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Properties

Label 2015.2.a.j
Level $2015$
Weight $2$
Character orbit 2015.a
Self dual yes
Analytic conductor $16.090$
Analytic rank $0$
Dimension $20$
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] = [2015,2,Mod(1,2015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, 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("2015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2015.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: \(16.0898560073\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} - 20 x^{18} + 130 x^{17} + 119 x^{16} - 1374 x^{15} + 129 x^{14} + 7595 x^{13} + \cdots - 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{10} + \beta_{5}) q^{6} + ( - \beta_{16} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{17} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{10} + \beta_{5}) q^{6} + ( - \beta_{16} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{17} + 1) q^{9} - \beta_1 q^{10} + \beta_{12} q^{11} + (\beta_{19} - \beta_{18} + \cdots + 2 \beta_1) q^{12}+ \cdots + ( - \beta_{19} + \beta_{18} + \beta_{16} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} + q^{3} + 25 q^{4} - 20 q^{5} + 9 q^{6} + 11 q^{7} + 15 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{2} + q^{3} + 25 q^{4} - 20 q^{5} + 9 q^{6} + 11 q^{7} + 15 q^{8} + 29 q^{9} - 5 q^{10} + 5 q^{11} + 20 q^{13} - 7 q^{14} - q^{15} + 39 q^{16} + 3 q^{17} + 30 q^{18} + 15 q^{19} - 25 q^{20} + 11 q^{22} - 5 q^{23} + 8 q^{24} + 20 q^{25} + 5 q^{26} - 8 q^{27} + 46 q^{28} - 2 q^{29} - 9 q^{30} - 20 q^{31} + 50 q^{32} + 43 q^{33} - q^{34} - 11 q^{35} + 10 q^{36} + 46 q^{37} + 7 q^{38} + q^{39} - 15 q^{40} - 12 q^{41} + 39 q^{42} + 13 q^{43} + 8 q^{44} - 29 q^{45} - 14 q^{46} + 21 q^{47} + 27 q^{48} + 55 q^{49} + 5 q^{50} - 9 q^{51} + 25 q^{52} + 7 q^{53} - 32 q^{54} - 5 q^{55} - 15 q^{56} + 70 q^{57} + 10 q^{58} + 5 q^{59} - 21 q^{61} - 5 q^{62} + 78 q^{63} + 51 q^{64} - 20 q^{65} - 25 q^{66} + 57 q^{67} + 23 q^{68} - 28 q^{69} + 7 q^{70} - 15 q^{71} + 16 q^{72} + 67 q^{73} + 18 q^{74} + q^{75} + 61 q^{76} - 22 q^{77} + 9 q^{78} + 12 q^{79} - 39 q^{80} + 68 q^{81} - 6 q^{82} + 21 q^{83} - 43 q^{84} - 3 q^{85} + 12 q^{86} + 20 q^{87} + 79 q^{88} - 30 q^{90} + 11 q^{91} + 16 q^{92} - q^{93} + 25 q^{94} - 15 q^{95} + 86 q^{96} + 105 q^{97} + 4 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} - 20 x^{18} + 130 x^{17} + 119 x^{16} - 1374 x^{15} + 129 x^{14} + 7595 x^{13} + \cdots - 80 \) : 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
\(\beta_{4}\)\(=\) \( ( 693422 \nu^{19} + 21603755 \nu^{18} - 139842005 \nu^{17} - 386865384 \nu^{16} + 3252214422 \nu^{15} + \cdots + 2717623652 ) / 11862129 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2432845 \nu^{19} + 10278457 \nu^{18} + 57998828 \nu^{17} - 274824340 \nu^{16} + \cdots - 428501262 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3007637 \nu^{19} + 22773821 \nu^{18} - 249346544 \nu^{17} - 334960374 \nu^{16} + \cdots + 4131335312 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6778453 \nu^{19} - 47714075 \nu^{18} - 61720720 \nu^{17} + 1121952558 \nu^{16} + \cdots - 1189586906 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7326607 \nu^{19} + 2133731 \nu^{18} + 315336628 \nu^{17} - 268256898 \nu^{16} + \cdots - 3840721600 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5116081 \nu^{19} - 25900864 \nu^{18} - 98287352 \nu^{17} + 655594759 \nu^{16} + 532671256 \nu^{15} + \cdots + 165764340 ) / 11862129 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16965341 \nu^{19} + 70294209 \nu^{18} + 399322572 \nu^{17} - 1864170586 \nu^{16} + \cdots - 1785728542 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22566867 \nu^{19} - 67442879 \nu^{18} - 666214072 \nu^{17} + 1998539908 \nu^{16} + \cdots + 5172742028 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11871897 \nu^{19} - 59004142 \nu^{18} - 230521646 \nu^{17} + 1493061572 \nu^{16} + \cdots + 226898044 ) / 11862129 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27722339 \nu^{19} - 143194135 \nu^{18} - 509889332 \nu^{17} + 3583598046 \nu^{16} + \cdots - 229714948 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27835499 \nu^{19} - 149290457 \nu^{18} - 485889358 \nu^{17} + 3708551726 \nu^{16} + \cdots - 673474626 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19833904 \nu^{19} - 127936470 \nu^{18} - 235303539 \nu^{17} + 3044449154 \nu^{16} + \cdots - 3067957936 ) / 11862129 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 8416450 \nu^{19} + 35423447 \nu^{18} + 196266720 \nu^{17} - 938425635 \nu^{16} + \cdots - 823605997 ) / 3954043 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 37944780 \nu^{19} - 174566858 \nu^{18} - 808628521 \nu^{17} + 4507675777 \nu^{16} + \cdots + 2058164753 ) / 11862129 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 81638869 \nu^{19} - 310781573 \nu^{18} - 2073541858 \nu^{17} + 8492678994 \nu^{16} + \cdots + 11764436086 ) / 23724258 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 15413987 \nu^{19} - 66348791 \nu^{18} - 351647916 \nu^{17} + 1745240773 \nu^{16} + \cdots + 1347189250 ) / 3954043 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - \beta_{10} + \beta_{7} + \beta_{3} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} - \beta_{16} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} - \beta_{17} + \beta_{13} - 11 \beta_{12} - 9 \beta_{10} + \beta_{9} + 9 \beta_{7} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{19} - 12 \beta_{18} - 12 \beta_{17} - 9 \beta_{16} - 2 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{19} - 13 \beta_{17} + 3 \beta_{16} - 3 \beta_{15} + 17 \beta_{13} - 90 \beta_{12} + \beta_{11} + \cdots + 547 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 158 \beta_{19} - 105 \beta_{18} - 109 \beta_{17} - 54 \beta_{16} - 33 \beta_{15} - 125 \beta_{14} + \cdots + 339 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 147 \beta_{19} + \beta_{18} - 126 \beta_{17} + 58 \beta_{16} - 55 \beta_{15} - 5 \beta_{14} + \cdots + 3500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1446 \beta_{19} - 820 \beta_{18} - 903 \beta_{17} - 234 \beta_{16} - 370 \beta_{15} - 1071 \beta_{14} + \cdots + 2795 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1399 \beta_{19} + 10 \beta_{18} - 1107 \beta_{17} + 727 \beta_{16} - 678 \beta_{15} - 116 \beta_{14} + \cdots + 22818 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12306 \beta_{19} - 6082 \beta_{18} - 7186 \beta_{17} - 374 \beta_{16} - 3539 \beta_{15} - 8652 \beta_{14} + \cdots + 21948 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12674 \beta_{19} - 9328 \beta_{17} + 7534 \beta_{16} - 7036 \beta_{15} - 1700 \beta_{14} + 17751 \beta_{13} + \cdots + 150882 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 100318 \beta_{19} - 43951 \beta_{18} - 56008 \beta_{17} + 6911 \beta_{16} - 31152 \beta_{15} + \cdots + 167941 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 111148 \beta_{19} - 1315 \beta_{18} - 76973 \beta_{17} + 70341 \beta_{16} - 66318 \beta_{15} + \cdots + 1009530 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 795479 \beta_{19} - 313246 \beta_{18} - 431255 \beta_{17} + 115962 \beta_{16} - 261113 \beta_{15} + \cdots + 1266004 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 950782 \beta_{19} - 24326 \beta_{18} - 627265 \beta_{17} + 616057 \beta_{16} - 588123 \beta_{15} + \cdots + 6825197 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 6190623 \beta_{19} - 2216289 \beta_{18} - 3295047 \beta_{17} + 1277729 \beta_{16} - 2121605 \beta_{15} + \cdots + 9458747 \) 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
