# Properties

 Label 650.2.b.a Level $650$ Weight $2$ Character orbit 650.b Analytic conductor $5.190$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(599,650)]

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([1, 0]))

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

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 650.b (of order $$2$$, degree $$1$$, 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: $$5.19027613138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
599.1
 − 1.00000i 1.00000i
1.00000i 3.00000i −1.00000 0 −3.00000 1.00000i 1.00000i −6.00000 0
599.2 1.00000i 3.00000i −1.00000 0 −3.00000 1.00000i 1.00000i −6.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.b.a 2
3.b odd 2 1 5850.2.e.v 2
5.b even 2 1 inner 650.2.b.a 2
5.c odd 4 1 26.2.a.b 1
5.c odd 4 1 650.2.a.g 1
15.d odd 2 1 5850.2.e.v 2
15.e even 4 1 234.2.a.b 1
15.e even 4 1 5850.2.a.bn 1
20.e even 4 1 208.2.a.d 1
20.e even 4 1 5200.2.a.c 1
35.f even 4 1 1274.2.a.o 1
35.k even 12 2 1274.2.f.a 2
35.l odd 12 2 1274.2.f.l 2
40.i odd 4 1 832.2.a.j 1
40.k even 4 1 832.2.a.a 1
45.k odd 12 2 2106.2.e.h 2
45.l even 12 2 2106.2.e.t 2
55.e even 4 1 3146.2.a.a 1
60.l odd 4 1 1872.2.a.m 1
65.f even 4 1 338.2.b.a 2
65.h odd 4 1 338.2.a.a 1
65.h odd 4 1 8450.2.a.y 1
65.k even 4 1 338.2.b.a 2
65.o even 12 2 338.2.e.d 4
65.q odd 12 2 338.2.c.c 2
65.r odd 12 2 338.2.c.g 2
65.t even 12 2 338.2.e.d 4
80.i odd 4 1 3328.2.b.g 2
80.j even 4 1 3328.2.b.k 2
80.s even 4 1 3328.2.b.k 2
80.t odd 4 1 3328.2.b.g 2
85.g odd 4 1 7514.2.a.i 1
95.g even 4 1 9386.2.a.f 1
120.q odd 4 1 7488.2.a.v 1
120.w even 4 1 7488.2.a.w 1
195.j odd 4 1 3042.2.b.f 2
195.s even 4 1 3042.2.a.l 1
195.u odd 4 1 3042.2.b.f 2
260.l odd 4 1 2704.2.f.j 2
260.p even 4 1 2704.2.a.n 1
260.s odd 4 1 2704.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 5.c odd 4 1
208.2.a.d 1 20.e even 4 1
234.2.a.b 1 15.e even 4 1
338.2.a.a 1 65.h odd 4 1
338.2.b.a 2 65.f even 4 1
338.2.b.a 2 65.k even 4 1
338.2.c.c 2 65.q odd 12 2
338.2.c.g 2 65.r odd 12 2
338.2.e.d 4 65.o even 12 2
338.2.e.d 4 65.t even 12 2
650.2.a.g 1 5.c odd 4 1
650.2.b.a 2 1.a even 1 1 trivial
650.2.b.a 2 5.b even 2 1 inner
832.2.a.a 1 40.k even 4 1
832.2.a.j 1 40.i odd 4 1
1274.2.a.o 1 35.f even 4 1
1274.2.f.a 2 35.k even 12 2
1274.2.f.l 2 35.l odd 12 2
1872.2.a.m 1 60.l odd 4 1
2106.2.e.h 2 45.k odd 12 2
2106.2.e.t 2 45.l even 12 2
2704.2.a.n 1 260.p even 4 1
2704.2.f.j 2 260.l odd 4 1
2704.2.f.j 2 260.s odd 4 1
3042.2.a.l 1 195.s even 4 1
3042.2.b.f 2 195.j odd 4 1
3042.2.b.f 2 195.u odd 4 1
3146.2.a.a 1 55.e even 4 1
3328.2.b.g 2 80.i odd 4 1
3328.2.b.g 2 80.t odd 4 1
3328.2.b.k 2 80.j even 4 1
3328.2.b.k 2 80.s even 4 1
5200.2.a.c 1 20.e even 4 1
5850.2.a.bn 1 15.e even 4 1
5850.2.e.v 2 3.b odd 2 1
5850.2.e.v 2 15.d odd 2 1
7488.2.a.v 1 120.q odd 4 1
7488.2.a.w 1 120.w even 4 1
7514.2.a.i 1 85.g odd 4 1
8450.2.a.y 1 65.h odd 4 1
9386.2.a.f 1 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(650, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2} + 169$$
$53$ $$T^{2} + 144$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 196$$
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