Properties

Label 4026.2.a.bb
Level $4026$
Weight $2$
Character orbit 4026.a
Self dual yes
Analytic conductor $32.148$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 22x^{6} + 42x^{5} + 182x^{4} - 111x^{3} - 538x^{2} - 256x - 32 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 22x^{6} + 42x^{5} + 182x^{4} - 111x^{3} - 538x^{2} - 256x - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{7} - 71\nu^{6} - 282\nu^{5} + 978\nu^{4} + 1918\nu^{3} - 3271\nu^{2} - 5518\nu - 808 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{7} - 71\nu^{6} - 282\nu^{5} + 978\nu^{4} + 1918\nu^{3} - 3127\nu^{2} - 5662\nu - 1672 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -17\nu^{7} + 47\nu^{6} + 402\nu^{5} - 738\nu^{4} - 3310\nu^{3} + 2455\nu^{2} + 9334\nu + 2440 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -35\nu^{7} + 101\nu^{6} + 798\nu^{5} - 1494\nu^{4} - 6730\nu^{3} + 4885\nu^{2} + 20170\nu + 4888 ) / 288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 22\nu^{5} - 42\nu^{4} - 182\nu^{3} + 119\nu^{2} + 530\nu + 176 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -21\nu^{7} + 67\nu^{6} + 466\nu^{5} - 1018\nu^{4} - 3878\nu^{3} + 3475\nu^{2} + 11718\nu + 2824 ) / 96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{5} + \beta_{4} + 2\beta_{3} - 2\beta_{2} + 11\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 6\beta_{6} - 7\beta_{5} + 16\beta_{3} - 15\beta_{2} + 25\beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} + 29\beta_{6} - 52\beta_{5} + 13\beta_{4} + 53\beta_{3} - 48\beta_{2} + 146\beta _1 + 135 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50\beta_{7} + 147\beta_{6} - 214\beta_{5} + \beta_{4} + 275\beta_{3} - 246\beta_{2} + 467\beta _1 + 723 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 284\beta_{7} + 637\beta_{6} - 1128\beta_{5} + 107\beta_{4} + 1074\beta_{3} - 919\beta_{2} + 2210\beta _1 + 2277 \) 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
−1.83085
−0.326029
4.26126
−2.29226
−2.68569
2.67396
−0.223274
3.42288
1.00000 1.00000 1.00000 −3.68259 1.00000 3.83085 1.00000 1.00000 −3.68259
1.2 1.00000 1.00000 1.00000 −2.31671 1.00000 2.32603 1.00000 1.00000 −2.31671
1.3 1.00000 1.00000 1.00000 −1.92862 1.00000 −2.26126 1.00000 1.00000 −1.92862
1.4 1.00000 1.00000 1.00000 0.723959 1.00000 4.29226 1.00000 1.00000 0.723959
1.5 1.00000 1.00000 1.00000 1.94066 1.00000 4.68569 1.00000 1.00000 1.94066
1.6 1.00000 1.00000 1.00000 2.72927 1.00000 −0.673957 1.00000 1.00000 2.72927
1.7 1.00000 1.00000 1.00000 3.42657 1.00000 2.22327 1.00000 1.00000 3.42657
1.8 1.00000 1.00000 1.00000 4.10747 1.00000 −1.42288 1.00000 1.00000 4.10747
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.bb 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{8} - 5T_{5}^{7} - 19T_{5}^{6} + 114T_{5}^{5} + 62T_{5}^{4} - 731T_{5}^{3} + 220T_{5}^{2} + 1400T_{5} - 888 \) Copy content Toggle raw display
\( T_{7}^{8} - 13T_{7}^{7} + 48T_{7}^{6} + 26T_{7}^{5} - 438T_{7}^{4} + 383T_{7}^{3} + 1020T_{7}^{2} - 900T_{7} - 864 \) Copy content Toggle raw display
\( T_{13}^{8} - 10T_{13}^{7} + 3T_{13}^{6} + 198T_{13}^{5} - 463T_{13}^{4} - 181T_{13}^{3} + 799T_{13}^{2} + 27T_{13} - 342 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} - 19 T^{6} + 114 T^{5} + \cdots - 888 \) Copy content Toggle raw display
$7$ \( T^{8} - 13 T^{7} + 48 T^{6} + \cdots - 864 \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + 3 T^{6} + 198 T^{5} + \cdots - 342 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} - 63 T^{6} + \cdots + 74928 \) Copy content Toggle raw display
$19$ \( T^{8} - 11 T^{7} - 34 T^{6} + \cdots - 68992 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} - 49 T^{6} + \cdots + 9472 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} - 78 T^{6} + \cdots - 316488 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} - 51 T^{6} + \cdots + 10512 \) Copy content Toggle raw display
$37$ \( T^{8} - 9 T^{7} - 198 T^{6} + \cdots + 1842288 \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} - 66 T^{6} + \cdots - 4632 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} - 139 T^{6} + \cdots + 3969216 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} - 170 T^{6} + \cdots + 854368 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{7} - 216 T^{6} + \cdots + 866448 \) Copy content Toggle raw display
$59$ \( T^{8} + 14 T^{7} + 17 T^{6} + \cdots - 4564 \) Copy content Toggle raw display
$61$ \( (T - 1)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} - 246 T^{6} + \cdots + 14042944 \) Copy content Toggle raw display
$71$ \( T^{8} - 11 T^{7} - 266 T^{6} + \cdots + 511104 \) Copy content Toggle raw display
$73$ \( T^{8} - 14 T^{7} - 241 T^{6} + \cdots - 389376 \) Copy content Toggle raw display
$79$ \( T^{8} - 22 T^{7} - 12 T^{6} + \cdots + 169088 \) Copy content Toggle raw display
$83$ \( T^{8} + 16 T^{7} - 379 T^{6} + \cdots - 493824 \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} - 491 T^{6} + \cdots - 158264 \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} - 431 T^{6} + \cdots - 91436722 \) Copy content Toggle raw display
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