# Properties

 Label 3850.2.a.o Level $3850$ Weight $2$ Character orbit 3850.a Self dual yes Analytic conductor $30.742$ Analytic rank $0$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - 2 * q^3 + q^4 - 2 * q^6 + q^7 + q^8 + q^9 $$q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} + q^{7} + q^{8} + q^{9} + q^{11} - 2 q^{12} + 4 q^{13} + q^{14} + q^{16} + q^{18} + 4 q^{19} - 2 q^{21} + q^{22} - 4 q^{23} - 2 q^{24} + 4 q^{26} + 4 q^{27} + q^{28} + 2 q^{29} - 10 q^{31} + q^{32} - 2 q^{33} + q^{36} + 6 q^{37} + 4 q^{38} - 8 q^{39} - 2 q^{42} + 4 q^{43} + q^{44} - 4 q^{46} - 10 q^{47} - 2 q^{48} + q^{49} + 4 q^{52} + 14 q^{53} + 4 q^{54} + q^{56} - 8 q^{57} + 2 q^{58} + 10 q^{59} - 8 q^{61} - 10 q^{62} + q^{63} + q^{64} - 2 q^{66} - 8 q^{67} + 8 q^{69} - 4 q^{71} + q^{72} - 4 q^{73} + 6 q^{74} + 4 q^{76} + q^{77} - 8 q^{78} + 16 q^{79} - 11 q^{81} - 4 q^{83} - 2 q^{84} + 4 q^{86} - 4 q^{87} + q^{88} + 10 q^{89} + 4 q^{91} - 4 q^{92} + 20 q^{93} - 10 q^{94} - 2 q^{96} - 6 q^{97} + q^{98} + q^{99}+O(q^{100})$$ q + q^2 - 2 * q^3 + q^4 - 2 * q^6 + q^7 + q^8 + q^9 + q^11 - 2 * q^12 + 4 * q^13 + q^14 + q^16 + q^18 + 4 * q^19 - 2 * q^21 + q^22 - 4 * q^23 - 2 * q^24 + 4 * q^26 + 4 * q^27 + q^28 + 2 * q^29 - 10 * q^31 + q^32 - 2 * q^33 + q^36 + 6 * q^37 + 4 * q^38 - 8 * q^39 - 2 * q^42 + 4 * q^43 + q^44 - 4 * q^46 - 10 * q^47 - 2 * q^48 + q^49 + 4 * q^52 + 14 * q^53 + 4 * q^54 + q^56 - 8 * q^57 + 2 * q^58 + 10 * q^59 - 8 * q^61 - 10 * q^62 + q^63 + q^64 - 2 * q^66 - 8 * q^67 + 8 * q^69 - 4 * q^71 + q^72 - 4 * q^73 + 6 * q^74 + 4 * q^76 + q^77 - 8 * q^78 + 16 * q^79 - 11 * q^81 - 4 * q^83 - 2 * q^84 + 4 * q^86 - 4 * q^87 + q^88 + 10 * q^89 + 4 * q^91 - 4 * q^92 + 20 * q^93 - 10 * q^94 - 2 * q^96 - 6 * q^97 + q^98 + 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
1.00000 −2.00000 1.00000 0 −2.00000 1.00000 1.00000 1.00000 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.o 1
5.b even 2 1 154.2.a.b 1
5.c odd 4 2 3850.2.c.d 2
15.d odd 2 1 1386.2.a.f 1
20.d odd 2 1 1232.2.a.c 1
35.c odd 2 1 1078.2.a.b 1
35.i odd 6 2 1078.2.e.l 2
35.j even 6 2 1078.2.e.h 2
40.e odd 2 1 4928.2.a.bf 1
40.f even 2 1 4928.2.a.d 1
55.d odd 2 1 1694.2.a.i 1
105.g even 2 1 9702.2.a.bz 1
140.c even 2 1 8624.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 5.b even 2 1
1078.2.a.b 1 35.c odd 2 1
1078.2.e.h 2 35.j even 6 2
1078.2.e.l 2 35.i odd 6 2
1232.2.a.c 1 20.d odd 2 1
1386.2.a.f 1 15.d odd 2 1
1694.2.a.i 1 55.d odd 2 1
3850.2.a.o 1 1.a even 1 1 trivial
3850.2.c.d 2 5.c odd 4 2
4928.2.a.d 1 40.f even 2 1
4928.2.a.bf 1 40.e odd 2 1
8624.2.a.z 1 140.c even 2 1
9702.2.a.bz 1 105.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3850))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{13} - 4$$ T13 - 4 $$T_{17}$$ T17 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T - 4$$
$17$ $$T$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T - 2$$
$31$ $$T + 10$$
$37$ $$T - 6$$
$41$ $$T$$
$43$ $$T - 4$$
$47$ $$T + 10$$
$53$ $$T - 14$$
$59$ $$T - 10$$
$61$ $$T + 8$$
$67$ $$T + 8$$
$71$ $$T + 4$$
$73$ $$T + 4$$
$79$ $$T - 16$$
$83$ $$T + 4$$
$89$ $$T - 10$$
$97$ $$T + 6$$