Embedded newform 925.2.bn.b.757.15

• Label: 925.2.bn.b.757.15
• Level: $925$
• Weight: $2$
• Character: 925.757
• Analytic conductor: $7.386$
• Analytic rank: $0$
• Dimension: $204$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.bn (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.38616218697\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{36})\)
Twist minimal:	no (minimal twist has level 185)
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			757.15
Character	\(\chi\)	\(=\)	925.757
Dual form			925.2.bn.b.468.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.321176 + 1.82148i) q^{2} +(2.27797 - 1.59505i) q^{3} +(-1.33525 + 0.485992i) q^{4} +(3.63698 + 3.63698i) q^{6} +(0.309940 + 3.54263i) q^{7} +(0.535504 + 0.927519i) q^{8} +(1.61890 - 4.44788i) q^{9} +(4.83320 - 2.79045i) q^{11} +(-2.26648 + 3.23687i) q^{12} +(-2.98008 + 1.08466i) q^{13} +(-6.35329 + 1.70236i) q^{14} +(-3.69448 + 3.10004i) q^{16} +(1.03247 - 2.83670i) q^{17} +(8.62168 + 1.52023i) q^{18} +(0.0922365 + 0.131727i) q^{19} +(6.35671 + 7.57564i) q^{21} +(6.63506 + 7.90736i) q^{22} +(-3.22285 + 5.58214i) q^{23} +(2.69930 + 1.25871i) q^{24} +(-2.93282 - 5.07980i) q^{26} +(-1.24756 - 4.65596i) q^{27} +(-2.13554 - 4.57968i) q^{28} +(2.02173 - 7.54520i) q^{29} +(-3.60337 + 3.60337i) q^{31} +(-5.19236 - 4.35691i) q^{32} +(6.55898 - 14.0658i) q^{33} +(5.49859 + 0.969550i) q^{34} +6.72581i q^{36} +(1.74988 - 5.82563i) q^{37} +(-0.210315 + 0.210315i) q^{38} +(-5.05845 + 7.22421i) q^{39} +(0.214432 + 0.589148i) q^{41} +(-11.7572 + 14.0117i) q^{42} +0.198014 q^{43} +(-5.09741 + 6.07486i) q^{44} +(-11.2029 - 4.07751i) q^{46} +(0.516958 - 0.138518i) q^{47} +(-3.47120 + 12.9547i) q^{48} +(-5.56052 + 0.980470i) q^{49} +(-2.17273 - 8.10875i) q^{51} +(3.45203 - 2.89659i) q^{52} +(0.212443 - 2.42823i) q^{53} +(8.08006 - 3.76779i) q^{54} +(-3.11989 + 2.18457i) q^{56} +(0.420224 + 0.152949i) q^{57} +(14.3928 + 1.25920i) q^{58} +(-0.179432 + 2.05092i) q^{59} +(-0.187756 + 0.402643i) q^{61} +(-7.72080 - 5.40616i) q^{62} +(16.2590 + 4.35657i) q^{63} +(1.44556 - 2.50378i) q^{64} +(27.7271 + 7.42946i) q^{66} +(14.0515 - 1.22935i) q^{67} +4.28948i q^{68} +(1.56225 + 17.8566i) q^{69} +(0.219398 - 1.24427i) q^{71} +(4.99242 - 0.880298i) q^{72} +(-6.87081 - 6.87081i) q^{73} +(11.1733 + 1.31631i) q^{74} +(-0.187177 - 0.131063i) q^{76} +(11.3835 + 16.2574i) q^{77} +(-14.7834 - 6.89362i) q^{78} +(-6.64942 + 0.581749i) q^{79} +(0.609426 + 0.511369i) q^{81} +(-1.00425 + 0.579805i) q^{82} +(-8.81283 + 4.10949i) q^{83} +(-12.1695 - 7.02607i) q^{84} +(0.0635975 + 0.360680i) q^{86} +(-7.42954 - 20.4125i) q^{87} +(5.17640 + 2.98859i) q^{88} +(-5.12449 - 0.448334i) q^{89} +(-4.76620 - 10.2212i) q^{91} +(1.59044 - 9.01985i) q^{92} +(-2.46081 + 13.9559i) q^{93} +(0.418343 + 0.897140i) q^{94} +(-18.7775 - 1.64282i) q^{96} +(-6.38433 - 3.68600i) q^{97} +(-3.57181 - 9.81348i) q^{98} +(-4.58714 - 26.0150i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	204 * q + 12 * q^2 + 6 * q^3 - 24 * q^6 + 12 * q^7 + 24 * q^8 - 36 * q^11 - 36 * q^12 + 12 * q^13 - 24 * q^14 - 24 * q^16 + 30 * q^17 + 90 * q^18 - 24 * q^21 + 24 * q^22 + 6 * q^23 + 36 * q^24 - 12 * q^26 + 42 * q^27 - 12 * q^31 - 12 * q^32 + 24 * q^33 + 36 * q^37 - 60 * q^38 + 24 * q^39 - 18 * q^41 + 42 * q^42 + 24 * q^43 - 180 * q^44 - 24 * q^46 + 6 * q^47 + 6 * q^48 + 36 * q^49 - 24 * q^51 - 60 * q^52 + 90 * q^53 - 24 * q^56 + 36 * q^57 + 60 * q^61 + 96 * q^62 + 6 * q^63 - 18 * q^64 + 120 * q^66 - 30 * q^67 + 132 * q^69 + 24 * q^71 - 216 * q^72 - 90 * q^73 - 66 * q^74 + 36 * q^76 - 24 * q^77 + 138 * q^78 - 96 * q^79 + 132 * q^81 + 18 * q^82 + 72 * q^83 + 96 * q^86 + 114 * q^87 - 198 * q^88 - 12 * q^89 - 96 * q^91 + 54 * q^92 - 42 * q^93 + 72 * q^94 + 60 * q^96 - 54 * q^97 + 60 * q^98 + 192 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\)	\(76\)	\(852\)
\(\chi(n)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

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

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.321176	+	1.82148i	0.227106	+	1.28798i	0.858618	+	0.512616i	\(0.171323\pi\)
							−0.631512	+	0.775366i	\(0.717566\pi\)
\(3\)	2.27797	−	1.59505i	1.31519	−	0.920903i	0.315578	−	0.948900i	\(-0.397802\pi\)
							0.999608	+	0.0279962i	\(0.00891262\pi\)
\(5\)		0			0
							\(7\)	0.309940	+	3.54263i	0.117146	+	1.33899i	0.796259	+	0.604955i	\(0.206809\pi\)
−0.679113	+	0.734034i	\(0.737635\pi\)
\(11\)	4.83320	−	2.79045i	1.45727	−	0.841353i	0.458390	−	0.888751i	\(-0.348426\pi\)
							0.998876	+	0.0473983i	\(0.0150930\pi\)
\(13\)	−2.98008	+	1.08466i	−0.826526	+	0.300831i	−0.720432	−	0.693525i	\(-0.756057\pi\)
							−0.106094	+	0.994356i	\(0.533834\pi\)
\(17\)	1.03247	−	2.83670i	0.250411	−	0.688000i	−0.749258	−	0.662279i	\(-0.769590\pi\)
							0.999669	−	0.0257213i	\(-0.00818824\pi\)
\(19\)	0.0922365	+	0.131727i	0.0211605	+	0.0302203i	0.829593	−	0.558369i	\(-0.188573\pi\)
							−0.808432	+	0.588589i	\(0.799684\pi\)
\(23\)	−3.22285	+	5.58214i	−0.672011	+	1.16396i	0.305322	+	0.952249i	\(0.401236\pi\)
							−0.977333	+	0.211708i	\(0.932097\pi\)
\(29\)	2.02173	−	7.54520i	0.375426	−	1.40111i	−0.477296	−	0.878743i	\(-0.658383\pi\)
							0.852722	−	0.522365i	\(-0.174950\pi\)
