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Properties

Label 2151.2.a.k
Level $2151$
Weight $2$
Character orbit 2151.a
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
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] = [2151,2,Mod(1,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2151.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.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: \(17.1758214748\)
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: \( x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{9} + 1) q^{5} - \beta_{12} q^{7} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{9} + 1) q^{5} - \beta_{12} q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{9} + \beta_{7} + \cdots + 2 \beta_1) q^{10}+ \cdots + (\beta_{19} + 3 \beta_{18} - \beta_{17} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8259046 \nu^{19} + 25715994 \nu^{18} + 170049029 \nu^{17} - 526870261 \nu^{16} + \cdots + 2324646198 ) / 615206542 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14412353 \nu^{19} + 53211591 \nu^{18} + 303862297 \nu^{17} - 1266785187 \nu^{16} + \cdots - 1169865603 ) / 615206542 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25152759 \nu^{19} + 155801032 \nu^{18} + 337358422 \nu^{17} - 3732397225 \nu^{16} + \cdots + 4109545279 ) / 615206542 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18194582 \nu^{19} + 67002479 \nu^{18} + 389694649 \nu^{17} - 1608799145 \nu^{16} + \cdots + 435415269 ) / 307603271 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 79748961 \nu^{19} - 301609179 \nu^{18} - 1707908090 \nu^{17} + 7153614242 \nu^{16} + \cdots - 5283572531 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32909587 \nu^{19} + 80534179 \nu^{18} + 838637985 \nu^{17} - 1930987718 \nu^{16} + \cdots - 1325315495 ) / 307603271 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 165811103 \nu^{19} - 636904335 \nu^{18} - 3626875370 \nu^{17} + 15370014952 \nu^{16} + \cdots - 234896285 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 167543794 \nu^{19} - 625032345 \nu^{18} - 3630490016 \nu^{17} + 14940203236 \nu^{16} + \cdots - 9584403627 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 191187961 \nu^{19} + 812153314 \nu^{18} + 3922186244 \nu^{17} - 19617511978 \nu^{16} + \cdots + 8319810742 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 230262066 \nu^{19} + 792118541 \nu^{18} + 5255031164 \nu^{17} - 19043970058 \nu^{16} + \cdots - 6936694175 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 148218686 \nu^{19} + 540557051 \nu^{18} + 3269848589 \nu^{17} - 12943039301 \nu^{16} + \cdots + 5574045831 ) / 615206542 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 349773159 \nu^{19} - 1538748210 \nu^{18} - 6932521542 \nu^{17} + 36955819636 \nu^{16} + \cdots - 7156850618 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 409558402 \nu^{19} + 1641156191 \nu^{18} + 8685638574 \nu^{17} - 39676028714 \nu^{16} + \cdots + 7398565297 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 598257155 \nu^{19} - 2412992855 \nu^{18} - 12540986368 \nu^{17} + 57939652746 \nu^{16} + \cdots - 14477391829 ) / 1230413084 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 339515531 \nu^{19} - 1337024169 \nu^{18} - 7181512587 \nu^{17} + 32022014849 \nu^{16} + \cdots - 10645201535 ) / 615206542 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 796840493 \nu^{19} + 3030854544 \nu^{18} + 17248139170 \nu^{17} - 72932486780 \nu^{16} + \cdots + 17183676468 ) / 1230413084 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} + \beta_{17} + \beta_{16} - \beta_{13} + \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{18} + 12 \beta_{17} + 11 \beta_{16} - 2 \beta_{14} - 10 \beta_{13} - \beta_{11} + 10 \beta_{10} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{19} + 11 \beta_{18} + 15 \beta_{17} - 12 \beta_{16} - 25 \beta_{15} - 15 \beta_{14} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 110 \beta_{18} + 113 \beta_{17} + 94 \beta_{16} - 3 \beta_{15} - 29 \beta_{14} - 84 \beta_{13} + \cdots + 442 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 122 \beta_{19} + 88 \beta_{18} + 156 \beta_{17} - 106 \beta_{16} - 230 \beta_{15} - 156 \beta_{14} + \cdots - 179 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{19} - 906 \beta_{18} + 962 \beta_{17} + 735 \beta_{16} - 54 \beta_{15} - 297 \beta_{14} + \cdots + 2674 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1017 \beta_{19} + 631 \beta_{18} + 1401 \beta_{17} - 842 \beta_{16} - 1891 \beta_{15} - 1400 \beta_{14} + \cdots - 1719 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 71 \beta_{19} - 7057 \beta_{18} + 7741 \beta_{17} + 5503 \beta_{16} - 638 \beta_{15} - 2651 \beta_{14} + \cdots + 16610 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8020 \beta_{19} + 4328 \beta_{18} + 11664 \beta_{17} - 6391 \beta_{16} - 14735 \beta_{15} - 11634 \beta_{14} + \cdots - 15143 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1031 \beta_{19} - 53193 \beta_{18} + 60120 \beta_{17} + 40187 \beta_{16} - 6269 \beta_{15} + \cdots + 105213 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 61387 \beta_{19} + 29207 \beta_{18} + 92831 \beta_{17} - 47505 \beta_{16} - 111466 \beta_{15} + \cdots - 126191 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11935 \beta_{19} - 392736 \beta_{18} + 455945 \beta_{17} + 288926 \beta_{16} - 55638 \beta_{15} + \cdots + 676831 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 461694 \beta_{19} + 196752 \beta_{18} + 717438 \beta_{17} - 349721 \beta_{16} - 828105 \beta_{15} + \cdots - 1012104 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 121329 \beta_{19} - 2860243 \beta_{18} + 3400864 \beta_{17} + 2055926 \beta_{16} - 463661 \beta_{15} + \cdots + 4409800 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3434226 \beta_{19} + 1333646 \beta_{18} + 5432860 \beta_{17} - 2563761 \beta_{16} - 6079625 \beta_{15} + \cdots - 7897579 \) 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
−2.66148
−2.12763
−2.02621
−1.88454
−1.38158
−1.25296
−0.921026
−0.510429
0.0242456
0.0888996
0.244320
0.854685
0.946263
1.22231
1.61701
1.66953
2.39589
2.41299
2.61567
2.67404
−2.66148 0 5.08345 2.46126 0 −5.08484 −8.20653 0 −6.55057
1.2 −2.12763 0 2.52683 2.10198 0 0.789580 −1.12089 0 −4.47224
1.3 −2.02621 0 2.10552 3.48073 0 1.37004 −0.213806 0 −7.05267
1.4 −1.88454 0 1.55150 −1.57704 0 −2.47195 0.845212 0 2.97201
1.5 −1.38158 0 −0.0912347 −1.35155 0 −1.50510 2.88921 0 1.86728
1.6 −1.25296 0 −0.430099 −1.67805 0 −0.437804 3.04481 0 2.10252
1.7 −0.921026 0 −1.15171 1.07676 0 3.99910 2.90281 0 −0.991725
1.8 −0.510429 0 −1.73946 1.97336 0 −1.30841 1.90873 0 −1.00726
1.9 0.0242456 0 −1.99941 −2.24441 0 −4.21532 −0.0969680 0 −0.0544170
1.10 0.0888996 0 −1.99210 0.674348 0 0.687902 −0.354896 0 0.0599493
1.11 0.244320 0 −1.94031 4.22460 0 0.0748971 −0.962697 0 1.03216
1.12 0.854685 0 −1.26951 −2.82017 0 3.33105 −2.79440 0 −2.41036
1.13 0.946263 0 −1.10459 4.23503 0 2.19590 −2.93776 0 4.00746
1.14 1.22231 0 −0.505952 0.403117 0 −4.45690 −3.06306 0 0.492736
1.15 1.61701 0 0.614711 1.46591 0 3.83563 −2.24002 0 2.37038
1.16 1.66953 0 0.787326 −3.19534 0 −2.48662 −2.02459 0 −5.33471
1.17 2.39589 0 3.74030 4.10699 0 −3.62464 4.16958 0 9.83991
1.18 2.41299 0 3.82251 −0.666388 0 0.549662 4.39769 0 −1.60799
1.19 2.61567 0 4.84173 2.80971 0 3.73688 7.43303 0 7.34928
1.20 2.67404 0 5.15050 0.519163 0 1.02094 8.42455 0 1.38826
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(239\) \(-1\)

