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Properties

Label 3360.2.d.c
Level $3360$
Weight $2$
Character orbit 3360.d
Analytic conductor $26.830$
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] = [3360,2,Mod(2911,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 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("3360.2911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.d (of order \(2\), degree \(1\), 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: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 222x^{12} + 1062x^{10} + 2899x^{8} + 4618x^{6} + 4143x^{4} + 1854x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{3} q^{5} + (\beta_{13} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_{3} q^{5} + (\beta_{13} - 1) q^{7} + q^{9} + \beta_{12} q^{11} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{13}+ \cdots + \beta_{12} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} - 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{3} - 8 q^{7} + 16 q^{9} - 8 q^{21} - 16 q^{25} + 16 q^{27} - 8 q^{29} + 8 q^{31} - 8 q^{37} - 16 q^{47} + 16 q^{49} + 16 q^{53} + 8 q^{55} - 8 q^{63} - 8 q^{65} - 16 q^{75} - 32 q^{77} + 16 q^{81} - 40 q^{83} - 8 q^{85} - 8 q^{87} + 24 q^{91} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 222x^{12} + 1062x^{10} + 2899x^{8} + 4618x^{6} + 4143x^{4} + 1854x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 6\nu^{12} - 147\nu^{10} - 1599\nu^{8} - 6019\nu^{6} - 10058\nu^{4} - 6996\nu^{2} - 1326 ) / 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78 \nu^{15} - 1889 \nu^{13} - 17418 \nu^{11} - 80337 \nu^{9} - 198939 \nu^{7} - 257881 \nu^{5} + \cdots - 25680 \nu ) / 986 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 206 \nu^{15} + 4281 \nu^{13} + 31894 \nu^{11} + 115089 \nu^{9} + 221715 \nu^{7} + 232769 \nu^{5} + \cdots + 21252 \nu ) / 986 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{15} - 14 \nu^{14} + 76 \nu^{13} - 316 \nu^{12} + 748 \nu^{11} - 2669 \nu^{10} + 3758 \nu^{9} + \cdots - 3389 ) / 58 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{14} - 433\nu^{12} - 3674\nu^{10} - 15352\nu^{8} - 34061\nu^{6} - 39448\nu^{4} - 21269\nu^{2} - 3748 ) / 29 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 385 \nu^{15} - 119 \nu^{14} - 8458 \nu^{13} - 2686 \nu^{12} - 68453 \nu^{11} - 22440 \nu^{10} + \cdots - 25602 ) / 986 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\nu^{14} + 1405\nu^{12} + 11093\nu^{10} + 42805\nu^{8} + 87526\nu^{6} + 93676\nu^{4} + 47047\nu^{2} + 7857 ) / 58 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -39\nu^{14} - 872\nu^{12} - 7230\nu^{10} - 29627\nu^{8} - 65090\nu^{6} - 75769\nu^{4} - 41545\nu^{2} - 7243 ) / 29 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 708 \nu^{15} + 238 \nu^{14} - 15819 \nu^{13} + 5372 \nu^{12} - 130911 \nu^{11} + 45373 \nu^{10} + \cdots + 57613 ) / 986 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 708 \nu^{15} - 238 \nu^{14} - 15819 \nu^{13} - 5372 \nu^{12} - 130911 \nu^{11} - 45373 \nu^{10} + \cdots - 57613 ) / 986 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 837 \nu^{15} - 18507 \nu^{13} - 151185 \nu^{11} - 609567 \nu^{9} - 1315156 \nu^{7} + \cdots - 138095 \nu ) / 986 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 385 \nu^{15} + 1224 \nu^{14} - 8458 \nu^{13} + 27557 \nu^{12} - 68453 \nu^{11} + 230741 \nu^{10} + \cdots + 241995 ) / 986 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 385 \nu^{15} - 1224 \nu^{14} - 8458 \nu^{13} - 27557 \nu^{12} - 68453 \nu^{11} - 230741 \nu^{10} + \cdots - 241009 ) / 986 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -22\nu^{15} - 494\nu^{13} - 4119\nu^{11} - 16972\nu^{9} - 37411\nu^{7} - 43473\nu^{5} - 23605\nu^{3} - 4000\nu ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 3 \beta_{11} - 2 \beta_{10} + \beta_{5} + 2 \beta_{4} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{14} + 10 \beta_{13} + 7 \beta_{11} - 7 \beta_{10} + 10 \beta_{9} - 2 \beta_{8} + 9 \beta_{6} + \cdots + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} - 11 \beta_{12} + 32 \beta_{11} + 25 \beta_{10} + \cdots + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 99 \beta_{14} - 92 \beta_{13} - 56 \beta_{11} + 56 \beta_{10} - 95 \beta_{9} + 27 \beta_{8} - 7 \beta_{7} + \cdots - 270 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 233 \beta_{15} + 142 \beta_{14} + 142 \beta_{13} + 104 \beta_{12} - 313 \beta_{11} - 261 \beta_{10} + \cdots - 142 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 969 \beta_{14} + 865 \beta_{13} + 500 \beta_{11} - 500 \beta_{10} + 907 \beta_{9} - 290 \beta_{8} + \cdots + 2448 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2295 \beta_{15} - 1442 \beta_{14} - 1442 \beta_{13} - 982 \beta_{12} + 3035 \beta_{11} + 2597 \beta_{10} + \cdots + 1442 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9428 \beta_{14} - 8277 \beta_{13} - 4698 \beta_{11} + 4698 \beta_{10} - 8726 \beta_{9} + 2917 \beta_{8} + \cdots - 23212 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22394 \beta_{15} + 14228 \beta_{14} + 14228 \beta_{13} + 9388 \beta_{12} - 29418 \beta_{11} + \cdots - 14228 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 91530 \beta_{14} + 79849 \beta_{13} + 45039 \beta_{11} - 45039 \beta_{10} + 84326 \beta_{9} + \cdots + 223386 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 217717 \beta_{15} - 138863 \beta_{14} - 138863 \beta_{13} - 90459 \beta_{12} + 285224 \beta_{11} + \cdots + 138863 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 443981 \beta_{14} - 386421 \beta_{13} - 217497 \beta_{11} + 217497 \beta_{10} - 408303 \beta_{9} + \cdots - 1080305 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2113542 \beta_{15} + 1349818 \beta_{14} + 1349818 \beta_{13} + 874906 \beta_{12} - 2765846 \beta_{11} + \cdots - 1349818 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2911.1
− 1.08637i
1.35301i
1.91530i
− 1.56835i
1.83821i
− 0.553274i
1.21565i
− 3.11418i
1.08637i
− 1.35301i
− 1.91530i
1.56835i
− 1.83821i
0.553274i
− 1.21565i
3.11418i
0 1.00000 0 − 1.00000i 0 −2.64358 + 0.107215i 0 1.00000 0
2911.2 0 1.00000 0 − 1.00000i 0 −2.63059 − 0.282858i 0 1.00000 0
2911.3 0 1.00000 0 − 1.00000i 0 −2.10227 − 1.60638i 0 1.00000 0
2911.4 0 1.00000 0 − 1.00000i 0 −1.15994 + 2.37793i 0 1.00000 0
2911.5 0 1.00000 0 − 1.00000i 0 −0.891429 + 2.49105i 0 1.00000 0
2911.6 0 1.00000 0 − 1.00000i 0 0.753683 − 2.53613i 0 1.00000 0
2911.7 0 1.00000 0 − 1.00000i 0 2.19554 − 1.47635i 0 1.00000 0
2911.8 0 1.00000 0 − 1.00000i 0 2.47859 + 0.925520i 0 1.00000 0
2911.9 0 1.00000 0 1.00000i 0 −2.64358 − 0.107215i 0 1.00000 0
2911.10 0 1.00000 0 1.00000i 0 −2.63059 + 0.282858i 0 1.00000 0
2911.11 0 1.00000 0 1.00000i 0 −2.10227 + 1.60638i 0 1.00000 0
2911.12 0 1.00000 0 1.00000i 0 −1.15994 − 2.37793i 0 1.00000 0
2911.13 0 1.00000 0 1.00000i 0 −0.891429 − 2.49105i 0 1.00000 0
2911.14 0 1.00000 0 1.00000i 0 0.753683 + 2.53613i 0 1.00000 0
2911.15 0 1.00000 0 1.00000i 0 2.19554 + 1.47635i 0 1.00000 0
2911.16 0 1.00000 0 1.00000i 0 2.47859 − 0.925520i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2911.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.d.c yes 16
4.b odd 2 1 3360.2.d.b ✓ 16
7.b odd 2 1 3360.2.d.b ✓ 16
28.d even 2 1 inner 3360.2.d.c yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.d.b ✓ 16 4.b odd 2 1
3360.2.d.b ✓ 16 7.b odd 2 1
3360.2.d.c yes 16 1.a even 1 1 trivial
3360.2.d.c yes 16 28.d 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 \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{16} + 92 T_{11}^{14} + 3236 T_{11}^{12} + 54208 T_{11}^{10} + 443776 T_{11}^{8} + 1706496 T_{11}^{6} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{19}^{8} - 50T_{19}^{6} + 112T_{19}^{5} + 432T_{19}^{4} - 1568T_{19}^{3} + 800T_{19}^{2} + 896T_{19} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 92 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 8667136 \) Copy content Toggle raw display
$17$ \( T^{16} + 152 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{8} - 50 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 8446345216 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{7} + \cdots - 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + \cdots + 29824)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots - 256)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 629407744 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 67649929216 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 8 T^{7} + \cdots - 30592)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 232 T^{6} + \cdots - 851456)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 56195547136 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 37953273856 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 257057112064 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 187509784576 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 35164710240256 \) Copy content Toggle raw display
$83$ \( (T^{8} + 20 T^{7} + \cdots - 6422528)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20967900381184 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 793439125504 \) Copy content Toggle raw display
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