# Properties

 Label 2600.1.b.b 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: $$x^{2} + 1$$ 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.

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

## 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.b 2
5.b even 2 1 inner 2600.1.b.b 2
5.c odd 4 1 104.1.h.a 1
5.c odd 4 1 2600.1.o.d 1
8.d odd 2 1 2600.1.b.a 2
13.b even 2 1 2600.1.b.a 2
15.e even 4 1 936.1.o.b 1
20.e even 4 1 416.1.h.b 1
40.e odd 2 1 2600.1.b.a 2
40.i odd 4 1 416.1.h.a 1
40.k even 4 1 104.1.h.b yes 1
40.k even 4 1 2600.1.o.b 1
60.l odd 4 1 3744.1.o.a 1
65.d even 2 1 2600.1.b.a 2
65.f even 4 1 1352.1.g.a 2
65.h odd 4 1 104.1.h.b yes 1
65.h odd 4 1 2600.1.o.b 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.b 2
65.r odd 12 2 1352.1.p.a 2
65.t even 12 2 1352.1.n.a 4
80.i odd 4 1 3328.1.c.a 2
80.j even 4 1 3328.1.c.e 2
80.s even 4 1 3328.1.c.e 2
80.t odd 4 1 3328.1.c.a 2
104.h odd 2 1 CM 2600.1.b.b 2
120.q odd 4 1 936.1.o.a 1
120.w even 4 1 3744.1.o.b 1
195.s even 4 1 936.1.o.a 1
260.p even 4 1 416.1.h.a 1
520.b odd 2 1 inner 2600.1.b.b 2
520.x odd 4 1 1352.1.g.a 2
520.bc even 4 1 104.1.h.a 1
520.bc even 4 1 2600.1.o.d 1
520.bg odd 4 1 416.1.h.b 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.a 2
520.cs even 12 2 1352.1.p.b 2
520.cv odd 12 2 1352.1.n.a 4
780.w odd 4 1 3744.1.o.b 1
1040.w odd 4 1 3328.1.c.e 2
1040.y even 4 1 3328.1.c.a 2
1040.co odd 4 1 3328.1.c.e 2
1040.cq even 4 1 3328.1.c.a 2
1560.bq even 4 1 3744.1.o.a 1
1560.cs odd 4 1 936.1.o.b 1

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

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