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Properties

Label 882.2.u.h
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: \( x^{18} - 3 x^{17} + 9 x^{16} - 9 x^{15} + 22 x^{14} - 268 x^{13} + 2427 x^{12} - 9710 x^{11} + \cdots + 3182656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 294)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{7} q^{4} + (\beta_{16} - \beta_{11} + \beta_{9}) q^{5} + \beta_{6} q^{7} + (\beta_{12} - \beta_{11} - \beta_{10} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{7} q^{4} + (\beta_{16} - \beta_{11} + \beta_{9}) q^{5} + \beta_{6} q^{7} + (\beta_{12} - \beta_{11} - \beta_{10} + \cdots + 1) q^{8}+ \cdots + (\beta_{17} + 2 \beta_{16} + \beta_{15} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} - 3 q^{4} - 3 q^{5} + q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{2} - 3 q^{4} - 3 q^{5} + q^{7} + 3 q^{8} + 3 q^{10} + 4 q^{11} - 5 q^{13} - q^{14} - 3 q^{16} + 7 q^{17} - 28 q^{19} + 4 q^{20} + 3 q^{22} - 9 q^{23} + 6 q^{25} - 2 q^{26} - 6 q^{28} + 5 q^{29} + 4 q^{31} + 3 q^{32} + 7 q^{34} + 2 q^{35} + 3 q^{37} + 3 q^{40} - 40 q^{41} - 15 q^{43} + 11 q^{44} - 5 q^{46} + 22 q^{47} - 43 q^{49} - 34 q^{50} + 9 q^{52} + 38 q^{53} - 63 q^{55} + 6 q^{56} - 26 q^{58} - 24 q^{59} + 22 q^{61} + 17 q^{62} - 3 q^{64} - 45 q^{65} + 106 q^{67} + 28 q^{68} + 19 q^{70} + 4 q^{71} - 23 q^{73} - 10 q^{74} + 14 q^{76} - 17 q^{77} - 10 q^{79} + 4 q^{80} + 40 q^{82} - q^{83} - 37 q^{85} + 22 q^{86} - 11 q^{88} - 18 q^{89} - q^{91} + 5 q^{92} + 6 q^{94} - 39 q^{95} - 56 q^{97} + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 9 x^{16} - 9 x^{15} + 22 x^{14} - 268 x^{13} + 2427 x^{12} - 9710 x^{11} + \cdots + 3182656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 91\!\cdots\!64 \nu^{17} + \cdots - 17\!\cdots\!44 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!86 \nu^{17} + \cdots + 70\!\cdots\!16 ) / 93\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!43 \nu^{17} + \cdots + 30\!\cdots\!96 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!45 \nu^{17} + \cdots + 43\!\cdots\!28 ) / 37\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!05 \nu^{17} + \cdots - 32\!\cdots\!72 ) / 37\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!65 \nu^{17} + \cdots - 15\!\cdots\!24 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28\!\cdots\!11 \nu^{17} + \cdots - 35\!\cdots\!12 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{17} + \cdots + 36\!\cdots\!64 ) / 13\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74\!\cdots\!07 \nu^{17} + \cdots - 28\!\cdots\!92 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!68 \nu^{17} + \cdots - 71\!\cdots\!32 ) / 13\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 98\!\cdots\!41 \nu^{17} + \cdots - 40\!\cdots\!16 ) / 53\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!49 \nu^{17} + \cdots + 32\!\cdots\!96 ) / 59\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49\!\cdots\!53 \nu^{17} + \cdots + 39\!\cdots\!24 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66\!\cdots\!29 \nu^{17} + \cdots - 16\!\cdots\!96 ) / 18\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 45\!\cdots\!35 \nu^{17} + \cdots - 92\!\cdots\!56 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 18\!\cdots\!19 \nu^{17} + \cdots - 29\!