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Properties

Label 110.5.e.a
Level $110$
Weight $5$
Character orbit 110.e
Analytic conductor $11.371$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,5,Mod(23,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.23");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 110.e (of order \(4\), degree \(2\), 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: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 595 x^{18} + 794 x^{17} + 135593 x^{16} + 104156 x^{15} - 14736226 x^{14} + \cdots + 68253675578125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{5}\cdot 11^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} - 2) q^{2} + (\beta_{6} + \beta_{3} - 1) q^{3} + 8 \beta_{3} q^{4} + ( - \beta_{4} - 3 \beta_{3}) q^{5} + (2 \beta_{7} - 2 \beta_{6} + 4) q^{6} + (\beta_{11} + \beta_{7} - 3 \beta_{3} - 3) q^{7}+ \cdots + (23 \beta_{19} + 12 \beta_{18} + \cdots + 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 40 q^{2} - 26 q^{3} + 8 q^{5} + 104 q^{6} - 48 q^{7} + 320 q^{8} - 132 q^{10} - 208 q^{12} + 20 q^{13} + 700 q^{15} - 1280 q^{16} + 1080 q^{17} - 376 q^{18} + 464 q^{20} - 1536 q^{21} - 1998 q^{23}+ \cdots + 16776 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 595 x^{18} + 794 x^{17} + 135593 x^{16} + 104156 x^{15} - 14736226 x^{14} + \cdots + 68253675578125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 76\!\cdots\!53 \nu^{19} + \cdots - 49\!\cdots\!75 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!55 \nu^{19} + \cdots - 35\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29\!\cdots\!59 \nu^{19} + \cdots - 10\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!45 \nu^{19} + \cdots + 36\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68\!\cdots\!88 \nu^{19} + \cdots - 43\!\cdots\!50 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 82\!\cdots\!56 \nu^{19} + \cdots - 93\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!42 \nu^{19} + \cdots + 44\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 67\!\cdots\!03 \nu^{19} + \cdots - 12\!\cdots\!25 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!22 \nu^{19} + \cdots - 43\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\!\cdots\!60 \nu^{19} + \cdots + 30\!\cdots\!50 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!73 \nu^{19} + \cdots + 71\!\cdots\!75 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25\!\cdots\!92 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 64\!\cdots\!69 \nu^{19} + \cdots + 11\!\cdots\!75 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 48\!\cdots\!56 \nu^{19} + \cdots - 19\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 56\!\cdots\!15 \nu^{19} + \cdots - 15\!\cdots\!75 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 91\!\cdots\!15 \nu^{19} + \cdots - 19\!\cdots\!75 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10\!\cdots\!54 \nu^{19} + \cdots - 37\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!11 \nu^{19} + \cdots + 33\!\cdots\!75 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 12\!