Embedded newform 28.5.h.a.5.1

• Label: 28.5.h.a.5.1
• Level: $28$
• Weight: $5$
• Character: 28.5
• Analytic conductor: $2.894$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	28.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89435896635\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{6})\)
Coefficient field:	6.0.11337408.1
Defining polynomial:	\( x^{6} + 18x^{4} + 81x^{2} + 12 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.1
Root			\(-3.17656i\) of defining polynomial
Character	\(\chi\)	\(=\)	28.5
Dual form			28.5.h.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.81152 + 4.50998i) q^{3} +(-26.9260 - 15.5457i) q^{5} +(-48.6720 + 5.65972i) q^{7} +(0.179888 - 0.311574i) q^{9} +(72.8605 + 126.198i) q^{11} -209.930i q^{13} +280.444 q^{15} +(-162.074 + 93.5735i) q^{17} +(153.531 + 88.6415i) q^{19} +(354.677 - 263.721i) q^{21} +(-166.842 + 288.979i) q^{23} +(170.838 + 295.901i) q^{25} -727.372i q^{27} -1387.57 q^{29} +(-1345.74 + 776.966i) q^{31} +(-1138.30 - 657.199i) q^{33} +(1398.53 + 604.248i) q^{35} +(1138.99 - 1972.78i) q^{37} +(946.779 + 1639.87i) q^{39} +781.977i q^{41} +837.131 q^{43} +(-9.68729 + 5.59296i) q^{45} +(-1296.36 - 748.454i) q^{47} +(2336.94 - 550.940i) q^{49} +(844.030 - 1461.90i) q^{51} +(2233.55 + 3868.63i) q^{53} -4530.67i q^{55} -1599.09 q^{57} +(-3815.50 + 2202.88i) q^{59} +(2257.75 + 1303.51i) q^{61} +(-6.99207 + 16.1831i) q^{63} +(-3263.51 + 5652.56i) q^{65} +(-2887.15 - 5000.70i) q^{67} -3009.82i q^{69} +1149.20 q^{71} +(-3194.47 + 1844.33i) q^{73} +(-2669.01 - 1540.96i) q^{75} +(-4260.52 - 5729.95i) q^{77} +(-1349.16 + 2336.81i) q^{79} +(3295.01 + 5707.12i) q^{81} +2689.70i q^{83} +5818.67 q^{85} +(10839.1 - 6257.93i) q^{87} +(5037.61 + 2908.47i) q^{89} +(1188.14 + 10217.7i) q^{91} +(7008.21 - 12138.6i) q^{93} +(-2755.99 - 4773.51i) q^{95} +7890.42i q^{97} +52.4268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q + 9 * q^3 - 27 * q^5 + 66 * q^7 + 90 * q^9 + 135 * q^11 - 486 * q^15 - 1107 * q^17 - 747 * q^19 + 2169 * q^21 + 243 * q^23 + 1878 * q^25 - 540 * q^29 - 5355 * q^31 - 1863 * q^33 + 6021 * q^35 + 2355 * q^37 + 6588 * q^39 - 948 * q^43 - 14418 * q^45 - 9747 * q^47 + 8430 * q^49 - 891 * q^51 + 6291 * q^53 - 6894 * q^57 - 2943 * q^59 + 4041 * q^61 + 15138 * q^63 + 7668 * q^65 + 1659 * q^67 + 2268 * q^71 - 8703 * q^73 - 3438 * q^75 - 10665 * q^77 - 7773 * q^79 - 7479 * q^81 - 702 * q^85 + 40014 * q^87 + 14985 * q^89 - 20952 * q^91 + 1449 * q^93 - 20655 * q^95 + 8100 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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
							\(3\)	−7.81152	+	4.50998i	−0.867947	+	0.501109i	−0.866665	−	0.498890i	\(-0.833741\pi\)
−0.00128125	+	0.999999i	\(0.500408\pi\)
\(5\)	−26.9260	−	15.5457i	−1.07704	−	0.621829i	−0.146943	−	0.989145i	\(-0.546943\pi\)
							−0.930096	+	0.367317i	\(0.880277\pi\)
\(7\)	−48.6720	+	5.65972i	−0.993307	+	0.115505i
							\(11\)	72.8605	+	126.198i	0.602153	+	1.04296i	0.992494	+	0.122290i	\(0.0390236\pi\)
−0.390341	+	0.920670i	\(0.627643\pi\)
\(13\)		−	209.930i		−	1.24219i	−0.783736	−	0.621094i	\(-0.786689\pi\)
							0.783736	−	0.621094i	\(-0.213311\pi\)
\(17\)	−162.074	+	93.5735i	−0.560810	+	0.323784i	−0.753470	−	0.657482i	\(-0.771622\pi\)
							0.192661	+	0.981265i	\(0.438288\pi\)
\(19\)	153.531	+	88.6415i	0.425295	+	0.245544i	0.697340	−	0.716740i	\(-0.254367\pi\)
