↑

Properties

Label 385.2.b.d
Level $385$
Weight $2$
Character orbit 385.b
Analytic conductor $3.074$
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] = [385,2,Mod(309,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.309");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.b (of order \(2\), degree \(1\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 30x^{14} + 367x^{12} + 2348x^{10} + 8359x^{8} + 16218x^{6} + 15449x^{4} + 5472x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{13} + \beta_{7}) q^{3} + (\beta_{2} - 2) q^{4} + \beta_{10} q^{5} + ( - \beta_{12} + \beta_{11} + \cdots + \beta_{2}) q^{6}+ \cdots + (\beta_{9} + \beta_{6} - \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{13} + \beta_{7}) q^{3} + (\beta_{2} - 2) q^{4} + \beta_{10} q^{5} + ( - \beta_{12} + \beta_{11} + \cdots + \beta_{2}) q^{6}+ \cdots + (\beta_{9} + \beta_{6} - \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 28 q^{4} - 4 q^{5} - 12 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 28 q^{4} - 4 q^{5} - 12 q^{6} - 20 q^{9} - 6 q^{10} + 16 q^{11} - 8 q^{14} + 10 q^{15} + 36 q^{16} + 12 q^{19} + 14 q^{20} + 12 q^{21} - 12 q^{24} + 14 q^{25} - 8 q^{26} - 8 q^{29} - 32 q^{30} - 28 q^{31} + 12 q^{34} - 6 q^{35} + 36 q^{36} - 32 q^{39} - 14 q^{40} - 8 q^{41} - 28 q^{44} - 18 q^{45} - 48 q^{46} - 16 q^{49} + 12 q^{50} + 32 q^{51} - 12 q^{54} - 4 q^{55} + 24 q^{56} + 80 q^{59} + 44 q^{60} - 20 q^{64} - 4 q^{65} - 12 q^{66} + 100 q^{69} + 6 q^{70} + 12 q^{71} - 6 q^{75} - 88 q^{76} - 24 q^{79} - 58 q^{80} + 80 q^{81} - 24 q^{84} - 40 q^{85} + 88 q^{86} - 44 q^{89} + 62 q^{90} + 12 q^{91} + 100 q^{94} - 4 q^{95} - 32 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30x^{14} + 367x^{12} + 2348x^{10} + 8359x^{8} + 16218x^{6} + 15449x^{4} + 5472x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 493 \nu^{15} + 2748 \nu^{14} + 13398 \nu^{13} + 64968 \nu^{12} + 146339 \nu^{11} + 595204 \nu^{10} + \cdots + 469312 ) / 220160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61 \nu^{14} - 1506 \nu^{12} - 14543 \nu^{10} - 69536 \nu^{8} - 171055 \nu^{6} - 200058 \nu^{4} + \cdots - 944 ) / 1720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53 \nu^{14} - 1238 \nu^{12} - 11099 \nu^{10} - 47900 \nu^{8} - 101987 \nu^{6} - 97842 \nu^{4} + \cdots - 1136 ) / 1376 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 493 \nu^{15} + 2748 \nu^{14} - 13398 \nu^{13} + 64968 \nu^{12} - 146339 \nu^{11} + 595204 \nu^{10} + \cdots + 469312 ) / 220160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71 \nu^{15} + 1282 \nu^{13} + 6249 \nu^{11} - 10876 \nu^{9} - 172911 \nu^{7} - 480314 \nu^{5} + \cdots - 118320 \nu ) / 22016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 77 \nu^{15} + 5652 \nu^{14} + 2902 \nu^{13} + 139032 \nu^{12} + 39491 \nu^{11} + 1333676 \nu^{10} + \cdots + 567488 ) / 110080 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1459 \nu^{14} - 34794 \nu^{12} - 319037 \nu^{10} - 1407124 \nu^{8} - 3035205 \nu^{6} + \cdots + 23664 ) / 27520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 77 \nu^{15} - 5652 \nu^{14} + 2902 \nu^{13} - 139032 \nu^{12} + 39491 \nu^{11} - 1333676 \nu^{10} + \cdots - 567488 ) / 110080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1545 \nu^{15} + 1564 \nu^{14} + 38750 \nu^{13} + 42504 \nu^{12} + 386375 \nu^{11} + 460452 \nu^{10} + \cdots + 508736 ) / 220160 