# Properties

 Label 825.4.c.m 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: 6.0.245110336.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 19x^{4} + 101x^{2} + 100$$ x^6 + 19*x^4 + 101*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + 3 \beta_1 q^{3} + (\beta_{5} - 2 \beta_{3} - 6) q^{4} - 3 \beta_{5} q^{6} + ( - \beta_{4} - 3 \beta_{2} + \beta_1) q^{7} + (2 \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q - b2 * q^2 + 3*b1 * q^3 + (b5 - 2*b3 - 6) * q^4 - 3*b5 * q^6 + (-b4 - 3*b2 + b1) * q^7 + (2*b4 + 3*b2 + 2*b1) * q^8 - 9 * q^9 $$q - \beta_{2} q^{2} + 3 \beta_1 q^{3} + (\beta_{5} - 2 \beta_{3} - 6) q^{4} - 3 \beta_{5} q^{6} + ( - \beta_{4} - 3 \beta_{2} + \beta_1) q^{7} + (2 \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{8} - 9 q^{9} + 11 q^{11} + (6 \beta_{4} - 3 \beta_{2} - 18 \beta_1) q^{12} + ( - 11 \beta_{4} + 13 \beta_{2} + 11 \beta_1) q^{13} + ( - 8 \beta_{3} - 48) q^{14} + (7 \beta_{5} - 6 \beta_{3} + 6) q^{16} + ( - 7 \beta_{4} - 5 \beta_{2} + 9 \beta_1) q^{17} + 9 \beta_{2} q^{18} + ( - 8 \beta_{5} + 2 \beta_{3} - 36) q^{19} + ( - 9 \beta_{5} - 3 \beta_{3} - 3) q^{21} - 11 \beta_{2} q^{22} + ( - 16 \beta_{4} - 80 \beta_1) q^{23} + (9 \beta_{5} + 6 \beta_{3} - 6) q^{24} + ( - 46 \beta_{5} + 4 \beta_{3} + 116) q^{26} - 27 \beta_1 q^{27} + (8 \beta_{4} + 40 \beta_{2} - 40 \beta_1) q^{28} + (26 \beta_{5} - 88) q^{29} + ( - 34 \beta_{5} - 22 \beta_{3} + 42) q^{31} + (14 \beta_{4} + 37 \beta_{2} + 78 \beta_1) q^{32} + 33 \beta_1 q^{33} + ( - 18 \beta_{5} - 24 \beta_{3} - 112) q^{34} + ( - 9 \beta_{5} + 18 \beta_{3} + 54) q^{36} + ( - 42 \beta_{4} + 66 \beta_{2} - 8 \beta_1) q^{37} + (12 \beta_{4} + 24 \beta_{2} - 100 \beta_1) q^{38} + (39 \beta_{5} - 33 \beta_{3} - 33) q^{39} + ( - 22 \beta_{5} - 8) q^{41} + (24 \beta_{4} - 144 \beta_1) q^{42} + ( - 27 \beta_{4} + 19 \beta_{2} + 51 \beta_1) q^{43} + (11 \beta_{5} - 22 \beta_{3} - 66) q^{44} + (48 \beta_{5} - 32 \beta_{3} - 96) q^{46} + (34 \beta_{4} + 58 \beta_{2} - 54 \beta_1) q^{47} + (18 \beta_{4} - 21 \beta_{2} + 18 \beta_1) q^{48} + ( - 14 \beta_{5} - 30 \beta_{3} + 157) q^{49} + ( - 15 \beta_{5} - 21 \beta_{3} - 27) q^{51} + ( - 4 \beta_{4} - 66 \beta_{2} - 532 \beta_1) q^{52} + ( - 56 \beta_{4} + 12 \beta_{2} + 270 \beta_1) q^{53} + 27 \beta_{5} q^{54} + (16 \beta_{5} + 32 \beta_{3} + 224) q^{56} + ( - 6 \beta_{4} + 24 \beta_{2} - 108 \beta_1) q^{57} + ( - 52 \beta_{4} + 114 \beta_{2} + 364 \beta_1) q^{58} + (60 \beta_{5} + 8 \beta_{3} - 432) q^{59} + ( - 78 \beta_{5} + 22 \beta_{3} + 140) q^{61} + (112 \beta_{4} - 32 \beta_{2} - 608 \beta_1) q^{62} + (9 \beta_{4} + 27 \beta_{2} - 9 \beta_1) q^{63} + ( - 31 \beta_{5} + 54 \beta_{3} + 650) q^{64} - 33 \beta_{5} q^{66} + ( - 104 \beta_{4} + 88 \beta_{2} + 284 \beta_1) q^{67} + (28 \beta_{4} + 102 \beta_{2} - 324 \beta_1) q^{68} + ( - 48 \beta_{3} + 240) q^{69} + (16 \beta_{5} + 28 \beta_{3} + 600) q^{71} + ( - 18 \beta_{4} - 27 \beta_{2} - 18 \beta_1) q^{72} + (23 \beta_{4} + 43 \beta_{2} - 19 \beta_1) q^{73} + ( - 142 \beta_{5} + 48 \beta_{3} + 672) q^{74} + (36 \beta_{5} + 88 \beta_{3} + 120) q^{76} + ( - 11 \beta_{4} - 33 \beta_{2} + 11 \beta_1) q^{77} + ( - 12 \beta_{4} + 138 \beta_{2} + 348 \beta_1) q^{78} + ( - 24 \beta_{5} + 154 \beta_{3} + 40) q^{79} + 81 q^{81} + (44 \beta_{4} - 14 \beta_{2} - 308 \beta_1) q^{82} + (195 \beta_{4} - 9 \beta_{2} - 289 \beta_1) q^{83} + (120 \beta_{5} + 24 \beta_{3} + 120) q^{84} + ( - 124 \beta_{5} - 16 \beta_{3} + 104) q^{86} + ( - 78 \beta_{2} - 264 \beta_1) q^{87} + (22 \beta_{4} + 33 \beta_{2} + 22 \beta_1) q^{88} + ( - 292 \beta_{5} - 182) q^{89} + ( - 154 \beta_{5} + 38 \beta_{3} + 162) q^{91} + ( - 160 \beta_{4} + 208 \beta_{2} - 160 \beta_1) q^{92} + (66 \beta_{4} + 102 \beta_{2} + 126 \beta_1) q^{93} + (64 \beta_{5} + 184 \beta_{3} + 1016) q^{94} + (111 \beta_{5} + 42 \beta_{3} - 234) q^{96} + ( - 148 \beta_{4} + 200 \beta_{2} - 374 \beta_1) q^{97} + (88 \beta_{4} - 111 \beta_{2} - 376 \beta_1) q^{98} - 99 q^{99}+O(q^{100})$$ q - b2 * q^2 + 3*b1 * q^3 + (b5 - 2*b3 - 6) * q^4 - 3*b5 * q^6 + (-b4 - 3*b2 + b1) * q^7 + (2*b4 + 3*b2 + 2*b1) * q^8 - 9 * q^9 + 11 * q^11 + (6*b4 - 3*b2 - 18*b1) * q^12 + (-11*b4 + 13*b2 + 11*b1) * q^13 + (-8*b3 - 48) * q^14 + (7*b5 - 6*b3 + 6) * q^16 + (-7*b4 - 5*b2 + 9*b1) * q^17 + 9*b2 * q^18 + (-8*b5 + 2*b3 - 36) * q^19 + (-9*b5 - 3*b3 - 3) * q^21 - 11*b2 * q^22 + (-16*b4 - 80*b1) * q^23 + (9*b5 + 6*b3 - 6) * q^24 + (-46*b5 + 4*b3 + 116) * q^26 - 27*b1 * q^27 + (8*b4 + 40*b2 - 40*b1) * q^28 + (26*b5 - 88) * q^29 + (-34*b5 - 22*b3 + 42) * q^31 + (14*b4 + 37*b2 + 78*b1) * q^32 + 33*b1 * q^33 + (-18*b5 - 24*b3 - 112) * q^34 + (-9*b5 + 18*b3 + 54) * q^36 + (-42*b4 + 66*b2 - 8*b1) * q^37 + (12*b4 + 24*b2 - 100*b1) * q^38 + (39*b5 - 33*b3 - 33) * q^39 + (-22*b5 - 8) * q^41 + (24*b4 - 144*b1) * q^42 + (-27*b4 + 19*b2 + 51*b1) * q^43 + (11*b5 - 22*b3 - 66) * q^44 + (48*b5 - 32*b3 - 96) * q^46 + (34*b4 + 58*b2 - 54*b1) * q^47 + (18*b4 - 21*b2 + 18*b1) * q^48 + (-14*b5 - 30*b3 + 157) * q^49 + (-15*b5 - 21*b3 - 27) * q^51 + (-4*b4 - 66*b2 - 532*b1) * q^52 + (-56*b4 + 12*b2 + 270*b1) * q^53 + 27*b5 * q^54 + (16*b5 + 32*b3 + 224) * q^56 + (-6*b4 + 24*b2 - 108*b1) * q^57 + (-52*b4 + 114*b2 + 364*b1) * q^58 + (60*b5 + 8*b3 - 432) * q^59 + (-78*b5 + 22*b3 + 140) * q^61 + (112*b4 - 32*b2 - 608*b1) * q^62 + (9*b4 + 27*b2 - 9*b1) * q^63 + (-31*b5 + 54*b3 + 650) * q^64 - 33*b5 * q^66 + (-104*b4 + 88*b2 + 