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Properties

Label 175.4.b.f
Level $175$
Weight $4$
Character orbit 175.b
Analytic conductor $10.325$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,4,Mod(99,175)]
 
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([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(10.3253342510\)
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: \( x^{8} + 64x^{6} + 1264x^{4} + 8905x^{2} + 14400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + (\beta_{7} - \beta_{6} - 9) q^{4} + ( - 2 \beta_{7} - 3 \beta_{6} + \cdots - 1) q^{6}+ \cdots + ( - 3 \beta_{6} + 6 \beta_{3} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + (\beta_{7} - \beta_{6} - 9) q^{4} + ( - 2 \beta_{7} - 3 \beta_{6} + \cdots - 1) q^{6}+ \cdots + ( - 52 \beta_{7} - 253 \beta_{6} + \cdots - 672) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41} - 876 q^{44} - 1994 q^{46} - 392 q^{49} - 1770 q^{51} + 5554 q^{54} + 378 q^{56} - 2036 q^{59} + 100 q^{61} + 4842 q^{64} - 7682 q^{66} - 2160 q^{69} + 430 q^{71} - 1246 q^{74} - 6514 q^{76} + 1902 q^{79} + 56 q^{81} + 2310 q^{84} - 198 q^{86} - 5638 q^{89} + 616 q^{91} + 6112 q^{94} - 2690 q^{96} - 4766 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 64x^{6} + 1264x^{4} + 8905x^{2} + 14400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 64\nu^{5} + 1144\nu^{3} + 5065\nu ) / 4200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 32\nu^{2} + 120 ) / 35 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} + 524\nu^{5} + 4724\nu^{3} - 7885\nu ) / 2100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{7} - 976\nu^{5} - 11956\nu^{3} - 29635\nu ) / 2100 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 50\nu^{4} + 556\nu^{2} + 830 ) / 35 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 52\nu^{4} + 655\nu^{2} + 1630 ) / 35 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - 2\beta_{3} - 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} - 4\beta_{4} - 26\beta_{2} - 26\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -32\beta_{7} + 32\beta_{6} + 99\beta_{3} + 392 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 131\beta_{5} + 163\beta_{4} + 1392\beta_{2} + 782\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1044\beta_{7} - 1009\beta_{6} - 3838\beta_{3} - 11534 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4952\beta_{5} - 5856\beta_{4} - 55144\beta_{2} - 25369\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
− 5.87199i
− 3.84167i
− 3.53510i
− 1.50478i
1.50478i
3.53510i
3.84167i
5.87199i
− 4.87199i 4.14916i −15.7363 0 20.2147 7.00000i 37.6910i 9.78444 0
99.2 − 4.84167i − 9.58084i −15.4418 0 −46.3873 − 7.00000i 36.0308i −64.7926 0
99.3 − 4.53510i 6.46622i −12.5671 0 29.3249 − 7.00000i 20.7124i −14.8119 0
99.4 − 0.504784i − 4.26379i 7.74519 0 −2.15229 7.00000i − 7.94792i 8.82008 0
99.5 0.504784i 4.26379i 7.74519 0 −2.15229 − 7.00000i 7.94792i 8.82008 0
99.6 4.53510i − 6.46622i −12.5671 0 29.3249 7.00000i − 20.7124i −14.8119 0
99.7 4.84167i 9.58084i −15.4418 0 −46.3873 7.00000i − 36.0308i −64.7926 0
99.8 4.87199i − 4.14916i −15.7363 0 20.2147 − 7.00000i − 37.6910i 9.78444 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
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 175.4.b.f 8
5.b even 2 1 inner 175.4.b.f 8
5.c odd 4 1 175.4.a.g ✓ 4
5.c odd 4 1 175.4.a.h yes 4
15.e even 4 1 1575.4.a.bg 4
15.e even 4 1 1575.4.a.bl 4
35.f even 4 1 1225.4.a.z 4
35.f even 4 1 1225.4.a.bd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.g ✓ 4 5.c odd 4 1
175.4.a.h yes 4 5.c odd 4 1
175.4.b.f 8 1.a even 1 1 trivial
175.4.b.f 8 5.b even 2 1 inner
1225.4.a.z 4 35.f even 4 1
1225.4.a.bd 4 35.f even 4 1
1575.4.a.bg 4 15.e even 4 1
1575.4.a.bl 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\):

\( T_{2}^{8} + 68T_{2}^{6} + 1544T_{2}^{4} + 11833T_{2}^{2} + 2916 \) Copy content Toggle raw display
\( T_{3}^{8} + 169T_{3}^{6} + 8880T_{3}^{4} + 177664T_{3}^{2} + 1201216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 68 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$3$ \( T^{8} + 169 T^{6} + \cdots + 1201216 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 100 T^{3} + \cdots - 6827031)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1082532040704 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 7332839424 \) Copy content Toggle raw display
$19$ \( (T^{4} - 29 T^{3} + \cdots + 56837320)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + 129 T^{3} + \cdots + 18445050)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 114 T^{3} + \cdots + 44772800)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{4} - 671 T^{3} + \cdots - 1082974824)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{4} + 1018 T^{3} + \cdots + 3039063360)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 50 T^{3} + \cdots + 1189016144)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{4} - 215 T^{3} + \cdots + 2950374906)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{4} - 951 T^{3} + \cdots - 178640145850)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2819 T^{3} + \cdots + 151094328480)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
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