# Properties

 Label 722.6.a.c Level $722$ Weight $6$ Character orbit 722.a Self dual yes Analytic conductor $115.797$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 722.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$115.797117905$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1441})$$ Defining polynomial: $$x^{2} - x - 360$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1441})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + ( -1 - \beta ) q^{3} + 16 q^{4} + ( -24 + 3 \beta ) q^{5} + ( -4 - 4 \beta ) q^{6} + ( 59 - 4 \beta ) q^{7} + 64 q^{8} + ( 118 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + 4 q^{2} + ( -1 - \beta ) q^{3} + 16 q^{4} + ( -24 + 3 \beta ) q^{5} + ( -4 - 4 \beta ) q^{6} + ( 59 - 4 \beta ) q^{7} + 64 q^{8} + ( 118 + 3 \beta ) q^{9} + ( -96 + 12 \beta ) q^{10} + ( 330 + \beta ) q^{11} + ( -16 - 16 \beta ) q^{12} + ( -809 + 5 \beta ) q^{13} + ( 236 - 16 \beta ) q^{14} + ( -1056 + 18 \beta ) q^{15} + 256 q^{16} + ( 27 + 10 \beta ) q^{17} + ( 472 + 12 \beta ) q^{18} + ( -384 + 48 \beta ) q^{20} + ( 1381 - 51 \beta ) q^{21} + ( 1320 + 4 \beta ) q^{22} + ( -1617 + 49 \beta ) q^{23} + ( -64 - 64 \beta ) q^{24} + ( 691 - 135 \beta ) q^{25} + ( -3236 + 20 \beta ) q^{26} + ( -955 + 119 \beta ) q^{27} + ( 944 - 64 \beta ) q^{28} + ( 1083 + 315 \beta ) q^{29} + ( -4224 + 72 \beta ) q^{30} + ( 748 - 316 \beta ) q^{31} + 1024 q^{32} + ( -690 - 332 \beta ) q^{33} + ( 108 + 40 \beta ) q^{34} + ( -5736 + 261 \beta ) q^{35} + ( 1888 + 48 \beta ) q^{36} + ( -5330 + 172 \beta ) q^{37} + ( -991 + 799 \beta ) q^{39} + ( -1536 + 192 \beta ) q^{40} + ( -8616 + 602 \beta ) q^{41} + ( 5524 - 204 \beta ) q^{42} + ( 5792 - 281 \beta ) q^{43} + ( 5280 + 16 \beta ) q^{44} + ( 408 + 291 \beta ) q^{45} + ( -6468 + 196 \beta ) q^{46} + ( -5520 - 1115 \beta ) q^{47} + ( -256 - 256 \beta ) q^{48} + ( -7566 - 456 \beta ) q^{49} + ( 2764 - 540 \beta ) q^{50} + ( -3627 - 47 \beta ) q^{51} + ( -12944 + 80 \beta ) q^{52} + ( -10593 + 601 \beta ) q^{53} + ( -3820 + 476 \beta ) q^{54} + ( -6840 + 969 \beta ) q^{55} + ( 3776 - 256 \beta ) q^{56} + ( 4332 + 1260 \beta ) q^{58} + ( 39327 - 73 \beta ) q^{59} + ( -16896 + 288 \beta ) q^{60} + ( 21398 + 825 \beta ) q^{61} + ( 2992 - 1264 \beta ) q^{62} + ( 2642 - 307 \beta ) q^{63} + 4096 q^{64} + ( 24816 - 2532 \beta ) q^{65} + ( -2760 - 1328 \beta ) q^{66} + ( -5453 + 3101 \beta ) q^{67} + ( 432 + 160 \beta ) q^{68} + ( -16023 + 1519 \beta ) q^{69} + ( -22944 + 1044 \beta ) q^{70} + ( 31878 - 1268 \beta ) q^{71} + ( 7552 + 192 \beta ) q^{72} + ( 6617 + 2984 \beta ) q^{73} + ( -21320 + 688 \beta ) q^{74} + ( 47909 - 421 \beta ) q^{75} + ( 18030 - 1265 \beta ) q^{77} + ( -3964 + 3196 \beta ) q^{78} + ( -33494 - 134 \beta ) q^{79} + ( -6144 + 768 \beta ) q^{80} + ( -70559 - 12 \beta ) q^{81} + ( -34464 + 2408 \beta ) q^{82} + ( -4134 - 2446 \beta ) q^{83} + ( 22096 - 816 \beta ) q^{84} + ( 10152 - 129 \beta ) q^{85} + ( 23168 - 1124 \beta ) q^{86} + ( -114483 - 1713 \beta ) q^{87} + ( 21120 + 64 \beta ) q^{88} + ( -61956 - 4276 \beta ) q^{89} + ( 1632 + 1164 \beta ) q^{90} + ( -54931 + 3511 \beta ) q^{91} + ( -25872 + 784 \beta ) q^{92} + ( 113012 - 116 \beta ) q^{93} + ( -22080 - 4460 \beta ) q^{94} + ( -1024 - 1024 \beta ) q^{96} + ( -90590 + 2622 \beta ) q^{97} + ( -30264 - 1824 \beta ) q^{98} + ( 40020 + 1111 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{2} - 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} + 128q^{8} + 239q^{9} + O(q^{10})$$ $$2q + 8q^{2} - 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} + 128q^{8} + 239q^{9} - 180q^{10} + 661q^{11} - 48q^{12} - 1613q^{13} + 456q^{14} - 2094q^{15} + 512q^{16} + 64q^{17} + 956q^{18} - 720q^{20} + 2711q^{21} + 2644q^{22} - 3185q^{23} - 192q^{24} + 1247q^{25} - 6452q^{26} - 1791q^{27} + 1824q^{28} + 2481q^{29} - 8376q^{30} + 1180q^{31} + 2048q^{32} - 1712q^{33} + 256q^{34} - 11211q^{35} + 3824q^{36} - 10488q^{37} - 1183q^{39} - 2880q^{40} - 16630q^{41} + 10844q^{42} + 11303q^{43} + 10576q^{44} + 1107q^{45} - 12740q^{46} - 12155q^{47} - 768q^{48} - 15588q^{49} + 4988q^{50} - 7301q^{51} - 25808q^{52} - 20585q^{53} - 7164q^{54} - 12711q^{55} + 7296q^{56} + 9924q^{58} + 78581q^{59} - 33504q^{60} + 43621q^{61} + 4720q^{62} + 4977q^{63} + 8192q^{64} + 47100q^{65} - 6848q^{66} - 7805q^{67} + 1024q^{68} - 30527q^{69} - 44844q^{70} + 62488q^{71} + 15296q^{72} + 16218q^{73} - 41952q^{74} + 95397q^{75} + 34795q^{77} - 4732q^{78} - 67122q^{79} - 11520q^{80} - 141130q^{81} - 66520q^{82} - 10714q^{83} + 43376q^{84} + 20175q^{85} + 45212q^{86} - 230679q^{87} + 42304q^{88} - 128188q^{89} + 4428q^{90} - 106351q^{91} - 50960q^{92} + 225908q^{93} - 48620q^{94} - 3072q^{96} - 178558q^{97} - 62352q^{98} + 81151q^{99} + O(q^{100})$$

## 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
 19.4803 −18.4803
4.00000 −20.4803 16.0000 34.4408 −81.9210 −18.9210 64.0000 176.441 137.763
1.2 4.00000 17.4803 16.0000 −79.4408 69.9210 132.921 64.0000 62.5592 −317.763
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 722.6.a.c 2
19.b odd 2 1 38.6.a.c 2
57.d even 2 1 342.6.a.i 2
76.d even 2 1 304.6.a.f 2
95.d odd 2 1 950.6.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.c 2 19.b odd 2 1
304.6.a.f 2 76.d even 2 1
342.6.a.i 2 57.d even 2 1
722.6.a.c 2 1.a even 1 1 trivial
950.6.a.d 2 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} - 358$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(722))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4 + T )^{2}$$
$3$ $$-358 + 3 T + T^{2}$$
$5$ $$-2736 + 45 T + T^{2}$$
$7$ $$-2515 - 114 T + T^{2}$$
$11$ $$108870 - 661 T + T^{2}$$
$13$ $$641436 + 1613 T + T^{2}$$
$17$ $$-35001 - 64 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$1671096 + 3185 T + T^{2}$$
$29$ $$-34206966 - 2481 T + T^{2}$$
$31$ $$-35625024 - 1180 T + T^{2}$$
$37$ $$16841900 + 10488 T + T^{2}$$
$41$ $$-61416816 + 16630 T + T^{2}$$
$43$ $$3493752 - 11303 T + T^{2}$$
$47$ $$-410935800 + 12155 T + T^{2}$$
$53$ $$-24187104 + 20585 T + T^{2}$$
$59$ $$1541823618 - 78581 T + T^{2}$$
$61$ $$230502754 - 43621 T + T^{2}$$
$67$ $$-3449006904 + 7805 T + T^{2}$$
$71$ $$396968940 - 62488 T + T^{2}$$
$73$ $$-3142002343 - 16218 T + T^{2}$$
$79$ $$1119872072 + 67122 T + T^{2}$$
$83$ $$-2126648040 + 10714 T + T^{2}$$
$89$ $$-2478833568 + 128188 T + T^{2}$$
$97$ $$5494062880 + 178558 T + T^{2}$$