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Properties

Label 110.5.d.a
Level $110$
Weight $5$
Character orbit 110.d
Analytic conductor $11.371$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,5,Mod(21,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.21");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 110.d (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: \(11.3706959392\)
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: \( x^{16} - 8 x^{15} - 676 x^{14} + 4616 x^{13} + 185738 x^{12} - 1040336 x^{11} - 26148232 x^{10} + \cdots + 28379889703121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{2} + 1) q^{3} - 8 q^{4} - \beta_{6} q^{5} + (\beta_{5} - \beta_1) q^{6} + ( - 4 \beta_{5} - \beta_{4}) q^{7} - 8 \beta_{5} q^{8} + ( - \beta_{3} - \beta_{2} + 11) q^{9}+ \cdots + (4 \beta_{15} + 40 \beta_{14} + \cdots + 645) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} - 128 q^{4} + 184 q^{9} + 336 q^{11} - 128 q^{12} + 576 q^{14} + 200 q^{15} + 1024 q^{16} + 480 q^{22} + 2832 q^{23} + 2000 q^{25} - 2496 q^{26} + 928 q^{27} - 2648 q^{31} + 2704 q^{33}+ \cdots + 10064 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 676 x^{14} + 4616 x^{13} + 185738 x^{12} - 1040336 x^{11} - 26148232 x^{10} + \cdots + 28379889703121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21\!\cdots\!24 \nu^{15} + \cdots - 70\!\cdots\!65 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23\!\cdots\!67 \nu^{15} + \cdots - 37\!\cdots\!10 ) / 95\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\!\cdots\!68 \nu^{15} + \cdots + 15\!\cdots\!61 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38\!\cdots\!66 \nu^{15} + \cdots - 41\!\cdots\!60 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 88\!\cdots\!74 \nu^{15} + \cdots - 39\!\cdots\!06 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!56 \nu^{15} + \cdots - 12\!\cdots\!82 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!08 \nu^{15} + \cdots - 17\!\cdots\!46 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!94 \nu^{15} + \cdots - 56\!\cdots\!90 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!81 \nu^{15} + \cdots + 59\!\cdots\!84 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23\!\cdots\!08 \nu^{15} + \cdots + 22\!\cdots\!53 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 89\!\cdots\!40 \nu^{15} + \cdots - 10\!\cdots\!81 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 97\!\cdots\!48 \nu^{15} + \cdots - 79\!\cdots\!09 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58\!\cdots\!66 \nu^{15} + \cdots + 39\!\cdots\!85 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 65\!\cdots\!34 \nu^{15} + \cdots - 54\!\cdots\!09 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 22\!\cdots\!18 \nu^{15} + \cdots - 83\!\cdots\!38 ) / 32\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + 5\beta_{5} + 10\beta_{2} + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + 2\beta_{7} - \beta_{6} + 5\beta_{5} - 8\beta_{3} + 18\beta_{2} + 10\beta _1 + 881 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 48 \beta_{15} - 70 \beta_{14} - 10 \beta_{13} - 96 \beta_{12} + 3 \beta_{10} + 48 \beta_{8} + \cdots + 3574 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 164 \beta_{15} - 90 \beta_{14} - 30 \beta_{13} - 288 \beta_{12} + 80 \beta_{11} + 120 \beta_{10} + \cdots + 124447 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2180 \beta_{15} - 2798 \beta_{14} - 866 \beta_{13} - 6880 \beta_{12} + 240 \beta_{11} - 1561 \beta_{10} + \cdots + 182344 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 51850 \beta_{15} - 21310 \beta_{14} - 8650 \beta_{13} - 95040 \beta_{12} + 53560 \beta_{11} + \cdots + 18766358 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 988400 \beta_{15} - 1193230 \beta_{14} - 494050 \beta_{13} - 4509400 \beta_{12} + 259000 \beta_{11} + \cdots + 86579553 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11429856 \beta_{15} - 4510990 \beta_{14} - 2217370 \beta_{13} - 26354752 \beta_{12} + \cdots + 2768912131 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 321155796 \beta_{15} - 371371530 \beta_{14} - 182795430 \beta_{13} - 2101520832 \beta_{12} + \cdots + 28220882710 ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 