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Properties

Label 175.3.c.e
Level $175$
Weight $3$
Character orbit 175.c
Analytic conductor $4.768$
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] = [175,3,Mod(174,175)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.174");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 175.c (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: \(4.76840462631\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1353778790400.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 30x^{6} + 269x^{4} + 660x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{3} q^{3} + (2 \beta_1 - 3) q^{4} - \beta_{6} q^{6} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + ( - 2 \beta_{5} - 3 \beta_{2}) q^{8} + ( - 2 \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{3} q^{3} + (2 \beta_1 - 3) q^{4} - \beta_{6} q^{6} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + ( - 2 \beta_{5} - 3 \beta_{2}) q^{8} + ( - 2 \beta_1 + 14) q^{9} + ( - \beta_1 - 7) q^{11} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_{3}) q^{12} + (\beta_{5} + 2 \beta_{4}) q^{13} + (\beta_{7} - \beta_{6} - 8) q^{14} + ( - 4 \beta_1 + 5) q^{16} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3}) q^{17} + (2 \beta_{5} + 18 \beta_{2}) q^{18} + (2 \beta_{7} + \beta_{6} - \beta_1) q^{19} + ( - \beta_{7} - \beta_{6} - 11 \beta_1 + 6) q^{21} + (\beta_{5} - 5 \beta_{2}) q^{22} + (\beta_{5} - 8 \beta_{2}) q^{23} + ( - 4 \beta_{7} + 3 \beta_{6} + 2 \beta_1) q^{24} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_1) q^{26} + ( - 2 \beta_{5} - 4 \beta_{4} - 5 \beta_{3}) q^{27} + ( - \beta_{5} + 3 \beta_{4} + \cdots - 5 \beta_{2}) q^{28}+ \cdots - 86 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{4} + 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{4} + 112 q^{9} - 56 q^{11} - 64 q^{14} + 40 q^{16} + 48 q^{21} - 112 q^{29} - 528 q^{36} - 96 q^{39} + 72 q^{44} + 464 q^{46} + 160 q^{49} + 456 q^{51} - 16 q^{56} + 40 q^{64} - 416 q^{71} + 304 q^{74} + 608 q^{79} + 104 q^{81} - 1248 q^{84} + 464 q^{86} - 552 q^{91} - 688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 30x^{6} + 269x^{4} + 660x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 15\nu^{2} + 22 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 10\nu^{5} + 181\nu^{3} + 1470\nu ) / 400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 45\nu^{4} - 253\nu^{2} - 210 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{7} - 10\nu^{6} + 195\nu^{5} - 225\nu^{4} + 1608\nu^{3} - 865\nu^{2} + 3460\nu + 1950 ) / 400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} - 195\nu^{5} - 1608\nu^{3} - 3460\nu ) / 200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 100\nu^{5} - 907\nu^{3} - 1370\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{7} - 125\nu^{5} + 5\nu^{4} - 1151\nu^{3} + 75\nu^{2} - 2910\nu + 110 ) / 80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{2} + \beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{7} - 7\beta_{6} - 31\beta_{5} + 59\beta_{2} - 12\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{5} - 30\beta_{4} + 15\beta_{3} + 16\beta _1 + 181 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -332\beta_{7} + 51\beta_{6} + 477\beta_{5} - 803\beta_{2} + 166\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 211\beta_{5} + 422\beta_{4} - 251\beta_{3} - 360\beta _1 - 2385 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4724\beta_{7} - 307\beta_{6} - 7441\beta_{5} + 10299\beta_{2} - 2362\beta_1 ) / 10 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
174.1
− 1.91623i
3.36572i
3.85201i
− 0.402520i
− 3.85201i
0.402520i
1.91623i
− 3.36572i
− 3.44949i −5.28195 −7.89898 0 18.2200i −6.46904 − 2.67423i 13.4495i 18.8990 0
174.2 − 3.44949i 5.28195 −7.89898 0 − 18.2200i 6.46904 − 2.67423i 13.4495i 18.8990 0
174.3 − 1.44949i −4.25453 1.89898 0 6.16690i 5.21071 − 4.67423i − 8.55051i 9.10102 0
174.4 − 1.44949i 4.25453 1.89898 0 − 6.16690i −5.21071 − 4.67423i − 8.55051i 9.10102 0
174.5 1.44949i −4.25453 1.89898 0 − 6.16690i 5.21071 + 4.67423i 8.55051i 9.10102 0
174.6 1.44949i 4.25453 1.89898 0 6.16690i −5.21071 + 4.67423i 8.55051i 9.10102 0
174.7 3.44949i −5.28195 −7.89898 0 − 18.2200i −6.46904 + 2.67423i − 13.4495i 18.8990 0
174.8 3.44949i 5.28195 −7.89898 0 18.2200i 6.46904 + 2.67423i − 13.4495i 18.8990 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 174.8
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 175.3.c.e 8
5.b even 2 1 inner 175.3.c.e 8
5.c odd 4 1 175.3.d.i ✓ 4
5.c odd 4 1 175.3.d.j yes 4
7.b odd 2 1 inner 175.3.c.e 8
35.c odd 2 1 inner 175.3.c.e 8
35.f even 4 1 175.3.d.i ✓ 4
35.f even 4 1 175.3.d.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.3.c.e 8 1.a even 1 1 trivial
175.3.c.e 8 5.b even 2 1 inner
175.3.c.e 8 7.b odd 2 1 inner
175.3.c.e 8 35.c odd 2 1 inner
175.3.d.i ✓ 4 5.c odd 4 1
175.3.d.i ✓ 4 35.f even 4 1
175.3.d.j yes 4 5.c odd 4 1
175.3.d.j yes 4 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 14T_{2}^{2} + 25 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 14 T^{2} + 25)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 46 T^{2} + 505)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 80 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} + 14 T + 43)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 276 T^{2} + 18180)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 546 T^{2} + 4545)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1230 T^{2} + 113625)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1004 T^{2} + 220900)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 28 T + 142)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2220 T^{2} + 454500)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3884 T^{2} + 3724900)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3330 T^{2} + 1022625)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3704 T^{2} + 250000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 7116 T^{2} + 6562980)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1076 T^{2} + 184900)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 4920 T^{2} + 1818000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8880 T^{2} + 7272000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 3374 T^{2} + 55225)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 104 T - 2342)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 14946 T^{2} + 41018625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 152 T + 2872)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 5406 T^{2} + 3822345)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 21570 T^{2} + 95558625)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 3336 T^{2} + 72720)^{2} \) Copy content Toggle raw display
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