↑

Properties

Label 25.13.c.b
Level $25$
Weight $13$
Character orbit 25.c
Analytic conductor $22.850$
Analytic rank $0$
Dimension $10$
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] = [25,13,Mod(7,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.7");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), 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: \(22.8498454319\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{10} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - 3 \beta_{7} + \beta_{4} + \cdots - 31) q^{3}+ \cdots + (\beta_{9} + 54 \beta_{8} + \cdots + 111342 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - 3 \beta_{7} + \beta_{4} + \cdots - 31) q^{3}+ \cdots + ( - 5332646 \beta_{9} + \cdots - 312689059782 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 318 q^{3} - 175080 q^{6} - 279598 q^{7} + 469980 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} - 318 q^{3} - 175080 q^{6} - 279598 q^{7} + 469980 q^{8} + 312620 q^{11} + 2359992 q^{12} - 5290738 q^{13} + 30547960 q^{16} + 41269502 q^{17} + 140573742 q^{18} + 107493420 q^{21} + 155490544 q^{22} + 510099842 q^{23} + 1475846420 q^{26} + 1993958640 q^{27} + 3562106488 q^{28} + 3077089820 q^{31} + 4623883832 q^{32} + 7503698004 q^{33} + 7760793660 q^{36} + 2599618502 q^{37} + 15310240920 q^{38} + 7412079020 q^{41} + 18593270064 q^{42} + 5784410402 q^{43} - 7382547880 q^{46} - 16053249598 q^{47} - 42572492208 q^{48} - 33139878180 q^{51} - 96763417228 q^{52} - 101763514618 q^{53} - 172002747600 q^{56} - 27733489920 q^{57} - 135238672320 q^{58} + 7731718220 q^{61} - 193287375176 q^{62} - 207465112158 q^{63} - 60815472960 q^{66} + 80010636002 q^{67} - 204699541412 q^{68} - 46557252580 q^{71} + 13986370620 q^{72} + 448527032342 q^{73} + 305095930800 q^{76} + 425580405844 q^{77} + 1690993241784 q^{78} + 1107831051810 q^{81} + 671946416464 q^{82} + 91118376722 q^{83} + 414117747320 q^{86} + 2078422804320 q^{87} + 1842434230560 q^{88} + 2737742572220 q^{91} - 906853941448 q^{92} + 91295366484 q^{93} - 3473259523680 q^{96} + 1409507601302 q^{97} + 1481746533298 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 26465 \nu^{9} - 176059482 \nu^{7} - 167682655437 \nu^{5} + 471206316514012 \nu^{3} + 74\!\cdots\!92 \nu ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2320992487 \nu^{8} + 18354612359011 \nu^{6} + \cdots + 42\!\cdots\!28 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 106451712085 \nu^{8} + 845493818346321 \nu^{6} + \cdots + 17\!\cdots\!56 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1460065153745 \nu^{9} + 56782570931856 \nu^{8} + \cdots - 65\!\cdots\!80 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1460065153745 \nu^{9} - 56782570931856 \nu^{8} + \cdots + 65\!\cdots\!80 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5235689538055 \nu^{9} - 55538770561995 \nu^{8} + \cdots - 59\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5235689538055 \nu^{9} - 55538770561995 \nu^{8} + \cdots - 59\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 965372918064815 \nu^{9} + \cdots - 14\!\cdots\!76 \nu ) / 73\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!85 \nu^{9} + \cdots - 18\!