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Properties

Label 441.4.c.a
Level $441$
Weight $4$
Character orbit 441.c
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(440,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.440");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), 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: \(16\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_1 - 4) q^{4} + \beta_{10} q^{5} + (\beta_{13} - 5 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_1 - 4) q^{4} + \beta_{10} q^{5} + (\beta_{13} - 5 \beta_{4}) q^{8} + \beta_{3} q^{10} + (\beta_{15} + 3 \beta_{9} - 2 \beta_{4}) q^{11} + (\beta_{11} - \beta_{7} - \beta_{5}) q^{13} + (\beta_{6} + \beta_{2} + 23) q^{16} + (\beta_{14} - 2 \beta_{10}) q^{17} + (\beta_{11} - \beta_{7} + \cdots - 2 \beta_{3}) q^{19}+ \cdots + ( - 18 \beta_{11} - 11 \beta_{7} + \cdots + 7 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( - 10313805 \nu^{14} + 461882826 \nu^{12} - 15525573175 \nu^{10} + 234921068610 \nu^{8} + \cdots - 278327304980712 ) / 24355954733376 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 131851797 \nu^{14} + 6000144674 \nu^{12} - 198479098895 \nu^{10} + \cdots - 301359467694888 ) / 73067864200128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5684446703 \nu^{14} - 263336690854 \nu^{12} + 8955427519661 \nu^{10} + \cdots - 62\!\cdots\!00 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + \cdots + 699777739755000 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2582250337 \nu^{14} - 122400945426 \nu^{12} + 4183683881139 \nu^{10} + \cdots - 25\!\cdots\!00 ) / 260956657857600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 774906315 \nu^{14} - 34871917166 \nu^{12} + 1166481691025 \nu^{10} - 17650306515630 \nu^{8} + \cdots + 23\!\cdots\!36 ) / 73067864200128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49963897373 \nu^{14} + 2354075825554 \nu^{12} - 80336766247031 \nu^{10} + \cdots + 49\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + \cdots + 80\!\cdots\!00 \nu ) / 521913315715200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 253797 \nu^{15} + 12085406 \nu^{13} - 413081109 \nu^{11} + 6837155440 \nu^{9} + \cdots + 69093405000 \nu ) / 49371246000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 144886104737 \nu^{15} - 6952944162226 \nu^{13} + 238669929280139 \nu^{11} + \cdots - 50\!\cdots\!00 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 277854990781 \nu^{14} + 13359750934538 \nu^{12} - 458260687617607 \nu^{10} + \cdots + 25\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 8720041741 \nu^{15} + 445313626418 \nu^{13} - 15626457197827 \nu^{11} + \cdots + 48\!\cdots\!00 \nu ) / 685011226876200 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + \cdots + 699777739755000 \nu ) / 173971105238400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 546527094713 \nu^{15} - 27021548295874 \nu^{13} + 936917477650511 \nu^{11} + \cdots - 24\!\cdots\!00 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 658131337369 \nu^{15} - 31288398347762 \nu^{13} + \cdots - 17\!\cdots\!00 \nu ) / 91\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} - 7\beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{7} + 12\beta_{5} - 2\beta_{3} + 7\beta _1 + 84 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 21\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{11} + 43\beta_{7} - 7\beta_{6} + 240\beta_{5} - 57\beta_{3} - 7\beta_{2} - 168\beta _1 - 1729 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{15} + 14 \beta_{14} + 189 \beta_{13} - 11 \beta_{12} - 542 \beta_{10} + 98 \beta_{9} + \cdots - 3241 \beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -31\beta_{6} - 55\beta_{2} - 542\beta _1 - 5437 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 602 \beta_{15} + 602 \beta_{14} - 4613 \beta_{13} + 149 \beta_{12} - 13386 \beta_{10} + \cdots + 72205 \beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 9524 \beta_{11} - 16801 \beta_{7} - 5817 \beta_{6} - 114324 \beta_{5} + 36499 \beta_{3} + \cdots - 849051 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -2814\beta_{15} - 15851\beta_{13} - 25458\beta_{9} + 231147\beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 226068 \beta_{11} - 325793 \beta_{7} + 150353 \beta_{6} - 2532900 \beta_{5} + 903195 \beta_{3} + \cdots + 19059467 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 577402 \beta_{15} - 