# Properties

 Label 403.2.h.b Level 403 Weight 2 Character orbit 403.h Analytic conductor 3.218 Analytic rank 0 Dimension 34 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$34$$ Relative dimension: $$17$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$34q + 6q^{2} - 2q^{3} + 34q^{4} - 5q^{5} - 2q^{7} + 36q^{8} - 23q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$34q + 6q^{2} - 2q^{3} + 34q^{4} - 5q^{5} - 2q^{7} + 36q^{8} - 23q^{9} - 7q^{10} - 5q^{11} - 28q^{12} - 17q^{13} - 7q^{14} + 8q^{15} + 18q^{16} - 8q^{17} + 6q^{18} + 3q^{19} - 8q^{20} + 13q^{21} + 12q^{22} - 14q^{23} - 6q^{24} - 26q^{25} - 3q^{26} + 28q^{27} - 7q^{28} - 18q^{29} - 60q^{30} - 9q^{31} + 58q^{32} - 14q^{33} - 15q^{34} + 50q^{35} - 49q^{36} - 6q^{37} + 2q^{38} + 4q^{39} - 29q^{40} - 5q^{41} + 8q^{42} - q^{43} - 22q^{44} + 13q^{45} + 34q^{46} + 16q^{47} - 49q^{48} + 3q^{49} - 35q^{51} - 17q^{52} + 30q^{53} - 2q^{54} + 21q^{55} - 7q^{56} + 34q^{58} - 9q^{59} - 38q^{60} - 28q^{61} - 62q^{62} + 88q^{63} + 56q^{64} - 5q^{65} + 140q^{66} - 31q^{67} - 39q^{68} + 5q^{69} + 56q^{70} + q^{71} - 32q^{72} - 10q^{73} - 39q^{74} - 2q^{75} - 16q^{76} + 76q^{77} - 23q^{79} - 22q^{80} - 29q^{81} - 10q^{82} + 3q^{83} + 52q^{84} - 32q^{85} + 4q^{86} + 18q^{87} - 10q^{88} + 26q^{89} + 35q^{90} + 4q^{91} - 94q^{92} - 41q^{93} + 70q^{94} + 28q^{95} - 23q^{96} + 32q^{97} - 38q^{98} - 70q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1 −2.49327 −1.44053 2.49508i 4.21641 0.778554 1.34849i 3.59165 + 6.22091i −0.277831 0.481217i −5.52611 −2.65028 + 4.59042i −1.94115 + 3.36216i
118.2 −2.00723 0.235463 + 0.407833i 2.02898 −0.859670 + 1.48899i −0.472628 0.818616i 1.82223 + 3.15619i −0.0581753 1.38911 2.40602i 1.72556 2.98875i
118.3 −1.85162 0.240159 + 0.415967i 1.42848 0.854909 1.48075i −0.444682 0.770212i −0.531962 0.921385i 1.05824 1.38465 2.39828i −1.58296 + 2.74177i
118.4 −1.72349 −1.49143 2.58324i 0.970414 −1.96711 + 3.40713i 2.57047 + 4.45218i 0.142448 + 0.246727i 1.77448 −2.94874 + 5.10737i 3.39029 5.87216i
118.5 −1.51349 1.68586 + 2.92000i 0.290659 1.24235 2.15182i −2.55154 4.41940i −1.03520 1.79302i 2.58707 −4.18427 + 7.24738i −1.88029 + 3.25676i
118.6 −0.837386 0.883556 + 1.53036i −1.29878 −2.21305 + 3.83311i −0.739877 1.28150i −1.99716 3.45918i 2.76236 −0.0613409 + 0.106246i 1.85318 3.20980i
118.7 −0.708536 −0.837990 1.45144i −1.49798 1.10081 1.90665i 0.593746 + 1.02840i 1.50089 + 2.59962i 2.47844 0.0955444 0.165488i −0.779961 + 1.35093i
118.8 −0.291491 −0.965220 1.67181i −1.91503 −0.817652 + 1.41622i 0.281353 + 0.487317i −0.566002 0.980344i 1.14120 −0.363299 + 0.629252i 0.238338 0.412814i
118.9 −0.272181 0.850363 + 1.47287i −1.92592 1.58639 2.74770i −0.231453 0.400888i 1.18195 + 2.04720i 1.06856 0.0537640 0.0931220i −0.431784 + 0.747872i
118.10 0.680422 0.896149 + 1.55218i −1.53703 −0.204382 + 0.354001i 0.609760 + 1.05613i 1.47430 + 2.55357i −2.40667 −0.106167 + 0.183887i −0.139066 + 0.240870i
118.11 1.28826 1.07551 + 1.86284i −0.340382 −1.69851 + 2.94190i 1.38554 + 2.39982i 0.278041 + 0.481581i −3.01502 −0.813438 + 1.40892i −2.18812 + 3.78994i
118.12 1.35626 0.0933728 + 0.161726i −0.160553 1.08899 1.88618i 0.126638 + 0.219343i −1.43912 2.49263i −2.93028 1.48256 2.56787i 1.47695 2.55816i
118.13 1.55799 −1.19850 2.07586i 0.427341 −0.908804 + 1.57410i −1.86725 3.23417i −1.80502 3.12639i −2.45019 −1.37279 + 2.37775i −1.41591 + 2.45243i
118.14 2.14110 −0.301124 0.521562i 2.58432 0.249792 0.432653i −0.644738 1.11672i 2.01488 + 3.48987i 1.25109 1.31865 2.28397i 0.534831 0.926355i
118.15 2.35132 1.34746 + 2.33387i 3.52869 −0.0458962 + 0.0794945i 3.16831 + 5.48767i −2.00349 3.47014i 3.59444 −2.13130 + 3.69152i −0.107916 + 0.186917i
118.16 2.53634 −1.60552 2.78084i 4.43304 1.34893 2.33641i −4.07215 7.05318i 0.661783 + 1.14624i 6.17102 −3.65540 + 6.33133i 3.42134 5.92593i
118.17 2.78699 −0.467575 0.809863i 5.76734 −2.03564 + 3.52583i −1.30313 2.25709i −0.420736 0.728737i 10.4996 1.06275 1.84073i −5.67332 + 9.82648i
222.1 −2.49327 −1.44053 + 2.49508i 4.21641 0.778554 + 1.34849i 3.59165 6.22091i −0.277831 + 0.481217i −5.52611 −2.65028 4.59042i −1.94115 3.36216i
222.2 −2.00723 0.235463 0.407833i 2.02898 −0.859670 1.48899i −0.472628 + 0.818616i 1.82223 3.15619i −0.0581753 1.38911 + 2.40602i 1.72556 + 2.98875i
222.3 −1.85162 0.240159 0.415967i 1.42848 0.854909 + 1.48075i −0.444682 + 0.770212i −0.531962 + 0.921385i 1.05824 1.38465 + 2.39828i −1.58296 2.74177i
See all 34 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 222.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.h.b 34
31.c even 3 1 inner 403.2.h.b 34

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.h.b 34 1.a even 1 1 trivial
403.2.h.b 34 31.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{17} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database