# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 3 q^{6} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 3 q^{6} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} + 2 \zeta_{12}^{2} q^{11} + 3 \zeta_{12} q^{12} + ( -1 - 2 \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{17} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{18} + ( -6 + 6 \zeta_{12}^{2} ) q^{19} + ( -9 + 3 \zeta_{12}^{2} ) q^{21} + 2 \zeta_{12}^{3} q^{22} -3 \zeta_{12} q^{23} + 3 \zeta_{12}^{2} q^{24} -9 \zeta_{12}^{3} q^{27} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} -9 q^{29} + 4 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 6 \zeta_{12} q^{33} + 4 q^{34} + 6 q^{36} -4 \zeta_{12} q^{37} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{38} -7 q^{41} + ( -9 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{42} -5 \zeta_{12}^{3} q^{43} + ( -2 + 2 \zeta_{12}^{2} ) q^{44} -3 \zeta_{12}^{2} q^{46} + 8 \zeta_{12} q^{47} + 3 \zeta_{12}^{3} q^{48} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + ( 12 - 12 \zeta_{12}^{2} ) q^{51} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( 9 - 9 \zeta_{12}^{2} ) q^{54} + ( 2 - 3 \zeta_{12}^{2} ) q^{56} + 18 \zeta_{12}^{3} q^{57} -9 \zeta_{12} q^{58} + 10 \zeta_{12}^{2} q^{59} + ( -1 + \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} + ( -12 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} - q^{64} + 6 \zeta_{12}^{2} q^{66} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{67} + 4 \zeta_{12} q^{68} -9 q^{69} + 2 q^{71} + 6 \zeta_{12} q^{72} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} -4 \zeta_{12}^{2} q^{74} -6 q^{76} + ( -2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + ( 10 - 10 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} -7 \zeta_{12} q^{82} -7 \zeta_{12}^{3} q^{83} + ( -3 - 6 \zeta_{12}^{2} ) q^{84} + ( 5 - 5 \zeta_{12}^{2} ) q^{86} + ( -27 \zeta_{12} + 27 \zeta_{12}^{3} ) q^{87} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{88} + ( 1 - \zeta_{12}^{2} ) q^{89} -3 \zeta_{12}^{3} q^{92} + 12 \zeta_{12} q^{93} + 8 \zeta_{12}^{2} q^{94} + ( -3 + 3 \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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 12q^{6} + 12q^{9} + O(q^{10})$$ $$4q + 2q^{4} + 12q^{6} + 12q^{9} + 4q^{11} - 8q^{14} - 2q^{16} - 12q^{19} - 30q^{21} + 6q^{24} - 36q^{29} + 8q^{31} + 16q^{34} + 24q^{36} - 28q^{41} - 4q^{44} - 6q^{46} + 26q^{49} + 24q^{51} + 18q^{54} + 2q^{56} + 20q^{59} - 2q^{61} - 4q^{64} + 12q^{66} - 36q^{69} + 8q^{71} - 8q^{74} - 24q^{76} + 20q^{79} - 18q^{81} - 24q^{84} + 10q^{86} + 2q^{89} + 16q^{94} - 6q^{96} + 48q^{99} + O(q^{100})$$

## 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 −2.59808 + 1.50000i 0.500000 + 0.866025i 0 3.00000 2.59808 + 0.500000i 1.00000i 3.00000 5.19615i 0
149.2 0.866025 + 0.500000i 2.59808 1.50000i 0.500000 + 0.866025i 0 3.00000 −2.59808 0.500000i 1.00000i 3.00000 5.19615i 0
249.1 −0.866025 + 0.500000i −2.59808 1.50000i 0.500000 0.866025i 0 3.00000 2.59808 0.500000i 1.00000i 3.00000 + 5.19615i 0
249.2 0.866025 0.500000i 2.59808 + 1.50000i 0.500000 0.866025i 0 3.00000 −2.59808 + 0.500000i 1.00000i 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
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.f 4
5.b even 2 1 inner 350.2.j.f 4
5.c odd 4 1 70.2.e.a 2
5.c odd 4 1 350.2.e.l 2
7.c even 3 1 inner 350.2.j.f 4
7.c even 3 1 2450.2.c.s 2
7.d odd 6 1 2450.2.c.a 2
15.e even 4 1 630.2.k.f 2
20.e even 4 1 560.2.q.i 2
35.f even 4 1 490.2.e.f 2
35.i odd 6 1 2450.2.c.a 2
35.j even 6 1 inner 350.2.j.f 4
35.j even 6 1 2450.2.c.s 2
35.k even 12 1 490.2.a.e 1
35.k even 12 1 490.2.e.f 2
35.k even 12 1 2450.2.a.q 1
35.l odd 12 1 70.2.e.a 2
35.l odd 12 1 350.2.e.l 2
35.l odd 12 1 490.2.a.k 1
35.l odd 12 1 2450.2.a.b 1
105.w odd 12 1 4410.2.a.h 1
105.x even 12 1 630.2.k.f 2
105.x even 12 1 4410.2.a.r 1
140.w even 12 1 560.2.q.i 2
140.w even 12 1 3920.2.a.b 1
140.x odd 12 1 3920.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 5.c odd 4 1
70.2.e.a 2 35.l odd 12 1
350.2.e.l 2 5.c odd 4 1
350.2.e.l 2 35.l odd 12 1
350.2.j.f 4 1.a even 1 1 trivial
350.2.j.f 4 5.b even 2 1 inner
350.2.j.f 4 7.c even 3 1 inner
350.2.j.f 4 35.j even 6 1 inner
490.2.a.e 1 35.k even 12 1
490.2.a.k 1 35.l odd 12 1
490.2.e.f 2 35.f even 4 1
490.2.e.f 2 35.k even 12 1
560.2.q.i 2 20.e even 4 1
560.2.q.i 2 140.w even 12 1
630.2.k.f 2 15.e even 4 1
630.2.k.f 2 105.x even 12 1
2450.2.a.b 1 35.l odd 12 1
2450.2.a.q 1 35.k even 12 1
2450.2.c.a 2 7.d odd 6 1
2450.2.c.a 2 35.i odd 6 1
2450.2.c.s 2 7.c even 3 1
2450.2.c.s 2 35.j even 6 1
3920.2.a.b 1 140.w even 12 1
3920.2.a.bk 1 140.x odd 12 1
4410.2.a.h 1 105.w odd 12 1
4410.2.a.r 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 $$S_{2}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{4} - 9 T_{3}^{2} + 81$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$81 - 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 - 13 T^{2} + T^{4}$$
$11$ $$( 4 - 2 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 36 + 6 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( 9 + T )^{4}$$
$31$ $$( 16 - 4 T + T^{2} )^{2}$$
$37$ $$256 - 16 T^{2} + T^{4}$$
$41$ $$( 7 + T )^{4}$$
$43$ $$( 25 + T^{2} )^{2}$$
$47$ $$4096 - 64 T^{2} + T^{4}$$
$53$ $$16 - 4 T^{2} + T^{4}$$
$59$ $$( 100 - 10 T + T^{2} )^{2}$$
$61$ $$( 1 + T + T^{2} )^{2}$$
$67$ $$6561 - 81 T^{2} + T^{4}$$
$71$ $$( -2 + T )^{4}$$
$73$ $$256 - 16 T^{2} + T^{4}$$
$79$ $$( 100 - 10 T + T^{2} )^{2}$$
$83$ $$( 49 + T^{2} )^{2}$$
$89$ $$( 1 - T + T^{2} )^{2}$$
$97$ $$( 196 + T^{2} )^{2}$$
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