Embedded newform 648.2.v.b.35.25

• Label: 648.2.v.b.35.25
• Level: $648$
• Weight: $2$
• Character: 648.35
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.v (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			35.25
Character	\(\chi\)	\(=\)	648.35
Dual form			648.2.v.b.611.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20377 - 0.742249i) q^{2} +(0.898133 - 1.78700i) q^{4} +(3.07462 + 1.11907i) q^{5} +(1.33825 + 0.235969i) q^{7} +(-0.245250 - 2.81777i) q^{8} +(4.53177 - 0.935029i) q^{10} +(0.982057 + 2.69818i) q^{11} +(-2.36880 - 2.82303i) q^{13} +(1.78609 - 0.709260i) q^{14} +(-2.38671 - 3.20992i) q^{16} +(-1.94027 + 1.12022i) q^{17} +(-1.22444 + 2.12079i) q^{19} +(4.76119 - 4.48926i) q^{20} +(3.18489 + 2.51906i) q^{22} +(1.08869 + 6.17426i) q^{23} +(4.37075 + 3.66750i) q^{25} +(-4.94689 - 1.64004i) q^{26} +(1.62360 - 2.17951i) q^{28} +(-5.00298 - 4.19800i) q^{29} +(4.65080 - 0.820062i) q^{31} +(-5.25562 - 2.09248i) q^{32} +(-1.50416 + 2.78865i) q^{34} +(3.85054 + 2.22311i) q^{35} +(3.70698 - 2.14023i) q^{37} +(0.100210 + 3.46179i) q^{38} +(2.39924 - 8.93804i) q^{40} +(-7.28211 - 8.67848i) q^{41} +(-3.24005 + 1.17928i) q^{43} +(5.70365 + 0.668391i) q^{44} +(5.89337 + 6.62433i) q^{46} +(2.21737 - 12.5753i) q^{47} +(-4.84262 - 1.76257i) q^{49} +(7.98359 + 1.17064i) q^{50} +(-7.17224 + 1.69759i) q^{52} -4.14210 q^{53} +9.39487i q^{55} +(0.336703 - 3.82875i) q^{56} +(-9.13841 - 1.33998i) q^{58} +(-3.92440 + 10.7822i) q^{59} +(9.59806 + 1.69240i) q^{61} +(4.98981 - 4.43922i) q^{62} +(-7.87971 + 1.38212i) q^{64} +(-4.12400 - 11.3306i) q^{65} +(-6.64683 + 5.57735i) q^{67} +(0.259201 + 4.47336i) q^{68} +(6.28527 - 0.181942i) q^{70} +(3.19451 + 5.53306i) q^{71} +(-2.87137 + 4.97336i) q^{73} +(2.87378 - 5.32784i) q^{74} +(2.69014 + 4.09283i) q^{76} +(0.677549 + 3.84257i) q^{77} +(5.31972 - 6.33979i) q^{79} +(-3.74612 - 12.5402i) q^{80} +(-15.2076 - 5.04177i) q^{82} +(1.56022 - 1.85940i) q^{83} +(-7.21919 + 1.27294i) q^{85} +(-3.02496 + 3.82451i) q^{86} +(7.36201 - 3.42894i) q^{88} +(-1.06132 - 0.612754i) q^{89} +(-2.50390 - 4.33688i) q^{91} +(12.0112 + 3.59982i) q^{92} +(-6.66482 - 16.7837i) q^{94} +(-6.13801 + 5.15040i) q^{95} +(16.9633 - 6.17414i) q^{97} +(-7.13768 + 1.47270i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	192 * q + 6 * q^2 - 6 * q^4 + 9 * q^8 - 3 * q^10 + 30 * q^11 - 9 * q^14 - 6 * q^16 + 18 * q^17 - 6 * q^19 + 27 * q^20 - 18 * q^22 - 12 * q^25 - 12 * q^28 + 36 * q^32 + 12 * q^34 + 18 * q^35 + 102 * q^38 + 9 * q^40 - 18 * q^41 - 42 * q^43 + 81 * q^44 - 3 * q^46 - 12 * q^49 - 57 * q^50 + 21 * q^52 + 69 * q^56 - 33 * q^58 + 84 * q^59 - 90 * q^62 - 51 * q^64 + 12 * q^65 + 30 * q^67 - 63 * q^68 - 33 * q^70 - 6 * q^73 - 51 * q^74 + 30 * q^76 - 12 * q^82 + 72 * q^83 - 42 * q^86 - 78 * q^88 - 144 * q^89 - 6 * q^91 + 3 * q^92 - 33 * q^94 - 42 * q^97 - 162 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{18}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20377	−	0.742249i	0.851195	−	0.524849i
							\(3\)		0			0
\(5\)	3.07462	+	1.11907i	1.37501	+	0.500464i								0.920662	−	0.390360i	\(-0.127649\pi\)
							0.454350	+	0.890823i	\(0.349872\pi\)
\(7\)	1.33825	+	0.235969i	0.505810	+	0.0891880i	0.420730	−	0.907186i	\(-0.361774\pi\)
							0.0850807	+	0.996374i	\(0.472885\pi\)
\(11\)	0.982057	+	2.69818i	0.296101	+	0.813531i	0.995142	+	0.0984497i	\(0.0313883\pi\)
							−0.699041	+	0.715082i	\(0.746389\pi\)
\(13\)	−2.36880	−	2.82303i	−0.656987	−	0.782967i	0.329963	−	0.943994i	\(-0.392964\pi\)
