# Properties

 Label 1950.4.a.h Level $1950$ Weight $4$ Character orbit 1950.a Self dual yes Analytic conductor $115.054$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,4,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1950.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1950.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: $$115.053724511$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 6 q^{6} + 32 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 6 * q^6 + 32 * q^7 - 8 * q^8 + 9 * q^9 $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} - 6 q^{6} + 32 q^{7} - 8 q^{8} + 9 q^{9} + 50 q^{11} + 12 q^{12} + 13 q^{13} - 64 q^{14} + 16 q^{16} + 30 q^{17} - 18 q^{18} - 120 q^{19} + 96 q^{21} - 100 q^{22} + 20 q^{23} - 24 q^{24} - 26 q^{26} + 27 q^{27} + 128 q^{28} + 82 q^{29} - 44 q^{31} - 32 q^{32} + 150 q^{33} - 60 q^{34} + 36 q^{36} + 306 q^{37} + 240 q^{38} + 39 q^{39} + 108 q^{41} - 192 q^{42} + 356 q^{43} + 200 q^{44} - 40 q^{46} + 178 q^{47} + 48 q^{48} + 681 q^{49} + 90 q^{51} + 52 q^{52} - 198 q^{53} - 54 q^{54} - 256 q^{56} - 360 q^{57} - 164 q^{58} + 94 q^{59} - 62 q^{61} + 88 q^{62} + 288 q^{63} + 64 q^{64} - 300 q^{66} + 140 q^{67} + 120 q^{68} + 60 q^{69} - 778 q^{71} - 72 q^{72} - 62 q^{73} - 612 q^{74} - 480 q^{76} + 1600 q^{77} - 78 q^{78} - 1096 q^{79} + 81 q^{81} - 216 q^{82} + 462 q^{83} + 384 q^{84} - 712 q^{86} + 246 q^{87} - 400 q^{88} + 1224 q^{89} + 416 q^{91} + 80 q^{92} - 132 q^{93} - 356 q^{94} - 96 q^{96} - 614 q^{97} - 1362 q^{98} + 450 q^{99}+O(q^{100})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 6 * q^6 + 32 * q^7 - 8 * q^8 + 9 * q^9 + 50 * q^11 + 12 * q^12 + 13 * q^13 - 64 * q^14 + 16 * q^16 + 30 * q^17 - 18 * q^18 - 120 * q^19 + 96 * q^21 - 100 * q^22 + 20 * q^23 - 24 * q^24 - 26 * q^26 + 27 * q^27 + 128 * q^28 + 82 * q^29 - 44 * q^31 - 32 * q^32 + 150 * q^33 - 60 * q^34 + 36 * q^36 + 306 * q^37 + 240 * q^38 + 39 * q^39 + 108 * q^41 - 192 * q^42 + 356 * q^43 + 200 * q^44 - 40 * q^46 + 178 * q^47 + 48 * q^48 + 681 * q^49 + 90 * q^51 + 52 * q^52 - 198 * q^53 - 54 * q^54 - 256 * q^56 - 360 * q^57 - 164 * q^58 + 94 * q^59 - 62 * q^61 + 88 * q^62 + 288 * q^63 + 64 * q^64 - 300 * q^66 + 140 * q^67 + 120 * q^68 + 60 * q^69 - 778 * q^71 - 72 * q^72 - 62 * q^73 - 612 * q^74 - 480 * q^76 + 1600 * q^77 - 78 * q^78 - 1096 * q^79 + 81 * q^81 - 216 * q^82 + 462 * q^83 + 384 * q^84 - 712 * q^86 + 246 * q^87 - 400 * q^88 + 1224 * q^89 + 416 * q^91 + 80 * q^92 - 132 * q^93 - 356 * q^94 - 96 * q^96 - 614 * q^97 - 1362 * q^98 + 450 * 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
 0
−2.00000 3.00000 4.00000 0 −6.00000 32.0000 −8.00000 9.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$13$$ $$-1$$

## 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 1950.4.a.h 1
5.b even 2 1 78.4.a.d 1
15.d odd 2 1 234.4.a.f 1
20.d odd 2 1 624.4.a.e 1
40.e odd 2 1 2496.4.a.i 1
40.f even 2 1 2496.4.a.r 1
60.h even 2 1 1872.4.a.r 1
65.d even 2 1 1014.4.a.d 1
65.g odd 4 2 1014.4.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.d 1 5.b even 2 1
234.4.a.f 1 15.d odd 2 1
624.4.a.e 1 20.d odd 2 1
1014.4.a.d 1 65.d even 2 1
1014.4.b.e 2 65.g odd 4 2
1872.4.a.r 1 60.h even 2 1
1950.4.a.h 1 1.a even 1 1 trivial
2496.4.a.i 1 40.e odd 2 1
2496.4.a.r 1 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7} - 32$$ T7 - 32 $$T_{11} - 50$$ T11 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 3$$
$5$ $$T$$
$7$ $$T - 32$$
$11$ $$T - 50$$
$13$ $$T - 13$$
$17$ $$T - 30$$
$19$ $$T + 120$$
$23$ $$T - 20$$
$29$ $$T - 82$$
$31$ $$T + 44$$
$37$ $$T - 306$$
$41$ $$T - 108$$
$43$ $$T - 356$$
$47$ $$T - 178$$
$53$ $$T + 198$$
$59$ $$T - 94$$
$61$ $$T + 62$$
$67$ $$T - 140$$
$71$ $$T + 778$$
$73$ $$T + 62$$
$79$ $$T + 1096$$
$83$ $$T - 462$$
$89$ $$T - 1224$$
$97$ $$T + 614$$