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Properties

Label 1210.2.b.k
Level $1210$
Weight $2$
Character orbit 1210.b
Analytic conductor $9.662$
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] = [1210,2,Mod(969,1210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1210, 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("1210.969");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1210.b (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: \(9.66189864457\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12904960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 59x^{4} - 86x^{3} + 72x^{2} + 132x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{4} - \beta_{3}) q^{3} - q^{4} + (\beta_{5} - \beta_1 + 1) q^{5} + ( - \beta_{5} - 1) q^{6} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{7} + \beta_{3} q^{8} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{4} - \beta_{3}) q^{3} - q^{4} + (\beta_{5} - \beta_1 + 1) q^{5} + ( - \beta_{5} - 1) q^{6} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{7} + \beta_{3} q^{8} + ( - \beta_{5} + 1) q^{9} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{10} + ( - \beta_{4} + \beta_{3}) q^{12} + (\beta_{6} + 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{7} - \beta_{5} - \beta_{2}) q^{14} + (\beta_{7} + \beta_{4} - \beta_{3} + 1) q^{15} + q^{16} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{17}+ \cdots + ( - 2 \beta_{6} - 4 \beta_{4} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 2 q^{5} - 4 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{5} - 4 q^{6} + 12 q^{9} + 2 q^{10} + 4 q^{14} + 6 q^{15} + 8 q^{16} + 12 q^{19} - 2 q^{20} - 8 q^{21} + 4 q^{24} + 12 q^{26} + 28 q^{29} - 6 q^{30} - 32 q^{31} - 4 q^{34} + 16 q^{35} - 12 q^{36} - 4 q^{39} - 2 q^{40} - 20 q^{41} - 2 q^{45} - 48 q^{49} - 28 q^{50} - 12 q^{51} - 8 q^{54} - 4 q^{56} - 20 q^{59} - 6 q^{60} - 20 q^{61} - 8 q^{64} + 36 q^{65} - 20 q^{69} + 8 q^{70} + 52 q^{71} + 32 q^{74} - 4 q^{75} - 12 q^{76} - 12 q^{79} + 2 q^{80} - 8 q^{81} + 8 q^{84} + 28 q^{85} - 20 q^{86} - 4 q^{89} - 2 q^{90} - 40 q^{91} + 32 q^{94} + 20 q^{95} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 59x^{4} - 86x^{3} + 72x^{2} + 132x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8456 \nu^{7} - 65554 \nu^{6} + 94660 \nu^{5} - 123504 \nu^{4} + 395647 \nu^{3} - 3352355 \nu^{2} + \cdots - 3658567 ) / 2173325 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1428\nu^{7} - 4902\nu^{6} + 5705\nu^{5} + 5873\nu^{4} + 74011\nu^{3} - 285465\nu^{2} + 211001\nu + 92554 ) / 310475 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11937 \nu^{7} + 16713 \nu^{6} - 44950 \nu^{5} - 50007 \nu^{4} - 646984 \nu^{3} + 239950 \nu^{2} + \cdots - 793771 ) / 2173325 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 226\nu^{7} - 254\nu^{6} - 370\nu^{5} + 1981\nu^{4} + 12322\nu^{3} - 9340\nu^{2} - 17688\nu + 11438 ) / 39515 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 186\nu^{7} - 259\nu^{6} + 245\nu^{5} + 931\nu^{4} + 14787\nu^{3} - 9835\nu^{2} + 8722\nu + 15708 ) / 28225 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17157 \nu^{7} + 75333 \nu^{6} - 80415 \nu^{5} - 170097 \nu^{4} - 870044 \nu^{3} + \cdots - 5475096 ) / 2173325 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{4} - 5\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 6\beta_{6} - 3\beta_{5} + 3\beta_{4} - 3\beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} - 7\beta_{5} - 10\beta_{2} - 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -47\beta_{5} - 30\beta_{4} + 37\beta_{3} - 17\beta_{2} - 43\beta _1 - 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 90\beta_{7} - 7\beta_{6} - 90\beta_{5} - 8\beta_{4} + 343\beta_{3} - 90\beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 202\beta_{7} - 335\beta_{6} + 263\beta_{5} - 263\beta_{4} + 403\beta_{3} + 403 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1091\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
