# Properties

 Label 403.2.bf.a Level 403 Weight 2 Character orbit 403.bf Analytic conductor 3.218 Analytic rank 0 Dimension 140 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$140$$ Relative dimension: $$35$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$140q - 8q^{2} - 6q^{3} - 12q^{4} - 2q^{5} + 12q^{6} - 12q^{7} - 10q^{8} + 62q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$140q - 8q^{2} - 6q^{3} - 12q^{4} - 2q^{5} + 12q^{6} - 12q^{7} - 10q^{8} + 62q^{9} - 12q^{11} - 26q^{12} - 6q^{13} - 24q^{14} - 18q^{15} + 48q^{16} + 20q^{18} + 4q^{19} - 2q^{20} - 14q^{21} + 12q^{22} - 18q^{24} - 6q^{26} + 42q^{28} - 36q^{31} - 10q^{32} - 30q^{33} + 30q^{34} - 8q^{35} + 10q^{37} - 72q^{38} - 8q^{39} - 12q^{40} - 8q^{41} + 52q^{43} - 36q^{44} - 6q^{45} - 24q^{46} + 12q^{47} + 40q^{50} - 36q^{51} + 2q^{52} + 24q^{53} + 18q^{54} - 6q^{55} - 14q^{57} + 42q^{58} - 58q^{59} + 18q^{60} - 36q^{61} - 18q^{62} - 58q^{63} - 108q^{65} + 16q^{66} + 36q^{67} - 18q^{68} + 30q^{70} - 26q^{71} + 8q^{72} - 50q^{73} - 164q^{75} - 22q^{76} + 48q^{77} - 6q^{78} - 48q^{79} - 148q^{80} - 66q^{81} + 54q^{82} + 6q^{83} + 14q^{84} - 42q^{85} + 6q^{86} + 28q^{87} + 48q^{88} - 36q^{89} + 90q^{90} - 46q^{91} + 16q^{93} + 4q^{94} + 48q^{95} - 66q^{96} + 26q^{97} + 20q^{98} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −0.725626 + 2.70807i 1.00391 + 0.579606i −5.07507 2.93009i −2.83136 + 0.758661i −2.29807 + 2.29807i −1.48686 + 1.48686i 7.65262 7.65262i −0.828115 1.43434i 8.21804i
37.2 −0.648505 + 2.42025i −0.739799 0.427123i −3.70501 2.13909i 3.15843 0.846300i 1.51351 1.51351i −3.05829 + 3.05829i 4.03636 4.03636i −1.13513 1.96611i 8.19303i
37.3 −0.626284 + 2.33732i −1.20623 0.696416i −3.33879 1.92765i 0.117630 0.0315189i 2.38319 2.38319i 1.27982 1.27982i 3.17450 3.17450i −0.530010 0.918004i 0.294679i
37.4 −0.616245 + 2.29986i 2.49411 + 1.43998i −3.17753 1.83455i 0.879300 0.235608i −4.84872 + 4.84872i 1.37504 1.37504i 2.81012 2.81012i 2.64706 + 4.58484i 2.16746i
37.5 −0.610803 + 2.27955i −2.48438 1.43436i −3.09121 1.78471i 1.20512 0.322912i 4.78716 4.78716i −0.172104 + 0.172104i 2.61896 2.61896i 2.61477 + 4.52891i 2.94437i
37.6 −0.529121 + 1.97471i −0.0550086 0.0317592i −1.88744 1.08972i −1.35641 + 0.363448i 0.0918213 0.0918213i 2.81815 2.81815i 0.259387 0.259387i −1.49798 2.59458i 2.87081i
37.7 −0.505205 + 1.88545i 2.41315 + 1.39323i −1.56764 0.905077i 0.127199 0.0340828i −3.84601 + 3.84601i −3.08429 + 3.08429i −0.262034 + 0.262034i 2.38220 + 4.12609i 0.257046i
37.8 −0.504814 + 1.88399i −2.36532 1.36562i −1.56254 0.902134i −3.95125 + 1.05873i 3.76686 3.76686i −1.17143 + 1.17143i −0.269949 + 0.269949i 2.22981 + 3.86215i 7.97858i
