↑

Properties

Label 4010.2.a.i
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 12x^{8} + 34x^{7} + 46x^{6} - 104x^{5} - 90x^{4} + 89x^{3} + 82x^{2} + 12x - 1 \) 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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 12x^{8} + 34x^{7} + 46x^{6} - 104x^{5} - 90x^{4} + 89x^{3} + 82x^{2} + 12x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 129 \nu^{9} + 529 \nu^{8} + 1066 \nu^{7} - 5745 \nu^{6} - 889 \nu^{5} + 16145 \nu^{4} - 1628 \nu^{3} + \cdots - 343 ) / 647 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 388 \nu^{9} - 1235 \nu^{8} - 4415 \nu^{7} + 14195 \nu^{6} + 14355 \nu^{5} - 45275 \nu^{4} + \cdots - 5975 ) / 1941 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 623 \nu^{9} + 1978 \nu^{8} + 7024 \nu^{7} - 21847 \nu^{6} - 24225 \nu^{5} + 63547 \nu^{4} + \cdots - 1817 ) / 1941 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 304 \nu^{9} + 1041 \nu^{8} + 3119 \nu^{7} - 11402 \nu^{6} - 8239 \nu^{5} + 32505 \nu^{4} + \cdots - 61 ) / 647 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 913 \nu^{9} + 2771 \nu^{8} + 10574 \nu^{7} - 31166 \nu^{6} - 36405 \nu^{5} + 94355 \nu^{4} + \cdots + 998 ) / 1941 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 983 \nu^{9} - 3364 \nu^{8} - 10360 \nu^{7} + 37699 \nu^{6} + 29640 \nu^{5} - 113839 \nu^{4} + \cdots - 205 ) / 1941 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1162 \nu^{9} + 4409 \nu^{8} + 10811 \nu^{7} - 48665 \nu^{6} - 19689 \nu^{5} + 142145 \nu^{4} + \cdots + 7799 ) / 1941 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1720 \nu^{9} + 5975 \nu^{8} + 18311 \nu^{7} - 68189 \nu^{6} - 52830 \nu^{5} + 213110 \nu^{4} + \cdots + 4916 ) / 1941 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{3} - \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 3\beta_{8} - \beta_{7} + 10\beta_{6} - 12\beta_{5} + \beta_{4} + 10\beta_{3} + 6\beta_{2} + 11\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{8} + 3\beta_{7} + 15\beta_{6} - 15\beta_{5} + \beta_{4} + 24\beta_{3} - 9\beta_{2} + 56\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{9} + 41 \beta_{8} - 2 \beta_{7} + 93 \beta_{6} - 114 \beta_{5} + 16 \beta_{4} + 97 \beta_{3} + \cdots + 135 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} + 142 \beta_{8} + 52 \beta_{7} + 171 \beta_{6} - 180 \beta_{5} + 29 \beta_{4} + 250 \beta_{3} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 74 \beta_{9} + 449 \beta_{8} + 70 \beta_{7} + 848 \beta_{6} - 1042 \beta_{5} + 195 \beta_{4} + \cdots + 1014 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25 \beta_{9} + 1462 \beta_{8} + 660 \beta_{7} + 1784 \beta_{6} - 1969 \beta_{5} + 445 \beta_{4} + \cdots + 848 \) 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
3.08731
1.47143
−1.54628
0.0585304
−0.557507
1.96171
−2.65028
−1.06458
−0.309245
2.54891
−1.00000 −3.13000 1.00000 1.00000 3.13000 2.49851 −1.00000 6.79692 −1.00000
1.2 −1.00000 −2.70072 1.00000 1.00000 2.70072 −1.21166 −1.00000 4.29386 −1.00000
1.3 −1.00000 −1.74362 1.00000 1.00000 1.74362 2.79885 −1.00000 0.0402221 −1.00000
1.4 −1.00000 −1.47624 1.00000 1.00000 1.47624 −2.32277 −1.00000 −0.820714 −1.00000
1.5 −1.00000 −1.25862 1.00000 1.00000 1.25862 1.88678 −1.00000 −1.41588 −1.00000
1.6 −1.00000 0.223737 1.00000 1.00000 −0.223737 −0.465545 −1.00000 −2.94994 −1.00000
1.7 −1.00000 0.675556 1.00000 1.00000 −0.675556 −1.96617 −1.00000 −2.54362 −1.00000
1.8 −1.00000 0.696154 1.00000 1.00000 −0.696154 2.65990 −1.00000 −2.51537 −1.00000
1.9 −1.00000 2.30797 1.00000 1.00000 −2.30797 −5.12569 −1.00000 2.32671 −1.00000
1.10 −1.00000 2.40579 1.00000 1.00000 −2.40579 −1.75221 −1.00000 2.78781 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(401\) \(-1\)

