# Properties

 Label 825.6.a.g 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

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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.18257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 26x + 8$$ x^3 - x^2 - 26*x + 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + (9 \beta_1 - 9) q^{6} + (6 \beta_{2} + 20 \beta_1 + 14) q^{7} + ( - 2 \beta_{2} - 37 \beta_1 + 21) q^{8} + 81 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + 9 * q^3 + (b2 - b1 - 14) * q^4 + (9*b1 - 9) * q^6 + (6*b2 + 20*b1 + 14) * q^7 + (-2*b2 - 37*b1 + 21) * q^8 + 81 * q^9 $$q + (\beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + (9 \beta_1 - 9) q^{6} + (6 \beta_{2} + 20 \beta_1 + 14) q^{7} + ( - 2 \beta_{2} - 37 \beta_1 + 21) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta_{2} - 9 \beta_1 - 126) q^{12} + ( - 44 \beta_{2} + 24 \beta_1 - 90) q^{13} + (14 \beta_{2} + 68 \beta_1 + 278) q^{14} + ( - 67 \beta_{2} + 35 \beta_1 - 186) q^{16} + (102 \beta_{2} - 236 \beta_1 - 100) q^{17} + (81 \beta_1 - 81) q^{18} + ( - 46 \beta_{2} + 172 \beta_1 - 994) q^{19} + (54 \beta_{2} + 180 \beta_1 + 126) q^{21} + (121 \beta_1 - 121) q^{22} + ( - 64 \beta_{2} - 688 \beta_1 + 464) q^{23} + ( - 18 \beta_{2} - 333 \beta_1 + 189) q^{24} + (68 \beta_{2} - 486 \beta_1 + 850) q^{26} + 729 q^{27} + ( - 138 \beta_{2} - 236 \beta_1 + 318) q^{28} + (330 \beta_{2} + 652 \beta_1 - 1840) q^{29} + ( - 44 \beta_{2} - 56 \beta_1 - 4956) q^{31} + (166 \beta_{2} + 395 \beta_1 + 645) q^{32} + 1089 q^{33} + ( - 338 \beta_{2} + 818 \beta_1 - 4728) q^{34} + (81 \beta_{2} - 81 \beta_1 - 1134) q^{36} + ( - 648 \beta_{2} + 448 \beta_1 + 2130) q^{37} + (218 \beta_{2} - 1408 \beta_1 + 4286) q^{38} + ( - 396 \beta_{2} + 216 \beta_1 - 810) q^{39} + ( - 330 \beta_{2} - 1804 \beta_1 - 2264) q^{41} + (126 \beta_{2} + 612 \beta_1 + 2502) q^{42} + ( - 618 \beta_{2} - 1340 \beta_1 + 11850) q^{43} + (121 \beta_{2} - 121 \beta_1 - 1694) q^{44} + ( - 624 \beta_{2} - 112 \beta_1 - 11648) q^{46} + ( - 364 \beta_{2} + 1544 \beta_1 + 7820) q^{47} + ( - 603 \beta_{2} + 315 \beta_1 - 1674) q^{48} + (280 \beta_{2} + 2832 \beta_1 - 5935) q^{49} + (918 \beta_{2} - 2124 \beta_1 - 900) q^{51} + (854 \beta_{2} + 694 \beta_1 - 6776) q^{52} + (1196 \beta_{2} + 360 \beta_1 + 7126) q^{53} + (729 \beta_1 - 729) q^{54} + ( - 546 \beta_{2} - 3100 \beta_1 - 12122) q^{56} + ( - 414 \beta_{2} + 1548 \beta_1 - 8946) q^{57} + (322 \beta_{2} + 1130 \beta_1 + 10284) q^{58} + ( - 2864 \beta_{2} - 4928 \beta_1 - 1988) q^{59} + ( - 616 \beta_{2} + 432 \beta_1 + 8262) q^{61} + ( - 12 \beta_{2} - 5352 \beta_1 + 4356) q^{62} + (486 \beta_{2} + 1620 \beta_1 + 1134) q^{63} + (2373 \beta_{2} + 1019 \beta_1 + 10694) q^{64} + (1089 \beta_1 - 1089) q^{66} + (64 \beta_{2} - 3360 \beta_1 - 4524) q^{67} + ( - 2108 \beta_{2} - 218 \beta_1 + 24538) q^{68} + ( - 576 \beta_{2} - 6192 \beta_1 + 4176) q^{69} + (3576 \beta_{2} - 3376 \beta_1 + 1552) q^{71} + ( - 162 \beta_{2} - 2997 \beta_1 + 1701) q^{72} + ( - 1028 \beta_{2} + 1656 \beta_1 - 846) q^{73} + (1096 \beta_{2} - 3702 \beta_1 + 10670) q^{74} + ( - 154 \beta_{2} + 744 \beta_1 + 1842) q^{76} + (726 \beta_{2} + 2420 \beta_1 + 1694) q^{77} + (612 \beta_{2} - 4374 \beta_1 + 7650) q^{78} + ( - 4202 \beta_{2} - 8252 \beta_1 - 8130) q^{79} + 6561 q^{81} + ( - 1474 \beta_{2} - 5234 \beta_1 - 25764) q^{82} + ( - 2800 \beta_{2} + 5744 \beta_1 + 15284) q^{83} + ( - 1242 \beta_{2} - 2124 \beta_1 + 2862) q^{84} + ( - 722 \beta_{2} + 6288 \beta_1 - 29686) q^{86} + (2970 \beta_{2} + 5868 \beta_1 - 16560) q^{87} + ( - 242 \beta_{2} - 4477 \beta_1 + 2541) q^{88} + (2640 \beta_{2} + 8896 \beta_1 - 66838) q^{89} + (1436 \beta_{2} - 5496 \beta_1 - 29716) q^{91} + (2560 \beta_{2} + 4752 \beta_1 - 112) q^{92} + ( - 396 \beta_{2} - 504 \beta_1 - 44604) q^{93} + (1908 \beta_{2} + 4544 \beta_1 + 21340) q^{94} + (1494 \beta_{2} + 3555 \beta_1 + 5805) q^{96} + (48 \beta_{2} + 13696 \beta_1 - 87090) q^{97} + (2552 \beta_{2} - 3415 \beta_1 + 51839) q^{98} + 9801 q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + 9 * q^3 + (b2 - b1 - 14) * q^4 + (9*b1 - 9) * q^6 + (6*b2 + 20*b1 + 14) * q^7 + (-2*b2 - 37*b1 + 21) * q^8 + 81 * q^9 + 121 * q^11 + (9*b2 - 9*b1 - 126) * q^12 + (-44*b2 + 24*b1 - 90) * q^13 + (14*b2 + 68*b1 + 278) * q^14 + (-67*b2 + 35*b1 - 186) * q^16 + (102*b2 - 236*b1 - 100) * q^17 + (81*b1 - 81) * q^18 + (-46*b2 + 172*b1 - 994) * q^19 + (54*b2 + 180*b1 + 126) * q^21 + (121*b1 - 121) * q^22 + (-64*b2 - 688*b1 + 464) * q^23 + (-18*b2 - 333*b1 + 189) * q^24 + (68*b2 - 486*b1 + 850) * q^26 + 729 * q^27 + (-138*b2 - 236*b1 + 318) * q^28 + (330*b2 + 652*b1 - 1840) * q^29 + (-44*b2 - 56*b1 - 4956) * q^31 + (166*b2 + 395*b1 + 645) * q^32 + 1089 * q^33 + (-338*b2 + 818*b1 - 4728) * q^34 + (81*b2 - 81*b1 - 1134) * q^36 + (-648*b2 + 448*b1 + 2130) * q^37 + (218*b2 - 1408*b1 + 4286) * q^38 + (-396*b2 + 216*b1 - 810) * q^39 + (-330*b2 - 1804*b1 - 2264) * q^41 + (126*b2 + 612*b1 + 2502) * q^42 + (-618*b2 - 1340*b1 + 11850) * q^43 + (121*b2 - 121*b1 - 1694) * q^44 + (-624*b2 - 112*b1 - 11648) * q^46 + (-364*b2 + 1544*b1 + 7820) * q^47 + (-603*b2 + 315*b1 - 1674) * q^48 + (280*b2 + 2832*b1 - 5935) * q^49 + (918*b2 - 