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Properties

Label 2015.2.a.i
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: \( x^{20} - 3 x^{19} - 28 x^{18} + 86 x^{17} + 319 x^{16} - 1012 x^{15} - 1903 x^{14} + 6323 x^{13} + \cdots - 184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 29 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{4} + \beta_1) q^{6} - \beta_{16} q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{6} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{4} + \beta_1) q^{6} - \beta_{16} q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{6} + \beta_{5} + 1) q^{9} - \beta_1 q^{10} + ( - \beta_{12} + \beta_1) q^{11} + ( - \beta_{17} - \beta_{13} + \beta_{7} + \cdots + 1) q^{12}+ \cdots + ( - \beta_{19} + 2 \beta_{18} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} + 7 q^{3} + 25 q^{4} - 20 q^{5} - 5 q^{6} + q^{7} + 9 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3 q^{2} + 7 q^{3} + 25 q^{4} - 20 q^{5} - 5 q^{6} + q^{7} + 9 q^{8} + 29 q^{9} - 3 q^{10} - 5 q^{11} + 34 q^{12} - 20 q^{13} - 7 q^{14} - 7 q^{15} + 39 q^{16} + 25 q^{17} + 20 q^{18} - 3 q^{19} - 25 q^{20} - 10 q^{21} + 35 q^{22} + 21 q^{23} + 2 q^{24} + 20 q^{25} - 3 q^{26} + 34 q^{27} - 16 q^{28} - 6 q^{29} + 5 q^{30} + 20 q^{31} + 16 q^{32} + 13 q^{33} + 11 q^{34} - q^{35} + 90 q^{36} + 20 q^{37} + 3 q^{38} - 7 q^{39} - 9 q^{40} - 12 q^{41} + 13 q^{42} + 37 q^{43} - 34 q^{44} - 29 q^{45} + 52 q^{46} + 9 q^{47} + 59 q^{48} + 51 q^{49} + 3 q^{50} + 27 q^{51} - 25 q^{52} + 43 q^{53} + 10 q^{54} + 5 q^{55} - 15 q^{56} - 6 q^{57} + 42 q^{58} - 19 q^{59} - 34 q^{60} + 3 q^{61} + 3 q^{62} - 8 q^{63} + 39 q^{64} + 20 q^{65} + 51 q^{66} + 35 q^{67} + 49 q^{68} - 12 q^{69} + 7 q^{70} - 9 q^{71} + 90 q^{72} + 17 q^{73} - 34 q^{74} + 7 q^{75} - 21 q^{76} + 74 q^{77} + 5 q^{78} + 60 q^{79} - 39 q^{80} + 68 q^{81} + 24 q^{82} + 15 q^{83} + 17 q^{84} - 25 q^{85} - 28 q^{86} - 4 q^{87} + 59 q^{88} - 12 q^{89} - 20 q^{90} - q^{91} + 54 q^{92} + 7 q^{93} - 7 q^{94} + 3 q^{95} - 62 q^{96} + 53 q^{97} + 56 q^{98} + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 28 x^{18} + 86 x^{17} + 319 x^{16} - 1012 x^{15} - 1903 x^{14} + 6323 x^{13} + \cdots - 184 \) : 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
\(\beta_{4}\)\(=\) \( ( - 302369651382 \nu^{19} + 1120536049882 \nu^{18} + 7111413788076 \nu^{17} + \cdots - 146383428353160 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 649195835252 \nu^{19} - 1344437144116 \nu^{18} - 18847970804065 \nu^{17} + \cdots + 11\!\cdots\!70 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 795562110615 \nu^{19} - 2084316680463 \nu^{18} - 23396275147102 \nu^{17} + \cdots + 516819647590073 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 939239068412 \nu^{19} + 5541855578123 \nu^{18} + 18173948409305 \nu^{17} + \cdots + 910716249853300 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1085605343775 \nu^{19} - 6281735114470 \nu^{18} - 22722252752342 \nu^{17} + \cdots - 11\!\cdots\!99 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1140633470828 \nu^{19} - 2146170457325 \nu^{18} - 34401206257041 \nu^{17} + \cdots + 11966682304007 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1264739825600 \nu^{19} + 4796137598808 \nu^{18} + 32637203628978 \nu^{17} + \cdots + 560175480343008 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1528015751477 \nu^{19} - 2809359856288 \nu^{18} - 47181875346229 \nu^{17} + \cdots + 13\!