Newform orbit 215.2.r.a

• Label: 215.2.r.a
• Level: $215$
• Weight: $2$
• Character orbit: 215.r
• Analytic conductor: $1.717$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 215 = 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	215.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.71678364346\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(20\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 14 q^{2} - 14 q^{3} - 14 q^{5} - 40 q^{6} - 14 q^{8}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−0.839494	+	2.39914i	0.538059	+	1.53769i	−3.48745	−	2.78114i	−0.127355	+	2.23244i	−4.14082			−1.34946	+	1.34946i	5.29568	−	3.32749i	0.270526	−	0.215737i	−5.24901	−	2.17966i
2.2	−0.800177	+	2.28677i	−0.424033	−	1.21182i	−3.02539	−	2.41266i	−2.22644	−	0.207268i	3.11045			2.04255	−	2.04255i	3.83529	−	2.40987i	1.05680	−	0.842767i	2.25552	−	4.92551i
2.3	−0.734553	+	2.09923i	−0.185233	−	0.529365i	−2.30355	−	1.83702i	2.20488	−	0.372173i	1.24732			3.08552	−	3.08552i	1.78211	−	1.11978i	2.09958	−	1.67436i	−0.838323	+	4.90193i
2.4	−0.611507	+	1.74759i	−0.544019	−	1.55472i	−1.11645	−	0.890342i	−1.03361	−	1.98284i	3.04967			−2.58784	+	2.58784i	−0.896728	+	0.563451i	0.224306	−	0.178878i	4.09724	−	0.593795i
2.5	−0.524511	+	1.49897i	0.495990	+	1.41746i	−0.408122	−	0.325467i	1.89592	−	1.18553i	−2.38487			−2.19429	+	2.19429i	−1.98741	+	1.24877i	0.582312	−	0.464378i	0.782639	+	3.46374i
2.6	−0.473261	+	1.35250i	0.873400	+	2.49603i	−0.0416212	−	0.0331918i	−2.10784	−	0.746326i	−3.78923			0.567558	−	0.567558i	−2.36197	+	1.48413i	−3.12186	+	2.48960i	2.00697	−	2.49765i
2.7	−0.394535	+	1.12752i	−0.555408	−	1.58727i	0.448030	+	0.357292i	−0.893566	+	2.04977i	2.00879			−1.06378	+	1.06378i	−2.60252	+	1.63527i	0.134562	−	0.107309i	−1.95860	−	1.81621i
2.8	−0.270849	+	0.774042i	0.524408	+	1.49867i	1.03788	+	0.827683i	0.380872	+	2.20339i	−1.30207			2.30836	−	2.30836i	−2.31050	+	1.45178i	0.374478	−	0.298636i	−1.80868	−	0.301975i
2.9	−0.112897	+	0.322642i	−0.714526	−	2.04200i	1.47231	+	1.17413i	2.23581	−	0.0337858i	0.739502			−0.158779	+	0.158779i	−1.12390	+	0.706195i	−1.31372	+	1.04766i	−0.241516	+	0.725181i
2.10	0.112062	−	0.320255i	−0.971532	−	2.77648i	1.47366	+	1.17520i	−2.13952	−	0.649977i	−0.998053			2.20332	−	2.20332i	1.11608	−	0.701282i	−4.41946	+	3.52441i	−0.447917	+	0.612353i
2.11	0.135189	−	0.386349i	0.0420108	+	0.120060i	1.43267	+	1.14252i	−0.422167	−	2.19585i	0.0520645			−0.279424	+	0.279424i	1.32825	−	0.834597i	2.33284	−	1.86038i	−0.905438	−	0.133752i
2.12	0.154217	−	0.440728i	0.466565	+	1.33337i	1.39320	+	1.11104i	−2.18249	+	0.486555i	0.659604			−3.50396	+	3.50396i	1.49525	−	0.939526i	0.785313	−	0.626266i	−0.122139	+	1.03692i
2.13	0.166349	−	0.475399i	1.01635	+	2.90457i	1.36533	+	1.08882i	1.33559	−	1.79338i	1.54990			0.184431	−	0.184431i	1.59767	−	1.00388i	−5.05807	+	4.03368i	−0.630395	−	0.933265i
2.14	0.332190	−	0.949343i	−0.172756	−	0.493708i	0.772761	+	0.616256i	−1.13785	+	1.92492i	−0.526086			1.39389	−	1.39389i	2.54499	−	1.59912i	2.13159	−	1.69989i	1.44942	+	1.71965i
2.15	0.549801	−	1.57124i	−0.0697227	−	0.199256i	−0.602851	−	0.480758i	1.98466	+	1.03012i	−0.351412			−1.23993	+	1.23993i	1.73217	−	1.08840i	2.31065	−	1.84268i	2.70973	−	2.55201i
2.16	0.592514	−	1.69331i	−0.742259	−	2.12125i	−0.952555	−	0.759637i	−0.0565456	−	2.23535i	−4.03173			−2.09032	+	2.09032i	1.18731	−	0.746037i	−1.60327	+	1.27857i	−3.81864	−	1.22873i
2.17	0.685541	−	1.95916i	0.611678	+	1.74807i	−1.80469	−	1.43919i	−1.85501	−	1.24857i	3.84409			2.62864	−	2.62864i	−0.541805	+	0.340438i	−0.336121	+	0.268047i	−3.71784	+	2.77833i
2.18	0.694028	−	1.98342i	0.945806	+	2.70296i	−1.88860	−	1.50611i	0.184637	+	2.22843i	6.01751			−0.996031	+	0.996031i	−0.739486	+	0.464650i	−4.06594	+	3.24248i	4.54805	+	1.18038i
2.19	0.813550	−	2.32499i	−0.996871	−	2.84889i	−3.18006	−	2.53601i	1.30450	+	1.81612i	−7.43465			2.75351	−	2.75351i	−4.31200	+	2.70941i	−4.77694	+	3.80948i	5.28373	−	1.55545i
2.20	0.931665	−	2.66255i	0.267410	+	0.764214i	−4.65749	−	3.71422i	1.60057	−	1.56147i	2.28389			−0.836210	+	0.836210i	−9.45155	+	5.93880i	1.83298	−	1.46175i	−2.66628	−	5.71635i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
43.f	odd	14	1	inner
215.r	even	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	215.2.r.a	✓	240
5.c	odd	4	1	inner	215.2.r.a	✓	240
43.f	odd	14	1	inner	215.2.r.a	✓	240
215.r	even	28	1	inner	215.2.r.a	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
215.2.r.a	✓	240	1.a	even	1	1	trivial
215.2.r.a	✓	240	5.c	odd	4	1	inner
215.2.r.a	✓	240	43.f	odd	14	1	inner
215.2.r.a	✓	240	215.r	even	28	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(215, [\chi])\).