Properties

Label 175.5.d.c
Level $175$
Weight $5$
Character orbit 175.d
Analytic conductor $18.090$
Analytic rank $0$
Dimension $2$
CM discriminant -35
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,5,Mod(76,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.76");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.d (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: \(18.0897435397\)
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 35)
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.

\(f(q)\) \(=\) \( q + 17 i q^{3} - 16 q^{4} + 49 i q^{7} - 208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 17 i q^{3} - 16 q^{4} + 49 i q^{7} - 208 q^{9} - 73 q^{11} - 272 i q^{12} - 23 i q^{13} + 256 q^{16} + 263 i q^{17} - 833 q^{21} - 2159 i q^{27} - 784 i q^{28} + 1153 q^{29} - 1241 i q^{33} + 3328 q^{36} + 391 q^{39} + 1168 q^{44} - 3457 i q^{47} + 4352 i q^{48} - 2401 q^{49} - 4471 q^{51} + 368 i q^{52} - 10192 i q^{63} - 4096 q^{64} - 4208 i q^{68} - 10078 q^{71} + 9502 i q^{73} - 3577 i q^{77} - 12167 q^{79} + 19855 q^{81} + 6382 i q^{83} + 13328 q^{84} + 19601 i q^{87} + 1127 q^{91} + 3383 i q^{97} + 15184 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 416 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 416 q^{9} - 146 q^{11} + 512 q^{16} - 1666 q^{21} + 2306 q^{29} + 6656 q^{36} + 782 q^{39} + 2336 q^{44} - 4802 q^{49} - 8942 q^{51} - 8192 q^{64} - 20156 q^{71} - 24334 q^{79} + 39710 q^{81} + 26656 q^{84} + 2254 q^{91} + 30368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.00000i
1.00000i
0 17.0000i −16.0000 0 0 49.0000i 0 −208.000 0
76.2 0 17.0000i −16.0000 0 0 49.0000i 0 −208.000 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.d.c 2
5.b even 2 1 inner 175.5.d.c 2
5.c odd 4 1 35.5.c.a 1
5.c odd 4 1 35.5.c.b yes 1
7.b odd 2 1 inner 175.5.d.c 2
15.e even 4 1 315.5.e.a 1
15.e even 4 1 315.5.e.b 1
20.e even 4 1 560.5.p.a 1
20.e even 4 1 560.5.p.b 1
35.c odd 2 1 CM 175.5.d.c 2
35.f even 4 1 35.5.c.a 1
35.f even 4 1 35.5.c.b yes 1
105.k odd 4 1 315.5.e.a 1
105.k odd 4 1 315.5.e.b 1
140.j odd 4 1 560.5.p.a 1
140.j odd 4 1 560.5.p.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 5.c odd 4 1
35.5.c.a 1 35.f even 4 1
35.5.c.b yes 1 5.c odd 4 1
35.5.c.b yes 1 35.f even 4 1
175.5.d.c 2 1.a even 1 1 trivial
175.5.d.c 2 5.b even 2 1 inner
175.5.d.c 2 7.b odd 2 1 inner
175.5.d.c 2 35.c odd 2 1 CM
315.5.e.a 1 15.e even 4 1
315.5.e.a 1 105.k odd 4 1
315.5.e.b 1 15.e even 4 1
315.5.e.b 1 105.k odd 4 1
560.5.p.a 1 20.e even 4 1
560.5.p.a 1 140.j odd 4 1
560.5.p.b 1 20.e even 4 1
560.5.p.b 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 289 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T + 73)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 529 \) Copy content Toggle raw display
$17$ \( T^{2} + 69169 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 1153)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) 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
$47$ \( T^{2} + 11950849 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 10078)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 90288004 \) Copy content Toggle raw display
$79$ \( (T + 12167)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 40729924 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 11444689 \) Copy content Toggle raw display
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