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Properties

Label 4005.2.a.u
Level $4005$
Weight $2$
Character orbit 4005.a
Self dual yes
Analytic conductor $31.980$
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] = [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: \(1\)
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: \( x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5 \nu^{11} + 8 \nu^{10} + 87 \nu^{9} - 121 \nu^{8} - 562 \nu^{7} + 666 \nu^{6} + 1618 \nu^{5} + \cdots - 432 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} - 33 \nu^{9} + 79 \nu^{8} + 196 \nu^{7} - 441 \nu^{6} - 496 \nu^{5} + 1021 \nu^{4} + \cdots + 153 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31 \nu^{11} - 82 \nu^{10} - 426 \nu^{9} + 1085 \nu^{8} + 2156 \nu^{7} - 5112 \nu^{6} - 4934 \nu^{5} + \cdots + 1512 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35 \nu^{11} + 83 \nu^{10} + 501 \nu^{9} - 1117 \nu^{8} - 2638 \nu^{7} + 5391 \nu^{6} + 6196 \nu^{5} + \cdots - 1728 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35 \nu^{11} - 83 \nu^{10} - 501 \nu^{9} + 1117 \nu^{8} + 2638 \nu^{7} - 5391 \nu^{6} - 6196 \nu^{5} + \cdots + 1782 ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43 \nu^{11} + 112 \nu^{10} + 597 \nu^{9} - 1505 \nu^{8} - 3035 \nu^{7} + 7245 \nu^{6} + 6884 \nu^{5} + \cdots - 2403 ) / 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 67 \nu^{11} - 172 \nu^{10} - 939 \nu^{9} + 2318 \nu^{8} + 4847 \nu^{7} - 11214 \nu^{6} - 11270 \nu^{5} + \cdots + 3726 ) / 27 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68 \nu^{11} + 170 \nu^{10} + 969 \nu^{9} - 2308 \nu^{8} - 5089 \nu^{7} + 11241 \nu^{6} + 11995 \nu^{5} + \cdots - 3789 ) / 9 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 217 \nu^{11} - 547 \nu^{10} - 3090 \nu^{9} + 7433 \nu^{8} + 16226 \nu^{7} - 36243 \nu^{6} + \cdots + 12312 ) / 27 \) 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
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 2\beta_{9} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + 8\beta_{2} + 2\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 9 \beta_{7} + 9 \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{11} - 21 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 14 \beta_{6} - 10 \beta_{5} - 21 \beta_{4} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 25 \beta_{11} + 10 \beta_{10} - 27 \beta_{9} + 16 \beta_{8} + 69 \beta_{7} + 73 \beta_{6} - 14 \beta_{5} + \cdots + 94 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{11} + 2 \beta_{10} - 172 \beta_{9} + 44 \beta_{8} + 62 \beta_{7} + 141 \beta_{6} - 83 \beta_{5} + \cdots + 469 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 234 \beta_{11} + 76 \beta_{10} - 270 \beta_{9} + 168 \beta_{8} + 508 \beta_{7} + 572 \beta_{6} + \cdots + 749 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 776 \beta_{11} + 34 \beta_{10} - 1311 \beta_{9} + 453 \beta_{8} + 666 \beta_{7} + 1248 \beta_{6} + \cdots + 3062 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1970 \beta_{11} + 528 \beta_{10} - 2398 \beta_{9} + 1508 \beta_{8} + 3711 \beta_{7} + 4404 \beta_{6} + \cdots + 5819 \) 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
2.75145
2.24327
2.11454
1.62987
1.02694
0.491103
0.478750
−0.551952
−1.26525
−1.76848
−1.85231
−2.29792
−2.75145 0 5.57047 1.00000 0 −4.05689 −9.82397 0 −2.75145
1.2 −2.24327 0 3.03225 1.00000 0 1.09855 −2.31562 0 −2.24327
1.3 −2.11454 0 2.47129 1.00000 0 −2.16289 −0.996554 0 −2.11454
1.4 −1.62987 0 0.656467 1.00000 0 −1.43414 2.18978 0 −1.62987
1.5 −1.02694 0 −0.945401 1.00000 0 2.58018 3.02474 0 −1.02694
1.6 −0.491103 0 −1.75882 1.00000 0 −1.39011 1.84597 0 −0.491103
1.7 −0.478750 0 −1.77080 1.00000 0 −0.0239408 1.80527 0 −0.478750
1.8 0.551952 0 −1.69535 1.00000 0 1.42988 −2.03966 0 0.551952
1.9 1.26525 0 −0.399141 1.00000 0 3.64120 −3.03551 0 1.26525
1.10 1.76848 0 1.12751 1.00000 0 −4.90565 −1.54298 0 1.76848
1.11 1.85231 0 1.43107 1.00000 0 −0.0119316 −1.05384 0 1.85231
1.12 2.29792 0 3.28045 1.00000 0 −2.76426 2.94239 0 2.29792
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.u ✓ 12
3.b odd 2 1 4005.2.a.v yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4005.2.a.u ✓ 12 1.a even 1 1 trivial
4005.2.a.v yes 12 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{12} + 3 T_{2}^{11} - 13 T_{2}^{10} - 41 T_{2}^{9} + 58 T_{2}^{8} + 202 T_{2}^{7} - 95 T_{2}^{6} + \cdots - 27 \) Copy content Toggle raw display
\( T_{7}^{12} + 8 T_{7}^{11} - 8 T_{7}^{10} - 184 T_{7}^{9} - 195 T_{7}^{8} + 1206 T_{7}^{7} + 2030 T_{7}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{11}^{12} - 68 T_{11}^{10} + 6 T_{11}^{9} + 1460 T_{11}^{8} - 160 T_{11}^{7} - 12884 T_{11}^{6} + \cdots - 10048 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots - 27 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 68 T^{10} + \cdots - 10048 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots - 31873 \) Copy content Toggle raw display
$17$ \( T^{12} - 76 T^{10} + \cdots + 58983 \) Copy content Toggle raw display
$19$ \( T^{12} + 24 T^{11} + \cdots + 454464 \) Copy content Toggle raw display
$23$ \( T^{12} + 24 T^{11} + \cdots - 2752 \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} + \cdots - 1493855 \) Copy content Toggle raw display
$31$ \( T^{12} + 12 T^{11} + \cdots + 4248896 \) Copy content Toggle raw display
$37$ \( T^{12} + 10 T^{11} + \cdots - 99145229 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 193162729 \) Copy content Toggle raw display
$43$ \( T^{12} + 42 T^{11} + \cdots - 175243 \) Copy content Toggle raw display
$47$ \( T^{12} + 22 T^{11} + \cdots + 14807017 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 147487831 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 52606196469 \) Copy content Toggle raw display
$61$ \( T^{12} + 52 T^{11} + \cdots - 3118592 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 3839515968 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 11944534080 \) Copy content Toggle raw display
$73$ \( T^{12} + 8 T^{11} + \cdots + 91641792 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 2475897617 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 407362496 \) Copy content Toggle raw display
$89$ \( (T - 1)^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41939735616 \) Copy content Toggle raw display
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