# Properties

 Label 245.6.a.c Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (3 \beta - 3) q^{3} + (\beta - 16) q^{4} + 25 q^{5} + 48 q^{6} + ( - 47 \beta + 16) q^{8} + ( - 9 \beta - 90) q^{9}+O(q^{10})$$ q + b * q^2 + (3*b - 3) * q^3 + (b - 16) * q^4 + 25 * q^5 + 48 * q^6 + (-47*b + 16) * q^8 + (-9*b - 90) * q^9 $$q + \beta q^{2} + (3 \beta - 3) q^{3} + (\beta - 16) q^{4} + 25 q^{5} + 48 q^{6} + ( - 47 \beta + 16) q^{8} + ( - 9 \beta - 90) q^{9} + 25 \beta q^{10} + (97 \beta - 349) q^{11} + ( - 48 \beta + 96) q^{12} + ( - 53 \beta + 315) q^{13} + (75 \beta - 75) q^{15} + ( - 63 \beta - 240) q^{16} + ( - 251 \beta + 105) q^{17} + ( - 99 \beta - 144) q^{18} + (86 \beta - 358) q^{19} + (25 \beta - 400) q^{20} + ( - 252 \beta + 1552) q^{22} + ( - 902 \beta + 230) q^{23} + (48 \beta - 2304) q^{24} + 625 q^{25} + (262 \beta - 848) q^{26} + ( - 999 \beta + 567) q^{27} + ( - 945 \beta + 3415) q^{29} + 1200 q^{30} + ( - 924 \beta + 660) q^{31} + (1201 \beta - 1520) q^{32} + ( - 1047 \beta + 5703) q^{33} + ( - 146 \beta - 4016) q^{34} + (45 \beta + 1296) q^{36} + ( - 1260 \beta - 3822) q^{37} + ( - 272 \beta + 1376) q^{38} + (945 \beta - 3489) q^{39} + ( - 1175 \beta + 400) q^{40} + (3818 \beta - 2796) q^{41} + (922 \beta - 14022) q^{43} + ( - 1804 \beta + 7136) q^{44} + ( - 225 \beta - 2250) q^{45} + ( - 672 \beta - 14432) q^{46} + (1575 \beta + 9857) q^{47} + ( - 720 \beta - 2304) q^{48} + 625 \beta q^{50} + (315 \beta - 12363) q^{51} + (1110 \beta - 5888) q^{52} + ( - 454 \beta - 27564) q^{53} + ( - 432 \beta - 15984) q^{54} + (2425 \beta - 8725) q^{55} + ( - 1074 \beta + 5202) q^{57} + (2470 \beta - 15120) q^{58} + ( - 5184 \beta - 27208) q^{59} + ( - 1200 \beta + 2400) q^{60} + ( - 5706 \beta + 28776) q^{61} + ( - 264 \beta - 14784) q^{62} + (1697 \beta + 26896) q^{64} + ( - 1325 \beta + 7875) q^{65} + (4656 \beta - 16752) q^{66} + ( - 4568 \beta - 20388) q^{67} + (3870 \beta - 5696) q^{68} + (690 \beta - 43986) q^{69} + (5304 \beta + 37720) q^{71} + (4509 \beta + 5328) q^{72} + (4192 \beta + 4670) q^{73} + ( - 5082 \beta - 20160) q^{74} + (1875 \beta - 1875) q^{75} + ( - 1648 \beta + 7104) q^{76} + ( - 2544 \beta + 15120) q^{78} + (17635 \beta - 34715) q^{79} + ( - 1575 \beta - 6000) q^{80} + (3888 \beta - 27783) q^{81} + (1022 \beta + 61088) q^{82} + (3924 \beta - 56876) q^{83} + ( - 6275 \beta + 2625) q^{85} + ( - 13100 \beta + 14752) q^{86} + (10245 \beta - 55605) q^{87} + (13396 \beta - 78528) q^{88} + (5722 \beta + 15964) q^{89} + ( - 2475 \beta - 3600) q^{90} + (13760 \beta - 18112) q^{92} + (1980 \beta - 46332) q^{93} + (11432 \beta + 25200) q^{94} + (2150 \beta - 8950) q^{95} + ( - 4560 \beta + 62208) q^{96} + ( - 13943 \beta + 55141) q^{97} + ( - 6462 \beta + 17442) q^{99}+O(q^{100})$$ q + b * q^2 + (3*b - 3) * q^3 + (b - 16) * q^4 + 25 * q^5 + 48 * q^6 + (-47*b + 16) * q^8 + (-9*b - 90) * q^9 + 25*b * q^10 + (97*b - 349) * q^11 + (-48*b + 96) * q^12 + (-53*b + 315) * q^13 + (75*b - 