# Properties

 Label 825.6.a.c Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 6) q^{2} - 9 q^{3} + (13 \beta + 12) q^{4} + (9 \beta + 54) q^{6} + (62 \beta - 104) q^{7} + ( - 71 \beta + 16) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b - 6) * q^2 - 9 * q^3 + (13*b + 12) * q^4 + (9*b + 54) * q^6 + (62*b - 104) * q^7 + (-71*b + 16) * q^8 + 81 * q^9 $$q + ( - \beta - 6) q^{2} - 9 q^{3} + (13 \beta + 12) q^{4} + (9 \beta + 54) q^{6} + (62 \beta - 104) q^{7} + ( - 71 \beta + 16) q^{8} + 81 q^{9} - 121 q^{11} + ( - 117 \beta - 108) q^{12} + ( - 74 \beta + 102) q^{13} + ( - 330 \beta + 128) q^{14} + (65 \beta + 88) q^{16} + (372 \beta + 178) q^{17} + ( - 81 \beta - 486) q^{18} + (852 \beta - 840) q^{19} + ( - 558 \beta + 936) q^{21} + (121 \beta + 726) q^{22} + ( - 330 \beta + 284) q^{23} + (639 \beta - 144) q^{24} + (416 \beta - 20) q^{26} - 729 q^{27} + (198 \beta + 5200) q^{28} + (1492 \beta - 398) q^{29} + ( - 1600 \beta - 4440) q^{31} + (1729 \beta - 1560) q^{32} + 1089 q^{33} + ( - 2782 \beta - 4044) q^{34} + (1053 \beta + 972) q^{36} + ( - 2816 \beta + 2362) q^{37} + ( - 5124 \beta - 1776) q^{38} + (666 \beta - 918) q^{39} + (8 \beta + 18238) q^{41} + (2970 \beta - 1152) q^{42} + ( - 3112 \beta - 3328) q^{43} + ( - 1573 \beta - 1452) q^{44} + (2026 \beta + 936) q^{46} + ( - 390 \beta - 21676) q^{47} + ( - 585 \beta - 792) q^{48} + ( - 9052 \beta + 24761) q^{49} + ( - 3348 \beta - 1602) q^{51} + ( - 524 \beta - 6472) q^{52} + ( - 7102 \beta + 9638) q^{53} + (729 \beta + 4374) q^{54} + (3974 \beta - 36880) q^{56} + ( - 7668 \beta + 7560) q^{57} + ( - 10046 \beta - 9548) q^{58} + ( - 1980 \beta - 404) q^{59} + ( - 2026 \beta - 11638) q^{61} + (15640 \beta + 39440) q^{62} + (5022 \beta - 8424) q^{63} + ( - 12623 \beta - 7288) q^{64} + ( - 1089 \beta - 6534) q^{66} + ( - 12704 \beta + 26612) q^{67} + (11614 \beta + 40824) q^{68} + (2970 \beta - 2556) q^{69} + (4354 \beta + 13516) q^{71} + ( - 5751 \beta + 1296) q^{72} + (5568 \beta + 20606) q^{73} + (17350 \beta + 8356) q^{74} + (10380 \beta + 78528) q^{76} + ( - 7502 \beta + 12584) q^{77} + ( - 3744 \beta + 180) q^{78} + ( - 11426 \beta - 2712) q^{79} + 6561 q^{81} + ( - 18294 \beta - 109492) q^{82} + (21960 \beta - 50700) q^{83} + ( - 1782 \beta - 46800) q^{84} + (25112 \beta + 44864) q^{86} + ( - 13428 \beta + 3582) q^{87} + (8591 \beta - 1936) q^{88} + ( - 26704 \beta - 13750) q^{89} + (9432 \beta - 47312) q^{91} + ( - 4558 \beta - 30912) q^{92} + (14400 \beta + 39960) q^{93} + (24406 \beta + 133176) q^{94} + ( - 15561 \beta + 14040) q^{96} + (9924 \beta + 115822) q^{97} + (38603 \beta - 76150) q^{98} - 9801 q^{99}+O(q^{100})$$ q + (-b - 6) * q^2 - 9 * q^3 + (13*b + 12) * q^4 + (9*b + 54) * q^6 + (62*b - 104) * q^7 + (-71*b + 16) * q^8 + 81 * q^9 - 121 * q^11 + (-117*b - 108) * q^12 + (-74*b + 102) * q^13 + (-330*b + 128) * q^14 + (65*b + 88) * q^16 + (372*b + 178) * q^17 + (-81*b - 486) * q^18 + (852*b - 840) * q^19 + (-558*b + 936) * q^21 + (121*b + 726) * q^22 + (-330*b + 284) * q^23 + (639*b - 144) * q^24 + (416*b - 20) * q^26 - 729 * q^27 + (198*b + 5200) * q^28 + (1492*b - 398) * q^29 + (-1600*b - 4440) * q^31 + (1729*b - 1560) * q^32 + 1089 * q^33 + (-2782*b - 4044) * q^34 + (1053*b + 972) * q^36 + (-2816*b + 2362) * q^37 + (-5124*b - 1776) * q^38 + (666*b - 918) * q^39 + (8*b + 18238) * q^41 + (2970*b - 1152) * q^42 + (-3112*b - 3328) * q^43 + (-1573*b - 1452) * q^44 + (2026*b + 936) * q^46 + (-390*b - 21676) * q^47 + (-585*b - 792) * q^48 + (-9052*b + 24761) * q^49 + (-3348*b - 1602) * q^51 + (-524*b - 6472) * q^52 + (-7102*b + 9638) * q^53 + (729*b + 4374) * q^54 + (3974*b - 36880) * q^56 + (-7668*b + 7560) * q^57 + (-10046*b - 9548) * q^58 + (-1980*b - 404) * q^59 + (-2026*b - 11638) * q^61 + (15640*b + 39440) * q^62 + (5022*b - 8424) * q^63 + (-12623*b - 7288) * q^64 + (-1089*b - 6534) * q^66 + (-12704*b + 26612) * q^67 + (11614*b + 40824) * q^68 + (2970*b - 2556) * q^69 + (4354*b + 13516) * q^71 + (-5751*b + 1296) * q^72 + (5568*b + 20606) * q^73 + (17350*b + 8356) * q^74 + (10380*b + 78528) * q^76 + (-7502*b + 12584) * q^77 + (-3744*b + 180) * q^78 + (-11426*b - 2712) * q^79 + 6561 * q^81 + (-18294*b - 109492) * q^82 + (21960*b - 50700) * q^83 + (-1782*b - 46800) * q^84 + (25112*b + 44864) * q^86 + (-13428*b + 3582) * q^87 + (8591*b - 1936) * q^88 + (-26704*b - 13750) * q^89 + (9432*b - 47312) * q^91 + (-4558*b - 30912) * q^92 + (14400*b + 39960) * q^93 + (24406*b + 133176) * q^94 + (-15561*b + 14040) * q^96 + (9924*b + 115822) * q^97 + (38603*b - 76150) * q^98 - 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 $$2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100})$$ 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 242 * q^11 - 333 * q^12 + 130 * q^13 - 74 * q^14 + 241 * q^16 + 728 * q^17 - 1053 * q^18 - 828 * q^19 + 1314 * q^21 + 1573 * q^22 + 238 * q^23 + 351 * q^24 + 376 * q^26 - 1458 * q^27 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 - 1391 * q^32 + 2178 * q^33 - 10870 * q^34 + 2997 * q^36 + 1908 * q^37 - 8676 * q^38 - 1170 * q^39 + 36484 * q^41 + 666 * q^42 - 9768 * q^43 - 4477 * q^44 + 3898 * q^46 - 43742 * q^47 - 2169 * q^48 + 40470 * q^49 - 6552 * q^51 - 13468 * q^52 + 12174 * q^53 + 9477 * q^54 - 69786 * q^56 + 7452 * q^57 - 29142 * q^58 - 2788 * q^59 - 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 - 14157 * q^66 + 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 + 34062 * q^74 + 167436 * q^76 + 17666 * q^77 - 3384 * q^78 - 16850 * q^79 + 13122 * q^81 - 237278 * q^82 - 79440 * q^83 - 95382 * q^84 + 114840 * q^86 - 6264 * q^87 + 4719 * q^88 - 54204 * q^89 - 85192 * q^91 - 66382 * q^92 + 94320 * q^93 + 290758 * q^94 + 12519 * q^96 + 241568 * q^97 - 113697 * q^98 - 19602 * 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
 3.37228 −2.37228
−9.37228 −9.00000 55.8397 0 84.3505 105.081 −223.432 81.0000 0
1.2 −3.62772 −9.00000 −18.8397 0 32.6495 −251.081 184.432 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.c 2
5.b even 2 1 33.6.a.e 2
15.d odd 2 1 99.6.a.d 2
20.d odd 2 1 528.6.a.o 2
55.d odd 2 1 363.6.a.f 2
165.d even 2 1 1089.6.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 5.b even 2 1
99.6.a.d 2 15.d odd 2 1
363.6.a.f 2 55.d odd 2 1
528.6.a.o 2 20.d odd 2 1
825.6.a.c 2 1.a even 1 1 trivial
1089.6.a.p 2 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 13T_{2} + 34$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 13T + 34$$
$3$ $$(T + 9)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 146T - 26384$$
$11$ $$(T + 121)^{2}$$
$13$ $$T^{2} - 130T - 40952$$
$17$ $$T^{2} - 728 T - 1009172$$
$19$ $$T^{2} + 828 T - 5817312$$
$23$ $$T^{2} - 238T - 884264$$
$29$ $$T^{2} - 696 T - 18243924$$
$31$ $$T^{2} + 10480 T + 6337600$$
$37$ $$T^{2} - 1908 T - 64511196$$
$41$ $$T^{2} - 36484 T + 332770036$$
$43$ $$T^{2} + 9768 T - 56044032$$
$47$ $$T^{2} + 43742 T + 477085816$$
$53$ $$T^{2} - 12174 T - 379065264$$
$59$ $$T^{2} + 2788 T - 30400064$$
$61$ $$T^{2} + 25302 T + 126184224$$
$67$ $$T^{2} - 40520 T - 921013232$$
$71$ $$T^{2} - 31386 T + 89872392$$
$73$ $$T^{2} - 46780 T + 291320452$$
$79$ $$T^{2} + 16850 T - 1006085552$$
$83$ $$T^{2} + 79440 T - 2400814800$$
$89$ $$T^{2} + 54204 T - 5148586428$$
$97$ $$T^{2} - 241568 T + 13776267004$$
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