Properties

Label 775.2.b.f
Level $775$
Weight $2$
Character orbit 775.b
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(249,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, 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("775.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4589249536.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 39x^{4} + 49x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 155)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_1 q^{3} + ( - \beta_{4} - 1) q^{4} + ( - \beta_{4} + \beta_{2} - 1) q^{6} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_1 q^{3} + ( - \beta_{4} - 1) q^{4} + ( - \beta_{4} + \beta_{2} - 1) q^{6} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{8} + \beta_{2} q^{9} + ( - \beta_{4} + \beta_{2} - 1) q^{11} + ( - \beta_{6} - 2 \beta_{5}) q^{12} + (\beta_{6} + \beta_{3} + \beta_1) q^{13} + (\beta_{7} + \beta_{4} + \beta_{2} + 3) q^{14} + (\beta_{7} + \beta_{2}) q^{16} + ( - \beta_{6} - \beta_{3}) q^{17} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{18} + (\beta_{7} - \beta_{4} + 2 \beta_{2}) q^{19} + ( - \beta_{7} + 2 \beta_{4}) q^{21} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{22} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{23} + 4 q^{24} + (\beta_{7} + \beta_{4} + 3 \beta_{2} - 1) q^{26} + ( - \beta_{3} - 2 \beta_1) q^{27} + ( - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{28} + ( - \beta_{7} + 2 \beta_{4} + 2) q^{29} - q^{31} + (\beta_{5} - \beta_{3} + 3 \beta_1) q^{32} + ( - \beta_{6} - 2 \beta_{5} - \beta_{3} + \beta_1) q^{33} + ( - \beta_{7} - 4 \beta_{2} + 2) q^{34} + ( - \beta_{7} - 3 \beta_{2}) q^{36} + ( - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} - 2 \beta_1) q^{37} + ( - \beta_{6} + 2 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{38} + (\beta_{7} - 2 \beta_{4} + 2 \beta_{2} + 2) q^{39} + ( - \beta_{7} + 3 \beta_{2} - 3) q^{41} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{42} + (2 \beta_{6} + \beta_{3} + 2 \beta_1) q^{43} + (2 \beta_{4} - 2 \beta_{2} + 6) q^{44} + (\beta_{7} + \beta_{4} + \beta_{2} + 3) q^{46} + (2 \beta_{6} + 4 \beta_{5} + 2 \beta_1) q^{47} - 2 \beta_{6} q^{48} + ( - 3 \beta_{7} + \beta_{4} - 3 \beta_{2} - 2) q^{49} + ( - \beta_{7} + 2 \beta_{4} - 3 \beta_{2} + 1) q^{51} + (6 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{52} + (\beta_{6} - 5 \beta_1) q^{53} + ( - \beta_{7} - 2 \beta_{4}) q^{54} + ( - 4 \beta_{4} - 4 \beta_{2} - 4) q^{56} + ( - 3 \beta_{6} - 2 \beta_{5} - \beta_{3}) q^{57} + ( - \beta_{6} + 4 \beta_{5} - 2 \beta_1) q^{58} + ( - 2 \beta_{7} - \beta_{4} + 2) q^{59} + (\beta_{7} - \beta_{4} + \beta_{2} + 5) q^{61} - \beta_{5} q^{62} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{63} + (\beta_{7} + 2 \beta_{4} - 3 \beta_{2} + 2) q^{64} + ( - \beta_{7} + 3 \beta_{4} - 5 \beta_{2} + 9) q^{66} + (3 \beta_{3} - 3 \beta_1) q^{67} + (\beta_{6} - 4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{68} + ( - \beta_{7} + 2 \beta_{4}) q^{69} + ( - \beta_{2} - 5) q^{71} + ( - 3 \beta_{5} + 3 \beta_{3} - \beta_1) q^{72} + (3 \beta_{6} - 2 \beta_{3} - \beta_1) q^{73} + (3 \beta_{7} + 4 \beta_{2} - 2) q^{74} + ( - \beta_{7} - \beta_{4} - 7 \beta_{2} + 3) q^{76} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{77} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{78} + (4 \beta_{7} - 3 \beta_{4} + \beta_{2} - 1) q^{79} + (\beta_{4} + 3 \beta_{2} - 6) q^{81} + ( - 4 \beta_{6} - 2 \beta_{5} - \beta_{3} + 3 \beta_1) q^{82} + ( - \beta_{6} - 2 \beta_{5} - 3 \beta_{3}) q^{83} + ( - 2 \beta_{7} - 8) q^{84} + (\beta_{7} + 2 \beta_{4} + 4 \beta_{2}) q^{86} + (4 \beta_{6} + 4 \beta_{5} - \beta_{3} + \beta_1) q^{87} + 4 \beta_{5} q^{88} + (\beta_{4} + 5 \beta_{2} + 1) q^{89} + (\beta_{7} - 5 \beta_{4} - 5 \beta_{2} + 1) q^{91} + ( - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{92} + \beta_1 q^{93} + ( - 2 \beta_{4} + 2 \beta_{2} - 10) q^{94} + ( - 4 \beta_{2} + 8) q^{96} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{97} + ( - 9 \beta_{5} + 8 \beta_{3} - 4 \beta_1) q^{98} + ( - \beta_{7} + \beta_{4} - 3 \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} - 8 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} - 8 q^{6} + 2 q^{9} - 8 q^{11} + 32 q^{14} + 6 q^{16} + 6 q^{19} + 32 q^{24} + 4 q^{26} + 16 q^{29} - 8 q^{31} + 4 q^{34} - 10 q^{36} + 20 q^{39} - 22 q^{41} + 48 q^{44} + 32 q^{46} - 32 q^{49} + 