Properties

Label 390.2.j.a
Level $390$
Weight $2$
Character orbit 390.j
Analytic conductor $3.114$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{2} q^{3} + q^{4} + ( - \beta_{10} - \beta_{7} + \cdots + \beta_1) q^{5}+ \cdots + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{2} q^{3} + q^{4} + ( - \beta_{10} - \beta_{7} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{9} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 12 q^{16} - 8 q^{17} + 4 q^{19} - 8 q^{21} - 4 q^{22} + 4 q^{25} - 4 q^{26} - 4 q^{30} + 12 q^{31} - 12 q^{32} + 16 q^{33} + 8 q^{34} + 12 q^{35} - 4 q^{38} + 12 q^{39} + 8 q^{41} + 8 q^{42} - 16 q^{43} + 4 q^{44} + 4 q^{45} + 20 q^{49} - 4 q^{50} + 4 q^{52} - 16 q^{53} - 12 q^{55} + 16 q^{57} - 20 q^{59} + 4 q^{60} + 16 q^{61} - 12 q^{62} + 12 q^{64} - 8 q^{65} - 16 q^{66} - 16 q^{67} - 8 q^{68} + 32 q^{69} - 12 q^{70} - 32 q^{71} - 24 q^{73} + 4 q^{76} - 16 q^{77} - 12 q^{78} - 12 q^{81} - 8 q^{82} - 8 q^{84} - 12 q^{85} + 16 q^{86} + 20 q^{87} - 4 q^{88} - 16 q^{89} - 4 q^{90} - 28 q^{91} + 8 q^{95} + 8 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47412266 \nu^{11} + 206226196 \nu^{10} + 498918647 \nu^{9} + 275321548 \nu^{8} + \cdots - 957747072 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 478873536 \nu^{11} - 47412266 \nu^{10} - 3079467412 \nu^{9} + 10994046217 \nu^{8} + \cdots + 16498970487 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 562969191 \nu^{11} - 625242846 \nu^{10} - 3845754997 \nu^{9} + 9502475052 \nu^{8} + \cdots - 278134328 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65564 \nu^{11} + 54701 \nu^{10} - 348926 \nu^{9} + 1928638 \nu^{8} - 2217584 \nu^{7} + \cdots + 1239927 ) / 1476181 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 642662513 \nu^{11} + 232175953 \nu^{10} + 3952388821 \nu^{9} - 13970332336 \nu^{8} + \cdots - 4015649046 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1158219438 \nu^{11} + 651192603 \nu^{10} + 7299225171 \nu^{9} - 23748128936 \nu^{8} + \cdots - 2779767646 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2007824523 \nu^{11} + 642662513 \nu^{10} + 12279123091 \nu^{9} - 44235399731 \nu^{8} + \cdots - 2642262891 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2024040388 \nu^{11} - 481621272 \nu^{10} + 11262741596 \nu^{9} - 52099333786 \nu^{8} + \cdots + 4694601129 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2281630388 \nu^{11} + 1661658328 \nu^{10} + 14577758021 \nu^{9} - 44298602736 \nu^{8} + \cdots + 27875512029 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 511017007 \nu^{11} - 120344967 \nu^{10} - 3187847414 \nu^{9} + 11403406359 \nu^{8} + \cdots + 1016773599 ) / 2782601185 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 771058311 \nu^{11} + 121340051 \nu^{10} + 4638436737 \nu^{9} - 17846443417 \nu^{8} + \cdots - 7445776057 ) / 2782601185 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{5} - \beta_{4} - 3\beta_{3} - 4\beta_{2} - 4\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{8} + 4\beta_{7} - 5\beta_{6} + 5\beta_{5} - \beta_{4} - 2\beta_{3} + \beta_{2} + 3\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 5 \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{11} + 17 \beta_{9} + 13 \beta_{8} - 25 \beta_{7} + 27 \beta_{6} - 72 \beta_{5} - 25 \beta_{4} + \cdots - 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 27 \beta_{11} + 59 \beta_{10} - 27 \beta_{9} - 59 \beta_{8} + 234 \beta_{7} - 218 \beta_{6} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 218 \beta_{11} - 157 \beta_{10} - 157 \beta_{9} - 301 \beta_{7} + 301 \beta_{6} + 519 \beta_{5} + \cdots + 301 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 301 \beta_{11} - 301 \beta_{10} + 710 \beta_{9} + 710 \beta_{8} - 1925 \beta_{7} + 1889 \beta_{6} + \cdots - 2036 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1889 \beta_{11} + 2660 \beta_{10} - 1889 \beta_{8} + 8736 \beta_{7} - 8557 \beta_{6} + 1674 \beta_{5} + \cdots + 1735 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8557 \beta_{11} - 3563 \beta_{10} - 8557 \beta_{9} - 3563 \beta_{8} + 31302 \beta_{5} + 19663 \beta_{4} + \cdots + 19663 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22745 \beta_{10} + 22745 \beta_{9} + 32134 \beta_{8} - 105363 \beta_{7} + 103116 \beta_{6} + \cdots - 73229 \) 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
