Properties

Label 525.4.a.i
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,Mod(1,525)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

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: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} - 7 q^{7} + ( - \beta - 25) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} - 3 q^{3} + (3 \beta + 3) q^{4} + (3 \beta + 3) q^{6} - 7 q^{7} + ( - \beta - 25) q^{8} + 9 q^{9} + ( - 2 \beta + 32) q^{11} + ( - 9 \beta - 9) q^{12} + (10 \beta - 2) q^{13} + (7 \beta + 7) q^{14} + (3 \beta + 11) q^{16} + (12 \beta - 26) q^{17} + ( - 9 \beta - 9) q^{18} + ( - 2 \beta - 60) q^{19} + 21 q^{21} + ( - 28 \beta - 12) q^{22} + (48 \beta - 32) q^{23} + (3 \beta + 75) q^{24} + ( - 18 \beta - 98) q^{26} - 27 q^{27} + ( - 21 \beta - 21) q^{28} + (12 \beta + 170) q^{29} + ( - 38 \beta + 52) q^{31} + ( - 9 \beta + 159) q^{32} + (6 \beta - 96) q^{33} + (2 \beta - 94) q^{34} + (27 \beta + 27) q^{36} + ( - 80 \beta + 134) q^{37} + (64 \beta + 80) q^{38} + ( - 30 \beta + 6) q^{39} + ( - 108 \beta + 62) q^{41} + ( - 21 \beta - 21) q^{42} + ( - 100 \beta + 248) q^{43} + (84 \beta + 36) q^{44} + ( - 64 \beta - 448) q^{46} + ( - 140 \beta + 164) q^{47} + ( - 9 \beta - 33) q^{48} + 49 q^{49} + ( - 36 \beta + 78) q^{51} + (54 \beta + 294) q^{52} + ( - 58 \beta - 462) q^{53} + (27 \beta + 27) q^{54} + (7 \beta + 175) q^{56} + (6 \beta + 180) q^{57} + ( - 194 \beta - 290) q^{58} + (76 \beta + 220) q^{59} + ( - 84 \beta - 398) q^{61} + (24 \beta + 328) q^{62} - 63 q^{63} + ( - 165 \beta - 157) q^{64} + (84 \beta + 36) q^{66} + (228 \beta + 64) q^{67} + ( - 6 \beta + 282) q^{68} + ( - 144 \beta + 96) q^{69} + (86 \beta + 112) q^{71} + ( - 9 \beta - 225) q^{72} + (38 \beta - 182) q^{73} + (26 \beta + 666) q^{74} + ( - 192 \beta - 240) q^{76} + (14 \beta - 224) q^{77} + (54 \beta + 294) q^{78} + ( - 8 \beta + 920) q^{79} + 81 q^{81} + (154 \beta + 1018) q^{82} + (224 \beta + 228) q^{83} + (63 \beta + 63) q^{84} + ( - 48 \beta + 752) q^{86} + ( - 36 \beta - 510) q^{87} + (20 \beta - 780) q^{88} + (336 \beta + 230) q^{89} + ( - 70 \beta + 14) q^{91} + (192 \beta + 1344) q^{92} + (114 \beta - 156) q^{93} + (116 \beta + 1236) q^{94} + (27 \beta - 477) q^{96} + ( - 278 \beta + 474) q^{97} + ( - 49 \beta - 49) q^{98} + ( - 18 \beta + 288) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 14 q^{7} - 51 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 14 q^{7} - 51 q^{8} + 18 q^{9} + 62 q^{11} - 27 q^{12} + 6 q^{13} + 21 q^{14} + 25 q^{16} - 40 q^{17} - 27 q^{18} - 122 q^{19} + 42 q^{21} - 52 q^{22} - 16 q^{23} + 153 q^{24} - 214 q^{26} - 54 q^{27} - 63 q^{28} + 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{33} - 186 q^{34} + 81 q^{36} + 188 q^{37} + 224 q^{38} - 18 q^{39} + 16 q^{41} - 63 q^{42} + 