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Properties

Label 882.2.u.i
Level $882$
Weight $2$
Character orbit 882.u
Analytic conductor $7.043$
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] = [882,2,Mod(127,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.u (of order \(7\), degree \(6\), 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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} + 72 x^{16} - 372 x^{15} + 1666 x^{14} - 5866 x^{13} + 17998 x^{12} - 46004 x^{11} + \cdots + 6056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} - \beta_{7} q^{4} + ( - \beta_{14} + \beta_{3}) q^{5} + (\beta_{15} - \beta_{12} + \cdots - \beta_{2}) q^{7}+ \cdots + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} - \beta_{7} q^{4} + ( - \beta_{14} + \beta_{3}) q^{5} + (\beta_{15} - \beta_{12} + \cdots - \beta_{2}) q^{7}+ \cdots + (2 \beta_{16} + 2 \beta_{15} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} - 3 q^{4} + 4 q^{5} + 7 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{2} - 3 q^{4} + 4 q^{5} + 7 q^{7} + 3 q^{8} - 4 q^{10} + 12 q^{11} - 12 q^{13} - 7 q^{14} - 3 q^{16} - 6 q^{17} - 12 q^{19} + 4 q^{20} - 5 q^{22} - 18 q^{23} + 9 q^{25} - 16 q^{26} - 7 q^{28} - 8 q^{29} + 14 q^{31} + 3 q^{32} + 6 q^{34} + 2 q^{37} + 12 q^{38} + 3 q^{40} - 16 q^{43} - 16 q^{44} - 10 q^{46} + 58 q^{47} - 35 q^{49} + 12 q^{50} - 12 q^{52} - 16 q^{53} + 5 q^{55} + q^{58} - 60 q^{59} - 4 q^{61} - 3 q^{64} + 18 q^{65} - 20 q^{67} + 8 q^{68} - 7 q^{70} - 14 q^{71} + 44 q^{73} + 12 q^{74} + 2 q^{76} + 35 q^{77} + 34 q^{79} - 10 q^{80} + 19 q^{83} - 12 q^{85} - 26 q^{86} + 16 q^{88} - 50 q^{89} - 14 q^{91} + 10 q^{92} + 12 q^{94} + 18 q^{95} + 134 q^{97} + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 9 x^{17} + 72 x^{16} - 372 x^{15} + 1666 x^{14} - 5866 x^{13} + 17998 x^{12} - 46004 x^{11} + \cdots + 6056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2562431 \nu^{16} - 20499448 \nu^{15} + 138503684 \nu^{14} - 610785448 \nu^{13} + \cdots + 9694671900 ) / 1154464364 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10915833 \nu^{16} - 87326664 \nu^{15} + 680167904 \nu^{14} - 3232958708 \nu^{13} + \cdots + 48449689624 ) / 1154464364 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6821995613 \nu^{17} + 127328155612 \nu^{16} - 1025588948238 \nu^{15} + \cdots + 242009542038840 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6821995613 \nu^{17} + 11354230191 \nu^{16} - 83870138186 \nu^{15} + 1911853246130 \nu^{14} + \cdots + 161873219220672 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9575228187 \nu^{17} - 207623923757 \nu^{16} + 1630322482948 \nu^{15} + \cdots - 238379649930820 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9575228187 \nu^{17} - 44845044578 \nu^{16} + 389429263732 \nu^{15} + \cdots - 240062215977732 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21959091724 \nu^{17} - 214261080159 \nu^{16} + 1657596789370 \nu^{15} + \cdots - 66513292550364 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21959091724 \nu^{17} - 159043479149 \nu^{16} + 1215855981290 \nu^{15} + \cdots + 51133979839084 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38985364927 \nu^{17} + 373107994276 \nu^{16} - 2906640395184 \nu^{15} + \cdots + 168945687726140 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 57173821228 \nu^{17} + 407047409323 \nu^{16} - 3128292038878 \nu^{15} + \cdots - 120223624369680 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 39109228304 \nu^{17} + 359148517625 \nu^{16} - 2805571365561 \nu^{15} + \cdots + 125128507585342 ) / 2764364919598 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 94943313275 \nu^{17} - 830662348986 \nu^{16} + 6425224910026 \nu^{15} + \cdots - 257173365376908 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 94943313275 \nu^{17} + 783373976689 \nu^{16} - 6046917931650 \nu^{15} + \cdots - 27104287558960 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 61469514303 \nu^{17} - 585281737323 \nu^{16} + 4525557132983 \nu^{15} + \cdots - 271225090762862 ) / 2764364919598 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 61469514303 \nu^{17} - 459700005828 \nu^{16} + 