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Properties

Label 441.4.p.a
Level $441$
Weight $4$
Character orbit 441.p
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(80,441)]
 
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([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.80");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.p (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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.9948826238976.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 36x^{6} + 935x^{4} + 12996x^{2} + 130321 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 63)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{4} q^{2} + ( - \beta_{5} + \beta_{2}) q^{5} + (16 \beta_{4} + 16 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{4} q^{2} + ( - \beta_{5} + \beta_{2}) q^{5} + (16 \beta_{4} + 16 \beta_{3}) q^{8} - 2 \beta_{7} q^{10} - 11 \beta_{3} q^{11} - \beta_{6} q^{13} + 64 \beta_1 q^{16} + 3 \beta_{2} q^{17} + ( - 3 \beta_{7} - 3 \beta_{6}) q^{19} - 44 q^{22} + 55 \beta_{4} q^{23} + (319 \beta_1 - 319) q^{25} + (4 \beta_{5} - 4 \beta_{2}) q^{26} + (89 \beta_{4} + 89 \beta_{3}) q^{29} + 8 \beta_{7} q^{31} + 6 \beta_{6} q^{34} + 184 \beta_1 q^{37} - 12 \beta_{2} q^{38} + (16 \beta_{7} + 16 \beta_{6}) q^{40} - 5 \beta_{5} q^{41} - 190 q^{43} + (220 \beta_1 - 220) q^{46} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{47} + (638 \beta_{4} + 638 \beta_{3}) q^{50} + 253 \beta_{3} q^{53} - 11 \beta_{6} q^{55} + 356 \beta_1 q^{58} - 4 \beta_{2} q^{59} + ( - 22 \beta_{7} - 22 \beta_{6}) q^{61} + 32 \beta_{5} q^{62} - 512 q^{64} - 444 \beta_{4} q^{65} + (296 \beta_1 - 296) q^{67} + (233 \beta_{4} + 233 \beta_{3}) q^{71} - 27 \beta_{7} q^{73} + 368 \beta_{3} q^{74} - 836 \beta_1 q^{79} + 64 \beta_{2} q^{80} + ( - 10 \beta_{7} - 10 \beta_{6}) q^{82} - 58 \beta_{5} q^{83} - 1332 q^{85} + 380 \beta_{4} q^{86} + ( - 352 \beta_1 + 352) q^{88} + ( - 33 \beta_{5} + 33 \beta_{2}) q^{89} - 4 \beta_{7} q^{94} + 1332 \beta_{3} q^{95} + 19 \beta_{6} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 256 q^{16} - 352 q^{22} - 1276 q^{25} + 736 q^{37} - 1520 q^{43} - 880 q^{46} + 1424 q^{58} - 4096 q^{64} - 1184 q^{67} - 3344 q^{79} - 10656 q^{85} + 1408 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 36x^{6} + 935x^{4} + 12996x^{2} + 130321 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 36\nu^{6} + 935\nu^{4} + 33660\nu^{2} + 467856 ) / 337535 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -148\nu^{6} + 33660\nu^{4} + 536690\nu^{2} + 10227852 ) / 337535 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 251\nu^{7} + 15895\nu^{5} + 234685\nu^{3} + 3261996\nu ) / 6413165 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2899\nu ) / 17765 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{6} + 72\nu^{4} + 1148\nu^{2} + 12996 ) / 361 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -148\nu^{7} - 6050\nu^{5} - 178090\nu^{3} - 4107458\nu ) / 377245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4682\nu^{7} - 102850\nu^{5} - 3027530\nu^{3} + 13410428\nu ) / 6413165 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + 6\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{2} + 108\beta _1 - 108 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -34\beta_{7} - 17\beta_{6} - 330\beta_{4} - 330\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{5} + 6\beta_{2} - 287\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 251\beta_{7} + 502\beta_{6} + 9714\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 935\beta_{5} - 1870\beta_{2} + 23004 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2899\beta_{7} - 2899\beta_{6} + 230574\beta_{4} ) / 12 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−1.53821 − 4.07847i
