# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{4} + 14 q^{6} + ( -7 - 14 \zeta_{6} ) q^{7} + 24 q^{8} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{4} + 14 q^{6} + ( -7 - 14 \zeta_{6} ) q^{7} + 24 q^{8} -22 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} -28 \zeta_{6} q^{12} + 14 q^{13} + ( 28 - 42 \zeta_{6} ) q^{14} + 16 \zeta_{6} q^{16} + ( -21 + 21 \zeta_{6} ) q^{17} + ( 44 - 44 \zeta_{6} ) q^{18} -49 \zeta_{6} q^{19} + ( -147 + 49 \zeta_{6} ) q^{21} + 10 q^{22} -159 \zeta_{6} q^{23} + ( 168 - 168 \zeta_{6} ) q^{24} + 28 \zeta_{6} q^{26} + 35 q^{27} + ( -84 + 28 \zeta_{6} ) q^{28} + 58 q^{29} + ( -147 + 147 \zeta_{6} ) q^{31} + ( 160 - 160 \zeta_{6} ) q^{32} -35 \zeta_{6} q^{33} -42 q^{34} -88 q^{36} + 219 \zeta_{6} q^{37} + ( 98 - 98 \zeta_{6} ) q^{38} + ( 98 - 98 \zeta_{6} ) q^{39} + 350 q^{41} + ( -98 - 196 \zeta_{6} ) q^{42} + 124 q^{43} -20 \zeta_{6} q^{44} + ( 318 - 318 \zeta_{6} ) q^{46} + 525 \zeta_{6} q^{47} + 112 q^{48} + ( -147 + 392 \zeta_{6} ) q^{49} + 147 \zeta_{6} q^{51} + ( 56 - 56 \zeta_{6} ) q^{52} + ( 303 - 303 \zeta_{6} ) q^{53} + 70 \zeta_{6} q^{54} + ( -168 - 336 \zeta_{6} ) q^{56} -343 q^{57} + 116 \zeta_{6} q^{58} + ( 105 - 105 \zeta_{6} ) q^{59} + 413 \zeta_{6} q^{61} -294 q^{62} + ( -308 + 462 \zeta_{6} ) q^{63} + 448 q^{64} + ( 70 - 70 \zeta_{6} ) q^{66} + ( 415 - 415 \zeta_{6} ) q^{67} + 84 \zeta_{6} q^{68} -1113 q^{69} -432 q^{71} -528 \zeta_{6} q^{72} + ( -1113 + 1113 \zeta_{6} ) q^{73} + ( -438 + 438 \zeta_{6} ) q^{74} -196 q^{76} + ( -105 + 35 \zeta_{6} ) q^{77} + 196 q^{78} + 103 \zeta_{6} q^{79} + ( 839 - 839 \zeta_{6} ) q^{81} + 700 \zeta_{6} q^{82} -1092 q^{83} + ( -392 + 588 \zeta_{6} ) q^{84} + 248 \zeta_{6} q^{86} + ( 406 - 406 \zeta_{6} ) q^{87} + ( 120 - 120 \zeta_{6} ) q^{88} + 329 \zeta_{6} q^{89} + ( -98 - 196 \zeta_{6} ) q^{91} -636 q^{92} + 1029 \zeta_{6} q^{93} + ( -1050 + 1050 \zeta_{6} ) q^{94} -1120 \zeta_{6} q^{96} + 882 q^{97} + ( -784 + 490 \zeta_{6} ) q^{98} -110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 7q^{3} + 4q^{4} + 28q^{6} - 28q^{7} + 48q^{8} - 22q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 7q^{3} + 4q^{4} + 28q^{6} - 28q^{7} + 48q^{8} - 22q^{9} + 5q^{11} - 28q^{12} + 28q^{13} + 14q^{14} + 16q^{16} - 21q^{17} + 44q^{18} - 49q^{19} - 245q^{21} + 20q^{22} - 159q^{23} + 168q^{24} + 28q^{26} + 70q^{27} - 140q^{28} + 116q^{29} - 147q^{31} + 160q^{32} - 35q^{33} - 84q^{34} - 176q^{36} + 219q^{37} + 98q^{38} + 98q^{39} + 700q^{41} - 392q^{42} + 248q^{43} - 20q^{44} + 318q^{46} + 525q^{47} + 224q^{48} + 98q^{49} + 147q^{51} + 56q^{52} + 303q^{53} + 70q^{54} - 672q^{56} - 686q^{57} + 116q^{58} + 105q^{59} + 413q^{61} - 588q^{62} - 154q^{63} + 896q^{64} + 