Embedded newform 980.2.o.f.31.9

• Label: 980.2.o.f.31.9
• Level: $980$
• Weight: $2$
• Character: 980.31
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.9
Character	\(\chi\)	\(=\)	980.31
Dual form			980.2.o.f.411.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.288532 + 1.38447i) q^{2} +(0.450639 - 0.780530i) q^{3} +(-1.83350 + 0.798926i) q^{4} +(0.866025 - 0.500000i) q^{5} +(1.21064 + 0.398687i) q^{6} +(-1.63511 - 2.30790i) q^{8} +(1.09385 + 1.89460i) q^{9} +(0.942109 + 1.05472i) q^{10} +(3.24107 + 1.87123i) q^{11} +(-0.202661 + 1.79113i) q^{12} +2.41990i q^{13} -0.901278i q^{15} +(2.72344 - 2.92966i) q^{16} +(0.505515 + 0.291859i) q^{17} +(-2.30740 + 2.06105i) q^{18} +(-3.07977 - 5.33433i) q^{19} +(-1.18839 + 1.60864i) q^{20} +(-1.65551 + 5.02706i) q^{22} +(3.73439 - 2.15605i) q^{23} +(-2.53823 + 0.236220i) q^{24} +(0.500000 - 0.866025i) q^{25} +(-3.35028 + 0.698219i) q^{26} +4.67556 q^{27} -0.435463 q^{29} +(1.24779 - 0.260047i) q^{30} +(-1.26933 + 2.19854i) q^{31} +(4.84181 + 2.92521i) q^{32} +(2.92110 - 1.68650i) q^{33} +(-0.258212 + 0.784080i) q^{34} +(-3.51922 - 2.59985i) q^{36} +(5.65039 + 9.78676i) q^{37} +(6.49659 - 5.80297i) q^{38} +(1.88881 + 1.09050i) q^{39} +(-2.57000 - 1.18115i) q^{40} +7.35068i q^{41} +5.80096i q^{43} +(-7.43747 - 0.841528i) q^{44} +(1.89460 + 1.09385i) q^{45} +(4.06247 + 4.54805i) q^{46} +(5.78826 + 10.0256i) q^{47} +(-1.05940 - 3.44594i) q^{48} +(1.34325 + 0.442358i) q^{50} +(0.455610 - 0.263046i) q^{51} +(-1.93332 - 4.43689i) q^{52} +(1.55746 - 2.69759i) q^{53} +(1.34905 + 6.47316i) q^{54} +3.74246 q^{55} -5.55147 q^{57} +(-0.125645 - 0.602884i) q^{58} +(-1.73534 + 3.00569i) q^{59} +(0.720054 + 1.65249i) q^{60} +(8.99597 - 5.19383i) q^{61} +(-3.41004 - 1.12299i) q^{62} +(-2.65284 + 7.54735i) q^{64} +(1.20995 + 2.09570i) q^{65} +(3.17773 + 3.55756i) q^{66} +(-8.52602 - 4.92250i) q^{67} +(-1.16004 - 0.131255i) q^{68} -3.88640i q^{69} -9.96771i q^{71} +(2.58400 - 5.62238i) q^{72} +(-8.48612 - 4.89946i) q^{73} +(-11.9191 + 10.6466i) q^{74} +(-0.450639 - 0.780530i) q^{75} +(9.90849 + 7.31997i) q^{76} +(-0.964785 + 2.92964i) q^{78} +(0.397549 - 0.229525i) q^{79} +(0.893735 - 3.89888i) q^{80} +(-1.17456 + 2.03439i) q^{81} +(-10.1768 + 2.12091i) q^{82} -2.59747 q^{83} +0.583719 q^{85} +(-8.03123 + 1.67376i) q^{86} +(-0.196236 + 0.339892i) q^{87} +(-0.980878 - 10.5397i) q^{88} +(8.55647 - 4.94008i) q^{89} +(-0.967745 + 2.93862i) q^{90} +(-5.12447 + 6.93662i) q^{92} +(1.14402 + 1.98149i) q^{93} +(-12.2100 + 10.9064i) q^{94} +(-5.33433 - 3.07977i) q^{95} +(4.46512 - 2.46097i) q^{96} +4.54044i