Newform orbit 79.5.h.a

• Label: 79.5.h.a
• Level: $79$
• Weight: $5$
• Character orbit: 79.h
• Analytic conductor: $8.166$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 79 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	79.h (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.16622708362\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(26\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 28 q^{2} - 32 q^{3} + 208 q^{4} - 25 q^{5} - 23 q^{6} - 146 q^{7} - 762 q^{8} - 666 q^{9}+O(q^{10}) \)

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	−7.57810	−	1.23156i	13.5439	+	0.545799i	40.7343	+	13.5991i	−32.4773	+	24.4051i	−101.965	−	20.8162i	−20.4053	+	32.2684i	−183.171	−	96.1354i	102.401	+	8.26665i	276.173	−	144.947i
3.2	−7.42221	−	1.20623i	−3.16323	−	0.127474i	38.4576	+	12.8390i	24.5046	−	18.4140i	23.3244	+	4.76171i	−26.4363	+	41.8057i	−163.422	−	85.7703i	−70.7476	−	5.71133i	−204.090	+	107.115i
3.3	−7.36604	−	1.19710i	−15.0891	−	0.608068i	37.6489	+	12.5690i	−26.2023	+	19.6897i	110.419	+	22.5421i	47.0317	−	74.3748i	−156.551	−	82.1641i	146.573	+	11.8326i	216.577	−	113.669i
3.4	−6.09199	−	0.990045i	14.6753	+	0.591393i	20.9556	+	6.99600i	34.5447	−	25.9587i	−88.8161	−	18.1319i	28.8284	−	45.5884i	−33.2954	−	17.4748i	134.277	+	10.8399i	−236.146	+	123.939i
3.5	−5.85679	−	0.951821i	1.89639	+	0.0764220i	18.2194	+	6.08254i	−5.55333	+	4.17306i	−11.0340	−	2.25261i	16.0269	−	25.3445i	−16.8544	−	8.84588i	−77.1469	−	6.22794i	36.4967	−	19.1550i
3.6	−5.22182	−	0.848628i	−10.7931	−	0.434945i	11.3706	+	3.79607i	−14.7407	+	11.0769i	55.9903	+	11.4305i	−48.0591	+	75.9994i	18.7957	+	9.86477i	35.5636	+	2.87099i	86.3733	−	45.3322i
3.7	−5.18740	−	0.843035i	4.47223	+	0.180225i	11.0219	+	3.67964i	−8.05694	+	6.05440i	−23.0473	−	4.70515i	10.2794	−	16.2556i	20.3829	+	10.6978i	−60.7690	−	4.90578i	46.8987	−	24.6143i
3.8	−4.66570	−	0.758249i	−13.1768	−	0.531007i	6.01720	+	2.00884i	26.0381	−	19.5663i	61.0764	+	12.4688i	10.2857	−	16.2656i	40.4163	+	21.2121i	92.6090	+	7.47617i	−136.322	+	71.5473i
3.9	−3.00808	−	0.488860i	11.8004	+	0.475541i	−6.36704	−	2.12563i	−0.0424632	+	0.0319090i	−35.2641	−	7.19922i	−41.0290	+	64.8822i	61.2888	+	32.1669i	58.2866	+	4.70538i	0.143332	−	0.0752263i
3.10	−2.13554	−	0.347059i	−6.49097	−	0.261577i	−10.7365	−	3.58437i	−31.7896	+	23.8884i	13.7710	+	2.81136i	3.37137	−	5.33139i	52.3360	+	27.4681i	−38.6731	−	3.12201i	76.1788	−	39.9817i
3.11	−1.90162	−	0.309044i	4.39192	+	0.176988i	−11.6559	−	3.89132i	28.6284	−	21.5128i	−8.29708	−	1.69386i	−8.09565	+	12.8022i	48.2569	+	25.3272i	−61.4797	−	4.96316i	−61.0888	+	32.0619i
3.12	−1.85445	−	0.301377i	17.0269	+	0.686160i	−11.8284	−	3.94891i	−21.0454	+	15.8146i	−31.3686	−	6.40395i	40.8072	−	64.5315i	47.3623	+	24.8576i	208.707	+	16.8486i	43.7936	−	22.9847i
3.13	−1.71903	−	0.279370i	−6.43716	−	0.259409i	−12.2996	−	4.10620i	9.85085	−	7.40244i	10.9932	+	2.24428i	43.9999	−	69.5803i	44.6697	+	23.4445i	−39.3676	−	3.17808i	−19.0019	+	9.97298i
3.14	−0.373071	−	0.0606300i	−7.06457	−	0.284692i	−15.0411	−	5.02145i	16.8763	−	12.6817i	2.61833	+	0.534535i	−20.3405	+	32.1659i	10.6617	+	5.59569i	−30.9103	−	2.49534i	−7.06495	+	3.70797i
3.15	0.997190	+	0.162059i	−17.0253	−	0.686098i	−14.2085	−	4.74348i	−8.57665	+	6.44493i	−16.8663	−	3.44328i	−1.91767	+	3.03255i	−27.7126	−	14.5447i	208.654	+	16.8443i	−9.59701	+	5.03690i
3.16	1.14553	+	0.186167i	6.18557	+	0.249270i	−13.8990	−	4.64017i	−18.6575	+	14.0202i	7.03935	+	1.43709i	5.66131	−	8.95265i	−31.4998	−	16.5324i	−42.5382	−	3.43404i	−23.9828	+	12.5871i
3.17	2.14364	+	0.348375i	10.5128	+	0.423652i	−10.7028	−	3.57311i	24.0150	−	18.0461i	22.3881	+	4.57056i	17.6191	−	27.8624i	−52.4661	−	27.5363i	29.6023	+	2.38975i	57.7663	−	30.3181i
3.18	3.27294	+	0.531904i	3.91043	+	0.157585i	−4.74738	−	1.58491i	−26.5175	+	19.9266i	12.7148	+	2.59574i	−33.4309	+	52.8668i	−61.6719	−	32.3679i	−65.4707	−	5.28534i	−97.3893	+	51.1139i
3.19	3.53861	+	0.575080i	−7.29649	−	0.294038i	−2.98554	−	0.996720i	19.8262	−	14.8984i	−25.6503	−	5.23655i	−35.3959	+	55.9741i	−60.7817	−	31.9007i	−27.5851	−	2.22690i	78.7249	−	41.3180i
3.20	4.01289	+	0.652159i	−5.88306	−	0.237079i	0.501413	+	0.167396i	0.888026	−	0.667308i	−23.4535	−	4.78806i	41.0711	−	64.9487i	−55.6948	−	29.2309i	−46.1832	−	3.72829i	3.99874	−	2.09870i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
79.h	odd	78	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	79.5.h.a	✓	624
79.h	odd	78	1	inner	79.5.h.a	✓	624

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
79.5.h.a	✓	624	1.a	even	1	1	trivial
79.5.h.a	✓	624	79.h	odd	78	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(79, [\chi])\).