Embedded newform 624.2.cn.d.401.2

• Label: 624.2.cn.d.401.2
• Level: $624$
• Weight: $2$
• Character: 624.401
• Analytic conductor: $4.983$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.cn (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.98266508613\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{12})\)
Coefficient field:	16.0.9349208943630483456.9
Defining polynomial:	x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			401.2
Root			\(0.500000 - 1.33108i\) of defining polynomial
Character	\(\chi\)	\(=\)	624.401
Dual form			624.2.cn.d.305.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.933998 + 1.45865i) q^{3} +(-2.76293 + 2.76293i) q^{5} +(0.657464 + 2.45369i) q^{7} +(-1.25529 - 2.72474i) q^{9} +(0.150860 - 0.563016i) q^{11} +(-1.20856 + 3.39697i) q^{13} +(-1.44957 - 6.61072i) q^{15} +(0.547000 - 0.947432i) q^{17} +(-1.32717 + 0.355613i) q^{19} +(-4.19313 - 1.33273i) q^{21} +(0.876460 + 1.51807i) q^{23} -10.2676i q^{25} +(5.14688 + 0.713876i) q^{27} +(5.12973 - 2.96165i) q^{29} +(-6.49983 - 6.49983i) q^{31} +(0.680339 + 0.745907i) q^{33} +(-8.59591 - 4.96285i) q^{35} +(-2.98942 - 0.801012i) q^{37} +(-3.82618 - 4.93562i) q^{39} +(-5.11781 - 1.37131i) q^{41} +(-3.26299 - 1.88389i) q^{43} +(10.9966 + 4.05999i) q^{45} +(5.51114 + 5.51114i) q^{47} +(0.473846 - 0.273575i) q^{49} +(0.871071 + 1.68278i) q^{51} +3.04435i q^{53} +(1.13876 + 1.97239i) q^{55} +(0.720857 - 2.26801i) q^{57} +(-8.19009 + 2.19453i) q^{59} +(-4.67266 + 8.09329i) q^{61} +(5.86037 - 4.87153i) q^{63} +(-6.04642 - 12.7248i) q^{65} +(-1.70856 + 6.37644i) q^{67} +(-3.03294 - 0.139433i) q^{69} +(-0.220122 - 0.821505i) q^{71} +(-5.18078 + 5.18078i) q^{73} +(14.9768 + 9.58993i) q^{75} +1.48065 q^{77} +13.1089 q^{79} +(-5.84847 + 6.84072i) q^{81} +(5.15394 - 5.15394i) q^{83} +(1.10637 + 4.12902i) q^{85} +(-0.471158 + 10.2486i) q^{87} +(-2.50797 + 9.35988i) q^{89} +(-9.12969 - 0.732051i) q^{91} +(15.5518 - 3.41012i) q^{93} +(2.68434 - 4.64941i) q^{95} +(-0.592450 + 0.158747i) q^{97} +(-1.72345 + 0.295697i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 8 * q^7 - 24 * q^13 + 16 * q^19 - 24 * q^21 - 16 * q^31 - 24 * q^33 + 16 * q^37 - 48 * q^39 + 24 * q^45 + 24 * q^49 + 24 * q^55 - 24 * q^57 - 24 * q^61 + 24 * q^63 - 32 * q^67 - 48 * q^69 + 56 * q^73 + 96 * q^79 + 24 * q^81 - 24 * q^85 - 48 * q^87 + 16 * q^91 - 24 * q^93 + 16 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\)	\(79\)	\(145\)	\(209\)	\(469\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{12}\right)\)	\(-1\)	\(1\)

