Embedded newform 175.3.d.b.76.1

• Label: 175.3.d.b.76.1
• Level: $175$
• Weight: $3$
• Character: 175.76
• Analytic conductor: $4.768$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	175.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.76840462631\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-10}) \)
Defining polynomial:	\( x^{2} + 10 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			76.1
Root			\(-3.16228i\) of defining polynomial
Character	\(\chi\)	\(=\)	175.76
Dual form			175.3.d.b.76.2

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.00000 q^{2} -3.16228i q^{3} +5.00000 q^{4} +9.48683i q^{6} +(3.00000 - 6.32456i) q^{7} -3.00000 q^{8} -1.00000 q^{9} +14.0000 q^{11} -15.8114i q^{12} +3.16228i q^{13} +(-9.00000 + 18.9737i) q^{14} -11.0000 q^{16} +6.32456i q^{17} +3.00000 q^{18} -28.4605i q^{19} +(-20.0000 - 9.48683i) q^{21} -42.0000 q^{22} -12.0000 q^{23} +9.48683i q^{24} -9.48683i q^{26} -25.2982i q^{27} +(15.0000 - 31.6228i) q^{28} -14.0000 q^{29} -37.9473i q^{31} +45.0000 q^{32} -44.2719i q^{33} -18.9737i q^{34} -5.00000 q^{36} -18.0000 q^{37} +85.3815i q^{38} +10.0000 q^{39} +18.9737i q^{41} +(60.0000 + 28.4605i) q^{42} -42.0000 q^{43} +70.0000 q^{44} +36.0000 q^{46} +44.2719i q^{47} +34.7851i q^{48} +(-31.0000 - 37.9473i) q^{49} +20.0000 q^{51} +15.8114i q^{52} +54.0000 q^{53} +75.8947i q^{54} +(-9.00000 + 18.9737i) q^{56} -90.0000 q^{57} +42.0000 q^{58} +9.48683i q^{59} +66.4078i q^{61} +113.842i q^{62} +(-3.00000 + 6.32456i) q^{63} -91.0000 q^{64} +132.816i q^{66} +102.000 q^{67} +31.6228i q^{68} +37.9473i q^{69} -16.0000 q^{71} +3.00000 q^{72} -63.2456i q^{73} +54.0000 q^{74} -142.302i q^{76} +(42.0000 - 88.5438i) q^{77} -30.0000 q^{78} +76.0000 q^{79} -89.0000 q^{81} -56.9210i q^{82} -72.7324i q^{83} +(-100.000 - 47.4342i) q^{84} +126.000 q^{86} +44.2719i q^{87} -42.0000 q^{88} +56.9210i q^{89} +(20.0000 + 9.48683i) q^{91} -60.0000 q^{92} -120.000 q^{93} -132.816i q^{94} -142.302i q^{96} -69.5701i q^{97} +(93.0000 + 113.842i) q^{98} -14.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^2 + 10 * q^4 + 6 * q^7 - 6 * q^8 - 2 * q^9 + 28 * q^11 - 18 * q^14 - 22 * q^16 + 6 * q^18 - 40 * q^21 - 84 * q^22 - 24 * q^23 + 30 * q^28 - 28 * q^29 + 90 * q^32 - 10 * q^36 - 36 * q^37 + 20 * q^39 + 120 * q^42 - 84 * q^43 + 140 * q^44 + 72 * q^46 - 62 * q^49 + 40 * q^51 + 108 * q^53 - 18 * q^56 - 180 * q^57 + 84 * q^58 - 6 * q^63 - 182 * q^64 + 204 * q^67 - 32 * q^71 + 6 * q^72 + 108 * q^74 + 84 * q^77 - 60 * q^78 + 152 * q^79 - 178 * q^81 - 200 * q^84 + 252 * q^86 - 84 * q^88 + 40 * q^91 - 120 * q^92 - 240 * q^93 + 186 * q^98 - 28 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00000			−1.50000			−0.750000	−	0.661438i	\(-0.769947\pi\)
							−0.750000	+	0.661438i	\(0.769947\pi\)
\(3\)		−	3.16228i		−	1.05409i	−0.849837	−	0.527046i	\(-0.823299\pi\)
							0.849837	−	0.527046i	\(-0.176701\pi\)
\(5\)		0			0
							\(7\)	3.00000	−	6.32456i	0.428571	−	0.903508i
\(11\)	14.0000			1.27273										0.636364	−	0.771389i	\(-0.280438\pi\)
							0.636364	+	0.771389i	\(0.280438\pi\)
\(13\)			3.16228i			0.243252i	0.992576	+	0.121626i	\(0.0388109\pi\)
							−0.992576	+	0.121626i	\(0.961189\pi\)
\(17\)			6.32456i			0.372033i	0.982547	+	0.186016i	\(0.0595577\pi\)
							−0.982547	+	0.186016i	\(0.940442\pi\)
\(19\)		−	28.4605i		−	1.49792i	−0.662615	−	0.748960i	\(-0.730553\pi\)
							0.662615	−	0.748960i	\(-0.269447\pi\)
\(23\)	−12.0000			−0.521739			−0.260870	−	0.965374i	\(-0.584009\pi\)
							−0.260870	+	0.965374i	\(0.584009\pi\)
\(29\)	−14.0000			−0.482759			−0.241379	−	0.970431i	\(-0.577600\pi\)
							−0.241379	+	0.970431i	\(0.577600\pi\)
\(31\)		−	37.9473i		−	1.22411i	−0.790816	−	0.612054i	\(-0.790344\pi\)
							0.790816	−	0.612054i	\(-0.209656\pi\)
\(37\)	−18.0000			−0.486486			−0.243243	−	0.969965i	\(-0.578211\pi\)
							−0.243243	+	0.969965i	\(0.578211\pi\)
\(41\)			18.9737i			0.462772i	0.972862	+	0.231386i	\(0.0743261\pi\)
							−0.972862	+	0.231386i	\(0.925674\pi\)
\(43\)	−42.0000			−0.976744			−0.488372	−	0.872635i	\(-0.662409\pi\)
							−0.488372	+	0.872635i	\(0.662409\pi\)
\(47\)			44.2719i			0.941955i	0.882145	+	0.470978i	\(0.156099\pi\)
							−0.882145	+	0.470978i	\(0.843901\pi\)
\(53\)	54.0000			1.01887			0.509434	−	0.860510i	\(-0.329855\pi\)
							0.509434	+	0.860510i	\(0.329855\pi\)
\(59\)			9.48683i			0.160794i	0.996763	+	0.0803969i	\(0.0256188\pi\)
							−0.996763	+	0.0803969i	\(0.974381\pi\)
\(61\)			66.4078i			1.08865i	0.838873	+	0.544326i	\(0.183215\pi\)
							−0.838873	+	0.544326i	\(0.816785\pi\)
\(67\)	102.000			1.52239			0.761194	−	0.648524i	\(-0.224613\pi\)
							0.761194	+	0.648524i	\(0.224613\pi\)
\(71\)	−16.0000			−0.225352			−0.112676	−	0.993632i	\(-0.535942\pi\)
							−0.112676	+	0.993632i	\(0.535942\pi\)
\(73\)		−	63.2456i		−	0.866377i	−0.901303	−	0.433189i	\(-0.857388\pi\)
							0.901303	−	0.433189i	\(-0.142612\pi\)
\(79\)	76.0000			0.962025			0.481013	−	0.876714i	\(-0.340269\pi\)
							0.481013	+	0.876714i	\(0.340269\pi\)
\(83\)		−	72.7324i		−	0.876294i	−0.898903	−	0.438147i	\(-0.855635\pi\)
							0.898903	−	0.438147i	\(-0.144365\pi\)
\(89\)			56.9210i			0.639562i	0.947492	+	0.319781i	\(0.103609\pi\)
							−0.947492	+	0.319781i	\(0.896391\pi\)
\(97\)		−	69.5701i		−	0.717218i	−0.933488	−	0.358609i	\(-0.883251\pi\)
							0.933488	−	0.358609i	\(-0.116749\pi\)

