↑

Properties

Label 38.10.c.b
Level $38$
Weight $10$
Character orbit 38.c
Analytic conductor $19.571$
Analytic rank $0$
Dimension $16$
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] = [38,10,Mod(7,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.7");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), 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: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 104960 x^{14} - 3899480 x^{13} + 8040649724 x^{12} - 270026959304 x^{11} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5}\cdot 19^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (16 \beta_{2} - 16) q^{2} + ( - 9 \beta_{2} - \beta_1 + 9) q^{3} - 256 \beta_{2} q^{4} + ( - \beta_{5} + 43 \beta_{2} - 43) q^{5} + (16 \beta_{3} + 144 \beta_{2} + 16 \beta_1) q^{6} + ( - \beta_{7} - \beta_{4} - 5 \beta_{3} + 230) q^{7} + 4096 q^{8} + (\beta_{9} + \beta_{8} + \cdots - 45 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (16 \beta_{2} - 16) q^{2} + ( - 9 \beta_{2} - \beta_1 + 9) q^{3} - 256 \beta_{2} q^{4} + ( - \beta_{5} + 43 \beta_{2} - 43) q^{5} + (16 \beta_{3} + 144 \beta_{2} + 16 \beta_1) q^{6} + ( - \beta_{7} - \beta_{4} - 5 \beta_{3} + 230) q^{7} + 4096 q^{8} + (\beta_{9} + \beta_{8} + \cdots - 45 \beta_1) q^{9}+ \cdots + (3055 \beta_{15} + \cdots + 1769662 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 128 q^{2} + 70 q^{3} - 2048 q^{4} - 341 q^{5} + 1120 q^{6} + 3704 q^{7} + 65536 q^{8} - 53064 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 128 q^{2} + 70 q^{3} - 2048 q^{4} - 341 q^{5} + 1120 q^{6} + 3704 q^{7} + 65536 q^{8} - 53064 q^{9} - 5456 q^{10} + 106682 q^{11} - 35840 q^{12} - 683 q^{13} - 29632 q^{14} + 2299 q^{15} - 524288 q^{16} - 611267 q^{17} + 1698048 q^{18} - 1609677 q^{19} + 174592 q^{20} - 941048 q^{21} - 853456 q^{22} - 416119 q^{23} + 286720 q^{24} - 3309027 q^{25} + 21856 q^{26} - 16903604 q^{27} - 474112 q^{28} - 5079157 q^{29} - 73568 q^{30} + 8618324 q^{31} - 8388608 q^{32} - 7592195 q^{33} - 9780272 q^{34} - 2035632 q^{35} - 13584384 q^{36} - 33091520 q^{37} + 18612624 q^{38} + 37931126 q^{39} - 1396736 q^{40} - 7394646 q^{41} - 15056768 q^{42} - 98675599 q^{43} - 13655296 q^{44} + 89231852 q^{45} + 13315808 q^{46} + 34129475 q^{47} + 4587520 q^{48} + 65004400 q^{49} + 105888864 q^{50} + 16027749 q^{51} - 174848 q^{52} - 110053995 q^{53} + 135228832 q^{54} + 135992300 q^{55} + 15171584 q^{56} + 14986101 q^{57} + 162533024 q^{58} - 2017760 q^{59} + 588544 q^{60} - 221861413 q^{61} - 68946592 q^{62} + 56812456 q^{63} + 268435456 q^{64} + 262635974 q^{65} - 121475120 q^{66} + 237580440 q^{67} + 312968704 q^{68} - 135348410 q^{69} - 32570112 q^{70} - 431190909 q^{71} - 217350144 q^{72} + 199873544 q^{73} + 264732160 q^{74} - 1822944486 q^{75} + 114275328 q^{76} + 2014342624 q^{77} - 303449008 q^{78} - 296762835 q^{79} - 22347776 q^{80} - 1069077456 q^{81} - 118314336 q^{82} + 1592138306 q^{83} + 481816576 q^{84} - 293558115 q^{85} - 1578809584 q^{86} - 1647860854 q^{87} + 436969472 q^{88} - 444394631 q^{89} - 713854816 q^{90} - 284099544 q^{91} - 106526464 q^{92} + 1322110374 q^{93} - 1092143200 q^{94} + 8705459 q^{95} - 146800640 q^{96} - 611719542 q^{97} - 520035200 q^{98} + 1201848626 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 104960 x^{14} - 3899480 x^{13} + 8040649724 x^{12} - 270026959304 x^{11} + \cdots + 13\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 92\!