396 q^{43} + 156 q^{44} - 960 q^{46} + 188 q^{47} - 75 q^{48} + 98 q^{49} + 120 q^{51} + 642 q^{52} - 982 q^{53} + 81 q^{54} + 357 q^{56} + 366 q^{57} - 774 q^{58} + 516 q^{59} - 880 q^{61} + 680 q^{62} - 126 q^{63} - 479 q^{64} + 156 q^{66} + 356 q^{67} + 558 q^{68} + 48 q^{69} + 310 q^{71} - 459 q^{72} - 326 q^{73} + 1358 q^{74} - 672 q^{76} - 434 q^{77} + 642 q^{78} + 1832 q^{79} + 162 q^{81} + 2190 q^{82} + 680 q^{83} + 189 q^{84} + 1456 q^{86} - 1056 q^{87} - 1540 q^{88} + 796 q^{89} - 42 q^{91} + 2880 q^{92} - 198 q^{93} + 2588 q^{94} - 927 q^{96} + 670 q^{97} - 147 q^{98} + 558 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.70156
−2.70156
−4.70156 −3.00000 14.1047 0 14.1047 −7.00000 −28.7016 9.00000 0
1.2 1.70156 −3.00000 −5.10469 0 −5.10469 −7.00000 −22.2984 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.a.i 2
3.b odd 2 1 1575.4.a.y 2
5.b even 2 1 105.4.a.g 2
5.c odd 4 2 525.4.d.j 4
15.d odd 2 1 315.4.a.g 2
20.d odd 2 1 1680.4.a.y 2
35.c odd 2 1 735.4.a.q 2
105.g even 2 1 2205.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 5.b even 2 1
315.4.a.g 2 15.d odd 2 1
525.4.a.i 2 1.a even 1 1 trivial
525.4.d.j 4 5.c odd 4 2
735.4.a.q 2 35.c odd 2 1
1575.4.a.y 2 3.b odd 2 1
1680.4.a.y 2 20.d odd 2 1
2205.4.a.v 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} + 3T_{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 62T_{11} + 920 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 62T + 920 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T - 1016 \) Copy content Toggle raw display
$17$ \( T^{2} + 40T - 1076 \) Copy content Toggle raw display
$19$ \( T^{2} + 122T + 3680 \) Copy content Toggle raw display
$23$ \( T^{2} + 16T - 23552 \) Copy content Toggle raw display
$29$ \( T^{2} - 352T + 29500 \) Copy content Toggle raw display
$31$ \( T^{2} - 66T - 13712 \) Copy content Toggle raw display
$37$ \( T^{2} - 188T - 56764 \) Copy content Toggle raw display
$41$ \( T^{2} - 16T - 119492 \) Copy content Toggle raw display
$43$ \( T^{2} - 396T - 63296 \) Copy content Toggle raw display
$47$ \( T^{2} - 188T - 192064 \) Copy content Toggle raw display
$53$ \( T^{2} + 982T + 206600 \) Copy content Toggle raw display
$59$ \( T^{2} - 516T + 7360 \) Copy content Toggle raw display
$61$ \( T^{2} + 880T + 121276 \) Copy content Toggle raw display
$67$ \( T^{2} - 356T - 501152 \) Copy content Toggle raw display
$71$ \( T^{2} - 310T - 51784 \) Copy content Toggle raw display
$73$ \( T^{2} + 326T + 11768 \) Copy content Toggle raw display
$79$ \( T^{2} - 1832 T + 838400 \) Copy content Toggle raw display
$83$ \( T^{2} - 680T - 398704 \) Copy content Toggle raw display
$89$ \( T^{2} - 796T - 998780 \) Copy content Toggle raw display
$97$ \( T^{2} - 670T - 679936 \) Copy content Toggle raw display
show more
show less