Newform orbit 12.22.b.a

• Label: 12.22.b.a
• Level: $12$
• Weight: $22$
• Character orbit: 12.b
• Analytic conductor: $33.537$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.5372813144\)
Analytic rank:	\(0\)
Dimension:	\(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 1212040 q^{4} - 73477608 q^{6} + 4862757768 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1448.15	−	1.26410i	80355.2	+	63272.4i	2.09715e6	+	3661.23i		−	1.80238e7i	−1.16287e8	−	9.17298e7i		−	1.08064e9i	−3.03699e9	−	7.95303e6i	2.45355e9	+	1.01685e10i	−2.27840e7	+	2.61013e10i
11.2	−1448.15	+	1.26410i	80355.2	−	63272.4i	2.09715e6	−	3661.23i			1.80238e7i	−1.16287e8	+	9.17298e7i			1.08064e9i	−3.03699e9	+	7.95303e6i	2.45355e9	−	1.01685e10i	−2.27840e7	−	2.61013e10i
11.3	−1422.90	−	269.282i	−24738.8	−	99238.8i	1.95213e6	+	766321.i			8.02021e6i	8.47766e6	+	1.47868e8i		−	9.13804e8i	−2.57132e9	−	1.61607e9i	−9.23633e9	+	4.91011e9i	2.15970e9	−	1.14119e10i
11.4	−1422.90	+	269.282i	−24738.8	+	99238.8i	1.95213e6	−	766321.i		−	8.02021e6i	8.47766e6	−	1.47868e8i			9.13804e8i	−2.57132e9	+	1.61607e9i	−9.23633e9	−	4.91011e9i	2.15970e9	+	1.14119e10i
11.5	−1309.28	−	618.823i	−101361.	+	13651.6i	1.33127e6	+	1.62042e6i		−	2.42790e7i	1.41157e8	+	4.48505e7i			2.63476e8i	−7.40247e8	−	2.94540e9i	1.00876e10	−	2.76748e9i	−1.50244e10	+	3.17879e10i
11.6	−1309.28	+	618.823i	−101361.	−	13651.6i	1.33127e6	−	1.62042e6i			2.42790e7i	1.41157e8	−	4.48505e7i		−	2.63476e8i	−7.40247e8	+	2.94540e9i	1.00876e10	+	2.76748e9i	−1.50244e10	−	3.17879e10i
11.7	−1218.33	−	782.825i	−54958.0	+	86255.2i	871523.	+	1.90748e6i			3.43031e7i	1.34480e8	−	6.20651e7i		−	6.03889e8i	4.31420e8	−	3.00620e9i	−4.41958e9	−	9.48084e9i	2.68533e10	−	4.17926e10i
11.8	−1218.33	+	782.825i	−54958.0	−	86255.2i	871523.	−	1.90748e6i		−	3.43031e7i	1.34480e8	+	6.20651e7i			6.03889e8i	4.31420e8	+	3.00620e9i	−4.41958e9	+	9.48084e9i	2.68533e10	+	4.17926e10i
11.9	−1118.98	−	919.257i	89974.0	+	48631.5i	407085.	+	2.05726e6i			1.11352e7i	−5.59744e7	−	1.37127e8i			4.14938e8i	1.43563e9	−	2.67625e9i	5.73030e9	+	8.75115e9i	1.02361e10	−	1.24600e10i
11.10	−1118.98	+	919.257i	89974.0	−	48631.5i	407085.	−	2.05726e6i		−	1.11352e7i	−5.59744e7	+	1.37127e8i		−	4.14938e8i	1.43563e9	+	2.67625e9i	5.73030e9	−	8.75115e9i	1.02361e10	+	1.24600e10i
11.11	−1006.34	−	1041.36i	68874.9	−	75608.2i	−71700.6	+	2.09593e6i		−	3.89323e7i	−1.48047e8	+	4.36439e6i		−	1.54048e8i	2.25476e9	−	2.03455e9i	−9.72855e8	−	1.04150e10i	−4.05424e10	+	3.91792e10i
11.12	−1006.34	+	1041.36i	68874.9	+	75608.2i	−71700.6	−	2.09593e6i			3.89323e7i	−1.48047e8	−	4.36439e6i			1.54048e8i	2.25476e9	+	2.03455e9i	−9.72855e8	+	1.04150e10i	−4.05424e10	−	3.91792e10i
11.13	−795.891	−	1209.84i	−53927.7	−	86903.1i	−830267.	+	1.92580e6i			1.55115e7i	−6.22182e7	+	1.34409e8i			1.05551e9i	2.99071e9	−	5.28237e8i	−4.64396e9	+	9.37297e9i	1.87665e10	−	1.23455e10i
11.14	−795.891	+	1209.84i	−53927.7	+	86903.1i	−830267.	−	1.92580e6i		−	1.55115e7i	−6.22182e7	−	1.34409e8i		−	1.05551e9i	2.99071e9	+	5.28237e8i	−4.64396e9	−	9.37297e9i	1.87665e10	+	1.23455e10i
11.15	−471.986	−	1369.08i	2769.76	+	102238.i	−1.65161e6	+	1.29237e6i		−	1.59334e7i	1.38665e8	−	5.20471e7i		−	2.91508e7i	2.54890e9	+	1.65120e9i	−1.04450e10	+	5.66352e8i	−2.18140e10	+	7.52033e9i
11.16	−471.986	+	1369.08i	2769.76	−	102238.i	−1.65161e6	−	1.29237e6i			1.59334e7i	1.38665e8	+	5.20471e7i			2.91508e7i	2.54890e9	−	1.65120e9i	−1.04450e10	−	5.66352e8i	−2.18140e10	−	7.52033e9i
11.17	−384.131	−	1396.28i	80301.8	−	63340.1i	−1.80204e6	+	1.07271e6i			3.38573e7i	−1.19287e8	−	8.77928e7i		−	1.23992e9i	2.19002e9	+	2.10409e9i	2.43641e9	−	1.01727e10i	4.72743e10	−	1.30057e10i
11.18	−384.131	+	1396.28i	80301.8	+	63340.1i	−1.80204e6	−	1.07271e6i		−	3.38573e7i	−1.19287e8	+	8.77928e7i			1.23992e9i	2.19002e9	−	2.10409e9i	2.43641e9	+	1.01727e10i	4.72743e10	+	1.30057e10i
11.19	−219.802	−	1431.38i	−101700.	−	10835.4i	−2.00053e6	+	629239.i		−	7.08257e6i	6.84438e6	+	1.47953e8i		−	8.25046e8i	1.34040e9	+	2.72520e9i	1.02255e10	+	2.20393e9i	−1.01378e10	+	1.55676e9i
11.20	−219.802	+	1431.38i	−101700.	+	10835.4i	−2.00053e6	−	629239.i			7.08257e6i	6.84438e6	−	1.47953e8i			8.25046e8i	1.34040e9	−	2.72520e9i	1.02255e10	−	2.20393e9i	−1.01378e10	−	1.55676e9i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	12.22.b.a	✓	40
3.b	odd	2	1	inner	12.22.b.a	✓	40
4.b	odd	2	1	inner	12.22.b.a	✓	40
12.b	even	2	1	inner	12.22.b.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.22.b.a	✓	40	1.a	even	1	1	trivial
12.22.b.a	✓	40	3.b	odd	2	1	inner
12.22.b.a	✓	40	4.b	odd	2	1	inner
12.22.b.a	✓	40	12.b	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(12, [\chi])\).