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Properties

Label 4970.2.a.z
Level $4970$
Weight $2$
Character orbit 4970.a
Self dual yes
Analytic conductor $39.686$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4970,2,Mod(1,4970)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4970, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4970.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4970 = 2 \cdot 5 \cdot 7 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4970.a (trivial)

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: yes
Analytic conductor: \(39.6856498046\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 9x^{7} + 47x^{6} + 6x^{5} - 151x^{4} + 80x^{3} + 79x^{2} - 54x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + q^{7} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + q^{7} + q^{8} + (\beta_{2} + 1) q^{9} - q^{10} + \beta_{8} q^{11} + \beta_1 q^{12} + (\beta_{3} + 1) q^{13} + q^{14} - \beta_1 q^{15} + q^{16} + (\beta_{7} - \beta_{4} + 1) q^{17} + (\beta_{2} + 1) q^{18} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{19} - q^{20} + \beta_1 q^{21} + \beta_{8} q^{22} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{23}+ \cdots + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 4 q^{3} + 9 q^{4} - 9 q^{5} + 4 q^{6} + 9 q^{7} + 9 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 4 q^{3} + 9 q^{4} - 9 q^{5} + 4 q^{6} + 9 q^{7} + 9 q^{8} + 7 q^{9} - 9 q^{10} - 2 q^{11} + 4 q^{12} + 11 q^{13} + 9 q^{14} - 4 q^{15} + 9 q^{16} + 9 q^{17} + 7 q^{18} + 9 q^{19} - 9 q^{20} + 4 q^{21} - 2 q^{22} + 2 q^{23} + 4 q^{24} + 9 q^{25} + 11 q^{26} + 7 q^{27} + 9 q^{28} + 2 q^{29} - 4 q^{30} + 18 q^{31} + 9 q^{32} + 8 q^{33} + 9 q^{34} - 9 q^{35} + 7 q^{36} + 15 q^{37} + 9 q^{38} - 7 q^{39} - 9 q^{40} + 15 q^{41} + 4 q^{42} + 7 q^{43} - 2 q^{44} - 7 q^{45} + 2 q^{46} + 12 q^{47} + 4 q^{48} + 9 q^{49} + 9 q^{50} + 4 q^{51} + 11 q^{52} + 3 q^{53} + 7 q^{54} + 2 q^{55} + 9 q^{56} + 9 q^{57} + 2 q^{58} + 24 q^{59} - 4 q^{60} + 25 q^{61} + 18 q^{62} + 7 q^{63} + 9 q^{64} - 11 q^{65} + 8 q^{66} - 4 q^{67} + 9 q^{68} + 3 q^{69} - 9 q^{70} + 9 q^{71} + 7 q^{72} + 32 q^{73} + 15 q^{74} + 4 q^{75} + 9 q^{76} - 2 q^{77} - 7 q^{78} + 20 q^{79} - 9 q^{80} - 7 q^{81} + 15 q^{82} + 11 q^{83} + 4 q^{84} - 9 q^{85} + 7 q^{86} + 26 q^{87} - 2 q^{88} + 10 q^{89} - 7 q^{90} + 11 q^{91} + 2 q^{92} + 32 q^{93} + 12 q^{94} - 9 q^{95} + 4 q^{96} + 19 q^{97} + 9 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 9x^{7} + 47x^{6} + 6x^{5} - 151x^{4} + 80x^{3} + 79x^{2} - 54x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 15\nu^{6} - 5\nu^{5} + 66\nu^{4} + 33\nu^{3} - 76\nu^{2} - 25\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 15\nu^{6} - 5\nu^{5} + 66\nu^{4} + 35\nu^{3} - 78\nu^{2} - 35\nu + 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{8} + 2\nu^{7} + 12\nu^{6} - 21\nu^{5} - 36\nu^{4} + 59\nu^{3} - 29\nu + 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - 3\nu^{7} - 11\nu^{6} + 35\nu^{5} + 27\nu^{4} - 114\nu^{3} + 20\nu^{2} + 72\nu - 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{8} + 12\nu^{7} + 29\nu^{6} - 143\nu^{5} - 46\nu^{4} + 473\nu^{3} - 132\nu^{2} - 289\nu + 84 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -2\nu^{8} + 7\nu^{7} + 21\nu^{6} - 83\nu^{5} - 46\nu^{4} + 273\nu^{3} - 54\nu^{2} - 166\nu + 47 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 8\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - \beta_{7} + 4\beta_{6} + 2\beta_{5} + 12\beta_{4} - 11\beta_{3} + 12\beta_{2} + 30\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{8} - 14\beta_{7} + 18\beta_{6} + 13\beta_{5} + 30\beta_{4} - 18\beta_{3} + 66\beta_{2} + 14\beta _1 + 191 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 35 \beta_{8} - 19 \beta_{7} + 64 \beta_{6} + 31 \beta_{5} + 125 \beta_{4} - 108 \beta_{3} + 127 \beta_{2} + \cdots + 295 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 184 \beta_{8} - 149 \beta_{7} + 224 \beta_{6} + 139 \beta_{5} + 345 \beta_{4} - 224 \beta_{3} + \cdots + 1508 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.47217
−2.22106
−0.848518
0.265854
0.341239
1.07442
2.37708
2.41697
3.06619
1.00000 −2.47217 1.00000 −1.00000 −2.47217 1.00000 1.00000 3.11163 −1.00000
1.2 1.00000 −2.22106 1.00000 −1.00000 −2.22106 1.00000 1.00000 1.93311 −1.00000
1.3 1.00000 −0.848518 1.00000 −1.00000 −0.848518 1.00000 1.00000 −2.28002 −1.00000
1.4 1.00000 0.265854 1.00000 −1.00000 0.265854 1.00000 1.00000 −2.92932 −1.00000
1.5 1.00000 0.341239 1.00000 −1.00000 0.341239 1.00000 1.00000 −2.88356 −1.00000
1.6 1.00000 1.07442 1.00000 −1.00000 1.07442 1.00000 1.00000 −1.84562 −1.00000
1.7 1.00000 2.37708 1.00000 −1.00000 2.37708 1.00000 1.00000 2.65049 −1.00000
1.8 1.00000 2.41697 1.00000 −1.00000 2.41697 1.00000 1.00000 2.84175 −1.00000
1.9 1.00000 3.06619 1.00000 −1.00000 3.06619 1.00000 1.00000 6.40153 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(71\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4970.2.a.z ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4970.2.a.z ✓ 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4970))\):

