Newform orbit 40.5.i.a

• Label: 40.5.i.a
• Level: $40$
• Weight: $5$
• Character orbit: 40.i
• Analytic conductor: $4.135$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	40.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.13479852335\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{2} + 32 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−3.99772	−	0.134931i	−9.27319	+	9.27319i	15.9636	+	1.07883i	−21.1329	+	13.3567i	38.3229	−	35.8204i	26.2670	−	26.2670i	−63.6724	−	6.46686i		−	90.9842i	86.2855	−	50.5451i
13.2	−3.98174	+	0.381759i	−3.10315	+	3.10315i	15.7085	−	3.04013i	18.7844	−	16.4969i	11.1713	−	13.5406i	−52.9781	+	52.9781i	−61.3867	+	18.1019i			61.7410i	−68.4966	+	72.8575i
13.3	−3.82786	+	1.16082i	12.1818	−	12.1818i	13.3050	−	8.88689i	−21.6293	−	12.5369i	−32.4892	+	60.7708i	−17.4020	+	17.4020i	−40.6136	+	49.4625i		−	215.790i	97.3469	+	22.8819i
13.4	−3.55430	−	1.83492i	6.54244	−	6.54244i	9.26616	+	13.0437i	20.3139	+	14.5721i	−35.2586	+	11.2490i	24.6615	−	24.6615i	−9.00061	−	63.3639i		−	4.60695i	−45.4630	−	89.0680i
13.5	−3.25500	+	2.32487i	1.65143	−	1.65143i	5.19000	−	15.1349i	4.54838	+	24.5828i	−1.53605	+	9.21475i	28.3940	−	28.3940i	18.2931	+	61.3300i			75.5456i	−71.9566	−	69.4424i
13.6	−2.86960	−	2.78664i	1.60861	−	1.60861i	0.469241	+	15.9931i	−24.8651	−	2.59384i	−9.09870	+	0.133450i	−23.8147	+	23.8147i	43.2206	−	47.2015i			75.8247i	64.1248	+	76.7334i
13.7	−2.32487	+	3.25500i	−1.65143	+	1.65143i	−5.19000	−	15.1349i	−4.54838	−	24.5828i	−1.53605	−	9.21475i	28.3940	−	28.3940i	61.3300	+	18.2931i			75.5456i	90.5912	+	42.3466i
13.8	−2.22419	−	3.32460i	−9.32929	+	9.32929i	−6.10592	+	14.7891i	13.0913	−	21.2983i	51.7663	+	10.2660i	43.0150	−	43.0150i	62.7486	−	12.5941i		−	93.0715i	−99.9259	+	3.84844i
13.9	−1.16082	+	3.82786i	−12.1818	+	12.1818i	−13.3050	−	8.88689i	21.6293	+	12.5369i	−32.4892	−	60.7708i	−17.4020	+	17.4020i	49.4625	−	40.6136i		−	215.790i	−73.0972	+	68.2408i
13.10	−0.401499	−	3.97980i	7.77558	−	7.77558i	−15.6776	+	3.19577i	13.6749	−	20.9284i	−34.0671	−	27.8233i	−5.91736	+	5.91736i	19.0131	+	61.1106i		−	39.9192i	−88.7812	−	46.0207i
13.11	−0.381759	+	3.98174i	3.10315	−	3.10315i	−15.7085	−	3.04013i	−18.7844	+	16.4969i	11.1713	+	13.5406i	−52.9781	+	52.9781i	18.1019	−	61.3867i			61.7410i	−58.5153	−	81.0923i
13.12	−0.133745	−	3.99776i	−4.11430	+	4.11430i	−15.9642	+	1.06936i	3.95777	+	24.6847i	16.9983	+	15.8977i	−12.8562	+	12.8562i	6.41019	+	63.6782i			47.1451i	98.1544	−	19.1237i
13.13	0.134931	+	3.99772i	9.27319	−	9.27319i	−15.9636	+	1.07883i	21.1329	−	13.3567i	38.3229	+	35.8204i	26.2670	−	26.2670i	−6.46686	−	63.6724i		−	90.9842i	56.2480	+	82.6811i
13.14	1.83492	+	3.55430i	−6.54244	+	6.54244i	−9.26616	+	13.0437i	−20.3139	−	14.5721i	−35.2586	−	11.2490i	24.6615	−	24.6615i	−63.3639	−	9.00061i		−	4.60695i	14.5195	−	98.9403i
13.15	2.00403	−	3.46177i	4.26672	−	4.26672i	−7.96776	−	13.8750i	−24.9456	−	1.64905i	−6.21980	−	23.3211i	50.3926	−	50.3926i	−63.9996	−	0.223240i			44.5901i	−55.7002	+	83.0511i
13.16	2.35645	−	3.23221i	−9.60402	+	9.60402i	−4.89431	−	15.2330i	−12.7978	−	21.4760i	8.41080	+	53.6735i	−60.7617	+	60.7617i	−60.7695	−	20.0765i		−	103.474i	−99.5720	−	9.24206i
13.17	2.78664	+	2.86960i	−1.60861	+	1.60861i	−0.469241	+	15.9931i	24.8651	+	2.59384i	−9.09870	−	0.133450i	−23.8147	+	23.8147i	−47.2015	+	43.2206i			75.8247i	61.8468	+	78.5810i
13.18	3.23221	−	2.35645i	9.60402	−	9.60402i	4.89431	−	15.2330i	12.7978	+	21.4760i	8.41080	−	53.6735i	−60.7617	+	60.7617i	−20.0765	−	60.7695i		−	103.474i	91.9720	+	39.2575i
13.19	3.32460	+	2.22419i	9.32929	−	9.32929i	6.10592	+	14.7891i	−13.0913	+	21.2983i	51.7663	−	10.2660i	43.0150	−	43.0150i	−12.5941	+	62.7486i		−	93.0715i	−90.8948	+	41.6909i
13.20	3.46177	−	2.00403i	−4.26672	+	4.26672i	7.96776	−	13.8750i	24.9456	+	1.64905i	−6.21980	+	23.3211i	50.3926	−	50.3926i	−0.223240	−	63.9996i			44.5901i	89.6606	−	44.2829i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
8.b	even	2	1	inner
40.i	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	40.5.i.a	✓	44
4.b	odd	2	1		160.5.m.a		44
5.c	odd	4	1	inner	40.5.i.a	✓	44
8.b	even	2	1	inner	40.5.i.a	✓	44
8.d	odd	2	1		160.5.m.a		44
20.e	even	4	1		160.5.m.a		44
40.i	odd	4	1	inner	40.5.i.a	✓	44
40.k	even	4	1		160.5.m.a		44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.5.i.a	✓	44	1.a	even	1	1	trivial
40.5.i.a	✓	44	5.c	odd	4	1	inner
40.5.i.a	✓	44	8.b	even	2	1	inner
40.5.i.a	✓	44	40.i	odd	4	1	inner
160.5.m.a		44	4.b	odd	2	1
160.5.m.a		44	8.d	odd	2	1
160.5.m.a		44	20.e	even	4	1
160.5.m.a		44	40.k	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(40, [\chi])\).