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Properties

Label 224.4.i.e
Level $224$
Weight $4$
Character orbit 224.i
Analytic conductor $13.216$
Analytic rank $0$
Dimension $12$
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] = [224,4,Mod(65,224)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.i (of order \(3\), 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: \(13.2164278413\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 4 x^{10} - 116 x^{9} - 217 x^{8} - 2018 x^{7} + 4474 x^{6} - 105024 x^{5} + \cdots + 292052964 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{7} - \beta_{2} + 1) q^{3} + (2 \beta_{2} - \beta_1) q^{5} - \beta_{9} q^{7} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{7} - \beta_{2} + 1) q^{3} + (2 \beta_{2} - \beta_1) q^{5} - \beta_{9} q^{7} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{9}+ \cdots + ( - 14 \beta_{11} + 18 \beta_{10} + \cdots - 990) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 10 q^{5} - 4 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 10 q^{5} - 4 q^{7} - 40 q^{9} + 42 q^{11} - 16 q^{13} + 52 q^{15} - 70 q^{17} + 158 q^{19} + 178 q^{21} - 158 q^{23} - 72 q^{25} - 492 q^{27} - 112 q^{29} + 2 q^{31} - 262 q^{33} + 418 q^{35} + 102 q^{37} - 280 q^{39} + 96 q^{41} - 368 q^{43} + 392 q^{45} + 766 q^{47} + 132 q^{49} - 394 q^{51} - 562 q^{53} - 372 q^{55} + 1316 q^{57} + 854 q^{59} - 106 q^{61} + 1864 q^{63} - 488 q^{65} - 906 q^{67} - 2788 q^{69} - 2976 q^{71} - 202 q^{73} + 2248 q^{75} + 590 q^{77} - 942 q^{79} - 674 q^{81} + 1232 q^{83} + 1996 q^{85} + 1656 q^{87} - 858 q^{89} + 4192 q^{91} + 1362 q^{93} - 2386 q^{95} + 3392 q^{97} - 11504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 4 x^{10} - 116 x^{9} - 217 x^{8} - 2018 x^{7} + 4474 x^{6} - 105024 x^{5} + \cdots + 292052964 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 53\!\cdots\!25 \nu^{11} + \cdots + 18\!\cdots\!88 ) / 21\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!67 \nu^{11} + \cdots + 28\!\cdots\!24 ) / 45\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41\!\cdots\!69 \nu^{11} + \cdots + 17\!\cdots\!96 ) / 77\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!26 \nu^{11} + \cdots + 47\!\cdots\!94 ) / 35\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 84\!\cdots\!52 \nu^{11} + \cdots - 65\!\cdots\!60 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 72\!\cdots\!61 \nu^{11} + \cdots + 78\!\cdots\!21 ) / 74\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!27 \nu^{11} + \cdots + 19\!\cdots\!92 ) / 14\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47\!\cdots\!25 \nu^{11} + \cdots + 25\!\cdots\!72 ) / 21\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!14 \nu^{11} + \cdots + 10\!\cdots\!06 ) / 71\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!27 \nu^{11} + \cdots + 22\!\cdots\!60 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!46 \nu^{11} + \cdots + 32\!\cdots\!12 ) / 70\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + 5 \beta_{4} + 4 \beta_{3} + \cdots + 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7 \beta_{11} + 4 \beta_{10} - 3 \beta_{9} - 22 \beta_{8} + 39 \beta_{7} - 48 \beta_{6} + 11 \beta_{5} + \cdots + 33 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 37 \beta_{11} - 24 \beta_{10} - \beta_{9} + 110 \beta_{8} - 113 \beta_{7} - 190 \beta_{6} + 25 \beta_{5} + \cdots + 1463 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 225 \beta_{11} + 639 \beta_{10} - 70 \beta_{9} + 930 \beta_{8} - 1223 \beta_{7} - 802 \beta_{6} + \cdots + 8900 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 937 \beta_{11} - 1200 \beta_{10} + 591 \beta_{9} + 4776 \beta_{8} - 2365 \beta_{7} - 4274 \beta_{6} + \cdots + 52779 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5427 \beta_{11} + 10841 \beta_{10} - 8020 \beta_{9} + 7550 \beta_{8} - 30069 \beta_{7} - 10338 \beta_{6} + \cdots + 484562 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 