Newform orbit 71.4.g.a

• Label: 71.4.g.a
• Level: $71$
• Weight: $4$
• Character orbit: 71.g
• Analytic conductor: $4.189$
• Analytic rank: $0$
• Dimension: $408$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	71.g (of order \(35\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.18913561041\)
Analytic rank:	\(0\)
Dimension:	\(408\)
Relative dimension:	\(17\) over \(\Q(\zeta_{35})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 408 q - 28 q^{2} - 35 q^{3} + 54 q^{4} - 26 q^{5} - 86 q^{6} - 11 q^{7} - 43 q^{8} + 142 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−0.233985	+	5.21007i	−0.293474	−	0.256400i	−19.1223	−	1.72104i	6.44948	+	19.8494i	1.40453	−	1.46902i	−7.96190	−	2.19734i	7.84049	−	57.8808i	−3.60391	−	26.6052i	−104.926	+	28.9578i
2.2	−0.206162	+	4.59054i	−5.28717	−	4.61926i	−13.0628	−	1.17567i	−2.15551	−	6.63398i	22.2949	−	23.3187i	23.1603	+	6.39183i	3.15542	−	23.2942i	2.99229	+	22.0900i	30.8979	−	8.52729i
2.3	−0.204814	+	4.56053i	7.10707	+	6.20926i	−12.7887	−	1.15100i	−1.66791	−	5.13330i	−29.7732	+	31.1403i	−6.39352	−	1.76450i	2.96615	−	21.8970i	8.33125	+	61.5038i	23.7522	−	6.55518i
2.4	−0.194134	+	4.32273i	−0.987708	−	0.862935i	−10.6805	−	0.961262i	−4.32650	−	13.3156i	3.92198	−	4.10207i	−27.2780	−	7.52824i	1.58201	−	11.6789i	−3.39339	−	25.0510i	58.3997	−	16.1173i
2.5	−0.130495	+	2.90570i	3.02552	+	2.64332i	−0.458276	−	0.0412456i	1.12187	+	3.45276i	−8.07550	+	8.44631i	13.0261	+	3.59499i	−2.94383	+	21.7322i	−1.45765	−	10.7608i	−10.1791	+	2.80925i
2.6	−0.108085	+	2.40670i	−5.80371	−	5.07055i	2.18726	+	0.196857i	2.99423	+	9.21530i	12.8306	−	13.4197i	2.66149	+	0.734524i	−3.29727	+	24.3414i	4.34828	+	32.1003i	−22.5021	+	6.21019i
2.7	−0.0382363	+	0.851398i	0.365902	+	0.319679i	7.24438	+	0.652006i	−5.47411	−	16.8476i	−0.286165	+	0.299305i	17.9695	+	4.95927i	−1.74732	+	12.8993i	−3.59261	−	26.5217i	14.5533	−	4.01646i
2.8	−0.0343485	+	0.764829i	−4.29278	−	3.75049i	7.38401	+	0.664573i	−1.39803	−	4.30270i	3.01594	−	3.15442i	−28.2721	−	7.80259i	−1.58407	+	11.6941i	0.737502	+	5.44446i	3.33885	−	0.921464i
2.9	−0.00752182	+	0.167486i	4.03117	+	3.52193i	7.93980	+	0.714595i	3.70169	+	11.3926i	−0.620197	+	0.648676i	−22.1553	−	6.11447i	−0.359446	+	2.65354i	0.222072	+	1.63940i	−1.93595	+	0.534289i
2.10	0.0520110	−	1.15812i	7.02849	+	6.14061i	6.62927	+	0.596645i	−3.82851	−	11.7829i	7.47709	−	7.82042i	1.16650	+	0.321933i	2.28069	−	16.8368i	8.06832	+	59.5627i	−13.8451	+	3.82101i
2.11	0.0602362	−	1.34126i	−2.56600	−	2.24185i	6.17244	+	0.555529i	4.11218	+	12.6560i	−3.16147	+	3.30664i	20.3457	+	5.61505i	2.55870	−	18.8891i	−2.06582	−	15.2505i	17.2227	−	4.75317i
2.12	0.105891	−	2.35785i	−1.52396	−	1.33145i	2.41956	+	0.217764i	−2.64887	−	8.15240i	−3.30072	+	3.45228i	−2.84258	−	0.784501i	3.30423	−	24.3928i	−3.07458	−	22.6975i	−19.5026	+	5.38238i
2.13	0.148054	−	3.29668i	−7.10626	−	6.20855i	−2.87837	−	0.259058i	−1.74093	−	5.35803i	−21.5197	+	22.5079i	−0.0785773	−	0.0216860i	2.26358	−	16.7104i	8.32852	+	61.4836i	−17.9214	+	4.94600i
2.14	0.173936	−	3.87298i	4.72249	+	4.12591i	−7.00192	−	0.630184i	3.47888	+	10.7069i	16.8010	−	17.5725i	17.3634	+	4.79198i	0.504678	−	3.72568i	1.65443	+	12.2135i	42.0727	−	11.6113i
2.15	0.182205	−	4.05711i	2.55957	+	2.23622i	−8.45917	−	0.761339i	−2.79069	−	8.58885i	9.53898	−	9.97699i	−16.7784	−	4.63054i	−0.268954	+	1.98550i	−2.07363	−	15.3081i	−35.3544	+	9.75719i
2.16	0.219654	−	4.89097i	−3.24713	−	2.83693i	−15.9055	−	1.43152i	6.12849	+	18.8616i	−14.5886	+	15.2584i	−30.0980	−	8.30653i	−5.23770	+	38.6662i	−1.12864	−	8.33193i	93.5974	−	25.8312i
2.17	0.247034	−	5.50065i	−1.86642	−	1.63064i	−22.2283	−	2.00058i	−3.68809	−	11.3508i	−9.43063	+	9.86367i	30.8114	+	8.50342i	−10.5827	+	78.1248i	−2.79977	−	20.6687i	−63.3476	+	17.4828i
3.1	−3.96369	−	3.46297i	0.139284	−	0.258832i	2.64479	+	19.5246i	−2.50994	−	7.72480i	−1.44840	+	0.543595i	12.8140	+	29.9799i	33.9336	−	51.4071i	14.8266	+	22.4614i	−16.8021	+	39.3105i
3.2	−3.57340	−	3.12198i	3.01247	−	5.59811i	1.94853	+	14.3846i	−1.12349	−	3.45774i	−28.2420	+	10.5994i	−14.2496	−	33.3386i	17.0331	−	25.8041i	−7.38964	−	11.1948i	−6.78035	+	15.8634i
3.3	−3.33929	−	2.91745i	−3.77604	+	7.01705i	1.56547	+	11.5568i	1.39764	+	4.30151i	33.0811	−	12.4156i	−5.08703	−	11.9017i	8.94636	−	13.5532i	−20.1063	−	30.4598i	7.88229	−	18.4415i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.g	even	35	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	71.4.g.a	✓	408
71.g	even	35	1	inner	71.4.g.a	✓	408

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.4.g.a	✓	408	1.a	even	1	1	trivial
71.4.g.a	✓	408	71.g	even	35	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(71, [\chi])\).