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Properties

Label 1519.2.a.k
Level $1519$
Weight $2$
Character orbit 1519.a
Self dual yes
Analytic conductor $12.129$
Analytic rank $0$
Dimension $13$
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] = [1519,2,Mod(1,1519)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1519, 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("1519.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1519 = 7^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1519.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: \(12.1292760670\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 9 x^{11} + 76 x^{10} - 17 x^{9} - 387 x^{8} + 332 x^{7} + 758 x^{6} - 875 x^{5} + \cdots + 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 217)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{9} q^{5} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{6}+ \cdots + (\beta_{10} - \beta_{7} + \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{9} q^{5} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{6}+ \cdots + ( - 2 \beta_{12} + \beta_{11} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 5 q^{2} + 17 q^{4} + q^{5} - 2 q^{6} + 12 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 5 q^{2} + 17 q^{4} + q^{5} - 2 q^{6} + 12 q^{8} + 25 q^{9} - 7 q^{10} + 15 q^{11} - 5 q^{12} + 4 q^{13} + 4 q^{15} + 29 q^{16} - 4 q^{17} + 16 q^{18} + 2 q^{19} + 26 q^{20} - 10 q^{22} + 14 q^{23} - 28 q^{24} + 24 q^{25} - 7 q^{26} + 12 q^{27} + 22 q^{29} + 6 q^{30} + 13 q^{31} + 19 q^{32} - 5 q^{33} + 20 q^{34} + 11 q^{36} + 12 q^{37} - 11 q^{38} + 11 q^{39} - 6 q^{40} + 4 q^{41} - 3 q^{43} + 52 q^{44} - 12 q^{45} - 3 q^{46} - 14 q^{47} + 48 q^{48} + 15 q^{50} + 16 q^{51} + 4 q^{52} + 19 q^{53} - 25 q^{54} + 18 q^{55} + 13 q^{57} + 24 q^{58} + 19 q^{59} + 6 q^{60} - 11 q^{61} + 5 q^{62} + 10 q^{64} + 68 q^{65} - 52 q^{66} - 25 q^{67} - 26 q^{68} + 52 q^{69} + 28 q^{71} + 52 q^{72} - 29 q^{73} + 54 q^{74} - 71 q^{75} + 37 q^{76} - 71 q^{78} + 30 q^{79} + 3 q^{80} + 25 q^{81} + 5 q^{82} - 10 q^{83} - q^{85} + 10 q^{86} + 50 q^{87} + 18 q^{88} - 11 q^{89} + 81 q^{90} + 35 q^{92} - 36 q^{94} - 20 q^{95} - 12 q^{96} - 3 q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 9 x^{11} + 76 x^{10} - 17 x^{9} - 387 x^{8} + 332 x^{7} + 758 x^{6} - 875 x^{5} + \cdots + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - \nu^{12} + 4 \nu^{11} + 13 \nu^{10} - 63 \nu^{9} - 45 \nu^{8} + 341 \nu^{7} - 6 \nu^{6} - 752 \nu^{5} + \cdots + 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( - 2 \nu^{12} + 2 \nu^{11} + 36 \nu^{10} - 27 \nu^{9} - 237 \nu^{8} + 107 \nu^{7} + 699 \nu^{6} + \cdots - 22 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( - \nu^{12} + 6 \nu^{11} + 9 \nu^{10} - 95 \nu^{9} + 14 \nu^{8} + 519 \nu^{7} - 292 \nu^{6} - 1166 \nu^{5} + \cdots + 53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 4 \nu^{12} + 8 \nu^{11} + 64 \nu^{10} - 119 \nu^{9} - 355 \nu^{8} + 586 \nu^{7} + 815 \nu^{6} + \cdots + 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 4 \nu^{12} - 10 \nu^{11} - 61 \nu^{10} + 152 \nu^{9} + 312 \nu^{8} - 777 \nu^{7} - 615 \nu^{6} + \cdots - 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 4 \nu^{12} - 11 \nu^{11} - 59 \nu^{10} + 168 \nu^{9} + 282 \nu^{8} - 866 \nu^{7} - 465 \nu^{6} + \cdots - 51 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( 2 \nu^{12} - 11 \nu^{11} - 19 \nu^{10} + 174 \nu^{9} - 15 \nu^{8} - 952 \nu^{7} + 532 \nu^{6} + \cdots - 108 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 4 \nu^{12} - 16 \nu^{11} - 50 \nu^{10} + 250 \nu^{9} + 149 \nu^{8} - 1339 \nu^{7} + 182 \nu^{6} + \cdots - 116 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( 4 \nu^{12} - 16 \nu^{11} - 49 \nu^{10} + 250 \nu^{9} + 133 \nu^{8} - 1342 \nu^{7} + 266 \nu^{6} + \cdots - 119 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} + 9\beta_{3} + \beta_{2} + 29\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{12} - 11 \beta_{11} + 2 \beta_{10} + \beta_{8} - 10 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{12} - 11 \beta_{11} - 12 \beta_{10} + 10 \beta_{9} - 12 \beta_{8} - 13 \beta_{6} - 2 \beta_{5} + \cdots + 68 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{12} - 92 \beta_{11} + 28 \beta_{10} - 2 \beta_{9} + 16 \beta_{8} - 78 \beta_{7} + \cdots + 617 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 106 \beta_{12} - 93 \beta_{11} - 103 \beta_{10} + 75 \beta_{9} - 104 \beta_{8} + 2 \beta_{7} + \cdots + 494 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 136 \beta_{12} - 703 \beta_{11} + 277 \beta_{10} - 35 \beta_{9} + 169 \beta_{8} - 562 \beta_{7} + \cdots + 4018 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 833 \beta_{12} - 725 \beta_{11} - 773 \beta_{10} + 506 \beta_{9} - 794 \beta_{8} + 35 \beta_{7} + \cdots + 3531 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1138 \beta_{12} - 5168 \beta_{11} + 2386 \beta_{10} - 408 \beta_{9} + 1507 \beta_{8} - 3917 \beta_{7} + \cdots + 26518 \) 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
−2.59466
−2.21883
−1.47057
−1.14823
−0.439068
0.238797
0.584832
0.602246
1.81576
1.84239
2.56957
2.57456
2.64321
−2.59466 2.07828 4.73227 1.79355 −5.39243 0 −7.08931 1.31925 −4.65366
1.2 −2.21883 −0.877744 2.92322 0.889927 1.94757 0 −2.04848 −2.22957 −1.97460
1.3 −1.47057 −1.55945 0.162576 4.37105 2.29329 0 2.70206 −0.568105 −6.42794
1.4 −1.14823 −3.07491 −0.681574 −2.50743 3.53069 0 3.07906 6.45505 2.87910
1.5 −0.439068 3.43224 −1.80722 −2.66715 −1.50699 0 1.67163 8.78030 1.17106
1.6 0.238797 0.303745 −1.94298 −3.39269 0.0725334 0 −0.941572 −2.90774 −0.810167
1.7 0.584832 2.71801 −1.65797 1.55108 1.58958 0 −2.13930 4.38760 0.907119
1.8 0.602246 −1.36230 −1.63730 0.673460 −0.820438 0 −2.19055 −1.14415 0.405588
1.9 1.81576 2.14096 1.29698 2.45773 3.88747 0 −1.27651 1.58372 4.46264
1.10 1.84239 −2.73867 1.39439 −3.57831 −5.04568 0 −1.11576 4.50029 −6.59263
1.11 2.56957 −2.85025 4.60270 3.57581 −7.32392 0 6.68783 5.12392 9.18831
1.12 2.57456 −0.536269 4.62836 −2.52145 −1.38066 0 6.76686 −2.71242 −6.49162
1.13 2.64321 2.32634 4.98654 0.354419 6.14899 0 7.89404 2.41186 0.936803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(31\) \(-1\)

