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Properties

Label 475.3.d.b
Level $475$
Weight $3$
Character orbit 475.d
Analytic conductor $12.943$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,3,Mod(474,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.474");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 19)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{3} q^{3} + 9 q^{4} + 13 q^{6} + \beta_1 q^{7} + 5 \beta_{3} q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{3} q^{3} + 9 q^{4} + 13 q^{6} + \beta_1 q^{7} + 5 \beta_{3} q^{8} + 4 q^{9} - 10 q^{11} + 9 \beta_{3} q^{12} - \beta_{3} q^{13} + \beta_{2} q^{14} + 29 q^{16} - 3 \beta_1 q^{17} + 4 \beta_{3} q^{18} + ( - \beta_{2} + 6) q^{19} + \beta_{2} q^{21} - 10 \beta_{3} q^{22} + 7 \beta_1 q^{23} + 65 q^{24} - 13 q^{26} - 5 \beta_{3} q^{27} + 9 \beta_1 q^{28} - \beta_{2} q^{29} - 2 \beta_{2} q^{31} + 9 \beta_{3} q^{32} - 10 \beta_{3} q^{33} - 3 \beta_{2} q^{34} + 36 q^{36} - 6 \beta_{3} q^{37} + (6 \beta_{3} - 13 \beta_1) q^{38} - 13 q^{39} + 2 \beta_{2} q^{41} + 13 \beta_1 q^{42} - 4 \beta_1 q^{43} - 90 q^{44} + 7 \beta_{2} q^{46} - 2 \beta_1 q^{47} + 29 \beta_{3} q^{48} + 24 q^{49} - 3 \beta_{2} q^{51} - 9 \beta_{3} q^{52} + 21 \beta_{3} q^{53} - 65 q^{54} + 5 \beta_{2} q^{56} + (6 \beta_{3} - 13 \beta_1) q^{57} - 13 \beta_1 q^{58} - \beta_{2} q^{59} - 40 q^{61} - 26 \beta_1 q^{62} + 4 \beta_1 q^{63} + q^{64} - 130 q^{66} + 11 \beta_{3} q^{67} - 27 \beta_1 q^{68} + 7 \beta_{2} q^{69} + 6 \beta_{2} q^{71} + 20 \beta_{3} q^{72} + 21 \beta_1 q^{73} - 78 q^{74} + ( - 9 \beta_{2} + 54) q^{76} - 10 \beta_1 q^{77} - 13 \beta_{3} q^{78} + 2 \beta_{2} q^{79} - 101 q^{81} + 26 \beta_1 q^{82} - 8 \beta_1 q^{83} + 9 \beta_{2} q^{84} - 4 \beta_{2} q^{86} - 13 \beta_1 q^{87} - 50 \beta_{3} q^{88} - \beta_{2} q^{91} + 63 \beta_1 q^{92} - 26 \beta_1 q^{93} - 2 \beta_{2} q^{94} + 117 q^{96} + 34 \beta_{3} q^{97} + 24 \beta_{3} q^{98} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{4} + 52 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{4} + 52 q^{6} + 16 q^{9} - 40 q^{11} + 116 q^{16} + 24 q^{19} + 260 q^{24} - 52 q^{26} + 144 q^{36} - 52 q^{39} - 360 q^{44} + 96 q^{49} - 260 q^{54} - 160 q^{61} + 4 q^{64} - 520 q^{66} - 312 q^{74} + 216 q^{76} - 404 q^{81} + 468 q^{96} - 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{3} + 20\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{3} + 50\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{2} + 5\beta_1 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1
2.30278i
− 2.30278i
− 1.30278i
1.30278i
−3.60555 −3.60555 9.00000 0 13.0000 − 5.00000i −18.0278 4.00000 0
474.2 −3.60555 −3.60555 9.00000 0 13.0000 5.00000i −18.0278 4.00000 0
474.3 3.60555 3.60555 9.00000 0 13.0000 − 5.00000i 18.0278 4.00000 0
474.4 3.60555 3.60555 9.00000 0 13.0000 5.00000i 18.0278 4.00000 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
19.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.d.b 4
5.b even 2 1 inner 475.3.d.b 4
5.c odd 4 1 19.3.b.b ✓ 2
5.c odd 4 1 475.3.c.b 2
15.e even 4 1 171.3.c.b 2
19.b odd 2 1 inner 475.3.d.b 4
20.e even 4 1 304.3.e.d 2
40.i odd 4 1 1216.3.e.g 2
40.k even 4 1 1216.3.e.h 2
60.l odd 4 1 2736.3.o.d 2
95.d odd 2 1 inner 475.3.d.b 4
95.g even 4 1 19.3.b.b ✓ 2
95.g even 4 1 475.3.c.b 2
95.l even 12 2 361.3.d.b 4
95.m odd 12 2 361.3.d.b 4
95.q odd 36 6 361.3.f.d 12
95.r even 36 6 361.3.f.d 12
285.j odd 4 1 171.3.c.b 2
380.j odd 4 1 304.3.e.d 2
760.t even 4 1 1216.3.e.g 2
760.y odd 4 1 1216.3.e.h 2
1140.w even 4 1 2736.3.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b ✓ 2 5.c odd 4 1
19.3.b.b ✓ 2 95.g even 4 1
171.3.c.b 2 15.e even 4 1
171.3.c.b 2 285.j odd 4 1
304.3.e.d 2 20.e even 4 1
304.3.e.d 2 380.j odd 4 1
361.3.d.b 4 95.l even 12 2
361.3.d.b 4 95.m odd 12 2
361.3.f.d 12 95.q odd 36 6
361.3.f.d 12 95.r even 36 6
475.3.c.b 2 5.c odd 4 1
475.3.c.b 2 95.g even 4 1
475.3.d.b 4 1.a even 1 1 trivial
475.3.d.b 4 5.b even 2 1 inner
475.3.d.b 4 19.b odd 2 1 inner
475.3.d.b 4 95.d odd 2 1 inner
1216.3.e.g 2 40.i odd 4 1
1216.3.e.g 2 760.t even 4 1
1216.3.e.h 2 40.k even 4 1
1216.3.e.h 2 760.y odd 4 1
2736.3.o.d 2 60.l odd 4 1
2736.3.o.d 2 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 13 \) acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T + 10)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12 T + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1225)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 325)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1300)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 468)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1300)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 5733)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 325)^{2} \) Copy content Toggle raw display
$61$ \( (T + 40)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 1573)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 11700)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1300)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1600)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 15028)^{2} \) Copy content Toggle raw display
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