Properties

Label 4020.2.q.j
Level $4020$
Weight $2$
Character orbit 4020.q
Analytic conductor $32.100$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 30 x^{10} - 53 x^{9} + 798 x^{8} - 1096 x^{7} + 4060 x^{6} - 915 x^{5} + 10392 x^{4} - 7038 x^{3} + 4869 x^{2} - 675 x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( - \beta_{9} - \beta_{6} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + ( - \beta_{9} - \beta_{6} + \beta_1) q^{7} + q^{9} + (\beta_{10} + \beta_{5} - 1) q^{11} + \beta_{5} q^{13} - q^{15} + (\beta_{10} + \beta_{8} + 2 \beta_{5} - \beta_{4} + \beta_{2}) q^{17} + (\beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{19} + ( - \beta_{9} - \beta_{6} + \beta_1) q^{21} + ( - \beta_{8} + 3 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{23} + q^{25} + q^{27} + (\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_1 + 1) q^{29} + (\beta_{11} + \beta_{10} + \beta_{5} - 1) q^{31} + (\beta_{10} + \beta_{5} - 1) q^{33} + (\beta_{9} + \beta_{6} - \beta_1) q^{35} + (\beta_{9} + 2 \beta_{8} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{37} + \beta_{5} q^{39} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{5} + \beta_1 - 1) q^{41} + ( - \beta_{7} + \beta_{6} - \beta_{3} - 1) q^{43} - q^{45} + (2 \beta_{11} + 2 \beta_{10} + \beta_{8} + 2 \beta_{5} - 2) q^{47} + ( - 2 \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{7} - 8 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{49} + (\beta_{10} + \beta_{8} + 2 \beta_{5} - \beta_{4} + \beta_{2}) q^{51} + ( - \beta_{7} + \beta_{4} - 2) q^{53} + ( - \beta_{10} - \beta_{5} + 1) q^{55} + (\beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{57} + ( - \beta_{6} - 2 \beta_{3} - 2 \beta_{2}) q^{59} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 4 \beta_{5} + \beta_{4} - \beta_{2}) q^{61} + ( - \beta_{9} - \beta_{6} + \beta_1) q^{63} - \beta_{5} q^{65} + ( - \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{67} + ( - \beta_{8} + 3 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{69} + (\beta_{10} + \beta_{9} + \beta_{6} - 4 \beta_{5} + \beta_1 + 4) q^{71} + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{73} + q^{75} + (\beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} - \beta_{4} - 4 \beta_{3} + \cdots + 4 \beta_1) q^{77}+ \cdots + (\beta_{10} + \beta_{5} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 12 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 12 q^{5} + 2 q^{7} + 12 q^{9} - 5 q^{11} + 6 q^{13} - 12 q^{15} + 9 q^{17} - 11 q^{19} + 2 q^{21} + 19 q^{23} + 12 q^{25} + 12 q^{27} + 8 q^{29} - 4 q^{31} - 5 q^{33} - 2 q^{35} + 5 q^{37} + 6 q^{39} - 9 q^{41} - 14 q^{43} - 12 q^{45} - 6 q^{47} - 46 q^{49} + 9 q^{51} - 20 q^{53} + 5 q^{55} - 11 q^{57} + 6 q^{59} + 27 q^{61} + 2 q^{63} - 6 q^{65} + 3 q^{67} + 19 q^{69} + 25 q^{71} - 8 q^{73} + 12 q^{75} + 10 q^{77} - 2 q^{79} + 12 q^{81} + 20 q^{83} - 9 q^{85} + 8 q^{87} + 12 q^{89} + 4 q^{91} - 4 q^{93} + 11 q^{95} - q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 30 x^{10} - 53 x^{9} + 798 x^{8} - 1096 x^{7} + 4060 x^{6} - 915 x^{5} + 10392 x^{4} - 7038 x^{3} + 4869 x^{2} - 675 x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3882223435 \nu^{11} + 10214206017 \nu^{10} + 117178408376 \nu^{9} + 172229907007 \nu^{8} + 2805124258771 \nu^{7} + \cdots - 213693056523513 ) / 41206785731685 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52241027 \nu^{11} - 8330484 \nu^{10} - 1527067746 \nu^{9} + 1015516512 \nu^{8} - 39224741894 \nu^{7} + 11206364358 \nu^{6} + \cdots + 52425010656 ) / 392445578397 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18208180335 \nu^{11} + 