Properties

Label 75.3.c.d
Level $75$
Weight $3$
Character orbit 75.c
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM discriminant -15
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(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: \(\Q(\sqrt{-1}) \)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{2} + 3 i q^{3} + 3 q^{4} - 3 q^{6} + 7 i q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + 3 i q^{3} + 3 q^{4} - 3 q^{6} + 7 i q^{8} - 9 q^{9} + 9 i q^{12} + 5 q^{16} - 14 i q^{17} - 9 i q^{18} + 22 q^{19} - 34 i q^{23} - 21 q^{24} - 27 i q^{27} + 2 q^{31} + 33 i q^{32} + 14 q^{34} - 27 q^{36} + 22 i q^{38} + 34 q^{46} - 14 i q^{47} + 15 i q^{48} - 49 q^{49} + 42 q^{51} + 86 i q^{53} + 27 q^{54} + 66 i q^{57} - 118 q^{61} + 2 i q^{62} - 13 q^{64} - 42 i q^{68} + 102 q^{69} - 63 i q^{72} + 66 q^{76} - 98 q^{79} + 81 q^{81} - 154 i q^{83} - 102 i q^{92} + 6 i q^{93} + 14 q^{94} - 99 q^{96} - 49 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} - 6 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} - 6 q^{6} - 18 q^{9} + 10 q^{16} + 44 q^{19} - 42 q^{24} + 4 q^{31} + 28 q^{34} - 54 q^{36} + 68 q^{46} - 98 q^{49} + 84 q^{51} + 54 q^{54} - 236 q^{61} - 26 q^{64} + 204 q^{69} + 132 q^{76} - 196 q^{79} + 162 q^{81} + 28 q^{94} - 198 q^{96}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 3.00000i 3.00000 0 −3.00000 0 7.00000i −9.00000 0
26.2 1.00000i 3.00000i 3.00000 0 −3.00000 0 7.00000i −9.00000 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
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 75.3.c.d 2
3.b odd 2 1 inner 75.3.c.d 2
4.b odd 2 1 1200.3.l.l 2
5.b even 2 1 inner 75.3.c.d 2
5.c odd 4 1 15.3.d.a 1
5.c odd 4 1 15.3.d.b yes 1
12.b even 2 1 1200.3.l.l 2
15.d odd 2 1 CM 75.3.c.d 2
15.e even 4 1 15.3.d.a 1
15.e even 4 1 15.3.d.b yes 1
20.d odd 2 1 1200.3.l.l 2
20.e even 4 1 240.3.c.a 1
20.e even 4 1 240.3.c.b 1
40.i odd 4 1 960.3.c.b 1
40.i odd 4 1 960.3.c.c 1
40.k even 4 1 960.3.c.a 1
40.k even 4 1 960.3.c.d 1
45.k odd 12 2 405.3.h.a 2
45.k odd 12 2 405.3.h.b 2
45.l even 12 2 405.3.h.a 2
45.l even 12 2 405.3.h.b 2
60.h even 2 1 1200.3.l.l 2
60.l odd 4 1 240.3.c.a 1
60.l odd 4 1 240.3.c.b 1
120.q odd 4 1 960.3.c.a 1
120.q odd 4 1 960.3.c.d 1
120.w even 4 1 960.3.c.b 1
120.w even 4 1 960.3.c.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 5.c odd 4 1
15.3.d.a 1 15.e even 4 1
15.3.d.b yes 1 5.c odd 4 1
15.3.d.b yes 1 15.e even 4 1
75.3.c.d 2 1.a even 1 1 trivial
75.3.c.d 2 3.b odd 2 1 inner
75.3.c.d 2 5.b even 2 1 inner
75.3.c.d 2 15.d odd 2 1 CM
240.3.c.a 1 20.e even 4 1
240.3.c.a 1 60.l odd 4 1
240.3.c.b 1 20.e even 4 1
240.3.c.b 1 60.l odd 4 1
405.3.h.a 2 45.k odd 12 2
405.3.h.a 2 45.l even 12 2
405.3.h.b 2 45.k odd 12 2
405.3.h.b 2 45.l even 12 2
960.3.c.a 1 40.k even 4 1
960.3.c.a 1 120.q odd 4 1
960.3.c.b 1 40.i odd 4 1
960.3.c.b 1 120.w even 4 1
960.3.c.c 1 40.i odd 4 1
960.3.c.c 1 120.w even 4 1
960.3.c.d 1 40.k even 4 1
960.3.c.d 1 120.q odd 4 1
1200.3.l.l 2 4.b odd 2 1
1200.3.l.l 2 12.b even 2 1
1200.3.l.l 2 20.d odd 2 1
1200.3.l.l 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 \(S_{3}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 196 \) Copy content Toggle raw display
$19$ \( (T - 22)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1156 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 196 \) Copy content Toggle raw display
$53$ \( T^{2} + 7396 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 118)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 98)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 23716 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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