Embedded newform 105.2.m.a.97.4

• Label: 105.2.m.a.97.4
• Level: $105$
• Weight: $2$
• Character: 105.97
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.838429221223\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			97.4
Root			\(-1.40927 + 0.118126i\) of defining polynomial
Character	\(\chi\)	\(=\)	105.97
Dual form			105.2.m.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.167056 - 0.167056i) q^{2} +(0.707107 + 0.707107i) q^{3} -1.94418i q^{4} +(2.23450 + 0.0836010i) q^{5} -0.236253i q^{6} +(-2.64501 - 0.0627175i) q^{7} +(-0.658899 + 0.658899i) q^{8} +1.00000i q^{9} +(-0.359321 - 0.387253i) q^{10} +3.98602 q^{11} +(1.37475 - 1.37475i) q^{12} +(0.500437 + 0.500437i) q^{13} +(0.431387 + 0.452341i) q^{14} +(1.52092 + 1.63915i) q^{15} -3.66822 q^{16} +(-1.67840 + 1.67840i) q^{17} +(0.167056 - 0.167056i) q^{18} -7.21850 q^{19} +(0.162536 - 4.34429i) q^{20} +(-1.82596 - 1.91465i) q^{21} +(-0.665888 - 0.665888i) q^{22} +(-5.16007 + 5.16007i) q^{23} -0.931824 q^{24} +(4.98602 + 0.373614i) q^{25} -0.167202i q^{26} +(-0.707107 + 0.707107i) q^{27} +(-0.121934 + 5.14238i) q^{28} -3.65191i q^{29} +(0.0197510 - 0.527908i) q^{30} -4.93821i q^{31} +(1.93060 + 1.93060i) q^{32} +(2.81854 + 2.81854i) q^{33} +0.560773 q^{34} +(-5.90504 - 0.361268i) q^{35} +1.94418 q^{36} +(0.292275 + 0.292275i) q^{37} +(1.20589 + 1.20589i) q^{38} +0.707725i q^{39} +(-1.52740 + 1.41723i) q^{40} +7.63184i q^{41} +(-0.0148172 + 0.624890i) q^{42} +(3.65191 - 3.65191i) q^{43} -7.74956i q^{44} +(-0.0836010 + 2.23450i) q^{45} +1.72404 q^{46} +(-0.305303 + 0.305303i) q^{47} +(-2.59383 - 2.59383i) q^{48} +(6.99213 + 0.331777i) q^{49} +(-0.770530 - 0.895358i) q^{50} -2.37361 q^{51} +(0.972943 - 0.972943i) q^{52} +(5.39653 - 5.39653i) q^{53} +0.236253 q^{54} +(8.90678 + 0.333235i) q^{55} +(1.78412 - 1.70147i) q^{56} +(-5.10425 - 5.10425i) q^{57} +(-0.610073 + 0.610073i) q^{58} +6.10959 q^{59} +(3.18681 - 2.95695i) q^{60} -7.11047i q^{61} +(-0.824957 + 0.824957i) q^{62} +(0.0627175 - 2.64501i) q^{63} +6.69141i q^{64} +(1.07639 + 1.16007i) q^{65} -0.941708i q^{66} +(0.944185 + 0.944185i) q^{67} +(3.26312 + 3.26312i) q^{68} -7.29744 q^{69} +(0.926119 + 1.04682i) q^{70} +1.19297 q^{71} +(-0.658899 - 0.658899i) q^{72} +(1.38298 + 1.38298i) q^{73} -0.0976524i q^{74} +(3.26147 + 3.78983i) q^{75} +14.0341i q^{76} +(-10.5431 - 0.249993i) q^{77} +(0.118230 - 0.118230i) q^{78} +8.64027i q^{79} +(-8.19666 - 0.306667i) q^{80} -1.00000 q^{81} +(1.27494 - 1.27494i) q^{82} +(-11.9895 - 11.9895i) q^{83} +(-3.72244 + 3.54999i) q^{84} +(-3.89070 + 3.61007i) q^{85} -1.22015 q^{86} +(2.58229 - 2.58229i) q^{87} +(-2.62639 + 2.62639i) q^{88} +7.82581 q^{89} +(0.387253 - 