↑

Properties

Label 275.2.e.d
Level $275$
Weight $2$
Character orbit 275.e
Analytic conductor $2.196$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,2,Mod(32,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.e (of order \(4\), degree \(2\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.6879707136000000000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + ( - \beta_{10} - \beta_{3}) q^{3} + ( - \beta_{7} + \beta_{4}) q^{4} + ( - \beta_{11} + \beta_{6}) q^{6} + ( - \beta_{9} + 2 \beta_1) q^{7} + ( - \beta_{15} + \beta_{12}) q^{8} + ( - 3 \beta_{7} + 3 \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + ( - \beta_{10} - \beta_{3}) q^{3} + ( - \beta_{7} + \beta_{4}) q^{4} + ( - \beta_{11} + \beta_{6}) q^{6} + ( - \beta_{9} + 2 \beta_1) q^{7} + ( - \beta_{15} + \beta_{12}) q^{8} + ( - 3 \beta_{7} + 3 \beta_{4}) q^{9} + (\beta_{6} + \beta_{5} - 1) q^{11} + ( - \beta_{14} + 2 \beta_{13}) q^{12} + ( - \beta_{15} - \beta_{12}) q^{13} + (5 \beta_{7} - 5 \beta_{4}) q^{14} + ( - \beta_{5} + 3) q^{16} + \beta_1 q^{17} + (3 \beta_{15} + 3 \beta_{12}) q^{18} + (\beta_{8} + \beta_{2}) q^{19} + (3 \beta_{11} - \beta_{6}) q^{21} + ( - 2 \beta_{14} + \beta_{13} + \cdots + \beta_1) q^{22}+ \cdots + ( - 3 \beta_{8} + 6 \beta_{7} - 9 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{11} + 56 q^{16} + 40 q^{26} - 48 q^{31} - 72 q^{36} + 40 q^{56} - 120 q^{66} + 112 q^{71} - 160 q^{86} - 120 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{13} - 155\nu^{9} - 875\nu^{5} - 1923\nu ) / 671 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\nu^{12} + 812\nu^{8} + 2441\nu^{4} - 1294 ) / 671 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -35\nu^{13} - 775\nu^{9} - 3704\nu^{5} - 2905\nu ) / 671 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 90\nu^{14} + 1897\nu^{10} + 7895\nu^{6} + 4115\nu^{2} ) / 671 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{12} - 102\nu^{8} - 381\nu^{4} - 110 ) / 61 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\nu^{14} + 257\nu^{10} + 1134\nu^{6} + 752\nu^{2} ) / 61 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -138\nu^{14} - 2864\nu^{10} - 11211\nu^{6} - 2731\nu^{2} ) / 671 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 111\nu^{12} + 2362\nu^{8} + 9849\nu^{4} + 2503 ) / 671 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 138\nu^{13} + 2864\nu^{9} + 11211\nu^{5} + 3402\nu ) / 671 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -138\nu^{13} - 2864\nu^{9} - 11211\nu^{5} - 2060\nu ) / 671 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -44\nu^{14} - 922\nu^{10} - 3731\nu^{6} - 1273\nu^{2} ) / 61 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 270\nu^{15} + 5691\nu^{11} + 23685\nu^{7} + 11674\nu^{3} ) / 671 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 443\nu^{15} + 9330\nu^{11} + 38600\nu^{7} + 17310\nu^{3} ) / 671 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -449\nu^{15} - 9367\nu^{11} - 37337\nu^{7} - 11098\nu^{3} ) / 671 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 629\nu^{15} + 13161\nu^{11} + 53127\nu^{7} + 19328\nu^{3} ) / 671 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} + 3\beta_{14} - 2\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{8} - 5\beta_{5} - 4\beta_{2} - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{10} - 5\beta_{9} + \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6\beta_{11} - 23\beta_{7} + 15\beta_{6} - 25\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -35\beta_{15} - 36\beta_{14} - 13\beta_{13} + 43\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 28\beta_{8} + 99\beta_{5} + 57\beta_{2} + 184 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 135\beta_{10} + 128\beta_{9} - 63\beta_{3} + 177\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -60\beta_{11} + 205\beta_{7} - 110\beta_{6} + 153\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 483\beta_{15} + 518\beta_{14} + 275\beta_{13} - 715\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -495\beta_{8} - 1663\beta_{5} - 858\beta_{2} - 2807 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -2014\beta_{10} - 1859\beta_{9} + 1145\beta_{3} - 2861\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2003\beta_{11} - 6670\beta_{7} + 3367\beta_{6} - 4379\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -3620\beta_{15} - 3945\beta_{14} - 2328\beta_{13} + 5695\beta_{12} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−0.294032 + 0.294032i
