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Properties

Label 5265.2.a.bi
Level $5265$
Weight $2$
Character orbit 5265.a
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5265,2,Mod(1,5265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5265, 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("5265.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.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: \(42.0412366642\)
Analytic rank: \(1\)
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: \( x^{14} - 2 x^{13} - 20 x^{12} + 36 x^{11} + 156 x^{10} - 242 x^{9} - 601 x^{8} + 750 x^{7} + 1188 x^{6} + \cdots + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} - \beta_{3} q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} - \beta_{3} q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1 - 1) q^{8} + \beta_1 q^{10} + (\beta_{10} - 1) q^{11} + q^{13} + ( - \beta_{12} - \beta_{9} + \beta_{7} - 1) q^{14} + (\beta_{12} + \beta_{8} + \beta_{6} + \cdots + 2) q^{16}+ \cdots + (\beta_{12} + \beta_{10} - \beta_{9} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8} + 2 q^{10} - 8 q^{11} + 14 q^{13} - 12 q^{14} + 16 q^{16} - 10 q^{17} + 10 q^{19} - 16 q^{20} - 10 q^{22} - 34 q^{23} + 14 q^{25} - 2 q^{26} + 2 q^{28} - 16 q^{29} + 6 q^{31} - 26 q^{32} + 8 q^{34} - 4 q^{35} + 4 q^{37} - 12 q^{38} + 12 q^{40} - 10 q^{41} - 8 q^{43} - 8 q^{44} - 16 q^{46} - 46 q^{47} + 2 q^{49} - 2 q^{50} + 16 q^{52} - 14 q^{53} + 8 q^{55} - 20 q^{56} - 28 q^{58} - 16 q^{59} - 30 q^{62} + 14 q^{64} - 14 q^{65} + 6 q^{67} - 4 q^{68} + 12 q^{70} - 34 q^{71} - 4 q^{73} - 8 q^{74} + 50 q^{76} - 24 q^{77} - 14 q^{79} - 16 q^{80} - 16 q^{82} - 4 q^{83} + 10 q^{85} - 6 q^{86} - 68 q^{88} - 18 q^{89} + 4 q^{91} - 90 q^{92} - 10 q^{95} + 12 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 20 x^{12} + 36 x^{11} + 156 x^{10} - 242 x^{9} - 601 x^{8} + 750 x^{7} + 1188 x^{6} + \cdots + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 129 \nu^{13} + 257 \nu^{12} + 2573 \nu^{11} - 4629 \nu^{10} - 19967 \nu^{9} + 31199 \nu^{8} + \cdots - 6526 ) / 116 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 139 \nu^{13} + 303 \nu^{12} + 2723 \nu^{11} - 5495 \nu^{10} - 20621 \nu^{9} + 37361 \nu^{8} + \cdots - 4122 ) / 116 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 687 \nu^{13} - 1513 \nu^{12} - 13437 \nu^{11} + 27455 \nu^{10} + 101677 \nu^{9} - 186875 \nu^{8} + \cdots + 20886 ) / 116 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 687 \nu^{13} + 1513 \nu^{12} + 13437 \nu^{11} - 27455 \nu^{10} - 101677 \nu^{9} + 186875 \nu^{8} + \cdots - 21002 ) / 116 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 681 \nu^{13} - 1497 \nu^{12} - 13347 \nu^{11} + 27179 \nu^{10} + 101331 \nu^{9} - 185115 \nu^{8} + \cdots + 23326 ) / 116 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 211 \nu^{13} + 466 \nu^{12} + 4122 \nu^{11} - 8459 \nu^{10} - 31124 \nu^{9} + 57611 \nu^{8} + \cdots - 5582 ) / 29 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 489 \nu^{13} + 1072 \nu^{12} + 9568 \nu^{11} - 19449 \nu^{10} - 72395 \nu^{9} + 132362 \nu^{8} + \cdots - 13826 ) / 58 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 541 \nu^{13} - 1201 \nu^{12} - 10551 \nu^{11} + 21783 \nu^{10} + 79531 \nu^{9} - 148205 \nu^{8} + \cdots + 14758 ) / 58 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22 \nu^{13} + 49 \nu^{12} + 429 \nu^{11} - 890 \nu^{10} - 3232 \nu^{9} + 6067 \nu^{8} + 11857 \nu^{7} + \cdots - 570 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 830 \nu^{13} - 1817 \nu^{12} - 16249 \nu^{11} + 32960 \nu^{10} + 123070 \nu^{9} - 224259 \nu^{8} + \cdots + 25044 ) / 58 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1018 \nu^{13} - 2241 \nu^{12} - 19910 \nu^{11} + 40651 \nu^{10} + 150654 \nu^{9} - 276583 \nu^{8} + \cdots + 30912 ) / 58 \) 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
