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Properties

Label 624.4.bv.h
Level $624$
Weight $4$
Character orbit 624.bv
Analytic conductor $36.817$
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] = [624,4,Mod(49,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
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: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{2} + 3) q^{3} + ( - \beta_{7} - \beta_{6} - 2 \beta_{2} + 1) q^{5} + ( - \beta_{9} - \beta_{8} + 2 \beta_{2} - 4) q^{7} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{2} + 3) q^{3} + ( - \beta_{7} - \beta_{6} - 2 \beta_{2} + 1) q^{5} + ( - \beta_{9} - \beta_{8} + 2 \beta_{2} - 4) q^{7} - 9 \beta_{2} q^{9} + (\beta_{8} + \beta_{6} + \beta_{5} + \cdots - 4) q^{11}+ \cdots + (9 \beta_{9} - 9 \beta_{7} - 9 \beta_{6} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 15 q^{3} - 30 q^{7} - 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 15 q^{3} - 30 q^{7} - 45 q^{9} - 60 q^{11} + 25 q^{13} - 45 q^{15} + 105 q^{17} - 180 q^{19} + 60 q^{23} - 960 q^{25} - 270 q^{27} - 495 q^{29} - 180 q^{33} - 60 q^{35} - 405 q^{37} - 345 q^{39} + 1065 q^{41} + 370 q^{43} - 135 q^{45} + 775 q^{49} + 630 q^{51} + 330 q^{53} + 260 q^{55} - 780 q^{59} - 1375 q^{61} + 270 q^{63} + 1605 q^{65} - 1590 q^{67} - 180 q^{69} - 1620 q^{71} - 1440 q^{75} - 4320 q^{77} - 1100 q^{79} - 405 q^{81} + 525 q^{85} + 1485 q^{87} + 2040 q^{89} - 4770 q^{91} - 990 q^{93} + 1380 q^{95} - 3750 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 35\nu^{3} + 210\nu + 96 ) / 192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 35\nu^{4} + 210\nu^{2} + 96\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{2} + 28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 49\nu^{5} + 700\nu^{3} + 96\nu^{2} + 2940\nu + 1344 ) / 96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 64\nu^{7} + 24\nu^{6} + 1309\nu^{5} + 1128\nu^{4} + 9030\nu^{3} + 13392\nu^{2} + 15336\nu + 24192 ) / 1152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 64\nu^{7} - 24\nu^{6} + 1309\nu^{5} - 1128\nu^{4} + 9030\nu^{3} - 13392\nu^{2} + 15336\nu - 24192 ) / 1152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 6 \nu^{8} + 70 \nu^{7} + 366 \nu^{6} + 1603 \nu^{5} + 7008 \nu^{4} + 12654 \nu^{3} + \cdots + 48384 ) / 1152 \) 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 \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{9} + 2\beta_{7} + 2\beta_{6} + 2\beta_{5} - \beta_{4} - 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} + 4\beta_{6} - 29\beta_{4} - 8\beta_{3} + 4\beta _1 + 644 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -70\beta_{9} - 70\beta_{7} - 70\beta_{6} - 70\beta_{5} + 35\beta_{4} + 384\beta_{2} + 280\beta _1 - 192 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 140\beta_{7} - 140\beta_{6} + 805\beta_{4} + 376\beta_{3} - 188\beta _1 - 16660 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2030 \beta_{9} + 2030 \beta_{7} + 2030 \beta_{6} + 2222 \beta_{5} - 1111 \beta_{4} - 18816 \beta_{2} + \cdots + 9408 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 192 \beta_{9} + 384 \beta_{8} - 3868 \beta_{7} + 3868 \beta_{6} - 22373 \beta_{4} - 13592 \beta_{3} + \cdots + 447860 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 56350 \beta_{9} - 55198 \beta_{7} - 55198 \beta_{6} - 68638 \beta_{5} + 34319 \beta_{4} + \cdots - 350784 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
3.27897i
− 2.04224i
− 5.36472i
− 0.917374i
5.04537i
− 5.04537i
0.917374i
5.36472i
2.04224i
− 3.27897i
0 1.50000 + 2.59808i 0 − 17.5414i 0 −23.1228 − 13.3499i 0 −4.50000 + 7.79423i 0
49.2 0 1.50000 + 2.59808i 0 − 12.0825i 0 25.7533 + 14.8686i 0 −4.50000 + 7.79423i 0
49.3 0 1.50000 + 2.59808i 0 2.69631i 0 −13.1657 − 7.60123i 0 −4.50000 + 7.79423i 0
49.4 0 1.50000 + 2.59808i 0 15.4704i 0 −17.8257 − 10.2917i 0 −4.50000 + 7.79423i 0
49.5 0 1.50000 + 2.59808i 0 20.1174i 0 13.3609 + 7.71395i 0 −4.50000 + 7.79423i 0
433.1 0 1.50000 − 2.59808i 0 − 20.1174i 0 13.3609 − 7.71395i 0 −4.50000 − 7.79423i 0
433.2 0 1.50000 − 2.59808i 0 − 15.4704i 0 −17.8257 + 10.2917i 0 −4.50000 − 7.79423i 0
433.3 0 1.50000 − 2.59808i 0 − 2.69631i 0 −13.1657 + 7.60123i 0 −4.50000 − 7.79423i 0
433.4 0 1.50000 − 2.59808i 0 12.0825i 0 25.7533 − 14.8686i 0 −4.50000 − 7.79423i 0
433.5 0 1.50000 − 2.59808i 0 17.5414i 0 −23.1228 + 13.3499i 0 −4.50000 − 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bv.h 10
4.b odd 2 1 39.4.j.c ✓ 10
12.b even 2 1 117.4.q.e 10
13.e even 6 1 inner 624.4.bv.h 10
52.i odd 6 1 39.4.j.c ✓ 10
52.i odd 6 1 507.4.b.i 10
52.j odd 6 1 507.4.b.i 10
52.l even 12 2 507.4.a.r 10
156.r even 6 1 117.4.q.e 10
156.v odd 12 2 1521.4.a.bk 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c ✓ 10 4.b odd 2 1
39.4.j.c ✓ 10 52.i odd 6 1
117.4.q.e 10 12.b even 2 1
117.4.q.e 10 156.r even 6 1
507.4.a.r 10 52.l even 12 2
507.4.b.i 10 52.i odd 6 1
507.4.b.i 10 52.j odd 6 1
624.4.bv.h 10 1.a even 1 1 trivial
624.4.bv.h 10 13.e even 6 1 inner
1521.4.a.bk 10 156.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 1105T_{5}^{8} + 441955T_{5}^{6} + 76029795T_{5}^{4} + 4880780280T_{5}^{2} + 31632011568 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{5} \) 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
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 332127005819904 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 66\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 18\!\cdots\!96 \) 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
$43$ \( T^{10} + \cdots + 51\!\cdots\!16 \) 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
$61$ \( T^{10} + \cdots + 87\!\cdots\!25 \) 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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