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Properties

Label 1911.4.a.be
Level $1911$
Weight $4$
Character orbit 1911.a
Self dual yes
Analytic conductor $112.753$
Analytic rank $1$
Dimension $14$
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] = [1911,4,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(112.752650021\)
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: \( x^{14} - 6 x^{13} - 63 x^{12} + 408 x^{11} + 1393 x^{10} - 10374 x^{9} - 12229 x^{8} + 122556 x^{7} + \cdots - 43904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} + \beta_{6} q^{5} - 3 \beta_1 q^{6} + ( - \beta_{11} - \beta_{9} + \beta_{5} + \cdots - 1) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} + \beta_{6} q^{5} - 3 \beta_1 q^{6} + ( - \beta_{11} - \beta_{9} + \beta_{5} + \cdots - 1) q^{8}+ \cdots + (9 \beta_{12} - 9 \beta_{7} + \cdots - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 6 q^{2} + 42 q^{3} + 50 q^{4} + 4 q^{5} - 18 q^{6} - 30 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 6 q^{2} + 42 q^{3} + 50 q^{4} + 4 q^{5} - 18 q^{6} - 30 q^{8} + 126 q^{9} - 32 q^{10} - 68 q^{11} + 150 q^{12} - 182 q^{13} + 12 q^{15} - 50 q^{16} - 54 q^{18} - 24 q^{19} - 96 q^{20} - 300 q^{22} - 64 q^{23} - 90 q^{24} - 118 q^{25} + 78 q^{26} + 378 q^{27} - 792 q^{29} - 96 q^{30} - 524 q^{31} - 126 q^{32} - 204 q^{33} + 88 q^{34} + 450 q^{36} - 344 q^{37} - 436 q^{38} - 546 q^{39} - 704 q^{40} - 44 q^{41} + 144 q^{43} - 1248 q^{44} + 36 q^{45} - 972 q^{46} + 236 q^{47} - 150 q^{48} - 714 q^{50} - 650 q^{52} - 1556 q^{53} - 162 q^{54} - 396 q^{55} - 72 q^{57} + 92 q^{58} - 1244 q^{59} - 288 q^{60} - 984 q^{61} - 236 q^{62} - 714 q^{64} - 52 q^{65} - 900 q^{66} - 1396 q^{67} - 960 q^{68} - 192 q^{69} - 1216 q^{71} - 270 q^{72} - 1768 q^{73} - 1212 q^{74} - 354 q^{75} - 496 q^{76} + 234 q^{78} - 1888 q^{79} - 1300 q^{80} + 1134 q^{81} - 1920 q^{82} + 1008 q^{83} + 56 q^{85} - 2128 q^{86} - 2376 q^{87} - 3148 q^{88} + 864 q^{89} - 288 q^{90} + 2856 q^{92} - 1572 q^{93} - 4860 q^{94} - 4984 q^{95} - 378 q^{96} - 1368 q^{97} - 612 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 63 x^{12} + 408 x^{11} + 1393 x^{10} - 10374 x^{9} - 12229 x^{8} + 122556 x^{7} + \cdots - 43904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1441879 \nu^{13} - 5183881 \nu^{12} - 57257672 \nu^{11} + 294023024 \nu^{10} + \cdots - 75859424384 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 208118 \nu^{13} - 1851177 \nu^{12} - 14002120 \nu^{11} + 126899616 \nu^{10} + \cdots + 16208102016 ) / 1889920896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12859563 \nu^{13} + 44786941 \nu^{12} + 916906920 \nu^{11} - 2938218608 \nu^{10} + \cdots - 237258238336 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8211189 \nu^{13} + 30478559 \nu^{12} + 582158952 \nu^{11} - 1992880720 \nu^{10} + \cdots - 258237275264 ) / 7559683584 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21058747 \nu^{13} - 77354973 \nu^{12} - 1519349384 \nu^{11} + 5132765040 \nu^{10} + \cdots + 447877051776 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14444211 \nu^{13} + 62891401 \nu^{12} + 1039908264 \nu^{11} - 4194643280 \nu^{10} + \cdots - 384946232704 ) / 7559683584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24577 \nu^{13} - 98839 \nu^{12} - 1751192 \nu^{11} + 6537680 \nu^{10} + 47690081 \nu^{9} + \cdots + 254131840 ) / 12212736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37828163 \nu^{13} + 136952037 \nu^{12} + 2696682856 \nu^{11} - 9015542064 \nu^{10} + \cdots - 550670171520 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43285889 \nu^{13} + 167149623 \nu^{12} + 3084882616 \nu^{11} - 11031866448 \nu^{10} + \cdots - 566992823424 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57659429 \nu^{13} + 212433323 \nu^{12} + 4132599160 \nu^{11} - 13999270096 \nu^{10} + \cdots - 1898192027264 ) / 15119367168 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36961217 \nu^{13} - 146367419 \nu^{12} - 2637740872 \nu^{11} + 9668605744 \nu^{10} + \cdots + 697785266816 ) / 7559683584 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{5} + 18\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + 2\beta_{11} - \beta_{10} - 2\beta_{8} - \beta_{7} - 2\beta_{6} - 4\beta_{5} + \beta_{4} + 25\beta_{2} + 219 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{12} + 35 \beta_{11} - 8 \beta_{10} + 27 \beta_{9} - 2 \beta_{8} - 6 \beta_{7} - 4 \beta_{6} + \cdots + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 31 \beta_{13} + 16 \beta_{12} + 94 \beta_{11} - 63 \beta_{10} - 4 \beta_{9} - 82 \beta_{8} + \cdots + 4565 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16 \beta_{13} + 102 \beta_{12} + 1027 \beta_{11} - 392 \beta_{10} + 635 \beta_{9} - 134 \beta_{8} + \cdots + 1977 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 807 \beta_{13} + 760 \beta_{12} + 3398 \beta_{11} - 2511 \beta_{10} - 108 \beta_{9} - 2626 \beta_{8} + \cdots + 101541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 912 \beta_{13} + 3726 \beta_{12} + 28987 \beta_{11} - 13904 \beta_{10} + 14723 \beta_{9} + \cdots + 77225 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20487 \beta_{13} + 26176 \beta_{12} + 110606 \beta_{11} - 84823 \beta_{10} - 588 \beta_{9} + \cdots + 2361149 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 36400 \beta_{13} + 120294 \beta_{12} + 809771 \beta_{11} - 439944 \beta_{10} + 345219 \beta_{9} + \cdots + 2763889 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 523663 \beta_{13} + 804584 \beta_{12} + 3411558 \beta_{11} - 2648743 \beta_{10} + 79060 \beta_{9} + \cdots + 56909421 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1260880 \beta_{13} + 3666382 \beta_{12} + 22579531 \beta_{11} - 13238848 \beta_{10} + 8245235 \beta_{9} + \cdots + 93082233 \) 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
5.31718
4.29566
4.21197
3.96203
2.30041
2.24751
2.06054
−0.0267877
−1.11764
−2.00578
−2.27160
−3.63893
−4.61525
−4.71932
−5.31718 3.00000 20.2724 6.77406 −15.9515 0 −65.2547 9.00000 −36.0189
1.2 −4.29566 3.00000 10.4527 −14.9360 −12.8870 0 −10.5360 9.00000 64.1599
1.3 −4.21197 3.00000 9.74066 −8.49708 −12.6359 0 −7.33162 9.00000 35.7894
1.4 −3.96203 3.00000 7.69771 16.3547 −11.8861 0 1.19769 9.00000 −64.7978
1.5 −2.30041 3.00000 −2.70812 −6.49112 −6.90123 0 24.6331 9.00000 14.9322
1.6 −2.24751 3.00000 −2.94870 6.25819 −6.74253 0 24.6073 9.00000 −14.0653
1.7 −2.06054 3.00000 −3.75416 12.1609 −6.18163 0 24.2200 9.00000 −25.0581
1.8 0.0267877 3.00000 −7.99928 −15.2159 0.0803631 0 −0.428584 9.00000 −0.407598
1.9 1.11764 3.00000 −6.75088 −1.89518 3.35291 0 −16.4862 9.00000 −2.11813
1.10 2.00578 3.00000 −3.97684 18.3233 6.01734 0 −24.0229 9.00000 36.7526
1.11 2.27160 3.00000 −2.83982 −4.25461 6.81481 0 −24.6238 9.00000 −9.66478
1.12 3.63893 3.00000 5.24182 9.58367 10.9168 0 −10.0368 9.00000 34.8743
1.13 4.61525 3.00000 13.3005 −4.52975 13.8457 0 24.4632 9.00000 −20.9059
1.14 4.71932 3.00000 14.2720 −9.63526 14.1580 0 29.5994 9.00000 −45.4719
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 1911.4.a.be yes 14
7.b odd 2 1 1911.4.a.bd ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.4.a.bd ✓ 14 7.b odd 2 1
1911.4.a.be yes 14 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{14} + 6 T_{2}^{13} - 63 T_{2}^{12} - 408 T_{2}^{11} + 1393 T_{2}^{10} + 10374 T_{2}^{9} + \cdots - 43904 \) Copy content Toggle raw display
\( T_{5}^{14} - 4 T_{5}^{13} - 808 T_{5}^{12} + 1604 T_{5}^{11} + 248520 T_{5}^{10} - 33428 T_{5}^{9} + \cdots + 6531484301176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 6 T^{13} + \cdots - 43904 \) Copy content Toggle raw display
$3$ \( (T - 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 6531484301176 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( (T + 13)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 60\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 48\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 27\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 24\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 28\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 44\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 63\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 64\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
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