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
							\(3\)	−0.933998	+	1.45865i	−0.539244	+	0.842150i
\(5\)	−2.76293	+	2.76293i	−1.23562	+	1.23562i								−0.273849	+	0.961773i	\(0.588297\pi\)
							−0.961773	+	0.273849i	\(0.911703\pi\)
\(7\)	0.657464	+	2.45369i	0.248498	+	0.927407i	0.971593	+	0.236659i	\(0.0760523\pi\)
							−0.723095	+	0.690749i	\(0.757281\pi\)
\(11\)	0.150860	−	0.563016i	0.0454859	−	0.169756i	−0.939446	−	0.342696i	\(-0.888660\pi\)
							0.984932	+	0.172940i	\(0.0553267\pi\)
\(13\)	−1.20856	+	3.39697i	−0.335195	+	0.942149i
							\(17\)	0.547000	−	0.947432i	0.132667	−	0.229786i	−0.792037	−	0.610473i	\(-0.790979\pi\)
0.924704	+	0.380687i	\(0.124313\pi\)
\(19\)	−1.32717	+	0.355613i	−0.304473	+	0.0815832i	−0.407821	−	0.913062i	\(-0.633711\pi\)
							0.103348	+	0.994645i	\(0.467044\pi\)
\(23\)	0.876460	+	1.51807i	0.182755	+	0.316540i	0.942818	−	0.333309i	\(-0.108165\pi\)
							−0.760063	+	0.649849i	\(0.774832\pi\)
\(29\)	5.12973	−	2.96165i	0.952566	−	0.549965i	0.0586892	−	0.998276i	\(-0.481308\pi\)
							0.893877	+	0.448312i	\(0.147975\pi\)
\(31\)	−6.49983	−	6.49983i	−1.16740	−	1.16740i	−0.982816	−	0.184588i	\(-0.940905\pi\)
							−0.184588	−	0.982816i	\(-0.559095\pi\)
\(37\)	−2.98942	−	0.801012i	−0.491457	−	0.131686i	0.00457534	−	0.999990i	\(-0.498544\pi\)
							−0.496032	+	0.868304i	\(0.665210\pi\)
\(41\)	−5.11781	−	1.37131i	−0.799268	−	0.214163i	−0.164005	−	0.986459i	\(-0.552441\pi\)
							−0.635262	+	0.772296i	\(0.719108\pi\)
\(43\)	−3.26299	−	1.88389i	−0.497602	−	0.287290i	0.230121	−	0.973162i	\(-0.426088\pi\)
							−0.727723	+	0.685872i	\(0.759421\pi\)
\(47\)	5.51114	+	5.51114i	0.803883	+	0.803883i	0.983700	−	0.179817i	\(-0.0575506\pi\)
							−0.179817	+	0.983700i	\(0.557551\pi\)
\(53\)			3.04435i			0.418173i	0.977897	+	0.209087i	\(0.0670490\pi\)
							−0.977897	+	0.209087i	\(0.932951\pi\)
\(59\)	−8.19009	+	2.19453i	−1.06626	+	0.285703i	−0.748955	−	0.662621i	\(-0.769444\pi\)
							−0.317304	+	0.948324i	\(0.602777\pi\)
\(61\)	−4.67266	+	8.09329i	−0.598273	+	1.03624i	0.394803	+	0.918766i	\(0.370813\pi\)
							−0.993076	+	0.117474i	\(0.962520\pi\)
\(67\)	−1.70856	+	6.37644i	−0.208734	+	0.779006i	0.779545	+	0.626346i	\(0.215450\pi\)
							−0.988279	+	0.152659i	\(0.951216\pi\)
\(71\)	−0.220122	−	0.821505i	−0.0261236	−	0.0974947i	0.951633	−	0.307237i	\(-0.0994043\pi\)
							−0.977757	+	0.209742i	\(0.932738\pi\)
\(73\)	−5.18078	+	5.18078i	−0.606365	+	0.606365i	−0.941994	−	0.335629i	\(-0.891051\pi\)
							0.335629	+	0.941994i	\(0.391051\pi\)
\(79\)	13.1089			1.47486			0.737431	−	0.675422i	\(-0.236039\pi\)
							0.737431	+	0.675422i	\(0.236039\pi\)
\(83\)	5.15394	−	5.15394i	0.565719	−	0.565719i	−0.365208	−	0.930926i	\(-0.619002\pi\)
							0.930926	+	0.365208i	\(0.119002\pi\)
\(89\)	−2.50797	+	9.35988i	−0.265844	+	0.992145i	0.695887	+	0.718151i	\(0.255011\pi\)
							−0.961731	+	0.273994i	\(0.911655\pi\)
\(97\)	−0.592450	+	0.158747i	−0.0601542	+	0.0161183i	−0.288771	−	0.957398i	\(-0.593247\pi\)
							0.228616	+	0.973517i	\(0.426580\pi\)

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	624.2.cn.d.401.2		16
3.2	odd	2	inner	624.2.cn.d.401.4		16
4.3	odd	2		78.2.k.a.11.4	yes	16
12.11	even	2		78.2.k.a.11.1	✓	16
13.6	odd	12	inner	624.2.cn.d.305.4		16
39.32	even	12	inner	624.2.cn.d.305.2		16
52.3	odd	6		1014.2.g.c.437.3		16
52.11	even	12		1014.2.g.d.239.3		16
52.15	even	12		1014.2.g.c.239.7		16
52.19	even	12		78.2.k.a.71.1	yes	16
52.23	odd	6		1014.2.g.d.437.7		16
156.11	odd	12		1014.2.g.d.239.7		16
156.23	even	6		1014.2.g.d.437.3		16
156.71	odd	12		78.2.k.a.71.4	yes	16
156.107	even	6		1014.2.g.c.437.7		16
156.119	odd	12		1014.2.g.c.239.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.k.a.11.1	✓	16	12.11	even	2
78.2.k.a.11.4	yes	16	4.3	odd	2
78.2.k.a.71.1	yes	16	52.19	even	12
78.2.k.a.71.4	yes	16	156.71	odd	12
624.2.cn.d.305.2		16	39.32	even	12	inner
624.2.cn.d.305.4		16	13.6	odd	12	inner
624.2.cn.d.401.2		16	1.1	even	1	trivial
624.2.cn.d.401.4		16	3.2	odd	2	inner
1014.2.g.c.239.3		16	156.119	odd	12
1014.2.g.c.239.7		16	52.15	even	12
1014.2.g.c.437.3		16	52.3	odd	6
1014.2.g.c.437.7		16	156.107	even	6
1014.2.g.d.239.3		16	52.11	even	12
1014.2.g.d.239.7		16	156.11	odd	12
1014.2.g.d.437.3		16	156.23	even	6
1014.2.g.d.437.7		16	52.23	odd	6