Newform orbit 21.30.e.a

• Label: 21.30.e.a
• Level: $21$
• Weight: $30$
• Character orbit: 21.e
• Analytic conductor: $111.884$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(111.883889004\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 25025 q^{2} + 90876411 q^{3} - 5448583733 q^{4} - 2101591778 q^{5} + 239387598450 q^{6} - 1147384421099 q^{7} - 53840580588510 q^{8} - 434659056644259 q^{9}+O(q^{10}) \)

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	−21682.7	−	37555.5i	2.39148e6	−	4.14217e6i	−6.71844e8	+	1.16367e9i	−3.79032e9	−	6.56503e9i	−2.07415e11			−1.42775e12	+	1.08693e12i	3.49879e13			−1.14384e13	−	1.98119e13i	−1.64369e14	+	2.84695e14i
4.2	−19318.8	−	33461.1i	2.39148e6	−	4.14217e6i	−4.77996e8	+	8.27914e8i	−7.79808e8	−	1.35067e9i	−1.84802e11			−1.66873e11	−	1.78663e12i	1.61939e13			−1.14384e13	−	1.98119e13i	−3.01299e13	+	5.21865e13i
4.3	−19095.2	−	33073.8i	2.39148e6	−	4.14217e6i	−4.60816e8	+	7.98156e8i	9.98170e9	+	1.72888e10i	−1.82663e11			1.32001e12	+	1.21552e12i	1.46941e13			−1.14384e13	−	1.98119e13i	3.81205e14	−	6.60266e14i
4.4	−13945.7	−	24154.6i	2.39148e6	−	4.14217e6i	−1.20528e8	+	2.08761e8i	−1.12432e10	−	1.94737e10i	−1.33403e11			1.74806e12	+	4.05197e11i	−8.25066e12			−1.14384e13	−	1.98119e13i	−3.13587e14	+	5.43149e14i
4.5	−13747.8	−	23811.9i	2.39148e6	−	4.14217e6i	−1.09569e8	+	1.89779e8i	−5.45988e9	−	9.45678e9i	−1.31511e11			−1.44405e12	+	1.06518e12i	−8.73628e12			−1.14384e13	−	1.98119e13i	−1.50123e14	+	2.60020e14i
4.6	−10085.3	−	17468.3i	2.39148e6	−	4.14217e6i	6.50070e7	−	1.12595e8i	4.83495e9	+	8.37438e9i	−9.64758e10			−7.77849e11	+	1.61705e12i	−1.34515e13			−1.14384e13	−	1.98119e13i	9.75244e13	−	1.68917e14i
4.7	−9506.59	−	16465.9i	2.39148e6	−	4.14217e6i	8.76848e7	−	1.51874e8i	4.01268e9	+	6.95016e9i	−9.09395e10			1.51909e12	−	9.55123e11i	−1.35420e13			−1.14384e13	−	1.98119e13i	7.62938e13	−	1.32145e14i
4.8	−5312.73	−	9201.91i	2.39148e6	−	4.14217e6i	2.11985e8	−	3.67169e8i	6.21129e8	+	1.07583e9i	−5.08212e10			−1.19318e12	−	1.34023e12i	−1.02094e13			−1.14384e13	−	1.98119e13i	6.59978e12	−	1.14312e13i
4.9	−803.934	−	1392.45i	2.39148e6	−	4.14217e6i	2.67143e8	−	4.62705e8i	−1.04266e10	−	1.80594e10i	−7.69038e9			−3.52235e11	−	1.75950e12i	−1.72228e12			−1.14384e13	−	1.98119e13i	−1.67646e13	+	2.90371e13i
4.10	1314.29	+	2276.42i	2.39148e6	−	4.14217e6i	2.64981e8	−	4.58960e8i	1.11285e10	+	1.92751e10i	1.25724e10			−1.79440e12	+	6.33421e9i	2.80425e12			−1.14384e13	−	1.98119e13i	−2.92521e13	+	5.06661e13i
4.11	3171.13	+	5492.56i	2.39148e6	−	4.14217e6i	2.48323e8	−	4.30109e8i	−5.04465e9	−	8.73759e9i	3.03349e10			7.64524e11	+	1.62339e12i	6.55484e12			−1.14384e13	−	1.98119e13i	3.19945e13	−	5.54161e13i
4.12	4961.84	+	8594.16i	2.39148e6	−	4.14217e6i	2.19196e8	−	3.79658e8i	8.80219e9	+	1.52458e10i	4.74646e10			1.78765e12	+	1.55597e11i	9.67819e12			−1.14384e13	−	1.98119e13i	−8.73501e13	+	1.51295e14i
4.13	10661.0	+	18465.4i	2.39148e6	−	4.14217e6i	4.11213e7	−	7.12241e7i	−5.16991e9	−	8.95455e9i	1.01983e11			−1.66601e12	+	6.66564e11i	1.32007e13			−1.14384e13	−	1.98119e13i	1.10233e14	−	1.90929e14i
4.14	11213.9	+	19423.1i	2.39148e6	−	4.14217e6i	1.69302e7	−	2.93240e7i	4.19413e9	+	7.26445e9i	1.07272e11			4.56481e11	−	1.73538e12i	1.28003e13			−1.14384e13	−	1.98119e13i	−9.40656e13	+	1.62926e14i
4.15	14667.6	+	25405.0i	2.39148e6	−	4.14217e6i	−1.61841e8	+	2.80316e8i	−8.78577e9	−	1.52174e10i	1.40309e11			1.75086e12	−	3.92932e11i	6.25395e12			−1.14384e13	−	1.98119e13i	2.57732e14	−	4.46405e14i
4.16	17773.2	+	30784.0i	2.39148e6	−	4.14217e6i	−3.63334e8	+	6.29314e8i	6.21359e9	+	1.07623e10i	1.70017e11			−3.33772e10	+	1.79410e12i	−6.74661e12			−1.14384e13	−	1.98119e13i	−2.20870e14	+	3.82558e14i
4.17	20203.6	+	34993.6i	2.39148e6	−	4.14217e6i	−5.47933e8	+	9.49048e8i	5.22782e7	+	9.05485e7i	1.93266e11			1.78686e12	+	1.64388e11i	−2.25874e13			−1.14384e13	−	1.98119e13i	−2.11241e12	+	3.65880e12i
4.18	20330.6	+	35213.6i	2.39148e6	−	4.14217e6i	−5.58229e8	+	9.66881e8i	9.23100e9	+	1.59886e10i	1.94481e11			−1.57660e12	−	8.56870e11i	−2.35667e13			−1.14384e13	−	1.98119e13i	−3.75343e14	+	6.50113e14i
4.19	21714.1	+	37610.0i	2.39148e6	−	4.14217e6i	−6.74573e8	+	1.16839e9i	−9.42284e9	−	1.63208e10i	2.07716e11			−1.27489e12	−	1.26276e12i	−3.52757e13			−1.14384e13	−	1.98119e13i	4.09218e14	−	7.08786e14i
16.1	−21682.7	+	37555.5i	2.39148e6	+	4.14217e6i	−6.71844e8	−	1.16367e9i	−3.79032e9	+	6.56503e9i	−2.07415e11			−1.42775e12	−	1.08693e12i	3.49879e13			−1.14384e13	+	1.98119e13i	−1.64369e14	−	2.84695e14i

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	21.30.e.a	✓	38
7.c	even	3	1	inner	21.30.e.a	✓	38

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.30.e.a	✓	38	1.a	even	1	1	trivial
21.30.e.a	✓	38	7.c	even	3	1	inner