Embedded newform 58.2.e.a.51.2

• Label: 58.2.e.a.51.2
• Level: $58$
• Weight: $2$
• Character: 58.51
• Analytic conductor: $0.463$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	58.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.463132331723\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{14})\)
Coefficient field:	\(\Q(\zeta_{28})\)
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			51.2
Root			\(-0.781831 - 0.623490i\) of defining polynomial
Character	\(\chi\)	\(=\)	58.51
Dual form			58.2.e.a.33.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.974928 - 0.222521i) q^{2} +(-0.626980 + 1.30194i) q^{3} +(0.900969 - 0.433884i) q^{4} +(-0.136585 - 0.598418i) q^{5} +(-0.321552 + 1.40881i) q^{6} +(-2.59161 - 1.24805i) q^{7} +(0.781831 - 0.623490i) q^{8} +(0.568532 + 0.712916i) q^{9} +(-0.266321 - 0.553021i) q^{10} +(-3.39687 - 2.70891i) q^{11} +1.44504i q^{12} +(-0.298199 + 0.373929i) q^{13} +(-2.80435 - 0.640075i) q^{14} +(0.864739 + 0.197371i) q^{15} +(0.623490 - 0.781831i) q^{16} +0.259558i q^{17} +(0.712916 + 0.568532i) q^{18} +(3.65470 + 7.58906i) q^{19} +(-0.382702 - 0.479894i) q^{20} +(3.24978 - 2.59161i) q^{21} +(-3.91449 - 1.88512i) q^{22} +(0.0317259 - 0.139000i) q^{23} +(0.321552 + 1.40881i) q^{24} +(4.16540 - 2.00595i) q^{25} +(-0.207515 + 0.430910i) q^{26} +(-5.51107 + 1.25786i) q^{27} -2.87647 q^{28} +(1.07561 - 5.27665i) q^{29} +0.886977 q^{30} +(6.46089 - 1.47465i) q^{31} +(0.433884 - 0.900969i) q^{32} +(5.65660 - 2.72407i) q^{33} +(0.0577572 + 0.253051i) q^{34} +(-0.392883 + 1.72133i) q^{35} +(0.821552 + 0.395639i) q^{36} +(-7.50895 + 5.98819i) q^{37} +(5.25179 + 6.58554i) q^{38} +(-0.299868 - 0.622683i) q^{39} +(-0.479894 - 0.382702i) q^{40} +4.28236i q^{41} +(2.59161 - 3.24978i) q^{42} +(3.17741 + 0.725223i) q^{43} +(-4.23582 - 0.966799i) q^{44} +(0.348969 - 0.437593i) q^{45} -0.142575i q^{46} +(-3.97456 - 3.16960i) q^{47} +(0.626980 + 1.30194i) q^{48} +(0.794384 + 0.996126i) q^{49} +(3.61460 - 2.88254i) q^{50} +(-0.337929 - 0.162738i) q^{51} +(-0.106426 + 0.466282i) q^{52} +(-2.06111 - 9.03032i) q^{53} +(-5.09299 + 2.45265i) q^{54} +(-1.15710 + 2.40274i) q^{55} +(-2.80435 + 0.640075i) q^{56} -12.1719 q^{57} +(-0.125524 - 5.38370i) q^{58} -10.2463 q^{59} +(0.864739 - 0.197371i) q^{60} +(-4.31279 + 8.95559i) q^{61} +(5.97076 - 2.87536i) q^{62} +(-0.583655 - 2.55716i) q^{63} +(0.222521 - 0.974928i) q^{64} +(0.264495 + 0.127374i) q^{65} +(4.90861 - 3.91449i) q^{66} +(-1.16176 - 1.45680i) q^{67} +(0.112618 + 0.233854i) q^{68} +(0.161078 + 0.128456i) q^{69} +1.76560i q^{70} +(5.97581 - 7.49342i) q^{71} +(0.888992 + 0.202907i) q^{72} +(2.90704 + 0.663513i) q^{73} +(-5.98819 + 7.50895i) q^{74} +6.68078i q^{75} +(6.58554 + 