Properties

Label 390.2.j.b
Level $390$
Weight $2$
Character orbit 390.j
Analytic conductor $3.114$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(73,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.j (of order \(4\), 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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{8} q^{3} + q^{4} + \beta_{14} q^{5} + \beta_{8} q^{6} + (\beta_{13} - \beta_{12} + \cdots - \beta_1) q^{7}+ \cdots - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{8} q^{3} + q^{4} + \beta_{14} q^{5} + \beta_{8} q^{6} + (\beta_{13} - \beta_{12} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{15} - \beta_{10} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{4} + 16 q^{8} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 16 q^{16} + 4 q^{17} + 4 q^{19} - 8 q^{21} + 4 q^{22} + 16 q^{23} - 16 q^{25} + 4 q^{26} + 4 q^{30} + 12 q^{31} + 16 q^{32} + 4 q^{34} - 12 q^{35} + 4 q^{38} - 4 q^{39} + 4 q^{41} - 8 q^{42} - 16 q^{43} + 4 q^{44} + 16 q^{46} - 80 q^{49} - 16 q^{50} + 4 q^{52} - 44 q^{53} - 20 q^{55} - 16 q^{57} + 12 q^{59} + 4 q^{60} - 32 q^{61} + 12 q^{62} + 16 q^{64} - 44 q^{65} - 32 q^{67} + 4 q^{68} - 16 q^{69} - 12 q^{70} + 16 q^{71} + 4 q^{76} - 32 q^{77} - 4 q^{78} - 16 q^{81} + 4 q^{82} - 8 q^{84} - 64 q^{85} - 16 q^{86} + 28 q^{87} + 4 q^{88} + 4 q^{89} + 76 q^{91} + 16 q^{92} + 40 q^{95} - 8 q^{97} - 80 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 217 \nu^{15} + 584 \nu^{14} - 5720 \nu^{13} + 17472 \nu^{12} - 49324 \nu^{11} + 194656 \nu^{10} + \cdots - 701696 ) / 113152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 210 \nu^{15} + 93 \nu^{14} + 7072 \nu^{13} + 4056 \nu^{12} + 92728 \nu^{11} + 67196 \nu^{10} + \cdots + 205728 ) / 56576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{15} - 483 \nu^{14} + 208 \nu^{13} - 15080 \nu^{12} - 656 \nu^{11} - 179204 \nu^{10} + \cdots - 58464 ) / 56576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 725 \nu^{15} - 22776 \nu^{13} - 273788 \nu^{11} - 1589116 \nu^{9} - 4674452 \nu^{7} + \cdots - 697504 \nu ) / 113152 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 247 \nu^{15} - 527 \nu^{14} + 7384 \nu^{13} - 15912 \nu^{12} + 82420 \nu^{11} - 180404 \nu^{10} + \cdots - 147424 ) / 56576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 182 \nu^{15} + 147 \nu^{14} - 5200 \nu^{13} + 4472 \nu^{12} - 53976 \nu^{11} + 51348 \nu^{10} + \cdots + 44128 ) / 28288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 515 \nu^{15} - 37 \nu^{14} + 15704 \nu^{13} - 312 \nu^{12} + 181060 \nu^{11} + 10308 \nu^{10} + \cdots - 4768 ) / 56576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 247 \nu^{15} - 527 \nu^{14} - 7384 \nu^{13} - 15912 \nu^{12} - 82420 \nu^{11} - 180404 \nu^{10} + \cdots - 147424 ) / 56576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 515 \nu^{15} - 37 \nu^{14} - 15704 \nu^{13} - 312 \nu^{12} - 181060 \nu^{11} + 10308 \nu^{10} + \cdots - 4768 ) / 56576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 527 \nu^{15} + 144 \nu^{14} - 15912 \nu^{13} + 4160 \nu^{12} - 180404 \nu^{11} + 43296 \nu^{10} + \cdots - 190208 ) / 56576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 515 \nu^{15} + 249 \nu^{14} - 15704 \nu^{13} + 6968 \nu^{12} - 181060 \nu^{11} + 69484 \nu^{10} + \cdots + 97568 ) / 56576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 527 \nu^{15} - 144 \nu^{14} - 15912 \nu^{13} - 4160 \nu^{12} - 180404 \nu^{11} - 43296 \nu^{10} + \cdots + 190208 ) / 56576 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 515 \nu^{15} - 249 \nu^{14} - 15704 \nu^{13} - 6968 \nu^{12} - 