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Properties

Label 22.11.b.a
Level $22$
Weight $11$
Character orbit 22.b
Analytic conductor $13.978$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [22,11,Mod(21,22)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(22, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.21");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 22.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: \(13.9778595588\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 135903 x^{8} - 6427236 x^{7} + 6935435151 x^{6} + 631292713590 x^{5} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{2}\cdot 11^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - 11) q^{3} - 512 q^{4} + ( - \beta_{4} - 2 \beta_{2} + 113) q^{5} + (\beta_{6} + 11 \beta_1) q^{6} + (\beta_{7} + \beta_{6} + \cdots - 30 \beta_1) q^{7}+ \cdots + (3 \beta_{4} - \beta_{3} - 74 \beta_{2} + 7775) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - 11) q^{3} - 512 q^{4} + ( - \beta_{4} - 2 \beta_{2} + 113) q^{5} + (\beta_{6} + 11 \beta_1) q^{6} + (\beta_{7} + \beta_{6} + \cdots - 30 \beta_1) q^{7}+ \cdots + (1569 \beta_{9} - 81939 \beta_{8} + \cdots + 4881790889) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 106 q^{3} - 5120 q^{4} + 1138 q^{5} + 78044 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 106 q^{3} - 5120 q^{4} + 1138 q^{5} + 78044 q^{9} + 95414 q^{11} + 54272 q^{12} - 156288 q^{14} + 1441618 q^{15} + 2621440 q^{16} - 582656 q^{20} - 6002304 q^{22} + 17496838 q^{23} - 1494468 q^{25} + 9714816 q^{26} + 54656930 q^{27} - 91050970 q^{31} - 12170158 q^{33} - 6879360 q^{34} - 39958528 q^{36} - 82676974 q^{37} - 55302528 q^{38} - 128221824 q^{42} - 48851968 q^{44} - 124619384 q^{45} + 352507996 q^{47} - 27787264 q^{48} - 374605478 q^{49} + 571129876 q^{53} + 1363103126 q^{55} + 80019456 q^{56} + 1594048512 q^{58} - 1508647610 q^{59} - 738108416 q^{60} - 1342177280 q^{64} + 1288087680 q^{66} + 3146811782 q^{67} + 5332296166 q^{69} - 1491609984 q^{70} - 328577450 q^{71} - 18684358968 q^{75} + 4256837904 q^{77} + 4919767680 q^{78} + 298319872 q^{80} - 16957790722 q^{81} + 4545650304 q^{82} - 12971187456 q^{86} + 3073179648 q^{88} + 17791426978 q^{89} + 40311734544 q^{91} - 8958381056 q^{92} - 11674310138 q^{93} - 62585189614 q^{97} + 48880194572 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 135903 x^{8} - 6427236 x^{7} + 6935435151 x^{6} + 631292713590 x^{5} + \cdots + 88\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25\!\cdots\!01 \nu^{9} + \cdots - 16\!\cdots\!24 ) / 17\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\!\cdots\!54 \nu^{9} + \cdots + 23\!\cdots\!76 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 80\!\cdots\!66 \nu^{9} + \cdots + 57\!\cdots\!44 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 80\!\cdots\!14 \nu^{9} + \cdots - 51\!\cdots\!16 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 89\!\cdots\!31 \nu^{9} + \cdots + 59\!\cdots\!84 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!11 \nu^{9} + \cdots - 15\!\cdots\!68 ) / 76\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!81 \nu^{9} + \cdots - 16\!\cdots\!24 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 81\!\cdots\!62 \nu^{9} + \cdots + 52\!\cdots\!84 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84\!\cdots\!19 \nu^{9} + \cdots - 55\!