Properties

 Label 475.4.b.c Level $475$ Weight $4$ Character orbit 475.b Analytic conductor $28.026$ 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] = [475,4,Mod(324,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (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: $$28.0259072527$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} - 5 i q^{3} - q^{4} + 15 q^{6} - 11 i q^{7} + 21 i q^{8} + 2 q^{9} +O(q^{10})$$ q + 3*i * q^2 - 5*i * q^3 - q^4 + 15 * q^6 - 11*i * q^7 + 21*i * q^8 + 2 * q^9 $$q + 3 i q^{2} - 5 i q^{3} - q^{4} + 15 q^{6} - 11 i q^{7} + 21 i q^{8} + 2 q^{9} - 54 q^{11} + 5 i q^{12} + 11 i q^{13} + 33 q^{14} - 71 q^{16} + 93 i q^{17} + 6 i q^{18} - 19 q^{19} - 55 q^{21} - 162 i q^{22} + 183 i q^{23} + 105 q^{24} - 33 q^{26} - 145 i q^{27} + 11 i q^{28} + 249 q^{29} + 56 q^{31} - 45 i q^{32} + 270 i q^{33} - 279 q^{34} - 2 q^{36} + 250 i q^{37} - 57 i q^{38} + 55 q^{39} + 240 q^{41} - 165 i q^{42} - 196 i q^{43} + 54 q^{44} - 549 q^{46} + 168 i q^{47} + 355 i q^{48} + 222 q^{49} + 465 q^{51} - 11 i q^{52} + 435 i q^{53} + 435 q^{54} + 231 q^{56} + 95 i q^{57} + 747 i q^{58} - 195 q^{59} - 358 q^{61} + 168 i q^{62} - 22 i q^{63} - 433 q^{64} - 810 q^{66} + 961 i q^{67} - 93 i q^{68} + 915 q^{69} - 246 q^{71} + 42 i q^{72} + 353 i q^{73} - 750 q^{74} + 19 q^{76} + 594 i q^{77} + 165 i q^{78} + 34 q^{79} - 671 q^{81} + 720 i q^{82} + 234 i q^{83} + 55 q^{84} + 588 q^{86} - 1245 i q^{87} - 1134 i q^{88} + 168 q^{89} + 121 q^{91} - 183 i q^{92} - 280 i q^{93} - 504 q^{94} - 225 q^{96} - 758 i q^{97} + 666 i q^{98} - 108 q^{99} +O(q^{100})$$ q + 3*i * q^2 - 5*i * q^3 - q^4 + 15 * q^6 - 11*i * q^7 + 21*i * q^8 + 2 * q^9 - 54 * q^11 + 5*i * q^12 + 11*i * q^13 + 33 * q^14 - 71 * q^16 + 93*i * q^17 + 6*i * q^18 - 19 * q^19 - 55 * q^21 - 162*i * q^22 + 183*i * q^23 + 105 * q^24 - 33 * q^26 - 145*i * q^27 + 11*i * q^28 + 249 * q^29 + 56 * q^31 - 45*i * q^32 + 270*i * q^33 - 279 * q^34 - 2 * q^36 + 250*i * q^37 - 57*i * q^38 + 55 * q^39 + 240 * q^41 - 165*i * q^42 - 196*i * q^43 + 54 * q^44 - 549 * q^46 + 168*i * q^47 + 355*i * q^48 + 222 * q^49 + 465 * q^51 - 11*i * q^52 + 435*i * q^53 + 435 * q^54 + 231 * q^56 + 95*i * q^57 + 747*i * q^58 - 195 * q^59 - 358 * q^61 + 168*i * q^62 - 22*i * q^63 - 433 * q^64 - 810 * q^66 + 961*i * q^67 - 93*i * q^68 + 915 * q^69 - 246 * q^71 + 42*i * q^72 + 353*i * q^73 - 750 * q^74 + 19 * q^76 + 594*i * q^77 + 165*i * q^78 + 34 * q^79 - 671 * q^81 + 720*i * q^82 + 234*i * q^83 + 55 * q^84 + 588 * q^86 - 1245*i * q^87 - 1134*i * q^88 + 168 * q^89 + 121 * q^91 - 183*i * q^92 - 280*i * q^93 - 504 * q^94 - 225 * q^96 - 758*i * q^97 + 666*i * q^98 - 108 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 30 