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Properties

Label 4029.2.a.e
Level $4029$
Weight $2$
Character orbit 4029.a
Self dual yes
Analytic conductor $32.172$
Analytic rank $1$
Dimension $18$
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] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, 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("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 10 x^{16} + 120 x^{15} - 56 x^{14} - 921 x^{13} + 1181 x^{12} + 3316 x^{11} + \cdots + 138 \) 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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 10 x^{16} + 120 x^{15} - 56 x^{14} - 921 x^{13} + 1181 x^{12} + 3316 x^{11} + \cdots + 138 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14 \nu^{17} - 113 \nu^{16} + 69 \nu^{15} + 1603 \nu^{14} - 3539 \nu^{13} - 6970 \nu^{12} + 25822 \nu^{11} + \cdots + 579 ) / 65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 43 \nu^{17} + 309 \nu^{16} + 49 \nu^{15} - 4878 \nu^{14} + 7019 \nu^{13} + 27126 \nu^{12} + \cdots + 752 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{17} - 44 \nu^{16} + 268 \nu^{15} - 57 \nu^{14} - 4166 \nu^{13} + 8732 \nu^{12} + 19508 \nu^{11} + \cdots + 3901 ) / 65 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29 \nu^{17} - 14 \nu^{16} - 1249 \nu^{15} + 2391 \nu^{14} + 14200 \nu^{13} - 37290 \nu^{12} + \cdots - 8052 ) / 65 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 135 \nu^{17} + 747 \nu^{16} + 1306 \nu^{15} - 13046 \nu^{14} + 2862 \nu^{13} + 86395 \nu^{12} + \cdots + 5672 ) / 65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 112 \nu^{17} - 553 \nu^{16} - 1554 \nu^{15} + 10770 \nu^{14} + 4760 \nu^{13} - 82553 \nu^{12} + \cdots - 8173 ) / 65 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 122 \nu^{17} + 604 \nu^{16} + 1683 \nu^{15} - 11759 \nu^{14} - 4938 \nu^{13} + 89970 \nu^{12} + \cdots + 9013 ) / 65 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 155 \nu^{17} + 862 \nu^{16} + 1603 \nu^{15} - 15648 \nu^{14} + 2259 \nu^{13} + 109679 \nu^{12} + \cdots + 9380 ) / 65 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 144 \nu^{17} - 620 \nu^{16} - 2609 \nu^{15} + 13641 \nu^{14} + 15402 \nu^{13} - 118549 \nu^{12} + \cdots - 14371 ) / 65 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 216 \nu^{17} + 982 \nu^{16} + 3465 \nu^{15} - 20078 \nu^{14} - 16733 \nu^{13} + 162347 \nu^{12} + \cdots + 16948 ) / 65 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 196 \nu^{17} + 880 \nu^{16} + 3311 \nu^{15} - 18659 \nu^{14} - 17313 \nu^{13} + 156626 \nu^{12} + \cdots + 18934 ) / 65 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 22 \nu^{17} - 100 \nu^{16} - 359 \nu^{15} + 2075 \nu^{14} + 1769 \nu^{13} - 17036 \nu^{12} + 363 \nu^{11} + \cdots - 1881 ) / 5 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 277 \nu^{17} + 1206 \nu^{16} + 4872 \nu^{15} - 26003 \nu^{14} - 27548 \nu^{13} + 221606 \nu^{12} + \cdots + 26492 ) / 65 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 432 \nu^{17} + 2146 \nu^{16} + 5890 \nu^{15} - 41521 \nu^{14} - 16670 \nu^{13} + 315542 \nu^{12} + \cdots + 30464 ) / 65 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 428 \nu^{17} - 1759 \nu^{16} - 8233 \nu^{15} + 39932 \nu^{14} + 53407 \nu^{13} - 356679 \nu^{12} + \cdots - 45468 ) / 65 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{17} - \beta_{15} + \beta_{14} + \beta_{7} + \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} - \beta_{15} + \beta_{14} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} + 8\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{17} + \beta_{16} - 8 \beta_{15} + 9 \beta_{14} + 2 \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{17} - \beta_{16} - 11 \beta_{15} + 13 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 11 \beta_{10} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 68 \beta_{17} + 9 \beta_{16} - 55 \beta_{15} + 72 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 97 \beta_{17} - 14 \beta_{16} - 91 \beta_{15} + 128 \beta_{14} + 33 \beta_{13} + 31 \beta_{12} + \cdots + 423 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 496 \beta_{17} + 59 \beta_{16} - 369 \beta_{15} + 564 \beta_{14} + 74 \beta_{13} + 68 \beta_{12} + \cdots + 320 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 808 \beta_{17} - 135 \beta_{16} - 691 \beta_{15} + 1146 \beta_{14} + 380 \beta_{13} + 339 \beta_{12} + \cdots + 2563 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3610 \beta_{17} + 329 \beta_{16} - 2486 \beta_{15} + 4402 \beta_{14} + 917 \beta_{13} + 797 \beta_{12} + \cdots + 2811 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6584 \beta_{17} - 1131 \beta_{16} - 5089 \beta_{15} + 9796 \beta_{14} + 3786 \beta_{13} + \cdots + 16233 