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Properties

Label 22.15.b.a
Level $22$
Weight $15$
Character orbit 22.b
Analytic conductor $27.352$
Analytic rank $0$
Dimension $14$
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] = [22,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.21");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(27.3523729934\)
Analytic rank: \(0\)
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: \( x^{14} - 2 x^{13} - 38299509 x^{12} + 1255603312 x^{11} + 548839279225666 x^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{6}\cdot 11^{7} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_1 + 314) q^{3} - 8192 q^{4} + (\beta_{3} + 4983) q^{5} + ( - \beta_{5} + 314 \beta_{2}) q^{6} + ( - \beta_{6} - 548 \beta_{2}) q^{7} - 8192 \beta_{2} q^{8} + (\beta_{4} + 19 \beta_{3} + \cdots + 787001) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_1 + 314) q^{3} - 8192 q^{4} + (\beta_{3} + 4983) q^{5} + ( - \beta_{5} + 314 \beta_{2}) q^{6} + ( - \beta_{6} - 548 \beta_{2}) q^{7} - 8192 \beta_{2} q^{8} + (\beta_{4} + 19 \beta_{3} + \cdots + 787001) q^{9}+ \cdots + ( - 350537 \beta_{13} + \cdots + 8489274531513) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4394 q^{3} - 114688 q^{4} + 69758 q^{5} + 11016572 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4394 q^{3} - 114688 q^{4} + 69758 q^{5} + 11016572 q^{9} + 20143042 q^{11} - 35995648 q^{12} + 62814720 q^{14} - 1359602 q^{15} + 939524096 q^{16} - 571457536 q^{20} - 2107666944 q^{22} - 7305755542 q^{23} + 19291879452 q^{25} - 6388480512 q^{26} + 34093422830 q^{27} - 33569873942 q^{31} + 2885838062 q^{33} + 167764701696 q^{34} - 90247757824 q^{36} + 73167823966 q^{37} + 71236111872 q^{38} - 222695314944 q^{42} - 165011800064 q^{44} + 2000205168616 q^{45} - 1612717386124 q^{47} + 294876348416 q^{48} + 3424602524990 q^{49} - 3530064068164 q^{53} - 3715439610854 q^{55} - 514578186240 q^{56} - 1374208002048 q^{58} - 818496564070 q^{59} + 11137859584 q^{60} - 7696581394432 q^{64} - 5938395621888 q^{66} + 16485465276922 q^{67} - 11394452631206 q^{69} - 392146020864 q^{70} - 19380879179878 q^{71} + 23016770893992 q^{75} + 60534793808304 q^{77} + 17335823992320 q^{78} + 4681380134912 q^{80} - 10394309810662 q^{81} - 79417078012416 q^{82} + 6375532305408 q^{86} + 17266007605248 q^{88} - 117770741987650 q^{89} + 150621364097712 q^{91} + 59848749400064 q^{92} + 27345122803162 q^{93} + 123398138843566 q^{97} + 118861332531788 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 38299509 x^{12} + 1255603312 x^{11} + 548839279225666 x^{10} + \cdots + 61\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 55\!\cdots\!75 \nu^{13} + \cdots - 60\!\cdots\!40 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 55\!\cdots\!75 \nu^{13} + \cdots - 60\!\cdots\!40 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!14 \nu^{13} + \cdots - 14\!\cdots\!20 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!50 \nu^{13} + \cdots - 57\!\cdots\!00 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 83\!\cdots\!63 \nu^{13} + \cdots - 10\!\cdots\!20 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 84\!\cdots\!67 \nu^{13} + \cdots - 37\!\cdots\!00 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!31 \nu^{13} + \cdots + 10\!\cdots\!60 ) / 63\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 82\!\cdots\!43 \nu^{13} + \cdots - 58\!\cdots\!60 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 75\!\cdots\!07 \nu^{13} + \cdots + 26\!\cdots\!00 ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 73\!\cdots\!21 \nu^{13} + \cdots + 25\!\cdots\!60 ) / 69\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!49 \nu^{13} + \cdots + 35\!\cdots\!80 ) / 82\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44\!\cdots\!41 \nu^{13} + \cdots - 35\!\cdots\!80 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38\!\cdots\!35 \nu^{13} + \cdots - 31\!\cdots\!20 ) / 69\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 64\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 32\beta_{4} + 608\beta_{3} + 1664\beta _1 + 175083904 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5936 \beta_{13} + 13584 \beta_{12} + 39 \beta_{11} + 9600 \beta_{10} - 10576 \beta_{9} + \cdots - 16257981440 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4501152 \beta_{13} + 5113024 \beta_{12} - 30616 \beta_{11} - 3157536 \beta_{10} + \cdots + 843813781718112 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 85739493712 \beta_{13} + 188174153200 \beta_{12} + 570270485 \beta_{11} + 124144932864 \beta_{10} + \cdots + 37\!