Newform orbit 153.4.n.a

• Label: 153.4.n.a
• Level: $153$
• Weight: $4$
• Character orbit: 153.n
• Analytic conductor: $9.027$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.02729223088\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(52\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 6 q^{3} + 396 q^{4} - 2 q^{5} - 40 q^{6} - 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−4.54352	+	2.62320i	−0.603478	−	5.16099i	9.76239	−	16.9090i	2.70511	+	10.0956i	16.2802	+	21.8660i	1.10879	−	4.13804i			60.4637i	−26.2716	+	6.22909i	−38.7736	−	38.7736i
4.2	−4.50464	+	2.60076i	5.10455	−	0.971362i	9.52786	−	16.5027i	−1.29791	−	4.84385i	−20.4679	+	17.6513i	−1.26468	+	4.71986i			57.5064i	25.1129	−	9.91674i	18.4443	+	18.4443i
4.3	−4.50259	+	2.59957i	−4.28169	+	2.94400i	9.51555	−	16.4814i	−1.94292	−	7.25109i	11.6256	−	24.3862i	2.93555	−	10.9556i			57.3523i	9.66574	−	25.2106i	27.5979	+	27.5979i
4.4	−4.45771	+	2.57366i	2.48939	+	4.56102i	9.24743	−	16.0170i	3.74830	+	13.9888i	−22.8355	−	13.9249i	6.60159	−	24.6375i			54.0204i	−14.6058	+	22.7084i	−52.7113	−	52.7113i
4.5	−4.16345	+	2.40377i	1.44826	+	4.99025i	7.55622	−	13.0878i	−2.82806	−	10.5544i	−18.0252	−	17.2954i	−5.33656	+	19.9163i			34.1934i	−22.8051	+	14.4543i	37.1449	+	37.1449i
4.6	−4.02974	+	2.32657i	−4.83579	−	1.90134i	6.82586	−	11.8227i	−0.472341	−	1.76280i	23.9106	−	3.58890i	−7.79590	+	29.0947i			26.2982i	19.7698	+	18.3890i	6.00469	+	6.00469i
4.7	−3.78020	+	2.18250i	2.27640	−	4.67098i	5.52661	−	9.57237i	−4.63503	−	17.2982i	1.58918	+	22.6255i	3.35702	−	12.5286i			13.3273i	−16.6360	−	21.2660i	55.2746	+	55.2746i
4.8	−3.50903	+	2.02594i	−5.19155	−	0.218568i	4.20886	−	7.28996i	4.97053	+	18.5503i	18.6601	−	9.75081i	2.80902	−	10.4834i			1.69255i	26.9045	+	2.26942i	−55.0234	−	55.0234i
4.9	−3.42463	+	1.97721i	5.19412	+	0.145183i	3.81872	−	6.61421i	5.11196	+	19.0781i	−18.0750	+	9.77267i	−7.35002	+	27.4307i		−	1.43375i	26.9578	+	1.50820i	−55.2279	−	55.2279i
4.10	−3.33157	+	1.92349i	−3.76912	−	3.57683i	3.39959	−	5.88826i	−3.05328	−	11.3950i	19.4371	+	4.66663i	6.11850	−	22.8346i		−	4.61951i	1.41256	+	26.9630i	32.0903	+	32.0903i
4.11	−3.04640	+	1.75884i	2.51232	−	4.54844i	2.18702	−	3.78802i	1.16904	+	4.36293i	0.346446	+	18.2751i	−4.35330	+	16.2467i		−	12.7550i	−14.3765	−	22.8542i	−11.2350	−	11.2350i
4.12	−3.00334	+	1.73398i	0.0205841	+	5.19611i	2.01337	−	3.48726i	1.85836	+	6.93548i	−9.07177	−	15.5700i	−4.34656	+	16.2216i		−	13.7791i	−26.9992	+	0.213915i	−17.6073	−	17.6073i
4.13	−2.97769	+	1.71917i	4.64954	+	2.31987i	1.91108	−	3.31009i	−3.74912	−	13.9919i	−17.8331	+	1.08548i	0.656617	−	2.45053i		−	14.3648i	16.2364	+	21.5727i	35.2181	+	35.2181i
4.14	−2.67737	+	1.54578i	−3.10816	+	4.16405i	0.778861	−	1.34903i	1.00527	+	3.75172i	1.88499	−	15.9532i	2.20209	−	8.21829i		−	19.9167i	−7.67865	−	25.8851i	−8.49080	−	8.49080i
4.15	−2.57011	+	1.48385i	4.73792	+	2.13356i	0.403646	−	0.699135i	0.0115990	+	0.0432882i	−15.3429	+	1.54691i	6.40143	−	23.8905i		−	21.3459i	17.8959	+	20.2173i	−0.0940442	−	0.0940442i
4.16	−2.26256	+	1.30629i	3.63487	−	3.71318i	−0.587222	+	1.01710i	2.85989	+	10.6733i	−3.37361	+	13.1495i	8.10598	−	30.2519i		−	23.9689i	−0.575485	−	26.9939i	−20.4130	−	20.4130i
4.17	−1.96930	+	1.13697i	−4.57176	+	2.46962i	−1.41458	+	2.45012i	−5.74261	−	21.4317i	6.19526	−	10.0614i	−4.65014	+	17.3546i		−	24.6249i	14.8020	−	22.5810i	35.6762	+	35.6762i
4.18	−1.81981	+	1.05067i	−1.90274	−	4.83524i	−1.79218	+	3.10415i	−1.06267	−	3.96594i	8.54288	+	6.80009i	−4.17816	+	15.5931i		−	24.3427i	−19.7592	+	18.4004i	6.10075	+	6.10075i
4.19	−1.10545	+	0.638233i	−1.79529	−	4.87616i	−3.18532	+	5.51713i	4.60025	+	17.1684i	5.09673	+	4.24455i	1.48258	−	5.53306i		−	18.3436i	−20.5539	+	17.5082i	−16.0428	−	16.0428i
4.20	−1.10271	+	0.636652i	−5.19383	+	0.155237i	−3.18935	+	5.52411i	2.91760	+	10.8886i	5.62848	−	3.47785i	−4.61264	+	17.2146i		−	18.3085i	26.9518	−	1.61255i	−10.1495	−	10.1495i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
17.c	even	4	1	inner
153.n	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	153.4.n.a	✓	208
9.c	even	3	1	inner	153.4.n.a	✓	208
17.c	even	4	1	inner	153.4.n.a	✓	208
153.n	even	12	1	inner	153.4.n.a	✓	208

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
153.4.n.a	✓	208	1.a	even	1	1	trivial
153.4.n.a	✓	208	9.c	even	3	1	inner
153.4.n.a	✓	208	17.c	even	4	1	inner
153.4.n.a	✓	208	153.n	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(153, [\chi])\).