Newform orbit 78.5.j.a

• Label: 78.5.j.a
• Level: $78$
• Weight: $5$
• Character orbit: 78.j
• Analytic conductor: $8.063$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	78.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.06285712054\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 144 q^{4} + 126 q^{7} + 68 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1	−1.41421	+	2.44949i	−8.78653	−	1.94856i	−4.00000	−	6.92820i	39.2302			17.1990	−	18.7668i	−52.7864	+	30.4762i	22.6274			73.4062	+	34.2422i	−55.4799	+	96.0940i
17.2	−1.41421	+	2.44949i	−8.72759	−	2.19755i	−4.00000	−	6.92820i	−33.7996			17.7256	−	18.2703i	−4.93081	+	2.84681i	22.6274			71.3415	+	38.3587i	47.7999	−	82.7919i
17.3	−1.41421	+	2.44949i	−6.21191	+	6.51246i	−4.00000	−	6.92820i	26.8504			−7.16725	−	24.4260i	79.2711	−	45.7672i	22.6274			−3.82437	−	80.9097i	−37.9722	+	65.7698i
17.4	−1.41421	+	2.44949i	−5.11510	+	7.40512i	−4.00000	−	6.92820i	−7.76453			−10.9049	−	23.0018i	−33.8439	+	19.5398i	22.6274			−28.6716	−	75.7558i	10.9807	−	19.0191i
17.5	−1.41421	+	2.44949i	−0.979831	−	8.94650i	−4.00000	−	6.92820i	−9.70905			23.3001	+	10.2522i	19.4920	−	11.2537i	22.6274			−79.0799	+	17.5321i	13.7307	−	23.7822i
17.6	−1.41421	+	2.44949i	5.49171	−	7.13030i	−4.00000	−	6.92820i	26.0371			9.69915	+	23.5356i	−19.6435	+	11.3412i	22.6274			−20.6824	−	78.3150i	−36.8221	+	63.7777i
17.7	−1.41421	+	2.44949i	6.73962	+	5.96469i	−4.00000	−	6.92820i	25.8074			−24.1417	+	8.07327i	47.1333	−	27.2124i	22.6274			9.84486	+	80.3995i	−36.4972	+	63.2150i
17.8	−1.41421	+	2.44949i	8.75797	+	2.07317i	−4.00000	−	6.92820i	−1.51787			−17.4638	+	18.5206i	−67.5211	+	38.9833i	22.6274			72.4039	+	36.3135i	2.14659	−	3.71800i
17.9	−1.41421	+	2.44949i	8.83167	−	1.73253i	−4.00000	−	6.92820i	−42.5067			−8.24605	+	24.0832i	64.3293	−	37.1405i	22.6274			74.9967	−	30.6023i	60.1135	−	104.120i
17.10	1.41421	−	2.44949i	−8.53538	−	2.85433i	−4.00000	−	6.92820i	−25.8074			−19.0625	+	16.8707i	47.1333	−	27.2124i	−22.6274			64.7056	+	48.7256i	−36.4972	+	63.2150i
17.11	1.41421	−	2.44949i	−6.17440	−	6.54804i	−4.00000	−	6.92820i	1.51787			−24.7713	+	5.86381i	−67.5211	+	38.9833i	−22.6274			−4.75353	+	80.8604i	2.14659	−	3.71800i
17.12	1.41421	−	2.44949i	−3.85547	+	8.13236i	−4.00000	−	6.92820i	7.76453			14.4677	+	20.9448i	−33.8439	+	19.5398i	−22.6274			−51.2707	−	62.7082i	10.9807	−	19.0191i
17.13	1.41421	−	2.44949i	−2.91542	−	8.51471i	−4.00000	−	6.92820i	42.5067			−24.9797	−	4.90034i	64.3293	−	37.1405i	−22.6274			−64.0007	+	49.6479i	60.1135	−	104.120i
17.14	1.41421	−	2.44949i	−2.53400	+	8.63590i	−4.00000	−	6.92820i	−26.8504			17.5699	+	18.4200i	79.2711	−	45.7672i	−22.6274			−68.1576	−	43.7668i	−37.9722	+	65.7698i
17.15	1.41421	−	2.44949i	3.42917	−	8.32111i	−4.00000	−	6.92820i	−26.0371			−15.5329	−	20.1675i	−19.6435	+	11.3412i	−22.6274			−57.4816	−	57.0689i	−36.8221	+	63.7777i
17.16	1.41421	−	2.44949i	6.08077	+	6.63508i	−4.00000	−	6.92820i	−39.2302			24.8521	−	5.51136i	−52.7864	+	30.4762i	−22.6274			−7.04854	+	80.6927i	−55.4799	+	96.0940i
17.17	1.41421	−	2.44949i	6.26693	+	6.45953i	−4.00000	−	6.92820i	33.7996			24.6853	−	6.21562i	−4.93081	+	2.84681i	−22.6274			−2.45116	+	80.9629i	47.7999	−	82.7919i
17.18	1.41421	−	2.44949i	8.23782	−	3.62469i	−4.00000	−	6.92820i	9.70905			2.77138	−	25.3045i	19.4920	−	11.2537i	−22.6274			54.7232	−	59.7191i	13.7307	−	23.7822i
23.1	−1.41421	−	2.44949i	−8.78653	+	1.94856i	−4.00000	+	6.92820i	39.2302			17.1990	+	18.7668i	−52.7864	−	30.4762i	22.6274			73.4062	−	34.2422i	−55.4799	−	96.0940i
23.2	−1.41421	−	2.44949i	−8.72759	+	2.19755i	−4.00000	+	6.92820i	−33.7996			17.7256	+	18.2703i	−4.93081	−	2.84681i	22.6274			71.3415	−	38.3587i	47.7999	+	82.7919i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.e	even	6	1	inner
39.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	78.5.j.a	✓	36
3.b	odd	2	1	inner	78.5.j.a	✓	36
13.e	even	6	1	inner	78.5.j.a	✓	36
39.h	odd	6	1	inner	78.5.j.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.5.j.a	✓	36	1.a	even	1	1	trivial
78.5.j.a	✓	36	3.b	odd	2	1	inner
78.5.j.a	✓	36	13.e	even	6	1	inner
78.5.j.a	✓	36	39.h	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(78, [\chi])\).