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Properties

Label 927.2.a.h
Level $927$
Weight $2$
Character orbit 927.a
Self dual yes
Analytic conductor $7.402$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [927,2,Mod(1,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, 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("927.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 24x^{12} + 221x^{10} - 980x^{8} + 2160x^{6} - 2203x^{4} + 808x^{2} - 75 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{5} + (\beta_{6} + 1) q^{7} + (\beta_{11} + \beta_{10} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{5} + (\beta_{6} + 1) q^{7} + (\beta_{11} + \beta_{10} + 2 \beta_1) q^{8} + ( - \beta_{8} + \beta_{3} - \beta_{2} + 1) q^{10} + ( - \beta_{13} - \beta_{12} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{13} - 3 \beta_{11} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 20 q^{4} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 20 q^{4} + 12 q^{7} + 4 q^{10} + 20 q^{13} + 36 q^{16} + 8 q^{19} + 32 q^{22} + 58 q^{25} + 26 q^{28} - 4 q^{31} - 16 q^{34} + 28 q^{37} - 32 q^{40} + 40 q^{43} + 14 q^{46} + 26 q^{49} + 28 q^{55} - 4 q^{58} + 24 q^{61} + 64 q^{64} - 8 q^{67} - 64 q^{70} + 56 q^{73} + 4 q^{79} - 26 q^{82} + 24 q^{85} + 28 q^{88} + 44 q^{91} - 60 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 24x^{12} + 221x^{10} - 980x^{8} + 2160x^{6} - 2203x^{4} + 808x^{2} - 75 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{12} - 521\nu^{10} + 4336\nu^{8} - 16042\nu^{6} + 25210\nu^{4} - 13457\nu^{2} + 1333 ) / 206 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -79\nu^{13} + 1691\nu^{11} - 13084\nu^{9} + 43520\nu^{7} - 56900\nu^{5} + 22917\nu^{3} - 8197\nu ) / 1030 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{12} + 290\nu^{10} - 2406\nu^{8} + 9112\nu^{6} - 15615\nu^{4} + 10620\nu^{2} - 1873 ) / 103 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\nu^{12} - 757\nu^{10} + 5931\nu^{8} - 20032\nu^{6} + 26424\nu^{4} - 8642\nu^{2} + 376 ) / 103 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\nu^{12} - 698\nu^{10} + 5558\nu^{8} - 19395\nu^{6} + 27614\nu^{4} - 11571\nu^{2} + 847 ) / 103 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32\nu^{12} - 698\nu^{10} + 5558\nu^{8} - 19395\nu^{6} + 27717\nu^{4} - 12395\nu^{2} + 1671 ) / 103 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\nu^{13} - 290\nu^{11} + 2406\nu^{9} - 9112\nu^{7} + 15615\nu^{5} - 10620\nu^{3} + 1873\nu ) / 103 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -66\nu^{13} + 1504\nu^{11} - 12841\nu^{9} + 50785\nu^{7} - 93815\nu^{5} + 73788\nu^{3} - 18478\nu ) / 515 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 66\nu^{13} - 1504\nu^{11} + 12841\nu^{9} - 50785\nu^{7} + 93815\nu^{5} - 73273\nu^{3} + 15388\nu ) / 515 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 271\nu^{13} - 5879\nu^{11} + 46226\nu^{9} - 156800\nu^{7} + 207340\nu^{5} - 63503\nu^{3} - 4437\nu ) / 1030 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -161\nu^{13} + 3544\nu^{11} - 28601\nu^{9} + 102200\nu^{7} - 153810\nu^{5} + 78028\nu^{3} - 10258\nu ) / 515 \) 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
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - \beta_{12} + 9\beta_{11} + 9\beta_{10} - \beta_{9} - \beta_{4} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{8} - 9\beta_{7} - 2\beta_{6} + 2\beta_{5} + 59\beta_{2} + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{13} - 14\beta_{12} + 68\beta_{11} + 71\beta_{10} - 11\beta_{9} - 10\beta_{4} + 277\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 110\beta_{8} - 64\beta_{7} - 31\beta_{6} + 28\beta_{5} - 2\beta_{3} + 429\beta_{2} + 664 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -154\beta_{13} - 141\beta_{12} + 491\beta_{11} + 539\beta_{10} - 88\beta_{9} - 81\beta_{4} + 1949\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 915\beta_{8} - 