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Properties

Label 4010.2.a.j
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $1$
Dimension $12$
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] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 16 x^{10} + 30 x^{9} + 93 x^{8} - 162 x^{7} - 238 x^{6} + 391 x^{5} + 240 x^{4} + \cdots - 8 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 16 x^{10} + 30 x^{9} + 93 x^{8} - 162 x^{7} - 238 x^{6} + 391 x^{5} + 240 x^{4} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{11} - 6 \nu^{10} - 190 \nu^{9} + 80 \nu^{8} + 1171 \nu^{7} - 374 \nu^{6} - 3046 \nu^{5} + \cdots + 278 ) / 58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 53 \nu^{11} - 108 \nu^{10} - 752 \nu^{9} + 1382 \nu^{8} + 3765 \nu^{7} - 5804 \nu^{6} - 8254 \nu^{5} + \cdots + 596 ) / 232 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25 \nu^{11} + 76 \nu^{10} - 648 \nu^{9} - 994 \nu^{8} + 5361 \nu^{7} + 4196 \nu^{6} - 17542 \nu^{5} + \cdots + 732 ) / 232 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 105 \nu^{11} + 52 \nu^{10} + 1840 \nu^{9} - 558 \nu^{8} - 11705 \nu^{7} + 1540 \nu^{6} + 32334 \nu^{5} + \cdots - 12 ) / 232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 113 \nu^{11} + 88 \nu^{10} + 1936 \nu^{9} - 1038 \nu^{8} - 12177 \nu^{7} + 3784 \nu^{6} + 34022 \nu^{5} + \cdots + 60 ) / 232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33 \nu^{11} - 47 \nu^{10} - 512 \nu^{9} + 588 \nu^{8} + 2875 \nu^{7} - 2369 \nu^{6} - 7166 \nu^{5} + \cdots + 312 ) / 58 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 223 \nu^{11} - 148 \nu^{10} - 3952 \nu^{9} + 1954 \nu^{8} + 25511 \nu^{7} - 8684 \nu^{6} - 72138 \nu^{5} + \cdots + 1908 ) / 232 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 139 \nu^{11} - 176 \nu^{10} - 2248 \nu^{9} + 2250 \nu^{8} + 13247 \nu^{7} - 9424 \nu^{6} - 34578 \nu^{5} + \cdots + 1388 ) / 116 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 165 \nu^{11} + 148 \nu^{10} + 2792 \nu^{9} - 1838 \nu^{8} - 17217 \nu^{7} + 7292 \nu^{6} + \cdots - 1328 ) / 116 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{6} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{11} - 8 \beta_{10} - \beta_{8} + \beta_{7} + 7 \beta_{6} + \beta_{5} + 11 \beta_{4} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{10} + \beta_{9} + 10 \beta_{8} + 12 \beta_{7} - 2 \beta_{6} + 10 \beta_{5} + 3 \beta_{4} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 53 \beta_{11} - 56 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} + 14 \beta_{7} + 41 \beta_{6} + 14 \beta_{5} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16 \beta_{10} + 15 \beta_{9} + 79 \beta_{8} + 107 \beta_{7} - 26 \beta_{6} + 82 \beta_{5} + \cdots + 449 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 336 \beta_{11} - 384 \beta_{10} + 31 \beta_{9} - 41 \beta_{8} + 141 \beta_{7} + 228 \beta_{6} + \cdots + 550 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - \beta_{11} - 180 \beta_{10} + 155 \beta_{9} + 584 \beta_{8} + 852 \beta_{7} - 245 \beta_{6} + \cdots + 2799 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2109 \beta_{11} - 2634 \beta_{10} + 332 \beta_{9} - 123 \beta_{8} + 1243 \beta_{7} + 1231 \beta_{6} + \cdots + 3857 \) 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
2.67561
2.34926
2.12920
1.16326
1.11338
