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Properties

Label 770.2.i.l
Level $770$
Weight $2$
Character orbit 770.i
Analytic conductor $6.148$
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] = [770,2,Mod(221,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.i (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: \(6.14848095564\)
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: \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 50x^{4} - 12x^{3} + 11x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 8x^{6} + 3x^{5} + 50x^{4} - 12x^{3} + 11x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 84\nu^{7} - 31\nu^{6} + 636\nu^{5} + 456\nu^{4} + 4685\nu^{3} + 156\nu^{2} + 36\nu - 3755 ) / 1381 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 371\nu^{7} - 252\nu^{6} + 2809\nu^{5} + 2014\nu^{4} + 19196\nu^{3} + 689\nu^{2} + 159\nu + 793 ) / 4143 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 634\nu^{7} - 1056\nu^{6} + 5984\nu^{5} - 1885\nu^{4} + 34144\nu^{3} - 26048\nu^{2} + 36375\nu - 2464 ) / 4143 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 793\nu^{7} - 1164\nu^{6} + 6596\nu^{5} - 430\nu^{4} + 37636\nu^{3} - 28712\nu^{2} + 8034\nu - 2716 ) / 4143 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2716\nu^{7} - 1923\nu^{6} + 20564\nu^{5} + 14744\nu^{4} + 135370\nu^{3} + 5044\nu^{2} + 1164\nu + 9323 ) / 4143 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2801\nu^{7} - 4404\nu^{6} + 23575\nu^{5} - 3734\nu^{4} + 131348\nu^{3} - 111394\nu^{2} + 27834\nu + 4915 ) / 4143 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 4\beta_{5} + \beta_{3} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 8\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} + 30\beta_{5} + \beta_{4} - 8\beta_{2} - 13\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{7} + 8\beta_{6} + 43\beta_{5} + 8\beta_{4} - 66\beta_{3} - 66\beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{6} - 140\beta_{3} + 66\beta_{2} + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 140\beta_{7} - 481\beta_{5} - 66\beta_{4} + 140\beta_{2} + 568\beta _1 - 140 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−1.15777 + 2.00531i
−0.132681 + 0.229810i
0.267083 − 0.462601i
1.52336 − 2.63854i
−1.15777 − 2.00531i
−0.132681 − 0.229810i
0.267083 + 0.462601i
1.52336 + 2.63854i
−0.500000 − 0.866025i −1.15777 + 2.00531i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.31553 −0.873699 − 2.49733i 1.00000 −1.18085 − 2.04528i 0.500000 − 0.866025i
221.2 −0.500000 − 0.866025i −0.132681 + 0.229810i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.265362 −1.51690 + 2.16772i 1.00000 1.46479 + 2.53709i 0.500000 − 0.866025i
221.3 −0.500000 − 0.866025i 0.267083 − 0.462601i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.534166 1.70312 − 2.02469i 1.00000 1.35733 + 2.35097i 0.500000 − 0.866025i
221.4 −0.500000 − 0.866025i 1.52336 − 2.63854i −0.500000 + 0.866025i 0.500000 + 0.866025i −3.04673 2.18747 + 1.48827i 1.00000 −3.14128 − 5.44086i 0.500000 − 0.866025i
331.1 −0.500000 + 0.866025i −1.15777 − 2.00531i −0.500000 − 0.866025i 0.500000 − 0.866025i 2.31553 −0.873699 + 2.49733i 1.00000 −1.18085 + 2.04528i 0.500000 + 0.866025i
331.2 −0.500000 + 0.866025i −0.132681 − 0.229810i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.265362 −1.51690 − 2.16772i 1.00000 1.46479 − 2.53709i 0.500000 + 0.866025i
331.3 −0.500000 + 0.866025i 0.267083 + 0.462601i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.534166 1.70312 + 2.02469i 1.00000 1.35733 − 2.35097i 0.500000 + 0.866025i
331.4 −0.500000 + 0.866025i 1.52336 + 2.63854i −0.500000 − 0.866025i 0.500000 − 0.866025i −3.04673 2.18747 − 1.48827i 1.00000 −3.14128 + 5.44086i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.i.l ✓ 8
7.c even 3 1 inner 770.2.i.l ✓ 8
7.c even 3 1 5390.2.a.ce 4
7.d odd 6 1 5390.2.a.cf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.l ✓ 8 1.a even 1 1 trivial
770.2.i.l ✓ 8 7.c even 3 1 inner
5390.2.a.ce 4 7.c even 3 1
5390.2.a.cf 4 7.d odd 6 1

Hecke kernels

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

\( T_{3}^{8} - T_{3}^{7} + 8T_{3}^{6} + 3T_{3}^{5} + 50T_{3}^{4} - 12T_{3}^{3} + 11T_{3}^{2} + 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + T_{13}^{3} - 42T_{13}^{2} - 56T_{13} + 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + 8 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} - 42 T^{2} + \cdots + 40)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{7} + \cdots + 678976 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + \cdots + 147456 \) Copy content Toggle raw display
$29$ \( (T^{4} - 9 T^{3} + 19 T^{2} + \cdots - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + \cdots + 1175056 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 28 T^{2} + \cdots - 96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 26 T^{3} + \cdots + 320)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 1016064 \) Copy content Toggle raw display
$53$ \( T^{8} + 14 T^{7} + \cdots + 4235364 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} + \cdots + 921600 \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} + \cdots + 192721 \) Copy content Toggle raw display
$67$ \( T^{8} + 30 T^{7} + \cdots + 2621161 \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + \cdots - 4458)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 17 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
$79$ \( T^{8} - 15 T^{7} + \cdots + 40000 \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 6 T^{7} + \cdots + 146458404 \) Copy content Toggle raw display
$97$ \( (T^{4} + 7 T^{3} + \cdots + 1832)^{2} \) Copy content Toggle raw display
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