Properties

Label 4020.2.q.k
Level $4020$
Weight $2$
Character orbit 4020.q
Analytic conductor $32.100$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4020,2,Mod(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.q (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 11 x^{12} - 8 x^{11} + 88 x^{10} - 57 x^{9} + 270 x^{8} + 17 x^{7} + 458 x^{6} - 101 x^{5} + 189 x^{4} - 30 x^{3} + 54 x^{2} - 12 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - \beta_{6} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - \beta_{6} q^{7} + q^{9} + ( - \beta_{12} - \beta_{10} - \beta_{5} + 1) q^{11} + (\beta_{5} - \beta_{3} - \beta_1) q^{13} - q^{15} + (\beta_{13} - \beta_{11} - 2 \beta_{5} - \beta_{2}) q^{17} + ( - \beta_{11} + 2 \beta_{5} - \beta_{2}) q^{19} + \beta_{6} q^{21} + ( - \beta_{13} - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{2}) q^{23} + q^{25} - q^{27} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} - 1) q^{29} + (\beta_{12} - \beta_{11} - \beta_{10} - \beta_{6} - \beta_{5} + \beta_1 + 1) q^{31} + (\beta_{12} + \beta_{10} + \beta_{5} - 1) q^{33} - \beta_{6} q^{35} + ( - \beta_{12} - 2 \beta_{11} + \beta_{9} - 2 \beta_{2}) q^{37} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{39} + (\beta_{13} + \beta_{12} + 2 \beta_{5} - \beta_{4} - 2) q^{41} + (2 \beta_{9} - 2 \beta_{8} - \beta_{2} + 1) q^{43} + q^{45} + ( - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_1 + 2) q^{47} + (\beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2}) q^{49} + ( - \beta_{13} + \beta_{11} + 2 \beta_{5} + \beta_{2}) q^{51} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2}) q^{53} + ( - \beta_{12} - \beta_{10} - \beta_{5} + 1) q^{55} + (\beta_{11} - 2 \beta_{5} + \beta_{2}) q^{57} + (\beta_{9} + 2 \beta_{8} - \beta_{3} - 1) q^{59} + (\beta_{13} - \beta_{12} - 2 \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{61}+ \cdots + ( - \beta_{12} - \beta_{10} - \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{3} + 14 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{3} + 14 q^{5} + 3 q^{7} + 14 q^{9} + 6 q^{11} + 9 q^{13} - 14 q^{15} - 12 q^{17} + 14 q^{19} - 3 q^{21} + 6 q^{23} + 14 q^{25} - 14 q^{27} - q^{29} + 7 q^{31} - 6 q^{33} + 3 q^{35} - 2 q^{37} - 9 q^{39} - 18 q^{41} - 6 q^{43} + 14 q^{45} + 7 q^{47} + 12 q^{51} - 12 q^{53} + 6 q^{55} - 14 q^{57} - 2 q^{59} + 3 q^{63} + 9 q^{65} + 25 q^{67} - 6 q^{69} + 22 q^{71} - 15 q^{73} - 14 q^{75} + q^{77} + 9 q^{79} + 14 q^{81} - q^{83} - 12 q^{85} + q^{87} - 12 q^{89} - 38 q^{91} - 7 q^{93} + 14 q^{95} + 16 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 11 x^{12} - 8 x^{11} + 88 x^{10} - 57 x^{9} + 270 x^{8} + 17 x^{7} + 458 x^{6} - 101 x^{5} + 189 x^{4} - 30 x^{3} + 54 x^{2} - 12 x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2680061185 \nu^{13} + 4118856992 \nu^{12} - 29747186705 \nu^{11} + 35610122429 \nu^{10} - 234940121607 \nu^{9} + \cdots + 970249939441 ) / 325648644013 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3347996299 \nu^{13} - 8708118669 \nu^{12} + 45065673273 \nu^{11} - 86278343802 \nu^{10} + 365843919170 \nu^{9} - 660716032257 \nu^{8} + \cdots - 194090246132 ) / 325648644013 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10892153560 \nu^{13} + 9758640678 \nu^{12} - 105683331260 \nu^{11} + 64289364146 \nu^{10} - 815634907035 \nu^{9} + \cdots + 861786952374 ) / 325648644013 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48522561533 \nu^{13} + 45174565234 \nu^{12} - 525040058194 \nu^{11} + 343114818991 \nu^{10} - 4183707071102 \nu^{9} + \cdots + 410860145420 ) / 651297288026 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 809674271 \nu^{13} - 252390382 \nu^{12} - 8122293536 \nu^{11} - 4713678201 \nu^{10} - 65999965440 \nu^{9} - 42770485013 \nu^{8} + \cdots - 7944658082 ) / 8458406338 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37194535803 \nu^{13} - 47919288537 \nu^{12} + 395084796401 \nu^{11} - 400339609502 \nu^{10} + 3098569859104 \nu^{9} + \cdots - 841108812233 ) / 325648644013 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 45711237420 \nu^{13} + 74034574851 \nu^{12} - 518899664240 \nu^{11} + 668330017670 \nu^{10} - 4117338117621 \nu^{9} + \cdots + 974875365383 ) / 325648644013 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58435287502 \nu^{13} + 104277787850 \nu^{12} - 683843870764 \nu^{11} + 962775171505 \nu^{10} - 5449809996738 \nu^{9} + \cdots + 2201747399207 ) / 325648644013 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 126618741863 \nu^{13} - 49216475610 \nu^{12} + 1335886940970 \nu^{11} - 183514685193 \nu^{10} + 10733169032550 \nu^{9} + \cdots + 903683328390 ) / 651297288026 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20029651747 \nu^{13} + 18183711674 \nu^{12} - 216517971596 \nu^{11} + 136874887445 \nu^{10} - 1725891567156 \nu^{9} + \cdots - 101131348946 ) / 93042469718 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 156907511177 \nu^{13} - 63210384410 \nu^{12} + 1644341611148 \nu^{11} - 250245646301 \nu^{10} + 13207909689396 \nu^{9} + \cdots + 1100661461810 ) / 651297288026 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 87738052524 \nu^{13} + 71009258532 \nu^{12} - 933363507764 \nu^{11} + 502153407798 \nu^{10} - 7424278972227 \nu^{9} + \cdots + 438504860088 ) / 325648644013 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 3\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + 3\beta_{3} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - 8\beta_{11} - \beta_{6} + 18\beta_{5} - \beta_{4} + \beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{12} + 10 \beta_{11} + \beta_{10} - 8 \beta_{9} + 9 \beta_{8} + \beta_{7} + 9 \beta_{6} - 12 \beta_{5} - 20 \beta_{3} + 10 \beta_{2} - 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{9} - 10\beta_{8} + \beta_{7} + 10\beta_{4} + 13\beta_{3} - 58\beta_{2} + 119 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{13} - 58\beta_{12} - 84\beta_{11} - 11\beta_{10} - 68\beta_{6} + 117\beta_{5} + \beta_{4} + 136\beta _1 - 117 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 80 \beta_{13} + 84 \beta_{12} + 415 \beta_{11} - 12 \beta_{10} - 84 \beta_{9} + 83 \beta_{8} - 12 \beta_{7} + 83 \beta_{6} - 812 \beta_{5} - 127 \beta_{3} + 415 \beta_{2} - 127 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 415\beta_{9} - 495\beta_{8} - 92\beta_{7} - 9\beta_{4} + 939\beta_{3} - 673\beta_{2} + 1027 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 596 \beta_{13} - 673 \beta_{12} - 2975 \beta_{11} + 101 \beta_{10} - 664 \beta_{6} + 5649 \beta_{5} - 596 \beta_{4} + 1114 \beta _1 - 5649 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 33 \beta_{13} + 2975 \beta_{12} + 5280 \beta_{11} + 697 \beta_{10} - 2975 \beta_{9} + 3571 \beta_{8} + 697 \beta_{7} + 3571 \beta_{6} - 8509 \beta_{5} - 6568 \beta_{3} + 5280 \beta_{2} - 6568 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5280\beta_{9} - 5247\beta_{8} + 730\beta_{7} + 4301\beta_{4} + 9250\beta_{3} - 21424\beta_{2} + 39857 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 216 \beta_{13} - 21424 \beta_{12} - 40903 \beta_{11} - 5031 \beta_{10} - 25725 \beta_{6} + 68139 \beta_{5} - 216 \beta_{4} + 46436 \beta _1 - 68139 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
0.295122 0.511167i
−0.288886 + 0.500366i
1.10425 1.91261i
0.150493 0.260662i
1.24942 2.16407i
−1.36264 + 2.36015i
−0.647766 + 1.12196i
0.295122 + 0.511167i
−0.288886 0.500366i
1.10425 + 1.91261i
0.150493 + 0.260662i
1.24942 + 2.16407i
−1.36264 2.36015i
−0.647766 1.12196i
0 −1.00000 0 1.00000 0 −1.59463 + 2.76199i 0 1.00000 0
841.2 0 −1.00000 0 1.00000 0 −1.13810 + 1.97125i 0 1.00000 0
841.3 0 −1.00000 0 1.00000 0 −0.952306 + 1.64944i 0 1.00000 0
