Embedded newform 175.4.x.a.3.55

• Label: 175.4.x.a.3.55
• Level: $175$
• Weight: $4$
• Character: 175.3
• Analytic conductor: $10.325$
• Analytic rank: $0$
• Dimension: $928$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	175.x (of order $60$, degree $16$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$928$
Relative dimension:	$58$ over $\Q(\zeta_{60})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			3.55
Character	$\chi$	$=$	175.3
Dual form			175.4.x.a.117.55

$q$-expansion

$f(q)$	$=$	$q+(2.78061 - 4.28176i) q^{2} +(2.17796 - 5.67379i) q^{3} +(-7.34780 - 16.5034i) q^{4} +(11.0343 + 1.80104i) q^{5} +(-18.2377 - 25.1021i) q^{6} +(-13.3513 + 12.8352i) q^{7} +(-50.7546 - 8.03874i) q^{8} +(-7.38346 - 6.64810i) q^{9} +(38.3938 - 42.2383i) q^{10} +(-14.9911 - 16.6493i) q^{11} +(-109.640 + 5.74600i) q^{12} +(-33.3392 - 16.9872i) q^{13} +(17.8326 + 92.8568i) q^{14} +(34.2511 - 58.6838i) q^{15} +(-78.8444 + 87.5656i) q^{16} +(51.2162 - 63.2467i) q^{17} +(-48.9961 + 13.1285i) q^{18} +(28.9974 + 12.9105i) q^{19} +(-51.3546 - 195.338i) q^{20} +(43.7457 + 103.707i) q^{21} +(-112.973 + 17.8932i) q^{22} +(104.969 + 68.1680i) q^{23} +(-156.152 + 270.463i) q^{24} +(118.512 + 39.7466i) q^{25} +(-165.438 + 95.5158i) q^{26} +(92.4055 - 47.0830i) q^{27} +(309.928 + 126.031i) q^{28} +(-55.2822 + 76.0894i) q^{29} +(-156.031 - 309.832i) q^{30} +(98.5579 - 10.3589i) q^{31} +(49.2996 + 183.988i) q^{32} +(-127.115 + 48.7949i) q^{33} +(-128.395 - 395.160i) q^{34} +(-170.439 + 117.582i) q^{35} +(-55.4642 + 170.701i) q^{36} +(-17.0667 - 325.653i) q^{37} +(135.910 - 88.2610i) q^{38} +(-168.993 + 152.162i) q^{39} +(-545.564 - 180.113i) q^{40} +(32.4428 - 10.5413i) q^{41} +(565.689 + 101.060i) q^{42} +(359.144 + 359.144i) q^{43} +(-164.619 + 369.741i) q^{44} +(-69.4979 - 86.6552i) q^{45} +(583.758 - 259.906i) q^{46} +(-310.823 + 251.699i) q^{47} +(325.109 + 638.062i) q^{48} +(13.5141 - 342.734i) q^{49} +(499.722 - 396.922i) q^{50} +(-247.302 - 428.339i) q^{51} +(-35.3768 + 675.029i) q^{52} +(197.600 + 75.8516i) q^{53} +(55.3455 - 526.578i) q^{54} +(-135.431 - 210.714i) q^{55} +(780.818 - 544.119i) q^{56} +(136.407 - 136.407i) q^{57} +(172.078 + 448.280i) q^{58} +(-247.333 - 52.5722i) q^{59} +(-1220.15 - 134.064i) q^{60} +(176.398 + 829.886i) q^{61} +(229.697 - 450.805i) q^{62} +(183.909 - 6.00765i) q^{63} +(28.3642 + 9.21610i) q^{64} +(-337.281 - 247.487i) q^{65} +(-144.529 + 679.956i) q^{66} +(-249.643 - 202.157i) q^{67} +(-1420.11 - 380.518i) q^{68} +(615.390 - 447.107i) q^{69} +(29.5319 + 1056.73i) q^{70} +(-541.294 - 393.273i) q^{71} +(321.302 + 396.775i) q^{72} +(-296.558 - 15.5419i) q^{73} +(-1441.82 - 832.436i) q^{74} +(483.630 - 585.848i) q^{75} -573.420i q^{76} +(413.849 + 29.8757i) q^{77} +(181.618 + 1146.69i) q^{78} +(907.955 + 95.4299i) q^{79} +(-1027.70 + 824.225i) q^{80} +(-93.9234 - 893.621i) q^{81} +(45.0753 - 168.223i) q^{82} +(-224.817 + 1419.44i) q^{83} +(1390.09 - 1483.97i) q^{84} +(679.046 - 605.642i) q^{85} +(2536.41 - 539.130i) q^{86} +(311.313 + 479.379i) q^{87} +(627.029 + 965.540i) q^{88} +(-549.439 + 116.787i) q^{89} +(-564.283 + 56.6195i) q^{90} +(663.156 - 201.115i) q^{91} +(353.710 - 2233.24i) q^{92} +(155.882 - 581.758i) q^{93} +(213.440 + 2030.75i) q^{94} +(296.715 + 194.684i) q^{95} +(1151.28 + 121.005i) q^{96} +(-4.41746 - 27.8907i) q^{97} +(-1429.93 - 1010.87i) q^{98} +222.592i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	928 * q - 8 * q^2 - 24 * q^3 - 10 * q^4 - 24 * q^7 + 84 * q^8 - 10 * q^9 - 96 * q^10 - 6 * q^11 - 72 * q^12 - 20 * q^14 - 368 * q^15 - 1670 * q^16 + 120 * q^17 - 14 * q^18 - 30 * q^19 - 12 * q^21 - 880 * q^22 + 296 * q^23 + 32 * q^25 - 48 * q^26 + 226 * q^28 - 200 * q^29 - 38 * q^30 - 18 * q^31 - 964 * q^32 - 1092 * q^33 + 288 * q^35 + 7400 * q^36 - 392 * q^37 + 5424 * q^38 + 2430 * q^39 + 2172 * q^40 - 2098 * q^42 + 1560 * q^43 - 10 * q^44 - 4224 * q^45 - 6 * q^46 + 96 * q^47 + 6232 * q^50 - 16 * q^51 - 8928 * q^52 - 2384 * q^53 - 30 * q^54 + 244 * q^56 + 1556 * q^57 + 640 * q^58 + 4890 * q^59 + 3676 * q^60 - 18 * q^61 + 224 * q^63 - 9700 * q^64 - 1116 * q^65 - 2610 * q^66 - 2404 * q^67 - 13614 * q^68 - 1700 * q^70 - 24 * q^71 - 518 * q^72 - 4200 * q^73 - 16104 * q^75 - 722 * q^77 - 356 * q^78 - 10 * q^79 + 6414 * q^80 - 6810 * q^81 + 1692 * q^82 + 20620 * q^84 + 2712 * q^85 - 6 * q^86 + 9102 * q^87 + 1650 * q^88 + 20370 * q^89 - 12 * q^91 + 1612 * q^92 - 4604 * q^93 - 30 * q^94 + 1652 * q^95 - 2610 * q^96 - 19478 * q^98

