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Properties

Label 770.2.c.f
Level $770$
Weight $2$
Character orbit 770.c
Analytic conductor $6.148$
Analytic rank $0$
Dimension $10$
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] = [770,2,Mod(309,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.309");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.c (of order \(2\), degree \(1\), 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: \(10\)
Coefficient field: 10.0.3545719128064.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 11x^{8} + 41x^{6} + 58x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{7} q^{3} - q^{4} + \beta_{5} q^{5} + \beta_1 q^{6} - \beta_{4} q^{7} - \beta_{4} q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{7} q^{3} - q^{4} + \beta_{5} q^{5} + \beta_1 q^{6} - \beta_{4} q^{7} - \beta_{4} q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - 3) q^{9}+ \cdots + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 2 q^{5} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{4} + 2 q^{5} - 26 q^{9} + 10 q^{11} + 10 q^{14} - 4 q^{15} + 10 q^{16} - 12 q^{19} - 2 q^{20} - 2 q^{25} + 20 q^{26} - 16 q^{29} - 4 q^{30} + 24 q^{31} - 4 q^{34} + 26 q^{36} + 8 q^{39} + 40 q^{41} - 10 q^{44} + 30 q^{45} - 16 q^{46} - 10 q^{49} - 12 q^{50} - 32 q^{51} + 2 q^{55} - 10 q^{56} - 72 q^{59} + 4 q^{60} + 20 q^{61} - 10 q^{64} + 52 q^{65} - 48 q^{69} + 2 q^{70} + 32 q^{71} - 12 q^{74} + 40 q^{75} + 12 q^{76} + 20 q^{79} + 2 q^{80} + 50 q^{81} - 16 q^{85} - 24 q^{86} - 28 q^{89} + 40 q^{90} - 20 q^{91} - 20 q^{94} + 8 q^{95} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 11x^{8} + 41x^{6} + 58x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 8\nu^{6} + 21\nu^{4} + 23\nu^{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + \nu^{8} + 12\nu^{7} + 8\nu^{6} + 49\nu^{5} + 17\nu^{4} + 79\nu^{3} + 7\nu^{2} + 43\nu + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - \nu^{8} + 12\nu^{7} - 8\nu^{6} + 49\nu^{5} - 17\nu^{4} + 79\nu^{3} - 7\nu^{2} + 43\nu - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 10\nu^{7} - 33\nu^{5} - 39\nu^{3} - 11\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - \nu^{8} + 12\nu^{7} - 12\nu^{6} + 45\nu^{5} - 45\nu^{4} + 55\nu^{3} - 55\nu^{2} + 15\nu - 11 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + \nu^{8} + 12\nu^{7} + 12\nu^{6} + 45\nu^{5} + 45\nu^{4} + 55\nu^{3} + 55\nu^{2} + 15\nu + 11 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 12\nu^{7} + 49\nu^{5} + 75\nu^{3} + 31\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 12\nu^{7} - 10\nu^{6} - 47\nu^{5} - 31\nu^{4} - 63\nu^{3} - 27\nu^{2} - 13\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} + \nu^{8} - 12\nu^{7} + 10\nu^{6} - 47\nu^{5} + 31\nu^{4} - 63\nu^{3} + 27\nu^{2} - 13\nu - 1 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{9} - 3\beta_{8} - 8\beta_{7} - 3\beta_{6} - 3\beta_{5} + 5\beta_{3} + 5\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{9} - 2\beta_{8} - 2\beta_{6} + 2\beta_{5} + 3\beta_{3} - 3\beta_{2} + \beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{9} + 11\beta_{8} + 34\beta_{7} + 9\beta_{6} + 9\beta_{5} - 21\beta_{3} - 21\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{9} + 8\beta_{8} + 9\beta_{6} - 9\beta_{5} - 14\beta_{3} + 14\beta_{2} - 7\beta _1 - 54 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{9} - 11\beta_{8} - 36\beta_{7} - 7\beta_{6} - 7\beta_{5} + \beta_{4} + 22\beta_{3} + 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 67\beta_{9} - 67\beta_{8} - 83\beta_{6} + 83\beta_{5} + 121\beta_{3} - 121\beta_{2} + 78\beta _1 + 432 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 183 \beta_{9} + 183 \beta_{8} + 608 \beta_{7} + 89 \beta_{6} + 89 \beta_{5} - 48 \beta_{4} + \cdots - 371 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
