Properties

Label 525.4.d.l
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: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} - 3 \beta_1 q^{3} + ( - 2 \beta_{3} - 1) q^{4} + (3 \beta_{3} + 3) q^{6} + 7 \beta_1 q^{7} + (5 \beta_{2} - 9 \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} - 3 \beta_1 q^{3} + ( - 2 \beta_{3} - 1) q^{4} + (3 \beta_{3} + 3) q^{6} + 7 \beta_1 q^{7} + (5 \beta_{2} - 9 \beta_1) q^{8} - 9 q^{9} + (20 \beta_{3} - 8) q^{11} + (6 \beta_{2} + 3 \beta_1) q^{12} + ( - 2 \beta_{2} - 38 \beta_1) q^{13} + ( - 7 \beta_{3} - 7) q^{14} + ( - 12 \beta_{3} - 39) q^{16} + (2 \beta_{2} + 62 \beta_1) q^{17} + ( - 9 \beta_{2} - 9 \beta_1) q^{18} + (16 \beta_{3} + 48) q^{19} + 21 q^{21} + (12 \beta_{2} + 152 \beta_1) q^{22} + ( - 34 \beta_{2} - 8 \beta_1) q^{23} + (15 \beta_{3} - 27) q^{24} + (40 \beta_{3} + 54) q^{26} + 27 \beta_1 q^{27} + ( - 14 \beta_{2} - 7 \beta_1) q^{28} + (54 \beta_{3} - 94) q^{29} + (18 \beta_{3} - 60) q^{31} + ( - 11 \beta_{2} - 207 \beta_1) q^{32} + ( - 60 \beta_{2} + 24 \beta_1) q^{33} + ( - 64 \beta_{3} - 78) q^{34} + (18 \beta_{3} + 9) q^{36} + (66 \beta_{2} + 66 \beta_1) q^{37} + (64 \beta_{2} + 176 \beta_1) q^{38} + ( - 6 \beta_{3} - 114) q^{39} + (80 \beta_{3} + 50) q^{41} + (21 \beta_{2} + 21 \beta_1) q^{42} + (62 \beta_{2} - 268 \beta_1) q^{43} + ( - 4 \beta_{3} - 312) q^{44} + (42 \beta_{3} + 280) q^{46} + (42 \beta_{2} + 464 \beta_1) q^{47} + (36 \beta_{2} + 117 \beta_1) q^{48} - 49 q^{49} + (6 \beta_{3} + 186) q^{51} + (78 \beta_{2} + 70 \beta_1) q^{52} + (64 \beta_{2} + 442 \beta_1) q^{53} + ( - 27 \beta_{3} - 27) q^{54} + ( - 35 \beta_{3} + 63) q^{56} + ( - 48 \beta_{2} - 144 \beta_1) q^{57} + ( - 40 \beta_{2} + 338 \beta_1) q^{58} + (204 \beta_{3} - 52) q^{59} + ( - 42 \beta_{3} - 234) q^{61} + ( - 42 \beta_{2} + 84 \beta_1) q^{62} - 63 \beta_1 q^{63} + (122 \beta_{3} - 17) q^{64} + (36 \beta_{3} + 456) q^{66} + ( - 38 \beta_{2} + 844 \beta_1) q^{67} + ( - 126 \beta_{2} - 94 \beta_1) q^{68} + ( - 102 \beta_{3} - 24) q^{69} + ( - 150 \beta_{3} - 68) q^{71} + ( - 45 \beta_{2} + 81 \beta_1) q^{72} + (322 \beta_{2} + 254 \beta_1) q^{73} + ( - 132 \beta_{3} - 594) q^{74} + ( - 112 \beta_{3} - 304) q^{76} + (140 \beta_{2} - 56 \beta_1) q^{77} + ( - 120 \beta_{2} - 162 \beta_1) q^{78} + (232 \beta_{3} + 216) q^{79} + 81 q^{81} + (130 \beta_{2} + 690 \beta_1) q^{82} + ( - 84 \beta_{2} - 292 \beta_1) q^{83} + ( - 42 \beta_{3} - 21) q^{84} + (206 \beta_{3} - 228) q^{86} + ( - 162 \beta_{2} + 282 \beta_1) q^{87} + ( - 220 \beta_{2} + 872 \beta_1) q^{88} + ( - 112 \beta_{3} + 702) q^{89} + (14 \beta_{3} + 266) q^{91} + (50 \beta_{2} + 552 \beta_1) q^{92} + ( - 54 \beta_{2} + 180 \beta_1) q^{93} + ( - 506 \beta_{3} - 800) q^{94} + ( - 33 \beta_{3} - 621) q^{96} + ( - 46 \beta_{2} + 594 \beta_1) q^{97} + ( - 49 \beta_{2} - 49 \beta_1) q^{98} + ( - 180 \beta_{3} + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 12 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 12 q^{6} - 36 q^{9} - 32 q^{11} - 28 q^{14} - 156 q^{16} + 192 q^{19} + 84 q^{21} - 108 q^{24} + 216 q^{26} - 376 q^{29} - 240 q^{31} - 312 q^{34} + 36 q^{36} - 456 q^{39} + 200 q^{41} - 1248 q^{44} + 1120 q^{46} - 196 q^{49} + 744 q^{51} - 108 q^{54} + 252 q^{56} - 208 q^{59} - 936 q^{61} - 68 q^{64} + 1824 q^{66} - 96 q^{69} - 272 q^{71} - 2376 q^{74} - 1216 q^{76} + 864 q^{79} + 324 q^{81} - 84 q^{84} - 912 q^{86} + 2808 q^{89} + 1064 q^{91} - 3200 q^{94} - 2484 q^{96} + 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 4 \) 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