−2.56905
−2.43900
−2.29096
−1.78883
−1.63778
−0.916284
−0.899481
−0.885749
−0.0607265
0.308698
0.799007
0.807918
1.01360
1.21802
1.90503
1.92666
2.52249
2.54458
2.70849
2.73336
−2.56905 −2.16838 4.60002 −1.00000 5.57068 4.34365 −6.67960 1.70187 2.56905
1.2 −2.43900 −0.0141075 3.94871 −1.00000 0.0344082 0.0183389 −4.75289 −2.99980 2.43900
1.3 −2.29096 2.29631 3.24849 −1.00000 −5.26075 3.53342 −2.86023 2.27303 2.29096
1.4 −1.78883 0.482314 1.19993 −1.00000 −0.862780 −3.70049 1.43119 −2.76737 1.78883
1.5 −1.63778 −1.87018 0.682336 −1.00000 3.06295 3.24684 2.15805 0.497577 1.63778
1.6 −0.916284 1.57440 −1.16042 −1.00000 −1.44259 −1.05034 2.89585 −0.521279 0.916284
1.7 −0.899481 −1.08917 −1.19093 −1.00000 0.979690 −0.297472 2.87018 −1.81370 0.899481
1.8 −0.885749 −3.32663 −1.21545 −1.00000 2.94656 0.422895 2.84808 8.06645 0.885749
1.9 −0.0607265 3.02743 −1.99631 −1.00000 −0.183845 −3.13718 0.242682 6.16534 0.0607265
1.10 0.308698 −1.06882 −1.90471 −1.00000 −0.329943 4.78148 −1.20537 −1.85762 −0.308698
1.11 0.799007 −0.0837664 −1.36159 −1.00000 −0.0669300 −5.11409 −2.68593 −2.99298 −0.799007
1.12 0.807918 3.36442 −1.34727 −1.00000 2.71817 5.15557 −2.70432 8.31931 −0.807918
1.13 1.01360 1.11008 −0.972613 −1.00000 1.12518 2.47289 −3.01304 −1.76772 −1.01360
1.14 1.21802 −2.92578 −0.516432 −1.00000 −3.56366 −3.07657 −3.06506 5.56021 −1.21802
1.15 1.90503 −1.87332 1.62915 −1.00000 −3.56873 −2.03676 −0.706474 0.509320 −1.90503
1.16 1.92666 2.73986 1.71202 −1.00000 5.27878 −0.124190 −0.554839 4.50682 −1.92666
1.17 2.52249 −3.22743 4.36296 −1.00000 −8.14118 4.11207 5.96055 7.41633 −2.52249
1.18 2.54458 1.69062 4.47491 −1.00000 4.30194 2.93829 6.29761 −0.141787 −2.54458
1.19 2.70849 2.19501 5.33594 −1.00000 5.94517 1.01771 9.03537 1.81806 −2.70849
1.20 2.73336 0.167152 5.47126 −1.00000 0.456888 −2.50605 9.48819 −2.97206 −2.73336
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(13\) \(-1\)
\(31\) \(1\)

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 2015.2.a.j ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.2.a.j ✓ 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):

\( T_{2}^{20} - 5 T_{2}^{19} - 20 T_{2}^{18} + 130 T_{2}^{17} + 119 T_{2}^{16} - 1374 T_{2}^{15} + \cdots - 80 \) Copy content Toggle raw display
\( T_{3}^{20} - T_{3}^{19} - 44 T_{3}^{18} + 45 T_{3}^{17} + 795 T_{3}^{16} - 822 T_{3}^{15} - 7644 T_{3}^{14} + \cdots + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 5 T^{19} + \cdots - 80 \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 11 T^{19} + \cdots - 10480 \) Copy content Toggle raw display
$11$ \( T^{20} - 5 T^{19} + \cdots - 48528 \) Copy content Toggle raw display
$13$ \( (T - 1)^{20} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1229552735 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 4491702272 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 19501670176 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 2800153313280 \) Copy content Toggle raw display
$31$ \( (T + 1)^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 19\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 48\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 13736099536569 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 8927167815068 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 21347237220235 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 297399235432448 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 66\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 84\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 51\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 81\!\cdots\!36 \) Copy content Toggle raw display
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