Properties

Label 525.4.d.k
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 2 \beta_1) q^{2} - 3 \beta_1 q^{3} + (4 \beta_{3} - 1) q^{4} + (3 \beta_{3} - 6) q^{6} - 7 \beta_1 q^{7} + ( - \beta_{2} + 6 \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 2 \beta_1) q^{2} - 3 \beta_1 q^{3} + (4 \beta_{3} - 1) q^{4} + (3 \beta_{3} - 6) q^{6} - 7 \beta_1 q^{7} + ( - \beta_{2} + 6 \beta_1) q^{8} - 9 q^{9} + ( - 2 \beta_{3} - 46) q^{11} + ( - 12 \beta_{2} + 3 \beta_1) q^{12} + (38 \beta_{2} - 4 \beta_1) q^{13} + (7 \beta_{3} - 14) q^{14} + (24 \beta_{3} + 9) q^{16} + (44 \beta_{2} - 22 \beta_1) q^{17} + ( - 9 \beta_{2} + 18 \beta_1) q^{18} + ( - 26 \beta_{3} + 54) q^{19} - 21 q^{21} + ( - 42 \beta_{2} + 82 \beta_1) q^{22} + (20 \beta_{2} + 160 \beta_1) q^{23} + ( - 3 \beta_{3} + 18) q^{24} + (80 \beta_{3} - 198) q^{26} + 27 \beta_1 q^{27} + ( - 28 \beta_{2} + 7 \beta_1) q^{28} + ( - 12 \beta_{3} + 118) q^{29} + (102 \beta_{3} - 30) q^{31} + ( - 47 \beta_{2} + 150 \beta_1) q^{32} + (6 \beta_{2} + 138 \beta_1) q^{33} + (110 \beta_{3} - 264) q^{34} + ( - 36 \beta_{3} + 9) q^{36} + (24 \beta_{2} + 102 \beta_1) q^{37} + (106 \beta_{2} - 238 \beta_1) q^{38} + (114 \beta_{3} - 12) q^{39} + ( - 80 \beta_{3} + 22) q^{41} + ( - 21 \beta_{2} + 42 \beta_1) q^{42} + ( - 128 \beta_{2} - 68 \beta_1) q^{43} + ( - 182 \beta_{3} + 6) q^{44} + ( - 120 \beta_{3} + 220) q^{46} + ( - 168 \beta_{2} + 200 \beta_1) q^{47} + ( - 72 \beta_{2} - 27 \beta_1) q^{48} - 49 q^{49} + (132 \beta_{3} - 66) q^{51} + ( - 54 \beta_{2} + 764 \beta_1) q^{52} + ( - 86 \beta_{2} - 8 \beta_1) q^{53} + ( - 27 \beta_{3} + 54) q^{54} + ( - 7 \beta_{3} + 42) q^{56} + (78 \beta_{2} - 162 \beta_1) q^{57} + (142 \beta_{2} - 296 \beta_1) q^{58} + (36 \beta_{3} + 232) q^{59} + (84 \beta_{3} - 342) q^{61} + ( - 234 \beta_{2} + 570 \beta_1) q^{62} + 63 \beta_1 q^{63} + ( - 52 \beta_{3} + 607) q^{64} + ( - 126 \beta_{3} + 246) q^{66} + (164 \beta_{2} + 368 \beta_1) q^{67} + ( - 132 \beta_{2} + 902 \beta_1) q^{68} + (60 \beta_{3} + 480) q^{69} + ( - 138 \beta_{3} - 370) q^{71} + (9 \beta_{2} - 54 \beta_1) q^{72} + (122 \beta_{2} - 212 \beta_1) q^{73} + ( - 54 \beta_{3} + 84) q^{74} + (242 \beta_{3} - 574) q^{76} + (14 \beta_{2} + 322 \beta_1) q^{77} + ( - 240 \beta_{2} + 594 \beta_1) q^{78} + (484 \beta_{3} + 204) q^{79} + 81 q^{81} + (182 \beta_{2} - 444 \beta_1) q^{82} + (84 \beta_{2} - 304 \beta_1) q^{83} + ( - 84 \beta_{3} + 21) q^{84} + ( - 188 \beta_{3} + 504) q^{86} + (36 \beta_{2} - 354 \beta_1) q^{87} + (34 \beta_{2} - 266 \beta_1) q^{88} + (112 \beta_{3} + 666) q^{89} + (266 \beta_{3} - 28) q^{91} + (620 \beta_{2} + 240 \beta_1) q^{92} + ( - 306 \beta_{2} + 90 \beta_1) q^{93} + ( - 536 \beta_{3} + 1240) q^{94} + ( - 141 \beta_{3} + 450) q^{96} + ( - 86 \beta_{2} - 1224 \beta_1) q^{97} + ( - 49 \beta_{2} + 98 \beta_1) q^{98} + (18 \beta_{3} + 414) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44} + 880 q^{46} - 196 q^{49} - 264 q^{51} + 216 q^{54} + 168 q^{56} + 928 q^{59} - 1368 q^{61} + 2428 q^{64} + 984 q^{66} + 1920 q^{69} - 1480 q^{71} + 336 q^{74} - 2296 q^{76} + 816 q^{79} + 324 q^{81} + 84 q^{84} + 2016 q^{86} + 2664 q^{89} - 112 q^{91} + 4960 q^{94} + 1800 q^{96} + 1656 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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
