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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{2} + 12 q^{4} - 13 q^{7} + 24 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{2} + 12 q^{4} - 13 q^{7} + 24 q^{8} - 36 q^{9} + 2 q^{11} + 11 q^{12} + 4 q^{13} + 2 q^{14} + 6 q^{15} + 12 q^{16} - 8 q^{17} - 18 q^{18} + 12 q^{19} - 29 q^{21} + 20 q^{22} + 10 q^{25} + 8 q^{26} - 18 q^{27} + 4 q^{28} - 8 q^{29} - 18 q^{30} - 12 q^{32} + 50 q^{33} - 4 q^{34} - 12 q^{35} - 4 q^{36} + 92 q^{37} + 6 q^{38} + 44 q^{39} + 2 q^{41} + 42 q^{42} - 14 q^{43} - 4 q^{44} - 2 q^{45} - 44 q^{46} + 10 q^{47} - 22 q^{48} + 43 q^{49} - 43 q^{50} - 90 q^{51} - 18 q^{52} + 16 q^{53} - 29 q^{54} - 8 q^{55} + 13 q^{56} + 7 q^{57} + 8 q^{58} - 84 q^{59} + 9 q^{60} - 53 q^{61} - 66 q^{62} + 11 q^{63} - 24 q^{64} + 79 q^{65} - 110 q^{66} + 46 q^{67} - 40 q^{68} - 55 q^{69} - 6 q^{70} - 14 q^{71} - 19 q^{72} + 16 q^{73} + 46 q^{74} - 44 q^{75} - 6 q^{76} - 96 q^{77} + 20 q^{78} + 20 q^{79} - 22 q^{80} - 56 q^{81} + 26 q^{82} + 57 q^{83} + 5 q^{84} - 2 q^{85} - 8 q^{86} - 198 q^{87} - 2 q^{88} + 60 q^{89} - 22 q^{90} - 252 q^{91} - 218 q^{93} - 10 q^{94} + 6 q^{95} - q^{97} + 20 q^{98} + 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \(S_{2}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display