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Properties

Label 98.4.c.h
Level $98$
Weight $4$
Character orbit 98.c
Analytic conductor $5.782$
Analytic rank $0$
Dimension $4$
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] = [98,4,Mod(67,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not 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: \(5.78218718056\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + 2) q^{2} + (5 \beta_{3} + 5 \beta_1) q^{3} + 4 \beta_{2} q^{4} - 14 \beta_1 q^{5} + 10 \beta_{3} q^{6} - 8 q^{8} + ( - 23 \beta_{2} - 23) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + 2) q^{2} + (5 \beta_{3} + 5 \beta_1) q^{3} + 4 \beta_{2} q^{4} - 14 \beta_1 q^{5} + 10 \beta_{3} q^{6} - 8 q^{8} + ( - 23 \beta_{2} - 23) q^{9} + ( - 28 \beta_{3} - 28 \beta_1) q^{10} - 14 \beta_{2} q^{11} - 20 \beta_1 q^{12} + 36 \beta_{3} q^{13} + 140 q^{15} + ( - 16 \beta_{2} - 16) q^{16} + ( - \beta_{3} - \beta_1) q^{17} - 46 \beta_{2} q^{18} - \beta_1 q^{19} - 56 \beta_{3} q^{20} + 28 q^{22} + ( - 140 \beta_{2} - 140) q^{23} + ( - 40 \beta_{3} - 40 \beta_1) q^{24} + 267 \beta_{2} q^{25} - 72 \beta_1 q^{26} + 20 \beta_{3} q^{27} - 286 q^{29} + (280 \beta_{2} + 280) q^{30} + (66 \beta_{3} + 66 \beta_1) q^{31} - 32 \beta_{2} q^{32} + 70 \beta_1 q^{33} - 2 \beta_{3} q^{34} + 92 q^{36} + (38 \beta_{2} + 38) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} - 360 \beta_{2} q^{39} + 112 \beta_1 q^{40} - 89 \beta_{3} q^{41} - 34 q^{43} + (56 \beta_{2} + 56) q^{44} + (322 \beta_{3} + 322 \beta_1) q^{45} - 280 \beta_{2} q^{46} + 370 \beta_1 q^{47} - 80 \beta_{3} q^{48} - 534 q^{50} + (10 \beta_{2} + 10) q^{51} + ( - 144 \beta_{3} - 144 \beta_1) q^{52} - 74 \beta_{2} q^{53} - 40 \beta_1 q^{54} + 196 \beta_{3} q^{55} + 10 q^{57} + ( - 572 \beta_{2} - 572) q^{58} + ( - 307 \beta_{3} - 307 \beta_1) q^{59} + 560 \beta_{2} q^{60} + 10 \beta_1 q^{61} + 132 \beta_{3} q^{62} + 64 q^{64} + (1008 \beta_{2} + 1008) q^{65} + (140 \beta_{3} + 140 \beta_1) q^{66} + 684 \beta_{2} q^{67} + 4 \beta_1 q^{68} - 700 \beta_{3} q^{69} + 588 q^{71} + (184 \beta_{2} + 184) q^{72} + (191 \beta_{3} + 191 \beta_1) q^{73} + 76 \beta_{2} q^{74} - 1335 \beta_1 q^{75} - 4 \beta_{3} q^{76} + 720 q^{78} + ( - 1220 \beta_{2} - 1220) q^{79} + (224 \beta_{3} + 224 \beta_1) q^{80} - 821 \beta_{2} q^{81} + 178 \beta_1 q^{82} + 299 \beta_{3} q^{83} - 28 q^{85} + ( - 68 \beta_{2} - 68) q^{86} + ( - 1430 \beta_{3} - 1430 \beta_1) q^{87} + 112 \beta_{2} q^{88} + 437 \beta_1 q^{89} + 644 \beta_{3} q^{90} + 560 q^{92} + ( - 660 \beta_{2} - 660) q^{93} + (740 \beta_{3} + 740 \beta_1) q^{94} + 28 \beta_{2} q^{95} + 160 \beta_1 q^{96} + 1049 \beta_{3} q^{97} - 322 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 46 q^{9} + 28 q^{11} + 560 q^{15} - 32 q^{16} + 92 q^{18} + 112 q^{22} - 280 q^{23} - 534 q^{25} - 1144 q^{29} + 560 q^{30} + 64 q^{32} + 368 q^{36} + 76 q^{37} + 720 q^{39} - 136 q^{43} + 112 q^{44} + 560 q^{46} - 2136 q^{50} + 20 q^{51} + 148 q^{53} + 40 q^{57} - 1144 q^{58} - 1120 q^{60} + 256 q^{64} + 2016 q^{65} - 1368 q^{67} + 2352 q^{71} + 368 q^{72} - 152 q^{74} + 2880 q^{78} - 2440 q^{79} + 1642 q^{81} - 112 q^{85} - 136 q^{86} - 224 q^{88} + 2240 q^{92} - 1320 q^{93} - 56 q^{95} - 1288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
