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Properties

Label 125.4.b.c
Level $125$
Weight $4$
Character orbit 125.b
Analytic conductor $7.375$
Analytic rank $0$
Dimension $12$
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] = [125,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.124");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(7.37523875072\)
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: \( x^{12} + 59x^{10} + 1261x^{8} + 11844x^{6} + 45376x^{4} + 43840x^{2} + 6400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{5} q^{3} + ( - \beta_{2} - 6) q^{4} + (\beta_{7} + \beta_{3} - \beta_{2} - 3) q^{6} + (\beta_{9} + \beta_{5}) q^{7} + (\beta_{11} - \beta_{9} + \cdots - 4 \beta_{4}) q^{8}+ \cdots + ( - \beta_{2} - \beta_1 - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{5} q^{3} + ( - \beta_{2} - 6) q^{4} + (\beta_{7} + \beta_{3} - \beta_{2} - 3) q^{6} + (\beta_{9} + \beta_{5}) q^{7} + (\beta_{11} - \beta_{9} + \cdots - 4 \beta_{4}) q^{8}+ \cdots + (6 \beta_{8} - 13 \beta_{7} + \cdots + 418) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 66 q^{4} - 26 q^{6} - 184 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 66 q^{4} - 26 q^{6} - 184 q^{9} + 54 q^{11} + 28 q^{14} + 82 q^{16} + 210 q^{19} - 296 q^{21} + 950 q^{24} - 316 q^{26} + 60 q^{29} - 336 q^{31} - 502 q^{34} + 2392 q^{36} - 68 q^{39} + 774 q^{41} - 872 q^{44} - 2646 q^{46} - 426 q^{49} - 3616 q^{51} - 2090 q^{54} + 2790 q^{56} + 1910 q^{59} + 2484 q^{61} + 2584 q^{64} + 3058 q^{66} - 4848 q^{69} - 2856 q^{71} + 978 q^{74} - 8470 q^{76} + 1820 q^{79} + 732 q^{81} + 1208 q^{84} + 10334 q^{86} + 3430 q^{89} + 3504 q^{91} - 2162 q^{94} - 7256 q^{96} + 5022 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 59x^{10} + 1261x^{8} + 11844x^{6} + 45376x^{4} + 43840x^{2} + 6400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5101\nu^{10} - 592341\nu^{8} - 39940739\nu^{6} - 726710976\nu^{4} - 4279657664\nu^{2} - 2709570240 ) / 62556160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 109\nu^{10} - 1819\nu^{8} - 246101\nu^{6} - 4225534\nu^{4} - 20667976\nu^{2} - 13480640 ) / 977440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 251\nu^{10} + 13869\nu^{8} + 266091\nu^{6} + 2090304\nu^{4} + 5896896\nu^{2} + 2837696 ) / 305152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4891\nu^{11} - 273149\nu^{9} - 5068571\nu^{7} - 31643184\nu^{5} + 10451264\nu^{3} + 365935680\nu ) / 78195200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69679 \nu^{11} - 4664281 \nu^{9} - 115477599 \nu^{7} - 1243477696 \nu^{5} + \cdots + 853741120 \nu ) / 625561600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 134607 \nu^{11} - 2450873 \nu^{9} + 91324033 \nu^{7} + 2416970432 \nu^{5} + \cdots + 21537885760 \nu ) / 625561600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38463\nu^{10} + 1912137\nu^{8} + 31656303\nu^{6} + 197560512\nu^{4} + 314388928\nu^{2} - 143733312 ) / 12511232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 279223 \nu^{10} + 12874257 \nu^{8} + 187980903 \nu^{6} + 962044352 \nu^{4} + 1605883328 \nu^{2} + 3555021760 ) / 62556160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1113321 \nu^{11} + 62330319 \nu^{9} + 1232137401 \nu^{7} + 10218035904 \nu^{5} + \cdots + 1013011520 \nu ) / 625561600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -6111\nu^{11} - 362079\nu^{9} - 7744891\nu^{7} - 72009564\nu^{5} - 263811706\nu^{3} - 201632120\nu ) / 2443600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2356811 \nu^{11} + 134878429 \nu^{9} + 2767756091 \nu^{7} + 24616777664 \nu^{5} + \cdots + 60893104320 \nu ) / 625561600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{11} - 4\beta_{10} + \beta_{9} - 2\beta_{6} + 2\beta_{5} + 11\beta_{4} ) / 25 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{8} - 4\beta_{7} - 12\beta_{2} + 3\beta _1 - 252 ) / 25 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 48\beta_{11} + 49\beta_{10} - 41\beta_{9} + 17\beta_{6} - 32\beta_{5} - 236\beta_{4} ) / 25 