Embedded newform 925.2.bq.b.357.8

• Label: 925.2.bq.b.357.8
• Level: $925$
• Weight: $2$
• Character: 925.357
• 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.bq (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			357.8
Character	\(\chi\)	\(=\)	925.357
Dual form			925.2.bq.b.868.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.400107 - 0.0705497i) q^{2} +(0.536320 - 0.765944i) q^{3} +(-1.72428 - 0.627585i) q^{4} +(-0.268623 + 0.268623i) q^{6} +(4.90270 + 0.428931i) q^{7} +(1.34932 + 0.779029i) q^{8} +(0.727029 + 1.99750i) q^{9} +(0.406725 + 0.234823i) q^{11} +(-1.40546 + 0.984113i) q^{12} +(-0.362737 + 0.996610i) q^{13} +(-1.93135 - 0.517503i) q^{14} +(2.32637 + 1.95206i) q^{16} +(-3.25401 + 1.18436i) q^{17} +(-0.149967 - 0.850504i) q^{18} +(-3.78685 + 5.40818i) q^{19} +(2.95795 - 3.52515i) q^{21} +(-0.146167 - 0.122649i) q^{22} +(0.516620 - 0.298271i) q^{23} +(1.32036 - 0.615693i) q^{24} +(0.215444 - 0.373160i) q^{26} +(4.62944 + 1.24046i) q^{27} +(-8.18442 - 3.81646i) q^{28} +(1.09712 + 4.09451i) q^{29} +(1.70594 + 1.70594i) q^{31} +(-2.79608 - 3.33224i) q^{32} +(0.397996 - 0.185589i) q^{33} +(1.38551 - 0.244303i) q^{34} -3.90051i q^{36} +(-1.07055 + 5.98781i) q^{37} +(1.89669 - 1.89669i) q^{38} +(0.568805 + 0.812338i) q^{39} +(0.559051 - 1.53598i) q^{41} +(-1.43220 + 1.20176i) q^{42} -2.50977i q^{43} +(-0.553935 - 0.660155i) q^{44} +(-0.227746 + 0.0828929i) q^{46} +(1.69432 - 6.32327i) q^{47} +(2.74285 - 0.734945i) q^{48} +(16.9588 + 2.99030i) q^{49} +(-0.838036 + 3.12759i) q^{51} +(1.25092 - 1.49078i) q^{52} +(13.6811 - 1.19694i) q^{53} +(-1.76476 - 0.822922i) q^{54} +(6.28115 + 4.39811i) q^{56} +(2.11140 + 5.80103i) q^{57} +(-0.150100 - 1.71565i) q^{58} +(-0.467014 - 5.33799i) q^{59} +(4.91565 + 10.5416i) q^{61} +(-0.562207 - 0.802915i) q^{62} +(2.70762 + 10.1050i) q^{63} +(-2.15322 - 3.72949i) q^{64} +(-0.172334 + 0.0461769i) q^{66} +(0.403093 - 4.60737i) q^{67} +6.35411 q^{68} +(0.0486149 - 0.555671i) q^{69} +(-1.87200 - 10.6167i) q^{71} +(-0.575113 + 3.26163i) q^{72} +(-1.78336 - 1.78336i) q^{73} +(0.850775 - 2.32024i) q^{74} +(9.92367 - 6.94863i) q^{76} +(1.89333 + 1.32572i) q^{77} +(-0.170273 - 0.365152i) q^{78} +(-3.11433 - 0.272468i) q^{79} +(-1.45214 + 1.21849i) q^{81} +(-0.332043 + 0.575116i) q^{82} +(2.23009 - 4.78245i) q^{83} +(-7.31266 + 4.22197i) q^{84} +(-0.177064 + 1.00418i) q^{86} +(3.72458 + 1.35564i) q^{87} +(0.365868 + 0.633701i) q^{88} +(4.57828 - 0.400548i) q^{89} +(-2.20587 + 4.73049i) q^{91} +(-1.07799 + 0.190078i) q^{92} +(2.22159 - 0.391726i) q^{93} +(-1.12401 + 2.41045i) q^{94} +(-4.05191 + 0.354496i) q^{96} +(-5.20922 - 9.02263i) q^{97} +(-6.57439 - 2.39288i) q^{98} +(-0.173357 + 0.983155i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	204 * q + 12 * q^2 + 18 * q^3 - 24 * q^6 + 12 * q^7 + 18 * q^8 - 36 * q^11 + 12 * q^12 + 12 * q^13 + 24 * q^14 - 24 * q^16 - 12 * q^17 - 66 * q^18 - 24 * q^21 + 24 * q^22 + 18 * q^23 - 36 * q^24 - 12 * q^26 - 6 * q^27 + 24 * q^28 - 12 * q^31 + 36 * q^32 + 36 * q^33 - 12 * q^37 - 60 * q^38 - 24 * q^39 - 18 * q^41 + 150 * q^42 + 180 * q^44 - 24 * q^46 + 6 * q^47 + 30 * q^48 - 36 * q^49 - 24 * q^51 - 60 * q^52 - 30 * q^53 - 24 * q^56 + 36 * q^57 + 138 * q^58 + 60 * q^61 - 72 * q^62 + 6 * q^63 + 18 * q^64 + 120 * q^66 - 30 * q^67 + 24 * q^68 - 132 * q^69 + 24 * q^71 + 192 * q^72 + 90 * q^73 + 66 * q^74 + 36 * q^76 + 132 * q^77 - 150 * q^78 + 96 * q^79 + 132 * q^81 + 6 * q^82 - 48 * q^83 + 96 * q^86 - 258 * q^87 + 174 * q^88 + 12 * q^89 - 96 * q^91 + 102 * q^92 + 42 * q^93 - 72 * q^94 + 60 * q^96 + 18 * q^97 - 36 * q^98 - 192 * q^99