−1.05097 + 1.05097i
0.575212 − 0.575212i
−1.40647 + 1.40647i
1.40647 − 1.40647i
−0.575212 + 0.575212i
1.05097 − 1.05097i
0.294032 − 0.294032i
−0.294032 − 0.294032i
−1.05097 − 1.05097i
0.575212 + 0.575212i
−1.40647 − 1.40647i
1.40647 + 1.40647i
−0.575212 − 0.575212i
1.05097 + 1.05097i
0.294032 + 0.294032i
−1.34500 + 1.34500i −1.98168 + 1.98168i − 1.61803i 0 − 5.33070i 3.00750 − 3.00750i −0.513743 − 0.513743i − 4.85410i 0
32.2 −1.34500 + 1.34500i 1.98168 − 1.98168i − 1.61803i 0 5.33070i 3.00750 − 3.00750i −0.513743 − 0.513743i − 4.85410i 0
32.3 −0.831254 + 0.831254i −0.756934 + 0.756934i 0.618034i 0 − 1.25841i −1.85874 + 1.85874i −2.17625 − 2.17625i 1.85410i 0
32.4 −0.831254 + 0.831254i 0.756934 − 0.756934i 0.618034i 0 1.25841i −1.85874 + 1.85874i −2.17625 − 2.17625i 1.85410i 0
32.5 0.831254 − 0.831254i −0.756934 + 0.756934i 0.618034i 0 1.25841i 1.85874 − 1.85874i 2.17625 + 2.17625i 1.85410i 0
32.6 0.831254 − 0.831254i 0.756934 − 0.756934i 0.618034i 0 − 1.25841i 1.85874 − 1.85874i 2.17625 + 2.17625i 1.85410i 0
32.7 1.34500 − 1.34500i −1.98168 + 1.98168i − 1.61803i 0 5.33070i −3.00750 + 3.00750i 0.513743 + 0.513743i − 4.85410i 0
32.8 1.34500 − 1.34500i 1.98168 − 1.98168i − 1.61803i 0 − 5.33070i −3.00750 + 3.00750i 0.513743 + 0.513743i − 4.85410i 0
43.1 −1.34500 − 1.34500i −1.98168 − 1.98168i 1.61803i 0 5.33070i 3.00750 + 3.00750i −0.513743 + 0.513743i 4.85410i 0
43.2 −1.34500 − 1.34500i 1.98168 + 1.98168i 1.61803i 0 − 5.33070i 3.00750 + 3.00750i −0.513743 + 0.513743i 4.85410i 0
43.3 −0.831254 − 0.831254i −0.756934 − 0.756934i − 0.618034i 0 1.25841i −1.85874 − 1.85874i −2.17625 + 2.17625i − 1.85410i 0
43.4 −0.831254 − 0.831254i 0.756934 + 0.756934i − 0.618034i 0 − 1.25841i −1.85874 − 1.85874i −2.17625 + 2.17625i − 1.85410i 0
43.5 0.831254 + 0.831254i −0.756934 − 0.756934i − 0.618034i 0 − 1.25841i 1.85874 + 1.85874i 2.17625 − 2.17625i − 1.85410i 0
43.6 0.831254 + 0.831254i 0.756934 + 0.756934i − 0.618034i 0 1.25841i 1.85874 + 1.85874i 2.17625 − 2.17625i − 1.85410i 0
43.7 1.34500 + 1.34500i −1.98168 − 1.98168i 1.61803i 0 − 5.33070i −3.00750 − 3.00750i 0.513743 − 0.513743i 4.85410i 0
43.8 1.34500 + 1.34500i 1.98168 + 1.98168i 1.61803i 0 5.33070i −3.00750 − 3.00750i 0.513743 − 0.513743i 4.85410i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
11.b odd 2 1 inner
55.d odd 2 1 inner
55.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.e.d ✓ 16
5.b even 2 1 inner 275.2.e.d ✓ 16
5.c odd 4 2 inner 275.2.e.d ✓ 16
11.b odd 2 1 inner 275.2.e.d ✓ 16
55.d odd 2 1 inner 275.2.e.d ✓ 16
55.e even 4 2 inner 275.2.e.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.e.d ✓ 16 1.a even 1 1 trivial
275.2.e.d ✓ 16 5.b even 2 1 inner
275.2.e.d ✓ 16 5.c odd 4 2 inner
275.2.e.d ✓ 16 11.b odd 2 1 inner
275.2.e.d ✓ 16 55.d odd 2 1 inner
275.2.e.d ✓ 16 55.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 15T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 63 T^{4} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 375 T^{4} + 15625)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 90 T^{4} + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 30 T^{2} + 45)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 63 T^{4} + 81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 75 T^{2} + 45)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 11)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + 9378 T^{4} + 10556001)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 150 T^{2} + 5445)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 6000 T^{4} + 4000000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 9378 T^{4} + 10556001)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 10143 T^{4} + 10556001)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 195 T^{2} + 5445)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2304)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 14 T + 4)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} + 55215 T^{4} + 635292025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 195 T^{2} + 45)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 4215 T^{4} + 25)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 127 T^{2} + 3481)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 19863 T^{4} + 81)^{2} \) Copy content Toggle raw display
show more
show less