# Properties

 Label 175.6.a.c Level 175 Weight 6 Character orbit 175.a Self dual yes Analytic conductor 28.067 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 175.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.0671684673$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{2} + 6 \beta q^{3} + ( -2 + 9 \beta ) q^{4} + ( -84 - 30 \beta ) q^{6} -49 q^{7} + ( 10 - 11 \beta ) q^{8} + ( 261 + 36 \beta ) q^{9} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{2} + 6 \beta q^{3} + ( -2 + 9 \beta ) q^{4} + ( -84 - 30 \beta ) q^{6} -49 q^{7} + ( 10 - 11 \beta ) q^{8} + ( 261 + 36 \beta ) q^{9} + ( 260 - 124 \beta ) q^{11} + ( 756 + 42 \beta ) q^{12} + ( 238 - 126 \beta ) q^{13} + ( 196 + 49 \beta ) q^{14} + ( 178 - 243 \beta ) q^{16} + ( -938 + 76 \beta ) q^{17} + ( -1548 - 441 \beta ) q^{18} + ( -1624 - 18 \beta ) q^{19} -294 \beta q^{21} + ( 696 + 360 \beta ) q^{22} + ( -760 - 568 \beta ) q^{23} + ( -924 - 6 \beta ) q^{24} + ( 812 + 392 \beta ) q^{26} + ( 3024 + 324 \beta ) q^{27} + ( 98 - 441 \beta ) q^{28} + ( 3222 + 252 \beta ) q^{29} + ( -280 + 540 \beta ) q^{31} + ( 2370 + 1389 \beta ) q^{32} + ( -10416 + 816 \beta ) q^{33} + ( 2688 + 558 \beta ) q^{34} + ( 4014 + 2601 \beta ) q^{36} + ( -2846 - 540 \beta ) q^{37} + ( 6748 + 1714 \beta ) q^{38} + ( -10584 + 672 \beta ) q^{39} + ( -2478 - 1092 \beta ) q^{41} + ( 4116 + 1470 \beta ) q^{42} + ( -884 + 4788 \beta ) q^{43} + ( -16144 + 1472 \beta ) q^{44} + ( 10992 + 3600 \beta ) q^{46} + ( -3976 - 3748 \beta ) q^{47} + ( -20412 - 390 \beta ) q^{48} + 2401 q^{49} + ( 6384 - 5172 \beta ) q^{51} + ( -16352 + 1260 \beta ) q^{52} + ( -4838 + 208 \beta ) q^{53} + ( -16632 - 4644 \beta ) q^{54} + ( -490 + 539 \beta ) q^{56} + ( -1512 - 9852 \beta ) q^{57} + ( -16416 - 4482 \beta ) q^{58} + ( -20944 - 2050 \beta ) q^{59} + ( -29974 - 4806 \beta ) q^{61} + ( -6440 - 2420 \beta ) q^{62} + ( -12789 - 1764 \beta ) q^{63} + ( -34622 - 1539 \beta ) q^{64} + ( 30240 + 6336 \beta ) q^{66} + ( -13364 + 1944 \beta ) q^{67} + ( 11452 - 7910 \beta ) q^{68} + ( -47712 - 7968 \beta ) q^{69} + ( 50808 - 4200 \beta ) q^{71} + ( -2934 - 2907 \beta ) q^{72} + ( -11354 + 5256 \beta ) q^{73} + ( 18944 + 5546 \beta ) q^{74} + ( 980 - 14742 \beta ) q^{76} + ( -12740 + 6076 \beta ) q^{77} + ( 32928 + 7224 \beta ) q^{78} + ( 18176 + 14904 \beta ) q^{79} + ( -36207 + 11340 \beta ) q^{81} + ( 25200 + 7938 \beta ) q^{82} + ( -50904 - 15750 \beta ) q^{83} + ( -37044 - 2058 \beta ) q^{84} + ( -63496 - 23056 \beta ) q^{86} + ( 21168 + 20844 \beta ) q^{87} + ( 21696 - 2736 \beta ) q^{88} + ( 53242 - 22208 \beta ) q^{89} + ( -11662 + 6174 \beta ) q^{91} + ( -70048 - 10816 \beta ) q^{92} + ( 45360 + 1560 \beta ) q^{93} + ( 68376 + 22716 \beta ) q^{94} + ( 116676 + 22554 \beta ) q^{96} + ( -5978 - 8820 \beta ) q^{97} + ( -9604 - 2401 \beta ) q^{98} + ( 5364 - 27468 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{2} + 6q^{3} + 5q^{4} - 198q^{6} - 98q^{7} + 9q^{8} + 558q^{9} + O(q^{10})$$ $$2q - 9q^{2} + 6q^{3} + 5q^{4} - 198q^{6} - 98q^{7} + 9q^{8} + 558q^{9} + 396q^{11} + 1554q^{12} + 350q^{13} + 441q^{14} + 113q^{16} - 1800q^{17} - 3537q^{18} - 3266q^{19} - 294q^{21} + 1752q^{22} - 2088q^{23} - 1854q^{24} + 2016q^{26} + 6372q^{27} - 245q^{28} + 6696q^{29} - 20q^{31} + 6129q^{32} - 20016q^{33} + 5934q^{34} + 10629q^{36} - 6232q^{37} + 15210q^{38} - 20496q^{39} - 6048q^{41} + 9702q^{42} + 3020q^{43} - 30816q^{44} + 25584q^{46} - 11700q^{47} - 41214q^{48} + 4802q^{49} + 7596q^{51} - 31444q^{52} - 