# Properties

 Label 2600.1.o.d Level $2600$ Weight $1$ Character orbit 2600.o Self dual yes Analytic conductor $1.298$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -104 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2600 = 2^{3} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2600.o (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.29756903285$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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: $D_6$ Artin field: Galois closure of 6.0.10816000.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 - q^7 + q^8 $$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{12} + q^{13} - q^{14} + q^{16} + q^{17} - q^{21} + q^{24} + q^{26} - q^{27} - q^{28} - 2 q^{31} + q^{32} + q^{34} - q^{37} + q^{39} - q^{42} + q^{43} - q^{47} + q^{48} + q^{51} + q^{52} - q^{54} - q^{56} - 2 q^{62} + q^{64} + q^{68} + q^{71} - q^{74} + q^{78} - q^{81} - q^{84} + q^{86} - q^{91} - 2 q^{93} - q^{94} + q^{96}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - q^7 + q^8 + q^12 + q^13 - q^14 + q^16 + q^17 - q^21 + q^24 + q^26 - q^27 - q^28 - 2 * q^31 + q^32 + q^34 - q^37 + q^39 - q^42 + q^43 - q^47 + q^48 + q^51 + q^52 - q^54 - q^56 - 2 * q^62 + q^64 + q^68 + q^71 - q^74 + q^78 - q^81 - q^84 + q^86 - q^91 - 2 * q^93 - q^94 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0
1.00000 1.00000 1.00000 0 1.00000 −1.00000 1.00000 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 $$\Q(\sqrt{-26})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.1.o.d 1
5.b even 2 1 104.1.h.a 1
5.c odd 4 2 2600.1.b.b 2
8.d odd 2 1 2600.1.o.b 1
13.b even 2 1 2600.1.o.b 1
15.d odd 2 1 936.1.o.b 1
20.d odd 2 1 416.1.h.b 1
40.e odd 2 1 104.1.h.b yes 1
40.f even 2 1 416.1.h.a 1
40.k even 4 2 2600.1.b.a 2
60.h even 2 1 3744.1.o.a 1
65.d even 2 1 104.1.h.b yes 1
65.g odd 4 2 1352.1.g.a 2
65.h odd 4 2 2600.1.b.a 2
65.l even 6 2 1352.1.p.a 2
65.n even 6 2 1352.1.p.b 2
65.s odd 12 4 1352.1.n.a 4
80.k odd 4 2 3328.1.c.e 2
80.q even 4 2 3328.1.c.a 2
104.h odd 2 1 CM 2600.1.o.d 1
120.i odd 2 1 3744.1.o.b 1
120.m even 2 1 936.1.o.a 1
195.e odd 2 1 936.1.o.a 1
260.g odd 2 1 416.1.h.a 1
520.b odd 2 1 104.1.h.a 1
520.p even 2 1 416.1.h.b 1
520.t even 4 2 1352.1.g.a 2
520.bc even 4 2 2600.1.b.b 2
520.bx odd 6 2 1352.1.p.a 2
520.cd odd 6 2 1352.1.p.b 2
520.cz even 12 4 1352.1.n.a 4
780.d even 2 1 3744.1.o.b 1
1040.be even 4 2 3328.1.c.e 2
1040.cb odd 4 2 3328.1.c.a 2
1560.n even 2 1 936.1.o.b 1
1560.y odd 2 1 3744.1.o.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.1.h.a 1 5.b even 2 1
104.1.h.a 1 520.b odd 2 1
104.1.h.b yes 1 40.e odd 2 1
104.1.h.b yes 1 65.d even 2 1
416.1.h.a 1 40.f even 2 1
416.1.h.a 1 260.g odd 2 1
416.1.h.b 1 20.d odd 2 1
416.1.h.b 1 520.p even 2 1
936.1.o.a 1 120.m even 2 1
936.1.o.a 1 195.e odd 2 1
936.1.o.b 1 15.d odd 2 1
936.1.o.b 1 1560.n even 2 1
1352.1.g.a 2 65.g odd 4 2
1352.1.g.a 2 520.t even 4 2
1352.1.n.a 4 65.s odd 12 4
1352.1.n.a 4 520.cz even 12 4
1352.1.p.a 2 65.l even 6 2
1352.1.p.a 2 520.bx odd 6 2
1352.1.p.b 2 65.n even 6 2
1352.1.p.b 2 520.cd odd 6 2
2600.1.b.a 2 40.k even 4 2
2600.1.b.a 2 65.h odd 4 2
2600.1.b.b 2 5.c odd 4 2
2600.1.b.b 2 520.bc even 4 2
2600.1.o.b 1 8.d odd 2 1
2600.1.o.b 1 13.b even 2 1
2600.1.o.d 1 1.a even 1 1 trivial
2600.1.o.d 1 104.h odd 2 1 CM
3328.1.c.a 2 80.q even 4 2
3328.1.c.a 2 1040.cb odd 4 2
3328.1.c.e 2 80.k odd 4 2
3328.1.c.e 2 1040.be even 4 2
3744.1.o.a 1 60.h even 2 1
3744.1.o.a 1 1560.y odd 2 1
3744.1.o.b 1 120.i odd 2 1
3744.1.o.b 1 780.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2600, [\chi])$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

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