Embedded newform 224.2.bd.a.109.7

• Label: 224.2.bd.a.109.7
• Level: $224$
• Weight: $2$
• Character: 224.109
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $240$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.bd (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			109.7
Character	\(\chi\)	\(=\)	224.109
Dual form			224.2.bd.a.37.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13266 + 0.846802i) q^{2} +(0.0986144 + 0.128517i) q^{3} +(0.565853 - 1.91828i) q^{4} +(2.39507 + 1.83780i) q^{5} +(-0.220525 - 0.0620594i) q^{6} +(0.134369 - 2.64234i) q^{7} +(0.983485 + 2.65193i) q^{8} +(0.769665 - 2.87243i) q^{9} +(-4.26906 - 0.0534610i) q^{10} +(-0.230980 - 0.0304091i) q^{11} +(0.302333 - 0.116449i) q^{12} +(0.451311 + 0.186939i) q^{13} +(2.08534 + 3.10666i) q^{14} +0.489041i q^{15} +(-3.35962 - 2.17093i) q^{16} +(5.15284 + 2.97500i) q^{17} +(1.56061 + 3.90525i) q^{18} +(0.456233 + 3.46543i) q^{19} +(4.88068 - 3.55450i) q^{20} +(0.352835 - 0.243304i) q^{21} +(0.287373 - 0.161151i) q^{22} +(0.792513 - 2.95770i) q^{23} +(-0.243832 + 0.387913i) q^{24} +(1.06475 + 3.97371i) q^{25} +(-0.669484 + 0.170432i) q^{26} +(0.894039 - 0.370323i) q^{27} +(-4.99272 - 1.75293i) q^{28} +(-3.01331 + 7.27478i) q^{29} +(-0.414121 - 0.553919i) q^{30} +(-1.65406 + 2.86492i) q^{31} +(5.64367 - 0.385998i) q^{32} +(-0.0188699 - 0.0326836i) q^{33} +(-8.35567 + 0.993769i) q^{34} +(5.17792 - 6.08164i) q^{35} +(-5.07462 - 3.10181i) q^{36} +(3.48959 + 2.67766i) q^{37} +(-3.45129 - 3.53883i) q^{38} +(0.0204809 + 0.0764359i) q^{39} +(-2.51822 + 8.15902i) q^{40} +(-7.62359 - 7.62359i) q^{41} +(-0.193614 + 0.574363i) q^{42} +(-3.49397 - 8.43518i) q^{43} +(-0.189034 + 0.425878i) q^{44} +(7.12236 - 5.46518i) q^{45} +(1.60693 + 4.02118i) q^{46} +(-3.11151 + 1.79643i) q^{47} +(-0.0523058 - 0.645853i) q^{48} +(-6.96389 - 0.710095i) q^{49} +(-4.57095 - 3.59924i) q^{50} +(0.125808 + 0.955605i) q^{51} +(0.613978 - 0.759962i) q^{52} +(0.617371 + 0.0812784i) q^{53} +(-0.699055 + 1.17652i) q^{54} +(-0.497328 - 0.497328i) q^{55} +(7.13945 - 2.24236i) q^{56} +(-0.400375 + 0.400375i) q^{57} +(-2.74723 - 10.7916i) q^{58} +(1.44362 - 10.9654i) q^{59} +(0.938119 + 0.276725i) q^{60} +(-1.36236 + 0.179358i) q^{61} +(-0.552525 - 4.64566i) q^{62} +(-7.48651 - 2.41968i) q^{63} +(-6.06551 + 5.21628i) q^{64} +(0.737364 + 1.27715i) q^{65} +(0.0490498 + 0.0210405i) q^{66} +(-1.13554 - 1.47986i) q^{67} +(8.62264 - 8.20120i) q^{68} +(0.458267 - 0.189821i) q^{69} +(-0.714890 + 11.2731i) q^{70} +(-6.13645 + 6.13645i) q^{71} +(8.37445 - 0.783890i) q^{72} +(5.34815 - 1.43303i) q^{73} +(-6.21998 - 0.0778921i) q^{74} +(-0.405688 + 0.528703i) q^{75} +(6.90584 + 1.08574i) q^{76} +(-0.111388 + 0.606241i) q^{77} +(-0.0879241 - 0.0692329i) q^{78} +(-12.7597 + 7.36679i) q^{79} +(-4.05679 - 11.3739i) q^{80} +(-7.59029 - 4.38226i) q^{81} +(15.0906 + 2.17929i) q^{82} +(5.17520 + 2.14364i) q^{83} +(-0.267073 - 0.814512i) q^{84} +(6.87397 + 16.5952i) q^{85} +(11.1004 + 6.59552i) q^{86} +(-1.23209 + 0.330137i) q^{87} +(-0.146523 - 0.642451i) q^{88} +(11.0131 + 2.95095i) q^{89} +(-3.43931 + 12.2214i) q^{90} +(0.554598 - 1.16740i) q^{91} +(-5.22526 - 3.19389i) q^{92} +(-0.531306 + 0.0699477i) q^{93} +(2.00307 - 4.66958i) q^{94} +(-5.27607 + 9.13843i) q^{95} +(0.606154 + 0.687241i) q^{96} -12.8059 q^{97} +(8.48905 - 5.09274i) q^{98} +(-0.265125 + 0.640069i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	240 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 - 16 * q^6 - 8 * q^7 - 16 * q^8 - 4 * q^9 - 4 * q^10 - 4 * q^11 - 4 * q^12 - 16 * q^13 + 24 * q^14 - 24 * q^16 - 24 * q^18 - 4 * q^19 - 48 * q^20 - 8 * q^21 - 24 * q^22 - 12 * q^23 + 36 * q^24 - 4 * q^25 - 4 * q^26 - 16 * q^27 + 12 * q^28 - 16 * q^29 + 44 * q^30 - 56 * q^31 - 4 * q^32 - 8 * q^33 - 32 * q^34 - 32 * q^35 - 96 * q^36 - 4 * q^37 + 36 * q^38 - 4 * q^39 - 68 * q^40 - 16 * q^41 - 28 * q^42 + 44 * q^44 + 8 * q^45 - 4 * q^46 - 16 * q^48 + 8 * q^50 - 28 * q^51 - 28 * q^52 - 20 * q^53 - 92 * q^54 - 16 * q^55 - 48 * q^56 - 16 * q^57 + 28 * q^58 - 36 * q^59 + 60 * q^60 - 4 * q^61 - 16 * q^63 + 56 * q^64 - 8 * q^65 - 36 * q^66 + 36 * q^67 - 36 * q^68 - 16 * q^69 - 44 * q^70 + 48 * q^71 - 4 * q^72 - 4 * q^73 + 68 * q^74 + 16 * q^75 - 16 * q^76 - 8 * q^77 - 120 * q^78 - 16 * q^80 - 44 * q^82 - 96 * q^83 - 28 * q^84 - 56 * q^85 - 4 * q^86 - 4 * q^87 + 40 * q^88 - 4 * q^89 + 224 * q^90 - 56 * q^91 - 80 * q^92 + 20 * q^93 + 4 * q^94 - 8 * q^95 - 32 * q^96 - 32 * q^97 + 96 * q^98 + 8 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13266	+	0.846802i	−0.800914	+	0.598779i
							\(3\)	0.0986144	+	0.128517i	0.0569351	+	0.0741992i	0.820953	−	0.570996i	\(-0.193443\pi\)
−0.764018	+	0.645195i	\(0.776776\pi\)
\(5\)	2.39507	+	1.83780i	1.07111	+	0.821890i	0.984681	−	0.174368i	\(-0.0557882\pi\)
							0.0864277	+	0.996258i	\(0.472455\pi\)
\(7\)	0.134369	−	2.64234i	0.0507866	−	0.998710i
							\(11\)	−0.230980	−	0.0304091i	−0.0696431	−	0.00916869i	0.0956241	−	0.995418i	\(-0.469515\pi\)
−0.165267	+	0.986249i	\(0.552849\pi\)
\(13\)	0.451311	+	0.186939i	0.125171	+	0.0518476i	0.444390	−	0.895833i	\(-0.353420\pi\)
							−0.319219	+	0.947681i	\(0.603420\pi\)
\(17\)	5.15284	+	2.97500i	1.24975	+	0.721542i	0.971059	−	0.238839i	\(-0.0767668\pi\)
							0.278689	+	0.960381i	\(0.410100\pi\)
\(19\)	0.456233	+	3.46543i	0.104667	+	0.795025i	0.960599	+	0.277937i	\(0.0896507\pi\)
							−0.855932	+	0.517088i	\(0.827016\pi\)
\(23\)	0.792513	−	2.95770i	0.165250	−	0.616723i	−0.832758	−	0.553638i	\(-0.813239\pi\)
							0.998008	−	0.0630853i	\(-0.0200940\pi\)
\(29\)	−3.01331	+	7.27478i	−0.559558	+	1.35089i	0.350558	+	0.936541i	\(0.385992\pi\)
							−0.910117	+	0.414352i	\(0.864008\pi\)
\(31\)	−1.65406	+	2.86492i	−0.297079	+	0.514556i	−0.975466	−	0.220149i	\(-0.929346\pi\)
							0.678388	+	0.734704i	\(0.262679\pi\)
\(37\)	3.48959	+	2.67766i	0.573685	+	0.440204i	0.854421	−	0.519581i	\(-0.173912\pi\)
							−0.280736	+	0.959785i	\(0.590579\pi\)
\(41\)	−7.62359	−	7.62359i	−1.19060	−	1.19060i	−0.976898	−	0.213707i	\(-0.931446\pi\)
							−0.213707	−	0.976898i	\(-0.568554\pi\)
\(43\)	−3.49397	−	8.43518i	−0.532825	−	1.28635i	−0.929645	−	0.368456i	\(-0.879886\pi\)
							0.396820	−	0.917896i	\(-0.370114\pi\)
\(47\)	−3.11151	+	1.79643i	−0.453860	+	0.262036i	−0.709459	−	0.704747i	\(-0.751061\pi\)
							0.255599	+	0.966783i	\(0.417727\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.bd.a.109.7	yes	240
4.3	odd	2		896.2.bh.a.529.16		240
7.2	even	3	inner	224.2.bd.a.205.27	yes	240
28.23	odd	6		896.2.bh.a.401.15		240
32.5	even	8	inner	224.2.bd.a.165.27	yes	240
32.27	odd	8		896.2.bh.a.753.15		240
224.37	even	24	inner	224.2.bd.a.37.7	✓	240
224.219	odd	24		896.2.bh.a.625.16		240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.bd.a.37.7	✓	240	224.37	even	24	inner
224.2.bd.a.109.7	yes	240	1.1	even	1	trivial
224.2.bd.a.165.27	yes	240	32.5	even	8	inner
224.2.bd.a.205.27	yes	240	7.2	even	3	inner
896.2.bh.a.401.15		240	28.23	odd	6
896.2.bh.a.529.16		240	4.3	odd	2
896.2.bh.a.625.16		240	224.219	odd	24
896.2.bh.a.753.15		240	32.27	odd	8