Properties

Label 525.4.d.j
Level $525$
Weight $4$
Character orbit 525.d
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (3 \beta_{3} - 6) q^{4} + ( - 3 \beta_{3} + 6) q^{6} - 7 \beta_{2} q^{7} + (25 \beta_{2} - \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + 3 \beta_{2} q^{3} + (3 \beta_{3} - 6) q^{4} + ( - 3 \beta_{3} + 6) q^{6} - 7 \beta_{2} q^{7} + (25 \beta_{2} - \beta_1) q^{8} - 9 q^{9} + (2 \beta_{3} + 30) q^{11} + ( - 9 \beta_{2} + 9 \beta_1) q^{12} + (2 \beta_{2} + 10 \beta_1) q^{13} + (7 \beta_{3} - 14) q^{14} + ( - 3 \beta_{3} + 14) q^{16} + ( - 26 \beta_{2} - 12 \beta_1) q^{17} + (9 \beta_{2} - 9 \beta_1) q^{18} + ( - 2 \beta_{3} + 62) q^{19} + 21 q^{21} + ( - 12 \beta_{2} + 28 \beta_1) q^{22} + (32 \beta_{2} + 48 \beta_1) q^{23} + (3 \beta_{3} - 78) q^{24} + (18 \beta_{3} - 116) q^{26} - 27 \beta_{2} q^{27} + (21 \beta_{2} - 21 \beta_1) q^{28} + (12 \beta_{3} - 182) q^{29} + (38 \beta_{3} + 14) q^{31} + (159 \beta_{2} + 9 \beta_1) q^{32} + (96 \beta_{2} + 6 \beta_1) q^{33} + (2 \beta_{3} + 92) q^{34} + ( - 27 \beta_{3} + 54) q^{36} + (134 \beta_{2} + 80 \beta_1) q^{37} + ( - 80 \beta_{2} + 64 \beta_1) q^{38} + ( - 30 \beta_{3} + 24) q^{39} + (108 \beta_{3} - 46) q^{41} + ( - 21 \beta_{2} + 21 \beta_1) q^{42} + ( - 248 \beta_{2} - 100 \beta_1) q^{43} + (84 \beta_{3} - 120) q^{44} + (64 \beta_{3} - 512) q^{46} + (164 \beta_{2} + 140 \beta_1) q^{47} + (33 \beta_{2} - 9 \beta_1) q^{48} - 49 q^{49} + (36 \beta_{3} + 42) q^{51} + (294 \beta_{2} - 54 \beta_1) q^{52} + (462 \beta_{2} - 58 \beta_1) q^{53} + (27 \beta_{3} - 54) q^{54} + ( - 7 \beta_{3} + 182) q^{56} + (180 \beta_{2} - 6 \beta_1) q^{57} + (290 \beta_{2} - 194 \beta_1) q^{58} + (76 \beta_{3} - 296) q^{59} + (84 \beta_{3} - 482) q^{61} + (328 \beta_{2} - 24 \beta_1) q^{62} + 63 \beta_{2} q^{63} + ( - 165 \beta_{3} + 322) q^{64} + ( - 84 \beta_{3} + 120) q^{66} + (64 \beta_{2} - 228 \beta_1) q^{67} + ( - 282 \beta_{2} - 6 \beta_1) q^{68} + ( - 144 \beta_{3} + 48) q^{69} + ( - 86 \beta_{3} + 198) q^{71} + ( - 225 \beta_{2} + 9 \beta_1) q^{72} + (182 \beta_{2} + 38 \beta_1) q^{73} + (26 \beta_{3} - 692) q^{74} + (192 \beta_{3} - 432) q^{76} + ( - 224 \beta_{2} - 14 \beta_1) q^{77} + ( - 294 \beta_{2} + 54 \beta_1) q^{78} + ( - 8 \beta_{3} - 912) q^{79} + 81 q^{81} + (1018 \beta_{2} - 154 \beta_1) q^{82} + ( - 228 \beta_{2} + 224 \beta_1) q^{83} + (63 \beta_{3} - 126) q^{84} + (48 \beta_{3} + 704) q^{86} + ( - 510 \beta_{2} + 36 \beta_1) q^{87} + (780 \beta_{2} + 20 \beta_1) q^{88} + (336 \beta_{3} - 566) q^{89} + (70 \beta_{3} - 56) q^{91} + (1344 \beta_{2} - 192 \beta_1) q^{92} + (156 \beta_{2} + 114 \beta_1) q^{93} + (116 \beta_{3} - 1352) q^{94} + ( - 27 \beta_{3} - 450) q^{96} + (474 \beta_{2} + 278 \beta_1) q^{97} + (49 \beta_{2} - 49 \beta_1) q^{98} + ( - 18 \beta_{3} - 270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{4} + 18 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{4} + 18 q^{6} - 36 q^{9} + 124 q^{11} - 42 q^{14} + 50 q^{16} + 244 q^{19} + 84 q^{21} - 306 q^{24} - 428 q^{26} - 704 q^{29} + 132 q^{31} + 372 q^{34} + 162 q^{36} + 36 q^{39} + 32 q^{41} - 312 q^{44} - 1920 q^{46} - 196 q^{49} + 240 q^{51} - 162 q^{54} + 714 q^{56} - 1032 q^{59} - 1760 q^{61} + 958 q^{64} + 312 q^{66} - 96 q^{69} + 620 q^{71} - 2716 q^{74} - 1344 q^{76} - 3664 q^{79} + 324 q^{81} - 378 q^{84} + 2912 q^{86} - 1592 q^{89} - 84 q^{91} - 5176 q^{94} - 1854 q^{96} - 1116 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{2} - 11\beta_1 \) 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
