Properties

Label 370.2.m.d
Level $370$
Weight $2$
Character orbit 370.m
Analytic conductor $2.954$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(159,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.159");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.m (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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 37x^{14} + 559x^{12} + 4431x^{10} + 19684x^{8} + 48248x^{6} + 58656x^{4} + 25392x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} - \beta_{10} q^{3} + \beta_1 q^{4} + \beta_{8} q^{5} + ( - \beta_{11} - \beta_{10}) q^{6} + (\beta_{2} - \beta_1 - 1) q^{7} - q^{8} + ( - \beta_{15} - \beta_{14} + \beta_{12} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} - \beta_{10} q^{3} + \beta_1 q^{4} + \beta_{8} q^{5} + ( - \beta_{11} - \beta_{10}) q^{6} + (\beta_{2} - \beta_1 - 1) q^{7} - q^{8} + ( - \beta_{15} - \beta_{14} + \beta_{12} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 3 q^{3} - 8 q^{4} - 12 q^{7} - 16 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 3 q^{3} - 8 q^{4} - 12 q^{7} - 16 q^{8} + 13 q^{9} - 6 q^{10} - 6 q^{11} + 3 q^{12} - 6 q^{13} + 9 q^{15} - 8 q^{16} - 13 q^{18} - 3 q^{19} - 6 q^{20} - 6 q^{21} - 3 q^{22} - 22 q^{23} + 3 q^{24} - 6 q^{25} - 12 q^{26} + 12 q^{28} - 9 q^{30} + 8 q^{32} + 6 q^{33} + 18 q^{35} - 26 q^{36} + 16 q^{37} + 15 q^{39} + 7 q^{41} + 6 q^{42} - 22 q^{43} + 3 q^{44} + 4 q^{45} - 11 q^{46} + 4 q^{49} + 6 q^{50} - 6 q^{52} + 3 q^{53} + 9 q^{54} - 35 q^{55} + 12 q^{56} + 18 q^{57} + 36 q^{58} + 15 q^{59} - 18 q^{60} + 12 q^{61} - 33 q^{62} + 16 q^{64} + 46 q^{65} + 24 q^{67} + 42 q^{69} + 12 q^{70} - 4 q^{71} - 13 q^{72} + 5 q^{74} - 10 q^{75} + 3 q^{76} - 24 q^{77} + 15 q^{78} + 6 q^{80} + 10 q^{81} + 14 q^{82} + 6 q^{83} + 12 q^{84} - 26 q^{85} - 11 q^{86} + 50 q^{87} + 6 q^{88} + 9 q^{89} - q^{90} - 24 q^{91} + 11 q^{92} - 25 q^{93} - 27 q^{94} - 53 q^{95} - 3 q^{96} + 68 q^{97} - 4 q^{98} + 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 37x^{14} + 559x^{12} + 4431x^{10} + 19684x^{8} + 48248x^{6} + 58656x^{4} + 25392x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 31\nu^{13} + 381\nu^{11} + 2361\nu^{9} + 7734\nu^{7} + 12700\nu^{5} + 8872\nu^{3} + 2272\nu - 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 31 \nu^{13} + 54 \nu^{12} - 381 \nu^{11} + 546 \nu^{10} - 2361 \nu^{9} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} - 31 \nu^{13} - 54 \nu^{12} - 381 \nu^{11} - 546 \nu^{10} - 2361 \nu^{9} + \cdots + 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 162 \nu^{12} - 273 \nu^{11} + 1670 \nu^{10} - 1237 \nu^{9} + \cdots + 5632 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} - 6 \nu^{14} - 27 \nu^{13} - 162 \nu^{12} - 273 \nu^{11} - 1670 \nu^{10} - 1237 \nu^{9} + \cdots + 5632 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} + 27 \nu^{13} + 170 \nu^{12} + 289 \nu^{11} + 1870 \nu^{10} + 1637 \nu^{9} + \cdots + 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 170 \nu^{12} - 289 \nu^{11} + 1870 \nu^{10} - 1637 \nu^{9} + \cdots + 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 170 \nu^{12} - 381 \nu^{11} + 1870 \nu^{10} - 2361 \nu^{9} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 170 \nu^{12} + 381 \nu^{11} + 1870 \nu^{10} + 2361 \nu^{9} + \cdots - 384 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{14} + 89 \nu^{12} + 1035 \nu^{10} + 5975 \nu^{8} + 17774 \nu^{6} + 24892 \nu^{4} + 11560 \nu^{2} + \cdots + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{14} - 89 \nu^{12} - 1035 \nu^{10} - 5975 \nu^{8} - 17774 \nu^{6} - 24892 \nu^{4} + \cdots - 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{15} + 6 \nu^{14} + 81 \nu^{13} + 