# Properties

 Label 350.2.e.c Level $350$ Weight $2$ Character orbit 350.e Analytic conductor $2.795$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} -5 q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + 5 \zeta_{6} q^{19} + 3 q^{22} + 7 \zeta_{6} q^{23} + 5 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} -4 q^{29} + ( 2 - 2 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -2 q^{34} -3 q^{36} -\zeta_{6} q^{37} + ( 5 - 5 \zeta_{6} ) q^{38} + 3 q^{41} + 2 q^{43} -3 \zeta_{6} q^{44} + ( 7 - 7 \zeta_{6} ) q^{46} + 7 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + ( -3 + 2 \zeta_{6} ) q^{56} + 4 \zeta_{6} q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} -2 q^{62} + ( -6 - 3 \zeta_{6} ) q^{63} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} -6 q^{71} + 3 \zeta_{6} q^{72} + ( 16 - 16 \zeta_{6} ) q^{73} + ( -1 + \zeta_{6} ) q^{74} -5 q^{76} + ( 3 - 9 \zeta_{6} ) q^{77} -14 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -3 \zeta_{6} q^{82} -6 q^{83} -2 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} -2 \zeta_{6} q^{89} + ( 15 - 10 \zeta_{6} ) q^{91} -7 q^{92} + ( 7 - 7 \zeta_{6} ) q^{94} -12 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} -9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 4q^{7} + 2q^{8} + 3q^{9} - 3q^{11} - 10q^{13} + 5q^{14} - q^{16} + 2q^{17} + 3q^{18} + 5q^{19} + 6q^{22} + 7q^{23} + 5q^{26} - q^{28} - 8q^{29} + 2q^{31} - q^{32} - 4q^{34} - 6q^{36} - q^{37} + 5q^{38} + 6q^{41} + 4q^{43} - 3q^{44} + 7q^{46} + 7q^{47} + 2q^{49} + 5q^{52} - 9q^{53} - 4q^{56} + 4q^{58} + 4q^{59} - 6q^{61} - 4q^{62} - 15q^{63} + 2q^{64} - 2q^{67} + 2q^{68} - 12q^{71} + 3q^{72} + 16q^{73} - q^{74} - 10q^{76} - 3q^{77} - 14q^{79} - 9q^{81} - 3q^{82} - 12q^{83} - 2q^{86} - 3q^{88} - 2q^{89} + 20q^{91} - 14q^{92} + 7q^{94} - 24q^{97} - 13q^{98} - 18q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.00000 1.73205i 1.00000 1.50000 2.59808i 0
151.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.00000 + 1.73205i 1.00000 1.50000 + 2.59808i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.e.c 2
5.b even 2 1 350.2.e.j 2
5.c odd 4 2 70.2.i.b 4
7.c even 3 1 inner 350.2.e.c 2
7.c even 3 1 2450.2.a.ba 1
7.d odd 6 1 2450.2.a.bb 1
15.e even 4 2 630.2.u.a 4
20.e even 4 2 560.2.bw.d 4
35.f even 4 2 490.2.i.a 4
35.i odd 6 1 2450.2.a.j 1
35.j even 6 1 350.2.e.j 2
35.j even 6 1 2450.2.a.k 1
35.k even 12 2 490.2.c.d 2
35.k even 12 2 490.2.i.a 4
35.l odd 12 2 70.2.i.b 4
35.l odd 12 2 490.2.c.a 2
105.x even 12 2 630.2.u.a 4
140.w even 12 2 560.2.bw.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 5.c odd 4 2
70.2.i.b 4 35.l odd 12 2
350.2.e.c 2 1.a even 1 1 trivial
350.2.e.c 2 7.c even 3 1 inner
350.2.e.j 2 5.b even 2 1
350.2.e.j 2 35.j even 6 1
490.2.c.a 2 35.l odd 12 2
490.2.c.d 2 35.k even 12 2
490.2.i.a 4 35.f even 4 2
490.2.i.a 4 35.k even 12 2
560.2.bw.d 4 20.e even 4 2
560.2.bw.d 4 140.w even 12 2
630.2.u.a 4 15.e even 4 2
630.2.u.a 4 105.x even 12 2
2450.2.a.j 1 35.i odd 6 1
2450.2.a.k 1 35.j even 6 1
2450.2.a.ba 1 7.c even 3 1
2450.2.a.bb 1 7.d odd 6 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13} + 5$$ $$T_{17}^{2} - 2 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( 5 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$49 - 7 T + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$1 + T + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$49 - 7 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$36 + 6 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$256 - 16 T + T^{2}$$
$79$ $$196 + 14 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$4 + 2 T + T^{2}$$
$97$ $$( 12 + T )^{2}$$