Newform orbit 89.11.d.a

• Label: 89.11.d.a
• Level: $89$
• Weight: $11$
• Character orbit: 89.d
• Analytic conductor: $56.547$
• Analytic rank: $0$
• Dimension: $296$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	89.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(56.5467954880\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 8 q^{2} - 92 q^{3} + 149496 q^{4} - 15584 q^{5} + 20220 q^{6} - 4 q^{7} - 8200 q^{8} + 174632 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−61.8757			−102.733	−	248.019i	2804.60			621.425	+	621.425i	6356.66	+	15346.3i	9519.08	+	22981.1i	−110176.			−9205.28	+	9205.28i	−38451.1	−	38451.1i
12.2	−61.4795			−130.763	−	315.689i	2755.73			−866.524	−	866.524i	8039.21	+	19408.4i	−11546.5	−	27875.8i	−106466.			−40806.6	+	40806.6i	53273.4	+	53273.4i
12.3	−61.0438			128.985	+	311.396i	2702.35			−1334.97	−	1334.97i	−7873.71	−	19008.8i	6390.26	+	15427.4i	−102453.			−38576.7	+	38576.7i	81491.8	+	81491.8i
12.4	−60.7229			0.965913	+	2.33192i	2663.27			2932.56	+	2932.56i	−58.6531	−	141.601i	−5018.44	−	12115.6i	−99541.5			41749.4	−	41749.4i	−178073.	−	178073.i
12.5	−55.0037			107.297	+	259.038i	2001.41			2728.74	+	2728.74i	−5901.73	−	14248.0i	−101.952	−	246.133i	−53761.1			−13834.0	+	13834.0i	−150091.	−	150091.i
12.6	−54.5144			14.7343	+	35.5717i	1947.82			−1224.81	−	1224.81i	−803.230	−	1939.17i	−2094.50	−	5056.56i	−50361.5			40705.7	−	40705.7i	66769.9	+	66769.9i
12.7	−54.3551			−25.1648	−	60.7532i	1930.48			−2685.15	−	2685.15i	1367.84	+	3302.25i	5655.34	+	13653.2i	−49272.0			38696.3	−	38696.3i	145952.	+	145952.i
12.8	−54.3166			74.4041	+	179.627i	1926.29			−3581.07	−	3581.07i	−4041.38	−	9756.75i	−10662.6	−	25741.8i	−49009.3			15023.9	−	15023.9i	194512.	+	194512.i
12.9	−50.9923			180.361	+	435.430i	1576.22			1319.05	+	1319.05i	−9197.02	−	22203.6i	−9335.43	−	22537.7i	−28158.8			−115315.	+	115315.i	−67261.2	−	67261.2i
12.10	−50.8358			−141.393	−	341.353i	1560.27			−2823.88	−	2823.88i	7187.82	+	17352.9i	1504.36	+	3631.84i	−27261.9			−54775.8	+	54775.8i	143554.	+	143554.i
12.11	−49.2143			−156.152	−	376.984i	1398.05			3117.23	+	3117.23i	7684.91	+	18553.0i	−309.008	−	746.012i	−18408.7			−75979.6	+	75979.6i	−153412.	−	153412.i
12.12	−47.7194			−30.5789	−	73.8241i	1253.14			1798.60	+	1798.60i	1459.21	+	3522.84i	−642.744	−	1551.72i	−10934.5			37239.0	−	37239.0i	−85828.3	−	85828.3i
12.13	−45.2177			−76.1075	−	183.740i	1020.64			1735.82	+	1735.82i	3441.41	+	8308.29i	12176.2	+	29396.0i	151.852			13786.0	−	13786.0i	−78489.6	−	78489.6i
12.14	−44.1554			93.9726	+	226.870i	925.702			3954.46	+	3954.46i	−4149.40	−	10017.5i	10323.4	+	24922.9i	4340.39			−885.149	+	885.149i	−174611.	−	174611.i
12.15	−40.3385			125.274	+	302.439i	603.198			−958.251	−	958.251i	−5053.38	−	12199.9i	8333.27	+	20118.3i	16974.5			−34021.5	+	34021.5i	38654.4	+	38654.4i
12.16	−39.6260			−102.819	−	248.228i	546.222			1645.17	+	1645.17i	4074.32	+	9836.28i	−9226.36	−	22274.4i	18932.4			−9291.30	+	9291.30i	−65191.5	−	65191.5i
12.17	−38.8224			−108.638	−	262.276i	483.182			−2987.64	−	2987.64i	4217.61	+	10182.2i	−3627.71	−	8758.07i	20995.9			−15232.6	+	15232.6i	115987.	+	115987.i
12.18	−38.6611			150.604	+	363.590i	470.678			−3704.58	−	3704.58i	−5822.51	−	14056.8i	−620.670	−	1498.43i	21392.0			−67762.1	+	67762.1i	143223.	+	143223.i
12.19	−38.0235			10.4189	+	25.1535i	421.787			−542.809	−	542.809i	−396.164	−	956.424i	930.294	+	2245.93i	22898.2			41229.8	−	41229.8i	20639.5	+	20639.5i
12.20	−34.4975			44.7582	+	108.056i	166.077			1875.42	+	1875.42i	−1544.05	−	3727.66i	−11768.9	−	28412.7i	29596.2			32081.2	−	32081.2i	−64697.2	−	64697.2i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
89.d	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	89.11.d.a	✓	296
89.d	odd	8	1	inner	89.11.d.a	✓	296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
89.11.d.a	✓	296	1.a	even	1	1	trivial
89.11.d.a	✓	296	89.d	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(89, [\chi])\).