q^{97} +8.18738i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 - 16 * q^9 + 30 * q^12 - 14 * q^16 - 8 * q^22 - 36 * q^24 + 16 * q^25 - 30 * q^26 - 40 * q^29 + 2 * q^32 + 60 * q^36 + 8 * q^37 + 60 * q^38 - 18 * q^44 - 12 * q^45 + 2 * q^46 + 4 * q^50 + 36 * q^52 - 8 * q^53 - 12 * q^54 + 48 * q^57 + 2 * q^58 + 14 * q^60 - 24 * q^61 + 4 * q^64 + 4 * q^65 - 24 * q^66 - 60 * q^68 + 4 * q^72 + 72 * q^73 + 38 * q^74 + 120 * q^78 - 36 * q^81 - 42 * q^82 + 28 * q^86 + 4 * q^88 + 60 * q^89 - 4 * q^92 - 8 * q^93 - 18 * q^94 + 60 * q^96

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.288532	+	1.38447i	0.204023	+	0.978966i
							\(3\)	0.450639	−	0.780530i	0.260177	−	0.450639i	−0.706112	−	0.708100i	\(-0.749553\pi\)
0.966289	+	0.257461i	\(0.0828859\pi\)
\(5\)	0.866025	−	0.500000i	0.387298	−	0.223607i
							\(7\)		0			0
\(11\)	3.24107	+	1.87123i	0.977218	+	0.564197i								0.901429	−	0.432927i	\(-0.142519\pi\)
							0.0757892	+	0.997124i	\(0.475852\pi\)
\(13\)			2.41990i			0.671161i	0.942012	+	0.335580i	\(0.108932\pi\)
							−0.942012	+	0.335580i	\(0.891068\pi\)
\(17\)	0.505515	+	0.291859i	0.122605	+	0.0707863i	0.560048	−	0.828460i	\(-0.310782\pi\)
							−0.437443	+	0.899246i	\(0.644116\pi\)
\(19\)	−3.07977	−	5.33433i	−0.706549	−	1.22378i	−0.966130	−	0.258057i	\(-0.916918\pi\)
							0.259581	−	0.965721i	\(-0.416416\pi\)
\(23\)	3.73439	−	2.15605i	0.778674	−	0.449568i	−0.0572861	−	0.998358i	\(-0.518245\pi\)
							0.835960	+	0.548790i	\(0.184911\pi\)
\(29\)	−0.435463			−0.0808634			−0.0404317	−	0.999182i	\(-0.512873\pi\)
							−0.0404317	+	0.999182i	\(0.512873\pi\)
\(31\)	−1.26933	+	2.19854i	−0.227978	+	0.394869i	−0.957209	−	0.289399i	\(-0.906545\pi\)
							0.729231	+	0.684268i	\(0.239878\pi\)
\(37\)	5.65039	+	9.78676i	0.928918	+	1.60893i	0.785136	+	0.619324i	\(0.212593\pi\)
							0.143782	+	0.989609i	\(0.454073\pi\)
\(41\)			7.35068i			1.14798i	0.818861	+	0.573992i	\(0.194606\pi\)
							−0.818861	+	0.573992i	\(0.805394\pi\)
\(43\)			5.80096i			0.884637i	0.896858	+	0.442319i	\(0.145844\pi\)
							−0.896858	+	0.442319i	\(0.854156\pi\)
\(47\)	5.78826	+	10.0256i	0.844305	+	1.46238i	0.886223	+	0.463258i	\(0.153320\pi\)
							−0.0419181	+	0.999121i	\(0.513347\pi\)
\(53\)	1.55746	−	2.69759i	0.213933	−	0.370543i	−0.739009	−	0.673696i	\(-0.764706\pi\)
							0.952942	+	0.303153i	\(0.0980393\pi\)
\(59\)	−1.73534	+	3.00569i	−0.225922	+	0.391308i	−0.956596	−	0.291419i	\(-0.905873\pi\)
							0.730674	+	0.682727i	\(0.239206\pi\)
\(61\)	8.99597	−	5.19383i	1.15182	−	0.665001i	0.202487	−	0.979285i	\(-0.435098\pi\)
							0.949329	+	0.314284i	\(0.101764\pi\)
