# Properties

 Label 418.2.u.a Level $418$ Weight $2$ Character orbit 418.u Analytic conductor $3.338$ Analytic rank $0$ Dimension $216$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.u (of order $$45$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$9$$ over $$\Q(\zeta_{45})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$216 q - 3 q^{7} - 27 q^{8}+O(q^{10})$$ 216 * q - 3 * q^7 - 27 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$216 q - 3 q^{7} - 27 q^{8} + 3 q^{11} + 6 q^{13} + 18 q^{14} - 9 q^{15} - 15 q^{17} + 42 q^{18} + 45 q^{19} - 48 q^{21} - 6 q^{23} + 6 q^{25} - 30 q^{26} + 18 q^{27} - 6 q^{28} - 18 q^{29} - 3 q^{31} + 45 q^{33} - 12 q^{34} - 30 q^{35} - 36 q^{37} + 15 q^{38} - 36 q^{39} - 51 q^{41} + 6 q^{42} - 36 q^{43} - 30 q^{44} + 132 q^{45} + 12 q^{46} - 36 q^{47} - 48 q^{49} - 24 q^{50} - 36 q^{51} - 3 q^{52} + 75 q^{53} + 120 q^{55} + 24 q^{56} - 36 q^{57} + 12 q^{58} - 42 q^{59} - 72 q^{60} - 108 q^{61} - 27 q^{62} + 81 q^{63} + 27 q^{64} - 102 q^{65} + 165 q^{66} - 84 q^{67} - 3 q^{68} - 30 q^{69} - 15 q^{70} + 24 q^{71} - 36 q^{73} - 6 q^{74} + 6 q^{76} + 6 q^{77} - 18 q^{78} - 36 q^{79} - 96 q^{81} - 39 q^{82} - 24 q^{84} + 6 q^{85} + 27 q^{86} - 132 q^{87} - 3 q^{88} - 42 q^{89} + 24 q^{90} + 108 q^{91} - 18 q^{92} - 75 q^{93} - 156 q^{94} + 66 q^{95} + 63 q^{97} - 12 q^{98} - 120 q^{99}+O(q^{100})$$ 216 * q - 3 * q^7 - 27 * q^8 + 3 * q^11 + 6 * q^13 + 18 * q^14 - 9 * q^15 - 15 * q^17 + 42 * q^18 + 45 * q^19 - 48 * q^21 - 6 * q^23 + 6 * q^25 - 30 * q^26 + 18 * q^27 - 6 * q^28 - 18 * q^29 - 3 * q^31 + 45 * q^33 - 12 * q^34 - 30 * q^35 - 36 * q^37 + 15 * q^38 - 36 * q^39 - 51 * q^41 + 6 * q^42 - 36 * q^43 - 30 * q^44 + 132 * q^45 + 12 * q^46 - 36 * q^47 - 48 * q^49 - 24 * q^50 - 36 * q^51 - 3 * q^52 + 75 * q^53 + 120 * q^55 + 24 * q^56 - 36 * q^57 + 12 * q^58 - 42 * q^59 - 72 * q^60 - 108 * q^61 - 27 * q^62 + 81 * q^63 + 27 * q^64 - 102 * q^65 + 165 * q^66 - 84 * q^67 - 3 * q^68 - 30 * q^69 - 15 * q^70 + 24 * q^71 - 36 * q^73 - 6 * q^74 + 6 * q^76 + 6 * q^77 - 18 * q^78 - 36 * q^79 - 96 * q^81 - 39 * q^82 - 24 * q^84 + 6 * q^85 + 27 * q^86 - 132 * q^87 - 3 * q^88 - 42 * q^89 + 24 * q^90 + 108 * q^91 - 18 * q^92 - 75 * q^93 - 156 * q^94 + 66 * q^95 + 63 * q^97 - 12 * q^98 - 120 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0.615661 + 0.788011i −1.77121 + 1.71044i −0.241922 + 0.970296i −2.96594 + 1.85333i −2.43831 0.342682i 4.46328 + 1.98718i −0.913545 + 0.406737i 0.106889 3.06091i −3.28646 1.19617i
5.2 0.615661 + 0.788011i −1.53524 + 1.48257i −0.241922 + 0.970296i −1.71626 + 1.07244i −2.11347 0.297029i −2.79496 1.24439i −0.913545 + 0.406737i 0.0542680 1.55403i −1.90173 0.692174i
5.3 0.615661 + 0.788011i −1.28208 + 1.23809i −0.241922 + 0.970296i 1.84514 1.15297i −1.76495 0.248048i 1.49870 + 0.667264i −0.913545 + 0.406737i 0.00616292 0.176483i 2.04454 + 0.744151i
5.4 0.615661 + 0.788011i −1.06998 + 1.03327i −0.241922 + 0.970296i 1.17323 0.733116i −1.47297 0.207013i −4.20324 1.87140i −0.913545 + 0.406737i −0.0274833 + 0.787018i 1.30002 + 0.473167i
5.5 0.615661 + 0.788011i 0.343250 0.331472i −0.241922 + 0.970296i −1.37828 + 0.861248i 0.472530 + 0.0664097i 1.46222 + 0.651023i −0.913545 + 0.406737i −0.0967521 + 2.77062i −1.52723 0.555866i
5.6 0.615661 + 0.788011i 0.516073 0.498366i −0.241922 + 0.970296i 1.38205 0.863602i 0.710443 + 0.0998463i 2.58409 + 1.15051i −0.913545 + 0.406737i −0.0867357 + 2.48379i 1.53140 + 0.557385i
