# Properties

 Label 825.6.a.i 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3368.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 15x + 11$$ x^3 - x^2 - 15*x + 11 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} - 9 q^{3} + (4 \beta_{2} + 8) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (11 \beta_{2} - 14 \beta_1 + 69) q^{7} + (8 \beta_{2} - 4 \beta_1 - 12) q^{8} + 81 q^{9}+O(q^{10})$$ q + (b1 + 1) * q^2 - 9 * q^3 + (4*b2 + 8) * q^4 + (-9*b1 - 9) * q^6 + (11*b2 - 14*b1 + 69) * q^7 + (8*b2 - 4*b1 - 12) * q^8 + 81 * q^9 $$q + (\beta_1 + 1) q^{2} - 9 q^{3} + (4 \beta_{2} + 8) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (11 \beta_{2} - 14 \beta_1 + 69) q^{7} + (8 \beta_{2} - 4 \beta_1 - 12) q^{8} + 81 q^{9} + 121 q^{11} + ( - 36 \beta_{2} - 72) q^{12} + (31 \beta_{2} + 25 \beta_1 - 152) q^{13} + ( - 34 \beta_{2} + 138 \beta_1 - 444) q^{14} + ( - 128 \beta_{2} + 32 \beta_1 - 400) q^{16} + (57 \beta_{2} - 34 \beta_1 + 81) q^{17} + (81 \beta_1 + 81) q^{18} + (136 \beta_{2} + 119 \beta_1 - 1351) q^{19} + ( - 99 \beta_{2} + 126 \beta_1 - 621) q^{21} + (121 \beta_1 + 121) q^{22} + ( - 164 \beta_{2} - 372 \beta_1 + 2284) q^{23} + ( - 72 \beta_{2} + 36 \beta_1 + 108) q^{24} + (162 \beta_{2} - 22 \beta_1 + 916) q^{26} - 729 q^{27} + (132 \beta_{2} - 304 \beta_1 + 2628) q^{28} + (580 \beta_{2} + 93 \beta_1 + 1185) q^{29} + (764 \beta_{2} - 920 \beta_1 - 764) q^{31} + ( - 384 \beta_{2} - 944 \beta_1 + 848) q^{32} - 1089 q^{33} + ( - 22 \beta_{2} + 400 \beta_1 - 1074) q^{34} + (324 \beta_{2} + 648) q^{36} + (1312 \beta_{2} - 410 \beta_1 - 1324) q^{37} + (748 \beta_{2} - 790 \beta_1 + 3698) q^{38} + ( - 279 \beta_{2} - 225 \beta_1 + 1368) q^{39} + (140 \beta_{2} - 521 \beta_1 + 3331) q^{41} + (306 \beta_{2} - 1242 \beta_1 + 3996) q^{42} + (37 \beta_{2} + 2 \beta_1 + 11831) q^{43} + (484 \beta_{2} + 968) q^{44} + ( - 1816 \beta_{2} + 1836 \beta_1 - 12716) q^{46} + ( - 1004 \beta_{2} + 640 \beta_1 + 1248) q^{47} + (1152 \beta_{2} - 288 \beta_1 + 3600) q^{48} + (1510 \beta_{2} - 3622 \beta_1 + 845) q^{49} + ( - 513 \beta_{2} + 306 \beta_1 - 729) q^{51} + ( - 756 \beta_{2} + 948 \beta_1 + 5408) q^{52} + (2746 \beta_{2} + 2226 \beta_1 + 4102) q^{53} + ( - 729 \beta_1 - 729) q^{54} + (136 \beta_{2} - 824 \beta_1 + 5376) q^{56} + ( - 1224 \beta_{2} - 1071 \beta_1 + 12159) q^{57} + (1532 \beta_{2} + 3992 \beta_1 + 6552) q^{58} + (844 \beta_{2} + 3292 \beta_1 - 26476) q^{59} + (916 \beta_{2} - 3326 \beta_1 - 22180) q^{61} + ( - 2152 \beta_{2} + 3976 \beta_1 - 34352) q^{62} + (891 \beta_{2} - 1134 \beta_1 + 5589) q^{63} + ( - 448 \beta_{2} - 1152 \beta_1 - 