Properties

Label 105.4.p.a
Level $105$
Weight $4$
Character orbit 105.p
Analytic conductor $6.195$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88q - 164q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 88q - 164q^{4} + 30q^{10} + 114q^{15} - 548q^{16} - 228q^{19} + 300q^{21} + 72q^{24} + 160q^{25} - 318q^{30} + 816q^{31} - 336q^{36} - 336q^{39} + 30q^{40} + 1377q^{45} - 1296q^{46} + 688q^{49} + 138q^{51} + 2268q^{54} - 1866q^{60} + 1392q^{61} + 2248q^{64} - 2772q^{66} + 150q^{70} - 765q^{75} - 1528q^{79} + 2844q^{81} - 5160q^{84} - 1084q^{85} - 5632q^{91} + 4416q^{94} - 3456q^{96} - 660q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −2.65960 4.60657i 1.84211 + 4.85867i −10.1470 + 17.5751i 0.975062 11.1377i 17.4825 21.4079i 7.48937 16.9384i 65.3942 −20.2133 + 17.9004i −53.9000 + 25.1303i
59.2 −2.65960 4.60657i 5.12878 0.834020i −10.1470 + 17.5751i 10.1331 + 4.72444i −17.4825 21.4079i −7.48937 + 16.9384i 65.3942 25.6088 8.55501i −5.18656 59.2440i
59.3 −2.57122 4.45348i −4.78955 + 2.01500i −9.22230 + 15.9735i −9.15758 + 6.41394i 21.2887 + 16.1492i −17.8760 4.84251i 53.7107 18.8796 19.3018i 52.1105 + 24.2914i
59.4 −2.57122 4.45348i −0.649738 5.15537i −9.22230 + 15.9735i −10.1334 + 4.72373i −21.2887 + 16.1492i 17.8760 + 4.84251i 53.7107 −26.1557 + 6.69928i 47.0922 + 32.9833i
59.5 −2.26674 3.92610i −5.16785 + 0.541618i −6.27618 + 10.8707i 11.1613 + 0.652852i 13.8406 + 19.0618i 18.5202 + 0.0273336i 20.6380 26.4133 5.59799i −22.7365 45.3001i
59.6 −2.26674 3.92610i −2.11487 4.74630i −6.27618 + 10.8707i 5.01525 9.99236i −13.8406 + 19.0618i −18.5202 0.0273336i 20.6380 −18.0547 + 20.0756i −50.5993 + 2.95968i
59.7 −1.89916 3.28944i 4.02025 + 3.29205i −3.21362 + 5.56616i −5.19426 + 9.90049i 3.19390 19.4765i 6.82394 17.2173i −5.97382 5.32484 + 26.4697i 42.4318 1.71639i
59.8 −1.89916 3.28944i 4.86112 + 1.83562i −3.21362 + 5.56616i −11.1712 0.451881i −3.19390 19.4765i −6.82394 + 17.2173i −5.97382 20.2610 + 17.8463i 19.7295 + 37.6052i
59.9 −1.82858 3.16720i −2.45762 + 4.57822i −2.68744 + 4.65479i −6.67931 8.96587i 18.9941 0.587877i 2.93091 + 18.2869i −9.60048 −14.9202 22.5031i −16.1830 + 37.5496i
59.10 −1.82858 3.16720i 2.73604 4.41747i −2.68744 + 4.65479i 4.42501 + 10.2674i −18.9941 0.587877i −2.93091 18.2869i −9.60048 −12.0281 24.1728i 24.4274 32.7897i
59.11 −1.76958 3.06500i −0.764269 + 5.13964i −2.26281 + 3.91930i 9.87152 + 5.24910i 17.1054 6.75251i −18.4952 0.962729i −12.2964 −25.8318 7.85614i −1.37994 39.5449i
59.12 −1.76958 3.06500i 4.06892 3.23170i −2.26281 + 3.91930i 0.389908 11.1735i −17.1054 6.75251i 18.4952 + 0.962729i −12.2964 6.11228 26.2991i −34.9368 + 18.5774i
59.13 −1.29175 2.23737i −4.90178 1.72412i 0.662782 1.14797i −9.00517 6.62623i 2.47436 + 13.1942i 1.81286 18.4313i −24.0925 21.0548 + 16.9025i −3.19292 + 28.7073i
59.14 −1.29175 2.23737i −3.94402 3.38300i 0.662782 1.14797i 1.23589 + 11.1118i −2.47436 + 13.1942i −1.81286 + 18.4313i −24.0925 4.11059 + 26.6853i 23.2648 17.1188i
59.15 −0.878214 1.52111i 3.00499 + 4.23911i 2.45748 4.25648i 11.1091 1.25980i 3.80914 8.29376i 15.1670 + 10.6284i −22.6842 −8.94012 + 25.4769i −11.6725 15.7919i
59.16 −0.878214 1.52111i 5.17367 + 0.482839i 2.45748 4.25648i 6.64558 8.99090i −3.80914 8.29376i −15.1670 10.6284i −22.6842 26.5337 + 4.99610i −19.5124 2.21274i
59.17 −0.850362 1.47287i −3.03689 + 4.21632i 2.55377 4.42326i −6.34140 + 9.20796i 8.79254 + 0.887546i 17.5269 5.98407i −22.2923 −8.55463 25.6089i 18.9546 + 1.50996i
59.18 −0.850362 1.47287i 2.13299 4.73818i 2.55377 4.42326i −11.1450 + 0.887834i −8.79254 + 0.887546i −17.5269 + 5.98407i −22.2923 −17.9007 20.2130i 10.7850 + 15.6602i
59.19 −0.388071 0.672158i −5.13489 + 0.795580i 3.69880 6.40651i 5.93654 9.47404i 2.52746 + 3.14272i −10.2225 + 15.4435i −11.9507 25.7341 8.17043i −8.67186 0.313697i
59.20 −0.388071 0.672158i −1.87845 4.84473i 3.69880 6.40651i 11.1730 0.404175i −2.52746 + 3.14272i 10.2225 15.4435i −11.9507 −19.9428 + 18.2012i −4.60760 7.35320i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.p.a 88
3.b odd 2 1 inner 105.4.p.a 88
5.b even 2 1 inner 105.4.p.a 88
7.d odd 6 1 inner 105.4.p.a 88
15.d odd 2 1 inner 105.4.p.a 88
21.g even 6 1 inner 105.4.p.a 88
35.i odd 6 1 inner 105.4.p.a 88
105.p even 6 1 inner 105.4.p.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.p.a 88 1.a even 1 1 trivial
105.4.p.a 88 3.b odd 2 1 inner
105.4.p.a 88 5.b even 2 1 inner
105.4.p.a 88 7.d odd 6 1 inner
105.4.p.a 88 15.d odd 2 1 inner
105.4.p.a 88 21.g even 6 1 inner
105.4.p.a 88 35.i odd 6 1 inner
105.4.p.a 88 105.p even 6 1 inner

Hecke kernels

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