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Properties

Label 2340.2.bp.h
Level $2340$
Weight $2$
Character orbit 2340.bp
Analytic conductor $18.685$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2340,2,Mod(1477,2340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2340, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2340.1477");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.bp (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: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 16x^{14} + 184x^{12} - 952x^{10} + 3559x^{8} - 6400x^{6} + 8200x^{4} - 2500x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{5} + (\beta_{4} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{5} + (\beta_{4} + 1) q^{7} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{7}) q^{11}+ \cdots + ( - 2 \beta_{10} + \beta_{4} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{7} + 8 q^{13} + 24 q^{25} - 32 q^{31} - 16 q^{37} + 16 q^{43} - 48 q^{49} + 8 q^{55} - 16 q^{61} - 16 q^{85} + 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 16x^{14} + 184x^{12} - 952x^{10} + 3559x^{8} - 6400x^{6} + 8200x^{4} - 2500x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 35491 \nu^{14} - 363396 \nu^{12} + 2507859 \nu^{10} + 13474208 \nu^{8} - 158354076 \nu^{6} + \cdots - 753159375 ) / 762132375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 138859 \nu^{14} - 1810009 \nu^{12} + 19266396 \nu^{10} - 59090128 \nu^{8} + 137416111 \nu^{6} + \cdots + 1582232750 ) / 762132375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 139996 \nu^{14} + 2221671 \nu^{12} - 26648649 \nu^{10} + 147956932 \nu^{8} + \cdots + 1507898625 ) / 762132375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 608 \nu^{14} + 8953 \nu^{12} - 98572 \nu^{10} + 426816 \nu^{8} - 1338722 \nu^{6} + \cdots - 1758750 ) / 1054125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 427606 \nu^{15} + 5430176 \nu^{13} - 57187959 \nu^{11} + 166947382 \nu^{9} + \cdots - 4755565375 \nu ) / 762132375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 556 \nu^{14} + 9121 \nu^{12} - 105204 \nu^{10} + 559012 \nu^{8} - 2064454 \nu^{6} + \cdots + 778750 ) / 813375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89611 \nu^{15} + 1709795 \nu^{13} - 20766378 \nu^{11} + 133587643 \nu^{9} + \cdots + 1015025450 \nu ) / 152426475 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 461964 \nu^{15} - 7122169 \nu^{13} + 79976796 \nu^{11} - 378494808 \nu^{9} + \cdots + 2954721500 \nu ) / 762132375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55008 \nu^{14} - 859376 \nu^{12} + 9821440 \nu^{10} - 49003198 \nu^{8} + 181205568 \nu^{6} + \cdots - 75362425 ) / 50808825 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 182924 \nu^{14} + 2712091 \nu^{12} - 30446913 \nu^{10} + 138348596 \nu^{8} - 487892095 \nu^{6} + \cdots - 24935825 ) / 152426475 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 267525 \nu^{15} + 4220264 \nu^{13} - 48199329 \nu^{11} + 242676081 \nu^{9} + \cdots - 222178975 \nu ) / 152426475 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1687511 \nu^{15} - 27136706 \nu^{13} + 311553204 \nu^{11} - 1615943192 \nu^{9} + \cdots - 3933546500 \nu ) / 762132375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 439301 \nu^{15} + 6928048 \nu^{13} - 79209951 \nu^{11} + 399619370 \nu^{9} + \cdots + 540702175 \nu ) / 152426475 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 88391 \nu^{15} + 1374933 \nu^{13} - 15713520 \nu^{11} + 78083489 \nu^{9} - 289914444 \nu^{7} + \cdots + 39283800 \nu ) / 30485295 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 458934 \nu^{15} - 7082714 \nu^{13} + 80332371 \nu^{11} - 389916768 \nu^{9} + \cdots + 280723825 \nu ) / 152426475 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 4\beta_{9} + 3\beta_{6} + \beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{15} - 7\beta_{14} + 7\beta_{13} + 9\beta_{12} + 7\beta_{11} + 9\beta_{8} + 2\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22\beta_{10} + 34\beta_{9} + 27\beta_{6} - 17\beta_{4} + 8\beta_{3} + 2\beta_{2} - 14\beta _1 - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 25 \beta_{15} - 61 \beta_{14} + 94 \beta_{13} + 78 \beta_{12} - 39 \beta_{11} + 5 \beta_{8} + \cdots + 58 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 147\beta_{10} - 228\beta_{4} - 147\beta_{3} + 132\beta_{2} - 132\beta _1 - 470 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 261 \beta_{15} + 48 \beta_{14} + 408 \beta_{13} - 18 \beta_{12} - 889 \beta_{11} - 757 \beta_{8} + \cdots + 514 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 580 \beta_{10} - 2983 \beta_{9} - 2532 \beta_{6} - 268 \beta_{4} - 2024 \beta_{3} + 1444 \beta_{2} + \cdots - 1255 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5152 \beta_{15} + 5827 \beta_{14} - 4411 \beta_{13} - 7299 \beta_{12} - 