\cdots\!80 ) / 37\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 2 \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + 2 \beta_{8} + \cdots - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + \beta_{15} - 3 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - 6 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} + 20 \beta_{13} - 9 \beta_{12} + 46 \beta_{11} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{17} + 64 \beta_{16} - 10 \beta_{15} - 6 \beta_{14} - 126 \beta_{13} + 45 \beta_{12} + \cdots + 122 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{17} - 310 \beta_{16} - 145 \beta_{15} + 165 \beta_{14} + 265 \beta_{13} - 486 \beta_{12} + \cdots - 1164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 343 \beta_{17} + 1899 \beta_{16} + 280 \beta_{15} + 184 \beta_{14} - 663 \beta_{13} + 343 \beta_{12} + \cdots + 2402 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2662 \beta_{17} - 6356 \beta_{16} - 1906 \beta_{15} - 1158 \beta_{14} + 1910 \beta_{13} + 2884 \beta_{12} + \cdots - 10650 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1398 \beta_{17} + 9825 \beta_{16} + 699 \beta_{15} - 2063 \beta_{14} + 8427 \beta_{13} - 13708 \beta_{12} + \cdots + 21508 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 36791 \beta_{17} - 12054 \beta_{16} - 10079 \beta_{15} - 12054 \beta_{14} - 59603 \beta_{13} + \cdots - 14029 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 81409 \beta_{17} - 113916 \beta_{16} - 4531 \beta_{15} + 76878 \beta_{14} + 355366 \beta_{13} + \cdots - 353966 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 54565 \beta_{17} + 1150424 \beta_{16} + 336096 \beta_{15} - 190198 \beta_{14} - 1396864 \beta_{13} + \cdots + 2638314 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 548766 \beta_{17} - 6641528 \beta_{16} - 1137805 \beta_{15} + 40273 \beta_{14} + 5231575 \beta_{13} + \cdots - 11750980 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 4115181 \beta_{17} + 25522789 \beta_{16} + 5992476 \beta_{15} - 1762014 \beta_{14} - 13376155 \beta_{13} + \cdots + 54341162 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 22548258 \beta_{17} - 85613724 \beta_{16} - 19159502 \beta_{15} + 282754 \beta_{14} + 11132752 \beta_{13} + \cdots - 158535934 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 139250194 \beta_{17} + 234978497 \beta_{16} + 69625097 \beta_{15} + 49146377 \beta_{14} + \cdots + 351299812 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 379140831 \beta_{17} - 183436484 \beta_{16} - 63526147 \beta_{15} - 183436484 \beta_{14} + \cdots - 303346821 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\beta_{10}\) \(1\)

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
0.724029 − 3.17218i
0.388817 − 1.70352i
−0.612847 + 2.68506i
2.89690 + 1.39507i
1.19656 + 0.576234i
−3.59346 − 1.73052i
1.35577 − 1.70008i
0.746429 − 0.935993i
−1.60220 + 2.00909i
1.35577 + 1.70008i
0.746429 + 0.935993i
−1.60220 − 2.00909i
2.89690 − 1.39507i
1.19656 − 0.576234i
−3.59346 + 1.73052i
0.724029 + 3.17218i
0.388817 + 1.70352i
−0.612847 − 2.68506i
0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.724029 − 3.17218i 0 1.12772 + 2.39338i 0.222521 + 0.974928i 0 0.724029 − 3.17218i
127.2 0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.388817 − 1.70352i 0 0.188473 − 2.63903i 0.222521 + 0.974928i 0 0.388817 − 1.70352i
127.3 0.900969 + 0.433884i 0 0.623490 + 0.781831i 0.612847 + 2.68506i 0 −2.61813 − 0.381328i 0.222521 + 0.974928i 0 −0.612847 + 2.68506i
253.1 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i −2.89690 + 1.39507i 0 −0.0919846 + 2.64415i 0.900969 − 0.433884i 0 2.89690 + 1.39507i
253.2 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i −1.19656 + 0.576234i 0 0.327963 − 2.62535i 0.900969 − 0.433884i 0 1.19656 + 0.576234i
253.3 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i 3.59346 − 1.73052i 0 1.51100 + 2.17184i 0.900969 − 0.433884i 0 −3.59346 − 1.73052i
379.1 0.222521 + 0.974928i 0 −0.900969 + 0.433884i −1.35577 − 1.70008i 0 1.73662 + 1.99603i −0.623490 − 0.781831i 0 1.35577 − 1.70008i
