Embedded newform 10.11.c.b.3.1

• Label: 10.11.c.b.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.b.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 - 16.0000i) q^{2} +(57.0000 + 57.0000i) q^{3} -512.000i q^{4} +(2925.00 - 1100.00i) q^{5} +1824.00 q^{6} +(6953.00 - 6953.00i) q^{7} +(-8192.00 - 8192.00i) q^{8} -52551.0i q^{9} +(29200.0 - 64400.0i) q^{10} +75242.0 q^{11} +(29184.0 - 29184.0i) q^{12} +(109857. + 109857. i) q^{13} -222496. i q^{14} +(229425. + 104025. i) q^{15} -262144. q^{16} +(-1.52893e6 + 1.52893e6i) q^{17} +(-840816. - 840816. i) q^{18} +4.03868e6i q^{19} +(-563200. - 1.49760e6i) q^{20} +792642. q^{21} +(1.20387e6 - 1.20387e6i) q^{22} +(-712423. - 712423. i) q^{23} -933888. i q^{24} +(7.34562e6 - 6.43500e6i) q^{25} +3.51542e6 q^{26} +(6.36120e6 - 6.36120e6i) q^{27} +(-3.55994e6 - 3.55994e6i) q^{28} -446120. i q^{29} +(5.33520e6 - 2.00640e6i) q^{30} -2.90807e7 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} +(4.28879e6 + 4.28879e6i) q^{33} +4.89257e7i q^{34} +(1.26892e7 - 2.79858e7i) q^{35} -2.69061e7 q^{36} +(-911847. + 911847. i) q^{37} +(6.46189e7 + 6.46189e7i) q^{38} +1.25237e7i q^{39} +(-3.29728e7 - 1.49504e7i) q^{40} -1.63946e8 q^{41} +(1.26823e7 - 1.26823e7i) q^{42} +(1.18423e8 + 1.18423e8i) q^{43} -3.85239e7i q^{44} +(-5.78061e7 - 1.53712e8i) q^{45} -2.27975e7 q^{46} +(2.76320e8 - 2.76320e8i) q^{47} +(-1.49422e7 - 1.49422e7i) q^{48} +1.85787e8i q^{49} +(1.45700e7 - 2.20490e8i) q^{50} -1.74298e8 q^{51} +(5.62468e7 - 5.62468e7i) q^{52} +(3.08460e8 + 3.08460e8i) q^{53} -2.03558e8i q^{54} +(2.20083e8 - 8.27662e7i) q^{55} -1.13918e8 q^{56} +(-2.30205e8 + 2.30205e8i) q^{57} +(-7.13792e6 - 7.13792e6i) q^{58} +9.40888e8i q^{59} +(5.32608e7 - 1.17466e8i) q^{60} -1.35361e9 q^{61} +(-4.65291e8 + 4.65291e8i) q^{62} +(-3.65387e8 - 3.65387e8i) q^{63} +1.34218e8i q^{64} +(4.42174e8 + 2.00489e8i) q^{65} +1.37241e8 q^{66} +(8.53571e8 - 8.53571e8i) q^{67} +(7.82811e8 + 7.82811e8i) q^{68} -8.12162e7i q^{69} +(-2.44746e8 - 6.50801e8i) q^{70} +2.82701e9 q^{71} +(-4.30498e8 + 4.30498e8i) q^{72} +(-2.75330e9 - 2.75330e9i) q^{73} +2.91791e7i q^{74} +(7.85496e8 + 5.19056e7i) q^{75} +2.06780e9 q^{76} +(5.23158e8 - 5.23158e8i) q^{77} +(2.00379e8 + 2.00379e8i) q^{78} -3.32450e9i q^{79} +(-7.66771e8 + 2.88358e8i) q^{80} -2.37791e9 q^{81} +(-2.62313e9 + 2.62313e9i) q^{82} +(1.34634e9 + 1.34634e9i) q^{83} -4.05833e8i q^{84} +(-2.79029e9 + 6.15393e9i) q^{85} +3.78953e9 q^{86} +(2.54288e7 - 2.54288e7i) q^{87} +(-6.16382e8 - 6.16382e8i) q^{88} +2.66745e9i q^{89} +(-3.38428e9 - 1.53449e9i) q^{90} +1.52767e9 q^{91} +(-3.64761e8 + 3.64761e8i) q^{92} +(-1.65760e9 - 1.65760e9i) q^{93} -8.84225e9i q^{94} +(4.44255e9 + 1.18131e10i) q^{95} -4.78151e8 q^{96} +(-5.26563e8 + 5.26563e8i) q^{97} +(2.97259e9 + 2.97259e9i) q^{98} -3.95404e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 32 * q^2 + 114 * q^3 + 5850 * q^5 + 3648 * q^6 + 13906 * q^7 - 16384 * q^8 + 58400 * q^10 + 150484 * q^11 + 58368 * q^12 + 219714 * q^13 + 458850 * q^15 - 524288 * q^16 - 3057854 * q^17 - 1681632 * q^18 - 1126400 * q^20 + 