# Properties

 Label 350.2.e.k 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 q^{6} + ( 2 + \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 q^{6} + ( 2 + \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} + 2 \zeta_{6} q^{12} + 2 q^{13} + ( -1 + 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} -8 \zeta_{6} q^{19} + ( 6 - 4 \zeta_{6} ) q^{21} + 9 \zeta_{6} q^{23} + ( -2 + 2 \zeta_{6} ) q^{24} + 2 \zeta_{6} q^{26} + 4 q^{27} + ( -3 + 2 \zeta_{6} ) q^{28} -6 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 3 q^{34} + q^{36} -8 \zeta_{6} q^{37} + ( 8 - 8 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} -3 q^{41} + ( 4 + 2 \zeta_{6} ) q^{42} -10 q^{43} + ( -9 + 9 \zeta_{6} ) q^{46} + 3 \zeta_{6} q^{47} -2 q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + 4 \zeta_{6} q^{54} + ( -2 - \zeta_{6} ) q^{56} -16 q^{57} -6 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} -5 q^{62} + ( 1 - 3 \zeta_{6} ) q^{63} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + 18 q^{69} -9 q^{71} + \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 8 - 8 \zeta_{6} ) q^{74} + 8 q^{76} + 4 q^{78} -5 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -3 \zeta_{6} q^{82} -6 q^{83} + ( -2 + 6 \zeta_{6} ) q^{84} -10 \zeta_{6} q^{86} + ( -12 + 12 \zeta_{6} ) q^{87} -3 \zeta_{6} q^{89} + ( 4 + 2 \zeta_{6} ) q^{91} -9 q^{92} + 10 \zeta_{6} q^{93} + ( -3 + 3 \zeta_{6} ) q^{94} -2 \zeta_{6} q^{96} + 5 q^{97} + ( -5 + 8 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 2q^{3} - q^{4} + 4q^{6} + 5q^{7} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} + 2q^{3} - q^{4} + 4q^{6} + 5q^{7} - 2q^{8} - q^{9} + 2q^{12} + 4q^{13} + q^{14} - q^{16} + 3q^{17} + q^{18} - 8q^{19} + 8q^{21} + 9q^{23} - 2q^{24} + 2q^{26} + 8q^{27} - 4q^{28} - 12q^{29} - 5q^{31} + q^{32} + 6q^{34} + 2q^{36} - 8q^{37} + 8q^{38} + 4q^{39} - 6q^{41} + 10q^{42} - 20q^{43} - 9q^{46} + 3q^{47} - 4q^{48} + 11q^{49} - 6q^{51} - 2q^{52} - 6q^{53} + 4q^{54} - 5q^{56} - 32q^{57} - 6q^{58} - 12q^{59} + 4q^{61} - 10q^{62} - q^{63} + 2q^{64} - 2q^{67} + 3q^{68} + 36q^{69} - 18q^{71} + q^{72} + 10q^{73} + 8q^{74} + 16q^{76} + 8q^{78} - 5q^{79} + 11q^{81} - 3q^{82} - 12q^{83} + 2q^{84} - 10q^{86} - 12q^{87} - 3q^{89} + 10q^{91} - 18q^{92} + 10q^{93} - 3q^{94} - 2q^{96} + 10q^{97} - 2q^{98} + O(q^{100})$$

## 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 1.00000 + 1.73205i −0.500000 0.866025i 0 2.00000 2.50000 0.866025i −1.00000 −0.500000 + 0.866025i 0
151.1 0.500000 + 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i 0 2.00000 2.50000 + 0.866025i −1.00000 −0.500000 0.866025i 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.k yes 2
5.b even 2 1 350.2.e.a 2
5.c odd 4 2 350.2.j.e 4
7.c even 3 1 inner 350.2.e.k yes 2
7.c even 3 1 2450.2.a.e 1
7.d odd 6 1 2450.2.a.o 1
35.i odd 6 1 2450.2.a.u 1
35.j even 6 1 350.2.e.a 2
35.j even 6 1 2450.2.a.be 1
35.k even 12 2 2450.2.c.d 2
35.l odd 12 2 350.2.j.e 4
35.l odd 12 2 2450.2.c.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.e.a 2 5.b even 2 1
350.2.e.a 2 35.j even 6 1
350.2.e.k yes 2 1.a even 1 1 trivial
350.2.e.k yes 2 7.c even 3 1 inner
350.2.j.e 4 5.c odd 4 2
350.2.j.e 4 35.l odd 12 2
2450.2.a.e 1 7.c even 3 1
2450.2.a.o 1 7.d odd 6 1
2450.2.a.u 1 35.i odd 6 1
2450.2.a.be 1 35.j even 6 1
2450.2.c.d 2 35.k even 12 2
2450.2.c.o 2 35.l odd 12 2

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ 
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 10 T + 43 T^{2} )^{2}$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 9 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$
$79$ $$1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 6 T + 83 T^{2} )^{2}$$
$89$ $$1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 5 T + 97 T^{2} )^{2}$$