# Properties

 Label 1850.2.b.p Level $1850$ Weight $2$ Character orbit 1850.b Analytic conductor $14.772$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.37161216.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 15x^{4} + 51x^{2} + 1$$ x^6 + 15*x^4 + 51*x^2 + 1 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} + ( - \beta_{4} - \beta_1) q^{7} + \beta_{4} q^{8} + (\beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q - b4 * q^2 + b1 * q^3 - q^4 + b2 * q^6 + (-b4 - b1) * q^7 + b4 * q^8 + (b3 - b2 - 2) * q^9 $$q - \beta_{4} q^{2} + \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} + ( - \beta_{4} - \beta_1) q^{7} + \beta_{4} q^{8} + (\beta_{3} - \beta_{2} - 2) q^{9} + (\beta_{3} - 2) q^{11} - \beta_1 q^{12} + (3 \beta_{4} - \beta_1) q^{13} + ( - \beta_{2} - 1) q^{14} + q^{16} + (2 \beta_{4} + \beta_1) q^{17} + ( - \beta_{5} + 2 \beta_{4} + \beta_1) q^{18} + ( - \beta_{3} - \beta_{2} + 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} + 5) q^{21} + ( - \beta_{5} + 2 \beta_{4}) q^{22} + (2 \beta_{4} - 2 \beta_1) q^{23} - \beta_{2} q^{24} + ( - \beta_{2} + 3) q^{26} + (\beta_{5} - 4 \beta_{4} - 2 \beta_1) q^{27} + (\beta_{4} + \beta_1) q^{28} + (2 \beta_{2} + 4) q^{29} + (\beta_{3} + 2 \beta_{2} + 1) q^{31} - \beta_{4} q^{32} + (\beta_{4} - 4 \beta_1) q^{33} + (\beta_{2} + 2) q^{34} + ( - \beta_{3} + \beta_{2} + 2) q^{36} - \beta_{4} q^{37} + (\beta_{5} - \beta_{4} + \beta_1) q^{38} + ( - \beta_{3} - 2 \beta_{2} + 5) q^{39} + ( - 2 \beta_{3} + 7) q^{41} + (\beta_{5} - 5 \beta_{4} - 2 \beta_1) q^{42} + (\beta_{5} + 6 \beta_{4} - 3 \beta_1) q^{43} + ( - \beta_{3} + 2) q^{44} + ( - 2 \beta_{2} + 2) q^{46} + \beta_1 q^{48} + (\beta_{3} - 3 \beta_{2} + 1) q^{49} + (\beta_{3} - 3 \beta_{2} - 5) q^{51} + ( - 3 \beta_{4} + \beta_1) q^{52} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{53} + (\beta_{3} - 2 \beta_{2} - 4) q^{54} + (\beta_{2} + 1) q^{56} + (\beta_{5} - 6 \beta_{4} + 2 \beta_1) q^{57} + ( - 4 \beta_{4} - 2 \beta_1) q^{58} + ( - \beta_{3} + 3 \beta_{2} + 2) q^{59} + ( - 3 \beta_{2} + 5) q^{61} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{62} + ( - 2 \beta_{5} + 6 \beta_{4} + 6 \beta_1) q^{63} - q^{64} + ( - 4 \beta_{2} + 1) q^{66} + ( - 2 \beta_{5} - \beta_1) q^{67} + ( - 2 \beta_{4} - \beta_1) q^{68} + ( - 2 \beta_{3} + 10) q^{69} + ( - 2 \beta_{3} + \beta_{2} + 3) q^{71} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{72} + ( - \beta_{5} + 5 \beta_{4} - \beta_1) q^{73} - q^{74} + (\beta_{3} + \beta_{2} - 1) q^{76} + ( - \beta_{5} + \beta_{4} + 4 \beta_1) q^{77} + (\beta_{5} - 5 \beta_{4} + 2 \beta_1) q^{78} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{79} + (\beta_{3} + 5 \beta_{2} + 3) q^{81} + (2 \beta_{5} - 7 \beta_{4}) q^{82} + (\beta_{5} - 6 \beta_{4} - 2 \beta_1) q^{83} + (\beta_{3} - 2 \beta_{2} - 5) q^{84} + (\beta_{3} - 3 \beta_{2} + 6) q^{86} + ( - 2 \beta_{5} + 10 \beta_{4} + 6 \beta_1) q^{87} + (\beta_{5} - 2 \beta_{4}) q^{88} + (3 \beta_{3} + 2 \beta_{2} - 2) q^{89} + (\beta_{3} + \beta_{2} - 2) q^{91} + ( - 