Properties

Label 4026.2.a.z
Level $4026$
Weight $2$
Character orbit 4026.a
Self dual yes
Analytic conductor $32.148$
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] = [4026,2,Mod(1,4026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, 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("4026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4026.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: \(32.1477718538\)
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: \( x^{7} - 18x^{5} - 10x^{4} + 91x^{3} + 90x^{2} - 66x - 56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 18x^{5} - 10x^{4} + 91x^{3} + 90x^{2} - 66x - 56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - \nu^{5} - 13\nu^{4} + 7\nu^{3} + 40\nu^{2} - 2\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 13\nu^{4} - 25\nu^{3} - 52\nu^{2} + 22\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} - 17\nu^{4} + 7\nu^{3} + 84\nu^{2} + 6\nu - 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 18\nu^{3} + 48\nu^{2} - 28\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 15\nu^{4} + 18\nu^{3} + 66\nu^{2} - 16\nu - 40 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - \beta_{2} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{6} + \beta_{5} - 4\beta_{4} + 5\beta_{3} - \beta_{2} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 11\beta_{5} - 3\beta_{4} + 12\beta_{3} - 9\beta_{2} - 12\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 31\beta_{6} + 15\beta_{5} - 31\beta_{4} + 50\beta_{3} - 11\beta_{2} - 28\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{6} + 111\beta_{5} - 43\beta_{4} + 132\beta_{3} - 73\beta_{2} - 124\beta _1 + 223 \) 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
3.27906
0.868643
−0.647395
−1.70776
−2.63895
−2.20695
3.05336
−1.00000 −1.00000 1.00000 −2.92143 1.00000 −1.10194 −1.00000 1.00000 2.92143
1.2 −1.00000 −1.00000 1.00000 −2.19576 1.00000 −0.720188 −1.00000 1.00000 2.19576
1.3 −1.00000 −1.00000 1.00000 −0.757998 1.00000 −3.37265 −1.00000 1.00000 0.757998
1.4 −1.00000 −1.00000 1.00000 −0.745947 1.00000 −1.28209 −1.00000 1.00000 0.745947
1.5 −1.00000 −1.00000 1.00000 2.19404 1.00000 1.89767 −1.00000 1.00000 −2.19404
1.6 −1.00000 −1.00000 1.00000 3.06294 1.00000 4.73255 −1.00000 1.00000 −3.06294
1.7 −1.00000 −1.00000 1.00000 3.36415 1.00000 −4.15335 −1.00000 1.00000 −3.36415
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(1\)

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 4026.2.a.z 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{7} - 2T_{5}^{6} - 18T_{5}^{5} + 25T_{5}^{4} + 104T_{5}^{3} - 57T_{5}^{2} - 195T_{5} - 82 \) Copy content Toggle raw display
\( T_{7}^{7} + 4T_{7}^{6} - 21T_{7}^{5} - 105T_{7}^{4} - 36T_{7}^{3} + 284T_{7}^{2} + 368T_{7} + 128 \) Copy content Toggle raw display
\( T_{13}^{7} + 7T_{13}^{6} - 27T_{13}^{5} - 242T_{13}^{4} - 192T_{13}^{3} + 565T_{13}^{2} - 52T_{13} - 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( (T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 2 T^{6} - 18 T^{5} + 25 T^{4} + \cdots - 82 \) Copy content Toggle raw display
$7$ \( T^{7} + 4 T^{6} - 21 T^{5} - 105 T^{4} + \cdots + 128 \) Copy content Toggle raw display
$11$ \( (T - 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + 7 T^{6} - 27 T^{5} - 242 T^{4} + \cdots - 44 \) Copy content Toggle raw display
$17$ \( T^{7} + 4 T^{6} - 92 T^{5} + \cdots - 92224 \) Copy content Toggle raw display
$19$ \( T^{7} + 4 T^{6} - 87 T^{5} + \cdots + 2048 \) Copy content Toggle raw display
$23$ \( T^{7} + T^{6} - 107 T^{5} - 136 T^{4} + \cdots + 9856 \) Copy content Toggle raw display
$29$ \( T^{7} - 6 T^{6} - 56 T^{5} + \cdots - 20356 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} - 77 T^{5} + \cdots + 1000 \) Copy content Toggle raw display
$37$ \( T^{7} + 15 T^{6} + 17 T^{5} + \cdots - 2792 \) Copy content Toggle raw display
$41$ \( T^{7} - T^{6} - 118 T^{5} + \cdots + 146564 \) Copy content Toggle raw display
$43$ \( T^{7} + 13 T^{6} - 79 T^{5} + \cdots + 21952 \) Copy content Toggle raw display
$47$ \( T^{7} - 11 T^{6} - 36 T^{5} + \cdots - 47104 \) Copy content Toggle raw display
$53$ \( T^{7} - 14 T^{6} - 65 T^{5} + \cdots + 22208 \) Copy content Toggle raw display
$59$ \( T^{7} - 39 T^{6} + 509 T^{5} + \cdots - 10936 \) Copy content Toggle raw display
$61$ \( (T + 1)^{7} \) Copy content Toggle raw display
$67$ \( T^{7} + 3 T^{6} - 332 T^{5} + \cdots + 1109504 \) Copy content Toggle raw display
$71$ \( T^{7} - 12 T^{6} - 279 T^{5} + \cdots + 6494768 \) Copy content Toggle raw display
$73$ \( T^{7} + 21 T^{6} + 59 T^{5} + \cdots + 56288 \) Copy content Toggle raw display
$79$ \( T^{7} - 15 T^{6} - 74 T^{5} + \cdots + 142336 \) Copy content Toggle raw display
$83$ \( T^{7} - 5 T^{6} - 205 T^{5} + \cdots + 196864 \) Copy content Toggle raw display
$89$ \( T^{7} + 8 T^{6} - 190 T^{5} + \cdots - 131482 \) Copy content Toggle raw display
$97$ \( T^{7} + 20 T^{6} - 2 T^{5} + \cdots + 4738 \) Copy content Toggle raw display
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