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Properties

Label 441.4.e.y
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
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] = [441,4,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("441.226");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), 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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.5922408960000.19
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 54x^{6} + 176x^{5} + 1307x^{4} - 2912x^{3} - 15314x^{2} + 16800x + 86044 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 49)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{2} + 1) q^{2} + (\beta_{6} - 8 \beta_1 - 8) q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{2} - 17) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{2} + 1) q^{2} + (\beta_{6} - 8 \beta_1 - 8) q^{4} + (\beta_{7} + \beta_{4}) q^{5} + (\beta_{2} - 17) q^{8} + (4 \beta_{4} + 2 \beta_{3}) q^{10} + (6 \beta_{6} + 28 \beta_1 + 28) q^{11} + ( - \beta_{7} - 6 \beta_{5} + 6 \beta_{3}) q^{13} + ( - 9 \beta_{6} + 9 \beta_{2} + \cdots - 9) q^{16}+ \cdots + ( - 91 \beta_{7} + \cdots - 189 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 34 q^{4} - 132 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 34 q^{4} - 132 q^{8} + 100 q^{11} + 174 q^{16} - 680 q^{22} + 352 q^{23} + 128 q^{25} - 520 q^{29} - 30 q^{32} - 212 q^{37} + 1080 q^{43} + 460 q^{44} - 696 q^{46} - 2732 q^{50} + 16 q^{53} + 780 q^{58} - 3356 q^{64} - 756 q^{65} + 1944 q^{67} - 4496 q^{71} - 284 q^{74} + 1048 q^{79} - 6568 q^{85} + 4820 q^{86} - 1260 q^{88} - 7024 q^{92} + 2192 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 54x^{6} + 176x^{5} + 1307x^{4} - 2912x^{3} - 15314x^{2} + 16800x + 86044 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 149\nu^{4} - 308\nu^{3} - 3293\nu^{2} + 3448\nu + 22372 ) / 8946 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 188 \nu^{7} - 658 \nu^{6} - 11168 \nu^{5} + 29565 \nu^{4} + 311680 \nu^{3} - 497414 \nu^{2} + \cdots + 4518059 ) / 3072951 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 94 \nu^{7} + 558 \nu^{6} + 4897 \nu^{5} - 31843 \nu^{4} - 120574 \nu^{3} + 1137914 \nu^{2} + \cdots - 11478684 ) / 1024317 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1034 \nu^{7} - 2932 \nu^{6} - 63485 \nu^{5} + 111426 \nu^{4} + 1820038 \nu^{3} - 68156 \nu^{2} + \cdots - 24320296 ) / 3072951 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 685 \nu^{7} + 10069 \nu^{6} + 3667 \nu^{5} - 388145 \nu^{4} + 210780 \nu^{3} + 6478590 \nu^{2} + \cdots - 36898988 ) / 2048634 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2744 \nu^{7} - 8917 \nu^{6} - 155726 \nu^{5} + 356991 \nu^{4} + 3179236 \nu^{3} - 3891986 \nu^{2} + \cdots + 2554244 ) / 6145902 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5467 \nu^{7} + 2506 \nu^{6} - 235570 \nu^{5} - 369522 \nu^{4} + 3410879 \nu^{3} + 8909612 \nu^{2} + \cdots - 44176664 ) / 6145902 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - 7\beta_{2} + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{4} + 10\beta_{3} - 7\beta_{2} + 14\beta _1 + 119 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 42\beta_{6} - 2\beta_{5} + 54\beta_{4} - 31\beta_{3} - 77\beta_{2} + 189 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{7} - 84\beta_{6} + 40\beta_{5} + 200\beta_{4} + 236\beta_{3} - 147\beta_{2} + 1316\beta _1 + 2023 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 356\beta_{7} - 2240\beta_{6} - 216\beta_{5} + 1686\beta_{4} - 315\beta_{3} + 245\beta_{2} + 2240\beta _1 + 3451 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1952 \beta_{7} - 6510 \beta_{6} + 2640 \beta_{5} + 6780 \beta_{4} + 3222 \beta_{3} + 1099 \beta_{2} + \cdots + 26397 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22148 \beta_{7} - 73010 \beta_{6} - 6566 \beta_{5} + 44318 \beta_{4} + 2477 \beta_{3} + 49287 \beta_{2} + \cdots + 28329 ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−2.82402 + 1.22474i
