Newform orbit 71.5.f.a

• Label: 71.5.f.a
• Level: $71$
• Weight: $5$
• Character orbit: 71.f
• Analytic conductor: $7.339$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	71.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.33926737895\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 138 q - 11 q^{2} + 20 q^{3} - 187 q^{4} - 18 q^{5} + 93 q^{6} - 7 q^{7} - 123 q^{8} - 329 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−4.68169	−	5.87066i	0.00874582	−	0.0383179i	−8.98603	+	39.3704i	39.2520			−0.265897	+	0.128049i	−61.9071	−	49.3693i	164.956	−	79.4386i	72.9771	+	35.1439i	−183.766	−	230.435i
23.2	−4.48058	−	5.61847i	1.42816	−	6.25719i	−7.93129	+	34.7493i	−11.1292			−41.5549	+	20.0118i	50.6279	+	40.3744i	127.181	−	61.2469i	35.8657	+	17.2720i	49.8655	+	62.5293i
23.3	−4.11859	−	5.16455i	−3.55260	+	15.5649i	−6.14945	+	26.9425i	5.91332			95.0176	−	45.7581i	26.1576	+	20.8600i	69.2484	−	33.3483i	−156.668	−	75.4474i	−24.3545	−	30.5396i
23.4	−4.10776	−	5.15096i	−1.17598	+	5.15232i	−6.09843	+	26.7190i	−44.7408			31.3701	−	15.1070i	−16.4570	−	13.1240i	67.7052	−	32.6051i	47.8150	+	23.0265i	183.784	+	230.458i
23.5	−3.49936	−	4.38806i	3.34247	−	14.6443i	−3.44921	+	15.1120i	1.36574			−75.9566	+	36.5788i	−20.8620	−	16.6369i	−2.52525	+	1.21609i	−130.305	−	62.7516i	−4.77922	−	5.99295i
23.6	−2.72594	−	3.41822i	0.432416	−	1.89454i	−0.693146	+	3.03687i	29.6157			−7.65468	+	3.68630i	58.5699	+	46.7079i	−50.7554	+	24.4425i	69.5762	+	33.5061i	−80.7305	−	101.233i
23.7	−2.63729	−	3.30706i	−1.77636	+	7.78275i	−0.420999	+	1.84452i	10.9887			30.4228	−	14.6509i	−33.6678	−	26.8492i	−53.7657	+	25.8922i	15.5627	+	7.49461i	−28.9805	−	36.3404i
23.8	−2.17711	−	2.73001i	1.34069	−	5.87396i	0.847186	−	3.71177i	−26.6260			−18.9548	+	9.12816i	−10.5428	−	8.40764i	−62.3139	+	30.0088i	40.2726	+	19.3942i	57.9679	+	72.6894i
23.9	−1.48993	−	1.86831i	−2.57376	+	11.2764i	2.28964	−	10.0315i	−12.0321			24.9025	−	11.9924i	10.5962	+	8.45016i	−56.6016	+	27.2579i	−47.5536	−	22.9006i	17.9270	+	22.4798i
23.10	−0.427887	−	0.536553i	1.26506	−	5.54258i	3.45553	−	15.1397i	29.4241			−3.51519	+	1.69283i	−35.3401	−	28.1828i	−19.4948	+	9.38822i	43.8587	+	21.1212i	−12.5902	−	15.7876i
23.11	−0.288161	−	0.361343i	2.41740	−	10.5913i	3.51280	−	15.3906i	−24.3752			−4.52369	+	2.17849i	−19.5244	−	15.5702i	−13.2360	+	6.37413i	−33.3534	−	16.0622i	7.02398	+	8.80779i
23.12	−0.112155	−	0.140638i	−1.10955	+	4.86126i	3.55313	−	15.5673i	−34.8609			0.808119	−	0.389169i	63.8589	+	50.9258i	−5.18095	+	2.49502i	50.5778	+	24.3570i	3.90982	+	4.90276i
23.13	0.429236	+	0.538244i	−3.07903	+	13.4901i	3.45487	−	15.1368i	42.2305			−8.58260	+	4.13316i	15.1287	+	12.0647i	19.5545	−	9.41693i	−99.5240	−	47.9282i	18.1268	+	22.7303i
23.14	0.456857	+	0.572880i	3.63135	−	15.9100i	3.44086	−	15.0754i	20.6289			10.7735	−	5.18825i	57.9494	+	46.2131i	20.7712	−	10.0029i	−166.962	−	80.4047i	9.42447	+	11.8179i
23.15	1.14448	+	1.43513i	−2.33687	+	10.2385i	2.81057	−	12.3139i	−19.2615			−17.3681	+	8.36402i	−74.1615	−	59.1418i	47.3497	−	22.8024i	−26.3877	−	12.7076i	−22.0443	−	27.6427i
23.16	1.85133	+	2.32150i	0.206468	−	0.904597i	1.59842	−	7.00313i	−5.26120			2.48226	−	1.19539i	20.3761	+	16.2494i	62.0210	−	29.8677i	72.2028	+	34.7710i	−9.74022	−	12.2139i
23.17	2.69393	+	3.37808i	−0.104908	+	0.459630i	−0.593834	+	2.60176i	23.1448			−1.83528	+	0.883824i	3.90425	+	3.11353i	51.8967	−	24.9922i	72.7782	+	35.0481i	62.3504	+	78.1849i
23.18	2.71340	+	3.40250i	2.71378	−	11.8898i	−0.654112	+	2.86585i	−39.0978			47.8187	−	23.0283i	−18.0489	−	14.3935i	51.2098	−	24.6613i	−61.0251	−	29.3881i	−106.088	−	133.030i
23.19	3.17214	+	3.97774i	−3.47190	+	15.2114i	−2.19960	+	9.63708i	−16.5304			−71.5203	+	34.4424i	33.3695	+	26.6113i	28.0308	−	13.4989i	−146.354	−	70.4802i	−52.4368	−	65.7537i
23.20	3.54779	+	4.44878i	3.11416	−	13.6440i	−3.64455	+	15.9678i	32.8227			71.7478	−	34.5519i	−74.5506	−	59.4521i	−1.94036	+	0.934428i	−103.483	−	49.8349i	116.448	+	146.021i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
71.f	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	71.5.f.a	✓	138
71.f	odd	14	1	inner	71.5.f.a	✓	138

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
71.5.f.a	✓	138	1.a	even	1	1	trivial
71.5.f.a	✓	138	71.f	odd	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(71, [\chi])\).