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 26459 \beta_{17} + 1550 \beta_{16} - 16953 \beta_{15} + 34359 \beta_{14} + 9597 \beta_{13} + \cdots + 22809 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 52969 \beta_{17} - 8906 \beta_{16} - 37087 \beta_{15} + 81465 \beta_{14} + 34992 \beta_{13} + \cdots + 106330 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 195665 \beta_{17} + 5142 \beta_{16} - 117226 \beta_{15} + 268286 \beta_{14} + 91728 \beta_{13} + \cdots + 177289 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 422173 \beta_{17} - 68279 \beta_{16} - 269753 \beta_{15} + 665226 \beta_{14} + 309301 \beta_{13} + \cdots + 714456 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1459248 \beta_{17} - 4804 \beta_{16} - 821669 \beta_{15} + 2094864 \beta_{14} + 829624 \beta_{13} + \cdots + 1344775 \) 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
2.77821
2.62992
2.53461
1.99378
1.64567
1.59625
1.47280
1.44853
0.615823
0.498176
−0.239152
−0.540222
−0.823313
−0.926931
−2.08696
−2.08785
−2.10151
−2.40783
−2.77821 1.00000 5.71847 1.71746 −2.77821 −2.08361 −10.3307 1.00000 −4.77148
1.2 −2.62992 1.00000 4.91649 0.484241 −2.62992 0.944316 −7.67014 1.00000 −1.27352
1.3 −2.53461 1.00000 4.42426 −2.65587 −2.53461 1.73204 −6.14455 1.00000 6.73160
1.4 −1.99378 1.00000 1.97517 1.24008 −1.99378 −4.06333 0.0495043 1.00000 −2.47246
1.5 −1.64567 1.00000 0.708234 −0.419467 −1.64567 1.49188 2.12582 1.00000 0.690305
1.6 −1.59625 1.00000 0.548008 −4.15667 −1.59625 −4.26341 2.31774 1.00000 6.63507
1.7 −1.47280 1.00000 0.169132 4.03939 −1.47280 0.912411 2.69650 1.00000 −5.94920
1.8 −1.44853 1.00000 0.0982249 1.34868 −1.44853 −1.91511 2.75477 1.00000 −1.95360
1.9 −0.615823 1.00000 −1.62076 −3.91500 −0.615823 4.24139 2.22975 1.00000 2.41095
1.10 −0.498176 1.00000 −1.75182 2.85615 −0.498176 −0.756391 1.86907 1.00000 −1.42286
1.11 0.239152 1.00000 −1.94281 −0.863011 0.239152 −3.18045 −0.942930 1.00000 −0.206391
1.12 0.540222 1.00000 −1.70816 3.09244 0.540222 −3.94821 −2.00323 1.00000 1.67061
1.13 0.823313 1.00000 −1.32216 0.963284 0.823313 4.26360 −2.73517 1.00000 0.793084
1.14 0.926931 1.00000 −1.14080 −2.07435 0.926931 1.24239 −2.91130 1.00000 −1.92278
1.15 2.08696 1.00000 2.35542 −2.36854 2.08696 −0.791085 0.741746 1.00000 −4.94306
1.16 2.08785 1.00000 2.35911 −3.16874 2.08785 0.308283 0.749773 1.00000 −6.61585
1.17 2.10151 1.00000 2.41637 −0.968130 2.10151 −2.79477 0.874998 1.00000 −2.03454
1.18 2.40783 1.00000 3.79762 −0.151954 2.40783 −4.33995 4.32836 1.00000 −0.365880
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

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 4029.2.a.e ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.e ✓ 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\( T_{2}^{18} + 6 T_{2}^{17} - 10 T_{2}^{16} - 120 T_{2}^{15} - 56 T_{2}^{14} + 921 T_{2}^{13} + 1181 T_{2}^{12} + \cdots + 138 \) Copy content Toggle raw display
\( T_{5}^{18} + 5 T_{5}^{17} - 39 T_{5}^{16} - 211 T_{5}^{15} + 543 T_{5}^{14} + 3345 T_{5}^{13} + \cdots + 1713 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 6 T^{17} + \cdots + 138 \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 5 T^{17} + \cdots + 1713 \) Copy content Toggle raw display
$7$ \( T^{18} + 13 T^{17} + \cdots + 97147 \) Copy content Toggle raw display
$11$ \( T^{18} + 27 T^{17} + \cdots + 4078016 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 136690556 \) Copy content Toggle raw display
$17$ \( (T - 1)^{18} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 369538843 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 100811587 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 251640617656 \) Copy content Toggle raw display
$31$ \( T^{18} + 18 T^{17} + \cdots - 69241568 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 174779303 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 38519458368 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 897551424 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 19505082249 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 81150029667 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 76376045403261 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 62\!\cdots\!33 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 3243294847644 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 28\!\cdots\!66 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 101768249578318 \) Copy content Toggle raw display
$79$ \( (T + 1)^{18} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 210574413039302 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 808427660064 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 2755203110338 \) Copy content Toggle raw display
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