# Properties

 Label 825.4.c.a Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 32 i q^{7} - 45 i q^{8} - 9 q^{9} +O(q^{10})$$ q + 5*i * q^2 + 3*i * q^3 - 17 * q^4 - 15 * q^6 + 32*i * q^7 - 45*i * q^8 - 9 * q^9 $$q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 32 i q^{7} - 45 i q^{8} - 9 q^{9} - 11 q^{11} - 51 i q^{12} - 38 i q^{13} - 160 q^{14} + 89 q^{16} + 2 i q^{17} - 45 i q^{18} - 72 q^{19} - 96 q^{21} - 55 i q^{22} + 68 i q^{23} + 135 q^{24} + 190 q^{26} - 27 i q^{27} - 544 i q^{28} + 54 q^{29} - 152 q^{31} + 85 i q^{32} - 33 i q^{33} - 10 q^{34} + 153 q^{36} - 174 i q^{37} - 360 i q^{38} + 114 q^{39} + 94 q^{41} - 480 i q^{42} - 528 i q^{43} + 187 q^{44} - 340 q^{46} + 340 i q^{47} + 267 i q^{48} - 681 q^{49} - 6 q^{51} + 646 i q^{52} - 438 i q^{53} + 135 q^{54} + 1440 q^{56} - 216 i q^{57} + 270 i q^{58} - 20 q^{59} + 570 q^{61} - 760 i q^{62} - 288 i q^{63} + 287 q^{64} + 165 q^{66} + 460 i q^{67} - 34 i q^{68} - 204 q^{69} - 1092 q^{71} + 405 i q^{72} + 562 i q^{73} + 870 q^{74} + 1224 q^{76} - 352 i q^{77} + 570 i q^{78} + 16 q^{79} + 81 q^{81} + 470 i q^{82} + 372 i q^{83} + 1632 q^{84} + 2640 q^{86} + 162 i q^{87} + 495 i q^{88} + 966 q^{89} + 1216 q^{91} - 1156 i q^{92} - 456 i q^{93} - 1700 q^{94} - 255 q^{96} + 526 i q^{97} - 3405 i q^{98} + 99 q^{99} +O(q^{100})$$ q + 5*i * q^2 + 3*i * q^3 - 17 * q^4 - 15 * q^6 + 32*i * q^7 - 45*i * q^8 - 9 * q^9 - 11 * q^11 - 51*i * q^12 - 38*i * q^13 - 160 * q^14 + 89 * q^16 + 2*i * q^17 - 45*i * q^18 - 72 * q^19 - 96 * q^21 - 55*i * q^22 + 68*i * q^23 + 135 * q^24 + 190 * q^26 - 27*i * q^27 - 544*i * q^28 + 54 * q^29 - 152 * q^31 + 85*i * q^32 - 33*i * q^33 - 10 * q^34 + 153 * q^36 - 174*i * q^37 - 360*i * q^38 + 114 * q^39 + 94 * q^41 - 480*i * q^42 - 528*i * q^43 + 187 * q^44 - 340 * q^46 + 340*i * q^47 + 267*i * q^48 - 681 * q^49 - 6 * q^51 + 646*i * q^52 - 438*i * q^53 + 135 * q^54 + 1440 * q^56 - 216*i * q^57 + 270*i * q^58 - 20 * q^59 + 570 * q^61 - 760*i * q^62 - 288*i * q^63 + 287 * q^64 + 165 * q^66 + 460*i * q^67 - 34*i * q^68 - 204 * q^69 - 1092 * q^71 + 405*i * q^72 + 562*i * q^73 + 870 * q^74 + 1224 * q^76 - 352*i * q^77 + 570*i * q^78 + 16 * q^79 + 81 * q^81 + 470*i * q^82 + 372*i * q^83 + 1632 * q^84 + 2640 * q^86 + 162*i * q^87 + 495*i * q^88 + 966 * q^89 + 1216 * q^91 - 1156*i * q^92 - 456*i * q^93 - 1700 * q^94 - 255 * q^96 + 526*i * q^97 - 3405*i * q^98 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 34 q^{4} - 30 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9 $$2 q - 34 q^{4} - 30 q^{6} - 18 q^{9} - 22 q^{11} - 320 q^{14} + 178 q^{16} - 144 q^{19} - 192 q^{21} + 270 q^{24} + 380 q^{26} + 108 q^{29} - 304 q^{31} - 20 q^{34} + 306 q^{36} + 228 q^{39} + 188 q^{41} + 374 q^{44} - 680 q^{46} - 1362 q^{49} - 12 q^{51} + 270 q^{54} + 2880 q^{56} - 40 q^{59} + 1140 q^{61} + 574 q^{64} + 330 q^{66} - 408 q^{69} - 2184 q^{71} + 1740 q^{74} + 2448 q^{76} + 32 q^{79} + 162 q^{81} + 3264 q^{84} + 5280 q^{86} + 1932 q^{89} + 2432 q^{91} - 3400 q^{94} - 510 q^{96} + 198 q^{99}+O(q^{100})$$ 2 * q - 34 * q^4 - 30 * q^6 - 18 * q^9 - 22 * q^11 - 320 * q^14 + 178 * q^16 - 144 * q^19 - 192 * q^21 + 270 * q^24 + 380 * q^26 + 108 * q^29 - 304 * q^31 - 20 * q^34 + 306 * q^36 + 228 * q^39 + 188 * q^41 + 374 * q^44 - 680 * q^46 - 1362 * q^49 - 12 * q^51 + 270 * q^54 + 2880 * q^56 - 40 * q^59 + 1140 * q^61 + 574 * q^64 + 330 * q^66 - 408 * q^69 - 2184 * q^71 + 1740 * q^74 + 2448 * q^76 + 32 * q^79 + 162 * q^81 + 3264 * q^84 + 5280 * q^86 + 1932 * q^89 + 2432 * q^91 - 3400 * q^94 - 510 * q^96 + 198 * q^99