0.359321i) q^{90} +(-1.29227 - 1.35505i) q^{91} +(10.0321 + 10.0321i) q^{92} +(3.49184 - 3.49184i) q^{93} +0.102005 q^{94} +(-16.1298 - 0.603474i) q^{95} +2.73028i q^{96} +(7.43671 - 7.43671i) q^{97} +(-1.11265 - 1.22350i) q^{98} +3.98602i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 8 * q^7 + 24 * q^8 - 16 * q^11 + 8 * q^15 - 48 * q^16 + 8 * q^21 - 16 * q^22 - 40 * q^23 + 24 * q^28 - 8 * q^30 + 48 * q^32 - 8 * q^35 - 16 * q^36 + 32 * q^37 - 16 * q^42 - 16 * q^43 + 64 * q^46 - 72 * q^50 - 16 * q^51 + 24 * q^53 + 24 * q^56 + 8 * q^57 + 32 * q^58 + 40 * q^60 + 8 * q^63 + 40 * q^65 - 32 * q^67 - 40 * q^70 + 64 * q^71 + 24 * q^72 - 24 * q^77 - 8 * q^78 - 16 * q^81 + 48 * q^85 + 64 * q^86 - 64 * q^88 - 48 * q^91 - 40 * q^92 + 24 * q^93 - 72 * q^95 - 96 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.167056	−	0.167056i	−0.118126	−	0.118126i	0.645573	−	0.763699i	\(-0.276619\pi\)
							−0.763699	+	0.645573i	\(0.776619\pi\)
\(3\)	0.707107	+	0.707107i	0.408248	+	0.408248i
							\(5\)	2.23450	+	0.0836010i	0.999301	+	0.0373875i
\(7\)	−2.64501	−	0.0627175i	−0.999719	−	0.0237050i
							\(11\)	3.98602			1.20183			0.600915	−	0.799313i	\(-0.294803\pi\)
0.600915	+	0.799313i	\(0.294803\pi\)
\(13\)	0.500437	+	0.500437i	0.138796	+	0.138796i	0.773091	−	0.634295i	\(-0.218709\pi\)
							−0.634295	+	0.773091i	\(0.718709\pi\)
\(17\)	−1.67840	+	1.67840i	−0.407071	+	0.407071i	−0.880716	−	0.473645i	\(-0.842938\pi\)
							0.473645	+	0.880716i	\(0.342938\pi\)
\(19\)	−7.21850			−1.65604			−0.828019	−	0.560700i	\(-0.810532\pi\)
							−0.828019	+	0.560700i	\(0.810532\pi\)
\(23\)	−5.16007	+	5.16007i	−1.07595	+	1.07595i	−0.0790800	+	0.996868i	\(0.525198\pi\)
							−0.996868	+	0.0790800i	\(0.974802\pi\)
\(29\)		−	3.65191i		−	0.678143i	−0.940761	−	0.339071i	\(-0.889887\pi\)
							0.940761	−	0.339071i	\(-0.110113\pi\)
\(31\)		−	4.93821i		−	0.886929i	−0.896292	−	0.443465i	\(-0.853749\pi\)
							0.896292	−	0.443465i	\(-0.146251\pi\)
\(37\)	0.292275	+	0.292275i	0.0480497	+	0.0480497i	0.730723	−	0.682674i	\(-0.239183\pi\)
							−0.682674	+	0.730723i	\(0.739183\pi\)
\(41\)			7.63184i			1.19189i	0.803024	+	0.595947i	\(0.203223\pi\)
							−0.803024	+	0.595947i	\(0.796777\pi\)
\(43\)	3.65191	−	3.65191i	0.556911	−	0.556911i	−0.371516	−	0.928427i	\(-0.621162\pi\)
							0.928427	+	0.371516i	\(0.121162\pi\)
\(47\)	−0.305303	+	0.305303i	−0.0445331	+	0.0445331i	−0.729023	−	0.684490i	\(-0.760025\pi\)
							0.684490	+	0.729023i	\(0.260025\pi\)
\(53\)	5.39653	−	5.39653i	0.741270	−	0.741270i	−0.231553	−	0.972822i	\(-0.574381\pi\)
							0.972822	+	0.231553i	\(0.0743805\pi\)
\(59\)	6.10959			0.795401			0.397701	−	0.917515i	\(-0.369808\pi\)
							0.397701	+	0.917515i	\(0.369808\pi\)