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 1519.2.a.k 13
7.b odd 2 1 1519.2.a.j 13
7.c even 3 2 217.2.f.b ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
217.2.f.b ✓ 26 7.c even 3 2
1519.2.a.j 13 7.b odd 2 1
1519.2.a.k 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1519))\):

\( T_{2}^{13} - 5 T_{2}^{12} - 9 T_{2}^{11} + 76 T_{2}^{10} - 17 T_{2}^{9} - 387 T_{2}^{8} + 332 T_{2}^{7} + \cdots + 21 \) Copy content Toggle raw display
\( T_{3}^{13} - 32 T_{3}^{11} - 4 T_{3}^{10} + 391 T_{3}^{9} + 99 T_{3}^{8} - 2281 T_{3}^{7} - 941 T_{3}^{6} + \cdots + 704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 5 T^{12} + \cdots + 21 \) Copy content Toggle raw display
$3$ \( T^{13} - 32 T^{11} + \cdots + 704 \) Copy content Toggle raw display
$5$ \( T^{13} - T^{12} + \cdots + 4647 \) Copy content Toggle raw display
$7$ \( T^{13} \) Copy content Toggle raw display
$11$ \( T^{13} - 15 T^{12} + \cdots + 343872 \) Copy content Toggle raw display
$13$ \( T^{13} - 4 T^{12} + \cdots - 75328 \) Copy content Toggle raw display
$17$ \( T^{13} + 4 T^{12} + \cdots - 10737984 \) Copy content Toggle raw display
$19$ \( T^{13} - 2 T^{12} + \cdots + 19657 \) Copy content Toggle raw display
$23$ \( T^{13} - 14 T^{12} + \cdots - 13105344 \) Copy content Toggle raw display
$29$ \( T^{13} - 22 T^{12} + \cdots + 19255104 \) Copy content Toggle raw display
$31$ \( (T - 1)^{13} \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 965580352 \) Copy content Toggle raw display
$41$ \( T^{13} - 4 T^{12} + \cdots - 3394167 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 177675968 \) Copy content Toggle raw display
$47$ \( T^{13} + 14 T^{12} + \cdots - 80540589 \) Copy content Toggle raw display
$53$ \( T^{13} - 19 T^{12} + \cdots - 50428224 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 297485648229 \) Copy content Toggle raw display
$61$ \( T^{13} + 11 T^{12} + \cdots + 44109824 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 69681546193 \) Copy content Toggle raw display
$71$ \( T^{13} - 28 T^{12} + \cdots - 72039 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 18890194112 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 3933944512 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 2613683136 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 55783604928 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 2765443775 \) Copy content Toggle raw display
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