170 \nu^{12} + 835 \nu^{11} + 1870 \nu^{10} + 4143 \nu^{9} + \cdots + 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -3\nu^{14} - 85\nu^{12} - 935\nu^{10} - 5051\nu^{8} - 13938\nu^{6} - 18092\nu^{4} - 8056\nu^{2} - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{14} + 87 \nu^{12} + 985 \nu^{10} + 5513 \nu^{8} + 8 \nu^{7} + 15872 \nu^{6} + 112 \nu^{5} + \cdots + 320 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3 \nu^{14} - 87 \nu^{12} - 985 \nu^{10} - 5513 \nu^{8} + 8 \nu^{7} - 15872 \nu^{6} + 112 \nu^{5} + \cdots - 320 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{7} + \beta_{6} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - 6\beta_{11} - 6\beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{13} - 2\beta_{9} - 2\beta_{8} - 9\beta_{7} - 9\beta_{6} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{15} - 11 \beta_{14} + 42 \beta_{11} + 42 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} + 101 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 24 \beta_{9} + \cdots - 192 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 103 \beta_{15} + 103 \beta_{14} - 320 \beta_{11} - 320 \beta_{10} - 99 \beta_{9} + 99 \beta_{8} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28 \beta_{15} - 28 \beta_{14} - 871 \beta_{13} - 32 \beta_{11} + 32 \beta_{10} - 230 \beta_{9} + \cdots + 1510 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 919 \beta_{15} - 919 \beta_{14} + 4 \beta_{13} + 8 \beta_{12} + 2546 \beta_{11} + 2546 \beta_{10} + \cdots + 294 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 282 \beta_{15} + 282 \beta_{14} + 7329 \beta_{13} + 366 \beta_{11} - 366 \beta_{10} + 2056 \beta_{9} + \cdots - 12252 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8039 \beta_{15} + 8039 \beta_{14} - 100 \beta_{13} - 200 \beta_{12} - 20692 \beta_{11} - 20692 \beta_{10} + \cdots - 2800 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2500 \beta_{15} - 2500 \beta_{14} - 61043 \beta_{13} - 3692 \beta_{11} + 3692 \beta_{10} - 17886 \beta_{9} + \cdots + 100922 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 69671 \beta_{15} - 69671 \beta_{14} + 1576 \beta_{13} + 3152 \beta_{12} + 169910 \beta_{11} + \cdots + 25586 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 20794 \beta_{15} + 20794 \beta_{14} + 506205 \beta_{13} + 35122 \beta_{11} - 35122 \beta_{10} + \cdots - 837320 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 600927 \beta_{15} + 600927 \beta_{14} - 20200 \beta_{13} - 40400 \beta_{12} - 1402224 \beta_{11} + \cdots - 229220 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
2.90925i
2.88937i
1.76701i
0.926756i
0.101618i
1.93403i
2.06794i
2.85998i
2.90925i
2.88937i
1.76701i
0.926756i
0.101618i
1.93403i
2.06794i
2.85998i
0.500000 + 0.866025i −2.51948 1.45462i −0.500000 + 0.866025i −2.23418 0.0919631i 2.90925i 0.191824 + 0.110750i −1.00000 2.73187 + 4.73173i −1.03745 1.98083i
159.2 0.500000 + 0.866025i −2.50227 1.44468i −0.500000 + 0.866025i 1.41509 1.73134i 2.88937i −0.668346 0.385870i −1.00000 2.67422 + 4.63189i 2.20693 + 0.359832i
159.3 0.500000 + 0.866025i −1.53027 0.883503i −0.500000 + 0.866025i 0.209495 + 2.22623i 1.76701i −3.45647 1.99560i −1.00000 0.0611550 + 0.105924i −1.82323 + 1.29454i
159.4 0.500000 + 0.866025i −0.802594 0.463378i −0.500000 + 0.866025i −0.214614 + 2.22574i 0.926756i 3.13584 + 1.81048i −1.00000 −1.07056 1.85427i −2.03486 + 0.927011i
159.5 0.500000 + 0.866025i −0.0880035 0.0508088i −0.500000 + 0.866025i −2.11174 0.735216i 0.101618i −1.94895 1.12523i −1.00000 −1.49484 2.58913i −0.419156 2.19643i
159.6 0.500000 + 0.866025i 1.67492 + 0.967016i −0.500000 + 0.866025i 1.59185 1.57036i 1.93403i −0.358580 0.207027i −1.00000 0.370239 + 0.641273i 2.15589 + 0.593399i
