# Properties

 Label 175.10.b.d Level $175$ Weight $10$ Character orbit 175.b Analytic conductor $90.131$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$90.1312713287$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 853x^{4} + 185508x^{2} + 4064256$$ x^6 + 853*x^4 + 185508*x^2 + 4064256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - 7 \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + 28 \beta_1) q^{3} + ( - 7 \beta_{3} + 8 \beta_{2} - 519) q^{4} + ( - 6 \beta_{3} + 36 \beta_{2} + 1638) q^{6} - 2401 \beta_1 q^{7} + (147 \beta_{5} + 470 \beta_{4} + 4685 \beta_1) q^{8} + (126 \beta_{3} - 90 \beta_{2} + 8667) q^{9}+O(q^{10})$$ q + (-b4 - 7*b1) * q^2 + (-b5 + b4 + 28*b1) * q^3 + (-7*b3 + 8*b2 - 519) * q^4 + (-6*b3 + 36*b2 + 1638) * q^6 - 2401*b1 * q^7 + (147*b5 + 470*b4 + 4685*b1) * q^8 + (126*b3 - 90*b2 + 8667) * q^9 $$q + ( - \beta_{4} - 7 \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + 28 \beta_1) q^{3} + ( - 7 \beta_{3} + 8 \beta_{2} - 519) q^{4} + ( - 6 \beta_{3} + 36 \beta_{2} + 1638) q^{6} - 2401 \beta_1 q^{7} + (147 \beta_{5} + 470 \beta_{4} + 4685 \beta_1) q^{8} + (126 \beta_{3} - 90 \beta_{2} + 8667) q^{9} + (658 \beta_{3} + 650 \beta_{2} - 1148) q^{11} + ( - 182 \beta_{5} - 700 \beta_{4} + 35462 \beta_1) q^{12} + ( - 175 \beta_{5} - 3017 \beta_{4} - 6594 \beta_1) q^{13} + (2401 \beta_{2} - 16807) q^{14} + (1617 \beta_{3} - 10614 \beta_{2} + 160987) q^{16} + ( - 1574 \beta_{5} + 3030 \beta_{4} - 338898 \beta_1) q^{17} + ( - 2268 \beta_{5} - 16947 \beta_{4} - 91089 \beta_1) q^{18} + ( - 2437 \beta_{3} + 15371 \beta_{2} - 74284) q^{19} + ( - 2401 \beta_{3} - 2401 \beta_{2} + 67228) q^{21} + ( - 4004 \beta_{5} - 40972 \beta_{4} + 949016 \beta_1) q^{22} + ( - 3808 \beta_{5} + 24200 \beta_{4} + 628544 \beta_1) q^{23} + ( - 10338 \beta_{3} - 4500 \beta_{2} + 483210) q^{24} + ( - 23394 \beta_{3} + 20986 \beta_{2} - 2928352) q^{26} + ( - 15786 \beta_{5} + 33930 \beta_{4} - 183960 \beta_1) q^{27} + (16807 \beta_{5} + 19208 \beta_{4} + 1246119 \beta_1) q^{28} + ( - 18914 \beta_{3} - 54866 \beta_{2} - 1360606) q^{29} + ( - 70302 \beta_{3} + 55698 \beta_{2} + 956480) q^{31} + ( - 20055 \beta_{5} - 36066 \beta_{4} - 8407317 \beta_1) q^{32} + (65640 \beta_{5} + 76152 \beta_{4} - 6753264 \beta_1) q^{33} + (748 \beta_{3} + 438178 \beta_{2} + 1327214) q^{34} + ( - 83601 \beta_{3} + 209376 \beta_{2} - 11798793) q^{36} + ( - 60522 \beta_{5} + 209418 \beta_{4} - 465206 \beta_1) q^{37} + (139278 \beta_{5} + 248060 \beta_{4} + 14493290 \beta_1) q^{38} + (13748 \beta_{3} + 110012 \beta_{2} + 2996896) q^{39} + ( - 131894 \beta_{3} - 163478 \beta_{2} - 4806886) q^{41} + (14406 \beta_{5} + 86436 \beta_{4} - 3932838 \beta_1) q^{42} + (65366 \beta_{5} - 121982 \beta_{4} - 20543724 \beta_1) q^{43} + ( - 1960 \beta_{3} - 314984 \beta_{2} - 32337328) q^{44} + (119896 \beta_{3} - 405224 \beta_{2} + 29915888) q^{46} + ( - 83238 \beta_{5} - 534778 \beta_{4} + 3456320 \beta_1) q^{47} + (9710 \beta_{5} - 174140 \beta_{4} + 5599594 \beta_1) q^{48} - 