284*b1) * q^67 + (28*b4 + 102*b2 - 324*b1) * q^68 + (-48*b3 + 240) * q^69 + (16*b5 + 28*b3 + 600) * q^71 + (-18*b4 - 27*b2 - 18*b1) * q^72 + (23*b4 + 43*b2 - 19*b1) * q^73 + (-142*b5 + 48*b3 + 672) * q^74 + (36*b5 + 88*b3 + 120) * q^76 + (-11*b4 - 33*b2 + 11*b1) * q^77 + (-12*b4 + 138*b2 + 348*b1) * q^78 + (-24*b5 + 154*b3 + 40) * q^79 + 81 * q^81 + (44*b4 - 14*b2 - 308*b1) * q^82 + (195*b4 - 9*b2 - 289*b1) * q^83 + (120*b5 + 24*b3 + 120) * q^84 + (-124*b5 - 16*b3 + 104) * q^86 + (-78*b2 - 264*b1) * q^87 + (22*b4 + 33*b2 + 22*b1) * q^88 + (-292*b5 - 182) * q^89 + (-154*b5 + 38*b3 + 162) * q^91 + (-160*b4 + 208*b2 - 160*b1) * q^92 + (66*b4 + 102*b2 + 126*b1) * q^93 + (64*b5 + 184*b3 + 1016) * q^94 + (111*b5 + 42*b3 - 234) * q^96 + (-148*b4 + 200*b2 - 374*b1) * q^97 + (88*b4 - 111*b2 - 376*b1) * q^98 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 34 q^{4} - 6 q^{6} - 54 q^{9}+O(q^{10})$$ 6 * q - 34 * q^4 - 6 * q^6 - 54 * q^9 $$6 q - 34 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} - 288 q^{14} + 50 q^{16} - 232 q^{19} - 36 q^{21} - 18 q^{24} + 604 q^{26} - 476 q^{29} + 184 q^{31} - 708 q^{34} + 306 q^{36} - 120 q^{39} - 92 q^{41} - 374 q^{44} - 480 q^{46} + 914 q^{49} - 192 q^{51} + 54 q^{54} + 1376 q^{56} - 2472 q^{59} + 684 q^{61} + 3838 q^{64} - 66 q^{66} + 1440 q^{69} + 3632 q^{71} + 3748 q^{74} + 792 q^{76} + 192 q^{79} + 486 q^{81} + 960 q^{84} + 376 q^{86} - 1676 q^{89} + 664 q^{91} + 6224 q^{94} - 1182 q^{96} - 594 q^{99}+O(q^{100})$$ 6 * q - 34 * q^4 - 6 * q^6 - 54 * q^9 + 66 * q^11 - 288 * q^14 + 50 * q^16 - 232 * q^19 - 36 * q^21 - 18 * q^24 + 604 * q^26 - 476 * q^29 + 184 * q^31 - 708 * q^34 + 306 * q^36 - 120 * q^39 - 92 * q^41 - 374 * q^44 - 480 * q^46 + 914 * q^49 - 192 * q^51 + 54 * q^54 + 1376 * q^56 - 2472 * q^59 + 684 * q^61 + 3838 * q^64 - 66 * q^66 + 1440 * q^69 + 3632 * q^71 + 3748 * q^74 + 792 * q^76 + 192 * q^79 + 486 * q^81 + 960 * q^84 + 376 * q^86 - 1676 * q^89 + 664 * q^91 + 6224 * q^94 - 1182 * q^96 - 594 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 19x^{4} + 101x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 9\nu^{3} + \nu ) / 10$$ (v^5 + 9*v^3 + v) / 10 $$\beta_{2}$$ $$=$$ $$( 3\nu^{5} + 37\nu^{3} + 83\nu ) / 10$$ (3*v^5 + 37*v^3 + 83*v) / 10 $$\beta_{3}$$ $$=$$ $$-\nu^{4} - 9\nu^{2} - 4$$ -v^4 - 9*v^2 - 4 $$\beta_{4}$$ $$=$$ $$( 2\nu^{5} + 23\nu^{3} + 52\nu ) / 5$$ (2*v^5 + 23*v^3 + 52*v) / 5 $$\beta_{5}$$ $$=$$ $$\nu^{4} + 11\nu^{2} + 17$$ v^4 + 11*v^2 + 17
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{2} - \beta_1 ) / 2$$ (b4 - b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{3} - 13 ) / 2$$ (b5 + b3 - 13) / 2 $$\nu^{3}$$ $$=$$ $$-4\beta_{4} + 5\beta_{2} + \beta_1$$ -4*b4 + 5*b2 + b1 $$\nu^{4}$$ $$=$$ $$( -9\beta_{5} - 11\beta_{3} + 109 ) / 2$$ (-9*b5 - 11*b3 + 109) / 2 $$\nu^{5}$$ $$=$$ $$( 71\beta_{4} - 89\beta_{2} + 3\beta_1 ) / 2$$ (71*b4 - 89*b2 + 3*b1) / 2

