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Properties

Label 6336.2.m.d
Level $6336$
Weight $2$
Character orbit 6336.m
Analytic conductor $50.593$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6336,2,Mod(5345,6336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("6336.5345");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6336.m (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: \(50.5932147207\)
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: \( x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{67}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + \beta_{4} q^{7} + (\beta_{12} + \beta_{2}) q^{11} + ( - \beta_{7} - \beta_{2}) q^{13} - \beta_1 q^{17} + (\beta_{6} + 2 \beta_{5}) q^{19} + \beta_{10} q^{23} + \beta_{9} q^{25} - 3 \beta_{3} q^{29} + ( - \beta_{8} - \beta_1) q^{31} + (\beta_{13} - 2 \beta_{12}) q^{35} + (\beta_{13} - 4 \beta_{12}) q^{37} + ( - \beta_{8} - \beta_1) q^{41} + (\beta_{6} - 4 \beta_{5}) q^{43} + ( - \beta_{14} + 2 \beta_{10}) q^{47} + \beta_{9} q^{49} + (2 \beta_{6} + 3 \beta_{5}) q^{53} + (\beta_{8} - \beta_{4} - 2 \beta_1) q^{55} + 2 \beta_{2} q^{59} + ( - \beta_{7} - \beta_{2}) q^{61} + ( - \beta_{8} + 4 \beta_1) q^{65} + (2 \beta_{15} - \beta_{3}) q^{67} + (2 \beta_{14} + \beta_{10}) q^{71} + (\beta_{14} - 3 \beta_{10}) q^{73} + ( - 2 \beta_{6} + \beta_{5} + 3 \beta_{3}) q^{77} + (6 \beta_{11} + \beta_{4}) q^{79} - 3 \beta_{13} q^{83} + (2 \beta_{7} + 2 \beta_{2}) q^{85} + ( - \beta_{11} + 4 \beta_{4}) q^{89} + (2 \beta_{15} - 3 \beta_{3}) q^{91} + (3 \beta_{9} + 9) q^{95} + (\beta_{9} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 144 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 355 \nu^{14} - 10004 \nu^{12} - 98898 \nu^{10} - 452577 \nu^{8} - 1013592 \nu^{6} - 1059069 \nu^{4} + \cdots - 31001 ) / 5193 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 546 \nu^{14} - 15115 \nu^{12} - 145023 \nu^{10} - 634108 \nu^{8} - 1316949 \nu^{6} - 1189412 \nu^{4} + \cdots - 30203 ) / 6924 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1484 \nu^{15} + 41644 \nu^{13} + 409389 \nu^{11} + 1864335 \nu^{9} + 4163457 \nu^{7} + \cdots + 175579 \nu ) / 10386 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1015 \nu^{15} + 28091 \nu^{13} + 269128 \nu^{11} + 1170461 \nu^{9} + 2389964 \nu^{7} + \cdots - 76556 \nu ) / 6924 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 993 \nu^{14} - 27892 \nu^{12} - 274609 \nu^{10} - 1252161 \nu^{8} - 2786567 \nu^{6} - 2824359 \nu^{4} + \cdots - 41833 ) / 6924 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1023 \nu^{14} + 28583 \nu^{12} + 278570 \nu^{10} + 1246815 \nu^{8} + 2679334 \nu^{6} + 2534013 \nu^{4} + \cdots + 10166 ) / 6924 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{14} - 139\nu^{12} - 1344\nu^{10} - 5955\nu^{8} - 12660\nu^{6} - 11913\nu^{4} - 3725\nu^{2} - 130 ) / 18 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1564 \nu^{14} + 43679 \nu^{12} + 425337 \nu^{10} + 1902009 \nu^{8} + 4095993 \nu^{6} + 3936261 \nu^{4} + \cdots + 33626 ) / 5193 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 552 \nu^{14} + 15484 \nu^{12} + 151816 \nu^{10} + 684738 \nu^{8} + 1487528 \nu^{6} + 1433460 \nu^{4} + \cdots + 15907 ) / 1731 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11513 \nu^{15} + 321748 \nu^{13} + 3138027 \nu^{11} + 14080611 \nu^{9} + 30534933 \nu^{7} + \cdots + 746923 \nu ) / 20772 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1446 \nu^{15} + 40461 \nu^{13} + 395409 \nu^{11} + 1778902 \nu^{9} + 3867651 \nu^{7} + \cdots + 48399 \nu ) / 2308 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4805 \nu^{15} - 134236 \nu^{13} - 1308260 \nu^{11} - 5860606 \nu^{9} - 12650074 \nu^{7} + \cdots - 111674 \nu ) / 6924 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2849 \nu^{15} + 79720 \nu^{13} + 779159 \nu^{11} + 3506728 \nu^{9} + 7632103 \nu^{7} + \cdots + 149246 \nu ) / 3462 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10535 \nu^{15} + 294490 \nu^{13} + 2872611 \nu^{11} + 12881169 \nu^{9} + 27832941 \nu^{7} + \cdots + 277567 \nu ) / 6924 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15991 \nu^{15} - 447125 \nu^{13} - 4364280 \nu^{11} - 19598358 \nu^{9} - 42480270 \nu^{7} + \cdots - 467573 \nu ) / 10386 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + 2\beta_{11} + \beta_{10} - 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{7} - 2\beta_{6} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{15} + 9\beta_{14} - 12\beta_{12} - 16\beta_{11} - 7\beta_{10} - 6\beta_{4} + 14\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{9} + 14\beta_{8} + 18\beta_{7} + 24\beta_{6} - 4\beta_{5} - 12\beta_{2} + 2\beta _1 + 59 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 45 \beta_{15} - 53 \beta_{14} + 10 \beta_{13} + 100 \beta_{12} + 89 \beta_{11} + 34 \beta_{10} + \cdots - 71 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 169\beta_{9} - 193\beta_{8} - 260\beta_{7} - 298\beta_{6} + 74\beta_{5} + 210\beta_{2} - 47\beta _1 - 661 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1218 \beta_{15} + 1365 \beta_{14} - 378 \beta_{13} - 2828 \beta_{12} - 2270 \beta_{11} + \cdots + 1696 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2235 \beta_{9} + 2624 \beta_{8} + 3564 \beta_{7} + 3848 \beta_{6} - 1088 \beta_{5} - 3024 \beta_{2} + \cdots + 8291 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16278 \beta_{15} - 18033 \beta_{14} + 5592 \beta_{13} + 38496 \beta_{12} + 30008 \beta_{11} + \cdots - 21704 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 29764 \beta_{9} - 35358 \beta_{8} - 48079 \beta_{7} - 50724 \beta_{6} + 14998 \beta_{5} + 41430 \beta_{2} + \cdots - 108382 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 108702 \beta_{15} + 120066 \beta_{14} - 38665 \beta_{13} - 259028 \beta_{12} - 200111 \beta_{11} + \cdots + 142759 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 198704 \beta_{9} + 237195 \beta_{8} + 322542 \beta_{7} + 337298 \beta_{6} - 101290 \beta_{5} + \cdots + 718897 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2905344 \beta_{15} - 3206435 \beta_{14} + 1046240 \beta_{13} + 6944600 \beta_{12} + \cdots - 3795586 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5311585 \beta_{9} - 6352931 \beta_{8} - 8638253 \beta_{7} - 9003282 \beta_{6} + 2718232 \beta_{5} + \cdots - 19173931 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 38843454 \beta_{15} + 42858411 \beta_{14} - 14049796 \beta_{13} - 92962276 \beta_{12} + \cdots + 50653210 \beta_{3} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times\).

