Embedded newform 175.2.x.a.117.11

• Label: 175.2.x.a.117.11
• Level: $175$
• Weight: $2$
• Character: 175.117
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$1.39738203537$
Analytic rank:	$0$
Dimension:	$288$
Relative dimension:	$18$ over $\Q(\zeta_{60})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			117.11
Character	$\chi$	$=$	175.117
Dual form			175.2.x.a.3.11

$q$-expansion

$f(q)$	$=$	$q+(0.437609 + 0.673859i) q^{2} +(1.12462 + 2.92974i) q^{3} +(0.550889 - 1.23732i) q^{4} +(-2.22451 - 0.227106i) q^{5} +(-1.48209 + 2.03992i) q^{6} +(-0.782993 + 2.52724i) q^{7} +(2.66204 - 0.421625i) q^{8} +(-5.08915 + 4.58229i) q^{9} +(-0.820427 - 1.59839i) q^{10} +(3.35615 - 3.72738i) q^{11} +(4.24455 + 0.222448i) q^{12} +(0.816789 - 0.416175i) q^{13} +(-2.04565 + 0.578315i) q^{14} +(-1.83636 - 6.77262i) q^{15} +(-0.363508 - 0.403716i) q^{16} +(-0.326203 - 0.402827i) q^{17} +(-5.31488 - 1.42412i) q^{18} +(-1.01710 + 0.452842i) q^{19} +(-1.50646 + 2.62731i) q^{20} +(-8.28471 + 0.548217i) q^{21} +(3.98041 + 0.630435i) q^{22} +(4.33381 - 2.81441i) q^{23} +(4.22903 + 7.32490i) q^{24} +(4.89685 + 1.01040i) q^{25} +(0.637878 + 0.368279i) q^{26} +(-10.7599 - 5.48243i) q^{27} +(2.69565 + 2.36104i) q^{28} +(-3.39698 - 4.67555i) q^{29} +(3.76018 - 4.20121i) q^{30} +(-0.340506 - 0.0357886i) q^{31} +(1.50812 - 5.62839i) q^{32} +(14.6946 + 5.64074i) q^{33} +(0.128699 - 0.396095i) q^{34} +(2.31572 - 5.44403i) q^{35} +(2.86619 + 8.82122i) q^{36} +(-0.334068 + 6.37440i) q^{37} +(-0.750244 - 0.487214i) q^{38} +(2.13786 + 1.92494i) q^{39} +(-6.01747 + 0.333344i) q^{40} +(6.75808 + 2.19583i) q^{41} +(-3.99489 - 5.34282i) q^{42} +(-3.02167 + 3.02167i) q^{43} +(-2.76308 - 6.20599i) q^{44} +(12.3615 - 9.03756i) q^{45} +(3.79303 + 1.68877i) q^{46} +(-9.91961 - 8.03274i) q^{47} +(0.773974 - 1.51901i) q^{48} +(-5.77384 - 3.95762i) q^{49} +(1.46204 + 3.74194i) q^{50} +(0.813322 - 1.40871i) q^{51} +(-0.0649800 - 1.23989i) q^{52} +(-5.47316 + 2.10095i) q^{53} +(-1.01424 - 9.64981i) q^{54} +(-8.31228 + 7.52937i) q^{55} +(-1.01881 + 7.05773i) q^{56} +(-2.47056 - 2.47056i) q^{57} +(1.66411 - 4.33515i) q^{58} +(8.60248 - 1.82851i) q^{59} +(-9.39151 - 