379.2 0.222521 + 0.974928i 0 −0.900969 + 0.433884i −0.746429 − 0.935993i 0 0.856101 − 2.50342i −0.623490 − 0.781831i 0 0.746429 − 0.935993i
379.3 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 1.60220 + 2.00909i 0 −2.53776 + 0.748170i −0.623490 − 0.781831i 0 −1.60220 + 2.00909i
505.1 0.222521 − 0.974928i 0 −0.900969 − 0.433884i −1.35577 + 1.70008i 0 1.73662 − 1.99603i −0.623490 + 0.781831i 0 1.35577 + 1.70008i
505.2 0.222521 − 0.974928i 0 −0.900969 − 0.433884i −0.746429 + 0.935993i 0 0.856101 + 2.50342i −0.623490 + 0.781831i 0 0.746429 + 0.935993i
505.3 0.222521 − 0.974928i 0 −0.900969 − 0.433884i 1.60220 − 2.00909i 0 −2.53776 − 0.748170i −0.623490 + 0.781831i 0 −1.60220 − 2.00909i
631.1 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i −2.89690 − 1.39507i 0 −0.0919846 − 2.64415i 0.900969 + 0.433884i 0 2.89690 − 1.39507i
631.2 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i −1.19656 − 0.576234i 0 0.327963 + 2.62535i 0.900969 + 0.433884i 0 1.19656 − 0.576234i
631.3 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i 3.59346 + 1.73052i 0 1.51100 − 2.17184i 0.900969 + 0.433884i 0 −3.59346 + 1.73052i
757.1 0.900969 − 0.433884i 0 0.623490 − 0.781831i −0.724029 + 3.17218i 0 1.12772 − 2.39338i 0.222521 − 0.974928i 0 0.724029 + 3.17218i
757.2 0.900969 − 0.433884i 0 0.623490 − 0.781831i −0.388817 + 1.70352i 0 0.188473 + 2.63903i 0.222521 − 0.974928i 0 0.388817 + 1.70352i
757.3 0.900969 − 0.433884i 0 0.623490 − 0.781831i 0.612847 − 2.68506i 0 −2.61813 + 0.381328i 0.222521 − 0.974928i 0 −0.612847 − 2.68506i
\(n\): 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.h 18
3.b odd 2 1 294.2.i.d ✓ 18
49.e even 7 1 inner 882.2.u.h 18
147.l odd 14 1 294.2.i.d ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.i.d ✓ 18 3.b odd 2 1
294.2.i.d ✓ 18 147.l odd 14 1
882.2.u.h 18 1.a even 1 1 trivial
882.2.u.h 18 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 3 T_{5}^{17} + 9 T_{5}^{16} + 9 T_{5}^{15} + 22 T_{5}^{14} + 268 T_{5}^{13} + \cdots + 3182656 \) 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
$5$ \( T^{18} + 3 T^{17} + \cdots + 3182656 \) Copy content Toggle raw display
$7$ \( T^{18} - T^{17} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 815216704 \) Copy content Toggle raw display
$13$ \( T^{18} + 5 T^{17} + \cdots + 9834496 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 1855197184 \) Copy content Toggle raw display
$19$ \( (T^{9} + 14 T^{8} + \cdots + 51176)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 484704256 \) Copy content Toggle raw display
$29$ \( T^{18} - 5 T^{17} + \cdots + 5053504 \) Copy content Toggle raw display
$31$ \( (T^{9} - 2 T^{8} + \cdots - 1683829)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} - 3 T^{17} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{18} + 40 T^{17} + \cdots + 9834496 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 75235361564224 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 163617253359616 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1150419275776 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 4228120576 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 758111524416 \) Copy content Toggle raw display
$67$ \( (T^{9} - 53 T^{8} + \cdots + 4846528)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 5094446239744 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 925434138060601 \) Copy content Toggle raw display
$79$ \( (T^{9} + 5 T^{8} + \cdots + 2402771)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 132938993664 \) Copy content Toggle raw display
$97$ \( (T^{9} + 28 T^{8} + \cdots + 6050602369)^{2} \) Copy content Toggle raw display
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