4411 \beta_{11} + \cdots - 2576 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19198 \beta_{10} - 28276 \beta_{9} - 24345 \beta_{6} + 19571 \beta_{4} - 5303 \beta_{3} + \cdots + 28276 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 24874 \beta_{15} + 49861 \beta_{14} - 80038 \beta_{13} - 69432 \beta_{12} + 38769 \beta_{11} + \cdots - 44185 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -66351\beta_{10} + 105285\beta_{4} + 66351\beta_{3} - 49488\beta_{2} + 49488\beta _1 + 185176 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 237987 \beta_{15} - 55797 \beta_{14} - 370689 \beta_{13} - 6309 \beta_{12} + 760126 \beta_{11} + \cdots - 416854 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 466342 \beta_{10} + 2548768 \beta_{9} + 2216091 \beta_{6} + 207247 \beta_{4} + 1729859 \beta_{3} + \cdots + 954418 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4535878 \beta_{15} - 5023954 \beta_{14} + 3688699 \beta_{13} + 6215733 \beta_{12} + \cdots + 2267939 \beta_{7} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(-\beta_{6}\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1477.1
−2.66925 + 1.54109i
1.21376 − 0.700763i
1.77869 + 1.02692i
0.488060 + 0.281782i
−0.488060 − 0.281782i
−1.77869 − 1.02692i
−1.21376 + 0.700763i
2.66925 − 1.54109i
−2.66925 − 1.54109i
1.21376 + 0.700763i
1.77869 − 1.02692i
0.488060 − 0.281782i
−0.488060 + 0.281782i
−1.77869 + 1.02692i
−1.21376 − 0.700763i
2.66925 + 1.54109i
0 0 0 −2.22422 + 0.229890i 0 2.73205 0 0 0
1477.2 0 0 0 −2.03738 − 0.921452i 0 2.73205 0 0 0
1477.3 0 0 0 −1.64926 + 1.50995i 0 −0.732051 0 0 0
1477.4 0 0 0 −1.08714 − 1.95400i 0 −0.732051 0 0 0
1477.5 0 0 0 1.08714 + 1.95400i 0 −0.732051 0 0 0
1477.6 0 0 0 1.64926 − 1.50995i 0 −0.732051 0 0 0
1477.7 0 0 0 2.03738 + 0.921452i 0 2.73205 0 0 0
1477.8 0 0 0 2.22422 − 0.229890i 0 2.73205 0 0 0
1513.1 0 0 0 −2.22422 − 0.229890i 0 2.73205 0 0 0
1513.2 0 0 0 −2.03738 + 0.921452i 0 2.73205 0 0 0
1513.3 0 0 0 −1.64926 − 1.50995i 0 −0.732051 0 0 0
1513.4 0 0 0 −1.08714 + 1.95400i 0 −0.732051 0 0 0
1513.5 0 0 0 1.08714 − 1.95400i 0 −0.732051 0 0 0
1513.6 0 0 0 1.64926 + 1.50995i 0 −0.732051 0 0 0
1513.7 0 0 0 2.03738 − 0.921452i 0 2.73205 0 0 0
1513.8 0 0 0 2.22422 + 0.229890i 0 2.73205 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1477.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
65.f even 4 1 inner
195.u odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.bp.h yes 16
3.b odd 2 1 inner 2340.2.bp.h yes 16
5.c odd 4 1 2340.2.u.h ✓ 16
13.d odd 4 1 2340.2.u.h ✓ 16
15.e even 4 1 2340.2.u.h ✓ 16
39.f even 4 1 2340.2.u.h ✓ 16
65.f even 4 1 inner 2340.2.bp.h yes 16
195.u odd 4 1 inner 2340.2.bp.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.u.h ✓ 16 5.c odd 4 1
2340.2.u.h ✓ 16 13.d odd 4 1
2340.2.u.h ✓ 16 15.e even 4 1
2340.2.u.h ✓ 16 39.f even 4 1
2340.2.bp.h yes 16 1.a even 1 1 trivial
2340.2.bp.h yes 16 3.b odd 2 1 inner
2340.2.bp.h yes 16 65.f even 4 1 inner
2340.2.bp.h yes 16 195.u odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\):

\( T_{7}^{2} - 2T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{16} + 1768T_{11}^{12} + 828504T_{11}^{8} + 47888800T_{11}^{4} + 146410000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 12 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 2)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 146410000 \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 4096 T^{12} + \cdots + 40960000 \) Copy content Toggle raw display
$19$ \( (T^{8} - 48 T^{5} + \cdots + 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 1312 T^{12} + \cdots + 2560000 \) Copy content Toggle raw display
$29$ \( (T^{8} + 160 T^{6} + \cdots + 774400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 16 T^{7} + \cdots + 327184)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 292)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 283982410000 \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots + 524176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 220 T^{6} + \cdots + 1464100)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 73116160000 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 106174476810000 \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 484)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 432 T^{6} + \cdots + 7420176)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 8157307210000 \) Copy content Toggle raw display
$73$ \( (T^{8} + 552 T^{6} + \cdots + 98010000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 184 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 252 T^{6} + \cdots + 656100)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 121173610000 \) Copy content Toggle raw display
$97$ \( (T^{8} + 232 T^{6} + \cdots + 327184)^{2} \) Copy content Toggle raw display
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