# Properties

 Label 370.2.b.c Level $370$ Weight $2$ Character orbit 370.b Analytic conductor $2.954$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(149,370)]

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([1, 0]))

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

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

chi := DirichletCharacter("370.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.b (of order $$2$$, degree $$1$$, 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: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} + (\beta_1 - 2) q^{5} - \beta_{3} q^{6} + (\beta_{2} + 2 \beta_1) q^{7} - \beta_1 q^{8} - 3 q^{9}+O(q^{10})$$ q + b1 * q^2 + b2 * q^3 - q^4 + (b1 - 2) * q^5 - b3 * q^6 + (b2 + 2*b1) * q^7 - b1 * q^8 - 3 * q^9 $$q + \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} + (\beta_1 - 2) q^{5} - \beta_{3} q^{6} + (\beta_{2} + 2 \beta_1) q^{7} - \beta_1 q^{8} - 3 q^{9} + ( - 2 \beta_1 - 1) q^{10} + 2 \beta_{3} q^{11} - \beta_{2} q^{12} - 4 \beta_1 q^{13} + ( - \beta_{3} - 2) q^{14} + ( - \beta_{3} - 2 \beta_{2}) q^{15} + q^{16} - 2 \beta_{2} q^{17} - 3 \beta_1 q^{18} + (\beta_{3} - 6) q^{19} + ( - \beta_1 + 2) q^{20} + ( - 2 \beta_{3} - 6) q^{21} + 2 \beta_{2} q^{22} + (2 \beta_{2} + 4 \beta_1) q^{23} + \beta_{3} q^{24} + ( - 4 \beta_1 + 3) q^{25} + 4 q^{26} + ( - \beta_{2} - 2 \beta_1) q^{28} + (2 \beta_{3} - \beta_{2}) q^{30} + (\beta_{3} - 4) q^{31} + \beta_1 q^{32} + 12 \beta_1 q^{33} + 2 \beta_{3} q^{34} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{35} + 3 q^{36} + \beta_1 q^{37} + (\beta_{2} - 6 \beta_1) q^{38} + 4 \beta_{3} q^{39} + (2 \beta_1 + 1) q^{40} + 2 q^{41} + ( - 2 \beta_{2} - 6 \beta_1) q^{42} - 4 \beta_1 q^{43} - 2 \beta_{3} q^{44} + ( - 3 \beta_1 + 6) q^{45} + ( - 2 \beta_{3} - 4) q^{46} + (\beta_{2} + 2 \beta_1) q^{47} + \beta_{2} q^{48} + ( - 4 \beta_{3} - 3) q^{49} + (3 \beta_1 + 4) q^{50} + 12 q^{51} + 4 \beta_1 q^{52} + (4 \beta_{2} + 2 \beta_1) q^{53} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{55} + (\beta_{3} + 2) q^{56} + ( - 6 \beta_{2} + 6 \beta_1) q^{57} + ( - \beta_{3} + 6) q^{59} + (\beta_{3} + 2 \beta_{2}) q^{60} + 12 q^{61} + (\beta_{2} - 4 \beta_1) q^{62} + ( - 3 \beta_{2} - 6 \beta_1) q^{63} - q^{64} + (8 \beta_1 + 4) q^{65} - 12 q^{66} + (\beta_{2} - 8 \beta_1) q^{67} + 2 \beta_{2} q^{68} + ( - 4 \beta_{3} - 12) q^{69} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{70} + 2 \beta_{3} q^{71} + 3 \beta_1 q^{72} - 4 \beta_1 q^{73} - q^{74} + (4 \beta_{3} + 3 \beta_{2}) q^{75} + ( - \beta_{3} + 6) q^{76} + (4 \beta_{2} + 12 \beta_1) q^{77} + 4 \beta_{2} q^{78} + ( - \beta_{3} - 4) q^{79} + (\beta_1 - 2) q^{80} - 9 