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Properties

Label 402.2.n.b
Level $402$
Weight $2$
Character orbit 402.n
Analytic conductor $3.210$
Analytic rank $0$
Dimension $220$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [402,2,Mod(11,402)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(402, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 59]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("402.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 402.n (of order \(66\), degree \(20\), 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: \(3.20998616126\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(11\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 220 q + 11 q^{2} - q^{3} + 11 q^{4} - q^{6} + 6 q^{7} - 22 q^{8} - q^{9} - 6 q^{11} - 9 q^{12} - 6 q^{13} + 22 q^{14} - 4 q^{15} + 11 q^{16} - 30 q^{17} - 9 q^{18} + 25 q^{19} + 99 q^{21} + 12 q^{22} - 8 q^{23} + 10 q^{24} - 10 q^{25} - 16 q^{26} - 28 q^{27} - 6 q^{28} - 6 q^{29} + 5 q^{30} - 2 q^{31} + 11 q^{32} - 20 q^{33} - 3 q^{34} - 170 q^{35} - q^{36} + 10 q^{37} + 3 q^{38} - 3 q^{39} + 74 q^{41} + 16 q^{42} - 22 q^{43} - 6 q^{44} + 37 q^{45} + 30 q^{46} + 12 q^{47} + 10 q^{48} - 27 q^{49} + 27 q^{50} - 3 q^{51} + 22 q^{52} + 2 q^{53} + 25 q^{54} - 144 q^{55} + 6 q^{56} + 16 q^{57} - 44 q^{58} + 33 q^{59} - q^{60} - 148 q^{61} + 22 q^{62} + 38 q^{63} - 22 q^{64} + 5 q^{66} - 24 q^{67} - 80 q^{69} - 154 q^{70} - 18 q^{71} - 12 q^{72} - 167 q^{73} - 122 q^{74} - 32 q^{75} - 50 q^{76} + 44 q^{77} - 18 q^{78} + 12 q^{79} + 47 q^{81} - 38 q^{82} + 165 q^{83} + 39 q^{84} - 30 q^{85} + 3 q^{86} - 98 q^{87} - 6 q^{88} + 22 q^{89} + 6 q^{90} + 4 q^{91} + 124 q^{93} + 22 q^{94} - 142 q^{95} + 2 q^{96} - 69 q^{97} - 27 q^{98} - 57 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.327068 − 0.945001i −1.58847 − 0.690488i −0.786053 + 0.618159i 1.93542 − 0.568291i −0.132975 + 1.72694i −0.303367 + 1.57402i 0.841254 + 0.540641i 2.04645 + 2.19363i −1.17005 − 1.64310i
11.2 −0.327068 − 0.945001i −1.44867 + 0.949398i −0.786053 + 0.618159i −2.76497 + 0.811870i 1.37100 + 1.05848i −0.155035 + 0.804400i 0.841254 + 0.540641i 1.19729 − 2.75073i 1.67155 + 2.34737i
11.3 −0.327068 − 0.945001i −0.685433 − 1.59065i −0.786053 + 0.618159i 0.798523 − 0.234468i −1.27899 + 1.16799i 0.0582625 − 0.302295i 0.841254 + 0.540641i −2.06036 + 2.18057i −0.482743 − 0.677918i
11.4 −0.327068 − 0.945001i −0.663010 + 1.60013i −0.786053 + 0.618159i 0.141454 − 0.0415348i 1.72897 + 0.103193i −0.267898 + 1.38999i 0.841254 + 0.540641i −2.12084 − 2.12180i −0.0855156 − 0.120090i
11.5 −0.327068 − 0.945001i −0.187659 + 1.72185i −0.786053 + 0.618159i 2.83825 − 0.833385i 1.68853 − 0.385826i 0.584073 − 3.03046i 0.841254 + 0.540641i −2.92957 − 0.646243i −1.71585 − 2.40957i
11.6 −0.327068 − 0.945001i 0.0660585 − 1.73079i −0.786053 + 0.618159i −2.79086 + 0.819470i −1.65720 + 0.503661i −0.548725 + 2.84706i 0.841254 + 0.540641i −2.99127 − 0.228667i 1.68720 + 2.36934i
11.7 −0.327068 − 0.945001i 1.33877 + 1.09896i −0.786053 + 0.618159i −1.51180 + 0.443905i 0.600648 − 1.62457i 0.543896 − 2.82200i 0.841254 + 0.540641i 0.584588 + 2.94249i 0.913953 + 1.28347i
