# Properties

 Label 128.4.k.a Level $128$ Weight $4$ Character orbit 128.k Analytic conductor $7.552$ Analytic rank $0$ Dimension $752$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 128.k (of order $$32$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.55224448073$$ Analytic rank: $$0$$ Dimension: $$752$$ Relative dimension: $$47$$ over $$\Q(\zeta_{32})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$752q - 16q^{2} - 16q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 16q^{7} - 16q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$752q - 16q^{2} - 16q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 16q^{7} - 16q^{8} - 16q^{9} - 16q^{10} - 16q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 16q^{15} - 16q^{16} - 16q^{17} - 16q^{18} - 16q^{19} - 16q^{20} - 16q^{21} - 16q^{22} - 16q^{23} - 16q^{24} - 16q^{25} - 16q^{26} - 16q^{27} - 16q^{28} - 16q^{29} - 16q^{30} - 16q^{31} - 16q^{32} - 16q^{33} - 16q^{34} - 16q^{35} - 16q^{36} - 16q^{37} - 16q^{38} - 16q^{39} - 16q^{40} - 16q^{41} - 16q^{42} - 16q^{43} - 16q^{44} - 16q^{45} - 16q^{46} - 16q^{47} - 16q^{48} - 16q^{49} + 5696q^{50} - 16q^{51} + 6608q^{52} - 16q^{53} + 3440q^{54} - 16q^{55} - 800q^{56} - 16q^{57} - 4768q^{58} - 16q^{59} - 9808q^{60} - 16q^{61} - 5872q^{62} - 12112q^{64} - 10960q^{66} - 16q^{67} - 4144q^{68} - 16q^{69} - 4048q^{70} - 16q^{71} + 1280q^{72} - 16q^{73} + 5248q^{74} - 16q^{75} + 11888q^{76} - 16q^{77} + 14096q^{78} - 16q^{79} + 10016q^{80} - 16q^{81} - 16q^{82} - 16q^{83} - 16q^{84} - 16q^{85} - 16q^{86} - 16q^{87} - 16q^{88} - 16q^{89} - 16q^{90} - 16q^{91} - 16q^{92} - 16q^{93} - 16q^{94} - 16q^{95} - 16q^{96} - 16q^{97} - 16q^{98} - 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −2.82809 + 0.0437786i 3.90803 + 1.18549i 7.99617 0.247619i −3.72183 0.366569i −11.1042 3.18158i −11.6508 17.4367i −22.6030 + 1.05035i −8.58237 5.73455i 10.5417 + 0.873752i
5.2 −2.82184 + 0.192933i −9.75553 2.95931i 7.92555 1.08885i −0.385980 0.0380157i 28.0995 + 6.46853i 8.69824 + 13.0178i −22.1546 + 4.60167i 63.9633 + 42.7389i 1.09651 + 0.0328059i
5.3 −2.73119 + 0.735238i −2.99224 0.907685i 6.91885 4.01616i 20.1386 + 1.98348i 8.83975 + 0.279059i −4.42679 6.62516i −15.9439 + 16.0559i −14.3201 9.56838i −56.4608 + 9.38941i
5.4 −2.72237 0.767264i −3.00383 0.911201i 6.82261 + 4.17756i 1.47948 + 0.145716i 7.47840 + 4.78536i 3.74690 + 5.60763i −15.3684 16.6076i −14.2570 9.52622i −3.91588 1.53184i
5.5 −2.71781 + 0.783272i −1.31785 0.399764i 6.77297 4.25757i −18.9129 1.86275i 3.89478 + 0.0542507i 16.8694 + 25.2468i −15.0728 + 16.8763i −20.8728 13.9467i 52.8606 9.75131i
5.6 −2.60432 1.10341i 9.60431 + 2.91344i 5.56498 + 5.74726i −17.8404 1.75713i −21.7980 18.1850i 10.3895 + 15.5490i −8.15141 21.1082i 61.3050 + 40.9627i 44.5233 + 24.2614i
