Embedded newform 112.5.s.c.33.3

• Label: 112.5.s.c.33.3
• Level: $112$
• Weight: $5$
• Character: 112.33
• Analytic conductor: $11.577$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5774358654\)
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:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			33.3
Root			\(-3.17656i\) of defining polynomial
Character	\(\chi\)	\(=\)	112.33
Dual form			112.5.s.c.17.3

$q$-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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)	\(85\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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			0
							\(3\)	7.81152	−	4.50998i	0.867947	−	0.501109i	0.00128125	−	0.999999i	\(-0.499592\pi\)
0.866665	+	0.498890i	\(0.166259\pi\)
\(5\)	−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	\(-0.960976\pi\)
0.390341	−	0.920670i	\(-0.372357\pi\)
\(13\)		−	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	0.272045	−	0.962284i	\(-0.412300\pi\)
							−0.697340	+	0.716740i	\(0.745633\pi\)
\(23\)	166.842	−	288.979i	0.315392	−	0.546275i	−0.664129	−	0.747618i	\(-0.731197\pi\)
							0.979521	+	0.201343i	\(0.0645307\pi\)
\(29\)	−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	0.405930	−	0.913904i	\(-0.366948\pi\)
							0.994429	+	0.105407i	\(0.0336145\pi\)
\(37\)	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	\(-0.572687\pi\)
							−0.226374	+	0.974040i	\(0.572687\pi\)
\(47\)	1296.36	+	748.454i	0.586854	+	0.338820i	0.763853	−	0.645391i	\(-0.223305\pi\)
							−0.176999	+	0.984211i	\(0.556639\pi\)
\(53\)	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	0.160900	−	0.986971i	\(-0.448560\pi\)
							0.935192	+	0.354142i	\(0.115227\pi\)
\(61\)	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	\(0.0557111\pi\)
							−0.341561	+	0.939860i	\(0.610956\pi\)
\(71\)	−1149.20			−0.227971			−0.113986	−	0.993482i	\(-0.536362\pi\)
							−0.113986	+	0.993482i	\(0.536362\pi\)
\(73\)	−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	−0.737459	−	0.675392i	\(-0.763975\pi\)
							0.953636	+	0.300963i	\(0.0973079\pi\)
\(83\)		−	2689.70i		−	0.390433i	−0.980760	−	0.195217i	\(-0.937459\pi\)
							0.980760	−	0.195217i	\(-0.0625410\pi\)
\(89\)	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	112.5.s.c.33.3		6
4.3	odd	2		28.5.h.a.5.1	✓	6
7.2	even	3		784.5.c.e.97.5		6
7.3	odd	6	inner	112.5.s.c.17.3		6
7.5	odd	6		784.5.c.e.97.2		6
12.11	even	2		252.5.z.f.145.3		6
20.3	even	4		700.5.o.a.649.2		12
20.7	even	4		700.5.o.a.649.5		12
20.19	odd	2		700.5.s.a.201.3		6
28.3	even	6		28.5.h.a.17.1	yes	6
28.11	odd	6		196.5.h.c.129.3		6
28.19	even	6		196.5.b.a.97.5		6
28.23	odd	6		196.5.b.a.97.2		6
28.27	even	2		196.5.h.c.117.3		6
84.59	odd	6		252.5.z.f.73.3		6
140.3	odd	12		700.5.o.a.549.5		12
140.59	even	6		700.5.s.a.101.3		6
140.87	odd	12		700.5.o.a.549.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.h.a.5.1	✓	6	4.3	odd	2
28.5.h.a.17.1	yes	6	28.3	even	6
112.5.s.c.17.3		6	7.3	odd	6	inner
112.5.s.c.33.3		6	1.1	even	1	trivial
196.5.b.a.97.2		6	28.23	odd	6
196.5.b.a.97.5		6	28.19	even	6
196.5.h.c.117.3		6	28.27	even	2
196.5.h.c.129.3		6	28.11	odd	6
252.5.z.f.73.3		6	84.59	odd	6
252.5.z.f.145.3		6	12.11	even	2
700.5.o.a.549.2		12	140.87	odd	12
700.5.o.a.549.5		12	140.3	odd	12
700.5.o.a.649.2		12	20.3	even	4
700.5.o.a.649.5		12	20.7	even	4
700.5.s.a.101.3		6	140.59	even	6
700.5.s.a.201.3		6	20.19	odd	2
784.5.c.e.97.2		6	7.5	odd	6
784.5.c.e.97.5		6	7.2	even	3