# Properties

 Label 350.2.j.c 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + 2 \zeta_{12}^{2} q^{11} + ( 2 + \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{17} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{18} + 2 \zeta_{12}^{3} q^{22} -3 \zeta_{12} q^{23} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{28} -6 q^{29} + 7 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 7 q^{34} -3 q^{36} -4 \zeta_{12} q^{37} -7 q^{41} -8 \zeta_{12}^{3} q^{43} + ( -2 + 2 \zeta_{12}^{2} ) q^{44} -3 \zeta_{12}^{2} q^{46} -7 \zeta_{12} q^{47} + ( 8 - 3 \zeta_{12}^{2} ) q^{49} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + ( -1 + 3 \zeta_{12}^{2} ) q^{56} -6 \zeta_{12} q^{58} -14 \zeta_{12}^{2} q^{59} + ( 14 - 14 \zeta_{12}^{2} ) q^{61} + 7 \zeta_{12}^{3} q^{62} + ( -3 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{63} - q^{64} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{67} + 7 \zeta_{12} q^{68} - q^{71} -3 \zeta_{12} q^{72} + ( 14 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{73} -4 \zeta_{12}^{2} q^{74} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{77} + ( -11 + 11 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} -7 \zeta_{12} q^{82} + 14 \zeta_{12}^{3} q^{83} + ( 8 - 8 \zeta_{12}^{2} ) q^{86} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{88} + ( 7 - 7 \zeta_{12}^{2} ) q^{89} -3 \zeta_{12}^{3} q^{92} -7 \zeta_{12}^{2} q^{94} + 7 \zeta_{12}^{3} q^{97} + ( 8 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{98} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 6q^{9} + 4q^{11} + 10q^{14} - 2q^{16} - 24q^{29} + 14q^{31} + 28q^{34} - 12q^{36} - 28q^{41} - 4q^{44} - 6q^{46} + 26q^{49} + 2q^{56} - 28q^{59} + 28q^{61} - 4q^{64} - 4q^{71} - 8q^{74} - 22q^{79} - 18q^{81} + 16q^{86} + 14q^{89} - 14q^{94} - 24q^{99} + O(q^{100})$$

## 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 0.500000 + 0.866025i 0 0 −2.59808 + 0.500000i 1.00000i −1.50000 + 2.59808i 0
149.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 2.59808 0.500000i 1.00000i −1.50000 + 2.59808i 0
249.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.59808 0.500000i 1.00000i −1.50000 2.59808i 0
249.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.59808 + 0.500000i 1.00000i −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
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.c 4
5.b even 2 1 inner 350.2.j.c 4
5.c odd 4 1 350.2.e.d 2
5.c odd 4 1 350.2.e.i yes 2
7.c even 3 1 inner 350.2.j.c 4
7.c even 3 1 2450.2.c.i 2
7.d odd 6 1 2450.2.c.j 2
35.i odd 6 1 2450.2.c.j 2
35.j even 6 1 inner 350.2.j.c 4
35.j even 6 1 2450.2.c.i 2
35.k even 12 1 2450.2.a.h 1
35.k even 12 1 2450.2.a.z 1
35.l odd 12 1 350.2.e.d 2
35.l odd 12 1 350.2.e.i yes 2
35.l odd 12 1 2450.2.a.i 1
35.l odd 12 1 2450.2.a.y 1

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

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 + 3 T^{2} + 9 T^{4} )^{2}$$
$5$ 
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 13 T^{2} )^{4}$$
$17$ $$1 - 15 T^{2} - 64 T^{4} - 4335 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 19 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 7 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} )$$
$59$ $$( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 13 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2}$$
$67$ $$1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + T + 71 T^{2} )^{4}$$
$73$ $$1 - 50 T^{2} - 2829 T^{4} - 266450 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 30 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 7 T - 40 T^{2} - 623 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 145 T^{2} + 9409 T^{4} )^{2}$$