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Properties

Label 98.3.b.b
Level $98$
Weight $3$
Character orbit 98.b
Analytic conductor $2.670$
Analytic rank $0$
Dimension $4$
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] = [98,3,Mod(97,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 98.b (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: \(2.67030659073\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + 2 q^{4} + ( - 2 \beta_{3} + \beta_{2}) q^{5} + (\beta_{3} + 2 \beta_{2}) q^{6} + 2 \beta_1 q^{8} + 6 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + 2 q^{4} + ( - 2 \beta_{3} + \beta_{2}) q^{5} + (\beta_{3} + 2 \beta_{2}) q^{6} + 2 \beta_1 q^{8} + 6 \beta_1 q^{9} + ( - \beta_{3} + 4 \beta_{2}) q^{10} + (3 \beta_1 - 9) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{12} + (2 \beta_{3} - 6 \beta_{2}) q^{13} + (3 \beta_1 - 9) q^{15} + 4 q^{16} + ( - 2 \beta_{3} - 5 \beta_{2}) q^{17} + 12 q^{18} + (\beta_{3} - \beta_{2}) q^{19} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{20} + ( - 9 \beta_1 + 6) q^{22} + ( - 9 \beta_1 - 15) q^{23} + (2 \beta_{3} + 4 \beta_{2}) q^{24} + ( - 12 \beta_1 - 2) q^{25} + (6 \beta_{3} - 4 \beta_{2}) q^{26} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{27} + ( - 6 \beta_1 + 12) q^{29} + ( - 9 \beta_1 + 6) q^{30} + (15 \beta_{3} - 7 \beta_{2}) q^{31} + 4 \beta_1 q^{32} + (12 \beta_{3} + 15 \beta_{2}) q^{33} + (5 \beta_{3} + 4 \beta_{2}) q^{34} + 12 \beta_1 q^{36} + ( - 24 \beta_1 + 31) q^{37} + (\beta_{3} - 2 \beta_{2}) q^{38} + (12 \beta_1 - 6) q^{39} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{40} + ( - 10 \beta_{3} + 2 \beta_{2}) q^{41} + ( - 6 \beta_1 - 2) q^{43} + (6 \beta_1 - 18) q^{44} + ( - 6 \beta_{3} + 24 \beta_{2}) q^{45} + ( - 15 \beta_1 - 18) q^{46} + (\beta_{3} - 29 \beta_{2}) q^{47} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{48} + ( - 2 \beta_1 - 24) q^{50} + (21 \beta_1 - 27) q^{51} + (4 \beta_{3} - 12 \beta_{2}) q^{52} + (12 \beta_1 + 39) q^{53} + ( - 3 \beta_{3} + 6 \beta_{2}) q^{54} + (15 \beta_{3} + 3 \beta_{2}) q^{55} + 3 q^{57} + (12 \beta_1 - 12) q^{58} + ( - 25 \beta_{3} - 13 \beta_{2}) q^{59} + (6 \beta_1 - 18) q^{60} + ( - 32 \beta_{3} + 7 \beta_{2}) q^{61} + (7 \beta_{3} - 30 \beta_{2}) q^{62} + 8 q^{64} + (42 \beta_1 + 42) q^{65} + ( - 15 \beta_{3} - 24 \beta_{2}) q^{66} + ( - 45 \beta_1 + 29) q^{67} + ( - 4 \beta_{3} - 10 \beta_{2}) q^{68} + (6 \beta_{3} - 3 \beta_{2}) q^{69} + (30 \beta_1 - 6) q^{71} + 24 q^{72} + (16 \beta_{3} + 53 \beta_{2}) q^{73} + (31 \beta_1 - 48) q^{74} + ( - 10 \beta_{3} - 22 \beta_{2}) q^{75} + (2 \beta_{3} - 2 \beta_{2}) q^{76} + ( - 6 \beta_1 + 24) q^{78} + (15 \beta_1 - 55) q^{79} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{80} + (54 \beta_1 - 9) q^{81} + ( - 2 \beta_{3} + 20 \beta_{2}) q^{82} + ( - 4 \beta_{3} + 68 \beta_{2}) q^{83} + (24 \beta_1 - 9) q^{85} + ( - 2 \beta_1 - 12) q^{86} + ( - 18 \beta_{3} - 24 \beta_{2}) q^{87} + ( - 18 \beta_1 + 12) q^{88} + (24 \beta_{3} + 63 \beta_{2}) q^{89} + ( - 24 \beta_{3} + 12 \beta_{2}) q^{90} + ( - 18 \beta_1 - 30) q^{92} + ( - 24 \beta_1 + 69) q^{93} + (29 \beta_{3} - 2 \beta_{2}) q^{94} + (9 \beta_1 + 15) q^{95} + (4 \beta_{3} + 8 \beta_{2}) q^{96} + (26 \beta_{3} - 22 \beta_{2}) q^{97} + ( - 54 \beta_1 + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 36 q^{11} - 36 q^{15} + 16 q^{16} + 48 q^{18} + 24 q^{22} - 60 q^{23} - 8 q^{25} + 48 q^{29} + 24 q^{30} + 124 q^{37} - 24 q^{39} - 8 q^{43} - 72 q^{44} - 72 q^{46} - 96 q^{50} - 108 q^{51} + 156 q^{53} + 12 q^{57} - 48 q^{58} - 72 q^{60} + 32 q^{64} + 168 q^{65} + 116 q^{67} - 24 q^{71} + 96 q^{72} - 192 q^{74} + 96 q^{78} - 220 q^{79} - 36 q^{81} - 36 q^{85} - 48 q^{86} + 48 q^{88} - 120 q^{92} + 276 q^{93} + 60 q^{95} + 144 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
