# Properties

 Label 350.10.a.c Level $350$ Weight $10$ Character orbit 350.a Self dual yes Analytic conductor $180.263$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$180.262542657$$ 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

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 6 q^{3} + 256 q^{4} + 96 q^{6} + 2401 q^{7} + 4096 q^{8} - 19647 q^{9} + O(q^{10})$$ $$q + 16 q^{2} + 6 q^{3} + 256 q^{4} + 96 q^{6} + 2401 q^{7} + 4096 q^{8} - 19647 q^{9} - 54152 q^{11} + 1536 q^{12} + 113172 q^{13} + 38416 q^{14} + 65536 q^{16} - 6262 q^{17} - 314352 q^{18} + 257078 q^{19} + 14406 q^{21} - 866432 q^{22} + 266000 q^{23} + 24576 q^{24} + 1810752 q^{26} - 235980 q^{27} + 614656 q^{28} + 1574714 q^{29} - 4637484 q^{31} + 1048576 q^{32} - 324912 q^{33} - 100192 q^{34} - 5029632 q^{36} + 11946238 q^{37} + 4113248 q^{38} + 679032 q^{39} + 21909126 q^{41} + 230496 q^{42} - 27520592 q^{43} - 13862912 q^{44} + 4256000 q^{46} - 52927836 q^{47} + 393216 q^{48} + 5764801 q^{49} - 37572 q^{51} + 28972032 q^{52} - 16221222 q^{53} - 3775680 q^{54} + 9834496 q^{56} + 1542468 q^{57} + 25195424 q^{58} - 140509618 q^{59} - 202963560 q^{61} - 74199744 q^{62} - 47172447 q^{63} + 16777216 q^{64} - 5198592 q^{66} - 153734572 q^{67} - 1603072 q^{68} + 1596000 q^{69} + 279655936 q^{71} - 80474112 q^{72} + 404022830 q^{73} + 191139808 q^{74} + 65811968 q^{76} - 130018952 q^{77} + 10864512 q^{78} - 130689816 q^{79} + 385296021 q^{81} + 350546016 q^{82} - 420134014 q^{83} + 3687936 q^{84} - 440329472 q^{86} + 9448284 q^{87} - 221806592 q^{88} - 469542390 q^{89} + 271725972 q^{91} + 68096000 q^{92} - 27824904 q^{93} - 846845376 q^{94} + 6291456 q^{96} + 872501690 q^{97} + 92236816 q^{98} + 1063924344 q^{99} + O(q^{100})$$

## 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 6.00000 256.000 0 96.0000 2401.00 4096.00 −19647.0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 350.10.a.c 1
5.b even 2 1 14.10.a.a 1
5.c odd 4 2 350.10.c.b 2
15.d odd 2 1 126.10.a.e 1
20.d odd 2 1 112.10.a.b 1
35.c odd 2 1 98.10.a.a 1
35.i odd 6 2 98.10.c.e 2
35.j even 6 2 98.10.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 5.b even 2 1
98.10.a.a 1 35.c odd 2 1
98.10.c.e 2 35.i odd 6 2
98.10.c.f 2 35.j even 6 2
112.10.a.b 1 20.d odd 2 1
126.10.a.e 1 15.d odd 2 1
350.10.a.c 1 1.a even 1 1 trivial
350.10.c.b 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 6$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(350))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-16 + T$$
$3$ $$-6 + T$$
$5$ $$T$$
$7$ $$-2401 + T$$
$11$ $$54152 + T$$
$13$ $$-113172 + T$$
$17$ $$6262 + T$$
$19$ $$-257078 + T$$
$23$ $$-266000 + T$$
$29$ $$-1574714 + T$$
$31$ $$4637484 + T$$
$37$ $$-11946238 + T$$
$41$ $$-21909126 + T$$
$43$ $$27520592 + T$$
$47$ $$52927836 + T$$
$53$ $$16221222 + T$$
$59$ $$140509618 + T$$
$61$ $$202963560 + T$$
$67$ $$153734572 + T$$
$71$ $$-279655936 + T$$
$73$ $$-404022830 + T$$
$79$ $$130689816 + T$$
$83$ $$420134014 + T$$
$89$ $$469542390 + T$$
$97$ $$-872501690 + T$$