# Properties

 Label 1300.2.bs.b Level $1300$ Weight $2$ Character orbit 1300.bs Analytic conductor $10.381$ Analytic rank $0$ Dimension $4$ 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.bs (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{3} + (2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{7} + ( - \zeta_{12}^{2} - \zeta_{12} - 1) q^{9}+O(q^{10})$$ q + (z^3 - z^2 + 1) * q^3 + (2*z^2 - z + 2) * q^7 + (-z^2 - z - 1) * q^9 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{3} + (2 \zeta_{12}^{2} - \zeta_{12} + 2) q^{7} + ( - \zeta_{12}^{2} - \zeta_{12} - 1) q^{9} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 3) q^{11} + (3 \zeta_{12}^{3} + 2) q^{13} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{17} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} - 1) q^{19} + (5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 5) q^{21} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 4 \zeta_{12} + 3) q^{23} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{27} + (\zeta_{12}^{2} - 2) q^{29} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{31} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12} - 4) q^{33} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 5 \zeta_{12} + 4) q^{37} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{39} + (5 \zeta_{12} + 5) q^{41} + ( - \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{43} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{47} + ( - 4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 4 \zeta_{12}) q^{49} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{51} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{53} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 5) q^{57} + (\zeta_{12}^{3} + 6 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{59} + ( - 9 \zeta_{12}^{2} + 9) q^{61} + ( - \zeta_{12}^{3} - 5 \zeta_{12}^{2} - \zeta_{12}) q^{63} + (\zeta_{12}^{3} - 14 \zeta_{12}^{2} + \zeta_{12}) q^{67} + (8 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 4 \zeta_{12} + 7) q^{69} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{71} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 8) q^{73} + (8 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 8) q^{77} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{79} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 5 \zeta_{12}) q^{81} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{83} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 1) q^{87} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 9 \zeta_{12} - 5) q^{89} + (12 \zeta_{12}^{3} + \zeta_{12}^{2} - 8 \zeta_{12} + 7) q^{91} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 9 \zeta_{12} + 6) q^{93} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - \zeta_{12} - 6) q^{97} + ( - 7 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 7) q^{99}+O(q^{100})$$ q + (z^3 - z^2 + 1) * q^3 + (2*z^2 - z + 2) * q^7 + (-z^2 - z - 1) * q^9 + (2*z^3 + 2*z^2 + z - 3) * q^11 + (3*z^3 + 2) * q^13 + (-z^3 + z^2 + 2*z - 2) * q^17 + (3*z^3 - 2*z^2 - z - 1) * q^19 + (5*z^3 - 3*z^2 - 3*z + 5) * q^21 + (z^3 + z^2 - 4*z + 3) * q^23 + (2*z^3 - 3*z^2 - 3*z + 2) * q^27 + (z^2 - 2) * q^29 + (3*z^3 + 6*z^2 - 6*z - 3) * q^31 + (-2*z^3 + 4*z^2 + z - 4) * q^33 + (5*z^3 - 2*z^2 - 5*z + 4) * q^37 + (2*z^3 - 2*z^2 + 3*z - 1) * q^39 + (5*z + 5) * q^41 + (-z^3 - 3*z^2 + 4*z + 4) * q^43 + (4*z^3 + 8*z^2 - 4) * q^47 + (-4*z^3 + 6*z^2 - 4*z) * q^49 + (-3*z^3 + 4*z^2 - 2) * q^51 + (z^3 + 4*z^2 - 4*z - 1) * q^53 + (-2*z^3 + 4*z - 5) * q^57 + (z^3 + 6*z^2 + 5*z - 5) * q^59 + (-9*z^2 + 9) * q^61 + (-z^3 - 5*z^2 - z) * q^63 + (z^3 - 14*z^2 + z) * q^67 + (8*z^3 - 7*z^2 - 4*z + 7) * q^69 + (-z^3 + 2*z^2 + 3*z - 3) * q^71 + (-4*z^3 + 8*z + 8) * q^73 + (8*z^3 - z^2 + z - 8) * q^77 + (4*z^3 - 4*z^2 + 2) * q^79 + (5*z^3 - 2*z^2 + 5*z) * q^81 + (-4*z^3 + 8*z^2 - 4) * q^83 + (-z^3 + 2*z^2 - z - 1) * q^87 + (4*z^3 - 4*z^2 - 9*z - 5) * q^89 + (12*z^3 + z^2 - 8*z + 7) * q^91 + (9*z^3 - 3*z^2 - 9*z + 6) * q^93 + (2*z^3 + 6*z^2 - z - 6) * q^97 + (-7*z^3 - 4*z^2 + 4*z + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 12 q^{7} - 6 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9 $$4 q + 2 q^{3} + 12 q^{7} - 6 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 8 q^{19} + 14 q^{21} + 14 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 20 q^{41} + 10 q^{43} + 12 q^{49} + 4 q^{53} - 20 q^{57} - 8 q^{59} + 18 q^{61} - 10 q^{63} - 28 q^{67} + 14 q^{69} - 8 q^{71} + 32 q^{73} - 34 q^{77} - 4 q^{81} - 28 q^{89} + 30 q^{91} + 18 q^{93} - 12 q^{97} + 20 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 8 * q^19 + 14 * q^21 + 14 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 20 * q^41 + 10 * q^43 + 12 * q^49 + 4 * q^53 - 20 * q^57 - 8 * q^59 + 18 * q^61 - 10 * q^63 - 28 * q^67 + 14 * q^69 - 8 * q^71 + 32 * q^73 - 34 * q^77 - 4 * q^81 - 28 * q^89 + 30 * q^91 + 18 * q^93 - 12 * q^97 + 20 * q^99

