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Properties

Label 507.4.b.i
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
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] = [507,4,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not 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: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} - 6) q^{4} + ( - \beta_{8} - \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} + \beta_{8} + \cdots - 7 \beta_1) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} - 6) q^{4} + ( - \beta_{8} - \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} + \beta_{8} + \cdots - 7 \beta_1) q^{8}+ \cdots + ( - 9 \beta_{9} + 9 \beta_{8} + \cdots - 45 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 35\nu^{3} + 210\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 35\nu^{4} + 210\nu^{2} ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 51\nu^{4} + 706\nu^{2} + 1792 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 49\nu^{5} + 700\nu^{3} + 2940\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 61\nu^{6} + 1168\nu^{4} + 7140\nu^{2} + 8064 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 64\nu^{7} + 1309\nu^{5} + 9030\nu^{3} + 15912\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} - 70\nu^{7} - 1603\nu^{5} - 12654\nu^{3} - 20304\nu ) / 576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{6} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 31\beta_{4} - 6\beta_{3} + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -35\beta_{9} - 35\beta_{8} - 35\beta_{6} + 96\beta_{2} + 595\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -70\beta_{5} + 875\beta_{4} + 306\beta_{3} - 8330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1015\beta_{9} + 1015\beta_{8} + 1111\beta_{6} - 4704\beta_{2} - 15995\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{7} + 1934\beta_{5} - 24307\beta_{4} - 11658\beta_{3} + 223930 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -28175\beta_{9} - 27599\beta_{8} - 34319\beta_{6} + 175392\beta_{2} + 436603\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 5.36472i
− 5.04537i
− 3.27897i
− 2.04224i
− 0.917374i
0.917374i
2.04224i
3.27897i
5.04537i
5.36472i
− 5.36472i 3.00000 −20.7803 2.69631i − 16.0942i − 15.2025i 68.5626i 9.00000 14.4650
337.2 − 5.04537i 3.00000 −17.4557 − 20.1174i − 15.1361i − 15.4279i 47.7076i 9.00000 −101.500
337.3 − 3.27897i 3.00000 −2.75167 17.5414i − 9.83692i 26.6999i − 17.2091i 9.00000 57.5178
337.4 − 2.04224i 3.00000 3.82924 − 12.0825i − 6.12673i 29.7373i − 24.1582i 9.00000 −24.6753
337.5 − 0.917374i 3.00000 7.15843 15.4704i − 2.75212i − 20.5833i − 13.9059i 9.00000 14.1922
337.6 0.917374i 3.00000 7.15843 − 15.4704i 2.75212i 20.5833i 13.9059i 9.00000 14.1922
337.7 2.04224i 3.00000 3.82924 12.0825i 6.12673i − 29.7373i 24.1582i 9.00000 −24.6753
337.8 3.27897i 3.00000 −2.75167 − 17.5414i 9.83692i − 26.6999i 17.2091i 9.00000 57.5178
337.9 5.04537i 3.00000 −17.4557 20.1174i 15.1361i 15.4279i − 47.7076i 9.00000 −101.500
337.10 5.36472i 3.00000 −20.7803 − 2.69631i 16.0942i 15.2025i − 68.5626i 9.00000 14.4650
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.b.i 10
13.b even 2 1 inner 507.4.b.i 10
13.c even 3 1 39.4.j.c ✓ 10
13.d odd 4 2 507.4.a.r 10
13.e even 6 1 39.4.j.c ✓ 10
39.f even 4 2 1521.4.a.bk 10
39.h odd 6 1 117.4.q.e 10
39.i odd 6 1 117.4.q.e 10
52.i odd 6 1 624.4.bv.h 10
52.j odd 6 1 624.4.bv.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c ✓ 10 13.c even 3 1
39.4.j.c ✓ 10 13.e even 6 1
117.4.q.e 10 39.h odd 6 1
117.4.q.e 10 39.i odd 6 1
507.4.a.r 10 13.d odd 4 2
507.4.b.i 10 1.a even 1 1 trivial
507.4.b.i 10 13.b even 2 1 inner
624.4.bv.h 10 52.i odd 6 1
624.4.bv.h 10 52.j odd 6 1
1521.4.a.bk 10 39.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{10} + 70T_{2}^{8} + 1645T_{2}^{6} + 14700T_{2}^{4} + 44100T_{2}^{2} + 27648 \) Copy content Toggle raw display
\( T_{5}^{10} + 1105T_{5}^{8} + 441955T_{5}^{6} + 76029795T_{5}^{4} + 4880780280T_{5}^{2} + 31632011568 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 70 T^{8} + \cdots + 27648 \) Copy content Toggle raw display
$3$ \( (T - 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 31632011568 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 14692478786352 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( (T^{5} + 105 T^{4} + \cdots - 18224352)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T^{5} - 60 T^{4} + \cdots + 8153671248)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 495 T^{4} + \cdots + 427627836)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 48\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 36\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{5} - 370 T^{4} + \cdots + 227329236796)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 21\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( (T^{5} - 165 T^{4} + \cdots - 46733997168)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 41\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{5} - 1375 T^{4} + \cdots - 933851008945)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 20\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( (T^{5} - 550 T^{4} + \cdots - 920208867136)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 28\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
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