# Properties

 Label 175.4.a.d Level $175$ Weight $4$ Character orbit 175.a Self dual yes Analytic conductor $10.325$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{3} + ( 2 + \beta ) q^{4} + ( -10 - 3 \beta ) q^{6} -7 q^{7} + ( -10 + 5 \beta ) q^{8} + ( -13 + 5 \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{2} + ( 2 + \beta ) q^{3} + ( 2 + \beta ) q^{4} + ( -10 - 3 \beta ) q^{6} -7 q^{7} + ( -10 + 5 \beta ) q^{8} + ( -13 + 5 \beta ) q^{9} + ( -23 - 10 \beta ) q^{11} + ( 14 + 5 \beta ) q^{12} + ( -30 + 8 \beta ) q^{13} + 7 \beta q^{14} + ( -66 - 3 \beta ) q^{16} + ( 54 - 5 \beta ) q^{17} + ( -50 + 8 \beta ) q^{18} + ( -32 + 7 \beta ) q^{19} + ( -14 - 7 \beta ) q^{21} + ( 100 + 33 \beta ) q^{22} + ( 13 + 5 \beta ) q^{23} + ( 30 + 5 \beta ) q^{24} + ( -80 + 22 \beta ) q^{26} + ( -30 - 25 \beta ) q^{27} + ( -14 - 7 \beta ) q^{28} + ( -193 - 27 \beta ) q^{29} + ( -114 + 66 \beta ) q^{31} + ( 110 + 29 \beta ) q^{32} + ( -146 - 53 \beta ) q^{33} + ( 50 - 49 \beta ) q^{34} + ( 24 + 2 \beta ) q^{36} + ( -9 - 57 \beta ) q^{37} + ( -70 + 25 \beta ) q^{38} + ( 20 - 6 \beta ) q^{39} + ( -222 - 61 \beta ) q^{41} + ( 70 + 21 \beta ) q^{42} + ( 75 - 77 \beta ) q^{43} + ( -146 - 53 \beta ) q^{44} + ( -50 - 18 \beta ) q^{46} + ( -58 - 108 \beta ) q^{47} + ( -162 - 75 \beta ) q^{48} + 49 q^{49} + ( 58 + 39 \beta ) q^{51} + ( 20 - 6 \beta ) q^{52} + ( 174 - 86 \beta ) q^{53} + ( 250 + 55 \beta ) q^{54} + ( 70 - 35 \beta ) q^{56} + ( 6 - 11 \beta ) q^{57} + ( 270 + 220 \beta ) q^{58} + ( -10 + 210 \beta ) q^{59} + ( 468 + 54 \beta ) q^{61} + ( -660 + 48 \beta ) q^{62} + ( 91 - 35 \beta ) q^{63} + ( 238 - 115 \beta ) q^{64} + ( 530 + 199 \beta ) q^{66} + ( -537 + 166 \beta ) q^{67} + ( 58 + 39 \beta ) q^{68} + ( 76 + 28 \beta ) q^{69} + ( 215 - 303 \beta ) q^{71} + ( 380 - 90 \beta ) q^{72} + ( 34 + 269 \beta ) q^{73} + ( 570 + 66 \beta ) q^{74} + ( 6 - 11 \beta ) q^{76} + ( 161 + 70 \beta ) q^{77} + ( 60 - 14 \beta ) q^{78} + ( -539 - 41 \beta ) q^{79} + ( 41 - 240 \beta ) q^{81} + ( 610 + 283 \beta ) q^{82} + ( 724 + 69 \beta ) q^{83} + ( -98 - 35 \beta ) q^{84} + ( 770 + 2 \beta ) q^{86} + ( -656 - 274 \beta ) q^{87} + ( -270 - 65 \beta ) q^{88} + ( -848 - 17 \beta ) q^{89} + ( 210 - 56 \beta ) q^{91} + ( 76 + 28 \beta ) q^{92} + ( 432 + 84 \beta ) q^{93} + ( 1080 + 166 \beta ) q^{94} + ( 510 + 197 \beta ) q^{96} + ( -1018 + 272 \beta ) q^{97} -49 \beta q^{98} + ( -201 - 35 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{3} + 5 q^{4} - 23 q^{6} - 14 q^{7} - 15 q^{8} - 21 