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Properties

Label 2002.2.g.b
Level $2002$
Weight $2$
Character orbit 2002.g
Analytic conductor $15.986$
Analytic rank $0$
Dimension $18$
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] = [2002,2,Mod(155,2002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2002.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2002.g (of order \(2\), degree \(1\), 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: \(15.9860504847\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 33 x^{16} + 430 x^{14} + 2853 x^{12} + 10462 x^{10} + 21621 x^{8} + 24505 x^{6} + 14080 x^{4} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{2} q^{3} - q^{4} - \beta_{9} q^{5} + \beta_1 q^{6} - \beta_{5} q^{7} - \beta_{5} q^{8} + (\beta_{12} - \beta_{11} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{2} q^{3} - q^{4} - \beta_{9} q^{5} + \beta_1 q^{6} - \beta_{5} q^{7} - \beta_{5} q^{8} + (\beta_{12} - \beta_{11} + 1) q^{9} - \beta_{8} q^{10} + \beta_{5} q^{11} + \beta_{2} q^{12} + (\beta_{14} + \beta_{12} + \beta_{4}) q^{13} + q^{14} + ( - \beta_{17} + \beta_{9} + \cdots + \beta_1) q^{15}+ \cdots + (\beta_{16} + \beta_{7} + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{3} - 18 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{3} - 18 q^{4} + 12 q^{9} - 6 q^{10} + 2 q^{12} + 18 q^{14} + 18 q^{16} - 10 q^{17} - 18 q^{22} - 8 q^{23} - 16 q^{25} - 2 q^{26} - 8 q^{27} + 16 q^{29} - 12 q^{30} + 6 q^{35} - 12 q^{36} - 10 q^{38} + 2 q^{39} + 6 q^{40} - 2 q^{42} - 30 q^{43} - 2 q^{48} - 18 q^{49} + 26 q^{51} + 38 q^{53} - 6 q^{55} - 18 q^{56} + 14 q^{61} - 28 q^{62} - 18 q^{64} + 44 q^{65} + 2 q^{66} + 10 q^{68} - 40 q^{69} + 4 q^{74} + 60 q^{75} + 18 q^{77} + 12 q^{78} + 26 q^{79} + 26 q^{81} - 40 q^{82} - 120 q^{87} + 18 q^{88} + 16 q^{90} + 2 q^{91} + 8 q^{92} + 56 q^{94} - 114 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 33 x^{16} + 430 x^{14} + 2853 x^{12} + 10462 x^{10} + 21621 x^{8} + 24505 x^{6} + 14080 x^{4} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 621 \nu^{16} + 22109 \nu^{14} + 316726 \nu^{12} + 2348329 \nu^{10} + 9633974 \nu^{8} + \cdots + 1029696 ) / 442448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37317 \nu^{16} + 1096391 \nu^{14} + 12002090 \nu^{12} + 60867617 \nu^{10} + 148056382 \nu^{8} + \cdots + 47692288 ) / 5862436 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 379727 \nu^{17} - 13031447 \nu^{15} - 178394698 \nu^{13} - 1256301995 \nu^{11} + \cdots - 1492353088 \nu ) / 46899488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16089 \nu^{17} - 528453 \nu^{15} - 6829834 \nu^{13} - 44635013 \nu^{11} - 158929802 \nu^{9} + \cdots - 16969376 \nu ) / 1769792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 300381 \nu^{16} + 10802181 \nu^{14} + 154972910 \nu^{12} + 1133377049 \nu^{10} + 4483729502 \nu^{8} + \cdots + 498268208 ) / 23449744 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 519571 \nu^{17} - 17344043 \nu^{15} - 227567842 \nu^{13} - 1497393039 \nu^{11} + \cdots + 67378880 \nu ) / 46899488 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49937 \nu^{16} + 1492701 \nu^{14} + 16809849 \nu^{12} + 89566530 \nu^{10} + 237433365 \nu^{8} + \cdots + 11908076 ) / 2931218 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 176425 \nu^{17} - 5351034 \nu^{15} - 61507567 \nu^{13} - 336842939 \nu^{11} + \cdots + 80146408 \nu ) / 11724872 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 236885 \nu^{17} + 3116834 \nu^{16} + 9514289 \nu^{15} + 98824290 \nu^{14} + 151703138 \nu^{13} + \cdots + 1157582592 ) / 93798976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 200213 \nu^{16} + 6244385 \nu^{14} + 74972792 \nu^{12} + 440997591 \nu^{10} + 1356232312 \nu^{8} + \cdots + 51314512 ) / 5862436 