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Properties

Label 407.2.e.d
Level $407$
Weight $2$
Character orbit 407.e
Analytic conductor $3.250$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(100,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.e (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.24991136227\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 2 q^{3} - 16 q^{4} + q^{5} - 4 q^{6} + 9 q^{7} + 12 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} - 2 q^{3} - 16 q^{4} + q^{5} - 4 q^{6} + 9 q^{7} + 12 q^{8} - 16 q^{9} - 24 q^{10} - 28 q^{11} + 15 q^{12} - 2 q^{14} - q^{15} - 8 q^{16} + 5 q^{17} - 13 q^{18} - 11 q^{19} + 23 q^{20} + 3 q^{21} + 2 q^{22} + 14 q^{23} + 34 q^{24} - 5 q^{25} + 24 q^{26} - 56 q^{27} + q^{28} - 10 q^{29} - 21 q^{30} - 4 q^{31} - 19 q^{32} + 2 q^{33} + 14 q^{34} + 3 q^{35} + 56 q^{36} + 19 q^{37} + 42 q^{38} - 10 q^{39} + 9 q^{40} + 21 q^{41} + 24 q^{42} - 18 q^{43} + 16 q^{44} - 58 q^{45} - 5 q^{46} + 22 q^{47} - 166 q^{48} + 5 q^{49} - 35 q^{50} + 52 q^{51} - 25 q^{52} + 13 q^{53} + 7 q^{54} - q^{55} - 20 q^{56} + 14 q^{57} - 5 q^{58} - 4 q^{59} + 184 q^{60} - 27 q^{61} + 18 q^{62} - 46 q^{63} - 4 q^{64} + 22 q^{65} + 4 q^{66} + 2 q^{67} - 40 q^{68} - 8 q^{69} + 48 q^{70} + 33 q^{71} - 40 q^{72} - 6 q^{73} - 2 q^{74} + 52 q^{75} + 5 q^{76} - 9 q^{77} + 2 q^{78} + 34 q^{79} - 84 q^{80} + 14 q^{81} - 128 q^{82} - 7 q^{83} - 8 q^{84} - 22 q^{85} - 9 q^{86} + 19 q^{87} - 12 q^{88} - 23 q^{89} + 121 q^{90} + 4 q^{91} + 23 q^{92} - 25 q^{93} - 4 q^{94} - 22 q^{95} + 59 q^{96} + 48 q^{97} + 38 q^{98} + 16 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1 −1.37963 − 2.38958i 1.52355 − 2.63886i −2.80673 + 4.86140i 1.82594 − 3.16262i −8.40770 −0.500708 + 0.867252i 9.97046 −3.14239 − 5.44278i −10.0764
100.2 −1.19250 − 2.06548i 0.318918 − 0.552382i −1.84413 + 3.19412i −0.554857 + 0.961041i −1.52124 0.137472 − 0.238108i 4.02650 1.29658 + 2.24575i 2.64668
100.3 −1.01375 − 1.75587i −1.10261 + 1.90978i −1.05540 + 1.82800i 0.851373 − 1.47462i 4.47111 −0.671965 + 1.16388i 0.224629 −0.931508 − 1.61342i −3.45233
100.4 −0.964232 − 1.67010i −0.834443 + 1.44530i −0.859485 + 1.48867i −1.61683 + 2.80044i 3.21839 1.99877 − 3.46196i −0.541954 0.107409 + 0.186038i 6.23601
100.5 −0.733220 − 1.26997i 0.931171 − 1.61284i −0.0752219 + 0.130288i 0.716935 − 1.24177i −2.73101 0.647662 − 1.12178i −2.71226 −0.234158 − 0.405574i −2.10268
100.6 −0.549144 − 0.951146i −1.35897 + 2.35380i 0.396881 − 0.687419i 1.88020 − 3.25659i 2.98507 1.05841 − 1.83323i −3.06836 −2.19357 − 3.79938i −4.12999
