# Properties

 Label 560.2.bb.c Level 560 Weight 2 Character orbit 560.bb Analytic conductor 4.472 Analytic rank 0 Dimension 70 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.bb (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$70$$ Relative dimension: $$35$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$70q - 2q^{2} - 2q^{3} - 4q^{5} - 70q^{7} - 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$70q - 2q^{2} - 2q^{3} - 4q^{5} - 70q^{7} - 8q^{8} + 6q^{10} - 2q^{11} + 4q^{12} - 6q^{13} + 2q^{14} - 6q^{15} + 4q^{16} + 18q^{18} + 14q^{19} + 20q^{20} + 2q^{21} + 12q^{22} + 20q^{24} - 6q^{25} - 36q^{26} - 8q^{27} + 2q^{29} + 28q^{30} + 16q^{31} + 8q^{32} + 4q^{34} + 4q^{35} - 40q^{36} - 10q^{37} + 12q^{38} + 44q^{40} - 2q^{43} - 24q^{44} + 22q^{45} - 16q^{46} + 44q^{48} + 70q^{49} - 74q^{50} + 8q^{51} - 28q^{52} + 30q^{53} - 32q^{54} - 6q^{55} + 8q^{56} + 76q^{57} - 56q^{58} + 2q^{59} - 64q^{60} + 30q^{61} - 48q^{62} + 12q^{64} - 10q^{65} + 80q^{66} - 6q^{67} + 36q^{68} - 16q^{69} - 6q^{70} - 4q^{72} + 36q^{73} - 32q^{74} - 98q^{75} + 44q^{76} + 2q^{77} + 84q^{78} - 40q^{79} - 24q^{80} - 82q^{81} - 24q^{82} - 10q^{83} - 4q^{84} - 32q^{85} + 32q^{86} + 4q^{87} - 32q^{88} + 158q^{90} + 6q^{91} + 92q^{92} + 56q^{93} - 20q^{94} + 6q^{95} + 16q^{96} - 2q^{98} - 34q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.41420 0.00610828i 1.35224 1.35224i 1.99993 + 0.0172767i −0.516149 + 2.17568i −1.92059 + 1.90407i −1.00000 −2.82819 0.0366488i 0.657087i 0.743228 3.07370i
29.2 −1.40637 0.148759i −0.279102 + 0.279102i 1.95574 + 0.418421i 2.19030 0.450071i 0.434038 0.351001i −1.00000 −2.68825 0.879388i 2.84420i −3.14733 + 0.307138i
29.3 −1.36233 + 0.379542i 2.24950 2.24950i 1.71190 1.03412i 0.718756 2.11740i −2.21079 + 3.91835i −1.00000 −1.93968 + 2.05856i 7.12050i −0.175541 + 3.15740i
29.4 −1.32540 + 0.493263i −2.08897 + 2.08897i 1.51338 1.30754i −2.21442 0.310380i 1.73832 3.79914i −1.00000 −1.36088 + 2.47952i 5.72761i 3.08810 0.680914i
29.5 −1.31948 0.508904i 1.51552 1.51552i 1.48203 + 1.34297i −2.14663 0.626067i −2.77094 + 1.22844i −1.00000 −1.27206 2.52623i 1.59360i 2.51382 + 1.91851i
29.6 −1.30469 0.545695i −1.97585 + 1.97585i 1.40443 + 1.42393i 1.90686 + 1.16786i 3.65608 1.49966i −1.00000 −1.05532 2.62418i 4.80794i −1.85056 2.56426i
29.7 −1.25189 0.657851i −1.16502 + 1.16502i 1.13446 + 1.64712i −1.61836 + 1.54302i 2.22489 0.692068i −1.00000 −0.336668 2.80832i 0.285460i 3.04109 0.867061i
29.8 −1.16286 + 0.804830i 0.441211 0.441211i 0.704499 1.87181i −2.22133 0.256291i −0.157968 + 0.868166i −1.00000 0.687255 + 2.74366i 2.61067i 2.78937 1.48976i
29.9 −1.13277 + 0.846660i −1.23809 + 1.23809i 0.566334 1.91814i 1.09503 + 1.94959i 0.354228 2.45070i −1.00000 0.982488 + 2.65230i 0.0657107i −2.89106 1.28132i
29.10 −0.954551 1.04347i −0.512166 + 0.512166i −0.177666 + 1.99209i −0.780394 2.09547i 1.02332 + 0.0455422i −1.00000 2.24828 1.71616i 2.47537i −1.44164 + 2.81455i
29.11 −0.795266 1.16942i 0.629715 0.629715i −0.735102 + 1.86001i 2.03423 0.928390i −1.23720 0.235612i −1.00000 2.75974 0.619555i 2.20692i −2.70344 1.64056i
29.12 −0.734791 + 1.20834i 0.510771 0.510771i −0.920165 1.77575i 2.08147 0.816996i 0.241875 + 0.992494i −1.00000 2.82184 + 0.192935i 2.47823i −0.542236 + 3.11544i
29.13 −0.543124 + 1.30576i −0.0768379 + 0.0768379i −1.41003 1.41838i −1.44155 + 1.70937i −0.0585996 0.142065i −1.00000 2.61789 1.07081i 2.98819i −1.44908 2.81072i
29.14 −0.536830 + 1.30836i −1.88282 + 1.88282i −1.42363 1.40474i 0.665150 2.13485i −1.45266 3.47417i −1.00000 2.60215 1.10852i 4.09005i 2.43608 + 2.01631i
29.15 −0.525764 1.31285i 1.37854 1.37854i −1.44714 + 1.38050i −1.88536 + 1.20226i −2.53459 1.08502i −1.00000 2.57324 + 1.17407i 0.800723i 2.56964 + 1.84308i
29.16 −0.435695 + 1.34543i 2.28818 2.28818i −1.62034 1.17239i 1.30264 + 1.81745i 2.08163 + 4.07553i −1.00000 2.28334 1.66924i 7.47157i −3.01280 + 0.960759i
29.17 −0.125099 1.40867i −1.73140 + 1.73140i −1.96870 + 0.352445i 1.43768 1.71263i 2.65556 + 2.22237i −1.00000 0.742760 + 2.72916i 2.99546i −2.59238 1.81096i
29.18 0.168286 1.40417i 0.0271728 0.0271728i −1.94336 0.472603i 1.37117 + 1.76632i −0.0335823 0.0427279i −1.00000 −0.990653 + 2.64927i 2.99852i 2.71096 1.62810i
29.19 0.171434 + 1.40378i −0.104638 + 0.104638i −1.94122 + 0.481312i −0.0634380 2.23517i −0.164828 0.128951i −1.00000 −1.00845 2.64254i 2.97810i 3.12682 0.472237i
29.20 0.239753 1.39374i −0.826272 + 0.826272i −1.88504 0.668309i −1.90570 + 1.16974i 0.953509 + 1.34971i −1.00000 −1.38339 + 2.46703i 1.63455i 1.17342 + 2.93651i
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 309.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.q even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bb.c 70
5.b even 2 1 560.2.bb.d yes 70
16.e even 4 1 560.2.bb.d yes 70
80.q even 4 1 inner 560.2.bb.c 70

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bb.c 70 1.a even 1 1 trivial
560.2.bb.c 70 80.q even 4 1 inner
560.2.bb.d yes 70 5.b even 2 1
560.2.bb.d yes 70 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{70} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database