# Properties

 Label 825.6.a.m Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576$$ x^7 - 2*x^6 - 152*x^5 + 358*x^4 + 5771*x^3 - 13444*x^2 - 51316*x + 92576 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 5$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 11) q^{7} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 11 \beta_1 - 25) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + 9 * q^3 + (b2 + b1 + 13) * q^4 + (-9*b1 - 9) * q^6 + (b4 + 2*b3 + b2 + 4*b1 - 11) * q^7 + (-2*b6 - 2*b5 - 2*b4 - 2*b3 - 2*b2 - 11*b1 - 25) * q^8 + 81 * q^9 $$q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (\beta_{2} + \beta_1 + 13) q^{4} + ( - 9 \beta_1 - 9) q^{6} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 11) q^{7} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 11 \beta_1 - 25) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta_{2} + 9 \beta_1 + 117) q^{12} + (5 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 42 \beta_1 + 4) q^{13} + ( - 2 \beta_{6} + 5 \beta_{5} - 13 \beta_{4} - 6 \beta_{3} - 19 \beta_{2} + \cdots - 173) q^{14}+ \cdots + 9801 q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + 9 * q^3 + (b2 + b1 + 13) * q^4 + (-9*b1 - 9) * q^6 + (b4 + 2*b3 + b2 + 4*b1 - 11) * q^7 + (-2*b6 - 2*b5 - 2*b4 - 2*b3 - 2*b2 - 11*b1 - 25) * q^8 + 81 * q^9 + 121 * q^11 + (9*b2 + 9*b1 + 117) * q^12 + (5*b6 + b5 - 2*b4 - b3 - 2*b2 + 42*b1 + 4) * q^13 + (-2*b6 + 5*b5 - 13*b4 - 6*b3 - 19*b2 + 20*b1 - 173) * q^14 + (16*b4 + 4*b3 + 21*b2 + 41*b1 + 53) * q^16 + (9*b4 - 5*b3 - 15*b2 - 11*b1 - 139) * q^17 + (-81*b1 - 81) * q^18 + (13*b6 - 9*b5 + 3*b4 + 2*b3 + 5*b2 - 23*b1 - 539) * q^19 + (9*b4 + 18*b3 + 9*b2 + 36*b1 - 99) * q^21 + (-121*b1 - 121) * q^22 + (-14*b6 + 36*b5 - 12*b4 + 11*b3 - 10*b2 + 111*b1 - 107) * q^23 + (-18*b6 - 18*b5 - 18*b4 - 18*b3 - 18*b2 - 99*b1 - 225) * q^24 + (-2*b6 + 5*b5 + 19*b4 + 16*b3 - 38*b2 - 35*b1 - 1712) * q^26 + 729 * q^27 + (62*b6 + 15*b5 + 67*b4 + 46*b3 + 58*b2 + 356*b1 - 87) * q^28 + (-43*b6 - 19*b5 + 6*b4 - 30*b3 - 50*b2 + 21*b1 - 364) * q^29 + (37*b6 + b5 + 4*b4 + 43*b3 + 4*b2 - 142*b1 - 2434) * q^31 + (22*b6 + 22*b5 - 42*b4 - 42*b3 - 138*b2 - 99*b1 - 1353) * q^32 + 1089 * q^33 + (30*b6 + b5 + 23*b4 - 6*b3 - 31*b2 + 618*b1 + 413) * q^34 + (81*b2 + 81*b1 + 1053) * q^36 + (9*b6 - 17*b5 + 32*b4 - 49*b3 + 46*b2 - 580*b1 - 3151) * q^37 + (-72*b6 + 26*b5 - 14*b4 - 58*b3 + 25*b2 + 423*b1 + 1593) * q^38 + (45*b6 + 9*b5 - 18*b4 - 9*b3 - 18*b2 + 378*b1 + 36) * q^39 + (-38*b6 + 110*b5 - 51*b4 - 57*b3 - 167*b2 + 715*b1 - 2467) * q^41 + (-18*b6 + 45*b5 - 117*b4 - 54*b3 - 171*b2 + 180*b1 - 1557) * q^42 + (59*b6 - 43*b5 + 42*b4 - 55*b3 + 292*b2 + 256*b1 + 456) * q^43 + (121*b2 + 121*b1 + 1573) * q^44 + (192*b6 - 10*b5 - 2*b4 + 212*b3 - 262*b2 + 195*b1 - 4147) * q^46 + (79*b6 - 27*b5 - 12*b4 - 63*b3 - 214*b2 - 88*b1 - 995) * q^47 + (144*b4 + 36*b3 + 189*b2 + 369*b1 + 477) * q^48 + (-191*b6 + 87*b5 - 106*b4 - 206*b3 - 120*b2 + 1387*b1 + 4988) * q^49 + (81*b4 - 45*b3 - 135*b2 - 99*b1 - 1251) * q^51 + (-60*b6 + 77*b5 + 17*b4 + 52*b3 - 94*b2 + 1909*b1 + 2884) * q^52 + (186*b6 - 62*b5 - 278*b4 - 6*b3 - 70*b2 + 756*b1 - 3152) * q^53 + (-729*b1 - 729) * q^54 + (-116*b6 - 127*b5 - 23*b4 - 132*b3 - 709*b2 - 1620*b1 - 9749) * q^56 + (117*b6 - 81*b5 + 27*b4 + 18*b3 + 45*b2 - 207*b1 - 4851) * q^57 + (110*b6 - 31*b5 + 159*b4 + 290*b2 + 2087*b1 - 1938) * q^58 + (-51*b6 - 241*b5 - 66*b4 - 33*b3 - 472*b2 - 1028*b1 + 423) * q^59 + (-352*b6 + 60*b5 - 106*b4 - 199*b3 - 242*b2 - 563*b1 - 3718) * q^61 + (-78*b6 + 195*b5 - 155*b4 - 20*b3 - 104*b2 + 2617*b1 + 9542) * q^62 + (81*b4 + 162*b3 + 81*b2 + 324*b1 - 891) * q^63 + (320*b6 + 128*b5 + 80*b4 + 404*b3 - 171*b2 + 2713*b1 + 5349) * q^64 + (-1089*b1 - 1089) * q^66 + (-352*b6 - 252*b5 + 226*b4 + 109*b3 + 554*b2 + 129*b1 + 2946) * q^67 + (6*b6 + 43*b5 - 241*b4 + 134*b3 - 340*b2 + 758*b1 - 23097) * q^68 + (-126*b6 + 324*b5 - 108*b4 + 99*b3 - 90*b2 + 999*b1 - 963) * q^69 + (378*b6 - 32*b5 - 16*b4 + 99*b3 - 338*b2 - 3429*b1 - 24285) * q^71 + (-162*b6 - 162*b5 - 162*b4 - 162*b3 - 162*b2 - 891*b1 - 2025) * q^72 + (-552*b6 + 92*b5 - 546*b4 - 374*b3 + 266*b2 - 950*b1 - 10924) * q^73 + (-178*b6 - 277*b5 + 17*b4 - 288*b3 + 526*b2 + 1526*b1 + 27627) * q^74 + (-218*b6 - 104*b5 + 56*b4 + 46*b3 - 360*b2 - 2185*b1 - 3905) * q^76 + (121*b4 + 242*b3 + 121*b2 + 484*b1 - 1331) * q^77 + (-18*b6 + 45*b5 + 171*b4 + 144*b3 - 342*b2 - 315*b1 - 15408) * q^78 + (590*b6 + 70*b5 + 111*b4 + 517*b3 - 357*b2 + 241*b1 - 26911) * q^79 + 6561 * q^81 + (850*b6 - 101*b5 + 677*b4 + 978*b3 - 663*b2 + 5360*b1 - 27239) * q^82 + (-1587*b6 - 301*b5 - 596*b4 - 1102*b3 + 262*b2 - 789*b1 + 6948) * q^83 + (558*b6 + 135*b5 + 603*b4 + 414*b3 + 522*b2 + 3204*b1 - 783) * q^84 + (-874*b6 - 701*b5 - 431*b4 - 924*b3 - 536*b2 - 9625*b1 - 12516) * q^86 + (-387*b6 - 171*b5 + 54*b4 - 270*b3 - 450*b2 + 189*b1 - 3276) * q^87 + (-242*b6 - 242*b5 - 242*b4 - 242*b3 - 242*b2 - 1331*b1 - 3025) * q^88 + (-435*b6 - 149*b5 + 348*b4 - 36*b3 - 58*b2 + 405*b1 - 21606) * q^89 + (892*b6 - 248*b5 + 522*b4 + 567*b3 + 1426*b2 - 3841*b1 - 32362) * q^91 + (548*b6 + 434*b5 + 258*b4 + 140*b3 - 547*b2 + 9773*b1 + 3599) * q^92 + (333*b6 + 9*b5 + 36*b4 + 387*b3 + 36*b2 - 1278*b1 - 21906) * q^93 + (162*b6 + 321*b5 + 795*b4 + 368*b3 + 618*b2 + 