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Properties

Label 125.6.a.c
Level $125$
Weight $6$
Character orbit 125.a
Self dual yes
Analytic conductor $20.048$
Analytic rank $0$
Dimension $10$
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] = [125,6,Mod(1,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 125.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: \(20.0479774766\)
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: \( x^{10} - x^{9} - 228 x^{8} + 141 x^{7} + 17708 x^{6} - 903 x^{5} - 532436 x^{4} - 421653 x^{3} + \cdots + 2593196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} + 5) q^{3} + (\beta_{2} - 2 \beta_1 + 15) q^{4} + (\beta_{8} + 2 \beta_{7} + \beta_{6} + \cdots + 11) q^{6}+ \cdots + ( - \beta_{9} + 9 \beta_{7} + \cdots + 46) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{7} + 5) q^{3} + (\beta_{2} - 2 \beta_1 + 15) q^{4} + (\beta_{8} + 2 \beta_{7} + \beta_{6} + \cdots + 11) q^{6}+ \cdots + (47 \beta_{9} + 512 \beta_{8} + \cdots + 2137) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9 q^{2} + 47 q^{3} + 145 q^{4} + 95 q^{6} + 394 q^{7} + 540 q^{8} + 455 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 9 q^{2} + 47 q^{3} + 145 q^{4} + 95 q^{6} + 394 q^{7} + 540 q^{8} + 455 q^{9} + 270 q^{11} + 1384 q^{12} + 1547 q^{13} + 1240 q^{14} + 785 q^{16} + 2284 q^{17} + 4277 q^{18} + 4175 q^{19} + 2795 q^{21} + 7408 q^{22} + 3802 q^{23} + 2525 q^{24} - 1880 q^{26} + 13640 q^{27} + 22288 q^{28} - 9300 q^{29} + 2395 q^{31} + 38589 q^{32} + 32289 q^{33} - 26085 q^{34} - 24340 q^{36} + 42884 q^{37} + 62540 q^{38} - 10415 q^{39} + 7020 q^{41} + 75773 q^{42} + 72082 q^{43} + 24040 q^{44} + 6395 q^{46} + 31269 q^{47} + 44197 q^{48} + 55070 q^{49} + 32695 q^{51} + 55849 q^{52} + 1757 q^{53} - 14075 q^{54} - 5325 q^{56} + 29585 q^{57} - 9125 q^{58} - 50725 q^{59} - 71855 q^{61} - 148162 q^{62} + 40087 q^{63} + 26220 q^{64} - 17935 q^{66} + 121409 q^{67} - 190427 q^{68} + 2160 q^{69} + 55770 q^{71} - 283545 q^{72} + 56297 q^{73} + 66765 q^{74} + 166225 q^{76} - 72922 q^{77} - 405046 q^{78} - 100550 q^{79} + 33210 q^{81} - 192162 q^{82} - 51408 q^{83} + 49340 q^{84} - 297305 q^{86} - 91215 q^{87} - 93795 q^{88} + 84850 q^{89} - 284055 q^{91} - 423451 q^{92} - 149946 q^{93} - 304685 q^{94} + 523220 q^{96} + 124629 q^{97} - 658457 q^{98} + 13785 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 228 x^{8} + 141 x^{7} + 17708 x^{6} - 903 x^{5} - 532436 x^{4} - 421653 x^{3} + \cdots + 2593196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6244 \nu^{9} + 55278 \nu^{8} + 418129 \nu^{7} - 29805808 \nu^{6} + 80211621 \nu^{5} + \cdots + 54424918116 ) / 14271437120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26491 \nu^{9} + 20247 \nu^{8} + 6095226 \nu^{7} - 3317102 \nu^{6} - 498908436 \nu^{5} + \cdots - 149913601876 ) / 14271437120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5670523 \nu^{9} + 156147984 \nu^{8} - 1307725268 \nu^{7} - 33312881189 \nu^{6} + \cdots + 288566883261468 ) / 1626943831680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11096934 \nu^{9} + 22101068 \nu^{8} + 2873235639 \nu^{7} - 4161842488 \nu^{6} + \cdots - 36512057011204 ) / 813471915840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49350869 \nu^{9} - 193840358 \nu^{8} - 10919584114 \nu^{7} + 35764124583 \nu^{6} + \cdots + 122425053381724 ) / 3253887663360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16685801 \nu^{9} - 27795202 \nu^{8} - 3498662371 \nu^{7} + 5753497657 \nu^{6} + \cdots + 12237002942816 ) / 406735957920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 86259489 \nu^{9} - 1387822 \nu^{8} + 19011945444 \nu^{7} - 288207643 \nu^{6} + \cdots - 245685823657444 ) / 1626943831680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} - 3\beta_{4} + \beta_{3} - \beta_{2} + 73\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{8} + 14\beta_{7} + 3\beta_{6} + 5\beta_{5} - 5\beta_{4} - 16\beta_{3} + 98\beta_{2} - 26\beta _1 + 3388 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 104 \beta_{9} - 117 \beta_{8} - 10 \beta_{7} + 135 \beta_{6} - 119 \beta_{5} - 937 \beta_{4} + \cdots + 1392 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 29 \beta_{9} - 630 \beta_{8} + 1542 \beta_{7} + 316 \beta_{6} + 540 \beta_{5} - 1200 \beta_{4} + \cdots + 275966 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9318 \beta_{9} - 11192 \beta_{8} - 1212 \beta_{7} + 16264 \beta_{6} - 11792 \beta_{5} - 134984 \beta_{4} + \cdots + 93718 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9096 \beta_{9} - 64196 \beta_{8} + 124832 \beta_{7} + 26400 \beta_{6} + 49592 \beta_{5} + \cdots + 23408022 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 827665 \beta_{9} - 1000529 \beta_{8} - 133176 \beta_{7} + 1831101 \beta_{6} - 1121141 \beta_{5} + \cdots + 6869360 \) Copy content Toggle raw display

