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Properties

Label 390.2.bd.b
Level $390$
Weight $2$
Character orbit 390.bd
Analytic conductor $3.114$
Analytic rank $0$
Dimension $16$
CM no
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(7,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bd (of order \(12\), degree \(4\), 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: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{5}) q^{2} + \beta_{2} q^{3} + (\beta_{14} + 1) q^{4} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{5}) q^{2} + \beta_{2} q^{3} + (\beta_{14} + 1) q^{4} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + ( - 2 \beta_{14} - 2 \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{11} - 8 q^{13} + 12 q^{15} - 8 q^{16} + 16 q^{17} - 16 q^{18} + 4 q^{19} - 4 q^{20} - 12 q^{22} - 4 q^{23} - 4 q^{25} + 4 q^{26} - 4 q^{28} + 48 q^{29} - 8 q^{30} + 16 q^{31} + 20 q^{34} + 12 q^{35} + 20 q^{37} - 4 q^{38} - 4 q^{39} - 12 q^{41} + 4 q^{43} + 12 q^{44} - 4 q^{46} - 64 q^{47} + 4 q^{49} + 16 q^{50} - 16 q^{52} + 32 q^{53} - 44 q^{55} - 24 q^{56} - 12 q^{58} - 4 q^{59} + 12 q^{60} + 4 q^{61} - 20 q^{62} - 24 q^{63} - 16 q^{64} - 8 q^{65} + 32 q^{66} - 36 q^{67} - 4 q^{68} - 4 q^{69} - 36 q^{71} - 8 q^{72} - 12 q^{74} + 8 q^{76} - 4 q^{77} - 12 q^{78} - 8 q^{80} + 8 q^{81} - 12 q^{82} + 24 q^{83} - 52 q^{85} + 16 q^{86} - 12 q^{87} + 24 q^{89} - 4 q^{90} - 8 q^{92} - 36 q^{94} - 44 q^{95} + 60 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} + \cdots + 46\!\cdots\!94 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 123434729989149 \nu^{15} + 492304781090243 \nu^{14} + \cdots + 195044141171618 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 137809854283897 \nu^{15} + 666808683532056 \nu^{14} + \cdots + 69\!\cdots\!44 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} + \cdots - 16575048540328 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + \cdots + 3631882693804 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!27 \nu^{15} + \cdots + 35\!\cdots\!44 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + \cdots - 77929949288 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + \cdots + 65633529574178 ) / 16530327389030 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!87 \nu^{15} + \cdots - 28\!\cdots\!14 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + \cdots - 330526984 ) / 63923990 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 567889621787826 \nu^{15} + \cdots - 77\!\cdots\!96 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!58 \nu^{15} + \cdots - 60\!\cdots\!06 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} + \cdots + 258618081490 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} + \cdots + 791533522 ) / 63923990 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 69\!\cdots\!34 \nu^{15} + \cdots - 83\!\cdots\!68 ) / 51\!\cdots\!90 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 4\beta_{14} + 2\beta_{13} + 3\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{6} - 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} + \cdots - 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} + \cdots + 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} + \cdots + 985 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} + \cdots - 5566 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} + \cdots + 17408 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} + \cdots + 77085 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + \cdots - 166538 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + \cdots - 469625 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + \cdots + 2650257 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} + \cdots + 3840272 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-\beta_{13}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.792206 − 1.03242i
−0.709944 + 0.925217i
0.277956 + 0.213283i
−1.09227 − 0.838128i
0.117630 − 0.893490i
−0.424637 + 3.22544i
2.69978 + 0.355433i
0.339278 + 0.0446668i
0.792206 + 1.03242i
−0.709944 − 0.925217i
0.277956 − 0.213283i
−1.09227 + 0.838128i
0.117630 + 0.893490i
−0.424637 − 3.22544i
2.69978 − 0.355433i
0.339278 − 0.0446668i
0.866025 + 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i −1.54620 + 1.61532i 0.258819 − 0.965926i 0.954985 + 1.65408i 1.00000i −0.866025 + 0.500000i −2.14671 + 0.625808i
7.2 0.866025 + 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i 2.23567 − 0.0421887i 0.258819 − 0.965926i 1.27707 + 2.21194i 1.00000i −0.866025 + 0.500000i 1.95724 + 1.08130i