1146781944 \nu^{10} - 532350662473 \nu^{9} + 458507766964 \nu^{8} - 13717814681058 \nu^{7} + \cdots - 55721026347906 ) / 13735595243895 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5825001184 \nu^{11} + 5354831941 \nu^{10} - 174825009876 \nu^{9} + 294981453038 \nu^{8} - 4639211296224 \nu^{7} + \cdots + 4077726580611 ) / 3532010205573 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17210840888 \nu^{11} + 1079200536 \nu^{10} - 502794995191 \nu^{9} + 433270280680 \nu^{8} - 12966377092280 \nu^{7} + \cdots - 37541561931450 ) / 8241357146337 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99006849920 \nu^{11} - 12877775961 \nu^{10} + 2892833484967 \nu^{9} - 2664598755331 \nu^{8} + 74666534698007 \nu^{7} + \cdots + 136395667228734 ) / 41206785731685 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 340088114552 \nu^{11} - 310530618669 \nu^{10} + 10196741983762 \nu^{9} - 17126290059559 \nu^{8} + 270520940621483 \nu^{7} + \cdots - 31631837094135 ) / 41206785731685 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 208917470212 \nu^{11} + 225445934212 \nu^{10} - 6274826024079 \nu^{9} + 11552472669332 \nu^{8} + \cdots + 136216640185254 ) / 24724071439011 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1061664495556 \nu^{11} + 934960473697 \nu^{10} - 31728339975486 \nu^{9} + 52500424408727 \nu^{8} + \cdots + 96443745926355 ) / 123620357195055 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 361920964801 \nu^{11} + 304890086297 \nu^{10} - 10768625400391 \nu^{9} + 17453392390817 \nu^{8} + \cdots + 31816447355970 ) / 41206785731685 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 2\beta_{10} - \beta_{8} - \beta_{7} - 10\beta_{5} - 2\beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + 5\beta_{6} - 25\beta_{3} - \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 28\beta_{11} - 57\beta_{10} + 2\beta_{9} + 17\beta_{8} + 2\beta_{6} + 220\beta_{5} + 18\beta _1 - 220 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 103 \beta_{11} + 22 \beta_{10} + 144 \beta_{9} + 8 \beta_{8} - 103 \beta_{7} - 446 \beta_{5} - 22 \beta_{4} + 623 \beta_{3} + 8 \beta_{2} - 623 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 748\beta_{7} + 3\beta_{6} + 1463\beta_{4} - 849\beta_{3} + 376\beta_{2} + 5536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3060 \beta_{11} - 1362 \beta_{10} - 3671 \beta_{9} + 98 \beta_{8} - 3671 \beta_{6} + 14744 \beta_{5} + 15847 \beta _1 - 14744 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 20269 \beta_{11} + 37161 \beta_{10} + 2113 \beta_{9} - 9116 \beta_{8} - 20269 \beta_{7} - 144691 \beta_{5} - 37161 \beta_{4} + 30930 \beta_{3} - 9116 \beta_{2} - 30930 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 88360\beta_{7} + 92478\beta_{6} + 56197\beta_{4} - 409535\beta_{3} + 8548\beta_{2} + 455798 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 554092 \beta_{11} - 952259 \beta_{10} - 108276 \beta_{9} + 228697 \beta_{8} - 108276 \beta_{6} + 3844291 \beta_{5} + 1018959 \beta _1 - 3844291 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2525310 \beta_{11} + 1976724 \beta_{10} + 2350334 \beta_{9} - 356591 \beta_{8} - 2525310 \beta_{7} - 13648472 \beta_{5} - 1976724 \beta_{4} + 10725256 \beta_{3} + \cdots - 10725256 \beta_1 \) 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
−2.64313 + 4.57803i
1.25129 2.16730i
0.307496 0.532598i
−0.818190 + 1.41715i
0.0725333 0.125631i
2.33000 4.03568i
−2.64313 4.57803i
1.25129 + 2.16730i
0.307496 + 0.532598i
−0.818190 1.41715i
0.0725333 + 0.125631i
2.33000 + 4.03568i
0 1.00000 0 −1.00000 0 −2.33998 + 4.05297i 0 1.00000 0
841.2 0 1.00000 0 −1.00000 0 −1.97947 + 3.42854i 0 1.00000 0
