# Properties

 Label 2600.1.b.a Level $2600$ Weight $1$ Character orbit 2600.b Analytic conductor $1.298$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -104 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2600 = 2^{3} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2600.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.29756903285$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ 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 104) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.104.1 Artin image: $C_2\times C_4\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{2} - i q^{3} - q^{4} - q^{6} + i q^{7} + i q^{8} +O(q^{10})$$ q - z * q^2 - z * q^3 - q^4 - q^6 + z * q^7 + z * q^8 $$q - i q^{2} - i q^{3} - q^{4} - q^{6} + i q^{7} + i q^{8} + i q^{12} + i q^{13} + q^{14} + q^{16} + i q^{17} + q^{21} + q^{24} + q^{26} - i q^{27} - i q^{28} + q^{31} - i q^{32} + q^{34} + i q^{37} + q^{39} - i q^{42} - i q^{43} + i q^{47} - i q^{48} + q^{51} - i q^{52} - q^{54} - q^{56} - 2 i q^{62} - q^{64} - i q^{68} - q^{71} + q^{74} - i q^{78} - q^{81} - q^{84} - q^{86} - q^{91} - 2 i q^{93} + q^{94} - q^{96} +O(q^{100})$$ q - z * q^2 - z * q^3 - q^4 - q^6 + z * q^7 + z * q^8 + z * q^12 + z * q^13 + q^14 + q^16 + z * q^17 + q^21 + q^24 + q^26 - z * q^27 - z * q^28 + q^31 - z * q^32 + q^34 + z * q^37 + q^39 - z * q^42 - z * q^43 + z * q^47 - z * q^48 + q^51 - z * q^52 - q^54 - q^56 - 2*z * q^62 - q^64 - z * q^68 - q^71 + q^74 - z * q^78 - q^81 - q^84 - q^86 - q^91 - 2*z * q^93 + q^94 - q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 $$2 q - 2 q^{4} - 2 q^{6} + 2 q^{14} + 2 q^{16} + 2 q^{21} + 2 q^{24} + 2 q^{26} + 4 q^{31} + 2 q^{34} + 2 q^{39} + 2 q^{51} - 2 q^{54} - 2 q^{56} - 2 q^{64} - 2 q^{71} + 2 q^{74} - 2 q^{81} - 2 q^{84} - 2 q^{86} - 2 q^{91} + 2 q^{94} - 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 + 2 * q^14 + 2 * q^16 + 2 * q^21 + 2 * q^24 + 2 * q^26 + 4 * q^31 + 2 * q^34 + 2 * q^39 + 2 * q^51 - 2 * q^54 - 2 * q^56 - 2 * q^64 - 2 * q^71 + 2 * q^74 - 2 * q^81 - 2 * q^84 - 2 * q^86 - 2 * q^91 + 2 * q^94 - 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1301$$ $$1601$$ $$1951$$ $$1977$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1299.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 0 0
1299.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 0 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
104.h odd 2 1 CM by $$\Q(\sqrt{-26})$$
5.b even 2 1 inner
520.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.b.a 2
5.b even 2 1 inner 2600.1.b.a 2
5.c odd 4 1 104.1.h.b yes 1
5.c odd 4 1 2600.1.o.b 1
8.d odd 2 1 2600.1.b.b 2
13.b even 2 1 2600.1.b.b 2
15.e even 4 1 936.1.o.a 1
20.e even 4 1 416.1.h.a 1
40.e odd 2 1 2600.1.b.b 2
