# Properties

 Label 315.6.a.c Level $315$ Weight $6$ Character orbit 315.a Self dual yes Analytic conductor $50.521$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.5209032411$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{65})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 16) q^{4} + 25 q^{5} - 49 q^{7} + (47 \beta - 16) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 16) * q^4 + 25 * q^5 - 49 * q^7 + (47*b - 16) * q^8 $$q - \beta q^{2} + (\beta - 16) q^{4} + 25 q^{5} - 49 q^{7} + (47 \beta - 16) q^{8} - 25 \beta q^{10} + ( - 97 \beta + 349) q^{11} + (53 \beta - 315) q^{13} + 49 \beta q^{14} + ( - 63 \beta - 240) q^{16} + ( - 251 \beta + 105) q^{17} + ( - 86 \beta + 358) q^{19} + (25 \beta - 400) q^{20} + ( - 252 \beta + 1552) q^{22} + (902 \beta - 230) q^{23} + 625 q^{25} + (262 \beta - 848) q^{26} + ( - 49 \beta + 784) q^{28} + (945 \beta - 3415) q^{29} + (924 \beta - 660) q^{31} + ( - 1201 \beta + 1520) q^{32} + (146 \beta + 4016) q^{34} - 1225 q^{35} + ( - 1260 \beta - 3822) q^{37} + ( - 272 \beta + 1376) q^{38} + (1175 \beta - 400) q^{40} + (3818 \beta - 2796) q^{41} + (922 \beta - 14022) q^{43} + (1804 \beta - 7136) q^{44} + ( - 672 \beta - 14432) q^{46} + (1575 \beta + 9857) q^{47} + 2401 q^{49} - 625 \beta q^{50} + ( - 1110 \beta + 5888) q^{52} + (454 \beta + 27564) q^{53} + ( - 2425 \beta + 8725) q^{55} + ( - 2303 \beta + 784) q^{56} + (2470 \beta - 15120) q^{58} + ( - 5184 \beta - 27208) q^{59} + (5706 \beta - 28776) q^{61} + ( - 264 \beta - 14784) q^{62} + (1697 \beta + 26896) q^{64} + (1325 \beta - 7875) q^{65} + ( - 4568 \beta - 20388) q^{67} + (3870 \beta - 5696) q^{68} + 1225 \beta q^{70} + ( - 5304 \beta - 37720) q^{71} + ( - 4192 \beta - 4670) q^{73} + (5082 \beta + 20160) q^{74} + (1648 \beta - 7104) q^{76} + (4753 \beta - 17101) q^{77} + (17635 \beta - 34715) q^{79} + ( - 1575 \beta - 6000) q^{80} + ( - 1022 \beta - 61088) q^{82} + (3924 \beta - 56876) q^{83} + ( - 6275 \beta + 2625) q^{85} + (13100 \beta - 14752) q^{86} + (13396 \beta - 78528) q^{88} + (5722 \beta + 15964) q^{89} + ( - 2597 \beta + 15435) q^{91} + ( - 13760 \beta + 18112) q^{92} + ( - 11432 \beta - 25200) q^{94} + ( - 2150 \beta + 8950) q^{95} + (13943 \beta - 55141) q^{97} - 2401 \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b - 16) * q^4 + 25 * q^5 - 49 * q^7 + (47*b - 16) * q^8 - 25*b * q^10 + (-97*b + 349) * q^11 + (53*b - 315) * q^13 + 49*b * q^14 + (-63*b - 240) * q^16 + (-251*b + 105) * q^17 + (-86*b + 358) * q^19 + (25*b - 400) * q^20 + (-252*b + 1552) * q^22 + (902*b - 230) * q^23 + 625 * q^25 + (262*b - 848) * q^26 + (-49*b + 784) * q^28 + (945*b - 3415) * q^29 + (924*b - 660) * q^31 + (-1201*b + 1520) * q^32 + (146*b + 4016) * q^34 - 1225 * q^35 + (-1260*b - 3822) * q^37 + (-272*b + 1376) * q^38 + (1175*b - 400) * q^40 + (3818*b - 2796) * q^41 + (922*b - 14022) * q^43 + (1804*b - 7136) * q^44 + (-672*b - 14432) * q^46 + (1575*b + 9857) * q^47 + 2401 * q^49 - 625*b * q^50 + (-1110*b + 5888) * q^52 + (454*b + 27564) * q^53 + (-2425*b + 8725) * q^55 + (-2303*b + 784) * q^56 + (2470*b - 15120) * q^58 + (-5184*b - 27208) * q^59 + (5706*b - 28776) * q^61 + (-264*b - 14784) * q^62 + (1697*b + 26896) * q^64 + (1325*b - 7875) * q^65 + (-4568*b - 20388) * q^67 + (3870*b - 5696) * q^68 + 1225*b * q^70 + (-5304*b - 37720) * q^71 + (-4192*b - 4670) * q^73 + (5082*b + 20160) * q^74 + (1648*b - 7104) * q^76 + (4753*b - 17101) * q^77 + (17635*b - 34715) * q^79 + (-1575*b - 6000) * q^80 + (-1022*b - 61088) * q^82 + (3924*b - 56876) * q^83 + (-6275*b + 2625) * q^85 + (13100*b - 14752) * q^86 + (13396*b - 78528) * q^88 + (5722*b + 15964) * q^89 + (-2597*b + 15435) * q^91 + (-13760*b + 