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Properties

Label 6036.2.a.g
Level $6036$
Weight $2$
Character orbit 6036.a
Self dual yes
Analytic conductor $48.198$
Analytic rank $1$
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] = [6036,2,Mod(1,6036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6036, 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("6036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.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: \(48.1977026600\)
Analytic rank: \(1\)
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: \( x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_1 - 1) q^{5} - \beta_{8} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_1 - 1) q^{5} - \beta_{8} q^{7} + q^{9} + (\beta_{10} + \beta_{4}) q^{11} + (\beta_{14} - \beta_{13} - \beta_{12} + \cdots - 2) q^{13}+ \cdots + (\beta_{10} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 358340218 \nu^{14} - 3455243067 \nu^{13} + 1639898571 \nu^{12} + 78821293300 \nu^{11} + \cdots - 546137578798 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 423712226 \nu^{14} - 628785145 \nu^{13} - 16928063565 \nu^{12} + 21911983318 \nu^{11} + \cdots + 400272956308 ) / 19481355031 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3567036137 \nu^{14} - 21649170771 \nu^{13} - 62131751904 \nu^{12} + 551620921145 \nu^{11} + \cdots - 198959454251 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1533791708 \nu^{14} - 8034154554 \nu^{13} - 33188007874 \nu^{12} + 208529518638 \nu^{11} + \cdots + 31492118482 ) / 19481355031 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1731551174 \nu^{14} - 8250353036 \nu^{13} - 38548196723 \nu^{12} + 207446618172 \nu^{11} + \cdots - 117449952898 ) / 19481355031 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5568687106 \nu^{14} - 19093754208 \nu^{13} - 162031083048 \nu^{12} + 502149347719 \nu^{11} + \cdots + 319535662097 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1924356769 \nu^{14} - 11454090727 \nu^{13} - 34583644911 \nu^{12} + 292039374175 \nu^{11} + \cdots - 142936166389 ) / 19481355031 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6083709676 \nu^{14} - 20946481836 \nu^{13} - 170476323909 \nu^{12} + 529474125973 \nu^{11} + \cdots - 157050312484 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7260291644 \nu^{14} - 32010037530 \nu^{13} - 175831046238 \nu^{12} + 814474288403 \nu^{11} + \cdots - 54571509779 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7262151155 \nu^{14} + 34249747695 \nu^{13} + 171462500097 \nu^{12} - 888312598286 \nu^{11} + \cdots - 461279859325 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 8881419661 \nu^{14} + 37650120882 \nu^{13} + 223317906615 \nu^{12} - 963258026887 \nu^{11} + \cdots + 92979418225 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9385042712 \nu^{14} + 47363234112 \nu^{13} + 204278546022 \nu^{12} - 1208912683499 \nu^{11} + \cdots + 716312928815 ) / 58444065093 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4057390393 \nu^{14} + 20201794850 \nu^{13} + 89153456075 \nu^{12} - 515653560454 \nu^{11} + \cdots + 164727575441 ) / 19481355031 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + \beta_{12} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 10\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 11 \beta_{11} - 11 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - \beta_{7} + 8 \beta_{6} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{14} - 13 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + 12 \beta_{9} - \beta_{8} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{14} - 9 \beta_{13} + 15 \beta_{12} - 123 \beta_{11} - 122 \beta_{10} + 43 \beta_{9} + \cdots + 515 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{14} - 162 \beta_{13} + 65 \beta_{12} - 182 \beta_{11} - 217 \beta_{10} + 126 \beta_{9} + \cdots + 864 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 165 \beta_{14} - 229 \beta_{13} + 140 \beta_{12} - 1450 \beta_{11} - 1449 \beta_{10} + 508 \beta_{9} + \cdots + 5723 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 172 \beta_{14} - 2064 \beta_{13} + 322 \beta_{12} - 3074 \beta_{11} - 3546 \beta_{10} + 1307 \beta_{9} + \cdots + 11937 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2910 \beta_{14} - 4243 \beta_{13} + 685 \beta_{12} - 17859 \beta_{11} - 18104 \beta_{10} + 5564 \beta_{9} + \cdots + 64963 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5604 \beta_{14} - 26986 \beta_{13} - 1347 \beta_{12} - 46638 \beta_{11} - 52501 \beta_{10} + \cdots + 156951 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 46561 \beta_{14} - 69367 \beta_{13} - 7687 \beta_{12} - 227013 \beta_{11} - 233283 \beta_{10} + \cdots + 748880 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 117637 \beta_{14} - 360309 \beta_{13} - 80333 \beta_{12} - 672857 \beta_{11} - 743383 \beta_{10} + \cdots + 2003516 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 713312 \beta_{14} - 1063396 \beta_{13} - 319108 \beta_{12} - 2948055 \beta_{11} - 3058630 \beta_{10} + \cdots + 8730595 \) 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
−3.01525
−2.82901
−2.61011
−1.04745
−1.01946
−0.927807
0.117275
0.357651
0.546893
1.03500
1.33620
1.40631
3.48442
3.52096
3.64439
0 1.00000 0 −4.01525 0 −3.41124 0 1.00000 0
1.2 0 1.00000 0 −3.82901 0 3.22079 0 1.00000 0
1.3 0 1.00000 0 −3.61011 0 0.982804 0 1.00000 0
1.4 0 1.00000 0 −2.04745 0 3.89716 0 1.00000 0
1.5 0 1.00000 0 −2.01946 0 −2.53250 0 1.00000 0
1.6 0 1.00000 0 −1.92781 0 2.55035 0 1.00000 0
1.7 0 1.00000 0 −0.882725 0 −2.85663 0 1.00000 0
1.8 0 1.00000 0 −0.642349 0 −2.57693 0 1.00000 0
1.9 0 1.00000 0 −0.453107 0 2.17063 0 1.00000 0
1.10 0 1.00000 0 0.0349978 0 0.637505 0 1.00000 0
1.11 0 1.00000 0 0.336200 0 0.568501 0 1.00000 0
1.12 0 1.00000 0 0.406311 0 −3.65793 0 1.00000 0
1.13 0 1.00000 0 2.48442 0 1.05465 0 1.00000 0
1.14 0 1.00000 0 2.52096 0 −1.02518 0 1.00000 0
1.15 0 1.00000 0 2.64439 0 −3.02199 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(503\) \(1\)