3520903281023 \nu^{15} + \cdots + 93248487801010 ) / 2764364919598 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 139712472639 \nu^{17} + 1213693979550 \nu^{16} - 9330170050364 \nu^{15} + \cdots + 251264603124444 ) / 5528729839196 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 186934845447 \nu^{17} + 1671840871771 \nu^{16} - 12940209910602 \nu^{15} + \cdots + 486374840410844 ) / 5528729839196 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} - 4 \beta_{14} - 5 \beta_{13} + 5 \beta_{12} + 2 \beta_{11} + \cdots + 8 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{17} + 2 \beta_{16} + 5 \beta_{15} - 11 \beta_{14} - 12 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + \cdots - 27 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{17} - 22 \beta_{16} + 15 \beta_{15} - 5 \beta_{14} + 27 \beta_{13} - 48 \beta_{12} - 22 \beta_{11} + \cdots - 81 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 17 \beta_{17} - 46 \beta_{16} - 59 \beta_{15} + 64 \beta_{14} + 129 \beta_{13} - 31 \beta_{12} + \cdots + 110 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 29 \beta_{17} + 211 \beta_{16} - 176 \beta_{15} + 180 \beta_{14} - 118 \beta_{13} + 398 \beta_{12} + \cdots + 746 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45 \beta_{17} + 107 \beta_{16} + 86 \beta_{15} - 35 \beta_{14} - 183 \beta_{13} + 96 \beta_{12} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 859 \beta_{17} - 1594 \beta_{16} + 2406 \beta_{15} - 2174 \beta_{14} - 19 \beta_{13} - 3117 \beta_{12} + \cdots - 6761 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3089 \beta_{17} - 9980 \beta_{16} - 4902 \beta_{15} - 837 \beta_{14} + 12441 \beta_{13} - 9445 \beta_{12} + \cdots - 4605 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 14398 \beta_{17} + 7160 \beta_{16} - 31037 \beta_{15} + 21051 \beta_{14} + 11570 \beta_{13} + \cdots + 59573 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 19688 \beta_{17} + 116126 \beta_{16} + 26121 \beta_{15} + 35680 \beta_{14} - 115063 \beta_{13} + \cdots + 100637 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 188593 \beta_{17} + 41448 \beta_{16} + 365511 \beta_{15} - 175240 \beta_{14} - 227079 \beta_{13} + \cdots - 494259 ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1717 \beta_{17} - 172945 \beta_{16} + 9264 \beta_{15} - 82044 \beta_{14} + 139202 \beta_{13} + \cdots - 211116 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2117887 \beta_{17} - 1674699 \beta_{16} - 3915830 \beta_{15} + 1195141 \beta_{14} + 3307215 \beta_{13} + \cdots + 3630446 ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 2153209 \beta_{17} + 11304918 \beta_{16} - 4555294 \beta_{15} + 7324082 \beta_{14} - 7018303 \beta_{13} + \cdots + 18662589 ) / 7 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 20927124 \beta_{17} + 29356802 \beta_{16} + 37858486 \beta_{15} - 4839756 \beta_{14} - 41720547 \beta_{13} + \cdots - 19788122 ) / 7 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 45529334 \beta_{17} - 91586562 \beta_{16} + 86858615 \beta_{15} - 82395641 \beta_{14} + 33350054 \beta_{13} + \cdots - 213988637 ) / 7 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 179603551 \beta_{17} - 406318884 \beta_{16} - 321324081 \beta_{15} - 33501463 \beta_{14} + \cdots + 689077 ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 3.23359i
0.500000 − 2.54694i
0.500000 + 0.636226i
0.500000 + 2.41526i
0.500000 − 1.39348i
0.500000 + 0.301097i
0.500000 − 1.45872i
0.500000 − 1.72081i
0.500000 + 1.85665i
0.500000 + 1.45872i
0.500000 + 1.72081i
0.500000 − 1.85665i
0.500000 − 2.41526i
0.500000 + 1.39348i
0.500000 − 0.301097i
0.500000 − 3.23359i
0.500000 + 2.54694i
0.500000 − 0.636226i
0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.305593 − 1.33889i 0 −2.33987 − 1.23491i 0.222521 + 0.974928i 0 0.305593 − 1.33889i
127.2 0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.0335766 − 0.147109i 0 2.17954 − 1.49987i 0.222521 + 0.974928i 0 0.0335766 − 0.147109i
127.3 0.900969 + 0.433884i 0 0.623490 + 0.781831i 0.383243 + 1.67910i 0 −0.309166 + 2.62763i 0.222521 + 0.974928i 0 −0.383243 + 1.67910i