577402 \beta_{14} - 2658957 \beta_{13} - 477677 \beta_{12} + \cdots + 36440285 \beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 544823\beta_{6} + 1732295\beta_{2} + 6262822\beta _1 + 61427333 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 15939826 \beta_{15} - 15939826 \beta_{14} + 63593341 \beta_{13} - 15069541 \beta_{12} + \cdots - 824549117 \beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 127461460 \beta_{11} + 117556481 \beta_{7} + 95472993 \beta_{6} + 1263507492 \beta_{5} + \cdots + 9745702827 ) / 14 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 60540774\beta_{15} + 216949675\beta_{13} + 590099298\beta_{9} - 2677172379\beta_{4} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
−4.21355 + 2.43270i
4.21355 + 2.43270i
3.91663 + 2.26127i
−3.91663 + 2.26127i
1.57646 + 0.910170i
−1.57646 + 0.910170i
0.648633 + 0.374489i
−0.648633 + 0.374489i
0.648633 − 0.374489i
−0.648633 − 0.374489i
1.57646 − 0.910170i
−1.57646 − 0.910170i
3.91663 − 2.26127i
−3.91663 − 2.26127i
−4.21355 − 2.43270i
4.21355 − 2.43270i
− 4.86539i 0 −15.6720 −12.7643 0 0 37.3274i 0 62.1035i
440.2 − 4.86539i 0 −15.6720 12.7643 0 0 37.3274i 0 − 62.1035i
440.3 − 4.52254i 0 −12.4534 −1.26570 0 0 20.1405i 0 5.72419i
440.4 − 4.52254i 0 −12.4534 1.26570 0 0 20.1405i 0 − 5.72419i
440.5 − 1.82034i 0 4.68636 −15.0874 0 0 − 23.0935i 0 27.4643i
440.6 − 1.82034i 0 4.68636 15.0874 0 0 − 23.0935i 0 − 27.4643i
440.7 − 0.748977i 0 7.43903 −10.8554 0 0 − 11.5635i 0 8.13042i
440.8 − 0.748977i 0 7.43903 10.8554 0 0 − 11.5635i 0 − 8.13042i
440.9 0.748977i 0 7.43903 −10.8554 0 0 11.5635i 0 − 8.13042i
440.10 0.748977i 0 7.43903 10.8554 0 0 11.5635i 0 8.13042i
440.11 1.82034i 0 4.68636 −15.0874 0 0 23.0935i 0 − 27.4643i
440.12 1.82034i 0 4.68636 15.0874 0 0 23.0935i 0 27.4643i
440.13 4.52254i 0 −12.4534 −1.26570 0 0 − 20.1405i 0 − 5.72419i
440.14 4.52254i 0 −12.4534 1.26570 0 0 − 20.1405i 0 5.72419i
440.15 4.86539i 0 −15.6720 −12.7643 0 0 − 37.3274i 0 − 62.1035i
440.16 4.86539i 0 −15.6720 12.7643 0 0 − 37.3274i 0 62.1035i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 440.16
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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.c.a 16
3.b odd 2 1 inner 441.4.c.a 16
7.b odd 2 1 inner 441.4.c.a 16
7.c even 3 1 63.4.p.a ✓ 16
7.c even 3 1 441.4.p.c 16
7.d odd 6 1 63.4.p.a ✓ 16
7.d odd 6 1 441.4.p.c 16
21.c even 2 1 inner 441.4.c.a 16
21.g even 6 1 63.4.p.a ✓ 16
21.g even 6 1 441.4.p.c 16
21.h odd 6 1 63.4.p.a ✓ 16
21.h odd 6 1 441.4.p.c 16
28.f even 6 1 1008.4.bt.a 16
28.g odd 6 1 1008.4.bt.a 16
84.j odd 6 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.c even 3 1
63.4.p.a ✓ 16 7.d odd 6 1
63.4.p.a ✓ 16 21.g even 6 1
63.4.p.a ✓ 16 21.h odd 6 1
441.4.c.a 16 1.a even 1 1 trivial
441.4.c.a 16 3.b odd 2 1 inner
441.4.c.a 16 7.b odd 2 1 inner
441.4.c.a 16 21.c even 2 1 inner
441.4.p.c 16 7.c even 3 1
441.4.p.c 16 7.d odd 6 1
441.4.p.c 16 21.g even 6 1
441.4.p.c 16 21.h odd 6 1
1008.4.bt.a 16 28.f even 6 1
1008.4.bt.a 16 28.g odd 6 1
1008.4.bt.a 16 84.j odd 6 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 \( T_{2}^{8} + 48T_{2}^{6} + 657T_{2}^{4} + 1958T_{2}^{2} + 900 \) acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 48 T^{6} + \cdots + 900)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 510 T^{6} + \cdots + 7001316)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 1333068831396)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 2752420357764)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 938202780969216)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 32994 T^{6} + \cdots + 567106596)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 10\!\cdots\!76)^{2} \) 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
$37$ \( (T^{4} - 598 T^{3} + \cdots + 192019982)^{4} \) 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
$47$ \( (T^{8} + \cdots + 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 49\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 154 T^{3} + \cdots + 4555177972)^{4} \) 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
$79$ \( (T^{4} - 1088 T^{3} + \cdots + 91919817907)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 75\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 68\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 64\!\cdots\!56)^{2} \) Copy content Toggle raw display
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