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

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
9.66805
8.68046
7.39170
4.41229
−0.413530
−1.47959
−3.32656
−5.47882
−8.80602
−9.64798
−8.66805 4.28370 43.1350 0 −37.1313 249.186 −96.5187 −224.650 0
1.2 −7.68046 21.5600 26.9895 0 −165.591 −187.485 38.4832 221.834 0
1.3 −6.39170 −9.87841 8.85381 0 63.1398 164.268 147.943 −145.417 0
1.4 −3.41229 −8.23734 −20.3563 0 28.1082 −112.651 178.655 −175.146 0
1.5 1.41353 20.4822 −30.0019 0 28.9522 −97.0073 −87.6416 176.519 0
1.6 2.47959 −10.7120 −25.8516 0 −26.5613 −46.0617 −143.448 −128.254 0
1.7 4.32656 29.8593 −13.2809 0 129.188 203.263 −195.910 648.576 0
1.8 6.47882 −22.3059 9.97505 0 −144.516 48.4610 −142.696 254.555 0
1.9 9.80602 16.9790 64.1580 0 166.497 176.093 315.342 45.2878 0
1.10 10.6480 4.96946 81.3794 0 52.9147 −4.06644 525.791 −218.305 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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 125.6.a.c yes 10
5.b even 2 1 125.6.a.b ✓ 10
5.c odd 4 2 125.6.b.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.6.a.b ✓ 10 5.b even 2 1
125.6.a.c yes 10 1.a even 1 1 trivial
125.6.b.c 20 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 9 T_{2}^{9} - 192 T_{2}^{8} + 1599 T_{2}^{7} + 12437 T_{2}^{6} - 95664 T_{2}^{5} + \cdots + 14895424 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(125))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 9 T^{9} + \cdots + 14895424 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 92663327376 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 27\!\cdots\!51 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 14\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 95\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 21\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 13\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 71\!\cdots\!51 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 22\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 18\!\cdots\!99 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 49\!\cdots\!51 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 73\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 22\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 85\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 95\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 31\!\cdots\!76 \) Copy content Toggle raw display
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