\cdots\!36 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 175917531938176 \beta_{13} + 202257049447552 \beta_{12} - 2039243909968 \beta_{11} + \cdots + 18\!\cdots\!28 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10\!\cdots\!64 \beta_{13} + \cdots + 11\!\cdots\!88 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 64\!\cdots\!88 \beta_{13} + \cdots + 53\!\cdots\!32 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 13\!\cdots\!88 \beta_{13} + \cdots + 22\!\cdots\!56 ) / 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 34\!\cdots\!52 \beta_{13} + \cdots + 25\!\cdots\!76 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 17\!\cdots\!48 \beta_{13} + \cdots + 36\!\cdots\!00 ) / 64 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 22\!\cdots\!12 \beta_{13} + \cdots + 15\!\cdots\!36 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22\!\cdots\!04 \beta_{13} + \cdots + 55\!\cdots\!60 ) / 64 \) 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
3563.38 − 1.41421i
2199.62 − 1.41421i
1021.72 − 1.41421i
320.191 − 1.41421i
−1087.11 − 1.41421i
−2599.55 − 1.41421i
−3417.25 − 1.41421i
3563.38 + 1.41421i
2199.62 + 1.41421i
1021.72 + 1.41421i
320.191 + 1.41421i
−1087.11 + 1.41421i
−2599.55 + 1.41421i
−3417.25 + 1.41421i
− 90.5097i −3249.38 −8192.00 −6683.13 294100.i 286020.i 741455.i 5.77548e6 604888.i
21.2 − 90.5097i −1885.62 −8192.00 124487. 170667.i 166636.i 741455.i −1.22739e6 − 1.12672e7i
21.3 − 90.5097i −707.723 −8192.00 −8968.27 64055.8i − 531246.i 741455.i −4.28210e6 811715.i
21.4 − 90.5097i −6.19137 −8192.00 −140247. 560.379i 1.02360e6i 741455.i −4.78293e6 1.26937e7i
21.5 − 90.5097i 1401.11 −8192.00 44552.9 − 126814.i − 500199.i 741455.i −2.81985e6 − 4.03247e6i
21.6 − 90.5097i 2913.55 −8192.00 −75323.6 − 263704.i − 867659.i 741455.i 3.70579e6 6.81751e6i
21.7 − 90.5097i 3731.25 −8192.00 97060.9 − 337715.i 769857.i 741455.i 9.13929e6 − 8.78495e6i
21.8 90.5097i −3249.38 −8192.00 −6683.13 − 294100.i − 286020.i − 741455.i 5.77548e6 − 604888.i
21.9 90.5097i −1885.62 −8192.00 124487. − 170667.i − 166636.i − 741455.i −1.22739e6 1.12672e7i
21.10 90.5097i −707.723 −8192.00 −8968.27 − 64055.8i 531246.i − 741455.i −4.28210e6 − 811715.i
21.11 90.5097i −6.19137 −8192.00 −140247. − 560.379i − 1.02360e6i − 741455.i −4.78293e6 − 1.26937e7i
21.12 90.5097i 1401.11 −8192.00 44552.9 126814.i 500199.i − 741455.i −2.81985e6 4.03247e6i
21.13 90.5097i 2913.55 −8192.00 −75323.6 263704.i 867659.i − 741455.i 3.70579e6 − 6.81751e6i
21.14 90.5097i 3731.25 −8192.00 97060.9 337715.i − 769857.i − 741455.i 9.13929e6 8.78495e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.14
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 22.15.b.a ✓ 14
4.b odd 2 1 176.15.h.e 14
11.b odd 2 1 inner 22.15.b.a ✓ 14
44.c even 2 1 176.15.h.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.15.b.a ✓ 14 1.a even 1 1 trivial
22.15.b.a ✓ 14 11.b odd 2 1 inner
176.15.h.e 14 4.b odd 2 1
176.15.h.e 14 44.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8192)^{7} \) Copy content Toggle raw display
$3$ \( (T^{7} + \cdots - 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{7} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 74\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 25\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 28\!\cdots\!92)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 27\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 20\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 63\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 47\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 59\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 29\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 77\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 72\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 96\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 62\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 23\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 43\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 97\!\cdots\!16)^{2} \) Copy content Toggle raw display
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