# Properties

 Label 350.2.c.c Level 350 Weight 2 Character orbit 350.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} -i q^{3} - q^{4} + q^{6} + i q^{7} -i q^{8} + 2 q^{9} +O(q^{10})$$ $$q + i q^{2} -i q^{3} - q^{4} + q^{6} + i q^{7} -i q^{8} + 2 q^{9} + 3 q^{11} + i q^{12} -2 i q^{13} - q^{14} + q^{16} + 3 i q^{17} + 2 i q^{18} + 7 q^{19} + q^{21} + 3 i q^{22} - q^{24} + 2 q^{26} -5 i q^{27} -i q^{28} + 6 q^{29} -4 q^{31} + i q^{32} -3 i q^{33} -3 q^{34} -2 q^{36} + 8 i q^{37} + 7 i q^{38} -2 q^{39} -9 q^{41} + i q^{42} -8 i q^{43} -3 q^{44} -6 i q^{47} -i q^{48} - q^{49} + 3 q^{51} + 2 i q^{52} + 12 i q^{53} + 5 q^{54} + q^{56} -7 i q^{57} + 6 i q^{58} -12 q^{59} -10 q^{61} -4 i q^{62} + 2 i q^{63} - q^{64} + 3 q^{66} -7 i q^{67} -3 i q^{68} + 6 q^{71} -2 i q^{72} -5 i q^{73} -8 q^{74} -7 q^{76} + 3 i q^{77} -2 i q^{78} -14 q^{79} + q^{81} -9 i q^{82} + 9 i q^{83} - q^{84} + 8 q^{86} -6 i q^{87} -3 i q^{88} + 15 q^{89} + 2 q^{91} + 4 i q^{93} + 6 q^{94} + q^{96} -10 i q^{97} -i q^{98} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} + 4q^{9} + 6q^{11} - 2q^{14} + 2q^{16} + 14q^{19} + 2q^{21} - 2q^{24} + 4q^{26} + 12q^{29} - 8q^{31} - 6q^{34} - 4q^{36} - 4q^{39} - 18q^{41} - 6q^{44} - 2q^{49} + 6q^{51} + 10q^{54} + 2q^{56} - 24q^{59} - 20q^{61} - 2q^{64} + 6q^{66} + 12q^{71} - 16q^{74} - 14q^{76} - 28q^{79} + 2q^{81} - 2q^{84} + 16q^{86} + 30q^{89} + 4q^{91} + 12q^{94} + 2q^{96} + 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i 2.00000 0
99.2 1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i 2.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.c.c 2
3.b odd 2 1 3150.2.g.f 2
4.b odd 2 1 2800.2.g.i 2
5.b even 2 1 inner 350.2.c.c 2
5.c odd 4 1 350.2.a.a 1
5.c odd 4 1 350.2.a.e yes 1
7.b odd 2 1 2450.2.c.h 2
15.d odd 2 1 3150.2.g.f 2
15.e even 4 1 3150.2.a.m 1
15.e even 4 1 3150.2.a.x 1
20.d odd 2 1 2800.2.g.i 2
20.e even 4 1 2800.2.a.h 1
20.e even 4 1 2800.2.a.x 1
35.c odd 2 1 2450.2.c.h 2
35.f even 4 1 2450.2.a.m 1
35.f even 4 1 2450.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 5.c odd 4 1
350.2.a.e yes 1 5.c odd 4 1
350.2.c.c 2 1.a even 1 1 trivial
350.2.c.c 2 5.b even 2 1 inner
2450.2.a.m 1 35.f even 4 1
2450.2.a.x 1 35.f even 4 1
2450.2.c.h 2 7.b odd 2 1
2450.2.c.h 2 35.c odd 2 1
2800.2.a.h 1 20.e even 4 1
2800.2.a.x 1 20.e even 4 1
2800.2.g.i 2 4.b odd 2 1
2800.2.g.i 2 20.d odd 2 1
3150.2.a.m 1 15.e even 4 1
3150.2.a.x 1 15.e even 4 1
3150.2.g.f 2 3.b odd 2 1
3150.2.g.f 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 - 5 T^{2} + 9 T^{4}$$
$5$ 
$7$ $$1 + T^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 - 58 T^{2} + 2209 T^{4}$$
$53$ $$1 + 38 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$1 - 85 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 121 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 14 T + 79 T^{2} )^{2}$$
$83$ $$1 - 85 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 15 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$