Properties

 Label 825.6.a.j Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.34253.1 Defining polynomial: $$x^{3} - x^{2} - 52x + 48$$ x^3 - x^2 - 52*x + 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 2) q^{2} - 9 q^{3} + (\beta_{2} + 4 \beta_1 + 7) q^{4} + ( - 9 \beta_1 - 18) q^{6} + ( - \beta_{2} - 11 \beta_1 + 61) q^{7} + (7 \beta_{2} + 77) q^{8} + 81 q^{9}+O(q^{10})$$ q + (b1 + 2) * q^2 - 9 * q^3 + (b2 + 4*b1 + 7) * q^4 + (-9*b1 - 18) * q^6 + (-b2 - 11*b1 + 61) * q^7 + (7*b2 + 77) * q^8 + 81 * q^9 $$q + (\beta_1 + 2) q^{2} - 9 q^{3} + (\beta_{2} + 4 \beta_1 + 7) q^{4} + ( - 9 \beta_1 - 18) q^{6} + ( - \beta_{2} - 11 \beta_1 + 61) q^{7} + (7 \beta_{2} + 77) q^{8} + 81 q^{9} - 121 q^{11} + ( - 9 \beta_{2} - 36 \beta_1 - 63) q^{12} + (28 \beta_{2} + 24 \beta_1 + 210) q^{13} + ( - 14 \beta_{2} + 22 \beta_1 - 250) q^{14} + ( - 11 \beta_{2} + 68 \beta_1 - 161) q^{16} + ( - 39 \beta_{2} - 37 \beta_1 + 801) q^{17} + (81 \beta_1 + 162) q^{18} + ( - 29 \beta_{2} - 67 \beta_1 - 935) q^{19} + (9 \beta_{2} + 99 \beta_1 - 549) q^{21} + ( - 121 \beta_1 - 242) q^{22} + (20 \beta_{2} - 220 \beta_1 - 684) q^{23} + ( - 63 \beta_{2} - 693) q^{24} + (108 \beta_{2} + 734 \beta_1 + 896) q^{26} - 729 q^{27} + (12 \beta_{2} - 92 \beta_1 - 1500) q^{28} + (77 \beta_{2} + 107 \beta_1 - 2615) q^{29} + (38 \beta_{2} + 130 \beta_1 + 146) q^{31} + ( - 189 \beta_{2} - 212 \beta_1 - 263) q^{32} + 1089 q^{33} + ( - 154 \beta_{2} + 64 \beta_1 + 814) q^{34} + (81 \beta_{2} + 324 \beta_1 + 567) q^{36} + ( - 272 \beta_{2} + 528 \beta_1 + 2866) q^{37} + ( - 154 \beta_{2} - 1562 \beta_1 - 3838) q^{38} + ( - 252 \beta_{2} - 216 \beta_1 - 1890) q^{39} + ( - 473 \beta_{2} + 977 \beta_1 - 3245) q^{41} + (126 \beta_{2} - 198 \beta_1 + 2250) q^{42} + ( - 341 \beta_{2} + 809 \beta_1 + 4621) q^{43} + ( - 121 \beta_{2} - 484 \beta_1 - 847) q^{44} + ( - 160 \beta_{2} - 784 \beta_1 - 9328) q^{46} + (422 \beta_{2} - 222 \beta_1 + 6538) q^{47} + (99 \beta_{2} - 612 \beta_1 + 1449) q^{48} + (4 \beta_{2} - 964 \beta_1 - 8555) q^{49} + (351 \beta_{2} + 333 \beta_1 - 7209) q^{51} + (162 \beta_{2} + 3432 \beta_1 + 19358) q^{52} + ( - 586 \beta_{2} + 962 \beta_1 + 1212) q^{53} + ( - 729 \beta_1 - 1458) q^{54} + (392 \beta_{2} - 2184 \beta_1 + 1624) q^{56} + (261 \beta_{2} + 603 \beta_1 + 8415) q^{57} + (338 \beta_{2} - 1092 \beta_1 - 2486) q^{58} + (356 \beta_{2} - 2748 \beta_1 - 2200) q^{59} + (364 \beta_{2} - 3300 \beta_1 - 18926) q^{61} + (244 \beta_{2} + 1052 \beta_1 + 4348) q^{62} + ( - 81 \beta_{2} - 891 \beta_1 + 4941) q^{63} + ( - 427 \beta_{2} - 6076 \beta_1 - 337) q^{64} + (1089 \beta_1 + 2178) q^{66} + ( - 680 \beta_{2} + 2144 \beta_1 + 12108) q^{67} + (850 \beta_{2} - 492 \beta_1 - 19762) q^{68} + ( - 180 \beta_{2} + 1980 \beta_1 + 6156) q^{69} + (980 \beta_{2} + 2612 \beta_1 - 25548) q^{71} + (567 \beta_{2} + 6237) q^{72} + ( - 428 \beta_{2} - 2808 \beta_1 + 15750) q^{73} + ( - 288 \beta_{2} - 702 \beta_1 + 27748) q^{74} + ( - 1096 \beta_{2} - 7436 \beta_1 - 30424) q^{76} + (121 \beta_{2} + 1331 \beta_1 - 7381) q^{77} + ( - 972 \beta_{2} - 6606 \beta_1 - 8064) q^{78} + ( - 775 \beta_{2} - 5417 \beta_1 - 34233) q^{79} + 6561 q^{81} + ( - 442 \beta_{2} - 