Properties

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

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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: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 7 i q^{7} - 45 i q^{8} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 7 i q^{7} - 45 i q^{8} - 9 q^{9} + 12 q^{11} - 51 i q^{12} - 30 i q^{13} - 35 q^{14} + 89 q^{16} - 134 i q^{17} - 45 i q^{18} + 92 q^{19} - 21 q^{21} + 60 i q^{22} - 112 i q^{23} + 135 q^{24} + 150 q^{26} - 27 i q^{27} - 119 i q^{28} + 58 q^{29} - 224 q^{31} + 85 i q^{32} + 36 i q^{33} + 670 q^{34} + 153 q^{36} - 146 i q^{37} + 460 i q^{38} + 90 q^{39} + 18 q^{41} - 105 i q^{42} - 340 i q^{43} - 204 q^{44} + 560 q^{46} + 208 i q^{47} + 267 i q^{48} - 49 q^{49} + 402 q^{51} + 510 i q^{52} + 754 i q^{53} + 135 q^{54} + 315 q^{56} + 276 i q^{57} + 290 i q^{58} - 380 q^{59} + 718 q^{61} - 1120 i q^{62} - 63 i q^{63} + 287 q^{64} - 180 q^{66} + 412 i q^{67} + 2278 i q^{68} + 336 q^{69} - 960 q^{71} + 405 i q^{72} - 1066 i q^{73} + 730 q^{74} - 1564 q^{76} + 84 i q^{77} + 450 i q^{78} - 896 q^{79} + 81 q^{81} + 90 i q^{82} - 436 i q^{83} + 357 q^{84} + 1700 q^{86} + 174 i q^{87} - 540 i q^{88} + 1038 q^{89} + 210 q^{91} + 1904 i q^{92} - 672 i q^{93} - 1040 q^{94} - 255 q^{96} - 702 i q^{97} - 245 i q^{98} - 108 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9} + 24 q^{11} - 70 q^{14} + 178 q^{16} + 184 q^{19} - 42 q^{21} + 270 q^{24} + 300 q^{26} + 116 q^{29} - 448 q^{31} + 1340 q^{34} + 306 q^{36} + 180 q^{39} + 36 q^{41} - 408 q^{44} + 1120 q^{46} - 98 q^{49} + 804 q^{51} + 270 q^{54} + 630 q^{56} - 760 q^{59} + 1436 q^{61} + 574 q^{64} - 360 q^{66} + 672 q^{69} - 1920 q^{71} + 1460 q^{74} - 3128 q^{76} - 1792 q^{79} + 162 q^{81} + 714 q^{84} + 3400 q^{86} + 2076 q^{89} + 420 q^{91} - 2080 q^{94} - 510 q^{96} - 216 q^{99}+O(q^{100}) \) 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
5.00000i 3.00000i −17.0000 0 −15.0000 7.00000i 45.0000i −9.00000 0
274.2 5.00000i 3.00000i −17.0000 0 −15.0000 7.00000i 45.0000i −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.a 2
5.b even 2 1 inner 525.4.d.a 2
5.c odd 4 1 105.4.a.b 1
5.c odd 4 1 525.4.a.a 1
15.e even 4 1 315.4.a.a 1
15.e even 4 1 1575.4.a.l 1
20.e even 4 1 1680.4.a.u 1
35.f even 4 1 735.4.a.j 1
105.k odd 4 1 2205.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 5.c odd 4 1
315.4.a.a 1 15.e even 4 1
525.4.a.a 1 5.c odd 4 1
525.4.d.a 2 1.a even 1 1 trivial
525.4.d.a 2 5.b even 2 1 inner
735.4.a.j 1 35.f even 4 1
1575.4.a.l 1 15.e even 4 1
1680.4.a.u 1 20.e even 4 1
2205.4.a.b 1 105.k odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{11} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 25 \) 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
$11$ \( (T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 900 \) Copy content Toggle raw display
$17$ \( T^{2} + 17956 \) Copy content Toggle raw display
$19$ \( (T - 92)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12544 \) Copy content Toggle raw display
$29$ \( (T - 58)^{2} \) Copy content Toggle raw display
$31$ \( (T + 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 21316 \) Copy content Toggle raw display
$41$ \( (T - 18)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 115600 \) Copy content Toggle raw display
$47$ \( T^{2} + 43264 \) Copy content Toggle raw display
$53$ \( T^{2} + 568516 \) Copy content Toggle raw display
$59$ \( (T + 380)^{2} \) Copy content Toggle raw display
$61$ \( (T - 718)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 169744 \) Copy content Toggle raw display
$71$ \( (T + 960)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1136356 \) Copy content Toggle raw display
$79$ \( (T + 896)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 190096 \) Copy content Toggle raw display
$89$ \( (T - 1038)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 492804 \) Copy content Toggle raw display
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