Newform orbit 315.2.t.b

• Label: 315.2.t.b
• Level: $315$
• Weight: $2$
• Character orbit: 315.t
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 4 q^{3} - 30 q^{4} + 15 q^{5} - q^{6} - 3 q^{7} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
101.1		−	2.80758i	0.0857668	−	1.72993i	−5.88252			0.500000	−	0.866025i	−4.85691	−	0.240797i	1.63885	−	2.07705i			10.9005i	−2.98529	−	0.296741i	−2.43144	−	1.40379i
101.2		−	2.34202i	−1.11747	+	1.32335i	−3.48504			0.500000	−	0.866025i	3.09931	+	2.61714i	−2.07734	−	1.63849i			3.47799i	−0.502517	−	2.95761i	−2.02825	−	1.17101i
101.3		−	1.82056i	1.12156	−	1.31989i	−1.31444			0.500000	−	0.866025i	−2.40293	−	2.04187i	−1.92980	+	1.80994i		−	1.24811i	−0.484202	−	2.96067i	−1.57665	−	0.910280i
101.4		−	1.34690i	−1.39768	−	1.02299i	0.185856			0.500000	−	0.866025i	−1.37786	+	1.88253i	1.86235	−	1.87927i		−	2.94413i	0.907000	+	2.85961i	−1.16645	−	0.673451i
101.5		−	1.34681i	1.51214	+	0.844642i	0.186101			0.500000	−	0.866025i	1.13757	−	2.03657i	−0.829875	−	2.51223i		−	2.94426i	1.57316	+	2.55444i	−1.16637	−	0.673405i
101.6		−	0.929301i	0.479000	+	1.66450i	1.13640			0.500000	−	0.866025i	1.54682	−	0.445136i	0.0655497	+	2.64494i		−	2.91466i	−2.54112	+	1.59459i	−0.804799	−	0.464651i
101.7		−	0.0264028i	1.26158	−	1.18677i	1.99930			0.500000	−	0.866025i	−0.0313339	−	0.0333092i	2.64251	+	0.130871i		−	0.105593i	0.183169	−	2.99440i	−0.0228655	−	0.0132014i
101.8			0.509846i	1.71720	+	0.226350i	1.74006			0.500000	−	0.866025i	−0.115404	+	0.875506i	−2.47153	−	0.944226i			1.90685i	2.89753	+	0.777376i	0.441540	+	0.254923i
101.9			0.692853i	−1.66836	+	0.465377i	1.51995			0.500000	−	0.866025i	−0.322438	−	1.15593i	−0.669425	−	2.55966i			2.43881i	2.56685	−	1.55283i	0.600029	+	0.346427i
101.10			0.917882i	−0.321911	−	1.70187i	1.15749			0.500000	−	0.866025i	1.56212	−	0.295477i	0.697139	+	2.55225i			2.89821i	−2.79275	+	1.09570i	0.794909	+	0.458941i
101.11			1.23569i	0.460303	+	1.66977i	0.473066			0.500000	−	0.866025i	−2.06332	+	0.568792i	2.47654	−	0.930987i			3.05595i	−2.57624	+	1.53720i	1.07014	+	0.617846i
101.12			1.75842i	−1.12723	+	1.31505i	−1.09206			0.500000	−	0.866025i	−2.31241	−	1.98215i	−1.42571	+	2.22875i			1.59655i	−0.458709	−	2.96472i	1.52284	+	0.879212i
101.13			2.16248i	1.73205		0.000611122i	−2.67633			0.500000	−	0.866025i	0.00132154	+	3.74553i	−0.841895	+	2.50823i		−	1.46255i	3.00000	−	0.00211699i	1.87276	+	1.08124i
101.14			2.34636i	0.822112	−	1.52451i	−3.50542			0.500000	−	0.866025i	3.57705	+	1.92897i	2.00080	−	1.73112i		−	3.53226i	−1.64826	−	2.50664i	2.03201	+	1.17318i
101.15			2.72808i	−1.55907	−	0.754526i	−5.44243			0.500000	−	0.866025i	2.05841	−	4.25326i	−2.63818	−	0.200022i		−	9.39122i	1.86138	+	2.35271i	2.36259	+	1.36404i
131.1		−	2.72808i	−1.55907	+	0.754526i	−5.44243			0.500000	+	0.866025i	2.05841	+	4.25326i	−2.63818	+	0.200022i			9.39122i	1.86138	−	2.35271i	2.36259	−	1.36404i
131.2		−	2.34636i	0.822112	+	1.52451i	−3.50542			0.500000	+	0.866025i	3.57705	−	1.92897i	2.00080	+	1.73112i			3.53226i	−1.64826	+	2.50664i	2.03201	−	1.17318i
131.3		−	2.16248i	1.73205		0.000611122i	−2.67633			0.500000	+	0.866025i	0.00132154	−	3.74553i	−0.841895	−	2.50823i			1.46255i	3.00000	+	0.00211699i	1.87276	−	1.08124i
131.4		−	1.75842i	−1.12723	−	1.31505i	−1.09206			0.500000	+	0.866025i	−2.31241	+	1.98215i	−1.42571	−	2.22875i		−	1.59655i	−0.458709	+	2.96472i	1.52284	−	0.879212i
131.5		−	1.23569i	0.460303	−	1.66977i	0.473066			0.500000	+	0.866025i	−2.06332	−	0.568792i	2.47654	+	0.930987i		−	3.05595i	−2.57624	−	1.53720i	1.07014	−	0.617846i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.t.b	✓	30
3.b	odd	2	1		945.2.t.b		30
7.d	odd	6	1		315.2.be.b	yes	30
9.c	even	3	1		945.2.be.b		30
9.d	odd	6	1		315.2.be.b	yes	30
21.g	even	6	1		945.2.be.b		30
63.i	even	6	1	inner	315.2.t.b	✓	30
63.t	odd	6	1		945.2.t.b		30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.t.b	✓	30	1.a	even	1	1	trivial
315.2.t.b	✓	30	63.i	even	6	1	inner
315.2.be.b	yes	30	7.d	odd	6	1
315.2.be.b	yes	30	9.d	odd	6	1
945.2.t.b		30	3.b	odd	2	1
945.2.t.b		30	63.t	odd	6	1
945.2.be.b		30	9.c	even	3	1
945.2.be.b		30	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^30 + 45*T2^28 + 897*T2^26 + 10463*T2^24 + 79503*T2^22 + 414717*T2^20 + 1525404*T2^18 + 4002126*T2^16 + 7492806*T2^14 + 9913508*T2^12 + 9074679*T2^10 + 5537526*T2^8 + 2112877*T2^6 + 446751*T2^4 + 39042*T2^2 + 27