Newform orbit 13.40.c.a

• Label: 13.40.c.a
• Level: $13$
• Weight: $40$
• Character orbit: 13.c
• Analytic conductor: $125.241$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 40 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(125.241416807\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(45\) 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.

\(\operatorname{Tr}(f)(q) = \) 90 * q - 524289 * q^2 - 2324522935 * q^3 - 12369505812481 * q^4 - 75655124623776 * q^5 - 302320992434730 * q^6 - 27963311342034467 * q^7 + 1866166850055213858 * q^8 - 58131653240326890198 * q^9

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−730479.	−	1.26523e6i	−4.44460e7	−	7.69827e7i	−7.92322e11	+	1.37234e12i	−6.68517e13			−6.49337e13	+	1.12468e14i	6.81316e15	−	1.18007e16i	1.51193e18			2.02233e18	−	3.50277e18i	4.88338e19	+	8.45826e19i
3.2	−717015.	−	1.24191e6i	−1.97112e9	−	3.41408e9i	−7.53343e11	+	1.30483e12i	9.41849e12			−2.82665e15	+	4.89590e15i	−2.20896e15	+	3.82603e15i	1.37227e18			−5.74436e18	+	9.94952e18i	−6.75320e18	−	1.16969e19i
3.3	−693620.	−	1.20139e6i	1.08467e9	+	1.87871e9i	−6.87340e11	+	1.19051e12i	2.56987e13			1.50470e15	−	2.60622e15i	−1.81088e16	+	3.13653e16i	1.14437e18			−3.26760e17	+	5.65965e17i	−1.78252e19	−	3.08741e19i
3.4	−658050.	−	1.13978e6i	9.13432e7	+	1.58211e8i	−5.91181e11	+	1.02396e12i	5.69263e13			1.20217e14	−	2.08222e14i	2.26197e16	−	3.91785e16i	8.32573e17			2.00959e18	−	3.48071e18i	−3.74604e19	−	6.48833e19i
3.5	−600609.	−	1.04028e6i	1.56297e9	+	2.70714e9i	−4.46583e11	+	7.73505e11i	2.06832e13			1.87747e15	−	3.25187e15i	1.21658e16	−	2.10717e16i	4.12511e17			−2.85947e18	+	4.95275e18i	−1.24225e19	−	2.15164e19i
3.6	−572089.	−	990887.i	−8.07986e8	−	1.39947e9i	−3.79693e11	+	6.57647e11i	5.96589e13			−9.24479e14	+	1.60124e15i	−2.67068e16	+	4.62575e16i	2.39854e17			7.20596e17	−	1.24811e18i	−3.41302e19	−	5.91152e19i
3.7	−563945.	−	976781.i	1.19066e9	+	2.06228e9i	−3.61190e11	+	6.25599e11i	−2.05825e13			1.34293e15	−	2.32603e15i	−1.62097e16	+	2.80761e16i	1.94700e17			−8.09062e17	+	1.40134e18i	1.16074e19	+	2.01045e19i
3.8	−562795.	−	974789.i	−8.02875e8	−	1.39062e9i	−3.58598e11	+	6.21110e11i	2.25832e12			−9.03708e14	+	1.56527e15i	−3.32165e13	+	5.75327e13i	1.88469e17			7.37060e17	−	1.27663e18i	−1.27097e18	−	2.20139e18i
3.9	−554003.	−	959562.i	−4.03327e8	−	6.98583e8i	−3.38962e11	+	5.87099e11i	−5.38069e13			−4.46889e14	+	7.74035e14i	−7.71068e15	+	1.33553e16i	1.42011e17			1.70093e18	−	2.94610e18i	2.98092e19	+	5.16310e19i