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 200213 \nu^{16} + 6244385 \nu^{14} + 74972792 \nu^{12} + 440997591 \nu^{10} + 1356232312 \nu^{8} + \cdots + 27864768 ) / 5862436 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 755705 \nu^{17} + 3234198 \nu^{16} + 23340281 \nu^{15} + 99996774 \nu^{14} + 277186718 \nu^{13} + \cdots + 915578304 ) / 93798976 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 236885 \nu^{17} - 3116834 \nu^{16} + 9514289 \nu^{15} - 98824290 \nu^{14} + 151703138 \nu^{13} + \cdots - 1157582592 ) / 93798976 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 755705 \nu^{17} - 3234198 \nu^{16} + 23340281 \nu^{15} - 99996774 \nu^{14} + 277186718 \nu^{13} + \cdots - 915578304 ) / 93798976 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2159179 \nu^{17} + 71016507 \nu^{15} + 917957290 \nu^{13} + 5979781543 \nu^{11} + \cdots + 1622227200 \nu ) / 46899488 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 5660697 \nu^{17} - 179333693 \nu^{15} - 2203881986 \nu^{13} - 13433918229 \nu^{11} + \cdots - 3395413536 \nu ) / 93798976 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{11} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{10} + 2\beta_{9} + \beta_{5} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{13} + 11\beta_{12} - 9\beta_{11} - \beta_{8} + 3\beta_{3} - \beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{17} + 9 \beta_{15} + 11 \beta_{14} + 9 \beta_{13} + 11 \beta_{10} - 26 \beta_{9} + \cdots + 56 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{15} + 4 \beta_{14} + 13 \beta_{13} - 103 \beta_{12} + 80 \beta_{11} - 4 \beta_{10} + \cdots - 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 126 \beta_{17} + 2 \beta_{16} - 80 \beta_{15} - 103 \beta_{14} - 80 \beta_{13} - 103 \beta_{10} + \cdots - 485 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 131 \beta_{15} - 73 \beta_{14} - 131 \beta_{13} + 940 \beta_{12} - 729 \beta_{11} + 73 \beta_{10} + \cdots + 2169 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1264 \beta_{17} - 66 \beta_{16} + 729 \beta_{15} + 940 \beta_{14} + 729 \beta_{13} + 940 \beta_{10} + \cdots + 4369 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1211 \beta_{15} + 944 \beta_{14} + 1211 \beta_{13} - 8588 \beta_{12} + 6761 \beta_{11} - 944 \beta_{10} + \cdots - 20328 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12431 \beta_{17} + 1268 \beta_{16} - 6761 \beta_{15} - 8588 \beta_{14} - 6761 \beta_{13} + \cdots - 40200 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10752 \beta_{15} - 10759 \beta_{14} - 10752 \beta_{13} + 78907 \beta_{12} - 63410 \beta_{11} + \cdots + 193025 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 121150 \beta_{17} - 19296 \beta_{16} + 63410 \beta_{15} + 78907 \beta_{14} + 63410 \beta_{13} + \cdots + 374543 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 93142 \beta_{15} + 115770 \beta_{14} + 93142 \beta_{13} - 728891 \beta_{12} + 599085 \beta_{11} + \cdots - 1846106 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1176027 \beta_{17} + 258822 \beta_{16} - 599085 \beta_{15} - 728891 \beta_{14} - 599085 \beta_{13} + \cdots - 3517575 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 791419 \beta_{15} - 1210014 \beta_{14} - 791419 \beta_{13} + 6762133 \beta_{12} - 5688959 \beta_{11} + \cdots + 17737136 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 11400268 \beta_{17} - 3214200 \beta_{16} + 5688959 \beta_{15} + 6762133 \beta_{14} + \cdots + 33217552 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2002\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(1431\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
3.15056i
2.32899i
1.29698i
0.886255i
0.344030i
− 0.656332i
− 1.64412i
− 1.69936i
− 3.00701i
− 3.15056i
− 2.32899i
− 1.29698i
− 0.886255i
− 0.344030i
0.656332i
1.64412i
