Embedded newform 121.4.a.c.1.2

• Label: 121.4.a.c.1.2
• Level: $121$
• Weight: $4$
• Character: 121.1
• Self dual: yes
• Analytic conductor: $7.139$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(7.13923111069\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{12})^+\)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(1.73205\) of defining polynomial
Character	\(\chi\)	\(=\)	121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} +5.92820 q^{3} -7.46410 q^{4} -12.8564 q^{5} +4.33975 q^{6} -16.9282 q^{7} -11.3205 q^{8} +8.14359 q^{9} -9.41154 q^{10} -44.2487 q^{12} -74.6410 q^{13} -12.3923 q^{14} -76.2154 q^{15} +51.4256 q^{16} +82.7846 q^{17} +5.96152 q^{18} +67.9230 q^{19} +95.9615 q^{20} -100.354 q^{21} +13.3538 q^{23} -67.1103 q^{24} +40.2872 q^{25} -54.6410 q^{26} -111.785 q^{27} +126.354 q^{28} -168.995 q^{29} -55.7935 q^{30} -65.4974 q^{31} +128.210 q^{32} +60.6025 q^{34} +217.636 q^{35} -60.7846 q^{36} +40.8564 q^{37} +49.7231 q^{38} -442.487 q^{39} +145.541 q^{40} -274.928 q^{41} -73.4641 q^{42} +2.28719 q^{43} -104.697 q^{45} +9.77568 q^{46} +71.8461 q^{47} +304.862 q^{48} -56.4359 q^{49} +29.4923 q^{50} +490.764 q^{51} +557.128 q^{52} -149.005 q^{53} -81.8320 q^{54} +191.636 q^{56} +402.662 q^{57} -123.713 q^{58} +545.631 q^{59} +568.879 q^{60} -101.303 q^{61} -47.9474 q^{62} -137.856 q^{63} -317.549 q^{64} +959.615 q^{65} +411.641 q^{67} -617.913 q^{68} +79.1642 q^{69} +159.321 q^{70} -470.636 q^{71} -92.1896 q^{72} -610.600 q^{73} +29.9090 q^{74} +238.831 q^{75} -506.985 q^{76} -323.923 q^{78} +978.225 q^{79} -661.149 q^{80} -882.559 q^{81} -201.261 q^{82} -26.1539 q^{83} +749.051 q^{84} -1064.31 q^{85} +1.67434 q^{86} -1001.84 q^{87} -352.887 q^{89} -76.6438 q^{90} +1263.54 q^{91} -99.6743 q^{92} -388.282 q^{93} +52.5950 q^{94} -873.246 q^{95} +760.056 q^{96} +847.585 q^{97} -41.3140 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 - 2 * q^3 - 8 * q^4 + 2 * q^5 + 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 - 50 * q^10 - 40 * q^12 - 80 * q^13 - 4 * q^14 - 194 * q^15 - 8 * q^16 + 124 * q^17 - 92 * q^18 - 72 * q^19 + 88 * q^20 - 76 * q^21 - 98 * q^23 - 252 * q^24 + 136 * q^25 - 40 * q^26 - 182 * q^27 + 128 * q^28 - 144 * q^29 + 266 * q^30 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 - 80 * q^36 + 54 * q^37 + 432 * q^38 - 400 * q^39 + 492 * q^40 - 536 * q^41 - 140 * q^42 + 60 * q^43 + 428 * q^45 + 314 * q^46 - 272 * q^47 + 776 * q^48 - 390 * q^49 - 232 * q^50 + 164 * q^51 + 560 * q^52 - 492 * q^53 + 110 * q^54 + 120 * q^56 + 1512 * q^57 - 192 * q^58 + 634 * q^59 + 632 * q^60 - 840 * q^61 - 134 * q^62 - 248 * q^63 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 962 * q^69 + 284 * q^70 - 678 * q^71 + 744 * q^72 + 400 * q^73 - 6 * q^74 - 520 * q^75 - 432 * q^76 - 440 * q^78 - 316 * q^79 - 1544 * q^80 - 1294 * q^81 + 512 * q^82 - 468 * q^83 + 736 * q^84 - 452 * q^85 - 156 * q^86 - 1200 * q^87 - 1842 * q^89 - 1532 * q^90 + 1280 * q^91 - 40 * q^92 - 638 * q^93 + 992 * q^94 - 2952 * q^95 + 952 * q^96 + 2194 * q^97 + 870 * q^98

