# Properties

 Label 350.2.e.f 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: $$\Q(\sqrt{-3})$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -3 q^{6} + ( 3 - \zeta_{6} ) q^{7} + q^{8} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -3 q^{6} + ( 3 - \zeta_{6} ) q^{7} + q^{8} -6 \zeta_{6} q^{9} + 3 \zeta_{6} q^{12} -2 q^{13} + ( -1 - 2 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} + 2 \zeta_{6} q^{19} + ( 6 - 9 \zeta_{6} ) q^{21} + \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + 2 \zeta_{6} q^{26} -9 q^{27} + ( -2 + 3 \zeta_{6} ) q^{28} - q^{29} + ( -10 + 10 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -2 q^{34} + 6 q^{36} + 8 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + ( -6 + 6 \zeta_{6} ) q^{39} -3 q^{41} + ( -9 + 3 \zeta_{6} ) q^{42} + 5 q^{43} + ( 1 - \zeta_{6} ) q^{46} -8 \zeta_{6} q^{47} -3 q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + 9 \zeta_{6} q^{54} + ( 3 - \zeta_{6} ) q^{56} + 6 q^{57} + \zeta_{6} q^{58} + ( -2 + 2 \zeta_{6} ) q^{59} + 9 \zeta_{6} q^{61} + 10 q^{62} + ( -6 - 12 \zeta_{6} ) q^{63} + q^{64} + ( 7 - 7 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 3 q^{69} + 6 q^{71} -6 \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 8 - 8 \zeta_{6} ) q^{74} -2 q^{76} + 6 q^{78} + 10 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 3 \zeta_{6} q^{82} -9 q^{83} + ( 3 + 6 \zeta_{6} ) q^{84} -5 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} + 7 \zeta_{6} q^{89} + ( -6 + 2 \zeta_{6} ) q^{91} - q^{92} + 30 \zeta_{6} q^{93} + ( -8 + 8 \zeta_{6} ) q^{94} + 3 \zeta_{6} q^{96} + ( -5 - 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 3q^{3} - q^{4} - 6q^{6} + 5q^{7} + 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q - q^{2} + 3q^{3} - q^{4} - 6q^{6} + 5q^{7} + 2q^{8} - 6q^{9} + 3q^{12} - 4q^{13} - 4q^{14} - q^{16} + 2q^{17} - 6q^{18} + 2q^{19} + 3q^{21} + q^{23} + 3q^{24} + 2q^{26} - 18q^{27} - q^{28} - 2q^{29} - 10q^{31} - q^{32} - 4q^{34} + 12q^{36} + 8q^{37} + 2q^{38} - 6q^{39} - 6q^{41} - 15q^{42} + 10q^{43} + q^{46} - 8q^{47} - 6q^{48} + 11q^{49} - 6q^{51} + 2q^{52} + 6q^{53} + 9q^{54} + 5q^{56} + 12q^{57} + q^{58} - 2q^{59} + 9q^{61} + 20q^{62} - 24q^{63} + 2q^{64} + 7q^{67} + 2q^{68} + 6q^{69} + 12q^{71} - 6q^{72} + 10q^{73} + 8q^{74} - 4q^{76} + 12q^{78} + 10q^{79} - 9q^{81} + 3q^{82} - 18q^{83} + 12q^{84} - 5q^{86} - 3q^{87} + 7q^{89} - 10q^{91} - 2q^{92} + 30q^{93} - 8q^{94} + 3q^{96} - 13q^{98} + O(q^{100})$$

## 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 1.50000 + 2.59808i −0.500000 0.866025i 0 −3.00000 2.50000 + 0.866025i 1.00000 −3.00000 + 5.19615i 0
151.1 −0.500000 0.866025i 1.50000 2.59808i −0.500000 + 0.866025i 0 −3.00000 2.50000 0.866025i 1.00000 −3.00000 5.19615i 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.f 2
5.b even 2 1 350.2.e.g 2
5.c odd 4 2 70.2.i.a 4
7.c even 3 1 inner 350.2.e.f 2
7.c even 3 1 2450.2.a.s 1
7.d odd 6 1 2450.2.a.bh 1
15.e even 4 2 630.2.u.b 4
20.e even 4 2 560.2.bw.c 4
35.f even 4 2 490.2.i.b 4
35.i odd 6 1 2450.2.a.c 1
35.j even 6 1 350.2.e.g 2
35.j even 6 1 2450.2.a.r 1
35.k even 12 2 490.2.c.b 2
35.k even 12 2 490.2.i.b 4
35.l odd 12 2 70.2.i.a 4
35.l odd 12 2 490.2.c.c 2
105.x even 12 2 630.2.u.b 4
140.w even 12 2 560.2.bw.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a 4 5.c odd 4 2
70.2.i.a 4 35.l odd 12 2
350.2.e.f 2 1.a even 1 1 trivial
350.2.e.f 2 7.c even 3 1 inner
350.2.e.g 2 5.b even 2 1
350.2.e.g 2 35.j even 6 1
490.2.c.b 2 35.k even 12 2
490.2.c.c 2 35.l odd 12 2
490.2.i.b 4 35.f even 4 2
490.2.i.b 4 35.k even 12 2
560.2.bw.c 4 20.e even 4 2
560.2.bw.c 4 140.w even 12 2
630.2.u.b 4 15.e even 4 2
630.2.u.b 4 105.x even 12 2
2450.2.a.c 1 35.i odd 6 1
2450.2.a.r 1 35.j even 6 1
2450.2.a.s 1 7.c even 3 1
2450.2.a.bh 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 $$S_{2}^{\mathrm{new}}(350, [\chi])$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} )$$
$5$ 
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4}$$
$23$ $$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$
$29$ $$( 1 + T + 29 T^{2} )^{2}$$
$31$ $$1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{2}$$
$47$ $$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4}$$
$67$ $$1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$
$79$ $$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 9 T + 83 T^{2} )^{2}$$
$89$ $$1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 97 T^{2} )^{2}$$