Embedded newform 700.2.t.d.299.11

• Label: 700.2.t.d.299.11
• Level: $700$
• Weight: $2$
• Character: 700.299
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
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			299.11
Character	\(\chi\)	\(=\)	700.299
Dual form			700.2.t.d.199.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.942109 + 1.05472i) q^{2} +(0.780530 - 0.450639i) q^{3} +(-0.224860 + 1.98732i) q^{4} +(1.21064 + 0.398687i) q^{6} +(-1.30833 + 2.29962i) q^{7} +(-2.30790 + 1.63511i) q^{8} +(-1.09385 + 1.89460i) q^{9} +(-3.24107 + 1.87123i) q^{11} +(0.720054 + 1.65249i) q^{12} +2.41990 q^{13} +(-3.65805 + 0.786573i) q^{14} +(-3.89888 - 0.893735i) q^{16} +(-0.291859 - 0.505515i) q^{17} +(-3.02880 + 0.631220i) q^{18} +(3.07977 - 5.33433i) q^{19} +(0.0151060 + 2.38451i) q^{21} +(-5.02706 - 1.65551i) q^{22} +(-2.15605 + 3.73439i) q^{23} +(-1.06454 + 2.31628i) q^{24} +(2.27981 + 2.55232i) q^{26} +4.67556i q^{27} +(-4.27589 - 3.11717i) q^{28} +0.435463 q^{29} +(-1.26933 - 2.19854i) q^{31} +(-2.73053 - 4.95421i) q^{32} +(-1.68650 + 2.92110i) q^{33} +(0.258212 - 0.784080i) q^{34} +(-3.51922 - 2.59985i) q^{36} +(9.78676 + 5.65039i) q^{37} +(8.52769 - 1.77723i) q^{38} +(1.88881 - 1.09050i) q^{39} +7.35068i q^{41} +(-2.50075 + 2.26240i) q^{42} +5.80096 q^{43} +(-2.98995 - 6.86180i) q^{44} +(-5.96996 + 1.24418i) q^{46} +(10.0256 + 5.78826i) q^{47} +(-3.44594 + 1.05940i) q^{48} +(-3.57652 - 6.01735i) q^{49} +(-0.455610 - 0.263046i) q^{51} +(-0.544138 + 4.80912i) q^{52} +(2.69759 - 1.55746i) q^{53} +(-4.93139 + 4.40489i) q^{54} +(-0.740624 - 7.44657i) q^{56} -5.55147i q^{57} +(0.410254 + 0.459290i) q^{58} +(1.73534 + 3.00569i) q^{59} +(-8.99597 - 5.19383i) q^{61} +(1.12299 - 3.41004i) q^{62} +(-2.92575 - 4.99421i) q^{63} +(2.65284 - 7.54735i) q^{64} +(-4.66981 + 0.973217i) q^{66} +(4.92250 + 8.52602i) q^{67} +(1.07025 - 0.466348i) q^{68} +3.88640i q^{69} -9.96771i q^{71} +(-0.573383 - 6.16112i) q^{72} +(-4.89946 - 8.48612i) q^{73} +(3.26063 + 15.6456i) q^{74} +(9.90849 + 7.31997i) q^{76} +(-0.0627260 - 9.90142i) q^{77} +(2.92964 + 0.964785i) q^{78} +(0.397549 + 0.229525i) q^{79} +(-1.17456 - 2.03439i) q^{81} +(-7.75290 + 6.92515i) q^{82} +2.59747i q^{83} +(-4.74218 - 0.506159i) q^{84} +(5.46514 + 6.11837i) q^{86} +(0.339892 - 0.196236i) q^{87} +(4.42040 - 9.61812i) q^{88} +(8.55647 + 4.94008i) q^{89} +(-3.16604 + 5.56486i) q^{91} +(-6.93662 - 5.12447i) q^{92} +(-1.98149 - 1.14402i) q^{93} +(3.34020 + 16.0273i) q^{94} +(-4.36382 - 2.63643i) q^{96} -4.54044 q^{97} +(2.97713 - 9.44122i) q^{98} -8.18738i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q + 2 * q^4 + 16 * q^9 + 14 * q^12 + 8 * q^13 - 2 * q^14 - 14 * q^16 - 54 * q^18 - 12 * q^21 - 36 * q^24 + 30 * q^26 - 32 * q^28 + 40 * q^29 - 60 * q^32 + 24 * q^33 + 60 * q^36 + 60 * q^37 + 46 * q^38 - 78 * q^42 + 18 * q^44 + 2 * q^46 + 28 * q^48 + 16 * q^49 + 46 * q^52 + 12 * q^53 - 12 * q^54 - 4 * q^56 - 42 * q^58 + 24 * q^61 - 8 * q^62 - 4 * q^64 + 24 * q^66 + 4 * q^68 - 90 * q^72 + 24 * q^73 - 38 * q^74 - 20 * q^77 - 36 * q^81 - 8 * q^82 + 20 * q^84 + 28 * q^86 + 78 * q^88 + 60 * q^89 - 72 * q^93 - 18 * q^94 - 60 * q^96 + 48 * q^97 + 124 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(-1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.942109	+	1.05472i	0.666172	+	0.745798i
							\(3\)	0.780530	−	0.450639i	0.450639	−	0.260177i	−0.257461	−	0.966289i	\(-0.582886\pi\)
0.708100	+	0.706112i	\(0.249553\pi\)
\(5\)		0			0
							\(7\)	−1.30833	+	2.29962i	−0.494504	+	0.869175i
\(11\)	−3.24107	+	1.87123i	−0.977218	+	0.564197i								−0.901429	−	0.432927i	\(-0.857481\pi\)
							−0.0757892	+	0.997124i	\(0.524148\pi\)
\(13\)	2.41990			0.671161			0.335580	−	0.942012i	\(-0.391068\pi\)
							0.335580	+	0.942012i	\(0.391068\pi\)
\(17\)	−0.291859	−	0.505515i	−0.0707863	−	0.122605i	0.828460	−	0.560048i	\(-0.189218\pi\)
							−0.899246	+	0.437443i	\(0.855884\pi\)
\(19\)	3.07977	−	5.33433i	0.706549	−	1.22378i	−0.259581	−	0.965721i	\(-0.583584\pi\)
							0.966130	−	0.258057i	\(-0.0830822\pi\)
\(23\)	−2.15605	+	3.73439i	−0.449568	+	0.778674i	−0.998358	−	0.0572861i	\(-0.981755\pi\)
							0.548790	+	0.835960i	\(0.315089\pi\)
\(29\)	0.435463			0.0808634			0.0404317	−	0.999182i	\(-0.487127\pi\)
							0.0404317	+	0.999182i	\(0.487127\pi\)
\(31\)	−1.26933	−	2.19854i	−0.227978	−	0.394869i	0.729231	−	0.684268i	\(-0.239878\pi\)
							−0.957209	+	0.289399i	\(0.906545\pi\)
\(37\)	9.78676	+	5.65039i	1.60893	+	0.928918i	0.989609	+	0.143782i	\(0.0459265\pi\)
							0.619324	+	0.785136i	\(0.287407\pi\)
\(41\)			7.35068i			1.14798i	0.818861	+	0.573992i	\(0.194606\pi\)
							−0.818861	+	0.573992i	\(0.805394\pi\)
\(43\)	5.80096			0.884637			0.442319	−	0.896858i	\(-0.354156\pi\)
							0.442319	+	0.896858i	\(0.354156\pi\)
\(47\)	10.0256	+	5.78826i	1.46238	+	0.844305i	0.999121	−	0.0419181i	\(-0.0133469\pi\)
							0.463258	+	0.886223i	\(0.346680\pi\)
\(53\)	2.69759	−	1.55746i	0.370543	−	0.213933i	−0.303153	−	0.952942i	\(-0.598039\pi\)
							0.673696	+	0.739009i	\(0.264706\pi\)
\(59\)	1.73534	+	3.00569i	0.225922	+	0.391308i	0.956596	−	0.291419i	\(-0.0941273\pi\)
							−0.730674	+	0.682727i	\(0.760794\pi\)
\(61\)	−8.99597	−	5.19383i	−1.15182	−	0.665001i	−0.202487	−	0.979285i	\(-0.564902\pi\)
							−0.949329	+	0.314284i	\(0.898236\pi\)