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
3.82843i 3.00000i −6.65685 0 11.4853 7.00000i 5.14214i −9.00000 0
274.2 1.82843i 3.00000i 4.65685 0 −5.48528 7.00000i 23.1421i −9.00000 0
274.3 1.82843i 3.00000i 4.65685 0 −5.48528 7.00000i 23.1421i −9.00000 0
274.4 3.82843i 3.00000i −6.65685 0 11.4853 7.00000i 5.14214i −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.l 4
5.b even 2 1 inner 525.4.d.l 4
5.c odd 4 1 105.4.a.e 2
5.c odd 4 1 525.4.a.l 2
15.e even 4 1 315.4.a.k 2
15.e even 4 1 1575.4.a.q 2
20.e even 4 1 1680.4.a.bo 2
35.f even 4 1 735.4.a.o 2
105.k odd 4 1 2205.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 5.c odd 4 1
315.4.a.k 2 15.e even 4 1
525.4.a.l 2 5.c odd 4 1
525.4.d.l 4 1.a even 1 1 trivial
525.4.d.l 4 5.b even 2 1 inner
735.4.a.o 2 35.f even 4 1
1575.4.a.q 2 15.e even 4 1
1680.4.a.bo 2 20.e even 4 1
2205.4.a.bb 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 \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{4} + 18T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{2} + 16T_{11} - 3136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 18T^{2} + 49 \) 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
$11$ \( (T^{2} + 16 T - 3136)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 2952 T^{2} + 1993744 \) Copy content Toggle raw display
$17$ \( T^{4} + 7752 T^{2} + 14531344 \) Copy content Toggle raw display
$19$ \( (T^{2} - 96 T + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18624 T^{2} + 84345856 \) Copy content Toggle raw display
$29$ \( (T^{2} + 188 T - 14492)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 120 T + 1008)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 78408 T^{2} + 929762064 \) Copy content Toggle raw display
$41$ \( (T^{2} - 100 T - 48700)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1686909184 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 40475001856 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 26437459216 \) Copy content Toggle raw display
$59$ \( (T^{2} + 104 T - 330224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 468 T + 40644)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 491098214656 \) Copy content Toggle raw display
$71$ \( (T^{2} + 136 T - 175376)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 585157681936 \) Copy content Toggle raw display
$79$ \( (T^{2} - 432 T - 383936)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 283424 T^{2} + 830361856 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1404 T + 392452)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 112834184464 \) Copy content Toggle raw display
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