Newform orbit 13.13.f.a

• Label: 13.13.f.a
• Level: $13$
• Weight: $13$
• Character orbit: 13.f
• Analytic conductor: $11.882$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8819196246\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(13\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 5152 q^{5} - 64742 q^{6} - 91524 q^{7} + 262074 q^{8} - 3897236 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−120.154	−	32.1952i	257.966	+	446.810i	9853.23	+	5688.77i	−4670.68	+	4670.68i	−16610.5	−	61991.3i	54898.7	−	14710.1i	−640475.	−	640475.i	132628.	−	229718.i	711574.	−	410828.i
2.2	−95.5902	−	25.6133i	−610.410	−	1057.26i	4934.21	+	2848.77i	−7207.18	+	7207.18i	31269.3	+	116699.i	−122301.	+	32770.4i	−112070.	−	112070.i	−479481.	+	830486.i	873535.	−	504336.i
2.3	−72.2023	−	19.3465i	360.185	+	623.859i	1291.64	+	745.730i	12624.7	−	12624.7i	−13936.7	−	52012.4i	−207680.	+	55647.8i	137665.	+	137665.i	6253.68	−	10831.7i	−1.15578e6	+	667288.i
2.4	−71.8315	−	19.2472i	−134.875	−	233.610i	1242.07	+	717.109i	6783.90	−	6783.90i	5191.93	+	19376.5i	110310.	−	29557.4i	139968.	+	139968.i	229338.	−	397225.i	−617869.	+	356727.i
2.5	−50.6414	−	13.5693i	549.131	+	951.122i	−1166.81	−	673.661i	−19544.2	+	19544.2i	−14902.7	−	55617.5i	41955.9	−	11242.1i	201795.	+	201795.i	−337369.	+	584340.i	1.25495e6	−	724544.i
2.6	−11.5857	−	3.10438i	−235.597	−	408.067i	−3422.65	−	1976.07i	−10174.2	+	10174.2i	1462.77	+	5459.13i	1465.24	−	392.611i	68258.9	+	68258.9i	154708.	−	267962.i	149460.	−	86291.0i
2.7	8.21877	+	2.20221i	570.416	+	987.990i	−3484.54	−	2011.80i	15478.9	−	15478.9i	2512.36	+	9376.24i	192917.	−	51692.0i	−48852.1	−	48852.1i	−385028.	+	666889.i	161305.	−	93129.6i
2.8	11.4354	+	3.06412i	−641.689	−	1111.44i	−3425.86	−	1977.92i	20754.5	−	20754.5i	−3932.42	−	14676.0i	−36977.6	+	9908.12i	−67404.6	−	67404.6i	−557808.	+	966151.i	300932.	−	173743.i
2.9	34.3703	+	9.20949i	237.371	+	411.139i	−2450.74	−	1414.93i	1045.50	−	1045.50i	4372.13	+	16317.0i	−154611.	+	41428.0i	−174260.	−	174260.i	153030.	−	265057.i	45562.5	−	26305.5i
2.10	67.8174	+	18.1716i	−141.476	−	245.044i	721.753	+	416.704i	1529.25	−	1529.25i	−5141.70	−	19189.1i	123755.	−	33160.1i	−161974.	−	161974.i	225689.	−	390906.i	131499.	−	75920.9i
2.11	89.3467	+	23.9404i	−589.975	−	1021.87i	3862.45	+	2229.98i	−15775.1	+	15775.1i	−28248.4	−	105425.i	−26939.4	+	7218.38i	23805.8	+	23805.8i	−430420.	+	745508.i	−1.78711e6	+	1.03179e6i
2.12	95.2253	+	25.5155i	487.469	+	844.321i	4869.57	+	2811.45i	−9168.64	+	9168.64i	24876.1	+	92838.7i	23385.6	−	6266.16i	106440.	+	106440.i	−209531.	+	362919.i	−1.10703e6	+	639144.i
2.13	113.725	+	30.4726i	−109.016	−	188.821i	8457.62	+	4883.01i	18181.0	−	18181.0i	−6643.99	−	24795.7i	−92411.2	+	24761.5i	472045.	+	472045.i	241952.	−	419073.i	2.62166e6	−	1.51362e6i
6.1	−29.2050	−	108.995i	52.9243	−	91.6676i	−7479.66	+	4318.39i	8243.68	−	8243.68i	−11536.9	−	3091.31i	−57109.8	+	213137.i	362307.	+	362307.i	260119.	+	450539.i	−1.13927e6	−	657760.i
6.2	−27.7087	−	103.410i	−405.190	+	701.809i	−6378.68	+	3682.73i	−15562.5	+	15562.5i	83801.6	+	22454.6i	45796.3	−	170914.i	247504.	+	247504.i	−62637.0	−	108490.i	2.04054e6	+	1.17811e6i
6.3	−21.6871	−	80.9374i	311.902	−	540.230i	−2533.29	+	1462.60i	10878.6	−	10878.6i	−50489.1	−	13528.5i	51975.1	−	193974.i	−69370.7	−	69370.7i	71154.9	+	123244.i	−1.11641e6	−	644561.i
6.4	−18.7953	−	70.1451i	533.836	−	924.631i	−1019.83	+	588.798i	−17226.5	+	17226.5i	−74892.0	−	20067.2i	−19491.8	+	72744.2i	−149859.	−	149859.i	−304242.	−	526962.i	1.53213e6	+	884577.i
6.5	−13.6554	−	50.9625i	−509.536	+	882.542i	1136.53	−	656.176i	7573.57	−	7573.57i	51934.5	+	13915.8i	−13725.9	+	51225.8i	−201770.	−	201770.i	−253533.	−	439132.i	−489388.	−	282548.i
6.6	−5.92432	−	22.1098i	−126.341	+	218.829i	3093.49	−	1786.03i	−4688.57	+	4688.57i	5586.75	+	1496.97i	−5757.24	+	21486.3i	−124112.	−	124112.i	233796.	+	404947.i	131440.	+	75887.0i
6.7	0.757328	+	2.82639i	456.198	−	790.159i	3539.83	−	2043.72i	10156.9	−	10156.9i	2578.79	+	690.984i	−7644.80	+	28530.8i	16932.0	+	16932.0i	−150513.	−	260697.i	36399.5	+	21015.3i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.f	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	13.13.f.a	✓	52
13.f	odd	12	1	inner	13.13.f.a	✓	52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.13.f.a	✓	52	1.a	even	1	1	trivial
13.13.f.a	✓	52	13.f	odd	12	1	inner

Hecke kernels

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