# Properties

 Label 354.2.e.c Level 354 Weight 2 Character orbit 354.e Analytic conductor 2.827 Analytic rank 0 Dimension 84 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 354.e (of order $$29$$, degree $$28$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.82670423155$$ Analytic rank: $$0$$ Dimension: $$84$$ Relative dimension: $$3$$ over $$\Q(\zeta_{29})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$84q - 3q^{2} + 3q^{3} - 3q^{4} + 3q^{6} + q^{7} - 3q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q - 3q^{2} + 3q^{3} - 3q^{4} + 3q^{6} + q^{7} - 3q^{8} - 3q^{9} - 26q^{11} + 3q^{12} - 3q^{13} + q^{14} - 3q^{16} + 3q^{17} - 3q^{18} + 4q^{19} - q^{21} + 3q^{22} - 2q^{23} + 3q^{24} + 41q^{25} + 26q^{26} + 3q^{27} + q^{28} - 2q^{29} + 8q^{31} - 3q^{32} - 3q^{33} - 26q^{34} + 83q^{35} - 3q^{36} - 53q^{37} + 4q^{38} + 3q^{39} - 7q^{41} - q^{42} + 119q^{43} + 3q^{44} - 31q^{46} - 12q^{47} + 3q^{48} - 38q^{49} - 133q^{50} - 3q^{51} - 32q^{52} - 83q^{53} + 3q^{54} - 83q^{55} + q^{56} - 4q^{57} + 56q^{58} - 57q^{59} - 48q^{61} - 21q^{62} + q^{63} - 3q^{64} - 33q^{65} - 3q^{66} - 88q^{67} - 26q^{68} + 89q^{69} - 62q^{70} - 35q^{71} - 3q^{72} - 71q^{73} - 24q^{74} + 17q^{75} + 33q^{76} + 113q^{77} + 3q^{78} - 5q^{79} - 3q^{81} - 7q^{82} - 51q^{83} - q^{84} + 125q^{85} + 61q^{86} + 31q^{87} + 32q^{88} - 58q^{89} + 173q^{91} - 2q^{92} + 21q^{93} + 17q^{94} + 26q^{95} + 3q^{96} + 20q^{98} + 3q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −0.561187 0.827689i −0.0541389 + 0.998533i −0.370138 + 0.928977i −3.82834 + 1.77118i 0.856857 0.515554i 1.02375 3.68721i 0.976621 0.214970i −0.994138 0.108119i 3.61440 + 2.17471i
7.2 −0.561187 0.827689i −0.0541389 + 0.998533i −0.370138 + 0.928977i −0.0493609 + 0.0228368i 0.856857 0.515554i −0.676801 + 2.43762i 0.976621 0.214970i −0.994138 0.108119i 0.0466025 + 0.0280398i
7.3 −0.561187 0.827689i −0.0541389 + 0.998533i −0.370138 + 0.928977i 3.87770 1.79401i 0.856857 0.515554i 0.929941 3.34934i 0.976621 0.214970i −0.994138 0.108119i −3.66100 2.20275i
19.1 0.468408 0.883512i 0.725995 0.687699i −0.561187 0.827689i −3.74342 + 0.823990i −0.267528 0.963550i −3.11942 + 2.37132i −0.994138 + 0.108119i 0.0541389 0.998533i −1.02545 + 3.69332i
19.2 0.468408 0.883512i 0.725995 0.687699i −0.561187 0.827689i 1.70467 0.375225i −0.267528 0.963550i 1.66436 1.26521i −0.994138 + 0.108119i 0.0541389 0.998533i 0.466964 1.68185i
19.3 0.468408 0.883512i 0.725995 0.687699i −0.561187 0.827689i 2.03876 0.448764i −0.267528 0.963550i −0.284870 + 0.216552i −0.994138 + 0.108119i 0.0541389 0.998533i 0.558482 2.01147i
25.1 0.796093 + 0.605174i −0.468408 0.883512i 0.267528 + 0.963550i −2.86336 2.71232i 0.161782 0.986827i −2.64683 + 3.11609i −0.370138 + 0.928977i −0.561187 + 0.827689i −0.638077 3.89210i