7.3 0.866025 + 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i −0.860320 + 2.06394i −0.258819 + 0.965926i 0.259847 + 0.450069i 1.00000i −0.866025 + 0.500000i −1.77703 + 1.35727i
7.4 0.866025 + 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i 1.17085 − 1.90502i −0.258819 + 0.965926i 1.97220 + 3.41596i 1.00000i −0.866025 + 0.500000i 1.96650 − 1.06437i
37.1 −0.866025 + 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i −2.13554 + 0.662933i 0.965926 − 0.258819i 0.648516 − 1.12326i 1.00000i 0.866025 + 0.500000i 1.51796 − 1.64189i
37.2 −0.866025 + 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i −0.00342112 − 2.23607i 0.965926 − 0.258819i −1.88057 + 3.25724i 1.00000i 0.866025 + 0.500000i 1.12100 + 1.93478i
37.3 −0.866025 + 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i 1.48905 − 1.66815i −0.965926 + 0.258819i 0.328411 − 0.568824i 1.00000i 0.866025 + 0.500000i −0.455475 + 2.18919i
37.4 −0.866025 + 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i 1.64991 + 1.50924i −0.965926 + 0.258819i −1.56046 + 2.70280i 1.00000i 0.866025 + 0.500000i −2.18348 − 0.482081i
223.1 0.866025 − 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i −1.54620 − 1.61532i 0.258819 + 0.965926i 0.954985 − 1.65408i − 1.00000i −0.866025 − 0.500000i −2.14671 − 0.625808i
223.2 0.866025 − 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i 2.23567 + 0.0421887i 0.258819 + 0.965926i 1.27707 − 2.21194i − 1.00000i −0.866025 − 0.500000i 1.95724 − 1.08130i
223.3 0.866025 − 0.500000i 0.258819 − 0.965926i 0.500000 − 0.866025i −0.860320 − 2.06394i −0.258819 − 0.965926i 0.259847 − 0.450069i − 1.00000i −0.866025 − 0.500000i −1.77703 − 1.35727i
223.4 0.866025 − 0.500000i 0.258819 − 0.965926i 0.500000 − 0.866025i 1.17085 + 1.90502i −0.258819 − 0.965926i 1.97220 − 3.41596i − 1.00000i −0.866025 − 0.500000i 1.96650 + 1.06437i
253.1 −0.866025 − 0.500000i −0.965926 + 0.258819i 0.500000 + 0.866025i −2.13554 − 0.662933i 0.965926 + 0.258819i 0.648516 + 1.12326i − 1.00000i 0.866025 − 0.500000i 1.51796 + 1.64189i
253.2 −0.866025 − 0.500000i −0.965926 + 0.258819i 0.500000 + 0.866025i −0.00342112 + 2.23607i 0.965926 + 0.258819i −1.88057 − 3.25724i − 1.00000i 0.866025 − 0.500000i 1.12100 − 1.93478i
253.3 −0.866025 − 0.500000i 0.965926 − 0.258819i 0.500000 + 0.866025i 1.48905 + 1.66815i −0.965926 − 0.258819i 0.328411 + 0.568824i − 1.00000i 0.866025 − 0.500000i −0.455475 − 2.18919i
253.4 −0.866025 − 0.500000i 0.965926 − 0.258819i 0.500000 + 0.866025i 1.64991 − 1.50924i −0.965926 − 0.258819i −1.56046 − 2.70280i − 1.00000i 0.866025 − 0.500000i −2.18348 + 0.482081i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bd.b ✓ 16
5.c odd 4 1 390.2.bn.b yes 16
13.f odd 12 1 390.2.bn.b yes 16
65.t even 12 1 inner 390.2.bd.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.b ✓ 16 1.a even 1 1 trivial
390.2.bd.b ✓ 16 65.t even 12 1 inner
390.2.bn.b yes 16 5.c odd 4 1
390.2.bn.b yes 16 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 4 T_{7}^{15} + 34 T_{7}^{14} - 112 T_{7}^{13} + 695 T_{7}^{12} - 2152 T_{7}^{11} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{15} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( T^{16} - 12 T^{15} + \cdots + 10000 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 16 T^{15} + \cdots + 160000 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{16} + 4 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{16} - 48 T^{15} + \cdots + 24010000 \) Copy content Toggle raw display
$31$ \( T^{16} - 16 T^{15} + \cdots + 21827584 \) Copy content Toggle raw display
$37$ \( T^{16} - 20 T^{15} + \cdots + 418609 \) Copy content Toggle raw display
$41$ \( T^{16} + 12 T^{15} + \cdots + 85264 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 151388416 \) Copy content Toggle raw display
$47$ \( (T^{8} + 32 T^{7} + \cdots - 24698204)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14952153841 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2922403926016 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1536975103504 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2412728463616 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 7487094784 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 243571706822656 \) Copy content Toggle raw display
$83$ \( (T^{8} - 12 T^{7} + \cdots - 8048)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 30073246952464 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 344958409179136 \) Copy content Toggle raw display
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