Properties

Label 4014.2.a.w
Level $4014$
Weight $2$
Character orbit 4014.a
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $7$
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] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, 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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 446)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{6} + 37\nu^{5} + 92\nu^{4} - 388\nu^{3} - 287\nu^{2} + 703\nu - 98 ) / 239 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{6} - 37\nu^{5} - 92\nu^{4} + 388\nu^{3} + 526\nu^{2} - 703\nu - 858 ) / 239 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{6} - 28\nu^{5} - 89\nu^{4} + 313\nu^{3} - 151\nu^{2} - 771\nu + 817 ) / 239 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -32\nu^{6} + 38\nu^{5} + 411\nu^{4} - 476\nu^{3} - 1212\nu^{2} + 1439\nu + 513 ) / 239 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -47\nu^{6} + 11\nu^{5} + 641\nu^{4} - 12\nu^{3} - 2049\nu^{2} - 269\nu + 985 ) / 239 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 4\beta_{4} + 10\beta_{3} + 12\beta_{2} + \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{6} - 22\beta_{5} - 12\beta_{4} - 13\beta_{3} + 4\beta_{2} + 55\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{6} + 9\beta_{5} + 52\beta_{4} + 90\beta_{3} + 121\beta_{2} + 19\beta _1 + 188 \) Copy content Toggle raw display

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
0.409721
1.26499
−2.01250
3.08762
−0.922829
2.08198
−2.90898
−1.00000 0 1.00000 −4.32652 0 2.79833 −1.00000 0 4.32652
1.2 −1.00000 0 1.00000 −1.88711 0 −3.39932 −1.00000 0 1.88711
1.3 −1.00000 0 1.00000 −1.52490 0 0.908231 −1.00000 0 1.52490
1.4 −1.00000 0 1.00000 −0.580657 0 −1.39541 −1.00000 0 0.580657
1.5 −1.00000 0 1.00000 −0.437167 0 3.40247 −1.00000 0 0.437167
1.6 −1.00000 0 1.00000 3.23274 0 4.23794 −1.00000 0 −3.23274
1.7 −1.00000 0 1.00000 3.52361 0 −0.552246 −1.00000 0 −3.52361
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(223\) \(1\)

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 4014.2.a.w 7
3.b odd 2 1 446.2.a.e 7
12.b even 2 1 3568.2.a.l 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
446.2.a.e 7 3.b odd 2 1
3568.2.a.l 7 12.b even 2 1
4014.2.a.w 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{7} + 2T_{5}^{6} - 22T_{5}^{5} - 42T_{5}^{4} + 92T_{5}^{3} + 256T_{5}^{2} + 174T_{5} + 36 \) Copy content Toggle raw display
\( T_{7}^{7} - 6T_{7}^{6} - 8T_{7}^{5} + 88T_{7}^{4} - 48T_{7}^{3} - 224T_{7}^{2} + 80T_{7} + 96 \) Copy content Toggle raw display
\( T_{11}^{7} + 9T_{11}^{6} - 14T_{11}^{5} - 254T_{11}^{4} - 58T_{11}^{3} + 2028T_{11}^{2} + 394T_{11} - 3494 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 2 T^{6} - 22 T^{5} - 42 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{7} - 6 T^{6} - 8 T^{5} + 88 T^{4} + \cdots + 96 \) Copy content Toggle raw display
$11$ \( T^{7} + 9 T^{6} - 14 T^{5} + \cdots - 3494 \) Copy content Toggle raw display
$13$ \( T^{7} + 2 T^{6} - 64 T^{5} + \cdots + 7056 \) Copy content Toggle raw display
$17$ \( T^{7} - 7 T^{6} - 44 T^{5} + \cdots - 1212 \) Copy content Toggle raw display
$19$ \( T^{7} + 2 T^{6} - 56 T^{5} + \cdots - 2304 \) Copy content Toggle raw display
$23$ \( T^{7} + 15 T^{6} + 44 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{7} + 9 T^{6} - 96 T^{5} + \cdots - 12096 \) Copy content Toggle raw display
$31$ \( T^{7} + 2 T^{6} - 92 T^{5} + \cdots + 10888 \) Copy content Toggle raw display
$37$ \( T^{7} - 5 T^{6} - 208 T^{5} + \cdots - 114624 \) Copy content Toggle raw display
$41$ \( T^{7} - 33 T^{6} + 368 T^{5} + \cdots + 45248 \) Copy content Toggle raw display
$43$ \( T^{7} - 20 T^{6} + 72 T^{5} + \cdots - 55296 \) Copy content Toggle raw display
$47$ \( T^{7} - 2 T^{6} - 216 T^{5} + \cdots + 536856 \) Copy content Toggle raw display
$53$ \( T^{7} - 13 T^{6} - 112 T^{5} + \cdots + 395712 \) Copy content Toggle raw display
$59$ \( T^{7} + 9 T^{6} - 210 T^{5} + \cdots - 346162 \) Copy content Toggle raw display
$61$ \( T^{7} - 8 T^{6} - 188 T^{5} + \cdots + 14636 \) Copy content Toggle raw display
$67$ \( T^{7} - 29 T^{6} + 78 T^{5} + \cdots - 1778034 \) Copy content Toggle raw display
$71$ \( T^{7} - 424 T^{5} + 48 T^{4} + \cdots + 4460544 \) Copy content Toggle raw display
$73$ \( T^{7} + 37 T^{6} + 472 T^{5} + \cdots - 206064 \) Copy content Toggle raw display
$79$ \( T^{7} - 32 T^{6} + 272 T^{5} + \cdots - 371712 \) Copy content Toggle raw display
$83$ \( T^{7} - 6 T^{6} - 200 T^{5} + \cdots + 263424 \) Copy content Toggle raw display
$89$ \( T^{7} - 17 T^{6} - 176 T^{5} + \cdots - 186084 \) Copy content Toggle raw display
$97$ \( T^{7} - 12 T^{6} - 428 T^{5} + \cdots + 5015808 \) Copy content Toggle raw display
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