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Properties

Label 891.2.d.b
Level $891$
Weight $2$
Character orbit 891.d
Analytic conductor $7.115$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(890,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.890");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.d (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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 99)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} + 2) q^{4} - \beta_{7} q^{5} + \beta_{6} q^{7} + ( - \beta_{5} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} + 2) q^{4} - \beta_{7} q^{5} + \beta_{6} q^{7} + ( - \beta_{5} - \beta_1) q^{8} + (\beta_{12} - \beta_{10}) q^{10} + ( - \beta_{13} + \beta_{9}) q^{11} - \beta_{10} q^{13} + (\beta_{11} + 3 \beta_{9} + \cdots + 2 \beta_{4}) q^{14}+ \cdots + (2 \beta_{15} - \beta_{13} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 28 q^{4} + 4 q^{16} - 12 q^{22} + 24 q^{25} + 8 q^{31} - 28 q^{37} - 20 q^{49} + 20 q^{55} + 12 q^{58} + 40 q^{64} - 16 q^{67} + 12 q^{82} + 12 q^{88} - 12 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2126395 \nu^{15} - 29523650 \nu^{13} - 288594750 \nu^{11} - 1489331264 \nu^{9} + \cdots - 9265241250 \nu ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2126395 \nu^{15} + 29523650 \nu^{13} + 288594750 \nu^{11} + 1489331264 \nu^{9} + \cdots + 35527461216 \nu ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 76525 \nu^{14} - 979500 \nu^{12} - 9369721 \nu^{10} - 43435590 \nu^{8} - 162472650 \nu^{6} + \cdots + 1101072567 ) / 423584193 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4252790 \nu^{14} - 59047300 \nu^{12} - 577189500 \nu^{10} - 2978662528 \nu^{8} + \cdots - 31661592483 ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10631975 \nu^{15} + 147618250 \nu^{13} + 1442973750 \nu^{11} + 7446656320 \nu^{9} + \cdots + 46326206250 \nu ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39997934 \nu^{15} - 420539398 \nu^{13} - 3462808572 \nu^{11} - 9232850001 \nu^{9} + \cdots + 461039893902 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37221750 \nu^{14} - 547049101 \nu^{12} - 5404599456 \nu^{10} - 29598749160 \nu^{8} + \cdots - 288885367575 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 443825 \nu^{14} + 5726971 \nu^{12} + 54341933 \nu^{10} + 251915070 \nu^{8} + 883763751 \nu^{6} + \cdots - 575177208 ) / 423584193 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66638462 \nu^{14} - 1077731743 \nu^{12} - 10858856544 \nu^{10} - 63069608934 \nu^{8} + \cdots - 595332875052 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 97658143 \nu^{15} + 1101746546 \nu^{13} + 9511770288 \nu^{11} + 32726649222 \nu^{9} + \cdots - 499515011694 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 72701927 \nu^{14} + 1184448644 \nu^{12} + 11940134553 \nu^{10} + 69899905212 \nu^{8} + \cdots + 654437481345 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 103243543 \nu^{15} + 1140072897 \nu^{13} + 9772062471 \nu^{11} + 31661080332 \nu^{9} + \cdots - 721903310019 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 55631891 \nu^{15} + 829143978 \nu^{13} + 8104912470 \nu^{11} + 43634691359 \nu^{9} + \cdots + 260205597450 \nu ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1637375 \nu^{14} - 21557623 \nu^{12} - 200480195 \nu^{10} - 929374050 \nu^{8} + \cdots + 967409211 ) / 423584193 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 73802611 \nu^{15} - 1088524229 \nu^{13} - 10640363835 \nu^{11} - 57054676113 \nu^{9} + \cdots - 341605444725 \nu ) / 13131109983 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{9} + \beta_{7} - 3\beta_{4} + \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{11} - 9\beta_{9} + \beta_{8} - 6\beta_{7} + 12\beta_{4} + 8\beta_{3} - 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + \beta_{13} + 3\beta_{12} + 6\beta_{10} + 18\beta_{6} + 8\beta_{5} - 30\beta_{2} + 29\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{14} - 13\beta_{8} - 54\beta_{3} + 125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13 \beta_{15} + 14 \beta_{13} - 39 \beta_{12} - 27 \beta_{10} - 135 \beta_{6} + 54 \beta_{5} + \cdots + 179 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 15\beta_{14} + 459\beta_{11} + 504\beta_{9} + 120\beta_{8} + 99\beta_{7} - 294\beta_{4} + 354\beta_{3} - 783 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -120\beta_{15} - 135\beta_{13} - 354\beta_{5} - 1137\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 150 \beta_{14} - 3138 \beta_{11} - 3588 \beta_{9} + 969 \beta_{8} - 231 \beta_{7} + 1584 \beta_{4} + \cdots - 5022 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 969 \beta_{15} + 1119 \beta_{13} + 2907 \beta_{12} + 231 \beta_{10} + 6726 \beta_{6} + 2319 \beta_{5} + \cdots + 7341 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -1269\beta_{14} - 7314\beta_{8} - 15267\beta_{3} + 32652 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7314 \beta_{15} + 8583 \beta_{13} - 21942 \beta_{12} + 630 \beta_{10} - 46431 \beta_{6} + \cdots + 47919 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 9852 \beta_{14} + 144288 \beta_{11} + 173844 \beta_{9} + 53106 \beta_{8} - 15030 \beta_{7} + \cdots - 214257 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -53106\beta_{15} - 62958\beta_{13} - 101034\beta_{5} - 315291\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