q^{81} + 2 \beta_1 q^{82} + ( - \beta_{2} + 12 \beta_1) q^{83} + (2 \beta_{3} + 6) q^{84} + (2 \beta_{3} + 4 \beta_{2}) q^{85} + 4 q^{86} - 2 \beta_{2} q^{88} + ( - 4 \beta_{3} - 6) q^{89} + (6 \beta_1 + 3) q^{90} + (4 \beta_{3} + 8) q^{91} + ( - 2 \beta_{2} - 4 \beta_1) q^{92} + ( - 4 \beta_{2} + 6 \beta_1) q^{93} + ( - \beta_{3} - 2) q^{94} + ( - 2 \beta_{3} + \beta_{2} - 6 \beta_1 + 12) q^{95} - \beta_{3} q^{96} - 2 \beta_1 q^{97} + ( - 4 \beta_{2} - 3 \beta_1) q^{98} - 6 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^2 + b2 * q^3 - q^4 + (b1 - 2) * q^5 - b3 * q^6 + (b2 + 2*b1) * q^7 - b1 * q^8 - 3 * q^9 + (-2*b1 - 1) * q^10 + 2*b3 * q^11 - b2 * q^12 - 4*b1 * q^13 + (-b3 - 2) * q^14 + (-b3 - 2*b2) * q^15 + q^16 - 2*b2 * q^17 - 3*b1 * q^18 + (b3 - 6) * q^19 + (-b1 + 2) * q^20 + (-2*b3 - 6) * q^21 + 2*b2 * q^22 + (2*b2 + 4*b1) * q^23 + b3 * q^24 + (-4*b1 + 3) * q^25 + 4 * q^26 + (-b2 - 2*b1) * q^28 + (2*b3 - b2) * q^30 + (b3 - 4) * q^31 + b1 * q^32 + 12*b1 * q^33 + 2*b3 * q^34 + (-b3 - 2*b2 - 4*b1 - 2) * q^35 + 3 * q^36 + b1 * q^37 + (b2 - 6*b1) * q^38 + 4*b3 * q^39 + (2*b1 + 1) * q^40 + 2 * q^41 + (-2*b2 - 6*b1) * q^42 - 4*b1 * q^43 - 2*b3 * q^44 + (-3*b1 + 6) * q^45 + (-2*b3 - 4) * q^46 + (b2 + 2*b1) * q^47 + b2 * q^48 + (-4*b3 - 3) * q^49 + (3*b1 + 4) * q^50 + 12 * q^51 + 4*b1 * q^52 + (4*b2 + 2*b1) * q^53 + (-4*b3 + 2*b2) * q^55 + (b3 + 2) * q^56 + (-6*b2 + 6*b1) * q^57 + (-b3 + 6) * q^59 + (b3 + 2*b2) * q^60 + 12 * q^61 + (b2 - 4*b1) * q^62 + (-3*b2 - 6*b1) * q^63 - q^64 + (8*b1 + 4) * q^65 - 12 * q^66 + (b2 - 8*b1) * q^67 + 2*b2 * q^68 + (-4*b3 - 12) * q^69 + (2*b3 - b2 - 2*b1 + 4) * q^70 + 2*b3 * q^71 + 3*b1 * q^72 - 4*b1 * q^73 - q^74 + (4*b3 + 3*b2) * q^75 + (-b3 + 6) * q^76 + (4*b2 + 12*b1) * q^77 + 4*b2 * q^78 + (-b3 - 4) * q^79 + (b1 - 2) * q^80 - 9 * q^81 + 2*b1 * q^82 + (-b2 + 12*b1) * q^83 + (2*b3 + 6) * q^84 + (2*b3 + 4*b2) * q^85 + 4 * q^86 - 2*b2 * q^88 + (-4*b3 - 6) * q^89 + (6*b1 + 3) * q^90 + (4*b3 + 8) * q^91 + (-2*b2 - 4*b1) * q^92 + (-4*b2 + 6*b1) * q^93 + (-b3 - 2) * q^94 + (-2*b3 + b2 - 6*b1 + 12) * q^95 - b3 * q^96 - 2*b1 * q^97 + (-4*b2 - 3*b1) * q^98 - 6*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 8 q^{5} - 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 8 * q^5 - 12 * q^9 $$4 q - 4 q^{4} - 8 q^{5} - 12 q^{9} - 4 q^{10} - 8 q^{14} + 4 q^{16} - 24 q^{19} + 8 q^{20} - 24 q^{21} + 12 q^{25} + 16 q^{26} - 16 q^{31} - 8 q^{35} + 12 q^{36} + 4 q^{40} + 8 q^{41} + 24 q^{45} - 16 q^{46} - 12 q^{49} + 16 q^{50} + 48 q^{51} + 8 q^{56} + 24 q^{59} + 48 q^{61} - 4 q^{64} + 16 q^{65} - 48 q^{66} - 48 q^{69} + 16 q^{70} - 4 q^{74} + 24 q^{76} - 16 q^{79} - 8 q^{80} - 36 q^{81} + 24 q^{84} + 