Properties

Label 770.2.m.d
Level $770$
Weight $2$
Character orbit 770.m
Analytic conductor $6.148$
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] = [770,2,Mod(43,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.m (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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{5} q^{3} - \beta_{2} q^{4} + ( - \beta_{2} + 2) q^{5} + \beta_{4} q^{6} + \beta_{7} q^{7} + \beta_1 q^{8} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{5} q^{3} - \beta_{2} q^{4} + ( - \beta_{2} + 2) q^{5} + \beta_{4} q^{6} + \beta_{7} q^{7} + \beta_1 q^{8} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{9} + ( - 2 \beta_{7} + \beta_1) q^{10} + ( - \beta_{7} + 2 \beta_{4} + \beta_1 - 1) q^{11} - \beta_{3} q^{12} + (2 \beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{13} + \beta_{2} q^{14} + (2 \beta_{5} - \beta_{3}) q^{15} - q^{16} + ( - 4 \beta_{7} - \beta_{6} + \beta_{4} + \beta_1) q^{17} + ( - \beta_{6} - \beta_{4}) q^{18} + ( - 3 \beta_{7} + \beta_{6} - 4 \beta_1) q^{19} + ( - 2 \beta_{2} - 1) q^{20} - \beta_{4} q^{21} + (\beta_{7} - 2 \beta_{3} - \beta_{2} - 1) q^{22} + 3 \beta_{5} q^{23} + (\beta_{6} - \beta_1) q^{24} + ( - 4 \beta_{2} + 3) q^{25} + ( - 2 \beta_{5} - 2 \beta_{3} - 4) q^{26} + (2 \beta_{3} + 2 \beta_{2} + 2) q^{27} - \beta_1 q^{28} + (\beta_{7} + 5 \beta_{6} - 4 \beta_1) q^{29} + (\beta_{6} + 2 \beta_{4} - \beta_1) q^{30} + (3 \beta_{5} + 3 \beta_{3} + 2) q^{31} + \beta_{7} q^{32} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 5 \beta_1) q^{33} + (\beta_{5} - \beta_{3} - 4 \beta_{2}) q^{34} + (2 \beta_{7} - \beta_1) q^{35} + (\beta_{5} + \beta_{3} + 1) q^{36} + ( - \beta_{3} + 6 \beta_{2} + 6) q^{37} + ( - \beta_{5} - 3 \beta_{2} + 3) q^{38} + (4 \beta_{7} + 4 \beta_1) q^{39} + (\beta_{7} + 2 \beta_1) q^{40} + (7 \beta_{7} - \beta_{4} - 7 \beta_1) q^{41} + \beta_{3} q^{42} - 6 \beta_1 q^{43} + (\beta_{7} + 2 \beta_{6} + \beta_{2} - \beta_1) q^{44} + ( - \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 1) q^{45} + 3 \beta_{4} q^{46} + ( - 6 \beta_{3} - 5 \beta_{2} - 5) q^{47} - \beta_{5} q^{48} - \beta_{2} q^{49} + ( - 3 \beta_{7} + 4 \beta_1) q^{50} + ( - 2 \beta_{7} + 2 \beta_{4} + 2 \beta_1) q^{51} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_1) q^{52} + ( - \beta_{5} + 6 \beta_{2} - 6) q^{53} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{54} + ( - \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{55} + q^{56} + (2 \beta_{7} - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_1) q^{57} + ( - 5 \beta_{5} + \beta_{2} - 1) q^{58} + ( - \beta_{5} + \beta_{3} - 4 \beta_{2}) q^{59} + ( - \beta_{5} - 2 \beta_{3}) q^{60} + (2 \beta_{7} - 2 \beta_1) q^{61} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{4} + 3 \beta_1) q^{62} + (\beta_{6} + \beta_{4}) q^{63} + \beta_{2} q^{64} + (4 \beta_{7} + 6 \beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{65} + (\beta_{5} - \beta_{4} + \beta_{3} - 4) q^{66} + (2 \beta_{3} - 3 \beta_{2} - 3) q^{67} + (\beta_{6} + \beta_{4} + 3 \beta_1) q^{68} + ( - 3 \beta_{5} + 3 \beta_{3} - 6 \beta_{2}) q^{69} + (2 \beta_{2} + 1) q^{70} + 12 q^{71} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_1) q^{72} + ( - 4 \beta_{6} - 4 \beta_{4} - 2 \beta_1) q^{73} + ( - 6 \beta_{7} + \beta_{6} - 7 \beta_1) q^{74} + (3 \beta_{5} - 4 \beta_{3}) q^{75} + ( - 3 \beta_{7} - \beta_{4} + 3 \beta_1) q^{76} + ( - \beta_{7} + 2 \beta_{3} + \beta_{2} + 1) q^{77} + (4 \beta_{2} - 4) q^{78} + (3 \beta_{7} + 5 \beta_{6} - 2 \beta_1) q^{79} + (\beta_{2} - 2) q^{80} + ( - 3 \beta_{5} - 3 \beta_{3} + 1) q^{81} + (\beta_{3} + 7 \beta_{2} + 7) q^{82} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{83} + ( - \beta_{6} + \beta_1) q^{84} + ( - 8 \beta_{7} - \beta_{6} + 3 \beta_{4} + 5 \beta_1) q^{85} + 6 q^{86} + (10 \beta_{7} - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_1) q^{87} + ( - 2 \beta_{5} + \beta_{2} - \beta_1 - 1) q^{88} + ( - 6 \beta_{5} + 6 \beta_{3} + 10 \beta_{2}) q^{89} + ( - \beta_{7} - 3 \beta_{6} - \beta_{4} + \beta_1) q^{90} + (2 \beta_{5} + 2 \beta_{3} + 4) q^{91} - 3 \beta_{3} q^{92} + ( - 4 \beta_{5} - 6 \beta_{2} + 6) q^{93} + (5 \beta_{7} + 6 \beta_{6} - \beta_1) q^{94} + ( - 9 \beta_{7} + 2 \beta_{6} - \beta_{4} - 5 \beta_1) q^{95} - \beta_{4} q^{96} + (7 \beta_{3} - 2 \beta_{2} - 2) q^{97} + \beta_1 q^{98} + ( - 5 \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 16 q^{5} - 8 q^{11} + 4 q^{12} - 4 q^{15} - 8 q^{16} - 8 q^{20} - 12 q^{23} + 24 q^{25} - 16 q^{26} + 8 q^{27} - 8 q^{31} + 4 q^{33} + 52 q^{37} + 28 q^{38} - 4 q^{42} - 16 q^{47} + 4 q^{48} - 44 q^{53} - 16 q^{55} + 8 q^{56} + 12 q^{58} + 12 q^{60} - 40 q^{66} - 32 q^{67} + 8 q^{70} + 96 q^{71} + 4 q^{75} - 32 q^{78} - 16 q^{80} + 32 q^{81} + 52 q^{82} + 48 q^{86} + 16 q^{91} + 12 q^{92} + 64 q^{93} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{4} + 5\nu^{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + \nu^{4} - 5\nu^{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 13\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 3\beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{6} - 5\beta_{4} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{5} + 4\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -16\beta_{7} + 13\beta_{6} - 13\beta_{4} - 13\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.14412 + 1.14412i
0.437016 0.437016i
1.14412 1.14412i
−0.437016 + 0.437016i
−1.14412 1.14412i
0.437016 + 0.437016i
1.14412 + 1.14412i
−0.437016 0.437016i
−0.707107 0.707107i −1.61803 1.61803i 1.00000i 2.00000 + 1.00000i 2.28825i 0.707107 + 0.707107i 0.707107 0.707107i 2.23607i −0.707107 2.12132i
43.2 −0.707107 0.707107i 0.618034 + 0.618034i 1.00000i 2.00000 + 1.00000i 0.874032i 0.707107 + 0.707107i 0.707107 0.707107i 2.23607i −0.707107 2.12132i
43.3 0.707107 + 0.707107i −1.61803 1.61803i 1.00000i 2.00000 + 1.00000i 2.28825i −0.707107 0.707107i −0.707107 + 0.707107i 2.23607i 0.707107 + 2.12132i
43.4 0.707107 + 0.707107i 0.618034 + 0.618034i 1.00000i 2.00000 + 1.00000i 0.874032i −0.707107 0.707107i −0.707107 + 0.707107i 2.23607i 0.707107 + 2.12132i
197.1 −0.707107 + 0.707107i −1.61803 + 1.61803i 1.00000i 2.00000 1.00000i 2.28825i 0.707107 0.707107i 0.707107 + 0.707107i 2.23607i −0.707107 + 2.12132i
197.2 −0.707107 + 0.707107i 0.618034 0.618034i 1.00000i 2.00000 1.00000i 0.874032i 0.707107 0.707107i 0.707107 + 0.707107i 2.23607i −0.707107 + 2.12132i
197.3 0.707107 0.707107i −1.61803 + 1.61803i 1.00000i 2.00000 1.00000i 2.28825i −0.707107 + 0.707107i −0.707107 0.707107i 2.23607i 0.707107 2.12132i
197.4 0.707107 0.707107i 0.618034 0.618034i 1.00000i 2.00000 1.00000i 0.874032i −0.707107 + 0.707107i −0.707107 0.707107i 2.23607i 0.707107 2.12132i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.m.d 8
5.c odd 4 1 inner 770.2.m.d 8
11.b odd 2 1 inner 770.2.m.d 8
55.e even 4 1 inner 770.2.m.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.m.d 8 1.a even 1 1 trivial
770.2.m.d 8 5.c odd 4 1 inner
770.2.m.d 8 11.b odd 2 1 inner
770.2.m.d 8 55.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{17}^{8} + 752T_{17}^{4} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 1792 T^{4} + 65536 \) Copy content Toggle raw display
$17$ \( T^{8} + 752T^{4} + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} - 54 T^{2} + 484)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + 18 T^{2} - 108 T + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 134 T^{2} + 3364)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 44)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 26 T^{3} + 338 T^{2} - 2132 T + 6724)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 174 T^{2} + 6724)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 656 T + 6724)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 22 T^{3} + 242 T^{2} + 1276 T + 3364)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 60 T^{2} + 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 16 T^{3} + 128 T^{2} + 352 T + 484)^{2} \) Copy content Toggle raw display
$71$ \( (T - 12)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 16672 T^{4} + \cdots + 33362176 \) Copy content Toggle raw display
$79$ \( (T^{4} - 126 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 1792 T^{4} + 65536 \) Copy content Toggle raw display
$89$ \( (T^{4} + 392 T^{2} + 26896)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 22 T^{3} + 242 T^{2} - 1364 T + 3844)^{2} \) Copy content Toggle raw display
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