Newform orbit 150.3.k.a

• Label: 150.3.k.a
• Level: $150$
• Weight: $3$
• Character orbit: 150.k
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.k (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.08720396540\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

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	1.39680	−	0.221232i	−0.786335	+	1.54327i	1.90211	−	0.618034i	1.86127	+	4.64065i	−0.756934	+	2.32960i	−6.90199	+	6.90199i	2.52015	−	1.28408i	−1.76336	−	2.42705i	3.62649	+	6.07030i
13.2	1.39680	−	0.221232i	−0.786335	+	1.54327i	1.90211	−	0.618034i	1.98098	−	4.59083i	−0.756934	+	2.32960i	3.83711	−	3.83711i	2.52015	−	1.28408i	−1.76336	−	2.42705i	1.75141	−	6.85074i
13.3	1.39680	−	0.221232i	0.786335	−	1.54327i	1.90211	−	0.618034i	−3.06706	−	3.94881i	0.756934	−	2.32960i	2.82371	−	2.82371i	2.52015	−	1.28408i	−1.76336	−	2.42705i	−5.15768	−	4.83718i
13.4	1.39680	−	0.221232i	0.786335	−	1.54327i	1.90211	−	0.618034i	4.42349	+	2.33083i	0.756934	−	2.32960i	0.284481	−	0.284481i	2.52015	−	1.28408i	−1.76336	−	2.42705i	6.69439	+	2.27709i
37.1	0.221232	+	1.39680i	−1.54327	−	0.786335i	−1.90211	+	0.618034i	−1.48914	−	4.77310i	0.756934	−	2.32960i	9.35986	+	9.35986i	−1.28408	−	2.52015i	1.76336	+	2.42705i	6.33763	−	3.13599i
37.2	0.221232	+	1.39680i	−1.54327	−	0.786335i	−1.90211	+	0.618034i	3.55879	−	3.51212i	0.756934	−	2.32960i	−5.48204	−	5.48204i	−1.28408	−	2.52015i	1.76336	+	2.42705i	5.69306	+	4.19393i
37.3	0.221232	+	1.39680i	1.54327	+	0.786335i	−1.90211	+	0.618034i	−3.00822	+	3.99382i	−0.756934	+	2.32960i	4.19292	+	4.19292i	−1.28408	−	2.52015i	1.76336	+	2.42705i	−6.24409	−	3.31834i
37.4	0.221232	+	1.39680i	1.54327	+	0.786335i	−1.90211	+	0.618034i	4.68416	−	1.74889i	−0.756934	+	2.32960i	2.83023	+	2.83023i	−1.28408	−	2.52015i	1.76336	+	2.42705i	3.47914	+	6.15594i
67.1	0.642040	−	1.26007i	−0.270952	+	1.71073i	−1.17557	−	1.61803i	−3.75559	−	3.30084i	1.98168	+	1.43977i	−5.04938	−	5.04938i	−2.79360	+	0.442463i	−2.85317	−	0.927051i	−6.57054	+	2.61306i
67.2	0.642040	−	1.26007i	−0.270952	+	1.71073i	−1.17557	−	1.61803i	−2.43941	+	4.36455i	1.98168	+	1.43977i	4.46938	+	4.46938i	−2.79360	+	0.442463i	−2.85317	−	0.927051i	3.93346	+	5.87604i
67.3	0.642040	−	1.26007i	0.270952	−	1.71073i	−1.17557	−	1.61803i	−3.90603	+	3.12137i	−1.98168	−	1.43977i	−6.00700	−	6.00700i	−2.79360	+	0.442463i	−2.85317	−	0.927051i	1.42533	+	6.92592i
67.4	0.642040	−	1.26007i	0.270952	−	1.71073i	−1.17557	−	1.61803i	2.55121	−	4.30016i	−1.98168	−	1.43977i	−1.41591	−	1.41591i	−2.79360	+	0.442463i	−2.85317	−	0.927051i	−3.78054	−	5.97558i