969.1
2.16751 − 2.16751i
−0.549472 + 0.549472i
1.24163 − 1.24163i
−1.85966 + 1.85966i
1.24163 + 1.24163i
−1.85966 − 1.85966i
2.16751 + 2.16751i
−0.549472 − 0.549472i
− 1.00000i − 1.61803i −1.00000 −0.549472 + 2.16751i −1.61803 − 5.01420i 1.00000i 0.381966 2.16751 + 0.549472i
969.2 − 1.00000i − 1.61803i −1.00000 2.16751 − 0.549472i −1.61803 3.77813i 1.00000i 0.381966 −0.549472 − 2.16751i
969.3 − 1.00000i 0.618034i −1.00000 −1.85966 + 1.24163i 0.618034 3.53474i 1.00000i 2.61803 1.24163 + 1.85966i
969.4 − 1.00000i 0.618034i −1.00000 1.24163 − 1.85966i 0.618034 − 0.298672i 1.00000i 2.61803 −1.85966 − 1.24163i
969.5 1.00000i − 0.618034i −1.00000 −1.85966 − 1.24163i 0.618034 − 3.53474i − 1.00000i 2.61803 1.24163 − 1.85966i
969.6 1.00000i − 0.618034i −1.00000 1.24163 + 1.85966i 0.618034 0.298672i − 1.00000i 2.61803 −1.85966 + 1.24163i
969.7 1.00000i 1.61803i −1.00000 −0.549472 − 2.16751i −1.61803 5.01420i − 1.00000i 0.381966 2.16751 − 0.549472i
969.8 1.00000i 1.61803i −1.00000 2.16751 + 0.549472i −1.61803 − 3.77813i − 1.00000i 0.381966 −0.549472 + 2.16751i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 969.8
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 1210.2.b.k 8
5.b even 2 1 inner 1210.2.b.k 8
5.c odd 4 1 6050.2.a.dd 4
5.c odd 4 1 6050.2.a.di 4
11.b odd 2 1 1210.2.b.l 8
11.c even 5 2 110.2.j.b ✓ 16
33.h odd 10 2 990.2.ba.h 16
44.h odd 10 2 880.2.cd.b 16
55.d odd 2 1 1210.2.b.l 8
55.e even 4 1 6050.2.a.da 4
55.e even 4 1 6050.2.a.dl 4
55.j even 10 2 110.2.j.b ✓ 16
55.k odd 20 2 550.2.h.j 8
55.k odd 20 2 550.2.h.n 8
165.o odd 10 2 990.2.ba.h 16
220.n odd 10 2 880.2.cd.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.j.b ✓ 16 11.c even 5 2
110.2.j.b ✓ 16 55.j even 10 2
550.2.h.j 8 55.k odd 20 2
550.2.h.n 8 55.k odd 20 2
880.2.cd.b 16 44.h odd 10 2
880.2.cd.b 16 220.n odd 10 2
990.2.ba.h 16 33.h odd 10 2
990.2.ba.h 16 165.o odd 10 2
1210.2.b.k 8 1.a even 1 1 trivial
1210.2.b.k 8 5.b even 2 1 inner
1210.2.b.l 8 11.b odd 2 1
1210.2.b.l 8 55.d odd 2 1
6050.2.a.da 4 55.e even 4 1
6050.2.a.dd 4 5.c odd 4 1
6050.2.a.di 4 5.c odd 4 1
6050.2.a.dl 4 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1210, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} + 48T_{13}^{6} + 556T_{13}^{4} + 1840T_{13}^{2} + 400 \) Copy content Toggle raw display
\( T_{19}^{4} - 6T_{19}^{3} - 17T_{19}^{2} + 28T_{19} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 52 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 48 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{8} + 62 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} - 17 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 76 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$29$ \( (T^{4} - 14 T^{3} + \cdots - 380)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{3} + \cdots - 1780)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 140 T^{6} + \cdots + 633616 \) Copy content Toggle raw display
$41$ \( (T^{4} + 10 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 15 T^{2} + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 188 T^{6} + \cdots + 3168400 \) Copy content Toggle raw display
$53$ \( T^{8} + 172 T^{6} + \cdots + 144400 \) Copy content Toggle raw display
$59$ \( (T^{4} + 10 T^{3} + \cdots - 239)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + \cdots + 1100)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 294 T^{6} + \cdots + 5285401 \) Copy content Toggle raw display
$71$ \( (T^{4} - 26 T^{3} + \cdots - 5380)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 50 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + \cdots - 316)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 222 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$89$ \( (T^{2} + T - 31)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 506 T^{6} + \cdots + 104427961 \) Copy content Toggle raw display
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