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Properties

Label 4012.2.a.i
Level $4012$
Weight $2$
Character orbit 4012.a
Self dual yes
Analytic conductor $32.036$
Analytic rank $0$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(0\)
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: \( x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{5} - \beta_{8} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{9} q^{5} - \beta_{8} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + (\beta_{4} + 1) q^{11} + \beta_{7} q^{13} + ( - \beta_{17} + \beta_{13} + \beta_{12} + \cdots - 1) q^{15}+ \cdots + ( - 3 \beta_{17} + \beta_{16} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 721980238514 \nu^{17} - 8792294540969 \nu^{16} + 19922000071949 \nu^{15} + \cdots + 926602245958234 ) / 221648864308394 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3830861910649 \nu^{17} + 47591230451132 \nu^{16} - 573472883791908 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 886595457233576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3229865338098 \nu^{17} - 28341808006461 \nu^{16} + 7224068968562 \nu^{15} + \cdots - 773148687785764 ) / 443297728616788 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3998990215870 \nu^{17} + 48822273723019 \nu^{16} - 95914145833494 \nu^{15} + \cdots - 25\!\cdots\!76 ) / 443297728616788 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13022680202683 \nu^{17} + 118335650003618 \nu^{16} - 44771164178744 \nu^{15} + \cdots - 51\!\cdots\!36 ) / 886595457233576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10740292040922 \nu^{17} + 79117903472021 \nu^{16} + 92870769217390 \nu^{15} + \cdots - 813298060986948 ) / 443297728616788 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33959676594129 \nu^{17} - 244226734403362 \nu^{16} - 339891834715816 \nu^{15} + \cdots + 37\!\cdots\!08 ) / 886595457233576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25214595273041 \nu^{17} + 189190565282336 \nu^{16} + 190733503488708 \nu^{15} + \cdots - 25\!\cdots\!80 ) / 443297728616788 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60700263486941 \nu^{17} - 478320509610002 \nu^{16} - 288622338255540 \nu^{15} + \cdots + 38\!\cdots\!44 ) / 886595457233576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 64223237008785 \nu^{17} + 539084204709562 \nu^{16} + 81357560610632 \nu^{15} + \cdots - 99\!\cdots\!64 ) / 886595457233576 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37688647737868 \nu^{17} + 267971533358753 \nu^{16} + 388679230699790 \nu^{15} + \cdots - 17\!\cdots\!72 ) / 443297728616788 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20738042771486 \nu^{17} - 150148962527149 \nu^{16} - 193241956632582 \nu^{15} + \cdots + 150607842162690 ) / 221648864308394 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2295821306687 \nu^{17} - 17156001995758 \nu^{16} - 17887868681760 \nu^{15} + \cdots + 240505445116088 ) / 20618499005432 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 83524578661994 \nu^{17} + 599721831065959 \nu^{16} + 816991773268750 \nu^{15} + \cdots - 49\!\cdots\!72 ) / 443297728616788 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 180069207323109 \nu^{17} + \cdots - 17\!\cdots\!32 ) / 886595457233576 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + \beta_{16} + 3 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{17} + 2 \beta_{16} - 7 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 10 \beta_{11} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{17} + 15 \beta_{16} + 41 \beta_{15} - 7 \beta_{14} + 17 \beta_{13} + 17 \beta_{12} + \cdots + 246 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 127 \beta_{17} + 37 \beta_{16} - 29 \beta_{15} + 52 \beta_{14} + 57 \beta_{13} + 146 \beta_{12} + \cdots + 227 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 222 \beta_{17} + 185 \beta_{16} + 456 \beta_{15} - 20 \beta_{14} + 238 \beta_{13} + 240 \beta_{12} + \cdots + 2115 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1349 \beta_{17} + 511 \beta_{16} + 86 \beta_{15} + 637 \beta_{14} + 824 \beta_{13} + 1582 \beta_{12} + \cdots + 2618 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2892 \beta_{17} + 2146 \beta_{16} + 4803 \beta_{15} + 250 \beta_{14} + 3098 \beta_{13} + 3110 \beta_{12} + \cdots + 18881 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 14580 \beta_{17} + 6321 \beta_{16} + 4094 \beta_{15} + 6863 \beta_{14} + 