Properties

Label 2013.2.a.g
Level $2013$
Weight $2$
Character orbit 2013.a
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8 \nu^{12} + 90 \nu^{11} - 26 \nu^{10} - 1165 \nu^{9} + 1132 \nu^{8} + 4883 \nu^{7} - 5564 \nu^{6} - 6816 \nu^{5} + 8767 \nu^{4} + 884 \nu^{3} - 4652 \nu^{2} + 1882 \nu + 410 ) / 359 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{12} + 11 \nu^{11} - 67 \nu^{10} - 282 \nu^{9} - 93 \nu^{8} + 2324 \nu^{7} + 2894 \nu^{6} - 8037 \nu^{5} - 10588 \nu^{4} + 11612 \nu^{3} + 12866 \nu^{2} - 5506 \nu - 3500 ) / 359 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29 \nu^{12} - 57 \nu^{11} - 534 \nu^{10} + 1037 \nu^{9} + 3615 \nu^{8} - 6886 \nu^{7} - 10884 \nu^{6} + 19682 \nu^{5} + 14037 \nu^{4} - 21334 \nu^{3} - 7010 \nu^{2} + 5294 \nu + 1655 ) / 359 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 66 \nu^{12} + 204 \nu^{11} + 683 \nu^{10} - 2521 \nu^{9} - 1431 \nu^{8} + 10039 \nu^{7} - 3900 \nu^{6} - 13511 \nu^{5} + 13003 \nu^{4} + 3344 \nu^{3} - 8582 \nu^{2} + 2064 \nu + 690 ) / 359 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 102 \nu^{12} + 250 \nu^{11} + 1284 \nu^{10} - 3276 \nu^{9} - 4953 \nu^{8} + 14601 \nu^{7} + 4449 \nu^{6} - 25874 \nu^{5} + 6682 \nu^{4} + 17733 \nu^{3} - 8694 \nu^{2} - 3109 \nu + 381 ) / 359 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 71 \nu^{12} + 350 \nu^{11} + 577 \nu^{10} - 4730 \nu^{9} + 533 \nu^{8} + 22021 \nu^{7} - 13660 \nu^{6} - 41824 \nu^{5} + 31900 \nu^{4} + 32078 \nu^{3} - 22798 \nu^{2} + \cdots + 3190 ) / 359 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 96 \nu^{12} - 362 \nu^{11} - 1124 \nu^{10} + 5005 \nu^{9} + 4007 \nu^{8} - 24132 \nu^{7} - 4314 \nu^{6} + 48405 \nu^{5} + 2496 \nu^{4} - 39687 \nu^{3} - 4488 \nu^{2} + 10085 \nu + 1901 ) / 359 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 196 \nu^{12} + 410 \nu^{11} + 2953 \nu^{10} - 5746 \nu^{9} - 16423 \nu^{8} + 28627 \nu^{7} + 42464 \nu^{6} - 61446 \nu^{5} - 54279 \nu^{4} + 56481 \nu^{3} + 31780 \nu^{2} + \cdots - 6469 ) / 359 \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{3} + 27\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} + \beta_{3} + 32 \beta_{2} + 12 \beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{12} + 15 \beta_{11} + 2 \beta_{10} + 12 \beta_{8} - 11 \beta_{7} + 15 \beta_{6} + 12 \beta_{5} + 14 \beta_{4} + 42 \beta_{3} + 153 \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{12} + 31 \beta_{11} + 2 \beta_{10} + 14 \beta_{9} + 13 \beta_{8} - 14 \beta_{7} + 42 \beta_{6} + 30 \beta_{5} + 80 \beta_{4} + 14 \beta_{3} + 166 \beta_{2} + 109 \beta _1 + 414 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15 \beta_{12} + 153 \beta_{11} + 28 \beta_{10} + 4 \beta_{9} + 104 \beta_{8} - 91 \beta_{7} + 153 \beta_{6} + 115 \beta_{5} + 137 \beta_{4} + 246 \beta_{3} - 2 \beta_{2} + 900 \beta _1 + 233 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 19 \beta_{12} + 328 \beta_{11} + 34 \beta_{10} + 134 \beta_{9} + 123 \beta_{8} - 137 \beta_{7} + 415 \beta_{6} + 312 \beta_{5} + 599 \beta_{4} + 135 \beta_{3} + 852 \beta_{2} + 890 \beta _1 + 2406 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 153 \beta_{12} + 1334 \beta_{11} + 273 \beta_{10} + 74 \beta_{9} + 797 \beta_{8} - 686 \beta_{7} + 1337 \beta_{6} + 1001 \beta_{5} + 1165 \beta_{4} + 1451 \beta_{3} - 36 \beta_{2} + 5466 \beta _1 + 1943 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 227 \beta_{12} + 2960 \beta_{11} + 387 \beta_{10} + 1100 \beta_{9} + 1038 \beta_{8} - 1168 \beta_{7} + 3572 \beta_{6} + 2782 \beta_{5} + 4363 \beta_{4} + 1129 \beta_{3} + 4344 \beta_{2} + 6882 \beta _1 + 14427 \) 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
−2.33467
−2.17077
−1.38103
−1.17594
−0.638829
−0.312603
0.669508
0.931518
1.31287
1.88247
2.25716
