# Properties

 Label 1300.2.bb.e Level $1300$ Weight $2$ Character orbit 1300.bb Analytic conductor $10.381$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.bb (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12} q^{3} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + 6 \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + 3*z * q^3 + (3*z^3 - 3*z) * q^7 + 6*z^2 * q^9 $$q + 3 \zeta_{12} q^{3} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + 6 \zeta_{12}^{2} q^{9} + (3 \zeta_{12}^{2} - 3) q^{11} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{13} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{17} + \zeta_{12}^{2} q^{19} - 9 q^{21} + 7 \zeta_{12} q^{23} + 9 \zeta_{12}^{3} q^{27} + (5 \zeta_{12}^{2} - 5) q^{29} - 4 q^{31} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{33} - 3 \zeta_{12} q^{37} + ( - 3 \zeta_{12}^{2} - 9) q^{39} + (7 \zeta_{12}^{2} - 7) q^{41} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{43} - 8 \zeta_{12}^{3} q^{47} + ( - 2 \zeta_{12}^{2} + 2) q^{49} + 21 q^{51} - 6 \zeta_{12}^{3} q^{53} + 3 \zeta_{12}^{3} q^{57} + 5 \zeta_{12}^{2} q^{59} + 5 \zeta_{12}^{2} q^{61} - 18 \zeta_{12} q^{63} + 13 \zeta_{12} q^{67} + 21 \zeta_{12}^{2} q^{69} + 3 \zeta_{12}^{2} q^{71} - 14 \zeta_{12}^{3} q^{73} - 9 \zeta_{12}^{3} q^{77} + 8 q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} + 12 \zeta_{12}^{3} q^{83} + (15 \zeta_{12}^{3} - 15 \zeta_{12}) q^{87} + ( - 7 \zeta_{12}^{2} + 7) q^{89} + ( - 9 \zeta_{12}^{2} + 12) q^{91} - 12 \zeta_{12} q^{93} + ( - 11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{97} - 18 q^{99} +O(q^{100})$$ q + 3*z * q^3 + (3*z^3 - 3*z) * q^7 + 6*z^2 * q^9 + (3*z^2 - 3) * q^11 + (3*z^3 - 4*z) * q^13 + (-7*z^3 + 7*z) * q^17 + z^2 * q^19 - 9 * q^21 + 7*z * q^23 + 9*z^3 * q^27 + (5*z^2 - 5) * q^29 - 4 * q^31 + (9*z^3 - 9*z) * q^33 - 3*z * q^37 + (-3*z^2 - 9) * q^39 + (7*z^2 - 7) * q^41 + (9*z^3 - 9*z) * q^43 - 8*z^3 * q^47 + (-2*z^2 + 2) * q^49 + 21 * q^51 - 6*z^3 * q^53 + 3*z^3 * q^57 + 5*z^2 * q^59 + 5*z^2 * q^61 - 18*z * q^63 + 13*z * q^67 + 21*z^2 * q^69 + 3*z^2 * q^71 - 14*z^3 * q^73 - 9*z^3 * q^77 + 8 * q^79 + (9*z^2 - 9) * q^81 + 12*z^3 * q^83 + (15*z^3 - 15*z) * q^87 + (-7*z^2 + 7) * q^89 + (-9*z^2 + 12) * q^91 - 12*z * q^93 + (-11*z^3 + 11*z) * q^97 - 18 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} - 6 q^{11} + 2 q^{19} - 36 q^{21} - 10 q^{29} - 16 q^{31} - 42 q^{39} - 14 q^{41} + 4 q^{49} + 84 q^{51} + 10 q^{59} + 10 q^{61} + 42 q^{69} + 6 q^{71} + 32 q^{79} - 18 q^{81} + 14 q^{89} + 30 q^{91} - 72 q^{99}+O(q^{100})$$ 4 * q + 12 * q^9 - 6 * q^11 + 2 * q^19 - 36 * q^21 - 10 * q^29 - 16 * q^31 - 42 * q^39 - 14 * q^41 + 4 * q^49 + 84 * q^51 + 10 * q^59 + 10 * q^61 + 42 * q^69 + 6 * q^71 + 32 * q^79 - 18 * q^81 + 14 * q^89 + 30 * q^91 - 72 * q^99

