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Properties

Label 441.4.p.c
Level $441$
Weight $4$
Character orbit 441.p
Analytic conductor $26.020$
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] = [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: \(16\)
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: \( x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{9} - 4 \beta_{3} + \beta_{2} + 4) q^{4} + (\beta_{13} - \beta_{12}) q^{5} + ( - 2 \beta_{13} + \beta_{12} + \cdots - 4 \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{9} - 4 \beta_{3} + \beta_{2} + 4) q^{4} + (\beta_{13} - \beta_{12}) q^{5} + ( - 2 \beta_{13} + \beta_{12} + \cdots - 4 \beta_{6}) q^{8}+ \cdots + ( - 29 \beta_{10} + 94 \beta_{9} + \cdots + 401) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 72 q^{10} - 188 q^{16} + 612 q^{19} + 528 q^{22} - 20 q^{25} - 1128 q^{31} - 1196 q^{37} + 3204 q^{40} + 328 q^{43} - 1392 q^{46} - 4452 q^{52} - 3372 q^{58} + 1632 q^{61} + 5432 q^{64} + 308 q^{67} - 4068 q^{73} - 2176 q^{79} + 10188 q^{82} - 4608 q^{85} + 708 q^{88} + 2916 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10313805 \nu^{14} + 461882826 \nu^{12} - 15525573175 \nu^{10} + 234921068610 \nu^{8} + \cdots - 278327304980712 ) / 24355954733376 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2582250337 \nu^{14} + 122400945426 \nu^{12} - 4183683881139 \nu^{10} + \cdots + 43\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 131851797 \nu^{14} - 6000144674 \nu^{12} + 198479098895 \nu^{10} - 3003233534994 \nu^{8} + \cdots + 301359467694888 ) / 73067864200128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20174289101 \nu^{14} + 1199936429098 \nu^{12} - 44140804007447 \nu^{10} + \cdots + 21\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + \cdots + 699777739755000 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6165622857 \nu^{14} - 304531357386 \nu^{12} + 10574100606979 \nu^{10} + \cdots - 11\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 937699527 \nu^{15} + 42257710318 \nu^{13} - 1411537509445 \nu^{11} + \cdots - 41\!\cdots\!12 \nu ) / 730678642001280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4906655549 \nu^{14} + 233254820202 \nu^{12} - 7979228432903 \nu^{10} + \cdots + 83\!\cdots\!00 ) / 608898868334400 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 151546793651 \nu^{14} - 7122898433398 \nu^{12} + 241898414566697 \nu^{10} + \cdots + 56\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 253797 \nu^{15} - 12085406 \nu^{13} + 413081109 \nu^{11} - 6837155440 \nu^{9} + \cdots - 69093405000 \nu ) / 49371246000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 144886104737 \nu^{15} + 6952944162226 \nu^{13} - 238669929280139 \nu^{11} + \cdots + 50\!\cdots\!00 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 383305735249 \nu^{15} - 18114452956802 \nu^{13} + 618137711412403 \nu^{11} + \cdots + 36\!\cdots\!00 \nu ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1280411313 \nu^{15} + 57618445232 \nu^{13} - 1927428289955 \nu^{11} + \cdots - 76\!\cdots\!68 \nu ) / 91334830250160 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 530090206069 \nu^{15} - 25526553824562 \nu^{13} + 876699628157143 \nu^{11} + \cdots - 94\!\cdots\!00 \nu ) / 91\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 12\beta_{3} + \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{13} + \beta_{12} + \beta_{11} - 20\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + 23\beta_{9} - 2\beta_{7} - 2\beta_{5} - 247\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{15} - 27\beta_{13} + 54\beta_{12} + 13\beta_{11} + 13\beta_{8} - 436\beta_{6} - 436\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -31\beta_{10} - 31\beta_{5} + 86\beta_{4} - 511\beta_{2} - 5437 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86\beta_{14} + 659\beta_{13} + 659\beta_{12} - 63\beta_{8} - 9656\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1662 \beta_{10} - 11375 \beta_{9} + 2814 \beta_{7} + 831 \beta_{5} + 2814 \beta_{4} + 122445 \beta_{3} + \cdots - 120462 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -2814\beta_{15} + 2814\beta_{14} + 31702\beta_{13} - 15851\beta_{12} + 9607\beta_{11} + 215296\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -21479\beta_{10} - 254407\beta_{9} + 82486\beta_{7} + 42958\beta_{5} + 2762309\beta_{3} + 21479 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 82486 \beta_{15} + 379851 \beta_{13} - 759702 \beta_{12} + 395191 \beta_{11} + 395191 \beta_{8} + \cdots + 4825904 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 544823\beta_{10} + 544823\beta_{5} - 2277118\beta_{4} + 5717999\beta_{2} + 61427333 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -2277118\beta_{14} - 9084763\beta_{13} - 9084763\beta_{12} + 12792423\beta_{8} + 108707968\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 27277998 \beta_{10} + 129130903 \beta_{9} - 60540774 \beta_{7} - 13638999 \beta_{5} - 60540774 \beta_{4} + \cdots + 1378604262 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 60540774 \beta_{15} - 60540774 \beta_{14} - 433899350 \beta_{13} + 216949675 \beta_{12} + \cdots - 2460222704 \beta_{6} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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