\(31\)	−3.60337	+	3.60337i	−0.647185	+	0.647185i	−0.952312	−	0.305127i	\(-0.901301\pi\)
							0.305127	+	0.952312i	\(0.401301\pi\)
\(37\)	1.74988	−	5.82563i	0.287678	−	0.957727i
							\(41\)	0.214432	+	0.589148i	0.0334887	+	0.0920094i	0.955310	−	0.295605i	\(-0.0955213\pi\)
−0.921821	+	0.387615i	\(0.873299\pi\)
\(43\)	0.198014			0.0301969			0.0150985	−	0.999886i	\(-0.495194\pi\)
							0.0150985	+	0.999886i	\(0.495194\pi\)
\(47\)	0.516958	−	0.138518i	0.0754061	−	0.0202050i	−0.220919	−	0.975292i	\(-0.570906\pi\)
							0.296325	+	0.955087i	\(0.404239\pi\)
\(53\)	0.212443	−	2.42823i	0.0291813	−	0.333544i	−0.967545	−	0.252698i	\(-0.918682\pi\)
							0.996727	−	0.0808460i	\(-0.0257622\pi\)
\(59\)	−0.179432	+	2.05092i	−0.0233601	+	0.267007i	0.975528	+	0.219873i	\(0.0705644\pi\)
							−0.998889	+	0.0471341i	\(0.984991\pi\)
\(61\)	−0.187756	+	0.402643i	−0.0240396	+	0.0515532i	−0.917961	−	0.396671i	\(-0.870165\pi\)
							0.893921	+	0.448224i	\(0.147943\pi\)
\(67\)	14.0515	−	1.22935i	1.71667	−	0.150189i	0.814095	−	0.580731i	\(-0.197233\pi\)
							0.902570	+	0.430542i	\(0.141678\pi\)
\(71\)	0.219398	−	1.24427i	0.0260378	−	0.147668i	−0.969017	−	0.246993i	\(-0.920557\pi\)
							0.995055	+	0.0993256i	\(0.0316685\pi\)
\(73\)	−6.87081	−	6.87081i	−0.804168	−	0.804168i	0.179576	−	0.983744i	\(-0.442527\pi\)
							−0.983744	+	0.179576i	\(0.942527\pi\)
\(79\)	−6.64942	+	0.581749i	−0.748118	+	0.0654519i	−0.454837	−	0.890574i	\(-0.650303\pi\)
							−0.293281	+	0.956026i	\(0.594747\pi\)
\(83\)	−8.81283	+	4.10949i	−0.967333	+	0.451075i	−0.841073	−	0.540922i	\(-0.818075\pi\)
							−0.126261	+	0.991997i	\(0.540298\pi\)
\(89\)	−5.12449	−	0.448334i	−0.543194	−	0.0475234i	−0.187743	−	0.982218i	\(-0.560117\pi\)
							−0.355451	+	0.934695i	\(0.615673\pi\)
\(97\)	−6.38433	−	3.68600i	−0.648231	−	0.374256i	0.139547	−	0.990215i	\(-0.455435\pi\)
							−0.787778	+	0.615959i	\(0.788769\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.bn.b.757.15		204
5.2	odd	4		185.2.bc.a.128.3	yes	204
5.3	odd	4		925.2.bq.b.868.15		204
5.4	even	2		185.2.z.a.17.3	✓	204
37.24	odd	36		925.2.bq.b.357.15		204
185.24	odd	36		185.2.bc.a.172.3	yes	204
185.98	even	36	inner	925.2.bn.b.468.15		204
185.172	even	36		185.2.z.a.98.3	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.z.a.17.3	✓	204	5.4	even	2
185.2.z.a.98.3	yes	204	185.172	even	36
185.2.bc.a.128.3	yes	204	5.2	odd	4
185.2.bc.a.172.3	yes	204	185.24	odd	36
925.2.bn.b.468.15		204	185.98	even	36	inner
925.2.bn.b.757.15		204	1.1	even	1	trivial
925.2.bq.b.357.15		204	37.24	odd	36
925.2.bq.b.868.15		204	5.3	odd	4