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Properties

Label 1638.2.dt.b
Level $1638$
Weight $2$
Character orbit 1638.dt
Analytic conductor $13.079$
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] = [1638,2,Mod(1297,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.1297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.dt (of order \(6\), degree \(2\), 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: \(13.0794958511\)
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: \( x^{20} + 56 x^{18} + 1306 x^{16} + 16508 x^{14} + 123139 x^{12} + 552164 x^{10} + 1447090 x^{8} + \cdots + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{10}) q^{2} - q^{4} + \beta_{4} q^{5} + ( - \beta_{19} + \beta_{6}) q^{7} + (\beta_{11} - \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{10}) q^{2} - q^{4} + \beta_{4} q^{5} + ( - \beta_{19} + \beta_{6}) q^{7} + (\beta_{11} - \beta_{10}) q^{8} + (\beta_{5} + \beta_1) q^{10} + ( - \beta_{18} - \beta_{17} + \beta_{14} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{19} + \beta_{18} + \beta_{17} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{4} + 4 q^{10} - 6 q^{11} + 8 q^{13} - 4 q^{14} + 20 q^{16} + 8 q^{17} - 12 q^{19} - 10 q^{22} + 16 q^{23} + 6 q^{25} - 8 q^{26} - 8 q^{29} + 12 q^{31} - 10 q^{35} - 6 q^{38} - 4 q^{40} + 18 q^{41} + 18 q^{43} + 6 q^{44} + 6 q^{47} - 20 q^{49} - 12 q^{50} - 8 q^{52} - 18 q^{53} - 12 q^{55} + 4 q^{56} + 24 q^{58} - 6 q^{61} - 20 q^{64} + 6 q^{65} + 24 q^{67} - 8 q^{68} + 42 q^{70} + 6 q^{71} + 24 q^{73} - 36 q^{74} + 12 q^{76} + 34 q^{77} + 18 q^{82} + 36 q^{86} + 10 q^{88} - 10 q^{91} - 16 q^{92} - 16 q^{94} + 80 q^{95} - 96 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 56 x^{18} + 1306 x^{16} + 16508 x^{14} + 123139 x^{12} + 552164 x^{10} + 1447090 x^{8} + \cdots + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3022 \nu^{18} + 132131 \nu^{16} + 2192815 \nu^{14} + 17182550 \nu^{12} + 65265673 \nu^{10} + \cdots + 8506128 ) / 137229096 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16974619 \nu^{19} + 3676068 \nu^{18} - 1283960340 \nu^{17} + 460750880 \nu^{16} + \cdots + 635189599008 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 68783 \nu^{18} + 3326870 \nu^{16} + 64093833 \nu^{14} + 629361751 \nu^{12} + 3334559671 \nu^{10} + \cdots + 49665168 ) / 1220236560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 68783 \nu^{18} - 3326870 \nu^{16} - 64093833 \nu^{14} - 629361751 \nu^{12} - 3334559671 \nu^{10} + \cdots - 49665168 ) / 1220236560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14856982 \nu^{19} - 4850310 \nu^{18} + 790524490 \nu^{17} - 212070255 \nu^{16} + \cdots - 13652335440 ) / 440505398160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29415555 \nu^{19} - 35398716 \nu^{18} - 1659317510 \nu^{17} - 1937602550 \nu^{16} + \cdots - 2399382291456 ) / 881010796320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29415555 \nu^{19} + 35398716 \nu^{18} - 1659317510 \nu^{17} + 1937602550 \nu^{16} + \cdots + 2399382291456 ) / 881010796320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 81018389 \nu^{19} - 34968720 \nu^{18} - 4920211380 \nu^{17} - 1891987600 \nu^{16} + \cdots + 4195815050880 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 81018389 \nu^{19} - 34968720 \nu^{18} + 4920211380 \nu^{17} - 1891987600 \nu^{16} + \cdots + 4195815050880 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 94807885 \nu^{19} - 59427928 \nu^{18} - 5289840320 \nu^{17} - 3162097960 \nu^{16} + \cdots - 1582858934208 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 94807885 \nu^{19} - 59427928 \nu^{18} + 5289840320 \nu^{17} - 3162097960 \nu^{16} + \cdots - 1582858934208 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 344897 \nu^{19} - 19039100 \nu^{17} - 437128002 \nu^{15} - 5437184344 \nu^{13} + \cdots - 2440473120 ) / 4880946240 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 123568043 \nu^{19} - 7693436 \nu^{18} + 6859712900 \nu^{17} - 252356260 \nu^{16} + \cdots + 1824337931904 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 131492007 \nu^{19} + 51179864 \nu^{18} + 7688673660 \nu^{17} + 2757223100 \nu^{16} + \cdots + 3119514604704 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 172641151 \nu^{19} - 3676068 \nu^{18} + 9295720300 \nu^{17} - 460750880 \nu^{16} + \cdots - 635189599008 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 201918419 \nu^{19} + 170259720 \nu^{18} - 11398391100 \nu^{17} + 8870380840 \nu^{16} + \cdots + 3867753991680 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 70734202 \nu^{19} + 51307110 \nu^{18} + 4079650620 \nu^{17} + 2690592110 \nu^{16} + \cdots - 82015264800 ) / 440505398160 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 349867935 \nu^{19} + 18978576 \nu^{18} + 19838226880 \nu^{17} + 1367677040 \nu^{16} + \cdots + 2806319104416 ) / 1762021592640 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 349867935 \nu^{19} - 18978576 \nu^{18} + 19838226880 \nu^{17} - 1367677040 \nu^{16} + \cdots - 2806319104416 ) / 1762021592640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{18} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{16} + 2 \beta_{15} + 8 \beta_{12} + \beta_{9} + 2 \beta_{5} - 10 \beta_{4} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{19} + 16 \beta_{18} - \beta_{17} - \beta_{16} - 11 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6 \beta_{19} - 8 \beta_{18} - 11 \beta_{17} + 11 \beta_{16} - 38 \beta_{15} + 2 \beta_{14} + \cdots - 68 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51 \beta_{19} - 235 \beta_{18} + 27 \beta_{17} + 27 \beta_{16} + 126 \beta_{15} + 184 \beta_{14} + \cdots - 559 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 142 \beta_{19} + 194 \beta_{18} + 114 \beta_{17} - 114 \beta_{16} + 596 \beta_{15} - 52 \beta_{14} + \cdots + 984 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 726 \beta_{19} + 3361 \beta_{18} - 534 \beta_{17} - 534 \beta_{16} - 1501 \beta_{15} - 2635 \beta_{14} + \cdots + 7091 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2478 \beta_{19} - 3528 \beta_{18} - 1267 \beta_{17} + 1267 \beta_{16} - 8894 \beta_{15} + \cdots - 13786 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10101 \beta_{19} - 47776 \beta_{18} + 9241 \beta_{17} + 9241 \beta_{16} + 18425 \beta_{15} + \cdots - 94044 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 38802 \beta_{19} + 57794 \beta_{18} + 15473 \beta_{17} - 15473 \beta_{16} + 130466 \beta_{15} + \cdots + 192188 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 141291 \beta_{19} + 679267 \beta_{18} - 149061 \beta_{17} - 149061 \beta_{16} - 232152 \beta_{15} + \cdots + 1279681 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 578836 \beta_{19} - 900656 \beta_{18} - 204786 \beta_{17} + 204786 \beta_{16} - 1899968 \beta_{15} + \cdots - 2686110 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1992900 \beta_{19} - 9676477 \beta_{18} + 2308476 \beta_{17} + 2308476 \beta_{16} + 2994721 \beta_{15} + \cdots - 17694713 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 8433696 \beta_{19} + 13655280 \beta_{18} + 2859385 \beta_{17} - 2859385 \beta_{16} + 27567590 \beta_{15} + \cdots + 37711000 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 28294599 \beta_{19} + 138146536 \beta_{18} - 34870729 \beta_{17} - 34870729 \beta_{16} + \cdots + 247305396 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 121462470 \beta_{19} - 203639060 \beta_{18} - 41125079 \beta_{17} + 41125079 \beta_{16} + \cdots - 531868544 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 403541787 \beta_{19} - 1976053159 \beta_{18} + 518481447 \beta_{17} + 518481447 \beta_{16} + \cdots - 3482371615 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1739867998 \beta_{19} + 3004912406 \beta_{18} + 599680470 \beta_{17} - 599680470 \beta_{16} + \cdots + 7532117496 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\beta_{12}\) \(\beta_{12}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1297.1
− 3.31964i
− 0.508531i
1.05091i
1.77962i
2.99764i
− 3.79415i
− 2.62249i
− 0.0521119i
1.91536i
2.55339i
3.79415i
2.62249i
0.0521119i
− 1.91536i
− 2.55339i
3.31964i
0.508531i
− 1.05091i
− 1.77962i
− 2.99764i
− 1.00000i 0 −1.00000 −2.87489 − 1.65982i 0 −0.744278 − 2.53891i 1.00000i 0 −1.65982 + 2.87489i
