# Properties

 Label 273.2.c.c Level $273$ Weight $2$ Character orbit 273.c Analytic conductor $2.180$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 15 x^{6} + 67 x^{4} + 77 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( -2 + \beta_{2} ) q^{4} + ( -\beta_{3} + \beta_{7} ) q^{5} -\beta_{1} q^{6} -\beta_{3} q^{7} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( -2 + \beta_{2} ) q^{4} + ( -\beta_{3} + \beta_{7} ) q^{5} -\beta_{1} q^{6} -\beta_{3} q^{7} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{8} + q^{9} + ( 1 - 3 \beta_{4} - \beta_{6} ) q^{10} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + ( 2 - \beta_{2} ) q^{12} + ( -1 - \beta_{1} - \beta_{3} - \beta_{6} ) q^{13} -\beta_{4} q^{14} + ( \beta_{3} - \beta_{7} ) q^{15} + ( 5 - \beta_{2} - \beta_{4} + \beta_{6} ) q^{16} + ( 2 + 2 \beta_{4} ) q^{17} + \beta_{1} q^{18} + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{7} ) q^{19} + ( \beta_{1} + 11 \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{20} + \beta_{3} q^{21} + ( -3 + \beta_{4} + \beta_{6} ) q^{22} + ( \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{23} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{24} + ( -5 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{25} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{26} - q^{27} + ( 2 \beta_{3} - \beta_{7} ) q^{28} + ( -2 - \beta_{2} + \beta_{4} + \beta_{6} ) q^{29} + ( -1 + 3 \beta_{4} + \beta_{6} ) q^{30} + ( \beta_{1} - \beta_{5} + \beta_{7} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{32} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{33} + ( 2 \beta_{1} - 8 \beta_{3} + 2 \beta_{7} ) q^{34} + ( -1 + \beta_{2} ) q^{35} + ( -2 + \beta_{2} ) q^{36} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{37} + ( 6 - 2 \beta_{2} - 4 \beta_{4} ) q^{38} + ( 1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{39} + ( -3 + 9 \beta_{4} + \beta_{6} ) q^{40} + ( \beta_{1} - 7 \beta_{3} - \beta_{5} ) q^{41} + \beta_{4} q^{42} + ( 4 + \beta_{2} - \beta_{4} - \beta_{6} ) q^{43} + ( -\beta_{1} - 3 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{44} + ( -\beta_{3} + \beta_{7} ) q^{45} + ( -2 \beta_{1} + 10 \beta_{3} - 2 \beta_{7} ) q^{46} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{47} + ( -5 + \beta_{2} + \beta_{4} - \beta_{6} ) q^{48} - q^{49} + ( -7 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{50} + ( -2 - 2 \beta_{4} ) q^{51} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{52} + ( -2 - \beta_{2} + \beta_{4} - 3 \beta_{6} ) q^{53} -\beta_{1} q^{54} + ( 2 - 2 \beta_{2} - 2 \beta_{6} ) q^{55} + ( -1 + 2 \beta_{4} + \beta_{6} ) q^{56} + ( \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{57} + ( -4 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{58} + ( \beta_{1} + 7 \beta_{3} - \beta_{5} ) q^{59} + ( -\beta_{1} - 11 \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{60} + ( -2 + 2 \beta_{6} ) q^{61} + ( -4 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{62} -\beta_{3} q^{63} + ( -2 + \beta_{2} + 2 \beta_{6} ) q^{64} + ( -2 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{65} + ( 3 - \beta_{4} - \beta_{6} ) q^{66} + ( -4 \beta_{1} - 6 \beta_{3} - 2 \beta_{7} ) q^{67} + ( -2 + 2 \beta_{2} - 8 \beta_{4} - 2 \beta_{6} ) q^{68} + ( -\beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{69} + ( -3 \beta_{1} - \beta_{3} + \beta_{5} ) q^{70} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{71} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{72} + ( -\beta_{1} - 8 \beta_{3} - \beta_{5} - \beta_{7} ) q^{73} + ( 6 - 2 \beta_{4} - 2 \beta_{6} ) q^{74} + ( 5 - \beta_{2} - \beta_{4} + \beta_{6} ) q^{75} + ( 8 \beta_{1} + 14 \beta_{3} - 2 \beta_{7} ) q^{76} + ( 1 - \beta_{4} + \beta_{6} ) q^{77} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{78} + ( \beta_{2} - \beta_{4} + \beta_{6} ) q^{79} + ( -\beta_{1} - 15 \beta_{3} + \beta_{5} + 6 \beta_{7} ) q^{80} + q^{81} + ( -5 + 2 \beta_{2} - 7 \beta_{4} - \beta_{6} ) q^{82} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{83} + ( -2 \beta_{3} + \beta_{7} ) q^{84} + ( 6 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{85} + ( 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{86} + ( 2 + \beta_{2} - \beta_{4} - \beta_{6} ) q^{87} + ( 1 - 2 \beta_{2} - 5 \beta_{4} + \beta_{6} ) q^{88} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{89} + ( 1 - 3 \beta_{4} - \beta_{6} ) q^{90} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{91} + ( 6 + 8 \beta_{4} + 4 \beta_{6} ) q^{92} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{93} + ( -11 + 4 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{94} + ( -12 + \beta_{2} + 7 \beta_{4} - \beta_{6} ) q^{95} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{96} + ( -\beta_{1} + 10 \beta_{3} - \beta_{5} + \beta_{7} ) q^{97} -\beta_{1} q^{98} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{3} - 14q^{4} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{3} - 14q^{4} + 8q^{9} + 4q^{10} + 14q^{12} - 6q^{13} - 2q^{14} + 34q^{16} + 20q^{17} - 24q^{22} - 6q^{23} - 34q^{25} + 28q^{26} - 8q^{27} - 18q^{29} - 4q^{30} - 6q^{35} - 14q^{36} + 36q^{38} + 6q^{39} - 8q^{40} + 2q^{42} + 34q^{43} - 34q^{48} - 8q^{49} - 20q^{51} + 18q^{52} - 10q^{53} + 16q^{55} - 6q^{56} - 20q^{61} - 28q^{62} - 18q^{64} - 10q^{65} + 24q^{66} - 24q^{68} + 6q^{69} + 48q^{74} + 34q^{75} + 4q^{77} - 28q^{78} - 2q^{79} + 8q^{81} - 48q^{82} + 18q^{87} - 8q^{88} + 4q^{90} - 6q^{91} + 56q^{92} - 72q^{94} - 78q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 15 x^{6} + 67 x^{4} + 77 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 8 \nu^{3} + 9 \nu$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{4} + 9 \nu^{2}$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + 10 \nu^{3} + 21 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 10 \nu^{4} + 23 \nu^{2} + 6$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 12 \nu^{5} + 41 \nu^{3} + 36 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - \beta_{3} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{4} - 7 \beta_{2} + 25$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 10 \beta_{3} + 39 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-8 \beta_{6} + 10 \beta_{4} + 47 \beta_{2} - 164$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{7} + 55 \beta_{5} - 79 \beta_{3} - 258 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 2.60520i − 2.54814i − 1.29051i − 0.233455i 0.233455i 1.29051i 2.54814i 2.60520i