11.8 −0.327068 − 0.945001i 1.36651 − 1.06427i −0.786053 + 0.618159i −1.20281 + 0.353177i −1.45267 − 0.943262i 0.645513 − 3.34924i 0.841254 + 0.540641i 0.734679 − 2.90865i 0.727153 + 1.02114i
11.9 −0.327068 − 0.945001i 1.46270 − 0.927637i −0.786053 + 0.618159i 1.96820 − 0.577915i −1.35502 − 1.07885i −0.624160 + 3.23845i 0.841254 + 0.540641i 1.27898 − 2.71371i −1.18986 − 1.67093i
11.10 −0.327068 − 0.945001i 1.63470 + 0.572487i −0.786053 + 0.618159i −3.32765 + 0.977087i 0.00634093 − 1.73204i −0.598560 + 3.10562i 0.841254 + 0.540641i 2.34452 + 1.87169i 2.01172 + 2.82506i
11.11 −0.327068 − 0.945001i 1.66086 + 0.491485i −0.786053 + 0.618159i 3.91626 − 1.14992i −0.0787591 − 1.73026i 0.198051 − 1.02759i 0.841254 + 0.540641i 2.51689 + 1.63257i −2.36755 − 3.32477i
41.1 0.580057 + 0.814576i −1.67678 + 0.434046i −0.327068 + 0.945001i −0.344939 + 2.39911i −1.32619 − 1.11410i 0.101387 + 1.06177i −0.959493 + 0.281733i 2.62321 − 1.45560i −2.15434 + 1.11064i
41.2 0.580057 + 0.814576i −1.50614 − 0.855298i −0.327068 + 0.945001i 0.165614 − 1.15187i −0.176943 − 1.72299i 0.0681730 + 0.713940i −0.959493 + 0.281733i 1.53693 + 2.57640i 1.03435 − 0.533245i
41.3 0.580057 + 0.814576i −1.08359 + 1.35123i −0.327068 + 0.945001i 0.302916 − 2.10683i −1.72923 − 0.0988747i −0.489148 − 5.12259i −0.959493 + 0.281733i −0.651663 − 2.92837i 1.89188 − 0.975333i
41.4 0.580057 + 0.814576i −0.149877 − 1.72555i −0.327068 + 0.945001i −0.239393 + 1.66501i 1.31866 − 1.12301i 0.377283 + 3.95109i −0.959493 + 0.281733i −2.95507 + 0.517242i −1.49514 + 0.770799i
41.5 0.580057 + 0.814576i 0.0305166 − 1.73178i −0.327068 + 0.945001i 0.112737 − 0.784103i 1.42837 − 0.979674i −0.435574 − 4.56154i −0.959493 + 0.281733i −2.99814 − 0.105696i 0.704105 − 0.362992i
41.6 0.580057 + 0.814576i 0.0614732 + 1.73096i −0.327068 + 0.945001i −0.00661232 + 0.0459897i −1.37434 + 1.05413i 0.170378 + 1.78427i −0.959493 + 0.281733i −2.99244 + 0.212815i −0.0412977 + 0.0212904i
41.7 0.580057 + 0.814576i 0.640714 + 1.60919i −0.327068 + 0.945001i −0.151342 + 1.05261i −0.939155 + 1.45533i −0.0483075 − 0.505899i −0.959493 + 0.281733i −2.17897 + 2.06206i −0.945217 + 0.487293i
41.8 0.580057 + 0.814576i 1.37048 − 1.05914i −0.327068 + 0.945001i −0.00800979 + 0.0557093i 1.65771 + 0.501997i −0.135504 − 1.41906i −0.959493 + 0.281733i 0.756432 − 2.90307i −0.0500256 + 0.0257900i
41.9 0.580057 + 0.814576i 1.53595 + 0.800536i −0.327068 + 0.945001i 0.620565 − 4.31612i 0.238841 + 1.71550i −0.207958 − 2.17784i −0.959493 + 0.281733i 1.71828 + 2.45917i 3.87577 − 1.99810i
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.p even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 402.2.n.b yes 220
3.b odd 2 1 402.2.n.a ✓ 220
67.h odd 66 1 402.2.n.a ✓ 220
201.p even 66 1 inner 402.2.n.b yes 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
402.2.n.a ✓ 220 3.b odd 2 1
402.2.n.a ✓ 220 67.h odd 66 1
402.2.n.b yes 220 1.a even 1 1 trivial
402.2.n.b yes 220 201.p even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{220} + 60 T_{5}^{218} + 18 T_{5}^{217} + 2280 T_{5}^{216} + 1436 T_{5}^{215} + \cdots + 17\!\cdots\!64 \) acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\). Copy content Toggle raw display