70q^{66} + 415q^{67} + 84q^{68} - 2226q^{69} - 864q^{71} - 528q^{72} - 1113q^{73} - 438q^{74} - 392q^{76} - 175q^{77} + 392q^{78} + 103q^{79} + 839q^{81} + 700q^{82} - 2184q^{83} - 196q^{84} + 248q^{86} + 406q^{87} + 120q^{88} + 329q^{89} - 392q^{91} - 1272q^{92} + 1029q^{93} - 1050q^{94} - 1120q^{96} + 1764q^{97} - 1078q^{98} - 220q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i 3.50000 + 6.06218i 2.00000 + 3.46410i 0 14.0000 −14.0000 + 12.1244i 24.0000 −11.0000 + 19.0526i 0
151.1 1.00000 + 1.73205i 3.50000 6.06218i 2.00000 3.46410i 0 14.0000 −14.0000 12.1244i 24.0000 −11.0000 19.0526i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.e.a 2
5.b even 2 1 7.4.c.a 2
5.c odd 4 2 175.4.k.a 4
7.c even 3 1 inner 175.4.e.a 2
7.c even 3 1 1225.4.a.c 1
7.d odd 6 1 1225.4.a.d 1
15.d odd 2 1 63.4.e.b 2
20.d odd 2 1 112.4.i.c 2
35.c odd 2 1 49.4.c.a 2
35.i odd 6 1 49.4.a.c 1
35.i odd 6 1 49.4.c.a 2
35.j even 6 1 7.4.c.a 2
35.j even 6 1 49.4.a.d 1
35.l odd 12 2 175.4.k.a 4
40.e odd 2 1 448.4.i.a 2
40.f even 2 1 448.4.i.f 2
105.g even 2 1 441.4.e.k 2
105.o odd 6 1 63.4.e.b 2
105.o odd 6 1 441.4.a.d 1
105.p even 6 1 441.4.a.e 1
105.p even 6 1 441.4.e.k 2
140.p odd 6 1 112.4.i.c 2
140.p odd 6 1 784.4.a.b 1
140.s even 6 1 784.4.a.r 1
280.bf even 6 1 448.4.i.f 2
280.bi odd 6 1 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 5.b even 2 1
7.4.c.a 2 35.j even 6 1
49.4.a.c 1 35.i odd 6 1
49.4.a.d 1 35.j even 6 1
49.4.c.a 2 35.c odd 2 1
49.4.c.a 2 35.i odd 6 1
63.4.e.b 2 15.d odd 2 1
63.4.e.b 2 105.o odd 6 1
112.4.i.c 2 20.d odd 2 1
112.4.i.c 2 140.p odd 6 1
175.4.e.a 2 1.a even 1 1 trivial
175.4.e.a 2 7.c even 3 1 inner
175.4.k.a 4 5.c odd 4 2
175.4.k.a 4 35.l odd 12 2
441.4.a.d 1 105.o odd 6 1
441.4.a.e 1 105.p even 6 1
441.4.e.k 2 105.g even 2 1
441.4.e.k 2 105.p even 6 1
448.4.i.a 2 40.e odd 2 1
448.4.i.a 2 280.bi odd 6 1
448.4.i.f 2 40.f even 2 1
448.4.i.f 2 280.bf even 6 1
784.4.a.b 1 140.p odd 6 1
784.4.a.r 1 140.s even 6 1
1225.4.a.c 1 7.c even 3 1
1225.4.a.d 1 7.d odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 + 28 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( -14 + T )^{2}$$
$17$ $$441 + 21 T + T^{2}$$
$19$ $$2401 + 49 T + T^{2}$$
$23$ $$25281 + 159 T + T^{2}$$
$29$ $$( -58 + T )^{2}$$
$31$ $$21609 + 147 T + T^{2}$$
$37$ $$47961 - 219 T + T^{2}$$
$41$ $$( -350 + T )^{2}$$
$43$ $$( -124 + T )^{2}$$
$47$ $$275625 - 525 T + T^{2}$$
$53$ $$91809 - 303 T + T^{2}$$
$59$ $$11025 - 105 T + T^{2}$$
$61$ $$170569 - 413 T + T^{2}$$
$67$ $$172225 - 415 T + T^{2}$$
$71$ $$( 432 + T )^{2}$$
$73$ $$1238769 + 1113 T + T^{2}$$
$79$ $$10609 - 103 T + T^{2}$$
$83$ $$( 1092 + T )^{2}$$
$89$ $$108241 - 329 T + T^{2}$$
$97$ $$( -882 + T )^{2}$$