# Properties

 Label 414.2.m.a Level $414$ Weight $2$ Character orbit 414.m Analytic conductor $3.306$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.m (of order $$33$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.30580664368$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$12$$ over $$\Q(\zeta_{33})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 12 q^{2} + 12 q^{4} - 13 q^{7} + 24 q^{8} - 36 q^{9}+O(q^{10})$$ 240 * q - 12 * q^2 + 12 * q^4 - 13 * q^7 + 24 * q^8 - 36 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 12 q^{2} + 12 q^{4} - 13 q^{7} + 24 q^{8} - 36 q^{9} + 2 q^{11} + 11 q^{12} + 4 q^{13} + 2 q^{14} + 6 q^{15} + 12 q^{16} - 8 q^{17} - 18 q^{18} + 12 q^{19} - 29 q^{21} + 20 q^{22} + 10 q^{25} + 8 q^{26} - 18 q^{27} + 4 q^{28} - 8 q^{29} - 18 q^{30} - 12 q^{32} + 50 q^{33} - 4 q^{34} - 12 q^{35} - 4 q^{36} + 92 q^{37} + 6 q^{38} + 44 q^{39} + 2 q^{41} + 42 q^{42} - 14 q^{43} - 4 q^{44} - 2 q^{45} - 44 q^{46} + 10 q^{47} - 22 q^{48} + 43 q^{49} - 43 q^{50} - 90 q^{51} - 18 q^{52} + 16 q^{53} - 29 q^{54} - 8 q^{55} + 13 q^{56} + 7 q^{57} + 8 q^{58} - 84 q^{59} + 9 q^{60} - 53 q^{61} - 66 q^{62} + 11 q^{63} - 24 q^{64} + 79 q^{65} - 110 q^{66} + 46 q^{67} - 40 q^{68} - 55 q^{69} - 6 q^{70} - 14 q^{71} - 19 q^{72} + 16 q^{73} + 46 q^{74} - 44 q^{75} - 6 q^{76} - 96 q^{77} + 20 q^{78} + 20 q^{79} - 22 q^{80} - 56 q^{81} + 26 q^{82} + 57 q^{83} + 5 q^{84} - 2 q^{85} - 8 q^{86} - 198 q^{87} - 2 q^{88} + 60 q^{89} - 22 q^{90} - 252 q^{91} - 218 q^{93} - 10 q^{94} + 6 q^{95} - q^{97} + 20 q^{98} + 84 q^{99}+O(q^{100})$$ 240 * q - 12 * q^2 + 12 * q^4 - 13 * q^7 + 24 * q^8 - 36 * q^9 + 2 * q^11 + 11 * q^12 + 4 * q^13 + 2 * q^14 + 6 * q^15 + 12 * q^16 - 8 * q^17 - 18 * q^18 + 12 * q^19 - 29 * q^21 + 20 * q^22 + 10 * q^25 + 8 * q^26 - 18 * q^27 + 4 * q^28 - 8 * q^29 - 18 * q^30 - 12 * q^32 + 50 * q^33 - 4 * q^34 - 12 * q^35 - 4 * q^36 + 92 * q^37 + 6 * q^38 + 44 * q^39 + 2 * q^41 + 42 * q^42 - 14 * q^43 - 4 * q^44 - 2 * q^45 - 44 * q^46 + 10 * q^47 - 22 * q^48 + 43 * q^49 - 43 * q^50 - 90 * q^51 - 18 * q^52 + 16 * q^53 - 29 * q^54 - 8 * q^55 + 13 * q^56 + 7 * q^57 + 8 * q^58 - 84 * q^59 + 9 * q^60 - 53 * q^61 - 66 * q^62 + 11 * q^63 - 24 * q^64 + 79 * q^65 - 110 * q^66 + 46 * q^67 - 40 * q^68 - 55 * q^69 - 6 * q^70 - 14 * q^71 - 19 * q^72 + 16 * q^73 + 46 * q^74 - 44 * q^75 - 6 * q^76 - 96 * q^77 + 20 * q^78 + 20 * q^79 - 22 * q^80 - 56 * q^81 + 26 * q^82 + 57 * q^83 + 5 * q^84 - 2 * q^85 - 8 * q^86 - 198 * q^87 - 2 * q^88 + 60 * q^89 - 22 * q^90 - 252 * q^91 - 218 * q^93 - 10 * q^94 + 6 * q^95 - q^97 + 20 * q^98 + 84 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 0.327068 0.945001i −1.60283 0.656460i −0.786053 0.618159i 0.587955 + 0.825667i −1.14459 + 1.29997i 0.631846 + 2.60451i −0.841254 + 0.540641i 2.13812 + 2.10439i 0.972558 0.285569i
13.2 0.327068 0.945001i −1.51415 + 0.841046i −0.786053 0.618159i 1.67302 + 2.34942i 0.299560 + 1.70595i 0.0366105 + 0.150910i −0.841254 + 0.540641i 1.58528 2.54693i 2.76740 0.812581i
13.3 0.327068 0.945001i −1.47171 0.913273i −0.786053 0.618159i −1.18105 1.65856i −1.34439 + 1.09207i −0.591460 2.43803i −0.841254 + 0.540641i 1.33186 + 2.68815i −1.95362 + 0.573635i
13.4 0.327068 0.945001i −1.08222 + 1.35233i −0.786053 0.618159i 0.204372 + 0.287001i 0.923996 + 1.46500i −0.726527 2.99479i −0.841254 + 0.540641i −0.657606 2.92704i 0.338059 0.0992632i
13.5 0.327068 0.945001i −0.431117 1.67754i −0.786053 0.618159i 2.15343 + 3.02407i −1.72628 0.141264i −1.21686 5.01596i −0.841254 + 0.540641i −2.62828 + 1.44643i 3.56206 1.04592i
