Properties

Label 75.22.b.a
Level $75$
Weight $22$
Character orbit 75.b
Analytic conductor $209.608$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(49,75)]
 
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([0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2844 i q^{2} - 59049 i q^{3} - 5991184 q^{4} + 167935356 q^{6} - 363303920 i q^{7} - 11074627008 i q^{8} - 3486784401 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2844 i q^{2} - 59049 i q^{3} - 5991184 q^{4} + 167935356 q^{6} - 363303920 i q^{7} - 11074627008 i q^{8} - 3486784401 q^{9} + 14581833156 q^{11} + 353773424016 i q^{12} + 113350790702 i q^{13} + 1033236348480 q^{14} + 18931815702784 q^{16} + 8589389597982 i q^{17} - 9916414836444 i q^{18} + 29202939273796 q^{19} - 21452733172080 q^{21} + 41470733495664 i q^{22} - 155899214954280 i q^{23} - 653945650195392 q^{24} - 322369648756488 q^{26} + 205891132094649 i q^{27} + 21\!\cdots\!80 i q^{28} + \cdots - 50\!\cdots\!56 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 11982368 q^{4} + 335870712 q^{6} - 6973568802 q^{9} + 29163666312 q^{11} + 2066472696960 q^{14} + 37863631405568 q^{16} + 58405878547592 q^{19} - 42905466344160 q^{21} - 13\!\cdots\!84 q^{24}+ \cdots - 10\!\cdots\!12 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
2844.00i 59049.0i −5.99118e6 0 1.67935e8 3.63304e8i 1.10746e10i −3.48678e9 0
49.2 2844.00i 59049.0i −5.99118e6 0 1.67935e8 3.63304e8i 1.10746e10i −3.48678e9 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 75.22.b.a 2
5.b even 2 1 inner 75.22.b.a 2
5.c odd 4 1 3.22.a.a 1
5.c odd 4 1 75.22.a.c 1
15.e even 4 1 9.22.a.d 1
20.e even 4 1 48.22.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.a 1 5.c odd 4 1
9.22.a.d 1 15.e even 4 1
48.22.a.e 1 20.e even 4 1
75.22.a.c 1 5.c odd 4 1
75.22.b.a 2 1.a even 1 1 trivial
75.22.b.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 8088336 \) acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8088336 \) Copy content Toggle raw display
$3$ \( T^{2} + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T - 14581833156)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 12\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{2} + 73\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T - 29202939273796)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T + 24\!\cdots\!58)^{2} \) Copy content Toggle raw display
$31$ \( (T - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 94\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T + 10\!\cdots\!30)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 27\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + 44\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T + 55\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T + 71\!\cdots\!02)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 24\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T - 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T - 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 29\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T - 31\!\cdots\!86)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 90\!\cdots\!24 \) Copy content Toggle raw display
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