\(67\)	−8.52602	−	4.92250i	−1.04162	−	0.601379i	−0.121327	−	0.992613i	\(-0.538715\pi\)
							−0.920291	+	0.391234i	\(0.872048\pi\)
\(71\)		−	9.96771i		−	1.18295i	−0.806324	−	0.591475i	\(-0.798546\pi\)
							0.806324	−	0.591475i	\(-0.201454\pi\)
\(73\)	−8.48612	−	4.89946i	−0.993225	−	0.573439i	−0.0869881	−	0.996209i	\(-0.527724\pi\)
							−0.906237	+	0.422771i	\(0.861058\pi\)
\(79\)	0.397549	−	0.229525i	0.0447278	−	0.0258236i	−0.477469	−	0.878648i	\(-0.658446\pi\)
							0.522197	+	0.852825i	\(0.325113\pi\)
\(83\)	−2.59747			−0.285109			−0.142554	−	0.989787i	\(-0.545532\pi\)
							−0.142554	+	0.989787i	\(0.545532\pi\)
\(89\)	8.55647	−	4.94008i	0.906984	−	0.523648i	0.0275247	−	0.999621i	\(-0.491237\pi\)
							0.879460	+	0.475973i	\(0.157904\pi\)
\(97\)			4.54044i			0.461011i	0.973071	+	0.230506i	\(0.0740380\pi\)
							−0.973071	+	0.230506i	\(0.925962\pi\)

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	980.2.o.f.31.9		32
4.3	odd	2	inner	980.2.o.f.31.13		32
7.2	even	3		140.2.o.a.131.13	yes	32
7.3	odd	6		980.2.g.a.391.6		32
7.4	even	3		980.2.g.a.391.5		32
7.5	odd	6	inner	980.2.o.f.411.13		32
7.6	odd	2		140.2.o.a.31.9	✓	32
28.3	even	6		980.2.g.a.391.7		32
28.11	odd	6		980.2.g.a.391.8		32
28.19	even	6	inner	980.2.o.f.411.9		32
28.23	odd	6		140.2.o.a.131.9	yes	32
28.27	even	2		140.2.o.a.31.13	yes	32
35.2	odd	12		700.2.t.d.299.11		32
35.9	even	6		700.2.p.c.551.4		32
35.13	even	4		700.2.t.d.199.16		32
35.23	odd	12		700.2.t.c.299.6		32
35.27	even	4		700.2.t.c.199.1		32
35.34	odd	2		700.2.p.c.451.8		32
140.23	even	12		700.2.t.c.299.1		32
140.27	odd	4		700.2.t.c.199.6		32
140.79	odd	6		700.2.p.c.551.8		32
140.83	odd	4		700.2.t.d.199.11		32
140.107	even	12		700.2.t.d.299.16		32
140.139	even	2		700.2.p.c.451.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.o.a.31.9	✓	32	7.6	odd	2
140.2.o.a.31.13	yes	32	28.27	even	2
140.2.o.a.131.9	yes	32	28.23	odd	6
140.2.o.a.131.13	yes	32	7.2	even	3
700.2.p.c.451.4		32	140.139	even	2
700.2.p.c.451.8		32	35.34	odd	2
700.2.p.c.551.4		32	35.9	even	6
700.2.p.c.551.8		32	140.79	odd	6
700.2.t.c.199.1		32	35.27	even	4
700.2.t.c.199.6		32	140.27	odd	4
700.2.t.c.299.1		32	140.23	even	12
700.2.t.c.299.6		32	35.23	odd	12
700.2.t.d.199.11		32	140.83	odd	4
700.2.t.d.199.16		32	35.13	even	4
700.2.t.d.299.11		32	35.2	odd	12
700.2.t.d.299.16		32	140.107	even	12
980.2.g.a.391.5		32	7.4	even	3
980.2.g.a.391.6		32	7.3	odd	6
980.2.g.a.391.7		32	28.3	even	6
980.2.g.a.391.8		32	28.11	odd	6
980.2.o.f.31.9		32	1.1	even	1	trivial
980.2.o.f.31.13		32	4.3	odd	2	inner
980.2.o.f.411.9		32	28.19	even	6	inner
980.2.o.f.411.13		32	7.5	odd	6	inner