							−0.272045	+	0.962284i	\(0.587700\pi\)
\(23\)	−166.842	+	288.979i	−0.315392	+	0.546275i	−0.979521	−	0.201343i	\(-0.935469\pi\)
							0.664129	+	0.747618i	\(0.268803\pi\)
\(29\)	−1387.57			−1.64991			−0.824954	−	0.565199i	\(-0.808799\pi\)
							−0.824954	+	0.565199i	\(0.808799\pi\)
\(31\)	−1345.74	+	776.966i	−1.40036	+	0.808497i	−0.994429	−	0.105407i	\(-0.966385\pi\)
							−0.405930	+	0.913904i	\(0.633052\pi\)
\(37\)	1138.99	−	1972.78i	0.831985	−	1.44104i	−0.0644762	−	0.997919i	\(-0.520538\pi\)
							0.896462	−	0.443122i	\(-0.146129\pi\)
\(41\)			781.977i			0.465185i	0.972574	+	0.232593i	\(0.0747209\pi\)
							−0.972574	+	0.232593i	\(0.925279\pi\)
\(43\)	837.131			0.452748			0.226374	−	0.974040i	\(-0.427313\pi\)
							0.226374	+	0.974040i	\(0.427313\pi\)
\(47\)	−1296.36	−	748.454i	−0.586854	−	0.338820i	0.176999	−	0.984211i	\(-0.443361\pi\)
							−0.763853	+	0.645391i	\(0.776695\pi\)
\(53\)	2233.55	+	3868.63i	0.795141	+	1.37723i	0.922749	+	0.385400i	\(0.125937\pi\)
							−0.127608	+	0.991825i	\(0.540730\pi\)
\(59\)	−3815.50	+	2202.88i	−1.09609	+	0.632829i	−0.935192	−	0.354142i	\(-0.884773\pi\)
							−0.160900	+	0.986971i	\(0.551440\pi\)
\(61\)	2257.75	+	1303.51i	0.606758	+	0.350312i	0.771695	−	0.635992i	\(-0.219409\pi\)
							−0.164938	+	0.986304i	\(0.552742\pi\)
\(67\)	−2887.15	−	5000.70i	−0.643162	−	1.11399i	−0.984723	−	0.174129i	\(-0.944289\pi\)
							0.341561	−	0.939860i	\(-0.389044\pi\)
\(71\)	1149.20			0.227971			0.113986	−	0.993482i	\(-0.463638\pi\)
							0.113986	+	0.993482i	\(0.463638\pi\)
\(73\)	−3194.47	+	1844.33i	−0.599450	+	0.346093i	−0.768825	−	0.639459i	\(-0.779158\pi\)
							0.169375	+	0.985552i	\(0.445825\pi\)
\(79\)	−1349.16	+	2336.81i	−0.216177	+	0.374429i	−0.953636	−	0.300963i	\(-0.902692\pi\)
							0.737459	+	0.675392i	\(0.236025\pi\)
\(83\)			2689.70i			0.390433i	0.980760	+	0.195217i	\(0.0625410\pi\)
							−0.980760	+	0.195217i	\(0.937459\pi\)
\(89\)	5037.61	+	2908.47i	0.635982	+	0.367184i	0.783065	−	0.621940i	\(-0.213655\pi\)
							−0.147083	+	0.989124i	\(0.546988\pi\)
\(97\)			7890.42i			0.838604i	0.907847	+	0.419302i	\(0.137725\pi\)
							−0.907847	+	0.419302i	\(0.862275\pi\)

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	28.5.h.a.5.1	✓	6
3.2	odd	2		252.5.z.f.145.3		6
4.3	odd	2		112.5.s.c.33.3		6
5.2	odd	4		700.5.o.a.649.5		12
5.3	odd	4		700.5.o.a.649.2		12
5.4	even	2		700.5.s.a.201.3		6
7.2	even	3		196.5.b.a.97.2		6
7.3	odd	6	inner	28.5.h.a.17.1	yes	6
7.4	even	3		196.5.h.c.129.3		6
7.5	odd	6		196.5.b.a.97.5		6
7.6	odd	2		196.5.h.c.117.3		6
21.17	even	6		252.5.z.f.73.3		6
28.3	even	6		112.5.s.c.17.3		6
28.19	even	6		784.5.c.e.97.2		6
28.23	odd	6		784.5.c.e.97.5		6
35.3	even	12		700.5.o.a.549.5		12
35.17	even	12		700.5.o.a.549.2		12
35.24	odd	6		700.5.s.a.101.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.h.a.5.1	✓	6	1.1	even	1	trivial
28.5.h.a.17.1	yes	6	7.3	odd	6	inner
112.5.s.c.17.3		6	28.3	even	6
112.5.s.c.33.3		6	4.3	odd	2
196.5.b.a.97.2		6	7.2	even	3
196.5.b.a.97.5		6	7.5	odd	6
196.5.h.c.117.3		6	7.6	odd	2
196.5.h.c.129.3		6	7.4	even	3
252.5.z.f.73.3		6	21.17	even	6
252.5.z.f.145.3		6	3.2	odd	2
700.5.o.a.549.2		12	35.17	even	12
700.5.o.a.549.5		12	35.3	even	12
700.5.o.a.649.2		12	5.3	odd	4
700.5.o.a.649.5		12	5.2	odd	4
700.5.s.a.101.3		6	35.24	odd	6
700.5.s.a.201.3		6	5.4	even	2
784.5.c.e.97.2		6	28.19	even	6
784.5.c.e.97.5		6	28.23	odd	6