\cdots\!97 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\!\cdots\!77 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51\!\cdots\!21 \nu^{15} + \cdots - 21\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 55\!\cdots\!53 \nu^{15} + \cdots + 72\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!31 \nu^{15} + \cdots + 44\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!61 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!41 \nu^{15} + \cdots - 23\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 57\!\cdots\!16 \nu^{15} + \cdots + 53\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 91\!\cdots\!13 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 94\!\cdots\!57 \nu^{15} + \cdots + 50\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!89 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!67 \nu^{15} + \cdots - 89\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!45 \nu^{15} + \cdots + 59\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65\!\cdots\!77 \nu^{15} + \cdots - 12\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + 2\beta_{6} - 27\beta_{3} - 26247\beta_{2} - 27\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 20 \beta_{15} + 12 \beta_{14} - 22 \beta_{13} + 20 \beta_{12} - 68 \beta_{11} - 34 \beta_{10} + \cdots + 703754 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 778 \beta_{14} + 2676 \beta_{13} - 1230 \beta_{12} + 2676 \beta_{11} + 5352 \beta_{10} + \cdots - 1263399644 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1309040 \beta_{15} - 2907028 \beta_{14} - 919192 \beta_{13} + 2907028 \beta_{11} + \cdots - 2725068052 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 143536500 \beta_{15} + 330697892 \beta_{14} + 5797748 \beta_{13} + 143536500 \beta_{12} + \cdots + 71321822224984 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 139606515776 \beta_{14} + 211189868296 \beta_{13} - 81567747560 \beta_{12} + 211189868296 \beta_{11} + \cdots - 76\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12601294630920 \beta_{15} - 29424601708560 \beta_{14} - 25368495885000 \beta_{13} + \cdots - 23\!\cdots\!92 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 52\!\cdots\!60 \beta_{15} + \cdots + 62\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 56\!\cdots\!16 \beta_{14} + \cdots - 27\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 35\!\cdots\!40 \beta_{15} + \cdots - 70\!\cdots\!88 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 74\!\cdots\!20 \beta_{15} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 39\!\cdots\!36 \beta_{14} + \cdots - 36\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 54\!\cdots\!00 \beta_{15} + \cdots - 10\!\cdots\!80 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 16\!\cdots\!00 \beta_{15} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
113.220 + 196.103i
95.0097 + 164.562i
48.3490 + 83.7429i
26.7983 + 46.4161i
−24.6030 − 42.6137i
−34.9668 − 60.5642i
−89.8616 − 155.645i
−132.946 − 230.268i
113.220 − 196.103i
95.0097 − 164.562i
48.3490 − 83.7429i
26.7983 − 46.4161i
−24.6030 + 42.6137i
−34.9668 + 60.5642i
−89.8616 + 155.645i
−132.946 + 230.268i
−8.00000 − 13.8564i −108.720 − 188.309i −128.000 + 221.703i −480.308 − 831.918i −1739.52 + 3012.94i −4308.78 4096.00 −13798.6 + 23899.8i −7684.93 + 13310.7i
