Embedded newform 10.11.c.a.3.1

• Label: 10.11.c.a.3.1
• Level: $10$
• Weight: $11$
• Character: 10.3
• Analytic conductor: $6.354$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.1
Root			\(-1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	10.3
Dual form			10.11.c.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 - 16.0000i) q^{2} +(-183.000 - 183.000i) q^{3} -512.000i q^{4} +(-1875.00 + 2500.00i) q^{5} -5856.00 q^{6} +(-8407.00 + 8407.00i) q^{7} +(-8192.00 - 8192.00i) q^{8} +7929.00i q^{9} +(10000.0 + 70000.0i) q^{10} -173398. q^{11} +(-93696.0 + 93696.0i) q^{12} +(-232623. - 232623. i) q^{13} +269024. i q^{14} +(800625. - 114375. i) q^{15} -262144. q^{16} +(1.88003e6 - 1.88003e6i) q^{17} +(126864. + 126864. i) q^{18} +1.10156e6i q^{19} +(1.28000e6 + 960000. i) q^{20} +3.07696e6 q^{21} +(-2.77437e6 + 2.77437e6i) q^{22} +(-5.22826e6 - 5.22826e6i) q^{23} +2.99827e6i q^{24} +(-2.73438e6 - 9.37500e6i) q^{25} -7.44394e6 q^{26} +(-9.35496e6 + 9.35496e6i) q^{27} +(4.30438e6 + 4.30438e6i) q^{28} +2.47908e7i q^{29} +(1.09800e7 - 1.46400e7i) q^{30} -1.00660e7 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} +(3.17318e7 + 3.17318e7i) q^{33} -6.01611e7i q^{34} +(-5.25438e6 - 3.67806e7i) q^{35} +4.05965e6 q^{36} +(5.63879e7 - 5.63879e7i) q^{37} +(1.76250e7 + 1.76250e7i) q^{38} +8.51400e7i q^{39} +(3.58400e7 - 5.12000e6i) q^{40} -1.53004e8 q^{41} +(4.92314e7 - 4.92314e7i) q^{42} +(5.93725e7 + 5.93725e7i) q^{43} +8.87798e7i q^{44} +(-1.98225e7 - 1.48669e7i) q^{45} -1.67304e8 q^{46} +(1.72339e8 - 1.72339e8i) q^{47} +(4.79724e7 + 4.79724e7i) q^{48} +1.41120e8i q^{49} +(-1.93750e8 - 1.06250e8i) q^{50} -6.88092e8 q^{51} +(-1.19103e8 + 1.19103e8i) q^{52} +(1.96386e8 + 1.96386e8i) q^{53} +2.99359e8i q^{54} +(3.25121e8 - 4.33495e8i) q^{55} +1.37740e8 q^{56} +(2.01585e8 - 2.01585e8i) q^{57} +(3.96653e8 + 3.96653e8i) q^{58} -6.94069e8i q^{59} +(-5.85600e7 - 4.09920e8i) q^{60} +9.06186e8 q^{61} +(-1.61056e8 + 1.61056e8i) q^{62} +(-6.66591e7 - 6.66591e7i) q^{63} +1.34218e8i q^{64} +(1.01773e9 - 1.45389e8i) q^{65} +1.01542e9 q^{66} +(-9.62074e8 + 9.62074e8i) q^{67} +(-9.62577e8 - 9.62577e8i) q^{68} +1.91354e9i q^{69} +(-6.72560e8 - 5.04420e8i) q^{70} -3.12088e9 q^{71} +(6.49544e7 - 6.49544e7i) q^{72} +(-6.36339e8 - 6.36339e8i) q^{73} -1.80441e9i q^{74} +(-1.21523e9 + 2.21602e9i) q^{75} +5.63999e8 q^{76} +(1.45776e9 - 1.45776e9i) q^{77} +(1.36224e9 + 1.36224e9i) q^{78} +1.96800e9i q^{79} +(4.91520e8 - 6.55360e8i) q^{80} +3.89211e9 q^{81} +(-2.44806e9 + 2.44806e9i) q^{82} +(-5.18382e9 - 5.18382e9i) q^{83} -1.57540e9i q^{84} +(1.17502e9 + 8.22514e9i) q^{85} +1.89992e9 q^{86} +(4.53672e9 - 4.53672e9i) q^{87} +(1.42048e9 + 1.42048e9i) q^{88} -7.77138e9i q^{89} +(-5.55030e8 + 7.92900e7i) q^{90} +3.91132e9 q^{91} +(-2.67687e9 + 2.67687e9i) q^{92} +(1.84208e9 + 1.84208e9i) q^{93} -5.51486e9i q^{94} +(-2.75390e9 - 2.06542e9i) q^{95} +1.53512e9 q^{96} +(-6.40361e8 + 6.40361e8i) q^{97} +(2.25792e9 + 2.25792e9i) q^{98} -1.37487e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 32 * q^2 - 366 * q^3 - 3750 * q^5 - 11712 * q^6 - 16814 * q^7 - 16384 * q^8 + 20000 * q^10 - 346796 * q^11 - 187392 * q^12 - 465246 * q^13 + 1601250 * q^15 - 524288 * q^16 + 3760066 * q^17 + 253728 * q^18 + 2560000 * q^20 + 6153924 * q^21 - 5548736 * q^22 - 10456526 * q^23 - 5468750 * q^25 - 14887872 * q^26 - 18709920 * q^27 + 8608768 * q^28 + 21960000 * q^30 - 20131996 * q^31 - 8388608 * q^32 + 63463668 * q^33 - 10508750 * q^35 + 8119296 * q^36 + 112775826 * q^37 + 35249920 * q^38 + 71680000 * q^40 - 306007196 * q^41 + 98462784 * q^42 + 118744914 * q^43 - 39645000 * q^45 - 334608832 * q^46 + 344678706 * q^47 + 95944704 * q^48 - 387500000 * q^50 - 1376184156 * q^51 - 238205952 * q^52 + 392771234 * q^53 + 650242500 * q^55 + 275480576 * q^56 + 403170960 * q^57 + 793306880 * q^58 - 117120000 * q^60 + 1812371604 * q^61 - 322111936 * q^62 - 133318206 * q^63 + 2035451250 * q^65 + 2030837376 * q^66 - 1924147934 * q^67 - 1925153792 * q^68 - 1345120000 * q^70 - 6241755196 * q^71 + 129908736 * q^72 - 1272678526 * q^73 - 2430468750 * q^75 + 1127997440 * q^76 + 2915513972 * q^77 + 2724480576 * q^78 + 983040000 * q^80 + 7784229762 * q^81 - 4896115136 * q^82 - 10367644206 * q^83 + 2350041250 * q^85 + 3799837248 * q^86 + 9073447440 * q^87 + 2840952832 * q^88 - 1110060000 * q^90 + 7822646244 * q^91 - 5353741312 * q^92 + 3684155268 * q^93 - 5507800000 * q^95 + 3070230528 * q^96 - 1280722494 * q^97 + 4515838432 * 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
							\(3\)	−183.000	−	183.000i	−0.753086	−	0.753086i	0.221968	−	0.975054i	\(-0.428752\pi\)
−0.975054	+	0.221968i	\(0.928752\pi\)
\(5\)	−1875.00	+	2500.00i	−0.600000	+	0.800000i
							\(7\)	−8407.00	+	8407.00i	−0.500208	+	0.500208i	−0.911503	−	0.411294i	\(-0.865077\pi\)
0.411294	+	0.911503i	\(0.365077\pi\)
\(11\)	−173398.			−1.07667			−0.538333	−	0.842732i	\(-0.680946\pi\)
							−0.538333	+	0.842732i	\(0.680946\pi\)
\(13\)	−232623.	−	232623.i	−0.626521	−	0.626521i	0.320670	−	0.947191i	\(-0.396092\pi\)
							−0.947191	+	0.320670i	\(0.896092\pi\)
\(17\)	1.88003e6	−	1.88003e6i	1.32410	−	1.32410i	0.413676	−	0.910424i	\(-0.364245\pi\)
							0.910424	−	0.413676i	\(-0.135755\pi\)
\(19\)			1.10156e6i			0.444877i	0.974947	+	0.222439i	\(0.0714017\pi\)
							−0.974947	+	0.222439i	\(0.928598\pi\)
\(23\)	−5.22826e6	−	5.22826e6i	−0.812303	−	0.812303i	0.172675	−	0.984979i	\(-0.444759\pi\)
							−0.984979	+	0.172675i	\(0.944759\pi\)
\(29\)			2.47908e7i			1.20865i	0.796737	+	0.604326i	\(0.206558\pi\)
							−0.796737	+	0.604326i	\(0.793442\pi\)
\(31\)	−1.00660e7			−0.351600			−0.175800	−	0.984426i	\(-0.556251\pi\)
							−0.175800	+	0.984426i	\(0.556251\pi\)
\(37\)	5.63879e7	−	5.63879e7i	0.813163	−	0.813163i	−0.171944	−	0.985107i	\(-0.555005\pi\)
							0.985107	+	0.171944i	\(0.0550048\pi\)
\(41\)	−1.53004e8			−1.32063			−0.660317	−	0.750987i	\(-0.729578\pi\)
							−0.660317	+	0.750987i	\(0.729578\pi\)
\(43\)	5.93725e7	+	5.93725e7i	0.403871	+	0.403871i	0.879595	−	0.475724i	\(-0.157814\pi\)
							−0.475724	+	0.879595i	\(0.657814\pi\)
\(47\)	1.72339e8	−	1.72339e8i	0.751441	−	0.751441i	−0.223307	−	0.974748i	\(-0.571685\pi\)
							0.974748	+	0.223307i	\(0.0716852\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10.11.c.a.3.1	✓	2
3.2	odd	2		90.11.g.b.73.1		2
4.3	odd	2		80.11.p.b.33.1		2
5.2	odd	4	inner	10.11.c.a.7.1	yes	2
5.3	odd	4		50.11.c.c.7.1		2
5.4	even	2		50.11.c.c.43.1		2
15.2	even	4		90.11.g.b.37.1		2
20.7	even	4		80.11.p.b.17.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.11.c.a.3.1	✓	2	1.1	even	1	trivial
10.11.c.a.7.1	yes	2	5.2	odd	4	inner
50.11.c.c.7.1		2	5.3	odd	4
50.11.c.c.43.1		2	5.4	even	2
80.11.p.b.17.1		2	20.7	even	4
80.11.p.b.33.1		2	4.3	odd	2
90.11.g.b.37.1		2	15.2	even	4
90.11.g.b.73.1		2	3.2	odd	2