Properties

Label 5265.2.a.bk
Level $5265$
Weight $2$
Character orbit 5265.a
Self dual yes
Analytic conductor $42.041$
Analytic rank $0$
Dimension $15$
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] = [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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 25 x^{13} + 24 x^{12} + 244 x^{11} - 226 x^{10} - 1170 x^{9} + 1051 x^{8} + 2842 x^{7} - 2478 x^{6} - 3252 x^{5} + 2697 x^{4} + 1497 x^{3} - 1107 x^{2} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 585)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + \beta_1 q^{10} + (\beta_{13} - 1) q^{11} + q^{13} + (\beta_{11} - \beta_1) q^{14} + (\beta_{9} - \beta_{8} + \beta_{2} + 2) q^{16} + (\beta_{5} - \beta_{3}) q^{17} + ( - \beta_{9} + 1) q^{19} + ( - \beta_{2} - 1) q^{20} + (\beta_{14} - \beta_{12} - \beta_{10} + \beta_{8} + \beta_{4} + \beta_1 + 1) q^{22} + (\beta_{12} + \beta_{2}) q^{23} + q^{25} - \beta_1 q^{26} + (\beta_{12} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} + 2) q^{28} + (\beta_{7} - 1) q^{29} + (\beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} + 1) q^{31} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} + \cdots - 1) q^{32}+ \cdots + ( - \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{4} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} + 21 q^{4} - 15 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} + 21 q^{4} - 15 q^{5} + 10 q^{7} + q^{10} - 9 q^{11} + 15 q^{13} - 3 q^{14} + 33 q^{16} + 3 q^{17} + 15 q^{19} - 21 q^{20} + 10 q^{22} + 6 q^{23} + 15 q^{25} - q^{26} + 35 q^{28} - 8 q^{29} + 22 q^{31} - 21 q^{32} + 9 q^{34} - 10 q^{35} + 4 q^{37} + 14 q^{38} - 13 q^{41} + 24 q^{43} + 5 q^{44} - 3 q^{46} + q^{47} + 37 q^{49} - q^{50} + 21 q^{52} + 7 q^{53} + 9 q^{55} - 17 q^{56} + 22 q^{58} - 19 q^{59} + 16 q^{61} + 13 q^{62} + 36 q^{64} - 15 q^{65} + 11 q^{67} + 28 q^{68} + 3 q^{70} - 28 q^{71} + 26 q^{73} - 8 q^{74} + 18 q^{76} + 24 q^{77} + 44 q^{79} - 33 q^{80} + 35 q^{82} + 3 q^{83} - 3 q^{85} - 40 q^{86} + 37 q^{88} - 4 q^{89} + 10 q^{91} + 74 q^{92} + 2 q^{94} - 15 q^{95} + 33 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 25 x^{13} + 24 x^{12} + 244 x^{11} - 226 x^{10} - 1170 x^{9} + 1051 x^{8} + 2842 x^{7} - 2478 x^{6} - 3252 x^{5} + 2697 x^{4} + 1497 x^{3} - 1107 x^{2} + \cdots + 144 \) : 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
\(\beta_{4}\)\(=\) \( ( 5719 \nu^{14} + 4109 \nu^{13} - 154636 \nu^{12} - 124671 \nu^{11} + 1667800 \nu^{10} + 1415087 \nu^{9} - 9140766 \nu^{8} - 7517408 \nu^{7} + 26823064 \nu^{6} + \cdots - 4402476 ) / 276852 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4405 \nu^{14} - 7316 \nu^{13} + 117715 \nu^{12} + 179847 \nu^{11} - 1244317 \nu^{10} - 1705067 \nu^{9} + 6609387 \nu^{8} + 7823414 \nu^{7} - 18528802 \nu^{6} + \cdots + 3216906 ) / 138426 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4836 \nu^{14} - 3051 \nu^{13} + 122293 \nu^{12} + 72520 \nu^{11} - 1196073 \nu^{10} - 642550 \nu^{9} + 5652535 \nu^{8} + 2594854 \nu^{7} - 13099408 \nu^{6} + \cdots + 497516 ) / 92284 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18125 \nu^{14} + 3836 \nu^{13} + 454973 \nu^{12} - 62739 \nu^{11} - 4495667 \nu^{10} + 313295 \nu^{9} + 22206429 \nu^{8} - 218102 \nu^{7} - 57679784 \nu^{6} + \cdots + 8513436 ) / 276852 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20158 \nu^{14} - 28129 \nu^{13} - 494767 \nu^{12} + 672840 \nu^{11} + 4723849 \nu^{10} - 6256954 \nu^{9} - 21984117 \nu^{8} + 28230844 \nu^{7} + 50746798 \nu^{6} + \cdots - 941400 ) / 276852 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20158 \nu^{14} - 28129 \nu^{13} - 494767 \nu^{12} + 672840 \nu^{11} + 4723849 \nu^{10} - 6256954 \nu^{9} - 21984117 \nu^{8} + 28230844 \nu^{7} + 50746798 \nu^{6} + \cdots + 442860 ) / 276852 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11117 \nu^{14} + 9803 \nu^{13} + 281132 \nu^{12} - 229887 \nu^{11} - 2767724 \nu^{10} + 2088959 \nu^{9} + 13296870 \nu^{8} - 9152588 \nu^{7} - 31900520 \nu^{6} + \cdots + 1636692 ) / 138426 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7887 \nu^{14} + 1393 \nu^{13} + 188584 \nu^{12} - 16089 \nu^{11} - 1735486 \nu^{10} - 5585 \nu^{9} + 7677490 \nu^{8} + 644504 \nu^{7} - 16587868 \nu^{6} + \cdots + 696384 ) / 92284 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27133 \nu^{14} - 12209 \nu^{13} + 661834 \nu^{12} + 320877 \nu^{11} - 6221350 \nu^{10} - 3127097 \nu^{9} + 28149600 \nu^{8} + 14002148 \nu^{7} + \cdots + 2271240 ) / 276852 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38305 \nu^{14} + 8737 \nu^{13} + 954256 \nu^{12} - 179217 \nu^{11} - 9234718 \nu^{10} + 1445125 \nu^{9} + 43620546 \nu^{8} - 5890210 \nu^{7} - 103554514 \nu^{6} + \cdots + 7246596 ) / 276852 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22838 \nu^{14} + 17393 \nu^{13} + 566699 \nu^{12} - 399384 \nu^{11} - 5468321 \nu^{10} + 3544496 \nu^{9} + 25749939 \nu^{8} - 15162380 \nu^{7} + \cdots + 2547864 ) / 138426 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) 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
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{5} + 9\beta_{3} + \beta_{2} + 30\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + 11 \beta_{9} - 11 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + 47 \beta_{2} + \beta _1 + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{14} - 13 \beta_{13} + 15 \beta_{12} - 14 \beta_{11} + \beta_{10} + 13 \beta_{9} + 12 \beta_{7} - 3 \beta_{6} - 12 \beta_{5} + 69 \beta_{3} + 13 \beta_{2} + 192 \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{14} + 2 \beta_{13} + 16 \beta_{12} - 17 \beta_{11} - 15 \beta_{10} + 96 \beta_{9} - 94 \beta_{8} - 13 \beta_{7} - 17 \beta_{6} - 2 \beta_{5} - 14 \beta_{4} + 2 \beta_{3} + 320 \beta_{2} + 17 \beta _1 + 648 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 121 \beta_{14} - 123 \beta_{13} + 159 \beta_{12} - 142 \beta_{11} + 17 \beta_{10} + 129 \beta_{9} - 8 \beta_{8} + 108 \beta_{7} - 49 \beta_{6} - 111 \beta_{5} - 4 \beta_{4} + 507 \beta_{3} + 130 \beta_{2} + 1270 \beta _1 + 174 