\( T_{3}^{9} - 4T_{3}^{8} - 9T_{3}^{7} + 47T_{3}^{6} + 6T_{3}^{5} - 151T_{3}^{4} + 80T_{3}^{3} + 79T_{3}^{2} - 54T_{3} + 8 \) Copy content Toggle raw display
\( T_{11}^{9} + 2 T_{11}^{8} - 54 T_{11}^{7} - 58 T_{11}^{6} + 842 T_{11}^{5} + 158 T_{11}^{4} - 3167 T_{11}^{3} + \cdots + 120 \) Copy content Toggle raw display
\( T_{13}^{9} - 11 T_{13}^{8} + 5 T_{13}^{7} + 272 T_{13}^{6} - 699 T_{13}^{5} - 933 T_{13}^{4} + \cdots - 758 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 4 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 2 T^{8} + \cdots + 120 \) Copy content Toggle raw display
$13$ \( T^{9} - 11 T^{8} + \cdots - 758 \) Copy content Toggle raw display
$17$ \( T^{9} - 9 T^{8} + \cdots + 2496 \) Copy content Toggle raw display
$19$ \( T^{9} - 9 T^{8} + \cdots + 216 \) Copy content Toggle raw display
$23$ \( T^{9} - 2 T^{8} + \cdots + 55344 \) Copy content Toggle raw display
$29$ \( T^{9} - 2 T^{8} + \cdots - 46272 \) Copy content Toggle raw display
$31$ \( T^{9} - 18 T^{8} + \cdots + 1181520 \) Copy content Toggle raw display
$37$ \( T^{9} - 15 T^{8} + \cdots + 688 \) Copy content Toggle raw display
$41$ \( T^{9} - 15 T^{8} + \cdots + 3120 \) Copy content Toggle raw display
$43$ \( T^{9} - 7 T^{8} + \cdots + 416728 \) Copy content Toggle raw display
$47$ \( T^{9} - 12 T^{8} + \cdots + 4050 \) Copy content Toggle raw display
$53$ \( T^{9} - 3 T^{8} + \cdots + 52704 \) Copy content Toggle raw display
$59$ \( T^{9} - 24 T^{8} + \cdots + 122832 \) Copy content Toggle raw display
$61$ \( T^{9} - 25 T^{8} + \cdots + 69056 \) Copy content Toggle raw display
$67$ \( T^{9} + 4 T^{8} + \cdots + 11920 \) Copy content Toggle raw display
$71$ \( (T - 1)^{9} \) Copy content Toggle raw display
$73$ \( T^{9} - 32 T^{8} + \cdots + 2218861 \) Copy content Toggle raw display
$79$ \( T^{9} - 20 T^{8} + \cdots + 495456 \) Copy content Toggle raw display
$83$ \( T^{9} - 11 T^{8} + \cdots + 3322728 \) Copy content Toggle raw display
$89$ \( T^{9} - 10 T^{8} + \cdots - 2486784 \) Copy content Toggle raw display
$97$ \( T^{9} - 19 T^{8} + \cdots - 15729984 \) Copy content Toggle raw display
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