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	175.3.d.b.76.1		2
5.2	odd	4		35.3.c.c.34.1	✓	4
5.3	odd	4		35.3.c.c.34.4	yes	4
5.4	even	2		175.3.d.h.76.2		2
7.6	odd	2	inner	175.3.d.b.76.2		2
15.2	even	4		315.3.e.c.244.3		4
15.8	even	4		315.3.e.c.244.2		4
20.3	even	4		560.3.p.f.209.1		4
20.7	even	4		560.3.p.f.209.3		4
35.2	odd	12		245.3.i.c.129.2		8
35.3	even	12		245.3.i.c.19.2		8
35.12	even	12		245.3.i.c.129.1		8
35.13	even	4		35.3.c.c.34.3	yes	4
35.17	even	12		245.3.i.c.19.3		8
35.18	odd	12		245.3.i.c.19.1		8
35.23	odd	12		245.3.i.c.129.3		8
35.27	even	4		35.3.c.c.34.2	yes	4
35.32	odd	12		245.3.i.c.19.4		8
35.33	even	12		245.3.i.c.129.4		8
35.34	odd	2		175.3.d.h.76.1		2
105.62	odd	4		315.3.e.c.244.4		4
105.83	odd	4		315.3.e.c.244.1		4
140.27	odd	4		560.3.p.f.209.2		4
140.83	odd	4		560.3.p.f.209.4		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.c.c.34.1	✓	4	5.2	odd	4
35.3.c.c.34.2	yes	4	35.27	even	4
35.3.c.c.34.3	yes	4	35.13	even	4
35.3.c.c.34.4	yes	4	5.3	odd	4
175.3.d.b.76.1		2	1.1	even	1	trivial
175.3.d.b.76.2		2	7.6	odd	2	inner
175.3.d.h.76.1		2	35.34	odd	2
175.3.d.h.76.2		2	5.4	even	2
245.3.i.c.19.1		8	35.18	odd	12
245.3.i.c.19.2		8	35.3	even	12
245.3.i.c.19.3		8	35.17	even	12
245.3.i.c.19.4		8	35.32	odd	12
245.3.i.c.129.1		8	35.12	even	12
245.3.i.c.129.2		8	35.2	odd	12
245.3.i.c.129.3		8	35.23	odd	12
245.3.i.c.129.4		8	35.33	even	12
315.3.e.c.244.1		4	105.83	odd	4
315.3.e.c.244.2		4	15.8	even	4
315.3.e.c.244.3		4	15.2	even	4
315.3.e.c.244.4		4	105.62	odd	4
560.3.p.f.209.1		4	20.3	even	4
560.3.p.f.209.2		4	140.27	odd	4
560.3.p.f.209.3		4	20.7	even	4
560.3.p.f.209.4		4	140.83	odd	4