\(67\)	4.92250	+	8.52602i	0.601379	+	1.04162i	0.992613	+	0.121327i	\(0.0387151\pi\)
							−0.391234	+	0.920291i	\(0.627952\pi\)
\(71\)		−	9.96771i		−	1.18295i	−0.806324	−	0.591475i	\(-0.798546\pi\)
							0.806324	−	0.591475i	\(-0.201454\pi\)
\(73\)	−4.89946	−	8.48612i	−0.573439	−	0.993225i	−0.996209	−	0.0869881i	\(-0.972276\pi\)
							0.422771	−	0.906237i	\(-0.361058\pi\)
\(79\)	0.397549	+	0.229525i	0.0447278	+	0.0258236i	0.522197	−	0.852825i	\(-0.325113\pi\)
							−0.477469	+	0.878648i	\(0.658446\pi\)
\(83\)			2.59747i			0.285109i	0.989787	+	0.142554i	\(0.0455316\pi\)
							−0.989787	+	0.142554i	\(0.954468\pi\)
\(89\)	8.55647	+	4.94008i	0.906984	+	0.523648i	0.879460	−	0.475973i	\(-0.157904\pi\)
							0.0275247	+	0.999621i	\(0.491237\pi\)
\(97\)	−4.54044			−0.461011			−0.230506	−	0.973071i	\(-0.574038\pi\)
							−0.230506	+	0.973071i	\(0.574038\pi\)

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	700.2.t.d.299.11		32
4.3	odd	2	inner	700.2.t.d.299.16		32
5.2	odd	4		700.2.p.c.551.4		32
5.3	odd	4		140.2.o.a.131.13	yes	32
5.4	even	2		700.2.t.c.299.6		32
7.3	odd	6		700.2.t.c.199.1		32
20.3	even	4		140.2.o.a.131.9	yes	32
20.7	even	4		700.2.p.c.551.8		32
20.19	odd	2		700.2.t.c.299.1		32
28.3	even	6		700.2.t.c.199.6		32
35.3	even	12		140.2.o.a.31.9	✓	32
35.13	even	4		980.2.o.f.411.13		32
35.17	even	12		700.2.p.c.451.8		32
35.18	odd	12		980.2.o.f.31.9		32
35.23	odd	12		980.2.g.a.391.5		32
35.24	odd	6	inner	700.2.t.d.199.16		32
35.33	even	12		980.2.g.a.391.6		32
140.3	odd	12		140.2.o.a.31.13	yes	32
140.23	even	12		980.2.g.a.391.8		32
140.59	even	6	inner	700.2.t.d.199.11		32
140.83	odd	4		980.2.o.f.411.9		32
140.87	odd	12		700.2.p.c.451.4		32
140.103	odd	12		980.2.g.a.391.7		32
140.123	even	12		980.2.o.f.31.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.o.a.31.9	✓	32	35.3	even	12
140.2.o.a.31.13	yes	32	140.3	odd	12
140.2.o.a.131.9	yes	32	20.3	even	4
140.2.o.a.131.13	yes	32	5.3	odd	4
700.2.p.c.451.4		32	140.87	odd	12
700.2.p.c.451.8		32	35.17	even	12
700.2.p.c.551.4		32	5.2	odd	4
700.2.p.c.551.8		32	20.7	even	4
700.2.t.c.199.1		32	7.3	odd	6
700.2.t.c.199.6		32	28.3	even	6
700.2.t.c.299.1		32	20.19	odd	2
700.2.t.c.299.6		32	5.4	even	2
700.2.t.d.199.11		32	140.59	even	6	inner
700.2.t.d.199.16		32	35.24	odd	6	inner
700.2.t.d.299.11		32	1.1	even	1	trivial
700.2.t.d.299.16		32	4.3	odd	2	inner
980.2.g.a.391.5		32	35.23	odd	12
980.2.g.a.391.6		32	35.33	even	12
980.2.g.a.391.7		32	140.103	odd	12
980.2.g.a.391.8		32	140.23	even	12
980.2.o.f.31.9		32	35.18	odd	12
980.2.o.f.31.13		32	140.123	even	12
980.2.o.f.411.9		32	140.83	odd	4
980.2.o.f.411.13		32	35.13	even	4