2 q^{51} - 8 q^{54} - 48 q^{56} + 6 q^{59} + 44 q^{61} + 18 q^{64} + 64 q^{66} - 42 q^{71} + 4 q^{74} + 4 q^{76} + 4 q^{79} - 40 q^{81} - 72 q^{84} + 16 q^{86} + 20 q^{89} - 8 q^{91} - 80 q^{94} + 56 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 39x^{4} + 49x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 6\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 11\nu^{5} - 35\nu^{3} - 29\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 15\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{6} + 9\nu^{4} + 22\nu^{2} + 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 6\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} - 2\beta_{5} - 6\beta_{3} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - 9\beta_{4} + 32\beta_{2} - 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{6} + 18\beta_{5} + 31\beta_{3} - 76\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
0.704624i
2.27841i
1.31743i
1.89122i
1.89122i
1.31743i
2.27841i
0.704624i
2.50350i 0.704624i −4.26753 0 −1.76403 4.77104i 5.67678i 2.50350 0
249.2 2.19117i 2.27841i −2.80122 0 −4.99239 1.38995i 1.75561i −2.19117 0
249.3 1.26438i 1.31743i 0.401352 0 1.66573 1.13698i 3.03621i 1.26438 0
249.4 0.576713i 1.89122i 1.66740 0 1.09069 4.24412i 2.11504i −0.576713 0
249.5 0.576713i 1.89122i 1.66740 0 1.09069 4.24412i 2.11504i −0.576713 0
249.6 1.26438i 1.31743i 0.401352 0 1.66573 1.13698i 3.03621i 1.26438 0
249.7 2.19117i 2.27841i −2.80122 0 −4.99239 1.38995i 1.75561i −2.19117 0
249.8 2.50350i 0.704624i −4.26753 0 −1.76403 4.77104i 5.67678i 2.50350 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.8
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 775.2.b.f 8
5.b even 2 1 inner 775.2.b.f 8
5.c odd 4 1 155.2.a.e 4
5.c odd 4 1 775.2.a.e 4
15.e even 4 1 1395.2.a.l 4
15.e even 4 1 6975.2.a.bn 4
20.e even 4 1 2480.2.a.x 4
35.f even 4 1 7595.2.a.s 4
40.i odd 4 1 9920.2.a.cb 4
40.k even 4 1 9920.2.a.cg 4
155.f even 4 1 4805.2.a.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.e 4 5.c odd 4 1
775.2.a.e 4 5.c odd 4 1
775.2.b.f 8 1.a even 1 1 trivial
775.2.b.f 8 5.b even 2 1 inner
1395.2.a.l 4 15.e even 4 1
2480.2.a.x 4 20.e even 4 1
4805.2.a.n 4 155.f even 4 1
6975.2.a.bn 4 15.e even 4 1
7595.2.a.s 4 35.f even 4 1
9920.2.a.cb 4 40.i odd 4 1
9920.2.a.cg 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 13T_{2}^{6} + 52T_{2}^{4} + 64T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 13 T^{6} + 52 T^{4} + 64 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} + 11 T^{6} + 39 T^{4} + 49 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 44 T^{6} + 544 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} - 8 T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{6} + 1168 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$17$ \( T^{8} + 51 T^{6} + 823 T^{4} + \cdots + 3364 \) Copy content Toggle raw display
$19$ \( (T^{4} - 3 T^{3} - 33 T^{2} + 107 T + 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 44 T^{6} + 544 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 20 T^{2} + 292 T - 584)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 171 T^{6} + 10495 T^{4} + \cdots + 2365444 \) Copy content Toggle raw display
$41$ \( (T^{4} + 11 T^{3} - 31 T^{2} - 359 T + 506)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 143 T^{6} + 5571 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
$47$ \( T^{8} + 204 T^{6} + 13040 T^{4} + \cdots + 1982464 \) Copy content Toggle raw display
$53$ \( T^{8} + 311 T^{6} + 28011 T^{4} + \cdots + 1705636 \) Copy content Toggle raw display
$59$ \( (T^{4} - 3 T^{3} - 97 T^{2} - 129 T - 44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 22 T^{3} + 160 T^{2} - 432 T + 352)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 288 T^{6} + 15552 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$71$ \( (T^{4} + 21 T^{3} + 159 T^{2} + 511 T + 584)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 191 T^{6} + 2619 T^{4} + \cdots + 1156 \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} - 260 T^{2} - 404 T + 6592)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 351 T^{6} + \cdots + 11316496 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} - 152 T^{2} + 420 T + 3688)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 512 T^{6} + 68512 T^{4} + \cdots + 215296 \) Copy content Toggle raw display
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