## 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
 − 1.00000i 1.00000i
5.00000i 3.00000i −17.0000 0 −15.0000 32.0000i 45.0000i −9.00000 0
199.2 5.00000i 3.00000i −17.0000 0 −15.0000 32.0000i 45.0000i −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.a 2
5.b even 2 1 inner 825.4.c.a 2
5.c odd 4 1 33.4.a.a 1
5.c odd 4 1 825.4.a.i 1
15.e even 4 1 99.4.a.b 1
15.e even 4 1 2475.4.a.b 1
20.e even 4 1 528.4.a.a 1
35.f even 4 1 1617.4.a.a 1
40.i odd 4 1 2112.4.a.l 1
40.k even 4 1 2112.4.a.y 1
55.e even 4 1 363.4.a.h 1
60.l odd 4 1 1584.4.a.t 1
165.l odd 4 1 1089.4.a.a 1

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

 $$T_{2}^{2} + 25$$ T2^2 + 25 $$T_{7}^{2} + 1024$$ T7^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 25$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1024$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 72)^{2}$$
$23$ $$T^{2} + 4624$$
$29$ $$(T - 54)^{2}$$
$31$ $$(T + 152)^{2}$$
$37$ $$T^{2} + 30276$$
$41$ $$(T - 94)^{2}$$
$43$ $$T^{2} + 278784$$
$47$ $$T^{2} + 115600$$
$53$ $$T^{2} + 191844$$
$59$ $$(T + 20)^{2}$$
$61$ $$(T - 570)^{2}$$
$67$ $$T^{2} + 211600$$
$71$ $$(T + 1092)^{2}$$
$73$ $$T^{2} + 315844$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 138384$$
$89$ $$(T - 966)^{2}$$
$97$ $$T^{2} + 276676$$