3.10	−495543.	−	858306.i	1.46302e9	+	2.53402e9i	−2.16249e11	+	3.74553e11i	−5.78487e13			1.44998e15	−	2.51143e15i	1.93092e16	−	3.34445e16i	−1.16214e17			−2.25455e18	+	3.90500e18i	2.86665e19	+	4.96519e19i
3.11	−486152.	−	842040.i	−1.24712e9	−	2.16007e9i	−1.97809e11	+	3.42616e11i	−8.27837e12			−1.21258e15	+	2.10025e15i	1.67784e16	−	2.90611e16i	−1.49868e17			−1.08433e18	+	1.87812e18i	4.02454e18	+	6.97071e18i
3.12	−361061.	−	625376.i	−1.83712e9	−	3.18198e9i	1.41482e10	−	2.45053e10i	−7.93428e13			−1.32662e15	+	2.29778e15i	−1.64472e16	+	2.84875e16i	−4.17424e17			−4.72373e18	+	8.18174e18i	2.86476e19	+	4.96191e19i
3.13	−355545.	−	615822.i	4.73994e8	+	8.20981e8i	2.20532e10	−	3.81973e10i	6.16919e13			3.37052e14	−	5.83792e14i	4.17737e15	−	7.23542e15i	−4.22290e17			1.57694e18	−	2.73134e18i	−2.19343e19	−	3.79913e19i
3.14	−301251.	−	521782.i	6.37118e8	+	1.10352e9i	9.33734e10	−	1.61727e11i	6.01790e12			3.83865e14	−	6.64874e14i	5.52912e15	−	9.57671e15i	−4.43745e17			1.21444e18	−	2.10347e18i	−1.81290e18	−	3.14003e18i
3.15	−289314.	−	501106.i	1.96851e9	+	3.40955e9i	1.07473e11	−	1.86148e11i	6.73229e13			1.13903e15	−	1.97286e15i	−1.35054e16	+	2.33920e16i	−4.42478e17			−5.72375e18	+	9.91383e18i	−1.94775e19	−	3.37360e19i
3.16	−273335.	−	473431.i	−1.65431e9	−	2.86534e9i	1.25453e11	−	2.17292e11i	6.90253e13			−9.04361e14	+	1.56640e15i	1.09716e16	−	1.90033e16i	−4.37699e17			−3.44718e18	+	5.97069e18i	−1.88671e19	−	3.26787e19i
3.17	−266870.	−	462232.i	4.31419e8	+	7.47239e8i	1.32439e11	−	2.29391e11i	−3.92658e13			2.30265e14	−	3.98831e14i	−1.99757e16	+	3.45989e16i	−4.34802e17			1.65403e18	−	2.86487e18i	1.04789e19	+	1.81499e19i
3.18	−210193.	−	364065.i	−3.60382e8	−	6.24201e8i	1.86516e11	−	3.23055e11i	−5.38975e13			−1.51500e14	+	2.62405e14i	2.89221e16	−	5.00946e16i	−3.87927e17			1.76653e18	−	3.05971e18i	1.13289e19	+	1.96222e19i
3.19	−175951.	−	304756.i	−9.02657e8	−	1.56345e9i	2.12960e11	−	3.68858e11i	2.52648e13			−3.17647e14	+	5.50181e14i	−2.52884e16	+	4.38008e16i	−3.43343e17			3.96699e17	−	6.87102e17i	−4.44536e18	−	7.69959e18i
3.20	−162970.	−	282273.i	−1.40643e9	−	2.43600e9i	2.21759e11	−	3.84098e11i	3.37737e12			−4.58412e14	+	7.93993e14i	−1.21190e15	+	2.09908e15i	−3.23749e17			−1.92980e18	+	3.34251e18i	−5.50411e17	−	9.53341e17i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	13.40.c.a	✓	90
13.c	even	3	1	inner	13.40.c.a	✓	90

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.40.c.a	✓	90	1.a	even	1	1	trivial
13.40.c.a	✓	90	13.c	even	3	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{40}^{\mathrm{new}}(13, [\chi])\).