# Properties

 Label 350.2.j.d 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: $$\Q(\zeta_{12})$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} -3 \zeta_{12}^{2} q^{11} + 2 \zeta_{12} q^{12} + \zeta_{12}^{3} q^{13} + ( 3 - \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{18} + ( -1 + \zeta_{12}^{2} ) q^{19} + ( 4 - 6 \zeta_{12}^{2} ) q^{21} -3 \zeta_{12}^{3} q^{22} + 9 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{24} + ( -1 + \zeta_{12}^{2} ) q^{26} + 4 \zeta_{12}^{3} q^{27} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{28} -6 q^{29} -8 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} -6 \zeta_{12} q^{33} -6 q^{34} + q^{36} + 7 \zeta_{12} q^{37} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{38} + 2 \zeta_{12}^{2} q^{39} + 3 q^{41} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{42} -2 \zeta_{12}^{3} q^{43} + ( 3 - 3 \zeta_{12}^{2} ) q^{44} + 9 \zeta_{12}^{2} q^{46} -9 \zeta_{12} q^{47} + 2 \zeta_{12}^{3} q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -12 + 12 \zeta_{12}^{2} ) q^{51} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{52} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{53} + ( -4 + 4 \zeta_{12}^{2} ) q^{54} + ( 1 + 2 \zeta_{12}^{2} ) q^{56} + 2 \zeta_{12}^{3} q^{57} -6 \zeta_{12} q^{58} + ( -8 + 8 \zeta_{12}^{2} ) q^{61} -8 \zeta_{12}^{3} q^{62} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} - q^{64} -6 \zeta_{12}^{2} q^{66} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12} q^{68} + 18 q^{69} + \zeta_{12} q^{72} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + 7 \zeta_{12}^{2} q^{74} - q^{76} + ( -9 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{77} + 2 \zeta_{12}^{3} q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + 11 \zeta_{12}^{2} q^{81} + 3 \zeta_{12} q^{82} + ( 6 - 2 \zeta_{12}^{2} ) q^{84} + ( 2 - 2 \zeta_{12}^{2} ) q^{86} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{87} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{88} + ( 6 - 6 \zeta_{12}^{2} ) q^{89} + ( 1 + 2 \zeta_{12}^{2} ) q^{91} + 9 \zeta_{12}^{3} q^{92} -16 \zeta_{12} q^{93} -9 \zeta_{12}^{2} q^{94} + ( -2 + 2 \zeta_{12}^{2} ) q^{96} -10 \zeta_{12}^{3} q^{97} + ( 3 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 8q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} + 8q^{6} + 2q^{9} - 6q^{11} + 10q^{14} - 2q^{16} - 2q^{19} + 4q^{21} + 4q^{24} - 2q^{26} - 24q^{29} - 16q^{31} - 24q^{34} + 4q^{36} + 4q^{39} + 12q^{41} + 6q^{44} + 18q^{46} - 4q^{49} - 24q^{51} - 8q^{54} + 8q^{56} - 16q^{61} - 4q^{64} - 12q^{66} + 72q^{69} + 14q^{74} - 4q^{76} - 20q^{79} + 22q^{81} + 20q^{84} + 4q^{86} + 12q^{89} + 8q^{91} - 18q^{94} - 4q^{96} - 12q^{99} + O(q^{100})$$

## Character values

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

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

## 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 −1.73205 + 1.00000i 0.500000 + 0.866025i 0 2.00000 −1.73205 + 2.00000i 1.00000i 0.500000 0.866025i 0
149.2 0.866025 + 0.500000i 1.73205 1.00000i 0.500000 + 0.866025i 0 2.00000 1.73205 2.00000i 1.00000i 0.500000 0.866025i 0
249.1 −0.866025 + 0.500000i −1.73205 1.00000i 0.500000 0.866025i 0 2.00000 −1.73205 2.00000i 1.00000i 0.500000 + 0.866025i 0
249.2 0.866025 0.500000i 1.73205 + 1.00000i 0.500000 0.866025i 0 2.00000 1.73205 + 2.00000i 1.00000i 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
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.d 4
5.b even 2 1 inner 350.2.j.d 4
5.c odd 4 1 70.2.e.d 2
5.c odd 4 1 350.2.e.b 2
7.c even 3 1 inner 350.2.j.d 4
7.c even 3 1 2450.2.c.q 2
7.d odd 6 1 2450.2.c.e 2
15.e even 4 1 630.2.k.d 2
20.e even 4 1 560.2.q.b 2
35.f even 4 1 490.2.e.g 2
35.i odd 6 1 2450.2.c.e 2
35.j even 6 1 inner 350.2.j.d 4
35.j even 6 1 2450.2.c.q 2
35.k even 12 1 490.2.a.d 1
35.k even 12 1 490.2.e.g 2
35.k even 12 1 2450.2.a.v 1
35.l odd 12 1 70.2.e.d 2
35.l odd 12 1 350.2.e.b 2
35.l odd 12 1 490.2.a.a 1
35.l odd 12 1 2450.2.a.bf 1
105.w odd 12 1 4410.2.a.bg 1
105.x even 12 1 630.2.k.d 2
105.x even 12 1 4410.2.a.x 1
140.w even 12 1 560.2.q.b 2
140.w even 12 1 3920.2.a.bh 1
140.x odd 12 1 3920.2.a.e 1

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

## Hecke kernels

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

 $$T_{3}^{4} - 4 T_{3}^{2} + 16$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13}^{2} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8}$$
$5$ 
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 25 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 3 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 82 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 94 T^{2} + 9409 T^{4} )^{2}$$