# Properties

 Label 350.4.c.b Level $350$ Weight $4$ Character orbit 350.c Analytic conductor $20.651$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.6506685020$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 8 i q^{3} -4 q^{4} -16 q^{6} + 7 i q^{7} -8 i q^{8} -37 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} + 8 i q^{3} -4 q^{4} -16 q^{6} + 7 i q^{7} -8 i q^{8} -37 q^{9} -28 q^{11} -32 i q^{12} + 18 i q^{13} -14 q^{14} + 16 q^{16} -74 i q^{17} -74 i q^{18} -80 q^{19} -56 q^{21} -56 i q^{22} -112 i q^{23} + 64 q^{24} -36 q^{26} -80 i q^{27} -28 i q^{28} -190 q^{29} + 72 q^{31} + 32 i q^{32} -224 i q^{33} + 148 q^{34} + 148 q^{36} + 346 i q^{37} -160 i q^{38} -144 q^{39} + 162 q^{41} -112 i q^{42} -412 i q^{43} + 112 q^{44} + 224 q^{46} -24 i q^{47} + 128 i q^{48} -49 q^{49} + 592 q^{51} -72 i q^{52} + 318 i q^{53} + 160 q^{54} + 56 q^{56} -640 i q^{57} -380 i q^{58} + 200 q^{59} -198 q^{61} + 144 i q^{62} -259 i q^{63} -64 q^{64} + 448 q^{66} + 716 i q^{67} + 296 i q^{68} + 896 q^{69} + 392 q^{71} + 296 i q^{72} + 538 i q^{73} -692 q^{74} + 320 q^{76} -196 i q^{77} -288 i q^{78} -240 q^{79} -359 q^{81} + 324 i q^{82} -1072 i q^{83} + 224 q^{84} + 824 q^{86} -1520 i q^{87} + 224 i q^{88} -810 q^{89} -126 q^{91} + 448 i q^{92} + 576 i q^{93} + 48 q^{94} -256 q^{96} -1354 i q^{97} -98 i q^{98} + 1036 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} - 32q^{6} - 74q^{9} + O(q^{10})$$ $$2q - 8q^{4} - 32q^{6} - 74q^{9} - 56q^{11} - 28q^{14} + 32q^{16} - 160q^{19} - 112q^{21} + 128q^{24} - 72q^{26} - 380q^{29} + 144q^{31} + 296q^{34} + 296q^{36} - 288q^{39} + 324q^{41} + 224q^{44} + 448q^{46} - 98q^{49} + 1184q^{51} + 320q^{54} + 112q^{56} + 400q^{59} - 396q^{61} - 128q^{64} + 896q^{66} + 1792q^{69} + 784q^{71} - 1384q^{74} + 640q^{76} - 480q^{79} - 718q^{81} + 448q^{84} + 1648q^{86} - 1620q^{89} - 252q^{91} + 96q^{94} - 512q^{96} + 2072q^{99} + O(q^{100})$$

## 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
2.00000i 8.00000i −4.00000 0 −16.0000 7.00000i 8.00000i −37.0000 0
99.2 2.00000i 8.00000i −4.00000 0 −16.0000 7.00000i 8.00000i −37.0000 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 350.4.c.b 2
5.b even 2 1 inner 350.4.c.b 2
5.c odd 4 1 14.4.a.a 1
5.c odd 4 1 350.4.a.l 1
15.e even 4 1 126.4.a.h 1
20.e even 4 1 112.4.a.a 1
35.f even 4 1 98.4.a.a 1
35.f even 4 1 2450.4.a.bo 1
35.k even 12 2 98.4.c.f 2
35.l odd 12 2 98.4.c.d 2
40.i odd 4 1 448.4.a.b 1
40.k even 4 1 448.4.a.o 1
55.e even 4 1 1694.4.a.g 1
60.l odd 4 1 1008.4.a.s 1
65.h odd 4 1 2366.4.a.h 1
105.k odd 4 1 882.4.a.i 1
105.w odd 12 2 882.4.g.k 2
105.x even 12 2 882.4.g.b 2
140.j odd 4 1 784.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 5.c odd 4 1
98.4.a.a 1 35.f even 4 1
98.4.c.d 2 35.l odd 12 2
98.4.c.f 2 35.k even 12 2
112.4.a.a 1 20.e even 4 1
126.4.a.h 1 15.e even 4 1
350.4.a.l 1 5.c odd 4 1
350.4.c.b 2 1.a even 1 1 trivial
350.4.c.b 2 5.b even 2 1 inner
448.4.a.b 1 40.i odd 4 1
448.4.a.o 1 40.k even 4 1
784.4.a.s 1 140.j odd 4 1
882.4.a.i 1 105.k odd 4 1
882.4.g.b 2 105.x even 12 2
882.4.g.k 2 105.w odd 12 2
1008.4.a.s 1 60.l odd 4 1
1694.4.a.g 1 55.e even 4 1
2366.4.a.h 1 65.h odd 4 1
2450.4.a.bo 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{2} + 64$$ $$T_{11} + 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$64 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( 28 + T )^{2}$$
$13$ $$324 + T^{2}$$
$17$ $$5476 + T^{2}$$
$19$ $$( 80 + T )^{2}$$
$23$ $$12544 + T^{2}$$
$29$ $$( 190 + T )^{2}$$
$31$ $$( -72 + T )^{2}$$
$37$ $$119716 + T^{2}$$
$41$ $$( -162 + T )^{2}$$
$43$ $$169744 + T^{2}$$
$47$ $$576 + T^{2}$$
$53$ $$101124 + T^{2}$$
$59$ $$( -200 + T )^{2}$$
$61$ $$( 198 + T )^{2}$$
$67$ $$512656 + T^{2}$$
$71$ $$( -392 + T )^{2}$$
$73$ $$289444 + T^{2}$$
$79$ $$( 240 + T )^{2}$$
$83$ $$1149184 + T^{2}$$
$89$ $$( 810 + T )^{2}$$
$97$ $$1833316 + T^{2}$$