5764801 q^{49} + ( - 186102 \beta_{3} - 587238 \beta_{2} - 8715576) q^{51} + (361424 \beta_{5} + 2925244 \beta_{4} + 26969348 \beta_1) q^{52} + ( - 450352 \beta_{5} - 1553376 \beta_{4} + \cdots + 22500870 \beta_1) q^{53}+ \cdots + ( - 376362 \beta_{3} + 7689150 \beta_{2} + \cdots + 633659724) q^{99}+O(q^{100})$$ q + (-b4 - 7*b1) * q^2 + (-b5 + b4 + 28*b1) * q^3 + (-7*b3 + 8*b2 - 519) * q^4 + (-6*b3 + 36*b2 + 1638) * q^6 - 2401*b1 * q^7 + (147*b5 + 470*b4 + 4685*b1) * q^8 + (126*b3 - 90*b2 + 8667) * q^9 + (658*b3 + 650*b2 - 1148) * q^11 + (-182*b5 - 700*b4 + 35462*b1) * q^12 + (-175*b5 - 3017*b4 - 6594*b1) * q^13 + (2401*b2 - 16807) * q^14 + (1617*b3 - 10614*b2 + 160987) * q^16 + (-1574*b5 + 3030*b4 - 338898*b1) * q^17 + (-2268*b5 - 16947*b4 - 91089*b1) * q^18 + (-2437*b3 + 15371*b2 - 74284) * q^19 + (-2401*b3 - 2401*b2 + 67228) * q^21 + (-4004*b5 - 40972*b4 + 949016*b1) * q^22 + (-3808*b5 + 24200*b4 + 628544*b1) * q^23 + (-10338*b3 - 4500*b2 + 483210) * q^24 + (-23394*b3 + 20986*b2 - 2928352) * q^26 + (-15786*b5 + 33930*b4 - 183960*b1) * q^27 + (16807*b5 + 19208*b4 + 1246119*b1) * q^28 + (-18914*b3 - 54866*b2 - 1360606) * q^29 + (-70302*b3 + 55698*b2 + 956480) * q^31 + (-20055*b5 - 36066*b4 - 8407317*b1) * q^32 + (65640*b5 + 76152*b4 - 6753264*b1) * q^33 + (748*b3 + 438178*b2 + 1327214) * q^34 + (-83601*b3 + 209376*b2 - 11798793) * q^36 + (-60522*b5 + 209418*b4 - 465206*b1) * q^37 + (139278*b5 + 248060*b4 + 14493290*b1) * q^38 + (13748*b3 + 110012*b2 + 2996896) * q^39 + (-131894*b3 - 163478*b2 - 4806886) * q^41 + (14406*b5 + 86436*b4 - 3932838*b1) * q^42 + (65366*b5 - 121982*b4 - 20543724*b1) * q^43 + (-1960*b3 - 314984*b2 - 32337328) * q^44 + (119896*b3 - 405224*b2 + 29915888) * q^46 + (-83238*b5 - 534778*b4 + 3456320*b1) * q^47 + (9710*b5 - 174140*b4 + 5599594*b1) * q^48 - 5764801 * q^49 + (-186102*b3 - 587238*b2 - 8715576) * q^51 + (361424*b5 + 2925244*b4 + 26969348*b1) * q^52 + (-450352*b5 - 1553376*b4 + 22500870*b1) * q^53 + (32292*b3 + 1176120*b2 + 39293100) * q^54 + (352947*b3 - 1128470*b2 + 11248685) * q^56 + (-182350*b5 + 403894*b4 - 2823704*b1) * q^57 + (-138180*b5 + 2535150*b4 - 53054610*b1) * q^58 + (49659*b3 + 2231195*b2 + 14196700) * q^59 + (-844773*b3 + 589107*b2 + 63915614) * q^61 + (1303812*b5 + 3668848*b4 + 15661156*b1) * q^62 + (-302526*b5 - 216090*b4 - 20809467*b1) * q^63 + (314727*b3 + 4312590*b2 - 2617387) * q^64 + (1386384*b3 + 2410512*b2 - 2685984) * q^66 + (-47712*b5 + 3939816*b4 + 85058596*b1) * q^67 + (2251634*b5 + 613704*b4 + 247828602*b1) * q^68 + (981336*b3 - 697656*b2 - 85967952) * q^69 + (-526260*b3 + 3499356*b2 + 98838168) * q^71 + (1391229*b5 + 8765370*b4 + 203104755*b1) * q^72 + (118516*b5 + 9544844*b4 + 114737770*b1) * q^73 + (679140*b3 + 4189718*b2 + 230232154) * q^74 + (2299290*b3 - 15924468*b2 + 242946662) * q^76 + (-1579858*b5 + 1560650*b4 + 2756348*b1) * q^77 + (591360*b5 - 3780504*b4 + 93377592*b1) * q^78 + (-1679412*b3 + 7475532*b2 + 320137552) * q^79 + (3824982*b3 - 4598370*b2 - 11942559) * q^81 + (570276*b5 + 13216518*b4 - 187558434*b1) * q^82 + (-1977367*b5 + 4479559*b4 - 366839060*b1) * q^83 + (-436982*b3 + 1680700*b2 + 85144262) * q^84 + (-4116*b3 + 16416916*b2 - 293660752) * q^86 + (-457014*b5 - 5138394*b4 + 207273864*b1) * q^87 + (-4229456*b5 + 11172080*b4 + 402041600*b1) * q^88 + (-1815976*b3 + 10612104*b2 - 168938826) * q^89 + (-420175*b3 + 7243817*b2 - 15832194) * q^91 + (-6344912*b5 - 25723952*b4 - 230374496*b1) * q^92 + (-7972076*b5 - 1921156*b4 + 564419504*b1) * q^93 + (-4825540*b3 + 2488928*b2 - 462668276) * q^94 + (-6385806*b3 - 8360604*b2 + 111128598) * q^96 + (864850*b5 + 33276782*b4 + 215832750*b1) * q^97 + (5764801*b4 + 40353607*b1) * q^98 + (-376362*b3 + 7689150*b2 + 633659724) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3114 q^{4} + 9828 q^{6} + 52002 q^{9}+O(q^{10})$$ 6 * q - 3114 * q^4 + 9828 * q^6 + 52002 * q^9 $$6 q - 3114 q^{4} + 9828 q^{6} + 52002 q^{9} - 6888 q^{11} - 100842 q^{14} + 965922 q^{16} - 445704 q^{19} + 403368 q^{21} + 2899260 q^{24} - 17570112 q^{26} - 8163636 q^{29} + 5738880 q^{31} + 7963284 q^{34} - 70792758 q^{36} + 17981376 q^{39} - 28841316 q^{41} - 194023968 q^{44} + 179495328 q^{46} - 34588806 q^{49} - 52293456 q^{51} + 235758600 q^{54} + 67492110 q^{56} + 85180200 q^{59} + 383493684 q^{61} - 15704322 q^{64} - 16115904 q^{66} - 515807712 q^{69} + 593029008 q^{71} + 1381392924 q^{74} + 1457679972 q^{76} + 1920825312 q^{79} - 71655354 q^{81} + 510865572 q^{84} - 1761964512 q^{86} - 1013632956 q^{89} - 94993164 q^{91} - 2776009656 q^{94} + 666771588 q^{96} + 3801958344 q^{99}+O(q^{100})$$ 6 * q - 3114 * q^4 + 9828 * q^6 + 52002 * q^9 - 6888 * q^11 - 100842 * q^14 + 965922 * q^16 - 445704 * q^19 + 403368 * q^21 + 2899260 * q^24 - 17570112 * q^26 - 8163636 * q^29 + 5738880 * q^31 + 7963284 * q^34 - 70792758 * q^36 + 17981376 * q^39 - 28841316 * q^41 - 194023968 * q^44 + 179495328 * q^46 - 34588806 * q^49 - 52293456 * q^51 + 235758600 * q^54 + 67492110 * q^56 + 85180200 * q^59 + 383493684 * q^61 - 15704322 * q^64 - 16115904 * q^66 - 515807712 * q^69 + 593029008 * q^71 + 1381392924 * q^74 + 1457679972 * q^76 + 1920825312 * q^79 - 71655354 * q^81 + 510865572 * q^84 - 1761964512 * q^86 - 1013632956 * q^89 - 94993164 * q^91 - 2776009656 * q^94 + 666771588 * q^96 + 3801958344 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 853x^{4} + 185508x^{2} + 4064256$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 1163\nu^{3} + 675324\nu ) / 3205440$$ (-v^5 + 1163*v^3 + 675324*v) / 3205440 $$\beta_{2}$$ $$=$$ $$( -11\nu^{4} - 6287\nu^{2} - 480096 ) / 9540$$ (-11*v^4 - 6287*v^2 - 480096) / 9540 $$\beta_{3}$$ $$=$$ $$( -5\nu^{4} - 1817\nu^{2} + 77688 ) / 1908$$ (-5*v^4 - 1817*v^2 + 77688) / 1908 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 692\nu^{3} + 94501\nu ) / 11130$$ (v^5 + 692*v^3 + 94501*v) / 11130 $$\beta_{5}$$ $$=$$ $$( 29\nu^{5} + 19697\nu^{3} + 4509828\nu ) / 320544$$ (29*v^5 + 19697*v^3 + 4509828*v) / 320544
