Properties

 Label 370.2.a.g Level $370$ Weight $2$ Character orbit 370.a Self dual yes Analytic conductor $2.954$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.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: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.892.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 8x + 10$$ x^3 - x^2 - 8*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - \beta_{2} q^{3} + q^{4} - q^{5} - \beta_{2} q^{6} - \beta_1 q^{7} + q^{8} + (2 \beta_1 + 3) q^{9}+O(q^{10})$$ q + q^2 - b2 * q^3 + q^4 - q^5 - b2 * q^6 - b1 * q^7 + q^8 + (2*b1 + 3) * q^9 $$q + q^{2} - \beta_{2} q^{3} + q^{4} - q^{5} - \beta_{2} q^{6} - \beta_1 q^{7} + q^{8} + (2 \beta_1 + 3) q^{9} - q^{10} + (\beta_{2} - \beta_1 + 4) q^{11} - \beta_{2} q^{12} + 2 \beta_{2} q^{13} - \beta_1 q^{14} + \beta_{2} q^{15} + q^{16} + ( - \beta_{2} + \beta_1) q^{17} + (2 \beta_1 + 3) q^{18} - \beta_{2} q^{19} - q^{20} + (2 \beta_{2} + 2) q^{21} + (\beta_{2} - \beta_1 + 4) q^{22} + (2 \beta_{2} - 2 \beta_1) q^{23} - \beta_{2} q^{24} + q^{25} + 2 \beta_{2} q^{26} + ( - 4 \beta_{2} - 4) q^{27} - \beta_1 q^{28} + (\beta_{2} + \beta_1 - 2) q^{29} + \beta_{2} q^{30} + 3 \beta_1 q^{31} + q^{32} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{33} + ( - \beta_{2} + \beta_1) q^{34} + \beta_1 q^{35} + (2 \beta_1 + 3) q^{36} - q^{37} - \beta_{2} q^{38} + ( - 4 \beta_1 - 12) q^{39} - q^{40} + (\beta_{2} - 3 \beta_1 - 2) q^{41} + (2 \beta_{2} + 2) q^{42} + (\beta_{2} + \beta_1 - 4) q^{43} + (\beta_{2} - \beta_1 + 4) q^{44} + ( - 2 \beta_1 - 3) q^{45} + (2 \beta_{2} - 2 \beta_1) q^{46} + ( - \beta_{2} + 2 \beta_1) q^{47} - \beta_{2} q^{48} + (\beta_{2} - \beta_1 - 1) q^{49} + q^{50} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{51} + 2 \beta_{2} q^{52} + ( - \beta_{2} + 3 \beta_1 + 6) q^{53} + ( - 4 \beta_{2} - 4) q^{54} + ( - \beta_{2} + \beta_1 - 4) q^{55} - \beta_1 q^{56} + (2 \beta_1 + 6) q^{57} + (\beta_{2} + \beta_1 - 2) q^{58} + ( - \beta_{2} + 2 \beta_1 + 4) q^{59} + \beta_{2} q^{60} + (\beta_{2} + \beta_1 + 2) q^{61} + 3 \beta_1 q^{62} + ( - 2 \beta_{2} - \beta_1 - 12) q^{63} + q^{64} - 2 \beta_{2} q^{65} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{66} + ( - \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{2} + \beta_1) q^{68} + (4 \beta_{2} - 4 \beta_1 - 8) q^{69} + \beta_1 q^{70} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{71} + (2 \beta_1 + 3) q^{72} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{73} - q^{74} - \beta_{2} q^{75} - \beta_{2} q^{76} + ( - \beta_{2} - 5 \beta_1 + 4) q^{77} + ( - 4 \beta_1 - 12) q^{78} + ( - \beta_{2} - 2 \beta_1 - 8) q^{79} - q^{80} + (4 \beta_{2} + 2 \beta_1 + 15) q^{81} + (\beta_{2} - 3 \beta_1 - 2) q^{82} + (3 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{2} + 2) q^{84} + (\beta_{2} - \beta_1) q^{85} + (\beta_{2} + \beta_1 - 4) q^{86} + ( - 2 \beta_1 - 8) q^{87} + (\beta_{2} - \beta_1 + 4) q^{88} + 6 q^{89} + ( - 2 \beta_1 - 3) q^{90} + ( - 4 \beta_{2} - 4) q^{91} + (2 \beta_{2} - 2 \beta_1) q^{92} + ( - 6 \beta_{2} - 6) q^{93} + ( - \beta_{2} + 2 \beta_1) q^{94} + \beta_{2} q^{95} - \beta_{2} q^{96} + (\beta_{2} - 3 \beta_1 - 6) q^{97} + (\beta_{2} - \beta_1 - 1) q^{98} + (5 \beta_{2} + 7 \beta_1 + 4) q^{99}+O(q^{100})$$ q + q^2 - b2 * q^3 + q^4 - q^5 - b2 * q^6 - b1 * q^7 + q^8 + (2*b1 + 3) * q^9 - q^10 + (b2 - b1 + 4) * q^11 - b2 * q^12 + 2*b2 * q^13 - b1 * q^14 + b2 * q^15 + q^16 + (-b2 + b1) * q^17 + (2*b1 + 3) * q^18 - b2 * q^19 - q^20 + (2*b2 + 2) * q^21 + (b2 - b1 + 4) * q^22 + (2*b2 - 2*b1) * q^23 - b2 * q^24 + q^25 + 2*b2 * q^26 + (-4*b2 - 4) * q^27 - b1 * q^28 + (b2 + b1 - 2) * q^29 + b2 * q^30 + 3*b1 * q^31 + q^32 + (-2*b2 - 2*b1 - 4) * q^33 + (-b2 + b1) * q^34 + b1 * q^35 + (2*b1 + 3) * q^36 - q^37 - b2 * q^38 + (-4*b1 - 12) * q^39 - q^40 + (b2 - 3*b1 - 2) * q^41 + (2*b2 + 2) * q^42 + (b2 + b1 - 4) * q^43 + (b2 - b1 + 4) * q^44 + (-2*b1 - 3) * q^45 + (2*b2 - 2*b1) * q^46 + (-b2 + 2*b1) * q^47 - b2 * q^48 + (b2 - b1 - 1) * q^49 + q^50 + (-2*b2 + 2*b1 + 4) * q^51 + 2*b2 * q^52 + (-b2 + 3*b1 + 6) * q^53 + (-4*b2 - 4) * q^54 + (-b2 + b1 - 4) * q^55 - b1 * q^56 + (2*b1 + 6) * q^57 + (b2 + b1 - 2) * q^58 + (-b2 + 2*b1 + 4) * q^59 + b2 * q^60 + (b2 + b1 + 2) * q^61 + 3*b1 * q^62 + (-2*b2 - b1 - 12) * q^63 + q^64 - 2*b2 * q^65 + (-2*b2 - 2*b1 - 4) * q^66 + (-b2 + 2*b1) * q^67 + (-b2 + b1) * q^68 + (4*b2 - 4*b1 - 8) * q^69 + b1 * q^70 + (-2*b2 - 2*b1 + 8) * q^71 + (2*b1 + 3) * q^72 + (-2*b2 + 2*b1 - 6) * q^73 - q^74 - b2 * q^75 - b2 * q^76 + (-b2 - 5*b1 + 4) * q^77 + (-4*b1 - 12) * q^78 + (-b2 - 2*b1 - 8) * q^79 - q^80 + (4*b2 + 2*b1 + 15) * q^81 + (b2 - 3*b1 - 2) * q^82 + (3*b2 + 2*b1) * q^83 + (2*b2 + 2) * q^84 + (b2 - b1) * q^85 + (b2 + b1 - 4) * q^86 + (-2*b1 - 8) * q^87 + (b2 - b1 + 4) * q^88 + 6 * q^89 + (-2*b1 - 3) * q^90 + (-4*b2 - 4) * q^91 + (2*b2 - 2*b1) * q^92 + (-6*b2 - 6) * q^93 + (-b2 + 2*b1) * q^94 + b2 * q^95 - b2 * q^96 + (b2 - 3*b1 - 6) * q^97 + (b2 - b1 - 1) * q^98 + (5*b2 + 7*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - q^7 + 3 * q^8 + 11 * q^9 $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} - 3 q^{10} + 11 q^{11} - q^{14} + 3 q^{16} + q^{17} + 11 q^{18} - 3 q^{20} + 6 q^{21} + 11 q^{22} - 2 q^{23} + 3 q^{25} - 12 q^{27} - q^{28} - 5 q^{29} + 3 q^{31} + 3 q^{32} - 14 q^{33} + q^{34} + q^{35} + 11 q^{36} - 3 q^{37} - 40 q^{39} - 3 q^{40} - 9 q^{41} + 6 q^{42} - 11 q^{43} + 11 q^{44} - 11 q^{45} - 2 q^{46} + 2 q^{47} - 4 q^{49} + 3 q^{50} + 14 q^{51} + 21 q^{53} - 12 q^{54} - 11 q^{55} - q^{56} + 20 q^{57} - 5 q^{58} + 14 q^{59} + 7 q^{61} + 3 q^{62} - 37 q^{63} + 3 q^{64} - 14 q^{66} + 2 q^{67} + q^{68} - 28 q^{69} + q^{70} + 22 q^{71} + 11 q^{72} - 16 q^{73} - 3 q^{74} + 7 q^{77} - 40 q^{78} - 26 q^{79} - 3 q^{80} + 47 q^{81} - 9 q^{82} + 2 q^{83} + 6 q^{84} - q^{85} - 11 q^{86} - 26 q^{87} + 11 q^{88} + 18 q^{89} - 11 q^{90} - 12 q^{91} - 2 q^{92} - 18 q^{93} + 2 q^{94} - 21 q^{97} - 4 q^{98} + 19 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - q^7 + 3 * q^8 + 11 * q^9 - 3 * q^10 + 11 * q^11 - q^14 + 3 * q^16 + q^17 + 11 * q^18 - 3 * q^20 + 6 * q^21 + 11 * q^22 - 2 * q^23 + 3 * q^25 - 12 * q^27 - q^28 - 5 * q^29 + 3 * q^31 + 3 * q^32 - 14 * q^33 + q^34 + q^35 + 11 * q^36 - 3 * q^37 - 40 * q^39 - 3 * q^40 - 9 * q^41 + 6 * q^42 - 11 * q^43 + 11 * q^44 - 11 * q^45 - 2 * q^46 + 2 * q^47 - 4 * q^49 + 3 * q^50 + 14 * q^51 + 21 * q^53 - 12 * q^54 - 11 * q^55 - q^56 + 20 * q^57 - 5 * q^58 + 14 * q^59 + 7 * q^61 + 3 * q^62 - 37 * q^63 + 3 * q^64 - 14 * q^66 + 2 * q^67 + q^68 - 28 * q^69 + q^70 + 22 * q^71 + 11 * q^72 - 16 * q^73 - 3 * q^74 + 7 * q^77 - 40 * q^78 - 26 * q^79 - 3 * q^80 + 47 * q^81 - 9 * q^82 + 2 * q^83 + 6 * q^84 - q^85 - 11 * q^86 - 26 * q^87 + 11 * q^88 + 18 * q^89 - 11 * q^90 - 12 * q^91 - 2 * q^92 - 18 * q^93 + 2 * q^94 - 21 * q^97 - 4 * q^98 + 19 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 8x + 10$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 6$$ v^2 + v - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 6$$ b2 - b1 + 6