## 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
 1.12946i − 2.91150i − 3.04096i 3.04096i 2.91150i − 1.12946i
4.59486i 3.00000i −13.1127 0 −13.7846 20.6383i 23.4921i −9.00000 0
199.2 4.38835i 3.00000i −11.2577 0 13.1651 11.7304i 14.2958i −9.00000 0
199.3 0.793499i 3.00000i 7.37036 0 −2.38050 2.90793i 12.1964i −9.00000 0
199.4 0.793499i 3.00000i 7.37036 0 −2.38050 2.90793i 12.1964i −9.00000 0
199.5 4.38835i 3.00000i −11.2577 0 13.1651 11.7304i 14.2958i −9.00000 0
199.6 4.59486i 3.00000i −13.1127 0 −13.7846 20.6383i 23.4921i −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.m 6
5.b even 2 1 inner 825.4.c.m 6
5.c odd 4 1 165.4.a.g 3
5.c odd 4 1 825.4.a.p 3
15.e even 4 1 495.4.a.i 3
15.e even 4 1 2475.4.a.z 3
55.e even 4 1 1815.4.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.g 3 5.c odd 4 1
495.4.a.i 3 15.e even 4 1
825.4.a.p 3 5.c odd 4 1
825.4.c.m 6 1.a even 1 1 trivial
825.4.c.m 6 5.b even 2 1 inner
1815.4.a.q 3 55.e even 4 1
2475.4.a.z 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 $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{6} + 41T_{2}^{4} + 432T_{2}^{2} + 256$$ T2^6 + 41*T2^4 + 432*T2^2 + 256 $$T_{7}^{6} + 572T_{7}^{4} + 63376T_{7}^{2} + 495616$$ T7^6 + 572*T7^4 + 63376*T7^2 + 495616

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 41 T^{4} + 432 T^{2} + \cdots + 256$$
$3$ $$(T^{2} + 9)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 572 T^{4} + 63376 T^{2} + \cdots + 495616$$
$11$ $$(T - 11)^{6}$$
$13$ $$T^{6} + 10240 T^{4} + \cdots + 6100234816$$
$17$ $$T^{6} + 6384 T^{4} + \cdots + 502835776$$
$19$ $$(T^{3} + 116 T^{2} + 3356 T + 80)^{2}$$
$23$ $$T^{6} + 38144 T^{4} + \cdots + 32480690176$$
$29$ $$(T^{3} + 238 T^{2} + 5136 T - 428416)^{2}$$
$31$ $$(T^{3} - 92 T^{2} - 53552 T + 6769664)^{2}$$
$37$ $$T^{6} + 199500 T^{4} + \cdots + 40502634332224$$
$41$ $$(T^{3} + 46 T^{2} - 9136 T - 245888)^{2}$$
$43$ $$T^{6} + 54092 T^{4} + \cdots + 5670875449600$$
$47$ $$T^{6} + \cdots + 120565319090176$$
$53$ $$T^{6} + \cdots + 298010064169024$$
$59$ $$(T^{3} + 1236 T^{2} + 424064 T + 32923904)^{2}$$
$61$ $$(T^{3} - 342 T^{2} - 68308 T - 2655176)^{2}$$
$67$ $$T^{6} + 943792 T^{4} + \cdots + 23\!\cdots\!56$$
$71$ $$(T^{3} - 1816 T^{2} + 1056112 T - 198158720)^{2}$$
$73$ $$T^{6} + 157232 T^{4} + \cdots + 26347524744256$$
$79$ $$(T^{3} - 96 T^{2} - 812212 T + 167159872)^{2}$$
$83$ $$T^{6} + 2992332 T^{4} + \cdots + 29\!\cdots\!76$$
$89$ $$(T^{3} + 838 T^{2} - 1499620 T - 693013592)^{2}$$
$97$ $$T^{6} + 2646124 T^{4} + \cdots + 12\!\cdots\!36$$