↑

Properties

Label 1122.2.b.a
Level $1122$
Weight $2$
Character orbit 1122.b
Analytic conductor $8.959$
Analytic rank $0$
Dimension $16$
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] = [1122,2,Mod(1055,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1122.1055");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.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: \(8.95921510679\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 18x^{14} + 124x^{12} + 420x^{10} + 746x^{8} + 681x^{6} + 288x^{4} + 42x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{9} q^{3} + q^{4} + \beta_{13} q^{5} + \beta_{9} q^{6} + (\beta_{9} + \beta_{4}) q^{7} - q^{8} + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{9} q^{3} + q^{4} + \beta_{13} q^{5} + \beta_{9} q^{6} + (\beta_{9} + \beta_{4}) q^{7} - q^{8} + \beta_{8} q^{9} - \beta_{13} q^{10} - \beta_{15} q^{11} - \beta_{9} q^{12} + (\beta_{13} - \beta_{2}) q^{13} + ( - \beta_{9} - \beta_{4}) q^{14} + ( - \beta_{14} + \beta_{7} + \beta_{6} + \cdots + 1) q^{15}+ \cdots + (\beta_{14} - \beta_{13} + 2 \beta_{11} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + q^{3} + 16 q^{4} - q^{6} - 16 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + q^{3} + 16 q^{4} - q^{6} - 16 q^{8} - 3 q^{9} + 6 q^{11} + q^{12} + 16 q^{16} + 16 q^{17} + 3 q^{18} + 24 q^{21} - 6 q^{22} - q^{24} - 22 q^{25} + 13 q^{27} - 20 q^{29} + 24 q^{31} - 16 q^{32} - 3 q^{33} - 16 q^{34} + 12 q^{35} - 3 q^{36} + 6 q^{37} - q^{39} + 22 q^{41} - 24 q^{42} + 6 q^{44} - 15 q^{45} + q^{48} - 54 q^{49} + 22 q^{50} + q^{51} - 13 q^{54} - 10 q^{55} + 8 q^{57} + 20 q^{58} - 24 q^{62} - 36 q^{63} + 16 q^{64} - 68 q^{65} + 3 q^{66} + 12 q^{67} + 16 q^{68} - 40 q^{69} - 12 q^{70} + 3 q^{72} - 6 q^{74} + q^{75} + 10 q^{77} + q^{78} + 17 q^{81} - 22 q^{82} - 10 q^{83} + 24 q^{84} + 12 q^{87} - 6 q^{88} + 15 q^{90} + 4 q^{91} + 6 q^{93} - 48 q^{95} - q^{96} + 54 q^{98} + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 18x^{14} + 124x^{12} + 420x^{10} + 746x^{8} + 681x^{6} + 288x^{4} + 42x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 61 \nu^{15} - 79 \nu^{14} + 1070 \nu^{13} - 1364 \nu^{12} + 7033 \nu^{11} - 8996 \nu^{10} + \cdots - 1285 ) / 442 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{15} - 172\nu^{13} - 3393\nu^{11} - 21815\nu^{9} - 63211\nu^{7} - 86275\nu^{5} - 50340\nu^{3} - 9236\nu ) / 221 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -89\nu^{14} - 1601\nu^{12} - 10946\nu^{10} - 36080\nu^{8} - 59227\nu^{6} - 43741\nu^{4} - 10264\nu^{2} + 256 ) / 221 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 163 \nu^{15} + 25 \nu^{14} - 2957 \nu^{13} + 482 \nu^{12} - 20514 \nu^{11} + 3549 \nu^{10} + \cdots + 678 ) / 442 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -98\nu^{14} - 1748\nu^{12} - 11817\nu^{10} - 38482\nu^{8} - 62969\nu^{6} - 48569\nu^{4} - 15049\nu^{2} - 1429 ) / 221 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 239 \nu^{15} - 184 \nu^{14} + 4051 \nu^{13} - 3079 \nu^{12} + 25389 \nu^{11} - 19084 \nu^{10} + \cdots - 3116 ) / 442 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{14} + 147\nu^{12} + 871\nu^{10} + 2324\nu^{8} + 2780\nu^{6} + 1214\nu^{4} + 79\nu^{2} + 8 ) / 13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 248 \nu^{15} - 42 \nu^{14} - 4419 \nu^{13} - 686 \nu^{12} - 30017 \nu^{11} - 3991 \nu^{10} + \cdots + 19 ) / 442 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 309 \nu^{15} - 25 \nu^{14} + 5489 \nu^{13} - 482 \nu^{12} + 37050 \nu^{11} - 3549 \nu^{10} + \cdots - 678 ) / 442 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 91 \nu^{15} + 329 \nu^{14} + 1781 \nu^{13} + 5742 \nu^{12} + 13546 \nu^{11} + 37635 \nu^{10} + \cdots + 2540 ) / 442 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 211 \nu^{15} - 3741 \nu^{13} - 25233 \nu^{11} - 83030 \nu^{9} - 141604 \nu^{7} - 120802 \nu^{5} + \cdots - 2646 \nu ) / 221 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 