75) * q^15 + (-63*b - 240) * q^16 + (-251*b + 105) * q^17 + (-99*b - 144) * q^18 + (86*b - 358) * q^19 + (25*b - 400) * q^20 + (-252*b + 1552) * q^22 + (-902*b + 230) * q^23 + (48*b - 2304) * q^24 + 625 * q^25 + (262*b - 848) * q^26 + (-999*b + 567) * q^27 + (-945*b + 3415) * q^29 + 1200 * q^30 + (-924*b + 660) * q^31 + (1201*b - 1520) * q^32 + (-1047*b + 5703) * q^33 + (-146*b - 4016) * q^34 + (45*b + 1296) * q^36 + (-1260*b - 3822) * q^37 + (-272*b + 1376) * q^38 + (945*b - 3489) * q^39 + (-1175*b + 400) * q^40 + (3818*b - 2796) * q^41 + (922*b - 14022) * q^43 + (-1804*b + 7136) * q^44 + (-225*b - 2250) * q^45 + (-672*b - 14432) * q^46 + (1575*b + 9857) * q^47 + (-720*b - 2304) * q^48 + 625*b * q^50 + (315*b - 12363) * q^51 + (1110*b - 5888) * q^52 + (-454*b - 27564) * q^53 + (-432*b - 15984) * q^54 + (2425*b - 8725) * q^55 + (-1074*b + 5202) * q^57 + (2470*b - 15120) * q^58 + (-5184*b - 27208) * q^59 + (-1200*b + 2400) * q^60 + (-5706*b + 28776) * q^61 + (-264*b - 14784) * q^62 + (1697*b + 26896) * q^64 + (-1325*b + 7875) * q^65 + (4656*b - 16752) * q^66 + (-4568*b - 20388) * q^67 + (3870*b - 5696) * q^68 + (690*b - 43986) * q^69 + (5304*b + 37720) * q^71 + (4509*b + 5328) * q^72 + (4192*b + 4670) * q^73 + (-5082*b - 20160) * q^74 + (1875*b - 1875) * q^75 + (-1648*b + 7104) * q^76 + (-2544*b + 15120) * q^78 + (17635*b - 34715) * q^79 + (-1575*b - 6000) * q^80 + (3888*b - 27783) * q^81 + (1022*b + 61088) * q^82 + (3924*b - 56876) * q^83 + (-6275*b + 2625) * q^85 + (-13100*b + 14752) * q^86 + (10245*b - 55605) * q^87 + (13396*b - 78528) * q^88 + (5722*b + 15964) * q^89 + (-2475*b - 3600) * q^90 + (13760*b - 18112) * q^92 + (1980*b - 46332) * q^93 + (11432*b + 25200) * q^94 + (2150*b - 8950) * q^95 + (-4560*b + 62208) * q^96 + (-13943*b + 55141) * q^97 + (-6462*b + 17442) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9}+O(q^{10})$$ 2 * q + q^2 - 3 * q^3 - 31 * q^4 + 50 * q^5 + 96 * q^6 - 15 * q^8 - 189 * q^9 $$2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9} + 25 q^{10} - 601 q^{11} + 144 q^{12} + 577 q^{13} - 75 q^{15} - 543 q^{16} - 41 q^{17} - 387 q^{18} - 630 q^{19} - 775 q^{20} + 2852 q^{22} - 442 q^{23} - 4560 q^{24} + 1250 q^{25} - 1434 q^{26} + 135 q^{27} + 5885 q^{29} + 2400 q^{30} + 396 q^{31} - 1839 q^{32} + 10359 q^{33} - 8178 q^{34} + 2637 q^{36} - 8904 q^{37} + 2480 q^{38} - 6033 q^{39} - 375 q^{40} - 1774 q^{41} - 27122 q^{43} + 12468 q^{44} - 4725 q^{45} - 29536 q^{46} + 21289 q^{47} - 5328 q^{48} + 625 q^{50} - 24411 q^{51} - 10666 q^{52} - 55582 q^{53} - 32400 q^{54} - 15025 q^{55} + 9330 q^{57} - 27770 q^{58} - 59600 q^{59} + 3600 q^{60} + 51846 q^{61} - 29832 q^{62} + 55489 q^{64} + 14425 q^{65} - 28848 q^{66} - 45344 q^{67} - 7522 q^{68} - 87282 q^{69} + 80744 q^{71} + 15165 q^{72} + 13532 q^{73} - 45402 q^{74} - 1875 q^{75} + 12560 q^{76} + 27696 q^{78} - 51795 q^{79} - 13575 q^{80} - 51678 q^{81} + 123198 q^{82} - 109828 q^{83} - 1025 q^{85} + 16404 q^{86} - 100965 