# Properties

 Label 825.4.c.f Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $0$ 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: $$0$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - 3 i q^{3} + 7 q^{4} + 3 q^{6} + 26 i q^{7} + 15 i q^{8} - 9 q^{9} +O(q^{10})$$ q + i * q^2 - 3*i * q^3 + 7 * q^4 + 3 * q^6 + 26*i * q^7 + 15*i * q^8 - 9 * q^9 $$q + i q^{2} - 3 i q^{3} + 7 q^{4} + 3 q^{6} + 26 i q^{7} + 15 i q^{8} - 9 q^{9} + 11 q^{11} - 21 i q^{12} - 32 i q^{13} - 26 q^{14} + 41 q^{16} - 74 i q^{17} - 9 i q^{18} + 60 q^{19} + 78 q^{21} + 11 i q^{22} - 182 i q^{23} + 45 q^{24} + 32 q^{26} + 27 i q^{27} + 182 i q^{28} + 90 q^{29} - 8 q^{31} + 161 i q^{32} - 33 i q^{33} + 74 q^{34} - 63 q^{36} + 66 i q^{37} + 60 i q^{38} - 96 q^{39} + 422 q^{41} + 78 i q^{42} + 408 i q^{43} + 77 q^{44} + 182 q^{46} + 506 i q^{47} - 123 i q^{48} - 333 q^{49} - 222 q^{51} - 224 i q^{52} + 348 i q^{53} - 27 q^{54} - 390 q^{56} - 180 i q^{57} + 90 i q^{58} + 200 q^{59} + 132 q^{61} - 8 i q^{62} - 234 i q^{63} + 167 q^{64} + 33 q^{66} + 1036 i q^{67} - 518 i q^{68} - 546 q^{69} + 762 q^{71} - 135 i q^{72} - 542 i q^{73} - 66 q^{74} + 420 q^{76} + 286 i q^{77} - 96 i q^{78} + 550 q^{79} + 81 q^{81} + 422 i q^{82} - 132 i q^{83} + 546 q^{84} - 408 q^{86} - 270 i q^{87} + 165 i q^{88} - 570 q^{89} + 832 q^{91} - 1274 i q^{92} + 24 i q^{93} - 506 q^{94} + 483 q^{96} - 14 i q^{97} - 333 i q^{98} - 99 q^{99} +O(q^{100})$$ q + i * q^2 - 3*i * q^3 + 7 * q^4 + 3 * q^6 + 26*i * q^7 + 15*i * q^8 - 9 * q^9 + 11 * q^11 - 21*i * q^12 - 32*i * q^13 - 26 * q^14 + 41 * q^16 - 74*i * q^17 - 9*i * q^18 + 60 * q^19 + 78 * q^21 + 11*i * q^22 - 182*i * q^23 + 45 * q^24 + 32 * q^26 + 27*i * q^27 + 182*i * q^28 + 90 * q^29 - 8 * q^31 + 161*i * q^32 - 33*i * q^33 + 74 * q^34 - 63 * q^36 + 66*i * q^37 + 60*i * q^38 - 96 * q^39 + 422 * q^41 + 78*i * q^42 + 408*i * q^43 + 77 * q^44 + 182 * q^46 + 506*i * q^47 - 123*i * q^48 - 333 * q^49 - 222 * q^51 - 224*i * q^52 + 348*i * q^53 - 27 * q^54 - 390 * q^56 - 180*i * q^57 + 90*i * q^58 + 200 * q^59 + 132 * q^61 - 8*i * q^62 - 234*i * q^63 + 167 * q^64 + 33 * q^66 + 1036*i * q^67 - 518*i * q^68 - 546 * q^69 + 762 * q^71 - 135*i * q^72 - 542*i * q^73 - 66 * q^74 + 420 * q^76 + 286*i * q^77 - 96*i * q^78 + 550 * q^79 + 81 * q^81 + 422*i * q^82 - 132*i * q^83 + 546 * q^84 - 408 * q^86 - 270*i * q^87 + 165*i * q^88 - 570 * q^89 + 832 * q^91 - 1274*i * q^92 + 24*i * q^93 - 506 * q^94 + 483 * q^96 - 14*i * q^97 - 333*i * q^98 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 22 q^{11} - 52 q^{14} + 82 q^{16} + 120 q^{19} + 156 q^{21} + 90 q^{24} + 64 q^{26} + 180 q^{29} - 16 q^{31} + 148 q^{34} - 126 q^{36} - 192 q^{39} + 844 q^{41} + 154 q^{44} + 364 q^{46} - 666 q^{49} - 444 q^{51} - 54 q^{54} - 780 q^{56} + 400 q^{59} + 264 q^{61} + 334 q^{64} + 66 q^{66} - 1092 q^{69} + 1524 q^{71} - 132 q^{74} + 840 q^{76} + 1100 q^{79} + 162 q^{81} + 1092 q^{84} - 816 q^{86} - 1140 q^{89} + 1664 q^{91} - 1012 q^{94} + 966 q^{96} - 198 q^{99}+O(q^{100})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 22 * q^11 - 52 * q^14 + 82 * q^16 + 120 * q^19 + 156 * q^21 + 90 * q^24 + 64 * q^26 + 180 * q^29 - 16 * q^31 + 148 * q^34 - 126 * q^36 - 192 * q^39 + 844 * q^41 + 154 * q^44 + 364 * q^46 - 666 * q^49 - 444 * q^51 - 54 * q^54 - 780 * q^56 + 400 * q^59 + 264 * q^61 + 334 * q^64 + 66 * q^66 - 1092 * q^69 + 1524 * q^71 - 132 * q^74 + 840 * q^76 + 1100 * q^79 + 162 * q^81 + 1092 * q^84 - 816 * q^86 - 1140 * q^89 + 1664 * q^91 - 1012 * q^94 + 966 * 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
1.00000i 3.00000i 7.00000 0 3.00000 26.0000i 15.0000i −9.00000 0
199.2 1.00000i 3.00000i 7.00000 0 3.00000 26.0000i 15.0000i −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.f 2
5.b even 2 1 inner 825.4.c.f 2
5.c odd 4 1 33.4.a.b 1
5.c odd 4 1 825.4.a.f 1
15.e even 4 1 99.4.a.a 1
15.e even 4 1 2475.4.a.e 1
20.e even 4 1 528.4.a.h 1
35.f even 4 1 1617.4.a.d 1
40.i odd 4 1 2112.4.a.u 1
40.k even 4 1 2112.4.a.h 1
55.e even 4 1 363.4.a.d 1
60.l odd 4 1 1584.4.a.l 1
165.l odd 4 1 1089.4.a.e 1

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 676$$ T7^2 + 676

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 676$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} + 1024$$
$17$ $$T^{2} + 5476$$
$19$ $$(T - 60)^{2}$$
$23$ $$T^{2} + 33124$$
$29$ $$(T - 90)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4356$$
$41$ $$(T - 422)^{2}$$
$43$ $$T^{2} + 166464$$
$47$ $$T^{2} + 256036$$
$53$ $$T^{2} + 121104$$
$59$ $$(T - 200)^{2}$$
$61$ $$(T - 132)^{2}$$
$67$ $$T^{2} + 1073296$$
$71$ $$(T - 762)^{2}$$
$73$ $$T^{2} + 293764$$
$79$ $$(T - 550)^{2}$$
$83$ $$T^{2} + 17424$$
$89$ $$(T + 570)^{2}$$
$97$ $$T^{2} + 196$$