Newform orbit 72.4.l.b

• Label: 72.4.l.b
• Level: $72$
• Weight: $4$
• Character orbit: 72.l
• Analytic conductor: $4.248$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24813752041\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 3 q^{2} + 6 q^{3} - 17 q^{4} - 3 q^{6} + 42 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−2.81730	−	0.250612i	2.93713	−	4.28640i	7.87439	+	1.41210i	3.52304	+	6.10209i	−9.34902	+	11.3400i	16.9049	+	9.76005i	−21.8306	−	5.95172i	−9.74649	−	25.1795i	−8.39622	−	18.0744i
11.2	−2.78774	−	0.478024i	−3.91093	−	3.42120i	7.54299	+	2.66521i	−7.74992	−	13.4233i	9.26725	+	11.4069i	4.08207	+	2.35678i	−19.7539	−	11.0356i	3.59080	+	26.7602i	15.1881	+	41.1252i
11.3	−2.75189	−	0.653547i	3.42450	+	3.90805i	7.14575	+	3.59697i	7.56234	+	13.0984i	−6.86974	−	12.9926i	−23.1409	−	13.3604i	−17.3135	−	14.5685i	−3.54563	+	26.7662i	−12.2503	−	40.9875i
11.4	−2.69888	+	0.846204i	5.16561	+	0.562595i	6.56788	−	4.56760i	−8.75283	−	15.1603i	−14.4174	+	2.85278i	−14.7005	−	8.48731i	−13.8608	+	17.8852i	26.3670	+	5.81229i	36.4516	+	33.5092i
11.5	−2.55734	+	1.20831i	−4.70468	+	2.20589i	5.08000	−	6.18010i	−1.01248	−	1.75366i	9.36610	−	11.3259i	−16.3356	−	9.43138i	−5.52384	+	21.9428i	17.2681	−	20.7560i	4.70820	+	3.26133i
11.6	−2.55635	+	1.21041i	0.355853	+	5.18395i	5.06981	−	6.18846i	2.33412	+	4.04281i	−7.18439	−	12.8213i	28.2737	+	16.3238i	−5.46963	+	21.9564i	−26.7467	+	3.68945i	−10.8603	−	7.50958i
11.7	−2.26617	+	1.69248i	−1.59153	−	4.94642i	2.27101	−	7.67089i	4.74971	+	8.22674i	11.9784	+	8.51575i	−16.7522	−	9.67189i	7.83636	+	21.2271i	−21.9341	+	15.7448i	−24.6872	−	10.6043i
11.8	−1.94193	−	2.05643i	3.42450	+	3.90805i	−0.457807	+	7.98689i	−7.56234	−	13.0984i	1.38648	−	14.6314i	23.1409	+	13.3604i	17.3135	−	14.5685i	−3.54563	+	26.7662i	−12.2503	+	40.9875i
11.9	−1.80785	−	2.17524i	−3.91093	−	3.42120i	−1.46335	+	7.86502i	7.74992	+	13.4233i	−0.371551	+	14.6922i	−4.08207	−	2.35678i	19.7539	−	11.0356i	3.59080	+	26.7602i	15.1881	−	41.1252i
11.10	−1.62569	−	2.31455i	2.93713	−	4.28640i	−2.71428	+	7.52547i	−3.52304	−	6.10209i	−14.6960	+	0.170208i	−16.9049	−	9.76005i	21.8306	−	5.95172i	−9.74649	−	25.1795i	−8.39622	+	18.0744i
11.11	−1.47499	+	2.41338i	5.12181	−	0.875794i	−3.64882	−	7.11942i	3.60771	+	6.24874i	−5.44099	+	13.6527i	6.44991	+	3.72386i	22.5638	+	1.69509i	25.4660	−	8.97131i	−20.4019	−	0.510040i
11.12	−1.16891	+	2.57559i	−4.87955	−	1.78605i	−5.26729	−	6.02127i	−2.83055	−	4.90265i	10.3039	−	10.4800i	25.8587	+	14.9296i	21.6653	−	6.52801i	20.6201	+	17.4302i	15.9359	−	1.55955i
11.13	−0.867121	+	2.69223i	0.0478653	+	5.19593i	−6.49620	−	4.66898i	−8.51655	−	14.7511i	−14.0301	−	4.37664i	−9.57807	−	5.52990i	18.2030	−	13.4407i	−26.9954	+	0.497410i	47.0983	−	10.1375i
11.14	−0.616605	−	2.76040i	5.16561	+	0.562595i	−7.23960	+	3.40415i	8.75283	+	15.1603i	−1.63215	−	14.6060i	14.7005	+	8.48731i	13.8608	+	17.8852i	26.3670	+	5.81229i	36.4516	−	33.5092i
11.15	−0.250335	+	2.81733i	−3.80768	+	3.53576i	−7.87466	−	1.41055i	10.8653	+	18.8192i	−9.00820	−	11.6126i	−10.5583	−	6.09581i	5.94529	−	21.8324i	1.99678	−	26.9261i	−55.7398	+	25.8999i
11.16	−0.232248	−	2.81888i	−4.70468	+	2.20589i	−7.89212	+	1.30935i	1.01248	+	1.75366i	7.31078	+	12.7496i	16.3356	+	9.43138i	5.52384	+	21.9428i	17.2681	−	20.7560i	4.70820	−	3.26133i
