Embedded newform 464.2.y.c.33.2

• Label: 464.2.y.c.33.2
• Level: $464$
• Weight: $2$
• Character: 464.33
• Analytic conductor: $3.705$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	464.y (of order \(14\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70505865379\)
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:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			33.2
Root			\(0.433884 + 0.900969i\) of defining polynomial
Character	\(\chi\)	\(=\)	464.33
Dual form			464.2.y.c.225.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.626980 + 1.30194i) q^{3} +(-0.136585 + 0.598418i) q^{5} +(2.59161 - 1.24805i) q^{7} +(0.568532 - 0.712916i) q^{9} +(3.39687 - 2.70891i) q^{11} +(-0.298199 - 0.373929i) q^{13} +(-0.864739 + 0.197371i) q^{15} -0.259558i q^{17} +(-3.65470 + 7.58906i) q^{19} +(3.24978 + 2.59161i) q^{21} +(-0.0317259 - 0.139000i) q^{23} +(4.16540 + 2.00595i) q^{25} +(5.51107 + 1.25786i) q^{27} +(1.07561 + 5.27665i) q^{29} +(-6.46089 - 1.47465i) q^{31} +(5.65660 + 2.72407i) q^{33} +(0.392883 + 1.72133i) q^{35} +(-7.50895 - 5.98819i) q^{37} +(0.299868 - 0.622683i) q^{39} -4.28236i q^{41} +(-3.17741 + 0.725223i) q^{43} +(0.348969 + 0.437593i) q^{45} +(3.97456 - 3.16960i) q^{47} +(0.794384 - 0.996126i) q^{49} +(0.337929 - 0.162738i) q^{51} +(-2.06111 + 9.03032i) q^{53} +(1.15710 + 2.40274i) q^{55} -12.1719 q^{57} +10.2463 q^{59} +(-4.31279 - 8.95559i) q^{61} +(0.583655 - 2.55716i) q^{63} +(0.264495 - 0.127374i) q^{65} +(1.16176 - 1.45680i) q^{67} +(0.161078 - 0.128456i) q^{69} +(-5.97581 - 7.49342i) q^{71} +(2.90704 - 0.663513i) q^{73} +6.68078i q^{75} +(5.42249 - 11.2599i) q^{77} +(-10.5977 - 8.45137i) q^{79} +(1.20895 + 5.29674i) q^{81} +(0.950401 + 0.457689i) q^{83} +(0.155324 + 0.0354518i) q^{85} +(-6.19549 + 4.70873i) q^{87} +(2.51138 + 0.573205i) q^{89} +(-1.23950 - 0.596912i) q^{91} +(-2.13094 - 9.33625i) q^{93} +(-4.04225 - 3.22359i) q^{95} +(-7.22194 + 14.9965i) q^{97} -3.96178i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q - 2 * q^5 - 4 * q^7 - 4 * q^9 - 26 * q^13 + 14 * q^15 + 14 * q^21 + 16 * q^23 + 22 * q^25 + 18 * q^29 - 28 * q^31 - 10 * q^33 - 4 * q^35 - 28 * q^37 - 28 * q^39 + 28 * q^43 - 4 * q^45 + 14 * q^47 - 8 * q^49 + 4 * q^51 + 30 * q^53 - 28 * q^55 - 20 * q^57 + 48 * q^59 - 28 * q^61 - 22 * q^63 + 2 * q^65 + 16 * q^67 - 28 * q^69 - 30 * q^71 + 42 * q^73 + 14 * q^77 - 28 * q^79 + 26 * q^81 + 28 * q^85 - 4 * q^87 + 14 * q^89 + 4 * q^91 + 66 * q^93 - 28 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\)	\(117\)	\(175\)	\(321\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{1}{14}\right)\)

