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Properties

Label 105.2.s.c
Level $105$
Weight $2$
Character orbit 105.s
Analytic conductor $0.838$
Analytic rank $0$
Dimension $8$
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] = [105,2,Mod(26,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.s (of order \(6\), 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: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + (\beta_{6} + \beta_{4} - \beta_1 + 1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{4} + \beta_{2} q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + (\beta_{6} + \beta_{4} - \beta_1 + 1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{4} + \beta_{2} q^{5} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 2) q^{6}+ \cdots + ( - 3 \beta_{7} - \beta_{6} + \cdots - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + q^{3} + 3 q^{4} + 4 q^{5} + 5 q^{6} + 2 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + q^{3} + 3 q^{4} + 4 q^{5} + 5 q^{6} + 2 q^{7} - 5 q^{9} - 3 q^{10} - 9 q^{12} - 12 q^{14} - q^{15} + q^{16} - 12 q^{17} - 19 q^{18} + 9 q^{19} + 6 q^{20} + 19 q^{21} - 40 q^{22} + 27 q^{23} + 16 q^{24} - 4 q^{25} - 6 q^{26} + 4 q^{27} + 3 q^{28} - 5 q^{30} - 21 q^{31} + 21 q^{32} + 2 q^{33} + q^{35} + 9 q^{36} + 7 q^{37} - 12 q^{38} - 3 q^{39} + 3 q^{40} - 30 q^{41} + 26 q^{42} + 16 q^{43} - 4 q^{45} - 7 q^{46} - 6 q^{47} - 25 q^{48} - 4 q^{49} - 6 q^{51} + 30 q^{52} + 24 q^{53} + 17 q^{54} - 21 q^{56} + 6 q^{57} - 13 q^{58} - 12 q^{59} - 18 q^{60} + 15 q^{61} + 24 q^{62} - 2 q^{63} + 38 q^{64} - 3 q^{65} + 22 q^{66} + 4 q^{67} - 13 q^{69} + 9 q^{70} - 14 q^{72} + 15 q^{73} + 54 q^{74} - 2 q^{75} - 36 q^{77} - 6 q^{78} - 29 q^{79} - q^{80} - 41 q^{81} + 27 q^{82} + 30 q^{83} - 3 q^{84} - 24 q^{85} + 9 q^{86} + 32 q^{87} - 2 q^{88} - 3 q^{89} + 7 q^{90} - 3 q^{91} - 9 q^{93} - 24 q^{94} + 9 q^{95} - 3 q^{96} - 39 q^{98} - 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 10\nu^{2} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 2\nu^{3} + 22\nu^{2} + 10\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 9\nu^{4} - 2\nu^{3} + 22\nu^{2} - 10\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 10\nu^{5} - 9\nu^{4} - 29\nu^{3} - 20\nu^{2} - 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 29\nu^{3} - 20\nu^{2} + 20\nu - 6 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} - 6\beta_{6} - 5\beta_{5} - 5\beta_{4} - 2\beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} - 7\beta_{4} + 4\beta_{2} + 25\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32\beta_{7} + 32\beta_{6} + 25\beta_{5} + 25\beta_{4} + 18\beta_{3} + 9\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 41\beta_{5} + 41\beta_{4} - 40\beta_{2} - 125\beta _1 + 20 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 2.33086i
− 1.07834i
− 0.385731i
2.06288i
2.33086i
1.07834i
0.385731i
− 2.06288i
−2.01859 − 1.16543i −1.21646 + 1.23297i 1.71646 + 2.97300i 0.500000 − 0.866025i 3.89248 − 1.07116i 1.11699 − 2.39840i − 3.33995i −0.0404447 − 2.99973i −2.01859 + 1.16543i
26.2 −0.933868 − 0.539169i 0.918594 − 1.46840i −0.418594 − 0.725026i 0.500000 − 0.866025i −1.64956 + 0.876010i −2.47720 + 0.929227i 3.05945i −1.31237 − 2.69772i −0.933868 + 0.539169i
26.3 −0.334053 − 0.192865i 1.42561 + 0.983691i −0.925606 − 1.60320i 0.500000 − 0.866025i −0.286507 − 0.603555i 2.36975 − 1.17656i 1.48553i 1.06470 + 2.80471i −0.334053 + 0.192865i
