↑

Properties

Label 417.2.m.a
Level $417$
Weight $2$
Character orbit 417.m
Analytic conductor $3.330$
Analytic rank $0$
Dimension $484$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [417,2,Mod(4,417)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(417, base_ring=CyclotomicField(138))
 
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("417.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 417 = 3 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 417.m (of order \(69\), degree \(44\), 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: \(3.32976176429\)
Analytic rank: \(0\)
Dimension: \(484\)
Relative dimension: \(11\) over \(\Q(\zeta_{69})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{69}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 484 q + q^{2} - 11 q^{3} + 11 q^{4} - 3 q^{5} + 2 q^{6} - q^{7} + 40 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 484 q + q^{2} - 11 q^{3} + 11 q^{4} - 3 q^{5} + 2 q^{6} - q^{7} + 40 q^{8} + 11 q^{9} - 26 q^{10} - 51 q^{11} - 11 q^{12} + 2 q^{13} - 52 q^{14} + 3 q^{15} - q^{16} + 46 q^{17} + q^{18} + 2 q^{19} - q^{20} + q^{21} + 12 q^{22} + 2 q^{23} - 3 q^{24} + 8 q^{25} + 135 q^{26} + 22 q^{27} - 3 q^{28} - 6 q^{29} - 105 q^{30} - 29 q^{31} - 269 q^{32} - 10 q^{33} - 50 q^{34} + 10 q^{35} - 22 q^{36} - 5 q^{37} + 136 q^{38} + 4 q^{39} - 117 q^{40} - 16 q^{41} - 3 q^{42} - 88 q^{43} - 70 q^{44} + 6 q^{45} - 40 q^{46} - 186 q^{47} - 2 q^{48} - 16 q^{49} - 11 q^{50} + 23 q^{51} + 220 q^{52} - 31 q^{53} - q^{54} - 182 q^{55} + 15 q^{56} - 42 q^{57} + 203 q^{58} + 68 q^{59} - 2 q^{60} - 15 q^{61} + 32 q^{62} + 2 q^{63} - 78 q^{64} - 2 q^{65} + 34 q^{66} + 51 q^{67} - 10 q^{68} - 91 q^{69} - 195 q^{70} - 36 q^{71} + 3 q^{72} - 14 q^{73} - 48 q^{74} + 39 q^{75} + 4 q^{76} + 26 q^{77} - 89 q^{78} + 14 q^{79} + 36 q^{80} + 11 q^{81} - 82 q^{82} + 80 q^{83} - 6 q^{84} - 62 q^{85} - 381 q^{86} - 81 q^{87} - 111 q^{88} + 90 q^{89} + 59 q^{90} - 2 q^{91} + 162 q^{92} - 17 q^{93} - 290 q^{94} + 2 q^{95} - 7 q^{96} - 96 q^{97} - 82 q^{98} - 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.47165 − 0.112613i −0.291646 + 0.956526i 4.10466 + 0.374809i −0.877516 − 0.857763i 0.828564 − 2.33136i 3.23030 − 3.79092i −5.20073 − 0.714824i −0.829885 − 0.557934i 2.07232 + 2.21891i
4.2 −2.25105 − 0.102562i −0.291646 + 0.956526i 3.06500 + 0.279875i −2.12112 − 2.07338i 0.754613 − 2.12328i −2.75576 + 3.23403i −2.40597 − 0.330693i −0.829885 − 0.557934i 4.56211 + 4.88483i
4.3 −1.42260 − 0.0648164i −0.291646 + 0.956526i 0.0278871 + 0.00254646i 0.621346 + 0.607360i 0.476895 − 1.34185i −0.539831 + 0.633520i 2.78212 + 0.382394i −0.829885 − 0.557934i −0.844562 − 0.904305i
4.4 −0.857796 − 0.0390828i −0.291646 + 0.956526i −1.25743 − 0.114820i −0.0328337 − 0.0320946i 0.287556 − 0.809106i 0.133890 − 0.157127i 2.77551 + 0.381484i −0.829885 − 0.557934i 0.0269103 + 0.0288139i
4.5 0.130690 + 0.00595449i −0.291646 + 0.956526i −1.97467 − 0.180313i 2.30287 + 2.25103i −0.0438110 + 0.123272i 3.27381 − 3.84199i −0.516212 − 0.0709517i −0.829885 − 0.557934i 0.287559 + 0.307901i
4.6 0.195329 + 0.00889954i −0.291646 + 0.956526i −1.95364 − 0.178393i −1.19848 − 1.17150i −0.0654795 + 0.184242i 0.175468 − 0.205921i −0.767435 − 0.105482i −0.829885 − 0.557934i −0.223671 − 0.239493i
