Properties

Label 75.3.f.b
Level $75$
Weight $3$
Character orbit 75.f
Analytic conductor $2.044$
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] = [75,3,Mod(7,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), 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: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{4} + 6 q^{6} - 5 \beta_1 q^{7} + 8 \beta_{3} q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{4} + 6 q^{6} - 5 \beta_1 q^{7} + 8 \beta_{3} q^{8} - 3 \beta_{2} q^{9} + 6 q^{11} + 8 \beta_1 q^{12} + 3 \beta_{3} q^{13} - 30 \beta_{2} q^{14} - 16 q^{16} - 14 \beta_1 q^{17} - 6 \beta_{3} q^{18} + 23 \beta_{2} q^{19} - 15 q^{21} + 12 \beta_1 q^{22} - 10 \beta_{3} q^{23} + 24 \beta_{2} q^{24} - 18 q^{26} - 3 \beta_1 q^{27} - 40 \beta_{3} q^{28} + 6 \beta_{2} q^{29} + 25 q^{31} - 6 \beta_{3} q^{33} - 84 \beta_{2} q^{34} + 24 q^{36} + 20 \beta_1 q^{37} + 46 \beta_{3} q^{38} + 9 \beta_{2} q^{39} - 60 q^{41} - 30 \beta_1 q^{42} + 49 \beta_{3} q^{43} + 48 \beta_{2} q^{44} + 60 q^{46} - 6 \beta_1 q^{47} + 16 \beta_{3} q^{48} + 26 \beta_{2} q^{49} - 42 q^{51} - 24 \beta_1 q^{52} - 20 \beta_{3} q^{53} - 18 \beta_{2} q^{54} + 120 q^{56} + 23 \beta_1 q^{57} + 12 \beta_{3} q^{58} - 18 \beta_{2} q^{59} - 37 q^{61} + 50 \beta_1 q^{62} + 15 \beta_{3} q^{63} + 64 \beta_{2} q^{64} + 36 q^{66} + 21 \beta_1 q^{67} - 112 \beta_{3} q^{68} - 30 \beta_{2} q^{69} + 132 q^{71} + 24 \beta_1 q^{72} - 20 \beta_{3} q^{73} + 120 \beta_{2} q^{74} - 184 q^{76} - 30 \beta_1 q^{77} + 18 \beta_{3} q^{78} + 10 \beta_{2} q^{79} - 9 q^{81} - 120 \beta_1 q^{82} + 2 \beta_{3} q^{83} - 120 \beta_{2} q^{84} - 294 q^{86} + 6 \beta_1 q^{87} + 48 \beta_{3} q^{88} - 132 \beta_{2} q^{89} + 45 q^{91} + 80 \beta_1 q^{92} - 25 \beta_{3} q^{93} - 36 \beta_{2} q^{94} - 19 \beta_1 q^{97} + 52 \beta_{3} q^{98} - 18 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{6} + 24 q^{11} - 64 q^{16} - 60 q^{21} - 72 q^{26} + 100 q^{31} + 96 q^{36} - 240 q^{41} + 240 q^{46} - 168 q^{51} + 480 q^{56} - 148 q^{61} + 144 q^{66} + 528 q^{71} - 736 q^{76} - 36 q^{81} - 1176 q^{86} + 180 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−2.44949 2.44949i −1.22474 + 1.22474i 8.00000i 0 6.00000 6.12372 + 6.12372i 9.79796 9.79796i 3.00000i 0
7.2 2.44949 + 2.44949i 1.22474 1.22474i 8.00000i 0 6.00000 −6.12372 6.12372i −9.79796 + 9.79796i 3.00000i 0
43.1 −2.44949 + 2.44949i −1.22474 1.22474i 8.00000i 0 6.00000 6.12372 6.12372i 9.79796 + 9.79796i 3.00000i 0
43.2 2.44949 2.44949i 1.22474 + 1.22474i 8.00000i 0 6.00000 −6.12372 + 6.12372i −9.79796 9.79796i 3.00000i 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
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.f.b 4
3.b odd 2 1 225.3.g.b 4
4.b odd 2 1 1200.3.bg.g 4
5.b even 2 1 inner 75.3.f.b 4
5.c odd 4 2 inner 75.3.f.b 4
15.d odd 2 1 225.3.g.b 4
15.e even 4 2 225.3.g.b 4
20.d odd 2 1 1200.3.bg.g 4
20.e even 4 2 1200.3.bg.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.f.b 4 1.a even 1 1 trivial
75.3.f.b 4 5.b even 2 1 inner
75.3.f.b 4 5.c odd 4 2 inner
225.3.g.b 4 3.b odd 2 1
225.3.g.b 4 15.d odd 2 1
225.3.g.b 4 15.e even 4 2
1200.3.bg.g 4 4.b odd 2 1
1200.3.bg.g 4 20.d odd 2 1
1200.3.bg.g 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 144 \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 144 \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 5625 \) Copy content Toggle raw display
$11$ \( (T - 6)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 729 \) Copy content Toggle raw display
$17$ \( T^{4} + 345744 \) Copy content Toggle raw display
$19$ \( (T^{2} + 529)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 90000 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T - 25)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 1440000 \) Copy content Toggle raw display
$41$ \( (T + 60)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 51883209 \) Copy content Toggle raw display
$47$ \( T^{4} + 11664 \) Copy content Toggle raw display
$53$ \( T^{4} + 1440000 \) Copy content Toggle raw display
$59$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$61$ \( (T + 37)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 1750329 \) Copy content Toggle raw display
$71$ \( (T - 132)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 1440000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 144 \) Copy content Toggle raw display
$89$ \( (T^{2} + 17424)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1172889 \) Copy content Toggle raw display
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