73.1	0.221232	−	1.39680i	−1.54327	+	0.786335i	−1.90211	−	0.618034i	−1.48914	+	4.77310i	0.756934	+	2.32960i	9.35986	−	9.35986i	−1.28408	+	2.52015i	1.76336	−	2.42705i	6.33763	+	3.13599i
73.2	0.221232	−	1.39680i	−1.54327	+	0.786335i	−1.90211	−	0.618034i	3.55879	+	3.51212i	0.756934	+	2.32960i	−5.48204	+	5.48204i	−1.28408	+	2.52015i	1.76336	−	2.42705i	5.69306	−	4.19393i
73.3	0.221232	−	1.39680i	1.54327	−	0.786335i	−1.90211	−	0.618034i	−3.00822	−	3.99382i	−0.756934	−	2.32960i	4.19292	−	4.19292i	−1.28408	+	2.52015i	1.76336	−	2.42705i	−6.24409	+	3.31834i
73.4	0.221232	−	1.39680i	1.54327	−	0.786335i	−1.90211	−	0.618034i	4.68416	+	1.74889i	−0.756934	−	2.32960i	2.83023	−	2.83023i	−1.28408	+	2.52015i	1.76336	−	2.42705i	3.47914	−	6.15594i
97.1	−1.26007	+	0.642040i	−1.71073	+	0.270952i	1.17557	−	1.61803i	−4.55798	−	2.05543i	1.98168	−	1.43977i	3.78399	+	3.78399i	−0.442463	+	2.79360i	2.85317	−	0.927051i	7.06306	−	0.336405i
97.2	−1.26007	+	0.642040i	−1.71073	+	0.270952i	1.17557	−	1.61803i	−0.888938	+	4.92034i	1.98168	−	1.43977i	−2.71277	−	2.71277i	−0.442463	+	2.79360i	2.85317	−	0.927051i	−2.03893	−	6.77073i
97.3	−1.26007	+	0.642040i	1.71073	−	0.270952i	1.17557	−	1.61803i	0.270329	+	4.99269i	−1.98168	+	1.43977i	0.734007	+	0.734007i	−0.442463	+	2.79360i	2.85317	−	0.927051i	−3.54614	−	6.11759i
97.4	−1.26007	+	0.642040i	1.71073	−	0.270952i	1.17557	−	1.61803i	3.78214	−	3.27039i	−1.98168	+	1.43977i	−0.746595	−	0.746595i	−0.442463	+	2.79360i	2.85317	−	0.927051i	−2.66605	+	6.54921i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.f	odd	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	150.3.k.a	✓	32
25.f	odd	20	1	inner	150.3.k.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
150.3.k.a	✓	32	1.a	even	1	1	trivial
150.3.k.a	✓	32	25.f	odd	20	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^32 - 8*T7^31 + 32*T7^30 + 284*T7^29 + 25448*T7^28 - 142288*T7^27 + 364296*T7^26 - 1854300*T7^25 + 166638564*T7^24 - 1025374732*T7^23 + 3196275472*T7^22 - 16158882444*T7^21 + 466723416510*T7^20 - 3207442469676*T7^19 + 11708421404408*T7^18 - 35543587433072*T7^17 + 520885885975198*T7^16 - 3820444601576048*T7^15 + 15237353540809448*T7^14 - 27365458011627264*T7^13 + 102565118295437700*T7^12 - 704144985634328076*T7^11 + 3108429779364139912*T7^10 - 4615821805971911968*T7^9 + 2215541712759687909*T7^8 - 3674078155579683000*T7^7 + 71043557766745632696*T7^6 - 41852960322497705512*T7^5 + 12447955605293321678*T7^4 - 4961459032142572564*T7^3 + 81835693100405150792*T7^2 - 45809399400441629132*T7 + 12821429097291977761