Properties

Label 525.4.d.m
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: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) 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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} - 3 \beta_{2} q^{3} + 3 \beta_{3} q^{4} + (3 \beta_{3} - 6) q^{6} + 7 \beta_{2} q^{7} + (\beta_{2} + 5 \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} - 3 \beta_{2} q^{3} + 3 \beta_{3} q^{4} + (3 \beta_{3} - 6) q^{6} + 7 \beta_{2} q^{7} + (\beta_{2} + 5 \beta_1) q^{8} - 9 q^{9} + ( - 5 \beta_{3} - 13) q^{11} + ( - 9 \beta_{2} - 9 \beta_1) q^{12} + (11 \beta_{2} - 17 \beta_1) q^{13} + ( - 7 \beta_{3} + 14) q^{14} + (33 \beta_{3} - 28) q^{16} + (53 \beta_{2} + 27 \beta_1) q^{17} + (9 \beta_{2} - 9 \beta_1) q^{18} + 56 \beta_{3} q^{19} + 21 q^{21} + ( - 2 \beta_{2} - 8 \beta_1) q^{22} + (115 \beta_{2} - 24 \beta_1) q^{23} + (15 \beta_{3} - 12) q^{24} + ( - 45 \beta_{3} + 124) q^{26} + 27 \beta_{2} q^{27} + (21 \beta_{2} + 21 \beta_1) q^{28} + (84 \beta_{3} - 11) q^{29} + (13 \beta_{3} - 74) q^{31} + (135 \beta_{2} - 21 \beta_1) q^{32} + (54 \beta_{2} + 15 \beta_1) q^{33} + (\beta_{3} - 56) q^{34} - 27 \beta_{3} q^{36} + ( - 66 \beta_{2} - 19 \beta_1) q^{37} + (168 \beta_{2} - 56 \beta_1) q^{38} + ( - 51 \beta_{3} + 84) q^{39} + ( - 45 \beta_{3} + 140) q^{41} + ( - 21 \beta_{2} + 21 \beta_1) q^{42} + (373 \beta_{2} - 58 \beta_1) q^{43} + ( - 54 \beta_{3} - 60) q^{44} + ( - 163 \beta_{3} + 374) q^{46} + ( - 120 \beta_{2} - 88 \beta_1) q^{47} + ( - 15 \beta_{2} - 99 \beta_1) q^{48} - 49 q^{49} + (81 \beta_{3} + 78) q^{51} + ( - 171 \beta_{2} + 33 \beta_1) q^{52} + (119 \beta_{2} + 89 \beta_1) q^{53} + ( - 27 \beta_{3} + 54) q^{54} + ( - 35 \beta_{3} + 28) q^{56} + ( - 168 \beta_{2} - 168 \beta_1) q^{57} + (263 \beta_{2} - 95 \beta_1) q^{58} + (209 \beta_{3} + 116) q^{59} + (273 \beta_{3} - 248) q^{61} + (113 \beta_{2} - 87 \beta_1) q^{62} - 63 \beta_{2} q^{63} + (87 \beta_{3} + 172) q^{64} + ( - 24 \beta_{3} + 18) q^{66} + ( - 640 \beta_{2} - 123 \beta_1) q^{67} + (483 \beta_{2} + 159 \beta_1) q^{68} + ( - 72 \beta_{3} + 417) q^{69} + ( - 235 \beta_{3} + 427) q^{71} + ( - 9 \beta_{2} - 45 \beta_1) q^{72} + (90 \beta_{2} - 88 \beta_1) q^{73} + (28 \beta_{3} - 18) q^{74} + (168 \beta_{3} + 672) q^{76} + ( - 126 \beta_{2} - 35 \beta_1) q^{77} + ( - 237 \beta_{2} + 135 \beta_1) q^{78} + ( - 103 \beta_{3} + 265) q^{79} + 81 q^{81} + ( - 275 \beta_{2} + 185 \beta_1) q^{82} + (821 \beta_{2} + 431 \beta_1) q^{83} + 63 \beta_{3} q^{84} + ( - 489 \beta_{3} + 1094) q^{86} + ( - 219 \beta_{2} - 252 \beta_1) q^{87} + ( - 118 \beta_{2} - 70 \beta_1) q^{88} + ( - 522 \beta_{3} + 28) q^{89} + (119 \beta_{3} - 196) q^{91} + (57 \beta_{2} + 345 \beta_1) q^{92} + (183 \beta_{2} - 39 \beta_1) q^{93} + ( - 56 \beta_{3} + 288) q^{94} + ( - 63 \beta_{3} + 468) q^{96} + (266 \beta_{2} + 704 \beta_1) q^{97} + (49 \beta_{2} - 49 \beta_1) q^{98} + (45 \beta_{3} + 117) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} - 18 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} - 18 q^{6} - 36 q^{9} - 