1.45880i) q^{60} +(0.301099 - 1.41656i) q^{61} +(-0.124892 - 0.245114i) q^{62} +(-7.59576 - 16.4494i) q^{63} +(3.41938 - 1.11102i) q^{64} +(-1.91147 + 0.740286i) q^{65} +(2.62944 + 12.3706i) q^{66} +(-1.09328 + 0.885317i) q^{67} +(-0.678125 + 0.181703i) q^{68} +(13.1194 + 9.53177i) q^{69} +(4.68189 - 0.821886i) q^{70} +(-6.16371 + 4.47820i) q^{71} +(-11.6155 + 14.3439i) q^{72} +(-1.20435 + 0.0631171i) q^{73} +(-4.44164 + 2.56438i) q^{74} +(2.54690 + 15.4828i) q^{75} +1.50794i q^{76} +(6.79213 + 11.4003i) q^{77} +(-0.361590 + 2.28299i) q^{78} +(-4.83574 + 0.508257i) q^{79} +(0.716939 + 0.980624i) q^{80} +(1.81383 - 17.2574i) q^{81} +(1.47772 + 5.51491i) q^{82} +(-0.543025 - 3.42852i) q^{83} +(-3.88563 + 10.5528i) q^{84} +(0.634155 + 0.970172i) q^{85} +(-3.35849 - 0.713869i) q^{86} +(9.87780 - 15.2105i) q^{87} +(7.36263 - 11.3375i) q^{88} +(-3.07374 - 0.653343i) q^{89} +(11.4996 + 4.37500i) q^{90} +(0.412231 + 2.39008i) q^{91} +(-1.09487 - 6.91272i) q^{92} +(-0.278088 - 1.03784i) q^{93} +(1.07202 - 10.1996i) q^{94} +(2.36538 - 0.776360i) q^{95} +(18.1858 - 1.91140i) q^{96} +(-0.557021 + 3.51689i) q^{97} +(0.140190 - 5.62265i) q^{98} +34.3480i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	288 * q - 8 * q^2 - 24 * q^3 - 10 * q^4 - 30 * q^5 - 10 * q^7 - 36 * q^8 - 10 * q^9 - 36 * q^10 - 6 * q^11 - 36 * q^12 - 20 * q^14 - 28 * q^15 - 30 * q^16 - 42 * q^17 - 14 * q^18 - 30 * q^19 - 12 * q^21 + 32 * q^22 - 40 * q^23 + 2 * q^25 - 48 * q^26 + 22 * q^28 - 58 * q^30 - 18 * q^31 + 8 * q^32 - 30 * q^33 - 2 * q^35 + 40 * q^36 - 10 * q^37 + 72 * q^38 + 30 * q^39 - 48 * q^40 + 6 * q^42 - 108 * q^43 - 10 * q^44 + 186 * q^45 - 6 * q^46 - 54 * q^47 - 248 * q^50 - 16 * q^51 + 216 * q^52 + 50 * q^53 - 30 * q^54 + 4 * q^56 - 216 * q^57 - 4 * q^58 + 90 * q^59 + 96 * q^60 - 18 * q^61 - 66 * q^63 - 100 * q^64 + 14 * q^65 - 90 * q^66 + 4 * q^67 + 342 * q^68 - 60 * q^70 - 24 * q^71 + 58 * q^72 - 6 * q^73 + 216 * q^75 - 80 * q^77 - 132 * q^78 - 10 * q^79 - 6 * q^80 - 10 * q^81 + 216 * q^82 + 20 * q^84 - 48 * q^85 - 6 * q^86 - 48 * q^87 - 122 * q^88 + 120 * q^89 - 12 * q^91 - 4 * q^92 + 106 * q^93 - 30 * q^94 - 98 * q^95 - 90 * q^96 + 222 * q^98

