# Properties

 Label 224.2.u.b Level $224$ Weight $2$ Character orbit 224.u Analytic conductor $1.789$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.u (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 4 q^{2} + 4 q^{3} + 8 q^{5} + 12 q^{6} - 8 q^{8} + 8 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40 q + 4 q^{2} + 4 q^{3} + 8 q^{5} + 12 q^{6} - 8 q^{8} + 8 q^{9} - 16 q^{10} + 12 q^{11} - 36 q^{12} + 8 q^{16} + 4 q^{19} - 4 q^{21} - 52 q^{22} + 16 q^{23} - 8 q^{24} - 16 q^{25} + 12 q^{26} - 32 q^{27} + 8 q^{28} + 16 q^{29} + 36 q^{30} + 24 q^{31} - 36 q^{32} + 8 q^{33} - 8 q^{34} - 8 q^{36} + 16 q^{37} - 12 q^{38} - 24 q^{39} + 36 q^{40} + 8 q^{41} + 52 q^{43} - 44 q^{44} - 64 q^{45} - 32 q^{46} - 36 q^{48} + 52 q^{50} + 16 q^{51} - 16 q^{52} - 32 q^{54} - 8 q^{55} + 12 q^{56} - 8 q^{57} + 40 q^{58} + 20 q^{59} + 52 q^{60} - 16 q^{61} - 12 q^{62} - 24 q^{63} - 48 q^{64} - 80 q^{65} - 40 q^{66} - 4 q^{67} - 4 q^{68} - 40 q^{69} + 32 q^{70} + 72 q^{72} - 24 q^{73} - 12 q^{74} + 20 q^{75} - 72 q^{76} + 4 q^{77} + 12 q^{78} + 60 q^{80} + 16 q^{82} + 4 q^{83} + 8 q^{84} - 64 q^{85} - 48 q^{86} - 32 q^{87} + 16 q^{88} + 32 q^{89} + 20 q^{90} + 108 q^{92} - 24 q^{93} + 68 q^{94} - 80 q^{95} - 40 q^{96} + 56 q^{97} + 4 q^{98} - 60 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.39530 0.230541i −3.03470 1.25702i 1.89370 + 0.643346i 0.551446 + 1.33131i 3.94452 + 2.45353i −0.707107 + 0.707107i −2.49396 1.33423i 5.50802 + 5.50802i −0.462509 1.98470i
29.2 −1.32480 + 0.494880i 0.0422763 + 0.0175114i 1.51019 1.31123i 1.39396 + 3.36532i −0.0646737 0.00227744i −0.707107 + 0.707107i −1.35179 + 2.48448i −2.11984 2.11984i −3.51215 3.76853i
29.3 −0.533524 + 1.30971i −0.910708 0.377228i −1.43070 1.39753i −0.227513 0.549265i 0.979945 0.991507i −0.707107 + 0.707107i 2.59368 1.12820i −1.43423 1.43423i 0.840764 0.00493097i
29.4 −0.420598 1.35022i 2.28471 + 0.946358i −1.64619 + 1.13580i 0.446012 + 1.07677i 0.316848 3.48290i −0.707107 + 0.707107i 2.22597 + 1.74501i 2.20299 + 2.20299i 1.26628 1.05510i
29.5 0.146191 + 1.40664i 2.41040 + 0.998422i −1.95726 + 0.411276i 0.0734640 + 0.177358i −1.05204 + 3.53652i −0.707107 + 0.707107i −0.864649 2.69302i 2.69188 + 2.69188i −0.238738 + 0.129265i
29.6 0.270256 1.38815i 0.350600 + 0.145223i −1.85392 0.750313i −1.27761 3.08441i 0.296344 0.447438i −0.707107 + 0.707107i −1.54258 + 2.37075i −2.01949 2.01949i −4.62691 + 0.939927i
29.7 0.461048 + 1.33695i −1.66728 0.690611i −1.57487 + 1.23280i 0.923458 + 2.22943i 0.154616 2.54748i −0.707107 + 0.707107i −2.37428 1.53715i 0.181564 + 0.181564i −2.55487 + 2.26249i
