Properties

Label 525.6.d.c
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 5 i q^{2} - 9 i q^{3} + 7 q^{4} + 45 q^{6} - 49 i q^{7} + 195 i q^{8} - 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 5 i q^{2} - 9 i q^{3} + 7 q^{4} + 45 q^{6} - 49 i q^{7} + 195 i q^{8} - 81 q^{9} + 52 q^{11} - 63 i q^{12} + 770 i q^{13} + 245 q^{14} - 751 q^{16} - 2022 i q^{17} - 405 i q^{18} - 1732 q^{19} - 441 q^{21} + 260 i q^{22} + 576 i q^{23} + 1755 q^{24} - 3850 q^{26} + 729 i q^{27} - 343 i q^{28} - 5518 q^{29} + 6336 q^{31} + 2485 i q^{32} - 468 i q^{33} + 10110 q^{34} - 567 q^{36} - 7338 i q^{37} - 8660 i q^{38} + 6930 q^{39} - 3262 q^{41} - 2205 i q^{42} - 5420 i q^{43} + 364 q^{44} - 2880 q^{46} + 864 i q^{47} + 6759 i q^{48} - 2401 q^{49} - 18198 q^{51} + 5390 i q^{52} - 4182 i q^{53} - 3645 q^{54} + 9555 q^{56} + 15588 i q^{57} - 27590 i q^{58} + 11220 q^{59} - 45602 q^{61} + 31680 i q^{62} + 3969 i q^{63} - 36457 q^{64} + 2340 q^{66} + 1396 i q^{67} - 14154 i q^{68} + 5184 q^{69} + 18720 q^{71} - 15795 i q^{72} - 46362 i q^{73} + 36690 q^{74} - 12124 q^{76} - 2548 i q^{77} + 34650 i q^{78} - 97424 q^{79} + 6561 q^{81} - 16310 i q^{82} + 81228 i q^{83} - 3087 q^{84} + 27100 q^{86} + 49662 i q^{87} + 10140 i q^{88} + 3182 q^{89} + 37730 q^{91} + 4032 i q^{92} - 57024 i q^{93} - 4320 q^{94} + 22365 q^{96} + 4914 i q^{97} - 12005 i q^{98} - 4212 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 90 q^{6} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} + 90 q^{6} - 162 q^{9} + 104 q^{11} + 490 q^{14} - 1502 q^{16} - 3464 q^{19} - 882 q^{21} + 3510 q^{24} - 7700 q^{26} - 11036 q^{29} + 12672 q^{31} + 20220 q^{34} - 1134 q^{36} + 13860 q^{39} - 6524 q^{41} + 728 q^{44} - 5760 q^{46} - 4802 q^{49} - 36396 q^{51} - 7290 q^{54} + 19110 q^{56} + 22440 q^{59} - 91204 q^{61} - 72914 q^{64} + 4680 q^{66} + 10368 q^{69} + 37440 q^{71} + 73380 q^{74} - 24248 q^{76} - 194848 q^{79} + 13122 q^{81} - 6174 q^{84} + 54200 q^{86} + 6364 q^{89} + 75460 q^{91} - 8640 q^{94} + 44730 q^{96} - 8424 q^{99}+O(q^{100}) \) 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
5.00000i 9.00000i 7.00000 0 45.0000 49.0000i 195.000i −81.0000 0
274.2 5.00000i 9.00000i 7.00000 0 45.0000 49.0000i 195.000i −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.c 2
5.b even 2 1 inner 525.6.d.c 2
5.c odd 4 1 21.6.a.c 1
5.c odd 4 1 525.6.a.b 1
15.e even 4 1 63.6.a.b 1
20.e even 4 1 336.6.a.i 1
35.f even 4 1 147.6.a.f 1
35.k even 12 2 147.6.e.d 2
35.l odd 12 2 147.6.e.c 2
60.l odd 4 1 1008.6.a.a 1
105.k odd 4 1 441.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 5.c odd 4 1
63.6.a.b 1 15.e even 4 1
147.6.a.f 1 35.f even 4 1
147.6.e.c 2 35.l odd 12 2
147.6.e.d 2 35.k even 12 2
336.6.a.i 1 20.e even 4 1
441.6.a.c 1 105.k odd 4 1
525.6.a.b 1 5.c odd 4 1
525.6.d.c 2 1.a even 1 1 trivial
525.6.d.c 2 5.b even 2 1 inner
1008.6.a.a 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 25 \) acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 25 \) 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
$11$ \( (T - 52)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 592900 \) Copy content Toggle raw display
$17$ \( T^{2} + 4088484 \) Copy content Toggle raw display
$19$ \( (T + 1732)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 331776 \) Copy content Toggle raw display
$29$ \( (T + 5518)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6336)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 53846244 \) Copy content Toggle raw display
$41$ \( (T + 3262)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 29376400 \) Copy content Toggle raw display
$47$ \( T^{2} + 746496 \) Copy content Toggle raw display
$53$ \( T^{2} + 17489124 \) Copy content Toggle raw display
$59$ \( (T - 11220)^{2} \) Copy content Toggle raw display
$61$ \( (T + 45602)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1948816 \) Copy content Toggle raw display
$71$ \( (T - 18720)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2149435044 \) Copy content Toggle raw display
$79$ \( (T + 97424)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6597987984 \) Copy content Toggle raw display
$89$ \( (T - 3182)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 24147396 \) Copy content Toggle raw display
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