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9 $$2 q - 2 q^{4} + 30 q^{6} + 4 q^{9} - 108 q^{11} + 66 q^{14} - 142 q^{16} - 38 q^{19} - 110 q^{21} + 210 q^{24} - 66 q^{26} + 498 q^{29} + 112 q^{31} - 558 q^{34} - 4 q^{36} + 110 q^{39} + 480 q^{41} + 108 q^{44} - 1098 q^{46} + 444 q^{49} + 930 q^{51} + 870 q^{54} + 462 q^{56} - 390 q^{59} - 716 q^{61} - 866 q^{64} - 1620 q^{66} + 1830 q^{69} - 492 q^{71} - 1500 q^{74} + 38 q^{76} + 68 q^{79} - 1342 q^{81} + 110 q^{84} + 1176 q^{86} + 336 q^{89} + 242 q^{91} - 1008 q^{94} - 450 q^{96} - 216 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9 - 108 * q^11 + 66 * q^14 - 142 * q^16 - 38 * q^19 - 110 * q^21 + 210 * q^24 - 66 * q^26 + 498 * q^29 + 112 * q^31 - 558 * q^34 - 4 * q^36 + 110 * q^39 + 480 * q^41 + 108 * q^44 - 1098 * q^46 + 444 * q^49 + 930 * q^51 + 870 * q^54 + 462 * q^56 - 390 * q^59 - 716 * q^61 - 866 * q^64 - 1620 * q^66 + 1830 * q^69 - 492 * q^71 - 1500 * q^74 + 38 * q^76 + 68 * q^79 - 1342 * q^81 + 110 * q^84 + 1176 * q^86 + 336 * q^89 + 242 * q^91 - 1008 * q^94 - 450 * q^96 - 216 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 − 1.00000i 1.00000i
3.00000i 5.00000i −1.00000 0 15.0000 11.0000i 21.0000i 2.00000 0
324.2 3.00000i 5.00000i −1.00000 0 15.0000 11.0000i 21.0000i 2.00000 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 475.4.b.c 2
5.b even 2 1 inner 475.4.b.c 2
5.c odd 4 1 19.4.a.a 1
5.c odd 4 1 475.4.a.e 1
15.e even 4 1 171.4.a.d 1
20.e even 4 1 304.4.a.b 1
35.f even 4 1 931.4.a.a 1
40.i odd 4 1 1216.4.a.f 1
40.k even 4 1 1216.4.a.a 1
55.e even 4 1 2299.4.a.b 1
95.g even 4 1 361.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 5.c odd 4 1
171.4.a.d 1 15.e even 4 1
304.4.a.b 1 20.e even 4 1
361.4.a.b 1 95.g even 4 1
475.4.a.e 1 5.c odd 4 1
475.4.b.c 2 1.a even 1 1 trivial
475.4.b.c 2 5.b even 2 1 inner
931.4.a.a 1 35.f even 4 1
1216.4.a.a 1 40.k even 4 1
1216.4.a.f 1 40.i odd 4 1
2299.4.a.b 1 55.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 $$S_{4}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}^{2} + 9$$ T2^2 + 9 $$T_{3}^{2} + 25$$ T3^2 + 25 $$T_{7}^{2} + 121$$ T7^2 + 121

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$T^{2} + 25$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 121$$
$11$ $$(T + 54)^{2}$$
$13$ $$T^{2} + 121$$
$17$ $$T^{2} + 8649$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} + 33489$$
$29$ $$(T - 249)^{2}$$
$31$ $$(T - 56)^{2}$$
$37$ $$T^{2} + 62500$$
$41$ $$(T - 240)^{2}$$
$43$ $$T^{2} + 38416$$
$47$ $$T^{2} + 28224$$
$53$ $$T^{2} + 189225$$
$59$ $$(T + 195)^{2}$$
$61$ $$(T + 358)^{2}$$
$67$ $$T^{2} + 923521$$
$71$ $$(T + 246)^{2}$$
$73$ $$T^{2} + 124609$$
$79$ $$(T - 34)^{2}$$
$83$ $$T^{2} + 54756$$
$89$ $$(T - 168)^{2}$$
$97$ $$T^{2} + 574564$$