# Properties

 Label 6422.2.a.bg Level $6422$ Weight $2$ Character orbit 6422.a Self dual yes Analytic conductor $51.280$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6422 = 2 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6422.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.2799281781$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 4 x^{6} + 22 x^{5} + 3 x^{4} - 28 x^{3} + 7 x^{2} + 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 494) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 4 x^{6} + 22 x^{5} + 3 x^{4} - 28 x^{3} + 7 x^{2} + 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} - 11 \nu^{5} + 12 \nu^{4} + 36 \nu^{3} - 7 \nu^{2} - 19 \nu + 1$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} - 7 \nu^{6} - 10 \nu^{5} + 33 \nu^{4} + 18 \nu^{3} - 20 \nu^{2} + 4 \nu - 4$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} + 14 \nu^{6} + 23 \nu^{5} - 78 \nu^{4} - 45 \nu^{3} + 91 \nu^{2} + \nu - 19$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{7} + 26 \nu^{6} + 35 \nu^{5} - 141 \nu^{4} - 66 \nu^{3} + 169 \nu^{2} + 19 \nu - 34$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 28 \nu^{6} + 46 \nu^{5} - 153 \nu^{4} - 99 \nu^{3} + 170 \nu^{2} + 26 \nu - 29$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$16 \nu^{7} - 56 \nu^{6} - 92 \nu^{5} + 306 \nu^{4} + 201 \nu^{3} - 349 \nu^{2} - 58 \nu + 70$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + 2 \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{2} + 8 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$-10 \beta_{7} - 6 \beta_{6} - 13 \beta_{5} - 2 \beta_{4} + 13 \beta_{2} + 24 \beta_{1} + 11$$ $$\nu^{5}$$ $$=$$ $$-29 \beta_{7} - 10 \beta_{6} - 44 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 44 \beta_{2} + 83 \beta_{1} + 25$$ $$\nu^{6}$$ $$=$$ $$-108 \beta_{7} - 38 \beta_{6} - 159 \beta_{5} - 34 \beta_{4} + 7 \beta_{3} + 161 \beta_{2} + 270 \beta_{1} + 91$$ $$\nu^{7}$$ $$=$$ $$-350 \beta_{7} - 85 \beta_{6} - 545 \beta_{5} - 121 \beta_{4} + 36 \beta_{3} + 552 \beta_{2} + 910 \beta_{1} + 266$$

## 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
 3.36639 2.47489 0.898616 0.533995 0.394661 −0.516521 −1.40354 −1.74849
−1.00000 −3.36639 1.00000 −0.659053 3.36639 1.55483 −1.00000 8.33259 0.659053
1.2 −1.00000 −2.47489 1.00000 2.86363 2.47489 1.77068 −1.00000 3.12506 −2.86363
1.3 −1.00000 −0.898616 1.00000 0.684128 0.898616 −3.14167 −1.00000 −2.19249 −0.684128
1.4 −1.00000 −0.533995 1.00000 −2.79724 0.533995 0.904890 −1.00000 −2.71485 2.79724
1.5 −1.00000 −0.394661 1.00000 0.507765 0.394661 1.24955 −1.00000 −2.84424 −0.507765
1.6 −1.00000 0.516521 1.00000 −1.53223 −0.516521 3.77281 −1.00000 −2.73321 1.53223
1.7 −1.00000 1.40354 1.00000 0.425845 −1.40354 −2.92512 −1.00000 −1.03007 −0.425845
1.8 −1.00000 1.74849 1.00000 2.50715 −1.74849 −1.18596 −1.00000 0.0572044 −2.50715
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$13$$ $$-1$$
$$19$$ $$-1$$