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{8} + \beta_{6} + \beta_{3} + 6\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{13} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - \beta_{4} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{13} + 11 \beta_{12} + \beta_{11} + 2 \beta_{10} - \beta_{9} + 10 \beta_{8} + 2 \beta_{7} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{13} + \beta_{12} + \beta_{11} + 14 \beta_{10} - 15 \beta_{9} + 14 \beta_{8} + 14 \beta_{7} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 31 \beta_{13} + 93 \beta_{12} + 15 \beta_{11} + 30 \beta_{10} - 18 \beta_{9} + 83 \beta_{8} + \cdots + 627 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 143 \beta_{13} + 20 \beta_{12} + 18 \beta_{11} + 142 \beta_{10} - 155 \beta_{9} + 140 \beta_{8} + \cdots + 676 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 339 \beta_{13} + 721 \beta_{12} + 155 \beta_{11} + 318 \beta_{10} - 213 \beta_{9} + 654 \beta_{8} + \cdots + 4200 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1298 \beta_{13} + 266 \beta_{12} + 213 \beta_{11} + 1273 \beta_{10} - 1380 \beta_{9} + 1240 \beta_{8} + \cdots + 5220 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3222 \beta_{13} + 5398 \beta_{12} + 1380 \beta_{11} + 2944 \beta_{10} - 2113 \beta_{9} + 5051 \beta_{8} + \cdots + 28752 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11119 \beta_{13} + 2936 \beta_{12} + 2113 \beta_{11} + 10733 \beta_{10} - 11411 \beta_{9} + \cdots + 39783 \) 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
2.74898
2.56616
2.25591
1.69419
1.41487
1.17566
−0.169293
−0.199325
−0.223920
−1.03609
−1.68974
−1.88407
−2.11355
−2.53978
−2.74898 0 5.55687 −1.00000 0 0.361779 −9.77774 0 2.74898
1.2 −2.56616 0 4.58517 −1.00000 0 −0.771466 −6.63395 0 2.56616
1.3 −2.25591 0 3.08913 −1.00000 0 4.21559 −2.45697 0 2.25591
1.4 −1.69419 0 0.870271 −1.00000 0 −3.19374 1.91397 0 1.69419
1.5 −1.41487 0 0.00186077 −1.00000 0 3.72094 2.82711 0 1.41487
1.6 −1.17566 0 −0.617819 −1.00000 0 0.892617 3.07767 0 1.17566
1.7 0.169293 0 −1.97134 −1.00000 0 3.42913 −0.672318 0 −0.169293
1.8 0.199325 0 −1.96027 −1.00000 0 −0.890678 −0.789381 0 −0.199325
1.9 0.223920 0 −1.94986 −1.00000 0 −3.21584 −0.884454 0 −0.223920
1.10 1.03609 0 −0.926527 −1.00000 0 −0.502440 −3.03213 0 −1.03609
1.11 1.68974 0 0.855216 −1.00000 0 −1.05947 −1.93439 0 −1.68974
1.12 1.88407 0 1.54972 −1.00000 0 1.46255 −0.848359 0 −1.88407
1.13 2.11355 0 2.46710 −1.00000 0 3.65830 0.987234 0 −2.11355
1.14 2.53978 0 4.45049 −1.00000 0 −4.10727 6.22371 0 −2.53978
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 5265.2.a.bi ✓ 14
3.b odd 2 1 5265.2.a.bj yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5265.2.a.bi ✓ 14 1.a even 1 1 trivial
5265.2.a.bj yes 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5265))\):

\( T_{2}^{14} + 2 T_{2}^{13} - 20 T_{2}^{12} - 36 T_{2}^{11} + 156 T_{2}^{10} + 242 T_{2}^{9} - 601 T_{2}^{8} + \cdots + 6 \) Copy content Toggle raw display
\( T_{7}^{14} - 4 T_{7}^{13} - 42 T_{7}^{12} + 162 T_{7}^{11} + 638 T_{7}^{10} - 2266 T_{7}^{9} + \cdots - 1434 \) Copy content Toggle raw display
\( T_{11}^{14} + 8 T_{11}^{13} - 48 T_{11}^{12} - 540 T_{11}^{11} + 185 T_{11}^{10} + 12044 T_{11}^{9} + \cdots - 79131 \) Copy content Toggle raw display
\( T_{17}^{14} + 10 T_{17}^{13} - 95 T_{17}^{12} - 1126 T_{17}^{11} + 2512 T_{17}^{10} + 45502 T_{17}^{9} + \cdots - 109056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots - 1434 \) Copy content Toggle raw display
$11$ \( T^{14} + 8 T^{13} + \cdots - 79131 \) Copy content Toggle raw display
$13$ \( (T - 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 10 T^{13} + \cdots - 109056 \) Copy content Toggle raw display
$19$ \( T^{14} - 10 T^{13} + \cdots - 19472639 \) Copy content Toggle raw display
$23$ \( T^{14} + 34 T^{13} + \cdots - 12288 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 1242771903 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 866845696 \) Copy content Toggle raw display
$37$ \( T^{14} - 4 T^{13} + \cdots + 690912 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 496588032 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 36140394592 \) Copy content Toggle raw display
$47$ \( T^{14} + 46 T^{13} + \cdots - 6002208 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 84400701894 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 1689122709 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 2153186584 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 1140755822336 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 940638087 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 138964589664 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 5188062016 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 428019715314 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 14835686352 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 3766290642432 \) Copy content Toggle raw display
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