\cdots\!30 \nu^{19} + \cdots - 45\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{18} - \beta_{17} + 4 \beta_{16} - 7 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \cdots + 27 ) / 110 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 8 \beta_{18} - 18 \beta_{17} - 8 \beta_{16} - \beta_{15} - 6 \beta_{14} - 18 \beta_{13} + \cdots + 6703 ) / 110 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 90 \beta_{19} + 525 \beta_{18} - 476 \beta_{17} + 494 \beta_{16} + 171 \beta_{15} - 108 \beta_{14} + \cdots + 28399 ) / 110 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 510 \beta_{19} - 1576 \beta_{18} - 4616 \beta_{17} - 454 \beta_{16} + 241 \beta_{15} + \cdots + 1071241 ) / 110 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20050 \beta_{19} + 95687 \beta_{18} - 114154 \beta_{17} + 90915 \beta_{16} + 59060 \beta_{15} + \cdots + 9341289 ) / 110 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 69930 \beta_{19} - 93887 \beta_{18} - 1205001 \beta_{17} + 235661 \beta_{16} + 324821 \beta_{15} + \cdots + 230280397 ) / 110 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4695915 \beta_{19} + 17565178 \beta_{18} - 25659125 \beta_{17} + 18380691 \beta_{16} + \cdots + 2691618956 ) / 110 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2782490 \beta_{19} + 25697658 \beta_{18} - 302762678 \beta_{17} + 116098112 \beta_{16} + \cdots + 54232336047 ) / 110 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1158594990 \beta_{19} + 3241817264 \beta_{18} - 5733292625 \beta_{17} + 3974755623 \beta_{16} + \cdots + 728992594972 ) / 110 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5767557590 \beta_{19} + 12125552459 \beta_{18} - 73145958459 \beta_{17} + 37856654925 \beta_{16} + \cdots + 13202713841216 ) / 110 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27083420965 \beta_{19} + 54199037061 \beta_{18} - 115983335119 \beta_{17} + 82394744493 \beta_{16} + \cdots + 17363158707413 ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1313305756730 \beta_{19} + 1644380766046 \beta_{18} - 8533915667975 \beta_{17} + 5395409386739 \beta_{16} + \cdots + 16\!\cdots\!79 ) / 55 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 79243560682000 \beta_{19} + 106476835092411 \beta_{18} - 280682055698121 \beta_{17} + \cdots + 49\!\cdots\!15 ) / 110 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 929988509666600 \beta_{19} + 691652313274360 \beta_{18} + \cdots + 80\!\cdots\!55 ) / 110 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 21\!\cdots\!40 \beta_{19} + \cdots + 12\!\cdots\!55 ) / 110 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 29\!\cdots\!30 \beta_{19} + \cdots + 20\!\cdots\!19 ) / 110 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 59\!\cdots\!50 \beta_{19} + \cdots + 31\!\cdots\!13 ) / 110 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 89\!\cdots\!10 \beta_{19} + \cdots + 49\!\cdots\!69 ) / 110 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 16\!\cdots\!75 \beta_{19} + \cdots + 77\!\cdots\!42 ) / 110 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(67\) \(101\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−2.71474 + 0.500000i
6.84791 + 0.500000i
−12.1344 + 0.500000i
3.44529 + 0.500000i
−5.83819 + 0.500000i
−2.08888 + 0.500000i
6.53821 + 0.500000i
16.0782 + 0.500000i
−0.956717 + 0.500000i
−7.17662 + 0.500000i
−2.71474 − 0.500000i
6.84791 − 0.500000i
−12.1344 − 0.500000i
3.44529 − 0.500000i
−5.83819 − 0.500000i
−2.08888 − 0.500000i
6.53821 − 0.500000i
16.0782 − 0.500000i
−0.956717 − 0.500000i
−7.17662 − 0.500000i
−2.00000 + 2.00000i −10.6476 − 10.6476i − 8.00000i 23.4587 + 8.64237i 42.5905 31.6644 − 31.6644i 16.0000 + 16.0000i 145.744i −64.2021 + 29.6326i
23.2 −2.00000 + 2.00000i −10.0850 − 10.0850i − 8.00000i −15.1413 + 19.8932i 40.3400 −50.3115 + 50.3115i 16.0000 + 16.0000i 122.415i −9.50371 − 70.0691i
