Properties

Label 105.4.m.a
Level $105$
Weight $4$
Character orbit 105.m
Analytic conductor $6.195$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 4q^{7} - 168q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 4q^{7} - 168q^{8} + 112q^{11} + 168q^{15} - 544q^{16} - 96q^{21} - 192q^{22} + 400q^{23} + 520q^{25} + 1052q^{28} - 48q^{30} - 1344q^{32} + 392q^{35} - 1728q^{36} - 456q^{37} + 1068q^{42} + 192q^{43} - 208q^{46} + 3528q^{50} + 672q^{51} - 1728q^{53} - 48q^{56} + 696q^{57} + 3016q^{58} + 840q^{60} - 36q^{63} - 4720q^{65} - 4784q^{67} + 2220q^{70} - 3088q^{71} - 1512q^{72} + 2352q^{77} + 1416q^{78} - 3888q^{81} - 472q^{85} + 10832q^{86} + 2128q^{88} - 5664q^{91} + 10600q^{92} - 1368q^{93} - 6912q^{95} - 3888q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −3.58711 + 3.58711i −2.12132 + 2.12132i 17.7347i 10.0305 4.93863i 15.2188i 2.02215 + 18.4095i 34.9193 + 34.9193i 9.00000i −18.2649 + 53.6957i
13.2 −3.58711 + 3.58711i 2.12132 2.12132i 17.7347i −10.0305 + 4.93863i 15.2188i −18.4095 2.02215i 34.9193 + 34.9193i 9.00000i 18.2649 53.6957i
13.3 −2.96540 + 2.96540i −2.12132 + 2.12132i 9.58718i −8.64856 + 7.08536i 12.5811i −4.10485 + 18.0596i 4.70662 + 4.70662i 9.00000i 4.63550 46.6574i
13.4 −2.96540 + 2.96540i 2.12132 2.12132i 9.58718i 8.64856 7.08536i 12.5811i −18.0596 + 4.10485i 4.70662 + 4.70662i 9.00000i −4.63550 + 46.6574i
13.5 −2.50091 + 2.50091i −2.12132 + 2.12132i 4.50911i −9.59495 5.73906i 10.6105i 16.0890 9.17301i −8.73040 8.73040i 9.00000i 38.3490 9.64324i
13.6 −2.50091 + 2.50091i 2.12132 2.12132i 4.50911i 9.59495 + 5.73906i 10.6105i 9.17301 16.0890i −8.73040 8.73040i 9.00000i −38.3490 + 9.64324i
13.7 −2.16465 + 2.16465i −2.12132 + 2.12132i 1.37142i 9.30469 6.19861i 9.18383i −8.45914 16.4755i −14.3486 14.3486i 9.00000i −6.72356 + 33.5592i
13.8 −2.16465 + 2.16465i 2.12132 2.12132i 1.37142i −9.30469 + 6.19861i 9.18383i 16.4755 + 8.45914i −14.3486 14.3486i 9.00000i 6.72356 33.5592i
13.9 −0.817511 + 0.817511i −2.12132 + 2.12132i 6.66335i 8.83013 + 6.85775i 3.46841i 18.0193 + 4.27846i −11.9875 11.9875i 9.00000i −12.8250 + 1.61244i
13.10 −0.817511 + 0.817511i 2.12132 2.12132i 6.66335i −8.83013 6.85775i 3.46841i −4.27846 18.0193i −11.9875 11.9875i 9.00000i 12.8250 1.61244i
13.11 −0.811442 + 0.811442i −2.12132 + 2.12132i 6.68312i −4.54645 + 10.2142i 3.44266i −15.2496 10.5095i −11.9145 11.9145i 9.00000i −4.59905 11.9774i
13.12 −0.811442 + 0.811442i 2.12132 2.12132i 6.68312i 4.54645 10.2142i 3.44266i 10.5095 + 15.2496i −11.9145 11.9145i 9.00000i 4.59905 + 11.9774i
13.13 −0.216837 + 0.216837i −2.12132 + 2.12132i 7.90596i −1.17774 11.1181i 0.919960i −3.57938 + 18.1711i −3.44900 3.44900i 9.00000i 2.66620 + 2.15544i
13.14 −0.216837 + 0.216837i 2.12132 2.12132i 7.90596i 1.17774 + 11.1181i 0.919960i −18.1711 + 3.57938i −3.44900 3.44900i 9.00000i −2.66620 2.15544i
13.15 1.52903 1.52903i −2.12132 + 2.12132i 3.32413i −1.55846 11.0712i 6.48713i 15.0664 10.7705i 17.3149 + 17.3149i 9.00000i −19.3111 14.5453i
13.16 1.52903 1.52903i 2.12132 2.12132i 3.32413i 1.55846 + 11.0712i 6.48713i 10.7705 15.0664i 17.3149 + 17.3149i 9.00000i 19.3111 + 14.5453i
13.17 1.79945 1.79945i −2.12132 + 2.12132i 1.52398i −11.1587 0.695951i 7.63441i −12.8255 13.3607i 17.1379 + 17.1379i 9.00000i −21.3317 + 18.8271i
13.18 1.79945 1.79945i 2.12132 2.12132i 1.52398i 11.1587 + 0.695951i 7.63441i 13.3607 + 12.8255i 17.1379 + 17.1379i 9.00000i 21.3317 18.8271i
13.19 2.53770 2.53770i −2.12132 + 2.12132i 4.87986i −2.35304 + 10.9299i 10.7666i 8.62377 + 16.3900i 7.91797 + 7.91797i 9.00000i 21.7656 + 33.7082i
13.20 2.53770 2.53770i 2.12132 2.12132i 4.87986i 2.35304 10.9299i 10.7666i −16.3900 8.62377i 7.91797 + 7.91797i 9.00000i −21.7656 33.7082i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.m.a 48
5.c odd 4 1 inner 105.4.m.a 48
7.b odd 2 1 inner 105.4.m.a 48
35.f even 4 1 inner 105.4.m.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.m.a 48 1.a even 1 1 trivial
105.4.m.a 48 5.c odd 4 1 inner
105.4.m.a 48 7.b odd 2 1 inner
105.4.m.a 48 35.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(105, [\chi])\).