# Properties

 Label 136.4.h.a Level $136$ Weight $4$ Character orbit 136.h Analytic conductor $8.024$ Analytic rank $0$ Dimension $52$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.02425976078$$ Analytic rank: $$0$$ Dimension: $$52$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$52 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + 428 q^{9}+O(q^{10})$$ 52 * q - 2 * q^2 + 10 * q^4 - 2 * q^8 + 428 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$52 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + 428 q^{9} - 232 q^{15} - 78 q^{16} - 28 q^{17} - 2 q^{18} + 1052 q^{25} + 448 q^{26} - 368 q^{30} + 958 q^{32} - 344 q^{33} - 198 q^{34} + 138 q^{36} - 524 q^{38} + 936 q^{47} - 1964 q^{49} - 1038 q^{50} - 1424 q^{52} - 1384 q^{55} + 2320 q^{60} - 2078 q^{64} - 1888 q^{66} - 874 q^{68} + 2472 q^{70} - 4010 q^{72} + 436 q^{76} + 1884 q^{81} - 2264 q^{84} - 1420 q^{86} + 1976 q^{87} - 224 q^{89} + 80 q^{94} + 5746 q^{98}+O(q^{100})$$ 52 * q - 2 * q^2 + 10 * q^4 - 2 * q^8 + 428 * q^9 - 232 * q^15 - 78 * q^16 - 28 * q^17 - 2 * q^18 + 1052 * q^25 + 448 * q^26 - 368 * q^30 + 958 * q^32 - 344 * q^33 - 198 * q^34 + 138 * q^36 - 524 * q^38 + 936 * q^47 - 1964 * q^49 - 1038 * q^50 - 1424 * q^52 - 1384 * q^55 + 2320 * q^60 - 2078 * q^64 - 1888 * q^66 - 874 * q^68 + 2472 * q^70 - 4010 * q^72 + 436 * q^76 + 1884 * q^81 - 2264 * q^84 - 1420 * q^86 + 1976 * q^87 - 224 * q^89 + 80 * q^94 + 5746 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 −2.76967 0.573530i −3.72680 7.34213 + 3.17697i 7.30821 10.3220 + 2.13743i 5.81849i −18.5132 13.0101i −13.1110 −20.2413 4.19147i
101.2 −2.76967 0.573530i 3.72680 7.34213 + 3.17697i −7.30821 −10.3220 2.13743i 5.81849i −18.5132 13.0101i −13.1110 20.2413 + 4.19147i
101.3 −2.76967 + 0.573530i −3.72680 7.34213 3.17697i 7.30821 10.3220 2.13743i 5.81849i −18.5132 + 13.0101i −13.1110 −20.2413 + 4.19147i
101.4 −2.76967 + 0.573530i 3.72680 7.34213 3.17697i −7.30821 −10.3220 + 2.13743i 5.81849i −18.5132 + 13.0101i −13.1110 20.2413 4.19147i
101.5 −2.64160 1.01092i −6.38011 5.95609 + 5.34088i −16.9716 16.8537 + 6.44977i 31.4494i −10.3344 20.1296i 13.7058 44.8322 + 17.1569i
101.6 −2.64160 1.01092i 6.38011 5.95609 + 5.34088i 16.9716 −16.8537 6.44977i 31.4494i −10.3344 20.1296i 13.7058 −44.8322 17.1569i
101.7 −2.64160 + 1.01092i −6.38011 5.95609 5.34088i −16.9716 16.8537 6.44977i 31.4494i −10.3344 + 20.1296i 13.7058 44.8322 17.1569i
101.8 −2.64160 + 1.01092i 6.38011 5.95609 5.34088i 16.9716 −16.8537 + 6.44977i 31.4494i −10.3344 + 20.1296i 13.7058 −44.8322 + 17.1569i
101.9 −2.37821 1.53106i −9.79402 3.31172 + 7.28234i 7.09681 23.2922 + 14.9952i 25.9378i 3.27375 22.3893i 68.9229 −16.8777 10.8656i
101.10 −2.37821 1.53106i 9.79402 3.31172 + 7.28234i −7.09681 −23.2922 14.9952i 25.9378i 3.27375 22.3893i 68.9229 16.8777 + 10.8656i
101.11 −2.37821 + 1.53106i −9.79402 3.31172 7.28234i 7.09681 23.2922 14.9952i 25.9378i 3.27375 + 22.3893i 68.9229 −16.8777 + 10.8656i
101.12 −2.37821 + 1.53106i 9.79402 3.31172 7.28234i −7.09681 −23.2922 + 14.9952i 25.9378i 3.27375 + 22.3893i 68.9229 16.8777 10.8656i
101.13 −1.98225 2.01760i −0.0741700 −0.141403 + 7.99875i 17.3656 0.147023 + 0.149645i 22.3324i 16.4186 15.5702i −26.9945 −34.4229 35.0368i
101.14 −1.98225 2.01760i 0.0741700 −0.141403 + 7.99875i −17.3656 −0.147023 0.149645i 22.3324i 16.4186 15.5702i −26.9945 34.4229 + 35.0368i
101.15 −1.98225 + 2.01760i −0.0741700 −0.141403 7.99875i 17.3656 0.147023 0.149645i 22.3324i 16.4186 + 15.5702i −26.9945 −34.4229 + 35.0368i
101.16 −1.98225 + 2.01760i 0.0741700 −0.141403 7.99875i −17.3656 −0.147023 + 0.149645i 22.3324i 16.4186 + 15.5702i −26.9945 34.4229 35.0368i
101.17 −1.64670 2.29965i −3.96792 −2.57674 + 7.57367i 5.58507 6.53398 + 9.12481i 3.60918i 21.6599 6.54599i −11.2556 −9.19694 12.8437i
101.18 −1.64670 2.29965i 3.96792 −2.57674 + 7.57367i −5.58507 −6.53398 9.12481i 3.60918i 21.6599 6.54599i −11.2556 9.19694 + 12.8437i
101.19 −1.64670 + 2.29965i −3.96792 −2.57674 7.57367i 5.58507 6.53398 9.12481i 3.60918i 21.6599 + 6.54599i −11.2556 −9.19694 + 12.8437i
101.20 −1.64670 + 2.29965i 3.96792 −2.57674 7.57367i −5.58507 −6.53398 + 9.12481i 3.60918i 21.6599 + 6.54599i −11.2556 9.19694 12.8437i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.52 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.b even 2 1 inner
136.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.h.a 52
4.b odd 2 1 544.4.h.a 52
8.b even 2 1 inner 136.4.h.a 52
8.d odd 2 1 544.4.h.a 52
17.b even 2 1 inner 136.4.h.a 52
68.d odd 2 1 544.4.h.a 52
136.e odd 2 1 544.4.h.a 52
136.h even 2 1 inner 136.4.h.a 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.h.a 52 1.a even 1 1 trivial
136.4.h.a 52 8.b even 2 1 inner
136.4.h.a 52 17.b even 2 1 inner
136.4.h.a 52 136.h even 2 1 inner
544.4.h.a 52 4.b odd 2 1
544.4.h.a 52 8.d odd 2 1
544.4.h.a 52 68.d odd 2 1
544.4.h.a 52 136.e odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(136, [\chi])$$.