# Properties

 Label 825.6.a.h Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.788.1 Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + 9 q^{3} + ( - 4 \beta_1 - 8) q^{4} + ( - 9 \beta_{2} - 9) q^{6} + (7 \beta_{2} + 31 \beta_1 - 38) q^{7} + (28 \beta_{2} + 8 \beta_1 + 20) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^2 + 9 * q^3 + (-4*b1 - 8) * q^4 + (-9*b2 - 9) * q^6 + (7*b2 + 31*b1 - 38) * q^7 + (28*b2 + 8*b1 + 20) * q^8 + 81 * q^9 $$q + ( - \beta_{2} - 1) q^{2} + 9 q^{3} + ( - 4 \beta_1 - 8) q^{4} + ( - 9 \beta_{2} - 9) q^{6} + (7 \beta_{2} + 31 \beta_1 - 38) q^{7} + (28 \beta_{2} + 8 \beta_1 + 20) q^{8} + 81 q^{9} - 121 q^{11} + ( - 36 \beta_1 - 72) q^{12} + ( - 2 \beta_{2} + 77 \beta_1 + 207) q^{13} + (138 \beta_{2} - 34 \beta_1 + 32) q^{14} + (32 \beta_{2} + 224 \beta_1 - 368) q^{16} + ( - 119 \beta_{2} + 219 \beta_1 + 138) q^{17} + ( - 81 \beta_{2} - 81) q^{18} + (149 \beta_{2} + 206 \beta_1 + 721) q^{19} + (63 \beta_{2} + 279 \beta_1 - 342) q^{21} + (121 \beta_{2} + 121) q^{22} + ( - 150 \beta_{2} + 190 \beta_1 - 1416) q^{23} + (252 \beta_{2} + 72 \beta_1 + 180) q^{24} + (22 \beta_{2} - 162 \beta_1 + 224) q^{26} + 729 q^{27} + ( - 220 \beta_{2} - 372 \beta_1 - 2160) q^{28} + (129 \beta_{2} - 308 \beta_1 + 1801) q^{29} + ( - 32 \beta_{2} - 242 \beta_1 + 2018) q^{31} + (176 \beta_{2} - 576 \beta_1 + 112) q^{32} - 1089 q^{33} + (400 \beta_{2} - 914 \beta_1 + 3694) q^{34} + ( - 324 \beta_1 - 648) q^{36} + ( - 1076 \beta_{2} - 928 \beta_1 - 6290) q^{37} + (46 \beta_{2} + 184 \beta_1 - 3118) q^{38} + ( - 18 \beta_{2} + 693 \beta_1 + 1863) q^{39} + ( - 781 \beta_{2} + 1892 \beta_1 + 8131) q^{41} + (1242 \beta_{2} - 306 \beta_1 + 288) q^{42} + ( - 1903 \beta_{2} - 949 \beta_1 - 7808) q^{43} + (484 \beta_1 + 968) q^{44} + (1836 \beta_{2} - 980 \beta_1 + 5816) q^{46} + ( - 520 \beta_{2} + 598 \beta_1 - 1118) q^{47} + (288 \beta_{2} + 2016 \beta_1 - 3312) q^{48} + ( - 10 \beta_{2} - 196 \beta_1 + 3775) q^{49} + ( - 1071 \beta_{2} + 1971 \beta_1 + 1242) q^{51} + ( - 624 \beta_{2} - 2052 \beta_1 - 8164) q^{52} + (3174 \beta_{2} + 2146 \beta_1 - 3606) q^{53} + ( - 729 \beta_{2} - 729) q^{54} + ( - 3592 \beta_{2} + 952 \beta_1 + 4336) q^{56} + (1341 \beta_{2} + 1854 \beta_1 + 6489) q^{57} + ( - 2596 \beta_{2} + 1132 \beta_1 - 6308) q^{58} + ( - 1822 \beta_{2} - 2954 \beta_1 + 5732) q^{59} + ( - 2786 \beta_{2} + 1694 \beta_1 + 2410) q^{61} + ( - 2776 \beta_{2} + 356 \beta_1 - 2492) q^{62} + (567 \beta_{2} + 2511 \beta_1 - 3078) q^{63} + ( - 2688 \beta_{2} - 5312 \beta_1 + 4736) q^{64} + (1089 \beta_{2} + 1089) q^{66} + (5936 \beta_{2} + 4430 \beta_1 + 31566) q^{67} + ( - 2228 \beta_{2} - 3580 \beta_1 - 21880) q^{68} + ( - 1350 \beta_{2} + 1710 \beta_1 - 12744) q^{69} + ( - 9528 \beta_{2} + 3010 \beta_1 + 14670) q^{71} + (2268 \beta_{2} + 648 \beta_1 + 1620) q^{72} + ( - 1244 \beta_{2} - 12157 \beta_1 + 13479) q^{73} + (2430 \beta_{2} - 2448 \beta_1 + 26398) q^{74} + ( - 1052 \beta_{2} - 6776 \beta_1 - 20092) q^{76} + ( - 847 \beta_{2} - 3751 \beta_1 + 4598) q^{77} + (198 \beta_{2} - 1458 \beta_1 + 2016) q^{78} + (3973 \beta_{2} - 770 \beta_1 + 3189) q^{79} + 6561 q^{81} + ( - 3236 \beta_{2} - 6908 \beta_1 + 19292) q^{82} + ( - 1382 \beta_{2} - 10151 \beta_1 - 35935) q^{83} + ( - 1980 \beta_{2} - 3348 \beta_1 - 19440) q^{84} + (3058 \beta_{2} - 5714 \beta_1 + 46832) q^{86} + (1161 \beta_{2} - 2772 \beta_1 + 16209) q^{87} + ( - 3388 \beta_{2} - 968 \beta_1 - 2420) q^{88} + ( - 10502 \beta_{2} + 6468 \beta_1 - 14324) q^{89} + (4896 \beta_{2} + 8798 \beta_1 + 39554) q^{91} + ( - 2120 \beta_{2} + 3224 \beta_1 - 7632) q^{92} + ( - 288 \beta_{2} - 2178 \beta_1 + 18162) q^{93} + (2392 \beta_{2} - 3276 \beta_1 + 16068) q^{94} + (1584 \beta_{2} - 5184 \beta_1 + 1008) q^{96} + ( - 3184 \beta_{2} + 20686 \beta_1 + 40248) q^{97} + ( - 4373 \beta_{2} + 352 \beta_1 - 4525) q^{98} - 9801 q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^2 + 9 * q^3 + (-4*b1 - 8) * q^4 + (-9*b2 - 9) * q^6 + (7*b2 + 31*b1 - 38) * q^7 + (28*b2 + 8*b1 + 20) * q^8 + 81 * q^9 - 121 * q^11 + (-36*b1 - 72) * q^12 + (-2*b2 + 77*b1 + 207) * q^13 + (138*b2 - 34*b1 + 32) * q^14 + (32*b2 + 224*b1 - 368) * q^16 + (-119*b2 + 219*b1 + 138) * q^17 + (-81*b2 - 81) * q^18 + (149*b2 + 206*b1 + 721) * q^19 + (63*b2 + 279*b1 - 342) * q^21 + (121*b2 + 121) * q^22 + (-150*b2 + 190*b1 - 1416) * q^23 + (252*b2 + 72*b1 + 180) * q^24 + (22*b2 - 162*b1 + 224) * q^26 + 729 * q^27 + (-220*b2 - 372*b1 - 2160) * q^28 + (129*b2 - 308*b1 + 1801) * q^29 + (-32*b2 - 242*b1 + 2018) * q^31 + (176*b2 - 576*b1 + 112) * q^32 - 1089 * q^33 + (400*b2 - 914*b1 + 3694) * q^34 + (-324*b1 - 648) * q^36 + (-1076*b2 - 928*b1 - 6290) * q^37 + (46*b2 + 184*b1 - 3118) * q^38 + (-18*b2 + 693*b1 + 1863) * q^39 + (-781*b2 + 1892*b1 + 8131) * q^41 + (1242*b2 - 306*b1 + 288) * q^42 + (-1903*b2 - 949*b1 - 7808) * q^43 + (484*b1 + 968) * q^44 + (1836*b2 - 980*b1 + 5816) * q^46 + (-520*b2 + 598*b1 - 1118) * q^47 + (288*b2 + 2016*b1 - 3312) * q^48 + (-10*b2 - 196*b1 + 3775) * q^49 + (-1071*b2 + 1971*b1 + 1242) * q^51 + (-624*b2 - 2052*b1 - 8164) * q^52 + (3174*b2 + 2146*b1 - 3606) * q^53 + (-729*b2 - 729) * q^54 + (-3592*b2 + 952*b1 + 4336) * q^56 + (1341*b2 + 1854*b1 + 6489) * q^57 + (-2596*b2 + 1132*b1 - 6308) * q^58 + (-1822*b2 - 2954*b1 + 5732) * q^59 + (-2786*b2 + 1694*b1 + 2410) * q^61 + (-2776*b2 + 356*b1 - 2492) * q^62 + (567*b2 + 2511*b1 - 3078) * q^63 + (-2688*b2 - 5312*b1 + 4736) * q^64 + (1089*b2 + 1089) * q^66 + (5936*b2 + 4430*b1 + 31566) * q^67 + (-2228*b2 - 3580*b1 - 21880) * q^68 + (-1350*b2 + 1710*b1 - 12744) * q^69 + (-9528*b2 + 3010*b1 + 14670) * q^71 + (2268*b2 + 648*b1 + 1620) * q^72 + (-1244*b2 - 12157*b1 + 13479) * q^73 + (2430*b2 - 2448*b1 + 26398) * q^74 + (-1052*b2 - 6776*b1 - 20092) * q^76 + (-847*b2 - 3751*b1 + 4598) * q^77 + (198*b2 - 1458*b1 + 2016) * q^78 + (3973*b2 - 770*b1 + 3189) * q^79 + 6561 * q^81 + (-3236*b2 - 6908*b1 + 19292) * q^82 + (-1382*b2 - 10151*b1 - 35935) * q^83 + (-1980*b2 - 3348*b1 - 19440) * q^84 + (3058*b2 - 5714*b1 + 46832) * q^86 + (1161*b2 - 2772*b1 + 16209) * q^87 + (-3388*b2 - 968*b1 - 2420) * q^88 + (-10502*b2 + 6468*b1 - 14324) * q^89 + (4896*b2 + 8798*b1 + 39554) * q^91 + (-2120*b2 + 3224*b1 - 7632) * q^92 + (-288*b2 - 2178*b1 + 18162) * q^93 + (2392*b2 - 3276*b1 + 16068) * q^94 + (1584*b2 - 5184*b1 + 1008) * q^96 + (-3184*b2 + 20686*b1 + 40248) * q^97 + (-4373*b2 + 352*b1 - 4525) * q^98 - 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 $$3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * q^98 - 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 4\nu - 9$$ 2*v^2 - 4*v - 9
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 11 ) / 2$$ (b2 + 2*b1 + 11) / 2

## 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
 −1.87740 3.35386 −0.476452
−6.55890 9.00000 11.0192 0 −59.0301 −146.487 137.611 81.0000 0
1.2 −1.08127 9.00000 −30.8308 0 −9.73147 139.508 67.9374 81.0000 0
1.3 5.64018 9.00000 −0.188384 0 50.7616 −145.021 −181.548 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.h 3
5.b even 2 1 165.6.a.d 3
15.d odd 2 1 495.6.a.c 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2T_{2}^{2} - 36T_{2} - 40$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 36 T - 40$$
$3$ $$(T - 9)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 152 T^{2} - 19424 T - 2963664$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} - 546 T^{2} - 76748 T + 7691848$$
$17$ $$T^{3} - 314 T^{2} + \cdots + 1079472216$$
$19$ $$T^{3} - 1808 T^{2} + \cdots + 729480096$$
$23$ $$T^{3} + 4288 T^{2} + \cdots + 857355136$$
$29$ $$T^{3} - 5582 T^{2} + \cdots - 329440872$$
$31$ $$T^{3} - 6328 T^{2} + \cdots - 5126546304$$
$37$ $$T^{3} + 16866 T^{2} + \cdots - 244700027368$$
$41$ $$T^{3} - 23282 T^{2} + \cdots + 945181300968$$
$43$ $$T^{3} + 20572 T^{2} + \cdots - 1240285492944$$
$47$ $$T^{3} + 3432 T^{2} + \cdots + 18016103040$$
$53$ $$T^{3} + 16138 T^{2} + \cdots + 985333601848$$
$59$ $$T^{3} - 21972 T^{2} + \cdots + 2567903224000$$
$61$ $$T^{3} - 8322 T^{2} + \cdots + 4408473611240$$
$67$ $$T^{3} - 84332 T^{2} + \cdots + 41154720036800$$
$71$ $$T^{3} + \cdots + 117803610062464$$
$73$ $$T^{3} + \cdots + 163971863295832$$
$79$ $$T^{3} - 6364 T^{2} + \cdots - 554174036768$$
$83$ $$T^{3} + 96272 T^{2} + \cdots - 3029676562224$$
$89$ $$T^{3} + 38938 T^{2} + \cdots + 96218735089528$$
$97$ $$T^{3} + \cdots + 251540556847112$$