↑

Properties

Label 2340.2.dj.e
Level $2340$
Weight $2$
Character orbit 2340.dj
Analytic conductor $18.685$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2340,2,Mod(361,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.dj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 48x^{16} + 528x^{12} + 1620x^{8} + 1368x^{4} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{7} q^{7} + (\beta_{17} + \beta_{15}) q^{11} + (\beta_{14} - \beta_{13} + \beta_{11} + \cdots - 1) q^{13} + (\beta_{16} - \beta_{12} + \cdots - \beta_{4}) q^{17} + ( - \beta_{19} - \beta_{18} - \beta_{13} + \cdots - 1) q^{19}+ \cdots + ( - 2 \beta_{19} - 3 \beta_{18} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{7} - 2 q^{13} - 12 q^{19} - 20 q^{25} - 12 q^{37} - 2 q^{43} + 8 q^{49} + 14 q^{61} + 30 q^{67} + 44 q^{79} - 30 q^{91} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 48x^{16} + 528x^{12} + 1620x^{8} + 1368x^{4} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -889\nu^{18} - 42755\nu^{14} - 473471\nu^{10} - 1489140\nu^{6} - 1451508\nu^{2} ) / 232698 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 129 \nu^{19} - 3151 \nu^{17} + 6684 \nu^{16} - 1667 \nu^{15} - 145042 \nu^{13} + 317268 \nu^{12} + \cdots + 803844 ) / 930792 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 129 \nu^{19} - 8722 \nu^{18} - 3151 \nu^{17} + 1667 \nu^{15} - 412752 \nu^{14} - 145042 \nu^{13} + \cdots + 525312 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7692 \nu^{19} + 1778 \nu^{18} + 2441 \nu^{17} - 367273 \nu^{15} + 85510 \nu^{14} + \cdots + 1481076 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7692 \nu^{19} + 2441 \nu^{17} + 367273 \nu^{15} + 114822 \nu^{13} + 3971028 \nu^{11} + \cdots - 465396 ) / 930792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10585 \nu^{19} + 664 \nu^{17} - 7464 \nu^{16} + 508981 \nu^{15} + 32632 \nu^{13} - 347190 \nu^{12} + \cdots - 2689884 ) / 930792 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10585 \nu^{19} - 664 \nu^{17} - 7464 \nu^{16} - 508981 \nu^{15} - 32632 \nu^{13} + \cdots - 2689884 ) / 930792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10585 \nu^{19} + 16094 \nu^{18} - 664 \nu^{17} + 508981 \nu^{15} + 764766 \nu^{14} + \cdots - 2536344 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10585 \nu^{19} + 16094 \nu^{18} + 664 \nu^{17} - 508981 \nu^{15} + 764766 \nu^{14} + \cdots + 2536344 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12943 \nu^{19} + 14224 \nu^{18} + 2939 \nu^{17} - 619724 \nu^{15} + 684080 \nu^{14} + \cdots + 2685240 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12943 \nu^{19} - 2939 \nu^{17} + 498 \nu^{16} - 619724 \nu^{15} - 139296 \nu^{13} + \cdots + 3531144 ) / 930792 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15172 \nu^{19} + 16286 \nu^{18} - 2229 \nu^{17} + 728800 \nu^{15} + 781678 \nu^{14} + \cdots - 3295440 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15172 \nu^{19} - 2229 \nu^{17} + 3296 \nu^{16} - 728800 \nu^{15} - 109076 \nu^{13} + \cdots + 3315768 ) / 930792 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15172 \nu^{19} - 2229 \nu^{17} - 3296 \nu^{16} - 728800 \nu^{15} - 109076 \nu^{13} + \cdots - 3315768 ) / 930792 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15172 \nu^{19} + 16286 \nu^{18} + 2229 \nu^{17} - 728800 \nu^{15} + 781678 \nu^{14} + \cdots + 3295440 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20506 \nu^{19} - 11404 \nu^{18} + 2229 \nu^{17} - 985330 \nu^{15} - 552034 \nu^{14} + \cdots + 4691628 