# Properties

 Label 950.2.a.g Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ Analytic rank $1$ 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] = [950,2,Mod(1,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, 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("950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.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: $$7.58578819202$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} + ( - \beta - 3) q^{7} + q^{8} - 2 \beta q^{9} +O(q^{10})$$ q + q^2 + (b - 1) * q^3 + q^4 + (b - 1) * q^6 + (-b - 3) * q^7 + q^8 - 2*b * q^9 $$q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} + ( - \beta - 3) q^{7} + q^{8} - 2 \beta q^{9} - \beta q^{11} + (\beta - 1) q^{12} + ( - 2 \beta - 3) q^{13} + ( - \beta - 3) q^{14} + q^{16} - q^{17} - 2 \beta q^{18} - q^{19} + ( - 2 \beta + 1) q^{21} - \beta q^{22} + (3 \beta - 5) q^{23} + (\beta - 1) q^{24} + ( - 2 \beta - 3) q^{26} + ( - \beta - 1) q^{27} + ( - \beta - 3) q^{28} + (2 \beta - 3) q^{29} + (3 \beta + 2) q^{31} + q^{32} + (\beta - 2) q^{33} - q^{34} - 2 \beta q^{36} + 6 \beta q^{37} - q^{38} + ( - \beta - 1) q^{39} - 3 \beta q^{41} + ( - 2 \beta + 1) q^{42} + (3 \beta - 6) q^{43} - \beta q^{44} + (3 \beta - 5) q^{46} + (\beta - 1) q^{48} + (6 \beta + 4) q^{49} + ( - \beta + 1) q^{51} + ( - 2 \beta - 3) q^{52} + ( - 6 \beta + 3) q^{53} + ( - \beta - 1) q^{54} + ( - \beta - 3) q^{56} + ( - \beta + 1) q^{57} + (2 \beta - 3) q^{58} + (7 \beta - 3) q^{59} + (3 \beta + 10) q^{61} + (3 \beta + 2) q^{62} + (6 \beta + 4) q^{63} + q^{64} + (\beta - 2) q^{66} + (3 \beta - 9) q^{67} - q^{68} + ( - 8 \beta + 11) q^{69} + ( - \beta - 12) q^{71} - 2 \beta q^{72} + ( - 6 \beta - 3) q^{73} + 6 \beta q^{74} - q^{76} + (3 \beta + 2) q^{77} + ( - \beta - 1) q^{78} + ( - 6 \beta + 2) q^{79} + (6 \beta - 1) q^{81} - 3 \beta q^{82} + ( - 6 \beta - 6) q^{83} + ( - 2 \beta + 1) q^{84} + (3 \beta - 6) q^{86} + ( - 5 \beta + 7) q^{87} - \beta q^{88} + 5 \beta q^{89} + (9 \beta + 13) q^{91} + (3 \beta - 5) q^{92} + ( - \beta + 4) q^{93} + (\beta - 1) q^{96} + ( - 4 \beta + 6) q^{97} + (6 \beta + 4) q^{98} + 4 q^{99} +O(q^{100})$$ q + q^2 + (b - 1) * q^3 + q^4 + (b - 1) * q^6 + (-b - 3) * q^7 + q^8 - 2*b * q^9 - b * q^11 + (b - 1) * q^12 + (-2*b - 3) * q^13 + (-b - 3) * q^14 + q^16 - q^17 - 2*b * q^18 - q^19 + (-2*b + 1) * q^21 - b * q^22 + (3*b - 5) * q^23 + (b - 1) * q^24 + (-2*b - 3) * q^26 + (-b - 1) * q^27 + (-b - 3) * q^28 + (2*b - 3) * q^29 + (3*b + 2) * q^31 + q^32 + (b - 2) * q^33 - q^34 - 2*b * q^36 + 6*b * q^37 - q^38 + (-b - 1) * q^39 - 3*b * q^41 + (-2*b + 1) * q^42 + (3*b - 6) * q^43 - b * q^44 + (3*b - 5) * q^46 + (b - 1) * q^48 + (6*b + 4) * q^49 + (-b + 1) * q^51 + (-2*b - 3) * q^52 + (-6*b + 3) * q^53 + (-b - 1) * q^54 + (-b - 3) * q^56 + (-b + 1) * q^57 + (2*b - 3) * q^58 + (7*b - 3) * q^59 + (3*b + 10) * q^61 + (3*b + 2) * q^62 + (6*b + 4) * q^63 + q^64 + (b - 2) * q^66 + (3*b - 9) * q^67 - q^68 + (-8*b + 11) * q^69 + (-b - 12) * q^71 - 2*b * q^72 + (-6*b - 3) * q^73 + 6*b * q^74 - q^76 + (3*b + 2) * q^77 + (-b - 1) * q^78 + (-6*b + 2) * q^79 + (6*b - 1) * q^81 - 3*b * q^82 + (-6*b - 6) * q^83 + (-2*b + 1) * q^84 + (3*b - 6) * q^86 + (-5*b + 7) * q^87 - b * q^88 + 5*b * q^89 + (9*b + 13) * q^91 + (3*b - 5) * q^92 + (-b + 4) * q^93 + (b - 1) * q^96 + (-4*b + 6) * q^97 + (6*b + 4) * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8} - 2 q^{12} - 6 q^{13} - 6 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{24} - 6 q^{26} - 2 q^{27} - 6 q^{28} - 6 q^{29} + 4 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} - 2 q^{38} - 2 q^{39} + 2 q^{42} - 12 q^{43} - 10 q^{46} - 2 q^{48} + 8 q^{49} + 2 q^{51} - 6 q^{52} + 6 q^{53} - 2 q^{54} - 6 q^{56} + 2 q^{57} - 6 q^{58} - 6 q^{59} + 20 q^{61} + 4 q^{62} + 8 q^{63} + 2 q^{64} - 4 q^{66} - 18 q^{67} - 2 q^{68} + 22 q^{69} - 24 q^{71} - 6 q^{73} - 2 q^{76} + 4 q^{77} - 2 q^{78} + 4 q^{79} - 2 q^{81} - 12 q^{83} + 2 q^{84} - 12 q^{86} + 14 q^{87} + 26 q^{91} - 10 q^{92} + 8 q^{93} - 2 q^{96} + 12 q^{97} + 8 q^{98} + 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8 - 2 * q^12 - 6 * q^13 - 6 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^24 - 6 * q^26 - 2 * q^27 - 6 * q^28 - 6 * q^29 + 4 * q^31 + 2 * q^32 - 4 * q^33 - 2 * q^34 - 2 * q^38 - 2 * q^39 + 2 * q^42 - 12 * q^43 - 10 * q^46 - 2 * q^48 + 8 * q^49 + 2 * q^51 - 6 * q^52 + 6 * q^53 - 2 * q^54 - 6 * q^56 + 2 * q^57 - 6 * q^58 - 6 * q^59 + 20 * q^61 + 4 * q^62 + 8 * q^63 + 2 * q^64 - 4 * q^66 - 18 * q^67 - 2 * q^68 + 22 * q^69 - 24 * q^71 - 6 * q^73 - 2 * q^76 + 4 * q^77 - 2 * q^78 + 4 * q^79 - 2 * q^81 - 12 * q^83 + 2 * q^84 - 12 * q^86 + 14 * q^87 + 26 * q^91 - 10 * q^92 + 8 * q^93 - 2 * q^96 + 12 * q^97 + 8 * q^98 + 8 * 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
 −1.41421 1.41421
1.00000 −2.41421 1.00000 0 −2.41421 −1.58579 1.00000 2.82843 0
1.2 1.00000 0.414214 1.00000 0 0.414214 −4.41421 1.00000 −2.82843 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$+1$$

## 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 950.2.a.g 2
3.b odd 2 1 8550.2.a.bn 2
4.b odd 2 1 7600.2.a.bg 2
5.b even 2 1 950.2.a.f 2
5.c odd 4 2 190.2.b.a 4
15.d odd 2 1 8550.2.a.cb 2
15.e even 4 2 1710.2.d.c 4
20.d odd 2 1 7600.2.a.v 2
20.e even 4 2 1520.2.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 5.c odd 4 2
950.2.a.f 2 5.b even 2 1
950.2.a.g 2 1.a even 1 1 trivial
1520.2.d.e 4 20.e even 4 2
1710.2.d.c 4 15.e even 4 2
7600.2.a.v 2 20.d odd 2 1
7600.2.a.bg 2 4.b odd 2 1
8550.2.a.bn 2 3.b odd 2 1
8550.2.a.cb 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{7}^{2} + 6T_{7} + 7$$ T7^2 + 6*T7 + 7 $$T_{11}^{2} - 2$$ T11^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 7$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2} + 6T + 1$$
$17$ $$(T + 1)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 10T + 7$$
$29$ $$T^{2} + 6T + 1$$
$31$ $$T^{2} - 4T - 14$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} - 18$$
$43$ $$T^{2} + 12T + 18$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 6T - 63$$
$59$ $$T^{2} + 6T - 89$$
$61$ $$T^{2} - 20T + 82$$
$67$ $$T^{2} + 18T + 63$$
$71$ $$T^{2} + 24T + 142$$
$73$ $$T^{2} + 6T - 63$$
$79$ $$T^{2} - 4T - 68$$
$83$ $$T^{2} + 12T - 36$$
$89$ $$T^{2} - 50$$
$97$ $$T^{2} - 12T + 4$$