5.25179i) q^{76} +(5.42249 + 11.2599i) q^{77} +(-0.430910 - 0.540344i) q^{78} +(10.5977 - 8.45137i) q^{79} +(-0.553021 - 0.266321i) q^{80} +(1.20895 - 5.29674i) q^{81} +(0.952915 + 4.17499i) q^{82} +(-0.950401 + 0.457689i) q^{83} +(1.80349 - 3.74499i) q^{84} +(0.155324 - 0.0354518i) q^{85} +3.25912 q^{86} +(6.19549 + 4.70873i) q^{87} -4.34475 q^{88} +(2.51138 - 0.573205i) q^{89} +(0.242846 - 0.504275i) q^{90} +(1.23950 - 0.596912i) q^{91} +(-0.0317259 - 0.139000i) q^{92} +(-2.13094 + 9.33625i) q^{93} +(-4.58021 - 2.20571i) q^{94} +(4.04225 - 3.22359i) q^{95} +(0.900969 + 1.12978i) q^{96} +(-7.22194 - 14.9965i) q^{97} +(0.996126 + 0.794384i) q^{98} -3.96178i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q + 2 * q^4 - 2 * q^5 - 12 * q^6 + 4 * q^7 - 4 * q^9 - 26 * q^13 - 14 * q^15 - 2 * q^16 + 2 * q^20 + 14 * q^21 + 4 * q^22 - 16 * q^23 + 12 * q^24 + 22 * q^25 + 14 * q^26 - 4 * q^28 + 18 * q^29 + 16 * q^30 + 28 * q^31 - 10 * q^33 - 4 * q^34 + 4 * q^35 + 18 * q^36 - 28 * q^37 + 22 * q^38 + 28 * q^39 - 14 * q^40 - 4 * q^42 - 28 * q^43 - 28 * q^44 - 4 * q^45 - 14 * q^47 - 8 * q^49 - 28 * q^50 - 4 * q^51 - 2 * q^52 + 30 * q^53 - 32 * q^54 + 28 * q^55 - 20 * q^57 - 10 * q^58 - 48 * q^59 - 14 * q^60 - 28 * q^61 + 10 * q^62 + 22 * q^63 + 2 * q^64 + 2 * q^65 - 16 * q^67 - 14 * q^68 - 28 * q^69 + 30 * q^71 + 42 * q^73 + 2 * q^74 + 28 * q^76 + 14 * q^77 - 16 * q^78 + 28 * q^79 - 2 * q^80 + 26 * q^81 - 20 * q^82 + 28 * q^85 + 52 * q^86 + 4 * q^87 - 4 * q^88 + 14 * q^89 + 70 * q^90 - 4 * q^91 + 16 * q^92 + 66 * q^93 - 8 * q^94 + 2 * q^96 - 28 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.974928	−	0.222521i	0.689378	−	0.157346i
							\(3\)	−0.626980	+	1.30194i	−0.361987	+	0.751674i	−0.999829	−	0.0184933i	\(-0.994113\pi\)
0.637842	+	0.770167i	\(0.279827\pi\)
\(5\)	−0.136585	−	0.598418i	−0.0610826	−	0.267621i	0.935160	−	0.354225i	\(-0.115255\pi\)
							−0.996243	+	0.0866048i	\(0.972398\pi\)
\(7\)	−2.59161	−	1.24805i	−0.979537	−	0.471720i	−0.125591	−	0.992082i	\(-0.540083\pi\)
							−0.853946	+	0.520362i	\(0.825797\pi\)
\(11\)	−3.39687	−	2.70891i	−1.02419	−	0.816767i	−0.0409677	−	0.999160i	\(-0.513044\pi\)
							−0.983226	+	0.182394i	\(0.941616\pi\)
\(13\)	−0.298199	+	0.373929i	−0.0827054	+	0.103709i	−0.821463	−	0.570262i	\(-0.806842\pi\)
							0.738757	+	0.673972i	\(0.235413\pi\)
\(17\)			0.259558i			0.0629522i	0.999505	+	0.0314761i	\(0.0100208\pi\)
							−0.999505	+	0.0314761i	\(0.989979\pi\)
\(19\)	3.65470	+	7.58906i	0.838446	+	1.74105i	0.651453	+	0.758689i	\(0.274160\pi\)
							0.186992	+	0.982361i	\(0.440126\pi\)
\(23\)	0.0317259	−	0.139000i	0.00661531	−	0.0289836i	−0.971513	−	0.236988i	\(-0.923840\pi\)
							0.978128	+	0.208005i	\(0.0666969\pi\)
\(29\)	1.07561	−	5.27665i	0.199736	−	0.979850i
