# Properties

 Label 350.2.a.h Level $350$ Weight $2$ Character orbit 350.a Self dual yes Analytic conductor $2.795$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{7} + q^{8} + 3 q^{9} +O(q^{10})$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{7} + q^{8} + 3 q^{9} -2 \beta q^{11} + \beta q^{12} + ( -2 - \beta ) q^{13} + q^{14} + q^{16} -2 q^{17} + 3 q^{18} + ( 4 - \beta ) q^{19} + \beta q^{21} -2 \beta q^{22} + ( 2 - 2 \beta ) q^{23} + \beta q^{24} + ( -2 - \beta ) q^{26} + q^{28} + ( 2 + 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + q^{32} -12 q^{33} -2 q^{34} + 3 q^{36} -2 q^{37} + ( 4 - \beta ) q^{38} + ( -6 - 2 \beta ) q^{39} + ( -6 + 2 \beta ) q^{41} + \beta q^{42} + ( -4 + 2 \beta ) q^{43} -2 \beta q^{44} + ( 2 - 2 \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + \beta q^{48} + q^{49} -2 \beta q^{51} + ( -2 - \beta ) q^{52} + ( 6 + 2 \beta ) q^{53} + q^{56} + ( -6 + 4 \beta ) q^{57} + ( 2 + 2 \beta ) q^{58} + ( -4 + \beta ) q^{59} + ( 6 - \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + 3 q^{63} + q^{64} -12 q^{66} + 8 q^{67} -2 q^{68} + ( -12 + 2 \beta ) q^{69} + ( -6 + 2 \beta ) q^{71} + 3 q^{72} + ( 2 - 2 \beta ) q^{73} -2 q^{74} + ( 4 - \beta ) q^{76} -2 \beta q^{77} + ( -6 - 2 \beta ) q^{78} + ( 2 + 2 \beta ) q^{79} -9 q^{81} + ( -6 + 2 \beta ) q^{82} + \beta q^{83} + \beta q^{84} + ( -4 + 2 \beta ) q^{86} + ( 12 + 2 \beta ) q^{87} -2 \beta q^{88} -10 q^{89} + ( -2 - \beta ) q^{91} + ( 2 - 2 \beta ) q^{92} + ( 12 + 4 \beta ) q^{93} + ( -4 - 2 \beta ) q^{94} + \beta q^{96} + ( -6 - 4 \beta ) q^{97} + q^{98} -6 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{7} + 2q^{8} + 6q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{7} + 2q^{8} + 6q^{9} - 4q^{13} + 2q^{14} + 2q^{16} - 4q^{17} + 6q^{18} + 8q^{19} + 4q^{23} - 4q^{26} + 2q^{28} + 4q^{29} + 8q^{31} + 2q^{32} - 24q^{33} - 4q^{34} + 6q^{36} - 4q^{37} + 8q^{38} - 12q^{39} - 12q^{41} - 8q^{43} + 4q^{46} - 8q^{47} + 2q^{49} - 4q^{52} + 12q^{53} + 2q^{56} - 12q^{57} + 4q^{58} - 8q^{59} + 12q^{61} + 8q^{62} + 6q^{63} + 2q^{64} - 24q^{66} + 16q^{67} - 4q^{68} - 24q^{69} - 12q^{71} + 6q^{72} + 4q^{73} - 4q^{74} + 8q^{76} - 12q^{78} + 4q^{79} - 18q^{81} - 12q^{82} - 8q^{86} + 24q^{87} - 20q^{89} - 4q^{91} + 4q^{92} + 24q^{93} - 8q^{94} - 12q^{97} + 2q^{98} + O(q^{100})$$

## 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
 −2.44949 2.44949
1.00000 −2.44949 1.00000 0 −2.44949 1.00000 1.00000 3.00000 0
1.2 1.00000 2.44949 1.00000 0 2.44949 1.00000 1.00000 3.00000 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 350.2.a.h 2
3.b odd 2 1 3150.2.a.bs 2
4.b odd 2 1 2800.2.a.bl 2
5.b even 2 1 350.2.a.g 2
5.c odd 4 2 70.2.c.a 4
7.b odd 2 1 2450.2.a.bq 2
15.d odd 2 1 3150.2.a.bt 2
15.e even 4 2 630.2.g.g 4
20.d odd 2 1 2800.2.a.bm 2
20.e even 4 2 560.2.g.e 4
35.c odd 2 1 2450.2.a.bl 2
35.f even 4 2 490.2.c.e 4
35.k even 12 4 490.2.i.f 8
35.l odd 12 4 490.2.i.c 8
40.i odd 4 2 2240.2.g.j 4
40.k even 4 2 2240.2.g.i 4
60.l odd 4 2 5040.2.t.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 5.c odd 4 2
350.2.a.g 2 5.b even 2 1
350.2.a.h 2 1.a even 1 1 trivial
490.2.c.e 4 35.f even 4 2
490.2.i.c 8 35.l odd 12 4
490.2.i.f 8 35.k even 12 4
560.2.g.e 4 20.e even 4 2
630.2.g.g 4 15.e even 4 2
2240.2.g.i 4 40.k even 4 2
2240.2.g.j 4 40.i odd 4 2
2450.2.a.bl 2 35.c odd 2 1
2450.2.a.bq 2 7.b odd 2 1
2800.2.a.bl 2 4.b odd 2 1
2800.2.a.bm 2 20.d odd 2 1
3150.2.a.bs 2 3.b odd 2 1
3150.2.a.bt 2 15.d odd 2 1
5040.2.t.t 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(350))$$:

 $$T_{3}^{2} - 6$$ $$T_{13}^{2} + 4 T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-6 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-24 + T^{2}$$
$13$ $$-2 + 4 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$10 - 8 T + T^{2}$$
$23$ $$-20 - 4 T + T^{2}$$
$29$ $$-20 - 4 T + T^{2}$$
$31$ $$-8 - 8 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$12 + 12 T + T^{2}$$
$43$ $$-8 + 8 T + T^{2}$$
$47$ $$-8 + 8 T + T^{2}$$
$53$ $$12 - 12 T + T^{2}$$
$59$ $$10 + 8 T + T^{2}$$
$61$ $$30 - 12 T + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$12 + 12 T + T^{2}$$
$73$ $$-20 - 4 T + T^{2}$$
$79$ $$-20 - 4 T + T^{2}$$
$83$ $$-6 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$-60 + 12 T + T^{2}$$