238 \nu^{15} + 152 \nu^{14} - 4182 \nu^{13} + 2630 \nu^{12} - 27846 \nu^{11} + 17043 \nu^{10} + \cdots + 931 ) / 221 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 222 \nu^{15} + 3847 \nu^{13} + 24947 \nu^{11} + 76561 \nu^{9} + 114992 \nu^{7} + 77248 \nu^{5} + \cdots + 89 \nu ) / 221 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 557 \nu^{15} - 17 \nu^{14} + 9908 \nu^{13} - 204 \nu^{12} + 67067 \nu^{11} - 442 \nu^{10} + 221317 \nu^{9} + \cdots + 697 ) / 442 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 238 \nu^{15} - 152 \nu^{14} - 4182 \nu^{13} - 2630 \nu^{12} - 27846 \nu^{11} - 17043 \nu^{10} + \cdots - 931 ) / 221 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{15} + \beta_{14} - 3 \beta_{13} + 6 \beta_{11} - 3 \beta_{10} + 7 \beta_{9} + 3 \beta_{8} + \cdots + 4 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{15} - 4 \beta_{14} - 3 \beta_{12} + 5 \beta_{9} + 3 \beta_{7} + 6 \beta_{6} - 14 \beta_{5} + \cdots - 34 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 13 \beta_{14} + 21 \beta_{13} - 9 \beta_{12} - 15 \beta_{11} + 12 \beta_{10} - 28 \beta_{9} + \cdots - 16 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{15} + 8 \beta_{14} + 6 \beta_{12} - 10 \beta_{9} - 3 \beta_{8} - 9 \beta_{7} - 18 \beta_{6} + \cdots + 41 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21 \beta_{15} + 101 \beta_{14} - 132 \beta_{13} + 90 \beta_{12} + 39 \beta_{11} - 69 \beta_{10} + \cdots + 80 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 123 \beta_{15} - 128 \beta_{14} - 123 \beta_{12} + 160 \beta_{9} + 90 \beta_{8} + 204 \beta_{7} + \cdots - 548 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 246 \beta_{15} - 689 \beta_{14} + 825 \beta_{13} - 693 \beta_{12} - 57 \beta_{11} + 447 \beta_{10} + \cdots - 461 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 288 \beta_{15} + 236 \beta_{14} + 288 \beta_{12} - 295 \beta_{9} - 231 \beta_{8} - 483 \beta_{7} + \cdots + 908 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1923 \beta_{15} + 4540 \beta_{14} - 5187 \beta_{13} + 4923 \beta_{12} - 381 \beta_{11} - 3000 \beta_{10} + \cdots + 2842 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6006 \beta_{15} - 4096 \beta_{14} - 6006 \beta_{12} + 5147 \beta_{9} + 4923 \beta_{8} + 9966 \beta_{7} + \cdots - 14575 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13587 \beta_{15} - 29650 \beta_{14} + 32889 \beta_{13} - 33786 \beta_{12} + 5547 \beta_{11} + \cdots - 18061 ) / 18 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4566 \beta_{15} + 2730 \beta_{14} + 4566 \beta_{12} - 3460 \beta_{9} - 3754 \beta_{8} - 7479 \beta_{7} + \cdots + 9153 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 92487 \beta_{15} + 193373 \beta_{14} - 210273 \beta_{13} + 227763 \beta_{12} - 49092 \beta_{11} + \cdots + 116411 ) / 18 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 277467 \beta_{15} - 151394 \beta_{14} - 277467 \beta_{12} + 193612 \beta_{9} + 227763 \beta_{8} + \cdots - 485390 ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 619167 \beta_{15} - 1261994 \beta_{14} + 1353741 \beta_{13} - 1519650 \beta_{12} + 375882 \beta_{11} + \cdots - 755606 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1122\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(749\) \(1057\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1055.1
1.47766i
− 1.47766i
1.07893i
− 1.07893i
0.886946i
− 0.886946i
− 0.469643i
0.469643i
2.55987i
− 2.55987i
0.171179i
− 0.171179i
2.08458i
− 2.08458i
− 1.64848i
1.64848i
−1.00000 −1.65816 − 0.500521i 1.00000 0.650255i 1.65816 + 0.500521i 1.64351i −1.00000 2.49896 + 1.65988i − 0.650255i
1055.2 −1.00000 −1.65816 + 0.500521i 1.00000 − 0.650255i 1.65816 − 0.500521i − 1.64351i −1.00000 2.49896 − 1.65988i 0.650255i
1055.3 −1.00000 −1.33295 − 1.10600i 1.00000 − 2.54202i 1.33295 + 1.10600i 4.37874i −1.00000 0.553527 + 2.94849i 2.54202i