1.00000 + 1.73205i −3.53553 + 6.12372i −2.00000 + 3.46410i −9.89949 − 17.1464i −14.1421 0 −8.00000 −11.5000 − 19.9186i 19.7990 − 34.2929i
67.2 1.00000 + 1.73205i 3.53553 − 6.12372i −2.00000 + 3.46410i 9.89949 + 17.1464i 14.1421 0 −8.00000 −11.5000 − 19.9186i −19.7990 + 34.2929i
79.1 1.00000 − 1.73205i −3.53553 − 6.12372i −2.00000 − 3.46410i −9.89949 + 17.1464i −14.1421 0 −8.00000 −11.5000 + 19.9186i 19.7990 + 34.2929i
79.2 1.00000 − 1.73205i 3.53553 + 6.12372i −2.00000 − 3.46410i 9.89949 − 17.1464i 14.1421 0 −8.00000 −11.5000 + 19.9186i −19.7990 − 34.2929i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.c.h 4
3.b odd 2 1 882.4.g.ba 4
7.b odd 2 1 inner 98.4.c.h 4
7.c even 3 1 98.4.a.g ✓ 2
7.c even 3 1 inner 98.4.c.h 4
7.d odd 6 1 98.4.a.g ✓ 2
7.d odd 6 1 inner 98.4.c.h 4
21.c even 2 1 882.4.g.ba 4
21.g even 6 1 882.4.a.bg 2
21.g even 6 1 882.4.g.ba 4
21.h odd 6 1 882.4.a.bg 2
21.h odd 6 1 882.4.g.ba 4
28.f even 6 1 784.4.a.y 2
28.g odd 6 1 784.4.a.y 2
35.i odd 6 1 2450.4.a.bx 2
35.j even 6 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g ✓ 2 7.c even 3 1
98.4.a.g ✓ 2 7.d odd 6 1
98.4.c.h 4 1.a even 1 1 trivial
98.4.c.h 4 7.b odd 2 1 inner
98.4.c.h 4 7.c even 3 1 inner
98.4.c.h 4 7.d odd 6 1 inner
784.4.a.y 2 28.f even 6 1
784.4.a.y 2 28.g odd 6 1
882.4.a.bg 2 21.g even 6 1
882.4.a.bg 2 21.h odd 6 1
882.4.g.ba 4 3.b odd 2 1
882.4.g.ba 4 21.c even 2 1
882.4.g.ba 4 21.g even 6 1
882.4.g.ba 4 21.h odd 6 1
2450.4.a.bx 2 35.i odd 6 1
2450.4.a.bx 2 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 50T_{3}^{2} + 2500 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$5$ \( T^{4} + 392 T^{2} + 153664 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2592)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T^{2} + 140 T + 19600)^{2} \) Copy content Toggle raw display
$29$ \( (T + 286)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 8712 T^{2} + 75898944 \) Copy content Toggle raw display
$37$ \( (T^{2} - 38 T + 1444)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15842)^{2} \) Copy content Toggle raw display
$43$ \( (T + 34)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 74966440000 \) Copy content Toggle raw display
$53$ \( (T^{2} - 74 T + 5476)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 35531496004 \) Copy content Toggle raw display
$61$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$67$ \( (T^{2} + 684 T + 467856)^{2} \) Copy content Toggle raw display
$71$ \( (T - 588)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 5323453444 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1220 T + 1488400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 178802)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 145876635844 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2200802)^{2} \) Copy content Toggle raw display
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