25.2 0.796093 + 0.605174i −0.468408 0.883512i 0.267528 + 0.963550i 0.697351 + 0.660566i 0.161782 0.986827i −1.30774 + 1.53959i −0.370138 + 0.928977i −0.561187 + 0.827689i 0.155399 + 0.947890i
25.3 0.796093 + 0.605174i −0.468408 0.883512i 0.267528 + 0.963550i 2.16601 + 2.05176i 0.161782 0.986827i 2.55368 3.00642i −0.370138 + 0.928977i −0.561187 + 0.827689i 0.482678 + 2.94421i
49.1 −0.370138 + 0.928977i 0.994138 + 0.108119i −0.725995 0.687699i −1.72394 + 2.02958i −0.468408 + 0.883512i −1.98744 1.19580i 0.907575 0.419889i 0.976621 + 0.214970i −1.24733 2.35272i
49.2 −0.370138 + 0.928977i 0.994138 + 0.108119i −0.725995 0.687699i 0.271121 0.319189i −0.468408 + 0.883512i 3.97639 + 2.39251i 0.907575 0.419889i 0.976621 + 0.214970i 0.196166 + 0.370009i
49.3 −0.370138 + 0.928977i 0.994138 + 0.108119i −0.725995 0.687699i 1.45282 1.71039i −0.468408 + 0.883512i −1.71569 1.03229i 0.907575 0.419889i 0.976621 + 0.214970i 1.05117 + 1.98271i
79.1 0.907575 + 0.419889i 0.947653 0.319302i 0.647386 + 0.762162i −2.35642 + 1.41781i 0.994138 + 0.108119i 0.260327 + 4.80144i 0.267528 + 0.963550i 0.796093 0.605174i −2.73396 + 0.297336i
79.2 0.907575 + 0.419889i 0.947653 0.319302i 0.647386 + 0.762162i 0.401618 0.241646i 0.994138 + 0.108119i −0.0924092 1.70439i 0.267528 + 0.963550i 0.796093 0.605174i 0.465963 0.0506766i
79.3 0.907575 + 0.419889i 0.947653 0.319302i 0.647386 + 0.762162i 1.95481 1.17617i 0.994138 + 0.108119i 0.00134066 + 0.0247270i 0.267528 + 0.963550i 0.796093 0.605174i 2.26799 0.246659i
85.1 0.796093 0.605174i −0.468408 + 0.883512i 0.267528 0.963550i −2.86336 + 2.71232i 0.161782 + 0.986827i −2.64683 3.11609i −0.370138 0.928977i −0.561187 0.827689i −0.638077 + 3.89210i
85.2 0.796093 0.605174i −0.468408 + 0.883512i 0.267528 0.963550i 0.697351 0.660566i 0.161782 + 0.986827i −1.30774 1.53959i −0.370138 0.928977i −0.561187 0.827689i 0.155399 0.947890i
85.3 0.796093 0.605174i −0.468408 + 0.883512i 0.267528 0.963550i 2.16601 2.05176i 0.161782 + 0.986827i 2.55368 + 3.00642i −0.370138 0.928977i −0.561187 0.827689i 0.482678 2.94421i
121.1 0.907575 0.419889i 0.947653 + 0.319302i 0.647386 0.762162i −2.35642 1.41781i 0.994138 0.108119i 0.260327 4.80144i 0.267528 0.963550i 0.796093 + 0.605174i −2.73396 0.297336i
121.2 0.907575 0.419889i 0.947653 + 0.319302i 0.647386 0.762162i 0.401618 + 0.241646i 0.994138 0.108119i −0.0924092 + 1.70439i 0.267528 0.963550i 0.796093 + 0.605174i 0.465963 + 0.0506766i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.2.e.c 84
59.c even 29 1 inner 354.2.e.c 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.e.c 84 1.a even 1 1 trivial
354.2.e.c 84 59.c even 29 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{84} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(354, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database