Properties

 Label 361.4.a.b Level $361$ Weight $4$ Character orbit 361.a Self dual yes Analytic conductor $21.300$ 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] = [361,4,Mod(1,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, 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("361.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 361.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: $$21.2996895121$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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 $$q + 3 q^{2} + 5 q^{3} + q^{4} - 12 q^{5} + 15 q^{6} + 11 q^{7} - 21 q^{8} - 2 q^{9} - 36 q^{10} - 54 q^{11} + 5 q^{12} - 11 q^{13} + 33 q^{14} - 60 q^{15} - 71 q^{16} - 93 q^{17} - 6 q^{18} - 12 q^{20} + 55 q^{21} - 162 q^{22} + 183 q^{23} - 105 q^{24} + 19 q^{25} - 33 q^{26} - 145 q^{27} + 11 q^{28} + 249 q^{29} - 180 q^{30} - 56 q^{31} - 45 q^{32} - 270 q^{33} - 279 q^{34} - 132 q^{35} - 2 q^{36} + 250 q^{37} - 55 q^{39} + 252 q^{40} - 240 q^{41} + 165 q^{42} - 196 q^{43} - 54 q^{44} + 24 q^{45} + 549 q^{46} - 168 q^{47} - 355 q^{48} - 222 q^{49} + 57 q^{50} - 465 q^{51} - 11 q^{52} - 435 q^{53} - 435 q^{54} + 648 q^{55} - 231 q^{56} + 747 q^{58} - 195 q^{59} - 60 q^{60} - 358 q^{61} - 168 q^{62} - 22 q^{63} + 433 q^{64} + 132 q^{65} - 810 q^{66} + 961 q^{67} - 93 q^{68} + 915 q^{69} - 396 q^{70} + 246 q^{71} + 42 q^{72} + 353 q^{73} + 750 q^{74} + 95 q^{75} - 594 q^{77} - 165 q^{78} + 34 q^{79} + 852 q^{80} - 671 q^{81} - 720 q^{82} + 234 q^{83} + 55 q^{84} + 1116 q^{85} - 588 q^{86} + 1245 q^{87} + 1134 q^{88} + 168 q^{89} + 72 q^{90} - 121 q^{91} + 183 q^{92} - 280 q^{93} - 504 q^{94} - 225 q^{96} - 758 q^{97} - 666 q^{98} + 108 q^{99}+O(q^{100})$$ 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 - 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 - 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 + 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 - 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 - 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 361.4.a.b 1
19.b odd 2 1 19.4.a.a 1
57.d even 2 1 171.4.a.d 1
76.d even 2 1 304.4.a.b 1
95.d odd 2 1 475.4.a.e 1
95.g even 4 2 475.4.b.c 2
133.c even 2 1 931.4.a.a 1
152.b even 2 1 1216.4.a.a 1
152.g odd 2 1 1216.4.a.f 1
209.d even 2 1 2299.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 19.b odd 2 1
171.4.a.d 1 57.d even 2 1
304.4.a.b 1 76.d even 2 1
361.4.a.b 1 1.a even 1 1 trivial
475.4.a.e 1 95.d odd 2 1
475.4.b.c 2 95.g even 4 2
931.4.a.a 1 133.c even 2 1
1216.4.a.a 1 152.b even 2 1
1216.4.a.f 1 152.g odd 2 1
2299.4.a.b 1 209.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(361))$$.

Hecke characteristic polynomials

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