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Properties

Label 1911.4.a.bb
Level $1911$
Weight $4$
Character orbit 1911.a
Self dual yes
Analytic conductor $112.753$
Analytic rank $1$
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] = [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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 75 x^{11} + 220 x^{10} + 2024 x^{9} - 5757 x^{8} - 23683 x^{7} + 64922 x^{6} + \cdots + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 273)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{8} - 1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{8} + \beta_{7} + 5 \beta_1 - 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{8} - 1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{8} + \beta_{7} + 5 \beta_1 - 2) q^{8} + 9 q^{9} + (\beta_{12} + \beta_{8} + \beta_{6} + \cdots - 1) q^{10}+ \cdots + ( - 9 \beta_{12} - 9 \beta_{5} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 3 q^{2} - 39 q^{3} + 55 q^{4} - 15 q^{5} - 9 q^{6} - 6 q^{8} + 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 3 q^{2} - 39 q^{3} + 55 q^{4} - 15 q^{5} - 9 q^{6} - 6 q^{8} + 117 q^{9} - 11 q^{10} + 57 q^{11} - 165 q^{12} - 169 q^{13} + 45 q^{15} + 311 q^{16} - 162 q^{17} + 27 q^{18} - 138 q^{19} - 12 q^{20} - 194 q^{22} - 54 q^{23} + 18 q^{24} + 558 q^{25} - 39 q^{26} - 351 q^{27} + 303 q^{29} + 33 q^{30} - 549 q^{31} - 171 q^{33} - 25 q^{34} + 495 q^{36} + 476 q^{37} - 444 q^{38} + 507 q^{39} - 1442 q^{40} - 534 q^{41} + 430 q^{43} + 414 q^{44} - 135 q^{45} + 273 q^{46} - 414 q^{47} - 933 q^{48} + 633 q^{50} + 486 q^{51} - 715 q^{52} - 1305 q^{53} - 81 q^{54} + 369 q^{55} + 414 q^{57} - 279 q^{58} - 1959 q^{59} + 36 q^{60} - 1262 q^{61} + 669 q^{62} + 342 q^{64} + 195 q^{65} + 582 q^{66} + 2456 q^{67} - 1620 q^{68} + 162 q^{69} + 198 q^{71} - 54 q^{72} - 1284 q^{73} - 2481 q^{74} - 1674 q^{75} - 748 q^{76} + 117 q^{78} + 1015 q^{79} - 372 q^{80} + 1053 q^{81} - 1604 q^{82} + 69 q^{83} + 2824 q^{85} - 1131 q^{86} - 909 q^{87} + 465 q^{88} - 4974 q^{89} - 99 q^{90} + 4122 q^{92} + 1647 q^{93} - 3316 q^{94} + 1848 q^{95} - 1269 q^{97} + 513 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 3 x^{12} - 75 x^{11} + 220 x^{10} + 2024 x^{9} - 5757 x^{8} - 23683 x^{7} + 64922 x^{6} + \cdots + 1152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2581927 \nu^{12} - 10029897 \nu^{11} - 194353649 \nu^{10} + 750975632 \nu^{9} + \cdots - 119303777312 ) / 2843310400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2270836 \nu^{12} + 4948421 \nu^{11} + 173831532 \nu^{10} - 355016651 \nu^{9} + \cdots + 91142254816 ) / 1421655200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4771329 \nu^{12} - 19261269 \nu^{11} - 356793423 \nu^{10} + 1436114514 \nu^{9} + \cdots - 216349766624 ) / 2843310400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15741139 \nu^{12} - 38844129 \nu^{11} - 1188932493 \nu^{10} + 2816180524 \nu^{9} + \cdots - 214203650784 ) / 8529931200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1683484 \nu^{12} - 4696359 \nu^{11} - 126304248 \nu^{10} + 343007449 \nu^{9} + \cdots - 40948280064 ) / 852993120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1683484 \nu^{12} - 4696359 \nu^{11} - 126304248 \nu^{10} + 343007449 \nu^{9} + \cdots - 42654266304 ) / 852993120 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 408713 \nu^{12} + 717143 \nu^{11} + 31347131 \nu^{10} - 51851008 \nu^{9} - 870612548 \nu^{8} + \cdots - 2156129472 ) / 203093600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3013949 \nu^{12} - 9407414 \nu^{11} - 226843463 \nu^{10} + 694882009 \nu^{9} + \cdots - 64853479744 ) / 1421655200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 447794 \nu^{12} - 956799 \nu^{11} - 33839598 \nu^{10} + 69396899 \nu^{9} + 920626669 \nu^{8} + \cdots - 13976180544 ) / 121856160 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21674518 \nu^{12} + 71581023 \nu^{11} + 1624382466 \nu^{10} - 5253478063 \nu^{9} + \cdots + 588286317408 ) / 4264965600 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} + 21\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + 28\beta_{2} - 2\beta _1 + 248 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} + \beta_{11} + 2 \beta_{10} + \beta_{9} - 36 \beta_{8} + 33 \beta_{7} - \beta_{6} + \cdots - 67 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 42 \beta_{12} + 6 \beta_{11} - 36 \beta_{9} - 51 \beta_{8} + 3 \beta_{7} - 48 \beta_{6} - 40 \beta_{5} + \cdots + 5860 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 63 \beta_{12} + 58 \beta_{11} + 96 \beta_{10} + 41 \beta_{9} - 1102 \beta_{8} + 923 \beta_{7} + \cdots - 2128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1382 \beta_{12} + 309 \beta_{11} + 102 \beta_{10} - 1044 \beta_{9} - 1882 \beta_{8} + 158 \beta_{7} + \cdots + 145195 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2528 \beta_{12} + 2303 \beta_{11} + 3494 \beta_{10} + 1228 \beta_{9} - 31948 \beta_{8} + 24840 \beta_{7} + \cdots - 65163 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 41470 \beta_{12} + 11405 \beta_{11} + 6654 \beta_{10} - 28218 \beta_{9} - 60555 \beta_{8} + \cdots + 3671781 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 84693 \beta_{12} + 78649 \beta_{11} + 115378 \beta_{10} + 33087 \beta_{9} - 900401 \beta_{8} + \cdots - 1907833 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1188139 \beta_{12} + 370424 \beta_{11} + 292788 \beta_{10} - 740253 \beta_{9} - 1815868 \beta_{8} + \cdots + 93881726 \) 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
−5.12729
−5.06600
−3.49615
−2.63538
−1.79237
0.0326692
0.0742846
1.66197
2.33790
2.57763
4.37089
4.85485
5.20699
−5.12729 −3.00000 18.2891 16.9992 15.3819 0 −52.7551 9.00000 −87.1596
1.2 −5.06600 −3.00000 17.6644 −10.1362 15.1980 0 −48.9599 9.00000 51.3498
1.3 −3.49615 −3.00000 4.22303 −5.93791 10.4884 0 13.2048 9.00000 20.7598
1.4 −2.63538 −3.00000 −1.05479 −9.75661 7.90613 0 23.8628 9.00000 25.7123
1.5 −1.79237 −3.00000 −4.78741 −0.476581 5.37711 0 22.9198 9.00000 0.854209
1.6 0.0326692 −3.00000 −7.99893 −21.8230 −0.0980076 0 −0.522672 9.00000 −0.712939
1.7 0.0742846 −3.00000 −7.99448 14.0690 −0.222854 0 −1.18814 9.00000 1.04511
1.8 1.66197 −3.00000 −5.23785 14.7667 −4.98591 0 −22.0009 9.00000 24.5418
1.9 2.33790 −3.00000 −2.53424 4.09505 −7.01369 0 −24.6280 9.00000 9.57381
1.10 2.57763 −3.00000 −1.35581 −16.4531 −7.73290 0 −24.1158 9.00000 −42.4101
1.11 4.37089 −3.00000 11.1047 17.0366 −13.1127 0 13.5704 9.00000 74.4652
1.12 4.85485 −3.00000 15.5695 −4.24373 −14.5645 0 36.7489 9.00000 −20.6026
1.13 5.20699 −3.00000 19.1127 −13.1394 −15.6210 0 57.8639 9.00000 −68.4168
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