1585284 * q^21 + 2407744 * q^22 - 1424846 * q^23 + 14691250 * q^25 + 7030848 * q^26 + 12722400 * q^27 - 7119872 * q^28 + 10670400 * q^30 - 58161436 * q^31 - 8388608 * q^32 + 8577588 * q^33 + 25378450 * q^35 - 53812224 * q^36 - 1823694 * q^37 + 129237760 * q^38 - 65945600 * q^40 - 327891356 * q^41 + 25364544 * q^42 + 236845554 * q^43 - 115612200 * q^45 - 45595072 * q^46 + 552640626 * q^47 - 29884416 * q^48 + 29140000 * q^50 - 348595356 * q^51 + 112493568 * q^52 + 616920194 * q^53 + 440165700 * q^55 - 227835904 * q^56 - 460409520 * q^57 - 14275840 * q^58 + 106521600 * q^60 - 2707220076 * q^61 - 930582976 * q^62 - 730774206 * q^63 + 884348850 * q^65 + 274482816 * q^66 + 1707141826 * q^67 + 1565621248 * q^68 - 489491200 * q^70 + 5654029124 * q^71 - 860995584 * q^72 - 5506594366 * q^73 + 1570991250 * q^75 + 4135608320 * q^76 + 1046315252 * q^77 + 400758336 * q^78 - 1533542400 * q^80 - 4755814398 * q^81 - 5246261696 * q^82 + 2692678194 * q^83 - 5580583550 * q^85 + 7579057728 * q^86 + 50857680 * q^87 - 1232764928 * q^88 - 6768568800 * q^90 + 3055342884 * q^91 - 729521152 * q^92 - 3315201852 * q^93 + 8885096000 * q^95 - 956301312 * q^96 - 1053125694 * q^97 + 5945178592 * 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
							\(3\)	57.0000	+	57.0000i	0.234568	+	0.234568i	0.814596	−	0.580028i	\(-0.196959\pi\)
−0.580028	+	0.814596i	\(0.696959\pi\)
\(5\)	2925.00	−	1100.00i	0.936000	−	0.352000i
							\(7\)	6953.00	−	6953.00i	0.413697	−	0.413697i	−0.469328	−	0.883024i	\(-0.655504\pi\)
0.883024	+	0.469328i	\(0.155504\pi\)
\(11\)	75242.0			0.467194			0.233597	−	0.972334i	\(-0.424950\pi\)
							0.233597	+	0.972334i	\(0.424950\pi\)
\(13\)	109857.	+	109857.i	0.295877	+	0.295877i	0.839396	−	0.543520i	\(-0.182909\pi\)
							−0.543520	+	0.839396i	\(0.682909\pi\)
\(17\)	−1.52893e6	+	1.52893e6i	−1.07682	+	1.07682i	−0.0800247	+	0.996793i	\(0.525500\pi\)
							−0.996793	+	0.0800247i	\(0.974500\pi\)
\(19\)			4.03868e6i			1.63107i	0.578711	+	0.815533i	\(0.303556\pi\)
							−0.578711	+	0.815533i	\(0.696444\pi\)
\(23\)	−712423.	−	712423.i	−0.110688	−	0.110688i	0.649594	−	0.760281i	\(-0.274939\pi\)
							−0.760281	+	0.649594i	\(0.774939\pi\)
\(29\)		−	446120.i		−	0.0217501i	−0.999941	−	0.0108751i	\(-0.996538\pi\)
							0.999941	−	0.0108751i	\(-0.00346171\pi\)
\(31\)	−2.90807e7			−1.01577			−0.507886	−	0.861424i	\(-0.669573\pi\)
							−0.507886	+	0.861424i	\(0.669573\pi\)
\(37\)	−911847.	+	911847.i	−0.0131496	+	0.0131496i	−0.713651	−	0.700501i	\(-0.752960\pi\)
							0.700501	+	0.713651i	\(0.252960\pi\)
\(41\)	−1.63946e8			−1.41508			−0.707540	−	0.706674i	\(-0.750195\pi\)
							−0.707540	+	0.706674i	\(0.750195\pi\)
\(43\)	1.18423e8	+	1.18423e8i	0.805551	+	0.805551i	0.983957	−	0.178406i	\(-0.0570941\pi\)
							−0.178406	+	0.983957i	\(0.557094\pi\)
\(47\)	2.76320e8	−	2.76320e8i	1.20482	−	1.20482i	0.232142	−	0.972682i	\(-0.425427\pi\)
							0.972682	−	0.232142i	\(-0.0745733\pi\)

(See \(a_n\) instead)

Twists

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

    

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