62 q^{11} + 42 q^{14} - 46 q^{16} + 112 q^{19} + 84 q^{21} - 18 q^{24} + 406 q^{26} + 124 q^{29} - 270 q^{31} - 222 q^{34} - 54 q^{36} + 234 q^{39} + 470 q^{41} - 348 q^{44} + 1170 q^{46} - 196 q^{49} + 474 q^{51} + 162 q^{54} + 42 q^{56} + 882 q^{59} - 446 q^{61} + 862 q^{64} + 24 q^{66} + 1524 q^{69} + 1238 q^{71} - 16 q^{74} + 3024 q^{76} + 854 q^{79} + 324 q^{81} + 126 q^{84} + 3398 q^{86} - 932 q^{89} - 546 q^{91} + 1040 q^{94} + 1746 q^{96} + 558 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) 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
2.56155i
1.56155i
1.56155i
2.56155i
3.56155i 3.00000i −4.68466 0 −10.6847 7.00000i 11.8078i −9.00000 0
274.2 0.561553i 3.00000i 7.68466 0 1.68466 7.00000i 8.80776i −9.00000 0
274.3 0.561553i 3.00000i 7.68466 0 1.68466 7.00000i 8.80776i −9.00000 0
274.4 3.56155i 3.00000i −4.68466 0 −10.6847 7.00000i 11.8078i −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.m 4
5.b even 2 1 inner 525.4.d.m 4
5.c odd 4 1 525.4.a.j 2
5.c odd 4 1 525.4.a.m yes 2
15.e even 4 1 1575.4.a.o 2
15.e even 4 1 1575.4.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 5.c odd 4 1
525.4.a.m yes 2 5.c odd 4 1
525.4.d.m 4 1.a even 1 1 trivial
525.4.d.m 4 5.b even 2 1 inner
1575.4.a.o 2 15.e even 4 1
1575.4.a.x 2 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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 13T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 31T_{11} + 134 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 13T^{2} + 4 \) 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
$11$ \( (T^{2} + 31 T + 134)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 3217 T^{2} + 719104 \) Copy content Toggle raw display
$17$ \( T^{4} + 9317 T^{2} + 2365444 \) Copy content Toggle raw display
$19$ \( (T^{2} - 56 T - 12544)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 37154 T^{2} + 187169761 \) Copy content Toggle raw display
$29$ \( (T^{2} - 62 T - 29027)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 135 T + 3838)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 9453 T^{2} + 2748964 \) Copy content Toggle raw display
$41$ \( (T^{2} - 235 T + 5200)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 21699352249 \) Copy content Toggle raw display
$47$ \( T^{4} + 77376 T^{2} + 736362496 \) Copy content Toggle raw display
$53$ \( T^{4} + 78429 T^{2} + 790396996 \) Copy content Toggle raw display
$59$ \( (T^{2} - 441 T - 137024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 223 T - 304316)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 73096692496 \) Copy content Toggle raw display
$71$ \( (T^{2} - 619 T - 138916)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 101736 T^{2} + 223681936 \) Copy content Toggle raw display
$79$ \( (T^{2} - 427 T + 494)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 178805505316 \) Copy content Toggle raw display
$89$ \( (T^{2} + 466 T - 1103768)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 4405683456784 \) Copy content Toggle raw display
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