2124*b1 - 900) * q^51 + (854*b2 + 694*b1 - 6776) * q^52 + (1196*b2 + 360*b1 + 7126) * q^53 + (729*b1 - 729) * q^54 + (-546*b2 - 3100*b1 - 12122) * q^56 + (-414*b2 + 1548*b1 - 8946) * q^57 + (322*b2 + 1130*b1 + 10284) * q^58 + (-2864*b2 - 4928*b1 - 1988) * q^59 + (-616*b2 + 432*b1 + 8262) * q^61 + (-12*b2 - 5352*b1 + 4356) * q^62 + (486*b2 + 1620*b1 + 1134) * q^63 + (2373*b2 + 1019*b1 + 10694) * q^64 + (1089*b1 - 1089) * q^66 + (64*b2 - 3360*b1 - 4524) * q^67 + (-2108*b2 - 218*b1 + 24538) * q^68 + (-576*b2 - 6192*b1 + 4176) * q^69 + (3576*b2 - 3376*b1 + 1552) * q^71 + (-162*b2 - 2997*b1 + 1701) * q^72 + (-1028*b2 + 1656*b1 - 846) * q^73 + (1096*b2 - 3702*b1 + 10670) * q^74 + (-154*b2 + 744*b1 + 1842) * q^76 + (726*b2 + 2420*b1 + 1694) * q^77 + (612*b2 - 4374*b1 + 7650) * q^78 + (-4202*b2 - 8252*b1 - 8130) * q^79 + 6561 * q^81 + (-1474*b2 - 5234*b1 - 25764) * q^82 + (-2800*b2 + 5744*b1 + 15284) * q^83 + (-1242*b2 - 2124*b1 + 2862) * q^84 + (-722*b2 + 6288*b1 - 29686) * q^86 + (2970*b2 + 5868*b1 - 16560) * q^87 + (-242*b2 - 4477*b1 + 2541) * q^88 + (2640*b2 + 8896*b1 - 66838) * q^89 + (1436*b2 - 5496*b1 - 29716) * q^91 + (2560*b2 + 4752*b1 - 112) * q^92 + (-396*b2 - 504*b1 - 44604) * q^93 + (1908*b2 + 4544*b1 + 21340) * q^94 + (1494*b2 + 3555*b1 + 5805) * q^96 + (48*b2 + 13696*b1 - 87090) * q^97 + (2552*b2 - 3415*b1 + 51839) * q^98 + 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9 $$3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22} + 640 q^{23} + 216 q^{24} + 2132 q^{26} + 2187 q^{27} + 580 q^{28} - 4538 q^{29} - 14968 q^{31} + 2496 q^{32} + 3267 q^{33} - 13704 q^{34} - 3402 q^{36} + 6190 q^{37} + 11668 q^{38} - 2610 q^{39} - 8926 q^{41} + 8244 q^{42} + 33592 q^{43} - 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 5310 q^{48} - 14693 q^{49} - 3906 q^{51} - 18780 q^{52} + 22934 q^{53} - 1458 q^{54} - 40012 q^{56} - 25704 q^{57} + 32304 q^{58} - 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 5508 q^{63} + 35474 q^{64} - 2178 q^{66} - 16868 q^{67} + 71288 q^{68} + 5760 q^{69} + 4856 q^{71} + 1944 q^{72} - 1910 q^{73} + 29404 q^{74} + 6116 q^{76} + 8228 q^{77} + 19188 q^{78} - 36844 q^{79} + 19683 q^{81} - 84000 q^{82} + 48796 q^{83} + 5220 q^{84} - 83492 q^{86} - 40842 q^{87} + 2904 q^{88} - 188978 q^{89} - 93208 q^{91} + 6976 q^{92} - 134712 q^{93} + 70472 q^{94} + 22464 q^{96} - 247526 q^{97} + 154654 q^{98} + 29403 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 27 * q^3 - 42 * q^4 - 18 * q^6 + 68 * q^7 + 24 * q^8 + 243 * q^9 + 363 * q^11 - 378 * q^12 - 290 * q^13 + 916 * q^14 - 590 * q^16 - 434 * q^17 - 162 * q^18 - 2856 * q^19 + 612 * q^21 - 242 * q^22 + 640 * q^23 + 216 * q^24 + 2132 * q^26 + 2187 * q^27 + 580 * q^28 - 4538 * q^29 - 14968 * q^31 + 2496 * q^32 + 3267 * q^33 - 13704 * q^34 - 3402 * q^36 + 6190 * q^37 + 11668 * q^38 - 2610 * q^39 - 8926 * q^41 + 8244 * q^42 + 33592 * q^43 - 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 5310 * q^48 - 14693 * q^49 - 3906 * q^51 - 18780 * q^52 + 22934 * q^53 - 1458 * q^54 - 40012 * q^56 - 25704 * q^57 + 32304 * q^58 - 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 5508 * q^63 + 35474 * q^64 - 2178 * q^66 - 16868 * q^67 + 71288 * q^68 + 5760 * q^69 + 4856 * q^71 + 1944 * q^72 - 1910 * q^73 + 29404 * q^74 + 6116 * q^76 + 8228 * q^77 + 19188 * q^78 - 36844 * q^79 + 19683 * q^81 - 84000 * q^82 + 48796 * q^83 + 5220 * q^84 - 83492 * q^86 - 40842 * q^87 + 2904 * q^88 - 188978 * q^89 - 93208 * q^91 + 6976 * q^92 - 134712 * q^93 + 70472 * q^94 + 22464 * q^96 - 247526 * q^97 + 154654 * q^98 + 29403 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 26x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 17$$ v^2 - v - 17
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 17$$ b2 + b1 + 17

## 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
 −4.78415 0.305203 5.47894
−5.78415 9.00000 1.45634 0 −52.0573 −17.6498 176.669 81.0000 0
1.2 −0.694797 9.00000 −31.5173 0 −6.25317 −83.1683 44.1316 81.0000 0
1.3 4.47894 9.00000 −11.9391 0 40.3105 168.818 −196.801 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.g 3
5.b even 2 1 165.6.a.c 3
15.d odd 2 1 495.6.a.b 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2T_{2}^{2} - 25T_{2} - 18$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} + \cdots - 18$$
$3$ $$(T - 9)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 68 T^{2} + \cdots - 247808$$
$11$ $$(T - 121)^{3}$$
$13$ $$T^{3} + 290 T^{2} + \cdots - 132063592$$
$17$ $$T^{3} + \cdots - 2547052488$$
$19$ $$T^{3} + 2856 T^{2} + \cdots + 137703680$$
$23$ $$T^{3} + \cdots + 15777349632$$
$29$ $$T^{3} + \cdots - 44413548456$$
$31$ $$T^{3} + \cdots + 121645522944$$
$37$ $$T^{3} + \cdots - 28013661736$$
$41$ $$T^{3} + \cdots + 119305168392$$
$43$ $$T^{3} + \cdots + 38997547520$$
$47$ $$T^{3} + \cdots + 679997104128$$
$53$ $$T^{3} + \cdots + 4393759072056$$
$59$ $$T^{3} + \cdots - 20798004639936$$
$61$ $$T^{3} + \cdots + 43064794504$$
$67$ $$T^{3} + \cdots - 1826752720192$$
$71$ $$T^{3} + \cdots + 34155066048000$$
$73$ $$T^{3} + \cdots + 163103734088$$
$79$ $$T^{3} + \cdots - 70772253539328$$
$83$ $$T^{3} + \cdots + 70386077185728$$
$89$ $$T^{3} + \cdots - 16088649675432$$
$97$ $$T^{3} + \cdots + 148869121092488$$