↑

Properties

Label 8384.2.a.ca
Level $8384$
Weight $2$
Character orbit 8384.a
Self dual yes
Analytic conductor $66.947$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8384,2,Mod(1,8384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8384 = 2^{6} \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8384.a (trivial)

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: yes
Analytic conductor: \(66.9465770546\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 22 x^{13} + 67 x^{12} + 178 x^{11} - 556 x^{10} - 679 x^{9} + 2232 x^{8} + 1220 x^{7} + \cdots + 76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 4192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + 1) q^{5} + \beta_{7} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + 1) q^{5} + \beta_{7} q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{14} - \beta_{10} + \beta_{8} + \cdots + 1) q^{11}+ \cdots + (3 \beta_{14} + \beta_{13} - \beta_{12} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 3 q^{3} + 9 q^{5} + 5 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 3 q^{3} + 9 q^{5} + 5 q^{7} + 8 q^{9} + 7 q^{11} + 7 q^{13} + 10 q^{15} - 22 q^{17} + 12 q^{19} + 16 q^{21} + 10 q^{23} + 4 q^{25} + 6 q^{27} + 18 q^{29} + 26 q^{31} - 18 q^{33} + 10 q^{35} + 18 q^{37} + 14 q^{39} - 17 q^{41} + 19 q^{43} + 21 q^{45} + 8 q^{47} + 2 q^{49} + 12 q^{51} + 37 q^{53} + 18 q^{55} - 42 q^{57} + 15 q^{59} + 17 q^{61} + 23 q^{63} - 46 q^{65} + 18 q^{67} + 12 q^{69} + 10 q^{71} - 26 q^{73} + 31 q^{75} + 44 q^{77} + 56 q^{79} - 17 q^{81} - 2 q^{83} + 12 q^{85} - 4 q^{87} - 27 q^{89} + 42 q^{91} + 38 q^{93} + 22 q^{95} - 14 q^{97} + 41 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 22 x^{13} + 67 x^{12} + 178 x^{11} - 556 x^{10} - 679 x^{9} + 2232 x^{8} + 1220 x^{7} + \cdots + 76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14286 \nu^{14} - 60957 \nu^{13} - 234684 \nu^{12} + 1246517 \nu^{11} + 918006 \nu^{10} + \cdots + 710950 ) / 7834 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 100860 \nu^{14} - 440280 \nu^{13} - 1621643 \nu^{12} + 8985293 \nu^{11} + 5758948 \nu^{10} + \cdots + 5764454 ) / 7834 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 129823 \nu^{14} + 567438 \nu^{13} + 2086027 \nu^{12} - 11582584 \nu^{11} - 7385039 \nu^{10} + \cdots - 7439542 ) / 7834 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 146753 \nu^{14} - 638280 \nu^{13} - 2367893 \nu^{12} + 13029761 \nu^{11} + 8551018 \nu^{10} + \cdots + 8301487 ) / 3917 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 404794 \nu^{14} - 1761077 \nu^{13} - 6530402 \nu^{12} + 35950687 \nu^{11} + 23568284 \nu^{10} + \cdots + 22885570 ) / 7834 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 501694 \nu^{14} + 2179352 \nu^{13} + 8104407 \nu^{12} - 44492863 \nu^{11} - 29429810 \nu^{10} + \cdots - 28173518 ) / 7834 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 618909 \nu^{14} - 2693822 \nu^{13} - 9979762 \nu^{12} + 54989753 \nu^{11} + 35931887 \nu^{10} + \cdots + 35075730 ) / 7834 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 631081 \nu^{14} + 2743477 \nu^{13} + 10188378 \nu^{12} - 56008930 \nu^{11} - 36894093 \nu^{10} + \cdots - 35534016 ) / 7834 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 854051 \nu^{14} - 3709270 \nu^{13} - 13797903 \nu^{12} + 75724570 \nu^{11} + 50133023 \nu^{10} + \cdots + 47975722 ) / 7834 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 863717 \nu^{14} + 3758264 \nu^{13} + 13931311 \nu^{12} - 76721126 \nu^{11} - 50224095 \nu^{10} + \cdots - 48907620 ) / 7834 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 878559 \nu^{14} + 3819996 \nu^{13} + 14180296 \nu^{12} - 77984441 \nu^{11} - 51285985 \nu^{10} + \cdots - 49593594 ) / 7834 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 900557 \nu^{14} + 3915261 \nu^{13} + 14537341 \nu^{12} - 79929965 \nu^{11} - 52613265 \nu^{10} + \cdots - 50777190 ) / 7834 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 3 \beta_{6} + \beta_{5} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{14} - 11 \beta_{13} + 11 \beta_{12} - 14 \beta_{11} + 12 \beta_{10} - 15 \beta_{8} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{14} + \beta_{13} + 12 \beta_{12} - 2 \beta_{11} - 31 \beta_{10} + 26 \beta_{9} + 40 \beta_{8} + \cdots + 201 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 99 \beta_{14} - 106 \beta_{13} + 102 \beta_{12} - 152 \beta_{11} + 114 \beta_{10} - 3 \beta_{9} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 47 \beta_{14} + 15 \beta_{13} + 112 \beta_{12} - 35 \beta_{11} - 364 \beta_{10} + 263 \beta_{9} + \cdots + 1685 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 840 \beta_{14} - 992 \beta_{13} + 908 \beta_{12} - 1519 \beta_{11} + 1000 \beta_{10} - 66 \beta_{9} + \cdots + 591 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 727 \beta_{14} + 172 \beta_{13} + 951 \beta_{12} - 440 \beta_{11} - 3883 \beta_{10} + 2453 \beta_{9} + \cdots + 14719 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6955 \beta_{14} - 9203 \beta_{13} + 7984 \beta_{12} - 14654 \beta_{11} + 8495 \beta_{10} + \cdots + 4359 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9379 \beta_{14} + 1804 \beta_{13} + 7689 \beta_{12} - 4857 \beta_{11} - 39606 \beta_{10} + 22159 \beta_{9} + \cdots + 131393 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 56927 \beta_{14} - 85126 \beta_{13} + 69982 \beta_{12} - 138977 \beta_{11} + 71302 \beta_{10} + \cdots + 31927 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 109480 \beta_{14} + 18199 \beta_{13} + 60276 \beta_{12} - 50180 \beta_{11} - 393903 \beta_{10} + \cdots + 1186740 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.04466
−2.02841
−2.00631
−1.34743
−1.34738
−1.17906
−0.0956383
0.621761
0.851429
1.11387
1.25063
2.15026
2.21264
2.81709
3.03121
0 −3.04466 0 −0.967755 0 −2.39846 0 6.26998 0
1.2 0 −2.02841 0 1.48227 0 0.0562566 0 1.11445 0
1.3 0 −2.00631 0 3.27570 0 4.67675 0 1.02529 0
1.4 0 −1.34743 0 −1.11255 0 −1.66943 0 −1.18442 0
1.5 0 −1.34738 0 −1.21228 0 1.27314 0 −1.18457 0
1.6 0 −1.17906 0 0.321884 0 −2.01956 0 −1.60982 0
1.7 0 −0.0956383 0 2.59248 0 1.45395 0 −2.99085 0
1.8 0 0.621761 0 3.95184 0 −4.68160 0 −2.61341 0
1.9 0 0.851429 0 −3.12575 0 −1.75203 0 −2.27507 0
1.10 0 1.11387 0 −2.21428 0 3.74305 0 −1.75929 0
1.11 0 1.25063 0 −1.75728 0 2.03246 0 −1.43592 0
1.12 0 2.15026 0 4.06328 0 1.98190 0 1.62362 0
1.13 0 2.21264 0 1.01908 0 −2.61171 0 1.89579 0
1.14 0 2.81709 0 0.607916 0 1.27138 0 4.93599 0
1.15 0 3.03121 0 2.07543 0 3.64390 0 6.18825 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(131\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8384.2.a.ca 15
4.b odd 2 1 8384.2.a.bz 15
8.b even 2 1 4192.2.a.n ✓ 15
8.d odd 2 1 4192.2.a.o yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4192.2.a.n ✓ 15 8.b even 2 1
4192.2.a.o yes 15 8.d odd 2 1
8384.2.a.bz 15 4.b odd 2 1
8384.2.a.ca 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8384))\):

