# Properties

 Label 825.4.c.l Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.2230106176.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 41x^{4} + 452x^{2} + 676$$ x^6 + 41*x^4 + 452*x^2 + 676 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (\beta_{5} + \beta_{3} - 7) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{7} + (4 \beta_{4} + 15 \beta_{2} - 3 \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 + 3*b2 * q^3 + (b5 + b3 - 7) * q^4 + (-3*b3 + 3) * q^6 + (-2*b4 - 2*b2 - 2*b1) * q^7 + (4*b4 + 15*b2 - 3*b1) * q^8 - 9 * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (\beta_{5} + \beta_{3} - 7) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{7} + (4 \beta_{4} + 15 \beta_{2} - 3 \beta_1) q^{8} - 9 q^{9} + 11 q^{11} + ( - 3 \beta_{4} - 21 \beta_{2} + 3 \beta_1) q^{12} + ( - 2 \beta_{4} + 4 \beta_{2} + 12 \beta_1) q^{13} + (4 \beta_{5} + 10 \beta_{3} + 22) q^{14} + ( - 7 \beta_{5} - 23 \beta_{3} + 9) q^{16} + (6 \beta_{4} - 76 \beta_{2} - 10 \beta_1) q^{17} + (9 \beta_{2} - 9 \beta_1) q^{18} + (4 \beta_{5} + 2 \beta_{3} - 48) q^{19} + ( - 6 \beta_{5} + 6 \beta_{3} + 6) q^{21} + ( - 11 \beta_{2} + 11 \beta_1) q^{22} + (8 \beta_{4} + 60 \beta_{2} - 20 \beta_1) q^{23} + (12 \beta_{5} + 9 \beta_{3} - 45) q^{24} + (18 \beta_{5} + 4 \beta_{3} - 168) q^{26} - 27 \beta_{2} q^{27} + (6 \beta_{4} + 110 \beta_{2} - 10 \beta_1) q^{28} + (20 \beta_{5} - 34 \beta_{3} - 34) q^{29} + (16 \beta_{5} - 4 \beta_{3} - 24) q^{31} + ( - 12 \beta_{4} - 225 \beta_{2} + 13 \beta_1) q^{32} + 33 \beta_{2} q^{33} + ( - 28 \beta_{5} + 52 \beta_{3} + 76) q^{34} + ( - 9 \beta_{5} - 9 \beta_{3} + 63) q^{36} + ( - 4 \beta_{4} - 122 \beta_{2} + 24 \beta_1) q^{37} + (14 \beta_{4} + 84 \beta_{2} - 64 \beta_1) q^{38} + ( - 6 \beta_{5} - 36 \beta_{3} - 12) q^{39} + (20 \beta_{5} - 2 \beta_{3} - 66) q^{41} + ( - 12 \beta_{4} + 66 \beta_{2} + 30 \beta_1) q^{42} + ( - 14 \beta_{4} + 150 \beta_{2} - 74 \beta_1) q^{43} + (11 \beta_{5} + 11 \beta_{3} - 77) q^{44} + ( - 44 \beta_{5} - 92 \beta_{3} + 356) q^{46} + (28 \beta_{4} - 36 \beta_{2} - 48 \beta_1) q^{47} + (21 \beta_{4} + 27 \beta_{2} - 69 \beta_1) q^{48} + ( - 20 \beta_{5} - 4 \beta_{3} + 51) q^{49} + (18 \beta_{5} + 30 \beta_{3} + 228) q^{51} + (42 \beta_{4} + 292 \beta_{2} - 144 \beta_1) q^{52} + ( - 52 \beta_{4} + 42 \beta_{2} - 32 \beta_1) q^{53} + (27 \beta_{3} - 27) q^{54} + (4 \beta_{5} - 54 \beta_{3} + 438) q^{56} + ( - 12 \beta_{4} - 144 \beta_{2} + 6 \beta_1) q^{57} + (26 \beta_{4} - 402 \beta_{2} - 114 \beta_1) q^{58} + ( - 4 \beta_{5} - 120 \beta_{3} + 308) q^{59} + (44 \beta_{5} + 12 \beta_{3} + 218) q^{61} + (44 \beta_{4} - 88 \beta_1) q^{62} + (18 \beta_{4} + 18 \beta_{2} + 