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Properties

Label 4005.2.a.t
Level $4005$
Weight $2$
Character orbit 4005.a
Self dual yes
Analytic conductor $31.980$
Analytic rank $0$
Dimension $10$
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] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, 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("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
Analytic rank: \(0\)
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: \( x^{10} - 19x^{8} - x^{7} + 128x^{6} + 14x^{5} - 358x^{4} - 59x^{3} + 344x^{2} + 71x - 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1335)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_{4} + 1) q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_{4} + 1) q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{8} - \beta_1 q^{10} + \beta_{8} q^{11} + ( - \beta_{5} + 2) q^{13} + (\beta_{9} - \beta_{8} - 2 \beta_1) q^{14} + (\beta_{9} + \beta_{7} + \beta_{3} + \cdots + 3) q^{16}+ \cdots + (\beta_{9} - 3 \beta_{8} + 2 \beta_{7} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 19x^{8} - x^{7} + 128x^{6} + 14x^{5} - 358x^{4} - 59x^{3} + 344x^{2} + 71x - 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14 \nu^{9} + 41 \nu^{8} - 212 \nu^{7} - 582 \nu^{6} + 1039 \nu^{5} + 2512 \nu^{4} - 1884 \nu^{3} + \cdots + 696 ) / 185 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{9} - 72 \nu^{8} - 106 \nu^{7} + 1004 \nu^{6} + 612 \nu^{5} - 4109 \nu^{4} - 1867 \nu^{3} + \cdots + 163 ) / 185 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{9} + 28 \nu^{8} + 144 \nu^{7} - 411 \nu^{6} - 238 \nu^{5} + 1896 \nu^{4} - 1052 \nu^{3} + \cdots + 1098 ) / 185 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13 \nu^{9} - 28 \nu^{8} - 144 \nu^{7} + 411 \nu^{6} + 238 \nu^{5} - 1896 \nu^{4} + 1237 \nu^{3} + \cdots - 1098 ) / 185 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28 \nu^{9} - 103 \nu^{8} - 424 \nu^{7} + 1426 \nu^{6} + 2078 \nu^{5} - 5706 \nu^{4} - 3768 \nu^{3} + \cdots + 97 ) / 185 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6\nu^{9} + 7\nu^{8} - 75\nu^{7} - 112\nu^{6} + 218\nu^{5} + 548\nu^{4} + 144\nu^{3} - 778\nu^{2} - 595\nu - 3 ) / 37 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42 \nu^{9} + 62 \nu^{8} + 636 \nu^{7} - 844 \nu^{6} - 3117 \nu^{5} + 3379 \nu^{4} + 5652 \nu^{3} + \cdots + 132 ) / 185 \) 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
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{3} + 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 10\beta_{6} + 10\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 28\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{9} - 2\beta_{8} + 11\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} + 13\beta_{3} + 47\beta_{2} + 144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{9} + \beta_{8} - 13 \beta_{7} + 84 \beta_{6} + 82 \beta_{5} + 28 \beta_{4} + 14 \beta_{3} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{9} - 28 \beta_{8} + 95 \beta_{7} + 14 \beta_{6} - 28 \beta_{5} - 14 \beta_{4} + 126 \beta_{3} + \cdots + 947 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 27 \beta_{9} + 14 \beta_{8} - 123 \beta_{7} + 665 \beta_{6} + 633 \beta_{5} + 275 \beta_{4} + 143 \beta_{3} + \cdots - 112 \) 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
2.72551
2.28323
2.20255
1.37466
0.167560
−0.395751
−1.33301
−2.11901
−2.19546
−2.71029
−2.72551 0 5.42842 1.00000 0 0.930190 −9.34419 0 −2.72551
1.2 −2.28323 0 3.21313 1.00000 0 0.569221 −2.76986 0 −2.28323
1.3 −2.20255 0 2.85124 1.00000 0 4.87392 −1.87489 0 −2.20255
1.4 −1.37466 0 −0.110304 1.00000 0 −2.28939 2.90096 0 −1.37466
1.5 −0.167560 0 −1.97192 1.00000 0 −2.76137 0.665537 0 −0.167560
1.6 0.395751 0 −1.84338 1.00000 0 1.55828 −1.52102 0 0.395751
1.7 1.33301 0 −0.223096 1.00000 0 4.48223 −2.96340 0 1.33301
1.8 2.11901 0 2.49018 1.00000 0 1.69488 1.03870 0 2.11901
1.9 2.19546 0 2.82005 1.00000 0 −3.75700 1.80038 0 2.19546
1.10 2.71029 0 5.34569 1.00000 0 3.69905 9.06779 0 2.71029
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(-1\)

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 4005.2.a.t 10
3.b odd 2 1 1335.2.a.i ✓ 10
15.d odd 2 1 6675.2.a.ba 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.i ✓ 10 3.b odd 2 1
4005.2.a.t 10 1.a even 1 1 trivial
6675.2.a.ba 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{10} - 19T_{2}^{8} + T_{2}^{7} + 128T_{2}^{6} - 14T_{2}^{5} - 358T_{2}^{4} + 59T_{2}^{3} + 344T_{2}^{2} - 71T_{2} - 21 \) Copy content Toggle raw display
\( T_{7}^{10} - 9 T_{7}^{9} - 5 T_{7}^{8} + 228 T_{7}^{7} - 358 T_{7}^{6} - 1567 T_{7}^{5} + 4023 T_{7}^{4} + \cdots - 2684 \) Copy content Toggle raw display
\( T_{11}^{10} + 4 T_{11}^{9} - 64 T_{11}^{8} - 296 T_{11}^{7} + 1148 T_{11}^{6} + 6832 T_{11}^{5} + \cdots + 92160 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 19 T^{8} + \cdots - 21 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 9 T^{9} + \cdots - 2684 \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 92160 \) Copy content Toggle raw display
$13$ \( T^{10} - 15 T^{9} + \cdots + 28 \) Copy content Toggle raw display
$17$ \( T^{10} + 11 T^{9} + \cdots - 378 \) Copy content Toggle raw display
$19$ \( T^{10} - 14 T^{9} + \cdots - 640 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} + \cdots + 11570688 \) Copy content Toggle raw display
$29$ \( T^{10} + 5 T^{9} + \cdots - 1350 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots - 22400 \) Copy content Toggle raw display
$37$ \( T^{10} - 23 T^{9} + \cdots + 40255028 \) Copy content Toggle raw display
$41$ \( T^{10} + 13 T^{9} + \cdots - 1220850 \) Copy content Toggle raw display
$43$ \( T^{10} - 25 T^{9} + \cdots + 27722936 \) Copy content Toggle raw display
$47$ \( T^{10} - T^{9} + \cdots + 37205568 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 187650378 \) Copy content Toggle raw display
$59$ \( T^{10} - 15 T^{9} + \cdots - 345594 \) Copy content Toggle raw display
$61$ \( T^{10} - 8 T^{9} + \cdots - 60061568 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1474948864 \) Copy content Toggle raw display
$71$ \( T^{10} - 22 T^{9} + \cdots + 11516160 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 200985728 \) Copy content Toggle raw display
$79$ \( T^{10} + 3 T^{9} + \cdots - 15102184 \) Copy content Toggle raw display
$83$ \( T^{10} - 10 T^{9} + \cdots + 1367040 \) Copy content Toggle raw display
$89$ \( (T - 1)^{10} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 113769472 \) Copy content Toggle raw display
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