−4.21355 + 2.43270i
−3.91663 + 2.26127i
−1.57646 + 0.910170i
−0.648633 + 0.374489i
0.648633 − 0.374489i
1.57646 − 0.910170i
3.91663 − 2.26127i
4.21355 − 2.43270i
−4.21355 − 2.43270i
−3.91663 − 2.26127i
−1.57646 − 0.910170i
−0.648633 − 0.374489i
0.648633 + 0.374489i
1.57646 + 0.910170i
3.91663 + 2.26127i
4.21355 + 2.43270i
−4.21355 + 2.43270i 0 7.83601 − 13.5724i −6.38217 − 11.0542i 0 0 37.3274i 0 53.7832 + 31.0517i
80.2 −3.91663 + 2.26127i 0 6.22668 − 10.7849i 0.632851 + 1.09613i 0 0 20.1405i 0 −4.95730 − 2.86210i
80.3 −1.57646 + 0.910170i 0 −2.34318 + 4.05851i 7.54372 + 13.0661i 0 0 − 23.0935i 0 −23.7848 − 13.7321i
80.4 −0.648633 + 0.374489i 0 −3.71952 + 6.44239i 5.42768 + 9.40102i 0 0 − 11.5635i 0 −7.04115 − 4.06521i
80.5 0.648633 − 0.374489i 0 −3.71952 + 6.44239i −5.42768 − 9.40102i 0 0 11.5635i 0 −7.04115 − 4.06521i
80.6 1.57646 − 0.910170i 0 −2.34318 + 4.05851i −7.54372 − 13.0661i 0 0 23.0935i 0 −23.7848 − 13.7321i
80.7 3.91663 − 2.26127i 0 6.22668 − 10.7849i −0.632851 − 1.09613i 0 0 − 20.1405i 0 −4.95730 − 2.86210i
80.8 4.21355 − 2.43270i 0 7.83601 − 13.5724i 6.38217 + 11.0542i 0 0 − 37.3274i 0 53.7832 + 31.0517i
215.1 −4.21355 − 2.43270i 0 7.83601 + 13.5724i −6.38217 + 11.0542i 0 0 − 37.3274i 0 53.7832 − 31.0517i
215.2 −3.91663 − 2.26127i 0 6.22668 + 10.7849i 0.632851 − 1.09613i 0 0 − 20.1405i 0 −4.95730 + 2.86210i
215.3 −1.57646 − 0.910170i 0 −2.34318 − 4.05851i 7.54372 − 13.0661i 0 0 23.0935i 0 −23.7848 + 13.7321i
215.4 −0.648633 − 0.374489i 0 −3.71952 − 6.44239i 5.42768 − 9.40102i 0 0 11.5635i 0 −7.04115 + 4.06521i
215.5 0.648633 + 0.374489i 0 −3.71952 − 6.44239i −5.42768 + 9.40102i 0 0 − 11.5635i 0 −7.04115 + 4.06521i
215.6 1.57646 + 0.910170i 0 −2.34318 − 4.05851i −7.54372 + 13.0661i 0 0 − 23.0935i 0 −23.7848 + 13.7321i
215.7 3.91663 + 2.26127i 0 6.22668 + 10.7849i −0.632851 + 1.09613i 0 0 20.1405i 0 −4.95730 + 2.86210i
215.8 4.21355 + 2.43270i 0 7.83601 + 13.5724i 6.38217 − 11.0542i 0 0 37.3274i 0 53.7832 − 31.0517i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 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.c 16
3.b odd 2 1 inner 441.4.p.c 16
7.b odd 2 1 63.4.p.a ✓ 16
7.c even 3 1 63.4.p.a ✓ 16
7.c even 3 1 441.4.c.a 16
7.d odd 6 1 441.4.c.a 16
7.d odd 6 1 inner 441.4.p.c 16
21.c even 2 1 63.4.p.a ✓ 16
21.g even 6 1 441.4.c.a 16
21.g even 6 1 inner 441.4.p.c 16
21.h odd 6 1 63.4.p.a ✓ 16
21.h odd 6 1 441.4.c.a 16
28.d even 2 1 1008.4.bt.a 16
28.g odd 6 1 1008.4.bt.a 16
84.h odd 2 1 1008.4.bt.a 16
84.n even 6 1 1008.4.bt.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.p.a ✓ 16 7.b odd 2 1
63.4.p.a ✓ 16 7.c even 3 1
63.4.p.a ✓ 16 21.c even 2 1
63.4.p.a ✓ 16 21.h odd 6 1
441.4.c.a 16 7.c even 3 1
441.4.c.a 16 7.d odd 6 1
441.4.c.a 16 21.g even 6 1
441.4.c.a 16 21.h odd 6 1
441.4.p.c 16 1.a even 1 1 trivial
441.4.p.c 16 3.b odd 2 1 inner
441.4.p.c 16 7.d odd 6 1 inner
441.4.p.c 16 21.g even 6 1 inner
1008.4.bt.a 16 28.d even 2 1
1008.4.bt.a 16 28.g odd 6 1
1008.4.bt.a 16 84.h odd 2 1
1008.4.bt.a 16 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 48 T_{2}^{14} + 1647 T_{2}^{12} - 27620 T_{2}^{10} + 336765 T_{2}^{8} - 1200006 T_{2}^{6} + \cdots + 810000 \) acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 48 T^{14} + \cdots + 810000 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 49018425731856 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 2752420357764)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} - 306 T^{7} + \cdots + 567106596)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 20\!\cdots\!41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 32\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 82 T^{3} + \cdots + 144774182)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 84\!\cdots\!49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 75\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 64\!\cdots\!56)^{2} \) Copy content Toggle raw display
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