416802370 \beta_{15} - 169075742 \beta_{14} - 99203978 \beta_{13} - 1333188864 \beta_{12} + \cdots + 75513122529 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 46717297264 \beta_{15} - 51239114110 \beta_{14} - 28738197010 \beta_{13} - 456715047888 \beta_{12} + \cdots + 4000126406448 ) / 20 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 162467950298 \beta_{15} - 67123146490 \beta_{14} - 45671274190 \beta_{13} - 784852312896 \beta_{12} + \cdots + 21666294917387 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2856400205360 \beta_{15} - 2769881474480 \beta_{14} - 1792143866480 \beta_{13} + \cdots + 223242920810593 ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 41909965730072 \beta_{15} - 15732927123020 \beta_{14} - 12546164511140 \beta_{13} + \cdots + 28\!\cdots\!21 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 90631880885304 \beta_{15} - 37243431725358 \beta_{14} - 43198020091746 \beta_{13} + \cdots + 36\!\cdots\!10 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
13.9261 − 1.41421i
13.0009 − 1.41421i
5.29112 − 1.41421i
2.97133 − 1.41421i
−0.757543 − 1.41421i
−7.97762 − 1.41421i
−9.23959 − 1.41421i
−13.2147 − 1.41421i
13.9261 + 1.41421i
13.0009 + 1.41421i
5.29112 + 1.41421i
2.97133 + 1.41421i
−0.757543 + 1.41421i
−7.97762 + 1.41421i
−9.23959 + 1.41421i
−13.2147 + 1.41421i
− 2.82843i −13.5441 −8.00000 −11.1803 38.3086i 48.2066i 22.6274i 102.443 31.6228i
21.2 − 2.82843i −10.3829 −8.00000 11.1803 29.3673i 40.4137i 22.6274i 26.8045 − 31.6228i
21.3 − 2.82843i −4.90915 −8.00000 −11.1803 13.8852i − 52.2014i 22.6274i −56.9002 31.6228i
21.4 − 2.82843i −0.353300 −8.00000 11.1803 0.999282i 34.1317i 22.6274i −80.8752 − 31.6228i
21.5 − 2.82843i 3.37558 −8.00000 11.1803 − 9.54757i − 59.8276i 22.6274i −69.6055 − 31.6228i
21.6 − 2.82843i 8.35959 −8.00000 −11.1803 − 23.6445i 92.1688i 22.6274i −11.1173 31.6228i
21.7 − 2.82843i 9.62155 −8.00000 −11.1803 − 27.2139i − 37.2623i 22.6274i 11.5742 31.6228i
21.8 − 2.82843i 15.8328 −8.00000 11.1803 − 44.7818i 36.1938i 22.6274i 169.676 − 31.6228i
21.9 2.82843i −13.5441 −8.00000 −11.1803 − 38.3086i − 48.2066i − 22.6274i 102.443 − 31.6228i
21.10 2.82843i −10.3829 −8.00000 11.1803 − 29.3673i − 40.4137i − 22.6274i 26.8045 31.6228i
21.11 2.82843i −4.90915 −8.00000 −11.1803 − 13.8852i 52.2014i − 22.6274i −56.9002 − 31.6228i
21.12 2.82843i −0.353300 −8.00000 11.1803 − 0.999282i − 34.1317i − 22.6274i −80.8752 31.6228i
21.13 2.82843i 3.37558 −8.00000 11.1803 9.54757i 59.8276i − 22.6274i −69.6055 31.6228i
21.14 2.82843i 8.35959 −8.00000 −11.1803 23.6445i − 92.1688i − 22.6274i −11.1173 − 31.6228i
21.15 2.82843i 9.62155 −8.00000 −11.1803 27.2139i 37.2623i − 22.6274i 11.5742 − 31.6228i
21.16 2.82843i 15.8328 −8.00000 11.1803 44.7818i − 36.1938i − 22.6274i 169.676 31.6228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.16
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 110.5.d.a ✓ 16
3.b odd 2 1 990.5.b.a 16
4.b odd 2 1 880.5.j.a 16
5.b even 2 1 550.5.d.b 16
5.c odd 4 2 550.5.c.c 32
11.b odd 2 1 inner 110.5.d.a ✓ 16
33.d even 2 1 990.5.b.a 16
44.c even 2 1 880.5.j.a 16
55.d odd 2 1 550.5.d.b 16
55.e even 4 2 550.5.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.5.d.a ✓ 16 1.a even 1 1 trivial
110.5.d.a ✓ 16 11.b odd 2 1 inner
550.5.c.c 32 5.c odd 4 2
550.5.c.c 32 55.e even 4 2
550.5.d.b 16 5.b even 2 1
550.5.d.b 16 55.d odd 2 1
880.5.j.a 16 4.b odd 2 1
880.5.j.a 16 44.c even 2 1
990.5.b.a 16 3.b odd 2 1
990.5.b.a 16 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(110, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - 8 T^{7} + \cdots + 1048464)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 81\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 49\!\cdots\!24)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 95\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 50\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 96\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 35\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 24\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 19\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 39\!\cdots\!24)^{2} \) Copy content Toggle raw display
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