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			0
							\(3\)	0.626980	+	1.30194i	0.361987	+	0.751674i	0.999829	−	0.0184933i	\(-0.00588693\pi\)
−0.637842	+	0.770167i	\(0.720173\pi\)
\(5\)	−0.136585	+	0.598418i	−0.0610826	+	0.267621i	−0.996243	−	0.0866048i	\(-0.972398\pi\)
							0.935160	+	0.354225i	\(0.115255\pi\)
\(7\)	2.59161	−	1.24805i	0.979537	−	0.471720i	0.125591	−	0.992082i	\(-0.459917\pi\)
							0.853946	+	0.520362i	\(0.174203\pi\)
\(11\)	3.39687	−	2.70891i	1.02419	−	0.816767i	0.0409677	−	0.999160i	\(-0.486956\pi\)
							0.983226	+	0.182394i	\(0.0583845\pi\)
\(13\)	−0.298199	−	0.373929i	−0.0827054	−	0.103709i	0.738757	−	0.673972i	\(-0.235413\pi\)
							−0.821463	+	0.570262i	\(0.806842\pi\)
\(17\)		−	0.259558i		−	0.0629522i	−0.999505	−	0.0314761i	\(-0.989979\pi\)
							0.999505	−	0.0314761i	\(-0.0100208\pi\)
\(19\)	−3.65470	+	7.58906i	−0.838446	+	1.74105i	−0.186992	+	0.982361i	\(0.559874\pi\)
							−0.651453	+	0.758689i	\(0.725840\pi\)
\(23\)	−0.0317259	−	0.139000i	−0.00661531	−	0.0289836i	0.971513	−	0.236988i	\(-0.0761603\pi\)
							−0.978128	+	0.208005i	\(0.933303\pi\)
\(29\)	1.07561	+	5.27665i	0.199736	+	0.979850i
							\(31\)	−6.46089	−	1.47465i	−1.16041	−	0.264856i	−0.401380	−	0.915912i	\(-0.631469\pi\)
−0.759030	+	0.651056i	\(0.774326\pi\)
\(37\)	−7.50895	−	5.98819i	−1.23446	−	0.984452i	−0.999924	−	0.0123461i	\(-0.996070\pi\)
							−0.234541	−	0.972106i	\(-0.575359\pi\)
\(41\)		−	4.28236i		−	0.668792i	−0.942433	−	0.334396i	\(-0.891468\pi\)
							0.942433	−	0.334396i	\(-0.108532\pi\)
\(43\)	−3.17741	+	0.725223i	−0.484550	+	0.110595i	−0.457816	−	0.889047i	\(-0.651368\pi\)
							−0.0267346	+	0.999643i	\(0.508511\pi\)
\(47\)	3.97456	−	3.16960i	0.579749	−	0.462334i	−0.289178	−	0.957275i	\(-0.593382\pi\)
							0.868926	+	0.494941i	\(0.164810\pi\)
\(53\)	−2.06111	+	9.03032i	−0.283116	+	1.24041i	0.610659	+	0.791894i	\(0.290905\pi\)
							−0.893774	+	0.448517i	\(0.851952\pi\)
\(59\)	10.2463			1.33395			0.666977	−	0.745078i	\(-0.267588\pi\)
							0.666977	+	0.745078i	\(0.267588\pi\)
\(61\)	−4.31279	−	8.95559i	−0.552196	−	1.14665i	−0.971113	−	0.238622i	\(-0.923304\pi\)
							0.418917	−	0.908025i	\(-0.362410\pi\)
\(67\)	1.16176	−	1.45680i	0.141931	−	0.177976i	−0.705785	−	0.708426i	\(-0.749406\pi\)
							0.847716	+	0.530450i	\(0.177977\pi\)
\(71\)	−5.97581	−	7.49342i	−0.709198	−	0.889306i	0.288475	−	0.957487i	\(-0.406852\pi\)
							−0.997673	+	0.0681816i	\(0.978280\pi\)
\(73\)	2.90704	−	0.663513i	0.340243	−	0.0776583i	−0.0489853	−	0.998799i	\(-0.515599\pi\)
							0.389229	+	0.921141i	\(0.372742\pi\)
\(79\)	−10.5977	−	8.45137i	−1.19233	−	0.950853i	−0.192794	−	0.981239i	\(-0.561755\pi\)
							−0.999538	+	0.0303860i	\(0.990326\pi\)
\(83\)	0.950401	+	0.457689i	0.104320	+	0.0502379i	0.485316	−	0.874339i	\(-0.338705\pi\)
							−0.380996	+	0.924577i	\(0.624419\pi\)
\(89\)	2.51138	+	0.573205i	0.266205	+	0.0607596i	0.353540	−	0.935419i	\(-0.384978\pi\)
							−0.0873347	+	0.996179i	\(0.527835\pi\)
\(97\)	−7.22194	+	14.9965i	−0.733277	+	1.52267i	0.115147	+	0.993348i	\(0.463266\pi\)
							−0.848424	+	0.529317i	\(0.822448\pi\)

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	464.2.y.c.33.2		12
4.3	odd	2		58.2.e.a.33.2	✓	12
12.11	even	2		522.2.n.a.91.1		12
29.22	even	14	inner	464.2.y.c.225.2		12
116.15	even	28		1682.2.a.s.1.4		6
116.23	odd	14		1682.2.b.j.1681.3		12
116.35	odd	14		1682.2.b.j.1681.9		12
116.43	even	28		1682.2.a.r.1.4		6
116.51	odd	14		58.2.e.a.51.2	yes	12
348.167	even	14		522.2.n.a.109.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.e.a.33.2	✓	12	4.3	odd	2
58.2.e.a.51.2	yes	12	116.51	odd	14
464.2.y.c.33.2		12	1.1	even	1	trivial
464.2.y.c.225.2		12	29.22	even	14	inner
522.2.n.a.91.1		12	12.11	even	2
522.2.n.a.109.1		12	348.167	even	14
1682.2.a.r.1.4		6	116.43	even	28
1682.2.a.s.1.4		6	116.15	even	28
1682.2.b.j.1681.3		12	116.23	odd	14
1682.2.b.j.1681.9		12	116.35	odd	14