# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - z * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + ( - 4 \zeta_{6} + 1) q^{13} - 3 \zeta_{6} q^{17} - 5 \zeta_{6} q^{19} - q^{21} + ( - 9 \zeta_{6} + 9) q^{23} + 5 q^{27} + ( - 9 \zeta_{6} + 9) q^{29} + 8 q^{31} + 3 \zeta_{6} q^{33} + (7 \zeta_{6} - 7) q^{37} + ( - \zeta_{6} - 3) q^{39} + (3 \zeta_{6} - 3) q^{41} - \zeta_{6} q^{43} + ( - 6 \zeta_{6} + 6) q^{49} - 3 q^{51} - 6 q^{53} - 5 q^{57} - 9 \zeta_{6} q^{59} + \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{63} + ( - 5 \zeta_{6} + 5) q^{67} - 9 \zeta_{6} q^{69} - 9 \zeta_{6} q^{71} - 2 q^{73} + 3 q^{77} + 8 q^{79} + (\zeta_{6} - 1) q^{81} - 9 \zeta_{6} q^{87} + (3 \zeta_{6} - 3) q^{89} + (3 \zeta_{6} - 4) q^{91} + ( - 8 \zeta_{6} + 8) q^{93} + 17 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 - z * q^7 + 2*z * q^9 + (3*z - 3) * q^11 + (-4*z + 1) * q^13 - 3*z * q^17 - 5*z * q^19 - q^21 + (-9*z + 9) * q^23 + 5 * q^27 + (-9*z + 9) * q^29 + 8 * q^31 + 3*z * q^33 + (7*z - 7) * q^37 + (-z - 3) * q^39 + (3*z - 3) * q^41 - z * q^43 + (-6*z + 6) * q^49 - 3 * q^51 - 6 * q^53 - 5 * q^57 - 9*z * q^59 + z * q^61 + (-2*z + 2) * q^63 + (-5*z + 5) * q^67 - 9*z * q^69 - 9*z * q^71 - 2 * q^73 + 3 * q^77 + 8 * q^79 + (z - 1) * q^81 - 9*z * q^87 + (3*z - 3) * q^89 + (3*z - 4) * q^91 + (-8*z + 8) * q^93 + 17*z * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - q^7 + 2 * q^9 $$2 q + q^{3} - q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{17} - 5 q^{19} - 2 q^{21} + 9 q^{23} + 10 q^{27} + 9 q^{29} + 16 q^{31} + 3 q^{33} - 7 q^{37} - 7 q^{39} - 3 q^{41} - q^{43} + 6 q^{49} - 6 q^{51} - 12 q^{53} - 10 q^{57} - 9 q^{59} + q^{61} + 2 q^{63} + 5 q^{67} - 9 q^{69} - 9 q^{71} - 4 q^{73} + 6 q^{77} + 16 q^{79} - q^{81} - 9 q^{87} - 3 q^{89} - 5 q^{91} + 8 q^{93} + 17 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^7 + 2 * q^9 - 3 * q^11 - 2 * q^13 - 3 * q^17 - 5 * q^19 - 2 * q^21 + 9 * q^23 + 10 * q^27 + 9 * q^29 + 16 * q^31 + 3 * q^33 - 7 * q^37 - 7 * q^39 - 3 * q^41 - q^43 + 6 * q^49 - 6 * q^51 - 12 * q^53 - 10 * q^57 - 9 * q^59 + q^61 + 2 * q^63 + 5 * q^67 - 9 * q^69 - 9 * q^71 - 4 * q^73 + 6 * q^77 + 16 * q^79 - q^81 - 9 * q^87 - 3 * q^89 - 5 * q^91 + 8 * q^93 + 17 * q^97 - 12 * q^99

## Character values

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

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0 1.00000 1.73205i 0
1101.1 0 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0 1.00000 + 1.73205i 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
13.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{19}^{2} + 5T_{19} + 25$$ T19^2 + 5*T19 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 9T + 81$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$T^{2} + 9T + 81$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} - 17T + 289$$
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