Character values

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

$n$	$101$	$127$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{13}{20}\right)$

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)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$	0.437609	+	0.673859i	0.309437	+	0.476491i	0.958837	−	0.283956i	$-0.0916469\pi$
							−0.649401	+	0.760446i	$0.724980\pi$
$3$	1.12462	+	2.92974i	0.649300	+	1.69148i	0.719989	+	0.693986i	$0.244147\pi$
							−0.0706887	+	0.997498i	$0.522520\pi$
$5$	−2.22451	−	0.227106i	−0.994829	−	0.101565i
							$7$	−0.782993	+	2.52724i	−0.295944	+	0.955205i
$11$	3.35615	−	3.72738i	1.01192	−	1.12385i								0.0196392	−	0.999807i	$-0.493748\pi$
							0.992277	−	0.124040i	$-0.0395851\pi$
$13$	0.816789	−	0.416175i	0.226536	−	0.115426i	−0.337039	−	0.941491i	$-0.609425\pi$
							0.563575	+	0.826065i	$0.309425\pi$
$17$	−0.326203	−	0.402827i	−0.0791157	−	0.0976998i	0.736051	−	0.676926i	$-0.236688\pi$
							−0.815167	+	0.579226i	$0.803355\pi$
$19$	−1.01710	+	0.452842i	−0.233339	+	0.103889i	−0.520075	−	0.854120i	$-0.674096\pi$
							0.286737	+	0.958009i	$0.407429\pi$
$23$	4.33381	−	2.81441i	0.903661	−	0.586844i	−0.00701045	−	0.999975i	$-0.502232\pi$
							0.910672	+	0.413131i	$0.135565\pi$
$29$	−3.39698	−	4.67555i	−0.630804	−	0.868227i	0.367279	−	0.930111i	$-0.380289\pi$
							−0.998083	+	0.0618834i	$0.980289\pi$
$31$	−0.340506	−	0.0357886i	−0.0611566	−	0.00642782i	0.0739007	−	0.997266i	$-0.476455\pi$
							−0.135057	+	0.990838i	$0.543122\pi$
$37$	−0.334068	+	6.37440i	−0.0549205	+	1.04794i	0.822666	+	0.568526i	$0.192486\pi$
							−0.877586	+	0.479419i	$0.840847\pi$
$41$	6.75808	+	2.19583i	1.05543	+	0.342931i	0.784799	−	0.619750i	$-0.212766\pi$
							0.270635	+	0.962682i	$0.412766\pi$
$43$	−3.02167	+	3.02167i	−0.460800	+	0.460800i	−0.898918	−	0.438118i	$-0.855645\pi$
							0.438118	+	0.898918i	$0.355645\pi$
$47$	−9.91961	−	8.03274i	−1.44692	−	1.17170i	−0.953302	−	0.302020i	$-0.902339\pi$
							−0.493622	−	0.869676i	$-0.664327\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.x.a.117.11	yes	288
5.2	odd	4		875.2.bb.a.768.8		288
5.3	odd	4		875.2.bb.b.768.11		288
5.4	even	2		875.2.bb.c.607.8		288
7.3	odd	6	inner	175.2.x.a.17.8	yes	288
25.3	odd	20	inner	175.2.x.a.103.8	yes	288
25.4	even	10		875.2.bb.a.257.8		288
25.21	even	5		875.2.bb.b.257.11		288
25.22	odd	20		875.2.bb.c.243.11		288
35.3	even	12		875.2.bb.b.143.11		288
35.17	even	12		875.2.bb.a.143.8		288
35.24	odd	6		875.2.bb.c.857.11		288
175.3	even	60	inner	175.2.x.a.3.11	✓	288
175.122	even	60		875.2.bb.c.493.8		288
175.129	odd	30		875.2.bb.a.507.8		288
175.171	odd	30		875.2.bb.b.507.11		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.x.a.3.11	✓	288	175.3	even	60	inner
175.2.x.a.17.8	yes	288	7.3	odd	6	inner
175.2.x.a.103.8	yes	288	25.3	odd	20	inner
175.2.x.a.117.11	yes	288	1.1	even	1	trivial
875.2.bb.a.143.8		288	35.17	even	12
875.2.bb.a.257.8		288	25.4	even	10
875.2.bb.a.507.8		288	175.129	odd	30
875.2.bb.a.768.8		288	5.2	odd	4
875.2.bb.b.143.11		288	35.3	even	12
875.2.bb.b.257.11		288	25.21	even	5
875.2.bb.b.507.11		288	175.171	odd	30
875.2.bb.b.768.11		288	5.3	odd	4
875.2.bb.c.243.11		288	25.22	odd	20
875.2.bb.c.493.8		288	175.122	even	60
875.2.bb.c.607.8		288	5.4	even	2
875.2.bb.c.857.11		288	35.24	odd	6