# Properties

 Label 175.4.a.b Level $175$ Weight $4$ Character orbit 175.a Self dual yes Analytic conductor $10.325$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.3253342510$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + 2q^{3} - 7q^{4} + 2q^{6} + 7q^{7} - 15q^{8} - 23q^{9} + O(q^{10})$$ $$q + q^{2} + 2q^{3} - 7q^{4} + 2q^{6} + 7q^{7} - 15q^{8} - 23q^{9} - 8q^{11} - 14q^{12} - 28q^{13} + 7q^{14} + 41q^{16} - 54q^{17} - 23q^{18} - 110q^{19} + 14q^{21} - 8q^{22} - 48q^{23} - 30q^{24} - 28q^{26} - 100q^{27} - 49q^{28} - 110q^{29} + 12q^{31} + 161q^{32} - 16q^{33} - 54q^{34} + 161q^{36} + 246q^{37} - 110q^{38} - 56q^{39} + 182q^{41} + 14q^{42} - 128q^{43} + 56q^{44} - 48q^{46} - 324q^{47} + 82q^{48} + 49q^{49} - 108q^{51} + 196q^{52} + 162q^{53} - 100q^{54} - 105q^{56} - 220q^{57} - 110q^{58} + 810q^{59} - 488q^{61} + 12q^{62} - 161q^{63} - 167q^{64} - 16q^{66} - 244q^{67} + 378q^{68} - 96q^{69} - 768q^{71} + 345q^{72} + 702q^{73} + 246q^{74} + 770q^{76} - 56q^{77} - 56q^{78} + 440q^{79} + 421q^{81} + 182q^{82} + 1302q^{83} - 98q^{84} - 128q^{86} - 220q^{87} + 120q^{88} + 730q^{89} - 196q^{91} + 336q^{92} + 24q^{93} - 324q^{94} + 322q^{96} - 294q^{97} + 49q^{98} + 184q^{99} + O(q^{100})$$

## 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
1.00000 2.00000 −7.00000 0 2.00000 7.00000 −15.0000 −23.0000 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
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 175.4.a.b 1
3.b odd 2 1 1575.4.a.e 1
5.b even 2 1 7.4.a.a 1
5.c odd 4 2 175.4.b.b 2
7.b odd 2 1 1225.4.a.j 1
15.d odd 2 1 63.4.a.b 1
20.d odd 2 1 112.4.a.f 1
35.c odd 2 1 49.4.a.b 1
35.i odd 6 2 49.4.c.b 2
35.j even 6 2 49.4.c.c 2
40.e odd 2 1 448.4.a.e 1
40.f even 2 1 448.4.a.i 1
55.d odd 2 1 847.4.a.b 1
60.h even 2 1 1008.4.a.c 1
65.d even 2 1 1183.4.a.b 1
85.c even 2 1 2023.4.a.a 1
105.g even 2 1 441.4.a.i 1
105.o odd 6 2 441.4.e.h 2
105.p even 6 2 441.4.e.e 2
140.c even 2 1 784.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 5.b even 2 1
49.4.a.b 1 35.c odd 2 1
49.4.c.b 2 35.i odd 6 2
49.4.c.c 2 35.j even 6 2
63.4.a.b 1 15.d odd 2 1
112.4.a.f 1 20.d odd 2 1
175.4.a.b 1 1.a even 1 1 trivial
175.4.b.b 2 5.c odd 4 2
441.4.a.i 1 105.g even 2 1
441.4.e.e 2 105.p even 6 2
441.4.e.h 2 105.o odd 6 2
448.4.a.e 1 40.e odd 2 1
448.4.a.i 1 40.f even 2 1
784.4.a.g 1 140.c even 2 1
847.4.a.b 1 55.d odd 2 1
1008.4.a.c 1 60.h even 2 1
1183.4.a.b 1 65.d even 2 1
1225.4.a.j 1 7.b odd 2 1
1575.4.a.e 1 3.b odd 2 1
2023.4.a.a 1 85.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$-7 + T$$
$11$ $$8 + T$$
$13$ $$28 + T$$
$17$ $$54 + T$$
$19$ $$110 + T$$
$23$ $$48 + T$$
$29$ $$110 + T$$
$31$ $$-12 + T$$
$37$ $$-246 + T$$
$41$ $$-182 + T$$
$43$ $$128 + T$$
$47$ $$324 + T$$
$53$ $$-162 + T$$
$59$ $$-810 + T$$
$61$ $$488 + T$$
$67$ $$244 + T$$
$71$ $$768 + T$$
$73$ $$-702 + T$$
$79$ $$-440 + T$$
$83$ $$-1302 + T$$
$89$ $$-730 + T$$
$97$ $$294 + T$$