Embedded newform 121.4.a.c.1.1

• Label: 121.4.a.c.1.1
• 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			1.1
Root			\(-1.73205\) of defining polynomial
Character	\(\chi\)	\(=\)	121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} +14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} -40.5885 q^{10} +4.24871 q^{12} -5.35898 q^{13} +8.39230 q^{14} -117.785 q^{15} -59.4256 q^{16} +41.2154 q^{17} -97.9615 q^{18} -139.923 q^{19} -7.96152 q^{20} +24.3538 q^{21} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} +14.6410 q^{26} -70.2154 q^{27} +1.64617 q^{28} +24.9948 q^{29} +321.794 q^{30} +31.4974 q^{31} -24.2102 q^{32} -112.603 q^{34} -45.6359 q^{35} -19.2154 q^{36} +13.1436 q^{37} +382.277 q^{38} +42.4871 q^{39} +346.459 q^{40} -261.072 q^{41} -66.5359 q^{42} +57.7128 q^{43} +532.697 q^{45} +304.224 q^{46} -343.846 q^{47} +471.138 q^{48} -333.564 q^{49} -261.492 q^{50} -326.764 q^{51} +2.87187 q^{52} -342.995 q^{53} +191.832 q^{54} -71.6359 q^{56} +1109.34 q^{57} -68.2872 q^{58} +88.3693 q^{59} +63.1206 q^{60} -738.697 q^{61} -86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} -79.6152 q^{65} +342.359 q^{67} -22.0873 q^{68} +882.836 q^{69} +124.679 q^{70} -207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} -35.9090 q^{74} -758.831 q^{75} +74.9845 q^{76} -116.077 q^{78} -1294.23 q^{79} -882.851 q^{80} -411.441 q^{81} +713.261 q^{82} -441.846 q^{83} -13.0512 q^{84} +612.313 q^{85} -157.674 q^{86} -198.164 q^{87} -1489.11 q^{89} -1455.36 q^{90} +16.4617 q^{91} +59.6743 q^{92} -249.718 q^{93} +939.405 q^{94} -2078.75 q^{95} +191.944 q^{96} +1346.42 q^{97} +911.314 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.73205	−0.965926	−0.482963	−	0.875641i	\(-0.660439\pi\)
			−0.482963	+	0.875641i	\(0.660439\pi\)
\(3\)	−7.92820	−1.52578	−0.762892	−	0.646526i	\(-0.776221\pi\)
			−0.762892	+	0.646526i	\(0.776221\pi\)
\(5\)	14.8564	1.32880	0.664399	−	0.747378i	\(-0.268688\pi\)
			0.664399	+	0.747378i	\(0.268688\pi\)
\(7\)	−3.07180	−0.165861	−0.0829307	−	0.996555i	\(-0.526428\pi\)
			−0.0829307	+	0.996555i	\(0.526428\pi\)
\(11\)	0	0
			\(13\)	−5.35898	−0.114332	−0.0571659	−	0.998365i	\(-0.518206\pi\)
−0.0571659	+	0.998365i				\(0.518206\pi\)
\(17\)	41.2154	0.588012	0.294006	−	0.955804i	\(-0.405011\pi\)
			0.294006	+	0.955804i	\(0.405011\pi\)
\(19\)	−139.923	−1.68950	−0.844751	−	0.535159i	\(-0.820252\pi\)
			−0.844751	+	0.535159i	\(0.820252\pi\)
\(23\)	−111.354	−1.00952	−0.504758	−	0.863261i	\(-0.668418\pi\)
			−0.504758	+	0.863261i	\(0.668418\pi\)
\(29\)	24.9948	0.160049	0.0800246	−	0.996793i	\(-0.474500\pi\)
			0.0800246	+	0.996793i	\(0.474500\pi\)
\(31\)	31.4974	0.182487	0.0912436	−	0.995829i	\(-0.470916\pi\)
			0.0912436	+	0.995829i	\(0.470916\pi\)
\(37\)	13.1436	0.0583998	0.0291999	−	0.999574i	\(-0.490704\pi\)
			0.0291999	+	0.999574i	\(0.490704\pi\)