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 20506 \nu^{19} - 11404 \nu^{18} - 2229 \nu^{17} + 985330 \nu^{15} - 552034 \nu^{14} + \cdots - 4691628 \nu ) / 930792 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 20506 \nu^{19} + 2229 \nu^{17} + 590 \nu^{16} + 985330 \nu^{15} + 109076 \nu^{13} + 19650 \nu^{12} + \cdots - 2761944 ) / 930792 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 20506 \nu^{19} - 2229 \nu^{17} + 590 \nu^{16} - 985330 \nu^{15} - 109076 \nu^{13} + \cdots - 2761944 ) / 930792 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{19} + \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{17} - \beta_{15} - \beta_{12} - \beta_{10} - \beta_{4} - \beta_{3} - 8\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{19} + 2 \beta_{18} + 2 \beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{13} - \beta_{11} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{19} + 8 \beta_{18} - 7 \beta_{14} + 7 \beta_{13} + 6 \beta_{11} + \beta_{7} + \beta_{6} + \cdots - 35 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} - 6 \beta_{18} + 6 \beta_{17} - 2 \beta_{16} + 6 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17 \beta_{17} + 6 \beta_{16} + 15 \beta_{15} + 15 \beta_{12} + 11 \beta_{10} - 3 \beta_{9} + \cdots + 59 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{19} - 35 \beta_{18} - 35 \beta_{17} + 9 \beta_{16} - 36 \beta_{15} + 16 \beta_{14} - 36 \beta_{13} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 41 \beta_{19} - 102 \beta_{18} + 92 \beta_{14} - 92 \beta_{13} - 61 \beta_{11} - 21 \beta_{7} + \cdots + 319 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 44 \beta_{19} + 203 \beta_{18} - 203 \beta_{17} + 44 \beta_{16} - 214 \beta_{15} - 104 \beta_{14} + \cdots - 158 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 600 \beta_{17} - 256 \beta_{16} - 550 \beta_{15} - 550 \beta_{12} - 344 \beta_{10} + \cdots - 1784 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 229 \beta_{19} + 1177 \beta_{18} + 1177 \beta_{17} - 229 \beta_{16} + 1261 \beta_{15} - 635 \beta_{14} + \cdots - 936 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1540 \beta_{19} + 3504 \beta_{18} - 3244 \beta_{14} + 3244 \beta_{13} + 1964 \beta_{11} + 818 \beta_{7} + \cdots - 10154 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1245 \beta_{19} - 6828 \beta_{18} + 6828 \beta_{17} - 1245 \beta_{16} + 7386 \beta_{15} + 3780 \beta_{14} + \cdots + 5487 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20404 \beta_{17} + 9106 \beta_{16} + 18998 \beta_{15} + 18998 \beta_{12} + 11298 \beta_{10} + \cdots + 58336 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6960 \beta_{19} - 39630 \beta_{18} - 39630 \beta_{17} + 6960 \beta_{16} - 43104 \beta_{15} + \cdots + 32016 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 53388 \beta_{19} - 118662 \beta_{18} + 110838 \beta_{14} - 110838 \beta_{13} - 65274 \beta_{11} + \cdots + 336834 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 39564 \beta_{19} + 230088 \beta_{18} - 230088 \beta_{17} + 39564 \beta_{16} - 251022 \beta_{15} + \cdots - 186396 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 689676 \beta_{17} - 311646 \beta_{16} - 645336 \beta_{15} - 645336 \beta_{12} - 378030 \beta_{10} + \cdots - 1950174 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 227070 \beta_{19} + 1336122 \beta_{18} + 1336122 \beta_{17} - 227070 \beta_{16} + 1460136 \beta_{15} + \cdots - 1083990 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(1\) \(-\beta_{5}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.25666 + 1.25666i
