# Properties

 Label 175.4.b.b Level $175$ Weight $4$ Character orbit 175.b Analytic conductor $10.325$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} -2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} +O(q^{10})$$ $$q + i q^{2} -2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} -8 q^{11} -14 i q^{12} + 28 i q^{13} -7 q^{14} + 41 q^{16} -54 i q^{17} + 23 i q^{18} + 110 q^{19} + 14 q^{21} -8 i q^{22} + 48 i q^{23} + 30 q^{24} -28 q^{26} -100 i q^{27} + 49 i q^{28} + 110 q^{29} + 12 q^{31} + 161 i q^{32} + 16 i q^{33} + 54 q^{34} + 161 q^{36} + 246 i q^{37} + 110 i q^{38} + 56 q^{39} + 182 q^{41} + 14 i q^{42} + 128 i q^{43} -56 q^{44} -48 q^{46} -324 i q^{47} -82 i q^{48} -49 q^{49} -108 q^{51} + 196 i q^{52} -162 i q^{53} + 100 q^{54} -105 q^{56} -220 i q^{57} + 110 i q^{58} -810 q^{59} -488 q^{61} + 12 i q^{62} + 161 i q^{63} + 167 q^{64} -16 q^{66} -244 i q^{67} -378 i q^{68} + 96 q^{69} -768 q^{71} + 345 i q^{72} -702 i q^{73} -246 q^{74} + 770 q^{76} -56 i q^{77} + 56 i q^{78} -440 q^{79} + 421 q^{81} + 182 i q^{82} -1302 i q^{83} + 98 q^{84} -128 q^{86} -220 i q^{87} -120 i q^{88} -730 q^{89} -196 q^{91} + 336 i q^{92} -24 i q^{93} + 324 q^{94} + 322 q^{96} -294 i q^{97} -49 i q^{98} -184 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{4} + 4q^{6} + 46q^{9} + O(q^{10})$$ $$2q + 14q^{4} + 4q^{6} + 46q^{9} - 16q^{11} - 14q^{14} + 82q^{16} + 220q^{19} + 28q^{21} + 60q^{24} - 56q^{26} + 220q^{29} + 24q^{31} + 108q^{34} + 322q^{36} + 112q^{39} + 364q^{41} - 112q^{44} - 96q^{46} - 98q^{49} - 216q^{51} + 200q^{54} - 210q^{56} - 1620q^{59} - 976q^{61} + 334q^{64} - 32q^{66} + 192q^{69} - 1536q^{71} - 492q^{74} + 1540q^{76} - 880q^{79} + 842q^{81} + 196q^{84} - 256q^{86} - 1460q^{89} - 392q^{91} + 648q^{94} + 644q^{96} - 368q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

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

## 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 2.00000i 7.00000 0 2.00000 7.00000i 15.0000i 23.0000 0
99.2 1.00000i 2.00000i 7.00000 0 2.00000 7.00000i 15.0000i 23.0000 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 175.4.b.b 2
5.b even 2 1 inner 175.4.b.b 2
5.c odd 4 1 7.4.a.a 1
5.c odd 4 1 175.4.a.b 1
15.e even 4 1 63.4.a.b 1
15.e even 4 1 1575.4.a.e 1
20.e even 4 1 112.4.a.f 1
35.f even 4 1 49.4.a.b 1
35.f even 4 1 1225.4.a.j 1
35.k even 12 2 49.4.c.b 2
35.l odd 12 2 49.4.c.c 2
40.i odd 4 1 448.4.a.i 1
40.k even 4 1 448.4.a.e 1
55.e even 4 1 847.4.a.b 1
60.l odd 4 1 1008.4.a.c 1
65.h odd 4 1 1183.4.a.b 1
85.g odd 4 1 2023.4.a.a 1
105.k odd 4 1 441.4.a.i 1
105.w odd 12 2 441.4.e.e 2
105.x even 12 2 441.4.e.h 2
140.j odd 4 1 784.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 5.c odd 4 1
49.4.a.b 1 35.f even 4 1
49.4.c.b 2 35.k even 12 2
49.4.c.c 2 35.l odd 12 2
63.4.a.b 1 15.e even 4 1
112.4.a.f 1 20.e even 4 1
175.4.a.b 1 5.c odd 4 1
175.4.b.b 2 1.a even 1 1 trivial
175.4.b.b 2 5.b even 2 1 inner
441.4.a.i 1 105.k odd 4 1
441.4.e.e 2 105.w odd 12 2
441.4.e.h 2 105.x even 12 2
448.4.a.e 1 40.k even 4 1
448.4.a.i 1 40.i odd 4 1
784.4.a.g 1 140.j odd 4 1
847.4.a.b 1 55.e even 4 1
1008.4.a.c 1 60.l odd 4 1
1183.4.a.b 1 65.h odd 4 1
1225.4.a.j 1 35.f even 4 1
1575.4.a.e 1 15.e even 4 1
2023.4.a.a 1 85.g odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(175, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{3}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( 8 + T )^{2}$$
$13$ $$784 + T^{2}$$
$17$ $$2916 + T^{2}$$
$19$ $$( -110 + T )^{2}$$
$23$ $$2304 + T^{2}$$
$29$ $$( -110 + T )^{2}$$
$31$ $$( -12 + T )^{2}$$
$37$ $$60516 + T^{2}$$
$41$ $$( -182 + T )^{2}$$
$43$ $$16384 + T^{2}$$
$47$ $$104976 + T^{2}$$
$53$ $$26244 + T^{2}$$
$59$ $$( 810 + T )^{2}$$
$61$ $$( 488 + T )^{2}$$
$67$ $$59536 + T^{2}$$
$71$ $$( 768 + T )^{2}$$
$73$ $$492804 + T^{2}$$
$79$ $$( 440 + T )^{2}$$
$83$ $$1695204 + T^{2}$$
$89$ $$( 730 + T )^{2}$$
$97$ $$86436 + T^{2}$$
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