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Properties

Label 775.2.o.c
Level $775$
Weight $2$
Character orbit 775.o
Analytic conductor $6.188$
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] = [775,2,Mod(149,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.o (of order \(6\), degree \(2\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.592240896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{4} - 1) q^{4} + ( - 2 \beta_{6} + 5 \beta_{3}) q^{6} + (2 \beta_{2} - \beta_1) q^{7} - 3 \beta_{7} q^{8} + (3 \beta_{6} + 3 \beta_{4} - 4 \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{4} - 1) q^{4} + ( - 2 \beta_{6} + 5 \beta_{3}) q^{6} + (2 \beta_{2} - \beta_1) q^{7} - 3 \beta_{7} q^{8} + (3 \beta_{6} + 3 \beta_{4} - 4 \beta_{3} + 1) q^{9} + (2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{11} + (4 \beta_{2} + 3 \beta_1) q^{12} + (\beta_{7} + \beta_{5} - \beta_{2} + \beta_1) q^{13} + (\beta_{6} + 2 \beta_{3}) q^{14} + (\beta_{4} - 2) q^{16} - 2 \beta_1 q^{17} + (9 \beta_{7} - 4 \beta_{5} + \cdots - 4 \beta_1) q^{18}+ \cdots + (8 \beta_{6} - 26 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{4} + 16 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{4} + 16 q^{6} + 10 q^{9} + 4 q^{11} + 10 q^{14} - 12 q^{16} + 18 q^{19} + 4 q^{21} - 18 q^{24} + 12 q^{26} - 36 q^{29} + 44 q^{31} + 28 q^{34} - 54 q^{36} - 20 q^{39} - 36 q^{41} - 32 q^{44} - 68 q^{46} - 6 q^{49} + 32 q^{51} + 140 q^{54} + 18 q^{56} - 32 q^{59} + 16 q^{64} + 136 q^{66} + 50 q^{69} - 12 q^{71} + 26 q^{74} - 14 q^{76} - 20 q^{79} - 76 q^{81} - 6 q^{84} + 18 q^{86} + 4 q^{89} - 32 q^{91} - 28 q^{94} - 34 q^{96} - 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 97\nu ) / 120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{6} + 40\nu^{4} - 280\nu^{2} + 441 ) / 360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 40\nu^{5} - 280\nu^{3} + 81\nu ) / 360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{6} + 160\nu^{4} - 760\nu^{2} + 1197 ) / 360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 40\nu^{5} - 190\nu^{3} + 81\nu ) / 270 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 4\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{6} - 19\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21\beta_{7} - 19\beta_{5} - 21\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{4} - 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -120\beta_{2} - 97\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.99426 − 1.15139i
1.12824 − 0.651388i
−1.12824 + 0.651388i
−1.99426 + 1.15139i
−1.99426 − 1.15139i
−1.12824 − 0.651388i
1.12824 + 0.651388i
1.99426 + 1.15139i
− 2.30278i −2.86029 + 1.65139i −3.30278 0 3.80278 + 6.58660i −0.262211 + 0.151388i 3.00000i 3.95416 − 6.84881i 0
149.2 − 1.30278i −0.262211 + 0.151388i 0.302776 0 0.197224 + 0.341603i −2.86029 + 1.65139i − 3.00000i −1.45416 + 2.51868i 0
149.3 1.30278i 0.262211 − 0.151388i 0.302776 0 0.197224 + 0.341603i 2.86029 − 1.65139i 3.00000i −1.45416 + 2.51868i 0
149.4 2.30278i 2.86029 − 1.65139i −3.30278 0 3.80278 + 6.58660i 0.262211 − 0.151388i − 3.00000i 3.95416 − 6.84881i 0
749.1 − 2.30278i 2.86029 + 1.65139i −3.30278 0 3.80278 − 6.58660i 0.262211 + 0.151388i 3.00000i 3.95416 + 6.84881i 0
749.2 − 1.30278i 0.262211 + 0.151388i 0.302776 0 0.197224 − 0.341603i 2.86029 + 1.65139i − 3.00000i −1.45416 − 2.51868i 0
749.3 1.30278i −0.262211 − 0.151388i 0.302776 0 0.197224 − 0.341603i −2.86029 − 1.65139i 3.00000i −1.45416 − 2.51868i 0
749.4 2.30278i −2.86029 − 1.65139i −3.30278 0 3.80278 − 6.58660i −0.262211 − 0.151388i − 3.00000i 3.95416 + 6.84881i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.o.c 8
5.b even 2 1 inner 775.2.o.c 8
5.c odd 4 1 775.2.e.c ✓ 4
5.c odd 4 1 775.2.e.d yes 4
31.c even 3 1 inner 775.2.o.c 8
155.j even 6 1 inner 775.2.o.c 8
155.o odd 12 1 775.2.e.c ✓ 4
155.o odd 12 1 775.2.e.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.e.c ✓ 4 5.c odd 4 1
775.2.e.c ✓ 4 155.o odd 12 1
775.2.e.d yes 4 5.c odd 4 1
775.2.e.d yes 4 155.o odd 12 1
775.2.o.c 8 1.a even 1 1 trivial
775.2.o.c 8 5.b even 2 1 inner
775.2.o.c 8 31.c even 3 1 inner
775.2.o.c 8 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2}^{4} + 7T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{8} - 11T_{7}^{6} + 120T_{7}^{4} - 11T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 7 T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 7 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} - 28 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 58 T^{2} + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T - 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11 T + 31)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 143 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} - 19 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( (T^{4} + 7 T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16 T^{3} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 52)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} - 203 T^{6} + \cdots + 3418801 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots + 729)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 319 T^{6} + \cdots + 639128961 \) Copy content Toggle raw display
$89$ \( (T^{2} - T - 81)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 154 T^{2} + 729)^{2} \) Copy content Toggle raw display
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