# Properties

 Label 126.10.a.e Level $126$ Weight $10$ Character orbit 126.a Self dual yes Analytic conductor $64.895$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 126.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.8945153566$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 256 q^{4} - 560 q^{5} - 2401 q^{7} + 4096 q^{8} + O(q^{10})$$ $$q + 16 q^{2} + 256 q^{4} - 560 q^{5} - 2401 q^{7} + 4096 q^{8} - 8960 q^{10} + 54152 q^{11} - 113172 q^{13} - 38416 q^{14} + 65536 q^{16} - 6262 q^{17} + 257078 q^{19} - 143360 q^{20} + 866432 q^{22} + 266000 q^{23} - 1639525 q^{25} - 1810752 q^{26} - 614656 q^{28} - 1574714 q^{29} - 4637484 q^{31} + 1048576 q^{32} - 100192 q^{34} + 1344560 q^{35} - 11946238 q^{37} + 4113248 q^{38} - 2293760 q^{40} - 21909126 q^{41} + 27520592 q^{43} + 13862912 q^{44} + 4256000 q^{46} - 52927836 q^{47} + 5764801 q^{49} - 26232400 q^{50} - 28972032 q^{52} - 16221222 q^{53} - 30325120 q^{55} - 9834496 q^{56} - 25195424 q^{58} + 140509618 q^{59} - 202963560 q^{61} - 74199744 q^{62} + 16777216 q^{64} + 63376320 q^{65} + 153734572 q^{67} - 1603072 q^{68} + 21512960 q^{70} - 279655936 q^{71} - 404022830 q^{73} - 191139808 q^{74} + 65811968 q^{76} - 130018952 q^{77} - 130689816 q^{79} - 36700160 q^{80} - 350546016 q^{82} - 420134014 q^{83} + 3506720 q^{85} + 440329472 q^{86} + 221806592 q^{88} + 469542390 q^{89} + 271725972 q^{91} + 68096000 q^{92} - 846845376 q^{94} - 143963680 q^{95} - 872501690 q^{97} + 92236816 q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
16.0000 0 256.000 −560.000 0 −2401.00 4096.00 0 −8960.00
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.10.a.e 1
3.b odd 2 1 14.10.a.a 1
12.b even 2 1 112.10.a.b 1
15.d odd 2 1 350.10.a.c 1
15.e even 4 2 350.10.c.b 2
21.c even 2 1 98.10.a.a 1
21.g even 6 2 98.10.c.e 2
21.h odd 6 2 98.10.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 3.b odd 2 1
98.10.a.a 1 21.c even 2 1
98.10.c.e 2 21.g even 6 2
98.10.c.f 2 21.h odd 6 2
112.10.a.b 1 12.b even 2 1
126.10.a.e 1 1.a even 1 1 trivial
350.10.a.c 1 15.d odd 2 1
350.10.c.b 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 560$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(126))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-16 + T$$
$3$ $$T$$
$5$ $$560 + T$$
$7$ $$2401 + T$$
$11$ $$-54152 + T$$
$13$ $$113172 + T$$
$17$ $$6262 + T$$
$19$ $$-257078 + T$$
$23$ $$-266000 + T$$
$29$ $$1574714 + T$$
$31$ $$4637484 + T$$
$37$ $$11946238 + T$$
$41$ $$21909126 + T$$
$43$ $$-27520592 + T$$
$47$ $$52927836 + T$$
$53$ $$16221222 + T$$
$59$ $$-140509618 + T$$
$61$ $$202963560 + T$$
$67$ $$-153734572 + T$$
$71$ $$279655936 + T$$
$73$ $$404022830 + T$$
$79$ $$130689816 + T$$
$83$ $$420134014 + T$$
$89$ $$-469542390 + T$$
$97$ $$872501690 + T$$
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