Embedded newform 112.5.s.c.33.2

• Label: 112.5.s.c.33.2
• 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			33.2
Root			\(0.391571i\) of defining polynomial
Character	\(\chi\)	\(=\)	112.33
Dual form			112.5.s.c.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35901 - 0.784623i) q^{3} +(38.3327 + 22.1314i) q^{5} +(-42.3967 + 24.5667i) q^{7} +(-39.2687 + 68.0154i) q^{9} +(11.7557 + 20.3615i) q^{11} -136.269i q^{13} +69.4592 q^{15} +(-227.333 + 131.251i) q^{17} +(387.162 + 223.528i) q^{19} +(-38.3418 + 66.6517i) q^{21} +(-374.585 + 648.800i) q^{23} +(667.098 + 1155.45i) q^{25} +250.353i q^{27} +406.524 q^{29} +(584.199 - 337.287i) q^{31} +(31.9521 + 18.4476i) q^{33} +(-2168.87 + 3.40909i) q^{35} +(-372.666 + 645.476i) q^{37} +(-106.920 - 185.191i) q^{39} -2476.20i q^{41} +2636.68 q^{43} +(-3010.55 + 1738.14i) q^{45} +(579.099 + 334.343i) q^{47} +(1193.96 - 2083.09i) q^{49} +(-205.964 + 356.741i) q^{51} +(1014.61 + 1757.36i) q^{53} +1040.68i q^{55} +701.541 q^{57} +(1014.22 - 585.561i) q^{59} +(-2022.05 - 1167.43i) q^{61} +(-6.04891 - 3848.33i) q^{63} +(3015.83 - 5223.58i) q^{65} +(-3779.79 - 6546.78i) q^{67} +1175.63i q^{69} -7575.36 q^{71} +(3284.18 - 1896.12i) q^{73} +(1813.18 + 1046.84i) q^{75} +(-998.615 - 574.460i) q^{77} +(3897.33 - 6750.37i) q^{79} +(-2984.33 - 5169.02i) q^{81} +2181.41i q^{83} -11619.0 q^{85} +(552.469 - 318.968i) q^{87} +(8050.38 + 4647.89i) q^{89} +(3347.69 + 5777.37i) q^{91} +(529.287 - 916.751i) q^{93} +(9893.98 + 17136.9i) q^{95} +9981.90i q^{97} -1846.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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
							\(3\)	1.35901	−	0.784623i	0.151001	−	0.0871803i	−0.422596	−	0.906318i	\(-0.638881\pi\)
0.573597	+	0.819138i	\(0.305548\pi\)
\(5\)	38.3327	+	22.1314i	1.53331	+	0.885256i	0.999206	+	0.0398360i	\(0.0126836\pi\)
							0.534102	+	0.845420i	\(0.320650\pi\)
\(7\)	−42.3967	+	24.5667i	−0.865238	+	0.501361i
							\(11\)	11.7557	+	20.3615i	0.0971545	+	0.168276i	0.910506	−	0.413496i	\(-0.135693\pi\)
−0.813351	+	0.581773i	\(0.802359\pi\)
\(13\)		−	136.269i		−	0.806328i	−0.915128	−	0.403164i	\(-0.867910\pi\)
							0.915128	−	0.403164i	\(-0.132090\pi\)
\(17\)	−227.333	+	131.251i	−0.786618	+	0.454154i	−0.838771	−	0.544485i	\(-0.816725\pi\)
							0.0521523	+	0.998639i	\(0.483392\pi\)
\(19\)	387.162	+	223.528i	1.07247	+	0.619191i	0.928855	−	0.370442i	\(-0.120794\pi\)
							0.143615	+	0.989634i	\(0.454127\pi\)
\(23\)	−374.585	+	648.800i	−0.708100	+	1.22646i	0.257462	+	0.966288i	\(0.417114\pi\)
							−0.965561	+	0.260176i	\(0.916219\pi\)
\(29\)	406.524			0.483382			0.241691	−	0.970353i	\(-0.422298\pi\)
							0.241691	+	0.970353i	\(0.422298\pi\)
\(31\)	584.199	−	337.287i	0.607907	−	0.350975i	−0.164239	−	0.986421i	\(-0.552517\pi\)
							0.772146	+	0.635445i	\(0.219183\pi\)
\(37\)	−372.666	+	645.476i	−0.272218	+	0.471495i	−0.969429	−	0.245370i	\(-0.921090\pi\)