16 q^{86} - 24 q^{89} + 12 q^{90} + 32 q^{91} - 8 q^{94} + 48 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 - 8 * q^5 - 12 * q^9 - 4 * q^10 - 8 * q^14 + 4 * q^16 - 24 * q^19 + 8 * q^20 - 24 * q^21 + 12 * q^25 + 16 * q^26 - 16 * q^31 - 8 * q^35 + 12 * q^36 + 4 * q^40 + 8 * q^41 + 24 * q^45 - 16 * q^46 - 12 * q^49 + 16 * q^50 + 48 * q^51 + 8 * q^56 + 24 * q^59 + 48 * q^61 - 4 * q^64 + 16 * q^65 - 48 * q^66 - 48 * q^69 + 16 * q^70 - 4 * q^74 + 24 * q^76 - 16 * q^79 - 8 * q^80 - 36 * q^81 + 24 * q^84 + 16 * q^86 - 24 * q^89 + 12 * q^90 + 32 * q^91 - 8 * q^94 + 48 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 3\nu ) / 3$$ (v^3 + 3*v) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 3\nu ) / 3$$ (-v^3 + 3*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$3\beta_1$$ 3*b1 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 3\beta_{2} ) / 2$$ (-3*b3 + 3*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 1.22474 − 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
1.00000i 2.44949i −1.00000 −2.00000 1.00000i −2.44949 4.44949i 1.00000i −3.00000 −1.00000 + 2.00000i
149.2 1.00000i 2.44949i −1.00000 −2.00000 1.00000i 2.44949 0.449490i 1.00000i −3.00000 −1.00000 + 2.00000i
149.3 1.00000i 2.44949i −1.00000 −2.00000 + 1.00000i 2.44949 0.449490i 1.00000i −3.00000 −1.00000 2.00000i
149.4 1.00000i 2.44949i −1.00000 −2.00000 + 1.00000i −2.44949 4.44949i 1.00000i −3.00000 −1.00000 2.00000i
 $$n$$: 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 370.2.b.c 4
3.b odd 2 1 3330.2.d.m 4
5.b even 2 1 inner 370.2.b.c 4
5.c odd 4 1 1850.2.a.s 2
5.c odd 4 1 1850.2.a.v 2
15.d odd 2 1 3330.2.d.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.c 4 1.a even 1 1 trivial
370.2.b.c 4 5.b even 2 1 inner
1850.2.a.s 2 5.c odd 4 1
1850.2.a.v 2 5.c odd 4 1
3330.2.d.m 4 3.b odd 2 1
3330.2.d.m 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 6$$ T3^2 + 6 $$T_{7}^{4} + 20T_{7}^{2} + 4$$ T7^4 + 20*T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 6)^{2}$$
$5$ $$(T^{2} + 4 T + 5)^{2}$$
$7$ $$T^{4} + 20T^{2} + 4$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$(T^{2} + 24)^{2}$$
$19$ $$(T^{2} + 12 T + 30)^{2}$$
$23$ $$T^{4} + 80T^{2} + 64$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 8 T + 10)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T - 2)^{4}$$
$43$ $$(T^{2} + 16)^{2}$$
$47$ $$T^{4} + 20T^{2} + 4$$
$53$ $$T^{4} + 200T^{2} + 8464$$
$59$ $$(T^{2} - 12 T + 30)^{2}$$
$61$ $$(T - 12)^{4}$$
$67$ $$T^{4} + 140T^{2} + 3364$$
$71$ $$(T^{2} - 24)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + 8 T + 10)^{2}$$
$83$ $$T^{4} + 300 T^{2} + 19044$$
$89$ $$(T^{2} + 12 T - 60)^{2}$$
$97$ $$(T^{2} + 4)^{2}$$