11.17	−0.229927	−	2.81907i	0.355853	+	5.18395i	−7.89427	+	1.29636i	−2.33412	−	4.04281i	14.5321	−	2.19510i	−28.2737	−	16.3238i	5.46963	+	21.9564i	−26.7467	+	3.68945i	−10.8603	+	7.50958i
11.18	−0.0437026	+	2.82809i	1.76134	−	4.88853i	−7.99618	−	0.247190i	−4.95891	−	8.58909i	13.7482	+	5.19486i	−11.4514	−	6.61147i	1.04853	−	22.6031i	−20.7954	−	17.2207i	24.5074	−	13.6489i
11.19	0.332650	−	2.80880i	−1.59153	−	4.94642i	−7.77869	−	1.86869i	−4.74971	−	8.22674i	−14.4229	+	2.82487i	16.7522	+	9.67189i	−7.83636	+	21.2271i	−21.9341	+	15.7448i	−24.6872	+	10.6043i
11.20	0.960011	+	2.66052i	4.10749	+	3.18254i	−6.15676	+	5.10826i	1.46312	+	2.53420i	−4.52397	+	13.9833i	4.48829	+	2.59132i	−19.5012	−	11.4762i	6.74292	+	26.1445i	−5.33769	+	6.32553i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
9.d	odd	6	1	inner
72.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	72.4.l.b	✓	64
3.b	odd	2	1		216.4.l.b		64
4.b	odd	2	1		288.4.p.b		64
8.b	even	2	1		288.4.p.b		64
8.d	odd	2	1	inner	72.4.l.b	✓	64
9.c	even	3	1		216.4.l.b		64
9.d	odd	6	1	inner	72.4.l.b	✓	64
12.b	even	2	1		864.4.p.b		64
24.f	even	2	1		216.4.l.b		64
24.h	odd	2	1		864.4.p.b		64
36.f	odd	6	1		864.4.p.b		64
36.h	even	6	1		288.4.p.b		64
72.j	odd	6	1		288.4.p.b		64
72.l	even	6	1	inner	72.4.l.b	✓	64
72.n	even	6	1		864.4.p.b		64
72.p	odd	6	1		216.4.l.b		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.4.l.b	✓	64	1.a	even	1	1	trivial
72.4.l.b	✓	64	8.d	odd	2	1	inner
72.4.l.b	✓	64	9.d	odd	6	1	inner
72.4.l.b	✓	64	72.l	even	6	1	inner
216.4.l.b		64	3.b	odd	2	1
216.4.l.b		64	9.c	even	3	1
216.4.l.b		64	24.f	even	2	1
216.4.l.b		64	72.p	odd	6	1
288.4.p.b		64	4.b	odd	2	1
288.4.p.b		64	8.b	even	2	1
288.4.p.b		64	36.h	even	6	1
288.4.p.b		64	72.j	odd	6	1
864.4.p.b		64	12.b	even	2	1
864.4.p.b		64	24.h	odd	2	1
864.4.p.b		64	36.f	odd	6	1
864.4.p.b		64	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^64 + 2451*T5^62 + 3328326*T5^60 + 3108844719*T5^58 + 2205690843492*T5^56 + 1247931844241193*T5^54 + 580891813526360313*T5^52 + 226852367383603045494*T5^50 + 75353158194809438011851*T5^48 + 21485122714593194963682303*T5^46 + 5292346811242857158070674568*T5^44 + 1130800676305753272977739943425*T5^42 + 210109660665849343788841871389296*T5^40 + 33983869060102527963436340989113579*T5^38 + 4785191439029269019493732539028636831*T5^36 + 585986156556349928397028001735213216574*T5^34 + 62302633308931475599757120803741823298573*T5^32 + 5735809368657799197683845754375976522966597*T5^30 + 455736554185564720252464659081327113180593084*T5^28 + 31111278539969518582393848577981239199088277687*T5^26 + 1815300849229240665070279974714630956842170995478*T5^24 + 89916963089255662992342883698192866175597263327871*T5^22 + 3751797624747007014832199032259484953536736053552845*T5^20 + 130481911666910620566820600169760734931147971767012092*T5^18 + 3737466086819162585770207287303708845543624395236358560*T5^16 + 86628857131570330197518754292365992551090163709489286848*T5^14 + 1594239535163809307987563563639427780974498000770703997440*T5^12 + 22541474037071353257666395046400372558984910837508345990144*T5^10 + 237967783937941679200217660211744707838891148679800231759872*T5^8 + 1745596325293847820917254366310921329901990898879208544010240*T5^6 + 8991172089035538414921951668988300387400466499090592454148096*T5^4 + 26493850539150216480297620126847064217452267425490726079692800*T5^2 + 49635371541145740490635260402280294519678770459825972183040000