181060 \nu^{11} - 69484 \nu^{10} + \cdots - 97568 ) / 56576 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 87 \nu^{15} - 80 \nu^{14} - 2664 \nu^{13} - 2368 \nu^{12} - 30740 \nu^{11} - 25984 \nu^{10} + \cdots - 9728 ) / 8704 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1859 \nu^{15} + 452 \nu^{14} - 55432 \nu^{13} + 12896 \nu^{12} - 615524 \nu^{11} + 132400 \nu^{10} + \cdots + 63104 ) / 113152 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} + \beta_{11} - \beta_{9} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - 2 \beta_{13} - \beta_{12} - 3 \beta_{11} + 4 \beta_{9} - \beta_{8} - 4 \beta_{7} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{15} + 10 \beta_{14} + 12 \beta_{13} - 10 \beta_{12} - \beta_{11} - \beta_{10} - 12 \beta_{9} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{14} + 13 \beta_{13} + 11 \beta_{12} + 27 \beta_{11} - 3 \beta_{10} - 36 \beta_{9} + \cdots + 14 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 79 \beta_{15} - 90 \beta_{14} - 128 \beta_{13} + 94 \beta_{12} + 15 \beta_{11} + 19 \beta_{10} + \cdots - 251 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{15} + 86 \beta_{14} - 111 \beta_{13} - 106 \beta_{12} - 273 \beta_{11} + 56 \beta_{10} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 716 \beta_{15} + 822 \beta_{14} + 1318 \beta_{13} - 898 \beta_{12} - 154 \beta_{11} - 266 \beta_{10} + \cdots + 2302 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 160 \beta_{15} - 738 \beta_{14} + 1034 \beta_{13} + 1036 \beta_{12} + 2772 \beta_{11} - 702 \beta_{10} + \cdots + 160 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6676 \beta_{15} - 7712 \beta_{14} - 13368 \beta_{13} + 8788 \beta_{12} + 1308 \beta_{11} + 3272 \beta_{10} + \cdots - 21604 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2236 \beta_{15} + 6552 \beta_{14} - 9828 \beta_{13} - 10446 \beta_{12} - 27776 \beta_{11} + 7502 \beta_{10} + \cdots - 2236 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 63498 \beta_{15} + 73944 \beta_{14} + 134552 \beta_{13} - 87536 \beta_{12} - 9346 \beta_{11} + \cdots + 205078 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 27224 \beta_{15} - 60312 \beta_{14} + 93686 \beta_{13} + 107620 \beta_{12} + 275254 \beta_{11} + \cdots + 27224 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 611880 \beta_{15} - 719500 \beta_{14} - 1348420 \beta_{13} + 880988 \beta_{12} + 49844 \beta_{11} + \cdots - 1959780 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 309968 \beta_{15} + 571020 \beta_{14} - 892700 \beta_{13} - 1120228 \beta_{12} - 2708552 \beta_{11} + \cdots - 309968 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
1.63314i
1.09662i
0.304860i
3.03462i
3.14581i
0.821039i
1.54171i
2.42515i
1.63314i
1.09662i
0.304860i
3.03462i
3.14581i
0.821039i
1.54171i
2.42515i
1.00000 −0.707107 0.707107i 1.00000 −1.15480 + 1.91479i −0.707107 0.707107i 4.24302i 1.00000 1.00000i −1.15480 + 1.91479i
73.2 1.00000 −0.707107 0.707107i 1.00000 −0.775429 2.09731i −0.707107 0.707107i 0.946681i 1.00000 1.00000i −0.775429 2.09731i
73.3 1.00000 −0.707107 0.707107i 1.00000 −0.215569 + 2.22565i −0.707107 0.707107i 2.86620i 1.00000 1.00000i −0.215569 + 2.22565i
73.4 1.00000 −0.707107 0.707107i 1.00000 2.14580 0.628922i −0.707107 0.707107i 5.15192i 1.00000 1.00000i 2.14580 0.628922i
73.5 1.00000 0.707107 + 0.707107i 1.00000 −2.22443 + 0.227886i 0.707107 + 0.707107i 4.05336i 1.00000 1.00000i −2.22443 + 0.227886i
73.6 1.00000 0.707107 + 0.707107i 1.00000 −0.580562 2.15939i 0.707107 + 0.707107i 3.80685i 1.00000 1.00000i −0.580562 2.15939i