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 6036.2.a.g ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6036.2.a.g ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):

\( T_{5}^{15} + 11 T_{5}^{14} + 22 T_{5}^{13} - 154 T_{5}^{12} - 659 T_{5}^{11} + 195 T_{5}^{10} + 4497 T_{5}^{9} + \cdots + 9 \) Copy content Toggle raw display
\( T_{7}^{15} + 4 T_{7}^{14} - 40 T_{7}^{13} - 164 T_{7}^{12} + 609 T_{7}^{11} + 2530 T_{7}^{10} + \cdots + 18812 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( (T - 1)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + 11 T^{14} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{15} + 4 T^{14} + \cdots + 18812 \) Copy content Toggle raw display
$11$ \( T^{15} + 5 T^{14} + \cdots - 264042 \) Copy content Toggle raw display
$13$ \( T^{15} + 14 T^{14} + \cdots - 117773 \) Copy content Toggle raw display
$17$ \( T^{15} + 5 T^{14} + \cdots - 104004 \) Copy content Toggle raw display
$19$ \( T^{15} - 3 T^{14} + \cdots - 12872 \) Copy content Toggle raw display
$23$ \( T^{15} + 32 T^{14} + \cdots - 81398862 \) Copy content Toggle raw display
$29$ \( T^{15} + 23 T^{14} + \cdots + 4218813 \) Copy content Toggle raw display
$31$ \( T^{15} + 13 T^{14} + \cdots - 53402374 \) Copy content Toggle raw display
$37$ \( T^{15} + 10 T^{14} + \cdots + 24255872 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 778226211 \) Copy content Toggle raw display
$43$ \( T^{15} - 4 T^{14} + \cdots - 75008 \) Copy content Toggle raw display
$47$ \( T^{15} + 20 T^{14} + \cdots - 48408336 \) Copy content Toggle raw display
$53$ \( T^{15} + 30 T^{14} + \cdots + 33788673 \) Copy content Toggle raw display
$59$ \( T^{15} + 14 T^{14} + \cdots + 55210986 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 3919129973 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 3683932484 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 921049300224 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 1179286194559 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 188319965276 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 2332890702 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 5436316061352 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 6645518722399 \) Copy content Toggle raw display
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