37.9 −0.495867 + 1.85060i 0.720276 + 0.415851i −1.44679 0.835304i −2.76191 + 0.740052i −1.12674 + 1.12674i −0.764137 + 0.764137i −0.446239 + 0.446239i −1.15414 1.99902i 5.47817i
37.10 −0.379919 + 1.41788i 0.690619 + 0.398729i −0.133982 0.0773543i 3.27203 0.876737i −0.827727 + 0.827727i −1.67111 + 1.67111i −1.91533 + 1.91533i −1.18203 2.04734i 4.97241i
37.11 −0.361997 + 1.35099i −2.03266 1.17356i 0.0379121 + 0.0218885i 3.14674 0.843165i 2.32129 2.32129i 1.04445 1.04445i −2.02129 + 2.02129i 1.25448 + 2.17283i 4.55644i
37.12 −0.330760 + 1.23441i 1.39622 + 0.806108i 0.317680 + 0.183412i 2.41406 0.646845i −1.45688 + 1.45688i 1.56840 1.56840i −2.13879 + 2.13879i −0.200379 0.347066i 3.19389i
37.13 −0.216378 + 0.807533i −1.68759 0.974331i 1.12676 + 0.650535i 0.191653 0.0513533i 1.15196 1.15196i −1.16887 + 1.16887i −1.95145 + 1.95145i 0.398643 + 0.690470i 0.165878i
37.14 −0.167077 + 0.623539i −0.512327 0.295792i 1.37116 + 0.791642i −2.05116 + 0.549606i 0.270036 0.270036i 3.10623 3.10623i −1.63563 + 1.63563i −1.32501 2.29499i 1.37080i
37.15 −0.137083 + 0.511599i 1.59451 + 0.920592i 1.48911 + 0.859737i −3.24004 + 0.868165i −0.689555 + 0.689555i −0.996271 + 0.996271i −1.39301 + 1.39301i 0.194981 + 0.337717i 1.77661i
37.16 −0.135765 + 0.506680i 2.44353 + 1.41078i 1.49376 + 0.862422i −0.993746 + 0.266274i −1.04656 + 1.04656i 1.98004 1.98004i −1.38160 + 1.38160i 2.48057 + 4.29648i 0.539662i
37.17 −0.133986 + 0.500044i −0.954633 0.551157i 1.49996 + 0.866002i −1.11945 + 0.299955i 0.403510 0.403510i −2.88943 + 2.88943i −1.36613 + 1.36613i −0.892451 1.54577i 0.599962i
37.18 0.0225085 0.0840028i −0.432116 0.249482i 1.72550 + 0.996218i 2.81785 0.755041i −0.0306835 + 0.0306835i 1.87837 1.87837i 0.245512 0.245512i −1.37552 2.38247i 0.253702i
37.19 0.0759990 0.283632i 1.58651 + 0.915973i 1.65738 + 0.956888i 1.30906 0.350763i 0.380373 0.380373i −1.54266 + 1.54266i 0.812630 0.812630i 0.178014 + 0.308329i 0.397950i
37.20 0.107992 0.403030i −2.45601 1.41798i 1.58128 + 0.912952i 3.58852 0.961540i −0.836717 + 0.836717i −1.10712 + 1.10712i 1.12879 1.12879i 2.52133 + 4.36707i 1.55012i
See next 80 embeddings (of 140 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 305.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bf even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bf.a yes 140
13.f odd 12 1 403.2.ba.a 140
31.e odd 6 1 403.2.ba.a 140
403.bf even 12 1 inner 403.2.bf.a yes 140

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.ba.a 140 13.f odd 12 1
403.2.ba.a 140 31.e odd 6 1
403.2.bf.a yes 140 1.a even 1 1 trivial
403.2.bf.a yes 140 403.bf even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database