73.7 1.00000 0.707107 + 0.707107i 1.00000 1.09015 + 1.95232i 0.707107 + 0.707107i 0.720033i 1.00000 1.00000i 1.09015 + 1.95232i
73.8 1.00000 0.707107 + 0.707107i 1.00000 1.71484 1.43504i 0.707107 + 0.707107i 3.30195i 1.00000 1.00000i 1.71484 1.43504i
187.1 1.00000 −0.707107 + 0.707107i 1.00000 −1.15480 1.91479i −0.707107 + 0.707107i 4.24302i 1.00000 1.00000i −1.15480 1.91479i
187.2 1.00000 −0.707107 + 0.707107i 1.00000 −0.775429 + 2.09731i −0.707107 + 0.707107i 0.946681i 1.00000 1.00000i −0.775429 + 2.09731i
187.3 1.00000 −0.707107 + 0.707107i 1.00000 −0.215569 2.22565i −0.707107 + 0.707107i 2.86620i 1.00000 1.00000i −0.215569 2.22565i
187.4 1.00000 −0.707107 + 0.707107i 1.00000 2.14580 + 0.628922i −0.707107 + 0.707107i 5.15192i 1.00000 1.00000i 2.14580 + 0.628922i
187.5 1.00000 0.707107 0.707107i 1.00000 −2.22443 0.227886i 0.707107 0.707107i 4.05336i 1.00000 1.00000i −2.22443 0.227886i
187.6 1.00000 0.707107 0.707107i 1.00000 −0.580562 + 2.15939i 0.707107 0.707107i 3.80685i 1.00000 1.00000i −0.580562 + 2.15939i
187.7 1.00000 0.707107 0.707107i 1.00000 1.09015 1.95232i 0.707107 0.707107i 0.720033i 1.00000 1.00000i 1.09015 1.95232i
187.8 1.00000 0.707107 0.707107i 1.00000 1.71484 + 1.43504i 0.707107 0.707107i 3.30195i 1.00000 1.00000i 1.71484 + 1.43504i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.j.b 16
3.b odd 2 1 1170.2.m.i 16
5.b even 2 1 1950.2.j.e 16
5.c odd 4 1 390.2.t.b yes 16
5.c odd 4 1 1950.2.t.e 16
13.d odd 4 1 390.2.t.b yes 16
15.e even 4 1 1170.2.w.i 16
39.f even 4 1 1170.2.w.i 16
65.f even 4 1 1950.2.j.e 16
65.g odd 4 1 1950.2.t.e 16
65.k even 4 1 inner 390.2.j.b 16
195.j odd 4 1 1170.2.m.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.b 16 1.a even 1 1 trivial
390.2.j.b 16 65.k even 4 1 inner
390.2.t.b yes 16 5.c odd 4 1
390.2.t.b yes 16 13.d odd 4 1
1170.2.m.i 16 3.b odd 2 1
1170.2.m.i 16 195.j odd 4 1
1170.2.w.i 16 15.e even 4 1
1170.2.w.i 16 39.f even 4 1
1950.2.j.e 16 5.b even 2 1
1950.2.j.e 16 65.f even 4 1
1950.2.t.e 16 5.c odd 4 1
1950.2.t.e 16 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 96 T_{7}^{14} + 3760 T_{7}^{12} + 77336 T_{7}^{10} + 890304 T_{7}^{8} + 5594816 T_{7}^{6} + \cdots + 4734976 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 96 T^{14} + \cdots + 4734976 \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} + \cdots + 58003456 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + \cdots + 2534464 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 77275104256 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21484523776 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 837825347584 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1092810981376 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 79370665984 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( T^{16} + 176 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 50182602625024 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 48729781571584 \) Copy content Toggle raw display
$61$ \( (T^{8} + 16 T^{7} + \cdots - 3982336)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots + 220928)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 846046756864 \) Copy content Toggle raw display
$73$ \( (T^{8} - 384 T^{6} + \cdots + 2967568)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 604860841984 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 509407657984 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 17644340224 \) Copy content Toggle raw display
$97$ \( (T^{8} + 4 T^{7} + \cdots + 12563008)^{2} \) Copy content Toggle raw display
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