1.69936i
3.00701i
− 1.00000i −3.15056 −1.00000 3.29231i 3.15056i 1.00000i 1.00000i 6.92603 3.29231
155.2 − 1.00000i −2.32899 −1.00000 − 3.88035i 2.32899i 1.00000i 1.00000i 2.42421 −3.88035
155.3 − 1.00000i −1.29698 −1.00000 − 0.485139i 1.29698i 1.00000i 1.00000i −1.31785 −0.485139
155.4 − 1.00000i −0.886255 −1.00000 1.53631i 0.886255i 1.00000i 1.00000i −2.21455 1.53631
155.5 − 1.00000i −0.344030 −1.00000 − 1.81148i 0.344030i 1.00000i 1.00000i −2.88164 −1.81148
155.6 − 1.00000i 0.656332 −1.00000 0.181366i − 0.656332i 1.00000i 1.00000i −2.56923 0.181366
155.7 − 1.00000i 1.64412 −1.00000 − 3.35483i − 1.64412i 1.00000i 1.00000i −0.296871 −3.35483
155.8 − 1.00000i 1.69936 −1.00000 2.85623i − 1.69936i 1.00000i 1.00000i −0.112181 2.85623
155.9 − 1.00000i 3.00701 −1.00000 − 1.33441i − 3.00701i 1.00000i 1.00000i 6.04209 −1.33441
155.10 1.00000i −3.15056 −1.00000 − 3.29231i − 3.15056i − 1.00000i − 1.00000i 6.92603 3.29231
155.11 1.00000i −2.32899 −1.00000 3.88035i − 2.32899i − 1.00000i − 1.00000i 2.42421 −3.88035
155.12 1.00000i −1.29698 −1.00000 0.485139i − 1.29698i − 1.00000i − 1.00000i −1.31785 −0.485139
155.13 1.00000i −0.886255 −1.00000 − 1.53631i − 0.886255i − 1.00000i − 1.00000i −2.21455 1.53631
155.14 1.00000i −0.344030 −1.00000 1.81148i − 0.344030i − 1.00000i − 1.00000i −2.88164 −1.81148
155.15 1.00000i 0.656332 −1.00000 − 0.181366i 0.656332i − 1.00000i − 1.00000i −2.56923 0.181366
155.16 1.00000i 1.64412 −1.00000 3.35483i 1.64412i − 1.00000i − 1.00000i −0.296871 −3.35483
155.17 1.00000i 1.69936 −1.00000 − 2.85623i 1.69936i − 1.00000i − 1.00000i −0.112181 2.85623
155.18 1.00000i 3.00701 −1.00000 1.33441i 3.00701i − 1.00000i − 1.00000i 6.04209 −1.33441
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2002.2.g.b ✓ 18
13.b even 2 1 inner 2002.2.g.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2002.2.g.b ✓ 18 1.a even 1 1 trivial
2002.2.g.b ✓ 18 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} + T_{3}^{8} - 16T_{3}^{7} - 13T_{3}^{6} + 74T_{3}^{5} + 45T_{3}^{4} - 113T_{3}^{3} - 60T_{3}^{2} + 40T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(2002, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$3$ \( (T^{9} + T^{8} - 16 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + 53 T^{16} + \cdots + 1600 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10604499373 \) Copy content Toggle raw display
$17$ \( (T^{9} + 5 T^{8} + \cdots + 143680)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 538240000 \) Copy content Toggle raw display
$23$ \( (T^{9} + 4 T^{8} + \cdots + 1536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} - 8 T^{8} + \cdots + 1814752)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 351260099584 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 143233024 \) Copy content Toggle raw display
$41$ \( T^{18} + 288 T^{16} + \cdots + 5760000 \) Copy content Toggle raw display
$43$ \( (T^{9} + 15 T^{8} + \cdots + 2120)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 19182157505536 \) Copy content Toggle raw display
$53$ \( (T^{9} - 19 T^{8} + \cdots + 424)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 67731583206400 \) Copy content Toggle raw display
$61$ \( (T^{9} - 7 T^{8} + \cdots - 450000)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 135768358273024 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 1071937198336 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 1399507928064 \) Copy content Toggle raw display
$79$ \( (T^{9} - 13 T^{8} + \cdots + 153128)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 412595536896 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 491221560384 \) Copy content Toggle raw display
$97$ \( T^{18} + 444 T^{16} + \cdots + 73547776 \) Copy content Toggle raw display
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