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Properties

Label 230.5.c.a
Level $230$
Weight $5$
Character orbit 230.c
Analytic conductor $23.775$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 230.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.7750915093\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48q - 384q^{4} - 64q^{6} - 1608q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 384q^{4} - 64q^{6} - 1608q^{9} + 3072q^{16} + 512q^{24} + 1488q^{25} - 384q^{26} - 2940q^{29} + 4732q^{31} + 2556q^{35} + 12864q^{36} + 2736q^{39} + 828q^{41} - 64q^{46} + 32572q^{49} - 3264q^{50} + 10112q^{54} + 12824q^{55} + 7092q^{59} - 24576q^{64} + 14000q^{69} - 6592q^{70} - 12708q^{71} + 24728q^{75} - 13040q^{81} - 15700q^{85} - 9728q^{94} - 46608q^{95} - 4096q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 − 2.82843i − 9.09812i −8.00000 −21.5641 + 12.6486i −25.7334 94.0651 22.6274i −1.77586 35.7757 + 60.9926i
229.2 2.82843i 9.09812i −8.00000 −21.5641 − 12.6486i −25.7334 94.0651 − 22.6274i −1.77586 35.7757 − 60.9926i
229.3 − 2.82843i − 15.3728i −8.00000 −24.9424 + 1.69556i −43.4808 −85.1277 22.6274i −155.323 4.79578 + 70.5479i
229.4 2.82843i 15.3728i −8.00000 −24.9424 − 1.69556i −43.4808 −85.1277 − 22.6274i −155.323 4.79578 − 70.5479i
229.5 − 2.82843i 13.4757i −8.00000 20.1787 − 14.7587i 38.1149 78.6574 22.6274i −100.593 −41.7439 − 57.0741i
229.6 2.82843i − 13.4757i −8.00000 20.1787 + 14.7587i 38.1149 78.6574 − 22.6274i −100.593 −41.7439 + 57.0741i
229.7 − 2.82843i − 0.832711i −8.00000 19.4576 − 15.6972i −2.35526 66.8572 22.6274i 80.3066 −44.3985 − 55.0343i
229.8 2.82843i 0.832711i −8.00000 19.4576 + 15.6972i −2.35526 66.8572 − 22.6274i 80.3066 −44.3985 + 55.0343i
229.9 − 2.82843i 6.81504i −8.00000 6.15914 + 24.2294i 19.2758 68.2555 22.6274i 34.5552 68.5312 − 17.4207i
229.10 2.82843i − 6.81504i −8.00000 6.15914 − 24.2294i 19.2758 68.2555 − 22.6274i 34.5552 68.5312 + 17.4207i
229.11 − 2.82843i 7.41746i −8.00000 23.4452 + 8.67897i 20.9797 −42.4567 22.6274i 25.9814 24.5478 − 66.3129i
229.12 2.82843i − 7.41746i −8.00000 23.4452 − 8.67897i 20.9797 −42.4567 − 22.6274i 25.9814 24.5478 + 66.3129i
229.13 − 2.82843i 10.7328i −8.00000 12.7779 − 21.4878i 30.3568 −32.1609 22.6274i −34.1923 −60.7766 − 36.1415i
229.14 2.82843i − 10.7328i −8.00000 12.7779 + 21.4878i 30.3568 −32.1609 − 22.6274i −34.1923 −60.7766 + 36.1415i
229.15 − 2.82843i 15.5407i −8.00000 22.6151 + 10.6563i 43.9558 −25.0660 22.6274i −160.514 30.1404 − 63.9653i
229.16 2.82843i − 15.5407i −8.00000 22.6151 − 10.6563i 43.9558 −25.0660 − 22.6274i −160.514 30.1404 + 63.9653i
229.17 − 2.82843i − 15.7740i −8.00000 −12.1414 − 21.8538i −44.6155 29.1236 22.6274i −167.818 −61.8118 + 34.3409i
229.18 2.82843i 15.7740i −8.00000 −12.1414 + 21.8538i −44.6155 29.1236 − 22.6274i −167.818 −61.8118 − 34.3409i
229.19 − 2.82843i − 1.19309i −8.00000 −1.65984 − 24.9448i −3.37456 28.6899 22.6274i 79.5765 −70.5547 + 4.69474i
229.20 2.82843i 1.19309i −8.00000 −1.65984 + 24.9448i −3.37456 28.6899 − 22.6274i 79.5765 −70.5547 − 4.69474i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.5.c.a ✓ 48
5.b even 2 1 inner 230.5.c.a ✓ 48
23.b odd 2 1 inner 230.5.c.a ✓ 48
115.c odd 2 1 inner 230.5.c.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.5.c.a ✓ 48 1.a even 1 1 trivial
230.5.c.a ✓ 48 5.b even 2 1 inner
230.5.c.a ✓ 48 23.b odd 2 1 inner
230.5.c.a ✓ 48 115.c odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(230, [\chi])\).