Properties

Label 390.2.x.a
Level $390$
Weight $2$
Character orbit 390.x
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(49,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("390.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.x (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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - 1) q^{2} + \beta_{4} q^{3} + \beta_{6} q^{4} + (\beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{5}+ \cdots - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - 1) q^{2} + \beta_{4} q^{3} + \beta_{6} q^{4} + (\beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{5}+ \cdots + (\beta_{11} - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 6 q^{4} - 2 q^{5} - 2 q^{7} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} - 6 q^{4} - 2 q^{5} - 2 q^{7} + 12 q^{8} + 6 q^{9} - 2 q^{10} + 6 q^{11} - 8 q^{13} + 4 q^{14} + 6 q^{15} - 6 q^{16} + 18 q^{17} - 12 q^{18} - 6 q^{19} + 4 q^{20} - 6 q^{22} + 6 q^{23} - 10 q^{25} - 2 q^{26} - 2 q^{28} + 14 q^{29} - 6 q^{30} - 6 q^{32} + 6 q^{33} + 26 q^{35} + 6 q^{36} - 12 q^{37} - 2 q^{39} - 2 q^{40} - 18 q^{41} - 12 q^{42} - 36 q^{43} - 4 q^{45} - 6 q^{46} + 16 q^{47} + 8 q^{49} - 10 q^{50} + 16 q^{51} + 10 q^{52} - 28 q^{55} - 2 q^{56} - 8 q^{57} + 14 q^{58} - 36 q^{59} + 10 q^{61} + 6 q^{62} + 2 q^{63} + 12 q^{64} + 6 q^{65} - 12 q^{66} + 4 q^{67} - 18 q^{68} + 16 q^{69} - 4 q^{70} - 12 q^{71} + 6 q^{72} + 28 q^{73} - 12 q^{74} - 8 q^{75} + 6 q^{76} - 2 q^{78} + 4 q^{79} - 2 q^{80} - 6 q^{81} + 18 q^{82} + 72 q^{83} + 12 q^{84} + 18 q^{85} + 6 q^{87} + 6 q^{88} + 18 q^{89} + 2 q^{90} + 2 q^{91} - 16 q^{93} - 8 q^{94} - 42 q^{95} - 48 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + \cdots + 2197 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 203419 \nu^{11} - 163110633 \nu^{10} + 591783880 \nu^{9} + 97338749 \nu^{8} + \cdots + 81183629852 ) / 63907274600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60968787 \nu^{11} - 2097063441 \nu^{10} + 2300362460 \nu^{9} + 17147379373 \nu^{8} + \cdots + 1269272187404 ) / 830794569800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 120243408 \nu^{11} + 454030419 \nu^{10} + 2051209685 \nu^{9} - 7036618932 \nu^{8} + \cdots + 618192439839 ) / 830794569800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16426431 \nu^{11} + 83789417 \nu^{10} - 226795620 \nu^{9} + 267354099 \nu^{8} + \cdots + 20321135952 ) / 63907274600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + \cdots + 11867687 ) / 11796200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20638 \nu^{11} + 28887 \nu^{10} - 18833 \nu^{9} - 91862 \nu^{8} - 291763 \nu^{7} + \cdots + 21764327 ) / 47610004 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5665538 \nu^{11} - 7145579 \nu^{10} - 9226575 \nu^{9} + 110201552 \nu^{8} - 247897203 \nu^{7} + \cdots - 40106339 ) / 12781454920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 680246389 \nu^{11} - 845264698 \nu^{10} + 11881898255 \nu^{9} - 20291622981 \nu^{8} + \cdots + 1108736562037 ) / 830794569800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1202455216 \nu^{11} + 4870077513 \nu^{10} - 1280213355 \nu^{9} - 35058222714 \nu^{8} + \cdots - 1182431604497 ) / 830794569800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 78524401 \nu^{11} + 161812093 \nu^{10} + 556127095 \nu^{9} - 2253760279 \nu^{8} + \cdots + 16572567908 ) / 31953637300 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 2\beta_{10} + 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{5} - 3\beta_{4} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{11} + 7\beta_{10} + 12\beta_{9} + \beta_{8} + 2\beta_{7} + 9\beta_{5} - 6\beta_{4} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{11} - 5 \beta_{10} - 8 \beta_{9} + 4 \beta_{8} + 29 \beta_{7} - 12 \beta_{6} - 7 \beta_{5} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 24 \beta_{6} + 18 \beta_{5} - 20 \beta_{4} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 50 \beta_{11} - 70 \beta_{10} - 162 \beta_{9} - 50 \beta_{8} + 54 \beta_{7} - 36 \beta_{6} - 90 \beta_{5} + \cdots - 104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81 \beta_{11} + 80 \beta_{10} - 90 \beta_{9} - 155 \beta_{8} - 382 \beta_{7} + 114 \beta_{6} + \cdots + 388 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 404 \beta_{11} - 223 \beta_{10} - 627 \beta_{9} - 213 \beta_{8} - 540 \beta_{7} + 459 \beta_{6} + \cdots + 249 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 327 \beta_{11} + 649 \beta_{10} + 866 \beta_{9} - 57 \beta_{8} - 2616 \beta_{7} + 1480 \beta_{6} + \cdots + 2981 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1120 \beta_{11} - 2139 \beta_{10} - 614 \beta_{9} + 1672 \beta_{8} + 195 \beta_{7} + 1784 \beta_{6} + \cdots - 288 \) Copy content Toggle raw display