4.7 0.625411 + 0.0284948i −0.291646 + 0.956526i −1.60139 − 0.146228i 2.77965 + 2.71708i −0.209655 + 0.589911i −2.08170 + 2.44299i −2.23781 − 0.307581i −0.829885 − 0.557934i 1.66100 + 1.77850i
4.8 1.05801 + 0.0482049i −0.291646 + 0.956526i −0.874652 − 0.0798673i −0.780362 − 0.762796i −0.354674 + 0.997956i −1.91463 + 2.24692i −3.02003 − 0.415093i −0.829885 − 0.557934i −0.788860 − 0.844663i
4.9 1.22619 + 0.0558677i −0.291646 + 0.956526i −0.491281 − 0.0448605i −2.01526 − 1.96990i −0.411054 + 1.15659i 2.22664 − 2.61308i −3.03197 − 0.416734i −0.829885 − 0.557934i −2.36105 − 2.52806i
4.10 2.07196 + 0.0944023i −0.291646 + 0.956526i 2.29240 + 0.209326i 1.43914 + 1.40674i −0.694577 + 1.95435i 1.03433 − 1.21384i 0.620412 + 0.0852737i −0.829885 − 0.557934i 2.84904 + 3.05058i
4.11 2.44956 + 0.111606i −0.291646 + 0.956526i 3.99618 + 0.364904i 0.435681 + 0.425874i −0.821159 + 2.31052i −2.68862 + 3.15523i 4.88964 + 0.672066i −0.829885 − 0.557934i 1.01970 + 1.09183i
7.1 −1.16838 − 2.53061i −0.898128 + 0.439735i −3.74174 + 4.39113i 1.08476 + 0.593784i 2.16215 + 1.75904i −0.199555 − 0.0760813i 10.1161 + 2.83440i 0.613267 − 0.789876i 0.235226 − 3.43888i
7.2 −0.942984 − 2.04243i −0.898128 + 0.439735i −1.98515 + 2.32968i −3.68297 − 2.01600i 1.74505 + 1.41970i −0.414703 − 0.158107i 2.29779 + 0.643812i 0.613267 − 0.789876i −0.644570 + 9.42327i
7.3 −0.764131 − 1.65505i −0.898128 + 0.439735i −0.858130 + 1.00706i 3.81271 + 2.08702i 1.41407 + 1.15043i −2.69136 − 1.02609i −1.18821 − 0.332921i 0.613267 − 0.789876i 0.540716 − 7.90499i
7.4 −0.417084 − 0.903373i −0.898128 + 0.439735i 0.655041 − 0.768725i −0.771860 − 0.422505i 0.771839 + 0.627938i 3.51683 + 1.34081i −2.88387 − 0.808023i 0.613267 − 0.789876i −0.0597488 + 0.873497i
7.5 −0.405706 − 0.878729i −0.898128 + 0.439735i 0.689597 − 0.809278i −0.301375 − 0.164968i 0.750784 + 0.610808i −2.25782 − 0.860805i −2.85486 − 0.799893i 0.613267 − 0.789876i −0.0226927 + 0.331756i
7.6 0.103679 + 0.224560i −0.898128 + 0.439735i 1.25749 − 1.47573i 3.27021 + 1.79006i −0.191863 − 0.156092i 1.33334 + 0.508341i 0.938097 + 0.262842i 0.613267 − 0.789876i −0.0629263 + 0.919949i
7.7 0.150974 + 0.326998i −0.898128 + 0.439735i 1.21303 − 1.42355i 0.528330 + 0.289200i −0.279386 − 0.227297i −3.62062 − 1.38038i 1.34226 + 0.376083i 0.613267 − 0.789876i −0.0148038 + 0.216424i
7.8 0.516556 + 1.11882i −0.898128 + 0.439735i 0.312233 − 0.366421i −1.07241 − 0.587019i −0.955918 − 0.777697i −0.821113 − 0.313053i 2.94447 + 0.825003i 0.613267 − 0.789876i 0.102812 − 1.50306i
7.9 0.749718 + 1.62383i −0.898128 + 0.439735i −0.777589 + 0.912541i 0.792371 + 0.433732i −1.38740 − 1.12873i 3.32621 + 1.26813i 1.37966 + 0.386563i 0.613267 − 0.789876i −0.110254 + 1.61185i
See next 80 embeddings (of 484 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
139.g even 69 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 417.2.m.a ✓ 484
139.g even 69 1 inner 417.2.m.a ✓ 484
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
417.2.m.a ✓ 484 1.a even 1 1 trivial
417.2.m.a ✓ 484 139.g even 69 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{484} - T_{2}^{483} - 16 T_{2}^{482} - 25 T_{2}^{481} + 131 T_{2}^{480} + 832 T_{2}^{479} + \cdots + 12\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(417, [\chi])\). Copy content Toggle raw display