Newform orbit 700.2.ba.a

• Label: 700.2.ba.a
• Level: $700$
• Weight: $2$
• Character orbit: 700.ba
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.ba (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(116\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 10 q^{2} - 6 q^{4} - 40 q^{8} + 96 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
139.1	−1.41376	−	0.0358504i	−0.953175	−	0.309705i	1.99743	+	0.101368i	−0.446577	−	2.19102i	1.33646	+	0.472021i	1.83097	−	1.90985i	−2.82025	−	0.214918i	−1.61443	−	1.17295i	0.552803	+	3.11358i
139.2	−1.41376	−	0.0358504i	0.953175	+	0.309705i	1.99743	+	0.101368i	0.446577	+	2.19102i	−1.33646	−	0.472021i	1.83097	+	1.90985i	−2.82025	−	0.214918i	−1.61443	−	1.17295i	−0.552803	−	3.11358i
139.3	−1.41179	+	0.0828293i	−2.08547	−	0.677611i	1.98628	−	0.233874i	1.96875	+	1.06021i	3.00037	+	0.783903i	−1.57428	−	2.12642i	−2.78483	+	0.494703i	1.46299	+	1.06292i	−2.86726	−	1.33371i
139.4	−1.41179	+	0.0828293i	2.08547	+	0.677611i	1.98628	−	0.233874i	−1.96875	−	1.06021i	−3.00037	−	0.783903i	−1.57428	+	2.12642i	−2.78483	+	0.494703i	1.46299	+	1.06292i	2.86726	+	1.33371i
139.5	−1.40841	−	0.127948i	−2.16710	−	0.704134i	1.96726	+	0.360406i	−2.23578	+	0.0356787i	2.96208	+	1.26899i	0.831537	+	2.51168i	−2.72460	−	0.759307i	1.77347	+	1.28850i	3.15347	+	0.235813i
139.6	−1.40841	−	0.127948i	2.16710	+	0.704134i	1.96726	+	0.360406i	2.23578	−	0.0356787i	−2.96208	−	1.26899i	0.831537	−	2.51168i	−2.72460	−	0.759307i	1.77347	+	1.28850i	−3.15347	−	0.235813i
139.7	−1.40041	+	0.197085i	−0.491097	−	0.159567i	1.92232	−	0.552000i	−2.06230	+	0.864250i	0.719187	+	0.126672i	−0.347937	−	2.62277i	−2.58324	+	1.15189i	−2.21134	−	1.60663i	2.71774	−	1.61675i
139.8	−1.40041	+	0.197085i	0.491097	+	0.159567i	1.92232	−	0.552000i	2.06230	−	0.864250i	−0.719187	−	0.126672i	−0.347937	+	2.62277i	−2.58324	+	1.15189i	−2.21134	−	1.60663i	−2.71774	+	1.61675i
139.9	−1.36917	−	0.354065i	−2.42780	−	0.788842i	1.74928	+	0.969554i	0.599283	+	2.15427i	3.04479	+	1.93966i	−2.32817	+	1.25682i	−2.05178	−	1.94685i	2.84491	+	2.06695i	−0.0577727	−	3.16175i
139.10	−1.36917	−	0.354065i	2.42780	+	0.788842i	1.74928	+	0.969554i	−0.599283	−	2.15427i	−3.04479	−	1.93966i	−2.32817	−	1.25682i	−2.05178	−	1.94685i	2.84491	+	2.06695i	0.0577727	+	3.16175i
139.11	−1.35159	+	0.416189i	−3.21505	−	1.04463i	1.65357	−	1.12503i	1.64240	−	1.51741i	4.78019	+	0.0738455i	2.52797	+	0.780605i	−1.76672	+	2.20877i	6.81825	+	4.95375i	−1.58832	+	2.73446i
