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Properties

Label 3648.2.k.l
Level $3648$
Weight $2$
Character orbit 3648.k
Analytic conductor $29.129$
Analytic rank $0$
Dimension $20$
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] = [3648,2,Mod(2431,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3648.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.k (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: \(29.1294266574\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 52 x^{18} + 986 x^{16} + 8824 x^{14} + 42757 x^{12} + 118964 x^{10} + 190576 x^{8} + 166880 x^{6} + \cdots + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{27} \)
Twist minimal: no (minimal twist has level 1824)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{5} q^{5} - \beta_{17} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{5} q^{5} - \beta_{17} q^{7} + q^{9} - \beta_{6} q^{11} - \beta_{12} q^{13} - \beta_{5} q^{15} + \beta_{9} q^{17} - \beta_{13} q^{19} - \beta_{17} q^{21} + \beta_{18} q^{23} + (\beta_{4} + 1) q^{25} + q^{27} + ( - \beta_{16} + \beta_{6}) q^{29} - \beta_{7} q^{31} - \beta_{6} q^{33} + (\beta_{18} - \beta_{15} + \cdots + \beta_{2}) q^{35}+ \cdots - \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{3} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{3} + 20 q^{9} + 4 q^{19} + 20 q^{25} + 20 q^{27} + 8 q^{31} - 44 q^{49} + 4 q^{57} + 8 q^{61} + 8 q^{67} - 64 q^{71} + 8 q^{73} + 20 q^{75} - 40 q^{77} + 16 q^{79} + 20 q^{81} - 40 q^{85} + 8 q^{93} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 52 x^{18} + 986 x^{16} + 8824 x^{14} + 42757 x^{12} + 118964 x^{10} + 190576 x^{8} + 166880 x^{6} + \cdots + 784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6129489 \nu^{18} - 503270162 \nu^{16} - 15292280756 \nu^{14} - 218329932812 \nu^{12} + \cdots - 431065515824 ) / 13184980080 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 324 \nu^{19} - 63673 \nu^{17} - 3678159 \nu^{15} - 67090708 \nu^{13} - 530209678 \nu^{11} + \cdots - 161227716 \nu ) / 2377508 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 44135573 \nu^{18} - 2105321874 \nu^{16} - 34485914292 \nu^{14} - 242796585004 \nu^{12} + \cdots - 85089320368 ) / 13184980080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 58112993 \nu^{18} - 2843282924 \nu^{16} - 48448470662 \nu^{14} - 358362966234 \nu^{12} + \cdots - 16819268808 ) / 3296245020 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 288998187 \nu^{18} - 14408246486 \nu^{16} - 254218219028 \nu^{14} - 2012914564196 \nu^{12} + \cdots - 193279155632 ) / 13184980080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13086829 \nu^{19} - 736973237 \nu^{17} - 15700936466 \nu^{15} - 164231677537 \nu^{13} + \cdots - 88201295524 \nu ) / 6592490040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 277198363 \nu^{18} + 13683725974 \nu^{16} + 237181223792 \nu^{14} + 1817350153464 \nu^{12} + \cdots + 4070955568 ) / 4394993360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 245795037 \nu^{19} - 17100852196 \nu^{17} - 454572431098 \nu^{15} + \cdots - 9452957030152 \nu ) / 92294860560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71365541 \nu^{18} - 3527867058 \nu^{16} - 61281933829 \nu^{14} - 470917124223 \nu^{12} + \cdots + 7980644984 ) / 1098748340 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1230756631 \nu^{18} - 61726355998 \nu^{16} - 1099881873184 \nu^{14} - 8846727713208 \nu^{12} + \cdots - 1015854488016 ) / 13184980080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 757475159 \nu^{18} - 37817077832 \nu^{16} - 668960567636 \nu^{14} - 5323249597992 \nu^{12} + \cdots - 740748756024 ) / 6592490040 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33725473 \nu^{19} + 1819048722 \nu^{17} + 36387985482 \nu^{15} + 349165808690 \nu^{13} + \cdots + 89024887496 \nu ) / 4614743028 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1455402533 \nu^{19} + 6689311006 \nu^{18} - 66900502534 \nu^{17} + 333183335128 \nu^{16} + \cdots + 2876008082976 ) / 184589721120 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1455402533 \nu^{19} + 6689311006 \nu^{18} + 