18112) * q^92 + (-11432*b - 25200) * q^94 + (-2150*b + 8950) * q^95 + (13943*b - 55141) * q^97 - 2401*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 31 q^{4} + 50 q^{5} - 98 q^{7} + 15 q^{8}+O(q^{10})$$ 2 * q - q^2 - 31 * q^4 + 50 * q^5 - 98 * q^7 + 15 * q^8 $$2 q - q^{2} - 31 q^{4} + 50 q^{5} - 98 q^{7} + 15 q^{8} - 25 q^{10} + 601 q^{11} - 577 q^{13} + 49 q^{14} - 543 q^{16} - 41 q^{17} + 630 q^{19} - 775 q^{20} + 2852 q^{22} + 442 q^{23} + 1250 q^{25} - 1434 q^{26} + 1519 q^{28} - 5885 q^{29} - 396 q^{31} + 1839 q^{32} + 8178 q^{34} - 2450 q^{35} - 8904 q^{37} + 2480 q^{38} + 375 q^{40} - 1774 q^{41} - 27122 q^{43} - 12468 q^{44} - 29536 q^{46} + 21289 q^{47} + 4802 q^{49} - 625 q^{50} + 10666 q^{52} + 55582 q^{53} + 15025 q^{55} - 735 q^{56} - 27770 q^{58} - 59600 q^{59} - 51846 q^{61} - 29832 q^{62} + 55489 q^{64} - 14425 q^{65} - 45344 q^{67} - 7522 q^{68} + 1225 q^{70} - 80744 q^{71} - 13532 q^{73} + 45402 q^{74} - 12560 q^{76} - 29449 q^{77} - 51795 q^{79} - 13575 q^{80} - 123198 q^{82} - 109828 q^{83} - 1025 q^{85} - 16404 q^{86} - 143660 q^{88} + 37650 q^{89} + 28273 q^{91} + 22464 q^{92} - 61832 q^{94} + 15750 q^{95} - 96339 q^{97} - 2401 q^{98}+O(q^{100})$$ 2 * q - q^2 - 31 * q^4 + 50 * q^5 - 98 * q^7 + 15 * q^8 - 25 * q^10 + 601 * q^11 - 577 * q^13 + 49 * q^14 - 543 * q^16 - 41 * q^17 + 630 * q^19 - 775 * q^20 + 2852 * q^22 + 442 * q^23 + 1250 * q^25 - 1434 * q^26 + 1519 * q^28 - 5885 * q^29 - 396 * q^31 + 1839 * q^32 + 8178 * q^34 - 2450 * q^35 - 8904 * q^37 + 2480 * q^38 + 375 * q^40 - 1774 * q^41 - 27122 * q^43 - 12468 * q^44 - 29536 * q^46 + 21289 * q^47 + 4802 * q^49 - 625 * q^50 + 10666 * q^52 + 55582 * q^53 + 15025 * q^55 - 735 * q^56 - 27770 * q^58 - 59600 * q^59 - 51846 * q^61 - 29832 * q^62 + 55489 * q^64 - 14425 * q^65 - 45344 * q^67 - 7522 * q^68 + 1225 * q^70 - 80744 * q^71 - 13532 * q^73 + 45402 * q^74 - 12560 * q^76 - 29449 * q^77 - 51795 * q^79 - 13575 * q^80 - 123198 * q^82 - 109828 * q^83 - 1025 * q^85 - 16404 * q^86 - 143660 * q^88 + 37650 * q^89 + 28273 * q^91 + 22464 * q^92 - 61832 * q^94 + 15750 * q^95 - 96339 * q^97 - 2401 * q^98

## 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
 4.53113 −3.53113
−4.53113 0 −11.4689 25.0000 0 −49.0000 196.963 0 −113.278
1.2 3.53113 0 −19.5311 25.0000 0 −49.0000 −181.963 0 88.2782
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 315.6.a.c 2
3.b odd 2 1 35.6.a.b 2
12.b even 2 1 560.6.a.l 2
15.d odd 2 1 175.6.a.d 2
15.e even 4 2 175.6.b.d 4
21.c even 2 1 245.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 3.b odd 2 1
175.6.a.d 2 15.d odd 2 1
175.6.b.d 4 15.e even 4 2
245.6.a.c 2 21.c even 2 1
315.6.a.c 2 1.a even 1 1 trivial
560.6.a.l 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 16$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(315))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 16$$
$3$ $$T^{2}$$
$5$ $$(T - 25)^{2}$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} - 601T - 62596$$
$13$ $$T^{2} + 577T + 37586$$
$17$ $$T^{2} + 41T - 1023346$$
$19$ $$T^{2} - 630T - 20960$$
$23$ $$T^{2} - 442 T - 13172224$$
$29$ $$T^{2} + 5885 T - 5853350$$
$31$ $$T^{2} + 396 T - 13834656$$
$37$ $$T^{2} + 8904 T - 5978196$$
$41$ $$T^{2} + 1774 T - 236091496$$
$43$ $$T^{2} + 27122 T + 170086856$$
$47$ $$T^{2} - 21289 T + 72995224$$
$53$ $$T^{2} - 55582 T + 768990296$$
$59$ $$T^{2} + 59600 T + 451339840$$
$61$ $$T^{2} + 51846 T + 142927344$$
$67$ $$T^{2} + 45344 T + 174936944$$
$71$ $$T^{2} + 80744 T + 1172746624$$
$73$ $$T^{2} + 13532 T - 239780284$$
$79$ $$T^{2} + 51795 T - 4382959400$$
$83$ $$T^{2} + 109828 T + 2765333536$$
$89$ $$T^{2} - 37650 T - 177665240$$
$97$ $$T^{2} + 96339 T - 838817066$$