Newform orbit 90.3.k.a

• Label: 90.3.k.a
• Level: $90$
• Weight: $3$
• Character orbit: 90.k
• Analytic conductor: $2.452$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45232237924\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} - 6 q^{7} - 48 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.36603	+	0.366025i	−2.54067	+	1.59530i	1.73205	−	1.00000i	−4.00416	−	2.99444i	2.88670	−	3.10917i	6.73232	−	1.80392i	−2.00000	+	2.00000i	3.91003	−	8.10627i	6.56583	+	2.62486i
7.2	−1.36603	+	0.366025i	−2.17538	+	2.06584i	1.73205	−	1.00000i	4.54899	+	2.07525i	2.21548	−	3.61824i	−9.78534	+	2.62197i	−2.00000	+	2.00000i	0.464594	−	8.98800i	−6.97363	−	1.16980i
7.3	−1.36603	+	0.366025i	−1.07491	−	2.80082i	1.73205	−	1.00000i	−4.68508	+	1.74643i	2.49352	+	3.43254i	−1.30080	+	0.348547i	−2.00000	+	2.00000i	−6.68914	+	6.02125i	5.76070	−	4.10052i
7.4	−1.36603	+	0.366025i	1.11865	−	2.78364i	1.73205	−	1.00000i	3.61223	−	3.45714i	−0.509217	+	4.21197i	−8.95134	+	2.39851i	−2.00000	+	2.00000i	−6.49726	−	6.22781i	−3.66899	+	6.04471i
7.5	−1.36603	+	0.366025i	1.72031	+	2.45775i	1.73205	−	1.00000i	3.16601	−	3.86994i	−3.24958	−	2.72767i	5.91251	−	1.58425i	−2.00000	+	2.00000i	−3.08107	+	8.45618i	−2.90835	+	6.44527i
7.6	−1.36603	+	0.366025i	2.95201	−	0.534441i	1.73205	−	1.00000i	−0.0399035	+	4.99984i	−3.83690	+	1.81057i	3.29458	−	0.882779i	−2.00000	+	2.00000i	8.42875	−	3.15535i	−1.77556	−	6.84452i
13.1	−1.36603	−	0.366025i	−2.54067	−	1.59530i	1.73205	+	1.00000i	−4.00416	+	2.99444i	2.88670	+	3.10917i	6.73232	+	1.80392i	−2.00000	−	2.00000i	3.91003	+	8.10627i	6.56583	−	2.62486i
13.2	−1.36603	−	0.366025i	−2.17538	−	2.06584i	1.73205	+	1.00000i	4.54899	−	2.07525i	2.21548	+	3.61824i	−9.78534	−	2.62197i	−2.00000	−	2.00000i	0.464594	+	8.98800i	−6.97363	+	1.16980i
13.3	−1.36603	−	0.366025i	−1.07491	+	2.80082i	1.73205	+	1.00000i	−4.68508	−	1.74643i	2.49352	−	3.43254i	−1.30080	−	0.348547i	−2.00000	−	2.00000i	−6.68914	−	6.02125i	5.76070	+	4.10052i
13.4	−1.36603	−	0.366025i	1.11865	+	2.78364i	1.73205	+	1.00000i	3.61223	+	3.45714i	−0.509217	−	4.21197i	−8.95134	−	2.39851i	−2.00000	−	2.00000i	−6.49726	+	6.22781i	−3.66899	−	6.04471i
13.5	−1.36603	−	0.366025i	1.72031	−	2.45775i	1.73205	+	1.00000i	3.16601	+	3.86994i	−3.24958	+	2.72767i	5.91251	+	1.58425i	−2.00000	−	2.00000i	−3.08107	−	8.45618i	−2.90835	−	6.44527i
13.6	−1.36603	−	0.366025i	2.95201	+	0.534441i	1.73205	+	1.00000i	−0.0399035	−	4.99984i	−3.83690	−	1.81057i	3.29458	+	0.882779i	−2.00000	−	2.00000i	8.42875	+	3.15535i	−1.77556	+	6.84452i
43.1	0.366025	+	1.36603i	−2.45775	+	1.72031i	−1.73205	+	1.00000i	−4.93447	+	0.806874i	−3.24958	−	2.72767i	−1.58425	−	5.91251i	−2.00000	−	2.00000i	3.08107	−	8.45618i	−2.90835	−	6.44527i
