# Properties

 Label 825.4.c.i Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + (8 \beta_{2} + 7 \beta_1) q^{8} - 9 q^{9}+O(q^{10})$$ q + b1 * q^2 + 3*b2 * q^3 + (b3 - 1) * q^4 + (-3*b3 + 3) * q^6 + (-2*b2 - 2*b1) * q^7 + (8*b2 + 7*b1) * q^8 - 9 * q^9 $$q + \beta_1 q^{2} + 3 \beta_{2} q^{3} + (\beta_{3} - 1) q^{4} + ( - 3 \beta_{3} + 3) q^{6} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + (8 \beta_{2} + 7 \beta_1) q^{8} - 9 q^{9} + 11 q^{11} + 3 \beta_1 q^{12} + ( - 42 \beta_{2} - 8 \beta_1) q^{13} + 16 q^{14} + (7 \beta_{3} - 63) q^{16} + (28 \beta_{2} + 30 \beta_1) q^{17} - 9 \beta_1 q^{18} + ( - 18 \beta_{3} + 36) q^{19} + 6 \beta_{3} q^{21} + 11 \beta_1 q^{22} + 112 \beta_{2} q^{23} + ( - 21 \beta_{3} - 3) q^{24} + (34 \beta_{3} + 30) q^{26} - 27 \beta_{2} q^{27} - 16 \beta_{2} q^{28} + (46 \beta_{3} - 134) q^{29} + ( - 104 \beta_{3} + 32) q^{31} + (120 \beta_{2} - 7 \beta_1) q^{32} + 33 \beta_{2} q^{33} + (2 \beta_{3} - 242) q^{34} + ( - 9 \beta_{3} + 9) q^{36} + (46 \beta_{2} + 44 \beta_1) q^{37} + ( - 144 \beta_{2} + 36 \beta_1) q^{38} + (24 \beta_{3} + 102) q^{39} + ( - 2 \beta_{3} - 246) q^{41} + 48 \beta_{2} q^{42} + (10 \beta_{2} + 86 \beta_1) q^{43} + (11 \beta_{3} - 11) q^{44} + ( - 112 \beta_{3} + 112) q^{46} + (8 \beta_{2} - 48 \beta_1) q^{47} + ( - 168 \beta_{2} + 21 \beta_1) q^{48} + ( - 4 \beta_{3} + 311) q^{49} + ( - 90 \beta_{3} + 6) q^{51} + ( - 64 \beta_{2} - 34 \beta_1) q^{52} + (22 \beta_{2} + 128 \beta_1) q^{53} + (27 \beta_{3} - 27) q^{54} + (16 \beta_{3} + 112) q^{56} + (54 \beta_{2} - 54 \beta_1) q^{57} + (368 \beta_{2} - 134 \beta_1) q^{58} - 196 q^{59} + (52 \beta_{3} - 578) q^{61} + ( - 832 \beta_{2} + 32 \beta_1) q^{62} + (18 \beta_{2} + 18 \beta_1) q^{63} + ( - 71 \beta_{3} - 321) q^{64} + ( - 33 \beta_{3} + 33) q^{66} + ( - 388 \beta_{2} + 152 \beta_1) q^{67} + (240 \beta_{2} - 2 \beta_1) q^{68} - 336 q^{69} + ( - 184 \beta_{3} + 320) q^{71} + ( - 72 \beta_{2} - 63 \beta_1) q^{72} + ( - 170 \beta_{2} + 252 \beta_1) q^{73} + ( - 2 \beta_{3} - 350) q^{74} + (36 \beta_{3} - 180) q^{76} + ( - 22 \beta_{2} - 22 \beta_1) q^{77} + (192 \beta_{2} + 102 \beta_1) q^{78} + ( - 74 \beta_{3} + 152) q^{79} + 81 q^{81} + ( - 16 \beta_{2} - 246 \beta_1) q^{82} + (336 \beta_{2} + 324 \beta_1) q^{83} + 48 q^{84} + (76 \beta_{3} - 764) q^{86} + ( - 264 \beta_{2} + 138 \beta_1) q^{87} + (88 \beta_{2} + 77 \beta_1) q^{88} + (8 \beta_{3} - 490) q^{89} + ( - 84 \beta_{3} - 128) q^{91} + 112 \beta_1 q^{92} + ( - 216 \beta_{2} - 312 \beta_1) q^{93} + ( - 56 \beta_{3} + 440) q^{94} + (21 \beta_{3} - 381) q^{96} + (802 \beta_{2} + 420 \beta_1) q^{97} + ( - 32 \beta_{2} + 311 \beta_1) q^{98} - 99 q^{99}+O(q^{100})$$ q + b1 * q^2 + 3*b2 * q^3 + (b3 - 1) * q^4 + (-3*b3 + 3) * q^6 + (-2*b2 - 2*b1) * q^7 + (8*b2 + 7*b1) * q^8 - 9 * q^9 + 11 * q^11 + 3*b1 * q^12 + (-42*b2 - 8*b1) * q^13 + 16 * q^14 + (7*b3 - 63) * q^16 + (28*b2 + 30*b1) * q^17 - 9*b1 * q^18 + (-18*b3 + 36) * q^19 + 6*b3 * q^21 + 11*b1 * q^22 + 112*b2 * q^23 + (-21*b3 - 3) * q^24 + (34*b3 + 30) * q^26 - 27*b2 * q^27 - 16*b2 * q^28 + (46*b3 - 134) * q^29 + (-104*b3 + 32) * q^31 + (120*b2 - 7*b1) * q^32 + 33*b2 * q^33 + (2*b3 - 242) * q^34 + (-9*b3 + 9) * q^36 + (46*b2 + 44*b1) * q^37 + (-144*b2 + 36*b1) * q^38 + (24*b3 + 102) * q^39 + (-2*b3 - 246) * q^41 + 48*b2 * q^42 + (10*b2 + 86*b1) * q^43 + (11*b3 - 11) * q^44 + (-112*b3 + 112) * q^46 + (8*b2 - 48*b1) * q^47 + (-168*b2 + 21*b1) * q^48 + (-4*b3 + 311) * q^49 + (-90*b3 + 6) * q^51 + (-64*b2 - 34*b1) * q^52 + (22*b2 + 128*b1) * q^53 + (27*b3 - 27) * q^54 + (16*b3 + 112) * q^56 + (54*b2 - 54*b1) * q^57 + (368*b2 - 134*b1) * q^58 - 196 * q^59 + (52*b3 - 578) * q^61 + (-832*b2 + 32*b1) * q^62 + (18*b2 + 18*b1) * q^63 + (-71*b3 - 321) * q^64 + (-33*b3 + 33) * q^66 + (-388*b2 + 152*b1) * q^67 + (240*b2 - 2*b1) * q^68 - 336 * q^69 + (-184*b3 + 320) * q^71 + (-72*b2 - 63*b1) * q^72 + (-170*b2 + 252*b1) * q^73 + (-2*b3 - 350) * q^74 + (36*b3 - 180) * q^76 + (-22*b2 - 22*b1) * q^77 + (192*b2 + 102*b1) * q^78 + (-74*b3 + 152) * q^79 + 81 * q^81 + (-16*b2 - 246*b1) * q^82 + (336*b2 + 324*b1) * q^83 + 48 * q^84 + (76*b3 - 764) * q^86 + (-264*b2 + 138*b1) * q^87 + (88*b2 + 77*b1) * q^88 + (8*b3 - 490) * q^89 + (-84*b3 - 128) * q^91 + 112*b1 * q^92 + (-216*b2 - 312*b1) * q^93 + (-56*b3 + 440) * q^94 + (21*b3 - 381) * q^96 + (802*b2 + 420*b1) * q^97 + (-32*b2 + 311*b1) * q^98 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 $$4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44} + 224 q^{46} + 1236 q^{49} - 156 q^{51} - 54 q^{54} + 480 q^{56} - 784 q^{59} - 2208 q^{61} - 1426 q^{64} + 66 q^{66} - 1344 q^{69} + 912 q^{71} - 1404 q^{74} - 648 q^{76} + 460 q^{79} + 324 q^{81} + 192 q^{84} - 2904 q^{86} - 1944 q^{89} - 680 q^{91} + 1648 q^{94} - 1482 q^{96} - 396 q^{99}+O(q^{100})$$ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 + 44 * q^11 + 64 * q^14 - 238 * q^16 + 108 * q^19 + 12 * q^21 - 54 * q^24 + 188 * q^26 - 444 * q^29 - 80 * q^31 - 964 * q^34 + 18 * q^36 + 456 * q^39 - 988 * q^41 - 22 * q^44 + 224 * q^46 + 1236 * q^49 - 156 * q^51 - 54 * q^54 + 480 * q^56 - 784 * q^59 - 2208 * q^61 - 1426 * q^64 + 66 * q^66 - 1344 * q^69 + 912 * q^71 - 1404 * q^74 - 648 * q^76 + 460 * q^79 + 324 * q^81 + 192 * q^84 - 2904 * q^86 - 1944 * q^89 - 680 * q^91 + 1648 * q^94 - 1482 * q^96 - 396 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 9\nu ) / 8$$ (v^3 + 9*v) / 8 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$ v^2 + 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ b3 - 9 $$\nu^{3}$$ $$=$$ $$8\beta_{2} - 9\beta_1$$ 8*b2 - 9*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
