Newform orbit 21.22.c.b

• Label: 21.22.c.b
• Level: $21$
• Weight: $22$
• Character orbit: 21.c
• Analytic conductor: $58.690$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(58.6902423003\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 58720260 q^{4} + 1422287188 q^{7} + 10150644696 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
20.1		−	1223.76i	−94027.5	−	40239.2i	599559.			3.45471e7			−4.92432e7	+	1.15067e8i	−6.99978e8	+	2.61872e8i		−	3.30013e9i	7.22197e9	+	7.56717e9i		−	4.22775e10i
20.2			1223.76i	−94027.5	+	40239.2i	599559.			3.45471e7			−4.92432e7	−	1.15067e8i	−6.99978e8	−	2.61872e8i			3.30013e9i	7.22197e9	−	7.56717e9i			4.22775e10i
20.3		−	247.072i	16209.6	+	100983.i	2.03611e6			−3.48005e7			2.49501e7	−	4.00494e6i	7.14474e8	−	2.19256e8i		−	1.02121e9i	−9.93485e9	+	3.27379e9i			8.59823e9i
20.4			247.072i	16209.6	−	100983.i	2.03611e6			−3.48005e7			2.49501e7	+	4.00494e6i	7.14474e8	+	2.19256e8i			1.02121e9i	−9.93485e9	−	3.27379e9i		−	8.59823e9i
20.5		−	2329.72i	52907.9	−	87527.8i	−3.33046e6			−3.45099e7			−2.03916e8	−	1.23261e8i	4.88555e7	−	7.45761e8i			2.87327e9i	−4.86187e9	−	9.26181e9i			8.03985e10i
20.6			2329.72i	52907.9	+	87527.8i	−3.33046e6			−3.45099e7			−2.03916e8	+	1.23261e8i	4.88555e7	+	7.45761e8i		−	2.87327e9i	−4.86187e9	+	9.26181e9i		−	8.03985e10i
20.7		−	367.125i	−91325.3	−	46043.9i	1.96237e6			−3.23352e7			−1.69039e7	+	3.35278e7i	−4.79207e8	−	5.73504e8i		−	1.49035e9i	6.22027e9	+	8.40995e9i			1.18711e10i
20.8			367.125i	−91325.3	+	46043.9i	1.96237e6			−3.23352e7			−1.69039e7	−	3.35278e7i	−4.79207e8	+	5.73504e8i			1.49035e9i	6.22027e9	−	8.40995e9i		−	1.18711e10i
20.9		−	1925.57i	78349.6	−	65739.6i	−1.61068e6			2.77549e7			−1.26586e8	−	1.50868e8i	7.43581e8	+	7.50540e7i		−	9.36744e8i	1.81697e9	−	1.03013e10i		−	5.34441e10i
20.10			1925.57i	78349.6	+	65739.6i	−1.61068e6			2.77549e7			−1.26586e8	+	1.50868e8i	7.43581e8	−	7.50540e7i			9.36744e8i	1.81697e9	+	1.03013e10i			5.34441e10i
20.11		−	1704.62i	−65800.8	−	78298.2i	−808569.			−2.49846e7			−1.33469e8	+	1.12165e8i	7.21028e7	+	7.43873e8i		−	2.19654e9i	−1.80087e9	+	1.03042e10i			4.25893e10i
20.12			1704.62i	−65800.8	+	78298.2i	−808569.			−2.49846e7			−1.33469e8	−	1.12165e8i	7.21028e7	−	7.43873e8i			2.19654e9i	−1.80087e9	−	1.03042e10i		−	4.25893e10i
20.13		−	2426.55i	100608.	+	18395.7i	−3.79100e6			−2.12980e7			4.46381e7	−	2.44130e8i	5.03992e8	+	5.51850e8i			4.11022e9i	9.78355e9	+	3.70150e9i			5.16808e10i
20.14			2426.55i	100608.	−	18395.7i	−3.79100e6			−2.12980e7			4.46381e7	+	2.44130e8i	5.03992e8	−	5.51850e8i		−	4.11022e9i	9.78355e9	−	3.70150e9i		−	5.16808e10i
20.15		−	2502.20i	−102208.	+	3714.28i	−4.16388e6			−1.82569e7			9.29389e6	+	2.55746e8i	−7.47262e8	+	1.20672e7i			5.17136e9i	1.04328e10	−	7.59261e8i			4.56824e10i
20.16			2502.20i	−102208.	−	3714.28i	−4.16388e6			−1.82569e7			9.29389e6	−	2.55746e8i	−7.47262e8	−	1.20672e7i		−	5.17136e9i	1.04328e10	+	7.59261e8i		−	4.56824e10i
20.17		−	2061.69i	−759.014	−	102273.i	−2.15341e6			1.95223e7			−2.10855e8	+	1.56485e6i	−6.67328e8	−	3.36481e8i			1.15978e8i	−1.04592e10	+	1.55253e8i		−	4.02489e10i
20.18			2061.69i	−759.014	+	102273.i	−2.15341e6			1.95223e7			−2.10855e8	−	1.56485e6i	−6.67328e8	+	3.36481e8i		−	1.15978e8i	−1.04592e10	−	1.55253e8i			4.02489e10i
20.19		−	1077.74i	64855.1	−	79083.3i	935630.			−7.14030e6			−8.52312e7	−	6.98969e7i	−3.76236e8	+	6.45749e8i		−	3.26855e9i	−2.04799e9	−	1.02579e10i			7.69538e9i
20.20			1077.74i	64855.1	+	79083.3i	935630.			−7.14030e6			−8.52312e7	+	6.98969e7i	−3.76236e8	−	6.45749e8i			3.26855e9i	−2.04799e9	+	1.02579e10i		−	7.69538e9i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	21.22.c.b	✓	52
3.b	odd	2	1	inner	21.22.c.b	✓	52
7.b	odd	2	1	inner	21.22.c.b	✓	52
21.c	even	2	1	inner	21.22.c.b	✓	52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.22.c.b	✓	52	1.a	even	1	1	trivial
21.22.c.b	✓	52	3.b	odd	2	1	inner
21.22.c.b	✓	52	7.b	odd	2	1	inner
21.22.c.b	✓	52	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^26 + 41943041*T2^24 + 771725025208368*T2^22 + 8200573001238507022896*T2^20 + 55753162288214348171106520320*T2^18 + 253810999677452961302607392789630976*T2^16 + 786830464457415137346047247320989191045120*T2^14 + 1656156013489291564948949527664674464841385115648*T2^12 + 2316949629053522120725696512494102497023139847746879488*T2^10 + 2062401159748063391840111846667618496499066953071235243704320*T2^8 + 1081044190459548322835287541354705732884760820503240650612113670144*T2^6 + 291144201805040949056467934335771078695290260472997615360706806301065216*T2^4 + 32048395943737439328304609939526673938311574893794800691164782333674192371712*T2^2 + 1090605960526774096005250554363860739773735207539225154348758442405950572724224000