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 4010.2.a.i ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.i ✓ 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{10} + 4T_{3}^{9} - 10T_{3}^{8} - 50T_{3}^{7} + 17T_{3}^{6} + 188T_{3}^{5} + 50T_{3}^{4} - 207T_{3}^{3} - 32T_{3}^{2} + 88T_{3} - 16 \) Copy content Toggle raw display
\( T_{7}^{10} + 3 T_{7}^{9} - 28 T_{7}^{8} - 60 T_{7}^{7} + 257 T_{7}^{6} + 480 T_{7}^{5} - 929 T_{7}^{4} + \cdots + 812 \) Copy content Toggle raw display
\( T_{11}^{10} + 11 T_{11}^{9} - 7 T_{11}^{8} - 473 T_{11}^{7} - 1561 T_{11}^{6} + 1601 T_{11}^{5} + \cdots - 14784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 4 T^{9} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots + 812 \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + \cdots - 14784 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots + 98 \) Copy content Toggle raw display
$17$ \( T^{10} - 9 T^{9} + \cdots + 3654 \) Copy content Toggle raw display
$19$ \( T^{10} + 13 T^{9} + \cdots - 6062 \) Copy content Toggle raw display
$23$ \( T^{10} + 3 T^{9} + \cdots - 176 \) Copy content Toggle raw display
$29$ \( T^{10} + 4 T^{9} + \cdots - 214108 \) Copy content Toggle raw display
$31$ \( T^{10} + 17 T^{9} + \cdots - 11703698 \) Copy content Toggle raw display
$37$ \( T^{10} + 15 T^{9} + \cdots + 9866 \) Copy content Toggle raw display
$41$ \( T^{10} + 11 T^{9} + \cdots + 38208216 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{9} + \cdots - 48356848 \) Copy content Toggle raw display
$47$ \( T^{10} - 3 T^{9} + \cdots - 6631952 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 470777142 \) Copy content Toggle raw display
$59$ \( T^{10} + 46 T^{9} + \cdots + 54211074 \) Copy content Toggle raw display
$61$ \( T^{10} + 54 T^{9} + \cdots - 20056568 \) Copy content Toggle raw display
$67$ \( T^{10} + 26 T^{9} + \cdots - 3464576 \) Copy content Toggle raw display
$71$ \( T^{10} + 16 T^{9} + \cdots + 109086 \) Copy content Toggle raw display
$73$ \( T^{10} - 4 T^{9} + \cdots - 4511176 \) Copy content Toggle raw display
$79$ \( T^{10} + 19 T^{9} + \cdots + 5121482 \) Copy content Toggle raw display
$83$ \( T^{10} - 19 T^{9} + \cdots - 6310164 \) Copy content Toggle raw display
$89$ \( T^{10} + 30 T^{9} + \cdots - 956788 \) Copy content Toggle raw display
$97$ \( T^{10} + 16 T^{9} + \cdots + 794554 \) Copy content Toggle raw display
show more
show less