Newform orbit 197.4.g.a

• Label: 197.4.g.a
• Level: $197$
• Weight: $4$
• Character orbit: 197.g
• Analytic conductor: $11.623$
• Analytic rank: $0$
• Dimension: $2058$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 197 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	197.g (of order \(49\), degree \(42\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.6233762711\)
Analytic rank:	\(0\)
Dimension:	\(2058\)
Relative dimension:	\(49\) over \(\Q(\zeta_{49})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{49}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2058 q - 42 q^{2} - 42 q^{3} - 84 q^{4} - 42 q^{5} - 119 q^{6} - 84 q^{7} - 98 q^{8} - 42 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
16.1	−5.49911	−	0.353054i	−2.78314	+	3.98984i	22.1812	+	2.85995i	3.27206	−	2.12988i	16.7134	−	20.9579i	−12.3005	+	0.789718i	−77.6965	−	15.1314i	1.15190	+	3.13009i	−18.7454	+	10.5572i
16.2	−5.24248	−	0.336578i	1.28903	−	1.84793i	19.4359	+	2.50599i	−6.80345	+	4.42855i	−7.37970	+	9.25385i	25.8222	−	1.65784i	−59.7978	−	11.6456i	7.57163	+	20.5746i	37.1575	−	20.9267i
16.3	−4.92908	−	0.316457i	4.70953	−	6.75146i	16.2613	+	2.09667i	6.63060	−	4.31604i	−25.3502	+	31.7881i	4.33945	−	0.278602i	−40.7048	−	7.92726i	−14.0777	−	38.2538i	−34.0486	+	19.1758i
16.4	−4.78000	−	0.306886i	2.41641	−	3.46411i	14.8199	+	1.91082i	−12.6224	+	8.21628i	−12.6135	+	15.8169i	−32.4079	+	2.08066i	−32.6408	−	6.35680i	3.16387	+	8.59725i	62.8567	−	35.4002i
16.5	−4.56883	−	0.293328i	2.48580	−	3.56358i	12.8538	+	1.65732i	14.1644	−	9.22000i	−12.4025	+	15.5522i	−25.2149	+	1.61885i	−22.2903	−	4.34104i	2.80495	+	7.62194i	−67.4192	+	37.9698i
16.6	−4.52040	−	0.290219i	−4.67204	+	6.69773i	12.4154	+	1.60079i	−14.7302	+	9.58833i	23.0633	−	28.9205i	−1.07991	+	0.0693322i	−20.0887	−	3.91227i	−13.7067	−	37.2455i	69.3692	−	39.0680i
16.7	−4.37837	−	0.281101i	−5.11136	+	7.32752i	11.1568	+	1.43851i	16.4461	−	10.7052i	24.4392	−	30.6458i	21.4024	−	1.37408i	−13.9925	−	2.72503i	−18.2417	−	49.5685i	−75.0164	+	42.2485i
16.8	−4.27791	−	0.274651i	−1.17628	+	1.68628i	10.2908	+	1.32685i	4.43621	−	2.88765i	5.49515	−	6.89071i	4.20983	−	0.270280i	−9.99729	−	1.94697i	7.86494	+	21.3716i	−19.7708	+	11.1347i
16.9	−4.01883	−	0.258017i	0.971817	−	1.39317i	8.15008	+	1.05084i	10.9750	−	7.14395i	−4.26503	+	5.34818i	23.5076	−	1.50924i	−0.859906	−	0.167467i	8.32835	+	22.6308i	−45.9500	+	25.8785i
16.10	−3.76828	−	0.241932i	0.210785	−	0.302176i	6.20707	+	0.800313i	−13.3929	+	8.71782i	−0.867403	+	1.08769i	11.5033	−	0.738537i	6.45487	+	1.25709i	9.27798	+	25.2113i	52.5773	−	29.6110i
16.11	−3.74801	−	0.240630i	−3.64412	+	5.22412i	6.05538	+	0.780755i	−1.27116	+	0.827435i	14.9153	−	18.7032i	−8.42351	+	0.540807i	6.98400	+	1.36013i	−4.68697	−	12.7360i	4.96344	−	2.79536i
16.12	−3.70875	−	0.238110i	5.81579	−	8.33737i	5.76384	+	0.743165i	−14.4885	+	9.43098i	−23.5546	+	29.5365i	15.8184	−	1.01557i	7.98313	+	1.55471i	−26.3635	−	71.6382i	55.9800	−	31.5273i
16.13	−2.99065	−	0.192006i	−1.86534	+	2.67411i	0.972802	+	0.125429i	10.2745	−	6.68798i	6.09203	−	7.63916i	−26.7473	+	1.71723i	20.6471	+	4.02103i	5.65350	+	15.3624i	−32.0116	+	18.0286i
16.14	−2.58467	−	0.165941i	4.03463	−	5.78395i	−1.28136	−	0.165212i	−0.871393	+	0.567214i	−11.3880	+	14.2801i	−0.270413	+	0.0173611i	23.6223	+	4.60044i	−7.85095	−	21.3336i	2.34638	−	1.32146i
16.15	−2.49414	−	0.160129i	3.29572	−	4.72466i	−1.73924	−	0.224250i	−1.21612	+	0.791603i	−8.97654	+	11.2562i	−20.1994	+	1.29684i	23.9275	+	4.65987i	−2.13581	−	5.80370i	3.15992	−	1.77963i
16.16	−2.34806	−	0.150751i	−0.632213	+	0.906324i	−2.44364	−	0.315072i	−8.66853	+	5.64259i	1.62111	−	2.03280i	6.62437	−	0.425298i	24.1664	+	4.70641i	8.90313	+	24.1927i	21.2049	−	11.9424i
16.17	−2.27370	−	0.145976i	−5.59996	+	8.02796i	−2.78591	−	0.359203i	1.36479	−	0.888378i	13.9045	−	17.4357i	−24.8008	+	1.59227i	24.1728	+	4.70766i	−23.7638	−	64.5739i	−3.23280	+	1.82068i
16.18	−2.15260	−	0.138202i	2.56671	−	3.67957i	−3.31972	−	0.428031i	10.8326	−	7.05122i	−6.03363	+	7.56593i	33.2091	−	2.13210i	24.0249	+	4.67886i	2.37361	+	6.44986i	−24.2927	+	13.6814i
16.19	−1.84830	−	0.118665i	−3.90867	+	5.60337i	−4.53219	−	0.584361i	−3.25835	+	2.12095i	7.88932	−	9.89289i	29.1744	−	1.87306i	22.8511	+	4.45025i	−6.79524	−	18.4649i	6.27408	−	3.53350i
16.20	−1.27364	−	0.0817702i	0.0894625	−	0.128251i	−6.31886	−	0.814726i	15.2502	−	9.92680i	−0.124430	+	0.156030i	−5.68391	+	0.364919i	18.0031	+	3.50610i	9.31641	+	25.3157i	−20.2350	+	11.3961i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
197.g	even	49	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	197.4.g.a	✓	2058
197.g	even	49	1	inner	197.4.g.a	✓	2058

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
197.4.g.a	✓	2058	1.a	even	1	1	trivial
197.4.g.a	✓	2058	197.g	even	49	1	inner

Hecke kernels

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