# Properties

 Label 750.2.l.a Level $750$ Weight $2$ Character orbit 750.l Analytic conductor $5.989$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$750 = 2 \cdot 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 750.l (of order $$20$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.98878015160$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{20})$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 4q^{3} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 4q^{3} - 4q^{7} - 16q^{12} + 20q^{16} + 8q^{18} + 40q^{19} - 4q^{22} + 56q^{27} - 4q^{28} + 96q^{33} + 40q^{34} + 64q^{37} + 40q^{39} + 4q^{42} + 24q^{43} - 16q^{48} + 64q^{57} - 20q^{58} - 4q^{63} + 104q^{67} - 140q^{69} - 8q^{72} + 60q^{73} + 60q^{78} - 80q^{79} - 40q^{81} - 96q^{82} - 60q^{84} - 80q^{87} - 24q^{88} - 12q^{93} + 12q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −0.891007 0.453990i −1.65131 + 0.522656i 0.587785 + 0.809017i 0 1.70861 + 0.283990i −0.712495 0.712495i −0.156434 0.987688i 2.45366 1.72614i 0
107.2 −0.891007 0.453990i −0.974581 1.43185i 0.587785 + 0.809017i 0 0.218312 + 1.71824i −3.13589 3.13589i −0.156434 0.987688i −1.10038 + 2.79091i 0
107.3 −0.891007 0.453990i 0.102145 + 1.72904i 0.587785 + 0.809017i 0 0.693954 1.58696i 2.97677 + 2.97677i −0.156434 0.987688i −2.97913 + 0.353226i 0
107.4 −0.891007 0.453990i 0.646378 + 1.60692i 0.587785 + 0.809017i 0 0.153600 1.72523i −2.03922 2.03922i −0.156434 0.987688i −2.16439 + 2.07736i 0
107.5 −0.891007 0.453990i 1.59860 0.666690i 0.587785 + 0.809017i 0 −1.72703 0.131724i 1.51403 + 1.51403i −0.156434 0.987688i 2.11105 2.13154i 0
107.6 0.891007 + 0.453990i −1.31434 1.12805i 0.587785 + 0.809017i 0 −0.658960 1.60180i 1.51403 + 1.51403i 0.156434 + 0.987688i 0.454984 + 2.96530i 0
107.7 0.891007 + 0.453990i −1.11131 + 1.32853i 0.587785 + 0.809017i 0 −1.59332 + 0.679206i −2.03922 2.03922i 0.156434 + 0.987688i −0.529988 2.95281i 0
107.8 0.891007 + 0.453990i −0.631448 + 1.61285i 0.587785 + 0.809017i 0 −1.29484 + 1.15039i 2.97677 + 2.97677i 0.156434 + 0.987688i −2.20255 2.03686i 0
107.9 0.891007 + 0.453990i 1.36935 1.06061i 0.587785 + 0.809017i 0 1.70160 0.323338i −3.13589 3.13589i 0.156434 + 0.987688i 0.750223 2.90468i 0
107.10 0.891007 + 0.453990i 1.40898 + 1.00736i 0.587785 + 0.809017i 0 0.798080 + 1.53723i −0.712495 0.712495i 0.156434 + 0.987688i 0.970457 + 2.83870i 0
143.1 −0.453990 + 0.891007i −1.51894 0.832356i −0.587785 0.809017i 0 1.43122 0.975505i −0.0556476 + 0.0556476i 0.987688 0.156434i 1.61437 + 2.52860i 0
143.2 −0.453990 + 0.891007i −0.333634 + 1.69961i −0.587785 0.809017i 0 −1.36290 1.06888i −2.58285 + 2.58285i 0.987688 0.156434i −2.77738 1.13410i 0
143.3 −0.453990 + 0.891007i 0.00452789 1.73204i −0.587785 0.809017i 0 1.54121 + 0.790366i 0.152718 0.152718i 0.987688 0.156434i −2.99996 0.0156850i 0
143.4 −0.453990 + 0.891007i 1.05251 + 1.37558i −0.587785 0.809017i 0 −1.70348 + 0.313291i −0.462249 + 0.462249i 0.987688 0.156434i −0.784450 + 2.89562i 0
143.5 −0.453990 + 0.891007i 1.69234 0.368756i −0.587785 0.809017i 0 −0.439743 + 1.67530i 2.72680 2.72680i 0.987688 0.156434i 2.72804 1.24812i 0
