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Properties

Label 1503.2.a.i
Level $1503$
Weight $2$
Character orbit 1503.a
Self dual yes
Analytic conductor $12.002$
Analytic rank $0$
Dimension $14$
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] = [1503,2,Mod(1,1503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1503, 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("1503.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1503 = 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1503.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: \(12.0015154238\)
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: \( x^{14} - 6 x^{13} - 4 x^{12} + 82 x^{11} - 72 x^{10} - 394 x^{9} + 586 x^{8} + 766 x^{7} - 1589 x^{6} + \cdots + 9 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{11} + 1) q^{5} - \beta_{10} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{11} + 1) q^{5} - \beta_{10} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{13} - \beta_{12} + \cdots + \beta_1) q^{10}+ \cdots + ( - \beta_{13} + \beta_{9} + \beta_{6} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{2} + 16 q^{4} + 12 q^{5} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{2} + 16 q^{4} + 12 q^{5} + 18 q^{8} + 4 q^{10} + 12 q^{11} - 2 q^{13} + 12 q^{14} + 16 q^{16} + 26 q^{17} + 30 q^{20} - 10 q^{22} + 24 q^{23} + 12 q^{25} + 10 q^{26} - 2 q^{28} + 16 q^{29} + 8 q^{31} + 42 q^{32} + 4 q^{34} + 16 q^{35} - 8 q^{37} + 20 q^{38} + 6 q^{40} + 26 q^{41} - 6 q^{43} + 36 q^{44} - 4 q^{46} + 24 q^{47} - 6 q^{49} + 2 q^{50} + 8 q^{52} + 38 q^{53} + 4 q^{55} - 12 q^{56} - 6 q^{58} + 10 q^{59} - 4 q^{61} + 24 q^{62} + 8 q^{64} + 36 q^{65} - 6 q^{67} + 20 q^{68} - 16 q^{70} + 32 q^{71} - 8 q^{74} + 12 q^{77} - 28 q^{79} + 40 q^{80} - 2 q^{82} + 10 q^{83} + 14 q^{85} - 70 q^{86} - 20 q^{88} + 44 q^{89} - 32 q^{91} + 18 q^{92} + 32 q^{94} + 8 q^{95} - 2 q^{97} + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 4 x^{12} + 82 x^{11} - 72 x^{10} - 394 x^{9} + 586 x^{8} + 766 x^{7} - 1589 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{13} + 32 \nu^{12} - 83 \nu^{11} - 239 \nu^{10} + 1277 \nu^{9} - 159 \nu^{8} - 5417 \nu^{7} + \cdots + 262 ) / 163 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46 \nu^{13} - 219 \nu^{12} - 466 \nu^{11} + 3067 \nu^{10} + 1392 \nu^{9} - 15655 \nu^{8} - 2786 \nu^{7} + \cdots - 51 ) / 978 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{13} + 63 \nu^{12} + 76 \nu^{11} - 985 \nu^{10} + 553 \nu^{9} + 5611 \nu^{8} - 6605 \nu^{7} + \cdots - 941 ) / 163 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112 \nu^{13} - 597 \nu^{12} - 922 \nu^{11} + 8977 \nu^{10} - 1926 \nu^{9} - 49810 \nu^{8} + \cdots - 3696 ) / 489 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 443 \nu^{13} + 2226 \nu^{12} + 3935 \nu^{11} - 32630 \nu^{10} + 903 \nu^{9} + 176522 \nu^{8} + \cdots + 5328 ) / 978 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 605 \nu^{13} - 2976 \nu^{12} - 5321 \nu^{11} + 42602 \nu^{10} - 423 \nu^{9} - 223682 \nu^{8} + \cdots - 5784 ) / 978 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 93 \nu^{13} - 503 \nu^{12} - 687 \nu^{11} + 7246 \nu^{10} - 2097 \nu^{9} - 38266 \nu^{8} + 29703 \nu^{7} + \cdots - 624 ) / 163 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 788 \nu^{13} - 4113 \nu^{12} - 6452 \nu^{11} + 59789 \nu^{10} - 8556 \nu^{9} - 320585 \nu^{8} + \cdots - 10845 ) / 978 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1240 \nu^{13} + 6435 \nu^{12} + 10138 \nu^{11} - 93625 \nu^{10} + 15246 \nu^{9} + 501031 \nu^{8} + \cdots + 15003 ) / 978 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 471 \nu^{13} - 2416 \nu^{12} - 3921 \nu^{11} + 35078 \nu^{10} - 4563 \nu^{9} - 187426 \nu^{8} + \cdots - 5600 ) / 326 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} + 6\beta_{2} + 7\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{12} + 11 \beta_{11} - \beta_{10} + 11 \beta_{9} + 11 \beta_{8} + 13 \beta_{7} - 11 \beta_{6} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8 \beta_{13} + 16 \beta_{12} + 23 \beta_{11} - 10 \beta_{10} + 23 \beta_{9} + 13 \beta_{8} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{13} + 95 \beta_{12} + 90 \beta_{11} - 12 \beta_{10} + 90 \beta_{9} + 