0.563963 1.36153i
0.0572576 0.138232i
−1.32833 + 3.20687i
1.52752 + 0.632721i
−1.46953 0.608701i
0.649118 + 0.268874i
0.563963 + 1.36153i
0.0572576 + 0.138232i
−1.32833 3.20687i
1.52752 0.632721i
−1.46953 + 0.608701i
0.649118 0.268874i
−1.00000 −0.707107 0.707107i 1.00000 −1.97581 + 1.04698i 0.707107 + 0.707107i 2.54214i −1.00000 1.00000i 1.97581 1.04698i
73.2 −1.00000 −0.707107 0.707107i 1.00000 0.383740 2.20289i 0.707107 + 0.707107i 1.52873i −1.00000 1.00000i −0.383740 + 2.20289i
73.3 −1.00000 −0.707107 0.707107i 1.00000 1.59207 + 1.57013i 0.707107 + 0.707107i 1.24244i −1.00000 1.00000i −1.59207 1.57013i
73.4 −1.00000 0.707107 + 0.707107i 1.00000 −2.14724 0.623976i −0.707107 0.707107i 1.64083i −1.00000 1.00000i 2.14724 + 0.623976i
73.5 −1.00000 0.707107 + 0.707107i 1.00000 −0.0440169 2.23563i −0.707107 0.707107i 4.35328i −1.00000 1.00000i 0.0440169 + 2.23563i
73.6 −1.00000 0.707107 + 0.707107i 1.00000 2.19126 + 0.445397i −0.707107 0.707107i 0.115977i −1.00000 1.00000i −2.19126 0.445397i
187.1 −1.00000 −0.707107 + 0.707107i 1.00000 −1.97581 1.04698i 0.707107 0.707107i 2.54214i −1.00000 1.00000i 1.97581 + 1.04698i
187.2 −1.00000 −0.707107 + 0.707107i 1.00000 0.383740 + 2.20289i 0.707107 0.707107i 1.52873i −1.00000 1.00000i −0.383740 2.20289i
187.3 −1.00000 −0.707107 + 0.707107i 1.00000 1.59207 1.57013i 0.707107 0.707107i 1.24244i −1.00000 1.00000i −1.59207 + 1.57013i
187.4 −1.00000 0.707107 0.707107i 1.00000 −2.14724 + 0.623976i −0.707107 + 0.707107i 1.64083i −1.00000 1.00000i 2.14724 0.623976i
187.5 −1.00000 0.707107 0.707107i 1.00000 −0.0440169 + 2.23563i −0.707107 + 0.707107i 4.35328i −1.00000 1.00000i 0.0440169 2.23563i
187.6 −1.00000 0.707107 0.707107i 1.00000 2.19126 0.445397i −0.707107 + 0.707107i 0.115977i −1.00000 1.00000i −2.19126 + 0.445397i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.6
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.a 12
3.b odd 2 1 1170.2.m.f 12
5.b even 2 1 1950.2.j.c 12
5.c odd 4 1 390.2.t.a yes 12
5.c odd 4 1 1950.2.t.b 12
13.d odd 4 1 390.2.t.a yes 12
15.e even 4 1 1170.2.w.f 12
39.f even 4 1 1170.2.w.f 12
65.f even 4 1 1950.2.j.c 12
65.g odd 4 1 1950.2.t.b 12
65.k even 4 1 inner 390.2.j.a 12
195.j odd 4 1 1170.2.m.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.a 12 1.a even 1 1 trivial
390.2.j.a 12 65.k even 4 1 inner
390.2.t.a yes 12 5.c odd 4 1
390.2.t.a yes 12 13.d odd 4 1
1170.2.m.f 12 3.b odd 2 1
1170.2.m.f 12 195.j odd 4 1
1170.2.w.f 12 15.e even 4 1
1170.2.w.f 12 39.f even 4 1
1950.2.j.c 12 5.b even 2 1
1950.2.j.c 12 65.f even 4 1
1950.2.t.b 12 5.c odd 4 1
1950.2.t.b 12 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 32T_{7}^{10} + 304T_{7}^{8} + 1176T_{7}^{6} + 1984T_{7}^{4} + 1216T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 32 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 141376 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 446224 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots + 23309584 \) Copy content Toggle raw display
$23$ \( T^{12} - 104 T^{9} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 932203024 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 632220736 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 3237154816 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 289272064 \) Copy content Toggle raw display
$47$ \( T^{12} + 304 T^{10} + \cdots + 85525504 \) Copy content Toggle raw display
$53$ \( T^{12} + 16 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$59$ \( T^{12} + 20 T^{11} + \cdots + 94789696 \) Copy content Toggle raw display
$61$ \( (T^{6} - 8 T^{5} + \cdots - 33056)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 8 T^{5} + \cdots - 122632)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 209295270144 \) Copy content Toggle raw display
$73$ \( (T^{6} + 12 T^{5} + \cdots + 6628)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 485409024 \) Copy content Toggle raw display
$83$ \( T^{12} + 320 T^{10} + \cdots + 2027776 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 24551129344 \) Copy content Toggle raw display
$97$ \( (T^{6} - 4 T^{5} + \cdots - 1546972)^{2} \) Copy content Toggle raw display
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