Properties

 Label 825.4.c.e 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 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - 3 i q^{3} + 7 q^{4} + 3 q^{6} + 36 i q^{7} + 15 i q^{8} - 9 q^{9} +O(q^{10})$$ q + i * q^2 - 3*i * q^3 + 7 * q^4 + 3 * q^6 + 36*i * q^7 + 15*i * q^8 - 9 * q^9 $$q + i q^{2} - 3 i q^{3} + 7 q^{4} + 3 q^{6} + 36 i q^{7} + 15 i q^{8} - 9 q^{9} + 11 q^{11} - 21 i q^{12} - 2 i q^{13} - 36 q^{14} + 41 q^{16} + 66 i q^{17} - 9 i q^{18} - 140 q^{19} + 108 q^{21} + 11 i q^{22} + 68 i q^{23} + 45 q^{24} + 2 q^{26} + 27 i q^{27} + 252 i q^{28} - 150 q^{29} - 128 q^{31} + 161 i q^{32} - 33 i q^{33} - 66 q^{34} - 63 q^{36} - 314 i q^{37} - 140 i q^{38} - 6 q^{39} - 118 q^{41} + 108 i q^{42} - 172 i q^{43} + 77 q^{44} - 68 q^{46} - 324 i q^{47} - 123 i q^{48} - 953 q^{49} + 198 q^{51} - 14 i q^{52} - 82 i q^{53} - 27 q^{54} - 540 q^{56} + 420 i q^{57} - 150 i q^{58} + 740 q^{59} + 122 q^{61} - 128 i q^{62} - 324 i q^{63} + 167 q^{64} + 33 q^{66} - 124 i q^{67} + 462 i q^{68} + 204 q^{69} - 988 q^{71} - 135 i q^{72} - 2 i q^{73} + 314 q^{74} - 980 q^{76} + 396 i q^{77} - 6 i q^{78} - 1100 q^{79} + 81 q^{81} - 118 i q^{82} + 868 i q^{83} + 756 q^{84} + 172 q^{86} + 450 i q^{87} + 165 i q^{88} + 470 q^{89} + 72 q^{91} + 476 i q^{92} + 384 i q^{93} + 324 q^{94} + 483 q^{96} + 1186 i q^{97} - 953 i q^{98} - 99 q^{99} +O(q^{100})$$ q + i * q^2 - 3*i * q^3 + 7 * q^4 + 3 * q^6 + 36*i * q^7 + 15*i * q^8 - 9 * q^9 + 11 * q^11 - 21*i * q^12 - 2*i * q^13 - 36 * q^14 + 41 * q^16 + 66*i * q^17 - 9*i * q^18 - 140 * q^19 + 108 * q^21 + 11*i * q^22 + 68*i * q^23 + 45 * q^24 + 2 * q^26 + 27*i * q^27 + 252*i * q^28 - 150 * q^29 - 128 * q^31 + 161*i * q^32 - 33*i * q^33 - 66 * q^34 - 63 * q^36 - 314*i * q^37 - 140*i * q^38 - 6 * q^39 - 118 * q^41 + 108*i * q^42 - 172*i * q^43 + 77 * q^44 - 68 * q^46 - 324*i * q^47 - 123*i * q^48 - 953 * q^49 + 198 * q^51 - 14*i * q^52 - 82*i * q^53 - 27 * q^54 - 540 * q^56 + 420*i * q^57 - 150*i * q^58 + 740 * q^59 + 122 * q^61 - 128*i * q^62 - 324*i * q^63 + 167 * q^64 + 33 * q^66 - 124*i * q^67 + 462*i * q^68 + 204 * q^69 - 988 * q^71 - 135*i * q^72 - 2*i * q^73 + 314 * q^74 - 980 * q^76 + 396*i * q^77 - 6*i * q^78 - 1100 * q^79 + 81 * q^81 - 118*i * q^82 + 868*i * q^83 + 756 * q^84 + 172 * q^86 + 450*i * q^87 + 165*i * q^88 + 470 * q^89 + 72 * q^91 + 476*i * q^92 + 384*i * q^93 + 324 * q^94 + 483 * q^96 + 1186*i * q^97 - 953*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 $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 22 q^{11} - 72 q^{14} + 82 q^{16} - 280 q^{19} + 216 q^{21} + 90 q^{24} + 4 q^{26} - 300 q^{29} - 256 q^{31} - 132 q^{34} - 126 q^{36} - 12 q^{39} - 236 q^{41} + 154 q^{44} - 136 q^{46} - 1906 q^{49} + 396 q^{51} - 54 q^{54} - 1080 q^{56} + 1480 q^{59} + 244 q^{61} + 334 q^{64} + 66 q^{66} + 408 q^{69} - 1976 q^{71} + 628 q^{74} - 1960 q^{76} - 2200 q^{79} + 162 q^{81} + 1512 q^{84} + 344 q^{86} + 940 q^{89} + 144 q^{91} + 648 q^{94} + 966 q^{96} - 198 q^{99}+O(q^{100})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 22 * q^11 - 72 * q^14 + 82 * q^16 - 280 * q^19 + 216 * q^21 + 90 * q^24 + 4 * q^26 - 300 * q^29 - 256 * q^31 - 132 * q^34 - 126 * q^36 - 12 * q^39 - 236 * q^41 + 154 * q^44 - 136 * q^46 - 1906 * q^49 + 396 * q^51 - 54 * q^54 - 1080 * q^56 + 1480 * q^59 + 244 * q^61 + 334 * q^64 + 66 * q^66 + 408 * q^69 - 1976 * q^71 + 628 * q^74 - 1960 * q^76 - 2200 * q^79 + 162 * q^81 + 1512 * q^84 + 344 * q^86 + 940 * q^89 + 144 * q^91 + 648 * 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 36.0000i 15.0000i −9.00000 0
199.2 1.00000i 3.00000i 7.00000 0 3.00000 36.0000i 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.e 2
5.b even 2 1 inner 825.4.c.e 2
5.c odd 4 1 165.4.a.b 1
5.c odd 4 1 825.4.a.d 1
15.e even 4 1 495.4.a.b 1
15.e even 4 1 2475.4.a.g 1
55.e even 4 1 1815.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.b 1 5.c odd 4 1
495.4.a.b 1 15.e even 4 1
825.4.a.d 1 5.c odd 4 1
825.4.c.e 2 1.a even 1 1 trivial
825.4.c.e 2 5.b even 2 1 inner
1815.4.a.c 1 55.e even 4 1
2475.4.a.g 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 $$T_{7}^{2} + 1296$$ T7^2 + 1296

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1296$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4356$$
$19$ $$(T + 140)^{2}$$
$23$ $$T^{2} + 4624$$
$29$ $$(T + 150)^{2}$$
$31$ $$(T + 128)^{2}$$
$37$ $$T^{2} + 98596$$
$41$ $$(T + 118)^{2}$$
$43$ $$T^{2} + 29584$$
$47$ $$T^{2} + 104976$$
$53$ $$T^{2} + 6724$$
$59$ $$(T - 740)^{2}$$
$61$ $$(T - 122)^{2}$$
$67$ $$T^{2} + 15376$$
$71$ $$(T + 988)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 1100)^{2}$$
$83$ $$T^{2} + 753424$$
$89$ $$(T - 470)^{2}$$
$97$ $$T^{2} + 1406596$$