↑

Properties

Label 125.6.b.b
Level $125$
Weight $6$
Character orbit 125.b
Analytic conductor $20.048$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [125,6,Mod(124,125)]
 
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([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.124");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 125.b (of order \(2\), degree \(1\), not 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: \(20.0479774766\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 198x^{10} + 15334x^{8} + 583824x^{6} + 11056294x^{4} + 87101298x^{2} + 87447271 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 5^{12} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{6} q^{3} + ( - \beta_{2} + \beta_1 - 26) q^{4} + (\beta_{4} + 10 \beta_1 + 8) q^{6} + ( - \beta_{10} + \beta_{8} - \beta_{7}) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 26 \beta_{7}) q^{8}+ \cdots + ( - \beta_{5} + \beta_{3} - 5 \beta_{2} + \cdots - 163) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{6} q^{3} + ( - \beta_{2} + \beta_1 - 26) q^{4} + (\beta_{4} + 10 \beta_1 + 8) q^{6} + ( - \beta_{10} + \beta_{8} - \beta_{7}) q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - 26 \beta_{7}) q^{8}+ \cdots + ( - 123 \beta_{5} + 253 \beta_{4} + \cdots + 10748) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 304 q^{4} + 154 q^{6} - 1926 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 304 q^{4} + 154 q^{6} - 1926 q^{9} + 1104 q^{11} + 522 q^{14} + 8212 q^{16} - 5940 q^{19} + 764 q^{21} - 4950 q^{24} + 6464 q^{26} - 21290 q^{29} + 8864 q^{31} - 79268 q^{34} + 150462 q^{36} - 53812 q^{39} + 26114 q^{41} - 101168 q^{44} + 75414 q^{46} - 17794 q^{49} + 78964 q^{51} + 2260 q^{54} - 129690 q^{56} - 59640 q^{59} + 108874 q^{61} + 45456 q^{64} - 327132 q^{66} + 117448 q^{69} + 36324 q^{71} + 538692 q^{74} - 14180 q^{76} - 363180 q^{79} + 40132 q^{81} + 655592 q^{84} - 735726 q^{86} + 245530 q^{89} + 401684 q^{91} - 472458 q^{94} - 997526 q^{96} + 220908 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 198x^{10} + 15334x^{8} + 583824x^{6} + 11056294x^{4} + 87101298x^{2} + 87447271 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 843485 \nu^{10} - 144093485 \nu^{8} - 9009961725 \nu^{6} - 246447816085 \nu^{4} + \cdots - 2738246403393 ) / 9677174074 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 302462347 \nu^{10} + 50170967224 \nu^{8} + 3030320965377 \nu^{6} + 79746372093121 \nu^{4} + \cdots + 860444212371573 ) / 561276096292 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 991065876 \nu^{10} + 164863207037 \nu^{8} + 9972437089671 \nu^{6} + 262087284462118 \nu^{4} + \cdots + 26\!\cdots\!41 ) / 561276096292 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2085135337 \nu^{10} + 351408399109 \nu^{8} + 21629931403952 \nu^{6} + 581340252491581 \nu^{4} + \cdots + 60\!\cdots\!78 ) / 561276096292 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2354372045 \nu^{10} + 396995923805 \nu^{8} + 24439654308230 \nu^{6} + 656859667865985 \nu^{4} + \cdots + 70\!\cdots\!24 ) / 561276096292 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7252937747 \nu^{11} + 1229896134524 \nu^{9} + 76047292332014 \nu^{7} + \cdots + 16\!\cdots\!36 \nu ) / 213565554639106 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14525436156 \nu^{11} + 2457866740933 \nu^{9} + 151977965759883 \nu^{7} + \cdots + 48\!\cdots\!29 \nu ) / 427131109278212 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29909298078 \nu^{11} - 4716175984177 \nu^{9} - 264991940200307 \nu^{7} + \cdots - 27\!\cdots\!25 \nu ) / 427131109278212 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39568055781 \nu^{11} - 6608855079479 \nu^{9} - 402151599440724 \nu^{7} + \cdots - 12\!\cdots\!10 \nu ) / 427131109278212 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 98147701217 \nu^{11} - 16565642339702 \nu^{9} + \cdots - 27\!\cdots\!79 \nu ) / 213565554639106 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 306181073726 \nu^{11} - 51397694217665 \nu^{9} + \cdots - 84\!