Properties

 Label 671.2.a.b Level $671$ Weight $2$ Character orbit 671.a Self dual yes Analytic conductor $5.358$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$671 = 11 \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 671.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$5.35796197563$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.2661761.1 Defining polynomial: $$x^{6} - x^{5} - 6 x^{4} + 3 x^{3} + 9 x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 6 x^{4} + 3 x^{3} + 9 x^{2} - x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 5 \nu + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 5 \nu^{3} + 2 \nu^{2} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + \beta_{4} + 7 \beta_{3} + 8 \beta_{2} + 19 \beta_{1} + 8$$

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
 1.58234 −1.68584 0.387870 −1.34697 2.36588 −0.303283
−2.28896 −1.58234 3.23932 0.631975 3.62191 0.582341 −2.83675 −0.496198 −1.44656
1.2 −1.57580 1.68584 0.483130 −0.593176 −2.65654 −2.68584 2.39028 −0.157941 0.934724
1.3 −0.255699 −0.387870 −1.93462 2.57819 0.0991779 −0.612130 1.00608 −2.84956 −0.659240
1.4 0.782747 1.34697 −1.38731 −0.742408 1.05434 −2.34697 −2.65140 −1.18568 −0.581118
1.5 1.54773 −2.36588 0.395474 0.422675 −3.66175 1.36588 −2.48338 2.59739 0.654188
1.6 1.78997 0.303283 1.20400 −3.29725 0.542868 −1.30328 −1.42482 −2.90802 −5.90199
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$61$$ $$1$$

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 671.2.a.b 6
3.b odd 2 1 6039.2.a.b 6
11.b odd 2 1 7381.2.a.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.b 6 1.a even 1 1 trivial
6039.2.a.b 6 3.b odd 2 1
7381.2.a.h 6 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 7 T_{2}^{4} + 2 T_{2}^{3} + 12 T_{2}^{2} - 5 T_{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(671))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 - 5 T + 12 T^{2} + 2 T^{3} - 7 T^{4} + T^{6}$$
$3$ $$-1 + T + 9 T^{2} - 3 T^{3} - 6 T^{4} + T^{5} + T^{6}$$
$5$ $$-1 + T + 6 T^{2} - 3 T^{3} - 9 T^{4} + T^{5} + T^{6}$$
$7$ $$4 + 3 T - 13 T^{2} - 11 T^{3} + 4 T^{4} + 5 T^{5} + T^{6}$$
$11$ $$( 1 + T )^{6}$$
$13$ $$2 + 55 T - 36 T^{2} - 63 T^{3} - 12 T^{4} + 4 T^{5} + T^{6}$$
$17$ $$-926 + 563 T + 328 T^{2} - 143 T^{3} - 39 T^{4} + 5 T^{5} + T^{6}$$
$19$ $$-2 - 17 T - 43 T^{2} - 40 T^{3} - 9 T^{4} + 3 T^{5} + T^{6}$$
$23$ $$131 + 298 T + 136 T^{2} - 74 T^{3} - 43 T^{4} + 3 T^{5} + T^{6}$$
$29$ $$-2 + 7 T + 7 T^{2} - 12 T^{3} - 9 T^{4} + T^{5} + T^{6}$$
$31$ $$7 - 124 T - 142 T^{2} - 21 T^{3} + 25 T^{4} + 10 T^{5} + T^{6}$$
$37$ $$-2993 - 15211 T - 8111 T^{2} - 1233 T^{3} + 32 T^{4} + 19 T^{5} + T^{6}$$
$41$ $$724 + 1331 T + 497 T^{2} - 204 T^{3} - 89 T^{4} + 7 T^{5} + T^{6}$$
$43$ $$-2 - 299 T + 108 T^{2} + 155 T^{3} - 74 T^{4} + 2 T^{5} + T^{6}$$
$47$ $$-26800 - 7745 T + 2959 T^{2} + 469 T^{3} - 110 T^{4} - 5 T^{5} + T^{6}$$
$53$ $$-1714 + 6301 T + 2143 T^{2} - 579 T^{3} - 82 T^{4} + 9 T^{5} + T^{6}$$
$59$ $$-4793 + 782 T + 1176 T^{2} - 174 T^{3} - 59 T^{4} + 5 T^{5} + T^{6}$$
$61$ $$( 1 + T )^{6}$$
$67$ $$-27503 - 31690 T - 12473 T^{2} - 1956 T^{3} - 57 T^{4} + 14 T^{5} + T^{6}$$
$71$ $$-10877 + 6679 T + 911 T^{2} - 726 T^{3} - 56 T^{4} + 14 T^{5} + T^{6}$$
$73$ $$467006 + 186509 T + 5248 T^{2} - 3277 T^{3} - 200 T^{4} + 14 T^{5} + T^{6}$$
$79$ $$-80764 - 24793 T + 7457 T^{2} + 726 T^{3} - 163 T^{4} - 5 T^{5} + T^{6}$$
$83$ $$-59872 - 71253 T + 8471 T^{2} + 2774 T^{3} - 171 T^{4} - 17 T^{5} + T^{6}$$
$89$ $$-547 + 1298 T - 740 T^{2} - 144 T^{3} + 115 T^{4} + 25 T^{5} + T^{6}$$
$97$ $$24269 + 7232 T - 7791 T^{2} - 1267 T^{3} + 96 T^{4} + 24 T^{5} + T^{6}$$