# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{3} -4 \zeta_{12}^{2} q^{4} + 14 q^{6} + ( -14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{7} -24 \zeta_{12}^{3} q^{8} + ( 22 - 22 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{2} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{3} -4 \zeta_{12}^{2} q^{4} + 14 q^{6} + ( -14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{7} -24 \zeta_{12}^{3} q^{8} + ( 22 - 22 \zeta_{12}^{2} ) q^{9} + 5 \zeta_{12}^{2} q^{11} -28 \zeta_{12} q^{12} -14 \zeta_{12}^{3} q^{13} + ( 14 - 42 \zeta_{12}^{2} ) q^{14} + ( 16 - 16 \zeta_{12}^{2} ) q^{16} + ( 21 \zeta_{12} - 21 \zeta_{12}^{3} ) q^{17} + ( 44 \zeta_{12} - 44 \zeta_{12}^{3} ) q^{18} + ( 49 - 49 \zeta_{12}^{2} ) q^{19} + ( -98 - 49 \zeta_{12}^{2} ) q^{21} + 10 \zeta_{12}^{3} q^{22} + 159 \zeta_{12} q^{23} -168 \zeta_{12}^{2} q^{24} + ( 28 - 28 \zeta_{12}^{2} ) q^{26} + 35 \zeta_{12}^{3} q^{27} + ( -28 \zeta_{12} + 84 \zeta_{12}^{3} ) q^{28} -58 q^{29} -147 \zeta_{12}^{2} q^{31} + ( -160 \zeta_{12} + 160 \zeta_{12}^{3} ) q^{32} + 35 \zeta_{12} q^{33} + 42 q^{34} -88 q^{36} + 219 \zeta_{12} q^{37} + ( 98 \zeta_{12} - 98 \zeta_{12}^{3} ) q^{38} -98 \zeta_{12}^{2} q^{39} + 350 q^{41} + ( -196 \zeta_{12} - 98 \zeta_{12}^{3} ) q^{42} -124 \zeta_{12}^{3} q^{43} + ( 20 - 20 \zeta_{12}^{2} ) q^{44} + 318 \zeta_{12}^{2} q^{46} + 525 \zeta_{12} q^{47} -112 \zeta_{12}^{3} q^{48} + ( -245 + 392 \zeta_{12}^{2} ) q^{49} + ( 147 - 147 \zeta_{12}^{2} ) q^{51} + ( -56 \zeta_{12} + 56 \zeta_{12}^{3} ) q^{52} + ( 303 \zeta_{12} - 303 \zeta_{12}^{3} ) q^{53} + ( -70 + 70 \zeta_{12}^{2} ) q^{54} + ( -504 + 336 \zeta_{12}^{2} ) q^{56} -343 \zeta_{12}^{3} q^{57} -116 \zeta_{12} q^{58} -105 \zeta_{12}^{2} q^{59} + ( 413 - 413 \zeta_{12}^{2} ) q^{61} -294 \zeta_{12}^{3} q^{62} + ( -462 \zeta_{12} + 308 \zeta_{12}^{3} ) q^{63} -448 q^{64} + 70 \zeta_{12}^{2} q^{66} + ( -415 \zeta_{12} + 415 \zeta_{12}^{3} ) q^{67} -84 \zeta_{12} q^{68} + 1113 q^{69} -432 q^{71} -528 \zeta_{12} q^{72} + ( -1113 \zeta_{12} + 1113 \zeta_{12}^{3} ) q^{73} + 438 \zeta_{12}^{2} q^{74} -196 q^{76} + ( 35 \zeta_{12} - 105 \zeta_{12}^{3} ) q^{77} -196 \zeta_{12}^{3} q^{78} + ( -103 + 103 \zeta_{12}^{2} ) q^{79} + 839 \zeta_{12}^{2} q^{81} + 700 \zeta_{12} q^{82} + 1092 \zeta_{12}^{3} q^{83} + ( -196 + 588 \zeta_{12}^{2} ) q^{84} + ( 248 - 248 \zeta_{12}^{2} ) q^{86} + ( -406 \zeta_{12} + 406 \zeta_{12}^{3} ) q^{87} + ( 