Newform orbit 30.10.e.a

• Label: 30.10.e.a
• Level: $30$
• Weight: $10$
• Character orbit: 30.e
• Analytic conductor: $15.451$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	30.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4510750849\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 296 q^{3} + 4544 q^{6} + 9756 q^{7}+O(q^{10}) \)

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	−11.3137	+	11.3137i	−138.928	+	19.5470i		−	256.000i	442.457	+	1325.65i	1350.64	−	1792.94i	4229.25	+	4229.25i	2896.31	+	2896.31i	18918.8	−	5431.23i	−20003.9	−	9992.22i
17.2	−11.3137	+	11.3137i	−125.652	+	62.4065i		−	256.000i	−1151.16	−	792.434i	715.542	−	2127.64i	−1314.95	−	1314.95i	2896.31	+	2896.31i	11893.9	−	15683.0i	21989.3	−	4058.54i
17.3	−11.3137	+	11.3137i	−89.2589	−	108.240i		−	256.000i	1142.31	−	805.147i	2234.44	+	214.744i	2602.86	+	2602.86i	2896.31	+	2896.31i	−3748.70	+	19322.7i	−3814.52	+	22032.9i
17.4	−11.3137	+	11.3137i	−65.1884	−	124.232i		−	256.000i	−1015.91	+	959.718i	2143.04	+	667.997i	−7930.01	−	7930.01i	2896.31	+	2896.31i	−11184.0	+	16196.9i	635.693	−	22351.6i
17.5	−11.3137	+	11.3137i	43.1765	+	133.487i		−	256.000i	−445.093	+	1324.77i	−1998.72	−	1021.75i	−3280.20	−	3280.20i	2896.31	+	2896.31i	−15954.6	+	11527.0i	−9952.42	−	20023.7i
17.6	−11.3137	+	11.3137i	73.5417	−	119.476i		−	256.000i	−1371.94	+	266.282i	519.692	+	2183.75i	8083.49	+	8083.49i	2896.31	+	2896.31i	−8866.24	−	17573.0i	12509.1	−	18534.4i
17.7	−11.3137	+	11.3137i	85.7313	+	111.055i		−	256.000i	−520.827	−	1296.87i	−2226.38	−	286.501i	5788.99	+	5788.99i	2896.31	+	2896.31i	−4983.28	+	19041.7i	20564.9	+	8779.89i
17.8	−11.3137	+	11.3137i	100.077	−	98.3235i		−	256.000i	257.681	−	1373.58i	−19.8436	+	2244.65i	−6195.32	−	6195.32i	2896.31	+	2896.31i	347.984	−	19679.9i	12625.0	+	18455.6i
17.9	−11.3137	+	11.3137i	140.295	−	0.428453i		−	256.000i	1192.41	+	728.894i	−1582.41	+	1592.11i	454.888	+	454.888i	2896.31	+	2896.31i	19682.6	−	120.220i	−21737.1	+	5244.06i
17.10	11.3137	−	11.3137i	−133.487	−	43.1765i		−	256.000i	445.093	−	1324.77i	−1998.72	+	1021.75i	−3280.20	−	3280.20i	−2896.31	−	2896.31i	15954.6	+	11527.0i	−9952.42	−	20023.7i
17.11	11.3137	−	11.3137i	−111.055	−	85.7313i		−	256.000i	520.827	+	1296.87i	−2226.38	+	286.501i	5788.99	+	5788.99i	−2896.31	−	2896.31i	4983.28	+	19041.7i	20564.9	+	8779.89i
17.12	11.3137	−	11.3137i	−62.4065	+	125.652i		−	256.000i	1151.16	+	792.434i	715.542	+	2127.64i	−1314.95	−	1314.95i	−2896.31	−	2896.31i	−11893.9	−	15683.0i	21989.3	−	4058.54i
17.13	11.3137	−	11.3137i	−19.5470	+	138.928i		−	256.000i	−442.457	−	1325.65i	1350.64	+	1792.94i	4229.25	+	4229.25i	−2896.31	−	2896.31i	−18918.8	−	5431.23i	−20003.9	−	9992.22i
17.14	11.3137	−	11.3137i	0.428453	−	140.295i		−	256.000i	−1192.41	−	728.894i	−1582.41	−	1592.11i	454.888	+	454.888i	−2896.31	−	2896.31i	−19682.6	−	120.220i	−21737.1	+	5244.06i
17.15	11.3137	−	11.3137i	98.3235	−	100.077i		−	256.000i	−257.681	+	1373.58i	−19.8436	−	2244.65i	−6195.32	−	6195.32i	−2896.31	−	2896.31i	−347.984	−	19679.9i	12625.0	+	18455.6i
17.16	11.3137	−	11.3137i	108.240	+	89.2589i		−	256.000i	−1142.31	+	805.147i	2234.44	−	214.744i	2602.86	+	2602.86i	−2896.31	−	2896.31i	3748.70	+	19322.7i	−3814.52	+	22032.9i
17.17	11.3137	−	11.3137i	119.476	−	73.5417i		−	256.000i	1371.94	−	266.282i	519.692	−	2183.75i	8083.49	+	8083.49i	−2896.31	−	2896.31i	8866.24	−	17573.0i	12509.1	−	18534.4i
17.18	11.3137	−	11.3137i	124.232	+	65.1884i		−	256.000i	1015.91	−	959.718i	2143.04	−	667.997i	−7930.01	−	7930.01i	−2896.31	−	2896.31i	11184.0	+	16196.9i	635.693	−	22351.6i
23.1	−11.3137	−	11.3137i	−138.928	−	19.5470i			256.000i	442.457	−	1325.65i	1350.64	+	1792.94i	4229.25	−	4229.25i	2896.31	−	2896.31i	18918.8	+	5431.23i	−20003.9	+	9992.22i
23.2	−11.3137	−	11.3137i	−125.652	−	62.4065i			256.000i	−1151.16	+	792.434i	715.542	+	2127.64i	−1314.95	+	1314.95i	2896.31	−	2896.31i	11893.9	+	15683.0i	21989.3	+	4058.54i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	30.10.e.a	✓	36
3.b	odd	2	1	inner	30.10.e.a	✓	36
5.c	odd	4	1	inner	30.10.e.a	✓	36
15.e	even	4	1	inner	30.10.e.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.10.e.a	✓	36	1.a	even	1	1	trivial
30.10.e.a	✓	36	3.b	odd	2	1	inner
30.10.e.a	✓	36	5.c	odd	4	1	inner
30.10.e.a	✓	36	15.e	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(30, [\chi])\).