↑

Properties

Label 2366.2.d.r
Level $2366$
Weight $2$
Character orbit 2366.d
Analytic conductor $18.893$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2366,2,Mod(337,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2366.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.d (of order \(2\), degree \(1\), not 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: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{2} q^{3} - q^{4} + ( - \beta_{10} + \beta_{7} - \beta_{6}) q^{5} - \beta_{6} q^{6} + \beta_{7} q^{7} + \beta_{7} q^{8} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{2} q^{3} - q^{4} + ( - \beta_{10} + \beta_{7} - \beta_{6}) q^{5} - \beta_{6} q^{6} + \beta_{7} q^{7} + \beta_{7} q^{8} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots - 4 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 12 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} - 12 q^{4} + 12 q^{9} + 4 q^{10} - 4 q^{12} + 12 q^{14} + 12 q^{16} - 8 q^{17} + 4 q^{22} + 12 q^{23} - 24 q^{25} + 40 q^{27} + 20 q^{29} - 28 q^{30} - 4 q^{35} - 12 q^{36} + 8 q^{38} - 4 q^{40} + 4 q^{42} - 52 q^{43} + 4 q^{48} - 12 q^{49} - 36 q^{51} + 36 q^{53} + 12 q^{55} - 12 q^{56} + 56 q^{61} + 4 q^{62} - 12 q^{64} + 8 q^{68} - 64 q^{69} + 96 q^{75} - 4 q^{77} + 44 q^{79} + 68 q^{81} - 12 q^{82} - 4 q^{87} - 4 q^{88} + 12 q^{90} - 12 q^{92} - 16 q^{94} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16 \nu^{10} + 80 \nu^{9} - 453 \nu^{8} + 1332 \nu^{7} - 3246 \nu^{6} + 5412 \nu^{5} - 4778 \nu^{4} + \cdots + 2977 ) / 143 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24 \nu^{10} + 120 \nu^{9} - 751 \nu^{8} + 2284 \nu^{7} - 6728 \nu^{6} + 12694 \nu^{5} - 20323 \nu^{4} + \cdots - 2041 ) / 286 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31 \nu^{10} + 155 \nu^{9} - 976 \nu^{8} + 2974 \nu^{7} - 8881 \nu^{6} + 16885 \nu^{5} - 28044 \nu^{4} + \cdots - 5252 ) / 286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{10} + 25 \nu^{9} - 157 \nu^{8} + 478 \nu^{7} - 1419 \nu^{6} + 2689 \nu^{5} - 4397 \nu^{4} + \cdots - 559 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6 \nu^{10} - 30 \nu^{9} + 185 \nu^{8} - 560 \nu^{7} + 1616 \nu^{6} - 3014 \nu^{5} + 4671 \nu^{4} + \cdots + 337 ) / 22 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 324 \nu^{11} + 1782 \nu^{10} - 10668 \nu^{9} + 34641 \nu^{8} - 98512 \nu^{7} + 195608 \nu^{6} + \cdots - 3573 ) / 7898 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4646 \nu^{11} + 25553 \nu^{10} - 167161 \nu^{9} + 560577 \nu^{8} - 1816484 \nu^{7} + \cdots + 828295 ) / 102674 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 502 \nu^{11} - 2761 \nu^{10} + 15505 \nu^{9} - 49065 \nu^{8} + 124356 \nu^{7} - 225603 \nu^{6} + \cdots + 4439 ) / 7898 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 324 \nu^{11} + 1782 \nu^{10} - 10668 \nu^{9} + 34641 \nu^{8} - 98512 \nu^{7} + 195608 \nu^{6} + \cdots - 7522 ) / 3949 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11954 \nu^{11} + 65747 \nu^{10} - 444641 \nu^{9} + 1507782 \nu^{8} - 5082768 \nu^{7} + \cdots + 3069716 ) / 102674 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17422 \nu^{11} - 95821 \nu^{10} + 585775 \nu^{9} - 1917330 \nu^{8} + 5639200 \nu^{7} + \cdots - 689598 ) / 102674 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - 2\beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 2\beta_{6} + 2\beta_{4} - 2\beta_{3} - 2\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} - 3 \beta_{10} - 7 \beta_{9} + \beta_{8} + 9 \beta_{7} + 10 \beta_{6} + 3 \beta_{4} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} - 6 \beta_{10} - 15 \beta_{9} + 2 \beta_{8} + 18 \beta_{7} + 22 \beta_{6} - 4 \beta_{5} + \cdots + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15 \beta_{11} + 29 \beta_{10} + 58 \beta_{9} - 8 \beta_{8} - 84 \beta_{7} - 56 \beta_{6} - 10 \beta_{5} + \cdots + 111 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 50 \beta_{11} + 102 \beta_{10} + 212 \beta_{9} - 29 \beta_{8} - 297 \beta_{7} - 224 \beta_{6} + \cdots - 211 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 135 \beta_{11} - 192 \beta_{10} - 389 \beta_{9} + 60 \beta_{8} + 519 \beta_{7} + 285 \beta_{6} + \cdots - 1140 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 778 \beta_{11} - 1258 \beta_{10} - 2581 \beta_{9} + 380 \beta_{8} + 3504 \beta_{7} + 2238 \beta_{6} + \cdots + 943 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 673 \beta_{11} + 649 \beta_{10} + 1423 \beta_{9} - 264 \beta_{8} - 1526 \beta_{7} - 630 \beta_{6} + \cdots + 11572 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9560 \beta_{11} + 13424 \beta_{10} + 28033 \beta_{9} - 4383 \beta_{8} - 36079 \beta_{7} - 21616 \beta_{6} + \cdots + 1476 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3186 \beta_{11} + 6937 \beta_{10} + 13572 \beta_{9} - 1759 \beta_{8} - 20504 \beta_{7} - 13811 \beta_{6} + \cdots - 111781 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.500000 + 1.69027i