							\(31\)	6.46089	−	1.47465i	1.16041	−	0.264856i	0.401380	−	0.915912i	\(-0.368531\pi\)
0.759030	+	0.651056i	\(0.225674\pi\)
\(37\)	−7.50895	+	5.98819i	−1.23446	+	0.984452i	−0.234541	+	0.972106i	\(0.575359\pi\)
							−0.999924	+	0.0123461i	\(0.996070\pi\)
\(41\)			4.28236i			0.668792i	0.942433	+	0.334396i	\(0.108532\pi\)
							−0.942433	+	0.334396i	\(0.891468\pi\)
\(43\)	3.17741	+	0.725223i	0.484550	+	0.110595i	0.457816	−	0.889047i	\(-0.348632\pi\)
							0.0267346	+	0.999643i	\(0.491489\pi\)
\(47\)	−3.97456	−	3.16960i	−0.579749	−	0.462334i	0.289178	−	0.957275i	\(-0.406618\pi\)
							−0.868926	+	0.494941i	\(0.835190\pi\)
\(53\)	−2.06111	−	9.03032i	−0.283116	−	1.24041i	−0.893774	−	0.448517i	\(-0.851952\pi\)
							0.610659	−	0.791894i	\(-0.290905\pi\)
\(59\)	−10.2463			−1.33395			−0.666977	−	0.745078i	\(-0.732412\pi\)
							−0.666977	+	0.745078i	\(0.732412\pi\)
\(61\)	−4.31279	+	8.95559i	−0.552196	+	1.14665i	0.418917	+	0.908025i	\(0.362410\pi\)
							−0.971113	+	0.238622i	\(0.923304\pi\)
\(67\)	−1.16176	−	1.45680i	−0.141931	−	0.177976i	0.705785	−	0.708426i	\(-0.250594\pi\)
							−0.847716	+	0.530450i	\(0.822023\pi\)
\(71\)	5.97581	−	7.49342i	0.709198	−	0.889306i	−0.288475	−	0.957487i	\(-0.593148\pi\)
							0.997673	+	0.0681816i	\(0.0217197\pi\)
\(73\)	2.90704	+	0.663513i	0.340243	+	0.0776583i	0.389229	−	0.921141i	\(-0.372742\pi\)
							−0.0489853	+	0.998799i	\(0.515599\pi\)
\(79\)	10.5977	−	8.45137i	1.19233	−	0.950853i	0.192794	−	0.981239i	\(-0.438245\pi\)
							0.999538	+	0.0303860i	\(0.00967364\pi\)
\(83\)	−0.950401	+	0.457689i	−0.104320	+	0.0502379i	−0.485316	−	0.874339i	\(-0.661295\pi\)
							0.380996	+	0.924577i	\(0.375581\pi\)
\(89\)	2.51138	−	0.573205i	0.266205	−	0.0607596i	−0.0873347	−	0.996179i	\(-0.527835\pi\)
							0.353540	+	0.935419i	\(0.384978\pi\)
\(97\)	−7.22194	−	14.9965i	−0.733277	−	1.52267i	−0.848424	−	0.529317i	\(-0.822448\pi\)
							0.115147	−	0.993348i	\(-0.463266\pi\)

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	58.2.e.a.51.2	yes	12
3.2	odd	2		522.2.n.a.109.1		12
4.3	odd	2		464.2.y.c.225.2		12
29.2	odd	28		1682.2.a.s.1.4		6
29.4	even	14	inner	58.2.e.a.33.2	✓	12
29.5	even	14		1682.2.b.j.1681.3		12
29.24	even	7		1682.2.b.j.1681.9		12
29.27	odd	28		1682.2.a.r.1.4		6
87.62	odd	14		522.2.n.a.91.1		12
116.91	odd	14		464.2.y.c.33.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.e.a.33.2	✓	12	29.4	even	14	inner
58.2.e.a.51.2	yes	12	1.1	even	1	trivial
464.2.y.c.33.2		12	116.91	odd	14
464.2.y.c.225.2		12	4.3	odd	2
522.2.n.a.91.1		12	87.62	odd	14
522.2.n.a.109.1		12	3.2	odd	2
1682.2.a.r.1.4		6	29.27	odd	28
1682.2.a.s.1.4		6	29.2	odd	28
1682.2.b.j.1681.3		12	29.5	even	14
1682.2.b.j.1681.9		12	29.24	even	7