# Properties

 Label 1950.2.i.m Level $1950$ Weight $2$ Character orbit 1950.i Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ 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(451,1950)]

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, 0, 4]))

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

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

chi := DirichletCharacter("1950.451");

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.i (of order $$3$$, degree $$2$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 - z * q^4 + z * q^6 + 2*z * q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + q^{8} - \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} - q^{12} + (\zeta_{6} + 3) q^{13} - 2 q^{14} + (\zeta_{6} - 1) q^{16} - 3 \zeta_{6} q^{17} + q^{18} - 2 \zeta_{6} q^{19} + 2 q^{21} - 6 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} + ( - \zeta_{6} + 1) q^{24} + (3 \zeta_{6} - 4) q^{26} - q^{27} + ( - 2 \zeta_{6} + 2) q^{28} + (3 \zeta_{6} - 3) q^{29} - 4 q^{31} - \zeta_{6} q^{32} + 6 \zeta_{6} q^{33} + 3 q^{34} + (\zeta_{6} - 1) q^{36} + (7 \zeta_{6} - 7) q^{37} + 2 q^{38} + ( - 3 \zeta_{6} + 4) q^{39} + ( - 3 \zeta_{6} + 3) q^{41} + (2 \zeta_{6} - 2) q^{42} - 10 \zeta_{6} q^{43} + 6 q^{44} - 6 \zeta_{6} q^{46} - 6 q^{47} + \zeta_{6} q^{48} + ( - 3 \zeta_{6} + 3) q^{49} - 3 q^{51} + ( - 4 \zeta_{6} + 1) q^{52} - 3 q^{53} + ( - \zeta_{6} + 1) q^{54} + 2 \zeta_{6} q^{56} - 2 q^{57} - 3 \zeta_{6} q^{58} + 7 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + ( - 2 \zeta_{6} + 2) q^{63} + q^{64} - 6 q^{66} + (10 \zeta_{6} - 10) q^{67} + (3 \zeta_{6} - 3) q^{68} + 6 \zeta_{6} q^{69} - 6 \zeta_{6} q^{71} - \zeta_{6} q^{72} + 13 q^{73} - 7 \zeta_{6} q^{74} + (2 \zeta_{6} - 2) q^{76} - 12 q^{77} + (4 \zeta_{6} - 1) q^{78} - 4 q^{79} + (\zeta_{6} - 1) q^{81} + 3 \zeta_{6} q^{82} + 6 q^{83} - 2 \zeta_{6} q^{84} + 10 q^{86} + 3 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + (18 \zeta_{6} - 18) q^{89} + (8 \zeta_{6} - 2) q^{91} + 6 q^{92} + (4 \zeta_{6} - 4) q^{93} + ( - 6 \zeta_{6} + 6) q^{94} - q^{96} + 14 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} + 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 - z * q^4 + z * q^6 + 2*z * q^7 + q^8 - z * q^9 + (6*z - 6) * q^11 - q^12 + (z + 3) * q^13 - 2 * q^14 + (z - 1) * q^16 - 3*z * q^17 + q^18 - 2*z * q^19 + 2 * q^21 - 6*z * q^22 + (6*z - 6) * q^23 + (-z + 1) * q^24 + (3*z - 4) * q^26 - q^27 + (-2*z + 2) * q^28 + (3*z - 3) * q^29 - 4 * q^31 - z * q^32 + 6*z * q^33 + 3 * q^34 + (z - 1) * q^36 + (7*z - 7) * q^37 + 2 * q^38 + (-3*z + 4) * q^39 + (-3*z + 3) * q^41 + (2*z - 2) * q^42 - 10*z * q^43 + 6 * q^44 - 6*z * q^46 - 6 * q^47 + z * q^48 + (-3*z + 3) * q^49 - 3 * q^51 + (-4*z + 1) * q^52 - 3 * q^53 + (-z + 1) * q^54 + 2*z * q^56 - 2 * q^57 - 3*z * q^58 + 7*z * q^61 + (-4*z + 4) * q^62 + (-2*z + 2) * q^63 + q^64 - 6 * q^66 + (10*z - 10) * q^67 + (3*z - 3) * q^68 + 6*z * q^69 - 6*z * q^71 - z * q^72 + 13 * q^73 - 7*z * q^74 + (2*z - 2) * q^76 - 12 * q^77 + (4*z - 1) * q^78 - 4 * q^79 + (z - 1) * q^81 + 3*z * q^82 + 6 * q^83 - 2*z * q^84 + 10 * q^86 + 3*z * q^87 + (6*z - 6) * q^88 + (18*z - 18) * q^89 + (8*z - 2) * q^91 + 6 * q^92 + (4*z - 4) * q^93 + (-6*z + 6) * q^94 - q^96 + 14*z * q^97 + 3*z * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 - q^4 + q^6 + 2 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{8} - q^{9} - 6 q^{11} - 2 q^{12} + 7 q^{13} - 4 q^{14} - q^{16} - 3 q^{17} + 2 q^{18} - 2 q^{19} + 4 q^{21} - 6 q^{22} - 6 q^{23} + q^{24} - 5 q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} - 8 q^{31} - q^{32} + 6 q^{33} + 6 q^{34} - q^{36} - 7 q^{37} + 4 q^{38} + 5 q^{39} + 3 q^{41} - 2 q^{42} - 10 q^{43} + 12 q^{44} - 6 q^{46} - 12 q^{47} + q^{48} + 3 q^{49} - 6 q^{51} - 2 q^{52} - 6 q^{53} + q^{54} + 2 q^{56} - 4 q^{57} - 3 q^{58} + 7 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 12 q^{66} - 10 q^{67} - 3 q^{68} + 6 q^{69} - 6 q^{71} - q^{72} + 26 q^{73} - 7 q^{74} - 2 q^{76} - 24 q^{77} + 2 q^{78} - 8 q^{79} - q^{81} + 3 q^{82} + 12 q^{83} - 2 q^{84} + 20 q^{86} + 3 q^{87} - 6 q^{88} - 18 q^{89} + 4 q^{91} + 12 q^{92} - 4 q^{93} + 6 q^{94} - 2 q^{96} + 14 q^{97} + 3 q^{98} + 12 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 + q^6 + 2 * q^7 + 2 * q^8 - q^9 - 6 * q^11 - 2 * q^12 + 7 * q^13 - 4 * q^14 - q^16 - 3 * q^17 + 2 * q^18 - 2 * q^19 + 4 * q^21 - 6 * q^22 - 6 * q^23 + q^24 - 5 * q^26 - 2 * q^27 + 2 * q^28 - 3 * q^29 - 8 * q^31 - q^32 + 6 * q^33 + 6 * q^34 - q^36 - 7 * q^37 + 4 * q^38 + 5 * q^39 + 3 * q^41 - 2 * q^42 - 10 * q^43 + 12 * q^44 - 6 * q^46 - 12 * q^47 + q^48 + 3 * q^49 - 6 * q^51 - 2 * q^52 - 6 * q^53 + q^54 + 2 * q^56 - 4 * q^57 - 3 * q^58 + 7 * q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 - 12 * q^66 - 10 * q^67 - 3 * q^68 + 6 * q^69 - 6 * q^71 - q^72 + 26 * q^73 - 7 * q^74 - 2 * q^76 - 24 * q^77 + 2 * q^78 - 8 * q^79 - q^81 + 3 * q^82 + 12 * q^83 - 2 * q^84 + 20 * q^86 + 3 * q^87 - 6 * q^88 - 18 * q^89 + 4 * q^91 + 12 * q^92 - 4 * q^93 + 6 * q^94 - 2 * q^96 + 14 * q^97 + 3 * q^98 + 12 * q^99

