Embedded newform 700.4.a.d.1.1

• Label: 700.4.a.d.1.1
• Level: $700$
• Weight: $4$
• Character: 700.1
• Self dual: yes
• Analytic conductor: $41.301$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(41.3013370040\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	700.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-5.00000 q^{3} -7.00000 q^{7} -2.00000 q^{9} -65.0000 q^{11} +13.0000 q^{13} -113.000 q^{17} +16.0000 q^{19} +35.0000 q^{21} +186.000 q^{23} +145.000 q^{27} -57.0000 q^{29} -258.000 q^{31} +325.000 q^{33} -134.000 q^{37} -65.0000 q^{39} -414.000 q^{41} -284.000 q^{43} +419.000 q^{47} +49.0000 q^{49} +565.000 q^{51} -130.000 q^{53} -80.0000 q^{57} +334.000 q^{59} -56.0000 q^{61} +14.0000 q^{63} +534.000 q^{67} -930.000 q^{69} +952.000 q^{71} +502.000 q^{73} +455.000 q^{77} -371.000 q^{79} -671.000 q^{81} +1220.00 q^{83} +285.000 q^{87} +880.000 q^{89} -91.0000 q^{91} +1290.00 q^{93} -241.000 q^{97} +130.000 q^{99} +O(q^{100})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−5.00000	−0.962250	−0.481125	−	0.876652i	\(-0.659772\pi\)
−0.481125	+	0.876652i				\(0.659772\pi\)
\(5\)	0	0
			\(7\)	−7.00000	−0.377964
\(11\)	−65.0000	−1.78166				−0.890829	−	0.454339i	\(-0.849876\pi\)
			−0.890829	+	0.454339i	\(0.849876\pi\)
\(13\)	13.0000	0.277350	0.138675	−	0.990338i	\(-0.455716\pi\)
			0.138675	+	0.990338i	\(0.455716\pi\)
\(17\)	−113.000	−1.61215	−0.806074	−	0.591814i	\(-0.798412\pi\)
			−0.806074	+	0.591814i	\(0.798412\pi\)
\(19\)	16.0000	0.193192	0.0965961	−	0.995324i	\(-0.469204\pi\)
			0.0965961	+	0.995324i	\(0.469204\pi\)
\(23\)	186.000	1.68625	0.843124	−	0.537720i	\(-0.180714\pi\)
			0.843124	+	0.537720i	\(0.180714\pi\)
\(29\)	−57.0000	−0.364987	−0.182494	−	0.983207i	\(-0.558417\pi\)
			−0.182494	+	0.983207i	\(0.558417\pi\)
\(31\)	−258.000	−1.49478	−0.747390	−	0.664386i	\(-0.768693\pi\)
			−0.747390	+	0.664386i	\(0.768693\pi\)
\(37\)	−134.000	−0.595391	−0.297695	−	0.954661i	\(-0.596218\pi\)
			−0.297695	+	0.954661i	\(0.596218\pi\)
\(41\)	−414.000	−1.57697	−0.788487	−	0.615051i	\(-0.789135\pi\)
			−0.788487	+	0.615051i	\(0.789135\pi\)
\(43\)	−284.000	−1.00720	−0.503600	−	0.863937i	\(-0.667991\pi\)
			−0.503600	+	0.863937i	\(0.667991\pi\)
\(47\)	419.000	1.30037	0.650185	−	0.759776i	\(-0.274691\pi\)
			0.650185	+	0.759776i	\(0.274691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.4.a.d.1.1		1
5.2	odd	4		140.4.e.a.29.2	yes	2
5.3	odd	4		140.4.e.a.29.1	✓	2
5.4	even	2		700.4.a.l.1.1		1
15.2	even	4		1260.4.k.c.1009.2		2
15.8	even	4		1260.4.k.c.1009.1		2
20.3	even	4		560.4.g.a.449.2		2
20.7	even	4		560.4.g.a.449.1		2
35.13	even	4		980.4.e.c.589.2		2
35.27	even	4		980.4.e.c.589.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.e.a.29.1	✓	2	5.3	odd	4
140.4.e.a.29.2	yes	2	5.2	odd	4
560.4.g.a.449.1		2	20.7	even	4
560.4.g.a.449.2		2	20.3	even	4
700.4.a.d.1.1		1	1.1	even	1	trivial
700.4.a.l.1.1		1	5.4	even	2
980.4.e.c.589.1		2	35.27	even	4
980.4.e.c.589.2		2	35.13	even	4
1260.4.k.c.1009.1		2	15.8	even	4
1260.4.k.c.1009.2		2	15.2	even	4