Embedded newform 50.3.c.a.7.1

• Label: 50.3.c.a.7.1
• Level: $50$
• Weight: $3$
• Character: 50.7
• Analytic conductor: $1.362$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36240132180\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.1
Root			\(1.00000i\) of defining polynomial
Character	\(\chi\)	\(=\)	50.7
Dual form			50.3.c.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +(3.00000 - 3.00000i) q^{3} +2.00000i q^{4} -6.00000 q^{6} +(-3.00000 - 3.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} -9.00000i q^{9} +12.0000 q^{11} +(6.00000 + 6.00000i) q^{12} +(-12.0000 + 12.0000i) q^{13} +6.00000i q^{14} -4.00000 q^{16} +(12.0000 + 12.0000i) q^{17} +(-9.00000 + 9.00000i) q^{18} +20.0000i q^{19} -18.0000 q^{21} +(-12.0000 - 12.0000i) q^{22} +(3.00000 - 3.00000i) q^{23} -12.0000i q^{24} +24.0000 q^{26} +(6.00000 - 6.00000i) q^{28} -30.0000i q^{29} -8.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(36.0000 - 36.0000i) q^{33} -24.0000i q^{34} +18.0000 q^{36} +(-48.0000 - 48.0000i) q^{37} +(20.0000 - 20.0000i) q^{38} +72.0000i q^{39} -48.0000 q^{41} +(18.0000 + 18.0000i) q^{42} +(-27.0000 + 27.0000i) q^{43} +24.0000i q^{44} -6.00000 q^{46} +(27.0000 + 27.0000i) q^{47} +(-12.0000 + 12.0000i) q^{48} -31.0000i q^{49} +72.0000 q^{51} +(-24.0000 - 24.0000i) q^{52} +(-12.0000 + 12.0000i) q^{53} -12.0000 q^{56} +(60.0000 + 60.0000i) q^{57} +(-30.0000 + 30.0000i) q^{58} -60.0000i q^{59} +32.0000 q^{61} +(8.00000 + 8.00000i) q^{62} +(-27.0000 + 27.0000i) q^{63} -8.00000i q^{64} -72.0000 q^{66} +(-3.00000 - 3.00000i) q^{67} +(-24.0000 + 24.0000i) q^{68} -18.0000i q^{69} -48.0000 q^{71} +(-18.0000 - 18.0000i) q^{72} +(-12.0000 + 12.0000i) q^{73} +96.0000i q^{74} -40.0000 q^{76} +(-36.0000 - 36.0000i) q^{77} +(72.0000 - 72.0000i) q^{78} +40.0000i q^{79} +81.0000 q^{81} +(48.0000 + 48.0000i) q^{82} +(93.0000 - 93.0000i) q^{83} -36.0000i q^{84} +54.0000 q^{86} +(-90.0000 - 90.0000i) q^{87} +(24.0000 - 24.0000i) q^{88} +30.0000i q^{89} +72.0000 q^{91} +(6.00000 + 6.00000i) q^{92} +(-24.0000 + 24.0000i) q^{93} -54.0000i q^{94} +24.0000 q^{96} +(12.0000 + 12.0000i) q^{97} +(-31.0000 + 31.0000i) q^{98} -108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 + 6 * q^3 - 12 * q^6 - 6 * q^7 + 4 * q^8 + 24 * q^11 + 12 * q^12 - 24 * q^13 - 8 * q^16 + 24 * q^17 - 18 * q^18 - 36 * q^21 - 24 * q^22 + 6 * q^23 + 48 * q^26 + 12 * q^28 - 16 * q^31 + 8 * q^32 + 72 * q^33 + 36 * q^36 - 96 * q^37 + 40 * q^38 - 96 * q^41 + 36 * q^42 - 54 * q^43 - 12 * q^46 + 54 * q^47 - 24 * q^48 + 144 * q^51 - 48 * q^52 - 24 * q^53 - 24 * q^56 + 120 * q^57 - 60 * q^58 + 64 * q^61 + 16 * q^62 - 54 * q^63 - 144 * q^66 - 6 * q^67 - 48 * q^68 - 96 * q^71 - 36 * q^72 - 24 * q^73 - 80 * q^76 - 72 * q^77 + 144 * q^78 + 162 * q^81 + 96 * q^82 + 186 * q^83 + 108 * q^86 - 180 * q^87 + 48 * q^88 + 144 * q^91 + 12 * q^92 - 48 * q^93 + 48 * q^96 + 24 * q^97 - 62 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\)	\(27\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.00000i	−0.500000	−	0.500000i
							\(3\)	3.00000	−	3.00000i	1.00000	−	1.00000i		−	1.00000i	\(-0.5\pi\)
1.00000			\(0\)
\(5\)		0			0
							\(7\)	−3.00000	−	3.00000i	−0.428571	−	0.428571i	0.459570	−	0.888142i	\(-0.348004\pi\)
−0.888142	+	0.459570i	\(0.848004\pi\)
\(11\)	12.0000			1.09091			0.545455	−	0.838140i	\(-0.316357\pi\)