1297.2 − 1.00000i 0 −1.00000 −0.440400 − 0.254265i 0 −0.212907 − 2.63717i 1.00000i 0 −0.254265 + 0.440400i
1297.3 − 1.00000i 0 −1.00000 0.910115 + 0.525455i 0 0.963812 + 2.46395i 1.00000i 0 0.525455 − 0.910115i
1297.4 − 1.00000i 0 −1.00000 1.54119 + 0.889808i 0 −2.51386 + 0.824914i 1.00000i 0 0.889808 − 1.54119i
1297.5 − 1.00000i 0 −1.00000 2.59603 + 1.49882i 0 2.50724 − 0.844840i 1.00000i 0 1.49882 − 2.59603i
1297.6 1.00000i 0 −1.00000 −3.28583 − 1.89707i 0 2.13806 + 1.55842i − 1.00000i 0 1.89707 − 3.28583i
1297.7 1.00000i 0 −1.00000 −2.27114 − 1.31124i 0 −0.116897 − 2.64317i − 1.00000i 0 1.31124 − 2.27114i
1297.8 1.00000i 0 −1.00000 −0.0451302 − 0.0260560i 0 −1.11788 + 2.39799i − 1.00000i 0 0.0260560 − 0.0451302i
1297.9 1.00000i 0 −1.00000 1.65875 + 0.957680i 0 −2.64520 − 0.0538842i − 1.00000i 0 −0.957680 + 1.65875i
1297.10 1.00000i 0 −1.00000 2.21130 + 1.27669i 0 1.74192 − 1.99141i − 1.00000i 0 −1.27669 + 2.21130i
1369.1 − 1.00000i 0 −1.00000 −3.28583 + 1.89707i 0 2.13806 − 1.55842i 1.00000i 0 1.89707 + 3.28583i
1369.2 − 1.00000i 0 −1.00000 −2.27114 + 1.31124i 0 −0.116897 + 2.64317i 1.00000i 0 1.31124 + 2.27114i
1369.3 − 1.00000i 0 −1.00000 −0.0451302 + 0.0260560i 0 −1.11788 − 2.39799i 1.00000i 0 0.0260560 + 0.0451302i
1369.4 − 1.00000i 0 −1.00000 1.65875 − 0.957680i 0 −2.64520 + 0.0538842i 1.00000i 0 −0.957680 − 1.65875i
1369.5 − 1.00000i 0 −1.00000 2.21130 − 1.27669i 0 1.74192 + 1.99141i 1.00000i 0 −1.27669 − 2.21130i
1369.6 1.00000i 0 −1.00000 −2.87489 + 1.65982i 0 −0.744278 + 2.53891i − 1.00000i 0 −1.65982 − 2.87489i
1369.7 1.00000i 0 −1.00000 −0.440400 + 0.254265i 0 −0.212907 + 2.63717i − 1.00000i 0 −0.254265 − 0.440400i
1369.8 1.00000i 0 −1.00000 0.910115 − 0.525455i 0 0.963812 − 2.46395i − 1.00000i 0 0.525455 + 0.910115i
1369.9 1.00000i 0 −1.00000 1.54119 − 0.889808i 0 −2.51386 − 0.824914i − 1.00000i 0 0.889808 + 1.54119i
1369.10 1.00000i 0 −1.00000 2.59603 − 1.49882i 0 2.50724 + 0.844840i − 1.00000i 0 1.49882 + 2.59603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1297.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.dt.b 20
3.b odd 2 1 546.2.bm.b yes 20
7.c even 3 1 1638.2.cr.b 20
13.e even 6 1 1638.2.cr.b 20
21.h odd 6 1 546.2.bd.b ✓ 20
39.h odd 6 1 546.2.bd.b ✓ 20
91.k even 6 1 inner 1638.2.dt.b 20
273.bp odd 6 1 546.2.bm.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bd.b ✓ 20 21.h odd 6 1
546.2.bd.b ✓ 20 39.h odd 6 1
546.2.bm.b yes 20 3.b odd 2 1
546.2.bm.b yes 20 273.bp odd 6 1
1638.2.cr.b 20 7.c even 3 1
1638.2.cr.b 20 13.e even 6 1
1638.2.dt.b 20 1.a even 1 1 trivial
1638.2.dt.b 20 91.k even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 28 T_{5}^{18} + 523 T_{5}^{16} - 318 T_{5}^{15} - 5416 T_{5}^{14} + 5544 T_{5}^{13} + \cdots + 576 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 28 T^{18} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + 6 T^{19} + \cdots + 11664 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( (T^{10} - 4 T^{9} + \cdots + 445776)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 12 T^{19} + \cdots + 82791801 \) Copy content Toggle raw display
$23$ \( (T^{10} - 8 T^{9} + \cdots - 1526664)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 4446755856 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 6038441240976 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 222697212379849 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 15776364816 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 237795695449 \) Copy content Toggle raw display
$47$ \( T^{20} - 6 T^{19} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 274366440000 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 149267013030144 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 979103150309376 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 39520987511184 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 60267010291489 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 3052036952064 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 26\!\cdots\!29 \) Copy content Toggle raw display
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