5960*b1 + 5719) * q^94 + (198*b6 + 198*b5 - 378*b4 - 378*b3 - 1242*b2 - 891*b1 - 12177) * q^96 + (-1120*b6 - 504*b5 - 664*b4 - 530*b3 - 1480*b2 + 2782*b1 + 7877) * q^97 + (970*b6 - 843*b5 + 1191*b4 + 1012*b3 - 28*b2 - 4589*b1 - 67178) * q^98 + 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10})$$ 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9 $$7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100})$$ 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9 + 847 * q^11 + 855 * q^12 + 113 * q^13 - 1212 * q^14 + 499 * q^16 - 1030 * q^17 - 729 * q^18 - 3803 * q^19 - 585 * q^21 - 1089 * q^22 - 514 * q^23 - 1863 * q^24 - 12111 * q^26 + 5103 * q^27 + 342 * q^28 - 2698 * q^29 - 17233 * q^31 - 9943 * q^32 + 7623 * q^33 + 4090 * q^34 + 7695 * q^36 - 23182 * q^37 + 11943 * q^38 + 1017 * q^39 - 16158 * q^41 - 10908 * q^42 + 4249 * q^43 + 11495 * q^44 - 28769 * q^46 - 7580 * q^47 + 4491 * q^48 + 37140 * q^49 - 9270 * q^51 + 23887 * q^52 - 20574 * q^53 - 6561 * q^54 - 73276 * q^56 - 34227 * q^57 - 8733 * q^58 - 364 * q^59 - 28127 * q^61 + 71917 * q^62 - 5265 * q^63 + 43379 * q^64 - 9801 * q^66 + 21493 * q^67 - 160660 * q^68 - 4626 * q^69 - 177084 * q^71 - 16767 * q^72 - 78670 * q^73 + 196750 * q^74 - 32701 * q^76 - 7865 * q^77 - 108999 * q^78 - 187432 * q^79 + 45927 * q^81 - 179552 * q^82 + 44592 * q^83 + 3078 * q^84 - 110433 * q^86 - 24282 * q^87 - 25047 * q^88 - 151168 * q^89 - 230153 * q^91 + 44767 * q^92 - 155097 * q^93 + 54040 * q^94 - 89487 * q^96 + 55589 * q^97 - 478341 * q^98 + 68607 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 44$$ v^2 + v - 44 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + \nu^{4} - 123\nu^{3} - 57\nu^{2} + 2642\nu + 464 ) / 48$$ (v^5 + v^4 - 123*v^3 - 57*v^2 + 2642*v + 464) / 48 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 11\nu^{4} + 171\nu^{3} - 1275\nu^{2} - 5642\nu + 21136 ) / 192$$ (-v^5 + 11*v^4 + 171*v^3 - 1275*v^2 - 5642*v + 21136) / 192 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} - 3\nu^{5} + 133\nu^{4} + 351\nu^{3} - 3716\nu^{2} - 9276\nu + 14048 ) / 192$$ (-v^6 - 3*v^5 + 133*v^4 + 351*v^3 - 3716*v^2 - 9276*v + 14048) / 192 $$\beta_{6}$$ $$=$$ $$( \nu^{6} - 148\nu^{4} + 66\nu^{3} + 5315\nu^{2} - 2754\nu - 37040 ) / 192$$ (v^6 - 148*v^4 + 66*v^3 + 5315*v^2 - 2754*v - 37040) / 192
