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Properties

Label 6026.2.a.f
Level $6026$
Weight $2$
Character orbit 6026.a
Self dual yes
Analytic conductor $48.118$
Analytic rank $1$
Dimension $20$
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] = [6026,2,Mod(1,6026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6026, 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("6026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6026.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: \(48.1178522580\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1083 x^{15} + 248 x^{14} + 5359 x^{13} + \cdots - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1083 x^{15} + 248 x^{14} + 5359 x^{13} + \cdots - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 99716990 \nu^{19} - 165884442 \nu^{18} - 2663674000 \nu^{17} + 3599346675 \nu^{16} + \cdots - 5029558767 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 152897921 \nu^{19} - 788021802 \nu^{18} - 2581287385 \nu^{17} + 17942676336 \nu^{16} + \cdots - 1903874094 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 309851458 \nu^{19} - 1383955827 \nu^{18} - 6233624999 \nu^{17} + 33136669317 \nu^{16} + \cdots + 10210385484 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 322669970 \nu^{19} - 1360457148 \nu^{18} - 6450050938 \nu^{17} + 31439546223 \nu^{16} + \cdots + 11277017919 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 322669970 \nu^{19} - 1360457148 \nu^{18} - 6450050938 \nu^{17} + 31439546223 \nu^{16} + \cdots + 7853033133 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 146343330 \nu^{19} - 627942667 \nu^{18} - 2750855046 \nu^{17} + 13927919237 \nu^{16} + \cdots + 3805064997 ) / 570664131 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 477486794 \nu^{19} + 2377631685 \nu^{18} + 8657638597 \nu^{17} - 56024761614 \nu^{16} + \cdots - 21496095405 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 550477613 \nu^{19} - 2275865718 \nu^{18} - 11409071752 \nu^{17} + 54163977555 \nu^{16} + \cdots + 18210231207 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 562516600 \nu^{19} - 2270260821 \nu^{18} - 11766008126 \nu^{17} + 53784538836 \nu^{16} + \cdots + 12725619606 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 840747436 \nu^{19} + 3451389795 \nu^{18} + 17340713522 \nu^{17} - 81372015369 \nu^{16} + \cdots - 15043298778 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 306455260 \nu^{19} + 1193388237 \nu^{18} + 6440353895 \nu^{17} - 27863463012 \nu^{16} + \cdots - 6583539255 ) / 570664131 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 306841820 \nu^{19} - 1142759006 \nu^{18} - 6603875713 \nu^{17} + 26675284060 \nu^{16} + \cdots + 3636815862 ) / 570664131 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1084667243 \nu^{19} - 4546314924 \nu^{18} - 21817310038 \nu^{17} + 105828203667 \nu^{16} + \cdots + 21903959223 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1089425851 \nu^{19} - 4793379879 \nu^{18} - 21455653031 \nu^{17} + 112365319416 \nu^{16} + \cdots + 26342084538 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 4456799 \nu^{19} - 18879387 \nu^{18} - 90366943 \nu^{17} + 445558719 \nu^{16} + 685127538 \nu^{15} + \cdots + 133089489 ) / 6661449 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 1571933387 \nu^{19} - 6592015260 \nu^{18} - 31890712168 \nu^{17} + 154671482871 \nu^{16} + \cdots + 44758625463 ) / 1711992393 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 600429471 \nu^{19} - 2519966401 \nu^{18} - 12156366042 \nu^{17} + 59005753619 \nu^{16} + \cdots + 16459377834 ) / 570664131 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} + \beta_{14} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 7\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + 3 \beta_{14} + 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + 2 \beta_{18} - 12 \beta_{17} + \beta_{16} - 2 \beta_{15} + 12 \beta_{14} + \beta_{13} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 9 \beta_{18} - 17 \beta_{17} + 13 \beta_{16} - 4 \beta_{15} + 37 \beta_{14} + 22 \beta_{13} + \cdots + 145 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 22 \beta_{19} + 28 \beta_{18} - 114 \beta_{17} + 17 \beta_{16} - 31 \beta_{15} + 109 \beta_{14} + \cdots + 673 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 150 \beta_{19} - 51 \beta_{18} - 202 \beta_{17} + 129 \beta_{16} - 73 \beta_{15} + 343 \beta_{14} + \cdots + 1229 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 300 \beta_{19} + 281 \beta_{18} - 1009 \beta_{17} + 209 \beta_{16} - 349 \beta_{15} + 917 \beta_{14} + \cdots + 4730 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1457 \beta_{19} - 161 \beta_{18} - 2077 \beta_{17} + 1166 \beta_{16} - 898 \beta_{15} + 2892 \beta_{14} + \cdots + 9882 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3342 \beta_{19} + 2525 \beta_{18} - 8679 \beta_{17} + 2212 \beta_{16} - 3470 \beta_{15} + 7512 \beta_{14} + \cdots + 34160 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13444 \beta_{19} + 740 \beta_{18} - 19767 \beta_{17} + 10096 \beta_{16} - 9373 \beta_{15} + 23467 \beta_{14} + \cdots + 77594 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 33431 \beta_{19} + 21740 \beta_{18} - 73552 \beta_{17} + 21449 \beta_{16} - 32345 \beta_{15} + \cdots + 251135 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 120065 \beta_{19} + 20697 \beta_{18} - 179378 \beta_{17} + 85390 \beta_{16} - 89704 \beta_{15} + \cdots + 602574 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 313513 \beta_{19} + 183868 \beta_{18} - 617232 \beta_{17} + 196626 \beta_{16} - 289842 \beta_{15} + \cdots + 1869005 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1047731 \beta_{19} + 267647 \beta_{18} - 1576771 \beta_{17} + 711696 \beta_{16} - 815151 \beta_{15} + \cdots + 4656292 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2819553 \beta_{19} + 1541732 \beta_{18} - 5140078 \beta_{17} + 1735208 \beta_{16} - 2528798 \beta_{15} + \cdots + 14033583 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 8984716 \beta_{19} + 2827347 \beta_{18} - 13552001 \beta_{17} + 5871457 \beta_{16} - 7162439 \beta_{15} + \cdots + 35913688 \) 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
2.80474
2.65160
2.60929
1.87740
1.76837
1.71702
1.51806
1.09509
0.700104
0.686421
0.178107
−0.100365
−0.435939
−0.590088
−1.29878
−1.82988
−1.91911
−1.95611
−2.02882
−2.44709
1.00000 −2.80474 1.00000 −0.947662 −2.80474 2.44948 1.00000 4.86656 −0.947662
1.2 1.00000 −2.65160 1.00000 0.347024 −2.65160 −2.28847 1.00000 4.03099 0.347024
1.3 1.00000 −2.60929 1.00000 1.96845 −2.60929 −1.34895 1.00000 3.80840 1.96845
1.4 1.00000 −1.87740 1.00000 3.35129 −1.87740 −2.95261 1.00000 0.524644 3.35129
1.5 1.00000 −1.76837 1.00000 1.49209 −1.76837 3.65436 1.00000 0.127121 1.49209
1.6 1.00000 −1.71702 1.00000 −4.22974 −1.71702 −1.91461 1.00000 −0.0518545 −4.22974
1.7 1.00000 −1.51806 1.00000 −3.34509 −1.51806 1.42565 1.00000 −0.695496 −3.34509
1.8 1.00000 −1.09509 1.00000 −0.379683 −1.09509 −3.84052 1.00000 −1.80078 −0.379683
1.9 1.00000 −0.700104 1.00000 1.01839 −0.700104 0.757569 1.00000 −2.50985 1.01839
1.10 1.00000 −0.686421 1.00000 1.74349 −0.686421 −1.83413 1.00000 −2.52883 1.74349
1.11 1.00000 −0.178107 1.00000 −1.41285 −0.178107 −1.81585 1.00000 −2.96828 −1.41285
1.12 1.00000 0.100365 1.00000 −2.25871 0.100365 1.56060 1.00000 −2.98993 −2.25871
1.13 1.00000 0.435939 1.00000 −0.796234 0.435939 3.50309 1.00000 −2.80996 −0.796234
1.14 1.00000 0.590088 1.00000 0.936829 0.590088 1.40799 1.00000 −2.65180 0.936829
1.15 1.00000 1.29878 1.00000 2.87965 1.29878 −3.57798 1.00000 −1.31316 2.87965
1.16 1.00000 1.82988 1.00000 0.255245 1.82988 −4.20483 1.00000 0.348474 0.255245
1.17 1.00000 1.91911 1.00000 −2.55109 1.91911 0.891228 1.00000 0.682993 −2.55109
1.18 1.00000 1.95611 1.00000 −3.19910 1.95611 0.785940 1.00000 0.826384 −3.19910
1.19 1.00000 2.02882 1.00000 −0.795313 2.02882 −4.29820 1.00000 1.11611 −0.795313
1.20 1.00000 2.44709 1.00000 −0.0770010 2.44709 −0.359767 1.00000 2.98826 −0.0770010
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(-1\)
\(131\) \(1\)