2 \beta_{4} + 2 \beta_1) q^{92} + ( - 2 \beta_{5} + 11 \beta_{4} + \beta_1) q^{93} + \beta_{2} q^{96} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{97} + ( - \beta_{5} - \beta_{4} + 3 \beta_1) q^{98} + ( - \beta_{3} + 3 \beta_{2} + 14) q^{99}+O(q^{100})$$ q - b4 * q^2 + b1 * q^3 - q^4 + b2 * q^6 + (-b4 - b1) * q^7 + b4 * q^8 + (b3 - b2 - 2) * q^9 + (b3 - 2) * q^11 - b1 * q^12 + (3*b4 - b1) * q^13 + (-b2 - 1) * q^14 + q^16 + (2*b4 + b1) * q^17 + (-b5 + 2*b4 + b1) * q^18 + (-b3 - b2 + 1) * q^19 + (-b3 + 2*b2 + 5) * q^21 + (-b5 + 2*b4) * q^22 + (2*b4 - 2*b1) * q^23 - b2 * q^24 + (-b2 + 3) * q^26 + (b5 - 4*b4 - 2*b1) * q^27 + (b4 + b1) * q^28 + (2*b2 + 4) * q^29 + (b3 + 2*b2 + 1) * q^31 - b4 * q^32 + (b4 - 4*b1) * q^33 + (b2 + 2) * q^34 + (-b3 + b2 + 2) * q^36 - b4 * q^37 + (b5 - b4 + b1) * q^38 + (-b3 - 2*b2 + 5) * q^39 + (-2*b3 + 7) * q^41 + (b5 - 5*b4 - 2*b1) * q^42 + (b5 + 6*b4 - 3*b1) * q^43 + (-b3 + 2) * q^44 + (-2*b2 + 2) * q^46 + b1 * q^48 + (b3 - 3*b2 + 1) * q^49 + (b3 - 3*b2 - 5) * q^51 + (-3*b4 + b1) * q^52 + (2*b5 - 2*b4 - 2*b1) * q^53 + (b3 - 2*b2 - 4) * q^54 + (b2 + 1) * q^56 + (b5 - 6*b4 + 2*b1) * q^57 + (-4*b4 - 2*b1) * q^58 + (-b3 + 3*b2 + 2) * q^59 + (-3*b2 + 5) * q^61 + (-b5 - b4 - 2*b1) * q^62 + (-2*b5 + 6*b4 + 6*b1) * q^63 - q^64 + (-4*b2 + 1) * q^66 + (-2*b5 - b1) * q^67 + (-2*b4 - b1) * q^68 + (-2*b3 + 10) * q^69 + (-2*b3 + b2 + 3) * q^71 + (b5 - 2*b4 - b1) * q^72 + (-b5 + 5*b4 - b1) * q^73 - q^74 + (b3 + b2 - 1) * q^76 + (-b5 + b4 + 4*b1) * q^77 + (b5 - 5*b4 + 2*b1) * q^78 + (2*b3 + 2*b2 - 2) * q^79 + (b3 + 5*b2 + 3) * q^81 + (2*b5 - 7*b4) * q^82 + (b5 - 6*b4 - 2*b1) * q^83 + (b3 - 2*b2 - 5) * q^84 + (b3 - 3*b2 + 6) * q^86 + (-2*b5 + 10*b4 + 6*b1) * q^87 + (b5 - 2*b4) * q^88 + (3*b3 + 2*b2 - 2) * q^89 + (b3 + b2 - 2) * q^91 + (-2*b4 + 2*b1) * q^92 + (-2*b5 + 11*b4 + b1) * q^93 + b2 * q^96 + (-2*b5 - 4*b4) * q^97 + (-b5 - b4 + 3*b1) * q^98 + (-b3 + 3*b2 + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 2 q^{6} - 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 + 2 * q^6 - 12 * q^9 $$6 q - 6 q^{4} + 2 q^{6} - 12 q^{9} - 10 q^{11} - 8 q^{14} + 6 q^{16} + 2 q^{19} + 32 q^{21} - 2 q^{24} + 16 q^{26} + 28 q^{29} + 12 q^{31} + 14 q^{34} + 12 q^{36} + 24 q^{39} + 38 q^{41} + 10 q^{44} + 8 q^{46} + 2 q^{49} - 34 q^{51} - 26 q^{54} + 8 q^{56} + 16 q^{59} + 24 q^{61} - 6 q^{64} - 2 q^{66} + 56 q^{69} + 16 q^{71} - 6 q^{74} - 2 q^{76} - 4 q^{79} + 30 q^{81} - 32 q^{84} + 32 q^{86} - 2 q^{89} - 8 q^{91} + 2 q^{96} + 88 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 + 2 * q^6 - 12 * q^9 - 10 * q^11 - 8 * q^14 + 6 * q^16 + 2 * q^19 + 32 * q^21 - 2 * q^24 + 16 * q^26 + 28 * q^29 + 12 * q^31 + 14 * q^34 + 12 * q^36 + 24 * q^39 + 38 * q^41 + 10 * q^44 + 8 * q^46 + 2 * q^49 - 34 * q^51 - 26 * q^54 + 8 * q^56 + 16 * q^59 + 24 * q^61 - 6 * q^64 - 2 * q^66 + 56 * q^69 + 16 * q^71 - 6 * q^74 - 2 * q^76 - 4 * q^79 + 30 * q^81 - 32 * q^84 + 32 * q^86 - 2 * q^89 - 8 * q^91 + 2 * q^96 + 88 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 15x^{4} + 