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Properties

Label 210.2.n.b
Level $210$
Weight $2$
Character orbit 210.n
Analytic conductor $1.677$
Analytic rank $0$
Dimension $12$
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] = [210,2,Mod(79,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.n (of order \(6\), 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: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.7652750400000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{8} + \beta_{2}) q^{3} + (\beta_{10} + 1) q^{4} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 1) q^{5}+ \cdots - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{8} + \beta_{2}) q^{3} + (\beta_{10} + 1) q^{4} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 1) q^{5}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{9} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} + 12 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} + 12 q^{6} + 6 q^{9} - 6 q^{11} + 6 q^{14} - 6 q^{16} + 6 q^{19} + 6 q^{21} + 6 q^{24} - 6 q^{26} - 48 q^{29} - 24 q^{34} - 30 q^{35} + 12 q^{36} + 6 q^{39} - 36 q^{41} + 6 q^{44} - 18 q^{46} + 24 q^{49} - 12 q^{51} + 6 q^{54} + 60 q^{55} - 24 q^{59} - 12 q^{61} - 12 q^{64} - 30 q^{65} - 6 q^{66} - 36 q^{69} - 30 q^{70} + 72 q^{71} + 18 q^{74} + 12 q^{76} + 48 q^{79} - 6 q^{81} + 6 q^{84} + 12 q^{86} - 12 q^{89} - 6 q^{94} - 6 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 21 x^{8} - 108 x^{7} + 368 x^{6} - 216 x^{5} + 84 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10967 \nu^{11} + 32246 \nu^{10} - 320538 \nu^{9} + 1200782 \nu^{8} + 26475 \nu^{7} + \cdots + 11668096 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11057 \nu^{11} + 124978 \nu^{10} - 554210 \nu^{9} + 1179630 \nu^{8} - 841061 \nu^{7} + \cdots + 15302400 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25665 \nu^{11} + 172184 \nu^{10} - 569522 \nu^{9} + 611754 \nu^{8} - 480425 \nu^{7} + \cdots + 4417824 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 34905 \nu^{11} + 255324 \nu^{10} - 866962 \nu^{9} + 1148906 \nu^{8} - 1016073 \nu^{7} + \cdots + 6873568 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43053 \nu^{11} - 242212 \nu^{10} + 665338 \nu^{9} - 227874 \nu^{8} + 407037 \nu^{7} + \cdots - 2193056 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 65619 \nu^{11} + 412146 \nu^{10} - 1257142 \nu^{9} + 1078106 \nu^{8} - 1258503 \nu^{7} + \cdots + 17709568 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 65883 \nu^{11} + 372468 \nu^{10} - 1055638 \nu^{9} + 532062 \nu^{8} - 1079355 \nu^{7} + \cdots + 6409568 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 112417 \nu^{11} - 805130 \nu^{10} + 2628966 \nu^{9} - 2969054 \nu^{8} + 1600117 \nu^{7} + \cdots - 33288832 ) / 6321728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 227 \nu^{11} + 1290 \nu^{10} - 3622 \nu^{9} + 1778 \nu^{8} - 3655 \nu^{7} + 23804 \nu^{6} + \cdots + 16768 ) / 8768 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 213873 \nu^{11} + 1049078 \nu^{10} - 2530334 \nu^{9} - 669562 \nu^{8} - 2913901 \nu^{7} + \cdots - 13456256 ) / 6321728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{10} - 4\beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{10} + 2\beta_{9} - 9\beta_{8} - 6\beta_{6} - 2\beta_{5} - \beta_{4} + 3\beta_{3} + 3\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{11} + 4 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} - 14 \beta_{6} - 11 \beta_{5} - 11 \beta_{4} + \cdots - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{11} - 62 \beta_{10} + 21 \beta_{8} + 40 \beta_{7} - 20 \beta_{6} - 35 \beta_{4} + 15 \beta_{3} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 70 \beta_{11} - 334 \beta_{10} - 70 \beta_{9} + 338 \beta_{8} + 127 \beta_{7} + 46 \beta_{6} + 127 \beta_{5} + \cdots - 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 820 \beta_{10} - 394 \beta_{9} + 1305 \beta_{8} + 590 \beta_{6} + 566 \beta_{5} + 197 \beta_{4} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 960 \beta_{11} - 960 \beta_{9} + 2716 \beta_{8} - 1521 \beta_{7} + 2098 \beta_{6} + 1521 \beta_{5} + \cdots + 1741 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4962 \beta_{11} + 10290 \beta_{10} - 2183 \beta_{8} - 7248 \beta_{7} + 3624 \beta_{6} + 4653 \beta_{4} + \cdots + 6482 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12210 \beta_{11} + 51306 \beta_{10} + 12210 \beta_{9} - 44234 \beta_{8} - 18405 \beta_{7} + \cdots + 13443 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 126436 \beta_{10} + 61230 \beta_{9} - 189895 \beta_{8} - 80174 \beta_{6} - 89626 \beta_{5} + \cdots - 13900 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(\beta_{10}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.593239 + 0.593239i
2.45845 + 2.45845i
−0.685661 − 0.685661i
0.406761 − 0.406761i
−1.45845 + 1.45845i
1.68566 − 1.68566i
0.593239 − 0.593239i
2.45845 − 2.45845i
−0.685661 + 0.685661i
0.406761 + 0.406761i
−1.45845 − 1.45845i
1.68566 + 1.68566i
