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Properties

Label 6040.2.a.p
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $1$
Dimension $19$
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] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.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: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + \beta_{3} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + \beta_{3} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{18} - 1) q^{11} + ( - \beta_{14} + \beta_1) q^{13} - \beta_1 q^{15} - \beta_{13} q^{17} + ( - \beta_{17} + \beta_{12} + \cdots + \beta_{3}) q^{19}+ \cdots + ( - 4 \beta_{18} - \beta_{17} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55395572792623 \nu^{18} + \cdots + 10\!\cdots\!24 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 825471425164009 \nu^{18} + \cdots + 34\!\cdots\!28 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 831149248013573 \nu^{18} + \cdots + 16\!\cdots\!72 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 893262017742841 \nu^{18} + \cdots + 17\!\cdots\!48 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56636909905056 \nu^{18} - 466859136090342 \nu^{17} - 472631481848911 \nu^{16} + \cdots - 98\!\cdots\!92 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!17 \nu^{18} + \cdots - 66\!\cdots\!40 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 265246968233581 \nu^{18} + \cdots - 60\!\cdots\!72 ) / 57\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!03 \nu^{18} + \cdots - 16\!\cdots\!04 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!21 \nu^{18} + \cdots + 11\!\cdots\!96 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 648499956243255 \nu^{18} + \cdots + 59\!\cdots\!36 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!33 \nu^{18} + \cdots + 81\!\cdots\!08 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 30\!\cdots\!01 \nu^{18} + \cdots + 66\!\cdots\!36 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!75 \nu^{18} + \cdots - 10\!\cdots\!12 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 41\!\cdots\!03 \nu^{18} + \cdots - 17\!\cdots\!04 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 57\!\cdots\!01 \nu^{18} + \cdots + 33\!\cdots\!28 ) / 45\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 60\!\cdots\!01 \nu^{18} + \cdots - 19\!\cdots\!20 ) / 45\!\cdots\!92 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{16} - \beta_{15} - \beta_{14} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - 17 \beta_{16} - 12 \beta_{15} - 14 \beta_{14} - \beta_{13} - 3 \beta_{12} + 2 \beta_{11} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{18} - \beta_{17} + 4 \beta_{15} - 32 \beta_{14} - 34 \beta_{13} + 32 \beta_{12} + 2 \beta_{11} + \cdots + 323 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{18} + 14 \beta_{17} - 222 \beta_{16} - 128 \beta_{15} - 174 \beta_{14} - 22 \beta_{13} + \cdots + 351 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 64 \beta_{18} - 25 \beta_{17} - 6 \beta_{16} + 92 \beta_{15} - 411 \beta_{14} - 444 \beta_{13} + \cdots + 3380 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 121 \beta_{18} + 146 \beta_{17} - 2662 \beta_{16} - 1350 \beta_{15} - 2074 \beta_{14} - 321 \beta_{13} + \cdots + 3583 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 959 \beta_{18} - 409 \beta_{17} - 118 \beta_{16} + 1463 \beta_{15} - 4922 \beta_{14} - 5324 \beta_{13} + \cdots + 36456 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1920 \beta_{18} + 1412 \beta_{17} - 30868 \beta_{16} - 14337 \beta_{15} - 24113 \beta_{14} + \cdots + 36668 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12470 \beta_{18} - 5672 \beta_{17} - 1427 \beta_{16} + 20174 \beta_{15} - 57021 \beta_{14} + \cdots + 399005 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 25505 \beta_{18} + 13564 \beta_{17} - 352686 \beta_{16} - 153886 \beta_{15} - 275852 \beta_{14} + \cdots + 376952 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 150682 \beta_{18} - 72638 \beta_{17} - 12258 \beta_{16} + 259294 \beta_{15} - 648026 \beta_{14} + \cdots + 4400719 