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Properties

Label 575.2.b.d
Level $575$
Weight $2$
Character orbit 575.b
Analytic conductor $4.591$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.b (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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - 2) q^{6} + (2 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - 2) q^{6} + (2 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - 2 q^{9} + ( - 2 \beta_{2} - 4) q^{11} + (3 \beta_{3} + \beta_1) q^{12} - 3 \beta_{3} q^{13} - 2 q^{14} + 3 \beta_{2} q^{16} + (2 \beta_{3} - 2 \beta_1) q^{17} - 2 \beta_1 q^{18} + 2 q^{19} + ( - 2 \beta_{2} - 6) q^{21} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} - \beta_{3} q^{23} - 5 q^{24} + 3 \beta_{2} q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + (4 \beta_{3} + 2 \beta_1) q^{28} + 3 q^{29} + (6 \beta_{2} + 3) q^{31} + (5 \beta_{3} + \beta_1) q^{32} + ( - 8 \beta_{3} - 6 \beta_1) q^{33} + ( - 4 \beta_{2} + 2) q^{34} + ( - 2 \beta_{2} - 2) q^{36} - 2 \beta_1 q^{37} + 2 \beta_1 q^{38} + (6 \beta_{2} + 3) q^{39} + ( - 4 \beta_{2} - 1) q^{41} + ( - 2 \beta_{3} - 4 \beta_1) q^{42} + ( - 4 \beta_{2} - 6) q^{44} + \beta_{2} q^{46} + ( - \beta_{3} - 2 \beta_1) q^{47} + (6 \beta_{3} - 3 \beta_1) q^{48} + ( - 4 \beta_{2} - 1) q^{49} + ( - 6 \beta_{2} + 2) q^{51} + ( - 3 \beta_{3} - 3 \beta_1) q^{52} + (2 \beta_{3} - 4 \beta_1) q^{53} + (\beta_{2} - 2) q^{54} + ( - 2 \beta_{2} - 6) q^{56} + (2 \beta_{3} + 4 \beta_1) q^{57} + 3 \beta_1 q^{58} + ( - 4 \beta_{2} - 4) q^{59} + ( - 8 \beta_{2} - 2) q^{61} + (6 \beta_{3} - 3 \beta_1) q^{62} + ( - 4 \beta_{3} - 4 \beta_1) q^{63} + (2 \beta_{2} - 1) q^{64} + (2 \beta_{2} + 6) q^{66} + ( - 4 \beta_{3} + 2 \beta_1) q^{67} + 2 \beta_1 q^{68} + (2 \beta_{2} + 1) q^{69} + (2 \beta_{2} + 11) q^{71} + ( - 2 \beta_{3} - 4 \beta_1) q^{72} + ( - 9 \beta_{3} + 4 \beta_1) q^{73} + ( - 2 \beta_{2} + 2) q^{74} + (2 \beta_{2} + 2) q^{76} + ( - 12 \beta_{3} - 8 \beta_1) q^{77} + (6 \beta_{3} - 3 \beta_1) q^{78} + (8 \beta_{2} + 6) q^{79} - 11 q^{81} + ( - 4 \beta_{3} + 3 \beta_1) q^{82} + (10 \beta_{3} - 2 \beta_1) q^{83} + ( - 6 \beta_{2} - 8) q^{84} + (3 \beta_{3} + 6 \beta_1) q^{87} + ( - 8 \beta_{3} - 6 \beta_1) q^{88} + (4 \beta_{2} + 8) q^{89} + (6 \beta_{2} + 6) q^{91} + ( - \beta_{3} - \beta_1) q^{92} + 15 \beta_{3} q^{93} + ( - \beta_{2} + 2) q^{94} + ( - 9 \beta_{2} - 7) q^{96} + (14 \beta_{3} + 6 \beta_1) q^{97} + ( - 4 \beta_{3} + 3 \beta_1) q^{98} + (4 \beta_{2} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9} - 12 q^{11} - 8 q^{14} - 6 q^{16} + 8 q^{19} - 20 q^{21} - 20 q^{24} - 6 q^{26} + 12 q^{29} + 16 q^{34} - 4 q^{36} + 4 q^{41} - 16 q^{44} - 2 q^{46} + 4 q^{49} + 20 q^{51} - 10 q^{54} - 20 q^{56} - 8 q^{59} + 8 q^{61} - 8 q^{64} + 20 q^{66} + 40 q^{71} + 12 q^{74} + 4 q^{76} + 8 q^{79} - 44 q^{81} - 20 q^{84} + 24 q^{89} + 12 q^{91} + 10 q^{94} - 10 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 1.61803i
− 0.618034i
0.618034i
1.61803i
− 1.61803i − 2.23607i −0.618034 0 −3.61803 − 1.23607i − 2.23607i −2.00000 0
24.2 − 0.618034i − 2.23607i 1.61803 0 −1.38197 − 3.23607i − 2.23607i −2.00000 0
24.3 0.618034i 2.23607i 1.61803 0 −1.38197 3.23607i 2.23607i −2.00000 0
24.4 1.61803i 2.23607i −0.618034 0 −3.61803 1.23607i 2.23607i −2.00000 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 575.2.b.d 4
5.b even 2 1 inner 575.2.b.d 4
5.c odd 4 1 23.2.a.a ✓ 2
5.c odd 4 1 575.2.a.f 2
15.e even 4 1 207.2.a.d 2
15.e even 4 1 5175.2.a.be 2
20.e even 4 1 368.2.a.h 2
20.e even 4 1 9200.2.a.bt 2
35.f even 4 1 1127.2.a.c 2
40.i odd 4 1 1472.2.a.t 2
40.k even 4 1 1472.2.a.s 2
55.e even 4 1 2783.2.a.c 2
60.l odd 4 1 3312.2.a.ba 2
65.h odd 4 1 3887.2.a.i 2
85.g odd 4 1 6647.2.a.b 2
95.g even 4 1 8303.2.a.e 2
115.e even 4 1 529.2.a.a 2
115.k odd 44 10 529.2.c.o 20
115.l even 44 10 529.2.c.n 20
345.l odd 4 1 4761.2.a.w 2
460.k odd 4 1 8464.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a ✓ 2 5.c odd 4 1
207.2.a.d 2 15.e even 4 1
368.2.a.h 2 20.e even 4 1
529.2.a.a 2 115.e even 4 1
529.2.c.n 20 115.l even 44 10
529.2.c.o 20 115.k odd 44 10
575.2.a.f 2 5.c odd 4 1
575.2.b.d 4 1.a even 1 1 trivial
575.2.b.d 4 5.b even 2 1 inner
1127.2.a.c 2 35.f even 4 1
1472.2.a.s 2 40.k even 4 1
1472.2.a.t 2 40.i odd 4 1
2783.2.a.c 2 55.e even 4 1
3312.2.a.ba 2 60.l odd 4 1
3887.2.a.i 2 65.h odd 4 1
4761.2.a.w 2 345.l odd 4 1
5175.2.a.be 2 15.e even 4 1
6647.2.a.b 2 85.g odd 4 1
8303.2.a.e 2 95.g even 4 1
8464.2.a.bb 2 460.k odd 4 1
9200.2.a.bt 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T - 2)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 19)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$71$ \( (T^{2} - 20 T + 95)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 282 T^{2} + 10201 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 76)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 252 T^{2} + 13456 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 332T^{2} + 5776 \) Copy content Toggle raw display
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