890.1
1.29716 + 2.24675i
1.29716 − 2.24675i
1.10617 + 1.91594i
1.10617 − 1.91594i
0.679041 − 1.17613i
0.679041 + 1.17613i
0.618600 − 1.07145i
0.618600 + 1.07145i
−0.618600 + 1.07145i
−0.618600 − 1.07145i
−0.679041 + 1.17613i
−0.679041 − 1.17613i
−1.10617 − 1.91594i
−1.10617 + 1.91594i
−1.29716 − 2.24675i
−1.29716 + 2.24675i
−2.59432 0 4.73051 − 0.158660i 0 − 3.36958i −7.08384 0 0.411616i
890.2 −2.59432 0 4.73051 0.158660i 0 3.36958i −7.08384 0 − 0.411616i
890.3 −2.21234 0 2.89443 − 2.94127i 0 1.98826i −1.97879 0 6.50707i
890.4 −2.21234 0 2.89443 2.94127i 0 − 1.98826i −1.97879 0 − 6.50707i
890.5 −1.35808 0 −0.155613 − 1.04055i 0 − 0.693931i 2.92750 0 1.41315i
890.6 −1.35808 0 −0.155613 1.04055i 0 0.693931i 2.92750 0 − 1.41315i
890.7 −1.23720 0 −0.469335 − 2.05938i 0 − 4.14864i 3.05506 0 2.54787i
890.8 −1.23720 0 −0.469335 2.05938i 0 4.14864i 3.05506 0 − 2.54787i
890.9 1.23720 0 −0.469335 − 2.05938i 0 4.14864i −3.05506 0 − 2.54787i
890.10 1.23720 0 −0.469335 2.05938i 0 − 4.14864i −3.05506 0 2.54787i
890.11 1.35808 0 −0.155613 − 1.04055i 0 0.693931i −2.92750 0 − 1.41315i
890.12 1.35808 0 −0.155613 1.04055i 0 − 0.693931i −2.92750 0 1.41315i
890.13 2.21234 0 2.89443 − 2.94127i 0 − 1.98826i 1.97879 0 − 6.50707i
890.14 2.21234 0 2.89443 2.94127i 0 1.98826i 1.97879 0 6.50707i
890.15 2.59432 0 4.73051 − 0.158660i 0 3.36958i 7.08384 0 − 0.411616i
890.16 2.59432 0 4.73051 0.158660i 0 − 3.36958i 7.08384 0 0.411616i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 890.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.d.b 16
3.b odd 2 1 inner 891.2.d.b 16
9.c even 3 1 99.2.g.b ✓ 16
9.c even 3 1 297.2.g.b 16
9.d odd 6 1 99.2.g.b ✓ 16
9.d odd 6 1 297.2.g.b 16
11.b odd 2 1 inner 891.2.d.b 16
33.d even 2 1 inner 891.2.d.b 16
99.g even 6 1 99.2.g.b ✓ 16
99.g even 6 1 297.2.g.b 16
99.h odd 6 1 99.2.g.b ✓ 16
99.h odd 6 1 297.2.g.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.g.b ✓ 16 9.c even 3 1
99.2.g.b ✓ 16 9.d odd 6 1
99.2.g.b ✓ 16 99.g even 6 1
99.2.g.b ✓ 16 99.h odd 6 1
297.2.g.b 16 9.c even 3 1
297.2.g.b 16 9.d odd 6 1
297.2.g.b 16 99.g even 6 1
297.2.g.b 16 99.h odd 6 1
891.2.d.b 16 1.a even 1 1 trivial
891.2.d.b 16 3.b odd 2 1 inner
891.2.d.b 16 11.b odd 2 1 inner
891.2.d.b 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 15T_{2}^{6} + 75T_{2}^{4} - 144T_{2}^{2} + 93 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 15 T^{6} + \cdots + 93)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 14 T^{6} + 51 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 33 T^{6} + \cdots + 372)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} + 51 T^{6} + \cdots + 372)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 93 T^{6} + \cdots + 214272)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 63 T^{6} + \cdots + 3348)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 86 T^{6} + \cdots + 119716)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 30 T^{6} + \cdots + 1488)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} - 15 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 7 T^{3} - 51 T^{2} + \cdots + 31)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 132 T^{6} + \cdots + 134292)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 183 T^{6} + \cdots + 232500)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 125 T^{6} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 146 T^{6} + \cdots + 105625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 182 T^{6} + \cdots + 265225)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 477 T^{6} + \cdots + 129054612)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots - 137)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 53 T^{6} + \cdots + 361)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 408 T^{6} + \cdots + 64686708)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 279 T^{6} + \cdots + 95232)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 294 T^{6} + \cdots + 12731700)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 308 T^{6} + \cdots + 1893376)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + \cdots + 751)^{4} \) Copy content Toggle raw display
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