\( T_{3}^{15} - 3 T_{3}^{14} - 22 T_{3}^{13} + 67 T_{3}^{12} + 178 T_{3}^{11} - 556 T_{3}^{10} - 679 T_{3}^{9} + \cdots + 76 \) Copy content Toggle raw display
\( T_{5}^{15} - 9 T_{5}^{14} + T_{5}^{13} + 186 T_{5}^{12} - 323 T_{5}^{11} - 1331 T_{5}^{10} + 3167 T_{5}^{9} + \cdots - 1328 \) Copy content Toggle raw display
\( T_{7}^{15} - 5 T_{7}^{14} - 41 T_{7}^{13} + 222 T_{7}^{12} + 540 T_{7}^{11} - 3433 T_{7}^{10} + \cdots - 5893 \) Copy content Toggle raw display
\( T_{11}^{15} - 7 T_{11}^{14} - 55 T_{11}^{13} + 466 T_{11}^{12} + 649 T_{11}^{11} - 10161 T_{11}^{10} + \cdots + 45296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 3 T^{14} + \cdots + 76 \) Copy content Toggle raw display
$5$ \( T^{15} - 9 T^{14} + \cdots - 1328 \) Copy content Toggle raw display
$7$ \( T^{15} - 5 T^{14} + \cdots - 5893 \) Copy content Toggle raw display
$11$ \( T^{15} - 7 T^{14} + \cdots + 45296 \) Copy content Toggle raw display
$13$ \( T^{15} - 7 T^{14} + \cdots + 3650156 \) Copy content Toggle raw display
$17$ \( T^{15} + 22 T^{14} + \cdots - 131072 \) Copy content Toggle raw display
$19$ \( T^{15} - 12 T^{14} + \cdots - 10805248 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 1198727168 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 155844608 \) Copy content Toggle raw display
$31$ \( T^{15} - 26 T^{14} + \cdots + 5373952 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 77851990016 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 1259210597 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 1133885296 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 10309206016 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 74059510016 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 2112407732 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 203004143872 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 915193856 \) Copy content Toggle raw display
$71$ \( T^{15} - 10 T^{14} + \cdots + 82313216 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 670661378048 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 338967298048 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 2311498891264 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 912103223488 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 1453505773568 \) Copy content Toggle raw display
show more
show less