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i 0
601.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.i.m 2
5.b even 2 1 78.2.e.a 2
5.c odd 4 2 1950.2.z.g 4
13.c even 3 1 inner 1950.2.i.m 2
15.d odd 2 1 234.2.h.a 2
20.d odd 2 1 624.2.q.g 2
60.h even 2 1 1872.2.t.c 2
65.d even 2 1 1014.2.e.a 2
65.g odd 4 2 1014.2.i.b 4
65.l even 6 1 1014.2.a.f 1
65.l even 6 1 1014.2.e.a 2
65.n even 6 1 78.2.e.a 2
65.n even 6 1 1014.2.a.c 1
65.q odd 12 2 1950.2.z.g 4
65.s odd 12 2 1014.2.b.c 2
65.s odd 12 2 1014.2.i.b 4
195.x odd 6 1 234.2.h.a 2
195.x odd 6 1 3042.2.a.i 1
195.y odd 6 1 3042.2.a.h 1
195.bh even 12 2 3042.2.b.h 2
260.v odd 6 1 624.2.q.g 2
260.v odd 6 1 8112.2.a.m 1
260.w odd 6 1 8112.2.a.c 1
780.br even 6 1 1872.2.t.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 5.b even 2 1
78.2.e.a 2 65.n even 6 1
234.2.h.a 2 15.d odd 2 1
234.2.h.a 2 195.x odd 6 1
624.2.q.g 2 20.d odd 2 1
624.2.q.g 2 260.v odd 6 1
1014.2.a.c 1 65.n even 6 1
1014.2.a.f 1 65.l even 6 1
1014.2.b.c 2 65.s odd 12 2
1014.2.e.a 2 65.d even 2 1
1014.2.e.a 2 65.l even 6 1
1014.2.i.b 4 65.g odd 4 2
1014.2.i.b 4 65.s odd 12 2
1872.2.t.c 2 60.h even 2 1
1872.2.t.c 2 780.br even 6 1
1950.2.i.m 2 1.a even 1 1 trivial
1950.2.i.m 2 13.c even 3 1 inner
1950.2.z.g 4 5.c odd 4 2
1950.2.z.g 4 65.q odd 12 2
3042.2.a.h 1 195.y odd 6 1
3042.2.a.i 1 195.x odd 6 1
3042.2.b.h 2 195.bh even 12 2
8112.2.a.c 1 260.w odd 6 1
8112.2.a.m 1 260.v odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36 $$T_{17}^{2} + 3T_{17} + 9$$ T17^2 + 3*T17 + 9 $$T_{19}^{2} + 2T_{19} + 4$$ T19^2 + 2*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$T^{2} - 7T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} + 10T + 100$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 10T + 100$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$(T - 13)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 18T + 324$$
$97$ $$T^{2} - 14T + 196$$