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Properties

Label 624.4.bv.g
Level $624$
Weight $4$
Character orbit 624.bv
Analytic conductor $36.817$
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] = [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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 98x^{5} + 34501x^{4} - 71304x^{3} + 78408x^{2} + 5248584x + 175668516 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 78)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta_{2} q^{3} + (\beta_{7} - \beta_{3} - 5 \beta_{2} + \cdots + 2) q^{5}+ \cdots + (9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_{2} q^{3} + (\beta_{7} - \beta_{3} - 5 \beta_{2} + \cdots + 2) q^{5}+ \cdots + (18 \beta_{7} - 9 \beta_{5} + \cdots + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 24 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 24 q^{7} - 36 q^{9} - 114 q^{11} + 110 q^{13} - 90 q^{15} - 76 q^{17} + 258 q^{19} - 182 q^{23} - 564 q^{25} + 216 q^{27} - 38 q^{29} + 342 q^{33} + 478 q^{35} - 540 q^{37} - 84 q^{39} + 144 q^{41} + 456 q^{43} + 270 q^{45} + 82 q^{49} + 456 q^{51} + 1212 q^{53} + 1434 q^{55} - 1980 q^{59} + 366 q^{61} + 216 q^{63} + 1076 q^{65} - 2004 q^{67} - 546 q^{69} - 2046 q^{71} + 846 q^{75} - 2904 q^{77} - 1924 q^{79} - 324 q^{81} + 2556 q^{85} - 114 q^{87} + 84 q^{89} + 130 q^{91} + 2394 q^{93} - 150 q^{95} + 366 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 98x^{5} + 34501x^{4} - 71304x^{3} + 78408x^{2} + 5248584x + 175668516 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 257358012279 \nu^{7} + 50969737533002 \nu^{6} - 649708769124688 \nu^{5} + \cdots - 43\!\cdots\!12 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7895 \nu^{7} - 1283102 \nu^{6} - 1354444 \nu^{5} - 84502 \nu^{4} - 45079295 \nu^{3} + \cdots + 781004382252 ) / 1586230820616 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6738505543559 \nu^{7} + 188249950580658 \nu^{6} + \cdots + 12\!\cdots\!60 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 640237538 \nu^{7} + 34920099036 \nu^{6} - 41710127654 \nu^{5} - 20367726882 \nu^{4} + \cdots - 17\!\cdots\!52 ) / 93\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23815465093839 \nu^{7} + 75900848627726 \nu^{6} + \cdots + 81\!\cdots\!16 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2200134832947 \nu^{7} - 20728365840436 \nu^{6} - 157171749987928 \nu^{5} + \cdots - 18\!\cdots\!20 ) / 10\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 109225170145549 \nu^{7} + 34836935659830 \nu^{6} + 433291347694712 \nu^{5} + \cdots + 64\!\cdots\!12 ) / 12\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{6} - 5\beta_{4} - 12\beta_{2} + 12\beta _1 + 12 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{7} + 61\beta_{4} + 6\beta_{3} + 294\beta_{2} + 6\beta _1 - 144 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 288\beta_{7} + 270\beta_{6} - 750\beta_{5} - 139\beta_{4} + 750\beta_{3} - 444\beta_{2} - 144\beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -597\beta_{7} - 3386\beta_{6} - 450\beta_{5} - 1693\beta_{4} - 147\beta_{3} - 147\beta_{2} + 147\beta _1 - 51378 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 26104\beta_{6} + 17781\beta_{5} + 35334\beta_{4} + 17781\beta_{3} + 118428\beta_{2} - 52572\beta _1 - 81747 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 100647 \beta_{7} - 54462 \beta_{5} - 677915 \beta_{4} - 46185 \beta_{3} - 6956541 \beta_{2} + \cdots + 3455178 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6910356 \beta_{7} - 4676788 \beta_{6} + 7946763 \beta_{5} + 1531263 \beta_{4} - 7946763 \beta_{3} + \cdots - 7121373 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
−5.66647 + 5.66647i
−9.30355 − 9.30355i
8.93752 + 8.93752i
7.03249 − 7.03249i
7.03249 + 7.03249i
8.93752 − 8.93752i
−9.30355 + 9.30355i
−5.66647 − 5.66647i
0 −1.50000 − 2.59808i 0 − 19.6772i 0 −3.73773 − 2.15798i 0 −4.50000 + 7.79423i 0
49.2 0 −1.50000 − 2.59808i 0 − 13.0068i 0 23.3426 + 13.4768i 0 −4.50000 + 7.79423i 0
49.3 0 −1.50000 − 2.59808i 0 0.346567i 0 −19.8163 − 11.4409i 0 −4.50000 + 7.79423i 0
49.4 0 −1.50000 − 2.59808i 0 15.0170i 0 −11.7885 − 6.80612i 0 −4.50000 + 7.79423i 0
433.1 0 −1.50000 + 2.59808i 0 − 15.0170i 0 −11.7885 + 6.80612i 0 −4.50000 − 7.79423i 0
433.2 0 −1.50000 + 2.59808i 0 − 0.346567i 0 −19.8163 + 11.4409i 0 −4.50000 − 7.79423i 0
433.3 0 −1.50000 + 2.59808i 0 13.0068i 0 23.3426 − 13.4768i 0 −4.50000 − 7.79423i 0
433.4 0 −1.50000 + 2.59808i 0 19.6772i 0 −3.73773 + 2.15798i 0 −4.50000 − 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
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.g 8
4.b odd 2 1 78.4.i.b ✓ 8
12.b even 2 1 234.4.l.c 8
13.e even 6 1 inner 624.4.bv.g 8
52.i odd 6 1 78.4.i.b ✓ 8
52.i odd 6 1 1014.4.b.o 8
52.j odd 6 1 1014.4.b.o 8
52.l even 12 1 1014.4.a.z 4
52.l even 12 1 1014.4.a.ba 4
156.r even 6 1 234.4.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.i.b ✓ 8 4.b odd 2 1
78.4.i.b ✓ 8 52.i odd 6 1
234.4.l.c 8 12.b even 2 1
234.4.l.c 8 156.r even 6 1
624.4.bv.g 8 1.a even 1 1 trivial
624.4.bv.g 8 13.e even 6 1 inner
1014.4.a.z 4 52.l even 12 1
1014.4.a.ba 4 52.l even 12 1
1014.4.b.o 8 52.i odd 6 1
1014.4.b.o 8 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 782T_{5}^{6} + 191065T_{5}^{4} + 14794776T_{5}^{2} + 1774224 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 782 T^{6} + \cdots + 1774224 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1312902756 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 1461685824 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 218270130465024 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 50391484900416 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 95\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} - 606 T^{3} + \cdots - 6144065136)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 82\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 33\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 37\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( (T^{4} + 962 T^{3} + \cdots + 368623918104)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
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