Newform orbit 21.22.e.b

• Label: 21.22.e.b
• Level: $21$
• Weight: $22$
• Character orbit: 21.e
• Analytic conductor: $58.690$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(58.6902423003\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 1311 q^{2} - 885735 q^{3} - 18627957 q^{4} + 15225465 q^{5} + 154826478 q^{6} - 771749103 q^{7} + 3249361014 q^{8} - 52301766015 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−1433.07	−	2482.15i	−29524.5	+	51137.9i	−3.05881e6	+	5.29802e6i	8.93257e6	+	1.54717e7i	1.69243e8			−7.41769e8	+	9.12359e7i	1.15233e10			−1.74339e9	−	3.01964e9i	2.56020e10	−	4.43440e10i
4.2	−1250.05	−	2165.14i	−29524.5	+	51137.9i	−2.07666e6	+	3.59688e6i	−1.55907e7	−	2.70039e7i	1.47628e8			5.57824e8	+	4.97371e8i	5.14061e9			−1.74339e9	−	3.01964e9i	−3.89782e10	+	6.75123e10i
4.3	−1093.39	−	1893.81i	−29524.5	+	51137.9i	−1.34245e6	+	2.32519e6i	−5.93104e6	−	1.02729e7i	1.29128e8			1.46994e7	−	7.47215e8i	1.28526e9			−1.74339e9	−	3.01964e9i	−1.29699e10	+	2.24646e10i
4.4	−890.974	−	1543.21i	−29524.5	+	51137.9i	−539094.	+	933738.i	1.39071e7	+	2.40879e7i	1.05222e8			2.87280e8	−	6.89939e8i	−1.81574e9			−1.74339e9	−	3.01964e9i	2.47818e10	−	4.29233e10i
4.5	−773.092	−	1339.03i	−29524.5	+	51137.9i	−146766.	+	254207.i	1.49130e7	+	2.58301e7i	9.13006e7			−3.13405e8	+	6.78471e8i	−2.78873e9			−1.74339e9	−	3.01964e9i	2.30582e10	−	3.99380e10i
4.6	−547.151	−	947.694i	−29524.5	+	51137.9i	449827.	−	779123.i	−1.24348e7	−	2.15378e7i	6.46175e7			−7.07061e8	+	2.42097e8i	−3.27941e9			−1.74339e9	−	3.01964e9i	−1.36075e10	+	2.35689e10i
4.7	−401.632	−	695.647i	−29524.5	+	51137.9i	725959.	−	1.25740e6i	−8.65528e6	−	1.49914e7i	4.74320e7			7.45493e8	−	5.27847e7i	−2.85084e9			−1.74339e9	−	3.01964e9i	−6.95248e9	+	1.20421e10i
4.8	−16.4448	−	28.4832i	−29524.5	+	51137.9i	1.04804e6	−	1.81525e6i	6.49673e6	+	1.12527e7i	1.94210e6			6.29530e8	+	4.02788e8i	−1.37913e8			−1.74339e9	−	3.01964e9i	2.13675e8	−	3.70096e8i
4.9	295.896	+	512.508i	−29524.5	+	51137.9i	873467.	−	1.51289e6i	1.68562e7	+	2.91957e7i	−3.49448e7			−1.57336e8	−	7.30610e8i	2.27490e9			−1.74339e9	−	3.01964e9i	−9.97536e9	+	1.72778e10i
4.10	433.225	+	750.369i	−29524.5	+	51137.9i	673207.	−	1.16603e6i	−1.89915e7	−	3.28942e7i	−5.11631e7			1.06042e8	−	7.39798e8i	2.98368e9			−1.74339e9	−	3.01964e9i	1.64552e10	−	2.85012e10i
4.11	460.926	+	798.348i	−29524.5	+	51137.9i	623670.	−	1.08023e6i	2.15410e6	+	3.73101e6i	−5.44345e7			−1.41722e8	+	7.33799e8i	3.08313e9			−1.74339e9	−	3.01964e9i	−1.98576e9	+	3.43944e9i
4.12	863.711	+	1495.99i	−29524.5	+	51137.9i	−443419.	+	768024.i	−989819.	−	1.71442e6i	−1.02003e8			−7.41689e8	+	9.18894e7i	2.09073e9			−1.74339e9	−	3.01964e9i	1.70984e9	−	2.96152e9i
4.13	1030.06	+	1784.11i	−29524.5	+	51137.9i	−1.07345e6	+	1.85928e6i	2.46274e6	+	4.26559e6i	−1.21648e8			6.83102e8	−	3.03180e8i	−1.02499e8			−1.74339e9	−	3.01964e9i	−5.07352e9	+	8.78759e9i
4.14	1273.11	+	2205.10i	−29524.5	+	51137.9i	−2.19305e6	+	3.79848e6i	−1.48466e7	−	2.57151e7i	−1.50352e8			1.39979e8	+	7.34133e8i	−5.82819e9			−1.74339e9	−	3.01964e9i	3.78028e10	−	6.54764e10i
4.15	1393.38	+	2413.41i	−29524.5	+	51137.9i	−2.83444e6	+	4.90939e6i	1.93301e7	+	3.34807e7i	−1.64555e8			−7.46842e8	−	2.77956e7i	−9.95355e9			−1.74339e9	−	3.01964e9i	−5.38683e10	+	9.33026e10i
16.1	−1433.07	+	2482.15i	−29524.5	−	51137.9i	−3.05881e6	−	5.29802e6i	8.93257e6	−	1.54717e7i	1.69243e8			−7.41769e8	−	9.12359e7i	1.15233e10			−1.74339e9	+	3.01964e9i	2.56020e10	+	4.43440e10i
16.2	−1250.05	+	2165.14i	−29524.5	−	51137.9i	−2.07666e6	−	3.59688e6i	−1.55907e7	+	2.70039e7i	1.47628e8			5.57824e8	−	4.97371e8i	5.14061e9			−1.74339e9	+	3.01964e9i	−3.89782e10	−	6.75123e10i
16.3	−1093.39	+	1893.81i	−29524.5	−	51137.9i	−1.34245e6	−	2.32519e6i	−5.93104e6	+	1.02729e7i	1.29128e8			1.46994e7	+	7.47215e8i	1.28526e9			−1.74339e9	+	3.01964e9i	−1.29699e10	−	2.24646e10i
16.4	−890.974	+	1543.21i	−29524.5	−	51137.9i	−539094.	−	933738.i	1.39071e7	−	2.40879e7i	1.05222e8			2.87280e8	+	6.89939e8i	−1.81574e9			−1.74339e9	+	3.01964e9i	2.47818e10	+	4.29233e10i
16.5	−773.092	+	1339.03i	−29524.5	−	51137.9i	−146766.	−	254207.i	1.49130e7	−	2.58301e7i	9.13006e7			−3.13405e8	−	6.78471e8i	−2.78873e9			−1.74339e9	+	3.01964e9i	2.30582e10	+	3.99380e10i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	21.22.e.b	✓	30
7.c	even	3	1	inner	21.22.e.b	✓	30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.22.e.b	✓	30	1.a	even	1	1	trivial
21.22.e.b	✓	30	7.c	even	3	1	inner