24320) q^{64} + ( - 1089 \beta_1 - 1089) q^{66} + (4474 \beta_{2} - 496 \beta_1 + 13726) q^{67} + ( - 268 \beta_{2} - 496 \beta_1 + 11868) q^{68} + (1476 \beta_{2} + 3348 \beta_1 - 20556) q^{69} + ( - 26 \beta_{2} + 1080 \beta_1 - 21206) q^{71} + (648 \beta_{2} - 324 \beta_1 - 972) q^{72} + ( - 3773 \beta_{2} + 8443 \beta_1 + 3802) q^{73} + (984 \beta_{2} + 5646 \beta_1 - 13378) q^{74} + ( - 6016 \beta_{2} + 4420 \beta_1 + 18364) q^{76} + (1331 \beta_{2} - 1694 \beta_1 + 8349) q^{77} + ( - 1458 \beta_{2} + 198 \beta_1 - 8244) q^{78} + (10742 \beta_{2} - 5831 \beta_1 + 9101) q^{79} + 6561 q^{81} + ( - 1804 \beta_{2} + 4552 \beta_1 - 16568) q^{82} + (2095 \beta_{2} + 4559 \beta_1 + 34496) q^{83} + ( - 1188 \beta_{2} + 2736 \beta_1 - 23652) q^{84} + (82 \beta_{2} + 12014 \beta_1 + 12020) q^{86} + ( - 5220 \beta_{2} - 837 \beta_1 - 10665) q^{87} + (968 \beta_{2} - 484 \beta_1 - 1452) q^{88} + ( - 12390 \beta_{2} + 2204 \beta_1 + 24872) q^{89} + ( - 2456 \beta_{2} + 4440 \beta_1 - 7224) q^{91} + (8960 \beta_{2} - 11728 \beta_1 - 19648) q^{92} + ( - 6876 \beta_{2} + 8280 \beta_1 + 6876) q^{93} + (552 \beta_{2} - 4412 \beta_1 + 23196) q^{94} + (3456 \beta_{2} + 8496 \beta_1 - 7632) q^{96} + ( - 3142 \beta_{2} - 10816 \beta_1 + 104024) q^{97} + ( - 11468 \beta_{2} + 12017 \beta_1 - 135883) q^{98} + 9801 q^{99}+O(q^{100})$$ q + (b1 + 1) * q^2 - 9 * q^3 + (4*b2 + 8) * q^4 + (-9*b1 - 9) * q^6 + (11*b2 - 14*b1 + 69) * q^7 + (8*b2 - 4*b1 - 12) * q^8 + 81 * q^9 + 121 * q^11 + (-36*b2 - 72) * q^12 + (31*b2 + 25*b1 - 152) * q^13 + (-34*b2 + 138*b1 - 444) * q^14 + (-128*b2 + 32*b1 - 400) * q^16 + (57*b2 - 34*b1 + 81) * q^17 + (81*b1 + 81) * q^18 + (136*b2 + 119*b1 - 1351) * q^19 + (-99*b2 + 126*b1 - 621) * q^21 + (121*b1 + 121) * q^22 + (-164*b2 - 372*b1 + 2284) * q^23 + (-72*b2 + 36*b1 + 108) * q^24 + (162*b2 - 22*b1 + 916) * q^26 - 729 * q^27 + (132*b2 - 304*b1 + 2628) * q^28 + (580*b2 + 93*b1 + 1185) * q^29 + (764*b2 - 920*b1 - 764) * q^31 + (-384*b2 - 944*b1 + 848) * q^32 - 1089 * q^33 + (-22*b2 + 400*b1 - 1074) * q^34 + (324*b2 + 648) * q^36 + (1312*b2 - 410*b1 - 1324) * q^37 + (748*b2 - 790*b1 + 3698) * q^38 + (-279*b2 - 225*b1 + 1368) * q^39 + (140*b2 - 521*b1 + 3331) * q^41 + (306*b2 - 1242*b1 + 3996) * q^42 + (37*b2 + 2*b1 + 11831) * q^43 + (484*b2 + 968) * q^44 + (-1816*b2 + 1836*b1 - 12716) * q^46 + (-1004*b2 + 640*b1 + 1248) * q^47 + (1152*b2 - 288*b1 + 3600) * q^48 + (1510*b2 - 3622*b1 + 845) * q^49 + (-513*b2 + 306*b1 - 729) * q^51 + (-756*b2 + 948*b1 + 5408) * q^52 + (2746*b2 + 2226*b1 + 4102) * q^53 + (-729*b1 - 729) * q^54 + (136*b2 - 824*b1 + 5376) * q^56 + (-1224*b2 - 