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.39378 + 0.0429626i
1.75374 + 1.62986i
2.00607 1.30680i
−0.330925 + 1.46916i
−1.44229 + 0.433312i
1.40719 0.536449i
−2.39378 0.0429626i
1.75374 1.62986i
2.00607 + 1.30680i
−0.330925 1.46916i
−1.44229 0.433312i
1.40719 + 0.536449i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −2.10012 0.767774i 0.866025 + 0.500000i −0.823063 + 1.42559i 1.00000 0.500000 0.866025i 0.385150 + 2.20265i
49.2 −0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −1.40066 + 1.74303i 0.866025 + 0.500000i 0.763837 1.32301i 1.00000 0.500000 0.866025i 2.20984 + 0.341491i
49.3 −0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 1.26873 1.84128i 0.866025 + 0.500000i −2.17283 + 3.76344i 1.00000 0.500000 0.866025i −2.22896 0.178114i
49.4 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i −0.571769 2.16173i −0.866025 0.500000i −0.603137 + 1.04466i 1.00000 0.500000 0.866025i −1.58623 + 1.57603i
49.5 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i −0.230377 + 2.22417i −0.866025 0.500000i 0.432713 0.749482i 1.00000 0.500000 0.866025i 2.04138 0.912572i
49.6 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 2.03420 0.928463i −0.866025 0.500000i 1.40247 2.42916i 1.00000 0.500000 0.866025i −1.82117 1.29743i
199.1 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i −2.10012 + 0.767774i 0.866025 0.500000i −0.823063 1.42559i 1.00000 0.500000 + 0.866025i 0.385150 2.20265i
199.2 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i −1.40066 1.74303i 0.866025 0.500000i 0.763837 + 1.32301i 1.00000 0.500000 + 0.866025i 2.20984 0.341491i
199.3 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 1.26873 + 1.84128i 0.866025 0.500000i −2.17283 3.76344i 1.00000 0.500000 + 0.866025i −2.22896 + 0.178114i
199.4 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i −0.571769 + 2.16173i −0.866025 + 0.500000i −0.603137 1.04466i 1.00000 0.500000 + 0.866025i −1.58623 1.57603i
199.5 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i −0.230377 2.22417i −0.866025 + 0.500000i 0.432713 + 0.749482i 1.00000 0.500000 + 0.866025i 2.04138 + 0.912572i
199.6 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 2.03420 + 0.928463i −0.866025 + 0.500000i 1.40247 + 2.42916i 1.00000 0.500000 + 0.866025i −1.82117 + 1.29743i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.x.a 12
3.b odd 2 1 1170.2.bj.d 12
5.b even 2 1 390.2.x.b yes 12
5.c odd 4 1 1950.2.bc.i 12
5.c odd 4 1 1950.2.bc.j 12
13.e even 6 1 390.2.x.b yes 12
15.d odd 2 1 1170.2.bj.c 12
39.h odd 6 1 1170.2.bj.c 12
65.l even 6 1 inner 390.2.x.a 12
65.r odd 12 1 1950.2.bc.i 12
65.r odd 12 1 1950.2.bc.j 12
195.y odd 6 1 1170.2.bj.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 1.a even 1 1 trivial
390.2.x.a 12 65.l even 6 1 inner
390.2.x.b yes 12 5.b even 2 1
390.2.x.b yes 12 13.e even 6 1
1170.2.bj.c 12 15.d odd 2 1
1170.2.bj.c 12 39.h odd 6 1
1170.2.bj.d 12 3.b odd 2 1
1170.2.bj.d 12 195.y odd 6 1
1950.2.bc.i 12 5.c odd 4 1
1950.2.bc.i 12 65.r odd 12 1
1950.2.bc.j 12 5.c odd 4 1
1950.2.bc.j 12 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 2 T_{7}^{11} + 19 T_{7}^{10} - 6 T_{7}^{9} + 205 T_{7}^{8} + 20 T_{7}^{7} + 708 T_{7}^{6} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 1982464 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 190660864 \) Copy content Toggle raw display
$29$ \( T^{12} - 14 T^{11} + \cdots + 21904 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 177209344 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 227195329 \) Copy content Toggle raw display
$41$ \( T^{12} + 18 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots + 5956)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2473271824 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4983230464 \) Copy content Toggle raw display
$61$ \( T^{12} - 10 T^{11} + \cdots + 89718784 \) Copy content Toggle raw display
$67$ \( T^{12} - 4 T^{11} + \cdots + 83759104 \) Copy content Toggle raw display
$71$ \( T^{12} + 12 T^{11} + \cdots + 4194304 \) Copy content Toggle raw display
$73$ \( (T^{6} - 14 T^{5} + \cdots - 230528)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + \cdots - 29312)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 36 T^{5} + \cdots - 6912)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1341001056256 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 415519473664 \) Copy content Toggle raw display
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