\(61\)		−	7.11047i		−	0.910402i	−0.890389	−	0.455201i	\(-0.849567\pi\)
							0.890389	−	0.455201i	\(-0.150433\pi\)
\(67\)	0.944185	+	0.944185i	0.115351	+	0.115351i	0.762426	−	0.647075i	\(-0.224008\pi\)
							−0.647075	+	0.762426i	\(0.724008\pi\)
\(71\)	1.19297			0.141579			0.0707897	−	0.997491i	\(-0.477448\pi\)
							0.0707897	+	0.997491i	\(0.477448\pi\)
\(73\)	1.38298	+	1.38298i	0.161865	+	0.161865i	0.783393	−	0.621527i	\(-0.213487\pi\)
							−0.621527	+	0.783393i	\(0.713487\pi\)
\(79\)			8.64027i			0.972106i	0.873929	+	0.486053i	\(0.161564\pi\)
							−0.873929	+	0.486053i	\(0.838436\pi\)
\(83\)	−11.9895	−	11.9895i	−1.31602	−	1.31602i	−0.916898	−	0.399122i	\(-0.869315\pi\)
							−0.399122	−	0.916898i	\(-0.630685\pi\)
\(89\)	7.82581			0.829534			0.414767	−	0.909928i	\(-0.363863\pi\)
							0.414767	+	0.909928i	\(0.363863\pi\)
\(97\)	7.43671	−	7.43671i	0.755083	−	0.755083i	−0.220340	−	0.975423i	\(-0.570717\pi\)
							0.975423	+	0.220340i	\(0.0707167\pi\)

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	105.2.m.a.97.4	yes	16
3.2	odd	2		315.2.p.e.307.5		16
4.3	odd	2		1680.2.cz.d.97.4		16
5.2	odd	4		525.2.m.b.118.6		16
5.3	odd	4	inner	105.2.m.a.13.3	✓	16
5.4	even	2		525.2.m.b.307.5		16
7.2	even	3		735.2.v.a.472.3		32
7.3	odd	6		735.2.v.a.607.5		32
7.4	even	3		735.2.v.a.607.6		32
7.5	odd	6		735.2.v.a.472.4		32
7.6	odd	2	inner	105.2.m.a.97.3	yes	16
15.8	even	4		315.2.p.e.118.6		16
20.3	even	4		1680.2.cz.d.433.5		16
21.20	even	2		315.2.p.e.307.6		16
28.27	even	2		1680.2.cz.d.97.5		16
35.3	even	12		735.2.v.a.313.3		32
35.13	even	4	inner	105.2.m.a.13.4	yes	16
35.18	odd	12		735.2.v.a.313.4		32
35.23	odd	12		735.2.v.a.178.5		32
35.27	even	4		525.2.m.b.118.5		16
35.33	even	12		735.2.v.a.178.6		32
35.34	odd	2		525.2.m.b.307.6		16
105.83	odd	4		315.2.p.e.118.5		16
140.83	odd	4		1680.2.cz.d.433.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.m.a.13.3	✓	16	5.3	odd	4	inner
105.2.m.a.13.4	yes	16	35.13	even	4	inner
105.2.m.a.97.3	yes	16	7.6	odd	2	inner
105.2.m.a.97.4	yes	16	1.1	even	1	trivial
315.2.p.e.118.5		16	105.83	odd	4
315.2.p.e.118.6		16	15.8	even	4
315.2.p.e.307.5		16	3.2	odd	2
315.2.p.e.307.6		16	21.20	even	2
525.2.m.b.118.5		16	35.27	even	4
525.2.m.b.118.6		16	5.2	odd	4
525.2.m.b.307.5		16	5.4	even	2
525.2.m.b.307.6		16	35.34	odd	2
735.2.v.a.178.5		32	35.23	odd	12
735.2.v.a.178.6		32	35.33	even	12
735.2.v.a.313.3		32	35.3	even	12
735.2.v.a.313.4		32	35.18	odd	12
735.2.v.a.472.3		32	7.2	even	3
735.2.v.a.472.4		32	7.5	odd	6
735.2.v.a.607.5		32	7.3	odd	6
735.2.v.a.607.6		32	7.4	even	3
1680.2.cz.d.97.4		16	4.3	odd	2
1680.2.cz.d.97.5		16	28.27	even	2
1680.2.cz.d.433.4		16	140.83	odd	4
1680.2.cz.d.433.5		16	20.3	even	4