Newform orbit 29.6.e.a

• Label: 29.6.e.a
• Level: $29$
• Weight: $6$
• Character orbit: 29.e
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−10.9467	−	2.49852i	−2.67824	−	5.56142i	84.7573	+	40.8170i	−8.95546	+	39.2364i	15.4226	+	67.5710i	52.7478	−	25.4020i	−544.918	−	434.558i	127.752	−	160.195i	196.066	−	407.135i
4.2	−7.77560	−	1.77473i	11.9139	+	24.7394i	28.4793	+	13.7149i	3.78229	−	16.5713i	−48.7317	−	213.507i	−72.5021	+	34.9152i	2.43427	+	1.94126i	−318.589	+	399.498i	−58.8191	+	122.139i
4.3	−6.94954	−	1.58619i	−8.33949	−	17.3171i	16.9491	+	8.16224i	19.2287	−	84.2463i	30.4874	+	133.574i	−150.109	+	72.2888i	73.4978	+	58.6125i	−78.8281	+	98.8473i	−267.261	+	554.972i
4.4	−5.87269	−	1.34040i	1.71185	+	3.55470i	3.86078	+	1.85925i	−4.47412	+	19.6024i	−5.28844	−	23.1702i	52.6961	−	25.3771i	130.524	+	104.089i	141.803	−	177.815i	52.5502	−	109.122i
4.5	−3.64383	−	0.831681i	−10.9557	−	22.7497i	−16.2452	−	7.82326i	−15.4292	+	67.5998i	21.0001	+	92.0076i	150.354	−	72.4064i	146.196	+	116.588i	−246.012	+	308.490i	112.443	−	233.490i
4.6	0.286206	+	0.0653246i	2.72061	+	5.64941i	−28.7534	−	13.8469i	17.1591	−	75.1791i	0.409610	+	1.79462i	89.3198	−	43.0142i	−14.6695	−	11.6985i	126.994	−	159.245i	9.82209	−	20.3958i
4.7	0.435057	+	0.0992990i	6.82610	+	14.1745i	−28.6516	−	13.7979i	−13.0294	+	57.0855i	1.56223	+	6.84456i	−84.1454	+	40.5223i	−22.2594	−	17.7513i	−2.81388	+	3.52849i	−11.3371	+	23.5417i
4.8	2.42781	+	0.554133i	−6.28143	−	13.0435i	−23.2438	−	11.1936i	−7.07385	+	30.9925i	−8.02230	−	35.1480i	−210.131	+	101.194i	−112.531	−	89.7408i	20.8309	−	26.1212i	−34.3480	+	71.3243i
4.9	6.13837	+	1.40104i	−8.68182	−	18.0280i	6.88566	+	3.31596i	7.61495	−	33.3633i	−28.0342	−	122.826i	98.2215	−	47.3010i	−119.902	−	95.6186i	−98.1262	+	123.046i	93.4868	−	194.127i
4.10	6.78331	+	1.54825i	11.8503	+	24.6073i	14.7853	+	7.12021i	10.0145	−	43.8764i	42.2859	+	185.266i	−23.5325	+	11.3326i	−84.8041	−	67.6290i	−313.584	+	393.222i	135.863	−	282.123i
4.11	7.72990	+	1.76430i	2.56407	+	5.32435i	27.8075	+	13.3914i	−19.7320	+	86.4514i	10.4263	+	45.6804i	144.832	−	69.7474i	−7.04176	−	5.61562i	129.734	−	162.681i	−305.052	+	633.447i
4.12	10.3328	+	2.35839i	−1.02664	−	2.13184i	72.3732	+	34.8531i	8.32499	−	36.4742i	−5.58033	−	24.4490i	−152.906	+	73.6357i	400.459	+	319.355i	148.017	−	185.608i	172.040	−	357.246i
5.1	−7.89410	+	6.29533i	−11.7224	+	2.67555i	15.5649	−	68.1942i	−29.9441	−	37.5487i	75.6940	−	94.9173i	30.8102	+	134.988i	166.246	+	345.213i	−88.6801	+	42.7061i	472.763	+	107.905i
5.2	−6.92324	+	5.52110i	8.06872	−	1.84163i	10.3280	−	45.2501i	67.2058	+	84.2734i	−45.6939	+	57.2983i	−7.78087	−	34.0902i	55.3795	+	114.997i	−157.223	+	75.7145i	−930.564	−	212.395i
5.3	−6.69148	+	5.33628i	27.9311	−	6.37510i	9.17938	−	40.2175i	−54.7694	−	68.6786i	−152.881	+	191.707i	−33.0581	−	144.837i	34.3563	+	71.3416i	520.570	−	250.693i	732.976	+	167.297i
5.4	−3.89730	+	3.10800i	−28.1891	+	6.43399i	−1.59134	+	6.97210i	16.0572	+	20.1351i	89.8648	−	112.687i	−13.4237	−	58.8131i	−84.6782	−	175.836i	534.296	−	257.303i	−125.160	−	28.5668i
5.5	−3.43267	+	2.73746i	−3.06970	+	0.700638i	−2.83115	+	12.4041i	−10.7196	−	13.4419i	8.61928	−	10.8082i	−21.5568	−	94.4466i	−85.1969	−	176.913i	−210.003	+	101.132i	73.5934	+	16.7972i
5.6	−2.31975	+	1.84994i	6.76376	−	1.54378i	−5.16171	+	22.6149i	−15.1268	−	18.9684i	−12.8343	+	16.0937i	39.5350	+	173.214i	−71.0579	−	147.553i	−175.570	+	84.5502i	70.1808	+	16.0183i
5.7	0.785335	−	0.626284i	21.6940	−	4.95152i	−6.89615	+	30.2140i	26.6272	+	33.3895i	13.9360	−	17.4752i	−11.9527	−	52.3684i	27.4532	+	57.0073i	227.177	−	109.403i	41.8226	+	9.54572i
5.8	2.90960	−	2.32033i	−9.65881	+	2.20456i	−4.03882	+	17.6952i	−63.6862	−	79.8599i	−22.9880	+	28.8260i	−47.5303	−	208.244i	80.9780	+	168.153i	−130.503	+	62.8469i	−370.603	−	84.5877i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	29.6.e.a	✓	72
29.e	even	14	1	inner	29.6.e.a	✓	72
29.f	odd	28	2		841.6.a.l		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.6.e.a	✓	72	1.a	even	1	1	trivial
29.6.e.a	✓	72	29.e	even	14	1	inner
841.6.a.l		72	29.f	odd	28	2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(29, [\chi])\).