18 \beta_1) q^{63} + ( - 7 \beta_{5} + 89 \beta_{3} - 359) q^{64} + ( - 33 \beta_{3} + 33) q^{66} + ( - 28 \beta_{4} - 128 \beta_{2} - 148 \beta_1) q^{67} + (16 \beta_{4} - 12 \beta_{2} + 108 \beta_1) q^{68} + (24 \beta_{5} + 60 \beta_{3} - 180) q^{69} + (16 \beta_{5} - 104 \beta_{3} - 216) q^{71} + ( - 36 \beta_{4} - 135 \beta_{2} + 27 \beta_1) q^{72} + ( - 14 \beta_{4} - 356 \beta_{2} - 168 \beta_1) q^{73} + (36 \beta_{5} + 138 \beta_{3} - 466) q^{74} + ( - 74 \beta_{5} - 124 \beta_{3} + 624) q^{76} + ( - 22 \beta_{4} - 22 \beta_{2} - 22 \beta_1) q^{77} + ( - 54 \beta_{4} - 504 \beta_{2} + 12 \beta_1) q^{78} + ( - 52 \beta_{5} - 14 \beta_{3} + 524) q^{79} + 81 q^{81} + (58 \beta_{4} + 78 \beta_{2} - 146 \beta_1) q^{82} + ( - 70 \beta_{4} + 582 \beta_{2} + 164 \beta_1) q^{83} + (18 \beta_{5} + 30 \beta_{3} - 330) q^{84} + ( - 32 \beta_{5} - 94 \beta_{3} + 1158) q^{86} + ( - 60 \beta_{4} - 102 \beta_{2} - 102 \beta_1) q^{87} + (44 \beta_{4} + 165 \beta_{2} - 33 \beta_1) q^{88} + (68 \beta_{3} + 730) q^{89} + ( - 4 \beta_{5} + 176 \beta_{3} + 56) q^{91} + ( - 160 \beta_{4} - 1252 \beta_{2} + 372 \beta_1) q^{92} + ( - 48 \beta_{4} - 72 \beta_{2} - 12 \beta_1) q^{93} + ( - 132 \beta_{5} - 76 \beta_{3} + 692) q^{94} + ( - 36 \beta_{5} - 39 \beta_{3} + 675) q^{96} + (64 \beta_{4} + 286 \beta_{2} + 240 \beta_1) q^{97} + ( - 64 \beta_{4} - 147 \beta_{2} + 131 \beta_1) q^{98} - 99 q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 + 3*b2 * q^3 + (b5 + b3 - 7) * q^4 + (-3*b3 + 3) * q^6 + (-2*b4 - 2*b2 - 2*b1) * q^7 + (4*b4 + 15*b2 - 3*b1) * q^8 - 9 * q^9 + 11 * q^11 + (-3*b4 - 21*b2 + 3*b1) * q^12 + (-2*b4 + 4*b2 + 12*b1) * q^13 + (4*b5 + 10*b3 + 22) * q^14 + (-7*b5 - 23*b3 + 9) * q^16 + (6*b4 - 76*b2 - 10*b1) * q^17 + (9*b2 - 9*b1) * q^18 + (4*b5 + 2*b3 - 48) * q^19 + (-6*b5 + 6*b3 + 6) * q^21 + (-11*b2 + 11*b1) * q^22 + (8*b4 + 60*b2 - 20*b1) * q^23 + (12*b5 + 9*b3 - 45) * q^24 + (18*b5 + 4*b3 - 168) * q^26 - 27*b2 * q^27 + (6*b4 + 110*b2 - 10*b1) * q^28 + (20*b5 - 34*b3 - 34) * q^29 + (16*b5 - 4*b3 - 24) * q^31 + (-12*b4 - 225*b2 + 13*b1) * q^32 + 33*b2 * q^33 + (-28*b5 + 52*b3 + 76) * q^34 + (-9*b5 - 9*b3 + 63) * q^36 + (-4*b4 - 122*b2 + 24*b1) * q^37 + (14*b4 + 84*b2 - 64*b1) * q^38 + (-6*b5 - 36*b3 - 12) * q^39 + (20*b5 - 2*b3 - 66) * q^41 + (-12*b4 + 66*b2 + 30*b1) * q^42 + (-14*b4 + 150*b2 - 74*b1) * q^43 + (11*b5 + 11*b3 - 77) * q^44 + (-44*b5 - 92*b3 + 356) * q^46 + (28*b4 - 36*b2 - 48*b1) * q^47 + (21*b4 + 27*b2 - 69*b1) * q^48 + (-20*b5 - 4*b3 + 51) * q^49 + (18*b5 + 30*b3 + 228) * q^51 + (42*b4 + 292*b2 - 144*b1) * q^52 + (-52*b4 + 42*b2 - 32*b1) * q^53 + (27*b3 - 27) * q^54 + (4*b5 - 54*b3 + 438) * q^56 + (-12*b4 - 144*b2 + 6*b1) * q^57 + (26*b4 - 402*b2 - 114*b1) * q^58 + (-4*b5 - 120*b3 + 308) * q^59 + (44*b5 + 12*b3 + 218) * q^61 + (44*b4 - 88*b1) * q^62 + (18*b4 + 18*b2 + 18*b1) * q^63 + (-7*b5 + 89*b3 - 359) * q^64 + (-33*b3 + 33) * q^66 + (-28*b4 - 128*b2 - 148*b1) * q^67 + (16*b4 - 12*b2 + 108*b1) * q^68 + (24*b5 + 60*b3 - 180) * q^69 + (16*b5 - 104*b3 - 216) * q^71 + (-36*b4 - 135*b2 + 27*b1) * q^72 + (-14*b4 - 356*b2 - 168*b1) * q^73 + (36*b5 + 138*b3 - 466) * q^74 + (-74*b5 - 124*b3 + 624) * q^76 + (-22*b4 - 22*b2 - 22*b1) * q^77 + (-54*b4 - 504*b2 + 12*b1) * q^78 + (-52*b5 - 14*b3 + 524) * q^79 + 81 * q^81 + (58*b4 + 78*b2 - 146*b1) * q^82 + (-70*b4 + 582*b2 + 164*b1) * q^83 + (18*b5 + 30*b3 - 330) * q^84 + (-32*b5 - 94*b3 + 1158) * q^86 + (-60*b4 - 102*b2 - 102*b1) * q^87 + (44*b4 + 165*b2 - 33*b1) * q^88 + (68*b3 + 730) * q^89 + (-4*b5 + 176*b3 + 56) * q^91 + (-160*b4 - 1252*b2 + 372*b1) * q^92 + (-48*b4 - 72*b2 - 12*b1) * q^93 + (-132*b5 - 76*b3 + 692) * q^94 + (-36*b5 - 39*b3 + 675) * q^96 + (64*b4 + 286*b2 + 240*b1) * q^97 + (-64*b4 - 147*b2 + 131*b1) * q^98 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 44 q^{4} + 24 q^{6} - 54 q^{9}+O(q^{10})$$ 6 * q - 44 * q^4 + 24 * q^6 - 54 * q^9 $$6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44} + 2320 q^{46} + 314 q^{49} + 1308 q^{51} - 216 q^{54} + 2736 q^{56} + 2088 q^{59} + 1284 q^{61} - 2332 q^{64} + 264 q^{66} - 1200 q^{69} - 1088 q^{71} - 3072 q^{74} + 3992 q^{76} + 3172 q^{79} + 486 q^{81} - 2040 q^{84} + 7136 q^{86} + 4244 q^{89} - 16 q^{91} + 4304 q^{94} + 4128 q^{96} - 594 q^{99}+O(q^{100})$$ 6 * q - 44 * q^4 + 24 * q^6 - 54 * q^9 + 66 * q^11 + 112 * q^14 + 100 * q^16 - 292 * q^19 + 24 * q^21 - 288 * q^24 - 1016 * q^26 - 136 * q^29 - 136 * q^31 + 352 * q^34 + 396 * q^36 - 392 * q^41 - 484 * q^44 + 2320 * q^46 + 314 * q^49 + 1308 * q^51 - 216 * q^54 + 2736 * q^56 + 2088 * q^59 + 1284 * q^61 - 2332 * q^64 + 264 * q^66 - 1200 * q^69 - 1088 * q^71 - 3072 * q^74 + 3992 * q^76 + 3172 * q^79 + 486 * q^81 - 2040 * q^84 + 7136 * q^86 + 4244 * q^89 - 16 * q^91 + 4304 * q^94 + 4128 * q^96 - 594 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 41x^{4} + 452x^{2} + 676$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 15\nu^{3} - 94\nu ) / 156$$ (v^5 + 15*v^3 - 94*v) / 156 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 21\nu^{2} + 26 ) / 6$$ (v^4 + 21*v^2 + 26) / 6 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 28\nu^{3} + 153\nu ) / 13$$ (v^5 + 28*v^3 + 153*v) / 13 $$\beta_{5}$$ $$=$$ $$( \nu^{4} + 27\nu^{2} + 110 ) / 6$$ (v^4 + 27*v^2 + 110) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{3} - 14$$ b5 - b3 - 14 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 12\beta_{2} - 19\beta_1$$ b4 - 12*b2 - 19*b1 $$\nu^{4}$$ $$=$$ $$-21\beta_{5} + 27\beta_{3} + 268$$ -21*b5 + 27*b3 + 268 $$\nu^{5}$$ $$=$$ $$-15\beta_{4} + 336\beta_{2} + 379\beta_1$$ -15*b4 + 336*b2 + 379*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 4.26150i − 4.59056i − 1.32906i 1.32906i 4.59056i 4.26150i