−0.707107 − 1.22474i
−1.41421 − 4.18154i 2.00000 − 3.16693i 5.91359i 0 −2.82843 −8.48528 4.47871i
97.2 −1.41421 4.18154i 2.00000 3.16693i − 5.91359i 0 −2.82843 −8.48528 − 4.47871i
97.3 1.41421 − 0.717439i 2.00000 − 6.63103i − 1.01461i 0 2.82843 8.48528 − 9.37769i
97.4 1.41421 0.717439i 2.00000 6.63103i 1.01461i 0 2.82843 8.48528 9.37769i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.3.b.b 4
3.b odd 2 1 882.3.c.f 4
4.b odd 2 1 784.3.c.e 4
7.b odd 2 1 inner 98.3.b.b 4
7.c even 3 1 14.3.d.a ✓ 4
7.c even 3 1 98.3.d.a 4
7.d odd 6 1 14.3.d.a ✓ 4
7.d odd 6 1 98.3.d.a 4
21.c even 2 1 882.3.c.f 4
21.g even 6 1 126.3.n.c 4
21.g even 6 1 882.3.n.b 4
21.h odd 6 1 126.3.n.c 4
21.h odd 6 1 882.3.n.b 4
28.d even 2 1 784.3.c.e 4
28.f even 6 1 112.3.s.b 4
28.f even 6 1 784.3.s.c 4
28.g odd 6 1 112.3.s.b 4
28.g odd 6 1 784.3.s.c 4
35.i odd 6 1 350.3.k.a 4
35.j even 6 1 350.3.k.a 4
35.k even 12 2 350.3.i.a 8
35.l odd 12 2 350.3.i.a 8
56.j odd 6 1 448.3.s.d 4
56.k odd 6 1 448.3.s.c 4
56.m even 6 1 448.3.s.c 4
56.p even 6 1 448.3.s.d 4
84.j odd 6 1 1008.3.cg.l 4
84.n even 6 1 1008.3.cg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a ✓ 4 7.c even 3 1
14.3.d.a ✓ 4 7.d odd 6 1
98.3.b.b 4 1.a even 1 1 trivial
98.3.b.b 4 7.b odd 2 1 inner
98.3.d.a 4 7.c even 3 1
98.3.d.a 4 7.d odd 6 1
112.3.s.b 4 28.f even 6 1
112.3.s.b 4 28.g odd 6 1
126.3.n.c 4 21.g even 6 1
126.3.n.c 4 21.h odd 6 1
350.3.i.a 8 35.k even 12 2
350.3.i.a 8 35.l odd 12 2
350.3.k.a 4 35.i odd 6 1
350.3.k.a 4 35.j even 6 1
448.3.s.c 4 56.k odd 6 1
448.3.s.c 4 56.m even 6 1
448.3.s.d 4 56.j odd 6 1
448.3.s.d 4 56.p even 6 1
784.3.c.e 4 4.b odd 2 1
784.3.c.e 4 28.d even 2 1
784.3.s.c 4 28.f even 6 1
784.3.s.c 4 28.g odd 6 1
882.3.c.f 4 3.b odd 2 1
882.3.c.f 4 21.c even 2 1
882.3.n.b 4 21.g even 6 1
882.3.n.b 4 21.h odd 6 1
1008.3.cg.l 4 84.j odd 6 1
1008.3.cg.l 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 18T_{3}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 18T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 54T^{2} + 441 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18 T + 63)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 264T^{2} + 7056 \) Copy content Toggle raw display
$17$ \( T^{4} + 198T^{2} + 2601 \) Copy content Toggle raw display
$19$ \( T^{4} + 18T^{2} + 9 \) Copy content Toggle raw display
$23$ \( (T^{2} + 30 T + 63)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 24 T + 72)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2994 T^{2} + 1447209 \) Copy content Toggle raw display
$37$ \( (T^{2} - 62 T - 191)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 1224 T^{2} + 345744 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 5058 T^{2} + 6335289 \) Copy content Toggle raw display
$53$ \( (T^{2} - 78 T + 1233)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8514 T^{2} + 10517049 \) Copy content Toggle raw display
$61$ \( T^{4} + 12582 T^{2} + 35964009 \) Copy content Toggle raw display
$67$ \( (T^{2} - 58 T - 3209)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 1764)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 19926 T^{2} + 47485881 \) Copy content Toggle raw display
$79$ \( (T^{2} + 110 T + 2575)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 27936 T^{2} + 189778176 \) Copy content Toggle raw display
$89$ \( T^{4} + 30726 T^{2} + 71419401 \) Copy content Toggle raw display
$97$ \( T^{4} + 11016 T^{2} + 6780816 \) Copy content Toggle raw display
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