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Properties

Label 105.3.n.a
Level $105$
Weight $3$
Character orbit 105.n
Analytic conductor $2.861$
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] = [105,3,Mod(31,105)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.n (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: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.523596960000.16
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{4}+ \cdots - 3 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{4}+ \cdots + (3 \beta_{7} - 6 \beta_{5} - 3 \beta_{4} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 12 q^{3} - 6 q^{4} - 16 q^{7} - 32 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 12 q^{3} - 6 q^{4} - 16 q^{7} - 32 q^{8} + 12 q^{9} + 20 q^{11} + 18 q^{12} - 16 q^{14} - 2 q^{16} - 18 q^{17} - 6 q^{18} + 48 q^{21} - 16 q^{22} + 62 q^{23} + 48 q^{24} + 20 q^{25} + 120 q^{26} - 120 q^{28} - 100 q^{29} - 126 q^{31} + 36 q^{32} - 60 q^{33} - 36 q^{36} - 80 q^{37} + 114 q^{38} - 12 q^{39} + 90 q^{40} + 90 q^{42} + 352 q^{43} - 18 q^{44} - 82 q^{46} - 72 q^{47} + 38 q^{49} + 20 q^{50} + 18 q^{51} - 48 q^{52} - 76 q^{53} + 18 q^{54} + 196 q^{56} - 40 q^{58} - 54 q^{59} - 60 q^{60} - 396 q^{61} - 96 q^{63} - 4 q^{64} - 60 q^{65} + 24 q^{66} + 184 q^{67} - 312 q^{68} + 164 q^{71} - 48 q^{72} + 348 q^{73} - 140 q^{74} - 60 q^{75} + 152 q^{77} - 240 q^{78} - 206 q^{79} - 36 q^{81} + 204 q^{82} + 132 q^{84} - 60 q^{85} + 178 q^{86} + 150 q^{87} + 124 q^{88} + 282 q^{89} - 114 q^{91} - 288 q^{92} + 126 q^{93} + 30 q^{94} - 120 q^{95} - 108 q^{96} - 592 q^{98} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 20\nu^{6} - 51\nu^{5} + 304\nu^{4} - 193\nu^{3} + 1752\nu^{2} - 2510\nu + 2630 ) / 630 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -87\nu^{7} + 24\nu^{6} - 841\nu^{5} - 1276\nu^{4} - 10117\nu^{3} - 4640\nu^{2} - 2900\nu - 13700 ) / 21630 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 137\nu^{7} - 361\nu^{6} + 1805\nu^{5} - 1115\nu^{4} + 11191\nu^{3} - 16967\nu^{2} + 21390\nu - 10830 ) / 21630 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 472\nu^{7} - 1249\nu^{6} + 6966\nu^{5} - 8699\nu^{4} + 48092\nu^{3} - 74565\nu^{2} + 78220\nu - 131200 ) / 64890 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 661\nu^{7} - 2047\nu^{6} + 8793\nu^{5} - 5927\nu^{4} + 44711\nu^{3} - 64485\nu^{2} + 84520\nu + 126050 ) / 64890 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -977\nu^{7} + 1985\nu^{6} - 15693\nu^{5} + 4657\nu^{4} - 114889\nu^{3} + 25521\nu^{2} - 381050\nu - 55810 ) / 64890 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} + \beta_{5} - 5\beta_{4} - \beta_{3} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} - 9\beta_{3} - 2\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 11\beta_{6} - 24\beta_{5} + 41\beta_{4} - 2\beta_{2} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -26\beta_{7} - 42\beta_{6} - 8\beta_{5} + 72\beta_{4} + 95\beta_{3} + 13\beta_{2} - 95\beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{7} - 121\beta_{6} + 189\beta_{5} + 34\beta_{4} + 243\beta_{3} + 68\beta_{2} + 475 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 155\beta_{7} + 311\beta_{6} + 777\beta_{5} - 905\beta_{4} + 155\beta_{2} + 1081\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.26021 − 2.18275i
−0.336732 − 0.583237i
0.836732 + 1.44926i
1.76021 + 3.04878i
−1.26021 + 2.18275i
−0.336732 + 0.583237i
0.836732 − 1.44926i
1.76021 − 3.04878i
