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Properties

Label 1075.2.b.i
Level $1075$
Weight $2$
Character orbit 1075.b
Analytic conductor $8.584$
Analytic rank $0$
Dimension $10$
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] = [1075,2,Mod(474,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1075.474");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.600538203136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 10x^{8} + 29x^{6} + 32x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{8} - \beta_{6} + \beta_{2}) q^{4} + (\beta_{8} + \beta_{2} - 1) q^{6} + ( - \beta_{7} - \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots + (\beta_{6} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{8} - \beta_{6} + \beta_{2}) q^{4} + (\beta_{8} + \beta_{2} - 1) q^{6} + ( - \beta_{7} - \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - \beta_{8} - 3 \beta_{6} - \beta_{5} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{6} + 10 q^{9} - 18 q^{11} + 14 q^{14} - 4 q^{16} + 22 q^{19} - 10 q^{21} + 18 q^{24} - 10 q^{26} + 44 q^{29} - 10 q^{31} + 14 q^{34} + 6 q^{36} + 6 q^{39} - 42 q^{41} + 40 q^{44} + 26 q^{46} + 26 q^{49} - 10 q^{51} - 8 q^{54} + 8 q^{56} + 16 q^{59} - 32 q^{61} + 34 q^{64} + 38 q^{66} + 30 q^{69} - 20 q^{71} - 6 q^{76} - 24 q^{79} - 30 q^{81} - 10 q^{84} + 72 q^{89} - 26 q^{91} + 56 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 10x^{8} + 29x^{6} + 32x^{4} + 12x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{8} + 9\nu^{6} + 20\nu^{4} + 12\nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{9} + 9\nu^{7} + 20\nu^{5} + 12\nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{7} - 16\nu^{5} - 25\nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\nu^{8} - 16\nu^{6} - 25\nu^{4} - 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{8} + 17\nu^{6} + 33\nu^{4} + 18\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} + 10\nu^{7} + 29\nu^{5} + 32\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{8} + 18\nu^{6} + 41\nu^{4} + 31\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{9} + 18\nu^{7} + 41\nu^{5} + 31\nu^{3} + 6\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 2\beta_{6} - \beta_{5} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + 2\beta_{7} + \beta_{4} + 2\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{8} + 14\beta_{6} + 7\beta_{5} - 2\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{9} - 14\beta_{7} - 7\beta_{4} - 16\beta_{3} + 36\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{8} - 87\beta_{6} - 43\beta_{5} + 16\beta_{2} - 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -95\beta_{9} + 87\beta_{7} + 43\beta_{4} + 103\beta_{3} - 216\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -216\beta_{8} + 527\beta_{6} + 259\beta_{5} - 103\beta_{2} + 242 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 579\beta_{9} - 527\beta_{7} - 259\beta_{4} - 630\beta_{3} + 1296\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times\).

\(n\) \(302\) \(476\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1
0.714533i
2.44983i
− 1.26073i
0.339328i
− 1.33536i
1.33536i
− 0.339328i
1.26073i
− 2.44983i
− 0.714533i
− 2.17442i − 2.04314i −2.72812 0 −4.44266 − 0.553698i 1.58325i −1.17442 0
474.2 − 1.96003i − 0.199933i −1.84170 0 −0.391875 3.80173i − 0.310264i 2.96003 0
474.3 − 0.878095i 1.69649i 1.22895 0 1.48968 2.10704i − 2.83532i 0.121905 0
474.4 − 0.722813i 1.13013i 1.47754 0 0.816870 − 0.754729i − 2.51361i 1.72281 0
474.5 − 0.369680i − 1.27684i 1.86334 0 −0.472023 − 1.49366i − 1.42820i 1.36968 0
474.6 0.369680i 1.27684i 1.86334 0 −0.472023 1.49366i 1.42820i 1.36968 0
474.7 0.722813i − 1.13013i 1.47754 0 0.816870 0.754729i 2.51361i 1.72281 0
474.8 0.878095i − 1.69649i 1.22895 0 1.48968 − 2.10704i 2.83532i 0.121905 0
474.9 1.96003i 0.199933i −1.84170 0 −0.391875 − 3.80173i 0.310264i 2.96003 0
474.10 2.17442i 2.04314i −2.72812 0 −4.44266 0.553698i − 1.58325i −1.17442 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.10
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 1075.2.b.i 10
5.b even 2 1 inner 1075.2.b.i 10
5.c odd 4 1 1075.2.a.n ✓ 5
5.c odd 4 1 1075.2.a.o yes 5
15.e even 4 1 9675.2.a.cb 5
15.e even 4 1 9675.2.a.cc 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1075.2.a.n ✓ 5 5.c odd 4 1
1075.2.a.o yes 5 5.c odd 4 1
1075.2.b.i 10 1.a even 1 1 trivial
1075.2.b.i 10 5.b even 2 1 inner
9675.2.a.cb 5 15.e even 4 1
9675.2.a.cc 5 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 \(S_{2}^{\mathrm{new}}(1075, [\chi])\):

\( T_{2}^{10} + 10T_{2}^{8} + 31T_{2}^{6} + 31T_{2}^{4} + 11T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{10} + 10T_{3}^{8} + 35T_{3}^{6} + 51T_{3}^{4} + 27T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{10} + 22T_{7}^{8} + 125T_{7}^{6} + 240T_{7}^{4} + 144T_{7}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 10 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} + 10 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 22 T^{8} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{5} + 9 T^{4} + \cdots - 191)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 81 T^{8} + \cdots + 93025 \) Copy content Toggle raw display
$17$ \( T^{10} + 87 T^{8} + \cdots + 3721 \) Copy content Toggle raw display
$19$ \( (T^{5} - 11 T^{4} + \cdots + 311)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 42 T^{8} + \cdots + 3721 \) Copy content Toggle raw display
$29$ \( (T^{5} - 22 T^{4} + \cdots - 307)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 5 T^{4} + \cdots + 4657)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 89 T^{8} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( (T^{5} + 21 T^{4} + \cdots + 13985)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$47$ \( T^{10} + 315 T^{8} + \cdots + 120409 \) Copy content Toggle raw display
$53$ \( T^{10} + 275 T^{8} + \cdots + 2449225 \) Copy content Toggle raw display
$59$ \( (T^{5} - 8 T^{4} + \cdots - 25087)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 16 T^{4} + \cdots + 9437)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 322 T^{8} + \cdots + 10975969 \) Copy content Toggle raw display
$71$ \( (T^{5} + 10 T^{4} + \cdots + 13577)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 203 T^{8} + \cdots + 1394761 \) Copy content Toggle raw display
$79$ \( (T^{5} + 12 T^{4} + \cdots - 557)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 364 T^{8} + \cdots + 5359225 \) Copy content Toggle raw display
$89$ \( (T^{5} - 36 T^{4} + \cdots + 202379)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 336 T^{8} + \cdots + 395641 \) Copy content Toggle raw display
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