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 2151.2.a.k yes 20
3.b odd 2 1 2151.2.a.j ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.2.a.j ✓ 20 3.b odd 2 1
2151.2.a.k yes 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):

\( T_{2}^{20} - 4 T_{2}^{19} - 21 T_{2}^{18} + 96 T_{2}^{17} + 164 T_{2}^{16} - 936 T_{2}^{15} - 540 T_{2}^{14} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{20} - 16 T_{5}^{19} + 67 T_{5}^{18} + 212 T_{5}^{17} - 2271 T_{5}^{16} + 2548 T_{5}^{15} + \cdots - 78784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 4 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 16 T^{19} + \cdots - 78784 \) Copy content Toggle raw display
$7$ \( T^{20} + 4 T^{19} + \cdots - 24064 \) Copy content Toggle raw display
$11$ \( T^{20} - 12 T^{19} + \cdots - 735404 \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + \cdots - 55588096 \) Copy content Toggle raw display
$17$ \( T^{20} - 24 T^{19} + \cdots - 31861696 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 3041439616 \) Copy content Toggle raw display
$23$ \( T^{20} - 12 T^{19} + \cdots - 83221504 \) Copy content Toggle raw display
$29$ \( T^{20} - 24 T^{19} + \cdots + 3203296 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10178867008 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1484607744 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 14710729952 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 14800745004544 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 168178221056 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 1686381404704 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 599715727156736 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 78953104580608 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 49\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 82\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 27\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 979929064644044 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 34\!\cdots\!28 \) Copy content Toggle raw display
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