Embedded newform 10.11.c.c.3.1

• Label: 10.11.c.c.3.1
• Level: $10$
• Weight: $11$
• Character: 10.3
• Analytic conductor: $6.354$
• Analytic rank: $0$
• Dimension: $6$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.35357252674\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{8}\cdot 5^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.1
Root			\(2.29143 - 2.29143i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.3
Dual form			10.11.c.c.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0000 + 16.0000i) q^{2} +(-266.441 - 266.441i) q^{3} -512.000i q^{4} +(2583.39 + 1758.32i) q^{5} +8526.10 q^{6} +(-13021.6 + 13021.6i) q^{7} +(8192.00 + 8192.00i) q^{8} +82932.3i q^{9} +(-69467.5 + 13201.2i) q^{10} +239729. q^{11} +(-136418. + 136418. i) q^{12} +(287980. + 287980. i) q^{13} -416690. i q^{14} +(-219833. - 1.15681e6i) q^{15} -262144. q^{16} +(-884527. + 884527. i) q^{17} +(-1.32692e6 - 1.32692e6i) q^{18} +197744. i q^{19} +(900261. - 1.32270e6i) q^{20} +6.93894e6 q^{21} +(-3.83567e6 + 3.83567e6i) q^{22} +(-3.55115e6 - 3.55115e6i) q^{23} -4.36536e6i q^{24} +(3.58223e6 + 9.08488e6i) q^{25} -9.21535e6 q^{26} +(6.36348e6 - 6.36348e6i) q^{27} +(6.66703e6 + 6.66703e6i) q^{28} +3.57499e7i q^{29} +(2.20263e7 + 1.49916e7i) q^{30} +9.65516e6 q^{31} +(4.19430e6 - 4.19430e6i) q^{32} +(-6.38737e7 - 6.38737e7i) q^{33} -2.83049e7i q^{34} +(-5.65359e7 + 1.07437e7i) q^{35} +4.24613e7 q^{36} +(1.34842e7 - 1.34842e7i) q^{37} +(-3.16390e6 - 3.16390e6i) q^{38} -1.53459e8i q^{39} +(6.75899e6 + 3.55673e7i) q^{40} +1.08475e8 q^{41} +(-1.11023e8 + 1.11023e8i) q^{42} +(8.62678e7 + 8.62678e7i) q^{43} -1.22741e8i q^{44} +(-1.45822e8 + 2.14247e8i) q^{45} +1.13637e8 q^{46} +(-1.76759e8 + 1.76759e8i) q^{47} +(6.98458e7 + 6.98458e7i) q^{48} -5.66463e7i q^{49} +(-2.02674e8 - 8.80424e7i) q^{50} +4.71348e8 q^{51} +(1.47446e8 - 1.47446e8i) q^{52} +(-1.48530e8 - 1.48530e8i) q^{53} +2.03631e8i q^{54} +(6.19316e8 + 4.21522e8i) q^{55} -2.13345e8 q^{56} +(5.26870e7 - 5.26870e7i) q^{57} +(-5.71999e8 - 5.71999e8i) q^{58} -7.98638e8i q^{59} +(-5.92287e8 + 1.12554e8i) q^{60} -4.24449e8 q^{61} +(-1.54483e8 + 1.54483e8i) q^{62} +(-1.07991e9 - 1.07991e9i) q^{63} +1.34218e8i q^{64} +(2.37604e8 + 1.25033e9i) q^{65} +2.04396e9 q^{66} +(1.12768e9 - 1.12768e9i) q^{67} +(4.52878e8 + 4.52878e8i) q^{68} +1.89234e9i q^{69} +(7.32675e8 - 1.07647e9i) q^{70} -2.60778e9 q^{71} +(-6.79381e8 + 6.79381e8i) q^{72} +(8.51903e8 + 8.51903e8i) q^{73} +4.31493e8i q^{74} +(1.46613e9 - 3.37503e9i) q^{75} +1.01245e8 q^{76} +(-3.12165e9 + 3.12165e9i) q^{77} +(2.45534e9 + 2.45534e9i) q^{78} +4.56153e8i q^{79} +(-6.77221e8 - 4.60934e8i) q^{80} +1.50609e9 q^{81} +(-1.73559e9 + 1.73559e9i) q^{82} +(7.60916e8 + 7.60916e8i) q^{83} -3.55274e9i q^{84} +(-3.84037e9 + 7.29799e8i) q^{85} -2.76057e9 q^{86} +(9.52524e9 - 9.52524e9i) q^{87} +(1.96386e9 + 1.96386e9i) q^{88} +2.03858e9i q^{89} +(-1.09480e9 - 5.76110e9i) q^{90} -7.49988e9 q^{91} +(-1.81819e9 + 1.81819e9i) q^{92} +(-2.57253e9 - 2.57253e9i) q^{93} -5.65629e9i q^{94} +(-3.47698e8 + 5.10851e8i) q^{95} -2.23507e9 q^{96} +(8.73239e9 - 8.73239e9i) q^{97} +(9.06341e8 + 9.06341e8i) q^{98} +1.98813e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	6 * q - 96 * q^2 + 128 * q^3 + 5460 * q^5 - 4096 * q^6 + 13512 * q^7 + 49152 * q^8 - 173280 * q^10 + 647832 * q^11 + 65536 * q^12 - 742902 * q^13 + 1577720 * q^15 - 1572864 * q^16 - 755118 * q^17 - 5683744 * q^18 + 2749440 * q^20 + 12277112 * q^21 - 10365312 * q^22 - 15052992 * q^23 + 42644850 * q^25 + 23772864 * q^26 - 47998120 * q^27 - 6918144 * q^28 - 20357760 * q^30 + 153847152 * q^31 + 25165824 * q^32 - 173025784 * q^33 - 208942440 * q^35 + 181879808 * q^36 - 32574498 * q^37 - 17493120 * q^38 + 737280 * q^40 + 226153272 * q^41 - 196433792 * q^42 - 63628752 * q^43 - 174000230 * q^45 + 481695744 * q^46 - 19700448 * q^47 - 33554432 * q^48 - 788800800 * q^50 + 2388638032 * q^51 - 380365824 * q^52 - 950001042 * q^53 - 135145080 * q^55 + 221380608 * q^56 - 338012560 * q^57 - 310652160 * q^58 - 156344320 * q^60 + 1603984392 * q^61 - 2461554432 * q^62 - 5135206432 * q^63 + 1398176070 * q^65 + 5536825088 * q^66 + 2502647712 * q^67 + 386620416 * q^68 + 174645120 * q^70 - 7137533808 * q^71 - 2910076928 * q^72 + 2304462438 * q^73 + 9497972200 * q^75 + 559779840 * q^76 + 597969864 * q^77 + 11288293632 * q^78 - 1431306240 * q^80 - 26068931054 * q^81 - 3618452352 * q^82 - 88180632 * q^83 + 1729841610 * q^85 + 2036120064 * q^86 + 35176248320 * q^87 + 5307039744 * q^88 - 25065537760 * q^90 - 43718253408 * q^91 - 7707131904 * q^92 + 22042053176 * q^93 - 34456200 * q^95 + 1073741824 * q^96 + 33281088582 * q^97 + 12874047264 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0000	+	16.0000i	−0.500000	+	0.500000i
							\(3\)	−266.441	−	266.441i	−1.09646	−	1.09646i	−0.994821	−	0.101643i	\(-0.967590\pi\)
−0.101643	−	0.994821i	\(-0.532410\pi\)
\(5\)	2583.39	+	1758.32i	0.826686	+	0.562663i
							\(7\)	−13021.6	+	13021.6i	−0.774769	+	0.774769i	−0.978936	−	0.204167i	\(-0.934552\pi\)
0.204167	+	0.978936i	\(0.434552\pi\)
\(11\)	239729.			1.48853			0.744266	−	0.667884i	\(-0.232800\pi\)
							0.744266	+	0.667884i	\(0.232800\pi\)
\(13\)	287980.	+	287980.i	0.775613	+	0.775613i	0.979082	−	0.203468i	\(-0.0652214\pi\)
							−0.203468	+	0.979082i	\(0.565221\pi\)
\(17\)	−884527.	+	884527.i	−0.622969	+	0.622969i	−0.946290	−	0.323320i	\(-0.895201\pi\)
							0.323320	+	0.946290i	\(0.395201\pi\)
\(19\)			197744.i			0.0798611i	0.999202	+	0.0399305i	\(0.0127137\pi\)
							−0.999202	+	0.0399305i	\(0.987286\pi\)
\(23\)	−3.55115e6	−	3.55115e6i	−0.551734	−	0.551734i	0.375207	−	0.926941i	\(-0.377572\pi\)
							−0.926941	+	0.375207i	\(0.877572\pi\)
\(29\)			3.57499e7i			1.74295i	0.490439	+	0.871476i	\(0.336837\pi\)
							−0.490439	+	0.871476i	\(0.663163\pi\)
\(31\)	9.65516e6			0.337249			0.168625	−	0.985680i	\(-0.446067\pi\)
							0.168625	+	0.985680i	\(0.446067\pi\)
\(37\)	1.34842e7	−	1.34842e7i	0.194453	−	0.194453i	−0.603164	−	0.797617i	\(-0.706094\pi\)
							0.797617	+	0.603164i	\(0.206094\pi\)
\(41\)	1.08475e8			0.936286			0.468143	−	0.883653i	\(-0.344923\pi\)
							0.468143	+	0.883653i	\(0.344923\pi\)
\(43\)	8.62678e7	+	8.62678e7i	0.586822	+	0.586822i	0.936769	−	0.349947i	\(-0.113801\pi\)
							−0.349947	+	0.936769i	\(0.613801\pi\)
\(47\)	−1.76759e8	+	1.76759e8i	−0.770713	+	0.770713i	−0.978231	−	0.207518i	\(-0.933461\pi\)
							0.207518	+	0.978231i	\(0.433461\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.11.c.c.3.1	✓	6
3.2	odd	2		90.11.g.c.73.2		6
4.3	odd	2		80.11.p.c.33.3		6
5.2	odd	4	inner	10.11.c.c.7.1	yes	6
5.3	odd	4		50.11.c.e.7.3		6
5.4	even	2		50.11.c.e.43.3		6
15.2	even	4		90.11.g.c.37.2		6
20.7	even	4		80.11.p.c.17.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.c.3.1	✓	6	1.1	even	1	trivial
10.11.c.c.7.1	yes	6	5.2	odd	4	inner
50.11.c.e.7.3		6	5.3	odd	4
50.11.c.e.43.3		6	5.4	even	2
80.11.p.c.17.3		6	20.7	even	4
80.11.p.c.33.3		6	4.3	odd	2
90.11.g.c.37.2		6	15.2	even	4
90.11.g.c.73.2		6	3.2	odd	2