# Properties

 Label 950.2.b.g Level $950$ Weight $2$ Character orbit 950.b Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(799,950)]

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([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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.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: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.4227136.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 22x^{2} + 9$$ x^6 + 9*x^4 + 22*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{5} + \beta_{2} + 1) q^{6} + (\beta_{4} - 2 \beta_{3} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{5} - 4) q^{9}+O(q^{10})$$ q + b4 * q^2 + (-b4 - b3 + b1) * q^3 - q^4 + (-b5 + b2 + 1) * q^6 + (b4 - 2*b3 - b1) * q^7 - b4 * q^8 + (-b5 - 4) * q^9 $$q + \beta_{4} q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{5} + \beta_{2} + 1) q^{6} + (\beta_{4} - 2 \beta_{3} - \beta_1) q^{7} - \beta_{4} q^{8} + ( - \beta_{5} - 4) q^{9} + (2 \beta_{5} + 2 \beta_{2}) q^{11} + (\beta_{4} + \beta_{3} - \beta_1) q^{12} + ( - \beta_{3} + 2 \beta_1) q^{13} + (\beta_{5} + 2 \beta_{2} - 1) q^{14} + q^{16} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{17} + ( - 4 \beta_{4} - \beta_1) q^{18} + q^{19} + ( - 5 \beta_{5} - 3 \beta_{2} - 2) q^{21} + (2 \beta_{3} + 2 \beta_1) q^{22} + ( - 5 \beta_{4} + \beta_{3} + \beta_1) q^{23} + (\beta_{5} - \beta_{2} - 1) q^{24} + ( - 2 \beta_{5} + \beta_{2}) q^{26} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_1) q^{27} + ( - \beta_{4} + 2 \beta_{3} + \beta_1) q^{28} + ( - 2 \beta_{5} + \beta_{2} - 4) q^{29} + (2 \beta_{5} - 2 \beta_{2} - 2) q^{31} + \beta_{4} q^{32} + ( - 6 \beta_{3} - 4 \beta_1) q^{33} + ( - 2 \beta_{5} - \beta_{2} + 2) q^{34} + (\beta_{5} + 4) q^{36} + ( - 5 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{37} + \beta_{4} q^{38} + ( - 2 \beta_{5} + 3 \beta_{2} - 9) q^{39} + (4 \beta_{5} - 4) q^{41} + ( - 2 \beta_{4} - 3 \beta_{3} - 5 \beta_1) q^{42} + (2 \beta_{3} - 2 \beta_1) q^{43} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{44} + ( - \beta_{5} - \beta_{2} + 5) q^{46} + (3 \beta_{4} + 2 \beta_{3} + 4 \beta_1) q^{47} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{48} + (2 \beta_{5} - 3 \beta_{2} - 9) q^{49} + (4 \beta_{5} + 3 \beta_{2} - 5) q^{51} + (\beta_{3} - 2 \beta_1) q^{52} + (3 \beta_{4} + 2 \beta_{3} + \beta_1) q^{53} + (\beta_{5} - 3 \beta_{2} + 