51x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 8\nu^{2} + 1 ) / 6$$ (v^4 + 8*v^2 + 1) / 6 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 14\nu^{2} + 31 ) / 6$$ (v^4 + 14*v^2 + 31) / 6 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 14\nu^{3} + 43\nu ) / 6$$ (v^5 + 14*v^3 + 43*v) / 6 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} + 31\nu^{3} + 110\nu ) / 3$$ (2*v^5 + 31*v^3 + 110*v) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2} - 5$$ b3 - b2 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 8\beta_1$$ b5 - 4*b4 - 8*b1 $$\nu^{4}$$ $$=$$ $$-8\beta_{3} + 14\beta_{2} + 39$$ -8*b3 + 14*b2 + 39 $$\nu^{5}$$ $$=$$ $$-14\beta_{5} + 62\beta_{4} + 69\beta_1$$ -14*b5 + 62*b4 + 69*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
 − 2.27307i 0.140435i 3.13264i − 3.13264i − 0.140435i 2.27307i
1.00000i 2.27307i −1.00000 0 −2.27307 1.27307i 1.00000i −2.16686 0
149.2 1.00000i 0.140435i −1.00000 0 0.140435 1.14044i 1.00000i 2.98028 0
149.3 1.00000i 3.13264i −1.00000 0 3.13264 4.13264i 1.00000i −6.81342 0
149.4 1.00000i 3.13264i −1.00000 0 3.13264 4.13264i 1.00000i −6.81342 0
149.5 1.00000i 0.140435i −1.00000 0 0.140435 1.14044i 1.00000i 2.98028 0
149.6 1.00000i 2.27307i −1.00000 0 −2.27307 1.27307i 1.00000i −2.16686 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.6 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 1850.2.b.p 6
5.b even 2 1 inner 1850.2.b.p 6
5.c odd 4 1 1850.2.a.y 3
5.c odd 4 1 1850.2.a.bc yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.y 3 5.c odd 4 1
1850.2.a.bc yes 3 5.c odd 4 1
1850.2.b.p 6 1.a even 1 1 trivial
1850.2.b.p 6 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1850, [\chi])$$:

 $$T_{3}^{6} + 15T_{3}^{4} + 51T_{3}^{2} + 1$$ T3^6 + 15*T3^4 + 51*T3^2 + 1 $$T_{7}^{6} + 20T_{7}^{4} + 52T_{7}^{2} + 36$$ T7^6 + 20*T7^4 + 52*T7^2 + 36 $$T_{13}^{6} + 36T_{13}^{4} + 228T_{13}^{2} + 4$$ T13^6 + 36*T13^4 + 228*T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6} + 15 T^{4} + 51 T^{2} + 1$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 20 T^{4} + 52 T^{2} + 36$$
$11$ $$(T^{3} + 5 T^{2} - 9 T - 51)^{2}$$
$13$ $$T^{6} + 36 T^{4} + 228 T^{2} + 4$$
$17$ $$T^{6} + 31 T^{4} + 123 T^{2} + 9$$
$19$ $$(T^{3} - T^{2} - 25 T - 11)^{2}$$
$23$ $$T^{6} + 64 T^{4} + 960 T^{2} + \cdots + 2304$$
$29$ $$(T^{3} - 14 T^{2} + 36 T + 24)^{2}$$
$31$ $$(T^{3} - 6 T^{2} - 36 T + 214)^{2}$$
$37$ $$(T^{2} + 1)^{3}$$
$41$ $$(T^{3} - 19 T^{2} + 51 T + 399)^{2}$$
$43$ $$T^{6} + 248 T^{4} + 18064 T^{2} + \cdots + 318096$$
$47$ $$T^{6}$$
$53$ $$T^{6} + 204 T^{4} + 9648 T^{2} + \cdots + 46656$$
$59$ $$(T^{3} - 8 T^{2} - 60 T - 84)^{2}$$
$61$ $$(T^{3} - 12 T^{2} - 18 T + 238)^{2}$$
$67$ $$T^{6} + 159 T^{4} + 5523 T^{2} + \cdots + 289$$
$71$ $$(T^{3} - 8 T^{2} - 54 T + 378)^{2}$$
$73$ $$T^{6} + 107 T^{4} + 1195 T^{2} + \cdots + 81$$
$79$ $$(T^{3} + 2 T^{2} - 100 T + 88)^{2}$$
$83$ $$T^{6} + 211 T^{4} + 3231 T^{2} + \cdots + 3969$$
$89$ $$(T^{3} + T^{2} - 189 T - 147)^{2}$$
$97$ $$T^{6} + 204 T^{4} + 240 T^{2} + \cdots + 64$$