 − 3.37228i − 2.37228i 2.37228i 3.37228i
3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −9.00000 0
199.2 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.3 2.37228i 3.00000i 2.37228 0 −7.11684 6.74456i 24.6060i −9.00000 0
199.4 3.37228i 3.00000i −3.37228 0 10.1168 4.74456i 15.6060i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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.i 4
5.b even 2 1 inner 825.4.c.i 4
5.c odd 4 1 33.4.a.d 2
5.c odd 4 1 825.4.a.k 2
15.e even 4 1 99.4.a.e 2
15.e even 4 1 2475.4.a.o 2
20.e even 4 1 528.4.a.o 2
35.f even 4 1 1617.4.a.j 2
40.i odd 4 1 2112.4.a.ba 2
40.k even 4 1 2112.4.a.bh 2
55.e even 4 1 363.4.a.j 2
60.l odd 4 1 1584.4.a.x 2
165.l odd 4 1 1089.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 5.c odd 4 1
99.4.a.e 2 15.e even 4 1
363.4.a.j 2 55.e even 4 1
528.4.a.o 2 20.e even 4 1
825.4.a.k 2 5.c odd 4 1
825.4.c.i 4 1.a even 1 1 trivial
825.4.c.i 4 5.b even 2 1 inner
1089.4.a.t 2 165.l odd 4 1
1584.4.a.x 2 60.l odd 4 1
1617.4.a.j 2 35.f even 4 1
2112.4.a.ba 2 40.i odd 4 1
2112.4.a.bh 2 40.k even 4 1
2475.4.a.o 2 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 $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} + 17T_{2}^{2} + 64$$ T2^4 + 17*T2^2 + 64 $$T_{7}^{4} + 68T_{7}^{2} + 1024$$ T7^4 + 68*T7^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 17T^{2} + 64$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 68T^{2} + 1024$$
$11$ $$(T - 11)^{4}$$
$13$ $$T^{4} + 3944 T^{2} + 839056$$
$17$ $$T^{4} + 15188 T^{2} + \cdots + 52649536$$
$19$ $$(T^{2} - 54 T - 1944)^{2}$$
$23$ $$(T^{2} + 12544)^{2}$$
$29$ $$(T^{2} + 222 T - 5136)^{2}$$
$31$ $$(T^{2} + 40 T - 88832)^{2}$$
$37$ $$T^{4} + 33096 T^{2} + \cdots + 237036816$$
$41$ $$(T^{2} + 494 T + 60976)^{2}$$
$43$ $$T^{4} + 124212 T^{2} + \cdots + 3591365184$$
$47$ $$T^{4} + 40064 T^{2} + \cdots + 323424256$$
$53$ $$T^{4} + 273864 T^{2} + \cdots + 17796627216$$
$59$ $$(T + 196)^{4}$$
$61$ $$(T^{2} + 1104 T + 282396)^{2}$$
$67$ $$T^{4} + 811808 T^{2} + \cdots + 609497344$$
$71$ $$(T^{2} - 456 T - 227328)^{2}$$
$73$ $$T^{4} + 1223048 T^{2} + \cdots + 190350709264$$
$79$ $$(T^{2} - 230 T - 31952)^{2}$$
$83$ $$T^{4} + 1792656 T^{2} + \cdots + 698521522176$$
$89$ $$(T^{2} + 972 T + 235668)^{2}$$
$97$ $$T^{4} + 3611528 T^{2} + \cdots + 1220662586896$$