Properties

Label 175.3.d.c
Level $175$
Weight $3$
Character orbit 175.d
Analytic conductor $4.768$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.76");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(4.76840462631\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta q^{3} - 2 \beta q^{6} + ( - 3 \beta + 2) q^{7} + 8 q^{8} + 4 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + \beta q^{3} - 2 \beta q^{6} + ( - 3 \beta + 2) q^{7} + 8 q^{8} + 4 q^{9} - q^{11} + 9 \beta q^{13} + (6 \beta - 4) q^{14} - 16 q^{16} + 3 \beta q^{17} - 8 q^{18} + 6 \beta q^{19} + (2 \beta + 15) q^{21} + 2 q^{22} - 8 q^{23} + 8 \beta q^{24} - 18 \beta q^{26} + 13 \beta q^{27} + 41 q^{29} + 18 \beta q^{31} - \beta q^{33} - 6 \beta q^{34} + 28 q^{37} - 12 \beta q^{38} - 45 q^{39} + 6 \beta q^{41} + ( - 4 \beta - 30) q^{42} + 82 q^{43} + 16 q^{46} - 9 \beta q^{47} - 16 \beta q^{48} + ( - 12 \beta - 41) q^{49} - 15 q^{51} - 74 q^{53} - 26 \beta q^{54} + ( - 24 \beta + 16) q^{56} - 30 q^{57} - 82 q^{58} - 42 \beta q^{59} + 36 \beta q^{61} - 36 \beta q^{62} + ( - 12 \beta + 8) q^{63} + 64 q^{64} + 2 \beta q^{66} - 2 q^{67} - 8 \beta q^{69} + 14 q^{71} + 32 q^{72} + 30 \beta q^{73} - 56 q^{74} + (3 \beta - 2) q^{77} + 90 q^{78} - 19 q^{79} - 29 q^{81} - 12 \beta q^{82} - 42 \beta q^{83} - 164 q^{86} + 41 \beta q^{87} - 8 q^{88} + 48 \beta q^{89} + (18 \beta + 135) q^{91} - 90 q^{93} + 18 \beta q^{94} + 27 \beta q^{97} + (24 \beta + 82) q^{98} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 4 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 4 q^{7} + 16 q^{8} + 8 q^{9} - 2 q^{11} - 8 q^{14} - 32 q^{16} - 16 q^{18} + 30 q^{21} + 4 q^{22} - 16 q^{23} + 82 q^{29} + 56 q^{37} - 90 q^{39} - 60 q^{42} + 164 q^{43} + 32 q^{46} - 82 q^{49} - 30 q^{51} - 148 q^{53} + 32 q^{56} - 60 q^{57} - 164 q^{58} + 16 q^{63} + 128 q^{64} - 4 q^{67} + 28 q^{71} + 64 q^{72} - 112 q^{74} - 4 q^{77} + 180 q^{78} - 38 q^{79} - 58 q^{81} - 328 q^{86} - 16 q^{88} + 270 q^{91} - 180 q^{93} + 164 q^{98} - 8 q^{99}+O(q^{100}) \) 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
2.23607i
2.23607i
−2.00000 2.23607i 0 0 4.47214i 2.00000 + 6.70820i 8.00000 4.00000 0
76.2 −2.00000 2.23607i 0 0 4.47214i 2.00000 6.70820i 8.00000 4.00000 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
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 175.3.d.c 2
5.b even 2 1 35.3.d.b 2
5.c odd 4 2 175.3.c.c 4
7.b odd 2 1 inner 175.3.d.c 2
15.d odd 2 1 315.3.h.a 2
20.d odd 2 1 560.3.f.b 2
35.c odd 2 1 35.3.d.b 2
35.f even 4 2 175.3.c.c 4
35.i odd 6 2 245.3.h.a 4
35.j even 6 2 245.3.h.a 4
105.g even 2 1 315.3.h.a 2
140.c even 2 1 560.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 5.b even 2 1
35.3.d.b 2 35.c odd 2 1
175.3.c.c 4 5.c odd 4 2
175.3.c.c 4 35.f even 4 2
175.3.d.c 2 1.a even 1 1 trivial
175.3.d.c 2 7.b odd 2 1 inner
245.3.h.a 4 35.i odd 6 2
245.3.h.a 4 35.j even 6 2
315.3.h.a 2 15.d odd 2 1
315.3.h.a 2 105.g even 2 1
560.3.f.b 2 20.d odd 2 1
560.3.f.b 2 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{3}^{2} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 49 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 405 \) Copy content Toggle raw display
$17$ \( T^{2} + 45 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 41)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1620 \) Copy content Toggle raw display
$37$ \( (T - 28)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 180 \) Copy content Toggle raw display
$43$ \( (T - 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 405 \) Copy content Toggle raw display
$53$ \( (T + 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8820 \) Copy content Toggle raw display
$61$ \( T^{2} + 6480 \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T - 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4500 \) Copy content Toggle raw display
$79$ \( (T + 19)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8820 \) Copy content Toggle raw display
$89$ \( T^{2} + 11520 \) Copy content Toggle raw display
$97$ \( T^{2} + 3645 \) Copy content Toggle raw display
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