\cdots\!56 ) / 32\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{3} - 88\beta_{2} - 33\beta _1 + 70 ) / 528 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{9} + 61 \beta_{8} + 5 \beta_{6} - \beta_{5} + 229 \beta_{4} + 69 \beta_{3} - 4068 \beta_{2} + \cdots + 7174143 ) / 264 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 138 \beta_{9} + 9509 \beta_{8} - 576 \beta_{7} + 429 \beta_{6} - 549 \beta_{5} + 45093 \beta_{4} + \cdots + 353150786 ) / 176 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 128980 \beta_{9} + 4968014 \beta_{8} - 326784 \beta_{7} + 483538 \beta_{6} - 291050 \beta_{5} + \cdots + 244094220135 ) / 264 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43958610 \beta_{9} + 2349551673 \beta_{8} - 253078080 \beta_{7} + 185555865 \beta_{6} + \cdots + 69950945011466 ) / 528 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2532775550 \beta_{9} + 105074317979 \beta_{8} - 13043341376 \beta_{7} + 12095714355 \beta_{6} + \cdots + 33\!\cdots\!31 ) / 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3251689222362 \beta_{9} + 148569530440461 \beta_{8} - 23260430561088 \beta_{7} + 16537551071349 \beta_{6} + \cdots + 38\!\cdots\!26 ) / 528 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 455825661961928 \beta_{9} + \cdots + 46\!\cdots\!59 ) / 264 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 71\!\cdots\!98 \beta_{9} + \cdots + 66\!\cdots\!78 ) / 176 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
−134.982 − 1.41421i
−138.673 − 1.41421i
171.602 − 1.41421i
230.003 − 1.41421i
−126.950 − 1.41421i
−134.982 + 1.41421i
−138.673 + 1.41421i
171.602 + 1.41421i
230.003 + 1.41421i
−126.950 + 1.41421i
− 22.6274i −289.396 −512.000 −5145.87 6548.28i − 3108.94i 11585.2i 24700.9 116438.i
21.2 − 22.6274i −251.838 −512.000 3507.04 5698.45i 17033.0i 11585.2i 4373.51 − 79355.3i
21.3 − 22.6274i −81.3085 −512.000 2188.57 1839.80i − 24594.3i 11585.2i −52437.9 − 49521.7i
21.4 − 22.6274i 189.131 −512.000 −1492.29 − 4279.54i 21897.3i 11585.2i −23278.6 33766.7i
21.5 − 22.6274i 380.412 −512.000 1511.55 − 8607.74i − 14680.5i 11585.2i 85664.2 − 34202.4i
21.6 22.6274i −289.396 −512.000 −5145.87 − 6548.28i 3108.94i − 11585.2i 24700.9 − 116438.i
21.7 22.6274i −251.838 −512.000 3507.04 − 5698.45i − 17033.0i − 11585.2i 4373.51 79355.3i
21.8 22.6274i −81.3085 −512.000 2188.57 − 1839.80i 24594.3i − 11585.2i −52437.9 49521.7i
21.9 22.6274i 189.131 −512.000 −1492.29 4279.54i − 21897.3i − 11585.2i −23278.6 − 33766.7i
21.10 22.6274i 380.412 −512.000 1511.55 8607.74i 14680.5i − 11585.2i 85664.2 34202.4i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.10
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 22.11.b.a ✓ 10
3.b odd 2 1 198.11.d.a 10
4.b odd 2 1 176.11.h.e 10
11.b odd 2 1 inner 22.11.b.a ✓ 10
33.d even 2 1 198.11.d.a 10
44.c even 2 1 176.11.h.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.11.b.a ✓ 10 1.a even 1 1 trivial
22.11.b.a ✓ 10 11.b odd 2 1 inner
176.11.h.e 10 4.b odd 2 1
176.11.h.e 10 44.c even 2 1
198.11.d.a 10 3.b odd 2 1
198.11.d.a 10 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 512)^{5} \) Copy content Toggle raw display
$3$ \( (T^{5} + 53 T^{4} + \cdots + 426349422396)^{2} \) Copy content Toggle raw display
$5$ \( (T^{5} + \cdots - 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 17\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 26\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 23\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 33\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 37\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 44\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 36\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 28\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 21\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 57\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 67\!\cdots\!44)^{2} \) Copy content Toggle raw display
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