↑

Properties

Label 11.15.b.b
Level $11$
Weight $15$
Character orbit 11.b
Analytic conductor $13.676$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,15,Mod(10,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 11.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(13.6761864967\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 163566 x^{10} + 10224581640 x^{8} + 305915789698560 x^{6} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6}\cdot 11^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 486) q^{3} + (\beta_{2} - 10877) q^{4} + (\beta_{4} - 3 \beta_{3} - 11307) q^{5} + ( - \beta_{6} - 417 \beta_1) q^{6} + (\beta_{9} + \beta_{6} + 319 \beta_1) q^{7} + (\beta_{7} - \beta_{6} - 5777 \beta_1) q^{8} + ( - \beta_{5} - 25 \beta_{4} + \cdots + 1100073) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 486) q^{3} + (\beta_{2} - 10877) q^{4} + (\beta_{4} - 3 \beta_{3} - 11307) q^{5} + ( - \beta_{6} - 417 \beta_1) q^{6} + (\beta_{9} + \beta_{6} + 319 \beta_1) q^{7} + (\beta_{7} - \beta_{6} - 5777 \beta_1) q^{8} + ( - \beta_{5} - 25 \beta_{4} + \cdots + 1100073) q^{9}+ \cdots + (1769328 \beta_{11} + \cdots - 11890821127838) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 5832 q^{3} - 130524 q^{4} - 135680 q^{5} + 13200780 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 5832 q^{3} - 130524 q^{4} - 135680 q^{5} + 13200780 q^{9} - 2225212 q^{11} + 40811892 q^{12} - 104225352 q^{14} - 135328320 q^{15} - 248620440 q^{16} + 3077616220 q^{20} - 2263930680 q^{22} + 5617677688 q^{23} + 4534941420 q^{25} - 22421106552 q^{26} - 16619859936 q^{27} + 73906813656 q^{31} + 154503019968 q^{33} - 189661264872 q^{34} - 15998386272 q^{36} - 32170218528 q^{37} - 450190787160 q^{38} + 1387253675640 q^{42} + 1658278243820 q^{44} - 1829643576360 q^{45} + 637903655272 q^{47} - 1287941662872 q^{48} - 5577020124564 q^{49} + 8629370688088 q^{53} + 6705223275960 q^{55} - 759657326064 q^{56} + 1308649899360 q^{58} - 319057951208 q^{59} - 23502810498420 q^{60} + 5684019519024 q^{64} + 7079527990920 q^{66} - 689205931848 q^{67} - 20471225254152 q^{69} + 41009932132680 q^{70} + 7251314650744 q^{71} - 50761855847880 q^{75} - 14619293932320 q^{77} + 81513511801800 q^{78} - 77504933615720 q^{80} + 6526391210604 q^{81} + 75766956787080 q^{82} - 124764637159152 q^{86} - 161556660706320 q^{88} + 227512316886784 q^{89} - 215922115692192 q^{91} + 60239663314612 q^{92} + 381197540672856 q^{93} + 23746610676192 q^{97} - 142689964303524 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 163566 x^{10} + 10224581640 x^{8} + 305915789698560 x^{6} + \cdots + 66\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 27261 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 131749231169 \nu^{10} + \cdots + 77\!\cdots\!60 ) / 42\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24702136310669 \nu^{10} + \cdots - 95\!\cdots\!00 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 67815560133773 \nu^{10} + \cdots + 16\!\cdots\!80 ) / 63\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 131749231169 \nu^{11} + \cdots - 77\!\cdots\!20 \nu ) / 42\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 131749231169 \nu^{11} + \cdots + 86\!\cdots\!80 \nu ) / 42\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6359742858197 \nu^{10} + \cdots + 42\!\cdots\!60 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 357378531105971 \nu^{11} + \cdots + 18\!\cdots\!80 \nu ) / 30\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20\!\cdots\!95 \nu^{11} + \cdots + 22\!\cdots\!00 ) / 30\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 642084548645307 \nu^{11} + \cdots + 27\!