\cdots\!69 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1716680758524 \nu^{19} + 4152674981933 \nu^{18} + 49747930892140 \nu^{17} + \cdots - 712108381188532 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1753652168678 \nu^{19} - 3867970825574 \nu^{18} - 51164780581486 \nu^{17} + \cdots - 100034177996257 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4523679004550 \nu^{19} + 18988066623627 \nu^{18} + 109128440217472 \nu^{17} + \cdots + 715707121630940 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 661804913231 \nu^{19} - 2132629393932 \nu^{18} - 17665231753858 \nu^{17} + \cdots - 145357913978964 ) / 10034670845719 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 5460977010158 \nu^{19} - 21623534961195 \nu^{18} - 135367385056436 \nu^{17} + \cdots - 29\!\cdots\!02 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 5741740591416 \nu^{19} + 18878128481935 \nu^{18} + 155522743029043 \nu^{17} + \cdots - 126327639667403 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 6930568896814 \nu^{19} + 22834907778030 \nu^{18} + 183336880819885 \nu^{17} + \cdots + 12\!\cdots\!15 ) / 70242695920033 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10483226650938 \nu^{19} + 39465601540942 \nu^{18} + 266728501875693 \nu^{17} + \cdots + 38\!\cdots\!12 ) / 70242695920033 \) 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
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} + \beta_{16} - \beta_{15} + 2 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + \beta_{16} - \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{19} + 12 \beta_{18} + 2 \beta_{17} + 12 \beta_{16} - 11 \beta_{15} - 2 \beta_{14} + 24 \beta_{13} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{19} - \beta_{17} + 15 \beta_{16} + \beta_{15} - 15 \beta_{14} - 14 \beta_{13} - 31 \beta_{12} + \cdots + 551 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15 \beta_{19} + 110 \beta_{18} + 32 \beta_{17} + 110 \beta_{16} - 93 \beta_{15} - 30 \beta_{14} + \cdots + 157 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 174 \beta_{19} + \beta_{18} - 18 \beta_{17} + 154 \beta_{16} + 21 \beta_{15} - 159 \beta_{14} + \cdots + 3566 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 158 \beta_{19} + 914 \beta_{18} + 349 \beta_{17} + 920 \beta_{16} - 722 \beta_{15} - 313 \beta_{14} + \cdots + 1430 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1610 \beta_{19} + 26 \beta_{18} - 219 \beta_{17} + 1358 \beta_{16} + 272 \beta_{15} - 1460 \beta_{14} + \cdots + 23559 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1454 \beta_{19} + 7222 \beta_{18} + 3252 \beta_{17} + 7374 \beta_{16} - 5408 \beta_{15} - 2826 \beta_{14} + \cdots + 12164 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13673 \beta_{19} + 408 \beta_{18} - 2263 \beta_{17} + 11093 \beta_{16} + 2838 \beta_{15} - 12400 \beta_{14} + \cdots + 157963 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12520 \beta_{19} + 55363 \beta_{18} + 27883 \beta_{17} + 57728 \beta_{16} - 39836 \beta_{15} + \cdots + 99497 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 110321 \beta_{19} + 5053 \beta_{18} - 21397 \beta_{17} + 86831 \beta_{16} + 26238 \beta_{15} + \cdots + 1071424 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 103767 \beta_{19} + 416107 \beta_{18} + 227332 \beta_{17} + 445358 \beta_{16} - 291117 \beta_{15} + \cdots + 793634 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 861393 \beta_{19} + 54496 \beta_{18} - 191238 \beta_{17} + 662932 \beta_{16} + 224859 \beta_{15} + \cdots + 7335904 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 839088 \beta_{19} + 3085720 \beta_{18} + 1792896 \beta_{17} + 3402030 \beta_{16} - 2119930 \beta_{15} + \cdots + 6219984 \) 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
−2.67609
−2.50197
−2.36793
−1.98949
−1.44089
−1.39625
−0.904083
−0.618511
−0.590723
0.177457
0.345780
0.775363
1.05756
1.31426
1.47281
2.17056
2.35090
2.43841
2.67608
2.70675
−2.67609 3.31537 5.16146 −1.00000 −8.87224 0.653385 −8.46034 7.99171 2.67609
