Newform orbit 109.5.j.a

• Label: 109.5.j.a
• Level: $109$
• Weight: $5$
• Character orbit: 109.j
• Analytic conductor: $11.267$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	109.j (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2673259761\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(35\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 420 q - 12 q^{2} - 12 q^{3} - 18 q^{4} - 12 q^{5} + 72 q^{6} + 138 q^{7} - 378 q^{8} - 216 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−7.56526	−	2.02711i	−2.60058	−	14.7486i	39.2676	+	22.6712i	8.87301	+	3.22951i	−10.2230	+	116.849i	28.1620	+	10.2501i	−162.502	−	162.502i	−134.644	+	49.0065i	−60.5800	−	42.4186i
2.2	−7.09419	−	1.90088i	1.94730	+	11.0437i	32.8578	+	18.9705i	7.17603	+	2.61186i	7.17824	−	82.0476i	−65.7690	−	23.9380i	−113.946	−	113.946i	−42.0558	+	15.3071i	−45.9433	−	32.1699i
2.3	−6.80446	−	1.82325i	1.03056	+	5.84460i	29.1200	+	16.8124i	−44.1382	−	16.0650i	3.64375	−	41.6483i	47.4668	+	17.2765i	−87.7930	−	87.7930i	43.0178	−	15.6572i	271.046	+	189.789i
2.4	−6.59480	−	1.76707i	0.399352	+	2.26484i	26.5125	+	15.3070i	29.0535	+	10.5746i	1.36848	−	15.6418i	57.6645	+	20.9882i	−70.5523	−	70.5523i	71.1451	−	25.8947i	−172.916	−	121.077i
2.5	−6.09567	−	1.63333i	−0.853925	−	4.84285i	20.6330	+	11.9125i	−6.70964	−	2.44211i	−2.70473	+	30.9151i	−18.4983	−	6.73282i	−34.9177	−	34.9177i	53.3911	−	19.4328i	36.9110	+	25.8454i
2.6	−5.52143	−	1.47946i	−2.07924	−	11.7919i	14.4410	+	8.33752i	−33.8929	−	12.3360i	−5.96538	+	68.1846i	−15.0840	−	5.49013i	−2.72845	−	2.72845i	−58.6116	+	21.3329i	168.887	+	118.256i
2.7	−5.35729	−	1.43548i	−1.36245	−	7.72686i	12.7836	+	7.38061i	30.8331	+	11.2223i	−3.79271	+	43.3509i	−75.2440	−	27.3866i	4.85828	+	4.85828i	18.2670	−	6.64864i	−149.072	−	104.382i
2.8	−4.56344	−	1.22277i	1.73252	+	9.82563i	5.47344	+	3.16009i	−8.20659	−	2.98695i	4.10822	−	46.9572i	18.9211	+	6.88673i	32.3371	+	32.3371i	−17.4262	+	6.34262i	33.7979	+	23.6656i
2.9	−4.51567	−	1.20997i	2.88047	+	16.3360i	5.07084	+	2.92765i	−11.6587	−	4.24340i	6.75877	−	77.2531i	2.47161	+	0.899591i	33.5353	+	33.5353i	−182.451	+	66.4069i	47.5122	+	33.2684i
2.10	−4.08179	−	1.09371i	2.03067	+	11.5165i	1.60841	+	0.928616i	41.0829	+	14.9529i	4.30697	−	49.2289i	16.1812	+	5.88948i	42.2597	+	42.2597i	−52.3908	+	19.0687i	−151.337	−	105.968i
2.11	−3.75814	−	1.00699i	−2.98063	−	16.9040i	−0.746838	−	0.431187i	19.4501	+	7.07925i	−5.82054	+	66.5290i	30.4982	+	11.1004i	46.3909	+	46.3909i	−200.746	+	73.0656i	−65.9673	−	46.1908i
2.12	−3.25924	−	0.873311i	−1.18016	−	6.69303i	−3.99643	−	2.30734i	−7.44044	−	2.70810i	−1.99867	+	22.8449i	75.8902	+	27.6218i	49.1852	+	49.1852i	32.7112	−	11.9059i	21.8852	+	15.3242i
2.13	−2.81604	−	0.754556i	0.501495	+	2.84412i	−6.49566	−	3.75027i	−30.2545	−	11.0118i	0.733818	−	8.38758i	−80.3347	−	29.2394i	48.4461	+	48.4461i	68.2776	−	24.8510i	76.8891	+	53.8383i
2.14	−2.60913	−	0.699114i	0.477429	+	2.70764i	−7.53760	−	4.35184i	7.17086	+	2.60998i	0.647272	−	7.39835i	−1.26129	−	0.459073i	47.1844	+	47.1844i	69.0118	−	25.1182i	−16.8850	−	11.8230i
2.15	−2.06658	−	0.553739i	−1.19584	−	6.78194i	−9.89228	−	5.71131i	37.3383	+	13.5900i	−1.28412	+	14.6776i	−20.9688	−	7.63202i	41.4861	+	41.4861i	31.5504	−	11.4834i	−69.6372	−	48.7605i
2.16	−1.17373	−	0.314500i	−2.74263	−	15.5542i	−12.5777	−	7.26173i	−15.5759	−	5.66916i	−1.67270	+	19.1190i	−56.3716	−	20.5176i	26.2266	+	26.2266i	−158.298	+	57.6156i	16.4989	+	11.5527i
2.17	−0.212520	−	0.0569444i	1.95441	+	11.0840i	−13.8145	−	7.97580i	−34.6205	−	12.6008i	0.215822	−	2.46686i	30.5676	+	11.1257i	4.97087	+	4.97087i	−42.9203	+	15.6217i	6.63998	+	4.64936i
2.18	−0.0508662	−	0.0136296i	0.693113	+	3.93084i	−13.8540	−	7.99861i	19.9045	+	7.24466i	0.0183196	−	0.209394i	−7.59613	−	2.76476i	1.19147	+	1.19147i	61.1440	−	22.2546i	−0.913727	−	0.639798i
2.19	0.513747	+	0.137658i	−1.37614	−	7.80445i	−13.6114	−	7.85856i	−31.8100	−	11.5779i	0.367361	−	4.19895i	19.6224	+	7.14198i	−11.9285	−	11.9285i	17.0994	−	6.22367i	−14.7485	−	10.3270i
2.20	0.633946	+	0.169865i	2.63967	+	14.9703i	−13.4834	−	7.78463i	18.9528	+	6.89827i	−0.869530	+	9.93878i	−77.1534	−	28.0815i	−14.6507	−	14.6507i	−141.028	+	51.3300i	10.8433	+	7.59257i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
109.j	odd	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	109.5.j.a	✓	420
109.j	odd	36	1	inner	109.5.j.a	✓	420

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
109.5.j.a	✓	420	1.a	even	1	1	trivial
109.5.j.a	✓	420	109.j	odd	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(109, [\chi])\).