29.8 1.16944 0.795236i 1.29558 + 0.536647i 0.735199 1.85997i 0.937867 + 2.26421i 1.94187 0.402713i −0.707107 + 0.707107i −0.619339 2.75979i −0.730784 0.730784i 2.89737 + 1.90204i
29.9 1.27165 + 0.618795i 2.04641 + 0.847650i 1.23419 + 1.57378i −1.31439 3.17322i 2.07779 + 2.34422i −0.707107 + 0.707107i 0.595604 + 2.76501i 1.34795 + 1.34795i 0.292129 4.84857i
29.10 1.35563 0.402828i −1.81729 0.752744i 1.67546 1.09217i −0.920909 2.22327i −2.76679 0.288389i −0.707107 + 0.707107i 1.83135 2.15550i 0.614581 + 0.614581i −2.14401 2.64296i
85.1 −1.39530 + 0.230541i −3.03470 + 1.25702i 1.89370 0.643346i 0.551446 1.33131i 3.94452 2.45353i −0.707107 0.707107i −2.49396 + 1.33423i 5.50802 5.50802i −0.462509 + 1.98470i
85.2 −1.32480 0.494880i 0.0422763 0.0175114i 1.51019 + 1.31123i 1.39396 3.36532i −0.0646737 + 0.00227744i −0.707107 0.707107i −1.35179 2.48448i −2.11984 + 2.11984i −3.51215 + 3.76853i
85.3 −0.533524 1.30971i −0.910708 + 0.377228i −1.43070 + 1.39753i −0.227513 + 0.549265i 0.979945 + 0.991507i −0.707107 0.707107i 2.59368 + 1.12820i −1.43423 + 1.43423i 0.840764 + 0.00493097i
85.4 −0.420598 + 1.35022i 2.28471 0.946358i −1.64619 1.13580i 0.446012 1.07677i 0.316848 + 3.48290i −0.707107 0.707107i 2.22597 1.74501i 2.20299 2.20299i 1.26628 + 1.05510i
85.5 0.146191 1.40664i 2.41040 0.998422i −1.95726 0.411276i 0.0734640 0.177358i −1.05204 3.53652i −0.707107 0.707107i −0.864649 + 2.69302i 2.69188 2.69188i −0.238738 0.129265i
85.6 0.270256 + 1.38815i 0.350600 0.145223i −1.85392 + 0.750313i −1.27761 + 3.08441i 0.296344 + 0.447438i −0.707107 0.707107i −1.54258 2.37075i −2.01949 + 2.01949i −4.62691 0.939927i
85.7 0.461048 1.33695i −1.66728 + 0.690611i −1.57487 1.23280i 0.923458 2.22943i 0.154616 + 2.54748i −0.707107 0.707107i −2.37428 + 1.53715i 0.181564 0.181564i −2.55487 2.26249i
85.8 1.16944 + 0.795236i 1.29558 0.536647i 0.735199 + 1.85997i 0.937867 2.26421i 1.94187 + 0.402713i −0.707107 0.707107i −0.619339 + 2.75979i −0.730784 + 0.730784i 2.89737 1.90204i
85.9 1.27165 0.618795i 2.04641 0.847650i 1.23419 1.57378i −1.31439 + 3.17322i 2.07779 2.34422i −0.707107 0.707107i 0.595604 2.76501i 1.34795 1.34795i 0.292129 + 4.84857i
85.10 1.35563 + 0.402828i −1.81729 + 0.752744i 1.67546 + 1.09217i −0.920909 + 2.22327i −2.76679 + 0.288389i −0.707107 0.707107i 1.83135 + 2.15550i 0.614581 0.614581i −2.14401 + 2.64296i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.u.b 40
4.b odd 2 1 896.2.u.b 40
32.g even 8 1 inner 224.2.u.b 40
32.h odd 8 1 896.2.u.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.b 40 1.a even 1 1 trivial
224.2.u.b 40 32.g even 8 1 inner
896.2.u.b 40 4.b odd 2 1
896.2.u.b 40 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.