0.803872
0.0694414
−0.660223
−1.58108
−1.73269
−1.82424
−2.50579
1.00000 −2.67561 1.00000 −1.00000 −2.67561 0.990876 1.00000 4.15891 −1.00000
1.2 1.00000 −2.34926 1.00000 −1.00000 −2.34926 −0.317129 1.00000 2.51901 −1.00000
1.3 1.00000 −2.12920 1.00000 −1.00000 −2.12920 −4.06902 1.00000 1.53348 −1.00000
1.4 1.00000 −1.16326 1.00000 −1.00000 −1.16326 −3.35475 1.00000 −1.64681 −1.00000
1.5 1.00000 −1.11338 1.00000 −1.00000 −1.11338 3.90936 1.00000 −1.76038 −1.00000
1.6 1.00000 −0.803872 1.00000 −1.00000 −0.803872 1.44710 1.00000 −2.35379 −1.00000
1.7 1.00000 −0.0694414 1.00000 −1.00000 −0.0694414 0.700507 1.00000 −2.99518 −1.00000
1.8 1.00000 0.660223 1.00000 −1.00000 0.660223 0.752610 1.00000 −2.56411 −1.00000
1.9 1.00000 1.58108 1.00000 −1.00000 1.58108 0.389201 1.00000 −0.500177 −1.00000
1.10 1.00000 1.73269 1.00000 −1.00000 1.73269 −1.89534 1.00000 0.00220907 −1.00000
1.11 1.00000 1.82424 1.00000 −1.00000 1.82424 −2.93972 1.00000 0.327838 −1.00000
1.12 1.00000 2.50579 1.00000 −1.00000 2.50579 −4.61369 1.00000 3.27901 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.j ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.j ✓ 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{12} + 2 T_{3}^{11} - 16 T_{3}^{10} - 30 T_{3}^{9} + 93 T_{3}^{8} + 162 T_{3}^{7} - 238 T_{3}^{6} + \cdots - 8 \) Copy content Toggle raw display
\( T_{7}^{12} + 9 T_{7}^{11} - 186 T_{7}^{9} - 329 T_{7}^{8} + 886 T_{7}^{7} + 1587 T_{7}^{6} - 2463 T_{7}^{5} + \cdots + 128 \) Copy content Toggle raw display
\( T_{11}^{12} - T_{11}^{11} - 58 T_{11}^{10} + 20 T_{11}^{9} + 1186 T_{11}^{8} + 442 T_{11}^{7} + \cdots - 8132 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + \cdots + 128 \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} + \cdots - 8132 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + \cdots - 10606 \) Copy content Toggle raw display
$17$ \( T^{12} + 11 T^{11} + \cdots + 357296 \) Copy content Toggle raw display
$19$ \( T^{12} + 13 T^{11} + \cdots - 1182784 \) Copy content Toggle raw display
$23$ \( T^{12} + 21 T^{11} + \cdots + 325112 \) Copy content Toggle raw display
$29$ \( T^{12} + 10 T^{11} + \cdots + 4139936 \) Copy content Toggle raw display
$31$ \( T^{12} + 11 T^{11} + \cdots + 7812944 \) Copy content Toggle raw display
$37$ \( T^{12} + 29 T^{11} + \cdots + 19965248 \) Copy content Toggle raw display
$41$ \( T^{12} + T^{11} + \cdots - 2177276 \) Copy content Toggle raw display
$43$ \( T^{12} + 23 T^{11} + \cdots + 67168 \) Copy content Toggle raw display
$47$ \( T^{12} + 17 T^{11} + \cdots - 33121792 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 134366642 \) Copy content Toggle raw display
$59$ \( T^{12} - 14 T^{11} + \cdots + 628768 \) Copy content Toggle raw display
$61$ \( T^{12} + 22 T^{11} + \cdots + 1379044 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1010262352 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 60597469952 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 554505232 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 125669072 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 3599698696 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 309735304 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10812751496 \) Copy content Toggle raw display
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