Properties

Label 224.3.o.d
Level $224$
Weight $3$
Character orbit 224.o
Analytic conductor $6.104$
Analytic rank $0$
Dimension $12$
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(79,224)]
 
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([3, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.o (of order \(6\), degree \(2\), not 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} - 116 x^{3} + 60 x^{2} - 20 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{3} + 1) q^{3} + ( - \beta_{8} - \beta_{4} - \beta_1) q^{5} + ( - 2 \beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} + 2 \beta_{2}) q^{7} + ( - 4 \beta_{10} + 5 \beta_{6} + 4 \beta_{5} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{3} + 1) q^{3} + ( - \beta_{8} - \beta_{4} - \beta_1) q^{5} + ( - 2 \beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} + 2 \beta_{2}) q^{7} + ( - 4 \beta_{10} + 5 \beta_{6} + 4 \beta_{5} + \beta_{3}) q^{9} + ( - 3 \beta_{10} - 6 \beta_{6} - 6) q^{11} + ( - 5 \beta_{11} - 2 \beta_{9} + 5 \beta_{8} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{11} - 3 \beta_{9} - 2 \beta_{8} - 3 \beta_{4}) q^{15} + (2 \beta_{10} + \beta_{7} + 6 \beta_{6} - \beta_{3} + 6) q^{17} + (\beta_{10} + 12 \beta_{6} - \beta_{5} + 4 \beta_{3}) q^{19} + (7 \beta_{11} - 3 \beta_{9} - 7 \beta_{8} - 8 \beta_{4} - \beta_{2} + 2 \beta_1) q^{21} + (5 \beta_{8} - 7 \beta_{4}) q^{23} + (2 \beta_{10} + 2 \beta_{7} - 14 \beta_{6} - 2 \beta_{3} - 14) q^{25} + (2 \beta_{7} + 13 \beta_{5} - 8) q^{27} + ( - 3 \beta_{11} + 2 \beta_{9} + 3 \beta_{8} + 2 \beta_{4} + 8 \beta_{2} + 8 \beta_1) q^{29} + (6 \beta_{11} - 9 \beta_{9}) q^{31} + ( - 6 \beta_{10} + 9 \beta_{6} + 6 \beta_{5} + 6 \beta_{3}) q^{33} + ( - \beta_{10} + 3 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} - 2 \beta_{3} + 23) q^{35} + ( - 4 \beta_{8} + \beta_{4} - 9 \beta_1) q^{37} + ( - 17 \beta_{11} + 8 \beta_{9} - 14 \beta_{2}) q^{39} + (\beta_{7} - 18 \beta_{5} - 25) q^{41} + ( - 10 \beta_{7} - 10 \beta_{5} + 10) q^{43} + (9 \beta_{11} + 2 \beta_{9} + 4 \beta_{2}) q^{45} + ( - 2 \beta_{8} + 5 \beta_{4} - 16 \beta_1) q^{47} + ( - 10 \beta_{10} + 9 \beta_{7} - 24 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} + \cdots + 20) q^{49}+ \cdots + ( - 21 \beta_{7} + 12 \beta_{5} - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} - 40 q^{9} - 30 q^{11} + 30 q^{17} - 78 q^{19} - 92 q^{25} - 156 q^{27} - 78 q^{33} + 222 q^{35} - 232 q^{41} + 200 q^{43} + 372 q^{49} - 10 q^{51} + 332 q^{57} + 110 q^{59} - 32 q^{65} - 434 q^{67} + 102 q^{73} + 60 q^{75} - 82 q^{81} + 536 q^{83} + 214 q^{89} + 8 q^{91} - 152 q^{97} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} - 116 x^{3} + 60 x^{2} - 20 x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2848753 \nu^{11} + 128409 \nu^{10} - 36949178 \nu^{9} - 33927641 \nu^{8} + 280693408 \nu^{7} + 324275527 \nu^{6} - 906643427 \nu^{5} + \cdots + 114476732 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 908144 \nu^{11} - 6494043 \nu^{10} + 3252536 \nu^{9} + 28087343 \nu^{8} + 91567337 \nu^{7} - 469374709 \nu^{6} + 376525973 \nu^{5} - 267919561 \nu^{4} + \cdots - 26684466 ) / 5128417 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 775023 \nu^{11} - 2790290 \nu^{10} - 2275565 \nu^{9} + 5234055 \nu^{8} + 68919723 \nu^{7} - 163955841 \nu^{6} + 148312850 \nu^{5} - 164484671 \nu^{4} + \cdots - 23024344 ) / 2930524 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 894552 \nu^{11} + 880921 \nu^{10} + 8410783 \nu^{9} + 6680758 \nu^{8} - 80688819 \nu^{7} - 13990590 \nu^{6} + 103354841 \nu^{5} - 1783409 \nu^{4} + \cdots - 16276324 ) / 2930524 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8673537 \nu^{11} - 20159299 \nu^{10} - 48705482 \nu^{9} - 5607853 \nu^{8} + 742433540 \nu^{7} - 912321093 \nu^{6} + 701452097 \nu^{5} + \cdots - 29235872 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8864783 \nu^{11} - 20455402 \nu^{10} - 48834261 \nu^{9} - 8785933 \nu^{8} + 751559539 \nu^{7} - 927115697 \nu^{6} + 797312818 \nu^{5} + \cdots - 70375800 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4885386 \nu^{11} + 11354967 \nu^{10} + 27428562 \nu^{9} + 3157071 \nu^{8} - 418178829 \nu^{7} + 513874517 \nu^{6} - 395095635 \nu^{5} + \cdots - 20957728 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5752921 \nu^{11} - 9678616 \nu^{10} - 40188533 \nu^{9} - 25630209 \nu^{8} + 486470615 \nu^{7} - 292655047 \nu^{6} + 125254932 \nu^{5} + \cdots + 18987776 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1785337 \nu^{11} + 6633082 \nu^{10} + 5117731 \nu^{9} - 14794777 \nu^{8} - 159689841 \nu^{7} + 400659219 \nu^{6} - 329059530 \nu^{5} + \cdots + 22371428 ) / 2930524 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15221487 \nu^{11} + 28755150 \nu^{10} + 95322325 \nu^{9} + 55362465 \nu^{8} - 1263522623 \nu^{7} + 1071722261 \nu^{6} - 959373046 \nu^{5} + \cdots + 55637788 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14292759 \nu^{11} + 36030049 \nu^{10} + 73017973 \nu^{9} - 5911956 \nu^{8} - 1221850366 \nu^{7} + 1754115116 \nu^{6} - 1489733269 \nu^{5} + \cdots + 95337586 ) / 10256834 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - 7\beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} + 9\beta_{9} + 3\beta_{8} + 3\beta_{7} + 3\beta_{5} + 9\beta_{4} + 8\beta_{2} + 8\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{11} - 16\beta_{10} + 19\beta_{9} - 14\beta_{7} - 51\beta_{6} + 14\beta_{3} + 17\beta_{2} - 51 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -45\beta_{10} + 20\beta_{8} - 146\beta_{6} + 45\beta_{5} + 61\beta_{4} + 40\beta_{3} + 54\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 75 \beta_{11} + 231 \beta_{9} + 75 \beta_{8} - 85 \beta_{7} - 96 \beta_{5} + 231 \beta_{4} + 205 \beta_{2} + 205 \beta _1 - 310 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 101 \beta_{11} - 497 \beta_{10} - 311 \beta_{9} - 441 \beta_{7} - 1609 \beta_{6} + 441 \beta_{3} - 276 \beta_{2} - 1609 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 356\beta_{10} + 768\beta_{8} + 1153\beta_{6} - 356\beta_{5} + 2367\beta_{4} - 316\beta_{3} + 2101\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 130 \beta_{11} - 401 \beta_{9} - 130 \beta_{8} - 4242 \beta_{7} - 4779 \beta_{5} - 401 \beta_{4} - 356 \beta_{2} - 356 \beta _1 - 15478 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6965 \beta_{11} - 1610 \beta_{10} - 21471 \beta_{9} - 1429 \beta_{7} - 5214 \beta_{6} + 1429 \beta_{3} - 19059 \beta_{2} - 5214 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 40931 \beta_{10} + 5807 \beta_{8} + 132571 \beta_{6} - 40931 \beta_{5} + 17901 \beta_{4} - 36333 \beta_{3} + 15890 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.378279 + 0.358951i
0.121721 + 0.507075i
−2.29733 + 1.90372i
2.79733 1.03769i
0.907369 + 0.0534805i
−0.407369 + 0.812545i
0.378279 0.358951i
0.121721 0.507075i
−2.29733 1.90372i
2.79733 + 1.03769i
0.907369 0.0534805i
−0.407369 0.812545i
0 −1.99052 3.44767i 0 −1.63031 0.941260i 0 −5.14749 + 4.74377i 0 −3.42430 + 5.93106i 0
79.2 0 −1.99052 3.44767i 0 1.63031 + 0.941260i 0 5.14749 4.74377i 0 −3.42430 + 5.93106i 0
79.3 0 0.824388 + 1.42788i 0 −3.95004 2.28056i 0 −6.75545 + 1.83408i 0 3.14077 5.43997i 0
79.4 0 0.824388 + 1.42788i 0 3.95004 + 2.28056i 0 6.75545 1.83408i 0 3.14077 5.43997i 0
79.5 0 2.66613 + 4.61787i 0 −1.86796 1.07847i 0 −6.91861 1.06433i 0 −9.71647 + 16.8294i 0
79.6 0 2.66613 + 4.61787i 0 1.86796 + 1.07847i 0 6.91861 + 1.06433i 0 −9.71647 + 16.8294i 0
207.1 0 −1.99052 + 3.44767i 0 −1.63031 + 0.941260i 0 −5.14749 4.74377i 0 −3.42430 5.93106i 0
207.2 0 −1.99052 + 3.44767i 0 1.63031 0.941260i 0 5.14749 + 4.74377i 0 −3.42430 5.93106i 0
207.3 0 0.824388 1.42788i 0 −3.95004 + 2.28056i 0 −6.75545 1.83408i 0 3.14077 + 5.43997i 0
