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Properties

Label 800.6.c.p
Level $800$
Weight $6$
Character orbit 800.c
Analytic conductor $128.307$
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] = [800,6,Mod(449,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.449");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), 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: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 423x^{10} + 57217x^{8} + 2682817x^{6} + 43719927x^{4} + 136382881x^{2} + 69355584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{34}\cdot 5^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( - \beta_{8} - \beta_{7}) q^{7} + (\beta_{2} - 38) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + ( - \beta_{8} - \beta_{7}) q^{7} + (\beta_{2} - 38) q^{9} + ( - \beta_{5} - 3 \beta_1) q^{11} + ( - \beta_{11} + \beta_{10} - 21 \beta_{6}) q^{13} + (2 \beta_{11} + \beta_{10} + 135 \beta_{6}) q^{17} + (2 \beta_{5} - 8 \beta_{4} + 19 \beta_1) q^{19} + ( - \beta_{3} - \beta_{2} + 147) q^{21} + ( - 4 \beta_{9} + 18 \beta_{8} + 12 \beta_{7}) q^{23} + ( - 7 \beta_{9} - 2 \beta_{8} - 121 \beta_{7}) q^{27} + ( - 3 \beta_{3} + 21 \beta_{2} - 693) q^{29} + ( - 6 \beta_{5} - 44 \beta_{4} + 70 \beta_1) q^{31} + (6 \beta_{11} + 20 \beta_{10} + 841 \beta_{6}) q^{33} + ( - 9 \beta_{11} + 21 \beta_{10} - 55 \beta_{6}) q^{37} + ( - 8 \beta_{5} - 25 \beta_{4} + 133 \beta_1) q^{39} + (2 \beta_{3} - 20 \beta_{2} + 1723) q^{41} + (11 \beta_{9} + 100 \beta_{8} + 472 \beta_{7}) q^{43} + (30 \beta_{9} - 79 \beta_{8} - 997 \beta_{7}) q^{47} + (16 \beta_{3} - 52 \beta_{2} - 237) q^{49} + (37 \beta_{5} + 56 \beta_{4} + 805 \beta_1) q^{51} + ( - 6 \beta_{11} + \cdots - 5652 \beta_{6}) q^{53}+ \cdots + ( - 13 \beta_{5} + 202 \beta_{4} + 7664 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 456 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 456 q^{9} + 1768 q^{21} - 8304 q^{29} + 20668 q^{41} - 2908 q^{49} + 51592 q^{61} - 11592 q^{69} + 293996 q^{81} - 166900 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 423x^{10} + 57217x^{8} + 2682817x^{6} + 43719927x^{4} + 136382881x^{2} + 69355584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 24853 \nu^{10} - 9930458 \nu^{8} - 1206170864 \nu^{6} - 43186800083 \nu^{4} + \cdots - 616368408219 ) / 139586494059 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 49706 \nu^{10} - 19860916 \nu^{8} - 2412341728 \nu^{6} - 86373600166 \nu^{4} + \cdots + 38130654508200 ) / 139586494059 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6667342 \nu^{10} + 2731452848 \nu^{8} + 342676849496 \nu^{6} + 12451001674718 \nu^{4} + \cdots - 985748713883379 ) / 1535451434649 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 118947509 \nu^{10} - 50825092502 \nu^{8} - 6996589318544 \nu^{6} - 338444138757719 \nu^{4} + \cdots - 87\!\cdots\!31 ) / 27126308678799 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 386055038 \nu^{10} + 157068388704 \nu^{8} + 19615902937416 \nu^{6} + 748007970931554 \nu^{4} + \cdots + 12\!\cdots\!44 ) / 27126308678799 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9541745 \nu^{11} + 4001662171 \nu^{9} + 532166547961 \nu^{7} + 23924590536433 \nu^{5} + \cdots + 703792715737641 \nu ) / 387492107507784 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9541745 \nu^{11} + 4001662171 \nu^{9} + 532166547961 \nu^{7} + 23924590536433 \nu^{5} + \cdots + 14\!\cdots\!09 \nu ) / 387492107507784 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 300009608147 \nu^{11} + 127889838020545 \nu^{9} + \cdots + 66\!\cdots\!03 \nu ) / 22\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66602944235 \nu^{11} - 28091602173319 \nu^{9} + \cdots - 11\!\cdots\!45 \nu ) / 28\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 105571031 \nu^{11} - 44388812711 \nu^{9} - 5929582199673 \nu^{7} + \cdots - 80\!\cdots\!01 \nu ) / 16145504479491 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19035342899 \nu^{11} + 8094656797841 \nu^{9} + \cdots + 16\!