# Properties

 Label 3850.2.a.bj Level $3850$ Weight $2$ Character orbit 3850.a Self dual yes Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.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: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 154) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} - q^{7} - q^{8} + (2 \beta + 3) q^{9}+O(q^{10})$$ q - q^2 + (b + 1) * q^3 + q^4 + (-b - 1) * q^6 - q^7 - q^8 + (2*b + 3) * q^9 $$q - q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} - q^{7} - q^{8} + (2 \beta + 3) q^{9} + q^{11} + (\beta + 1) q^{12} + ( - \beta + 1) q^{13} + q^{14} + q^{16} + (2 \beta + 2) q^{17} + ( - 2 \beta - 3) q^{18} + (\beta - 5) q^{19} + ( - \beta - 1) q^{21} - q^{22} - 4 q^{23} + ( - \beta - 1) q^{24} + (\beta - 1) q^{26} + (2 \beta + 10) q^{27} - q^{28} - 2 \beta q^{29} + 2 q^{31} - q^{32} + (\beta + 1) q^{33} + ( - 2 \beta - 2) q^{34} + (2 \beta + 3) q^{36} + (4 \beta + 2) q^{37} + ( - \beta + 5) q^{38} - 4 q^{39} + (2 \beta + 2) q^{41} + (\beta + 1) q^{42} + ( - 2 \beta + 6) q^{43} + q^{44} + 4 q^{46} + 2 q^{47} + (\beta + 1) q^{48} + q^{49} + (4 \beta + 12) q^{51} + ( - \beta + 1) q^{52} + (2 \beta - 4) q^{53} + ( - 2 \beta - 10) q^{54} + q^{56} - 4 \beta q^{57} + 2 \beta q^{58} + (\beta + 5) q^{59} + ( - \beta - 3) q^{61} - 2 q^{62} + ( - 2 \beta - 3) q^{63} + q^{64} + ( - \beta - 1) q^{66} + (6 \beta + 2) q^{67} + (2 \beta + 2) q^{68} + ( - 4 \beta - 4) q^{69} + ( - 2 \beta + 2) q^{71} + ( - 2 \beta - 3) q^{72} + (4 \beta - 4) q^{73} + ( - 4 \beta - 2) q^{74} + (\beta - 5) q^{76} - q^{77} + 4 q^{78} + (6 \beta + 11) q^{81} + ( - 2 \beta - 2) q^{82} + ( - 5 \beta + 1) q^{83} + ( - \beta - 1) q^{84} + (2 \beta - 6) q^{86} + ( - 2 \beta - 10) q^{87} - q^{88} + 10 q^{89} + (\beta - 1) q^{91} - 4 q^{92} + (2 \beta + 2) q^{93} - 2 q^{94} + ( - \beta - 1) q^{96} + (2 \beta - 8) q^{97} - q^{98} + (2 \beta + 3) q^{99} +O(q^{100})$$ q - q^2 + (b + 1) * q^3 + q^4 + (-b - 1) * q^6 - q^7 - q^8 + (2*b + 3) * q^9 + q^11 + (b + 1) * q^12 + (-b + 1) * q^13 + q^14 + q^16 + (2*b + 2) * q^17 + (-2*b - 3) * q^18 + (b - 5) * q^19 + (-b - 1) * q^21 - q^22 - 4 * q^23 + (-b - 1) * q^24 + (b - 1) * q^26 + (2*b + 10) * q^27 - q^28 - 2*b * q^29 + 2 * q^31 - q^32 + (b + 1) * q^33 + (-2*b - 2) * q^34 + (2*b + 3) * q^36 + (4*b + 2) * q^37 + (-b + 5) * q^38 - 4 * q^39 + (2*b + 2) * q^41 + (b + 1) * q^42 + (-2*b + 6) * q^43 + q^44 + 4 * q^46 + 2 * q^47 + (b + 1) * q^48 + q^49 + (4*b + 12) * q^51 + (-b + 1) * q^52 + (2*b - 4) * q^53 + (-2*b - 10) * q^54 + q^56 - 4*b * q^57 + 2*b * q^58 + (b + 5) * q^59 + (-b - 3) * q^61 - 2 * q^62 + (-2*b - 3) * q^63 + q^64 + (-b - 1) * q^66 + (6*b + 2) * q^67 + (2*b + 2) * q^68 + (-4*b - 4) * q^69 + (-2*b + 2) * q^71 + (-2*b - 3) * q^72 + (4*b - 4) * q^73 + (-4*b - 2) * q^74 + (b - 5) * q^76 - q^77 + 4 * q^78 + (6*b + 11) * q^81 + (-2*b - 2) * q^82 + (-5*b + 1) * q^83 + (-b - 1) * q^84 + (2*b - 6) * q^86 + (-2*b - 10) * q^87 - q^88 + 10 * q^89 + (b - 1) * q^91 - 4 * q^92 + (2*b + 2) * q^93 - 2 * q^94 + (-b - 1) * q^96 + (2*b - 8) * q^97 - q^98 + (2*b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} - 10 q^{19} - 2 q^{21} - 2 q^{22} - 8 