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

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.bb 13
7.b odd 2 1 1911.4.a.bc 13
7.d odd 6 2 273.4.i.e ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.4.i.e ✓ 26 7.d odd 6 2
1911.4.a.bb 13 1.a even 1 1 trivial
1911.4.a.bc 13 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{13} - 3 T_{2}^{12} - 75 T_{2}^{11} + 220 T_{2}^{10} + 2024 T_{2}^{9} - 5757 T_{2}^{8} + \cdots + 1152 \) Copy content Toggle raw display
\( T_{5}^{13} + 15 T_{5}^{12} - 979 T_{5}^{11} - 14663 T_{5}^{10} + 340748 T_{5}^{9} + 5461238 T_{5}^{8} + \cdots - 1380530799648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 3 T^{12} + \cdots + 1152 \) Copy content Toggle raw display
$3$ \( (T + 3)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 1380530799648 \) Copy content Toggle raw display
$7$ \( T^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 85\!\cdots\!35 \) Copy content Toggle raw display
$13$ \( (T + 13)^{13} \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 10\!\cdots\!86 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots - 34\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 53\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 28\!\cdots\!43 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 20\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 43\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 39\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 88\!\cdots\!99 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 22\!\cdots\!97 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 89\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 51\!\cdots\!70 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 71\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 42\!\cdots\!92 \) Copy content Toggle raw display
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