Properties

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

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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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} - 3 i q^{3} - q^{4} + 9 q^{6} - 7 i q^{7} + 21 i q^{8} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} - 3 i q^{3} - q^{4} + 9 q^{6} - 7 i q^{7} + 21 i q^{8} - 9 q^{9} - 6 q^{11} + 3 i q^{12} + 41 i q^{13} + 21 q^{14} - 71 q^{16} - 27 i q^{17} - 27 i q^{18} + 4 q^{19} - 21 q^{21} - 18 i q^{22} + 75 i q^{23} + 63 q^{24} - 123 q^{26} + 27 i q^{27} + 7 i q^{28} + 123 q^{29} - 205 q^{31} - 45 i q^{32} + 18 i q^{33} + 81 q^{34} + 9 q^{36} + 262 i q^{37} + 12 i q^{38} + 123 q^{39} + 57 q^{41} - 63 i q^{42} + 407 i q^{43} + 6 q^{44} - 225 q^{46} + 60 i q^{47} + 213 i q^{48} - 49 q^{49} - 81 q^{51} - 41 i q^{52} + 327 i q^{53} - 81 q^{54} + 147 q^{56} - 12 i q^{57} + 369 i q^{58} - 33 q^{59} - 427 q^{61} - 615 i q^{62} + 63 i q^{63} - 433 q^{64} - 54 q^{66} + 628 i q^{67} + 27 i q^{68} + 225 q^{69} + 300 q^{71} - 189 i q^{72} + 98 i q^{73} - 786 q^{74} - 4 q^{76} + 42 i q^{77} + 369 i q^{78} - 686 q^{79} + 81 q^{81} + 171 i q^{82} + 1401 i q^{83} + 21 q^{84} - 1221 q^{86} - 369 i q^{87} - 126 i q^{88} - 714 q^{89} + 287 q^{91} - 75 i q^{92} + 615 i q^{93} - 180 q^{94} - 135 q^{96} - 494 i q^{97} - 147 i q^{98} + 54 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 18 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 18 q^{6} - 18 q^{9} - 12 q^{11} + 42 q^{14} - 142 q^{16} + 8 q^{19} - 42 q^{21} + 126 q^{24} - 246 q^{26} + 246 q^{29} - 410 q^{31} + 162 q^{34} + 18 q^{36} + 246 q^{39} + 114 q^{41} + 12 q^{44} - 450 q^{46} - 98 q^{49} - 162 q^{51} - 162 q^{54} + 294 q^{56} - 66 q^{59} - 854 q^{61} - 866 q^{64} - 108 q^{66} + 450 q^{69} + 600 q^{71} - 1572 q^{74} - 8 q^{76} - 1372 q^{79} + 162 q^{81} + 42 q^{84} - 2442 q^{86} - 1428 q^{89} + 574 q^{91} - 360 q^{94} - 270 q^{96} + 108 q^{99}+O(q^{100}) \) 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
3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −9.00000 0
274.2 3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −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.d 2
5.b even 2 1 inner 525.4.d.d 2
5.c odd 4 1 525.4.a.c 1
5.c odd 4 1 525.4.a.h yes 1
15.e even 4 1 1575.4.a.a 1
15.e even 4 1 1575.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.c 1 5.c odd 4 1
525.4.a.h yes 1 5.c odd 4 1
525.4.d.d 2 1.a even 1 1 trivial
525.4.d.d 2 5.b even 2 1 inner
1575.4.a.a 1 15.e even 4 1
1575.4.a.j 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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) 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
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1681 \) Copy content Toggle raw display
$17$ \( T^{2} + 729 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5625 \) Copy content Toggle raw display
$29$ \( (T - 123)^{2} \) Copy content Toggle raw display
$31$ \( (T + 205)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 68644 \) Copy content Toggle raw display
$41$ \( (T - 57)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 165649 \) Copy content Toggle raw display
$47$ \( T^{2} + 3600 \) Copy content Toggle raw display
$53$ \( T^{2} + 106929 \) Copy content Toggle raw display
$59$ \( (T + 33)^{2} \) Copy content Toggle raw display
$61$ \( (T + 427)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 394384 \) Copy content Toggle raw display
$71$ \( (T - 300)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9604 \) Copy content Toggle raw display
$79$ \( (T + 686)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1962801 \) Copy content Toggle raw display
$89$ \( (T + 714)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 244036 \) Copy content Toggle raw display
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