# Properties

 Label 403.2.cb.a Level 403 Weight 2 Character orbit 403.cb Analytic conductor 3.218 Analytic rank 0 Dimension 576 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.cb (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$576$$ Relative dimension: $$36$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$576q - 12q^{2} - 10q^{3} - 18q^{4} - 32q^{5} - 4q^{7} - 22q^{8} - 74q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$576q - 12q^{2} - 10q^{3} - 18q^{4} - 32q^{5} - 4q^{7} - 22q^{8} - 74q^{9} - 18q^{10} - 30q^{13} - 80q^{14} + 10q^{15} - 78q^{16} - 30q^{17} + 2q^{18} - 16q^{19} + 34q^{20} - 70q^{21} - 60q^{22} - 30q^{23} + 20q^{24} + 80q^{27} - 16q^{28} - 10q^{29} + 24q^{31} - 112q^{32} + 4q^{33} - 20q^{34} - 38q^{35} - 48q^{36} + 28q^{39} + 8q^{40} - 22q^{41} - 10q^{42} - 120q^{43} - 60q^{44} + 18q^{45} - 100q^{46} + 4q^{47} - 10q^{48} - 78q^{49} - 120q^{50} + 20q^{52} - 80q^{54} - 10q^{55} + 432q^{56} + 70q^{58} + 52q^{59} + 160q^{60} - 72q^{62} - 316q^{63} - 30q^{65} + 40q^{66} - 44q^{67} + 174q^{69} + 66q^{70} - 20q^{71} - 264q^{72} - 20q^{73} - 10q^{74} + 210q^{75} + 26q^{76} + 96q^{78} + 40q^{79} - 18q^{80} + 54q^{81} - 138q^{82} - 290q^{83} + 220q^{84} - 30q^{85} - 20q^{86} - 8q^{87} - 10q^{89} + 70q^{91} - 134q^{93} - 24q^{94} + 102q^{95} - 70q^{96} - 110q^{97} + 200q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1 −2.56739 + 0.985528i −0.915361 0.0962083i 4.13393 3.72220i 0.706575 0.706575i 2.44490 0.655110i −0.0270244 + 0.515657i −4.44807 + 8.72982i −2.10581 0.447604i −1.11770 + 2.51040i
15.2 −2.42084 + 0.929275i 2.30609 + 0.242379i 3.51064 3.16099i −2.31557 + 2.31557i −5.80791 + 1.55622i 0.246948 4.71205i −3.20682 + 6.29373i 2.32484 + 0.494160i 3.45383 7.75743i
15.3 −2.31845 + 0.889969i 1.00099 + 0.105208i 3.09687 2.78844i −0.241939 + 0.241939i −2.41437 + 0.646928i −0.164948 + 3.14739i −2.44344 + 4.79553i −1.94354 0.413111i 0.345605 0.776242i
15.4 −2.22333 + 0.853455i 2.95877 + 0.310980i 2.72851 2.45676i 3.03274 3.03274i −6.84373 + 1.83377i 0.0323199 0.616700i −1.80727 + 3.54696i 5.72319 + 1.21650i −4.15446 + 9.33107i
15.5 −2.18231 + 0.837709i −1.82986 0.192326i 2.57441 2.31801i 2.00072 2.00072i 4.15442 1.11317i 0.230296 4.39430i −1.55387 + 3.04965i 0.376947 + 0.0801226i −2.69016 + 6.04221i
15.6 −2.13502 + 0.819558i −2.70347 0.284147i 2.40035 2.16129i −1.50038 + 1.50038i 6.00485 1.60899i −0.159342 + 3.04042i −1.27702 + 2.50630i 4.29359 + 0.912632i 1.97370 4.43299i
15.7 −1.74124 + 0.668401i 3.03251 + 0.318730i 1.09888 0.989437i −1.90134 + 1.90134i −5.49338 + 1.47195i −0.213728 + 4.07817i 0.441419 0.866333i 6.16010 + 1.30937i 2.03984 4.58157i
15.8 −1.73427 + 0.665722i −2.17361 0.228456i 1.07820 0.970817i 2.39448 2.39448i 3.92171 1.05082i −0.177207 + 3.38131i 0.463117 0.908919i 1.73795 + 0.369412i −2.55860 + 5.74672i