2.60520i −1.00000 −4.78706 3.78706i 2.60520i 1.00000i 7.26084i 1.00000 9.86604
64.2 2.54814i −1.00000 −4.49301 3.49301i 2.54814i 1.00000i 6.35254i 1.00000 −8.90068
64.3 1.29051i −1.00000 0.334573 1.33457i 1.29051i 1.00000i 3.01280i 1.00000 1.72229
64.4 0.233455i −1.00000 1.94550 2.94550i 0.233455i 1.00000i 0.921097i 1.00000 −0.687642
64.5 0.233455i −1.00000 1.94550 2.94550i 0.233455i 1.00000i 0.921097i 1.00000 −0.687642
64.6 1.29051i −1.00000 0.334573 1.33457i 1.29051i 1.00000i 3.01280i 1.00000 1.72229
64.7 2.54814i −1.00000 −4.49301 3.49301i 2.54814i 1.00000i 6.35254i 1.00000 −8.90068
64.8 2.60520i −1.00000 −4.78706 3.78706i 2.60520i 1.00000i 7.26084i 1.00000 9.86604
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.8 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.c.c 8
3.b odd 2 1 819.2.c.d 8
4.b odd 2 1 4368.2.h.q 8
7.b odd 2 1 1911.2.c.l 8
13.b even 2 1 inner 273.2.c.c 8
13.d odd 4 1 3549.2.a.v 4
13.d odd 4 1 3549.2.a.x 4
39.d odd 2 1 819.2.c.d 8
52.b odd 2 1 4368.2.h.q 8
91.b odd 2 1 1911.2.c.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.c 8 1.a even 1 1 trivial
273.2.c.c 8 13.b even 2 1 inner
819.2.c.d 8 3.b odd 2 1
819.2.c.d 8 39.d odd 2 1
1911.2.c.l 8 7.b odd 2 1
1911.2.c.l 8 91.b odd 2 1
3549.2.a.v 4 13.d odd 4 1
3549.2.a.x 4 13.d odd 4 1
4368.2.h.q 8 4.b odd 2 1
4368.2.h.q 8 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 15 T_{2}^{6} + 67 T_{2}^{4} + 77 T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 77 T^{2} + 67 T^{4} + 15 T^{6} + T^{8}$$
$3$ $$( 1 + T )^{8}$$
$5$ $$2704 + 2240 T^{2} + 468 T^{4} + 37 T^{6} + T^{8}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$1024 + 2320 T^{2} + 576 T^{4} + 44 T^{6} + T^{8}$$
$13$ $$28561 + 13182 T - 1014 T^{3} - 306 T^{4} - 78 T^{5} + 6 T^{7} + T^{8}$$
$17$ $$( -160 + 112 T + 8 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$19$ $$1024 + 10752 T^{2} + 1984 T^{4} + 97 T^{6} + T^{8}$$
$23$ $$( 1352 - 140 T - 78 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$29$ $$( -440 - 244 T - 14 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$31$ $$4096 + 6912 T^{2} + 2032 T^{4} + 89 T^{6} + T^{8}$$
$37$ $$262144 + 148480 T^{2} + 9216 T^{4} + 176 T^{6} + T^{8}$$
$41$ $$1784896 + 324304 T^{2} + 17376 T^{4} + 248 T^{6} + T^{8}$$
$43$ $$( -896 + 160 T + 64 T^{2} - 17 T^{3} + T^{4} )^{2}$$
$47$ $$6739216 + 605408 T^{2} + 18984 T^{4} + 241 T^{6} + T^{8}$$
$53$ $$( 712 - 356 T - 142 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$59$ $$1364224 + 365968 T^{2} + 18384 T^{4} + 248 T^{6} + T^{8}$$
$61$ $$( 320 - 136 T - 28 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$67$ $$66064384 + 3824640 T^{2} + 67072 T^{4} + 452 T^{6} + T^{8}$$
$71$ $$1024 + 2320 T^{2} + 576 T^{4} + 44 T^{6} + T^{8}$$
$73$ $$5234944 + 595776 T^{2} + 22384 T^{4} + 305 T^{6} + T^{8}$$
$79$ $$( -80 - 104 T - 32 T^{2} + T^{3} + T^{4} )^{2}$$
$83$ $$6739216 + 605408 T^{2} + 18984 T^{4} + 241 T^{6} + T^{8}$$
$89$ $$150544 + 379392 T^{2} + 19060 T^{4} + 253 T^{6} + T^{8}$$
$97$ $$82882816 + 4847616 T^{2} + 87232 T^{4} + 553 T^{6} + T^{8}$$