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 162 \beta_{14} + 34 \beta_{13} + 182 \beta_{12} - 199 \beta_{11} - 154 \beta_{10} + 782 \beta_{9} - 738 \beta_{8} - 121 \beta_{7} - 197 \beta_{6} - 42 \beta_{5} - 140 \beta_{4} + 44 \beta_{3} + 2215 \beta_{2} + 198 \beta _1 + 4375 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 998 \beta_{14} - 1037 \beta_{13} + 1472 \beta_{12} - 1271 \beta_{11} + 199 \beta_{10} + 1165 \beta_{9} - 166 \beta_{8} + 873 \beta_{7} - 556 \beta_{6} - 940 \beta_{5} - 89 \beta_{4} + 3674 \beta_{3} + 1184 \beta_{2} + \cdots + 1642 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1536 \beta_{14} + 381 \beta_{13} + 1799 \beta_{12} - 1995 \beta_{11} - 1354 \beta_{10} + 6199 \beta_{9} - 5594 \beta_{8} - 995 \beta_{7} - 1942 \beta_{6} - 565 \beta_{5} - 1246 \beta_{4} + 613 \beta_{3} + 15552 \beta_{2} + \cdots + 30162 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7801 \beta_{14} - 8285 \beta_{13} + 12723 \beta_{12} - 10689 \beta_{11} + 1993 \beta_{10} + 10053 \beta_{9} - 2228 \beta_{8} + 6697 \beta_{7} - 5444 \beta_{6} - 7653 \beta_{5} - 1256 \beta_{4} + 26529 \beta_{3} + \cdots + 14512 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13590 \beta_{14} + 3559 \beta_{13} + 16490 \beta_{12} - 18402 \beta_{11} - 10971 \beta_{10} + 48527 \beta_{9} - 41782 \beta_{8} - 7718 \beta_{7} - 17569 \beta_{6} - 6244 \beta_{5} - 10549 \beta_{4} + \cdots + 211211 \) Copy content Toggle raw display

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
2.77390
2.52515
2.35360
1.81907
1.25728
1.12404
0.601514
0.420191
−0.528800
−0.664482
−1.28835
−2.02417
−2.18666
−2.52000
−2.66230
−2.77390 0 5.69451 −1.00000 0 4.30891 −10.2482 0 2.77390
1.2 −2.52515 0 4.37640 −1.00000 0 −1.59871 −6.00077 0 2.52515
1.3 −2.35360 0 3.53946 −1.00000 0 −0.0779966 −3.62327 0 2.35360
1.4 −1.81907 0 1.30903 −1.00000 0 2.90773 1.25693 0 1.81907
1.5 −1.25728 0 −0.419240 −1.00000 0 4.56890 3.04167 0 1.25728
1.6 −1.12404 0 −0.736533 −1.00000 0 −3.82866 3.07597 0 1.12404
1.7 −0.601514 0 −1.63818 −1.00000 0 −2.59900 2.18842 0 0.601514
1.8 −0.420191 0 −1.82344 −1.00000 0 −0.843791 1.60658 0 0.420191
1.9 0.528800 0 −1.72037 −1.00000 0 3.29922 −1.96733 0 −0.528800
1.10 0.664482 0 −1.55846 −1.00000 0 2.80920 −2.36453 0 −0.664482
1.11 1.28835 0 −0.340163 −1.00000 0 −2.81777 −3.01494 0 −1.28835
1.12 2.02417 0 2.09727 −1.00000 0 3.66640 0.196897 0 −2.02417
1.13 2.18666 0 2.78147 −1.00000 0 −4.25337 1.70880 0 −2.18666
1.14 2.52000 0 4.35041 −1.00000 0 3.01221 5.92303 0 −2.52000
1.15 2.66230 0 5.08784 −1.00000 0 1.44674 8.22077 0 −2.66230
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.bk 15
3.b odd 2 1 5265.2.a.bl 15
9.c even 3 2 585.2.i.h 30
9.d odd 6 2 1755.2.i.h 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.h 30 9.c even 3 2
1755.2.i.h 30 9.d odd 6 2
5265.2.a.bk 15 1.a even 1 1 trivial
5265.2.a.bl 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5265))\):