\cdots\!91 \nu ) / 427131109278212 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - 12\beta_{9} + 2\beta_{8} - 25\beta_{7} + 4\beta_{6} ) / 100 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} - 5\beta_{4} + 3\beta_{3} + 21\beta_{2} - 215\beta _1 - 3190 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + 232\beta_{9} - 58\beta_{8} + 655\beta_{7} - 108\beta_{6} ) / 50 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 417\beta_{5} + 153\beta_{4} - 288\beta_{3} - 1084\beta_{2} + 14053\beta _1 + 135090 ) / 100 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1834\beta_{11} + 2216\beta_{10} - 20708\beta_{9} + 6667\beta_{8} - 63015\beta_{7} + 11637\beta_{6} ) / 100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7739\beta_{5} - 737\beta_{4} + 4494\beta_{3} + 15910\beta_{2} - 214447\beta _1 - 1579365 ) / 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 161037 \beta_{11} - 221401 \beta_{10} + 1016748 \beta_{9} - 382036 \beta_{8} + 3125885 \beta_{7} - 741358 \beta_{6} ) / 100 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2067398\beta_{5} - 78593\beta_{4} - 991867\beta_{3} - 4024261\beta_{2} + 51170177\beta _1 + 319598170 ) / 100 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5586217 \beta_{11} + 8373643 \beta_{10} - 26796184 \beta_{9} + 11031136 \beta_{8} + \cdots + 24945266 \beta_{6} ) / 50 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 132108199 \beta_{5} + 15512035 \beta_{4} + 52393018 \beta_{3} + 260142186 \beta_{2} + \cdots - 17149422330 ) / 100 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 715056092 \beta_{11} - 1146582466 \beta_{10} + 2964551068 \beta_{9} - 1287595471 \beta_{8} + \cdots - 3355557613 \beta_{6} ) / 100 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
− 1.07997i
6.77553i
− 5.43493i
5.27990i
7.80974i
5.70247i
− 5.70247i
− 7.80974i
− 5.27990i
5.43493i
− 6.77553i
1.07997i
− 10.5692i − 27.1829i −79.7074 0 −287.301 17.8847i 504.228i −495.913 0
124.2 − 10.1950i 26.6801i −71.9380 0 272.004 168.897i 407.167i −468.828 0
124.3 − 8.52240i 20.5341i −40.6314 0 175.000 − 83.1243i 73.5599i −178.650 0
124.4 − 7.00425i − 3.77591i −17.0595 0 −26.4474 − 199.649i − 104.647i 228.743 0
124.5 − 1.83032i − 20.8887i 28.6499 0 −38.2331 88.1835i − 111.009i −193.339 0
124.6 − 1.82035i − 9.90017i 28.6863 0 −18.0218 162.318i − 110.471i 144.987 0
124.7 1.82035i 9.90017i 28.6863 0 −18.0218 − 162.318i 110.471i 144.987 0
124.8 1.83032i 20.8887i 28.6499 0 −38.2331 − 88.1835i 111.009i −193.339 0
124.9 7.00425i 3.77591i −17.0595 0 −26.4474 199.649i 104.647i 228.743 0
124.10 8.52240i − 20.5341i −40.6314 0 175.000 83.1243i − 73.5599i −178.650 0
124.11 10.1950i − 26.6801i −71.9380 0 272.004 − 168.897i − 407.167i −468.828 0
124.12 10.5692i 27.1829i −79.7074 0 −287.301 − 17.8847i − 504.228i −495.913 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.12
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 125.6.b.b 12
5.b even 2 1 inner 125.6.b.b 12
5.c odd 4 2 125.6.a.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.6.a.d ✓ 12 5.c odd 4 2
125.6.b.b 12 1.a even 1 1 trivial
125.6.b.b 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 344T_{2}^{10} + 43675T_{2}^{8} + 2461040T_{2}^{6} + 56367200T_{2}^{4} + 299906304T_{2}^{2} + 459271936 \) acting on \(S_{6}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 459271936 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 135229754530431 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 51\!\cdots\!31 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 151826210798656)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 84\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 30\!\cdots\!31 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 32\!\cdots\!75)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 74\!\cdots\!71)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 57\!\cdots\!91 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 68\!\cdots\!91 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 17\!\cdots\!89)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 23\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!91 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 42\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
show more
show less