q^{23} - 2 q^{24} - 2 q^{26} + 20 q^{27} - 2 q^{28} + 4 q^{31} - 2 q^{32} + 2 q^{33} - 4 q^{34} + 6 q^{36} + 4 q^{37} + 10 q^{38} - 8 q^{39} + 4 q^{41} + 2 q^{42} + 12 q^{43} + 2 q^{44} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} + 24 q^{51} + 2 q^{52} - 8 q^{53} - 20 q^{54} + 2 q^{56} + 10 q^{59} - 6 q^{61} - 4 q^{62} - 6 q^{63} + 2 q^{64} - 2 q^{66} + 4 q^{67} + 4 q^{68} - 8 q^{69} + 4 q^{71} - 6 q^{72} - 8 q^{73} - 4 q^{74} - 10 q^{76} - 2 q^{77} + 8 q^{78} + 22 q^{81} - 4 q^{82} + 2 q^{83} - 2 q^{84} - 12 q^{86} - 20 q^{87} - 2 q^{88} + 20 q^{89} - 2 q^{91} - 8 q^{92} + 4 q^{93} - 4 q^{94} - 2 q^{96} - 16 q^{97} - 2 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 - 10 * q^19 - 2 * q^21 - 2 * q^22 - 8 * q^23 - 2 * q^24 - 2 * q^26 + 20 * q^27 - 2 * q^28 + 4 * q^31 - 2 * q^32 + 2 * q^33 - 4 * q^34 + 6 * q^36 + 4 * q^37 + 10 * q^38 - 8 * q^39 + 4 * q^41 + 2 * q^42 + 12 * q^43 + 2 * q^44 + 8 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^49 + 24 * q^51 + 2 * q^52 - 8 * q^53 - 20 * q^54 + 2 * q^56 + 10 * q^59 - 6 * q^61 - 4 * q^62 - 6 * q^63 + 2 * q^64 - 2 * q^66 + 4 * q^67 + 4 * q^68 - 8 * q^69 + 4 * q^71 - 6 * q^72 - 8 * q^73 - 4 * q^74 - 10 * q^76 - 2 * q^77 + 8 * q^78 + 22 * q^81 - 4 * q^82 + 2 * q^83 - 2 * q^84 - 12 * q^86 - 20 * q^87 - 2 * q^88 + 20 * q^89 - 2 * q^91 - 8 * q^92 + 4 * q^93 - 4 * q^94 - 2 * q^96 - 16 * q^97 - 2 * q^98 + 6 * 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
 −0.618034 1.61803
−1.00000 −1.23607 1.00000 0 1.23607 −1.00000 −1.00000 −1.47214 0
1.2 −1.00000 3.23607 1.00000 0 −3.23607 −1.00000 −1.00000 7.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 3850.2.a.bj 2
5.b even 2 1 154.2.a.d 2
5.c odd 4 2 3850.2.c.q 4
15.d odd 2 1 1386.2.a.m 2
20.d odd 2 1 1232.2.a.p 2
35.c odd 2 1 1078.2.a.w 2
35.i odd 6 2 1078.2.e.n 4
35.j even 6 2 1078.2.e.q 4
40.e odd 2 1 4928.2.a.bk 2
40.f even 2 1 4928.2.a.bt 2
55.d odd 2 1 1694.2.a.l 2
105.g even 2 1 9702.2.a.cu 2
140.c even 2 1 8624.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 5.b even 2 1
1078.2.a.w 2 35.c odd 2 1
1078.2.e.n 4 35.i odd 6 2
1078.2.e.q 4 35.j even 6 2
1232.2.a.p 2 20.d odd 2 1
1386.2.a.m 2 15.d odd 2 1
1694.2.a.l 2 55.d odd 2 1
3850.2.a.bj 2 1.a even 1 1 trivial
3850.2.c.q 4 5.c odd 4 2
4928.2.a.bk 2 40.e odd 2 1
4928.2.a.bt 2 40.f even 2 1
8624.2.a.bf 2 140.c even 2 1
9702.2.a.cu 2 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3850))$$:

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{13}^{2} - 2T_{13} - 4$$ T13^2 - 2*T13 - 4 $$T_{17}^{2} - 4T_{17} - 16$$ T17^2 - 4*T17 - 16 $$T_{19}^{2} + 10T_{19} + 20$$ T19^2 + 10*T19 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} - 4T - 16$$
$19$ $$T^{2} + 10T + 20$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} - 4T - 76$$
$41$ $$T^{2} - 4T - 16$$
$43$ $$T^{2} - 12T + 16$$
$47$ $$(T - 2)^{2}$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} - 10T + 20$$
$61$ $$T^{2} + 6T + 4$$
$67$ $$T^{2} - 4T - 176$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$T^{2} + 8T - 64$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 2T - 124$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 16T + 44$$