\( T_{2}^{15} + T_{2}^{14} - 25 T_{2}^{13} - 24 T_{2}^{12} + 244 T_{2}^{11} + 226 T_{2}^{10} - 1170 T_{2}^{9} - 1051 T_{2}^{8} + 2842 T_{2}^{7} + 2478 T_{2}^{6} - 3252 T_{2}^{5} - 2697 T_{2}^{4} + 1497 T_{2}^{3} + 1107 T_{2}^{2} + \cdots - 144 \) Copy content Toggle raw display
\( T_{7}^{15} - 10 T_{7}^{14} - 21 T_{7}^{13} + 490 T_{7}^{12} - 569 T_{7}^{11} - 8719 T_{7}^{10} + 21032 T_{7}^{9} + 68052 T_{7}^{8} - 231515 T_{7}^{7} - 214807 T_{7}^{6} + 1101950 T_{7}^{5} + 116660 T_{7}^{4} + \cdots + 106368 \) Copy content Toggle raw display
\( T_{11}^{15} + 9 T_{11}^{14} - 70 T_{11}^{13} - 742 T_{11}^{12} + 1580 T_{11}^{11} + 21374 T_{11}^{10} - 20260 T_{11}^{9} - 289253 T_{11}^{8} + 229474 T_{11}^{7} + 1817025 T_{11}^{6} - 1667166 T_{11}^{5} - 3794205 T_{11}^{4} + \cdots - 285264 \) Copy content Toggle raw display
\( T_{17}^{15} - 3 T_{17}^{14} - 144 T_{17}^{13} + 389 T_{17}^{12} + 8171 T_{17}^{11} - 18798 T_{17}^{10} - 236702 T_{17}^{9} + 424337 T_{17}^{8} + 3803893 T_{17}^{7} - 4484916 T_{17}^{6} - 34584564 T_{17}^{5} + \cdots - 261146880 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + T^{14} - 25 T^{13} - 24 T^{12} + \cdots - 144 \) Copy content Toggle raw display
$3$ \( T^{15} \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 10 T^{14} - 21 T^{13} + \cdots + 106368 \) Copy content Toggle raw display
$11$ \( T^{15} + 9 T^{14} - 70 T^{13} + \cdots - 285264 \) Copy content Toggle raw display
$13$ \( (T - 1)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 3 T^{14} - 144 T^{13} + \cdots - 261146880 \) Copy content Toggle raw display
$19$ \( T^{15} - 15 T^{14} - 29 T^{13} + \cdots + 27202816 \) Copy content Toggle raw display
$23$ \( T^{15} - 6 T^{14} - 154 T^{13} + \cdots - 70668 \) Copy content Toggle raw display
$29$ \( T^{15} + 8 T^{14} - 146 T^{13} + \cdots - 3440394 \) Copy content Toggle raw display
$31$ \( T^{15} - 22 T^{14} + \cdots + 1264815616 \) Copy content Toggle raw display
$37$ \( T^{15} - 4 T^{14} + \cdots - 3358658304 \) Copy content Toggle raw display
$41$ \( T^{15} + 13 T^{14} + \cdots - 61544363328 \) Copy content Toggle raw display
$43$ \( T^{15} - 24 T^{14} + \cdots - 1483131392 \) Copy content Toggle raw display
$47$ \( T^{15} - T^{14} + \cdots - 3634105450752 \) Copy content Toggle raw display
$53$ \( T^{15} - 7 T^{14} + \cdots + 2779484544 \) Copy content Toggle raw display
$59$ \( T^{15} + 19 T^{14} - 116 T^{13} + \cdots - 41637672 \) Copy content Toggle raw display
$61$ \( T^{15} - 16 T^{14} + \cdots + 8592905275 \) Copy content Toggle raw display
$67$ \( T^{15} - 11 T^{14} + \cdots + 347744128 \) Copy content Toggle raw display
$71$ \( T^{15} + 28 T^{14} + \cdots + 2803793184 \) Copy content Toggle raw display
$73$ \( T^{15} - 26 T^{14} + \cdots - 513711307776 \) Copy content Toggle raw display
$79$ \( T^{15} - 44 T^{14} + \cdots - 58061076934400 \) Copy content Toggle raw display
$83$ \( T^{15} - 3 T^{14} + \cdots + 2197025856 \) Copy content Toggle raw display
$89$ \( T^{15} + 4 T^{14} + \cdots + 112704050880 \) Copy content Toggle raw display
$97$ \( T^{15} - 33 T^{14} + \cdots - 21794978952 \) Copy content Toggle raw display
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