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 308282 \beta_{18} + 132598 \beta_{17} - 4000805 \beta_{16} - 1669352 \beta_{15} - 3122385 \beta_{14} + \cdots + 3889559 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1743576 \beta_{18} - 891156 \beta_{17} - 53088 \beta_{16} + 3203127 \beta_{15} - 7274769 \beta_{14} + \cdots + 48752267 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3518378 \beta_{18} + 1327790 \beta_{17} - 45212269 \beta_{16} - 18279525 \beta_{15} - 35092958 \beta_{14} + \cdots + 40226199 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 19618273 \beta_{18} - 10663514 \beta_{17} + 691708 \beta_{16} + 38617512 \beta_{15} + \cdots + 541608246 \) 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
3.33655
3.31016
2.89238
2.44352
2.34269
1.90812
1.68584
1.40330
1.02734
−0.237601
−0.435585
−0.815820
−0.942024
−1.62033
−1.76710
−1.97307
−2.06929
−2.11583
−3.37325
0 −3.33655 0 1.00000 0 2.35293 0 8.13258 0
1.2 0 −3.31016 0 1.00000 0 −0.477457 0 7.95713 0
1.3 0 −2.89238 0 1.00000 0 0.451178 0 5.36583 0
1.4 0 −2.44352 0 1.00000 0 −4.90700 0 2.97081 0
1.5 0 −2.34269 0 1.00000 0 −1.53409 0 2.48818 0
1.6 0 −1.90812 0 1.00000 0 1.15158 0 0.640908 0
1.7 0 −1.68584 0 1.00000 0 −4.78926 0 −0.157936 0
1.8 0 −1.40330 0 1.00000 0 3.27194 0 −1.03074 0
1.9 0 −1.02734 0 1.00000 0 3.96391 0 −1.94457 0
1.10 0 0.237601 0 1.00000 0 4.27350 0 −2.94355 0
1.11 0 0.435585 0 1.00000 0 −2.87898 0 −2.81027 0
1.12 0 0.815820 0 1.00000 0 −2.49382 0 −2.33444 0
1.13 0 0.942024 0 1.00000 0 1.85968 0 −2.11259 0
1.14 0 1.62033 0 1.00000 0 1.05807 0 −0.374546 0
1.15 0 1.76710 0 1.00000 0 0.0387940 0 0.122653 0
1.16 0 1.97307 0 1.00000 0 −0.188446 0 0.892989 0
1.17 0 2.06929 0 1.00000 0 −0.835530 0 1.28196 0
1.18 0 2.11583 0 1.00000 0 −4.51128 0 1.47674 0
1.19 0 3.37325 0 1.00000 0 −3.80572 0 8.37885 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(151\) \(-1\)

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 6040.2.a.p ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.p ✓ 19 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{19} + 5 T_{3}^{18} - 29 T_{3}^{17} - 165 T_{3}^{16} + 325 T_{3}^{15} + 2208 T_{3}^{14} + \cdots + 5628 \) Copy content Toggle raw display
\( T_{7}^{19} + 8 T_{7}^{18} - 46 T_{7}^{17} - 465 T_{7}^{16} + 660 T_{7}^{15} + 10488 T_{7}^{14} + \cdots - 1728 \) Copy content Toggle raw display
\( T_{11}^{19} + 18 T_{11}^{18} + 38 T_{11}^{17} - 1110 T_{11}^{16} - 6616 T_{11}^{15} + 18635 T_{11}^{14} + \cdots - 71561024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} \) Copy content Toggle raw display
$3$ \( T^{19} + 5 T^{18} + \cdots + 5628 \) Copy content Toggle raw display
$5$ \( (T - 1)^{19} \) Copy content Toggle raw display
$7$ \( T^{19} + 8 T^{18} + \cdots - 1728 \) Copy content Toggle raw display
$11$ \( T^{19} + 18 T^{18} + \cdots - 71561024 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 302102468 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 3277832192 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 1174750448 \) Copy content Toggle raw display
$23$ \( T^{19} + 25 T^{18} + \cdots + 12767064 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 4354518300450 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 1069253719488 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 13613688741888 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 983121641472 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 109439134728192 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 4551635331072 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 440292906606192 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 20\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 11935851696 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 184683113759286 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots + 36\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 238547559642064 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 23416063524864 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 340899631005696 \) Copy content Toggle raw display
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