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Properties

Label 1098.2.k.f
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: 8.0.2512515625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 14x^{6} - 22x^{5} + 61x^{4} - 50x^{3} + 160x^{2} + 400x + 400 \) 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{5} q^{4} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{5} + (\beta_{7} + \beta_{5} + 2 \beta_{3} - \beta_1) q^{7} - \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{5} q^{4} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{5} + (\beta_{7} + \beta_{5} + 2 \beta_{3} - \beta_1) q^{7} - \beta_{2} q^{8} + (\beta_{6} + \beta_{4} - \beta_{3} + \cdots - 1) q^{10}+ \cdots + ( - 2 \beta_{7} - 5 \beta_{5} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{5} - 7 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{5} - 7 q^{7} - 2 q^{8} - 2 q^{10} - 12 q^{11} + 6 q^{13} - 2 q^{14} - 2 q^{16} + 12 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} + 17 q^{23} + 10 q^{25} - 9 q^{26} - 7 q^{28} - 14 q^{29} - 11 q^{31} + 8 q^{32} + 12 q^{34} + q^{35} + 4 q^{37} - 16 q^{38} - 2 q^{40} - 27 q^{41} + 3 q^{43} + 3 q^{44} + 17 q^{46} + 2 q^{47} - 21 q^{49} + 10 q^{50} + 6 q^{52} - 20 q^{53} + 5 q^{55} + 8 q^{56} - 4 q^{58} + 13 q^{59} - 10 q^{61} + 44 q^{62} - 2 q^{64} - 27 q^{65} - 14 q^{67} - 3 q^{68} + 26 q^{70} + 12 q^{71} - 9 q^{73} + 34 q^{74} - 16 q^{76} - 9 q^{77} - 26 q^{79} + 3 q^{80} + 18 q^{82} + 4 q^{83} - q^{85} + 8 q^{86} + 3 q^{88} + 38 q^{89} - 11 q^{91} - 3 q^{92} + 7 q^{94} + 22 q^{95} - 24 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 14x^{6} - 22x^{5} + 61x^{4} - 50x^{3} + 160x^{2} + 400x + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2103 \nu^{7} + 14653 \nu^{6} - 305414 \nu^{5} + 1249622 \nu^{4} - 3081331 \nu^{3} + \cdots + 15494280 ) / 25309640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9501 \nu^{7} + 39121 \nu^{6} - 174803 \nu^{5} + 423889 \nu^{4} - 1067537 \nu^{3} + \cdots - 593850 ) / 6327410 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10618 \nu^{7} - 41789 \nu^{6} + 214867 \nu^{5} - 487976 \nu^{4} + 2104228 \nu^{3} - 1707805 \nu^{2} + \cdots + 3800400 ) / 6327410 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27386 \nu^{7} + 114695 \nu^{6} - 484345 \nu^{5} + 1207580 \nu^{4} - 2165920 \nu^{3} + \cdots - 11229820 ) / 12654820 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 49601 \nu^{7} + 46533 \nu^{6} - 433144 \nu^{5} - 5198 \nu^{4} - 1865321 \nu^{3} + \cdots - 18134600 ) / 12654820 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 81159 \nu^{7} + 299214 \nu^{6} - 1519167 \nu^{5} + 2678906 \nu^{4} - 7286973 \nu^{3} + \cdots - 29785020 ) / 12654820 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + \beta_{4} + 4\beta_{3} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} + 7\beta_{4} + 2\beta_{3} + 2\beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} + 3\beta_{6} + 36\beta_{5} - 18\beta_{3} + 18\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -54\beta_{7} + 11\beta_{6} + 100\beta_{5} - 54\beta_{4} - 46\beta_{2} + 11\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -89\beta_{6} - 89\beta_{4} + 152\beta_{3} - 346\beta_{2} - 40\beta _1 + 152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 435\beta_{7} - 435\beta_{6} - 889\beta_{5} + 112\beta_{4} + 18\beta_{3} - 323\beta _1 - 454 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(245\) \(307\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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
0.932212 + 2.86905i