1055.4 −1.00000 −1.33295 + 1.10600i 1.00000 2.54202i 1.33295 − 1.10600i − 4.37874i −1.00000 0.553527 − 2.94849i − 2.54202i
1055.5 −1.00000 −0.423385 − 1.67951i 1.00000 4.07146i 0.423385 + 1.67951i 4.79476i −1.00000 −2.64149 + 1.42216i − 4.07146i
1055.6 −1.00000 −0.423385 + 1.67951i 1.00000 − 4.07146i 0.423385 − 1.67951i − 4.79476i −1.00000 −2.64149 − 1.42216i 4.07146i
1055.7 −1.00000 −0.240909 − 1.71522i 1.00000 2.17582i 0.240909 + 1.71522i − 0.914768i −1.00000 −2.88393 + 0.826423i − 2.17582i
1055.8 −1.00000 −0.240909 + 1.71522i 1.00000 − 2.17582i 0.240909 − 1.71522i 0.914768i −1.00000 −2.88393 − 0.826423i 2.17582i
1055.9 −1.00000 −0.189982 − 1.72160i 1.00000 − 3.14128i 0.189982 + 1.72160i − 0.746826i −1.00000 −2.92781 + 0.654145i 3.14128i
1055.10 −1.00000 −0.189982 + 1.72160i 1.00000 3.14128i 0.189982 − 1.72160i 0.746826i −1.00000 −2.92781 − 0.654145i − 3.14128i
1055.11 −1.00000 1.13094 − 1.31186i 1.00000 − 0.273490i −1.13094 + 1.31186i 0.751563i −1.00000 −0.441972 − 2.96726i 0.273490i
1055.12 −1.00000 1.13094 + 1.31186i 1.00000 0.273490i −1.13094 − 1.31186i − 0.751563i −1.00000 −0.441972 + 2.96726i − 0.273490i
1055.13 −1.00000 1.55715 − 0.758481i 1.00000 − 0.724190i −1.55715 + 0.758481i − 3.18674i −1.00000 1.84941 − 2.36213i 0.724190i
1055.14 −1.00000 1.55715 + 0.758481i 1.00000 0.724190i −1.55715 − 0.758481i 3.18674i −1.00000 1.84941 + 2.36213i − 0.724190i
1055.15 −1.00000 1.65730 − 0.503338i 1.00000 − 3.51246i −1.65730 + 0.503338i 5.10109i −1.00000 2.49330 − 1.66837i 3.51246i
1055.16 −1.00000 1.65730 + 0.503338i 1.00000 3.51246i −1.65730 − 0.503338i − 5.10109i −1.00000 2.49330 + 1.66837i − 3.51246i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1055.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1122.2.b.a ✓ 16
3.b odd 2 1 1122.2.b.c yes 16
11.b odd 2 1 1122.2.b.c yes 16
33.d even 2 1 inner 1122.2.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1122.2.b.a ✓ 16 1.a even 1 1 trivial
1122.2.b.a ✓ 16 33.d even 2 1 inner
1122.2.b.c yes 16 3.b odd 2 1
1122.2.b.c yes 16 11.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 \(S_{2}^{\mathrm{new}}(1122, [\chi])\):

\( T_{5}^{16} + 51 T_{5}^{14} + 1006 T_{5}^{12} + 9679 T_{5}^{10} + 46740 T_{5}^{8} + 102704 T_{5}^{6} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{29}^{8} + 10T_{29}^{7} - 52T_{29}^{6} - 592T_{29}^{5} + 608T_{29}^{4} + 8448T_{29}^{3} + 1088T_{29}^{2} - 22272T_{29} + 11520 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 51 T^{14} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{16} + 83 T^{14} + \cdots + 82944 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 123 T^{14} + \cdots + 51840000 \) Copy content Toggle raw display
$17$ \( (T - 1)^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 175 T^{14} + \cdots + 6718464 \) Copy content Toggle raw display
$23$ \( T^{16} + 160 T^{14} + \cdots + 1638400 \) Copy content Toggle raw display
$29$ \( (T^{8} + 10 T^{7} + \cdots + 11520)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 12 T^{7} + \cdots - 6400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 3 T^{7} + \cdots + 63328)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 11 T^{7} + \cdots + 107424)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 846111744 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30471193600 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 519110656 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 8278187769856 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 41108373504 \) Copy content Toggle raw display
$67$ \( (T^{8} - 6 T^{7} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 609288797618176 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 108362105856 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{8} + 5 T^{7} + \cdots - 1704960)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 6544162816 \) Copy content Toggle raw display
$97$ \( (T^{8} - 384 T^{6} + \cdots - 15104)^{2} \) Copy content Toggle raw display
show more
show less