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Properties

Label 1098.2.k.g
Level $1098$
Weight $2$
Character orbit 1098.k
Analytic conductor $8.768$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1098,2,Mod(217,1098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1098, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1098.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1098.k (of order \(5\), degree \(4\), 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: \(8.76757414194\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.9048765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 13x^{6} - 25x^{5} + 91x^{4} + 10x^{3} + 152x^{2} + 528x + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 366)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{2} + \beta_{3} q^{4} + \beta_1 q^{5} + (\beta_{5} + 1) q^{7} - \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{2} + \beta_{3} q^{4} + \beta_1 q^{5} + (\beta_{5} + 1) q^{7} - \beta_{5} q^{8} - \beta_{7} q^{10} + (\beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots - 2) q^{11}+ \cdots + (\beta_{5} + 6 \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{4} + q^{5} + 6 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 2 q^{4} + q^{5} + 6 q^{7} + 2 q^{8} - q^{10} - 4 q^{11} - 14 q^{13} + 4 q^{14} - 2 q^{16} - 2 q^{17} - q^{19} + q^{20} - 11 q^{22} - 3 q^{23} - 15 q^{25} - 16 q^{26} + 6 q^{28} - 12 q^{29} + 5 q^{31} - 8 q^{32} + 2 q^{34} - 3 q^{35} + 12 q^{37} - 9 q^{38} + 4 q^{40} - q^{41} + 9 q^{43} - 9 q^{44} + 3 q^{46} + 2 q^{47} + 16 q^{49} - 20 q^{50} - 9 q^{52} + 29 q^{53} + 9 q^{55} + 4 q^{56} - 23 q^{58} - 3 q^{59} - 15 q^{61} + 30 q^{62} - 2 q^{64} - 29 q^{65} - 16 q^{67} - 2 q^{68} - 2 q^{70} + 14 q^{71} - 5 q^{73} + 28 q^{74} + 9 q^{76} - 13 q^{77} - 19 q^{79} + q^{80} - 4 q^{82} + 16 q^{83} - 48 q^{85} + 16 q^{86} + 9 q^{88} + 3 q^{89} - 23 q^{91} - 3 q^{92} + 13 q^{94} + 27 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 13x^{6} - 25x^{5} + 91x^{4} + 10x^{3} + 152x^{2} + 528x + 1936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4813 \nu^{7} + 1185893 \nu^{6} - 5916019 \nu^{5} + 23932495 \nu^{4} - 73512421 \nu^{3} + \cdots + 276643708 ) / 2322038444 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1070517 \nu^{7} + 5415963 \nu^{6} + 15713109 \nu^{5} - 25342913 \nu^{4} + 49673483 \nu^{3} + \cdots - 2026019160 ) / 4644076888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 54123 \nu^{7} - 271754 \nu^{6} + 1093310 \nu^{5} - 3361382 \nu^{4} + 22904346 \nu^{3} + \cdots - 423544 ) / 105547202 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1607181 \nu^{7} + 3108582 \nu^{6} - 18103060 \nu^{5} + 83733376 \nu^{4} - 132768098 \nu^{3} + \cdots - 973977092 ) / 2322038444 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 244737 \nu^{7} + 289845 \nu^{6} - 6146061 \nu^{5} + 5496821 \nu^{4} - 53365131 \nu^{3} + \cdots - 282016768 ) / 211094404 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 539577 \nu^{7} + 208191 \nu^{6} - 6210607 \nu^{5} + 7666983 \nu^{4} - 37407101 \nu^{3} + \cdots - 187811272 ) / 211094404 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - 2\beta_{5} + 6\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + 6\beta_{5} + 9\beta_{4} + 8\beta_{3} + 6\beta_{2} - 2\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -17\beta_{7} + 4\beta_{6} + 46\beta_{5} - 30\beta_{3} - 30\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34\beta_{7} - 93\beta_{6} - 58\beta_{5} - 93\beta_{4} - 58\beta_{3} + 34\beta _1 - 144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 92\beta_{7} + 92\beta_{4} + 812\beta_{3} + 390\beta_{2} - 145\beta _1 + 390 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1049\beta_{7} + 1049\beta_{6} + 1026\beta_{5} + 482\beta_{4} - 1054\beta_{2} - 567\beta _1 + 1026 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times\).

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
−1.10626 + 3.40473i
0.797247 − 2.45367i
−1.10626 − 3.40473i
0.797247 + 2.45367i
−1.40703 + 1.02227i
2.21604 − 1.61005i
−1.40703 − 1.02227i
2.21604 + 1.61005i
0.809017 − 0.587785i 0 0.309017 − 0.951057i −1.10626 + 3.40473i 0 1.30902 + 0.951057i −0.309017 − 0.951057i 0 1.10626 + 3.40473i
217.2 0.809017 − 0.587785i 0 0.309017 − 0.951057i 0.797247 − 2.45367i 0 1.30902 + 0.951057i −0.309017 − 0.951057i 0 −0.797247 − 2.45367i
253.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i −1.10626 − 3.40473i 0 1.30902 − 0.951057i −0.309017 + 0.951057i 0 1.10626 − 3.40473i
253.2 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0.797247 + 2.45367i 0 1.30902 − 0.951057i −0.309017 + 0.951057i 0 −0.797247 + 2.45367i
325.1 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i −1.40703 + 1.02227i 0 0.190983 − 0.587785i 0.809017 + 0.587785i 0 1.40703 + 1.02227i
325.2 −0.309017 − 0.951057i 0 −0.809017 + 0.587785i 2.21604 − 1.61005i 0 0.190983 − 0.587785i 0.809017 + 0.587785i 0 −2.21604 − 1.61005i
973.1 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −1.40703 − 1.02227i 0 0.190983 + 0.587785i 0.809017 − 0.587785i 0 1.40703 − 1.02227i
973.2 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 2.21604 + 1.61005i 0 0.190983 + 0.587785i 0.809017 − 0.587785i 0 −2.21604 + 1.61005i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1098.2.k.g 8
3.b odd 2 1 366.2.h.b ✓ 8
61.e even 5 1 inner 1098.2.k.g 8
183.n odd 10 1 366.2.h.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
366.2.h.b ✓ 8 3.b odd 2 1
366.2.h.b ✓ 8 183.n odd 10 1
1098.2.k.g 8 1.a even 1 1 trivial
1098.2.k.g 8 61.e even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{8} - T_{5}^{7} + 13T_{5}^{6} - 25T_{5}^{5} + 91T_{5}^{4} + 10T_{5}^{3} + 152T_{5}^{2} + 528T_{5} + 1936 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} + 4T_{7}^{2} - 2T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 23 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 7 T^{3} + \cdots - 205)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 495616 \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} + \cdots + 1681 \) Copy content Toggle raw display
$23$ \( T^{8} + 3 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 5 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( T^{8} - 9 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( (T^{4} - T^{3} - 51 T^{2} + \cdots - 220)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 29 T^{7} + \cdots + 774400 \) Copy content Toggle raw display
$59$ \( T^{8} + 3 T^{7} + \cdots + 20647936 \) Copy content Toggle raw display
$61$ \( T^{8} + 15 T^{7} + \cdots + 13845841 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + \cdots + 15241216 \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$73$ \( T^{8} + 5 T^{7} + \cdots + 25816561 \) Copy content Toggle raw display
$79$ \( T^{8} + 19 T^{7} + \cdots + 3168400 \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{8} - 3 T^{7} + \cdots + 4393216 \) Copy content Toggle raw display
$97$ \( T^{8} - 27 T^{7} + \cdots + 57214096 \) Copy content Toggle raw display
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