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + 2\beta_1 ) / 6$$ (b5 - b4 + 2*b1) / 6 $$\nu^{2}$$ $$=$$ $$( 11\beta_{3} - 25\beta_{2} - 1706 ) / 6$$ (11*b3 - 25*b2 - 1706) / 6 $$\nu^{3}$$ $$=$$ $$( -415\beta_{5} + 451\beta_{4} + 9538\beta_1 ) / 6$$ (-415*b5 + 451*b4 + 9538*b1) / 6 $$\nu^{4}$$ $$=$$ $$( -6287\beta_{3} + 9085\beta_{2} + 713186 ) / 6$$ (-6287*b3 + 9085*b2 + 713186) / 6 $$\nu^{5}$$ $$=$$ $$( 192679\beta_{5} - 150811\beta_{4} - 6789298\beta_1 ) / 6$$ (192679*b5 - 150811*b4 - 6789298*b1) / 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 4.96128i − 18.2745i − 22.2358i 22.2358i 18.2745i − 4.96128i
41.8019i 0.232339i −1235.40 0 9.71222 2401.00i 30239.6i 19682.9 0
99.2 34.1627i 79.6469i −655.088 0 2720.95 2401.00i 4888.28i 13339.4 0
99.3 13.3607i 163.415i 333.491 0 2183.34 2401.00i 11296.4i −7021.32 0
99.4 13.3607i 163.415i 333.491 0 2183.34 2401.00i 11296.4i −7021.32 0
99.5 34.1627i 79.6469i −655.088 0 2720.95 2401.00i 4888.28i 13339.4 0
99.6 41.8019i 0.232339i −1235.40 0 9.71222 2401.00i 30239.6i 19682.9 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.10.b.d 6
5.b even 2 1 inner 175.10.b.d 6
5.c odd 4 1 7.10.a.b 3
5.c odd 4 1 175.10.a.d 3
15.e even 4 1 63.10.a.e 3
20.e even 4 1 112.10.a.h 3
35.f even 4 1 49.10.a.c 3
35.k even 12 2 49.10.c.e 6
35.l odd 12 2 49.10.c.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 5.c odd 4 1
49.10.a.c 3 35.f even 4 1
49.10.c.d 6 35.l odd 12 2
49.10.c.e 6 35.k even 12 2
63.10.a.e 3 15.e even 4 1
112.10.a.h 3 20.e even 4 1
175.10.a.d 3 5.c odd 4 1
175.10.b.d 6 1.a even 1 1 trivial
175.10.b.d 6 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3093T_{2}^{4} + 2559636T_{2}^{2} + 364046400$$ acting on $$S_{10}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3093 T^{4} + \cdots + 364046400$$
$3$ $$T^{6} + 33048 T^{4} + \cdots + 9144576$$
$5$ $$T^{6}$$
$7$ $$(T^{2} + 5764801)^{3}$$
$11$ $$(T^{3} + 3444 T^{2} + \cdots + 108859759460352)^{2}$$
$13$ $$T^{6} + 26253893652 T^{4} + \cdots + 17\!\cdots\!00$$
$17$ $$T^{6} + 446677667052 T^{4} + \cdots + 48\!\cdots\!24$$
$19$ $$(T^{3} + 222852 T^{2} + \cdots + 43\!\cdots\!60)^{2}$$
$23$ $$T^{6} + 3527218647168 T^{4} + \cdots + 94\!\cdots\!96$$
$29$ $$(T^{3} + 4081818 T^{2} + \cdots - 44\!\cdots\!00)^{2}$$
$31$ $$(T^{3} - 2869440 T^{2} + \cdots - 74\!\cdots\!84)^{2}$$
$37$ $$T^{6} + 256366243984332 T^{4} + \cdots + 11\!\cdots\!84$$
$41$ $$(T^{3} + 14420658 T^{2} + \cdots - 19\!\cdots\!12)^{2}$$
$43$ $$T^{6} + \cdots + 47\!\cdots\!00$$
$47$ $$T^{6} + 928644247418784 T^{4} + \cdots + 19\!\cdots\!56$$
$53$ $$T^{6} + \cdots + 57\!\cdots\!84$$
$59$ $$(T^{3} - 42590100 T^{2} + \cdots - 42\!\cdots\!00)^{2}$$
$61$ $$(T^{3} - 191746842 T^{2} + \cdots + 51\!\cdots\!08)^{2}$$
$67$ $$T^{6} + \cdots + 41\!\cdots\!96$$
$71$ $$(T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80)^{2}$$
$73$ $$T^{6} + \cdots + 38\!\cdots\!04$$
$79$ $$(T^{3} - 960412656 T^{2} + \cdots + 11\!\cdots\!00)^{2}$$
$83$ $$T^{6} + \cdots + 36\!\cdots\!04$$
$89$ $$(T^{3} + 506816478 T^{2} + \cdots - 19\!\cdots\!40)^{2}$$
$97$ $$T^{6} + \cdots + 24\!\cdots\!56$$