Properties

Label 3040.2.j.e
Level $3040$
Weight $2$
Character orbit 3040.j
Analytic conductor $24.275$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(2431,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.j (of order \(2\), degree \(1\), 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: \(24.2745222145\)
Analytic rank: \(0\)
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: \( x^{20} - 4 x^{19} - 14 x^{18} + 64 x^{17} + 130 x^{16} - 376 x^{15} - 56 x^{14} + 68 x^{13} - 162 x^{12} + \cdots + 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + q^{5} + \beta_{7} q^{7} + ( - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + q^{5} + \beta_{7} q^{7} + ( - \beta_{3} + 1) q^{9} - \beta_{7} q^{11} - \beta_{12} q^{13} - \beta_{4} q^{15} + \beta_{13} q^{17} + ( - \beta_{4} + \beta_{2}) q^{19} - \beta_{6} q^{21} + (\beta_{7} - \beta_{2} + \beta_1) q^{23} + q^{25} + (\beta_{17} + \beta_{10} - \beta_{4}) q^{27} + (\beta_{12} - \beta_{5}) q^{29} - \beta_{18} q^{31} + \beta_{6} q^{33} + \beta_{7} q^{35} - \beta_{15} q^{37} + (\beta_{8} + \beta_{7} + 2 \beta_1) q^{39} + (\beta_{14} - \beta_{12}) q^{41} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{43} + ( - \beta_{3} + 1) q^{45} + (\beta_{8} + \beta_{7}) q^{47} + (\beta_{9} - 3) q^{49} + ( - \beta_{18} - \beta_{10} - \beta_{4}) q^{51} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{6}) q^{53}+ \cdots + ( - 2 \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{5} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{5} + 28 q^{9} + 8 q^{17} + 20 q^{25} + 28 q^{45} - 52 q^{49} + 88 q^{57} - 112 q^{61} - 24 q^{73} + 192 q^{77} + 44 q^{81} + 8 q^{85} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 14 x^{18} + 64 x^{17} + 130 x^{16} - 376 x^{15} - 56 x^{14} + 68 x^{13} - 162 x^{12} + \cdots + 162 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16\!\cdots\!70 \nu^{19} + \cdots - 89\!\cdots\!38 ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!58 \nu^{19} + \cdots - 24\!\cdots\!66 ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!82 \nu^{19} + \cdots + 47\!\cdots\!46 ) / 13\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26\!\cdots\!36 \nu^{19} + \cdots + 66\!\cdots\!84 ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!82 \nu^{19} + \cdots - 79\!\cdots\!44 ) / 75\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58\!\cdots\!44 \nu^{19} + \cdots + 32\!\cdots\!22 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 58\!\cdots\!98 \nu^{19} + \cdots + 72\!\cdots\!28 ) / 12\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!84 \nu^{19} + \cdots - 28\!\cdots\!92 ) / 39\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 57\!\cdots\!66 \nu^{19} + \cdots + 24\!\cdots\!62 ) / 69\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 83\!\cdots\!06 \nu^{19} + \cdots + 56\!\cdots\!96 ) / 75\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 81\!\cdots\!58 \nu^{19} + \cdots + 93\!\cdots\!68 ) / 66\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!46 \nu^{19} + \cdots - 26\!\cdots\!16 ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 96\!\cdots\!60 \nu^{19} + \cdots - 43\!\cdots\!44 ) / 69\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!74 \nu^{19} + \cdots - 33\!\cdots\!04 ) / 75\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 42\!\cdots\!98 \nu^{19} + \cdots - 42\!\cdots\!88 ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 24\!\cdots\!74 \nu^{19} + \cdots - 93\!\cdots\!70 ) / 13\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!94 \nu^{19} + \cdots + 13\!\cdots\!76 ) / 18\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 56\!\cdots\!63 \nu^{19} + \cdots - 92\!\cdots\!20 ) / 19\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 21\!\cdots\!24 \nu^{19} + \cdots + 69\!\cdots\!44 ) / 19\!\cdots\!82 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{13} + \beta_{9} - \beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - \beta_{18} - 5 \beta_{17} - 2 \beta_{15} - 3 \beta_{14} + 5 \beta_{12} + \beta_{11} + \cdots + 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{17} - \beta_{16} - 4 \beta_{15} - 14 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} + 9 \beta_{11} + \cdots + 10 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20 \beta_{15} - 32 \beta_{14} + 46 \beta_{12} + 15 \beta_{11} - 45 \beta_{8} - 33 \beta_{7} + \cdots + 33 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{19} + 12 \beta_{18} + 84 \beta_{17} + 28 \beta_{16} - 112 \beta_{15} - 238 \beta_{14} + \cdots - 248 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 165 \beta_{19} + 285 \beta_{18} + 975 \beta_{17} + 210 \beta_{16} - 510 \beta_{15} - 805 \beta_{14} + \cdots - 3276 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 408 \beta_{19} + 840 \beta_{18} + 3984 \beta_{17} + 1377 \beta_{16} - 970 \beta_{15} - 1755 \beta_{14} + \cdots - 12002 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1764 \beta_{19} + 3276 \beta_{18} + 11424 \beta_{17} + 3026 \beta_{16} + 221 \beta_{13} + 8796 \beta_{10} + \cdots - 36584 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8897 \beta_{19} + 18327 \beta_{18} + 75891 \beta_{17} + 25520 \beta_{16} + 18428 \beta_{15} + \cdots - 225968 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 40397 \beta_{19} + 78097 \beta_{18} + 277849 \beta_{17} + 80524 \beta_{16} + 157006 \beta_{15} + \cdots - 859688 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 72558 \beta_{19} + 148770 \beta_{18} + 581334 \beta_{17} + 191511 \beta_{16} + 820260 \beta_{15} + \cdots - 1728150 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1967420 \beta_{15} + 3106320 \beta_{14} - 4172630 \beta_{12} - 1334817 \beta_{11} + 3135231 \beta_{8} + \cdots - 4840011 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1363824 \beta_{19} - 2784078 \beta_{18} - 10586070 \beta_{17} - 3441620 \beta_{16} + 14906908 \beta_{15} + \cdots + 31516520 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 11782511 \beta_{19} - 23551671 \beta_{18} - 85846761 \beta_{17} - 26657582 \beta_{16} + \cdots + 259021508 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 60489264 \beta_{19} - 123091152 \beta_{18} - 461735232 \beta_{17} - 148851575 \beta_{16} + \cdots + 1376683918 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 146293704 \beta_{19} - 294199416 \beta_{18} - 1078679376 \beta_{17} - 338981730 \beta_{16} + \cdots + 3240487392 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 1098016927 \beta_{19} - 2229782065 \beta_{18} - 8306371241 \beta_{17} - 2664003420 \beta_{16} + \cdots + 24792599720 ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 3663718335 \beta_{19} - 7391167155 \beta_{18} - 27193918415 \beta_{17} - 8599206308 \beta_{16} + \cdots + 81511640168 ) / 8 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 8202739190 \beta_{19} - 16636407690 \beta_{18} - 61749127950 \beta_{17} - 19743987781 \beta_{16} + \cdots + 184440112066 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
3.90833 + 1.61889i
3.90833 1.61889i
−0.496938 + 1.19971i
−0.496938 1.19971i
2.54532 1.05431i
2.54532 + 1.05431i
−0.213257 0.514849i
−0.213257 + 0.514849i
−0.104184 0.251522i
−0.104184 + 0.251522i
−0.607227 + 0.251522i
−0.607227 0.251522i
−1.24295 + 0.514849i
−1.24295 0.514849i
0.436708 + 1.05431i
0.436708 1.05431i
−2.89637 1.19971i
−2.89637 + 1.19971i
0.670564 1.61889i
0.670564 + 1.61889i
0 −3.23777 0 1.00000 0 1.90426i 0 7.48316 0
2431.2 0 −3.23777 0 1.00000 0 1.90426i 0 7.48316 0
2431.3 0 −2.39943 0 1.00000 0 5.15057i 0 2.75726 0
2431.4 0 −2.39943 0 1.00000 0 5.15057i 0 2.75726 0
2431.5 0 −2.10861 0 1.00000 0 2.53578i 0 1.44625 0
2431.6 0 −2.10861 0 1.00000 0 2.53578i 0 1.44625 0
2431.7 0 −1.02970 0 1.00000 0 1.48351i 0 −1.93972 0
2431.8 0 −1.02970 0 1.00000 0 1.48351i 0 −1.93972 0
2431.9 0 −0.503043 0 1.00000 0 3.03554i 0 −2.74695 0
2431.10 0 −0.503043 0 1.00000 0 3.03554i 0 −2.74695 0
2431.11 0 0.503043 0 1.00000 0 3.03554i 0 −2.74695 0
2431.12 0 0.503043 0 1.00000 0 3.03554i 0 −2.74695 0
2431.13 0 1.02970 0 1.00000 0 1.48351i 0 −1.93972 0
2431.14 0 1.02970 0 1.00000 0 1.48351i 0 −1.93972 0
2431.15 0 2.10861 0 1.00000 0 2.53578i 0 1.44625 0
2431.16 0 2.10861 0 1.00000 0 2.53578i 0 1.44625 0
2431.17 0 2.39943 0 1.00000 0 5.15057i 0 2.75726 0
2431.18 0 2.39943 0 1.00000 0 5.15057i 0 2.75726 0
2431.19 0 3.23777 0 1.00000 0 1.90426i 0 7.48316 0
2431.20 0 3.23777 0 1.00000 0 1.90426i 0 7.48316 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.j.e 20
4.b odd 2 1 inner 3040.2.j.e 20
19.b odd 2 1 inner 3040.2.j.e 20
76.d even 2 1 inner 3040.2.j.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.j.e 20 1.a even 1 1 trivial
3040.2.j.e 20 4.b odd 2 1 inner
3040.2.j.e 20 19.b odd 2 1 inner
3040.2.j.e 20 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 22T_{3}^{8} + 160T_{3}^{6} - 448T_{3}^{4} + 388T_{3}^{2} - 72 \) acting on \(S_{2}^{\mathrm{new}}(3040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - 22 T^{8} + \cdots - 72)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 48 T^{8} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 48 T^{8} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 110 T^{8} + \cdots + 449352)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 2 T^{4} + \cdots + 128)^{4} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} + 112 T^{8} + \cdots + 831744)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{10} \) Copy content Toggle raw display
$31$ \( (T^{10} - 112 T^{8} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 190 T^{8} + \cdots + 1912968)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 216 T^{8} + \cdots + 14623232)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 112 T^{8} + \cdots + 831744)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 128 T^{8} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 334 T^{8} + \cdots + 8306888)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 192 T^{8} + \cdots - 401408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 28 T^{4} + \cdots - 992)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} - 166 T^{8} + \cdots - 29768)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 576 T^{8} + \cdots - 469895168)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 6 T^{4} + \cdots - 76192)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} - 184 T^{8} + \cdots - 512)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 128 T^{8} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 392 T^{8} + \cdots + 189267968)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 270 T^{8} + \cdots + 31904072)^{2} \) Copy content Toggle raw display
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