Properties

Label 525.6.d.b
Level $525$
Weight $6$
Character orbit 525.d
Analytic conductor $84.202$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(84.2015054018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 i q^{2} - 9 i q^{3} - 4 q^{4} + 54 q^{6} - 49 i q^{7} + 168 i q^{8} - 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 6 i q^{2} - 9 i q^{3} - 4 q^{4} + 54 q^{6} - 49 i q^{7} + 168 i q^{8} - 81 q^{9} + 444 q^{11} + 36 i q^{12} - 442 i q^{13} + 294 q^{14} - 1136 q^{16} + 126 i q^{17} - 486 i q^{18} - 2684 q^{19} - 441 q^{21} + 2664 i q^{22} + 4200 i q^{23} + 1512 q^{24} + 2652 q^{26} + 729 i q^{27} + 196 i q^{28} + 5442 q^{29} + 80 q^{31} - 1440 i q^{32} - 3996 i q^{33} - 756 q^{34} + 324 q^{36} + 5434 i q^{37} - 16104 i q^{38} - 3978 q^{39} + 7962 q^{41} - 2646 i q^{42} - 11524 i q^{43} - 1776 q^{44} - 25200 q^{46} + 13920 i q^{47} + 10224 i q^{48} - 2401 q^{49} + 1134 q^{51} + 1768 i q^{52} - 9594 i q^{53} - 4374 q^{54} + 8232 q^{56} + 24156 i q^{57} + 32652 i q^{58} - 27492 q^{59} + 49478 q^{61} + 480 i q^{62} + 3969 i q^{63} - 27712 q^{64} + 23976 q^{66} + 59356 i q^{67} - 504 i q^{68} + 37800 q^{69} + 32040 q^{71} - 13608 i q^{72} - 61846 i q^{73} - 32604 q^{74} + 10736 q^{76} - 21756 i q^{77} - 23868 i q^{78} + 65776 q^{79} + 6561 q^{81} + 47772 i q^{82} + 40188 i q^{83} + 1764 q^{84} + 69144 q^{86} - 48978 i q^{87} + 74592 i q^{88} + 7974 q^{89} - 21658 q^{91} - 16800 i q^{92} - 720 i q^{93} - 83520 q^{94} - 12960 q^{96} + 143662 i q^{97} - 14406 i q^{98} - 35964 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 108 q^{6} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 108 q^{6} - 162 q^{9} + 888 q^{11} + 588 q^{14} - 2272 q^{16} - 5368 q^{19} - 882 q^{21} + 3024 q^{24} + 5304 q^{26} + 10884 q^{29} + 160 q^{31} - 1512 q^{34} + 648 q^{36} - 7956 q^{39} + 15924 q^{41} - 3552 q^{44} - 50400 q^{46} - 4802 q^{49} + 2268 q^{51} - 8748 q^{54} + 16464 q^{56} - 54984 q^{59} + 98956 q^{61} - 55424 q^{64} + 47952 q^{66} + 75600 q^{69} + 64080 q^{71} - 65208 q^{74} + 21472 q^{76} + 131552 q^{79} + 13122 q^{81} + 3528 q^{84} + 138288 q^{86} + 15948 q^{89} - 43316 q^{91} - 167040 q^{94} - 25920 q^{96} - 71928 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
6.00000i 9.00000i −4.00000 0 54.0000 49.0000i 168.000i −81.0000 0
274.2 6.00000i 9.00000i −4.00000 0 54.0000 49.0000i 168.000i −81.0000 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 525.6.d.b 2
5.b even 2 1 inner 525.6.d.b 2
5.c odd 4 1 21.6.a.a 1
5.c odd 4 1 525.6.a.d 1
15.e even 4 1 63.6.a.d 1
20.e even 4 1 336.6.a.r 1
35.f even 4 1 147.6.a.b 1
35.k even 12 2 147.6.e.i 2
35.l odd 12 2 147.6.e.j 2
60.l odd 4 1 1008.6.a.c 1
105.k odd 4 1 441.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 5.c odd 4 1
63.6.a.d 1 15.e even 4 1
147.6.a.b 1 35.f even 4 1
147.6.e.i 2 35.k even 12 2
147.6.e.j 2 35.l odd 12 2
336.6.a.r 1 20.e even 4 1
441.6.a.j 1 105.k odd 4 1
525.6.a.d 1 5.c odd 4 1
525.6.d.b 2 1.a even 1 1 trivial
525.6.d.b 2 5.b even 2 1 inner
1008.6.a.c 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 36 \) acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 36 \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 444)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 195364 \) Copy content Toggle raw display
$17$ \( T^{2} + 15876 \) Copy content Toggle raw display
$19$ \( (T + 2684)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17640000 \) Copy content Toggle raw display
$29$ \( (T - 5442)^{2} \) Copy content Toggle raw display
$31$ \( (T - 80)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 29528356 \) Copy content Toggle raw display
$41$ \( (T - 7962)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 132802576 \) Copy content Toggle raw display
$47$ \( T^{2} + 193766400 \) Copy content Toggle raw display
$53$ \( T^{2} + 92044836 \) Copy content Toggle raw display
$59$ \( (T + 27492)^{2} \) Copy content Toggle raw display
$61$ \( (T - 49478)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 3523134736 \) Copy content Toggle raw display
$71$ \( (T - 32040)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3824927716 \) Copy content Toggle raw display
$79$ \( (T - 65776)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1615075344 \) Copy content Toggle raw display
$89$ \( (T - 7974)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 20638770244 \) Copy content Toggle raw display
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