Character values

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

\(n\)	\(76\)	\(852\)
\(\chi(n)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{1}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.400107	−	0.0705497i	−0.282919	−	0.0498862i	0.0303881	−	0.999538i	\(-0.490326\pi\)
							−0.313307	+	0.949652i	\(0.601437\pi\)
\(3\)	0.536320	−	0.765944i	0.309645	−	0.442218i	−0.633924	−	0.773395i	\(-0.718557\pi\)
							0.943569	+	0.331177i	\(0.107446\pi\)
\(5\)		0			0
							\(7\)	4.90270	+	0.428931i	1.85305	+	0.162121i	0.958544	−	0.284943i	\(-0.0919748\pi\)
0.894502	+	0.447064i	\(0.147530\pi\)
\(11\)	0.406725	+	0.234823i	0.122632	+	0.0708018i	0.560061	−	0.828451i	\(-0.310778\pi\)
							−0.437429	+	0.899253i	\(0.644111\pi\)
\(13\)	−0.362737	+	0.996610i	−0.100605	+	0.276410i	−0.979776	−	0.200096i	\(-0.935875\pi\)
							0.879171	+	0.476506i	\(0.158097\pi\)
\(17\)	−3.25401	+	1.18436i	−0.789214	+	0.287251i	−0.705009	−	0.709198i	\(-0.749057\pi\)
							−0.0842049	+	0.996448i	\(0.526835\pi\)
\(19\)	−3.78685	+	5.40818i	−0.868763	+	1.24072i	0.100560	+	0.994931i	\(0.467936\pi\)
							−0.969323	+	0.245791i	\(0.920952\pi\)
\(23\)	0.516620	−	0.298271i	0.107723	−	0.0621937i	−0.445171	−	0.895446i	\(-0.646857\pi\)
							0.552893	+	0.833252i	\(0.313524\pi\)
\(29\)	1.09712	+	4.09451i	0.203730	+	0.760332i	0.989833	+	0.142236i	\(0.0454291\pi\)
							−0.786102	+	0.618096i	\(0.787904\pi\)
\(31\)	1.70594	+	1.70594i	0.306397	+	0.306397i	0.843510	−	0.537113i	\(-0.180485\pi\)
							−0.537113	+	0.843510i	\(0.680485\pi\)
\(37\)	−1.07055	+	5.98781i	−0.175998	+	0.984391i
							\(41\)	0.559051	−	1.53598i	0.0873090	−	0.239880i	−0.888352	−	0.459163i	\(-0.848150\pi\)
0.975661	+	0.219283i	\(0.0703718\pi\)
\(43\)		−	2.50977i		−	0.382736i	−0.981518	−	0.191368i	\(-0.938708\pi\)
							0.981518	−	0.191368i	\(-0.0612924\pi\)
\(47\)	1.69432	−	6.32327i	0.247141	−	0.922344i	−0.725154	−	0.688587i	\(-0.758231\pi\)
							0.972295	−	0.233757i	\(-0.0751020\pi\)