2.76295 + 3.37136i
1.53821 + 4.07847i
−2.76295 − 3.37136i
−1.53821 + 4.07847i
2.76295 − 3.37136i
1.53821 − 4.07847i
−2.76295 + 3.37136i
−2.44949 + 1.41421i 0 0 −10.5357 − 18.2483i 0 0 − 22.6274i 0 51.6140 + 29.7993i
80.2 −2.44949 + 1.41421i 0 0 10.5357 + 18.2483i 0 0 − 22.6274i 0 −51.6140 − 29.7993i
80.3 2.44949 − 1.41421i 0 0 −10.5357 − 18.2483i 0 0 22.6274i 0 −51.6140 − 29.7993i
80.4 2.44949 − 1.41421i 0 0 10.5357 + 18.2483i 0 0 22.6274i 0 51.6140 + 29.7993i
215.1 −2.44949 − 1.41421i 0 0 −10.5357 + 18.2483i 0 0 22.6274i 0 51.6140 − 29.7993i
215.2 −2.44949 − 1.41421i 0 0 10.5357 − 18.2483i 0 0 22.6274i 0 −51.6140 + 29.7993i
215.3 2.44949 + 1.41421i 0 0 −10.5357 + 18.2483i 0 0 − 22.6274i 0 −51.6140 + 29.7993i
215.4 2.44949 + 1.41421i 0 0 10.5357 − 18.2483i 0 0 − 22.6274i 0 51.6140 − 29.7993i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h 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.p.a 8
3.b odd 2 1 inner 441.4.p.a 8
7.b odd 2 1 inner 441.4.p.a 8
7.c even 3 1 63.4.c.b ✓ 4
7.c even 3 1 inner 441.4.p.a 8
7.d odd 6 1 63.4.c.b ✓ 4
7.d odd 6 1 inner 441.4.p.a 8
21.c even 2 1 inner 441.4.p.a 8
21.g even 6 1 63.4.c.b ✓ 4
21.g even 6 1 inner 441.4.p.a 8
21.h odd 6 1 63.4.c.b ✓ 4
21.h odd 6 1 inner 441.4.p.a 8
28.f even 6 1 1008.4.k.b 4
28.g odd 6 1 1008.4.k.b 4
84.j odd 6 1 1008.4.k.b 4
84.n even 6 1 1008.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.b ✓ 4 7.c even 3 1
63.4.c.b ✓ 4 7.d odd 6 1
63.4.c.b ✓ 4 21.g even 6 1
63.4.c.b ✓ 4 21.h odd 6 1
441.4.p.a 8 1.a even 1 1 trivial
441.4.p.a 8 3.b odd 2 1 inner
441.4.p.a 8 7.b odd 2 1 inner
441.4.p.a 8 7.c even 3 1 inner
441.4.p.a 8 7.d odd 6 1 inner
441.4.p.a 8 21.c even 2 1 inner
441.4.p.a 8 21.g even 6 1 inner
441.4.p.a 8 21.h odd 6 1 inner
1008.4.k.b 4 28.f even 6 1
1008.4.k.b 4 28.g odd 6 1
1008.4.k.b 4 84.j odd 6 1
1008.4.k.b 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 8T_{2}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 444 T^{2} + 197136)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 242 T^{2} + 58564)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 888)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 3996 T^{2} + 15968016)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 7992 T^{2} + 63872064)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6050 T^{2} + 36602500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 15842)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56832 T^{2} + 3229876224)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 184 T + 33856)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 11100)^{4} \) Copy content Toggle raw display
$43$ \( (T + 190)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1776 T^{2} + 3154176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 128018 T^{2} + 16388608324)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 7104 T^{2} + 50466816)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 429792 T^{2} + 184721163264)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 296 T + 87616)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 108578)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 647352 T^{2} + 419064611904)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 836 T + 698896)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1493616)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 483516 T^{2} + 233787722256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 320568)^{4} \) Copy content Toggle raw display
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