13.6 0.327068 0.945001i 0.0157937 + 1.73198i −0.786053 0.618159i −1.24523 1.74869i 1.64189 + 0.551550i −0.351646 1.44951i −0.841254 + 0.540641i −2.99950 + 0.0547087i −2.05979 + 0.604808i
13.7 0.327068 0.945001i 0.240116 1.71533i −0.786053 0.618159i −2.36816 3.32562i −1.54245 0.787938i 0.386739 + 1.59416i −0.841254 + 0.540641i −2.88469 0.823754i −3.91726 + 1.15021i
13.8 0.327068 0.945001i 0.754592 + 1.55904i −0.786053 0.618159i 2.12858 + 2.98918i 1.72009 0.203180i −0.231123 0.952703i −0.841254 + 0.540641i −1.86118 + 2.35287i 3.52097 1.03385i
13.9 0.327068 0.945001i 1.23371 + 1.21571i −0.786053 0.618159i −0.213728 0.300139i 1.55235 0.768242i 1.08956 + 4.49122i −0.841254 + 0.540641i 0.0441053 + 2.99968i −0.353536 + 0.103807i
13.10 0.327068 0.945001i 1.32227 1.11875i −0.786053 0.618159i −0.518620 0.728300i −0.624745 1.61545i −0.945394 3.89697i −0.841254 + 0.540641i 0.496807 2.95858i −0.857868 + 0.251893i
13.11 0.327068 0.945001i 1.35455 1.07944i −0.786053 0.618159i 1.79492 + 2.52061i −0.577038 1.63310i 0.738833 + 3.04551i −0.841254 + 0.540641i 0.669632 2.92431i 2.96903 0.871787i
13.12 0.327068 0.945001i 1.72251 + 0.181502i −0.786053 0.618159i −1.09565 1.53862i 0.734899 1.56841i −0.216031 0.890490i −0.841254 + 0.540641i 2.93411 + 0.625279i −1.81235 + 0.532155i
25.1 0.888835 0.458227i −1.73199 0.0143219i 0.580057 0.814576i 0.132094 + 0.544499i −1.54602 + 0.780915i −0.382138 1.10411i 0.142315 0.989821i 2.99959 + 0.0496107i 0.366914 + 0.423441i
25.2 0.888835 0.458227i −1.33859 1.09917i 0.580057 0.814576i 0.448804 + 1.84999i −1.69346 0.363606i 1.59879 + 4.61941i 0.142315 0.989821i 0.583642 + 2.94268i 1.24663 + 1.43869i
25.3 0.888835 0.458227i −0.909501 1.47404i 0.580057 0.814576i −0.721543 2.97424i −1.48384 0.893426i 0.0997971 + 0.288345i 0.142315 0.989821i −1.34562 + 2.68129i −2.00421 2.31298i
25.4 0.888835 0.458227i −0.818784 + 1.52630i 0.580057 0.814576i −0.448629 1.84927i −0.0283727 + 1.73182i −1.08663 3.13963i 0.142315 0.989821i −1.65919 2.49942i −1.24614 1.43813i
25.5 0.888835 0.458227i −0.301182 + 1.70566i 0.580057 0.814576i −0.115473 0.475985i 0.513879 + 1.65406i 1.23894 + 3.57969i 0.142315 0.989821i −2.81858 1.02743i −0.320745 0.370160i
25.6 0.888835 0.458227i 0.143725 1.72608i 0.580057 0.814576i −0.196850 0.811428i −0.663187 1.60006i −1.10397 3.18970i 0.142315 0.989821i −2.95869 0.496161i −0.546785 0.631024i
25.7 0.888835 0.458227i 0.694471 1.58673i 0.580057 0.814576i −0.0619918 0.255534i −0.109811 1.72857i 0.632608 + 1.82780i 0.142315 0.989821i −2.03542 2.20388i −0.172193 0.198721i
25.8 0.888835 0.458227i 0.715945 + 1.57716i 0.580057 0.814576i 0.904639 + 3.72897i 1.35905 + 1.07377i −0.0597010 0.172495i 0.142315 0.989821i −1.97485 + 2.25831i 2.51279 + 2.89991i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 409.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
23.c even 11 1 inner
207.m even 33 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.m.a 240
9.c even 3 1 inner 414.2.m.a 240
23.c even 11 1 inner 414.2.m.a 240
207.m even 33 1 inner 414.2.m.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.m.a 240 1.a even 1 1 trivial
414.2.m.a 240 9.c even 3 1 inner
414.2.m.a 240 23.c even 11 1 inner
414.2.m.a 240 207.m even 33 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{240} - 35 T_{5}^{238} + 64 T_{5}^{237} + 306 T_{5}^{236} - 2438 T_{5}^{235} + 6225 T_{5}^{234} + 35180 T_{5}^{233} - 218266 T_{5}^{232} + 12834 T_{5}^{231} + 2351760 T_{5}^{230} - 10323448 T_{5}^{229} + \cdots + 24\!\cdots\!01$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.