Newform orbit 680.2.bw.a

• Label: 680.2.bw.a
• Level: $680$
• Weight: $2$
• Character orbit: 680.bw
• Analytic conductor: $5.430$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.bw (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.42982733745\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(104\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 8 q^{3}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−1.41122	+	0.0919263i	−0.532730	−	1.28612i	1.98310	−	0.259457i	1.43197	−	1.71740i	0.870029	+	1.76603i	−1.07592	+	2.59749i	−2.77474	+	0.548451i	0.751009	−	0.751009i	−1.86296	+	2.55527i
43.2	−1.41059	+	0.101160i	0.925634	+	2.23468i	1.97953	−	0.285390i	0.833587	+	2.07488i	−1.53175	−	3.05858i	1.46719	−	3.54211i	−2.76344	+	0.602818i	−2.01566	+	2.01566i	−1.38575	−	2.84248i
43.3	−1.41051	−	0.102287i	−0.0345520	−	0.0834159i	1.97907	+	0.288553i	2.13365	+	0.668991i	0.0402036	+	0.121193i	0.337784	−	0.815483i	−2.76199	−	0.609439i	2.11556	−	2.11556i	−2.94110	−	1.16186i
43.4	−1.40972	+	0.112635i	−1.15905	−	2.79820i	1.97463	−	0.317569i	2.02285	−	0.952933i	1.94912	+	3.81414i	1.43661	−	3.46829i	−2.74790	+	0.670096i	−4.36522	+	4.36522i	−2.74432	+	1.57121i
43.5	−1.40213	+	0.184471i	−1.29603	−	3.12889i	1.93194	−	0.517305i	−0.402426	+	2.19956i	2.39439	+	4.14803i	−1.01180	+	2.44271i	−2.61341	+	1.08172i	−5.98892	+	5.98892i	0.158500	−	3.15830i
43.6	−1.39969	−	0.202187i	0.326326	+	0.787822i	1.91824	+	0.565997i	−1.62071	−	1.54055i	−0.297467	−	1.16868i	−0.371745	+	0.897471i	−2.57050	−	1.18006i	1.60715	−	1.60715i	1.95700	+	2.48398i
43.7	−1.38293	−	0.295809i	−0.522842	−	1.26225i	1.82499	+	0.818166i	−2.19476	+	0.427831i	0.349669	+	1.90027i	1.75164	−	4.22882i	−2.28182	−	1.67132i	0.801403	−	0.801403i	3.16175	+	0.0575689i
43.8	−1.37868	−	0.315012i	1.03921	+	2.50889i	1.80153	+	0.868604i	1.20534	−	1.88339i	−0.642418	−	3.78633i	0.785931	−	1.89741i	−2.21012	−	1.76504i	−3.09322	+	3.09322i	−2.25507	+	2.21691i
43.9	−1.37082	+	0.347624i	0.853522	+	2.06059i	1.75832	−	0.953062i	1.17808	−	1.90056i	−1.88634	−	2.52799i	−1.66073	+	4.00935i	−2.07903	+	1.91771i	−1.39619	+	1.39619i	−0.954256	+	3.01486i
43.10	−1.36950	−	0.352800i	0.179117	+	0.432427i	1.75106	+	0.966319i	−1.19085	+	1.89258i	−0.0927409	−	0.655402i	−0.439667	+	1.06145i	−2.05717	−	1.94115i	1.96641	−	1.96641i	2.29858	−	2.17176i
43.11	−1.34648	+	0.432434i	0.340105	+	0.821085i	1.62600	−	1.16453i	1.22719	+	1.86922i	−0.813008	−	0.958500i	−1.24369	+	3.00255i	−1.68579	+	2.27115i	1.56281	−	1.56281i	−2.46071	−	1.98618i
43.12	−1.33544	+	0.465415i	−0.417438	−	1.00778i	1.56678	−	1.24306i	−1.57359	+	1.58865i	1.02650	+	1.15155i	−0.0111683	+	0.0269627i	−1.51379	+	2.38923i	1.27995	−	1.27995i	1.36204	−	2.85392i
43.13	−1.32319	+	0.499174i	−0.768929	−	1.85636i	1.50165	−	1.32100i	−1.74486	−	1.39838i	1.94408	+	2.07248i	0.430802	−	1.04005i	−1.32756	+	2.49752i	−0.733497	+	0.733497i	3.00681	+	0.979325i
43.14	−1.31804	+	0.512620i	1.16805	+	2.81993i	1.47444	−	1.35130i	−2.00465	+	0.990645i	−2.98509	−	3.11800i	−0.958464	+	2.31394i	−1.25066	+	2.53690i	−4.46633	+	4.46633i	2.13438	−	2.33333i
43.15	−1.31800	−	0.512703i	−0.902788	−	2.17952i	1.47427	+	1.35149i	−0.186653	−	2.22826i	0.0724296	+	3.33548i	−0.856085	+	2.06677i	−1.25018	−	2.53713i	−1.81398	+	1.81398i	−0.896429	+	3.03256i
43.16	−1.24967	−	0.662063i	−0.671792	−	1.62185i	1.12334	+	1.65472i	1.63134	+	1.52929i	−0.234249	+	2.47154i	−0.772054	+	1.86390i	−0.308281	−	2.81158i	−0.0577730	+	0.0577730i	−1.02616	−	2.99115i
43.17	−1.21233	−	0.728189i	−0.0135844	−	0.0327956i	0.939481	+	1.76561i	0.252941	−	2.22172i	−0.00741267	+	0.0496510i	2.00134	−	4.83165i	0.146738	−	2.82462i	2.12043	−	2.12043i	−1.92448	+	2.50926i
43.18	−1.21179	−	0.729091i	1.06685	+	2.57561i	0.936853	+	1.76700i	1.70634	+	1.44513i	0.585056	−	3.89892i	−1.80996	+	4.36962i	0.153041	−	2.82428i	−3.37428	+	3.37428i	−1.01409	−	2.99526i
43.19	−1.20819	+	0.735039i	−0.125586	−	0.303191i	0.919434	−	1.77613i	0.408921	−	2.19836i	0.374589	+	0.274001i	0.307523	−	0.742425i	0.194678	+	2.82172i	2.04517	−	2.04517i	1.12183	+	2.95660i
43.20	−1.19515	−	0.756053i	1.18551	+	2.86208i	0.856767	+	1.80719i	−2.23573	+	0.0388262i	0.747020	−	4.31693i	0.497135	−	1.20019i	0.342370	−	2.80763i	−4.66474	+	4.66474i	2.70139	+	1.64393i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
85.k	odd	8	1	inner
680.bw	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	680.2.bw.a	✓	416
5.c	odd	4	1		680.2.bz.a	yes	416
8.d	odd	2	1	inner	680.2.bw.a	✓	416
17.d	even	8	1		680.2.bz.a	yes	416
40.k	even	4	1		680.2.bz.a	yes	416
85.k	odd	8	1	inner	680.2.bw.a	✓	416
136.p	odd	8	1		680.2.bz.a	yes	416
680.bw	even	8	1	inner	680.2.bw.a	✓	416

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
680.2.bw.a	✓	416	1.a	even	1	1	trivial
680.2.bw.a	✓	416	8.d	odd	2	1	inner
680.2.bw.a	✓	416	85.k	odd	8	1	inner
680.2.bw.a	✓	416	680.bw	even	8	1	inner
680.2.bz.a	yes	416	5.c	odd	4	1
680.2.bz.a	yes	416	17.d	even	8	1
680.2.bz.a	yes	416	40.k	even	4	1
680.2.bz.a	yes	416	136.p	odd	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(680, [\chi])\).