139.12	−1.35159	+	0.416189i	3.21505	+	1.04463i	1.65357	−	1.12503i	−1.64240	+	1.51741i	−4.78019	−	0.0738455i	2.52797	−	0.780605i	−1.76672	+	2.20877i	6.81825	+	4.95375i	1.58832	−	2.73446i
139.13	−1.34906	+	0.424305i	−0.905941	−	0.294358i	1.63993	−	1.14483i	0.509500	−	2.17725i	1.34707	+	0.0127112i	−2.23272	+	1.41949i	−1.72661	+	2.24027i	−1.69297	−	1.23001i	0.236472	+	3.15342i
139.14	−1.34906	+	0.424305i	0.905941	+	0.294358i	1.63993	−	1.14483i	−0.509500	+	2.17725i	−1.34707	−	0.0127112i	−2.23272	−	1.41949i	−1.72661	+	2.24027i	−1.69297	−	1.23001i	−0.236472	−	3.15342i
139.15	−1.28665	−	0.586966i	−0.187929	−	0.0610618i	1.31094	+	1.51044i	1.24687	−	1.85616i	0.205958	+	0.188873i	−2.16677	−	1.51826i	−0.800148	−	2.71289i	−2.39546	−	1.74041i	−2.69378	+	1.65636i
139.16	−1.28665	−	0.586966i	0.187929	+	0.0610618i	1.31094	+	1.51044i	−1.24687	+	1.85616i	−0.205958	−	0.188873i	−2.16677	+	1.51826i	−0.800148	−	2.71289i	−2.39546	−	1.74041i	2.69378	−	1.65636i
139.17	−1.25523	−	0.651450i	−1.57146	−	0.510598i	1.15123	+	1.63545i	2.22077	−	0.261078i	1.63992	+	1.66465i	2.58780	−	0.550719i	−0.379646	−	2.80283i	−0.218276	−	0.158587i	−2.95767	−	1.11901i
139.18	−1.25523	−	0.651450i	1.57146	+	0.510598i	1.15123	+	1.63545i	−2.22077	+	0.261078i	−1.63992	−	1.66465i	2.58780	+	0.550719i	−0.379646	−	2.80283i	−0.218276	−	0.158587i	2.95767	+	1.11901i
139.19	−1.22689	+	0.703370i	−2.48783	−	0.808344i	1.01054	−	1.72592i	−1.69256	−	1.46125i	3.62087	−	0.758110i	−2.64443	−	0.0837129i	−0.0258677	+	2.82831i	3.10882	+	2.25869i	3.10439	+	0.602300i
139.20	−1.22689	+	0.703370i	2.48783	+	0.808344i	1.01054	−	1.72592i	1.69256	+	1.46125i	−3.62087	+	0.758110i	−2.64443	+	0.0837129i	−0.0258677	+	2.82831i	3.10882	+	2.25869i	−3.10439	−	0.602300i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
25.e	even	10	1	inner
28.d	even	2	1	inner
100.h	odd	10	1	inner
175.m	odd	10	1	inner
700.ba	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.ba.a	✓	464
4.b	odd	2	1	inner	700.2.ba.a	✓	464
7.b	odd	2	1	inner	700.2.ba.a	✓	464
25.e	even	10	1	inner	700.2.ba.a	✓	464
28.d	even	2	1	inner	700.2.ba.a	✓	464
100.h	odd	10	1	inner	700.2.ba.a	✓	464
175.m	odd	10	1	inner	700.2.ba.a	✓	464
700.ba	even	10	1	inner	700.2.ba.a	✓	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.ba.a	✓	464	1.a	even	1	1	trivial
700.2.ba.a	✓	464	4.b	odd	2	1	inner
700.2.ba.a	✓	464	7.b	odd	2	1	inner
700.2.ba.a	✓	464	25.e	even	10	1	inner
700.2.ba.a	✓	464	28.d	even	2	1	inner
700.2.ba.a	✓	464	100.h	odd	10	1	inner
700.2.ba.a	✓	464	175.m	odd	10	1	inner
700.2.ba.a	✓	464	700.ba	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(700, [\chi])\).