−0.741229 − 2.28127i
0.932212 − 2.86905i
−0.741229 + 2.28127i
−0.841285 − 0.611229i
2.15030 + 1.56229i
−0.841285 + 0.611229i
2.15030 − 1.56229i
−0.809017 + 0.587785i 0 0.309017 − 0.951057i −0.767122 + 2.36096i 0 −2.50835 − 1.82242i 0.309017 + 0.951057i 0 −0.767122 − 2.36096i
217.2 −0.809017 + 0.587785i 0 0.309017 − 0.951057i 0.267122 − 0.822117i 0 0.199334 + 0.144825i 0.309017 + 0.951057i 0 0.267122 + 0.822117i
253.1 −0.809017 − 0.587785i 0 0.309017 + 0.951057i −0.767122 − 2.36096i 0 −2.50835 + 1.82242i 0.309017 − 0.951057i 0 −0.767122 + 2.36096i
253.2 −0.809017 − 0.587785i 0 0.309017 + 0.951057i 0.267122 + 0.822117i 0 0.199334 − 0.144825i 0.309017 − 0.951057i 0 0.267122 − 0.822117i
325.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −2.67024 + 1.94005i 0 −1.51994 + 4.67790i −0.809017 − 0.587785i 0 −2.67024 − 1.94005i
325.2 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 2.17024 − 1.57678i 0 0.328960 − 1.01243i −0.809017 − 0.587785i 0 2.17024 + 1.57678i
973.1 0.309017 − 0.951057i 0 −0.809017 − 0.587785i −2.67024 − 1.94005i 0 −1.51994 − 4.67790i −0.809017 + 0.587785i 0 −2.67024 + 1.94005i
973.2 0.309017 − 0.951057i 0 −0.809017 − 0.587785i 2.17024 + 1.57678i 0 0.328960 + 1.01243i −0.809017 + 0.587785i 0 2.17024 − 1.57678i
\(n\): 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.f 8
3.b odd 2 1 366.2.h.c ✓ 8
61.e even 5 1 inner 1098.2.k.f 8
183.n odd 10 1 366.2.h.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
366.2.h.c ✓ 8 3.b odd 2 1
366.2.h.c ✓ 8 183.n odd 10 1
1098.2.k.f 8 1.a even 1 1 trivial
1098.2.k.f 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 \(S_{2}^{\mathrm{new}}(1098, [\chi])\):

\( T_{5}^{8} + 2T_{5}^{7} + 2T_{5}^{6} - 10T_{5}^{5} + 41T_{5}^{4} + 40T_{5}^{3} + 473T_{5}^{2} - 209T_{5} + 361 \) Copy content Toggle raw display
\( T_{7}^{8} + 7T_{7}^{7} + 42T_{7}^{6} + 110T_{7}^{5} + 141T_{7}^{4} - 50T_{7}^{3} + 268T_{7}^{2} - 104T_{7} + 16 \) 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
$5$ \( T^{8} + 2 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$7$ \( T^{8} + 7 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} - 7 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 3 T^{3} - 14 T^{2} + \cdots - 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} - 17 T^{7} + \cdots + 309136 \) Copy content Toggle raw display
$29$ \( (T^{4} + 7 T^{3} + \cdots + 1255)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 11 T^{7} + \cdots + 102400 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 1263376 \) Copy content Toggle raw display
$41$ \( T^{8} + 27 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$43$ \( T^{8} - 3 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
$47$ \( (T^{4} - T^{3} - 17 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 20 T^{7} + \cdots + 26512201 \) Copy content Toggle raw display
$59$ \( T^{8} - 13 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + \cdots + 13845841 \) Copy content Toggle raw display
$67$ \( T^{8} + 14 T^{7} + \cdots + 144400 \) Copy content Toggle raw display
$71$ \( T^{8} - 12 T^{7} + \cdots + 2624400 \) Copy content Toggle raw display
$73$ \( T^{8} + 9 T^{7} + \cdots + 524176 \) Copy content Toggle raw display
$79$ \( T^{8} + 26 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$83$ \( T^{8} - 4 T^{7} + \cdots + 10137856 \) Copy content Toggle raw display
$89$ \( T^{8} - 38 T^{7} + \cdots + 77281681 \) Copy content Toggle raw display
$97$ \( T^{8} + 24 T^{7} + \cdots + 870250000 \) Copy content Toggle raw display
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