# Properties

 Label 1950.2.y.f Level $1950$ Weight $2$ Character orbit 1950.y Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(49,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1950.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.y (of order $$6$$, degree $$2$$, 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: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1 - \zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −1.00000 + 1.73205i −1.00000 0.500000 0.866025i 0
49.2 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −1.00000 + 1.73205i −1.00000 0.500000 0.866025i 0
199.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i −1.00000 1.73205i −1.00000 0.500000 + 0.866025i 0
199.2 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −1.00000 1.73205i −1.00000 0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.y.f 4
5.b even 2 1 1950.2.y.c 4
5.c odd 4 1 390.2.bb.b 4
5.c odd 4 1 1950.2.bc.b 4
13.e even 6 1 1950.2.y.c 4
15.e even 4 1 1170.2.bs.e 4
65.l even 6 1 inner 1950.2.y.f 4
65.o even 12 1 5070.2.a.y 2
65.q odd 12 1 5070.2.b.o 4
65.r odd 12 1 390.2.bb.b 4
65.r odd 12 1 1950.2.bc.b 4
65.r odd 12 1 5070.2.b.o 4
65.t even 12 1 5070.2.a.bg 2
195.bf even 12 1 1170.2.bs.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 5.c odd 4 1
390.2.bb.b 4 65.r odd 12 1
1170.2.bs.e 4 15.e even 4 1
1170.2.bs.e 4 195.bf even 12 1
1950.2.y.c 4 5.b even 2 1
1950.2.y.c 4 13.e even 6 1
1950.2.y.f 4 1.a even 1 1 trivial
1950.2.y.f 4 65.l even 6 1 inner
1950.2.bc.b 4 5.c odd 4 1
1950.2.bc.b 4 65.r odd 12 1
5070.2.a.y 2 65.o even 12 1
5070.2.a.bg 2 65.t even 12 1
5070.2.b.o 4 65.q odd 12 1
5070.2.b.o 4 65.r odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 2T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 2 T + 4)^{2}$$
$11$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$13$ $$T^{4} - 22T^{2} + 169$$
$17$ $$T^{4} - 16T^{2} + 256$$
$19$ $$T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16$$
$23$ $$T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1$$
$29$ $$T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121$$
$41$ $$T^{4} - 4T^{2} + 16$$
$43$ $$T^{4} + 24 T^{3} + 215 T^{2} + \cdots + 529$$
$47$ $$(T^{2} - 14 T + 37)^{2}$$
$53$ $$T^{4} + 168T^{2} + 144$$
$59$ $$T^{4} + 12 T^{3} + 35 T^{2} + \cdots + 169$$
$61$ $$T^{4} + 108 T^{2} + 11664$$
$67$ $$T^{4} - 16 T^{3} + 204 T^{2} + \cdots + 2704$$
$71$ $$T^{4} + 36 T^{3} + 536 T^{2} + \cdots + 10816$$
$73$ $$(T - 2)^{4}$$
$79$ $$(T^{2} - 14 T + 1)^{2}$$
$83$ $$(T^{2} - 4 T - 44)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16$$
$97$ $$T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16$$