Newform orbit 49.5.h.a

• Label: 49.5.h.a
• Level: $49$
• Weight: $5$
• Character orbit: 49.h
• Analytic conductor: $5.065$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	49.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.06512819111\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 13 q^{2} - 20 q^{3} + 131 q^{4} + 16 q^{5} - 238 q^{6} - 14 q^{7} - 132 q^{8} - 344 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.72301	+	6.93812i	−0.982273	−	0.0736112i	−28.9939	−	26.9024i	15.8384	−	23.2306i	3.18546	−	6.61468i	37.8623	−	31.1038i	158.159	−	76.1656i	−79.1359	−	11.9278i	118.049	+	173.146i
3.2	−2.32518	+	5.92447i	15.1761	+	1.13729i	−17.9640	−	16.6682i	−4.62981	+	6.79068i	−42.0250	+	87.2659i	17.8963	+	45.6149i	48.7732	−	23.4879i	148.925	+	22.4469i	−29.4660	−	43.2187i
3.3	−2.24432	+	5.71844i	−15.2923	−	1.14600i	−15.9348	−	14.7853i	−27.0641	+	39.6957i	40.8743	−	84.8765i	44.9382	−	19.5336i	31.7561	−	15.2929i	152.447	+	22.9777i	−166.257	−	243.854i
3.4	−2.14973	+	5.47743i	1.99608	+	0.149586i	−13.6521	−	12.6673i	−8.56377	+	12.5607i	−5.11040	+	10.6118i	−46.4305	−	15.6592i	13.9089	−	6.69818i	−76.1333	−	11.4753i	−50.3908	−	73.9097i
3.5	−1.86238	+	4.74527i	−14.8662	−	1.11407i	−7.32032	−	6.79227i	22.8541	−	33.5207i	32.9731	−	68.4693i	−33.9489	+	35.3337i	−27.6208	+	13.3015i	139.667	+	21.0515i	116.502	+	170.877i
3.6	−1.14133	+	2.90807i	−2.55973	−	0.191826i	4.57460	+	4.24461i	−1.62748	+	2.38708i	3.47935	−	7.22495i	24.4257	+	42.4780i	−62.5991	+	30.1461i	−73.5799	−	11.0904i	−5.08430	−	7.45729i
3.7	−1.09504	+	2.79013i	8.72443	+	0.653806i	5.14315	+	4.77214i	23.3611	−	34.2645i	−11.3778	+	23.6263i	27.0222	−	40.8754i	−62.1547	+	29.9321i	−4.40705	−	0.664256i	70.0207	+	102.702i
3.8	−0.510728	+	1.30131i	12.9530	+	0.970695i	10.2963	+	9.55353i	−23.7768	+	34.8741i	−7.87866	+	16.3602i	8.91228	−	48.1827i	−37.8429	+	18.2242i	86.7433	+	13.0744i	−33.2387	−	48.7523i
3.9	−0.401994	+	1.02426i	−8.02931	−	0.601714i	10.8413	+	10.0593i	−2.64119	+	3.87391i	3.84405	−	7.98225i	−34.2223	−	35.0690i	−30.5232	+	14.6992i	−15.9875	−	2.40972i	−2.90616	−	4.26256i
3.10	0.184016	−	0.468866i	10.9476	+	0.820407i	11.5429	+	10.7102i	10.2657	−	15.0571i	2.39919	−	4.98198i	−33.7025	+	35.5688i	14.4066	−	6.93784i	39.0810	+	5.89051i	−5.17069	−	7.58401i
3.11	0.422075	−	1.07543i	−11.5688	−	0.866963i	10.7504	+	9.97494i	4.14613	−	6.08126i	−5.81527	+	12.0755i	43.2863	−	22.9630i	31.9189	−	15.3713i	52.9906	+	7.98704i	−4.78999	−	7.02562i
3.12	0.946535	−	2.41173i	−1.40486	−	0.105280i	6.80831	+	6.31719i	−24.2278	+	35.5356i	−1.58366	+	3.28850i	−13.1689	+	47.1972i	59.0277	−	28.4263i	−78.1327	−	11.7766i	62.7700	+	92.0667i
3.13	1.53988	−	3.92356i	7.96712	+	0.597053i	−1.29424	−	1.20088i	0.799328	−	1.17240i	14.6110	−	30.3401i	48.2739	−	8.40397i	54.0555	−	26.0318i	−16.9768	−	2.55884i	−3.36910	−	4.94156i
3.14	1.62311	−	4.13561i	−3.01597	−	0.226016i	−2.73994	−	2.54230i	19.7774	−	29.0081i	−5.82995	+	12.1060i	−44.7740	−	19.9070i	49.0828	−	23.6370i	−71.0503	−	10.7091i	−87.8652	−	128.875i
3.15	1.85330	−	4.72213i	−16.7294	−	1.25370i	−7.13500	−	6.62031i	0.0385359	−	0.0565218i	−36.9248	+	76.6752i	−7.36102	+	48.4439i	28.6416	−	13.7931i	198.207	+	29.8749i	−0.195485	−	0.286724i
3.16	2.40248	−	6.12141i	16.4638	+	1.23379i	−19.9709	−	18.5303i	−4.83568	+	7.09264i	47.1064	−	97.8175i	−48.2951	−	8.28179i	−66.6154	+	32.0803i	189.439	+	28.5534i	31.7994	+	46.6411i
3.17	2.42912	−	6.18931i	−5.42427	−	0.406493i	−20.6780	−	19.1864i	−21.7101	+	31.8428i	−15.6921	+	32.5850i	−13.6594	−	47.0576i	−73.1327	+	35.2188i	−50.8378	−	7.66257i	144.349	+	211.721i
3.18	2.78627	−	7.09929i	−1.52536	−	0.114310i	−30.9078	−	28.6783i	14.4646	−	21.2156i	−5.06157	+	10.5105i	31.3829	+	37.6313i	−179.773	+	86.5743i	−77.7816	−	11.7237i	−110.314	−	161.800i
5.1	−7.66292	+	1.15500i	6.58188	−	9.65384i	42.0971	−	12.9852i	−10.5843	−	0.793181i	−39.2862	+	81.5786i	48.8288	+	4.09296i	−195.876	+	94.3289i	−20.2829	−	51.6801i	82.0225	−	6.14674i
5.2	−6.64063	+	1.00091i	−6.56555	+	9.62989i	27.8069	−	8.57730i	38.8860	+	2.91410i	33.9607	−	70.5200i	17.4908	+	45.7720i	−79.2610	+	38.1701i	−20.0357	−	51.0502i	−261.144	+	19.5701i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.h	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	49.5.h.a	✓	216
49.h	odd	42	1	inner	49.5.h.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.5.h.a	✓	216	1.a	even	1	1	trivial
49.5.h.a	✓	216	49.h	odd	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(49, [\chi])\).