1.2 −2.50197 0.751133 4.25988 −1.00000 −1.87931 −3.91116 −5.65416 −2.43580 2.50197
1.3 −2.36793 0.0841374 3.60711 −1.00000 −0.199232 3.39164 −3.80553 −2.99292 2.36793
1.4 −1.98949 −2.14692 1.95806 −1.00000 4.27126 0.413435 0.0834304 1.60925 1.98949
1.5 −1.44089 1.16823 0.0761522 −1.00000 −1.68328 −1.92488 2.77205 −1.63525 1.44089
1.6 −1.39625 3.20392 −0.0504940 −1.00000 −4.47346 1.19002 2.86300 7.26507 1.39625
1.7 −0.904083 −2.37331 −1.18263 −1.00000 2.14567 2.52455 2.87737 2.63258 0.904083
1.8 −0.618511 0.849329 −1.61744 −1.00000 −0.525319 −5.09105 2.23743 −2.27864 0.618511
1.9 −0.590723 1.39472 −1.65105 −1.00000 −0.823894 4.78385 2.15676 −1.05475 0.590723
1.10 0.177457 −2.91866 −1.96851 −1.00000 −0.517935 3.66787 −0.704238 5.51857 −0.177457
1.11 0.345780 2.43136 −1.88044 −1.00000 0.840716 −2.61745 −1.34178 2.91151 −0.345780
1.12 0.775363 −1.54084 −1.39881 −1.00000 −1.19471 0.0170260 −2.63531 −0.625800 −0.775363
1.13 1.05756 −0.620888 −0.881574 −1.00000 −0.656624 −2.42880 −3.04743 −2.61450 −1.05756
1.14 1.31426 0.514319 −0.272716 −1.00000 0.675950 3.44604 −2.98694 −2.73548 −1.31426
1.15 1.47281 −1.21718 0.169180 −1.00000 −1.79268 −2.14821 −2.69646 −1.51847 −1.47281
1.16 2.17056 2.78003 2.71134 −1.00000 6.03423 1.25736 1.54402 4.72858 −2.17056
1.17 2.35090 2.17542 3.52674 −1.00000 5.11419 4.23188 3.58921 1.73244 −2.35090
1.18 2.43841 −2.98491 3.94584 −1.00000 −7.27844 −3.60260 4.74477 5.90969 −2.43841
1.19 2.67608 −1.20349 5.16142 −1.00000 −3.22065 2.04281 8.46022 −1.55160 −2.67608
1.20 2.70675 3.33823 5.32648 −1.00000 9.03576 −4.89571 9.00395 8.14381 −2.70675
\(n\): 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.i ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.2.a.i ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):

\( T_{2}^{20} - 3 T_{2}^{19} - 28 T_{2}^{18} + 86 T_{2}^{17} + 319 T_{2}^{16} - 1012 T_{2}^{15} + \cdots - 184 \) Copy content Toggle raw display
\( T_{3}^{20} - 7 T_{3}^{19} - 20 T_{3}^{18} + 229 T_{3}^{17} + 5 T_{3}^{16} - 2990 T_{3}^{15} + \cdots + 1459 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots - 184 \) Copy content Toggle raw display
$3$ \( T^{20} - 7 T^{19} + \cdots + 1459 \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - T^{19} + \cdots + 284356 \) Copy content Toggle raw display
$11$ \( T^{20} + 5 T^{19} + \cdots - 44407784 \) Copy content Toggle raw display
$13$ \( (T + 1)^{20} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 372611679073 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 365898117120 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 150408141216 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 5014650880 \) Copy content Toggle raw display
$31$ \( (T - 1)^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 6987075584 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 16230051840 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 16\!\cdots\!67 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 315268607983744 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 18\!\cdots\!87 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 38\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 3339316314112 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 64\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 348918218256384 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 23\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 4585561817088 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 251738990522368 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 2402184960532 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 22\!\cdots\!52 \) Copy content Toggle raw display
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