Newform orbit 197.4.e.a

• Label: 197.4.e.a
• Level: $197$
• Weight: $4$
• Character orbit: 197.e
• Analytic conductor: $11.623$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 197 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	197.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.6233762711\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(48\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 7 q^{2} - 7 q^{3} + 159 q^{4} - 7 q^{5} + 26 q^{6} - 53 q^{7} + 49 q^{8} + 391 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
6.1	−5.28937	+	1.20727i	−2.14603	−	0.489818i	19.3122	−	9.30029i	−4.36572	+	9.06550i	11.9425			5.60970	−	24.5777i	−56.9878	+	45.4462i	−19.9606	−	9.61253i	12.1474	−	53.2214i
6.2	−5.05771	+	1.15439i	−0.802239	−	0.183106i	17.0401	−	8.20607i	5.54130	−	11.5066i	4.26887			−5.20062	+	22.7854i	−44.2630	+	35.2986i	−23.7161	−	11.4211i	−14.7432	+	64.5940i
6.3	−4.98986	+	1.13890i	5.41261	+	1.23539i	16.3939	−	7.89487i	−8.43757	+	17.5208i	−28.4152			−6.51769	+	28.5559i	−40.7992	+	32.5362i	3.44396	+	1.65853i	22.1478	−	97.0359i
6.4	−4.93087	+	1.12544i	8.18390	+	1.86792i	15.8391	−	7.62772i	3.18930	−	6.62265i	−42.4560			1.15067	−	5.04140i	−37.8821	+	30.2100i	39.1609	+	18.8589i	−8.27264	+	36.2448i
6.5	−4.65379	+	1.06220i	−8.36922	−	1.91022i	13.3217	−	6.41540i	3.98034	−	8.26527i	40.9776			1.20188	−	5.26576i	−25.3257	+	20.1966i	42.0688	+	20.2593i	−9.74433	+	42.6927i
6.6	−4.35477	+	0.993947i	−4.58291	−	1.04602i	10.7683	−	5.18574i	−2.83248	+	5.88170i	20.9972			−1.18458	+	5.18997i	−13.8011	+	11.0060i	−4.41721	−	2.12722i	6.48867	−	28.4287i
6.7	−4.23718	+	0.967109i	5.25614	+	1.19968i	9.81066	−	4.72457i	0.445098	−	0.924256i	−23.4315			3.51884	−	15.4170i	−9.81675	+	7.82859i	1.86164	+	0.896520i	−0.992106	+	4.34670i
6.8	−3.71046	+	0.846888i	−1.81491	−	0.414242i	5.84252	−	2.81361i	−5.04290	+	10.4717i	7.08497			−0.120547	+	0.528152i	4.50883	−	3.59567i	−21.2038	−	10.2112i	9.84312	−	43.1255i
6.9	−3.46189	+	0.790154i	−6.86422	−	1.56671i	4.15258	−	1.99978i	−5.36100	+	11.1322i	25.0011			−6.97041	+	30.5394i	9.41411	−	7.50750i	20.3367	+	9.79366i	9.76302	−	42.7746i
6.10	−3.45017	+	0.787478i	−1.28875	−	0.294149i	4.07577	−	1.96279i	8.46205	−	17.5716i	4.67804			7.50032	−	32.8610i	9.61810	−	7.67018i	−22.7518	−	10.9567i	−15.3582	+	67.2887i
6.11	−3.42780	+	0.782374i	1.54660	+	0.353002i	3.92998	−	1.89258i	5.76545	−	11.9721i	−5.57762			−3.44993	+	15.1151i	10.0006	−	7.97521i	−22.0588	−	10.6230i	−10.3962	+	45.5487i
6.12	−3.40114	+	0.776287i	4.02676	+	0.919081i	3.75735	−	1.80945i	−4.50770	+	9.36034i	−14.4090			2.93112	−	12.8421i	10.4454	−	8.32991i	−8.95610	−	4.31303i	8.06500	−	35.3351i
6.13	−2.92366	+	0.667307i	5.97020	+	1.36266i	0.894767	−	0.430897i	4.38015	−	9.09547i	−18.3642			−4.77484	+	20.9199i	16.4283	−	13.1011i	9.46029	+	4.55583i	−6.73661	+	29.5150i
6.14	−2.77864	+	0.634207i	−8.80685	−	2.01011i	0.110876	−	0.0533948i	−8.61001	+	17.8789i	25.7459			7.61772	−	33.3754i	17.5522	−	13.9974i	49.1939	+	23.6905i	12.5852	−	55.1395i
6.15	−2.31003	+	0.527249i	9.83877	+	2.24564i	−2.14950	+	1.03515i	−6.73449	+	13.9843i	−23.9119			0.275979	−	1.20914i	19.2396	−	15.3431i	67.4324	+	32.4737i	8.18365	−	35.8549i
6.16	−2.17232	+	0.495818i	−4.25639	−	0.971492i	−2.73461	+	1.31692i	1.77717	−	3.69033i	9.72791			−2.18202	+	9.56005i	19.2240	−	15.3306i	−7.15314	−	3.44477i	−2.03085	+	8.89772i
6.17	−2.00454	+	0.457523i	−8.04978	−	1.83731i	−3.39891	+	1.63683i	4.47490	−	9.29222i	16.9767			−0.638388	+	2.79696i	18.9245	−	15.0918i	37.0970	+	17.8650i	−4.71870	+	20.6740i
6.18	−1.77835	+	0.405896i	3.55906	+	0.812332i	−4.20998	+	2.02742i	−2.71301	+	5.63362i	−6.65897			−6.79789	+	29.7835i	18.0729	−	14.4126i	−12.3191	−	5.93258i	2.53801	−	11.1197i
6.19	−1.69384	+	0.386608i	8.91279	+	2.03429i	−4.48812	+	2.16137i	8.48300	−	17.6151i	−15.8833			0.935927	−	4.10056i	17.6334	−	14.0622i	50.9733	+	24.5474i	−7.55869	+	33.1168i
6.20	−1.38176	+	0.315378i	−3.51241	−	0.801685i	−5.39795	+	2.59951i	−1.22448	+	2.54267i	5.10615			5.06822	−	22.2053i	15.5035	−	12.3637i	−12.6318	−	6.08317i	0.890043	−	3.89953i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
197.e	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	197.4.e.a	✓	288
197.e	even	14	1	inner	197.4.e.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
197.4.e.a	✓	288	1.a	even	1	1	trivial
197.4.e.a	✓	288	197.e	even	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(197, [\chi])\).