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Properties

Label 775.2.e.g
Level $775$
Weight $2$
Character orbit 775.e
Analytic conductor $6.188$
Analytic rank $0$
Dimension $8$
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] = [775,2,Mod(501,775)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.501");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.e (of order \(3\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 23x^{4} + x^{3} + 16x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{6} + \beta_{4} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{4} + ( - 2 \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{5} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{6} + \beta_{4} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{4} + ( - 2 \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 3 q^{3} + 6 q^{4} + 10 q^{6} + q^{7} + 18 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 3 q^{3} + 6 q^{4} + 10 q^{6} + q^{7} + 18 q^{8} - q^{9} + 4 q^{11} + 8 q^{12} - q^{13} - 8 q^{14} + 10 q^{16} + 12 q^{17} + 11 q^{18} - 5 q^{19} + 9 q^{21} - 10 q^{23} + 2 q^{24} - 12 q^{26} + 6 q^{27} - 4 q^{28} - 26 q^{29} + 19 q^{31} + 28 q^{32} - 8 q^{33} + 24 q^{34} - 5 q^{36} - 16 q^{37} - 9 q^{38} - 22 q^{39} - 4 q^{41} - 6 q^{42} + q^{43} - 7 q^{44} - 22 q^{46} - 20 q^{47} + 27 q^{48} - 37 q^{49} - 12 q^{51} - 21 q^{52} + q^{53} + 24 q^{54} - 29 q^{56} + 4 q^{57} - 26 q^{58} + 6 q^{59} - 2 q^{61} + 17 q^{62} - 24 q^{63} + 34 q^{64} + 2 q^{66} + 7 q^{67} + 5 q^{68} - 6 q^{69} + 12 q^{71} - 14 q^{72} - 20 q^{73} + 18 q^{74} - 23 q^{76} - 58 q^{77} - 22 q^{78} - 2 q^{79} + 12 q^{81} + 18 q^{82} + 4 q^{83} + 41 q^{84} - 13 q^{86} + 10 q^{88} + 54 q^{89} - 44 q^{91} - 86 q^{92} - 55 q^{93} - 48 q^{94} + 13 q^{96} + 18 q^{97} + 46 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 23x^{4} + x^{3} + 16x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\nu^{7} + 857\nu^{6} - 474\nu^{5} + 4932\nu^{4} + 5726\nu^{3} + 14569\nu^{2} + 4405\nu + 2370 ) / 5817 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -146\nu^{7} - 352\nu^{6} + 792\nu^{5} - 4191\nu^{4} + 1232\nu^{3} - 5984\nu^{2} + 20203\nu - 3960 ) / 5817 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -166\nu^{7} + 556\nu^{6} - 1251\nu^{5} + 1530\nu^{4} - 1946\nu^{3} + 9452\nu^{2} - 1174\nu + 438 ) / 5817 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 197\nu^{7} - 800\nu^{6} + 1800\nu^{5} - 3708\nu^{4} + 2800\nu^{3} - 13600\nu^{2} - 3793\nu - 9000 ) / 5817 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -130\nu^{7} + 85\nu^{6} - 676\nu^{5} - 624\nu^{4} - 3206\nu^{3} - 494\nu^{2} - 312\nu - 498 ) / 1939 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -565\nu^{7} + 444\nu^{6} - 2938\nu^{5} - 2712\nu^{4} - 11249\nu^{3} - 2147\nu^{2} - 1356\nu - 822 ) / 1939 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + 2\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + \beta_{3} - 9\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{7} + 30\beta_{6} - 11\beta_{4} + 6\beta_{3} + 10\beta_{2} - 30\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{7} + 67\beta_{6} + 36\beta_{5} + 36\beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 77\beta_{5} + 88\beta_{4} - 36\beta_{3} + 193\beta _1 + 88 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
501.1
−0.844316 − 1.46240i
−0.392238 − 0.679376i
0.434993 + 0.753430i
1.30156 + 2.25437i
−0.844316 + 1.46240i
−0.392238 + 0.679376i
0.434993 − 0.753430i
1.30156 − 2.25437i
−1.68863 −1.34432 + 2.32842i 0.851477 0 2.27005 − 3.93185i 2.52642 − 4.37588i 1.93943 −2.11437 − 3.66220i 0
501.2 −0.784476 −0.892238 + 1.54540i −1.38460 0 0.699939 − 1.21233i −2.44756 + 4.23929i 2.65514 −0.0921773 − 0.159656i 0
501.3 0.869986 −0.0650072 + 0.112596i −1.24312 0 −0.0565553 + 0.0979567i 1.58680 − 2.74842i −2.82147 1.49155 + 2.58344i 0
501.4 2.60312 0.801561 − 1.38834i 4.77625 0 2.08656 − 3.61403i −1.16566 + 2.01898i 7.22690 0.215000 + 0.372390i 0
676.1 −1.68863 −1.34432 − 2.32842i 0.851477 0 2.27005 + 3.93185i 2.52642 + 4.37588i 1.93943 −2.11437 + 3.66220i 0
676.2 −0.784476 −0.892238 − 1.54540i −1.38460 0 0.699939 + 1.21233i −2.44756 − 4.23929i 2.65514 −0.0921773 + 0.159656i 0
676.3 0.869986 −0.0650072 − 0.112596i −1.24312 0 −0.0565553 − 0.0979567i 1.58680 + 2.74842i −2.82147 1.49155 − 2.58344i 0
676.4 2.60312 0.801561 + 1.38834i 4.77625 0 2.08656 + 3.61403i −1.16566 − 2.01898i 7.22690 0.215000 − 0.372390i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.e.g 8
5.b even 2 1 155.2.e.c ✓ 8
5.c odd 4 2 775.2.o.e 16
31.c even 3 1 inner 775.2.e.g 8
155.i odd 6 1 4805.2.a.k 4
155.j even 6 1 155.2.e.c ✓ 8
155.j even 6 1 4805.2.a.i 4
155.o odd 12 2 775.2.o.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.c ✓ 8 5.b even 2 1
155.2.e.c ✓ 8 155.j even 6 1
775.2.e.g 8 1.a even 1 1 trivial
775.2.e.g 8 31.c even 3 1 inner
775.2.o.e 16 5.c odd 4 2
775.2.o.e 16 155.o odd 12 2
4805.2.a.i 4 155.j even 6 1
4805.2.a.k 4 155.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} - 5 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 33489 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + \cdots + 225 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 1067089 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 35721 \) Copy content Toggle raw display
$23$ \( (T^{4} + 5 T^{3} + \cdots + 157)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 13 T^{3} + 29 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 19 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 45369 \) Copy content Toggle raw display
$43$ \( T^{8} - T^{7} + \cdots + 32761 \) Copy content Toggle raw display
$47$ \( (T^{4} + 10 T^{3} + \cdots + 321)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - T^{7} + \cdots + 73441 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} - 151 T^{2} + \cdots + 321)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 7 T^{7} + \cdots + 3613801 \) Copy content Toggle raw display
$71$ \( T^{8} - 12 T^{7} + \cdots + 164025 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + \cdots + 112896 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 80089 \) Copy content Toggle raw display
$83$ \( T^{8} - 4 T^{7} + \cdots + 233998209 \) Copy content Toggle raw display
$89$ \( (T^{4} - 27 T^{3} + \cdots - 459)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 9 T^{3} + \cdots - 2177)^{2} \) Copy content Toggle raw display
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