# Properties

 Label 650.4.a.b Level $650$ Weight $4$ Character orbit 650.a Self dual yes Analytic conductor $38.351$ 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] = [650,4,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("650.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 650.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: $$38.3512415037$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 8 q^{6} - 20 q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10})$$ q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 8 * q^6 - 20 * q^7 - 8 * q^8 - 11 * q^9 $$q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 8 q^{6} - 20 q^{7} - 8 q^{8} - 11 q^{9} - 48 q^{11} - 16 q^{12} - 13 q^{13} + 40 q^{14} + 16 q^{16} - 66 q^{17} + 22 q^{18} - 16 q^{19} + 80 q^{21} + 96 q^{22} - 168 q^{23} + 32 q^{24} + 26 q^{26} + 152 q^{27} - 80 q^{28} + 6 q^{29} + 20 q^{31} - 32 q^{32} + 192 q^{33} + 132 q^{34} - 44 q^{36} - 254 q^{37} + 32 q^{38} + 52 q^{39} - 390 q^{41} - 160 q^{42} + 124 q^{43} - 192 q^{44} + 336 q^{46} + 468 q^{47} - 64 q^{48} + 57 q^{49} + 264 q^{51} - 52 q^{52} - 558 q^{53} - 304 q^{54} + 160 q^{56} + 64 q^{57} - 12 q^{58} - 96 q^{59} - 826 q^{61} - 40 q^{62} + 220 q^{63} + 64 q^{64} - 384 q^{66} + 160 q^{67} - 264 q^{68} + 672 q^{69} - 420 q^{71} + 88 q^{72} - 362 q^{73} + 508 q^{74} - 64 q^{76} + 960 q^{77} - 104 q^{78} + 776 q^{79} - 311 q^{81} + 780 q^{82} + 320 q^{84} - 248 q^{86} - 24 q^{87} + 384 q^{88} + 1626 q^{89} + 260 q^{91} - 672 q^{92} - 80 q^{93} - 936 q^{94} + 128 q^{96} + 1294 q^{97} - 114 q^{98} + 528 q^{99}+O(q^{100})$$ q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 8 * q^6 - 20 * q^7 - 8 * q^8 - 11 * q^9 - 48 * q^11 - 16 * q^12 - 13 * q^13 + 40 * q^14 + 16 * q^16 - 66 * q^17 + 22 * q^18 - 16 * q^19 + 80 * q^21 + 96 * q^22 - 168 * q^23 + 32 * q^24 + 26 * q^26 + 152 * q^27 - 80 * q^28 + 6 * q^29 + 20 * q^31 - 32 * q^32 + 192 * q^33 + 132 * q^34 - 44 * q^36 - 254 * q^37 + 32 * q^38 + 52 * q^39 - 390 * q^41 - 160 * q^42 + 124 * q^43 - 192 * q^44 + 336 * q^46 + 468 * q^47 - 64 * q^48 + 57 * q^49 + 264 * q^51 - 52 * q^52 - 558 * q^53 - 304 * q^54 + 160 * q^56 + 64 * q^57 - 12 * q^58 - 96 * q^59 - 826 * q^61 - 40 * q^62 + 220 * q^63 + 64 * q^64 - 384 * q^66 + 160 * q^67 - 264 * q^68 + 672 * q^69 - 420 * q^71 + 88 * q^72 - 362 * q^73 + 508 * q^74 - 64 * q^76 + 960 * q^77 - 104 * q^78 + 776 * q^79 - 311 * q^81 + 780 * q^82 + 320 * q^84 - 248 * q^86 - 24 * q^87 + 384 * q^88 + 1626 * q^89 + 260 * q^91 - 672 * q^92 - 80 * q^93 - 936 * q^94 + 128 * q^96 + 1294 * q^97 - 114 * q^98 + 528 * 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 −4.00000 4.00000 0 8.00000 −20.0000 −8.00000 −11.0000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$13$$ $$+1$$

## 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 650.4.a.b 1
5.b even 2 1 26.4.a.c 1
5.c odd 4 2 650.4.b.f 2
15.d odd 2 1 234.4.a.e 1
20.d odd 2 1 208.4.a.b 1
35.c odd 2 1 1274.4.a.d 1
40.e odd 2 1 832.4.a.o 1
40.f even 2 1 832.4.a.d 1
60.h even 2 1 1872.4.a.q 1
65.d even 2 1 338.4.a.c 1
65.g odd 4 2 338.4.b.d 2
65.l even 6 2 338.4.c.e 2
65.n even 6 2 338.4.c.a 2
65.s odd 12 4 338.4.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 5.b even 2 1
208.4.a.b 1 20.d odd 2 1
234.4.a.e 1 15.d odd 2 1
338.4.a.c 1 65.d even 2 1
338.4.b.d 2 65.g odd 4 2
338.4.c.a 2 65.n even 6 2
338.4.c.e 2 65.l even 6 2
338.4.e.a 4 65.s odd 12 4
650.4.a.b 1 1.a even 1 1 trivial
650.4.b.f 2 5.c odd 4 2
832.4.a.d 1 40.f even 2 1
832.4.a.o 1 40.e odd 2 1
1274.4.a.d 1 35.c odd 2 1
1872.4.a.q 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(650))$$:

 $$T_{3} + 4$$ T3 + 4 $$T_{7} + 20$$ T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 4$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T + 48$$
$13$ $$T + 13$$
$17$ $$T + 66$$
$19$ $$T + 16$$
$23$ $$T + 168$$
$29$ $$T - 6$$
$31$ $$T - 20$$
$37$ $$T + 254$$
$41$ $$T + 390$$
$43$ $$T - 124$$
$47$ $$T - 468$$
$53$ $$T + 558$$
$59$ $$T + 96$$
$61$ $$T + 826$$
$67$ $$T - 160$$
$71$ $$T + 420$$
$73$ $$T + 362$$
$79$ $$T - 776$$
$83$ $$T$$
$89$ $$T - 1626$$
$97$ $$T - 1294$$