Properties

Label 2500.1.t.a
Level $2500$
Weight $1$
Character orbit 2500.t
Analytic conductor $1.248$
Analytic rank $0$
Dimension $100$
Projective image $D_{125}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2500,1,Mod(11,2500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2500, base_ring=CyclotomicField(250))
 
chi = DirichletCharacter(H, H._module([125, 238]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2500.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2500.t (of order \(250\), degree \(100\), 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: \(1.24766253158\)
Analytic rank: \(0\)
Dimension: \(100\)
Coefficient field: \(\Q(\zeta_{250})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{100} - x^{75} + x^{50} - x^{25} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{125}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{125} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{250}^{81} q^{2} - \zeta_{250}^{37} q^{4} + \zeta_{250}^{58} q^{5} + \zeta_{250}^{118} q^{8} + \zeta_{250}^{84} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{250}^{81} q^{2} - \zeta_{250}^{37} q^{4} + \zeta_{250}^{58} q^{5} + \zeta_{250}^{118} q^{8} + \zeta_{250}^{84} q^{9} + \zeta_{250}^{14} q^{10} + (\zeta_{250}^{86} - \zeta_{250}^{57}) q^{13} + \zeta_{250}^{74} q^{16} + (\zeta_{250}^{22} + \zeta_{250}^{4}) q^{17} + \zeta_{250}^{40} q^{18} - \zeta_{250}^{95} q^{20} + \zeta_{250}^{116} q^{25} + (\zeta_{250}^{42} - \zeta_{250}^{13}) q^{26} + ( - \zeta_{250}^{73} + \zeta_{250}^{46}) q^{29} + \zeta_{250}^{30} q^{32} + ( - \zeta_{250}^{103} - \zeta_{250}^{85}) q^{34} - \zeta_{250}^{121} q^{36} + (\zeta_{250}^{38} + \zeta_{250}^{10}) q^{37} - \zeta_{250}^{51} q^{40} + (\zeta_{250}^{76} + \zeta_{250}^{52}) q^{41} - \zeta_{250}^{17} q^{45} + \zeta_{250}^{70} q^{49} + \zeta_{250}^{72} q^{50} + ( - \zeta_{250}^{123} + \zeta_{250}^{94}) q^{52} + ( - \zeta_{250}^{55} + \zeta_{250}^{24}) q^{53} + ( - \zeta_{250}^{29} + \zeta_{250}^{2}) q^{58} + ( - \zeta_{250}^{69} - \zeta_{250}^{53}) q^{61} - \zeta_{250}^{111} q^{64} + ( - \zeta_{250}^{115} - \zeta_{250}^{19}) q^{65} + ( - \zeta_{250}^{59} - \zeta_{250}^{41}) q^{68} - \zeta_{250}^{77} q^{72} + (\zeta_{250}^{78} + \zeta_{250}^{64}) q^{73} + ( - \zeta_{250}^{119} - \zeta_{250}^{91}) q^{74} - \zeta_{250}^{7} q^{80} - \zeta_{250}^{43} q^{81} + (\zeta_{250}^{32} + \zeta_{250}^{8}) q^{82} + (\zeta_{250}^{80} + \zeta_{250}^{62}) q^{85} + ( - \zeta_{250}^{65} - \zeta_{250}^{27}) q^{89} + \zeta_{250}^{98} q^{90} + ( - \zeta_{250}^{93} - \zeta_{250}) q^{97} + \zeta_{250}^{26} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q+O(q^{10}) \) Copy content Toggle raw display \( 100 q+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(-1\) \(-\zeta_{250}^{37}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.285019 0.958522i
0.837528 0.546394i
0.379779 0.925077i
0.332820 0.942991i
0.962028 0.272952i
0.675333 0.737513i
−0.693653 + 0.720309i
0.514440 + 0.857527i
0.745941 + 0.666012i
0.470704 0.882291i
0.910106 0.414376i
−0.988652 0.150226i
−0.162637 + 0.986686i
0.236499 0.971632i
−0.998737 + 0.0502443i
0.745941 0.666012i
−0.920232 + 0.391374i
−0.0125660 0.999921i
0.947098 + 0.320944i
0.0376902 + 0.999289i
0.988652 0.150226i 0 0.954865 0.297042i 0.492727 + 0.870184i 0 0 0.899405 0.437116i 0.656586 0.754251i 0.617860 + 0.786288i
31.1 0.954865 + 0.297042i 0 0.823533 + 0.567269i −0.514440 0.857527i 0 0 0.617860 + 0.786288i −0.137790 + 0.990461i −0.236499 0.971632i
71.1 −0.137790 + 0.990461i 0 −0.962028 0.272952i 0.823533 + 0.567269i 0 0 0.402906 0.915241i 0.260842 + 0.965382i −0.675333 + 0.737513i
91.1 −0.711536 0.702650i 0 0.0125660 + 0.999921i −0.675333 0.737513i 0 0 0.693653 0.720309i −0.974527 0.224271i −0.0376902 + 0.999289i
111.1 0.920232 0.391374i 0 0.693653 0.720309i −0.947098 + 0.320944i 0 0 0.356412 0.934329i −0.332820 + 0.942991i −0.745941 + 0.666012i
131.1 0.356412 0.934329i 0 −0.745941 0.666012i −0.556876 + 0.830596i 0 0 −0.888136 + 0.459580i 0.850994 0.525175i 0.577573 + 0.816339i
171.1 −0.675333 0.737513i 0 −0.0878512 + 0.996134i −0.888136 0.459580i 0 0 0.793990 0.607930i 0.0125660 + 0.999921i 0.260842 + 0.965382i
