Properties

Label 6032.2.a.bd
Level $6032$
Weight $2$
Character orbit 6032.a
Self dual yes
Analytic conductor $48.166$
Analytic rank $1$
Dimension $12$
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] = [6032,2,Mod(1,6032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6032, 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("6032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.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: \(48.1657624992\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} - 1567 x^{3} + 273 x^{2} + 424 x + 68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 3016)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + \beta_{6} q^{5} + ( - \beta_{5} - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + \beta_{6} q^{5} + ( - \beta_{5} - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{8} - 1) q^{11} + q^{13} + ( - \beta_{6} + \beta_{4}) q^{15} + (\beta_{11} - \beta_{8} + \beta_{5} - \beta_1 + 1) q^{17} + ( - \beta_{10} - \beta_{9} - \beta_{5} - \beta_{2} - 1) q^{19} + ( - \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5}) q^{21} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_1 - 1) q^{23} + ( - \beta_{11} + \beta_{4} + 1) q^{25} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + 2 \beta_1 - 3) q^{27} + q^{29} + (\beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 - 1) q^{31} + (\beta_{11} - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - \beta_1) q^{33} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{35}+ \cdots + ( - \beta_{11} + \beta_{9} + 4 \beta_{8} + \beta_{7} - \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} - 1567 x^{3} + 273 x^{2} + 424 x + 68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2282 \nu^{11} + 16131 \nu^{10} + 9692 \nu^{9} - 248259 \nu^{8} + 163630 \nu^{7} + 1388641 \nu^{6} - 1280151 \nu^{5} - 3383822 \nu^{4} + 3048679 \nu^{3} + \cdots - 852468 ) / 65776 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6401 \nu^{11} + 33417 \nu^{10} + 92699 \nu^{9} - 597451 \nu^{8} - 381637 \nu^{7} + 3756664 \nu^{6} + 14221 \nu^{5} - 9634183 \nu^{4} + 2385714 \nu^{3} + \cdots - 1171164 ) / 131552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16367 \nu^{11} + 101865 \nu^{10} + 132713 \nu^{9} - 1588671 \nu^{8} + 265149 \nu^{7} + 8922602 \nu^{6} - 4612487 \nu^{5} - 21371169 \nu^{4} + \cdots - 3423820 ) / 131552 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17223 \nu^{11} - 109739 \nu^{10} - 138813 \nu^{9} + 1780553 \nu^{8} - 425221 \nu^{7} - 10457092 \nu^{6} + 6357337 \nu^{5} + 26262073 \nu^{4} - 17298418 \nu^{3} + \cdots + 4767844 ) / 131552 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10119 \nu^{11} - 60913 \nu^{10} - 97717 \nu^{9} + 985031 \nu^{8} + 31695 \nu^{7} - 5750850 \nu^{6} + 2040411 \nu^{5} + 14264261 \nu^{4} - 6191772 \nu^{3} + \cdots + 2366212 ) / 65776 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26939 \nu^{11} - 166263 \nu^{10} - 237481 \nu^{9} + 2671453 \nu^{8} - 268833 \nu^{7} - 15551452 \nu^{6} + 7449581 \nu^{5} + 38773061 \nu^{4} - 21268834 \nu^{3} + \cdots + 6458996 ) / 131552 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38513 \nu^{11} + 239513 \nu^{10} + 332907 \nu^{9} - 3856379 \nu^{8} + 492779 \nu^{7} + 22531384 \nu^{6} - 11171843 \nu^{5} - 56665175 \nu^{4} + \cdots - 10216572 ) / 131552 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46969 \nu^{11} - 299085 \nu^{10} - 354899 \nu^{9} + 4688943 \nu^{8} - 1186075 \nu^{7} - 26615124 \nu^{6} + 15634207 \nu^{5} + 65014311 \nu^{4} + \cdots + 11916636 ) / 131552 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25420 \nu^{11} - 159523 \nu^{10} - 208810 \nu^{9} + 2535167 \nu^{8} - 428768 \nu^{7} - 14597111 \nu^{6} + 7551271 \nu^{5} + 36085832 \nu^{4} - 20112733 \nu^{3} + \cdots + 6086412 ) / 65776 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + 3\beta_{2} + 8\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{8} + 5 \beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{3} + 17 \beta_{2} + 19 \beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{11} + 15 \beta_{10} - 6 \beta_{9} - \beta_{8} + 31 \beta_{7} + 7 \beta_{6} + 7 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} + 68 \beta_{2} + 97 \beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 129 \beta_{11} + 47 \beta_{10} - 42 \beta_{9} - 18 \beta_{8} + 140 \beta_{7} + 40 \beta_{6} - 4 \beta_{5} + 67 \beta_{4} - 13 \beta_{3} + 308 \beta_{2} + 335 \beta _1 + 371 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 608 \beta_{11} + 236 \beta_{10} - 163 \beta_{9} - 45 \beta_{8} + 675 \beta_{7} + 199 \beta_{6} + 21 \beta_{5} + 332 \beta_{4} - 31 \beta_{3} + 1302 \beta_{2} + 1498 \beta _1 + 1246 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2645 \beta_{11} + 897 \beta_{10} - 813 \beta_{9} - 308 \beta_{8} + 2957 \beta_{7} + 945 \beta_{6} - 150 \beta_{5} + 1541 \beta_{4} - 135 \beta_{3} + 5640 \beta_{2} + 5962 \beta _1 + 5478 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11734 \beta_{11} + 4015 \beta_{10} - 3407 \beta_{9} - 1083 \beta_{8} + 13217 \beta_{7} + 4287 \beta_{6} - 488 \beta_{5} + 7028 \beta_{4} - 354 \beta_{3} + 24076 \beta_{2} + 25752 \beta _1 + 21400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 50638 \beta_{11} + 16440 \beta_{10} - 15395 \beta_{9} - 5457 \beta_{8} + 57179 \beta_{7} + 19154 \beta_{6} - 3790 \beta_{5} + 31050 \beta_{4} - 1316 \beta_{3} + 103252 \beta_{2} + 107549 \beta _1 + 91361 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 219785 \beta_{11} + 71157 \beta_{10} - 65762 \beta_{9} - 21803 \beta_{8} + 248754 \beta_{7} + 83997 \beta_{6} - 16064 \beta_{5} + 136569 \beta_{4} - 3677 \beta_{3} + 441426 \beta_{2} + \cdots + 377635 \) Copy content Toggle raw display

