Properties

 Label 19.4.a.a Level $19$ Weight $4$ Character orbit 19.a Self dual yes Analytic conductor $1.121$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [19,4,Mod(1,19)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(19, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("19.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 19.a (trivial)

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: yes Analytic conductor: $$1.12103629011$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} - 5 q^{3} + q^{4} - 12 q^{5} + 15 q^{6} + 11 q^{7} + 21 q^{8} - 2 q^{9}+O(q^{10})$$ q - 3 * q^2 - 5 * q^3 + q^4 - 12 * q^5 + 15 * q^6 + 11 * q^7 + 21 * q^8 - 2 * q^9 $$q - 3 q^{2} - 5 q^{3} + q^{4} - 12 q^{5} + 15 q^{6} + 11 q^{7} + 21 q^{8} - 2 q^{9} + 36 q^{10} - 54 q^{11} - 5 q^{12} + 11 q^{13} - 33 q^{14} + 60 q^{15} - 71 q^{16} - 93 q^{17} + 6 q^{18} + 19 q^{19} - 12 q^{20} - 55 q^{21} + 162 q^{22} + 183 q^{23} - 105 q^{24} + 19 q^{25} - 33 q^{26} + 145 q^{27} + 11 q^{28} - 249 q^{29} - 180 q^{30} + 56 q^{31} + 45 q^{32} + 270 q^{33} + 279 q^{34} - 132 q^{35} - 2 q^{36} - 250 q^{37} - 57 q^{38} - 55 q^{39} - 252 q^{40} + 240 q^{41} + 165 q^{42} - 196 q^{43} - 54 q^{44} + 24 q^{45} - 549 q^{46} - 168 q^{47} + 355 q^{48} - 222 q^{49} - 57 q^{50} + 465 q^{51} + 11 q^{52} + 435 q^{53} - 435 q^{54} + 648 q^{55} + 231 q^{56} - 95 q^{57} + 747 q^{58} + 195 q^{59} + 60 q^{60} - 358 q^{61} - 168 q^{62} - 22 q^{63} + 433 q^{64} - 132 q^{65} - 810 q^{66} - 961 q^{67} - 93 q^{68} - 915 q^{69} + 396 q^{70} - 246 q^{71} - 42 q^{72} + 353 q^{73} + 750 q^{74} - 95 q^{75} + 19 q^{76} - 594 q^{77} + 165 q^{78} - 34 q^{79} + 852 q^{80} - 671 q^{81} - 720 q^{82} + 234 q^{83} - 55 q^{84} + 1116 q^{85} + 588 q^{86} + 1245 q^{87} - 1134 q^{88} - 168 q^{89} - 72 q^{90} + 121 q^{91} + 183 q^{92} - 280 q^{93} + 504 q^{94} - 228 q^{95} - 225 q^{96} + 758 q^{97} + 666 q^{98} + 108 q^{99}+O(q^{100})$$ q - 3 * q^2 - 5 * q^3 + q^4 - 12 * q^5 + 15 * q^6 + 11 * q^7 + 21 * q^8 - 2 * q^9 + 36 * q^10 - 54 * q^11 - 5 * q^12 + 11 * q^13 - 33 * q^14 + 60 * q^15 - 71 * q^16 - 93 * q^17 + 6 * q^18 + 19 * q^19 - 12 * q^20 - 55 * q^21 + 162 * q^22 + 183 * q^23 - 105 * q^24 + 19 * q^25 - 33 * q^26 + 145 * q^27 + 11 * q^28 - 249 * q^29 - 180 * q^30 + 56 * q^31 + 45 * q^32 + 270 * q^33 + 279 * q^34 - 132 * q^35 - 2 * q^36 - 250 * q^37 - 57 * q^38 - 55 * q^39 - 252 * q^40 + 240 * q^41 + 165 * q^42 - 196 * q^43 - 54 * q^44 + 24 * q^45 - 549 * q^46 - 168 * q^47 + 355 * q^48 - 222 * q^49 - 57 * q^50 + 465 * q^51 + 11 * q^52 + 435 * q^53 - 435 * q^54 + 648 * q^55 + 231 * q^56 - 95 * q^57 + 747 * q^58 + 195 * q^59 + 60 * q^60 - 358 * q^61 - 168 * q^62 - 22 * q^63 + 433 * q^64 - 132 * q^65 - 810 * q^66 - 961 * q^67 - 93 * q^68 - 915 * q^69 + 396 * q^70 - 246 * q^71 - 42 * q^72 + 353 * q^73 + 750 * q^74 - 95 * q^75 + 19 * q^76 - 594 * q^77 + 165 * q^78 - 34 * q^79 + 852 * q^80 - 671 * q^81 - 720 * q^82 + 234 * q^83 - 55 * q^84 + 1116 * q^85 + 588 * q^86 + 1245 * q^87 - 1134 * q^88 - 168 * q^89 - 72 * q^90 + 121 * q^91 + 183 * q^92 - 280 * q^93 + 504 * q^94 - 228 * q^95 - 225 * q^96 + 758 * q^97 + 666 * q^98 + 108 * q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−3.00000 −5.00000 1.00000 −12.0000 15.0000 11.0000 21.0000 −2.00000 36.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$19$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.4.a.a 1
3.b odd 2 1 171.4.a.d 1
4.b odd 2 1 304.4.a.b 1
5.b even 2 1 475.4.a.e 1
5.c odd 4 2 475.4.b.c 2
7.b odd 2 1 931.4.a.a 1
8.b even 2 1 1216.4.a.f 1
8.d odd 2 1 1216.4.a.a 1
11.b odd 2 1 2299.4.a.b 1
19.b odd 2 1 361.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 1.a even 1 1 trivial
171.4.a.d 1 3.b odd 2 1
304.4.a.b 1 4.b odd 2 1
361.4.a.b 1 19.b odd 2 1
475.4.a.e 1 5.b even 2 1
475.4.b.c 2 5.c odd 4 2
931.4.a.a 1 7.b odd 2 1
1216.4.a.a 1 8.d odd 2 1
1216.4.a.f 1 8.b even 2 1
2299.4.a.b 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(19))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T + 5$$
$5$ $$T + 12$$
$7$ $$T - 11$$
$11$ $$T + 54$$
$13$ $$T - 11$$
$17$ $$T + 93$$
$19$ $$T - 19$$
$23$ $$T - 183$$
$29$ $$T + 249$$
$31$ $$T - 56$$
$37$ $$T + 250$$
$41$ $$T - 240$$
$43$ $$T + 196$$
$47$ $$T + 168$$
$53$ $$T - 435$$
$59$ $$T - 195$$
$61$ $$T + 358$$
$67$ $$T + 961$$
$71$ $$T + 246$$
$73$ $$T - 353$$
$79$ $$T + 34$$
$83$ $$T - 234$$
$89$ $$T + 168$$
$97$ $$T - 758$$