# Properties

 Label 1690.4.a.a Level $1690$ Weight $4$ Character orbit 1690.a Self dual yes Analytic conductor $99.713$ 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] = [1690,4,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, 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("1690.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1690 = 2 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1690.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: $$99.7132279097$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 8 q^{3} + 4 q^{4} - 5 q^{5} + 16 q^{6} + 4 q^{7} - 8 q^{8} + 37 q^{9}+O(q^{10})$$ q - 2 * q^2 - 8 * q^3 + 4 * q^4 - 5 * q^5 + 16 * q^6 + 4 * q^7 - 8 * q^8 + 37 * q^9 $$q - 2 q^{2} - 8 q^{3} + 4 q^{4} - 5 q^{5} + 16 q^{6} + 4 q^{7} - 8 q^{8} + 37 q^{9} + 10 q^{10} - 12 q^{11} - 32 q^{12} - 8 q^{14} + 40 q^{15} + 16 q^{16} + 66 q^{17} - 74 q^{18} + 100 q^{19} - 20 q^{20} - 32 q^{21} + 24 q^{22} + 132 q^{23} + 64 q^{24} + 25 q^{25} - 80 q^{27} + 16 q^{28} - 90 q^{29} - 80 q^{30} - 152 q^{31} - 32 q^{32} + 96 q^{33} - 132 q^{34} - 20 q^{35} + 148 q^{36} + 34 q^{37} - 200 q^{38} + 40 q^{40} + 438 q^{41} + 64 q^{42} + 32 q^{43} - 48 q^{44} - 185 q^{45} - 264 q^{46} + 204 q^{47} - 128 q^{48} - 327 q^{49} - 50 q^{50} - 528 q^{51} + 222 q^{53} + 160 q^{54} + 60 q^{55} - 32 q^{56} - 800 q^{57} + 180 q^{58} - 420 q^{59} + 160 q^{60} + 902 q^{61} + 304 q^{62} + 148 q^{63} + 64 q^{64} - 192 q^{66} + 1024 q^{67} + 264 q^{68} - 1056 q^{69} + 40 q^{70} - 432 q^{71} - 296 q^{72} - 362 q^{73} - 68 q^{74} - 200 q^{75} + 400 q^{76} - 48 q^{77} - 160 q^{79} - 80 q^{80} - 359 q^{81} - 876 q^{82} - 72 q^{83} - 128 q^{84} - 330 q^{85} - 64 q^{86} + 720 q^{87} + 96 q^{88} - 810 q^{89} + 370 q^{90} + 528 q^{92} + 1216 q^{93} - 408 q^{94} - 500 q^{95} + 256 q^{96} - 1106 q^{97} + 654 q^{98} - 444 q^{99}+O(q^{100})$$ q - 2 * q^2 - 8 * q^3 + 4 * q^4 - 5 * q^5 + 16 * q^6 + 4 * q^7 - 8 * q^8 + 37 * q^9 + 10 * q^10 - 12 * q^11 - 32 * q^12 - 8 * q^14 + 40 * q^15 + 16 * q^16 + 66 * q^17 - 74 * q^18 + 100 * q^19 - 20 * q^20 - 32 * q^21 + 24 * q^22 + 132 * q^23 + 64 * q^24 + 25 * q^25 - 80 * q^27 + 16 * q^28 - 90 * q^29 - 80 * q^30 - 152 * q^31 - 32 * q^32 + 96 * q^33 - 132 * q^34 - 20 * q^35 + 148 * q^36 + 34 * q^37 - 200 * q^38 + 40 * q^40 + 438 * q^41 + 64 * q^42 + 32 * q^43 - 48 * q^44 - 185 * q^45 - 264 * q^46 + 204 * q^47 - 128 * q^48 - 327 * q^49 - 50 * q^50 - 528 * q^51 + 222 * q^53 + 160 * q^54 + 60 * q^55 - 32 * q^56 - 800 * q^57 + 180 * q^58 - 420 * q^59 + 160 * q^60 + 902 * q^61 + 304 * q^62 + 148 * q^63 + 64 * q^64 - 192 * q^66 + 1024 * q^67 + 264 * q^68 - 1056 * q^69 + 40 * q^70 - 432 * q^71 - 296 * q^72 - 362 * q^73 - 68 * q^74 - 200 * q^75 + 400 * q^76 - 48 * q^77 - 160 * q^79 - 80 * q^80 - 359 * q^81 - 876 * q^82 - 72 * q^83 - 128 * q^84 - 330 * q^85 - 64 * q^86 + 720 * q^87 + 96 * q^88 - 810 * q^89 + 370 * q^90 + 528 * q^92 + 1216 * q^93 - 408 * q^94 - 500 * q^95 + 256 * q^96 - 1106 * q^97 + 654 * q^98 - 444 * 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 −8.00000 4.00000 −5.00000 16.0000 4.00000 −8.00000 37.0000 10.0000
 $$n$$: 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 1690.4.a.a 1
13.b even 2 1 10.4.a.a 1
39.d odd 2 1 90.4.a.a 1
52.b odd 2 1 80.4.a.f 1
65.d even 2 1 50.4.a.c 1
65.h odd 4 2 50.4.b.a 2
91.b odd 2 1 490.4.a.o 1
91.r even 6 2 490.4.e.i 2
91.s odd 6 2 490.4.e.a 2
104.e even 2 1 320.4.a.m 1
104.h odd 2 1 320.4.a.b 1
117.n odd 6 2 810.4.e.w 2
117.t even 6 2 810.4.e.c 2
143.d odd 2 1 1210.4.a.b 1
156.h even 2 1 720.4.a.j 1
195.e odd 2 1 450.4.a.q 1
195.s even 4 2 450.4.c.d 2
208.o odd 4 2 1280.4.d.g 2
208.p even 4 2 1280.4.d.j 2
260.g odd 2 1 400.4.a.b 1
260.p even 4 2 400.4.c.c 2
455.h odd 2 1 2450.4.a.b 1
520.b odd 2 1 1600.4.a.bx 1
520.p even 2 1 1600.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 13.b even 2 1
50.4.a.c 1 65.d even 2 1
50.4.b.a 2 65.h odd 4 2
80.4.a.f 1 52.b odd 2 1
90.4.a.a 1 39.d odd 2 1
320.4.a.b 1 104.h odd 2 1
320.4.a.m 1 104.e even 2 1
400.4.a.b 1 260.g odd 2 1
400.4.c.c 2 260.p even 4 2
450.4.a.q 1 195.e odd 2 1
450.4.c.d 2 195.s even 4 2
490.4.a.o 1 91.b odd 2 1
490.4.e.a 2 91.s odd 6 2
490.4.e.i 2 91.r even 6 2
720.4.a.j 1 156.h even 2 1
810.4.e.c 2 117.t even 6 2
810.4.e.w 2 117.n odd 6 2
1210.4.a.b 1 143.d odd 2 1
1280.4.d.g 2 208.o odd 4 2
1280.4.d.j 2 208.p even 4 2
1600.4.a.d 1 520.p even 2 1
1600.4.a.bx 1 520.b odd 2 1
1690.4.a.a 1 1.a even 1 1 trivial
2450.4.a.b 1 455.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1690))$$:

 $$T_{3} + 8$$ T3 + 8 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 8$$
$5$ $$T + 5$$
$7$ $$T - 4$$
$11$ $$T + 12$$
$13$ $$T$$
$17$ $$T - 66$$
$19$ $$T - 100$$
$23$ $$T - 132$$
$29$ $$T + 90$$
$31$ $$T + 152$$
$37$ $$T - 34$$
$41$ $$T - 438$$
$43$ $$T - 32$$
$47$ $$T - 204$$
$53$ $$T - 222$$
$59$ $$T + 420$$
$61$ $$T - 902$$
$67$ $$T - 1024$$
$71$ $$T + 432$$
$73$ $$T + 362$$
$79$ $$T + 160$$
$83$ $$T + 72$$
$89$ $$T + 810$$
$97$ $$T + 1106$$