100.7 −0.199866 − 0.346178i −0.985556 + 1.70703i 0.920107 − 1.59367i −0.202463 + 0.350677i 0.787917 −1.19763 + 2.07435i −1.53506 −0.442643 − 0.766680i 0.161862
100.8 0.0881832 + 0.152738i −0.241318 + 0.417975i 0.984447 − 1.70511i −0.907389 + 1.57164i −0.0851208 2.13461 − 3.69726i 0.699980 1.38353 + 2.39635i −0.320066
100.9 0.200398 + 0.347099i 1.50269 − 2.60273i 0.919681 − 1.59293i −1.24790 + 2.16142i 1.20454 0.632747 − 1.09595i 1.53880 −3.01612 − 5.22408i −1.00030
100.10 0.567140 + 0.982315i 0.192563 − 0.333529i 0.356705 − 0.617831i −0.848833 + 1.47022i 0.436841 −2.25088 + 3.89864i 3.07777 1.42584 + 2.46963i −1.92563
100.11 0.817321 + 1.41564i −0.830740 + 1.43888i −0.336026 + 0.582013i 1.51541 − 2.62476i −2.71592 1.63268 − 2.82788i 2.17072 0.119742 + 0.207400i 4.95430
100.12 0.967579 + 1.67590i −1.28622 + 2.22780i −0.872417 + 1.51107i −1.45116 + 2.51349i −4.97807 0.156669 − 0.271358i 0.493788 −1.80872 − 3.13279i −5.61645
100.13 1.15094 + 1.99349i 1.32589 − 2.29651i −1.64933 + 2.85673i 0.147414 − 0.255329i 6.10410 1.49806 − 2.59471i −2.98938 −2.01598 − 3.49178i 0.678660
100.14 1.24078 + 2.14910i −0.154923 + 0.268334i −2.07908 + 3.60108i 0.392171 − 0.679260i −0.768902 −0.775890 + 1.34388i −5.35563 1.45200 + 2.51493i 1.94640
232.1 −1.37963 + 2.38958i 1.52355 + 2.63886i −2.80673 − 4.86140i 1.82594 + 3.16262i −8.40770 −0.500708 − 0.867252i 9.97046 −3.14239 + 5.44278i −10.0764
232.2 −1.19250 + 2.06548i 0.318918 + 0.552382i −1.84413 − 3.19412i −0.554857 − 0.961041i −1.52124 0.137472 + 0.238108i 4.02650 1.29658 − 2.24575i 2.64668
232.3 −1.01375 + 1.75587i −1.10261 − 1.90978i −1.05540 − 1.82800i 0.851373 + 1.47462i 4.47111 −0.671965 − 1.16388i 0.224629 −0.931508 + 1.61342i −3.45233
232.4 −0.964232 + 1.67010i −0.834443 − 1.44530i −0.859485 − 1.48867i −1.61683 − 2.80044i 3.21839 1.99877 + 3.46196i −0.541954 0.107409 − 0.186038i 6.23601
232.5 −0.733220 + 1.26997i 0.931171 + 1.61284i −0.0752219 − 0.130288i 0.716935 + 1.24177i −2.73101 0.647662 + 1.12178i −2.71226 −0.234158 + 0.405574i −2.10268
232.6 −0.549144 + 0.951146i −1.35897 − 2.35380i 0.396881 + 0.687419i 1.88020 + 3.25659i 2.98507 1.05841 + 1.83323i −3.06836 −2.19357 + 3.79938i −4.12999
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.14
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.e.d ✓ 28
37.c even 3 1 inner 407.2.e.d ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.e.d ✓ 28 1.a even 1 1 trivial
407.2.e.d ✓ 28 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 2 T_{2}^{27} + 24 T_{2}^{26} + 36 T_{2}^{25} + 322 T_{2}^{24} + 419 T_{2}^{23} + 2828 T_{2}^{22} + \cdots + 576 \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\). Copy content Toggle raw display