425\beta_{7} - 330\beta_{6} + 277\beta_{5} - 46\beta_{3} + 3113\beta_{2} + 4529 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1359\beta_{13} - 1245\beta_{12} + 3492\beta_{11} + 4028\beta_{10} - 610\beta_{9} - 631\beta_{4} + 13822\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7263\beta_{8} - 2743\beta_{7} - 3026\beta_{6} + 2391\beta_{5} - 656\beta_{3} + 22608\beta_{2} + 31321 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11127 \beta_{13} - 10289 \beta_{12} + 24695 \beta_{11} + 29871 \beta_{10} - 3822 \beta_{9} + \cdots + 98489 \beta_1 \) 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
−2.71631
−2.61204
−2.19418
−1.49630
−1.46308
−0.683098
−0.371990
0.371990
0.683098
1.46308
1.49630
2.19418
2.61204
2.71631
−2.71631 0 5.37832 4.18121 0 1.30376 −9.17656 0 −11.3575
1.2 −2.61204 0 4.82276 −2.84138 0 0.528127 −7.37317 0 7.42179
1.3 −2.19418 0 2.81440 −0.965737 0 1.91080 −1.78694 0 2.11900
1.4 −1.49630 0 0.238906 −3.96554 0 −3.70540 2.63512 0 5.93362
1.5 −1.46308 0 0.140605 −0.979247 0 4.33523 2.72044 0 1.43272
1.6 −0.683098 0 −1.53338 4.01547 0 4.16945 2.41364 0 −2.74296
1.7 −0.371990 0 −1.86162 2.16865 0 −2.54197 1.43649 0 −0.806715
1.8 0.371990 0 −1.86162 −2.16865 0 −2.54197 −1.43649 0 −0.806715
1.9 0.683098 0 −1.53338 −4.01547 0 4.16945 −2.41364 0 −2.74296
1.10 1.46308 0 0.140605 0.979247 0 4.33523 −2.72044 0 1.43272
1.11 1.49630 0 0.238906 3.96554 0 −3.70540 −2.63512 0 5.93362
1.12 2.19418 0 2.81440 0.965737 0 1.91080 1.78694 0 2.11900
1.13 2.61204 0 4.82276 2.84138 0 0.528127 7.37317 0 7.42179
1.14 2.71631 0 5.37832 −4.18121 0 1.30376 9.17656 0 −11.3575
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(103\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.a.h ✓ 14
3.b odd 2 1 inner 927.2.a.h ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.a.h ✓ 14 1.a even 1 1 trivial
927.2.a.h ✓ 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 24T_{2}^{12} + 221T_{2}^{10} - 980T_{2}^{8} + 2160T_{2}^{6} - 2203T_{2}^{4} + 808T_{2}^{2} - 75 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(927))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 24 T^{12} + \cdots - 75 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 64 T^{12} + \cdots - 150528 \) Copy content Toggle raw display
$7$ \( (T^{7} - 6 T^{6} + \cdots - 224)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} - 104 T^{12} + \cdots - 196608 \) Copy content Toggle raw display
$13$ \( (T^{7} - 10 T^{6} + \cdots - 722)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 140685312 \) Copy content Toggle raw display
$19$ \( (T^{7} - 4 T^{6} + \cdots + 560)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} - 154 T^{12} + \cdots - 972 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 564056832 \) Copy content Toggle raw display
$31$ \( (T^{7} + 2 T^{6} + \cdots + 19328)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 14 T^{6} + \cdots - 34240)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 25026794688 \) Copy content Toggle raw display
$43$ \( (T^{7} - 20 T^{6} + \cdots + 44800)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} - 296 T^{12} + \cdots - 49152 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 1420824686592 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 439665708 \) Copy content Toggle raw display
$61$ \( (T^{7} - 12 T^{6} + \cdots + 262)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 4 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 1480367013888 \) Copy content Toggle raw display
$73$ \( (T^{7} - 28 T^{6} + \cdots - 738304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} - 2 T^{6} + \cdots + 19744)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 214678380108 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 397977366528 \) Copy content Toggle raw display
$97$ \( (T^{7} - 6 T^{6} + \cdots + 994)^{2} \) Copy content Toggle raw display
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