159.7 0.500000 + 0.866025i 1.79089 + 1.03397i −0.500000 + 0.866025i 1.99857 + 1.00285i 2.06794i 1.25580 + 0.725037i −1.00000 0.638185 + 1.10537i 0.130790 + 2.23224i
159.8 0.500000 + 0.866025i 2.47681 + 1.42999i −0.500000 + 0.866025i −0.654464 + 2.13815i 2.85998i −4.15112 2.39665i −1.00000 2.58973 + 4.48555i −2.17892 + 0.502291i
249.1 0.500000 0.866025i −2.51948 + 1.45462i −0.500000 0.866025i −2.23418 + 0.0919631i 2.90925i 0.191824 0.110750i −1.00000 2.73187 4.73173i −1.03745 + 1.98083i
249.2 0.500000 0.866025i −2.50227 + 1.44468i −0.500000 0.866025i 1.41509 + 1.73134i 2.88937i −0.668346 + 0.385870i −1.00000 2.67422 4.63189i 2.20693 0.359832i
249.3 0.500000 0.866025i −1.53027 + 0.883503i −0.500000 0.866025i 0.209495 2.22623i 1.76701i −3.45647 + 1.99560i −1.00000 0.0611550 0.105924i −1.82323 1.29454i
249.4 0.500000 0.866025i −0.802594 + 0.463378i −0.500000 0.866025i −0.214614 2.22574i 0.926756i 3.13584 1.81048i −1.00000 −1.07056 + 1.85427i −2.03486 0.927011i
249.5 0.500000 0.866025i −0.0880035 + 0.0508088i −0.500000 0.866025i −2.11174 + 0.735216i 0.101618i −1.94895 + 1.12523i −1.00000 −1.49484 + 2.58913i −0.419156 + 2.19643i
249.6 0.500000 0.866025i 1.67492 0.967016i −0.500000 0.866025i 1.59185 + 1.57036i 1.93403i −0.358580 + 0.207027i −1.00000 0.370239 0.641273i 2.15589 0.593399i
249.7 0.500000 0.866025i 1.79089 1.03397i −0.500000 0.866025i 1.99857 1.00285i 2.06794i 1.25580 0.725037i −1.00000 0.638185 1.10537i 0.130790 2.23224i
249.8 0.500000 0.866025i 2.47681 1.42999i −0.500000 0.866025i −0.654464 2.13815i 2.85998i −4.15112 + 2.39665i −1.00000 2.58973 4.48555i −2.17892 0.502291i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.m.d yes 16
5.b even 2 1 370.2.m.c 16
37.e even 6 1 370.2.m.c 16
185.l even 6 1 inner 370.2.m.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.m.c 16 5.b even 2 1
370.2.m.c 16 37.e even 6 1
370.2.m.d yes 16 1.a even 1 1 trivial
370.2.m.d yes 16 185.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 3 T_{3}^{15} - 14 T_{3}^{14} - 51 T_{3}^{13} + 154 T_{3}^{12} + 561 T_{3}^{11} - 771 T_{3}^{10} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 12 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots - 11840)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 6 T^{15} + \cdots + 1849600 \) Copy content Toggle raw display
$17$ \( T^{16} + 58 T^{14} + \cdots + 226576 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 1435500544 \) Copy content Toggle raw display
$23$ \( (T^{8} + 11 T^{7} + \cdots - 9224)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 192 T^{14} + \cdots + 14500864 \) Copy content Toggle raw display
$31$ \( T^{16} + 121 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3512479453921 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23754764740996 \) Copy content Toggle raw display
$43$ \( (T^{8} + 11 T^{7} + \cdots + 177208)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 35452370944 \) Copy content Toggle raw display
$53$ \( T^{16} - 3 T^{15} + \cdots + 802816 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 159386189824 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 899128754176 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1893990400 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 451477504 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10676686950400 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 5065884565504 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1363677184 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 5611687210000 \) Copy content Toggle raw display
$97$ \( (T^{8} - 34 T^{7} + \cdots + 3678976)^{2} \) Copy content Toggle raw display
show more
show less