# Properties

 Label 128.12.e.b Level 128 Weight 12 Character orbit 128.e Analytic conductor 98.348 Analytic rank 0 Dimension 42 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 128.e (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$98.3479271116$$ Analytic rank: $$0$$ Dimension: $$42$$ Relative dimension: $$21$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 16) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$42q + 2q^{3} + 2q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$42q + 2q^{3} + 2q^{5} + 540846q^{11} + 2q^{13} - 6075004q^{15} - 4q^{17} + 11291290q^{19} - 354292q^{21} + 66463304q^{27} - 77673206q^{29} + 343549808q^{31} - 4q^{33} + 434731684q^{35} + 522762058q^{37} - 3824193658q^{43} - 97301954q^{45} - 4586900144q^{47} - 8474257474q^{49} - 7074245796q^{51} + 2100608058q^{53} - 955824746q^{59} - 2150827022q^{61} + 27758037828q^{63} - 1884965292q^{65} + 3186519018q^{67} + 16193060732q^{69} - 28890034486q^{75} + 22711870540q^{77} + 48011833792q^{79} - 90656394430q^{81} - 55713221118q^{83} + 84575506252q^{85} + 147369662716q^{91} + 69689773328q^{93} + 375702304500q^{95} - 4q^{97} + 286271331106q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1 0 −585.540 585.540i 0 6859.07 6859.07i 0 19914.8i 0 508568.i 0
33.2 0 −457.250 457.250i 0 −5575.43 + 5575.43i 0 55452.7i 0 241008.i 0
33.3 0 −442.035 442.035i 0 −2497.66 + 2497.66i 0 39392.3i 0 213643.i 0
33.4 0 −396.720 396.720i 0 −7616.23 + 7616.23i 0 16490.4i 0 137626.i 0
33.5 0 −367.824 367.824i 0 502.344 502.344i 0 33063.2i 0 93442.1i 0
33.6 0 −255.917 255.917i 0 4214.49 4214.49i 0 80345.3i 0 46160.2i 0
33.7 0 −221.481 221.481i 0 7918.20 7918.20i 0 76836.1i 0 79039.1i 0
33.8 0 −219.312 219.312i 0 4575.66 4575.66i 0 2778.29i 0 80951.6i 0
33.9 0 −137.041 137.041i 0 −3899.66 + 3899.66i 0 37938.5i 0 139586.i 0
33.10 0 −13.6026 13.6026i 0 3501.81 3501.81i 0 37649.1i 0 176777.i 0
33.11 0 4.32776 + 4.32776i 0 −2233.99 + 2233.99i 0 71414.8i 0 177110.i 0
33.12 0 56.4628 + 56.4628i 0 −6260.45 + 6260.45i 0 32944.0i 0 170771.i 0
33.13 0 145.072 + 145.072i 0 4652.81 4652.81i 0 43253.0i 0 135056.i 0
33.14 0 179.701 + 179.701i 0 −2961.29 + 2961.29i 0 5179.04i 0 112562.i 0
33.15 0 199.508 + 199.508i 0 −8937.54 + 8937.54i 0 55653.3i 0 97540.0i 0
33.16 0 291.167 + 291.167i 0 9753.88 9753.88i 0 16447.0i 0 7590.38i 0
33.17 0 296.568 + 296.568i 0 2867.98 2867.98i 0 8380.90i 0 1242.24i 0
33.18 0 435.014 + 435.014i 0 1517.34 1517.34i 0 59722.7i 0 201326.i 0
33.19 0 475.976 + 475.976i 0 −2805.14 + 2805.14i 0 76033.6i 0 275960.i 0
33.20 0 491.248 + 491.248i 0 −6777.57 + 6777.57i 0 47511.7i 0 305503.i 0
See all 42 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.21 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.12.e.b 42
4.b odd 2 1 128.12.e.a 42
8.b even 2 1 16.12.e.a 42
8.d odd 2 1 64.12.e.a 42
16.e even 4 1 16.12.e.a 42
16.e even 4 1 inner 128.12.e.b 42
16.f odd 4 1 64.12.e.a 42
16.f odd 4 1 128.12.e.a 42

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.12.e.a 42 8.b even 2 1
16.12.e.a 42 16.e even 4 1
64.12.e.a 42 8.d odd 2 1
64.12.e.a 42 16.f odd 4 1
128.12.e.a 42 4.b odd 2 1
128.12.e.a 42 16.f odd 4 1
128.12.e.b 42 1.a even 1 1 trivial
128.12.e.b 42 16.e even 4 1 inner

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database