# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1699,1950)]

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, 4]))

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

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

chi := DirichletCharacter("1950.1699");

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.z (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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + (\zeta_{12}^{2} - 1) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10})$$ q + (z^3 - z) * q^2 + (-z^3 + z) * q^3 + (-z^2 + 1) * q^4 + (z^2 - 1) * q^6 + 2*z * q^7 + z^3 * q^8 + (-z^2 + 1) * q^9 $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + (\zeta_{12}^{2} - 1) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} - 2 \zeta_{12}^{2} q^{11} - \zeta_{12}^{3} q^{12} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{13} - 2 q^{14} - \zeta_{12}^{2} q^{16} - 5 \zeta_{12} q^{17} + \zeta_{12}^{3} q^{18} + (2 \zeta_{12}^{2} - 2) q^{19} + 2 q^{21} + 2 \zeta_{12} q^{22} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{23} + \zeta_{12}^{2} q^{24} + (\zeta_{12}^{2} + 3) q^{26} - \zeta_{12}^{3} q^{27} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} - 9 \zeta_{12}^{2} q^{29} - 4 q^{31} + \zeta_{12} q^{32} - 2 \zeta_{12} q^{33} + 5 q^{34} - \zeta_{12}^{2} q^{36} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{37} - 2 \zeta_{12}^{3} q^{38} + ( - \zeta_{12}^{2} - 3) q^{39} - 5 \zeta_{12}^{2} q^{41} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{42} + 10 \zeta_{12} q^{43} - 2 q^{44} + ( - 6 \zeta_{12}^{2} + 6) q^{46} + 2 \zeta_{12}^{3} q^{47} - \zeta_{12} q^{48} - 3 \zeta_{12}^{2} q^{49} - 5 q^{51} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{52} + \zeta_{12}^{3} q^{53} + \zeta_{12}^{2} q^{54} + (2 \zeta_{12}^{2} - 2) q^{56} + 2 \zeta_{12}^{3} q^{57} + 9 \zeta_{12} q^{58} + (8 \zeta_{12}^{2} - 8) q^{59} + ( - 11 \zeta_{12}^{2} + 11) q^{61} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{62} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{63} - q^{64} + 2 q^{66} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{67} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{68} + (6 \zeta_{12}^{2} - 6) q^{69} + ( - 14 \zeta_{12}^{2} + 14) q^{71} + \zeta_{12} q^{72} + 13 \zeta_{12}^{3} q^{73} + ( - 11 \zeta_{12}^{2} + 11) q^{74} + 2 \zeta_{12}^{2} q^{76} - 4 \zeta_{12}^{3} q^{77} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{78} + 4 q^{79} - \zeta_{12}^{2} q^{81} + 5 \zeta_{12} q^{82} - 6 \zeta_{12}^{3} q^{83} + ( - 2 \zeta_{12}^{2} + 2) q^{84} - 10 q^{86} - 9 \zeta_{12} q^{87} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{88} + 2 \zeta_{12}^{2} q^{89} + ( - 8 \zeta_{12}^{2} + 2) q^{91} + 6 \zeta_{12}^{3} q^{92} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{93} - 2 \zeta_{12}^{2} q^{94} + q^{96} + 2 \zeta_{12} q^{97} + 3 \zeta_{12} q^{98} - 2 q^{99} +O(q^{100})$$ q + (z^3 - z) * q^2 + (-z^3 + z) * q^3 + (-z^2 + 1) * q^4 + (z^2 - 1) * q^6 + 2*z * q^7 + z^3 * q^8 + (-z^2 + 1) * q^9 - 2*z^2 * q^11 - z^3 * q^12 + (-z^3 - 3*z) * q^13 - 2 * q^14 - z^2 * q^16 - 5*z * q^17 + z^3 * q^18 + (2*z^2 - 2) * q^19 + 2 * q^21 + 2*z * q^22 + (6*z^3 - 6*z) * q^23 + z^2 * q^24 + (z^2 + 3) * q^26 - z^3 * q^27 + (-2*z^3 + 2*z) * q^28 - 9*z^2 * q^29 - 4 * q^31 + z * q^32 - 2*z * q^33 + 5 * q^34 - z^2 * q^36 + (11*z^3 - 11*z) * q^37 - 2*z^3 * q^38 + (-z^2 - 3) * q^39 - 5*z^2 * q^41 + (2*z^3 - 2*z) * q^42 + 10*z * q^43 - 2 * q^44 + (-6*z^2 + 6) * q^46 + 2*z^3 * q^47 - z * q^48 - 3*z^2 * q^49 - 5 * q^51 + (3*z^3 - 4*z) * q^52 + z^3 * q^53 + z^2 * q^54 + (2*z^2 - 2) * q^56 + 2*z^3 * q^57 + 9*z * q^58 + (8*z^2 - 8) * q^59 + (-11*z^2 + 11) * q^61 + (-4*z^3 + 4*z) * q^62 + (-2*z^3 + 2*z) * q^63 - q^64 + 2 * q^66 + (-2*z^3 + 2*z) * q^67 + (5*z^3 - 5*z) * q^68 + (6*z^2 - 6) * q^69 + (-14*z^2 + 14) * q^71 + z * q^72 + 13*z^3 * q^73 + (-11*z^2 + 11) * q^74 + 2*z^2 * q^76 - 4*z^3 * q^77 + (-3*z^3 + 4*z) * q^78 + 4 * q^79 - z^2 * q^81 + 5*z * q^82 - 6*z^3 * q^83 + (-2*z^2 + 2) * q^84 - 10 * q^86 - 9*z * q^87 + (-2*z^3 + 2*z) * q^88 + 2*z^2 * q^89 + (-8*z^2 + 2) * q^91 + 6*z^3 * q^92 + (4*z^3 - 4*z) * q^93 - 2*z^2 * q^94 + q^96 + 2*z * q^97 + 3*z * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 2 * q^6 + 2 * q^9 $$4 q + 2 q^{4} - 2 q^{6} + 2 q^{9} - 4 q^{11} - 8 q^{14} - 2 q^{16} - 4 q^{19} + 8 q^{21} + 2 q^{24} + 14 q^{26} - 18 q^{29} - 16 q^{31} + 20 q^{34} - 2 q^{36} - 14 q^{39} - 10 q^{41} - 8 q^{44} + 12 q^{46} - 6 q^{49} - 20 q^{51} + 2 q^{54} - 4 q^{56} - 16 q^{59} + 22 q^{61} - 4 q^{64} + 8 q^{66} - 12 q^{69} + 28 q^{71} + 22 q^{74} + 4 q^{76} + 16 q^{79} - 2 q^{81} + 4 q^{84} - 40 q^{86} + 4 q^{89} - 8 q^{91} - 4 q^{94} + 4 q^{96} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^6 + 2 * q^9 - 4 * q^11 - 8 * q^14 - 2 * q^16 - 4 * q^19 + 8 * q^21 + 2 * q^24 + 14 * q^26 - 18 * q^29 - 16 * q^31 + 20 * q^34 - 2 * q^36 - 14 * q^39 - 10 * q^41 - 8 * q^44 + 12 * q^46 - 6 * q^49 - 20 * q^51 + 2 * q^54 - 4 * q^56 - 16 * q^59 + 22 * q^61 - 4 * q^64 + 8 * q^66 - 12 * q^69 + 28 * q^71 + 22 * q^74 + 4 * q^76 + 16 * q^79 - 2 * q^81 + 4 * q^84 - 40 * q^86 + 4 * q^89 - 8 * q^91 - 4 * q^94 + 4 * q^96 - 8 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1699.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 1.73205 1.00000i 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i −1.73205 + 1.00000i 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 1.73205 + 1.00000i 1.00000i 0.500000 0.866025i 0
