Embedded newform 672.2.bd.a.431.21

• Label: 672.2.bd.a.431.21
• Level: $672$
• Weight: $2$
• Character: 672.431
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.bd (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			431.21
Character	\(\chi\)	\(=\)	672.431
Dual form			672.2.bd.a.527.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31637 - 1.12568i) q^{3} +(-1.86088 - 3.22314i) q^{5} +(-2.46429 + 0.962962i) q^{7} +(0.465685 - 2.96364i) q^{9} +(-2.26958 - 1.31034i) q^{11} +3.57182i q^{13} +(-6.07784 - 2.14810i) q^{15} +(-0.186192 - 0.107498i) q^{17} +(1.14103 + 1.97631i) q^{19} +(-2.15994 + 4.04162i) q^{21} +(-2.33586 - 4.04583i) q^{23} +(-4.42574 + 7.66560i) q^{25} +(-2.72309 - 4.42547i) q^{27} -2.57415 q^{29} +(-4.26196 - 2.46064i) q^{31} +(-4.46265 + 0.829921i) q^{33} +(7.68949 + 6.15077i) q^{35} +(7.31104 - 4.22103i) q^{37} +(4.02073 + 4.70186i) q^{39} -0.909442i q^{41} -3.73541 q^{43} +(-10.4188 + 4.01400i) q^{45} +(-0.586728 - 1.01624i) q^{47} +(5.14541 - 4.74603i) q^{49} +(-0.366107 + 0.0680851i) q^{51} +(1.06171 - 1.83894i) q^{53} +9.75355i q^{55} +(3.72672 + 1.31714i) q^{57} +(-6.79199 - 3.92136i) q^{59} +(-0.301301 + 0.173956i) q^{61} +(1.70629 + 7.75168i) q^{63} +(11.5125 - 6.64673i) q^{65} +(4.98736 - 8.63836i) q^{67} +(-7.62919 - 2.69640i) q^{69} +10.0808 q^{71} +(4.45173 - 7.71062i) q^{73} +(2.80309 + 15.0728i) q^{75} +(6.85470 + 1.04354i) q^{77} +(9.70950 - 5.60578i) q^{79} +(-8.56628 - 2.76024i) q^{81} -1.73937i q^{83} +0.800163i q^{85} +(-3.38855 + 2.89767i) q^{87} +(-15.4694 + 8.93127i) q^{89} +(-3.43953 - 8.80199i) q^{91} +(-8.38024 + 1.55848i) q^{93} +(4.24662 - 7.35536i) q^{95} -8.88553 q^{97} +(-4.94029 + 6.11600i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 2 q^{3} - 2 q^{9} + 4 q^{19} - 16 q^{25} + 8 q^{27} - 14 q^{33} + 16 q^{43} - 16 q^{49} + 34 q^{51} + 4 q^{57} + 36 q^{67} + 4 q^{73} - 10 q^{81} - 72 q^{91} - 32 q^{97} + 44 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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			0
							\(3\)	1.31637	−	1.12568i	0.760009	−	0.649912i
\(5\)	−1.86088	−	3.22314i	−0.832210	−	1.44143i								−0.896282	−	0.443485i	\(-0.853742\pi\)
							0.0640716	−	0.997945i	\(-0.479591\pi\)
\(7\)	−2.46429	+	0.962962i	−0.931412	+	0.363965i
							\(11\)	−2.26958	−	1.31034i	−0.684304	−	0.395083i	0.117171	−	0.993112i	\(-0.462618\pi\)
−0.801475	+	0.598029i	\(0.795951\pi\)
\(13\)			3.57182i			0.990645i	0.868709	+	0.495323i	\(0.164950\pi\)
							−0.868709	+	0.495323i	\(0.835050\pi\)
\(17\)	−0.186192	−	0.107498i	−0.0451582	−	0.0260721i	0.477251	−	0.878767i	\(-0.341633\pi\)
							−0.522409	+	0.852695i	\(0.674967\pi\)
\(19\)	1.14103	+	1.97631i	0.261769	+	0.453398i	0.966712	−	0.255866i	\(-0.0823607\pi\)
							−0.704943	+	0.709264i	\(0.749027\pi\)
\(23\)	−2.33586	−	4.04583i	−0.487061	−	0.843615i	0.512828	−	0.858491i	\(-0.328598\pi\)
							−0.999889	+	0.0148767i	\(0.995264\pi\)
\(29\)	−2.57415			−0.478008			−0.239004	−	0.971019i	\(-0.576821\pi\)
							−0.239004	+	0.971019i	\(0.576821\pi\)
\(31\)	−4.26196	−	2.46064i	−0.765471	−	0.441945i	0.0657857	−	0.997834i	\(-0.479045\pi\)
