Newform orbit 315.2.z.b

• Label: 315.2.z.b
• Level: $315$
• Weight: $2$
• Character orbit: 315.z
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
104.1	−1.37508	−	2.38171i	−1.72249	−	0.181752i	−2.78170	+	4.81804i	2.23561	+	0.0452693i	1.93568	+	4.35239i	0.187702	−	2.63908i	9.79992			2.93393	+	0.626130i	−2.96633	−	5.38682i
104.2	−1.37508	−	2.38171i	1.72249	+	0.181752i	−2.78170	+	4.81804i	−2.23561	−	0.0452693i	−1.93568	−	4.35239i	2.37937	+	1.15699i	9.79992			2.93393	+	0.626130i	2.96633	+	5.38682i
104.3	−1.19580	−	2.07119i	−0.122865	+	1.72769i	−1.85987	+	3.22139i	0.810771	−	2.08390i	3.72528	−	1.81149i	−1.82814	+	1.91257i	4.11294			−2.96981	−	0.424546i	−5.28567	+	0.812672i
104.4	−1.19580	−	2.07119i	0.122865	−	1.72769i	−1.85987	+	3.22139i	−0.810771	+	2.08390i	−3.72528	+	1.81149i	−2.57040	+	0.626932i	4.11294			−2.96981	−	0.424546i	5.28567	−	0.812672i
104.5	−1.18498	−	2.05244i	−0.887757	−	1.48724i	−1.80833	+	3.13213i	−0.317361	−	2.21343i	−2.00050	+	3.58441i	0.783979	+	2.52693i	3.83142			−1.42378	+	2.64062i	−4.16687	+	3.27423i
104.6	−1.18498	−	2.05244i	0.887757	+	1.48724i	−1.80833	+	3.13213i	0.317361	+	2.21343i	2.00050	−	3.58441i	−1.79640	−	1.94241i	3.83142			−1.42378	+	2.64062i	4.16687	−	3.27423i
104.7	−1.05869	−	1.83370i	−1.03202	+	1.39102i	−1.24164	+	2.15058i	−1.87597	−	1.21685i	3.64330	+	0.419758i	1.87249	−	1.86916i	1.02327			−0.869874	−	2.87112i	−0.245270	+	4.72824i
104.8	−1.05869	−	1.83370i	1.03202	−	1.39102i	−1.24164	+	2.15058i	1.87597	+	1.21685i	−3.64330	−	0.419758i	2.55499	−	0.687047i	1.02327			−0.869874	−	2.87112i	0.245270	−	4.72824i
104.9	−0.938003	−	1.62467i	−1.62365	−	0.603134i	−0.759700	+	1.31584i	−1.09295	+	1.95076i	0.543094	+	3.20363i	2.27915	+	1.34367i	−0.901609			2.27246	+	1.95855i	4.19453	−	0.0541402i
104.10	−0.938003	−	1.62467i	1.62365	+	0.603134i	−0.759700	+	1.31584i	1.09295	−	1.95076i	−0.543094	−	3.20363i	−0.0240802	−	2.64564i	−0.901609			2.27246	+	1.95855i	−4.19453	+	0.0541402i
104.11	−0.804619	−	1.39364i	−1.60125	+	0.660311i	−0.294824	+	0.510650i	2.20682	+	0.360486i	2.20863	+	1.70026i	−1.06757	+	2.42081i	−2.26959			2.12798	−	2.11464i	−1.27326	−	3.36557i
104.12	−0.804619	−	1.39364i	1.60125	−	0.660311i	−0.294824	+	0.510650i	−2.20682	−	0.360486i	−2.20863	−	1.70026i	−2.63026	−	0.285863i	−2.26959			2.12798	−	2.11464i	1.27326	+	3.36557i
104.13	−0.639446	−	1.10755i	−0.624436	−	1.61557i	0.182218	−	0.315610i	2.10896	−	0.743174i	−1.39004	+	1.72467i	−2.47149	−	0.944330i	−3.02386			−2.22016	+	2.01765i	−2.17167	−	1.86056i
104.14	−0.639446	−	1.10755i	0.624436	+	1.61557i	0.182218	−	0.315610i	−2.10896	+	0.743174i	1.39004	−	1.72467i	−0.417929	+	2.61253i	−3.02386			−2.22016	+	2.01765i	2.17167	+	1.86056i
