Newform orbit 36.8.h.a

• Label: 36.8.h.a
• Level: $36$
• Weight: $8$
• Character orbit: 36.h
• Analytic conductor: $11.246$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2458609174\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 3 q^{2} - q^{4} - 6 q^{5} + 213 q^{6} - 1122 q^{9}+O(q^{10}) \)

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	−11.3077	−	0.368978i	−40.2700	+	23.7766i	127.728	+	8.34459i	85.7671	−	49.5177i	464.134	−	253.999i	−1324.40	−	764.644i	−1441.23	−	141.487i	1056.35	−	1914.97i	−988.099	+	528.284i
11.2	−11.3044	−	0.458089i	46.2108	+	7.18041i	127.580	+	10.3569i	470.092	−	271.407i	−519.098	−	102.339i	−125.431	−	72.4176i	−1437.48	−	175.522i	2083.88	+	663.626i	−5438.45	+	2852.76i
11.3	−11.1881	+	1.68133i	36.5819	−	29.1336i	122.346	−	37.6218i	−315.564	+	182.191i	−360.298	+	387.455i	331.882	+	191.612i	−1305.56	+	626.620i	489.471	−	2131.52i	3224.23	−	2568.93i
11.4	−10.8853	+	3.08391i	−22.0162	−	41.2588i	108.979	−	67.1385i	162.727	−	93.9503i	366.891	+	381.217i	598.144	+	345.338i	−979.218	+	1066.90i	−1217.57	+	1816.72i	−1481.59	+	1524.51i
11.5	−10.6008	−	3.95258i	−43.4991	−	17.1704i	96.7543	+	83.8010i	−282.514	+	163.110i	393.259	+	353.954i	791.631	+	457.048i	−694.444	−	1270.79i	1597.35	+	1493.80i	3639.58	−	612.436i
11.6	−10.2657	+	4.75548i	15.7283	+	44.0411i	82.7709	−	97.6370i	−168.386	+	97.2179i	−370.899	−	377.319i	−267.560	−	154.476i	−385.394	+	1395.93i	−1692.24	+	1385.38i	1266.29	−	1798.77i
11.7	−10.1453	−	5.00724i	−2.83407	+	46.6794i	77.8552	+	101.600i	58.0808	−	33.5330i	262.487	−	459.387i	1011.03	+	583.718i	−281.130	−	1420.60i	−2170.94	−	264.585i	−757.156	+	49.3785i
11.8	−9.40906	−	6.28249i	2.83407	−	46.6794i	49.0607	+	118.225i	58.0808	−	33.5330i	−319.929	+	421.404i	−1011.03	−	583.718i	281.130	−	1420.60i	−2170.94	−	264.585i	−757.156	−	49.3785i
11.9	−8.72344	−	7.20428i	43.4991	+	17.1704i	24.1966	+	125.692i	−282.514	+	163.110i	−255.761	−	463.165i	−791.631	−	457.048i	694.444	−	1270.79i	1597.35	+	1493.80i	3639.58	+	612.436i
11.10	−8.23458	+	7.75833i	−42.1945	+	20.1650i	7.61675	−	127.773i	135.559	−	78.2652i	191.007	−	493.409i	815.008	+	470.545i	928.585	+	1111.25i	1373.75	−	1701.70i	−509.067	+	1696.19i
11.11	−7.13475	+	8.78040i	18.0495	−	43.1418i	−26.1907	−	125.292i	206.733	−	119.357i	250.024	+	466.288i	−662.470	−	382.477i	1286.98	+	663.961i	−1535.43	−	1557.38i	−426.984	+	2666.78i
11.12	−6.82651	+	9.02212i	−39.1591	−	25.5649i	−34.7974	−	123.179i	−442.134	+	255.266i	497.970	−	178.779i	−1224.84	−	707.163i	1348.88	+	526.939i	879.873	+	2002.20i	715.190	−	5731.56i
11.13	−6.04893	−	9.56088i	−46.2108	−	7.18041i	−54.8208	+	115.666i	470.092	−	271.407i	210.875	+	485.250i	125.431	+	72.4176i	1437.48	−	175.522i	2083.88	+	663.626i	−5438.45	−	2852.76i
11.14	−5.97339	−	9.60826i	40.2700	−	23.7766i	−56.6372	+	114.788i	85.7671	−	49.5177i	−469.000	−	244.898i	1324.40	+	764.644i	1441.23	−	141.487i	1056.35	−	1914.97i	−988.099	−	528.284i
11.15	−5.76846	+	9.73267i	45.5303	+	10.6765i	−61.4497	−	112.285i	25.5922	−	14.7757i	−366.551	+	381.545i	486.771	+	281.037i	1447.30	+	49.6429i	1959.02	+	972.212i	−3.82102	+	334.314i
11.16	−4.13796	−	10.5298i	−36.5819	+	29.1336i	−93.7545	+	87.1441i	−315.564	+	182.191i	458.146	+	264.648i	−331.882	−	191.612i	1305.56	+	626.620i	489.471	−	2131.52i	3224.23	+	2568.93i
11.17	−2.77190	−	10.9689i	22.0162	+	41.2588i	−112.633	+	60.8093i	162.727	−	93.9503i	391.536	−	355.859i	−598.144	−	345.338i	979.218	+	1066.90i	−1217.57	+	1816.72i	−1481.59	−	1524.51i
11.18	−1.91846	+	11.1499i	−4.07212	+	46.5877i	−120.639	−	42.7812i	345.583	−	199.523i	−511.635	−	134.780i	−1342.20	−	774.918i	708.446	−	1263.03i	−2153.84	−	379.422i	1561.66	+	4235.98i
11.19	−1.01451	−	11.2681i	−15.7283	−	44.0411i	−125.942	+	22.8632i	−168.386	+	97.2179i	−480.305	+	221.909i	267.560	+	154.476i	385.394	+	1395.93i	−1692.24	+	1385.38i	1266.29	+	1798.77i
11.20	−0.719044	+	11.2908i	−19.9236	+	42.3090i	−126.966	−	16.2372i	−369.143	+	213.125i	−463.378	−	255.376i	737.762	+	425.947i	274.626	−	1421.88i	−1393.10	−	1685.89i	−2140.93	−	4321.18i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
36.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	36.8.h.a	✓	80
3.b	odd	2	1		108.8.h.a		80
4.b	odd	2	1	inner	36.8.h.a	✓	80
9.c	even	3	1		108.8.h.a		80
9.d	odd	6	1	inner	36.8.h.a	✓	80
12.b	even	2	1		108.8.h.a		80
36.f	odd	6	1		108.8.h.a		80
36.h	even	6	1	inner	36.8.h.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.8.h.a	✓	80	1.a	even	1	1	trivial
36.8.h.a	✓	80	4.b	odd	2	1	inner
36.8.h.a	✓	80	9.d	odd	6	1	inner
36.8.h.a	✓	80	36.h	even	6	1	inner
108.8.h.a		80	3.b	odd	2	1
108.8.h.a		80	9.c	even	3	1
108.8.h.a		80	12.b	even	2	1
108.8.h.a		80	36.f	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(36, [\chi])\).