Embedded newform 58.2.e.a.5.2

• Label: 58.2.e.a.5.2
• Level: $58$
• Weight: $2$
• Character: 58.5
• 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			5.2
Root			\(0.433884 - 0.900969i\) of defining polynomial
Character	\(\chi\)	\(=\)	58.5
Dual form			58.2.e.a.35.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 - 0.623490i) q^{2} +(-0.240787 + 0.0549581i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-0.892482 - 1.11914i) q^{5} +(-0.153989 + 0.193096i) q^{6} +(0.904459 + 3.96269i) q^{7} +(-0.433884 - 0.900969i) q^{8} +(-2.64795 + 1.27518i) q^{9} +(-1.39554 - 0.318523i) q^{10} +(0.815852 - 1.69413i) q^{11} +0.246980i q^{12} +(-3.39903 - 1.63689i) q^{13} +(3.17783 + 2.53424i) q^{14} +(0.276404 + 0.220425i) q^{15} +(-0.900969 - 0.433884i) q^{16} +1.78568i q^{17} +(-1.27518 + 2.64795i) q^{18} +(3.79673 + 0.866579i) q^{19} +(-1.28967 + 0.621074i) q^{20} +(-0.435565 - 0.904459i) q^{21} +(-0.418416 - 1.83320i) q^{22} +(2.97489 - 3.73039i) q^{23} +(0.153989 + 0.193096i) q^{24} +(0.656661 - 2.87702i) q^{25} +(-3.67805 + 0.839492i) q^{26} +(1.14680 - 0.914542i) q^{27} +4.06460 q^{28} +(1.25499 - 5.23689i) q^{29} +0.353534 q^{30} +(-5.35487 + 4.27036i) q^{31} +(-0.974928 + 0.222521i) q^{32} +(-0.103340 + 0.452764i) q^{33} +(1.11335 + 1.39610i) q^{34} +(3.62759 - 4.54885i) q^{35} +(0.653989 + 2.86531i) q^{36} +(2.21049 + 4.59014i) q^{37} +(3.50870 - 1.68970i) q^{38} +(0.908404 + 0.207337i) q^{39} +(-0.621074 + 1.28967i) q^{40} -10.2017i q^{41} +(-0.904459 - 0.435565i) q^{42} +(2.35311 + 1.87654i) q^{43} +(-1.47011 - 1.17238i) q^{44} +(3.79035 + 1.82534i) q^{45} -4.77135i q^{46} +(-5.65094 + 11.7343i) q^{47} +(0.240787 + 0.0549581i) q^{48} +(-8.57812 + 4.13101i) q^{49} +(-1.28039 - 2.65877i) q^{50} +(-0.0981376 - 0.429969i) q^{51} +(-2.35220 + 2.94957i) q^{52} +(5.32989 + 6.68347i) q^{53} +(0.326396 - 1.43004i) q^{54} +(-2.62410 + 0.598934i) q^{55} +(3.17783 - 2.53424i) q^{56} -0.961830 q^{57} +(-2.28396 - 4.87684i) q^{58} -5.64006 q^{59} +(0.276404 - 0.220425i) q^{60} +(-11.9210 + 2.72090i) q^{61} +(-1.52408 + 6.67741i) q^{62} +(-7.44813 - 9.33966i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(1.20167 + 5.26488i) q^{65} +(0.201499 + 0.418416i) q^{66} +(10.6285 - 5.11840i) q^{67} +(1.74091 + 0.397351i) q^{68} +(-0.511300 + 1.06173i) q^{69} -5.81820i q^{70} +(-3.36755 - 1.62173i) q^{71} +(2.29780 + 1.83244i) q^{72} +(3.62836 + 2.89352i) q^{73} +(4.59014 + 2.21049i) q^{74} +0.728839i q^{75} +(1.68970 - 3.50870i) q^{76} +(7.45124 + 1.70070i) q^{77} +(0.839492 - 0.404278i) q^{78} +(4.80754 + 9.98295i) q^{79} +(0.318523 + 1.39554i) q^{80} +(5.27144 - 6.61017i) q^{81} +(-6.36063 - 7.97598i) q^{82} +(0.807254 - 3.53681i) q^{83} +(-0.978705 + 0.223383i) q^{84} +(1.99842 - 1.59369i) q^{85} +3.00974 q^{86} +(-0.0143755 + 1.32995i) q^{87} -1.88035 q^{88} +(1.33024 - 1.06083i) q^{89} +(4.10150 - 0.936140i) q^{90} +(3.41220 - 14.9498i) q^{91} +(-2.97489 - 3.73039i) q^{92} +(1.05469 - 1.32254i) q^{93} +(2.89813 + 12.6975i) q^{94} +(-2.41869 - 5.02247i) q^{95} +(0.222521 - 0.107160i) q^{96} +(12.5268 + 2.85915i) q^{97} +(-4.13101 + 8.57812i) q^{98} +5.52634i 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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{11}{14}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.781831	−	0.623490i	0.552838	−	0.440874i
							\(3\)	−0.240787	+	0.0549581i	−0.139019	+	0.0317301i	−0.291464	−	0.956582i	\(-0.594142\pi\)
0.152445	+	0.988312i	\(0.451285\pi\)
\(5\)	−0.892482	−	1.11914i	−0.399130	−	0.500493i	0.541135	−	0.840936i	\(-0.317995\pi\)