−0.762807 − 0.762807i
0.923198 + 0.923198i
−1.70417 − 1.70417i
0.287118 + 0.287118i
−1.25666 + 1.25666i
0.762807 − 0.762807i
−0.923198 + 0.923198i
1.70417 − 1.70417i
−0.287118 + 0.287118i
−1.25666 − 1.25666i
0.762807 + 0.762807i
−0.923198 − 0.923198i
1.70417 + 1.70417i
−0.287118 − 0.287118i
1.25666 − 1.25666i
−0.762807 + 0.762807i
0.923198 − 0.923198i
−1.70417 + 1.70417i
0.287118 − 0.287118i
0 0 0 − 1.00000i 0 −3.58237 − 2.06828i 0 0 0
361.2 0 0 0 − 1.00000i 0 −2.67172 − 1.54252i 0 0 0
361.3 0 0 0 − 1.00000i 0 1.01451 + 0.585730i 0 0 0
361.4 0 0 0 − 1.00000i 0 1.07697 + 0.621786i 0 0 0
361.5 0 0 0 − 1.00000i 0 2.66261 + 1.53726i 0 0 0
361.6 0 0 0 1.00000i 0 −3.58237 − 2.06828i 0 0 0
361.7 0 0 0 1.00000i 0 −2.67172 − 1.54252i 0 0 0
361.8 0 0 0 1.00000i 0 1.01451 + 0.585730i 0 0 0
361.9 0 0 0 1.00000i 0 1.07697 + 0.621786i 0 0 0
361.10 0 0 0 1.00000i 0 2.66261 + 1.53726i 0 0 0
901.1 0 0 0 − 1.00000i 0 −3.58237 + 2.06828i 0 0 0
901.2 0 0 0 − 1.00000i 0 −2.67172 + 1.54252i 0 0 0
901.3 0 0 0 − 1.00000i 0 1.01451 − 0.585730i 0 0 0
901.4 0 0 0 − 1.00000i 0 1.07697 − 0.621786i 0 0 0
901.5 0 0 0 − 1.00000i 0 2.66261 − 1.53726i 0 0 0
901.6 0 0 0 1.00000i 0 −3.58237 + 2.06828i 0 0 0
901.7 0 0 0 1.00000i 0 −2.67172 + 1.54252i 0 0 0
901.8 0 0 0 1.00000i 0 1.01451 − 0.585730i 0 0 0
901.9 0 0 0 1.00000i 0 1.07697 − 0.621786i 0 0 0
901.10 0 0 0 1.00000i 0 2.66261 − 1.53726i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.dj.e ✓ 20
3.b odd 2 1 inner 2340.2.dj.e ✓ 20
13.e even 6 1 inner 2340.2.dj.e ✓ 20
39.h odd 6 1 inner 2340.2.dj.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.dj.e ✓ 20 1.a even 1 1 trivial
2340.2.dj.e ✓ 20 3.b odd 2 1 inner
2340.2.dj.e ✓ 20 13.e even 6 1 inner
2340.2.dj.e ✓ 20 39.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 3 T_{7}^{9} - 15 T_{7}^{8} - 54 T_{7}^{7} + 225 T_{7}^{6} + 423 T_{7}^{5} - 1251 T_{7}^{4} + \cdots + 3267 \) acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + 3 T^{9} + \cdots + 3267)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 2176782336 \) Copy content Toggle raw display
$13$ \( (T^{10} + T^{9} + \cdots + 371293)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 6198727824 \) Copy content Toggle raw display
$19$ \( (T^{10} + 6 T^{9} + \cdots + 432)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 108 T^{18} + \cdots + 2985984 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 410025070224 \) Copy content Toggle raw display
$31$ \( (T^{10} + 63 T^{8} + \cdots + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 6 T^{9} + \cdots + 7527168)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{10} + T^{9} + \cdots + 1666681)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 144 T^{8} + \cdots + 5391684)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 288 T^{8} + \cdots - 442368)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 241471833524496 \) Copy content Toggle raw display
$61$ \( (T^{10} - 7 T^{9} + \cdots + 644809)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 15 T^{9} + \cdots + 27)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 55788550416 \) Copy content Toggle raw display
$73$ \( (T^{10} + 195 T^{8} + \cdots + 29403)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 11 T^{4} + \cdots + 137)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + 288 T^{8} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{10} + 9 T^{9} + \cdots + 22211523)^{2} \) Copy content Toggle raw display
show more
show less