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Properties

Label 275.2.e.c
Level $275$
Weight $2$
Character orbit 275.e
Analytic conductor $2.196$
Analytic rank $0$
Dimension $8$
CM discriminant -55
Inner twists $8$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1499238400.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{6} - 2 \beta_{4}) q^{4} + \beta_{5} q^{7} + (\beta_{7} - 3 \beta_{3}) q^{8} + 3 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{6} - 2 \beta_{4}) q^{4} + \beta_{5} q^{7} + (\beta_{7} - 3 \beta_{3}) q^{8} + 3 \beta_{4} q^{9} + \beta_1 q^{11} + ( - \beta_{7} + 2 \beta_{3}) q^{13} + ( - \beta_{6} + \beta_{4}) q^{14} + (2 \beta_1 - 7) q^{16} + ( - \beta_{5} - 2 \beta_{2}) q^{17} + 3 \beta_{3} q^{18} + ( - \beta_{5} + 3 \beta_{2}) q^{22} + ( - 3 \beta_1 + 7) q^{26} + ( - \beta_{7} - 2 \beta_{3}) q^{28} + 2 \beta_1 q^{31} + 7 \beta_{2} q^{32} + ( - \beta_{6} - 9 \beta_{4}) q^{34} + ( - 3 \beta_1 + 6) q^{36} + (\beta_{7} + 4 \beta_{3}) q^{43} + (2 \beta_{6} + 11 \beta_{4}) q^{44} + (4 \beta_{6} - 7 \beta_{4}) q^{49} + (\beta_{5} - 12 \beta_{2}) q^{52} + ( - \beta_1 - 11) q^{56} - 4 \beta_{4} q^{59} + ( - 2 \beta_{5} + 6 \beta_{2}) q^{62} - 3 \beta_{7} q^{63} + (3 \beta_{6} + 14 \beta_{4}) q^{64} + (3 \beta_{7} - 8 \beta_{3}) q^{68} - 8 q^{71} + (3 \beta_{5} - 9 \beta_{2}) q^{72} + (\beta_{7} - 6 \beta_{3}) q^{73} + (3 \beta_{5} + 2 \beta_{2}) q^{77} - 9 q^{81} + (3 \beta_{7} - 4 \beta_{3}) q^{83} + ( - 3 \beta_1 + 17) q^{86} + 11 \beta_{3} q^{88} + 4 \beta_{6} q^{89} + (2 \beta_1 + 12) q^{91} + ( - 4 \beta_{7} + 5 \beta_{3}) q^{98} - 3 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 56 q^{16} + 56 q^{26} + 48 q^{36} - 88 q^{56} - 64 q^{71} - 72 q^{81} + 136 q^{86} + 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 19\nu^{2} + 11\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 34\nu^{5} - 50\nu^{4} + 82\nu^{3} - 62\nu^{2} + 51\nu - 15 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 13\nu^{6} + 49\nu^{5} - 105\nu^{4} + 167\nu^{3} - 167\nu^{2} + 116\nu - 35 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 83\nu^{5} + 155\nu^{4} - 249\nu^{3} + 229\nu^{2} - 157\nu + 45 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\nu^{7} - 37\nu^{6} + 161\nu^{5} - 275\nu^{4} + 503\nu^{3} - 443\nu^{2} + 379\nu - 110 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} + 228\nu^{5} - 430\nu^{4} + 724\nu^{3} - 684\nu^{2} + 492\nu - 145 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\nu^{7} - 47\nu^{6} + 191\nu^{5} - 395\nu^{4} + 693\nu^{3} - 723\nu^{2} + 569\nu - 190 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 2\beta_{2} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 3\beta_{6} + \beta_{5} - 11\beta_{4} - 10\beta_{3} - 4\beta_{2} + 3\beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{6} + 2\beta_{5} - 12\beta_{4} - 2\beta_{3} - 14\beta_{2} - 4\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{7} + 15\beta_{6} + \beta_{5} + 47\beta_{4} + 58\beta_{3} - 12\beta_{2} - 25\beta _1 + 87 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{7} + 30\beta_{6} - 12\beta_{5} + 101\beta_{4} + 35\beta_{3} + 75\beta_{2} + 11\beta _1 - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24\beta_{7} - 14\beta_{6} - 18\beta_{5} - 43\beta_{4} - 149\beta_{3} + 117\beta_{2} + 84\beta _1 - 279 ) / 2 \) 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
0.500000 − 1.41267i
0.500000 − 0.0845405i
0.500000 + 1.08454i
0.500000 + 2.41267i
0.500000 + 1.41267i
0.500000 + 0.0845405i
0.500000 − 1.08454i
0.500000 − 2.41267i
−1.91267 + 1.91267i 0 − 5.31662i 0 0 −0.605599 + 0.605599i 6.34361 + 6.34361i 3.00000i 0
32.2 −0.584541 + 0.584541i 0 1.31662i 0 0 3.69232 − 3.69232i −1.93870 − 1.93870i 3.00000i 0
32.3 0.584541 − 0.584541i 0 1.31662i 0 0 −3.69232 + 3.69232i 1.93870 + 1.93870i 3.00000i 0
32.4 1.91267 − 1.91267i 0 − 5.31662i 0 0 0.605599 − 0.605599i −6.34361 − 6.34361i 3.00000i 0
43.1 −1.91267 − 1.91267i 0 5.31662i 0 0 −0.605599 − 0.605599i 6.34361 − 6.34361i − 3.00000i 0
43.2 −0.584541 − 0.584541i 0 − 1.31662i 0 0 3.69232 + 3.69232i −1.93870 + 1.93870i − 3.00000i 0
43.3 0.584541 + 0.584541i 0 − 1.31662i 0 0 −3.69232 − 3.69232i 1.93870 − 1.93870i − 3.00000i 0
43.4 1.91267 + 1.91267i 0 5.31662i 0 0 0.605599 + 0.605599i −6.34361 + 6.34361i − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
11.b 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.c ✓ 8
5.b even 2 1 inner 275.2.e.c ✓ 8
5.c odd 4 2 inner 275.2.e.c ✓ 8
11.b odd 2 1 inner 275.2.e.c ✓ 8
55.d odd 2 1 CM 275.2.e.c ✓ 8
55.e even 4 2 inner 275.2.e.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.e.c ✓ 8 1.a even 1 1 trivial
275.2.e.c ✓ 8 5.b even 2 1 inner
275.2.e.c ✓ 8 5.c odd 4 2 inner
275.2.e.c ✓ 8 11.b odd 2 1 inner
275.2.e.c ✓ 8 55.d odd 2 1 CM
275.2.e.c ✓ 8 55.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 54T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 54T^{4} + 25 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 744T^{4} + 400 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 1704 T^{4} + 250000 \) Copy content Toggle raw display
$17$ \( T^{8} + 2664 T^{4} + 960400 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 44)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 15144 T^{4} + 52128400 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T + 8)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 85224 T^{4} + 400 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 55464 T^{4} + 749664400 \) Copy content Toggle raw display
$89$ \( (T^{2} + 176)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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