# Properties

 Label 175.3.c.a Level $175$ Weight $3$ Character orbit 175.c Analytic conductor $4.768$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 175.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.76840462631$$ 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 7) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} -5 q^{4} + 7 i q^{7} -3 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{2} -5 q^{4} + 7 i q^{7} -3 i q^{8} -9 q^{9} -6 q^{11} -21 q^{14} -11 q^{16} -27 i q^{18} -18 i q^{22} + 18 i q^{23} -35 i q^{28} + 54 q^{29} -45 i q^{32} + 45 q^{36} + 38 i q^{37} + 58 i q^{43} + 30 q^{44} -54 q^{46} -49 q^{49} -6 i q^{53} + 21 q^{56} + 162 i q^{58} -63 i q^{63} + 91 q^{64} + 118 i q^{67} + 114 q^{71} + 27 i q^{72} -114 q^{74} -42 i q^{77} + 94 q^{79} + 81 q^{81} -174 q^{86} + 18 i q^{88} -90 i q^{92} -147 i q^{98} + 54 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{4} - 18q^{9} + O(q^{10})$$ $$2q - 10q^{4} - 18q^{9} - 12q^{11} - 42q^{14} - 22q^{16} + 108q^{29} + 90q^{36} + 60q^{44} - 108q^{46} - 98q^{49} + 42q^{56} + 182q^{64} + 228q^{71} - 228q^{74} + 188q^{79} + 162q^{81} - 348q^{86} + 108q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
174.1
 − 1.00000i 1.00000i
3.00000i 0 −5.00000 0 0 7.00000i 3.00000i −9.00000 0
174.2 3.00000i 0 −5.00000 0 0 7.00000i 3.00000i −9.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
5.b even 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.3.c.a 2
5.b even 2 1 inner 175.3.c.a 2
5.c odd 4 1 7.3.b.a 1
5.c odd 4 1 175.3.d.a 1
7.b odd 2 1 CM 175.3.c.a 2
15.e even 4 1 63.3.d.a 1
20.e even 4 1 112.3.c.a 1
35.c odd 2 1 inner 175.3.c.a 2
35.f even 4 1 7.3.b.a 1
35.f even 4 1 175.3.d.a 1
35.k even 12 2 49.3.d.a 2
35.l odd 12 2 49.3.d.a 2
40.i odd 4 1 448.3.c.a 1
40.k even 4 1 448.3.c.b 1
60.l odd 4 1 1008.3.f.a 1
105.k odd 4 1 63.3.d.a 1
105.w odd 12 2 441.3.m.a 2
105.x even 12 2 441.3.m.a 2
140.j odd 4 1 112.3.c.a 1
140.w even 12 2 784.3.s.a 2
140.x odd 12 2 784.3.s.a 2
280.s even 4 1 448.3.c.a 1
280.y odd 4 1 448.3.c.b 1
420.w even 4 1 1008.3.f.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 5.c odd 4 1
7.3.b.a 1 35.f even 4 1
49.3.d.a 2 35.k even 12 2
49.3.d.a 2 35.l odd 12 2
63.3.d.a 1 15.e even 4 1
63.3.d.a 1 105.k odd 4 1
112.3.c.a 1 20.e even 4 1
112.3.c.a 1 140.j odd 4 1
175.3.c.a 2 1.a even 1 1 trivial
175.3.c.a 2 5.b even 2 1 inner
175.3.c.a 2 7.b odd 2 1 CM
175.3.c.a 2 35.c odd 2 1 inner
175.3.d.a 1 5.c odd 4 1
175.3.d.a 1 35.f even 4 1
441.3.m.a 2 105.w odd 12 2
441.3.m.a 2 105.x even 12 2
448.3.c.a 1 40.i odd 4 1
448.3.c.a 1 280.s even 4 1
448.3.c.b 1 40.k even 4 1
448.3.c.b 1 280.y odd 4 1
784.3.s.a 2 140.w even 12 2
784.3.s.a 2 140.x odd 12 2
1008.3.f.a 1 60.l odd 4 1
1008.3.f.a 1 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 16 T^{4}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ 1
$7$ $$1 + 49 T^{2}$$
$11$ $$( 1 + 6 T + 121 T^{2} )^{2}$$
$13$ $$( 1 + 169 T^{2} )^{2}$$
$17$ $$( 1 + 289 T^{2} )^{2}$$
$19$ $$( 1 - 19 T )^{2}( 1 + 19 T )^{2}$$
$23$ $$1 - 734 T^{2} + 279841 T^{4}$$
$29$ $$( 1 - 54 T + 841 T^{2} )^{2}$$
$31$ $$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$
$37$ $$1 - 1294 T^{2} + 1874161 T^{4}$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$1 - 334 T^{2} + 3418801 T^{4}$$
$47$ $$( 1 + 2209 T^{2} )^{2}$$
$53$ $$1 - 5582 T^{2} + 7890481 T^{4}$$
$59$ $$( 1 - 59 T )^{2}( 1 + 59 T )^{2}$$
$61$ $$( 1 - 61 T )^{2}( 1 + 61 T )^{2}$$
$67$ $$1 + 4946 T^{2} + 20151121 T^{4}$$
$71$ $$( 1 - 114 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 + 5329 T^{2} )^{2}$$
$79$ $$( 1 - 94 T + 6241 T^{2} )^{2}$$
$83$ $$( 1 + 6889 T^{2} )^{2}$$
$89$ $$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$
$97$ $$( 1 + 9409 T^{2} )^{2}$$