1071*b1 + 12159) * q^57 + (1532*b2 + 3992*b1 + 6552) * q^58 + (844*b2 + 3292*b1 - 26476) * q^59 + (916*b2 - 3326*b1 - 22180) * q^61 + (-2152*b2 + 3976*b1 - 34352) * q^62 + (891*b2 - 1134*b1 + 5589) * q^63 + (-448*b2 - 1152*b1 - 24320) * q^64 + (-1089*b1 - 1089) * q^66 + (4474*b2 - 496*b1 + 13726) * q^67 + (-268*b2 - 496*b1 + 11868) * q^68 + (1476*b2 + 3348*b1 - 20556) * q^69 + (-26*b2 + 1080*b1 - 21206) * q^71 + (648*b2 - 324*b1 - 972) * q^72 + (-3773*b2 + 8443*b1 + 3802) * q^73 + (984*b2 + 5646*b1 - 13378) * q^74 + (-6016*b2 + 4420*b1 + 18364) * q^76 + (1331*b2 - 1694*b1 + 8349) * q^77 + (-1458*b2 + 198*b1 - 8244) * q^78 + (10742*b2 - 5831*b1 + 9101) * q^79 + 6561 * q^81 + (-1804*b2 + 4552*b1 - 16568) * q^82 + (2095*b2 + 4559*b1 + 34496) * q^83 + (-1188*b2 + 2736*b1 - 23652) * q^84 + (82*b2 + 12014*b1 + 12020) * q^86 + (-5220*b2 - 837*b1 - 10665) * q^87 + (968*b2 - 484*b1 - 1452) * q^88 + (-12390*b2 + 2204*b1 + 24872) * q^89 + (-2456*b2 + 4440*b1 - 7224) * q^91 + (8960*b2 - 11728*b1 - 19648) * q^92 + (-6876*b2 + 8280*b1 + 6876) * q^93 + (552*b2 - 4412*b1 + 23196) * q^94 + (3456*b2 + 8496*b1 - 7632) * q^96 + (-3142*b2 - 10816*b1 + 104024) * q^97 + (-11468*b2 + 12017*b1 - 135883) * q^98 + 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 $$3 q + 2 q^{2} - 27 q^{3} + 28 q^{4} - 18 q^{6} + 232 q^{7} - 24 q^{8} + 243 q^{9} + 363 q^{11} - 252 q^{12} - 450 q^{13} - 1504 q^{14} - 1360 q^{16} + 334 q^{17} + 162 q^{18} - 4036 q^{19} - 2088 q^{21} + 242 q^{22} + 7060 q^{23} + 216 q^{24} + 2932 q^{26} - 2187 q^{27} + 8320 q^{28} + 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3267 q^{33} - 3644 q^{34} + 2268 q^{36} - 2250 q^{37} + 12632 q^{38} + 4050 q^{39} + 10654 q^{41} + 13536 q^{42} + 35528 q^{43} + 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 12240 q^{48} + 7667 q^{49} - 3006 q^{51} + 14520 q^{52} + 12826 q^{53} - 1458 q^{54} + 17088 q^{56} + 36324 q^{57} + 17196 q^{58} - 81876 q^{59} - 62298 q^{61} - 109184 q^{62} + 18792 q^{63} - 72256 q^{64} - 2178 q^{66} + 46148 q^{67} + 35832 q^{68} - 63540 q^{69} - 64724 q^{71} - 1944 q^{72} - 810 q^{73} - 44796 q^{74} + 44656 q^{76} + 28072 q^{77} - 26388 q^{78} + 43876 q^{79} + 19683 q^{81} - 56060 q^{82} + 101024 q^{83} - 74880 q^{84} + 24128 q^{86} - 36378 q^{87} - 2904 q^{88} + 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 5472 q^{93} + 74552 q^{94} - 27936 q^{96} + 319746 q^{97} - 431134 q^{98} + 29403 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 - 27 * q^3 + 28 * q^4 - 18 * q^6 + 232 * q^7 - 24 * q^8 + 243 * q^9 + 363 * q^11 - 252 * q^12 - 450 * q^13 - 1504 * q^14 - 1360 * q^16 + 334 * q^17 + 162 * q^18 - 4036 * q^19 - 2088 * q^21 + 242 * q^22 + 7060 * q^23 + 216 * q^24 + 2932 * q^26 - 2187 * q^27 + 8320 * q^28 + 4042 * q^29 - 608 * q^31 + 3104 * q^32 - 3267 * q^33 - 3644 * q^34 + 2268 * q^36 - 2250 * q^37 + 12632 * q^38 + 4050 * q^39 + 10654 * q^41 + 13536 * q^42 + 35528 * q^43 + 3388 * q^44 - 41800 * q^46 + 2100 * q^47 + 12240 * q^48 + 7667 * q^49 - 3006 * q^51 + 14520 * q^52 + 12826 * q^53 - 1458 * q^54 + 17088 * q^56 + 36324 * q^57 + 17196 * q^58 - 81876 * q^59 - 62298 * q^61 - 109184 * q^62 + 18792 * q^63 - 72256 * q^64 - 2178 * q^66 + 46148 * q^67 + 35832 * q^68 - 63540 * q^69 - 64724 * q^71 - 1944 * q^72 - 810 * q^73 - 44796 * q^74 + 44656 * q^76 + 28072 * q^77 - 26388 * q^78 + 43876 * q^79 + 19683 * q^81 - 56060 * q^82 + 101024 * q^83 - 74880 * q^84 + 24128 * q^86 - 36378 * q^87 - 2904 * q^88 + 60022 * q^89 - 28568 * q^91 - 38256 * q^92 + 5472 * q^93 + 74552 * q^94 - 27936 * q^96 + 319746 * q^97 - 431134 * q^98 + 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15x + 11$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 10$$ v^2 - 10
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 10$$ b2 + 10

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.76300 0.723686 4.03932
−7.52601 −9.00000 24.6408 0 67.7341 234.126 55.3856 81.0000 0
1.2 1.44737 −9.00000 −29.9051 0 −13.0263 −41.5023 −89.5997 81.0000 0
1.3 8.07863 −9.00000 33.2643 0 −72.7077 39.3760 10.2141 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.i 3
5.b even 2 1 165.6.a.b 3
15.d odd 2 1 495.6.a.d 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2T_{2}^{2} - 60T_{2} + 88$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} + \cdots + 88$$
$3$ $$(T + 9)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 232 T^{2} + \cdots + 382608$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} + 450 T^{2} + \cdots - 22659488$$
$17$ $$T^{3} - 334 T^{2} + \cdots + 57782448$$
$19$ $$T^{3} + \cdots - 1630951200$$
$23$ $$T^{3} + \cdots + 24275701568$$
$29$ $$T^{3} + \cdots + 65949214584$$
$31$ $$T^{3} + \cdots - 211578448896$$
$37$ $$T^{3} + \cdots + 431879868536$$
$41$ $$T^{3} + \cdots - 7803557208$$
$43$ $$T^{3} + \cdots - 1659712050000$$
$47$ $$T^{3} + \cdots - 52162385088$$
$53$ $$T^{3} + \cdots - 2687939232856$$
$59$ $$T^{3} + \cdots - 3633753791296$$
$61$ $$T^{3} + \cdots - 12904038746056$$
$67$ $$T^{3} + \cdots + 40648408406912$$
$71$ $$T^{3} + \cdots + 8578136735360$$
$73$ $$T^{3} + \cdots + 144432126809632$$
$79$ $$T^{3} + \cdots + 351884592248992$$
$83$ $$T^{3} + \cdots - 5794291383408$$
$89$ $$T^{3} + \cdots - 246103360939432$$
$97$ $$T^{3} + \cdots - 179909862970168$$