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -58\beta_{8} + 49\beta_{7} + 95\beta_{3} + 327\beta_{2} - 78\beta _1 + 4107 ) / 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -835\beta_{11} - 683\beta_{10} + 1042\beta_{9} - 65\beta_{6} + 968\beta_{5} + 5161\beta_{4} ) / 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1191\beta_{8} - 479\beta_{7} - 3761\beta_{3} - 7941\beta_{2} + 1663\beta _1 - 75701 ) / 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 16101\beta_{11} + 10620\beta_{10} - 25125\beta_{9} - 2626\beta_{6} - 27536\beta_{5} - 111611\beta_{4} ) / 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -25847\beta_{8} + 1864\beta_{7} + 111688\beta_{3} + 184604\beta_{2} - 34523\beta _1 + 1502344 ) / 25 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 333376 \beta_{11} - 183145 \beta_{10} + 593025 \beta_{9} + 116831 \beta_{6} + \cdots + 2430196 \beta_{4} ) / 25 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 578106\beta_{8} + 90711\beta_{7} - 2944951\beta_{3} - 4223151\beta_{2} + 723678\beta _1 - 31324531 ) / 25 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7212875 \beta_{11} + 3446803 \beta_{10} - 13835842 \beta_{9} - 3496135 \beta_{6} + \cdots - 53527681 \beta_{4} ) / 25 \) 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
− 4.78516i
− 1.07081i
− 3.90965i
2.74652i
− 0.420609i
3.45688i
− 3.45688i
0.420609i
− 2.74652i
3.90965i
1.07081i
4.78516i
− 5.19564i − 6.10690i −18.9947 0 −31.7292 − 29.3201i 57.1243i −10.2942 0
124.2 − 4.37562i − 8.30353i −11.1460 0 −36.3331 29.0783i 13.7658i −41.9487 0
124.3 − 4.23279i 5.51148i −9.91650 0 23.3289 − 2.28738i 8.11216i −3.37638 0
124.4 − 3.61169i 9.75552i −5.04433 0 35.2339 14.1661i − 10.6750i −68.1701 0
124.5 − 1.95323i − 1.11531i 4.18489 0 −2.17846 1.57558i − 23.7999i 25.7561 0
124.6 − 0.288727i − 4.57894i 7.91664 0 −1.32206 − 18.9048i − 4.59556i 6.03327 0
124.7 0.288727i 4.57894i 7.91664 0 −1.32206 18.9048i 4.59556i 6.03327 0
124.8 1.95323i 1.11531i 4.18489 0 −2.17846 − 1.57558i 23.7999i 25.7561 0
124.9 3.61169i − 9.75552i −5.04433 0 35.2339 − 14.1661i 10.6750i −68.1701 0
124.10 4.23279i − 5.51148i −9.91650 0 23.3289 2.28738i − 8.11216i −3.37638 0
124.11 4.37562i 8.30353i −11.1460 0 −36.3331 − 29.0783i − 13.7658i −41.9487 0
124.12 5.19564i 6.10690i −18.9947 0 −31.7292 29.3201i − 57.1243i −10.2942 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 125.4.b.c 12
5.b even 2 1 inner 125.4.b.c 12
5.c odd 4 1 125.4.a.b ✓ 6
5.c odd 4 1 125.4.a.c yes 6
15.e even 4 1 1125.4.a.f 6
15.e even 4 1 1125.4.a.k 6
20.e even 4 1 2000.4.a.k 6
20.e even 4 1 2000.4.a.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.4.a.b ✓ 6 5.c odd 4 1
125.4.a.c yes 6 5.c odd 4 1
125.4.b.c 12 1.a even 1 1 trivial
125.4.b.c 12 5.b even 2 1 inner
1125.4.a.f 6 15.e even 4 1
1125.4.a.k 6 15.e even 4 1
2000.4.a.k 6 20.e even 4 1
2000.4.a.n 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 81T_{2}^{10} + 2480T_{2}^{8} + 35305T_{2}^{6} + 225905T_{2}^{4} + 479416T_{2}^{2} + 38416 \) acting on \(S_{4}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 81 T^{10} + \cdots + 38416 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 193877776 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 677129848641 \) Copy content Toggle raw display
$11$ \( (T^{6} - 27 T^{5} + \cdots - 47445831)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 51\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} - 105 T^{5} + \cdots + 29290968875)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} - 30 T^{5} + \cdots - 369868315500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 3206882909804)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 18008321955939)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 92\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 10\!\cdots\!75)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 153598743998924)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 39\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
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