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 44$$ b2 - b1 + 44 $$\nu^{3}$$ $$=$$ $$2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + 75\beta _1 - 44$$ 2*b6 + 2*b5 + 2*b4 + 2*b3 - b2 + 75*b1 - 44 $$\nu^{4}$$ $$=$$ $$-8\beta_{6} - 8\beta_{5} + 8\beta_{4} - 4\beta_{3} + 115\beta_{2} - 161\beta _1 + 3260$$ -8*b6 - 8*b5 + 8*b4 - 4*b3 + 115*b2 - 161*b1 + 3260 $$\nu^{5}$$ $$=$$ $$254\beta_{6} + 254\beta_{5} + 238\beta_{4} + 298\beta_{3} - 181\beta_{2} + 6687\beta _1 - 6628$$ 254*b6 + 254*b5 + 238*b4 + 298*b3 - 181*b2 + 6687*b1 - 6628 $$\nu^{6}$$ $$=$$ $$-1124\beta_{6} - 1316\beta_{5} + 1052\beta_{4} - 724\beta_{3} + 11771\beta_{2} - 20709\beta _1 + 288564$$ -1124*b6 - 1316*b5 + 1052*b4 - 724*b3 + 11771*b2 - 20709*b1 + 288564

## 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
 9.48549 6.35371 4.42881 1.60892 −3.33276 −6.33680 −10.2074
−10.4855 9.00000 77.9456 0 −94.3694 152.972 −481.762 81.0000 0
1.2 −7.35371 9.00000 22.0770 0 −66.1834 −251.987 72.9708 81.0000 0
1.3 −5.42881 9.00000 −2.52798 0 −48.8593 29.0935 187.446 81.0000 0
1.4 −2.60892 9.00000 −25.1935 0 −23.4803 174.969 149.214 81.0000 0
1.5 2.33276 9.00000 −26.5582 0 20.9949 −146.260 −136.602 81.0000 0
1.6 5.33680 9.00000 −3.51852 0 48.0312 73.9998 −189.555 81.0000 0
1.7 9.20737 9.00000 52.7757 0 82.8663 −97.7871 191.289 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.m 7
5.b even 2 1 825.6.a.o yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.m 7 1.a even 1 1 trivial
825.6.a.o yes 7 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} + 9T_{2}^{6} - 119T_{2}^{5} - 1053T_{2}^{4} + 2894T_{2}^{3} + 27140T_{2}^{2} - 9288T_{2} - 125184$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} + 9 T^{6} - 119 T^{5} + \cdots - 125184$$
$3$ $$(T - 9)^{7}$$
$5$ $$T^{7}$$
$7$ $$T^{7} + \cdots + 207675630158484$$
$11$ $$(T - 121)^{7}$$
$13$ $$T^{7} - 113 T^{6} + \cdots + 30\!\cdots\!36$$
$17$ $$T^{7} + 1030 T^{6} + \cdots + 11\!\cdots\!04$$
$19$ $$T^{7} + 3803 T^{6} + \cdots + 29\!\cdots\!25$$
$23$ $$T^{7} + 514 T^{6} + \cdots + 56\!\cdots\!54$$
$29$ $$T^{7} + 2698 T^{6} + \cdots - 44\!\cdots\!00$$
$31$ $$T^{7} + 17233 T^{6} + \cdots + 17\!\cdots\!76$$
$37$ $$T^{7} + 23182 T^{6} + \cdots + 50\!\cdots\!44$$
$41$ $$T^{7} + 16158 T^{6} + \cdots - 11\!\cdots\!52$$
$43$ $$T^{7} - 4249 T^{6} + \cdots - 15\!\cdots\!00$$
$47$ $$T^{7} + 7580 T^{6} + \cdots - 20\!\cdots\!28$$
$53$ $$T^{7} + 20574 T^{6} + \cdots - 27\!\cdots\!52$$
$59$ $$T^{7} + 364 T^{6} + \cdots - 47\!\cdots\!00$$
$61$ $$T^{7} + 28127 T^{6} + \cdots + 24\!\cdots\!24$$
$67$ $$T^{7} - 21493 T^{6} + \cdots + 12\!\cdots\!16$$
$71$ $$T^{7} + 177084 T^{6} + \cdots - 27\!\cdots\!20$$
$73$ $$T^{7} + 78670 T^{6} + \cdots + 22\!\cdots\!88$$
$79$ $$T^{7} + 187432 T^{6} + \cdots + 77\!\cdots\!00$$
$83$ $$T^{7} - 44592 T^{6} + \cdots - 36\!\cdots\!52$$
$89$ $$T^{7} + 151168 T^{6} + \cdots - 42\!\cdots\!00$$
$97$ $$T^{7} - 55589 T^{6} + \cdots - 15\!\cdots\!23$$