\cdots\!60 \nu ) / 30\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 27261 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - 38545\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -54\beta_{8} + 29\beta_{5} - 976\beta_{4} - 23242\beta_{3} - 56478\beta_{2} + 1050779163 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 882 \beta_{11} + 928 \beta_{10} - 21636 \beta_{9} + 464 \beta_{8} - 63502 \beta_{7} + 61474 \beta_{6} + \cdots + 464 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4956156 \beta_{8} - 2092878 \beta_{5} + 77263992 \beta_{4} + 1558744548 \beta_{3} + \cdots - 46097292286314 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 80011836 \beta_{11} - 66972096 \beta_{10} + 2028594744 \beta_{9} - 33486048 \beta_{8} + \cdots - 33486048 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 331203087000 \beta_{8} + 119545893276 \beta_{5} - 4518539905296 \beta_{4} - 70840940372904 \beta_{3} + \cdots + 21\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5108449822776 \beta_{11} + 3825468584832 \beta_{10} - 137503653102576 \beta_{9} + 1912734292416 \beta_{8} + \cdots + 1912734292416 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 19\!\cdots\!60 \beta_{8} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 28\!\cdots\!16 \beta_{11} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
− 226.210i
− 216.434i
− 178.057i
− 136.965i
− 103.833i
− 65.6203i
65.6203i
103.833i
136.965i
178.057i
216.434i
226.210i
− 226.210i 1187.31 −34786.9 5228.25 − 268580.i 1.51120e6i 4.16291e6i −3.37327e6 − 1.18268e6i
10.2 − 216.434i −3231.82 −30459.8 −114987. 699477.i − 1.23221e6i 3.04649e6i 5.66171e6 2.48871e7i
10.3 − 178.057i 13.3237 −15320.5 106860. − 2372.38i − 1.22809e6i − 189373.i −4.78279e6 − 1.90272e7i
10.4 − 136.965i 3325.48 −2375.55 −75755.4 − 455476.i − 300954.i − 1.91867e6i 6.27587e6 1.03759e7i
10.5 − 103.833i −3432.09 5602.73 70302.4 356363.i 1.20102e6i − 2.28295e6i 6.99624e6 − 7.29970e6i
10.6 − 65.6203i −778.203 12078.0 −59488.4 51066.0i 120663.i − 1.86768e6i −4.17737e6 3.90365e6i
10.7 65.6203i −778.203 12078.0 −59488.4 − 51066.0i − 120663.i 1.86768e6i −4.17737e6 − 3.90365e6i
10.8 103.833i −3432.09 5602.73 70302.4 − 356363.i − 1.20102e6i 2.28295e6i 6.99624e6 7.29970e6i
10.9 136.965i 3325.48 −2375.55 −75755.4 455476.i 300954.i 1.91867e6i 6.27587e6 − 1.03759e7i
10.10 178.057i 13.3237 −15320.5 106860. 2372.38i 1.22809e6i 189373.i −4.78279e6 1.90272e7i
10.11 216.434i −3231.82 −30459.8 −114987. − 699477.i 1.23221e6i − 3.04649e6i 5.66171e6 − 2.48871e7i
10.12 226.210i 1187.31 −34786.9 5228.25 268580.i − 1.51120e6i − 4.16291e6i −3.37327e6 1.18268e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.15.b.b ✓ 12
3.b odd 2 1 99.15.c.b 12
4.b odd 2 1 176.15.h.d 12
11.b odd 2 1 inner 11.15.b.b ✓ 12
33.d even 2 1 99.15.c.b 12
44.c even 2 1 176.15.h.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.15.b.b ✓ 12 1.a even 1 1 trivial
11.15.b.b ✓ 12 11.b odd 2 1 inner
99.15.c.b 12 3.b odd 2 1
99.15.c.b 12 33.d even 2 1
176.15.h.d 12 4.b odd 2 1
176.15.h.d 12 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 163566 T_{2}^{10} + 10224581640 T_{2}^{8} + 305915789698560 T_{2}^{6} + \cdots + 66\!\cdots\!00 \) acting on \(S_{15}^{\mathrm{new}}(11, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( (T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 29\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 50\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 25\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 86\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 35\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less