841.3 0 1.00000 0 −1.00000 0 −0.550200 + 0.952975i 0 1.00000 0
841.4 0 1.00000 0 −1.00000 0 1.22030 2.11362i 0 1.00000 0
841.5 0 1.00000 0 −1.00000 0 2.26952 3.93092i 0 1.00000 0
841.6 0 1.00000 0 −1.00000 0 2.37984 4.12200i 0 1.00000 0
3781.1 0 1.00000 0 −1.00000 0 −2.33998 4.05297i 0 1.00000 0
3781.2 0 1.00000 0 −1.00000 0 −1.97947 3.42854i 0 1.00000 0
3781.3 0 1.00000 0 −1.00000 0 −0.550200 0.952975i 0 1.00000 0
3781.4 0 1.00000 0 −1.00000 0 1.22030 + 2.11362i 0 1.00000 0
3781.5 0 1.00000 0 −1.00000 0 2.26952 + 3.93092i 0 1.00000 0
3781.6 0 1.00000 0 −1.00000 0 2.37984 + 4.12200i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.6
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.j 12
67.c even 3 1 inner 4020.2.q.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.q.j 12 1.a even 1 1 trivial
4020.2.q.j 12 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 \(S_{2}^{\mathrm{new}}(4020, [\chi])\):

\( T_{7}^{12} - 2 T_{7}^{11} + 46 T_{7}^{10} - 56 T_{7}^{9} + 1415 T_{7}^{8} - 1559 T_{7}^{7} + 22138 T_{7}^{6} - 11920 T_{7}^{5} + 234221 T_{7}^{4} - 130675 T_{7}^{3} + 856300 T_{7}^{2} + 618125 T_{7} + 1155625 \) Copy content Toggle raw display
\( T_{11}^{12} + 5 T_{11}^{11} + 56 T_{11}^{10} + 181 T_{11}^{9} + 1633 T_{11}^{8} + 4932 T_{11}^{7} + 28842 T_{11}^{6} + 64899 T_{11}^{5} + 301761 T_{11}^{4} + 644112 T_{11}^{3} + 1767096 T_{11}^{2} + 1705860 T_{11} + 1476225 \) Copy content Toggle raw display
\( T_{17}^{12} - 9 T_{17}^{11} + 106 T_{17}^{10} - 257 T_{17}^{9} + 2470 T_{17}^{8} - 1082 T_{17}^{7} + 58966 T_{17}^{6} + 30097 T_{17}^{5} + 329050 T_{17}^{4} - 477462 T_{17}^{3} + 662989 T_{17}^{2} - 349377 T_{17} + 154449 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + 46 T^{10} + \cdots + 1155625 \) Copy content Toggle raw display
$11$ \( T^{12} + 5 T^{11} + 56 T^{10} + \cdots + 1476225 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + 106 T^{10} + \cdots + 154449 \) Copy content Toggle raw display
$19$ \( T^{12} + 11 T^{11} + \cdots + 202350625 \) Copy content Toggle raw display
$23$ \( T^{12} - 19 T^{11} + 258 T^{10} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} + \cdots + 4576928409 \) Copy content Toggle raw display
$31$ \( T^{12} + 4 T^{11} + 77 T^{10} + \cdots + 140625 \) Copy content Toggle raw display
$37$ \( T^{12} - 5 T^{11} + 187 T^{10} + \cdots + 173896969 \) Copy content Toggle raw display
$41$ \( T^{12} + 9 T^{11} + 134 T^{10} + \cdots + 45369 \) Copy content Toggle raw display
$43$ \( (T^{6} + 7 T^{5} - 56 T^{4} - 201 T^{3} + \cdots - 196)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + \cdots + 27503900649 \) Copy content Toggle raw display
$53$ \( (T^{6} + 10 T^{5} - 26 T^{4} - 251 T^{3} + \cdots - 378)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 3 T^{5} - 284 T^{4} + 787 T^{3} + \cdots - 31104)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} - 27 T^{11} + 517 T^{10} + \cdots + 2313441 \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{11} + \cdots + 90458382169 \) Copy content Toggle raw display
$71$ \( T^{12} - 25 T^{11} + \cdots + 1462374081 \) Copy content Toggle raw display
$73$ \( T^{12} + 8 T^{11} + \cdots + 1128576899025 \) Copy content Toggle raw display
$79$ \( T^{12} + 2 T^{11} + \cdots + 7024283721 \) Copy content Toggle raw display
$83$ \( T^{12} - 20 T^{11} + \cdots + 5425577928369 \) Copy content Toggle raw display
$89$ \( (T^{6} - 6 T^{5} - 256 T^{4} + 1469 T^{3} + \cdots + 46518)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + T^{11} + \cdots + 149669170641 \) Copy content Toggle raw display
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