−0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.40280 + 1.74131i 1.00000 −1.45550 − 2.20942i − 1.00000i 0.500000 − 0.866025i 2.08551 − 0.806615i
79.2 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.806615 − 2.08551i 1.00000 2.24325 + 1.40280i − 1.00000i 0.500000 − 0.866025i −0.344208 + 2.20942i
79.3 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.20942 + 0.344208i 1.00000 −2.51980 + 0.806615i − 1.00000i 0.500000 − 0.866025i −1.74131 − 1.40280i
79.4 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −1.40280 − 1.74131i 1.00000 2.51980 − 0.806615i 1.00000i 0.500000 − 0.866025i −0.344208 − 2.20942i
79.5 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i −0.806615 + 2.08551i 1.00000 1.45550 + 2.20942i 1.00000i 0.500000 − 0.866025i −1.74131 + 1.40280i
79.6 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 2.20942 − 0.344208i 1.00000 −2.24325 − 1.40280i 1.00000i 0.500000 − 0.866025i 2.08551 + 0.806615i
109.1 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −1.40280 − 1.74131i 1.00000 −1.45550 + 2.20942i 1.00000i 0.500000 + 0.866025i 2.08551 + 0.806615i
109.2 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i −0.806615 + 2.08551i 1.00000 2.24325 − 1.40280i 1.00000i 0.500000 + 0.866025i −0.344208 − 2.20942i
109.3 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 2.20942 − 0.344208i 1.00000 −2.51980 − 0.806615i 1.00000i 0.500000 + 0.866025i −1.74131 + 1.40280i
109.4 0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i −1.40280 + 1.74131i 1.00000 2.51980 + 0.806615i − 1.00000i 0.500000 + 0.866025i −0.344208 + 2.20942i
109.5 0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i −0.806615 − 2.08551i 1.00000 1.45550 − 2.20942i − 1.00000i 0.500000 + 0.866025i −1.74131 − 1.40280i
109.6 0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i 2.20942 + 0.344208i 1.00000 −2.24325 + 1.40280i − 1.00000i 0.500000 + 0.866025i 2.08551 − 0.806615i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.n.b ✓ 12
3.b odd 2 1 630.2.u.f 12
4.b odd 2 1 1680.2.di.c 12
5.b even 2 1 inner 210.2.n.b ✓ 12
5.c odd 4 1 1050.2.i.u 6
5.c odd 4 1 1050.2.i.v 6
7.b odd 2 1 1470.2.n.j 12
7.c even 3 1 inner 210.2.n.b ✓ 12
7.c even 3 1 1470.2.g.i 6
7.d odd 6 1 1470.2.g.h 6
7.d odd 6 1 1470.2.n.j 12
15.d odd 2 1 630.2.u.f 12
20.d odd 2 1 1680.2.di.c 12
21.h odd 6 1 630.2.u.f 12
28.g odd 6 1 1680.2.di.c 12
35.c odd 2 1 1470.2.n.j 12
35.i odd 6 1 1470.2.g.h 6
35.i odd 6 1 1470.2.n.j 12
35.j even 6 1 inner 210.2.n.b ✓ 12
35.j even 6 1 1470.2.g.i 6
35.k even 12 1 7350.2.a.do 3
35.k even 12 1 7350.2.a.dp 3
35.l odd 12 1 1050.2.i.u 6
35.l odd 12 1 1050.2.i.v 6
35.l odd 12 1 7350.2.a.dn 3
35.l odd 12 1 7350.2.a.dq 3
105.o odd 6 1 630.2.u.f 12
140.p odd 6 1 1680.2.di.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b ✓ 12 1.a even 1 1 trivial
210.2.n.b ✓ 12 5.b even 2 1 inner
210.2.n.b ✓ 12 7.c even 3 1 inner
210.2.n.b ✓ 12 35.j even 6 1 inner
630.2.u.f 12 3.b odd 2 1
630.2.u.f 12 15.d odd 2 1
630.2.u.f 12 21.h odd 6 1
630.2.u.f 12 105.o odd 6 1
1050.2.i.u 6 5.c odd 4 1
1050.2.i.u 6 35.l odd 12 1
1050.2.i.v 6 5.c odd 4 1
1050.2.i.v 6 35.l odd 12 1
1470.2.g.h 6 7.d odd 6 1
1470.2.g.h 6 35.i odd 6 1
1470.2.g.i 6 7.c even 3 1
1470.2.g.i 6 35.j even 6 1
1470.2.n.j 12 7.b odd 2 1
1470.2.n.j 12 7.d odd 6 1
1470.2.n.j 12 35.c odd 2 1
1470.2.n.j 12 35.i odd 6 1
1680.2.di.c 12 4.b odd 2 1
1680.2.di.c 12 20.d odd 2 1
1680.2.di.c 12 28.g odd 6 1
1680.2.di.c 12 140.p odd 6 1
7350.2.a.dn 3 35.l odd 12 1
7350.2.a.do 3 35.k even 12 1
7350.2.a.dp 3 35.k even 12 1
7350.2.a.dq 3 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{6} + 3T_{11}^{5} + 36T_{11}^{4} + 17T_{11}^{3} + 876T_{11}^{2} + 1323T_{11} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{6} - 20 T^{3} + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 33 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 5473632256 \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 57 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{3} + 12 T^{2} + \cdots + 24)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 15 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 57 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} + \cdots - 128)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26639462656 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 436880018961 \) Copy content Toggle raw display
$59$ \( (T^{6} + 12 T^{5} + \cdots + 190096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + \cdots + 506944)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23612624896 \) Copy content Toggle raw display
$71$ \( (T - 6)^{12} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{2} + 256)^{3} \) Copy content Toggle raw display
$79$ \( (T^{6} - 24 T^{5} + \cdots + 161604)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 198 T^{4} + \cdots + 7396)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 6 T^{5} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 342 T^{4} + \cdots + 12544)^{2} \) Copy content Toggle raw display
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