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Properties

Label 624.4.q.k
Level $624$
Weight $4$
Character orbit 624.q
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(289,624)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 60x^{6} - 4x^{5} + 2787x^{4} - 120x^{3} + 48784x^{2} + 1626x + 660969 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \beta_1 - 3) q^{3} + (\beta_{5} + \beta_{4} + 1) q^{5} + ( - \beta_{3} - 4 \beta_1) q^{7} - 9 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_1 - 3) q^{3} + (\beta_{5} + \beta_{4} + 1) q^{5} + ( - \beta_{3} - 4 \beta_1) q^{7} - 9 \beta_1 q^{9} + (\beta_{7} + \beta_{2} - 3 \beta_1 + 3) q^{11} + (\beta_{7} - \beta_{5} - \beta_{4} + \cdots - 10) q^{13}+ \cdots + ( - 18 \beta_{7} - 9 \beta_{5} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 6 q^{5} - 15 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} + 6 q^{5} - 15 q^{7} - 36 q^{9} + 12 q^{11} - 72 q^{13} - 9 q^{15} - 15 q^{17} + 46 q^{19} + 90 q^{21} - 156 q^{23} + 342 q^{25} + 216 q^{27} + 225 q^{29} + 582 q^{31} + 36 q^{33} - 36 q^{35} + 239 q^{37} + 135 q^{39} - 615 q^{41} + 153 q^{43} - 27 q^{45} - 960 q^{47} + 111 q^{49} + 90 q^{51} + 594 q^{53} - 146 q^{55} - 276 q^{57} - 726 q^{59} - 894 q^{61} - 135 q^{63} - 1497 q^{65} + 1425 q^{67} - 468 q^{69} - 708 q^{71} + 2452 q^{73} - 513 q^{75} + 540 q^{77} + 794 q^{79} - 324 q^{81} + 324 q^{83} + 63 q^{85} + 675 q^{87} + 2106 q^{89} + 2585 q^{91} - 873 q^{93} + 1038 q^{95} + 2761 q^{97} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 60x^{6} - 4x^{5} + 2787x^{4} - 120x^{3} + 48784x^{2} + 1626x + 660969 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16000 \nu^{7} + 5035180 \nu^{6} - 743200 \nu^{5} + 233948111 \nu^{4} - 54662360 \nu^{3} + \cdots + 245616080400 ) / 190173798243 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5642746 \nu^{7} - 390103145 \nu^{6} + 1850936651 \nu^{5} - 23380325608 \nu^{4} + \cdots - 9710715299193 ) / 380347596486 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22287563 \nu^{7} - 144062245 \nu^{6} + 21263800 \nu^{5} - 11885146535 \nu^{4} + \cdots - 7027356311100 ) / 380347596486 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7780121 \nu^{7} + 62417533 \nu^{6} - 9212920 \nu^{5} + 5395630795 \nu^{4} + \cdots + 3044727259740 ) / 126782532162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1125727 \nu^{7} - 11257340 \nu^{6} - 86371345 \nu^{5} - 518400535 \nu^{4} + \cdots - 125842978143 ) / 12269277306 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 56958257 \nu^{7} + 254376860 \nu^{6} - 3702232139 \nu^{5} + 12043638175 \nu^{4} + \cdots + 2709639520407 ) / 380347596486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19186 \nu^{7} - 4065 \nu^{6} + 1338963 \nu^{5} + 853870 \nu^{4} + 40046313 \nu^{3} + \cdots + 1131596001 ) / 80599194 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + 3\beta_{3} - 2\beta_{2} + 2\beta _1 - 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 3\beta_{6} + 7\beta_{5} + \beta_{2} + 359\beta _1 - 359 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{7} - 14\beta_{6} - 9\beta_{5} + 5\beta_{4} - 14\beta_{3} + 7\beta_{2} - 7\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 31\beta_{7} + 31\beta_{5} + 180\beta_{4} - 87\beta_{3} - 62\beta_{2} - 5860\beta _1 - 62 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1709\beta_{7} + 2595\beta_{6} - 79\beta_{5} + 1709\beta_{2} + 1171\beta _1 - 1171 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1994 \beta_{7} + 2611 \beta_{6} - 7548 \beta_{5} - 5554 \beta_{4} + 2611 \beta_{3} + 997 \beta_{2} + \cdots + 140341 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34259\beta_{7} + 34259\beta_{5} - 41085\beta_{4} + 43530\beta_{3} - 68518\beta_{2} - 15092\beta _1 - 68518 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
3.10787 − 5.38298i
−3.16158 + 5.47601i
−2.24736 + 3.89255i
2.30108 − 3.98558i
3.10787 + 5.38298i
−3.16158 − 5.47601i
−2.24736 − 3.89255i
2.30108 + 3.98558i
0 −1.50000 − 2.59808i 0 −13.3405 0 −9.68077 + 16.7676i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 − 2.59808i 0 −5.89537 0 −2.21146 + 3.83036i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 − 2.59808i 0 0.845261 0 12.3695 − 21.4246i 0 −4.50000 + 7.79423i 0
289.4 0 −1.50000 − 2.59808i 0 21.3907 0 −7.97729 + 13.8171i 0 −4.50000 + 7.79423i 0
529.1 0 −1.50000 + 2.59808i 0 −13.3405 0 −9.68077 − 16.7676i 0 −4.50000 − 7.79423i 0
529.2 0 −1.50000 + 2.59808i 0 −5.89537 0 −2.21146 − 3.83036i 0 −4.50000 − 7.79423i 0
529.3 0 −1.50000 + 2.59808i 0 0.845261 0 12.3695 + 21.4246i 0 −4.50000 − 7.79423i 0
529.4 0 −1.50000 + 2.59808i 0 21.3907 0 −7.97729 − 13.8171i 0 −4.50000 − 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.q.k 8
4.b odd 2 1 156.4.i.b ✓ 8
12.b even 2 1 468.4.l.e 8
13.c even 3 1 inner 624.4.q.k 8
52.i odd 6 1 2028.4.a.j 4
52.j odd 6 1 156.4.i.b ✓ 8
52.j odd 6 1 2028.4.a.l 4
52.l even 12 2 2028.4.b.i 8
156.p even 6 1 468.4.l.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.i.b ✓ 8 4.b odd 2 1
156.4.i.b ✓ 8 52.j odd 6 1
468.4.l.e 8 12.b even 2 1
468.4.l.e 8 156.p even 6 1
624.4.q.k 8 1.a even 1 1 trivial
624.4.q.k 8 13.c even 3 1 inner
2028.4.a.j 4 52.i odd 6 1
2028.4.a.l 4 52.j odd 6 1
2028.4.b.i 8 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{4} - 3T_{5}^{3} - 331T_{5}^{2} - 1401T_{5} + 1422 \) Copy content Toggle raw display
\( T_{7}^{8} + 15 T_{7}^{7} + 743 T_{7}^{6} + 12510 T_{7}^{5} + 454224 T_{7}^{4} + 6266520 T_{7}^{3} + \cdots + 1142440000 \) 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
$5$ \( (T^{4} - 3 T^{3} + \cdots + 1422)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1142440000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 10482483456 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 47829690000 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 86\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 422740821878244 \) Copy content Toggle raw display
$31$ \( (T^{4} - 291 T^{3} + \cdots - 263956160)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{4} + 480 T^{3} + \cdots - 20740410000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 297 T^{3} + \cdots - 3881015208)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 16\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} - 1226 T^{3} + \cdots - 50237794661)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 397 T^{3} + \cdots + 925243009216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 162 T^{3} + \cdots + 1814661792)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
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