43.2	0.366025	+	1.36603i	−2.06584	−	2.17538i	−1.73205	+	1.00000i	−0.477275	+	4.97717i	2.21548	−	3.61824i	2.62197	+	9.78534i	−2.00000	−	2.00000i	−0.464594	+	8.98800i	−6.97363	+	1.16980i
43.3	0.366025	+	1.36603i	−1.59530	−	2.54067i	−1.73205	+	1.00000i	−0.591182	−	4.96493i	2.88670	−	3.10917i	−1.80392	−	6.73232i	−2.00000	−	2.00000i	−3.91003	+	8.10627i	6.56583	−	2.62486i
43.4	0.366025	+	1.36603i	0.534441	+	2.95201i	−1.73205	+	1.00000i	4.34994	+	2.46536i	−3.83690	+	1.81057i	−0.882779	−	3.29458i	−2.00000	−	2.00000i	−8.42875	+	3.15535i	−1.77556	+	6.84452i
43.5	0.366025	+	1.36603i	2.78364	+	1.11865i	−1.73205	+	1.00000i	−4.80008	+	1.39971i	−0.509217	+	4.21197i	2.39851	+	8.95134i	−2.00000	−	2.00000i	6.49726	+	6.22781i	−3.66899	−	6.04471i
43.6	0.366025	+	1.36603i	2.80082	−	1.07491i	−1.73205	+	1.00000i	3.85499	−	3.18419i	2.49352	+	3.43254i	0.348547	+	1.30080i	−2.00000	−	2.00000i	6.68914	−	6.02125i	5.76070	+	4.10052i
67.1	0.366025	−	1.36603i	−2.45775	−	1.72031i	−1.73205	−	1.00000i	−4.93447	−	0.806874i	−3.24958	+	2.72767i	−1.58425	+	5.91251i	−2.00000	+	2.00000i	3.08107	+	8.45618i	−2.90835	+	6.44527i
67.2	0.366025	−	1.36603i	−2.06584	+	2.17538i	−1.73205	−	1.00000i	−0.477275	−	4.97717i	2.21548	+	3.61824i	2.62197	−	9.78534i	−2.00000	+	2.00000i	−0.464594	−	8.98800i	−6.97363	−	1.16980i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
9.c	even	3	1	inner
45.k	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	90.3.k.a	✓	24
3.b	odd	2	1		270.3.l.b		24
5.c	odd	4	1	inner	90.3.k.a	✓	24
9.c	even	3	1	inner	90.3.k.a	✓	24
9.c	even	3	1		810.3.g.k		12
9.d	odd	6	1		270.3.l.b		24
9.d	odd	6	1		810.3.g.i		12
15.e	even	4	1		270.3.l.b		24
45.k	odd	12	1	inner	90.3.k.a	✓	24
45.k	odd	12	1		810.3.g.k		12
45.l	even	12	1		270.3.l.b		24
45.l	even	12	1		810.3.g.i		12

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.3.k.a	✓	24	1.a	even	1	1	trivial
90.3.k.a	✓	24	5.c	odd	4	1	inner
90.3.k.a	✓	24	9.c	even	3	1	inner
90.3.k.a	✓	24	45.k	odd	12	1	inner
270.3.l.b		24	3.b	odd	2	1
270.3.l.b		24	9.d	odd	6	1
270.3.l.b		24	15.e	even	4	1
270.3.l.b		24	45.l	even	12	1
810.3.g.i		12	9.d	odd	6	1
810.3.g.i		12	45.l	even	12	1
810.3.g.k		12	9.c	even	3	1
810.3.g.k		12	45.k	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^24 + 6*T7^23 + 18*T7^22 + 1196*T7^21 - 3891*T7^20 - 86784*T7^19 + 264542*T7^18 - 6533676*T7^17 + 6812778*T7^16 + 398196074*T7^15 - 752665806*T7^14 + 24919991784*T7^13 - 80858340983*T7^12 - 1266619287492*T7^11 + 5716568968056*T7^10 - 49036549194236*T7^9 + 432645584162106*T7^8 - 1054216635208830*T7^7 + 2383901657917088*T7^6 - 11066035390916880*T7^5 + 2587325236375965*T7^4 + 43246021288307356*T7^3 + 13964271119124552*T7^2 + 56561139537325932*T7 + 114548137832252881