q^{44} + ( 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{45} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{46} + ( -7 + 2 \beta_{2} - \beta_{3} ) q^{47} + ( \beta_{3} + \beta_{4} ) q^{48} + ( 1 + \beta_{1} - \beta_{5} ) q^{49} + ( 2 - 4 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} ) q^{50} + ( -2 + 2 \beta_{2} - 3 \beta_{3} ) q^{51} + ( -3 + 3 \beta_{2} - 7 \beta_{3} ) q^{53} + ( -1 + 4 \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{54} + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( -2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{56} + ( 1 + \beta_{2} - \beta_{3} ) q^{57} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{58} + ( -4 \beta_{1} + 4 \beta_{2} + 5 \beta_{5} ) q^{59} + ( 2 - \beta_{3} ) q^{60} + ( 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} ) q^{61} + ( 3 - 6 \beta_{1} - 4 \beta_{4} - 3 \beta_{5} ) q^{62} + ( 1 + 3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{63} + ( -6 \beta_{2} + 5 \beta_{3} ) q^{64} + ( 4 - 3 \beta_{2} + 4 \beta_{3} ) q^{66} + ( -1 - \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{67} + ( 4 + 6 \beta_{1} + 3 \beta_{4} - 4 \beta_{5} ) q^{68} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{69} + ( -1 + 4 \beta_{2} - \beta_{3} ) q^{70} + ( 3 \beta_{1} - 3 \beta_{2} + 10 \beta_{5} ) q^{71} + ( 5 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{72} + ( 7 - 9 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -6 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{74} + ( -7 + 2 \beta_{1} + 4 \beta_{4} + 7 \beta_{5} ) q^{75} + ( 3 + 7 \beta_{1} + 5 \beta_{4} - 3 \beta_{5} ) q^{76} + ( 6 - 5 \beta_{2} + 5 \beta_{3} ) q^{77} + ( 4 - 9 \beta_{2} + 8 \beta_{3} ) q^{79} + ( 2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{80} + ( 4 - 4 \beta_{1} - 7 \beta_{4} - 4 \beta_{5} ) q^{81} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{82} + ( -4 + 7 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{84} + ( 4 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( -3 - 5 \beta_{2} + 4 \beta_{3} ) q^{86} + ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 8 \beta_{5} ) q^{87} + ( -6 - \beta_{1} + 4 \beta_{4} + 6 \beta_{5} ) q^{88} + ( 6 - 7 \beta_{1} - 6 \beta_{5} ) q^{89} + ( -2 - 3 \beta_{2} + 3 \beta_{3} ) q^{90} + ( 1 - 4 \beta_{2} - \beta_{3} ) q^{92} + ( -10 + 3 \beta_{1} + 5 \beta_{4} + 10 \beta_{5} ) q^{93} + ( 1 - 3 \beta_{1} - 5 \beta_{4} - \beta_{5} ) q^{94} + ( 5 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{95} + ( -9 + 2 \beta_{2} - 4 \beta_{3} ) q^{96} + ( 6 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{97} + ( 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{98} + ( -3 + 4 \beta_{2} - 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{2} + 2q^{3} - 8q^{5} + q^{6} + 3q^{7} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$6q + 2q^{2} + 2q^{3} - 8q^{5} + q^{6} + 3q^{7} - 6q^{8} + 3q^{9} - 5q^{10} + 8q^{11} - 10q^{14} + 2q^{15} - 2q^{16} + 2q^{17} + 18q^{18} + 4q^{19} + 4q^{21} - 3q^{22} + 5q^{23} - 9q^{24} - 10q^{25} + 2q^{27} - 14q^{28} + q^{29} + q^{30} - 10q^{31} - 7q^{32} - 10q^{33} + 26q^{34} - 4q^{35} + 7q^{36} - 12q^{37} + 24q^{38} - 6q^{40} + 7q^{41} - 8q^{42} - 13q^{43} - 11q^{45} - 8q^{46} - 36q^{47} - q^{48} + 4q^{49} - q^{50} - 2q^{51} + 2q^{53} + 3q^{54} - 6q^{55} + 4q^{56} + 10q^{57} - 3q^{58} + 19q^{59} + 14q^{60} - 4q^{61} - q^{62} + 4q^{63} - 22q^{64} + 10q^{66} + q^{67} + 21q^{68} + 6q^{69} + 4q^{70} + 27q^{71} + 4q^{72} + 18q^{73} + 8q^{74} - 15q^{75} + 21q^{76} + 16q^{77} - 10q^{79} + 5q^{80} + q^{81} + 14q^{82} - 14q^{83} + 7q^{84} - 5q^{85} - 36q^{86} + 18q^{87} - 15q^{88} + 11q^{89} - 24q^{90} - 22q^{93} - 5q^{94} - 3q^{95} - 42q^{96} - 7q^{97} - 5q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.222521 − 0.385418i −0.623490 + 1.07992i 0.900969 − 1.56052i 0.222521 + 0.385418i −0.623490 − 1.07992i 0.900969 + 1.56052i