## Character values

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

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
549.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −2.59808 + 1.50000i 0 0 0 2.59808 + 1.50000i 0 3.00000 5.19615i 0
549.2 0 2.59808 1.50000i 0 0 0 −2.59808 1.50000i 0 3.00000 5.19615i 0
1049.1 0 −2.59808 1.50000i 0 0 0 2.59808 1.50000i 0 3.00000 + 5.19615i 0
1049.2 0 2.59808 + 1.50000i 0 0 0 −2.59808 + 1.50000i 0 3.00000 + 5.19615i 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
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bb.e 4
5.b even 2 1 inner 1300.2.bb.e 4
5.c odd 4 1 260.2.i.d 2
5.c odd 4 1 1300.2.i.a 2
13.c even 3 1 inner 1300.2.bb.e 4
15.e even 4 1 2340.2.q.a 2
20.e even 4 1 1040.2.q.b 2
65.n even 6 1 inner 1300.2.bb.e 4
65.o even 12 1 3380.2.f.a 2
65.q odd 12 1 260.2.i.d 2
65.q odd 12 1 1300.2.i.a 2
65.q odd 12 1 3380.2.a.b 1
65.r odd 12 1 3380.2.a.a 1
65.t even 12 1 3380.2.f.a 2
195.bl even 12 1 2340.2.q.a 2
260.bj even 12 1 1040.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.d 2 5.c odd 4 1
260.2.i.d 2 65.q odd 12 1
1040.2.q.b 2 20.e even 4 1
1040.2.q.b 2 260.bj even 12 1
1300.2.i.a 2 5.c odd 4 1
1300.2.i.a 2 65.q odd 12 1
1300.2.bb.e 4 1.a even 1 1 trivial
1300.2.bb.e 4 5.b even 2 1 inner
1300.2.bb.e 4 13.c even 3 1 inner
1300.2.bb.e 4 65.n even 6 1 inner
2340.2.q.a 2 15.e even 4 1
2340.2.q.a 2 195.bl even 12 1
3380.2.a.a 1 65.r odd 12 1
3380.2.a.b 1 65.q odd 12 1
3380.2.f.a 2 65.o even 12 1
3380.2.f.a 2 65.t even 12 1

## Hecke kernels

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

 $$T_{3}^{4} - 9T_{3}^{2} + 81$$ T3^4 - 9*T3^2 + 81 $$T_{7}^{4} - 9T_{7}^{2} + 81$$ T7^4 - 9*T7^2 + 81 $$T_{19}^{2} - T_{19} + 1$$ T19^2 - T19 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 9T^{2} + 81$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$(T^{2} + 3 T + 9)^{2}$$
$13$ $$T^{4} - 22T^{2} + 169$$
$17$ $$T^{4} - 49T^{2} + 2401$$
$19$ $$(T^{2} - T + 1)^{2}$$
$23$ $$T^{4} - 49T^{2} + 2401$$
$29$ $$(T^{2} + 5 T + 25)^{2}$$
$31$ $$(T + 4)^{4}$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$(T^{2} + 7 T + 49)^{2}$$
$43$ $$T^{4} - 81T^{2} + 6561$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 5 T + 25)^{2}$$
$61$ $$(T^{2} - 5 T + 25)^{2}$$
$67$ $$T^{4} - 169 T^{2} + 28561$$
$71$ $$(T^{2} - 3 T + 9)^{2}$$
$73$ $$(T^{2} + 196)^{2}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T^{2} - 7 T + 49)^{2}$$
$97$ $$T^{4} - 121 T^{2} + 14641$$