## 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 6422.2.a.bg 8
13.b even 2 1 6422.2.a.bh 8
13.f odd 12 2 494.2.m.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.m.a 16 13.f odd 12 2
6422.2.a.bg 8 1.a even 1 1 trivial
6422.2.a.bh 8 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

 $$T_{3}^{8} + \cdots$$ $$T_{5}^{8} - \cdots$$ $$T_{7}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$-2 - 6 T + 7 T^{2} + 28 T^{3} + 3 T^{4} - 22 T^{5} - 4 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$-3 + 12 T - 2 T^{2} - 36 T^{3} + 25 T^{4} + 20 T^{5} - 11 T^{6} - 2 T^{7} + T^{8}$$
$7$ $$-128 + 240 T + 5 T^{2} - 220 T^{3} + 86 T^{4} + 38 T^{5} - 19 T^{6} - 2 T^{7} + T^{8}$$
$11$ $$214 - 602 T + 455 T^{2} + 102 T^{3} - 251 T^{4} + 70 T^{5} + 18 T^{6} - 10 T^{7} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$-512 + 2560 T - 2816 T^{2} - 1120 T^{3} + 920 T^{4} + 92 T^{5} - 59 T^{6} - 2 T^{7} + T^{8}$$
$19$ $$( -1 + T )^{8}$$
$23$ $$976 - 912 T - 803 T^{2} + 608 T^{3} + 213 T^{4} - 128 T^{5} - 22 T^{6} + 8 T^{7} + T^{8}$$
$29$ $$-10067 + 23216 T + 9008 T^{2} - 7800 T^{3} - 605 T^{4} + 782 T^{5} - 75 T^{6} - 8 T^{7} + T^{8}$$
$31$ $$-46016 + 14416 T + 20681 T^{2} - 6910 T^{3} - 1734 T^{4} + 772 T^{5} - 29 T^{6} - 12 T^{7} + T^{8}$$
$37$ $$191808 + 58176 T - 68384 T^{2} - 7888 T^{3} + 5980 T^{4} + 196 T^{5} - 155 T^{6} + T^{8}$$
$41$ $$2597437 - 150928 T - 298871 T^{2} + 12620 T^{3} + 11803 T^{4} - 304 T^{5} - 188 T^{6} + 2 T^{7} + T^{8}$$
$43$ $$-12200 + 52620 T + 45937 T^{2} + 5380 T^{3} - 3597 T^{4} - 880 T^{5} + 11 T^{6} + 16 T^{7} + T^{8}$$
$47$ $$-498 + 6906 T + 12049 T^{2} + 3616 T^{3} - 2036 T^{4} - 868 T^{5} - 41 T^{6} + 12 T^{7} + T^{8}$$
$53$ $$-36195267 - 6670512 T + 1893013 T^{2} + 368178 T^{3} - 20757 T^{4} - 6040 T^{5} - 108 T^{6} + 24 T^{7} + T^{8}$$
$59$ $$989862 + 961590 T - 149681 T^{2} - 120656 T^{3} + 20303 T^{4} + 1500 T^{5} - 294 T^{6} - 4 T^{7} + T^{8}$$
$61$ $$-15483 - 80340 T - 137936 T^{2} - 95036 T^{3} - 28899 T^{4} - 3318 T^{5} + 47 T^{6} + 26 T^{7} + T^{8}$$
$67$ $$-195584 - 423936 T - 98560 T^{2} + 83264 T^{3} + 9392 T^{4} - 1960 T^{5} - 202 T^{6} + 10 T^{7} + T^{8}$$
$71$ $$745566 - 851190 T + 139 T^{2} + 131052 T^{3} - 29319 T^{4} + 734 T^{5} + 357 T^{6} - 36 T^{7} + T^{8}$$
$73$ $$-228467 - 1370940 T + 996920 T^{2} + 301276 T^{3} - 3574 T^{4} - 4820 T^{5} - 160 T^{6} + 20 T^{7} + T^{8}$$
$79$ $$63006 + 55686 T - 25853 T^{2} - 31764 T^{3} - 8999 T^{4} - 640 T^{5} + 118 T^{6} + 22 T^{7} + T^{8}$$
$83$ $$-2176946 - 1968854 T - 8015 T^{2} + 163596 T^{3} + 5174 T^{4} - 3164 T^{5} - 141 T^{6} + 18 T^{7} + T^{8}$$
$89$ $$-17288 + 16012 T + 46229 T^{2} - 30286 T^{3} - 8154 T^{4} + 3934 T^{5} - 173 T^{6} - 18 T^{7} + T^{8}$$
$97$ $$-117248 + 731904 T - 24768 T^{2} - 99584 T^{3} + 12512 T^{4} + 2160 T^{5} - 308 T^{6} - 8 T^{7} + T^{8}$$