1849.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i −1.73205 1.00000i 1.00000i 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n 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.z.b 4
5.b even 2 1 inner 1950.2.z.b 4
5.c odd 4 1 78.2.e.b 2
5.c odd 4 1 1950.2.i.b 2
13.c even 3 1 inner 1950.2.z.b 4
15.e even 4 1 234.2.h.b 2
20.e even 4 1 624.2.q.b 2
60.l odd 4 1 1872.2.t.i 2
65.f even 4 1 1014.2.i.e 4
65.h odd 4 1 1014.2.e.d 2
65.k even 4 1 1014.2.i.e 4
65.n even 6 1 inner 1950.2.z.b 4
65.o even 12 1 1014.2.b.a 2
65.o even 12 1 1014.2.i.e 4
65.q odd 12 1 78.2.e.b 2
65.q odd 12 1 1014.2.a.a 1
65.q odd 12 1 1950.2.i.b 2
65.r odd 12 1 1014.2.a.e 1
65.r odd 12 1 1014.2.e.d 2
65.t even 12 1 1014.2.b.a 2
65.t even 12 1 1014.2.i.e 4
195.bc odd 12 1 3042.2.b.d 2
195.bf even 12 1 3042.2.a.d 1
195.bl even 12 1 234.2.h.b 2
195.bl even 12 1 3042.2.a.m 1
195.bn odd 12 1 3042.2.b.d 2
260.bg even 12 1 8112.2.a.bb 1
260.bj even 12 1 624.2.q.b 2
260.bj even 12 1 8112.2.a.x 1
780.cj odd 12 1 1872.2.t.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 5.c odd 4 1
78.2.e.b 2 65.q odd 12 1
234.2.h.b 2 15.e even 4 1
234.2.h.b 2 195.bl even 12 1
624.2.q.b 2 20.e even 4 1
624.2.q.b 2 260.bj even 12 1
1014.2.a.a 1 65.q odd 12 1
1014.2.a.e 1 65.r odd 12 1
1014.2.b.a 2 65.o even 12 1
1014.2.b.a 2 65.t even 12 1
1014.2.e.d 2 65.h odd 4 1
1014.2.e.d 2 65.r odd 12 1
1014.2.i.e 4 65.f even 4 1
1014.2.i.e 4 65.k even 4 1
1014.2.i.e 4 65.o even 12 1
1014.2.i.e 4 65.t even 12 1
1872.2.t.i 2 60.l odd 4 1
1872.2.t.i 2 780.cj odd 12 1
1950.2.i.b 2 5.c odd 4 1
1950.2.i.b 2 65.q odd 12 1
1950.2.z.b 4 1.a even 1 1 trivial
1950.2.z.b 4 5.b even 2 1 inner
1950.2.z.b 4 13.c even 3 1 inner
1950.2.z.b 4 65.n even 6 1 inner
3042.2.a.d 1 195.bf even 12 1
3042.2.a.m 1 195.bl even 12 1
3042.2.b.d 2 195.bc odd 12 1
3042.2.b.d 2 195.bn odd 12 1
8112.2.a.x 1 260.bj even 12 1
8112.2.a.bb 1 260.bg even 12 1

## Hecke kernels

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

 $$T_{7}^{4} - 4T_{7}^{2} + 16$$ T7^4 - 4*T7^2 + 16 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{17}^{4} - 25T_{17}^{2} + 625$$ T17^4 - 25*T17^2 + 625

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$T^{4} - T^{2} + 169$$
$17$ $$T^{4} - 25T^{2} + 625$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} + 9 T + 81)^{2}$$
$31$ $$(T + 4)^{4}$$
$37$ $$T^{4} - 121 T^{2} + 14641$$
$41$ $$(T^{2} + 5 T + 25)^{2}$$
$43$ $$T^{4} - 100 T^{2} + 10000$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$(T^{2} + 1)^{2}$$
$59$ $$(T^{2} + 8 T + 64)^{2}$$
$61$ $$(T^{2} - 11 T + 121)^{2}$$
$67$ $$T^{4} - 4T^{2} + 16$$
$71$ $$(T^{2} - 14 T + 196)^{2}$$
$73$ $$(T^{2} + 169)^{2}$$
$79$ $$(T - 4)^{4}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} - 2 T + 4)^{2}$$
$97$ $$T^{4} - 4T^{2} + 16$$