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1545 \nu^{15} - 1564 \nu^{14} + 38750 \nu^{13} - 42504 \nu^{12} + 386375 \nu^{11} - 460452 \nu^{10} + \cdots - 508736 ) / 220160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1357 \nu^{15} - 32022 \nu^{13} - 287811 \nu^{11} - 1219372 \nu^{9} - 2389755 \nu^{7} + \cdots + 1023632 \nu ) / 110080 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 331 \nu^{15} - 8186 \nu^{13} - 78533 \nu^{11} - 365236 \nu^{9} - 826605 \nu^{7} + \cdots + 98416 \nu ) / 13760 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 141 \nu^{15} - 3670 \nu^{13} - 38147 \nu^{11} - 202476 \nu^{9} - 580859 \nu^{7} + \cdots - 122864 \nu ) / 5504 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} - 9 \beta_{13} + \beta_{12} + \beta_{11} - 11 \beta_{10} - 11 \beta_{8} + \cdots + 37 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{12} + 14 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} - 12 \beta_{8} + \beta_{6} - 9 \beta_{4} + \cdots - 131 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17 \beta_{15} + 11 \beta_{14} + 76 \beta_{13} - 19 \beta_{12} - 19 \beta_{11} + 94 \beta_{10} + \cdots - 230 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 135 \beta_{12} - 135 \beta_{11} - 103 \beta_{10} + 114 \beta_{9} + 103 \beta_{8} - \beta_{6} + \cdots + 823 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 190 \beta_{15} - 92 \beta_{14} - 628 \beta_{13} + 220 \beta_{12} + 220 \beta_{11} - 748 \beta_{10} + \cdots + 1453 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1142 \beta_{12} + 1142 \beta_{11} + 786 \beta_{10} - 858 \beta_{9} - 786 \beta_{8} - 100 \beta_{6} + \cdots - 5376 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1804 \beta_{15} + 714 \beta_{14} + 5091 \beta_{13} - 2104 \beta_{12} - 2104 \beta_{11} + \cdots - 9362 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9109 \beta_{12} - 9109 \beta_{11} - 5707 \beta_{10} + 5985 \beta_{9} + 5707 \beta_{8} + 1696 \beta_{6} + \cdots + 36156 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 15779 \beta_{15} - 5429 \beta_{14} - 40609 \beta_{13} + 18317 \beta_{12} + 18317 \beta_{11} + \cdots + 61561 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 70536 \beta_{12} + 70536 \beta_{11} + 40550 \beta_{10} - 39983 \beta_{9} - 40550 \beta_{8} + \cdots - 248703 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 131665 \beta_{15} + 41117 \beta_{14} + 319774 \beta_{13} - 151363 \beta_{12} - 151363 \beta_{11} + \cdots - 412766 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
309.1
− 2.74464i
− 2.46327i
− 2.43000i
− 2.23341i
− 1.73030i
− 1.34939i
− 0.797933i
− 0.234058i
0.234058i
0.797933i
1.34939i
1.73030i
2.23341i
2.43000i
2.46327i
2.74464i
− 2.74464i 1.05459i −5.53304 −0.948436 + 2.02496i 2.89448 − 1.00000i 9.69692i 1.88783 5.55779 + 2.60311i
309.2 − 2.46327i − 2.19787i −4.06768 −2.17998 − 0.497690i −5.41394 1.00000i 5.09324i −1.83063 −1.22594 + 5.36986i
309.3 − 2.43000i − 2.50028i −3.90488 1.86626 − 1.23169i −6.07568 − 1.00000i 4.62886i −3.25142 −2.99300 − 4.53502i
309.4 − 2.23341i 3.01818i −2.98812 −1.37828 − 1.76078i 6.74083 − 1.00000i 2.20688i −6.10941 −3.93255 + 3.07826i
309.5 − 1.73030i − 0.375719i −0.993931 2.21549 + 0.302653i −0.650106 1.00000i − 1.74080i 2.85884 0.523681 − 3.83346i
309.6 − 1.34939i − 0.423229i 0.179138 −1.26242 − 1.84561i −0.571102 − 1.00000i − 2.94051i 2.82088 −2.49046 + 1.70350i