							−0.986950	+	0.161027i	\(0.948519\pi\)
\(17\)	−1.94027	+	1.12022i	−0.470585	+	0.271692i	−0.716484	−	0.697603i	\(-0.754250\pi\)
							0.245900	+	0.969295i	\(0.420917\pi\)
\(19\)	−1.22444	+	2.12079i	−0.280906	+	0.486544i	−0.971608	−	0.236596i	\(-0.923968\pi\)
							0.690702	+	0.723139i	\(0.257302\pi\)
\(23\)	1.08869	+	6.17426i	0.227007	+	1.28742i	0.858810	+	0.512294i	\(0.171204\pi\)
							−0.631803	+	0.775129i	\(0.717685\pi\)
\(29\)	−5.00298	−	4.19800i	−0.929030	−	0.779549i	0.0466128	−	0.998913i	\(-0.485157\pi\)
							−0.975643	+	0.219364i	\(0.929602\pi\)
\(31\)	4.65080	−	0.820062i	0.835309	−	0.147287i	0.260396	−	0.965502i	\(-0.416147\pi\)
							0.574913	+	0.818215i	\(0.305036\pi\)
\(37\)	3.70698	−	2.14023i	0.609424	−	0.351851i	−0.163316	−	0.986574i	\(-0.552219\pi\)
							0.772740	+	0.634723i	\(0.218886\pi\)
\(41\)	−7.28211	−	8.67848i	−1.13727	−	1.35535i	−0.925821	−	0.377963i	\(-0.876625\pi\)
							−0.211454	−	0.977388i	\(-0.567820\pi\)
\(43\)	−3.24005	+	1.17928i	−0.494103	+	0.179839i	−0.577040	−	0.816716i	\(-0.695792\pi\)
							0.0829367	+	0.996555i	\(0.473570\pi\)
\(47\)	2.21737	−	12.5753i	0.323437	−	1.83430i	−0.197002	−	0.980403i	\(-0.563121\pi\)
							0.520439	−	0.853899i	\(-0.325768\pi\)
\(53\)	−4.14210			−0.568961			−0.284481	−	0.958682i	\(-0.591821\pi\)
							−0.284481	+	0.958682i	\(0.591821\pi\)
\(59\)	−3.92440	+	10.7822i	−0.510913	+	1.40372i	0.369373	+	0.929281i	\(0.379572\pi\)
							−0.880287	+	0.474442i	\(0.842650\pi\)
\(61\)	9.59806	+	1.69240i	1.22891	+	0.216689i	0.750155	−	0.661262i	\(-0.229979\pi\)
							0.478750	+	0.877951i	\(0.341090\pi\)
\(67\)	−6.64683	+	5.57735i	−0.812039	+	0.681382i	−0.951094	−	0.308903i	\(-0.900038\pi\)
							0.139054	+	0.990285i	\(0.455594\pi\)
\(71\)	3.19451	+	5.53306i	0.379119	+	0.656653i	0.990934	−	0.134348i	\(-0.0428939\pi\)
							−0.611816	+	0.791000i	\(0.709561\pi\)
\(73\)	−2.87137	+	4.97336i	−0.336069	+	0.582088i	−0.983690	−	0.179875i	\(-0.942431\pi\)
							0.647621	+	0.761963i	\(0.275764\pi\)
\(79\)	5.31972	−	6.33979i	0.598515	−	0.713283i	−0.378703	−	0.925518i	\(-0.623630\pi\)
							0.977219	+	0.212236i	\(0.0680744\pi\)
\(83\)	1.56022	−	1.85940i	0.171257	−	0.204096i	−0.673589	−	0.739106i	\(-0.735248\pi\)
							0.844845	+	0.535011i	\(0.179692\pi\)
\(89\)	−1.06132	−	0.612754i	−0.112500	−	0.0649518i	0.442694	−	0.896673i	\(-0.354023\pi\)
							−0.555194	+	0.831721i	\(0.687356\pi\)
\(97\)	16.9633	−	6.17414i	1.72236	−	0.626889i	0.724324	−	0.689460i	\(-0.242152\pi\)
							0.998041	+	0.0625705i	\(0.0199298\pi\)

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	648.2.v.b.35.25		192
3.2	odd	2		216.2.v.b.11.8	✓	192
8.3	odd	2	inner	648.2.v.b.35.22		192
12.11	even	2		864.2.bh.b.335.7		192
24.5	odd	2		864.2.bh.b.335.8		192
24.11	even	2		216.2.v.b.11.11	yes	192
27.5	odd	18	inner	648.2.v.b.611.22		192
27.22	even	9		216.2.v.b.59.11	yes	192
108.103	odd	18		864.2.bh.b.815.8		192
216.59	even	18	inner	648.2.v.b.611.25		192
216.157	even	18		864.2.bh.b.815.7		192
216.211	odd	18		216.2.v.b.59.8	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.b.11.8	✓	192	3.2	odd	2
216.2.v.b.11.11	yes	192	24.11	even	2
216.2.v.b.59.8	yes	192	216.211	odd	18
216.2.v.b.59.11	yes	192	27.22	even	9
648.2.v.b.35.22		192	8.3	odd	2	inner
648.2.v.b.35.25		192	1.1	even	1	trivial
648.2.v.b.611.22		192	27.5	odd	18	inner
648.2.v.b.611.25		192	216.59	even	18	inner
864.2.bh.b.335.7		192	12.11	even	2
864.2.bh.b.335.8		192	24.5	odd	2
864.2.bh.b.815.7		192	216.157	even	18
864.2.bh.b.815.8		192	108.103	odd	18