Properties

Label 525.4.d.e
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,Mod(274,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 3 i q^{3} + 4 q^{4} - 6 q^{6} + 7 i q^{7} + 24 i q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} + 3 i q^{3} + 4 q^{4} - 6 q^{6} + 7 i q^{7} + 24 i q^{8} - 9 q^{9} - 21 q^{11} + 12 i q^{12} + 24 i q^{13} - 14 q^{14} - 16 q^{16} + 22 i q^{17} - 18 i q^{18} - 16 q^{19} - 21 q^{21} - 42 i q^{22} - 25 i q^{23} - 72 q^{24} - 48 q^{26} - 27 i q^{27} + 28 i q^{28} - 167 q^{29} + 10 q^{31} + 160 i q^{32} - 63 i q^{33} - 44 q^{34} - 36 q^{36} + 133 i q^{37} - 32 i q^{38} - 72 q^{39} - 168 q^{41} - 42 i q^{42} - 97 i q^{43} - 84 q^{44} + 50 q^{46} + 400 i q^{47} - 48 i q^{48} - 49 q^{49} - 66 q^{51} + 96 i q^{52} - 182 i q^{53} + 54 q^{54} - 168 q^{56} - 48 i q^{57} - 334 i q^{58} - 488 q^{59} + 28 q^{61} + 20 i q^{62} - 63 i q^{63} - 448 q^{64} + 126 q^{66} + 967 i q^{67} + 88 i q^{68} + 75 q^{69} - 285 q^{71} - 216 i q^{72} - 838 i q^{73} - 266 q^{74} - 64 q^{76} - 147 i q^{77} - 144 i q^{78} + 469 q^{79} + 81 q^{81} - 336 i q^{82} - 406 i q^{83} - 84 q^{84} + 194 q^{86} - 501 i q^{87} - 504 i q^{88} - 324 q^{89} - 168 q^{91} - 100 i q^{92} + 30 i q^{93} - 800 q^{94} - 480 q^{96} + 114 i q^{97} - 98 i q^{98} + 189 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} - 12 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} - 12 q^{6} - 18 q^{9} - 42 q^{11} - 28 q^{14} - 32 q^{16} - 32 q^{19} - 42 q^{21} - 144 q^{24} - 96 q^{26} - 334 q^{29} + 20 q^{31} - 88 q^{34} - 72 q^{36} - 144 q^{39} - 336 q^{41} - 168 q^{44} + 100 q^{46} - 98 q^{49} - 132 q^{51} + 108 q^{54} - 336 q^{56} - 976 q^{59} + 56 q^{61} - 896 q^{64} + 252 q^{66} + 150 q^{69} - 570 q^{71} - 532 q^{74} - 128 q^{76} + 938 q^{79} + 162 q^{81} - 168 q^{84} + 388 q^{86} - 648 q^{89} - 336 q^{91} - 1600 q^{94} - 960 q^{96} + 378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
1.00000i
1.00000i
2.00000i 3.00000i 4.00000 0 −6.00000 7.00000i 24.0000i −9.00000 0
274.2 2.00000i 3.00000i 4.00000 0 −6.00000 7.00000i 24.0000i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
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 525.4.d.e 2
5.b even 2 1 inner 525.4.d.e 2
5.c odd 4 1 525.4.a.d 1
5.c odd 4 1 525.4.a.f yes 1
15.e even 4 1 1575.4.a.d 1
15.e even 4 1 1575.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 5.c odd 4 1
525.4.a.f yes 1 5.c odd 4 1
525.4.d.e 2 1.a even 1 1 trivial
525.4.d.e 2 5.b even 2 1 inner
1575.4.a.d 1 15.e even 4 1
1575.4.a.h 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 21)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 576 \) Copy content Toggle raw display
$17$ \( T^{2} + 484 \) Copy content Toggle raw display
$19$ \( (T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 625 \) Copy content Toggle raw display
$29$ \( (T + 167)^{2} \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17689 \) Copy content Toggle raw display
$41$ \( (T + 168)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 9409 \) Copy content Toggle raw display
$47$ \( T^{2} + 160000 \) Copy content Toggle raw display
$53$ \( T^{2} + 33124 \) Copy content Toggle raw display
$59$ \( (T + 488)^{2} \) Copy content Toggle raw display
$61$ \( (T - 28)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 935089 \) Copy content Toggle raw display
$71$ \( (T + 285)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 702244 \) Copy content Toggle raw display
$79$ \( (T - 469)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 164836 \) Copy content Toggle raw display
$89$ \( (T + 324)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 12996 \) Copy content Toggle raw display
show more
show less