\(n\) \(1729\) \(3521\) \(4159\) \(4357\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5345.1
0.151206i
1.26301i
− 0.151206i
− 1.26301i
− 3.65718i
2.24296i
3.65718i
− 2.24296i
0.242964i
− 1.65718i
− 0.242964i
1.65718i
− 0.736993i
2.15121i
0.736993i
− 2.15121i
0 0 0 −2.93352 0 − 1.84240i 0 0 0
5345.2 0 0 0 −2.93352 0 − 1.84240i 0 0 0
5345.3 0 0 0 −2.93352 0 1.84240i 0 0 0
5345.4 0 0 0 −2.93352 0 1.84240i 0 0 0
5345.5 0 0 0 −1.18087 0 − 3.25662i 0 0 0
5345.6 0 0 0 −1.18087 0 − 3.25662i 0 0 0
5345.7 0 0 0 −1.18087 0 3.25662i 0 0 0
5345.8 0 0 0 −1.18087 0 3.25662i 0 0 0
5345.9 0 0 0 1.18087 0 − 3.25662i 0 0 0
5345.10 0 0 0 1.18087 0 − 3.25662i 0 0 0
5345.11 0 0 0 1.18087 0 3.25662i 0 0 0
5345.12 0 0 0 1.18087 0 3.25662i 0 0 0
5345.13 0 0 0 2.93352 0 − 1.84240i 0 0 0
5345.14 0 0 0 2.93352 0 − 1.84240i 0 0 0
5345.15 0 0 0 2.93352 0 1.84240i 0 0 0
5345.16 0 0 0 2.93352 0 1.84240i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5345.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
33.d even 2 1 inner
44.c even 2 1 inner
88.g even 2 1 inner
264.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6336.2.m.d yes 16
3.b odd 2 1 6336.2.m.c ✓ 16
4.b odd 2 1 6336.2.m.c ✓ 16
8.b even 2 1 inner 6336.2.m.d yes 16
8.d odd 2 1 6336.2.m.c ✓ 16
11.b odd 2 1 6336.2.m.c ✓ 16
12.b even 2 1 inner 6336.2.m.d yes 16
24.f even 2 1 inner 6336.2.m.d yes 16
24.h odd 2 1 6336.2.m.c ✓ 16
33.d even 2 1 inner 6336.2.m.d yes 16
44.c even 2 1 inner 6336.2.m.d yes 16
88.b odd 2 1 6336.2.m.c ✓ 16
88.g even 2 1 inner 6336.2.m.d yes 16
132.d odd 2 1 6336.2.m.c ✓ 16
264.m even 2 1 inner 6336.2.m.d yes 16
264.p odd 2 1 6336.2.m.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6336.2.m.c ✓ 16 3.b odd 2 1
6336.2.m.c ✓ 16 4.b odd 2 1
6336.2.m.c ✓ 16 8.d odd 2 1
6336.2.m.c ✓ 16 11.b odd 2 1
6336.2.m.c ✓ 16 24.h odd 2 1
6336.2.m.c ✓ 16 88.b odd 2 1
6336.2.m.c ✓ 16 132.d odd 2 1
6336.2.m.c ✓ 16 264.p odd 2 1
6336.2.m.d yes 16 1.a even 1 1 trivial
6336.2.m.d yes 16 8.b even 2 1 inner
6336.2.m.d yes 16 12.b even 2 1 inner
6336.2.m.d yes 16 24.f even 2 1 inner
6336.2.m.d yes 16 33.d even 2 1 inner
6336.2.m.d yes 16 44.c even 2 1 inner
6336.2.m.d yes 16 88.g even 2 1 inner
6336.2.m.d yes 16 264.m 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 \(S_{2}^{\mathrm{new}}(6336, [\chi])\):

\( T_{5}^{4} - 10T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{167}^{2} + 6T_{167} - 108 \) 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
$5$ \( (T^{4} - 10 T^{2} + 12)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 14 T^{2} + 36)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 12 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 38 T^{2} + 36)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 20 T^{2} + 48)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 48 T^{2} + 108)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 22 T^{2} + 108)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 60 T^{2} + 432)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 132 T^{2} + 3888)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 60 T^{2} + 432)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 192 T^{2} + 8748)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 142 T^{2} + 3468)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 130 T^{2} + 2028)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 56 T^{2} + 576)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 38 T^{2} + 36)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 52)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 118 T^{2} + 3468)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 264 T^{2} + 15552)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 182 T^{2} + 6084)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 180 T^{2} + 3888)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 212 T^{2} + 10404)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T - 12)^{8} \) Copy content Toggle raw display
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