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Properties

Label 224.3.s.a
Level $224$
Weight $3$
Character orbit 224.s
Analytic conductor $6.104$
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] = [224,3,Mod(33,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.s (of order \(6\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{10} q^{5} - \beta_{13} q^{7} + (\beta_{8} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{10} q^{5} - \beta_{13} q^{7} + (\beta_{8} + \beta_1 + 1) q^{9} + ( - \beta_{12} - \beta_{6} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - 9 \beta_{13} + 9 \beta_{12} + \cdots + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} + 48 q^{17} + 56 q^{21} + 16 q^{25} + 112 q^{29} + 120 q^{33} + 8 q^{37} - 72 q^{45} - 128 q^{49} - 24 q^{53} - 528 q^{57} - 360 q^{61} - 8 q^{65} + 72 q^{73} + 32 q^{81} + 720 q^{85} + 408 q^{89} - 232 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4 \nu^{15} - 158 \nu^{13} - 2599 \nu^{11} - 22087 \nu^{9} - 102335 \nu^{7} - 245347 \nu^{5} + \cdots - 128520 ) / 257040 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1349 \nu^{15} + 1130 \nu^{14} + 51424 \nu^{13} + 38180 \nu^{12} + 794798 \nu^{11} + \cdots - 5574870 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1349 \nu^{15} - 1130 \nu^{14} + 51424 \nu^{13} - 38180 \nu^{12} + 794798 \nu^{11} + \cdots + 5574870 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3289 \nu^{15} - 17168 \nu^{14} - 108644 \nu^{13} - 624062 \nu^{12} - 1373628 \nu^{11} + \cdots + 48849570 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3289 \nu^{15} - 17168 \nu^{14} + 108644 \nu^{13} - 624062 \nu^{12} + 1373628 \nu^{11} + \cdots + 48849570 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1954 \nu^{15} - 2193 \nu^{14} + 69214 \nu^{13} - 84762 \nu^{12} + 981808 \nu^{11} + \cdots + 2607885 ) / 13558860 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1954 \nu^{15} - 4845 \nu^{14} - 69214 \nu^{13} - 169060 \nu^{12} - 981808 \nu^{11} + \cdots + 143606925 ) / 13558860 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10806 \nu^{15} + 2260 \nu^{14} + 412836 \nu^{13} + 76360 \nu^{12} + 6515547 \nu^{11} + \cdots - 255209220 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10806 \nu^{15} - 2260 \nu^{14} + 412836 \nu^{13} - 76360 \nu^{12} + 6515547 \nu^{11} + \cdots + 255209220 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9238 \nu^{15} + 14310 \nu^{14} - 334664 \nu^{13} + 512910 \nu^{12} - 4891055 \nu^{11} + \cdots - 173692260 ) / 54235440 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 977 \nu^{15} + 34607 \nu^{13} + 490904 \nu^{11} + 3304813 \nu^{9} + 10370438 \nu^{7} + \cdots - 22202004 \nu ) / 3389715 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7670 \nu^{15} + 3982 \nu^{14} + 279280 \nu^{13} + 151848 \nu^{12} + 4108875 \nu^{11} + \cdots - 2544570 ) / 27117720 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7670 \nu^{15} + 3982 \nu^{14} - 279280 \nu^{13} + 151848 \nu^{12} - 4108875 \nu^{11} + \cdots - 2544570 ) / 27117720 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9238 \nu^{15} - 334664 \nu^{13} - 4891055 \nu^{11} - 34544651 \nu^{9} - 118066943 \nu^{7} + \cdots - 61542453 \nu ) / 27117720 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26438 \nu^{15} - 2260 \nu^{14} + 966548 \nu^{13} - 76360 \nu^{12} + 14370011 \nu^{11} + \cdots + 255209220 ) / 54235440 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{11} - \beta_{9} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} - 2\beta_{3} + 2\beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{15} - 6 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} + 13 \beta_{11} + 9 \beta_{9} + 2 \beta_{8} + \cdots + 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 10 \beta_{9} + 9 \beta_{8} + 2 \beta_{7} + \cdots + 63 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20 \beta_{15} + 15 \beta_{14} - 45 \beta_{13} + 45 \beta_{12} - 92 \beta_{11} - 33 \beta_{9} + \cdots - 140 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11 \beta_{15} + 6 \beta_{14} + 13 \beta_{13} + 13 \beta_{12} + 52 \beta_{11} - 12 \beta_{10} + \cdots - 303 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 49 \beta_{15} + 1176 \beta_{13} - 1176 \beta_{12} + 2239 \beta_{11} + 201 \beta_{9} + 152 \beta_{8} + \cdots + 1400 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21 \beta_{15} - 36 \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 816 \beta_{11} + 72 \beta_{10} + \cdots - 1401 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3575 \beta_{15} - 2730 \beta_{14} - 13362 \beta_{13} + 13362 \beta_{12} - 24133 \beta_{11} + \cdots + 16920 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1577 \beta_{15} - 630 \beta_{14} - 1291 \beta_{13} - 1291 \beta_{12} + 9806 \beta_{11} + 1260 \beta_{10} + \cdots + 70056 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 40984 \beta_{15} + 27165 \beta_{14} + 64251 \beta_{13} - 64251 \beta_{12} + 112052 \beta_{11} + \cdots - 308440 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 19787 \beta_{15} + 10410 \beta_{14} + 8812 \beta_{13} + 8812 \beta_{12} - 45437 \beta_{11} + \cdots - 671586 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1304123 \beta_{15} - 810468 \beta_{14} - 925548 \beta_{13} + 925548 \beta_{12} - 1569697 \beta_{11} + \cdots + 11304592 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 644022 \beta_{15} - 363636 \beta_{14} - 133956 \beta_{13} - 133956 \beta_{12} + 522108 \beta_{11} + \cdots + 19796271 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17302151 \beta_{15} + 10397226 \beta_{14} + 1830030 \beta_{13} - 1830030 \beta_{12} + 2693803 \beta_{11} + \cdots - 159843144 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.707107 + 3.42121i
0.707107 − 2.60548i
0.707107 − 1.17406i
−0.707107 + 0.358323i
0.707107 + 0.358323i
−0.707107 − 1.17406i
−0.707107 − 2.60548i
0.707107 + 3.42121i
−0.707107 − 3.42121i
0.707107 + 2.60548i
0.707107 + 1.17406i
−0.707107 − 0.358323i
0.707107 − 0.358323i
−0.707107 + 1.17406i
−0.707107 + 2.60548i
0.707107 − 3.42121i
0 −4.19011 + 2.41916i 0 −0.0446470 − 0.0257769i 0 −6.12357 − 3.39144i 0 7.20469 − 12.4789i 0
33.2 0 −3.19104 + 1.84235i 0 −2.63938 − 1.52385i 0 0.812549 + 6.95268i 0 2.28850 − 3.96380i 0
33.3 0 −1.43792 + 0.830185i 0 7.27622 + 4.20093i 0 3.99843 − 5.74565i 0 −3.12159 + 5.40674i 0
33.4 0 −0.438854 + 0.253372i 0 −4.59219 − 2.65130i 0 5.27770 + 4.59846i 0 −4.37160 + 7.57184i 0
33.5 0 0.438854 − 0.253372i 0 −4.59219 − 2.65130i 0 −5.27770 − 4.59846i 0 −4.37160 + 7.57184i 0
33.6 0 1.43792 − 0.830185i 0 7.27622 + 4.20093i 0 −3.99843 + 5.74565i 0 −3.12159 + 5.40674i 0
33.7 0 3.19104 − 1.84235i 0 −2.63938 − 1.52385i 0 −0.812549 − 6.95268i 0 2.28850 − 3.96380i 0