9468q^{53} - 37908q^{54} - 441q^{56} - 12876q^{57} - 37314q^{58} - 43938q^{59} - 64754q^{61} - 15300q^{62} - 27342q^{63} - 70783q^{64} + 66816q^{66} - 24784q^{67} + 14994q^{68} - 103392q^{69} + 97416q^{71} - 8775q^{72} - 17452q^{73} + 43434q^{74} - 12782q^{76} - 19404q^{77} + 73080q^{78} + 51256q^{79} - 61074q^{81} + 58338q^{82} - 117558q^{83} - 76146q^{84} - 150048q^{86} + 63180q^{87} + 40656q^{88} + 84276q^{89} - 17150q^{91} - 150912q^{92} + 92280q^{93} + 159468q^{94} + 255906q^{96} - 20776q^{97} - 21609q^{98} - 16740q^{99} + O(q^{100})$$

## 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
 4.27492 −3.27492
−8.27492 25.6495 36.4743 0 −212.248 −49.0000 −37.0241 414.897 0
1.2 −0.725083 −19.6495 −31.4743 0 14.2475 −49.0000 46.0241 143.103 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 175.6.a.c 2
5.b even 2 1 7.6.a.b 2
5.c odd 4 2 175.6.b.c 4
15.d odd 2 1 63.6.a.f 2
20.d odd 2 1 112.6.a.h 2
35.c odd 2 1 49.6.a.f 2
35.i odd 6 2 49.6.c.d 4
35.j even 6 2 49.6.c.e 4
40.e odd 2 1 448.6.a.u 2
40.f even 2 1 448.6.a.w 2
55.d odd 2 1 847.6.a.c 2
60.h even 2 1 1008.6.a.bq 2
105.g even 2 1 441.6.a.l 2
140.c even 2 1 784.6.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 5.b even 2 1
49.6.a.f 2 35.c odd 2 1
49.6.c.d 4 35.i odd 6 2
49.6.c.e 4 35.j even 6 2
63.6.a.f 2 15.d odd 2 1
112.6.a.h 2 20.d odd 2 1
175.6.a.c 2 1.a even 1 1 trivial
175.6.b.c 4 5.c odd 4 2
441.6.a.l 2 105.g even 2 1
448.6.a.u 2 40.e odd 2 1
448.6.a.w 2 40.f even 2 1
784.6.a.v 2 140.c even 2 1
847.6.a.c 2 55.d odd 2 1
1008.6.a.bq 2 60.h even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 9 T_{2} + 6$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 9 T + 70 T^{2} + 288 T^{3} + 1024 T^{4}$$
$3$ $$1 - 6 T - 18 T^{2} - 1458 T^{3} + 59049 T^{4}$$
$5$ 1
$7$ $$( 1 + 49 T )^{2}$$
$11$ $$1 - 396 T + 142198 T^{2} - 63776196 T^{3} + 25937424601 T^{4}$$
$13$ $$1 - 350 T + 546978 T^{2} - 129952550 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 1800 T + 3567406 T^{2} + 2555742600 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 3266 T + 7614270 T^{2} + 8086939334 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 2088 T + 9365230 T^{2} + 13439084184 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 + 20 T + 53103102 T^{2} + 572583020 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 + 6232 T + 144242070 T^{2} + 432151540024 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 + 6048 T + 223864366 T^{2} + 700698303648 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 - 3020 T - 30383466 T^{2} - 443965497860 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 + 11700 T + 292735582 T^{2} + 2683336581900 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 + 9468 T + 858185230 T^{2} + 3959474927724 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 + 43938 T + 1852599934 T^{2} + 31412343849462 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 + 24784 T + 2799959190 T^{2} + 33461500651888 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 + 17452 T + 3828622374 T^{2} + 36179245441036 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 51256 T + 3645565854 T^{2} - 157717602787144 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 + 117558 T + 7798161502 T^{2} + 463065739909794 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 - 84276 T + 5915697430 T^{2} - 470602194123924 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 + 20776 T + 16174049358 T^{2} + 178410581179432 T^{3} + 73742412689492826049 T^{4}$$