143.6 0.453990 0.891007i −1.70181 + 0.322239i −0.587785 0.809017i 0 −0.485490 + 1.66262i −0.0556476 + 0.0556476i −0.987688 + 0.156434i 2.79232 1.09678i 0
143.7 0.453990 0.891007i −0.530925 + 1.64867i −0.587785 0.809017i 0 1.22794 + 1.22154i 0.152718 0.152718i −0.987688 + 0.156434i −2.43624 1.75064i 0
143.8 0.453990 0.891007i 0.207905 1.71953i −0.587785 0.809017i 0 −1.43772 0.965894i −2.58285 + 2.58285i −0.987688 + 0.156434i −2.91355 0.714995i 0
143.9 0.453990 0.891007i 1.42607 0.983013i −0.587785 0.809017i 0 −0.228447 1.71692i −0.462249 + 0.462249i −0.987688 + 0.156434i 1.06737 2.80370i 0
143.10 0.453990 0.891007i 1.49556 + 0.873670i −0.587785 0.809017i 0 1.45742 0.935916i 2.72680 2.72680i −0.987688 + 0.156434i 1.47340 + 2.61325i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 743.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.l.a 80
3.b odd 2 1 inner 750.2.l.a 80
5.b even 2 1 750.2.l.c 80
5.c odd 4 1 150.2.l.a 80
5.c odd 4 1 750.2.l.b 80
15.d odd 2 1 750.2.l.c 80
15.e even 4 1 150.2.l.a 80
15.e even 4 1 750.2.l.b 80
25.d even 5 1 750.2.l.b 80
25.e even 10 1 150.2.l.a 80
25.f odd 20 1 inner 750.2.l.a 80
25.f odd 20 1 750.2.l.c 80
75.h odd 10 1 150.2.l.a 80
75.j odd 10 1 750.2.l.b 80
75.l even 20 1 inner 750.2.l.a 80
75.l even 20 1 750.2.l.c 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.l.a 80 5.c odd 4 1
150.2.l.a 80 15.e even 4 1
150.2.l.a 80 25.e even 10 1
150.2.l.a 80 75.h odd 10 1
750.2.l.a 80 1.a even 1 1 trivial
750.2.l.a 80 3.b odd 2 1 inner
750.2.l.a 80 25.f odd 20 1 inner
750.2.l.a 80 75.l even 20 1 inner
750.2.l.b 80 5.c odd 4 1
750.2.l.b 80 15.e even 4 1
750.2.l.b 80 25.d even 5 1
750.2.l.b 80 75.j odd 10 1
750.2.l.c 80 5.b even 2 1
750.2.l.c 80 15.d odd 2 1
750.2.l.c 80 25.f odd 20 1
750.2.l.c 80 75.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(750, [\chi])$$:

 $$35\!\cdots\!30$$$$T_{7}^{17} +$$$$10\!\cdots\!37$$$$T_{7}^{16} -$$$$13\!\cdots\!84$$$$T_{7}^{15} +$$$$84\!\cdots\!08$$$$T_{7}^{14} -$$$$15\!\cdots\!36$$$$T_{7}^{13} +$$$$12\!\cdots\!04$$$$T_{7}^{12} -$$$$17\!\cdots\!76$$$$T_{7}^{11} +$$$$10\!\cdots\!08$$$$T_{7}^{10} +$$$$48\!\cdots\!28$$$$T_{7}^{9} +$$$$46\!\cdots\!16$$$$T_{7}^{8} -$$$$24\!\cdots\!48$$$$T_{7}^{7} + 603823112768 T_{7}^{6} - 45403189408 T_{7}^{5} + 834922128 T_{7}^{4} + 128231520 T_{7}^{3} + 44783648 T_{7}^{2} - 719264 T_{7} + 5776$$">$$T_{7}^{40} + \cdots$$ $$23\!\cdots\!50$$$$T_{13}^{22} -$$$$41\!\cdots\!00$$$$T_{13}^{21} -$$$$10\!\cdots\!00$$$$T_{13}^{20} +$$$$27\!\cdots\!00$$$$T_{13}^{19} +$$$$43\!\cdots\!00$$$$T_{13}^{18} +$$$$13\!\cdots\!00$$$$T_{13}^{17} +$$$$38\!\cdots\!00$$$$T_{13}^{16} +$$$$17\!\cdots\!00$$$$T_{13}^{15} +$$$$38\!\cdots\!50$$$$T_{13}^{14} +$$$$51\!\cdots\!00$$$$T_{13}^{13} +$$$$14\!\cdots\!50$$$$T_{13}^{12} +$$$$24\!\cdots\!00$$$$T_{13}^{11} +$$$$54\!\cdots\!00$$$$T_{13}^{10} +$$$$41\!\cdots\!00$$$$T_{13}^{9} +$$$$10\!\cdots\!25$$$$T_{13}^{8} -$$$$29\!\cdots\!00$$$$T_{13}^{7} +$$$$89\!\cdots\!00$$$$T_{13}^{6} +$$$$44\!\cdots\!00$$$$T_{13}^{5} +$$$$13\!\cdots\!00$$$$T_{13}^{4} +$$$$16\!\cdots\!00$$$$T_{13}^{3} +$$$$50\!\cdots\!00$$$$T_{13}^{2} +$$$$92\!\cdots\!00$$$$T_{13} +$$$$14\!\cdots\!00$$">$$T_{13}^{40} - \cdots$$