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Properties

Label 6040.2.a.o
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $0$
Dimension $15$
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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 \) 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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} - \beta_{4} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} - \beta_{4} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} - \beta_{9} q^{11} + \beta_{11} q^{13} + \beta_1 q^{15} - \beta_{10} q^{17} + (\beta_{9} - \beta_{8} - \beta_{7} + 1) q^{19} + ( - \beta_{13} + \beta_{12} + \beta_{10} + \cdots - 1) q^{21}+ \cdots + (\beta_{14} - 2 \beta_{13} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51463483 \nu^{14} + 296143482 \nu^{13} - 3862202797 \nu^{12} - 2330841226 \nu^{11} + \cdots - 143292511468 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27789758 \nu^{14} + 300288361 \nu^{13} - 435211827 \nu^{12} - 5214821099 \nu^{11} + \cdots - 30335382422 ) / 6040684094 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 188871903 \nu^{14} + 1585090827 \nu^{13} + 256024289 \nu^{12} - 32706135385 \nu^{11} + \cdots - 220043410060 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 242873093 \nu^{14} + 1541595868 \nu^{13} + 2637655539 \nu^{12} - 33195175548 \nu^{11} + \cdots - 97377907560 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 251785062 \nu^{14} - 222305117 \nu^{13} - 9395987162 \nu^{12} + 9538824983 \nu^{11} + \cdots - 227271071436 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 415665579 \nu^{14} - 2446939882 \nu^{13} - 5421483309 \nu^{12} + 52091499282 \nu^{11} + \cdots + 128826450508 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 479001604 \nu^{14} - 2804333197 \nu^{13} - 6084620332 \nu^{12} + 59209221923 \nu^{11} + \cdots + 173737611300 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 487587165 \nu^{14} + 2274354782 \nu^{13} + 9272430227 \nu^{12} - 50346781486 \nu^{11} + \cdots + 152024446264 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 506155106 \nu^{14} + 2103881269 \nu^{13} + 10476774266 \nu^{12} - 46165780471 \nu^{11} + \cdots + 159337477636 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 684215577 \nu^{14} + 3616262202 \nu^{13} + 11062015193 \nu^{12} - 79740997314 \nu^{11} + \cdots - 221398514376 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 742080601 \nu^{14} + 3659749286 \nu^{13} + 13526986483 \nu^{12} - 82608253774 \nu^{11} + \cdots - 100634081820 ) / 12081368188 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 773085981 \nu^{14} + 3240047422 \nu^{13} + 17400445159 \nu^{12} - 79691289118 \nu^{11} + \cdots - 106981323844 ) / 12081368188 \) 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
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{10} + \beta_{8} + 2\beta_{3} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} + 10\beta_{2} + 11\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{14} - 12 \beta_{13} - 4 \beta_{12} + \beta_{11} + 11 \beta_{10} - 2 \beta_{9} + 9 \beta_{8} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} + 3 \beta_{13} - 20 \beta_{12} + 12 \beta_{10} - 17 \beta_{9} + 14 \beta_{8} - 2 \beta_{7} + \cdots + 199 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 124 \beta_{14} - 118 \beta_{13} - 75 \beta_{12} + 20 \beta_{11} + 104 \beta_{10} - 36 \beta_{9} + \cdots + 116 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27 \beta_{14} + 54 \beta_{13} - 283 \beta_{12} + 4 \beta_{11} + 119 \beta_{10} - 207 \beta_{9} + \cdots + 1662 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1232 \beta_{14} - 1086 \beta_{13} - 1028 \beta_{12} + 290 \beta_{11} + 940 \beta_{10} - 464 \beta_{9} + \cdots + 1256 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 459 \beta_{14} + 704 \beta_{13} - 3513 \beta_{12} + 142 \beta_{11} + 1099 \beta_{10} - 2240 \beta_{9} + \cdots + 14627 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12015 \beta_{14} - 9653 \beta_{13} - 12485 \beta_{12} + 3653 \beta_{11} + 8306 \beta_{10} - 5241 \beta_{9} + \cdots + 13786 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6395 \beta_{14} + 8160 \beta_{13} - 40841 \beta_{12} + 2923 \beta_{11} + 9710 \beta_{10} - 22904 \beta_{9} + \cdots + 133140 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 115881 \beta_{14} - 83676 \beta_{13} - 142654 \beta_{12} + 42692 \beta_{11} + 72263 \beta_{10} + \cdots + 151248 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 80022 \beta_{14} + 89726 \beta_{13} - 457023 \beta_{12} + 46968 \beta_{11} + 82775 \beta_{10} + \cdots + 1240185 \) 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
−2.95828
−2.53958
−1.77658
−1.76219
−1.07149
−0.532237
−0.422275
0.266097
1.14431
1.18532
2.06490
2.38022
2.79329
3.03078
3.19771
0 −2.95828 0 1.00000 0 2.63281 0 5.75141 0
1.2 0 −2.53958 0 1.00000 0 −0.798381 0 3.44946 0
1.3 0 −1.77658 0 1.00000 0 2.60627 0 0.156227 0
1.4 0 −1.76219 0 1.00000 0 −1.99837 0 0.105317 0
1.5 0 −1.07149 0 1.00000 0 1.66476 0 −1.85192 0
1.6 0 −0.532237 0 1.00000 0 −2.74903 0 −2.71672 0
1.7 0 −0.422275 0 1.00000 0 −2.22380 0 −2.82168 0
1.8 0 0.266097 0 1.00000 0 3.77999 0 −2.92919 0
1.9 0 1.14431 0 1.00000 0 0.357411 0 −1.69055 0
1.10 0 1.18532 0 1.00000 0 1.34515 0 −1.59502 0
1.11 0 2.06490 0 1.00000 0 −3.39301 0 1.26381 0
1.12 0 2.38022 0 1.00000 0 −1.25626 0 2.66546 0
1.13 0 2.79329 0 1.00000 0 4.28460 0 4.80245 0
1.14 0 3.03078 0 1.00000 0 3.09463 0 6.18562 0
1.15 0 3.19771 0 1.00000 0 −0.346742 0 7.22533 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.o ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.o ✓ 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{15} - 5 T_{3}^{14} - 19 T_{3}^{13} + 119 T_{3}^{12} + 106 T_{3}^{11} - 1063 T_{3}^{10} + \cdots + 272 \) Copy content Toggle raw display
\( T_{7}^{15} - 7 T_{7}^{14} - 21 T_{7}^{13} + 220 T_{7}^{12} + 67 T_{7}^{11} - 2631 T_{7}^{10} + \cdots + 3968 \) Copy content Toggle raw display
\( T_{11}^{15} - 7 T_{11}^{14} - 53 T_{11}^{13} + 351 T_{11}^{12} + 1308 T_{11}^{11} - 6583 T_{11}^{10} + \cdots + 97792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 5 T^{14} + \cdots + 272 \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 7 T^{14} + \cdots + 3968 \) Copy content Toggle raw display
$11$ \( T^{15} - 7 T^{14} + \cdots + 97792 \) Copy content Toggle raw display
$13$ \( T^{15} - 2 T^{14} + \cdots + 439984 \) Copy content Toggle raw display
$17$ \( T^{15} + 3 T^{14} + \cdots + 65152 \) Copy content Toggle raw display
$19$ \( T^{15} - 8 T^{14} + \cdots + 15260288 \) Copy content Toggle raw display
$23$ \( T^{15} - 15 T^{14} + \cdots + 101432 \) Copy content Toggle raw display
$29$ \( T^{15} - 5 T^{14} + \cdots + 5899616 \) Copy content Toggle raw display
$31$ \( T^{15} - 27 T^{14} + \cdots + 9996796 \) Copy content Toggle raw display
$37$ \( T^{15} + 4 T^{14} + \cdots + 52581888 \) Copy content Toggle raw display
$41$ \( T^{15} - 20 T^{14} + \cdots + 38226944 \) Copy content Toggle raw display
$43$ \( T^{15} - 25 T^{14} + \cdots + 27873536 \) Copy content Toggle raw display
$47$ \( T^{15} - 35 T^{14} + \cdots + 58443264 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 5781594112 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 464229568 \) Copy content Toggle raw display
$61$ \( T^{15} - 23 T^{14} + \cdots + 8839296 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 9923979956 \) Copy content Toggle raw display
$71$ \( T^{15} - 30 T^{14} + \cdots + 69632 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 25905868628 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 2688421888 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 4524880844 \) Copy content Toggle raw display
$89$ \( T^{15} - 19 T^{14} + \cdots + 23431424 \) Copy content Toggle raw display
$97$ \( T^{15} + 8 T^{14} + \cdots - 4850432 \) Copy content Toggle raw display
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