309.7 − 0.797933i − 3.35029i 1.36330 1.56336 + 1.59872i −2.67331 1.00000i − 2.68369i −8.22444 1.27567 − 1.24745i
309.8 − 0.234058i − 1.07314i 1.94522 −1.87600 + 1.21681i −0.251177 − 1.00000i − 0.923409i 1.84837 0.284804 + 0.439093i
309.9 0.234058i 1.07314i 1.94522 −1.87600 − 1.21681i −0.251177 1.00000i 0.923409i 1.84837 0.284804 − 0.439093i
309.10 0.797933i 3.35029i 1.36330 1.56336 − 1.59872i −2.67331 − 1.00000i 2.68369i −8.22444 1.27567 + 1.24745i
309.11 1.34939i 0.423229i 0.179138 −1.26242 + 1.84561i −0.571102 1.00000i 2.94051i 2.82088 −2.49046 − 1.70350i
309.12 1.73030i 0.375719i −0.993931 2.21549 − 0.302653i −0.650106 − 1.00000i 1.74080i 2.85884 0.523681 + 3.83346i
309.13 2.23341i − 3.01818i −2.98812 −1.37828 + 1.76078i 6.74083 1.00000i − 2.20688i −6.10941 −3.93255 − 3.07826i
309.14 2.43000i 2.50028i −3.90488 1.86626 + 1.23169i −6.07568 1.00000i − 4.62886i −3.25142 −2.99300 + 4.53502i
309.15 2.46327i 2.19787i −4.06768 −2.17998 + 0.497690i −5.41394 − 1.00000i − 5.09324i −1.83063 −1.22594 − 5.36986i
309.16 2.74464i − 1.05459i −5.53304 −0.948436 − 2.02496i 2.89448 1.00000i − 9.69692i 1.88783 5.55779 − 2.60311i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 309.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.b.d ✓ 16
5.b even 2 1 inner 385.2.b.d ✓ 16
5.c odd 4 1 1925.2.a.be 8
5.c odd 4 1 1925.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.b.d ✓ 16 1.a even 1 1 trivial
385.2.b.d ✓ 16 5.b even 2 1 inner
1925.2.a.be 8 5.c odd 4 1
1925.2.a.bf 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\):

\( T_{2}^{16} + 30T_{2}^{14} + 367T_{2}^{12} + 2348T_{2}^{10} + 8359T_{2}^{8} + 16218T_{2}^{6} + 15449T_{2}^{4} + 5472T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{16} + 152 T_{13}^{14} + 9604 T_{13}^{12} + 328504 T_{13}^{10} + 6618016 T_{13}^{8} + \cdots + 2955879424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 30 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} + 34 T^{14} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T - 1)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 2955879424 \) Copy content Toggle raw display
$17$ \( T^{16} + 132 T^{14} + \cdots + 287296 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots - 14560)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 76200289936 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{7} + \cdots + 320)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 14 T^{7} + \cdots - 3926)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 240103920016 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + \cdots - 107720)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 6461587456 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 135862336 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2050807300096 \) Copy content Toggle raw display
$59$ \( (T^{8} - 40 T^{7} + \cdots + 12730)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 170 T^{6} + \cdots - 301352)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 497978616976 \) Copy content Toggle raw display
$71$ \( (T^{8} - 6 T^{7} + \cdots + 16820656)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + 12 T^{7} + \cdots - 1118800)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 700555608064 \) Copy content Toggle raw display
$89$ \( (T^{8} + 22 T^{7} + \cdots - 193400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
show more
show less