9332 \beta_1 + 33854) q^{82} + ( - 1474 \beta_{2} - 14234 \beta_1 + 32042) q^{83} + ( - 108 \beta_{2} + 828 \beta_1 + 13500) q^{84} + ( - 214 \beta_{2} + 442 \beta_1 + 41990) q^{86} + ( - 693 \beta_{2} - 963 \beta_1 + 23535) q^{87} + ( - 847 \beta_{2} - 9317) q^{88} + ( - 132 \beta_{2} - 2140 \beta_1 + 56494) q^{89} + (1378 \beta_{2} - 6602 \beta_1 - 8410) q^{91} + ( - 1904 \beta_{2} - 6576 \beta_1 - 22128) q^{92} + ( - 342 \beta_{2} - 1170 \beta_1 - 1314) q^{93} + (1044 \beta_{2} + 13268 \beta_1 - 180) q^{94} + (1701 \beta_{2} + 1908 \beta_1 + 2367) q^{96} + (1664 \beta_{2} + 8760 \beta_1 - 56154) q^{97} + ( - 952 \beta_{2} - 10415 \beta_1 - 50902) q^{98} - 9801 q^{99}+O(q^{100})$$ q + (b1 + 2) * q^2 - 9 * q^3 + (b2 + 4*b1 + 7) * q^4 + (-9*b1 - 18) * q^6 + (-b2 - 11*b1 + 61) * q^7 + (7*b2 + 77) * q^8 + 81 * q^9 - 121 * q^11 + (-9*b2 - 36*b1 - 63) * q^12 + (28*b2 + 24*b1 + 210) * q^13 + (-14*b2 + 22*b1 - 250) * q^14 + (-11*b2 + 68*b1 - 161) * q^16 + (-39*b2 - 37*b1 + 801) * q^17 + (81*b1 + 162) * q^18 + (-29*b2 - 67*b1 - 935) * q^19 + (9*b2 + 99*b1 - 549) * q^21 + (-121*b1 - 242) * q^22 + (20*b2 - 220*b1 - 684) * q^23 + (-63*b2 - 693) * q^24 + (108*b2 + 734*b1 + 896) * q^26 - 729 * q^27 + (12*b2 - 92*b1 - 1500) * q^28 + (77*b2 + 107*b1 - 2615) * q^29 + (38*b2 + 130*b1 + 146) * q^31 + (-189*b2 - 212*b1 - 263) * q^32 + 1089 * q^33 + (-154*b2 + 64*b1 + 814) * q^34 + (81*b2 + 324*b1 + 567) * q^36 + (-272*b2 + 528*b1 + 2866) * q^37 + (-154*b2 - 1562*b1 - 3838) * q^38 + (-252*b2 - 216*b1 - 1890) * q^39 + (-473*b2 + 977*b1 - 3245) * q^41 + (126*b2 - 198*b1 + 2250) * q^42 + (-341*b2 + 809*b1 + 4621) * q^43 + (-121*b2 - 484*b1 - 847) * q^44 + (-160*b2 - 784*b1 - 9328) * q^46 + (422*b2 - 222*b1 + 6538) * q^47 + (99*b2 - 612*b1 + 1449) * q^48 + (4*b2 - 964*b1 - 8555) * q^49 + (351*b2 + 333*b1 - 7209) * q^51 + (162*b2 + 3432*b1 + 19358) * q^52 + (-586*b2 + 962*b1 + 1212) * q^53 + (-729*b1 - 1458) * q^54 + (392*b2 - 2184*b1 + 1624) * q^56 + (261*b2 + 603*b1 + 8415) * q^57 + (338*b2 - 1092*b1 - 2486) * q^58 + (356*b2 - 2748*b1 - 2200) * q^59 + (364*b2 - 3300*b1 - 18926) * q^61 + (244*b2 + 1052*b1 + 4348) * q^62 + (-81*b2 - 891*b1 + 4941) * q^63 + (-427*b2 - 6076*b1 - 337) * q^64 + (1089*b1 + 2178) * q^66 + (-680*b2 + 2144*b1 + 12108) * q^67 + (850*b2 - 492*b1 - 19762) * q^68 + (-180*b2 + 1980*b1 + 6156) * q^69 + (980*b2 + 2612*b1 - 25548) * q^71 + (567*b2 + 6237) * q^72 + (-428*b2 - 2808*b1 + 15750) * q^73 + (-288*b2 - 702*b1 + 27748) * q^74 + (-1096*b2 - 7436*b1 - 30424) * q^76 + (121*b2 + 1331*b1 - 7381) * q^77 + (-972*b2 - 6606*b1 - 8064) * q^78 + (-775*b2 - 5417*b1 - 34233) * q^79 + 6561 * q^81 + (-442*b2 - 9332*b1 + 33854) * q^82 + (-1474*b2 - 14234*b1 + 32042) * q^83 + (-108*b2 + 828*b1 + 13500) * q^84 + (-214*b2 + 442*b1 + 41990) * q^86 + (-693*b2 - 963*b1 + 23535) * q^87 + (-847*b2 - 9317) * q^88 + (-132*b2 - 2140*b1 + 56494) * q^89 + (1378*b2 - 6602*b1 - 8410) * q^91 + (-1904*b2 - 6576*b1 - 22128) * q^92 + (-342*b2 - 1170*b1 - 1314) * q^93 + (1044*b2 + 13268*b1 - 180) * q^94 + (1701*b2 + 1908*b1 + 2367) * q^96 + (1664*b2 + 8760*b1 - 56154) * q^97 + (-952*b2 - 10415*b1 - 50902) * q^98 - 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 $$3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100})$$ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 - 363 * q^11 - 225 * q^12 + 654 * q^13 - 728 * q^14 - 415 * q^16 + 2366 * q^17 + 567 * q^18 - 2872 * q^19 - 1548 * q^21 - 847 * q^22 - 2272 * q^23 - 2079 * q^24 + 3422 * q^26 - 2187 * q^27 - 4592 * q^28 - 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 3267 * q^33 + 2506 * q^34 + 2025 * q^36 + 9126 * q^37 - 13076 * q^38 - 5886 * q^39 - 8758 * q^41 + 6552 * q^42 + 14672 * q^43 - 3025 * q^44 - 28768 * q^46 + 19392 * q^47 + 3735 * q^48 - 26629 * q^49 - 21294 * q^51 + 61506 * q^52 + 4598 * q^53 - 5103 * q^54 + 2688 * q^56 + 25848 * q^57 - 8550 * q^58 - 9348 * q^59 - 60078 * q^61 + 14096 * q^62 + 13932 * q^63 - 7087 * q^64 + 7623 * q^66 + 38468 * q^67 - 59778 * q^68 + 20448 * q^69 - 74032 * q^71 + 18711 * q^72 + 44442 * q^73 + 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 30798 * q^78 - 108116 * q^79 + 19683 * q^81 + 92230 * q^82 + 81892 * q^83 + 41328 * q^84 + 126412 * q^86 + 69642 * q^87 - 27951 * q^88 + 167342 * q^89 - 31832 * q^91 - 72960 * q^92 - 5112 * q^93 + 12728 * q^94 + 9009 * q^96 - 159702 * q^97 - 163121 * q^98 - 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 35$$ v^2 - 35
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 35$$ b2 + 35

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
 −7.17710 0.921799 7.25531
−5.17710 −9.00000 −5.19759 0 46.5939 123.437 192.576 81.0000 0
1.2 2.92180 −9.00000 −23.4631 0 −26.2962 85.0105 −162.052 81.0000 0
1.3 9.25531 −9.00000 53.6607 0 −83.2977 −36.4478 200.476 81.0000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$1$$

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 825.6.a.j 3
5.b even 2 1 165.6.a.a 3
15.d odd 2 1 495.6.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.a 3 5.b even 2 1
495.6.a.e 3 15.d odd 2 1
825.6.a.j 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 7T_{2}^{2} - 36T_{2} + 140$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 7 T^{2} - 36 T + 140$$
$3$ $$(T + 9)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 172 T^{2} + 2896 T + 382464$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} - 654 T^{2} + \cdots + 317918392$$
$17$ $$T^{3} - 2366 T^{2} + \cdots + 137826264$$
$19$ $$T^{3} + 2872 T^{2} + \cdots + 11543616$$
$23$ $$T^{3} + 2272 T^{2} + \cdots - 3706904576$$
$29$ $$T^{3} + 7738 T^{2} + \cdots + 5220625848$$
$31$ $$T^{3} - 568 T^{2} + \cdots - 289787904$$
$37$ $$T^{3} - 9126 T^{2} + \cdots + 129972509048$$
$41$ $$T^{3} + 8758 T^{2} + \cdots - 1122652557432$$
$43$ $$T^{3} - 14672 T^{2} + \cdots + 518908872384$$
$47$ $$T^{3} - 19392 T^{2} + \cdots + 1508908531200$$
$53$ $$T^{3} - 4598 T^{2} + \cdots - 728896505288$$
$59$ $$T^{3} + 9348 T^{2} + \cdots - 6267836310080$$
$61$ $$T^{3} + 60078 T^{2} + \cdots - 13500896397400$$
$67$ $$T^{3} - 38468 T^{2} + \cdots + 8479952260160$$
$71$ $$T^{3} + 74032 T^{2} + \cdots - 17011990639616$$
$73$ $$T^{3} - 44442 T^{2} + \cdots + 9750515676328$$
$79$ $$T^{3} + 108116 T^{2} + \cdots + 9069370346752$$
$83$ $$T^{3} + \cdots + 739830059345664$$
$89$ $$T^{3} + \cdots - 158914472576552$$
$97$ $$T^{3} + \cdots - 352998320493112$$