7.2 −8.00000 − 13.8564i −90.5097 − 156.767i −128.000 + 221.703i −153.299 − 265.521i −1448.15 + 2508.28i 12144.4 4096.00 −6542.50 + 11331.9i −2452.78 + 4248.34i
7.3 −8.00000 − 13.8564i −43.8490 − 75.9487i −128.000 + 221.703i 993.746 + 1721.22i −701.584 + 1215.18i −8244.38 4096.00 5996.03 − 10385.4i 15899.9 − 27539.5i
7.4 −8.00000 − 13.8564i −22.2983 − 38.6218i −128.000 + 221.703i 397.433 + 688.374i −356.773 + 617.949i 2271.77 4096.00 8847.07 − 15323.6i 6358.93 − 11014.0i
7.5 −8.00000 − 13.8564i 29.1030 + 50.4079i −128.000 + 221.703i −1358.85 − 2353.60i 465.648 − 806.526i 3587.38 4096.00 8147.53 − 14111.9i −21741.6 + 37657.5i
7.6 −8.00000 − 13.8564i 39.4668 + 68.3585i −128.000 + 221.703i −134.126 − 232.313i 631.468 − 1093.74i −5487.23 4096.00 6726.25 − 11650.2i −2146.01 + 3717.01i
7.7 −8.00000 − 13.8564i 94.3616 + 163.439i −128.000 + 221.703i 1091.71 + 1890.90i 1509.79 − 2615.03i 6918.71 4096.00 −7966.73 + 13798.8i 17467.4 − 30254.4i
7.8 −8.00000 − 13.8564i 137.446 + 238.063i −128.000 + 221.703i −526.809 − 912.460i 2199.13 − 3809.00i −5029.91 4096.00 −27941.1 + 48395.4i −8428.95 + 14599.4i
11.1 −8.00000 + 13.8564i −108.720 + 188.309i −128.000 − 221.703i −480.308 + 831.918i −1739.52 − 3012.94i −4308.78 4096.00 −13798.6 − 23899.8i −7684.93 − 13310.7i
11.2 −8.00000 + 13.8564i −90.5097 + 156.767i −128.000 − 221.703i −153.299 + 265.521i −1448.15 − 2508.28i 12144.4 4096.00 −6542.50 − 11331.9i −2452.78 − 4248.34i
11.3 −8.00000 + 13.8564i −43.8490 + 75.9487i −128.000 − 221.703i 993.746 − 1721.22i −701.584 − 1215.18i −8244.38 4096.00 5996.03 + 10385.4i 15899.9 + 27539.5i
11.4 −8.00000 + 13.8564i −22.2983 + 38.6218i −128.000 − 221.703i 397.433 − 688.374i −356.773 − 617.949i 2271.77 4096.00 8847.07 + 15323.6i 6358.93 + 11014.0i
11.5 −8.00000 + 13.8564i 29.1030 − 50.4079i −128.000 − 221.703i −1358.85 + 2353.60i 465.648 + 806.526i 3587.38 4096.00 8147.53 + 14111.9i −21741.6 − 37657.5i
11.6 −8.00000 + 13.8564i 39.4668 − 68.3585i −128.000 − 221.703i −134.126 + 232.313i 631.468 + 1093.74i −5487.23 4096.00 6726.25 + 11650.2i −2146.01 − 3717.01i
11.7 −8.00000 + 13.8564i 94.3616 − 163.439i −128.000 − 221.703i 1091.71 − 1890.90i 1509.79 + 2615.03i 6918.71 4096.00 −7966.73 − 13798.8i 17467.4 + 30254.4i
11.8 −8.00000 + 13.8564i 137.446 − 238.063i −128.000 − 221.703i −526.809 + 912.460i 2199.13 + 3809.00i −5029.91 4096.00 −27941.1 − 48395.4i −8428.95 − 14599.4i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.10.c.b ✓ 16
19.c even 3 1 inner 38.10.c.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.c.b ✓ 16 1.a even 1 1 trivial
38.10.c.b ✓ 16 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 70 T_{3}^{15} + 107714 T_{3}^{14} + 45968 T_{3}^{13} + 7876434392 T_{3}^{12} + \cdots + 13\!\cdots\!25 \) acting on \(S_{10}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16 T + 256)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 67\!\cdots\!28)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots - 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 14\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 96\!\cdots\!52)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 35\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 46\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 65\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
show more
show less