2) q^{54} + ( - \beta_{5} - 2 \beta_{2} + 1) q^{56} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{57} + ( - 4 \beta_{4} + \beta_{3} - 2 \beta_1) q^{58} + (3 \beta_{5} + \beta_{2} + 1) q^{59} + ( - 2 \beta_{5} - 4 \beta_{2} + 8) q^{61} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{62} + ( - \beta_{4} + 9 \beta_{3} + \beta_1) q^{63} - q^{64} + (4 \beta_{5} + 6 \beta_{2}) q^{66} + ( - \beta_{4} - 4 \beta_{3} + 3 \beta_1) q^{67} + (2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{68} + (7 \beta_{5} - 2 \beta_{2} - 5) q^{69} - 2 \beta_{5} q^{71} + (4 \beta_{4} + \beta_1) q^{72} + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{73} + (2 \beta_{5} + 2 \beta_{2} + 5) q^{74} - q^{76} + ( - 18 \beta_{4} - 6 \beta_{3} + 4 \beta_1) q^{77} + ( - 9 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{78} + (4 \beta_{2} - 8) q^{79} + (5 \beta_{5} - \beta_{2} - 2) q^{81} + ( - 4 \beta_{4} + 4 \beta_1) q^{82} + ( - 2 \beta_{4} - 4 \beta_{3} - 6 \beta_1) q^{83} + (5 \beta_{5} + 3 \beta_{2} + 2) q^{84} + (2 \beta_{5} - 2 \beta_{2}) q^{86} + ( - 5 \beta_{4} + 7 \beta_{3} - 6 \beta_1) q^{87} + ( - 2 \beta_{3} - 2 \beta_1) q^{88} + 10 q^{89} - 7 \beta_{5} q^{91} + (5 \beta_{4} - \beta_{3} - \beta_1) q^{92} + (14 \beta_{4} + 2 \beta_1) q^{93} + ( - 4 \beta_{5} - 2 \beta_{2} - 3) q^{94} + ( - \beta_{5} + \beta_{2} + 1) q^{96} - 6 \beta_{3} q^{97} + ( - 9 \beta_{4} - 3 \beta_{3} + 2 \beta_1) q^{98} + ( - 6 \beta_{5} - 8 \beta_{2} - 6) q^{99}+O(q^{100})$$ q + b4 * q^2 + (-b4 - b3 + b1) * q^3 - q^4 + (-b5 + b2 + 1) * q^6 + (b4 - 2*b3 - b1) * q^7 - b4 * q^8 + (-b5 - 4) * q^9 + (2*b5 + 2*b2) * q^11 + (b4 + b3 - b1) * q^12 + (-b3 + 2*b1) * q^13 + (b5 + 2*b2 - 1) * q^14 + q^16 + (-2*b4 + b3 + 2*b1) * q^17 + (-4*b4 - b1) * q^18 + q^19 + (-5*b5 - 3*b2 - 2) * q^21 + (2*b3 + 2*b1) * q^22 + (-5*b4 + b3 + b1) * q^23 + (b5 - b2 - 1) * q^24 + (-2*b5 + b2) * q^26 + (-2*b4 + 3*b3 - b1) * q^27 + (-b4 + 2*b3 + b1) * q^28 + (-2*b5 + b2 - 4) * q^29 + (2*b5 - 2*b2 - 2) * q^31 + b4 * q^32 + (-6*b3 - 4*b1) * q^33 + (-2*b5 - b2 + 2) * q^34 + (b5 + 4) * q^36 + (-5*b4 - 2*b3 - 2*b1) * q^37 + b4 * q^38 + (-2*b5 + 3*b2 - 9) * q^39 + (4*b5 - 4) * q^41 + (-2*b4 - 3*b3 - 5*b1) * q^42 + (2*b3 - 2*b1) * q^43 + (-2*b5 - 2*b2) * q^44 + (-b5 - b2 + 5) * q^46 + (3*b4 + 2*b3 + 4*b1) * q^47 + (-b4 - b3 + b1) * q^48 + (2*b5 - 3*b2 - 9) * q^49 + (4*b5 + 3*b2 - 5) * q^51 + (b3 - 2*b1) * q^52 + (3*b4 + 2*b3 + b1) * q^53 + (b5 - 3*b2 + 2) * q^54 + (-b5 - 2*b2 + 1) * q^56 + (-b4 - b3 + b1) * q^57 + (-4*b4 + b3 - 2*b1) * q^58 + (3*b5 + b2 + 1) * q^59 + (-2*b5 - 4*b2 + 8) * q^61 + (-2*b4 - 2*b3 + 2*b1) * q^62 + (-b4 + 9*b3 + b1) * q^63 - q^64 + (4*b5 + 6*b2) * q^66 + (-b4 - 4*b3 + 3*b1) * q^67 + (2*b4 - b3 - 2*b1) * q^68 + (7*b5 - 2*b2 - 5) * q^69 - 2*b5 * q^71 + (4*b4 + b1) * q^72 + (3*b4 + 3*b3 + 3*b1) * q^73 + (2*b5 + 2*b2 + 5) * q^74 - q^76 + (-18*b4 - 6*b3 + 4*b1) * q^77 + (-9*b4 + 3*b3 - 2*b1) * q^78 + (4*b2 - 8) * q^79 + (5*b5 - b2 - 2) * q^81 + (-4*b4 + 4*b1) * q^82 + (-2*b4 - 4*b3 - 6*b1) * q^83 + (5*b5 + 3*b2 + 2) * q^84 + (2*b5 - 2*b2) * q^86 + (-5*b4 + 7*b3 - 6*b1) * q^87 + (-2*b3 - 2*b1) * q^88 + 10 * q^89 - 7*b5 * q^91 + (5*b4 - b3 - b1) * q^92 + (14*b4 + 2*b1) * q^93 + (-4*b5 - 2*b2 - 3) * q^94 + (-b5 + b2 + 1) * q^96 - 6*b3 * q^97 + (-9*b4 - 3*b3 + 2*b1) * q^98 + (-6*b5 - 8*b2 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 4 q^{6} - 26 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 + 4 * q^6 - 26 * q^9 $$6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + 4 q^{11} - 4 q^{14} + 6 q^{16} + 6 q^{19} - 22 q^{21} - 4 q^{24} - 4 q^{26} - 28 q^{29} - 8 q^{31} + 8 q^{34} + 26 q^{36} - 58 q^{39} - 16 q^{41} - 4 q^{44} + 28 q^{46} - 50 q^{49} - 22 q^{51} + 14 q^{54} + 4 q^{56} + 12 q^{59} + 44 q^{61} - 6 q^{64} + 8 q^{66} - 16 q^{69} - 4 q^{71} + 34 q^{74} - 6 q^{76} - 48 q^{79} - 2 q^{81} + 22 q^{84} + 4 q^{86} + 60 q^{89} - 14 q^{91} - 26 q^{94} + 4 q^{96} - 48 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 + 4 * q^6 - 26 * q^9 + 4 * q^11 - 4 * q^14 + 6 * q^16 + 6 * q^19 - 22 * q^21 - 4 * q^24 - 4 * q^26 - 28 * q^29 - 8 * q^31 + 8 * q^34 + 26 * q^36 - 58 * q^39 - 16 * q^41 - 4 * q^44 + 28 * q^46 - 50 * q^49 - 22 * q^51 + 14 * q^54 + 4 * q^56 + 12 * q^59 + 44 * q^61 - 6 * q^64 + 8 * q^66 - 16 * q^69 - 4 * q^71 + 34 * q^74 - 6 * q^76 - 48 * q^79 - 2 * q^81 + 22 * q^84 + 4 * q^86 + 60 * q^89 - 14 * q^91 - 26 * q^94 + 4 * q^96 - 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9x^{4} + 22x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 6\nu^{3} + 7\nu ) / 3$$ (v^5 + 6*v^3 + 7*v) / 3 $$\beta_{5}$$ $$=$$ $$\nu^{4} + 5\nu^{2} + 3$$ v^4 + 5*v^2 + 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} - 5\beta_{2} + 12$$ b5 - 5*b2 + 12 $$\nu^{5}$$ $$=$$ $$3\beta_{4} - 6\beta_{3} + 17\beta_1$$ 3*b4 - 6*b3 + 17*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 − 2.19869i 1.91223i − 0.713538i 0.713538i − 1.91223i 2.19869i