23.3 −2.00000 + 2.00000i −6.46303 − 6.46303i − 8.00000i 2.31362 − 24.8927i 25.8521 16.9732 − 16.9732i 16.0000 + 16.0000i 2.54158i 45.1582 + 54.4127i
23.4 −2.00000 + 2.00000i −5.95135 − 5.95135i − 8.00000i −22.9098 + 10.0069i 23.8054 50.1600 − 50.1600i 16.0000 + 16.0000i − 10.1628i 25.8058 − 65.8336i
23.5 −2.00000 + 2.00000i −2.75325 − 2.75325i − 8.00000i −18.1773 − 17.1635i 11.0130 −46.1754 + 46.1754i 16.0000 + 16.0000i − 65.8392i 70.6816 − 2.02774i
23.6 −2.00000 + 2.00000i 1.50963 + 1.50963i − 8.00000i 22.0125 − 11.8512i −6.03852 −11.3467 + 11.3467i 16.0000 + 16.0000i − 76.4420i −20.3225 + 67.7274i
23.7 −2.00000 + 2.00000i 3.05105 + 3.05105i − 8.00000i 2.59098 + 24.8654i −12.2042 −36.3023 + 36.3023i 16.0000 + 16.0000i − 62.3822i −54.9127 − 44.5488i
23.8 −2.00000 + 2.00000i 3.29205 + 3.29205i − 8.00000i 10.5135 + 22.6819i −13.1682 57.4046 − 57.4046i 16.0000 + 16.0000i − 59.3248i −66.3907 − 24.3368i
23.9 −2.00000 + 2.00000i 4.65055 + 4.65055i − 8.00000i −24.7566 + 3.47983i −18.6022 −7.08897 + 7.08897i 16.0000 + 16.0000i − 37.7447i 42.5536 − 56.4729i
23.10 −2.00000 + 2.00000i 10.3970 + 10.3970i − 8.00000i 24.0960 − 6.66219i −41.5880 −28.9774 + 28.9774i 16.0000 + 16.0000i 135.195i −34.8675 + 61.5163i
67.1 −2.00000 − 2.00000i −10.6476 + 10.6476i 8.00000i 23.4587 − 8.64237i 42.5905 31.6644 + 31.6644i 16.0000 − 16.0000i − 145.744i −64.2021 − 29.6326i
67.2 −2.00000 − 2.00000i −10.0850 + 10.0850i 8.00000i −15.1413 − 19.8932i 40.3400 −50.3115 − 50.3115i 16.0000 − 16.0000i − 122.415i −9.50371 + 70.0691i
67.3 −2.00000 − 2.00000i −6.46303 + 6.46303i 8.00000i 2.31362 + 24.8927i 25.8521 16.9732 + 16.9732i 16.0000 − 16.0000i − 2.54158i 45.1582 − 54.4127i
67.4 −2.00000 − 2.00000i −5.95135 + 5.95135i 8.00000i −22.9098 − 10.0069i 23.8054 50.1600 + 50.1600i 16.0000 − 16.0000i 10.1628i 25.8058 + 65.8336i
67.5 −2.00000 − 2.00000i −2.75325 + 2.75325i 8.00000i −18.1773 + 17.1635i 11.0130 −46.1754 − 46.1754i 16.0000 − 16.0000i 65.8392i 70.6816 + 2.02774i
67.6 −2.00000 − 2.00000i 1.50963 − 1.50963i 8.00000i 22.0125 + 11.8512i −6.03852 −11.3467 − 11.3467i 16.0000 − 16.0000i 76.4420i −20.3225 − 67.7274i
67.7 −2.00000 − 2.00000i 3.05105 − 3.05105i 8.00000i 2.59098 − 24.8654i −12.2042 −36.3023 − 36.3023i 16.0000 − 16.0000i 62.3822i −54.9127 + 44.5488i
67.8 −2.00000 − 2.00000i 3.29205 − 3.29205i 8.00000i 10.5135 − 22.6819i −13.1682 57.4046 + 57.4046i 16.0000 − 16.0000i 59.3248i −66.3907 + 24.3368i
67.9 −2.00000 − 2.00000i 4.65055 − 4.65055i 8.00000i −24.7566 − 3.47983i −18.6022 −7.08897 − 7.08897i 16.0000 − 16.0000i 37.7447i 42.5536 + 56.4729i
67.10 −2.00000 − 2.00000i 10.3970 − 10.3970i 8.00000i 24.0960 + 6.66219i −41.5880 −28.9774 − 28.9774i 16.0000 − 16.0000i − 135.195i −34.8675 − 61.5163i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.5.e.a ✓ 20
5.c odd 4 1 inner 110.5.e.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.5.e.a ✓ 20 1.a even 1 1 trivial
110.5.e.a ✓ 20 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 26 T_{3}^{19} + 338 T_{3}^{18} + 1006 T_{3}^{17} + 46267 T_{3}^{16} + 1239772 T_{3}^{15} + \cdots + 71\!\cdots\!56 \) acting on \(S_{5}^{\mathrm{new}}(110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 8)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{2} - 1331)^{10} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 30\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
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