q^{87} - 143660 q^{88} + 37650 q^{89} - 9675 q^{90} - 22464 q^{92} - 90684 q^{93} + 61832 q^{94} - 15750 q^{95} + 119856 q^{96} + 96339 q^{97} + 28422 q^{99}+O(q^{100})$$ 2 * q + q^2 - 3 * q^3 - 31 * q^4 + 50 * q^5 + 96 * q^6 - 15 * q^8 - 189 * q^9 + 25 * q^10 - 601 * q^11 + 144 * q^12 + 577 * q^13 - 75 * q^15 - 543 * q^16 - 41 * q^17 - 387 * q^18 - 630 * q^19 - 775 * q^20 + 2852 * q^22 - 442 * q^23 - 4560 * q^24 + 1250 * q^25 - 1434 * q^26 + 135 * q^27 + 5885 * q^29 + 2400 * q^30 + 396 * q^31 - 1839 * q^32 + 10359 * q^33 - 8178 * q^34 + 2637 * q^36 - 8904 * q^37 + 2480 * q^38 - 6033 * q^39 - 375 * q^40 - 1774 * q^41 - 27122 * q^43 + 12468 * q^44 - 4725 * q^45 - 29536 * q^46 + 21289 * q^47 - 5328 * q^48 + 625 * q^50 - 24411 * q^51 - 10666 * q^52 - 55582 * q^53 - 32400 * q^54 - 15025 * q^55 + 9330 * q^57 - 27770 * q^58 - 59600 * q^59 + 3600 * q^60 + 51846 * q^61 - 29832 * q^62 + 55489 * q^64 + 14425 * q^65 - 28848 * q^66 - 45344 * q^67 - 7522 * q^68 - 87282 * q^69 + 80744 * q^71 + 15165 * q^72 + 13532 * q^73 - 45402 * q^74 - 1875 * q^75 + 12560 * q^76 + 27696 * q^78 - 51795 * q^79 - 13575 * q^80 - 51678 * q^81 + 123198 * q^82 - 109828 * q^83 - 1025 * q^85 + 16404 * q^86 - 100965 * q^87 - 143660 * q^88 + 37650 * q^89 - 9675 * q^90 - 22464 * q^92 - 90684 * q^93 + 61832 * q^94 - 15750 * q^95 + 119856 * q^96 + 96339 * q^97 + 28422 * q^99

## 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
 −3.53113 4.53113
−3.53113 −13.5934 −19.5311 25.0000 48.0000 0 181.963 −58.2198 −88.2782
1.2 4.53113 10.5934 −11.4689 25.0000 48.0000 0 −196.963 −130.780 113.278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 245.6.a.c 2
7.b odd 2 1 35.6.a.b 2
21.c even 2 1 315.6.a.c 2
28.d even 2 1 560.6.a.l 2
35.c odd 2 1 175.6.a.d 2
35.f even 4 2 175.6.b.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 7.b odd 2 1
175.6.a.d 2 35.c odd 2 1
175.6.b.d 4 35.f even 4 2
245.6.a.c 2 1.a even 1 1 trivial
315.6.a.c 2 21.c even 2 1
560.6.a.l 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{2} - T_{2} - 16$$ T2^2 - T2 - 16 $$T_{3}^{2} + 3T_{3} - 144$$ T3^2 + 3*T3 - 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 16$$
$3$ $$T^{2} + 3T - 144$$
$5$ $$(T - 25)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 601T - 62596$$
$13$ $$T^{2} - 577T + 37586$$
$17$ $$T^{2} + 41T - 1023346$$
$19$ $$T^{2} + 630T - 20960$$
$23$ $$T^{2} + 442 T - 13172224$$
$29$ $$T^{2} - 5885 T - 5853350$$
$31$ $$T^{2} - 396 T - 13834656$$
$37$ $$T^{2} + 8904 T - 5978196$$
$41$ $$T^{2} + 1774 T - 236091496$$
$43$ $$T^{2} + 27122 T + 170086856$$
$47$ $$T^{2} - 21289 T + 72995224$$
$53$ $$T^{2} + 55582 T + 768990296$$
$59$ $$T^{2} + 59600 T + 451339840$$
$61$ $$T^{2} - 51846 T + 142927344$$
$67$ $$T^{2} + 45344 T + 174936944$$
$71$ $$T^{2} - 80744 T + 1172746624$$
$73$ $$T^{2} - 13532 T - 239780284$$
$79$ $$T^{2} + 51795 T - 4382959400$$
$83$ $$T^{2} + 109828 T + 2765333536$$
$89$ $$T^{2} - 37650 T - 177665240$$
$97$ $$T^{2} - 96339 T - 838817066$$