33.8 0 4.19011 − 2.41916i 0 −0.0446470 − 0.0257769i 0 6.12357 + 3.39144i 0 7.20469 − 12.4789i 0
129.1 0 −4.19011 − 2.41916i 0 −0.0446470 + 0.0257769i 0 −6.12357 + 3.39144i 0 7.20469 + 12.4789i 0
129.2 0 −3.19104 − 1.84235i 0 −2.63938 + 1.52385i 0 0.812549 − 6.95268i 0 2.28850 + 3.96380i 0
129.3 0 −1.43792 − 0.830185i 0 7.27622 − 4.20093i 0 3.99843 + 5.74565i 0 −3.12159 − 5.40674i 0
129.4 0 −0.438854 − 0.253372i 0 −4.59219 + 2.65130i 0 5.27770 − 4.59846i 0 −4.37160 − 7.57184i 0
129.5 0 0.438854 + 0.253372i 0 −4.59219 + 2.65130i 0 −5.27770 + 4.59846i 0 −4.37160 − 7.57184i 0
129.6 0 1.43792 + 0.830185i 0 7.27622 − 4.20093i 0 −3.99843 − 5.74565i 0 −3.12159 − 5.40674i 0
129.7 0 3.19104 + 1.84235i 0 −2.63938 + 1.52385i 0 −0.812549 + 6.95268i 0 2.28850 + 3.96380i 0
129.8 0 4.19011 + 2.41916i 0 −0.0446470 + 0.0257769i 0 6.12357 − 3.39144i 0 7.20469 + 12.4789i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.s.a ✓ 16
4.b odd 2 1 inner 224.3.s.a ✓ 16
7.c even 3 1 1568.3.c.h 16
7.d odd 6 1 inner 224.3.s.a ✓ 16
7.d odd 6 1 1568.3.c.h 16
8.b even 2 1 448.3.s.g 16
8.d odd 2 1 448.3.s.g 16
28.f even 6 1 inner 224.3.s.a ✓ 16
28.f even 6 1 1568.3.c.h 16
28.g odd 6 1 1568.3.c.h 16
56.j odd 6 1 448.3.s.g 16
56.m even 6 1 448.3.s.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.a ✓ 16 1.a even 1 1 trivial
224.3.s.a ✓ 16 4.b odd 2 1 inner
224.3.s.a ✓ 16 7.d odd 6 1 inner
224.3.s.a ✓ 16 28.f even 6 1 inner
448.3.s.g 16 8.b even 2 1
448.3.s.g 16 8.d odd 2 1
448.3.s.g 16 56.j odd 6 1
448.3.s.g 16 56.m even 6 1
1568.3.c.h 16 7.c even 3 1
1568.3.c.h 16 7.d odd 6 1
1568.3.c.h 16 28.f even 6 1
1568.3.c.h 16 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 40 T_{3}^{14} + 1170 T_{3}^{12} - 15232 T_{3}^{10} + 145315 T_{3}^{8} - 405120 T_{3}^{6} + \cdots + 50625 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 40 T^{14} + \cdots + 50625 \) Copy content Toggle raw display
$5$ \( (T^{8} - 54 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 19775390625 \) Copy content Toggle raw display
$13$ \( (T^{8} + 1160 T^{6} + \cdots + 243360000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 24 T^{7} + \cdots + 308880625)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{4} - 28 T^{3} + \cdots + 1155216)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( (T^{8} - 4 T^{7} + \cdots + 54931640625)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10159773753600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 3616 T^{6} + \cdots + 34857216)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 26\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( (T^{8} + 12 T^{7} + \cdots + 234667580625)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 45\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} + 180 T^{7} + \cdots + 371063169)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 32\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 19561159840000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 115300885730625)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 38\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 179911788134400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 41805935994001)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 161218887840000)^{2} \) Copy content Toggle raw display
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