# Properties

 Label 1300.2.d.c Level $1300$ Weight $2$ Character orbit 1300.d Analytic conductor $10.381$ Analytic rank $0$ Dimension $6$ 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.9144576.1 Defining polynomial: $$x^{6} + 12 x^{4} + 36 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{7} + ( -1 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{7} + ( -1 + \beta_{3} ) q^{9} + ( \beta_{4} - \beta_{5} ) q^{11} + ( -1 - \beta_{2} - \beta_{5} ) q^{13} + ( -2 \beta_{1} + \beta_{5} ) q^{19} + ( -2 \beta_{1} + \beta_{4} ) q^{21} + ( -\beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{23} + \beta_{4} q^{27} + ( 2 + 2 \beta_{2} + \beta_{3} ) q^{29} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} ) q^{31} + ( -2 + 4 \beta_{2} + \beta_{3} ) q^{33} + ( -4 + 2 \beta_{2} - \beta_{3} ) q^{37} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{39} + ( \beta_{4} + 2 \beta_{5} ) q^{41} + ( 3 \beta_{1} - \beta_{4} ) q^{43} + ( -4 - 4 \beta_{2} - \beta_{3} ) q^{47} + ( 1 + 2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -4 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{53} + ( 10 - 2 \beta_{2} - 3 \beta_{3} ) q^{57} + ( 4 \beta_{4} - \beta_{5} ) q^{59} + ( -2 + 3 \beta_{3} ) q^{61} + ( 8 + 2 \beta_{2} + \beta_{3} ) q^{63} + ( -4 + 2 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 6 \beta_{2} + \beta_{3} ) q^{69} + ( 6 \beta_{1} + 3 \beta_{5} ) q^{71} + ( 8 + 2 \beta_{2} - \beta_{3} ) q^{73} -3 \beta_{4} q^{77} + ( 4 - 4 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -3 + 2 \beta_{2} + 3 \beta_{3} ) q^{81} + ( -4 + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 2 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -6 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{91} + ( 6 - \beta_{3} ) q^{93} + ( 6 + 2 \beta_{3} ) q^{97} + ( -4 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{9} + O(q^{10})$$ $$6q - 6q^{9} - 6q^{13} + 12q^{29} - 12q^{33} - 24q^{37} - 12q^{39} - 24q^{47} + 6q^{49} + 60q^{57} - 12q^{61} + 48q^{63} - 24q^{67} + 48q^{73} + 24q^{79} - 18q^{81} - 24q^{83} + 12q^{91} + 36q^{93} + 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 12 x^{4} + 36 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 6 \nu^{2}$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{4}$$ $$=$$ $$\nu^{3} + 6 \nu$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + 10 \nu^{3} + 22 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-6 \beta_{3} + 2 \beta_{2} + 24$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} - 10 \beta_{4} + 38 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 2.60168i − 2.26180i − 0.339877i 0.339877i 2.26180i 2.60168i
0 2.60168i 0 0 0 −2.76873 0 −3.76873 0
649.2 0 2.26180i 0 0 0 −1.11575 0 −2.11575 0
649.3 0 0.339877i 0 0 0 3.88448 0 2.88448 0
649.4 0 0.339877i 0 0 0 3.88448 0 2.88448 0
649.5 0 2.26180i 0 0 0 −1.11575 0 −2.11575 0
649.6 0 2.60168i 0 0 0 −2.76873 0 −3.76873 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d 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.d.c 6
5.b even 2 1 1300.2.d.d 6
5.c odd 4 1 260.2.f.a 6
5.c odd 4 1 1300.2.f.e 6
13.b even 2 1 1300.2.d.d 6
15.e even 4 1 2340.2.c.d 6
20.e even 4 1 1040.2.k.c 6
65.d even 2 1 inner 1300.2.d.c 6
65.f even 4 1 3380.2.a.m 3
65.h odd 4 1 260.2.f.a 6
65.h odd 4 1 1300.2.f.e 6
65.k even 4 1 3380.2.a.n 3
195.s even 4 1 2340.2.c.d 6
260.p even 4 1 1040.2.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 5.c odd 4 1
260.2.f.a 6 65.h odd 4 1
1040.2.k.c 6 20.e even 4 1
1040.2.k.c 6 260.p even 4 1
1300.2.d.c 6 1.a even 1 1 trivial
1300.2.d.c 6 65.d even 2 1 inner
1300.2.d.d 6 5.b even 2 1
1300.2.d.d 6 13.b even 2 1
1300.2.f.e 6 5.c odd 4 1
1300.2.f.e 6 65.h odd 4 1
2340.2.c.d 6 15.e even 4 1
2340.2.c.d 6 195.s even 4 1
3380.2.a.m 3 65.f even 4 1
3380.2.a.n 3 65.k even 4 1

## Hecke kernels

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

 $$T_{3}^{6} + 12 T_{3}^{4} + 36 T_{3}^{2} + 4$$ $$T_{7}^{3} - 12 T_{7} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$4 + 36 T^{2} + 12 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$( -12 - 12 T + T^{3} )^{2}$$
$11$ $$324 + 216 T^{2} + 36 T^{4} + T^{6}$$
$13$ $$2197 + 1014 T + 351 T^{2} + 132 T^{3} + 27 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$12996 + 2304 T^{2} + 96 T^{4} + T^{6}$$
$23$ $$30276 + 2988 T^{2} + 96 T^{4} + T^{6}$$
$29$ $$( 84 - 12 T - 6 T^{2} + T^{3} )^{2}$$
$31$ $$4356 + 936 T^{2} + 60 T^{4} + T^{6}$$
$37$ $$( -252 + 12 T^{2} + T^{3} )^{2}$$
$41$ $$5184 + 2160 T^{2} + 108 T^{4} + T^{6}$$
$43$ $$2116 + 2316 T^{2} + 120 T^{4} + T^{6}$$
$47$ $$( -468 - 36 T + 12 T^{2} + T^{3} )^{2}$$
$53$ $$576 + 6768 T^{2} + 204 T^{4} + T^{6}$$
$59$ $$171396 + 12528 T^{2} + 216 T^{4} + T^{6}$$
$61$ $$( -532 - 96 T + 6 T^{2} + T^{3} )^{2}$$
$67$ $$( -588 - 48 T + 12 T^{2} + T^{3} )^{2}$$
$71$ $$492804 + 46656 T^{2} + 432 T^{4} + T^{6}$$
$73$ $$( -252 + 144 T - 24 T^{2} + T^{3} )^{2}$$
$79$ $$( 1696 - 144 T - 12 T^{2} + T^{3} )^{2}$$
$83$ $$( -252 + 12 T^{2} + T^{3} )^{2}$$
$89$ $$4064256 + 96768 T^{2} + 576 T^{4} + T^{6}$$
$97$ $$( -24 + 60 T - 18 T^{2} + T^{3} )^{2}$$