# Properties

 Label 350.2.e.i 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.

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

## 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 −0.500000 2.59808i −1.00000 1.50000 2.59808i 0
151.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 2.59808i −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.i yes 2
5.b even 2 1 350.2.e.d 2
5.c odd 4 2 350.2.j.c 4
7.c even 3 1 inner 350.2.e.i yes 2
7.c even 3 1 2450.2.a.i 1
7.d odd 6 1 2450.2.a.h 1
35.i odd 6 1 2450.2.a.z 1
35.j even 6 1 350.2.e.d 2
35.j even 6 1 2450.2.a.y 1
35.k even 12 2 2450.2.c.j 2
35.l odd 12 2 350.2.j.c 4
35.l odd 12 2 2450.2.c.i 2

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

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13}$$ $$T_{17}^{2} + 7 T_{17} + 49$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$5$ 
$7$ $$1 + T + 7 T^{2}$$
$11$ $$1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4}$$
$19$ $$1 - 19 T^{2} + 361 T^{4}$$
$23$ $$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 7 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4}$$
$53$ $$1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4}$$
$59$ $$1 - 14 T + 137 T^{2} - 826 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$
$67$ $$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + T + 71 T^{2} )^{2}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 14 T + 83 T^{2} )^{2}$$
$89$ $$1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$