\(41\)	−261.072	−0.994453	−0.497226	−	0.867621i	\(-0.665648\pi\)
			−0.497226	+	0.867621i	\(0.665648\pi\)
\(43\)	57.7128	0.204677	0.102339	−	0.994750i	\(-0.467367\pi\)
			0.102339	+	0.994750i	\(0.467367\pi\)
\(47\)	−343.846	−1.06713	−0.533565	−	0.845759i	\(-0.679148\pi\)
			−0.533565	+	0.845759i	\(0.679148\pi\)
\(53\)	−342.995	−0.888943	−0.444471	−	0.895793i	\(-0.646608\pi\)
			−0.444471	+	0.895793i	\(0.646608\pi\)
\(59\)	88.3693	0.194995	0.0974975	−	0.995236i	\(-0.468916\pi\)
			0.0974975	+	0.995236i	\(0.468916\pi\)
\(61\)	−738.697	−1.55050	−0.775250	−	0.631654i	\(-0.782376\pi\)
			−0.775250	+	0.631654i	\(0.782376\pi\)
\(67\)	342.359	0.624266	0.312133	−	0.950038i	\(-0.398957\pi\)
			0.312133	+	0.950038i	\(0.398957\pi\)
\(71\)	−207.364	−0.346614	−0.173307	−	0.984868i	\(-0.555445\pi\)
			−0.173307	+	0.984868i	\(0.555445\pi\)
\(73\)	1010.60	1.62030	0.810149	−	0.586224i	\(-0.199386\pi\)
			0.810149	+	0.586224i	\(0.199386\pi\)
\(79\)	−1294.23	−1.84319	−0.921593	−	0.388157i	\(-0.873112\pi\)
			−0.921593	+	0.388157i	\(0.873112\pi\)
\(83\)	−441.846	−0.584324	−0.292162	−	0.956369i	\(-0.594375\pi\)
			−0.292162	+	0.956369i	\(0.594375\pi\)
\(89\)	−1489.11	−1.77355	−0.886773	−	0.462205i	\(-0.847058\pi\)
			−0.886773	+	0.462205i	\(0.847058\pi\)
\(97\)	1346.42	1.40936	0.704679	−	0.709526i	\(-0.251091\pi\)
			0.704679	+	0.709526i	\(0.251091\pi\)

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.1.1		2
3.2	odd	2		1089.4.a.v.1.2		2
4.3	odd	2		1936.4.a.w.1.2		2
11.2	odd	10		121.4.c.c.81.1		8
11.3	even	5		121.4.c.f.9.1		8
11.4	even	5		121.4.c.f.27.1		8
11.5	even	5		121.4.c.f.3.2		8
11.6	odd	10		121.4.c.c.3.1		8
11.7	odd	10		121.4.c.c.27.2		8
11.8	odd	10		121.4.c.c.9.2		8
11.9	even	5		121.4.c.f.81.2		8
11.10	odd	2		11.4.a.a.1.2	✓	2
33.32	even	2		99.4.a.c.1.1		2
44.43	even	2		176.4.a.i.1.2		2
55.32	even	4		275.4.b.c.199.4		4
55.43	even	4		275.4.b.c.199.1		4
55.54	odd	2		275.4.a.b.1.1		2
77.76	even	2		539.4.a.e.1.2		2
88.21	odd	2		704.4.a.p.1.2		2
88.43	even	2		704.4.a.n.1.1		2
132.131	odd	2		1584.4.a.bc.1.1		2
143.142	odd	2		1859.4.a.a.1.1		2
165.164	even	2		2475.4.a.q.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	11.10	odd	2
99.4.a.c.1.1		2	33.32	even	2
121.4.a.c.1.1		2	1.1	even	1	trivial
121.4.c.c.3.1		8	11.6	odd	10
121.4.c.c.9.2		8	11.8	odd	10
121.4.c.c.27.2		8	11.7	odd	10
121.4.c.c.81.1		8	11.2	odd	10
121.4.c.f.3.2		8	11.5	even	5
121.4.c.f.9.1		8	11.3	even	5
121.4.c.f.27.1		8	11.4	even	5
121.4.c.f.81.2		8	11.9	even	5
176.4.a.i.1.2		2	44.43	even	2
275.4.a.b.1.1		2	55.54	odd	2
275.4.b.c.199.1		4	55.43	even	4
275.4.b.c.199.4		4	55.32	even	4
539.4.a.e.1.2		2	77.76	even	2
704.4.a.n.1.1		2	88.43	even	2
704.4.a.p.1.2		2	88.21	odd	2
1089.4.a.v.1.2		2	3.2	odd	2
1584.4.a.bc.1.1		2	132.131	odd	2
1859.4.a.a.1.1		2	143.142	odd	2
1936.4.a.w.1.2		2	4.3	odd	2
2475.4.a.q.1.2		2	165.164	even	2