# Properties

 Label 403.2.f.c Level 403 Weight 2 Character orbit 403.f Analytic conductor 3.218 Analytic rank 0 Dimension 36 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 5q^{2} - 19q^{4} + 12q^{5} - 6q^{6} - 4q^{7} + 30q^{8} - 22q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 5q^{2} - 19q^{4} + 12q^{5} - 6q^{6} - 4q^{7} + 30q^{8} - 22q^{9} - 3q^{10} - 17q^{11} + 8q^{12} + 8q^{13} + 4q^{14} - 12q^{15} - 17q^{16} + 7q^{17} - 14q^{18} - 4q^{19} - 15q^{20} + 28q^{21} - 30q^{22} + 4q^{23} - 48q^{24} + 24q^{25} + 25q^{26} - 6q^{27} - q^{28} + 15q^{29} + 35q^{30} + 36q^{31} - 35q^{32} - 17q^{33} - 6q^{34} - 17q^{35} - 35q^{36} - 3q^{37} + 14q^{38} + 3q^{39} - 2q^{40} - 44q^{41} + 57q^{42} + 4q^{43} + 64q^{44} + 5q^{45} - 13q^{46} + 24q^{47} - 89q^{48} - 44q^{49} - 84q^{50} - 28q^{51} + 50q^{52} - 28q^{53} - 21q^{54} + 29q^{55} + 11q^{56} + 32q^{57} + 49q^{58} - 11q^{59} + 54q^{60} - 4q^{61} - 5q^{62} - 9q^{63} + 34q^{64} - 5q^{65} + 52q^{66} - 16q^{67} + 53q^{68} + 4q^{69} - 44q^{70} - 5q^{71} - 27q^{72} + 64q^{73} + q^{74} - 98q^{75} - 42q^{76} - 22q^{77} + 143q^{78} - 6q^{79} - 2q^{80} + 10q^{81} + 22q^{82} + 36q^{83} + 38q^{84} + 2q^{85} + 84q^{86} - 34q^{87} - 69q^{88} - 54q^{89} - 32q^{90} - 43q^{91} - 86q^{92} + 44q^{94} - 2q^{95} + 170q^{96} - 28q^{97} - 29q^{98} + 154q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1 −1.39316 2.41302i −1.13229 1.96118i −2.88179 + 4.99141i −3.38130 −3.15492 + 5.46447i −0.923334 + 1.59926i 10.4866 −1.06415 + 1.84316i 4.71069 + 8.15916i
94.2 −1.33157 2.30635i −1.12120 1.94198i −2.54616 + 4.41008i 4.27464 −2.98592 + 5.17177i 1.44001 2.49418i 8.23530 −1.01419 + 1.75663i −5.69198 9.85880i
94.3 −1.19633 2.07211i 1.01324 + 1.75498i −1.86241 + 3.22579i 3.38808 2.42434 4.19908i −2.00851 + 3.47885i 4.12693 −0.553314 + 0.958368i −4.05326 7.02046i
94.4 −1.00222 1.73589i −0.398982 0.691057i −1.00888 + 1.74743i −2.51167 −0.799734 + 1.38518i 2.36824 4.10191i 0.0356005 1.18163 2.04664i 2.51724 + 4.35999i
94.5 −0.944860 1.63655i 0.990948 + 1.71637i −0.785520 + 1.36056i −1.89350 1.87261 3.24346i −0.456547 + 0.790763i −0.810613 −0.463958 + 0.803598i 1.78910 + 3.09881i
94.6 −0.748359 1.29620i 0.472887 + 0.819064i −0.120081 + 0.207987i 2.74524 0.707778 1.22591i 1.47907 2.56182i −2.63398 1.05276 1.82343i −2.05442 3.55837i
94.7 −0.685864 1.18795i −0.770575 1.33468i 0.0591818 0.102506i 3.15415 −1.05702 + 1.83081i −1.71049 + 2.96266i −2.90582 0.312428 0.541141i −2.16332 3.74698i
94.8 −0.683087 1.18314i 1.43249 + 2.48115i 0.0667843 0.115674i −1.11791 1.95703 3.38968i −1.05496 + 1.82724i −2.91483 −2.60407 + 4.51038i 0.763627 + 1.32264i
94.9 −0.0573543 0.0993406i −1.57377 2.72585i 0.993421 1.72066i −1.85062 −0.180525 + 0.312678i −1.95903 + 3.39313i −0.457325 −3.45350 + 5.98164i 0.106141 + 0.183841i
94.10 −0.0496225 0.0859486i 0.816280 + 1.41384i 0.995075 1.72352i −0.352370 0.0810117 0.140316i 2.05703 3.56288i −0.396002 0.167373 0.289899i 0.0174854 + 0.0302857i
94.11 0.105154 + 0.182132i 0.0693408 + 0.120102i 0.977885 1.69375i 1.89472 −0.0145830 + 0.0252584i −2.28636 + 3.96010i 0.831932 1.49038 2.58142i 0.199238 + 0.345090i
94.12 0.137984 + 0.238996i −0.732250 1.26829i 0.961921 1.66610i −1.37475 0.202078 0.350009i 0.239780 0.415311i 1.08286 0.427621 0.740662i −0.189694 0.328559i
94.13 0.497248 + 0.861259i 1.12055 + 1.94084i 0.505488 0.875531i 0.441402 −1.11438 + 1.93016i 0.363263 0.629189i 2.99441 −1.01125 + 1.75153i 0.219486 + 0.380161i
94.14 0.646873 + 1.12042i −1.49698 2.59285i 0.163110 0.282515i 2.87150 1.93671 3.35449i 0.577201 0.999742i 3.00954 −2.98190 + 5.16480i 1.85749 + 3.21727i
94.15 0.809863 + 1.40272i −0.263517 0.456425i −0.311757 + 0.539980i −1.44443 0.426825 0.739283i 1.00072 1.73330i 2.22953 1.36112 2.35752i −1.16979 2.02614i
94.16 1.06256 + 1.84040i 0.793339 + 1.37410i −1.25806 + 2.17902i −1.24766 −1.68594 + 2.92013i −1.62133 + 2.80822i −1.09681 0.241228 0.417818i −1.32571 2.29620i
94.17 1.12604 + 1.95037i 1.66294 + 2.88030i −1.53595 + 2.66034i −0.820579 −3.74509 + 6.48669i 1.96872 3.40992i −2.41402 −4.03076 + 6.98148i −0.924008 1.60043i
94.18 1.20670 + 2.09007i −0.882453 1.52845i −1.91225 + 3.31212i 3.22507 2.12971 3.68877i −1.47346 + 2.55211i −4.40325 −0.0574470 + 0.0995011i 3.89169 + 6.74061i
373.1 −1.39316 + 2.41302i −1.13229 + 1.96118i −2.88179 4.99141i −3.38130 −3.15492 5.46447i −0.923334 1.59926i 10.4866 −1.06415 1.84316i 4.71069 8.15916i
373.2 −1.33157 + 2.30635i −1.12120 + 1.94198i −2.54616 4.41008i 4.27464 −2.98592 5.17177i 1.44001 + 2.49418i 8.23530 −1.01419 1.75663i −5.69198 + 9.85880i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.18 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.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.f.c 36
13.c even 3 1 inner 403.2.f.c 36
13.c even 3 1 5239.2.a.p 18
13.e even 6 1 5239.2.a.o 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.f.c 36 1.a even 1 1 trivial
403.2.f.c 36 13.c even 3 1 inner
5239.2.a.o 18 13.e even 6 1
5239.2.a.p 18 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database