5.26150i 3.00000i −19.6833 0 15.7845 10.3207i 61.4719i −9.00000 0
199.2 3.59056i 3.00000i −4.89212 0 −10.7717 16.1465i 11.1590i −9.00000 0
199.3 2.32906i 3.00000i 2.57547 0 6.98719 22.4672i 24.6309i −9.00000 0
199.4 2.32906i 3.00000i 2.57547 0 6.98719 22.4672i 24.6309i −9.00000 0
199.5 3.59056i 3.00000i −4.89212 0 −10.7717 16.1465i 11.1590i −9.00000 0
199.6 5.26150i 3.00000i −19.6833 0 15.7845 10.3207i 61.4719i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.4.c.l 6
5.b even 2 1 inner 825.4.c.l 6
5.c odd 4 1 165.4.a.d 3
5.c odd 4 1 825.4.a.s 3
15.e even 4 1 495.4.a.l 3
15.e even 4 1 2475.4.a.s 3
55.e even 4 1 1815.4.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 5.c odd 4 1
495.4.a.l 3 15.e even 4 1
825.4.a.s 3 5.c odd 4 1
825.4.c.l 6 1.a even 1 1 trivial
825.4.c.l 6 5.b even 2 1 inner
1815.4.a.s 3 55.e even 4 1
2475.4.a.s 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{6} + 46T_{2}^{4} + 577T_{2}^{2} + 1936$$ T2^6 + 46*T2^4 + 577*T2^2 + 1936 $$T_{7}^{6} + 872T_{7}^{4} + 213136T_{7}^{2} + 14017536$$ T7^6 + 872*T7^4 + 213136*T7^2 + 14017536

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 46 T^{4} + 577 T^{2} + \cdots + 1936$$
$3$ $$(T^{2} + 9)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 872 T^{4} + \cdots + 14017536$$
$11$ $$(T - 11)^{6}$$
$13$ $$T^{6} + 7120 T^{4} + \cdots + 1165812736$$
$17$ $$T^{6} + 28164 T^{4} + \cdots + 55273890816$$
$19$ $$(T^{3} + 146 T^{2} + 5376 T + 30960)^{2}$$
$23$ $$T^{6} + 45344 T^{4} + \cdots + 2768896$$
$29$ $$(T^{3} + 68 T^{2} - 54364 T - 3163056)^{2}$$
$31$ $$(T^{3} + 68 T^{2} - 23232 T - 1812096)^{2}$$
$37$ $$T^{6} + 79180 T^{4} + \cdots + 383101578304$$
$41$ $$(T^{3} + 196 T^{2} - 26076 T - 4364208)^{2}$$
$43$ $$T^{6} + \cdots + 978058072166400$$
$47$ $$T^{6} + \cdots + 439614082584576$$
$53$ $$T^{6} + 546668 T^{4} + \cdots + 15\!\cdots\!04$$
$59$ $$(T^{3} - 1044 T^{2} + 64144 T + 84227264)^{2}$$
$61$ $$(T^{3} - 642 T^{2} - 60548 T + 22757384)^{2}$$
$67$ $$T^{6} + 980112 T^{4} + \cdots + 76\!\cdots\!96$$
$71$ $$(T^{3} + 544 T^{2} - 129728 T + 6553600)^{2}$$
$73$ $$T^{6} + 1409152 T^{4} + \cdots + 31464740110336$$
$79$ $$(T^{3} - 1586 T^{2} + 562208 T + 14694992)^{2}$$
$83$ $$T^{6} + 3118012 T^{4} + \cdots + 85\!\cdots\!56$$
$89$ $$(T^{3} - 2122 T^{2} + 1406940 T - 293444632)^{2}$$
$97$ $$T^{6} + 2965324 T^{4} + \cdots + 49\!\cdots\!96$$