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Properties

Label 175.5.d.g
Level $175$
Weight $5$
Character orbit 175.d
Analytic conductor $18.090$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 110x^{6} + 7113x^{4} - 190880x^{2} + 4177936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + 3 \beta_{3} q^{3} + (\beta_1 + 22) q^{4} - 3 \beta_{2} q^{6} + ( - \beta_{5} - 4 \beta_{3}) q^{7} + (\beta_{7} - 2 \beta_{5} + \cdots + \beta_{3}) q^{8}+ \cdots - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + 3 \beta_{3} q^{3} + (\beta_1 + 22) q^{4} - 3 \beta_{2} q^{6} + ( - \beta_{5} - 4 \beta_{3}) q^{7} + (\beta_{7} - 2 \beta_{5} + \cdots + \beta_{3}) q^{8}+ \cdots + ( - 54 \beta_1 + 396) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 172 q^{4} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 172 q^{4} - 72 q^{9} - 376 q^{11} + 24 q^{14} + 596 q^{16} + 1020 q^{21} + 64 q^{29} - 1548 q^{36} - 6360 q^{39} + 2104 q^{44} - 28440 q^{46} + 7828 q^{49} + 24240 q^{51} + 27300 q^{56} + 5092 q^{64} - 18016 q^{71} - 39720 q^{74} + 8624 q^{79} - 57672 q^{81} + 47400 q^{84} + 18000 q^{86} + 34480 q^{91} + 3384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 110x^{6} + 7113x^{4} - 190880x^{2} + 4177936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 125\nu^{4} - 4064\nu^{2} - 47852 ) / 8449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 125\nu^{4} - 12513\nu^{2} + 188720 ) / 16898 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{7} - 2166\nu^{5} + 141325\nu^{3} - 1118204\nu ) / 4934216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{7} + 2166\nu^{5} - 141325\nu^{3} + 6052420\nu ) / 4934216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -41\nu^{7} + 17650\nu^{5} - 1112883\nu^{3} + 38227222\nu ) / 2467108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 436\nu^{4} - 22497\nu^{2} + 517664 ) / 1988 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -154\nu^{7} + 16867\nu^{5} - 782451\nu^{3} + 11977428\nu ) / 1233554 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{2} + \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 2\beta_{5} + 13\beta_{4} + 105\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 132\beta_{2} + 15\beta _1 - 524 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 250\beta_{7} + 50\beta_{5} - 1769\beta_{4} + 4555\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 500\beta_{6} - 4186\beta_{2} - 5319\beta _1 - 113572 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -952\beta_{7} + 15638\beta_{5} - 182027\beta_{4} + 43177\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−7.21587 − 3.16228i
−7.21587 + 3.16228i
−4.78865 − 3.16228i
−4.78865 + 3.16228i
4.78865 − 3.16228i
4.78865 + 3.16228i
7.21587 − 3.16228i
7.21587 + 3.16228i
−7.21587 − 9.48683i 36.0688 0 68.4558i −32.7197 + 36.4750i −144.814 −9.00000 0
76.2 −7.21587 9.48683i 36.0688 0 − 68.4558i −32.7197 − 36.4750i −144.814 −9.00000 0
76.3 −4.78865 − 9.48683i 6.93120 0 45.4292i 48.0513 − 9.59562i 43.4273 −9.00000 0
76.4 −4.78865 9.48683i 6.93120 0 − 45.4292i 48.0513 + 9.59562i 43.4273 −9.00000 0
76.5 4.78865 − 9.48683i 6.93120 0 − 45.4292i −48.0513 − 9.59562i −43.4273 −9.00000 0
76.6 4.78865 9.48683i 6.93120 0 45.4292i −48.0513 + 9.59562i −43.4273 −9.00000 0
76.7 7.21587 − 9.48683i 36.0688 0 − 68.4558i 32.7197 + 36.4750i 144.814 −9.00000 0
76.8 7.21587 9.48683i 36.0688 0 68.4558i 32.7197 − 36.4750i 144.814 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c 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.g 8
5.b even 2 1 inner 175.5.d.g 8
5.c odd 4 2 35.5.c.e ✓ 8
7.b odd 2 1 inner 175.5.d.g 8
15.e even 4 2 315.5.e.e 8
20.e even 4 2 560.5.p.g 8
35.c odd 2 1 inner 175.5.d.g 8
35.f even 4 2 35.5.c.e ✓ 8
105.k odd 4 2 315.5.e.e 8
140.j odd 4 2 560.5.p.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.e ✓ 8 5.c odd 4 2
35.5.c.e ✓ 8 35.f even 4 2
175.5.d.g 8 1.a even 1 1 trivial
175.5.d.g 8 5.b even 2 1 inner
175.5.d.g 8 7.b odd 2 1 inner
175.5.d.g 8 35.c odd 2 1 inner
315.5.e.e 8 15.e even 4 2
315.5.e.e 8 105.k odd 4 2
560.5.p.g 8 20.e even 4 2
560.5.p.g 8 140.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 75T_{2}^{2} + 1194 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 75 T^{2} + 1194)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} + 94 T - 5432)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 52250 T^{2} + 145926400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 221000 T^{2} + 8745990400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 347850 T^{2} + 10320458400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1011930 T^{2} + 933058464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 16 T - 3332)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 1373776281600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2666340 T^{2} + 250900015104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1115820 T^{2} + 61553565600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 18960156666024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 899330 T^{2} + 169299331600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1107108 T^{2} + 23031858816)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 40047069962400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 499402073322600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 357890658714024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4504 T - 2114432)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 268409242240000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2156 T - 7670912)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 403700 T^{2} + 2403940900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 17469797990400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 160123400 T^{2} + 608524806400)^{2} \) Copy content Toggle raw display
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