120 \zeta_{12} - 120 \zeta_{12}^{3} ) q^{88} + ( -329 + 329 \zeta_{12}^{2} ) q^{89} + ( -294 + 196 \zeta_{12}^{2} ) q^{91} -636 \zeta_{12}^{3} q^{92} -1029 \zeta_{12} q^{93} + 1050 \zeta_{12}^{2} q^{94} + ( -1120 + 1120 \zeta_{12}^{2} ) q^{96} + 882 \zeta_{12}^{3} q^{97} + ( -490 \zeta_{12} + 784 \zeta_{12}^{3} ) q^{98} + 110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + 56q^{6} + 44q^{9} + O(q^{10})$$ $$4q - 8q^{4} + 56q^{6} + 44q^{9} + 10q^{11} - 28q^{14} + 32q^{16} + 98q^{19} - 490q^{21} - 336q^{24} + 56q^{26} - 232q^{29} - 294q^{31} + 168q^{34} - 352q^{36} - 196q^{39} + 1400q^{41} + 40q^{44} + 636q^{46} - 196q^{49} + 294q^{51} - 140q^{54} - 1344q^{56} - 210q^{59} + 826q^{61} - 1792q^{64} + 140q^{66} + 4452q^{69} - 1728q^{71} + 876q^{74} - 784q^{76} - 206q^{79} + 1678q^{81} + 392q^{84} + 496q^{86} - 658q^{89} - 784q^{91} + 2100q^{94} - 2240q^{96} + 440q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i −6.06218 3.50000i −2.00000 + 3.46410i 0 14.0000 12.1244 14.0000i 24.0000i 11.0000 + 19.0526i 0
74.2 1.73205 1.00000i 6.06218 + 3.50000i −2.00000 + 3.46410i 0 14.0000 −12.1244 + 14.0000i 24.0000i 11.0000 + 19.0526i 0
149.1 −1.73205 1.00000i −6.06218 + 3.50000i −2.00000 3.46410i 0 14.0000 12.1244 + 14.0000i 24.0000i 11.0000 19.0526i 0
149.2 1.73205 + 1.00000i 6.06218 3.50000i −2.00000 3.46410i 0 14.0000 −12.1244 14.0000i 24.0000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4 T_{2}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} + T^{4}$$
$3$ $$2401 - 49 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$117649 + 98 T^{2} + T^{4}$$
$11$ $$( 25 - 5 T + T^{2} )^{2}$$
$13$ $$( 196 + T^{2} )^{2}$$
$17$ $$194481 - 441 T^{2} + T^{4}$$
$19$ $$( 2401 - 49 T + T^{2} )^{2}$$
$23$ $$639128961 - 25281 T^{2} + T^{4}$$
$29$ $$( 58 + T )^{4}$$
$31$ $$( 21609 + 147 T + T^{2} )^{2}$$
$37$ $$2300257521 - 47961 T^{2} + T^{4}$$
$41$ $$( -350 + T )^{4}$$
$43$ $$( 15376 + T^{2} )^{2}$$
$47$ $$75969140625 - 275625 T^{2} + T^{4}$$
$53$ $$8428892481 - 91809 T^{2} + T^{4}$$
$59$ $$( 11025 + 105 T + T^{2} )^{2}$$
$61$ $$( 170569 - 413 T + T^{2} )^{2}$$
$67$ $$29661450625 - 172225 T^{2} + T^{4}$$
$71$ $$( 432 + T )^{4}$$
$73$ $$1534548635361 - 1238769 T^{2} + T^{4}$$
$79$ $$( 10609 + 103 T + T^{2} )^{2}$$
$83$ $$( 1192464 + T^{2} )^{2}$$
$89$ $$( 108241 + 329 T + T^{2} )^{2}$$
$97$ $$( 777924 + T^{2} )^{2}$$
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