Character values

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

$n$	$101$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{20}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	2.78061	−	4.28176i	0.983093	−	1.51383i	0.130372	−	0.991465i	$-0.458383\pi$
							0.852722	−	0.522366i	$-0.174950\pi$
$3$	2.17796	−	5.67379i	0.419149	−	1.09192i	−0.547723	−	0.836660i	$-0.684505\pi$
							0.966872	−	0.255262i	$-0.0821616\pi$
$5$	11.0343	+	1.80104i	0.986940	+	0.161090i
							$7$	−13.3513	+	12.8352i	−0.720902	+	0.693037i
$11$	−14.9911	−	16.6493i	−0.410909	−	0.456361i								0.501792	−	0.864988i	$-0.332674\pi$
							−0.912701	+	0.408628i	$0.866008\pi$
$13$	−33.3392	−	16.9872i	−0.711279	−	0.362415i	0.0606190	−	0.998161i	$-0.480693\pi$
							−0.771898	+	0.635746i	$0.780693\pi$
$17$	51.2162	−	63.2467i	0.730691	−	0.902329i	−0.267431	−	0.963577i	$-0.586175\pi$
							0.998123	+	0.0612484i	$0.0195082\pi$
$19$	28.9974	+	12.9105i	0.350130	+	0.155888i	0.574263	−	0.818671i	$-0.305289\pi$
							−0.224134	+	0.974558i	$0.571955\pi$
$23$	104.969	+	68.1680i	0.951637	+	0.618000i	0.924317	−	0.381627i	$-0.124636\pi$
							0.0273201	+	0.999627i	$0.491303\pi$
$29$	−55.2822	+	76.0894i	−0.353988	+	0.487222i	−0.948461	−	0.316893i	$-0.897360\pi$
							0.594474	+	0.804115i	$0.297360\pi$
$31$	98.5579	−	10.3589i	0.571017	−	0.0600163i	0.185380	−	0.982667i	$-0.440648\pi$
							0.385637	+	0.922651i	$0.373982\pi$
$37$	−17.0667	−	325.653i	−0.0758312	−	1.44694i	−0.726282	−	0.687397i	$-0.758753\pi$
							0.650451	−	0.759548i	$-0.274580\pi$
$41$	32.4428	−	10.5413i	0.123578	−	0.0401530i	−0.246575	−	0.969124i	$-0.579305\pi$
							0.370153	+	0.928971i	$0.379305\pi$
$43$	359.144	+	359.144i	1.27370	+	1.27370i	0.944133	+	0.329564i	$0.106902\pi$
							0.329564	+	0.944133i	$0.393098\pi$
$47$	−310.823	+	251.699i	−0.964642	+	0.781152i	−0.975773	−	0.218786i	$-0.929790\pi$
							0.0111305	+	0.999938i	$0.496457\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.x.a.3.55	✓	928
7.5	odd	6	inner	175.4.x.a.103.4	yes	928
25.17	odd	20	inner	175.4.x.a.17.4	yes	928
175.117	even	60	inner	175.4.x.a.117.55	yes	928

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.x.a.3.55	✓	928	1.1	even	1	trivial
175.4.x.a.17.4	yes	928	25.17	odd	20	inner
175.4.x.a.103.4	yes	928	7.5	odd	6	inner
175.4.x.a.117.55	yes	928	175.117	even	60	inner