−1.26021 − 2.18275i −1.50000 − 0.866025i −1.17628 + 2.03737i −1.93649 + 1.11803i 4.36551i −6.18050 + 3.28656i −4.15226 1.50000 + 2.59808i 4.88079 + 2.81792i
31.2 −0.336732 − 0.583237i −1.50000 − 0.866025i 1.77322 − 3.07131i 1.93649 − 1.11803i 1.16647i −6.82455 − 1.55742i −5.08226 1.50000 + 2.59808i −1.30416 − 0.752955i
31.3 0.836732 + 1.44926i −1.50000 − 0.866025i 0.599760 − 1.03881i 1.93649 − 1.11803i − 2.89852i 4.76104 + 5.13152i 8.70121 1.50000 + 2.59808i 3.24065 + 1.87099i
31.4 1.76021 + 3.04878i −1.50000 − 0.866025i −4.19671 + 7.26891i −1.93649 + 1.11803i − 6.09756i 0.244004 + 6.99575i −15.4667 1.50000 + 2.59808i −6.81728 − 3.93596i
61.1 −1.26021 + 2.18275i −1.50000 + 0.866025i −1.17628 − 2.03737i −1.93649 − 1.11803i − 4.36551i −6.18050 − 3.28656i −4.15226 1.50000 − 2.59808i 4.88079 − 2.81792i
61.2 −0.336732 + 0.583237i −1.50000 + 0.866025i 1.77322 + 3.07131i 1.93649 + 1.11803i − 1.16647i −6.82455 + 1.55742i −5.08226 1.50000 − 2.59808i −1.30416 + 0.752955i
61.3 0.836732 − 1.44926i −1.50000 + 0.866025i 0.599760 + 1.03881i 1.93649 + 1.11803i 2.89852i 4.76104 − 5.13152i 8.70121 1.50000 − 2.59808i 3.24065 − 1.87099i
61.4 1.76021 − 3.04878i −1.50000 + 0.866025i −4.19671 − 7.26891i −1.93649 − 1.11803i 6.09756i 0.244004 − 6.99575i −15.4667 1.50000 − 2.59808i −6.81728 + 3.93596i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.n.a ✓ 8
3.b odd 2 1 315.3.w.a 8
5.b even 2 1 525.3.o.l 8
5.c odd 4 2 525.3.s.h 16
7.c even 3 1 735.3.h.a 8
7.d odd 6 1 inner 105.3.n.a ✓ 8
7.d odd 6 1 735.3.h.a 8
21.g even 6 1 315.3.w.a 8
35.i odd 6 1 525.3.o.l 8
35.k even 12 2 525.3.s.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a ✓ 8 1.a even 1 1 trivial
105.3.n.a ✓ 8 7.d odd 6 1 inner
315.3.w.a 8 3.b odd 2 1
315.3.w.a 8 21.g even 6 1
525.3.o.l 8 5.b even 2 1
525.3.o.l 8 35.i odd 6 1
525.3.s.h 16 5.c odd 4 2
525.3.s.h 16 35.k even 12 2
735.3.h.a 8 7.c even 3 1
735.3.h.a 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + 13T_{2}^{6} - 2T_{2}^{5} + 91T_{2}^{4} - 50T_{2}^{3} + 190T_{2}^{2} + 100T_{2} + 100 \) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 16 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1230957225 \) Copy content Toggle raw display
$17$ \( T^{8} + 18 T^{7} + \cdots + 138297600 \) Copy content Toggle raw display
$19$ \( T^{8} - 846 T^{6} + \cdots + 9054081 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 7138560100 \) Copy content Toggle raw display
$29$ \( (T^{4} + 50 T^{3} + \cdots - 1825400)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 385089749136 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 5596891350625 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13887679716 \) Copy content Toggle raw display
$43$ \( (T^{4} - 176 T^{3} + \cdots + 762376)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 72 T^{7} + \cdots + 26010000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 98219560000 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2582886122496 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 84471609600 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 273278017600 \) Copy content Toggle raw display
$71$ \( (T^{4} - 82 T^{3} + \cdots + 22760224)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 4446784387600 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 111959592561216 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 580473805464576 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 2211287961600 \) Copy content Toggle raw display
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