↑

Properties

Label 624.4.bv.i
Level $624$
Weight $4$
Character orbit 624.bv
Analytic conductor $36.817$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 1323 x^{18} + 5334 x^{17} + 725427 x^{16} - 2648080 x^{15} - 215057242 x^{14} + \cdots + 15\!\cdots\!13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{36} \)
Twist minimal: no (minimal twist has level 312)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta_1 + 3) q^{3} + ( - \beta_{5} - \beta_{2}) q^{5} + (\beta_{17} + \beta_{12} - \beta_1 - 2) q^{7} + 9 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_1 + 3) q^{3} + ( - \beta_{5} - \beta_{2}) q^{5} + (\beta_{17} + \beta_{12} - \beta_1 - 2) q^{7} + 9 \beta_1 q^{9} + ( - \beta_{3} + \beta_1 - 2) q^{11} + ( - \beta_{15} - \beta_{14} - 8 \beta_1 - 3) q^{13} + ( - 3 \beta_{5} - 3) q^{15} + ( - \beta_{18} - \beta_{17} + \beta_{14} + \cdots - 1) q^{17}+ \cdots + ( - 9 \beta_{14} - 27 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 30 q^{3} - 12 q^{7} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 30 q^{3} - 12 q^{7} - 90 q^{9} - 42 q^{11} + 12 q^{13} - 36 q^{15} - 110 q^{17} - 102 q^{19} - 54 q^{23} - 332 q^{25} - 540 q^{27} + 152 q^{29} - 126 q^{33} - 42 q^{35} - 414 q^{37} + 378 q^{39} - 90 q^{41} - 308 q^{43} - 108 q^{45} + 928 q^{49} - 660 q^{51} + 904 q^{53} + 34 q^{55} - 756 q^{59} - 556 q^{61} + 108 q^{63} + 162 q^{65} + 816 q^{67} + 162 q^{69} - 702 q^{71} - 498 q^{75} - 2072 q^{77} - 2284 q^{79} - 810 q^{81} - 2538 q^{85} - 456 q^{87} - 2532 q^{89} + 2622 q^{91} + 1314 q^{93} - 3510 q^{95} - 3882 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 1323 x^{18} + 5334 x^{17} + 725427 x^{16} - 2648080 x^{15} - 215057242 x^{14} + \cdots + 15\!\cdots\!13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 67\!\cdots\!53 \nu^{19} + \cdots + 44\!\cdots\!65 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!83 \nu^{19} + \cdots + 12\!\cdots\!78 ) / 23\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 67\!\cdots\!94 \nu^{19} + \cdots - 24\!\cdots\!71 ) / 42\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\!\cdots\!77 \nu^{19} + \cdots + 25\!\cdots\!59 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!38 \nu^{19} + \cdots + 95\!\cdots\!81 ) / 78\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76\!\cdots\!00 \nu^{19} + \cdots - 24\!\cdots\!61 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\!\cdots\!34 \nu^{19} + \cdots + 86\!\cdots\!25 ) / 85\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!59 \nu^{19} + \cdots + 66\!\cdots\!21 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!43 \nu^{19} + \cdots - 66\!\cdots\!13 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29\!\cdots\!91 \nu^{19} + \cdots - 55\!\cdots\!66 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!71 \nu^{19} + \cdots - 11\!\cdots\!58 ) / 85\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!92 \nu^{19} + \cdots + 11\!\cdots\!69 ) / 85\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 49\!\cdots\!32 \nu^{19} + \cdots + 35\!\cdots\!81 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 51\!\cdots\!08 \nu^{19} + \cdots - 29\!\cdots\!97 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 58\!\cdots\!25 \nu^{19} + \cdots - 48\!\cdots\!06 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 57\!\cdots\!85 \nu^{19} + \cdots + 46\!\cdots\!81 ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 76\!\cdots\!77 \nu^{19} + \cdots + 36\!\cdots\!88 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 96\!\cdots\!29 \nu^{19} + \cdots - 51\!\cdots\!26 ) / 25\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 41\!\cdots\!60 \nu^{19} + \cdots + 30\!\cdots\!61 ) / 25\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{9} - 4\beta_{6} - \beta_{4} - 4\beta_{3} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 8 \beta_{17} + 24 \beta_{16} + 8 \beta_{15} + 8 \beta_{14} - 16 \beta_{12} - 8 \beta_{10} + \cdots + 1072 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 60 \beta_{19} + 96 \beta_{18} + 192 \beta_{17} - 192 \beta_{16} - 136 \beta_{15} - 272 \beta_{14} + \cdots - 724 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 156 \beta_{19} - 276 \beta_{18} - 1176 \beta_{17} + 3712 \beta_{16} + 956 \beta_{15} + 988 \beta_{14} + \cdots + 118008 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 15916 \beta_{19} + 40960 \beta_{18} + 65112 \beta_{17} - 95096 \beta_{16} - 37512 \beta_{15} + \cdots - 1209436 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 136572 \beta_{19} - 303792 \beta_{18} - 659600 \beta_{17} + 2218312 \beta_{16} + 475576 \beta_{15} + \cdots + 59493620 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3157112 \beta_{19} + 14853088 \beta_{18} + 18949136 \beta_{17} - 40263872 \beta_{16} + \cdots - 625951832 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11511764 \beta_{19} - 31524706 \beta_{18} - 47608412 \beta_{17} + 170094068 \beta_{16} + \cdots + 4057398944 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 457485732 \beta_{19} + 5274046464 \beta_{18} + 5460218392 \beta_{17} - 15753282776 \beta_{16} + \cdots - 258393605380 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14305823708 \beta_{19} - 47745364848 \beta_{18} - 56689315320 \beta_{17} + 215347364576 \beta_{16} + \cdots + 4671760907716 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 8269770572 \beta_{19} + 1869975707456 \beta_{18} + 1611790960944 \beta_{17} - 5882962019552 \beta_{16} + \cdots - 96870070547028 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 2161566795908 \beta_{19} - 8700285987216 \beta_{18} - 8598778053560 \beta_{17} + \cdots + 697494298843840 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 30249050597312 \beta_{19} + 662594533419008 \beta_{18} + 488277927074816 \beta_{17} + \cdots - 34\!\cdots\!72 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 12\!\cdots\!32 \beta_{19} + \cdots + 42\!\cdots\!16 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17\!\cdots\!20 \beta_{19} + \cdots - 11\!\cdots\!20 ) / 16 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 48\!\cdots\!50 \beta_{19} + \cdots + 16\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 72\!\cdots\!52 \beta_{19} + \cdots - 39\!\cdots\!76 ) / 16 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 11\!\cdots\!64 \beta_{19} + \cdots + 41\!\cdots\!20 ) / 8 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 26\!\cdots\!04 \beta_{19} + \cdots - 12\!\cdots\!72 ) / 16 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−18.7214 + 0.500000i
15.0503 − 0.500000i
12.9507 − 0.500000i
−2.50969 + 0.500000i
−2.23060 + 0.500000i
8.27097 + 0.500000i
−3.79012 − 0.500000i
−8.90958 − 0.500000i
15.3247 + 0.500000i
−13.4352 − 0.500000i
−13.4352 + 0.500000i
15.3247 − 0.500000i
−8.90958 + 0.500000i
−3.79012 + 0.500000i
8.27097 − 0.500000i
−2.23060 − 0.500000i
−2.50969 − 0.500000i
12.9507 + 0.500000i
15.0503 + 0.500000i
−18.7214 − 0.500000i
0 1.50000 + 2.59808i 0 − 20.8553i 0 0.795742 + 0.459422i 0 −4.50000 + 7.79423i 0
49.2 0 1.50000 + 2.59808i 0 − 11.1843i 0 5.37673 + 3.10426i 0 −4.50000 + 7.79423i 0
49.3 0 1.50000 + 2.59808i 0 − 9.08463i 0 −26.6581 − 15.3911i 0 −4.50000 + 7.79423i 0
49.4 0 1.50000 + 2.59808i 0 − 4.64366i 0 28.4896 + 16.4485i 0 −4.50000 + 7.79423i 0
49.5 0 1.50000 + 2.59808i 0 − 4.36458i 0 −20.5672 − 11.8745i 0 −4.50000 + 7.79423i 0
49.6 0 1.50000 + 2.59808i 0 6.13699i 0 1.57703 + 0.910496i 0 −4.50000 + 7.79423i 0
49.7 0 1.50000 + 2.59808i 0 7.65614i 0 8.61342 + 4.97296i 0 −4.50000 + 7.79423i 0
49.8 0 1.50000 + 2.59808i 0 12.7756i 0 23.8172 + 13.7509i 0 −4.50000 + 7.79423i 0
49.9 0 1.50000 + 2.59808i 0 13.1907i 0 −2.03684 − 1.17597i 0 −4.50000 + 7.79423i 0
49.10 0 1.50000 + 2.59808i 0 17.3013i 0 −25.4076 − 14.6691i 0 −4.50000 + 7.79423i 0
433.1 0 1.50000 − 2.59808i 0 − 17.3013i 0 −25.4076 + 14.6691i 0 −4.50000 − 7.79423i 0
433.2 0 1.50000 − 2.59808i 0 − 13.1907i 0 −2.03684 + 1.17597i 0 −4.50000 − 7.79423i 0
433.3 0 1.50000 − 2.59808i 0 − 12.7756i 0 23.8172 − 13.7509i 0 −4.50000 − 7.79423i 0
433.4 0 1.50000 − 2.59808i 0 − 7.65614i 0 8.61342 − 4.97296i 0 −4.50000 − 7.79423i 0
433.5 0 1.50000 − 2.59808i 0 − 6.13699i 0 1.57703 − 0.910496i 0 −4.50000 − 7.79423i 0
433.6 0 1.50000 − 2.59808i 0 4.36458i 0 −20.5672 + 11.8745i 0 −4.50000 − 7.79423i 0
433.7 0 1.50000 − 2.59808i 0 4.64366i 0 28.4896 − 16.4485i 0 −4.50000 − 7.79423i 0
433.8 0 1.50000 − 2.59808i 0 9.08463i 0 −26.6581 + 15.3911i 0 −4.50000 − 7.79423i 0
433.9 0 1.50000 − 2.59808i 0 11.1843i 0 5.37673 − 3.10426i 0 −4.50000 − 7.79423i 0
433.10 0 1.50000 − 2.59808i 0 20.8553i 0 0.795742 − 0.459422i 0 −4.50000 − 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
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.i 20
4.b odd 2 1 312.4.bf.a ✓ 20
13.e even 6 1 inner 624.4.bv.i 20
52.i odd 6 1 312.4.bf.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.bf.a ✓ 20 4.b odd 2 1
312.4.bf.a ✓ 20 52.i odd 6 1
624.4.bv.i 20 1.a even 1 1 trivial
624.4.bv.i 20 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 1416 T_{5}^{18} + 820612 T_{5}^{16} + 256103136 T_{5}^{14} + 47596465086 T_{5}^{12} + \cdots + 34\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 65\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 39\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 65\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
show more
show less