# Properties

 Label 350.2.j.b Level $350$ Weight $2$ Character orbit 350.j Analytic conductor $2.795$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} - q^{6} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -2 + 2 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} - q^{6} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -2 + 2 \zeta_{12}^{2} ) q^{9} + 6 \zeta_{12}^{2} q^{11} -\zeta_{12} q^{12} + 4 \zeta_{12}^{3} q^{13} + ( -1 - 2 \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{18} + ( 2 - 2 \zeta_{12}^{2} ) q^{19} + ( 3 - \zeta_{12}^{2} ) q^{21} + 6 \zeta_{12}^{3} q^{22} -3 \zeta_{12} q^{23} -\zeta_{12}^{2} q^{24} + ( -4 + 4 \zeta_{12}^{2} ) q^{26} -5 \zeta_{12}^{3} q^{27} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + 3 q^{29} -8 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} -6 \zeta_{12} q^{33} -2 q^{36} + 4 \zeta_{12} q^{37} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} -4 \zeta_{12}^{2} q^{39} + 9 q^{41} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{42} + 7 \zeta_{12}^{3} q^{43} + ( -6 + 6 \zeta_{12}^{2} ) q^{44} -3 \zeta_{12}^{2} q^{46} -\zeta_{12}^{3} q^{48} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( 5 - 5 \zeta_{12}^{2} ) q^{54} + ( 2 - 3 \zeta_{12}^{2} ) q^{56} + 2 \zeta_{12}^{3} q^{57} + 3 \zeta_{12} q^{58} -6 \zeta_{12}^{2} q^{59} + ( -5 + 5 \zeta_{12}^{2} ) q^{61} -8 \zeta_{12}^{3} q^{62} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{63} - q^{64} -6 \zeta_{12}^{2} q^{66} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + 3 q^{69} -6 q^{71} -2 \zeta_{12} q^{72} + ( 16 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{73} + 4 \zeta_{12}^{2} q^{74} + 2 q^{76} + ( -6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{77} -4 \zeta_{12}^{3} q^{78} + ( 2 - 2 \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} + 9 \zeta_{12} q^{82} -3 \zeta_{12}^{3} q^{83} + ( 1 + 2 \zeta_{12}^{2} ) q^{84} + ( -7 + 7 \zeta_{12}^{2} ) q^{86} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{87} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{88} + ( -15 + 15 \zeta_{12}^{2} ) q^{89} + ( 8 - 12 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12}^{3} q^{92} + 8 \zeta_{12} q^{93} + ( 1 - \zeta_{12}^{2} ) q^{96} + 14 \zeta_{12}^{3} q^{97} + ( 5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 4q^{6} - 4q^{9} + 12q^{11} - 8q^{14} - 2q^{16} + 4q^{19} + 10q^{21} - 2q^{24} - 8q^{26} + 12q^{29} - 16q^{31} - 8q^{36} - 8q^{39} + 36q^{41} - 12q^{44} - 6q^{46} + 26q^{49} + 10q^{54} + 2q^{56} - 12q^{59} - 10q^{61} - 4q^{64} - 12q^{66} + 12q^{69} - 24q^{71} + 8q^{74} + 8q^{76} + 4q^{79} - 2q^{81} + 8q^{84} - 14q^{86} - 30q^{89} + 8q^{91} + 2q^{96} - 48q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.00000 2.59808 + 0.500000i 1.00000i −1.00000 + 1.73205i 0
149.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 −2.59808 0.500000i 1.00000i −1.00000 + 1.73205i 0
249.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.00000 2.59808 0.500000i 1.00000i −1.00000 1.73205i 0
249.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 −1.00000 −2.59808 + 0.500000i 1.00000i −1.00000 1.73205i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.j.b 4
5.b even 2 1 inner 350.2.j.b 4
5.c odd 4 1 70.2.e.c 2
5.c odd 4 1 350.2.e.e 2
7.c even 3 1 inner 350.2.j.b 4
7.c even 3 1 2450.2.c.g 2
7.d odd 6 1 2450.2.c.l 2
15.e even 4 1 630.2.k.b 2
20.e even 4 1 560.2.q.g 2
35.f even 4 1 490.2.e.h 2
35.i odd 6 1 2450.2.c.l 2
35.j even 6 1 inner 350.2.j.b 4
35.j even 6 1 2450.2.c.g 2
35.k even 12 1 490.2.a.b 1
35.k even 12 1 490.2.e.h 2
35.k even 12 1 2450.2.a.bc 1
35.l odd 12 1 70.2.e.c 2
35.l odd 12 1 350.2.e.e 2
35.l odd 12 1 490.2.a.c 1
35.l odd 12 1 2450.2.a.w 1
105.w odd 12 1 4410.2.a.bd 1
105.x even 12 1 630.2.k.b 2
105.x even 12 1 4410.2.a.bm 1
140.w even 12 1 560.2.q.g 2
140.w even 12 1 3920.2.a.p 1
140.x odd 12 1 3920.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 5.c odd 4 1
70.2.e.c 2 35.l odd 12 1
350.2.e.e 2 5.c odd 4 1
350.2.e.e 2 35.l odd 12 1
350.2.j.b 4 1.a even 1 1 trivial
350.2.j.b 4 5.b even 2 1 inner
350.2.j.b 4 7.c even 3 1 inner
350.2.j.b 4 35.j even 6 1 inner
490.2.a.b 1 35.k even 12 1
490.2.a.c 1 35.l odd 12 1
490.2.e.h 2 35.f even 4 1
490.2.e.h 2 35.k even 12 1
560.2.q.g 2 20.e even 4 1
560.2.q.g 2 140.w even 12 1
630.2.k.b 2 15.e even 4 1
630.2.k.b 2 105.x even 12 1
2450.2.a.w 1 35.l odd 12 1
2450.2.a.bc 1 35.k even 12 1
2450.2.c.g 2 7.c even 3 1
2450.2.c.g 2 35.j even 6 1
2450.2.c.l 2 7.d odd 6 1
2450.2.c.l 2 35.i odd 6 1
3920.2.a.p 1 140.w even 12 1
3920.2.a.bc 1 140.x odd 12 1
4410.2.a.bd 1 105.w odd 12 1
4410.2.a.bm 1 105.x even 12 1

## Hecke kernels

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

 $$T_{3}^{4} - T_{3}^{2} + 1$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 - 13 T^{2} + T^{4}$$
$11$ $$( 36 - 6 T + T^{2} )^{2}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 4 - 2 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( -3 + T )^{4}$$
$31$ $$( 64 + 8 T + T^{2} )^{2}$$
$37$ $$256 - 16 T^{2} + T^{4}$$
$41$ $$( -9 + T )^{4}$$
$43$ $$( 49 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 36 + 6 T + T^{2} )^{2}$$
$61$ $$( 25 + 5 T + T^{2} )^{2}$$
$67$ $$625 - 25 T^{2} + T^{4}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$65536 - 256 T^{2} + T^{4}$$
$79$ $$( 4 - 2 T + T^{2} )^{2}$$
$83$ $$( 9 + T^{2} )^{2}$$
$89$ $$( 225 + 15 T + T^{2} )^{2}$$
$97$ $$( 196 + T^{2} )^{2}$$