							−0.831257	+	0.555889i	\(0.812378\pi\)
\(37\)	7.31104	−	4.22103i	1.20193	−	0.693933i	0.240944	−	0.970539i	\(-0.422543\pi\)
							0.960983	+	0.276606i	\(0.0892098\pi\)
\(41\)		−	0.909442i		−	0.142031i	−0.997475	−	0.0710155i	\(-0.977376\pi\)
							0.997475	−	0.0710155i	\(-0.0226240\pi\)
\(43\)	−3.73541			−0.569644			−0.284822	−	0.958580i	\(-0.591935\pi\)
							−0.284822	+	0.958580i	\(0.591935\pi\)
\(47\)	−0.586728	−	1.01624i	−0.0855831	−	0.148234i	0.820056	−	0.572283i	\(-0.193942\pi\)
							−0.905640	+	0.424048i	\(0.860609\pi\)
\(53\)	1.06171	−	1.83894i	0.145837	−	0.252598i	−0.783848	−	0.620953i	\(-0.786746\pi\)
							0.929685	+	0.368355i	\(0.120079\pi\)
\(59\)	−6.79199	−	3.92136i	−0.884242	−	0.510517i	−0.0121872	−	0.999926i	\(-0.503879\pi\)
							−0.872055	+	0.489408i	\(0.837213\pi\)
\(61\)	−0.301301	+	0.173956i	−0.0385776	+	0.0222728i	−0.519165	−	0.854674i	\(-0.673757\pi\)
							0.480587	+	0.876947i	\(0.340424\pi\)
\(67\)	4.98736	−	8.63836i	0.609303	−	1.05534i	−0.382053	−	0.924140i	\(-0.624783\pi\)
							0.991356	−	0.131203i	\(-0.0418839\pi\)
\(71\)	10.0808			1.19637			0.598183	−	0.801359i	\(-0.295889\pi\)
							0.598183	+	0.801359i	\(0.295889\pi\)
\(73\)	4.45173	−	7.71062i	0.521036	−	0.902460i	−0.478665	−	0.877998i	\(-0.658879\pi\)
							0.999701	−	0.0244626i	\(-0.00778746\pi\)
\(79\)	9.70950	−	5.60578i	1.09240	−	0.630700i	0.158189	−	0.987409i	\(-0.449434\pi\)
							0.934216	+	0.356709i	\(0.116101\pi\)
\(83\)		−	1.73937i		−	0.190921i	−0.995433	−	0.0954603i	\(-0.969568\pi\)
							0.995433	−	0.0954603i	\(-0.0304323\pi\)
\(89\)	−15.4694	+	8.93127i	−1.63975	+	0.946712i	−0.658837	+	0.752286i	\(0.728951\pi\)
							−0.980917	+	0.194426i	\(0.937716\pi\)
\(97\)	−8.88553			−0.902189			−0.451094	−	0.892476i	\(-0.648966\pi\)
							−0.451094	+	0.892476i	\(0.648966\pi\)

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	672.2.bd.a.431.21		56
3.2	odd	2	inner	672.2.bd.a.431.16		56
4.3	odd	2		168.2.v.a.11.23	yes	56
7.2	even	3	inner	672.2.bd.a.527.15		56
8.3	odd	2	inner	672.2.bd.a.431.22		56
8.5	even	2		168.2.v.a.11.26	yes	56
12.11	even	2		168.2.v.a.11.6	yes	56
21.2	odd	6	inner	672.2.bd.a.527.22		56
24.5	odd	2		168.2.v.a.11.3	✓	56
24.11	even	2	inner	672.2.bd.a.431.15		56
28.23	odd	6		168.2.v.a.107.3	yes	56
56.37	even	6		168.2.v.a.107.6	yes	56
56.51	odd	6	inner	672.2.bd.a.527.16		56
84.23	even	6		168.2.v.a.107.26	yes	56
168.107	even	6	inner	672.2.bd.a.527.21		56
168.149	odd	6		168.2.v.a.107.23	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.v.a.11.3	✓	56	24.5	odd	2
168.2.v.a.11.6	yes	56	12.11	even	2
168.2.v.a.11.23	yes	56	4.3	odd	2
168.2.v.a.11.26	yes	56	8.5	even	2
168.2.v.a.107.3	yes	56	28.23	odd	6
168.2.v.a.107.6	yes	56	56.37	even	6
168.2.v.a.107.23	yes	56	168.149	odd	6
168.2.v.a.107.26	yes	56	84.23	even	6
672.2.bd.a.431.15		56	24.11	even	2	inner
672.2.bd.a.431.16		56	3.2	odd	2	inner
672.2.bd.a.431.21		56	1.1	even	1	trivial
672.2.bd.a.431.22		56	8.3	odd	2	inner
672.2.bd.a.527.15		56	7.2	even	3	inner
672.2.bd.a.527.16		56	56.51	odd	6	inner
672.2.bd.a.527.21		56	168.107	even	6	inner
672.2.bd.a.527.22		56	21.2	odd	6	inner