q^{9} + O(q^{10})$$ $$2 q - q^{2} + 5 q^{3} + 5 q^{4} - 23 q^{6} - 14 q^{7} - 15 q^{8} - 21 q^{9} - 56 q^{11} + 33 q^{12} - 52 q^{13} + 7 q^{14} - 135 q^{16} + 103 q^{17} - 92 q^{18} - 57 q^{19} - 35 q^{21} + 233 q^{22} + 31 q^{23} + 65 q^{24} - 138 q^{26} - 85 q^{27} - 35 q^{28} - 413 q^{29} - 162 q^{31} + 249 q^{32} - 345 q^{33} + 51 q^{34} + 50 q^{36} - 75 q^{37} - 115 q^{38} + 34 q^{39} - 505 q^{41} + 161 q^{42} + 73 q^{43} - 345 q^{44} - 118 q^{46} - 224 q^{47} - 399 q^{48} + 98 q^{49} + 155 q^{51} + 34 q^{52} + 262 q^{53} + 555 q^{54} + 105 q^{56} + q^{57} + 760 q^{58} + 190 q^{59} + 990 q^{61} - 1272 q^{62} + 147 q^{63} + 361 q^{64} + 1259 q^{66} - 908 q^{67} + 155 q^{68} + 180 q^{69} + 127 q^{71} + 670 q^{72} + 337 q^{73} + 1206 q^{74} + q^{76} + 392 q^{77} + 106 q^{78} - 1119 q^{79} - 158 q^{81} + 1503 q^{82} + 1517 q^{83} - 231 q^{84} + 1542 q^{86} - 1586 q^{87} - 605 q^{88} - 1713 q^{89} + 364 q^{91} + 180 q^{92} + 948 q^{93} + 2326 q^{94} + 1217 q^{96} - 1764 q^{97} - 49 q^{98} - 437 q^{99} + O(q^{100})$$

## 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
 3.70156 −2.70156
−3.70156 5.70156 5.70156 0 −21.1047 −7.00000 8.50781 5.50781 0
1.2 2.70156 −0.701562 −0.701562 0 −1.89531 −7.00000 −23.5078 −26.5078 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.d 2
3.b odd 2 1 1575.4.a.v 2
5.b even 2 1 175.4.a.e yes 2
5.c odd 4 2 175.4.b.d 4
7.b odd 2 1 1225.4.a.r 2
15.d odd 2 1 1575.4.a.s 2
35.c odd 2 1 1225.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.d 2 1.a even 1 1 trivial
175.4.a.e yes 2 5.b even 2 1
175.4.b.d 4 5.c odd 4 2
1225.4.a.r 2 7.b odd 2 1
1225.4.a.t 2 35.c odd 2 1
1575.4.a.s 2 15.d odd 2 1
1575.4.a.v 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 10$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-10 + T + T^{2}$$
$3$ $$-4 - 5 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 7 + T )^{2}$$
$11$ $$-241 + 56 T + T^{2}$$
$13$ $$20 + 52 T + T^{2}$$
$17$ $$2396 - 103 T + T^{2}$$
$19$ $$310 + 57 T + T^{2}$$
$23$ $$-16 - 31 T + T^{2}$$
$29$ $$35170 + 413 T + T^{2}$$
$31$ $$-38088 + 162 T + T^{2}$$
$37$ $$-31896 + 75 T + T^{2}$$
$41$ $$25616 + 505 T + T^{2}$$
$43$ $$-59440 - 73 T + T^{2}$$
$47$ $$-107012 + 224 T + T^{2}$$
$53$ $$-58648 - 262 T + T^{2}$$
$59$ $$-443000 - 190 T + T^{2}$$
$61$ $$215136 - 990 T + T^{2}$$
$67$ $$-76333 + 908 T + T^{2}$$
$71$ $$-937010 - 127 T + T^{2}$$
$73$ $$-713308 - 337 T + T^{2}$$
$79$ $$295810 + 1119 T + T^{2}$$
$83$ $$526522 - 1517 T + T^{2}$$
$89$ $$730630 + 1713 T + T^{2}$$
$97$ $$19588 + 1764 T + T^{2}$$