## Character values

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

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$\zeta_{12}$$ $$1$$ $$\zeta_{12}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0.500000 1.86603i 0 0 0 3.86603 + 2.23205i 0 −0.633975 0.366025i 0
293.1 0 0.500000 0.133975i 0 0 0 2.13397 1.23205i 0 −2.36603 + 1.36603i 0
357.1 0 0.500000 + 1.86603i 0 0 0 3.86603 2.23205i 0 −0.633975 + 0.366025i 0
457.1 0 0.500000 + 0.133975i 0 0 0 2.13397 + 1.23205i 0 −2.36603 1.36603i 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
65.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bs.b 4
5.b even 2 1 260.2.bk.a yes 4
5.c odd 4 1 260.2.bf.b 4
5.c odd 4 1 1300.2.bn.a 4
13.f odd 12 1 1300.2.bn.a 4
65.o even 12 1 inner 1300.2.bs.b 4
65.s odd 12 1 260.2.bf.b 4
65.t even 12 1 260.2.bk.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bf.b 4 5.c odd 4 1
260.2.bf.b 4 65.s odd 12 1
260.2.bk.a yes 4 5.b even 2 1
260.2.bk.a yes 4 65.t even 12 1
1300.2.bn.a 4 5.c odd 4 1
1300.2.bn.a 4 13.f odd 12 1
1300.2.bs.b 4 1.a even 1 1 trivial
1300.2.bs.b 4 65.o even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121$$
$11$ $$T^{4} + 8 T^{3} + 41 T^{2} + 130 T + 169$$
$13$ $$(T^{2} - 4 T + 13)^{2}$$
$17$ $$T^{4} + 6 T^{3} + 9 T^{2} + 9$$
$19$ $$T^{4} + 8 T^{3} + 41 T^{2} + 130 T + 169$$
$23$ $$T^{4} - 14 T^{3} + 53 T^{2} - 4 T + 1$$
$29$ $$(T^{2} + 3 T + 3)^{2}$$
$31$ $$T^{4} + 2916$$
$37$ $$T^{4} - 12 T^{3} + 35 T^{2} + \cdots + 169$$
$41$ $$T^{4} - 20 T^{3} + 125 T^{2} + \cdots + 625$$
$43$ $$T^{4} - 10 T^{3} + 29 T^{2} + \cdots + 529$$
$47$ $$T^{4} + 128T^{2} + 1024$$
$53$ $$T^{4} - 4 T^{3} + 8 T^{2} + 88 T + 484$$
$59$ $$T^{4} + 8 T^{3} + 65 T^{2} + \cdots + 3481$$
$61$ $$(T^{2} - 9 T + 81)^{2}$$
$67$ $$T^{4} + 28 T^{3} + 591 T^{2} + \cdots + 37249$$
$71$ $$T^{4} + 8 T^{3} + 17 T^{2} + 22 T + 121$$
$73$ $$(T^{2} - 16 T + 16)^{2}$$
$79$ $$T^{4} + 56T^{2} + 16$$
$83$ $$T^{4} + 128T^{2} + 1024$$
$89$ $$T^{4} + 28 T^{3} + 197 T^{2} + \cdots + 2209$$
$97$ $$T^{4} + 12 T^{3} + 111 T^{2} + \cdots + 1089$$