Properties

Label 75.3.c.f
Level $75$
Weight $3$
Character orbit 75.c
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
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] = [75,3,Mod(26,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta + 1) q^{2} + ( - \beta + 3) q^{3} - 7 q^{4} + ( - 5 \beta - 3) q^{6} + (6 \beta - 3) q^{8} + ( - 5 \beta + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta + 1) q^{2} + ( - \beta + 3) q^{3} - 7 q^{4} + ( - 5 \beta - 3) q^{6} + (6 \beta - 3) q^{8} + ( - 5 \beta + 6) q^{9} + (10 \beta - 5) q^{11} + (7 \beta - 21) q^{12} + 10 q^{13} + 5 q^{16} + ( - 2 \beta + 1) q^{17} + ( - 7 \beta - 24) q^{18} + 7 q^{19} + 55 q^{22} + ( - 12 \beta + 6) q^{23} + (15 \beta + 9) q^{24} + ( - 20 \beta + 10) q^{26} + ( - 16 \beta + 3) q^{27} + (20 \beta - 10) q^{29} + 42 q^{31} + (14 \beta - 7) q^{32} + (25 \beta + 15) q^{33} - 11 q^{34} + (35 \beta - 42) q^{36} - 40 q^{37} + ( - 14 \beta + 7) q^{38} + ( - 10 \beta + 30) q^{39} + ( - 10 \beta + 5) q^{41} - 50 q^{43} + ( - 70 \beta + 35) q^{44} - 66 q^{46} + (28 \beta - 14) q^{47} + ( - 5 \beta + 15) q^{48} - 49 q^{49} + ( - 5 \beta - 3) q^{51} - 70 q^{52} + (28 \beta - 14) q^{53} + (10 \beta - 93) q^{54} + ( - 7 \beta + 21) q^{57} + 110 q^{58} + (40 \beta - 20) q^{59} - 8 q^{61} + ( - 84 \beta + 42) q^{62} + 97 q^{64} + ( - 55 \beta + 165) q^{66} + 45 q^{67} + (14 \beta - 7) q^{68} + ( - 30 \beta - 18) q^{69} + (20 \beta - 10) q^{71} + (21 \beta + 72) q^{72} - 35 q^{73} + (80 \beta - 40) q^{74} - 49 q^{76} + ( - 50 \beta - 30) q^{78} + 12 q^{79} + ( - 35 \beta - 39) q^{81} - 55 q^{82} + ( - 42 \beta + 21) q^{83} + (100 \beta - 50) q^{86} + (50 \beta + 30) q^{87} - 165 q^{88} + ( - 90 \beta + 45) q^{89} + (84 \beta - 42) q^{92} + ( - 42 \beta + 126) q^{93} + 154 q^{94} + (35 \beta + 21) q^{96} - 70 q^{97} + (98 \beta - 49) q^{98} + (35 \beta + 120) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} - 14 q^{4} - 11 q^{6} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} - 14 q^{4} - 11 q^{6} + 7 q^{9} - 35 q^{12} + 20 q^{13} + 10 q^{16} - 55 q^{18} + 14 q^{19} + 110 q^{22} + 33 q^{24} - 10 q^{27} + 84 q^{31} + 55 q^{33} - 22 q^{34} - 49 q^{36} - 80 q^{37} + 50 q^{39} - 100 q^{43} - 132 q^{46} + 25 q^{48} - 98 q^{49} - 11 q^{51} - 140 q^{52} - 176 q^{54} + 35 q^{57} + 220 q^{58} - 16 q^{61} + 194 q^{64} + 275 q^{66} + 90 q^{67} - 66 q^{69} + 165 q^{72} - 70 q^{73} - 98 q^{76} - 110 q^{78} + 24 q^{79} - 113 q^{81} - 110 q^{82} + 110 q^{87} - 330 q^{88} + 210 q^{93} + 308 q^{94} + 77 q^{96} - 140 q^{97} + 275 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 + 1.65831i
0.500000 1.65831i
3.31662i 2.50000 1.65831i −7.00000 0 −5.50000 8.29156i 0 9.94987i 3.50000 8.29156i 0
26.2 3.31662i 2.50000 + 1.65831i −7.00000 0 −5.50000 + 8.29156i 0 9.94987i 3.50000 + 8.29156i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.f yes 2
3.b odd 2 1 inner 75.3.c.f yes 2
4.b odd 2 1 1200.3.l.f 2
5.b even 2 1 75.3.c.c 2
5.c odd 4 2 75.3.d.c 4
12.b even 2 1 1200.3.l.f 2
15.d odd 2 1 75.3.c.c 2
15.e even 4 2 75.3.d.c 4
20.d odd 2 1 1200.3.l.s 2
20.e even 4 2 1200.3.c.d 4
60.h even 2 1 1200.3.l.s 2
60.l odd 4 2 1200.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 5.b even 2 1
75.3.c.c 2 15.d odd 2 1
75.3.c.f yes 2 1.a even 1 1 trivial
75.3.c.f yes 2 3.b odd 2 1 inner
75.3.d.c 4 5.c odd 4 2
75.3.d.c 4 15.e even 4 2
1200.3.c.d 4 20.e even 4 2
1200.3.c.d 4 60.l odd 4 2
1200.3.l.f 2 4.b odd 2 1
1200.3.l.f 2 12.b even 2 1
1200.3.l.s 2 20.d odd 2 1
1200.3.l.s 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 11 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 11 \) Copy content Toggle raw display
$3$ \( T^{2} - 5T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 275 \) Copy content Toggle raw display
$13$ \( (T - 10)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 11 \) Copy content Toggle raw display
$19$ \( (T - 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 396 \) Copy content Toggle raw display
$29$ \( T^{2} + 1100 \) Copy content Toggle raw display
$31$ \( (T - 42)^{2} \) Copy content Toggle raw display
$37$ \( (T + 40)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 275 \) Copy content Toggle raw display
$43$ \( (T + 50)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2156 \) Copy content Toggle raw display
$53$ \( T^{2} + 2156 \) Copy content Toggle raw display
$59$ \( T^{2} + 4400 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T - 45)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 1100 \) Copy content Toggle raw display
$73$ \( (T + 35)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4851 \) Copy content Toggle raw display
$89$ \( T^{2} + 22275 \) Copy content Toggle raw display
$97$ \( (T + 70)^{2} \) Copy content Toggle raw display
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