# Properties

 Label 164.2.a.a Level 164 Weight 2 Character orbit 164.a Self dual yes Analytic conductor 1.310 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$164 = 2^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 164.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.30954659315$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.25808.1 Defining polynomial: $$x^{4} - 10 x^{2} - 6 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} + ( 2 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 3 - \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + 2 \beta_{1} q^{13} + ( -1 - 2 \beta_{1} + 3 \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} + ( 2 - \beta_{1} - \beta_{2} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{21} + ( -2 - 2 \beta_{2} ) q^{23} + ( 3 - 2 \beta_{1} ) q^{25} + ( -5 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{27} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -8 - 2 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{33} + ( -7 + 4 \beta_{1} - \beta_{3} ) q^{35} + ( 4 - \beta_{2} - \beta_{3} ) q^{37} + ( -6 + 2 \beta_{1} ) q^{39} - q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( 6 + 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( 4 + 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{55} + ( -4 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{57} + ( 2 + 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + ( -6 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{63} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 7 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( -6 + 2 \beta_{3} ) q^{69} + ( -4 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{71} + ( 4 - \beta_{2} + 3 \beta_{3} ) q^{73} + ( 6 - 5 \beta_{1} + 3 \beta_{2} ) q^{75} + ( 2 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{77} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{79} + ( 7 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{81} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 12 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{87} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 12 - 2 \beta_{1} + 6 \beta_{3} ) q^{91} + ( -2 - 4 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 7 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{95} + ( 2 - 4 \beta_{3} ) q^{97} + ( 16 + 7 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 4q^{5} + 12q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 4q^{5} + 12q^{9} + 4q^{11} - 10q^{15} - 4q^{17} + 6q^{19} - 12q^{23} + 12q^{25} - 10q^{27} - 4q^{29} - 8q^{31} - 20q^{33} - 26q^{35} + 16q^{37} - 24q^{39} - 4q^{41} + 4q^{43} + 4q^{45} - 6q^{47} + 16q^{49} - 4q^{51} - 16q^{53} - 2q^{55} + 4q^{57} + 12q^{59} + 24q^{61} - 10q^{63} + 4q^{65} + 28q^{67} - 28q^{69} - 2q^{71} + 8q^{73} + 30q^{75} + 8q^{77} - 18q^{79} + 28q^{81} - 12q^{83} + 32q^{85} + 44q^{87} + 4q^{89} + 36q^{91} - 28q^{93} + 14q^{95} + 16q^{97} + 58q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 10 x^{2} - 6 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 7 \nu - 3$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu^{2} + 7 \nu - 12$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + 7 \beta_{1} + 3$$

## 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
 0.707500 3.31526 −2.46810 −1.55466
0 −3.24028 0 2.56613 0 −0.858626 0 7.49944 0
1.2 0 0.0950939 0 1.17025 0 3.14501 0 −2.99096 0
1.3 0 2.21551 0 3.59669 0 −5.06479 0 1.90849 0
1.4 0 2.92968 0 −3.33307 0 2.77840 0 5.58303 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$41$$ $$1$$

## 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 164.2.a.a 4
3.b odd 2 1 1476.2.a.g 4
4.b odd 2 1 656.2.a.i 4
5.b even 2 1 4100.2.a.c 4
5.c odd 4 2 4100.2.d.c 8
7.b odd 2 1 8036.2.a.i 4
8.b even 2 1 2624.2.a.v 4
8.d odd 2 1 2624.2.a.y 4
12.b even 2 1 5904.2.a.bp 4
41.b even 2 1 6724.2.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 1.a even 1 1 trivial
656.2.a.i 4 4.b odd 2 1
1476.2.a.g 4 3.b odd 2 1
2624.2.a.v 4 8.b even 2 1
2624.2.a.y 4 8.d odd 2 1
4100.2.a.c 4 5.b even 2 1
4100.2.d.c 8 5.c odd 4 2
5904.2.a.bp 4 12.b even 2 1
6724.2.a.c 4 41.b even 2 1
8036.2.a.i 4 7.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(164))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 2 T^{2} + 4 T^{3} - 8 T^{4} + 12 T^{5} + 18 T^{6} - 54 T^{7} + 81 T^{8}$$
$5$ $$1 - 4 T + 12 T^{2} - 16 T^{3} + 34 T^{4} - 80 T^{5} + 300 T^{6} - 500 T^{7} + 625 T^{8}$$
$7$ $$1 + 6 T^{2} + 26 T^{3} + 24 T^{4} + 182 T^{5} + 294 T^{6} + 2401 T^{8}$$
$11$ $$1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 1254 T^{5} + 3146 T^{6} - 5324 T^{7} + 14641 T^{8}$$
$13$ $$1 + 12 T^{2} - 48 T^{3} + 118 T^{4} - 624 T^{5} + 2028 T^{6} + 28561 T^{8}$$
$17$ $$1 + 4 T + 20 T^{2} + 124 T^{3} + 534 T^{4} + 2108 T^{5} + 5780 T^{6} + 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 6 T + 62 T^{2} - 208 T^{3} + 1448 T^{4} - 3952 T^{5} + 22382 T^{6} - 41154 T^{7} + 130321 T^{8}$$
$23$ $$1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 16100 T^{5} + 57132 T^{6} + 146004 T^{7} + 279841 T^{8}$$
$29$ $$1 + 4 T + 76 T^{2} + 348 T^{3} + 2870 T^{4} + 10092 T^{5} + 63916 T^{6} + 97556 T^{7} + 707281 T^{8}$$
$31$ $$1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 22072 T^{5} + 88412 T^{6} + 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 64380 T^{5} + 290228 T^{6} - 810448 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + T )^{4}$$
$43$ $$1 - 4 T + 124 T^{2} - 244 T^{3} + 6678 T^{4} - 10492 T^{5} + 229276 T^{6} - 318028 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 6 T + 126 T^{2} + 640 T^{3} + 8608 T^{4} + 30080 T^{5} + 278334 T^{6} + 622938 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 16 T + 212 T^{2} + 1824 T^{3} + 15558 T^{4} + 96672 T^{5} + 595508 T^{6} + 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 117764 T^{5} + 877212 T^{6} - 2464548 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 294264 T^{5} + 1562820 T^{6} - 5447544 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 28 T + 538 T^{2} - 6638 T^{3} + 64208 T^{4} - 444746 T^{5} + 2415082 T^{6} - 8421364 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 2 T + 98 T^{2} - 268 T^{3} + 3408 T^{4} - 19028 T^{5} + 494018 T^{6} + 715822 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 77380 T^{5} + 1129748 T^{6} - 3112136 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 333696 T^{5} + 2284206 T^{6} + 8874702 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 12 T + 252 T^{2} + 1644 T^{3} + 24598 T^{4} + 136452 T^{5} + 1736028 T^{6} + 6861444 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 4 T + 228 T^{2} - 796 T^{3} + 29014 T^{4} - 70844 T^{5} + 1805988 T^{6} - 2819876 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 16 T + 268 T^{2} - 3376 T^{3} + 38118 T^{4} - 327472 T^{5} + 2521612 T^{6} - 14602768 T^{7} + 88529281 T^{8}$$