Properties

Label 105.10.a.a
Level $105$
Weight $10$
Character orbit 105.a
Self dual yes
Analytic conductor $54.079$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,10,Mod(1,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

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: yes
Analytic conductor: \(54.0787627972\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 988x^{2} - 844x + 192256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 10) q^{2} - 81 q^{3} + ( - \beta_{2} - 21 \beta_1 + 120) q^{4} + 625 q^{5} + ( - 81 \beta_1 + 810) q^{6} + 2401 q^{7} + (4 \beta_{3} + 37 \beta_{2} + \cdots - 7360) q^{8}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 10) q^{2} - 81 q^{3} + ( - \beta_{2} - 21 \beta_1 + 120) q^{4} + 625 q^{5} + ( - 81 \beta_1 + 810) q^{6} + 2401 q^{7} + (4 \beta_{3} + 37 \beta_{2} + \cdots - 7360) q^{8}+ \cdots + ( - 255879 \beta_{3} - 898857 \beta_{2} + \cdots - 53321247) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 41 q^{2} - 324 q^{3} + 501 q^{4} + 2500 q^{5} + 3321 q^{6} + 9604 q^{7} - 29367 q^{8} + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 41 q^{2} - 324 q^{3} + 501 q^{4} + 2500 q^{5} + 3321 q^{6} + 9604 q^{7} - 29367 q^{8} + 26244 q^{9} - 25625 q^{10} - 32854 q^{11} - 40581 q^{12} - 133882 q^{13} - 98441 q^{14} - 202500 q^{15} - 90479 q^{16} + 334344 q^{17} - 269001 q^{18} + 72046 q^{19} + 313125 q^{20} - 777924 q^{21} + 990556 q^{22} + 47460 q^{23} + 2378727 q^{24} + 1562500 q^{25} + 4203458 q^{26} - 2125764 q^{27} + 1202901 q^{28} + 7010312 q^{29} + 2075625 q^{30} - 3711690 q^{31} + 18115265 q^{32} + 2661174 q^{33} + 12757214 q^{34} + 6002500 q^{35} + 3287061 q^{36} + 5222716 q^{37} - 759964 q^{38} + 10844442 q^{39} - 18354375 q^{40} - 5689292 q^{41} + 7973721 q^{42} - 71286384 q^{43} + 6098004 q^{44} + 16402500 q^{45} + 13833560 q^{46} - 100832160 q^{47} + 7328799 q^{48} + 23059204 q^{49} - 16015625 q^{50} - 27081864 q^{51} - 7238698 q^{52} - 149250454 q^{53} + 21789081 q^{54} - 20533750 q^{55} - 70510167 q^{56} - 5835726 q^{57} - 191838758 q^{58} + 24779068 q^{59} - 25363125 q^{60} - 301336048 q^{61} - 107946280 q^{62} + 63011844 q^{63} - 71492607 q^{64} - 83676250 q^{65} - 80235036 q^{66} - 10978016 q^{67} - 527886134 q^{68} - 3844260 q^{69} - 61525625 q^{70} - 507837170 q^{71} - 192676887 q^{72} - 158796526 q^{73} - 252589374 q^{74} - 126562500 q^{75} - 604814036 q^{76} - 78882454 q^{77} - 340480098 q^{78} - 36676584 q^{79} - 56549375 q^{80} + 172186884 q^{81} - 1841487282 q^{82} + 201009048 q^{83} - 97434981 q^{84} + 208965000 q^{85} + 162200156 q^{86} - 567835272 q^{87} - 1446483548 q^{88} + 161323024 q^{89} - 168125625 q^{90} - 321450682 q^{91} + 563662840 q^{92} + 300646890 q^{93} - 918128840 q^{94} + 45028750 q^{95} - 1467336465 q^{96} - 1913030882 q^{97} - 236356841 q^{98} - 215555094 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 988x^{2} - 844x + 192256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 11\nu^{2} + 542\nu - 4240 ) / 168 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 11\nu^{2} + 1382\nu - 4408 ) / 84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 431\nu^{2} - 718\nu - 211468 ) / 84 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 2 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 3\beta_{2} - 10\beta _1 + 4940 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 22\beta_{3} + 575\beta_{2} - 2874\beta _1 + 13024 ) / 10 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−17.8249
28.4297
15.2587
−24.8635
−38.2301 −81.0000 949.541 625.000 3096.64 2401.00 −16727.3 6561.00 −23893.8
1.2 −27.3730 −81.0000 237.282 625.000 2217.21 2401.00 7519.86 6561.00 −17108.1
1.3 8.08732 −81.0000 −446.595 625.000 −655.073 2401.00 −7752.47 6561.00 5054.57
1.4 16.5158 −81.0000 −239.228 625.000 −1337.78 2401.00 −12407.1 6561.00 10322.4
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.10.a.a 4
3.b odd 2 1 315.10.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.10.a.a 4 1.a even 1 1 trivial
315.10.a.i 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 41T_{2}^{3} - 434T_{2}^{2} - 16984T_{2} + 139776 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 41 T^{3} + \cdots + 139776 \) Copy content Toggle raw display
$3$ \( (T + 81)^{4} \) Copy content Toggle raw display
$5$ \( (T - 625)^{4} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 55\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 96\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 79\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 33\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 84\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 45\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 43\!\cdots\!24 \) Copy content Toggle raw display
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