309.1
− 1.97382i
1.44272i
− 0.776378i
2.09118i
0.216294i
− 0.216294i
− 2.09118i
0.776378i
− 1.44272i
1.97382i
− 1.00000i − 3.27510i −1.00000 −0.130920 − 2.23223i −3.27510 1.00000i 1.00000i −7.72630 −2.23223 + 0.130920i
309.2 − 1.00000i − 1.63209i −1.00000 −0.509179 + 2.17732i −1.63209 1.00000i 1.00000i 0.336288 2.17732 + 0.509179i
309.3 − 1.00000i − 0.426870i −1.00000 2.07460 + 0.834294i −0.426870 1.00000i 1.00000i 2.81778 0.834294 − 2.07460i
309.4 − 1.00000i 2.34949i −1.00000 1.69655 − 1.45661i 2.34949 1.00000i 1.00000i −2.52013 −1.45661 − 1.69655i
309.5 − 1.00000i 2.98457i −1.00000 −2.13105 + 0.677228i 2.98457 1.00000i 1.00000i −5.90764 0.677228 + 2.13105i
309.6 1.00000i − 2.98457i −1.00000 −2.13105 − 0.677228i 2.98457 − 1.00000i − 1.00000i −5.90764 0.677228 − 2.13105i
309.7 1.00000i − 2.34949i −1.00000 1.69655 + 1.45661i 2.34949 − 1.00000i − 1.00000i −2.52013 −1.45661 + 1.69655i
309.8 1.00000i 0.426870i −1.00000 2.07460 − 0.834294i −0.426870 − 1.00000i − 1.00000i 2.81778 0.834294 + 2.07460i
309.9 1.00000i 1.63209i −1.00000 −0.509179 − 2.17732i −1.63209 − 1.00000i − 1.00000i 0.336288 2.17732 − 0.509179i
309.10 1.00000i 3.27510i −1.00000 −0.130920 + 2.23223i −3.27510 − 1.00000i − 1.00000i −7.72630 −2.23223 − 0.130920i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 309.10
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 770.2.c.f ✓ 10
5.b even 2 1 inner 770.2.c.f ✓ 10
5.c odd 4 1 3850.2.a.bz 5
5.c odd 4 1 3850.2.a.ca 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.c.f ✓ 10 1.a even 1 1 trivial
770.2.c.f ✓ 10 5.b even 2 1 inner
3850.2.a.bz 5 5.c odd 4 1
3850.2.a.ca 5 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{10} + 28T_{3}^{8} + 276T_{3}^{6} + 1120T_{3}^{4} + 1600T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{10} + 72T_{13}^{8} + 1412T_{13}^{6} + 6320T_{13}^{4} + 3008T_{13}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{10} + 124T_{17}^{8} + 4912T_{17}^{6} + 77888T_{17}^{4} + 520192T_{17}^{2} + 1183744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 28 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( (T - 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 72 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{10} + 124 T^{8} + \cdots + 1183744 \) Copy content Toggle raw display
$19$ \( (T^{5} + 6 T^{4} + \cdots + 2768)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 80 T^{8} + \cdots + 295936 \) Copy content Toggle raw display
$29$ \( (T^{5} + 8 T^{4} - 16 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 12 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 124 T^{8} + \cdots + 295936 \) Copy content Toggle raw display
$41$ \( (T^{5} - 20 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 416 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{10} + 276 T^{8} + \cdots + 5456896 \) Copy content Toggle raw display
$53$ \( T^{10} + 236 T^{8} + \cdots + 23348224 \) Copy content Toggle raw display
$59$ \( (T^{5} + 36 T^{4} + \cdots - 26752)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots + 3664)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 172 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$71$ \( (T^{5} - 16 T^{4} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 268 T^{8} + \cdots + 5914624 \) Copy content Toggle raw display
$79$ \( (T^{5} - 10 T^{4} + \cdots - 46112)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 172 T^{8} + \cdots + 4734976 \) Copy content Toggle raw display
$89$ \( (T^{5} + 14 T^{4} + \cdots - 14752)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 151638590464 \) Copy content Toggle raw display
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