−0.400969 + 0.694498i 1.12349 1.94594i 0.678448 + 1.17511i 0.246980 0.900969 + 1.56052i 1.17845 + 2.04113i −2.69202 −1.02446 1.77441i −0.0990311 + 0.171527i
22.2 0.277479 0.480608i −0.400969 + 0.694498i 0.846011 + 1.46533i −2.80194 0.222521 + 0.385418i 1.34601 + 2.33136i 2.04892 1.17845 + 2.04113i −0.777479 + 1.34663i
22.3 1.12349 1.94594i 0.277479 0.480608i −1.52446 2.64044i −1.44504 −0.623490 1.07992i −1.02446 1.77441i −2.35690 1.34601 + 2.33136i −1.62349 + 2.81197i
146.1 −0.400969 0.694498i 1.12349 + 1.94594i 0.678448 1.17511i 0.246980 0.900969 1.56052i 1.17845 2.04113i −2.69202 −1.02446 + 1.77441i −0.0990311 0.171527i
146.2 0.277479 + 0.480608i −0.400969 0.694498i 0.846011 1.46533i −2.80194 0.222521 0.385418i 1.34601 2.33136i 2.04892 1.17845 2.04113i −0.777479 1.34663i
146.3 1.12349 + 1.94594i 0.277479 + 0.480608i −1.52446 + 2.64044i −1.44504 −0.623490 + 1.07992i −1.02446 + 1.77441i −2.35690 1.34601 2.33136i −1.62349 2.81197i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 146.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.c.c 6
13.b even 2 1 169.2.c.b 6
13.c even 3 1 169.2.a.b 3
13.c even 3 1 inner 169.2.c.c 6
13.d odd 4 2 169.2.e.b 12
13.e even 6 1 169.2.a.c yes 3
13.e even 6 1 169.2.c.b 6
13.f odd 12 2 169.2.b.b 6
13.f odd 12 2 169.2.e.b 12
39.h odd 6 1 1521.2.a.o 3
39.i odd 6 1 1521.2.a.r 3
39.k even 12 2 1521.2.b.l 6
52.i odd 6 1 2704.2.a.ba 3
52.j odd 6 1 2704.2.a.z 3
52.l even 12 2 2704.2.f.o 6
65.l even 6 1 4225.2.a.bb 3
65.n even 6 1 4225.2.a.bg 3
91.n odd 6 1 8281.2.a.bf 3
91.t odd 6 1 8281.2.a.bj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 13.c even 3 1
169.2.a.c yes 3 13.e even 6 1
169.2.b.b 6 13.f odd 12 2
169.2.c.b 6 13.b even 2 1
169.2.c.b 6 13.e even 6 1
169.2.c.c 6 1.a even 1 1 trivial
169.2.c.c 6 13.c even 3 1 inner
169.2.e.b 12 13.d odd 4 2
169.2.e.b 12 13.f odd 12 2
1521.2.a.o 3 39.h odd 6 1
1521.2.a.r 3 39.i odd 6 1
1521.2.b.l 6 39.k even 12 2
2704.2.a.z 3 52.j odd 6 1
2704.2.a.ba 3 52.i odd 6 1
2704.2.f.o 6 52.l even 12 2
4225.2.a.bb 3 65.l even 6 1
4225.2.a.bg 3 65.n even 6 1
8281.2.a.bf 3 91.n odd 6 1
8281.2.a.bj 3 91.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 2 T_{2}^{5} + 5 T_{2}^{4} + 3 T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} + 5 T^{4} - 2 T^{5} + T^{6}$$
$3$ $$1 - T + 3 T^{2} + 5 T^{4} - 2 T^{5} + T^{6}$$
$5$ $$( -1 + 3 T + 4 T^{2} + T^{3} )^{2}$$
$7$ $$169 - 52 T + 55 T^{2} - 14 T^{3} + 13 T^{4} - 3 T^{5} + T^{6}$$
$11$ $$169 - 247 T + 257 T^{2} - 126 T^{3} + 45 T^{4} - 8 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$169 + 195 T + 199 T^{2} + 56 T^{3} + 19 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$1 - 11 T + 125 T^{2} + 42 T^{3} + 27 T^{4} - 4 T^{5} + T^{6}$$
$23$ $$169 - 13 T + 66 T^{2} - 21 T^{3} + 26 T^{4} - 5 T^{5} + T^{6}$$
$29$ $$6889 + 3652 T + 1853 T^{2} + 210 T^{3} + 45 T^{4} - T^{5} + T^{6}$$
$31$ $$( -167 - 36 T + 5 T^{2} + T^{3} )^{2}$$
$37$ $$841 + 1189 T + 1333 T^{2} + 434 T^{3} + 103 T^{4} + 12 T^{5} + T^{6}$$
$41$ $$2401 + 2401 T + 2058 T^{2} + 441 T^{3} + 98 T^{4} - 7 T^{5} + T^{6}$$
$43$ $$169 - 520 T + 1769 T^{2} + 546 T^{3} + 129 T^{4} + 13 T^{5} + T^{6}$$
$47$ $$( 167 + 101 T + 18 T^{2} + T^{3} )^{2}$$
$53$ $$( 337 - 86 T - T^{2} + T^{3} )^{2}$$
$59$ $$1 - 83 T + 6870 T^{2} - 1575 T^{3} + 278 T^{4} - 19 T^{5} + T^{6}$$
$61$ $$57121 + 16013 T + 5445 T^{2} + 210 T^{3} + 83 T^{4} + 4 T^{5} + T^{6}$$
$67$ $$1681 + 2952 T + 5143 T^{2} + 154 T^{3} + 73 T^{4} - T^{5} + T^{6}$$
$71$ $$299209 - 121434 T + 34515 T^{2} - 4900 T^{3} + 507 T^{4} - 27 T^{5} + T^{6}$$
$73$ $$( 911 - 120 T - 9 T^{2} + T^{3} )^{2}$$
$79$ $$( 127 - 162 T + 5 T^{2} + T^{3} )^{2}$$
$83$ $$( 203 - 140 T + 7 T^{2} + T^{3} )^{2}$$
$89$ $$78961 - 20794 T + 8567 T^{2} + 252 T^{3} + 195 T^{4} - 11 T^{5} + T^{6}$$
$97$ $$90601 + 25284 T + 9163 T^{2} + 14 T^{3} + 133 T^{4} + 7 T^{5} + T^{6}$$