1.00000i 3.03293i −1.00000 0 −3.03293 2.46980i 1.00000i −6.19869 0
799.2 1.00000i 2.25561i −1.00000 0 2.25561 4.22547i 1.00000i −2.08777 0
799.3 1.00000i 2.77733i −1.00000 0 2.77733 4.69527i 1.00000i −4.71354 0
799.4 1.00000i 2.77733i −1.00000 0 2.77733 4.69527i 1.00000i −4.71354 0
799.5 1.00000i 2.25561i −1.00000 0 2.25561 4.22547i 1.00000i −2.08777 0
799.6 1.00000i 3.03293i −1.00000 0 −3.03293 2.46980i 1.00000i −6.19869 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 799.6 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 950.2.b.g 6
5.b even 2 1 inner 950.2.b.g 6
5.c odd 4 1 950.2.a.k 3
5.c odd 4 1 950.2.a.m yes 3
15.e even 4 1 8550.2.a.cj 3
15.e even 4 1 8550.2.a.co 3
20.e even 4 1 7600.2.a.bm 3
20.e even 4 1 7600.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 5.c odd 4 1
950.2.a.m yes 3 5.c odd 4 1
950.2.b.g 6 1.a even 1 1 trivial
950.2.b.g 6 5.b even 2 1 inner
7600.2.a.bm 3 20.e even 4 1
7600.2.a.cb 3 20.e even 4 1
8550.2.a.cj 3 15.e even 4 1
8550.2.a.co 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{6} + 22T_{3}^{4} + 157T_{3}^{2} + 361$$ T3^6 + 22*T3^4 + 157*T3^2 + 361 $$T_{7}^{6} + 46T_{7}^{4} + 637T_{7}^{2} + 2401$$ T7^6 + 46*T7^4 + 637*T7^2 + 2401 $$T_{11}^{3} - 2T_{11}^{2} - 32T_{11} + 24$$ T11^3 - 2*T11^2 - 32*T11 + 24 $$T_{13}^{6} + 50T_{13}^{4} + 445T_{13}^{2} + 441$$ T13^6 + 50*T13^4 + 445*T13^2 + 441

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6} + 22 T^{4} + \cdots + 361$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 46 T^{4} + \cdots + 2401$$
$11$ $$(T^{3} - 2 T^{2} - 32 T + 24)^{2}$$
$13$ $$T^{6} + 50 T^{4} + \cdots + 441$$
$17$ $$T^{6} + 46 T^{4} + \cdots + 49$$
$19$ $$(T - 1)^{6}$$
$23$ $$T^{6} + 82 T^{4} + \cdots + 3969$$
$29$ $$(T^{3} + 14 T^{2} + \cdots + 25)^{2}$$
$31$ $$(T^{3} + 4 T^{2} + \cdots - 152)^{2}$$
$37$ $$T^{6} + 163 T^{4} + \cdots + 81$$
$41$ $$(T^{3} + 8 T^{2} - 48 T - 64)^{2}$$
$43$ $$T^{6} + 84 T^{4} + \cdots + 5184$$
$47$ $$T^{6} + 219 T^{4} + \cdots + 275625$$
$53$ $$T^{6} + 78 T^{4} + \cdots + 9$$
$59$ $$(T^{3} - 6 T^{2} + \cdots + 175)^{2}$$
$61$ $$(T^{3} - 22 T^{2} + \cdots + 536)^{2}$$
$67$ $$T^{6} + 262 T^{4} + \cdots + 219961$$
$71$ $$(T^{3} + 2 T^{2} - 16 T - 24)^{2}$$
$73$ $$T^{6} + 198 T^{4} + \cdots + 59049$$
$79$ $$(T^{3} + 24 T^{2} + \cdots - 320)^{2}$$
$83$ $$T^{6} + 472 T^{4} + \cdots + 2383936$$
$89$ $$(T - 10)^{6}$$
$97$ $$T^{6} + 360 T^{4} + \cdots + 419904$$