15.9 −1.67538 + 0.643119i 0.0849006 + 0.00892341i 0.907016 0.816681i −1.60006 + 1.60006i −0.147980 + 0.0396511i −0.0146284 + 0.279127i 0.635067 1.24639i −2.92731 0.622220i 1.65168 3.70974i
15.10 −1.64926 + 0.633093i 0.350555 + 0.0368448i 0.832971 0.750010i 0.166326 0.166326i −0.601483 + 0.161167i 0.0307024 0.585837i 0.705077 1.38379i −2.81291 0.597903i −0.169015 + 0.379614i
15.11 −1.50027 + 0.575900i −1.72223 0.181014i 0.432864 0.389753i −2.07725 + 2.07725i 2.68806 0.720264i 0.206300 3.93644i 1.03418 2.02969i −0.00112424 0.000238966i 1.92015 4.31273i
15.12 −1.28262 + 0.492351i 1.16310 + 0.122247i −0.0835907 + 0.0752654i 2.24725 2.24725i −1.55200 + 0.415857i −0.137685 + 2.62719i 1.31761 2.58595i −1.59659 0.339365i −1.77593 + 3.98880i
15.13 −1.00504 + 0.385798i 2.51527 + 0.264366i −0.625026 + 0.562776i 0.441644 0.441644i −2.62994 + 0.704690i 0.134189 2.56048i 1.38854 2.72516i 3.32227 + 0.706170i −0.273484 + 0.614254i
15.14 −0.857856 + 0.329300i −2.24412 0.235866i −0.858811 + 0.773277i 1.02734 1.02734i 2.00280 0.536649i 0.00452335 0.0863108i 1.31643 2.58364i 2.04600 + 0.434890i −0.543006 + 1.21961i
15.15 −0.632627 + 0.242843i −2.36395 0.248461i −1.14505 + 1.03100i −1.89446 + 1.89446i 1.55583 0.416884i −0.154670 + 2.95128i 1.08929 2.13786i 2.59206 + 0.550960i 0.738430 1.65854i
15.16 −0.448407 + 0.172127i 0.358465 + 0.0376761i −1.31485 + 1.18390i 2.11069 2.11069i −0.167223 + 0.0448073i 0.209190 3.99157i 0.821917 1.61310i −2.80737 0.596724i −0.583140 + 1.30975i
15.17 −0.378014 + 0.145106i 1.58507 + 0.166597i −1.36445 + 1.22856i −2.82295 + 2.82295i −0.623351 + 0.167027i 0.00967852 0.184677i 0.705159 1.38395i −0.449762 0.0956000i 0.657488 1.47674i
15.18 −0.0732518 + 0.0281187i −0.541154 0.0568776i −1.48171 + 1.33414i 0.197759 0.197759i 0.0412399 0.0110502i −0.150499 + 2.87169i 0.142267 0.279215i −2.64483 0.562176i −0.00892549 + 0.0200470i
15.19 0.0302615 0.0116163i 2.11154 + 0.221931i −1.48551 + 1.33756i −0.761137 + 0.761137i 0.0664762 0.0178122i −0.0268833 + 0.512964i −0.0588479 + 0.115496i 1.47489 + 0.313497i −0.0141915 + 0.0318747i
15.20 0.185296 0.0711283i 3.02116 + 0.317537i −1.45701 + 1.31190i 0.944103 0.944103i 0.582394 0.156052i −0.137953 + 2.63230i −0.356880 + 0.700416i 6.09216 + 1.29493i 0.107786 0.242091i
See next 80 embeddings (of 576 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner
31.f odd 10 1 inner
403.cb even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.cb.a 576
13.f odd 12 1 inner 403.2.cb.a 576
31.f odd 10 1 inner 403.2.cb.a 576
403.cb even 60 1 inner 403.2.cb.a 576

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.cb.a 576 1.a even 1 1 trivial
403.2.cb.a 576 13.f odd 12 1 inner
403.2.cb.a 576 31.f odd 10 1 inner
403.2.cb.a 576 403.cb even 60 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