Newform orbit 69.5.h.a

• Label: 69.5.h.a
• Level: $69$
• Weight: $5$
• Character orbit: 69.h
• Analytic conductor: $7.133$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13252745279\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(30\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 300 q - 19 q^{3} + 234 q^{4} - 10 q^{6} - 18 q^{7} - 111 q^{9}+O(q^{10}) \)

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	−4.11610	+	6.40478i	8.97488	−	0.671914i	−17.4323	−	38.1713i	6.99421	−	23.8201i	−32.6381	+	60.2478i	−31.6181	+	36.4893i	195.658	+	28.1313i	80.0971	−	12.0607i	123.774	+	142.842i
2.2	−4.08077	+	6.34980i	−1.82217	+	8.81361i	−17.0207	−	37.2701i	−1.39862	+	4.76328i	−48.5289	−	47.5367i	54.0931	−	62.4268i	186.576	+	26.8256i	−74.3594	−	32.1197i	−24.5384	−	28.3188i
2.3	−3.63251	+	5.65229i	−6.90793	−	5.76893i	−12.1066	−	26.5099i	2.68502	−	9.14434i	57.7008	−	18.0899i	3.93980	−	4.54677i	87.4108	+	12.5678i	14.4389	+	79.7027i	41.9331	+	48.3934i
2.4	−3.39074	+	5.27609i	3.68875	−	8.20933i	−9.69338	−	21.2255i	−7.37394	+	25.1133i	30.8056	+	47.2979i	−0.916617	+	1.05783i	45.5297	+	6.54619i	−53.7862	−	60.5644i	−107.497	−	124.058i
2.5	−3.15045	+	4.90219i	−7.93709	+	4.24295i	−7.45954	−	16.3341i	−12.0836	+	41.1529i	4.20561	−	52.2764i	−46.0495	+	53.1439i	11.2870	+	1.62282i	44.9947	−	67.3534i	−163.671	−	188.886i
2.6	−2.72930	+	4.24687i	2.47487	+	8.65304i	−3.94020	−	8.62784i	4.64119	−	15.8064i	−43.5030	−	13.1063i	−37.0863	+	42.7998i	−32.5548	−	4.68067i	−68.7501	+	42.8302i	54.4607	+	62.8510i
2.7	−2.52768	+	3.93314i	7.84620	+	4.40876i	−2.43382	−	5.32931i	−7.84851	+	26.7296i	−37.1729	+	19.7163i	16.6015	−	19.1591i	−46.9311	−	6.74767i	42.1256	+	69.1841i	−85.2928	−	98.4331i
2.8	−2.30231	+	3.58247i	−7.95344	+	4.21222i	−0.886802	−	1.94183i	8.56609	−	29.1734i	3.22114	−	38.1908i	24.5435	−	28.3247i	−58.4441	−	8.40299i	45.5143	−	67.0033i	84.7911	+	97.8541i
2.9	−2.18249	+	3.39602i	8.74164	−	2.14095i	−0.123067	−	0.269480i	7.29897	−	24.8580i	−11.8079	+	34.3594i	47.9514	−	55.3388i	−62.7486	−	9.02189i	71.8327	−	37.4308i	68.4884	+	79.0399i
2.10	−1.85051	+	2.87944i	0.149878	−	8.99875i	1.77982	+	3.89726i	13.0909	−	44.5836i	25.6340	+	17.0838i	−46.3292	+	53.4668i	−68.7229	−	9.88086i	−80.9551	−	2.69742i	104.151	+	120.197i
2.11	−1.18989	+	1.85151i	−8.40626	−	3.21477i	4.63439	+	10.1479i	−4.40450	+	15.0003i	15.9547	−	11.7391i	8.45761	−	9.76060i	−59.1593	−	8.50582i	60.3306	+	54.0483i	−22.5324	−	26.0038i
2.12	−1.16876	+	1.81863i	−1.49992	−	8.87413i	4.70524	+	10.3030i	−4.27625	+	14.5636i	17.8918	+	7.64395i	33.8030	−	39.0108i	−58.4735	−	8.40722i	−76.5005	+	26.6209i	−21.4878	−	24.7982i
2.13	−0.842718	+	1.31130i	7.53602	−	4.92020i	5.63732	+	12.3440i	−5.53880	+	18.8634i	0.101095	+	14.0283i	−41.7057	+	48.1309i	−45.6233	−	6.55964i	32.5832	−	74.1575i	−20.0679	−	23.1596i
2.14	−0.386142	+	0.600848i	−1.66494	+	8.84466i	6.43473	+	14.0901i	−11.2060	+	38.1641i	−4.67140	−	4.41567i	37.6947	−	43.5020i	−22.2621	−	3.20080i	−75.4560	−	29.4516i	−18.6037	−	21.4699i
2.15	−0.152702	+	0.237609i	−5.50899	+	7.11696i	6.61350	+	14.4815i	1.03882	−	3.53791i	−0.849819	−	2.39576i	−24.2369	+	27.9709i	−8.92398	−	1.28308i	−20.3021	−	78.4144i	0.682008	+	0.787079i
2.16	0.152702	−	0.237609i	7.29091	+	5.27661i	6.61350	+	14.4815i	−1.03882	+	3.53791i	2.36711	−	0.926638i	−24.2369	+	27.9709i	8.92398	+	1.28308i	25.3148	+	76.9426i	0.682008	+	0.787079i
2.17	0.386142	−	0.600848i	4.08932	+	8.01732i	6.43473	+	14.0901i	11.2060	−	38.1641i	6.39625	+	0.638760i	37.6947	−	43.5020i	22.2621	+	3.20080i	−47.5549	+	65.5708i	−18.6037	−	21.4699i
2.18	0.842718	−	1.31130i	−8.61694	−	2.59776i	5.63732	+	12.3440i	5.53880	−	18.8634i	−10.6681	+	9.11018i	−41.7057	+	48.1309i	45.6233	+	6.55964i	67.5033	+	44.7694i	−20.0679	−	23.1596i
2.19	1.16876	−	1.81863i	−1.06097	−	8.93724i	4.70524	+	10.3030i	4.27625	−	14.5636i	−17.4935	−	8.51599i	33.8030	−	39.0108i	58.4735	+	8.40722i	−78.7487	+	18.9643i	−21.4878	−	24.7982i
2.20	1.18989	−	1.85151i	7.16005	−	5.45286i	4.63439	+	10.1479i	4.40450	−	15.0003i	−1.57634	−	19.7452i	8.45761	−	9.76060i	59.1593	+	8.50582i	21.5326	−	78.0855i	−22.5324	−	26.0038i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
23.c	even	11	1	inner
69.h	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	69.5.h.a	✓	300
3.b	odd	2	1	inner	69.5.h.a	✓	300
23.c	even	11	1	inner	69.5.h.a	✓	300
69.h	odd	22	1	inner	69.5.h.a	✓	300

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
69.5.h.a	✓	300	1.a	even	1	1	trivial
69.5.h.a	✓	300	3.b	odd	2	1	inner
69.5.h.a	✓	300	23.c	even	11	1	inner
69.5.h.a	✓	300	69.h	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(69, [\chi])\).