40.i odd 4 1 416.1.h.b 1
40.k even 4 1 104.1.h.a 1
40.k even 4 1 2600.1.o.d 1
60.l odd 4 1 3744.1.o.b 1
65.d even 2 1 2600.1.b.b 2
65.f even 4 1 1352.1.g.a 2
65.h odd 4 1 104.1.h.a 1
65.h odd 4 1 2600.1.o.d 1
65.k even 4 1 1352.1.g.a 2
65.o even 12 2 1352.1.n.a 4
65.q odd 12 2 1352.1.p.a 2
65.r odd 12 2 1352.1.p.b 2
65.t even 12 2 1352.1.n.a 4
80.i odd 4 1 3328.1.c.e 2
80.j even 4 1 3328.1.c.a 2
80.s even 4 1 3328.1.c.a 2
80.t odd 4 1 3328.1.c.e 2
104.h odd 2 1 CM 2600.1.b.a 2
120.q odd 4 1 936.1.o.b 1
120.w even 4 1 3744.1.o.a 1
195.s even 4 1 936.1.o.b 1
260.p even 4 1 416.1.h.b 1
520.b odd 2 1 inner 2600.1.b.a 2
520.x odd 4 1 1352.1.g.a 2
520.bc even 4 1 104.1.h.b yes 1
520.bc even 4 1 2600.1.o.b 1
520.bg odd 4 1 416.1.h.a 1
520.bk odd 4 1 1352.1.g.a 2
520.ci odd 12 2 1352.1.n.a 4
520.cm even 12 2 1352.1.p.b 2
520.cs even 12 2 1352.1.p.a 2
520.cv odd 12 2 1352.1.n.a 4
780.w odd 4 1 3744.1.o.a 1
1040.w odd 4 1 3328.1.c.a 2
1040.y even 4 1 3328.1.c.e 2
1040.co odd 4 1 3328.1.c.a 2
1040.cq even 4 1 3328.1.c.e 2
1560.bq even 4 1 3744.1.o.b 1
1560.cs odd 4 1 936.1.o.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 40.k even 4 1
104.1.h.a 1 65.h odd 4 1
104.1.h.b yes 1 5.c odd 4 1
104.1.h.b yes 1 520.bc even 4 1
416.1.h.a 1 20.e even 4 1
416.1.h.a 1 520.bg odd 4 1
416.1.h.b 1 40.i odd 4 1
416.1.h.b 1 260.p even 4 1
936.1.o.a 1 15.e even 4 1
936.1.o.a 1 1560.cs odd 4 1
936.1.o.b 1 120.q odd 4 1
936.1.o.b 1 195.s even 4 1
1352.1.g.a 2 65.f even 4 1
1352.1.g.a 2 65.k even 4 1
1352.1.g.a 2 520.x odd 4 1
1352.1.g.a 2 520.bk odd 4 1
1352.1.n.a 4 65.o even 12 2
1352.1.n.a 4 65.t even 12 2
1352.1.n.a 4 520.ci odd 12 2
1352.1.n.a 4 520.cv odd 12 2
1352.1.p.a 2 65.q odd 12 2
1352.1.p.a 2 520.cs even 12 2
1352.1.p.b 2 65.r odd 12 2
1352.1.p.b 2 520.cm even 12 2
2600.1.b.a 2 1.a even 1 1 trivial
2600.1.b.a 2 5.b even 2 1 inner
2600.1.b.a 2 104.h odd 2 1 CM
2600.1.b.a 2 520.b odd 2 1 inner
2600.1.b.b 2 8.d odd 2 1
2600.1.b.b 2 13.b even 2 1
2600.1.b.b 2 40.e odd 2 1
2600.1.b.b 2 65.d even 2 1
2600.1.o.b 1 5.c odd 4 1
2600.1.o.b 1 520.bc even 4 1
2600.1.o.d 1 40.k even 4 1
2600.1.o.d 1 65.h odd 4 1
3328.1.c.a 2 80.j even 4 1
3328.1.c.a 2 80.s even 4 1
3328.1.c.a 2 1040.w odd 4 1
3328.1.c.a 2 1040.co odd 4 1
3328.1.c.e 2 80.i odd 4 1
3328.1.c.e 2 80.t odd 4 1
3328.1.c.e 2 1040.y even 4 1
3328.1.c.e 2 1040.cq even 4 1
3744.1.o.a 1 120.w even 4 1
3744.1.o.a 1 780.w odd 4 1
3744.1.o.b 1 60.l odd 4 1
3744.1.o.b 1 1560.bq even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{31} - 2$$ acting on $$S_{1}^{\mathrm{new}}(2600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 1$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$(T + 1)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$