							−0.940265	+	0.340442i	\(0.889423\pi\)
\(7\)	0.904459	+	3.96269i	0.341853	+	1.49776i	0.795158	+	0.606402i	\(0.207388\pi\)
							−0.453305	+	0.891356i	\(0.649755\pi\)
\(11\)	0.815852	−	1.69413i	0.245989	−	0.510800i	−0.741017	−	0.671486i	\(-0.765656\pi\)
							0.987006	+	0.160686i	\(0.0513706\pi\)
\(13\)	−3.39903	−	1.63689i	−0.942722	−	0.453991i	−0.101593	−	0.994826i	\(-0.532394\pi\)
							−0.841129	+	0.540835i	\(0.818108\pi\)
\(17\)			1.78568i			0.433091i	0.976273	+	0.216545i	\(0.0694789\pi\)
							−0.976273	+	0.216545i	\(0.930521\pi\)
\(19\)	3.79673	+	0.866579i	0.871029	+	0.198807i	0.634599	−	0.772842i	\(-0.281165\pi\)
							0.236430	+	0.971648i	\(0.424022\pi\)
\(23\)	2.97489	−	3.73039i	0.620307	−	0.777841i	−0.368080	−	0.929794i	\(-0.619985\pi\)
							0.988388	+	0.151953i	\(0.0485563\pi\)
\(29\)	1.25499	−	5.23689i	0.233045	−	0.972466i
							\(31\)	−5.35487	+	4.27036i	−0.961762	+	0.766980i	−0.972486	−	0.232963i	\(-0.925158\pi\)
0.0107233	+	0.999943i	\(0.496587\pi\)
\(37\)	2.21049	+	4.59014i	0.363403	+	0.754614i	0.999861	−	0.0166978i	\(-0.00531532\pi\)
							−0.636458	+	0.771312i	\(0.719601\pi\)
\(41\)		−	10.2017i		−	1.59323i	−0.604485	−	0.796616i	\(-0.706621\pi\)
							0.604485	−	0.796616i	\(-0.293379\pi\)
\(43\)	2.35311	+	1.87654i	0.358846	+	0.286170i	0.786272	−	0.617880i	\(-0.212008\pi\)
							−0.427427	+	0.904050i	\(0.640580\pi\)
\(47\)	−5.65094	+	11.7343i	−0.824274	+	1.71162i	−0.130479	+	0.991451i	\(0.541651\pi\)
							−0.693796	+	0.720172i	\(0.744063\pi\)
\(53\)	5.32989	+	6.68347i	0.732116	+	0.918045i	0.998955	−	0.0456963i	\(-0.0145507\pi\)
							−0.266839	+	0.963741i	\(0.585979\pi\)
\(59\)	−5.64006			−0.734273			−0.367137	−	0.930167i	\(-0.619662\pi\)
							−0.367137	+	0.930167i	\(0.619662\pi\)
\(61\)	−11.9210	+	2.72090i	−1.52633	+	0.348376i	−0.901637	−	0.432493i	\(-0.857634\pi\)
							−0.624696	+	0.780868i	\(0.714777\pi\)
\(67\)	10.6285	−	5.11840i	1.29847	−	0.625312i	0.348402	−	0.937345i	\(-0.386724\pi\)
							0.950071	+	0.312034i	\(0.101010\pi\)
\(71\)	−3.36755	−	1.62173i	−0.399654	−	0.192463i	0.223250	−	0.974761i	\(-0.428333\pi\)
							−0.622905	+	0.782298i	\(0.714048\pi\)
\(73\)	3.62836	+	2.89352i	0.424667	+	0.338661i	0.812389	−	0.583116i	\(-0.198167\pi\)
							−0.387722	+	0.921776i	\(0.626738\pi\)
\(79\)	4.80754	+	9.98295i	0.540890	+	1.12317i	0.974978	+	0.222299i	\(0.0713563\pi\)
							−0.434088	+	0.900870i	\(0.642929\pi\)
\(83\)	0.807254	−	3.53681i	0.0886077	−	0.388216i	−0.911105	−	0.412174i	\(-0.864770\pi\)
							0.999713	+	0.0239581i	\(0.00762682\pi\)
\(89\)	1.33024	−	1.06083i	0.141005	−	0.112448i	−0.550447	−	0.834870i	\(-0.685543\pi\)
							0.691452	+	0.722422i	\(0.256971\pi\)
\(97\)	12.5268	+	2.85915i	1.27190	+	0.290303i	0.804611	−	0.593803i	\(-0.202374\pi\)
							0.467288	+	0.884105i	\(0.345231\pi\)

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.5.2	✓	12
3.2	odd	2		522.2.n.a.469.1		12
4.3	odd	2		464.2.y.c.353.2		12
29.6	even	14	inner	58.2.e.a.35.2	yes	12
29.8	odd	28		1682.2.a.s.1.6		6
29.9	even	14		1682.2.b.j.1681.11		12
29.20	even	7		1682.2.b.j.1681.1		12
29.21	odd	28		1682.2.a.r.1.2		6
87.35	odd	14		522.2.n.a.325.1		12
116.35	odd	14		464.2.y.c.209.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.e.a.5.2	✓	12	1.1	even	1	trivial
58.2.e.a.35.2	yes	12	29.6	even	14	inner
464.2.y.c.209.2		12	116.35	odd	14
464.2.y.c.353.2		12	4.3	odd	2
522.2.n.a.325.1		12	87.35	odd	14
522.2.n.a.469.1		12	3.2	odd	2
1682.2.a.r.1.2		6	29.21	odd	28
1682.2.a.s.1.6		6	29.8	odd	28
1682.2.b.j.1681.1		12	29.20	even	7
1682.2.b.j.1681.11		12	29.9	even	14