−4.23824 − 1.22474i
5.23824 + 1.22474i
3.82402 − 1.22474i
−2.82402 − 1.22474i
−4.23824 + 1.22474i
5.23824 − 1.22474i
3.82402 + 1.22474i
−1.76556 + 3.05805i 0 −2.23444 − 3.87016i −1.03865 + 1.79899i 0 0 −12.4689 0 −3.66760 − 6.35247i
226.2 −1.76556 + 3.05805i 0 −2.23444 − 3.87016i 1.03865 − 1.79899i 0 0 −12.4689 0 3.66760 + 6.35247i
226.3 2.26556 − 3.92407i 0 −6.26556 − 10.8523i −6.73953 + 11.6732i 0 0 −20.5311 0 30.5377 + 52.8928i
226.4 2.26556 − 3.92407i 0 −6.26556 − 10.8523i 6.73953 − 11.6732i 0 0 −20.5311 0 −30.5377 − 52.8928i
361.1 −1.76556 − 3.05805i 0 −2.23444 + 3.87016i −1.03865 − 1.79899i 0 0 −12.4689 0 −3.66760 + 6.35247i
361.2 −1.76556 − 3.05805i 0 −2.23444 + 3.87016i 1.03865 + 1.79899i 0 0 −12.4689 0 3.66760 − 6.35247i
361.3 2.26556 + 3.92407i 0 −6.26556 + 10.8523i −6.73953 − 11.6732i 0 0 −20.5311 0 30.5377 − 52.8928i
361.4 2.26556 + 3.92407i 0 −6.26556 + 10.8523i 6.73953 + 11.6732i 0 0 −20.5311 0 −30.5377 + 52.8928i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.y 8
3.b odd 2 1 49.4.c.e 8
7.b odd 2 1 inner 441.4.e.y 8
7.c even 3 1 441.4.a.u 4
7.c even 3 1 inner 441.4.e.y 8
7.d odd 6 1 441.4.a.u 4
7.d odd 6 1 inner 441.4.e.y 8
21.c even 2 1 49.4.c.e 8
21.g even 6 1 49.4.a.e ✓ 4
21.g even 6 1 49.4.c.e 8
21.h odd 6 1 49.4.a.e ✓ 4
21.h odd 6 1 49.4.c.e 8
84.j odd 6 1 784.4.a.bf 4
84.n even 6 1 784.4.a.bf 4
105.o odd 6 1 1225.4.a.bb 4
105.p even 6 1 1225.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e ✓ 4 21.g even 6 1
49.4.a.e ✓ 4 21.h odd 6 1
49.4.c.e 8 3.b odd 2 1
49.4.c.e 8 21.c even 2 1
49.4.c.e 8 21.g even 6 1
49.4.c.e 8 21.h odd 6 1
441.4.a.u 4 7.c even 3 1
441.4.a.u 4 7.d odd 6 1
441.4.e.y 8 1.a even 1 1 trivial
441.4.e.y 8 7.b odd 2 1 inner
441.4.e.y 8 7.c even 3 1 inner
441.4.e.y 8 7.d odd 6 1 inner
784.4.a.bf 4 84.j odd 6 1
784.4.a.bf 4 84.n even 6 1
1225.4.a.bb 4 105.o odd 6 1
1225.4.a.bb 4 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 17T_{2}^{2} + 16T_{2} + 256 \) Copy content Toggle raw display
\( T_{5}^{8} + 186T_{5}^{6} + 33812T_{5}^{4} + 145824T_{5}^{2} + 614656 \) Copy content Toggle raw display
\( T_{13}^{4} - 3234T_{13}^{2} + 2458624 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + 17 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 186 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 50 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 3234 T^{2} + 2458624)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 95001747709456 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{4} - 176 T^{3} + \cdots + 44943616)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 130 T + 1040)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{4} + 106 T^{3} + \cdots + 4946176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 66836 T^{2} + 307721764)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 270 T - 61400)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} + \cdots + 442849936)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{4} - 972 T^{3} + \cdots + 20259536896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1124 T + 303104)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} - 524 T^{3} + \cdots + 255728444416)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 11466 T^{2} + 6492304)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 80\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} - 3082884 T^{2} + 841222821124)^{2} \) Copy content Toggle raw display
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