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
 2.59774 −2.91729 1.31955
1.00000 −3.34596 1.00000 −1.00000 −3.34596 −2.59774 1.00000 8.19547 −1.00000
1.2 1.00000 0.406728 1.00000 −1.00000 0.406728 2.91729 1.00000 −2.83457 −1.00000
1.3 1.00000 2.93923 1.00000 −1.00000 2.93923 −1.31955 1.00000 5.63910 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$37$$ $$1$$

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 370.2.a.g 3
3.b odd 2 1 3330.2.a.bg 3
4.b odd 2 1 2960.2.a.u 3
5.b even 2 1 1850.2.a.z 3
5.c odd 4 2 1850.2.b.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.g 3 1.a even 1 1 trivial
1850.2.a.z 3 5.b even 2 1
1850.2.b.o 6 5.c odd 4 2
2960.2.a.u 3 4.b odd 2 1
3330.2.a.bg 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(370))$$:

 $$T_{3}^{3} - 10T_{3} + 4$$ T3^3 - 10*T3 + 4 $$T_{7}^{3} + T_{7}^{2} - 8T_{7} - 10$$ T7^3 + T7^2 - 8*T7 - 10

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - 10T + 4$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + T^{2} - 8T - 10$$
$11$ $$T^{3} - 11 T^{2} + 28 T + 8$$
$13$ $$T^{3} - 40T - 32$$
$17$ $$T^{3} - T^{2} - 12T - 8$$
$19$ $$T^{3} - 10T + 4$$
$23$ $$T^{3} + 2 T^{2} - 48 T + 64$$
$29$ $$T^{3} + 5 T^{2} - 16 T - 76$$
$31$ $$T^{3} - 3 T^{2} - 72 T + 270$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3} + 9 T^{2} - 40 T - 364$$
$43$ $$T^{3} + 11 T^{2} + 16 T - 80$$
$47$ $$T^{3} - 2 T^{2} - 30 T + 56$$
$53$ $$T^{3} - 21 T^{2} + 80 T + 316$$
$59$ $$T^{3} - 14 T^{2} + 34 T + 80$$
$61$ $$T^{3} - 7 T^{2} - 8 T + 4$$
$67$ $$T^{3} - 2 T^{2} - 30 T + 56$$
$71$ $$T^{3} - 22 T^{2} + 64 T + 640$$
$73$ $$T^{3} + 16 T^{2} + 36 T - 208$$
$79$ $$T^{3} + 26 T^{2} + 170 T + 224$$
$83$ $$T^{3} - 2 T^{2} - 158 T - 664$$
$89$ $$(T - 6)^{3}$$
$97$ $$T^{3} + 21 T^{2} + 80 T - 316$$