66900502534 \nu^{17} + 333183335128 \nu^{16} + \cdots + 2876008082976 ) / 184589721120 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 455362351 \nu^{19} - 23920086668 \nu^{17} - 460847198354 \nu^{15} + \cdots - 4361057837176 \nu ) / 46147430280 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1156223519 \nu^{19} + 56802670262 \nu^{17} + 975068564726 \nu^{15} + 7312477178382 \nu^{13} + \cdots - 554341185456 \nu ) / 92294860560 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 437834077 \nu^{19} - 21798595706 \nu^{17} - 383671062794 \nu^{15} - 3024366583582 \nu^{13} + \cdots - 326173032928 \nu ) / 18458972112 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2978241002 \nu^{19} - 148923210876 \nu^{17} - 2642012848893 \nu^{15} + \cdots - 3745743785932 \nu ) / 46147430280 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1314740558 \nu^{19} + 65813389174 \nu^{17} + 1169449773067 \nu^{15} + 9375841863464 \nu^{13} + \cdots + 1754301590628 \nu ) / 15382476760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{16} - \beta_{15} + \beta_{12} + \beta_{8} + \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} - \beta_{13} - \beta_{11} - \beta_{9} - \beta_{7} + 2\beta_{5} + \beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{19} - 7 \beta_{18} + 18 \beta_{17} + 14 \beta_{16} + 12 \beta_{15} + 2 \beta_{14} + \cdots - 4 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{14} + 8 \beta_{13} + 10 \beta_{11} + \beta_{10} + 9 \beta_{9} + 10 \beta_{7} - 27 \beta_{5} + \cdots + 78 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27 \beta_{19} + 95 \beta_{18} - 185 \beta_{17} - 130 \beta_{16} - 108 \beta_{15} - 14 \beta_{14} + \cdots - 11 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 268 \beta_{14} - 268 \beta_{13} - 374 \beta_{11} - 62 \beta_{10} - 366 \beta_{9} - 424 \beta_{7} + \cdots - 2830 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1583 \beta_{19} - 4457 \beta_{18} + 7006 \beta_{17} + 5022 \beta_{16} + 4364 \beta_{15} + \cdots + 880 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2253 \beta_{14} + 2253 \beta_{13} + 3529 \beta_{11} + 754 \beta_{10} + 3849 \beta_{9} + 4589 \beta_{7} + \cdots + 27309 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 19963 \beta_{19} + 50215 \beta_{18} - 66853 \beta_{17} - 49405 \beta_{16} - 45239 \beta_{15} + \cdots - 12829 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 76540 \beta_{14} - 76540 \beta_{13} - 135670 \beta_{11} - 34268 \beta_{10} - 162878 \beta_{9} + \cdots - 1079854 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 941713 \beta_{19} - 2215847 \beta_{18} + 2596314 \beta_{17} + 1972418 \beta_{16} + \cdots + 655104 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 661238 \beta_{14} + 661238 \beta_{13} + 1327334 \beta_{11} + 378699 \beta_{10} + 1723114 \beta_{9} + \cdots + 10836300 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10696833 \beta_{19} + 24114525 \beta_{18} - 25625451 \beta_{17} - 19923029 \beta_{16} + \cdots - 7800831 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 23328118 \beta_{14} - 23328118 \beta_{13} - 52753076 \beta_{11} - 16489630 \beta_{10} + \cdots - 439888608 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 474786607 \beta_{19} - 1039981705 \beta_{18} + 1025968926 \beta_{17} + 812772462 \beta_{16} + \cdots + 356741624 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 210411071 \beta_{14} + 210411071 \beta_{13} + 530938539 \beta_{11} + 177782184 \beta_{10} + \cdots + 4503505983 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 5187753871 \beta_{19} + 11139468011 \beta_{18} - 10387973657 \beta_{17} - 8352955207 \beta_{16} + \cdots - 3975701445 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 7765711492 \beta_{14} - 7765711492 \beta_{13} - 21598662662 \beta_{11} - 7616570800 \beta_{10} + \cdots - 185685681486 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 224356709201 \beta_{19} - 475042246663 \beta_{18} + 424602754666 \beta_{17} + \cdots + 174272835232 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(1921\) \(2053\) \(2623\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
