Newform orbit 9.24.c.a

• Label: 9.24.c.a
• Level: $9$
• Weight: $24$
• Character orbit: 9.c
• Analytic conductor: $30.168$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.1683633611\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2049 q^{2} - 162336 q^{3} - 88080385 q^{4} - 121251630 q^{5} - 922041945 q^{6} + 670564216 q^{7} - 6691053918 q^{8} - 15707798892 q^{9}+O(q^{10}) \)

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	−2668.65	−	4622.24i	−232101.	+	200680.i	−1.00491e7	+	1.74055e7i	−6.19968e7	+	1.07382e8i	1.54699e9	+	5.37279e8i	9.15169e8	+	1.58512e9i	6.24973e10			1.35982e10	−	9.31559e10i	6.61791e11
4.2	−2624.02	−	4544.94i	298725.	−	70045.8i	−9.57665e6	+	1.65873e7i	−2.19852e7	+	3.80794e7i	−1.10221e9	−	1.17389e9i	−4.15089e9	−	7.18956e9i	5.64936e10			8.43303e10	−	4.18489e10i	2.30758e11
4.3	−2298.99	−	3981.97i	−20411.7	−	306148.i	−6.37643e6	+	1.10443e7i	−3.25040e6	+	5.62986e6i	−1.17215e9	+	7.85111e8i	3.99249e9	+	6.91520e9i	2.00668e10			−9.33099e10	+	1.24980e10i	2.98906e10
4.4	−2155.56	−	3733.55i	104678.	+	288419.i	−5.09860e6	+	8.83104e6i	7.28850e7	−	1.26241e8i	8.51188e8	−	1.01253e9i	−4.71607e7	−	8.16847e7i	7.79711e9			−7.22283e10	+	6.03821e10i	−6.28433e11
4.5	−2084.75	−	3610.89i	−297336.	−	75726.4i	−4.49804e6	+	7.79084e6i	9.18314e7	−	1.59057e8i	3.46431e8	+	1.23152e9i	−1.89346e9	−	3.27957e9i	2.53286e9			8.26742e10	+	4.50324e10i	−7.65781e11
4.6	−1437.74	−	2490.24i	−196795.	−	235404.i	60111.2	−	104116.i	−8.10132e7	+	1.40319e8i	−3.03271e8	+	8.28516e8i	−4.36457e9	−	7.55966e9i	−2.44670e10			−1.66867e10	+	9.26525e10i	4.65903e11
4.7	−1402.17	−	2428.63i	306375.	−	16654.6i	262137.	−	454034.i	−6.24766e6	+	1.08213e7i	−4.70038e8	−	7.20720e8i	2.78756e9	+	4.82820e9i	−2.49948e10			9.35884e10	−	1.02051e10i	3.50412e10
4.8	−1269.95	−	2199.62i	125596.	+	279944.i	968741.	−	1.67791e6i	−6.63944e7	+	1.14999e8i	4.56270e8	−	6.31781e8i	2.02711e7	+	3.51106e7i	−2.62273e10			−6.25943e10	+	7.03199e10i	3.37271e11
4.9	−962.814	−	1667.64i	−261364.	+	160724.i	2.34028e6	−	4.05349e6i	−1.21147e7	+	2.09833e7i	5.19675e8	+	2.81113e8i	1.29020e9	+	2.23468e9i	−2.51664e10			4.24786e10	−	8.40149e10i	4.66569e10
4.10	−711.068	−	1231.61i	159433.	−	262153.i	3.18307e6	−	5.51324e6i	5.54849e7	−	9.61027e7i	−4.36237e8	−	9.95028e6i	−1.97964e9	−	3.42884e9i	−2.09832e10			−4.33053e10	−	8.35918e10i	−1.57814e11
4.11	186.630	+	323.252i	−254964.	−	170695.i	4.12464e6	−	7.14409e6i	1.84635e7	−	3.19797e7i	7.59358e6	−	1.14274e8i	2.65089e9	+	4.59148e9i	6.21025e9			3.58699e10	+	8.70419e10i	1.37833e10
4.12	197.360	+	341.838i	−85021.5	+	294813.i	4.11640e6	−	7.12982e6i	4.71876e7	−	8.17313e7i	−1.17558e8	+	2.91207e7i	−2.85057e9	−	4.93733e9i	6.56082e9			−7.96859e10	−	5.01308e10i	3.72518e10
4.13	475.248	+	823.153i	130521.	−	277682.i	3.74258e6	−	6.48234e6i	−9.91305e7	+	1.71699e8i	2.90605e8	−	2.45290e7i	1.94392e9	+	3.36698e9i	1.50879e10			−6.00717e10	−	7.24868e10i	−1.88446e11
4.14	524.048	+	907.678i	291651.	+	95303.0i	3.64505e6	−	6.31341e6i	−4.37964e7	+	7.58576e7i	6.63349e7	+	3.14669e8i	−3.25039e9	−	5.62983e9i	1.64328e10			7.59779e10	+	5.55905e10i	−9.18057e10
4.15	705.867	+	1222.60i	268633.	+	148255.i	3.19781e6	−	5.53877e6i	8.66611e7	−	1.50101e8i	8.36347e6	+	4.33078e8i	3.12193e9	+	5.40734e9i	2.08714e10			5.01843e10	+	7.96522e10i	2.44685e11
4.16	1489.61	+	2580.08i	−24299.9	+	305864.i	−243573.	+	421881.i	−6.02610e7	+	1.04375e8i	−8.25351e8	+	3.92922e8i	4.52542e9	+	7.83826e9i	2.35402e10			−9.29622e10	−	1.48649e10i	−3.59062e11
4.17	1555.96	+	2695.00i	−303041.	+	48053.5i	−647700.	+	1.12185e6i	−4.91176e7	+	8.50743e7i	−6.01023e8	−	7.41926e8i	−2.86187e9	−	4.95691e9i	2.20735e10			8.95249e10	−	2.91244e10i	−3.05700e11
4.18	1633.18	+	2828.76i	−64023.2	−	300074.i	−1.14027e6	+	1.97501e6i	4.40994e7	−	7.63824e7i	7.44274e8	−	6.71181e8i	−1.71109e9	−	2.96369e9i	1.99512e10			−8.59452e10	+	3.84234e10i	2.88090e11
4.19	2120.67	+	3673.11i	273772.	−	138536.i	−4.80019e6	+	8.31417e6i	4.96074e6	−	8.59225e6i	1.08944e9	+	7.11805e8i	7.61092e8	+	1.31825e9i	−5.13950e9			5.57588e10	−	7.58544e10i	4.20804e10
4.20	2341.27	+	4055.20i	−265039.	+	154588.i	−6.76881e6	+	1.17239e7i	1.00559e8	−	1.74173e8i	−1.24741e9	−	7.12856e8i	2.00849e9	+	3.47880e9i	−2.41105e10			4.63485e10	−	8.19436e10i	9.41744e11

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9.24.c.a	✓	44
3.b	odd	2	1		27.24.c.a		44
9.c	even	3	1	inner	9.24.c.a	✓	44
9.d	odd	6	1		27.24.c.a		44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.24.c.a	✓	44	1.a	even	1	1	trivial
9.24.c.a	✓	44	9.c	even	3	1	inner
27.24.c.a		44	3.b	odd	2	1
27.24.c.a		44	9.d	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(9, [\chi])\).