89 \beta_{8} + 118 \beta_{7} + \cdots + 464 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 41 \beta_{13} + 170 \beta_{12} + 197 \beta_{11} - 78 \beta_{10} + 194 \beta_{9} + 121 \beta_{8} + \cdots + 682 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 59 \beta_{13} + 756 \beta_{12} + 669 \beta_{11} - 113 \beta_{10} + 659 \beta_{9} + 648 \beta_{8} + \cdots + 2961 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 116 \beta_{13} + 1532 \beta_{12} + 1518 \beta_{11} - 578 \beta_{10} + 1454 \beta_{9} + 988 \beta_{8} + \cdots + 5056 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 758 \beta_{13} + 5792 \beta_{12} + 4772 \beta_{11} - 992 \beta_{10} + 4572 \beta_{9} + 4508 \beta_{8} + \cdots + 19608 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 575 \beta_{13} + 12712 \beta_{12} + 11122 \beta_{11} - 4290 \beta_{10} + 10279 \beta_{9} + 7548 \beta_{8} + \cdots + 37237 \) 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
−2.36993
−1.81134
−1.63397
−1.52914
−0.595548
−0.0493127
0.633056
0.777145
0.872021
1.78945
2.14788
2.44994
2.61235
2.70739
−2.36993 0 3.61656 3.67340 0 −0.937045 −3.83114 0 −8.70571
1.2 −1.81134 0 1.28096 2.56153 0 1.55400 1.30243 0 −4.63980
1.3 −1.63397 0 0.669864 −1.17880 0 −1.31799 2.17340 0 1.92612
1.4 −1.52914 0 0.338260 −1.51816 0 −3.21207 2.54103 0 2.32147
1.5 −0.595548 0 −1.64532 1.57183 0 1.28118 2.17097 0 −0.936099
1.6 −0.0493127 0 −1.99757 −1.33250 0 −3.53213 0.197131 0 0.0657093
1.7 0.633056 0 −1.59924 2.32692 0 4.09048 −2.27852 0 1.47307
1.8 0.777145 0 −1.39605 4.16105 0 −0.413809 −2.63922 0 3.23374
1.9 0.872021 0 −1.23958 −2.81662 0 1.82137 −2.82498 0 −2.45615
1.10 1.78945 0 1.20214 −0.910437 0 −2.98103 −1.42773 0 −1.62918
1.11 2.14788 0 2.61339 1.02511 0 2.87041 1.31748 0 2.20181
1.12 2.44994 0 4.00220 1.90019 0 3.81250 4.90528 0 4.65535
1.13 2.61235 0 4.82439 3.97329 0 −3.25651 7.37831 0 10.3797
1.14 2.70739 0 5.32998 −1.43679 0 0.220643 9.01557 0 −3.88997
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(167\) \(-1\)

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 1503.2.a.i yes 14
3.b odd 2 1 1503.2.a.h ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1503.2.a.h ✓ 14 3.b odd 2 1
1503.2.a.i yes 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 6 T_{2}^{13} - 4 T_{2}^{12} + 82 T_{2}^{11} - 72 T_{2}^{10} - 394 T_{2}^{9} + 586 T_{2}^{8} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1503))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 6 T^{13} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 12 T^{13} + \cdots + 9738 \) Copy content Toggle raw display
$7$ \( T^{14} - 46 T^{12} + \cdots - 2016 \) Copy content Toggle raw display
$11$ \( T^{14} - 12 T^{13} + \cdots + 134172 \) Copy content Toggle raw display
$13$ \( T^{14} + 2 T^{13} + \cdots - 1173488 \) Copy content Toggle raw display
$17$ \( T^{14} - 26 T^{13} + \cdots - 63758 \) Copy content Toggle raw display
$19$ \( T^{14} - 158 T^{12} + \cdots + 99136 \) Copy content Toggle raw display
$23$ \( T^{14} - 24 T^{13} + \cdots - 98889856 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 1671719168 \) Copy content Toggle raw display
$31$ \( T^{14} - 8 T^{13} + \cdots + 22208 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 117862128 \) Copy content Toggle raw display
$41$ \( T^{14} - 26 T^{13} + \cdots - 3901654 \) Copy content Toggle raw display
$43$ \( T^{14} + 6 T^{13} + \cdots - 53183522 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 4526879868 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 23847186266 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 1587058016 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 50051772108 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1972020262 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 2117939328 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 36765383984 \) Copy content Toggle raw display
$79$ \( T^{14} + 28 T^{13} + \cdots + 54980982 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 3870696730016 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 194929231744 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 36017758368 \) Copy content Toggle raw display
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