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
−2.41502
−2.05683
−1.89571
−1.22268
−0.285855
−0.260195
0.848556
1.32893
2.19118
2.59227
2.89888
4.27647
0 −3.41502 0 1.12461 0 3.99026 0 8.66236 0
1.2 0 −3.05683 0 −3.01704 0 1.81391 0 6.34423 0
1.3 0 −2.89571 0 2.36446 0 −4.24816 0 5.38516 0
1.4 0 −2.22268 0 2.71513 0 −1.08088 0 1.94031 0
1.5 0 −1.28586 0 −3.44298 0 −3.94227 0 −1.34658 0
1.6 0 −1.26019 0 0.631127 0 −0.921981 0 −1.41191 0
1.7 0 −0.151444 0 3.67772 0 0.901102 0 −2.97706 0
1.8 0 0.328928 0 −1.23337 0 3.96198 0 −2.89181 0
1.9 0 1.19118 0 3.27457 0 −4.85960 0 −1.58109 0
1.10 0 1.59227 0 −1.41565 0 −0.109839 0 −0.464663 0
1.11 0 1.89888 0 −0.496327 0 −2.15081 0 0.605764 0
1.12 0 3.27647 0 −1.18224 0 0.646306 0 7.73529 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(-1\)
\(29\) \(-1\)

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 6032.2.a.bd 12
4.b odd 2 1 3016.2.a.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3016.2.a.j 12 4.b odd 2 1
6032.2.a.bd 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\( T_{3}^{12} + 6 T_{3}^{11} - 10 T_{3}^{10} - 112 T_{3}^{9} - 53 T_{3}^{8} + 635 T_{3}^{7} + 620 T_{3}^{6} - 1447 T_{3}^{5} - 1572 T_{3}^{4} + 1260 T_{3}^{3} + 1216 T_{3}^{2} - 272 T_{3} - 64 \) Copy content Toggle raw display
\( T_{5}^{12} - 3 T_{5}^{11} - 28 T_{5}^{10} + 78 T_{5}^{9} + 276 T_{5}^{8} - 663 T_{5}^{7} - 1220 T_{5}^{6} + 2077 T_{5}^{5} + 2767 T_{5}^{4} - 2249 T_{5}^{3} - 2502 T_{5}^{2} + 676 T_{5} + 584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} - 10 T^{10} - 112 T^{9} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} - 28 T^{10} + 78 T^{9} + \cdots + 584 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} - 32 T^{10} - 220 T^{9} + \cdots - 320 \) Copy content Toggle raw display
$11$ \( T^{12} + 14 T^{11} + 24 T^{10} + \cdots - 15616 \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} - 97 T^{10} + \cdots - 1952 \) Copy content Toggle raw display
$19$ \( T^{12} + 11 T^{11} - 70 T^{10} + \cdots + 269312 \) Copy content Toggle raw display
$23$ \( T^{12} + 15 T^{11} - 23 T^{10} + \cdots + 169984 \) Copy content Toggle raw display
$29$ \( (T - 1)^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 13 T^{11} - 169 T^{10} + \cdots + 69533696 \) Copy content Toggle raw display
$37$ \( T^{12} + 23 T^{11} + 83 T^{10} + \cdots - 491296 \) Copy content Toggle raw display
$41$ \( T^{12} + 2 T^{11} - 243 T^{10} + \cdots + 1871504 \) Copy content Toggle raw display
$43$ \( T^{12} + 26 T^{11} + \cdots + 261999616 \) Copy content Toggle raw display
$47$ \( T^{12} + 15 T^{11} - 92 T^{10} + \cdots + 75729664 \) Copy content Toggle raw display
$53$ \( T^{12} - 31 T^{11} + \cdots + 215535872 \) Copy content Toggle raw display
$59$ \( T^{12} + 7 T^{11} + \cdots - 24958242944 \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} - 349 T^{10} + \cdots - 61610320 \) Copy content Toggle raw display
$67$ \( T^{12} + 47 T^{11} + \cdots - 13200980224 \) Copy content Toggle raw display
$71$ \( T^{12} + 32 T^{11} + \cdots + 12084770816 \) Copy content Toggle raw display
$73$ \( T^{12} + 25 T^{11} + \cdots - 7690217456 \) Copy content Toggle raw display
$79$ \( T^{12} + 7 T^{11} - 304 T^{10} + \cdots + 456662464 \) Copy content Toggle raw display
$83$ \( T^{12} + 12 T^{11} + \cdots - 180385845632 \) Copy content Toggle raw display
$89$ \( T^{12} - 6 T^{11} + \cdots + 8969176544 \) Copy content Toggle raw display
$97$ \( T^{12} + 7 T^{11} - 642 T^{10} + \cdots - 550370336 \) Copy content Toggle raw display
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