207.4 0 0.824388 1.42788i 0 3.95004 2.28056i 0 6.75545 + 1.83408i 0 3.14077 + 5.43997i 0
207.5 0 2.66613 4.61787i 0 −1.86796 + 1.07847i 0 −6.91861 + 1.06433i 0 −9.71647 16.8294i 0
207.6 0 2.66613 4.61787i 0 1.86796 1.07847i 0 6.91861 1.06433i 0 −9.71647 16.8294i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.o.d 12
4.b odd 2 1 56.3.k.d 12
7.c even 3 1 inner 224.3.o.d 12
7.c even 3 1 1568.3.g.j 6
7.d odd 6 1 1568.3.g.l 6
8.b even 2 1 56.3.k.d 12
8.d odd 2 1 inner 224.3.o.d 12
28.d even 2 1 392.3.k.l 12
28.f even 6 1 392.3.g.i 6
28.f even 6 1 392.3.k.l 12
28.g odd 6 1 56.3.k.d 12
28.g odd 6 1 392.3.g.j 6
56.h odd 2 1 392.3.k.l 12
56.j odd 6 1 392.3.g.i 6
56.j odd 6 1 392.3.k.l 12
56.k odd 6 1 inner 224.3.o.d 12
56.k odd 6 1 1568.3.g.j 6
56.m even 6 1 1568.3.g.l 6
56.p even 6 1 56.3.k.d 12
56.p even 6 1 392.3.g.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d 12 4.b odd 2 1
56.3.k.d 12 8.b even 2 1
56.3.k.d 12 28.g odd 6 1
56.3.k.d 12 56.p even 6 1
224.3.o.d 12 1.a even 1 1 trivial
224.3.o.d 12 7.c even 3 1 inner
224.3.o.d 12 8.d odd 2 1 inner
224.3.o.d 12 56.k odd 6 1 inner
392.3.g.i 6 28.f even 6 1
392.3.g.i 6 56.j odd 6 1
392.3.g.j 6 28.g odd 6 1
392.3.g.j 6 56.p even 6 1
392.3.k.l 12 28.d even 2 1
392.3.k.l 12 28.f even 6 1
392.3.k.l 12 56.h odd 2 1
392.3.k.l 12 56.j odd 6 1
1568.3.g.j 6 7.c even 3 1
1568.3.g.j 6 56.k odd 6 1
1568.3.g.l 6 7.d odd 6 1
1568.3.g.l 6 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\):

\( T_{3}^{6} - 3T_{3}^{5} + 28T_{3}^{4} - 13T_{3}^{3} + 466T_{3}^{2} - 665T_{3} + 1225 \) Copy content Toggle raw display
\( T_{5}^{12} - 29T_{5}^{10} + 654T_{5}^{8} - 4737T_{5}^{6} + 25022T_{5}^{4} - 64141T_{5}^{2} + 117649 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} + 28 T^{4} - 13 T^{3} + \cdots + 1225)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 29 T^{10} + 654 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$7$ \( T^{12} - 186 T^{10} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} + 15 T^{5} + 234 T^{4} + \cdots + 263169)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 928 T^{4} + 259808 T^{2} + \cdots + 20420848)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 15 T^{5} + 202 T^{4} - 415 T^{3} + \cdots + 1225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 39 T^{5} + 1234 T^{4} + \cdots + 290521)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 2481 T^{10} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{6} + 1384 T^{4} + 517712 T^{2} + \cdots + 35753200)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 2205 T^{10} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{12} - 2429 T^{10} + \cdots + 41\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( (T^{3} + 58 T^{2} - 1904 T - 106036)^{4} \) Copy content Toggle raw display
$43$ \( (T^{3} - 50 T^{2} - 1500 T + 77000)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} - 9349 T^{10} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{12} - 6293 T^{10} + \cdots + 82\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( (T^{6} - 55 T^{5} + 3076 T^{4} + \cdots + 43020481)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} - 13125 T^{10} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( (T^{6} + 217 T^{5} + \cdots + 135936003025)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 7184 T^{4} + \cdots + 1372000000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 51 T^{5} + 6822 T^{4} + \cdots + 2144153025)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 20753 T^{10} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{3} - 134 T^{2} + 1916 T + 185080)^{4} \) Copy content Toggle raw display
$89$ \( (T^{6} - 107 T^{5} + \cdots + 825668447569)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 38 T^{2} - 3036 T - 117740)^{4} \) Copy content Toggle raw display
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