92807 \beta_{11} - 39628 \beta_{10} + 45037 \beta_{9} + 5324 \beta_{8} - 150491 \beta_{7} + \cdots + 906433 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9713 \beta_{11} + 111209 \beta_{10} - 482550 \beta_{9} + 532914 \beta_{8} - 518951 \beta_{7} + \cdots + 15819278 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4853049 \beta_{11} - 110402 \beta_{10} - 1489363 \beta_{9} + 3503536 \beta_{8} - 7773 \beta_{7} + \cdots + 19040297 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20503613 \beta_{11} - 10804051 \beta_{10} - 22879052 \beta_{9} + 37563118 \beta_{8} - 19373893 \beta_{7} + \cdots + 393043664 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.39293 + 3.69208i
−1.98748 − 4.22573i
−5.26741 − 2.41510i
6.27880 + 0.743762i
3.30710 − 4.23645i
0.0619223 + 4.70938i
−1.39293 − 3.69208i
−1.98748 + 4.22573i
−5.26741 + 2.41510i
6.27880 − 0.743762i
3.30710 + 4.23645i
0.0619223 − 4.70938i
0 −3.37895 − 5.85251i 0 −0.506132 + 0.876646i 0 −3.19229 − 18.2431i 0 −9.33457 + 16.1680i 0
65.2 0 −2.78574 − 4.82504i 0 2.27739 − 3.94456i 0 3.15583 + 18.2494i 0 −2.02069 + 3.49995i 0
65.3 0 0.772551 + 1.33810i 0 −2.73994 + 4.74571i 0 18.3748 + 2.31660i 0 12.3063 − 21.3152i 0
65.4 0 1.14804 + 1.98846i 0 9.99683 − 17.3150i 0 −14.3223 + 11.7419i 0 10.8640 − 18.8170i 0
65.5 0 2.36327 + 4.09330i 0 −8.51409 + 14.7468i 0 −18.5160 − 0.399106i 0 2.32992 − 4.03554i 0
65.6 0 4.88083 + 8.45385i 0 4.48594 − 7.76988i 0 12.4999 − 13.6657i 0 −34.1450 + 59.1409i 0
193.1 0 −3.37895 + 5.85251i 0 −0.506132 − 0.876646i 0 −3.19229 + 18.2431i 0 −9.33457 − 16.1680i 0
193.2 0 −2.78574 + 4.82504i 0 2.27739 + 3.94456i 0 3.15583 − 18.2494i 0 −2.02069 − 3.49995i 0
193.3 0 0.772551 − 1.33810i 0 −2.73994 − 4.74571i 0 18.3748 − 2.31660i 0 12.3063 + 21.3152i 0
193.4 0 1.14804 − 1.98846i 0 9.99683 + 17.3150i 0 −14.3223 − 11.7419i 0 10.8640 + 18.8170i 0
193.5 0 2.36327 − 4.09330i 0 −8.51409 − 14.7468i 0 −18.5160 + 0.399106i 0 2.32992 + 4.03554i 0
193.6 0 4.88083 − 8.45385i 0 4.48594 + 7.76988i 0 12.4999 + 13.6657i 0 −34.1450 − 59.1409i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.i.e yes 12
4.b odd 2 1 224.4.i.c ✓ 12
7.c even 3 1 inner 224.4.i.e yes 12
7.c even 3 1 1568.4.a.be 6
7.d odd 6 1 1568.4.a.bj 6
8.b even 2 1 448.4.i.o 12
8.d odd 2 1 448.4.i.q 12
28.f even 6 1 1568.4.a.bf 6
28.g odd 6 1 224.4.i.c ✓ 12
28.g odd 6 1 1568.4.a.bi 6
56.k odd 6 1 448.4.i.q 12
56.p even 6 1 448.4.i.o 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.i.c ✓ 12 4.b odd 2 1
224.4.i.c ✓ 12 28.g odd 6 1
224.4.i.e yes 12 1.a even 1 1 trivial
224.4.i.e yes 12 7.c even 3 1 inner
448.4.i.o 12 8.b even 2 1
448.4.i.o 12 56.p even 6 1
448.4.i.q 12 8.d odd 2 1
448.4.i.q 12 56.k odd 6 1
1568.4.a.be 6 7.c even 3 1
1568.4.a.bf 6 28.f even 6 1
1568.4.a.bi 6 28.g odd 6 1
1568.4.a.bj 6 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 6 T_{3}^{11} + 119 T_{3}^{10} - 262 T_{3}^{9} + 7858 T_{3}^{8} - 22398 T_{3}^{7} + \cdots + 37982569 \) acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots + 37982569 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 5955980625 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 38\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{6} + 8 T^{5} + \cdots - 534809600)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 39\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{6} + 56 T^{5} + \cdots - 1723104256)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{6} - 48 T^{5} + \cdots + 25630880768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 81331445805056)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 84\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 84\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 23\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 97\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 983066668723200)^{2} \) Copy content Toggle raw display
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