1.61803i
0.618034i
0.618034i
1.61803i
4.23607i 3.00000i −9.94427 0 −12.7082 7.00000i 8.23607i −9.00000 0
274.2 0.236068i 3.00000i 7.94427 0 0.708204 7.00000i 3.76393i −9.00000 0
274.3 0.236068i 3.00000i 7.94427 0 0.708204 7.00000i 3.76393i −9.00000 0
274.4 4.23607i 3.00000i −9.94427 0 −12.7082 7.00000i 8.23607i −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.k 4
5.b even 2 1 inner 525.4.d.k 4
5.c odd 4 1 105.4.a.d 2
5.c odd 4 1 525.4.a.o 2
15.e even 4 1 315.4.a.l 2
15.e even 4 1 1575.4.a.n 2
20.e even 4 1 1680.4.a.bd 2
35.f even 4 1 735.4.a.m 2
105.k odd 4 1 2205.4.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 5.c odd 4 1
315.4.a.l 2 15.e even 4 1
525.4.a.o 2 5.c odd 4 1
525.4.d.k 4 1.a even 1 1 trivial
525.4.d.k 4 5.b even 2 1 inner
735.4.a.m 2 35.f even 4 1
1575.4.a.n 2 15.e even 4 1
1680.4.a.bd 2 20.e even 4 1
2205.4.a.be 2 105.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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 18T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 92T_{11} + 2096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 92 T + 2096)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 14472 T^{2} + 51897616 \) Copy content Toggle raw display
$17$ \( T^{4} + 20328 T^{2} + 84566416 \) Copy content Toggle raw display
$19$ \( (T^{2} - 108 T - 464)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 55200 T^{2} + 556960000 \) Copy content Toggle raw display
$29$ \( (T^{2} - 236 T + 13204)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60 T - 51120)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 26568 T^{2} + 56610576 \) Copy content Toggle raw display
$41$ \( (T^{2} - 44 T - 31516)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 5974671616 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 10225254400 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 1362791056 \) Copy content Toggle raw display
$59$ \( (T^{2} - 464 T + 47344)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 684 T + 81684)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 539808 T^{2} + 891136 \) Copy content Toggle raw display
$71$ \( (T^{2} + 740 T + 41680)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 238728 T^{2} + 868834576 \) Copy content Toggle raw display
$79$ \( (T^{2} - 408 T - 1129664)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 3264522496 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1332 T + 380836)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 2135093750416 \) Copy content Toggle raw display
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