26.4 1.78651 + 1.03144i −0.627739 − 1.61429i 1.12774 + 1.95330i 0.500000 − 0.866025i 0.543588 − 3.53142i −0.00953166 + 2.64573i 0.527019i −2.21189 + 2.02671i 1.78651 − 1.03144i
101.1 −2.01859 + 1.16543i −1.21646 − 1.23297i 1.71646 − 2.97300i 0.500000 + 0.866025i 3.89248 + 1.07116i 1.11699 + 2.39840i 3.33995i −0.0404447 + 2.99973i −2.01859 − 1.16543i
101.2 −0.933868 + 0.539169i 0.918594 + 1.46840i −0.418594 + 0.725026i 0.500000 + 0.866025i −1.64956 − 0.876010i −2.47720 − 0.929227i − 3.05945i −1.31237 + 2.69772i −0.933868 − 0.539169i
101.3 −0.334053 + 0.192865i 1.42561 − 0.983691i −0.925606 + 1.60320i 0.500000 + 0.866025i −0.286507 + 0.603555i 2.36975 + 1.17656i − 1.48553i 1.06470 − 2.80471i −0.334053 − 0.192865i
101.4 1.78651 − 1.03144i −0.627739 + 1.61429i 1.12774 − 1.95330i 0.500000 + 0.866025i 0.543588 + 3.53142i −0.00953166 − 2.64573i − 0.527019i −2.21189 − 2.02671i 1.78651 + 1.03144i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.s.c ✓ 8
3.b odd 2 1 105.2.s.d yes 8
5.b even 2 1 525.2.t.g 8
5.c odd 4 2 525.2.q.f 16
7.b odd 2 1 735.2.s.k 8
7.c even 3 1 735.2.b.d 8
7.c even 3 1 735.2.s.l 8
7.d odd 6 1 105.2.s.d yes 8
7.d odd 6 1 735.2.b.c 8
15.d odd 2 1 525.2.t.f 8
15.e even 4 2 525.2.q.e 16
21.c even 2 1 735.2.s.l 8
21.g even 6 1 inner 105.2.s.c ✓ 8
21.g even 6 1 735.2.b.d 8
21.h odd 6 1 735.2.b.c 8
21.h odd 6 1 735.2.s.k 8
35.i odd 6 1 525.2.t.f 8
35.k even 12 2 525.2.q.e 16
105.p even 6 1 525.2.t.g 8
105.w odd 12 2 525.2.q.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c ✓ 8 1.a even 1 1 trivial
105.2.s.c ✓ 8 21.g even 6 1 inner
105.2.s.d yes 8 3.b odd 2 1
105.2.s.d yes 8 7.d odd 6 1
525.2.q.e 16 15.e even 4 2
525.2.q.e 16 35.k even 12 2
525.2.q.f 16 5.c odd 4 2
525.2.q.f 16 105.w odd 12 2
525.2.t.f 8 15.d odd 2 1
525.2.t.f 8 35.i odd 6 1
525.2.t.g 8 5.b even 2 1
525.2.t.g 8 105.p even 6 1
735.2.b.c 8 7.d odd 6 1
735.2.b.c 8 21.h odd 6 1
735.2.b.d 8 7.c even 3 1
735.2.b.d 8 21.g even 6 1
735.2.s.k 8 7.b odd 2 1
735.2.s.k 8 21.h odd 6 1
735.2.s.l 8 7.c even 3 1
735.2.s.l 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{7} - T_{2}^{6} - 12T_{2}^{5} + 6T_{2}^{4} + 48T_{2}^{3} + 56T_{2}^{2} + 24T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 28 T^{6} + \cdots + 21904 \) Copy content Toggle raw display
$13$ \( T^{8} + 21 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( T^{8} - 27 T^{7} + \cdots + 454276 \) Copy content Toggle raw display
$29$ \( T^{8} + 179 T^{6} + \cdots + 879844 \) Copy content Toggle raw display
$31$ \( T^{8} + 21 T^{7} + \cdots + 695556 \) Copy content Toggle raw display
$37$ \( T^{8} - 7 T^{7} + \cdots + 15376 \) Copy content Toggle raw display
$41$ \( (T^{4} + 15 T^{3} + \cdots - 378)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 6 T^{7} + \cdots + 147456 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + \cdots + 4194304 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 9216 \) Copy content Toggle raw display
$61$ \( T^{8} - 15 T^{7} + \cdots + 76176 \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$71$ \( T^{8} + 104 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$73$ \( T^{8} - 15 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$79$ \( T^{8} + 29 T^{7} + \cdots + 12166144 \) Copy content Toggle raw display
$83$ \( (T^{4} - 15 T^{3} + \cdots - 42)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 3 T^{7} + \cdots + 2643876 \) Copy content Toggle raw display
$97$ \( T^{8} + 408 T^{6} + \cdots + 49449024 \) Copy content Toggle raw display
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