# Properties

 Label 350.4.a.l Level $350$ Weight $4$ Character orbit 350.a Self dual yes Analytic conductor $20.651$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.6506685020$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 −8.00000 4.00000 0 −16.0000 7.00000 8.00000 37.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.a.l 1
5.b even 2 1 14.4.a.a 1
5.c odd 4 2 350.4.c.b 2
7.b odd 2 1 2450.4.a.bo 1
15.d odd 2 1 126.4.a.h 1
20.d odd 2 1 112.4.a.a 1
35.c odd 2 1 98.4.a.a 1
35.i odd 6 2 98.4.c.f 2
35.j even 6 2 98.4.c.d 2
40.e odd 2 1 448.4.a.o 1
40.f even 2 1 448.4.a.b 1
55.d odd 2 1 1694.4.a.g 1
60.h even 2 1 1008.4.a.s 1
65.d even 2 1 2366.4.a.h 1
105.g even 2 1 882.4.a.i 1
105.o odd 6 2 882.4.g.b 2
105.p even 6 2 882.4.g.k 2
140.c even 2 1 784.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 5.b even 2 1
98.4.a.a 1 35.c odd 2 1
98.4.c.d 2 35.j even 6 2
98.4.c.f 2 35.i odd 6 2
112.4.a.a 1 20.d odd 2 1
126.4.a.h 1 15.d odd 2 1
350.4.a.l 1 1.a even 1 1 trivial
350.4.c.b 2 5.c odd 4 2
448.4.a.b 1 40.f even 2 1
448.4.a.o 1 40.e odd 2 1
784.4.a.s 1 140.c even 2 1
882.4.a.i 1 105.g even 2 1
882.4.g.b 2 105.o odd 6 2
882.4.g.k 2 105.p even 6 2
1008.4.a.s 1 60.h even 2 1
1694.4.a.g 1 55.d odd 2 1
2366.4.a.h 1 65.d even 2 1
2450.4.a.bo 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(350))$$:

 $$T_{3} + 8$$ T3 + 8 $$T_{11} + 28$$ T11 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 8$$
$5$ $$T$$
$7$ $$T - 7$$
$11$ $$T + 28$$
$13$ $$T + 18$$
$17$ $$T + 74$$
$19$ $$T - 80$$
$23$ $$T - 112$$
$29$ $$T - 190$$
$31$ $$T - 72$$
$37$ $$T - 346$$
$41$ $$T - 162$$
$43$ $$T - 412$$
$47$ $$T + 24$$
$53$ $$T + 318$$
$59$ $$T + 200$$
$61$ $$T + 198$$
$67$ $$T - 716$$
$71$ $$T - 392$$
$73$ $$T + 538$$
$79$ $$T - 240$$
$83$ $$T - 1072$$
$89$ $$T - 810$$
$97$ $$T + 1354$$