1.70397i
− 1.70397i
1.59595i
− 1.59595i
2.15136i
− 2.15136i
0.478981i
− 0.478981i
− 3.94408i
3.94408i
− 0.350153i
0.350153i
1.52988i
− 1.52988i
1.45345i
− 1.45345i
− 4.58884i
4.58884i
0.709059i
− 0.709059i
0 1.00000 0 −3.84826 0 − 1.43848i 0 1.00000 0
2431.2 0 1.00000 0 −3.84826 0 1.43848i 0 1.00000 0
2431.3 0 1.00000 0 −2.90882 0 − 5.16584i 0 1.00000 0
2431.4 0 1.00000 0 −2.90882 0 5.16584i 0 1.00000 0
2431.5 0 1.00000 0 −1.91454 0 − 1.12794i 0 1.00000 0
2431.6 0 1.00000 0 −1.91454 0 1.12794i 0 1.00000 0
2431.7 0 1.00000 0 −1.19929 0 − 1.87667i 0 1.00000 0
2431.8 0 1.00000 0 −1.19929 0 1.87667i 0 1.00000 0
2431.9 0 1.00000 0 −0.835245 0 − 4.74253i 0 1.00000 0
2431.10 0 1.00000 0 −0.835245 0 4.74253i 0 1.00000 0
2431.11 0 1.00000 0 0.430587 0 − 0.0646038i 0 1.00000 0
2431.12 0 1.00000 0 0.430587 0 0.0646038i 0 1.00000 0
2431.13 0 1.00000 0 1.49782 0 − 0.665758i 0 1.00000 0
2431.14 0 1.00000 0 1.49782 0 0.665758i 0 1.00000 0
2431.15 0 1.00000 0 1.69641 0 − 3.75190i 0 1.00000 0
2431.16 0 1.00000 0 1.69641 0 3.75190i 0 1.00000 0
2431.17 0 1.00000 0 3.01679 0 − 3.47281i 0 1.00000 0
2431.18 0 1.00000 0 3.01679 0 3.47281i 0 1.00000 0
2431.19 0 1.00000 0 4.06455 0 − 3.06179i 0 1.00000 0
2431.20 0 1.00000 0 4.06455 0 3.06179i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.2.k.l 20
4.b odd 2 1 3648.2.k.k 20
8.b even 2 1 1824.2.k.a ✓ 20
8.d odd 2 1 1824.2.k.b yes 20
19.b odd 2 1 3648.2.k.k 20
24.f even 2 1 5472.2.k.b 20
24.h odd 2 1 5472.2.k.a 20
76.d even 2 1 inner 3648.2.k.l 20
152.b even 2 1 1824.2.k.a ✓ 20
152.g odd 2 1 1824.2.k.b yes 20
456.l odd 2 1 5472.2.k.a 20
456.p even 2 1 5472.2.k.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.k.a ✓ 20 8.b even 2 1
1824.2.k.a ✓ 20 152.b even 2 1
1824.2.k.b yes 20 8.d odd 2 1
1824.2.k.b yes 20 152.g odd 2 1
3648.2.k.k 20 4.b odd 2 1
3648.2.k.k 20 19.b odd 2 1
3648.2.k.l 20 1.a even 1 1 trivial
3648.2.k.l 20 76.d even 2 1 inner
5472.2.k.a 20 24.h odd 2 1
5472.2.k.a 20 456.l odd 2 1
5472.2.k.b 20 24.f even 2 1
5472.2.k.b 20 456.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3648, [\chi])\):

\( T_{5}^{10} - 30T_{5}^{8} - 4T_{5}^{7} + 281T_{5}^{6} + 60T_{5}^{5} - 956T_{5}^{4} - 224T_{5}^{3} + 1144T_{5}^{2} + 288T_{5} - 288 \) Copy content Toggle raw display
\( T_{31}^{10} - 4 T_{31}^{9} - 148 T_{31}^{8} + 416 T_{31}^{7} + 7080 T_{31}^{6} - 15072 T_{31}^{5} + \cdots + 139264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T - 1)^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 30 T^{8} + \cdots - 288)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + 92 T^{18} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{20} + 128 T^{18} + \cdots + 50176 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 2316304384 \) Copy content Toggle raw display
$17$ \( (T^{10} - 110 T^{8} + \cdots - 298432)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 4581380653056 \) Copy content Toggle raw display
$29$ \( T^{20} + 260 T^{18} + \cdots + 4194304 \) Copy content Toggle raw display
$31$ \( (T^{10} - 4 T^{9} + \cdots + 139264)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1241245548544 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 86973087744 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 6553600000000 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 1485551444224 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 86973087744 \) Copy content Toggle raw display
$59$ \( (T^{10} - 272 T^{8} + \cdots - 1605632)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 4 T^{9} + \cdots + 4267008)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 4 T^{9} + \cdots - 751321088)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 32 T^{9} + \cdots - 245235712)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 4 T^{9} + \cdots - 14381056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 8 T^{9} + \cdots - 174845952)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 527849522200576 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 2847831752704 \) Copy content Toggle raw display
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