191.1 0.212007 0.977268i 0 −0.910106 0.414376i −0.997159 0.0753268i 0 0 −0.597905 + 0.801567i 0.162637 0.986686i −0.285019 + 0.958522i
211.1 0.793990 0.607930i 0 0.260842 0.965382i −0.137790 0.990461i 0 0 −0.379779 0.925077i −0.0376902 0.999289i −0.711536 0.702650i
231.1 −0.910106 0.414376i 0 0.656586 + 0.754251i 0.988652 + 0.150226i 0 0 −0.285019 0.958522i −0.947098 0.320944i −0.837528 0.546394i
271.1 0.998737 0.0502443i 0 0.994951 0.100362i 0.938734 + 0.344643i 0 0 0.988652 0.150226i −0.236499 + 0.971632i 0.954865 + 0.297042i
291.1 0.938734 0.344643i 0 0.762443 0.647056i −0.778462 + 0.627691i 0 0 0.492727 0.870184i 0.994951 + 0.100362i −0.514440 + 0.857527i
311.1 0.617860 0.786288i 0 −0.236499 0.971632i 0.998737 + 0.0502443i 0 0 −0.910106 0.414376i 0.402906 + 0.915241i 0.656586 0.754251i
331.1 −0.470704 + 0.882291i 0 −0.556876 0.830596i −0.285019 0.958522i 0 0 0.994951 0.100362i 0.356412 + 0.934329i 0.979855 + 0.199710i
371.1 −0.597905 + 0.801567i 0 −0.285019 0.958522i −0.974527 0.224271i 0 0 0.938734 + 0.344643i −0.470704 + 0.882291i 0.762443 0.647056i
391.1 0.793990 + 0.607930i 0 0.260842 + 0.965382i −0.137790 + 0.990461i 0 0 −0.379779 + 0.925077i −0.0376902 + 0.999289i −0.711536 + 0.702650i
411.1 0.402906 0.915241i 0 −0.675333 0.737513i −0.236499 + 0.971632i 0 0 −0.947098 + 0.320944i −0.711536 0.702650i 0.793990 + 0.607930i
431.1 0.850994 + 0.525175i 0 0.448383 + 0.893841i −0.745941 + 0.666012i 0 0 −0.0878512 + 0.996134i 0.492727 0.870184i −0.984564 + 0.175023i
471.1 −0.236499 0.971632i 0 −0.888136 + 0.459580i 0.994951 + 0.100362i 0 0 0.656586 + 0.754251i −0.675333 + 0.737513i −0.137790 0.990461i
491.1 −0.0878512 + 0.996134i 0 −0.984564 0.175023i 0.577573 + 0.816339i 0 0 0.260842 0.965382i −0.999684 + 0.0251301i −0.863923 + 0.503623i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
625.j even 125 1 inner
2500.t odd 250 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2500.1.t.a 100
4.b odd 2 1 CM 2500.1.t.a 100
625.j even 125 1 inner 2500.1.t.a 100
2500.t odd 250 1 inner 2500.1.t.a 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2500.1.t.a 100 1.a even 1 1 trivial
2500.1.t.a 100 4.b odd 2 1 CM
2500.1.t.a 100 625.j even 125 1 inner
2500.1.t.a 100 2500.t odd 250 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{100} + T^{75} + T^{50} + T^{25} + 1 \) Copy content Toggle raw display
$3$ \( T^{100} \) Copy content Toggle raw display
$5$ \( T^{100} + T^{75} + T^{50} + T^{25} + 1 \) Copy content Toggle raw display
$7$ \( T^{100} \) Copy content Toggle raw display
$11$ \( T^{100} \) Copy content Toggle raw display
$13$ \( T^{100} + 25 T^{97} + 375 T^{94} + 4400 T^{91} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{100} + 125 T^{93} + 25 T^{92} - 600 T^{87} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{100} \) Copy content Toggle raw display
$23$ \( T^{100} \) Copy content Toggle raw display
$29$ \( T^{100} - 100 T^{87} + 650 T^{86} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{100} \) Copy content Toggle raw display
$37$ \( T^{100} + 5 T^{95} + 25 T^{94} + 15 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{100} + 25 T^{87} + 650 T^{86} + 5005 T^{85} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{100} \) Copy content Toggle raw display
$47$ \( T^{100} \) Copy content Toggle raw display
$53$ \( T^{100} + 5 T^{95} + 25 T^{94} + 15 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{100} \) Copy content Toggle raw display
$61$ \( T^{100} + 25 T^{97} + 375 T^{94} + 4400 T^{91} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{100} \) Copy content Toggle raw display
$71$ \( T^{100} \) Copy content Toggle raw display
$73$ \( T^{100} - 100 T^{87} + 650 T^{86} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{100} \) Copy content Toggle raw display
$83$ \( T^{100} \) Copy content Toggle raw display
$89$ \( T^{100} + 5 T^{95} + 25 T^{94} + 15 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{100} - 500 T^{93} + 25 T^{92} + 150 T^{87} + \cdots + 1 \) Copy content Toggle raw display
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