# Properties

 Label 350.2.e.j 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + ( - 2 \zeta_{6} + 3) q^{7} - q^{8} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^2 + (z - 1) * q^4 + (-2*z + 3) * q^7 - q^8 + 3*z * q^9 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + ( - 2 \zeta_{6} + 3) q^{7} - q^{8} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + 5 q^{13} + (\zeta_{6} + 2) q^{14} - \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + (3 \zeta_{6} - 3) q^{18} + 5 \zeta_{6} q^{19} - 3 q^{22} - 7 \zeta_{6} q^{23} + 5 \zeta_{6} q^{26} + (3 \zeta_{6} - 1) q^{28} - 4 q^{29} + ( - 2 \zeta_{6} + 2) q^{31} + ( - \zeta_{6} + 1) q^{32} - 2 q^{34} - 3 q^{36} + \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{38} + 3 q^{41} - 2 q^{43} - 3 \zeta_{6} q^{44} + ( - 7 \zeta_{6} + 7) q^{46} - 7 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + (5 \zeta_{6} - 5) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + (2 \zeta_{6} - 3) q^{56} - 4 \zeta_{6} q^{58} + ( - 4 \zeta_{6} + 4) q^{59} - 6 \zeta_{6} q^{61} + 2 q^{62} + (3 \zeta_{6} + 6) q^{63} + q^{64} + ( - 2 \zeta_{6} + 2) q^{67} - 2 \zeta_{6} q^{68} - 6 q^{71} - 3 \zeta_{6} q^{72} + (16 \zeta_{6} - 16) q^{73} + (\zeta_{6} - 1) q^{74} - 5 q^{76} + (9 \zeta_{6} - 3) q^{77} - 14 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + 3 \zeta_{6} q^{82} + 6 q^{83} - 2 \zeta_{6} q^{86} + ( - 3 \zeta_{6} + 3) q^{88} - 2 \zeta_{6} q^{89} + ( - 10 \zeta_{6} + 15) q^{91} + 7 q^{92} + ( - 7 \zeta_{6} + 7) q^{94} + 12 q^{97} + ( - 3 \zeta_{6} + 8) q^{98} - 9 q^{99} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^4 + (-2*z + 3) * q^7 - q^8 + 3*z * q^9 + (3*z - 3) * q^11 + 5 * q^13 + (z + 2) * q^14 - z * q^16 + (2*z - 2) * q^17 + (3*z - 3) * q^18 + 5*z * q^19 - 3 * q^22 - 7*z * q^23 + 5*z * q^26 + (3*z - 1) * q^28 - 4 * q^29 + (-2*z + 2) * q^31 + (-z + 1) * q^32 - 2 * q^34 - 3 * q^36 + z * q^37 + (5*z - 5) * q^38 + 3 * q^41 - 2 * q^43 - 3*z * q^44 + (-7*z + 7) * q^46 - 7*z * q^47 + (-8*z + 5) * q^49 + (5*z - 5) * q^52 + (-9*z + 9) * q^53 + (2*z - 3) * q^56 - 4*z * q^58 + (-4*z + 4) * q^59 - 6*z * q^61 + 2 * q^62 + (3*z + 6) * q^63 + q^64 + (-2*z + 2) * q^67 - 2*z * q^68 - 6 * q^71 - 3*z * q^72 + (16*z - 16) * q^73 + (z - 1) * q^74 - 5 * q^76 + (9*z - 3) * q^77 - 14*z * q^79 + (9*z - 9) * q^81 + 3*z * q^82 + 6 * q^83 - 2*z * q^86 + (-3*z + 3) * q^88 - 2*z * q^89 + (-10*z + 15) * q^91 + 7 * q^92 + (-7*z + 7) * q^94 + 12 * q^97 + (-3*z + 8) * q^98 - 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + 4 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 - q^4 + 4 * q^7 - 2 * q^8 + 3 * q^9 $$2 q + q^{2} - q^{4} + 4 q^{7} - 2 q^{8} + 3 q^{9} - 3 q^{11} + 10 q^{13} + 5 q^{14} - q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} - 6 q^{22} - 7 q^{23} + 5 q^{26} + q^{28} - 8 q^{29} + 2 q^{31} + q^{32} - 4 q^{34} - 6 q^{36} + q^{37} - 5 q^{38} + 6 q^{41} - 4 q^{43} - 3 q^{44} + 7 q^{46} - 7 q^{47} + 2 q^{49} - 5 q^{52} + 9 q^{53} - 4 q^{56} - 4 q^{58} + 4 q^{59} - 6 q^{61} + 4 q^{62} + 15 q^{63} + 2 q^{64} + 2 q^{67} - 2 q^{68} - 12 q^{71} - 3 q^{72} - 16 q^{73} - q^{74} - 10 q^{76} + 3 q^{77} - 14 q^{79} - 9 q^{81} + 3 q^{82} + 12 q^{83} - 2 q^{86} + 3 q^{88} - 2 q^{89} + 20 q^{91} + 14 q^{92} + 7 q^{94} + 24 q^{97} + 13 q^{98} - 18 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^4 + 4 * q^7 - 2 * q^8 + 3 * q^9 - 3 * q^11 + 10 * q^13 + 5 * q^14 - q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 - 6 * q^22 - 7 * q^23 + 5 * q^26 + q^28 - 8 * q^29 + 2 * q^31 + q^32 - 4 * q^34 - 6 * q^36 + q^37 - 5 * q^38 + 6 * q^41 - 4 * q^43 - 3 * q^44 + 7 * q^46 - 7 * q^47 + 2 * q^49 - 5 * q^52 + 9 * q^53 - 4 * q^56 - 4 * q^58 + 4 * q^59 - 6 * q^61 + 4 * q^62 + 15 * q^63 + 2 * q^64 + 2 * q^67 - 2 * q^68 - 12 * q^71 - 3 * q^72 - 16 * q^73 - q^74 - 10 * q^76 + 3 * q^77 - 14 * q^79 - 9 * q^81 + 3 * q^82 + 12 * q^83 - 2 * q^86 + 3 * q^88 - 2 * q^89 + 20 * q^91 + 14 * q^92 + 7 * q^94 + 24 * q^97 + 13 * q^98 - 18 * q^99

## 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 2.00000 + 1.73205i −1.00000 1.50000 2.59808i 0
151.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 2.00000 1.73205i −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.j 2
5.b even 2 1 350.2.e.c 2
5.c odd 4 2 70.2.i.b 4
7.c even 3 1 inner 350.2.e.j 2
7.c even 3 1 2450.2.a.k 1
7.d odd 6 1 2450.2.a.j 1
15.e even 4 2 630.2.u.a 4
20.e even 4 2 560.2.bw.d 4
35.f even 4 2 490.2.i.a 4
35.i odd 6 1 2450.2.a.bb 1
35.j even 6 1 350.2.e.c 2
35.j even 6 1 2450.2.a.ba 1
35.k even 12 2 490.2.c.d 2
35.k even 12 2 490.2.i.a 4
35.l odd 12 2 70.2.i.b 4
35.l odd 12 2 490.2.c.a 2
105.x even 12 2 630.2.u.a 4
140.w even 12 2 560.2.bw.d 4

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

 $$T_{3}$$ T3 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9 $$T_{13} - 5$$ T13 - 5 $$T_{17}^{2} + 2T_{17} + 4$$ T17^2 + 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T - 5)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} - 2T + 4$$
$37$ $$T^{2} - T + 1$$
$41$ $$(T - 3)^{2}$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} + 7T + 49$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 6T + 36$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 16T + 256$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 2T + 4$$
$97$ $$(T - 12)^{2}$$