5.7 −2.53887 + 1.24665i 7.58094 + 2.29965i 4.89175 6.33015i 9.48044 + 0.933742i −22.1139 + 3.61223i 6.95727 + 10.4123i −4.52806 + 22.1697i 29.7326 + 19.8667i −25.2337 + 9.44810i
5.8 −2.50388 1.31551i 3.59854 + 1.09161i 4.53886 + 6.58777i 17.1492 + 1.68905i −7.57431 7.46718i 15.6405 + 23.4076i −2.69848 22.4659i −10.6918 7.14401i −40.7177 26.7892i
5.9 −2.42367 1.45802i −4.84264 1.46900i 3.74833 + 7.06753i −16.3353 1.60888i 9.59512 + 10.6211i −11.5654 17.3089i 1.21991 22.5945i −1.15646 0.772719i 37.2455 + 27.7166i
5.10 −2.36116 + 1.55721i −6.77444 2.05500i 3.15018 7.35366i −5.70543 0.561936i 19.1956 5.69705i −18.2713 27.3450i 4.01313 + 22.2687i 19.2204 + 12.8426i 14.3465 7.55774i
5.11 −2.23791 1.72967i 7.71088 + 2.33907i 2.01649 + 7.74169i 13.5785 + 1.33737i −13.2105 18.5719i −18.9453 28.3537i 8.87784 20.8131i 31.5368 + 21.0722i −28.0744 26.4793i
5.12 −2.14725 + 1.84101i 3.92169 + 1.18963i 1.22134 7.90622i −5.31043 0.523032i −10.6110 + 4.66546i −0.627511 0.939136i 11.9329 + 19.2251i −8.48522 5.66964i 12.3657 8.65350i
5.13 −1.88791 + 2.10613i −3.74897 1.13724i −0.871580 7.95238i 4.16267 + 0.409988i 9.47289 5.74882i 10.4814 + 15.6865i 18.3942 + 13.1777i −9.68822 6.47346i −8.72225 + 7.99312i
5.14 −1.83459 2.15274i −7.96376 2.41578i −1.26855 + 7.89878i 9.56707 + 0.942274i 9.40970 + 21.5759i 2.63589 + 3.94489i 19.3313 11.7602i 35.1358 + 23.4770i −15.5232 22.3241i
5.15 −1.79489 2.18595i 2.18691 + 0.663393i −1.55674 + 7.84707i −5.13671 0.505922i −2.47513 5.97120i 4.12327 + 6.17091i 19.9475 10.6817i −18.1072 12.0988i 8.11391 + 12.1367i
5.16 −1.35278 + 2.48395i 6.26012 + 1.89899i −4.33998 6.72046i −18.2007 1.79261i −13.1855 + 12.9809i −7.34552 10.9934i 22.5643 1.68898i 13.1332 + 8.77534i 29.0742 42.7844i
5.17 −1.13660 2.59001i −2.41332 0.732073i −5.41626 + 5.88762i 13.6673 + 1.34611i 0.846916 + 7.08259i −16.0547 24.0276i 21.4051 + 7.33627i −17.1615 11.4669i −12.0478 36.9283i
5.18 −1.11449 + 2.59960i 0.572589 + 0.173693i −5.51584 5.79444i 8.36768 + 0.824145i −1.08968 + 1.29492i −6.62562 9.91594i 21.2105 7.88116i −22.1520 14.8015i −11.4681 + 20.8341i
5.19 −0.935141 2.66937i 3.85090 + 1.16816i −6.25102 + 4.99247i −11.1272 1.09593i −0.482895 11.3718i −0.831707 1.24474i 19.1723 + 12.0176i −8.98487 6.00350i 7.48005 + 30.7274i
5.20 −0.668572 + 2.74827i −6.55625 1.98882i −7.10602 3.67484i −14.4435 1.42256i 9.84915 16.6887i 0.534510 + 0.799951i 14.8504 17.0724i 16.5794 + 11.0780i 13.5661 38.7437i
See next 80 embeddings (of 752 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.47 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
128.k even 32 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.4.k.a 752
128.k even 32 1 inner 128.4.k.a 752

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.k.a 752 1.a even 1 1 trivial
128.4.k.a 752 128.k even 32 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database