\cdots\!84 \nu ) / 73\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - 9\beta_{8} - 116\beta_{7} + 116\beta_{6} - 16\beta_{5} - 16\beta_{4} + 642\beta_1 ) / 1500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -1279\beta_{7} - 1279\beta_{6} + 274\beta_{5} - 274\beta_{4} + \beta_{3} - 44\beta_{2} - 317607 ) / 200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7943 \beta_{9} + 62112 \beta_{8} + 1395763 \beta_{7} - 1395763 \beta_{6} + 69338 \beta_{5} + \cdots - 12083781 \beta_1 ) / 3000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1003369 \beta_{7} + 1003369 \beta_{6} - 209390 \beta_{5} + 209390 \beta_{4} - 907 \beta_{3} + \cdots + 234284201 ) / 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 30522287 \beta_{9} - 240986208 \beta_{8} - 5291172667 \beta_{7} + 5291172667 \beta_{6} + \cdots + 82803373629 \beta_1 ) / 3000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 18595471861 \beta_{7} - 18595471861 \beta_{6} + 3972910806 \beta_{5} - 3972910806 \beta_{4} + \cdots - 4481734021973 ) / 200 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 117270049583 \beta_{9} + 941412271872 \beta_{8} + 19729028239003 \beta_{7} + \cdots - 441721308677661 \beta_1 ) / 3000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 68672208918853 \beta_{7} + 68672208918853 \beta_{6} - 15091683522358 \beta_{5} + 15091683522358 \beta_{4} + \cdots + 17\!\cdots\!09 ) / 200 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 450926003132447 \beta_{9} + \cdots + 21\!\cdots\!49 \beta_1 ) / 3000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
14.0132i
− 5.61354i
− 62.7587i
60.9123i
− 8.55327i
− 14.0132i
5.61354i
62.7587i
− 60.9123i
8.55327i
−77.8857 − 77.8857i 451.057 − 451.057i 8036.37i 0 −70261.7 −107575. − 107575.i 306898. − 306898.i 124537.i 0
7.2 −34.6956 − 34.6956i −539.020 + 539.020i − 1688.43i 0 37403.2 −25919.1 − 25919.1i −200694. + 200694.i − 49644.2i 0
7.3 −3.05147 − 3.05147i 299.721 − 299.721i − 4077.38i 0 −1829.18 139850. + 139850.i −24940.8 + 24940.8i 351775.i 0
7.4 54.8437 + 54.8437i 506.429 − 506.429i 1919.66i 0 55548.9 −78559.8 − 78559.8i 119359. − 119359.i 18500.3i 0
7.5 61.7891 + 61.7891i −877.187 + 877.187i 3539.78i 0 −108401. −67595.3 − 67595.3i 34368.1 − 34368.1i − 1.00747e6i 0
18.1 −77.8857 + 77.8857i 451.057 + 451.057i − 8036.37i 0 −70261.7 −107575. + 107575.i 306898. + 306898.i − 124537.i 0
18.2 −34.6956 + 34.6956i −539.020 − 539.020i 1688.43i 0 37403.2 −25919.1 + 25919.1i −200694. − 200694.i 49644.2i 0
18.3 −3.05147 + 3.05147i 299.721 + 299.721i 4077.38i 0 −1829.18 139850. − 139850.i −24940.8 − 24940.8i − 351775.i 0
18.4 54.8437 − 54.8437i 506.429 + 506.429i − 1919.66i 0 55548.9 −78559.8 + 78559.8i 119359. + 119359.i − 18500.3i 0
18.5 61.7891 − 61.7891i −877.187 − 877.187i − 3539.78i 0 −108401. −67595.3 + 67595.3i 34368.1 + 34368.1i 1.00747e6i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.13.c.b 10
5.b even 2 1 5.13.c.a ✓ 10
5.c odd 4 1 5.13.c.a ✓ 10
5.c odd 4 1 inner 25.13.c.b 10
15.d odd 2 1 45.13.g.a 10
15.e even 4 1 45.13.g.a 10
20.d odd 2 1 80.13.p.c 10
20.e even 4 1 80.13.p.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.13.c.a ✓ 10 5.b even 2 1
5.13.c.a ✓ 10 5.c odd 4 1
25.13.c.b 10 1.a even 1 1 trivial
25.13.c.b 10 5.c odd 4 1 inner
45.13.g.a 10 15.d odd 2 1
45.13.g.a 10 15.e even 4 1
80.13.p.c 10 20.d odd 2 1
80.13.p.c 10 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 2 T_{2}^{9} + 2 T_{2}^{8} - 151200 T_{2}^{7} + 124044608 T_{2}^{6} - 2679039616 T_{2}^{5} + \cdots + 24\!\cdots\!32 \) acting on \(S_{13}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 33\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 33\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 29\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 82\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 91\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 61\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 48\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
show more
show less