Newform orbit 315.2.k.b

• Label: 315.2.k.b
• Level: $315$
• Weight: $2$
• Character orbit: 315.k
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{2} + q^{3} - 7 q^{4} - 24 q^{5} + 3 q^{6} + 7 q^{7} + 12 q^{8} + 9 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
16.1	−1.17004	−	2.02657i	1.32145	+	1.11972i	−1.73798	+	3.01027i	−1.00000			0.723033	−	3.98812i	−1.00344	+	2.44808i	3.45385			0.492465	+	2.95930i	1.17004	+	2.02657i
16.2	−1.16277	−	2.01398i	0.434379	−	1.67670i	−1.70408	+	2.95156i	−1.00000			−3.88192	+	1.07479i	0.459529	−	2.60554i	3.27475			−2.62263	−	1.45664i	1.16277	+	2.01398i
16.3	−1.08176	−	1.87367i	−1.55999	+	0.752618i	−1.34041	+	2.32167i	−1.00000			3.09769	+	2.10874i	2.14928	+	1.54292i	1.47299			1.86713	−	2.34815i	1.08176	+	1.87367i
16.4	−0.599034	−	1.03756i	1.72885	+	0.105192i	0.282317	−	0.488988i	−1.00000			−0.926499	−	1.85680i	2.04342	−	1.68061i	−3.07261			2.97787	+	0.363723i	0.599034	+	1.03756i
16.5	−0.259245	−	0.449026i	−1.70948	+	0.278682i	0.865584	−	1.49923i	−1.00000			0.568312	+	0.695356i	−1.91816	+	1.82226i	−1.93458			2.84467	−	0.952806i	0.259245	+	0.449026i
16.6	−0.148731	−	0.257610i	0.310785	−	1.70394i	0.955758	−	1.65542i	−1.00000			−0.485175	+	0.173367i	−2.64436	+	0.0857253i	−1.16353			−2.80683	−	1.05912i	0.148731	+	0.257610i
16.7	0.154039	+	0.266804i	1.05264	−	1.37548i	0.952544	−	1.64985i	−1.00000			0.529131	+	0.0689719i	2.20433	+	1.46319i	1.20307			−0.783880	−	2.89578i	−0.154039	−	0.266804i
16.8	0.304907	+	0.528114i	−1.22862	+	1.22086i	0.814064	−	1.41000i	−1.00000			−1.01937	−	0.276604i	1.83653	−	1.90451i	2.21248			0.0190166	−	2.99994i	−0.304907	−	0.528114i
16.9	0.517769	+	0.896802i	−1.26722	−	1.18074i	0.463831	−	0.803378i	−1.00000			0.402766	−	1.74780i	−1.07729	−	2.41649i	3.03170			0.211690	+	2.99252i	−0.517769	−	0.896802i
16.10	0.805191	+	1.39463i	1.04104	+	1.38428i	−0.296664	+	0.513837i	−1.00000			−1.09233	+	2.56648i	2.48389	+	0.911196i	2.26528			−0.832482	+	2.88218i	−0.805191	−	1.39463i
16.11	0.859635	+	1.48893i	−1.31121	−	1.13170i	−0.477944	+	0.827824i	−1.00000			0.557856	−	2.92514i	0.594106	+	2.57818i	1.79511			0.438533	+	2.96778i	−0.859635	−	1.48893i
16.12	1.28004	+	2.21710i	1.68737	−	0.390872i	−2.27701	+	3.94390i	−1.00000			3.02651	+	3.24073i	−1.62783	+	2.08571i	−6.53853			2.69444	−	1.31909i	−1.28004	−	2.21710i
256.1	−1.17004	+	2.02657i	1.32145	−	1.11972i	−1.73798	−	3.01027i	−1.00000			0.723033	+	3.98812i	−1.00344	−	2.44808i	3.45385			0.492465	−	2.95930i	1.17004	−	2.02657i
256.2	−1.16277	+	2.01398i	0.434379	+	1.67670i	−1.70408	−	2.95156i	−1.00000			−3.88192	−	1.07479i	0.459529	+	2.60554i	3.27475			−2.62263	+	1.45664i	1.16277	−	2.01398i
256.3	−1.08176	+	1.87367i	−1.55999	−	0.752618i	−1.34041	−	2.32167i	−1.00000			3.09769	−	2.10874i	2.14928	−	1.54292i	1.47299			1.86713	+	2.34815i	1.08176	−	1.87367i
256.4	−0.599034	+	1.03756i	1.72885	−	0.105192i	0.282317	+	0.488988i	−1.00000			−0.926499	+	1.85680i	2.04342	+	1.68061i	−3.07261			2.97787	−	0.363723i	0.599034	−	1.03756i
256.5	−0.259245	+	0.449026i	−1.70948	−	0.278682i	0.865584	+	1.49923i	−1.00000			0.568312	−	0.695356i	−1.91816	−	1.82226i	−1.93458			2.84467	+	0.952806i	0.259245	−	0.449026i
256.6	−0.148731	+	0.257610i	0.310785	+	1.70394i	0.955758	+	1.65542i	−1.00000			−0.485175	−	0.173367i	−2.64436	−	0.0857253i	−1.16353			−2.80683	+	1.05912i	0.148731	−	0.257610i
256.7	0.154039	−	0.266804i	1.05264	+	1.37548i	0.952544	+	1.64985i	−1.00000			0.529131	−	0.0689719i	2.20433	−	1.46319i	1.20307			−0.783880	+	2.89578i	−0.154039	+	0.266804i
256.8	0.304907	−	0.528114i	−1.22862	−	1.22086i	0.814064	+	1.41000i	−1.00000			−1.01937	+	0.276604i	1.83653	+	1.90451i	2.21248			0.0190166	+	2.99994i	−0.304907	+	0.528114i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.k.b	✓	24
3.b	odd	2	1		945.2.k.b		24
7.c	even	3	1		315.2.l.b	yes	24
9.c	even	3	1		315.2.l.b	yes	24
9.d	odd	6	1		945.2.l.b		24
21.h	odd	6	1		945.2.l.b		24
63.g	even	3	1	inner	315.2.k.b	✓	24
63.n	odd	6	1		945.2.k.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.k.b	✓	24	1.a	even	1	1	trivial
315.2.k.b	✓	24	63.g	even	3	1	inner
315.2.l.b	yes	24	7.c	even	3	1
315.2.l.b	yes	24	9.c	even	3	1
945.2.k.b		24	3.b	odd	2	1
945.2.k.b		24	63.n	odd	6	1
945.2.l.b		24	9.d	odd	6	1
945.2.l.b		24	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^24 + T2^23 + 16*T2^22 + 9*T2^21 + 157*T2^20 + 62*T2^19 + 930*T2^18 + 70*T2^17 + 3854*T2^16 - 284*T2^15 + 10997*T2^14 - 2832*T2^13 + 21841*T2^12 - 5217*T2^11 + 26201*T2^10 - 7238*T2^9 + 20866*T2^8 - 2978*T2^7 + 7004*T2^6 - 414*T2^5 + 1693*T2^4 - 36*T2^3 + 141*T2^2 + 9