253.1 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i −0.885662 + 0.426512i 0 −1.74345 − 1.99007i 0.900969 − 0.433884i 0 0.885662 + 0.426512i
253.2 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i 1.36925 − 0.659398i 0 1.96981 − 1.76631i 0.900969 − 0.433884i 0 −1.36925 − 0.659398i
253.3 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i 2.44184 − 1.17593i 0 0.317713 + 2.62661i 0.900969 − 0.433884i 0 −2.44184 − 1.17593i
379.1 0.222521 + 0.974928i 0 −0.900969 + 0.433884i −2.58198 − 3.23770i 0 1.58340 + 2.11963i −0.623490 − 0.781831i 0 2.58198 − 3.23770i
379.2 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 0.0413468 + 0.0518472i 0 1.69712 − 2.02973i −0.623490 − 0.781831i 0 −0.0413468 + 0.0518472i
379.3 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 1.57113 + 1.97014i 0 0.144913 + 2.64178i −0.623490 − 0.781831i 0 −1.57113 + 1.97014i
505.1 0.222521 − 0.974928i 0 −0.900969 − 0.433884i −2.58198 + 3.23770i 0 1.58340 − 2.11963i −0.623490 + 0.781831i 0 2.58198 + 3.23770i
505.2 0.222521 − 0.974928i 0 −0.900969 − 0.433884i 0.0413468 − 0.0518472i 0 1.69712 + 2.02973i −0.623490 + 0.781831i 0 −0.0413468 − 0.0518472i
505.3 0.222521 − 0.974928i 0 −0.900969 − 0.433884i 1.57113 − 1.97014i 0 0.144913 − 2.64178i −0.623490 + 0.781831i 0 −1.57113 − 1.97014i
631.1 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i −0.885662 − 0.426512i 0 −1.74345 + 1.99007i 0.900969 + 0.433884i 0 0.885662 − 0.426512i
631.2 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i 1.36925 + 0.659398i 0 1.96981 + 1.76631i 0.900969 + 0.433884i 0 −1.36925 + 0.659398i
631.3 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i 2.44184 + 1.17593i 0 0.317713 − 2.62661i 0.900969 + 0.433884i 0 −2.44184 + 1.17593i
757.1 0.900969 − 0.433884i 0 0.623490 − 0.781831i −0.305593 + 1.33889i 0 −2.33987 + 1.23491i 0.222521 − 0.974928i 0 0.305593 + 1.33889i
757.2 0.900969 − 0.433884i 0 0.623490 − 0.781831i −0.0335766 + 0.147109i 0 2.17954 + 1.49987i 0.222521 − 0.974928i 0 0.0335766 + 0.147109i
757.3 0.900969 − 0.433884i 0 0.623490 − 0.781831i 0.383243 − 1.67910i 0 −0.309166 − 2.62763i 0.222521 − 0.974928i 0 −0.383243 − 1.67910i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.u.i yes 18
3.b odd 2 1 882.2.u.f ✓ 18
49.e even 7 1 inner 882.2.u.i yes 18
147.l odd 14 1 882.2.u.f ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.u.f ✓ 18 3.b odd 2 1
882.2.u.f ✓ 18 147.l odd 14 1
882.2.u.i yes 18 1.a even 1 1 trivial
882.2.u.i yes 18 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 4 T_{5}^{17} + 11 T_{5}^{16} - 58 T_{5}^{15} + 334 T_{5}^{14} - 1126 T_{5}^{13} + 2475 T_{5}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 4 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{18} - 7 T^{17} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{18} - 12 T^{17} + \cdots + 12769 \) Copy content Toggle raw display
$13$ \( T^{18} + 12 T^{17} + \cdots + 5494336 \) Copy content Toggle raw display
$17$ \( T^{18} + 6 T^{17} + \cdots + 2483776 \) Copy content Toggle raw display
$19$ \( (T^{9} + 6 T^{8} + \cdots - 10984)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 18 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 122921569 \) Copy content Toggle raw display
$31$ \( (T^{9} - 7 T^{8} + \cdots + 4459)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 930113651776 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 96144645184 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 443692096 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 323424256 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 997043776 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 654519832576 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 37070111296 \) Copy content Toggle raw display
$67$ \( (T^{9} + 10 T^{8} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 18368141641249 \) Copy content Toggle raw display
$79$ \( (T^{9} - 17 T^{8} + \cdots + 6627461)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 19\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 4898880064 \) Copy content Toggle raw display
$97$ \( (T^{9} - 67 T^{8} + \cdots + 352790143)^{2} \) Copy content Toggle raw display
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