2.30577
2.65453
−2.33467 1.00000 3.45069 0.0293558 −2.33467 −2.70286 −3.38688 1.00000 −0.0685361
1.2 −2.17077 1.00000 2.71223 2.18891 −2.17077 3.77754 −1.54609 1.00000 −4.75162
1.3 −1.38103 1.00000 −0.0927606 −0.547866 −1.38103 1.24123 2.89016 1.00000 0.756618
1.4 −1.17594 1.00000 −0.617176 3.63888 −1.17594 −0.818481 3.07763 1.00000 −4.27908
1.5 −0.638829 1.00000 −1.59190 −2.55169 −0.638829 −2.98273 2.29461 1.00000 1.63009
1.6 −0.312603 1.00000 −1.90228 −0.566394 −0.312603 3.64999 1.21987 1.00000 0.177056
1.7 0.669508 1.00000 −1.55176 2.40049 0.669508 3.10183 −2.37793 1.00000 1.60715
1.8 0.931518 1.00000 −1.13227 1.13816 0.931518 −2.87358 −2.91777 1.00000 1.06022
1.9 1.31287 1.00000 −0.276371 −3.61438 1.31287 −0.837447 −2.98858 1.00000 −4.74521
1.10 1.88247 1.00000 1.54370 3.12160 1.88247 2.89419 −0.858967 1.00000 5.87632
1.11 2.25716 1.00000 3.09478 1.69786 2.25716 0.963803 2.47110 1.00000 3.83236
1.12 2.30577 1.00000 3.31659 −2.09861 2.30577 3.60849 3.03576 1.00000 −4.83892
1.13 2.65453 1.00000 5.04652 2.16368 2.65453 −2.02198 8.08709 1.00000 5.74356
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(-1\)

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 2013.2.a.g 13
3.b odd 2 1 6039.2.a.h 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.g 13 1.a even 1 1 trivial
6039.2.a.h 13 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 4 T_{2}^{12} - 11 T_{2}^{11} + 57 T_{2}^{10} + 28 T_{2}^{9} - 290 T_{2}^{8} + 51 T_{2}^{7} + 644 T_{2}^{6} - 259 T_{2}^{5} - 640 T_{2}^{4} + 274 T_{2}^{3} + 256 T_{2}^{2} - 74 T_{2} - 35 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 4 T^{12} - 11 T^{11} + 57 T^{10} + \cdots - 35 \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 7 T^{12} - 9 T^{11} + 162 T^{10} + \cdots + 44 \) Copy content Toggle raw display
$7$ \( T^{13} - 7 T^{12} - 21 T^{11} + \cdots - 17156 \) Copy content Toggle raw display
$11$ \( (T - 1)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} - 9 T^{12} - 27 T^{11} + \cdots - 18548 \) Copy content Toggle raw display
$17$ \( T^{13} - 19 T^{12} + 65 T^{11} + \cdots + 244288 \) Copy content Toggle raw display
$19$ \( T^{13} - 14 T^{12} + T^{11} + 768 T^{10} + \cdots - 3150 \) Copy content Toggle raw display
$23$ \( T^{13} - 5 T^{12} - 84 T^{11} + \cdots - 6016 \) Copy content Toggle raw display
$29$ \( T^{13} - 10 T^{12} - 106 T^{11} + \cdots - 242600 \) Copy content Toggle raw display
$31$ \( T^{13} + T^{12} - 115 T^{11} + \cdots - 8931814 \) Copy content Toggle raw display
$37$ \( T^{13} + 8 T^{12} - 206 T^{11} + \cdots - 15250532 \) Copy content Toggle raw display
$41$ \( T^{13} - 21 T^{12} + 7 T^{11} + \cdots - 5681822 \) Copy content Toggle raw display
$43$ \( T^{13} - 11 T^{12} + \cdots + 347470436 \) Copy content Toggle raw display
$47$ \( T^{13} - 22 T^{12} + 31 T^{11} + \cdots + 926644 \) Copy content Toggle raw display
$53$ \( T^{13} - 16 T^{12} + 14 T^{11} + \cdots - 483498 \) Copy content Toggle raw display
$59$ \( T^{13} - 19 T^{12} - 179 T^{11} + \cdots - 22000 \) Copy content Toggle raw display
$61$ \( (T - 1)^{13} \) Copy content Toggle raw display
$67$ \( T^{13} - 12 T^{12} + \cdots - 1402879138 \) Copy content Toggle raw display
$71$ \( T^{13} - 5 T^{12} - 256 T^{11} + \cdots - 5920072 \) Copy content Toggle raw display
$73$ \( T^{13} - 18 T^{12} + \cdots - 157498316 \) Copy content Toggle raw display
$79$ \( T^{13} + T^{12} - 435 T^{11} + \cdots - 6611280300 \) Copy content Toggle raw display
$83$ \( T^{13} - 48 T^{12} + \cdots + 129082984 \) Copy content Toggle raw display
$89$ \( T^{13} - 15 T^{12} + \cdots - 227388350 \) Copy content Toggle raw display
$97$ \( T^{13} + 17 T^{12} - 323 T^{11} + \cdots + 17107184 \) Copy content Toggle raw display
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