\(53\)	13.6811	−	1.19694i	1.87924	−	0.164412i	0.910666	−	0.413144i	\(-0.135569\pi\)
							0.968572	+	0.248732i	\(0.0800137\pi\)
\(59\)	−0.467014	−	5.33799i	−0.0608000	−	0.694948i	−0.964030	−	0.265792i	\(-0.914367\pi\)
							0.903230	−	0.429156i	\(-0.141189\pi\)
\(61\)	4.91565	+	10.5416i	0.629384	+	1.34972i	0.919637	+	0.392769i	\(0.128483\pi\)
							−0.290253	+	0.956950i	\(0.593739\pi\)
\(67\)	0.403093	−	4.60737i	0.0492456	−	0.562880i	−0.930990	−	0.365045i	\(-0.881054\pi\)
							0.980235	−	0.197835i	\(-0.0633909\pi\)
\(71\)	−1.87200	−	10.6167i	−0.222166	−	1.25996i	−0.868030	−	0.496512i	\(-0.834614\pi\)
							0.645864	−	0.763453i	\(-0.276497\pi\)
\(73\)	−1.78336	−	1.78336i	−0.208726	−	0.208726i	0.595000	−	0.803726i	\(-0.297152\pi\)
							−0.803726	+	0.595000i	\(0.797152\pi\)
\(79\)	−3.11433	−	0.272468i	−0.350389	−	0.0306551i	−0.0893971	−	0.995996i	\(-0.528494\pi\)
							−0.260992	+	0.965341i	\(0.584050\pi\)
\(83\)	2.23009	−	4.78245i	0.244785	−	0.524942i	−0.745077	−	0.666979i	\(-0.767587\pi\)
							0.989861	+	0.142036i	\(0.0453650\pi\)
\(89\)	4.57828	−	0.400548i	0.485297	−	0.0424580i	0.158118	−	0.987420i	\(-0.449457\pi\)
							0.327179	+	0.944962i	\(0.393902\pi\)
\(97\)	−5.20922	−	9.02263i	−0.528916	−	0.916109i	−0.999431	−	0.0337173i	\(-0.989265\pi\)
							0.470516	−	0.882392i	\(-0.344068\pi\)

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.bq.b.357.8		204
5.2	odd	4		185.2.z.a.98.10	yes	204
5.3	odd	4		925.2.bn.b.468.8		204
5.4	even	2		185.2.bc.a.172.10	yes	204
37.17	odd	36		925.2.bn.b.757.8		204
185.17	even	36		185.2.bc.a.128.10	yes	204
185.54	odd	36		185.2.z.a.17.10	✓	204
185.128	even	36	inner	925.2.bq.b.868.8		204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.z.a.17.10	✓	204	185.54	odd	36
185.2.z.a.98.10	yes	204	5.2	odd	4
185.2.bc.a.128.10	yes	204	185.17	even	36
185.2.bc.a.172.10	yes	204	5.4	even	2
925.2.bn.b.468.8		204	5.3	odd	4
925.2.bn.b.757.8		204	37.17	odd	36
925.2.bq.b.357.8		204	1.1	even	1	trivial
925.2.bq.b.868.8		204	185.128	even	36	inner