5.7 0.615661 + 0.788011i 0.736190 0.710930i −0.241922 + 0.970296i −2.96978 + 1.85573i 1.01346 + 0.142433i −1.76367 0.785235i −0.913545 + 0.406737i −0.0681449 + 1.95141i −3.29072 1.19772i
5.8 0.615661 + 0.788011i 1.77612 1.71518i −0.241922 + 0.970296i 2.38937 1.49305i 2.44506 + 0.343631i 1.14234 + 0.508603i −0.913545 + 0.406737i 0.108063 3.09453i 2.64758 + 0.963641i
5.9 0.615661 + 0.788011i 2.28688 2.20842i −0.241922 + 0.970296i −0.827792 + 0.517262i 3.14820 + 0.442451i −1.03766 0.461997i −0.913545 + 0.406737i 0.248029 7.10262i −0.917247 0.333851i
9.1 −0.990268 0.139173i −0.724333 + 2.90514i 0.961262 + 0.275637i −0.0381555 + 1.09263i 1.12160 2.77606i −3.43887 + 1.53108i −0.913545 0.406737i −5.26635 2.80017i 0.189849 1.07669i
9.2 −0.990268 0.139173i −0.551416 + 2.21161i 0.961262 + 0.275637i 0.0341180 0.977012i 0.853847 2.11334i 1.90848 0.849710i −0.913545 0.406737i −1.93832 1.03062i −0.169760 + 0.962755i
9.3 −0.990268 0.139173i −0.367573 + 1.47426i 0.961262 + 0.275637i −0.126277 + 3.61609i 0.569173 1.40875i −0.999064 + 0.444812i −0.913545 0.406737i 0.610521 + 0.324620i 0.628311 3.56333i
9.4 −0.990268 0.139173i −0.232391 + 0.932069i 0.961262 + 0.275637i 0.0720784 2.06405i 0.359848 0.890656i −2.22827 + 0.992090i −0.913545 0.406737i 1.83410 + 0.975206i −0.358638 + 2.03394i
9.5 −0.990268 0.139173i −0.0582954 + 0.233810i 0.961262 + 0.275637i 0.00205953 0.0589772i 0.0902681 0.223421i 3.37543 1.50284i −0.913545 0.406737i 2.59757 + 1.38115i −0.0102475 + 0.0581166i
9.6 −0.990268 0.139173i 0.352008 1.41183i 0.961262 + 0.275637i −0.0725922 + 2.07877i −0.545070 + 1.34910i −1.68057 + 0.748238i −0.913545 0.406737i 0.779498 + 0.414467i 0.361195 2.04844i
9.7 −0.990268 0.139173i 0.460826 1.84827i 0.961262 + 0.275637i 0.00305043 0.0873529i −0.713570 + 1.76615i 3.88390 1.72922i −0.913545 0.406737i −0.554900 0.295046i −0.0151779 + 0.0860782i
9.8 −0.990268 0.139173i 0.516596 2.07195i 0.961262 + 0.275637i −0.0850036 + 2.43418i −0.799928 + 1.97989i −1.17126 + 0.521480i −0.913545 0.406737i −1.37727 0.732308i 0.422949 2.39866i
9.9 −0.990268 0.139173i 0.604580 2.42484i 0.961262 + 0.275637i 0.0844542 2.41845i −0.936169 + 2.31710i −3.34132 + 1.48765i −0.913545 0.406737i −2.86548 1.52360i −0.420216 + 2.38316i
25.1 0.241922 0.970296i −0.0940796 + 2.69409i −0.882948 0.469472i 1.14013 2.33762i 2.59130 + 0.743043i 1.02329 + 1.13648i −0.669131 + 0.743145i −4.25656 0.297647i −1.99236 1.67179i
25.2 0.241922 0.970296i −0.0808654 + 2.31568i −0.882948 0.469472i −0.432270 + 0.886285i 2.22733 + 0.638678i 1.27217 + 1.41289i −0.669131 + 0.743145i −2.36315 0.165248i 0.755383 + 0.633841i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 405.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.e even 9 1 inner
209.u even 45 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.u.a 216
11.c even 5 1 inner 418.2.u.a 216
19.e even 9 1 inner 418.2.u.a 216
209.u even 45 1 inner 418.2.u.a 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.u.a 216 1.a even 1 1 trivial
418.2.u.a 216 11.c even 5 1 inner
418.2.u.a 216 19.e even 9 1 inner
418.2.u.a 216 209.u even 45 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{216} - 6 T_{3}^{213} + 24 T_{3}^{212} + 12 T_{3}^{211} - 953 T_{3}^{210} - 1410 T_{3}^{209} + 4107 T_{3}^{208} - 20934 T_{3}^{207} + 55374 T_{3}^{206} - 69486 T_{3}^{205} + 522581 T_{3}^{204} + 2183754 T_{3}^{203} + \cdots + 23\!\cdots\!81$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.