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0.732051	0.258819	0.129410	−	0.991591i	\(-0.458692\pi\)
			0.129410	+	0.991591i	\(0.458692\pi\)
\(3\)	5.92820	1.14088	0.570442	−	0.821338i	\(-0.306772\pi\)
			0.570442	+	0.821338i	\(0.306772\pi\)
\(5\)	−12.8564	−1.14991	−0.574956	−	0.818184i	\(-0.694981\pi\)
			−0.574956	+	0.818184i	\(0.694981\pi\)
\(7\)	−16.9282	−0.914037	−0.457019	−	0.889457i	\(-0.651083\pi\)
			−0.457019	+	0.889457i	\(0.651083\pi\)
\(11\)	0	0
			\(13\)	−74.6410	−1.59244	−0.796219	−	0.605009i	\(-0.793170\pi\)
−0.796219	+	0.605009i				\(0.793170\pi\)
\(17\)	82.7846	1.18107	0.590536	−	0.807011i	\(-0.298916\pi\)
			0.590536	+	0.807011i	\(0.298916\pi\)
\(19\)	67.9230	0.820138	0.410069	−	0.912055i	\(-0.365505\pi\)
			0.410069	+	0.912055i	\(0.365505\pi\)
\(23\)	13.3538	0.121064	0.0605319	−	0.998166i	\(-0.480720\pi\)
			0.0605319	+	0.998166i	\(0.480720\pi\)
\(29\)	−168.995	−1.08212	−0.541061	−	0.840983i	\(-0.681977\pi\)
			−0.541061	+	0.840983i	\(0.681977\pi\)
\(31\)	−65.4974	−0.379474	−0.189737	−	0.981835i	\(-0.560763\pi\)
			−0.189737	+	0.981835i	\(0.560763\pi\)
\(37\)	40.8564	0.181534	0.0907669	−	0.995872i	\(-0.471068\pi\)
			0.0907669	+	0.995872i	\(0.471068\pi\)
\(41\)	−274.928	−1.04723	−0.523617	−	0.851954i	\(-0.675418\pi\)
			−0.523617	+	0.851954i	\(0.675418\pi\)
\(43\)	2.28719	0.00811146	0.00405573	−	0.999992i	\(-0.498709\pi\)
			0.00405573	+	0.999992i	\(0.498709\pi\)
\(47\)	71.8461	0.222975	0.111488	−	0.993766i	\(-0.464438\pi\)
			0.111488	+	0.993766i	\(0.464438\pi\)
\(53\)	−149.005	−0.386178	−0.193089	−	0.981181i	\(-0.561851\pi\)
			−0.193089	+	0.981181i	\(0.561851\pi\)
\(59\)	545.631	1.20398	0.601992	−	0.798502i	\(-0.294374\pi\)
			0.601992	+	0.798502i	\(0.294374\pi\)
\(61\)	−101.303	−0.212631	−0.106315	−	0.994332i	\(-0.533905\pi\)
			−0.106315	+	0.994332i	\(0.533905\pi\)
\(67\)	411.641	0.750596	0.375298	−	0.926904i	\(-0.377540\pi\)
			0.375298	+	0.926904i	\(0.377540\pi\)
\(71\)	−470.636	−0.786679	−0.393339	−	0.919393i	\(-0.628680\pi\)
			−0.393339	+	0.919393i	\(0.628680\pi\)
\(73\)	−610.600	−0.978977	−0.489488	−	0.872010i	\(-0.662816\pi\)
			−0.489488	+	0.872010i	\(0.662816\pi\)
\(79\)	978.225	1.39315	0.696576	−	0.717483i	\(-0.254706\pi\)
			0.696576	+	0.717483i	\(0.254706\pi\)
\(83\)	−26.1539	−0.0345875	−0.0172938	−	0.999850i	\(-0.505505\pi\)
			−0.0172938	+	0.999850i	\(0.505505\pi\)
\(89\)	−352.887	−0.420292	−0.210146	−	0.977670i	\(-0.567394\pi\)
			−0.210146	+	0.977670i	\(0.567394\pi\)
\(97\)	847.585	0.887208	0.443604	−	0.896223i	\(-0.353700\pi\)
			0.443604	+	0.896223i	\(0.353700\pi\)

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	121.4.a.c.1.2		2
3.2	odd	2		1089.4.a.v.1.1		2
4.3	odd	2		1936.4.a.w.1.1		2
11.2	odd	10		121.4.c.c.81.2		8
11.3	even	5		121.4.c.f.9.2		8
11.4	even	5		121.4.c.f.27.2		8
11.5	even	5		121.4.c.f.3.1		8
11.6	odd	10		121.4.c.c.3.2		8
11.7	odd	10		121.4.c.c.27.1		8
11.8	odd	10		121.4.c.c.9.1		8
11.9	even	5		121.4.c.f.81.1		8
11.10	odd	2		11.4.a.a.1.1	✓	2
33.32	even	2		99.4.a.c.1.2		2
44.43	even	2		176.4.a.i.1.1		2
55.32	even	4		275.4.b.c.199.2		4
55.43	even	4		275.4.b.c.199.3		4
55.54	odd	2		275.4.a.b.1.2		2
77.76	even	2		539.4.a.e.1.1		2
88.21	odd	2		704.4.a.p.1.1		2
88.43	even	2		704.4.a.n.1.2		2
132.131	odd	2		1584.4.a.bc.1.2		2
143.142	odd	2		1859.4.a.a.1.2		2
165.164	even	2		2475.4.a.q.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	11.10	odd	2
99.4.a.c.1.2		2	33.32	even	2
121.4.a.c.1.2		2	1.1	even	1	trivial
121.4.c.c.3.2		8	11.6	odd	10
121.4.c.c.9.1		8	11.8	odd	10
121.4.c.c.27.1		8	11.7	odd	10
121.4.c.c.81.2		8	11.2	odd	10
121.4.c.f.3.1		8	11.5	even	5
121.4.c.f.9.2		8	11.3	even	5
121.4.c.f.27.2		8	11.4	even	5
121.4.c.f.81.1		8	11.9	even	5
176.4.a.i.1.1		2	44.43	even	2
275.4.a.b.1.2		2	55.54	odd	2
275.4.b.c.199.2		4	55.32	even	4
275.4.b.c.199.3		4	55.43	even	4
539.4.a.e.1.1		2	77.76	even	2
704.4.a.n.1.2		2	88.43	even	2
704.4.a.p.1.1		2	88.21	odd	2
1089.4.a.v.1.1		2	3.2	odd	2
1584.4.a.bc.1.2		2	132.131	odd	2
1859.4.a.a.1.2		2	143.142	odd	2
1936.4.a.w.1.1		2	4.3	odd	2
2475.4.a.q.1.1		2	165.164	even	2