\cdots\!99 \nu ) / 14\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 2\beta _1 - 282 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{10} - 7\beta_{9} - 2\beta_{8} - 610\beta_{7} + 844\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{5} + 2\beta_{4} - 11\beta_{3} - 183\beta_{2} + 608\beta _1 + 42993 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -55\beta_{11} - 910\beta_{10} + 1453\beta_{9} + 665\beta_{8} + 105151\beta_{7} - 213559\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4324\beta_{5} - 1985\beta_{4} + 4719\beta_{3} + 65381\beta_{2} - 312415\beta _1 - 14834487 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4081\beta_{11} + 56413\beta_{10} - 66597\beta_{9} - 32520\beta_{8} - 4659097\beta_{7} + 12793558\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 522703\beta_{5} + 255533\beta_{4} - 436183\beta_{3} - 5847659\beta_{2} + 36545225\beta _1 + 1317508941 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3827934 \beta_{11} - 51278835 \beta_{10} + 47999061 \beta_{9} + 23727792 \beta_{8} + \cdots - 11551430338 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 116092479 \beta_{5} - 57409920 \beta_{4} + 78887523 \beta_{3} + 1050669337 \beta_{2} + \cdots - 236381843961 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 832852317 \beta_{11} + 11089778898 \beta_{10} - 8654090683 \beta_{9} - 4288711715 \beta_{8} + \cdots + 2494872744181 \beta_{6} ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
− 13.7478i
− 12.7478i
− 5.29401i
− 6.29401i
− 0.794658i
− 1.79466i
0.794658i
1.79466i
5.29401i
6.29401i
13.7478i
12.7478i
0 − 26.4955i 0 0 0 35.9189i 0 −459.013 0
449.2 0 − 26.4955i 0 0 0 35.9189i 0 −459.013 0
449.3 0 − 11.5880i 0 0 0 − 89.6769i 0 108.718 0
449.4 0 − 11.5880i 0 0 0 − 89.6769i 0 108.718 0
449.5 0 − 2.58932i 0 0 0 204.489i 0 236.295 0
449.6 0 − 2.58932i 0 0 0 204.489i 0 236.295 0
449.7 0 2.58932i 0 0 0 − 204.489i 0 236.295 0
449.8 0 2.58932i 0 0 0 − 204.489i 0 236.295 0
449.9 0 11.5880i 0 0 0 89.6769i 0 108.718 0
449.10 0 11.5880i 0 0 0 89.6769i 0 108.718 0
449.11 0 26.4955i 0 0 0 − 35.9189i 0 −459.013 0
449.12 0 26.4955i 0 0 0 − 35.9189i 0 −459.013 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.c.p 12
4.b odd 2 1 inner 800.6.c.p 12
5.b even 2 1 inner 800.6.c.p 12
5.c odd 4 1 800.6.a.x ✓ 6
5.c odd 4 1 800.6.a.y yes 6
20.d odd 2 1 inner 800.6.c.p 12
20.e even 4 1 800.6.a.x ✓ 6
20.e even 4 1 800.6.a.y yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.6.a.x ✓ 6 5.c odd 4 1
800.6.a.x ✓ 6 20.e even 4 1
800.6.a.y yes 6 5.c odd 4 1
800.6.a.y yes 6 20.e even 4 1
800.6.c.p 12 1.a even 1 1 trivial
800.6.c.p 12 4.b odd 2 1 inner
800.6.c.p 12 5.b even 2 1 inner
800.6.c.p 12 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{6} + 843T_{3}^{4} + 99875T_{3}^{2} + 632025 \) Copy content Toggle raw display
\( T_{11}^{6} - 781187T_{11}^{4} + 150735355475T_{11}^{2} - 7280148286726225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 843 T^{4} + \cdots + 632025)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 51148 T^{4} + \cdots + 433859342400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 72\!\cdots\!25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 183809385447424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 10\!\cdots\!89)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 13\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2076 T^{2} + \cdots - 43086643200)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 5167 T^{2} + \cdots + 41293330275)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 14177606744200)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 59\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 12\!\cdots\!21)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 25\!\cdots\!25)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 74566113251375)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
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