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Properties

Label 810.4.c.b
Level $810$
Weight $4$
Character orbit 810.c
Analytic conductor $47.792$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,4,Mod(649,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.c (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: \(47.7915471046\)
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: \( x^{10} - 4 x^{9} + 8 x^{8} + 326 x^{7} + 17389 x^{6} - 26726 x^{5} + 20930 x^{4} + 60664 x^{3} + \cdots + 5604552 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 4 q^{4} + ( - \beta_{5} + 2) q^{5} + \beta_{7} q^{7} + 4 \beta_1 q^{8} + (\beta_{4} - 2 \beta_1) q^{10} + (\beta_{3} - 3) q^{11} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 1) q^{13}+ \cdots + ( - 14 \beta_{9} + 2 \beta_{8} + \cdots + 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 40 q^{4} + 22 q^{5} - 4 q^{10} - 28 q^{11} - 4 q^{14} + 160 q^{16} - 42 q^{19} - 88 q^{20} + 96 q^{25} - 108 q^{26} + 216 q^{29} + 88 q^{31} - 64 q^{34} - 22 q^{35} + 16 q^{40} + 38 q^{41} + 112 q^{44}+ \cdots + 2472 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} + 8 x^{8} + 326 x^{7} + 17389 x^{6} - 26726 x^{5} + 20930 x^{4} + 60664 x^{3} + \cdots + 5604552 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3069306257 \nu^{9} + 9839161208 \nu^{8} - 14292291865 \nu^{7} - 1055859400495 \nu^{6} + \cdots + 10\!\cdots\!88 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 126984661 \nu^{9} + 3338428935 \nu^{8} - 9319640328 \nu^{7} - 29990085728 \nu^{6} + \cdots + 71\!\cdots\!68 ) / 128391834330000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36511750 \nu^{9} + 256633707 \nu^{8} - 1158090966 \nu^{7} - 10310201786 \nu^{6} + \cdots + 397138179818268 ) / 25678366866000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7693431431 \nu^{9} - 34071112675 \nu^{8} + 25567729558 \nu^{7} + 2584536858358 \nu^{6} + \cdots - 97\!\cdots\!88 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74929422752 \nu^{9} + 224040810959 \nu^{8} - 514785758458 \nu^{7} + \cdots + 20\!\cdots\!92 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2948768567 \nu^{9} - 7163257166 \nu^{8} - 7371870266 \nu^{7} + 1027047802894 \nu^{6} + \cdots + 47\!\cdots\!80 ) / 770351005980000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 188930110468 \nu^{9} - 614441663143 \nu^{8} + 1113969161918 \nu^{7} + 59856351456698 \nu^{6} + \cdots - 69\!\cdots\!56 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 98781710869 \nu^{9} - 406287078952 \nu^{8} + 673324404908 \nu^{7} + 33188487895928 \nu^{6} + \cdots - 76\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34320285080 \nu^{9} - 118550902779 \nu^{8} + 145374880822 \nu^{7} + 11460236117462 \nu^{6} + \cdots - 17\!\cdots\!16 ) / 26\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 10\beta_{9} + 4\beta_{8} - 6\beta_{6} + 40\beta_{5} - 20\beta_{4} - 10\beta_{3} + 5\beta_{2} - 30\beta _1 + 46 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - 4\beta_{8} + 6\beta_{7} - 6\beta_{6} + 23\beta_{5} - \beta_{4} - 186\beta _1 + 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1610 \beta_{9} - 1336 \beta_{8} - 240 \beta_{7} - 1506 \beta_{6} - 340 \beta_{5} - 2620 \beta_{4} + \cdots - 13894 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -237\beta_{8} + 618\beta_{6} - 330\beta_{5} - 855\beta_{4} + 35\beta_{3} - 265\beta_{2} - 22950 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 220990 \beta_{9} + 20624 \beta_{8} + 11400 \beta_{7} + 191514 \beta_{6} - 783760 \beta_{5} + \cdots - 2756374 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 37181 \beta_{9} + 101482 \beta_{8} - 133734 \beta_{7} + 149010 \beta_{6} - 517757 \beta_{5} + \cdots - 64301 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 30358790 \beta_{9} + 31821184 \beta_{8} - 1787040 \beta_{7} + 23227614 \beta_{6} - 17187140 \beta_{5} + \cdots + 553595986 ) / 120 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5834631 \beta_{8} - 11326734 \beta_{6} + 12011790 \beta_{5} + 17161365 \beta_{4} - 4382105 \beta_{3} + \cdots + 456964542 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4251951010 \beta_{9} - 577521056 \beta_{8} + 690973800 \beta_{7} - 5576633766 \beta_{6} + \cdots + 93344792506 ) / 120 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
2.81091 − 2.81091i
8.80143 − 8.80143i
0.785134 − 0.785134i
−7.54004 + 7.54004i
−2.85744 + 2.85744i
2.81091 + 2.81091i
8.80143 + 8.80143i
0.785134 + 0.785134i
−7.54004 − 7.54004i
−2.85744 − 2.85744i
− 2.00000i 0 −4.00000 −9.35710 − 6.11921i 0 15.3362i 8.00000i 0 −12.2384 + 18.7142i
649.2 − 2.00000i 0 −4.00000 −4.01358 + 10.4351i 0 − 14.0253i 8.00000i 0 20.8702 + 8.02717i
649.3 − 2.00000i 0 −4.00000 3.36692 − 10.6613i 0 − 27.8276i 8.00000i 0 −21.3227 − 6.73383i
649.4 − 2.00000i 0 −4.00000 9.82343 + 5.33856i 0 − 8.56259i 8.00000i 0 10.6771 − 19.6469i
649.5 − 2.00000i 0 −4.00000 11.1803 + 0.00687881i 0 34.0793i 8.00000i 0 0.0137576 − 22.3607i
649.6 2.00000i 0 −4.00000 −9.35710 + 6.11921i 0 − 15.3362i − 8.00000i 0 −12.2384 − 18.7142i
649.7 2.00000i 0 −4.00000 −4.01358 − 10.4351i 0 14.0253i − 8.00000i 0 20.8702 − 8.02717i
649.8 2.00000i 0 −4.00000 3.36692 + 10.6613i 0 27.8276i − 8.00000i 0 −21.3227 + 6.73383i
649.9 2.00000i 0 −4.00000 9.82343 − 5.33856i 0 8.56259i − 8.00000i 0 10.6771 + 19.6469i
649.10 2.00000i 0 −4.00000 11.1803 − 0.00687881i 0 − 34.0793i − 8.00000i 0 0.0137576 + 22.3607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.c.b yes 10
3.b odd 2 1 810.4.c.a ✓ 10
5.b even 2 1 inner 810.4.c.b yes 10
15.d odd 2 1 810.4.c.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.4.c.a ✓ 10 3.b odd 2 1
810.4.c.a ✓ 10 15.d odd 2 1
810.4.c.b yes 10 1.a even 1 1 trivial
810.4.c.b yes 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{10} + 2441T_{7}^{8} + 1955296T_{7}^{6} + 608633056T_{7}^{4} + 76656025856T_{7}^{2} + 3050751289600 \) Copy content Toggle raw display
\( T_{11}^{5} + 14T_{11}^{4} - 4475T_{11}^{3} - 28900T_{11}^{2} + 4412500T_{11} - 43750000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 3050751289600 \) Copy content Toggle raw display
$11$ \( (T^{5} + 14 T^{4} + \cdots - 43750000)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{5} + 21 T^{4} + \cdots - 47258100)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} - 108 T^{4} + \cdots - 7402995000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 44 T^{4} + \cdots - 2754440224)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 1170493111492)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 83\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 10387813550000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 150160521759320)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 73647711587952)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 139713892015104)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 78592909601762)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
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