Newform orbit 15.12.e.a

• Label: 15.12.e.a
• Level: $15$
• Weight: $12$
• Character orbit: 15.e
• Analytic conductor: $11.525$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5251477084\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−61.8751	+	61.8751i	3.69942	−	420.872i		−	5609.07i	−4347.55	−	5470.55i	25812.6	+	26270.4i	−11712.7	−	11712.7i	220342.	+	220342.i	−177120.	−	3113.96i	607496.	+	69485.9i
2.2	−57.5586	+	57.5586i	345.894	+	239.801i		−	4577.97i	1295.31	+	6866.61i	−33711.7	+	6106.58i	53787.8	+	53787.8i	145622.	+	145622.i	62138.3	+	165891.i	−469788.	−	320675.i
2.3	−48.4604	+	48.4604i	−342.528	+	244.585i		−	2648.82i	−5084.66	+	4793.15i	4746.35	−	28451.7i	−47877.8	−	47877.8i	29116.1	+	29116.1i	57503.4	−	167554.i	14126.7	−	478683.i
2.4	−43.6110	+	43.6110i	−418.328	−	46.3532i		−	1755.85i	6653.83	−	2134.16i	20265.2	−	16222.2i	35410.1	+	35410.1i	−12741.2	−	12741.2i	172850.	+	38781.7i	−197108.	+	383253.i
2.5	−37.4902	+	37.4902i	369.789	−	201.005i		−	763.031i	6987.38	−	67.8135i	−6327.78	+	21399.2i	−46363.7	−	46363.7i	−48173.8	−	48173.8i	96341.3	−	148659.i	−259416.	+	264501.i
2.6	−37.2586	+	37.2586i	104.521	+	407.704i		−	728.404i	−1506.89	−	6823.30i	−19084.8	−	11296.2i	−5795.67	−	5795.67i	−49166.3	−	49166.3i	−155298.	+	85226.8i	310371.	+	198082.i
2.7	−26.1390	+	26.1390i	−159.262	−	389.593i			681.505i	−2800.46	+	6402.00i	14346.5	+	6020.60i	5513.09	+	5513.09i	−71346.5	−	71346.5i	−126418.	+	124095.i	−94140.6	−	240543.i
2.8	−21.4344	+	21.4344i	367.459	−	205.234i			1129.13i	−6189.39	−	3243.38i	−3477.19	+	12275.3i	45393.8	+	45393.8i	−68099.9	−	68099.9i	92905.0	−	150830.i	202186.	−	63146.0i
2.9	−3.46044	+	3.46044i	−124.374	+	402.092i			2024.05i	6126.08	+	3361.44i	−961.029	−	1821.80i	10635.2	+	10635.2i	−14091.1	−	14091.1i	−146209.	−	100019.i	−32831.0	+	9566.86i
2.10	−1.50709	+	1.50709i	357.838	+	221.583i			2043.46i	−3168.43	+	6228.10i	−873.238	+	205.348i	−31114.0	−	31114.0i	−6166.19	−	6166.19i	78948.9	+	158582.i	−4611.19	−	14161.4i
2.11	1.50709	−	1.50709i	−221.583	−	357.838i			2043.46i	3168.43	−	6228.10i	−873.238	−	205.348i	−31114.0	−	31114.0i	6166.19	+	6166.19i	−78948.9	+	158582.i	−4611.19	−	14161.4i
2.12	3.46044	−	3.46044i	−402.092	+	124.374i			2024.05i	−6126.08	−	3361.44i	−961.029	+	1821.80i	10635.2	+	10635.2i	14091.1	+	14091.1i	146209.	−	100019.i	−32831.0	+	9566.86i
2.13	21.4344	−	21.4344i	205.234	−	367.459i			1129.13i	6189.39	+	3243.38i	−3477.19	−	12275.3i	45393.8	+	45393.8i	68099.9	+	68099.9i	−92905.0	−	150830.i	202186.	−	63146.0i
2.14	26.1390	−	26.1390i	389.593	+	159.262i			681.505i	2800.46	−	6402.00i	14346.5	−	6020.60i	5513.09	+	5513.09i	71346.5	+	71346.5i	126418.	+	124095.i	−94140.6	−	240543.i
2.15	37.2586	−	37.2586i	−407.704	−	104.521i		−	728.404i	1506.89	+	6823.30i	−19084.8	+	11296.2i	−5795.67	−	5795.67i	49166.3	+	49166.3i	155298.	+	85226.8i	310371.	+	198082.i
2.16	37.4902	−	37.4902i	201.005	−	369.789i		−	763.031i	−6987.38	+	67.8135i	−6327.78	−	21399.2i	−46363.7	−	46363.7i	48173.8	+	48173.8i	−96341.3	−	148659.i	−259416.	+	264501.i
2.17	43.6110	−	43.6110i	46.3532	+	418.328i		−	1755.85i	−6653.83	+	2134.16i	20265.2	+	16222.2i	35410.1	+	35410.1i	12741.2	+	12741.2i	−172850.	+	38781.7i	−197108.	+	383253.i
2.18	48.4604	−	48.4604i	−244.585	+	342.528i		−	2648.82i	5084.66	−	4793.15i	4746.35	+	28451.7i	−47877.8	−	47877.8i	−29116.1	−	29116.1i	−57503.4	−	167554.i	14126.7	−	478683.i
2.19	57.5586	−	57.5586i	−239.801	−	345.894i		−	4577.97i	−1295.31	−	6866.61i	−33711.7	−	6106.58i	53787.8	+	53787.8i	−145622.	−	145622.i	−62138.3	+	165891.i	−469788.	−	320675.i
2.20	61.8751	−	61.8751i	420.872	−	3.69942i		−	5609.07i	4347.55	+	5470.55i	25812.6	−	26270.4i	−11712.7	−	11712.7i	−220342.	−	220342.i	177120.	−	3113.96i	607496.	+	69485.9i

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	15.12.e.a	✓	40
3.b	odd	2	1	inner	15.12.e.a	✓	40
5.b	even	2	1		75.12.e.d		40
5.c	odd	4	1	inner	15.12.e.a	✓	40
5.c	odd	4	1		75.12.e.d		40
15.d	odd	2	1		75.12.e.d		40
15.e	even	4	1	inner	15.12.e.a	✓	40
15.e	even	4	1		75.12.e.d		40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
15.12.e.a	✓	40	1.a	even	1	1	trivial
15.12.e.a	✓	40	3.b	odd	2	1	inner
15.12.e.a	✓	40	5.c	odd	4	1	inner
15.12.e.a	✓	40	15.e	even	4	1	inner
75.12.e.d		40	5.b	even	2	1
75.12.e.d		40	5.c	odd	4	1
75.12.e.d		40	15.d	odd	2	1
75.12.e.d		40	15.e	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(15, [\chi])\).