0.500000 + 1.73154i
0.500000 − 0.399480i
0.500000 + 0.613147i
0.500000 − 3.15681i
0.500000 − 2.47866i
0.500000 − 1.69027i
0.500000 − 1.73154i
0.500000 + 0.399480i
0.500000 − 0.613147i
0.500000 + 3.15681i
0.500000 + 2.47866i
− 1.00000i −2.55629 −1.00000 3.48754i 2.55629i 1.00000i 1.00000i 3.53463 3.48754
337.2 − 1.00000i −0.865515 −1.00000 3.71131i 0.865515i 1.00000i 1.00000i −2.25088 3.71131
337.3 − 1.00000i −0.466545 −1.00000 − 3.38938i 0.466545i 1.00000i 1.00000i −2.78234 −3.38938
337.4 − 1.00000i 0.252878 −1.00000 − 1.14776i − 0.252878i 1.00000i 1.00000i −2.93605 −1.14776
337.5 − 1.00000i 2.29079 −1.00000 0.901839i − 2.29079i 1.00000i 1.00000i 2.24770 0.901839
337.6 − 1.00000i 3.34469 −1.00000 − 1.56356i − 3.34469i 1.00000i 1.00000i 8.18694 −1.56356
337.7 1.00000i −2.55629 −1.00000 − 3.48754i − 2.55629i − 1.00000i − 1.00000i 3.53463 3.48754
337.8 1.00000i −0.865515 −1.00000 − 3.71131i − 0.865515i − 1.00000i − 1.00000i −2.25088 3.71131
337.9 1.00000i −0.466545 −1.00000 3.38938i − 0.466545i − 1.00000i − 1.00000i −2.78234 −3.38938
337.10 1.00000i 0.252878 −1.00000 1.14776i 0.252878i − 1.00000i − 1.00000i −2.93605 −1.14776
337.11 1.00000i 2.29079 −1.00000 − 0.901839i 2.29079i − 1.00000i − 1.00000i 2.24770 0.901839
337.12 1.00000i 3.34469 −1.00000 1.56356i 3.34469i − 1.00000i − 1.00000i 8.18694 −1.56356
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.d.r 12
13.b even 2 1 inner 2366.2.d.r 12
13.c even 3 1 182.2.m.b ✓ 12
13.d odd 4 1 2366.2.a.bf 6
13.d odd 4 1 2366.2.a.bh 6
13.e even 6 1 182.2.m.b ✓ 12
39.h odd 6 1 1638.2.bj.g 12
39.i odd 6 1 1638.2.bj.g 12
52.i odd 6 1 1456.2.cc.d 12
52.j odd 6 1 1456.2.cc.d 12
91.g even 3 1 1274.2.v.e 12
91.h even 3 1 1274.2.o.d 12
91.k even 6 1 1274.2.o.d 12
91.l odd 6 1 1274.2.o.e 12
91.m odd 6 1 1274.2.v.d 12
91.n odd 6 1 1274.2.m.c 12
91.p odd 6 1 1274.2.v.d 12
91.t odd 6 1 1274.2.m.c 12
91.u even 6 1 1274.2.v.e 12
91.v odd 6 1 1274.2.o.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b ✓ 12 13.c even 3 1
182.2.m.b ✓ 12 13.e even 6 1
1274.2.m.c 12 91.n odd 6 1
1274.2.m.c 12 91.t odd 6 1
1274.2.o.d 12 91.h even 3 1
1274.2.o.d 12 91.k even 6 1
1274.2.o.e 12 91.l odd 6 1
1274.2.o.e 12 91.v odd 6 1
1274.2.v.d 12 91.m odd 6 1
1274.2.v.d 12 91.p odd 6 1
1274.2.v.e 12 91.g even 3 1
1274.2.v.e 12 91.u even 6 1
1456.2.cc.d 12 52.i odd 6 1
1456.2.cc.d 12 52.j odd 6 1
1638.2.bj.g 12 39.h odd 6 1
1638.2.bj.g 12 39.i odd 6 1
2366.2.a.bf 6 13.d odd 4 1
2366.2.a.bh 6 13.d odd 4 1
2366.2.d.r 12 1.a even 1 1 trivial
2366.2.d.r 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2366, [\chi])\):

\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 12T_{3}^{3} + 21T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{12} + 42T_{5}^{10} + 643T_{5}^{8} + 4292T_{5}^{6} + 11827T_{5}^{4} + 13306T_{5}^{2} + 5041 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} - 10 T^{4} + \cdots - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 42 T^{10} + \cdots + 5041 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 88 T^{10} + \cdots + 495616 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots - 176)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 369869824 \) Copy content Toggle raw display
$23$ \( (T^{6} - 6 T^{5} + \cdots + 3142)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 10 T^{5} + \cdots - 368)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 88 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{12} + 310 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( T^{12} + 226 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( (T^{6} + 26 T^{5} + \cdots + 2944)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 272 T^{10} + \cdots + 31719424 \) Copy content Toggle raw display
$53$ \( (T^{6} - 18 T^{5} + \cdots + 44928)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 168 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{6} - 28 T^{5} + \cdots - 283487)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 280 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 750321664 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2708994304 \) Copy content Toggle raw display
$79$ \( (T^{6} - 22 T^{5} + \cdots + 8032)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 879478336 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10303062016 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 6400000000 \) Copy content Toggle raw display
show more
show less