# Properties

 Label 1300.2.c.c Level $1300$ Weight $2$ Character orbit 1300.c Analytic conductor $10.381$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{7} + 3 q^{9}+O(q^{10})$$ q + 2*i * q^7 + 3 * q^9 $$q + 2 i q^{7} + 3 q^{9} - 2 q^{11} - i q^{13} - 6 i q^{17} + 6 q^{19} + 8 i q^{23} - 2 q^{29} + 10 q^{31} + 6 i q^{37} - 6 q^{41} + 4 i q^{43} + 2 i q^{47} + 3 q^{49} + 6 i q^{53} + 10 q^{59} - 2 q^{61} + 6 i q^{63} - 10 i q^{67} + 10 q^{71} + 2 i q^{73} - 4 i q^{77} + 4 q^{79} + 9 q^{81} - 6 i q^{83} + 6 q^{89} + 2 q^{91} - 2 i q^{97} - 6 q^{99} +O(q^{100})$$ q + 2*i * q^7 + 3 * q^9 - 2 * q^11 - i * q^13 - 6*i * q^17 + 6 * q^19 + 8*i * q^23 - 2 * q^29 + 10 * q^31 + 6*i * q^37 - 6 * q^41 + 4*i * q^43 + 2*i * q^47 + 3 * q^49 + 6*i * q^53 + 10 * q^59 - 2 * q^61 + 6*i * q^63 - 10*i * q^67 + 10 * q^71 + 2*i * q^73 - 4*i * q^77 + 4 * q^79 + 9 * q^81 - 6*i * q^83 + 6 * q^89 + 2 * q^91 - 2*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9}+O(q^{10})$$ 2 * q + 6 * q^9 $$2 q + 6 q^{9} - 4 q^{11} + 12 q^{19} - 4 q^{29} + 20 q^{31} - 12 q^{41} + 6 q^{49} + 20 q^{59} - 4 q^{61} + 20 q^{71} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{91} - 12 q^{99}+O(q^{100})$$ 2 * q + 6 * q^9 - 4 * q^11 + 12 * q^19 - 4 * q^29 + 20 * q^31 - 12 * q^41 + 6 * q^49 + 20 * q^59 - 4 * q^61 + 20 * q^71 + 8 * q^79 + 18 * q^81 + 12 * q^89 + 4 * q^91 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 3.00000 0
1249.2 0 0 0 0 0 2.00000i 0 3.00000 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 1300.2.c.c 2
5.b even 2 1 inner 1300.2.c.c 2
5.c odd 4 1 52.2.a.a 1
5.c odd 4 1 1300.2.a.d 1
15.e even 4 1 468.2.a.b 1
20.e even 4 1 208.2.a.c 1
20.e even 4 1 5200.2.a.q 1
35.f even 4 1 2548.2.a.e 1
35.k even 12 2 2548.2.j.f 2
35.l odd 12 2 2548.2.j.e 2
40.i odd 4 1 832.2.a.e 1
40.k even 4 1 832.2.a.f 1
45.k odd 12 2 4212.2.i.d 2
45.l even 12 2 4212.2.i.i 2
55.e even 4 1 6292.2.a.g 1
60.l odd 4 1 1872.2.a.f 1
65.f even 4 1 676.2.d.c 2
65.h odd 4 1 676.2.a.c 1
65.k even 4 1 676.2.d.c 2
65.o even 12 2 676.2.h.c 4
65.q odd 12 2 676.2.e.c 2
65.r odd 12 2 676.2.e.b 2
65.t even 12 2 676.2.h.c 4
80.i odd 4 1 3328.2.b.q 2
80.j even 4 1 3328.2.b.e 2
80.s even 4 1 3328.2.b.e 2
80.t odd 4 1 3328.2.b.q 2
120.q odd 4 1 7488.2.a.bw 1
120.w even 4 1 7488.2.a.bn 1
195.j odd 4 1 6084.2.b.m 2
195.s even 4 1 6084.2.a.m 1
195.u odd 4 1 6084.2.b.m 2
260.l odd 4 1 2704.2.f.f 2
260.p even 4 1 2704.2.a.g 1
260.s odd 4 1 2704.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 5.c odd 4 1
208.2.a.c 1 20.e even 4 1
468.2.a.b 1 15.e even 4 1
676.2.a.c 1 65.h odd 4 1
676.2.d.c 2 65.f even 4 1
676.2.d.c 2 65.k even 4 1
676.2.e.b 2 65.r odd 12 2
676.2.e.c 2 65.q odd 12 2
676.2.h.c 4 65.o even 12 2
676.2.h.c 4 65.t even 12 2
832.2.a.e 1 40.i odd 4 1
832.2.a.f 1 40.k even 4 1
1300.2.a.d 1 5.c odd 4 1
1300.2.c.c 2 1.a even 1 1 trivial
1300.2.c.c 2 5.b even 2 1 inner
1872.2.a.f 1 60.l odd 4 1
2548.2.a.e 1 35.f even 4 1
2548.2.j.e 2 35.l odd 12 2
2548.2.j.f 2 35.k even 12 2
2704.2.a.g 1 260.p even 4 1
2704.2.f.f 2 260.l odd 4 1
2704.2.f.f 2 260.s odd 4 1
3328.2.b.e 2 80.j even 4 1
3328.2.b.e 2 80.s even 4 1
3328.2.b.q 2 80.i odd 4 1
3328.2.b.q 2 80.t odd 4 1
4212.2.i.d 2 45.k odd 12 2
4212.2.i.i 2 45.l even 12 2
5200.2.a.q 1 20.e even 4 1
6084.2.a.m 1 195.s even 4 1
6084.2.b.m 2 195.j odd 4 1
6084.2.b.m 2 195.u odd 4 1
6292.2.a.g 1 55.e even 4 1
7488.2.a.bn 1 120.w even 4 1
7488.2.a.bw 1 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1300, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4$$