104.15	−0.513079	−	0.888680i	−1.29309	+	1.15236i	0.473499	−	0.820124i	−0.229284	+	2.22428i	1.68753	+	0.557890i	−1.40254	−	2.24341i	−3.02409			0.344150	−	2.98019i	2.09431	−	0.937473i
104.16	−0.513079	−	0.888680i	1.29309	−	1.15236i	0.473499	−	0.820124i	0.229284	−	2.22428i	−1.68753	−	0.557890i	1.24158	+	2.33634i	−3.02409			0.344150	−	2.98019i	−2.09431	+	0.937473i
104.17	−0.430691	−	0.745979i	−0.0265195	−	1.73185i	0.629010	−	1.08948i	−2.22929	+	0.173963i	−1.28050	+	0.765674i	1.96991	−	1.76620i	−2.80640			−2.99859	+	0.0918554i	1.08991	+	1.58808i
104.18	−0.430691	−	0.745979i	0.0265195	+	1.73185i	0.629010	−	1.08948i	2.22929	−	0.173963i	1.28050	−	0.765674i	2.51453	−	0.822893i	−2.80640			−2.99859	+	0.0918554i	−1.08991	−	1.58808i
104.19	−0.139034	−	0.240814i	−1.62845	−	0.590047i	0.961339	−	1.66509i	0.326583	−	2.21209i	0.0843182	+	0.474190i	1.72305	−	2.00776i	−1.09077			2.30369	+	1.92172i	−0.578109	+	0.228910i
104.20	−0.139034	−	0.240814i	1.62845	+	0.590047i	0.961339	−	1.66509i	−0.326583	+	2.21209i	−0.0843182	−	0.474190i	2.60030	−	0.488328i	−1.09077			2.30369	+	1.92172i	0.578109	−	0.228910i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
9.d	odd	6	1	inner
35.c	odd	2	1	inner
45.h	odd	6	1	inner
63.o	even	6	1	inner
315.z	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.z.b	✓	80
3.b	odd	2	1		945.2.z.b		80
5.b	even	2	1	inner	315.2.z.b	✓	80
7.b	odd	2	1	inner	315.2.z.b	✓	80
9.c	even	3	1		945.2.z.b		80
9.d	odd	6	1	inner	315.2.z.b	✓	80
15.d	odd	2	1		945.2.z.b		80
21.c	even	2	1		945.2.z.b		80
35.c	odd	2	1	inner	315.2.z.b	✓	80
45.h	odd	6	1	inner	315.2.z.b	✓	80
45.j	even	6	1		945.2.z.b		80
63.l	odd	6	1		945.2.z.b		80
63.o	even	6	1	inner	315.2.z.b	✓	80
105.g	even	2	1		945.2.z.b		80
315.z	even	6	1	inner	315.2.z.b	✓	80
315.bg	odd	6	1		945.2.z.b		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.z.b	✓	80	1.a	even	1	1	trivial
315.2.z.b	✓	80	5.b	even	2	1	inner
315.2.z.b	✓	80	7.b	odd	2	1	inner
315.2.z.b	✓	80	9.d	odd	6	1	inner
315.2.z.b	✓	80	35.c	odd	2	1	inner
315.2.z.b	✓	80	45.h	odd	6	1	inner
315.2.z.b	✓	80	63.o	even	6	1	inner
315.2.z.b	✓	80	315.z	even	6	1	inner
945.2.z.b		80	3.b	odd	2	1
945.2.z.b		80	9.c	even	3	1
945.2.z.b		80	15.d	odd	2	1
945.2.z.b		80	21.c	even	2	1
945.2.z.b		80	45.j	even	6	1
945.2.z.b		80	63.l	odd	6	1
945.2.z.b		80	105.g	even	2	1
945.2.z.b		80	315.bg	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^40 + 33*T2^38 + 627*T2^36 + 8076*T2^34 + 78237*T2^32 + 590661*T2^30 + 3579858*T2^28 + 17599302*T2^26 + 70880757*T2^24 + 233723268*T2^22 + 631394847*T2^20 + 1385532684*T2^18 + 2456443260*T2^16 + 3457446795*T2^14 + 3823928757*T2^12 + 3209179095*T2^10 + 2003725548*T2^8 + 843386148*T2^6 + 224830053*T2^4 + 16572033*T2^2 + 962361