							0.697212	+	0.716865i	\(0.254424\pi\)
\(41\)		−	2476.20i		−	1.47305i	−0.676410	−	0.736526i	\(-0.736465\pi\)
							0.676410	−	0.736526i	\(-0.263535\pi\)
\(43\)	2636.68			1.42601			0.713003	−	0.701161i	\(-0.247335\pi\)
							0.713003	+	0.701161i	\(0.247335\pi\)
\(47\)	579.099	+	334.343i	0.262154	+	0.151355i	0.625317	−	0.780371i	\(-0.284970\pi\)
							−0.363163	+	0.931726i	\(0.618303\pi\)
\(53\)	1014.61	+	1757.36i	0.361200	+	0.625617i	0.988159	−	0.153436i	\(-0.0490338\pi\)
							−0.626959	+	0.779053i	\(0.715700\pi\)
\(59\)	1014.22	−	585.561i	0.291359	−	0.168216i	−0.347195	−	0.937793i	\(-0.612866\pi\)
							0.638555	+	0.769576i	\(0.279533\pi\)
\(61\)	−2022.05	−	1167.43i	−0.543416	−	0.313741i	0.203047	−	0.979169i	\(-0.434916\pi\)
							−0.746462	+	0.665428i	\(0.768249\pi\)
\(67\)	−3779.79	−	6546.78i	−0.842011	−	1.45841i	−0.888192	−	0.459472i	\(-0.848039\pi\)
							0.0461815	−	0.998933i	\(-0.485295\pi\)
\(71\)	−7575.36			−1.50275			−0.751375	−	0.659876i	\(-0.770609\pi\)
							−0.751375	+	0.659876i	\(0.770609\pi\)
\(73\)	3284.18	−	1896.12i	0.616284	−	0.355812i	−0.159137	−	0.987257i	\(-0.550871\pi\)
							0.775421	+	0.631445i	\(0.217538\pi\)
\(79\)	3897.33	−	6750.37i	0.624472	−	1.08162i	−0.364171	−	0.931332i	\(-0.618647\pi\)
							0.988643	−	0.150285i	\(-0.0480192\pi\)
\(83\)			2181.41i			0.316651i	0.987387	+	0.158326i	\(0.0506096\pi\)
							−0.987387	+	0.158326i	\(0.949390\pi\)
\(89\)	8050.38	+	4647.89i	1.01633	+	0.586781i	0.913040	−	0.407870i	\(-0.133728\pi\)
							0.103294	+	0.994651i	\(0.467062\pi\)
\(97\)			9981.90i			1.06089i	0.847720	+	0.530444i	\(0.177975\pi\)
							−0.847720	+	0.530444i	\(0.822025\pi\)

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.33.2		6
4.3	odd	2		28.5.h.a.5.2	✓	6
7.2	even	3		784.5.c.e.97.4		6
7.3	odd	6	inner	112.5.s.c.17.2		6
7.5	odd	6		784.5.c.e.97.3		6
12.11	even	2		252.5.z.f.145.1		6
20.3	even	4		700.5.o.a.649.3		12
20.7	even	4		700.5.o.a.649.4		12
20.19	odd	2		700.5.s.a.201.2		6
28.3	even	6		28.5.h.a.17.2	yes	6
28.11	odd	6		196.5.h.c.129.2		6
28.19	even	6		196.5.b.a.97.4		6
28.23	odd	6		196.5.b.a.97.3		6
28.27	even	2		196.5.h.c.117.2		6
84.59	odd	6		252.5.z.f.73.1		6
140.3	odd	12		700.5.o.a.549.4		12
140.59	even	6		700.5.s.a.101.2		6
140.87	odd	12		700.5.o.a.549.3		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.h.a.5.2	✓	6	4.3	odd	2
28.5.h.a.17.2	yes	6	28.3	even	6
112.5.s.c.17.2		6	7.3	odd	6	inner
112.5.s.c.33.2		6	1.1	even	1	trivial
196.5.b.a.97.3		6	28.23	odd	6
196.5.b.a.97.4		6	28.19	even	6
196.5.h.c.117.2		6	28.27	even	2
196.5.h.c.129.2		6	28.11	odd	6
252.5.z.f.73.1		6	84.59	odd	6
252.5.z.f.145.1		6	12.11	even	2
700.5.o.a.549.3		12	140.87	odd	12
700.5.o.a.549.4		12	140.3	odd	12
700.5.o.a.649.3		12	20.3	even	4
700.5.o.a.649.4		12	20.7	even	4
700.5.s.a.101.2		6	140.59	even	6
700.5.s.a.201.2		6	20.19	odd	2
784.5.c.e.97.3		6	7.5	odd	6
784.5.c.e.97.4		6	7.2	even	3