							0.545455	+	0.838140i	\(0.316357\pi\)
\(13\)	−12.0000	+	12.0000i	−0.923077	+	0.923077i	−0.997246	−	0.0741688i	\(-0.976370\pi\)
							0.0741688	+	0.997246i	\(0.476370\pi\)
\(17\)	12.0000	+	12.0000i	0.705882	+	0.705882i	0.965667	−	0.259784i	\(-0.0836515\pi\)
							−0.259784	+	0.965667i	\(0.583651\pi\)
\(19\)			20.0000i			1.05263i	0.850289	+	0.526316i	\(0.176427\pi\)
							−0.850289	+	0.526316i	\(0.823573\pi\)
\(23\)	3.00000	−	3.00000i	0.130435	−	0.130435i	−0.638875	−	0.769310i	\(-0.720600\pi\)
							0.769310	+	0.638875i	\(0.220600\pi\)
\(29\)		−	30.0000i		−	1.03448i	−0.855840	−	0.517241i	\(-0.826959\pi\)
							0.855840	−	0.517241i	\(-0.173041\pi\)
\(31\)	−8.00000			−0.258065			−0.129032	−	0.991640i	\(-0.541187\pi\)
							−0.129032	+	0.991640i	\(0.541187\pi\)
\(37\)	−48.0000	−	48.0000i	−1.29730	−	1.29730i	−0.930171	−	0.367126i	\(-0.880342\pi\)
							−0.367126	−	0.930171i	\(-0.619658\pi\)
\(41\)	−48.0000			−1.17073			−0.585366	−	0.810769i	\(-0.699049\pi\)
							−0.585366	+	0.810769i	\(0.699049\pi\)
\(43\)	−27.0000	+	27.0000i	−0.627907	+	0.627907i	−0.947541	−	0.319634i	\(-0.896440\pi\)
							0.319634	+	0.947541i	\(0.396440\pi\)
\(47\)	27.0000	+	27.0000i	0.574468	+	0.574468i	0.933374	−	0.358906i	\(-0.116850\pi\)
							−0.358906	+	0.933374i	\(0.616850\pi\)
\(53\)	−12.0000	+	12.0000i	−0.226415	+	0.226415i	−0.811193	−	0.584778i	\(-0.801182\pi\)
							0.584778	+	0.811193i	\(0.301182\pi\)
\(59\)		−	60.0000i		−	1.01695i	−0.861077	−	0.508475i	\(-0.830210\pi\)
							0.861077	−	0.508475i	\(-0.169790\pi\)
\(61\)	32.0000			0.524590			0.262295	−	0.964988i	\(-0.415521\pi\)
							0.262295	+	0.964988i	\(0.415521\pi\)
\(67\)	−3.00000	−	3.00000i	−0.0447761	−	0.0447761i	0.684364	−	0.729140i	\(-0.260080\pi\)
							−0.729140	+	0.684364i	\(0.760080\pi\)
\(71\)	−48.0000			−0.676056			−0.338028	−	0.941136i	\(-0.609760\pi\)
							−0.338028	+	0.941136i	\(0.609760\pi\)
\(73\)	−12.0000	+	12.0000i	−0.164384	+	0.164384i	−0.784505	−	0.620122i	\(-0.787083\pi\)
							0.620122	+	0.784505i	\(0.287083\pi\)
\(79\)			40.0000i			0.506329i	0.967423	+	0.253165i	\(0.0814714\pi\)
							−0.967423	+	0.253165i	\(0.918529\pi\)
\(83\)	93.0000	−	93.0000i	1.12048	−	1.12048i	0.128813	−	0.991669i	\(-0.458883\pi\)
							0.991669	−	0.128813i	\(-0.0411167\pi\)
\(89\)			30.0000i			0.337079i	0.985695	+	0.168539i	\(0.0539050\pi\)
							−0.985695	+	0.168539i	\(0.946095\pi\)
\(97\)	12.0000	+	12.0000i	0.123711	+	0.123711i	0.766252	−	0.642540i	\(-0.222120\pi\)
							−0.642540	+	0.766252i	\(0.722120\pi\)

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	50.3.c.a.7.1	✓	2
3.2	odd	2		450.3.g.e.307.1		2
4.3	odd	2		400.3.p.a.257.1		2
5.2	odd	4		50.3.c.b.43.1	yes	2
5.3	odd	4	inner	50.3.c.a.43.1	yes	2
5.4	even	2		50.3.c.b.7.1	yes	2
15.2	even	4		450.3.g.c.343.1		2
15.8	even	4		450.3.g.e.343.1		2
15.14	odd	2		450.3.g.c.307.1		2
20.3	even	4		400.3.p.a.193.1		2
20.7	even	4		400.3.p.g.193.1		2
20.19	odd	2		400.3.p.g.257.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.3.c.a.7.1	✓	2	1.1	even	1	trivial
50.3.c.a.43.1	yes	2	5.3	odd	4	inner
50.3.c.b.7.1	yes	2	5.4	even	2
50.3.c.b.43.1	yes	2	5.2	odd	4
400.3.p.a.193.1		2	20.3	even	4
400.3.p.a.257.1		2	4.3	odd	2
400.3.p.g.193.1		2	20.7	even	4
400.3.p.g.257.1		2	20.19	odd	2
450.3.g.c.307.1		2	15.14	odd	2
450.3.g.c.343.1		2	15.2	even	4
450.3.g.e.307.1		2	3.2	odd	2
450.3.g.e.343.1		2	15.8	even	4