841.4 0 −1.00000 0 1.00000 0 0.827889 1.43395i 0 1.00000 0
841.5 0 −1.00000 0 1.00000 0 1.07806 1.86725i 0 1.00000 0
841.6 0 −1.00000 0 1.00000 0 1.26463 2.19041i 0 1.00000 0
841.7 0 −1.00000 0 1.00000 0 2.01447 3.48916i 0 1.00000 0
3781.1 0 −1.00000 0 1.00000 0 −1.59463 2.76199i 0 1.00000 0
3781.2 0 −1.00000 0 1.00000 0 −1.13810 1.97125i 0 1.00000 0
3781.3 0 −1.00000 0 1.00000 0 −0.952306 1.64944i 0 1.00000 0
3781.4 0 −1.00000 0 1.00000 0 0.827889 + 1.43395i 0 1.00000 0
3781.5 0 −1.00000 0 1.00000 0 1.07806 + 1.86725i 0 1.00000 0
3781.6 0 −1.00000 0 1.00000 0 1.26463 + 2.19041i 0 1.00000 0
3781.7 0 −1.00000 0 1.00000 0 2.01447 + 3.48916i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4020.2.q.k 14
67.c even 3 1 inner 4020.2.q.k 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.q.k 14 1.a even 1 1 trivial
4020.2.q.k 14 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\):

\( T_{7}^{14} - 3 T_{7}^{13} + 29 T_{7}^{12} - 48 T_{7}^{11} + 443 T_{7}^{10} - 662 T_{7}^{9} + 3972 T_{7}^{8} - 4257 T_{7}^{7} + 22767 T_{7}^{6} - 21740 T_{7}^{5} + 86396 T_{7}^{4} - 55186 T_{7}^{3} + 196412 T_{7}^{2} + \cdots + 253009 \) Copy content Toggle raw display
\( T_{11}^{14} - 6 T_{11}^{13} + 66 T_{11}^{12} - 314 T_{11}^{11} + 2508 T_{11}^{10} - 10626 T_{11}^{9} + 52703 T_{11}^{8} - 150791 T_{11}^{7} + 526254 T_{11}^{6} - 1182652 T_{11}^{5} + 3345335 T_{11}^{4} - 4939116 T_{11}^{3} + \cdots + 112225 \) Copy content Toggle raw display
\( T_{17}^{14} + 12 T_{17}^{13} + 132 T_{17}^{12} + 900 T_{17}^{11} + 6523 T_{17}^{10} + 36897 T_{17}^{9} + 197164 T_{17}^{8} + 755693 T_{17}^{7} + 2356939 T_{17}^{6} + 4625169 T_{17}^{5} + 6676031 T_{17}^{4} + \cdots + 165649 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( (T - 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 3 T^{13} + 29 T^{12} + \cdots + 253009 \) Copy content Toggle raw display
$11$ \( T^{14} - 6 T^{13} + 66 T^{12} + \cdots + 112225 \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + 88 T^{12} + \cdots + 15625 \) Copy content Toggle raw display
$17$ \( T^{14} + 12 T^{13} + 132 T^{12} + \cdots + 165649 \) Copy content Toggle raw display
$19$ \( T^{14} - 14 T^{13} + 141 T^{12} + \cdots + 1225 \) Copy content Toggle raw display
$23$ \( T^{14} - 6 T^{13} + 106 T^{12} + \cdots + 10778089 \) Copy content Toggle raw display
$29$ \( T^{14} + T^{13} + 109 T^{12} + \cdots + 327429025 \) Copy content Toggle raw display
$31$ \( T^{14} - 7 T^{13} + \cdots + 35082413809 \) Copy content Toggle raw display
$37$ \( T^{14} + 2 T^{13} + 115 T^{12} + \cdots + 10465225 \) Copy content Toggle raw display
$41$ \( T^{14} + 18 T^{13} + 258 T^{12} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( (T^{7} + 3 T^{6} - 136 T^{5} - 149 T^{4} + \cdots + 64592)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} - 7 T^{13} + 176 T^{12} + \cdots + 63091249 \) Copy content Toggle raw display
$53$ \( (T^{7} + 6 T^{6} - 124 T^{5} - 763 T^{4} + \cdots - 91420)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + T^{6} - 124 T^{5} + 49 T^{4} + \cdots - 7504)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + 177 T^{12} + \cdots + 87853129 \) Copy content Toggle raw display
$67$ \( T^{14} - 25 T^{13} + \cdots + 6060711605323 \) Copy content Toggle raw display
$71$ \( T^{14} - 22 T^{13} + \cdots + 88534217209 \) Copy content Toggle raw display
$73$ \( T^{14} + 15 T^{13} + 206 T^{12} + \cdots + 94381225 \) Copy content Toggle raw display
$79$ \( T^{14} - 9 T^{13} + \cdots + 5828559025 \) Copy content Toggle raw display
$83$ \( T^{14} + T^{13} + 282 T^{12} + \cdots + 4133075521 \) Copy content Toggle raw display
$89$ \( (T^{7} + 6 T^{6} - 114 T^{5} - 811 T^{4} + \cdots + 12500)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} - 16 T^{13} + \cdots + 877554894841 \) Copy content Toggle raw display
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