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 6026.2.a.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6026.2.a.f ✓ 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6026))\):

\( T_{3}^{20} + 5 T_{3}^{19} - 17 T_{3}^{18} - 115 T_{3}^{17} + 78 T_{3}^{16} + 1083 T_{3}^{15} + 248 T_{3}^{14} + \cdots - 18 \) Copy content Toggle raw display
\( T_{5}^{20} + 6 T_{5}^{19} - 25 T_{5}^{18} - 189 T_{5}^{17} + 190 T_{5}^{16} + 2285 T_{5}^{15} + \cdots - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 5 T^{19} + \cdots - 18 \) Copy content Toggle raw display
$5$ \( T^{20} + 6 T^{19} + \cdots - 27 \) Copy content Toggle raw display
$7$ \( T^{20} + 12 T^{19} + \cdots - 270684 \) Copy content Toggle raw display
$11$ \( T^{20} + 3 T^{19} + \cdots - 424 \) Copy content Toggle raw display
$13$ \( T^{20} + 13 T^{19} + \cdots - 19568 \) Copy content Toggle raw display
$17$ \( T^{20} + 14 T^{19} + \cdots + 677991 \) Copy content Toggle raw display
$19$ \( T^{20} + 21 T^{19} + \cdots + 54469298 \) Copy content Toggle raw display
$23$ \( (T - 1)^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 25679638168 \) Copy content Toggle raw display
$31$ \( T^{20} + 27 T^{19} + \cdots - 14361232 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 3661515520 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 52756161123 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 318682312 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 2998099362 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 9024059268660 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 148011951389146 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 982687065453 \) Copy content Toggle raw display
$67$ \( T^{20} + 8 T^{19} + \cdots - 53620778 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13\!\cdots\!78 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 69613391890416 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 38\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 9032621002110 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 48833158321728 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
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