10740 \beta_{13} + \cdots + 29208 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 35985 \beta_{17} + 24210 \beta_{16} + 49797 \beta_{15} + 6043 \beta_{14} + 38481 \beta_{13} + \cdots + 174104 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 159475 \beta_{17} + 74086 \beta_{16} + 68613 \beta_{15} + 69730 \beta_{14} + 132398 \beta_{13} + \cdots + 320459 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 432410 \beta_{17} + 268942 \beta_{16} + 514908 \beta_{15} + 87030 \beta_{14} + 461917 \beta_{13} + \cdots + 1650712 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1753792 \beta_{17} + 842636 \beta_{16} + 926227 \beta_{15} + 688583 \beta_{14} + 1575225 \beta_{13} + \cdots + 3479739 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5060507 \beta_{17} + 2959032 \beta_{16} + 5336145 \beta_{15} + 1066828 \beta_{14} + 5404657 \beta_{13} + \cdots + 16021039 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 19305219 \beta_{17} + 9411007 \beta_{16} + 11390705 \beta_{15} + 6711080 \beta_{14} + 18284331 \beta_{13} + \cdots + 37517387 \) 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
−2.80641
−2.57211
−2.15409
−1.74374
−1.35164
−0.397638
−0.267006
0.0880346
0.154866
0.839537
0.877348
1.33821
1.56216
2.54274
2.60297
2.97166
3.05303
3.26206
0 −2.80641 0 3.19831 0 4.58875 0 4.87591 0
1.2 0 −2.57211 0 −0.393688 0 −1.17366 0 3.61576 0
1.3 0 −2.15409 0 −0.228433 0 2.56317 0 1.64008 0
1.4 0 −1.74374 0 0.394783 0 −1.78199 0 0.0406123 0
1.5 0 −1.35164 0 1.29976 0 −4.65863 0 −1.17306 0
1.6 0 −0.397638 0 3.51710 0 2.09256 0 −2.84188 0
1.7 0 −0.267006 0 −2.18930 0 1.68463 0 −2.92871 0
1.8 0 0.0880346 0 −2.20461 0 0.606938 0 −2.99225 0
1.9 0 0.154866 0 −1.21070 0 −0.213015 0 −2.97602 0
1.10 0 0.839537 0 −4.04363 0 −2.98927 0 −2.29518 0
1.11 0 0.877348 0 −1.13091 0 −4.50700 0 −2.23026 0
1.12 0 1.33821 0 3.45044 0 −1.09754 0 −1.20921 0
1.13 0 1.56216 0 −0.592812 0 4.71818 0 −0.559654 0
1.14 0 2.54274 0 3.21926 0 1.46547 0 3.46555 0
1.15 0 2.60297 0 0.183919 0 3.24534 0 3.77545 0
1.16 0 2.97166 0 −4.20066 0 0.274921 0 5.83079 0
1.17 0 3.05303 0 2.19982 0 −4.80385 0 6.32102 0
1.18 0 3.26206 0 2.73134 0 1.98500 0 7.64104 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

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 4012.2.a.i ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.i ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\( T_{3}^{18} - 8 T_{3}^{17} - 4 T_{3}^{16} + 178 T_{3}^{15} - 265 T_{3}^{14} - 1405 T_{3}^{13} + 3503 T_{3}^{12} + \cdots - 16 \) Copy content Toggle raw display
\( T_{5}^{18} - 4 T_{5}^{17} - 45 T_{5}^{16} + 195 T_{5}^{15} + 712 T_{5}^{14} - 3442 T_{5}^{13} + \cdots - 424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 8 T^{17} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{18} - 4 T^{17} + \cdots - 424 \) Copy content Toggle raw display
$7$ \( T^{18} - 2 T^{17} + \cdots + 45432 \) Copy content Toggle raw display
$11$ \( T^{18} - 12 T^{17} + \cdots + 1520296 \) Copy content Toggle raw display
$13$ \( T^{18} - 2 T^{17} + \cdots + 12694048 \) Copy content Toggle raw display
$17$ \( (T + 1)^{18} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 194725139 \) Copy content Toggle raw display
$23$ \( T^{18} - 21 T^{17} + \cdots - 286528 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 1558945656 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 132729344 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 1230214688 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 11787813684 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 1968086739584 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 3463958082816 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 4928999638031 \) Copy content Toggle raw display
$59$ \( (T + 1)^{18} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 33898516378592 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 8747554682016 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 628385280681984 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 515437217017384 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 58696261290172 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 829454018989248 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 513545658373376 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 27368271151968 \) Copy content Toggle raw display
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