3.70156i
2.70156i
2.70156i
3.70156i
4.70156i 3.00000i −14.1047 0 14.1047 7.00000i 28.7016i −9.00000 0
274.2 1.70156i 3.00000i 5.10469 0 −5.10469 7.00000i 22.2984i −9.00000 0
274.3 1.70156i 3.00000i 5.10469 0 −5.10469 7.00000i 22.2984i −9.00000 0
274.4 4.70156i 3.00000i −14.1047 0 14.1047 7.00000i 28.7016i −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.j 4
5.b even 2 1 inner 525.4.d.j 4
5.c odd 4 1 105.4.a.g 2
5.c odd 4 1 525.4.a.i 2
15.e even 4 1 315.4.a.g 2
15.e even 4 1 1575.4.a.y 2
20.e even 4 1 1680.4.a.y 2
35.f even 4 1 735.4.a.q 2
105.k odd 4 1 2205.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 5.c odd 4 1
315.4.a.g 2 15.e even 4 1
525.4.a.i 2 5.c odd 4 1
525.4.d.j 4 1.a even 1 1 trivial
525.4.d.j 4 5.b even 2 1 inner
735.4.a.q 2 35.f even 4 1
1575.4.a.y 2 15.e even 4 1
1680.4.a.y 2 20.e even 4 1
2205.4.a.v 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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 25T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} - 62T_{11} + 920 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 25T^{2} + 64 \) 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
$11$ \( (T^{2} - 62 T + 920)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 2068 T^{2} + \cdots + 1032256 \) Copy content Toggle raw display
$17$ \( T^{4} + 3752 T^{2} + \cdots + 1157776 \) Copy content Toggle raw display
$19$ \( (T^{2} - 122 T + 3680)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 47360 T^{2} + \cdots + 554696704 \) Copy content Toggle raw display
$29$ \( (T^{2} + 352 T + 29500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 66 T - 13712)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 148872 T^{2} + \cdots + 3222151696 \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T - 119492)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 283408 T^{2} + \cdots + 4006383616 \) Copy content Toggle raw display
$47$ \( T^{4} + 419472 T^{2} + \cdots + 36888580096 \) Copy content Toggle raw display
$53$ \( T^{4} + 551124 T^{2} + \cdots + 42683560000 \) Copy content Toggle raw display
$59$ \( (T^{2} + 516 T + 7360)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 880 T + 121276)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1129040 T^{2} + \cdots + 251153327104 \) Copy content Toggle raw display
$71$ \( (T^{2} - 310 T - 51784)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 82740 T^{2} + \cdots + 138485824 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1832 T + 838400)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1259808 T^{2} + \cdots + 158964879616 \) Copy content Toggle raw display
$89$ \( (T^{2} + 796 T - 998780)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1808772 T^{2} + \cdots + 462312964096 \) Copy content Toggle raw display
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